Thomas Benz_ Steinar Nordal-numerical Methods In Geotechnical Engineering _ Proceedings Of The Seventh European Conference On Numerical Methods In Geotechnical Engineering, Trondheim, Norway, 2-4 June 681e5s

[email protected]

NE & <EE 3 >NE <EE 4 >NE & >EE <EE 5 >EE >EE

where γw = water specific weight; µw = water viscosity; dh = equivalent grains diameter; and χ = numerical coefficient; t counter of temporal step t.

Partial Seals Seals or no seal with no with some with large erosion (I) erosion (II) erosion (III) hl el

u el

hl u

u u

el u

el l

hu

u

hl

NE = No-erosion; EE = Excessive-erosion boundaries; hl = highly likely; l = likely; el = equally likely; u = unlikely; hu = highly unlikely

Figure 4. Problem setting. (1) One-dimensional unsteady seepage flow through a heterogeneous base (B) – transition (T) system; (2) B and T are divided into elements; (3) a constant total piezometric head difference H is assigned.

293

To solve the problem, the unsteady state is decomposed according to a sequence of steady states, whose duration is t (“successive steady states” method, Harr 1962, Franciss 1985). This limited time interval may assume constant or variable value; for each t, the continuity equation holds: Figure 5. Forces acting on migrating particles; a) plugged particle (d = d0 ); b) unplugged particle (d < d0 ); d, particle diameter; d0 , average size of a pore channel.

effective weight of the particle (A) and the confining stresses (FS ) (Fig. 5). For a horizontal flow path, vcr is expressed as follows:

Therefore, the suspension rate Qt through the elements’ section , and the volume of the granular suspension Vm,t , composed by the scoured particles dragged by the seeping fluid, entering and washed out from each element, is the same during each temporal step:

So, the total volume of each element, composed of the original material (Vor,i,t ), accumulated material (Vacc,i,t , Vacc,0 = 0) and water saturating the ith element (Vw,i,t ), doesn’t vary during the interval time t. The specific weight of filtering suspension is valued rearranging the equation proposed by Indraratna & Vafai (1997):

Sacc,i,t = the fraction of accumulated material scoured from the (ith) element; Sor,i,t = fraction of the original material scoured from the (ith) element, at time t Sacc,i,t and Sor,i,t are computed through both hydraulic and geometric methods): The particles can be scoured only if subjected to a flow velocity greater than the local, critical flow rate (vcr ); vcr is computed by analysing the forces acting on a movable particle and imposing its dynamic equilibrium along the flow direction. A particle can migrate if the drag force FD (Stokes law) exceeds the local shear resistance induced by the

where β = coefficient that allows to take into the density of the granular matrix (0 < β < 4/π; β = 4/π for granular matrix composed by spherical particles arranged in hexagonal configuration, most dense state) (Kovacs 1981, Biswas 2005). The original material is generally subjected to strong confining actions (frictional forces, geometric hindrances); high values of the flow velocity need to mobilize the plugged particles (Eq. 5). Conversely, the accumulated, unplugged particles, may be easily scoured during simulation; the corresponding critical flow velocity thus assumes small values (Eq. 6). The hydraulic conditions allowing the migration of movable particles are first considered; the analysis of the geometric conditions follows, if the previous one are verified (v > vcr ). The particles (diameter dj ) composing the (ith) element can be scoured only if able to through the voids of the (i + 1)th element. The statistical distributions of the pore volumes (Vi ) and the corresponding constriction sizes, related to the geometry of the micro-configurations of suitably constrained granular masses composed of randomly disposed spheres (diameters dj ) are first theoretically determined. The analysis is carried out through the methods of Statistical Mechanics, by maximizing the configurational entropy associated with the distribution of these micro-variables (Musso & Federico 1983, Federico & Musso 1992). The theoretical pore size distribution (PSD) curves depend on the grain size Pj,t and porosity ni,t of each element of the system.

294

The material scoured from the ith element, at time t (Vs,out ), is function of the specific weight of filtering suspension, γm , Indraratna & Vafai (1997):

Table 4. Physical and mechanical parameters assigned to the analysed materials.

The scoured volume of material is composed both by Vor,out and Vacc,out (Federico & Montanaro 2006, 2009):

k0 m/s

φ

Materials

n0

◦

c’ kPa

S. Valentino (Base) S. Valentino (Transition) Suorva (Base) Suorva (Finer Transition) Suorva (Coarser Transition)

1 · 10−6 1 · 10−6 1 · 10−6 2.5 · 10−6 5 · 10−6

0.3 0.3 0.3 0.3 0.3

25 25 25 25 25

0 0 0 0 0

Vor,out and Vacc,out in turn are decomposed into their granular fractions (Indraratna & Vafai, 1997):

The element within which each scoured fraction is deposited, is determined through the corresponding length of the migration path, Lmig,j , this one depends on the probability of a particle not encountering a smaller constriction size. The length covered by an assigned particle up to its arrest is finally formulated on the base of concepts of stereology and it depends on the PSD as well as on the thickness of the filter (Musso & Federico 1983, Federico & Musso 1992). The length of the migration path is then compared to the length that the particles can cross during each temporal step t:

where mj = number of constrictions greater than the particle size encountered by the particle along its path; and s = unit step assigned to each comparison. At time t + t, the accumulated and original volume fractions within each element are:

By this way, the procedure allows to evaluate the time evolution of the migration phenomena within granular media. 5

RESULTS OF SIMULATIONS

Particles migration phenomena regarding the moraine materials of the S. Valentino and Suorva dams have been simulated through the proposed numerical procedure (Tab. 4).

Figure 6. Ratio Qt /Q0 versus time t (hours); Q0 , initial value of the suspension flow rate Qt .

The length of the B-T systems is 3 m (B: 1 m, T : 2 m); to study the displacements of the particles, from the B through the protective T , each system has been divided into 60 elements, 5 cm length. The constant, overall piezometric head difference H is equal to 6 m; this value takes into the available piezometric measures for the two dams. To promote their erosion, confining stresses on B particles are neglected (unplugged particles). Results of numerical simulations show that erosion mainly involves the finer fractions (d ≈ 0.002 mm) of the analyzed core materials. The evolution of the ratio Qt /Q0 versus time t is shown in Fig. 6 for the three analyzed case; Q0 is the initial value of the suspension flow rate. The ratio Qt /Q0 rapidly increases if the Suorva core material is protected by the coarser transition, due to the intense erosion of the finer fractions of B and the corresponding increase of permeability. After 12 hours, Qt /Q0 still increases: both granulometric and hydraulic stabilizations are not yet occurred. If the Suorva core material is protected by the finer transition, after 4 hours Qt /Q0 slowly decreases: the erosion of B is not still exhausted; the voids of T are progressively clogged, the process seems to reach a stable state. The finer protective transition of S. Valentino dam controls the washout of the finer particles of B; Qt very less increases respect to the Suorva analyzed cases. Stabilization occurs after 6 hours; Qt becomes only 1.2 times greater than Q0 .

295

REFERENCES

Figure 7. Particles accumulation in the protective transitions (d = 0.002 mm).

The lengths of the path crossed by the eroded particles through the examined granular transitions (T ) are very different (Fig. 7). Within the T material of the S. Valentino dam, the maximum particles accumulation occurs just few cm after the B-T interface; in the finer transition material of Suorva dam, the maximum particles accumulation occurs about 70 cm after the B-T interface; finally, in the coarser transition material of Suorva dam, about 150 cm after the B-T interface. 6

CONCLUDING REMARKS

The granulometric stability of cohesionless moraine materials, composing the core of earth dams, has been analyzed. Conventional criteria have been first taken into . The deposition–erosion process that takes place in proximity of the core–protective transition material has been then numerically simulated. The evolution of the migration phenomena has been carried out by taking into voids, constriction sizes and porosities of the particulate materials (geometric-probabilistic models) as well as the rate of the seeping suspension and piezometric gradients (hydraulic models), through an original numerical procedure. Results of numerical simulations put into evidence that the proposed numerical procedure allows to simulate the deposition–erosion process taking place at the interface core–granular transition; specifically, erosion phenomena mainly involve the finer fractions of these materials and are not negligible if the protective granular transitions are too coarse. In these cases, the particles migration phenomena can cause anomalies and malfunction, such as those ones often occurred in dams whose core is composed by broadly graded cohesionless materials protected in turn by coarse transitions.

Atmatzidis, D. K. 1989. A study of sand migration in gravel. 12th Int. Conf. On Soil Mech. and Found. Engrg.. Rio De Janeiro, Brazil, Session 8/3, pp. 683–686. Biswas, S. 2005. Study of cohesive soil granular filter interaction incorporating critical hydraulic gradient and clogging. Engineering-Research Master, University of Wollongong, NSW, Australia. I.T.C.O.L.D. 1981 – Materials for Earth and Rockfill Dams in Italy – Italian Committee on Large Dams (I.C.O.L.D.), Research Report n.2, November (in Italian). Federico, F. & Musso, A. 1992. Some advances in the geometric-probabilistic method for filter design. Int. Conf. on “Filters and Filtration Phenomena in Geotechnical Engineering”, 75–82, Karlsruhe, October. Federico, F. & Montanaro, A. 2006. Geotechnical design of granular transitions as protective filters.Thesis, University of Rome “Tor Vergata” (in Italian). Federico, F. & Montanaro, A. 2009. Internal erosion in embankment dams. Phenomena, Lab. Experiments, Numerical Simulations. Colloquium Lagrangianum, February, Maratea, Italy. Foster, M. & Fell R. 2001. Assessing Embankment Dam Filters That Do Not Satisfy Design Criteria. J. Geotech. Engrg., A..S.C.E., May, 398–407. Franciss, F.O. 1985. Soils & Rocks Hydraulics. Fundamentals, Num. Meth. & Tech. of Electrical Analogs. Balkema. Harr, M. E. 1962. Groundwater and Seepage. Dover Publications Inc. Indraratna, B. & Vafai, F. 1997. Analytical Model for Particle Migration Within Base Soil – Filter System. J. Geotech. Engrg., A..S.C.E., 123 (2), 100–109. Kovacs, G. 1981. Seepage hydraulics. Elsevier Publ.. U.S.A. Lafleur, J. 2007. Internal stability of particles in dam cores made of cohesionless broadly graded moraines. Internal Erosion of Dams and their Foundations – Fell & Fry (eds), Taylor & Francis Group, London, ISBN 978-0-415. Musso, A. & Federico, F. 1983. A geometrical probabilistic approach to the design of filters. Rivista Italiana di Geotecnica, Vol. XVII, n . 4, 173–193 (in Italian). Musso, A. & Federico, F. 1985. Pore size distribution in filtration analyses. XI I.C.S.M.F.E., S. Francisco, Vol. I, 1207–1212. Nilsson, Å. 2007. The susceptibility of internal erosion in the Suorva Dam. Internal Erosion of Dams and their Foundations – Fell & Fry (eds), Taylor & Francis Group, London, ISBN 978-0-415. Reddi, L. N., Xiao, M., Hajra, M. G. & Lee, M. 2000. Permeability Reduction of Soil Filters due to Physical Clogging. J. Geotech. Engrg., A.S.C.E., 126 (3), 236–246. Sherard, J. L. & Dunnigan, L. P. 1989. Critical Filter for Impervious Soils. J. Geotech. Engrg., A.S.C.E., 115 (7), 546–566. Silveira, A. 1965. An analysis of the problem of washing through in protective filters. Proc. 6th Int. Conf. on Soil Mech. and Found. Engrg., Montreal, Vol. 2, 551–557. Wittmann, L. 1979. The Process of Soil-Filtration – its Physics and the Approach in Engineering Practice. 7th E.C.S.M.F.E., Vol. l, Brighton, 303–310.

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Numerical Methods in Geotechnical Engineering – Benz & Nordal (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-59239-0

Numerical prediction of time-dependent rock swelling based on an example of a major tunnel project in Ontario/Canada Ansgar Kirsch* & Thomas Marcher ILF Consulting Engineers, Rum/Innsbruck, Austria

ABSTRACT: Especially long-term deformation of rock formations causes major problems in the design and construction of underground structures and often governs the final lining design of the tunnel. This paper systematically studies the initiating processes and the effects of rock swelling. As an example a tunnel excavation in the Silurian age cap rocks of the Niagara escarpment is presented. During and after construction of the tunnel the surrounding rock experiences a change in anion concentration through diffusion through the porewater towards the tunnel inner surface. It is assumed that swelling occurs in the shale formations. A reduction of chloride concentration by 2% is defined as the extent of swell initiation and is considered to be a “diffusion front”, which propagates into the surrounding rock formation. Based on the differential equations for diffusion a robust and fast numerical model for radially-symmetric boundary conditions has been developed using the Finite Difference Method. This enabled to carry out a parametric study incorporating various tunnel lining layers (representing the different materials and their time-related installation). In such a way the potential and order of magnitude of swelling (spatial and timely distribution) can be estimated.

1

INTRODUCTION

The success of design and construction of a tunnel is generally related to the knowledge of geological environment, rock mass parameters, overburden thickness, in-situ stress field, tunnel size and shape, etc. Especially long-term deformation of rock formations often governs the final lining design of the tunnel. The main subject of the present paper is to identify the effects of time-dependent deformations due to swelling. In many publications, the time dependent deformation is described as “rock squeeze” and “swelling”. However, these processes are interrelated, and the individual effects of each are difficult to distinguish. In the present paper “swelling of rock” is considered as a time dependent volume increase due to a physico-chemical reaction of the rock with water. As an example a tunnel excavation in the Silurian age cap rocks of the Niagara escarpment is considered in the following (1; 3). Ontario Power Generation has contracted to build a 10.4 km long water diversion tunnel to increase power production at the Sir Adam Beck Generating Complex in Niagara Falls, Canada. The tunnel is a two-, cast-in-place concrete lined *

Formerly: Division of Geotechnical und Tunnel Engineering, University of Innsbruck, Innsbruck, Austria

pressure tunnel with a nominal internal diameter of 12.7 m. The tunnel is predominantly situated in the Queenston Formation of South Ontario which is known to exhibit time-dependent swelling and aggressive ground water conditions and which is expected to present unique challenges in design and construction. In particular shale formations with considerable contents of chloride tend to swell if the concentration of chloride ions contained in the pore fluid of the rock mass is reduced (1). The investigations and resulting conclusions given in this paper are, thus, in particular applicable to these formations. The tunnel is excavated and ed by an initial lining consisting of shotcrete, steel ribs and rock dowels. Time after the excavation a waterproofing membrane and a final lining of cast in place concrete will be installed, thus forming a double shell lining system. All lining components influence the velocity of chloride diffusion and hence have an effect on the swelling potential of the rock mass at a given time. It was assumed that swelling occurs predominantly in the shale formations if there is a reduction of chloride concentration. The surrounding rock experiences a change in anion concentration through diffusion trough the porewater towards the tunnel inner surface. It was assumed that rock swelling commences at a reduction in chlorides of 2% (based on engineering judgement

297

and contractual basis). This reduction of chloride concentration was considered to define a “diffusion front”. The properties of the lining material with respect to geometry and chloride permeability were systematically varied in a parametric study. It was to determine which boundary conditions govern the penetration depth of the diffusion front into the surrounding rock formation over the 90 year design life of the tunnel. 2

PHYSICAL BACKGROUND

Diffusion is modeled as a chemical process of net mass transfer from the pore fluid with higher concentration in the in-situ rock mass to a substance with a lower concentration. Basic parameters are a chloride diffusion coefficient of D = 1.5 × 10−6 cm2 /s for the lower shale formation, and • a chloride diffusion coefficient of D = 1.5 × 10−5 cm2 /s for the upper shale formation and for the plastic zone developping in the vicinity of the tunnel.

3

NUMERICAL MODELLING

3.1 Governing differential equations The differential equation for diffusion, also known as Fick’s law, is

with concentration C, time t, coefficient of diffusion D and the nabla operator ∇. As a deep circular tunnel in more or less homogeneous rock can be considered a radially symmetric problem, equation (??) was expressed radial symmetry

•

The chloride diffusion coefficient was determined with the Nordtest Method, NT Build 443 (2). With this test concrete and similar materials, which are open to diffusion are being tested. For other materials (e.g. waterproofing membrane) the diffusion coefficients were varied in meaningful ranges (cf. e.g. (5; 6)). It was assumed that the distribution of chlorides in the rock mass prior to excavating the tunnel is uniform. The diffusion rate along bedding planes is taken the same as given for the plastic zone. It is further assumed that during/after excavation of the tunnel (before installation of the waterproofing membrane) there is a continuous supply of freshwater or high relative humidity maintaining zero chloride concentration at the tunnel circumferences. The surrounding rock experiences a change in anion concentration through diffusion from the chlorides towards the tunnel surface. It was assumed that a diffusion gradient (gradient of concentration) with a characteristic logarithmic shape develops with time. At the start of the chloride diffusion zero chloride concentration was prescribed at the tunnel circumference and the concentration of chlorides in the rock was taken to be 100% and to remain 100% at an infinite distance from the rock surface. A 2% reduction of chloride ion concentration was assumed to initiate swelling and, thus, defines a “diffusion front”. This diffusion front advances from the tunnel circumferences at time 0 into the surrounding rock formation. All structural materials applied during excavation of the tunnel have influence on the diffusion gradient. Therefore multiple-phase-medium calculations had to be performed to assess the development of the diffusion front. To accomodate for system changes, such as installation of waterproofing membrane and final lining, and to cover a long period of time with sufficient accuracy a flexible Finite Difference scheme was developed.

with radial coordinate r. Equation (2) served to numerical modelling of the diffusion process. 3.2 Finite difference method A two-dimensional Finite Difference Scheme was implemented in Matlab to perform a parametric study concerning the geometry of the problem and the diffusion coefficients for the multi-phase-material. Under the given initial and boundary conditions the Finite Difference Method (FDM) promised a high performance with sufficient control over input and output parameters. The solution of equation (2) is achieved by discretising the time variable tj (j = 1 . . . m) and the spatial variable ri (i = 1 . . . n) and expressing the derivatives in equation (2) by Finite Differences

and

This integration scheme is also called “Forward Time, Centred Space (FTCS)”. The integration of equation (2) can be performed explicitely, leading to

j+1

Equation (6) can be solved for Ci variables are known for time tj .

298

since all other

The choice of increment size, r and t, has an influence on the quality of the integration. E.g. Press et al. (4) put forward that the stability of the integration with an explicit scheme can only be achieved for

This requirement leads to narrow time intervals t, which are unsatisfactory for the purpose of this study. The right hand part of equation (6) can, however, be j+1 expressed in of Ci for a time step tj+1 . This leads to a system of n − 2 linear equations with the j+1 j+1 unknowns C2 , . . . , Cn−1 , which has to be solved in every time step. With this implicit integration scheme, equation (6) becomes

Figure 1. Geometry of the multiphase system.

Figure 2. Geometry of the model (not to scale).

The jump of the diffusion coefficient at the material layer interfaces has to be taken into as well. Therefore, the diffusion coefficient D is considered as Dl for the material to the left of ri , and consequently as Dr for the material to the right of ri . Finally, the following approximation to equation (2) was achieved

final lining: membrane: shotcrete lining: rock:

d1 = 0.600 m d2 = 0.003 m, i.e. 3.0 mm d3 = 0.130 m d4 = 14.000 m

The thickness of the, theoretically, infinite rock layer was increased in a preliminary study until no boundary effect could be detected any more. Thus, it was found that 14 m of rock mass represent the outer border of the model with sufficient accuracy. For the Finite Difference scheme radial increments of r = 1.0 cm were chosen. 4.2 Time discretisation

4 ANALYSIS After a thorough consistency check of the Finite Difference scheme for a homogeneous layer, four layers representing the different materials (final lining, waterproofing membrane, shotcrete and rock) were incorporated into the model.

The time variable t was discretised with time increments of t = 0.1 d. Calculations were performed from t1 = 0 (start of the tunnel excavation) to tend = 33,580 d (construction time + lifetime of the structure). For the first 2 years (=730 days) only layers 3 and 4 were active (stage 1). At that point layers 1 and 2 were activated and the change in chloride concentration was calculated for an additional period of 90 years (stage 2). 4.3 Initial and boundary conditions

4.1 Spatial discretisation The geometry of the problem is illustrated in Fig. 1. As the diffusion equation was solved for radial symmetry, the radius r plays an important role in the model; the inner radius of the final lining was r0 = 6.38 m. In a first step, the thicknesses of the different layers were chosen as

Providing initial conditions for the radial distribution of the cloride concentration (Ci1 for t1 = 0 at every j j location ri ) and boundary conditions (C1 and Cn for every time step tj at the left and right boundary (r1 and j rn , respectively)), the concentration Ci at time tj was calculated with equation (9).

299

Table 1.

Results before installation of final lining. Shale formation

Figure 3. Display of simulation results.

2

Shotcrete

No.

D (cm /s)

tR (m) D (cm2 /s)

A 1.1.1 A 1.1.2 A 1.1.3 A 1.1.4 A 1.1.5 A 1.2.1 A 1.2.2 A 1.2.3 A 1.2.4 A 1.2.5

1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5

14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0

1.0 × 10−4 1.5 × 10−5 1.0 × 10−6 1.0 × 10−7 1.0 × 10−8 1.0 × 10−4 1.5 × 10−5 1.0 × 10−6 1.0 × 10−7 1.0 × 10−8

2% lim. tS (m) tD (m) 0.13 0.13 0.13 0.13 0.13 0.16 0.16 0.16 0.16 0.16

0.95 0.91 0.69 0.42 0 0.94 0.90 0.68 0.41 0

(dashed line), the chloride ions have diffused towards the tunnel contour; the chloride concentration in the vicinity of the tunnel decreases. Fig. 4 shows a detail of the vicinity of the tunnel contour. It becomes obvious that the 2% reduction front penetrates well into the rock. In the given example the penetration depth is 0.73 m after two years. Within this zone, there is potential for swelling. 5 Figure 4. Detail of Fig. 3.

For the given problem the following boundary conditions were chosen: •

the chloride concentration of the minor (=tunnel) j surface is zero for all tj , i.e. C1 = 0. • the chloride concentration on the outer boundary (at unlimited distance in the rock mass) is always 100% j (=1.0), i.e. Cn = 1.0. 4.4

Material parameters

The only material parameter relevant for the diffusion process is the diffusion coefficient D. An initial set of parameters was chosen as final lining: D1 = 1.5 × 10−5 membrane: D2 = 1.5 × 10−8 shotcrete lining: D3 = 1.5 × 10−6 rock: D4 = 1.5 × 10−5 4.5

Evaluation of results

The results of the simulations can be visualised as plots of concentration C (in %) vs. radial distance r (in m) (cf. Fig. 3). The r

axis starts at r0 = 6.38 m and covers a total of r = d = 14.73 m. The calculation for the whole analysis (tend = 92 years) with the given incrementation took roughly 30 seconds. Fig. 3 shows the initial distribution of chloride concentration as full line. Two years after construction

RESULTS OF THE PARAMETRIC STUDY

A parametric study served to investigate the influence of the diffusion coefficients for all materials and the shotcrete thickness on the penetration depth of the 2% reduction front into the rock. 5.1 Results before installation of final lining In stage 1 only the rock mass and the shotcrete forming the initial lining were modelled. The duration of this construction stage was assumed to be 2 years (730 days) at maximum. The shotcrete lining was modelled with a thickness of 130 mm (calculation set A 1.1) and 160 mm (calculation set A 1.2). The variation in diffusion coefficients is expressed in Tab. 1. 5.2 Results after end of operational time In the second set of calculations the reduction in chlorides was analysed after installation of the final lining until the end of the operational time of 90 years. Case A considers shotcrete with D = 1.5 × 10−5 cm2 /s (same as rock mass). The coefficients for the waterproofing membrane and the final lining concrete were varied as expressed in Tab. 2. Case B (Tab. 3) considers the case that a very permeable shotcrete (D = 1.0 × 10−1 cm2 /s) is placed. Other parameters were varied according to Case A. Calculations B 4.1.1 to B 4.1.4 consider a lower chloride diffusion rate in the shale formation (D = 1.5 × 10−6 cm2 /s instead of 1.5 × 10−6 cm2 /s). Case C (Tab. 4) finally investigates the condition that no waterproofing membrane is present.

300

Table 2.

Results after end of operational time (case A). Shale formation

Shotcrete

Membrane

Final lining

No.

D (cm /s)

tR (m)

D (cm /s)

tS (m)

D (cm /s)

tM (m)

D (cm2 /s)

tF (m)

2% limit tD (m)

A 2.1.1 A 2.1.2 A 2.1.3 A 3.1.1 A 3.1.2 A 3.1.3

1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5

14.0 14.0 14.0 14.0 14.0 14.0

1.0 × 10−5 1.0 × 10−5 1.0 × 10−5 1.0 × 10−5 1.0 × 10−5 1.0 × 10−5

0.13 0.13 0.13 0.13 0.13 0.13

1.0 × 10−8 1.0 × 10−9 1.0 × 10−10 1.0 × 10−8 1.0 × 10−9 1.0 × 10−10

0.003 0.003 0.003 0.003 0.003 0.003

1.0 × 10−5 1.0 × 10−5 1.0 × 10−5 1.0 × 10-6 1.0 × 10-6 1.0 × 10-6

0.6 0.6 0.6 0.6 0.6 0.6

4.15 1.25 0.00 3.60 1.05 0.00

Table 3.

2

2

2

Results after end of operational time (case B). Shale formation

Shotcrete

No.

D (cm2 /s)

tR (m)

D (cm2 /s)

tS (m)

D (cm2 /s)

tM (m)

D (cm2 /s)

tF (m)

2% limit tD (m)

B 2.1.1 B 2.1.2 B 2.1.3 B 3.1.1 B 3.1.2 B 3.1.3 B 4.1.1 B 4.1.2 B 4.1.3 B 4.1.4

1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−6 1.5 × 10−6 1.5 × 10−6 1.5 × 10−6

14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0

1.0 × 10-1 1.0 × 10-1 1.0 × 10-1 1.0 × 10-1 1.0 × 10-1 1.0 × 10-1 1.0 × 10-1 1.0 × 10-1 1.0 × 10-1 1.0 × 10-1

0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13

1.0 × 10−8 1.0 × 10−9 1.0 × 10−10 1.0 × 10−8 1.0 × 10−9 1.0 × 10−10 1.0 × 10−8 1.0 × 10−9 1.0 × 10−10 1.0 × 10-11

0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003

1.0 × 10−5 1.0 × 10−5 1.0 × 10−5 1.0 × 10−6 1.0 × 10−6 1.0 × 10−6 1.0 × 10−5 1.0 × 10−5 1.0 × 10−5 1.0 × 10−5

0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6

4.15 1.25 0 3.60 1.05 0 1.70 0.90 0 0

Table 4.

Membrane

Final lining

Results after end of operational time (case C). Shale formation

Shotcrete

Membrane

No.

D (cm2 /s)

tR (m)

D (cm2 /s)

tS (m)

D (cm2 /s)

tM (m)

D (cm2 /s)

tF (m)

2% limit tD (m)

C 1.1.1 C 1.1.2 C 1.1.3 C 1.1.4 C 1.2.1 C 1.2.2 C 1.2.3 C 1.2.4 C 1.3.1

1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5 1.5 × 10−5

14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0 14.0

1.0 × 10−5 1.0 × 10−5 1.0 × 10−5 1.0 × 10−5 1.0 × 10−1 1.0 × 10−1 1.0 × 10−1 1.0 × 10−1 1.0 × 10−8

0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13 0.13

1.0 × 10−5 1.0 × 10−6 1.0 × 10−7 1.0 × 10−8 1.0 × 10−5 1.0 × 10−6 1.0 × 10−7 1.0 × 10−8 1.0 × 10−5

0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030 0.030

1.0 × 10−5 1.0 × 10−6 1.0 × 10−7 1.0 × 10−8 1.0 × 10−5 1.0 × 10−6 1.0 × 10−7 1.0 × 10−8 1.0 × 10−5

0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6

5.55 4.25 2.65 0.05 5.67 4.27 2.77 0.17 0

It was analysed, which coefficient of chloride diffusion for the final lining concrete is required to make the final lining concrete act as a barrier. The diffusion coefficient of the waterproofing membrane was set equal to the coefficient of the final lining concrete. Moreover, the overall lining thickness was adjusted to reflect exactly 600 mm of final lining thickness. All other parameters were set according to Case B. The results are presented for the assumption that diffusion through the shotcrete is the same as through the rock (C 1.1.x), shotcrete is open to diffusion (C 1.2.x) and shotcrete acts as a barrier (C 1.3.x).

6

Final lining

CONCLUSIONS

6.1

Results for the short term

The results of the analysis A 1.1.x and A 1.2.x are representing the construction stage of excavation and of the tunnel, which may last for two years (730 days). High humidity and water used in construction or flowing from the surface may trigger chloride diffusion in the tunnel before the final lining is installed. The analysis leads to the conclusion that:

301

•

Shotcrete with a Chloride Diffusion Coefficient <1.0 × 10−8 cm2 /s acts as a barrier to chloride

•

diffusion over the analysed exposure time of two years. • There is only marginal difference in chloride diffusion between a shotcrete thicknesses of 130 mm and 160 mm. • In case shotcrete is open to diffusion of chlorides, at a maximum one metre of rock mass around the tunnel will suffer a reduction of chlorides larger than 2%. 6.2

Results for the long term

The results of the analysis A 2.1.x, A 3.1.x, B and C consider the system behaviour in the lifetime of the tunnel, i.e. over a period of 90 years (32850 days). A combination of construction materials was analysed using different parameters of chloride diffusion. The analysis leads to the conclusion that: •

A waterproofing membrane with a chloride diffusion coefficient <1.0 × 10−10 cm2 /s can be considered a barrier for chloride diffusion. • In case of shotcrete with a diffusion coefficient <1.0 × 10−8 cm2 /s (i.e. barrier) the diffusion coefficient of the final lining is irrelevant also within 90 years and no reduction in chlorides ≥2% will occur in the rock mass. • In case the diffusion coefficient for the final lining is less than 1.0 × 10−8 cm2 /s and no waterproofing membrane is provided, the final lining concrete alone will be sufficient to stop the propagation of chloride diffusion over the 90 years operational time. There is no significantly different effect detectable with respect to chloride diffusion resistance of a final lining concrete with a coefficient of 1.0 × 10−5 cm2 /s or a final lining concrete with 1.0 × 10−6 cm2 /s.

The effect of two different chloride diffusion rates in the rock (diffusion coefficient of 1.5 × 10−6 cm2 /s versus 1.5 × 10−5 cm2 /s) is analysed for waterproofing membrane, shotcrete with a reduced diffusion coefficient of 1.0 × 10−1 cm2 /s and the final lining with a regular diffusion coefficient of 1.0 × 10−5 cm2 /s. There is marginal difference in the extent of chloride diffusion (zone of rock mass with chloride reduction ≥2%) when applying different diffusion rates for the rock mass. A waterproofing membrane with a chloride diffusion coefficient <1.0 × 10−10 cm2 /s can be considered a barrier for chloride diffusion in both cases.

REFERENCES Hughes, M. and Bonapace, P. and Rigbey, S. and Charalambu, H. (2007) An innovative approach to tunneling in the swelling Queenston formation of Southern Ontario, Rapid Excavation and Tunneling Conference 2007, (eds M. Traylor, and J. Townsend), pp. 901–912. Nordtest, Concrete, hardened: accelerated chloride penetration, NT Build 443. Pöttler, R. (2007), Das Niagara Tunnel Projekt, in Geotechnik-Kolloquium, Geotechnik im Verkehrsbau/ Infrastrukturmanahmen – national & international, (eds. H. Klapperich and H. Konietzky), pp. 205–212. Press H., Teukolsky S., Vetterling W. and Flannery B. (1992), Numerical Recipes in Fortran, Cambridge University Press, 2edn. Rowe P., Hrapovic L. and Kosaric N. (1995), Diffusion of chloride and dichloromethane through an HDPE geomembrane, Geosynthetics International, 2(3): pp. 507–536. Thoft-Christensen P. (2003), Stochastic modelling of the diffusion coeffcient for concrete, in Reliability and Optimization of Structural Systems (eds. H. Furuta, M. Dogaki and M. Sakano), pp.151–160, Swets & Zeitlinger.

302


Some features of the coupled consolidation models used for the evaluation of the dissipation test E. Imre Ybl Miklós Faculty of Architecture and Civil Engineering, Budapest, Hungary Geotechnical Department, Budapest University of Technology and Economics, Budapest, Hungary

P. Rózsa Department of Computer Science and Information Theory, Budapest University of Technology and Economics, Budapest, Hungary

ABSTRACT: Two families of the point-symmetric linear coupled consolidation models – differing in one boundary condition – are treated. The models are with space dimension one (oedometric), two (cylindrical) or three (spherical). Both qualitative and quantitative analyses are made. The numerical features of the analytical solution are tested for the cylindrical models. It is shown that the numerical features of the analytical solution are strongly influenced by the distance of the zero solution from the actual solution moreover, by the size of the displacement domain.

1

INTRODUCTION

The analytical consolidation models used for the evaluation of the dissipation test are the pointsymmetric linear coupled consolidation models with space dimensions two and three. Two sets of the boundary conditions are used. Constant displacement boundary condition is assumed at the penetrometer – soil interface by both models. The suggested model implies displacement type condition, the models available implies volumetric strain type condition in the soil. These two sets of boundary conditions (either displacement type at both boundaries; displacement and volumetric strain types at the two boundaries) may occur in the oedometric modelling (relaxation and compression tests) for space dimension one. The aim of the research was to study the main analytical and numerical properties of these models. The system of partial differential equations of the point-symmetric linear coupled consolidation was summarized into a single model in the function of the space dimension (Fig 1, Table 1). Closed form analytical solution was given in the special case of the two sets of boundary conditions. The solutions were analysed. It was shown that – independently of the space dimension – the transient solution is determined by the boundary conditions.The constants of the analytical solution were determined from the initial and from the boundary conditions. In the one dimensional case the “boundary condition equation” has closed form solution resulting in some dimensionless variables. In the two and three dimensional problems– using the asymptotic Bessel

Figure 1. The displacement domain for the point-symmetric models bounded by a (a) 0 dimensional sphere.

formulae – approximate closed form solutions were given resulting some approximate model laws. The numerical features of the analytical solution – including the Bessel function were approximation, the validity of the approximate root formulae and the resulting approximate model law; moreover the

303

Table 1. models.

Summary of point-symmetric consolidation

Model type 1D 2D 3D

v or ε boundary condition

Origin

No (uncoupled) v-v (coupled 1) v-ε (coupled 2) No (uncoupled) v-v (coupled 1) v-ε (coupled 2) No (uncoupled) v-v (coupled 1) v-ε (coupled 2)

Terzaghi (1923 ) Imre ( 1997–1999) Biot (1941) Soderberg (1962) Imre & Rózsa (1998) Randolph at al (1979) Torstensson (1975) Imre & Rózsa (2002) Imre & Rózsa (2005)

2.3 Boundary conditions In the following four boundary conditions are presented for m = 2. Three are common, one is different for the two models. (1) The (common) boundary condition Nr. 1 implies that the pore water pressure is zero at r = r1 : (2) The (common) boundary condition Nr. 2 entails that the flux is equal to zero at r = r0 :

(3) The (common) boundary condition Nr. 3 implies that the displacement equals to a constant at r = r0 :

convergence properties – were tested on the example of the cylindrical models. 2 2.1

(4) Boundary condition Nr. 4 – concerning the suggested model – implies that the displacement equals to zero at r = r1 :

POINT-SYMMETRIC CONSOLIDATION Models

The basic units of the two linear coupled consolidation model-families (coupled 1 and 2) – differing in one boundary condition – are presented in this chapter. The models are one dimensional with various space dimension m, as shown in Figure 1. 2.2

(5) Boundary condition Nr. 5 – concerning the models available – expresses that the volumetric strain ε is constant r = r1 :

System of differential equations

Two equations have been derived from the equilibrium condition and, from the continuity condition (Imre et al, 2007). Equation (1) compiles the equilibrium condition, the effective stress equality, the geometrical and, the constitutive equations, as follows:

These boundary conditions are equally usable for m = 3, in the case of m = 1 the half of the space domain is generally used and r0 = 0 is assumed. 3

SOLUTION

3.1 Structure of Solution and, Equation (2) compiles the continuity equation, the Darcy’s law and the geometrical equation, as follows:

where the volumetric strain and the Laplacian operator, containing the dimension m of the embedding space are as follows:

The solution is equal to the following sum for each variable (Imre, 1997–1999).:

where the superscripts has the following meaning: p indicates the drained continuum-mechanical problem (independent of t), L indicates the steady-state seepage problem (independent of t), t indicates the transient seepage problem, w concerns the self-weight component (independent of both t and r). 3.2 Analytical solution

v is the radial displacement, u is the excess pore water pressure (neglecting the gravitational component of the hydraulic head), r and t are the space and the time co-ordinates respectively, Eoed is the oedometric modulus:

G is the shear modulus, E is the is Young modulus, µ is the Poisson’s ratio in of the effective normal stress σ (σ = σ − u where σ is the total normal stress), k is the coefficient of permeability, γ v is the unit weight of water.

Steady-state solution part The solution of the drained continuum-mechanical problem for the displacement vp is the solution of the following part of Equation (1):

which is the cavity expansion model for m = 2, 3 and the oedometer (K0 ) compression model for m = 1. The solution has the following general form:

304

where the parameters can be determined from the inhomogeneous form of the boundary conditions. Transient solution part The transient part of the solution can be determined in different ways. Theoretically the best way is if vt is determined by the solution of Equation (16). The transient part of the displacement solution in the function of m (Imre et al, 2007):

where the mean pore water pressure:

Inserting this boundary condition function into the equilibrium Equation:

From this: where Jm/2 and Ym/2 are the Bessel functions of the first and second kinds, with the order of m/2, and λk , µk , Ck parameters of the solution, m is embedding space dimension. The volumetric strain and the pore water pressure solutions from this:

It follows that for a realistic u the change in ε with t is positive in the vicinity at the outer boundary (rebound) and negative in the vicinity of the pile (compression). By further integration:

The function u is determined using Equation (1): It follows that for a realistic u the transient part of v is non-negative and, monotonously decreases with t for any r. The initial condition for u and vt have the following relationships: 4 ANALYSIS OF THE MODELS A qualitative analysis of the solutions is made in this chapter on the basis of some explicit expressions derived from the system of differential equations (1) and (2) and, the boundary conditions. 4.1 The u and v solution of the models

The initial displacement function v0t (r) is zero for the coupled 1 model if u0 (r) is zero or constant.

Equation (1) is used to predict some features of v on the basis of u and, Equation (2) is used to characterise u on the basis of (the initial condition for) v. It is assumed that u is positive and monotonously decreases with t for any r (“realistic” u) with the time t. Instead of v the volumetric strain ε is used sometimes since their relation is unique due to the boundary conditions. According to the definition of the volumetric strain, the compression is negative, the swelling is positive.

Coupled 2 models (available) Inserting the inhomogeneous form of boundary condition Nr. 5 into the equilibrium Equation:

4.2 Analysis of Equation (1)

It follows that for a realistic u the change in ε with t is -negative (compression). By further integration:

From this:

Some explicit expressions are derived for v and u by integrating the equilibrium Equation (1) with respect to r including boundary condition Nr. 1:

In the coming expressions the homogeneous form of the boundary conditions is included.

It follows that for a realistic u the transient part of v is non-negative and, monotonously decreases with t for any r. The initial condition for u and vt have the following relationships for the coupled 2 models:

Coupled 1 (suggested) models A boundary condition function is derived by further integration between r0 and r1 using boundary condition Nr. 3 and boundary condition Nr. 4: It follows that the initial displacement function v0t (r) is zero for the coupled 2 model if u0 (r) is zero.

305

4.3 The total stress and the effective stress On the basis of the explicit expressions, the following total stress and the effective solutions can be derived for space dimension two m = 2:

The roots of the boundary condition equation for the coupled 1 and 2 model-families, respectively, for for space dimension one m = 1:

4.3.1 Coupled 1 (suggested) model, m=2 The radial effective stress for any r: Inserting this into the analytic pile solutions the following dimensionless arguments appear:

and, at the shaft-soil interface:

It follows that for a realistic u the effective stress increases with time at the shaft-soil interface; decreases with time at the outer boundary. The radial total stress at the shaft-soil interface:

It follows that for a realistic u, the radial total stress decreases with time by the value of the mean pore water pressure which variable has an important role.

The appearance of the time factor T ensures that the solutions related to different space domains can be transformed into each other. For the two-three dimensional models, the roots of the boundary condition equation have no closed form solution for the latter models. Therefore, the solution has to be computed for each boundary condition. However, approximate closed form root formulae can be suggested as follows. The asymptotical Bessel functions formulae:

4.3.2 Coupled 2 model (available), m = 2 The effective stress for any r:

Inserting these into the boundary condition equation, the roots for the coupled 1 and 2 model-families, respectively:

and, at the shaft-soil interface:

The radial total stress at the shaft-soil interface

It follows that the effective stress increases with time at the shaft-soil interface and it is constant at the outer boundary. The mean of the first invariant of the effective stress tensor on the displacement domain increases with time. 5

CONSTANTS OF THE SOLUTION

The parameters of the solution can be determined from the boundary conditions and from the initial condition.

5.1

Boundary conditions

For the coupled 1 or 2 model-families, the “boundary condition equation” (arisen from the homogeneous form of boundary conditions Nr.3 and Nr.4 or Nr.3 and Nr.5) can be written as follows, respectively:

Using these, approximate time factors can be derived and, the solutions related to different space domains can approximately be transformed into each other. It can be noted that – within a model-family -, the precise (1 dimensional case) and the approximate (2 and 3 dimensional cases) instances are identical (compare e.g. (38), (39) with (44), (45)). 5.2 Coefficients from the initial conditions The initial condition is generally given by a closed form function for u0 . From this, the initial displacement function v0 is to be determined then from this, the Bessel coefficients can be determined as follows. The coefficients Ck and Ek (k = 1 . . . ∞) can be computed from the initial displacement function v0t (r) with the same formula:

306

Table 2.

Displacement domains for the numerical tests.

1 2 3 4 5 6 7

r1 [cm]

r1 − r0 [cm]

n = r1 /r0 [-]

7 33.25 64.75 127.75 255.5 511 1022

5.25 31.5 63 126 253.75 509.25 1020.25

4 19 37 73 146 292 584

*r0 = 1.75 cm. Table 3.

Parameter D [–] for initial conditions 1 to 10.

u0

1

2

3

4

5

D [-]

0.001

0.016

0.053

0.135

0.222

u0

6

7

8

9

10

D [-]

0.291

0.408

0.591

0.74

0.97

6

SIMULATIONS, MODEL LAW

6.1 Simulations Seven space domains with the same r0 and varying r1 were assumed (Table 2). For the description of the initial pore water pressure distribution u0 , the following normalised function, varying between 0 and 1 was used to define a function series:

Being parameter F in one-to-one relation with D if the space domain is specified, an u0 series was defined prescribing D identically for the various space domains, using ten values for D (Table 3). The u0 series can be characterized as follows. At the limit D → 0 u0 (r) ≡ 0 (except in r0 ). At the limit D → 1 u0 (r) ≡ 1 (except in r1 ). According to the results (Fig 2(a)), if u0 is between the nearly constant zero function (except in r0 ) and the linear function then u0 is convex. If u0 varies between the linear u0 and the nearly constant 1 function (except in r1 ) then u0 is concave. The corresponding dissipation curve series were determined for the various space domains, the results can be summarized as follows. 6.1.1 Pore water pressure results, n = 37 The convergence of the dissipation curve series is controlled by the initial condition in a different way.

Figure 2. (a) Initial condition shape functions. (b) Coupled 1 and 2 dissipation curves, m = 3.

For the Randolph – Wroth model, there is one zero solution at initial condition D = 0. As D → 1, the dissipation curves “move away” from the zero solution, (see Fig 2(b), in dashed line). For the Imre – Rózsa model, there are two zero solution at initial condition D = 0 and 1. For convex initial distributions the dissipation curves “move away” from the zero solution at initial condition D = 0, for concave distributions the dissipation curves “move back” to the zero solution at initial condition D = 1 (Fig 2(b), in solid line). For not too extreme initial conditions (i.e. 3 to 7), the dissipation curve solutions are very similar. The time factor Tκ is about three times larger for the RandolphWroth’s model, than for the other model. 6.1.2 Model law If the solutions of the same initial condition and for the various space domains may coincide in of the normalised time coordinate (T ) then the model law is valid. This condition was tested for the dissipation curves (i.e. for u at r = r0 ). As it can be seen in Figure 2(a), for the one term cylindrical model solutions, the model law can be expected to be valid. However, for multi-term solutions, as it can be seen in Figure 2(b) and (c), with increasing space domain and r1, the dissipation curve solutions deviate increasingly. The one term solution seems to be a fixed

307

Figure 4. Variation of the argument r0 λk with k and n.

Figure 5. The validity of the approximate root formulae.

The separation of the two ranges can be seen in Figure 4 for the various space domains on the example of r0 λk . The arguments r0 λk and r0 γk are about the same for the two models on a given space domain. For example, for the smallest space domain (n = 4) r0 λk > 8 if k > 8. For the space domain of the dissipation test (n = 37) r0 λk > 8 if k>90. The separation of the two ranges for the various space domains on the example of argument r1 is as follows. Being the r1 λk and r1 γk about equal to kπ, they are in the large range if k > 3. Figure 3. Dissipation curve envelopes in the function of the space domain, m = 2. (a). One-term dissipation curve solution, cylindrical models. (b) Multiterm solution, Randolph-Wroth model. b. Multiterm solutions, Imre-Rózsa model.

element of the band, forming an the upper envelope for the Imre-Rózsa model. The model law seems to be valid for “not too extreme cases” (i.e. 4 to 7) which are close to the one term solution. 7 THE NUMERICAL FEATURES

7.3 Numerical tests for the convergence

In the case of the two cylindrical models (m = 2), 250 were considered for n = 37 and 40 otherwise. 7.1

7.2 Validity of the root formulae The precise roots of the boundary condition equations were determined with the secant method and, were transformed using the approximate closed form formulae. The results are shown in Figure 5. According to the results, the error of the approximate closed form formulae decreases with k and is with opposite sign for the two models. The effect of the size of the space domain shows a twisted mirror image for the two models.

Bessel function approximation

The rate of convergence of the Fourier-Bessel expansion of the pore water pressure at r = r0 was tested for ten initial conditions and seven space domains as follows. The series:

The Bessel functions were approximated according to Press et al (1986) in both the small range (r < 8) and the large range (r > 8) differently.

308

Figure 7. Imre-Rózsa model, the solution at r = r0 in the function of D and cut off at number k. a. n = 4. b. n = 37.

Figure 6. Randolph-Wroth model, the solution r = r0 in the function of D and cut off at number k. a. n = 4. b. n = 37.

were cut off at a certain term k and, the difference between the sum and 1 was computed. The results of the convergence tests are summarized in the function of the initial conditions, space domain and number of in the Figures 6 and 7, for the case of k < 41. According to the results, for small displacement domain (n = 4), the limit differs from 1. Considering the dependence on D, the error related to a certain k is rapidly increasing as the mean initial condition ordinate D varies ‘towards’ the zero solution (i.e. D → 0 for both models, D → 1 for the Imre – Rózsa model). For the not too extreme initial conditions (i.e. 3 to 7), the numerical error is not important. Considering the dependence on the number of , some oscillation was found for concave initial conditions (i.e. 7 to 10). This was always eliminatable in the “small range” by a summation procedure where the last term was halved. In the case of n=37, the results show that after the summation the series was not convergent for the Randolph-Wroth model in the large range and some smaller error was encountered for the Imre – Rózsa model (Figs 8–9). 8

DISCUSSION, CONCLUSION

8.1 Analytical features An analysis was made for the point-symmetric coupled linear consolidation models (see Table 1), resulting in a unified analytical solution, a unified qualitative

Figure 8. The unsmoothened sums of the Randolph-Wroth model, in the function of the cut off numbers k and the initial condition 1 to 10.

description and some mathematically derived time factors for two sets of the boundary conditions. Constant displacement boundary condition is assumed at the inner boundary r0 by both modelfamilies. Constant volumetric strain boundary condition is assumed at the outer boundary r1 in the consolidation theories available with space dimensions 1 to 3 (coupled 2 models) and constant displacement is assumed at the outer boundary r1 in the suggested models with space dimensions 1 to 3 (coupled 1 models). The results of the unified qualitative description showed that for a specified boundary condition the

309

Figure 9. The smoothened sums of the Imre-Rózsa model, in the function of the cut off numbers k and the initial condition 1 to 10.

model behaviour is the basically the same. An immediate consequence of the results is that the oedometric relaxation test can be used to study the phenomena after pile penetration and vice versa. The results of the unified qualitative description showed that for the two kinds of boundary conditions at r1 the model behaviour is different. For the models available, the volume of the displacement domain is decreasing, the total stress at r0 is constant at the shaft with time. This behaviour seems to be qualitatively not realistic for piles. The volume of the displacement domain is constant, the total stress is decreasing at the shaft with time for the suggested models. This behaviour is qualitatively more realistic for the pile case.

8.2

Numerical features

The precise roots of the boundary condition equation were determined for various space domains and initial conditions, the validity of the approximate root formulae and the resulting approximate model law was tested. The error of the Bessel function approximation and the convergence properties of the analytical solutions were characterized. The Bessel functions were approximated according to Press et al (1986) in both the small range (r<8) and the large range (r>8) differently. According to the results, the error of the Bessel function approximation was great for the large range type approximation in some cases, for example in the case of smaller space. Concerning the error of the approximate closed form root formulae, it was found that it decreases with k and is with opposite sign for the two models. The two models gave a twisted mirror image for the dependence on the size of the displacement domain. The model law was found to be approximately valid in the case of the one-term solutions and, in the case of “not too extreme initial conditions” (i.e. initial condition 4 to 7). These initial conditions are similar to the initial condition related to the undrained penetration

(see Fig 2(a)). It follows that the model law can be used to transform the solution concerning various space domains if the penetration is undrained. The convergence errors were not “important” if the initial pore water pressure was not “extreme” (n > 19, D > 0.1), which may occur for undrained penetration. In this case about 40 are suggested to be taken into at least. The numerical error was to large “close” to the zero solutions (i.e. D = 0 both models, and D = 1 for the Imre – Rózsa model). Approaching these limits, the error of the approximation increased. This can probably be explained as follows. If the solution series v0 (r, t) “converges” to the zero solution v0 (r, t) ≡ 0 then its coefficients Ck , Ek converge to zero, too. Being any other term constant in Equations (49)–(50), the sum of the numerical series for u0 (r0 ) will decrease at every k. The numerical features of the analytical solution – including the use of approximate root formulae; the large range Bessel function approximation moreover the convergence properties need some further research. ACKNOWLEDGEMENT The of the National Research Fund Jedlik Ányos NKFP B1 2006 08 and the Norwegian research fund HU-0121 was used for this research. REFERENCES Baligh, M. M. (1986). Undrained deep penetration, II. pore pressures. Geotechnique, 36(4): 487–503. Biot, M. A. (1941). General Theory Of Three Dimensional Consolidation. Jl. of Appl. Phys. 12: 155–164. Imre, E. & Rózsa, P. 1998. Consolidation around piles. Proc. of 3rd Seminar on Deep Foundations on Bored and Auger Piles. Ghent 385–391. Imre, E. (1997–1999) Consolidation models for incremental oedometric tests. Acta Tech. Acad. Sci. Hung. 369–398. Imre, E. and Rózsa, P. (2002). Modelling for consolidation around the pile tip. Proc. of the 9th Int. Conf. on Piling and Deep Foundations (DFI), Nizza. 513–519. Imre, E. (2002). Pile consolidation models and scale effect. Proc. of NUMGE 2002. Paris. 789–796. Imre, E., Rózsa, P (2005). Point-Symmetric Consolidation Models for the Evaluation of the Dissipation Test. 11th IACMAG 2005, Turin, Italy. 181–191. Imre, E., Farkas, M, Rózsa, P., (2007). A comment on the similarity of the coupled consolidation models with pointsymmetry NUMOG XI Greece p. 337–343. Randolph, M. F. & Wroth, C. P. (1979). An analytical solution for the consolidation around displacement piles. I. J. for Num. Anal. Meth. in Geom, 3: 217–229. Soderberg, L. O. (1962). Consolidation Theory Applied to Foundation Pile Time Effects. Geotechnique, 12: 217–232. Terzaghi, K. (1923). Die Berechnung der Durch. des Tones aus demVerlauf der hydrodyn. Spannung-sercsheinungen, Sitzber. Ak. Wiss. Wien, Abt.IIa, Vol. 123. Torstensson, B. A (1977). The pore pressure probe. Paper No. 34. NGI.

310


Steady state seepage flow through zoned earth structures affected by permeability defects F. Federico, F. Calzoletti & A. Montanaro University of Rome “Tor Vergata”, Rome, Italy

ABSTRACT: Permeability defects in earth structures may induce considerable leakages, shear resistance loss, internal erosion. These undesirable phenomena could evolve towards serviceability (settlements, heterogeneities related to changes of permeabilities, high piezometric gradients) or ultimate limit states (local or global instabilities, structural collapse, piping, hydrofracturing). The effects of defects in the core of zoned earth structures are investigated through several 2D finite element analyses of steady state seepage flows. In computations, the hydraulic losses in the pervious upstream material are neglected; the free surface is lowered by a filter – drain system between the core and the downstream shell. The discharge rate and the free surface profile are first parametrically computed for assigned values of the main geometrical and hydraulic variables. Numerical results are then worked out through a multiple linear regression law to define an original analytical relationships allowing to foresee the discharge rate for more general permeability defects. 1

INTRODUCTION

Permeability defects in earth structures and their foundation soils may occur due to many causes (Jappelli & Federico 1993, Jappelli 2004): granulometric heterogeneity of the quarried materials (intrinsic permeability defects); compaction errors and/or the use of inappropriate rollers; the climate conditions during placing; structural discontinuities at the of different geological formations; particle migration and internal erosion phenomena; discontinuities of displacements; dynamic actions; animal actions. Permeability defects are practically unavoidable in earth dams (Talbot & Ralston 1985), underground diaphragm walls (Sherard et al. 1963, Jappelli et al. 1988), grouted zones (Federico 1994, Federico et al. 2002) and foundation soils. They may induce undesirable phenomena, whose severity falls in a wide range. Specific attention deserves the redistribution of the interstitial pressures: soil shear strength decreases and instability phenomena may occur if these pressures increase, Casagrande (1961); internal erosion, Charles (2000), potential local increases of the hydraulic gradients and related drag forces may cause particles migration, that might evolve up to piping and dam collapse (Federico & Musso 1990). For example, with reference to the Nocelle earth dam, on the river Arvo, up to the Sila Mountains in the South Italy, fully operating since 1931, some drillings made between 1995 and 1998 showed sand layers in the clayey core whose thickness varied from 1 cm to 1 m (Catalano et al. 2004). Different effects followed the internal erosion phenomena occurred in the core of the Suorva dam, Nilsson (2007), and of dams of the James bay project,

Lafleur (2007), both made of cohesionless broadly graded moraines. These materials are internally unstable or suffosive; some sinkholes occurred in 1983 and 1989; turbid water and increased leakages were observed in the Suorva dam; strongly changes of the pore pressure dissipation, mainly concentrated near the downstream face of the cores, occurred in dams of the James bay project. The need to understand how a permeability defect modifies the seepage through an embankment dam and causes serviceability (change of discharge, turbid water, piping) or ultimate limit states (local or global instabilities, structural collapse, hydrofracturing) of the dam is well recognized (Jappelli et al. 1996, Jappelli 2004). To this purpose, the 2D steady state seepage flow through zoned earth structures, whose core houses a permeability defect, is carried out by means of a finite element (FE) approach, with the following goals: to evaluate the discharges changes caused by permeability dedefects within the core, as well as the related interstitial pressures and the hydraulic gradients variations; to define an analytical relation between the permeability defect features and the otained discharge rate, through the multiple linear regression of results of seepage flow simulations.

2

NUMERICAL MODELING OF THE SEEPAGE FLOW

2.1 Features of the SEEP/W code The seepage flow through permeable soils is governed by the Darcy’s law and the continuity equation. The

311

solution of the governing partial differential equation (PDE) depends on the boundary conditions (even on the initial conditions if unsteady flow is concerned) assigned to the seepage flow domain; these ones, in turn, are expressed in of piezometric head, interstitial pressure, discharge Harr (1962). The numerical modeling allows to analyse complex cases due to irregularly shaped flow domains and peculiar boundary conditions; the simplified hypotheses through which analytic models may be applied are thus removed. The SEEP/W code analyses the 2D seepage (plane or radial) flow through saturated or partially saturated soils. Steady or transient regimes may be ed for. In the paper, the seepage flow through a zoned earth dam with a central core housing a permeability defect (Aubertin & Chapuis, 2002) is studied. The dam rests on a homogeneous foundation soil layer. The computed variables are: the discharge (Q [m3 /s/m]), the interstitial pressures (u [kPa]), the piezometric gradients (grad h) and the free surface profile. The permeability functions which characterize the core material, the permeability defect, the foundation soil have to be first defined. The permeability functions allow to express the hydraulic conductivity coefficient in of the interstitial pressure (positive or negative). In saturated soils, the pores are filled by the flow and the seepage is maximised; conversely, if the water content decreases (partially saturated soil), the water seeps through a reduced (width and number of the void conduits) effective void section; therefore, the overall permeability reduces too. As a consequence, the hydraulic conductivity of the soil is not constant but it depends on the soil water content. If partially saturated soils are involved, as always occurs for unconfined seepage flows, the Laplace governing equation becomes a non linear PDE. Depending the water content, in turn, on the interstitial pressure, through the permeability function, the hydraulic conductivity can be expressed in function of the interstitial pressure. As a final result, the permeability assumes a constant value for saturated soils and values decreasing with the interstitial pressure reduction (negative pressures, partially saturated soils). The decrease of the permeability coefficient is related to the grain size distribution, as reported in literature. For steady state motion, it is not necessary to define the storage function, that is the volumetric water content in of the interstitial pressure (it shows the soil capacity to store water). 2.2 Problem setting Geometry: Referring to a typical cross section of a zoned earth structure, the geometric and hydraulic variables considered for parametric simulations are defined in figure 1. The height H of the structure ranges between 10 m (dykes) and 50 m (dam). The shapes of the cross

Figure 1. Cross section and definition of the main geometric and hydraulic variables assumed in computations.

Figure 2. Permeability functions of materials.

section and of the dam core are trapezoidal. The slope of the upstream and downstream shells is 3H:1V; the slope of the core is 1H:2V. The width at the core top (Lmin ) is 3 m (dike) or 10 m (dam). The thickness T of the foundation soil is 20 m. The width of the horizontal permeability defect varies from 25% to 100% of the corresponding core width. The permeability defect is alternately located at various heights within the core; its thickness s is constant, equal to 0.5 m (H = 10 m) and 2 m (H = 50 m); its permeability is larger than the permeability of the core material one. Physical properties: the high permeability of the coarse soils, compared to that one of the core, allows to neglect the upstream hydraulic head losses (horizontal free surface); the piezometric head is dissipated through the core. Furthermore, immediately downstream the core, an efficient filter-drain system lowers the free surface that falls into the drain, up to its exit point. So, it does not cross the downstream shell. Then, in 2D steady state seepage flow simulations, only the central core and the foundation soil offer hydraulic resistance (Fig. 2). The saturated conductivity ksat of the core material has been fixed equal to ksat = 1·10−8 m/s. Referring to the foundation soil, high permeabilities or large discharge would result unacceptable. Therefore, the assigned permeabilities (ksat = 0.5·10−8 m/s; ksat = 1·10−8 m/s; ksat = 5·10−8 m/s) refer to an original small permeability or take into the effects of in situ treatments (e.g., grouting) (Fig. 2). The permeability defect is characterized by a coarser grain size distribution. Its saturated conductivity (ksat ) ranges from 10 to 100 times larger than the core one (Fig. 2).

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Figure 3. Boundary conditions of the seepage flow domain. Table 1.

Figure 4. Location of points and sections to evaluate interstitial pressures and hydraulic gradients.

Parameters assumed in FE computations.

constant values kn m/s Dyke (H = 10 m) 1·10−8 Dam (H = 50 m) 1·10−8

H m

T m

8 45

20 0.5 3 20 2 10

s m

Lmin m

core side slope 1H:2V 1H:2V

variable parameters

Dyke (H = 10 m) Dam (H = 50 m)

a/H

Ld /Ldmax

rd

rfs

1/2, 1/4, 0 1/2, 1/4, 0

1/4, 1/2, 3/4, 0.85,1 1/4, 1/2, 3/4, 0.85,1

1, 10, 100 1, 10, 100

0, 0.5, 1.5 0, 0.5, 1.5

2.3 Boundary conditions The boundary conditions of the geotechnical system are described in Fig. 3: constant piezometric head (h) equal to the geometric elevation of the reservoir level, for all the nodes of the upstream shell and at the interface foundation soil-reservoir; discharge Q = 0 for all the nodes at the interface core-downstream side; through a trial and error procedure, it allows to define the position of the free surface exit-point; interstitial pressure u = 0 for all the nodes at the interface foundation soil-downstream drain; an efficient drainage ensures this condition; discharge Q = 0 without the condition of modifiable node for the remaining nodes on the domain border. 2.4

For a foundation soil with negligible permeability (rfs = 0), the discharge is computed along the vertical section, corresponding to the symmetry axis of the core (Sec. A-A , Fig. 4). If rfs = 0 (pervious foundation soil), the chosen section for computation is represented by the piece-wise line composed by the foundation soil-horizontal drainage line and the coredrainage line (Sections D-D , E-E , Fig. 4). Both the core and the foundation soil contribute to the discharge rate towards the horizontal drainage. Hydraulic gradients – The greatest hydraulic gradients occur at the core-downstream shell (sect. D-D ). This interface thus plays a significant role in erosion and suffusion phenomena, which could occur and induce serviceability or ultimate limit states, up to the dam collapse. Along this interface, the core material is not well confined; moreover, it is exposed to high hydraulic gradients; the corresponding drag forces can easily scour the smallest particles through the voids of the filter- drain system, if this one has not been correctly designed, Nilsson (2007). In absence of arching phenomena, the internal volumes of the core are less exposed to erosion risks, because their particles are surrounded by similar particles and exhibit an appreciable interlocking (high effective stresses and shear resistance). Therefore, they better resist to the erosive action carried out by the drag forces (Federico & Musso 1990).

3

Development of computations

The main parameters (Tab. 1) are: the geometric elevation of the defect compared to the core height (a/H ); the defect width compared to the core base width (Ld /Ldmax ), the (saturated) permeability of the defect compared to the core permeability (rd ); the saturated permeability of the foundation soil compared to the core permeability (rfs ). For each model, two schemes have been ed for: (a) negligible permeability of the foundation soil (rfs = 0): the seepage flow takes place only in the core (limit case); (b) permeable foundation soil (rfs = 0): the water seeps both through the foundation soil and the core.

RESULTS OF THE FE SIMULATION

Discharge variations - A permeability defect in the core always gets a discharge increase, in comparison to the case of a homogeneous core (absence of a defect). Results concerning the discharge are reported in figure 5. The presence of a foundation soil whose overall permeability is comparable to that of the core one, further increases the discharges, although the corresponding numerical results substantially do not modify (Fig. 6, dam height H = 50 m, for 3 permeability values of the foundation soil (rfs = 0.5; 1 ; 5). Defect location (a/H ): it plays a moderate influence in the discharge variation; if a/H decreases

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Table 3. Increase of the discharge rate, as a function of the adimensional ratio Ld /Ldmax . Relative length of the perm. defect

Discharge increase

Discharge increase

0,25 < Ld /Ldmax < 0,50 0,50 < Ld /Ldma x < 0,75 0,75 < Ld /Ldmax < 0,95 0,95 < Ld /Ldmax < 1,0

negligible appreciable high very high

0 ÷ 40% 40 ÷ 110% 110 ÷ 280% 280 ÷ 900%

Figure 5. Adimensional discharge rate (H height of the core) (dyke: height H = 10 m; dam: H = 50 m) as a function of the permeability defect characteristics. The permeability of the foundation soil has been neglected.

Figure 7. Interstitial pressure (u) distribution vs the elevation (y), along the cross section B-B (fig. 4).

Figure 6. Discharge rate (dam, height H = 50 m) as a function of the permeability defect characteristics as well as of the permeability of the foundation soils.

(reduction of the defect elevation), being unchanged the remaining parameters, the discharge slightly increases. Length of the defect (Ld /Ldmax ): it affects the discharge increase only if its values become very near to the unity; for smaller values, it does not play a remarkable role (Tab. 3). Permeability of the defect (rd ): it plays a very important role in the modification of discharges. If rd = 100, the increases of the discharge become very significant especially if appreciable rd values are coupled to a significant length of the defect; on the contrary, if rd = 10, the increases of the discharge are appreciable (maximum increase: 70% respect to the discharge obtained for a homogenous core). Permeability of the foundation soil (rfs ): the presence of the foundation soil causes an increase of the discharge which is proportional to its permeability. Two cases have been considered: rfs = 0.5 and rfs = 1: considerable increases of the discharges (up to 140% for H = 10 m) in comparison to the discharges without a pervious foundation soil; rfs = 5: marked increases of the discharges (up to 700% for H = 10 m). The overall discharge increases due to the contribution of the foundation soil, as expected.

Figure 8. Interstitial pressure (u) distribution vs the elevation (Y ), along the cross section C-C (fig. 4).

However, the effective most significant discharge increases occur in absence of the foundation soil, due to the greater volumetric percentage incidence of the permeability defect compared to the whole seepage domain volume. In presence of foundation soil, a not remarkable reduction of the interstitial pressures occurs in the core, proportional to the permeability parameter rfs of foundation soil. Figures 10 (related to a quite remarkable defect Ld /Ldmax = 0.75) and 11 (Ld /Ldmax = 1) highlight that if Ld /Ldmax = 0.75, high hydraulic gradient values localize at the extremities of the defect, which are thus extremely exposed to potential erosive phenomena (Kenney & Lau 1985, Indraratna & Vafai 1997). If Ld /Ldmax = 1, the intensity of the gradients at the edges of the defect appears smaller than that one corresponding to the case Ld /Ldmax = 0.75.

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Figure 12. Free surface as a function of the abscissa x (the origin of x coincides with the lower point of the upstream face). Dyke height H = 10 m; permeability defect elevation H /2; foundation soil permeability neglected (rfs = 0).

Figure 9. Interstitial pressure (u) distribution vs the abscissa (x), along the horizontal cross section E-E , F-F , G-G .

Figure 10. Hydraulic gradient distribution (defect elevation H /4; Ld /Ldmax = 0.75; rfs = 0).

that one of the homogenous core case (one only black curve); near the downstream shell, the permeability defect causes a general lifting of the free surfaces compared to the free surface of a homogenous core, as more marked as greater is the permeability of the defect (the light-grey curves are above the dark-grey curves). Lastly, the curves corresponding to a permeability defect whose width is equal to the core width always lay under the curve (black) related to that one of the homogenous core. The presence of a pervious foundation soil layer induces a free surface lowering, compared to the case (rfs = 0). 4 AN APPROXIMATE ANALYTICAL RELATIONSHIP OF THE DISCHARGE RATE

Figure 11. Hydraulic gradient distribution (defect elevation H /4; Ld /Ldmax = 1; rfs = 0).

Therefore, the most critical situation for the erosive effects often occurs just before than the permeability defect extends along the whole core width. Free surface generally slightly modifies due to the presence of a permeability defect in the core, as those ones considered in simulations. The profiles of the free surfaces (dyke, H = 10), referred to a permeability defect located at the elevation H /2, are represented in figure 12; the defect length and permeability vary. Near the upstream shell, the free surfaces does not significantly vary (dark-grey, light-grey curves); it practically matches the free surface corresponding to

The multiple linear regression model generalizes to the case of many variables the well-known model of the simple linear regression, which considers only one independent variable. The model applies the least square method to get the vector of the parameters to be inserted in the linear regression law of considered base values. Referring to the discharge rate, the main parameters are x1 = H /H ; x2 = rfs ; x3 = rd ; x4 = a/H ; x5 = α(rd ) · Ld /Ldmax ; x6 = β(rd ) · (Ld /Ldmax )3 . The α(rd ), β(rd ) functions allow to optimize the coefficients xi (i = 1, . . . 6); they are defined as follows:

the linearly decreasing function α(rd ) assumes the values 1 if rd = 10 and 0 if rd = 100;

the linearly increasing function β(rd ) assumes the values 1 if rd = 100 and 0 if rd = 10.

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The adimensional discharge rate y = Q/ksat · H is:

where c1 = 6.548; c2 = −7.061; c3 = 0.513; c4 = 5.247· 10−3 ; c5 = −0.371; c6 = 0.377; c7 = 3.236. 5

CONCLUDING REMARKS

The effects of permeability defects in the core of a zoned earth structure, resting on a permeable soil layer, have been numerically (FEM) analysed. Several cases have been taken into and simulated; numerical results allowed us to relate the features of the permeability defect (location, length, permeability) to computed discharge, interstitial pressures, piezometric gradients, leakage. The further re-organization of results, by applying the multiple linear regression technique, allowed us to work out an analytical relationship, referred to the discharge rate, in a adimensional form, for both cases of presence/absence of foundation soil layer. The law synthetically describes obtained results (336 simulations) and allows to foresee results pertaining to different cases, as in current seepage problems characterized by a permeability defect. REFERENCES Aubertin, M. & Chapuis, R.P. 2002. A simplified method to estimate saturated and unsaturated seepage through dikes under steady-state conditions, Can. Geot. J., 1321–1328. Casagrande,A. 1961. Control of Seepage through Foundation and Abutments of Dams, Geotechnique, XI, 161–181. Catalano, A., Federico F. & Jappelli, R. 2004. Analisi del comportamento della diga di Nocelle dopo 70 anni di esercizio. XXII Conv. Naz. di Geotecnica “Sicurezza ed Adeguamento delle Opere Esistenti”, settembre, Palermo. Charles, J. A. 2000. Internal Erosion in European Embankment Dams. Report of the WG on Internal Erosion in Embankment Dams, Euro Club I.C.O.L.D., March.

Federico, F. & Musso, A. 1990. Limit state design of s and transitions in embankment dams. L’Ingegnere, A.N.I.A.I., LXV, n. 1–4, pp. 49–56. Federico, F. 1994. Numerical analysis of the effectiveness of imperfect underground barriers. 1st Int. Congr. on “Environmental Geotechnics”, Edmonton. Federico, F., Jappelli, R. & Marchetti, A. 2002. Effectiveness of water barriers in dam foundations. 5th Eur. Conf. Num. Meth. in Geotech. Engrg, 4/6, September, Paris. Harr, M. E., 1962. Groundwater and seepage. McGraw-Hill. Indraratna, B. & Vafai F. 1997. Analytical Model for Particle Migration Within Base Soil – Filter System. J. of Geotech. Geoenv. Engrg., A.S.C.E., 123(2), pp. 100–109. Jappelli, R., Valore, C. & Federico, F. 1988. Imperfect underground barriers under transient seepage conditions. 6th Int. Conf. on Num. Meth. in Geomech., 1, pp. 637–641, Innsbruck. Jappelli, R. & Federico, F. 1993. I difetti delle costruzioni geotecniche. Identificazione e strategie di intervento. Corso C.I.A.S. “Evoluzione nella Sperimentazione per le Costruzioni”, Rel. gen. della sessione “Affidabilità delle indagini sulle fondazioni”, pp. 109–130, Rovinj. Jappelli, R., Federico, F. & Musso, A. 1996. – Analysis of seepage limit states in embankment dams. XVI Int. Conf. on Large Dams, Q. 73, Florence. Jappelli, R. 2004. Difetti delle grandi dighe e rimedi strategici. Convegno Problemi Strutturali nell’Ingegneria delle Dighe, Lincei, febbraio, Roma. Kenney, T.C. & Lau, D. 1985. Internal stability of granular filters. Canadian Geotechnical Journal, Vol. 22 (2), pp. 215–225. Lafleur, J. 2007. Internal stability of particles in dam cores made of cohesionless broadly graded moraines. Internal Erosion of Dams and their Foundations – Fell & Fry (eds), Taylor & Francis Group, London, ISBN 978-0-415. Nilsson, Å. 2007. The susceptibility of internal erosion in the Suorva Dam. Internal Erosion of Dams and their Foundations – Fell & Fry (eds), Taylor & Francis Group, London, ISBN 978-0-415. Sherard, J.L., Woodward, R. J., Gizienski, S. F. & Clevenger, A. 1963. Earth and Earth-Rock Dams. Wiley. Talbot, J. R. & Ralston, D. C. 1985. Earth Dam Seepage Control, SCS Experience. Symp. on “Seepage and Leakage from Dams and Impoundments”, A.S.C.E., Geotechnical Engineering Division, Denver, Colorado, May, 44–6.

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Unsaturated soil mechanics


A comparison between numerical integration algorithms for unsaturated soils constitutive models F. Cattaneo Università degli Studi di Brescia, Brescia, Italy

G. Della Vecchia Politecnico di Torino, Torino, Italy

C. Jommi & G. Maffioli Politecnico di Milano, Milano, Italy

ABSTRACT: Constitutive models for soils in unsaturated conditions must for the role of suction on the response of the volume element. Different constitutive formulations have been proposed to this aim in the recent past by different authors. Focusing the attention on elastic-plastic models with generalised hardening, a comparison between classical numerical algorithms, which can be adopted for the numerical integration of the soil skeleton constitutive law, is discussed. A refined Runge-Kutta-Dormand-Prince explicit formula and a fully implicit algorithm are compared one to the other in of convergence order and computational cost, on a typical stress path dependent on both strain and suction variations. Results are shown with specific reference to the axis-symmetric formulation of an elastic-plastic constitutive model, in which mixed isotropic and rotational hardening is ruled by volumetric plastic strain and degree of saturation. Iso-errors maps are presented to evaluate the performance of the implicit algorithm.

1

INTRODUCTION

The engineering applications involving soils above the water table, or soils used as construction ma-terials, take advantage of constitutive formulations able to tackle the non-linear, irreversible behaviour of the volume element in unsaturated conditions. In this case, the response of the soil skeleton is ruled by the coupled action of total stress, water pressure and gas pressure. In the last years, different constitutive formulations have been proposed to model the behaviour of soils in unsaturated conditions. With reference to the class of models developed in the framework of elastoplasticity, the constitutive equations for the soil skeleton may be written in the general form (Gens et al. 2006):

where σ˙ c is a suitable constitutive stress rate, Dc is the tangent elastic-plastic tensor, ε˙ is the total strain rate and ε˙ s is a measure of that part of the strain rate which is due to suction variations, where suction, s, is defined as the difference between the gas and the water pressures. The choice of the appropriate constitutive stress, and of the way in which the suction induced strains are calculated, in general distinguish the different models one from the other. All the appropriate constitutive models require numerical procedures to integrate the stress paths,

due to the high non linearities introduced by hydromechanical coupling. Alternative strategies may be pursued to this aim. Typically, explicit integration schemes may prove to be a valuable choice during the model development stage, as they do not require, in general, heavy analytical effort. In view of the application of the models to real scale problems more efficient numerical algorithms are required in general. Common numerical algorithms have been applied to integrate a constitutive model for unsaturated soils, conceived in the framework of elasto-plasticity with generalised hardening, dependent on plastic strain and degree of saturation. Convergence, iso-error maps and computational effort are analysed in the following, with reference to a coupled hydro-mechanical path involving plastic strains induced by both load and suction variations. 2

ELASTIC-PLASTIC CONSTITUTIVE FORMULATION

In the context of a general coupled finite element analysis, the integration is carried out at the Gauss point level, given the strain, the water pressure and the gas pressures increments. To describe the soil response, both the hydraulic and the mechanical constitutive relationships must be introduced. The hydraulic constitutive law, named

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water retention curve, describes the depen-dence of the amount of water stored in the soil pores as a function of suction. Adopting the degree of sa-turation as a useful measure of the amount of soil water, and limiting the attention to a wide class of models relating it to the suction and to the total void ratio, the retention curve can be described in the general functional form:

where h and η are hardening functions. The first term describes the changes in the internal variables associated with plastic strains, while the second term describes a reversible evolution of the elastic locus with the hydraulic history, ruled by the changes in the degree of saturation. The plastic multiplier λ is subjected to the classical Kuhn-Tucker conditions:

For this class of retention models, the variation of the degree of saturation in the incremental step depends directly on the suction increment and on the total volumetric strain increment. Being uncoupled from the plastic strain increment, the variation of the degree of saturation can be calculated by a suitable integration method, before entering the mechanical constitutive behaviour routine. Therefore, at the beginning of the latter numerical integration procedure, the total strain increment, the suction increment, and the degree of saturation increment are known. The unknowns to be found are the updated constitutive stress and the updated plastic variables. Adopting a linear kinematic description, the strain rate is decomposed additively in an elastic, reversible part, ε˙ e , and a plastic, irreversible part, ε˙ p :

stating that plastic strains may occur only for states on the yield surface. Let:

In the class of models considered herein the average ˆ soil skeleton stress, σ,

is adopted as constitutive stress. In the previous expression δ is the vector whose entries are 1 for the normal stress components and 0 for the shear components. The constitutive stress is the static variable linked to the elastic strains:

The elastic locus is bounded by a convex yield surface, f , defined in of internal variables q, depending on plastic strains and on the degree of saturation:

Plastic strains are defined prescribing a general non associated flow rule

denote the gradients of f with respect to σ and q. From the consistency condition, λf˙ = 0, the constitutive equation (Eq. 5) and the flow rule (Eq. 7), the following expression for the plastic multiplier is ˙ obtained, in of ε˙ and Sr:

provided that:

3

INTEGRATION ALGORITHMS

In a boundary value problem the load path is divided into steps. Let n + 1 be a time step bounded by tn and p tn+1 . The state of the material (σˆ n , qn , εn , εn , Sr n ) is assumed to be completely known at time tn . The p unknowns σˆ n+1 , qn+1 and εn+1 must be determined, given the assigned increments ε and Sr, through the integration of Equations 5-9. Two classical approaches (see e.g. Ortiz & Popov 1985) have been implemented in a constitutive driver, able to tackle any stress path under general mixed static and kinematic control conditions. A refined explicit algorithm of the Runge-Kutta family (e.g. Sloan 1987, Jakobsen & Lade 2002), and a fully implicit backward Euler scheme (Borja & Lee 1990, Tamagnini et al. 2002) were adopted. In both cases, the initial trial solution is calculated by a fully elastic prediction, which is checked against an initial trial yield surface, in which the internal variables have already been updated for the change in the degree of saturation. 3.1 Explicit algorithm

where g is the plastic potential, and λ is the plastic multiplier. The evolution of the internal variables q is provided by the following generalised hardening law:

The trial elastic state is calculated with an explicit approach:

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trial trial If fn+1 := f (σˆ trial n+1 , qn+1 ) ≤ 0, the trial state satisfies the Kuhn-Tucker conditions (Eq. 9), and it coincides with the converged solution in the step. Otherwise, if trial fn+1 > 0, a plastic integration step must be performed. In the latter case, if at the beginning of the step the stress state was strictly inside the yield locus, fn < 0, the intersection with the trial yield surface must be found before starting the plastic integration step. To find the stresses and the internal variable at the yield surface intersection, the scalar quantity α, which satisfies the equation:

must be found. In general, Equation 14 is non-linear with respect to the variable α and can be solved by a variety of numerical methods. As α is bounded in the interval 0 ÷ 1, Regula-Falsi procedure proves to be a valuable choice (Quarteroni et al. 2007). Stresses and internal variables on the yield surface become the initial values at the beginning of the plastic integration step, which consists in solving the non-linear system of ordinary differential Equations 5, 8. Several methods have been proposed to integrate the non-linear constitutive equations. All of them require a sub-stepping procedure, to limit the integration error. Among them, the algorithms belonging to the Runge-Kutta’s class, with automatic substepping and error control, have proven to be reliable and efficient. The basic idea of this class of algorithms is to modify the substep size by comparing two different order estimates of the problem solution. The Runge-Kutta-Dormand-Prince algorithm was chosen here. When an explicit algorithm is adopted, the plastic extension does not force the stress state to lie on the yield surface. The constraint fn+1 = 0 has to be checked at the end, and, in the case it were violated, consistency must be restored. The method proposed by Sloan (1987) can be adopted to this aim. Assuming that the internal variables qn+1 remain constant, the stress state is modified in order to restore the consistency condition. A scalar β must be found, so that

where σˆ ∗n+1 is the stress state at the end of the explicit integration procedure, and σˆ n+1 is the final, corrected stress state, satisfying the consistency condition, for the given internal variables. 3.2

Implicit algorithm

The classical elastic-plastic operator split (Borja & Lee 1990) of the original problem (OR) may be extended to hardening rules dependent on both the plastic strains and the degree of saturation, as the sum of an elastic predictor (EP) and a plastic corrector (PC) (Tamagnini et al. 2002):

=

OR ε˙ e = ε˙ tot − λQ ˙ q˙ = λh + Srη

EP

+

ε˙ e = ε˙ tot ˙ q˙ = Srη

PC ε˙ e = −λQ q˙ = λh

Trial values are calculated solving the elastic predictor problem:

A suitable accurate procedure must be envisaged to solve the non linear equations (Eqs. 16). In fact, the accuracy of the elastic prediction affects significantly the final accuracy of the whole integration procedure, as the results presented in the following will demontrial strate. A closed form evaluation of qn+1 is strongly suggested when the function η is sufficiently simple to be integrated analytically, especially for large increments of the degree of saturation. trial If the constraint fn+1 ≤ 0 is violated, the trial state lies outside the yield locus, and consistency needs to be restored. The trial state becomes the initial state for the following system:

where the unknowns are εe , qn+1 and λ. Newton’s algorithm can be adopted to find the solution of the non linear sysyem.

4

NUMERICAL IMPLEMENTATION

The two integration algorithms have been implemented in a constitutive driver, in which the momentum balance, the water mass balance and the air mass balance are solved simultaneously at the representative volume element level. The transition between a two-phase (saturated or dry) system and the general three-phase system is tackled with a physically based approach, by including water vapour and dissolved gas in the formulation. The hydro-mechanical model proposed by Romero & Jommi (2008) is adopted here to describe the behaviour of unsaturated soils. Basically, its formulation is an extension to unsaturated conditions of the mixed isotropic-rotational hardening model proposed by Dafalias (1987) for saturated soils. The relevant constitutive equations will be summarised here, for axis-symmetric stress and strain paths only, adopting the usual geotechnical triaxial variables.

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The elastic behaviour is described by:

The yield function,

depends on two internal variables, pˆ c and Mα ruling isotropic and rotational hardening, respectively. An associated flow rule is adopted to describe plastic strains (f = g). The hardening functions h, describing the evolution of the internal variables with plastic strains, are:

The original formulation of the model was adopted here, although the hardening law (Eq. 20b), depending on volumetric plastic strain only, does not lead to a unique inclination of the yield surface at critical state. The evolution of the preconsolidation stress pˆ c with the degree of saturation, i.e. the so called LoadingCollapse (LC) curve, is written in the finite form,

The LC curve is thus the closed form of the hardening function η for the degree of saturation. The model is able to reproduce quite well a series of typical tests on clayey soils (Romero & Jommi 2008). One of these tests is simulated here, to evaluate the performance of the numerical integration algorithms. The test chosen is characterised by hardening promoted by both plastic strains and changes in the degree of saturation. A sample of Boom clay was compacted in oedometer, on the dry side of the optimum Proctor. Oedometer compaction gave the soil an anisotropic structure, described by an initial yield surface rotated with respect to the hydrostatic axis. After compaction, the sample was mounted in a controlled-suction triaxial cell, and it was compressed isotropically to a final total external confining stress of p = 600 kPa. Afterwards, a wetting-drying-wetting cycle was performed. The experimental data showed that the initial anisotropy was completely erased along the first wetting path. In the following drying-wetting cycle the response of the soil was isotropic. Significant volumetric plastic strains were accumulated in the first wetting path, due to the tendency of the yield locus to reduce with increasing saturation, which is compensated by plastic strain hardening. Due to initial anisotropy, also irreversible shear strains were observed. Further volumetric plastic strains were recorded in the following

Figure 1. Wetting-drying-wetting cycle at constant total confining pressure on Boom clay sample: experimental data and exact numerical simulation for the evolution of volumetric strain.

drying stage, were the suction mainly acted as an increasing external stress (Romero & Jommi 2008). Relevant experimental data for the volumetric strains are reported in Figure 1. The exact numerical solution of the constitutive model prediction is also reported in the figure. The solution was obtained by means of the explicit or the implicit algorithm, by subdividing the whole stress path in a huge amount of steps (10,000 steps for the explicit procedure). To evaluate the performance of the two different algorithms, a set of numerical analyses were performed, changing the number of time intervals in which the whole stress path was subdivided. All the analyses were carried out on a Intel® Core™ 2 Duo processor (E4600 at 2.40 GHz). The results of the parametric analyses for the explicit and the implicit integration algorithm are reported in Figures 2a,b, respectively. The numerical simulations confirm that the refined explicit algorithm is rather efficient, although only conditionally stable. A minimum of 80 time steps are necessary to obtain a sufficiently accurate solution. On the contrary, the implicit algorithm is unconditionally stable, and a reliable solution can be obtained with a reduced number of steps. The error of the numerical solution with respect to the exact solution is shown in Figures 3, 4, as a function of the number of steps, and of the U time necessary for the Newton-Raphson procedure to integrate the whole wetting-drying-wetting cycle. The error is computed with reference to the value of the axial strain at the end of the whole cyclic path. A convergence order of one characterises both algorithms, although unconditional stability of the implicit algorithm allows for a lower global number of steps and lower U time. Convergence of the global Newton-Raphson algorithm for the solution of the balance equations is shown in Figure 5 for the two methods. The figure refers to

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Figure 3. Error on the final volumetric strain vs number of steps.

Figure 2. Numerical solutions for volumetric strain obtained with different number of steps: (a) refined explicit procedure and (b) fully implicit Euler scheme.

a time step in the first wetting path, where significant volumetric plastic strain is promoted by the reduction of suction. Adopting a consistent stiffness matrix guarantees convergence of order two for the implicit algorithm (Simo & Taylor 1985), while the explicit procedure convergence rate decreases rapidly as the number of iteration increases, as only the continuous stiffness matrix was evaluated. Iso-error maps for the implicit integration algorithm are presented in Figures 6, 7.The error is defined on the predicted stress state with respect to the exact solution:

The error was calculated by integrating the load paths with different assigned increments of the degree of saturation and of the volumetric strain, starting from a stress point at yield. In the first case, the internal

Figure 4. Error on the final volumetric strain vs U time.

variables were updated exactly at the beginning of the integration procedure, by means of the closed form of η (Sr) (Eq. 21). In the second case the initial closed form calculation of η (Sr) (Eq. 16b) was substituted by its numerical estimate. The results plotted in Figure 6 demonstrate that the error in the calculated stress keeps small even for relatively high assigned values of the volumetric strain increment. At fixed volumetric strain increment, the error is slightly higher when relevant plastic volumetric strain is accumulated at increasing degree of saturation. The relevance of the saturation hardening function on the global efficiency of the numerical algorithm is demonstrated in Figure 7. Linearisation introduced by the numerical estimate produces a much faster increase of the error, especially at decreasing degree of saturation.

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Figure 7. Iso-error map for the implicit procedure with a linearised numerical estimate of saturation hardening.

Figure 5. Convergence rate of the global Netwon–Raphson procedure.

The numerical results showed that the global efficiency of the algorithms is affected by the way in which the generalised hardening laws are tackled at the predictor level. Closed form solutions, if available, are clearly advantageous. REFERENCES

Figure 6. Iso-error map for the implicit procedure with a closed form calculation of saturation hardening.

5

CONCLUSIONS

The different formulations proposed to model the constitutive behaviour of unsaturated soils share high non linearities coming from strong hydro-mechanical coupling. Refined numerical algorithms are mandatory for their implementation in finite element codes. The use of a refined Runge-Kutta-Dormand-Prince explicit algorithm and of a fully implicit Euler scheme were tested to this aim, on a model developed in the framework of elastoplasticity with generalised hardening. Both procedures proved to be rather efficient in the integration of the hydro-mechanical laws at the Gauss point level, although conditional stability of the explicit one may pose severe limits on the maximum step size. The advantages of the implicit procedure emerge when the solution of the global balance equations is tackled, if a consistent tangent matrix approach is implemented.

Borja, R. I. & Lee, S. R. 1990. Cam-clay plasticity, part 1: Implicit integration of elasto-plastic constitutive relations. Computer Methods in Applied Mechanics and Engineering 78(1): 49–72. Dafalias, Y. 1987. An anisotropic critical state soil plasticity model. Mechanics Research Communications 13(6), 341– 347. Gens, A., Sanchez, M. & Sheng, D. 2006. On constitutive modelling of unsaturated soils. Acta Geotechnica 1(3): 137–147. Jakobsen, K. P. & Lade, P.V. 2002. Implementation algorithm for a single hardening constitutive model for frictional materials. International Journal for Numerical and Analytical Methods in Geomechanics 26(7): 661–681. Ortiz, M. & Popov, E.P. 1985. Accuracy and stability of integration algorithms for elastoplastic constitutive relations. International Journal for Numerical Methods in Engineering 21(9): 1561–1576. Quarteroni, A., Sacco, R. & Saleri, F. 2007. Numerical Mathematics (2 ed.). Milano:Springer. Romero, E. & Jommi, C. 2008. An insight into the role of hydraulic history on the volume changes of anisotropic clayey soils. Water Resour. Res. 44, doi 10.1029/2007WR006558. Simo, J. C. & Taylor, R.L. 1985. Consistent tangent operators for rate-independent elastoplasticity. Computer Methods in Applied Mechanics and Engineering 48(1): 101–118. Sloan, S. W. 1987. Substepping schemes for the numerical integration of elastoplastic stress-strain relations. International Journal for Numerical Methods in Engineering 24(5): 893–911. Tamagnini, C., Castellanza, R. & Nova, R. 2002. Numerical integration of elastoplastic constitutive equation for geomaterials with extended hardening laws. In Pande & Pietruszczak (eds), Numerical Models in Geomechanics – NUMOG VIII: 213–218.

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Comparison of stress update algorithms for partially saturated soil models Matthias Hofmann ALPINE BeMo Tunnelling GmbH, Austria

Günter Hofstetter Institute of Basic Sciences in Civil Engineering, University of Innsbruck, Austria

Alexander Ostermann Department of Mathematics, University of Innsbruck, Austria

ABSTRACT: In this paper different stress update algorithms of constitutive models for partially saturated soils are compared on the basis of the Barcelona Basic Model (BBM). They include both explicit and implicit integration schemes applying the Richardson extrapolation method for sub-stepping with error control. The comparison of the mentioned stress update algorithms is performed for prescribed ranges of volumetric and deviatoric strain increments on the basis of two sets of material parameters for the BBM. Finally, some tests of an extensive experimental program are selected for computing the constitutive response of an unsaturated soil and comparing the numerical results with the experimental data.

1

INTRODUCTION

The development of constitutive models for partially saturated soils and the implementation into FE-programs are ongoing research topics. The latter requires selection of a suitable stress update algorithm. In addition to accuracy, robustness and efficiency of the employed stress update algorithm play a decisive role especially for large-scale FE-analyses. This is the motivation for comparing different stress update algorithms of constitutive models for partially saturated soils. The comparison is performed on the basis of the Barcelona Basic Model (BBM), which is currently probably the most well-known constitutive model for partially saturated soils. It is formulated in of two independent stress parameters, consisting of net stress and matric suction. The investigated stress update algorithms include both explicit and implicit integration schemes. The former include a forward Euler integration scheme and a semi-explicit integration algorithm (Mittendorfer 2006). The Richardson extrapolation method, described in (Fellin et al. 2009), is used as the basis for sub-stepping with error control, which is an essential ingredient especially for the explicit stress update algorithms. The considered implicit schemes include a general return mapping algorithm (Simo and Hughes 1998) and a computationally efficient version of a return mapping algorithm. Whereas the former requires the solution of a system of several nonlinear equations at the integration point level, the latter is characterized by analytical integration of

the hardening law and by solving only a nonlinear scalar equation at the integration point level. Moreover, enhancements of implicit integration methods with sub-stepping and error control techniques are investigated. In addition, a fifth-order Runge–Kutta stress update algorithm with error control is included in this investigation (Hairer and Wanner 1996). The comparison of the mentioned stress update algorithms is performed for prescribed ranges of volumetric and deviatoric strain increments on the basis of two sets of material parameters for the BBM. Finally, some tests of an extensive experimental program are selected for computing the constitutive response of an unsaturated soil and comparing the numerical results with the experimental data.

2

UNSATURATED SOIL MODEL

For the present three-phase formulation the Barcelona Basic Model (BBM) is employed as constitutive model for describing unsaturated soil behavior. It is employed here in its original version, proposed in (Alonso et al. 1990), because the latter was agreed as the basis for extensive benchmark activities within the framework of the MUSE network (Marie Curie Research Training Network). The BBM is formulated in of the net stress tensor σ and the capillary pressure pc . The net stress

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is the total stress in excess of the pore air pressure pa , and the capillary pressure or matric suction

is the difference between the air pressure pa and the water pressure pw . For stress states located within the elastic domain, enclosed by the yield surface, the elastic volumetric and deviatoric strain rates are given as

Figure 1. Yield surface of the BBM for different values of the preconsolidation pressure.

with the material parameters κ and κs , representing the elastic stiffness for changes of the mean net pressure p = (σ : I)/3 and for changes of the capillary pressure pc , respectively. Further, e, patm , s˙ij and G denote the void ratio, the atmospheric air pressure, the deviatoric stress rate and the shear modulus, respectively. It follows from (31 ) that the elastic volumetric strain rate ε˙ ev depends on both the mean net pressure p and the capillary pressure pc . Within the elastic domain the stress point (p , e) lies on the unloading-reloading line (URL) with slope κ. For isotropic plastic conditions it lies on the isotropic compression line (ICL) with the suction-dependent slope

λ(pc ) describes the soil stiffness during plastic loading in a hydrostatic test for a given capillary pressure pc in of the respective stiffness λ(0) at saturated conditions and the material parameters r and β. The intersection point of the URL and the ICL is the preconsolidation pressure p0 . The ICL is defined by the slope λ(pc ) and the void ratio N (pc ) − 1 at p = 1 with N (pc ) denoting the respective specific volume. From the volumetric behavior of the BBM follows

In (7) M defines the slope of the critical state line. ps and p0 both depend on the capillary pressure according to (8). For negative values of p the intersection of the yield surface (7) with the plane J2 = 0 is given by ps according to (81 ) with the material parameter ks describing the increase in cohesion due to the capillary pressure. The preconsolidation pressure p0 and the one for saturated conditions (p0 )∗ are located on the so called loading collapse yield curve (LC curve) according to (82 ). This curve is the intersection of the yield surface with the plane J2 = 0 for positive values of p . Here, pref serves as a reference pressure such that for (p0 )∗ = pref (82 ) degenerates to p0 = pref = const. The plastic strain rate is determined from the nonassociated flow rule

with the flow potential

where α is a constant. The hardening law relates the rate of the preconsolidation pressure at saturated conditions (p0 )∗ , which serves as the hardening parameter, p to the volumetric plastic strain rate ε˙ v by

from which

(11) describes the evolution of the yield surface. The latter is shown for two different values of (p0 )∗ in Fig. 1.

is obtained. The yield surface is defined as

3 with the second invariant of the deviatoric stress tensor J2 = sij sij /2 and

STRESS UPDATE ALGORITHMS

Within the framework of a FE-analysis the plastic strains, the hardening variables and the stresses of the soil skeleton at a given time instant tn+1 are determined from the respective known values at tn and from given increments of the total strain of the soil skeleton and of the capillary pressure, ε and pc , for the current time step t = tn+1 − tn .

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To this end, the following stress update algorithms for the BBM are investigated regarding accuracy, robustness and efficiency:

quantities refer to the current values at tn+1 . In case of a constant value of G from (32 ) it follows

(a) an explicit stress update algorithm, characterized by forward integration of the constitutive rate equation Inserting (14) into (15) gives c

(b)

(c)

(d)

(e)

with Cep = ∂σ/∂ε and ,ep = ∂σ/∂pc denoting the constitutive tangent operators; it is combined with adaptive sub-stepping and error control based on the Richardson extrapolation method; a general return mapping algorithm (Simo and Hughes 1998), which is characterized by backward Euler integration of the rate equations for the plastic strains (9) and the hardening variable (11), and enforcing the condition f = 0 for the yield function (7) at tn+1 ; it requires solving a system of nonlinear equations, consisting of the consistency parameter, the net stress and the hardening variable; an optimized return mapping algorithm, which is described in the subsequent subsection. It is characterized by solving only a scalar nonlinear equation for the unknown first invariant of net stress I1 ; a semi-explicit stress update algorithm (Mittendorfer 2006), which is characterized by explicit integration of the rate equations for the plastic strains (9) and the hardening variable (11) and by enforcing the condition f = 0 for the yield function (7) at tn+1 for determining the consistency parameter; it is combined with sub-stepping and error control along the lines of the explicit algorithm; the implicit fifth-order Runge–Kutta integration algorithm with error control RADAU5, proposed in (Hairer and Wanner 1996).

The term enclosed by the brackets is a scalar quantity, hence, sij and sijTrial differ only by a scalar factor. Thus, from (16) it follows

Making use of γ = (εv − εev )/(3∂g/∂I1 ), resulting from (141 ), yields

In (18) the incremental volumetric strain εv is known from the current estimate of the displacement increment at tn+1 . Further, J2 and εev in (18) can be replaced by

and

3.1 Optimized return mapping algorithm For deriving a computationally efficient version of the return mapping algorithm the flow rule (9) is split into a volumetric and deviatoric part

following from (7) and from integration of the rate constitutive equation (31 ). The rate of the void ratio is given by

Backward Euler integration of (13) yields

Integration of (21) yields the value of the void ratio at tn+1

where γ = γt. ˙ Note that quantities with the subscript n refer to the converged values at tn , whereas all other

(18) together with (19), (20), (22) and (6) represents a nonlinear scalar equation for the unknown I1 (or p = I1 /3), which can be solved, e.g., by the Newton method. Once p has been determined from this equation, it is inserted into (6), yielding p0 , and the latter into the recast form of (82 ) yielding the hardening parameter (p0 )∗ .

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Figure 2. Integration errors for the general return mapping algorithm (top) and the optimized return mapping algorithm (bottom).

3.2

Comparison of the investigated stress update algorithms

The comparison of the investigated stress update algorithms is performed on the basis of two different sets of material parameters for the BBM, provided in (Alonso et al. 1990), for prescribed combinations of volumetric and deviatoric strain increments ranging from 0 up to 3%. For this relavitely large range of strain increments Fig. 2 shows a comparison of the integration errors obtained by means of the general return mapping algorithm and the optimized return mapping algorithm. A particular point of the diagrams shown in Fig. 2 indicates the error of a single step stress update for a particular combination of volumetric and deviatoric strain increment (εv , εs ). E.g., the point (εv = 2%, εs = 3%) represents the integration error for the strain increment εv = 0.02 and εs = 0.03, obtained by a single step backward Euler integration. The error of the computed stress is defined as a relative error, related to the "exact" value for the respective stress component computed by the RADAU5 algorithm (Hairer and Wanner 1996) prescribing an extremely small error tolerance of 10−10 . According to Fig. 2 the integration errors for the investigated

Figure 3. Work precision diagrams for two sets of material parameters: (a) explicit stress update, (b) general return mapping algorithm, (c) optimized return mapping algorithm, (d) semi-explicit stress update algorithm, (e) implicit fifth-order Runge–Kutta algorithm.

strain increments reach up to 40%. Contrary to the general return mapping algorithm the optimized return mapping algorithm gives the exact solution for hydrostatic strain paths due to the analytical integration of the hardening law. Because of the large integration errors, similar to the explicit and semi-explicit stress update algorithm, the return mapping algorithms are also enhanced by adaptive sub-stepping and error control. A comparison of the efficiency of the investigated stress update algorithms for prescribed maximum values of the integration error, ranging from 10−1 to 10−10 , is shown in Fig. 3. The diagrams are based on stress updates for 25 combinations of volumetric and deviatoric strain increments of 0.5%, 0.75%, 1.0%, 1.25% and 1.5%. The mean values of the computed errors and the mean values of the number of required arithmetic operations are shown in the diagrams of Fig. 3. It follows from Fig. 3 that for a prescribed error tolerance the optimized return mapping algorithm is by far more efficient than the general return mapping algorithm and it is even more efficient than the explicit integration method. The RADAU5 algorithm is very efficient for very small prescribed values of error tolerances.

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Table 1.

Identified set of material parameters.

Parameter

Dimension

Identified value

κ G κs (p0 )∗ M ks pc r β λ(0) N (0)

[−] [MPa] [−] [MPa] [−] [−] [MPa] [−] [MPa−1 ] [−] [−]

0.010 104.4 0.0001 0.094 1.013 0.627 10−11.9 0.953 99.08 0.079 1.436

4

COMPARISON OF MEASURED AND COMPUTED RESPONSE

An extensive experimental program for the constitutive behavior of unsaturated soils is documented in (Bucio 2002). This set of experimental results was also used for a parameter determination benchmark within the framework of the MUSE network (Marie Curie Research Training Network). The soil is a sandy silt with an initial value of matric suction of 0.8 MPa and compacted with an isotropic confining pressure of 0.6 MPa. From the extensive test data five tests are selected in the present work for a comparison of measured and computed constitutive response. They include a hydrostatic test at saturated conditions (SAT1), a suction controlled hydrostatic test (TISO-1), and three different triaxial tests (IS-OC-03, IS-NC-06, IWS-NC-02). The abbreviations “OC” and “NC” indicate over consolidated and normal consolidated tests, respectively, the numbers of the “IS” tests indicate the pressure at which shearing started and “IWS” denotes tests including wetting paths. In a first step the 11 constitutive parameters of the BBM are determined from the test results of the five selected tests by minimizing the sum of weighted least square errors between given experimental data and the results, predicted by the constitutive model. Since such optimization problems are commonly characterized by many local minima, global optimization techniques are a good choice. In the present context a particle swarm optimization method was applied. Details on the identification procedure are provided in (Hofmann et al. 2009), the identified material parameters are summarized in Table 1. Comparisons of the stress strain curves computed on the basis of the identified material parameters with the test results are shown in Figure 4. The calculations are conducted with the optimized return mapping algorithm described in Section 3.1. 5

CONCLUSIONS

In this paper several stress update algorithms were compared with respect to accuracy and efficiency: (a) an explicit stress update algorithm, (b) a general

Figure 4. Comparison of computed constitutive response with experimental data.

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return mapping algorithm, (c) an optimized return mapping algorithm, (d) a semi-explicit stress update algorithm, and (e) an implicit fifth-order RungeKutta stress update algorithm. Large integration errors were encountered for the return mapping algorithms when larger strain increments are integrated in one step. Hence, similar to the explicit and semi-explicit stress update algorithm, they are enhanced by adaptive sub-stepping and error control. For a prescribed error threshold value the optimized return mapping algorithm is by far more efficient than the general return mapping algorithm and it is even more efficient than the explicit integration method. The RADAU5 algorithm is very efficient for very small prescribed values of error tolerances. The application of the optimized return mapping algorithm for computing the stresses for five representative tests of the extensive experimental program (Bucio 2002) showed good correspondence between computed stresses and test results. The application of the optimized return mapping stress update algorithm in the context of the numerical simulation of the impoundment of an earth dam is documented in (Pertl et al. 2009). ACKNOWLEDGEMENTS

REFERENCES Alonso, E. E., A. Gens, and A. Josa (1990). A constitutive model for partially saturated soils. Géotechnique 40, 405–430. Bucio, M. B. (2002). Estudio experimental del comportamiento hidro-mecánico de suelos colapsables. Ph. D. thesis, Universitat Politècnica de Catalunya. Fellin, W., M. Mittendorfer, and A. Ostermann (2009). Adaptive integration of constitutive rate equations. Computers and Geotechnics 36(5), 698–708. Hairer, E. and G. Wanner (1996). Solving Ordinary Differential Equations II, Stiff and Differential-Algebraic Problems (2nd rev. ed.). Springer Berlin Heidelberg. Hofmann, M., T. Most, and G. Hofstetter (2009). Parameter identification for partially saturated soil models. In Proceedings of the Second International Conference on Computational Methods in Tunnelling, pp. 701–708. Marie Curie Research Training Network (2009). http://muse .dur.ac.uk. (April 17, 2009). Mittendorfer, M. (2006). Interne Differentiation nichtlinearer anelastischer Materialmodelle. Master’s thesis, LeopoldFranzens-Universität Innsbruck. Pertl, M., M. Hofmann, and G. Hofstetter (2009). Coupled numerical analysis of an embankment dam. In Proceedings of the Second International Conference on Long Term Behaviour of Dams, pp. 519–522. Simo, J. and T. Hughes (1998). Computational Inelasticity. Springer New York.

Financial by a scholarship for young researchers granted by the University of Innsbruck to the first author is gratefully acknowledged.

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Modelling of the hysteretic soil–water retention curve for unsaturated soils A. Tsiampousi, L. Zdravkovic & D.M. Potts Department of Civil and Environmental Engineering, Imperial College London, London, UK

ABSTRACT: One of the most important features in unsaturated soil mechanics is the soil-water retention curve, which defines the relationship between the degree of saturation and suction. It has long been shown that the retention curve exhibits significant hysteresis. Several attempts have been made in the past to model this hysteretic behaviour assuming bi-linear and parallel hydraulic paths. An alternative approach, based on simple geometric curves, is proposed herein. Even though the formulation is simple, implementation of the model into a numerical code is relatively demanding. In particular, the algorithm has to keep track of the reversal points where the path changes from drying to wetting and vice versa. Following the formulation and implementation of the new model, simulations of laboratory experiments on unsaturated soils are presented, demonstrating its effectiveness.

1

INTRODUCTION

Unsaturated soil behaviour has been proven to be significantly different from saturated soil behaviour. Not only do the principles of classical soil mechanics, such as the principle of effective stress, not apply, but the hydraulic behaviour of unsaturated soils also exhibits significant differences. Drying and wetting of the soil are irreversible processes. As the nonsolid phase of unsaturated soils consists of both water and air, the pore pressure is no longer the pore water pressure, uw , but the excess of air pressure, ua , over the water pressure, uw , called the matrix suction: Figure 1. Typical shape of the soil-water retention curve.

The volume of water, Vw , within the pores, over the total volume of the voids, Vv , is a measure of the soil saturation, called the degree of saturation, Sr :

The degree of saturation, Sr , generally reduces with the logarithm of suction, s, following the so called soil – water retention curve (SWRC).The retention curve has the typical s – shape shown schematically in Figure 1. In saturated conditions, the whole volume of voids is occupied by water and the degree of saturation, Sr , is 1. It is common for soil to withstand a significant amount of suction and still maintain a degree of saturation, Sr , of unity. Desaturation occurs only when air starts being present in the pores in the form of occluded bubbles.At that time, the degree of saturation, Sr , starts reducing.

On drying the soil further, the largest of the pores empty of water and fill with air. The corresponding value of suction is called the air – entry value of suction, sair . The switch from fully to partly saturated conditions is usually modelled to be at the air – entry value of suction, sair . As suction increases with further drying, air exists in a continuous form and water retreats to smaller voids, loosing its continuity. At very high values of suction water is present only in the form of meniscus at the interparticle s. Any further increase in suction is attributed to the meniscus water and does not cause a further decrease of the degree of saturation, Sr , which has reached its residual value. A subsequent decrease of suction to zero due to wetting brings the soil back towards saturated conditions. Nevertheless, drying and wetting are not reversible

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Figure 2. Hydraulic hysteresis in the soil-water retention curve.

processes and the wetting path followed to saturation lies beneath the drying path, demonstrating the hydraulic hysteresis shown in Figure 2. It is possible for air to be present in the form of occluded bubbles even when the soil is wetted to zero suction (Sr = 1.0 at s = 0 kPa). The retention state of a soil can exist anywhere between the abovementioned drying and wetting paths (point A in Fig. 2). These two paths are followed only when a reconstituted soil sample is dried from slurry to residual conditions and is subsequently wetted to full saturation and are, therefore, referred to as primary paths. The primary paths bound an infinite number of scanning paths, as illustrated in Figure 2. On wetting from an initial retention point A, the scanning path followed res the primary wetting path at lower values of suction. The reverse behaviour is observed during drying; the soil follows a scanning drying path which converges to the primary one at larger values of suction. A direct consequence of the non-uniqueness of the SWRC is that samples of the same soil can exhibit significantly different degrees of saturation under the same value of suction, depending on their drying and wetting history. The mechanical behaviour of unsaturated soils, however, is intrinsically related to the degree of saturation and therefore, identical samples subjected to similar stress states can display different mechanical response. Furthermore, the degree of saturation is indicative of the water storage and contributes to the evolution of water flow within unsaturated soils. The necessity of modelling the hydraulic hysteresis becomes obvious. Several authors have attempted to include the hydraulic hysteresis into numerical modelling of unsaturated soils (Wheeler 1996, Vaunat et al. 2000, Wheeler et al. 2003, Li 2005, Sun et al. 2007, Lloret et al 2009). The general trend is to model the hysteretic retention curve as a linear closed loop. A typical example is the work of Wheeler et al (2003) who proposed the basic shape illustrated in Figure 3. Even though the model is an improvement over the past

Figure 3. Hysteretic soil-water retention model proposed by Wheeler et al. (2003).

approaches, modelling the primary paths as linear is a crude approximation of the experimentally observed behaviour, presented above. Similarly, the scanning paths are also linear, reing the primary paths with an abrupt change in slope. An alternative approach is proposed here. The primary and the scanning paths are simple geometric curves which have a common tangent at the point of intersection. Despite its geometric simplicity, the model has demonstrated effectiveness in the representation of laboratory data. 2

HYSTERETIC MODEL FOR THE SWRC

The hysteretic SWRC model developed and presented herein was aimed at satisfying two fundamental requirements: the need for a realistic shape both for the primary and the scanning paths and the necessity for a smooth transition from scanning to primary paths. The first requirement arises from the fact that the slope of the retention curve is essential in a coupled analysis as it affects the flow of water. More specifically, the flow generated due to changes in water content, and therefore in the degree of saturation, is dependent on the gradient of the SWRC which controls the water storage within the soil. Clearly, a linear retention relationship with a constant gradient, results in an unrealistic constant water storage within the soil, independent of the suction level and the degree of saturation. Therefore, realistic shapes for the primary and the scanning paths are of importance. The second requirement, for a smooth transition from one type of path to the other, relates to numerical singularities. The robustness of the model is believed to be improved when abrupt changes on the slope of the retention curve are avoided. 2.1 Formulation 2.1.1 Primary drying and wetting paths The model is formulated in of degree of saturation, Sr , and equivalent suction, seq , where seq is the

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2.1.2 Scanning drying and wetting paths On drying from an initial retention point A(sA , Sr,A ) in between the two primary curves, the soil is assumed to follow the scanning path shown in Figure 4a. This scanning path is assumed to be the arc of a circle, centred on the vertical line ing through point A so that the suction corresponding to the centre of the circle is equal to the suction at point A, sA . The circle and the primary drying curve have a common tangent at point Bdr (sB dr , Sr,B dr ), also shown in Figure 4a. In this way, the slope of the scanning path is always zero at point A and a smooth transition from the scanning to the primary drying path is provided at point Bdr . The radius of the circle, rdr , and the suction at point Bdr , sB dr , need to be identified. The expression for the scanning drying path is:

The slope of the scanning drying path for the current value of suction is:

Figure 4. Primary and scanning paths assumed by the hysteretic SWRC model; (a) drying; (b) wetting.

excess of the current suction over the air – entry value of suction, sair :

Desaturation during drying and full saturation during wetting is assumed to occur at the air - entry value, sair , ensuring the two primary paths have a common point (seq , Sr ) = (0.0, 1.0). Furthermore, it is assumed that the residual point is also common for the two primary paths and that it occurs at 0.0 residual degree of saturation. The corresponding equivalent suction, s0 , is a model parameter. The above assumptions allow the following s-shape curve to be adopted:

where α is a fitting parameter, carrying the index d for drying and w for wetting. For the wetting path to lie beneath the drying path, αw has to be larger than αd , while if they are equal a monotonic curve is generated. The shape of the primary paths is shown schematically in Figure 4a. The slope of the primary paths for the current value of equivalent suction is:

As noted above, to define the scanning drying path, the radius rdr is required. As Bdr is a common point for the two curves given by Equations 4 and 6:

Furthermore, the two curves share a common tangent at point Bdr :

The above two Equations, 8 and 9, form a system where the suction at point Bdr , sdr B , and the radius rdr are the two unknown variables. The system can have the two possible solutions pictured in Figure 5. Care must, therefore, be taken in order to select and apply the correct one. Furthermore, solution of the system is not straightforward and requires a numerical approach. For the numerical instabilities associated with the presence of the square root to be eliminated, an equivalent system of equations was solved where both sides of the equations were squared. The Newton method was chosen and proved to be adequate and efficient. A limited number of iterations was generally required for convergence to be achieved. On wetting from the same initial point A(sA , Sr,A ), the soil follows the wetting scanning path shown in Figure 4b, reing the primary wetting path at point Bwet (sB wet , Sr,B wet ). Similar to the drying scanning

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Figure 5. Possible solutions of the system of equations when drying from a retention point A.

path, the wetting scanning path is the arc of a circle, centred on the vertical line ing through point A. The circle and the primary wetting curve have a common tangent at point Bwet (sB wet , Swet r,B ). The expressions for the scanning wetting path and its gradient at the current value of equivalent suction, seq , are given below:

and:

Similar to drying, the following system of equations needs to be solved, in of the radius, rwet , and the equivalent suction at point Bwet , sB wet :

and:

For the solution of the system the Newton method was once more employed. As for the drying scanning path, the suitability of the estimated solution was checked so that the appropriate circle was selected. 2.2

Model parameters

Four model parameters are required to define the hysteretic SWRC model described above: two fitting

parameters αd and αw , for the primary drying and wetting paths respectively, the suction at the air-entry value, sair , and the suction at zero residual degree of saturation, s0 . For the primary drying path to lie above the primary wetting one, αd needs to be larger than αw . In addition, the suction at zero degree of saturation, s0 , needs to be larger than the air-entry value of suction, sair . The model parameters dictate the shape and the position of the primary curves, which remain unvarying during the analysis. On the contrary, the scanning paths are not directly controlled by the model parameters; their shape is always circular and the actual path followed is determined primarily by the initial retention state (point A in Fig. 4) and indirectly by the model parameters through the necessity of ing the primary paths with a common tangent. This lack of explicit control over the scanning paths could be regarded as a limitation of the model which, however, it guarantees simplicity. 2.3 Implementation The abovementioned hysteretic SWRC model was implemented in the Imperial College Finite Element Program (ICFEP) (Potts & Zdravković 1999). Depending on the suction change and on the suction level, the appropriate path needs to be selected. The suction change indicates the direction of hydraulic loading (drying or wetting), while based on the suction level itself distinction is made between the corresponding primary and scanning paths. For this procedure to be feasible a number of variables need to be stored during the analysis. It is essential to information concerning the last retention point before a change in the direction of hydraulic loading is detected. This point is commonly referred to as the reversal point. If the soil is wetted from an initial point A, shown in Figure 2, to point B, point A is considered to be the reversal point for this wetting path. If the soil is subsequently dried to point C, point B is the new reversal point for this drying path. One drying and one wetting scanning path correspond to every reversal point and remain unaffected provided that the direction of hydraulic loading remains unchanged. To obtain the congruent scanning path the system of equations presented above needs to be resolved, in order to determine the point of intersection between the scanning and the corresponding primary path. The system, however, needs to be resolved only once for every reversal point and only for the applied direction of hydraulic loading, as long as information regarding the point of intersection are stored. The variables stored are herein referred to as reversal parameters and consist of the following quantities which require recalculation every time that a reversal in the direction of hydraulic loading occurs: the suction, srev , and the degree of saturation, Sr,rev , of the reversal retention point, the radius of the

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corresponding circle, r, and the suction at the intersection point, scommon . The initial soil state, consisting of the stress state as well as the degree of saturation, has also to be established. The initial equivalent suction is calculated based on the air-entry value of suction, sair , and together with the corresponding initial degree of saturation form the coordinates of the initial reversal point, srev and Sr,rev . Once an increment of suction occurs, the direction of hydraulic loading is identified; drying is detected in case the resulting equivalent suction, seq , is larger than the suction at the reversal point, srev , while wetting is detected in the opposite case. If drying is detected, the system of Equations 8 and 9 is resolved and the radius, r = rdr , and the suction at the intersection point, scommon = sB dr , are calculated and stored. Distinction in the employment of the scanning or the primary drying path (Eq. 4 and 6) is based on the comparison of the current suction, seq , with the reversal parameter scommon . For subsequent changes of suction and provided that there is no reversal in the direction of hydraulic loading, the same reversal parameters are used in Equations 4 and 6. If wetting is detected, the system of Equations 12 and 13 is solved and the corresponding reversal parameters r = rwet and scommon = sB wet are calculated and stored. For suction levels higher than scommon the scanning wetting path, given by Equation 11, is employed, while for suction levels lower than scommon the primary wetting path obtained from Equation 2 is used.

3

Figure 6. Hydraulic paths followed by specimens of compacted London clay fill (Melgarejo 2002); (a) numerical reproduction of the primary paths; (b) numerical reproduction of a cyclic hydraulic path.

COMPARISON WITH EXPERIMENTAL RESULTS

The hysteretic model for the SWRC presented above, was employed to reproduce results from laboratory tests carried out at Imperial College London (Melgarejo 2004). The tests were performed on intact and reconstituted specimens of compacted London clay fill. Two reconstituted samples, formed from a slurry, were prepared from the compacted material supplied and the primary drying path of the SWRC was obtained, as shown in Figure 6a. However, the maximum suction that could be measured was reached before the residual degree of saturation was achieved and it was, therefore, impossible to develop the full SWRC. One of the samples was then wetted from this point, forming the scanning wetting path shown in Figure 6a. The numerical reproduction of the primary drying and wetting paths, obtained for sair = 0.0 kPa, s0 = 1.0E+7 kPa, αd = 3.8E-5 and αw = 3.5E-3, is also illustrated in Figure 6a. It should be noted that parameter αw , controlling the primary wetting path, was fitted in such a way that the scanning wetting path, followed by the reconstituted sample upon wetting, converged to the primary path at the lower values of suction.

Two intact samples with a significantly different initial retention states were tested. The first sample (point A in Fig. 6a) demonstrated a degree of saturation of 88% (Sr,A = 0.88) at 1000 kPa of suction (sA = 1000 kPa) and was dried to 56% degree of saturation obtained at 18000 kPa suction. The scanning drying path was reproduced using the hysteretic model proposed herein and the comparison of experimental and numerical results is shown in Figure 6. It is evident that the model is capable of accurately simulating this scanning path which distinctively lies beneath the primary drying path and it finally converges to it at high suction levels. The second intact sample was dried from its initial retention state (point A in Fig. 6b), with sA = 150 kPa and Sr,A = 0.75, to point B (sB = 1500 kPa, SrB = 0.46). Subsequently, the sample was wetted to point C (sC = 170 kPa, Sr,C = 0.74) and dried to point D (sD = 22500 kPa. Sr,D = 0.28).The numerically reproduced scanning paths are shown in Figure 6b. The first scanning drying path was accurately reproduced, while the wetting scanning path was underestimated for low values of suction. As a result, the subsequent drying path was simulated initiating

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from a point lying slightly beneath the actual initial point. Nevertheless, the reproduced scanning path followed closely the experimentally observed behaviour for values of suction up to 10000 kPa, overestimating the degree of saturation thereafter. The overall performance of the hysteretic SWRC model is considered to be efficient, despite its simplicity in of the shape of the curves employed. The primary paths were effectively reproduced by the model, however, the air-entry value of suction, sair , assumed was 0.0 kPa. Melgarejo (2004) reported that the reconstituted samples remained fully saturated to values of suction of the order of 1000 kPa. Employing one fitting parameter for each primary curve and assuming that full saturation upon wetting occurs at the same value of suction as desaturation upon drying, is clearly a shortcoming of the model, even though it ensures simplicity. The capability of the model to reproduce the scanning paths is generally satisfying, despite the simple geometric shape assumed. As the scanning path followed entirely depends on the primary path to which it converges, it is expected that improvement of the expression used for the latter will also improve the performance of the former. 4

CONCLUSIONS

A new approach concerning the modelling of the hydraulic hysteresis, exhibited by the soil-water retention curve of unsaturated soils, was presented. An s-shape curve was adopted for the primary drying and wetting paths, while the scanning paths were assumed to be arcs of circles which re the corresponding primary path with a common tangent. A smooth transition from the scanning to the primary paths was obtained. The formulation and implementation of the proposed

model into ICFEP was presented. Even though the formulation is simple and based on 4 parameters, the model is capable of effectively reproducing the hydraulic paths obtained in the laboratory for reconstituted and intact samples of a compacted London clay fill. REFERENCES Li, X.S. 2005. Modelling of hysteresis response for arbitrary wetting/drying paths. Computers and Geotechnics 32: 133–137. Lloret ,M., Sanchez, M. & Wheeler, S.J. 2009. Generalised elasto-plastic stress-strain and modified suction-degree of saturation relations of a fully coupled model. In O.Buzzi, S. Fityus & D. Sheng (eds), Unsaturated soils; theoretical and numerical advances in unsaturated soil mechanics; Proc. 4th Asia Pacific conf., Newcastle, Australia, 23–25 November 2009. London: Taylor & Francis Group. Melgarejo ML (2002) Laboratory and numerical investigations of soil-water retention curves. PhD thesis, Imperial College London, UK. Potts, D.M. & Zdravkoviæ, L. 1999. Finite element analysis in geotechnical engineering: theory. London: Thomas Telford. Sun, D., Sheng, D. & Sloan, S.W. 2007. Elastoplastic modelling of of hydraulic and stress-strain behaviour of unsaturated soils. Mechanics of Geomaterials 39: 212–221. Vaunat, J., Romero, E. & Jommi, C. 2000. An elastoplastic hydro-mechanical model for unsaturated soils. Experimental evidence and theoretical approaches in unsaturated soils: 121–138. Rotterdam: Balkema. Wheeler, S.J. 1996. Inclusion of specific water volume within an elastoplastic model for unsaturated soil. Can Geotech J 33:42–57. Wheeler, S.J., Sharma, R.S. & Buisson, M.S.R. 2003. Coupling of hydraulic hysteresis and stress-starin behaviour in unsaturated soils. Geotechnique 53: 41–54.

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Numerical integration and analysis of equilibrium in unsaturated multiphase media R. Tamagnini, M. Mavroulidou & M.J. Gunn Department of Engineering Systems, London South Bank University, UK

ABSTRACT: Unsaturated soil mechanics has been largely focused on the phenomena of collapse upon wetting. Since the early studies this mechanical feature has influenced the way of representing experimental data and the constitutive modelling. Collapse has been generally modelled with a plastic amount of strains coupled with the suction rate. This constitutive approach is discussed. In the paper, the equilibrium and stability of the granular matrix is studied applying the criteria of thermodynamics. The system is considered as an open continua and the filtration of the wetting phase during collapse is taken into . The paper provides some remarks about the hardening in partially saturated conditions. The incremental constitutive system of equations is implicitly integrated adopting the Laplace transform and the Newton Raphson technique. This allows defining the problem in the domain of a physical quantity that can be measured in laboratory tests (namely the degree of saturation). Time evolution of the capillary stress is obtained with the inverse transformation. The proposed integration is a part of the research program on lime stabilized material in partially saturated condition. The integration for the partially saturated model is described in the present paper.

1

INTRODUCTION

Unsaturated soils are mixtures of a granular solid matrix, water (in liquid and gaseous form) air and interfaces between constituents. The solid matrix shows mechanical behaviours similar to those of the saturated soils in constant suction experiments but the peculiarity of unsaturated soil is the collapse (solid matrix compaction) during wetting. The latter feature has leaded to the revision of an interesting formulation (Bishop (1959)) for this kind of soils that generally speaking are the real soils ensemble into which the saturated soils belongs for the particular case of fully saturation. In this paper unsaturated soils mechanics is modelled starting from simple physical assumptions. The proposed study accords with the thermodynamically based framework proposed by Coussy (2004). This is the most simple and suitable mathematical framework that can be adopted among the others (see for example Hassanizadeh and Gray (1990) for a detailed description of the mixture with interfaces). The paper shows as the collapse can be physically explained as the impulse that the capillary stress applies to the solid matrix when the interfaces between the different fluids and the matrix is perturbed by the filtration of water. The abrupt jump that the stress field experiences when the interface between the fluids vanishes or moves is studied by the Laplace transform of the preconsolidation pressure. It is shown as the preconsolidation pressure also models the capillary stress,

namely the stress acting within the meniscus water and the interfaces between the fluids and the solid. The hardening forces are modelled as a function of the water saturation rate. The ‘softening’ induced by wetting is modelled by the Maxwell symmetry of a free energy quadratic form that is a function of the strains as well as the saturation degree. This framework differs from other works for unsaturated soils (Buscanera and Nova (2008)). It is remarkable that the assumption of a double dependency of the free energy of the skeleton on both the strains and the saturation degree allows for a simple numerical implementation because the cross effects of the saturation rate is implemented in the elastic predictor stage straightforwardly. It is to be remarked that the capillary forces are the thermodynamic force controlling the water retention properties (called Affinity in this paper) and at the same time it affects the Bishop’s stress and they are coupled with the collapse strains. The present framework is the developing research on lime treated soils in partially saturated conditions that is going on at London South Bank University.

2 THERMODYNAMICS Ulm and Coussy (1998) have proposed a thermodynamically based model for concrete in which the driving forces cementing the granular matrix are taken into . This framework fulfils the model by

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Tamagnini (2000;2004) and the entropy inequality reads:

in which the free energy is defined as:

in which the first term is the work done by the total stress tensor σ on the overall deformation ε , the second and third term are respectively the opposite in sing of the work that the skeleton do on the volume fractions of the fluids: vw and va (w represents the water phase a the air and s the solid); uw and ua are the pressures of the water and air:

and the in equation (1) are: equation (8) can be obtained by substituting the volume fractions of the fluids with: Am represents the driving force controlling the rate of Sr (water degree of saturation) and has the meaning of a Gibbs potential. It is the difference between the capillary stress (or energy) as defined by Hassanizadeh and Gray (1990) and Coussy (2004) and the suction (intended as the difference between the air and water pressure). χ and ζ are respectively: the hardening kinematic variable (volumetric plastic strains) and the hardening force controlling the frozen energy of the granular matrix; σ is the Bishop’s stress and εis the strain tensor. The first interesting Maxwell symmetry shows the following characteristic of the model:

and:

in the equations, n is the porosity and δij is the Kronecker delta. Note that in this equation the work of the interfaces is not taken into and the system is closed. The same expression has been obtained by Houlsby (1997).The internal energy of the solid skeleton can be written as:

in which T is the temperature and S the Entropy. The free energy of the solid matrix can be obtained by the Legendre transform:

The free energy rate for isothermal deformation is then:

with θ = ω:

Equation 4 and 6 show that in the energy balance there is a term, Am , that s for the retention properties of the matrix and another term defining the hardening/softening that depends on a kinematic variable. The thermodynamic force representing the preconsolidation pressure is equivalent to the wetability potential Am and the role of the kinematic internal variable χ and the saturation degree Sr can be interchanged. The latter feature will be adopted to describe the evolution of the preconsolidation pressure in the Laplace transform. The equation 4 states also a dependency of the wetability potential on the plastic strains, higher is the plastic strains higher is the wetability potential, even if it seems to be reasonable the feature deserves further investigation. The definition of the capillary stress can be obtained by the following considerations. If a Representative Element Volume REV is considered as a closed system during the deformation process, the work done on the solid matrix can be expressed in a simple way as:

with:

the first derivative in equation 13 is the Bishop’s stress and the second one is the scaled suction (Houlsby (1997)). In Sheng et al. (2004) they use this equations to model the solid skeleton even during collapse, but phenomenological evidences suggest that collapse occurs when the liquid phase enters the REV, then when the REV is an open system and the equation (12) is not strictly correct. The first law of thermodynamics for open system can be rewritten as follow:

in which G is the Gibbs free energy and dnin -dnout is the overall mass balance dN of the fluids, the Gibbs

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potential of the fluids is distributed on the surface of the grains and in the menisci water. Considering that the Gibbs free energy is defined as:

and assuming that the pressures of the fluids entering and exiting are the same and that the net mass balance of the fluids is:

The equation 14 reads:

in which D is the elastic stiffness and L defines the Laplace operator and pc is the hardening force ζ . For the equation 6 it represents also the evolution of the Gibbs potential of the capillary forces Am . The Laplace transform can be expressed through the following equation:

The Laplace transform of the capillary stress is defined as:

in which the work done by the interface stress is introduced in the last term according to Hassanizadeh and Gray (1990). Introducing equation 8 and applying the Legendre transform as in 11, the equation 12 results:

and then:

where pc is capillary stress (or Am ), only the first two of pc can be linked to the suction as defined by the difference of the bulk pressures of the fluids (in literature the suction) and Hassanizadeh and Gray (1990) suggest the this equivalence holds only at equilibrium. In the model proposed by Tamagnini (2000,2004) pc represents also the ‘preconsolidation pressure’ of the solid matrix that is a function of Sr and not necessarily equal to the suction measured in laboratory for the definition of the WRC (Water Retention Curve), Equation 18 also explain that there is nothing new in the formulation of Gallipoli et al. (2003), it is well known that the capillary stress depends on the three contributions and it is a function of Sr and not suction and Sr . The second interesting Maxwell symmetry is:

Equation (22) defines the evolution of the capillary stress in the domain of the variable Sr and the inverse transform states that the capillary stress is the sum of a unit step function u(t) and a delayed step function u(t-ω) :

The time evolution of the capillary stress can be represented as in figure 1. The figure shows that at time 0 and ω there are two abrupt jumps in the value of Am . These jumps are due to the energy input provided by Sr , these jumps reduce the number of menisci affecting the preconsoldation pressure ζ and they allow their motion. Moreover, they induce the softening and subsequent rearrangement of the solid grains. During water filtration the difference between the capillary forces and suction tends to the equilibrium and the rate of the Gibbs potential Am tends towards zero (after the time ω). During this stage the configuration of the menisci network changes and the subsequent variations of Sr produce the same mechanical phenomena. In figure 1 the first graph represents the step function u(t) the second is the delayed step function u(t-ω) and the third is the combination of the two. After time ω the difference between the capillary stress and the suction vanish and the Affinity is equal to zero representing the hydraulic equilibrium. 3

Equation 20 explains the variation of the Bishop’s stress during wetting (collapse) that is equal to the variation of the Affinity during compaction. The system of differential equations defining the problem of the stress integration is:

IMPLEMENTATION

The modified Cam clay is integrated enhancing the return mapping scheme proposed by Simo and Hughes (1998), the improvement is based on two points: the modification of the elastic trial step that describes the variation of the ‘effective’ Bishop’s stress and the effects of the capillary stress. At the timetn ∈ [0; T ] of the time domain the following state variables are known:

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in which h is the vector of the internal variables. System 25 has to satisfied the Kuhn-Tucker condition:

and the initial condition:

Figure 1. Time evolution of the Affinity Am during the work input due to Sr

Respectively: the total strain, the plastic strain, the degree of saturation at time n. The initial stress σ n and the elastic strain tensor are known (in the following the super script indicating the Bishop’s stress will be omitted). The integration algorithm updates these variables at the time tn+1 ∈ [0; T ]. The unknowns are defined as:

The problem is split in two parts; the first part is a trial elastic step that is called elastic predictor in which the deformation is considered totally elastic. In this stage the dimension of the elastic domain of equation 26 changes due to the changes in the saturation degree; this is a remarkable difference with respect to the original return mapping for saturated soils (see for example Borja and Lee (1990)). After the check of the discrete condition imposed by equation (27), if the trial stress belongs to the elastic domain the stress are updated and the stiffness matrix is the elastic tangent matrix on the other hand if the trial elastic stress drives the current stress outside or on the plastic yield surface the trial step is followed by the plastic corrector phase. The key issue for the extended hardening rule with saturation degree dependency is the variation of the boundary of the elastic domain during the elastic predictor stage; this variation is driven by the increment Sr n+1 this feature of the algorithm implies that an elastic unloading due to saturation could be compatible with an elastic-plastic condition, namely the elastic unloading could be compatible with the p condition f (σ trial n+1 ; q(εv n ; Sr n+1 )) > 0 triggering a plastic corrector stage that models the plastic shrinkage (collapse). At time tn , in the equilibrium condition at the general iteration k, the increments:

are known. Then, the trial state variables can be written as: in which εij = ∇(u) and u is the vector of the displacement rate that is computed during the linearization of the global iterative problem, λ is the plastic multiplier, S r is the rate of the saturation degree (that is known at the start of the local iterative linearization). The changes in the saturation degree is computed with an evolution law controlling the energy of the interfaces and the fluids pressure (the water retention curve), the rate of the preconsolidation pressure pc controls the dimension of the elastic domain, that is the closure of the following domain in the effective stress space :

Note that the second equation introduces the Laplace operator in the system. The hardening rule is integrated in a closed form and splitted adopting the Lie’s formula (see Simo and Hughes ((1998)); the trial invariants are:

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at this time the check for plasticity is applied: trial trial if : f (ptrial n+1 ; qn+1 ; pcn+1 )<0 trial then : (•)trial n+1 =(•)n+1 →EXIT else : Plastic corrector

The plastic corrector uses the trial stress and trial hardening as the initial condition and through a return mapping computes the stress and plastic strains. From the discrete Khun-Tucker condition results:

The aim of the plastic corrector is the definition of the plastic multiplier at time tn+1 and to compute then the stress through the equations:

Figure 2. Collapse in of the mean Bishop’s stress.

in which Ri is the residual. The problem is solved when the residual is smaller than a prescribed tolerance value. The non linear systems can be rewritten as:

if : A(xi + dxi ) < TOL then : xn+1 = xi + dxi → EXIT else : Perform another iteration

that is a non linear system of four equations in the four unknowns: pn+1 , qn+1 , pcn+1 and λn+1 ; note that Sr n+1 and εijn+1 are known at the start of the integration. The system can be rewritten as:

The numerical test is run with the following constitutive parameters, k = 0.04; λ = 0.17; M = 1.2; ω = 7.69 (note in this test the condition θ = ω is fulfilled if the dependency of θon v is disregarded), v = 1.6. The initial values of the state variables are: Sr = 0.71; s = 300 KPa the resulting initial Bishop stress tensor components are σ = (200; 200; 200 ; 0 ;0 ; 0) KPa and the initial net stress tensor = (4; 4; 4 ; 0 ;0 ; 0). The isotropic net stress is increased till 110 KPa and after the saturation is increased from 0.71 to 1.0 (full saturation) figure 2 reports the collapse in compressibility plane p : e and figure 3 in the plane of the isotropic net stress p and e.

4

in which:

The application of the Newton Raphson technique to system (36) provides:

CONCLUSION

The paper has shown as the mechanics of unsaturated soils can be written adopting the Bishop’s stress and enriching the formulation by the momentum of the interfaces between the different fluids. The Laplace transform is the suitable mathematical tool to describe the abrupt jump that the solid matrix experiences during the motion or vanishing of the capillary stress. This approach differs substantially from the ‘classic’ bitensorial approach in which the collapse is modelled by the simple rate of suction (defined as the difference between the fluids pressures). The impulsive force that the matrix receives during the transient variation of the water content is modeled as the composition of two phases.

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Figure 3. Collapse in of the mean net stress.

ACKNOWLEDGEMENTS This study is part of a larger research programme on the hydro-mechanical properties of lime-treated clays, funded by the UK Engineering and Physical Sciences Research Council (EPSRC) through grant EP/E037305/1 REFERENCES

Borja. R.I., Lee S.R., 1990, Cam-Clay plasticity, Part I: Implcit integration of elasto-plastic constitutive relations, Comp. Meth. Appl. Mech. Eng., 78 49–72 Buscanera G. and Nova R., (2009) An elastoplastic strainhardening model for soil allowing for hydraulic bondingdebonding effects Int. J. for Num. and Anal. Meth. in Geomech, Vol., 33, No., 8, Pages: 1055–1086 Coussy, O., (2004), Poro-mechanics, Ed. J. Wiley & Sons D. Gallipoli, A. Gens, R. Sharma, J. Vaunat An elastoplastic model for unsaturated soil incorporating the effects of suction and degree of satu Géotechnique 53(1) pp 123–136 Hassanizadeh, S. M., & Gray, W.G., (1990) Mechanics and thermodynamics of multiphase porous media including interphase boundaries, Adv. Water Res., Vol. 13 No.4, pp. 149–186 Houlsby, G.T. (1997) The Work Input to an Unsaturated Granular Material, Géotechnique, Vol. 47, No. 1, March, pp 193–196 Sheng, D., Smith D.W., Sloan S.W., Gens A. (2003), Finite element formulation and algorithms for unsaturated soils, Part II: Verification and Application, Int. J. Num. An. Meth. Geomech. Vol.-27, pp. 767–790 J.C. Simo and T.J.R. Hughes (1998) Computational Inelasticity (Interdisciplinary Applied Mathematics) SpringerVerlag Berlin and Heidelberg GmbH & Co. K Tamagnini, R., (2000), Modellazione dei terreni non saturi e implementazione agli elementi finiti, MSc Thesis, University of Rome La Sapienza (in italian) Tamagnini, R., (2004), An extended Cam-clay model for unsaturated soils with hydraulic hysteresis, Geotechnique 54, No.3, pp. 223–228 Ulm, F. & Coussy, O. (1998) Coupling in early-age concrete: from constitutive modeling to structural design Int. J. Solids Structure Vol. 35. No 31–32. pp. 4295–4311

Bishop,A.W. (1959),The principle of effective stress,Teknisk Ukeblad, 106(39), pp. 859–863

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Artificial intelligence


A genetic algorithm for solving slope stability problems: From Bishop to a free slip plane R. van der Meij & J.B. Sellmeijer Deltares, Delft, The Netherlands

ABSTRACT: Finding the safety factor of an embankment using a limit equilibrium method requires a search algorithm to find the representative slip circle. Because of the complex solution space, a grid based method is most often preferred. This paper presents a genetic algorithm as an alternative. This genetic algorithm gives accurate results faster then a traditional grid based method. Because of its efficiency, the genetic algorithm is even able to find a free slip surface using Spencer’s method with the lowest safety factor.

1

INTRODUCTION

Several computational programs are available, with which the stability of a soil body can be calculated with a limit equilibrium method. In such a program, a slip surface is analyzed with a certain methodology, for example, the method “Bishop” (Bishop 1995) to determine its stability. The enters an area in which the program needs to find the circle with the minimal stability factor. Searching such a space usually happens by calculating all possible slip circles with corresponding tangent lines and reporting the one with the minimal safety. This algorithm has several disadvantages: • •

It is sequential and therefore time consuming There is no guarantee the (global) minimal safety factor will be found. • A small displacement or change in boundary conditions of the grid can lead to fundamentally different answers. • A small change in boundary conditions can lead to fundamentally different answers. • Much experience and understanding of the method is required, even though this is not obvious. Other search routines (for example hill climbing) have great disadvantages as well. In the recent past Genetic Algorithms (Barricelli, Nils Aall 1957) are used more frequently as a search procedure and it seems to be a well-suited method to find the representative slip plane with the minimal safety factor. For “Flood Control 2015”, a genetic algorithm (GA) has been implemented in the stability program MStab. Genetic algorithms process a mathematical representation of a solution of an analyzed problem. For Bishop’s method, this representation is a vector containing the X and Y value of the centre of the circle, and the radius of the circle. This representation can be seen as an individual and a sum of individuals for a

population. An individual can be tested for its fitness, for example with Bishop’s method. The genetic algorithm improves the quality of a population is a similar way as nature does. Two individuals cross their DNA, there is a chance for mutations and a new individual is created. Two new individuals fight, and the fittest one continues to the next generation. The algorithm seems to be faster and better at finding a global minimum. A disadvantage is that the results are not always reproducible. On top of that, there will be a very strong tendency to find the global minimum, while sometimes, a local minimum is interesting as well. This can be overcome using penalties steering the result in the desired direction. Because of its high speed, a genetic algorithm makes it possible to find a free slip surface with Janbu’s or Spencer’s method. This paper will present in the second section that the GA fundamentally works using an analytical simplification of Bishop’s formula. The next section shows the efficiency of the GA by comparing calculation time and accuracy of a grid based method to the algorithm. Thereafter, the efficiency of the genetic algorithm is explained. Finally, it is shown that a GA can perform a free surface search using Spencer’s method. 2 TESTING THE GA IN MATLAB An analytical formulation of Bishop’s method is derived for a simplified embankment in order to have an analytical safety factor to test the genetic algorithm. Only the crest, slope and surface level of an embankment will be considered, as can be seen in Figure 1. H is the height of the embankment, L is the length of the slope. Different angles defining the slip circle are defined with α0 through α3 . The location of the centre of the slip circle is defined with X and Y using the outer crest as a reference point. The radius of the circle

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Figure 1. Slip circle entering in zone I and exiting in zone III.

is defined as R. The subsoil is divided in three area’s. Area I is underneath the crest, area II is underneath the slope and area III is underneath the ground level. For the purpose of simplicity, no water pressures are considered. The soil is cohesive and homogenous without internal friction. The explicit result depends on the zone (I, II or III) where the circle enters and exits the soil body. In total, four types or circles can be distinguished. A circle that enters through the crest and exits on the surface level, as shown in figure 1, has the safety factor of which the result of the derivation is shown in equation (1). The safety factor of the circle that enters through the crest and exits in the slope of the embankment is given in equation (2) and the safety factor of a circle that enters through the slope and exits on the surface level is given in equation (3). Finally, the safety factor of a circle that enters and exits in the slope of the embankment is given in equation (4). To calculate the safety of the embankment, one first needs to check which case is relevant, and the safety factor can be calculated directly. These formula’s are programmed in Matlab to compare Matlab’s genetic algorithm with the genetic algorithm we wish to implement in the stability program MStab.

Figure 2 shows the solution space for a fixed radius. Matlab has a complex GA tool. Because it is difficult to understand and reproduce, a simple GA specifically built to minimize the above equation is programmed as well. This GA is called the “MStab GA” as it will be used in MStab in the future. Because of the chaotic convergence procedure of a GA, 10000

Figure 2. Solution space of Bishop’s equation above a slope.

Table 1. Average and standard deviation of the results of the Matlab GA and the MStab GA. Simulation

Matlab GA

MStab GA

Pop

Gen

average

St. dev.

Avg.

St. dev.

50 100

50 100

2.7828 2.7792

4.30E-3 5.26E-4

2.7790 2.7789

1.6192e-4 3.1659e-5

runs have been performed to analyze the precision. “Pop” stands for the size of the population, “Gen” stands for the number of generations. The average value of the optimum and its standard deviation are presented in Table 1. A population of 50 individuals running 50 generations seems to be sufficient to get an answer with less then 1% error. The precision of the methods is alike although the deviation of the MStab GA is an order of magnitude lower. 3

IMPLEMENTATION OF BISHOP’S AND VAN’S METHOD IN MSTAB

The “MStab GA” as mentioned previously is implemented in the stability program MStab in order to find the representative slip circle.The grid and GA are compared with the limit equilibrium methods Bishop and Van (Van 2001). Figure 3 on the previous page shows the representative slip circle found with the grid search algorithm, figure 4 below shows the representative slip plane found with the GA. Figure 5 shows the slip plane found with the Van’s method. Table 2 compares the calculation time of the different search algorithms with Bishop’s method. The representative circle is found each time because it is already contained in the initial small search area. The calculation time of the grid method is directly proportional to the size of the grid. The calculation time of the GA only depends on the population size and the number of generations, so it does not vary.

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Table 3. Calculation time grid versus GA with increasing search area. Van

Small Small Larger Larger Large Large Full grid GA grid GA grid GA GA

Calc. t [s] 4,8 f [–] 1,11

17 1,12

19,6 1,09

16 1,08

263,4 13,5 1,08 1,08

10,5 1,09

Figure 3. Representative slip circle using grid and tangent lines.

Figure 6. Combination of calculations for bishop analysis.

The grid method is only faster if the specifies the location of the slip plane very well. If the search area increases, the GA becomes relatively faster.Absolutely, the calculation time also decreases. This is because more geometrically impossible slip planes are in the population and therefore not analyzed. The faster calculation leads to less precision but Table 3 shows it is still sufficient. Searching the entire area is impossible with a grid method and can be performed rapidly with the GA.

Figure 4. Representative slip circle found with a GA.

4

Figure 5. Representative slip plane method Van. Table 2. Calculation time grid versus GA with increasing search area. BHP

Grid small

Calc. time[s] 2,5 f [–] 1,10

GA small

Large grid

Large GA

full GA

5,0 1,10

31 1,10

5,0 1,10

5,0 1,10

One can see that for a small search area the grid method is the quickest. As the search area increases, the grid becomes relatively slower. This phenomenon is amplified with Van’s method as the search space is more complex.

CHOICE OF GA VERSUS GRID METHOD

The calculation time of a grid based method is a function of the calculation time of a single analysis times a * b * c (see figure 6) The optimization procedure of a GA is fundamentally different. In each dimension, a near value needs to be selected and through a number of generations (n) the right combination will be found. For bishop’s method, n * (a+b+c) calculations need to be performed for the optimization. Earlier in this paper, it has been shown that 50 is a good value for n. Van’s analysis (Figure 7) uses 5 parameters to describe the slip circle. A grid based method uses a*b*c*d*e calculations. A GA based method uses n * (a+b+c+d+e) calculations. With an increasing search area and more search dimensions, the GA becomes a more efficient alternative. Figure 8 shows an approach for an analysis of a free slip plane. An upper and a lower bound of the slip plane is defined, and in between 13 straight lines are defined. Including the surface lines, 15 points on these

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earth pressures can be cut off in such a case by the limit equilibrium method. This is common practice in Bishop’s method. Alternatively, unrealistic slip planes can also be avoided when defining the genome. This issue has not yet been addressed, but as the method is very robust, it already works. Figures 9, 10 and 11 present the representative slip plane of respectively a Bishop, Van and Spencer analysis. One can see that as the shape of the slip plane becomes more complex, the safety factor decreases. Because of the high pore water pressures in the bottom sand layer, the slip plane tends to be deep and long. It is difficult to describe this surface with a circle, and therefore Bishop’s method gives a relative high safety factor of 1,08. Van’s method is designed to analyze such problems and consequently gives a lower safety factor of 1,06. The fact that Spencer’s method combined with the genetic algorithm gives a significant lower safety factor of 0,97 is remarkable. Especially, if one takes into that the ive shear force is cut off in Bishop’s and Van’s method, but not in Spencer’s method. If this cut off is also implemented in Spencer’s method, the safety factor will be lower and the ive wedge can exit more steeply.

Figure 7. Combination of calculations for Van’s analysis.

Figure 8. Approach for a free slip plane.

lines have to be found that, together, have the lowest safety factor. Assuming we allow 10 points per line, with a grid based method, 1015 calculations have to be performed. With a GA based method, n * (10 + 10 + 10…) = 150 * n calculations need to be performed. This makes a free surface search feasible. Most other search algorithms have the curse of dimension (Bellman 1957) whereby the calculation time exponentially increases with the number of degrees of freedom in the problem. Because the search time increases with the sum of the number of degrees of freedom, this curse is overcome.

5

FREE SLIP SURFACE SEARCH

Figure 8 shows an approach for a free surface search. As a limit equilibrium method, one can choose for example Janbu’s or Spencer’s method. In this case, Spencer’s method is chosen. An upper and lower boundary is defined with 15 points. The first point is connected through the surface line on the crest, the second through 14th point is connected with a straight line in between, and the last point is again connected by the surface line. The genetic algorithm must find the combination of points on the lines that has the lowest safety factor. The optimization is by far not as straightforward as in Bishop’s method. Bishop will always be able to calculate a safety factor given a centre for the circle and a tangent line. Spencer is not able to produce a safety factor if a sudden increase of the slip surface slope comes across.There are two fundamental ways of addressing this issue. The unrealistically high ive

6

CONCLUSIONS

A Genetic Algorithm is an optimization procedure to find the representative slip circle that has several advantages above a grid based method. First, the genetic algorithm can find the correct minimum, even if the solution space is very complex. The method is good at finding the global minimum, even if there are several local minima. Even though the algorithm does not converge directly via the same path to the solution, the standard deviation of the solution is relatively small and therefore reliable. A much larger search space can be investigated in the same amount of time. One can also choose to have a quick answer with a relative good precision in very little time. The time of an analysis is known in advance as the number of generations are fixed. This makes it a good procedure when many automated calculations are performed. The genetic algorithm theoretically works for all limit equilibrium methods. Its relative efficiency increases with a larger search space and also with a larger number of parameters to be optimized. With Van’s method, the genetic algorithm is in general faster then a grid based method. Finding a free slip plane using a grid based method is not possible whereas the efficiency of the genetic algorithm does make it feasible as the genetic algorithm overcomes the curse of dimension. An analysis based on a free slip plane gives a significantly lower factor of safety with a better limit equilibrium model.

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Figure 9. Slope stability calculated with Bishop’s Method, f = 1,08.

Figure 10. Slope stability calculated with Van’s Method, f = 1,06.

Figure 11. Slope stability calculated with the genetic algorithm and Spencer’s Method, f = 0,97.

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REFERENCES Barricelli, Nils Aall (1957). “Symbiogenetic evolution processes realized by artificial methods”. Methodos: 143– 182 Bellman, R.E. (1957). Dynamic Programming. Princeton University Press, Princeton, NJ. Bishop, C. M. (1995). Neural Networks for Pattern Recognition. Oxford University Press, ISBN 0-19-853864-2

Bishop, W. (1955). “The use of the slip circle in the stability analysis of slopes”. Geotechnique, Vol 5, 7–17. Van, M. A. (2001). “New approach for uplift induced slope failure”. XVth International Conference on Soil Mechanics and Geotechnical Engineering, Istanbul. 2285–2288

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Simulation of the mechanical behavior of railway ballast by intelligent computing M.A. Shahin Department of Civil Engineering, Curtin University of Technology, Perth, WA, Australia

ABSTRACT: Ballast is one of the main components of railway track foundations, thus, an accurate prediction of its mechanical behavior is crucial for stability of railway tracks. In this paper, one of the most commonly used intelligent computing techniques, i.e. Artificial Neural Networks (ANNs), is utilized to model the mechanical behavior of ballast under static loading conditions. Experimental results from a series of large-scale consolidated drained triaxial compression tests collected from the literature are used for ANN model calibration and validation. The results indicate that predictions from the ANN model compare well with those obtained from the large-scale experiments. In particular, ANN predictions demonstrate a high degree of accuracy in simulating the stress-strain and volume change characteristics of ballast. The plastic dilation and contraction of ballast at various confining pressures, and the strain-hardening and post-peak strain-softening are also well simulated.

1

INTRODUCTION

Ballasted railway tracks are usually consisted of a granular medium of ballast and sub-ballast (capping) placed above a compacted sub-grade (formation soil). The stability and performance of a given railway track are often governed by the mechanical behavior of ballast. Based on experimental results, Indraratna et al. (2001) concluded that ballast can be responsible for more than 60% of the total deformation of railway tracks, which induces costly regular track maintenance. This necessitates accurate predictions of the constitutive relationships (i.e. stress-strain and volume change under loading) that govern ballast behavior. These constitutive relationships for ballast are very complex and highly nonlinear. Consequently, the development of constitutive models for ballast behavior using conventional analytical solutions requires rigorous mathematical procedures with various model simplifications, which can affect model reliability. An example of such sophisticated mathematical constitutive models for ballast behavior is developed by Salim and Indraratna (2004), which requires 11 ballast parameters that are difficult to determine in the laboratory. In this context, artificial intelligence using neural networks is more efficient as it provides the ballast constitutive model representation, with fewer model parameters, directly from raw experimental laboratory data without any need for problem simplifications or assumptions. The potential use of neural networks for constitutive modeling was first introduced by Ghaboussi et al. (1991) and since then, neural networks have been applied successfully in constitutive modeling of soils (e.g. Ellis et al., 1995; Penumadu and Zhao, 1999;

Zhu et al., 1998). In this study, the feasibility of using artificial neural network in developing accurate and parsimonious constitutive models for ballast behavior is investigated.

2

BRIEF OVERVIEW OF ARTIFICIAL NEURAL NETWORKS

The type of artificial neural networks (ANNs) used in this study are multilayer perceptrons (MLPs) that are trained with the back-propagation algorithm (Rumelhart et al., 1986). A comprehensive description of back-propagation MLPs is beyond the scope of this paper but can be found in Fausett (1994). The typical MLP consists of a number of processing elements or nodes that are arranged in layers: an input layer; an output layer; and one or more intermediate layers called hidden layers. Each processing element in a specific layer is linked to the processing element of the other layers via weighted connections. The input from each processing element in the previous layer is multiplied by an adjustable connection weight. The weighted inputs are summed at each processing element, and a threshold value (or bias) is either added or subtracted. The combined input is then ed through a nonlinear transfer function (e.g. sigmoidal or tanh function) to produce the output of the processing element. The output of one processing element provides the input to the processing elements in the next layer. The propagation of information in MLPs starts at the input layer, where the network is presented with a pattern of measured input data and the corresponding measured outputs. The outputs of the

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network are compared with the measured outputs, and an error is calculated. This error is used with a learning rule to adjust the connection weights to minimize the prediction error. The above procedure is repeated with presentation of new input and output data until some stopping criterion is met. Using the above procedure, the network can obtain a set of weights that produces input-output mapping with the smallest possible error. This process is called “training” or “learning”. Once training has been successful, the performance of the trained model has to be verified using an independent validation set. 3

CONSTITUTIVE MODELLING OF BALLAST USING ARTIFICIAL NEURAL NETWORKS

3.2 Data division and scaling

In this work, the artificial neural networks for constitutive modeling of ballast are developed using the computer-based software package Neuframe Version 4 (Neusciences, 2000). The data used to calibrate and validate the models consisted of results from six large-scale triaxial, isotropically consolidated compression tests that were reported by Indraratna et al. (1998). The ballast used was latite basalt, a quarried igneous aggregate that is commonly used as railway ballast in New South Wales, Australia. Latite basalt used is highly angular in shape and has coefficient of uniformity Cu = 1.5, coefficient of curvature Cc = 0.9, unit weight γ = 15.3 kN/m3 , maximum particle size dmax = 53 mm and effective particle size d10 = 27.1 mm. The triaxial compression tests were conducted under drained conditions at confining pressures between 15 to 240 kPa. 3.1

study, the following varying axial strain increments are chosen: 0.05, 0.1, 0.2, 0.3, . . . , 1.1, 1.2, 1.3, . . . , 1.8, 1.9, 2.0. As recommended by Penumadu and Zhao (1999), using varying strain increment values results in good modeling capability without the need for a large size of training data. Because the data needed for the ANN models at the above strain increments were not recorded in the original experiments of the triaxial tests, the curves of the deviator stress-axial strain and volumetric strain-axial strain of the available triaxial tests were digitized to obtain the required data. A set of 21 training patterns was used in representing a single triaxial test.

Model inputs and outputs

In simulations of the mechanical behavior of soils and rocks, e.g. ballast, the current state of stress and strain governs the next state of stress and strain. Thus, a typical neural network for constitutive modeling of ballast includes current state nodes, which are processing element that past activity (i.e. memory nodes). At the beginning of the training process, the inputs for current state of stress and/or strain are set to zero and training proceeds to predict the next expected state of stress and/or strain for an input strain or stress increment. The predicted stress and/or strain are then copied back to the current state nodes for the next pattern of input data. In this work, two single-output ANN models are developed to simulate the stress-strain and volume change characteristics of ballast. The inputs to the first ANN model are the current state of deviator stress (qi ), confining pressure (σ3 ), current axial strain (εa,i ) and axial strain increment (εa,i ). The single output is the next state of stress (qi+1 ). The inputs of the second ANN model are the current state of volumetric strain (εv,i ), confining pressure (σ3 ), current axial strain (εa,i ) and axial strain increment (εa,i ). The single output is the next state of volumetric strain (εv,i+1 ). In this

The six available triaxial tests were divided into two sets: training set for model calibration and an independent validation set for model verification. The training set includes the triaxial tests results related to confining pressures of 15, 60, 120 and 240 kPa; whereas the testing set has two triaxial test results at confining pressures of 30 and 90 kPa. Before presenting the data patterns to the neural networks, the inputs and outputs are scaled between 0.0 and 1.0 to eliminate their dimensions and to ensure that they all receive equal attention during training. 3.3 Model architecture and internal parameters Model architecture requires selection of the optimum number of hidden layers and the corresponding number of hidden nodes.As proposed by Hornik (1989) and Cybenko (1989), a network with one hidden layer can approximate any continuous function if sufficient connection weights are used. Therefore, one hidden layer was used in the current study. The optimal number of hidden nodes is obtained by a trial-and-error approach in which the network was trained using the software default values of learning rate of 0.2 and momentum term of 0.8, with a tanh transfer function in the hidden layer nodes and a sigmoidal transfer function in the output layer node. For each selected number of hidden layer nodes, training is terminated when the error between the actual and predicted values of outputs over all patterns has no significant improvement. This was achieved at 20,000 training cycles (epochs). As a result of training, a network with three hidden layer nodes was found to perform the best for the deviator stressaxial strain model; whereas a network with two hidden layer nodes was optimal for the volumetric strain-axial strain model. 3.4 Model performance and validation The performance of the developed ANN models in the training set is shown in Figure 1, and the predictive ability of the models in the validation set is depicted in Figure 2. It can be seen that excellent agreement has been achieved between ANN model predictions and

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Figure 1. Performance of the developed ANN models in the training set.

laboratory experimental data in the training and validation sets, with coefficients of correlation equal to unity in both sets. This demonstrates the strong capability of ANN models in generalizing the complex nonlinear constitutive relationships of ballast behavior. For example, the nonlinear relationships of deviator stress versus axial strain and volumetric strain (compression is considered positive and dilation is negative) versus axial strain are predicted accurately. The strain hardening and the gradual decrease of deviator stress beyond peak failure (post-peak strain softening) are very well simulated. The transition of ballast behavior from initial compression to dilation at low confining pressures and the change from dilative behavior at low confining pressure to overall compacting behavior at high confining pressure are also well captured. In conventional constitutive modeling, the strain softening region will result in negative soil modulus, which tends to increase the mathematical modeling effort significantly (Zhu et al., 1998). As mentioned earlier, the current state of stress and strain affects the next state of stress and strain. Consequently, in modeling the ANN constitutive relationships of ballast behavior, an approach was used to add incremental axial strain to the current stress and strain so that the next stress and strain are predicted, which are copied back to the current state of stress and

Figure 2. Predictive ability of the developed ANN models in the validation set.

strain for the next pattern of input data. The above procedure is applied using the developed ANN models at confining pressures of 30 and 90 kPa, and the virtual results, which are shown in Figure 3, are compared with the experimental laboratory data. It can be seen from Figure 3 that good agreement still exists between the measured and predicted deviator stress-axial strain, and volumetric strain-axial strain.

4

SUMMARY AND CONCLUSIONS

Artificial neural networks (ANNs) were used to model the constitutive relationships of the mechanical behavior of railway ballast. Two ANN models were developed; one to simulate the deviator stress-axial strain behavior and the other for volumetric strain-axial strain behavior. The type of ANNs used were multilayer perceptrons (MLPs) that were trained with the back-propagation algorithm. The scheme used for ANN model development was based on the well known plasticity theory that the current state of stress and/or strain influences the next state of stress and/or strain. The results of the ANN models were compared with the experimental tests data. The results indicate that the ANN based models were capable of accurately simulating the complex

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that can be implemented in a finite element analysis. The source code can be provided by the author upon request. REFERENCES

Figure 3. True predictions of the developed ANN models in the validation sets.

constitutive relationships of the mechanical behavior of railway ballast. The highly nonlinear relationships of deviator stress versus axial strain and of volumetric strain versus axial strain of ballast at various confining pressures were accurately predicted. Strain hardening and post-peak strain softening were well simulated, and the plastic shear dilation and contraction of ballast were also captured. To facilitate the use of the developed ANN models, they are translated into C++ code

Cybenko, G. 1989. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals, and Systems 3:303–314. Ellis, G.W., Yao, C., Zhao, R. & Penumadu, D. 1995. Stressstrain modeling of sands using artificial neural networks. Journal of Geotechnical Engineering 121 (5):429–435. Fausett, L.V. 1994. Fundamentals neural networks: Architecture, algorithms, and applications. Englewood Cliffs, New Jersey: Prentice-Hall. Ghaboussi, J., Garrett, J.J. & Wu, X. 1991. Knowledge-based modeling of material behavior with neural networks. Journal of Engineering Mechanics 117 (1):132–153. Hornik, K., Stinchcombe, M. & White, H. 1989. Multilayer feedforward networks are universal approximators. Neural Networks 2:359–366. Indraratna, B., Ionescu, D. & Christie, D. 1998. Shear behavior of railway ballast based on large-scale triaxial tests. Journal of Geotechnical and Geoenvironmental Engineering 124 (5):439–449. Indraratna, B., Salim, W., Ionescu, D. & Christie, D. 2001. Stress-strain and degradation behavior of railway ballast under static and dynamic loading, based on large-scale triaxial testing, Proceedings of the 15th International Conference of Soil Mechanics and Geotechnical Engineering, Istanbul: 2093–2096. Neusciences. 2000. Neuframe Version 4.0. Southampton, Hampshire: Neusciences Corp. Penumadu, D. & Zhao, R. 1999. Triaxial compression behavior of sand and gravel using artificial neural networks (ANN). Computers and Geotechnics 24 (3):207–230. Rumelhart, D.E., Hinton, G.E. & Williams, R.J. 1986. Learning internal representation by error propagation. In Parallel Distributed Processing, edited by D. E. Rumelhart & J. L. McClelland. Cambridge: MIT Press. Salim, W. & Indraratna, B. 2004. A new elasto-plastic constitutive model for granular aggregates incorporating particle breakage. Canadian Geotechnical Journal 41 (4):657–671. Zhu, J.H., Zaman, M.M. & Anderson, S.A. 1998. Modeling of soil behavior with a recurrent neural network. Canadian Geotechnical Journal 35 (5):858–872.

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Three dimensional site characterization model of Suurpelto (Finland) using vector machine A. Pijush Samui & Tim Länsivaara Tampere University of Technology, Tampere, Finland

ABSTRACT: Site characterization is an important task in Geotechnical Engineering. The main objective of site characterization models is to predict the subsurface soil properties with minimum in-situ test data. The success of random field method and geostatic for site characterization model is limited. This paper describes Vector Machine (SVM) applied for site characterization modelling of Suurpelto based on Cone Penetration Test (T) Data. Suurpelto is a new development area of 325 hectares that will be built during the following 15 years. The subsoil in the area consists of soft clay up to a depth of 20m. In three dimensional site characterization model, the functionqc = f (X , Y , Z). where X, Y and Z are the coordinates of a point corresponding to Cone Resistance(qc ) value, is to be approximated with which qc value at any half space point in Suurpelto can be determined. SVM model, which is firmly based on the theory of statistical learning theory, uses regression technique by introducing ε-insensitive loss function has been used in this study. This study shows that SVM can be used as a practical tool for site characterization model of Suurpelto.

1

INTRODUCTION

One of the most important steps in geotechnical engineering is site characterization. The basic objective of site characterization is to provide sufficient and reliable information and data on the site condition to a level compatible and consistent with the needs and requirements of the project. In situ tests based on standard penetration test (SPT), cone penetration test (T) and shear wave velocity method are popular among geotechnical engineering. These tests are time consuming and expensive. Modelling of spatial variation of soil properties based on limited finite number of in situ test data is an imperative task in probabilistic site characterization. It has been used to design future soil sampling programs for the site and to specify the soil stratification. It is never possible to know the geotechnical properties at every location beneath an actual site because, in order to do so, one would need to sample and/or test the entire subsurface profile. So one has to predict geotechnical properties at any point of a site based on a limited number of tests. The prediction of soil property is a difficult task due to uncertainty. Spatial variability, measurement ‘noise’, measurement and model bias, and statistical error due to limited measurements are the sources of uncertainties. In probabilistic site characterization, random field theory has been used by many researchers in geotechnical engineering (Yaglom 1962; Lumb 1975; Vanmarcke 1977; Tang 1979; Wu & Wong 1981; Asaoka & Grivas 1982; Vanmarcke 1983; Baecher 1984; Kulatilake & Miller 1987; Kulatilake 1989; Fenton 1998; Phoon

& Kulhawy 1999; Uzielli et al. 2005). Geostatistics (Matheron 1963; Journel & Huijbregts 1978) also have been used to model spatial variation of soil properties (Kulatilake & Ghosh 1988; Kulatilake 1989; Soulie et al. 1990; Chiasson et al. 1995; DeGroot 1996). The success of Random filed method and Geostatistic is very limited in site characterization (Juang et al. 2001). The objective of this paper is to use Vecor Machine(SVM) for three dimensional (3D) site characterization model for Suurpelto based on a large amount Cone Resistance(qc ) values in an area of 325 hectares. Cone penetration test(T) has been done at five points. The Vector Machine (SVM) based on statistical learning theory has been developed by Vapnik (1995). It provides a new, efficient novel approach to improve the generalization performance and can attain a global minimum. In general, SVMs have been used for pattern recognition problems. Recently it has been used to solve non-linear regression estimation and time series prediction by introducing ε-insensitive loss function (Mukherjee et al. 1997; Muller et al. 1997; Vapnik 1995; Vapnik et al. 1997). The SVM implements the structural risk minimization principle (SRMP), which has been shown to be superior to the more traditional Empirical Risk Minimization Principle (ERMP) employed by many of the other modelling techniques (Osuna et al. 1997; Gunn 1998). SRMP minimizes an upper bound of the generalization error whereas, ERMP minimizes the training error. In this way, it produces the better generalization than traditional techniques.

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2

VECTOR MACHINE

An interesting property of SVM approach is that it is an approximate implementation of the Structural Risk Minimization (SRM) induction principle which tells that the generalization ability of learning machines depend more on capacity concept than merely the dimensionality of the space or the number of free parameters of the loss function. This study uses the SVM as a regression technique by introducing a ε-insensitive loss function. In this section, a brief introduction on how to construct SVM for regression problem is presented. More details can be found elsewhere (Boser et al. 1992; Cortes & Vapnik 1995; Gualtieri et al. 1999; Vapnik 1998). The ε-insensitive loss function can be described in the following way:

The constant 0 < C < ∞ determines the trade-off between the flatness of f and the amount up to which deviations larger than ε are tolerated (Smola & Scholkopf 2004). In practice, the C value is selected by trail and error. The above constrained optimization problem (5) is solved by using the method of Lagrange multipliers. Lagrangian function is constructed in the following way

Where α, α∗ , γ and γ ∗ are the Lagrangian multipliers. The solution to the constrained optimization problem is determined by the saddle point of the Lagrangian function L(w, ξ, ξ ∗ , α, α∗ , γ, γ ∗ ) which has to be minimized with respect to w, b, ξ and ξ ∗ . The minimum with respect to w, b, ξ and ξ ∗ of the Lagrangian, L is given by,

consider the problem of approximating a set of data,

Where x is the input, y is the output, RN is the Ndimensional vector space and r is the one dimensional vector space.The main aim in SVM is to find a function that gives a deviation of ε from the actual output and at the same time is as flat as possible. Let us assume a linear function:

Where, w = is an adjustable weight vector and b = the scalar threshold. Flatness in the case of (3) means that one seeks a small w. One way of obtaining this is by minimizing the Euclidean norm w2 . This is equivalent to the following convex optimization problem:

In order to allow for some errors, the slack variables ξi and ξi∗ are introduced in (4). The formulation can then be restated as:

Substituting (7) into (6) yields the dual optimization problem Maximize:

The coefficients αi , α∗i are determined by solving the above optimization problem (8). From the KarushKuhn-Tucker (KKT) optimality condition, it is known that some of αi and α∗i , will be zero. The non-zero αi and α∗i are called vectors. So equation (3) can be written as

From (7) it is clear that w has been completely described as a linear combination of training patterns. So, the complexity of a function representation by

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vectors is independent of the dimensionality of input space and it depends only on the number of vectors. When linear regression is not appropriate, then input data has to be mapped into a high dimensional feature space through some nonlinear mapping (Boser et al. 1992). After replacing x by its mapping in the feature space (x) into the optimization problem (8) Maximize:

dataset: This is required to examine the model performance. In this study, the remaining 242 data is considered as testing dataset.The coordinates(X,Y and Z) of each data were prepared as input of the model, while N value was the output from this model. The data is normalized between 0 to 1. When applying SVM, in addition to the specific kernel parameters, the optimum values of the capacity factor C and the size of the error-insensitive zone ε should be determined during the modeling experiment. In this study, the radial basis function is used as the kernel function of the SVM. The program is constructed using the SVM toolbox in MATLAB.

3 The concept of kernel function [K(xi , xj ) = (xi ).(xj )] has been introduced to reduce the computational demand (Cristianini & ShwaeTaylor 2000, Cortes & Vapnik 1995). So optimization problem can be written as: Maximize:

Some common kernels have been used such as polynomial (homogeneous), polynomial (nonhomogeneous), radial basis function, Gaussian function, sigmoid etc for non-linear cases. The regression function (3) has been obtained by applying same procedure as in linear case. Figure 1 shows the SVM architecture for qc prediction. In the present study, SVM has been used for prediction of qc values in the subsurface of Suurpelto (Finland). ε-insensitive loss function has been used in this analysis. For implementing the SVM, the data has been divided into two sub-sets: (1) A training dataset: This is required to train the model. In this study, 2429 data is considered for training dataset. (2) A testing

Figure 1. SVM architecture for qc prediction.

RESULTS AND DISCUSSIONS

In this analysis as a first step, the free parameters of radial basis function σ, .. C and ε. have been chosen arbitrarily. So it is necessary to investigate the impact of these free parameters on the generalization error. Firstly, the influence of σ. on the prediction performance is studied. It is known to us that the level of predicting accuracy is greatly influenced by the value of σ.˜ using too small σ. (i.e., σ.˜ → 0) or too large σ. (i.e., σ. → ∝ ) will be not well suited for good model. Figure 2 represents the impacts of σ. on the testing results. Figure 2 represents the impacts of σ. on the testing results. The Mean Absolute Error (MAE) n 1

[MAE = |ai − pi | , where ai is the actual data, pi is n i=1 the predicted data and n is the number of data] achieve minimum value of 0.0271 at σ. = 3. It can be seen from ˜ Figure 2 the MAE values change sharply when σ. < 40 ˜ In this study, a σ value and tend to flatten after σ. ≥ 40. 3 have been used. Figure 3 shows the variation between the MAE and the C values. The MAE has a minimum value of 0.0271 at C . = 150. The numver of vector is 1020. Figure 4 depicts the variation MAE value with ε values. The MAE has minimum value at ε = 0.002. Figure 5 represents the performance of SVM model for training dataset (coefficient of correlation, R = 0.962) the result are almost identical to the original data. In order

Figure 2. Variation of σ with MAE.

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Figure 3. Variation of MAE with C values. Figure 6. Performance of testing dataset.

SVM technique. SVM technique has shown to be a promising tool for site characterization. SVM training consists of solving a – uniquely solvable – quadratic optimization problem and always finds a global minimum. In this study, C and ε. factors are considered in SVM method by using a kernel function. A detailed parametric analysis of these parameters on the predictive performance has been carried out. The SVM was found to generalize well by setting the capacity factor C as 150 and ε value as 0.002. The result obtained shows that the SVM model is accurate in predicting qc values. In general, SVM is shown to provide a general site characterization model of Suurpelto (Finland). The predicted qc values from the developed model can also be used to estimate the subsurface information, allowable bearing pressure of soils and elastic modulus of soils.

Figure 4. Variation of MAE with ε˜ .

REFERENCES

Figure 5. Performance of training dataset.

to evaluate the capabilities of the SVM model, the model is validated with new qc data that are not part of the training dataset. Figure 6 shows the performance of the SVM model for testing dataset(R = 0.946). From the Figure 6, it is clear that the SVM model has predicted the actual values of qc very well and it can be used for 3D site characterization model of Suurpelto (Finland).

4

CONCLUSIONS

The three dimensional site characterization model has been developed for Suurpelto (Finland) using

Asaoka, A., & Grivas, D.A. 1982. Spatial variability of the undrained strength of clays. Journal of Geotechnical. Engineering, ASCE. 108(5):743–745. Boser, B. E., Guyon, I. M., & Vapnik, V. N.1992. A training algorithm for optimal margin classifiers. In D. Haussler, editor, 5th Annual ACM Workshop on COLT. 144–152, Pittsburgh, PA, ACM Press. Chiasson, P., Lafleur, J., Soulie, M. & Law, K.T. 1995. Characterizing spatial variability of clay by geostatistics. Canadian Geotechnical Jornal. 32:1–10. Cortes, C., & Vapnik, V.N.1995. vector networks. Machine Learning. 20:273–297. Cristianini, N., & Shawe-Taylor, J. 2000. An introduction to vector machine. London:Cambridge University press. Degroot, D.J. 1996. Analyzing spatial variability of in situ soil properties. In: ASCE proceedings of uncertainty’96, uncertainty in the geologic environment: from theory to practice, ASCE. 58: 210–238. Fenton, G.A. 1998. Random field characterization NGES data. Paper presented at the workshop on Probabilistic Site Characterization at NGES, Seattle, Washington. Gualtieri, J.A., Chettri, S.R., Cromp, R.F., & Johnson, L.F. 1999. vector machine classifiers as applied to AVIRIS data. In the Summaries of the Eighth JPLAirbrone Earth Science Workshop.

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Gunn, R. 2003. Vector Machines for Classification and Regression. httD://www.ecs.soton.ac.uk/∼srg/ publications/pdf/SVM.pdf. Juang, C.H., Jiang, T., & Christopher, R.A. 2001. Three-dimensional site characterisation: neural network approach. Géotechnique. 51(9):799–809. Kulatilake, P.H.S.W. 1989. Probabilistic Potentiometric surface mapping. Journal of Geotechnical Engineering, ASCE. 115(11):1569–1587. Kulatilake, P.H.S.W., & Ghosh, A. .1988. An investigation into accuracy of sparial variation estimation using static cone penetrometer data. Proc. Of the First Int. Symp. On Penetration Testing. 815–821, Orlando, Fla. Kulatilake, P.H.S.W., and Miller, K.M. 1987. A scheme for estimating the spatial variation of soil properties in three dimensions. Proc. Of the Fifth Int. Conf. On Application of Statistics and Probabilities in Soil and Struct Engnieering. 669–677, Vancouver, British Columbia, Canada. Lumb. P. 1975. “Spatial variability of soil properties,” Proc. Second Int. Conf. On Application of Statistics and Probability in Soil and struct. Engrg. 397–421, Aachen, . Mukherjee, S., Osuna, E., & Girosi, F. 1997. Nonlinear prediction of chaotic time series using vector machine. Proc., IEEE Workshop on Neural Networks for Signal Processing 7, Institute of Electrical and Electronics Engineers. 511–519, New York. Muller, K. R., Smola, A., Ratsch, G., Scholkopf, B., Kohlmorgen, J., & Vapnik, V. 1997. Predicting time series with vector machines. Proc., Int. Conf. on Artificial Neural Networks. 999, Springer-Verlag, Berlin. Osuna, E., Freund, R., & Girosi, F.(1997). An improved training algorithm for vector machines. Proc., IEEE Workshop on Neural Networks for Signal Processing 7.

276–285, Institute of Electrical and Electronics Engineers, New York. Phoon, K.K, & Kulhawy, F.H. 1999. Characterization of geotechnical variability. Can. Geotech. J. 36(4):612–624. Smola,A.J. & Scholkopf, B.2004.A tutorial on vector regression. Statistics and Computing. 14: 199–222. Soulie, M., Montes, P. & Sivestri, V. 1990. Modelling spatial variability of soil parameters. Candian Geotechnical Journal. 27:617–630. Tang, W.H. 1979. Probabilistic evaluation of penetration resistance. Journal of Geotechnical Engineering, ASCE. 105(GT10):1173–1191. Uzielli, M., Vannucchi, G., & Phoon, K.K. 2005. Random filed characterization of strees-normalised cone penetration testing parameters. Géotechnique. 55(1):3–20. Vanmarcke, E.H. 1977. Probabilistic Modeling of soil profiles. Journal of Geotechnical Engineering, ASCE. 102(11): 1247–1265. Vanmarcke, E.H. 1983. Random fields: analysis and synthesis., Cambridge, Mass: The MIT Press. Vapnik, V., Golowich, S. and Smola, A. 1997. method for function approximation regression estimation and signal processing. Advance in neural information processing system 9, Mozer M. and Petsch T., Eds. Cambridge, Ma:MIT press. Vapnik, V.N. 1998. Statistical learning theory. New York: Wiley. Vapnik, V.N.1995. The nature of statistical learning theory. Springer:New York. Wu, T.H., & Wong, K. 1981. Probabilistic soil exploration: a case history. Journal of Geotechnical Engineering, ASCE. 107(GT12):1693–1711. Yaglom, A.M. 1962. Theory of stationary random functions. Englewood Cliffs, N.J: Prentice-Hall, Inc.

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Reliability and probability analysis


Evaluation of soil variability and its consequences M. Huber & P.A. Vermeer Institute of Geotechnical Engineering, University of Stuttgart,

A. Bárdossy Institute of Hydraulic Engineering – Department of Hydrology & Geohydrology, University of Stuttgart,

ABSTRACT: The focus of the present paper is on the incorporation of soil variability into a design process. One possibility offers the full probabilistic method. For this purpose a deep knowledge on stochastic soil properties is needed. Therefore field experiments have been carried out to evaluate stochastic soil properties, which are compared to data in literature. Within a case study the procedure of the full probabilistic methods using the finite element method is carried out by using different random field generators. The conventional Gaussian random field approach is compared to the sequential indicator method adapted from hydrology. Through this the impact of the different correlation structures is highlighted. The results of this contribution help to understand stochastic soil properties and offers different ways to describe soil variability.

1

INTRODUCTION

Whenever soil properties are measured at different locations within a “homogeneous” soil layer, a scattering of the measurement values appears. According to Phoon & Kulhway (1999) there are different reasons for spatially varying soil properties like geological processes, measurement errors as well as errors due to geotechnical modelling. One possibility is to cover these varying soil properties with a global safety factor. Standards like the Eurocode 7 offer a different way to capture the variability of soil properties. By using partial safety factors for solicitation and resistance parameters, one can take soil parameters into , which cannot be described by only one single value. The most exact way to consider stochastic soil properties is the full probabilistic method. Within this contribution this fully probabilistic approach is introduced within a Finite Element case study, whereas the standard approach is compared to a new concept called sequential indicator method adapted form hydrology. Moreover the results of experiments to evaluate stochastic soil properties are presented and compared to properties found within a literature study. 1.1

Full probabilistic method

The full probabilistic method can take into soil variability in the most accurate way via using random fields. Random fields are used for incorporating not only mean and standard deviation of a spatial distributed soil property. The spatial dependency of a random field can be expressed via an autocorrelation function ρ(τ) according to Journel & Huijbregts

(1978). Herein τ is the lag between the points. If a random field Xi has mean µX and variance σX 2 then the definition of the autocorrelation ρ(τ) is as shown in equation (1). Herein is E(X ) is the expected value operator. The well known Markov correlation function is a negative exponential function using the correlation distance θ as described in formula (1) and in Figure 1.

If the autocorrelation function only depends on the absolute separation distance of Xi and Xj the random field is called isotropic. Another assumption is ergodicity. Ergodicity means that the probabilistic properties of a random field can be completely estimated from observing on realization of that field. Like for many approaches in Natural Sciences, stationarity is an assumption of the model, and may only be approximately true. Also, stationarity usually depends upon scale. According to Baecher & Christian (2003), within small region, such as a construction site, soil properties may behave as if drawn from a stationary process; whereas, the same properties over a larger region may follow this assumption. By definition, autocorrelation functions are symmetric and bounded. Another assumption is the separability of the autocovariance function according to Vanmarcke (1983). Separable autocovariance function can be expressed as a product of autocovariance of lower dimension fields. Vanmarcke (1983) as well as Rackwitz (2000) offer various models for autocorrelation functions. In the field of geostatistics the spatial dependence is described via a so called variogram. The so-called semicovariance γ(τ) is defined as the

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as described in the German standard DIN EN 1990 (2002). Apart from the big computational effort the stochastic description of soil is demanding. Studying scientific publications like listed in Table 1 one can easily deduce that there is only a limited knowledge on characterizing spatial variability of soil properties. 1.2 Evaluation of spatial variability

Figure 1. Relationship between semivariance and autocorrelation according to CLARK (1979).

Figure 2. Scheme of probabilistic Finite Element calculation.

expected squared increment of the values between two locations according to Wackernagel (2003) and Baker et al. (2006). The full probabilistic approach is based on the principle of stochastic simulation. Stochastic simulation is the process of building alternative, equally probable realizations of the spatial distribution as described by Deutsch & Journel (1992). These random fields fulfil the requirements of the mean value, standard deviation as well as the correlation function. After mapping these random fields onto the Finite element (FE) mesh, the FE calculations are carried out. As along as the statistics of the output is not stable, the circle of generation of random fields and FE calculation is not stopped (Figure 2). The full probabilistic method has the ability to describe the stochastic behaviour of a geotechnical problem in the most accurate way compared to other techniques like First Order Reliability Method

In order to perform a full probabilistic analysis further knowledge of spatial variability of soil parameters is needed. To gain more knowledge of spatial behaviour of mechanical soil properties, experiments have been conducted within an urban tunnelling site. During the tunnelling construction process at the Fasanenhoftunnel in Stuttgart () 45 horizontal core borings have been carried out as shown in Figure 3. These horizontal borings were grouped within a geological homogeneous layer of mudstone with a separation distance of 2.5 m. The elevation of the boreholes is varied according to the gradient of the tunnel. Therefore the first and the last borehole have a difference in the elevation approximately 2.5 m. At an approximate depth of 1.35 m 45 borehole deformation tests have been carried out. Figure 5 shows the equipment of the borehole deformation test as described in the German standard DIN 4094-5 (2001). Within this test to half-shells are pressed diametrically against the walls of a borehole. Three different loading cycles have been executed. The pressure was raised up to three different levels of 1000 kN/m2 , 2000 kN/m2 and 3000 kN/m2 . During this loading process the deformation of the pressure plates was measured as illustrated as a schematic curve in Figure 5. The statistical evaluation correlation distance EB,3 by using the semicovariance and the correlation function is shown in Figure 6. The resulting correlation lengths for all loading cycles are shown in Table 2. The reason of the spatial variability can be deduced to the concentration of limestone inside the layer of mudstone. The results shown in Table 2 are an additional contribution to evaluate spatial variability of mechanical soil properties and are an extension of the presented results of the state of the art presented in Table 1. 2

CASE STUDY

A case study is presented in order to enlighten the scheme of a full probabilistic analysis by using the FEM program PLAXIS (Al-Khoury et. al. 2008). For the sake of simplicity, a strip footing on a spatially varying soil with undrained cohesion is chosen. In Figure 7 the geometry of the strip footing is shown. 15-noded elements have been used to calculate the bearing capacity using a linearly elastic, perfectly plastic model. The bearing capacity of the strip footing is evaluated by increasing the load until it cannot bear additional load increments.

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Table 1.

Summary of correlation distances of different soils presented in literature. Soil

Source

Type

Property

Correlation distance

Asoaka & Grivas (1982) Mulla (1988) Ronold (1990) Unlu et al. (1990) Soulie et al. (1990) Rehfeld et. al. (1992) Hess et al. (1992) Chaisson et al. (1995) Popescu et al. (1995) Jaksa et al. (2004) Vrouwenvelder & Calle (2003)

Clay Clay Clay Sand Clay Sand Sand Clay Clay Clay Clay

Undrained shear strength Penetrometer resistance Shear strength Permeability Shear strength Permeability Permeability Cone resistance Cone resistance Dilatometer Cone resistance

θh = 40–70 m θv = 2 m θh = 12 – 16 m θh = 20 m, θh /θv = 10 θh = 25 m, θh /θv = 8 θh = 25 m, θh /θv = 8 θv = 0.2–1.0 m, θh /θv = 10 θv = 1.5 m θh = 0.8–1.8 m θv = 0.5–2 m θh = 20–35 m

Figure 3. Experimental setup to evaluate the correlation length.

Figure 5. Typical curve of measured displacement vs. pressure and evaluation according to DIN 4094-5.

Figure 4. Testing Equipment as described in the German standard DIN 4094-5 (2001).

2.1 Mid-point method The mid-point method is used to generate the Gaussian random fields representing the undrained cohesion beneath the strip footing. As described by Baecher & Christian (2003), the following equation 2 is used to generate correlated numbers y. The correlation matrix was calculated by using an isotropic correlation function shown in formula (1). Herein is L the lower

Figure 6. Evaluation of the correlation distance of EB,3 using semicovariance and correlation function.

triangular resulting of the Cholesky decomposition of the correlation matrix. The vector of uncorrelated, random variables is x.

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Figure 7. Geometry of the strip footing with a single realisation of a random field for lognormal distributed, undrained cohesion (µ = 100 kN/m2 , C.O.V = 10 %, θ = B).

Figure 9. Influence of the correlation length on the mean bearing capacity for different correlation distances and coefficients of variation.

Table 2. Mean value µ, coefficient of variation C.O.V. of the lognormal distributed EB,3 and horizontal correlation distance θ of EB,3 .

Loading cycle 1 2 3

loading reloading loading reloading loading reloading

µ [MN/m2 ]

C.O.V = σ/µ [%]

θ [m]

125 660 229 432 229 397

54 64 53 57 59 61

10 10–15 10–15 10–15 15–20 15–20

Figure 10. Conventional Gaussian approach (a) and sequential indicator approach (b) using a mean value µ = 100 kN/m2 , a standard deviation σ = 10 kN/m2 and a correlation distance θ = 10 m.

bearing capacity is lower compared to other correlation distances. 2.2 Sequential indicator simulation

Figure 8. Convergence of the mean and the C.O.V due to additional realisations of the random field for undrained cohesion.

As mentioned above, there are lots of equi probable realizations of the random field necessary to capture variability, until the influence of additional realizations is negligible (Figure 8). In Figure 9 the influence of the spatial variability on the mean baring capacity can easily be seen. The more the coefficient of variation increases, the more the better the influence of the correlation distance can be seen. As shown by Fenton & Griffiths (2008) the most “critical” correlation distance is the width of the strip footing. Here the mean

As described above, Gaussian fields can describe spatial variability by using a mean value, a standard deviation and one single variogram or correlation function. The spatial dependence is an integral of the whole distribution of the parameters emphasising the mean value. But is this approach appropriate for geo-mechanical problems like the strip footing? In geostatistics there is a long tradition in simulation of random fields in geohydraulics. Deutsch & Journal (1992) as well as Seiffert & Jensen (1999) propose the Sequential Indicator Simulation as an extension of the Gaussian case. In Figure 10 one can see these two approaches in comparison. Both random fields share the same mean value, the same standard deviation and the same correlation distance. But they look different. This can be conducted to the correlation of extreme high and low values. By using the indicator approach the differences of these random fields can be

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Figure 12. Correlation distances at different thresholds of the cumulativ distribution function.

Figure 11. Indicator approach for the threshold (38% quantile of EB,3 ) and evaluation of the indicator - correlation distance θi .

enlightened. Basic principle of the indicator approach is the subdivision of the cumulative distribution function into certain thresholds. Therefore equation (3) is needed to take every value of the function z(x) up to a certain threshold q into as described in Journel (1983). Figure 13. Indicator correlation distances of the measurements.

This explanation becomes clearer by studying Figure 11. In the upper part of Figure 11 all measured values of the modulus of elasticity EB,3 are plotted in gray. Up to a threshold of EB,3 = 199 MN/m2 every value is taken into . Everything above this threshold is equal to zero. In the lower part of Figure 11 the correlation function of the indicator function is plotted in order to evaluate the correlation distance θind . In this way the random fields of Figure 10 are evaluated and the results are plotted in Figure 12 together with die cumulative distribution function of the random fields. One can clearly see that the extreme values of the distribution function have a shorter correlation length than the median value in the standard Gaussian case, which is visualized in grey. In the indicator approach one constant indicator correlation distance is detected. When analyzing the test results (Figure 13) it is apparent that a standard Gaussian model does not take the correlation of extreme values into . The correlation of extreme values of the soil should have an impact on the bearing capacity. Therefore an additional case study was carried out using the indicator approach as shown in Figure 12. For this purpose a mean value µ = 100 kN/m2 and a standard deviation σ = 10 kN/m2 have been used for

Figure 14. Comparison of the Standard approach and the Indicator Approach for the bearing capacity of a strip footing.

generation of random fields, which fulfil the indicator correlation distance θind = 10 m for all thresholds. The results of this are printed in Figure 14. The impact of the indicator approach on the mean value of the bearing capacity of the strip footing is apparent. A different correlation structure, which is introduced by the indicator approach an impact on the response of the system.

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3


This contribution examines soil variability. Description, evaluation and incorporation of soil variability into a design process are presented within a case study on the bearing capacity of a strip footing on variable soil.The traditional way of Gaussian random fields and a new indicator based approach are used to represent soil variability within a case study. It can be derived from this case study that there is a need for more investigations in evaluating stochastic soil properties and appropriate models to represent these properties apart from the traditional way. Within this contribution an alternative method is offered. It can be seen form the evaluations of the test results that the state of the art offers not a complete representation of stochastic soil properties. Moreover there should be more emphasis to gain knowledge in stochastic description of soil behaviour. Further knowledge in of stochastic description of different soil layers via random fields is necessary to make prediction of settlements due to construction processes. ACKNOWLEDGMENTS These in situ test are friendly ed by Prof. Dr.-Ing. habil H. Schad (Materialprüfungsanstalt Universität Stuttgart - Division 5 Geotechnics) through consulting, and fruitful discussions. Moreover we thank Dipl.-Ing. C.-D. Hauck (Stadt Stuttgart, Tiefbauamt), who allowed us to carry out the experiments on-site. Thanks also to Dr.-Ing. A. Möllmann and Dr.-Ing. S. Möller for discussions and helpful comments. REFERENCES Din 4094-5. 2001. Geotechnical field investigations - part 5: Borehole deformation tests. Din en 1990.2002. Eurocode: Basis of structural design. Din en 1997.2009: Eurocode 7: Geotechnical design – Part 1: General rules. Al-Khoury R. , Bakker K. J., Bonnier P. G., Burd H. J., Soltys G. & Vermeer P.A. 2008. PLAXIS 2D Version 9. Asoaka, A. & Grivas, D. A. 1982. Spatial variability of the undrained strength of clays.ASCE, Journal of Engineering Mechanics, 108(5): 743–756. Baecher, G. B. & Christian, J. T. 2003. Reliability and statistics in geotechnical engineering. John Wiley & Sons Inc. Baker, J. , Calle, E. & Rackwitz, R. 2006. t committee on structural safety probabilistic model code, section 3.7: Soil properties.

Chiasson, P., Lafleur, J. , Soulié, M. & Haw, K.T. 1995. Characterizing spatial variability of clay by geostatistics. Canadian Geotechnical Journal, 32: 1–10. Clark, I. 1979. Practical Geostatistics. Applied Science Publishers LTD. Deutsch, C.V. & Journel, A.G. 1992. GSLIB: Geostatistical software library and s’s guide. Oxford University Press, volume 340. Fenton, G.A. & Griffiths, D.V. 2008. Risk assessment in geotechnical engineering. John Wiley & Sons, New York. Hess, K. M., Wolf, S. H. & Celia, M. A. 1992. Large scale natural gradient tracer test in sand and gravel, cape cod, massachusetts 3. hydraulic conductivity variability and calculated macrodispersivities. Water Resources Research, 28: 2011–2017. Jaksa, M. B., Yeong, K. S., Wong, K. T. & Lee, S. L. 2004. Horizontal spatial variability of elastic modulus in sand from the dilatometer. In 9th Australia New Zealand Conference on Geomechanics, volume I, pages 289–294, Auckland. Journel, A. G. 1983. Nonparametric estimation of spatial distributions. Mathematical Geology, 15(3): 445–468. Journel, A. G. & Huijbregts, C. J. 1978. Mining geostatics. Academic Press, London. Mulla, D.J. 1988. Estimating spatial patterns in water content, matric suction and hydraulic conductivity. Soil Science Society, 52: 1547–1553. Phoon, K.-K. & Kulhawy, F. H. 1999. Characterization of geotechnical variability. Canadian Geotechnical Journal, 36: 612–624. Popescu, R., Prevost, J. H & Vanmarcke, E. H. 1995. Numerical simulations of soil liquefaction using stochastic input parameters. In Proceedings of the 3rd International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics. Rackwitz, R. 2000. Reviewing probabilistic soils modelling. Computers and Geotechnics, 26(3-4): 199–223. K. R. Rehfeldt, Boggs, J. M. & Gelhar, L . W. 1992. Field study of dispersion in a heterogeneous aquifer, 3-d geostatistical analysis of hydraulic conductivity. Water Resources Research, 28(12): 3309–3324. Ronold, M. 1990. Random field modeling of foundation failure modes. Journal of Geotechnical Engineering, 166(4). Seifert, D. & Jensen, J. L. 1990. Using Sequential Indicator Simulation as a tool in reservoir description: Issues and Uncertainties. Mathematical Geology, 31(5): 527–550. Soulié, M., Montes, P. & Silvestri, V. 1990. Modeling spatial variability of soil parameters. Canadian Geotechnical Journal, 27: 617–630. Unlu, K., Nielsen D.R., Biggar, J.W. & Morkoc, F. 1990. Statistical parameters characterizing variability of selected soil hydraulic properties. Soil Science Society American Journal, 54: 1537–1547. Vanmarcke, E. H. 1983. Random fields: analysis and synthesis. The M.I.T., 3rd edition. Vrouwenvelder, T. & Calle, E. 2003. Measuring spatial correlation of soil properties. Heron, 48(4): 297–311. Wackernagel, H. 2003. Multivariate geostatistics: An introduction with applications. Springer Verlag, 3rd edition.

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Inverse modelling including spatial variability applied to the construction of a road embankment A. Hommels & F. Molenkamp Delft University of Technology, Delft, The Netherlands

M. Huber & P.A. Vermeer Institute of Geotechnical Engineering, University of Stuttgart,

ABSTRACT: Almost all natural soils are highly variable in their properties and rarely homogeneous. There is a natural variation of soil properties from one point to another in space due to different deposition conditions and different loading conditions. This is called spatial heterogeneity and can be modeled using the Random Finite Element Method. To improve model predictions an inverse modeling technique can be implemented to incorporate measurements into a deterministic model. This enables the improvement of the poorly known parameters and consequently the model results. This allows for observations of on-going processes to be used for enhancing the quality of subsequent model predictions based on improved knowledge of the soil parameters. The Ensemble Kalman filter calculates an assimilated state each time measurements become available, using these measurements and the model predictions for that time step including both the measurement and its possible error range. In this way the uncertain parameters are improved. The combination of the Ensemble Kalman filter and the Random Finite Element method is a very powerful instrument to address the uncertainty of soil parameters and in this way improve the model prediction.

1 1.1

INTRODUCTION Inverse modelling

In geotechnical engineering context inverse modelling or back analysis consists in finding the values of the mechanical parameters, or of other quantities characterizing a soil or rock mass, that when introduced in the stress analysis of the problem under examination lead to results (e.g. displacements, stresses etc.) as close as possible to the corresponding in situ measurements. The optimal state of the system is obtained by minimizing the difference between the observed values in the system and the forecasted or modelled state of the system within a certain time interval. In the eighties and the nineties, several articles concerning inverse analysis in geo-mechanics using the Maximum Likelihood and the (Extended) Bayesian method were published (Gens et al. 1996, Honjo et al. 1994a,b). Recent developments in other fields of science have shown a new powerful technique indicated as the Ensemble Kalman filter. In a filter, the state of the system is analysed each time data becomes available.

constant throughout the mass or the layer. However, in nature, this is not really the case: properties of natural soils will vary through depth and often also in horizontal extent due to for example different loading conditions or different depositional conditions. One way to deal with this uncertainty is the Random Finite Element Method. This technique incorporates the spatial correlation between the properties using Monte Carlo simulations in order to represent the proper stochastic properties. The general formulations of the Ensemble Kalman filter and the Random Finite Element Method will be discussed in the next sections. In Hommels et al. 2006 the effectiveness of the Ensemble Kalman filter using the Random Finite Element Method for a soil column has been proven. In this paper the Ensemble Kalman filter will be combined using two Random Finite Element methods for the construction of a road embankment.

2 THEORY 2.1 Basic principles

1.2

Geological uncertainty

Usually in the field of geomechanics the Finite Element Method is used for numerical simulations. In this method, one particular value for a certain parameter is assigned to the soil mass or a soil layer, which remains

The true state of the subsurface at time step k can be described by a state vector xt (k). The elements of the state vector are filled with stresses but possibly also strains or other state parameters. The superscript ‘t’ denotes that xt (k) is the “true” state; the exact value is

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probably unknown. To obtain insight in the true state, a model is developed to make a forecast or to model xf (k + 1) at time step k + 1:

The superscript ‘f ’ denotes that xf (k) is a “forecast” of the true state xt (k) at time step k, which is in the best case a good approximation. In the context of the shallow subsurface, the state vector can partly be filled with displacements u. M denotes the dynamical model operator, which describes for instance the constitutive model of the soil, e.g. the soil parametersYoung’s modulus Eand Poisson’s ratio ν in the simplest elastic case. If there are uncertainties in the parameters, which have to be updated, the state vector xf (k)is also filled with the uncertain parameters. Since models are never perfect:

in which η(k) is the unknown model error in the kth forecast with E{η} = 0 (E denotes expectation) and E{η2 } = P, which is the model error covariance matrix. Some entities of the state are compared with data from an observational network, for example the measurements yo of the surface displacements. All available data for time step k are stored in an observation vector yo (k). The superscript ‘o’ denotes that yo (k) is an “observation”. There is a difference ε(k) between the “true” state xt (k) and the actual “observed” data yo (k):

where H (k) is the linear observational operator. The observations are assumed unbiased (E{ε} = 0)) and E(ε2 ) = R, which is the measurement error covariance matrix. The final goal of inverse modelling methods is to improve the state vector; at each time step k measurements become available, with an error ε(k) as small as possible. 2.2

Ensemble Kalman Filter

Evensen (1994, 2003) introduced the Ensemble Kalman Filter (EnKF). The EnKF was designed to resolve two major problems related to the use of the Extended Kalman filter (EKF). The first problem relates to the use of an approximate closure scheme in the EKF, and the other one to the huge computational requirements associated with the storage and forward integration of the error covariance matrix in the EKF. In the EnKF, an ensemble of N possible state vectors, which are randomly generated using a Monte Carlo approach, represents the statistical properties of the state vector. The algorithm does not require a tangent linear model, which is required for the EKF, and is very easy to implement.

At initialisation, an ensemble of N initial states (ξN )0 are generated to represent the uncertainty at time f step k = 0. The matrix Ek+1 defines an approximation of the covariance matrix Pk+1 . The time update equations for the Ensemble Kalman filter for each ensemble ξ are

in which Gk is the noise input matrix and wk is the process noise. The measurement update step equation is:

in which yo are the measurements, H is the observational operator, ε is a randomly added measurement noise, because the measurements are treated as random variables (Evensen, 2003) and K is the Kalman gain, which is defined as

The performance of the EnKF is dependent on several input parameters, which are amongst others the model- and measurement noise, the number of ensemble , the amount of observations and the initial parameter uncertainty. 2.3 Simulation approach Variability can be captured via the simulation approach. Herein equi-probable realisations of random fields are generated fulfilling a target mean, standard deviation and correlation function. This concept is realized in the Random Finite Element Method (RFEM). RFEM combines the finite element analysis with a random field theory. In the Finite Element Method the uncertainty of a material is defined by its mean µ and its standard deviation σ. For the introduction of more spatial variability, the introduction of an additional statistical parameter, the spatial correlation length θ, is required. The spatial correlation length defines the distance beyond which there is minimal correlation and can be determined from for example T-data. A large value of θ indicates a strongly correlated material, while a small value indicates a weakly correlated material. As described by Baecher & Christian (2003) stationarity and ergodicity are assumed. A random field is said to be second order stationary, if mean and covariance depend only upon vector separation. Ergodicity means that the probabilistic properties of a random field can be completely estimated from observing on realization of that field. With

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these assumptions the random field can be generated. There a several theories to generate a random field. In this paper the Local Average Subdivision theory (LAS) will be compared with the Midpoint method. A brief description of each random field generator will be given. 2.3.1 Local Average Subdivision In the Random Finite Element Method, this is done using the Local Average Subdivision (LAS) (Fenton & Vanmarcke 1990), based on a standard normal distribution (mean µ is zero and the standard deviation σ equals one) and a spatial correlation function ρ(τ). LAS generates a square random field by uniformly subdividing a square domain into smaller square cells, where each cell has a unique local average, which is correlated with surrounding cells. For an isotropic field, where θ is equal in all directions, the Gauss-Markov correlation function ρ(τ) is given by

in which τ is the lag vector. The random field for a certain parameter E for cell i, based on a standard normal distribution, can then be transformed into for example a normal distribution:

in which µE is the mean and σE is standard deviation of parameter E; Zi is the local average value for cell i. There are an infinite number of possibilities for the random field, based on the given set of statistics of µ, σ and θ. Again Monte Carlo simulations are used to express the spatial distribution (Hicks & Samy 2002). 2.3.2 Midpoint Method This Midpoint-Method is a robust method in generating correlated random fields, as described by Sudret (2007). This technique is based on the LU triangular decomposition of the matrix of covariances between the center points of each element. The covariance between two center points is calculated using a correlation function such as a Gauss-Markov correlation function ρ(τ) in equation 9. Covariance matrix is symmetric and positive-definite and therefore can be decomposed into the product of a lower and an upper triangular matrix as shown in equation 11. Through the multiplication of the vector of in-dependent random variables r∗ with the lower triangular matrix L a correlated vector r is calculated fulfilling the requirements of the correlation function ρ(τ) used for the covariance matrix. The advantages of the method are that it is simple to implement, is not limited to particular forms of covariance functions and automatically handles anisotropies. The major drawback of this method is the amount of storage required which, at least in its general form as presented. Dowd (2003) presents possibilities how to overcome this problem.

Figure 1. Road embankment based on a random field (correlation length θv = 5 m, θh = ∞); 10 observation points are indicated with rectangles.

2.4 Monte Carlo simulations Since both the Ensemble Kalman filter and the Random Finite Element Method are based on Monte Carlo simulation, these simulations are combined in order to save computational effort. However enough realizations are required to ensure a good representation of probability density of the state estimate. 3

CASE STUDY

3.1 Construction of a road embankment For this case study, a four meter high construction of a road embankment is considered, where the Young’s modulus E of the foundation is uncertain and modelled using LAS and Midpoint method (figure 1). Herein 8-noded quadric lateral elements have been used in combination with the using the Mohr-Coulomb model (Smith and Griffiths, 2004). The horizontal spatial correlation length is considered to be infinite, where the vertical spatial correlation length is considered to be 30% of the depth of the foundation. In figure 1, the foundation below the embankment, modelled using a Random Finite Element Method is shown. The observation locations at which the vertical displacement are measured and which are used for the input of the EnKF are indicated with blue rectangle. The embankment is constructed in four phases as shown in figure 2. 3.2 Results of the calculation 3First the influence of the amount of measurement noise is considered. In figure 3 the difference bet-ween a measurement noise with a standard deviation of 10−4

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(left) and 10−6 (right) using 100 ensemble in each case is shown. The influence of the different standard deviations of the measurement noise can clearly be seen in LAS as well as in Midpoint method. The difference between the results of the LAS and the Midpoint method shown in figure 3 are more or less negligible. From these subfigures it can be concluded that if the measurement noise decreases, the parameter uncertainty decreases and the speed of the assimilation process increases. In the right subfigure it is clearly shown that both the mean and the standard deviation of the Young’s modulus are improved. One can clearly see the influence of the amount of ensemble in figure 4; 50, 100 and 150

ensemble are used. From 4 it can be concluded that 50 ensemble is not enough for a correct assimilation process of the Young’s modulus E. Comparing the results of the LAS method (left column, figure 4) and the Midpoint method (right column, figure 4) for 100 and 150 ensemble , it can be concluded that the reduction of the parameter uncertainty, using the Midpoint method, of theYoung’s modulus shows small differences. The differences between the results of the two different random field generators can be explained via the distribution and the correlation parameter shown in figure 5. The Midpoint method matches the target distribution function in a better way than LAS. The non exact matching of the correlation function (figure 5) can be explained via ergodic fluctuations, which are related to the domain size of the random field as described by Deutsch & Journel (1998). Moreover the theory behind the random field generators offers another explanation. Using LAS for the generation of the random fields, there is top-to-bottom dependency (Fenton and Vanmarcke, 1990).

4 Figure 2. Embankment height at the left axis constructed in four phases; at the right axis the vertical displacement at one of the observation locations.


Within this contribution the concept of inverse modeling in geotechnics is presented by using the Ensemble

Figure 3. Differences between a measurement noise with a standard devitation of 10−4 (upper part) and 10−6 (lower part) using 100 ensemble and using LAS (left column) and Midpoint method (right column).

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Figure 4. Difference between 50 (first row), 100 (second row) and 150 (third row) ensemble using a measurement noise with a standard deviation of 10−4 together with LAS (left column) and Midpoint method (right column).

Kalman filter. This method enables the incorporation of measurements results and possible error ranges. A case study of the construction of a road embankment shows the influence of different measurement noises and different amounts of ensemble on the back analysis of the modulus of elasticity. If more ensemble are used, the parameter uncertainty will decrease more and faster. Also a decrease in measurement noise, will lead to better assimilation results. If more observations are available, further research on more suitable random field generators is needed. In general the combination of the Ensemble Kalman Filter with the Random Finite Element Method is a very powerful instrument to reduce the parameter uncertainty. It depends on the boundary conditions of the problem set, how spatial variability can be incorporated.

Figure 5. Evaluation of the correlation length of the random field realisations by using the autocorrelation funciton.

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REFERENCES Baecher, G.B. and Christian, J.T. 2003. Reliability and statistics in geotechnical engineering, John Wiley & Sons Inc. Deutsch, C. and Journel, A.: GSLIB-Geostatistical software Library and s’s guide, Oxford University Press, 1998. Dowd, P. A. 1992: A review of recent developments in geostatistics, Computers & Geosciences,17 (10), p. 1481– 1500. Evensen, G. 1994. Sequential data assimilation with nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics. Journal of Geoph. Research, vol. 99, C5/10, p. 143–162. Evensen, G. 2003. The Ensemble Kalman filter: theoretical formulation and practical implementation. Ocean dynamics, vol. 53, 4, p. 343–367. Fenton, G. A., and Vanmarcke, E.H. 1990. Simulation of random fields via local average subdivision. Journal of Geotech. Eng, ASCE, 116, 8, p. 1733–1749. Gens, A., Ledesma, A. and Alonso, E. 1996. Estimation of parameters in geotechnical back analysis – II. Application to a tunnel excavation problem. Computers and Geotechnics, vol. 18, 1, p. 29–46.

Hicks, M. A. and Samy, K. 2002. Influence of heterogeneity on undrained clay slope stability. The Quarterly Journal of Engineering Geology and Hydrogeology, vol. 35, 1, p. 41–49. Hommels, A. and Molenkamp, F., Inverse analysis of an embankment using the Ensemble Kalman Filter including heterogeneity of the soft soil. Proc. of the Sixth Eur. Conference on Numerical Methods in Geotechnical Engineering, 6–8 September 2006, Graz – Austria. Honjo, Y., Liu, W.T. and Soumitra, G. 1994a. Inverse analysis of an embankment on soft clay by extended Bayesian method. Int. Journal for Num. and Anal. Methods in Geo-mechanics, vol. 18, p. 709–734. Honjo, Y., Wen-Tsung, L. and Sakajo, S. 1994b. Application of Akaike information criterion statistics to geotechnical inverse analysis: the extended Bayesian method. Structural safety, vol. 14, p. 5–29. Smith, I. M. and D. V. Griffiths. 2004. Programming the finite elememt method. Chichester, Wiley and sons. Sudret, B. 2007. Uncertainty propagation and sensitivity analysis in mechanical models – Contributions to structural reliability and stochastic spectral methods, Habilitation à diriger des recherches, Université Blaise Pascal, Clermont-Ferrand.

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Reliability analysis of piping in embankment dam Ali Noorzad Power & Water University of Technology, Tehran, Iran

Mohsen Rohaninejad Iran University of Science & Technology, Tehran, Iran

ABSTRACT: The present paper is focused on the application of reliability analysis to assess the piping phenomenon in embankment dams. In order to run the reliability analysis, the effective parameters in the failure mode of piping are identified. Then the probability distribution curves of the parameters are drawn based on the experimental results of the site. Utilizing Monte Carlo simulation and repetitive finite element analysis, the probability of the occurrence of piping and exit gradients in critical zones are specified. To show the capability of the approach, a case study, namely, No. 4 Chahnimeh dam located in Sistan- Baluchistan province, Iran has been considered. The obtained results show that the dam demonstrates a high reliability in of piping phenomenon and seems to be safe. In order to the results, sensitive analysis has been implemented with the new hypotheses confirming the accuracy of the proposed hypotheses.

1

INTRODUCTION

The role of water resources projects in developing countries and the safety level of the infrastructure are important to decision makers. With respect to the failure mode in embankment, the uncertainties involved are required to be first identified and then prioritized using qualitative analysis. The significant risks should be quantitatively assessed. Reliability analysis is one of the methods which can be used in quantifying risks. New aspects of geotechnical analysis based on risk assessment have been widely considered due to the uncertainty in geotechnical parameters. These uncertainties are originated from physical uncertainty, human errors and uncertainty in modeling. It is believed that the combination of these unknown parameters leads to unreliable design. Currently, most of engineering analyses in conventional approaches are not considered as complete design process. Thus, any analytical method determining the failure probability needs to compute the entire of relevant mechanisms and their effective characteristics (Stewart, 2000). There are various methods to determine the probability of dam’s failure in quantitative risk assessment for all failure modes. For some failure modes like overtopping and liquefaction, the methods are well developed; in contrast, there is little work on other phenomenon such as piping (Fell et al., 2000). In conventional design, which utilizes deterministic methods, guidelines are available providing enough assurance to the project. However, in reliability analysis, different calibrations should be performed

(US Army Corps of Engineers, 2006). Calibrations include performing of comparative studies of deterministic analyses for a wide range of geotechnical issues and case studies. In this research, a case study, namely, Chahnimeh dam is selected. Base on the existing documents, piping phenomenon is expected to be the most serious failure mode of this dam. One of the significant problems in reliability theory is the investigation of reliability index changes due to changing in different parameters. In order to overcome this problem, two analyses are carried out. 2

RELIABILTIY ANANLYSIS IN GEOTECHNICS

From 1980, comprehensive studies on the development of reliability theory have been done. Among them, Filippas et al. (1988) and Phoon et al. (1995) studied the impact of geotechnical parameters in reliability of foundation. The results indicated that coefficient of variation (COV) of soil parameters is important in the structure reliability. Many researchers have investigated the problem of water seepage in soil (Elkateb et al., 2003). Applying the fundamental procedures into the geotechnical practice, Griffiths & Fenton (1993) considered the effect of soil spatial variability on seepage flow. To incorporate soil parameter variations, quantitative approaches in risk analysis are utilized. Classical statistical methods for soil variation is based on statistical outcome like the average, the coefficient

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of variation and the probability distribution; such as Schultze (1975) and Griffiths & Fenton (1998). They have used the statistical distribution like normal, lognormal and beta to interpret the field data. In an attempt, Tobutt & Richards (1979) logically used the variability of soil parameters in combination of limit state equation with Monte Carlo simulation for reliability analysis of earth slopes. In the same line of thought, Paice et al. (1994) integrated spatial correlation by performing the Monte Carlo simulation into deterministic numerical analysis. In addition, Auvinet et al. 1996 used the stochastic finite element method as an efficient procedure to consider the soil variability into a numerical analysis framework. In the probabilistic methods, variables are divided into two main categories of resistance and load. Under this division, a simple form of the limit state function can be defined according to Equation 1 in which there is an implicit or explicit relation between variables and the safety of a model. Accordingly, a limit state equation (LSE) can be defined whereas Z = 0.

Therefore, the limit state equation (LSE) clarifies two different regions where the LSE ≥ 0 or not. In this case, r is a vector of resistant variables, and s is a vector of load variables. However, in complex problems as well as in this research there is an implicit LSE in which the relation between stress and resistant is not explicitly known. The reliability-based approach relies on selecting design parameters that satisfy a desired degree of rliability or a certain probability of failure. To compute the reliability index, safety levels can be used as given in Equation 2. This index has been frequently used in slope stability analysis. For instance, Wolff (1996) suggested a reliability index of three for ordinary slopes and four for critical slopes. The reliability index can be computed through

where mFS is the mean factor of safety; L is a limit state value usually equal to one and σFS is the standard deviation of safety factor (Elkateb et al., 2003). As shown in Figure 1, by determining β, the failure mode of dam can be achieved (Phoon, 2004). The level of safety can be considered as bounded by the following two extremes; the high level of safety that is physically achievable at any cost, and a hazardous level below which the dam cannot withstand normal operating conditions. Between these bounds, safety decision-making involves striking a balance between the risks and the benefits and between social equity and economic efficiency. Based on random variables, the algorithm can be summarized as follows: 1. Identifying all parameters pertinent to risk and reliability analysis;

Figure 1. Assessment of dam safety using the reliability index.

2. Defining the parameters as random variables in the failure modes and asg statistical distributions to them; 3. Constructing suitable reliability model in accordance with statistical models controlling random variables; 4. Determining indefinite limits and making the probability distributions to compute the probability of failure modes. 2.1 Piping reliability assessment with finite element model There is no specific procedure to determine failure probability of internal erosion and piping, which are the main problems of most embankment dams. Foster et al. (2000) presented an approach for evaluation of failure probability due to piping. The probability of failure is estimated by adjusting the historical frequency of piping failure by weighting factors which take into the dam zoning. Lacasse et al. (2004) evaluated the risk of piping based on engineering judgment. In an attempt to investigate the piping risk, Badv and Sargordi [24] utilized the limit state equation incorporation the experimental relation of Sellmeijer and Koenders [25] that seems not to be reliable. The finite element analysis is the only method to evaluate the limit state equation without loss of accuracy. An explicit term for the limit state equation is unavailable. In addition to utilize the finite element analysis, there is a need to use probabilistic techniques in engineering problems as they provide a deeper understanding of failure mechanisms. Hence, to accomplish these advantages, a well-defined model of the structure together with a reliability technique is required (Rajabalinejad et al., 2009). Monte Carlo simulation is extensively considered to be among the most efficient and commonly applicable procedure. Therefore, the combination of finite element model together with Monte Carlo simulation is the appropriate alternative for the reliability analysis.

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Table 1. Technical specifications of No. 4 Chahnimeh dam Dam type: homogeneous embankment Crest width: 8 m Height from foundation: 15.5 m Upstream slope: 1 to 3.5 (V/H)

Crest length: 15250 m Foundation width: 105.5 m Reservoir volume: 820 MCM at normal level Downstream slop: 1 to 3 (V/H)

Monte Carlo simulation has progressively become more frequent in dam risk assessments. For the first time, Kim and Major (1978) used Monte Carlo simulation with limit equilibrium equation. Soil parameters were assumed as uncorrelated random variables. In order to get an acceptable estimation of the simulation, a large number of random variable should be produced. According to Equation 3, Nowak & Collins (2000) presented the required number of random samples as

where Pf is the approximate failure probability and COV is the coefficient of variables.

3 A CASE STUDY: NO. 4 CHAHNIMEH DAM To demonstrate the capability of the Monte Carlo finite element method, a case study has been considered. No.4 Chahnimeh dam is located in south of Iran and a summary of the dam characteristic is given in Table 1. The elements controlling seepage in all sections of the dam are vertical and horizontal drain, upstream blanket and relief wells. Downstream berm and clayey cutoff wall are also constructed in some specific sections. According to available documents prepared by dam consultant, piping phenomenon is expected to be the most serious failure mode of this dam. With the assumption of using suitable materials with enough compaction during construction and vertical and horizontal drains along with the proper filter, seepage from dam body does not seem to be an important issue. However, seepage from the foundation of dam can be significant because of the large thickness of alluvial layer, considerable length of dam, high variation of permeability coefficients, the presence of sand lens in the foundation and lack of cutoff wall. Therefore, the piping phenomenon and intensive seepage in some zones of dam foundation are likely to occur. In this study lognormal distribution for permeability random variable and normal distribution for density of foundation soil are considered [U.S. Army Corps of Engineers, (2006), Duncan, (2000) and Gui et al., (2000)]. The probability density function for one of input random variable is presented in Figure 2, which is appropriately interpreted with lognormal and gamma

Figure 2. PDF of various coefficient of permeability.

distribution. In addition, vertical coefficient of permeability, Ky , is taken as random variable and the permeability ratio, Kx /Ky is assumed to be 9. In our model, the normal water level is considered as the load. Also, the variations of two soil parameters are considered in the safety analysis of the dam. The variations are considered both in horizontal and vertical directions according to Table 2. As the main purpose of the investigation is to study piping and seepage failure mode in this dam, the reliability analysis will be focused on piping phenomenon from foundation to horizontal drain and to downstream, respectively. Based on erosion model proposed by Fell et al. 2001, only the piping within the foundation is assessed. Subsequently from Equation 3, the failure probability is computed as 10−4 ; then the required number of random variables is taken 2500.

4

RELIABILITY ANALYSIS IMPLEMENTATION

To accomplish the probabilistic analysis, a code is provided along with the FLAC software. Figure 3 shows a finite difference model of No.4 Chahnimeh dam. Because the dam’s geometry is somewhat regular, the most element of numerical model is rectangular elements. This is very helpful in reducing the calculation time and makes a simpler model by using the fewer meshes. According to simulation process, the variation of the input parameters provides a wide range of possible combinations of different variables, under the assumed probability distribution function (PDF). Results of stochastic numerical analysis may include total discharge, exit gradients in major parts of the dam and the safety factor against piping in horizontal drain and downstream. By evaluating the above results, the probability of failure due to piping can be determined. An overview of the process of determining failure probability is shown in Figure 4. In order to study the random numbers generation process, Figure 5 shows the probability density function (PDF) for one of the permeability random variable. It is properly fitted to lognormal distribution and in good agreement with the hypotheses.

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Table 2. The summary of field results.

Specifications Coefficient of permeability (cm/s) Saturated density (kN/m3 )

First layer Second layer Third layer

Maximum

Minimum

Average

Standard Deviation

Coefficient of Variation

0.0075 0.0032 0.0033 18.40

1.40 × 10−6 1.40 × 10−6 1.60 × 10−6 14.00

6.12 × 10−4 3.95 × 10−4 4.87 × 10−4 15.38

13.60 × 10−4 7.22 × 10−4 7.50 × 10−4 0.77

220 183 154 5

Figure 3. The finite difference model of No.4 Chahnimeh dam, modeled with FLAC.

Figure 5. PDF of random numbers for the first layer.

Figure 4. Process of determination failure probability.

5

DISCUSSION OF RESULTS

After obtaining the results of Monte Carlo simulation and determining the failure probability and reliability

index, new features can be achieved toward dam’s stability against piping. A summary of the simulation results is presented in Table 3. It can be stated that coefficient of the variation of less than 10 percent of safety factor shows the appropriate scattering of the output data (Nwaiwu, 2008). The average amounts of exit gradients are negligible and the safety factor and safety margin are evaluated to be very high. By substituting the average and standard deviation of the safety factors in Equation 1, the reliability index in horizontal drain is 5.48 and in downstream is 4.94. These quantities show that the safety level of the dam is high. Furthermore, with regard to the fact that the safety factor in none of the iterations has not been less than one, therefore the probability of failure will be less than 10−4 . Considering Figure 1, and the amount of β, the failure probability can be predicted to be about 3 × 10−7 being an acceptable level for dam construction. Figure 6 presents the cumulative density function (CDF) for safety factor against piping in downstream.

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Table 3.

Monte Carlo simulation results.

Outputs

Max.

Min.

Average

Coefficient of Variation

Total discharge (m3 ) Horizontal gradients at vertical drain Vertical gradients from foundation to horizontal drain Vertical gradients at downstream Horizontal gradients at relief wells Safety factor against piping within the dam Safety factor against piping downstream of dam Safety margin against piping within the dam (kPa) Safety margin against piping downstream of dam (kPa)

2.98 0.32 0.58 0.47 19.02 2.57 2.20 13.2 11.8

0.188 0.077 0.026 0.060 0.076 1.30 1.12 6.7 1.8

0.594 0.166 0.133 0.144 1.38 2.01 1.83 9.2 7.8

54.40 29.41 66.79 45.06 102.45 8.59 9.15 9.48 17.56

Figure 7. CDF of safety factor in downstream.


As observed, the probability of occurrence is higher than 1.2 indicating the safety factor is safe in different locations.

5.1

Sensitivity analysis

One of the significant problems in the reliability theory is to determine the changes in the reliability index due to the changes in different parameters. This process has been studied by sensitivity analysis. Two analyses have been performed. In the first analysis, normal distribution is employed based on Nwaiwu (2008) who introduced normal and gamma distribution for adjusting permeability data. In the second analysis, the horizontal coefficient of permeability, Kx , is considered as a random variable and the permeability ratio, Ky /Kx , is assumed to be 9. 5.1.1 Changing the type of random distribution function In this analysis, the distribution function of permeability is changed from lognormal to normal distribution. The average amount of exit gradients is negligible and the safety factors are evaluated to be high. The reliability indexes are 4.1 and 3.4 in horizontal drain and downstream, respectively. Also, the safety factors for different situations are greater than 1. The cumulative density function (CDF) for safety factor against piping in downstream is depicted in Figure 7. In this figure, the probability of existence of safety factor is

Figure 8. PDF of generated random number for the first layer.

not less than 1.1 indicating the dam is safe against piping. In order to explain the causes of decreasing reliability index, Figure 8 shows the PDF of one of the random generations with normal distribution. In this figure, the generation outputs cannot be even adjusted with normal distribution. It can be mentioned that the high coefficient of variation in the safety factor is due to generating inappropriate random variable. Therefore, it can be stated that the lognormal distribution can be fitted better than normal distribution for the coefficient of permeability.

379

REFERENCES


5.1.2 Changing random characters It can be observed that coefficient of variation of less than 10 percent for the safety factor shows the scattering of the output data. As mentioned in the previous section, the reliability indexes are 8.9 and 5.4 respectively in this analysis. These quantities illustrate that the dam safety level is high. In addition, the safety factor in none of the iterations has not been less than one. Figure 9 shows the cumulative density function (CDF) for safety factor against piping in downstream. In this figure, the probability of existence of safety factor is not less than 1.4 indicating the dam is safe against piping. Based on the obtained results, it is shown the capability and validity of the proposed hypotheses in examining the reliability against piping phenomenon in embankment dam such as No. 4 Chahnimeh dam.

6

CONCLUSIONS

In this study, using the finite element model and Monte Carlo simulation, reliability analysis of piping phenomenon for No. 4 Chahnimeh dam has been carried out. The results indicate that the dam demonstrates a high reliability in of piping phenomenon and seems to be safe. With particular attention to high reliability index, it is observed that the design of this dam appears to be on the conservative side. Sensitive analyses is confirmed the accuracy of the proposed hypotheses. The main purpose of using reliability models is its capability for becoming an effective mechanism in decision-making process. Reliability analysis can be utilized in evaluating the risks related to embankment dam. Probabilistic assessment is more efficient than deterministic methods which are only relied on the safety factor. Due to low failure probability of infrastructures like dams, safety factor cannot well explain the uncertainties. With respect to the effect of failure mode in embankment dams, it is necessary to use systematic analyses to reduce or minimize the uncertainty effects. Therefore, along with conventional analyses, it is better that these analyses are also accomplished.

Auvinet, G., Bouayed, A., Orlandi, S. and Lopez, A. 1996. Stochastic finite element method in geomechanics. Proceeding of the 1996 Conference on Uncertainty in the Geologic Environment, Uncertainty 96, Vol. 2, Madison, Wis. pp. 1239–1253. Badv, K. and Sargordi, F. 2001. An investigation into the risk of piping at dams in the Urmia region, Iran. Iranian Journal of Science & Technology, 25(B4), pp. 625–634. Duncan, M. 2000. Factors of safety and reliability in geotechnical engineering. J. of Geotechnical and Geoenvironmental Engineering, ASCE, 126(4), pp. 307–316. Elkateb, T., Chalaturnyk, R. and Robertson, P.K. 2003. An overview of soil heterogeneity: quantification and implications on geotechnical field problems. Canada Geotechnical Journal, Vol. 40, pp. 1–15. Fell, R., Bowles, D., Anderson, L. and Bell, G. 2000. The status of methods for estimation of the probability of failure of dams for use in quantitative risk assessment. Proceeding of the 20th Congress on Large Dams, International Commission on Large Dams (ICOLD), Beijing, China, pp. 76–96. Fell, R., Wan, F. C., Cyganiewicz, J. and Foster, M. 2001. The time for development and detect ability of internal erosion and piping in embankment dams and their foundation. UNICIV Report No. R-376, School of Civil and Environmental Engineering the University of New South Wales, Sydney Australia. ISBN: 85841 2663. Filippas, O.B., Kulhawy, F.H. and Grigoriu, M.D. 1988. Reliability based foundation design for transmission line structures: uncertainties in soil property measurement. Electric Power Research Institute, Palo Alto, Calif., Report No. EL-5507(3). Foster, M.A., Fell, R. and Spannagle, M. 2000. A method for estimating the relative likelihood of failure of embankment dams by Piping. Canadian Geotechnical Journal, 37(5), pp. 1025–1062. Griffiths, D. V. and Fenton, A. 1998. Probabilistic analysis of exit gradients due to steady seepage. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 124(9), pp. 789–797. Gui, S., Zhang, R. and Turner, J. 2000. Probabilistic slope stability analysis with stochastic soil hydraulic conductivity. Journal of Geotechnical and Geoenvironmental Eng., ASCE, 126(1), pp. 1–9. Kim, H. and Major, G. 1978. Application of Monte Carlo techniques to slope stability analysis. Proceedings of the 19th U.S. Rock Mechanics Symposium, Reno, Nev., pp. 28–39. Lacasse, S., Nadim, F., Hoeg, K. and Gregersen, O. 2004. Risk assessment in geotechnical engineering: The importance of engineering judgment. Proc. Advanced in Geotechnical Engineering: The Skempton Conference, 2, pp. 856–867. Nowak, A.S. and Collins, K.C. 2000. Reliability of structure. University of Michigan, McGraw-Hill International Edition. Nwaiwu, C. 2008. Statistical distributions of hydraulic conductivity from reliability analysis data. Journal of Geotechnical, Springer Science, DOI 10.1007/s107069221-4. Phoon, K.K. 2004. Towards reliability-based design for geotechnical engineering. Special Lecture for Korean Geotechnical Society, 15 p. Phoon, K.K., Kulhawy, F.H and Grigoriu, M.D. 1995. RBD of foundations for transmission line structures. Electric Power Research Institute (EPRI), Palo Alto, Report No. TR-105000.

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Rajabalinejad, M. (2009). “Reliability methods for finite element models”. Ph.D. thesis, Delft University of Technology, Delft, The Netherlands, 120 p. Schultze, E. 1975. Some aspects concerning the application of statistics and probability to foundation structures. Proceeding of the Second International Conference on the Applications of Statistics and Probability in Soil and Structure Engineering, Aachen, , pp. 457–494. Sellmeijer, J.B and Koenders, M. 1991. A mathematical model for piping. Applied Mathematical Modeling, 15, pp. 65–75. Stewart, R.A. 2000. Dam risk management. Proceeding of GeoEng. Conference, Melbourne, Australia, pp. 19–24.

Tobutt, D.C. and Richards, E. 1979. The reliability of earth slopes. International Journal for Numerical and Analytical Methods in Geomechanics, 3, pp. 323–354. US Army Corps of Engineers 2006. Reliability analysis and risk assessment for seepage and slope stability failure modes in embankment dams. Report No. 1110-2-561. Wolff, T.F. 1996. Probabilistic slope stability in theory and practice. Proceedings of the 1996 Conference on Uncertainty in the Geologic Environment, 96, Madison, pp. 419–433.

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Spatial variability of soil parameters in an analysis of a strip footing using hypoplastic model R. Suchomel & D. Mašín Charles University in Prague, Czech Republic

ABSTRACT: Advanced hypoplastic constitutive model is used in probabilistic analyses of a typical geotechnical problem, strip footing. Spatial variability of soil parameters, rather than state variables, is studied by means of random field Monte-Carlo simulations. The model, including correlation length, was calibrated using a comprehensive set of experimental data. Foundation displacement uy for given load follows closely lognormal distribution, even though some model parameters are distributed normally. The vertical correlation length θv was found to have minor effect on µ[uy ], but significant effect on σ[uy ], which decreases with decreasing θv due to spatial averaging. Applicability of a simpler probabilistic method (FOSM) is also discussed.

1

INTRODUCTION

Geomechanical properties measured in site investigation programs are highly variable. The causes for the parameter variability can be broadly divided into two groups (Helton 1997). Objective (aleatory) uncertainty results from inherent spatial variability of soil properties, whereas subjective (epistemic) uncertainty is caused by the lack of knowledge and measurement error. Both sources of uncertainty need to be considered in geotechnical design (Schweiger and Peschl 2005). In this work, we focus on the description of the aleatory uncertainty, which is inherent to the given soil deposit and cannot be reduced by additional experiments or improvement of experimental devices. Advanced constitutive models for soils distinguish between material parameters and state variables. In principle, soil parameters are specific to the given mineralogical properties of soil particles and soil granulometry. State variables (such as void ratio e) then allow us to predict the dependency of the soil behaviour on its state. In this respect, the sources of aleatory uncertainty can further be subdivided into two groups: 1. In some situations, soil mineralogy and granulometry may be regarded as spatially invariable, and the uncertainty in the mechanical properties of soil deposit come from variability in soil state. In this case, soil parameters may be considered as constants, and it is sufficient to consider spatial variability of state variables describing relative density of soil. 2. In other cases, soil properties are variable due to varying granulometry and mineralogy of soil grains. Such a situation is for example typical for soil deposits of sedimentary basins, where the granulometry varies due to the variable geological

conditions during the deposition. In such a case, it is necessary to consider spatially variable soil parameters. Most of the applications of probabilistic methods in combination with advanced soil constitutive models consider uncertainty in the state variable only. As an example, Hicks and Onisiphorou (2005) studied stability of underwater sandfill berms. Their aim was to study whether presence of ‘pockets’ of liquifiable material may be enough to cause instability in a predominantly dilative fill. They used a doublehardening constitutive model with probabilistic distribution of state variable ψ. In other applications, Tejchman (2006) studied the influence of the fluctuation of void ratio on formation of the shear zone in the biaxial specimen using the hypoplastic model by von Wolffersdorff (1996). The aim of the research project presented is a complete evaluation of the influence of parameter variability of an advanced constitutive model on predictions of typical geotechnical problems. Suchomel and Mašín (2009) performed a set of laboratory experiments on sandy material, that were used for evaluation of probabilistic distributions and spatial variability of parameters of a hypoplastic model for granular materials by von Wolffersdorff (1996). The influence of their variation on predictions of a typical geotechnical problem (strip footing) is presented in this contribution. 2

EXPERIMENTAL PROGRAM AND CALIBRATION OF A CONSTITUTIVE MODEL

For details of the experimental program and calibration of the models see Suchomel and Mašín (2009).

383

Figure 1. The wall of the sand pit in south part of the Tˇreboˇn basin. Black dots represent positions of specimens for the laboratory investigation. Figure 2. Typical experimental and simulated results of drained triaxial tests.

The material for investigation comes from the south part of upper Cretaceous Tˇreboˇn basin in the South Bohemia from the sand pit “Kolný”. The pit is located in the upper part of the so-called Klikovské layers, youngest (senon) strata of the South Bohemian basins. These fluvial layers are characterised by a rhythmical variation of gravely sands, sands and sands with dark grey clay inclusions. Altogether forty samples were taken from a ten meters high pit wall in a regular grid (Fig. 1). The laboratory program was selected to provide for each of the samples enough information to calibrate a hypoplastic model for granular materials by von Wolffersdorff (1996). The following tests were performed on each of the 40 samples:

Table 1. Characteristic values of statistical distributions of parameters of the hypoplastic model. param.

dist.

mean

st. dev.

φc hs n ec0 ei0 ed0 α β

log. log. log. norm. norm. norm. log. norm.

35.1◦ 3.82 GPa 0.289 0.847 1.016 0.318 0.074 1.261

1.62◦ 14.6 GPa 0.095 0.111 0.133 0.042 0.048 0.605

•

Oedometric compression test on initially very loose specimens. • Drained triaxial compression test on specimen dynamically compacted to void ratio corresponding to the dense in-situ conditions. One test per specimen at the cell pressure of 200 kPa. • Measurement of the angle of repose. The hypoplastic model by von Wolffersdorff (1996) has eight material parameters. The model was calibrated using procedures outlined by Herle and Gudehus (1999). The whole process of calibration was automated to reduce subjectivity of calibration. Examples of the measured and simulated results of triaxial experiments are shown in Figure 2 (specimens from one column of the sampling grid). Suitability of different statistical distributions (normal and lognormal) to represent the experimental data was evaluated using Kolmogorov-Smirnov tests. Characteristic values of statistical distributions of parameters of the hypoplastic model are given in Tab. 1 (note the results differ slightly from Suchomel and Mašín (2009), as two specimens c1 and e4 with unusual behaviour were not considered in the present evaluation). Statistical distributions of parameters hs and β are in Figure 3, as an example. As position of each of the 40 samples was known, Suchomel and Mašín (2009) could also evaluate the correlation length in the horizontal (θh ) and vertical (θv ) directions. The dependency of the correlation

Figure 3. Examples statistical distributions of hypoplastic parameters (hs and β).

coefficient ρ on distance was approximated using an exponential expression due to Markov

where τh is the horizontal distance between two specimens and τv is the vertical distance. The correlation length could successfully be evaluated using parameter ϕc only. This parameters depends directly on soil granulometry. The least square fit of Eq. (1) through the experimental data is shown in Figure 4, leading to θh = 242 m and θv = 5.1 m. Note that practically no correlation is observed in the vertical direction, therefore the obtained value θv = 5.1 m is implied by the adopted vertical sampling distance, rather than by the actual autocorrelation properties. Additional experiments on specimens obtained from the outcrop in a

384

Figure 4. Evaluation of the correlation coefficient ρ in horizontal (a) and vertical (b) directions for parameter φc , together with least square fit of Eq. (1).

Figure 6. Tornado diagram showing sensitivity of foundation displacements on different parameters.

4 Figure 5. The problem geometry and finite element mesh.

smaller vertical sampling distance are currently being performed to evaluate θv more precisely. In addition to the laboratory experiments, five in situ porosity tests with membrane porosimeter were performed at different locations within the area from which the samples were obtained. Average natural void ratio was 0.41. The sand was thus in a dense state.

3

STRIP FOOTING PROBLEM

The influence of spatial variation of parameters of the hypoplastic model was studied by simulations of a typical geotechnical problem – settlement of a strip footing. Simulations were performed using a finite element package Tochnog Professional. The problem geometry and finite element mesh are shown in Figure 5. The mesh consist of 1920 nine-noded quadrilateral elements. The foundation was analysed as rigid and perfectly smooth. Element size in the vicinity of the footing is 0.5 m. The soil unit weight is 18.7 kN/m3 . The initial K0 = 0.43 was calculated from Jáky formula K0 = 1 − sin ϕc , with average value of ϕc measured in the experiments. The initial value of void ratio e = 0.48 was used in simulations. The soil was thus slightly looser then in situ, in order to ensure that the void ratios do not sur the physical lower bound ed during Monte-Carlo simulations. Spatial variability of void ratio was not considered. The analyses thus focused on qualitative evaluation of the influence of the spatial variability of soil parameters. In all cases, foundation displacements corresponding to the load of 500 kPa were evaluated. Bearing capacity of the foundation was not evaluated, as the peak loads depend on the mesh density due to the localisation phenomena.

SENSITIVITY ANALYSIS

At first, sensitivity of the results on different material parameters was evaluated. In these simulations, spatial variability of soil parameters was not considered. Only one parameter was varied at a time, all other parameters were given their mean or median values (for normally and lognormally distributed parameters respectively). A Tornado diagram showing sensitivity of foundation displacements uy on different parameters is given in Figure 6. It shows uy for the mean value µ[X ] and for µ[X ] ± σ[X ], where X is parameter value in the case of normally distributed parameters and its logarithm in the case of lognormally distributed parameters. As expected, foundation settlements are influenced the most significantly by the parameters controlling soil bulk modulus (parameters hs and n) and parameter β that influences the shear stiffness. Less significant is the influence of the relative density, controlled through parameters ec0 , ei0 and ed0 . Note that ec0 and the other two reference void ratios ei0 and ed0 were varied simultaneously to ensure constant ratios between them imposed during calibration (Suchomel and Mašín 2009). The smallest influence on foundation settlements have parameters α and ϕc , which control soil strength.

5

PROBABILISTIC ANALYSES

The following probabilistic analyses of the strip footing were performed. First of all, the problem was simulated without considering spatial variability of the parameters (i.e. the correlation length was infinite). In the second step, spatial variability of the parameters was introduced through simulations based on random field theory by Vanmarcke (1983) (RFEM). Last, applicability of a simpler probabilistic method based on Taylor series expansion (first order second moment method, FOSM) was studied.

385

Figure 7. The dependency of µ[uy ] and σ[uy ] on the number of Monte-Carlo realisations.

Figure 8. Probabilistic distributions of uy for Monte-Carlo analyses with infinite correlation length. Table 2. Results of probabilistic simulations with infinite correlation length (in meters).

5.1

Simulations with infinite correlation length

If spatial variability of the soil parameters is neglected, the problem can be simulated using approximate analytical methods (for example, FOSM method). These methods have, however, a number of limitations, as discussed in Sec. 5.3. The probabilistic aspects of the problem analysed in this contribution are fairly complex. The constitutive model and thus also the dependency of uy on X are non-linear. Some of the model parameters follow Gaussian distribution, whereas other follow lognormal distribution. For this reason, analyses with spatially invariable fields of input variables were performed using Monte-Carlo method. This method is fully general, but depending on the problem solved it may require significantly large number of realisations and consequently a considerable computational effort. Figure 7 shows the dependency of the mean value µ[uy ] and standard deviation σ[uy ] for random field simulations from Sec. 5.2. At least 700 Monte-Carlo realisations is required to get a reasonably stable estimate of µ[uy ] and σ[uy ]. In all presented simulations, at least 1000 realisations were performed. Four analyses were performed. In three of them, only one parameters was varied at a time and the other parameters were given their mean (normal parameters) or median (lognormal parameters) values. These analyses were performed for the parameters hs , n and β. β follows normal, whereas hs and n follow lognormal distribution. In the last analysis, all paremeters were considered as random. All parameters were simulated as uncorrelated, with the exception of ec0 , ed0 and ei0 , which were perfectly correlated to preserve constant ratios between them. Figure 8 shows probabilistic distributions of uy and Tab. 2 gives the values of µ[uy ] and σ[uy ]. The distribution of the output variable is well described by the lognormal distribution, even in the case of β as single variable parameter, which itself follows the Gaussian distribution. Slight deviation from the log-normal distribution shows the analysis with n and all parameters random.

RFEM

FOSM

random param.

µ[uy ]

σ[uy ]

µ[uy ]

σ[uy ]

hs n β all param.

0.231 0.197 0.217 0.229

0.128 0.083 0.087 0.163

0.193 0.193 0.193 0.193

0.107 0.089 0.077 0.164

Figure 9. Typical random field simulations with θv = 5.1 m (bottom part of the mesh not shown).

5.2 Random field simulations Spatial variability of soil parameters was considered in the second set of analyses. Random fields were generated using method based on the Cholesky decomposition of the correlation matrix. Due to uncertainty in the correlation length in the vertical direction (discussed in Sec. 2), simulations were run with different values of θv . All parameters were considered as random, ec0 , ed0 and ei0 were perfectly correlated and other parameters were uncorrelated. Example random fields (parameters hs and β) for θv = 5.1 m are shown in Figure 9. The same figure

386

Table 3. Results of probabilistic simulations with variable vertical correlation length (in meters). RFEM

Figure 10. Probabilistic distributions of uy in random field analyses with finite θv and all parameters treated as random.

Figure 11. The dependency of µ[uy ] and σ[uy ] on θv predicted by the random field method.

shows also corresponding distribution of void ratio after 0.8 m of the foundation displacement. Study of this example, as was well as other simulations not presented here, reveals that the lowest void ratios occur in softer areas characterised by low values of the parameter β. Parameter hs , which also have a substantial influence on uy (Sec. 4), affects due to its highly skewed lognormal distribution (Fig. 3) the results in a global way, whereas the parameter β controls the local deformation pattern. Typical statistical distributions of the output variable uy are shown in Figure 10. In all studied cases, uy follows lognormal distribution. This agrees with the results of RFEM simulations with spatially invariable parameters (Sec. 5.1). Figure 11 and Tab. 3 show µ[uy ] and σ[uy ] predicted by the RFEM simulations with different values of θv . There is only slight change (decrease) of µ[uy ] with θv , whereas σ[uy ] decreases with decreasing θv substantially. This decrease is caused by the spatial averaging of soil properties, leading to the reduction of variance of the input variables and consequently of the performance function (see Sec. 5.3 for more details). 5.3 FOSM simulations One of the popular approximate analytical methods for probabilistic analyses is the first-order, secondmoment (FOSM) method. Unlike the Monte-Carlo method, the FOSM method has a number of limitations. First, it consideres linear dependency of the performance function (in our case uy ) on the input variables (in our case, material parameters X ). Also, it

FOSM

θv

µ[uy ]

σ[uy ]

γ

eff. vert. dist.

1m 2m 5.1 m 12.3 m

0.215 0.219 0.226 0.225

0.039 0.059 0.089 0.119

0.48 0.61 0.75 0.87

1.33 1.78 2.53 3.04

does not provide any information on the skewness of the probabilistic distribution of the output variable. Its applicability to solve the highly complex probabilistic problem from this work is studied in this section. Details of the method may be found elsewhere, see e.g. Suchomel and Mašín (2010). Parameter values (normally distributed parameters) or their logarithms (lognormal parameters) are used as an input into the FOSM method. Tab. 2 gives the values of µ[uy ] and σ[uy ] by the FOSM and RFEM methods for infinite correlation length. The FOSM method underestimates both µ[uy ] and σ[uy ] due to the non-linear dependency of the output variable uy on the parameters hs and β (see Fig. 6). The method does not provide any information on the skewness of the statistical distribution of uy .Therefore, its use requires a check of the distribution of uy through Monte-Carlo simulation (or other general probabilistic method). In our case, the distribution of uy is clearly lognormal (Figs. 8 and 10). As discussed by Suchomel and Mašín (2010), the FOSM method can indirectly consider spatial variability of the input variables through reduction of their variances due to spatial averaging. The reduction factor γ is defined as γ = (σ[Xi ]A /σ[Xi ])2 , where σ[Xi ] describes the global statistics of the variable Xi and σ[Xi ]A is the standard deviation of the spatially averaged field. It may be calculated by integration of the Markov function (Eq. (1)) (Vanmarcke 1983). Suchomel and Mašín (2010) have shown that in the case of a slope stability problem in spatially variable c-ϕ soil, γ can be estimated a priori by integrating the Markov function in 1D along the potential failure surface. The value of γ may be evaluated by comparing standard deviations of the FOSM and RFEM outputs (Tab. 3). As γ is a variance reduction factor of the input parameters, however, a linear dependency of the ouput variable on the input parameters (or its logarithms) must be assumed. The results are thus only approximate. As expected, γ decreases with θv . Tab. 3 gives also an effective vertical distance below the foundation that leads to the given γ, found by 2D rectangular integration of the Markov function. This distance depends on θv and it thus cannot be easily estimated a priori. This limits applicability of the FOSM method for estimation of σ[uy ] in the case of spatially variable parameters with finite correlation length.

387

6

CONCLUDING REMARKS

Advanced hypoplastic constitutive model was used in probabilistic analyses of a typical geotechnical problem, strip footing. In the analyses, spatial variability of soil parameters, rather than state variables, was emphasized. It was shown that the result are influenced the most by the soil parameters hs , n and β. The output variable uy was found to follow closely lognormal distribution, even in the case when normally distributed parameters (such as β) were varied. The vertical correlation length θv was found to have minor effect on µ[uy ], but significant effect on σ[uy ], which decreases with decreasing θv due to spatial averaging. Even though the problem is highly complex and non-linear, the FOSM method was found to provide satisfactory predictions for infinite correlation length. For finite correlation length, however, the variance reduction factor γ cannot be easily estimated a priori. ACKNOWLEDGEMENT Financial by the research grants GACR 205/08/0732, GAUK 31109 and MSM 0021620855 is greatly appreciated.

Herle, I. and G. Gudehus (1999). Determination of parameters of a hypoplastic constitutive model from properties of grain assemblies. Mechanics of Cohesive-Frictional Materials 4, 461–486. Hicks, M. A. and C. Onisiphorou (2005). Stochastic evaluation of static liquefaction in a predominantly dilative sand fill. Géotechnique 55(2), 123–133. Schweiger, H. F. and G. M. Peschl (2005). Reliability analysis in geotechnics with a random set finite element method. Computers and Geotechnics 32, 422–435. Suchomel, R. and D. Mašín (2009). Calibration of an advanced soil constitutive model for use in probabilistic numerical analysis. In P. et al. (Ed.), Proc. Int. Symposium on Computational Geomechanics (ComGeo I), Juan-les-Pins, , pp. 265–274. Suchomel, R. and D. Mašín (2010). Comparison of different probabilistic methods for predicting stability of a slope in spatially variable c-phi soil. Computers and Geotechnics 37, 132–140. Tejchman, J. (2006). Effect of fuctuation of current void ratio on the shear zone formation in granular bodies within micro-polar hypoplasticity. Computers and Geotechnics 33(1), 29–46. Vanmarcke, E. H. (1983). Random fields: anaylisis and synthesis. M.I.T. press, Cambridge, Mass. von Wolffersdorff, P. A. (1996). A hypoplastic relation for granular materials with a predefned limit state surface. Mechanics of Cohesive-Frictional Materials 1, 251–271.

REFERENCES Helton, J. C. (1997). Uncertainty and sensitivity analysis in the presence of stochastic and subjective uncertainty. Journal of Statistical Computation and Simulation 57, 3–76.

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Validating models against experience in foundation engineering, using the ROC curve A.M.J. Mens & A.F. van Tol Deltares, Delft, The Netherlands Delft University of Technology, Delft, The Netherlands

ABSTRACT: This paper introduces the area under the Receiver Operating Characteristic (ROC) curve, in short the AUC, as a useful measure to compare rules of thumb for vibratory driving. Using case histories from the Dutch ‘GeoBrain’ database and two rules of thumb for predicting the drivability of steel sheet piles, the AUC determines the performance of these rules. In addition to this measure, the theory behind this method provides us with a practical way for threshold optimization. An example is provided to explain both the database and the measures in more detail.

1

2

INTRODUCTION

Validation is an important, but often underrated part of the development process of models or new prediction methods. It is difficult to compare observations with predictions in a well-organized, objective manner. Furthermore, comparing different methods is not always fair, because the input variables can be different. The authors encountered this problem after the development of the GeoBrain Foundations prediction model (Hemmen & Bles, 2005) and the GeoBrain experience database. The validation appeared to be rather difficult. To simply validate the model with one-to-one observations versus predictions was impossible. Due to the fact that the classification of the observations is different than the prediction results it is not possible find a simple correlation. One solution is the use of so-called ‘Receiver Operating Characteristic’-curves (ROC-curves), as described in a previous paper (Mens et al., 2010, in press), to compare observations with model predictions. However, the ROC-curve by itself is not able to compare different models, only a derived measure from this graph can represent the performance of a model (or better: classifier). A common method in other scientific areas, such as medical and ecologic sciences and machine learning, is to calculate the area under the ROC curve, abbreviated AUC (Bradley, 1997). This paper first briefly explains the GeoBrain case history database and the rules of thumb for vibratory driving with a short example. It continues by describing the theoretical background of the ROC and the AUC measure. Using the theory, it shows some early results, proving that the method is a valuable contribution to the geotechnical engineering community. It concludes with an outlook for future research in this area.

DATA: GEOBRAIN DATABASE

The GeoBrain experiences database (Barends, 2005) contains case histories for foundation and drilling technology. Since 2005, different contractors have been filling this database with their recent experiences in the Netherlands. The total number of entries counted 1850 projects by the end of august 2009. In February 2009, 364 of them concerned the vibratory installation of steel sheet piles (www.geobrain.nl). An “experience” is uniquely defined by the type of element (for example sheet-pile or prefabricated concrete pile), the type of equipment used and the soil conditions. In addition to this numerical data, also details concerning the building pit, the crew and the surroundings are included. Although the database comprised 364 observations at the time, only 195 of them could be used for a first evaluation for both the rules of thumb. An observation was discarded when – essential data was lacking (like a Cone Penetration Test); – a combination of installation techniques was used (both hammering and driving); – unexpected obstacles were present; – large differences (>1.5 m) existed between the entered length of the sheet pile and the difference between the head and the toe of the pile1 – the head of the pile was deeper than 1.5 m below the surface. The example below shows one project (one observation) of all 195.

1

389

This would be a measure for an inaccurate entry.

2.1

– Concerning vibrations: no – Concerning noise level: no

Example 1a

Observation in Amsterdam ‘Bijlmer Station’: Sheet 1: Overview – – – –

– Damage to cables and pipes? No Sheet 7: Experiences

Profile (see also at the website: www.geobrain.nl) Type of sheetpile: AZ26 Length of the piles: 23 m Vibratory Equipment: DELMAG D 30

– Result: bad (see later) – Any delay: 0 days – Damage: – 5 piles that broke out of clutch – 1 skewed pile – 5 piles did not reach the predetermined depth

Sheet 2: Situation – – – – – – –

Project name: Station Bijlmer 1 Type Construction: Dry building pit Toelevel: NAP −19 m Head level: NAP 4 m Retaining length: 1 m Number of sheet piles: 330 Condition of the top layer: dry sand

3

3.1 Rule 1- CUR rule of thumb The CUR-rule calculates the free displacement amplitude (d), that is used to determine the appropriate vibration equipment (CUR166, 2005):

Sheet 3: Geotechnics: – – – – – – – – – – – – – – – – –

RULES OF THUMB

Representative T: station bijlmer.gef Resemblance of different Ts: uniform Mean surface level: NAP +2 m Mean ground water level: NAP +1 m Cohesive layers (e.g. clay) present? yes Weak layers present? yes Firm toplayer present? yes Gravel layers present? No Firm sand layers present? No Obstacles present? No

where d = the displacement amplitude in m, Me = the eccentric moment in kgm, mv = the vibrating mass of the vibrator in kg, and mp = the mass of the sheet pile in kg. The displacement amplitude should be larger than 0.005 m for the equipment to be sufficient.

Sheet 4: Sheet pile

3.2 Rule 2-AZZ rule of thumb

Sheet pile producer: Arcelor Moment of resistance: 2600 cm3 /cm Steel quality: S 240 GP Used sheet piles? No Single/double/triple installation: double Clutches punched or welded? Punched Coated piles? No

Based on the HYPERVIB1 model, (Azzouzi, 2003) developed a formula that calculates the required vertical cyclic force (Fc ) to be able to determine the most suitable vibrator. This formula uses the mean cone resistance over the considered sand layers, taken from a cone penetration test (T):

Sheet 5: Installation where Fc = the required vertical cyclic force from the vibrator that should be used in kN, L = length of the sheet pile in the soil in m, χ = the perimeter of the sheet pile (per unit of length) in m2 /m, qc,gem = the mean cone resistance over the sand layers in MPa, and At = the cross-sectional area of the toe of the sheet pile in cm2 .

– Vibratory driving – Low frequency – 1700 rpm – Eccentric moment: 500 Nm – Vertical Force: 1600 kN – Pull-down? no – Pile frame: no frame – Method: one-by-one – Reduction clutch friction applied? No – Guided piles? Yes – Experience-level crew: Good

3.3 Defining positive and negative

Sheet 6: Surrounding – – – –

Any adjacent objects? Yes Foundation adjacent objects: directly on soil Distance between piles and object: 0 m Damage class for adjacent buildings: severe: ignorable – Any settlement? Yes, 45 cm next to the sheet piles, 20 cm at 1 m and 0 cm at 3 m. – Any complaints from the neighbourhood?

To be able to compare predictions with observations (field-experiences, or cases) the ‘positive’ and ‘negative’ need to be defined for both the predictions and the observations. 3.3.1 Observations Every project in the database receives points that reflect the amount of damage (wtot ):

390

3.4

where

Example 1b (example 1a continued)

3.4.1 Observation The following numbers reflect the amount of damage, using the information in example 1a (sheet 2 and 7) and equations 3 and 4.

xi is the percentage of the sheet piles that suffered from a specific problem i: – i = 1: the sheet pile did not reach the predetermined depth, – i = 2: the sheet piles experienced irregularities along the clutches, – i = 3: the sheet pile experienced burned clutches, – i = 4: the sheet pile broke out of the vibratory equipment – i = 5: the sheet pile experienced any other damage than described above. and (wi,max ):is the maximum amount of points to be received per problem, based on weighted experts’ opinions:

Based on the damage (wtot ) and the percentages xi , experts distinguish two situations. The first one is less strict than the second one in the sense that type I needs more damage in a project to be classified as negative. Type I classifies the case as negative if there is any

According to these numbers, experts will classify this example in case of a type I situation as positive and in a type II situation as negative, for not reaching the predetermined depth. The website shows the Type II classification (sheet 7). 3.4.2 Prediction According to the information in example 1a and equations 1 and 2, the CUR-rule and the AZZ rule predict the following.The CUR-rule uses an eccentric moment of 500 Nm, a dynamic mass of 4000 kg and the mass of the pile is 4498.8 kg, resulting in

which is a positive prediction, because d > 0.005 m. The AZZ-prediction uses a mean cone resistance of 12.7 Mpa, the length of the pile is 23 m, the painting area 2.8 m2 , the cross-sectional area is 249,2 cm2 and the used vertical force is 1600 kN, resulting in

Type II classifies the case as negative if there is any

In case a project is already classified as a negative type I, automatically it is a negative type II as well. 3.3.2 Predictions According to the CUR handbook, the displacement amplitude d should be at least 0.005 m for the equipment to be sufficient. In case of the AZZ-rule, the actually used vertical force should exceed the required vertical cyclic force from the vibrator (Fc). This paper shows that theoretically it is possible to optimize these threshold values, using the cases from the GeoBrain experience database. The AUC-curve method enables to determine which rule of thumb performs best. Example 1b applies equations 3 and 4 to the data from example 1a.

which is a negative ‘prediction’, because the used vertical force (1600 kN) is smaller than the needed vertical force (1950 kN). 4 THEORY: ROC-SPACE AND AUC 4.1 ROC-Space The Receiver Operating Characteristic (ROC) Space is primarily a way to visualize and compare binary classification models. In fact a point in this space represents the relationship between the ‘true positive ratio’ (Eq. 8a) and the ‘false positive ratio’ (Eq. 8b). The true positive ratio (TPR) is defined as the fraction

391

Table 1. matrix.

Example of a contingency table, or confusion Predictions

− +

Observations Total

Table 2. matrix.

−

+

Total

TN FN P−

FP TP P+

O− O+ N

Example of a contingency table, or confusion Predictions

Observations. total

− +

−

+

Total

TN = 4 FN = 84 P− = 88

FP = 12 TP = 95 P+ = 107

O− = 16 O+ = 179 N = 195

Figure 1. Receiver Operating (ROC-space) with some examples.

of actually-positive cases correctly classified as ‘positive’ by the prediction model. The false positive ratio (FPR) is defined as the fraction of actually-negative cases incorrectly classified as ‘positive’ (Metz, 1978; Mens et al., 2010, in press). The input for a point in this space is a given prediction model (such as the CUR-rule) and a set with N observations. For these N observations, the binary result (positive or negative) is known. Using the information from these observations, it is possible to calculate the binary prediction results. These can be summarized in a two-by-two contingency table (or ‘confusion’-matrix), which serves as the base for a so-called sensitivity-pair, the point in the ROC-space. Table 1 provides an example of this contingency table. O− represents the total number of negative observations and O+ the positive ones. P − and P + represent the total number of negative and positive predictions respectively. The numbers from Table 1 enable us to calculate the following measures (amongst others):

These measures are all dependent on the used ‘threshold’ value in the model. Take the CUR-rule as an example. The rule predicts a pile to be vibratory drivable (a ‘positive’ outcome), if the calculated displacement amplitude d exceeds 0.005 m:

Table 2 shows the contingency table for the CURmodel, using all the 195 observations from the GeoBrain experience database.

Characteristic

space

Using this table, it is now possible to calculate the so-called sensitivity-pair (TPR,FPR) and to plot the outcome in the ROC-space. Figure 1 (Fawcett 2006) explains this in more detail. The coordinates in the ROC-space represent possible models. The sensitivity-pair (0,1) represents the perfect predictor. Roughly one could say that a point in the ROC space is better more to the top left of the graph. In Figure 1 model C’ is better than model A. Model B is considered to behave randomly and model C is the ’ reverse’ model of model C’. As stated previously, the input of a ROC space consists of a prediction model and a set of observations. Figure 2 shows the ROC-space for both the AZZ and the CUR rule, for both type I and type II classifications.

4.2 Area under the curve Equation 9 shows a ‘threshold’-value of 0.005 m at the right hand side. By changing this value, the contingency table will change (more or less predictions will be positive) and therefore the sensitivity-pair will change, resulting in a different point in the ROC-space. By changing the threshold and plotting all resulting sensitivity-pairs in one plot, a ROC curve appears. The points that make up the plot enable us to find the best threshold value for the models at hand. Figure 3 shows these ROC-curves for both rules with both classification types. The area under this curve (AUC) is a single-valued measure for the performance of a model (Bradley 1997). In this way, it is possible to compare different models, although their input variables and their behaviour might be different. Since the AUC is the portion of the area of the unit square, its value will always be between 0 and 1.0. However, because random guessing produces the diagonal line between (0,0)

392

5.2 Discussion part 1

Figure 2. ROC-space for type I labelled and type II labelled observations and their CUR and AZZ predictions.

where the TPR (Eq 8a), or P(P + |O+ ) is reliable in contrast with the FPR, it is possible to calculate the FPR, assuming the probability on a positive observation is for instance 95%. This results in 0.53*0.95=0.50. This means that in 50% of the positive predictions, the project will turn out to be negative. Secondly, one could suggest to ‘flip’ the > −sign in equation 9 to improve the results. Flipping the sign would move the points to the other side of the line of no discrimination (compare to model C in figure 4). Physically however, flipping the sign is impossible. Pushing the prediction result to the limits, a small displacement amplitude (say 0) would be better than a large one, which does not make sense. Changing the right hand side of equation 9 though, could improve the performance. Part two of the discussion will elaborate on that.

Figure 3. ROC-curve for type I labelled and type II labelled observations and their CUR and AZZ predictions.

5.2.2 AZZ Based on the same observations, the AZZ-rule generally performs better, but regarding the previous reasoning about the observations and considering the TPR of 25%, the AZZ-rule does not perform well.

and (1,1), no realistic classifier should have an AUC less than 0.5 (Fawcett 2006).

5

5.2.1 CUR-rule Based on the observations, the CUR-rule performs not very well. This means that either the observations do not reflect reality, or the rule of thumb is not reliable. Firstly, consider the amount of observations. Type I provides 16 negative observations and 179 positive ones. For Type II this is 19 versus 176. This means the percentages for negative observations compared to the total number is 8% and 10% respectively for type I en type II. These percentages reflect the common idea about the Dutch vibratory driving projects. However, the absolute number of negative observations is statistically low. In fact, this number is too small to conclude that the CUR-rule does not perform well. It is however possible to draw some conclusions from the positive observations. Contractors always are interested in the probability of a negative observation, given a positive prediction, in other the FPR (Eq. 8b), or P(O− |P + ). Using the fact that P(O− |P + ) = 1- P(O+ |P + ) and Bayes’ rule:

RESULTS AND DISCUSSION 5.3 Results part 2 (AUC)

5.1 Results part 1 (ROC-SPACE) Figure 2 shows some results for the CUR and the AZZ rules of thumb, using the cases from the GeoBrain experience database as present in February 2009. The pentagram and the circle at the left show the results for both type I (left) and type II (right) classifications for the AZZ-rule. The square and the diamond at the right show the results for de CUR-rule in case of a type I and a type II respectively.

Figure 3 shows the results for the CUR rule of thumb, using again the cases from the database. For the AZZrule it is more complicated, but for explaining purposes a threshold value α has been incorporated:

The dots show again the previous results for the current situation as denoted in Figure 2.

393

Table 3. AUC values for the curves, as shown in Figure 6.

could be improved. Changing two parameters as a ‘threshold’ variable provides for a ‘three-dimensional ROC-curve’, or in fact a ‘ROC-area’. The area under the curve then changes into a volume, resulting in a new measure: the ‘VUA’ or the ‘Volume Under the Area’. Although it is not possible to plot curves, this measure is easily extendable for more dimensions. Besides optimizing threshold parameters for more dimensions, the database seems to provide for enough data on prefabricated piles to validate the current rules of thumb and more complicated models.

AUC classifier

Type I

Type II

CUR AZZ

0.359 0.516

0.367 0.522

5.4

Discussion part 2

Figure 3 shows clearly the difference in performance for both rules. Table 3 provides the matching AUC values. In this (hypothetical) case, the CUR-rule seems not to be a realistic classifier and the AZZ-rule with classification type II performs best. As soon as more (negative) observations are available, a new analysis will provide a more valid outcome. Every point that makes up the ROC-curve reflects another threshold. In this way the ‘best’ threshold can be determined from the figure. In this case the result is not valid, because of a lack in (negative) observations. The authors expect to be able to perform a sound validation, as soon as it is possible to use 500 observations (in total). 6 6.1

CONCLUSIONS AND RECOMMENDATIONS Conclusions

The Receiver Operating Characteristic (ROC)-space enables us to judge models on their performance, using case histories for vibratory driving of steel sheet pile walls from the GeoBrain database. As soon as more (negative) observations are available, a validation of the current rules of thumb is possible. Furthermore, the ROC-curve in this space enables us to improve threshold parameters for practical use of the models. Comparing rules of thumb with for example different input-variables is possible using the Area under the Curve (AUC). Since this measure is single valued, it provides for an objective comparison between different models within geotechnical engineering, independent of the variables used. 6.2

ACKNOWLEDGEMENTS This research was sponsored by Deltares and Delft University of Technology. We kindly acknowledge the cooperation of the contractors from the NVAF (the Dutch foundation for contractors in foundation engineering). Without their continuous contribution to the GeoBrain database, this investigation would not have been possible. REFERENCES CUR 2005 Damwandconstructies, 4e druk (in Dutch). CUR, Civieltechnisch Centrum Uitvoering Research en Regelgeving. Nr 166. Azzouzi, S. 2003. Intrillen van stalen damwanden in nietcohesieve gronden – welke predictie is (on)juist? (in Dutch). MSc Thesis, Delft, Delft University of Technology. Barends, F. B. J. 2005. Associating with advancing insight – Terzaghi Oration 2005. XVI International Conference on Soil Mechanics and Geotechnical Engineering, Osaka. Bradley, A. P. 1997. The use of the area under the ROC Curve in the evaluation of machine learning algorithms. Pattern Recognition 30(7): 1145–1159. Fawcett, T. 2006. An Introduction to ROC analysis Pattern Recognition Letters 27: 861–874. Hemmen, B. and T. Bles 2005. GeoBrain Funderingstechniek: Ervaringsdatabase voorspelt uitvoeringsrisico’s (in Dutch). Civiele Techniek 2: 24–25. Mens, A. M. J., T. J. Bles, M Korff, A.F. van Tol. 2010 (in press). Validating models against experience in foundation engineering, using the experience database of GeoBrain. Geotechnical challenges in urban regeneration, 11th International Conference. London. Metz, C. E. 1978. Basic Principles of ROC Analysis. Seminars in Nuclear Medicine VIII(4): 283–298.

Recommendations for further research

The measures described above only for equations where one threshold value is involved. The AZZ-rule however, depends on two parameters that

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Dynamic problems and Geohazards


A 2.5D finite element model for simulation of unbounded domains under dynamic loading P. Alves Costa, R. Calçada, J. Couto Marques & A. Silva Cardoso Faculty of Engineering, University of Porto, Portugal

ABSTRACT: In this paper a 2.5D finite element model developed by the authors is presented. Apart from other features, the proposed model is appropriate to deal with unbounded domains submitted to dynamic loads, which is the aspect focused in the present work. The use of the 2.5D finite element method has been growing in the past few years, proving to be a good strategy to deal with very long structures (such as roads and railways) submitted to dynamic loads, whether of stationary or moving type. Within this approach a subject of particular interest resides in the formulation of special procedures to treat the boundary effects that are inherent in the truncation of the domain associated with the finite element discretization. Typically, the treatment of such artificial boundaries can be dealt with by local or global procedures. Due to their simplicity and compatibility with the conventional finite element formulation, the most popular techniques are based on local procedures such as the absorbing boundaries or the finite-infinite element coupling. In this paper the accuracy of both methods is compared and discussed for both dynamic loading conditions, i.e., stationary and moving loads.

1

INTRODUCTION

During the last decade, a considerable research effort has been devoted to the development of numerical and semi-analytical methods for the modelling of unbounded structures submitted to dynamic load conditions. The increasing significance of this issue is related to the current expansion of high-speed railway lines. This kind of structure is difficult to model due to several reasons, namely the infinite character of its geometry. However, this apparent drawback can be transformed into an advantage, since it confers to these structures the necessary properties to be dealt with by numerical models developed in the wavenumber-frequency domain, employing Fourier expansions along space and time. This is the main concept behind most of the more sophisticated modelling strategies that have been followed in recent years. The modelling strategy based on the wavenumberfrequency domain has been adapted to semi-analytical models, but also to numerical models like the boundary element or the finite element method. Due to its nature, the boundary element method is particularly adequate to deal with unbounded geomechanical domains (Lombaert et al. 2006). However, the modelling of complex geometries and the consideration of inhomogeneities of the medium is often very difficult. An alternative procedure is the finite element method or the coupling between the finite element and boundary element methods (Rastandi 2003; Muller 2007; Alves Costa 2008). Nevertheless, the coupling of these two methods is a complex task particularly for the problems considered in this paper.

The finite element method is a very popular and attractive alternative. An accurate procedure based on finite element formulation constitutes a valuable framework of analysis, since this method is known to be capable of dealing with irregularities in geometry and materials, including embedded structures, the natural layering of soil deposits and material inhomogeneities (at least in the cross section of the problem in the context of 2.5D modelling) (Alves Costa et al. 2009; Alves Costa et al. 2010). Within this approach a topic of particular interest is the formulation of special procedures to treat the boundary effects that are inherent in the truncation of the domain associated with the finite element discretization. For static problems, the contribution of the ground is reflected in of stiffness, so it is possible to truncate the domain at a sufficiently remote location where the ground deformation is so small that it can be neglected. However, in dynamic analyses the ground model should fulfil the requirements of representing not only the dynamic ground stiffness but also the radiation condition. The latter requirement demands a special treatment of the boundary conditions, since spurious reflection of the waves at the mesh boundary should not occur. A rigorous approach can be achieved using finite elements to represent the near-field domain and boundary elements to simulate the far-field domain. This approach has been used in the context of 2.5D modelling by several researchers with satisfactory results (Yang & Hung 2001; Sheng et al. 2006; Muller 2007; Alves Costa 2008). Another approach consists in the use of local procedures, such as absorbing boundaries (Lysmer & Kuhlemeyer 1969) or infinite elements

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(Yang & Hung 2001; Yang & Hung 2008). Due to their simplicity and compatibility with the conventional finite element formulation, the most popular techniques are based on local procedures, and several methods are available for static or dynamic conditions (White et al. 1977; Marques & Owen 1983; Kausel 1988; Bettess 1992). However, the reliability and accuracy of almost all these procedures have not been checked in the context of 2.5D modelling. In this paper, the reliability of a 2.5D finite element model developed by the authors and coupled with local procedures of boundary treatment is evaluated from the theoretical point of view. The model was implemented in the numerical platform Matlab 2009. Since the model is developed in the frequency domain, it is possible to take advantage of the numerical computational tools available in Matlab 2009 for parallel processing, which allows for a considerable reduction of the computation time. 2 2.1

NUMERICAL MODEL 2.5D finite element method

The application of 2.5D finite elements is confined to structures which can be assumed to have infinite development and invariant properties in one direction, as illustrated in Figure 1. In these cases the structure is two-dimensional, since the cross-section remains invariable in the longitudinal direction, but the loading is three-dimensional. The main concept behind the proposed solution to the problem is the use of a method which is between the two and the three dimensional domain. This method was first proposed by (Hwang & Lysmer 1981) for the study of underground structures under travelling seismic waves. Subsequently, the method has been applied by a few researchers to the study of vibrations induced by traffic. In this field, special attention should be dedicated to the works of (Yang & Hung 2001; Sheng et al. 2006; Muller 2007; Alves Costa 2008; Alves Costa et al. 2010). Assuming that the response of the structure is linear, the analysis can be carried out in the wavenumber/frequency domain. All the variables, i.e., loads (action) and displacements (response), must be transformed to the wavenumber/frequency domain by means of a double Fourier transform, related with the direction along the track (x direction) and with time. Transformed quantities are functions of the Fourier images of x and t, defined as wavenumber and frequency and are represented by k1 and ω, respectively. Following the usual steps of the finite element procedure, namely the strong and weak formulations, the following equilibrium equation can be derived for any point of a three dimensional domain:

Figure 1. Infinite structure invariant in one direction.

the displacement field, ρ is the mass density and p represents the applied loads. After the transformation, the cross-section of the domain remains on the untransformed domain and is discretized into finite elements. This approach enables to rewrite Equation 1 in of nodal variables. The concept of virtual work can be applied in the transformed domain by recourse to Parserval’s theorem (Kulhánek 1995; Grundmann & Dinkel 2000; Muller 2007):

Considering Equation 1, the virtual work of the internal stresses and inertial forces in the transformed domain is given by, respectively:

where: B is the matrix with the derivatives of the shape functions; N is the shape function matrix; D is the strain-stress matrix; un is the vector of nodal displacements (in the transformed domain). The virtual work done by the external loads is computed taking advantage of the fact that the geometry is only discretized on the ZY plane. So, considering a coordinate s, parallel to the edge of the element where traction is applied, the virtual work developed by the load system is given by,

Replacing and rearranging Equations 3, 4 and 5 in Equation 1 yields,

where δε is the virtual strain field, σ represents the stress field, δu is the virtual displacement field, u is

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Adopting the classic finite element notation leads to,

and

where [K] and [M] are the stiffness and mass matrices, respectively. As usual, matrix [B] is derived from the product of the differential operator matrix [L] (in the transformed domain) and matrix [N]. Since the direction x is transformed to the wavenumber domain, the derivatives with respect to k1 are analytically computed, as presented in the following expression.

Figure 2. Infinite element: a) global coordinates; b) local coordinates.

A hysteretic damping model is considered, i.e., with complex stiffness parameters. The computational efficiency can be improved by dividing matrix [K] into sub-matrices, independent of the wavenumber and frequency. This is achieved by considering the matrix [B] as the sum of two matrices, whereby the numerical and analytical derivatives are separated. Equation 6 can then be replaced by:

infinity (Bettess 1992). In elasto-dynamic harmonic conditions, this purpose can be reached by the combination of three functions: i) a standard shape function; ii) a decay function; iii) an oscillatory function. The issue is complex in elasto-dynamic problems since there is no longer a unique wave speed, even for a half-space problem. This problem can be overcome by means of special infinite elements, which represent the characteristics of multiple waves propagating out to the far field (Yun & Kim 2006); however, this procedure increases its complexity. Alternatively, as demonstrated by (Yang & Hung 2001), the use of conventional infinite elements combined with criteria for the choice of the decay and oscillatory factors can lead to accurate results even for moving load problems. So, in the present work, the authors decided to use the infinite elements proposed by the aforementioned authors. A schematic representation of the adopted infinite elements is presented in Figure 2. The displacement shape functions of the element are defined by:

The global system of equations is completely defined after the assembly of the individual element matrices and the introduction of the Neumann and Dirichlet boundary conditions. The results obtained after solving the equation system are in the transformed domain, requiring a double inverse Fourier transform for converting the solution to the space/time domain. The advantage of this method in relation to the fully three-dimensional finite element approach is evident: instead of solving an equation system with a large number of degrees of freedom, a smaller system of equations is solved many times, corresponding to a range of wavenumbers. This procedure provides a great reduction of computational time.

where α and k’ are the decay and oscillatory factors, respectively. For details regarding the value selection procedure for these factors, readers should refer to the works of (Yang et al. 1996; Yang & Hung 2001). Having defined the infinite element shape functions the usual finite element procedure is applied, i.e., the stiffness and mass matrices of each infinite element are computed and added to the global matrices, forming the global system of equations.

2.2 Treatment of artificial boundaries by the infinite element method

2.3 Treatment of artificial boundaries with absorbing boundaries

In both the infinite and finite element formulations the field variable is approximated by shape functions. However, the shape functions for infinite elements must be more elaborate, since they have to represent a “reasonable” behaviour of the field variable towards

Since the finite element itself cannot deal with the unbounded soil medium directly because of the limit of the discretized mesh size, special boundary conditions must be constructed in order to avoid the spurious wave reflection on the artificial edge of the finite element

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Figure 4. A homogeneous soil layer on a rigid base.

Figure 3. Absorbing boundary.

mesh. Regarding this subject, several procedures have been proposed during the last 40 years, with various degrees of complexity and, of course, of efficiency. In spite of the recent and not so recent developments in the field of “absorbing boundaries”, one of the most commonly used is the standard viscous boundary proposed by (Lysmer & Kuhlemeyer 1969). This consists of discrete dashpots attached to all degrees-of-freedom at the boundary, in which the dashpot coefficients are constant quantities depending only on the compression and shear wave velocities and the mass density of the soil adjacent to the boundary. In the frequency domain, the damping matrix associated to these dashpots enables to formulate a relationship between the stress and displacement fields along the artificial boundary. Assuming the local referential, as illustrated in Figure 3, this relationship in the frequency domain is given by:

normal angle of incidence. However, waves that strike the boundary at other angles will reflect some of the wave energy. To address or minimize this drawback, (White et al. 1977) have suggested a correction factor to the damping matrix based on a weighting of the absorbing properties for a generalized angle of incidence of the wave front. The damping matrix proposed by these authors is given by:

where the variable s represents the ratio between compression and shear wave velocities. To the authors’ knowledge, there is no experience in the use of absorbing boundary conditions in the context of 2.5D finite element modelling. Therefore the reliability and accuracy of both formulations previously presented will be investigated and discussed in the present paper. 3

where q corresponds to the stress field at the boundary, q = {σxy , σyy , σzy }T , u represents the displacement vector, u = {ux , uy , uz }T and [C] is the local damping matrix, given by:

where Vs and Vp represent the shear and compression wave velocities, respectively. Following the usual finite element procedure, the global damping matrix is computed by assembling the local damping matrices and is added to the remaining global matrices of the finite element problem, forming the global system of equations. Although (Kausel & Tassoulas 1981) have shown, in a two-dimensional benchmark problem, that the standard viscous boundary is a reasonable approximation with little extra computational effort, an important aspect is the fact that the standard viscous boundary perfectly absorbs waves that hit the boundary with a

NUMERICAL EXAMPLES

3.1 Example description In order to check the reliability and accuracy of the proposed approaches for the dynamic analysis of the ground response under stand-still or moving dynamic load actions, a small example, which was previously presented by (Takemiya 2001), is used to compare the numerical solutions with those obtained by the semianalytical approach presented by (Sheng et al. 1999). The geometry of the problem is depicted in Figure 4, as well as the properties of the ground. The load, with a magnitude of 1 N, is distributed over a 2 × 2 m2 surface (when the load is moving it is assumed that for t = 0 s, the load surface is centred with the origin of the referential). Three distinct situations are analyzed: i) nonmoving dynamic load with oscillatory frequency, f; ii) non-oscillatory load moving along the x direction; iii) oscillatory load moving along the x direction. In the 2.5 D formulation, only half of the domain was discretized, taking into the symmetry conditions of the problem. The adopted finite element mesh has a thickness of 10 m and a width of 20 m. In the lateral boundary opposite to the symmetry plane,

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Figure 5. Ground surface vertical displacement at t = 0s for f = 10 Hz (computed by the semi-analytical approach).

the methods described above were applied in order to avoid spurious wave reflection. The two described procedures, i.e., the infinite element method and the absorbing boundary techniques are analyzed and the results assessed by comparison with the theoretical ones.

Figure 6. Relative error of vertical displacement amplitude at the ground surface for f = 10 Hz: a) infinite elements; b) Lysmer viscous boundary.

3.2 Non-moving dynamic loads Two distinct situations are investigated. The first one corresponds to a frequency of excitation of 10 Hz, a little above the first cut-off frequency of the ground, while in the second example a higher excitation frequency is chosen, i.e., f = 40 Hz. Figure 5 displays the vertical displacement of the ground surface at time t = 0 s for the 10 Hz excitation frequency. These results were computed by the semi-analytical approach and are used to evaluate the accuracy of the local procedures in preventing spurious wave reflection at the artificial boundaries. The displacement is plotted in non-dimensional form with D* = µuz a (µ is the soil shear modulus and a is the half-width of the loading area). The relative error colour maps of the vertical displacement amplitude provided by the numerical methods are shown in Figure 6, for 10 Hz excitation frequency. The relative error is defined with reference to the semi-analytical solution. Two main conclusions can be drawn: i) the error associated to the use of infinite elements is below 5%, even in the vicinity of the artificial boundary (y = 20 m); ii) the Lysmer absorbing boundaries provide low accuracy, particularly at a transversal distance from the dynamic source above 16 m. To better discern the differences between the various approaches, Figure 7 presents the vertical ground surface displacement along the alignment y = 18 m at t = 0 s. Another aspect that must be mentioned is that the coupling between 2.5 D finite and infinite elements allows the simulation of the wave propagation even when the discretized domain on the transversal direction is close to, or even smaller than the wavelength of the propagated wave, contrarily to the Lysmer absorbing boundary. This aspect is very important, and is

Figure 7. Vertical displacements along the alignment y = 18 m at t = 0 s for f = 10 Hz.

one possible justification for the divergence of results between the semi-analytical solution and the numerical approach based on the absorbing boundaries proposed by Lysmer. From the presented results, it is possible to conclude that the approach based on the infinite element technique is the most accurate. It is also interesting to observe that White’s proposal performs better than Lysmer’s. This happens because: i) Lysmer viscous boundary is based on 1D wave propagation theory, generalized and adapted to 2D wave propagation problems; ii) in the Lysmer viscous boundary energy absorption is only fully accomplished if the wave front is normal to the boundary; this requirement is satisfied in 1D wave propagation problems but much more difficult to fulfil when the dimensionality of the problem increases; iii) White’s proposal is based on a weighting of the energy absorption, assuming arbitrary wave propagation direction, which makes it more adequate to deal with 3D problems.

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Figure 8. Relative error of the vertical displacement amplitude at the ground surface for f = 40 Hz: a) infinite elements; b) Lysmer viscous boundary.

Another aspect that has been investigated refers to the ability of the methods to simulate the system response when more than one P-SV mode is excited. In this case the excitation frequency of 40 Hz has been considered and Figure 8 displays the relative error of the vertical displacement amplitude at the ground surface computed using infinite elements and Lysmer’s viscous boundary. Once again infinite elements perform better. The t analysis of Figures 6 b and 8 b also shows that the absorbing boundary technique is more efficient for higher frequencies of excitation. 3.3

Figure 9. Vertical displacement at the ground surface for M = 1.5 at t = 0 s computed by the semi-analytical approach.

Non oscillatory moving loads

One of the most relevant features of the 2.5 D finite element method is its ability to deal with moving loads in a simple way. In this case, and when the load is non-oscillatory, the system excitation comes from the load speed and consequent change of the ground stress field over time. As previously discussed by several authors, “true dynamic” phenomena develop when the load speed reaches the velocity of propagation of Rayleigh waves on the ground. For that reason only an extremely high speed is considered here, namely for Mach number, M = c/Vs , equal to 1.5. Since the absorbing boundary technique proposed by White has been shown above to be superior to the Lysmer approach, in the following examples only the first one is employed. The qualitative comparison between the benchmark solution of Figure 9 and the numerical results of Figure 10 shows that the solution based on the infinite element approach fits the semi-analytical results very well. Regarding White’s absorbing boundary, although a visual qualitative evaluation shows a good global fit to the semi-analytical results, a more detailed investigation leads to the expected conclusion, i.e., in observation points close

Figure 10. Vertical displacement at the ground surface for M = 1.5 at t = 0 s: a) infinite element approach; b) White’s boundary technique.

to the artificial boundary the accuracy of the model decreases. This particular aspect is well illustrated by the vertical displacement time history at the location (0,18,0) presented in Figure 11, which shows that White’s technique cannot simulate accurately the oscillation tail subsequent to the peak displacement associated to Mach’s cone. On the other hand, the model based on the coupling between finite and infinite elements simulates with higher accuracy the whole vertical displacement time history at the observation point selected.

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Figure 11. Transient vertical displacements at (0,18,0) for M = 0.927.

3.4 Oscillatory moving loads In order to fulfil the requirement of generalization of the proposed model, another aspect worth analyzing is related to the system response when subjected to moving harmonic loads. Since the model is developed in the frequency-wavenumber domain, the system response associated to a moving load with time variable magnitude can be decomposed in several harmonic loads through a Fourier series or by a Fourier transform. So, in such conditions, the key parameter for an exact computation resides on the ability of the numerical model to deal with harmonic moving loads. With this purpose in mind a different example is now analyzed, considering a constant load speed M = 0.5 (c = 100 m/s) and a driving frequency value of 10 Hz. This problem is considerably more complex than the examples presented above, because the Doppler effect induced by the load movement excites several P-SV modes at distinct frequencies. The relative error of the vertical displacement amplitude at the ground surface for the 10 Hz driving frequency is illustrated in Figure 12 for the two numerical procedures under investigation. Note that the colour maps are not defined in of absolute coordinates x and y, but in a moving referential where the coordinate x is replaced by s = x − ct. The absorbing boundary technique proposed by White leads to significant errors which exceed 20% in substantial portions of the analyzed domain. On the other hand, the finite-infinite element coupling provides once again a very reasonable approximation to the exact theoretical solution, with associated errors below 7%.

Figure 12. Relative error of the vertical displacement amplitude at the ground surface for f = 10 Hz and M = 1.5: a) infinite elements; b) White’s boundary technique.

i.e., infinite elements and absorbing boundaries, a study was performed with the aim of checking the reliability and accuracy of different treatment procedures by comparison with benchmark solutions. Three types of excitation were considered: i) dynamic harmonic loads with fixed location; ii) moving non-harmonic loads; iii) moving dynamic harmonic loads. The main conclusion that can be drawn from the present study is that, in the conditions considered, i.e., assuming constant dimensions for the finite element mesh, a greater accuracy was achieved through the procedure based on the coupling between finite and infinite elements. This conclusion applies to the three different loading conditions that have been analyzed. On the other hand, the techniques based on absorbing conditions are associated to higher error levels, mainly for lower excitation frequencies. Nevertheless, the 3D features of the analyzed problem show that the White proposal can minimize some of these errors in comparison with the Lysmer proposal. In addition to the results presented here, other potentialities and features of the proposed model can be found in (Alves Costa et al. 2010).

ACKNOWLEDGMENTS 4

CONCLUSIONS

In this paper, a 2.5D model developed by the authors has been presented. Apart from other aspects, the main characteristics of the model are described, with special emphasis on the numerical techniques that have been implemented in order to deal with the modelling of unbounded domains through finite element procedures. Following a strategy based on local procedures,

This paper reports research developed under the financial of “FCT – Fundação para a Ciência e Tecnologia”, Portugal. The first author wishes to thank FCT for the financial provided by the grant SFRH / BD / 29747 / 2006. The authors also wish to acknowledge the of the project “Risk Assessment and Management for High-Speed Rail Systems” of the MIT – Portugal Program Transportation Systems Area.

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REFERENCES Alves Costa, P. (2008). Moving loads on the ground: a 2.5D transformed finite element code for train-track-soil interaction. Internal Report. Porto, FEUP. Alves Costa, P., R. Calçada, et al. (2009). Influence of soil non linearity on the dynamic response of high speed railway Lines. COMPDYN 2009. Rhodes. Alves Costa, P., R. Calçada, et al. (2010). Influence of soil non-linearity on the dynamic response of high-speed railway tracks. Soil Dynamics and Earthquake Engineering doi:10.1016/j.soildyn.2009.11.002. Bettess, P. (1992). Infinite Elements, Phenshaw Press. Grundmann, H. and J. Dinkel (2000). Moving oscillating loads acting on a dam over a layered half space. Wave 2000, Balkema. Hwang, R. and J. Lysmer (1981). Response of buried structures to travelling waves. Journal of Geotechnical Engineering Division 107. Kausel, E. (1988). Local transmitting boundaries. Journal of Engineering Mechanics 114(6): 1011–1027. Kausel, E. and J. Tassoulas (1981). Transmitting boundaries: a closed-form comparison. Bulletin of Seismological Society of America 71(1): 143–159. Kulhánek, O. (1995). Time Series Analysis – Lecture Notes. Uppsala, Uppsala University. Lombaert, G., G. DeGrande, et al. (2006). The experimental validation of a numerical model for the prediction of railway induced vibrations. Journal of Sound and Vibration 297: 512–535. Lysmer, J. and R. L. Kuhlemeyer (1969). Finite dynamic model for infinite media. Journal of Engineering Mechanics Division 95: 859–877. Marques, J. M. M. C. and D. R. Owen (1983). Infinite elements in quasi-static materially non linear problems. Computers and Structures 18(4): 739–751. Muller, K. (2007). Dreidimensionale dynamische TunnelHalbraum-Interaktion. Lehrstuhl fur Baumechanik. Munchen, Technische Universitat Munchen. PhD.

Rastandi, J. I. (2003). Modelization of Dynamic SoilStructure Interaction Using IntegralTransform-Finite Element Coupling. Lehrstuhl für Baumechanik. München, Technischen Universität München. PhD. Sheng, X., C. Jones, et al. (1999). Ground vibration generated by a harmonic load acting on a railway track. Journal of Sound and Vibration 225: 3–28. Sheng, X., C. Jones, et al. (2006). Prediction of ground vibration from trains using wavenumber finite and boundary element method. Journal of Sound and Vibration 293: 575–586. Takemiya, H. (2001). Ground vibrations alongside tracks induced by high-speed trains: prediction and mitigation. Noise and Vibration from High-Speed Trains. V. Krylov. Brighton, Thomas Telford Publishing. White, W., S. Valliappan, et al. (1977). Unified boundary for finite dynamic models. Journal of Engineering Mechanics 103: 949–964. Yang, Y. and H. Hung (2008). Soil Vibrations Caused by Underground MovingTrains. Journal of Geotechnical and Geoenvironmental Engineering 134(11): 1633–1644. Yang,Y., S. Kuo, et al. (1996). Frequency independent infinite elements for analysing semi-infinite problems. International Journal for Numerical Methods in Engineering 39: 3553–3569. Yang, Y. B. and H. H. Hung (2001). A 2.5D finite/infinite element approach for modelling visco-elastic body subjected to moving loads. International Journal for Numerical Methods in Engineering 51: 1317–1336. Yun, C. B. and J. M. Kim (2006). Dynamic Infinite Elements for Soil-Structure Interaction Analysis in a Layered Soil Medium. Computational Methods in Engineering and Science. Sanya, Hainan, Tsinghua University Press & Springer: 153167.

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A comparison of different approaches for the modelling of shallow foundations in seismic soil-structure interaction problems Stéphane Grange Laboratoire 3S–R, UJF, INPG, CNRS, Grenoble,

Diana Salciarini University of Perugia, Perugia, Italy

Panagiotis Kotronis Institut GeM, Ecole Centrale de Nantes, UMR CNRS 6183, Nantes,

Claudio Tamagnini University of Perugia, Perugia, Italy

ABSTRACT: In this work, the performance of two different macroelement models for shallow foundations on sands is assessed by considering the dynamic response of a RC bridge subject to earthquake loading. The first macroelement model is formulated within the framework of kinematic hardening elastoplasticity with prescribed bounding surface (Grange et al. 2009). The second macroelement model has been recently developed within the framework of the theory of hypoplasticity (Salciarini and Tamagnini 2009). The results of a series of FE simulations show that a significant reduction of the computed structural loads can be obtained by taking properly into the foundation–soil behavior, rather than assuming zero displacements and rotations at the pier bases. The two macroelements considered provide quite similar results, in spite of the large differences existing in their mathematical formulation.

1

INTRODUCTION

Recent developments in the analysis of the seismic response of slender structures such as tall buildings and bridge piers resting on shallow foundations have shown that the proper consideration of soil deformability is of primary importance for an accurate prediction of the deformation and loads experienced by the structure during the earthquake, see, e.g., (Grange et al. 2009). A substantial progress towards an efficient and reliable approach to the analysis of soil–foundation– structure interaction (SFSI) problems for such kind of structures has been recently achieved by the development of the so–called macroelement models for describing the overall behavior of the foundation–soils system (see, e.g., Nova and Montrasio 1991, Martin and Houlsby 2001, Crémer et al. 2001). In the macroelement approach, the mechanical response of the foundation–soil system is described by means of a constitutive equation relating the generalized load vector:

and the generalized displacement vector:

Figure 1. Notation adopted for generalized forces (a) and displacement (b) components.

In the above definitions, V , Hx , Hy , Mx and My are the resultant forces and moments acting on the foundation; w, ux , uy , θx and θy are the displacements and rotations (in the vertical yz and xz planes) of the foundation, and B is a characteristic length (i.e., the foundation diameter or width), introduced for dimensional consistency (see Fig. 1). To reproduce correctly some important features of the experimentally observed behavior of the foundation–soil system such as nonlinearity, irreversibility and dependence from past loading history, the constitutive equation for the macromodel must be formulated in rate–form:

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where d := u˙ is the generalized velocity vector, K is the tangent stiffness of the system, depending on the system state and loading direction, and q is a pseudo– vector of internal variables ing for the effects of previous loading history. The properties of the stiffness matrix K are selected according to the basic features of observed behavior. To reproduce a rate–independent response, K must be positively homogeneous of degree zero with respect to the generalized velocity vector d. In order to reproduce an inelastic behavior, K must depend on the loading direction d/ d (Kolymbas 1991). In this work, the performance of two different macroelement models is assessed with respect to the analysis of the dynamic response of a RC bridge subject to earthquake loading. The first macroelement model is formulated within the framework of kinematic hardening elastoplasticity with prescribed bounding surface (Grange et al. 2009). The second macroelement model has been recently developed within the framework of the theory of hypoplasticity (Salciarini and Tamagnini 2009).

is positive, and zero otherwise. The particular elastoplastic macroelement model considered in this study is a kinematic hardening elastoplastic macroelement specifically developed for cyclic loading conditions by Grange et al. (2009). The yield function is given by the following equation:

2 THE ELASTOPLASTIC MACROELEMENT

are non–dimensionalized components of the generalized force vector; Vf is the bearing capacity of the foundation under a vertical centered load; Bx and B is the footing size; γ and ρ are internal variables defining the size of the yield locus; τ is a kinematic internal variable defining the position of the yield locus in the generalized loading space; a, b, c, d, e and f are model constants controlling the shape of the yield locus. The failure locus of the foundation in the generalized loading space is found by setting τ = 0, and ρ = γ = 1 in eq. (6). As for the plastic potential function g, an associative flow rule is adopted in the (hx , hy , mx , my ) hyperplane, while, in agreement with available experimental observations, a non–associative flow rule is defined in the (hx , v), (hy , v), (mx , v) and (my , v) planes. A detailed description of the plastic potential function and of the evolution equations for the internal variables ρ, γ and τ is provided in Grange (2008) and Grange et al. (2008). A specific feature of this model is in the possibility of taking into the irreversible displacements associated to the uplift of the foundation which takes place at high values of the load eccentricity. However, as uplift has not been considered in this work, the interested reader is referred to Grange et al. (2009) for further details.

In the macromodels developed in the framework of the theory of elastoplasticity (Nova and Montrasio 1991; Martin and Houlsby 2001; Crémer et al. 2001; Grange et al. 2008), the constitutive equation is built starting from the fundamental assumptions of: i) elastic and plastic decomposition of the generalized velocity; ii) existence of a yield function f (t, q) in the generalized load space; iii) existence of a plastic potential function g(t, q) providing the plastic flow direction; iv) existence of a suitable hardening law for the internal variables; and, v) enforcement of Prager’s consistency condition. The resulting constitutive equation in rate form then reads:

where Ke is the elastic stiffness matrix; hq (t, q) is the hardening function controlling the evolution of the internal variables with plastic displacements; Kp is a strictly positive scalar given by:

and H (γ) ˙ is the Heaviside step function, equal to 1 if the plastic multiplier:

where, for a square footing:

3 THE HYPOPLASTIC MACROELEMENT In the development of a hypoplastic macrolement, Salciarini and Tamagnini (2009) have assumed from the outset that the tangent stiffness tensor appearing in eq. (1) possesses the following basic structure:

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Differently from elastoplasticity, the tangent stiffness K(t, q, η) varies continuously with the direction η of the generalized velocity. This property is known as incremental nonlinearity (see, e.g., Tamagnini et al. 2000), and is the key to the modeling of irreversibility of the model response. The construction of a specific hypoplastic macroelement requires the definition of the constitutive functions L(t, q) (a 5 × 5 matrix) and N (t, q) (a 5– dimensional vector). Upon load reversal, the incremental response of the hypoplastic macroelement is assumed to be almost elastic:

Figure 2. Plan view of the bridge.

while the plastic flow direction m can be derived from the plastic potential function of Nova and Montrasio (1991):

where Ke is the elastic stiffness matrix of the elastoplastic macroelement, and kv , kh and km define the vertical, horizontal and rotational stiffnesses of the foundation–soil system. Thus, the matrix L can be written as follows (Salciarini and Tamagnini 2009):

where mR is a material constant. The constitutive function N is obtained following the approach proposed by (Niemunis 2002), according to which N can be expressed as:

where 0 ≤ Y (t, Vf ) ≤ 1 is a scalar loading function and m(t) is a 5–dimensional unit vector. In the particular case of continued loading along a straight path of sufficient length, eq. (8) and (11) yield:

The scalar function 0 ≤ Y (t, Vf ) ≤ 1 controls the degree of nonlinearity of the model response. If Y = 0, eq. (12) reduces to a linear relation between t˙ and d. When Y (t, Vf ) → 1 the system reaches an ultimate failure state (t˙ = 0) for a collapse mechanism characterized by:

Thus m can be identified as the direction of the generalized velocity vector at bearing capacity failure (i.e., unconfined “plastic flow” direction). The loading function Y can be defined starting from a 5–dimensional generalization of the failure locus proposed by Nova and Montrasio (1991):

in which the nondimensional variables v , hx , hy , mx and my are obtained from the corresponding quantities of eq. (7) by replacing Vf with Vg , a dummy variable determined from the condition g(t, Vg ) = 0. Finally, in order to describe the foundation–system response under both monotonic and cyclic loading conditions, the hypoplastic macroelement is equipped with the following set of internal state variables:

where Vf is again the bearing capacity of the foundation under a vertical centered load, and δ is a vectorial quantity – the “internal displacement” vector – which keeps track of the previous displacement history, mimicking the concept of intergranular strain introduced by (Niemunis and Herle 1997) for continuum hypoplasticity. The details of the evolution equations for these internal variables are provided in Salciarini and Tamagnini (2009). 4 THE PROBLEM CONSIDERED The problem considered is a four–span RC bridge, whose geometry is shown in Fig. 2. This particular structure has been studied at the European research centre ELSA (JRC Ispra), where a series of 1 to 2.5 scale models of the bridge piers have been subject to pseudo–dynamic tests, see Pinto et al. (1996). The three piers are made of reinforced concrete with a hollow rectangular section shape. The bridge deck is composed of hollow prestressed concrete beams. Some geometrical characteristics of the piers and beams sections are given in Tab. 1. The FE model of the structure is shown in Fig. 3. Non–linear Timoshenko multifiber beam elements have been adopted to reproduce the behaviour of the piers (Kotronis and Mazars 2005). In detail, 40 concrete fibers and 80 steel fibers (representing the

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Table 1.

Deck Piers

Geometrical properties of structural elements. A (m2 )

Ix (m4 )

Iy (m4 )

Iz (m4 )

J (m4 )

1.11 0.66

0.13 0.056

– 0.19

2.26 –

2.39 0.20

Table 2. ment.

Figure 3. FE model of the bridge.

reinforcement bars at their actual position) have been used for each section. The mesh is refined at the base of the piers where inelastic behavior is more likely to occur. As for the prestressed concrete deck elements, linear elastic behavior has been assumed. The inertial characteristics of the structural elements have been simulated by means of lumped masses, as shown in Fig. 3. The material constants adopted for pier and deck elements are given in Grange (2008). In order to validate the FE model, a first comparison of the numerical and experimental results for the small–scale bridge model, under the hypothesis of fixed base has been presented in (Grange et al.), (Grange et al.). In this paper, the foundations of the three piers have been modeled using the two macroelements discussed in Sect. 2 and 3. The material parameters adopted for the elastoplastic macroelement are summarized in Tab. 2. They can be considered appropriate for a foundation resting on a medium– dense sand. The material constants for the hypoplastic macroelement have been selected by matching the predictions of the two macroelements on both monotonic and cyclic loading paths. The result of this calibration procedure is shown in Tab. 3. The seismic input adopted in the FE simulations is shown in Fig. 4. It is an artificial accelerogram, applied in the x direction, consistent with the 5% damping response spectrum provided by Eurocode 8 for a soil of Class B, with a peak horizontal acceleration of 0.35g. In the FE analyses, the accelerogram has been scaled by multiplying the accelerations by 2.5 and dividing the time scale by the same factor, in order to respect the similitude laws. The same input motion is applied at the base of the piles and at the bridge abutments.

5

RESULTS OF FE SIMULATIONS

Some results of the FE simulations performed with the two macroelement models are shown in Fig. 5 to 10. To assess the influence of soil–foundation deformability, the results of full SFSI analyses are also compared to those obtained assuming the soil as perfectly rigid

Material constants of the elastoplastic macroele-

kv (MN/m)

kh (MN/m)

km (MN/m)

Vf (MN)

298.68

244.37

108.65

11.26

a (–)

b (–)

c (–)

d (–)

e (–)

f (–)

0.48

0.33

1.00

0.95

1.00

0.95

Table 3. element.

Material constants of the hypoplastic macro-

µ (–)

ψ (–)

β (–)

λh (–)

λm (–)

κ (–)

0.48

0.33

0.95

1.75

1.50

0.25

ξ (MN/m)

mR (–)

mT (–)

R (mm)

βr (–)

χ (–)

0.0

1.1

1.05

5.0

1.0

1.5

Figure 4. Original and scaled accelerograms of the imposed seismic excitation.

(i.e., zero displacement and rotations at the base of the piers). The computed load–displacement curves for the pier foundations are shown in Fig. 5 (Hx vs. ux ) and 6 (My vs. θy ). Although the hypoplastic model tends to predict somewhat smaller horizontal displacements, the agreement between the two numerical solutions appears quite good. This is confirmed by the time histories of horizontal forces and moments at the foundations, as shown in Fig. 7 and 8 for pier P2. The results obtained with the two macroelements are almost coincident, whereas forces and moments obtained under the hypothesis of fixed base are much larger. It is interesting to note that in the particular case considered, the incorporation of the deformable macroelements in the FE analysis does not affect significantly the magnitude of horizontal displacements at the top of the piers, as shown in Fig. 9. This is a consequence of the fact that, in the fixed base case, the bending moments at the base of the piers are so

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Figure 7. Time–history of horizontal force at the foundation of pier P2.

Figure 5. Horizontal force vs. horizontal displacement at the foundations: a) piers P1 and P3; b) pier P2.

Figure 8. Time–history of rocking moment at the foundation of pier P2.

Figure 9. Time–history of horizontal displacement at the top of pier P2.

Figure 6. Rocking moment vs. rotation at the foundations: a) piers P1 and P3; b) pier P2.

large that the piers enter into the non–linear regime, and undergo significant plastic rotations, as demonstrated by the moment–curvature diagram for the base of pier P2, shown in Fig. 10. However, for different soil conditions or higher earthquake accelerations, a significant underestimation of the displacements at the top of the piles can occur if the deformability of the soil is not properly taken into (Grange et al. 2009), (Grange et al. 2010). Finally, Fig. 11 shows the time–histories of vertical displacements accumulated at the foundation of pier P2, for the two macroelements. Differently from

Figure 10. Bending moment vs. curvature at the base of pier P2.

computed horizontal displacements and rotations, in this case the response of the two macroelements is quite different. While the elastoplastic analysis predicts very small permanent settlements (about 1 mm),

409

REFERENCES

Figure 11. Time–history of vertical displacement at the foundation of pier P2.

the hypoplastic simulation yields a much larger permanent settlement (about 6 mm). This is most likely due to the different nature of the plastic potential functions adopted by the two models in the Hx : V and My /B : V planes. 6

CONCLUDING REMARKS

In this work, two recently developed macroelements for shallow foundations have been used to model the soil–foundation response in the seismic analysis of a RC bridge. In spite of the different mathematical structure of the two models, the results obtained in the two cases are surprisingly similar, both in of computed displacements and structural loads.The only exception is represented by the way the two macroelements predict a continuous accumulation of vertical settlements of the foundation under the earthquake excitation. It is worth noting that the good agreement between the computed cyclic response of the foundations provided by the two macroelements is mostly due to the presence in the set of internal state variables of a vectorial quantity (the back–stress in the elastoplastic model; the internal displacement in the hypoplastic model) which takes into the effects of the previous loading history. Finally, the comparison with the results obtained under the hypothesis of rigid soil indicates that this last assumption may lead to a significant overestimation of computed structural loads. Moreover, a proper consideration of the soil–foundation deformability is a key factor in the proper estimation of the displacements experienced by the structure under the seismic action.

Crémer, C., A. Pecker, and L. Davenne (2001). Cyclic macro–element for soil–structure interaction: material and geometrical non–linearities. Int. J. Num. Anal. Meth. Geomech. 25, 1257–1284. Grange, S. (2008). Modélisation simplifiée 3D de l’interaction sol-structure: application au génie parasismique. Ph. D. thesis, INP Grenoble. http://tel.archives-ouvertes.fr/tel00306842/fr. Grange, S., L. Botrugno, P. Kotronis, and C. Tamagnini (2009). The effect of soil–structure interaction on a reinforced concrete bridge. In G. Pande, S. Pietruszczak, C. Tamagnini, and R. Wang (Eds.), Computational Geomechanics – COMGEO I. Intl. Centre for Computational Engineering, Rhodes. Grange, S., L. Botrugno, P. Kotronis, and C. Tamagnini (2010). On the influence of soil structure interaction on a reinforced concrete bridge. Earthquake Engineering and Structural Dynamics. (In print). Grange, S., P. Kotronis, and J. Mazars (2008). A macro– element for a circular foundation to simulate 3d soil– structure interaction. Int. J. Num. Anal. Meth. Geomech. 32, 1205–1227. Grange, S., P. Kotronis, and J. Mazars (2009). A macro– element to simulate 3d soil–structure interaction considering plasticity and uplift. Int. Journal of Solids and Structures 46, 3651–3663. Kolymbas, D. (1991). An outline of hypoplasticity. Archive of Applied Mechanics 61, 143–151. Kotronis, P. and J. Mazars (2005). Simplified modelling strategies to simulate the dynamic behaviour of r/c walls. Journal of Earthquake Engineering 9(2), 285–306. Martin, C. M. and G. T. Houlsby (2001). Combined loading of spudcan foundations on clay: numerical modelling. Géotechnique 51, 687–700. Niemunis, A. (2002). Extended Hypoplastic Models for Soils. Habilitation Thesis, Bochum University. Niemunis, A. and I. Herle (1997). Hypoplastic model for cohesionless soils with elastic strain range. Mech. Cohesive–Frictional Materials 2, 279–299. Nova, R. and L. Montrasio (1991). Settlements of shallow foundations on sand. Géotechnique 41, 243–256. Pinto, A., G. Verzeletti, P. Pegon, G. Magonette, P. Negro, and J. Guedes (1996). Pseudo Dynamic Testing of Large-Scale R/C Bridges. HMC Grant Holder, Report EUR 16378 EN, JRC Ispra, Italy. Salciarini, D. and C. Tamagnini (2009). A hypoplastic macroelement model for shallow foundations under monotonic and cyclic loads. Acta Geotechnica 4(3), 163–176. Tamagnini, C., G. Viggiani, and R. Chambon (2000). A review of two different approaches to hypoplasticity. In D. Kolymbas (Ed.), Constitutive Modelling of Granular Materials, pp. 107–145. Springer, Berlin.

410


A finite element approach for dynamic seepage flows R. Stucchi, A. Cividini & G. Gioda Department of Structural Engineering, Politecnico di Milano, Milan, Italy

ABSTRACT: A finite element approach for the dynamic analysis of seepage flows is presented that represents the first step of a study on the effects of earthquakes on retaining or embedded structures in saturated granular soils. The equations governing the flow of a liquid within a porous skeleton under an acceleration field varying with time are recalled first. Then they are combined in a differential equation that, reduced in its weak form, leads to a finite formulation of the problem in term of discharge velocity only. This first approach shows some stability problem during the time integration process unless exceedingly small time increments are adopted. To overcome this drawback two alternative formulations, involving different sets of free variables, are outlined and commented upon.

1

INTRODUCTION

The solution of geotechnical problems involving saturated two phase soils requires the simultaneous analyses of the seepage flow of the pore water and of the stress distribution within the soil skeleton. In quasi static conditions, i.e. under an acceleration field constant with time (gravity), the literature provides exhaustive theoretical bases and broadly accepted methods for the numerical analysis of seepage and of the coupled effective stress-flow problem (e.g. Desai 1976; Sandhu & Wilson 1969; Zaman et al. 2000). In dynamic conditions however, e.g. during earthquakes, the analysis of seepage becomes less straightforward since recourse cannot be made anymore to the usual concept of hydraulic head (Bear 1988; Bird et al. 2007). This led to various numerical approaches for the dynamic coupled problem (Biot 1956) that involve different assumptions, different governing equations and different free variables (Zienkiewicz & Shiomi 1984; Zienkiewicz et al. 1999). The relatively complex mathematical structure of the problem does not permit a straightforward evaluation of the consequences of these assumptions and, hence, makes the choice of the most appropriate numerical approach somewhat controversial. The mentioned difficulties suggested undertaking a study on the coupled dynamic analysis of saturated granular deposits. Its initial part, discussed here, concerns only the finite element solution of dynamic seepage flows that represents a necessary first step towards the analysis of coupled problems. Note that a correct analysis of the dynamic seepage is particularly relevant when dealing with granular deposits. In fact their high hydraulic conductivity rules out the assumption of undrained conditions sometime adopted in engineering practice.

The fluid dynamic equations governing the flow of a liquid within a rigid porous skeleton are recalled first. Their weak form is then presented and a first finite element formulation is derived in which the discharge velocity of fluid, instead of its ‘pseudo-displacement’ (Zienkiewicz & Shiomi 1984), represents the nodal variable. The numerical results suggested some observation on the limits of this approach and on the possible computational advantages obtained using alternative sets of nodal variables. On these bases two alternative formulations are considered. The first one involves as free variables both discharge velocity and pore pressure, the second one only the pore pressure. Finally, the solution of a benchmark problem is compared with Westergaard results (1933). 2

GOVERNING EQUATIONS

The dynamic equations governing the seepage flow of a liquid within a saturated and rigid porous skeleton in isothermal conditions are recalled here. All variables are in general functions of time t. Uppercase and lowercase underlined letters denote, respectively, matrices and column vectors. Superscripts -1 and T denote inverse and transpose. In the following vˆ is the vector of the velocity components of the liquid particles while v represents the discharge velocity in Darcy sense, which pertains to the liquid phase. The two vectors are related through the matrix N S of the area, or surface, porosities the entries of which (nSx , nSy , nSz ) are the ratios between the area of pores and the total area of the sections,

The experimental difficulties met in determining the area porosities suggest using the volume porosity n

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in the formulation, which represents the ratio between the volume of voids and the total volume of a soil element. The volume porosity can be seen as the average value of the area porosities (Bear 1988). Consequently, the following approximated relationship will be used,

The relative velocity w of the fluid phase with respect to that of the solid one, u˙ , is expressed as,

Since a rigid porous skeleton is assumed here, the velocity u˙ is simply the time integral of the accelerations imposed by the earthquake and, hence, is known. As a consequence, the problem can be tackled considering a fixed solid phase and subjecting the fluid phase to a known acceleration field varying with time. This allows using v instead of w as a free variable. 2.1

Equation of compatibility

For a liquid this equation expresses the relationship between the strain rates, collected in vector ε˙ , and the velocity vˆ through a differential operator C analogous to that governing the strain-displacement relationship for solids,

Figure 1. Relevant quantities for the equation of motion in the x direction.

On these bases, the following relationship holds for a Newtonian fluid,

where µ is its deviatoric viscosity and I 0 is a diagonal matrix the entries of which are equal to 2 if they correspond to normal stresses, otherwise they are equal to 1. Substitution of eqs.(7) and (5) into eq.(8) leads to the strain rate vs. shear stress relationship,

where As usual, the first three entries of vector ε˙ correspond to normal strains and the remaining three to shear strains. The above equation, rewritten in of the discharge velocity v, becomes,

2.2

Shear stress-shear strain rate relationship

Assuming that the fluid phase behaves as a Newtonian liquid, a linear relationship can be established between the deviatoric strain rates e˙ and the deviatoric stresses τ. To this purpose the following quantities are introduced for convenience,

Here ε˙ and σ are the strain rate and the stress vectors; p is the pore pressure (positive if tensile); ε˙ vol is the volumetric strain rate and m is a vector the entries of which are equal to 1 if they correspond to normal strains/stresses, otherwise they vanish.

2.3 Equation of motion Adopting an Eulerian approach, the equation of motion expresses the momentum balance of the liquid contained within an infinitesimal volume of the porous medium. This implies that the rate of increase of momentum is equal to the difference between the inward and outward momentum rates plus the contribution of the external forces acting on the liquid. The equation of motion for a one dimensional flow in the x direction (cf. Figure 1) can be written as,

where ρ is the density, b∗x is the component of the imposed acceleration field (e.g. gravity) in the x direction and fDx is the drag force due to the interaction between the flowing liquid and the porous skeleton.

412

On these bases, the mass continuity for a three dimensional flow is expressed as,

In matrix form eq.(16) becomes,

2.5 Equation of state for the liquid phase Figure 2. Relevant quantities for the equation of mass continuity in the x direction.

Introducing the simplifying assumption expressed by eq.(2), the matrix form of the equation of motion for a three dimensional flow becomes,

Here vectors b∗ and f D collect, respectively, the given acceleration components in the Cartesian directions and the drag forces. Confining our attention to laminar flows, the following relationship holds between the drag forces and the discharge velocity,

To reach the balance between the number of unknowns and that of equations an additional scalar relationship is necessary.This is represented by the equation of state that expresses the variation of the density of liquid ρ with temperature and pore pressure. As previously mentioned, it is reasonable to treat the dynamic seepage flow under isothermal conditions, thus neglecting the variation of density with temperature. In addition, considering the high bulk modulus of water, also the change of density with the pore pressure is marginal. Consequently, the variation with time of the density of liquid ρ was neglected in eq.(17) and in the finite element formulation. 3

FINAL SYSTEM OF MATRIX EQUATIONS

Introducing the assumption of constant density, the governing equations (9, 6b) and (12) can be re-written in the following form,

where K is the intrinsic permeability matrix of the porous skeleton. 2.4

Mass continuity equation

If internal sources are neglected, the continuity of mass requires that the mass of fluid cumulated within an infinitesimal volume of the porous medium in a unit time coincides with the difference between the rate of masses entering and leaving it. Figure 2 shows the relevant quantities for a flow in the x direction. As to the rate of mass accumulation, ˙ , two contributions exist. The first one is the change M in mass due to the volumetric strain rate of the fluid phase that, neglecting the possible volumetric viscosity of the liquid, can be related to the pore pressure rate through the bulk modulus B of the liquid and the volume porosity,

The second contribution depends on the change in density of the liquid,

The final system of governing differential equations consists of eq.(17) and of the combination of eqs.(18) and (19),

It can be shown that, if b∗ represents the acceleration of gravity, under some simplifying assumptions eq.(21) reduces to Darcy law between discharge velocity and space derivative of the hydraulic head. Eqs.(20) and (21) form the final system of four scalar differential equations, involving as free

413

variables the discharge velocity components and the pore pressure. As to the boundary conditions, consider a porous domain having surface and volume . The surface can be subdivided into its impervious part, V , where the velocity normal to it vanishes and its pervious part, P , where the pore pressure p∗ is known, i.e.

Here α collects the direction cosines of the vector normal to .

The integration of eq.(24) is carried through a series of time increments ti , so that ti = ti−1 + ti , and assuming a linear variation of the nodal velocities within each increment. This leads to the following expressions for ve and v˙ e within the time interval,

4 VELOCITY APPROACH Let now derive an approach where the discharge velocity represents the only nodal variable. To this purpose eq.(21) and the boundary condition eq.(22b) are written in weak form introducing a virtual variation δv of the discharge velocity that fulfils eq.(22a). Note that the quadratic term on the right hand side of eq.(21), equivalent to the kinetic part of the hydraulic head in standard seepage analyses, can be disregarded because the small value of the discharge velocity makes its contribution marginal. In addition, the presence of this non linear term would increase the burden of the numerical solution.

where 0 ≤ β ≤ 1. Assuming β = 1/2, substitution of eqs.(26) into eq.(24) leads to,

The pore pressure at the end of the time interval pe (ti ) that appears in vector f ep (ti ), cf. eq.(25e), can be expresses as,

Integrating by parts the second term within brackets, and applying Green-Gauss theorem, introducing the interpolation or shape function matrix S ev , expressing the velocity distribution within the e-th element as a function of the nodal velocities ve , after some manipulations one obtains,

The following expressions hold for matrices and vectors in eq.(24), where V e is the volume of the element; eP represents its sides where a known pore pressure p∗e is imposed; b∗ is the vector of the imposed accelerations and pe is the unknown pore pressure distribution within the e-th element,

Considering that the pore pressure rate and the velocity vary linearly within the time step, cf. eq. (20), eq.(28) leads to the following relationship,

where the pore pressure pe is defined at the integration point of the element. On these bases, and knowing the flow velocities and pore pressures at time ti−1 , the following iterative process can be adopted to evaluate the quantities at the end of the step:

414

– The pore pressure pe (ti ) is approximated through eq.(29) assuming ve (ti ) = ve (ti−1 ); – The velocity ve (ti ) is calculated solving the system of linear equations (27); – The pore pressure pe (ti ) is updated through eq.(29); – The iterations end when ve (ti ) and pe (ti ) stabilize.

5 VELOCITY- PORE PRESSURE APPROACH

6

The above iterative approach, where the pore pressure does not represent a nodal variable and, hence, does not have a continuous distribution throughout the mesh, might show some stability problem unless very small time integration steps are adopted. To overcome this drawback an alternative approach can be easily formulated where both discharge velocity and pore pressure represent nodal variables. To this purpose eq.(20) is written in the following weak form where δp is a virtual pore pressure variation,

Taking into that in many cases of interest in geotechnical engineering the flow velocity is small, some of eq.(21) can be disregarded since their contribution is likely to be marginal. These are the term of the left hand side of the eq.(21), which depends on the acceleration, and the first term on the right hand side that depends on the square of velocity. Consider now the linearly depending on velocity. The second one on the right hand side of eq.(21), related to the curvature of the streamlines, plays a major role in standard flow problem. However, in the case of seepage flows its contribution is in general smaller than that of the last term on the right hand side that represents the interaction between the flowing liquid and the solid particles. Hence, eq.(21) reduces to,

Introducing the interpolation function vector for the pore pressure, sep , that relates the pore pressure within the element to its nodal values pe , and integrating over the volume V e of the e-th element, eq.(30) leads to,

PORE PRESSURE APPROACH

Upon substitution of eq.(35) into eq.(20) one obtains the governing equation in of the pore pressure only (Zienkiewicz et al. 1999) ,

where

Now the pore pressure pe in eq.(25e) is expressed in of the nodal pore pressures pe through the interpolation functions,

Writing eq.(36) in weak form, and following the same procedure previously outlined for the velocity approach, the pore pressure finite element formulation is arrived at,

where

Combining eqs.(24), (31) and (33) the following system is arrived at,

7 TEST EXAMPLE The time integration of eq.(34) is carried out assuming a linear variation of velocity during the time increment (cf. eqs.26). In addition, also a linear variation of the pore pressure should be assumed. This introduces an approximation since, as previously observed, the pore pressure rate should vary linearly with time. Note that, to be consistent with eq.(20), the shape functions of the pore pressure should coincide with the space derivatives of those of the flow velocity. This implies that higher order element should be used for the velocity and lower order elements for the pore pressure. The consequent non negligible computational burden and the above approximation suggest disregarding this approach for practical applications.

The velocity and the pore pressure approaches were used for determining the water pressure distribution along a vertical rigid wall due to a dynamic excitation in the horizontal direction (Westergaard 1933). The mesh consists of 200 four node quadrilateral elements and 231 nodes, 11 of which discretize the vertical wall. The numerical results are reported, and compared with Westergaard solution, in Figures 3 and 4. They show, respectively, the maximum excess pressure distribution, with respect to the hydrostatic one, along the wall and the variation with time of the excess pressure at its base. In these figures H is the wall height and pmax is the maximum excess pressure at the wall base from Westergaard solution.

415

8

CONCLUSIONS

Three finite element approaches have been presented for dynamic seepage analysis, which involve different sets of free variables. Two of them, which seem more convenient from the numerical standpoint, have been used for solving a benchmark problem obtaining an acceptable agreement with Westergaard (1933) solution. The study will now proceed towards the finite element formulation of dynamic two phase problems in view of the analysis of the effects of earthquakes on structures embedded in saturated granular deposits. ACKNOWLEDGEMENTS The writers wish to thank Prof. Gabriele Dubini for his friendly advice and stimulating comments. Figure 3. Excess pressure distribution along the vertical wall.

REFERENCES Bear, J. 1988. Dynamics of Fluids in Porous Media, New York: Dover Publications. Biot, M.A. 1956. The theory of propagation of elastic waves in a fluid saturated porous solid. J. Acou. Soc. Am., 168–191. Bird, R.B., Stewart, W.E. & Lightfoot, E.N. 2007. Transport Phenomena. New York: J.Wiley & Sons. Desai, C.S. 1976. Finite element residual schemes for unconfined flow. Int. J. Numer. Methods Eng., 10:1415–1418. Sandhu, R.S. & Wilson, E.L. 1969. Finite Element Analysis of Seepage in Elastic Media. ASCE EM3, 95:641–652. Westergaard, H.M. 1933. Water pressures on dams during earthquake. Transaction of ASCE, 98:418–434. Zaman, M., Gioda, G. & Booker, J. (eds.) 2000. Modeling in Geomechanics. Chichester: J.Wiley & Sons. Zienkiewicz, O.C. & Shiomi, T. 1984. Dynamic behaviour of saturated porous media; the generalized Biot formulation and its numerical solution. Int. J. Numer. Anal. Meth. Geomech., 8:71–96. Zienkiewicz, O.C, Chan, A.H.C., Pastor, M., Schrefler, B.A. & Shiomi, T. 1999. Computational Geomechanics. Chichester: J.Wiley & Sons.

Figure 4. Variation with time of the excess pressure at the wall base.

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A method to solve Biot’s u-U formulation for soil dynamics applications using the ABAQUS/explicit platform F.J. Ye, S.H. Goh & F.H. Lee Department of Civil Engineering, National University of Singapore, Singapore

ABSTRACT: This paper describes a numerical method for solving the coupled equations resulting from Biot’s u–U formulation for waves propagating in a saturated porous medium. The proposed ABAQUS Dual Phase Coupling (ADPC) method is implemented in the explicit solver module of the general purpose finite element program ABAQUS. This method features two overlapping meshes representing the fluid and solid phases of the saturated medium respectively. Interactions between the two phases occur via dashpots connecting the two meshes, and also by the use of -defined material subroutines. In this way, the coupling effects arising from complex dual-phase interaction can be incorporated into the analysis. In this paper, the ADPC method is restricted to linear material behavior of both the solid and fluid phases. Preliminary validation of the proposed method is carried out using a one-dimensional problem. The good agreement between the ADPC results and published analytical/numerical solutions suggests that the proposed method provides a feasible, alternative approach for studying dynamics problems involving wave propagation in saturated soil media.

1

INTRODUCTION

In geotechnical engineering, Biot’s two phase theory has been applied to solve practical problems such as earthquake-induced liquefaction. Numerical studies based on the fully-coupled dual-phase formulation were able to predict the pore pressure behavior during earthquakes. The numerical results compared quite favorably with centrifuge test measurements obtained as part of the VELACS project (Zienkiewicz et al., 1987; Yogachandran, 1990). On the other hand, transient phenomena, such as those associated with blast or impulsive loadings in saturated soils, have also been studied using the fully coupled method following Biot’s theory (Prevost, 1982). Various finite element implementations of Biot’s theory have been carried out by Ghaboussi & Dikman (1978), Zienkiewicz & Shiomi (1984), Zienkiewicz et al. (1987), and Prevost (1985). However, these implementations usually result in very specialized and non-commercial codes, due to the complexities associated with the discretization and solution of the dual phases in both the spatial and time domains. In this paper, an alternative numerical approach, the ABAQUS Dual Phase Coupling (ADPC) method, is proposed to solve the u-U formulation of Biot’s fully coupled equations. As the name suggests, this method is implemented in the general purpose finite element program ABAQUS. More specifically, the implementation is carried out using the explicit solver module of the program. However, ABAQUS itself cannot solve Biot’s coupled equations with its available intrinsic functions and modules. These limitations are

overcome in the ADPC method via the use of special ABAQUS features such as overlapping meshes, connector elements and -defined subroutines. The proposed method is used to analyze a one-dimensional transient problem involving a fluid-saturated porous medium, the results of which are presented in this paper. 2

BIOT’S THEORY ON DYNAMICS OF SATURATED POROUS MEDIA

Biot (1956a, 1956b) developed a theory which describes the wave propagation mechanism in the dual-phase medium for the linear elastic regime, by introducing the concepts of mass and viscous coupling in the fluid-solid interaction. It was found that the solution consists of two dilatational waves and one rotational wave propagating in the porous medium with frequency dependent damping factors. Biot’s original formulation was based on the concepts of partial solid stress and partial fluid pressure as defined in Section 2.1. This formulation, together with the elastic constitutive relations, is commonly recognized as the u-U formulation. 2.1 Assumptions Biot’s theory on the dynamics of fluid saturated porous media was established based on the following assumptions: 1) linear elastic, isotropic material behavior; 2) reversibility of stress strain relations; 3) small strain; 4) incompressibility of the pore fluid which may contain

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air bubbles; 5) the fluid flow through the porous skeleton follows Darcy’s Law; 6) the frequency range is low such that the assumption of Poiseuille flow is valid. The original formulation, which incorporates the mass and viscous coupling effects in the stress equilibrium for both the solid skeleton and fluid phases, can be written with respect to the three directions.

of fluid per unit volume of the aggregate. ρ11 and ρ22 can be regarded as the intrinsic density coefficients, and ρ12 or ρa the mass coupling parameter between fluid and solid. This parameter may be considered as a measure of the coupled inertial effect, which is reflected by the additional inertial coupling induced by the relative motion between the two phases. For porous media under small strain conditions, the porosity n may be assumed to be independent of the stress level. Therefore, ρ1 and ρ2 can be defined as

where

where ρs is the true mass density of the solid grain and ρf true mass density of the pore fluid. Therefore, the density of the porous aggregate is

2.3 Viscous damping effect

In the equations, the vectors u and U represent the average displacement components in the three directions of the solid skeleton and the fluid phase, respectively. L denotes the spatial derivative operator matrix. The average displacement of the fluid is defined as the volume of fluid flowing through a unit area perpendicular to each direction. π is the partial fluid pressure, which is defined as the average force ed by the fluid portion of the infinitesimal cubic face. σ s , also known as the partial solid stress vector, is the average force ed by the solid portion of an infinitesimal cubic face. The partial solid (fluid) stress in the y direction will be the average force in the corresponding direction ed by solid (fluid). However, the partial solid stress has three components with respect to the three directions while the partial fluid pressure in all directions preserves the same value. Both σ s and π are taken as positive in tension, and negative in compression, following ABAQUS convention. It is noted that the viscous damping and inertial effects are explicitly incorporated into Equation 1. The viscous damping component is a function of the relative velocity between the two phases, while the inertial component includes induced mass coupling effects arising from relative motion between the two phases. 2.2

Mass coupling

The viscous coefficient D in Equation 1 corresponds to the viscous drag imposed to each constituent per unit volume when the fluid flows by the walls of the pores due to external loading. The viscous coefficient D can be defined if the Poiseuille flow assumption is valid

where µ is the fluid viscosity and n is the porosity. K is the absolute permeability for isotropic condition. The opposite signs for the viscous term indicate that the viscous drag imposed on one phase will induce an equivalent drag of an opposite direction on the other phase. 2.4 Stress-strain compatibility relationship The fluid solid interaction in a saturated porous medium is a complex physical mechanism in which the volumetric behavior of one phase must be compatible with the other on an infinitesimal scale. A coupled linear elastic stress-strain relationship was proposed by Biot to capture the coupling in the constitutive model between the two phases. The partial solid stress of the solid aggregate σ s is related to the strain components of the solid skeleton e and volumetric strain of the fluid ζ via the following relations:

The partial pore fluid pressure π is related to the volumetric strain of fluid ζ and the volumetric strain ε of solid skeleton, as follows

In Equation 1, the densities are defined as where the strain vector e for the solid skeleton may be expressed as where ρ1 and ρ2 denote, respectively, the mass of solid per unit volume of the aggregate and the mass

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The volumetric strains, ε and ζ, may be defined as

where A, N , Q and R are the elastic material constants as defined in Section 2.4. ˆ is a square stiffness matrix which In Equation 6a, D defines the stress-strain relationship of the partial solid stress as follows

3.2 Finite element form Either 1-D or 2-D elements can be used to solve the coupled one-dimensional relations of Equation 11. In this study, four-node quadrilateral elements are adopted. For each element, the displacement variables ˆ can be written in of their nodal vectors uˆ and U as follows:

where N and A are two material constants which define the increment of partial solid stress resulting from the solid strain increment. The material constant Q reflects the nature of the interaction which maintains the volumetric equilibrium of the solid and fluid phases. The material constant R is the pressure needed to force a certain volume of fluid into the solid matrix while the total volume remains unchanged. The vector m is equivalent to the Kronecker delta, which is defined as:

Applying Galerkin’s method to Equation 11, the following finite element formulation can be obtained at the element level.

where

2.5 The u-U formulation Substituting Equation 6 into Equation 1, the resulting equations of motion can be reformulated into a form which contains the two independent displacement variables u and U:

Fes and Fef include all the body forces and prescribed tractions at the boundary.

3.3 Principle of explicit time-integration

where grad is a gradient operator. Equation (10) is commonly known as the u-U formulation.

The coupled equation system represented by Equation 13 can be simply expressed as

where 3

FINITE ELEMENT IMPLEMENTATION

3.1 Simplified one-dimensional form Equation 10 can be reduced to a one-dimensional form if the displacement components in the other two directions are neglected. In practice, the inertial effects associated with the density term ρ12 are also assumed to be negligible, and hence these can be dropped from Equation 10. On the other hand, a complete stress balance equation should include the solid and fluid phase body forces, i.e. bs and bf . Therefore the 1-D form of the u-U formulation can be written as

Equation 14 may be discretized in time using the well-known central-difference scheme, to obtain the incremental velocity and acceleration as follows:

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in which the subscripts t+ and t− indicate values at times t + t and t − t, respectively. t is the incremental time step. Substituting Equation 15 into the incremental form of Equation 14 yields:

Figure 1. Illustration of how the dashpots work in the overlapping system for one-dimensional analysis.

Equation 16 implies that the incremental displacement ui,t+ at time t + t can be calculated by inverting M + Ct without solving the equation. This explicit time-integration scheme is especially efficient when the M and C matrices are diagonal, which allows the current incremental displacements to be easily computed. By lumping the entire mass and viscous damping to the corresponding nodes, the computational costs can be considerably reduced. 4 ABAQUS DUAL PHASE COUPLING (ADPC) METHOD

Figure 2. Infinite fluid saturated porous column subject to step velocity loading in example problem.

In this section, the three key features of the ADPC method are introduced.These are: overlapping meshes, connector elements and defined subroutines. 4.1 Overlapping meshes In this method, two identical and overlapping discrete meshes are simultaneously present in the finite element model to simulate an integrated fluid saturated porous medium. One mesh contains the corresponding degrees of freedom associated with the solid phase, while the other mesh contains the corresponding degrees of freedom for the fluid phase. 4.2 Connector elements Given the two overlapping meshes, the viscous coupling effects discussed in Section 2.3 can be introduced by linking the two meshes at the corresponding nodes with connector elements in the form of dashpots. These connector elements are prescribed appropriate mechanical properties that control how the two meshes interact with each other. Figure 1 shows how the connector elements are assigned in the one-dimensional problem. 4.3 defined material subroutine As discussed in Section 2.4, the deformation behaviors of the two-phase medium are also dictated by the coupled dependency in stress-strain relations. These coupled stress strain relationships may follow a linear elastic manner or a nonlinear elasto-plastic manner, both of which are not available in the existing ABAQUS material model library. This limitation may be circumvented using the VUMAT feature in

ABAQUS, which allows non-conventional or special material behavior to be modeled using -defined subroutines. In this study, VUMAT subroutines were introduced to for the coupled elastic stressstrain relationship discussed in Section 2.4. 5 APPLICATION OF ADPC TO A ONE DIMENSIONAL EXAMPLE PROBLEM 5.1

Description of the problem

In this example, a transient step velocity loading is instantaneously applied over the surface of a fluidsaturated, porous half-space. As shown on Figure 2a, this is a one-dimensional problem which may be modeled using a fully-constrained, infinitely long column. The loading is applied in such a way that both the solid and fluid phases at x = 0 will be subjected to a step velocity loading at time t = 0, i.e. v(x = 0, t < 0) = 0 and v(x = 0, t ≥ 0) = 1, as shown on Figure 2b. 5.2 Analytical and finite difference solutions Garg et al. (1974) presented the solutions for this problem using (i) analytical techniques involving the Bessel function, (ii) numerical inversion of the Laplace transform, and (iii) a finite difference code POROUS. The field equations for the one dimensional problem considered by Garg and his associates follow the u-U formulation:

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Table 1.

Basic parameters for the example.

Porosity Bulk modulus of solid (Pa) Bulk modulus of fluid (Pa) Young’s modulus of skeleton (Pa) Poisson’s ratio Density of solid (kg/m3 ) Density of fluid (kg/m3 ) Magnitude of initial velocity (m/s) Low drag (kg/m3 s) Medium drag (kg/m3 s) High drag (kg/m3 s)

n KS KF E υ ρs ρf v0 Dl Dm Dh

0.18 3.600 × 1010 2.200 × 109 2.321 × 1010 0.171 2660 1000 1 2.19 × 104 2.19 × 106 2.19 × 1010

Figure 3. FE model for the example.

where the material constants a, b and c are given in Garg et al. (1974), u1 and u2 are the solid and fluid displacements; ρ1 , ρ2 and D are as defined previously. 5.3 ADPC implementation Figure 3 shows the overlapping finite element meshes representing the solid and fluid phases. Following Garg et al. (1974), the step velocity loading is applied to both meshes, that is, both phases. Comparing Equation 17 with Equation 11, the two are exactly identical such that a = A + 2N , b = R and c = Q. Garg et al. (1974) adopted values of a, b and c as

where a1 and b1 can be expressed as

Figure 4. Nodal velocity history at 10 cm below the column top for low drag condition.

where ex and ζ x are the strains in the direction of interest for the solid and fluid phases, respectively. Equation 20 can be coded into individualVUMAT subroutines for the solid and fluid phases respectively, as explained in Section 4.3. The infinite, saturated porous column was modeled using two overlapping rows of 200 four-node reduced integration elements, each with dimensions 0.005 m × 0.005 m. This results in a column length of 1.0 m, which is sufficiently long to minimize wave reflection effects at the output points of interest. The lateral displacements for both meshes were constrained using roller-type boundary conditions.

5.4 Results

In these relations, KS , KF and KB are the intrinsic bulk modulus of the non-porous solid, fluid and porous solid skeleton, respectively.All the relevant parameters are provided on Table 1, and are identical to those used by Garg et al. (1974). The amplitude of the step velocity loading H (t) at the column top is shown in Figure 3. The coupled stress strain relations incorporated in VUMAT can be written in one-dimensional form as:

To examine the influence of drag interaction effects on the propagating waves in the two phases, three viscous coefficients D of low, medium and high magnitude were studied. Figures 4–6 present the nodal velocity responses of both the solid and fluid phases at a point 10 cm below the top of the column. The ADPC results are plotted together with the results presented in Garg et al. (1974). As shown on Figure 4, the computed solid and fluid velocity responses from ADPC for the low drag condition are close to those predicted using POROUS and the analytical solutions. The figures clearly show the existence of two dilatational wavefronts (Biot, 1956a), especially in the fluid phase velocity history. It is noted that the analytical solution preserves the step nature of the wavefront (i.e. very short rise time of about

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Figure 5. Nodal velocity history at 10 cm below the column top for medium drag condition.

related to the dynamics of saturated porous media. This method is verified using a published onedimensional wave propagation problem that was previously analyzed using analytical and finite difference approaches. The good agreement between the ADPC results and those reported by Garg et al. (1974) serves as a preliminary first-order validation of the proposed method. The results also indicate that the magnitude of the viscous coefficient has a significant effect on the computed response of the two phases. A low viscous drag may result in minimal interaction arising from the relative motion of the two phases, while a high viscous drag may induce strong coupling between the two phases, so that the solid and fluid motions coalesce as one. Furthermore, two wavefronts can be observed in the ADPC results, which is consistent with Biot’s finding that two dilatational waves are present in a saturated porous medium. REFERENCES

Figure 6. Nodal velocity history at 10 cm below the column top for high drag condition.

0.1 µsec), whereas the numerical results from ADPC and POROUS indicate some smearing of the wavefronts, especially the second wavefront in the fluid phase. Figure 5 shows the solid and fluid nodal velocity histories at the same point for the medium drag condition. As before, there is good agreement between ADPC and Garg et al.’s results. The existence of two wavefronts is still present in the fluid phase velocity history. Unlike the low drag case, the particle velocity increases almost linearly between the two wavefronts. Figure 6 shows the solid and fluid nodal velocity histories at the same point for the high drag condition. Under high viscous coupling, the two wavefronts coalesce into a single front, the effect of which is very obvious for the fluid phase response. Due to artificial viscosity effects, it is noted that the ADPC and POROUS results show a rise time of about 4 µsec, which is much larger than the 0.15 µsec rise time given by the analytical solution. In principle, the numerical results can be improved by finer spatial zoning, or a more refined mesh. This effect is not discussed here. 6

CONCLUSIONS

Biot, M. A. 1956a. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. low-frequency range. The Journal of the Acoustical Society of America 28(2): 168–178. Biot, M. A. 1956b. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. high-frequency range. The Journal of the Acoustical Society of America 28(2): 179–191. Garg, S.K., Nayfeh, A.H. & Good, A.J. 1974. Compressional waves in fluid-saturated elastic porous media. Journal of Applied Physics 45(5): 1968–1974. Ghaboussi, J. & Dikman. S. U. 1978. Liquefaction analysis of horizontally layered sands. Journal of Geotechnical Division, ASCE GT3: 341–356. Prevost, J. H. 1982. Nonlinear transient phenomena in saturated porous media. Computer Methods in Applied Mechanics and Engineering. 20: 3–18. Prevost, J. H. 1985. Wave propagation in fluid-saturated porous media: An efficient finite element procedure. Soil Dynamics and Earthquake Engineering 4(4): 183–202. Yogachandran, C. 1990. Numerical and dynamic centrifuge modeling of initiation of flow failure and interface behavior. Department of Civil and Environmental Engineering, University of California, Davis. Ph.D thesis. Zienkiewicz, O. C. & Shiomi. T. 1984. Dynamic behaviour of saturated porous media; The generalized Biot formulation and its numerical solution. International Journal for Numerical and Analytical Methods in Geomechanics 8:71–96. Zienkiewicz, Chan, A.H.C., Pastor, M. & Shiomi, T. 1987. Computational approach to soil dynamics. Developments in Geotechnical Engineering 42: Soil Dynamics and Liquefaction, Elsevier: 3–17.

This paper describes a method implemented on the ABAQUS platform for solving Biot’s u-U formulation

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Alternative formulations for cyclic nonlinear elastic models: Parametric study and comparative analyses D. Taborda, L. Zdravkovic, S. Kontoe & D.M. Potts Department of Civil & Environmental Engineering, Imperial College London, UK

ABSTRACT: The utilisation of cyclic nonlinear elastic models presents numerous advantages when analysing simple dynamic problems, such as one-dimensional site response. This class of models requires a small number of parameters to be determined and can efficiently reproduce the strain dependency of the secant shear stiffness and hysteretic damping. However, it has been frequently noted that, for medium to large deformation levels, the predicted damping ratio can be significantly larger than the values evaluated for most soils through laboratory testing. To minimise the effects of this overestimation on the overall response of the system, which may lead to non conservative results, different formulations have been proposed. In this paper, two alternative expressions for the stress-strain behaviour of soils – hyperbolic and logarithmic – are presented and their ability to reproduce well-established empirical stiffness degradation and damping ratio curves is assessed. Finally, the results of a set of dynamic finite element analyses of a one-dimensional wave propagation problem are presented to illustrate the impact of the different formulations on the engineering behaviour of soil deposits.

1

2

INTRODUCTION

In earthquake engineering practice, a site response analysis is frequently necessary to assess how a soil deposit with a particular stratigraphy alters the bedrock seismic motion. As horizontal layering is often encountered, one-dimensional wave propagation is assumed when estimating the effects of soil amplification. Naturally, the accuracy of the predicted response greatly depends on the chosen method of analysis and, particularly, on its ability to reproduce the dynamic behaviour of the material. One of the possible approaches to study this type of problems is to perform a dynamic finite element analysis using a suitable constitutive model, such as those developed according to the cyclic nonlinear elastic framework (Finn et al. 1977, Taborda et al. 2007). In this paper, two alternative formulations of a model of this class, with clearly distinct levels of sophistication and flexibility, are presented. Particular emphasis is given to the evaluation of their ability to reproduce the most common aspects of dynamic soil behaviour, namely the secant shear stiffness degradation and damping ratio. Finally, the impact of the chosen stress-strain relation on the ed dynamic response of a soil deposit is explored through a series of one-dimensional analyses. The widely used equivalent-linear method (Bardet et al. 2000) is also employed in order to provide a basis for comparison and assessment of the nonlinear methods.

CYCLIC NONLINEAR ELASTIC MODELS

2.1 Basic concepts The cyclic nonlinear elastic models were proposed as a method of analysis for dynamic problems by Finn et al. (1977) and included three main components: the backbone curve, which determined the basic stress-strain relationship; the unloading and reloading rules defining how a change in shearing direction affected soil behaviour; and the hardening laws, which allowed the generation of excess pore water pressures during cyclic loading to be simulated. Note that the latter aspect of the model will be disregarded in this study, since it is mainly relevant for problems where the degradation of stiffness due to the reduction of effective stress is crucial, such as liquefaction. Analytically, a model of this class can be expressed by the following equation:

where τ, γ = current shear stress and strain, respectively; τ r , γ r = shear stress and strain at the last known reversal point, respectively; Fbb = chosen backbone function and n = a scaling factor. The latter variable, n, is used to define the unloading/reloading behaviour. As an example, the basic Masing rules (Kramer 1996) can be modelled by initialising n = 1, corresponding to Rule (A) – during initial loading, the stress-strain

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curve follows the backbone curve –, and changing its value to 2 when the first reversal is detected, which is equivalent to Rule (B) – when a strain reversal occurs, the curve retains the same shape as the backbone curve but is scaled up by a factor of 2. More complex laws, such as those defined in the extended set of Masing rules (Kramer 1996) require either the storage of multiple reversal points or the use of mechanical models (Iwan 1967). Although the one-dimensional form of Equation 1 is efficient to illustrate the main concepts of the cyclic nonlinear elastic framework, the implementation of a model in a general finite element code, such as ICFEP (Potts & Zdravkovic 1999), requires its components to be formulated in 3D stress-strain space. As proposed in Taborda et al. (2007), this can be achieved by rewriting Equation 1 in the following form:

where J and Ed = the second invariants of the stress and strain tensor, respectively. Further details on this aspect, including the procedure employed to detect the occurrence of shear reversals, can also be found in the same reference.

Figure 1. Dynamic behaviour reproduced by the hyperbolic degradation model for different values of parameter a.

2/π = 63.66%. This limit, which is related to the coupling of the chosen backbone function with the basic Masing rules, raises obvious concerns over the possible overestimation of the damping ratio, since much lower values are usually measured for soils (Guerreiro 2008). A possible solution to minimise this problem is to impose a minimum value for the tangent shear modulus, as discussed in Taborda et al. (2009). 2.3 Logarithmic function model

2.2

Hyperbolic degradation model

This version of the model employs a hyperbolic function as the backbone curve, as proposed by Kondner & Zelasko (1963):

where Gmax = initial stiffness and a = degradation parameter. Substituting the expression above in Equation (2) leads to:

The un-/reloading behaviour is defined by the basic Masing rules through the scaling factor n, as previously presented. It has been shown that this model accurately reproduces most secant shear stiffness degradation curves available in the literature (Guerreiro 2008) and has been successfully employed in the analysis of the Lotung seismic array (Taborda et al. 2009). In Figure 1, the results of a small parametric study illustrate the effect of the value of a on the modelled dynamic behaviour. It can be seen that, as a increases, the secant shear modulus degrades faster (i.e. for smaller deformation levels), while the damping ratio increases (i.e. the respective curve is displaced to the left). It is also evident that, independently of parameter a, the value of the damping ratio always tends to

This model, which was proposed by Puzrin & Shiran (2000), employs the following logarithmic function to define the backbone curve:

where JL = model parameter and α and R = auxiliary constants determined by:

with c a model parameter and xL calculated by:

where Ed,L is a third input material property. To obtain the stress-strain behaviour, Equation 5 can be substituted in Equation 2:

where

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Figure 2. Effect of parameter c on the stiffness degradation curve predicted by the logarithmic model.

Since an extensive description of the ability of this model to reproduce dynamic soil behaviour can be found in the original paper and in Lavda (2009), only the roles of the initial stiffness, Gmax , and of parameter c are going to be analysed in this text. Unlike the hyperbolic degradation model, the initial stiffness, Gmax , influences the shape of the stiffness degradation and damping ratio curves when employing the logarithmic function to describe the stress-strain behaviour of the material. Therefore, it is important to note that a given set of parameters is only valid for the value of Gmax used during the calibration procedure. However, by analysing the model’s equations, it becomes clear that, for the same behaviour to be obtained for a different value of Gmax , only the value of JL needs to be recalculated according to:

To investigate the role of parameter c, which defines the minimum value of the tangent stiffness, a small parametric study was performed. Values between c = 0.40 and 1.00 were chosen, while the remaining parameters were kept constant with the following magnitudes: Gmax = 100 MPa; Ed,L = 0.001 and JL = 20.0 kPa. The obtained stiffness degradation and damping ratio curves are illustrated in Figures 2 and 3, respectively. It can be seen that, as the value of c is reduced, the stiffness decreases for small deformation levels, while the opposite trend is ed for larger strain amplitudes. In of energy dissipation, the results in Figure 3 show that this parameter can be used to efficiently control the magnitude of the damping ratio, since ξ max decreases as the value of c is reduced. It should also be noted that ξ, for c < 1.0, suffers a sharp reduction as the deformation level increases. Therefore, a balance between limiting the magnitude of ξ max and severely underestimating the damping ratio at large strains must be sought. Often, this approach involves establishing a lower limit for c. Finally, Figure 3 also indicates that smaller values of c allow the damping ratio at lower strain levels to be increased. This is particularly important to avoid excessive amplification when analysing weaker ground motions.

Figure 3. Effect of parameter c on the damping ratio curve predicted by the logarithmic function model.

3

EQUIVALENT-LINEAR ANALYSIS

3.1 General aspects This method of analysis in the frequency domain is widely used in earthquake engineering practice, due to its reliability and to the vast experience accumulated during the last decades. The term “equivalentlinear” designates the central concept subjacent to this method: although the nonlinear behaviour of the material in of stiffness degradation and damping ratio is considered, the calculation procedure employs a constant value for these two properties for the complete duration of the excitation. Since these quantities are strain dependent, an iterative process is required to estimate the final (or “converged”) values of these two properties. More details on the exact algorithm can be found in Bardet et al. (2000). In the present case, an equivalent-linear analysis using EERA (Bardet et al. 2000) was performed to supply a term of comparison for the response determined by the truly nonlinear methods. Although the outcome of this calculation cannot be regarded as “exact”, it is widely accepted that this approach provides consistent and reasonably reliable results for simple one-dimensional wave propagation problems. 3.2 Geometry and input motion A 10 m-deep soil deposit of dry sand was subjected to the motion recorded in the EW direction by the instrument located at a depth of 47 m at the Lotung seismic array, Taiwan, during the LSST-7 event (20th May 1985). The time-history, with a total duration of about 35.50 s, is illustrated in Figure 4. 3.3 Material properties The modelled material was characterised by a unit weight of γ = 20 kN/m3 and a small-strain shear modulus of Gmax = 20 MPa, resulting in a value for the shear wave velocity, Vs , of about 99 m/s. The dynamic behaviour, defined by the stiffness degradation and damping ratio curves, was determined

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Table 1.

Material properties employed in the analyses.

Parameter

Hyperbolic

Logarithmic (stiffness)

Logarithmic (damping)

Gmax (MPa) ν a Ed,L JL (kPa) c

20 0.2 3398.0 – – –

20 0.2 – 8.02E-04 4.39 0.988

20 0.2 – 1.11E-02 12.23 0.744

Figure 4. Acceleration time history employed as input motion.

Figure 6. Stiffness degradation curves reproduced by the cyclic nonlinear elastic models.

Figure 5. Stiffness degradation and damping curves predicted by Darendeli (2001).

using the expressions proposed by Darendeli (2001) for p = 60 kPa (mean effective stress at the centre of the deposit assuming K0 = 0.4); PI = 0%; OCR = 1.0; f = 1 Hz and N = 10 (number of cycles). The resulting behaviour is illustrated in Figure 5. Note that the slight decrease in damping ed after its peak value is a consequence of the employed expressions and was disregarded in the equivalent-linear analysis by assuming a constant value of ξ = ξmax for this range of strain amplitudes (i.e γ > γ(ξ = ξ max )).

4 4.1

DYNAMIC FINITE ELEMENT ANALYSES General aspects

To highlight the impact of the different formulations of the cyclic nonlinear elastic model on the obtained ground response, the wave propagation problem described in the previous section was re-examined using the finite element method. Due to its onedimensional nature, plane strain analyses were performed using ICFEP on a 0.5 m × 10.0 m column, discretised into 20 eight-noded quadrilateral elements with a size of 0.5 m × 0.5 m each. In of boundary conditions the vertical movement was restricted

for the nodes placed on the lateral and bottom boundaries, since the only solicitation applied to the material was a vertically travelling shear wave. Additionally, the acceleration corresponding to the input motion (Fig. 4) was imposed at the nodes located on the bottom boundary. The initial stress state of the material was generated using γ = 20 kN/m3 and K0 = 0.4. The time-integration scheme employed in the dynamic analyses was the generalised-α method (Kontoe et al. 2008), with parameters δ = 0.60, α = 0.3052, αm = 0.35 and αf = 0.45 and constant time step of t = 0.01 s. 4.2 Material behaviour and properties The stiffness curve illustrated in Figure 5 was used to calibrate the hyperbolic degradation model using the least squares method. Subsequently, the same input data was supplied to a genetic algorithms-based optimisation software in order to evaluate the 3 parameters employed by the logarithmic function. Finally, since one of the advantages of this formulation is its increased flexibility, a third calibration procedure was performed using the same software to determine a set of parameters capable of reproducing, approximately, the reference damping ratio curve in Figure 5. The obtained values for the parameters required by the different expressions are presented in Table 1 while the modelled stiffness degradation and damping ratio curves are illustrated in Figures 6 and 7, respectively, together with those obtained using the expressions suggested by Darendeli (2001). It is evident that both

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Figure 7. Damping ratio curves reproduced by the cyclic nonlinear elastic models.

formulations can reproduce the reference stiffness values with a very high degree of accuracy. However, when the same parameters are used to determine the evolution of the damping ratio, it can be seen that both models highly overestimate this property for deformation levels above 0.03%. Conversely, the logarithmic function model, when calibrated using the damping ratio data, approximates accurately this curve, while underestimating the stiffness for strains below 0.1%. Furthermore, it has to be noted that the precise shape of the damping curve could not be obtained, resulting in slight overestimation of this property for deformation levels below 0.03% and underestimation above this value. 5

Figure 8. Damping values ed for the different analyses. Table 2.

Maximum surface acceleration values ed. Maximum acceleration (m/s2 )

Analysis

Positive

Negative

EERA Hyperbolic Logarithmic (stiffness) Logarithmic (damping)

1.05 0.51 0.55 0.96

−0.69 −0.42 −0.70 −0.86

RESULTS

5.1 Stress-strain behaviour To assess the impact of the chosen formulation and, for the logarithmic function, of the distinct values of the parameters, the stress-strain curves at a depth of 5 m were analysed. For the three studied cases, the stress-strain data was split into individual half-loops, which were subsequently integrated to estimate the equivalent secant shear stiffness and the corresponding damping ratio (Taborda et al. 2009). The results referring to the latter property are shown in Figure 8, together with the “converged” value of about 17.3% determined by EERA. It is interesting to note that the main difference between the three sets of results is not the maximum damping ratio (all the values are between 19% and 22%), but the strain at which it is ed, which is almost 10 times larger for the logarithmic function (damping) than for the remaining cases. 5.2 Ground response

Figure 9. Acceleration response spectra for 5% damping.

In of the resulting ground response, the two extreme values – one positive and one negative – of the acceleration ed at the surface are presented for each case in Table 2. It is evident that the two analyses where the employed cyclic nonlinear elastic

model was calibrated based only on the stiffness curve have yielded significantly lower values. Identical conclusion can be drawn from the acceleration response spectra for 5% damping (Fig. 9), where EERA and

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the logarithmic (damping) case exhibit much larger spectral accelerations for periods between 0.8 s and 2.0 s. Furthermore, the higher damping reproduced for the observed strain range by the hyperbolic model explains the weaker response ed at the surface, when compared to the logarithmic (stiffness) case. In fact, Figure 8 illustrated that the calculated strain amplitudes were limited to the range where the damping introduced by the hyperbolic model is larger than that predicted by the logarithmic function model (stiffness). 6

CONCLUSIONS

The cyclic nonlinear elastic models provide an efficient and simple tool to simulate wave propagation problems where the generation of pore water pressures is not relevant. However, their known inability to reproduce accurately and simultaneously the stiffness degradation and damping ratio curves observed in laboratory for most soils requires extreme caution when using this class of models in engineering practice. In this paper, the two alternative formulations of a cyclic nonlinear elastic model implemented in the finite element code ICFEP were presented and their modelling capabilities were explored in simple parametric studies. Finally, a one-dimensional wave propagation problem was simulated to study the impact of the chosen formulation on the ed ground response. In effect, the obtained results showed that the calibration of a given model relying entirely on the adequate reproduction of the stiffness degradation curve may lead to an overestimation of the damping ratio and, consequently, to the determination of weaker, nonconservative, ground responses. In order to overcome this problem, the superior flexibility of the logarithmic function model was explored and a more reasonable approximation of the reference damping ratio curve was obtained. The modified set of parameters resulted in a stronger ground response, very close to the one calculated using an equivalent-linear analysis. ACKNOWLEDGMENTS The authors would like to acknowledge the of FCT – Fundação para a Ciência e Tecnologia, Portugal, sponsor of the PhD programme of D. Taborda at

Imperial College London, UK. The Lotung strongmotion data used as the input motion in this paper was kindly supplied by the Institute of Earth Science, Taiwan. REFERENCES Bardet, J.P., Ichii, K. and Lin, C.H. 2000. EERA: a computer program for equivalent-linear analysis of layered soil deposits. University of Southern California. Darendeli, M.B. 2001. Development of a new family of normalized modulus reduction and material damping curves. PhD thesis. University of Texas, Austin. Finn, W.D.L, Lee, K.W. & Martin G.R. 1977. An effective stress model for liquefaction. Journal of the Geotechnical Engineering Division 103(6): 517–533. Guerreiro, P.G.H.M. 2008. Dynamic soil behaviour – test interpretation and numerical modelling. MSc thesis, Imperial College London. Iwan, W.D. 1967. On a class of models for the yielding behaviour of continuous composite systems. Journal of Applied Mechanics 34(3): 612–617. Kondner, R.L. & Zelasko, J.S. 1963. A hyperbolic stressstrain formulation for sands. Proc. of the 2nd Pan American Conference on Soil Mechanics and Foundation Engineering, Brazil, Vol. 1: 289–324. Kontoe, S., Zdravković, L. & Potts, D.M. 2008. An assessment of time integration schemes for dynamic geotechnical problems. Computers & Geotechnics 35(2): 253–264. Kramer, S. 1996. Geotechnical earthquake engineering. New Jersey: Prentice Hall. Lavda, A. 2009. Dynamic behaviour of gravelly soils – test interpretation and numerical modelling. MSc thesis, Imperial College London. Potts, D.M. & Zdravkoviæ, L. 1999. Finite element analysis in geotechnical engineering: theory. London: Thomas Telford. Puzrin, A.M. & Shiran, A. 2000. Effects of the constitutive relationship on seismic response of soils. Part I: Constitutive modeling of cyclic behavior of soils. Soil Dynamics and Earthquake Engineering 19(5): 305–318. Taborda, D., Kontoe, S., Zdravković, L. & Potts, D.M. 2009. Application of cyclic nonlinear elastic models to site response analysis. Proc. of the 1st Int. Symp. on Computational Geomechanics – COMGEO I, Juan-les-Pins, 29 April – 1 May 2009. Taborda, D., Zdravković, L., Kontoe, S. & Potts, D.M. 2007. The importance of cyclic nonlinear models in dynamic finite element analysis. Proc. of the 10th Intern. Symp. on Numerical Models in Geomechanics – NUMOG X, Rhodes, Greece.

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Analysis of the effect of pile length in a pile group on the transfer and impedance functions in soil-pile interaction models A. Mahboubi Power & Water University of Technology, Tehran, Iran

K. Panaghi Department of Civil Engineering, Power & Water University of Technology, Tehran, Iran

ABSTRACT: Numerous works is done to for deterministic parameters effects on the seismic soil-pile interaction phenomenon. To consider soil nonlinearity, one has to consider the whole system of soil-pile-structure in analyses with no effectual simplification. This approach, however, can lead to cumbersome calculations. Despite its effect on the degree of accuracy of the results, the soil-pile interaction problems can be considered with linear or slightly nonlinear behavior in soil. This methodology allows for the use of superposition law in the studies. Adopting this approach, this study considered the soil-pile interaction phenomenon as kinematic and inertial interactions. Pile groups of 2 × 2 with pile lengths of 16, 20 and 25 m were used in analyses. The piles had circular cross sections with the diameter of 1 m. Separation between pile and the surrounding soil was allowed in the models developed by the software ABAQUS. Finally the results are presented and discussed in detail.

1

INTRODUCTION

1.1 History The earlier approach in the past for structural analyses was to apply the seismic loads directly to the foundation of such systems. This idea was considered to be conservative since the soil beneath the structure was thought to lengthen the period of the seismic excitations on structures, hence reducing the amount of wave energy transferred from soil to the structural system. The experience from destructive earthquakes (Bhuj Earthquake of 2001, Chi-Chi Earthquake of 1999, Kobe Earthquake of 1995 and North Ridge Earthquake of 1994) showed that building codes were not mature enough to for this phenomenon. The period lengthening caused by this phenomenon could lead to maximum amplitudes happening later than predicted by response spectrums provided in some building provisions. Consequently more research was attracted towards soil-foundation-structure interaction issues and empirical and analytical methods as well as numerical simulations grew rapidly. More study showed that depending on site characteristics, wave propagation and structural system properties, soil-pile interaction may either magnify or decrease the ground motion. Although variety of the systems characteristics establishes a wide range of diversity in studies, this difficulty is sometimes overcome by normalization of parameters. To obtain a precise nonlinear analysis regarding soil-pile interaction problems, one has to consider the whole system of superstructure and substructure

together. However, this approach even in these days of highly developed computers can lead to cumbersome calculations and efficiency of such approach can be a matter of problem. To overcome the difficulty and since in dynamic soil-pile interaction analyses the displacements at the pile heads disappear with a great rate with depth, the slightly nonlinear behavior of the soil around the pile heads in flexible piles can be assumed. This allows for the use of superposition law which is applicable for linear problems. Adopting this methodology, the problem of soil-pile interaction can be divided into kinematic and inertial interactions which the data obtained from the former can be interpreted as input data for the latter case. The work presented here uses the same approach that deals with the effect of pile length on the functions used for analyzing this phenomenon. 1.2 Previous work Nogami & Konagai (1986, 1988) analyzed the dynamic response of pile foundations in the time domain using a Winkler approach. Fan et al. (1991) performed an extensive parametric study using an equivalent linear approach to develop dimensional graphs for pile head deflections versus the free-field response for various soil profiles subjected to vertically propagating harmonic waves. Makris & Gazetas (1992) applied free-field accelerations to a one dimensional beam on dynamic Winkler foundation model with frequency dependent springs and dashpots to analyze the response of single piles and pile groups. The

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results showed that interaction effects on kinematic loading are not significant at low frequencies but are significant for pile head loading which can be interpreted as inertial interaction. Their study was limited to equivalent linear elastic approach and one dimensional harmonic loading. Nogami et al. (1992) introduced material and geometrical nonlinearity in their analyses using discrete systems of mass, springs and dashpots. Bentley & El Naggar (2000) mentioned that elastic kinematic interaction for a single pile slightly amplifies the free-field transfer function. Cai et al. (2000) attempted to include plasticity and work-hardening of soil using a finite element technique in the time domain; however, they used fixed boundary conditions and neglected damping in the sub-system. Kim & Stewart (2003) performed an extensive empirical study of SSI effects using several strong motion data sets recorded at different building sites. They concluded that decrease in the natural frequency and increase in the damping associated with SSI might indeed affect the structural response. The response was influenced most by a parameter describing the relative stiffness between the foundation soil and the structure which is known as impedance factor. This parameter reflects dynamic characteristics of the soil, as well as the dynamic characteristics of the structure. Maheshwari et al. (2003) examined the effects of plasticity and work hardening of soil on the free field response of single piles and pile groups using the hierarchical single surface (HiSS) soil model. Maheshwari et al. (2004) extended their work to include the superstructure in order to evaluate the effects of SPSI for a fully coupled system. They conducted the analyses for both harmonic and transient excitations and compared both linear and nonlinear responses. They mentioned that the effects of nonlinearity on the responses are dependent on the frequency of excitation with nonlinearity causing an increase in response at low frequencies of excitation. The work presented in this paper is an attempt to evaluate the effects of change in piles length on the kinematic and inertial interaction of soil-pile systems which are assessed by transfer and impedance functions, respectively. The effect of foundation on the earthquake ground motion is termed kinematic interaction and the effect of foundation compliance on structural response plus the effects of inertial loads on the foundation is referred to as inertial interaction. The frequency dependent transfer functions are defined by the ratio of the foundation motion to the free field motion in the absence of structure. The flexibility of the foundation and the damping associated with foundation-soil interaction is described by a frequency dependent foundation impedance function which is defined by means of the stiffness and damping of the system and as research develops, more data are available that consider different aspects of the issue. 2 THE NUMERICAL SIMULATION Three types of 2 × 2 pile groups with different pile lengths were used to study the effect of pile length on

the results. The circular pile lengths were considered to be 16, 20 and 25 m. The pile diameters in all analyses were 1m and therefore the length to pile diameter ratios were 16, 20 and 25, respectively. The space to diameter ratio in the models was 2.5. The pile heads had a thickness of 1.2 m and were embedded in the soil with embedment depth of 0.6 m. The concrete considered for the pile group material type had elastic behavior and no damping was assigned to it. For the kinematic interaction study, pile groups were simulated with no structure on top and pile cap displacements acquired by the seismic loading were considered to calculate transfer functions. The secondary simulations had structures on top of the pile caps. The load initiated by the structure was 80 tons and for the purpose of simplicity in modeling, the existence of structure was simulated by change in the material density of the pile caps so that the structure’s weight could be ed for. The soil was considered to be of very dense type around the pile caps and for the depth of further than 1.2 m, dense sand was assumed. The two layer profile of the soil was adopted to ensure the pile heads fixed behavior. That is, the pile cap was restricted by the dense soil media in a way that partially fixed behavior was ensured. This method holds the advantage of consideration of the in-between interaction of piles which is ignored in completely fixed pile cap simulations. The soil extent was 30 m in every dimension and Drucker-Prager behavioral law was used as its failure criterion. Although soil behavioral laws with more reliability could be employed, the simplicity of the used constitutive law provided time efficient computations viable. In the software, a parameter named flow stress ratio can represent the degree of anisotropy in the soil which is dependent on the soil cohesion as well. The parameter is defined as the ratio of the triaxial tensile strength to the corresponding magnitude in compressive state. The magnitude for both soils in the calculations was considered as 0.8. Since the cohesionless soil(s) were considered in all analyses, the parameter in this study was not dependent on cohesion. The stiffness and mass proportional dampings of the soil(s) were modeled by Rayleigh type damping. In addition, material damping in concrete piles was not considered in analyses. Table 1 summarizes the soil(s), piles and structure properties in more detail.

3

BOUNDARY CONDITIONS

Silent boundary conditions at models sides were simulated by use of infinite elements. These elements were applied to the side boundaries to ensure completely transmitting boundaries. Since wave energy reflection can cause box effect in the models and hence getting false results, use of such elements seemed to be beneficial. On the other hand, since in practice wave energy is partially reflected and to have a better simulation, the boundaries at the bottom of the models were considered to be reflecting. Figure 1 shows the 25 m piles group developed for the analyses.

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Table 1. Material properties considered for model parts in all analyses. Model part

Material type

Material property

Top soil

Very dense sand

Bottom soil

Dense sand

E = 130 MPa ν = 0.35 γ = 21 kN/m3 φ = 36◦ = 6◦ E = 80 MPa ν = 0.35 γ = 19 kN/m3 φ = 35◦ = 5◦

Pile groups

Concrete

Structure

Fictional

E = 21 GPa ν = 0.20 γ = 24 kN/m3 W = 800 kN

Figure 2. The Acceleration-Time history used in the kinematic interaction study.

resultant response of the system at the pile caps was considered. The displacements of the pile caps plus free-field displacement data by which transfer functions for the kinematic interaction study could be calculated were converted to the corresponding magnitudes in the frequency domain. This was done by fast Fourier transform and hence, the analysis was conducted in the frequency domain. For the inertial interaction study, however, the harmonic loading with the magnitude of 50 ton and frequencies of excitation of 5, 7.5, 10, 12.5 and 15 Hz was applied to the pile caps in every model. The loading duration for the harmonic excitations was adjusted to have at least 50 cycles of loading in a way that steady state of the system response could be ensured. The results regarding changes of magnitude and direction of force and the corresponding displacements of pile caps with time were extracted to be used for the inertial interaction study. 5

Figure 1. The finite element model of soil-pile system.

The infinite elements outward directions can be seen at the sides of the model.

4

KINEMATIC INTERACTION

The input earthquake motion forces the structure and the soil in free-field to oscillate. Kinematic interaction in soil-structure systems is studied by means of frequency dependent transfer functions. The magnitude of transfer function is defined as the ratio of the maximum pile head displacement to the corresponding value of free-field. This function can be arithmetically shown as:

LOADING

The types of loading used in analyses regarding kinematic and inertial interactions were different. For the kinematic interaction study, the acceleration-time history of a site located in the Fars province in Iran was chosen (Figure 2). The duration of the event was 43.525s and for computational efficiency, only the duration of 5 to 15s was used in which the most noticeable amplitudes of the excitation occurred. This type of loading was initiated at the bedrock and the

Where Up (f ) and Uff (f ) are the maximum lateral response at the pile top and the free field for a given excitation frequency, respectively. Transfer functions for three sets of pile groups were calculated and the results are shown in Figure 3. It can be seen that in 16 m pile length case, the magnitude of transfer functions is generally higher than the other two cases. This result may be attributed to the fixity of pile heads and the

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Figure 3. Transfer function-Frequency diagrams obtained in the kinematic interaction study.

fact that loading was initiated at the bottom of models. The flexible behavior of longer piles makes the displacement of the pile groups be distributed along the piles and therefore, less pile cap displacement could occur. This trend changes as the frequency of excitation increases which then motion of the system reaches to a more steady state. As the frequency of excitation increases and gradually matches the system’s natural frequency, the pile cap displacements increase and the magnitude of transfer functions almost become the same as the others. The difference in the transfer function magnitudes is higher at low frequencies and as the frequency increases, it decreases. This implies that the effect of pile length in the transfer function magnitudes in this study is more noticeable at low frequencies when the excitation frequency magnitude is notably different from the system’s natural frequency. 6

INERTIAL INTERACTION

Inertial interaction was studied by means of impedance functions. The impedance function can be defined by virtue of the stiffness and damping of a system. In the frequency domain, this magnitude is represented by a complex number which is a function of the driving frequency, magnitude of the driving force as well as properties of the soil-pile system. The impedance functions in the present study were calculated using the following equations:

In which P0 is the amplitude of the exciting lateral force, U0 is the lateral peak amplitude response at the top of the pile at the steady state, tl is the time lag between the driving force and the pile top response and f is the driving frequency. The magnitudes were normalized by static state stiffness of the soil-pile group systems. The damping

Figure 4. Real parts of the normalized impedance functions versus frequency in the inertial interaction study.

Figure 5. Imaginary parts of the normalized impedance functions versus frequency in the inertial interaction study.

calculated in the study includes internal and radiation damping. Since this study focused on the interaction between two bodies with different stiffness’s; the radiation damping had to be considered. This type of damping occurs between two bodies with one being stiffer than the other. The seismic excitation forces the body of the pile foundation to oscillate in frequencies much higher than the soil around it. This difference in motion tends to force the soil to oscillate at the frequencies the same as the concrete piles. Due to this, a part of wave energy transmitted from the piles to the soil in the vicinity of them is dissipated. Since radiation damping is dependent on the surfaces in and to have a more accurate simulation, separation between the pile bodies and the surrounding soil was considered in modeling. Figures 4 and 5 show the results obtained for the inertial interaction study. The changes of real and imaginary parts of the impedance functions with frequency are depicted and it can be seen that the behavior of all systems are almost the same at high frequencies. However, at low frequencies, the real part of the normalized impedance function for the 16 m pile length seems to be more affected by the less flexible behavior of it compared to the two other pile groups.

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This is because the magnitude of the real part of the impedance function which represents the dynamic stiffness of the system regarding the 16 m pile length group is higher than the others. The trend in the diagrams regarding the imaginary parts of the normalized impedance functions is different, however. The imaginary parts for the 16 m pile length seem to be less than the two other sets. This is consistent with the results obtained for the kinematic interaction study, where the pile cap for the shortest piles had more displacements and hence higher transfer functions. The higher displacement at the cap leads to more separation between the pile bodies and the soil which causes a decrease in radiation damping. Due to this, the overall damping magnitude decreases and therefore the imaginary part of the impedance function is lessened.

the magnitude of damping. This is because radiation damping is dependent on the surfaces adjacent to each other and as this parameter increases, the magnitude of damping changes with the same trend. Besides the separation effect on the damping, the pile head displacements which seemed to be higher in shorter piles cause the soil around the caps to show more plastic behavior. The increase in plastic behavior of the soil causes damping to increase. However, since the plastic strain rate for this study was limited (Rayleigh damping was used), the effect of frequency of excitation is more highlighted compared to high strain levels. Because of this, the higher strain rate in shorter piles could not be the mere parameter influencing the damping magnitude in the system. REFERENCES

7

CONCLUSION

Three sets of 2 × 2 pile groups with different pile lengths for dynamic soil-pile interaction study were considered. The following conclusion could be drawn: Although the system lower stiffness can be a factor in obtaining higher transfer functions in free-head piles, this parameter does not necessarily affect the system response the same way for fixed head piles. One parameter that should be considered is the location of the load applied to the system. For transfer function calculation in this study, seismic excitation was applied to the bedrock and the more flexible behavior of longer piles led to the distribution of displacement along the piles, resulting less pile head displacements. This idea was also confirmed by the results obtained for the impedance function calculations. The harmonic loads applied to the pile caps in the inertial interaction study caused higher magnitudes in shorter piles for the real part of the impedance functions. This can be due to the less flexible behavior of such systems which leads to higher dynamic stiffness during seismic events. The imaginary parts for the normalized impedance functions presented here seem to be lesser for the shorter piles. The reason can be attributed to the higher stiffness of such systems which although in free-head piles lead to increase in damping represented by them, cause more pile head displacements as it was mentioned before. The increase in pile head displacement leads to the occurrence of more separation and as a result of that, radiation damping reduces. Besides the separation parameter, one has to consider the overall surfaces of the piles bodies and soil which can affect

Bentley, K. J. and El Naggar, M. H., 2000, Numerical analysis of kinematic response of single piles, Canadian Geotechnical Journal, 37, pp. 1368–1382. Cai, Y. X., Gould, P. L. and Desai, C. S., 2000, Nonlinear analysis of 3D seismic interaction of soil-pile-structure system and application, Engineering Structures, 22, pp. 191–199. Fan, K., Gazetas, G., Kaynia, A. M., Kausel, E. and Ahmad, S., 1991, Kinematic seismic response of single piles and pile groups, Journal of Geotechnical Engineering, ASCE, 117 (12), pp. 1860–1879. Maheshwari, B. K.,Truman, K. Z., Gould, P. L. and El Naggar, M. H., 2003,Three dimensional nonlinear seismic analysis of single piles using FEM: effects of plasticity of soil, International Journal of Geomechanics, ASCE. Maheshwari, B. K., Truman, K. Z., El Naggar, M. H. and Gould, P. L., 2004, Three dimensional finite element nonlinear dynamic analysis of pile groups for lateral transient and seismic excitations, Canadian Geotechnical Journal, 41, pp. 118–133. Kim, S. and Stewart, J. P., 2003, Kinematic soil-structure interaction from strong motion recordings, Journal of Geotechnical and geoenvironmental Engineering, ASCE, 129 (4), pp. 323–335. Makris, N. and Gazetas, G., 1992, Dynamic pile-soil-pile interaction, Part II: Lateral and seismic response, Earthquake Engineering and Structural Dynamics, 21, pp. 145–162. Nogami, T. and Konagai, K., 1986, Time domain axial response of dynamically loaded single piles, Journal of Engineering Mechanics, ASCE, 112 (11), pp. 1241–1252. Nogami, T. and Konagai, K., 1988, Time domain flexural response of dynamically loaded single piles, Journal of Engineering Mechanics, ASCE, 114 (9), pp. 1512–1525. Nogami, T., Otani, J., Konagai, K. and Chen, H. L., 1992, Nonlinear soil-pile interaction model for dynamic lateral motion, Journal of Geotechical Engineering, ASCE, 118 (1), pp. 89–106.

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Dynamic fragmentation in rock avalanches: A numerical model of micromechanical behaviour K.L. Rait & E.T. Bowman Department of Civil and Natural Resources Engineering, University of Canterbury, Christchurch, New Zealand

ABSTRACT: It is hypothesized that the dynamic fragmentation of rock clasts is a key mechanism to the long run out of rock avalanches. That is, large rock avalanches tend to travel further than expected in comparison with scaled-up grain flows and this so-called “size effect” is a function of the avalanche size. It is thought that the overburden pressure due to the avalanche size in a large rock avalanche promotes fragmentation of rock clasts and therefore influences run out. The discrete element method is used to investigate this micromechanical behaviour by placing a single brittle fragmenting cluster of particles within a group of non-fragmenting clusters. Using PFC3D this system is placed under a high strain rate to determine the effect of fragmentation on the behaviour of near particles. This paper shows that the overburden strain-rate is directly related to the fragmentation process and that so long as load is applied sufficiently quickly, particles will dynamically fragment rather than simply split or crush; the associated fragments will possess a kinetic energy and therefore collide with near particles. This force of collision can influence the behaviour of the near particles, and under dynamic fragmentation, kinetic energy is dispersed through the system as the near particles fragment. It is postulated that this energy movement produces an isotropic dispersive stress that could explain the long run-out of rock avalanches via the decrease in effective stress within the system. 1 1.1

INTRODUCTION Rock avalanches

Sturzstrom or giant rock avalanches are known to behave significantly differently to other avalanche and landslide phenomena. For mountainous areas around the world these catastrophic hazards are a threat to populations and their lifelines. A pre-fractured ground mass and a tectonically active region in a steep mountainous area are recognised as an initial condition for many events (Friedmann et al., 2003).A sturzstrom generally begins as a rock fall or rock slide and changes into a dynamically disintegrating rock mass that appears to behave as a granular flow. It can also entrain and/or deposit material as it falls. Sturzstrom are understood to travel upwards of 30 times in horizontal distance compared to the initial fall height and the momentum of the flow may cause the debris to surge upward especially in valleys where there are confining walls (e.g. Elm and Falling Mountain events, Hsu 1978, Davies and McSaveney 2002). These flows travel for 30–100 seconds and characteristically stop suddenly (Hsu 1975, 1978). The deposit from a sturzstrom can cover tens of square kilometres and can be only a few metres thick at the distal regions which are often deposited as a levee (Friedman et al., 2006). Typically a sturzstrom deposit shows a preserved stratigraphy and inverse grading – silt is present at the base and as a matrix material, with angular blocks ed within the matrix and large

boulders sitting on the top. The inverse grading of sturzstrom deposits is perhaps partly explained by the higher probability of fine particles or powder filling small voids as the sturzstrom flows – a sieving process (Friedmann et al., 2006). It may also be explained through greater crushing of rock deeper in the deposit, as discussed later. Many mechanical theories to describe the behaviour of sturzstrom have been advanced by researchers over the last century. The first of these theories involved mechanical fluidisation which was suggested by Albert Heim in 1882 after his investigation of the famous Elm sturzstrom (Hsu, 1978). This was later ed to some degree by Hsu (1975) who suggested that sturzstrom are likely to follow Bagnoldian (i.e. collisional) grain flow behaviour. Further theories advanced since 1882 include air cushioning (Kent, 1966), air fluidization (Shreve, 1968), the development of frictionite (Erismann, 1979) and acoustic fluidization (Melosh, 1983). Each of these theories appear relevant to specific sturzstrom events, however they do not appear able to fully explain the behaviours and deposits that are common to all sturzstrom. In particular, these theories are collectively unable to explain the flow of the rock debris as a dry granular mass, the angularity of the blocks and fine silt found in the deposits, and the inverse grading. McSaveney and co-authors have suggested that fragmentation of rock during a sturzstrom is responsible for producing the very fractured and angular rubble

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that commonly occurs in the deposit area (McSaveney 1978; Davies and McSaveney, 2002; Davies et al., 1999). Fragmentation according to Davies et al. (1999), occurs in a sturzstrom as the overburden stress, or pressure from the high volume of material from these events, exceeds the strength of unted rock. Therefore the rock material at the base of a sturzstrom is more likely to undergo fragmentation. The fragmentation of a rock in this situation is thought to result in an isotropic dispersive stress that dilates the debris – an effect that is additional to mechanical fluidisation for long run-out in sturzstrom. Davies et al. (1999) state that the process responsible for fragmentation is more likely to involve a crushing or grinding behaviour than grain collisions thus producing substantial silt. As rock clasts undergo fragmentation at the base of a sturzstrom, larger material will be found at the top of the flow, as it is under less pressure, with the finer material at the base – thus inverse grading. We suggest that the fragmentation of multiple grains occurring under this high overburden stress produces high velocity fragments of rock that impulsively load surrounding grains. As the fragmentation becomes more violent, these impacts produce an internal isotropic pressure that causes the fine rock material to act like a pressurised fluid, reducing effective stress and therefore reducing friction (i.e. following Terzaghi’s theory of effective stress and Coulomb-type friction) (Terzaghi and Peck, 1967). The reduction in friction allows the granular material to flow rapidly across terrain and produce long run out. 1.2

Discrete Element Modelling (DEM)

The Discrete Element Method utilizes discrete particles that only interact at points (Cundall and Strack, 1979). Calculations of forces and for the displacements of particles alternate between Newton’s second law and a force-displacement law. This is the basis for the software Particle Flow Code in Three Dimensions (PFC3D ) built by Itasca Consulting Group. PFC3D utilises spherical discrete particles that can be arranged and bonded as agglomerates to represent angular and breakable rock or grains, or retained as individuals to represent a granular medium such as sand at low stress. The PFC3D numerical code can show the effect of the applied macroscopic stress and strain on the micro mechanisms within the medium being tested, and give numerical and graphical results relating to particle-level behaviour, such as the number of bonds broken, kinetic energy of the system or individual particles and so on. Using the DEM method, numerical agglomerates can be impacted against a wall at high velocity or crushed between two platens to investigate comminution in a particle of sand or piece of rock (Thornton et al., 1996; McDowell and Harireche 2002). The strength of the agglomerate is represented by bonding the particles at their points. As these bonds break the agglomerate is said to fracture, and fragments occur once groups of bonded particles move away from the original agglomerate being tested.

At sufficiently high impact, there are generally no large fragments remaining due to the extensive shattering that occurs (Kafui and Thornton 2000). Dense agglomerates during high velocity impact will fracture or shatter, whereas loose or highly porous agglomerates simply disintegrate (Mishra and Thornton 2001). Thornton et al., (1996) found through their DEM experiments that a compressive wave propagates from the point of through the agglomerate when it impacts a wall at high velocity. If the rate of loading an agglomerate is rapid in comparison to crack growth many flaws will be activated to accommodate the unloading of the stress (Thornton et al., 1996), a conclusion ed by the earlier experimental and analytical work of Grady (1981) and Lundberg (1976). There are several methods of packing spherical particles to represent a grain or rock clast. The most strongly ed type of packing for crystalline structures is hexagonal close packing (H) where every other layer is the same. Random percentages of particles can then be removed to represent flaws and create the correct porosity required to simulate the targeted material. The removal of particles in this random manner from agglomerates produces a Weibull distribution of strengths (McDowell and Harireche 2002). Under slow isotropic compression agglomerates will undergo deformation through the bonds initially being broken by shear. At a critical point unstable fracture begins and the bonds are then broken rapidly through tensile stresses (Bolton, et al., 2008). In agglomerates there are critical bonds that once broken lead to the splitting or disintegration of the agglomerates (Cheng et al., 2003). Under triaxial compression, the frequency of breakage was found to increase as confining pressure increased with around 15% of energy dissipated through bond breakage (Bolton et al., 2008). This suggests that high overburden pressure may cause multiple fragmentation events which in turn could provide a mechanism towards the high mobility of sturzstrom. In this work we attempt to represent the behaviour that might be seen within a microscopic area of the basal region of a sturzstrom during the fragmentation of rock clasts. We utilize PFC3D to model the resultant influence a fragmenting clast has on surrounding clasts and compare this to the behaviour seen by Cheng et al (2003) and Robertson (2000) in single agglomerate testing. 2

SINGLE AGGLOMERATE TEST

2.1 Agglomerate creation H agglomerates built from discrete spherical particles in PFC3D are created following the method outlined by Robertson (2000) and Cheng et al (2003). In Table 1 are listed the properties assigned to the discrete particles. An agglomerate is built from a maximum of 1150 particles and brought together into an H form. This minimizes the space between spheres and eliminates

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Table 1.

Numerical model details of agglomerate.

Parameter Agglomerate radius Particle radius Particle density Normal and shear bond strength Particle normal and shear stiffness Particle friction coefficient Percentage removed for flaws

Numerical Dimensions Value m m kgm−3 N

0.25 0.002 2650 4 and 4 × 103

Nm−1

4 × 106 0.5 20%

overlapping which can cause locked-in forces to occur (Cheng et al., 2003). Statistical variability of strength and shape is introduced by randomly removing particles. In these tests 20% of the particles were removed to reduce the regularity of the packing and introduce flaws (Cheng et al., 2003). Following Robertson (2000) only bonds were used. The high coordination number from the use of bonds in an H packed agglomerate allows the generation of resistance to moment without mechanical complexity. The use of bonds only is also ed by the ‘Block Caving’ example in the Itasca manuals (Itasca, 2008) which states that bond values control fragmentation. The agglomerate is then brought to equilibrium such that the mean unbalanced forces acting on the agglomerate are virtually zero. Single ball drop experiments similar to those in the Itasca Verification Problems manual (Itasca, 2008) were performed to obtain a critical damping factor of 0.2 which corresponds to a coefficient of restitution of approximately 0.55. Platens (represented by smooth and stiff walls in PFC3D ) are placed at the base and top of the agglomerate and testing is undertaken by altering the speed of descent of the top platen to reach 40% strain. The intention behind this method is to represent the degree to which rock avalanche material may be crushed via the overburden pressure due to the velocity of loading as the sturzstrom moves at high speed over a changing terrain.

2.2

Results of single agglomerate testing

Bond strengths of 4 N and 4 kN were tested to determine what differences may occur with weaker or stronger materials. A randomly chosen agglomerate was crushed under varying platen speeds with only the bond strength changed between tests. Once the prescribed strain was met gravity was turned off so that there was no increase in kinetic energy from the relaxing of grains under the effects of gravity. From Figure 1A it can be seen that at high strain rates “inertia-induced dynamic impact effects” have an effect on crushing behaviour as also found by Cheng et al (2003). This dynamic regime is suggested as being of the dominant influence on the run out of sturzstrom.

Figure 1. A – Peak applied stress by strain rate for bond strengths; B – Power law relationship for bond strengths.

Both the 4 N and 4 kN data collapse to power law relationships (see Figure 1B) as also reported by Grady (2008) in highly brittle material. The power law relationship for the stronger material appears to begin under a greater strain rate compared to that for the weaker material. Curiously the data from both bond strengths converge to a similar peak stress at high strain. We find that at low strain rates, after the peak stress is reached and the agglomerate is fractured, the platens continue to approach one another until another is found on the agglomerate where another fracture may occur. This behaviour was also reported by Cheng et al (2003). Under high strain rates the material in with the load applying platen is crushed quickly into individual particles or fines, with the base of the agglomerate showing multiple fractures. Once the full strain is reached and gravity removed, the base of the agglomerate separates quickly into small fragments and all material disperses quickly. The stronger material was chosen as the most realistic representation of rock, given that tests on this stronger agglomerate revealed a more brittle nature similar to that seen in real rock. The peak kinetic energy produced from crushing a randomly oriented and flawed agglomerate with 4 kN bond strengths at

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Figure 4. The agglomerate system.

3 AGGLOMERATE SYSTEM TESTING 3.1 Agglomerate system creation Figure 2. Peak kinetic energy by approximate real time for bond strength of 4 kN by strain rate.

Figure 3. Kinetic energy histories of a 4 kN bond strength H agglomerate model.

varying strain rates is plotted against approximate real time in Figure 2. The peak kinetic energies indicate a logarithmic relationship. This suggests fairly sensibly that a low strain rate will take longer to cause fracturing and therefore peak kinetic energy in this synthetic rock sample. Conversely high strain rates appear to obtain peak kinetic energy almost instantaneously suggesting that explosive fragmentation may occur as discussed by Grady (1981). The kinetic energy history of four of the strain rates tested with the 4 kN model is shown in Figure 3. The residual kinetic energy declines in all cases however remains the highest where the original agglomerate was crushed with high strain rate. We suggest that a very high strain rate, arising from a voluminous load progressing at high speed over a rough terrain, can cause high values of residual internal kinetic energy leading to high numbers of particle impacts within the sturzstrom body. This leads to a high internal pressure, analogous to a pore pressure, and an associated reduction in effective stress which reduces the resistance to sturzstrom motion.

The stronger synthetic material was also chosen for the agglomerate system testing. Due to hardware limitations however, the particle radius size was increased to 0.09 m in order to reduce the total number of particles in the system. At a maximum of 69 particles and 282 bonds per agglomerate with 27 agglomerates overall in the system, the maximum total number of particles was 1863 with a maximum of 7614 bonds of which a maximum of 282 can break. Pre-determined central coordinates were chosen for the 27 agglomerates.At each set of coordinates an H agglomerate was created following Robertson (2000), randomly rotated and flawed by removing 20% of the particles. A cubical arrangement of walls set at low velocity was then used in order to bring the agglomerates into close (see Figure 4). All bonds were set to extremely high levels initially so that the agglomerates would not unduly break during this compression and subsequent cycling to equilibrium under gravity.The close of the agglomerates was used to represent the assumed close of rock clasts at the base of a sturzstrom. Once brought to equilibrium, a top platen was placed above the system and this platen set to descend at varying velocities to compress the system to 40% strain. For these tests, all other confining walls were kept stationary in order to test the system behaviour under one dimensional loading at varying strain rates. Once 40% strain had been achieved in these tests, as with the single agglomerate tests, gravity was turned off so as to remove any effects on the kinetic energy of the system relaxing. The stress on each wall and kinetic energy of the system was logged and analysed. Stress is calculated as the ratio of the force on the wall (or platen) by the crosssectional area of agglomerates that are in – 9πr2 (9 agglomerates in with each platen). The peak applied stress is then defined as the maximum stress that occurs on the top platen as it descends. 3.2 Results of agglomerate system testing At high strain rate the central agglomerate quickly separates into individual particles or fines – all bonds are broken. As the system relaxes the fine particles move throughout the system settling into the spaces between

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Figure 5. Resulting graphic of the central agglomerate behaviour at 40% strain after compression under high strain rate (A) and low strain rate (B) of the agglomerate system.

Figure 7. Kinetic energy of agglomerate system.

Figure 6. Peak kinetic energy by strain rate versus time.

the surrounding agglomerates. These fines move both horizontally and vertically as the surrounding agglomerates push into the space left void from the dynamic breakage of the central agglomerate (see Figure 5A). In comparison, at low strain rate, the central agglomerate slowly fractures into a few fragments surrounded by fines (around 90% of the bonds are broken at 40% strain). As the system relaxes, the fines drift to the base of the system and the fragments are held in the central void of the system by the surrounding agglomerates (see Figure 5B). As seen in the single agglomerate testing and in the work completed by Cheng et al (2003), the value of the peak applied stress grows exponentially as strain rate increases. As discussed in the single agglomerate testing above this collapses to a power law relationship where the power law begins at a higher value of strain rate due to the use of the stronger material in the breakable agglomerate. Similarly the peak kinetic energy follows the same logarithmic trend indicated in the single agglomerate testing. The highest peak kinetic energy occurs almost instantaneously at high strain rate and the lowest peak kinetic energy occurs after approximately 6.5 seconds for the lowest strain rate of 10% (see Figure 6). From Figure 7 it can be seen that the higher strain rates produce higher kinetic energy. Over time all kinetic energy histories dissipate towards zero after gravity is removed and the particles and agglomerates settle. These tests on a system of agglomerates show the kinetic energy from a higher strain rate dissipating faster than that of a low strain rate. This is likely to be due to a higher frequency of impacts with adjacent

Figure 8. Average horizontal wall stress.

particles, leading to faster energy dissipation in the particle system overall. Compare this graph with Figure 8 of the average wall stress in the horizontal directions. The high strain rate case shown here retains a much greater proportion of wall stress after the particles have settled than is seen for the low strain rate. Although the high strain rate may lose kinetic energy at a faster rate after a fragmentation event, the high average stress on the horizontal walls (in comparison to the test at low strain rate) suggests that there is substantial internal pressure applied to the surrounding agglomerates from both the strain and impulsive loading of the fragmented grains from the central agglomerate.

4

DISCUSSION

The complex nature of sturzstrom – for example their temporal unpredictability, high speed, large volume of material and short duration – means that it is virtually impossible to study these phenomena in situ. This paper presents a study utilizing discrete numerical modelling in PFC3D to investigate the micromechanical behaviour postulated to occur during a sturzstrom which may influence both run out and deposit type. From the results presented above it is clear that modelling in PFC3D is capable of providing useful information about the likely internal mechanisms involved in sturzstrom.

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From the results shown of the single agglomerate and multi agglomerate system tests it is found that a high strain rate modelled by the high velocity of a top loading platen produces high peak kinetic energy levels in the material tested. This high strain rate also represents the fast application of a load which results in explosive fragmentation which separates the material into fines. At low strain rate (or low platen velocity), the material separates into fragments with fines drifting toward the base platen. Similar results have been reported by Mishra and Thornton (2001) when impacting single agglomerates at varying velocities into a stiff platen. The agglomerate system results indicate that pressure increases on surrounding agglomerates when a neighbouring agglomerate explosively fragments. The fines from a fragmented particle appear to travel throughout the available spaces between other agglomerates with the resultant void being filled by these other agglomerates.This behaviour suggests that towards the base of a sturzstrom, rock clasts may fragment explosively causing fine rock powder and dilation of the flow and as the effective stress of the whole system is reduced, produce long run out. 5

FURTHER WORK

In future work to further investigate the micromechanical behaviour of sturzstrom, it is intended to continue discrete numerical testing examining the impact of an explosively fragmenting agglomerate on its neighbours where the neighbouring agglomerates are also able to fragment. This work will utilise a larger particle system and more highly powered computational hardware. ACKNOWLEDGEMENTS Funding for this work has been provided by the Department of Civil and Natural Resources Engineering at the University of Canterbury. REFERENCES Bolton, M. D., Nakata, Y., Cheng, Y. P. (2008). Micro- and macro-mechanical behaviour of DEM crushable materials. Geotechnique 58(6): 471–480. Cheng, Y. P., Bolton, M. D., Nakata, Y. (2003). Crushing and plastic deformation of soils simulated using DEM. Geotechnique 54(2): 131–141. Cundall, P. A. and Strack O. D. L. (1979). Discrete numerical model for granular assemblies. Geotechnique 29(1): 47–65. Davies, T. R., McSaveney, M. J., Hodgson, K. A. (1999). A fragmentation-spreading model for long-runout rock avalanches. Canadian Geotechnical Journal 36(6): 1096–1110.

Davies, T. R. and McSaveney M. J. (2002). Dynamic simulation of the motion of fragmenting rock avalanches. Canadian Geotechnical Journal 39(4): 789–798. Erismann, T. H. (1979). Mechanisms of large landslides. Rock Mechanics 12: 15–46. Friedmann, S. J., Kwon, G., Losert, W. (2003). Granular memory and its effect on the triggering and distribution of rock avalanche events. Journal of Geophysical Research-Solid Earth 108(B8). Friedmann, S. J., Taberlet, N., Losert, W. (2006). Rockavalanche dynamics: insights from granular physics experiments. International Journal of Earth Sciences 95(5): 911–919. Grady, D. E. (1981). Fragmentation of solids under impulsive stress loading. Journal of Geophysical Research 86(NB2): 1047–1054. Grady, D. E. (2008). Fragment size distributions from the dynamic fragmentation of brittle solids. 10th Hypervelocity Impact Symposium (HVIS 2007), Williamsburg, VA. Hsu, K. J. (1975). Catastrophic debris streams (sturzstroms) generated by rockfalls. Geological Society of America Bulletin 86(1): 129–140. Hsu, K. J. (1978). Albert Heim: Observations on Landslides and Relevance to Modern Interpretations. Rockslides and Avalanches I. B. Voight (ed). Itasca (2008). Particle flow code in three dimensions, Itasca Consulting Group Inc. Kafui, K. D. and Thornton C. (2000). Numerical simulations of impact breakage of a spherical crystalline agglomerate. Powder Technology 109(1–3): 113–132. Kent, P. E. (1966). Transport mechanism in catastrophic rock falls. Journal of Geology 74(1): 79–83. Lundberg, B. (1976). Split Hopkinson bar study of energy absorption in dynamic rock fragmentation. International Journal of Rock Mechanics and Mining Sciences 13(6): 187–197. McDowell, G. R. and Harireche O. (2002). Discrete element modelling of soil particle fracture. Geotechnique 52(2): 131–135. McSaveney, M. J. (1978). Sherman Glacier Rock Avalanche, Alaska, U.S.A. Rockslides and Avalanches I. B. Voight (ed). Melosh, H. J. (1983). Acoustic fluidization. American Scientist 71(2): 158–165. Mishra, B. K. and Thornton C. (2001). Impact breakage of particle agglomerates. International Journal of Mineral Processing 61(4): 225–239. Robertson, D. (2000). Computer simulations of crushable aggregates. PhD Dissertation. University of Cambridge. Shreve, R. L. (1968). Leakage and fluidization in air-layer lubricated avalanches. Geological Society of America Bulletin 79(5): 653–657. Terzaghi, K. and R. B. Peck (1967). Soil Mechanics in Engineering Practice. New York, John Wiley & Sons, Inc. Thornton, C., Yin, K. K., Adams, M. J. (1996). Numerical simulation of the impact fracture and fragmentation of agglomerates. Journal of Physics D-Applied Physics 29(2): 424–435.

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Evaluation of the efficiency of a model of rockfall protection structures based on real-scale experiments F. Bourrier L3S-R, UJF-INPG-CNRS, Grenoble Universités, Cemagref, Grenoble,

Ph. Gotteland L3S-R, UJF-INPG-CNRS, Grenoble Universités, Grenoble,

A. Heymann RAZEL, Saint Quentin, Cemagref, L3S-R, UJF-INPG-CNRS, Grenoble Universités, Grenoble,

S. Lambert Cemagref, Grenoble,

ABSTRACT: A model for the design of rockfall protection sandwich structures is presented and evaluated using results from real-scale experiments. The experiments consist of the impact by a 260 kg spherical projectile on a structure composed of gabion cages filled with coarse materials in the front part and fine granular material in the kernel part. This structure stands against a rigid concrete wall. The model allows ing for the mechanisms occurring in the individual layers of the structure. The comparison between the simulations and the experiments shows that the model correctly predicts the time evolution of the force on the projectile. However, the model partially s for the time evolution of the stress on the rigid concrete wall due to the simplicity of the formulation of the constitutive model used to characterize the kernel layer.

1

INTRODUCTION

The current climate changes could cause an increase in the frequency of natural hazards, justifying increased research to improve the efficiency of protective techniques. In particular, frequent rockfall occurrence has led to the development of ive protection methods such as rockfall restraining nets and reinforced soil structures. Whereas several studies were held to improve the design of restraining nets both from the experimental and from the numerical point of view, the design of reinforced dams generally rests on an empirical approach. Even though several experimental campaigns and numerical studies were carried out (Labiouse et al., 1994; Hearn et al., 1995 ; Peila et al., 2007 ; Aminata et al., 2008), a thorough analysis of the mechanical response of these structures during rock impacts remains to be done. The paper focuses on cellular sandwich structures (Nicot 2007). Such structures constitute innovative developments aiming at using the cellular technology, classical in the field of civil engineering, for rockfall protection. The main interest of such structures is that they allow strongly reducing the stress transmitted to the back part of the structure. The structure is built with an assembly of layers. The front layer is composed of cubic boxes filled with a granular material, the kernel layer is made of fine granular material and the back

part is either a rigid concrete wall or an embankment. In order to explore the mechanical behaviour of these systems under impact loadings, an experimental campaign of impacts on reference structures (Figure 1–3) is performed and a numerical model of the structure is developed. In a second time, the experimental and the simulation results are compared. 2

EXPERIMENTAL INVESTIGATIONS AT THE STRUCTURE SCALE

To study the dynamic mechanisms occurring during the impact, an experimental campaign consisting of impacts on a sandwich structure was held. The structure (Figure 1–3) consisted of a front layer composed with 15 cells filled with coarse material, a kernel layer composed of Seine sand and a 3 m height concrete wall leaned against a ground compacted embankment. The structure is laid on a concrete slab considered as a rigid interface. The cells composing the front layer were cubic in shape and 500 mm in height. They were made up of a hexagonal wire mesh and filled up with crushed quarry limestone, 80 to 150 mm in grain size. The material of the kernel layer consisted of Seine sand which is a wellgraded sand whose size distribution ranges from 0.2 to 5 mm. For practical purposes, the material of the kernel

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Figure 1. Sandwich structure used in the experiments and measurement devices (a1 : accelerometer on the projectile; a2 : accelerometer at the interface between the front and the kernel layers; a3 : accelerometer in the middle of the kernel layer; F1 : stress tensor on the back part wall). Figure 4. Time evolution of the impact force Fimp on the projectile.

Figure 2. Principle of the experimental device.

Figure 5. Time evolution of the stress σtran on the back part of the structure measured in the experiments.

Figure 3. Overviews of the experimental device (a) and of the impacted structure (b).

layer is dumped in bulk (Figure 3) and maintained in a geotextile container. The structure is subjected to a “pendular” impact by a 260 kg spherical boulder made of steel shell filled with concrete. The maximal impact energy is 10 kJ. The projectile is hanged from a 7 m high crossbar by means of two chains (Figure 2). The projectile can be lifted up to a maximal height of 4.75 m using a hand cable winch. A tri-axial piezoresistive accelerometer is placed on the projectile. In addition, accelerometers are placed at the interface between the front and the kernel layers and in the middle of the kernel layer. All sensors are placed along the direction of the velocity of the projectile before impact at the same height as the impact

point. The accelerometer on the projectile allowed determining the time evolution of the impact force Fimp applied to the boulder on the structure. The other accelerometers are used to measure the time elapsed between the beginning of the impact and the beginning of the displacements at the location of the accelerometer considered. The stress sensor F1 provides the time evolution of the stress σtran , normal to the impact direction at the same height as the impact point on the back part of the structure. Finally, the displacement of the front face of the structure is measured after each impact. The results presented in the following concern a 10 kJ impact obtained by dropping the spherical boulder from a 4.75 m height which corresponds to an incident velocity of 8.8 m/s. They are the time evolution of the impact force Fimp on the projectile (Figure 4) and of the stress σtran on the back part of the structure (Figure 5). The experimental results obtained for the impact force exhibit large variability for times smaller than 0.01 s. Although a smoothing of the results was performed (Figure 4), the interpretation of the results

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Figure 7. Cell strains definition. Figure 6. Principle of the model of the structure.

for the impact force should therefore be done with caution. 3

DISCRETE MODEL OF THE STRUCTURE

The structure is modelled as an assembly of rectangular cells of the same size as the wire netting cages used in the experiments (Figure 7). Each cell is supposed to be a regular cubic system at the structure scale. The different cells of the structure interact at their interfaces by means of forces applied at the cells gravity centre. For the calculation of the interaction forces, each cell is divided into two sub-cells and a constitutive model is associated with each of these sub-cells (Figure 6). Depending on the sub-cells position, the constitutive models are developed from dynamic or static experimental investigations. Front sub-cells that are directly impacted by the boulders require defining constitutive models from experiments in which sub-cells are subjected to dynamic loadings (sub-cell 1 on Figure 6). On the contrary, inner sub-cells inside the structure are considered to be only subjected to static loadings. 3.1 Front sub-cells: dynamic constitutive models As the impacting boulder stiffness is larger than the front sub-cell stiffness the interaction force between them is only depending on the sub-cell strain. In the x-direction, the increment dF i of the interaction force Fi between the boulder and the cell i of the front layer is given by (Figure 7):

where dui is the increment of the penetration of the boulder inside the cell i:

g

where ki is the stiffness of the front sub-cell of the cell g i. The stiffness ki is defined from a constitutive model based on the function fid which relates the force Fi on the projectile and the projectile penetration ui :

Figure 8. Constitutive models for front sub-cells.

As the impact direction of the velocity of the boulder is not normal to the front face of the structure, a tangential force at the point also exists. This force is depending on the normal force Fi following a Coulomb’s friction model. Modelling this tangential interaction allows ing for the changes in the boulder tangential and rotational velocities during the impact. Mechanical tests have been carried out to get information about the constitutive models associated with the front sub-cells. The dynamic tests (Lambert et al., 2009) consisted in the impact of the 260 kg projectile by a vertical fall on a front cell surrounded with the same material as the filling material of the cells in order to provide lateral boundary conditions similar to those observed in the structure. These conditions will be called Confined Material MC conditions (Lambert et al. 2009). These results (Figure 8) showed that the constitutive model for front sub-cells is defined by a linear relation (Fi = k l ui ) for ui < ulim . When the projectile penetration reaches values larger than ulim , the interaction force remains constant (Fi = F lim ). Finally, the unloading phase is characterized by a linear relation (Fi = k ul ui ). 3.2 Inner sub-cells: static constitutive models The interaction force between inner sub-cells depends on the strains of both sub-cells in along the

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Figure 9. Constitutive models for inner sub-cells associated with the front layer.

x-axis. The equilibrium condition between the two ading sub-cells i and j yields a relation between the increment dF i/j of the interaction force Fi/j and the relative displacement duij between the two cells along the x-direction, (Figure 7):

with

g0

where lid0 , lj are the lengths of the sub-cells associated with the cell iand cell j at the first time step; and Sc is the area of the interaction surface. Eij is the equivalent g modulus ing for both modulii Eid and Ej of the sub-cells:

Assuming that the interface between ading cells exhibits only little changes in size and shape over the g loading, it is possible to express the modulii Eid andEj in of stresses and strains. For inner sub-cells associated with the front layer, the constitutive models are characterized from static compression tests (Lambert, 2007) of cells under RC (Rigid Conditions) and FD (Free to Deform) conditions (Figure 9). For FD conditions, the evolution of the axial stress σixx between sub-cells in the x-direction, depending on the sub-cell axial strain εxx i in the xdirection, is characterized by a linear increase in the stress until the threshold value εlim is reached. For lim lim εxx i larger than ε , the stress is equal to σ . On the contrary, no threshold value is observed for RC conditions. Interestingly, the coefficient of the linear relation is different depending on the confinement conditions l (Figure 9): Erc (for RC conditions) and Efdl (for FD conditions). Finally, the unloading phase is also characterized by a linear decrease of σixx for decreasing εxx i

Figure 10. Constitutive models for inner sub-cells associated with kernel cells. l using the coefficient Erc for RC conditions, and Efdl for FD conditions (Figure 9). The constitutive model associated with MC conditions is comprised between these two extreme cases. For the inner sub-cells associated with the kernel layer, as no experimental results were available, a bilinear constitutive model fully characterized by a loading l ul l Eke and an unloading Eke = 3Eke modulus was chosen. The values of the loading modulus were evaluated from oedometric tests assuming that the sand cells were loaded over a simple oedometric path. In the simulations, two types of material were modelled: a loose l sand associated with a loading modulus Eke = 10 MPa and a dense sand for which the loading modulus is l Eke = 200 MPa.

3.3 Lateral forces and back part of the structure The cell strain in the x-direction entails lateral cell strains that induce normal forces Filat at cell interfaces in the y- and z-directions. Flat i are calculated with Kδ , a constant coefficient, as follows (Nicot et al., 2007):

g

where ldi , lj are the lengths of the sub-cells associated with the cell i and cell j at the current time step. f The Filat forces induce tangential forces Fi that counter the cells displacements along the x-direction:

where µi is the Coulomb’s friction coefficient associated with the cell i. The values of K δ and µi were determined from numerical simulations performed at the cell scale (Nicot et al., 2007). One can also note that, when the cell length reaches a limit value l lim corresponding to the compaction limit of the cell, the cell is considered as a rigid body that stills interacts with its neighbouring cells.

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Table 1. Values of the parameters of the constitutive models. Constitutive model

Parameters

Front sub-cells

ulim = 0.05 m; k l = 2 × 106 N/m; k ul = 2 × 107 N/m; F lim = 90000 N/m

Inner sub-cells Front layer

FD: εlim = 3%; σ lim = 60 kN/m2 ; Efdl = 2 MPa; Efdul = 20MPa l ul = 360 MPa = 36 MPa; Erc RC: Erc

Inner sub-cells Kernel layer

l Dense sand: Eke = 200 MPa; ul Eke = 600 MPa l Loose sand: Eke = 10 MPa; ul Eke = 30 MPa

Figure 12. Time evolution of the impact force Fimp on the projectile for a kernel layer composed of dense sand.

Figure 11. Time evolution of the impact force Fimp on the projectile for a kernel layer composed of loose sand.

Finally, the back part of the structure is modelled as an elastic boundary condition associated with the modulus Ebo . The values of the parameters of the different constitutive models are summarized in Table 1.

4

Figure 13. Time evolution of the stress σtran on the back part of the structure for a kernel layer composed of loose sand.

COMPARISONS BETWEEN SIMULATIONS AND EXPERIMENTS

The experimental tests were simulated using the model presented in the previous section. The comparison is performed using experimental results from a 10 kJ impact which corresponds to an incident velocity of 8.8 m/s. Simulations were held using constitutive models for kernel layers associated with either dense or loose sands. In addition, for both cases, the constitutive models associated with the inner sub-cells of the front layer under free-to-deform and rigid lateral boundaries were both used to get information on the more appropriate model. The comparison between the experiments and the simulations focuses on the impact force Fimp on the projectile (accelerometer a1 – Figure 1) and on the stress σtran on the back part of the structure (stress tensor F1 – Figure 1). The simulation results (Figure 11–14) first show that the model used for the inner sub-cells associated

Figure 14. Time evolution of the stress σtran on the back part of the structure for a kernel layer composed of dense sand.

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with the front layer does not strongly influences the results obtained neither for the impact force nor for the stress on the back part of the structure. Given the weak differences observed and the reduced amount of experimental result currently available, it seems therefore not possible to determine the more appropriate constitutive models for inner sub-cells associated with the front layer. In addition, the simulations provide time evolution of the impact force very close to the experimental results. The adequacy is particularly marked during the loading phase corresponding to times smaller than 0.02 s. However, the unloading phase, characterized by a decrease in the impact force, is better predicted by simulations using a constitutive model associated with a dense sand made kernel layer. Indeed, if the kernel layer is modelled as a loose sand, the impact force largely faster decreases than what is observed in the experiments. Finally, the simulation results show that the model is not able to predict both the time evolution and the quantitative values of the stress on the back part of the structure whatever the constitutive model used for the kernel layer. For dense sands models, the maximum value of σtran is observed at the same time as in the experiments. However, this maximum value is larger than the experimental value. On the contrary, for loose sand models, the maximum value is of the same order of magnitude order as in the experiments but its occurrence is time delayed compared to the experiments. These differences may be due to the simple formulation of the constitutive model used for sub-cells associated with the kernel layer. Using a bi-linear model is certainly not sufficient to model adequately both the energy dissipation and the dynamic mechanisms occurring in the kernel layer during the impact. 5

CONCLUSION

In this paper, experimental results by the impact of a projectile on a rockfall protection sandwich structure were used for the evaluation of a structure model based on a multi-scale approach. The comparison between the experimental results and the simulations shows that the model allows for a correct prediction of the impact force on the projectile both in a qualitative and in a quantitative point of view. On the contrary, the prediction of the stress on the back part of the structure is less accurate because of the simple formulation

of the model used to characterize the kernel layer. These results therefore emphasizes that the constitutive model for the kernel layer has to be improved before using the structure model for design purposes. The improvement of this constitutive model is currently in progress by means of experimental studies on the energy transfer inside a sand layer during rock impacts. ACKNOWLEDGEMENTS The results presented in this paper were obtained in the framework of the French research development project REMPARe (www.rempare.fr) ed by the French National Research Agency (ANR). All the partners of project REMPARe, especially partners CER-LC, as well as, for their financial , the research consortium VOR-RNVO, and the PGRN (Natural Hazard Pole of Grenoble) from the Isère General Council are gratefully acknowledged by the authors. REFERENCES Aminata, D., Yashima, A., Sawada, K., Sung, E., Sugimori, K., & Inoue, S. 2008. New Protection Wall Against Rockfall Using a Ductile Cast Iron . Journal of Natural Disaster Science 30 (1): 25–33. Bertrand, D., Nicot, F., Gotteland, P., & Lambert, S. 2006. Modelling a geo-composite cell using discrete analysis. Computers and Geotechnics 32: 564–577. Hearn, G., Barrett, R., & Henson, H. 1995. Development of effective rockfall barriers. Journal of transportation engineering 121 (6): 507–516. Labiouse, V., Descoeudres, F., Montani, S., & Schmidhalter, C. 1994. Experimental study of rock blocks falling down on a reinforced concrete slab covered by absorbing cushions. Revue française de géotechnique 69 : 41–61. Lambert, S., Gotteland, P., & Nicot, F. 2009. Experimental study of the impact response of geocells as components of rockfall protection embankments. Natural Hazards and Earth Systems Sciences 9: 459–467. Lambert, S., Gotteland, P., Plé, O., Bertrand, D., & Nicot, F. 2004. Modélisation du comportement mécanique de cellules de matériaux confinés. Journées Nationales de Géotechnique et de Géologie: 219–226. Peila, D., Oggeri, C., & Castiglia, C. 2007. Ground reinforced embankments for rockfall protection: design and evaluation of full scale tests. Landslides 4: 255–265. Nicot, F., Gotteland, P., Bertrand, D., & Lambert, S. 2007. Multi-scale approach to geo-composite cellular structures subjected to impact. International Journal for Numerical and Analytical Methods in Geomechanics 31: 1477–1515.

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Evaluation of viscous damping due to solid-fluid interaction in a poroelastic layer subjected to shear dynamic actions J. Grazina & P.L. Pinto Department of Civil Engineering, University of Coimbra, Portugal

D. Taborda Department of Civil and Environmental Engineering, Imperial College London, UK

ABSTRACT: Dynamic response of poroelastic saturated materials is largely dependent of factors such as the solid skeleton permeability and the frequency of the movement. Depending on these quantities, the behaviour is undrained for total coupled interaction or fully drained for null interaction. Between these limit cases, some relative movement occurs among solid skeleton and fluid, generating viscous damping, which, in turn, modifies the elastic response of the system. This paper presents results from FE analyses of coupled viscous solid-fluid interaction in poroelastic saturated layers, using a code with coupled formulation us − uw − p. Firstly, the natural frequencies are evaluated for free vibration responses considering different coupled levels. Secondly, the damping ratios are estimated by imposing harmonic shear actions, with the previously obtained frequencies, to the layers. It was observed that a sudden change of modal frequencies occurs, from nearly undrained values to nearly drained values. This was followed by a variation of modal damping ratios, ranging from almost null values, for permeabilities close to the above mentioned limit cases, to maximum values at intermediate coupled interaction.

1

INTRODUCTION

Dynamic behaviour of poroelastic saturated materials is usually described using the Lagrange classical mechanics - Biot theory (Biot 1941, 1956a, 1956b) or using the mechanics of continuum medium using the Porous Mixtures theory (Fillunger, 1913), which involves the concept of volume fractions (Schanz & Diebels 2003). Some simplifications are possible in both theories for common soil dynamic analyses, which result in similar equilibrium equations. In accordance with these theories, the interaction dynamic force between the solid skeleton and the porous fluid is expressed by:

where k = permeability; n = porosity; η = viscosity of fluid; γw = bulk unit weight of fluid; vs = velocity of solid; and vw = velocity of fluid. The interaction between the solid skeleton and the fluid in a poroelastic material is significantly dependent on the permeability and the frequency of the movement. Zienkiewicz et al. (1980) presented parametric analyses for wide ranges of permeabilities and frequencies of harmonic vertical excitations in a poroelastic column, where different coupled behaviours were observed. In those analyses, bound parameters were defined considering nearly

full drained (null coupled) and nearly undrained (full coupled) situations, as well as the limit values for consolidation analyses despite inertial forces. For nearly undrained behaviour, relative movements between solid and fluid are reduced and interaction forces developed are insignificant. In the opposite situation, for nearly full drained behaviour, permeabilities are very high and, consequently, interaction forces are very small, in spite the existence of large relative movements between both phases. In between these limit situations, relevant viscous interaction forces may be developed for common dynamic loadings, as those originated by foundations of industry equipment or earthquakes. This paper presents a numerical analysis, where the viscous damping due to interaction forces is evaluated, in poroelastic saturated layers subjected to harmonic shear accelerations.

2 2.1

NUMERICAL MODELLING General description

The analysis consists, firstly, on the determination of modal frequencies for the range of permeabilities considered. For this purpose, a free vibration horizontal movement was induced at the layer by releasing a pre-imposed displacement on the top. The modal frequencies were calculated for the response at the top

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using a Discrete Fourier Transformer (DFT) algorithm (Taborda 2008). Permeabilities of poroelastic materials related to limit cases of almost null and almost full coupled behaviour were defined and compared with responses of one-phase materials, respectively, with dry and undrained properties. Secondly, harmonic shear accelerations, with the previously calculated frequencies, were imposed at the base of the layer. The one-mode response with limited amplification allowed the evaluation of the viscous damping ratio for the viscous interaction level considered and related with poroelastic permeability. These analyses were carried using the FEMEPDYN finite element code (Grazina, 2009) with coupled formulation us − uw − p (Zienkiewicz et al. 1999, Arduino & Macari 2001), developed at the University of Coimbra. This formulation implemented enables computation of nodal displacements, velocities and accelerations of both solid and fluid phases, as well as pore pressures at the corner nodes of mesh elements. For the time integration, the trapezoidal Newmark algorithm was used, to avoid the introduction of any undesirable numerical damping effect.

2.2

where ρ = material density, calculated using properties from Table 1 as ρ = ρd = (1 − n)ρs for dry materials and as ρ = ρsat = (1 − n)ρs + nρw for saturated materials.

2.4 Time discretization For excitation of a specific vibration mode, accurate values of natural frequency modes must be obtained. Considering the inversely proportional relation between the total time t of computed response and the frequency step f = 5 × 10−3 Hz established, a t=200 s is required for calculation of modal frequencies. Consequently, considering the spatial discretizations adopted, the time step values of t = 5 × 10−3 s

Finite element models

The model consists in homogeneous layers with a thickness of 20 m, settled over a rigid bedrock material. The absence of vertical movements on the dynamic response of the layers subjected to shear horizontal loadings allows a major simplification of the FE mesh. In these conditions, a semi-infinite layer can be modelled by a single column with restrictions of vertical displacement at lateral boundaries. At the bottom, for the bedrock interface, vertical and horizontal displacement restrictions and an impervious boundary were considered (Fig. 1a). The columns are composed of 20 elements of materials with a shear modulus, G, of 20 MPa (G20) and 40 MPa (G40) and of 40 elements of material with a shear modulus of 80 MPa (G80). Quadrilateral hybrid elements of Q8/C4 type were used for poroelastic materials, with 8 nodes for computation of displacements, velocities and accelerations (dva) and 4 corner nodes for pore pressure (pwp) continuity, as presented in Figure 1b. For one-phase materials analyses Q8 isoparametric elements were used.

2.3

Properties of one-phase materials are presented at Table 2 considering theoretical modelling of null and full coupled interaction. Shear wave velocities, vS , are also presented for these situations, calculated accordingly to Equation 2:

Materials properties

Figure 1. FE model and hybrid elements used in the analyses. Table 1.

Properties of poroelastic materials.

G Es (MPa) (MPa) ν 20 40 80

Table 2.

Three values of the shear modulus were considered in the numerical analyses, for both poroelastic and onephase type materials. Major properties of poroelastic materials are presented in Table 1. In this case, the bulk unit weight of saturated material is γsat = 19.78 kN/m3 and an incompressible fluid was considered with a bulk modulus of Kw = 1 × 105 MPa. A wide range of permeability coefficients was used, some of them with unrealistic high values in order to achieve the almost null coupled situation.

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52 104 208

ρs (kg/m3 )

ρw (kg/m3 )

n

k (m/s)

0.3 2.6 × 103 1.0 × 103 0.365 1 × 102 to 1 × 10−5

Properties of one-phase materials.

G (MPa)

Es (MPa)

ν

Coupled Interaction

γ (kN/m3 )

vS (m/s)

20

52.0 59.6 104.0 119.2 208.0 238.4

0.3 0.49 0.3 0.49 0.3 0.49

Null / Dry Full / Saturated Null / Dry Full / Saturated Null / Dry Full / Saturated

16.20 19.78 16.20 19.78 16.20 19.78

110.1 99.6 155.7 140.9 220.1 199.2

40 80

and t = 2.5 × 10−3 s were used, respectively for G20/G40 and G80 materials. For the second part of the analyses, where accelerations with modal frequencies were applied at the base of the column, the time step values were defined accordingly with the natural period Tn imposed, using the relation t = Tn /80. For this value, a one-mode response can be achieved and related damping ratio computed.

3

k = 1 × 10−3 m/s to 100 m/s) it is visible a gradual drop of displacements with time, that reveals the presence of viscous damping. In the sequence of crescent permeabilities, this drop increase sharply until k = 1 m/s and

EVALUATION OF MODAL FREQUENCIES

3.1 Free vibration responses The movements of the layers in free vibration regime may exhibit the viscous damping effect and the frequencies content for each coupled level. Figures 2a-2i present the development of normalized top displacements with time, d/d0 , and frequencies spectra for a sequence of increasing permeabilities, from full to null interaction behaviours, referring to the G40 layer. Limit coupled interaction levels responses (Fig. 2a,2i) were obtained with one-phase materials. Similar responses are showed in Figures 2a and 2b, meaning that for k = 1 × 10−5 m/s the almost full coupled behaviour is achieved. At the opposite case, Figure 2h reveals that even for k=100 m/s some damping effect still remains and null interaction cannot be achieved. However, modal frequencies in this last case are similar to those of null interaction analysis (Fig. 2i). For the middle-range permeabilities (from

Figure 2. Evolution of normalized displacements and frequency content with increasing permeabilities for the G40 layer.

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Table 3. Modal frequencies of the G20 layer for different coupled interaction levels. Permeab.

Modal frequencies

Coupled interaction

k (m/s)

f1 (Hz)

f2 (Hz)

f3 (Hz)

Null / Analytical Null / One-phase Poroelastic

∞ ∞ 102 10 1 10−1 10−2 10−3 10−4 10−5 0 0

1.376 1.375 1.375 1.375 1.350 1.250 1.245 1.245 1.245 1.245 1.245 1.245

4.127 4.120 4.120 4.120 4.125 (3.908) (3.735) 3.730 3.730 3.730 3.735 3.730

6.878 6.850 6.845 6.850 6.870 – – – 6.205 6.205 6.224 6.205

Figure 3. Variation of free vibration response with permeability of poroelastic G20 layer.

Full / Analytical Full / One-phase

Table 4. Modal frequencies of the G40 layer for different coupled interaction levels.


Permeab.

Modal frequencies

Coupled interaction

k (m/s)

f1 (Hz)

f2 (Hz)

f3 (Hz)


∞ ∞ 102 10 1 10−1 10−2 10−3 10−4 10−5 0 0

1.946 1.945 1.945 1.945 1.925 1.780 1.760 1.760 1.760 1.760 1.761 1.760

5.937 5.820 5.820 5.820 5.825 (5.623) (5.280) 5.270 5.270 5.265 5.282 5.270

9.728 9.655 9.655 9.655 9.665 – – – (8.750) 8.750 8.804 8.750



tends to reduce for higher permeabilities as viscous interaction becomes lower. The frequencies spectra reveals well defined natural frequencies for higher and lower permeabilities, and more dispersive distributions where damping effect is more pronounced (k = 1 × 10−1 m/s and 1 m/s). In these last cases, higher natural frequency modes are strongly damped and not detectable. Figures 3–5 present the positive envelopes of the normalized displacements for the first t = 40 s of the analyses, for each values of k and G considered. Comparing these last figures, it is visible that for higher values of G and for lower permeabilities responses became more damped. This is noticeable for

the highest damped response, which is reached with k = 1 m/s for G20 and G40 and with k = 1 × 10−1 m/s for G80. The response decay is also more pronounced in this last case.

3.2 Modal frequencies The results of the first three natural frequencies detected from spectral distributions for the aforementioned values of k and G, are compiled in Tables 3-5. Cases where these values were not detected or with dispersive frequency distribution (poorly defined as shown in Fig. 2e) are also indicated on these tables. Analytical values for the limit coupled cases are also presented, according with (Kramer 1996):

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Table 5. Modal frequencies of the G80 layer for different coupled interaction levels. Permeab.

Modal frequencies

Coupled interaction

k (m/s)

f1 (Hz)

f2 (Hz)

f3 (Hz)


∞ ∞ 102 10 1 10−1 10−2 10−3 10−4 10−5 0 0

2.752 2.750 2.750 2.750 2.740 2.540 2.490 2.490 2.490 2.490 2.490 2.490

8.255 8.240 8.240 8.245 8.240 (8.375) (7.490) (7.460) 7.460 7.460 7.470 7.460

13.758 13.705 13.705 13.705 13.715 – – – (12.410) 12.410 12.450 12.410


Figure 6. Amplified acceleration responses for the G40 layer subjected to shear harmonic excitation with f1 = 1.78 Hz.

– Non-detectable values; () Inaccurate values.

where fn = frequency of mode n and H = the thickness of the layer. In these tables it is evident a good match between the analytical and the one-phase frequency values, for both coupled limits. This is a proof of good accuracy afforded by the spatial and time discretizations adopted, although computed frequencies tend to be lower than the analytical values, as had been noticed by Grazina (2009). For any value of G, it is visible that natural frequencies remain similar with those of coupled limit cases for a broad range of k values, approximately corresponding to two sets of frequency modes. Transition between these sets occurs suddenly in a narrow range of k values, when viscous damping effect is more notorious (as shown in Figures. 3–5).

4 4.1

Figure 7. Amplified acceleration responses for the G40 layer subjected to shear harmonic excitation with f2 = 5.27 Hz.

which turns the achievement of stationary response less accurate. Responses at the top of the layers are related with the imposed movement at the base with the amplification factor, D, expressed for a homogeneous layer by (Kramer, 1996):

EVALUATION OF VISCOUS DAMPING Methodology

The natural frequencies obtained were used to evaluate the respective modal damping ratios, ξ. For this purpose, each harmonic accelerations with a modal frequency was applied at the base and the forced vibration response was obtained at the top. These calculations proceeded until a stationary response was reached. The duration of time analyses depends on the viscous damping presented. Figures 6, 7 exemplify time evolutions of relative acceleration responses on the top and excited frequencies of the movement. These are particularly referred to the calculations of the 1st mode (f1 = 1.78 Hz) with k = 1 × 10−1 m/s and of the 2nd mode (f2 = 5.27 Hz) with k = 1 × 10−3 m/s, both for the G40 layer. Onemode response is easily obtained for the first mode calculations. However, for the second mode analyses, some influence of the first mode still persists,

where ω = 2πf is the angular frequency. Using this relation, Equation 3 becomes:

and substituting on Equation (4) the first term under square root becomes 0 and therefore:

Estimated values of D for each mode are then used to evaluate the respective damping ratio, ξ, using the previous equation.

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Table 6. layers.

For middle-range permeabilities, coupled effect is notorious and the values of k related to their maximum influence were identified. These values represent a transition of the coupled behaviour, dividing near full from near null coupled interactions. This is followed by a sudden transition of frequency modes, which can generally be divided in two sets of values. Each of these sets has very similar values to the respective limit cases. These limit cases, of null and full coupled interaction and without viscous damping, were calculated using one-phase materials with properties respectively of dry and saturated material. A comparison between values of damping ratio for the 1st and 2nd modes reveals that damping is higher for the 2nd mode at near full coupled situations and for the 1st mode at near null coupled situation.

Modal damping ratios for G20, G40 and G80

G20 layer

G40 layer

G80 layer

k (m/s)

ξ1 (%)

ξ2 (%)

ξ1 (%)

ξ2 (%)

ξ1 (%)

ξ2 (%)

100 10 1 10−1 10−2 10−3

<0.05 0.45 3.74 1.87 0.21 <0.05

– 0.18 1.51 4.68 – –

<0.05 0.32 2.92 2.65 0.28 <0.05

– (0.19) 1.06 5.25 0.82 (0.15)

<0.05 0.23 2.16 3.53 0.40 <0.05

– (0.12) 0.76 6.95 1.18 (0.14)

– Non-detectable values; () Inaccurate values.

4.2

Damping ratio values

For permeabilities near coupled limits, damping is very small and resonance hinders the system to achieve a stationary response. Also, in analyses for higher frequency modes, the presence of less order modes influenced significantly stationary responses. Consequently, only damping ratios for 1st and 2nd modes were calculated for the middle-range permeabilities. These values are presented on Table 6 for the 3 values of G considered. The results of the damping ratios, ξ, are in accordance with the free vibration responses presented above. Higher values of ξ exists with k = 1 m/s for G20 layer and with k = 1 × 10−1 m/s for G40/G80 layers (as noticed in Figs. 3-5). The results on the table also show that ξ1 > ξ2 for behaviours near null coupled interaction and the opposite (ξ1 < ξ2 ) for near full coupled interaction. It is noticeable the strong influence on damped free response of small values of ξ for k = 100 m/s, as can be seen in Figure 2h. 5

CONCLUSIONS

The method described was successfully able to evaluate damping ratios related with viscous coupled effect on poroelastic saturated layers. Permeability values for almost full coupled interaction were founded. However, for the values of shear modulus studied, even with unlikely permeabilities of k = 100 m/s, the almost null interaction behaviour was not achieved.

REFERENCES Arduino, P.; Macari, E.J. (2001). Implementation of a porous media formulation for geomaterials. Journal of Engineering Mechanics. ASCE, Vol. 127, No 2, pp. 157–166. Biot, M.A. 1941. General theory of three-dimensional consolidation. Journal of Applied Physics 12: 155–164. Biot, M.A. 1956a. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range. Journal of the Acoustical Society of America 28(2): 168–178. Biot, M.A. 1956b. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. Journal of the Acoustical Society of America 28(2): 179– 191. Grazina, J. 2009. Modelação dinâmica com acoplamento viscoso de maciços elastoplásticos. Aplicação a estruturas de e flexíveis submetidas a acções sísmicas. PhD thesis, University of Coimbra (in Portuguese). Kramer, S.L. 1996. Geotechnical Earthquake Engineering. Prentice Hall, New Jersey. Schanz, M.; Diebels, S. 2003. A comparative study of Biot’s theory and the linear theory of porous media for wave propagation problems, Acta Mechanica 161: 213–235. Taborda, D. 2008. DFTi – Improved Discrete Fourier Transform Algorithm software: version 1.1.1. Zienkiewicz, O.C.; Chang, C.T.; Bettess, P. 1980. Drained, undrained, consolidating and dynamic behaviour assumptions in soils, Geotéchnique, 30(4): 385–395. Zienkiewicz, O.C.; Chan, A.H.C.; Pastor, M.; Schrefler, B.A.; Shiomi, T. (1999). Computational Geomechanics with Special Reference of Earthquake Engineering. John Wiley & Sons, Chichester, pp. 398.

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Non linear numerical modeling of slopes stability under seismic loading – reinforcement effect F. Hage Chehade Université Libanaise – Centre de Modélisation, PRASE, EDST – IUT – Liban

M. Sadek & I. Shahrour Université des Sciences et Technologies de Lille,

ABSTRACT: This paper presents results of a numerical modeling of slope stability problems under seismic loading in Lebanon by using a global dynamic approach. This methodology offers several advantages when compared to simplified methods like pseudo-static approaches. So, we can investigate the effect of the governing parameters such as the non linear soil behavior, the presence of weak soil near the surface, the spatial and temporal variability of the seismic loading and the reinforcement element… Some of these parameters are critical in triggering instability under seismic loading. The analysis is focused on a parametric study of the reinforcement element along the slope (position, length, inclination, numbers) in order to give the best appropriate reinforcement scheme that minimize the earthquake effect. The present study is conducted by using measures recorded during real earthquakes (Turkey, 1999).

1

INTRODUCTION

Earthquake risk is a major concern in Lebanon which is located in an active seismic zone. Lebanon has witnessed several earthquakes, some of which caused massive destruction in the past. The topography of Lebanon is known by its high mountains and steep slopes. Because of large demographic expansion, new constructions are actually built over dangerous slopes. Moreover, since February 2008, abnormal seismic activity has been noted in southern Lebanon as well as a large number of minor earthquakes in a three-month period. In case of medium or severe earthquakes, extent of damage on Lebanese slopes could very severe in of both human and economic losses. So it is necessary to give some recommendations to improve the stability of Lebanese slopes under seismic events. The slopes represent a weak zone where the seismic consequences could be amplified. During the last century, there are more than 76 earthquakes generating important critical cases number of slope movements varying from one thousand to one million. The slope morphology controls the instability type: The steep slopes that have an inclination angle more than 35◦ are quite often the cause of superficial movements such as falling boulders, rock slides or soil collapse (Keefer, 1984; Rodriguez et al., 1999). Gentle slopes can be affected by soil flows and mudflows like the 1960 earthquake of Chili (Keefer, 1984). Nevertheless, most of the events which are produced in gentle slopes are the result of a combination of seismic vibration

and strong precipitation. Then except for slope movements over gentle slopes and reactivation of ancient deposits sliding, the majority of slope movements are produced in slopes steeper than 25◦ . Site observations coupled with some works of Jibson et al. (1994) show that the characteristics of the seismic input can largely influence the type of instability. Earthquakes of small magnitudes characterized by small accelerations and a high frequency content cause essentially bloc falls which belong to the category of superficial movements. On the other hand, earthquakes of bigger magnitudes (large acceleration and low frequency content) can cause more profound slides. This type of earthquakes is the origin of a huge number of slope movements. The stability of slopes depends on its initial state condition before the earthquake. When the slope is close to static failure, a small seismic shock is enough to trigger movement. The Ledu earthquake (China, 1984) having a magnitude of 2.9 was enough to trigger a sliding because of the small depth if its source (Feng & Guo, 1985). There are several real cases of slope landslides recorded during real earthquakes. The pseudo-static (Terzaghi, 1950) and the Newmark (Newmark, 1963) methods are largely used to assess slope stability subjected to seismic loading because of their simplicity and there is no need to sophistical softwares. The first one ignores the cyclic nature of the earthquake and applies for that an additional static force. A seismic coefficient is used where his value is provided in various reglementation. The

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Figure 1. Slope case study.

second method assumes that the soil is an undeformable rigid-perfectly plastic block. This assumption is not suitable for soils. During the earthquake in Salvador for example (2001), severe damage of buildings was obtained at the limit of Armenia village and a significant broken of slopes. These observations lead to the fact the French Parasismic Reglementation could underestimate the observations (AFPS, 2001). Nevertheless, the high cost of numerical methods and the technical knowledge required for their practical use (mechanical parameters for soils, seismic loading) lead to the fact that the traditional and simplified methods still widely used. In this work, we propose a numerical global dynamic analysis of the slope stability problems under seismic loading. This methodology has the advantage of taking into the main parameters, generally neglected by using simplified methods, such as the spatial and temporal variability of the loading, the initial soil state, the presence of water in the soil, the presence of weak soil layers near the surface, the non linear soil behavior under static or cyclic loading and the influence of reinforcement elements. The numerical analysis is performed by using the FLAC3D finite difference software (Itasca, 2005). For the seismic loading, a real earthquake record corresponding to the Kocaeli earthquake (1999) in Turkey (Chen and Scawthorn, 2003) is adopted. At first, a comparative study between the results given by respectively elastic modeling and elastoplastic modeling of the soil behavior shows the limitation of the elastic modeling. Then the analysis is focused on the stability of reinforced slopes. A parametric study according to the length, the position, the inclination, and the number of the reinforcement element is performed. The obtained results describe the evolution of the slope state and give recommendations about the most efficient scheme of the reinforcement element that could reduce the seismic hazard on slopes under seismic loading.

2 2.1

PROBLEM UNDER CONSIDERATION Geometry

Figure 1 represents typical slope geometry of two soil layers of equal depth (H) that has been modelled. The

Figure 2. Records of the Kocaeli earthquake a) Velocity, b) Acceleration and c) Fourier Spectra of Velocity Component.

rigid soil at the bottom has a lateral extent equal to 14H while the slope extent is equal to 8H. These values have been adopted in order to minimise the boundary effects (Bourdeau, 2005). At the lateral boundary, we use free field conditions which ensure the absorption of the outward waves. The results in this paper are presented in the case of H equal to 15 m and the slope angle (α) is equal to 30◦ . 2.2 Seismic loading Dynamic loading is applied at the bottom of the rigid soil layer as a velocity excitation. The slope is subjected to a real earthquake loading representative of the 1999 Kocaeli earthquake (Mw = 7.4) in Turkey. The estimated peak velocity is approximately 40 cm/s (peak acceleration 0.247 g), and the duration is equal to 30 s (Figures 2a-b). Fourier analysis of the earthquake velocity record reveals a dominant frequency of about 0.9 Hz (the second peak is observed at 1.3 Hz). Figure 2-c shows that the most of the power for the input history is contained in low frequencies (Parish et al., 2009). For comparison, the natural frequencies of the slope were determined by a Fourier analysis of its free response. Figure 3 shows a fundamental frequency equal to 1.1 Hz which is close to dominant frequency of seismic loading (0.9 Hz) while to 3.2 Hz. 2.3 Mesh For an accurate representation of the wave transmission through the soil model, the spatial element size, l, must be smaller than approximately onetenth to one-eighth of the wavelength associated with

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Table 1.

Soil mechanical properties.

Youngs Modulus Poisson’s ratio Elastic shear modulus Cohesion Friction angle Dilation angle Unit Weight

Rigid soil

Weak soil

1000 (MPa) 0.25 400 MPa 200 (kPa) 35◦ 3◦ 20 kN/m3

25 (MPa) 0.3 9.6 (MPa) 5 kPa 30◦ 10◦ 20 kN/m3

Figure 3. Response spectra of free motion of the slope.

Figure 4. Mesh of the slope domain.

the highest frequency component of the input wave (Kuhlemeyer and Lysmer, 1973):

Figure 5. Evolution of the lateral displacement at the crest and the node A of the slope.

where λ is the wave length associated with the highest frequency component that contains appreciable energy. So, reasonable analyses could be time and memory consuming. In such cases, it may be possible to adjust the input by recognizing that most of the second frequency is equal the power for the input history is contained in lower frequency components. Figure 4 shows, in the vertical plane, the mesh used for the numerical modeling. It contains 500 elements. 3

NUMERICAL ANALYSIS OF THE SLOPE BEHAVIOR

The behavior of the slopes under the seismic loading was analyzed in two cases of the soil behavior. In the first case, the soil behavior is assumed to be linear elastic. In the second case, the soil behavior is assumed to be non linear elastoplastic and it is described by using the non associated Mohr-Coulomb criterion where the properties are given in table 1. When elasticity is used, Rayleigh damping equal to 5% is adopted in order to compensate the energy dissipation through the medium. Figure 5 depicts the evolution of the lateral displacement at the crest and at node A (see figure 7) on the slope. We can see that the plasticity leads to very important lateral displacement and the discrepancy between the elastic and elastoplastic calculations starts after 4 s. Figure 6 shows the lateral velocity amplification at time equal to 15 s for the both calculations. We

Figure 6. Lateral velocity amplification at time equal to 15 s (origin at the bottom of the slope).

can see a significant decrease in velocity amplification with plasticity. This result could be explained by the energy dissipation attributed to the plastic deformation and by the influence of plasticity on the reduction of the frequency response of the slope. The previous results clearly illustrate the elastic calculation doesn’t lead to a realistic description of the slope behaviour and its evolution with time. So, in the case of reinforcement, the numerical modelling is carried out only in the case of elastoplastic soil behaviour.

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Figure 7. Reference position of the reinforcement element.

4 4.1

Figure 8. Effect of the length of the reinforcement element on the horizontal displacement at slope crest.

NUMERICAL ANALYSIS OF REINFORCED SLOPE Methodology

This section is focused on the effect of reinforcement (nail or micropile) on the slope stability under seismic loading. The reinforcement element has an axial rigidity (EA) equal to 1100 103 kN and a stiffness (EI) equal to 850 105 kN m2 . In order to study the performance of the reinforcement element a parametric study according to the length, position, inclination and number has been performed. 4.2

Effect of the reinforcement element length

The evolution of the slope stability has been investigated according to the reinforcement element length. The reinforcement element is perpendicular to the slope surface and it is located at 3 m below the crest (figure 7). Three lengths of reinforcement length have been respectively tested (7, 10 and 13 m). Figure 8 shows the evolution of the horizontal displacement at the slope crest respectively without and for the three tested lengths of reinforcement element. When the reinforcement is used, we can see a decrease in the horizontal displacement at crest about 12% when its reinforcement length is equal to 7 m and 15% when this is equal to 10 m. There is no significant decrease in horizontal displacement when the length of the reinforcement element increases from 10 to 13 m. Figures 9a and 9b depict at t = 15 s the distribution along the reinforcement element of the horizontal displacement and the bending moment. A significant decrease of the horizontal displacement is noted along the reinforcement element especially near the tip: the amount of reduction reaches 50% and 80% respectively where the reinforcement length increases from 7 m to 10 and 13 m (figure 9a). The bending moment slightly increases with length in the upper part and decreases around the tip (figure 9b). The highest lateral displacement at the slope is located under the reinforcement element (figure 10) at time equal to 15 s for the case of reinforcement element length equal to 10 m.

Figure 9. Numerical results along the reinforcement element (a) horizontal displacement and (b) bending moment (x position on the reinforcement element, see figure 7).

4.3 Effect of reinforcement element position In addition to the previous reference position, the effect of two more positions of the reinforcement element on the slope stability has been investigated (length = 10 m). In the first one, the head of the reinforcement element is located at the crest (upper case). In the second position, the head of the reinforcement element is located immediately at the mesh node on the slope under the reference position (see figure 7).

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Figure 10. Shape of lateral displacement at time equal to 15 s and reinforcement element length equal to 10 m.

Figure 13. Effect of the reinforcement inclination on the slope horizontal displacement.

4.4 Effect of inclination The efficiency of the reinforcement element on the slope stability according to its inclination has been also analyzed. Four inclination values (angle β, see figure 7) of the reinforcement element (length = 10 m) have been investigated: – The reinforcement element is horizontal (angle slope-reinforcement is equal to the slope angle) – The angle slope-reinforcement is equal to 60◦ – The reinforcement is perpendicular to slope surface (reference case) – The reinforcement is vertical

Figure 11. Effect of the position of the reinforcement element on the evolution with of the horizontal displacement at node A.

In this When the angle slope-reinforcement is equal to 60◦ , we obtain the lower values of the slope horizontal displacement slope as compared to the other cases (figure 13): this displacement is reduced by about 50% near the crest. 4.5 Effect of reinforcement elements number The effect of number of reinforcement elements on the slope stability has been tested (length = 10 m and reinforcement element perpendicular to the slope surface). So, three configurations have been analyzed:

Figure 12. Effect of the position of the reinforcement element on the slope horizontal displacement at time = 15 s.

The numerical results show that the reinforcement element at the crest is the less efficient position (figures 11 and 12). The horizontal displacement at node A is reduced by 13% at the end of the loading with the lower position of the reinforcement element. Moreover, at time equal to 15 s, it can be seen that the slope horizontal displacement is reduced by approximatively 20% while the reinforcement element is in the lower position (figure 12).

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– The slope is reinforced by one element (reference case, see figure 7) – The slope is reinforced by two elements (dashed lines in figure 15) – The slope is reinforced by three elements (figure 15 and the tip of the third element is at the slope crest). The reinforcement of the slope by using two elements leads to a significant improvement of its stability (figure 14): the horizontal displacement at crest is reduced by 42% as compared to the non reinforcement slope results. The reinforcement of the slope by using three elements has an insignificant effect on the stability according to the obtained values when using two reinforcement elements. The shape of horizontal displacement around the slope at time equal to

investigated. The elastoplastic modeling leads to more realistic results compared to those obtained when used elastic calculation. The plasticity leads to a significant residual displacement in one hand and to a reduction of the velocity amplification. A parametric study according to length, position, inclination and number of reinforcement elements has been performed. The following conclusions are related to the particular problem treated in this paper. The efficient length of the reinforcement element should be defined according to the failure circle shape obtained by a reference calculation without any reinforcement. The position of the reinforcement improves the stability (not at the crest). The optimal inclination is equal to 60◦ . Figure 14. Effect of the reinforcement element number on the slope horizontal displacement.

Figure 15. Shape of lateral displacement around the slope at time equal to 15 s in the case of two reinforcement elements.

15 s clearly shows an important reduction of this displacement comparing to the case on one reinforcement element (figures 10 and 15). ACKNOWLEDGMENT We think the Lebanese University and the Lebanese National Council of Scientific Research for partially funding of this work. 5

CONCLUSION

In this paper a finite difference numerical global dynamic modeling of slope stability has been presented. A typical problem of slope has been modeled. The effects of the soil behavior and of the reinforcement elements on the slope stability have been

REFERENCES Bourdeau C. 2005. Effets de site et mouvement de versant en zone sismique : apport de la modélisation numérique, thèse Ecole de Mines de Paris Chen W. F. and Scawthorn C. 2003. Earthquake Engineering Handbook, CRC Press LLC Feng X et Guo A. 1985. Earthquake landslide in China. Proceedings of the fourth international conference and field workshop on landslides, Tokyo, p. 339–346 Jibson R.W., Prentice C.S., Borissoff A., Rogozhin A. and Langer C.J. 1994. Some observations of landslides triggered by the 29 April 1991 Racha earthquake, Republic of Georgia. Bulletin of the Seismological Society of America, 84: 963–973 Keefer, D.K. 1984. Landslides caused by earthquakes. Bulletin of the seismological society of America, 95, p. 406–421 Kuhlmeyer R. L. and Lysmer J. 1973. Finite Element Method Accuracy for Wave Propagation Problems, J. Soil Mech. & Foundations Div., ASCE, 99(SM5): 421–42 Itasca Consulting Group, FLAC: Fast Lagrangian Analysis of Continua, vol. I. ’s Manual, vol. II. Verification Problems and Example Applications, Second Edition (FLAC3D Version 3.0), Minneapolis, Minnesota 55401 USA, 2005 Newmark N.M. 1963. Effects of earthquakes on dams and embankments, Fifth Rankine Lecture – Géotechnique 15(2): 139–193. Parish, Y., Sadek, M., and Shahrour, I. 2009. Review Article: Numerical analysis of the seismic behaviour of earth dam”, Nat. Hazards Earth Syst. Sci., 9: 451–458 Rodriguez C.E., Bommer J.J. and Chandler R.J. 1999. Earthquake-induced landslides: 1980-1997. Soil dynamics earthquake engineering, 18: 325–346. Terzaghi K. 1950 Mechanism of landslides. The geological survey of America Engineering Geology, Berkley

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Numerical analysis of blast impact on sealings of neighbouring structures W. Krajewski & O. Reul University of Applied Sciences Darmstadt/CDM Consult GmbH,

L. te Kamp ITASCA Consultants GmbH,

ABSTRACT: Two case histories of sealings of structures subjected to seismic loads are presented. The first case history deals with the base sealing of a landfill, which has been stressed dynamically during the construction of a drainage gallery. The proof procedure showing that the excavation blastings have been acceptable for the sealing is discussed. In the second case history, a pit for blast impacts will be sealed at the base by a plastic liner and a concrete raft to protect the groundwater. Extended investigations comprising large scale blasting tests and dynamic finite element and finite difference analysis, respectively, have been performed to investigate the influence of the blast impacts on the sealing. In summary, the case history shows, that numerical analyses are well suited for the investigation of seismic problems. However, the required scientific and technical knowledge as well as the expenditure for system modelling and calibration are rather high.

1

INTRODUCTION

An increasing number of underground structures are driven in the immediate vicinity of existing facilities and buildings. Therefore it is sometimes necessary to carry out blastings close to the sealings of those neighbouring structures. Sealings, usually comprising of clay and/or flexible plastic liners, are installed in order to protect the groundwater from substances within the structure or vice versa. In the past, studies concerning the integrity of the sealings have been carried out almost exclusively based on experimental methods or on empirical data, respectively. Recently, numerical methods are applied more frequently to investigate the performance of sealings under blast impacts. The first case history presented in the scope of this paper deals with the base sealing of a large domestic landfill site subjected to blast driving for a new drainage gallery in immediate vicinity. For this project the influence of the blastings on the sealing have been estimated conventionally. In the second case history the performance of the sealing of a pit for blasting operations has been investigated by means of dynamic numerical analysis and measurements carried out during a series of test blasts.

2

BASE SEALING OF A LANDFILL

2.1 System The drainage installations of a landfill for sludge and domestic waste within the Rhine-Main-area has been redeveloped by the construction of an underground

Figure 1. Rehabilitation of the drainage system of a landfill by a system of drainage galleries: Performance of blast impact tests.

drainage gallery system. The system comprises a 840 m long tunnel as well as several inclined galleries and shafts which reach up to the base of the deposit (Figure 1). The underground excavations have been carried out by means of blast driving. As the blastings have been executed in close neighbourhood of the existing clay liner, the compatibility of the dynamic impact had to be proofed. However, there existed no experiences

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concerning the magnitude of the allowable dynamic impact. 2.2

Blast impact tests

The magnitude of the seismic velocities due to excavation blastings has been measured during field tests in the shafts and inclined galleries. All tests have been performed at the faces of the approximately 1.5 m high bench. For each blasting eight holes have been bored. Each hole has been filled with detonating cord Dynacord 100 (Figure 1). The specific consumption of blasting agent amounted to approximately 300 g/m3 . During the blastings, the speed of vibrations was measured at the accessible top surface of the base sealing using geophones. The distance between the blastings and the geophones was approximately 12 m in average. The measured seismic velocities amounted to v = 1.5 mm/s to v = 2.7 mm/s while the frequency spectrum ranged between f = 4 Hz and f = 78 Hz. In a first step the seismic displacements have been derived from the available measurements to evaluate their compatibility for the base sealing. Assuming a sinus-shaped curve progression for the vibrations, the displacement si in i-direction is given by:

where vi = seismic velocity in i-direction, and f = frequency. In horizontal direction maximum seismic displacements of sx = 68 µm and sy = 44 µm, respectively, have been calculated. In vertical direction the maximum displacement amounts to sz = 57 µm. Under the assumption that – the maximum seismic displacements for mass points on the top surface of the base sealing interfere with each other and – that the corresponding mass points at the bottom surface of the base sealing have zero displacements It is possible to evaluate the strains occurring in the clay liner of the base sealing. The resulting strains and distortions can be simulated in a soil mechanical triaxial test. The distortion γ* of the base sealing with the thickness d is given by:

The vertical strain ε1 which has to be applied in the triaxial test can then be approximated by:

where ν = poisson ratio of the clay liner. Under consideration of the amplification and an additional safety margin the vertical strain amounts to ε1 ≈ ±95 µm/m.

In a laboratory test these vertical strains have been applied on a clay sample in approximately 2,000 cycles during the period of one week. Afterwards the water permeability of the sample has been measured. The test showed that the cyclic loads did not have a negative effect on the material behaviour of the sealing material. Thus, for the excavation works a seismic velocity of up to vi = 30 mm/s was allowed. However, during construction single blasting events with remarkably higher seismic velocities occurred without causing damage to the base sealing. This leads to the conclusion that the allowed seismic velocity still contains a large safety distance to ultimate limit conditions. 3

PIT FOR BLAST OPERATIONS

3.1 System In southern currently a pit for blast operations is built. The pit is about 10 m deep with a slope angle of 25◦ (Figure 2). The bedrock consists of weathered limestone. After installation of the blasting explosives, the pit is filled with broken granite. The blastings can be performed at the surface level, in 2 m depth or in 5 m depth, respectively. On the surface level the specific blasting agent amounts up to approximately 30 kg net explosive mass (NEM). The loadings sparked inside of the test pit are 250 kg (2 m depth) or 115 kg NEM (5 m depth), respectively. Due to the high water permeability of the limestone a pollution of the groundwater due to the explosives or their reaction products is possible. Therefore the slopes and the base of the pit will be sealed with flexible plastic liners (PE-HD). At the bottom, the liners are protected by a 30 cm thick reinforced concrete raft. First estimations of the seismic velocities, which may occur at the sealing showed that rather high values of approximately vi ≈ 300 mm/s to vi ≈ 500 mm/s have to be taken into . These values are more than one decimal power higher than the issible seismic velocities for the base sealing of the landfill as presented in section 2. Therefore detailed investigations concerning the influence of the blastings on the sealing system were necessary. 3.2 Preliminary numerical analysis In a first step, simplified dynamic calculations based on finite element analysis (Plaxis 2009) have been executed using an axis-symmetric model. The blasting pit, the concrete raft and the subsoil were modelled with triangle shaped 15-node-elements for which the strains are zero in the tangential direction due to the axis-symmetric conditions. The stress-strain behaviour of the subsoil is characterized by the socalled Hardening-Soil-model which models the nonlinear stress-strain behaviour of soils as well as the different behaviour during loading, reloading and unloading conditions. The concrete raft and the backfilling material of the test pit are considered to behave linear-elastically.

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Figure 2. Blasting pit: Simplified cross section.

Figure 3. Blasting pit: Dynamic Impact.

The blasting load is modelled by the seismic impulse given in Figure 3 (“Original Blast Input”). For all simulations carried out, the maximum impulse is reached after approximately 15 ms. After 50 ms the impulse has decreased to zero. In the finite element mesh the blasting load is modelled as a pressure at the surface of a spherical blasting chamber with a diameter of 1 m and its centre situated at the blasting point. The magnitude of the impulse is chosen based on test experiences, which show that the detonation of 250 kg of explosives lead to principal stresses of about 10 MPa at the edge of the blasting destruction zone. Further considerations show that an impulse of 3,500 MPams has to be used for an appropriate reproduction of the blasting load. For lower blasting loads the impulse is reduced linearly. The damping factors were taken to α = 2.0% and β = 2.0%. As an example of the extensive results achieved, the time dependent displacements are presented in Figure 4 for selected points on the concrete raft. The vertical displacements reach maximum values of uy ≈ 1.5 mm up to uy ≈ 3.8 mm between t = 0.05 s and

Figure 4. Test pit, finite element analysis: Horizontal and vertical displacements for selected points on the concrete raft.

t = 0.065 s. Afterwards, the oscillation of the system is damped. As an important result of the axis – symmetric finite element analysis it can be concluded, that the blasting causes no significant vertical strains in the sealing. The horizontal displacements of the sealing increase from ux ≈ 0 at the centre of the raft up to ux ≈ 0.072 mm at the edges. The maximum horizontal

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Figure 6. Test pit: Finite difference mesh.

Figure 5. Large scale test pit (depth: 5 m).

strain of the plastic liners amounts to a sufficiently low value of ε ≈ 10−4 . In spite of the simplifications made, the preliminary analysis provided useful information for the further design of the test pit: – The maximum strains of the plastic sealing liners will probably range between ε = 1 · 10−4 to ε = 3 · 10−4 . – The vertical pressure acting on the sealing system will amount to approximately σ ≈ 800 kPa. – The seismic velocity of the sealing systems will range between approximately v ≈ 150 mm/s to v ≈ 300 mm/s. – The maximum seismic acceleration will reach up to approximately a ≈ 8 m/s2 . 3.3

protection measures. The investigations show that even at small distances of less than 2 m between sealing and explosives (test 2 with 15 kg NEM) no mechanical or thermal damages could be observed. 3.4 Numerical analysis of the large scale tests To achieve a deeper understanding of the results of the large scale tests as well as to predict the seismic loads on the sealing system during the planned operational blastings, three-dimensional numerical analysis has been performed applying the finite difference method (Itasca 2006). At this stage the known input parameters were – Geometry and geomechanical characteristics of the different soil layers and sealing system, respectively. – Explosive charges (kg NEM) and geometrical position of the blasting points. – Seismic velocities measured in the large scale tests.

Large scale tests

In addition to the numerical analysis, large scale blastings tests have been performed in a 5 m deep blasting test pit (Figure 5). The test pit was designed similar to the planned operating blasting pit but with a smaller scale. Amongst others, a test with 35 kg NEM installed in a depth of 2 m has been carried out. The resulting seismic velocities and seismic accelerations have been measured with geophones, showing extremely high seismic velocities of v > 800 mm/s at the sealing system. After finishing the tests the sealing system has been uncovered and has been inspected carefully. In spite of the significant seismic velocities, the investigations showed that the plastic liners as well as the concrete raft were entirely intact. Moreover, the observations and the interpretation of the test results showed: – The seismic velocities, measured in the neighbourhood of the blasting point are higher than expected. However, a strong damping of the seismic impact in the back-filling of the test pit leads to rather small velocities at larger distances. – The seismic oscillation is of low frequency (<70 Hz). – The strains of the sealing and the concrete raft due to blasting impact are small and do not demand special

Unknown parameters were the sequence of the detonations as well as the correlation between blasting pressure and explosive charge. The detonations were modelled by a normal pressure loaded at the surface of a spherical blasting chamber (see section 3.2). In a first investigation step the numerical model was calibrated by reproducing the results of the field tests (large scale blasting test). The applied threedimensional model has geometrical dimensions of approximately x/y z = 30 m/30 m/16 m (Figure 6) and comprises 220,000 zones. The time dependent development of the normal pressure was first modelled according to Figure 3 (“Original Blast Input”). However, the calibration procedure showed best fittings of test results using the “Modified Blast Input A” shown also in Figure 3. The detonation pressure was evaluated to 8 MPa relating to an explosive charge of 15 kg NEM. Since no information concerning the correlation between detonation pressure and explosive load was available, the detonation pressure was assumed to be a linear function of the explosive load. Two different types of damping have been introduced in the numerical model. The following parameters have been evaluated in the calibration procedure:

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Figure 8. Operational blasting pit: Finite difference mesh.

Figure 7. Test pit, calibration analysis: Comparison between measurements and analysis results.

– Rayleigh-damping with 0.75% of the critical damping and an intermediate frequency of f = 2500 Hz. – Artificial viscosity, chosing Neumann-term and Landshoff-term to 0.6 each. Figure 7 shows the measured and calculated seismic velocities for one selected point of the sealing system. The velocities in y- and z- direction could be modelled with sufficient accuracy. However, only 50% of the measured values in the horizontal x-direction could be reproduced numerically. One reason for the discrepancy between measurements and numerical simulation is the fact that a spherical propagation of the blast was modelled. In reality a grenade which was covered with a carpet was the source of the blast. This yielded in a directional blast which was not modelled numerically because of insufficient knowledge of the blast direction. 3.5

Figure 9. Operational blasting pit, 115 kg NEM: Seismic velocities in the concrete raft.

Numerical analysis of the operational blastings

After calibration of the numerical model, the situation during the planned operational blastings has been investigated. For these investigations the numerical model had to be extended to geometrical dimensions of approximately x/y/z = 115 m/115 m/900 m (Figure 8). For the discretization of the system, 680,000 zones have been introduced. For the operational phase two blastings with different explosive load have been investigated: – 250 kg NEM, depth of blasting chamber 2 m below ground surface. – 115 kg NEM, depth of blasting chamber 5 m below ground surface. As an exemplary result of the numerical analyses for the 115 kg blasting, Figure 9 shows the seismic velocities which have been calculated for several points on the concrete raft. The maximum seismic velocities amount to approximately v ≈ 400 mm/s at point 17 at the centre of the raft. The other points which are located in a distance

Figure 10. Operational blasting pit, 115 kg NEM: Strains in the concrete raft.

between 5 m and 7 m from the centre of the raft show a significant decrease of the seismic velocities. The rather large seismic velocities at the centre of the raft are accompanied by maximum strains of ε ≈ −0.005 % (Figure 10), which are sufficiently small and are considered to be tolerable for the concrete raft. The maximum principal stresses increase up to σ ≈ 1.4 MPa (Figure 11). This load is small compared to the uniaxial stresses at failure of the concrete of σf > 20 MPa.

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Figure 11. Operational blasting pit, 115 kg NEM: Maximum principal stresses in the concrete raft.

the blasting pit will be rather small and will be tolerable for the sealing system. The large scale field tests and the numerical analyses have been essential in providing a reliable prediction of the strains and stresses in the sealing due to a blast impact. The investigations show the complexity when modelling seismic problems. The numerical modelling of dynamic loads and the evaluation of the relevant material parameters proved to be challenging. The calibration of the numerical model requires defined boundary conditions which can be provided in model test. However, for real structures or even large scale tests the identification of boundary conditions can be difficult. It is therefore one of the major future tasks to broaden the experience and knowledge of dynamic numerical analysis for the geotechnical engineering practice.

4

REFERENCES


It can be stated, that the operational blastings with explosive loads of 115 kg and 250 kg NEM, respectively, probably will cause seismic velocities of approximately v = 450 mm/s at the sealing system. Seismic velocities of this magnitude have already been measured in field tests and have been proofed as tolerable. In spite of this rather high seismic velocity, the strains and stresses occuring at the sealing system of

CDM Consult GmbH, 2009. project reports, unpublished, Stuttgart (). ITASCA. 2006. Fast Lagrangian Analysis of Continua in 3 Dimensions – FLAC3D , Vers. 3.1”. ITASCA Consulting Group, Minneapolis, Minnesota, US. Krajewski, W.; Weiß, J..; Ernst, D. 1998. Ertüchtigung der Dränage einer Deponie durch den Bau eines bergmännischen Stollensystems, geotechnik 21, Nr.3. Plaxis bv. 2009. PLAXIS 2D, Dynamics Module. Version 9.

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Numerical analysis of the seismic behavior of vertical shaft Sangseom Jeong, Yongmin Kim & Sungjun Lee Yonsei University, Korea

Jungbum Jang & Yonghee Lee Korea Electric Power Research Institute, Korea

ABSTRACT: Dynamic responses of a vertical shaft subjected to seismic loads in a layered soil were investigated by using a three-dimensional Finite Element(FE) approach. The emphasis was on quantifying the load distribution and deformation of the vertical shaft under earthquake loadings in a multi-layered soil. It was found that the dynamic behavior of the vertical shaft was significantly influenced by the soil stratigraphy and input motion. Based on the FE analysis results, the maximum values of shear force and bending moment occurred near the interface between the soil layers. The deformation and loading values of vertical shaft was highly influenced by the amplitude of earthquake for the case of the vertical shaft constructed in a multi-layered soil.

1

INTRODUCTION

South Korea suffers from serious lack of land space due to its high population of about 47 million people on a little less than 100,000 km2 of land and the fact that 75% of the land space is mountainous. There is a great demand for development of underground space. Recently, a number of huge underground construction projects such as subways, tunnels, and underground infrastructures have been frequently performed in urban areas. It is necessary to construct the underground structure deeper and larger than existing structure foundations and tunnels in the ground. Vertical shafts are so essential to underground construction such as a ventilation system, working place in underground structures and vertical approach to deep tunnel. However, vertical shaft has higher earthquake risk as compared to horizontal tunnel since vertical shaft is frequently constructed to multi-layered soil which has amplified deformation and loading by the earthquake waves. Much work has been carried out on horizontal tunnels under seismic loads (An et al. 1997, Hashash et al. 2001, Huo et al. 2005, Kawashima 2006). A relatively less work has been done on the vertical shaft, with most works (Kaizu 1990, Kato et al. 1991, Ohbo et al. 1992) focusing on the performance of vertical structures which are limited to site-specific characteristics. The seismic design of underground structure is characterized in of dynamic deformation and stress-strain relationships on the structure due to a fundamental deformation of the surrounding soil during earthquakes. In case of vertical shaft, the special attention is given to evaluate the fundamental deformation of the surrounding soil and the ground motion parameters such as peak accelerations, velocities, target

response spectra, and ground motion time histories. These relations are based on the premise that vertical shaft under seismic loading will tend to deform with the surrounding soil, and thus the structure is designed to accommodate the free-field deformations without loss of its structural integrity (Kaizu 1990, Kawashima 2006). However, it is necessary to investigate the influence of soil and loading conditions on the vertical shaft since there exist some uncertainties among researchers. The overall objective of this study is to perform the rational and economical seismic design for vertical shaft. Therefore, a series of dynamic finite element (FE) analyses were performed for the deformation characteristics, the shear force and the bending moment developed in circular Reinforced Concrete vertical shaft. 2

NUMERICAL ANALYSIS

In order to fully understand the seismic behavior of vertical shaft, a detailed structure and its surrounding soil are modeled by Finite Element Method(FEM) software package, ABAQUS (2008). This software is a powerful, general purpose FEM program which has been widely used in many seismic soil-structure interaction problems, and the code has been extensively verified for static and dynamic analyses of underground structures. It allows the frictional elements between the ground and the structure. Analyses and interpretations of the numerical results focus on the shear force and the bending moment of the vertical shaft. Issue including amplitude of normalized acceleration along the surrounding soil and the structure was considered in this paper.

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2.1

Model description

The three-dimensional model includes standard finite element techniques. The finite element mesh for a typical case is shown in Figure 1. The surrounding soil has been modeled using three-dimensional 8-noded solid hexahedral elements (Lee et al. 2006, Jeong & Won 2009). The cross-section of the vertical shaft has been represented by cylindrical shell elements with a thickness of 0.5 m. The vertical shaft element is assumed to remain elastic at all time, while the surrounding ground is idealized as a linear elastic material first and then as an elasto-plastic material. This model was selected among the soil models in the library of ABAQUS (2008). All the analyses were performed considering a 60 m-height and a 9 m outer diameter vertical shaft. The material properties of the reinforced

Figure 1. The finite-element model of the vertical section of the shaft and surrounding ground. Table 1.

Vertical Shaft Case A Case B Case C

concrete are: modulus of elasticity (E) of 28 MN/m2 , Poisson’s ratio (υ) of 0.2, and mass density equal to 2500 kg/m3 . The surrounding soil consists of two layer system. As shown in Figure 1, total depth (H), in a vertical layer is 60 m. The first layer (h1 ) and second layer (h2 ) of soil were varied from 20 m to 40 m. It was assumed that the bedrock is located at a depth of 60 m. Properties of material and ground conditions for numerical analyses were summarized in Table 1, where the parameters are listed: unit weight (γ), soil and structure modulus (E), mean shear wave velocity (Vs ), cohesion(c), internal friction angle (φ), Poisson’s ratio (υ). In dynamic analysis, both the structure and the surrounding ground are modeled by three-dimensional of FEM, an issue is the effect of the location and nature of the lateral boundaries on the response of the soilstructure system. This is needed because the model of the continuum requires the existence of a finite domain with well-defined boundaries. If the lateral boundaries are created artificially, it becomes necessary to determine appropriate conditions that simulate the physical behavior on the actual system. The appropriate boundary conditions should work as energy sinks rather than energy reflectors in the sense that the energy transmitted to the lateral boundary through the soil media should not be reflected back to the structure. Otherwise, the solution would be affected by the reflected energy between the structure and boundaries of the ground which does not exist in reality. In this paper, viscous dashpot (damper) boundaries have been placed on the right and left-hand artificial boundary. It is based on the absorbing boundaries in order to simulate the radiation of energy. Viscous dashpot boundary is achieved using horizontal and vertical viscous dashpots, which absorb the radiated energy from the P and S waves, respectively. The efficiency of the viscous dashpots is quite acceptable, but as it depends strongly on the angle of incidence of the impinging waves the dashpots were placed at the boundaries to improve the accuracy of the simulation. The interface between the structure and the ground were modeled as a frictional surface. The can open if there is a tensile normal stress or it can slip if the magnitude of the applied shear stress is larger than the shear strength, which is assumed to follow the Coulomb friction law. A coefficient of friction, µ,

Material Properties.

Model

Thickness (m)

γ (kN/m3 )

E (MPa)

Vs (m/s)

c (kPa)

φ (deg)

υ

– –

– –

– –

0.2 0.2

Side wall Top and bottom slab h1 h2 h1 h2

Elastic Elastic

0.6 0.5/1.0

25 25

28,000 28,000

Mohr-Coulomb Mohr-Coulomb

20 40

18 21

41 807

179 761

100 0

30 40

0.4 0.3


30 30

18 21

41 700

179 761

100 0

30 40

0.4 0.3

h1 h2


40 20

18 21

41 807

179 761

100 0

30 40

0.4 0.3

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Figure 3. Comparison of shear force and bending moment on vertical shaft (case A).

Figure 2. Input accelerations.

equal to 0.4 is assumed which corresponds to a friction angle of 22◦ ; no cohesion between structure and ground is used (Huo et al. 2005). In this study, a special attention was paid to the initial stress field of soil. In ABAQUS program, at first the initial stress is established in soil through initial condition command and the adding the gravity of soil. In the actual application for soil, the stress induced from the soil weight over the calculation point is considered as the vertical stress and the horizontal stress is obtained through the vertical stress multiplied by the lateral pressure coefficient K0 . In this paper, the initial horizontal stresses in the soil were set up according to a K0 value of 0.5. The ground motions imposed at the bottom of the model for the numerical analyses are the motions ed

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Figure 5. Comparison of shear force and bending moment on vertical shaft (case C).

Figure 4. Comparison of shear force and bending moment on vertical shaft (case B).

at the Hachinohe and Ofunato in Japan. Also artificial earthquake was made to evaluate the behavior of vertical shaft as a ground motion. The input accelerations are specified on the bottom of the soil and at rock level, to for the amplification effects of the soil layer and its influence on the results. The applied maximum horizontal accelerations were applied about 0.154 g (Figure 2). 3 3.1

NUMERICAL RESULTS Shear force and bending moments on vertical shaft

Figures 3–5 show the shear force and bending moment distributions of vertical shaft in case A, case B and

case C, respectively. It shows the maximum value in a certain time along the depth. It can be observed that shear force and bending moment of vertical shaft occurs maximum value at a point between the soil layers. The maximum shear force and bending moment due to relative displacement is too large between the soil layers; displacement of upper soil is larger than lower soil. Also, based on case B result, shear force = 1486.12 kN which is 1.6 times larger than the value of 981.11 kN can be obtained for the vertical shaft at point between the soil layers when the Ofunato earthquake was induced. Figure 6 shows the comparison of maximum shear force and bending moment on vertical shaft. It can be observed that shear force and bending moment occurs at the largest value in the case A, and then the smallest

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Figure 6. Comparison of maximum shear force and bending moment on vertical shaft.

values in the case C ground. Earthquake waves and ground conditions slightly affects the seismic response of the vertical shaft.

3.2 Acceleration amplitude ration between vertical shaft and ground The influence of the ground conditions on the vertical shaft response to the three earthquake was investigated through analyses conducted for a value of acceleration of the input motion (amax = 0.154 g). Figure 7 shows the characteristics of vertical shaft and surrounding soil under earthquake loading. The normalized acceleration(aeach_point /abottom ) at each point in the surrounding soil and the vertical shaft was obtained at points 0 m, 20 m, 40 m and 60 m respectively. It can be observed that the normalized acceleration of the vertical shaft is smaller than that of the surrounding soil in three ground cases. These results are based on characteristics of underground structure which tends to deform with the surrounding soil. Especially, normalized acceleration which had most significantly occurred was estimated in the case A, because earthquake wave was amplified through a layered soil.

Figure 7. Spatial distribution of normalized acceleration along the vertical shaft and the ground for three earthquakes.

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4

CONCLUSION

This paper is based on the analysis of the seismic behavior of vertical shaft. Numerical simulations were performed for three earthquakes using ABAQUS. The main objective of this study is to investigate the rational seismic design that can accommodate various factors influencing the deformation of the vertical shaft. A series of dynamic FE analyses were conducted to determine the behavior of the shaft under seismic loading. Based on the findings of this study, the following conclusions can be drawn: 1. Based on the FE results, it is shown that the dynamic behavior of the vertical shaft is significantly influenced by the soil stratigraphy and input motion. 2. The location of maximum shear force and bending moment is needed to be checked for seismic design of vertical shaft. It is important to note that almost all of the maximum values occur near the interface between the soil layers. 3. The deformation and loading values of vertical shaft and surrounding soil was highly influenced by the amplitude of earthquake for the case of the vertical shaft constructed in a multi-layered soil. REFERENCES

An, X., Shawky, A.A. & Maekawa, K. 1997. The collapse mechanism of a subway station during the great hanshin earthquake. Cement and concrete composites. Vol.19: 241–257. Jeong, S.S. & Won, J.H. 2009. Effects of vertical load on the lateral response of single pile and pile groups in clay. International journal of geo-engineerirng. Vol. 1: 11–20. Kaizu, N. 1990. Seismic response of shaft for underground transmission line. Proceedings from the Third Japan-U.S. workshop in earthquake resistant design of lifeline facilities and countermeasures for soil liquefaction. 513–525. Kato, K., Ohbo, K., Hayashi, K. & Ueno, K. 1991. Earthquake observation of shaft and ground(in Japanese). Proceeding of the 46th annual conference of the JSCE. 1261–1262. Kawashima, K. 2006. Seismic analysis of underground structures. Journal of disaster research. Vol.1: 378–390. Lee, C.J., Lee, J.H. & Jeong, S.S. The influence of soil slip on negative skin friction in pile groups connected to cap. Geotechnique. Vol. 56: 53–56. Ohbo, N., Hayashi, K. & Ueno, K. 1992. Dynamic behavior of super deep vertical shaft during earthquake. Earthquake engineering 10th world conference. 5031–5036. Hashash, Youssef M.A., Hook, Jeffrey J., Schmidt, B. & Yao, John I.C. 2001. Seismic design and analysis of underground structures. Tunnelling and underground space technology. Vol. 16: 247–293. Huo, H., Bobet, A., Fernández, F. & Ramirez, J. 2005. Load transfer mechanisms between underground structure and surrounding ground: Evaluation of the failure of the Daikai station. Journal of geotechnical and geoenvironmental engineering, ASCE. Vol. 131: 1522–1533.

ABAQUS theoretical ’s manual. 2008. Hibbit, Karlsson and Sorensen, Inc.

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Numerical and experimental study of the detection of underground heterogeneities P. Alfonsi, E. Bourgeois, F. Rocher-Lacoste & L. Lenti Université Paris Est, LC, MSRGI, Paris,

M. Froumentin Centre d’Etudes Techniques de l’Equipement Normandie-Centre, Rouen,

ABSTRACT: Ground investigation in urban areas is often limited to the identification of the main layers, but it is not common to undertake to locate underground heterogeneities, although they can have a significant influence on the delays, in case caves, buried foundations, or archaeological remains of great historic value are present in the subsoil. In order to assess the performance of the methods that can be used to detect the presence of such heterogeneities, a full scale experiment was carried out in which two anomalies with very different densities have been buried at different depths in a ground layer. Experimental wave propagation tests were performed and the results were compared with numerical simulations. We discuss to what extent numerical computations can provide a way of getting valuable information if little experimental data is available.

1

INTRODUCTION

Ground investigation is seen as essential for majorprojects (tunnels, railways) but generally very restricted in common urban sites. However, many research works in this field aim at the detection of voids in the subsoil, to prevent collapses caused by tunnelling or by vibrations induced in the ground by high speed trains (Picoux, 2002). When underground works or excavations are made urban areas, buried structures or archaeological remains may be discovered and damaged. It is generally thought that micro gravimetry (which is based on a accurate measurement of local variations of the gravity field) or radar are fruitful for cavities detection or archaeological prospection, but need fine tuning and the results depend on the anisotropy of the materials. Geophysical methods can be used to investigate the structure of the subsoil from the surface, but they do not seem to be very suitable in heterogeneous grounds: electromagnetic methods (radars), seismic methods (Baltazart and al, 2006), and other techniques have been tested (Lagabrielle and Grandsert, 2005, Léonard, 2001). Some researches aim at extending the applications of such methods to more general contexts, by improving the experimental devices (Leparoux and al, 2000, Abraham and al, 2006) or the numerical processing of the collected data (Chammas, 2002). Another view point consists in assuming that softwares performing three-dimensional computations can be used to improve the insight gained from experimental results. In order to this idea, we have compared experimental results with numerical

Figure 1. Side view of the numerical model.

simulations. The main objective is to check that numerical models can help analyzing experimental data, and to define the appropriate numerical treatment that can be applied to the signals in order to extract valuable information regarding the subsoil structure.

2

EXPERIMENTAL SETUP

An experimental excavation was filled with two different materials, and two anomalies were buried at different depths: one is a “cavity” (made of honeycomb plastic structure), the other is a square area filled with flint, denser than the surrounding ground. The dimensions of the excavation vary between 15 m × 15 m at the maximum depth of 5 m, and 30 × 15 m at the ground surface (fig. 1). It is enclosed in the space between two vertical concrete walls. The underlying layer is made of chalk. The excavation was filled with a layer of sandy fill (for depths smaller than 2 m), and a layer of limestone between 2 and 5 m in depth. A simple investigation campaign was undertaken, aiming at locating the heterogeneities on the basis of wave propagation tests in the ground: a shock is emitted at one point of the surface, and waves are measures

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Table 1.

Mechanical properties.

Materials

ρ (t/m3 )

E(GPa)

ν

Vs (m/s)

Vp (m/s)

Natural ground Fill Limestone Flint Concrete Chalk

2.4 2.4 2.2 2.8 3.4 2.4

6 10 3 40 50 20

0.3 0.26 0.32 0.2 0.18 0.28

310 407 227 772 1116 570

580 714 442 1260 1787 1032

Figure 2. Top view of the numerical model with the positions of the shocks and of the heterogeneities.

by means of geophones placed on the surface near the impact. Six 3D geophones were placed on a line, first in the vicinity of the cavity, then on a line located between the heterogeneities. Shocks were emitted on four different points, using a mace. For each test, geophones were placed in line with the shock. The position of the shocks are shown by the white dots on the top view of figure 2, the other dots showing the positions of the geophones. Tests 1 and 2 were carried out with the shocks and the geophones on the upper line of the figure, which goes above the cavity ; tests 3 and 4 with the lower line (between the void and the flint zone).

Figure 3. Close up on the first wave in the geophone signals.

4 3

NUMERICAL MODEL

RESULTS

4.1 Experimental results

Three dimensional finite element simulations (dynamic and linear wave propagation analyses) are carried out with the software package CESAR-LC. The dimensions of the mesh reproduce the geometry of the actual site. To simplify the mesh and to keep within the limits of the available computation tools, the sizes of the cavity and the flint zone are larger than the actual ones. To model the propagation of waves by finite elements, one has to choose a suitable size for the elements, with respect to the wavelength, keeping in mind that numerical tools do not make it possible to solve arbitrarily large linear systems. Newmark’s method of direct integration is applied in basic dynamic calculation for advancing the resolution in time. We use the implicit values δ = 2.α = 1/2 for the control parameters. The distance between nodes in the horizontal and vertical directions was roughly 50 cm. The mesh includes 45000 nodes and 45000 quadratic elements. Displacements are set to zero on the lateral and lower boundaries. Properties of the materials are given in table 1; it is recalled that wave velocity depends on the stiffness of the material they propagate in, and that elastic moduli of geomaterials in the range of very small strains are much larger than the moduli generally adopted in the analysis of the behaviour of geotechnical works under service or limit loads.

Geophones provide displacements along three axis every 0.001 second. The signals recorded by the six geophones A, B, C, D, E and F, show clearly four waves in the first seconds of the records of the vertical component. The velocity of waves across materials, and the time needed to get around a cavity results in differences in the measured vertical displacements for different points of the ground surface. Such differences can be seen by using a suitable time scale (fig. 3). For each geophone, we point out the time between the shock and the first minimum and the first maximum, for each of the four waves. An average value of the time lags is computed (grey lines in figures 4 and 5). The amplitude of the initial shock was not precisely controlled, so that reproducibility of the shocks is questionable. Note also that the absolute values of the vertical displacements depend on the shock, which prevents from comparing results obtained for two different shocks. However, the shape of the averaged signal is slightly different between a shock emitted above the cavity (test 1, fig. 4) and a shock emitted at a larger distance from the anomaly (test 2, fig. 5). Nevertheless, it remains difficult to define a detection procedure on the basis of the experimental signals. We tried to achieve this by means of numerical simulations.

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Figure 4. Results of test n◦ 1: time lags between geophones responses (vertical scale is not relevant). Figure 6. Numerical model for one single centered anomaly (top view).

Figure 7. Amplitude of the vertical signal at different points between y = 8.5 and y = 16.96 m. Figure 5. Results of test n◦ 2: time lags between geophones responses (vertical scale is not relevant).

4.2 Results of numerical simulations Several simulations were carried out in various configurations, some of them simpler than the actual experimental structure. Numerical simulations were performed at first for homogeneous ground layers in order to obtain reference signals, then with anomalies, either void or filled with a material (flint) in which wave velocities are higher. 4.2.1 Simulation with one centered anomaly In the first step, we performed simulations with only one anomaly placed in the center of the excavation, in the area (x, y, z) ∈ [7, 10 m] × [8.5, 10.5 m] × [−4, −2 m]. Depending on the simulations, it is made of limestone (i.e. identical to the surrounding ground) or flint, or void. A vertical force is applied on the surface at x = 8.5 m , y = 12.5 m ; the forces varies over a short time interval, following a bell-shaped curve.

The influence of the anomaly is discussed by comparing the displacements computed at the surface (at the positions of the black dots of figure 6). Results are compared with the vertical displacements obtained if the fill and limestone layers are homogeneous. Simulations reproduce 400 time steps of 0.1 ms, and last roughly 2.5 hours on standard hardware. The vertical displacements computed at the surface show a shift in the time for which the minimal value is obtained, depending on the distance to the source of the signal. This time lag is also clear for the first peak of vertical upward displacement, then becomes less distinct after 15 ms (fig. 7). It seems therefore preferable to limit the analysis to the first 15 ms of propagation. It can be noted also that the maximum amplitude in obtained at the location of the signal source, y = 12.5 m. Figure 8 (respectively 9) shows the time lags obtained for the first minimum (resp. maximum) of the vertical displacement, for different points of the surface.

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Figure 8. Time lags for the first minimum of the vertical displacement.

Figure 10. Ratio of the amplitude between the model containing a void zone and the homogeneous model, for frequencies in the range 70–90 Hz. White dots show the position of the void.

Figure 9. Time lags between the maximum vertical displacement at the point of impact and at the current point.

Figure 8 shows no difference between the time lags for the homogeneous material (limestone) and the void anomalies (fig. 8). For a faster anomaly (made of flint), there seems to be smaller lags on one side of the anomaly (above the anomaly) and larger ones on the other side. Time lags show a singularity for the simulations with a void (fig. 9), which might be simpler to detect. Another way of looking at the results consists in defining set of 642 nodes located at the surface in the area (x, y) ∈ [2, 13 m] × [8, 17 m] in order to perform a Fourier transform of the vertical displacements. We compare the amplitudes obtained for the frequencies in two 20 Hz intervals, for a homogeneous layer and for a layer containing a void or a dense anomaly, for frequencies in the range 70–90 Hz. Figures 9 and 10 show clearly the position of the impact, and one can see a dissymmetric pattern between the zone of the anomaly and the area above the impact (White dotted lines show the position of the anomalies). The results corresponding to frequencies between 90 and 200 hertz (not shown here) do not clearly indicate the position of the anomaly, but also reflect the fact that the presence of the anomaly modifies the signal in the zone above the impact. 4.2.2 Simulations with two anomalies The next step was to make a simulation with a model geometry closer to that of the actual experimental site,

Figure 11. Ratio of the amplitude between the model containing a flint anomaly and the homogeneous model, for 70–90 Hz range frequencies. White dots show the position of the anomaly.

with two anomalies relatively close to each other (and located at two different depths): – a void zone at (x, y, z) ∈ [5, 7 m] × [11.5, 13.5 m] × [−4, −2 m ]; – a flint zone at (x, y, z) ∈ [11, 14 m] x [ 9.5, 11.5 m] x [−5, −3 m ]. (note that the size of this dense anomaly has been exaggerated). In the simulations, a shock is emitted on a line situated between the anomalies, at (x, y, z) = (9 m, 11.5 m, 0). Displacements are plotted along the y direction, for three parallel lines (x = 6, 9 and 12.5 m, respectively above the void zone, the central homogeneous limestone zone, and the flint anomaly). No special feature of the curves makes it possible to detect either of the anomalies (fig. 12). Results obtained for x = 6 m shows a decrease of the time lags above the cavity (fig. 13), however it is

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Figure 12. Time lag between the impact and the first minimum of the vertical displacement.

Figure 14. Ratio of the amplitude obtained with the model containing anomalies and the homogeneous model, for the frequency range 70–90 Hz. White dotted lines show the positions of the anomalies.

Figure 13. Time lags between the first maximum at the impact point and the first maximum at the current point.

not so clear for the points at x = 12.5 m above the flint anomaly. In other words, the time lags between the first maximum of the displacement reflect the perturbation of wave propagation by heterogeneities, but the signals are difficult to interpret. Spectral analysis of the computed displacement has been performed on frequency intervals of 20 Hz. Figure 13 shows the results obtained for the interval [90–110 Hz], together with the position of the impact and of the anomalies. More precisely, the gray shades show the value of the Fourier transform of the vertical displacement (or of the vertical velocity, or acceleration) computed for the model with the heterogeneities, divided by the same value computed with the homogeneous model. This method could in principle give a interesting way of extracting relevant information from the numerical analysis, but the mesh is clearly too coarse to allow for a good spatial resolution of the anomalies. 4.3

Comparison between experimental results and simulations

Tests involve too few geophones to perform such a Fourier analysis of the results. Nevertheless, we have tried to compare the experimental results with numerical simulations, for the configuration with two anomalies. We compare the results for a shock placed above the cavity (x=6 m), then for x = 9 m and x = 11.5 m.

Figure 15. Time lags for the first minimum of the displacement, tests 1 and 2.

Numerical simulations show a good agreement between time lags between the emission of a shock and the reception by the geophones for test 2. Wave velocity is well reproduced. Besides, the slope is relatively constant on either side of the impact which leads to conclude that the velocity is almost uniform over the zone covered by the sensors. In other words, the number of geophones does not allow to detect the anomaly. For test 1, there is also a good agreement between measures and simulations. As before, it can be observed that:

475

– there is a good agreement between simulations and measures; – the time lags do not reveal the position of the anomalies, since the curves are almost symmetrical with respect to the position of the impact.

5

CONCLUSION

We have discussed the possibility of detecting a cavity in a simple, bilayered ground mass, on the basis of wave propagation tests. Emitting a shock at the ground surface proves unsuccesful: we assume that results could be more useful with a significantly larger number of geophones, in order to see local variations of the time of propagation of waves. On the other hand, there is a good agreement between the results of numerical simulations and the experimental data. This leads to the conclusion that more accurate numerical models (with more nodes and a better spatial discretization), and experimental setups including a larger number of geophones together with the appropriate acquisition devices might provide a way of detecting underground heterogeneities from the surface with relatively simple experimental means.

Figure 16. Time lags between the first maximum of the displacements, tests 1 and 2.

REFERENCES

Figure 17. Time lags for the first minimum of the vertical displacement: comparison between measures and simulations for test 4.

Other comparisons have been undertaken for a shock located on the line x = 9 m, (test 4): the agreement between the simulations and the experiment is not so good (fig.17). Detecting the heterogeneities on the basis of the time lags does not seem possible.

Baltazart, V., Abraham, O., Leparoux, D. (2006). Utilisation des ondes sismiques de surface pour la détection de cavités souterraines sous voies ferrées. Etudes et recherches des laboratoires des Ponts et Chaussées. LC publication, 41–60. Chammas, R. (2002). Caractérisation mécanique de sols hétérogènes par ondes de surface. Thèse de doctorat, université de Nantes, . Lagabrielle, R., Grandsert, P. (2005). Comparison of cavities detection methods in a silt layer. Example of the SNCF test site along the Paris-Lille high speed line.. Geoline Symp. Lyon, . Léonard, C.(2001). Détection des cavités souterraines par sismique réflexion haute résolution et par Impact – Echo. Thèse de doctorat, université de Lille, . Leparoux, D., Bitri, A., Grandjean, G. (2000). Underground cavity detection: a new method based on seismic Rayleigh waves. Eur. Journal of Environmental and Eng. Geophysics, 5, 33–53. Picoux, B. (2002). Etude théorique et expérimentale de la propagation dans le sol des vibrations émises par un trafic ferroviaire. Thèse de doctorat, université de Nantes, .

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Numerical modelling of impacts on granular materials with a combined discrete – continuum approach A. Breugnot EGIS Géotechnique, Seyssins, L3S-R, UJF-INPG-CNRS, Grenoble Universités, Grenoble,

Ph. Gotteland & P. Villard L3S-R, UJF-INPG-CNRS UMR 5521, Grenoble Universités, Grenoble,

ABSTRACT: This paper investigates modelling of granular material submitted to high energy impact due to block impact. An original combined discrete – continuum method is proposed which permits to use discrete element method to model precisely the complex behaviour of granular material in the vicinity of the impacted zone while a continuum approach is used in farther areas. Coupled methods proposed are validated through simple static and dynamic tests and, finally used to simulate high energy impact of a cubic impactant on a gravel layer.

1

INTRODUCTION

The anthropization of mountainous regions raises the problem of infrastructures (roads, railways, industrial areas . . .) and inhabitants’ protection against risk of rockfall. Zones defined as potentially risky, can be protected with civil engineering structures placed upward to stop or deviate the trajectory of a falling rock. Among structures protections, the dams (embankments) and cushions covering rock sheds, benefit from the capacity of geomaterials to absorb energy and distribute loading through the structure. The behaviour of ground structures submitted to local impact due to rockfall is quite complex to be characterized precisely. Indeed, high energy loading induces large and irreversible deformations, high strain rates and stresses in geomaterials which make difficult the prediction of mechanical behaviour of rockfall protection structures. Consequently, their design suffers from a lack of regulation and is often limited to static stability consideration, and only a few approaches (Tissières 1999, Ronco 2009) estimate the dynamical component of impact braking force. At this time the most sophisticated numerical codes may assist in the analysis of the dynamical phenomena induced by impact loading, using either continuum (Pichler et al. 2008, Peila et al. 2007) or discrete approaches (Plassiard 2007, Calvetti et al. 2005, Bertrand et al. 2005). Indeed, models are often calibrated with experimental results, and permit to enlarge analysis to other configurations tests (parametric study). Considering the granular nature of geomaterials, and numerous rearrangements and fractures that take place in the most solicited areas, the Distinct Element

Method (DEM) (Cundall & Strack 1979) seems to be particularly adapted to model local mechanical behaviours in geomaterials under dynamical impact. However, the use of this approach, to model large scale works, requires a large number of particles, which increases both the times of modelling and computation. In order to improve the design of larges structures, modelling needs to maintain accuracy in highly stressed areas while representing the mechanical behaviour of the global structure. Consequently, an innovative and original multiscale approach is developed (Xiao & Belytschko 2004) to improve the computational efficiency: a discrete element method is coupled with a continuum mechanical model, which resolution can be far coarser than in DEM. Thereby, continuum domain, adapted to materials with non significant discontinuities, is used as a boundary condition. The first part of the paper is devoted to the procedure of the combined discrete – continuum elements method.The use of such combined approach for granular material is validated through a simple configuration tests, in static and dynamic applications. In the second part, simulations, modelling an impact with a cubic impactant on a soil layer, are led to show the prospects of this combined approach. 2

COMBINED DISCRETE – CONTINUUM METHOD

2.1 General coupling method and methodology Research concerning combined discrete – continuum approaches began in the early eighties, and was led in physical domain in order to study material’s

477

Figure 1. Discrete and continuum scaling factors in overlapping domain.

behaviour at molecular scale. Since then, the coupling methods and domains application were widely developed (Munjiza 2004, Itasca 2006, Onate 2003, . . .). Xiao & Belytschko (2004) and Xiao & Hou (2007) proposed two different decompositions of domains which permitted to link continuum and discrete domains, either with an “edge-to-edge” method or with a “bridging” domain method, to study dynamical wave propagation or crack propagation in micromechanical structures. In the latter case, discrete and continuum domains are overlapped in a bridging subdomain, where Hamiltonian H is taken to be a linear combination of the discrete and continuum total energies (Fig. 1), respectively HDiscrete and HContinuum (Equation 1).

Figure 2. Discrete element position in a continuum volume belonging to bridging domain.

Figure 3. Projection of discrete element position on a continuum surface belonging to “edge-to-edge” domain.

In bridging zone, discrete and continuous displacements are linked to ensure continuity between the two domains. The displacement dj (Xj ) of a discrete particle j, localized by vector position Xj , is written dj (Equation 2). The continuum displacement u(Xj ), at the same localization Xj , can be expressed in of displacement ui of the 8 nodes i = a to g, which surround discrete particle j, by the mean of kinematic relations k ji (Equation 3). The two domains are finally constrained via Equation 4: discrete displacements are required to conform to the continuum displacements at the positions of particles. The difference between discrete and continuum displacements is characterized by vector of residual displacements g.

Originally formulated for ordered particles, the formulation of the method, proposed by Xiao and Belytschko, remains relevant for amorphous sample used to model geomaterials.

For the simpler “edged-to-edge” model, the Hamiltonian is defined as the sum of discrete and continuum ones (no scaling) because domains are dist. Aiming at formulating seamless method for non ordered discrete element sample, the kinematics constraints are calculating using displacement of fictive nodes j ∗ (localized by position vector X∗j ) obtained by orthogonal projection of position vector Xj in the vicinity of the junction, on the continuum plane border (Fig. 3). Kinematic relations and constrains become (Equation 5 & 6):

where nodes i = a to d, are the 4 nodes surrounding fictive node j ∗ in the plane junction. Combined problem is solved by minimizing modified Hamiltonian HL for the complete model, using the Lagrangian multiplier λ method to ensure continuity in bridging domain (Equation 7). An explicit algorithm is used for dynamical application and the

478

Table 1. model.

Characterization of elasticity in discrete element

Elastic’s Parameters

kn N/m

ks N/m

Gravel Natural Soil

1.77e6 × R 3.78e6 × R

0.35e6 × R 1.13e6 × R

Table 2. model.

Characterization of plasticity in discrete element

Figure 4. Representation of coupled samples used for static triaxial tests. (a) “bridging” model (b) “edge-to-edge” model.

Plastic’s Parameters

cf –

bn N

bs N

scheme of resolution is detailed in Xiao Belytschko 2004 and Frangin et al. 2006.

Gravel Natural Soil

1.0 ∞

1.78e5 × R2 ∞

1.78e5 × R2 ∞

In our approach, distinct element method (DEM) (PFC3D code, Itasca) is employed in the discrete domain while finite difference method (FLAC3D code, Itasca) is employed in the continuum domain. Implementation of the coupled approach in these two different codes is motivated by the fact they have T/IP socket connection ability which permits data transmission at each calculation step to control and correct displacements at combined boundary. 2.2 Validation of coupling methods 2.2.1 Triaxial quasi-static test In this section, these coupled methods are compared and validated through static and dynamic tests. First, a numerical triaxial test is performed on a cubic elastic sample (Fig. 4) (length = 3 m), characterized by a Young’s modulus E = 500 MPa and a Poisson coefficient υ = 0.32. When it is used, the granular material is modelled by spherical particles of various diameters (ratio of 2.0 between the greater and smaller particles) associated together by rigid elastic bonds. The properties in discrete domain are equal to those of “Natural Soil” material in section 3, and are summed up in Table 1 & 2. For quasi-static solicitation, local damping coefficient is set to 0.7 in both discrete and continuum approaches. The confining pressure σconf apply on the lateral faces is constant and equal to σconf = 10 kPa. The “bridging” or “edge-to-edge” models can be compared with continuum and discrete approaches. Vertical stresses and lateral strains measured numerically are very close whatever the model employed. Only lateral deformation curve for DE method presents a quite different behaviour. Apparent Young’s modulus (Fig. 5) and Poisson coefficient (Fig. 6) are overall unchanged for coupled models: material continuity is ensured at the transition areas. Implementation of ‘Edge-to-edge’and ‘bridging’coupling methods are validated for elastic material in quasi-static application.

Figure 5. Vertical stresses (Young modulus characterization) in sample during triaxial test: comparison of coupled models with full continuum and full discrete models.

Figure 6. Lateral strain (Poisson’s effect) in sample during triaxial test: comparison of coupled models with full continuum and full discrete models.

2.2.2 Dynamical compression test The coupling methods are also tested for dynamical application, in particular for wave propagation into an elastic medium. The same four models are handled for this validation test (Fig. 7). The material properties are identical to those established in the previous static test, except damping coefficient which is set to zero for dynamical application.

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Figure 7. Schematic representation of coupled samples used for dynamic tests. (a) “Bridging” model (b) “Edge-to-edge” model.

Figure 10. Scheme of the bridging coupled model, defined by two overlapping zones.

Figure 8. Imposed dynamical stress loading at the top of the discrete element model.

insignificant differences are partly inherent to coupled models: constraining degree of liberty can slightly modify local mechanical behaviour. These previous dynamical tests provide interesting results regarding the capability of combined model to describe dynamical behaviours in elastic domains.

3

IMPACT BLOCK SIMULATION

3.1 Model description

Figure 9. Displacements in samples during dynamical simulation: comparison between different approaches.

A dynamical force load σ(t) is applied on the head of each sample, and in order to evaluate their dynamical behaviour, vertical displacements are recorded at points A(0,0,3) and B(0,0,1.5). Amplitudes and frequencies numerically calculated are very similar for all samples. It should be noted that errors on peaks don’t exceed 3–5% (Fig. 9). These

In this section we employ coupling methods to investigate modelling possibilities of a rock impact on geomaterial layer or later on a global structure. The model presented thereafter is adapted from experimental research, described by Pichler et al. 2005, concerning the loading of layers of gravel subjected to rockfall. The experimental approach is based on measurements or estimations of penetration depth, impact force and impact duration when a cubic rock impacts onto the ground with a tip. The following model is composed by a trench of 4 m width and 2 m depth, dug in natural soil, and filled by well-graded gravel (Fig. 10). In the different approaches, mechanical behaviour of gravel is described by elasto-plastic behaviour to take into the absorbing energy capacity of gravel. The natural soil is not submitted to high load, and is modelled by elastic behaviour. It can be viewed as a boundary condition. Bloc impact on gravel is simulated with the “edgeto-edge” and “bridging” combined model, and entirely discrete one in order to validate the numerical processes. The transition between discrete and continuum approaches must take place in elastic domain (natural soil). Elastic parameters of the gravel cushion were estimated by the mean of tests on the embankment

480

Table 3. Penetration and impact force obtained experimentally and simulated numerically for an 850 kJ impact. Block Experimental V = 13 m/s & m = 104 kg Pichler et al. (2005) Numerical Penetration Max. Impact Force

0.51 m 3.5 MN

1.15 m 2.5 MN

The shape of the impactant block is quite cubic, has a masse of around m = 10,000 kg, and is supposed very stiffer compared to the impacted soil. Fall is not calculated but consequently, the velocity of the boulder was initialized to v = 13 m/s, which correspond to a 8.5 m high fall. The kinetic energy of the impact is almost E = 850 kJ which corresponds to the experimental work of Pichler et al. (2005).

3.2 Calculation results and comments

Figure 11. Penetration and impact force calculated by coupled and discrete approaches.

(E = 196 MPa, υ = 0.36), but no significant information were available to estimate plasticity’s parameters. First, to validate the combined discrete – continuum approach the discrete element model retained is rather simple, composed only by spherical element. The inter-particular interactions are characterized by local normal and shear stiffness’s (kn and ks ), normal and shear local bonds (bn and bs ), and a local friction coefficient (cf ). Using spherical elements tends to limit the macro friction angle to values around 30◦ (Chareyre 2003) which can be considered as a low value for gravel. The choice of a local friction coefficient cf = 1.0 (45◦ ) lead to a macro friction angle of φ = 29◦ , and the local bond normal bn and shear bs strength are fitted to obtained a macro cohesion c = 16 kPa. Micro-mechanical parameters (kn and ks ) of the natural soil insure an elastic behaviour characterized by E = 500 MPa and υ = 0.32). The mechanical parameters for discrete numerical simulation are summarized Table 1 & 2. It should be noted that values of kn , ks , bn and bs are not intrinsic to a material but depend on particle radius chosen for soil modelling. Note that damping coefficient is kept to zero for impact simulation.

During impact, penetration and resulting force on the cubic impactant are recorded to be compared to experimental measures (Pichler et al. 2005). The response of the model based on coupling methods is close to the entirely discrete one (Fig. 11), and draws the prospects of this approach to model large geometry of structure which need locally accurate description. However, due to simplifying assumptions made to model the gravel soil, the comparison with experimental and analytical results given by Pichler et al. (2005) show that dynamical resistance of the gravel layer is under-estimated. Indeed, experimental penetration and analytical impact force are respectively 50% lower and 40% upper (Table 3). These differences can be partially explained by both static characterisation of the mechanical parameters and no consideration of dynamical effect during impact (energy dissipation, local plasticity, braking of grains, etc.). First, the macro mechanical parameters (ϕ = 29◦ , c = 16 kPa), affected to the gravel material to describe plasticity, are determinate by the analysis of the peak resistance for triaxial. Post-failure, resistance is very weaker, mainly because of the low residual resistance retained for well-graded gravel in large deformations. Thus, mechanical behaviour needs to be updated to simulate upper friction angle in geomaterial, in introducing rolling resistance at each (Plassiard 2007), in inhibiting rotation degrees of freedom (Calvetti et al. 2005) or using non-spherical particles (Bertrand et al. 2005, Salot et al. 2009) for instance. Another explanation is issued from the phenomenon of particles ejection observed numerically around impact (Fig. 12). The model seems not to dissipate enough energy, and the elastic restitution is overestimated, for shallow discrete element, compared to experimental observations. The actual constitutive law, calibrated from quasi-static triaxial tests, can not

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REMPARe (www.rempare.fr) ed by the French National Research Agency (ANR). All the partners of project REMPARe, as well as the research consortium VOR-RNVO for the financial (Software), are gratefully acknowledged by the authors. REFERENCES

Figure 12. Cross section of the DE model after impact. Visualisation of particles ejection in the vicinity of impact area.

model phenomenon as viscosity, compaction and local breaking in high loaded areas. Dissipations, due to interparticle sliding (friction) or break of cohesive bond, are not sufficient to represent all sources of dissipation during impact. Aiming at improving the response of the granular material, and in the same time the validity of this model, additional dissipative laws need to be implemented. 4

CONCLUSION

An innovative combined discrete – continuum element method has been adapted to study mechanical behaviour of large civil and geotechnical engineering structures. Validated by means of static and dynamical elementary tests, prospects of coupling methods are then evaluated through the boulder impact on a gravel layer. In this case, the good accordance between discrete and continuum – discrete element methods shows the interest of such method to describe locally impact phenomenon. At this step, the limits of the numerical model are based on the description of the dissipative behaviour of high loading granular material. Local constitutive laws needs to be further developed to take into dynamical behaviour of granular under high energy impact. In the framework of REMPARe research project (www.rempare.fr), high energy impact experiments (2000 kJ) have been performed to test protection dams at real scale. Experimental data should permit us to feat our model in order to optimize and improve protection structure design. ACKNOWLEDGEMENTS The work presented in this paper was performed in the framework of the French research development project

Bertrand, D. Nicot, F. Gotteland, P. & Lambert, S. 2005. Modelling a geo-composite cell using discrete analysis. Computers and Geotechnics 32: 564–577. Calvetti, F. Di prisco, C. & Vecchiotti M. 2005. Experimental and numerical study of rock-fall impacts on granular soils. Rivista Italiana di Geotecnica 4. Cundall, P.A. & Strack, O.D.L. 1979. A discrete numerical model for granular assemblies. Géotechnique 29(1): 47–65. Chareyre, B. 2003 Modélisation du comportement d’ouvrages composites sol – géosynthétique par éléments discrets. Application aux ancrages en tranchées en tête de talus. Thèse de doctorat, université Joseph Fourier, Grenoble. Frangin, E. Marin, P. & Daudeville, L. 2006. On the use of combined finite/discrete element method for impacted concrete structures. Journal de Physique IV 134: 461–466. Itasca Consulting Group. 2006. Fish in PFC3D, AC/DC (Adaptive Continuum/Discontinuum) logic. Munjiza, A. 2004. The Combined Finite-Discrete Element Method. John Wiley (eds). Oñate, E. 2003. Multiscale computational analysis in mechanics using finite calculus: an introduction. Multiscale Computational Mechanics for Materials and Structures. Computer Methods in Applied Mechanics and Engineering 192 (28–30): 3043–3059. Peila, D. Oggeri, C. & Castiglia C. 2007. Ground reinforced embankments for rockfall protection design and evaluation of full scale tests. Landslides 4: 255–265. Pichler, B. Hellmich, Ch. & Mang, H.A. 2005. Impacts of rocks onto gravel. Design and Evaluation of experiments. International Journal of Impact Engineering 31: 559–578. Pichler, B. Hellmich, Ch. Mang H.A. & Eberhardsteiner, J. 2008. Semi-probalistic design of rockfall. Comput Mech 42: 327–336. Plassiard, J.-P. 2007. Modélisation par la méthode des éléments discrets d’impacts de blocs rocheux sur structures de protection type merlons. Thèse de doctorat, université Joseph Fourier, Grenoble. Ronco, C. Oggeri, C. & Peila D. 2009. Design of reinforced ground embankments used for rockfall protection. Natural Hazards and Earth System Sciences 9: 1189–1199. Salot, C. Gotteland, Ph. & Villard P. 2009. Influence of relative density on granular materials behavior: DEM simulations of triaxial tests. Granular Matter: 11: 221–236. Tissières, P. 1999. Ditches and reinforced ditches against falling rocks. t Japan-Swiss scientific seminar on impact load by rock falls and design of protection structures, 4–7 October 1999. Kanazawa, Japan. Xiao, S.P. & Belytschko, T. 2004. A bridging domain method for coupling continua with molecular dynamics. Computer methods in applied mechanics and engineering 193: 1645–1669. Xiao, S. Hou, W.2007. Studies of nanotube-based aluminium composites using the bridge domain coupling method. International journal for multiscale computational engineering 5(6).

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Numerical simulations of the dynamic impact force of fluidized debris flows onto structures F. Federico & A. Amoruso University of Rome “Tor Vergata”, Rome, Italy

ABSTRACT: Mechanical effects due to the impact of fluid-like debris flows against structures are analyzed. Current design criteria usually neglect the impulsive component of the phenomenon to evaluate the exerted thrust, although recent theoretical and experimental studies pointed out its importance. The results of experiments reported in literature are not easily comparable, since recorded impact pressures are widely scattered and seem to depend on several factors. Numerical analyses (FEA) of the impact against structures aiming to highlight the mechanics of this phenomenon have been carried out; results allow to analyze the impulsive phase, identify the most important physical and mechanical governing factors, forecast possible mechanical effects. The analytical and numerical evidences finally suggest a different, possible interpretation of lab tests. 1

INTRODUCTION

The analysis of the dynamic interaction between fluids and solids involves different areas of civil engineering; referring to debris flows, hydraulic and geotechnical engineers devoted specific attention to the triggering (Musso & Olivares 2003), fluidization (Musso et al. 2004, Hungr 2003) and runout phenomena (Iverson 1997) rather than to the impact against structures (Armanini 1997, Scotton & Deganutti 1997). Events like the ones occurred in Sarno and Quindici (Campania, Italy), in May 1998, prove the destructive power of the high-velocity fluid-like debris flows. They originated from the shallow pyroclastic soils of the Appennino Campano, covered a distance up to 4 km and reached the inhabited areas (Revellino et al. 2004), causing death and widespread damage to structures. Modeling of debris flow surges is difficult, due to the diversity of substances composing debris flows: water, mixtures of granular and fine particles in water and large solid particles such as boulders; solid concentration, for example, strongly affects the rheology of these mixtures. Solid fraction made of fine particles (silt and clay) can be incorporated into the fluid and the grains act as part of the fluid (Iverson 1997). Typical values of density of such flows range from 1000 to 1500 kg/m3 ; the viscosity lies in the interval 0.001–0.1 Pa · s (Iverson, 1997). The analysis of the impulsive phase of the impact force by debris flow should take into the different properties of the flow (Fig. 1). The selection of a suitable flow resistance law which describes the flow regime, by taking into the physical properties, still constitutes a complex problem. The propagation of a fluid-like debris flow is often modeled through a visco-plastic constitutive behaviour, and the Bingham

Figure 1. Impact of debris flows. (a) general scheme of the impact between a fast moving debris flow and a (solid or deformable) body (a structure or a block); (b) simplified schemes to study the impact phenomena; (1) blocks (isolated or in groups), eventually transported by fluid flow; (2) dry or saturated granular mass; (3) fluid-like debris flow (a wide range of solid concentration is allowed).

or Herschel-Bulkley approaches are assumed; thus, the fluid is described as a single-phase material that remains rigid until the deviatoric stress exceeds a threshold value (Iverson, 1997). Seminara & Tubino (1993) propose a general theoretical framework for mixtures of fluids and granular matter, based on the mass and momentum balances of a control volume, considering that the mixture and both the solid and fluid components behave like compressible fluids. Literature on impact shows that a modified value of the hydrodynamic pressure exerted by an incompressible fluid flow (proportional to the square of velocity, Hungr 2003, or a multiple of the hydrostatic force,

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Armanini 1997), usually approximate the unknown, impulsive action. By this way, the impact force assumes a constant value. Analytical or numerical studies regarding the evolution with time of the impact force acted by fluid-like debris flows against a structure, are not developed, yet. In coastal engineering, on the other hand, recent works (Peregrine 2003, Bullock et al. 2004) report original advanced theoretical and experimental results of the impact of sea waves against structures. The same AA. sustain that the dynamics of fluid-structure interaction is not completely understood, yet; geometrical (wave shape), kinematical (impact velocity) and physical and mechanical (presence of air in the fluid, solid concentration, . . .) variables control the phenomenon. Their role must be carefully evaluated to identify the actions which determine ultimate limit states of the structure under impact, their evolution with time and to distinguish the effects related to the impulsive from those ones related to the hydrodynamic phases. Water waves and debris flows are characterized by different pressure and velocity fields; however, if a uniformly progressive flow is considered (Hungr 2000), a surge with constant velocity can be considered. Results pertaining to coastal engineering appreciably differ from the ones obtained in the study of debris flows. To interpret this difference, FEA are carried out and results are compared to those ones obtained through theoretical models; the proposed procedure finally allowed some mechanical effects, such as the displacement of a free, rigid block resting on a rough surface or the collapse of a reinforced concrete pillar, to be numerically analyzed. 2

NUMERICAL ANALYSES

2.1 Overview The dynamic interaction between flow and structures often plays a key role in the efficiency and safety of engineering applications. Fluid flow is governed by non-linear partial differential equations; in many situations, the flow spans a huge range of length scales, the non-linearity of the governing equations resulting in the transfer of energy from a length scale to another one. Due to this complexity, to solve the governing equations of fluid mechanics problems, many innovative numerical methods have been proposed; generally, the computational domain needs to be divided into discrete components; different numerical methods are based on different discretization techniques, which can be roughly grouped into two main groups: grid-based methods (FEM, FDM) and meshless methods (SPH, DEM). Smoothed Particle Hydrodynamics (SPH) method allows the solution of the equations of dynamics, expressed in Lagrangian form, to be obtained. Although initially developed in astrophysics, the method has been applied to solve non trivial problem of fluid dynamics, otherwise treated through Eulerian

Figure 2. Impact of a 2D rectangular fluid domain against a rigid structure. (a) geometry of the problem (3D view: displacements are not allowed along the direction 3); (b) geometry of the problem (2D view; Walkden et al., 2001).

schemes: free surface flows (Monaghan 1992), waves, unsteady jets, fluid-structure interaction. The method is based on the discretization of the computational domain into a finite number of movable points which represent fluid particles; the field of motion is then obtained by interpolating, through suitable functions, the values of the state quantities (density, velocity, . . .) of each particle. As stated before, the acceleration of the particles is calculated by solving the equations of motion (Euler equations, Navier-Stokes equations, Shallow water equations) in Lagrangian form.The reduction of the whole number of movable points allows to reduce time computation; however, by this way, the problem of particle deficiency near or on the boundaries arises (Monaghan 2005). Different procedures have been developed to properly treat problems concerning and impact between bodies (Oger et al. 2006), but this aspect has not been univocally cleared. One of the main features of the SPH method lies in the introduction of an equation of state (EOS) describing the behaviour of a compressible fluid, through the celerity c0 of sound waves in the fluid. However, the actual values assumed by this parameter would lead to extremely small (thus, prohibitive) computational time steps. Being the maximum velocity (vmax ) of the flow in the numerical simulations a priori known (or estimated), to simulate the behaviour of an almost incompressible fluid (negligible density variations), a small value of the Mach number:

is imposed (Ma << 1); c0 is thus easily obtained, but its value, generally, have no physical meaning. 2.2 The FE code Numerical analyses of the impact between a fluid and a solid body have been carried out by means of FE Abaqus\Explicit code; the explicit time integration procedure is based on the implementation of a central

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difference rule (Bathe 1995). The meshes of both the fluid and the solid domains are made of C3D8R elements (8-node linear bricks, reduced integration with hourglass control). 2.3 Treatment of The implemented algorithm models the master and the slave interacting surfaces. The master surface pushes into the slave surface; as a reaction, forces are generated to prevent that the nodes of the slave surface penetrate the master surface. is established with pairs which use a kinematic algorithm that enforces constraints and conserves momentum. At the beginning of each time increment, the kinematic state of a model is modified according to a predicted configuration, without considering the conditions. The slave nodes that penetrate the master surface are then determined; the depth of each node’s penetration, the corresponding associated mass and the time increment are used to calculate the force necessary to resist the penetration. If this force had been applied during the increment, it would have caused the slave node to exactly the surface. The resisting force at each slave node is defined through a common hard condition (as opposed to a penalty condition); no pressure is transmitted between the considered surfaces if their nodes are not in . Any pressure can be transmitted between the surfaces in .

motion from the mesh motion. It has been proven that this technique has a high computational efficiency. In an adaptive meshing increment, the element formulations, boundary conditions, external loads, conditions, etc. are all handled first in a manner consistent with a pure Lagrangian analysis. Once the Lagrangian motion has been updated and the mesh sweeps have been performed to find the new mesh, the solution variables are remapped by performing an advection sweep. Both momentum and field variables are advected during an advection sweep. This procedure is performed following the work of Benson (1992), ensuring that the advecting momentum is properly conserved during remapping. The applied method, known as the Half-Shift Index method, first shifts each of the nodal momentum variables to the element centre; the shifted momentum is then advected from the old to the new mesh. Finally, the momentum variables at the element centres of the new mesh are shifted back to the nodes. 2.6 Constitutive model of the fluid In case of floods (5 ÷ 10 % concentration of sediment by volume) as well as of hyperconcentrated flows (up to 40% of sediment concentration), the fluid behaviour is controlled by water (USGS 2005). In computations, it is defined through the state equation [p = f (ρ)], according to the linear Hugoniot explicit form:

2.4 Adaptive meshing In the numerical computations, the fluid domain has been defined as an Adaptive meshing domain. Adaptive meshing technique makes it possible to maintain a high quality mesh throughout the analysis, even if large deformations occur. This is achieved by allowing the mesh to move independently of the underlying material. The adaptive meshing technique in the code follows the work of Van Leer (1977) and combines the features of a pure Lagrangian analysis and an Eulerian analysis. Therefore, it is often referred to as an ALE (Arbitrary Lagrangian Eulerian formulation). A smoother mesh is created by sweeping iteratively over the adaptive domain. During each mesh sweep, nodes in the domain are relocated based on the positions of neighboring nodes and element centres. A volume smoothing technique is used to improve the quality of the mesh and one mesh sweep is performed after each increment. 2.5 Advecting solution variables to the new mesh The ALE method of adaptive meshing introduces advective into the momentum balance and mass conservation equations. These for the independent mesh and material motion. The code solves these modified equations by decoupling the material

the variable εv and the parameters ρ0 , co being previously defined. By setting the material and kinematic parameters s = 0 and G0 = 0, it is obtained the bulk response p = Kεv = KU /c0 where K = ρ0 c02 . For water, if ρ0 = 1000 kg/m3 , c0 = 1450 m/s, U = 5 m/s, it results p = 7.25 MPa. The code allows the to define the deviatoric behaviour, through the Newtonian viscous fluid model: S = 2υ · e˙ ; where Sis the deviatoric stress, ν the fluid viscosity and e˙ the deviatoric component of the strain rate. The deviatoric and volumetric responses are thus uncoupled. A rigid behaviour for the fluid-bottom interface has been considered.

3

FEA RESULTS

The impact pressure increments (p) vs time (t), as a function of the position along the depth of the considered point, is represented in Figure 3. The maximum theoretical pressure (pmax ) is attained only in the lowest part of the fluid domain, (points B and C, Fig. 3); from the top to the bottom of the fluid front, the confinement action increases and

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Figure 3. Impact pressure p computed for three points on the fluid front vs the elapsed time t after the impact. Point A is located 5 cm below the free surface. The geometry of the impacting fluid mass is h = 1 m, b = 1 m, L = 2 m; the fluid properties are c0 = 1450 m/s, ρ0 = 1000 kg/m3 . The impact velocity is U = 5 m/s; pmax = 7.25 MPa.

Figure 5. Impact of a rectangular fluid front; c0 = 150 m/s; pressure development over the fluid domain.

new time increment (t = 0.266 ms after the impact, Fig. 4b), at the borders and on the whole lateral surface, the pressure vanishes. In Figures 4c, d (respectively 0.378 ms and 0.490 ms after the impact) it is shown that the maximum impact pressure (pmax = 5.80 MPa) is reached only in the central portion of the fluid domain; moving from the centre towards the border, the pressure progressively reduces and the area characterized by maximum pressure decreases with time. 3.2 Results obtained for c0,2 = 150 m/s

Figure 4. Impact of a rectangular fluid front; c0 = 1450 m/s; pressure development over the fluid domain.

the maximum pressure lasts longer at the bottom of the fluid front. The numerical analysis approximates the actual, continuous fluid pressure evolution through discretized time steps; thus, it doesn’t allow to model the almost instantaneous growth and subsequent decrease to nil values of the fluid pressure in those points belonging to both the free and the impact surfaces. A minor error can be observed in numerical results obtained for the fluid pressure excess in the point A (Fig. 3), located 5 cm below the free surface. Different simulations have been carried out taking into two different celerity of sound waves, c: c0,1 = 1450 m/s (pure water) and c0,2 = 150 m/s (water with a small volumetric fraction of entrained air). For both simulations, the impact velocity was equal to 4 m/s and the value of viscosity υ = 0 has been assigned to the fluid; this parameter doesn’t play a remarkable role on the time evolution of impact pressure increments (Federico et al. 2005). 3.1 Results obtained for c0,1 = 1450 m/s The distribution of the pressure at the fluid front, for some prefixed instants, is reported in Figure 4; at the impact, an increment of pressure involves the whole fluid front, including the free surface (Fig. 4a, t = 0.154 ms after the impact); however, after just the

The pressure evolution after the impact is reported in Figure 5. The dynamics of the impact is similar to the one obtained for c0,1 = 1450 m/s; however, as expected, the characteristic time is bigger. Larger times in fact are needed (Fig. 5a, b) to ensure an appreciable dissipation of the borders pressure; in this case, furthermore, the maximum pressure is less than the previous case. The maximum theoretical value is pmax = 0.60 MPa. Another set of computations have been carried out for the geometrical scheme reported in Figure 2a, in order to evaluate and to compare the impulse I to the values obtained by Walkden et al. (2001), under the hypothesis of incompressible fluid behavior. The results are reported in Figure 6. FEA results fall between the two extreme theoretical responses (elastic and inelastic impacts). The analysis of the impact of 2D waves characterized by a triangular cross section has been theoretically carried out by Cooker (2002), referring to both incompressible and compressible behaviours. In the numerical scheme a 3D domain has been set up; the lateral displacements (horizontal direction perpendicular to velocity vector U ) of the waves have been constrained, in order to force a 2D-like behaviour. The domain has been subdivided into about 22800 elements (tetrahedrons); the time integration step is approximately equal to 10−6 s. It can be shown (Federico et al. 2004) that, if the curvature of the fluid front increases, the impact lasts longer and the impact force decreases, due to the different time evolution of pressure over the zones progressively impacting the solid.

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4

Figure 6. FEA; impact of a rectangular fluid front: ratio I /4ρa2 U vs the dimensionless ratio b/2a describing the geometry of the fluid domain. (I , impulse).

CONCLUDING REMARKS

Design criteria in literature, taking into the mechanical effects due to the impact of fluid-like debris flows against structures, usually neglect the impulsive component of the phenomena to evaluate the exerted thrust. The results of experiments in literature are not easily comparable (Federico & Amoruso 2008, 2009), since recorded impact pressures are widely scattered and depend on several factors. Therefore, especially in lab experiments, a key role is played by the position of pressure transducers, coupled to the optimal frequency of data acquisition, that govern the recorded values of the impact time interval and maximum pressure. To better understand this phenomenon, FEA analyses of the impact against structures have been carried out; results allow to analyze the impulsive phase, to identify the most important governing factors, to foresee possible mechanical effects.

REFERENCES

Figure 7. Triangular isosceles water-wave (H = 1 m) impacting a vertical, rigid, fixed, impermeable wall (impact velocity U = 5 m/s). (a) – (c) pressure p within the cross section of the wave after the impact; (a) t1 = 0.5 ms after the impact; (b) t2 = 2.5 ms; (c) t3 = 4.5 ms. The black straight lines mark the theoretical, Cooker (2002), boundary of the maximum pressure domain (p = pmax ) at times t1 , t2 , t3 .

Pressure increment pmax and impulse I , instead, don’t vary appreciably. Time evolution of the fluid pressure, for some instants after the impact, is reported in Figure 7a, c. The zone characterized by the maximum pressure pmax (pmax = 7.25 MPa) changes with the time; theoretical Cooker (2002) and numerical results agree although a sharp transition between the pmax domain and the p = 0 zone is not obtained. These results are referred to a fluid domain characterized by a planar, vertical front; this geometry induces, at the impact, a simultaneous increment of fluid’s pressure over the whole front.

Armanini, A. 1997. On the dynamic impact of debris flows. Armanini & Michiue (eds), Recent developments on debris flow: 208–26. Springer. Bathe, K.J. 1995. Finite element procedures. Englewood Cliffs: Prentice-Hall. Benson, D.J. 1992. Momentum advection on a staggered mesh. J. Comput. Phys. (100): 143–62. Bullock, G., Obhrai, C., Muller G., Wolters, G., Peregrine, D.H. & Bredmose, H. 2004. Characteristics and design implications of breaking wave impacts. In: Proc. of 29th Int. Conf. Coastal Eng., ASCE, Lisbon. Cooker, M. J., 2002. Liquid impact, kinetic energy loss and compressibility: Lagrangian, Eulerian and acoustic viewpoints. J. Eng. Math.(44): pp. 259–276. Cooker, M.J. & Peregrine D.H. 1992. Wave impact pressure and its effect upon bodies lying on the sea bed. Coast Eng (18): pp. 205–229. Cooker, M.J. & Peregrine D.H. 1995. Pressure impulse theory for liquid impact problems. J. Fluid Mech, (297): 193–214. Federico. F., Musso, A. & Amoruso, A. 2005. Impact of a fluid-like debris flows on reinforced concrete pillars. Numerical simulations and back-analyses of a failure case. ICF Congress – XI – Post Symposium on “Damage and Repair of Historical and Monumental Building”. Federico, F., Amoruso, A. 2005. Numerical analysis of the dynamic impact of debris flows on structures. ISEC 03 – 3rd International “Structural Engineering and Construction Conference”, Shunan, Japan. Federico, F. & Amoruso, A. 2008. Simulation of mechanical effects due to the impact of fluid-like debris flows on structures. Italian Journal of Engineering Geology and Environment, 5–24, 2008. Federico, F., Amoruso, A. 2009. Impact between fluids and solids. Comparison between analytical and FEA results. Int. J. Impact Engrg. (36): 154–164. Hungr, O. 2003 Flow slides and flows in granular soils. Picarelli L. (ed), Proc. Int. Workshop on “Occurrence and mechanisms of flow-like landslides in natural slopes and earthfills”, Sorrento., 37–44. Iverson, R.M. 1997. The Physics of debris flows. Review of Geophysics, 35, (3): 245–296.

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Monaghan, J.J. 2005, Smoothed Particle Hydrodynamics, Reports on Progress in Physics, 68, 1703–1759. Musso, A., Federico, F. & Troiano, G. 2004. A mechanism of pore pressure accumulation in rapidly sliding submerged porous blocks. Computers and Geotechnics, 31, (3): 209–226. Musso A., Olivares L. 2003. Flowslides in pyroclastic soils: transition from “static liquefaction” to “fluidization”. Invited Lecture Proc. Int. Workshop on Occurance and mechanisms of flow in natural slopes and earthfills. Sorrento, Italy, May 2003. Oger G., Doring M., Alessandrini B. & Ferrant P. 2006. Two-dimensional SPH simulations of wedge water entries. Journal of Computational Physics, Volume 213, Issue 2, April, Pages: 803–822. Peregrine, D.H. 2003. Water-wave impact on walls. Ann Rev Fluid Mech, (35): 23–43.

Revellino, P., Hungr, O., Guadagno, F.M. & Evans, S.G. 2004. Velocity and runout simulation of destructive debris flows and debris avalanches in pyroclastic deposits, Campania region, Italy. Environ Geol., (45): 295–311. Seminara & Tubino 1993. Debris Flows: mechanics, prevention, forecasting, G.N.D.C.I.-C.N.R. Monograph (in Italian). USGS 2005. Distinguishing between Debris flows and floods from field evidence in small watersheds. USGS fact sheet 2004–3142. Van Leer, B. 1977. Towards the ultimate conservative difference scheme III. Upstream-centred finite-difference schemes for ideal compressible flow. J. Comput. Physiol,(23): 263–75. Walkden, M.J., Wood D.J., Bruce T. & Peregrine D.H. 2001. Impulsive seaward loads induced by wave overtopping on caisson breakwaters. Coastal Engrg., (42): 257–276.

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Three dimensional analysis of seismic performance of an earthfill dam in Ethiopia B.G. Tensay & W. Wu Institute of Geotechnical Engineering, BOKU, Vienna, Austria

ABSTRACT: The seismic performance of earth dams is usually studied by two dimensional space. However, considerable effort is required to estimate the overall three dimensional dynamic response of dams in a narrow canyon from plane strain analyses of the dam. This is so because the plane strain analysis normally ignores the arching effect of the valley. Researches reported in this paper represent 3D numerical study of an earthfill dam founded on a liquefiable foundation subjected to earthquake loading and effect of canyon geometry on its seismic performance. The shape of the canyon is varied to determine the related effects to the earth dam. A finite difference numerical scheme is used for the study. The assumed 3D model contains all details of the dam body and foundation materials of Tendaho earthfill dam in Ethiopia. Results and discussions related with the significance of these two factors for the seismic performance evaluation of earth dams are presented.

1

INTRODUCTION

In numerical modeling, it is usually intended to simplify the real structure by eliminating regions that are believed to have minor effect on the desired results. This is mainly due to the lack of proper simulating tools, as well as insufficient knowledge of the relevant affecting factors. One of the important stages in the design of earth dams is the exact evaluation of irrecoverable volume and porewater pressure through out the dam body as a result of shaking. This can be explained in of evaluating the seismic performance of the dam. The seismic performance of earthfill dams is often performed in two dimensional (2D) space, which demands selection of the critical cross sections of the dam. However, considerable judgment is required to estimate the overall three dimensional dynamic response of a dam in a narrow canyon from plane strain analyses of individual critical sections of the dam. In addition as the two dimensional consideration requires such simplifications as eliminating the effect of the canyon geometry, the two dimensional deformation analysis is believed to render inaccurate results. This is so because the plane strain analysis normally ignores the arching effect of the valley which is particularly relevant for dams in narrow valleys. For this purpose the effect of the canyon geometry is studied by carrying out 3D model of a real dam site. The assumed 3D model contains all details of the dam body and its foundation of Tendaho earthfill dam in Ethiopia but with variable valley configuration. Since the 1971 San Fernando earthquake in California (Ming and Li, 2003), major progress has been achieved in the understanding of the earthquake action

on dams. Gazetas (1987) discussed the historical developments of theoretical methods for estimating the dynamic response of earth dams to earthquake ground excitation. Progress in the area of geotechnical computation and numerical modeling offers interesting facilities for the analysis of the dam response in considering complex issues such as the soil non linearity, the evolution of the pore pressure during the dam construction procedure and real earthquake records. Detailed analysis techniques include equivalent linear (decoupled) solutions, and non linear finite element and finite difference coupled or decoupled formulations. Tendaho dam project over river Awash is located in the north-east of Addis Ababa, capital city of Ethiopia. It is among the largest dams in the country. It is a zoned earthfill dam with fill volume of about 4 Mm3 and expected to impound about 1.8 Bm3 water for irrigation purpose. The length of the dam crest is about 412 m and its maximum height at the river section is about 53 m. The width of the dam is about 10 m at the crest and 412 m at the widest point in the foundation level. The Tendaho dam is built on an alluvium consisting of alternating layers of mudstones siltstones, conglomerates and sandstones. This paper presents a numerical study that investigates the seismic performance of the Tendaho earthfill dam taking into the canyon geometry and ground conditions. 2

GEOLOGICAL SETTING OF DAM SITE

Tendaho dam site is located within an area known as the Tendaho graben, which forms the center of Afar

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Figure 1. Typical geological profile along the dam axis.

triangle. The volcanic rocks are composed of material of sea floor spreading as a result of crustal plate separation of Arabia and Africa during Tertiary times. Due to tensile tectonic strain acting along three rift lineations, the NW–SE oriented Tendaho graben, a fault-bounded basin, is formed. The cores from the boreholes show the lake deposit to be a mixed assemblage of silt, clay, sand, calcareous inclusions, mudstones, sand stones, and conglomerates. The typical geological profile of the dam site is shown in Figure 1. The dam site is located in western and southern parts of highly complex fault zone of Afar triangle. Many of the fault scarps are of recent date and the area is seismically active. From the regional seismicity review, earthquakes with magnitude, M, greater than 6 can be expected. The Tendaho dam design project recommends a peak acceleration of 0.18 g for the Operating Basis Earthquake, OBE. Moreover, the regional seismicity study required 0.3 g for the Maximum Credible Earthquake, MCE.

3

the elastoplastic analysis constitutes an efficient tool for the investigation of stability of dams under seismic loading when there exist few data like the case here. The seismically induced settlement could be used for the evaluation of the stability of the dam. 3.1

Numerical model

The numerical analyses are conducted using the finite difference program FLAC3D. The analyses are carried out within the framework of plasticity. This program is based on a continuum finite difference discretization using the Langrangian approach (FLAC3D, 2005). In this numerical model, the equations of motion are derived for a continuum media. And the equations of motion are used to obtain the velocities and displacements from stresses and forces. The strain rates are then calculated according to the new nodal velocities in each element. To analyze the problem, the strain-rate tensor and rotation rate tensor can be written as follows:

CONSTITUTIVE MODEL

The seismic concern of an earthfill dam is the development of large displacement that could endanger the safety and serviceability of the dam. Such movements depend on the earthquake loading, the geometry of the dam, and the strength properties of the materials of the dam and foundation, valley geometry and the ground water conditions. For the calculation of these movements, a 3D finite difference modeling FLAC3D is used during the dynamic analyses. The behaviors of the geomaterials are described by an elastic-plastic Mohr-Coulomb constitutive model. This is so because

where vi is the deformation velocity and vi, j is the velocity gradient. The equation of motion is written as:

The constitutive equation can be written out in general as:

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Figure 2. Dam geometry and its zone (not to scale).

"

where σ ij is the co-rotational stress-rate tensor, Hij is the constitutive function, and κ is a parameter, which takes into the loading history. The co-rotational stress rate tensor is defined as follows

dσij is the material time derivative of σ. dt The above equations are solved by finite difference method. A coupled calculation with dynamic groundwater flow is performed to determine the excess pore pressure. Regarding the mechanism of pore pressure generation, an empirical equation that relates the increment of volume decrease, εvd to the cyclic shear-strain amplitude, γ, and accumulated irreversible volume strain , εvd is used (Byrne 1991)

where C1 and C2 are material parameters, which vary according to the sand type. These parameters are estimated from the following relationships

to the recommendation by Kuhlemeyer and Lysmer (1973). Therefore xmax ≤ λ/10 is used in this study. From the seismic record analysis, the highest angular frequency of the input motion is about 15 rad/sec. Accordingly, the grid size or element size for the different materials of the dam body and the foundation are determined as shown in Figure 3. The Free-Field Boundaries are used to absorb the outward waves originating from the structure.This system of boundary condition involves the execution of free-field calculations in parallel with the main-grid analysis. The lateral boundaries of the main grid are coupled to the free-field grid by viscous dashpots to simulate a quiet boundary. The Sigmoidal hysteretic damping with four parameters is used in the analyses for the energy dissipation through the medium. 4

DAM GEOMETRY AND MATERIAL PARAMETERS

The basic general geometry, zones and slopes of the earthfill dam considered are shown in Figure 3. It consists of the following zones. Zone 1: Impervious core. Zones 2A & 2B: Shell (sandy gravel). Zone 3: Transition zone (fine sand) Zone 4: Filter Zone (coarse sand). The dynamic properties used in this work are taken from Seed et al. (1986). Except the lake deposit, the maximum shear modulus of all materials is considered to vary with the mean effective stress according to the formula:

where Dr and (N1 )60 are the relative density and SPT blow counts corrected for energy respectively. The dynamic loading is applied at the base of the foundation layer as an acceleration time history. The frequency content of the input motion and the velocity of the propagating waves affect the accuracy of the numerical solutions. For appropriate wave propagation through an element, the maximum element size, xmax has to be smaller than one-tenth to one-eighth of the wave length,λ. This wave length corresponds to the highest frequency component, f , that contains appreciable energy of the input motion. This is according

But for the lake deposit Gmax = ρ × (Vs )2 with the shear wave velocity Vs = 1 km/sec is used. The values used for the coefficient K2max are listed in Table 2 along with other soil parameter. The materials that are assumed to liquefy are modeled with the porewater pressure generation model proposed by Finn (1975). The other materials are modeled with the Mohr-Coulomb constitutive model. The shear strength envelope is specified by friction angle and cohesion. The model parameters for the different materials of the dam body as well as the foundation are given in Table 1 and 2.

491

Table 1.

Zone material property. Zone

Table 2.

Property

1

2A

2B

3

4

Specific gravity, Gs Dry density, (kg/m3 ) Porosity, n Permeability, k (m/s) Cohesion, c (kPa) Friction angle, φ (◦ )

2.72 1600 0.41 2.5 × 10−8 7 25

2.70 1800 0.33 5 × 10−5 0 34

2.70 1800 0.33 1 × 10−6 0 34

2.70 1800 0.33 1 × 10−5 0 34

2.70 1800 0.33 1 × 10−4 0 34

Material property.

Material

Poisson’s ratio, ν

K 2max

Mixed clay core Sandy gravel shell Alluvium foundation

0.34 0.3 0.3

40 90 70

5

SEISMIC LOADING

As there is no acceleration time history at / or around the area, the commonly used acceleration time history for earthquake resistant design, the 1940 El Centro (California) earthquake is used, Figure 4. This earthquake had a magnitude of 7.1. The base line correction and filtering of the raw acceleration record is carried out. So this modified and scaled to different magnitudes is applied to the considered earthfill dam model.

6

DYNAMIC ANALYSIS RESULTS

In order to know whether the canyon geometry has an effect on the seismic performance, different 3D seismic simulations for different shape of canyon and earthquake magnitudes are carried out. The obtained results are compared and correlated. From the results of the analyses and correlations created, the canyon geometry under which three dimensional behavior is of importance in the dynamic response of a dam are determined. And the resulting correlation is then applied to the seismic performance of the Tendaho earthfill dam. The results of the analysis give a clue which model (2D or 3D) to use for the problem at hand. In order to investigate the effect of the canyon geometry on the seismic performance of earth dams, different valley configuration are considered, Figure 3. The numerical model outlined above has been applied to four different cases of canyon geometry. The dynamic analysis is carried out for the horizontal El Centro earthquake scaled to different acceleration magnitudes, 0.15 g, 0.3 g and 0.6 g. For the sake of illustration crest settlement contours for one slope angle 20◦ is plotted in Figure 5. The crest

settlement time histories for the different slope considered are plotted in Figure 6. The parameter m is the tangent of the slope angle of the valley from the horizontal. In addition, the maximum crest settlements for different canyon geometries and different magnitude of earthquake acceleration are presented in Figure 7. The results of the analysis show that the canyon effect is highly pronounced for dams with tangent of slope angle of valley greater than about 0.5. An m value of 0.5 is equivalent to a slope angle of about 27◦ . This effect decreases with decrease in the magnitude of the acceleration. The canyon effect diminishes with decrease in the value of the slope angle below about 27◦ . In addition to the canyon effect of the dam, the danger of liquefaction of the alluvium foundation material of the dam was evaluated. Based on the preliminary evaluation of liquefaction susceptible of the alluvium material, there is a danger of liquefaction. In addition the shell materials are assumed to liquefy. So in the dynamic analysis of the dam, the alluvium and shell material are assumed to liquefy and the analysis is carried out for two different peak ground acceleration, PGA magnitudes, the MCE, and OBE. For the case considered, a maximum crest settlement of 0.80 m and maximum horizontal crest displacement of 2.11 m has been predicted under the action of MCE, Figure 8 and 9. The same case was analyzed for the OBE, and a maximum crest settlement of 0.63 m is predicted. 7

CONCLUSIONS

From the results of analyses for the different shapes of the valley with the data from Tendaho dam, the following important conclusions are drawn: 1. The canyon effect diminishes with decrease in the slope of the valley below about 27◦ , two dimensional analyses can suffice for dams with are constructed on valley with slope less than about 27◦ . 2. Plane strain analysis (2D) gives conservative results as compared to real 3D analysis. If plane strain analysis (2D) is carried out for dam to be constructed in valley with slope angle greater than about 27◦ , then the analysis will be on the safe side. In the case

492

Figure 3. 3D model of dam along with the valley.

Figure 4. Modified input acceleration time history.

Figure 7. Crest settlement versus slope of valley, m for different magnitude of accelerations.

Figure 5. Crest settlement contours for angle of slope of 20◦ .

Figure 8. Time history of horizontal and vertical displacement at the dam crest.

Figure 6. Crest settlement versus dynamic time for different slope of valley (0.3 g).

where carrying out 3D model analysis is expensive then a 2D model analysis with reduction factor that takes the arching or canyon effect into can be done.

3. As far as the Tendaho dam is concerned, the angle of the slope varies from about 26◦ to 30◦ with the vertical which corresponds to m values of 0.49 and 0.58 respectively. Based on this, the effect of the canyon shape on the dam is not significant. 4. The results of the dynamic analysis carried out using hysteretic damping based on the finite difference numerical method indicate that displacements are concentrated near the crest of the dam. 5. The peak ground accelerations are predicted to be amplified from 0.3 g at the base of the model to about 0.7 g at the crest for the Tendaho dam. This corresponds to an amplification value of about 2.4.

493

REFERENCES

Figure 9. Time history of vertical displacement at the dam crest.

6. A peak horizontal crest displacement of magnitude 2.11 m and peak crest settlements of 0.80 m are predicted. 7. From the results it was observed that with increasing level of earthquake magnitude the effect of the canyons shape becomes significant.

Bawson, E. M. et al. 2001. A practical oriented pore-pressure generation model. In: Billaux et al. (eds), Proc. of the 2nd Int. FLAC Symposium Lyon, , Balkema, Rotterdam. Fang, H.Y. 2002. Foundation Engineering Handbook. Kluwer Academic Publishers, USA, ISBN 0-412-98891-7. Finn, W.D.L 1979. Soil-dynamics-liquefaction of sands. In: Proc. Int. Conf. Microzonation, Seattle, Washington. Gazetas, G. 1991. Foundation Vibrations in Foundation Engineering Handbook, (H.Y. Fang, ed.) Van Nostrand Reinhold, New York, NY. Gopal Madabhushi, S. P. 2004. Modelling of earthquake damage using geotechnical centrifuges, Special section on Geotechnics and Earthquake hazards. Itasca Consulting Group, Inc. 2005. FLAC3D – Fast Lagrangian Analysis of Continua in 3 Dimensions.Version 3.0. Minneapolis, Minnesota, MN 55401. John, K. 2004. Dynamic Modeling with QUAKE/W. An Engineering Methodology, 1st edition, Geoslope international ltd, Canada. Kenji, I. 1996. Soil behaviour in earthquake geotechnics. Clarendon Press, Oxford. Prakash, S., Y. Wu & Rafnsson, E.A. 1995a. On Seismic Design Displacements of Rigid Retaining Walls, Proc. Third Intern. Conf. on Recent Adv. in Geo. Erthq. Engrg. and Soil Dyn., St. Louis, MO. Parish,Y., Sadek, M., & Shahrour, I. 2009. Numerical analysis of the seismic behaviour of earth dam. Natural Hazards Earth System Science, 9. Robert, W. D. 2002. Geotechnical Earthquake Engineering Handbook. The McGraw-Hill Companies, Inc, USA.

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Slopes and cuts


Effect of updated geometry in analyses of progressive failure A.S. Gylland Norwegian University of Science and Technology, Trondheim, Norway

H.P. Jostad Norwegian Geotechnical Institute, Oslo, Norway, Norwegian University of Science and Technology, Trondheim, Norway

ABSTRACT: Case records of progressive failure in Sweden and Norway shows excessive propagation of failure zones into horizontal ground accompanied by soil heave in the ive zone. According to static equilibrium analyses the ive resistance in these zones is theoretically exceeded. Thus, the horizontal propagation should stop due to this ive failure mode. This paper aims at investigating the mechanism of progressive failure with special attention to the effect of updated geometry and horizontal propagation by comparing FEM calculations with and without updated mesh together with case records and a simple analytical method. In the simulations, the effect of ive heave is limited whereas the case records show considerably higher values. This clearly demonstrates that this type of failure mechanisms is not fully understood and more detailed investigation of this problem is required.

1

INTRODUCTION

Progressive failure can occur in slopes of strain softening material and is often recognized by large slide events triggered by a limited local disturbance. In Norway and Sweden such slope failures are often associated with soft, sensitive and quick clays. The mechanisms involved in progressive failure have been widely discussed (Bishop 1967, Bjerrum 1967, Bernander 2000, Andresen et al. 2002, Andresen & Jostad 2004, Gylland et al. 2010) and the reader is referred to these publications for further details. One of the features with downward progressive failure is the occurrence of zones down slope with extensive ive heave, often spreading far out into horizontal ground. Some examples where this was highly apparent are the Surte landslide in 1950 (Jakobson 1952), the Tuve landslide in 1977 (Larsson & Jansson 1982), both occurring close to Gothenburg in Sweden, and the Bekkelaget landslide in 1953 in Oslo, Norway (Eide & Bjerrum 1954). The cases of Tuve and Surte have been analyzed in of progressive failure by (Bernander 2000) and it is found that the ive resistance in the slope theoretically is exceeded during the failure process. However, no ive failure mechanism occurs in the field. Instead the failure zone propagates near horizontally into almost flat terrain. To explain this feature, two hypotheses are proposed. First, the ive heave occurring during the failure process increases the ive resistance and thus forces the failure surface to propagate further horizontally. Second, it is known that these clay materials

have a rate dependent strength (Berre & Bjerrum 1973). This effect might become considerable in a sliding process and thereby increase the ive strength of the material. In both these cases, inertia effects should be included. This paper aims at investigating the mechanism of progressive failure with special attention to the effect of updated geometry.

2

METHOD AND PARAMETERS

2.1 FEM analysis The FEM analysis presented herein are performed using Plaxis 2D version 9.02. 2.1.1 Softening related issues The framework of rate independent elasto-plasticity has some issues when using strain softening materials. Such a material will locally develop negative second order internal work which makes it locally unstable (Hill 1958, Rudnicki & Rice 1975). This implies that strains might localize in thin zones (shear bands). In the FEM the shear band will shrink to the minimum size given by the element size, type and orientation (Pietruszczak & Mróz 1981). This implies that the results obtained from a FEM analysis with a softening material will depend on the mesh discretization (de Borst et al. 1993). To obtain mesh independent solutions a regularization technique must be used (Tikhonov & Arsenin 1977).

497

In this paper no regularization other than the element size is used to set the width of the shear band. The element scaling technique proposed by Pietruszczak & Mróz (1981) is utilized to have representative softening behavior in the shear band. 2.1.2 Updated mesh procedure In the updated mesh analysis the effect of geometry change on the equilibrium conditions is included. Plaxis 2D uses the co-rotational rate of Kirchhoff stress and an Updated Lagrangian formulation (Plaxis 2009). 2.1.3 Material model The material model used is the elastic-perfectly plastic Mohr-Coulomb model in Plaxis. Plane strain conditions are used. The material is set to undrained and effective stress parameters are used as input. Strain softening behavior is achieved by using a negative dilatancy angle. In the model the stress state will move downwards along the Mohr-Coulomb line once failure is reached. The tension cutoff criterion is used to control the residual strength. Figure 1, together with equations 1, 2 and 3, describes the situation in an s − t stress space. These expressions can be derived by using the basic equations of the material model.

Figure 1. MC model with negative dilatancy.

Here c is the cohesion, φ the friction angle, ψ the dilatancy angle, ν the Poisson’s ratio, E the Young’s modulus, σtens the tensile strength and σ0 the effective initial stress. The peak strength is controlled by the initial stress state and the effective stress parameters. The residual strength is constant with depth and controlled by the friction angle, cohesion and tension cut-off criterion. The softening modulus is given by the elastic stiffness, Poisson’s ratio, friction angle and negative dilatancy angle. 2.1.4 Geometrical model A 6 m thick flat terrain of sensitive clay with a 2 m high sand fill is analyzed. Load is added at the top of the fill and increased using incremental multipliers until failure occurs. Here, failure implies the formation of a definite failure mechanism. The geometry is shown in Figure 2. The flat terrain is chosen to better isolate the geometry effects and to avoid self propagating failure zones as can be seen in slopes (Bernander 2000). To generate constant strength with depth, a heavy material is added to the surface during the generation of initial stresses with the soil unit weight set to zero. Jaky’s formula (K0 = 1 − sin φ) sets the ratio between the initial major and minor principal stress. 5354 6noded triangular plane strain elements are used. This gives a sufficiently fine mesh to avoid severe alignment. The average equivalent shear band thickness in the model is 0.9 m. 2.1.5 Material parameters Table 1 gives the parameters used. Two stiffness values are chosen, one that can be regarded as stiff (models x.1) and one that is very soft (models x.2). The softening modulus is kept constant by balancing ψ according to the change in E. The sr /su -ratio is set to 0.75 for models 1.x while it is 0.25 for the others. The undrained shear strength (su ) is set constant equal to 41 kPa. The tensile cut-off criterion and the dilatancy angle are used to obtain the desired softening modulus (SM ) and residual strength (sr ).

Figure 2. Model geometry.

498

The parameters chosen give a ratio between the softening modulus and shear band thickness of approximately 330. The corresponding ratio for the sensitive Onsøy clay is 1500 for active loading, 1000 for direct shear and close to 0 for ive loading when using the sample size as length scale (Lacasse et al. 1985). This implies that the material simulated here can be regarded as mildly sensitive with respect to the rate of softening. The finite element model (used for both Models 1.x and 2.x) is shown in Figure 2. Models 3.x have the same geometry, but the softening material is only applied to a 0.6 m, one element thick layer at the base. The overlying material is modeled as perfectly plastic. These models are used for two reasons. First, softening is very limited in ive loading of sensitive clays (Lacasse et al. 1985). Second, when a ive failure zone starts to develop in a softening medium, resistance will be lost in this failure mode. Although there is some ive heave, this increase in resistance might be less than the softening induced loss. This results in a situation where the first ive failure surface will give the final situation almost unaffected by ive heave. 2.2 Analytical model A simple analytical model is developed in order to analyze the effect of ive heave from standard limit equilibrium based earth pressure theory. Figure 3 illustrates a simplified situation where the earth pressure in front of the ive heave zone (2) is given by P2 . P1 gives the earth pressure in the heave zone (1). The force T is the shear force at the base between these two zones. Table 1. Model ID

Material parameters. 1.1

1.2

2.1

2.2

φ [◦ ] 25 25 25 25 c [kPa] 5 5 5 5 ψ [◦ ] −3.73 −18.54 −3.75 −18.54 ν [−] 0.35 0.35 0.35 0.35 E [kPa] 8000 1000 8000 1000 γ [kN/m3 ] 20 20 20 20 σtens [kPa] −30.4 −30.4 −3.0 −3.0

SM [kPa] −300 su [kPa] 40 sr [kPa] 30

−300 −300 40 40 30 10

3.1

3.2

Fill

25 25 35 5 5 5 −3.75 −18.54 0 0.35 0.35 0.35 8000 1000 10000 20 20 17 −3.0 −3.0 –

−300 −300/0* −300/0* 40 40 40 10 10/– 10/–

Equilibrium requires P2 = P1 − T . By expressing these forces in of stresses and assuming constant undrained strength with depth and that Sr acts along L one obtains the expression given in Equation 4. It is here assumed that the ive resistance can be expressed as σ ive = Nr · su . With full friction along the vertical face where P1 and P2 acts, the value of Nr is 1 + π/2. The parameter M represents the ratio between σive1 and σive2 meaning that M > 1 implies failure along line 1 and M < 1 implies failure along line 2. The minimum size of the parameter L is given by the geometry of zone 1 for the assumptions made in √ this simplified model, L = H0 · 2. Inserting this in equation 4 one obtains the graph shown in Figure 10 where M is plotted as a function of H /H0 and sr /su . Note that no softening is assumed along the ive failure line, only on the horizontal propagation plane.

A mechanism that is not included in the model is the case where L approaches zero. This represents a mechanism where the ive failure zone is continuously pushed forward based on a surface slope gradient criterion. Since this information is not available for the case records studied, the presented formulation is used. 3

RESULTS

3.1 FEM analysis Figures 4 to 9 show the failure surfaces for the different models. Results from the calculations with and without update mesh are presented in the same figure. In all cases a failure zone developed horizontally before a more circular surface formed as the final failure mechanism. For models 1.x and 2.1, there is almost no difference in the obtained failure mechanisms. For model 2.2 the updated mesh analysis terminated before forming a

– – –

* Models 3.x have a softening layer under a non-softening. Figure 4. Model 1.1. E = 8000 kPa, sr /su = 0.75.

Figure 3. Analytical ive heave model.

Figure 5. Model 1.2. E = 1000 kPa, sr /su = 0.75.

499

Table 2.


Capacity loads. Without update mesh

Update mesh

Model

Load [kPa]

Disp. [m]

Load [kPa]

Disp. [m]

1.1 1.2 2.1 2.2 3.1 3.2

166 173 102 85 147 186

0.24 1.77 0.22 2.21 0.38 3.10

184 260 118 125 201/308 286

0.22 2.03 0.21 1.60 0.38/2.84 2.90




definite final failure mechanism because of a nearly singular stiffness matrix. Studying the incremental strains for the last steps a shear surface as illustrated by the dotted line in Figure 7 is indicated. For models 3.x, with softening material only in a thin layer at the base, longer horizontal propagation is seen. For model 3.2, the conventional analysis gave more horizontal propagation compared to the updated geometry calculation. It should be noted that the horizontal failure zone for model 3.2 propagated to the left boundary before ive failure occurred. Table 2 compares the maximum capacity loads and corresponding vertical displacement at the center of the fill. All calculations displayed two peaks in the load displacement curve. The first occurs approximately at 50% of the maximum capacity and represents the onset of the horizontal failure zone. The second peak is often followed by unloading and represents the formation of the final failure mechanism. The updated geometry analysis gives higher capacity loads. This is in accordance with results obtained by Van Langen (1991) and comes from the reduction in driving forces and increase in resisting forces as the fill is pushed down. It should be noted that large deformations were needed to obtain the maximum load for the calculations with low stiffness. For model 3.1, values for the forming of the first and second shear surface is given.

Figure 10. Comparison of FEM calculations and analytical model. Table 3.

ive heave.

Model

ive heave [m]

1.1 1.2 2.1 2.2 3.1 3.2

0.02–0.06 0.10–0.50 0.02–0.07 0.20–0.60 0.05–0.20 (0.10–1.50) 0.20–0.5

Ranges of ive heave obtained for the six models are shown in Table 3. It is seen that the heave for the low stiffness cases is larger compared to the cases with high stiffness. 3.2 Analytical model The analytical model together with results from the FEM calculations and the landslides at Surte, Tuve and Bekkelaget is presented in Figure 10. γH /su is set to 3 according to the simulations. For the line marked with *, γH /su is set to 13 which is more representative for Surte and Tuve. For Bekkelaget the value is close to 6.

500

Figure 11. Distribution of shear stresses for models 2.x.

4

DISCUSSION

4.1 Effect of update geometry The calculations show that all deformation and thus also the ive heave is higher for the models with low stiffness. However, the effect on the failure surface is limited, also for an updated mesh simulation. Studying the results from both the FEM calculations and the analytical model it seen that a low residual to peak strength ratio increases the ability of the failure surface to progress horizontally. This is expected as a low residual strength implies low resistance against horizontal sliding. Concerning the geometry effects, a low stiffness gives a higher ive heave and thus conditions for horizontal propagation. However, at the same time the stiffness affects the distribution of shear stresses along the failure surfaces. Figure 11 shows the distribution of shear stresses along the horizontal base for models 2.x. It is seen that low stiffness gives a narrower hardening zone and a longer residual zone compared to the high stiffness case. This implies that the low stiffness case has a reduced resistance for sliding along the horizontal plane and should thus intensify such a failure mode. For model 3.1, the effect of a non-softening top layer on the final failure mechanism is to increase the horizontal propagation some. In model 3.2, with low stiffness, excessive deformations occurred and the failure zone propagated first horizontally all the way to the left boundary. The obtained result implies that the boundary condition might affect the result and that ive failure could have occurred there if it had been a lowering in the terrain or similar. When comparing the FEM calculations to the analytical modes it is seen that extensive horizontal propagation should not be expected for models 1.x and 2.1. The ive heave is too low. For models 2.2 and 3.x however, horizontal propagation could be possible. But still, it does not happen to any high extent. Studies of the developing failure zones show that the first zone becomes the final. This is partly expected for models 2.x as the strain softening induced reduction of strength in the ive zone prevents formation of new modes. However, this is not expected for models 3.x.

Still, once a failure zone is formed, all deformation occurs within this mechanism. For model 3.1, failure zone 2 forms after extensive heave in this region thus indicating that for large deformations, the issible kinematics of the problems are governing. A possible explanation for this behavior is the resulting load application in the vertical section A in Figure 2. In the model, a moment is acting on this section together with the horizontal force. This enhances the circular ive failure mode. For the case studies, the ive heave occurs down slope, a considerable distance from the load application point. Here the force is more purely horizontal. It is interesting to see how the historical slides plot in Figure 10. Due to extensive ive heave, the three examples included here all plot in a region where horizontal propagation could be expected, indicating that this mechanism could be possible due to geometry effects. As noted above, the difference in application of the horizontal force might explain this nonconformity between the historical slides and the simulations. Further, the γH /su – parameter is higher for the case records compared to the simulations because of larger values of H and lower values of su . As shown in Figure 10, increasing values of γH /su – promotes horizontal propagation. This indicates that the depth of the simulated model and the undrained shear strength used prevents ive heave from being effective in the studied mechanism. 4.2 Stress rotations By studying stress points along the horizontal failure zone it is seen that the maximum shear stress is horizontal as the peak strength is reached. This explains why this failure mode develops first. As the load is increased the shear stresses can rotate further and finally give the ive failure mode. This aspect is discussed by e.g. Potts et al. (1996) and illustrates the importance in distinguishing between local and global failure when performing analysis with strain softening materials. A locally developing failure zone might not be the path of minimum resistance in a global sense. Jostad & Andresen (2004) have shown results where several local failure zones are initially developing before dying out. 5

LIMITATIONS AND FURTHER WORK

As mentioned, the FEM modeling performed here is not regularized with implications as discussed. Further, the simple tri-linear softening model used implies some uncertainties. The direction of propagation will depend on the degree of mobilization in the area close to the material point which es the peak strength first. The model used here will overestimate the stiffness before peak and the softening after peak. In addition, Norwegian quick clays have anisotropic strength properties (Lacasse et al. 1985) which might affect the propagation direction.

501

The co-rotational rate of Kirchhoff stress used in the update-mesh analysis is most adequate for moderate levels of shear strain. This might not be the case in the failure calculations performed here. The iteration process will also affect the results. For softening materials the initial failure will to a large extent govern the direction and the development of localized zones. This initial localization is affected by the size of the applied load step and is thus sensitive to the iteration settings. In order to overcome these issues, simulations should be done in a regularized model with smooth anisotropic material response. A Newton-Raphson iteration scheme as used by e.g. Jostad (1993) might resolve some of the iterative issues. Concerning the results from the updated geometry calculations it seems clear that full slopes with representative values of γH /su must be modeled in order to capture the full effect of ive heave. Even though the simple analytical model indicates that the ive heave in the case records could be sufficient to explain the horizontal propagation seen, the results remain inconclusive and it is believed that both effects of strain rate and inertia should be included to investigate these results further. 6

CONCLUSIONS

Progressive failure is investigated with special focus on the effect of updated geometry and its effect on horizontal propagation of failure zones by comparing FEM simulations with and without updated mesh. The effect updated geometry on horizontal propagation is limited in the simulations performed. This is mainly because of the low ive heave obtained and the governing kinematics in the final failure configuration. In historical slide events a much higher heave is seen. This indicates that the ive heave mechanism might partly explain the excessive horizontal propagation of progressive failures. However, the results are not conclusive and it is recommended that full slopes with realistic geometry and material parameters are investigated. Further, to complete the study, strain rate effects on the material strength and inertia effects should be included. ACKNOWLEDGEMENTS Professor Steinar Nordal and Professor Lars Grande at the Norwegian University of Science and Technology are greatly acknowledged for valuable discussions and comments. The work described in this paper is ed by the Norwegian Public Roads istration, the Norwegian National Rail istration, the Norwegian Water Resources and Energy Directorate and by the Research Council of Norway through the International Centre for Geohazards (ICG). Their is gratefully acknowledged. This is ICG contribution No 286.

REFERENCES Andresen, L. & Jostad, H.P. 2004. Analyses of progressive failure in long natural slopes. Proceedings of NUMOG04, IX, Ottawa, Canada. Andresen, L., Jostad, H.P. & Höeg, K. 2002. Numerical procedure for assessing the capacity of anisotropic and strain-softening clay. Proc. Of WCCM V. Vienna, Austria. Bernander, S. 2000. Progressive Landslides in Long Natural Slopes. Licentiate thesis, University of Luleå, Sweden. Berre, T. & Bjerrum, L. 1973. Shear strength of normally consolidated clays. Proc. of the Eight Int. Conf. on Soil Mech. and Found. Eng., Moscow. 3: 39–49. Bishop, A.W. 1967. Progressive failure- with special reference to the mechanism causing it. Proc. Of. Geo. Conf. Oslo 2: 142–150. Bjerrum, L. 1967. Progressive failure in slopes of overconsolidated plastic clay and clayey shales. Journal of soil mechanics and foundations, ASCE, SM5: 3–49. de Borst, R. Sluys, L.J. Mühlhaus, H.B. & Pamin, J. 1993. Fundamental issues in finite element analyses of localization of deformation. Engineering Computations 10: 99–121. Eide, O. & Bjerrum, L. 1954. The slide at Bekkelaget. Proc. of the European Conf. on Stability of Earth Slopes 2: 1–15. Gylland,A.S. Sayd, M.S. Jostad, H.P. & Bernander, S. in press. Investigation of soil property sensitivity in progressive failure. Submitted to NUMGE2010, Tr.heim., Norway Hill, R. 1958. A general theory of uniqueness and stability in elastic-plastic solids. Journal of the Mechanics and Physics of Solids 6(3): 236–249. Jakobson, B. 1952. The landslide at Surte on the Göta River. Proceedings of the Royal Swedish Geotechnical institute 5. Jostad, H.P. 1993. Bifurcation analysis of frictional materials. PhD thesis. Norwegian University of Science and Technology, Trondheim, Norway. Jostad, H.P. & Andresen, L. 2004. Modeling of shear band propagation in clays using interface elements with finite thickness. Proceedings of NUMOG04, IX, Ottawa, Canada. 121–128. Lacasse, S. Berre, T. & Lefevbre, G. 1985. Block sampling of sensitive clays. Proc. of the Eleventh Int. Conf. on Soil Mech. and Found. Eng, San Francisco. 2: 887–892. SGI Report No 18. 1982. The Landslide at Tuve November 30 1977. Swedish Geotechnical Insitute. Pietruszczak, St. & Mróz ,Z. 1981. Finite element analysis of deformation of strain-softening materials. International journal for numerical methods in engineering. 17: 327–334. Plaxis manuals. 2009. Plaxis BV Delft, the Netherlands. www.plaxis.nl. Potts, D.M., Dounias, G.T. & Vaughan, P.R. 1996. Finite element analysis of progressive failure of Carsington embankment. Géotechnique 40(1): 79–101. Rudnicki, J.W. & Rice, J.R. 1975. Conditions for the localization of deformation in pressure-sensitive dilatant materials. Jour. of the Mech. and Phys. of Solids. 23(6): 371–394. Tikhonov, A.N. & Arsenin, V.Y. 1977. Solutions of Ill-Posed Problems, New York: Vh Winston. Van Langen, H. (1991). Numerical analysis of soil-structure interaction. Dissertation thesis. Delft University of Technology, Delft, the Netherlands.

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Evaluation of the effective width method for strip footings on slopes under undrained loading K. Georgiadis & E. Skoufaki Department of Civil Engineering, Aristotle University of Thessaloniki, Thessaloniki, Greece

ABSTRACT: The effective width method is well established and commonly applied for the calculation of the bearing capacity of eccentrically loaded shallow foundations on horizontal ground. This paper examines the applicability of the method for the case of undrained loading of strip footings on or near slopes. Finite element analyses are presented, which consider various slope angles, slope/footing distances and load eccentricities. The calculated ultimate vertical loads and moments are presented in of failure envelopes and load interaction diagrams and are compared to those obtained with the application of the effective width method to analytical solutions based on finite element analyses and upper bound plasticity calculations for centrally loaded footings. It is observed that the results obtained directly from finite element analyses and those obtained with the application of the effective width method, diverge, especially at relatively low eccentricities. However, the effective width method is shown to be always more conservative. Finally, a comparison of load interaction diagrams is also presented for the problem of combined eccentric and inclined loading, for the case of a 45 degree slope and different slope/footing distances.

1

INTRODUCTION

The undrained bearing capacity of strip footings on slopes has been investigated by several authors. Hansen (1961) and Vesic (1975) proposed empirical factors to take of ground and load inclination, however their solutions apply only to footings at the crest of a slope and do not consider the case of “overall” slope stability type failure. Meyerhof (1957), Kusakabe et al. (1981) and Azzouz & Baligh (1983) presented solutions for vertical only loading which take of the distance of the footing from the slope, but either do not consider other affecting parameters or are generally not conservative (Georgiadis, 2009). A complete solution for vertical only loading was presented recently (Georgiadis, 2009), based on finite element analyses, and was subsequently confirmed with the upper bound plasticity method (Georgiadis, 2010a). The effect of inclined loading was investigated in a separate study (Georgiadis, 2010b) with the upper bound plasticity and finite element methods and an equation for the horizontal – vertical load interaction curve was proposed. This paper presents a series of finite element analyses performed in order to investigate the influence of eccentric loading on the undrained bearing capacity. The effective width method proposed by Meyerhof (1957) is combined with the above solutions for vertical and inclined loading and the results are compared to the finite element analysis results. Various slope angles, slope/footing distances and load eccentricities

are considered and both the analytical and numerical results are presented in of load interaction diagrams.

2

PROBLEM DEFINITION

The geometry of the problem is presented in Figure 1. Four slope angles β = 0◦ , 15◦ , 30◦ and 45◦ and two normalised footing/slope distances λ = 0 and 1 are considered. The footing width and slope height D are constant in all analyses equal to B = 2 m and D = 7.5 m, respectively. It is noted that the value of D did not affect the calculated failure loads in any of the analyses performed since “bearing capacity” failure as opposed to “overall” slope stability type failure was always predicted. In addition, although the analytical solutions presented consider the effect of footing depth db , finite element results are presented only for surface footings. The numerical analyses were performed with the finite element program Plaxis V8.6 (Brinkgreve & Broere, 2006). 15-noded triangular elements were used to model the soil, while the footing was modeled with beam elements. Interface elements were placed between the footing and the soil. The soil was modelled as a Tresca material with constant undrained shear strength cu , bulk unit weight γ and undrainedYoung’s modulus Eu = 30 MPa. A single ratio of cu /γB = 2.5 was considered. The value of cu /γB affects the failure loads but has a very limited effect on the normalised load interaction

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d/B and D/B and the slope angle β. Low values of cu /γB and/or large slope angles β lead to “overall” slope stability type failure, while other combinations lead to “bearing capacity” type failure independent of D/B. In the latter case the following equation can be used instead of the design charts:

Figure 1. Problem definition.

where Nco = undrained bearing capacity factor for a footing at the crest of a slope:

λ = dimensionless function of the distance of the footing from the slope and the footing depth (see Fig. 1):

and λo = critical value of λ, beyond which Nc = 5.14:

Finally, the depth factor gq in Equation (1) varies linearly with λ: Figure 2. Finite element mesh for 45◦ slope angle: a) whole mesh with boundary conditions and b) detail of mesh under footing.

diagrams provided that “bearing capacity” failure takes place (Georgiadis 2010b). The footing was modelled as linear elastic with bending stiffness EI = 2.4·106 kNm2 /m and axial stiffness of EA = 2.9·107 kN/m. Finally, a very thin 2.5 cm zone of zero tensile strength soil elements was modelled beneath the footing. As a consequence, the interface elements were also not allowed to sustain tension and therefore an effective gap was allowed to form between the soil and the footing due to footing rotation in the eccentric loading analyses. 3 ANALYTICAL SOLUTION 3.1 Vertical loading The solution for vertical only loading presented by Georgiadis (2009), extended to include foundation depth, db , (Fig. 1) is given by the following equation:

3.2 Inclined central loading The failure loads for the case of inclined loading can be calculated from the following expression (Georgiadis, 2010b):

where v = V /Vo the normalised vertical load, vo = normalised vertical load for horizontal ground surface (ξ = 1), Vo = ultimate load for vertical only loading, obtained from Equation (1), h = H /Ho the normalised horizontal load (positive when directed towards the slope as shown in Fig. 1), Ho = Bcu the ultimate horizontal load and ξ a parameter which depends on both the slope angle β and the normalized footing distance λ:

3.3 Inclined eccentric loading where Vo = ultimate vertical load for vertical only loading, Nc = undrained bearing capacity factor, po = overburden pressure and gq = depth factor. Nc can be obtained from the design charts provided in Georgiadis (2009) as a function of the ratios cu /γB,

The influence of eccentric loading can be taken into by applying the effective width principle to the above equations. In the case of vertical eccentric loading this can be done simply by substituting the footing width B with the effective width B = B-2e and the footing distance d with d = d + 2e (only for positive

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load eccentricities as shown in Fig. 1) in Equations (1), (3) and (4). For general inclined eccentric loading the load interaction curve can be obtained by modifying Equation (7):

where:

and m = M /(BV o ) is the normalised bending moment (M = eV ). 4

RESULTS

4.1 Vertical eccentric loading The Figure 3a illustrates the failure envelopes in M-V load space for the case of a footing at the crest of a slope (λ = 0), obtained from the finite element analysis results (FE) and Equation 1 with the application of the effective width principle (EQ). Both solutions reproduce the expected reduction in the size of the failure envelope due to the increase of the slope angle β. It can be observed that although Equation 1 compares excellently to the FE results for the case of zero load eccentricity (M = 0) and any slope angle, the two solutions diverge with increasing eccentricity, with the analytical solution underestimating the failure loads by up to 8%. Beyond a certain eccentricity all failure curves converge towards the horizontal ground surface failure curve, especially for positive load eccentricities, indicating that as the failure mechanism reduces in size it becomes identical to the horizontal ground surface mechanism and therefore the presence of the slope becomes insignificant. Similar observations can be made for the case of a footing at a normalised distance of λ = 1 from the slope (Figure 3b). In this case the transition from a failure mechanism involving the slope to the horizontal ground surface failure mechanism is more marked and occurs at lower eccentricity. The two different failure modes are shown in Figures 4a and 4b for e = 0.45 m and e = 0.15 m, respectively. As a consequence the overall influence of the slope angle on the calculated failure loads is less significant than the λ = 0 case. As seen in Figure 3b, similarly to the λ = 0 case, the effective width method provides a conservative estimate of the failure loads, compared to the FE results, with the maximum difference being less than 8%. The analytical and numerical results are presented in of normalised load interaction diagrams in Figures 5a and b for λ = 0 and λ = 1, respectively. It is clear from this figure that no unique load interaction

Figure 3. (a) Failure envelopes and (b) load interaction diagrams for λ = 0.

Figure 4. Failure mechanism for λ = 1: (a) e = 0.45 m and (b) e = 0.15 m.

diagram exists for footings at the top of slopes since the shape of the failure curve depends on both the slope angle and the distance of the footing from the slope. It can be observed by comparing the two figures that the influence of the slope on the load interaction diagrams reduces with the increase of the normalised distance. This is not surprising, since it is expected that beyond

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Figure 5. Load interaction diagrams in the v – m normalised load plane: (a) λ = 0 and (b) λ = 1.

a certain distance all curves should tend towards the unique load interaction curve for horizontal ground surface. The critical distance beyond which the presence of the slope is not relevant is given by Equation 5 for the case of central loading and appears to be less for the case of eccentric loading. As seen in Figure 5 the initial inclination of the load interaction curve (at low eccentricities) increases with the increase of slope angle. This indicates that the slope has a smaller effect on eccentric failure loads than on the ultimate failure load Vo .

4.2

Inclined eccentric loading

The horizontal failure loads and bending moments at different normalised vertical loads v (V /Vo ) = 0.25, 0.5 and 0.75, for the case of a 45◦ slope angle are shown in Figures 6a and 6b for normalised slopefooting distances λ = 0 and λ = 1, respectively. The finite element results are compared in these figures to the failure loads obtained from Equation 9. It can be observed that although the introduction of effective dimensions to the equations for inclined loading is in relatively good to excellent agreement with the FE results for either eccentric or inclined loading, it is does not perform as well in the case of combined loading. It is noted, however, that the analytical results

Figure 6. Failure envelopes in the h – m normalised load plane for different vertical load levels: (a) λ = 0 and (b) λ = 1.

are always conservative compared to the finite element analyses. As seen in Figure 6, the failure curves are not symmetrical about the H nor the M axis for the case of λ = 0 and become almost symmetrical for λ = 1. This can be attributed to the fact, as noted above, that the effect of the slope reduces with the increase of the slope/footing distance. For the non-symmetric λ = 0 case, the failure loads are higher for positive eccentricities (away from the slope) and negative horizontal loads (towards the horizontal ground surface), with the maximum bending moment obtained for negative horizontal load.

5

CONCLUSIONS

An analytical expression was presented for the calculation of the failure loads of footings on or near slopes under eccentric and combined loading. The expression was developed by applying the effective width principle to previous solutions for vertical central loading and inclined central loading. The predictions of this solution were compared to the results of a series of finite element analyses of footings on or near slopes for various slope angles. It was found that the application of the effective width method to

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footings on slopes is always conservative, yields sufficiently accurate results for vertically loaded footings but underestimates the bearing capacity in the case of combined loading. The results of the analyses were also used to investigate the influence of slope angle and slope/footing distance on the bearing capacity and the shape of the failure load envelopes. REFERENCES Azzouz A. S. & Baligh M. M. (1983). “Loaded areas on cohesive slopes”. Journal of Geotechnical Engineering,ASCE, Vol. 109, No. 5, pp.724–729. Brinkgreve R. B. J. & Broere W. (2006). “Plaxis ’s manual”, Plaxis B.V., Netherlands. Hansen J B. (1961). “A general formula for bearing capacity”. Danish Geotechnical Institute, Bulletin 11, Copenhagen, Denmark, pp 38–46. Georgiadis K. 2009. Undrained bearing capacity of strip footings on slopes”. Journal of Geotechnical and Geoenvironmental Engineering ASCE, (10/13/2009), 10.1061/(ASCE) GT.1943-5606.0000269.

Georgiadis K. 2010a. An upper bound solution for the undrained bearing capacity of strip footings at the top of a slope. Geotechnique (in press). Georgiadis K. 2010b. The influence of load inclination on the undrained bearing capacity of strip footings on slopes. Computers and Geotechnics (in press). Kusakabe O, Kimura T & Yamaguchi H. (1981). “Bearing capacity of slopes under strip loads on the top surfaces”. Soils and Foundations, Vol. 21, No. 4, pp 29–40. Meyerhof G G. 1953. The bearing capacity of foundations under eccentric and inclined loads. Proceedings 3rd ICSMFE, Zurich, pp 440–445. Meyerhof G G. 1957. The ultimate bearing capacity of foundations on slopes. Proceedings of the 4th ICSMFE, London, pp 384–386. Vesic A S. 1975. Chapter 3: Bearing capacity of shallow foundations. Foundation Engineering Handbook, Ed. Winterkorn H. F. and Fang H. Y., Van Nostrand Reinhold.

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Failure induced pore pressure by simple procedure in LEM T. Länsivaara Tampere University of Technology, Finland

ABSTRACT: Pore pressures play often a dominant role in stability evaluations. In addition to initial pore pressure conditions, also failure induced pore pressure may have a significant effect on the stability of embankments on soft sensitive Scandinavian clays. When analyzing the stability for such embankments using effective stress analysis, one should thus be able to incorporate also this effect into the calculations. In finite element analysis this is possible, although the results are depending on the applied soil model and its parameters. One should thus be very careful in assessing them. Especially the general shape of the yield surface and the parameters governing it are of importance. In calculations based on the limite equilibrium method (lem) the failure induced pore pressure is often ignored, resulting in an overestimation of safety. In the paper a new simple procedure to include failure induced pore pressure in lem calculations is introduced. The method is aimed for normally consolidated soft clays. Such conditions are dominant for many railway embankments on existing railway lines in southern Finland. Calculations will be made for two embankments, one existing embankment still in use, and one old embankment taken out of use where a full scale failure test was conducted.

1

INTRODUCTION

In failures occurring in soft normally consolidated clays, the pore pressures play often a dominant role. In addition to e.g. loading or ground water/precipitation caused pore pressure, the failure process itself induces an excess pore pressure. Therefore, if one desires to use effective stress analysis, pore pressure development both before and under the failure progress needs to be addressed. In finite element (fem) analysis failure induced pore pressure may be ed, although one needs to be very careful in choosing the soil model and its parameters. In limite equilibrium method (lem) the failure induced pore pressure is usually ignored, leading to an over prediction of safety. In this article both fem and lem analysis are addressed and a new simple way to incorporate failure induced pore pressure in lem is introduced.

2

well know phenomena that higher strain rates gives higher undrained shear strength in e.g. the vane tests. The strain rate does not however, influence the effective strength parameters; see e.g. Janbu & Senneset (1995). The effective strength parameters, i.e. the friction angle and cohesion are rate independent, while the phenomenon is caused by dissimilar pore pressure response. The higher the strain rate is, the lower is the pore pressure, resulting in a higher shear stress. From soft clay behavior, we also know that the lower the strain rate is, the more there will be creep strain. In of soil modeling, the rate dependency of failure shear stress can thus be explained by that a higher volumetric straining for lower strain rates needs to be compensated by a larger reduction of effective stress. For high strain rates, the creep tendency is much smaller, resulting in a smaller reduction of effective stresses, as a smaller tendency to volumetric hardening need to be compensated. This is illustrated in a principal sketch in Figure 1.

FAILURE INDUCED PORE PRESSURE

When soft normally consolidated clay is loaded, the break down of the soil skeleton results in a tendency to large volumetric compression. However, the low permeability of the clay hinders the water to dissipate, resulting in pore pressure build up and undrained conditions in most usual loading rates. In of soil modeling, the tendency to large positive plastic volumetric straining needs to be compensated by negative elastic straining, which is possible only by a reduction in effective stresses. It is also known, that the higher the strain rate is, the higher shear stress is obtained. This result in the

3

FINITE ELEMENT ANALYSIS

It has been shown, that for a drained failure analysis by the finite element method the factor of safety is not significantly influenced by the deformation properties (Cheng et al 2007). This is however true only for drained conditions, or an analysis done using a simple elastic-perfectly plastic soil model. From chapter 2 it is clear, that to model realistically undrained failure one needs to have a proper description of the yield surface, elastic and plastic deformation properties and

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Figure 1. Undrained shear for normally consolidated clay. The arrows correspond to stress paths for different loading rates. The higher rate, the higher stress level on the failure line is reached.

rate dependency. The shape of the yield/plastic potential surface, especially between the K0 line and failure state, is a key factor on the development of excess pore pressure. The ratio between elastic and plastic deformation parameters governs how closely the modeled yield surface is followed in a rate independent undrained analysis. The stiffer the elastic response is modeled, the more closely is the initial yield surface followed by the resulting stress path. One could thus be attempted to model the rate dependency by softer elastic parameters for high strain rates and stiffer elastic properties for low strain rates. However, to properly model the rate dependency of failure, one needs to incorporate also creep in the model. This is a rather difficult task, as the important area is just close to the initial yield surface. The model would need to describe well both the low creep area inside the initial yield surface, and the high tendency for creep when ing the initial yield surface. When applying finite element analysis in practical engineering, soft clay behavior is often described with cap type models like the soft soil and soft soil creep models in Plaxis. As discussed above, to describe the failure induced pore pressure development properly, one needs to address the properties of the model carefully. The M -parameter that in Plaxis governs the yield and plastic potential surfaces is normally set to give a proper K0 value in the normally consolidated range. However, near failure the surfaces might then be to steep resulting in a low excess pore pressure build up and high failure load. To incorporate the failure induced pore pressure in a failure analysis, one needs to asses what is the true shape of the yield surface near failure. Then it might be more appropriate to fit the M -parameter more to the friction angle, having then a yield surface more close to the modified cam clay model. If one wishes that the stress paths in the analysis follows closely to the given yield surface, the elastic properties should be modeled very stiff. For a lamda kappa relation of 20 or higher the relation did not have any influence to the development of excess pore pressure, and thus also to the failure load in the analysis (Mansikkamäki & Länsivaara 2009).

Figure 2. Over prediction of factor of safety in undrained effective stress analysis using traditional lem.

4

LEM ANALYSIS

In limite equilibrium based stability analysis the stress conditions are described in a somewhat simplified way. Stress distribution is not considered, while stresses e.g. from external loads are transferred solely to the bottom of the slice upon they act. Thus in undrained conditions one needs to compensate this stress increase by a pore pressure increase to avoid unrealistic increase of strength in undrained analysis with effective stresses. All general methods assume an equal factor of safety along the slip surface and give an equilibrium strength/shear stress needed to balance the unstabilising forces. Failure induced pore pressures are normally not ed for. The equilibrium shear stress obtained from the analysis is therefore compared to a strength level corresponding to drained analysis, leading to an over prediction of strength and safety, see Figure 2. Failure induced pore pressure was introduce to lime equilibrium analysis at least already in 1981 by Svanö (Svanö 1981). Herein a more simplified approach is introduced aiming to solve the problem for existing railway embankments of soft clays in Finland. The need for stability evaluations on the existing railway lines rises from the need to increase train loads. Therein the situation is that embankments have been built several decades ago on very soft clays. There might be some small overconsolidation in the clays due to aging effects, but under the embankments the clays are generally normally consolidated. If a failure state occurs, there will thus develop an excess pore pressure corresponding to a stress change from the in situ state at K0NC line to the failure state. It has been shown (Länsivaara 1996, Länsivaara 1999) that the shape of the initial yield surface can be approximated by an inclined elliptical yield surface by only knowing the friction angle of the clay. Then one only needs to assume that associated flow is valid for the yield point on the K0 -line, and fit the inclination accordingly. In Figure 3, some examples of estimations for yield surfaces are presented. One can thus describe both the initial hydrostatic stress pK0 and the failure hydrostatic stress pf with the aid of preconsolidation pressure and friction angle, i.e.:

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Figure 4. Effective pore pressure parameter ru as function of friction angle.

Figure 3. Estimations of yield surfaces using only strength and preconsolidation data (Länsivaara 1999).

Where σcv = preconsolidation pressure and φ = friction angle. For a normally consolidated soil, the preconsolidation pressure can be substituted by the effective in situ vertical pressure.An estimation for failure induced pore pressure can then be obtained from equation (3):

In limite equilibrium method this can be used by applying a similar pore pressure parameter as ru , with the exception that it now only stands for failure induced pore pressure and should be applied to effective vertical stress. This new pore pressure parameter is herein referred as ru and is defined as:

Where ue = failure induced excess pore pressure. An equation for ru can now be solved by using an inclined elliptical yield surface. For simplicity, the solution is herein presented in graphical form in Figure 4. As can be seen from Figure 4, the method gives a decreasing pore pressured development with increasing friction angle. The same conclusion can be drawn also by only looking at the yield surfaces presented in Figure 3. With higher friction angles the yield surfaces are more inclined, and the relative horizontal distance from the K0 -line to failure on top of the yield surfaces decreases.

The procedure described above is strictly valid only for the active case. So in limite equilibrium calculations, one should apply different solutions in shear and ive zones. Simply by looking at Figure 3, one might argue, that the failure induced pore pressure in the ive zone should be higher than in the active zone. However, as described earlier, the clays in southern Finland are often slightly overconsolidated due to aging. So while perfectly normally consolidated conditions usually occur under an existing embankment due its own loading, the clay next to the embankment in the ive zone is most likely slightly overconsolidated. Then, the development of excess pore pressure due to failure in ive case is less than might be assumed from Figure 3. It should also be ed, that the intention herein is to try and develop a simple way to for failure induced pore pressure for engineering practice.

5 APPLICATION The method to include failure induced pore pressures in LEM calculations described in chapter 4 will be tested for two cases, both related to railway embankment stability. Case 1 includes an existing track on very soft normally consolidated clay. Case 2 is a test site, where a full scale failure test was recently performed.

5.1 Case 1, Turku-Uusikaupunki track The Turku-Uusikaupunki track is located in southwest Finland. The subsoil consists of very soft clay with a water content ranging generally from 70 to 100% and clay content close to 80%. The undrained shear strength varies generally between 6 and 10 kPa. Due to large settlements, the old railway embankment stands almost fully inside the subsoil.

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Table 1. Calculation parameters used in lem and fem calculations for Turku-Uusikaupunki site. φ o

Embankment Dry crust Top soft clay

35 25 25

c kPa

γ kN/m3

0 4 4

20 16 15.3

Figure 6. Failure surface from the fem calculations by soft soil model for case 1, the Turku-Uusikapunki site.

Figure 5. Critical slips surface from the lem-calculations for case 1, the Turku-Uusikapunki site.

Effective strength parameters and unit weights used in the calculations are presented in Table 1. From Figure 4 it can be seen, that for a friction angle of 25◦ the effective pore pressure parameter for the limite equilibrium method calculations is close to ru = 0.2. The calculated critical failure surface by lem is presented in Figure 5. The minimum factor of safety varied between FOS = 1.55 to 1.62 depending on how the pore pressure from the train load was modeled. The pore pressure from the train load caused also some instability to the calculations. If the failure induced pore pressure would be left uned, the limite equilibrium method would yield a factor of safety equal to FOS = 1.91. Previous finite element calculations (Mansikkamäki 2008) using the Soft Soil model in Plaxis program had given a factor of safety equal to 1.56 if the shape of the yield surface was set to give an appropriate K0 value according to the standard procedures. If, however, the yield surface was set to better model the shape of the surface near failure by defining M according to the friction angle, the factor of safety dropped down to FOS = 1.26. In Figure 6 the developed failure surface is shown with the aid of developed shear strains. The shape and location of the failure surfaces from lem and fem calculations coincide reasonable well. The largest differences can be found under the train load. There, in that area also the greatest difficulties in the lem calculations were found. The excess pore pressures in the lem calculations where at maximum 11 kPa in the shear zone decreasing towards zero towards the end of slip surface. 5.2

Case 2; Perniö test site

The Perniö test site is located near the city of Salo in Southern Finland. There, a full scale failure test has

Figure 7. Typical cross section from the 50 m long failure test site in Perniö. Table 2. Calculation parameters used in lem and fem calculations for the Perniö site. φ o

Embankment Fill Dry crust Clay 1 Clay 2

34 34 0 15 16

c kPa

γ kN/m3

0 0 40 0 0

21 19 17 23 26

been recently performed on an old railway embankment, see (Mansikkamäki & Länsivaara 2010) in the same proceedings. The subsoil in Perniö site consists of a sandy fill on top of a dry crust layer. Under the dry crust a soft cay layer can be found followed by silt and moraine layers. The soft clay has a water content ranging typically from 70 to 90%. The undrained shear strength is the soft clay varies in the range of 9 to 14 kPa. A typical cross section from the site is shown in Figure 7, and the calculation parameters in Table 2. In the lem calculations the ru parameter was set to ru = 0.205 for the upper clay layer and ru = 0.19 for the lower clay layer. In fem calculations using the soft soil model, parameter M was adjusted according to the friction angle and the relation between elastic and plastic volumetric stiffness that is between κ and λ was to give high pore pressure response.

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Figure 8. Critical slips surface from the lem-calculations for case 2, the Perniö site. All analyzed slip surfaces with FOS < 1.02 are presented.

function. For example the Janbu simplified method with the correction factor f0 resulted in this case in a slightly lower failure load and a more deep seated critical slip surface. In general it can though be concluded, that the lem and fem gave very similar results. Greatest differences are again under the load as a result of different load and pore pressure distributions under the load. The pore pressures in the lem analysis using the ru parameter is about 10 kPa in the shear zone decreasing to zero towards the ditch. The measured excess pore pressure just before they started to accelerate where from 10 to 20 kPa in the shear zone and below 10 in the ive zone. 6

Figure 9. Failure surface from the fem calculations by soft soil model for case 2, the Perniö site.

The lem calculations using ru to for failure induced pore pressure gave a failure load of approximately 80 kPa. The fem calculations gave a very similar result, with a failure load of 80.5 kPa. Actual failure in the test occurred at a load of 87 kPa. It most though be noted, that there are no exact answer to what is the failure load. It is highly related to the loading rate. Now the loading was ended when the containers used where filled up to maximum level with sand, in more detail see Mansikkamäki & Länsivaara (2010). After the maximum load was added the pore pressures continued steadily to increase. The pore pressure increase accelerated about 1.5 hours after end of loading resulting in the final failure half an hour later. Obviously the embankment would have failed also with a smaller load if enough time would have been waited. In neither lem nor fem calculations have 3D effects been ed, which also results in a smaller failure load in the predictions. If the excess pore pressure would be left unnoticed in the lem calculations, the calculations would give a failure load of approximately 120 kPa that is a significant over prediction. In Figure 8 calculated critical slip surfaces for lem calculations are shown for 80 kPa load. All analyzed failure surfaces with FOS < 1.02 are presented. In Figure 9 the failure surface from fem is presented. The result from the lem calculation is also depending on the method used. The presented results correspond to the GLE method using sin(x) as the force

CONCLUSIONS

A simple approach to include failure induced pore pressure in lem calculations has been introduced. The goal has been to develop a tool for the practice to include this important effect in undrained effective analysis using straightforward lem calculations. The results have been compared to fem calculations using a cap-type hardening plasticity model, and the results of a full scale failure load test. The main idea in the method is to use an effective stress pore pressure parameter ru , which value is basically determined in relation to the clays friction angle and yield surface. As an approximation of the yield surface is used, only the friction angle is needed to determine the parameter. It can well be argued, that the method is strictly valid only in active shear. However, due to the stress-time history typically found in the subsoil beneath rail way track in southern Finland, the error in the generality of the assumptions is generally not much. Although the results are at this stage only preliminary, the method shows very good potential. The calculations are very much in line with the results from fem analysis. Also results from the full scale failure tests the new method. The calculated failure load equal to 80 kPa was surprisingly close to the actual value of 87 kPa. There are though many things left unnoticed. Also, the choice of which limite equilibrium method is used influences much on the failure load. For example, using Janbu simplified instead of GLE method drops the failure load to 70 kPa in the lem calculations. If, however, the failure induced pore pressure would not be ed, the lem calculation would significantly over predict the failure load. It can thus be concluded, that the new method show good potential. In the future more work will be but in analysis of the failure tests and in the development of both lem and fem methods. REFERENCES Plaxis 2D. Material Models Manual, version 9.0. Janbu, N., & Senneset, K. 1995. Soil parameters determined by triaxial testing. Proceedings of the 11th European

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Conference on Soil Mechanics and Foundation Engineering, Copenhagen. Vol. 3 pp. 101–106. Länsivaara, T. 1995. A critical state model for anisotropic soft soils. Proceedings of the 11th European Conference on Soil Mechanics and Foundation Engineering, Copenhagen. Vol. 6 pp. 101–106. Länsivaara,T. 1999.A study of the mechanical behavior of soft clay. Doctoral thesis, Norwegian University of Science and Technology. Mansikkamäki, J. 2008. Stability analysis of existing railway embankments based on finite element method. Master‘s Thesis (in Finnish). Tampere University of Technology. Mansikkamäki, J. & Länsivaara, T. 2009. Effective stress analysis of old railway embankments. 17th

International Conference on Soil Mechanics & Geotechnical Engineering. Mansikkamäki, J. & Länsivaara, T. 2010. Analysis of a full scale failure test on old railway embankment. 7th European Conference on Numerical Methods in Geotechnical Engineering (to be published). Cheng, Y. M., Länsivaara, T. & Wei, W. B. 2007 Twodimensional slope stability analysis by limit equilibrium and strength reduction methods. Computers and Geotechnics, vol 34 pp. 137–150. Svanö, G. 1981. Undrained effective stress analyses. Doctoral thesis. The Norwegian Institute of technology.

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Investigation of soil property sensitivity in progressive failure A.S. Gylland & M.S. Sayd Norwegian University of Science and Technology, Trondheim, Norway

H.P. Jostad Norwegian Geotechnical Institute, Oslo, Norway Norwegian University of Science and Technology, Trondheim, Norway

S. Bernander Luleå Technical University, Luleå, Sweden

ABSTRACT: Progressive failure in long natural slopes with strain softening clay is studied. A FEM approach which s for the non-linear stress-strain curve of the material, including the post peak softening behavior, is used. Sensitivity of the results to variations of key parameters like in-situ shear stress at failure plane, brittleness, stiffness of the soil mass, and geometry are investigated in of critical load for initiating the slide and critical length. The results show that the capacity of the slope is decreasing with increasing initial shear stress level, increasing brittleness and decreasing stiffness of the soil mass. Further, by studying variations in the critical length it is indicated that the validity of classical limit equilibrium methods is limited for steep slopes in soft and very sensitive clay.

1 1.1

INTRODUCTION Background

This paper deals with progressive failure as a result of strain softening material behavior. Strain softening is in this context defined as loss of strength when a material is sheared beyond a peak value. For soft sensitive clays this behavior is not only a rheological phenomenon. Such materials are relatively loosely packed and when sheared the behavior will tend to be contractant. If the shearing occurs under undrained conditions there is no possibility of volume change and the contractant behavior induces excess pore pressures. Following the principle of effective stresses this implies that the forces between the soil grains is reduced and the shear strength of the soil is decreasing toward a residual value (see Figure 2). When considering capacity analysis, strain softening leads to situations where the assumptions behind limit equilibrium (LE) are not longer fulfilled. In contrast to materials behaving perfectly plastic, the shear stress along a potential failure surface may adopt any value between the initial shear stress, peak shear strength and the residual shear strength during the failure process. This feature gives a situation where the failure zone, or strain softening zone, under certain conditions spreads progressively in the soil. Here, the maximum global capacity in most cases is obtained before the global failure mechanism used by limit equilibrium methods has formed. One of the characteristic

features of progressive failures it that a relative small and local disturbance in highly sensitive clay generates an extensive landslide, often spreading into areas with modest inclination. This process will be discussed in the following section. Among the first to include thoughts of the progressive failure process in slope stability analysis was Terzaghi (1936). Following that, this failure mechanism has been widely discussed and modeled for several materials showing strain softening behavior, e.g. (Bishop 1967, Bjerrum 1967, Christian & Whitman 1969, Palmer & Rice 1973, Bernander 1978). With the development of the finite element method, several authors have analyzed progressive failure of slopes by this approach. Some examples are (Wiberg et al. 1990, Potts et al. 1996, Andresen et al. 2002, Andresen & Jostad 2004, Andresen & Jostad 2007, Chai & Carter 2009, Gylland & Jostad, 2010). One simple way of incorporating the effect of strain softening behavior in limit equilibrium analyses is to reduce the peak shear strength measured in the laboratory or to simply use the residual value as proposed by Lefebvre & Rochelle (1974). Another alternative is to apply some correction factor (Dascal et al. 1972). 1.2 Progressive failure 1.2.1 Mechanism An analog to the progressive failure mechanism can be the process of falling dominoes. The first failure is

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Figure 2. Shear stress/deformation relationship.

1.3 Sensitivity study Figure 1. Progressive failure mechanism.

triggered locally and once started; it develops progressively from one to the next brick. When transferred to a slope with a strain softening material the individual falling brick may be compared with the local instability of a soil element when it is loaded into the post peak softening regime. This can be due to an applied load, displacement or increased pore pressure. The material point next to this one will then have to sustain the contribution due to softening induced loss of strength, in addition to the applied load. In turn this point will also be loaded into the softening regime and additional loads are transferred to the next soil element. In this way a failure zone develops progressively through the soil. This mechanism is illustrated in Figure 1, where a slope is loaded at the crest. N is the horizontal force due to the distributed load, q. Along the potential failure surface three points are studied in of stress and strain. At the considered stage of loading point a has reached the residual strength, point b is at the peak level and point c is in the hardening regime. This is also illustrated in the graph showing shear stress distribution along the failure surface. The plus-sign indicates increased resistance and the minus-sign indicates loss of resistance compared to the initial state. For a progressive failure to develop the material has to be strain softening and have strains high enough so that the peak is ed. In addition there is the condition used by Bernander (2000) and also emphasized by Andresen & Jostad (2004) that for a progressive failure to propagate in an infinitely long slope, the residual shear strength must be lower than the initial shear stress. This condition is seen in Figure 1 as the shaded minus-area. As mentioned in the introduction, the mechanism of progressive failure cannot be modeled well within limit equilibrium methods. This is for instance seen in Figure 1 where the assumption of constant mobilization along a potential shear surface is not fulfilled.

This study aims at investigating the relative importance of some of the parameters on the capacity of an infinitely long slope of sensitive clay. Undrained loading conditions are assumed. The parameters studied are the average Young’s modulus (E) of the overlying soil layer, the ratio between peak and residual strength and the value of the initial shear stresses. The effect of slope geometry and softening modulus is also commented.

2

METHOD

The finite element software BIFURC developed at the Norwegian Geotechnical Institute is used for the calculations presented herein (Jostad & Andresen 2002). A slope is modeled as a top layer of 3-noded truss elements with thickness H on a weak layer consisting of 6-noded interface elements. The top layer is modeled as linearly elastic and a strain softening material law is applied to the interface elements. The model in this study is fitted to the method of Bernander where a constitutive shear stress/deformation relationship as shown in Figure 2 is used. The method of Bernander is motivated by and designed for long slab-slides in Swedish sensitive clays caused by up slope triggering agents. It was developed during the eighties based on the model presented in (Bernander 1978, Bernander et al. 1988, Bernander et al. 1989). The model is summarized in Bernander (2000) and readers are referred to these references for further details. The main difference between the FEM calculations performed herein and the method of Bernander is the solution algorithm. Bernander’s method is a finite difference method (FDM) approach where soil deformations and a full stress/strain curve is incorporated. It is formulated so that different states are compared in an iterative process until the one meeting the applied critical load at its current location is found. In the FEM analysis the load is applied at a given location using a Newton-Raphson scheme and a robust arc-length based solution algorithm (Jostad & Nordal 1995).

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3

RESULTS

The main result of the analyses is the critical load (Ncr ) and the critical length (Lcr ). The former being the maximum value of the force N in Figure 1 which represents the state where any further attempts on increasing the load results in a progressively selfdeveloping failure. Lcr is defined as the length of positive resistance along the potential shear surface (see Figure 1) when Ncr is reached. If using limit equilibrium methods to analyze slope stability in strain softening materials, Lcr indicates the maximum size of the slip surface to be used if any reasonable compatibility between occurring mechanisms and the limit equilibrium assumptions is to be obtained (Bernander 2000). 3.1

Figure 3. Stiffness and initial stress sensitivity for Cr/Cu = 0.25.

Procedure

The modeling and choice of parameters are based on the example slope inAppendixA of (Bernander, 2000). Four values of the undrained Young’s modulus of the overlying soil mass (Eu ), initial shear stress (τ0 ) on the potential failure surface and the Cr/Cu ratio are chosen for the study. The parameters are normalized according to the values and results from appendix A in Bernander (2000). For simple estimates the initial shear stress can for instance be found from the shear surface inclination and the material weight, τ0 = W ·sinα, see Figure 1. The following parameters are defined for presenting the results. A Cu-value of 30 kPa is used. The value of δcr (see Figure 2) is kept constant at 0.30 m, meaning that the softening modulus is changed according to the variation in Cr/Cu.

3.2 Sensitivity plots The results in of n and l constitute a fifth order tensor. In order to simplify the interpretation, the results are presented parameter by parameter where the first presentation gives the sensitivity with one parameter fixed while the second gives the sensitivity of this parameter. Some results are then omitted, but the general trends are captured. The most sensitive combinations can be interpreted as those showing the steepest inclination in the graphs. The sensitivities of the parameters are compared in section 3.4.2. 3.2.1 Initial shear stress, t Sensitivity plots in of n (solid line) and l (dashed line) for variations of the stiffness and the initial shear stress for Cr/Cu = 0.25 is shown in Figure 3. Figure 4 complements Figure 3 by showing how the initial stress sensitivity varies for different Cr/Cu ratios for e = 1. This implies that it shows how the esensitivity graph is shifted with changes in theCr/Curatio. The graph is only for one value of e, but the trend is similar for other values of the stiffness.

Figure 4. Cr/Cu and initial shear stress sensitivity for e = 1.

Cr/Cu is varying some because it was not possible to run all calculations with the same parameters as the softening and balance of parameters did not through the numerical algorithm. It is seen from both graphs that the capacity and the critical length are reduced when the initial stresses are increased. It is further seen that the capacity is more sensitive for changes in the initial stress level if the stiffness is high. The initial stress sensitivity is rather moderately affected by changes in the Cr/Curatio. Compared to n, the critical length is relatively insensitive to changes in the initial stress level. 3.2.2 Stiffness, e Initial stress and stiffness sensitivity for Cr/Cu = 0.25 are shown in Figure 5. The results are complemented by Figure 6 showing the stiffness-sensitivity for varying Cr/Cu-ratio and fixed t. These results show a clear trend of decreasing capacity and critical length for decreasing stiffness. This is also seen in Figure 3. It is further seen that the stiffness sensitivity increases for lower values of the initial shear stress and higher values of the Cr/Cu-ratio. 3.2.3 Cu/Cr-ratio Sensitivities of n and l to variations in stiffness and Cr/Cu-ratio for t = 1 is shown in Figure 7. Figure 8 shows the Cr/Cu-sensitivity when varying initial stresses but with a fixed stiffness.

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Figure 5. Initial stress and stiffness sensitivity for Cr/Cu = 0.25.

Figure 8. Initial shear stress sensitivity for e = 1.

Figure 9. Parameter comparison, critical load.

Figure 6. Cr/Cu and stiffness sensitivity for t = 1.

Figure 10. Parameter comparison, critical length. Figure 7. Stiffness and Cr/Cu sensitivity for t = 1.

Both graphs show reduced capacity and critical length for decreasing values of the Cr/Cu-ratio. Higher stiffness and lower initial stress increase the Cr/Curatio sensitivity. The critical length is reduced with decreasing stiffness and almost unaffected by changes in the initial stresses as found in sections 3.2.1 and 3.2.2. 3.2.4 Parameter comparison Sensitivities of the different parameters are presented in Figures 9 and 10. The base parameters e = 1, t = 1, Cr/Cu = 0.5 are used. Cr/Cu is normalized with 0.5 to make all graphs through the same point and

thereby ease the interpretation. In addition the parameter δcr is studied. It is defined in Figure 2 and for a constant Cr/Cu-ratio this parameter defines the rate of softening or softening modulus. ing that the graphs with the steepest inclination represent the theoretically most sensitive parameters, one can see from Figure 9 that the most sensitive parameter, when considering the capacity, is the initial stress level. Then the Cr/Cu-ratio follows and is particularly decisive for high values. The stiffness and the rate of softening affect the result, but not as much as the other parameters. As far as the critical length is concerned, it is the stiffness together with the initial shear stress level that

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are the most sensitive parameters. However, the Cr/Cu ratio and the rate of softening are also important. It can be noted that increased rate of softening reduces both the capacity and the critical length.

4

DISCUSSION

4.1 Parametric analysis The parametric analyses show that the capacity, in of the crest loading, decreases for reducing values of the stiffness and falling values of the Cr/Cu ratio. It is also lowered for increased values of the initial shear stress level (or increased slope inclination). These effects become more pronounced for higher values of stiffness, initial stress level and Cr/Cu ratio. The critical length shows basically the same trends as the capacity. These results show the importance of assessing the initial stresses correctly. This is not unexpected as the margin between the initial stress level and the undrained shear strength is bound to have a decisive effect on the capacity. This can be seen from Figure 1. For the case of an infinite slope, as studied here, this value will be constant. However, for real slopes the initial stress level will vary and is best assessed using the finite element method. It is also interesting to see that the soil stiffness, which is omitted in the LE calculations, comes in as an important parameter and should be assessed correctly. It is also seen that the brittleness of the soil, both in of the Cr/Cu ratio and the rate of softening, affects the results. The more brittle, the lower the capacity and critical length. The above considerations show that the capacity of a slope, in the context of progressive failure, decreases the steeper the slope is and the softer and more brittle the material is. At the same time, the critical length is reduced for the same conditions. As mentioned, the critical length can be seen as a measure of the maximum LE-shear surface length that within reason can be applied in a strain softening material. This implies that for steep slopes in soft and brittle material, the validity of LE to evaluate the factor of safety for long shear surfaces is reduced. In such cases it is necessary to include the effect of strain softening in the analysis. An important point to note in the context of this study is that the presented results deal with theoretical variation only. In real cases, inherent variation of the different parameters will not be the same. E.g. the uncertainty in the stiffness parameter might be relatively high compared to the uncertainty of the initial stress level. This implies that the practical sensitivity of the initial stresses might be less than for the stiffness value because of the smaller variation range.

4.2 Effect of geometry The calculations presented herein are based on the special case of an infinite slope. However, for real slopes,

Figure 11. Slope A and B. Table 1.

Capacity and critical length for slope A and B.

Slope

Ncr kN/m

Lcr m

A B

690 188

298 117

the geometry will affect the capacity as exemplified in this section. Two slopes shown in Figure 11 are studied within the framework of the Bernander FDM method. Slope A is linearly decreasing from the top to the toe while slope B is curving according to the function h(x) = H (x/L)2 . The following parameters are used; L = 300 m, H = 21 m, D = 20 m, Cu = 30 kPa, Cr/Cu = 0.75 and Eu = 3600 kPa. Initial shear stresses are generated in a separate stage where plastic creep deformation of the clay is ed for. The results in of Ncr andLcr are presented in Table 1. It is seen that the capacity of the slope is 73% less for the curved slope compared to the linear while the critical length is 61% less. This example illustrates the importance of assessing the geometry correctly in progressive slope failure analysis.

4.3 Limitations and further work The analyses presented herein have several limitations and it is emphasized that the aim of the work is to illustrate trends in the parameter variations. One of the issues is the limited set of material parameters used. Further, the sensitivity analysis covers only the case of an infinite slope. An effect of geometry is exemplified in section 4.2. As shown by Lacasse et al. (1985), the softening branch of sensitive clay is non-linear. Considering the impact of varying the post peak parameters, this nonlinearity should be investigated further. It is shown by the analyses of Cr/Cu and δcr that the rate of softening, or softening modulus, may be of fundamental importance for the capacity of a slope. Within the strain based finite element method this value is set by the shear band thickness. This is a parameter yet to be determined precicely for sensitive clays and should be studied further.

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5

CONCLUSION

A parametric analysis of progressive failure in an infinite slope is performed. The sensitivity of changes in the Young’s modulus of the overlying soil mass, initial stress level and Cr/Cu ratio are studied with respect to the capacity of the slope under undrained loading and the critical length. The analyses show that the capacity is lowered for decreasing values of the Young’s modulus, Cr/Cu ratio and increased values of the initial shear stresses or slope inclination. The initial shear stress level is the most sensitive parameter followed by the Cr/Cu ratio. The critical length is reduced for the same conditions as the capacity and is most sensitive to variations in the initial shear stress level and the Young’s modulus. It is also illustrated that the geometrical shape of the failure surface is of importance. The results emphasize the initial shear stress level as an important parameter in slope stability analysis involving progressive failure. The results indicate further that limit equilibrium methods are less suitable for analysis of the most critical slopes – meaning slopes of high inclination in markedly strain-softening materials. ACKNOWLEDGEMENTS Professor Steinar Nordal and Professor Lars Grande at the Norwegian University of Science and Technology are greatly acknowledged for valuable discussions and comments. The work described in this paper is ed by the Norwegian Public Roads istration, the Norwegian National Rail istration, the Norwegian Water Resources and Energy Directorate and by the Research Council of Norway through the International Centre for Geohazards (ICG). Their is gratefully acknowledged. This is ICG contribution No 287. REFERENCES Andresen, L. & Jostad, H.P. 2007. Numerical modeling of failure mechanisms in sensitive soft clay – Application to offshore geohazards. 2007 OTC, Huston, Texas, U. S. A. Andresen, L. & Jostad, H.P. 2004. Analyses of progressive failure in long natural slopes. Proceedings of NUMOG04, IX, Ottawa, Canada. Andresen, L., Jostad, H.P. & Höeg, K. 2002. Numerical procedure for assessing the capacity of anisotropic and strain-softening clay. Proc. of the Fifth World Congr. on Comp. Mech.. Vienna, Austria.

Bernander, S. 1978. Brittle failures in normally consolidated soils. Vag- och Vattenbyggaren. Bernander, S. 2000. Progressive Landslides in Long Natural Slopes. Licentiate thesis, University of Luleå, Sweden. Bernander, S. Gustås, H. & Olofsson, J. 1988. Improved model for progressive failure analysis of slope stability. Nordic Geotechnical meeting, Oslo. Bernander, S. Gustås, H. & Olofsson, J. 1989. Improved model for progressive failure analysis of slope stability. Proc. of the twelft int. conf. on soil mech. and found. eng., Rio de Janeiro: 1539–1542. Bishop, A.W. 1967. Progressive failure- with special reference to the mechanism causing it. Proc. Of. Geo. Conf. Oslo 2: 142–150. Bjerrum, L. 1967. Progressive failure in slopes of overconsolidated plastic clay and clayey shales. Journal of soil mechanics and foundations, ASCE 93: SM5. Chai, J. & Carter, J.P. 2009. Simulation of the progressive failure of an embankment on soft soil. Computers and Geotechnics 36: 1024–1038. Christian, J.T. & Whitman, R.V. 1969. A one-dimensional model for progressive failure. Proc. 7th Int. Conf. Soil Mech. Mexico: 541–545. Dascal, O., Asce, M., Tournier, J.P., Tavenas, F., Asce, A.M. & La Rochelle, P: 1972. Failure of a test embankment on sensitive clay. Proc. Spec. Conf. performance Earth ed Struct. Am. Soc. Civ. Eng. 1: 1–54. Gylland, A.S. & Jostad, H.P. in press. On the effect of updated geometry in analyses of progressive failure. Submitted to NUMGE2010, Trondheim, Norway. Jostad, H.P. & Andresen, L.A. 2002. Capacity analysis of anisotropic and strain-softening clays. Proceedings of NUMOG VII .., Rome, Italy: 469–474. Jostad, H.P. & Nordal, S. 1995. Bifurcation analysis of frictional materials. Proc. Num. Mod. In Geomech. – NUMOG V :173–179. Lacasse, S. Berre, T. & Lefevbre, G. 1985. Block sampling of sensitive clays. Proc. of the eleventh int. conf. on soil mech. and found. eng., San Fransisco. 2: 887–892. Lefebvre, G. & Rochelle, P.L. 1974. The analysis of two slope failures in cemented champlain clays. Canadian Geotechnical Journal 11(1). Palmer, A.C. & Rice, J.R. 1973. The growth of slip surfaces in the progressive failure of over-consolidated clay. Proc. R. Soc. Lond. A. 332: 527–548. Potts, D.M., Dounias, G.T. & Vaughan, P.R. 1996. Finite element analysis of progressive failure of Carsington embankment. Géotechnique 40(1): 79–101. Terzaghi, K. 1936. Stability of slopes of natural clay. Proc. of Int. Conf. Soil Mech. and Found. Engr. 1. Wiberg, N.E. Koponen, M. & Runesson, K. 1990. Finite Element Analysis of Progressive Failure in Long Slopes. International journal of numerical and analytical methods in geomechanics. 14: 599–612.

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Short-term slope stability calculation according to Eurocode 7 V. Thakur The Norwegian Public Roads istration, Norway

S. Nordal The Norwegian University of Science and Technology, Norway

S. Hove The Norwegian Public Roads istration, Norway

ABSTRACT: It is the experience of the authors that Finite Element (FE) methods gives just as reliable results as Limit Equilibrium (LE) methods for slope stability analyses and that FE gives better insight in the slope stability problem at hand. Still, the application of finite element (FE) methods for slope stability analyses is not common in Norway compared to analyses with limit equilibrium (LE) based tools. This is mostly related to uncertainty in proper use of FE. In Norway, soft soils often require slope stability evaluations for undrained conditions. This paper addresses the problem that even though an undrained effective stress analysis would be preferable, there is a danger in using too simple effective stress based soil models in such an analysis. The undrained shear strength is easily overestimated. The paper describes how undrained stability calculations with the simple Mohr-Coulomb (MC) model are used in practice. FE results are compared to results from LE methods. Geotechnical design must comply with Eurocode 7 and Eurocode requirements relevant for slope stability evaluations are presented and discussed. 1

INTRODUCTION

Table 1.

Janbu 1973, Bjerrum & Kjærnsli 1957 were pioneers in developing and applying limit equilibrium (LE) methods in Norway for evaluating stability of natural slopes and manmade fills. In the later years finite element (FE) methods are used, but mainly in academia. The application of FE for practical design is still limited probably because of the need for more soil parameters in a FE simulation and uncertainty in the general use of FE. However, it is the experience of the authors that FE and LE in practice give very similar results if proper input parameters are given and if higher order elements and a sufficiently fine FE mesh are used. In undrained applications, unfortunate shortcomings of simple soil models seem to cause some confusion related to the risk of overestimating the undrained strength in an effective stress analysis. This relates to the fact that normally consolidated clays show significant contractancy upon undrained shearing and significant strength anisotropy. The simple soil models mostly available in commercial FE codes do not offer soil models that can reproduce such behaviour. Still the FE methods can be and should be used for design, but if an effective stress approach is used to an undrained situation, care must be taken to specify input parameters that predict correct undrained strength. From 2010 Geotechnical design in Norway must follow Eurocode 7 NS-EN 1997−1: 2004+NA: 2008 (hereafter, referred as EN 1997). According to the

Material factor (EN 1997).

Soil parameters

Symbol

γM

Friction angle (tan ϕ ) Effective cohesion Undrained shear strength Unit weight

γϕ γc γcu γγ

1.25 1.25 1.4 1.0

national annex (NA) to EN 1997 in Norway, the unified material factor or partial factor for soil parameters (γM ), shown in Table 1, will replace the factor of safety. In practice this does not imply a significant change. For undrained conditions we define:

Here Cu refers to undrained cohesion and subscript d refers to design value. For effective stressbased simulations we define:

Here C and tan ϕ refers to effective cohesion and effective friction angle, respectively. With one value for γM for cohesion and friction, we preferably write the design Mohr Coulomb strength as:

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Figure 1. The slope geometry.

where tan ϕd is the average mobilized strength. In the Norwegian National Annex (NA) to EN 1997 it is stated that the material factors should be increased relative to tabulated values in Table 1 for sensitive soils like quick clays. Further, it is stated that for natural slopes with initial marginal stability, one may accept lower material factors if the building activity in the area lead to an increase in stability and at no point in time decreases the stability. Keeping this in view, this paper presents LE and FE results for a natural slope consisting of materials known for progressive failures. The results are discussed in the view of the requirements and footnotes in NA to EN 1997. Figure 2. Undrained strength in compression and extension.

2 A REAL SLOPE CASE A real slope geometry taken from a road project is shown in Figure 1. The slope consists of 10–14 m thick sensitive-soft-clay-layer also know as quick clays. A sensitive clay layer is located at depth between 6 to 14 m from the surface. The side slopes are approximately 1:2 to 1:3, and a river or water-canal is located beside the road. The ground water level is about 2.5 m below the ground surface. Short and long term stability of the slope has been evaluated using the LE and FE methods.

2.1

Material behaviour

Quick clays are highly sensitive in nature and known for strain softening behaviour i.e. decreasing shear resistance after peak with increasing strain. This may lead to progressive failure. The quick clays are anisotropic in nature. Figure 2 illustrates typical results from an anisotropic consolidated undrained triaxial test on a quick clay. Figure 3 shows results from anisotropically consolidated triaxial tests on samples from the site in Figure 1. During the undrainded compression test the clay display contractancy with a tendency for volume decrease with a peak shear strength CCu .

Figure 3. Undrained triaxial compression test results for the quick clay samples taken from the slope area in Figure 1.

When the clay is subjected to an undrained triaxial extension test we again observe contractancy and find a peak strength, CEu equal to about 40 % of CCu , Figure 2. During undrained direct simple shear tests the quick clays may exhibit peak strength, CD u , which is around 70% of CCu . These values reflect the importance of taking anisotropy into for undrained strength of normally consolidated soft clays.

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Table 2.

Calculations methods applied.

Table 3.

Input parameters.

Description

Parameters

Average undrained shear strength (Cau ) based calculation using FE Anisotropic ADP calculation using LE Effective stress parameters (at failure, and at peak) based undrained analysis using the FE Undrained calculation based on the effective stress parameters obtained from Equation 8, using FE Drained analyses using LE and FE

Cau

Input parameters

E CCu , CD u , Cu C’, ϕ’

Upper soft clay Sensitive clay * Lower clay Upper soft clay (using Equation 7) Sensitive clay (using Equation 7) Lower clay (using Equation 7)

C’, ϕ’ C’, ϕ’

Unit CD u weight C’ ϕ’ Profile (kN/m3 ) (kPa) (degree) (kPa) 19 19 20 19

5 10 10 20

30 26p /30f 30 22

15 + 2.z 25 + 1.5.z 45 + 2.z ----

19

10

13

----

20

0

18

----

*p: ø’ at peak, f: ø’ at failure.

2.2 Calculation methods The stability of the slope shown in Figure 1 has been calculated using the FE tool Plaxis 2D V9.0 and LE tool Geosuite-Stability V10. The FE calculations are performed with 15-noded triangular elements using a simple MC elastic perfectly plastic soil model with zero dilatancy. Short-term stability has been investigated using both a total stress and an undrained effective stress based method. Different stability calculations, as listed in Table 2, have been performed and checked if the obtained material factors comply with the requirements according of EN 1997. At present Plaxis 2D V 9.0 does not have any commercially available anisotropic soil model, therefore an average shear strength based calculation is used for short-term stability analyses. The average undrained shear strength (Cau ) as referred to in Table 2 is defined by Equation 4.

C E C For CD u = 0.7Cu and Cu = 0.4Cu the average strength becomes equal to the direct simple shear D strength, CD u , which explains why Cu often can be used as an average strength , Cau .

Modified C’ and ϕ’ values are used in the effective stress based undrained analyses to fit the correct undrained strength profile with depth, as explained in Section 2.5.

stress based calculations (with friction angle zero) are presented in Figure 4 and Table 4. The results from the effective stress parameters based undrained analyses are presented and discussed in Section 2.5. The results show that that the slope is critical against short-term stability, whereas the long-term stability is rather good. The material factors in the range of 0.6 is certainly unrealistic, considering that the slope actually is a real and apparently stable slope. However, we do observe that the results obtained from the LE and FE calculations are in good agreement (0.62 and 0.67). Part of the reason for the low material factor is the lack of a soil model for anisotropic undrained strength in the FE simulations. However, the main reason for the low material factor for undrained conditions lies in the conservative estimate of the design strength. According to Eorocode the design strength is not the most probable strength, but a carefully estimated average, with only 5% probability of having less strength than the design value. However, it is evident that the slope in reality has a marginal stability wrt building activity. We also note that the FE material factor lies slightly below the LE result. Similar trend has been observed for several other slope geometries from the area. Average variation between the LE and FE results is 5% and FE gives slightly lower γM than LE. This is believed to relate to search routines for the most critical sliding surface in LE programs and assumptions in LE regarding interslice forces. 2.5 Effective stress parameters based undrained calculation in plaxis

2.3 Input parameters Soil input parameters are presented in Table 3. These parameters are suggested based on field and laboratory tests, as the one shown in Figure 3. Increasing CD u with depth, z, is expressed by MCu , (in kPa/m).

2.4 LE versus FE results The LE and FE results obtained from the long-term (drained) and the short-term stability using a total

All finite element calculations produce results that are largely dependent on the selection of soil models and input parameters. Therefore, proper understanding of constitutive models as well as the input parameters, is important for a safe design. The MC model in Plaxis gives a possibility to perform both drained and undrained effective stress calculations. However, the input parameters C’ and ϕ’ for the undrained effective stress simulations must be scaled down in order to produce useful results. The adjustments are needed due to the simplicity of the MC model.

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Figure 4. Results: Cau analyses performed using the LE tool (upper), and the FE tool Plaxis (lower). Table 4.

Table 5. Input parameters selected for the back calculation of an undrained triaxial compression test in SOIL TEST in Plaxis.

Obtained material factors for the slope. γM

Calculation methods FE (Plaxis)* LE (Geosuite-Stability)*

Drained 1.82 1.90

Parameters

Name

HS Model

MC Model

Soil unit weight (kN/m3 ) Secant stiffness (kPa) Tangent odometer stiffness (kPa) Unloading/reloading stiffness (kPa) Power for stress dependent stiffness Reference stress (kPa) Poisson’s ratio Lateral stress coefficient Cohesion (kPa) Frictional angle Dilatancy angle

γ Eref 50 Eref oed

19 8000 8000

19 8000 –

Eref ur

25000

–

m

1

–

Pref νur KNC o C’ ϕ’ ψ

100 0.3 0.5 10 30 −1

– 0.3 0.5 10 30 and 28 −1

Undrained 0.62a 0.67a , 0.83b

* a = based on Cau ; b = ADP analyses

Figure 5. Back calculation of an undrained triaxial compression test using the HS and MC model in Plaxis.

Figure 5 shows in principle for how the adjustments are made, illustrated by simulations of an undrained triaxial test. For the purpose of the illustration, the laboratory test results are assumed to be equal to simulated results from the Hardening Soil (HS) model, here shown together with results from using the MC model. The input parameters for the simulation of Figure 5 are presented in Table 5.

Figure 5 and Table 6 shows that the effective stress parameters (C’ and φ’) can be selected differently according to if we aim to fit the ultimate Coulomb line or the undrained strength. It we use effective stress parameters to fit the Coulomb line we over predict the real undrained strength by 34%. Even a Mohr Coulomb failure line through the peak of the effective stress path will over predict the undrained strength, actually by 15%. It is thus important to select low effective stress strength parameters corresponding to Ma = 6singϕa ’/ (3-sinϕa ’) or Ccu for stability calculations using the MC model. For the HS simulations in Figure 5 a negative dilatancy angle were used to simulate contractancy, see Table 5. In general negative dilatancy angles should

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Table 6.

Effective stress parameters at different stages.

Parameters at

c’ (kPa)

ϕ’ (o)

M

γm = (τ/τo )

% error

Failure (A) Peak (B) Actual (C)

10 10 10

30 28 23

Mf = 1.2 Mp = 1.1 Ma = 0.9

1.54 1.32 1.15

34% 15% –

Table 7. Undrained stability calculation based on effective stress parameters obtained from the lab and Equation 7. Effective stress parameters based undrained stability calculation for the slope Parameters selected at failure (FE) Parameters selected at peak (FE) Parameters selected from actual CD u based on Equation 8 (FE)

γM 1.41 1.25 0.89

be avoided in a complete boundary value problem unless appropriate regularisation techniques have been implemented to handle the associated strain softening behaviour as described by Thakur (2007). The effective stress parameters corresponding to design CuD for the slope have been obtained from Equation 7. The equation is based on the linearly elastic perfectly plastic Mohr Coulomb criterion with zero dilatancy:

Here σz is vertical effective stress. Input parameters for the effective stress based undrained calculation in Plaxis are shown in Table 3 and a corresponding CD u profile is presented in Figure 1 and Figure 6. In Figure 6 an alternative undrained strength profile resulting from using unadjusted effective stress values for C’ and ϕ’ is also shown. Note the serious over prediction this gives of the undrained strength with depth. Results from the undrained analyses are presented in Table 7. Variation in γM is significant. The results obtained using the parameters at the failure and at the peak are misleading. Thus in Table 7 only γM = 0.89 is realistic and comparable to the ADP result presented in Table 4. The FE Cau analysis in Table 4 gave γM = 0.62 and is hence lower than the γM = 0.89 found in the undrained effective stress analysis. This is attributed to the use of an inaccurate reference depth in Plaxis for the total stress analysis. The problem of inaccurate reference depths is avoided in the undrained effective stress analysis, which is recommended if properly reduced C’and ϕ’ are applied. 3

STABILITY IMPROVEMENT MEASURES

Table 8 summarizes computed material factors for the slope shown in Figure 4 compared to requirements in

Figure 6. CD u profile used in the calculations. Table 8.

Calculation summary and comparison.

Calculation methods Cau based calculations FE LE ADP analysis (LE) Undrained effective stress parameters based on Equation 7 in (FE) Undrained effective stress parameters based (FE) Drained analyses LE FE

γM Increased obtained from γM as per γM the calculation EN 1997 (NPRA) γM ≥ 1.4

γM ≥ 1.6

0.62 0.67 0.83

γM ≥ 1.4

γM ≥ 1.6

0.89

γM ≥ 1.4

γM ≥ 1.6

1.25 (peak) 1.41 (failure)

γM ≥ 1.25 γM ≥ 1.6 γM ≥ 1.25 γM ≥ 1.6

1.82 1.90

EN 1997. Material factors from the undrained effective stress simulation must be compared with the undrained factor γcu which is 1.4. As mentioned earlier, the result shows a marginally stable slope with material factors less than 1 for undrained loading. Thus building in this slope is a serious difficulty. The tabulated requirement in the NA to EN. 1997, is by no means fulfilled. Actually The Norwegian Public Roads istration (NPRA) recommends even higher material factors than Eorocode with a minimum γM 1.6 to for progressive failure in quick calys, see reference Hb016, 2010. Obviously the stability of the slope must be improved prior to any construction activities. Such

525

Figure 7. Influence of terrain modification on the location of the critical slip surfaces.

work is currently under planning. Alternatives for stability improvement are studied using FE. 20% improvement has been obtained by excavating 3–5 m of the top soil from the left slope and by adding a 5 m high counter fill in the valley at the toe of the slope. (The design process for improving the stability of the slope is not finished as by February 2010.) The following observations were made from these FE calculations: a) The material factor γM after the improvement is 1.40 and 1.41 using LE and FE, respectively, for the left slope. b) The FE calculations revealed that now the right hand slope was critical with material factors of 1.15 and 1.11, which means that stability measures are needed also in this slope. This highlights one of the advantages of FE that no presumptions have to be made about the location of the slip surface, see Figure 7. 4

CONCLUSIONS

Long term and short-term stability calculations of a real slope show that the LE and FE results are in good agreement. The material factors may still often be 5% less from FE, probably due to shortcomings in the LE methods to identify the most critical slip surface and assumptions on interslice forces. This leads to conclusion that FE methods should be used more often to analyse actual slopes. However, one must be aware of pitfalls in selection of input parameters for an effective stress based undrained analysis if a simple MC model is used. Effective stress parameters for undrained analyses should be selected such that Equation 7 gives the design value of the undrained shear strength.

LE and FE analyses can equally well be applied under Eurocode 7. For FE a c-ϕ reduction procedure should preferably be used. An exception is when material factors less than one occur. A gravity multiplicator can then be used to evaluate the margin against failure, but a simulation with an upscaled strength is preferred to see the upscaling factor needed and hence determine a material factor consistent with the one from a c-ϕ reduction procedure. A FE tool like Plaxis can be extremely useful when improvement measures like counter fills are studied, since the method automatically locates the new critical surface. In summary, these results show the capability and usefulness of the FE for real slope stability evaluations. FE has so far been used to a very little extent, especially in onshore geotechnical practice in Norway. However, examples as above do encourage us in further application of FE in geotechnical practice. ACKNOWLEDGEMENT The author would like to thank the NPRA for providing resources for this study. REFERENCES Bjerrum, L. & Kjærnsli, B. 1957. Analysis of the stability of some Norwegian natural clay slopes. Geotechnique 6(4):1–6. EuroCode 7. 2008. Geotechnical design. NS-EN 1990:2002 +NA:2008. Geosute Stabilitet. 2010. Geosute Toolbox. Vianova Novapoint. Hb016. 2010. Geoteknikk I vegbygning. Statens Vegvesen. Handbooks of recommended practice. Janbu, N. 1973. Slope stability computations. The embankment Dam engineering, Casagrande Volume. Editors Hirschfeld and Poulos. John Wiley &Sons: 47–86. Nordal, S. Alén, C. Emdal, A. Jendeby, L. Lyche, E. Madshus, C. 2009. Skredet i Kattmarkvegen I Namson I 13.mars 2009. Report by the investigation committee appointed by the Norwegain trasnportation ministry, NTNU. Plaxis 2D V 9. 2010. PLAXIS BV, the Netherlands. SVV Report. 2010. Geoteknisk Vurderingsrapport til Fv 900 Klett-Heimdal. Statens Vegvesen Region midt. Thakur, V. 2007. Strain Localization in sensitive soft clays. PhD Thesis, NTNU.

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Embankments, shallow foundations, and settlements


3D settlement analysis using GIS and FEM: A case study in Sliedrecht area, the Netherlands N.B. Yenigül MTI Holland BV, Kinderdijk, The Netherlands

A.S. Elkadi TNO DIANA BV, Delft, The Netherlands

ABSTRACT: A comparative regional settlement analysis is presented in this paper by using 3D Geosciences Information System (GIS) and 3D finite element analysis (midas GTS) approaches for the Sliedrecht area, Netherlands. For settlement analysis in GIS, Koppejan Formula is used. The compressibility coefficient involved in the formula has been determined based on the cone resistance values. Mohr-Coulomb constitutive model was adopted in nonlinear analysis for the needs of 3D finite element analysis. The results obtained from the GIS analysis are on average comparable to those of 3D finite element analysis.

1

INTRODUCTION

2.1 Geology

The application of 3D-GIS in geotechnical site characterization and engineering geology are scarce. There are some 3D modeling examples in dredging industry for offshore civil engineering projects (Orlic, 1997), for an underground powerhouse excavation (Houlding, 1994), and for aggregate resources assessment and foundation purposes (Orlic, 1997). But the implementation of 3D-GIS modeling for regional settlement analysis is not fully treated as found in the literature. Ground settlement is defined as vertical downward displacement due to a surcharge or other activities applied by engineering constructions. An accurate modeling of subsurface conditions and prediction of ground-structure interaction may help to prevent undesirable damages and related high repair cost. Hence, a comparative settlement analysis by using 3D Geosciences Information System (GIS) and 3D finite element analysis (midasGTS) approaches for the Sliedrecht area, Netherlands, is presented in this paper.

2

DESCRIPTION OF AREA

Sliedrecht, a small city with an area of 5 km2 situated about 9 km east of the city of Dordrecht, the Netherlands. It is built on the natural levee and floodplain of the river Beneden Merwede. Present ground elevation ranges from NAP (Normal Amsterdam Peil, i.e., Normal Amsterdam ordnance datum, which is comparable to Mean Sea Level, used as the reference level in the Netherlands) +4.2 m to NAP −3.6 m. The original ground surface was raised by placing fill material.

Upper Pleistocene Kreftenheye Formation and the Holocene Westland Formation are the two major geological formations encountered in the area. Complex subsurface structure is characterized by composition of clay, peat and sand package up to a depth of −25 m and the fill unit overlays them. The fill has been used in order to raise the ground level of the area for soil improvement. The water level is around NAP −2 m in the study area. 2.2 Geotechnical units The geological units have been categorized into four geotechnical groups (Figure 1) according to lithology and engineering properties characteristic for different lithology not related to geological age (Orlic, 1997). The four geotechnical units can be described as follow (Welidenya 1996 and Orlic, 1997). 1. Fill unit is man-made and overlays the other units all over the area (Fig. 1). It is composed of sand, some clay, peat and other debris. The elevation of the ground surface, as well the top of the fill, varies between +4.2 m and −4.2 m NAP. Its thickness of the fill unit varies from 0.1 to 7.6 m with an average thickness of approximately 1.5 m. It has a low compressibility (compressibility coefficient C = 200). The unit is partially saturated and has a unit weight of 20 kN/m3 . 2. Clay unit is present all over the study area (Figure 1). It has a unit weight of 16 kN/m3 and a C of 12. The clay has a cone resistance, qc between 0.1–2.1 MPa. The thickness of the unit ranges

529

Figure 1. Perspective view of the 3D geotechnical model looking from north.

between 0.1 and 11.6 m. It is fully saturated, medium to highly and deform plastically. 3. The peat unit occurs in most parts of the study area except in the southern part (Figure 1). Its thickness varies between 1.7 and 8.5 m. This unit presents the worst consolidation behavior among the other units in the area. It is rich in organic matter, very light, completely saturated with water, extremely compressible and plastic. Peat has a unit weight of 11 kN/m3 and a C of 6. The qc value varies between 0.05 and 1.6 MPa. 4. Sand unit has a weight of 20 kN/m3 and C of 200. In the southwest part of the area thickness of sand unit reaches up to approximately 15 m (Figure 1). The unit occurs at a depth of −9.8 to −14.3 m. The qc value of sand is greater than 4 MPa. 3

3D GIS MODELING

The digital database was created based on the raw data provided by the Royal Geological Survey of the Netherlands (Chowdhury, 1994, Somboon-Anek, 1995, and Orlic, 1997). These raw data have included the location map of boreholes and Ts, a settlement map, a geological map, geological cross-sections in east west and in the north-south directions, and T logs. The digital data base was updated, by entering the cone resistance data for the geotechnical units based on the interpretation of T logs. In the previous studies, the settlement values were calculated first at existing or dummy boreholes intersecting clay or peat and then the values were extrapolated to produce the settlement prediction maps. In this study settlement calculations are performed at the center of each 3D grid cell. 3.1

3D Grid model construction for the sliedrecht area

The 3D-grid model of area is created with an azimuth of 180◦ and zero inclination. The grid cell dimensions are 10 m × 10 m in horizontal (i.e. N-S and E-W) and 1 m in vertical direction (depth). The study area covers approximately 1420 m × 1310 m surface area and depth is limited to −20 m NAP. These dimensions, when intersected with the above defined grid, result in 372,040 cells.

Figure 2. Experimental semi-variogram and fitted model for qc values for top) clay and bottom) peat units.

Thereafter the 3D-grid model is intersected with the existing 3D volume geological model in order to determine volumes of geotechnical units (fill, clay, peat, and sand) enclosed in every grid cell separately and to interpolate qc values. This, in turn, will allow determining the C values of peat and clay units per grid cell. 3.2 Geostatistical modeling Statistical analysis in the form of histogram and frequency plots is performed to check the distribution of the qc values of peat and clay layers at sampled T locations and to select the model for geostatistical analysis. The analysis result indicated that the qc data shows a log-normal distribution both for clay and peat units. For qc in clay unit a mean value of 0.42 MPa and a standard deviation of 0.29 and in peat unit a mean value of 0.29 MPa and a standard deviation of 0.23 are determined. Kriging was chosen as a geostatistical model due to: 1) log normal distribution of qc data 2) its common use in geosciences, and 3) to be known as the Best Linear Unbiased Estimator. Thereafter a trend and a semi-variogram analysis (Figure 2) are performed to

530

(kPa), and p is the increase in vertical effective stress (kPa). The coefficient of compressibility beyond pre-consolidation pressure, C is estimated by:

where and Cs are the Koppejan’s primary and secondary coefficients of compressibility beyond the pre-consolidation pressure, respectively, and t is time interval (day) and td is time interval of one day. Since in practice settlement is often regarded as completed after 104 days (CUR, 1996) the equation 2 becomes

Figure 3. 3D perspective for estimated cone resistance values for top) clay and bottom) peat.

investigate the stationary and spatial variability and to ensure the applicability of Kriging as a geostatistical model. Then a cross validation is performed to investigate the accuracy of estimation compared to the original values. For both clay and peat units the Z-statistics show normal distributions with the averages close to zero and the standard deviations close to one. Figure 3 presents the three dimensional perspective for estimated cone resistance values for both clay and peat units. 4

3D SETTLEMENT ANALYSIS IN GIS

4.1 Settlement analysis procedure Settlement analysis is performed using modified Terzaghi or so-called Koppejan Formula (CUR, 1996) for long-term settlements. The C involved in the formula has been determined based on theqc values. Thereafter the settlement results are presented as settlement prediction map and as 3D iso-volume to reflect the real picture of settlement predictions. Modified Terzaghi’s or Koppejan equation: is given by

Where, S is settlement (m), H is the thickness of the compressible layer (m), C is the coefficient of compressibility, σv is the initial vertical effective stress at the mid-height of the compressible layer

The 3D settlement calculations were performed by dividing the area (by creating 3D Grid block model) into soil columns, each consists of blocks (cells) of 10 × 10 m in N-S and E-W directions and 1m depth. The settlement calculation is performed at the center of each cell separately in cumulative sequence from bottom to top. The calculations considered the volumetric contribution of the different soil types encountered within each cell by applying a volume weighted average algorithm. The top of the fill unit is considered as ground level and the total settlement under a uniformly distributed load of 50 kPa is calculated. This geometrical condition is considered as present situation in Sliedrecht area and the settlement calculation results may serve as a primary reference for future construction activities in the area. a constant phreatic groundwater level is considered at NAP-2. Representative soil parameters given in the Dutch standards (NEN 6740) are used to determine the C values in equation 1 in this study. The required parameters, which will be used in the calculations, are chosen according to description and consistency of the soil type (Table 1). The values are presented together with other representative values of unit weight and qc values, which can be used (if available) for confirmation of the selected soil type due to the fact that consistency of the soil type is not described unless boreholes and laboratory testing are conducted. The data available in this work are composed mainly of qc values and no laboratory test results were available. Therefore, the Cp and Cs values for peat and clay units are selected based on qc values as depicted inTable 1.Then by using equation (3) the C values for clay and peat units are calculated in each grid cell as the qc values that belong to these units are stored in the grid cells’ centroid. First, the total vertical stress (σv ), the hydrostatic pore pressure (u) and the vertical effective stress (σv ) at the center of each grid cell is calculated by using the following equations (Douwes Dekker, 1991): σv is given by

531

Table 1.

Relation between soil type and soil parameters according to NEN6740 (CUR, 1996).

Soil

ixture

Consistency

γs kN/m3

qc MPa

Cp —

Cs —

Clay

Clean

Soft Medium Stiff Soft Medium Stiff – Soft Medium

14 17 19–20 15 18 20–21 18–20 13 15–16

0.5 1 2 0.7 1.5 2.5 1 0.2 0.5

7 15 25–30 10 20 30–50 25–50 7.5 10–15

80 160 320–500 110 240 400–600 320–1680 30 40–60

Soft Medium

10–12 12–13

0.1–0.2 0.2

5–7.5 7.5–10

20–30 30–40

Slightly sandy

Very sandy Organic Peat

Not preloaded Moderately preloaded

* The table presents the low representative value for the soil properties of the concerning soil type. If the most unfavorable situation is created by application of the high representative value of the layer average, a value based on consistency/relative density of the soil type concerned must be chosen from the next row (i.e. from denser, respectively stiffer material) and in the case of dense, respectively stiffer material the value after “ – ” must be chosen (CUR, 1996).

where, zi is depth of layer in (m) [depth to the center of each grid cell] and γiI is the unit weight of the layer i (kN/m3 ) [average unit weights of soil units in each cell]. u (water pressure) is determined by,

where hw is water column (m) and γw is the unit weight of water (kN/m3 ). And finally σv is equal to σv − u. For calculating the vertical effective stress at each grid cell center, the volume weighted average of unit weight value that has been calculated for each cell, The volume weighted average unit weight per grid cell (kN/m3 ) is calculated based on the volume of each unit available in a grid cell as follows:

where, γs , γc , γp , and γf are the unit weight of sand, clay, peat and fill units respectively (kN/m3 ) and vs , vp , vc , and vf are volumes of sand, peat, clay and fill units available within a grid cell (m3 ) respectively. After calculating the σv , the settlement is calculated at each grid cell center by using equation 1. The soil thickness in the calculations is taken 1m, which correspond to the height of the grid cell. The volume weighted average of compressibility coefficient values are used in the calculations.The volume weighted average compressibility coefficient per grid cell Cwv is determined based on the volume of each unit available in a grid cell as follow:

where, Cs , Cp , Cc , Cf are the compressibility coefficients of sand, peat, clay and fill units available within a grid cell, respectively and vs , vp , vc and vf are the same as defined above for equation 6. The calculations start at the top of each soil column and proceed downwards cell by cell, until the top of the sand layer. The total settlement, both at the top of the soil column and at any cell center, can be obtained by upward cumulative summation of settlement values at grid cell centers. Settlement values at every required elevation are available as well as the total settlement at the ground level since the settlement values were determined at each grid cell. This enables more accurate settlement prediction for shallow and/or deep foundation designs. Moreover it gives an insight to which depth the excavation of top soil layers is needed to find a foundation level with only small or threshold settlement values.

4.2 Results Figure 4 presents the settlement analysis results as a 2D settlement map and Figure 5 shows the result in 3D iso-volume. The settlement values ranges between 0.0–3.5 m. As it is mentioned earlier in Section 2.2 the qc values for clay unit ranges between 0.1–2.1 MPa and for peat unit ranges between 0.05–1.6 MPa. This rather wide range of qc values results in C values of between 3.75–23.08 for clay unit and C values of between 2.5– 5.0 for peat unit (Table 1). Hence, the variation and distribution of the qc readings within the grid cells results in different C values per grid cell and this leads to a variation and densely distributed settlement zones. In the areas where the thickness of compressible layers, peat and clay, are increa to 10 m, the settlement values also increase up to 2.5 m. Thickness of compressible layers decrease towards south

532

Figure 6. 3D Mesh and uniform load in midasGTS.

Table 2.

Material properties.

Figure 4. Settlement prediction map of Sliedrecht area. Parameter

Fill

Clay

Peat

Sand

γ (dry) [kN/m3 ] γ (wet) [kN/m3 ] Young’s modulus [kPa] Poison’s ratio [–] Cohesion [kPa] Friction angle [◦ ] Dilatancy angle

18 20 25000 0.3 0.5 30 0

15 16 500 0.35 10 22.5 0

10 11 750 0.35 5 15 0

18 20 25000 0.3 0.5 30 0

the middle part of the area and towards east it reaches up to 1m.

5 Figure 5. 3D perspective view of settlement in Sliedrecht area.

and southwest. Therefore towards north and northeast direction, the settlement values are higher. In the west part of the area the thickness of peat is approximately 7 m and clay in total is about 3 m, which result in high settlement values. The thickness of fill is about 4 m in the south, southwest part even at some spots it reaches up to 6 m. In northern part, the fill thickness is about 2 m and in the north-east part it decreases up to 0.3 m. The thickness of the fill contributes in reducing surcharge effect within the compressible layer. Moreover fill layer, which is of higher stiffness, absorb and redistributes the strains resulting from the underlying compressible layers. Therefore the lower the fill thickness the higher the settlement value expected and this is witnessed in the settlement zones in the middle, west and northeast parts of the area (Figure 4). As peat is more compressible compared to clay, when the thinner clay layer is followed by a thicker peat layer in other words where peat is closer to the surface, the settlement values are expected to be higher than the case where thicker clay layer is overlaying. This is the case seen in the middle and the western parts of the area, as the thickness of upper clay layer is about 0.7 m at the west, decreasing to about 0.3 m in

3D SETTLEMENT ANALYSIS IN FINITE ELEMENT USING MidasGTS

Midas GTS software v3.0.0, a general 2D/3D Geotechnical and Tunnel analysis System, was used for the nonlinear finite element analysis. Based on the available soil parameters and subsequent assessment of soil conditions, the Mohr-Coulomb constitutive model was adopted in nonlinear phased analysis. An advanced model for soft soil would have probably been better suited but the lack of soil investigation data dos not grant the use of such models. The main aim of the analysis was to predict the settlement of the soil layers under own weight and a uniform surcharge load of 50 kPa (Fig. 6). A common practice in finite element analysis for geotechnical problems is to run a staged analysis by initializing the stresses under own weight in the first stage while resetting their resulting displacements. The following stage(s) would include the analysis due to new loads, excavations, etc. In this study, the GIS modeling do not for stress initialization and calculated displacements are total displacements. Therefore, for the sake of comparison, two analyses were performed, one with stress initialization and one without. The prior would calculate settlements due to both self-weight and surcharge load, while the latter would calculate settlement results due the surcharge load only.

533

Figure 7. Settlement results due to self-weight and surcharge load in nonlinear analysis.

The soil properties used for the Mohr-Coulomb model during the analysis are summarized in Table 2 and are based on literature values (Van Meurs et al. 1999) following the qc values from T results. The literature values are used in the analysis because these were based on the laboratory test result values from the area between Amsterdam and Utrecht, where the soil units show similarity with those available in the study area. A quarter-symmetry model with 50 × 50 m side dimensions was considered with a total of 21208 tetrahydron elements. The soil profile used from ground surface consisted of 1m fill, 0.9 m clay, 5 m peat, 1.3 m clay, 1m peat, 0.8 m clay, and 11m sand, respectively. The surcharge load is uniformly applied on top of the fill layer. All analyses took place in drained conditions. The total settlement value from the analysis without stress initialization is 0.92 m (Fig. 7), whereas the settlement value from the analysis with stress initialization and load application in a following stage resulted in 0.52 m. This latter value indicates the net settlement due to the surcharge load. 6

CONCLUSIONS

The comparative 3D settlement analysis using 3D GIS and 3D finite element approaches have been presented

in this study. The results of 3D GIS analysis are presented as settlement prediction map and iso-volumes. The C values used in 3D GIS analysis are determined using qc values both for peat and clay units. The settlement varies between 0–3.1 m in the area. The detailed settlement zones observed is mainly due to the spatial variation in qc values. The average value of the settlement is around 1.5 m. The 3D finite element analysis resulted in a settlement value comparable to the average value obtained from the GIS analysis of about 1 m. The use of the 3D GIS approach coupled with geostatistical distribution of properties could be used for preliminary urban planning on regional scale. A detailed 3D finite element analysis could then be used for detailed analysis on project scale. The advance in computer hardware and availability of -friendly full 3D geotechnical software, e.g. midas GTS, makes such analysis accessible for the geotechnical design office. REFERENCES Douwes D.1991. Soil Mechanics. Lecture Notes of M.Sc. Course in Engineering Geology. ITC-Delft. The Netherlands: 286. Chowdhury. M. A. 1994. Application of Geotechnical Database and GIS For The Preparation of Engineering Geological Maps In a Queternary Geological Area. MSc Thesis. ITC-Delft. The Netherlands: 107. Houlding. S.W. 1994. 3D Geoscience Modelling: Computer Techniques For GeologicalCharacterization. SpringerVerlag Berlin: 309. Somboon-anek. P. 1995. 3D-GIS Modelling For Geotechnical Purposes: Sliedrecht. Netherlands. MSc. Thesis. ITC. Delft. The Netherlands: 44. Center for Civil Engineering Research and Codes (CUR). 1996. Building on Soft Soils. Balkema. Rotterdam. The Netherlands: 389. Welideniya. H.S. 1996. GIS Application in Foundation Cost Zonation For an Area in Sliedrecht The Netherlands. Msc. Thesis. ITC- Delft. The Netherlands. Orlic. B. 1997. Predicting Subsurface Conditions for Geotechnical. Modelling. ITC Publication. No.55. The Netherlands. Meurs van. A.N.G. Berg van den. A. and Ven.ans. A.A.M. 1999. Embankment widening with the Gap-method. In Barendsen et. al. [ed]. Geotechnical Engineering for Transportation Infrastructure. Rotterdam: Balkema.

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A comparison of 1D, 2D, and 3D settlement analyses of the Tower of Pisa A.J. Klettke & L. Edgers Department of Civil and Environmental Engineering, Tufts University, Medford, MA.

ABSTRACT: This paper compares and evaluates settlement analyses of the Tower of Pisa by 1D, 2D, and 3D computer models. The Tower of Pisa has been an important settlement case study for more than a century. The Tower settlement has been previously modeled using 1D consolidation and, more recently, 2D and 3D finite element analysis. The recent work provides a large amount of subsurface data used in these analyses. The 1D analyses underestimate the average settlement of the Tower by neglecting shear induced deformations. In 2D, a plane strain analyses can model the Tower’s tilt but the magnitude of the settlement must be calibrated for the difference between plane strain and axisymmetric loading. The results of the calibrated 2D and 3D analyses show excellent agreement and both agree reasonably well with the estimated actual settlements and tilts.

1

INTRODUCTION

2.2 Estimated settlement

The development of advanced numerical techniques, especially two-dimensional and three-dimensional finite element analysis, has greatly enhanced our ability to model complex geotechnical problems. These advanced techniques, however, offer a number of challenges including: evaluating the effects of numerical error and instability; the time for setup, verification, and execution; and, computer memory requirements. Traditional models, such as one-dimensional consolidation theory offer the advantage of simplicity, although they are severely limited in their ability to model complex situations. This paper describes a comparison of results obtained from one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) models. The Tower of Pisa was selected as the case study for this comparison because of its settlement history and the well characterized soil conditions. It is beyond the scope of this paper to model the behavior of the Tower or to evaluate the causes of its settlement and tilt history in detail.

Original elevations of the Tower foundation in 1173 are unknown. Thus an absolute measured settlement history is not available. There are, however, some estimated final settlements reported in the literature. Mitchell et al. (1979) estimate the final settlement of the Tower to be approximately 140 cm on the north side, 220 cm in the center, and 310 cm on the south side. Bai et al. (2008) report observed final settlement values of 201 cm on the north side and 390 cm on the south, although the basis of these values is not known. 2.3 Tilting history Burland and Potts (1995) provide a reconstructed history of the tilt of the Tower based on adjustments made to its masonry layers and their relative inclinations. Table 1 summarizes the tilting history of the Tower. During the first phase of construction, the Tower tilted Table 1. Estimated Tilting and Load History (Costanzo et al., 1994). Year

Activity

Wt (kN)

Tilt0

1173 1178 1272 1278 1285 1360 1370 1550 1758 1817 1911 1990

Start Construction Complete 3-1/2 floors Resume Construction Complete to 7th floor Resume Construction Complete Construction – – – – – –

0 94.8 94.8 137.28 137.28 137.28 144.53 144.53 144.53 144.53 144.53 144.53

0 0 −0.200 0.103 1.112 1.112 1.611 4.684 4.831 5.103 5.246 5.469

2 TOWER HISTORY 2.1 Construction Construction of the Tower of Pisa began in August 1173. In 1178, after three and a half stories had been built, work was interrupted possibly because of political or construction difficulties. After nearly a century, work on the tower resumed in 1272. By 1278, construction reached the seventh cornice when work again was stopped. Work on the bell chamber began in 1360 and the Tower was finally completed in 1370.

535

Figure 1. North-South profile (Rampello & Callisto, 1998).

slightly to the north. After construction resumed in 1272, the Tower began to tilt distinctly towards the south. By the time construction was completed in 1370, the estimated tilt was 1.611◦ increasing to 4.684◦ by 1550. Miscellaneous construction activities since then have caused incremental increases in the Tower’s tilt. For example, in 1838 the excavation of a walkway known as the catino increased the Tower’s tilt by about 0.5◦ (Burland and Potts, 1995). By 1990, the tilting of the Tower had increased to 5.469◦ . This tilt overstressed the concrete masonry of the structure and threatened the stability of the foundation. As a result, over the next decade the Tower was stabilized by well-documented measures including the installation of pre-stressed tendons around the lower levels of theTower, a temporary concrete ring around the base of the Tower with lead ingots placed on the ring in calculated phases, and soil extraction (Burland et al., 2003). The analyses described in this paper do not consider the period after the Tower was stabilized.

3 3.1

SUBSURFACE CONDITIONS Soil conditions

The subsurface conditions beneath the Tower have been extensively studied from early in the 20th century. As many as 16 committees in the past century have investigated methods to stabilize the Tower of Pisa (Lo Presti et al., 2003). Figure 1 provides a northsouth cross section created by Rampello and Callisto (1998) based upon a large amount of the available borehole and T data.

The upper A horizon consists of varying thicknesses of mixed sand, silt, and clayey soils. The B horizon, known as the Pancone clay, consists of an upper clay (B1 , B2 , B3 ) and intermediate clay (B4 , B5 ). These are underlain by intermediate sand (B6 ) and lower clays (B7 , B8 , B9 , B10 ). The depression in the top boundary of the upper Pancone clay (B1 , B2 , B3 ) shows that this layer has contributed greatly to the overall settlement behavior of the Tower. In addition, variations in the upper A horizon may have caused the Tower’s initial tilt. This initial tilt would have produced non-uniform stresses in the foundation soil, causing the Tower’s inclination to increase over the following centuries. 3.2 Soil properties Mitchell et al. (1977), in a 1D settlement analysis, reported some of the soil properties data of earlier committees and provided some one dimensional consolidation parameters. Rampello and Callisto (1998) and Lo Presti et al. (2003) summarize the vast amount of data on the soils below the Pisa Tower from the past century. They include soil compressibility, creep, strength and stress history data obtained from very high quality samples and modern testing methods. Table 2 presents the soil properties based on these data used in this study. 4

NUMERICAL MODELING – METHODS AND ASSUMPTIONS

4.1 Introduction The numerical analyses were based on the loading history of Table 1, the soil profile of Figure 1 and the soil

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Table 2.

Input Soil Properties.

Soil Layer Fill A1-Clayey Silt A2-Upper Sand B1,B2,B3 – Upper Clay B4,B5 – Int. Clay B6 – Int. Sand B7,B8,B9,B10 – Lower Clay C – Lower sand

Cc – 0.225 – 0.853 0.296 – 0.518 –

Cr – 0.026 – 0.141 0.042 – 0.081 –

Cα – 0.004 – 0.015 0.004 – 0.01 –

Avg. OCR – 3.23 – 1.63 2 – 1.25 –

c, kPa 1 6.8 1 9.8 1 1 3.1 1

0 34 34 34 24 29 34 26 34

E, kPa 22000 – 13000 – – 18000 – 100000

ν 0.3 – 0.3 – – 0.3 – 0.3

Cc and Cr = compression and recompression indices, Cα = secondary compression index, OCR = overconsolidation ratio, c = cohesion, = friction angle, E = modulus of elasticity, ν = Poisson’s ratio.

properties data of Table 2. Klettke (2009) describes the details of parameter selection. WinSaf-I and WinSafTR (Prototype Engineering, 2001) were used for the 1D computations. WinSaf-I uses 1D compression theory but distributes the applied stresses throughout the subsurface using elastic theory. WinSaf-TR computes time rates using conventional 1D consolidation theory. The 2D and 3D analyses were performed using Plaxis 2D (1998) and Plaxis 3D Foundation (2007). Plaxis 2D and Plaxis 3D Foundation use the finite element method (FEM) with automated mesh-generation capability, interface elements, and a number of available soil models. A series of initial studies were performed to validate the performance of these computer programs. These included Plaxis 2D element type and Plaxis 2D and 3D mesh refinement and mesh size studies. The FEM programs were also used to perform a 1D analysis for a comparison with WinSaf and in order to their accuracy. 4.2 1D analysis The 1D profile used for this analysis is a simplification of a north-south profile of Figure 1 with layers of uniform thickness. The Tower was assumed to apply a net uniform vertical load over a circular area, using ramp loadings to approximate the construction time-history of Table 1. The loads ed for the 3 m deep excavation within the fill layer. Shear induced displacements were neglected in this simplified analysis. Thus, the settlement computed in this 1D analysis is an average settlement representative of the Tower’s center. WinSaf-TR performed a time-rate analysis for the ultimate settlement computed by WinSaf-I. Independently calculated secondary compression was added to the settlements due to primary consolidation after the final load was applied.

Plaxis 2D analyses were performed using both the axisymmetric and plane strain models. The axisymmetric model can represent the Tower’s circular foundation but cannot model the non-uniform stresses beneath the tower and its tilt. The plane strain model can for the Tower’s tilt but computes too much settlement because of the differences between plane strain and axisymmetric loading conditions. The Plaxis 2D axisymmetric analyses used a 60 m wide mesh with significant mesh refinement within the A1 and B1, B2, B3 soil clusters. The mesh consisted of 2,622 15-noded elements and 21,193 nodes. The loads applied for these analyses were the same as the loads applied for the 1D analysis. Plane strain mesh computations (2286 elements, 18525 nodes) were first made with variations in the mesh to capture some of the detailed subsurface variations in soil properties and type beneath the Tower, particularly in horizon A (Figure 1). These initial computations produced very uniform settlements, suggesting that subsurface variations beneath the Tower, by themselves, are not a major cause of the Tower’s present day tilt. A comparison of the plane strain computations with the axisymmetric computations with the same soil properties and loading provided a measure of the effect of plane strain versus axisymmetric conditions on settlements. The axisymmetric computation produced a final settlement at the centerline of 260 cm versus 380 cm. for the plane strain loading, suggesting a scaling factor of 0.683. An initial 2D plane strain analysis using uniform stresses produced no measurable differential settlement. Therefore, a series of 2D plane strain computations were made to investigate the extent to which non-uniform stresses contribute to the Tower’s inclination. An eleven phase calculation was established in Plaxis 2D based on the loading increments of Table 3. These stresses include the effects added to the weight of the Tower due to overturning moments caused by the Tower’s tilt to date as summarized by Table 1.

4.3 2D analysis The Plaxis analyses used a Soft Soil Creep (SSC) model to model the compressible clay layers and a Mohr-Coulomb (MC) model for the sand and fill layers. The clay layers experienced undrained loading followed by consolidation. The sand and fill layers beneath the Tower were assumed to be drained.

4.4

3D analyses

A series of initial computations were made with symmetrical loadings in order to evaluate the effects of mesh fineness and mesh size on the results, and the accuracy of Plaxis 3D Foundation by comparison with

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Table 3. kPa).

Non-uniform Applied Stresses (All stresses in

Year

Tilt

σavg

σnorth

σsouth

1173 1178 1272 1278 1285 1360 1370 1550 1758 1817 1911 1990

0◦ 0◦ −0.200◦ 0.103◦ 1.112◦ 1.112◦ 1.611◦ 4.684◦ 4.831◦ 5.103◦ 5.246◦ 5.469◦

0.0 314.2 314.2 455.0 455.0 455.0 479.1 479.1 479.1 479.1 479.1 479.1

0.0 314.2 324.4 447.6 374.0 374.0 346.9 93.9 82.0 59.5 47.5 29.2

0.0 314.2 304.1 462.5 536.1 536.1 611.2 864.2 876.2 898.6 910.6 928.9

Figure 3. 1D, 2D, and 3D Center Settlement vs. Time.

Figure 2. Non-uniform line loads for 3D analysis.

the Plaxis 2D results. The soil stratification and constitutive models were the same as described above for the Plaxis 2D analyses. Mesh design is particularly important in 3D finite element analyses in order to produce accurate results without excessive computer memory or execution time requirements. The mesh used for the final computations was 120 m × 120 m × 43 m and consisted of almost 13,882 6-noded elements and 37,602 nodes. Non-uniform stresses were applied in 11 phases as in the 2D analyses to model the effects of overturning moments caused by the tilt of the Tower. These stresses were approximated by a series of line loads as illustrated by Figure 2.

5 5.1

NUMERICAL RESULTS Uniform loadings

Figure 3 presents a plot of center settlement of the Tower versus time for the 1D, 2D, and 3D computations assuming uniform loading as described above. All three curves show large settlement rates at the times corresponding to the two major loading periods, 1173– 1178 and 1272–1278. After the completion of Tower construction in 1370, the settlement rates decrease rapidly. This suggests that primary consolidation is

largely completed by this time and much of the Tower settlement over the final 600 years has been due to other factors including secondary compression and shear induced creep The ultimate settlement under the center of the Tower computed by the 1D, 2D, and 3D analyses are 152 cm, 260 cm, and 240 cm respectively.The 1D computation shows the least settlement because it neglects the effects of shear induced displacements in the foundation soils. An approximate calculation by Mitchell et al. (1977) using elastic theory and an estimated modulus of elasticity for the clays estimates that these shear induced displacements may for about 26 cm, 24 cm, and 2 cm respectively for each of the three loading phases (52 cm total) bringing the 1D computation into better alignment with the 2D and 3D computations. The final values of 260 cm for the 2D analysis and 240 cm from the 3D analysis agree well with the range of actual settlements estimated by Mitchell et al. (1979) and Bai et al. (2008). The difference between the Plaxis 2D and Plaxis 3D computations occurs because of differences in element type, 15-noded elements in Plaxis 2D versus 6-noded elements in Plaxis 3D, and mesh fineness, an average element size of 0.99 m in Plaxis 2D versus 3.18 m in Plaxis 3D. 5.2 Non-uniform loadings Figure 4 presents a plot of center settlement of the Tower versus time for the 2D and 3D computations assuming non-uniform loading as described above. The Plaxis 2D results were obtained from the plane strain model, reduced by the calibration factor of 0.683 determined from a comparison of the axisymmetric and plane strain results as described in section 4.3. The 2D (calibrated) and 3D curves agree well. The small differences are due to element type and mesh fineness issues as described in the preceding section 5.1. Both computations show center settlements that agree well with the estimated final values. However, the computed settlements are low on the south side and high on the north side of the Tower. This suggests that the computed tilts are less than the measured tilts of the Tower. This point is illustrated in Figure 5 which compares the computed 2D and 3D inclinations to the estimated final settlement values.

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capacity failure during construction. Bai et al. (2008) perform a three-dimensional creep analysis and conclude that creep effects could for 1.5◦ of the tower’s tilt. Other factors that may contribute to the difference between the estimated and computed settlements include: •

Uncertainty in the measured settlements and inclinations. • Numerical error due to the FEM discretization. • The approximation of the non-symmetrical stress distribution. • Soil property estimates in material such as the Upper Pancone clay that is very sensitive to sample disturbance.

Figure 4. 1D, 2D, and 3D Center Settlement vs. Time for Non-Uniform Loading.

7

Figure 5. 2D, 3D and Estimated Measured Tilt vs. Time.

6

DISCUSSION

The 1D analysis computes settlements that are small compared to the estimated final settlements and the settlements computed by the 2D and 3D analyses. This occurs because the 1D analysis neglects both initial and long-term shear induced displacements. The 2D and 3D analyses with uniform loading show much better agreement with the estimated center settlement (Figure 3) because they better for the shear induced movements. The calibrated 2D plane strain analyses show that the non-uniform thicknesses in the soil layers, especially the A1 layer, were found to have little effect on the long-term tilting of the Tower. The results of the 2D and 3D analyses with nonuniform loading, Figure 4, correctly show the trend of increasing tilt from north to south over the centuries. The overturning moment generated from the Tower weight and the center of gravity moving horizontally cause the Tower’s increasing tilt with time. However, the computed differential settlements and tilts are smaller than the estimated actual values, Figure 5. These larger tilts may have developed because the foundation soils were very close to experiencing a bearing capacity failure during construction, not fully ed for in these analyses. In fact, Burland et al. (2003) note that had work continued without interruption, it is likely that soils beneath the foundation of the Tower would have experienced an undrained bearing

CONCLUSIONS

The 1D analyses underestimate the average settlement of the Tower because they neglect the shear induced deformations. In 2D, a plane strain analysis can model the Tower’s tilt but the magnitude of the settlement must be calibrated for the difference between the infinite strip plane strain loading and the actual axisymmetric loading. The results of the calibrated 2D and the 3D analyses show excellent agreement and both agree reasonably well with the estimated final settlements and tilts. Initial mesh refinement and mesh size studies are important to validate and compare the performance of these computer programs. Mesh design is particularly important in 3D finite element analyses in order to produce accurate results without excessive computer memory or execution time requirements. Although the 2D and 3D analyses agreed reasonably well, the 3D analysis requires much longer execution times than 2D and produces more discretization error because of a coarser mesh. REFERENCES Bai, J., Morgenstern, N., and Chan, D. (2008). “ThreeDimensional Creep Analysis of the Leaning Tower of Pisa”, Japanese Geotechnical Society, Soils and Foundations, Vol. 48, No. 2, pp. 195–205. Burland, J.B., Jamiolkowski, M., and Viggiani, C. (2003). “The Stabilisation of the Leaning Tower of Pisa”, Japanese Geotechnical Society, Soils and Foundations Vol. 43, No. 5, pp. 63–80. Burland, J.B and Potts, D.M. (1995). “Development and Application of a Numerical Model for the Leaning Tower of Pisa”, Pre-Failure Deformation of Geomaterials, Shibuya, Mitachi, & Miura (eds). Balkema. pp. 715–738. Costanzo, D., Jamiolkowski, M., Lancellotta, R. and Pepe, B. (1994). “Leaning Tower of Pisa-Description of the Behaviour”, International Symposium on Settlement 94, Invited Lecture, Austin, Texas. Klettke, A.J. (2009). “A Comparison of 1D, 2D, and 3D Settlement Analyses of the Tower of Pisa”, Master of Science Thesis, Tufts University, Medford, MA, USA.

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Lo Presti, D.C.F., Jamiolkowski, M., and Pepe, M. (2003). “Geotechnical characterization of the subsoil of Pisa Tower”, Characterisation and Engineering Properties of Natural Soils, Swets & Zeitlinger, Lisse. pp. 909–946. Mitchell, J.K., Vivatrat, V., and Lambe, T.W. (1977). “Foundation Performance of Tower of Pisa”, ASCE, Journal of the Geotechnical Engineering Division, Vol. 103, No. GT3, pp. 227–249. Mitchell, J.K., Vivatrat, V. and Lambe, T.W. (1979). Closure of “Foundation Performance of Tower of Pisa”. ASCE, Journal of Geotechnical Engineering Division, Vol. 105, No. GT11, pp. 1363–1365.

PLAXIS 8.2. (1998). Finite element code for soil and rock analysis, version 8.2, R.B.J. Brinkgreve and P.A. Vermeer, eds., Rotterdam, The Netherlands. PLAXIS 3D Foundation Version 2 (2007). Finite element code for soil and rock analysis, version 2, R.B.J. Brinkgreve and W.M. Swolfs, eds., Rotterdam, The Netherlands. Rampello, S. and Callisto, L. (1998). “A Study on the subsoil of the Tower of Pisa based on results from standard and high-quality samples”, Canadian Geotechnical Journal, Vol. 35, No. 6, pp. 1074–1092. WinSaf-I (2001), Prototype Engineering, Winchester, MA. WinSaf-TR (2001), Prototype Engineering, Winchester, MA.

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Analysis of a full scale failure test on old railway embankment J. Mansikkamäki & T. Länsivaara Tampere University of Technology, Finland

ABSTRACT: Tampere University of Technology has started a research project related to stability of railway embankments. One notable part of this project was a full scale stability test on an old railway embankment. The test site was an obsolete short rail track in southern part of Finland. Subsoil consists of very soft and sensitive clay with undrained strength varying between 9. . .12 kPa. The test was accomplished in October 2009. Extensive instrumentation was accomplished to get comprehensive information from the test. Measurements included pore water pressure and earth pressure gauges, automatic inclinometers, different types of settlement indicators and displacement measurements with automatic theodolites and laser scanners. The focus was on to determine the shape of slip surface and to measure pore pressure behavior under the load, beside the load and also attempting to measure failure induced pore pressure. Motives for this kind of extensive test are mainly in the development of stability calculation methods, comparison between undrained strength (Su or C u ) and effective strength parameters (c and ϕ ), improve LEM and FEM calculations, especially how to better for the failure induced pore pressure. Analysis will be accomplished using LEM-program and 2D/3D FEM-programs using material models which are particularly developed to soft soil calculations.

1

INTRODUCTION

Stability of railway embankments on soft clays is commonly calculated with limit equilibrium method using undrained strength parameters. However, calculations with undrained strength might for some cases produce too small factors of safety. The calculated total factor of safety might even be less than 1.0 for existing embankments. On the other hand LEM calculations with effective strength parameters tend to overestimate the safety factor for undrained conditions. A major problem in effective stress analysis is the assumptions for stress and pore pressure distribution and the difficulty in ing for failure induced pore pressure. According to the guidelines by Finnish railway authorities (2006), the failure induced pore pressure can be taken into by using reduced effective strength parameters. The reduced strength parameters should be applied in conventional LEM analysis (lem) and when applying simple elastic-perfectly plastic models in the finite element method (fem). Alternatively fem calculations with hardening plasticity models can be used in order to for the failure induced pore pressure. Selection of preliminary fem calculations are presented in this paper.

2 TEST SITE AND TEST PROCEDURE Finnish Rail istration and Tampere University of Technology accomplished a full scale embankment

Figure 1. Test site situated southern part of Finland near grain storage silos and Helsinki –Turku railway track.

failure test to gather more information about failure induced pore pressure and to be able to compare different calculation methods to an actual failure. The test embankment was an existing old railway embankment in southern part of Finland as shown in figure 1. The upmost soil layer in the test site was old embankment fill that consist of sand and gravel. Dry crust layer was 1.0–1.5 m thick and had partially settled under the groundwater level. Beneath the dry crust there is a 5 to 7 m thick soft clay layer which undrained strength varies between 9. . .12 kPa. Frictional soil layers under the soft soil consist of silt and moraine.

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prism system and laser scanning. The pore pressure gauges were mostly concentrated to one cross section to be able to capture the failure induced pore pressure. 3 3.1

Table 1. Basic soil parameters for preliminary “realistic” FEM calculations.

Embankment Fill (sand) Dry crust Clay1 Clay2

γ

ϕ

MC MC MC SS SS

21.0 19.0 17.0 15.0 16.0

34.0 0.2 34.0 0.2 0.0 40.0 23.0 0.2 26.0 0.2

Drained Drained Drained Undrain Undrain

c /cu

M – – – 0.90 1.03

One should notice that at least for now the SRM method is a standard procedure only for the MohrCoulomb model (Plaxis 2D). When more sophisticated models are used, the SRM needs to be applied separately. The simple Mohr-Coulomb model can’t take into the failure induced pore pressure while the more sophisticated hardening models can. A material model used in the preliminary FEM calculations was hardening plasticity model named Soft Soil model (SS). The yield surface of the Soft Soil model includes a cap which shape/height is set with parameter M. The failure criteria itself is similar to conventional Mohr-Coulomb.

Table 2. Basic soil parameters for the calculations using undrained shear strength.

Embankment Fill (sand) Dry crust Clay1 Clay2

γ

ϕ

cu

cu /m

20.0 19.0 17.0 15.5 16.5

34.0 34.0 0.0 0.0 0.0

0.0 0.0 30.0 9.5 14.0

0.0 0.0 0.0 0.5 5.0

Calculation method

Stability calculations with finite element method were performed using Strength reduction method (SRM). It is a well known method where the strength parameters tanϕ and c of the soil are reduced until failure of the structure occurs. The factor of safety is determined from the relation between the input soil strength and the limit strength which causes a failure in the structure as presented in equation 1:

Figure 2. Typical cross section from the test site.

model Type

PARAMETERS FOR THE PRELIMINARY CALCULATIONS

Typical cross section from the middle of the test structure is presented in figure 2 together with some of the soil investigation results. The main calculation parameters for the different soil layers are presented in the tables 1 and 2. Loading was accomplished by filling containers with gravel. Between rails and containers a framework of steel beams were laid to simulate real bogie units. Loading structure consisted of 4 units or “cars”, each 12 m long. The loading was made in two days on 20.–21. October 2009. During the first loading day the total load was raised to 24 kPa, which corresponded to a stress state close to the preconsolidation pressure of the clay. On the second day the load was raised to maximum in 5 kPa steps constantly observing the displacements and the measuring data from the instruments located in the subsoil. Maximum load 85. . .87 kPa was fully loaded at 7:34 pm. The embankment finally collapsed two hours later at 9:27 pm. Instrumentation was extensive including e.g. 40 strain-type pore pressure gauges, 9 strain-type earth pressure gauges, 9 inclinometer tubes, 3 settlement tubes with a total of 54 pressure gauges, automatic deformation monitoring using 2 tachymeters and 27

3.2 Assessing of M-parameter in soft soil –model In the Soft Soil model the M-parameter determines the shape of the cap yield surface. As associated flow is assumed for the cap, it governs also the plastic flow. According to standard procedure the M-parameter is set in a manner that the model yields a realistic coefficient of earth pressure at rest. However, in stability calculation it might be more important to match the yield surface near the failure line (Mansikkamäki, 2008). In Figure 3 examples of two calculated stress paths are shown together with the estimated yield surface and a triaxial stress path. In addition, an undrained stress path for Mohr-Coulomb model is also shown. One can see that the influence of M-parameter is essential. In the preliminary calculation the M-parameter was adjusted only to match with the friction angle ϕ. As the effective stress path for undrained conditions follows close to the initial yield surface, the shape of the yield surface much determines at which shear stress level the failure line is reached, influencing thus strongly also on the safety factor. If the shape of the yield surface is known, the M-parameter should preferably be adjusted to give a best fit of the actual true yield surface at the most relevant stress level.

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Figure 3. Failure line, investigated stress path and two different calculated stress paths caused by the difference of the M-parameter (Mansikkamäki and Länsivaara 2009).

3.3 Stiffness parameters For drained analysis, it has been shown that the stiffness parameters used in fem-calculations play an insignificant role to the factor of safety (Cheng et al., 2006; Sæterbø Glåmen et al., 2004). However, for undrained conditions applying effective strength and hardening plasticity the situation is quite different. Relation of the stiffness parameters λ∗ and κ∗ determines the hardening effect of the soil model. The larger difference in undrained conditions is between λ and κ, the closer stress path follows the initial yield surface. Previously made studies have shown that influence can be significant if the λ/κ –relation is 10 or less (Mansikkamäki et al., 2009). Influence is found to be practically negligible if the ratio is more than 20. Typical true values in Finnish soft clays are λ∗ = 10 × κ∗ . If the stress path should follow closely the yield surface of the model, which is a safe approximation, one should choose a rather high stiffness relation. In the FEM calculations concerning this study laboratory tested λ∗ and on the other hand κ∗ = λ∗ /20 –values were used. Effective strength parameters and unit weights used in the fem-calculations are presented in table 1.

4

PRELIMINARY CALCULATIONS

Preliminary calculation results are presented in figure 4. There two sets of calculations are presented for lem-calculations using undrained shear strength (Cu), lem-calculations using effective strength and fem calculations using hardening plasticity. The calculations made with undrained shear strength represent present guidelines with safe assessment of the parameters. Basic soil parameters for the undrained shear strength calculations without strength reduction are presented in the table 2. The overall safety factor of 1.0 is reached at a load of 27. . .35 kPa. This method produces very conservative results as failure occurred with 87 kPa train load.

Figure 4. Preliminary calculations with different methods.

Limit equilibrium calculations were made assuming an excess pore pressure development due to failure for the whole slip surface. The failure load was predicted to be 60. . .72 kPa with this method. Finite element calculations are presented with diamond and line with dots. Both calculations were carried out using SRM prescribed earlier. The first calculation was made using more conservative soil parameters and pre-overburden pressure according to new FEM calculation code for Finnish rail istration. The second calculation was made using more realistic soil parameters. Calculation with conservative soil parameters forecasts that failure will occur when train load is 65 kPa and failure load 80.5 kPa, when using realistic soil parameters as presented in the table 1.

5

PORE PRESSURE DEVELOPMENT

One of the main goals of failure load test was to get more information about pore pressure development during short-term static loading. This will then be used to develop calculation methods and models to be able to better for the failure induced pore pressure. Extensive pore pressure instrumentation was applied to the test site to reach that goal. Selection of pore pressure data with applied train load is presented in the figure 5. Train load increasing from 0 kPa to 24 kPa illustrates first loading day and steps from 24 kPa to over 80 kPa second loading day. Failure occurred 21.10.2009 9:27 PM, when load was 85. . .87 kPa. Excess pore pressures increased more slowly than train load during the whole test. An 80 kPa increase in train load corresponded to 22. . .32 kPa increase in the pore pressures under the embankment. Pore pressures started to increase in whole failure area when the load increased to 30 kPa. For the maximum train load pore pressures increased 12. . .20 kPa in direct simple shear zone and under 10 kPa near the ditch in the ive zone. Pore pressures started to increase rapidly an hour after the loading was stopped caused by yielding

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Figure 5. Train load and excess pore pressures in the test site during the loading and the failure.

in the soil skeleton. Total failure of the soil mass occurred approximately one hour later. As shown in figure 5, failure induced pore caused significant and rapid increase in excess pore pressure measurements during the collapse.

6


In figure 6 measured pore pressure data and preliminarily calculated pore pressures under the embankment are presented. As shown, the calculated data correspond quite well with measured excess pore pressure development; despite the calculation slightly overestimates the increase of pressures. Most of the difference developed on the first calculation steps, when train load was increasing from 0 kPa to 50 kPa. During final loading phases difference between calculated and measured pressures was negligible. Nevertheless, this result is very promising because one can easily correct pore pressure behaviour by adjusting M-parameter of Soft Soil model as mentioned before. The calculated pore pressure development presented in the figure represents “softest” behaviour one can model with Soft Soil model, since M-parameter was fully adjusted to match with friction angle. During these preliminary calculations only a limited amount of laboratory data was available. Hence the shape of the yield locus was not fully known and a bit too conservative assumptions were perhaps made. This is one of the reasons that the calculated failure load was 4. . .7 kPa less than the actual failure load.

Figure 6. Calculated pore pressure development versus train load and measured pore pressure data.

Also a great simplification was made in that the time effects were ignored in the preliminary calculations. It is a well known fact that the strength of the clay is strongly depending on the loading rate. The faster the loading is applied, the higher failure load is applied. This is explained by the pore pressure response of the clay, while the effective strength parameters are independent of loading rate. So to fully analyse the preformed failure test, one needs to include creep effects in the model. This will be done in connection with the more comprehensive analysis of the test.

7

CONCLUSIONS

In this paper some preliminary results from a full scale failure load test are presented. The test corresponds to a rather typical railways embankment on soft clay in southern Finland. As also known previously, present stability calculation guidelines by the Railway authorities using undrained shear strength often leads to a rather conservative estimate of the safety. To fully analyse the failure, calculations with FEM are needed. Therein one should apply a hardening plasticity model which includes also time effects. Preliminary calculations have been done using the Soft Soil model in Plaxis. The times effects have thus jet been disregarded. The failure induced pore pressure that develops in the analysis are much dependent on the shape of the yield surface. In the preliminary

544

calculations, the yield surface was modeled so, that the M parameter was adjusted according to the friction angle. This gives a yield surface more close to the modified cam clay model. This way a maximum pore pressure increase can be modeled i.e. representing the softest undrained response. Time effects are not included, but the modeling would correspond close to a slow undrained loading. In the future anisotropic models should be applied to better model the actual yield surface. In the failure test the load was applied rather rapidly. This is probably the main reason why the preliminary calculations underestimated slightly the failure load. Other possible reasons are conservative strength parameters, especially for the dry crust, and that 3D effects where not ed for. The preformed test gives valuable information about undrained failure and rated pore pressure development. Future studies will include among others more detailed analysis of the failure tests and analysis with more sophisticated models.

REFERENCES Cheng Y.M., Wei W.B., Länsivaara T. 2006. Factors of safety by limit equilibrium and strength reduction methods. NUMGE06. Finnish railway authorities (RHK). Ratahallintokeskuksen julkaisuja, A10/2006. Radan stabiliteetin laskenta, olemassa olevat penkereet. Kirjallisuus ja laskennallinen taustaaineisto. Helsinki 2006. 319 s. Mansikkamäki, J. 2008. Master’s Thesis: Stability analysis of existing railway embankments based on finite element method. Tampere University of Technology. Mansikkamäki, J. & Länsivaara, T. 2009. Effective stress analysis of old railway embankments. 17th International Conference on Soil Mechanics & Geotechnical Engineering. Plaxis 2D. Material Models Manual, version 9.0. 2008. Sæterbø Glåmen M.G., S. Nordal &A. Emdal. Slope Stability Evaluations using the Finite Element method, NGM 2004.

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Analysis of ground movements induced by diaphragm wall installation B. Garitte, M. Arroyo & A. Gens Department of Geotechnical Engineering, UPC, Barcelona, Spain

ABSTRACT: The construction process of diaphragm walls can lead to movements in the surrounding area that are seldom taken into . However, these movements may be important in situations where soft soils dominate. In this paper, after briefly reviewing the state of the art on this issue, we present results of a hydromechanical simulation of the problem in a case located in deltaic soils of Barcelona. The auscultation record of the settlements of a nearby building is employed to validate the computational model. length and bentonite slurry level were found to be the most influent parameters on induced displacements.

1

INTRODUCTION

available in the literature and present modeling results relevant to a real case.

Excavation protected by diaphragm walls may be considered as the default execution method for stations and other rail- and underground structures in urban areas. Diaphragm walls of reinforced concrete are built section by section. The principal construction phases of such a section, called , are as follows: (a) construction of guide wall (b) trench excavation under bentonite slurry or other ing fluid (c) bottom-up concrete filling of the trench using a tube (d) reinforcement installation. Diaphragm wall s may reach important dimensions (up to 1.2 m width, 8 m length and 60 m depth are not exceptional). There are several reasons why the study of the effects due to installation is interesting. First (direct effect) to assess the possible impact of the installation on its environment, as sometimes collapse occurs and, quite more frequently, significant movements in the vicinity of the are ed during construction. Second (indirect effect) because even if ground movements induced by execution are small, they may lead to stress changes in the environment relevant to later excavation stages. Direct effect of installation has more significance in the case of soft soils. The indirect effect is more important in hard overconsolidated soils. As a first approximation, ignoring the indirect effect tends to leave design on the side of safety, since the broad consequence of excavation is horizontal stress relaxation. Ignoring the direct effect instead, is not safe, since the movements induced by the diaphragm wall installation are added to those produced by the subsequent excavation of the structure. Given the increasing requirements regarding limitation of ground movements in urban areas, development of tools to evaluate the effect of the installation of diaphragm walls in soft soils is necessary. In the paper, we briefly review some observations on the problem

2

STATE OF THE ART

2.1 Field observations Although well documented cases of instrumented excavation of diaphragm wall s are rare, there are enough in the literature (Di Biagio and Myrvoll 1973, Poh and Wong, 1988, Tsai and others 2000, De Wit and Lengkeek, 2003) to obtain some general lessons. Table 1 lists the main geometric features of these cases, i.e. the width (W), length (L) and depth (D) of the studied . A description of soil type and a measure of the characteristic resistance (CR) are also included for each case. The correspondent values are also given for the case study presented later in this paper. Where installation does not lead to failure (a) final superficial settlements measured at the edge of the vary between 10 and 20 mm (b) those settlements attenuate rapidly with the distance from the , being negligible for distances greater than 10W (Fig. 1) (c) horizontal displacements at a certain depth tend to reach values several times higher than the surface settlements. Its attenuation with distance from the follows a law similar to the surface settlement. Some characteristics of the construction process are clearly reflected in the observations. For example, settlements and movements toward the that occur during the excavation phase decrease during concreting. Occasionally, the movement direction is even reversed (Fig. 2). Unloading (excavation) and loading (concreting) during the installation of a also produce pore water pressure changes in less permeable materials. In that case, measurements during the excavation correspond to a situation of partial drainage

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Table 1.

Cases of installation experiments in the litterature.

Case

Soil

CR

W(m)

L(m)

D(m)

Barcelona (this study)

Sand Clayey silt Soft clay Soft clay Silty sand Clay and turf Sand

qc = 5–10 MPa Su = 40–80 kPa Su = 30–40 kPa Su = 14–40 kPa qc = 20–40 MPa Su = 30–40 kPa qc = 20 MPa

1–1.2

2.8–6

25–35

1 1.2 0.9 0.8

1.8–5 2–6 8 3

28 55.5 15 35

Oslo (Di Biagio and Myrvoll, 1973) Singapur (Poh y Wong, 1998) Taiwan (Tsai et al., 2000) Amsterdam (De Wit and Lengkeek, 2002)

Figure 1. Effect of the installation of a on the superficial settlements (Poh and Wong, 1998).

Figure 3. Measured fresh concrete pressure in a 45 m deep (Schad et al., 2007).

Figure 2. Settlement at 3 m from the measured with an hydraulic cell (Di Biagio y Myrvoll, 1973).

and more movements may occur during the subsequent consolidation phase. Other interesting observations are those concerning the effect of variations of the basic construction process. In practice, such variations may be due to work incidents (a) excavation induced movements are very sensitive to the sustaining fluid level in the trench. Apparently small decreases (e.g. 1m for a 55 m deep ) can produce significant additional movements and even the collapse of the excavation (b) movements are not increased significantly during important waiting times (e.g. one day), as long as a constant fluid level is maintained inside the trench. (c) Once the impervious cake has developed satisfactorily on the wall, excavation induced movements are relatively insensitive to the density of the sustaining fluid (in the range 1 to 1.3 t/m3 ). One aspect that is not clear from the cases described in the literature is the influence of the size of the

on induced movements. Greater displacements are expected to be induced by s of larger length and depth, but there are few data to quantify this phenomenon. Finally, observations have repeatedly confirmed (Uriel and Oteo, 1977, Lings et al., 1994; Schad et Al., 2007) that the pressure applied by fresh concrete on the walls is not always hydrostatic. In fact, hydrostatic pressure seemed to be maintained only until a certain depth, referred to as the “critical depth” (Fig. 3). The observed critical depths vary between 1/3 and 1/5 of the depth. This characteristic is still poorly understood, but seems to depend on the interplay between concreting rate and hardening. 2.2 Analytical solutions There are a number of analytical solutions to the problem of stability (e.g. Fox, 2004; Tsai, 2000). Most of them were derived using limit equilibrium and the differences between them lie mainly in the degree of complexity of the alleged failure surface.

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A major drawback of these analytical solutions is their limited applicability for layered profiles. Moreover, only stability is dealt with and they are not useful for quantifying the settlement induced by excavation. However, it is interesting that these analytic solutions also indicate the enormous importance of the level of bentonite on the stability. For instance, Fox (2004) predicts a security factor reduction from 2 to 1, by a decrease of bentonite level of 2 m in a 15 m deep trench of 8 m length in sand (ϕ = 34◦ , γ = 20 kN/m3 ).

Figure 4. Plane view of the study area. The investigated diaphragm wall (R1–5) and reference measurement points (P1–7) are indicated.

2.3 Numerical models An interesting alternative to obtain quantitative answers to the problem is to use numerical models. However this approach is relatively costly. For practical reasons numerical modeling of excavations still takes place mostly in 2D. In such models, diaphragm wall construction is simultaneous for the entire wall length. Available examples (Ng and Yan, 1998; Gourvenec and Powrie, 1999, Schafer and Triantafyllidis, 2006) make clear that it is very difficult to obtain approximate results if the three-dimensionality of the problem is ignored. The cited authors also emphasized the importance of the initial earth pressure coefficient K0 , because of its influence on stress redistribution. Several common modeling features can be noticed from the precedents (1) guide wall construction is not considered; (2) excavation under bentonite is reproduced by removing the elements included in the volume of the and prescribing the hydrostatic pressure of bentonite on the new contour; (3) fresh concrete pouring is represented by changing the boundary condition of total stresses from the hydrostatic bentonite profile to a bilinear profile; (4) finally, to represent hardened concrete, the total stress boundary condition is removed and elements in the volume are re-activated, with material parameters corresponding to those of reinforced concrete. Note that, while this procedure is in accordance with the above mentioned field observations, it does neglect the tangential frictional stresses between soil and fresh concrete that will be necessary for equilibrium. The critical depth is usually taken equal to one third of depth.

Figure 5. Settlements measured during the execution of the R-diaphragm wall. Vertical dotted lines indicate the day in which each was built (R5 to R1).

potentially disturbing activities, like jet-grout treatment and micro-pile installation. However, the construction sequence began with the execution of a diaphragm wall section near the building (Fig. 4). During that period, which preceded all other construction activities, significant building movements were already ed (Fig. 5). Those records made clear that diaphragm wall construction had produced some movement, but they did leave open the magnitude, because settlement occurring after the diaphragm wall section was finished might have been due to consolidation or to other, later activities. Since similar diaphragm walls needed to be constructed in the vicinity a detailed study of this problem by means of numerical simulations seemed necessary. 3.2 Geotechnical site characterization

3 3.1

CASE STUDY Background

The case that inspired the studies described here can be considered typical of the excavations in soft soil of deltaic areas near Barcelona. The motivation arose from the observation of significant movements in a building near some excavation works during diaphragm wall installation. Construction activities were complex because most diaphragm wall installation was simultaneous to other

The geotechnical profile at the site might be described by five main levels. Made ground (2 m thick), clay (2 m), sand (11 m), silt (33 m) and gravel (undefined). The water table is found at the top of the sand layer. The geotechnical site characterization procedure cannot be described here other than it was heavily reliant on in situ tests. The in situ measurement campaign also provided information on the earth pressure coefficient K0 which was used to prescribe initial stress state. The silt package is slightly overconsolidated and has a relatively high deformability. It is likely that much of its deformation takes place under plastic

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Table 2a.

Geomechanical parameters (MC).

Made ground Clays Sand Gravel

Table 2b.

Silt

κ (m/d)

E (kPa)

ν

c (kPa)

φ (◦ )

0.22 0.005 31.3 105

3000 15,000 20,000 31,000

0.3 0.3 0.3 0.3

30 50 1 50

25 29 33 36

Geomechanical parameters (MCC). κ (m/d)

φ (◦ )

OCR

κ

λ

0.003

27

1.15

0.012

0.06

regime. Hence, the characterization of this regime has some importance and a modified Cam-clay (MCC) model was chosen. For the other materials a simpler Mohr-Coulomb (MC) model was selected. The most important geomechanical parameters employed in the calculations are shown in Table 2. 3.3

Characteristics of the numerical models

Precedent published analyses of this problem modeled clayey soils, for which undrained behaviour might be safely assumed. The geotechnical profile in this case includes layers of very different permeability and any generic assumption about drainage was not granted. For this reason fully coupled hydromechanical computations were performed. The program employed was Code_Bright, a finite element code developed in the Department of Geotechnical Engineering of UPC (Olivella, 1995). The modeling of the construction process of a required some modifications to the program, the most important being the implementation of a boundary condition with (bi)linear stress variation with depth. The new implementation was verified by benchmarking against a case reported in the literature (Gourvenec and Powrie, 1999). Two types of analysis were performed: a parametric study of the excavation of an isolated and a detailed modeling of a particular excavation sequence of five s, namely the R-diaphragm wall (Fig. 4). Figure 6 shows the mesh used for modeling the R-diaphragm wall excavation sequence (the mesh used to simulate the installation of one is similar). Advantage was taken of the vertical longitudinal symmetry plane of the s. Coupled hydro-mechanical computations require the explicit specification of construction times. After some consultation with the site managers and inspection of construction records, a site-representative construction sequence was established for the base case. Excavation under bentonite is modeled by removing meter by meter the elements of a during 3.2 hours in the base case (25 m deep ). Once

Figure 6. 3D mesh used for the analysis of the R-diaphragm wall excavation sequence.

a certain volume has been removed in a , hydrostatic bentonite pressure is applied as total stress on the new wall. After the excavation sequence finishes, a five hour waiting time represents bottom cleansing and reinforcement placement. The hydrostatic bentonite pressure profile is then replaced by the bilinear profile representing fresh concrete. Concrete hardening time was estimated as 12 hours, after which solid elements with concrete properties are placed in the . Concrete hardening is thus modeled as an instantaneous process, which is clearly unrealistic. 4

RESULTS

4.1 Base case Geometric features of the base case (depth D, length L and width W) are shown in Table 3. Bentonite level within the , nb , and the assumed critical depth Dc , are also indicated. Surface settlement histories at different distances from the edge are given in Figure 7. Times at which excavation ends, fresh concrete is poured and hardening is assumed are indicated by vertical dotted lines. Excavation produces settlement, reaching about 3.5 mm at 2 m from the edge. Most settlements induced by the excavation occur simultaneously to it and later consolidation has a moderate influence. Fresh concrete deposition results in heave and after concrete hardening, settlements are resumed. Final settlement values are similar to those ed after excavation. It is worth noting the qualitative similarity with the measurements by Di Biagio and Myrvoll (Fig. 2), especially during fresh concrete injection. 4.2 Parametric study of a single The parametric study includes five variations on the base case. As outlined in Table 3, a single parameter was changed from the base case for each variant. In

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Figure 7. Time evolution of settlements at different distances from the edge (base case). Table 3.

Modeling of single . Parametric study cases.

Case

D(m)

L(m)

W(m)

nb (m)

Dc (m)

Base (V1) V2 V3 V4 V5 V6

25 25 25 35 25 25

3.6 3.6 6 3.6 3.6 3.6

1.2 1.2 1.2 1.2 1.2 1

0 2 0 0 0 0

8 8 8 8 5 8

Figure 9. Superficial settlement simulated at a distance of 3 m from the wall for two s of 3.6 m length and one of 6 m length.

Figure 10. Comparison of measured and simulated settlements at observation points P1 and P2.

4.3 R-diaphragm wall construction sequence

Figure 8. Left graph: evolution of settlements at 3 m from the edge for the 6 cases of the parametric study. Right graph: horizontal displacements of the wall after the excavation phase for the 6 cases of the parametric study.

variant 2 (V2) the level of bentonite, nb , was lowered by 2 m; in variant 3 (V3) a length of 6m is considered for the ; in variant 4 (V4) a deeper was excavated up to 35 m; in variant 5 (V5) the critical depth was modified to 1/5 of the depth instead of 1/3 and in variant 6 (V6) the was made thinner. A comparison between the different cases is given in Figure 8, in of settlement evolution and horizontal displacement of the wall. According to the simulation results, the most damaging cases are V3 and V2, i.e. a longer and a lower bentonite level. Differences between the other cases are negligible for superficial settlement measurements, although, as expected, a deeper causes higher movements at depth.

The R-diaphragm wall s (Fig. 4) have similar dimensions to the base case : all s have a length of 3.6 m, excepted one (R1), which is 4.9 m long and the depth of the wall is 24 m instead of 25 m in the base case. The construction sequence was simulated according to work site information. Bentonite level and critical depth were taken as in the base case because no measurements were available. This computation confirmed some of the previous results. For instance, the installation of one 6 m long produces more settlement than the consecutive installation of two 3.6 m long s (Fig. 9). The difference in settlement behaviour appears mostly during the excavation phase. After concrete hardening, quite similar final settlement levels are predicted for the 6 m long and the t 7.2 m long excavation. Simulated and measured settlements at observation points P1 and P2 are compared in Figure 10. The position of those points was illustrated in Figure 4. Quite a good agreement between measurements and simulation is obtained. One remarkable difference is the smooth evolution of the measurements when compared with simulation results. This is likely due to the abrupt modeling approach adopted for concrete hardening (instantaneous).

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REFERENCES

Figure 11. Measured and simulated settlement profiles after construction.

Measured and simulated settlement profiles after construction are drawn in Figure 11. Two simulation profiles have been plotted: one at the level of R4, corresponding to observation point P1 and one at the height of R1-R2, corresponding with the remaining points. The fit is quite good for measurement points close to the wall, but its quality decreases with distance to the wall. The observed discrepancies are probably due to the fact that the building structure is not taken into in the computation. The structure may act as a stiffening element, hindering settlement recovery during fresh concrete injection. The overall good match between simulation results and measurements can be considered satisfactory and gives more credibility to the results of the parametric study. 5

CONCLUSIONS

Diaphragm wall installation in soft soils may produce settlement in its neighborhood. Numerical models may help to quantify and understand the problem. The presented parametric study allow for isolating two influent parameters: bentonite level and length. Other parameters, like width, depth and critical depth were found to be less important in the range considered in the parametric study.

De Wit, J. C. W. M. & Lengkeek, H. J. (2002). Full scale test on environmental impact of diaphragm wall trench installation in Amsterdam – the final results. Proceedings of the international symposium on geotechnical aspects of underground construction in soft ground, Toulouse, (eds R. Kastner, F. Emeriault, D. Dias and A. Guilloux), pp. 433–440. Lyon DiBiagio E, Myrvoll F. (1972) Full scale field test of a slurry trench excavation in soft clay. Proceedings of the 15th European Conference Soil Mechanics Foundation Engineering, Madrid 1972; 461–471. Fox, P.J. (2004) Analytical solutions for stability of slurry trench, ASCE Journal of Geotechnical and Geoenvironmental Engineering, Vol. 130, No. 7, 749–758 Gourvenec, S. M. & Powrie, W. (1999). Three-dimensional finite-element analysis of diaphragm wall installation. Géotechnique 49, No. 6, 801–823 Lings M, Ng CWW, Nash DFT. (1994) The lateral pressure of wet concrete in diaphragm wall s cast under bentonite. Proceedings of the Institution of Civil Engineers: Geotechnical Engineering; 107:163–172. Ng CWW, Yan RWM. (1998) Stress transfer and deformation mechanism around a diaphragm wall . Journal of Geotechnical and Geoenvironmental Engineering; 128(7):638–648. Olivella, S., 1995. Nonisothermal multiphase flow of brine and gas through saline media. Doctoral Thesis, Technical University of Catalonia (UPC), Barcelona, Spain. Poh TY, Wong IH. (1998) Effects of construction of diaphragm wall s on adjacent ground: field trial. Journal of Geotechnical and Geoenvironmental Engineering; 124(8):749–756 Schad, H., Vermeer, P.A., Lächler, A. (2007) Fresh concrete pressure in diaphragm wall s and resulting deformations. In: Grosse, Ch. U. (Ed.): Advances in Construction Materials, Berlin: Springer Verlag, 2007, pp. 505–512. Schafer R, Triantafyllidis T. (2006) The influence of the construction process on the deformation behaviour of diaphragm walls in soft clayey ground. International Journal for Numerical and Analytical Methods in Geomechanics; 30:563–576 Tsai, J.S., Jou, L.D., Hsieh, H.S. (2000) A full scale stability experiment on a diaphragm wall trench, Canadian Geotechnical Journal, 37, 379–392 Uriel S. y Oteo C. S. (1977) Stress and strain beside a circular trench wall. Proc. 9th ICSMFE, Tokyo, 1,781–788.

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Bearing capacity of a surface footing founded on a layered clay subsoil Z. Bournta* Aristotle University of Thessaloniki, Greece

L. Zdravkovic Imperial College London, UK

ABSTRACT: The ultimate bearing capacity of surface footings resting on homogeneous soils has been studied extensively. Real soil strength profiles beneath footings, however, are not homogeneous but may increase or decrease with depth, or consist of distinct layers having significantly different properties. The current study focuses on the application of the finite element method for the evaluation of the ultimate bearing capacity of a rough rigid footing resting on a layered clay profile. Computations are performed using the finite element analysis software ICFEP (Imperial College Finite Element Program). One set of parametric analyses is carried out for the case of strip and circular footings resting on soft clay overlying stiff clay, in order to examine the effect of the vicinity of the strong clay layer on the bearing capacity factor. Another set of parametric analyses is performed for the case of circular footings resting on a multi-layered soil profile that includes a layer of soft clay resting within the strong clay subsoil, in order to investigate the effect of both the vicinity and the thickness of the soft clay layer on the bearing capacity factor. In both cases comparison of the analyses results with the literature has been made.

1

INTRODUCTION

The ultimate bearing capacity of surface footings resting on a single layer of homogeneous soil has been studied by numerous investigators, with practitioners generally using Terzaghi’s (1943) expression to compute ultimate footing loads. In reality, however, soil strength profiles beneath footings are not homogeneous but may increase or decrease with depth or consist of distinct layers having significantly different properties. For such critical soil profiles the modified conventional procedures appear to be generally unrealistic and suffer from several limitations. The current study focuses on the application of the finite element method for the evaluation of the ultimate bearing capacity of a rough, rigid surface footing resting on layered clay profile. Both circular and strip footings are examined. 2

FINITE ELEMENT ANALYSES

Due to symmetry only half of the problem is discretized with the use of eight nodded quadrilateral isoparametric elements. Only the soil is discretized into finite elements. The foundation is represented by appropriate boundary conditions. A footing diameter of 15 m is modelled. The boundaries of the mesh are sufficiently distant from the footing, extending 100 m *

formerly Imperial College London

horizontally and 100 m vertically. The two vertical sides of the mesh are restrained in the horizontal direction, while the base of the mesh is not allowed to move either in the vertical or the horizontal direction. Along the top boundary of the mesh from the edge of the footing to the right hand side corner of the mesh, horizontal and vertical movements are allowed. At the underside of the footing 6-noded interface elements of zero thickness are used in order to model the movement of the soil in with the footing realistically. All the analyses are performed using the Imperial College Finite Element Program (ICFEP, Potts and Zdravkovic, 1999), which employs a modified Newton-Raphson approach with an error controlled sub-stepping stress point algorithm to solve the non-linear finite element equations. To model the rough interface between the side of the footing and the soil, interface elements along the foundation base are assigned an equal normal and shear stiffness of 105 kN/m3 and an undrained strength of either 20 kPa in the case of soft clay lying at the surface, or 200 kPa in the case of strong clay being present at the surface. The footing is considered to be rigid and the loading is modelled by applying increments of uniform vertical displacement to the nodes located at the top of the interface elements. The ultimate footing load is then obtained by summing the vertical reactions of the nodes which have been subjected to this displacement. The soil is modelled as an isotropic elastic-perfectly plastic material satisfying the Tresca failure criterion.

553

An undrained Young modulus, Eu, of 50 MPa and a Poisson’s ratio of 0.49 are assigned to the clay. The unit weight of the clay is taken to be 20 kN/m3 . Constant undrained strength Su is adopted within all clay layers. The undrained shear strength of the clay is 20 kPa for the soft layer and 200 kPa for the strong layer. 3 3.1

RESULTS Footings on homogeneous clay

In order for the accuracy of the finite element method to be established, analyses for the cases of both strip and circular rough footings lying on a homogeneous clay profile have firstly been performed. The Nc factor of 5.182 obtained for the strip footing lying on strong homogeneous clay is 0.8% in error compared with the 5.14 solution of Prandtl (1920). In a similar way, the Nc factor of 5.185 for the footing lying on soft homogeneous clay is 0.88% in error with the known solution. The Nc factor of 6.0903 obtained for the circular footing lying on strong homogeneous clay is 0.66% in error compared with the 6.05 solution of Eason & Shield (1960). In a similar way, the Nc factor of 6.094 for the footing lying on soft homogeneous clay is 0.73% in error with the known solution. The accuracy of the finite element analysis results was therefore proved to be excellent, being within 1% of the closed form solutions. 3.2

Figure 1. Mobilized bearing capacity factors against normalised displacement δ/B (strip footing).

Footings on a two layered soil profile

Parametric analyses were carried out in order to predict the bearing capacity of a surface strip footing (width B = 15 m) resting on a layered subsoil consisting of soft clay (Su1 = 20 kPa) overlying strong clay (Su2 = 200 kPa), by varying the thickness, H, of the top soft layer. The effect of the vicinity of the strong clay layer on the bearing capacity factor is examined. Mobilized bearing capacity factors against normalised displacement δ/B, with B being the width of the surface footing for different H/B ratios are shown in Figure 1, where a reduction in bearing capacity with increasing H/B ratios is indicated. A decrease of the mobilized bearing capacity factor with increasing relative thickness of the upper soft clay, H/B, is presented in Figure 2. By increasing the thickness of the upper layer the contribution of the underlying strong layer to the bearing capacity factor of the layered subsoil gradually diminishes and tends to reach the value for the case of soft homogeneous clay subsoil. For H/B = 0.267 the influence of the strong layer on the bearing capacity factor is in a range of 19% and reduces almost to 4% for H/B = 0.47 The values of bearing capacity factor Nc for rough strip footings obtained by the current study are compared with the results of Merifield and Sloan (1999), who had followed upper and lower bound numerical approach and bracketed the actual collapse load for the case of strip footings on a two layered clay profile. The results of the current numerical analysis are lying

Figure 2. Mobilized bearing capacity factors against relative thickness of the upper soft clay, H/B.

Figure 3. Comparison of the bearing capacity factor Nc for rough strip footings obtained by the current study with the results of Merifield and Sloan (1999).

within their upper and lower bound solutions for the range of H/B ratios that were examined (Fig. 3). Parametric study was also carried out in order to predict the bearing capacity of a surface circular footing (D = 15 m) resting on the same layered subsoil

554

Figure 4. Mobilized bearing capacity factors against normalised displacement δ/B (circular footing).

Figure 7. Comparison of the ultimate bearing capacity of circular footings obtained by the current study with the semi-empirical approach of Meyerhof (1974).

Figure 5. Mobilized bearing capacity factors against relative thickness of the upper soft clay, H/D.

Figure 8. Comparison of the bearing capacity factor Nc for circular footings obtained by the current study with the limit equilibrium approach of Button (1953).

Figure 6. Comparison of the mobilised bearing capacity factor for circular footings obtained by the current study with model footing tests performed by Brown and Meyerhof (1969).

(Fig. 4), where again a reduction in bearing capacity with increasing H/D ratios is indicated. Figure 5 presents the decrease of the mobilized bearing capacity factor with increasing the relative thickness of the upper soft clay, H/D. For H/D > 0.25 the influence of the strong layer on the bearing capacity factor is in the range of 8% and reduces almost to

0% for H/D = 0.47. Thus, from this value onwards the effect of the under layer can be neglected and the soil behaviour appears to depend only on the top soft soil. Comparison of the results of the current study with model footing tests performed by Brown and Meyerhof (1969) has shown that the empirical results lie very close to the ones of the present study over the range of H/D ratios that were investigated (Figure 6). A reduction in bearing capacity for a soft -over- strong clay system occurs up to a depth ratio of H/D = 0.2 and after that the bearing capacity factor for circular footings stabilizes at the value for a homogeneous subsoil. Results produced by the semi-empirical approach of Meyerhof (1974) appear to deviate significantly from the results of the current study and therefore are not considered to be reliable over a wide range of geometric scales. Button (1953), following the limit equilibrium approach, obtained results that seem to overestimate the bearing capacity factor for strip footings. (Fig. 8).

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the upper soft clay. For strip footings the results of the current numerical analysis are lying within the upper and lower bound solutions of Merifield and Sloan (1999) for the range of H/B ratios that were examined. For circular footings the results of the current study lie very close to the ones obtained by model footing tests (Brown and Meyerhof (1969)). Results produced by the semi-empirical approach of Meyerhof and Hanna (1974) appear, to some degree, to be valid only for the restricted case of relatively thin soft soil crust layer (H/D < 0.4). The results of Button (1953) seem to overestimate the bearing capacity factor with the difference ranging from 35% at H/B = 0.133 to 5%, as the ratio H/B increases to 0.4. – For circular footings lying on a multi- layered soil profile it was found that the bearing capacity of the surface foundation increases with increasing the thickness of the upper strong clay layer. When the thickness of the upper layer, H, however, reaches an average value of 1.2 diameter of the footing (1.2D), most of the cases show no or very little influence from the underlying soft layer. When the thickness of the soft clay layer reaches a value of 0.2D, any further increase in thickness has a minor effect on the bearing capacity of the footing.

Figure 9. Mobilized bearing capacity factors against relative depth of the soft layer, H/D, for different values of its relative thickness, t/D.

3.3

Circular footings on a multi-layered soil profile

Parametric analyses were performed for the case of a circular rough footing (D = 15 m) lying on a multi-layered soil profile that includes a layer of soft clay resting within the strong clay profile in order to investigate the effect of both the vicinity and the thickness of the soft clay layer. The parametric analyses were performed by changing both the depth of the soft clay layer, H/D, and its thickness, t/D. The study was carried out for ratios of H/D varying from 0.033 to 1.467 and t/D from 0.066 to 0.333. A ratio of undrained strength Su1 /Su2 = 10 was used. It was found that the bearing capacity of the surface foundation increases with increasing the thickness of the upper strong clay layer. When the thickness of the upper layer, H, reaches an average value of 1.2 diameter of the footing (1.2D), most of the cases show no or very little influence from the underlying soft layer (Fig. 9). However, when the thickness of the upper layer is less than 1.2D the bearing capacity reduces due to the soft soil underneath. For a given ratio of H/D, increasing the thickness of the soft clay layer results in a general decrease in the bearing capacity of the foundation. When the thickness of the soft clay layer reaches a value of 0.2D, any further increase in thickness has a minor effect on the bearing capacity of the footing as inferred from the closeness of lines in Figure 9 when t/D ≥ 0.2. 4

CONCLUSIONS

The following conclusions result from this numerical parametric study:

REFERENCES Brown, J. & Meyerhof, G. 1969. Experimental study of bearing capacity in layered clays, Proc. 7th int. conf. on soil mech. and foundation eng. 2. Button, S. J. 1953. The bearing capacity of footings on a two-layer cohesive subsoil, Proc. 3rd Int. Conf. on soil mechanics and foundation engineering, Zurich, 1: 332–335. Eason, G. & Shield, R. T. 1960. The plastic indentation of a semi infinite solid by a perfectly rough circular punch, J. Appl. Math.Phys. (ZAMP) 11: 33–43. Merifield, R. & Sloan, S. & Yu, H. 1999. Rigorous plasticity solutions for the bearing capacity of two-layered clays, Géotechnique, 49(4). Meyerhof, G. G. 1974. Ultimate bearing capacity of footings on sand layer overlaying clay, Can. Geotech. J., 11(2): 223–229. Potts, D.M. & Zdravkovic, L. 1999. Finite element analysis in geotechnical engineering. London: Thomas Telford Ltd. Prandtl, L. 1920. Uber die harte plastischer korper, Nachrichten von der koniglichen gesellschaft der wissenschaften, Gottingen Math. Phys. Klasse:74–85. Terzaghi, K. 1943. Theoretical soil mechanics, Wiley, New York.

– For footings lying on a two-layered soil profile there appears to be a decrease of the mobilized bearing capacity factor with increasing relative thickness of

556


Finite element analysis of the main embankment at Empingham dam A. Grammatikopoulou & N. Kovacevic Geotechnical Consulting Group, London, UK

L. Zdravkovic & D.M. Potts Imperial College, London, UK

ABSTRACT: Empingham dam is a 37 m high earth fill embankment which was constructed in the UK in the early 1970’s. The embankment was built on a brecciated Upper Lias Clay (ULC) foundation, of fill derived from it. The ULC is a typical stiff plastic clay which shows a post-peak strength loss and as such is prone to progressive failure. The paper describes the finite element back analyses of the embankment behaviour during construction, using a kinematic hardening “bubble” model which s for both pre-peak plasticity and post-peak strain softening.

1

INTRODUCTION

Empingham dam is one of the last large dams to be built on a stiff plastic clay in the UK, using a fill derived from the same clay (Bridle et al. 1985). It is a zoned earth-fill dam 37 m high and 1200 m long, founded on Upper Lias Clay (ULC) (Figure 1). The ULC in the valley slopes was extensively disturbed by cambering and in the valley floor by bulging (Horswill & Horton, 1976). The clay was then brecciated by periglacial ground freezing. Brecciated clays have severely disturbed fabric which is heterogeneous and therefore causes variation in the undrained shear strength. Hence, at the design stage there were uncertainties with respect to the properties of the ULC (and the fill derived from it). In order to obtain information with respect to the clay foundation and its strength, an instrumented trial embankment was built as part of the upstream fill of the main embankment (Figure 1). This was designed to induce shear stresses in excess of the assumed undrained strength. The trial embankment suffered large movements during its construction; however, there was no sign of strain softening and progressive failure despite the likelihood that the stiff plastic ULC would strain-soften.

Figure 1. Cross section of main embankment at Empingham and location of the trial bank.

The information obtained from the construction of the trial embankment, combined with slips that occurred in the temporary slopes of borrow pits upstream of the main embankment confirmed the undrained strength used for the initial design. In the main embankment the undrained strength of the foundation alone was too low to carry the full height of the embankment. Hence, vertical sand drains were installed below the centre of the embankment where it was more than 25 m high. However, for the ends of the embankment, where it was less than 25 m high, no drainage was provided and therefore these sections relied solely on the undrained strength of the ULC in the foundation. Just before completion of the main dam the north section (where there were no drains) started to spread laterally, quite suddenly, when the fill was some 2 m below the final crest level (Bridle et al. 1985). This suggested potential instability which was in contrast to the apparent stability of the trial embankment. The behaviour of the trial embankment was reproduced successfully by finite element back-analyses (Kovacevic et al. 2007a,b), by adopting a kinematic hardening model which s for pre-peak plasticity and non-linearity. Kovacevic et at. (2007a) used the same constitutive model in the back-analysis of the north section of the main embankment and attempted to recover the observed behaviour by varying the undrained strength. However, it was not possible to reproduce the sudden increase in movements as filling approached the crest level. Kovacevic et at. (2007a) concluded that this was only possible by using a model which ed for strain softening. However, at the time when the analyses were carried out the kinematic hardening model they used could not model strain-softening.

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Figure 2. North section of main embankment at Empingham with position of rod gauges.

This paper presents the back-analysis of the north section of the main embankment with a new kinematic hardening model which s for both pre-peak plasticity and post-peak strain softening. This backanalysis is also compared with a back-analysis using the same kinematic hardening model, but assuming no strain-softening.

2

MAIN EMBANKMENT

The north section of the main dam is shown in Figure 2. Lateral movements were measured by rod gauges. In addition there were piezometers in the foundation and in the fill. The ULC is the dominant foundation material at the Empingham Dam site; however, in general this is overlain by younger deposits of Northampton Sands and Estuarine Series and by more recent deposits of head and alluvium of variable thickness. These overlying materials were excavated before placement of the fill started. The pore water pressures in the ULC were found to be hydrostatic, with the ground water table close to the surface. A value of the coefficient of earth pressure at rest of Ko = 1 was deduced on the basis of suction measurements (Bridle at al. 1985). The permeability of the ULC was very low and the permeability of the fill derived from it was even lower; hence the overall dam response during construction was undrained. Figure 3 summarises a number of undrained strength measurements made in the ULC foundation. Results of conventional quick undrained triaxial tests on 100 mm samples showed great scatter. Quick plate loading tests showed less scatter but on average they both predicted similar undrained strength. Slow triaxial tests with pore pressure measurements were also carried out (Maguire 1976); the undrained strength was shown to be dependent on sample size. Figure 3 also shows the undrained strength profile assumed in design.

3 3.1

CONSTITUTIVE MODELS Foundation

The ULC foundation was modelled with the new constitutive model. This is an extension of an existing two-surface kinematic hardening model

Figure 3. Undrained strength profiles in the ULC foundation.

Figure 4. Representation of bounding and kinematic yield surfaces in triaxial stress space.

(Grammatikopoulou 2004, Grammatikopoulou et al. 2006). The latter is a version of the two-surface “bubble” model by Al-Tabbaa and Wood (1989). It employs a single kinematic yield surface, within the modified Cam Clay bounding surface, Figure 4. The kinematic yield surface encloses the region within which the behaviour is assumed to be elastic. The model has been generalized and implemented into the finite element code ICFEP (Grammatikopoulou 2004, Grammatikopoulou et al. 2006).A number of improvements have been made including a variety of shapes of

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the yield and plastic potential surfaces. The equation of the bounding surface in general stress space is:

The equation of the kinematic yield surface is:

where p is the mean effective stress, s is the deviatoric stress tensor (tensor quantities are represented by bold type), po defines the size of the bounding surface, pa and sa are the mean effective stress and the deviatoric stress tensor at the centre of the kinematic surface respectively, R is the ratio of the size of the kinematic surface to that of the bounding surface. The functions g(θ b ) and g(θ y ) define the shape of the bounding and kinematic surfaces in the deviatoric plane where θ b and θ y are the values of the Lode‘s angle for the bounding and kinematic surfaces respectively. In the analyses presented in this paper a Mohr-Coulomb hexagon was adopted as the deviatoric shape of the yield surface:

where φ is the critical state angle of shearing resistance. The plastic potential surface was assumed to have a circular shape. Moreover, the model requires the slopes of the isotropic compression line, λ∗ , and the elastic part of the swelling line, κ∗ , which are assumed to be straight in ln v-lv p space. In the analyses presented in this paper a constant Poisson’s ratio, µ, is assumed. The model is an improvement over the model used in the back-analyses of Kovacevic et al. (2007a, b) in that it employs a modified hardening modulus which ensures a smooth elasto-plastic transition (Grammatikopoulou et al. 2006). The parameters required by this version of the model are: λ∗ , κ∗ , µ, φ , R and α (parameter in the modified hardening modulus). An additional parameter, N , is also usually adopted in order to fix the model in ln ν-ln p space, taken here as the specific volume of the isotropic compression line at p = 1 kPa. In the new extension of the model, softening is achieved by allowing the critical state angle of shearing resistance, φ , to vary with the plastic deviatoric p strain, Ed , from the peak value, φ p , to the residual value, φ r , as shown in Figure 5, following the ideas put forward by Potts et al. (1990).The deviatoric plastic p strain invariant Ed is defined as follows:

p

where es is the plastic deviatoric strain tensor. In this extension of the model the peak angle of shearing resistance, φ p , replaces the angle of shearing resistance, φ ,

Figure 5. Variation of angle of shearing resistance with the p deviatoric plastic strain invariant, Ed .

as an input parameter and the model requires the following additional parameters: φ r , the residual angle of p shearing resistance, E d p , the plastic deviatoric strain at p peak and E d r , the plastic deviatoric strain at residual.

3.2 Fill For the clay fill the same two-surface kinematic hardening model adopted in the back-analyses of Kovacevic et al. (2007a, b) is employed. Kovacevic et al. (2007a, b) showed that the adopted model and parameters simulated successfully the behaviour of the fill. No softening was modeled for the fill. This model requires the same parameters as the model used for the foundation, without the softening part and with the exception that the parameter ψ replaces the parameter α in the hardening modulus (Grammatikopoulou et al. 2006). The partially saturated nature of the ULC fill was modelled by setting the compressibility of the pore fluid as a multiple of the soil skeleton compressibility using a coefficient β. This coefficient was chosen to give similar pore water pressure response and volume changes to those obtained from both laboratory and field measurements (Kovacevic et al. 2007a, b).

4

MODEL PARAMETERS

4.1 Foundation Kovacevic et al. (2007a) recovered the observed behaviour of the north section of the main embankment by using a Mohr-Coulomb strain softening elasto-plastic model, to simulate the behaviour of the ULC foundation, in which the peak undrained strength was reduced and a stiffer pre-peak response, as compared to the available laboratory data, was adopted. The peak undrained strength profile adopted in the Mohr-Coulomb model can be seen in Figure 3. Figure 6 shows the stress-strain curve and pore pressure response predicted by the same model in undrained triaxial compression. The peak and residual angles of shearing resistance were assumed to be

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Table 1. Parameters for kinematic hardening ‘bubble’ models for the Upper Lias Clay foundation and the fill derived from it. Foundation Parameter γ (kN/m3 ) N λ∗ κ∗ µ R α/ψ(1) φ p (◦ ) φ r (◦ ) p E d p (%) p E d r (%) OCR Ko

Strain softening 20.5 2.35 0.06 0.0022 0.2 0.85 6.0 21.0 13.0 1.0 5.0

n/a n/a n/a 1.85 1.0

Fill 20.0 5.18 0.16 0.03 0.3 0.5 0.75 20.0 n/a n/a n/a 3.5(2) 2.5(3)

(1) α in the model used for the foundation / ψ in the model used for the fill (2) defined in of mean effective stress (3) as constructed.

Figure 6. Predicted behaviour of ULC foundation in undrained triaxial compression.

equal to 21◦ and 13◦ respectively. Moreover, the plastic deviatoric strain at peak and residual were assumed to be equal to 1% and 5% respectively. In the new kinematic hardening model the parameters γ, N , λ∗ , µ, R and Ko were taken to be the same as in the analyses of Kovacevic et al. (2007a, b). The parameter κ∗ was reduced in order to achieve a stiffer pre-peak response similar to the one adopted in the Mohr-Coulomb model and a value of κ∗ = 0.0022 was assumed. In the analysis which did not model strain softening, φ was assumed to be equal to 21◦ . In the analysis which modelled strain-softening the peak and residual angles of shearing resistance, φ p and φ r , and the plastic deviatoric strain at peak and p p residual, E d p and E d r were taken to be the same as for the Mohr-Coulomb strain softening model. Finally, the OCR (which defines the size of the bounding surface) was reduced in order to model an undrained strength lower than the design strength and close to the strength predicted by the Mohr-Coulomb model. A value equal to 1.85 was adopted. The undrained strength predicted by the kinematic hardening model is plotted in Figure 3; this is the peak strength in the case where strain-softening is modelled. It should be noted that the assumption of the Mohr-Coulomb hexagon as the deviatoric shape of the yield surface and the circle in the case of the plastic potential results in the plane strain undrained strength in compression (or extension) being lower than the undrained strength in triaxial compression. The predicted stress-strain curve and pore pressure response in undrained triaxial compression are plotted in Figure 6. Two predictions are shown – one in which there is no softening and one with softening. The parameters adopted in the model for both cases are summarized in Table 1.

No strain softening

Figure 7. Finite element mesh of north section of main dam.

4.2 Fill The parameters for the clay fill are the same as those adopted by Kovacevic et al (2007a, b). These are also summarized in Table 1. 5


The analyses were carried out using the finite element code ICFEP (Potts and Zdravkovic, 1999). Figure 7 shows the finite element mesh adopted in the analyses. The analyses were plane strain and used eight noded isoparametric quadrilateral elements with 2 × 2 integration. A modified Newton-Raphson scheme, with an error controlled sub-stepping algorithm, was used as the non-linear solver. As mentioned before the permeability of the ULC foundation was low. Hence, this was modelled as undrained by setting a large value of the compressibility of the pore fluid. The partially saturated nature of the ULC fill was modelled as mentioned before, by adopting a value of β = 0.5. Appropriate values of suction were specified in the newly constructed elements, as described by Potts and Zdravkovic (2001). In this way it was possible to control the undrained strength of the clay fill material in the effective stress analyses.

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The analyses initiated with the excavation of the saturated ‘head’ material of a variable thickness (3.5 m to 9.0 m) and a shallow trench for the inclined clay core. The fill operations were then modelled with more placing of fill in the downstream berm. Subsequently, the fill was raised uniformly in layers. Fill layers were of variable thickness, the lower layers being thicker (3 to 4 m) than the upper ones (2 m). As mentioned before, two analyses were carried out, one in which no softening was modelled and one in which softening behaviour was simulated in the ULC foundation. In both analyses the kinematic surface was assumed to be centred around the current stress state at the beginning of the analysis. Modelling of the strain softening is not easy. A procedure similar to that employed by Potts et al. (1990) was used in the FE analyses reported herein. 6

RESULTS OF ANALYSES

Figure 8 shows the development of the horizontal displacement at points A and B (shown in Figure 2), as the fill of the main embankment is placed, together with the predictions of the two analyses. The analysis modelling strain softening predicts the observed increase in the horizontal displacements as the crest of the dam (last 2 m of fill) is placed, whereas the analysis modelling no strain-softening cannot capture this abruptness of movement. Figure 9 plots the observed and predicted horizontal displacements during construction of the main embankment for the strain-softening analysis. It can be seen that the analysis predicts well the sudden acceleration of movement as the fill approaches the crest level. Figure 10 plots the incremental displacement vectors predicted by the two analyses due to the placement of the last layer of fill (2 m thick). These vectors show the direction of movement. Their absolute magnitude is not important as it is their relative magnitude which indicates the current mechanism of behaviour. Figure 10 suggests that the strain softening analysis shows a different mechanism than the analysis with no strain softening, as the top of the dam is placed, with the former indicating an increase in the horizontal movements, which is not indicated in the latter. Figure 11 plots the predicted contours of plastic shear strain, Ed p , for the same stage of the analyses. Figure 11b indicates that there is some strain softening at the top of the ULC foundation. Hence, the sudden increase in the horizontal movements seems to be a consequence of the low undrained shear strength at the top of the ULC foundation combined with strainsoftening. 7

CONCLUSIONS

A new kinematic hardening model has been developed which s for pre-peak plasticity and post-peak strain softening. The model has been used in the backanalysis of the north section of the main embankment

Figure 8. Observed and predicted horizontal movement of the north section of the main dam (a) rod gauge point A (b) rod gauge point B.

at Empingham. The back-analysis shows that it is only possible to capture the observed sudden increase in horizontal movements of the embankment, as the fill placement approached the crest of the dam, by modelling strain-softening. This sudden increase seems to be a consequence of the low undrained shear strength at the top of the ULC foundation in combination with strain-softening. It should be noted that similar results were obtained by Kovacevic et al. (2007a) by using a Mohr-Coulomb

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strain softening model to characterize the behaviour of the ULC foundation. However, Kovacevic et al. (2007a) had to switch from a kinematic hardening model to a Mohr-Coulomb strain-softening model, as the kinematic hardening model that was available when they carried out their analyses did not model strain-softening. This paper shows that the new kinematic hardening model, which can simulate strain softening, is capable of reproducing the observed undrained behaviour of the north section of the main embankment at Empingham. REFERENCES Figure 9. Observed and predicted horizontal movements during construction of main dam (north section) – ‘bubble’ strain-softening analysis.

Figure 10. Predicted incremental displacement vectors at north section of main dam due to the last 2 m of fill placing (a) analysis with no-strain softening (b) analysis with strain softening.

Al-Tabbaa, A. & Wood, D.M. 1989. An experimentally based “bubble” model for clay. Int. Conf. Num. Models Geomech., NUMOG III, Edt.A. Pietruszczak G. N. Pande: 91–99. Bridle, R.C., Vaughan, P.R. & Jones, H.N. 1985. Empingham Dam: Design, construction and performance. Proc. Inst. Civ. Engrs, London, Part 1 (78): 247–289. Grammatikopoulou, A. 2004. Development, implementation and application of kinematic hardening models for overconsolidated clays. PhD thesis, Imperial College, London, UK. Grammatikopoulou, A., Zdravkovic, L. & Potts, D.M. 2006. General formulation of two kinematic hardening constitutive models with a smooth elasto-plastic transition. International Journal of Geomechanics, ASCE, 6(5): 291–310. Horswill, P. & Horton,A. 1976. The valley of the River Gwash with special reference to cambering and valley bulging. Phil Trans. Royal Soc., London, 283: 451–461. Kovacevic, N., Higgins, K.G., Potts, D.M. & Vaughan, P.R. 2007a. Undrained behaviour of brecciated Upper Lias Clay at Empingham Dam. Geotechnique 57(2): 181–195. Kovacevic, N., Higgins, K.G., Potts, D.M. 2007b. Finite element back-analysis of trial bank at Empingham Dam. Int. Conf. Num. Models Geomech., NUMOG X, Edt. G. N. Pande & A. Pietruszczak: 587–593. Maguire, W. M. 2004. The undrained strength and stressstrain behaviour of Upper Lias Clay. PhD thesis, University of London, UK. Potts, D.M., Dounias G.T. & Vaughan, P.R. 1990. Finite element analysis of progressive failure of Carsington embankment. Geotechnique 40(1): 79–101. Potts, D.M. & Zdravkovic L. 1999. Finite element analysis in geotechnical engineering: Theory. London: Thomas Telford. Potts, D.M. & Zdravkovic L. 2001. Finite element analysis in geotechnical engineering: Application. London: Thomas Telford.

Figure 11. Predicted contours of plastic shear strain, Ed p , for north section of main dam due to the last 2 m of fill placing (a) analysis with no-strain softening (b) analysis with strain softening.

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Forecasting of the stability of the tailing dam in permafrost region on the basis of numerical methods A.B. Lolaev, A.P. Akopov, A.Kh. Oganesian & M.N. Sumin North Caucasian Institute of Mining and Metallurgy (State Technological University), Vladikavkaz, Russia

V.V. Butygin Norilsk Institute of Industry, Norilsk, Russia

ABSTRACT: A complex of numerical methods was used for predicting the stability of the tailing dam in Norilsk industrial region (Northern Siberia). The carried out calculations include the calculation of the temperature conditions of the tailing dam on the base of UWay FEM package, the calculation of the stability of the tailing dam on the base of UniFos FEM package and of the fuzzy modelling. The forecast of geoecological processes on the basis of described methods will allow to increase reliability of the accepted design decisions.

1

INTRODUCTION

the construction and the bottom back, also the heat conditions of the soils (Lolaev, 2006).

Owing to the high density of the industrial enterprises of the mining and non-ferrous metallurgy industries as well as the growing amounts of waste products in Norilsk industrial region (Northern Siberia) there the changes in physical and mechanical properties of the frozen soils that cause the infringement of the reliability of the foundations. That is why the stability problem of engineering constructions in the area of cryolitozone becomes more and more urgent. Each hydrotechnical construction including tailing dams is unique if not in design, then in method of building and exploitation. That is why its safety must be ensured not by standard observations but by investigations including scientific researches and original methods of observations as well as searching of new technological processes and securing ecological safety of constructions. For this reason it is difficult to produce the adequate theoretical model for the forecasting of the thermic state of a dam body and of frozen basis of dam. The main objective of this paper is to develop special technique of forecasting of the stability of a dam. This technique includes the theory of fuzzy sets and the direct calculations using the licensed software. The main factors determining the stability of dam were revealed for fuzzy modelling, including lithology and physicomechanical characteristics of the dam composed soils, specifically their strength characteristics: angle of internal friction and the cohesion; the water presence in the dam body characterized by the depression curve; as well as the constructive parameters of the dam the main of which are the height of

2

MODELLING THE STABILITY OF THE TAILING DAM

For forecasting the stability of a tailing dam the prognostic model based on the theory of fuzzy models of logic and linguistic ones in particular (Lolaev, 2006) was adopted. The algorithm of construction of fuzzy model with use of linguistic variables looks as follows: 1. Definition of factor’s space of the investigated phenomenon; 2. Delimitation of opposite scale and under each factor; 3. Preparation of a matrix of interrogation; 4. Coding of the factors – LV (translation in to the metrics); 5. of polynomial factors using the method of the least squares;

6. Estimation of a mistake of numerical experiment; 7. Estimation of the importance of polynomial factors; 8. Estimation of polynomial adequacy to an expert estimation of the investigated phenomenon; 9. Estimation of adequacy of the received model to the investigated phenomenon; The algorithm of construction of adequate forecasting model is submitted in Figure 1.

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The coding of the factors consists in transformation of the names of scale in the metrics. The of polynomial factors consists in translation of dependent factors (Y) in a precise scale and performance of traditional actions with a matrix of interrogation accepted in the theory of experiment planning. The estimation of a mistake of numerical experiment originally consists in comparison of meaning of the polynomial’s free member b0 with an estimation of opinion of the expert at the centre of planning of factor’s space of the investigated phenomenon. The estimation of the importance of polynomial’s coefficients consists in exception of those coefficients, which meaning are lower than a mistake of their definition on t – Student’s criterion. The estimation of polynomial’s adequacy with important coefficients to an expert estimation of the investigated phenomenon consists in comparison of calculated polynomial factors and expert estimation among themselves (and also on algorithm “necessity – opportunity” – NEC – POS ); The estimation of adequacy of the received model to the investigated phenomenon consists in comparison of calculated and experimental data by traditional statistical methods. Six input linguistic variables were chosen describing investigated phenomenon in the most complete form. In a fuzzy kind variables are coded as follows: X1 is the angle of slope and determined as:

where x1 – means the angle of slope in degrees. X2 is the height of dam and determined as:

Figure 1. Algorithm of construction of forecasting model.

As the explanatory to algorithm of forecasting model construction we shall note, that the definition of factor’s space includes: •

definition of the maximum number of the influencing factors; • allocation of the essentially influencing factors; • choice of linear – independent and controlled factors. The delimitation of opposition scale includes: •

definition of quantitative estimation of the bottom and top borders of the chosen factor; • definition of quantity of divisions (term – sets) of splitting of a scale and their names; • definition of a degree of illegibility of concept. The preparation of a matrix of interrogation of the expert is carried out according to methods of the theory of experiment planning.

where x2 – means the height of a dam in meters. X3 is the granule composition of tails and determined as: X3 = −1 for clay fraction, X3 = 0 for mixed fraction, X3 = +1 – for sand fraction. X4 is determined as the thermal condition of the soils in dam’s body. X4 = −1 for thawed condition, X4 = 0 for semi frozen, X4 = +1 – for frozen condition. X5 is determined as the characteristic of underground water pressure. X5 = −1 for no pressure condition, X5 = 0 for gydrostatic pressure, X5 = +1 – for piezometric pressure. X6 is the width of dam’s beach and determined as:

where x6 – is the width of dam’s beach. Y – is the value of the stability coefficient of the dam’s body. The variables are shown in Figure 2.

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The resulting equation in a coded kind was found as:

Only substantial coefficients are presented in a equation (valuation of an error has made 0.1 at a level of the substantiality of 0.05). It should be noted, that the equation (4) has obviously a non-linear character. Even the threefold interactions having a physical sense proved to be significant. For example: X2 *X3 *X4 – it is possibly interpreted as the characteristic of complex influence of the height of dam, granule composition of tails and temperature condition of dam on its stability coefficient; X3 *X4 *X5 – is a parameter of complex influence of granule composition of tails, temperature condition of dam and type of underground water pressure on its stability coefficient. Cumulative action of the several factors (the double and threefold interactions) is commensurable on a degree of influence on the stability coefficient of the dam with the linear factors, and some even sur them. The submitted method permits to compare the degree of the influence of the various factors on the stability coefficient of the dam by the value of the coefficients of variables. Geometric characteristics X1 and X2 have the strongest effect on stability coefficient of the dam. It is important on the stage of acception of design decision. But then they have constant meaning in the equation (4). Factor X3 (granule composition of tails) depends upon construction parameters. Then factor X3 become constant too. Another factors X4 , X5 and X6 depend upon the exploitation conditions of the disposal area which can often change. In this case, the submitted method permits to forecast the influence of the changing factor on the stability coefficient of the dam. The forecast of the dam stability was done on the base of submitted method and equation (4). 3

CASE HISTORY

The climatic characteristics of Norilsk industrial region are: • • • • • •

Figure 2. Characteristics of tailing dam.

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average annual temperature of the air is −9.4◦ C; the maximum temperature of the air is +32◦ C and the minimum temperature is −56◦ C; the maximum speed of wind is 40 m/sec; winds with speed above 15 m/sec are observed during 90 days; the strong winds and snowfalls are observed up to 130 days a year; average amount of precipitations is 564.5 mm per year.

Figure 3. The situation plan of the tailing dump. 1. Tailing dump 1; 2. The pond; 3. Local tailing dam; 4. Spillway channel; 5. Magnetic pyrite depository 2.

Object of research of the present work is the tailing dump 1, located on the territory of Norilsk industrial region. The basic hydraulic engineering structures include: • • •

pond for reception of pulp and storage of tails local tailing dam; system of pipelines for turnaround water supply with coastal and floating pump stations • spillway system The constructive characteristics of dam are: 1. 2. 3. 4. 5. 6. 7.

the disposal area – 715 000 sq. ms the length of the tailing dam – 4.41 kms the height of dam – 56.7 m; the inclination of a top drain level – 1:20–1:30 the inclination of a bottom slope – 1: 3 the maximal depth of pool – 4.0 m the average depth of pool – 0.4 m.

The basis of pool and dam is layered by artificial, alluvial and moraine soils with gravel and pebble with sand, sandy loam and loam additions of 5–60 m capacity being in a frozen state. The underlaying layer is heterogenous rock. The situation plan of the tailing dump is shown in Figure 3. Problems of stability of the structure are connected with the fact that the levee of the tailing dump was erected by principle I, with application of freezing columns as the basic method of protection of a structure. During operation and preservation of the damp the freezing system has failed, and maintenance of the stability of object now consists in the ing of

flood waters and maintenance of a safe water level in pond zone. Thus during the summer-autumnal period on downstream side of a dam egresses and separate earthflows are observed. In the tailing dump 1 the tails of floatation ore-dressing of Norilsk concentration plant had been stored from 1948 to 1975. After commissioning of the tailing dump 2 in 1976, the dump 1 had been used from 1976 to 1987 as a backwater basin. Since 1987 the tailing dump has served as a construction for reception of drain, regulation and ing of flood waters and for accumulation of surface-water flow arriving on the water – collecting area of the tailing dump in the form of atmospheric precipitation. Now on the tailing dump 1 the hydromechanized lifting and reprocessing of stale tails with extraction of non-ferrous metals and precious metals is done. In the magnetic pyrite depository 2 which is located in the basis of a downstream side of the tailing dump 1 works on lifting of magnetic pyrite concentrate with the application of means of hydromechanization are being done. Condition of the local dam (3) of the tailing dump 1 (1) in the place where it ads the magnetic pyrite depository is unsatisfactory. Lifting of stale concentrate and tails in a hydromechanized way can entail destruction of the local dam (3), filling of the workedout area of the magnetic pyrite depository 2 (5) and loss of valuable raw materials.

4 TEST RESULTS AND DISCUSSION The carried out engineering-geological researches have shown complexity of a structure of a body of the tailing dam: the combination of thawed and frozen lays, presence of over-, inter- and underglacial waters, that has an effect on stability of the structure. Mathematical modelling of various schemes of working-off of stale tails and magnetic pyrite concentrate, with the forecast of stability of frame fillings, was done for the development of effective technique. Drilling and maintenance of inspection equipment, measuring of temperatures and water levels in a dam body were executed. Exploration drilling with full testing of core samples was done on the territory of the magnetic pyrite depository 2. At forecasting of stability of a levee the complex of programs was used. Program complex UWay FEM package (Vlasov et al. 2003) has been applied to temperature condition calculation on the basis of finite elements. Forecasting of stability of a levee is executed by means of program UniFos and on the basis of fuzzy sets method . The program complex UWay permits to forecast of the stress-strained state of soil and rocky massifs, and also to forecast of change of temperature and hydrological modes of soil massif and definition of the effective characteristics of composite materials, layered and ty massifs of rocks (Vlasov et al. 2003).

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Figure 4. Results of calculations of the temperature conditions. a) in winter time; b) in summer time.

Program UWay allows to reflect in calculations: •

changes of geometry of settlement area (for example, the registration of phases of buildings); • changes of properties of materials; • changes of boundary conditions (both power, and kinematic); • operation of materials according to various models of behaviour of soils under load At the first stage the calculations of the temperature conditions in tailing dam’s massif for winter and summer periods were made. The results of calculations are shown in Figure 4. The given results were used for the appointment of physical and mechanical characteristics of soils composing the dam’s body and has formed the basis of the calculations of dam’s coefficient stability. The calculation were carried out by means of program UniFos. The program UniFos is a part of UWay complex and is intended for calculations of stability of soil constructions. It is written in object-oriented language C++ with usage of optimising compiler Borland C++ Borland International v.5.02 with library OWL usage v.5.0 (Vlasov et al. 2003). At the next stage the check of adequacy of an equation (4) was carried out by comparison with the results of the numeric calculations with the software package “UniFos”. Calculations were made for the next parameters of tailing dam in equation (4): • • • • • •

angle of slope – 15◦ ; the height of dam – 40 m; granule composition of tails – sand fraction; thermal condition of dam – frozen; type of underground water pressure – no pressure; the width of dam’s beach – 100 m.

Figure 5. Results of calculations of the stability coefficient. a) in winter time prior to the beginning of excavating; b) in winter time on termination of excavating; c) in summer time on termination of excavating at flooding by water; d) the same without flooding by water.

Numeric calculations were made for the next parameters: • • •

angle of internal friction ϕ = 20◦ ; cohesion C = 0 mPa; density ρ = 16 kN/m3

The results of comparison show their good convergence: coefficient of stability Kst = 1.48 calculated on equation (4) and Kst = 1.20 on numeric calculations. At the last stage the calculations of the stability coefficient of the tailing dam under various conditions were made. The calculation were carried out for four conditions: a) in winter time before excavating works; b) in winter time on termination of excavating; c) in summer time on termination of excavating works at flooding by water; d) in summer time without flooding by water. The results of calculations of the dam’s stability coefficient are shown in Figure 5. The analysis of results of mathematical modelling shows (Fig. 5) that when excavating of magnetic pyrite concentrate by earlier suggested technology stability of a dam is not provided during the summer period

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(stability coefficient is less than 1). The given conclusion has formed the basis for alteration of work technique that has provided ,in its turn, safe and effective execution phase. 5

CONCLUSIONS

Program complex UWay FEM package has been applied to the dam’s temperature condition calculation on the basis of finite elements. Forecasting of stability of a levee is executed by means of program UniFos and on the basis of fuzzy sets method. The forecast of the dam’s stability on the basis of the described method and received equation (4) has shown their high adequacy. Carried out calculations permit to forecast the stability of tailing dam and has formed the basis for

alteration of work technique that has provided ,in its turn, safe and effective execution phase. REFERENCES Lolaev, A.B. 2006. Fuzzy modelling in the geoecological forecasting in cryolitic zone. In works of North Caucasian Institute of Mining & Metallurgy (State Technological University. Vladikavkaz, Russia (in Russian). Vlasov, A.N., Yanovsky, Yu.G., Mnushkin, M.G., Popov, A.A. 2003. Solving geomechanical problems with UWay FEM package. In Proceedings EPMESC’IX. International Conference on Enhancement and Promotion of Computational Methods in Engineering and Science. Macao. Lolaev, A.B., Akopov A.P., Oganesian A.Kh. & Sumin M.N. 2009. Forecasting of the stability of the tailing dam in permafrost region. In Proceedings of 5-th International Conference “Global Scientific Potential”, Tambov, Russia (in Russian).

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Numerical modeling of the mechanical response of recycled materials in embankments M.M. Villani, X. Liu & A. Scarpas Section of Structural Mechanics, Faculty of Civil Engineering and Geosciences, Delft University of Technology, The Netherlands

A. D’Andrea Department of Hydraulic, Transport and Roads, Sapienza University of Rome, Italy

ABSTRACT: Every year, in Europe, 3 billion tonnes of aggregates are required for construction applications and, at the same time, 1.3 billion tonnes of waste are disposed of. The idea of using some of them as new alternative materials could be economically beneficial but several studies have to be done in order to guarantee their bearing capacity and durability. In this study three waste products, clay and lime, pozzolan, sludge and lime and construction and demolition wastes, were selected in order to use them as new alternative materials for embankment construction. On the basis of the information obtained from laboratory and in situ tests, the embankment was simulated by means of the finite element method. Comparisons have been made between in situ and modeled results by considering material stress dependence and surface deflection profiles. Modeling with the finite element system CAPA-3D has shown good agreement with in situ response. The results indicate that stabilized materials can be characterized by a hyperelastic assumption while unbound materials can be appropriately modeled by using a Mr-θ model.

1

INTRODUCTIONS

Natural materials for construction applications are a limited and expensive resource. For this reason, European laws allow to recycle materials in which the mechanical and chemical compositions conform to predefined specifications. The goal of this investigation is to find a good compromise between the mechanical characteristics of the recycled materials and the total expense for recycling. In the framework of this project various materials were studied for their potential to be used for recycling purposes. They were clay (by-product of excavation activities) to which lime was added (AC), pozzolan (by-product of tunnelling activities), sludge (provided by an aggregate industry) and lime (3P30), construction and demolition waste (C&D) Because the mechanical response characteristics of these materials depend and vary with the selected percentage of water and stabilizer, extensive mechanical tests were performed. The test results were compared with those of standard aggregate material (MG). Various geotechnical characteristics tests, CBR, Uniaxial Compressive Strength and Resilient Modulus were performed. In order to know also the in situ response of the material an embankment was created in Nepi (VT) by the Italian Railway Association during a project that investigates the mechanical behavior of recycled material. During the construction of the embankment,

load cells were installed inside the embankment and different in-situ tests such as static plate load, FWD and LWD were considered. In the last phase of these studies, comparisons between Finite Element Modeling and in situ test results have been done for the different recycled materials. The analyses provided valuable insight into the various mechanisms and phenomena controlling the response of the embankment. 2

LAB MATERIAL CHARACTERIZATION

2.1 Introduction In order to obtain information of the geotechnical and mechanical behavior of the materials in relation to water content and load application, an extensive laboratory testing was performed for clay and lime and for pozzolan, sludge and lime (Villani 2009). The main results are presented in the following paragraph. 2.2 Index properties Index properties tests were performed according to American Society for Testing and Materials (ASTM) test procedures listed below:

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• •

Water content – ASTM D 2216-80 Liquid Limit and Plastic Limit and Plasticity Index – ASTM D 4318-84

Table 1.

Clay Pozzolan Sludge Lime C&D MG

Table 2. rials.

Geotechnical characteristics of the mixtures. w (%)

γs (kN/m3 )

wl (%)

Ip (%)

AASHO Class

24 24 68 – 10 9

27 25 27 22 26 26

44 – 43 – – –

27 – 20 – – –

A7-6 A4 A7-6 – A1-b A1-a

Mechanical characteristics of mixtures and mate-

Material

Maturation time (day)

UCS (daN/cm2)

CBR (%)

Clay 3P30 C&D MG

0 7 7 0

4.5 18 N.A. N.A.

37 147 62 108

• •

Figure 1. Geometrical characteristics of the embankment.

Particle Size Analysis – ASTM D 422-63 Specific Gravity – ASTM 854

2.3

Mechanical characteristics

2.3.1 CBR and Uniaxial compressive strength In order to obtain information about the bearing capacity of the embankment different tests (Californian Bearing Ratio and Uniaxial Compressive Strength) for different maturation time (7 days for stabilized materials) have been done. 2.3.2 Resilient modulus The resilient modulus is based on the conventional triaxial compression test in which the stress state due to the wheel load is reproduced by cyclic loading. It is expressed by the ratio between the maximum deviatoric stress and the elastic component of deformation recovered. Different relations between the resilient modulus and the first invariant stress have been reported in literature. In this study the following Mr − θ (Hicks & Monismith 1972) was considered:

where θ is the first invariant stress and k1 and k2 are regression coefficients. 3

IN SITU MECHANICAL CHARACTERIZATION

The total length of the entire embankment is eighty meters (twenty meters for each material). The dimensions of the entire embankment are reported in Figure 1.

Figure 2. Position of the pressure cells into the embankment and transversal section.

Considering the pressure cell it is possible to study the stress-depth relationship under low load (345 kPa, static plate load test) and high load application (truck age 700 kPa). For the purpose of this paper the results of load cell located at 72 cm are discussed. Figure 2 indicates the configuration adopted in order not to create interference between different load cells. The Static Plate Load Test has been done according to AASHTO T 222 and ASTM D 1196: Non repetitive Static Plate Load of Soils and Flexible Pavement Components, for Use in Evaluation and Design of Airport and Highway Pavements. From the results presented in Table 3–6 the relation between bearing capacity during a first (Md) and a second (Md ) load application and maturation time. Results are not smooth because they are affected by many factors such as weather conditions or heterogeneity of the material. Other used instruments are Falling Weight Deflectometer (Dynatest 8000) and Light Weight Deflectometer (Dynatest Light Falling Weight Deflectometer 3031). It is clearly shown that FWD and LWD moduli are usually higher than static plate load tests.

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Table 3. lime).

Deformation modulus – maturation time (clay and

Maturation time (day) Clay and lime

0

15

166

205

Md (N/mm2 ) Md (N/mm2 ) Md/Md

80 195 0.41

127 358 0.35

132 273 0.49

145 290 0.5

Table 4.

Deformation modulus – maturation time (3P30). Maturation time (day)

3P30

0

15

166

205


47 181 0.26

66 236 0.63

600 698 0.86

145 290 0.5

Table 5.

Deformation modulus – maturation time (C&D).

Figure 3. Dimension of the embankment finite element mesh.

Maturation time (day) C&D

0

15

166

205


44 191 0.23

66 264 0.25

132 391 0.34

138 300 0.46

Table 6.

4

4.1 Introduction

Deformation modulus – maturation time (MG). Maturation time (day)

M&G

0

15

166

205


94 224 0.42

114 335 0.34

170 450 0.38

205 360 0.57

Table 7.

Comparison between different in situ tests.

Material

E SPL (MPa)

E FWD (MPa)

E LWD (MPa)

Clay and lime 3P30 C&D MG Subgrade

159 454 126 139 30

388 1122 188 236 –

287 923 137 161 –

NUMERICAL MODELLING

The results show that the predicted modulus values depend on the type of the instruments utilized. Also, the bearing capacity of the stabilized material is strongly dependent on the load position indicating thus the in homogeneity of the material.

Different numerical techniques can be used in order to model the mechanical response of the embankment. At the beginning of this study, a multilayer elastic analysis was performed by means of the following programs: BISAR, KENLAYER (Huang Y. H., 1993) and mePADS. Among these programs, the results of mePADS were closer to the measured response. The use of Finite Element Analysis instead of Multilayer Elastic Analysis has several advantages since different types of load areas and constitutive models can be taken into and since different types of boundary conditions can be specified instead of only infinite boundaries. The finite element system CAPA-3D (Scarpas & Liu 2000) was utilized to perform the numerical simulations. The system is capable of solving large scale models like those typically encountered in pavement engineering and to consider various soil conditions such as fully or partially saturated condition under static and/or dynamic load conditions. The finite element mesh of the embankment and the loading area are inserted in Figure 3. The position of the load doesn’t allow considering a double symmetric geometry, for this reason the slope of the embankment has also been taken into . Due to the half symmetry, proper boundary conditions had to be introduced in the analyses. Constraints have been defined for each surface of the structure (in order to simulate far field confinement) except for the surface on top.

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Table 8. Comparison between modulus obtained from static plate load or using FEA.

Table 10. Comparison between stresses obtained from load cells and using FEA.

Material

δ (mm)

E FEM (MPa)

E SPL (MPa)

σv (kPa)

Material

σv measured (kPa)

σv CAPA3D (kPa)

Clay and lime 3P30 C&D MG

0.57 0.2 0.72 0.65

160 550 110 125

159 454 126 139

9 6 12 11


1 1 6 10

9 6 12 11

Table 9.

Stresses and stiffness in wet condition.

Material

δ (mm)

σv (kPa)

E SPL (MPa)

Bearing capacity loss (%)


0.67 0.26 0.83 0.67

12 7 14 12

128 425 93 125

20 23 15 0

The goal of the finite element modeling was the development of an analysis methodology for embankments built with recycled materials. For this reason, in the framework of this research different materials were compared with the predictions of measured in situ response for the purpose of identifying the model that gave best prediction. According to the available data two different approaches have been taken into ; in the first a hyperelastic neo-Hookean constitutive model was used while in the second, the model of Eq. (1). In the following section results from both models are compared. The static plate load and its elastic modulus were considered in order to obtain the relation between stress and depth while truck ages were utilized to determine the variation of stresses in the horizontal direction. 4.2

Stress modelling

Stress modeling was studied considering static plate load applications and truck ages modeling. 4.2.1 Static plate load application During the static plate load application, the load is applied (345 kPa) and deflections are measured. By means of CAPA-3D, the displacement measured from the Static Plate Load tests were applied on the mesh of Figure 3. On the basis of the finite element results, the modulus of the embankment was computed. As it is shown in Table 8 moduli obtained are very close to those obtained by formulae from literature. It can be concluded that the combination of the static plate load measurements and Finite Element Analyses can be used for the determination of material characteristics. This technique makes also possible to compare stresses and moduli in wet and dry conditions and to show a loss of bearing capacity (Table 9) caused by

Figure 4. Influence of E1/E2 ratio on the stress – depth relation.

bad weather condition. A comparison between in situ stresses and the ones obtained via modelling showed, in some cases, significant differences (Table 10). In case of clay and lime and 3P30, the differences between in situ results and modeling, can be attributed to the local discontinuity generated by the stiffness differences between the embankment material and the sand used in order to fill the hole created during the installation of the load cells. Considering the modulus of the embankment E2 and the modulus of the sand E1 it is possible to study the variation of the vertical stresses with the moduli ratio. Figure 4 shows the influence of the moduli ratio on the generated discontinuity of stresses in the vicinity of the load cell. 4.2.2 Truck age modeling Regarding modeling of truck age, the FEM was also utilized to evaluate the effects of tire pressure, wheel configuration and axle load of a truck on the structural response of the embankment. The truck age was aligned with the position of the load cells (Fig. 2). Load cells ed load transmitted by the truck in relation to time. The results in term of stresses on the first wheel of the track are shown in Fig. 5–7. The pressure under a single wheel (650 kPa) was calculated considering the load of the camion and the load percentage applied to a single wheel. The modulus calculated by the static plate load model was utilised for these analyses. The

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Figure 5. Relation stresses – space at 0.72 m. Figure 8. Relation vertical stress – depth for C&D.

Figure 6. Relation stresses – space at 0.72 m for C&D.

Figure 9. Relation between deflection and distance from the center plate for different FWD tests.

Figure 7. Relation stresses – space at 0.72 m for granular material.

unsymmetrical shape is due to the presence of the tandem wheel. Results regarding 3P30 are not presented because the differences between measured and computed values were even higher due to the high moduli ratio. 4.3 Deflection modeling In the previous chapter it was shown that the hyperelastic model was capable of predicting the variation of the stress. As suggested in literature (Desai 2002) and as it is also obtained from the analysis with the elastic material or with a material model according to Eq. (1) the variation of stress with the dept is not depending on the model used (Figure 8 shows the result obtained for C&D). On the other hand, it has been


also demonstrated that, depending on material type, the choice of material model can have some influence on the computed surface displacement profiles. In order to investigate the influence of material model on the predicted surface deflection, FWD displacements were studied. Finite Element Analyses of the tests showed that in case of the stabilized materials i.e. Clay and Lime and 3P30, the hyperelastic model was adequate in simulating both stresses and surface deflections, Figure 9–10.

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better than the traditional materials but particular attention must be taken to moisture penetration and different components mixing. Data from static plate load tests can be utilized for predicting the mechanical response of the embankment using Finite Element Analysis. In relation to the model that can be used, the response of stabilized materials can be simulated by means of hyperelastic models while, for the accurate simulation of the response of granular material and C&D the effect of lateral confinement must be taken into .


ACKNOWLEDGMENT The authors gratefully acknowledge the of the staff of Road Material Laboratory of Sapienza, University of Rome (N. Fiore, G.P. Rossi, A. Di Curzio), the “Istituto Sperimentale Ferrovie” of Italian Railway Network as well as DIC-Road and Transportation Division (University of Pisa) for soil testing data. REFERENCES


In the case of C&D and granular material the analyses showed that only the model of Eq. (1) was capable of simulating accurately stress (Fig. 11–12). 5

CONCLUSIONS

This paper presents a comprehensive methodology for characterization of recycled materials for road constructions by means of laboratory tests. On the basis of laboratory results appropriate material models can be calibrated and utilized for the prediction of the response of embankments. In this project the laboratory response was compared with actual in situ tests showing that mechanical response of the alternative materials is, for most cases,

Villani M. M. 2009. Modellazione di rilevati costituiti da materiali alternativi, Master thesis, Sapienza University of Rome. D’Andrea A. & Villani M. M. 2008, Recycling of material with high water content, Proc. International conference of solid waste management and construction. (pg. 1327– 1338). Scarpas A. and Liu X. 2002. CAPA-3D Finite Element System-’s Manual, Part I, II and III, Section of Structure Mechanics, Delft University of Technology, Netherlands. Scarpas A. 2005. Mechanics based Computational Platform for Pavement Engineering, PhD thesis, Delft University of Technology. Liu X., 2003, Numerical modelling of porous media response under static and dynamic load, PhD thesis, Delft University of Technology. Huang Y.H. 1993. Pavement Analysis and Design, PrenticeHall International UK, London. Hicks, R.G., and Monismith, C.L. (1971). Factors influencing the resilient modulus of granular materials. Highway Research Record 345: 15–31. Desai C. S. 2002. Mechanistic Pavement Analysis and Design using Unified Material and Computer Models. In A. Scarpas & S.N.Shoukry (eds.), Proc. of the Third International Symposium on 3D Finite Element Analysis, Design and Research, Amsterdam, 2–5 April 2002: 21–53.

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Rail track structural analysis using three-dimensional numerical models A. Paixão & E. Fortunato Laboratório Nacional de Engenharia Civil, Lisbon, Portugal

ABSTRACT: The current competitive growth of transport systems and the rising demands placed on railways have promoted the improvement of systems that comprise the railway infrastructure. From this perspective, the optimization of track design and the reduction of life-cycle costs assume greater relevance. Attaining a better comprehension of the role of the elements that form the structure is essential to achieve such objectives. Within this context, the use of numerical models is a valuable tool for better understanding of the track behaviour and thus to optimize its design. The present study focuses on some aspects of three-dimensional track modelling. Linear and nonlinear constitutive laws are used to reproduce the behaviour of granular material. Stress and deformation results obtained on different layers, are presented and the suitability of different track solutions is discussed. Finally a comparison of the results obtained with different software is also presented.

1

INTRODUCTION

Regarding the current life-cycle costs reduction paradigm for transport infrastructures, namely in what relates to the railway system, optimizing the design of these structures holds greater relevance. In order to contribute to the optimization of the design of structural solutions in respect to the physical and mechanical characteristics of trackbed layers, numerical models can be used for better understanding of the behaviour of the railway track. To this end, the present study makes use of three-dimensional numerical modelling to reproduce the structural behaviour of the track and its subgrade. The developed models aim at simulating the performance of various elements that comprise the railway track when submitted to vertical loadings. The ballasted track is the most widespread superstructure solution for railways. It is a rather complex structure, often showing a nonlinear behaviour when submitted to traffic loads. Many analytical models have been developed to either individually represent each response component associated with such loads or to simultaneously represent the different components. So far, most efforts focused on studying its response to vertical loads. Finite element and finite difference methods are particularly suited to model actions and to make use of nonlinear constitutive relations for some of the materials that comprise the track system. These methods allow the spatial variation of physical and mechanical characteristics, improving resemblance with the real structure. Previous studies regarding the track response due to loads imposed by trains have been carried out with

the help of different models (Tarabji & Thompson 1976, Prause & Kennedy 1977, Adegoke et al. 1979, Sauvage & Larible 1982, Chang et al. 1980, Huang et al. 1984) which have evolved in time in accordance with the available computational capacity. The intensive use of these models has made it possible to carry out parametric studies and to prepare diagrams that describe the relations between relevant variables. Some of these results have been used to develop design procedures for trackbed layers (Li & Selig 1998, ORE 1983 and Ministerio de Fomento 1999). Other authors (Sanguino et al. 1998, Williams & Pérez 1998, Aubry et al. 1999) have modelled the railway track using commercial software, making evidence of the distinctive capabilities. In general, the above studies have considered the linear elastic behaviour of the layers ing the track. Initially, this consideration was due to the limited computational potentialities of both software and hardware available at the time. In face of the developments made on computing performance and on the tools available, some authors suggested the use of more complex models in order to obtain a better estimate of the actual behaviour of these materials (Ministerio de Fomento 1999 and Fortunato 2005). Within this context, this study focuses on several aspects of rail track modelling and on the influence of physical and mechanical characteristics of the trackbed layers. Results are compared and discussed. Different constitutive laws to reproduce the behaviour of granular materials are taken into . A comparison of the results using different commercial software (FLAC3D and ANSYS) is also carried out.

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can be expected. Therefore, in the present study, a conservative approach has been adopted to comprise such scenarios. A value of 300 kN was used as reference for the analyses carried out. Since symmetry conditions were present, a quarter of loading (i.e. 75 kN) was applied on top of the rail, in the transverse vertical plane of symmetry. In the studies performed, the load was applied in two stages. Initially, gravity was activated so that the bulk weight of the materials could be taken into . After achieving equilibrium of the initial stress state, the load was applied on the rail. 2.2 Modelling rail, rail pads and sleepers Figure 1. Example of the track model used as reference.

2 2.1

RAILWAY TRACK MODELLING General aspects

In applications of this kind, the construction of the model usually takes into the symmetry of the problem, allowing the reduction of the domain to a quarter of its total (see Figure 1). As a result, two symmetry planes can be defined: a vertical plane containing the longitudinal axis of the track and a transverse vertical plane where the load is applied. The studied models were approximately 2.5 m long in the rail direction, 3.8 m wide in the transverse direction and showed a constant depth of 4.0 m from the sleepers’ bottom surface. It was decided to keep a constant depth for all the studies performed due to excessive influence of this parameter on the vertical displacements. Horizontal displacements were restricted in the vertical boundary planes and vertical displacements restricted in the lower horizontal boundary. The models were developed with 8-node hexahedral grid with about 10,000 elements/zones and with a total of 12,000 nodes to represent the following track elements: rail, rail pads, sleepers, ballast, subballast, capping layer and subgrade (see Figure 1). In FLAC3D, a mixed discretisation technique is used where each 8-node zone corresponds to an assembly of 2 overlays of five 4-node tetrahedral. In ANSYS, 8-node elements were used. As in similar static problems, only a single axis load was considered, as experience shows that in quasistatic problems, the influence of loading a second axis, at a distance of approximately 3.0 m, introduces very small variations in the results. Regarding the vertical loading, some authors (Alves Ribeiro, et al. 2009) have carried out dynamic analyses to evaluate the wheel-rail interaction forces when high-speed vehicles cross vertical stiffness transition zones. These studies indicate that when crossing abrupt subgrade deformability transitions (i.e. moduli of deformability transition between 50 and 1000 MPa) a dynamic increment of about 45% of the static load

When studying this type of structures, some of the relevant modelling aspects are related to how the superstructure is modelled. When the track is loaded with a single axle, the load distribution between sleepers is strongly dependent on the vertical stiffness of the system. Underestimating the load transferred to the sleeper located right under the axle induces lower stress levels at the sub-grade, leading to a non-conservative approach to the problem. In order to attain a better reproduction of the load distribution between sleepers, special attention needs to be paid to some modelling aspects. Within this context, several models were developed, aiming at evaluating the influence of different modelling assumptions. Regarding the results obtained, some considerations can be made, as follows. To reproduce the rail, a simplified rectangular shape can be adopted with the same width as the rail foot, using brick elements. The rail height and the Young modulus should be determined to obtain the same bending stiffness as the actual rail type being modelled. To assess the influence of such parameters two situations were compared: changing a UIC 60 by a UIC 54 rail (having 30% less moment of inertia) may induce an increase of vertical stress on top of the subgrade by about 4%, due to the axle load. Displacements at the rail may be increased by 5%. Although changing the rail induces stress and deformation variations, in the context of rail engineering, choosing between different rail types aims at dealing with issues outside the scope of this study. In the studies performed, rail type UIC60 was modelled with a parallelepiped shape. The rail was 0.15 m wide and an equivalent height was determined so that a similar moment of inertia could be attained. These elements were linear elastic, with Young’s modulus of 210 GPa and Poisson’s ratio of 0.3. An equivalent density was attributed to the rail elements to obtain the actual rail weight. Regarding the sleepers modelling, it is also important to adequately reproduce the flexural behaviour of these components. To reproduce the B70 monoblock sleepers, brick elements were used. The sleeper’s dimensions were determined to obtain similar plan area and approximate moment of inertia as the B70 sleeper. A Young’s

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modulus of 64 GPa and Poisson’s ratio of 0.25 were chosen to characterise the linear elastic behaviour and an equivalent density was considered to obtain the actual mass of 300 kg per sleeper. Some authors (Ministerio de Fomento 1999) recommend the use of higher-order elements, such as 20-node brick elements, to adequately reproduce the flexural behaviour of track framework components. However, experience shows that 8-node brick elements can also be used, as long as a minimum of 4 layers of elements are considered. Different approaches can be adopted regarding the modelling of rail pads. elements or brick elements between the rail and the sleeper surfaces usually bring good results, ensuring the vertical stiffness provided by the supplier is kept. In some situations the ’s choice depends also on the features available in the software. The load distribution between sleepers strongly depends on the rail pad vertical stiffness. Figure 2 illustrates the effects of including either relatively soft or stiff rail pads (average stresses are given at the centre of gravity of each element/zone). As it could be expected, using a stiffer rail pad induces high stresses at the top layers and higher concentration of load under the first sleeper. It is interesting to note that, in this situation, an additional 10% of the load is being transferred to the sleeper under the load. On the other hand, the vertical displacement at the top of the rail is decreased by 20 % (from 1.82 mm to 1.45 mm). In the following analyses, rail pads were placed over the full width of the sleeper and rail. They were modelled with 0.01 m deep brick elements. An equivalent Young modulus was determined so that a vertical stiffness of 100 kN/mm could be achieved. 2.3 Modelling trackbed layers Choosing the constitutive laws to reproduce the behaviour of the trackbed materials may be a key decision in order to obtain a reasonable approximation to the problem. In some situations, the consideration of a linear elastic behaviour of the materials that comprise the trackbed layers may lead to unrealistic load distribution between sleepers. As a result, inadequate stress distribution in depth may be obtained. In previous studies (Paixão & Fortunato 2009), the authors have compared the results of considering either linear or nonlinear elastic-perfectly plastic behaviour of ballast and sub-ballast material.Although the results suggested that the use of more complex constitutive relations, taking into the variable elasticity of materials (Fortunato & Resende 2006), gives more realistic results, regarding the distribution of loads in the three-dimensions, the adoption of linear elastierfectly plastic laws is a fair approximation to the real problem and has the advantage of requiring less computation resources. Special care needs to be taken in the modelling of ballast between sleepers (crib) in order to avoid

Figure 2. Vertical stress in the ballast layer, under the sleepers.

Figure 3. Vertical stress in the ballast layer, under the sleepers.

obtaining unrealistic results. Generally, in linear elastic models, when the ballast is modelled between sleepers and if no elements on the sleeper sides are included, poor results may be obtained. The consideration of yield criterion to reproduce the ballast behaviour of the crib is usually adopted with fairly good results. Since the presence of crib shows limited relevance to the vertical track structural behaviour, it may also be omitted. To illustrate the described behaviour, four models were developed regarding the different ballast behaviour and the crib was omitted in two of them. As depicted in Figure 3, the assumption of the ballast linear elastic behaviour with the presence of crib originates a case where the vertical load is partially transferred from the sleeper’s vertical sides to the crib, reducing stresses under the sleepers. Thus, if linear elastic behaviour is being considered, the modelling of ballast between sleepers can be avoided. Therefore, in the following analyses the crib was omitted so that a more realistic behaviour could be achieved with linear elastic models. Ballast shoulders were included (material placed laterally beside the sleepers); 0.5 m wide and a 2:3 slope. In order to evaluate the influence of the variation of some railway track characteristics, namely thickness and modulus of deformability (E) of the substructure layers, several three-dimensional numerical models

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Table 1. Values for the different parameters of the track layers. Layer/Value Parameter

Ballast

Sub-Ballast

Capping

E (MPa) Poisson ratio friction angle (◦ ) bulk weight (kN/m3 ) layer depth (m)

130 0.20 45 17.3 0.35

100 to 500 0.30 55 23.0 0.30 Sub-grade 25 to 60 0.30 19.5

40 to 80 0.30 45 23.0 0.4

E (MPa) Poisson ratio bulk weight (kN/m3 )

Figure 4. Vertical stress, under the load application point.

were developed, reproducing a variety of structural cases. Two situations were studied for the materials behaviour: 1) linear elastic; 2) linear elastic-perfectly plastic. Table 1 presents a summary of the physical and mechanical characteristics on the different ing layers used in this study. In cases where nonlinear behaviour was considered for the granular material layers, a Drucker-Prager (DP) criterion with nearly zero cohesion was adopted. The circumscribed alternative of the DP cone was used to fit to the Mohr-Coulomb (MC) yield surface outer edges. Although MC yield criterion usually brings a better reproduction of the granular material behaviour, DP criterion was used so that a comparison between different software could be carried out (see section 3.4). As could be expected, some tests showed that DP yield surface also allowed slightly faster calculation speeds.

3

RESULTS

Deformations and stresses observed in the trackbed were analysed and a brief description of the results is presented. The effects of the bulk weight were subtracted to show the results due to the axle loading.

3.1

Figure 5. Contours of total vertical stress (kPa): transverse (left) and longitudinal (right) alignments.

Models with linear elastic behaviour

From the series of parametric studies carried out, several considerations can be drawn. Definitely, one of the most relevant aspects that influence the track behaviour concerns the subgrade parameters, in particular its deformability. Take, for example, a structural solution which comprises ballast with E = 130 MPa, sub-ballast with E = 200 MPa and capping layer with E = 80 MPa. Increasing the subgrade modulus of deformability from 25 to 60 MPa leads to a vertical stress increment of about 30% at the upper part of the subgrade (see Figure 4). On the other hand, very small variations occur when changing the modulus of the upper layers. However, the results indicate that using

Figure 6. Contours of vertical displacement (m) due to the loading: transverse (left) and longitudinal (right) alignments.

higher values of E on the upper layers slightly reduces stresses at the subgrade. The subgrade deformability exhibits even greater influence on the displacements. Coming back to the above example, changing the subgrade deformability between 25 to 60 MPa, reduces vertical displacements at the ballast top by approximately 57%. To give an example of the deformation and stress distribution in depth, contours of these results are presented in Figures 5 and 6. In general, vertical stress concentration occurs under the outer edge of the sleeper, while maximum vertical displacement occurs under the centre of the sleeper. Note that, in some of the studied cases, the consideration of linear elastic behaviour leads to a situation that does not adequately represent the material behaviour since it allows the development of horizontal tensile

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stress in the upper ing layers. Such stress state is not ed by the granular materials that comprise the ballast or sub-ballast layers. This occurrence can be prevented with the introduction of elastierfectly plastic behaviour in some materials, as will be addressed next. 3.2 Models with elastic-perfectly plastic behaviour The same series of parametric studies mentioned in section 3.1 were carried out, though in this case, the elastic-perfectly plastic behaviour was introduced in the ballast, sub-ballast and capping layers, according to Table 1. In comparison with linear elastic models, slightly higher increments of stress and deformations were obtained with these models, mainly under the loaded area. Giving the previous example, increasing the subgrade modulus of deformability between 25 and 60 MPa, and considering instead a nonlinear behaviour, a vertical stress increment of about 27% at the upper part of the subgrade was obtained. Vertical displacements at the ballast top were reduced by approximately 61%. Once again, the results indicate that the subgrade deformability exhibits great influence in the obtained stresses and deformations. Despite the fact that horizontal tensile stresses do develop on the upper trackbed layers of the linear elastic models, which is not a fair reproduction of the material behaviour, the obtained vertical stress and deformation results with both material laws were relatively close. The maximum displacements obtained here were about 6% greater than those obtained with linear elastic models. 3.3 Sub-ballast layer thickness variation For both material behaviour approaches, described above, variations were implemented on the sub-ballast and capping layer thickness in order to evaluate the benefit of replacing capping layer material with subballast material. The sum of the depth of the two layers was kept constant. The results indicate that there is a limited benefit regarding the reduction of vertical stresses on the subgrade. Such benefit is relatively small and only relevant up to a certain point in the reduction of the capping layer depth. Taking as an example, a track structure with a 0.30 m deep sub-ballast layer and a 0.40 m deep capping layer: replacing the top 0.30 m of that capping layer with sub-ballast material only reduces the maximum stress at the subgrade by about 4%. 3.4 Comparison between different numerical tools Regarding the three-dimensional models developed for this study, different automatic calculation software was used, namely FLAC3D (Itasca 2006) and ANSYS (2009). The first software makes use of the finite difference method, while the second uses the finite element method. In both cases the modelling of the

Figure 7. Average vertical stress under the rail alignment on top of the ballast layer.

rail track followed the same geometry and parameters described in section 2.3 The explicit calculation (Lagrangian) and the method of discretisation used by FLAC3D provide an adequate modelling of continua, allowing the resolution of three-dimensional equilibrium problems with considerable geometric and physical complexity. One of the characteristics that led to the preference for this software is related to the straightforward implementation of constitutive models for materials with the development of routines in an internal programming language. Considering the resolution algorithm used (Itasca 2006), the implementation of constitutive models is relatively simple, whether they are linear or nonlinear. On the other hand, the ANSYS software makes use of the finite element method and, for the analysed cases, slightly smaller computation times could be achieved, since the studied problems showed relatively small strains, allowing a fast convergence. The studies presented in section 3 were performed with these two programs. The results obtained were quite consistent, either considering linear and nonlinear behaviour of the materials. The results were consistent under the loaded area, especially in deeper layers. Minor result differences occurred under the distant sleepers, at the ballast and sub-ballast layers, as illustrated in Figure 7.

4

CONCLUSIONS

In order to study the structural behaviour of rail tracks, a few aspects concerning the numerical modelling of such structures were introduced. A set of results obtained with different three-dimensional modelling tools was presented and discussed. To evaluate how different techniques, used to model track components, influence the load distribution between sleepers, some analyses were carried out and considerations were made. A wide rage of different track structures was modelled in order to evaluate the influence of some physical and mechanical characteristics of the trackbed layers. The results indicate that the stiffness of the

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subgrade has a great influence with regard to the vertical stress levels and displacements obtained at different depths. Considering the variations performed on the parameters of the ing layers, namely its modulus of deformability and depth, the replacement of capping layer material with sub-ballast material showed a limited influence on the output results, taking into the modulus of deformability of these materials. A comparison between linear and nonlinear behaviour models for granular materials was conduced. The analysis of the results evidenced that, when studying deeper layers, the use of linear elastic laws may lead to a close approximation to the results obtained with more complex constitutive laws. The above analyses were carried out using two numeric programs (FLAC3D and ANSYS) and a good consistency between both tools could be achieved. Given the results, it can be concluded that both software are powerful tools to study and model the track behaviour. In general, the development of these models has contributed to a better understanding of the structural behaviour of the track and its subgrade. The use of these models establishes a good base for further developments, such as analysing the benefit of including non-traditional structural elements in the track. At present day, studying how the inclusion of under-sleeper pads, bituminous sub-ballast layers or geo-synthetics influence the track behaviour presents relative importance, considering the current paradigm for life-cycle cost reduction. ACKNOWLEDGEMENT The authors acknowledge the financial of R&D project “PTDC/ECM/70571/2006 – Optimisation of High-Speed Railway Track Using Bituminous Sub-ballast” funded by Fundação para a Ciência e a Tecnologia (FCT), from Portuguese Ministry of Science, Technology and Higher Education. REFERENCES Adegoke, C., Chang, C. & Selig, E. 1979. Study of analytical models for track systems. Transport Research Record 733: 12–20. Alves Ribeiro, C., Dahlberg, T., Calçada, R. & Delgado, R. 2009.Análise dinâmica de zonas de transição em vias férreas de alta velocidade através de métodos de análise explícitos. 3as Jornadas Hispano Portuguesas sobre Geotecnia en las Infraestructuras Ferroviarias, Madrid, 25–26 June 2009: 14–19.

ANSYS 2009. Release 12.0 Documentation for ANSYS. Canonsburg, PA: ANSYS, Inc. Aubry, D., Baroni, A., Clouteau, A., Fodil, A. & Modaressi, A. 1999. Modélisation du comportement du ballast en voie. Proceedings of XII European Conference on Soil Mechanics and Geotechnical Engineering, Amsterdam: Balkema. Chang, C, Adegoke, C. & Selig, E. 1980. GEOTRACK model for railroad track performance. Journal of the Geotechnical Engineering Division, ASCE, 106(11): 1201–1218. ITASCA 2006. FLAC3D – Fast Lagrangian Analysis of Continua in 3D. ’s Manual. Minneapolis, MN: Itasca Consulting Group, Inc. Fortunato, E. 2005. Renovação de Plataformas Ferroviárias. Estudos Relativos à Capacidade de Carga. PhD Thesis. Porto: University of Porto. Fortunato, E. & Resende, R. 2006. Mechanical Behaviour of Railway Track Structure and Foundation – Threedimensional Numerical Modelling. Railway Foundations. RailFound 06, Birmingham, 11–13 Septembre 2006: 217–227. Huang, Y., Lin, C., Deng, X. & Rose, J. 1984. KENTRACK. A Computer Program for Hot Mix Asphalt and Conventional Ballast Railway Trackbeds. RR-84-1. Lexington, KY: Asphalt Institute. Li, D. & Selig, E. 1998. Method for railroad track foundation design. I: Development. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 124(4): 316–322. Ministerio de Fomento 1999. Recomendaciones para el proyecto de plataformas ferroviarias. Madrid: Ministerio de Fomento. Secretaría de Estado de Infraestructuras y Transportes. Office de Recherches et d’Essais de l’Union Internationale des Chemins de Fer ORE. 1983. Question D117. Rapport No. 27. Prause, R. & Kennedy, J. 1977. Parametric study of track response. Federal Railroad istration, Office of Research and Development – 77/75. Washington: US Dept. of Transportation. Paixão, A. & Fortunato, E. 2009. Análise estrutural de viaférrea com recurso a um modelo numérico tridimensional. 3as Jornadas Hispano Portuguesas sobre Geotecnia en las Infraestructuras Ferroviarias, Madrid, 25–26 June 2009: 136–148. Sanguino, M., Requejo, P. & Urroz, E. 1998. Cálculo de plataformas ferroviarias mediante el empleo de modelos matemáticos avanzados. Congreso Nacional de Ingeniería Ferroviaria, Coruña, 3–5 June 1998: 161–172. Sauvage, R. & Larible, G. 1982. La modélisation par éléments finis des couches d’assise de la voie ferrée. Revue Générale des Chemins de Fer (9): 475–484. Tarabji, S. & Thompson, M. 1976. Finite Element Analysis of a Railway Track System. Federal Railroad istration, Office of Research and Development – 76–257. Washington, DC: US Dept. of Transportation. Williams, P. & Pérez, M. 1998. Cálculo de plataformas ferroviarias mediante elementos finitos. Congreso Nacional de Ingeniería Ferroviaria, Coruña, 3–5 June 1998: 149–160.

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Three dimensional analyses of ring foundations M. Laman, A. Yildiz, M. Ornek & A. Demir Civil Engineering Department, Cukurova University, Adana, Turkey

ABSTRACT: Numerical predictions of the ultimate bearing capacity of ring foundations ed by a sand bed are presented. Three dimensional numerical analyses of the test models were carried out using the finite element package PLAXIS (Finite Element Code for Soil and Rock Analysis). It is shown that the behaviour of ring foundations on sand beds may be reasonably well represented by the hardening soil model available in the Plaxis package. The hardening soil model parameters were derived from the results of drained triaxial tests. The results from the finite element analysis are in very good agreement with the experimental observations.

1

INTRODUCTION

In recent decades, an exponential growth in the area of digital computers and computational mechanics has resulted in the application of the non-linear finite element method to almost all areas of geotechnical engineering, including shallow foundations. The finite element method (FEM) has also become a highly useful tool, and has been widely used for the numerical analysis of soil structures (Abdel-Baki and Raymond 1994; Yetimoglu et al. 1994; Ismail and Raymond 1995; Ismael, 1996; Ohri et al, 1997; Kurian et al. 1997; Chandrashekhara et al. 1998; Otani et al. 1998; Yoo 2001, Boushehrian and Hataf 2003; Laman and Yildiz, 2003; Zhao and Wang 2007, Laman and Yildiz 2007). It provides the advantage of idealising the material behaviour of soil, which is non-linear with plastic deformations and stress path dependent, in a more rational manner. The FEM can also be particularly useful for identifying the patterns of deformations and stress distribution in and around the reinforcing elements, during deformation and at ultimate state. Ring foundations are used in a variety of structures, such as cooling towers, smoke-stacks, transmission towers, radar stations, liquid storage tanks and TV antennae. Analyses for the ultimate bearing capacities of these foundations are not as advanced and as well understood as those for strip, rectangular, square and circular foundations. A few studies relating to ring foundations have been reported in the literature. Ismael (1996) investigated the behaviour of ring foundations on very dense cemented sands by using plate loading tests. The load–settlement curves and ultimate bearing capacities for solid and ring plates were compared. Ismael (1996) found that the ultimate bearing capacity of ring plates is close to that of the solid plates, and proposed that ring foundations can be used with different ratios of the inside to outside radii (ri /re ) up to 75% in practical applications. Ohri et al. (1997)

performed a series of laboratory tests on model ring footings and found that, for a ratio of internal to external diameter of the ring (n) equal to 0.38, the unit bearing capacity reaches its maximum for dune sand. Hataf and Razavi (2003) found that the value of n for the maximum unit bearing capacity of sand is not unique, but is in the range 0.2–0.4. Boushehrian and Hataf (2003) performed tests to investigate the bearing capacity of circular and ring footings on sand by conducting laboratory model tests together with numerical analysis. They found that n is 0.40 for ring foundations. Laman and Yildiz (2003) performed some experimental analysis and investigated the bearing capacity of ring foundations ed by sand beds; they showed that the optimum ring width ratio (r/R) is 0.30. They found that a ring foundation with optimum width gives similar performance to that of a full circular foundation with the same outer diameter. Zhao and Wang (2007) utilize a finite difference code FLAC to study bearing capacity factor Nc for ring footings in cohesionless, frictional and ponderable soil. Soil model employs Mohr–Coulomb yield criterion and associative flow rule. The value of Nc is found to decrease significantly with an increase in ri /ro , which is the ratio of internal radius to external radius of the ring. The value of Nc for a rough ring footing, especially for lager values of u, is obviously higher than that for a smooth footing. The results are compared with those available in the literature. In this study, three dimensional (3D) FE analyses were carried out on ring foundations resting on sand soil using the FE program Plaxis 3D Foundation V2.1 (Brinkgreve and Broere, 2006). The hardening soil model (Schanz et al. 1999), incorporating parameters from drained triaxial compression and oedometer tests, was used for the mathematical modelling of the non-linear soil in the numerical analyses. The results were compared with the results of model tests reported by Yildiz (2002), Laman and Yildiz (2003).

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Table 1.

Properties of sand paper.

Property

Value

Coarse sand fraction (%) Medium sand fraction (%) Fine sand fraction (%) D10 , D30 , D60 (mm) Cu , Cc Specific gravity γkmax , γkmin (kN/m3 ) γkmean (kN/m3 ) c (kPa) ϕ (◦ ) USCS

0 34 66 0.28, 0.36, 0.42 1.50, 1.10 2.68 17.80, 15.90 17.10 0 41 SP

2 2.1

EXPERIMENTAL INVESTIGATION General

The experimental programme was carried out using a test facility in the geotechnical laboratory of the Civil Engineering Department of the University of Cukurova. The experimental set-up has been used extensively for the bearing capacity of shallow foundations on sand (Yildiz 2002; Laman and Yildiz 2003). Details of the experimental programme, test procedures and analysis of the test results of model studies of the ultimate bearing capacity of ring foundations on sand have been presented in detail by Yildiz (2002) and Laman and Yildiz (2003). 2.2

Soil properties

Uniform, clean, fine sand obtained from the Seyhan riverbed in Southern Turkey was used for the model tests. The properties are summarized in Table 1. The angle of shearing resistance of the sand having a dry unit weight 17.1 kN/m3 and under normal pressures of 50, 100, and 200 kPa was determined by directshear tests. The measured average peak friction angle was 41◦ . 2.3

Figure 1. Ring foundation.

Model foundations

Loading tests were carried out on five different model rigid foundations fabricated from mild steel. All models were 20 mm thick and 85 mm in diameter (D). The first of the five model foundations was circular; the others were ring foundations. The diameter of the inner boundaries of the ring foundations (d) were 25.0, 45.0, 55.0 and 65.0 mm. Fig. 1 shows the geometry of the model ring foundation considered in this investigation Figure 2. An overview of the test setup.

2.4 Model tests Tests were conducted in a steel tank 700 mm long by 700 mm wide and 700 mm deep.The model foundation tests were performed at a unit weight of 17.1 kN/m3 . The facility of the test is shown in Fig. 2. To maintain consistency of the in-place density throughout the test pit, the same compactive effort was applied

to each layer. The model foundation was placed on the surface of the sand bed at predetermined locations in the test pit. Vertical compressive load was applied in small increments to the model foundation by means of a mechanical jack ed against a reaction beam. Constant load increments were applied

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until the foundation settlement during the load increment had stopped. The tests were continued until the applied vertical load clearly reduced, or a large settlement of the foundation resulted from a relatively small increase in vertical load. 3

Table 2.


The FE studies of the bearing capacity of ring foundations resting on sand bed with the same model geometries as in the tests were carried out using the program Plaxis 3D Foundation V.2.1. The program is a special purpose three-dimensional finite element computer program used to perform deformation analyses for various types of foundations in soil and rock (Brinkgreve and Broere, 2006). Stresses, strains and failure states of a given problem can be calculated. An elasto-plastic hyperbolic model called the hardening soil model (HSM) was selected for the nonlinear sand behaviour in this study. The HSM is an advanced model for simulating the behaviour of different types of soil, both soft and stiff (Schanz et al., 1999). When subjected to primarily deviatoric loading, sandy soil decreases in stiffness, and simultaneously irreversible plastic strains develop. The observed relationship between the pressure and the axial strain can be well approximated by a hyperbola, as used in the variable elastic hyperbolic model (Duncan and Chang 1970). The HSM is formulated in the framework of the classical theory of plasticity. The HSM supersedes the hyperbolic model: first by using the theory of plasticity rather than the theory of elasticity, second by including soil dilatancy, and third by introducing a yield cap (Schanz et al. 1999). Limiting states of stress are described by means of the friction angle ϕ, cohesion c and dilatancy angle ψ. Soil stiffness is described by using three different input stiffnesses: the triaxial loading stiffness E50 , the unloading–reloading stiffness Eur , and the oedometer loading stiffness Eoed . The HSM also s for stress dependence of stiffness moduli. This means that all stiffness values increase with pressure. Hence all three input stiffness values relate to a reference stress, usually taken as 100 kPa. (Brinkgreve and Broere, 2006). Under primary loading the behaviour is distinctly non-linear, and is assumed to be hyperbolic up to a failure stress. In contrast to E50 , which determines the magnitude of both the elastic and the plastic strains, Eur is a true elasticity modulus. In conjunction with a Poisson’s ratio υur , the elasticity modulus Eur determines the soil behaviour under unloading and reloading. Both the secant virgin loading modulus E50 and the unloading modulus Eur are stress-level dependent. For the HSM, these parameters are computed as

HSM parameters.

Property

Value

pref (kPa) γn (kN/m3 ) E50 (kPa) Eur (kPa) Eoed (kPa) m c (kPa) ϕ (◦ ) (◦ ) υ K0 Rf

100 17.10 28000 75000 28000 0.50 0.50 41 11 0.20 0.34 0.90

ref Eref 50 and Eur are input parameters for a particular reference pressure pref . The exponent m can be determined from both oedometer and triaxial test results. A value of m=0.5 is typical for sands, and m = 1.0 for clays. In the HSM the virgin oedometer stiffness is stress dependent according to

ref In addition to the moduli Eref 50 and Eur , the oedomeref ter modulus Eoed is also an input modulus for the HSM. Together with the parameters m, υur , c’, ϕ’ and the dilatancy angle ψ, there are a total of eight input parameters (Vermeer et al. 2001). The soil parameters in Table 2 represent sand used in the model tests. The initial stresses in the soil are generated using Jaky’s formula, expressed by Equation 4 (in Plaxis, the procedure to generate initial soil stresses are often known as the K0 procedure),

where K0 is the coefficient of lateral earth pressure and ϕ is the friction angle of the soil. The ring foundation is modeled as a rigid plate, and is considered to be very stiff and rough in the analyses. Values of Young’s modulus and Poisson’s ratio of 207 × 106 kPa and 0.25, respectively, were assumed for the foundation. The program incorporates a fully automatic mesh generation procedure, in which the geometry is divided into elements of the basic element type, and compatible structural elements. In the analysis, the number of elements was about 3000. A local mesh refinement is also applied to around the foundation. The sand medium was modeled using quadratic 15-node wedge elements. A typical graded FE mesh composed of soil and foundation together with the boundary conditions used is shown in Fig. 3. The boundaries of the mesh were based on the soil in dimensions used in the physical modelling. In previous supplementary analyses, Yildiz (2002) reported that these boundary distances did not influence the results. The program generates

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Figure 5. Curves of loading against settlement.

Figure 3. Typical finite element mesh.

Figure 6. Relationship between r/R and qu .

Figure 4. Total displacement fields below the ring foundations.

full fixity at the base of the geometry and smooth conditions at the vertical sides. Fig. 4 shows a typical example of the resultant vertical and horizontal displacement fields below the ring foundation.

4

COMPARISON BETWEEN NUMERICAL AND EXPERIMENTAL RESULTS

Typical plots for the load–settlement behaviour obtained from the experimental test and FE analysis are shown in Fig. 5. The vertical displacements predicted by the HSM are in very good agreement with the experimental results. The measured and predicted

ultimate bearing capacities are 220 kPa and 213 kPa, respectively. The settlement ratio (s/D) for the unreinforced sand at failure was approximately 4.9% in the test and 5.4% in the analysis. The failure loads were obtained from the load–settlement (q−s) graphs and used to calculate the ultimate bearing capacities. As seen from Fig. 6 that a good agreement is found between the experimental results and the FE modeling. It can be seen from Fig. 6 that q remains approximately constant when r/R changes from 0 to 0.40, and then decreases when r/R changes from 0.30 to 0.65. The numerical results show that an optimum value of r/R may visually be estimated from the curve as approximately 0.40. In the literature, the ratio of inside to outside radii (r/R) is generally recommended to be in the range 0.2–0.4 for ring foundations. Ohri et al. (1997) found that for a ratio of internal to external diameter of the ring (n) equal to 0.38. Hataf and Razavi (2003) also proposed that the value of n for sand is not unique but is in the range 0.2–0.4.

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vertical displacements with increasing r/R ratio when the ultimate bearing capacity is reached.

5

CONCLUSIONS

The ultimate bearing capacity of ring foundations ed by a sand bed was investigated using the three dimensional FE program Plaxis. The results of the finite element simulations were compared with the model test results. Based on comparisons between the field observations and the finite-element results the following main conclusions can be drawn: Numerical analyses, using an elasto-plastic hyperbolic model (hardening soil model), incorporating parameters derived from drained triaxial tests, gave results that closely match those from physical model tests. The optimum ring width ratio, r/R, for ring foundations is about 0.40. It was noted that the value of bearing capacity decreases after the r/R > 0.4. A ring foundation with the optimum width shows approximately similar performance to that of a full circular foundation with the same outer diameter as the ring foundations. This option can provide an economical solution in practical applications. REFERENCES

Figure 7. Displacement shading.

Fig. 7 shows some typical examples of the resultant vertical and horizontal displacement fields below the ring foundation at the ultimate bearing capacity. It can be seen that there is a reduction of horizontal and

Abdel-Baki, M. S. & Raymond, G. P. 1994. Reduction of settlement using soil geosynthetic reinforcement. Vertical and Horizontal Deformations of Foundations and Embankments, ASCE, Vol. 1, pp.525–537. Boushehrian, J. H. & Hataf, N. 2003. Experimental and numerical investigation of the bearing capacity of model circular and ring footings on reinforced sand. Geotextiles and Geomembranes, 21, No. 4, 241–256. Brinkgreve R.B.J. & Broere W., 2006. Plaxis Finite Element Code for Soil and Rock Analysis. 3D Foundation–Version 2.1. Chandrashekhara, K., Antony, S. J. & Mondal, D. 1998. Semianalytical finite element analysis of a strip footing on an elastic reinforced soil. Applied Mathematical Modelling, 22, 331–349. Duncan, M. & Chang, C. Y. 1970. Nonlinear analysis of stress and strain in soil. Journal of Soil Mechanics and Foundations, ASCE, 96, No. 5, 1629–1653. Hataf, N. & Razavi, M. R. 2003. Behavior of ring footing on sand. Iranian Journal of Science and Technology, Transaction B, 27, 47–56. Ismael, N. F. 1996. Loading tests on circular and ring plates in very dense cemented sands. Journal of Geotechnical Engineering, ASCE, 122, No. 4, 281–287. Ismail, I. & Raymond, G. P. 1995. Geosynthetic reinforcement of granular layered soil. Proceedings of Geosynthetics’95 Conference, Vol. 1, pp. 317–330. Kurian, N., Beena, K. S. & Kumar, R. K. 1997. Settlement of reinforced sand in foundations. Journal of Geotechnical Engineering, 123, No.9, 818–827. Laman, M. & Yildiz, A. 2003. Model studies of ring foundations on geogrid-reinforced sand. Geosynthetics International, 10, No. 5, 142–152. Laman, M. & Yildiz, A. 2007. Numerical studies of ring foundations on geogrid-reinforced sand. Geosynthetics International, 14, No. 2, 1–13.

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Ohri, M. L., Purhit, D. G. M. & Dubey, M. L. 1997. Behavior of ring footings on dune sand overlaying dense sand. International Conference of Civil Engineers, Tehran, Iran. Otani, J., Ochiai, H. & Yamamoto K. 1998. Bearing capacity analysis of reinforced foundation on cohesive soil. Geotextiles and Geomembranes, 16, No. 4, 195–206. Schanz, T., Vermeer, P. A. & Bonnier, P. G. 1999. The hardening soil model: formulation and verification. Beyond 2000 in Computational Geotechnics, A. A. Balkema Publishers, Rotterdam, pp. 281–296. Vermeer, P. A., Punlor, A. & Ruse, N. 2001. Arching effects behind a soldier pile wall. Computers and Geotechnics, 28, No. 6–7, 379–396.

Yetimoglu, T., Wu, J. T. H. & Saglamer, A. 1994. Bearing capacity of rectangular footings on geogrid-reinforced sand. Journal of Geotechnical Engineering, ASCE, 120, No. 12, 2083–2099. Yildiz, A. 2002. Bearing Capacity of Shallow Foundations on Geogrid-Reinforced Sand. PhD thesis, University of Cukurova, Turkey. Yoo, C. 2001. Laboratory investigation of bearing capacity behaviour of strip footing on geogrid-reinforced sand slope. Geotextiles and Geomembranes, 19, No. 5, 279–298. Zhao L. & Wang J.H. 2007. Vertical bearing capacity for ring foundations. Computers and Geotechnics, 35, 292–304.

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Piles


A back analysis of vertical load tests on bored piles in granular soil L. Tosini, A. Cividini & G. Gioda Politecnico di Milano, Department of Structural Engineering, Milan, Italy

ABSTRACT: Two back analyses are discussed of load tests carried out on bentonite slurry piles bored in granular soils. The first case concerns a test on a pile belonging to a 12 pile group. The back analysis permits calibrating an axisymmetric finite element model that reproduces, with reasonable accuracy, the experimental results. The calibrated model is then extended to three-dimensional conditions and applied to the analysis of the entire group. The results suggest some comments on the different assumptions that can be adopted in the calculations and on their effects on the global load-settlement curve of the pile group. The second case concerns a load test in which, in addition to the load-settlement data, also the axial strains along the pile were measured through electrical extensometers. The numerical back analyses highlight an apparent contradiction between the two sets of experimental data. On their bases some conclusions are drawn on the possible causes of the observed inconsistency and on the influence of the construction technology on the soil-pile interaction.

1

INTRODUCTION

When dealing with the design of deep foundations in granular deposits the prediction of their settlements is not straightforward due to the difficulties met in defining the values of the mechanical parameters influencing them. In most cases, in fact, the design is based on the results of penetrometer tests, which provide only approximated mechanical parameters of the granular soil. In addition, the pile settlements depend on the mechanical characteristics of the pile/soil interface that, in turn, are influenced by the adopted construction technology. Being aware of the possible limited accuracy of the computed settlements, load tests are customarily carried out to quantitatively assess the behaviour of the deep foundation under loading. In some instances, however, the load test does not directly provide the sought results, e.g. when the settlement of a pile group has to be evaluated on the basis of a load test on a single pile. In other instances the results present some apparent inconsistencies that make their interpretation somewhat controversial. Here the back analyses of two load tests on bored piles are presented based on twoand three-dimensional, elastic-plastic finite element calculations. Among the various approaches proposed in the literature for the analysis of piles under vertical loads, see e.g. (Coyle & Reese 1966; Poulos & Davis 1968; Butterfield & Banerjee 1971; Ottaviani 1975; Randolph & Wroth 1978), the finite element method was adopted here since it can be easily applied to inhomogeneous deposits ing for their non linear behaviour. The first examined case aims at evaluating the loadsettlement curve of a 12 pile group based on the results

of a load test on a single pile. The back analysis of the test permits an acceptable calibration of the numerical model in 2D axisymmetric regime. However, when extended to 3D conditions for the entire group, the calculations show the appreciable influence of the different assumptions that can be introduced in the analysis of the deep foundation. The second case concerns a load test in which, in addition to the load-settlement curve, also the axial strain within the pile was measured through extensometers applied to its steel reinforcement. In this case the experimental data present an apparent contradiction that leads to comments on the possible influence of details of the construction technique. 2

FIRST LOAD TEST

The test was carried out on a 17 m long bentonite slurry bored pile with a diameter of 120 cm. Due to the geometrical characteristics of the foundation mat, the pile head is located about 5 m below the ground surface. The soil deposit consists of sand and gravel with a marginal percentage of silt. The design was based on the results of standard penetration tests shown in Figure 1. Note that the high NSPT values reported at two depths are likely to depend on the presence of boulders. Figures 2a and 2b report, respectively, the estimated variation with depth of the friction angle ϕ and of the elastic modulus ratio E/E* (Mandolini & Viggiani 1997). In the present case the modulus E* coincides with that of the first layer and was evaluated on the basis of the load test data. The numerical results and the experimental data are shown by solid and dashed lines in Figure 3.

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Figure 1. Results of the standard penetration test. Figure 3. First load test: experimental results (solid line) and numerical simulation based on the calibrated finite element model (dashed line).

Figure 2. Estimated variation with depth of the friction angle ϕ (a) and of the elastic modulus ratio E/E* (b).

The numerical analyses were carried out in axisymmetric conditions modelling both the pile and the 9 layers of the surrounding soil through a mesh of four node isoparametric elements. To for the disturbance caused by the pile construction, thin interface elements (Desai et al. 1984) were placed along the pile contour and at its base. Figure 4 shows a detail of the finite element mesh in the vicinity of the pile. An elastic behaviour was assumed for the pile, while an elastic perfectly plastic constitutive model, obeying Mohr-Coulomb yield criterion with a non associated flow rule, was adopted for the soil layers. The modulus E* (cf. Figure 2b) was calibrated first by matching the slope of the initial part of the load settlement curve. Note that, since E* is calibrated on the basis of the load test, the evaluated variation of the soil elastic modulus with depth is already influenced by the disturbance due to construction. Consequently, it was assumed that the elastic modulus of each interface element coincides with that of the corresponding soil layer. The remaining mechanical parameters entering in the numerical

Figure 4. A detail of the axi-symmetric finite element meshes in the vicinity of the pile.

analyses are: the angle of plastic dilation ψ, governing the flow rule, that depends on the friction angle ϕ through a parameter δ, i.e. tan ψ = δ tan ϕ ; the reduction factor α of the interface friction angle ϕ*, i.e. tan ϕ* = α tan ϕ ; the coefficient of horizontal pressure K0 relating the normal stress between pile and soil to the vertical stress.

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Figure 5. Contour lines of the error function: δ dilatancy parameter, α friction angle reduction coefficient.

Figure 6. Detail of the horizontal section of the 3D finite element mesh.

The above parameters, collected in vector p, were evaluated through a back analysis that consists in minimizing the discrepancy F between the n measured settlements uˆ and the corresponding numerical results u(p). In order to express F in non dimensional form the difference between measured and calculated displacements is divided by the maximum measured displacement uˆ max ,

The back analysis was carried out asg different values to K0 and working out, for each of them, the values of α and δ that minimizes the function F. Comments on the minimization algorithms suitable for back analyses can be found e.g. in (Gioda & Sakurai, 1987). Figure 5 reports the contour line of the error function for one of such minimizations. The back analysis process led to the following values of the sought parameters: E* = 750 MPa; K0 = 0.5; α = 0.7; δ = 0.07. Having calibrated its material parameters, the numerical model was extended to three dimensional conditions to investigate the behaviour of the 12 pile group. Taking advantage of its double symmetry only 1/4 of the problem was discretized into a mesh of 8 node brick elements. A detail of the horizontal section of the mesh, in the vicinity of the pile group, is shown in Figure 6. The calculated load-settlement diagrams are summarized in Figure 7. Curve (a) represents the mere superposition of the load-settlement diagram calculated for a single pile (cf. Figure 3), i.e. the pile interaction is neglected. Curve (b) refers to the actual 3D case in which the pile group is connected to a foundation mat and the interaction between the mat and

Figure 7. Calculated load-settlement curves: a) simple superposition of 12 independent piles; b) pile group and interaction between soil and foundation mat; c) pile group without soil-mat interaction. (A dashed line represents the working load of the foundation).

the underlying soil is ed for. Finally, curve (c) represents the 3D case in which the mat-soil interaction is neglected. Case (c) could represent the case of an extremely severe erosion of the soil underneath a bridge pier. The numerical results show the increase of settlements due to the interaction between the piles (curves b and c) with respect to the case in which the interaction is neglected (curve a). In addition, the calculation permits a quantitative assessment of the effects of the interaction between the foundation mat and the underlying soil.

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Figure 8. Second load test: experimental results (solid line) and numerical simulation based on the calibrated finite element model (dashed line).

In particular, it can be observed that depending on the assumptions adopted in the calculations the expected settlement of the pile group under working loads (dashed line in Figure 7) varies between 6.5 and 10.5 mm.

3

SECOND LOAD TEST

This test was carried out on a 80 cm diameter bored pile having length of 11.5 m. Since the in situ investigation indicates that the soil profile is reasonably uniform, mechanical properties constant with depth were used in the design. The adopted finite element mesh is similar to the one used for the first load case and depicted in Figure 4. Applying the same procedure described for the previous case, the back analysis led to the following parameter characterizing the numerical model: E = 700 MPa; K0 = 0.6; α = 0.7; δ = 0.03. The consequent numerical results are compared in Figure 8 with the experimental ones. In this case, in addition to the load-settlement data, also the axial strains at three locations along the pile where measured through electrical extensometers. They were placed at 3.65 m, 7.10 m and 10.15 m from the pile top. The available strain measurements correspond to point A of the load-settlement diagram in Figure 8. The distribution of the axial load derived from the measured strain is compared with the corresponding finite element results in Figure 9. It can be observed that, while the calibrated numerical model provides an acceptable approximation of

Figure 9. Axial force from the extensometer measurements (solid line) and numerical results based on the calibrated numerical model (dashed line).

the load-settlement data (cf. Figure 8), a large difference exists between the computed axial force and that deriving from the experimental data (cf. Figure 9). According to the experimental data in Figure 9 a vanishing vertical load reaches the pile tip and, hence, the limit skin friction has not been reached yet. On the contrary, the calibrated numerical model indicates that about half of the applied load is carried by the base. This leads to conclude that the limit skin friction was reached, at least for the upper portion of the pile. In order to overcome the above apparent contradiction and, hence, to limit the axial force that reaches the pile base, various attempts were made by modifying the soil parameters. The most successful one (see Figure 10) consisted in introducing a limited adhesion (50 kPa) at the soilpile interface that could depend on the silty fraction present in the granular deposit. Unfortunately this provision eliminates any similarity between the experimental and calculated loadsettlement diagrams, as shown by the diagrams in Figure 11. Note, in particular, the difference existing between points B and A in Figure 11 that correspond, respectively, to the experimental and numerical diagrams in Figure 10. The difficulties met in modelling both sets of experimental data cannot merely depend on errors in the values of the soil parameters. They should rather depend on some aspects of the field problem that lead to the barely relevant force at the pile base. A possible cause of this effect could be the presence of a very soft zone at the pile tip that depends, for instance, on the partial cleaning of the excavation

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Figure 10. Axial force from the extensometer measurements (solid line) and numerical results with soil-pile adhesion of 50 kPa (dashed line).

Figure 12. Axial force from the extensometer measurements (solid line) and numerical results with a soft zone at the pile tip (dashed line).

Figure 11. Comparison between experimental results (solid line) and numerical ones with soil-pile adhesion of 50 kPa (dashed line).

Figure 13. Comparison between experimental results (solid line) and numerical ones with a soft zone at the pile tip (dashed line).

bottom or on the presence of a zone where the concrete was mixed with the bentonite slurry. The finite element model was then modified introducing a layer of soft material below the pile tip and reducing the soil-pile adhesion to 20 kPa. The corresponding numerical results are shown in Figures 12 and 13. Apparently the introduction of a soft zone improves the agreement between experimental and numerical

results, even though some discrepancy can still be observed (cf. Figure 13). This could depend on the assumed homogeneity of the soil deposit or on the limits of the relatively simple constitutive model adopted in the calculations. However, it seems reasonable to conclude that the presence of the mentioned soft zone is likely to be a possible cause of the measured marginal load transferred to the pile tip.

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4

CONCLUSIONS

The two discussed case histories show that the back analysis of load tests represents a practical procedure for calibrating the numerical models of deep foundations. When dealing with pile groups, the parameters obtained from the axisymmetric interpretation of the load test on a single pile can be adopted for the three dimensional analysis of the entire group. In this case, in addition to the interaction between the piles, the finite element model can also for the elastilastic interaction between the foundation mat and the underlying soil. In the second test the numerical analysis highlighted an apparent contradiction of the in situ measurements. In fact, the strains measured along the pile show that a limited load is transferred to its base. On the contrary, the back analysis of the load-settlement diagram indicates that at least half of the applied load reaches the base. This apparent contradiction is likely to depend on some field condition that is not properly ed for in the numerical model. In the present context it appears that the mentioned discrepancy between experimental and numerical results can be, at least partially, reduced by introducing in the calculations a soft zone underneath the pile tip. This zone could be due, for instance, to a poor cleaning of the excavation bottom or to the formation of a soft mixture of concrete and bentonite at the pile tip. If this explanation can be accepted, the back analysis provided an insight into an apparent weakness of the application of the construction technique at that specific site. It is worthwhile observing that a more effective interpretation of the second load test could have been reached if the vertical load at the pile tip were directly measured through a load cell or a flat jack. This additional information would have in fact validated and completed the measurements of the electrical extensometers.

It can be finally observed that numerical modelling is not only a useful design tool for analysing various aspects of the interaction between deep foundations and surrounding soil. Its use in the interpretation of in situ measurements could also lead to a deeper understanding of the effectiveness of a construction technology and, perhaps, to some suggestions for improving its application. ACKNOWLEDGEMENTS The authors wish to thank Michela Chiorboli and Mario De Miranda for providing the experimental data and for their technical comments. REFERENCES Butterfield, R. & Banerjee, P.K. 1971. The problem of pile group-pile cap interaction. Geotechnique 21(2): 135–142. Coyle, H.M. & Reese, L.C. 1966. Load transfer for axially loaded piles in clay. Journal of Soil Mechanics and Foundation Engineering ASCE, 92(2): 1–26. Desai, C.S., Zaman, M.M., Lightner, J.G. and Siriwardane, H.J. 1984. Thin-layer elements for interfaces and ts. International Journal for Numerical and Analytical Methods in Geomechanics, 8(1): 19–43. Gioda, G. & Sakurai, S. 1987. Back analysis procedures for the interpretation of field measurements in geomechanics. International Journal for Numerical and Analytical Methods in Geomechanics, 11(6): 555–583, Mandolini, A. & Viggiani, C. 1997. Settlement of piled foundations.Geotechnique, 47(4): 791–816. Ottaviani, M. 1975. Three-dimensional finite element analysis of vertically loaded pile groups. Geotechnique 25(2): 159–174. Poulos, H.G. & Davis, E.H. 1968. The settlement behaviour of single, axially loaded, incompressible piles and piers. Geotechnique 18: 351–371. Randolph, M.F. & Wroth, C.P. 1978. Analysis of the deformation of vertically loaded piles. Journal of Geotechnical Engineering ASCE, 104(12): 1465–1488.

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A numerical study on the effects of time on the axial load capacity of piles in soft clays K.P. Giannopoulos, L. Zdravkovic & D.M. Potts Imperial College London, UK

ABSTRACT: This paper investigates the axial load capacity of a pre-loaded pile installed in soft clay and subjected to vertical loading. Previous studies on pre-loaded shallow footings have shown that the undrained shear strength of soft clays is enhanced with time due to the dissipation of the excess pore water pressures generated during initial loading, as well as due to the soil ageing, after all pore pressures have dissipated, associated with creep. It remains a question whether the pile load capacity is enhanced with time as well, after the effects of installation have settled. The problem is investigated by means of a series of coupled finite element analyses, thus taking of consolidation processes in the ground, in which the soil is modelled using an elastic-plastic constitutive model. Subsequently a similar set of coupled analyses is performed, using an elastic-viscoplastic model, to examine the effect of creep on pile capacity.

1

INTRODUCTION

The bearing resistance of a pile is usually determined either by analytical or semi-empirical methods, or from the results of pile load tests. The ultimate axial load which can be ed by a pile is equal to the sum of the base resistance and shaft resistance. The base resistance is the product of the base area and the ultimate bearing capacity at base level, whereas the shaft resistance is the product of the perimeter area of the shaft and the shaft friction between the pile and the soil, i.e. the shearing resistance. The short term capacity of a pile is usually the most critical in pile design and several solutions have been put forward either analytically or empirically. However, a question remains if extra loading is applied on an existing piled foundation, which has been in place for a period of time. Such a situation may arise when new floors are added to an existing structure, resulting in a load increase on existing piles. Previous studies on the effects of preloading and soil ageing on the bearing capacity of shallow foundations on soft clay by Zdravkovic et al. (2003) and Bodas Freitas et al. (2007) respectively have shown that the capacity of shallow foundations is significantly enhanced with time, however it remains a question whether this is the case for deep foundations. Clearly, pre-loaded foundations have to sustain a certain load over a period of time during which the soil is subjected to time-related processes of consolidation, as well as creep. Both of these processes enhance the soil strength and stiffness compared to the initial ones, and it remains a question whether these processes affect the long-term load capacity of single piles.

It is evident that the method of installation of a single pile significantly affects the processes involved in the soil surrounding the pile. It has been described by several authors including Tomlinson (2008) and Jardine et al. (2005) that during pile installation and equilibration, the soil strength surrounding the pile may be enhanced or reduced depending on the method of installation or even the soil medium. However, this is out of the scope in this study. The effects of installation are omitted and the processes examined are those that take place after the effects of installation have settled. The problem is investigated by means of a series of coupled finite element analyses. Two constitutive models, the well known Modified Cam Clay (MCC) model and the Equivalent Time (ET) Creep model, are used in order to distinguish the time effects on pile capacity. The MCC model s for the consolidation processes in the ground and the ET Creep model takes of the creep effects as well. The software package used to perform the required analyses is the Imperial College Finite Element Program (ICFEP), Potts & Zdravkovic (1999), which employs a modified Newton-Raphson approach with an error-controlled sub-stepping algorithm as a non-linear solver. 2

CHOICE OF CONSTITUTIVE MODELS

The scope of this study is to distinguish the effects of consolidation and the effects of creep on the axial load capacity of preloaded singles piles in soft clays. Two constitutive models are required, which can predict these effects. The first one is a form of the MCC model (Potts & Zdravkovic, 1999), which can predict accurately the change of undrained shear strength during

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Table 1. (a) MCC model parameters, (b) ET Creep model parameters. ϕ

λ

G

Degrees – 32 0.022

– 0.22

kPa kN/m3 – m/s 1700 17.0 3.00 5 × 10−10

to

o

εvm vp limit

day 1.0

– – 0.00434 0.5

κ

γ

v1

k

Notes: ϕ = angle of shearing resistance; κ = slope of the swelling or instant time line in v-lnp’ space; λ = slope of the consolidation or the reference time line in v-lnp’ space; G = elastic shear modulus; γ = bulk unit weight of the soil; v1 = specific volume at p’= 1 kPa on the isotropic virgin consolidation line; k = permeability; to = reference time; o = time dependent parameter; εvm vp limit = limit for the amount of volumetric visco-plastic strain.

consolidation. The second one is the ET Creep model (Bodas Freitas, 2008), which can predict the change of undrained shear strength due to the development of viscoplastic strains, or creep strains, with time along with the associated changes due to consolidation. The MCC model is a critical state model, which is fully described in Potts & Zdravkovic (1999). The model parameters are given in Table 1. Undrained shear strength is not an input parameter; however it is possible to calculate it from the effective stress and input parameters and the input Ko and OCR profiles. The ET Creep model is a critical state model as well which is based on the overstress theory and the Equivalent Time (ET) concept (Yin et al. 2002). The time-dependent behaviour of the model is characterized by the coefficient of secondary consolidation Cα. It incorporates a non-logarithmic function of viscoplastic strain with time, with a limit for the amount of viscoplastic strain that is attained at infinite time. It can predict soil ageing by ing for viscoplastic strains, i.e. creep strains, however it is not able to determine any contribution to the soil strength and stiffness by the development of structure with time. Thus, this study is limited to the effects of consolidation and creep on the axial load capacity of a preloaded pile in soft clay. More details of this model may be found in Bodas Freitas (2008). The model parameters are given in Table 1.

3

SOIL CONDITIONS

The ground profile adopted for this study is that of a typical soft clay profile, which is more susceptible to creep. Similar soil conditions as those adopted by Zdravkovic et al. (2003) and Bodas Freitas et al. (2007) are used, in order to make this analysis consistent with the footing analysis. The material parameters are based on the site investigation and laboratory data from a site in Grimbsy, Yorkshire (Mair et al. 1992).

Figure 1. Su , OCR and Ko profiles.

The ground profile at the site under investigation consists of normally consolidated clay with 2 m thick stronger and overconsolidated crust above the ground water table.The soil above the water table is assumed to be saturated and able to sustain tensile pore water pressures. The results from undrained triaxial tests (Fig. 1) are used in order to obtain the OCR and Ko profiles above the water table for the MCC analysis. For depths below the ground water table, where the soil is normally consolidated, the ratio Su /σv ’ is set 0.3, which is typical of soft clays. For the ET Creep model analysis, the undrained shear strength is not only a function of the basic model parameters as it is in the MCC model. It is in addition a function of the shearing rate. Therefore, a realistic rate is incorporated in the analyses as described later.

4

PROBLEM GEOMETRY AND BOUNDARY CONDITIONS

The mesh used to perform the finite element analyses is shown in Figure 2. The analysis is axi-symmetric and the mesh domain is 50 m deep and 20 m wide. The single pile is 20 m long and its diameter is 1 m. The pile dimensions are chosen arbitrarily, however they are chosen such that the pile represent a typical friction pile. Full adhesion between the pile and soil is assumed. Both the pile and the soil are discretised. The pile is modelled as a very stiff elastic material which is able to transfer load from the pile head to the surrounding soil, whereas the soil is modelled either using the MCC or ET Creep model. The mesh consists of eight nodded isoparametric elements with four pore pressures degrees of freedom at the corner nodes, and two displacement degrees of freedom at both corner and mid-side nodes. The displacement boundary conditions set prevent the base of the mesh from moving in the vertical and horizontal directions, while the vertical sides of the mesh are fixed in the horizontal direction and can move in the vertical direction only. The seepage boundary conditions applied for the coupled analyses allow no flow of water to the base of

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Figure 3. Definition of failure load for ET Creep analysis.

Figure 2. Mesh of single pile.

the mesh, around the pile and to the left hand boundary of the mesh, while on the right hand boundary pore pressures are kept equal to their initial values, as determined by the water table 2 m below ground level. Along the ground surface, next to the pile no flow of water is allowed during the loading stages, aiming to maintain undrained conditions, while in the consolidation and creep periods the pore water pressures are set equal to their initial values. 5

DESCRIPTION OF THE ANALYSES

The approach adopted to determine the effects of time on the axial load capacity of single piles is as follows: (1) a single pile is first loaded undrained to failure in order to establish its initial short term (undrained) load capacity; (2) a further series of analyses is then performed, on the same single pile, in which the pile is first loaded undrained to a percentage (20, 40, 60, 80 or 100%) of its initial short term load capacity; (3) the load is then held constant at this value for a period of 50 years, during which it is ensured that all excess pore water pressures generated during initial loading have dissipated; and (4) additional load under undrained conditions is then applied until failure is reached to determine the new ultimate undrained load capacity. This approach is followed for both the MCC and ET Creep analyses, however due to the time dependent nature of the latter a realistic loading rate is required.

This rate is chosen to satisfy the assumption that the pile is loaded to failure under constant displacement rate over a period of six months. It should be noted though that due to the development of viscoplastic strains and consolidation during loading, the failure load cannot be well defined in the ET Creep model. A coupled analysis is performed in order to obtain the displacement rate at which the increase in the sustained load between a pile displacement δ, after 6 months, and a pile displacement four times δ is smaller than 5%, as shown in Figure 3. This rate is defined as the rate which gives failure in 6 months. It is equal to 0.0004 m/day and is kept constant for all ET Creep model analyses. The amount of viscoplastic strains in the ET Creep model is proportional to the OCR value. The soft clay is assumed normally consolidated, i.e. OCR equal to 1. However, such an OCR value represents the situation when the clay is first formed, resulting in unrealistic initial viscoplastic strains. In these analyses, OCR is chosen to be equal to 1.05, which is consistent with the observations made by Schmertmann (1991), where it is claimed that no real soil can be purely normally consolidated since it has been in place for a certain period of time during which creep has made the soil to appear “overconsolidated”. This is usually termed as “aging preconsolidation”. For the MCC model, analyses to obtain the load capacity are undrained since the interest is in the short term capacity, whereas in those where the load is held constant, until all excess pore water pressures dissipate, are coupled to allow consolidation. On other hand, the ET Creep analyses are all coupled. This is due to the time dependent nature of the model and the relation of strain with time. It should be noted though that rapid loading even in coupled conditions may be considered as undrained, since the period for full dissipation of excess pore water pressure is expected to be longer than the 6 month initial loading period.

6

DISCUSSION

6.1 MCC analysis The first MCC analysis is on the behaviour of the single pile under undrained conditions. In Figure 4, the load-settlement response is plotted for undrained loading. Failure is assumed to be reached when the curve reaches a plateau; this is at 4% of the pile diameter. The

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Figure 4. Load-settlement response for undrained loading.

Figure 6. MCC model: load-settlement response for different levels of pre-loading.

clear that there is no gain in capacity for all levels of preloading. It seems that pile capacity is not enhanced due to the dissipation of excess water pressures generated during initial loading, even though settlements are larger for higher levels of preloading. This s for the excess pore water pressures developed during initial loading. As shown in Figure 5, the magnitudes of the excess pore water pressures, as well as their concentration around the pile toe, indicate that the soil’s strength is enhanced only around the pile toe. Considering also the contribution of the base in respect to the shaft in the load capacity (Fig. 4), one would anticipate that the consolidation processes in the ground do not affect the long term capacity of piles founded in clays. 6.2 ET Creep analysis

Figure 5. Contours of excess pore water pressure at failure.

undrained load capacity of the single pile is 1615.3 kN. The contribution of the base and shaft resistances to the pile capacity is also presented, which shows that the pile works as a friction pile with the capacity mainly being provided by the shaft resistance. Contours of excess pore water pressure at failure are also plotted in Figure 5, which are indicative of the soil area whose strength is enhanced during consolidation. Subsequent analyses for different levels of preloading (20, 40, 60 and 80%) are performed, where the load settlement profiles are plotted in Figure 6. It is

The first analysis using the ET Creep model is to determine the initial load capacity. A constant displacement rate for a period of 6 months is applied, where the load capacity is calculated equal to 2265.6 kN. This value is quite different to the value calculated in the MCC analysis. This is due to the difference of the plastic potential in the deviatoric plane between the two models. In particular, the ET Creep model assumes that the shape of the yield and plastic potential surfaces in the deviatoric plane es through the corners of the Mohr Coulomb hexagon and follows closely its shape, whereas the MCC model assumes that the yield and plastic potential surfaces are given by a Mohr Coulomb hexagon and a circle respectively. Such differences in the plastic potential in the deviatoric plane can cause significant discrepancy in the predicted failure loads (Potts & Gens, 1984). Thus, comparisons can only be made in of the proportional increase in load capacity observed in each set of analyses. Analyses for different levels of preloading (20, 40, 60, 80 and 100%) are then performed. The load

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Figure 8. Gain in load capacity for different levels of pre-loading. Figure 7. ET Creep model: load-settlement response for different levels of pre-loading.

settlement profiles are plotted in Figure 7. The load capacity of the pile is enhanced; it is proportional to the level of pre-loading. 50 years of maintained load on the pile is beneficial to the pile capacity. This is due to the development of viscoplastic strains which enhance the soil’s strength. During that period, due to the coupled nature of the analyses, dissipation of excess pore water pressures generated during initial loading takes place as well; however it has been already shown that this is not enhancing the capacity of the pile due to their concentration around the pile toe. Viscoplastic strains enhance the undrained shear strength of the soil due to the changes in void ratio with time under constant load, making the soil to appear preconsolidated. Higher values of preloading give larger creep settlements and larger load capacities. This is due to the fact that the larger the stress in the soil, the larger the change in strain and void ratio with time due to creep. Settlement is due to consolidation and creep. The contribution of the consolidation settlement is very small in respect to the creep settlement in the soft clay for a preloading period of 50 years. This is indicated in Figures 6 and 7, where the settlement for different levels of preloading for the MCC model is of the order of mm, whereas for the ET Creep model of the order of cm. Creep settlement is due to: (a) settlement of the soil by its own weight, taking place in the entire soil volume, which is accumulated at the pile level, and (b) settlement depending on the level of preloading, where the more loaded the pile, the larger creep settlement. In Figure 8 the normalised load capacity, as predicted by the ET Creep model, is plotted against the initial level of pre-load. Figure 8 also includes the results produced by the analysis using the MCC model. The results show that there is an important increase in the load capacity of pre-loaded piles when soil hardening due to creep is considered. In addition the effect of

creep is found to be more significant at higher levels of pre-load, resulting in an additional increase in load capacity of 29.1% at 100% pre-load.

7

CONCLUSIONS

The results presented in this paper demonstrate the importance of considering the time dependent nature of soils in bearing capacity. Creep in piled foundations in soft clays can give a rise to increased pile capacity and settlement depending on the level of preloading. For instance, the capacity of a typical 20 m long friction pile with 1 m diameter may be enhanced by 29.1% after 50 years of loading at its full capacity. This allows the addition of floors on buildings founded on piles, whose piles have been preloaded for the life span of the building. On the other hand, it seems that consolidation processes in the ground due to initial loading do not enhance the load capacity of piles. REFERENCES Bodas Freitas, T.M. 2008. Numerical modelling of the time dependent behaviour of soils. PhD Thesis. University of London (Imperial College London) Bodas Freitas, T.M., Potts, D.M. & Zdravkovic, L. 2007. A numerical study on the effect of ageing on undrained bearing capacity. 10th Int. Symp. on Numerical Methods in Geomechanics: 419–424 Jardine, R., Chow, F., Overy, R. & Standing, J. 2005. I design methods for driven piles in sands and clays. London: Thomas Telford Mair, R.J., Hight, D.W. & Potts, D.M. 1992. Finite element analyses of settlements above a tunnel in soft ground. Contractors Rep. 265. Crowthorne, England: Transport and Road Research Laboratory Potts, D.M. & Gens, A. 1984. The effect of the plastic potential in boundary value problems involving plain strain deformations. Int. Journal Numerical Analytical Methods Geomechanics 8: 259–286

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Potts, D.M. & Zdravkovic, L. 1999. Finite element analysis in geotechnical engineering: Theory. London: Thomas Telford Schmertmann, J.H. 1991. The mechanical ageing of soils. Journal of Geotechnical Engineering 117(9).: 1288–1330 Tomlinson, M.J. 2008. Pile design and construction practice. 5th edition. London: E & FN Spon, an imprint of Chapman & Hall

Yin, J.-H., Zhu, J.-G. & Graham, J. 2002. A new elastic viscoplastic model for time-dependent behaviour of normally and overconsolidated clays: theory and verification. Canadian Geotechnical Journal 39: 157–173 Zdravkovic, L., Potts, D.M. & Jackson, C. 2003. Numerical study on the effect of preloading on undrained bearing capacity. Int. Journal of Geomechanics 3(1): 1–10

600


Analysis of foundation solution of new building in built-up area Ž. Arbanas, V. Jagodnik & S. Dugonjić Faculty of Civil Engineering, University of Rijeka, Rijeka, Croatia

ABSTRACT: Foundations of a new building in old urban or built-up areas are very demanding and conditioned by eliminating their influence on surrounding buildings. In this paper we are presenting the numerical analysis and design solutions of an accommodation business building foundations, nearby the centre of the city of Rijeka, Croatia. The site of future building was surrounded by old masonry buildings so as a new underground garage. In geological point of view the location is built on the alluvial deposits in the Rjeˇcina river mouth covered with an embankment. Extra complication is insisting on one store underground garage below the ground water level and shallow foundation of the nearby masonry buildings. In the process of selecting the foundation construction, the optimization of possible alternate constructions was conducted. Based on results of the analyses, solution with bored piles to the 22 m deep gravel layer was adopted for the foundation construction designed.

1

INTRODUCTION

Foundations of new building in old urban or built-up areas are very demanding and, as a rule, conditioned by eliminating of influence on surrounding buildings. The existence of free site in old urban area indicates that the unfavourable geotechnical conditions or some other problems on the site are present. Presence of surrounding buildings and the limitation of space for the construction site organization represent the major problems during the design and construction of a new building. The city of Rijeka has been founded in roman time in the estuary of the Rjeˇcina River in the northwest part of the Adriatic Sea. Geographical position affected by forming of an old urban nucleus during the 18th century. Expansion of the city area started in the second half of the 18th century by filing of the Rjeˇcina River estuary (Arbanas et al. 1994a). Intensive expanding of the Rijeka port during the 20th century influenced on the expansion of the built-up area on all disposed sites around the old city centre.The rarest free sites in the urban core of the city of Rijeka are situated on the embankments on border parts of the city. Intensive constructions on one of the free wider areas nearby the old city core started during the last years. The location is surrounded by the old cathedral with significantly leaning bell-tower and old masonry building from the second half of the 19th century, so as some relatively new buildings with contemporary safe foundations and constructions. The construction of the new underground garage under the future square pointed on geotechnical problems of this area, such as unfavourable deformability properties of unconsolidated soil layers, presence of the old waste material and high underground water level with numerous springs and underground flows. The construction of garage complex was stopped several times due to

Figure 1. Photo of the site.

requirements of additional redesign of foundation and underground construction. The last free site in this area was small and surrounded by two old masonry buildings and the new underground garage building, Figure 1. In geological point of view the location is built on the alluvial clayeysilty deposits in the Rjeˇcina River mouth covered with an artificial low compacted embankment filled 450 years ago. Limestone bedrock is positioned from 35 to 55 m below the surface. The ground water level is relatively high, only 1 m below the existing terrain surface. The designed future building is relatively small but because of unfavourable geotechnical conditions and possible influence on the nearby buildings, demands complex foundation analysis and construction. Extra complication is insisting on one store underground garage below the ground water level and below the shallow foundation of the nearby masonry building. The designed foundation construction predicts bored piles to the 22 m deep gravel layer to decrease the settlements and the influence on nearby buildings. In

601

the process of foundation construction optimization, mat foundation and foundation on the piles with different depths reaching the limestone bedrock. Stress strain analyses during the optimization process shown that the settlements caused by bored piles founded on the 22 m deep gravel layer have acceptable values without significant impact on surrounding masonry buildings. The pit construction for underground garage store predicts jet-grouting pile wall. The underpinning of nearby shallow foundation is the standard procedure during the undercutting of the foundation. Design predicts underpinning with jet-grouting technique as a part of open pit construction. The lowering of ground water level is predicted with system of drainage and wells below the pile cap construction during the construction works until the weight of construction overloads buoyancy. The construction works were started in July 2007. During the construction, the observational method according to Eurocode 7 was used (Arbanas et al. 2008a).

2

GEOTECHNICAL PROPERTIES OF THE SITE

The construction site is located in the estuary of the Rjeˇcina River, formed by filling of the river basin after 1550 (Benac & Arbanas 1990). The datum of the site plateau is about 2.30 m above the sea water level. Based on the field investigation and results of geotechnical investigations in the wider area, the geotechnical cross section was determined. The geotechnical cross section consists of thick cover and limestone bedrock. The thickness of the high compressible cover is from 35 to 50 m upwards limestone bedrock. The cover is formed of artificial fill so as gravel and silty-clayey sediment layers. The artificial fill layer consists of heterogeneous silty and clayey material mixed with construction waste from 3.0 to 7.0 m deep. The lower part of the cover is formed from upper well graded to clayey gravel layer (GC/GFs/GFc) with thickness of 5.0 to 7.0 m; high plasticity clays (CH) layer with thickness of 13.0 to 15.0 m, lower well graded gravel (GW) and silty gravel (GFs) layer with thickness of 6.0 to 7.0 m and layer of uniform graded sands (SU) with thickness of 8.0 to 15.0 m. The ground water level is relatively high and caused by the sea level and groundwater flows. Cover layers have unfavourable deformability characteristics in the wider area of the Rjeˇcina River estuary, especially in high plasticity clay layers and uniform graded sands (Pavlovec et al. 1992, Arbanas et al. 1994a, Pavlovec et al. 1998). The consistency of clay layers is liquid to plastic with very low values of deformability modules (Mv < 5 MPa) and low values of strength parameters (φ = 22◦ , c = 3 kPa). The layer of uniformly graded sands are relatively soft to medium compacted and the results of SPT in this layers are NSPT = 8 to 23. Gravel layers are formed in periods with high velocities and flows of Rjeˇcina River and have relatively favourable geotechnical properties

than silty and clayey layers (Arbanas et al. 1994b, Benac et al. 1992). Another adversity on the location is existence of steep slope of the bedrock and quickly increases of the cover thickness from 35 to 50 m towards the Rjeˇcina River Basin. Evident existence of unfavourable geotechnical conditions in the site wider area is clearly visible on the existing buildings from second half of the 19th century. These buildings were spread founded in upper gravel layer and evident damages due existence of significant different settlements are present. Total values of the settlements are also remarkable however these values aren’t wholly known. The confirmation of remarkable settlements is the fact that numerous basements are now below the water level and sink. 3

CONSTRUCTION AND FOUNDATION DESIGN

The five-story business building with layout dimensions of 26 × 34 m was designed on the described site. Building was designed to be placed in the with an old masonry five-story building from the second half of the 19th century, on the south border of the site and nearby the new underground garage facility on the west. According to design, an underground garage store was planed below the underground water level and below the shallow foundations of the nearby building. Regarding the position of the new building and positions of nearby buildings, designed underground level and adverse geotechnical characteristics of the location, it was necessary to analyse the following importance during the construction: •

Bottom of the basement pit lies below the level of the existing shallow foundation of the nearby facility so the design required the underpinning of the existing building before the basement pit excavation. • The basement pit is below the ordinary underground water level so substantial inflow is expected. It was necessary to prevent the lateral inflow in and to reduce the seepage through the bottom layers in the basement pit. • Geotechnical anchors from garage underground construction pervade in space of the future basement pit. • The effect of the construction weigh on soil with unfavourable deformation characteristics can affect significant deformations with adversely impacts on nearby facilities. New additional settlements of the nearby facility would provoke extra damages in masonry constructions that can endanger the stability of existing structures.

4

DESIGN SOLUTIONS

To overload described problems, numerous technical solutions were analysed (Arbanas et al. 2008b). To ensure the stability of the basement pit and to prevent the inflow of the ground water jet grouting curtain

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in walls, pipe axis). Performed analyses implied on necessity of serious stress-strain analysis in the process of selecting of adequate foundation structure to extenuate influence of the new structure on nearby facilities. Results of analyses implied on deep foundation solution. The optimization of foundation chosen process was conducted manageably by following influent factors: impact on nearby structures, technical feasibility and cost price.

5

Figure 2. Schematic ground plan of foundation.

Figure 3. Cross section of the foundation construction of the new building.

construction was designed. The jet-grouting curtain was constructed of the jet grouting piles with a diameter of 1.00 m on a distance of 0.80 m on the diameter of the basement pit, Figure 2. The jet-grouting curtain was reinforced by piling the steel profiles between the certain jet grouting piles to assume bending moments due to active soil pressure on. To reduce the ground water inflow through the bottom of the basement pit the construction of jet grouting curtain was performed up to −8.50 m to the layer of the high plasticity clay, 5 m below the designed pit bottom level. On the with the neighbouring building, jet-grouting piles were bored trough existing shallow foundations and so served to safely underpin the construction during the excavation of the basement pit. The ground water lowering below the level of the basement pit excavation was realized by two wells (drainage shafts) carried out to the datum −6.50 m, connected on drainage system below the mat foundation, Figure 3. Drainage shafts allowed the water extraction and water drainage until the higher buoyancy on the mat foundation was achieved (Arbanas et al. 2008a). Lightly performed analyses implied that the load of the new construction on the simple mat foundation can provoke the soil settlement in order of 20 cm, while the settlements of the nearby facilities could reach settlements up to 8 cm. These settlements can cause serious construction damage as a functioning damage in nearby existing buildings (tilting, cracking

FOUNDATION STRUCTURE MODELLING AND RESULTS

To optimize foundation construction, numerical modelling of possible foundation solutions was carried out. The considered possible foundation solutions can be seen in Table 1. Geotechnical modelling and analyses were performed using software package Geo Studio GEO–Slope, Sigma/W module (GEO Slope 1998). For numerical modelling, the same representative geotechnical cross section was used, Figure 4. Different foundation solutions were modelled using mat foundation model in combination with different piles depths. Geotechnical model have been analysed with finite element method using soil characteristics obtained from geotechnical research works. Using the stressstrain analyses, four solutions of foundation construction have been analysed in variations of shallow mat foundation and deep foundation on the bored piles. Elastic-plastic Mohr-Coulomb material model and rectangular finite element meshes were used in the plain strain models. Because of small distance between the piles, it was possible to use plain model. Number of elements varying from model to model is shown in Table 2. Settlements under shallow mat foundation below the basement on the elastic plate are reaching values from 14 to 26 cm, Figure 5. The most non-convenient situation is on the with existing neighbour building where displacements beneath the edge point reaches 12 to 15 cm. After eliminating of mat foundation, solutions with different depths of bored piles were analysed. Deep foundation solutions purported large diameter bored piles to different layers in geotechnical crosssection. Analyses of bored piles on 13 m depth clay layer resulted with decreasing settlements, reaching values from 14 to 20 cm, Figure 6. On the with existing neighbour building calculating displacements beneath the edge point reached unacceptable values of 8 to 10 cm. Analyses of bored piles in 22 m depth gravel layer resulted with decreasing settlements, reaching values from 2.5 to 5 cm, Figure 7. On the with existing neighbour building calculating displacements beneath the edge point reached acceptable values of 2.5 to 3 cm. Analyses of bored piles founded on limestone bedrock from 28 to 41 m depth resulted with expected small settlement values, from 0.1 to 1.2 mm and

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Table 1.

Possible foundation solutions.

Foundation solution

Depth

Mat foundation Piles to clay layer Piles to gravel layer Piles to bedrock

3m 13 m 22 m 28 to 41 m

Figure 6. Settlements of the bored piles founded in clay layer.

Figure 4. Numerical model for mat foundation. Table 2.

Number of finite elements in the model.

Model type Mat foundation Piles to clay layer (13 m deep) Piles to gravel layer (22 m deep) Piles to bedrock (28 to 41 m deep)

Total number of elements

Number of pile elements

10456 10768

0 225

10768

400

13149

1121

Figure 7. Settlements of bored piles founded in gravel layer.

Figure 8. Settlements of bored piles founded on limestone bedrock.

Figure 5. Settlements caused by mat foundation.

negligible settlement values on the with existing neighbour building, Figure 8. The review of obtained settlements is presented in Table 3. Based on performed analyses it was clear that shallow mat foundation and deep foundation founded in clay layer are unacceptable because of the impact on the neighbour buildings and expected damages. Selection of acceptable and sure solution was based on the cost-benefit analysis and comparison between cost price and cost

of the possible damages recovery on nearby facilities. Estimated cost prices of foundation construction for different solutions so as impact and estimated damages on nearby structures are presented in Table 4. The significant difference in cost between deep foundation on piles founded on the gravel layer and piles founded on the bedrock, excluding the possibility of damages on surrounding facilities, decided in type of foundation construction selection. After the foundation construction with 22 m deep bored piles with base in the gravel layer with plate cap construction was selected, detailed analyses were examined. The analysis of pile group capacity confirmed assumed dimensions and disposition of piles in the foundation structure. Foundation structure contains 55 bored piles with 1.0 m diameter positioned

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Table 3. Settlements estimation for different foundation solutions. Below construction

Below nearby building

Foundation solution

min (cm)

max (cm)

min (cm)

max (cm)

Mat foundation Piles to clay layer (13 m deep) Piles to gravel layer (22 m deep) Piles to bedrock (28 to 41 m deep)

14 14

26 20

12 8

15 10

2.5

5

2.5

3

0.02

0.12

0.02

0.04

Figure 9. Photo of the new building after construction.

Table 4. Estimation of cost price and impact on nearby building. Foundation solution

Cost price (in 1000 EUR)

Estimated damage of nearby building

Mat foundation Piles to clay layer (13 m deep) Piles to gravel layer (22 m deep) Piles to bedrock (28 to 41 m deep)

120 420

Structural damage Structural damage

655

Fissures in mortar

1.065

No damage

below the pillars and walls in the upper frame construction. Piles were in heads connected with plate cap construction to ensure commonly acting and soil-construction interaction. During the design process, it was necessary to analyse some others geotechnical problems such as stability of the jet-grouting curtain, the ground water inflow through the bottom of the basement pit and possible hydraulic heave, pumping impact on the water level lowering and uplift stability of structure after finishing the basement construction.

6

CONSTRUCTION AND OBSERVATIONS DURING CONSTRUCTION

The construction started in July 2007. Due to vicinity of the existing buildings, the works were carefully constructed in the next order (Arbanas et al. 2008a): • • •

Construction of the bored piles. Jet grouting curtain on the edges of the building pit. Construction of the drainage wells and extracting of the ground water. • Excavation of the building pit. • Construction of the drainage system below the cap plate. • Construction of the underground store. Bored piles with diameter of 1.00 m (totally 55) were constructed below the individual pillar point and perimeter walls of the future building reaching the lower gravel layer at depth of 20.00 m.

While building pit construction the jet-grouting curtain on the edges of the pit construction, was carried out. The height of the jet grouting curtain is 7.0 m, and it is extra reinforced with steel profiles I 160 between certain jet-grouting piles. Underpinning of nearby building with jet grouting piles before basement pit excavation was performed nearly vertical. Underpinning jet-grouting piles are the part of the curtain of the basement pit. Before beginning of the excavation works it was necessary to build drainage wells situated in the central part of the pit which enables adequate lowering of the underground water levels beneath the level of foundation plate. Drainage wells were made of perforate tubes with diameter of 1.0 m to 6.50 m under the excavation level. Extracting of the ground water from the drainage wells was taken out continuously to keep the ground water level 1.0 m under design construction. After the construction of jet grouting curtain it was possible to begin the excavation of the basement pit. On the bottom of the excavation, bellow the cap plate the drainage system was placed into the gravel material to drive drainage water towards the drainage wells. In final design, the usage of observational method during construction was predicted according to Eurocode 7: Geotechnical Design (BSI, 2004). For that purpose, in context of the main foundation design, scheme of monitoring and surveying has been done. Using that scheme the construction of monitoring equipment was defined: three vertical inclinometers – sliding deformeters and two clinometers on masonry wall of nearby existing building. Monitoring was conducted in construction phases (Nicholson et al. 1999). The new building was completed in September 2008, Figure 9. Conducted monitoring on installed monitoring equipment is in range with calculated stress-stain analyses. Slope increasing on nearby building is negligible which validates adequately design estimations. Monitoring is still in progress and it will be conducted in long time period of exploitation.

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7

REFERENCES

CONCLUSIONS

Constructions of new buildings on free locations in old urban areas are very rear. The existence of free site in old urban area indicates that the unfavourable geotechnical conditions or some other problems are present. Presence of surrounding buildings and the limitation of space for construction site organization represent the major problems during the design and construction of new buildings. Foundations of the new building in an old urban or built-up area are very demanding and conditioned by eliminating their influence on surrounding buildings. In this paper we are presenting the numerical analysis and design solutions of an accommodation business building foundation nearby the centre of Rijeka, Croatia. The site of future building is surrounded by an old masonry building from the second half of the 19th century and by the new underground garage. In geological point of view the location is built on the alluvial deposits in the Rjeˇcina River mouth covered with an artificial embankment filled 450 years ago. The ground water level is relatively high, only 1 m below the existing terrain surface. The building in construction is relatively small but because of unfavourable geotechnical conditions and unfavourable influence on the nearby buildings demands complex foundation analysis and construction. Extra complication is insisting on the one store underground garage below the shallow foundation of the nearby masonry building. After performed stressstrain analyses and optimization process, designed and constructed foundation assumed bored piles to the 22.0 m deep gravel layer to decrease possible settlements and influence on nearby buildings. The lowering of the ground water was executed with system of drainage and wells below the pile cap construction during the construction works until the weight of construction overload buoyancy. The construction works started in July 2007 and were completed in September 2008. During the construction, the observational method according to Eurocode 7 was used. Conducted monitoring on installed monitoring equipment is in range with calculated stress-stain analyses. Slope increasing on nearby building is negligible which validates adequately design estimations. Monitoring is still in progress and it will be conducted in long time period of exploitation.

ˇ & Jardas, B. 1994a. Geotechnical Arbanas, Ž., Benac, C. properties of the soil in the coastal area of the City of Rijeka. Maritime almanac 32: 467–480 (in Croatian). ˇ Jardas, B. & Marković, A. 1994b. Arbanas, Ž., Benac, C., Geotechnical aspects of expansion of Rijeka Port. Modern traffic 14(5–6): 204–208 (in Croatian). Arbanas, Ž., Jagodnik, V., Pavlić, V. & Goršić, D. 2008a. Conditions of building foundation in old urban areas. In J. Logar, A. Petkovšek and J. Klopˇciˇc (eds), Peto posvetovanje slovenskih geotehnikov, Proc., Nova Gorica, Slovenija, 12–14 June 2008: 175–184. Ljubljana: Slovenian Geotechnical Society. Arbanas, Ž., Jagodnik, V., Grošić, M. & Goršić, D. 2008b. Foundation of new buildings in old urban areas. In M.J. Brown, M.F. Bransby, A.J. Brennan and J.A. Knappett (eds), Second British Geotechnical Association (BGA) International Conference on Foundations – ICOF2008, Proc. intern. symp., Dundee, Scotland, UK, 24–27 June 2008: 975–984. Norfolk: HIS BRE Press. ˇ & Arbanas, Ž. 1990. Sedimentation in the area Benac, C. of the mouth of Rjeˇcina River. Maritime almanac 28: 593–609 (in Croatian). ˇ Arbanas, Ž. & Jardas, B. 1992. The morphoBenac, C., genesis and the evolution of the river mouths in the Kvarner area. In: Proceedings of International Symposium Geomorphology and Sea, Mali Lošinj, Croatia: 37–45. BSI 2004. Eurocode 7: Geotechnical design-Part 1: General Rules, BS EN 1997-1. London: British Standard Institution. GEO-Slope Int. Ltd. 1998. ’s Guide Sigma/W for Finite Element/Deformation Analysis, Version 4. Calgary: GEOSlope. Nicholson, D. P., Tse, C. M. & Penny, C. 1999. The observational method in ground engineering: Principles and applications, Report 185. London: CIRIA. ˇ Galić, D. 1992. SedPavlovec, E., Arbanas, Ž., Benac & C., imentation and deformational properties of the embankment south of the Old City of Rijeka. Maritime almanac 30: 655–677 (in Croatian). ˇ & Arbanas, Ž. 1998. Settlement Pavlovec, E., Benac, C. of coastal area in the City of Rijeka. Gradevinar 50(4): 203–208 (in Croatian).

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Collapse of thin-walled model piles during hard driving J. Bergan Norwegian University of Science and Technology/Det Norske Veritas AS

S. Øren Holo Norwegian University of Science and Technology/Myklebust AS

S. Nordal Norwegian University of Science and Technology

ABSTRACT: In 2002 five out of eight piles collapsed during pile driving at the Valhall-field in the North Sea. The steel pipe piles had an open end where the steel tapered off towards the pile tip. Two committees were established to determine the causes of pile refusal, and it was concluded that this tip geometry was a contributing factor to pile collapse. In this study laboratory work ed by a numerical study has been performed to validate the committees’ conclusions. Model piles with a diameter of 102 mm, and a wall-thickness of 2 mm were used. These piles had the same D/t-ratio as the piles used in the Valhall-field. Two different pile tips where investigated; a flat-ended pile tip and a tapered pile tip. A noncircular or oval flat-ended pile tip was also investigated. One flat ended model pile was instrumented with strain gauges, and the stresses from the measurements were compared to simulations using Abaqus. It was found from the laboratory results and from the numerical analyses that the maximum stress in the flat-ended pile tip during hard driving was 30%–50% higher than the incoming stress wave. Based on this, it is suggested that the incoming stress wave should be limited to 70% of the pile material’s yield strength if such hard driving is likely. A tapered pile tip is an unfortunate construction detail that should be avoided.

1

2 THEORETICAL BACKGROUND

INTRODUCTION

Offshore foundation piles are usually thin-walled open steel piles with diameters up to 2,5 meters. In 2002 the Valhall-field in the North Sea was being upgraded with another platform. The platform was a jacket with 8 piles, 2 in each jacket leg. During driving 5 out of the 8 piles refused in a very dense sand layer at 45 to 55 meters below seabed before reaching their target penetration. The piles in the existing platform were driven without experiencing any problems. The piles in the new platform had a different pile tip than the existing ones. The steel tapered off towards the tip, a construction detail referred to as a chamfered pile tip. Pile tip details are shown in Figure 3. The chamfered pile tip was made in an attempt to prevent plugging of the pile, and therefore make the driving easier. After the incident, two committees were appointed to determine the causes of the pile refusals as documented by BP Norge (2003).The committees did an extensive numerical study, but did not include any laboratory work. In the Master’s thesis of Bergan & Holo (2009) hard driving has been the subject and with special focus on pile tip collapse. The thesis includes laboratory work and back-calculation with the finite element code Abaqus.

The one dimensional stress wave equation is widely used for analyzing the stresses during pile driving:

A solution of eq. (1) was found by D’Alembert.He showed that the displacement, u, at a location, x, at a time, t, in a bar is given by any two functions where x and t is related through the wave velocity, c.

It can be showed that the stresses in the bar can be determined by:

where σ is the compressive stress ρ is the material density v is the material particle velocity The material particle velocity is given by the velocity of the falling ram and the initial value of the stress

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Figure 2. Photos of the rig.

Figure 1. Pile driving test rig.

wave is accordingly only dependent on the fall height, theoretically less or equal to:

After reflection from the pile tip an upward propagating wave will superpose on the downward propagating wave both satisfying eq. (1). Whether the pile tip will reflect a compression or tension wave is strictly determined by the boundary conditions.A free end will reflect a tension wave, while a fixed end will reflect a compression wave. Figure 3. Pile tip details.

3 TEST FACILITY/LABORATORY WORK The laboratory tests were performed in the Geotechnical Divison Laboratory at the Norwegian University of Science and Technology (NTNU). The pile driving test rig is illustrated in Figure 1 and 2, where a weak concrete block is buried in sand in a barrel. The weak concrete is supposed to represent a firm soil layer causing extremely hard driving. The model piles were 3 m long steel tubes with diameter D = 102 mm, wall thickness t = 2 mm and

with a yield strength fy = 235 MPa. The D/t ratio is similar to the piles used in the Valhall-field. Concrete blocks with a varying diameter of 30 to 40 cm over a height of 30 cm were made to represent a hard layer. The concrete’s compression strength was measured in the lab at the Department of Structural Engineering at NTNU before driving. One pile was instrumented with 2 strain gauges (1-LY61-6/120) from HBM to measure the stresses in the pile during driving. The gauges were set up midway down the pile, 1.5 meters from the top.

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Table 1.

Pile driving data.

Pile

Compression strength of the concrete layer

Penetration

no.

Pile tip

MPa

mm

1 2 3 4

Flat-ended Flat-ended Chamfered Ovalized flat-ended

14 34 9 9

230 90 119 150

Collapse

No Yes Yes Yes

Figure 4. Ovalization of pile 4.

Figure 5. Test result from strain gauges.

Two different pile tips were studied; a flat-ended tip and a chamfered tip, Figure 3. A third pile detail, an ovalized flat-ended pile tip, was also studied and the ovalization is illustrated in Figure 4.

4 TEST RESULTS The instrumented pile had a flat-ended pile tip, and a result from the strain gauges is shown in Figure 5. The theoretical maximum stress level of the incoming stress wave according to equation 3 is 200 MPa as indicated in the Figure. The peak measurements show reasonable agreement with the theoretical value of the incoming stress wave. After about 0,9 ms the strain gauges pick up the reflection from the pile tip. This fits well with steel’s wave speed of 5172 m/s. Because of the short pile the compression wave from the tip adds up with the remaining incoming wave. The maximum stress is found in a peak after about 1,7 ms to be about 250 MPa. This is higher than the given yield strength, but pile tip collapse did not occur during this peak. The given yield strength of 235 MPa is a characteristic value, and a real yield strength up to 260 MPa is likely. The reflected wave from the pile tip hitting the concrete is partly in compression and it can be assumed that the stress at the pile tip would be about 300 MPa. This is about 50% above the incoming stress wave and above the steel’s yield stress. The reason why collapse didn’t occur can be explained by the three-dimensional stress

Figure 6. a. Pile tips after pile driving.

state at the pile tip. Some data and test results from the different model piles are shown in table 1. Figure 6a and b shows the pile tips after driving. The piles collapsed in different manners. Pile number 2 experienced an unsymmetrical collapse after the concrete block cracked. Pile number 3’s collapse was symmetrical probably because of the chamfered pile tip and pile number 4 collapsed by gradually pressing the nonsymmetrical cross section together.

5

NUMERICAL MODELLING

The back-calculation was performed with the finite element code Abaqus/CAE ver. 6.7-1 (Simulia (2006)). The purpose of the back-calculation with

609

Figure 7. One of the numerical models used in Abaqus with a steel ram at the top and a concrete block at the lower end.

Figure 6. b. Pile tips after pile driving. Table 2.

Steel Concrete

Material data used in the back-calculation. E modulus

ρ

ν

Uniaxial strength

MPa

kg/m3

–

MPa

210 000 26 200

7 850 2 500

0.3 0.2

250 40

this dynamic, explicit 3D FEM-analysis was to compare selected laboratory results and to study details in the behavior closer. The steel pile was modeled with the material data shown in table 2. A linear elastic, perfectly plastic material model was used for the steel. The numerical model consisted of the ram weight, the helmet, the pile and the concrete block, as shown in Figure 7 (the pile is the long, straight rod shown with the ram at the top and the concrete block at the lower end). Low order 4 noded tetrahedral elements were used for the concrete in order to reduce numerical noise in the analysis to a minimum. (Cook et al (2002)). The condition was defined as general in Abaqus with a friction coefficient of one in the tangential direction. In most of the analyses only one fourth of the pile and concrete was modelled and the model consisted of approximately 450 000 elements.

The concrete was modelled elastic perfect plastic with a yield strength of 40 MPa for the test shown. Due to the high number of elements, only one stroke with the ram was analyzed. In order to validate the measured stresses in the pile the wave propagation caused by the impact of the falling ram was studied as shown in Figure 8. The figure shows the axial stress during the first 0,1 milliseconds after the ram touches the helmet, in a sequence from upper left to lower right. The pile still was completely linear elastic in the top for the particular fall height used for this simulation. The stresses found from the Abaqus simulation at a point half way down the 3 m long pile are shown in Figure 9. At this point strain gauges were positioned and the simulated stresses are shown to give acceptable agreement with the measurements. The numerical model was used to study the stresses at the pile tip, which could not be measured. A comparison of the stresses at a point half way down the pile and the stresses at the pile tip, 1,5 meter further down, is shown in Figure 10. The analysis shows that the stresses at the pile tip are about 30% higher than the stresses in the incoming stress wave. The results in Figures 9 and 10 are for a pile with a flat-ended pile tip. An analysis with the chamfered pile tip was also performed. Based on the result in Figure 5 a yield strength of 250 MPa was assumed for the pile, even though the characteristic value normally used as a design strength for this steel is 235 MPa. For a study of the shape of the permanent deformations this is of minor importance. Figure 11 shows the deformations and mobilization of the Mises yield strength of the steel in the pile tip for an analysis with a chamfered pile tip. The pile tip is bent inward and collapses apparently because of the inward tapered pile tip and the deformation agrees well with the observations in the laboratory as shown in Figure 6b – Pile 3. The picture of Pile 3 is taken at a stage when the deformation has developed

610

Figure 10. Simulated axial stresses at a point half way down the pile compared to the simulated stresses at the pile tip.

Figure 8. Stress wave propagation in the ram weight, the helmet and the pile due to the hammer impact for the first 0,1 ms.

Figure 11. Deformed mesh of chamfered pile tip. The shadings illustrate the utilization of the material in of an equivalent Mises stress.

Figure 9. Back-calculation of the axial stresses at a point half way down the pile.

much further than in the numerical simulation shown in Figure 11. Figure 12 shows a contour plot of the Mises yield stress in the concrete block into which the pile tip is partly embedded. The three dimensional stress state allows for vertical stresses higher than the yield strength of the concrete.

Figure 12. Deformed mesh of the concrete block. The shadings represent the utilization of the material strength.

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6

CONCLUSIONS

The model tests in the laboratory gave a pile tip collapse during hard driving. There can be several reasons for pile tip collapse. During hard driving yielding can occur simply due to too high stresses in the stress wave initiated by the ram. These stresses could theoretically double due to the reflection from a fixed pile tip. Based on our simulations and measurements the pile tip stress is not doubled at the pile tip since the tip is only partly fixed in the hard soil or here weak concrete. The intensity of the incoming stress wave from the impact is mainly related to the impact velocity of the ram or simply the fall height. For flat-ended piles the laboratory work as well as the numerical analyses indicate an increase in pile tip stresses by 30–50% compared with the incoming stress wave. Based on the results in this study, it is suggested that the incoming stress wave should be limited to about 70% of the pile material’s nominal yield strength in order to avoid pile tip collapse, if hard driving is expected. The stresses in the pile tip is found to be substantially higher in a chamfered (tapered) pile tip than in a flat -ended pile tip. The chamfered tip is an unfortunate construction detail which leads to premature pile collapse. As expected an out-of-roundness is also seen to cause premature pile collapse in our tests. This study validates the conclusions from the Valhall-committees that the chamfered pile was a contributing factor of pile collapse at the Valhall-field. A chamfered pile tip cause high inward soil stresses

normal to the chamfered surface, which will tend to bend the tip inward. Permanent deformation may occur and the pile tip may collapse during continuous hard driving. If the soil strength varies along the circumference of the pile tip, a chamfered pile tip also increases the chance of a nonsymmetrical response along the circumference of the tip leading to a non-circular cross section. An oval cross section will during further driving lead to pile collapse. A tapered pile tip is beyond doubt an unfortunate construction detail that should be avoided.

REFERENCES Bergan, J. & Holo, S.Ø. 2009. Kollaps av pelespiss ved hard ramming, MSc thesis NTNU. BP Norge, 2003. Valhall Water Injection Platform Jacket: Investigation into Premature Pile Refusal – A and B reports, BP Norge AS. Simulia, 2006. Abaqus manual version 6.7. Cook, R. D., Malkus, D. S., Plesha, M. E. & Witt, R. J. 2002. Concepts and Applications of Finite Element Analysis, Fourth Edition. University of Wisconsin – Madison: John Wiley & Sons, Inc. Alm, T. & Hamre, L. 1998. Soil model of driveability prediction. Houston: Offshore Technology Conference. API-RP 2a. 2003. API Recommended Practice for Planning, Deg and Constructing Fixed Offshore Platforms – Load and Resistance Factor Design.Pkt G.10.5. American Petroleum Institute. Brush, D. O., & Almroth, B. O. 1975. Buckling of bars, plates and shells. McGraw Hill.

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Dynamic analysis of large diameter piles Statnamic load test K.J. Bakker Delft University of Technology, Delft, The Netherlands WAD32 bv, IJsselstein, The Netherlands

F.J.M. Hoefsloot Fugro Ingenieursbureau, Leidschendam, The Netherlands

E. de Jong Volker Wessels Stevin Geotechniek bv, Woerden, The Netherlands

ABSTRACT: In order to check the bearing capacity of a newly introduced large diameter casing pile, both T analysis and static load testing was applied. In addition to that, two piles where tested using the Statnamic load testing technique. With respect to the interpretation, in addition to the standard Unloading Point method by Middendorp et al. (1992), a dynamic analysis with a dynamic finite element code, i.e. Plaxis, was done; further the static load test results were used to calibrate the soil parameters for the analysis. Comparing the Dynamic load testing results according to Middendorp with the Numerical results; it came forward that some additional mass below the pile tip, more or less moving with the pile, needs to be taken into . Further after Mullins et al. (2002), the damping is calculated for the stationary part of track 4. Overall this gives a better agreement with numerical analysis for the bearing capacity. Based on an extrapolation of the physical test up to deformation required by NEN 6743-1 (2006), an ultimate load bearing capacity of 15.5MN was established.

1

INTRODUCTION

Related to the city extension Leidsche Rijn near Utrecht, The Netherlands, and the extension of the railway between Utrecht and Gouda to four railway tracks, a total of 20 new viaducts need to be built. In order to improve the foundation concept of the classic multi pile pier foundation, an alternative was proposed that includes the use of bored casing piles with a diameter of 1.65 m. This innovation reduces the original design of 48 prefab concrete piles to 6 large diameter casing piles only. Since the bridges are intended to the railway track, the design requirements are very strict. On the one hand the piles should be capable to bear a design load of at the least 12,000 kN, on the other hand the deformation during train ing’s should be less than 10 mm. Since bored piles show a relatively less favourable load-displacement curve compared to driven piles, to improve the stiffness, the introduction of pile tip grouting device adopted. Both the large diameter casing pile in combination with the grout injection device at the tip was a first time application in the Netherlands. For the evaluation of the bearing capacity, as a start T’s after prestressing the grouting device, were taken and evaluated as a first step in checking the influence of the installation procedure. In addition a total of three piles were instrumented and tested on site by means of a static load test

and further the deformation of all piles were monitored during construction and the actual ing of the first trains in December 2007. Details are given in de Jong et al. (2010). In addition, after that it became clear that a suitable testing device would be available; two additional test piles were tested with a 16 MN statnamic device. In addition to the standard procedure of checking these tests using the UPM (Unloading Point Method), by Middendorp et al. (1992), a dynamic axisymmetric Finite Element analysis of the load tests was performed. The result of the static load tests both with respect to load and deformation was used to calibrate the soil parameters for the dynamic numerical analysis. Further the dynamic response as calculated with Plaxis was put into the same UPM evaluation procedure for further corroboration of the method. In this paper a comparison between the UPM evaluation procedure by Middendorp and the Numerical analysis with Plaxis will be described. Attention shall be given to the effect of drained bearing capacity for design and undrained behaviour during the test. Some typical modelling issues such as mesh finesse at pile tip and dynamic effects in interface elements will be discussed. In addition to understanding the dynamic behaviour of the pile, the calibrated numerical model opens up the possibility to evaluate the equivalent static load

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the piles was loaded up to 8,000 kN. More details and results of the Static Pile test can be found in de Jong et al. (2010). After that the Static load tests were done, the introduction of a 16 MN Statnamic device in the Netherlands by Fugro made it possible to test the piles to loads that would approach the ultimate bearing capacity of the piles. In addition to the field test it was decided to try for a dynamic finite element analysis to get a better understanding and to create added value to the field test. The Statnamic load test were done not too far from the site were the Static tests were done, and had a similar soil layering. Therefore the static tests were used as an additional source of information for calibration of the soil parameters for the dynamic analysis. The static analysis was back-analysed and soil parameters were calibrated and applied for the first prediction of the Statnamic test. 3

Figure 1. Stages in statnamic testing.

bearing capacity, which can be compared to the Dutch code for static load capacity NEN 6743 (2006). 2

STATIC LOAD TESTING AND SOIL PARAMETER ESTIMATION

Before it was decided to do statnamic testing a static pile test was done; although it was known in advance that the limit load with respect to bearing capacity could not be reached. At bridge number 6 a row of three, 17.85 m long, 1.65 m diameter piles at a centre to centre distance of 3.75 m where individually tested. The goal of these tests was verification of the load displacement behaviour up to 8,000 kN, i.e. the pile stiffness. Strains were measured at three levels with vibrating wire strain gauges, and a load cell was installed at the top. Further displacements were measured at the pile head by high accuracy levelling. In addition to the deadweight of 1,000 kN the reaction frame was anchored to the ground by 8 Gewi-anchors with a capacity of 1,250 kN each. Prior to pile installation one T was carried out at the centre of each pile. Based on the 3 T’s before pile installation the ultimate bearing capacity of the piles was estimated to be about 20,000 kN. During the static test, the piles were loaded in steps of 1,000 kN.The load was then kept constant for 1 hour, a time that was extended if the deformation rate was above 0.3 mm/h up to a maximum of 4 hours. At a load of 6,000 kN the piles were unloaded to 4,000 kN and reloaded to 6,000 kN in steps of 1,000 kN. Only one of

STATNAMIC TESTING

In November 2008 two Statnamic load tests were carried out on sacrificial test piles that had been installed in 2005. Originally it was intended to test these piles to refusal by means of a static load test. Such a test would have required a load of approximately 20,000 kN, even though the piles have a limited length of 13 m. In a Statnamic load test, pile loading is achieved by launching a reaction mass. Due to the high acceleration of the reaction mass, the effectiveness of the load is increased by a factor of about twenty. The loading is perfectly axial, the pile and the soil are compressed as a single unit and the static load-displacement behaviour of the pile can be determined if a series of tests is performed. Given the fact that the information of the static load test up to a load of 8,000 kN was already available, the two test piles were only tested once with the maximum Statnamic load of approximately 16 MN. The general description of the statnamic test has been given by Middendorp et al. (1992). The test procedure itself is according to the draft European guideline (Hölsher and van Tol, 2009). 4

EMPIRICAL EVALUATION OF BEARING CAPACITIY

According to Middendorp’s (1992) simplified evaluation procedure, that dynamic force balance may be written as: Where Fu (t) = static soil reaction (point and shaft) Fv (t) = damping force (depends on pile velocity) Fa (t) = inertia force, depending on mass and acceleration Both Fstat (t) and Fa (t) have been measured or can be inferred from the measuring data. Whereas Fu (t) is

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the unknown soil resistance we want to establish and further Fv (t) is also relatively unknown and depends on the known pile velocity. However, as the loading time is relatively long compared to the velocity of pressure waves in the pile, which is in the order of 3,800 m/s, whereas the loading time is in the order of 0,08 s, pressure waves may travel up and down the pile, 12 times within this period. For that reason the load may be judged to act semi-static. The effect will be that the pile will displace more or less as a rigid mass. For that reason it is assumed that the time of maximum displacement equals zero velocity of the pile. Realising that at this point; i.e. referred to as the “Unloading Point” the damping is zero, the pile resistance can be calculated as:

Figure 2. Soil resistance as back-analyzed from the Statnamic test, assuming averaged damping in track 4 of the test; concave loading curve is found.

According to the UP method it is assumed that the soil resistance between maximum statnamic force and standstill, has developed beyond the point of maximum bearing capacity commonly referred to as trajectory 4, is nearly a constant. If this is really the case may be disputed. However, if this is assumed equation 1 may be rewritten as

Where C4 = damping coefficient m = Pile mass a(t) = Pile acceleration (measured) Given that Statnamic loading as well as the pile accelerations are measured, the average damping factor can be calculated according to:

With

With the damping coefficient determined, the soil resistance can be back-calculated for the whole track between maximum statnamic loading and maximum displacement. It is customary to back analyse the damping for this whole trajectory and taking the average value. According to this standard interpretation method, the Soil resistance can be back analysed and will result in the curve given in Figure 2. Contrary to expectation, the curve has a concave form; moves upward whereas static loading tests normally show a more convex; i.e. hyperbolic shape. Evaluating this phenomenon in more detail the result seems to be negatively influenced by the averaging procedure for the damping ratio. According to Mullins et al. (2002), a better result can be found by taking the static value of the damping ratio as indicated in Fig. 3, instead of averaging the damping in

Figure 3. Interpretation of Damping coefficient C4, interpretation after Mullins et al. 2002.

the whole trajectory. With this adaptation the result is improved, corroborates better with the numerical results and shows the common observed hyperbolic shape, see also Fig. 10 with the improved curve. Based on the evaluation with the UP method, it was established that the largest soil resistance was reached for a deformation of 47 mm, which is further taken into . Before going into more detail into the final results of the Statnamic test, some details of the numerical analysis will be given, that shed some more light on the way the test results may be interpreted.

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NUMERICAL MODELLING OF STATNAMIC LOAD TEST

For the numerical analysis of the load test an axisymmetrical model was made using dynamics module of the Plaxis 2D V9 finite element code. With respect to the numerical analysis distinction must be made between the prediction for the first statnamic test and back-analysis on the first and the second test.

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Table 1.

Soil type

Soil parameters after calibration with static loading test. Level NAP [m]

γdry kg/m3

Holocene (loose) sand +0.50–−7.10 18 Pleistocene (dense) sand −7.10–down 18

γwet kg/m3

Eref 50 kN/m2

Eref ur kN/m2

19 19

10000 50000

30000 0.7 10−4 150000 0.7 10−4

m [–]

γ0,7 [–]

G0 kN/m2

C kN/m2

37500 1.00 187500 0.03

φ

◦

ψ K0 [–]

◦

28 0 38 8

0.50 0.75

m = power in hyperbolic relation between elastic stiffness and isotropic stress; e.g. 0,5 for sand and 1.0 for clay. γ0,7 = threshold for small strains in the Hssmall model; i.e. the strain at which the shear modulus has reduced to 70% of Gmax . G0 = Gmax = shear modulus at very small strain; may be compared to the shear modulus in dynamic analyses.

With respect to the dynamic prediction for the first Statnamic test, the displacements where overestimated to be between 130 and 180 mm, whereas the actual displacement was about 50 mm. Given this observation, and in order to improve the model the tested data was back-analyzed, to reduce the differences between numerical and field test. For that, a sensitivity analysis with the model, and the applied parameters was performed: – A first observation in this analysis was that the actual loading time during the test was a little shorter than assumed in the prediction, and subsequently contained a little less energy. – Secondly back-analysis indicated that the friction angle, of the Pleistocene sand layer at the pile toe, must have been higher than first assumed, i.e. a better match was found for a friction angle φ = 38◦ instead of 35◦ . – Subsequently the dilatancy angle was adjusted, assuming the common relation ψ = φ − 30◦ , and was increased up to ψ = 8◦ for this layer. – Introduction of the Hssmal, small strain material model, see Benz (2007), improved the results; further the parameters for this material model were evaluated, indicating that with γ0,7 = 10−4 the best agreement was found. – In addition to that, the damping factors where slightly increased up to Rayleigh α = 10−3 and Rayleigh β = 1.75 × 10−3 , see Zienkiewicz et al. (1991).

Figure 4. Simulation of Static analysis of the test pile configuration (13 m pile) with Plaxis; with (upper curve), or without base grouting of the pile toe (lower curve), (Plaxis output loading per radial).

For the prediction of the first statnamic test a relatively detailed soil layering was applied, based on a direct interpretation of the T data in combination with table 1 of NEN 6740 (2006). Further the parameters were calibrated with respect to the static load test, see section 2, carried out in 2005, recognizing that both the piles in the static tests (with a length of 18 m) and those in the statnamic tests (with a length of 13 m), were positioned in the same geologic layering. Further a first analysis of the static bearing capacity of the pile indicated the stiffness effect of the grout bag at the pile toe. From measurements during installation and grouting of the pile, it was known that a pile rise of about 8 mm was observed. In the finite element analysis the effect was modelled as a volume strain in a zone of 17% in a zone of 0.5 m below the pile tip, which gave the same rising of the pile. A first indication of the Static bearing capacity of the pile, with and without grouting at the pile toe is indicated in Fig. 4. The static model test has been performed up to a total settlement of the pile head of 0.1 D (0.165 m) according to the NEN 6743 (2006). The load capacity based on the static numerical analysis indicates a bearing capacity of: Please note that the output of Plaxis, Fig. 4, presents the load per radial, so this result needs to be multiplied with 2π, to get the full bearing capacity of the pile.

In addition to that, the mesh needed to be refined around the pile tip, further some variation in Holocene upper layers was ignored as some of the very soft layers near the soil surface seemed to destabilize the numerical analysis. Due to tensile stresses at the pile delaminating of the interface elements seemed to develop leading to numerical instability. As a solution averaged soil parameters for the Holocene layer were assumed which seemed not to affect the overall result to much, see Fig. 5 and table 1. 5.1 Drained and undrained analysis For the static analysis as indicated in Fig 4 drained analysis was assumed. For the statnamic test, given the large diameter and short duration of the test, only 0,08 sec for the test, undrained behaviour must be assumed. The effect is illustrated in Fig 7, where the

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Figure 7. Static analysis with the optimized Finite element model; drained and undrained.

Figure 5. Finite Element model for dynamic load test.

Figure 6. Characteristic displacement field in Plaxis dynamic analysis, indicating that a zone underneath the pile toe moves more or less with the pile.

results of both a drained and an undrained analysis are given. Due to undrained behaviour the bearing capacity at the toe will be limited due to excess water pressure, whereas due to dilation of the soil at the pile shaft the friction in undrained analysis leads to a higher soil resistance. Fig 7 indicates that undrained behaviour at the shaft friction is dominant. For the interpretation of the dynamic analysis the ratio between the two curves will be taken into . 5.2 Dynamic analysis with plaxis Using the same procedure as for the direct interpretation of the Statnamic test, the bearing capacity was back-analysed from the numerical analysis, i.e. the UPM method as explained in the previous paragraph. Based on several variations of the numerical analysis, and considering that the elements in the numerical model itself do not differentiate between pile elements and soil elements, it became clear that to explain the test properly it is necessary to for some moving soil mass underneath the pile toe. Here, to be conservative, a soil volume of 1 times the diameter and 0.6 times this depth was adopted. To get a full agreement

Figure 8. Results of the dynamic deformation analysis back-analyzed with Plaxis.

with the static analysis the size of this soil mass needed to be increased up to 2 times the pile diameter, which in itself is not unlikely, see Fig 6, but was not applied further, in order to have a conservative result. Compared to more slender piles, the effect of soil mass moving with the pile tip the effect here was more dominant due to the relative large diameter of the pile. The result of the back analysis is indicated in Fig. 8. and Fig. 9. In section 5.1 it was argued why the dynamic analysis was performed undrained whereas for the bearing capacity it is customary to assume drained soil behaviour. The difference between drained and undrained behaviour is partly contributing to what is known as the rate-effect. Here this effect is discounted for by taking the ratio between the drained and the undrained analysis as presented in Fig. 7, for the measured deformation of 47 mm; which gave a reduction factor of 0.84. In Fig. 9, the characteristic load displacement curve based on the back-analysis with Plaxis is given, that may be compared to Fig. 10. The agreement overall is reasonable, the only shortcoming is that it seems to be difficult to represent the elastic unloading at the end, that lags behind in the numerical analysis. This test result in itself may however also be disputed, there is

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correcting for the rate effect as indicated in Fig. 7, the bearing capacity of the pile is approximated as:

Comparing this value with the results of direct static analysis it is concluded that:

6 Figure 9. Results of Numerical simulation of the Statnamic load test on 17 nov. 2008 evaluated with the UP method. To compare with the physical result as indicated in Figure 10.

CONCLUDING REMARKS

– A dynamic analysis of the statnamic pile test helps to explain for the deformation behaviour. – In order to prevent delaminating of interface element at the pile in the softer upper layers it was deemed necessary to apply averaged soil parameters for the whole upper layer. – Rate effects seems to be partially explained by undrained behaviour of the soils and partially by soil mass moving with the pile that needs to be taken into in order to get agreement between numerical and physical test. – In order to get a reasonable corroboration between test and back-analysis one needs to realise that for the latter mean values of soil parameters are needed whereas for design it is customary to use characteristic i.e. conservative values. – Overall the agreement between field test and numerical model seems to be reasonably good. REFERENCES

Figure 10. Hyperbolic extrapolation of the load displacement curve acc to eq. no 8.

reason to assume that the pile tip has come loose in rebound. Based on the UPM evaluation of the numerical analysis a maximum pile load during statnamic testing was found of: Fr,I (δ = 47 mm) = 9.84 MN In order to compare this load with the ultimate bearing capacity according to a T or a static load test, it is necessary to consider not only the drainage effect, but also the fact that the deformations derived with the Statnamic test do not satisfy the necessary deformation for a static load test, that requires a displacement of 0.1 Deq , or in this case 0.165 m. Referring to the load development curves in Fig. 7, an exponential extrapolation is assumed. Further,

Benz, T. (2007), Small Strain Stiffness of Soils and its Numerical consequences, PhD. Thesis Stuttgart University. Middendorp, P., P. Bermingham and B. Kuiper, (1992). Statnamic load testing of foundation piles. Proc. 4th Int. Conf. on Stress Wave to Piles, The Hague. Balkema Rotterdam. de Jong, E., K.J. Bakker & F.J.M. Hoefsloot, (2010), Statnamic load tests on large diameter casing Piles, execution and interpretation, Proc. 11th Deep Foundations Institute Conference, London. Hölscher, P. and A.F. van Tol, (2009), Rapid Load Testing on Piles, CRC Press, Taylor and s, London UK. Mullins, G.C. Lewis & M.D. Justason, (2002). Advancements in Statnamic Data Regression Techniques. ASCE Geotechnical SpecialPublication 116, pp. 915–928. NEN 6743-1 (2006) Geotechniek- Calculation method for foundations on piles – pressure piles (in Dutch), NEN. Zienkiewicz, O.C., Taylor R.L. (1991), The Finite Element Method (4th edition), Volume 2, Solid and Fluid mechanics, Dynamics and Non Linearity. Mc Graw-Hill, Uk.

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Finite difference analysis of pile on sloping ground under ive loading K. Muthukkumaran & M. Gokul Khrishnan Department of Civil Engineering, National Institute of Technology, Tiruchirappalli, Tamilnadu, India

ABSTRACT: Pile foundations are slender structural elements used to transfer loads from structures into deep hard strata below the ground level. It is time consuming and expensive to carry out field test over the piles in larger lengths. Computer simulations of Finite Element/Finite Difference Modelling will allow for in depth studies to analyze the pile – soil interaction of laterally loaded piles on sloping ground under ive loading. This paper presents a three dimensional finite difference analysis for the lateral response of pile located at the crest of slopes under ive loading: 0 degrees, 33 degrees 40 min, 26 degrees 33 min. with relative densities: 30%, 45%, and 70%. Soil stratum is represented elastic-plastic Mohr Coulomb model. FDA and Model test results are compared and analyzed. Conclusions are drawn regarding application of the analytical method to study the effect of slope on laterally loaded pile.

1

INTRODUCTION

Pile foundations can be used to transmit both vertical and horizontal loads. Many pile foundations ing structures such as wharfs and jetties along the coast, offshore structures, bridge foundations, tall structures like chimneys, TV towers, and high rise buildings are subjected to significant lateral forces. The lateral forces are mainly due to the action of wind and earth pressures due to lateral soil movement in case of on land structures. While in the case of coastal and offshore structures, the predominant forces leading to lateral movements are mainly due to waves, currents, winds berthing forces, mooring forces, lateral earth pressure due to unstable slope as a result of dredging or siltation, etc. Many analytical approaches have been developed in recent years for the response analysis of laterally loaded piles. The various analytical and numerical methods that are commonly employed to study the static and dynamic behaviour of laterally loaded piles are (a) Beam – on Winkler foundation approach (b) Elastic Continuum approach; (c) Boundary Element method; (d) Finite Element approach; (e) Finite difference approaches, etc. Most of the approaches consider either the theory of subgrade reaction (Matlock and Ripperger, 1956) or the theory of elasticity (Pise, 1984). However, the load – deflection behaviour of laterally loaded piles is highly nonlinear in nature, and hence requires a nonlinear analysis. Several empirical and numerical methods have been proposed for analyzing the response of single and pile groups to lateral loading from horizontal soil movement. Most of the numerical methods that have been proposed utilize the finite-element method (Carter 1982; Broms et al. 1987, Springman 1989, Goh et al. 1997, Ellis et al. 1999) or the finite difference method (Poulos and Davis 1980). In the approach taken

by Poulos and Davies (1980), the solution is based on a point load in an elastic half-space (Mindlin’s solution) and empirically to for the presence of rigid bearing layer. Prakash and Kumar (1996) developed a method to predict the load deflection relationship for single piles embedded in sand and subjected to lateral load, considering soil nonlinearity based on the results of 14 full-scale lateral pile load tests. Kim and Barker (2002) studied the effect of live load surcharge on retaining walls and abutments. This paper describes a finite difference approach to determine the effect of slope on lateral response of pile in dry sand subjected to lateral soil movement induced by surcharge load.

2

NUMERICAL MODELLING

Numerical models involving FDA can offer several approximations to predict true solutions. Often the problem being modelled is complex and has to be simplified to obtain a solution. Two of the major factors which have a vast impact on both the real and model piles are 1. The constitutive properties of the sand and 2. The soil structure interaction at the interface over the structural surface. The finite difference program FLAC 3D is used for the study. The method of solution in FLAC 3D is characterized by the following three approaches

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•

Finite difference approach (First-order space and time derivatives of a variable are approximated by finite differences, assuming linear variations of the variable over finite space and time intervals, respectively.)

Figure 1. Maximum Unbalanced Force Vs Number of Steps.

Figure 2. 8 Nodded brick element.

•

Discrete-model approach (The continuous medium is replaced by a discrete equivalent—one in which all forces involved (applied and interactive) are concentrated at the nodes of a three-dimensional mesh used in the medium representation.) • Dynamic-solution approach (The inertial in the equations of motion are used as numerical means to reach the equilibrium state of the system under consideration.)

3

DESCRIPTION OF THE APPROACH

FLAC 3D uses an explicit time-marching finite difference solution scheme for every time step, the calculation sequence can be summarized as follows 1. Nodal forces are calculated from stresses, applied loads and body forces (velocity and displacement vary linearly and stress and strain are constant within an element) 2. The equations of motion are invoked to derive new nodal velocities and displacements 3. Element strain rates are derived from nodal velocities 4. New stresses are derived from strain rates, using the material constitutive law The sequence is repeated at every time step, and the maximum out-of-balance force in the model is monitored. This force will either approach zero, indicating that the system is reaching an equilibrium state, or it will approach a constant, nonzero value, indicating that a portion (or all) of the system is at steady-state (plastic) flow of material. In this.

Figure 3. Two nodded pile sel coordinate system and 12 degrees of freedom of the beam finite element.

equilibrium if the net nodal-force vector (the resultant force) at each grid point is zero. Figure 1 shows the variation of the unbalanced force Vs the time step. It is observed the state static equilibrium was reached at the 7000th step at which the unbalanced force is almost zero.

3.2 Modelling with FLAC 3D 3.1

Static solution in FLAC 3D

A static or steady-state solution is reached in FLAC 3D when the rate of change of kinetic energy in a model approaches a negligible value. This is accomplished by damping the equations of motion. A model is in exact

Soil block in FLAC 3D is modelled using brick shaped eight nodded elements (Figure 1) of elastic plastic Mohr-Coulomb model. Piles/beams are modelled as two nodded structural finite element segments (Figure 2).

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Table 3.

List of Analysis.

Soil surface

Relative density (%)

Horizontal 1V:2H 1V:1.5H

30,45,70 30,45,70 30,45,70

Figure 4. Discretized finite element model. Table 1.

Soil Properties. Sand

Material model

30%

45%

70%

Unit

Dry Soil Weight Young’s Modulus Poisson’s Ratio Cohesion Friction Angle Bulk modulus Shear modulus

16 30,000 0.333 0 30 3.0 × 107 1.1 × 107

17 45,000 0.319 0 32 4.14 × 107 1.7 × 107

18.5 60,000 0.291 0 36 4.7 × 107 2.3 × 107

kN/m3 kN/m2 – kN/m2 degree kN/m2 kN/m2

Figure 5. Bending momentVs Depth curve for 1V:2H slope.

5

LIST OF ANALYSIS CARRIED OUT

List of analysis carried out for the various parameters is shown in the table 3.

Ref. Muthukkumaran et al. 2008.

6

Table 2.

6.1 Model validation

Pile Properties.

Description

Pile

Unit

Young’s modulus Poisson’s ratio Cross sectional area Polar moment of inertia Moment of inertia Perimeter

7.65 × 104 0.33 7.54 × 10−5 1.09 × 10−8 5.43 × 10−9 0.07854

kN/m2 m2 m3 m4 m

Ref. Muthukkumaran et al. 2008.

The typical finite element discretization of the model is shown in Figure 4. 4

MATERIAL PROPERTIES

4.1 Soil and structural properties The analyses are conducted with sand of various relative densities of 30%, 45% and 70%. The input parameters for soil and structural elements are taken from Muthukkumaran et al. (2008). The input values of soil and structural elements are presented in Table-1 & 2 respectively.

RESULTS AND DISCUSSIONS

Figure 5 shows the bending moment variation of Muthukkumaran et al. and finite difference analysis. It is observed that the value of the Maximum bending moment obtained by FDA is having good agreement with Muthukkumaran et al. (2008) and the location of the maximum bending moment is also at the same depth. Figure 6 shows the maximum bending variation for the increase in relative density and it is observed that the variation of bending moment obtained through finite difference analysis is similar to that obtained by Muthukkumaran et al. (2008). From these two figures it is observed that the result obtained by finite difference analysis is having good agreement with Muthukkumaran et al. (2008) and hence the developed FD model can be used for further parametric study. The following parameters are considered to study the effect of slope on laterally loaded pile. 6.2 Parametric study From figure 7 & 8 it is seen that the increase in surcharge load increases the bending moment. This is

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Figure 6. Relative Density Vs Maximum Bending moment for 1V:2H for 20 kN load.

Figure 8. Bending moment Vs Depth curve for zero slope and 70% relative density.

Figure 7. Bending moment Vs Depth curve for 1V:1.5H and 30% relative density. Figure 9. Relative density Vs Max. Bending moment for 20 kN load.

due to increase in lateral soil movement. The maximum bending moment of 10900 Nmm is observed for 30% relative density with 1V:1.5H slope and minimum bending moment of 3040 Nmm is observed for 70% relative density with zero slope. The depth of maximum bending moment is occurred at a depth of 14D for zero slope and 16D for 1V: 1.5H and 1V: 2H slope. Figure 8 shows effect of relative density on maximum bending moment for zero, 1V:2H and 1V:1.5H slopes with 20 kN surcharge load. The increase in relative density decreases the maximum bending moment for all three slope angles. Increase of relative density from 30% to 70% decreases the maximum bending moment by 30%, 35% and 46% for 1V:1.5H, 1V:2H and 0 slopes respectively. Figure 10 shows the effect of slope on maximum bending moment for 30%, 45% and 70% relative density with 20 kN surcharge load. The increase in steepness of slope from zero to 1V:1.5H increase the maximum bending moment by 45%, 53% and 55% for 30%, 45% and 70% respectively.

From figure 11 & 12 it is seen that maximum lateral displacement of the pile is maximum at the top and as the slope of the soil increase the displacement of the pile is reduced due to the reduction in the ive resistance. Figure 13 shows the effect of relative density on displacement for zero, 1V:2H and 1V:1.5H slopes with 20kN surcharge load. The maximum displacement of 57 mm is observed for 30% relative density with 1V:1.5H slope and a minimum displacement of 27 mm is observed for 70% relative density with zero slope. Increase in relative density from 30% to 70% decreases the maximum displacement by 42%, 31% and 43% for 1V:1.5H, 1V:2H and 0 slopes respectively. Figure 14 shows the effect of slope on maximum displacement for 30%, 45% and 70% relative density with 20 kN surcharge load. The increase in steepness of slope from zero to 1V:1.5H increases the maximum displacement by 45%, 54% and 46% for 30%, 45% and 70% respectively.

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Figure 10. Steepness of slope angle vs maximum bending moment for 20 kN load.

Figure 11. Displacement Vs Depth curve for 1V:1.5H and 70% relative density.

7

Figure 12. Displacement Vs Depth curve for zero slope and 70% relative density.

Figure 13. Relative density vs Displacement for 20 kN load.

CONCLUSION

The results obtained through the finite difference analysis were compared with the existing results and found to be more reliable and can be applied to study the effect of slope on laterally loaded piles. This paper described a finite difference approach for the lateral response of pile located at the crest of slopes under ive loading under various slopes of 0 degree, 33 degree 40 min, 26degree 33min and for relative densities of 30%, 45%, and 70% were analysed. When the ground surface changes from horizontal to 1V:1.5H, the maximum bending moment was found to be increased by 45%, 53% and 55% for relative densities of 30%, 45% and 70% respectively and the maximum displacement was found to be increased by 45%, 54% and 46%. As relative density increases from 30% to 70%, the maximum bending moment decreases by 30%, 35% and 46% for 1V:1.5H, 1V:2H

Figure 14. Steepness of slope vs Displacement for 20 kN load.

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and 0 slopes respectively and maximum displacement decreases by 42%, 31% and 43%. ACKNOWLEDGEMENTS Funding for these studies was provided by Department of Science and Technology (DST) under Fast Track for Young Scientists Program (SR/FTP/ETA-08/2007), and this is gratefully acknowledged. REFERENCES Broms, B.B., Pandey, P.C. and Goh, A.T.C. (1987) The lateral displacement of piles from embankment loads, Proc., Japan society of Civil Engineers, Tokyo, Japan, 338 8(12), pg: 1–11 Carter, J.P. 1982 A numerical method for pile deformation due to nearby surface loads, Proceedings of 4th Interna tional Conference Numer. Methods in Geo-mech., Vol. 2, 811–817 Ellis, E.A. and Springman, S.M. 2001, Modeling of soilstructure interaction for piled bridge abutment in plane strain FEM analyses, Computers and Geotechnics 28 pg: 79–98

Goh, A.T.C., Teh, C.I. and Wong, K.S. 1996 Analysis of pile subjected to embankment induced lateral soil movement, journal of Geotechnical and geo Environmental Eng., ASCE, 123(9) 792–801 Kim, J.S. and Barker, R.M. 2002. Effect of live load surcharge on retaining walls and abutments, journal of Geotechnical and geo Environmental Eng., ASCE, 127(6), 499–509 Muthukkumaran, K., Sundaravadivelu, R. and Gandhi, S.R. 2008, Effect of slope on P-Y curves due to Surcharge load. Soils and Foundation Vol 28, No. Japanese geotechnical society Matlock, H. and Ripperger, E.A. 1956, Procedure and instruentation for tests on a laterally loaded pile, Proceedings of the 8th Texas Conference on Soil Mechanics and Foundation Engineering, Bureau of Engineering Research, University of Texas, Special Publication 29, 1–39 Prakash, S. and Kumar, S. 1996 Non linear lateral pile deflection prediction in sand, journal of geotechnical Engineers, ASCE, 112: 130–138 Pise, P.J. (1984) Lateral response of free-head pile, Journal of Geotechnical Engineering, ASCE, 110: 1805–1809 Poulos, H.G. and Davis, E.H. (1980) Pile foundation analysis and design, John Wiley and Sons, New York Springman, S.M. 1989 Lateral loading of piles due to Simulated embankment construction, Ph.D. thesis, Univ. of Cambridge, England

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Ground displacements due to pile driving in Gothenburg clay T. Edstam & A. Kullingsjö Skanska Sweden AB, Gothenburg, Sweden

ABSTRACT: A 1 km long highway bridge is under construction in Gothenburg, Sweden. The ground consists of soft clay to a depth of 80 to 100 m. Therefore, the bridge pier foundations will be ed by 50 to 80 m long precast concrete friction piles. During pile driving for one of the bridge piers vertical and horizontal ground displacements were measured on and below the ground surface. The measurements are compared with the response obtained from some empirical and semi-analytical theoretical models as well as the response obtained from three dimensional finite element analyses. Generally, there is a good agreement between the measurements and the models. However, the empirical and semi-analytical models are based on several idealised assumptions which make these methods less suitable for assessing ground displacements in more complex situations. Therefore, additional FE-analyses have been performed in order to estimate how some more complex field conditions affect the ground displacements.

1

INTRODUCTION

Table 1. Typical properties of the clay at the test site.

The Partihallen highway bridge is under construction in Gothenburg, Sweden. The 1 km long bridge will be ed by a large number of bridge piers, founded on precast concrete piles with typical lengths in the range of 50 to 80 m. The piles for the new bridge piers are partly driven close to an existing railway bridge being sensitive to (differential) displacements. In order to gain more insight in how ground displacements, due to pile driving, evolve and may be assessed the study described in this paper was undertaken. The study comprised extensive measurements of evolving ground displacements due to pile driving for one of the bridge piers. Theoretical analyses were also performed, based on some theoretical models reported in the literature. Furthermore, simulations were made in 3D FEM. At the test site the ground surface is almost flat, even though some minor excavations were done before pile driving. The ground consists of soft high plastic marine clay to a depth of more than 80 m, with the ground water level located about 1 m below the ground surface. In Table 1 some typical properties of the clay are given.

Unit weight [kN/m3 ] Water content [%] Liquid limit [%] Sensitivity [−] Undrained shear strength* [kPa] Over consolidation ratio [−]

2

Figure 1. Plan view showing the bridge piers, the measuring lines (N, S and W) and the measuring equipment (open symbol = settlement gauge; filled symbol = settlement gauge & bellow hose & inclinometer).

FIELD MEASUREMENTS

The field measurements were performed in connection to pile driving for the bridge pier foundations A10, A11 and A12, see Figure 1. Focus was put on ground displacements due to pile driving for bridge pier A11, even though the measurements also gave some information on ground displacements during piling for bridge piers A10 and A12.

15 to 17 60 to 90 70 to100 5 to 20 10 to 15 1.1 to 1.5

*Strength at the top of the clay; the increase with depth is typically 1.0 to 1.5 kPa/m.

The instrumentation was installed along three lines N(orth), S(outh) and W(est) and consisted of a total of 6 inclinometers, 6 bellow hoses and 18 settlement gauges. The inclinometers and bellow hoses captured

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Figure 2. Plan view showing the location of the 60 piles ing bridge pier A11. Filled symbols represent the last three piles that were driven.

relative ground displacements down to a depth of about 45 m while the settlement gauges captured displacements at ground surface. In order to capture the absolute displacements below ground surface, the data obtained from the bellow hoses and inclinometers (not reaching firm layers) were adjusted in order to fit the displacements at the ground surface, measured with the settlement gauges. The inclinometers at positions 3, 6, 10 and 12 were of an automatic type which made it possible to measure ground displacements with a time interval of 10 minutes. The other instruments were measured manually as the pile driving evolved, thus necessitating intermediate pausing of the pile driving every now and then. The foundation for bridge pier A11 consists of a total of 60 precast concrete friction piles with a length of 52 m. Each pile has a cross sectional area of 275 × 275 mm2 and a typical spacing of 1.3 m, see Figure 2. Most of the piles have an inclination of 9:1 to 7:1 (vertical to horizontal). The pile installation scheme was planned in such a way as to try to produce a symmetric displacement pattern around the bridge pier. Thus, the piles in row A were driven first, then the piles in row B, etc…, finishing with the piles in row E. Within each row piling started from the central part evolving to the south, then continuing from the central part and evolving to the north. Generally speaking this scheme was followed, but with some minor adjustments due to practical reasons. Some of the bellow hoses and inclinometers were installed and calibrated in the end of December 2008. In early January 2009 the piles for bridge pier A10 were installed. Then, the remaining instruments were installed and all instruments were calibrated. Thereafter, all 60 piles, but three, were driven for bridge pier A11 during a period of two weeks in the beginning of February 2009. During this period a major part of the measurements took place. In order to not delay the construction works too much some of the piles for bridge pier A12 were also installed during this period, even though most of the piles for bridge pier A12 were installed in the end of February. The manual instruments were read at a few occasions until the end of May, while the automatic inclinometers continued ing several times a day. However, no additional

Figure 3. Measured displacements at ground surface; (a) heave; (b) horizontal displacements in parallel with lines N, S and W.

ground displacements were ed after the end of the pile driving for bridge pier A12. As previously mentioned, measurements were made at a large number of occasions and locations before, during and after pile driving. However, in this paper only some of the ground displacements, due to the installation of piles for bridge foundation A11, will be discussed. The measured horizontal and vertical displacements at the ground surface along lines N, S and W are shown in Figure 3, at two stages of the pile installation. In the first stage 18 piles were installed while 57 piles were installed at the second stage. As may be seen the patterns along lines N and S are very similar, as expected. The displacements along line W are larger, at the same distance from the centre of the bridge pier, which again is as expected. The measured horizontal and vertical ground displacements at positions 3 and 6 (cf. Figure 1), after the installation of 57 piles, are shown in Figure 4. Two curves are shown for each measurement at positions 3 and 6. The difference between these curves reflects the accuracy of the measurements made with the settlement gauges.

3

MODELLING OF GROUND DISPLACEMENTS

In order to assess the evolving ground displacements due to pile driving three different approaches,

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Figure 5. The Rehnman-method, which is frequently used in Sweden (after Olsson & Holm, 1993).

Figure 4. Measured displacements at and below ground surface at positions 3 and 6; (a) heave; (b) horizontal displacements in parallel with lines N and S.

requiring very different amount of work effort, were adopted. A first approach, applied before the field measurements took place, was to use an empirical method, which is frequently used in Sweden. The method was originally suggested by Hellman (1981). However, it was developed further by S-E Rehnman at the Royal Institute of Technology, but not published until in Olsson & Holm (1993). In this method it is assumed that the ground surface heave is limited by a line starting at the pile tip and ending at the ground surface, with a 45◦ inclination to the vertical, see Figure 5. Furthermore, it is assumed that the horizontal displacement at the ground surface is equal to the ground surface heave. In the present study it is also assumed that the volume of the displaced soil corresponds to the total volume of the driven piles. The horizontal and vertical displacements at ground surface, δh and δv , can then be calculated as:

where V = total volume of the piles driven into the ground; L = pile length; a, b = horizontal extent of the pile group; and x = distance from the edge of the pile group (0 ≤ x ≤ L). A second approach, partly applied before the field measurements took place, was to use the shallow strain path method (SSPM) introduced by Sagaseta (1987)

and described in more detail in Sagaseta et al (1997). In this method the installation of a pile is approximated to a spherical point source penetrating an inviscid fluid from the ground surface down to the level of the pile tip. In order to for the incompressibility of the fluid a point sink is introduced, moving from the ground surface to a position above ground surface, corresponding to the pile length. In order to model the stress free ground surface corrective shear stresses must be applied radially along the ground surface. The analyses in the present study are based on the small strain version of the SSPM and the assumption that the strains and corrective shear stresses at the ground surface are linked by a linear-elastic relation. The horizontal (radial) and vertical displacements at ground surface, δr and δv , due to the installation of a single pile with a circular cross-section, can then be calculated as:

where R = pile radius; L = pile length; and r = radial distance from the pile. The effect of several piles is ed for by superposition. Unfortunately, closed form solutions do not exist for calculating displacements below ground surface. However, in the present study the equations presented in Sagaseta et al (1997) for a single pile were solved numerically using the engineering calculation software MathCad. Again, the resulting effect of several piles is ed for by superposition. As shown in Section 4 the calculated response, if applying the Sagaseta-method, and the measured response agrees quite well, especially at ground surface. Therefore, several simplifying approaches were tested in order to make the Sagaseta-method easier to apply in practice, as described below. A simplified way to superpose the effect of a large number of piles was tried out. In this approach the 60 piles for bridge pier A11 are replaced by three equivalent “super-piles”. Each super-pile replaces a cluster of piles, whose outside edge forms a square. Each superpile is horizontally located in the centre of the cluster it replaces and with its cross-sectional area being equal to the sum of the cross-sectional area of the piles it

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Figure 6. Calculated heave at and below ground surface at positions 3, 6, and 10, using the Sagaseta-method with various simplifications.

replaces. Of course, this approach cannot be used for assessing ground displacements very close to an individual pile or in between the piles. However, as shown in Figure 6 this approach works very well in the present case. Several piles ing bridge pier A11 are inclined, which is difficult to for when applying the Sagaseta-method. In order to assess the influence of this feature, calculations were made with two different assumptions regarding the horizontal location of the (vertical) super-piles; based on the horizontal location of the pile tops and the pile tips respectively. As shown in Figure 6 the calculated ground displacements differ somewhat, but within acceptable limits. A third approach, applied after the field tests were finished, was to use the finite element method (FEM). Since the situation at hand is three dimensional it was decided to use 3D FEM (PLAXIS 3D Foundation, Version 2.2), but still trying to keep things as simple as possible. Some preliminary tests were done for a single pile using the “volumetric strain” feature available in PLAXIS (Brinkgreve, 2007) and assuming the soil to behave as a linear-elastic solid. Since the results seemed promising further preliminary tests were done, including the introduction of “super-piles”, in the same way as in the Sagaseta-method. Eventually, the installation of the 60 piles for bridge pier A11 was simulated using three super-piles, as indicated in Figure 9. It should be noted that the governing boundary condition is of “displacement type” (volumetric expansion). Thus, the magnitude of Young’s modulus has no effect on the resulting displacements in the surrounding soil.A Poisson’s ratio of 0.495 was used since the soil consists of low permeable clay.

4

COMPARISON BETWEEN MEASURED AND MODELLED GROUND DISPLACEMENTS

The measured and calculated displacements at the ground surface are shown in Figure 7. Both the Sagaseta-method and the 3D FEM simulation seem to capture the measured ground displacements quite well,

Figure 7. Measured and calculated displacements at ground surface along line N and S; (a) heave; (b) in horizontal direction in parallel with lines N and S.

both in of magnitude and pattern. The Rehnmanmethod seems to capture the right order of magnitude fairly good, but not the right pattern. The measured and calculated ground displacements at positions 3 and 6 are shown in Figure 8. Again, both the Sagaseta-method and the 3D FEM simulation seem to capture the measured ground displacements quite well.

5

CALCULATED GROUND DISPLACEMENTS FOR MORE COMPLEX FIELD CONDITIONS

A great advantage of 3D FEM compared to the previously discussed analytical methods is the possibility to for the effect of more complex field conditions. If necessary, a non-linear stress-strain behaviour and time effects may be ed for, at least qualitatively. However, the previously described FE-analyses indicate that very simple assumptions, i.e. a linearelastic soil model, the introduction of super-piles and neglecting of time effects, will suffice for capturing the measured behaviour. Therefore, these simple assumptions were maintained when analysing the ground displacements for some more complex scenarios as described below. A construction is “wished in place” in the 3D FEmodel, see Figure 9, before the installation of the piles for bridge pier A11 is simulated. The construction has an area of 10 × 20 m2 with its centre being located

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Figure 10. Measured shear modulus in Gothenburg clay (from Kullingsjö, 2007).

Figure 8. Measured and calculated displacements at and below ground surface at positions 3 and 6; (a) heave; (b) horizontal displacements in parallel with lines N and S.

Figure 9. View of the 3D FE-model used for modelling pile installation for bridge pier A11, including the effect of a hypothetical bridge pier existing before pile driving.

36 m from the centre of bridge pier A11, along line N (cf. Figure 1). Three scenarios were studied, in which the construction consists of various combinations of the following features: a vertical surface load of 50 kPa; a 0.5 m thick concrete slab; 65 friction piles with a length of 52 m and a spacing of 1.5 m, see Figure 9. The concrete slab is modelled using the “floor” feature while the piles are modelled using the “embedded pile” feature, see Brinkgreve (2007). In both cases, the

Figure 11. FE-calculated ground surface displacements along line N for some hypothetical scenarios.

axial and bending stiffness of the structural are ed for. The introduction of a 50 kPa surface load has no influence on the displacements due to pile driving.This is a direct consequence of the use of the linear-elastic model for describing the soil. However, the deviatoric strain, due to pile driving for bridge pier A11, is in the order of 0.001 in the area of the surface load. For such a small strain the shear modulus of Gothenburg clay does not vary too much, see Figure 10. Thus, the effect of the surface load would probably be rather small if a non-linear material model had been used in the FE-analyses. The introduction of the concrete slab has a negligible effect on the heave at and below the ground surface, see Figure 11 and Figure 12a. The horizontal displacements at the location of the slab (e.g. position 3) will of course be affected at the ground surface, but this effect will decrease with increasing distance from the slab, see Figure 11 and Figure 12b. These effects are a consequence of the relatively low bending stiffness and relatively high axial stiffness of the concrete slab. Founding the concrete slab on friction piles will reduce the heave at ground surface at the location of the slab (e.g. position 3), see Figure 11. This reduction will successively decrease with depth, see Figure 12a. Below a certain depth the heave will increase compared to the situation without piles. This is due to the relatively high axial stiffness of the piles and the load transfer mechanism between the friction piles and the surrounding soil. On the other hand the piles will have

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This is a pragmatic way to make the Sagaseta- method easier to apply in practice, especially if a large number of piles should be ed for. However, super-piles should only be introduced when estimating ground displacements at some distance from the piling area. The ground displacements may also be estimated by 3D FEM, using the concept of “super-piles” and a simple linear-elastic material model. However, for ideal field conditions the benefit of using 3D FEM is limited compared to the Sagaseta-method. 3D FEM has its merits when the field conditions are more complex, as demonstrated for some hypothetical scenarios. The concept of super-piles and the use of a simple linear-elastic material model, will probably suffice also in such a case, if ground displacements at some distance from the piled area are of main interest. However, well-documented case histories are needed in order to judge if this hypothesis is correct. ACKNOWLEDGMENT

Figure 12. Calculated displacements at and below ground surface at positions 3 and 6 for some hypothetical scenarios; (a) heave; (b) horizontal displacements in parallel with line N.

a negligible effect on the horizontal displacements due to the high slenderness of the piles, see Figure 11 and Figure 12b. 6

CONCLUSIONS

Construction works in urban areas require reliable methods for assessing the environmental impact. In this paper the evolving ground displacements due to pile driving are treated. Based on the performed field measurements and analyses the following can be concluded: For ideal field conditions (flat ground surface, homogenous ground conditions, etc), the Rehnmanmethod may be used in order to make a rough estimate of the displacements at ground surface. In order to get a better estimate of ground displacements for ideal field conditions the Sagaseta- method may be used. This method may also be used to estimate displacements below ground surface. When applying the Sagaseta-method a cluster of piles may be approximated with a single “super-pile”.

The financial from the following organisations and companies is greatly acknowledged: The Development Fund of the Swedish Construction Industry, Skanska Sweden AB, The Swedish Road istration, The Swedish Pile Commission, The Swedish Geotechnical Society, Ruukki and finally Infra – Centre of Competence at Chalmers University of Technology. REFERENCES Brinkgreve, R. B. J. (ed.) 2007. PLAXIS 3D Foundation ’s Manual. Hellman, L. 1981. On foundation technique in urban areas (in Swedish; original title Om grundläggningsteknik i tätort). Byggnadskonst, Vol. 73 No. 10: 13–16. Kullingsjö, A. 2007. Effects of deep excavations in soft clay on the immediate surroundings. Analysis of the possibility to predict deformations and reactions against the retaining systems. Diss. Chalmers University of Technology, Gothenburg. Olsson, C. & Holm, G. 1993. Piled foundations (in Swedish; original title Pålgrundläggning). Swedish Geotechnical Institute, Linköping. Sagaseta, C. 1987. Analysis of undrained soil deformation due to ground loss. Geotechnique, Vol. 37 No. 3: 301–320. Sagaseta, C., Whittle, A. J. & Santagata, M. 1997. Deformation analysis of shallow penetration in clay. International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 21 No. 10: 687–719.

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Lateral loading of pile foundations due to embankment construction A. Feddema & J. Breedeveld Deltares Geo-engineering, Delft, The Netherlands

A.F. van Tol Delft University of Technology, Delft, The Netherlands Deltares Geo-engineering, Delft, The Netherlands

ABSTRACT: One of the research projects within the Dutch Delft Cluster research programme is “Lateral loading of pile foundations due to embankment construction”. As part of this research project, centrifuge tests have been performed in Deltares’ geotechnical centrifuge. The results of the centrifuge tests have been evaluated with several calculation methods, such as the Finite Element Method (2D and 3D), in order to validate prediction models for soil deformations and bending moments in foundation piles. In this paper the evaluation results of the centrifuge tests with the Finite Element Method (PLAXIS) are presented.

1

INTRODUCTION

In 2004 GeoDelft/Deltares performed a test series in the geocentrifuge. The objective of these tests was to gain insight in the behaviour of laterally loaded piles, installed next to an embankment at different stages of the embankment construction process. The geocentrifuge tests were conducted to generate an extra case among four field tests that have been used to validate calculation methods used in the Netherlands to predict (1) horizontal soil deformations induced by raising an embankment and (2) the resulting lateral loads on a pile foundation (Feddema et al. 2009). In this paper the measured horizontal soil deformations near a pile, installed in the embankment toe are compared with the horizontal soil deformations predicted with an empirical calculation method (Bourges et al. 1979) and with 2D and 3D FEM analyses with PLAXIS. Since horizontal soil deformations are hard to predict, also this empirical method has been used to asses its accuracy and to compared it with the FEM prediction of the horizontal soil deformations. Moreover, the measured bending moments in the piles are compared to the 2D and 3D FEM predictions. With regard to the dimensions, the prototype dimensions are used in the predictions. 2

CENTRIFUGE TESTS

2.1 Test layout and In the geocentrifuge a model scale of 1:100 has been applied. In order to achieve stress conditions in the geocentrifuge similar as in the prototype situation an acceleration of 100 g was applied. As a result, 1 mm in

Figure 1. Cross section of test lay out with instrumentation.

the scale model corresponds to 100 mm in prototype. Figure 1 shows a cross-section of the scale model. The scale model was built up in a strongbox, which has one side made of Perspex. First a foundation layer of Baskarp sand with a thickness of 50 mm has been applied. Then a 100 mm thick layer of Speswhite clay was installed, which represents the soft Holocene clay and peat layers in the Netherlands. At the start of the centrifuge test the groundwater table was set at a level equal to the top of the clay layer and was maintained at this level during the test. After reconsolidation of the clay at 100 g the actual test started. During the test in the geocentrifuge a 50 mm (5 m in prototype) thick embankment was constructed, by applying an embankment of Baskarp sand on top of the Speswhite clay in five construction stages (see Table 1 for construction stages). During the test two piles were installed in the toe of the embankment; the first pile is installed after reconsolidation of the clay and prior to the embankment construction, while the second one is installed directly after completion the embankment. The piles were installed during the test by pushing them through

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Table 1.

Construction stages in geocentrifuge test.

Table 2.

Embankment load∗ Stage

Description

[m]

[kPa]

Time [days]

1 2 3 4 5 6 7 8

installation pile 1 stage 1 stage 2 stage 3 stage 4 stage 5 installation pile 2 end of test

0 1.8 2.6 3.4 4.2 5 5 5

0 30.6 44.2 57.8 71.4 85 85 85

0 0 7 42 126 371 581 1825

Soil properties of Speswhite clay.

Parameter

Value

Unit

dry/wet unit weight γd /γsat compression ratio CR recompression ratio RR initial void ratio e0 vertical permeability kv effective cohesion c effective angle of internal friction ϕ undrained shear strength cu normally consolidated earth pressure coefficient at rest Knc 0

16/16 0.178 0.0233 1.551 1.8 × 10−4 0 22 11 0.64

kPa – – – m/day kPa ◦

kPa –

∗

the effective embankment load in kPa will decrease during the test due to settlement below the groundwater table. Table 3. PLAXIS parameters for soil properties of Speswhite.

the clay layer 4.1 m into the sand layer with a hydraulic plunger. The top of the piles was fixed between two pressure gauges that allowed for a limited rotation of the pile head. Brass square tubes were used to represent the piles with prototype measurements of the free length of 15.15 m and an outside diameter of 500 mm. These model piles have a bending stiffness (prototype) of approximately 1.39 × 105 kNm2 , which is comparable to an un-fractured prefabricated square concrete pile with a diameter of 500 mm. This type of pile is often used in the Netherlands for bridge abutments. During test the following sensors, of which the positions in the cross-section are indicated in Figure 1, collected measurement data:

Model

– 1 soil pressure meter (GDD1) – 7 piezometers (W1 t/m W7) – 2 model piles equipped with 6 strain gauges each (R1 t/m R6). The horizontal and vertical soil deformations in the clay layer were determined using video analysis of a marker grid that had been applied to the clay. 2.2

Production and properties of speswhite clay

The Speswhite clay used in the test was obtained by enforced consolidation of a 0.29 m thick slurry sample. Therefore an air pressure of 50 kPa was applied on top of the slurry in a consolidation cell.According to (Ladd 1986) this results in an undrained shear strength of the clay of approximately 11 kPa. In the consolidation cell the Speswhite clay slurry sample was compressed with 0.09 m at the end-ofconsolidation. It was assumed that 95% of this (i.e. 0.0855 m) comprised primary compression and only 5% creep deformation. Moreover, from oedometer tests on Speswhite clay from earlier tests it was concluded that the recompression ratio of Speswhite clay should be taken 7.64 times stiffer than the primary compression ratio. Based on results of triaxial tests on Speswhite clay conducted in the past, values for the effective cohesion

Parameter

SS

SSC

HS

dry/wet unit weight γd /γsat [kN/m3 ] modified compression index λ∗ [–] modified swelling index κ∗ [–] modified creep index µ∗ [–] triaxial stiffness Eref 50 [kN/m2 ] oedometer stiffness 2 Eref oed [kN/m ] unload-reload stiffness 2 Eref ur [kN/m ] normally consolidated earth pressure coefficient at rest Knc 0 [–] Poisson’s ratio unloadreload νur [–] cohesion c [kPa] angle of internal friction ϕ [◦ ] vertical permeability kv [m/day] horizontal permeability kh [m/day]

16/16

16/16

16/16

0.085

0.085

n/a

0.022

0.022

n/a

n/a

1.3 × 10−4 n/a

n/a

n/a

1,870

n/a

n/a

1,050

n/a

n/a

1,230

0.64

0.64

0.64

0.2

0.2

0.2

0.5 22

0.5 22

0.5 22

2 × 10−4 2 × 10−4

2 × 10−4

4 × 10−4 4 × 10−4

4 × 10−4

(c ) and angle of internal friction (ϕ ) were assessed. At first, the lateral earth pressure coefficient at rest for the normally consolidated Speswhite clay (Knc 0 ) was taken from (Al-Tabba 1987). Later on, the results from K0 -CRS tests confirmed this value. The soil parameters for the Speswhite clay as mentioned above are summarized in Table 2. In addition, a suitable set of Speswhite clay strength and stiffness parameter values for the FEM analyses (PLAXIS) have been derived from simulation of the clay production process (see Table 3).

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Table 4.

PLAXIS Mohr Coulomb parameters for sand.

Table 5.

Parameter for Bourges & Mieussens method.

Parameter

Value

Unit

Parameter

Value

Unit

dry/wet unit weight γd /γsat Young’s Modulus E Poisson’s ratio ν effective cohesion c effective angle of internal friction ϕ angle of dilatancy ψ horizontal and vertical permeability k

17/19 10,000 0.3 0.1 42 3 1

kN/m3 kN/m2 – kPa

X/L D/B Thickness D of compressible soil layer Total settlement St Initial settlement Si Stability factor F Movement factor λ∗

1.07 0.27 10 0.85 0.09 ≈1 2.38∗∗

– – m m m – –

3

◦ ◦

m/day

∗ λ depends on the factor of safety and the ratio of the length of the slope (L) and the distance between the crown of the embankment and the point of interest (X). ∗∗ extrapolated value.

CALCULATIONS

3.1 Introduction of methods used The following calculation methods are presented in this paper for the calculation of the soil deformations near the piles and the resulting bending moments in the piles, both at the end of the geocentrifuge test: – empirical method of Bourges & Mieussens (Bourges et al. 1979), to predict horizontal soil deformations; – 2D FEM analysis, to predict horizontal soil deformations; – 2D FEM analysis with the pile modeled as a wall using plate elements, to predict bending moments in the piles; – 3D FEM analysis, to predict bending moments in the piles. The Bourges & Mieussens method is prefered to other empirical or analytical methods because of the use of the stability factor as a parameter that determines the horizontal soil deformation. The presented FEM results have been calculated with PLAXIS 2D v.8.6 and PLAXIS 3D Tunnel v.2.0. 3.2 Material and model properties 3.2.1 Plaxis FEM analyses In the 2D PLAXIS analyses the behaviour of the Speswhite clay in the geocentrifuge tests is modeled with three different constitutive models: the Soft Soil (SS), Soft Soil Creep (SSC) and Hardening Soil (HS) model. Therefore the soil properties in Table 2 have been modified to the corresponding model parameter values (see Table 3). For the sand layers the PLAXIS Mohr-Coulomb model has been used (Table 4). The model piles have a bending stiffness of approximately 1.39 × 105 kNm2 and an axial stiffness of approximately 6.34 × 106 kN. For the fixation of the pile heads a rotation stiffness of 50,000 kNm/rad has been used. For the 2D FEM analysis with the piles modeled as plate elements, the bending stiffness of the plate elements is reduced with a model factor S in order to simulate 3D behaviour from:

where EImodel = bending stiffness of the beam element in FEM model (kNm2 /m ); EIpile = bending stiffness of pile (kNm2 ); S = 3D/2D model factor (–); Dpile = pile diameter (m). The bending moment in the piles can be derived from:

where Mpile = bending moment in the pile [kNm]; Mmodel = bending moment beam element [kNm/m ]; S = 3D/2D model factor; Dpile = pile diameter (m). In the Dutch design practice a value for the model factor S ranging from 1.5 till 3 is often used for single piles or pile groups with a large centre-to-centre distance. This value has also been used in the 2D FEM analyses presented in this paper. 3.2.2 Model properties empirical method The required embankment settlement, including creep, for the Bourges & Mieussens method is determined with the calculation program MSettle, in which the Dutch Koppejan method (Visschedijk et al. 2009) is used (see also paragraph 3.3). The values for the parameters required for the Bourges & Mieussens method are summarized in Table 5. In this paper only the horizontal soil deformations at the end of the test are presented. The maximum horizontal soil displacement can be derived from:

where St = total settlement (m); Si = initial settlement (m); ρh;max = maximum horizontal soil displacement (m); λ = movement factor (–); D = thickness of compressible soil layer (m). For compressible soil layers with a relatively uniform stiffness, the distribution of horizontal soil displacements within the compressible layer can be derived from:

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Figure 2. Measured versus calculated embankment settlement.

Figure 3. Measured versus calculated (PLAXIS) free field horizontal soil deformations at the pile location at the end of the test.

where ρh = horizontal soil displacement (m); ρh;max = maximum horizontal soil displacement (m); z = depth (m); D = thickness of compressible soil layer (m). 3.3

Settlement of embankment

In Figure 2 the time-settlement curves of the measured and calculated settlements are presented for the crown of the embankment. The measured settlement at the end of the test is 0.75 m. Figure 2 shows that all calculated settlements are larger than the measured settlements during the whole test. Furthermore the time-settlement curves from MSettle and the Soft Soil model overlap as well as the time-settlement curves for the 2D and 3D Soft Soil Creep model. The difference between measured and calculated values for the final settlement are: for MSettle and the Soft Soil model +15%, for the 2D and 3D Soft Soil Creep +21% and for the Hardening Soil model +32%. From these results it is concluded that for this case the MSettle and the Soft Soil model generate the best results concerning settlements. 3.4

Figure 4. Measured versus calculated free field horizontal soil deformations at the pile location at the end of the test.

and Mieussens method slightly over predicts the maximum horizontal displacement (0.37 m) and the total displacement curve equals the measured curve.

Horizontal soil displacements

In Figure 3 the measured and calculated (PLAXIS) free field horizontal soil deformations near the piles at the end of the test are presented. The measured maximum horizontal displacement is 0.35 m. All PLAXIS models overestimate the horizontal soil deformation at the bottom 3 to 4 m of the clay layer. With the Soft Soil model the smallest horizontal soil deformations are calculated whereas the Hardening Soil model over predicts the horizontal soil deformations. The 2D Soft Soil Creep model gives very good results. The difference between the curves for the 2D and 3D Soft Soil Creep model is probably caused by the difference in element size. Figure 4 shows the measured free field horizontal soil deformations near the piles at the end of the test from the PLAXIS 2D Soft Soil Creep model and the empirical Bourges & Mieussens method. The Bourges

3.5 Bending moments From paragraph 3.4 it is concluded that from the numerical models the Soft Soil Creep model is the most suitable model to predict the horizontal soil deformations. Therefore, this material model has been used to predict the bending moments in the piles with a 2D and 3D FEM model. 3.5.1 Pile installed prior to embankment construction Figure 5 shows measured and calculated bending moments at the end of the test for the pile that has been installed prior to embankment construction. With the 2D model the bending moments are underestimated; with the 3D model the bending moments are overestimated.

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Figure 5. Measured versus calculated bending moments for pile installed prior to embankment construction.

Figure 7. Effective soil pressures on front and back of pile installed prior to embankment construction.

pressures acting on the front (embankment side) and the back of the pile have been analysed.

4.1 Pile installed prior to embankment construction

Figure 6. Measured versus calculated bending moments for pile installed after completion of embankment.

3.5.2 Pile installed after completion of embankment Figure 6 shows measured and calculated bending moments at the end of the test for the pile that has been installed after completion of the embankment. The maximum free field horizontal soil displacement between pile installation and the end of the test is approximately 0.04 m. In the 2D model the bending moments are largely underestimated; with the 3D model the predicted bending moments are very close to the measured values.

4 ANALYSIS OF SOIL PRESSURES ON PILES In order to explain the (large) differences between the results of the 2D and 3D analysis, the effective soil

In Figure 7 the calculated effective soil pressures on both sides of the wall/pile are presented for the pile installed prior to the embankment construction at the end of the test. It shows that the pressure at the front of the pile is higher in the 2D analysis than in the 3D analysis. The largest difference, however, occurs at the back of the pile. In the 2D analysis a considerable counter pressures acts on the pile, whereas in the 3D analysis this counter pressure is very low. This results in a large difference in resulting effective pressure (= front pressure – backpressure) on the pile (see Figure 8). Figure 8 shows that the resulting effective soil pressure in the 3D situation is approximately twice the pressure in the 2D situation, which results in the same difference in the bending moments. This is primarily caused by the difference in pressure at the back of the pile between the 2D and 3D situation. In the 2D situation the wall representing the piles is preventing that the horizontal soil deformations can get past the piles. The piles/wall will bend due to the soil pressures at the front, but at the back of the pile a counter pressure can develop as large as the ive earth pressure. In the 3D situation however, the soil can move around the piles. The piles will also bend due to the soil pressures at the front, but the pile deformation is smaller than the horizontal soil deformations at some distance from the pile. Due to this “drag” the soil pressure at the back of the pile is much lower than in the 2D situation.

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5

Figure 8. Resulting effective soil pressures on pile installed prior to embankment construction.

CONCLUSIONS

1. Based on the results for this case, the empirical calculation method of Bourges & Mieussens seems to be useful in daily practice for predicting free field horizontal soil deformations. 2. In general, the measured horizontal soil deformations can be well predicted by the FEM with soil parameters calibrated from simulation of the clay production process. The PLAXIS Soft Soil Creep model generates the best results for this case. 3. The 2D FEM predictions of bending moments, in which the pile behavior is modeled as a wall using plate elements, are poor. This is caused by the fact that in the 2D FEM model the soil cannot flow around the structure, as it does around a pile. In the 2D FEM model the soil on the ive side of the plate elements will develop a pressure, through which deformation of the plate element is resisted. In real life situations the pressure can be considerably lower. 4. When the centre-to-centre distance between piles is relatively small, the 2D approach may be acceptable. In situations where the soil can move around the piles the model factor used for the reduction of the bending stiffness of the wall in a 2D analysis depends on the type of soil and the situation at hand. Therefore, it is concluded that a 2D FEM approach is not suitable for the prediction of bending moments in piles, that are horizontally loaded by displacing soil. 5. The bending moments in the piles calculated with the 3D FEM model vary from reasonable (±60%) to good (±15%). Therefore, the use of a 3D FEM model is recommended in situations (1) where the soil can move around the pile and (2) risks are considerable. REFERENCES

Figure 9. Resulting effective soil pressures on pile installed after completion of embankment.

4.2

Pile installed after completion of embankment

In Figure 9 the calculated resulting effective soil pressures are presented for the pile installed after completion of the embankment construction at the end of the test. Here the difference between the 2D and 3D calculations is even larger (2 to 4 times) than for the pile installed prior to embankment construction.

Al-Tabba, A 1987, Permeability and stress-strain response of speswhite kaolin, Ph.D. Thesis Cambridge University Bourges, F & Mieussens, C. 1979, Déplacements latéraux à proximité des remblais sur sols compressibles, Méthode de prevision, Bulletin liaison Laboratoire Central des Ponts et Chaussées, 101, mai–juin 1979 Feddema, A & Breedeveld, J. 2009, Lateral loading of pile constructions due to horizontal soil deformations, analysis of case study centrifuge test GeoDelft, report no. 4108220031-v02 Delft: Deltares Ladd, C.C. 1986, Stability Evaluation During Staged Construction, The Twenty-Second Terzaghi Lecture, ASCE 1986 Annual Convention, October 28 1986 Visschedijk, M.A.T. & Trompille, V. 2009, MSettle version 8.2, Embankment Design and Soil Settlement Prediction, Delft: Deltares

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Modelling of piled rafts with different pile models S.W. Lee Geotechnical Consulting Group (Asia) Ltd, Hong Kong

W.W.L. Cheang Plaxis (Asia), Singapore

W.M. Swolfs & R.B.J. Brinkgreve* Plaxis BV, & Delft University of Technology*, The Netherlands

ABSTRACT: There has been an increasing use of three-dimensional finite element analyses to analyse the behaviour of piled raft foundations. The raft-piles-soil interaction can be fully modelled for complex ground conditions and pile arrangements.This paper uses the Plaxis 3D Foundation programme to model the performance of two well-documented piled raft foundations in the Frankfurt clay, . The piles are modelled by solid elements with/without interface elements and embedded piles. The embedded pile approach predicts the raft settlements and the load sharing between the raft and the piles in good agreement with the interfaced solid pile approach. The predictions made by the two approaches fall within +/−10% of the measurements.

1

INTRODUCTION

In traditional piled foundation design it is commonly assumed that the entire design load is resisted by the piles only, although the pile cap/raft is also part of the foundation system. Over the past three decades there has been an increasing recognition of the concept of piled raft foundation, and an increasing number of tall buildings are ed by piled rafts (Katzenbach et al. 2000). A piled raft foundation consists of both a pile cap/raft and piles, where the raft transmits load directly to the competent bearing ground and the piles are used to reduce settlements and differential settlements/tilting, i.e. settlement reducers (Burland et al. 1977). Two key issues associated with the piled raft design are (1) the load sharing between the raft and the piles; and (2) the control of absolute and differential settlements by the settlement-reducer piles (Randolph 1994). The design of piled raft foundations requires methods of analysis which can consider the raft-piles-soil interaction and calculate the load-sharing between the raft and the piles (Poulos et al. 1997). Three broad categories of analysis are (1) simplified calculation methods (e.g. equivalent raft/pier method); (2) approximate computer-based analyses (e.g. plate on springs approach); and (3) more rigorous computerbased methods (e.g. boundary element methods and three-dimensional finite element analysis (3D FEA)). This paper uses the Plaxis 3D Foundation FE programme to investigate the behaviour of piled raft foundations using two well-documented case histories in Frankfurt. The piles are modelled by solid elements

(with or without interfaces) and embedded piles, and their predictions are compared to field measurements and results from other analysis approaches. Discussion is made on the performance of the embedded pile approach. 2

BACKGROUND INFORMATION OF FEA

The two case histories investigated are the Torhaus and Westend 1 high rise buildings on piled rafts in Frankfurt, . The ground conditions of engineering interest comprise quaternary terrace sands and gravels underlain successively by the tertiary Frankfurt clay and the rocky Frankfurt limestone, see Figure 1 (Katzenbach et al. 2005). The Frankfurt clay is a stiff, over-consolidated clay with an undrained shear strength (cu ) of 100 to 200 kPa (Sommer et al. 1985), suggesting that foundation behaviour is dominated by the soil stiffness instead of the soil strength. Figure 2 shows the Young’s modulus (E) profiles of the Frankfurt clay proposed by Amann & Breth (1975) and Reul (2000). In the Plaxis 3D analyses the mechanical behaviour of the Frankfurt clay is modelled by the Hardening Soil (HS) constitutive model. The initial (or input) stiffness profile of the Frankfurt Clay modelled by the HS model is shown in Figure 2, which lies in between Amann & Breth’s and Reul’s profiles. Table 1 summarises the soil input parameters used in the 3D analyses. Apart from the stiffness parameters associated with the HS model, the input parameters in Table 1 are based on Reul & Randolph (2003).

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results to the drained analysis for the long-term condition. Because the input stiffness parameters for the Frankfurt clay already represent a stiff soil, its overconsolidation ratio (OCR) is input as 1. The rafts and piles are modelled as a linear elastic material, with an E of 34 GPa for the rafts modelled using solid elements and E of 23.5 and 22 GPa for the piles ing Torhaus and Westend 1 respectively. The piles are modelled by three different approaches – (1) solid elements with interface elements (SPI) along the pile shafts; (2) solid elements with no interface element (SPNI); and (3) embedded piles (EP). In Case (1) the pile-soil interface strength is specified similar to the surrounding soil strength, considering that the input φ’ of 20◦ for the Frankfurt clay is close to its critical state strength. The embedded pile is a slender beam element connected to the surrounding soil by embedded skin (or shaft) and foot (or toe) interfaces, (Engin et al. 2008). The pile can cross the bulk soil elements in any directions, and new nodes are generated at the interaction of the pile and soil elements. An elastic behaviour is specified for the soil region within the pile diameter to minimise mesh-dependent effects. The embedded pile approach thus allows for modelling of a large number of piles installed in non-symmetrical arrangements with a reasonable/practicable mesh size.

Figure 1. Geology of Frankfurt am Main.

3 TORHAUS Figure 2. Soil stiffness profiles for Frankfurt clay. Table 1.

Soil input parameters.

Parameters

Quaternary sands & gravels

Frankfurt clay

Frankfurt limestone

Soil model Material type γ(kN/m3 ) Eref 50 (MPa) Eref oed (MPa) Eref ur (MPa) m [−] c’ (kPa) φ ’ (Deg) Knc 0 [−] νur or ν [−] pref (kPa)

MC Drained 18 75 – – – – 32.5 0.46 0.25 –

HS Drained 19 38 38 114 1 20 20 0.66 0.20 100

MC Drained 22 2000 – – – 1000 15 0.50 0.25 –

Notes: MC = Mohr Coulomb; γ = unit weight; Eref 50 = secant ref stiffness; Eref oed = oedometer stiffness; Eur = unload-reload stiffness; m = stress dependency stiffness; c’= effective cohesion; φ’= friction angle; Knc 0 = coefficient of earth pressure at rest; ν = Poisson’s ratio; pref = reference pressure.

A drained material with an effective c’ and φ’ is specified for the long-term behaviour of the Frankfurt clay, as Reul (2002) has demonstrated that for the similar subsoil and loading conditions a fully coupled elasto-plastic FE analysis would give similar

Figure 3 shows a schematic diagram of the Torhaus building located in Frankfurt am Main. Detailed information about the geotechnical aspects of the building foundation was given by Sommer et al. (1985) and Reul & Randolph (2003). The 130 m tall building was constructed between 1983 and 1986, and ed by two piled rafts sized 17.5 m × 24.5 m × 2.5 m thick. Each raft has 42 nos of 0.9 m diameter, 20 m long bored piles spaced at 3.8 times pile diameter c/c, and the edge-to-edge distance between the two rafts is 10 m. The groundwater table is located at 3 m below ground level (mbgl), which corresponds to the raft underside. The ground conditions comprise a 5.5 m thick quaternary terrace sands and gravels underlain by the Frankfurt clay and the Frankfurt limestone which is located outside the foundation influence zone. Figure 4 shows the 3D model set-up for the Touhaus piled raft foundation. Taking advantage of two symmetric planes, only one-half of a piled raft is modelled. The number of 15-noded wedge elements is 33 400. The 0.5 m thick soil above the raft top surface is modelled by a surcharge. A settlement-effective building load of 200 MN per raft is applied as a uniformly distributed load (UDL) on the raft top surface. In the embedded pile analysis the unit shaft and unit toe capacities are capped at 150 kPa and 1350 kPa respectively for the 20 m long piles. Figure 5 shows the predicted mesh deformation upon imposition of the 200 MN building load. Figure 6 compares the measured average raft settlement of 124 mm in 1988 with the Plaxis 3D Foundation

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Figure 5. Deformed mesh for piled raft of Torhaus.

Figure 3. Schematic view of Torhaus and piled rafts.

Figure 6. Comparison of raft settlements for Torhaus.

Figure 7. Comparison of Qp /Qt for Torhaus.

piles with interfaces) analysis predicts a raft settlement of 106 mm, which is the closest to the measured 124 mm. Figure 7 compares the measured ratio of the sum of all pile loads over the total load on the foundation system (Qp /Qt ) with the predictions made by Plaxis and Reul & Randolph. The Plaxis SPI, SPNI (solid piles with no interfaces) and EP (embedded piles) analyses predict a Qp /Qt of 0.74, 0.78 and 0.77 respectively, which are higher than the measured 0.67.

Figure 4. 3D model for piled raft of Torhaus.

predictions and Reul & Randolph’s (2003) predictions. Note that Reul and Randolph have used the Abaqus 3D FE programme with the piles modelled as equivalent square piles using solid elements with no interfaces along the pile shafts. The Plaxis SPI (solid

4 WESTEND 1 Figure 8 shows a schematic diagram of the Westend 1 building located in Frankfurt am Main. Detailed

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Figure 8. Schematic view of Westend 1 and piled raft.

information about the geotechnical aspects of building foundation was given by Katzenbach et al. (2000) and Reul & Randolph (2003). The 208 m high office tower was constructed between 1990 and 1993 and ed by a piled raft sized 47 m × 62 m × 3 to 4.65 m thick. The 40 nos of bored piles are 1.3 m diameter and 30 m long. The groundwater table is located at 7 mbgl, and the raft underside at 14.5 mbgl. The ground conditions comprise a 8.5 m thick quaternary terrace sands and gravels underlain by the Frankfurt clay of at least 63 m thickness and the Frankfurt limestone. Figure 9 shows the 3D model set-up for the Westend 1 piled raft foundation.The number of 15-noded wedge elements is 37,400. The 9.85 m thick soil above the raft top surface is modelled by a surcharge. A settlementeffective building load of 957 MN is applied as a UDL within the core area on the raft top surface. In the embedded pile analysis the unit shaft and unit toe capacities are capped at 200 kPa and 1800 kPa respectively for the 30 m long piles. Figure 10 shows the predicted mesh deformation upon imposition of the 957 MN building load. Figure 11 compares the measured raft settlement of 120 mm in mid-1994 with the predictions made by Plaxis, Clancy & Randolph (1993) and Reul & Randolph (2003). Note that the Clancy and Randolph prediction was based a hybrid approach of load transfer treatment of individual piles, together with elastic interaction between piles and between the various raft elements and the piles. The Plaxis SPI and EP analyses predict a settlement of 109 mm and 114 mm respectively, which are close to the measured 120 mm. Figure 12 compares the measured Qp /Qt of 0.50 with the different predictions. The Plaxis SPI and EP analyses predict a Qp /Qt of 0.54 and 0.51 respectively, which are in good agreement with the measured 0.50.

Figure 9. 3D model for piled raft of Westend 1.

Figure 10. Deformed mesh for piled raft of Westend 1.

Figure 11. Comparison of raft settlements for Westend 1.

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Table 4.

Predictability Analyses

Torhaus Westend1 Average

1 2 3 4

+10% +15% +13% +16%

Summary of predicted raft settlements and Qp /Qt Settlement (mm)

Qp /Qt [−]

Analyses

Torhaus

Westend1

Torhaus

Westend1

Measurement Plaxis SPI Plaxis SPNI Plaxis EP R & R (2003) C & R (1993)

124 106 104 105 96 –

120 109 98 114 109 117

0.67 0.74 0.78 0.77 0.76 –

0.50 0.54 0.72 0.51 0.66 0.55

6

Notes: R & R = Reul & Randolph; C & R = Clancy & Randolph.

Table 3. tions.

Predictability/deviation for raft settlement predic-

Predictability Analyses

Torhaus Westend1 Average

1 2 3 4

−15% −15% −23% −16%

5

Plaxis EP Plaxis SPI R & R (2003) Plaxis SPNI

−5% −9% −9% −18%

Plaxis SPI Plaxis EP R & R (2003) Plaxis SPNI

+7% +2% +32% +44%

+9% +9% +23% +30%

with both methods predicting average deviations in the range of +/− 10% from the measurements. For the soil parameters and reasonably fine 3D mesh sizes used herein, the 3D FEA tend to under-predict the raft settlements and over-predict the load portion transmitted to the piles (i.e. higher predicted Qp /Qt ).

Figure 12. Comparison of Qp /Qt for Westend 1.

Table 2. ratios.

Predictability/deviation for Qp /Qt predictions.

−10% −12% −16% −17%

CONCLUSION

This paper uses the Plaxis 3D Foundation programme to analyse the behaviour of the piled rafts for the Torhaus and Westend 1 high rise buildings in Frankfurt, . The piles have been modelled using solid elements with/without interface elements and embedded piles. The embedded pile approach gives very similar predictions to the interfaced solid pile approach, where both approaches predict the raft settlements and the load sharing between the raft and the piles within +/− 10% of the measurements. This demonstrates that the embedded pile approach is technically rigorous for the ground conditions and the vertically loaded piled raft systems considered. For other ground and loading conditions (e.g. combined vertical and horizontal loads), it is always recommended to calibrate the embedded pile model first before using it in the foundation design. REFERENCES

DISCUSSION

Table 2 summarises the predicted raft settlements and the ratios of the sum of all pile loads over the total load on the foundation system (Qp /Qt ). The deviation of the predictions from the measurements in Table 2 can be expressed in of percentage, where a positive/negative percentage represents an over-prediction/under-prediction respectively. A smaller deviation percentage represents a better prediction. The predictability is summarised in Table 3 for the raft settlements and in Table 4 for the Qp /Ot ratios. The Clancy & Randolph (1993) hybrid approach is not included in the comparison because it has not been used in the Torhaus prediction. The results in Tables 3 and 4 show that the performance of the embedded piles (Plaxis EP) is comparable to the solid piles with interfaces (Plaxis SPI),

Amann, P. & Breth, H. 1975. Uber den Einflub des Verformungsververhaltens des Frankfurter Tons auf die Tiefenwirkung eines Hochhauses und die Form der Setzungsmulde. Mitteilung der Versuchsanstalt fur Bodenmechanik und Grundbau TH Darmstadt, Heft 15. Burland, J.B., Broms, B.B. & De Mello, V.F.B. 1977. Behaviour of foundations and structures. 9th Soil. Mech. Found. Engng.; Proc. int. conf., Tokyo 2: 495–546. Clancy, P. & Randolph, M.F. 1993. An approximate analysis procedure for piled raft foundations. Int. J. Numer. Analy. Methods Geomech. 17: 849–869. Engin, H.K., Septanika, E.G. and Brinkgreve, R.B.J. 2008. Estimation of pile group behaviour using embedded piles. 12th Int. Assoc. Comp. Methods Advances Geomech.; Proc. int. conf., Goa, India: 3231–3238. Katzenbach, R., Arslan, U. & Moormann, C. 2000. Piled raft foundation projects in . In J.A. Hemsley (ed.), Design applications of raft foundations: 323–391. London: Thomas Telford. Katzenbach, R., Bachmann, G., Boled-Mekasha, G. & Ramm, H. 2005. The combined pile raft foundations (RF): an appropriate solution for the foundation of high

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rise buildings. 7th Geotechnics in urban areas; Proc. int. geot. conf., Bratislava, Slovak Republic: 47–60. Poulos, H.G., Small, J.C., Ta, L.D., Sinha, J. & Chen, L. 1997. Comparison of some methods for analysis of piled rafts. 14th Soil Mech. Found. Engng.; Proc. int. conf., Hamburg 2: 1119–1124. Randolph, M.F. 1994. Design methods for pile groups and piled rafts. 13th Soil Mech. Found. Engng.; Proc. int. conf., New Delhi 5: 61–82. Reul, O. 2000. In-situ measurements and numerical studies on the bearing behaviour of piled rafts. PhD thesis (in German). Darmstadt University of Technology, .

Reul, O. 2002. Study of the influence of the consolidation process on the calculated bearing behaviour of a piled raft. In P. Mestat (ed.), 5th Numerical Methods in Geotechnical Engineering; Proc. European conf., Paris: 383–388. Reul, O. & Randolph, M.F. 2003. Piled rafts in overconsolidated clay: comparison of in situ measurements and numerical analyses. Geotechnique 53(3): 301–315. Sommer, H., Wittmann, P. & Ripper, P. 1985. Piled raft foundation of a tall building in Frankfurt clay. 11th Soil Mech. Found. Engng.; Proc. int. conf., San Francisco 4: 2253–2257.

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Modelling performance of jack-in piles Sun Jie & Siew-Ann Tan National Uiversity of Singapore, Singapore

ABSTRACT: The installation of jack-in piles involved the quasi-static insertion of a solid cylindrical or square pile into the ground by means of large hydraulic pressures equal to at least twice the design working load of the pile. Instrumented field tests data from several Singapore sites have shown that the pile capacity usually increased significantly beyond three times working load with small displacements of pile head, even though the piles were installed with jack pressures of about twice working load.This paper attempt to develop a new model for Jack-in piles. The pile installation process is modeled as a spherical cavity expansion at the pile tip and the prescribed horizontal displacement at the pile shaft. The numerical cavity expansion model with updated mesh computation is first validated against a closed-form Tresca soil solution with very good agreement. Subsequently, several effective stress computations were made using the Hardening Soil model to examine lateral earth pressure, the shaft shear stress distribution at failure and load-settlement response after pile jacking.The results are compared with centrifuge pile load test with good agreement.

1

INTRODUCTION

The installation of Jack-in piles involved the quasistatic insertion of a solid cylindrical or square pile into the ground by means of large hydraulic pressures. During the installation, the soil around the pile is pushed away and compacted while the stresses surrounding the pile are significantly increased. These complicate the problem significantly. The complexity of the problem and their interaction explain why litte progress has been made to date in modelling of displacement piles in FEM (Broere & Tol, 2006). This paper describes the development of a new FEM numerical model for Jack-in pile in PLAXIS. Firstly, Broere & Tol (2006) method of modeling displacement pile is reviewed and the problem of their model is re-investigated. Then, the numerical cavity expansion model with updated mesh computation is validated against a closed-form Mohr-Coulomb soil and Tresca soil solutions. Finally, the new numerical model is proposed. This model combines the volumetric strain below the pile tip and the prescribed displacement around the pile shaft. Thereafter, the results are compared with centrifuge pile load test and a field pile load test with further discussion.

2

BROERE & TOL ‘S METHOD

Plaxis V9 with updated mesh capability does not allow simulating the actual installation process. In order to overcome the limitations of the code, Broere and Tol proposed a new modelling approach. The pile installation process is simulated directly after the initial stress

Figure 1. Normal and shear stresses after Pile installation (left) and at failure (right) ( Boere & Tol. 2006).

generation by increasing the volume of the pile cluster by the prescribed displacements at the pile-soil boundary. After that the material of the pile cluster is replaced by the linear elastic concrete material and then the interface elements between soil and pile are activated. The stress state obtained in this step is maintained and all displacements are set to zero. Their approach can predicte acceptable bearing capacity of a displacement pile in sand. The shear stress on the pile-soil interface along the pile shaft at failure is shown in Figure 1. As observed, the full shaft friction of the pile at failure increases with depth but decreases again some two meters above the pile tip. The distribution is different from findings by many authors (i.e. Lehane et al (1993), Nicola (1996), and Tomlinson (2001)), showing continued increase

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of shaft friction towards the pile tip. The only similar case is found by Klotz (2001). However, he did not give any explanation on the contrasting observations. Clearly, the stress state around the pile using Broere & Tol’s (2006) method is different from many experimental findings, and their numerical model does not capture the installation effects correctly, although it can reliably predict the total bearing capacity of the displacement pile. The behavior of shaft friction is described with an elastic-plastic model in Plaxis used in the calculation. To distinguish between elastic and plastic behavior the Coulomb criterion is used: Figure 2. Radial stress distribution along pile shaft.

where σh is the normal effective stress on the pile shaft. φi and ci are the friction angle and the cohesion of the interface element respectively. In Plaxis, they are calculated from the strength of the soil using following Equation.

Thus the shaft friction reduction near the pile tip is due to the decrease of normal effective stresses some distance above the pile tip. The following calculations make an attempt to explain why the normal effective stress reduces when Broere & Tol’s (2006) method is used. Three cases are considered: Case 1: the prescribed displacements ux = 15 mm and uy = 0 mm are applied Case 2: the prescribed displacements ux = 0 mm and uy = 400 mm are applied Case 3: the prescribed displacements ux = 15 mm and uy = 400 mm are applied The normal radial stress states along pile shaft obtained in Case1∼3 then are checked and the results are shown in Figure 2. As can be seen from the graphs, Case 1 where zero vertical displacement is applied shows reasonable distribution of normal effective stress along pile shaft. This is similar trend as Mahutka et al’s (2006) calculation. Very high horizontal stresses can be observed at the pile tip. Case 2 where there is only applied vertical displacement at the pile tip shows significant reduction on normal effective stress some distance above the pile tip, from initial value 60 kPa, decreasing to 6 kPa. It is due to the relaxation of the stress occuring as the prescribed displacement is deactived when the pile cluster material is activated owing to the numerical implementation in Plaxis. Thus when these two prescribed displacements are combined together (Case 3), it is not surprising that reduced normal effective stress near the pile tip can be observed (Figure 2). Clearly, applying horizontal prescribed displacement to the pile shaft gives reasonable distribution of normal radial effective stress, while applying simple vertical prescribed displacement to the pile tip to simulate the installation effect will give unreasonable behavior of shaft friction.

The pile/cone penetration can be simulated by expanding a cavity of an initial zero radius or finite radius. Many authors (Vesic.1977, Ladanyi.1961, Randolph et al. 1994 and Yasufuku et al. 2001) believed that the soil displacements in front of the pile/cone tip may be considered closer to those undergoing spherical expansion. Thus spherical cavity expansion will be applied to the soil cluster below the pile tip to simulate the installation effect to the soil below. Prior to that, the numerical model for spherical expansion in Plaxis was tested by comparing computed pressure-expansion curves with those given by closed-form solutions.

3

NUMERICAL SIMULATION OF SHERICAL EXPANSION VS THEORY

Xu (2007) applied volumetric strain to simulate the spherical cavity expansion. The numerical procedures are summarized here. The analysis was performed with an axi-symetric mesh using triangular elements with 15 nodes and 12 gauss stress points. The initial radius of the spherical cavity ‘ao ’ was set at a nominal value of 0.1 m. Xu (2007) suggested that this value was selected so that the variation in initial stresses adjacent to the cavity had minimal effect for the analyses performed. The radius of the mesh domain is 12 m and the height is 24 m. This mesh boundary is sufficient to represent an semiinfinite half-space soil mass for spherical cavity with initial radius is 0.1 m. The cavity pressure-expansion relationship can be obtained by selecting appropriate nodes and gauss points for output. For large strain problem, the Updated Mesh option in Plaxis was used. To validate the accuracy of the numerical model, the pressure-expansion curves derived using Plaxis are compared with the closed-form solutions of Yu and Houlsby (1991) for a Mohr-Coulomb soil model in drained condition and the closed-form solutions of Collins and Yu (1996) for Tresca soil model in undrained conditions. Four cases with the parameters listed in Table 1, were considered. Case 1∼3 are drained conditions while case 4 is undrained condition.

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Table 1.

Case1 Case2 Case3 Case4

Soil parameters in the verification calculations. Po: kPa

ao : m

E: Mpa

ν

φ

ψ

120 120 120 240

0.1 0.1 0.1 0.1

5 50 100 6.5

0.2 0.2 0.2 0.3

20 40 40 0

0 0 10 30

Figure 3. (Continued)

Figure 3. Comparison between closed-form and numerical results.

As can be seen in Fig. 2, the results from FEM calculations showed very good agreement with those from closed-form solutions. The difference between the FEM results and the closed-form solutions is within 5% error. Figure 4. Volumetric strain and prescirbed displacement.

4

NEW MODEL

The spherical cavity expansion will be applied to the soil cluster below the pile tip and the prescribed horizontal displacement is applied at the interface between pile and soil along the pile shaft (Fig. 4). The combination of the spherical cavity expansion at the pile tip and

the cylindrical cavity expansion by prescribed horizontal displacement at the pile shaft is used to simulate the pile installation process. Thereafter, the material of the pile cluster is replaced by the linear elastic concrete material and the interface elements between soil

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Figure 6. Lateral earth pressure after pile jacking along a vertical section. Figure 5. Geometry of the cavity and the pile.

and pile are activated, and the prescribed horizontal displacement deactivated. The relationship between geometry of the cavity and the pile is schematically shown in Figure 5. The ‘d’ is diameter of the pile and ‘a’ is radius of the spherical cavity. Based on the assumption that the angle of the soil wedge, ∠ABC = 45◦ + φ/2. The radius of spherical cavity ‘a’ equals d/2 × tan (45◦ + φ/2).

5

RESULTS AND DISCUSSION

The prediction of performance of a displacement pile is evaluated by comparisons with the centrifuge test. The parameters used in the FEM calculations are the same as those used by Broere & Tol (2006). In order to judge the correctness of the FEM results, the following issues are considered:

Figure 7. Shear stress distribution along the shaft at faiure.

(1) Lateral earth pressure after pile jacking. (2) The shear stress on the pile-soil interface along the pile shaft at failure. (3) The total load capacity of the pile at failure, distribution between the shaft friction and base resistance and the load-settlement curve. The results of the radial stress in the vertical cross section at a distance of 0.3 m from the pile shaft after installation are shown in Fig. 6. As can be seen, very high horizontal effective stresses can be found at the pile tip. Below the peak level, the radial stresses reduces to a value below Ko values. This distribution of radial stress is in accordance with findings by Mahutka et al. (2006). The distribution of shear stress on the pile-soil interface along the pile shaft at failure are shown in Fig. 7 compared with the results form Broere & Tol’s (2006) method and the design approach proposed by Randolph et al. (1994). It shows shear stress calculated from new model generally increases with depth and has a very high value near the pile tip. This pattern is similar to those from design approach proposed by Randolph et al. (1994). While Broere & Tol’s (2006)

Figure 8. Load-settlement curve with new model compared to Broere & Tol’s (2006) method.

method gives a different trend near the pile tip with the shear stress decreased to zero at the pile tip, unlike real experimental findings. Fig. 8 shows the load-settlement curve for the new model and Broere & Tol’s (2006) method, compared to centrifuge test. As can be seen, the load-settlement curve from the new model seems to agree better with the test results. Both the shaft friction and

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Table 2. FEM results from different models compared with test results. Ftotal : MN Fshaft : MN Fbase : MN Broere & Tol’s method 2.33 New model 2.30 Centrifuge test 2.34

1.07 1.10 1.12

1.26 1.20 1.22

REFERENCES

base resistance predictions are better than those from Broere & Tol’s (2006) method (Table 2). 6

conditions in order to the general applicability of this new model for Jack-in piles installations. This research is ongoing at NUS.

CONCLUSIONS

The stresses increase due to pile installation is modeled with a few relatively simple steps. Applying prescribed horizontal displacement cylindrical cavity expansion) to the pile shaft gives reasonable distribution of induced normal radial effective stress around the pile shaft, while applying simple vertical prescribed displacement to the pile tip to simulate the installation effect will give unreasonable behavior of shaft friction near the pile tip compared to experimental results. Contrasted to Broere & Tol’s (2006) method, the proposed method applies spherical cavity expansion to the soil cluster below the pile tip instead of prescribed vertical displacement; and the prescribed horizontal displacement is applied at the interface between pile and soil along the shaft similar to Broere & Tol’s procedure. The results of the new model showed reasonable stress state around the pile shaft and pile tip, and are in closer agreement with experimental findings. Prediction of bearing capacity can be obtained using this new model and the distribution between base resistance and shaft friction obtained is reasonably good. Further tests of the simulation scheme would be needed for different pile geometries and different soil

W. Broere, A. F. v. T. (2006). “Modelling the bearing capacity of displacement piles in sand.” Proceedings of the Institution of Civil Engineers: Geotechnical Engineering 159(3): 195–206. Ladanyi, M. B. 1961. Discussion. In Proceedings of the 5th International Conference on Soils Mechanics and Foundation Engineering. pp. 270–271. X. Xu , B. M. L. (2008). “Pile and penetrometer end bearing resistance in two-layered soil profiles.” Geotechnique 58(3): 187–197. K.-P. Mahutka, F. K., J. Grabe (2006). Numerical modelling of pile jacking, driving and vibratory driving. Proc. Int. Conf. Numerical Simulation of Construction Processes in Geotechnical Eng. for Urban Environment Bochum, . B. M. Lehane, R. J. J., R. Frank (1993). “Mechanisms of shaft friction in sand from instrumented pile tests.” Journal of Geotechnical Engineering 119(1): 19–35. De Nicola A., R. M. F. (1999). “Centrifuge modelling of pipe piles in sand under axial loads.” Geotechnique 49(3): 295– 318. Klotz, E. U., Coop, M.R. (2001). “An investigation of the effect of soil state on the capacity of driven piles in sands.” Geotechnique 51(9): 733–751. Randolph, M. F. (1994). “Design of driven piles in sand” Geotechnique 44(3): 427–448. Vesic, A. S. (1970). “Tests on instrumented piles, Ogeechee River site.” Journal of the Soil Mechanics and Foundations Division 96(2): 561–584. Yasufuku N., Ochiai H. & Ohno S. (2001) “Pile end-bearing capacity of sand related to soil compressibilty” Soil & Foundations. 41(4): 59–71. Yu, H.-S. H., G.T. (1991). “Finite cavity expansion in dilatant soils: loading analysis.” Geotechnique 41(2): 173–183. Yu, H.-S. (2000). Cavity expansion methods in geomechanics. Boston, Kluwer Academic Publishers.

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Numerical analyses of axial load capacity of rock socketed piles in Turkey M. Kirkit, H. Kılıç & C. Akgüner Yıldız Technical University, Istanbul, Turkey

ABSTRACT: High rise residential and commercial buildings are increasingly being constructed in urban areas of Turkey. Rock socketed piles are typically selected when large loads due to superstructures need to be transferred to competent bearing strata and restrict deformations within the serviceability limits of such structures. The axial load carrying capacity of rock socketed piles can be estimated utilizing various approaches: static analyses, information/data collected from pile load tests, numerical methods and empirical approaches that take into the engineering properties of rocks surrounding the pile. In this study pile capacity of two rock socketed pile from Turkey were estimated using the finite element approach. A geometric model was constructed for the pile foundation which involved the pile diameter and length and the socketed length. The Mohr-Coulomb and Hardening Soil model was utilized to investigate the effective soil parameters. The loaddisplacement behavior from pile load tests were compared with those obtained from finite element analyses.

1

INTRODUCTION

Rock socketed piles are typically selected when large loads of superstructures, such as high-rise buildings, tower structures, and bridge footings/abutments, need to be transferred to competent bearing strata so as to restrict deformations within the serviceability limits. Furthermore, the use of drilled piles socketed into rock as foundation structures is one of the best solutions when layers of loose soil overlie bedrock at shallow depths. In these cases, considerable bearing capacity can be ensured by shaft friction in rock, even with small pile displacements (Carrubba 1997). Piles can be classified based on the expected governing load-transfer mechanism (CFEM 2006): a) at the tip of the pile, b) on the pile shaft, c) both at the tip and on the shaft. The axial load carrying capacity of rock socketed piles can be estimated by applying static analyses, information/data collected from pile load tests, numerical methods and empirical approaches. Load tests are conducted to determine the in-situ bearing capacity and the load-deformation behavior of piles. Pile load testing provides the most reliable information for design because it is a large scale, if not full scale, model for the behavior of a design pile in actual soil conditions. The acceptance of numerical analyses in geotechnical problems is growing and finite element calculations are increasingly being used in the design of foundations. Pile behavior has been widely investigated using numerical analysis. Various modeling techniques of the pile–soil interface have been reported in literature (Seol et al. 2009).

In this study, two rock socketed piles constructed in Turkey were analyzed with the finite element method. A geometric model was developed to investigate the load-displacement behavior, which was then compared with the results of pile load tests. The effect of varying material properties and models were determined based on parametric studies and analyses involving MohrCoulomb and Hardening Soil models. 2


Two dimensional numerical analyses of piles with a circular cross-section were conducted with Plaxis 2D V9 finite element code. The pile behavior was assumed to be linear-elastic. The soil was described by the Mohr-Coulomb (MC) and by the Hardening-Soil (HS) models. The yield criterion for both models is an extension of Coulomb’s frictional law to general states of stress. The Mohr-Coulomb yield condition is specified by the friction angle φ and the cohesion c . Additionally, the dilatancy angle ψ is used to model dilatant behavior. In contrast to the MC model, a cap type yield surface is introduced in the HS model to calculate irreversible strains due to primary isotropic compression. This yield cap describes an ellipse in p - q modified plane for stress paths, where the top of the ellipse lies on the q-axis. Furthermore, the HS model involves a family of subsequent shear yield surfaces to estimate irreversible strains due to deviatoric loading, which may expand up to the Mohr-Coulomb yield surface (Schanz 1998; Wehnert and Vermeer 2004). The elastic parameters for the MC model require at least the two following parameters: E (Young’s modulus), ν (Poisson’s ratio). The HS model necessitates three stiffness parameters. The stiffness in the MC model is constant,

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Figure 2. Extended of interface element. Table 1.

Figure 1. Geometric model based on the pile diameter, pile length and socket length.

while the stiffness of the HS model is stress dependent. Details of the models are given by Wehnert and Vermeer (2004). 2.1

Geometric Model and Material Parameters

Fifteen-node triangle axisymetric elements were used in analyzing the concrete shaft and the soft rock mass, thus providing a fourth order interpolation for displacements. Interface elements were used to simulate the bonding and separation between the concrete and the rock mass. Fine meshes were used in regions of high stress gradient near the interface (Ls from the axis of the socket) as shown in Figure 1, in which D is the pile diameter, Ls is the socket length and L is the total length of the pile. The radial displacements are restricted to 30D from the axis of the socket. Both the radial and vertical displacements are restricted to 1.5L beneath the tip of the socket, as shown in Figure 1. The boundaries were selected because larger lateral and vertical geometries did not have any effect on the results. Distributed loads were applied to the piles. The critical point in numerical analyses of piles is to correctly model the pile-soil-pile interface. Interface elements were used in this study to separate the piles and the soil (clay, sand, silt, etc.) or rock environments. An elastic-plastic model based on the Coulomb criterion was used to describe and model the behavior of rock-pile and soil-pile interaction in Plaxis. The strength properties of interfaces were linked to the strength properties of the rock layers. Each data set had an associated strength reduction factor, Rinter for interfaces in Plaxis as follows:

Material parameters of pile.

Parameter

Description

Material Model Unit Weight (γ) Modulus of Elasticity (E) Poisson’s Ratio (ν)

Lineer – Elastic 20.4 kN/m3 30 GPa 0.15

Wehnert and Vermeer (2004) studied the effect of mesh “fineness” for bored piles. Very fine mesh and very coarse mesh were used in their analysis.They concluded that the mesh effect can be neglected for shaft resistance and a small change occurs for base resistance. In this study, a fine mesh was used in the area near the pile for increased precision and a medium mesh was utilized in other areas of the numerical model. Since Plaxis may cause stress oscillation at the edge of the socket base, an extended interface zone of two meters was used at the tip of pile as shown in Figure 2. These elements enhance the flexibility of the finite element mesh, and thus, prevent non physical stress results (Brinkgreve et al. 2004). The concrete pile was assumed to be an isotropic, homogeneous and elastic solid with a Poisson’s ratio of ν = 0.15, which is typical for drilled shafts (Nam 2004). An elasto-plastic Mohr-Coulomb model was selected to represent the rock mass because of its simplicity and easy application. The material properties of the piles for the numerical analyses are presented in Table 1. The material parameters for the soils (clay, sand, silt, etc.) and rock were obtained from site investigations conducted as part of the pile load testing. The modulus of elasticity was determined from pressiometer tests and the shear strength parameters were found from laboratory tests on undisturbed soil and rock samples. 2.2 Analysis phase Plaxis analyses were run in two phases. The at-rest stresses of the system under its own loads prior to pile construction were calculated in the first phase. Pile construction and applied loads were examined in the

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Table 2.

Properties of test piles.

Load Test

City

D (m)

L (m)

Ls (m)

LT – 1 LT – 2

Ankara Mersin

0.8 0.9

10 20

1.5 8.5

second phase. The at-rest earth pressure coefficient, Ko was calculated as Ko = 1 − sin φ (Brinkgreve et al. 2004). The structural loads, i.e., loads involved during pile testing, were gradually applied to piles by the “Total Multipliers” option. The displacements during pile load testing were measured at the pile head. Similarly, the calculated loads and displacements at the pile head were considered in numerical analyses. 2.3

Pile load tests, numerical model and analyses

The properties of the two test piles are presented in Table 2. The soils for pile load test LT-1 consist of a marl deposit underlying 8.5 m thick silty clays. The displacements began increasing at approximately 3100 kN applied load and reached failure at about 5400 kN (Çalı¸san & Özkan 1996). Slow maintained load testing (SML) was conducted in accordance with ASTM-D 1143. The soil layering for LT-2 consist of (from the surface down) 8.5 m alluvium overlying 2.0 m conglomerate followed by a basal claystone (Yıldırım 1994). The numerical model based on the dimensions of Pile LT-1 and the corresponding finite element mesh to analyze the rock socketed pile load testing is shown in Figure 3, which was similar for Pile LT-2. (Kirkit 2009). The material properties obtained from in-situ investigations were utilized in numerical analyses as summarized in Table 3. Various material models were selected in analyses to identify the most appropriate model corresponding to the encountered soil type. Initially the Mohr-Coulomb model was chosen for the material behavior of the soils and rocks, which was followed by analyses with the HS model for the soils and the MC model for the rocks. Analyses were repeated with and without interface elements to determine their effects on the outcome. Furthermore, parametric analyses were conducted for Pile LT-1 to identify how the changes in cohesion c, internal friction angle φ and modulus of elasticity E vary the obtained results when the MC model was applied to soils and rocks. The calculated and measured load-displacement curves for Pile LT-1 are compared in Figure 4. The final displacement was obtained when the MC model was used for the soils and rocks; however, the shapes of the curves are not similar. The load-displacement was linear especially when no interface elements are used. The final displacement was slightly overestimated when the HS model for soils and the MC model for rocks are utilized. The comparison of the results and pile load test data are shown in Figure 5 for Pile LT-2, where both curves have a linear behavior.

Figure 3. Numerical model of Pile LT-1 and the finite element mesh.

Varying c , φ , and E affect the results obtained from analyses of Pile LT-1 when MC model was utilized for the soils and rocks. The cohesion of the marl layer was estimated as c = 700 kPa; therefore the analyses were repeated for a range of 600 to 800 kPa (Figure 6). The internal friction for the marl layer was calculated as φ = 32◦ in the field. Results of the analyses within ±3◦ of this value are presented in Figure 7. The estimated in-situ modulus of elasticity for the marl layer was E = 255 MPa. Results of the analyses within ±50 MPa of this value are shown in Figure 8. When Figures 6 through 8 are considered, it can be concluded that while the variation of cohesion and modulus of elasticity have a significant effect on the outcome, the changes in internal friction are less important. In addition, an analysis was carried out with the “HS small strain” model. However, a logical load – displacement curve could not be obtained due to lack of data for soil parameters obtained from laboratory experiments.

3

RESULTS

In this study, numerical analyses of two rock socketed piles with a circular cross-section were conducted with the Plaxis 2D V9 finite element code. The material properties were obtained from in-situ investigations or laboratory tests. Different material models (MohrCoulomb and Hardening Soil) are considered to identify the appropriate material model for the analyses in accordance with the soil types. Analyses were conducted to determine the effect of incorporating interface elements. Parametric studies were run for varying material properties of the Mohr-Coulomb model. The

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Table 3.

Material parameters of soils.

Test

Elevetion (m)

Soil Type

Material Model

γsat (kN/m3 )

c (kPa)

φ (◦ )

E /Eref 50 (MPa)

ν /νu

Rinter

0.0–8.5 8.5–10.0 0–2.0 2.0–8.5 8.5–11.5 11.5–20.0

Silty Clay Marl Loose Alluvial Alluvial Conglomerate Claystone

MC/HS MC MC/HS MC/HS MC MC

19.0 27.5 16.5 17.0 23.0 24.0

75 700 1 1 1 800

20 35 30 32 35 33

4/3 255 4/3 20/15 65 125

0.35/ 0.20 0.25 0.30/0.20 0.30/0.20 0.20 0.20

0.75 0.50 0.60 0.60 0.50 0.50

LT – 1 LT – 2

Figure 4. Comparison of the load test and finite element results for Pile LT-1.

Figure 6. The influence of cohesion on the results from analyses.

Figure 7. The effect of varying the internal friction angle. Figure 5. Results of the load test and finite element calculations for Pile LT-2.

variation of many parameters of the numerical model was studied through back-analysis. The following conclusions are reached based on the findings: 1. In investigating the pile-soil interaction, interface elements and an appropriate interface element coefficient (Rinter ) should be used. 2. Varying material properties based on the soil type does not have a significant effect on the results. The commonly accepted Mohr-Coulomb model, which involves typical parameters obtained from in-situ and laboratory testing, can be selected for analyses involving rock socketed piles.

Figure 8. The effect of changes in modulus of elasticity.

3. When the Mohr-Coulomb model is used to model the soil and rock behavior, cohesion and modulus of elasticity are critical parameters. The internal friction angle is of lesser significance.

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ACKNOWLEDGEMENT The authors wish to acknowledge Prof. Dr. Sönmez Yıldırım, Prof. Dr. Mustafa Yıldırım and Müge ˙Inanır for their valuable contributions in preparing this paper. REFERENCES Bowles, J. E. 1997. Foundation Design and Analyses, Singapore: The McGraw Hill Companies. Brinkgreve, R.B.J., Broere, W. & Waterman, D. 2004. PLAXIS – 2D Version 8, Netherlands: Plaxis bv Delpht. Canadian Geotechnical Society 2006. Canadian Foundation Engineering Manual 4th Edition. Vancouer: Bitech Publishers Ltd. Carrubba, P. 1997. Skin Friction of Large-Diameter Piles Socketed into Rock, Canadian Geotechnical Journal, 34: 230–240. Çalı¸san, O. & Özkan, M.Y. 1996. The Evalution of the Pile Load Test for the Çayırhan Termoelectric Power Plant, Soil Mechanics and Foundation Engineering 6. National Conference: 218–226 (in Turkish). D 1143 \D 1143M 2007. Standard Test Method for Piles Under Static Axial Compressive Load, United States: American Society For Testing and Materials.

Kirkit, M. 2009. Establishing a Database of Pile Load Test Conducted in Turkey and Analysis of Rock-Socketed Piles within The Database, Master Thesis, Istanbul:Yıldız Technical University. Mayne, P.W., Christopher, B.R. & DeJong, J. 2001. Geotechnical Site Characterization – Manual of Subsurface Investigations, National Highway Institute, Publication No. FHWA NHI-01-031, Washington: Federal Highway istration. Nam, M. S. 2004. Improved Design For Drilled Shafts in Rock, PhD Thesis, Houston, TX: University of Houston. Schanz, T. 1998. Zur Modellierung des mechanichen Verhalten von Reibungsmaterialien, Mitteilungen des Institus für Geotechnik der Universtät Stuttgart: Heft 45. Seol, H., Jeong, S. & Kim, Y. 2009. Load transfer analysis of rock-socketed drilled shafts by coupled soil resistance, Computers and Geotechnics, 36: 446–453. Wehnert, M. & Vermeer, P.A. 2004. Numerical Analyses of Load Tests on Bored Piles, NUMOG 9th Ottawa, Canada: 1–6. Yıldırım, S. 1994. Load Test on a Pile Socketed into Claystone, First Technical Congress on the Developments in Civil Engineering, TRNC, 322–329 (in Turkish).

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Numerical simulation of low-strain integrity tests on model piles J. Fischer, C. Missal, M. Breustedt & J. Stahlmann Institute for Soil Mechanics and Foundation Engineering, Technische Universität Braunschweig,

ABSTRACT: Using the theory of wave propagation in piles, the characteristics of the measured pile head velocity during a low strain integrity test enable the tester to rate the quality of cast in place concrete piles. In reference to possible pile appearances in situ, several polyamide plastic piles with well-defined discontinuities were constructed and afterwards tested. To simulate the influence of the surrounded soil on the recorded data, piles were driven into different soil types at our test site. Correlations between the data from the model piles and cast in place piles on site were clearly visible. To better understand the interaction between pile, soil, and the initiated stress wave traveling axially through the pile, a three dimensional finite difference numerical model was developed. The model piles, the surrounding soil, and the impact of the hammer were simulated and the velocity-time graph of the pile head was theoretically calculated. For the specification of the pile-soil interaction, interface-elements were used. These elements were able to give a realistic reproduction of the damping effects in the radial and axial pile directions. The results showed a very good agreement between calculation and measurement with respect to pile discontinuities and damping effects from the surrounding soil. 1

INTRODUCTION

National and international standards for low strain integrity testing give recommendations for assessing the quality of piles by the recorded pile head velocity during testing. Depending on the size and the direction of the recorded velocity amplitude, the quality of the pile can be classified. Furthermore, the measured wave speed will allow the tester to rate the pile length and its concrete quality. Within the last few years the Institute for Soil Mechanics and Foundation Engineering at the Technische Universität Braunschweig (IGB-TUBS) made varying low strain integrity test studies on cast in place concrete piles all over Europe. Depending on the soil conditions and the pile system used, the quality of the recorded data was very different. In some cases, even though the pile producer could prove the correct installation, the recorded pile head velocity showed minor or major quality reductions on the basis of national and international recommendations for low strain integrity testing, listed for example in Beim (2008). To better understand the correlation between pile, soil, and the stress wave, an experimental test series on polyamide plastic piles was done. Additionally, the results of the experimental tests were recalculated in a three dimensional numerical model on the basis of finite differences. 2 THEORETICAL BACKGROUND 2.1 Basis of dynamic pile testing Low strain integrity testing is a worldwide used method for assessing the condition of piles. The low

strain method is based on the theory of one dimensional wave propagation. The smooth surface of the pile head is hit with a hand held hammer to generate a stress wave in the pile. A sensor is set on top of the pile head and measures the acceleration of the stress wave. Through integration of the measured acceleration signal, a velocity time trend will be developed whose characteristics enable inferences on the pile integrity to be made. The stress wave propagates through the pile from pile head to pile toe, where it will be reflected and will reach the pile head, and thus control section, after time:

where t = time; L = length of the pile; and c = wave speed. The return time of the wave is dependent on the pile length L and the wave speed c. In the one dimensional case, the wave speed c, neglecting the Poisson’s ratio ν, can be calculated using the following equation:

where E = dynamic Elastic Modulus; and ρ = density. Should there be causes for impedance variation (necking, bulging, rock pocket etc.) along the length of the pile, a part of the stress wave will be reflected and reach the control section before the pile toe reflex.

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Figure 2. Velocity-time graph of a model pile with a constant cross-sectional enlargement and an abrupt reduction. Figure 1. Wave propagation during low strain integrity testing.

Within the paper, downward directed particle velocities will be defined as positive. As a result the amplitudes in the velocity-time graph of figure 1 are positive.

The impedance Z of the pile is given by the following equation: 3

EXPERIMENTAL RESEARCH

3.1 Model piles where E = dynamic modulus of elasticity; A = crosssectional area; and c = wave speed. In addition to the reflections due to a change of the pile impedance, a change in the surrounding soil density will cause stress wave reflections. Figure 1 shows a pile on the left side and its stress wave propagation in reference to length and time on the right side. The pile shows a cross sectional reduction after approximately two thirds of its length. The downward travelling stress wave is divided in two parts at that point. One part travels back upwards to the pile head, the other part travels further downwards and is reflected at the pile toe. The pile head velocity which will be recorded during an integrity test is highlighted in black. The impact of the hammer causes the first velocity amplitude. Afterwards, while the stress wave travels through the pile, no pile head velocity is measured. The second amplitude is caused by the piles cross sectional reduction. After equation 1, based on a given wave speed for concrete, the location of the impedance reduction can be calculated. Likewise, the length or the wave speed of the pile can be calculated by the third amplitude, caused by the pile toe.

2.2 Sign conventions The velocity sign conventions are positive and negative for compressive and tensile waves. Through the impact of the hammer, a compressive wave with a downward directed particle velocity is initiated in the pile. At any point of impedance reduction, the reflected part of the wave will travel backwards to the pile top as a tensile wave. The particle velocity will still be directed downward. In contrast, at any point of impedance enlargement, the compressive wave will be reflected as compressive wave with an upward directed particle velocity.

As visual inspection of cast in place concrete piles is hardly possible, polyamide plastic piles with welldefined discontinuities were constructed and tested instead. On the basis of unclear velocity-time graph data we measured during the mentioned test series on cast in place concrete piles, the possible pile cross-sectional distributions were assumed and rebuilt as geometrically scaled down model piles. An example of a model pile is shown in figure 3. All figures shown in this chapter illustrate the results we received from the experimental research program on the model piles. 3.2 Piles with well-defined discontinuities After driving or boring into the ground and the placement of the concrete, the casing of a displacement cast in place concrete pile will be withdrawn. During withdrawal of the casing an enlargement of the pile cross-section is possible between stiff soil conditions in the deeper parts of the soil and the upper, softer conditions. Within further progress, the cross-sectional enlargement will possibly reduce to the regular diameter of the casing. Comparable soil conditions were often found during testing on site. Therefore, a model pile with the mentioned crosssectional conditions was developed. Figure 2 shows the result of the low strain integrity test on that model pile. Regarding the velocity-time graph of the recorded data, the signal shows a very strong reduction of the pile impedance after two-thirds of its length. The previous constant enlargement is not visible in the signal. Therefore, only by observing the velocity-time graph, the pile should be rated as damaged. The results of that test series showed that under certain soil conditions the recorded signal may lead the pile integrity tester to misread the true conditions of

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Figure 3. Full displacement bored pile – original, model pile and schematic drawing with a soil filled gap between the coils.

Figure 4. Wave speed in accordance to the depth of the coil and the soil between the coil and the pile.

the pile. Instead of an abrupt cross-sectional reduction, a constant enlargement is often not clearly visible in the velocity-time graph.

3.3 Full displacement bored piles According to equation 2 the wave speed for a cast in place concrete pile can be calculated. The German standard EA-Pfähle (2007) suggests a range of the wave speed for set concrete piles to be between 4100 m/s and 3500 m/s. If the measured wave speed, calculated by the recorded velocity-time graph form ‘peak to peak’ after equation 1 is higher, a shortage of the pile is likely. If the wave speed is slower than suggested in the recommendations, a weak quality of the concrete seems to be possible. In fact, while testing full displacement bored piles during our test studies on site, sometimes a very low wave speed under 3000 m/s was measured throughout an entire test site. A bad concrete quality for the piles was unlikely, especially as a sample was tested for quality from each supply. The production of longer piles was very doubtful as well. Therefore, a model pile was produced with exactly the same structure as the tested full displacement bored piles on site, scaled down geometrically. With a length of 200 cm and a diameter of 5.0 cm the original wave speed of the polyamide model pile was 1790 m/s. The gaps between the coils were cut into the pile in four increments, each increment 0.25 cm. The model pile is shown in figure 3. After every step, the wave speed of the model pile was tested.The result is given in the left part of figure 4. It is clearly visible that, with an increasing depth of the gap, the wave speed reduces significantly. After full installation of the pile coil, a wave speed reduction of 16% was measured. Further on, we filled the gap between the coil and the pile section by section (see figure 3). Each section had a length of 50 cm. After each section was filled the pile integrity was tested and the wave speed calculated. The result is shown on the right site of figure 4. An additional wave speed reduction of 30% was measured. In total a wave speed reduction of 41% could be generated though the mentioned processes.

Figure 5. Wave speed in accordance to the soil in the gap between coil and pile.

Additionally, we filled the gap between the coil and the pile with gravel and sand. As shown in figure 5 the wave speed reduction is dependent on the soil type. As a result of the second test series, a reduction of the traveling wave speed in accordance to the size of the coil and the soil in the gap between the pile and the coil was clearly visible. In relation to the data form the tests on site, the percentage change of the wave speed was congruent. 3.4 Influence of the surrounded soil According to chapter 2, the surrounded soil has an influence to the recorded velocity-time graph of a low strain integrity test. A change of the soil stiffness will be measured as a stress wave reflection on the pile head. Furthermore, the damping of the soil will lead to an energy reduction of the stress wave which will, for example, lead to a weaker toe reflex amplitude. To simulate the influence of the surrounding soil, model piles with a constant diameter were driven into different soil types at our test site. The damping effect and the change of impedance were clearly visible. 4

NUMERICAL MODEL

4.1 Numerical simulation To recalculate the results of the experimental researches numerically, the program FLAC3D (Fast Lagrangian Analysis of Continua in 3 Dimensions) was chosen. The program works on the basis of the

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explicit finite difference method. The materials, as for example the pile or the surrounded soil, are represented by polyhedrons in a three dimensional grid. The constitutive law of the elements can be chosen as linear or non-linear. The use of the dynamic analysis option permits three-dimensional, fully dynamic analysis with FLAC3D (2006).

4.2

Constitutive law

Because the ‘low strain’ impact of the small hand held hammer used for the experimental tests on the model pile will cause no plastic deformations, an ideal linear elastic constitutive law was chosen for the pile material. Mohr’s constitutive law was chosen for the soil, not only because no displacement of the soil was expected, but also because the soil parameters (φ, c, E and ν) are available for regular soils. More important than the constitutive law of the soil, the damping effects had to be included in the numerical model, as the stress wave looses energy by damping in radial and axial pile direction. To consider the damping effects, interface elements were used (see also chapter 5.2).

The optimum critical time-step can be calculated after Schmitt (2009) with:

where tcrit = critical time-step; and ωmax = largest unique frequency of the structure. To determine ωmax , the systems largest unique frequency, a complete dynamic analysis of the whole system has to be carried out. As this approach is very complex the critical timestep will be estimated in most cases. From Konietzky (2001) the critical time-step for a full or half-space can be estimated with the maximum of the longitudinal wave speed νP and the minimum grid-point distance X after the following equation and was used for our numerical model:

where tcrit = critical time-step; X = minimum gridpoint distance; and νp maximum longitudinal wave speed. 4.5 Grid discretization

4.3

Initiation pulse

As described in Plassmann (2002) the quality of a low strain integrity test depends on the force and the time of the impact pulse. For an ideal reflection of the stress wave at any change of impedance, a short time and a high intensity of the force is necessary. The requirements can be summarized in equation 4 as:

where d = pile diameter; L = pile length; c0 = wave speed; and t = time. From Elmer (1995) the described ideal impact pulse can be generated in a numerical model by the following cosine function:

where tI = time; T = runtime of the wave; and P0 = force. When using the cosine function, a very good agreement between the initiation pulse of the experimental research and the numerical simulation could be generated.

When building up a numerical model its discretization has to be considered. To avoid long calculation times and still receive satisfactory results in of accuracy, an optimized number of elements in radial and axial direction must be found. The main discretization criteria when building up the numerical model was to avoid a so called ‘numerical dissipation’ of the stress wave. When choosing a small number of elements in radial and axial direction, parts of the stress wave will be dissipated numerically while traveling through the pile. Therefore, sensitivity studies in accordance to the numbers of elements were carried out. The numerical result of two different sensitivity studies is shown in figure 6. As known from a free end pile with no damping effects around its pile shaft (e.g. ideal Hopkinson bar), the pile toe reflex, measured after equation 1 as pile head velocity, will be twice as high as the initiation pulse. Using an inadequate discretization for the numerical model, a delayed and declined pile toe reflex is calculated. When choosing a discretization with more than 40 elements per wavelength (equation 7) the described numerical dissipation could be eliminated. 5

NUMERICAL RESULTS

4.4 Time discretization

5.1 Simulation of gradual change in section

The quality of the numerical results depends on the chosen time-step where the critical time-step is the most influential factor. If the chosen time step is too large, no plausible numerical results can be calculated.

In the early stages, we started our numerical investigations with a model pile similar to the pile shown in figure 3. The length of the pile was 1.6 m. The cross sectional surface exhibited a gradual enlargement to

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Figure 6. Discretization studies of the numerical model. Figure 8. Rheological model of the interface-elements, FLAC3D (2006).

Figure 7. Comparison between an experimental result and a numerical calculation according to gradual change in section.

1.25 m, while the remaining 0.35 m of the diameter corresponds to the pile head diameter. The numerical investigations for the cross sectional enlargement within the first 1.25 m were carried out from 10% to 300%. A very good accordance between the velocity-time graph of the model pile and the numerical simulation could be achieved as specifically shown for a pile with a gradual cross sectional enlargement of about 66% in figure 7. As the initiation pulse varies during an experimental test, both the numerical and the experimental velocitytime graph were standardized to the initiation pulse.

5.2 Influence of the soil As described in chapter 3.4 a change of the surrounding soil condition causes stress wave reflections. Furthermore a reduction of the stress wave energy will occur through damping effects of the soil. These effects have to be considered in the numerical simulation. For the specification of the pile-soil interaction, interface-elements were used. These elements were able to give a realistic reproduction of the damping effects in the radial and axial pile directions. The rheological model of the interface elements in FLAC3D is illustrated in figure 8. The shear stiffness and normal stiffness both greatly affect the damping of the wave speed amplitude in the pile. For realistic reproduction of the soil damping in the numerical simulation, an

Figure 9. Comparison between an experimental result and a numerical calculation according to damping effects.

estimated value for the stiffness of ks and kn must be determined with sensitivity studies. Figure 9 shows the pile head velocity measured from a model pile driven into soil at the test site, compared with the result of a numerical simulation. The noticeable decrease in the velocity from approximately 1.4 m to 1.9 m is based on a change in impedance due to a higher ground resistance. The result is comparable in both the driven model pile and the numerical simulation. Figure 10 shows the decrease of the wave speed amplitude across pile length due to damping. The numerical simulations show that under consideration of the apparent stiffness for the interface elements, different soil types will lead to plausible results. Therefore a loose sand (e.g. hydraulic fill) has only minor damping effects over the pile length when compared to a dense sand.

6

OUTLOOK

Initial numerical simulations of the low strain integrity test on the full displacement bore pile model show that when choosing a group-modulus of elasticity for both the polyamide and the soil in the gaps between the coils, a realistic reduction of the wave speed is calculated. A numerical result is shown in Figure 11.

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results of any unclear data measured during a low strain integrity test on site. Through numerous variations of the possible interactions causing stress wave reflections the most likely pile appearance can be determined based on the best match between the measured signal and the numerical result. REFERENCES

Figure 10. Dependence between soil damping and pile length.

Figure 11. Wave speed reduction.

Different effects on the travelling stress wave, caused by any change of impedance within the length of the pile, can be simulated numerically with a very good agreement compared to the experimental results on the model piles. The applicability of the numerical model to cast in place concrete piles on site still needs to be proven. The need for more detailed analysis through the use of additional numerical simulation is, to clarify the

Beim, G. Likins, G. 2008. Worldwide dynamic foundation testing codes and standards, 8th International Conference on the Application of Stress-Wave Theory to Piles, Science, Technology and Practice, pp 689–697, Lisbon, J.A. dos Santos, Lisbon, Portugal. EA Pfähle. 2007. Empfehlungen des Arbeitskreises Pfähle, Deutsche Gesellschaft für Geotechnik (DGGT), Dortmund, Ernst & Sohn. Elmer, K.-H. 1995. Modellierung und Simulation einer Dehnwelle zur Risserkennung im Stab, Technische Mechanik, Band 15, Heft 1. FLAC3D. 2006. Fast Lagrangian Analysis of Continua in 3D, Itasca Consulting Group Inc., Minneapolis, Minnesota, Version 3.1. FLAC3D. 2006. Fast Lagrangian Analysis of Continua in 3D Manual – Optional Features, Itasca Consulting Group Inc., Minneapolis, Minnesota. Konietzky, H. 2001. Numerische Simulationen in der Geomechanik mittels expliziterVerfahren,Veröffentlichungen des Instituts für Geotechnik der TU Bergakademie Freiberg, Heft 2001–2. Plassmann, B. 2002. Zur Optimierung der Meßtechnik und derAuswertemethodik bei Pfahlintegritätsprüfungen, Dissertation, Institute for Soil Mechanics and Foundation Engineering, Technische Universität Braunschweig, Heft 67. Schmitt, J. 2009. Spannungsverformungsverhalten des Gebirges beim Vortrieb mit Tunnelbohrmaschinen mit Schild, Dissertation, Institute for Soil Mechanics and Foundation Engineering, Technische Universität Braunschweig, Heft 89.

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Response of pile groups in clays under lateral loading based on 3-D numerical experiments E.M. Comodromos & M.C. Papadopoulou Department of Civil Engineering, University of Thessaly, Volos, Greece

I.K. Rentzeperis Egnatia Odos S.A., Greece

ABSTRACT: The response of laterally loaded pile foundations may be significantly important in the design of structures for such loads. Although simplified numerical methods are reliable for evaluating the response of a single pile under horizontal load, their application is questionable for assessing the response of pile groups. The aim of this paper is to evaluate the influence of the interaction between the piles of a group fixed in a rigid pile cap, in clayey soils, on both the lateral load capacity and the stiffness of the group. For this purpose, a parametric three-dimensional nonlinear numerical analysis, allowing for the nonlinearities of the interaction between soil and piles as well as the effect of pile-soil separation, was carried out for different arrangements of pile groups. The response of the pile groups is compared to that of the single pile. The influence of the number of piles, the spacing and the deflection level to the group response is discussed. Furthermore, the contribution of the piles constituting the group to the total group resistance is examined.

1

INTRODUCTION

The response of laterally loaded pile foundations may be significantly important in the design of structures for such loads. In many cases the criterion for the design of piles to resist lateral loads is not the ultimate lateral capacity but the deflection of the piles, Poulos and Davis (1980). In the case of bridges or other structures founded on piles, only a few centimetres of displacement could cause significant stress development on these structures. The load-deflection curve of a single free-head pile can be determined using numerical methods and/or results from pile load tests, while full-scale pile group tests for determining the response of a pile group are very rare due to the extremely high cost required. Furthermore, for single piles various approaches have been proposed with the aim to take into nonlinearities arising from soil-pile interaction. Within this framework Reese (1977) proposed the well-known “p-y analysis”, according to which the soil response is described by a family of curves giving soil resistance as a function of deflection and depth below the ground surface. The simplicity of the method, in conjunction with the welldefined procedures for establishing the “p-y” curves, made the method the most widely used. Although the method is reliable for evaluating the response of a single pile under horizontal load, it is questionable if reasonably reliable simple methods could be applied to assess the response of pile groups. It is however

commonly accepted that for the same mean load, the piles of a pile group exhibit significantly greater deflection than an identical single pile. This behaviour should be attributed to the fact that the resisting zones behind the piles overlap. Clearly the effect of the overlapping becomes larger as spacing between piles decreases. The application of three-dimensional (3D) numerical analysis on the other hand, despite its complexity and high computational demands, is the most powerful tool for pile group response evaluation under horizontal or other loading, since it is able to predict both stiffness and ultimate resistance reduction factors, particularly in the case of sensitive soils undergoing plastification for even a low level of loading. The aim of this paper is to use numerical analysis tools to estimate the interaction level between soil and piles for various layouts of horizontally-loaded fixed head pile groups, in various soil conditions, and to determine the reduction factors for ultimate lateral load capacity and stiffness corresponding to the working load or any other load level. 2

SOIL-PILE INTERACTION EFFECT ON PILE GROUPS

According to Prakash and Sharma (1990), and Oteo (1972), the lateral group efficiency nL defined by Equation 1 may reach only 40%, depending on the

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number of piles in a group and the layout of the group:

3

NUMERICAL ANALYSES OF FIXED-HEAD PILE GROUPS

3.1 Group configurations and loadings

As mentioned in the introduction, the loaddeflection curve could be the determinant factor for the design of a project and, therefore, the group stiffness reduction factor caused by a lateral load is of greater importance than the group efficiency factor. The widely used “p-y” method could be considered as extremely effective for the prediction of a single pile under horizontal loading and this has been demonstrated by the application of the back-analysis procedure in many cases, where a pile load test was carried out. Even though available data from a single pile test under horizontal loading exists together with the results from a “p-y” analysis, further calculations are required to establish the response of a pile group due to the effect of pile-soil-pile interaction. Poulos (1971, 1989) introduced four different kinds of interaction and reduction factors for piles under lateral load, depending on the loading at the pile head and the type of deformation. Moreover, based on the elastic continuum approach Randolph (1981) proposed a relationship for estimating the interaction factors in fixed-head piles, demonstrating that the interaction under lateral loading decreases much more rapidly with spacing between piles than for axial loading. Wakai et al. (1999) used 3-D elasto-plastic finite element analysis to estimate the effect of soil-pile interaction within model tests for free or fixed-head pile groups. In that analysis thin frictional elements were inserted between the pile and the soil in order to consider slippage at the pile-soil interface. It must be mentioned however, that in many cases where the pilesoil interaction is governed by nonlinearities arising from the soil separation behind the pile and the yield of soil in front of the pile, a 3-D analysis including interface elements around the piles can be considered more accurate in providing the response of a pile group. Comodromos (2003) and Comodromos & Pitilakis (2005) utilised 3-D FDA (Finite Different Analysis) to evaluate the response of free-head and fixed-head pile groups respectively. A parametric 3-D analysis was then performed and the results have been compared with those of the pile test. The effect of the pile-soilpile interaction was then estimated for various group configurations and, finally, a relationship was proposed allowing the establishment of load-deflection curves for both free-head and fixed-head pile groups. As stated in those papers, the applicability of the proposed formulas to different soil profiles should be verified or readjusted for different soil profiles. With the aim to examine the response of fixed-head pile groups under lateral loading in clayey soils, an extensive parametric numerical analysis for various pile dispositions has been carried out in this paper, for different soil profiles covering the range from very soft to hard clays.

Configurations of 3 × 3 were examined with spacings of 2.0D, 3.0D, 6.0D and 9.0D, along with complementary 2 × 2 and 4 × 4 configurations in some cases, with the aim of drawing conclusions about the effect of number of piles and their axial distance on the group response. The piles have a diameter of D = 1.00 m and a length of 25 m. The three dimensional finite difference code FLAC3D has been used for a series of parametric analyses of fixed-head pile groups. The geometry of the mesh was parametrically defined in order to give the possibility for geometrical variations when needed. A mesh generator subroutine was implemented using the FISH built-in programming language providing the possibility of mesh refinement and geometry variation. The bottom elevation and the lateral sides of the computational domain were taken far enough from the group to avoid any significant boundary effect. More specifically, the distance between the piles’ tip and the bottom of the mesh was taken equal to 25.0D and the lateral sides of the domain were taken either 30 or 60 m away from the outmost side of the piles, depending on the distance between the two corner piles of the group. At the bottom level of the computational domain all movements were restrained while at the lateral external sides, lateral movements perpendicular to the boundary were prohibited. Based on the experience gained through this and previous numerical works, a mesh refinement around the piles leads to a more accurate distribution of stresses and displacements. This has been taken into when preparing the finite difference meshes, as characteristically illustrated in Figure 1, which corresponds to a 3 × 3 pile group configuration at a distance of 3.0D, consisting of 22872 elements and 24045 nodes. Four types of clayey soil were examined in the parametric solutions, corresponding to soft, medium stiff, stiff and very stiff clay, referred to as C1, C2, C3 and C4, respectively. The elastic perfectly-plastic Tresca constitutive model was used to simulate the behaviour of the soil.Table I summarises the properties of the four soil types. The shear strength and the Young’s modulus of the soil were considered to increase with depth, z, as presented in Table I. In the particular case of pile foundations, where the applied loads provoke shear strains in the immediate vicinity of the piles, partial drainage occurs in this area even in the case of short term loading. For this reason, values of Poisson’s ratio slightly lower than vu = 0.5, which corresponds to undrained conditions, have been adopted in the analysis. Pile behaviour was considered as linear elastic with Poisson’s ratio ν = 0.20 and a modulus of elasticity E = 32000 MPa. Due to the fact that soil has a limited capacity in sustaining tension, interface elements were introduced to

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Figure 2. Components of the interface constitutive model in FLAC3D .

The normal and shear forces are determined by the following equations: Figure 1. Finite difference mesh for a 3 × 3 pile group, spacing 3.0D, cross section at y = 0. Table 1. Geotechnical properties of soil types C1 (soft clay), C2 (medium stiff clay), C3 (stiff clay) and C4 (very stiff clay). C1 Young’s Modulus, E (MPa) Poisson’s Ratio, ν Undrained Shear Strength, cu (kPa) Soil – Pile adhesion, ca (kPa) Unit weight, γ (kN/m3 )

C2

C3

400Cu 300Cu 200Cu

C4 150Cu

0.45 0.45 0.45 0.45 25 + z 50 + z 100 + z 150 + z 25

50

72

75

0

20

20

20

allow pile separation from the surrounding soil. Separation occurs near the top and behind the pile generally no deeper that 20% of pile length, depending on pile and soil stiffness. Together with the local yield at the top of the soil where large compressive stresses are developed in front of the soil, separation is considered as the main reason for the nonlinear behaviour. According to Poulos and Davis (1980), separation is able to cause an increase in displacements up to the extreme level of 100%, while 30 to 40% appears to be more reasonable in the case of stiff piles. The constitutive model of the interface elements in FLAC3D is defined by a linear Coulomb shear-strength criterion that limits the shear force acting at an interface node, a dilation angle that causes an increase in effective normal force on the target face after the shear strength limit is reached, and a tensile strength limit. Figure 2 illustrates the components of the constitutive model acting at an interface node. The interface elements are allowed to separate if tension develops across the interface and exceeds the tension limit of the interface. Once gapping is formed between the pile-soil interface, the shear and normal forces are set to zero.

where Fn , Fsi = normal and shear force respectively; kn , ks = normal and shear stiffness respectively; A = area associated with an interface node; usi = incremental relative shear displacement vector; un = absolute normal penetration of the interface node into the target face; σ n = the additional normal stress added due to interface stress initialization; and σ si = the additional shear stress vector due to interface stress initialization. In many cases, particularly when linear elastic analysis is performed, values for interface stiffness are defined to simulate the nonlinear behaviour of a problem. In the present analysis, where nonlinear analysis is carried out and the use of interface elements covers the soil-pile separation, the value for the interface stiffness should be high enough, in comparison with the surrounding soil, in order to minimise the contribution of those elements to the accumulated displacements. To satisfy the above requirement the guidelines of FLAC3D manual (2006) propose values for kn and ks of the order of ten times the equivalent stiffness of the stiffest neighbouring zone. The use of considerably higher values is tempting as it could be considered as more appropriate, but in that case the solution convergence will be very slow. Based on this principle, for both kn and ks , a value of 1000 MPa/m was taken. Numerical analyses were carried out for all the aforementioned soil types. The simulation sequence included an initial step, in which the initial stress condition was established, followed by a number of loading steps in order to define the load-displacement curve up to a level of deflection of at least 10%D. The pile group total load, H , was applied to the top of the central pile for the 3 × 3 configuration and to the top of a corner pile for the 2 × 2 and 4 × 4 arrangements. In either cases, to simulate the fact that the piles were fixed in a rigid pile head, the pile heads of the rest of the piles where restrained to undertake the same deflection

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Figure 3. Displacement contours in the direction of loading for the case of a 3 × 3 layout with a spacing of 2.0D, soil type C1.

with the loaded pile (nodes slaved to that where the load was applied). In addition, the degrees of freedom of the nodes at the pile heads corresponding to the directions y-y and z-z were eliminated. The direction of loading was always the x direction, whereas a “ramp loading” procedure was also used to avoid numerical instabilities due to the high load value. The case of a fixed-head single pile was also considered since its response is to be used to compare the group responses with. The single pile response was derived by the 3 × 3 mesh and a preliminary analysis was carried out for the central pile of the first, the second and the third row in the direction of loading (position of piles P8 , P5 and P2 respectively at the pile group), confirming that the position of the single pile does not affect the results.

3.2



Numerical results

Figures 3 to 5 illustrate the displacement contours along the direction of loading at the plane y = 0 for the case of the 3 × 3 layout with spacing of 2, 3 and 6 diameters, for soil type C1. The displacement contours correspond to a mean load of 1.75 MN, where the mean load, Hm , is defined as the total load imposed at the pile cap divided by the number of piles.The level of interaction between piles and soil can be drawn qualitatively from the displacement contours. When spacing is too small (Figure 3), the displacement tends to be unified at the soil surface between the piles, while from a certain level of loading the resisting zones behind the piles overlap. When these zones are plastified, the lateral load capacity is rather the load capacity of an equivalent single pile containing the piles than the summation of the lateral load capacity of the piles. A comparison between Figures 3 to 5 demonstrates that as spacing increases the effect of overlapping between the resisting zones becomes less significant. A detailed comparison of the results demonstrated that the load-deflection curve is significantly affected by pile spacing while the number of rows and the total

Figure 6. Numerically established load-deflection curves, for the fixed-head single pile and various pile group configurations, soil type C1.

number of piles play also an important but less affecting role. Figure 6 illustrates the load-deflection curves at the top of the pile for various fixed-head pile groups together with that of the fixed-head single pile. The group 3 × 3 with a spacing of 9.0D is the most stiff, followed by the 3 × 3 group with the 6.0D spacing and the 4 × 4 group with a spacing of 9.0D.

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In addition, the lowest stiffness is exhibited by the 4 × 4 group with a spacing of 2.0D. When examining the groups in either the 3 × 3 or the 4 × 4 layout, it can be verified that when spacing decreases, the stiffness of the group declines and that the number of piles affects the response of the group as well. Despite the variation of the load-deflection curve of each group, it can be concluded that all curves have a form similar to that of the single pile. As it was previously stated, the criterion for the design of piles to resist lateral loads, in the majority of cases, is not the ultimate lateral capacity but the deflection of the piles under a specific load. From the results of the numerical analyses, it may be concluded that the piles in groups undergo considerably more deflection for a given mean load Hm per pile than a single pile under the same load.A comparison between the deflection of the single pile and that of the pile group for the same mean load provide the deflection amplification factor defined by the following equation:

in which ymG and ymLs stand for the deflection at the head of the piles and the single pile under the same horizontal mean load Hm , respectively. It can be seen that the deflection amplification factor, Ra , is the inverse of the stiffness efficiency factor, RG , defined by Equation (5), whereas the stiffness of a pile group for a given mean load Hm can be then calculated using Equation (6):

where KS = the stiffness of the single pile for a given load; and KG = the stiffness of the pile group for the same load. The total group stiffness is determined by multiplying KG with the number of piles of the group. Figure 7 illustrates the variation of the amplification factor Ra with the normalized deflection of various pile groups. More specifically, the amplification factor is plotted against the level of deflection for 3 × 3 and 4 × 4 pile group configurations in soft clay and pile spacings of 2.0, 3.0, 6.0 and 9.0D. It can be seen that the Ra factor exhibits its maximum effect at low levels of deflection. This shoud be attributed to the fact that at low level of displacements linear elastic behaviour or limited yielding occurs in the surrounding soil and the interaction between the piles of a group exhibits its maximum effect and as a result the Ra factor attains its maximum value. As displacement increases the Ra factor shows a gradual degradation up to a certain level of deflection, which depends on both the group arrangement and the soil strength. Beyond this level the Ra factor practically remains constant. The effect of the pile spacing can be seen in Figure 8. For the same layout, the group with the

Figure 7. Variation of amplification factor Ra with deflection level for a fixed-head pile group in soil type C1, 3 × 3 arrangement with various spacings.

Figure 8. Variation of amplification factor Ra with spacing, for a deflection of 2, 3, 5 and 7%D, for a 3 × 3 pile group arrangement, in soil type C1.

Figure 9. Variation of amplification factor Ra with spacing, for a deflection of 2, 3, 5 and 7%D, for a 3 × 3 and a 4 × 4 pile group arrangement, in soil type C1.

minimum spacing shows the maximum deflection amplification factor, Ra , which means that exhibits the maximum stiffness reduction. Figure 9 illustrates the variation of the amplification factor Ra with different configurations. It can be seen that the lateral displacement is amplified by more than four times in 4 × 4 groups with 2.0D spacing, whereas the effect becomes less significant in the case of a 3 × 3 group with the same spacing, where the amplification factor was found of the order of 3.0. Thus, for the same spacing, the greater the number of the piles in a group the greater the stiffness reduction.

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carries 115 or 110% of the mean load. This percentage gradually decreases with deflection level, becoming 102 or 101% when deflection increases to 10% of the pile diameter. Finally, the loads transferred to the other piles of the group remain within the limits of these two piles. 4

Figure 10. Variation of normalized load with normalized deflection for piles P2 , P3 , P5 , P6 , P8 and P9 in a 3 × 3 layout with a spacing of 6.0D, in soil type C1.

CONCLUSIONS

In this paper, the effects of the interaction to the response of a pile group fixed in a rigid cap were examined for various group configurations under horizontal loading. Based on the results of the parametric three dimensional nonlinear numerical analysis, the response of particular piles in the group was investigated and their contribution to the entire group behaviour was also quantified. REFERENCES

Figure 11. Variation of normalized load with normalized deflection for piles P2 , P3 , P5 , P6 , P8 and P9 in a 3 × 3 layout with a spacing of 9.0D, in soil type C1.

In order to investigate the effect of interaction within a pile group, the responses of the piles in a 3 × 3 layout were examined precisely. As anticipated, the central pile, P5 , carries the lowest load for the same deflection, presenting the minimum stiffness, while the two corner piles on the direction of loading (P7 and P9 ) carry the highest load, presenting the maximum stiffness. It should also be noted that the front piles (P8 and P9 ) resist more than the back row (P2 and P3 ). Figures 10 and 11 illustrate the normalized load undertaken by the piles of the group as a function of the normalized deflection in the case of spacing equal to 6.0D and 9.0D, respectively. The central pile, P5 , initially carries the 72% or 79% of the mean load for spacings of 6.0D and 9.0D, respectively. These percentages gradually increase to 99% when the deflection level becomes of the order of 10% of the pile diameter. On the other hand, pile P9 initially

Comodromos, E. 2003. Response prediction of horizontally loaded pile groups. Geotechnical Engineering Journal 34(2): 123–133. Comodromos, E. & Pitilakis, K. 2005. Response Evaluation of Horizontally Loaded Fixed-Head Pile Groups using 3-D Nonlinear Analysis. International Journal for Numerical and Analytical Methods in Geomechanics 29: 597–625. Itasca. FLAC3D 2006.Fast Lagrangian analysis of continua. Itasca Consulting Group; ’s manual ver 3.1. Minneapolis. Oteo, C.S. Displacement of a vertical pile group subjected to lateral loads. Proceedings of 5th European Conference of Soil Mechanics & Foundation engineering, Madrid, 1972: 397–405. Poulos, H.G. 1971. Behaviour of laterally loaded piles: Isingle pile, and II- pile group. Journal of Soil Mechanics & Foundation Division 97: 711–751. Poulos, H.G. 1989. Pile behaviour – theory and application. Géotechnique 39(3): 366–415. Poulos, H.G. & Davis, E.H. 1980. Pile foundation analysis and design. Singapore: J. Wiley & Sons Ltd. Prakash, S. & Sharma, D. 1990. Pile foundation in engineering practice. New York: J. Wiley & Sons Ltd. Randolph, M.F. 1981. The response of flexible piles to lateral loading. Géotechnique 31(2): 247–259. Reese, L.C. 1977. Laterally loaded piles: Program documentation. Journal of Geotechnical Engineering Division 103: 287–305. Wakai, A., Gose, S., Ugai, K. 1999. 3-D Elasto-plastic finite element analyses of pile foundations subjected to lateral loading. Soils and Foundations 39(1): 97–111.

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Selection of the proper hammer in pile driving and estimation of the total driving time A. Afshani, A. Fakher & M. Palassi Department of Civil Engineering, University of Tehran, Tehran, Iran

ABSTRACT: There are various methods for analyzing pile driving process such as dynamic formulas, wave equation analyses and dynamic measurements. The programs of the two latter methods are relatively expensive to purchase and also require trained engineers to interpret the data it collects. Meanwhile, the use of the site specific empirical formulas based on the real cases of pile driving and output analyses of the wave equation analysis programs can be beneficial. In the current study, pile driving data from three sites in the south of Iran were collected. Using the data of these cases and one-dimensional wave equation analysis program GRLWEAP, couple of experimental formulas which determine the proper range of hammer’s ID for driving a pile, are proposed. Finally, due to the importance of the time in marine projects, another experimental formula is also proposed for the estimation of the total driving time.

1

2

INTRODUCTION

Engineers and contractors have been deg and installing pile foundations for many years. The most common way of installing a pile is driving of it by a hammer especially in marine environments. A hammer which is too small may not be able to drive the pile to the required capacity; or may need an excessive number of blows. On the other hand, a hammer which is too large may damage the pile or fails to operate properly after a few blows. Three general methods are available for predicting the hammer performance: dynamic formulae, dynamic test and wave equation analysis. Dynamic test methods measure strain and acceleration of the produced wave by hammer impact near the pile head. These measurements can be used to evaluate the performance of the pile driving system, determine pile integrity and estimate static pile capacity. But these tests must be performed during pile driving and also need trained engineers to interpret the data it collects. Wave equation analysis performed in the design stage requires assumptions on the hammer type and performance level, the drive system components, as well as the soil response during driving. In the current paper, based on the field observation data and a parametric study, couples of experimental equations were presented to provide a proper hammer for driving a certain pile in a certain soil condition. Furthermore, an equation was also obtained which represents an estimation of the total driving time.

DRIVEABILITY ANALYSIS

2.1 The basics of driveability analysis By performing a static soil analysis the graph of ultimate soil capacity as function of depth can be plotted. The wave equation is then used to calculate the blow count for certain depth value. In this way, the blow count versus embedded length curve is obtained. This process is called a driveability study which can be performed: 1. To determine if the proposed hammer(s) are suitable for the pile installation. 2. To assess the ability of the hammer(s) to restart pile driving after an interruption.

2.2 GRLWEAP program GRLWEAP is the latest updated wave equation analysis program (Pile dynamic 2003). This program simulates motions and forces in foundation pile when driven by either impact or vibratory hammer. The program can compute the following: 1. The blow count of a pile under one or more assumed ultimate resistance values and other dynamic soil resistance parameters, given a hammer and a driving system (helmet, hammer cushion, pile cushion). 2. The energy transferred by the hammer to the pile for each capacity analyzed. 3. The expected blow count per meter along the penetrated depth of the pile (driveability analysis).

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Table 1.

Soil condition.

From (m)

To (m)

Site A 0 12 18 28 Site B 0 7 17 27 Site C 0 12 22 27

3 3.1

Table 2. Collected records from the field for calibration of the input information in GRLWEAP program.

Description

12 18 28 43

Dense silty sand Very dense clayey silty sand Hard silty clay Very dense sandy silt

7 17 27 –

Back fill material Sandy soil with some shell debris Clayey soil with layers of dense silt Clayey soil with layers of V. dense silt

12 22 27 -

Gray silty sand Very dense silty gravel with sand V. dense silty gravel with sand and gobble Very dense silty gravel with sand

Pile No

FIELD MEASURMENTS OF THE PILE DRIVING General description

During the process of obtaining an equation for selecting a proper hammer to drive a pile, a series of data was needed to calibrate the unknown parameters in the used wave equation program. In order to fulfill this goal, the field data from three recent pile installation projects were collected. All three projects were located in the south of Iran on the coastline of the Persian Gulf. The more details of these projects are presented in subsequent parts. 3.2

Site A

This site has located on Bahonar harbor in BandarAbbas, a port city on the southern coast of Iran, on the Persian Gulf. At this site, abundant numbers of piles were driven as a deep foundation which their characteristics are: piles type = open-ended steel pipe pile, piles total length = 24 to 36 m, piles embedded length = 16 to 30 m, piles diameter = 0.762 m, piles wall thickness = 18 mm. The Delmag D-36 was chosen as a typical diesel hammer with an efficiency of 0.8. The general soil profile was consisted of dense to hard silty sand as described in Table 1. In order to calibrate the input information of the GRLWEAP program for the next analyses, field data of blow counts per meter was collected in each of these projects. For instance, in site A, the blow counts per meter along the driven length of the 15 cases of pile driving were gathered which has been summarized in Table 2. 3.3

Site B

This site was in Kish, a resort island in the Persian Gulf and was performed as an expansion phase of Kish commercial harbor. The characteristics of used piles

Site A A1 B1 C1 B2 C2 B3 A4 B4 M1-1 M1-2 M1-3 M3-1 M3-3 M4-1 M4-3 Site B L3-AP3 L3-AP4 L3-AP5 L3-AP6

Total pile length (m)

Driven length of the pile (m)

Hammer model

Total number of the blow counts

31.45 31 31 35.1 35 36.31 24.3 24.3 32 34.65 32 32 32 32 32

16.45 16 16 23.1 23 29.6 22.3 22.3 18 23.65 18 18 18 18 18

Da -36 D-36 D-36 D-36 D-36 D-36 D-36 D-36 D-36 D-36 D-36 D-36 D-36 D-36 D-36

970 848 896 1632 1570 2925 1521 1367 1808 2115 2405 2009 1856 1837 1817

7 7 7 7

6.75 6.75 6.25 6.5

D-62-22 D-62-22 D-62-22 D-62-22

80 117 31 41

Pile No

Total pile length (m)

Recorded length of the pileb (m)

Hammer model

Total number of the blow counts

Site C B31 A32 A27 B32 A34 C34 B35 C31 B37 A37 E34 D43 D42 B31

38.6 38.6 38.6 38.6 38.6 38.6 38.6 38.6 24 24 39.6 39.6 39.6 39.6

9.48 5.52 7.34 8.54 8.6 8.33 7.79 7.2 9 8.95 8.64 8.96 8.71 8.2

D-100-13 D-100-13 D-100-13 D-100-13 D-100-13 D-100-13 D-100-13 D-100-13 D-46 D-100-13 D-100-13 D-100-13 D-100-13 D-100-13

1134 761 918 984 920 1089 1012 946 978 1293 843 1160 958 1016

a

D denotes Delmag diesel hammer. The blow count per meter has been recorded in this length which shows the last few meter of the driven length. b

are: piles type = open-ended steel pipe pile, piles total length = 7 m, piles embedded length = 6.75 m, piles diameter = 0.914 m, piles wall thickness = 16 mm. The hammer which was used to drive the piles was Delmag D-62-22 with approximate efficiency of 0.85. The soil condition of this site is summarized in Table 1. Based upon field observation, four records of blow counts per meter along the driven length of piles were gathered during the driving process. The total numbers

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of blow counts along the entire driven length of these piles have been shown in Table 2.

Table 3. The values of unknown parameters in GRLWEPA before and after calibration in different sites.

3.4 Site C This site locates in Kangan harbor, a small harbor in Kangan town in the coastline of the Persian Gulf on Bushehr province. Multitude numbers of large diameter piles were driven by Delmag D-100-13 and Delmag D-46 with estimate efficiency of 0.85. Piles characteristics were: pile type = open-ended steel pipe pile, pile total length = maximum 40 m, pile embedded length = 25 to 30 m, pile diameter = 1.422 m, pile wall thickness = 19.8 mm. The soil profile mainly consists of the normal to the very dense silty sand and gravel. The general soil profile is described in Table 1. Totally, 14 records of blow counts per meter along the penetrated length of pile were gathered. But in this site, these records are only pertaining to the last few meter of driving length which has been shown in Table 2. The total number of blow counts along the recorded driven length of the piles has presented in the last column of Table 2.

Site

Qs (mm)

Qt (mm)

Js (sec/m)

Jt (sec/m)

Stiffness of the hammer cushion (kN/mm)

The values of unknown parameters used for Starting estimates A 2.5 6.35 0.16 0.5 40000 B 2.5 7.6 0.16 0.5 40000 C 2.5 11.85 0.65 0.5 10000 The values of unknown parameters after calibration A 2.5 6.35 0.16 0.5 900000 B 2.5 7.6 0.16 0.5 648000 C 2.5 2.5 0.65 0.5 38180

4 ALIBRATION OF INGOING PARAMETERS IN GRLWEAP FROM FIELD TEST DATA 4.1 Calibration procedure The analyses have been carried out in this study using GRLWEAP program. The soil parameters needed for analyses are: toe quake, Qt , skin quake, Qs , which describe maximum elastic deformation at the toe and skin of the pile; toe damping, Jt , skin damping, Js , which describe dynamic behavior of the soil at the toe and skin of the pile respectively. The required driving system properties are hammer and pile cushions information such as cross sectional area, elastic modulus, thickness, coefficient of restitution, cushions’stiffness and helmet weight. Physical and mechanical properties of pile are also needed. All of the above information was provided from field data except soil properties and the hammer cushion’s stiffness. The unknown parameters were back-calculated from the set of pile per blow data, which were directly collected in the field as presented in Table 2. To initiate the iterative process, starting estimates are required. The typical values of smith damping and quake parameters recommended by others (Mcvay & Kuo 1999, Nath 1990 , Pile dynamic 2003) are used to obtain starting estimates for Qt , Qs , Js and Jt . These values have been shown in Table 3. The hammer cushion’s stiffness is calculated from EA/t whe E is the elastic modulus, A is the cross sectional area and t is the thickness of the cushion. The used hammer cushion material in the both of sites A and B were a few rounds of towing wire, while wood hammer cushion was exerted in site C. Table 3 shows the quantities of the unknown parameters in GRLWEAP before and after calibration. In order to evaluate the calibration result, measured blow counts

Figure 1. Comparison of the measured blow counts per meter in field with the calibrated model of site B in GRLWEAP program.

per meter of the field were compared with GRLWEAP program output after calibration process. For instance, the calibration result in the site B has been displayed in Figure 1.

5

PARAMETRIC STUDY

After calibration of driveability analyses for each of three above mentioned sites, a parametric study was undertaken for each site to produce more data. This parametric study also considers the effects of various involving parameters on the numbers of the blow counts per meter, n. These parameters are (1) coefficient of restitution of the hammer cushion, C.O.R, which shows amount of energy dissipation when ram impacts a pile cushion; (2) toe quake, Qt , which describes maximum elastic deformation of the soil at the pile toe; (3) skin damping, Js , which describes dynamic behavior of soil at the skin of the pile; (4) helmet weight; (5) stiffness of the hammer cushion, k; (6) embedded length of the pile. After conducting the parameter study, following results were obtained: C.O.R of commonly used material for cushions usually varies between 0.5 and 0.8. For increasing the

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Figure 2. Effect of the hammer cushion stiffness on the blow counts per meter in site C.

value of C.O.R, no notable changes was recognized in the numbers of blow counts per meter. According to the others recommendation (Mcvay & Kuo 1999, Nath 1990, Pile dynamic 2003), toe quake, Qt , can vary from 2.5 mm to D/120, where D is pile diameter in millimeter, but skin quake, Qs , can be taken 2.5 mm for all soil types. Also toe damping, Jt , can be chosen 0.5 sec/m for all soil types while skin damping, Js , is changing from 0.16 sec/m for non-cohesive soils to 0.65 sec/m for cohesive soil. So, here only Qt and Js were considered. This study showed a slight variation of blow counts per meter, while changing these parameters. Helmet weight was varied between 10 and 30 kN to investigate its effect on blow counts per meter. The helmet is usually heavy and so rigid so that it is considered as a lump mass in wave equation analyses. Increasing the helmet weight whiles the hammer, pile and soil properties are constant, resulted in decreasing and then increasing of the blow counts per meter values. The stiffness of hammer cushion, k, is calculated based on the k = E.A/t where E is elastic modulus, A is cross sectional area of the cushion and t is cushion thickness. To investigate the effect of hammer cushion stiffness, elastic modulus of cushion material was varied from 250 to 5000 MPa and the thickness was taken to 5 cm. As it can be seen in Figure 2, which depicts the effect of the hammer cushion stiffness in site C, for increasing the value of the k form 5000A to 100000A (MN/m3 × A), the blow counts per meter are decreased. It means that stiffer cushions transmit greater percentage of the hammer’s energy to the pile head. Figure 3 shows the effect of embedded length of the pile on the number of blow counts per meter in the site A for different Delmag diesel hammers. As the pile is driven into the soil, total soil resistance at the point and skin of the pile is accumulated. According to the Figure 3, as the embedded length of the pile is varied from 0 to 16.45 m, blow counts per meter tend to increase exponentially.

Figure 3. Effect of embedded length of the pile on the blow counts per meter in site A for various Delmag hammers.

6 APPROXIMATE EQUATIONS 6.1 Approximate equations for hammer selection In order to present an equation to determine which hammer is suitable for driving a certain pile in a known soil condition, a basic concept of energy transferring during pile driving and some of the more accurate pile driving formulas were considered. Then, using the results of parametric study, the final equations were obtained. As the ram descending to impact the cushion, potential energy or rated energy, Er , which is given by manufactures, is progressively converted to the kinetic energy. After losing some energy in the hammer and driving systems, the energy that actually arrives at the pile top is called transferred energy as follow:

where Et = transferred energy; Er = rated energy; eh = hammer efficiency; and ed = loss factors in driving system. In the current study, some of the more accurate pile driving formulas like Janbu, Gate and Hiley (Lowery et al. 1968, Fragaszy et al. 1985, Allen 2005) were considered in the of energy as presented in Table 4. Using the general form of Janbu formulas in the term of transferred energy and knowing the fact that energy is product of a force or resistance through a distance, leads to

where Et = transferred energy; Ru = total soil resistance; s = permanent set of the pile per blow and C = elastic compression of the pile which is computed by C = Ru L/AE where Ru is total soil resistance; L is total length of the pile; A is cross sectional area of the pile and E is elastic modulus of the pile. Mean value of diesel hammer efficiency, eh , is about 0.8 and according to the output results of GRLWEAP

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Table 4. Pile driving formulas. Formula

Equation for Et

Remarks

Janbu

Et = Qu sKu

Gate Hiley

Et = [Qu (104.5(2.4-log s))]2 Et = [Qu (s + 0.5(c1 + c2 + c3 ))]/α

Ku = Cd {1 + (1 + λe /Cd )0.5 } λe = WHL/AEs2 Cd = 0.75 + 0.15Wp /W Qu (kN), s(mm), Et (kN.m) α = {(W + n2 Wp )/W + Wp } c1 , c2 , c3 , and n are tabulated by (Chellis 1961)

analyses, average quantity of loss factor in driving systems, eh , was also obtained 0.8. Thus, for simplicity of final equation, the values of eh and ed were set to 0.8. So, inserting corresponding values for eh and ed into the Equation (1) and using Equations (1) and (2) gives

Using the results of the parameter study, it was observed that number of blow counts per meter, n, is varying with the embedded length of the pile exponentially as presented in Figure 3. As the set per blow, s, is the inverse of the n, the Equations of (4) and (5) were obtained by plotting the embedded length of the pile with blow counts per meter for driving the pile with small and large hammers respectively as follows:

Table 5. Comparison the output results of the proposed equations with GRLWEAP program. Data of the Pile No. A1 of Site A Total length of the pile: 31.45 m Embedded length of the pile: 16.45 m Cross sectional area of the pile: 420.7 cm2 Total soil resistance (skin plus toe) 3836 kN Elastic modulus of pile: 2 × 108 kN/m2 Min Set of pile per blow (smin ): 5.27 × 10−3 m Max Set of pile per blow (smax ): 14.64 × 10−3 m Output results of the proposed equations and GRLWEAP Min rated Energy (Eqs (6)&(7)): 115 kJ Max rated Energy (Eqs (6)&(7)): 264 kJ Range of hammer ID (Eqs (6)&(7)): Da -36 to D-80 Range of hammer ID (GRLWEAP): D-15 to D-80 Used hammer in filed: D-36 a

where smin = minimum set of pile per blow, which is usually created by small hammers; smax = maximum set of pile per blow, which is made by large hammers; nmin and nmax are minimum and maximum number of blow counts per meter respectively; andLe = embedded length of the pile. Putting the Equations (4) and (5) into the Equation (3) yields the Equations of (6) and (7) as follows:

D denotes Delmag diesel hammer.

Emax. As the rated energy of various hammers is given by manufactures, a proper hammer for a certain pile and a certain soil condition can be selected. Exerting the Equations (6) and (7) for the condition of 3 studied sites showed a very good agreement with GRLWEAP output result. Table 5 indicates the output results of the proposed equations and GRLWEAP program using the data of the pile No. A1 of site A. The used hammer for driving this pile in field was Delmag D-36. According to Table 5, Equations (6) and (7) are suggesting the range of Delmag D-36 to D-80 for driving of this pile. 6.2 Approximate equations for the estimation of driving time The driving time can be estimated from total number of blow counts, N , and hammer blow rate (blows per minute), B, as follow:

where Emin and Emax are minimum and maximum rated energy of diesel hammers. Ingoing parameters in Equations (6) and (7) are: total soil resistance, Ru , embedded length of the pile, Le , total length of the pile, L, cross sectional area of pile, A and elastic modulus of pile material, E. The two latter experimental equations present a range of rated energy between Emin and

where T = total driving time; N = total number of blow counts and B = the blow rate. In order to obtain an experimental equation for computing total driving time, a parametric study carried out based on the GRLWEAP calibrated models, but for brevity it is not included here. Considering the pattern of blow counts

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per meter, n, with embedded length as the pile is driven into the soil usually shows an exponential growth as presented in Figure 3. Thus, in this study using the general form of geometric progression formulae, N is computed based on parametric study as follow:

where N = total number of blow counts; n◦ = blow counts in the first meter of driving; Le = embedded length of the pile which shows how many number of must be added and q = common ratio which is related to the pile driving factors like the embedded length, total soil resistance and rated energy of hammer according to the parametric study results as given by

modeled in the wave equation analysis program, GRLWEAP. Unknown input parameters in GRLWEAP were obtained by calibration of the output result of this program with recoded blow counts per meter in the field. Then a parametric study was performed to produce more data. Based upon the measured and generated data, a series of equations were obtained: (i) Equations (6) and (7) represent the minimum and maximum of rated energy. As the rated energy of various hammers is given by hammer manufactures, a proper range of hammers are achieved for driving a certain pile in a certain soil condition. (ii) Equation (11) represents an estimation of total driving time. ACKNOWLEDGMENTS The research was ed by Pars Geometry Consultant and Karan Darya Construction Company. Dr A. Cheshmi was also very helpful. These s are acknowledged.

where q = common ratio; Le = embedded length; Er = rated energy of used hammer and Ru = total soil resistance. According to the collected data of the sites A, B and C, averagely 5 blows per meter in the first meter of driving were observed. Furthermore, mean values of blow rate, B, in the diesel hammers in the studied sites was about 45 blows per minute. Thus, to avoid the complexity of the final equation, n◦ and B are set to 5 and 45 respectively. Using Equations (8) and (9) with corresponding values of n◦ and B yields

where q, common ratio, must be computed from Equation (10). It should be noted that: (a) Equation (11) represents an estimation of total driving time which does not include interruption times (wait time). However, it shows a good agreement with GRLWEAP results. (b) If refusal occurs (blow counts goes high e.g. greater than 300 blows per foot according to (API 2000)), the driving time calculation is meaningless. 7

CONCLUSIONS

REFERENCES Allen, T. 2005. Development of the WSDOT pile driving formula and its calibration for load and resistance factor design. Report No: WA-RD-610.1, HQ Material labartobary, Geotechnical Div, Washington State Department of Transportation , Olympia, Washington. API. (21nd ed) 2000. Recommended practice planning, deg, and construction fixed offshore platform working stress design. API Recommended practice 2A-WSD (RP 2A-WSD). Chellis, R. (2nd ed.) 1961. Pile foundations. New York: McGraw-Hill. Fragaszy, R.J. & Higgins, J. 1985. Development of guidelines for constructive control of pile driving and estimation of pile capacity. Report No: WA-RD-68.1, Washington State department of Transportation, Washington. Lowery, L., Finley, J. & TJ, H. 1968.A comparison of dynamic pile driving formulas with the wave equation. Research report 33-12, The Texas Transportation Institute, Texas A&M University, The texas Highway Department, Texas. Mcvay, M. & Kuo, C. 1999. Estimate damping and quake by using the traditional soil testing. Report No. WPI0510838, Univeristy of Florida, Florida Department of Transportation Managment Center. Nath, B. 1990. A continuum method of pile driving analysis: Comparison with the wave equation method. Computers and Geotechnics 10 (4): 265–285. Pile dynamic, I. 2003. GRLWEAP Wave Equation Analysis Of Pile Driving. Procedures and Models, Cleveland, Ohio.

The pile driving data of three different sites were collected and the driving conditions in these sites were

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Settlement analysis of a large piled raft foundation M. Wehnert Wechselwirkung – Numerische Geotechnik GmbH, Stuttgart,

T. Benz Norwegian University of Science and Technology, Trondheim, Norway Wechselwirkung – Numerische Geotechnik GmbH, Stuttgart,

P. Gollub BAUER Spezialtiefbau GmbH, Schrobenhausen,

T. Cubaleski E.ON Engineering GmbH, Gelsenkirchen,

ABSTRACT: This paper presents a settlement analysis of a large piled raft foundation. Total raft area is about 12,600 m2 . The raft investigated is a monolithic raft ed by more than 500 large diameter bored piles. Due to problem size, all bored piles are discretized with embedded elements. Analyses taking into the full soil-structure interface between soil and pile are conducted for smaller pile groups and single piles: First, load tests on single piles are back calculated. Then, a small pile group is analyzed. From this, the element size in the 3D analysis of the complete raft is determined so that the embedded element approach yields reasonable results in the service load range. The analysis aims mainly for an estimate of differential settlements and quantification of load distribution between raft and piles.

1 1.1

INTRODUCTION Project description and aim of the investigation

This paper is concerned with three dimensional finite element analyses of a piled raft foundation. The piled raft carries all main structural components of the 1070 MW coal-fired ’Maasvlakte Power Plant 3’ (MPP3) in Rotterdam, the Netherlands: The boiler building (UHA), the steam turbine building (UMA), two stair cases (UHT and UMT), the unit control building (UCA), and the air preheater denox (UVA). The raft is ed by more than 500 large diame-ter bored piles with diameters of 120 or 150 cm. The piles have different lengths and the monolithic raft varies in thickness and foundation depth. Wechselwirkung–Numerische Geotechnik GmbH was mandated by BAUER Spezialtiefbau GmbH to perform settlement analyses of the piled raft foundation using the finite element method (FEM). The methodology for deriving the final 3D FEM model was chosen as follows: •

The single piles as well as a pile group underneath a highly loaded column of the UHA structure are modeled with the 3D program. Again, piles are discretized with volume elements; soil-structure interaction is ed for by interface elements. • Transfer 3D pile group to a 3D FE model using ABAQUS version 6.7. In the ABAQUS model, piles are discretized with embedded elements. In both codes, PLAXIS and ABAQUS, the same constitutive models are used. Concrete is assumed to be linear elastic. The Hardening-Soil model (Schanz et al. 2000) is adopted for sand and clay. • Analyze the complete raft using ABAQUS 6.7 considering the findings of the previous analyses.

Back analyze three pile load tests performed on the site with a 2D FE code (PLAXIS V8.6). Piles and pile-soil interaction are modeled with volume and interface elements, respectively. • Transfer findings from the axi-symmetric 2D model to a 3D FE model (PLAXIS 3D Foundation V2.1).

1.2 Soil profile and subsoil conditions The subsoil in the area of the power plant consists of thick layers of sand divided by thin layers of clay and peat. Soil profiles at four different locations A to D are shown in Figure 1. The location of points A to D in respect to the foundation is shown in Figure 7. The thicknesses of most layers vary only little so that in the calculation model they are assumed constant in thickness. An exempt is the uppermost clay layer. This layer’s boundary to the underlying sand layer is assumed as an inclined plane.

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Figure 1. Subsoil conditions in the area of the piled raft.

Figure 3. 2D (left) and 3D (right) FE mesh for the back analysis of the pile load tests.

Figure 2. Subsoil conditions in the area of the pile load tests.

2 2.1

•

Pre-stressing of pile base with closed lift-cell for pile 1: pile is assumed to be 0.75 m longer in the analysis. • Pre-stressing of pile base with open lift-cell for pile 2: no special measures are assumed in the analyses (base grouting compensates the loosening due to installation). • Disturbance underneath pile toe for pile 3: cluster (ø 2.4 m, height 3 m) with reduced stiffness parameters (40%) is placed underneath pile toe.

BACKCALCULATION OF PILE LOAD TESTS Pile load test data

Three bored piles were tested on the site (Körber 2009). Skin friction of all piles was eliminated from ground level to the uppermost clay and peat layer. All three piles have different embedment lengths in the dense sand. Diameters and pile base pre-stressing vary, too: Pile 1 has a diameter of 1.5 m, a total length of 31.0 m and is equipped with a closed lift-cell. Pile 2 has a diameter of 1.5 m, a total length of 36.0 m and is equipped with an open foot-grouting-cell. Pile 3 has a diameter of 1.2 m, a total length of 31.0 m and no special equipment at the pile toe. The geology in the load test area differs slightly from the one shown in Figure 1. Figure 2 illustrates the subsoil model as well as the three tested piles. 2.2

simulate soil disturbance underneath pile toe 3 due to installation processes. Disturbed and loosened soil conditions underneath pile toes are often reported for bored piles in granular soils (Stocker 1980, Feda 1986, Hartung 1994). Pile base pre-stressing can be a suitable countermeasure. Two different methods of pile base pre-stressing were investigated in the load tests: closed lift cells and base grouting. With regard to the size of the power plant’s raft, the consideration of piles and special measures at the pile toe in the numerical analysis should be as simple as possible.Thus, piles are wished in place.The following additional assumptions are made concerning the pile base:

Back analysis of load tests

In a first stage, axis-symmetric analyses of the pile load tests are performed. The aim of these analyses was twofold: First, to calibrate soil parameters. Secondly, to simulate pre-stressing in pile base 1 and 2, and to

The 2D axis-symmetric domain has a width of 25 m and a depth of 55 m. 15-noded triangular elements with a fourth order interpolation function are used. In a second stage, the findings from 2D calculations are transferred to a 3D model (Figure 3.) One half of the test pile is modeled here due to limitations in geometry pre processing – typically a quarter would do. The 3D domain has a width of 20 m and a depth of 55 m. 15-noded wedge elements with a quadratic interpolation function are used in the 3D analysis. In both analyses piles are modeled with volume elements. Interface elements are located along the pile shaft. The FE discretizations for both, the 2D and 3D analysis, are shown in Figure 3.

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Table 1.

γr c ϕ ψ Eref 50 Eref oed Eref ur νur m Rinter

Soil Parameters of Hardening Soil model used in all analyses.

[kN/m3 ] [kN/m2 ] [◦ ] [◦ ] [MN/m2 ] [MN/m2 ] [MN/m2 ] [–] [–] [–]

Sand

Clay and peat

Sand, dense 1

Clay 1

Sand, dense 2

Sand, with clay lamin.

Sand, clayey

Clay 2

Sand, dense 3

20.0 0.0 30.0 0.0 35.0 35.0 105.0 0.20 0.5 1.0

17.0 2.0 22.5 0.0 1.92 0.96 7.68 0.15 1.0 1.0

20.0 0.0 25.0 0.0 200.0 200.0 600.0 0.20 0.5 1.0

18.0 5.0 25.0 0.0 3.2 1.6 8.0 0.15 1.0 1.0

20.0 0.0 35.0 5.0 150.0 150.0 450.0 0.20 0.5 1.0

20.0 0.0 27.0 0.0 43.0 43.0 129.0 0.20 0.5 1.0

18.0 0.0 27.0 0.0 7.4 3.7 33.3 0.20 1.0 1.0

18.0 5.0 25.0 0.0 4.0 2.0 12.0 0.15 1.0 1.0

20.0 0.0 35.0 5.0 250.0 250.0 750.0 0.20 0.5 1.0

The piles are assumed linear elastic. Isotropic hardening plasticity is assumed in all soil layers (Hardening-Soil). The soil parameters used in all analyses are shown in Table 1. Elimination of skin friction in the upper sand layers is ed for by a reduced value of Rinter = 0.01. In all other layers Rinter = 1.0, which is a common setting for bored piles (Wehnert 2006). A comparison of numerical results and pile load test (PLT) data is shown in Figure 4. The measured load-settlement-curves of the three piles can be modeled reasonably well with the assumptions made concerning the pile toe. 3

FROM VOLUME PILES TO EMBEDDED PILES

For the large number of bored piles at hand, it is not feasible to consider the soil-structure interface of each individual pile in the analysis. Instead, pile elements are embedded in soil elements. Soil elements are the host for pile elements and constrain the translational degrees of freedom of pile nodes. The embedded pile technique is explored in more detail in Sadek, & Shahrour (2004). In contrast to this reference, here, a simplified form is used as no cut-off in skin friction and no point bearing failure is considered in the (rebar-) formulation. For service load conditions this seems to be acceptable. A further advantage of the embedded element approach is that geometry changes concerning piles e.g. optimization of pile layout or length, can be realized without regeneration of the entire problem geometry. To ensure that the use of embedded piles does not derogate the analysis results, one part of the raft is modeled with and without embedded elements first. In the latter case, piles are modeled same as in the previously discussed 3D calculation of a single pile. Due to constraints in the pre-processor of Plaxis 3D Foundations V2.1, again one half of the pile group is modeled. The calculation with embedded piles is performed with ABAQUS V6.7 as 64bit computing is inevitable in analysis of the complete raft. The area around a highly loaded boiler column of the UHA structure is selected for the comparison.

Figure 4. Back analysis of pile load test (PLT) data for pile 1 (top), 2 (middle) and 3 (bottom).

Underneath the column, a 5 × 5 pile group of pile type 1 (ø 1.5 m, with closed lift-cell) is located. Around this group a row of five piles of pile type 3 (ø 1.2 m, without pre-stressing) s the raft. A sketch of the pile layout is shown in Figure 5. The grey rectangle in the center of the pile group indicates the boiler column. In the pile group analysis the same assumptions are made as previously discussed in the back analyses of load tests, except for the additional

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length of pile type 1. Here, an additional length of 1.0 m instead of 0.75 m is assumed. This is due to higher pre-stressing of production piles compared to test pile type 1. The domain of all analyses has a width of 15.625 m, a length of 27.5 m and a depth of 75.15 m. For the study on the performance of the embedded elements a coarse and a fine mesh in the ABAQUS calculation is investigated. An overview of the finite element meshes used in the analyses is given in Table 2. The subsoil is layered according to Figure 1 and the soil parameters are again taken from Table 1. A raft thickness of three meters is assumed where the bottom of the raft is located at a depth of −7.975 m. Unloading due to excavation is taken into in the analyses. The ground water table is assumed at a depth of −3.65 m. During construction the groundwater table in the upper sand layers, above the clay and peat layer, is lowered to the bottom of the raft Hence, the analyses incorporate the following steps: Phase 1:

Phase 2: Phase 3:

Initial conditions considering the unloading due to excavation and ground water table lowering in the upper sand layers Pile installation (wished in place) Activation of foundation slab

Phase 4: Phase 5: Phase 6:

Calculated vertical deformations below the boiler column (center) are given in the last row of Table 2. Note that the raft is also subject to uniform settlements prior analysis phase 6: Uniform vertical deformations of about 38 mm are found due to self weight, basement load, and groundwater uplift. All FE discretizations listed in Table 2 can reproduce this result. However, the analysis results start to deviate with application of the boiler column point load and hence, with higher pile loads and bending in the raft. Both pile group models based on embedded piles show a stiffer behaviour than the one based on volume piles and interface elements. This result is to be expected as the embedded piles are smeared in neighbouring elements and there can be no localized deformation in the interface between soil and pile. However, for service loads and in particular for the service loads in the project at hand, a reasonable agreement between both calculation approaches can be obtained with a finer mesh. 4

FINAL ANALYSIS

The domain of the 3D FE mesh of the complete raft has a size of 320 m by 440 m by 75 m. A total of 857,608 nodes are defined, resulting in a total of 2,572,824 variables in the model. The characteristic element length in the raft is smaller than the one used in the analysis denoted ‘coarse mesh’ in Table 2. This holds also for all soil elements which are in with the structure.

Figure 5. Layout of selected pile group. Table 2.

Increase of ground water table to a level of −3.65 m Application of a distributed load of 75 kPa representing the basement Application of single loads representing the up-going structure

Pile group analyses. PLAXIS 3DF

ABAQUS fine

ABAQUS coarse

Type of element Nodes per element Interpolation No. of elements No. of nodes

wedge 15 quadratic 47,730 133,115

tetrahedron 10 quadratic 62,518 93,948

tetrahedron 10 quadratic 23,222 35,794

Max. settlement

55 mm

52 mm

49 mm

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All elements, except those discretizing piles, use a quadratic interpolation function. Figure 6 details raft and pile layout. The 3D FE mesh for soil and structures is illustrated in Figure 7.

Figure 6. Raft and pile layout of main structural components.

In the final design only piles of type 1 and 3 are considered. Additionally, shaft grouting (above first clay layer) is performed on the site for the piles underneath the staircase towers UHT and UMT. This grouting is ed for in the model by a 50% increase of pile diameter (Ø 2.25 m) above the first clay layer. Next to the main raft, the loads of the coal storage as well as of the structures UHQ and UVB are considered. The coal storage is modeled as distributed load. UHQ and UVB are founded on driven piles. In the analysis, the driven piles are assumed to have a quadratic 40 × 40 cm cross-section. Their spacing is assumed to be 4.0 m. Both structures are embedded in the soil cluster (embedded elements). The basement and the up-going structures are ed for with single and distributed loads. In total, more than 275 single loads are considered in the analysis. Load input is facilitated through external access of ASCII files containing load information. The point and area of load application is defined in node sets and surfaces, respectively. The excavated soil in the area of the raft is not considered in the analysis. In determination of initial conditions this material is replaced by a load on top of the raft. During all construction phases the excavation pit is ed by distributed loads which act on the vertical slope of the pit. The magnitude of these loads is equivalent to the initial horizontal stress in the

Table 3.

UMA UHT UCA UMT UHA UVA Raft Total ∗

Figure 7. A BAQUS FE mesh as used in the analysis.

Load distribution. # Load [MN]

Uplift [MN]

Pile Force [MN]

α∗ [–]

736 139 197 153 920 222 2367

217 27 43 38 158 0 483

375 132 96 190 607 162 1562

0.72 1.18 0.62 1.65 0.80 0.73 0.83

α = Pile Force/(# Load – Uplift).

Figure 8. The raft subdivided in UMA, UCA, UMT, UHA, UHT, and UVA. UHA boiler columns marked by a cross.

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Figure 9. Vertical deformations of piled raft.

subsoil. Thus, the following construction phases are considered in the 3D analysis: Phase 1: Initial conditions considering the ground water table lowering in the upper sand Phase 2: Unloading due to excavation Phase 3: Pile installation (wished in place) Phase 4: Activation of foundation slab Phase 5: Increase of ground water table to −3.65 m Phase 6: Application of distributed loads representing the basement and staircases Phase 7: Application of all loads representing the up-going structure and live loads Phase 8: Activation of surrounding structures and loads including coal storage 5

The analysis presented shows the possibility of analyzing a large and inhomogeneous raft (raft thickness & pile layout) by relatively simple means. The variation in load distribution factors presented in Table 3 proof it necessary to model the entire raft in the settlement analysis: Neither a 2D cross section nor a 3D calculation of the UHA assembly alone would be adequate to forecast differential settlements of the boiler columns. REFERENCES

RESULTS AND CONCLUSIONS

The piled raft foundation’s dead and live loads are partly carried by the piles and partly transferred to the subsoil directly via raft stress. The load transfer is analyzed in more detail in Table 3. Here all applicable dead and live loads at the end of Phase 8 are compared to calculated pile and buoyancy forces. The comparison is conducted for the components UMA, UCA, UMT, UHA, UHT, and UVA, separately. The geometry assumed for these components is sketched in Figure 8. This figure additionally indicates the areas and volumes assumed for calculating dead loads and buoyancy forces. Pile forces are derived from the FE analysis by integrating the stresses of the uppermost pile elements. Vertical deformations of the piled raft at the end of phase 8 are exemplarily shown in Figure 9. The calculated differential settlements of the four main columns in the boiler house (UHA) are below those specified in the design requirements.

Feda, J. 1986. Zulässige Belastung von Großbohrpfählen. Bautechnik 63(2): 42–45. Hartung, M. 1994: Einflüsse der Herstellung auf die Pfahltragfähigkeit in Sand. Mitteilung des Instituts für Grundbau und Bodenmechanik der Technischen Universität Braunschweig, 45. Körber, G. 2009: Vorwegnahme von Setzungen bei hochbelasteten Großbohrpfählen durch den Einbau von Hubkissen an der Pfahlsohle. In: Pfahl-Symposium 2009, 425–436, Braunschweig. Sadek, M. & Shahrour, I. 2004: A three dimensional embedded beam element for reinforced geomaterials. Int. J. Numerical and Analytical Methods in Geomechanics 28(9): 931–946. Schanz, T. & Vermeer, P.A. & Bonnier, P.G. 1999: The hardening soil model – formulation and verification. In Beyond 2000 in Computational Geotechnics, 281–296. Rotterdam: Balkema. Stocker, M. 1980: Vergleich der Tragfähigkeit unterschiedlich hergestellter Pfähle. In Vorträge der Baugrundtagung in Mainz, 565–590. Essen: Glückauf. Wehnert, M. 2006: Ein Beitrag zur drainierten und undrainierten Analyse in der Geotechnik, Universität Stuttgart, Mitteilung des Instituts für Geotechnik, 53.

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Study of a complex deep foundation system using 3D Finite Element analysis F. Tschuchnigg & H.F. Schweiger Computational Geotechnics Group, Institute for Soil Mechanics and Foundation Engineering, Graz University of Technology, Graz, Austria

ABSTRACT: This paper shows results from numerical analyses with the objective to assess the settlement behaviour of two towers situated close to each other. The aim of the study was to find the optimal layout of the foundation elements with respect to minimising vertical and differential displacements. Since a 2D representation of the problem is not possible a number of 3D analyses have been performed. Different arrangements of diaphragm wall s have been investigated using simplified finite element models. All calculations in the paper are performed with the Finite element code Plaxis 3D Foundation and the mechanical behaviour of the soil is described with both the Hardening Soil and the Hardening Soil Small model.

1

INTRODUCTION

In general high rise buildings cannot be ed by shallow foundations and a deep foundation system is required. Depending on the soil profile and the corresponding soil properties a pile, piled raft or diaphragm wall foundation is the solution for most cases. For these types of deep foundation systems assessment of settlements and differential settlements are the key issues. Thus ultimate limit state conditions are not considered in the proposed paper. To find the optimal layout of the foundation elements with respect to minimising vertical displacements, advanced numerical modelling is essential. Numerical analyses with the objective to assess the settlement behaviour of the towers are presented. Since a 2D representation of the problem is not possible a number of 3D analyses have been performed. Different arrangements of diaphragm wall s are investigated and finally a new 64 bit calculation kernel was used to run one big finite element model without any geometrical simplifications. All calculations in the paper are performed with the Finite element code Plaxis 3D Foundation and the diaphragm wall s are modelled as individual volume elements. The mechanical behaviour of the soil is described with the Hardening Soil model, a double hardening model and the Hardening Soil Small model. Both models are available in the Plaxis model library (Brinkgreve & Swolfs 2007). 2

Figure 1. Project overview.

tower II of 165 m. It is planned to build the foundations for both towers at the same time but to construct the superstructure of tower I first. Due to the fact that the towers will be located very close to each other it is necessary to take the loads from the later built tower II into for the design of the foundation system of tower I. Figure 1 shows a schematic overview of the construction and the designed layout of the diaphragm wall s. The barrettes have a unit length of 3.6 m and a unit width of 0.6 m. This layout is used for all calculations presented in this paper. The circumference of the foundation elements indicates at the same time the dimensions of the two towers. Next to the towers four stories of underground car parks are planned.

GENERAL INFORMATION

2.1 Project overview

2.2 Soil conditions and its numerical modelling

The projects discussed are two very high and slender towers. Tower I has a total height of about 220 m and

The soil profile for the finite element simulation is based on core drillings with depths down to −70.0 m

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Table 1.

Soil properties for the Hardening Soil model.

Parameter

Deposit

Gravel

Sandy silt

Fine sand

γunsat (kN/m3 ) γsat (kN/m3 ) Eref 50 (kPa) Eref oed (kPa) Eref ur (kPa) c (kPa) ϕ (◦ ) ψ (◦ ) υur (−) pref (kPa) m (−) Knc 0 (−)

17.5 20.5 2000 2000 6000 0 27.5 0.0 0.2 100 0.60 0.538

21.0 22.0 40000 40000 120000 0 35.0 5.0 0.2 100 0.00 0.426

20.0 20.0 20000 20000 50000 20.0 25.0 0 0.2 100 0.80 0.577

20.0 21.0 25000 25000 62500 2.0 32.5 2.5 0.2 100 0.65 0.463

Figure 2. Soil profile and geometry of towers.

Table 2. Additional parameters for the HSS model.

from the surface. Figure 2 shows the soil profile obtained with the borehole logs together with important levels of the construction. Due to the fact that the interaction between deep foundation elements and the surrounding soil is the key element for such a foundation the mechanical representation of the soil is the most important part of the analysis. For the calculations presented either the so called Hardening Soil model (Schanz et al. 1999) or the Hardening Soil Small model (Benz 2007) was used to model the soil behaviour. Both models are elasto – plastic constitutive models which enable to model both deviatoric and volumetric hardening and take the stress dependency of stiffness into . The Hardening Soil Small (HSS) model additionally allows for modelling the high stiffness at very low strains. As a consequence the obtained soil displacements at deeper depths are automatically reduced and a more realistic settlement profile with depth can be computed. Compared with the Hardening soil model (HS) the HSS model needs two additional parameters to describe the stiffness behaviour at small strains. Namely the initial shear modulus G0 and the shear strain level γ0.7 , which represents the amount of shear strains where the secant shear modulus is reduced to 70% of its initial value. The small strain shear modulus G0 is defined with the help of a correlation between very small strain stiffness and stiffness at larger strains after Alpan (1970). The therefore necessary static Young’s modulus is interpreted as unloading/reloading stiffness, which correlates to recent published experimental data (Wichtemann & Triantafyllidis 2009). The value of γ0.7 is taken from stiffness reduction curves after Vucetic and Dobrey (1991). The input parameters for the Hardening Soil model are given in table 1. The used parameters for the HSS model are given in table 2. The values for the initial shear modulus G0 are again reference values at a stress level of 100 kPa. The stress dependency is taken into according to equation 1. A similar power law is also utilized for the scaling of the other stiffnesses used in the constitutive models.

Parameter

Gravel

Sandy silt

Fine sand

G0 (kPa) γ0.7 (−)

150000 0.0001

62500 0.0002

78125 0.0002

All slabs behave linear elastic and the diaphragm wall elements are modelled with the Mohr-Coulomb model (linear elastic - ideal plastic model). In addition the tensile strength of the barrettes is limited to a value of 3000 kPa. 3

COMPUTATIONAL MODEL

3.1 Description of finite element models Due to the geometrical layout of the problem (Fig. 1) a two dimensional representation is not possible and full 3D modelling is required. In the 3D calculations 15 noded wedge elements with quadratic shape functions are used. Because of the small distance between the two towers an interaction is expected and thus it is necessary to model both towers. To reduce the complexity of the 3D model in a first step the barrettes of only one tower are modelled in full detail and the foundation system of the other tower is modelled as homogenized blocks, meaning that the zones of the sub-soil in which s are installed are defined with smeared properties. With this approach the global settlement behaviour of the entire structure is calculated because the interaction of the towers is taken into . All barrettes are modelled by means of volume elements. However, to validate this modelling assumption an analysis where both foundations are explicitly modelled is presented. Figure 3 shows exemplary one finite element model for the detailed analysis of tower 1. All models analysed consist of around 50000 finite elements.

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Figure 4. Contour lines of vertical displacements for detailed model of tower II with 25 m long barrettes.

Figure 3. Standard 3D Finite elment model.

3.2 Calculation procedure All calculations in this paper are drained analysis, thus final settlements are presented. To obtain realistic final deformations a reliable stress distribution in the soil after the excavation is required, consequently it is necessary to model the building process. This is done in the following phases: – – – – – – – – – –

Generation of initial stresses Activation of the sheet pile wall Excavation and groundwater lowering Activation of barrettes (wished in place) Activation of slabs Full loads of tower I and loads from basement floors of tower II Closing of settlement t – tower I Full loads of tower II Closing of settlement t – tower II End of ground water lowering

For the generation of the initial stress state it is important to take the overconsolidation of the soil into . This is done with the so called preoverburden-pressure (POP). See Equation 2 below:

where σp is the largest vertical effective stress earlier reached and σyy is the in-situ effective vertical stress. For all soil layers a value of 600 kPa is defined for the POP and the earth pressure coefficient K0 is increased from Knc 0 to a value of 0.7. All displacements discussed in the paper are worked out after the final calculation phase and deformations computed until the activation of the slabs are reset to zero.

4

RESULTS OF NUMERICAL ANALYSIS

4.1 Optimisation of the foundation concept The results shown in this paragraph are related to the HSS model and the diaphragm wall layout shown in Figure 1. The next section will compare the results with the HS model and will show the benefit of the HSS model.

Figure 5. Optimised barrett layout for tower I and II.

In the first analysis a constant length of 25.0 m for all barrettes is used. The calculation is performed once for a detailed geometry of tower I and in a separate calculation for a detailed model of tower II. For both calculations maximum vertical displacements of about 80 mm are calculated. And the assumption that tower II contributes to settlements also in the region of tower I are confirmed. Figure 4 shows the contour lines of vertical displacements for the model, where tower II is modelled in detail. Because of the eccentric loading of both towers the maximum settlements are also off-centre. Due to the fact that the differential settlements will lead to slightly leaning towers it is necessary to optimise the foundation system in a way that the expected maximum of deformation is in the centre of each tower. Additional it is required to design the deep foundation elements of tower I with prevision of the settlements coming from the later built tower II. For this optimisation procedure a number of 3D analyses have been performed for both towers. The result of this study is a final layout of the s with lengths between 20 and 30 m. The maximum settlements calculated are again about 80 mm for both towers, but this foundation set-up does not yield eccentric settlement troughs. Figure 5 shows the optimised layout for both towers.

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Figure 6. Top view of 3D FE mesh for 64 bit calculation kernel. Table 3.

Mesh details.

Model

Nr. El.

Nr.Nodes

Nr.El in 2D plane

Layers

32 bit model 64 bit model

49096 136710

131993 361243

2888 6510

17 21

Figure 7. Vertical displacements of entire 3D model (a) and structural elements (b).

Finally a calculation with a detailed geometry for both towers and a much finer mesh discretisation is performed. More than 300 diaphragm wall s are modelled. The top view of the mesh is shown in figure 6 and mesh details are given in table 3. The calculation phases and the soil parameters remain unchanged. Due to the size of the analysis a 64 bit calculation kernel is used, because the standard 32 bit kernel would not be able to solve such a big model. As expected the obtained results are similar to the ones calculated with the first approach, where only the foundation system of one tower is modelled in detail. Figure 7a shows the contour lines of vertical displacements of the entire model and figure 7b the settlements of the structural elements. The maximum settlements of both towers are again about 80 mm. 4.2

Differential settlements

Due to the fact that the two towers are located in a densely built-up region differential settlements are a key issue of the settlement prediction. Figure 8 shows schematically the situation in a top view. Additionally some selected points are presented for which the differential settlements are worked out. Table 4 presents the vertical displacements uy obtained in the selected points and the inclination tan α between two neighbouring points. For the distribution of settlements between the two towers the big model, where both towers are modelled in detail, is used. The railway lines are in the most critical area, where a maximum vertical deformation of 18 mm and a inclination of the settlement trough (tan α) up to 1/600 is

Figure 8. Schematic top view including adjacent buildings.

calculated. This value is after Bjerrum (1973) acceptable from a mechanical point of view. In the region of the highways the settlements are 14 mm and between the towers, where a road is situated, displacements up to 40 mm are computed. 4.3 HS vs HSS As mentioned above the so called Hardening Soil Small model (HSS) takes the very high stiffness at small strains into which automatically decreases the settlements at deeper depths. To show this effect the calculations with the final diaphragm wall layout are considered and the settlement distribution over depth, for a point in the middle of the towers,

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Table 4.

Differential settlements of selected points.

Selected points

uy (mm)

A1 A2 A3 A4 B1 B2 B3 B4 C1 C2 C3 D1 D2 D3

33 14 6 2 19 5 2 0 36 18 13 38 32 48

Distance (m)

tan α(−)

17.0 1/900 16.0 1/2000 16.0 1/4000 23.0 1/1600 16.0 1/5300 16.0 1/8000 11.5 5.0 1/1000 9.0 1/1500 15.0

1/600

Figure 10. Settlement troughs.

1/900

Figure 11. Normalized settlement troughs.

Figure 9. HS vs HSS – Settlements over depth.

is worked out. This is done for the analysis performed with the HS and the HSS model. Figure 9 shows the comparison of both constitutive models. Due to the fact that the settlements are almost the same for both towers, only the graph for tower I is shown. Until a depth of 36.6 m below ground level, which is the level of the longest barrettes, the distribution of settlements obtained with the HS model is similar to the one obtained with the HSS model, but the HSS model computes 25% less settlements. Beneath the foundation elements the difference between the HS and HSS model is increasing and at a depth of −75.0 m below the surface the settlements obtained with the HSS model are 51% smaller than the once calculated with the HS model This clearly shows the influence of small strain stiffness and indicates that once a model including small strain stiffness is used the effect of the position of the bottom boundary condition is reduced and the right depth of influence is taken into automatically by the constitutive model. Another significant difference are the settlement troughs computed. With the HS model the settlements at the surface are higher and the spread of relevant settlements are wider compared to the HSS model.

Figure 10 shows the settlement trough of cross section A–A (Fig. 8) at the surface for both constitutive models. The Hardening Soil Small model computes differential settlements between point A2 and A3 in the range of 1/2000. When using the HS model this value decreases significantly to a value of 1/1000. This decrease of tan α is related to the big difference in maximum vertical displacements obtained with the different models. If the settlements are normalized by their maximum values the behaviour changes and the settlement trough computed with the HSS model is steeper. Figure 11 shows a normalized settlement trough for cross section A-A. In this particular project the HS model gives conservative results for both the maximum settlements and the differential settlements, but for other applications it is possible that the HSS model yields steeper settlement troughs, which is the more critical scenario when considering differential settlements. 5

CONCLUSIONS

Results from numerical analyses with the objective to assess the settlement behaviour of two high towers founded on diaphragm wall s have been presented. The first calculations show that deep foundation elements having the same length yield to eccentric

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settlements troughs and due to the interaction of both towers to additional settlements in the region of tower I once tower II is built. As a consequence the s must have different lengths in different regions. The final concept with lengths of barrettes between 20– 30 m shows the improvement related to the settlement behaviour of the towers. In most calculations the finite element model was simplified in that way that only one tower is modelled in detail and the foundation of the second tower is modelled with a homogenized block. For the final foundation concept the calculated maximum vertical displacements for both towers are about 80 mm. The settlements of the highways and the railway lines are in the order of 15–20 mm and the maximum inclination of the settlement trough is less than 1/500. Finally a calculation using a 64 bit calculation kernel was performed. In this calculation with about 137000 elements both towers are modelled in detail. The maximum vertical displacements are similar to the calculations using the simplified models, but concerning the interaction of both towers and the settlement trough between the buildings a more accurate result is obtained. The comparison of the Hardening Soil model with the Hardening Soil Small model shows that once the high stiffness at small strains is taken into settlements from deeper levels are automatically reduced. Due to the difference in computed maximum deformations also the settlement troughs show less differential

settlements when the HSS model is used. Hence the influence of the model boundary conditions on the computed displacement is diminished and a more realistic settlement behaviour can be obtained.

REFERENCES Alpan, I. 1970. The geotechnical properties of soils. EarthScience Reviews 6(1): 5–49. Benz, T. 2007. Small-strain stiffness of soils and its numerical consequences. Dissertation. Mitteilung 55 des Instituts für Geotechnik. Universität Stuttgart. Bjerrum, L. 1963. Allowable settlements of structures. 3rd European Conference on Soil Mechanics and Foundation Engineering; Proc. int. conf., Wiesbaden, 15–18 October 1963. Vol. 3: 135–137. Brinkgreve, R.B.J. & Swolfs, W.M. 2007. Plaxis 3D Foundation, Finite element code for soil and rock analyses. s manual. Netherlands. Schanz, T., Vermeer, P.A., Bonnier, P.G. 1999. The Hardening-Soil Model: Formulation and Verification. In R.B.J. Brinkgreve (ed.), Beyond 2000 in Computational Geotechnics: 281–290. Rotterdam: Balkema. Vucetic, M., Dobry, R. 1991. Effect of Soil Plasticity on Cyclic Response. Journal of Geotechnical Engineering 117 (1): 89–107. Wichtemann T., Triantafyllidis T. 2009. On the correlation of “static” and “dynamic” stiffness moduli of non-cohesive soils. Bautechnik Special Issue 2009 – Geotechnical Engineering 86 (S1): 28–39.

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The influence of pile displacement on soil plug capacity of open-ended pipe pile in sand Li Sa Geotechnical Institute, Civil Engineering Department, Tianjin University, Tianjin, China

Lars Grande Civil and Transport Engineering Department, Norwegian University of Science and Technology, Trondheim, Norway

Huang Jianchuan & Liu Guohui Geotechnical Institute, Civil Engineering Department, Tianjin University, Tianjin, China

ABSTRACT: In order to study the bearing capacity of open-ended pipe piles in sand, the soil plug capacity has been calculated by the finite element method, and the results are compared with performed on 3.5 m long model piles. The analysis shows that the outside shaft friction is quickly mobilized to maximum whereas large pile displacement is needed to mobilize the inside shaft friction and the soil plug capacity. When the pile is loaded up to its ultimate bearing capacity as determined from the load settlement curve, the soil plug and the inside shaft friction are not in limit state according to the analysis. Therefore, the β value describing the magnitude of the inside shaft friction varies not only by the value of vertical position h/Ri but also by the pile displacement. The influence of displacement should be considered properly when estimating the inside shaft friction.

1

INTRODUCTION

Open-ended pipe piles are widely used for foundation of structures both onshore and offshore. Driving of full scale pipe piles in sands generally takes place in a partially plugged or fully coring manner. Compared to solid piles, the characteristics of the bearing capacity of an open-ended pipe pile is much more complex due to the soil plugs inside the piles. Many design criteria for open-ended piles have been suggested, based on field tests, chamber tests or analytical methods. In general, the bearing capacity of a pile comes from three parts, namely the external shaft friction force Qexter , the annulus resistance Qann , and either the internal shaft friction force Qinter or the plug resistance Qplug whichever is the lesser.

(1a) is considered to be closer to the reality (Hao et al 2001). The most difficult problem in using (1a) is how to calculate the internal shaft friction because the stress state in the soil plug is affected by many factors. In general, one may use a one-dimensional plug analysis, in which the soil plug is modeled as a series of horizontal thin discs (Figure. 1). Each of them is subjected to the vertical effective stress σv on the top,

Figure 1. The stress state of the soil plug.

a vertical effective stress σv + dσv at the bottom, and the shear stress τi between the soil plug and the pile wall. The effective unit weight of the soil plug is γ . The force equilibrium condition is applied to each disc. The shear stress acting between the pile wall and the soil plug is then characterized through the following assumption:

leading to:

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Here, K = σr /σv , β = K tan δ and δ is the friction angle between soil and pile wall, Ri is the inner radius of the pipe pile. To improve the accuracy of the one-dimensional plug analysis, the concept of the wedged soil plug (Muff et al. 1990, O’Neill and Raines 1991, Randolph et al 1991) is introduced. According to test results, they claim that the soil plug can be divided into a wedged plug zone below an unwedged plug zone. While the wedged plug zone transfers load from the pile to the soil plug, the unwedged plug zone transfers no load. It is merely providing a surcharge pressure on top of the wedged plug zone. According to the above-mentioned assumption, they (Randolph et al 1991, Lehane, 2001) integrated Eq.(3), subjected to a boundary condition of a surcharge pressure γ Lup acting at z = 0 and letting z = Lwp , to obtain

Where Lup = length of unwedged plug; Lwp = length of wedged plug. The plug resistance qplug is then dependent on Lwp and β. This equation is useful in indicating how the plug capacity is mobilized. However, it is not easy to apply the analysis to practical cases, due to the sensitivity of the method with respect to the lateral earth pressure coefficient, which is not easily estimated. Recent contributions examining plug resistance in sands have focused on determining expressions for K (lateral earth pressure coefficient). Paik and Lee (1993) used model tests to develop an expression for the average K value in a wedged plug in of the peak friction angle of the sand and the ratio of plug length to pile penetration (PLR). De Nicola and Randolph (1997) used centrifuge model tests to propose a profile of K along the soil plug length as a function of both the relative density of the sand and the soil plug height. Lehane(2001) pointed out that the combination factor β is strongly dependent on the interface friction angle and the dilatancy capacity of the sand. Although the authors used different methods to determine the soil plug capacity, they all assumed limit equilibrium condition when they did the analysis.

2 THE MODEL TESTS For studying pipe pile bearing capacity, the model tests were done at Norwegian University of Science and Technology (NTNU).The tests were carried out in a concrete bin which is 3 meters deep and 4 meters by 4 meters wide. From a top silo, sand was filled into a spreader wagon which automatically ed back and forth with a speed of 8 cm/sec. The sand was then “rained” out through holes in the spreader bottom. The porosity of the sand is controlled by the size of holes in the spreader bottom. Different size holes and hence porosities, may be obtained by exchanging 462 nozzles inserted in the bottom plate of the spreader,

Figure 2. The formation of the soil plug.

thus changing the nozzle diameter. In these tests, the size of the nozzles was 16 mm. The model piles were driven into the sand, then the load was applied in different levels for getting a pipe pile’s load-displacement curve (p-s curve). During installation of the test pile, the plug length for six different penetration levels was measured. From Figure 2, it can be seen that plugging occurred after the pile had penetrated 1.7 m. Based on results from the model tests, some analysis were performed for getting information about the β value and soil plug capacity by the finite element method.

3


3.1 The calculation model and parameters The finite element method is used to simulate the model tests procedure, and to get some information that is difficult to measure during the test. The analysis was performed with the FEM program PLAXIS using the hardening soil model. For single pile analysis, the geometry is simulated by means of an axisymmetric model in which the pile axis is positioned along the axis of symmetry. Both the soil and the pile are modeled with 15-noded elements. Interface elements are placed around the pile to model the interaction between the pile and the soil. The soil calculation scope was 3 meters deep and 1 meter by 1 meter wide which simulate the model test condition. The mesh was generated with a global coarseness set to medium. A local refinement was made in the pile cluster, the cluster around the pile whose width is 0.2m. Fixed boundary was assigned at the bottom. The two vertical sides were assigned horizontal fixities. The sketch of the finite element model is shown in Figure 3. The physical and the mechanical indexes of the sand used to do analysis are shown in Table 1.The characteristics of model pile are shown in the Table 2. They are all obtained from the model tests mentioned above.

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Table 1. The physical and mechanical index of sand. γ

e

Dr

kN/m3 16.0

0.67

0.73

ϕ

δ

E

◦

◦

40

30

kPa 8000

Table 2. The model pile characteristics. Pile length m

diameter mm

wall thickness mm

density kN/m3

E GPa

3.5

75

4

27.3

70

Figure 4. Comparing calculation result with model test result.

Figure 3. The finite element model.

The load acts on the top of the pile, and the load settlement curve for the pile is obtained under the load control. Initial effective stresses are generated by the K0 procedure. In the initial situation the pile does not exist and the soil properties are assigned to the corresponding clusters. Then, assigned the pile properties to the pile cluster, assigned non to the cluster above the soil plug. Finally the pile is subjected to the static load on the top. 3.2 Calculation results 3.2.1 Reliability of calculation The results from the FEM analysis is compared with the result from the NTNU model tests, see Figure 4. From Figure 4, we can see that the analysis results are close to the test results, and the discrepancy mainly appears at the tail of load settlement curve. Consequently the finite analysis may be regarded to simulate the model test with a fair accuracy, and the FEM results may describe fairly well the role that the plug plays in the test. 3.2.2 Analysis of calculation results As mentioned above, it is very important to assess the β value for calculating the internal shaft friction coming

Figure 5. The distribution of internal shaft friction along soil plug height.

from soil plug. Because the open-ended model pile got plugged and were given large deformations, the resistance provided by the soil plug should be the largest value that it can provide. The β value may be judged from the distribution of τi along the soil plug height as obtained by the FEM analysis, see Figure 5. From Figure 5, it can be seen that within the soil plug (−1.2 m∼ −2.5), the internal shaft friction only appears in the part extending from −2.3 m to −2.5 m. Above this part, the internal shaft friction is very small. At the same time, τi is not only related to the height of the soil plug height, but also to the displacement of the pile. The relation between β and the pile displacement is shown in Figure 6, and labels in the figure yield h/Ri values, here h is the position along the soil column. It can be seen that the β value changes with the pile displacement for a given value of h/Ri . For h/Ri = 0.3, β gets its peak value (β = 0.6) when the pile displacement is about 6 mm. For the h/Ri = 0.7, β gets its peak value (β = 0.7) at a pile displacement about 12 mm. In this analysis, when 4.0 ≥ h/Ri ≥ 1.0, the β value increases with increasing pile displacement. When the h/Ri is less than 1.0, β value has stabilized before the

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Figure 6. The relationship between β and pile displacement.

Figure 7. The relationship between β and h/Ri .

Figure 8. Comparing calculation results with De Nicola’s method.

end of loading step, and when h/Ri ≥ 4.0, β is a very small, almost constant value. Figure 7 shows the relationship between β and h/Ri for different pile displacements. The β value is clearly affected by h/Ri and the pile displacement. The results at the end of the loading were compared with the value calculated according to De Nicola’s (1997) method, see Figure 8. De Nicola give a schematic diagram of the proposed design variation of K along the length of the soil column for open-ended piles, and the method to decide the value of Kmax , Kmin . Based on this method,

Figure 9. Comparing τe with τi changing with pile displacement.

and the relationship of β and K: β = K tan δ, one may obtain the variation of β along the length of the soil column shown in Figure 8 where Dr = 0.73, δ = 30◦ . It can be seen that β is divided into three parts for varying values of h/Ri both in the FEM calculation and in De Nicola ‘s result. The largest β value appears in the height of approximately one inner diameter (1Ri ). In our calculation β is reduced nearly linearly over the height range of 1Ri ∼3.0Ri . In De Nicola ‘s result, this range changes to 1Ri ∼5.0Ri , and the reduction gradient is less than obtained in the FEM analysis. When exceeding this range, the β stays constant and the value is very small in two results. From the FEM analysis, one may also compare how the internal shaft friction τi and the external shaft friction τe at h/Ri = 1.0 varies with the pile displacement, see Figure 9. When the load is applied, both the internal and external shaft friction begins to increase with the pile displacement. In the initial stage, the external shaft friction is higher than the internal. When the pile displacement is more than 6 mm, the external shaft friction becomes stable and reaches its peak value, but the internal shaft friction continues to increase during the whole loading. At the end of analysis, the internal shaft friction is about twice as large as the external shaft friction. This implies that it takes larger deformations to mobilize the plug resistance and the inside shaft friction than to mobilize the outer shaft skin friction resistance. The development of the resistance below the soil plug(qplug ) and pile annulus (qann ) with the displacement of pile is shown in Figure 10. The figure shows that the resistance below the plug and the annulus increases with the pile displacement. Until in the end of the calculation, the resistance below annulus seems to get the peak value, while the resistance below the plug is still increasing.

4

CONCLUSION

The bearing capacity of an open-ended pipe pile is a complicated problem. According to the performed

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the soil plug is not yet in limit state, and the influence of the pile displacement should therefore be considered properly in simplified analyses of the soil plug capacity for open-ended piles. REFERENCES

Figure 10. The development of qplug qann with pile displacement.

FEM analysis, it can be concluded that the soil plug capacity is mobilized gradually with the pile displacement. The analysis shows that during the loading process, the soil plug resistance increases all the time. Although the external shaft friction and the pile annulus capacity reach their ultimate value, the whole soil plug is far away from its limit state. Large pile displacements are needed in order to mobilize the soil plug capacity. When the pile reaches its ultimate bearing capacity determined by the load settlement curve,

Hao X.Y., Pang Y.S., Yang S.P. 2001. Calculation method for vertical bearing capacity of single open-ended pipe pile. Journal of Hohai university 29(12):124–127 Kyuho Paik and Rodrigo Salgado 2003. Determination of bearing capacity of open-ended piles in sand. Journal of geotechnical and geoenvironment engineering 129(1):46– 57 Lehane, B.M. and Gavin, K. G. 2001. Base resistance of jacked pipe piles in sand. Journal of geotechnical and geoenvironment engineering 127(6): 473–480 Lehane, B. M. and Randolph, M. F. 2002. Evaluation of a minimum base resistance for driven pipe piles in siliceous sand. Journal of Geotechnical and Geoenvironmental Engineering 128(3): 198–206 Nicola, A.D. and Randolph, M.F. 1997. The plugging behavior of driven and jacked piles in sand. Geotechnique 47(4): 841–856 Nicola, A.D. and Randolph, M.F. 1999. Centrifuge modeling of pile in sand under axial load . Geotechnique 49(3): 295–318. Randolph M.F. et al. 1991. One-dimensional analysis of soil plugs in pipe piles. Geotechnique 41(4): 587–598

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Deep excavations and retaining walls


3D modelling of a deep excavation in a sloping site for the assessment of induced ground movements O.J. Gastebled & S. Baghery Coyne & Bellier Consulting Engineers, Tractebel Engineering, Gennevilliers,

ABSTRACT: In densely built area with high land value, the trend is towards building higher and deeper, often adjacent to existing structures. Such projects involve the adoption of tight ground movement criteria which are nowadays commonly checked using 2D finite element analysis. For the excavation discussed here, however, the constraints in of site configuration and projects requirements lead to adopting a retaining system which takes advantage of its 3D arching effects. 3D finite element analysis was thus adopted at the design stage. The project concerns a high-rise 49 floors tower to be constructed in Monaco, on a sloping site. The tower basement and foundations involve a deep excavation, mainly in marlstone, with depths of 70 m on the uphill side and 30 m on the downhill side. The results obtained using 3D modelling are discussed in term of induced ground movements and plasticity in so far that it provides better insight into the way the retaining system fulfils its role.

1

INTRODUCTION

A high-rise tower, which will be the highest building in Monaco, is currently under construction in a densely built area. The 150 m high tower consists of a 10 level deep basement, excavated on a 1/3 slope. The exceptional depth of this excavation (72 m on the uphill side), the steep sloping terrain, the site’s exiguity, the built environment and the presence of a poor marlstone layer at depth all contributed to adopting a 3D approach at the early design stage of the retaining solution to ensure: – Safety against loss of overall stability (GEO limit state), – Limitation of the induced displacements and settlements at the foundations of adjacent buildings (SLS limit state). While the retaining wall design could be checked against structural failure (STR limit state) using standard analysis tools based on the elasto-plastic subgrade reaction method and 2D modelling (see German Geotechnical Society, 2008), the 3D configuration of the site and the 3D nature of the design rendered inapplicable the standard analysis approach for the overall stability check (such as 2D limit equilibrium method) and for the SLS (such as 2D plane strain finite element analysis). To avoid over-design and to validate a safe and economical retaining solution, a detailed 3D numerical model was therefore developed using Midas GTS, a finite element package dedicated to geotechnical analysis.

Figure 1. Location and project dimensions. Impressions of tower and excavation.

2

PROJECT CHARACTERISTICS

2.1 Excavation geometry The construction site is a mostly terrace hillside located in Monaco, bounded downhill by a street and uphill by the French border. The tower basement and podium are to occupy the great majority of the area of the available construction site, creating difficult access conditions in this steep slope, Figure 1. The four-sided excavation is 70 m long in the downhill-uphill direction, 30 m wide on the uphill side

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Figure 2. Bird-eye view of full 3D model. Natural terrain and adjacent building foundation levels, before tower excavation.

Figure 3. Zoom on bird-eye view of 3D model. Tower excavation completed and adjacent building foundation levels.

and 65 m wide on the downhill side, at street level, Figure 4. The maximum difference in altitude of the natural terrain over the excavated area is 39 m, Figure 2. With an excavation level at 37 m NGM (Ordnance Survey Datum of Monaco), the excavated depth is 33 m on the downhill side and 72 m on the uphill side, see Figure 3. The embedded depths of the diaphragm wall and foundation barrettes extend a further 15 m below the excavation level (toe level of the in-situ concrete walls is at 22 m NGM). 2.2

Built environment

The following buildings are within immediate proximity of the excavated area, Figures 4 and 6: – The 10 storey high “Charles III” high school, founded on piles, is at a close distance, south of the excavation, – A 7 storey high and 3 basement deep building immediately adjacent to the excavation, on the downhill side. – A 7 storey and 10 storey buildings are at a close distance from the excavation, on the uphill side. 2.3

Figure 4. Zoom on top view of 3D model. Tower excavation completed and adjacent building foundations.

Figure 5. Top view of 3D mesh. Excavation boundary in grey. Outcrops: scree in white and marlstones in darker colours.

dilatometer tests, Menard tests, core sampling and various laboratory tests. A relatively thick scree layer (>10 m) exists uphill and thins and then vanishes at mid-length of the planned excavated area (light grey top layer in Figure 2). Underlying the scree layer is a heterogeneous marlstone, tending slightly towards a lime-marlstone, to depths of at least 20 m below the planned excavation level. Particularly poor clayey marlstones are found locally, on the uphill side of the excavation. These anomalies are located at excavation level (37 m NGM), with thicknesses varying between 10 and 20 m. The bedding plane of the marlstone is favourable, tilting in the uphill direction. After due consideration of the various identified t families and taking into the scale effect, a continuum mechanics approach with an equivalent modulus and shear strength parameters was chosen to represent the behaviour of the marlstones in the SLS analysis. Reduced shear strength parameters were adopted for the stability analysis (GEO limit state). 2.4 Construction method and system

Geological-geotechnical characterization

Extensive site investigations have been carried out; these include borehole logging, down-hole video,

The presence of the scree layer in a state of near limit equilibrium associated with the exiguity of the

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Figure 6. Zoom on bird-eye view of 3D mesh. Planned excavation and neighbouring foundations. Excavation boundary in white.

site precluded a sloped excavation. Thus in order to undertake the construction of the main retaining wall a micro-pile wall, consisting of grouted steel tube soldier piles, shotcrete infilling, horizontal steel girder beams and prestressed cable anchors, is designed to enable a 12 m deep excavation on the uphill side in order to establish a working platform for the construction of the second retaining wall, of bored concrete piles, shotcrete infilling and prestressed cable anchors enabling a further 20 m to be excavated down to the main work platform at 74 m NGM. From the main work platform s of in-situ concrete, 54 m deep, are constructed to create a buttressed diaphragm wall and foundation barrettes. An up/down construction method is then adopted, whereby excavation progresses below basement slabs in parallel to the erection of the tower super-structure. The diaphragm wall is anchored in its upper part and braced on basement slabs over the full excavation depth. Buttresses are used to increase the diaphragm wall’s stiffness and the barrettes are used as basement walls and deep foundation system. The soil enclosed by the diaphragm wall is considered reinforced due to the presence of buttresses and barrettes. In zones of low barrette density, on the uphill side, the soil shear strength is further improved by installing vertical fibreglass nails.

3

NUMERICAL MODELLING

3.1 General aspects A detailed 3D finite element model of the ground, retraining structures, basement and tower foundations

was built. The model boundaries were chosen far enough from the excavation to minimise boundary effects: 375 m long in uphill-downhill direction and 250 m wide with a base at 7 m NGM, i.e. a model thickness varying between 60 m and 100 m over the excavation area, see Figure 2. The use of an automatic tetrahedron mesher allowed total freedom in defining the geometry of the ground solids and the grading of mesh sizes, see Figures 5 and 6. The tetrahedron mesh size ranges from 2 m on the retaining wall surfaces to 25 m on the model boundary.A total of 154 400 linear tetrahedron elements were generated, for a total of 30 900 nodes. The accuracy on surface displacements proved satisfactory when comparing the results of a linear elastic analysis carried out with the same mesh of linear elements and of quadratic elements (difference <10%). 3.2

Retaining walls, basement and foundations

The micro-pile wall was modelled by a combination of shell elements, representing the shotcrete, horizontal beam elements, representing the steel girders, and vertical beam elements, representing groups of micro-piles on the basis of 1 for 2 or 1 for 4. The bored pile wall was modelled by a combination of shell elements, representing the shotcrete, and vertical beam elements, representing the bored concrete piles on the basis of 1 for 1, figure 7. The diaphragm wall, the foundation barrettes and the infrastructure slabs were modelled by shell elements, Figure 8. A shear-free interface was defined between the buttresses and the diaphragm wall, beam elements representing buttresses were connected at

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Figure 9. Top view of retaining walls.

Figure 7. Front view (facing west) of the three-level retaining system. The enclosing diaphragm wall is shown transparent.

Figure 10. Top view of cable anchor system. Free length in light grey, grouted length in dark grey.

their neutral axis to the shell elements representing the diaphragm wall, see Figure 9. 3.3 Prestressed anchors

Figure 8. Birds-eye view of barrette foundations and basement slabs enclosed in diaphragm wall (transparent).

A total of 520 prestressed anchors, distributed on 19 rows, were modelled, see Figures 10 and 11. Each anchor had its orientation, free length and grouted length accurately represented. An approach based on the elasto-plastic subgrade reaction method was applied to a number of characteristic sections were used to determine the design pre-stress force for each anchor. The grouted length of the anchors was modelled using the so called “embedded truss elements” which allow the positioning of truss elements independently of the 3D solid mesh of the ground, while keeping full mechanical bond between the two element types. Nodal connectivity between anchor and ground was enforced at the connection point between grouted and free length. The free length of the anchor was modelled using a single truss element linking the end of

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3.5 Initial stress state The sloping terrain produces in-situ stress rotation, stress release under the slope, and local stress concentration at the slope toe. The earth pressure acting on the retaining walls and the ground movements induced by the stress release can be significantly influenced by this complex in-situ stress state. Very little is known, however, on the actual insitu stress state, especially for the deeper layers, which are influenced by geological history (e.g. overconsolidation, erosion and tectonic activity), terrain geometry and very long term ground behaviour. A pragmatic approach was adopted: – Ensure that the obtained in-situ stress distributions are fully compatible with the plastic laws of the ground layers and with static equilibrium under self weight of the ground, – Carry out a parametric study on K0 , using a lower bound and upper bound value.

Figure 11. Side view (facing north) of cable anchor system.

the grouted part to the anchored point on the retaining wall. During the prestressing stage of an anchor, only the grouted length was activated and an opposite prestress forces was applied to the end of the grouted length (action) and on the wall (reaction). The free length element was activated at the next analysis stage to take into the anchor’s stiffness.

The values of K0 obtained under normal conditions of consolidation were indirectly controlled by setting specific “long term” Poisson ratios, valid only for the initialisation stage. This approach allowed investigation into the behaviour for an upper and lower bound K0 in the deeper layers of the model while the stress state in the zone of slope influence remained mainly controlled by the terrain geometry.

3.4 Material models All structural elements (steel and concrete) are assumed to behave linear-elastically. This is a good approximation as the design is checked, independently from this detailed 3D model, against structural failure (STR) using a conservative 2D approach. As described in section 2.3, the behaviour of all ground layers were modelled with a continuum mechanics approach, using an isotropic linear-elastic perfectly plastic Mohr-Coulomb model. The inability of this model to represent the significant stiffness changes between the primary loading and unloading/reloading stress paths (De Vos & Whenham 2005 and Schweiger 2002) is partly compensated by specifying the unloading/reloading stiffness for the bulk modulus, K, while specifying the primary loading stiffness for the shear modulus, G.This approach delivers a good approximation of soft rock behaviour when subjected to an excavation loading path. The equivalent isotropic shear strength properties of the rock mass were adopted for the analysis aimed at calculating displacements (SLS limit state). For global stability analysis (GEO limit state), a conservative approach was adopted using the isotropic Mohr-Coulomb model with two sets of shear strength parameters for the marlstones: – Equivalent shear strength properties of the rock mass was reduced using regular partial safety factors, – Characteristic values (not factored) of the shear strength properties of the rock ts were assumed to be ubiquitous and unfavourably oriented.

3.6 Construction stages The non-linear analysis was carried out with a total of 50 construction stages. Two stages were used for stress initialisation, including the application of adjacent building loads on their respective foundations. Two analysis stages were considered for each of the 19 rows of anchors: – Excavation down to anchor level, – Installation of the anchor and prestress force. A single analysis stage was used per basement slab, below the anchor levels. 4

RESULTS

4.1 Induced displacements The results of one of the SLS analysis cases considered are presented in of induced displacements at ground level in Figures 12 and 14. Diaphragm wall displacements results are presented in Figure 15. Induced plastic strains in the marlstone are shown in Figure 13. The induced displacements at the foundation of adjacent buildings are shown to remain limited. The plastic strain of the marlstone in the grouted zone of the anchors remains relatively small. The strong influence of the poor marlstone layer is clearly visible on the diaphragm wall displacements and on the plastic volumes at the excavation level (uphill side).

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Figure 12. Top view of natural terrain with imprint of adjacent buildings. Contour plot of horizontal displacements after completion of construction (max = 9 mm, contour every mm).

Figure 15. Side view (facing north) of the outer face of the diaphragm wall. Contour plot of horizontal displacements of wall after completion (max = 28 mm, contour every 2 mm).

the terrain strength properties c and φ using safety coefficients deduced from national recommendation CLOUTERRE. The strength reduction was applied just before starting excavation. 5

Figure 13. Bird-eye view of ground (transparent), excavation and anchors. 3D colour plot of plastic volumes (colour = value of plastic shear strain, max = 1‰, contour every 0.1‰).

Figure 14. Top view of natural terrain with imprint of adjacent buildings. Contour plot of vertical displacements after completion of construction (min = −5 mm settlement, contour every mm).

4.2

Global stability

The overall stability of the excavation in the slope, taking into the complete construction history was checked under varied assumption of initial stresses. This check was carried out by reducing

CONCLUSIONS

The numerical analysis presented here was carried out as part of a recent design. The excavation is currently under construction. The retaining system and the adjacent buildings are being closely monitored in term of induced movements. The displacement fields predicted by this model were instrumental in defining the threshold values and action values which allow early detection of warning signs and timely response to geotechnical hazard. For the presented case, satisfactory understanding of the soil-structure interaction and of the way the retaining system fulfils its role could only be gained using 3D numerical analysis. This in turn allowed optimizing and validating the design. Thanks to continuous improvement in software and hardware, detailed 3D analysis has become a viable option and a valuable tool available to the engineer in the design office. It should be noted however that successful 3D analysis requires sound theoretical understanding and experience. REFERENCES De Vos, M. & Whenham, V. 2005. Final Report - part 2: The use of finite element and finite difference methods in geotechnical engineering. Geotechnet WP3- Innovative Design Tools in Geotechnics. www.geotechnet.org. German Geotechnical Society (DGGT). 2008. Recommendations on Excavations EAB. Berlin: Ernst & Sohn. Schweiger, H.F. 2002. Results from numerical benchmark exercises in geotechnics. In P. Mestat (ed.), Proc. of 5th European Conf. Numerical Methods in Geotechnical Engineering: 305-314. Paris: Presses Ponts et chausses.

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Analysis of an excavation in asymmetrical soil conditions: The Marquês station A. Pedro & J. Almeida e Sousa Department of Civil Engineering, University of Coimbra, Portugal

D. Taborda Department of Civil and Environmental Engineering, Imperial College London, UK

P. França CJC / Figueiredo Ferraz, São Paulo, Brasil

ABSTRACT: The Marquês station belongs to line D of the Oporto Metro, Portugal. The main access to the station platforms consists of an elliptical shaft of approximately 48 m and 40 m in the principal directions, reaching a depth of 27 m. The performed geotechnical survey revealed asymmetrical soil conditions, due to the existence of a fault. Thus, on one side of the fault, the excavation was carried out in weathered residual granitic soil, while on the other half the works were performed in a slightly weathered granitic rock mass. The monitoring results showed a great difference in the displacements measured on the two sides. This paper presents a 3D finite element back-analysis of the shaft, modelling the complexity of its geology and the entire construction sequence. A parametric study will also be presented in order to evaluate the influence of the system and the geological conditions on the behaviour of the structure.

1

INTRODUCTION

The Oporto Metro, in Portugal, is a network composed of 4 major lines with a total combined length of approximately 70 km, including almost 20 km of completely new lines. In the centre of the city, due to the density of buildings and infrastructures, the network is mostly underground and includes nearly 7 km of tunnels and 12 stations (Fig. 1). The Marquês station belongs to line D and its design by CJC / Figueiredo Ferraz faced multiple challenges due to the geological conditions in

which the excavation was performed (Andrade et al. 2003). In this paper, after a brief description of the project, a 3D numerical analysis of the Marquês station is presented and the obtained results are compared with those ed by the instrumentation. Finally, a parametric study was carried out in order to evaluate the influence of the geological conditions on the behaviour of the structure.

2

PROJECT CHARACTERISTICS

2.1 Location and geometry

Figure 1. Oporto Metro network (adapted from UrbanRail.Net).

The Marquês underground station is located beneath the Marquês do Pombal square in the centre of Oporto city. The square has a centenary garden and is surrounded on all sides by buildings which strongly restrained the location and the geometry of the station (Figs. 2, 5). Consequently, it was decided that the main access to the station should be performed through the excavation of an elliptical shaft of approximately 48 m and 40 m in the principal directions, reaching a depth of 27 m (Andrade et al. 2003). In order to minimise the impact of the works, the major axis of the shaft was aligned with the main direction of the square and complementary tunnels were excavated at the bottom of the shaft to accommodate the platforms (Fig. 2).

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Figure 4. Excavation of the 11th level. Figure 2. Location and geometry of the Marquês station.

Figure 5. Aerial view of the shaft when the excavation was concluded.

Figure 3. Vertical and horizontal construction sequence.

2.2

Geological-geotechnical characterisation

The characterisation of the local soil conditions included the execution of 15 rotary boreholes. These allowed the recovering of samples for laboratory testing, the definition of the ground profile and the execution of several in-situ tests. Generally, SPT tests were performed with 1.5 m spacing in depth, and 16 Lugeon and 1 Lefranc permeability tests were conducted. With the collected samples, besides the usual characterisation tests, it was possible to carry out several uniaxial and point load tests (Geodata/Normetro, 2002). The geotechnical survey revealed both the typical heterogeneity of this type of soils, which led to the definition of 7 distinct layers, and the unexpected asymmetrical soil conditions (Santos et al. 2004). These were caused by a fault which intersected the shaft close to its centre, exhibiting some obliquity relatively to the shaft’s principal axis. Therefore, on one side of the fault (South side), the excavation was carried out in weathered residual granitic soil, while on the other half (North side) the works were performed in a slightly weathered granitic rock mass (Fig. 6). During the initial survey, the water table was identified, for both sides of the fault, at approximately 6 m depth. 2.3

Construction method and system

The shaft was designed according to the NATM principles (Rabcewicz, 1964). The excavation was executed in steps of 1.8 m of depth (14 levels) and 6 to 12 m of length, depending on the type of soil (Figs. 3-4).

The used along the entire contour of the shaft consisted of fibre-reinforced shotcrete, except for the first metre where a beam of concrete with 60 cm of thickness was built. Due to the expected increase of the magnitude of stresses with depth, the thickness of the shotcrete varied from 30 cm, for the first 5.4 m, to 45 cm, for the next 3.6 m, and finally to 60 cm until the base of the excavation (Fig. 3). The areas of the shaft near the platform tunnels were reinforced in order to assure the stability of the entire structure during their excavation. A fine mesh of drains was installed along the entire contour of the shaft to avoid large water pressures acting on the lining and also to prevent the occurrence of local instabilities. During the first stages of excavation, the rate of displacements on the South side of the shaft increased sharply, requiring the design of jet-grouting columns with the objective of preventing potential problems and, eventually, failure. The columns with diameter of 1 m were only executed along the South perimeter of the excavation, at a depth of 8 m and were, in average, 6 m long (Figs. 2-3). 2.4 Instrumentation and observation Considering the specific aspects of the project, in particular the geotechnical and geological conditions at the site, and the spatial configuration of the shaft, an instrumentation plan was defined in order to allow both the evaluation of safety during the construction works and the extrapolation of the behaviour from the

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early stages of excavation to the later ones in order to modify and adapt, if necessary, the construction methodology and the structural solutions (Santos et al. 2004). In order to achieve those purposes, several instrumentation devices were installed on the shaft and on the surrounding buildings. In this paper only the values ed by the survey marks (Fig. 2), located at different depths will be analysed, due to space limitations. Figure 6. Vertical cut of the 3D model (stage 21).

3

Table 1.

NUMERICAL MODELLING

3.1 General aspects The numerical analyses were performed with the finite element software UCGeoCode (UCGC), which has been developed at the University of Coimbra since 1999 (Almeida e Sousa 1999, França et al. 2006). This code, in its most recent version, has several constitutive models implemented and a 3D formulation which allowed the modelling of the complete construction sequence used in the excavation of the Marquês station. It should be noted, however, that the platform and the TBM tunnels were not considered since they were constructed after the excavation of the entire shaft. The employed mesh consists of 5648 20-noded elements and enables the reproduction of the vertical and horizontal construction sequences using a total of 86 stages. The complexity of the geological ground profile was taken into in the model, though some simplifications were required. As a result, the fault was simulated by a plane of solid elements, dividing the model in two sides, while the stratification of the excavated materials was considered to be horizontal. In Figure 6 a part of the geometry of the model used in the analyses can be visualised for stage 21. The analyses were performed assuming drained conditions, due to the specific nature of the soils and to the employed drainage scheme. Finally, the behaviour of the materials was simulated using the linear elastierfectly plastic Mohr-Coulomb constitutive model (the dilatancy, ψ, was considered to be 0◦ ). In order to achieve a good agreement between the results of the numerical simulations and the monitoring data, a back-analysis process was carried out. In this study, the soil characteristics – in of deformability and strength – were adjusted, until the calculated displacements agreed reasonably well with the instrumentation data. During this study, particular attention was also given to the influence of the lining and the jet-grouting properties on the behaviour of the excavation. The methodology employed for the back-analysis process consisted, firstly, in the evaluation of the impact of the stiffness, cohesion and K0 value of each individual layer on the computed deformed shape of the shaft. The obtained results were then analysed and a final study was carried out in order to assess the relative influence of the stiffness of the lining and of the jet grout columns.

Geotechnical parameters.

Complex

γ (kN/m3 )

c (kPa)

φ (◦ )

E (MPa)

ν

K0

G7 G6 G5 G4 G3 G2 G1 Fault

19 19 20 21 23 23 23 19

0 0 20 50 200 250 350 0

28 32 36 40 45 45 45 28

20 30 90 300 500 800 1500 20

0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

0.50 0.50 0.65 0.65 0.80 0.80 0.80 0.50

Table 2.

Lining

Lining and jet-grouting parameters. γ c φ E 3 Model (kN/m ) (kPa) (◦ ) (MPa) ν

Beam LE Shotcrete LE Jet-grouting MC

25 25 21

– – 250

– – 40

20000 10000 350

0.25 0.25 0.20

In this paper, due to space limitations, only the results corresponding to the set of parameters leading to the best reproduction of the behaviour observed in situ will be presented. The final values determined by the back-analysis procedure are listed in Tables 1 and 2 and are within the ranges initially proposed by the geological and geotechnical study performed prior to the design of the shaft. Moreover, they are also ed by a recent study carried out by Topa Gomes (2009). In the next section, the main results in of displacements are presented and compared with the monitoring data.

3.2 Results of the back-analysis Figures 7–8 present the evolution with time of the vertical displacement of the extremities of the shaft’s axes, for levels 1 and 6. The graphs illustrate the significant difference between the displacements measured at opposite sides of the fault. This contrast is nearly 4 times greater in level 1 and tends to decrease slowly with depth.

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Figure 7. Evolution of the vertical displacements – Level 1.

Figure 9. Evolution of the convergence sections – Level 1.

Figure 8. Evolution of the vertical displacements – Level 6.

Figure 10. Evolution of the convergence sections – Level 6.

4 The magnitudes of the vertical displacements obtained by the reference calculation of the backanalysis study (CB) show a reasonable agreement which tends to improve with depth. For level 1, there are some discrepancies on the North side (mostly after the excavation of level 8) and, particularly, for the survey mark P1. The reason for this difference is probably related o the poor geological conditions likely to be found at such short distance from the fault, which could not be reproduced in the numerical model, since the discontinuity was simulated as a plane of limited thickness. The effect of the jet-grouting columns is quite visible in Figure 7, mostly between levels 4 and 7 on the South side, when the rate of vertical displacements indicates a dramatic reduction. In Figures 9–10 the results obtained for the convergence sections are presented. It can be seen that in level 1 the analysis shows a good agreement with the values of convergence measured along the principal axes (1-3, 2-4). For the sections measured within the same side, it can be seen that the numerical simulation is unable to reproduce neither the magnitude nor, for section 1-2, the direction. The agreement between the analysis and the monitoring data is better for level 6. At this depth, only the magnitude of the convergence section 3-4 is highly underestimated.

PARAMETRIC STUDY

4.1 Methodology In order to evaluate the influence of the system and of the geological conditions on the behaviour of the shaft, two parametric studies were carried out. The results of the numerical simulations were subsequently compared with those obtained in the analysis presented in the previous section (CB). The first study evaluates the impact of the jetgrouting columns on the control of the displacements and on the stability of the shaft. The second study consists of two additional analyses where the geological conditions are assumed to vary only with depth. Therefore, in one case the geology corresponds to that previously attributed to the North side, while in the other calculation the stratigraphy proposed for the South side was employed for the whole model. In Table 3, the designations and respective description of all the performed analyses are listed. 4.2 Influence of the jet-grouting columns (PS1) Figure 11 shows the evolution of the vertical displacements in level 1 for the analyses with (CB) and without (NJ) the modelling of the jet-grouting columns. As expected, the results on the South side of the two

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Table 3.

Description of the analyses performed.

Analysis

Description

CB NJ NS SS

reference analysis without modelling the jet-grouting columns entire model with the North side geology entire model with the South side geology

Figure 13. PS1 – Hoop force along the contour of the shaft for the last stage.

Figure 11. PS1 – Evolution of the vertical displacements in the level 1.

Figure 14. PS2 – Vertical displacements along the contour of the shaft for the last stage.

Figure 12. PS1 – Vertical displacements along the contour of the shaft for the last stage.

analyses are only similar until level 4 is reached, since the columns were executed at this depth. After this point, the rates at which the vertical displacements evolve are clearly distinct, with the case labelled as NJ ing much larger movements. It is also possible to observe that the majority of this difference is accumulated between levels 4 and 7, since the displacement rate after this stage is similar in both analyses. Figure 12 presents the vertical displacements along the contour of the shaft at the end of the analyses for levels 1 and 6. In the North side the values obtained are relatively small and approximately constant, which corresponds to the expected behaviour since no jet-grouting exists on that side. On the South side, the displacements in level 1 are higher when no jet-grouting is used and the peak is located approximately at the extremity of the smaller axis (θ = 270◦ ). The displacements reached in level 6 are smaller, when

compared with the ones in level 1, and approximately identical for both analyses. Furthermore, it is also possible to visualise that, in the analysis without jetgrouting, the displacements presented important fluctuations, with heaves being reached near the fault due to the horizontal construction sequence. That aspect is apparently smoother for the case CB probably due to the stabilising effect of the jet-grouting. Figure 13 shows the hoop forces along the contour of the shaft for the last stage of the analyses. It can be seen that the forces tend to increase with depth, reaching a maximum value of nearly 2000 kN in the South side, level 6. There is also a significant difference between the forces in the North and South sides, with higher magnitudes being ed on the more deformable side. The absence of the jet-grouting columns does not change significantly the magnitude of the hoop forces. It is also possible to observe in the same diagram the apparently stabilising effect of the columns on the South side. 4.3 Influence of the geological conditions (PS2) In Figure 14, the vertical displacements obtained along the contour of the shaft corresponding to last stage are illustrated for the second parametric study. As expected, the displacements obtained when geology

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Figure 15. PS2 – Hoop force along the contour of the shaft for the last stage.

only varied in depth are symmetric and approximately constant. The deformations on the NS model are much smaller when compared with those on the SS model. In both analyses the displacements tend to decrease with depth, as previously observed for the CB and NJ cases. It is possible to conclude from the figure that the vertical displacements of the NS model are similar to those ed on both levels of the North side for cases CB and NJ. However, on the South side the results, in particular those for level 1, are quite different. At this depth, the displacements of the SS model are smaller than the ones obtained for the NJ model, implying that the stiffness differential between the two sides of the fault tends to amplify the movements of the more deformable side (in this case, a factor of almost 2 is observed). This effect is not evident for level 6 where the displacements of the SS and NJ analyses are of the same magnitude. In of hoop forces, which are presented in Figure 15, it can be observed that, for the NS and SS models, the obtained distributions are symmetrical. However, the amplification effect ed for the displacements on the South side of the model does not seem to globally apply to the obtained forces. In fact, although for level 1 the NJ case yielded slightly larger values than those calculated for the SS case (i.e. amplification), for level 6 the hoop forces determined using the asymmetrical geology were slightly lower (i.e. no amplification). On the North side, the hoop forces are similar for all the cases analysed. 5

CONCLUSIONS

the excavation revealed an asymmetrical behaviour in of displacements caused by the presence of a fault dividing the shaft. The mechanical properties of the different formations were estimated by performing a 3D backanalysis, allowing the influence of several aspects on the behaviour of the shaft to be effectively assessed. Furthermore, it was observed that the execution of the jet-grouting columns occupied a central role in the stabilisation of the vertical displacements. A parametric study concerning the geological conditions revealed that the stiffness differential between the two sides of the fault resulted in an amplification of the vertical displacements and, to a lesser extent, in the reduction of the hoop forces determined on the more deformable side. ACKNOWLEDGEMENTS The authors would like to acknowledge Metro do Porto SA, Normetro ACE and CJC/ Figueiredo Ferraz for authorising the publication of this article. REFERENCES Almeida e Sousa, J. 1999. Tunnels in Soils – Behaviour and numerical modelling. PhD thesis, University of Coimbra (in Portuguese). Andrade, J. C., Campanhã, C., Mota, A, Jordão, P. 2003. Estação subterrânea em poço e túnel no Metro do Porto. Proc. of the Jornadas Hispano-Lusas sobre obras subterrâneas, Madrid: 391–408 (in Portuguese). França, P., Taborda, D., Pedro, A., Almeida e Sousa, J., Topa Gomes, A. 2006. Estação Salgueiros do Metro do Porto: aspectos executivos e estudo do comportamento. Proc. of the III Congresso Luso-Brasileiro de Geotecnia, Curitiba, 369–374 (in Portuguese). Geodata/Normetro 2002. Relatório geotécnico-geomecânico geral dos troços enterrados (linhas C, S e ramal de ligação C-S). Projecto de execução. Technical report (in Portuguese). Rabcewicz L. 1964. The New Austrian Tunnelling Method, Part one, Water Power, November: 453–457. Santos, L., Jordão, P., Mota, A., Gaspar, A., Andrade, J.C. 2004. Estação do Marquês. Observação e acompanhamento durante a obra. Proc. of the 9◦ Congresso Nacional de Geotecnia, Aveiro, Vol. III: 235–244 (in Portuguese). Topa Gomes, A. 2009. Elliptical shafts open by the sequential excavation method – Oporto Metro. PhD thesis, University of Porto (in Portuguese).

The analysed project, the Marquês shaft, presented a unique set of geological conditions. Consequently,

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Comparison of finite element predictions with results from a centrifuge test representing a double anchor wall in sand P.J. Bourne-Webb formerly Imperial College, UK

D.M. Potts Imperial College, UK

D. König Ruhr-Universität Bochum,

ABSTRACT: A study examining the effect of plastic hinging on the response of embedded retaining walls was undertaken. Double anchored walls embedded in dry sand were modelled in a geotechnical centrifuge at 1/30th scale. Finite element calculations were undertaken using Lade’s double hardening cap model to represent the behaviour of the sand. Analyses yielded good accord with the test results in many aspects of the wall behaviour. The consequences of testing with a dig-accelerate versus an accelerate-dig sequence, and the effect of introducing a hinge zone into the wall section was examined. Comparisons between tests and their associated numerical simulation are of interest as they provide insight into the influence of these factors in the observed behaviour; it was found that the consequence of the dig-accelerate sequence appears minimal and other factors had a greater influence, while the introduction of a hinge zone into the wall system resulted in significant changes in the response with reference to the intact wall system.

1

2

INTRODUCTION

As part of the work undertaken in order to develop design guidance for the application of EN 1993-5 Steel piling to the plastic design of steel sheet pile retaining walls, both physical and numerical modelling was undertaken. Physical modelling in the form of centrifuge tests undertaken at a scale of 1:30 was undertaken at the Institute of Geotechnical Engineering and Soil Mechanics of Ruhr-Universität (RU) Bochum, and numerical predictions using the finite element method were undertaken at Imperial College, London. The intention of the physical modelling was to provide a dataset based on a realistic construction process for an embedded retaining wall, from which the finite element model could be verified in order to provide confidence in subsequent calculations. The following paper presents part of this study in which an embedded wall with two levels of was modelled. This follows on from worked reported elsewhere which examined the results of testing and numerical predictions for the case of a wall with a single level of , Bourne-Webb et al. (in print).

CENTRIFUGE MODELLING

A total of ten centrifuge tests were performed at RU Bochum. Seven of these simulated embedded walls with a single row of anchors and the remaining three two anchor levels. The intention of the tests was to measure the earth pressure acting on a wall in dry sand, forming a kinematic mechanism associated with plastic hinge formation, while taking into realistic construction processes. The physical modelling allowed comparison of responses between walls that remain elastic and those in which the wall had a plastic hinge, and provided a means for ing the finite element models. Only the three double anchor tests (SPWFG17 to 19) are presented in detail here. In these tests, a 10.5 m long (35 cm model) wall with two levels of , in dry sand was modelled. In each of these tests, the upper anchor was located at a depth of 0.54 m (1.8 cm), and the lower anchor 4.5 m (15 cm) depth, Fig. 1. Elastic walls were modelled in tests SPWFG17 and 18, and a plastic hinge zone was introduced just below the lower anchor level in test SPWFG19. In order to achieve exact similitude between model and prototype, impractically thin steel wall sections would have been required and, therefore, the model

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Figure 1. Layout of centrifuge test models.

Figure 3. Observed anchor response and equivalent linear stiffness values.

Figure 2. Wall hinge zone plastic moment-curvature response.

wall was formed from aluminium sheet. When a test was intended to study the effect of plastic hinging, a notch was introduced into the wall section at a level where the maximum bending moment was observed in the elastic tests. The use of the notch was a compromise which had a number of problems, not least of which was that the section strain hardens and does not exhibit the softening of resistance due to section buckling observed in real steel sheet piles, Fig. 2. In test SPWFG19, the notch was further modified by drilling a series of holes along its length in order to reduce the available bending resistance. The effect was to reduce the resistance by about half compared to the notch alone. Each test specimen was constructed by applying greased-plastic sheeting to the side walls and then

raining sand into the strongbox. During the placement of the sand, the process was halted at appropriate stages to allow the instrumentation and anchor cables to be placed. The instrumentation included displacement transducers (LVDTs) at various levels on the wall, load cells on the anchors and strain gauges over the height of the wall on its centreline, Fig. 1. The anchor system comprised wire cables which ran from a waling beam on the wall, inside a tube, through the sand, ing through the rear wall of the strongbox and onto an axle mechanism. Anchor pre-stressing could be simulated via a counterweight system. A brake system allowed the anchors to be fixed and a force transducer measured each of the anchor loads, König (2002). Equivalent linear effective stiffness values for each anchor have been assessed from the various tests, Fig. 3. It can be seen that the effective anchor stiffness values changed from test to test, as well as during each test. The variation from the theoretical stiffness value of 7330 kN/m/m was greatest in the first test and improved as the tests proceeded possibly as a result of the action of loading and unloading of the anchor strands, Bourne-Webb (2004). In all the tests, the anchors were fixed from the outset of the test with no pre-loading applied. In tests SPWFG17 and 19, excavation was undertaken prior to

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Table 2. sand.

Lade cap model parameters inferred for Bochum

Parameter

Value

Modulus number, Kur Elastic exponent, n Poisson’s ratio, µ Collapse modulus, C Collapse exponent, p Failure constant, η1 Failure exponent, m Plastic potential const., R Plastic potential const., S Plastic potential const., t Work hardening const., α Work hardening const., β Work hardening const., P Work hardening expt., l

1315 0.512 0.2 3.45 × 10−4 0.807 74.41 0.246 0 0.527 −2.37 2.618 −0.0112 0.132 0.976

3.1 Soil model

Figure 4. Inferred bending moment profiles for double anchor wall tests at the end of each test. Table 1. Anchor load values. Test

Anchor

At test end kN/m

Predicted kN/m

SPWFG17

Upper Lower Upper Lower Upper Lower

0.71 1.73 1.02 1.96 0.83 1.57

1.00 1.73 1.20 1.70 1.02 1.63

SPWFG18 SPWFG19

acceleration in the centrifuge, allowing higher accelerations (up to 60 g) to be used, as the 30 g rated excavation equipment was not required. In test SPWFG 18, excavation was undertaken at 30 g acceleration. In this case, excavation was simulated by lowering the front wall of the strong-box and then using a scraper mechanism to remove the sand in front of the model sheet pile wall. The system for carrying out the test is described in König (2002). Data obtained from the tests included a bending strain envelope, from which the bending moment distribution on the centre-line of the wall was inferred (Fig. 4); anchor loads (Fig. 3 and Table 1), and displacements at the level of the wall toe, the hinge zone (when present), anchor level and the head of the wall (Fig. 7). 3

NUMERICAL ANALYSES

Numerical modelling was undertaken at Imperial College, London using the program ICFEP (Potts and Zdravkovic, 1999).

The finite element (FE) analyses undertaken and reported here used Lade’s double hardening cap model, Lade (1977); Lade and Nelson (1987). Lade’s cap model has been implemented within ICFEP, validated and applied to boundary value problems, Kovacevic (1994). The model was chosen for these calculations because it reproduces many aspects of granular soil behaviour including a stress dependent non-linear elastic stiffness, the influence of the intermediate principal stress, σ’2 on the strength and the stress-strain response, and curvature of the failure envelope. In order to derive suitable parameters for the model (Table 2), laboratory testing of the sand used in the centrifuge tests was undertaken at RU Bochum and comprised four drained triaxial compression tests and two oedometer tests. The parameters determined from the Bochum test data were used in the analyses presented here on the basis that the element tests were reproduced by FEA simulation, Fig. 5. Good accord in of mobilised shear strength and volumetric strain response is predicted up to about 8% to 10% axial strain. Beyond this point, dilation is predicted to continue whereas in the tests, the rate of dilation appears to reduce beyond this point. This divergence is likely to be due to a combination of shear banding effects and the shearing of the soil towards its critical state, effects the model does not capture. It is not expected that the results of the analyses would have been greatly affected by this divergence because strains outside this range occurred only in small areas of the model. 3.2 Wall model One-dimensional, curved, 3-node iso-parametric Mindlin beam elements have been implemented in ICFEP in order to be able to model structural elements (Day, 1990; Potts & Zdravkovic, 1999). As part of the implementation of the beam elements, an elastoplastic constitutive model which allows hardening or

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Figure 6. Geometry and displacement boundary conditions for finite element mesh.

Numerical simulation of four point bending tests were undertaken to that the moment-curvature characteristics were replicated, Fig. 2. The sand-wall and sand-strong box interfaces were modelled with the zero thickness interface elements which use an elastic-perfectly plastic Coulomb type soil model. Interface shear tests yielded a value of interface angle of friction of 26◦ which was assigned to these elements. 3.3 Initialisation and boundary conditions Figure 5. Verification of Lade’s cap model against element tests. Table 3.

Model wall section properties.

Parameter

Main wall

Hinge zone

Density, γ kN/m3 Young’s modulus, E GPa Wall thickness, t cm 2nd Moment of area, I cm4 /m Yield stress, fy MPa Plastic moment, MP kNm/m

26.5 69 0.30 0.2250 118 –

26.5 69 0.085 0.0010 118 0.0028

softening of the element force parameters to a residual value was included. The form of the model is piecewise linear and the force parameters are uncoupled. A check of the axial and shear force values during the calculations showed that they were sufficiently small not to have impacted on the available moment resistance. To model the hinge mechanism in the centrifuge tests, beam elements at the position of the notch have been given characteristics appropriate to the particular test. The plastic bending characteristics for the wall sections were based on both physical and numerical simulation of bending tests. Elements over the remainder of the wall are elastic with properties associated with the full aluminium sheet section, Table 3.

The geometry for the finite element mesh is well defined by the strong box used for the centrifuge modelling, which had internal plan dimensions of 0.63 m long by 0.36 m wide and 0.41 m deep. The twodimensional finite element mesh used is illustrated in Fig. 6. Initial stresses have been based on a coefficient of earth pressure ‘at-rest’, K0 = (1 − sin φ ) = 0.38 and a bulk unit weight for the sand of about 16.5 kN/m3 . As the sand was dry, pore water pressures throughout the problem were set to zero. All the boundaries representing the strongbox container are fixed in the horizontal direction and the strongbox base and rear wall boundaries are also fixed in the vertical direction, Fig. 6. A bar element was used to represent the anchor(s), and in these tests the end was fixed in the vertical and horizontal direction throughout the analysis. It should be noted that the bar as defined, is not attached to the solid elements surrounding it, in order to replicate the test set-up where the anchor es through a duct. In order to simulate the centrifuge test, where gravity was increased to about 30g, vertical body forces were applied to the entire mesh from an initial stress state (based on 1g), until an equivalent of 30g acceleration has been applied. The application of body forces modified the K0 state; in this case to an average value of about 0.34 but with a non geo-static variation with depth. This variation is at least qualitatively similar to profiles reported elsewhere, Garnier (2002).

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Figure 7. Observed test wall response at 30g compared to numerical predictions for lateral displacement and bending moment.

3.4 Test and analysis sequence The test sequence modelled in each of the backanalyses was fairly simple; with the anchors locked off and no pre-stress having been applied, excavation proceeded to the required depth. In tests SPWFG 17 & 19, excavation of the sand in front of the wall, was carried out at 1g, before body forces equivalent to a total of 60 g acceleration were applied – for brevity and because the comparison does not alter significantly at higher acceleration values, only the results at 30 g acceleration are presented here. In test SPWFG 18, the model was ‘accelerated’ to 30 g and then excavation followed. 4

COMPARISON AND DISCUSSION

The back-analyses presented here generally provide reasonable agreement with the wall responses observed in the physical modelling, Fig. 7 (note that all results are at model scale). Bending moments and wall deflection at the lower anchor level were particularly well reproduced however displacements elsewhere were not so well replicated. Similarly, anchor loads predicted at the lower level show much better accord with those observed than at the upper anchor level where loads tend to be overpredicted by about a quarter to a third, Table 1. In order to be compatible with the recorded anchor loads, pressures in the uppermost part of the wall would have to be half those predicted in the FEA. It is thought that this discrepancy is due to the proximity of the upper free-boundary surface and possible over-prediction of resistance by the soil model at low confining pressures. Also, the sand density achieved in this zone may not be as compact as at deeper levels and thus, the model parameters used may not be entirely compatible with the actual state.

The effect of the decision to carry out the excavation before accelerating the model in the centrifuge can be assessed by comparing tests SPWFG17 (at 30 g) and 18, Fig. 7. The argument for using the sequence used in tests SPWFG17 and 19 was that the predicted response would be similar to that for the standard sequence i.e. that used in SPWFG 18, but there would be added flexibility to enable the centrifuge to be spun up to higher accelerations, increasing the bending moment demand in the wall. When the two physical test results are compared, the results for the wall forces and displacements show significant differences. In both cases, the mode of deformation is towards the excavation as expected however it is the top of the wall that leads in SPWFG17 and the toe in SPWFG18. A consequence of the mode of movement in the latter case is that the bending about the lower anchor is reduced to about half of the value in SPWFG17. Clearly, in this case there are factors at play that mean that the comparison between the two loading sequences is inconclusive. However, when the two FEA calculations are compared, there appears to be broad agreement in of the wall forces and displacements that might be expected, Fig. 7 and Fig. 8. In particular, the moment over the lower anchor is very similar, as are deflections at each anchor level and the general mechanism of movement, Fig. 8. If it is assumed that the FEA provides a reasonable indicator of the likely mechanisms involved in the two sequences, then all things being equal, better agreement between the two actual tests would be expected. It is interesting to note that as for the other two tests, the FEA suggests that the toe of the wall should not have kicked out in the manner observed. Wall plasticity was simulated by incorporating a weakened hinge zone in the wall. In this case, the wall was notched and drilled out just below the level of the

709

anchor walls but the flexibility effect reduced moment demand by only about 20%, Bourne-Webb (2004); Bourne-Webb, Potts & König (in print). 5

CONCLUSIONS

In this study, the finite element method has been successfully used to capture many aspects of the observed response of a series of centrifuge model tests representing a double anchored wall system embedded in dry sand. The calculation used Lade’s double hardening cap model to represent the sand and zero-thickness beam and bar elements to model the structure. Because of this ability to capture key characteristics of the physical model tests, more extensive parametric studies have been done with some degree of confidence. The comparison of FE simulations suggests that in an ideal situation, physical modelling based on a digaccelerate sequence should yield similar results to that for an accelerate-dig sequence. However, it is apparent from the physical tests that such accord may not always be readily achievable in practise. ACKNOWLEDGEMENTS The authors would like to thank our sponsors Corus, Arbed and HSP, and colleagues at RWTH Aachen who helped in developing the model wall used in the centrifuge modelling. REFERENCES

Figure 8. Incremental displacement vectors illustrating movement mechanism.

lower anchor position. By comparing tests SPWFG17 and 19, Fig. 7, the effect of the hinge zone can be assessed. The predicted and observed effect of the hinge zone in SPWFG19 is apparent in that the bending moment at this location is reduced by about 70%, compared with the intact wall in SPWFG17 and there is a clear discontinuity in the deflected wall shape at the hinge zone. This is a consequence of the use of a notch to provide a weakened zone which was present throughout the test resulting in a much more flexible wall system. As a result, the moment demand generated in the wall (SPWFG19) reduced substantially compared to that mobilised in the intact wall (SPWFG17), Fig. 7. This is not a surprise; the effect of flexibility is well known, Rowe (1952) however the magnitude of the reduction was unexpected. Similar results were seen in the single

Bourne-Webb, P.J. 2004. Ultimate limit state analysis for embedded retaining walls, PhD thesis, Imperial College, University of London. Bourne-Webb, P.J., Potts, D.M. and König, D. In print. Analysis of model sheet pile walls with plastic hinges, accepted for publication in Géotechnique. Day, R.A. 1990. Finite element analysis of sheet pile retaining walls, PhD thesis, Imperial College, University of London. Garnier, J. (2002). Properties of soil samples used in centrifuge tests. Physical modeling in Geotechnics: IMG ’02. Phillips, Guo & Popescu (eds), Newfoundland, Canada, 5–19. König, D. 2002. Modeling of deep excavations. Intl. Conf. Physical Modeling in Geotechnics, IMG ’02, Newfoundland, Canada, 83–88. Kovacevic, N. 1994. Numerical analyses of rockfill dams, cut slopes and road embankments, PhD Thesis, Imperial College, University of London. Lade, P.V. 1977. Elasto-plastic stress-strain theory for cohesionless soil with curved yield surfaces, Intl. J. Solids Struct., 13, 1019–1035. Lade, P.V. & Nelson, R.B. 1987. Modelling the elastic behaviour of granular materials, Intl. J. Num. Anal. Meth. Geomech., 11, 521–542. Potts, D.M. & Zdravkovic, L. 1999. Finite element analysis in geotechnical engineering: Theory, Thomas Telford, London, 440 pages. Rowe, P.W. 1952. ‘Anchored sheet pile walls’, Proc. Institution of Civil Engineers Part 1, 1, 27–70.

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Crane monopile foundation analysis A. Mar Coffey Geotechnics Limited, Manchester, UK

ABSTRACT: This article describes the use of three-dimensional finite element analysis to analyse the deformation and stability of a crane monopile foundation ing a 24.4 m high tower crane in close proximity to an existing two-level basement structure. The crane monopile is a composite structure composed of concentric structural and geotechnical elements whose pile head is pinned to the top of the basement floor slab by a reinforced concrete tie-slab. The author was engaged to carry out a Category III check of this temporary works structure. This case study illustrates the importance of identifying the issues associated with the problem before any analysis is carried out and the best practice of performing ing calculations on simplified idealisations of the problem to gain an idea of orders of magnitude of the results and to develop confidence in the numerical analysis predictions. 1 1.1

INTRODUCTION Background

Three-dimensional finite element analysis predictions on simplified representations of the pile foundation were compared with Coffey in-house developed analytical tools. The crane monopile is a composite structure composed of concentric elements and the effect of explicitly modelling the various rings of different material using solid finite elements was explored. 1.2

Description of the problem considered

monopile is to be pinned to the top of the basement floor slab by a 500 mm thick concrete tie-slab. As well as ensuring that the proposed design of the monopile and tie-slab was structurally capable of carrying the applied loads from the tower crane, the close proximity of the foundation to the basement warranted investigation of the loads and deformations induced in these neighbouring structures by the activities of the tower crane. Such predictions would not have been possible via simplistic calculations hence the recourse to finite element modelling to gain insight into this three-dimensional soil-structure interaction problem. 1.3 Tower crane and ground details

The proposed foundation is a monopile situated very close to an existing two-level basement structure as illustrated in Figure 1. As shown, the top of the

Figure 1. Plan and elevation (A-A) showing the crane monopile and adjacent basement structure.

The base of the crane platform grillage (1000 mm thick) is elevated at 9.5 m above the top of concrete capping and tie slab. The tower crane is 24.4 m high with a 60 m long jib, a schematic of which is shown in Figure 2. The soil parameters used in the analysis were derived in the first instance from data in a comprehensive geotechnical investigation interpretative report. In summary, the site is underlain by: 2 m of made ground, 4 m of weathered London Clay, 14.5 m of

Figure 2. Schematic of tower crane.

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Table 1.

Material properties for the soil and pile. α∗int –

γ Material kN/m3

K0 –

φu ◦

su ψ kPa ◦

Eu νu MPa –

London Clay

1

0

150 0

30

Material

γ kN/m3

E MPa

ν –

Concrete 20:1 Sand/Cement Mix Steel Cement-Bentonite Grout

24 20 77 24

2.1 × 104 30 2.0 × 105 200

0.15 0.25 0.30 0.15

20

0.495 0.85

*Interface reduction factor.

Figure 3. Plan cross-section showing the concentric components of the crane monopile.

London Clay, 19.5 m of Lambeth Clay and 3.5 m of Thanet Sands founded on Upper Chalk. Given the transitory nature of the crane loading a short-term undrained response of the ground was considered appropriate; consequently an undrained analysis was performed using undrained soil parameters. A range of soil parameters were considered for the detailed analysis and because of the extreme nature of the problem in of the close proximity of the surrounding structures and uncertainties with regard to soil strength and stiffness due to construction stage effects; the final analyses assumed a single London Clay layer with very conservative values for the undrained Young’s modulus and undrained shear strength of: 30 MPa and 150 kPa respectively. The pile-soil interaction was analysed as an undrained load case with the London Clay modelled as an undrained, cohesive linear elastic-perfectly plastic (Tresca) material. The Mohr-Coulomb strength model was used with the friction and dilatancy angles equal to zero (φ = ψ = 0), cohesion equal to the undrained shear strength (c = su ) and zero tensile strength tension cut-off criterion in place which restricts the development of tensile stresses in the soil. The monopile is a composite structure as shown in Figure 3 which comprises a 16.825 m length of steel pipe (outer diameter 2.2 m with a wall thickness of 40 mm) surrounded by a 6.825 m length of concrete caisson (inner diameter of 3 m with a wall thickness of 160 mm) in the upper portion of the pile where it is known that the high lateral loads will develop. The infill between the concrete and steel is a cementbentonite grout mixture and the infill in the steel pipe is a 20:1 sand-cement mixture. The basement structure comprises three floors and a contiguous piled wall composed of 28.825 m length; 900 mm diameter piles at 1050 mm centres. The horizontal distance between the bored pile wall and crane monopile centrelines is 3.2 m. The top of the crane monopile is connected to the top of the basement structure by a 500 mm thick reinforced concrete slab as shown in Figure 1.

The material properties used for this problem are summarised in Table 1. Interface elements have been used along the outside surfaces of the monopile for all the finite element models in this study. These elements are used to improve the results by allowing for slip between the monopile and the soil and to model a possibly reduced strength su,int = αint .su along the sides of the monopile to for reduced soil strength due to the effects of pile installation. 2 2.1

PRELIMINARY ANALYSES ERCAP analyses

The monopile was first analysed using the boundary element program ERCAP (Poulos 1992). The program implements the method described by Poulos & Davis (1980). This program can analyse a pile subjected to lateral loading and/or lateral soil movements. ERCAP (Earth Retention CAPacity of piles) can analyse the effects of the proximity of a pile to a slope or cutting in an approximate manner. It has the facility to enable the assessment of the stabilising force which a pile or row of piles can develop in a potentially unstable soil mass. In this problem it was used to model the lateral interaction of the monopile with the surrounding soil when subjected to the horizontal load and overturning moment at the pile head. The objective of the preliminary analyses was to compare results from PLAXIS 3D Foundation (PLAXIS BV 2007) with ERCAP. 2.2 Analysis details The ERCAP program restricts the to a single uniform pile geometry. For this reason, two separate analyses were performed with uniform cross-sectional representations of the actual crane monopile. To bound the predictions of lateral pile deflection in the London Clay; the performance of the steel pipe alone and the composite pile were considered. The first analysis modelled the steel tubular section whereas the second analysis modelled the composite pile; each over the full 16.825 m length of pile. For the latter, a composite Young’s modulus of 12.4 GPa for a solid circular

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Table 2.

Evaluation of composite Young’s modulus of pile.

Component Caisson 20:1 Sand/ Cement Mix Steel Pipe CementBentonite Grout

I = 5.964,

Table 3.

do m

di m

UCS MPa

I m4

EI kNm2

3.32 2.12

3.00 0.00

20.0 2.00

1.9880 0.9915

4.174 × 107 2.975 × 104

2.20 3.00

2.12 2.20

– 2.00

0.1584 2.8260

3.167 × 107 5.652 × 105

EI = 7.401 × 107 .

Loads acting at the base of the crane.

Load Case

Horiz. Thrust H kN

Vert. Load V kN

Overturning Moment M kNm

In-service crane Out-of-service crane

40 191

−1622 −1565

6242 3104

Figure 4. Steel pipe pile deflection predictions from ERCAP.

pile of 3.32 m diameter was calculated

on the basis of the ratio of the summations EI/ I (see Table 2) where do , di , UCS, I and E are the outer/inner diameter, unconfined compressive strength, second moment of area and Young’s modulus respectively. Both the in-service and out-of-service crane loads were considered (Table 3) to identify the worst case combination which would develop the highest deflections, shear forces and bending moments in the monopile. This was for the situation without any horizontal restraint offered by the tie-slab. Both the in-service and out-of-service load cases were performed undrained and this is applicable as the key stratum is London Clay with an average coefficient of consolidation cv of 0.3 m2 /year. For drainage paths, D, in the range of 5.5 m to 30 m and an out-of-service time, t, of 1 year say, the dimensionless time factor T ( = cv t/D2 ) is less than 0.01 and Duncan (1996) has suggested that the soil can then be considered to behave in an undrained manner under the loading specified. With reference to Figure 1, it can be seen that the crane grillage soffit level is elevated at 9.5 m above the top of concrete and tie slab. Thus for a 1 m thick crane platform grillage the lever arm will be 10 m – inducing an additional bending moment equal to the horizontal thrust multiplied by this lever arm. For the in-service crane loads this produces an overturning moment of 6642 kNm at the top of concrete capping and tie-slab. Figures 4-5 show the ERCAP predictions of pile deflection in the London Clay for the steel pipe alone and composite pile respectively under the action of the in-service and out-of-service load conditions. It can be seen that the in-service loads produce slightly higher horizontal pile head displacement. From the deflected pile shapes it can be seen that the composite pile is behaving more like a short rigid pile than the steel pipe alone. The actual crane monopile is a combination of

Figure 5. Composite pile deflection predictions from ERCAP.

these two simplifications and so it is expected that the deflection predictions will fall within the range of deflection predictions shown for this extreme case without a tie-slab in place. Figure 6 shows the ERCAP prediction of bending moment developed in the pile for these two load conditions. It can be seen that the in-service loads induce higher bending moments in the pile than their out-of-service counterparts. 3 THREE-DIMENSIONAL ANALYSES OF CRANE MONOPILE AND BASEMENT 3.1 Outline The three-dimensional finite element analyses (3DFEA) considered two idealisations of the pile: (1) a ‘simplified composite pile’ consisting of a solid circular pile with a: Young’s modulus of 12.4 GPa, diameter of 3.32 m and length of 16.825 m (as used

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Figure 6. Pile bending moment distribution predictions from ERCAP.

Figure 8. Pile bending moment comparison between PLAXIS and ERCAP.

Figure 7. Pile deflection comparison between PLAXIS and ERCAP.

Figure 9. Pile shear force distribution predictions from ERCAP.

in the ERCAP analyses) and (2) a ‘complex composite pile’ comprising individual concentric elements as summarised in Table 2 (see the next section for further details on modelling). The horizontal deflections, bending moments and shear force predictions are broadly similar as shown in Figures 7–9 thus confirming the idealisation approaches and different analysis methodologies used.

between each concentric component. Through the pile designer, interface elements were specified along the outside boundaries of the monopile to allow for the simulation of slippage and separation between the soil and monopile. The use of the pile designer creates an equivalent structural line element along the centreline of the pile which enables the pile: displacements, bending moments and shear forces to be output in a convenient manner.

3.2

Modelling of the complex composite pile

The series of concentric elements (Figure 3) forming the crane monopile were explicitly modelled in PLAXIS 3D Foundation using the pile designer. This was achieved by selecting the circular tube pile type which is defined by its wall thickness and internal diameter. Tubular piles were specified for the concrete caisson, cement-bentonite grout and steel pipe and each of these components were centred on plan at the same (x,z) coordinates. This created a mesh of solid elements with full connection at the mating boundaries

3.3 Three-dimensional analyses of crane monopile and basement In order to predict the interaction of the monopile with the ground and the adjacent basement structure, a 3D finite element analysis using PLAXIS 3D Foundation was performed. A serviceability limit state analysis (no partial factors applied to materials) was performed with the unfactored working loads applied to the pile head at ground level. The analysis did not consider the

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Table 4. Analysis phases for the crane monopile and basement. Phase Description

Notes

0

K0 -procedure.

1 2 3

Generation of initial stresses in the virgin ground. Wish-in-place of basement and contiguous pile wall. Installation of the crane monopile. Application of crane loads.

Table 6.

Model load conditions considered.

Model

Description

1

Jib load, moment and tailwind applied away from the pile wall without the surface tie-slab in place Jib load, moment and tailwind applied away from the contiguous pile wall with the surface tie-slab in place Jib load, moment and tailwind applied in the direction towards the contiguous pile wall with the surface tie slab propping against the basement wall in place Jib load, moment and tailwind applied parallel to the contiguous pile wall with the surface tie-slab providing restraint

2 No basement construction details were available.

3

Previous displacements reset to zero. – 4

Table 5. Physical and material properties for the tie-slab, basement floors and wall. Component Tie-slab 500 mm thick Floor 400 mm thick Floor 1000 mm thick Contiguous pile wall 900 mm diameter at 1050 mm centres

γ kN/m3

t m

E GPa

ν –

24 24 24 24

0.5 0.4 1 0.742∗

21 21 21 21

0.15 0.15 0.15 0.15

*Equivalent plate thickness refer to Eq.(1).

detailed stages of excavation and construction of the basement. The phases considered are summarised in Table 4. The initial stresses in the ground were computed using the K0 -procedure with K0 = 1 for the London Clay layer. As the analysis was in of total stress parameters no pore water was modelled so the phreatic level was set below the base of the model. The basement floors were idealised using 6-noded triangular plate elements and the basement walls were modelled using 8-noded quadrilateral plate elements. The contiguous bored pile wall was idealised as a continuous plate with a reduced thickness, deq , to for the spacing of the piles:

where d is the pile diameter and s is the centre-to-centre pile spacing. The properties of the tie-slab, basement floors and wall are summarised in Table 5. The soil and monopile were modelled using 15noded wedge elements. The horizontal, vertical and moment loads applied to the monopile were rationalised into equivalent horizontal and vertical pressures acting over the steel pipe cross-section. Interface elements were inserted between the soil, walls, floors and outside surfaces of the monopile to simulate the reduced strength between the soil and these structures. For these analyses an interface reduction factor of 0.85 was assumed, resulting in a reduced undrained shear

Figure 10. Deformed mesh shown to an exaggerated scale (scaled up 500x); Max. 3.73 × 10−3 m, Min. 0, (Model 2, Phase 3).

strength of 127.5 kPa between the soil and non-soil structures. Four 3D finite element analyses were undertaken to model the excavation and ‘wished-in-place’ construction of the basement and the subsequent monopile with loading based on the tower crane load specifications. The soil-structure interaction between the monopile, walls and floors was simulated in these analyses. These analyses considered the in-service load combination and the results confirmed that the worst case corresponded to case 2 of Table 6.

3.4 Results This section summarises a selection of results from the 3D finite element analyses. Figure 10 shows the deformed shape for Model 2 (Table 6) Phase 3 (Table 4) – note that the London Clay has been hidden from view. Figure 1a shows the horizontal deflection of the crane monopile. The shear force and bending moments developed in the pile are shown in Figures 11b, c. These plots are direct outputs from the program with no additional post-processing

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Figure 11. 3DFEA pile outputs (Model 2, Phase 3).

made. The resolution of the shear force diagram is a result of the relatively coarse mesh being used. However, independent checks described in Section 2.2 on similar mesh refinements of the simplified pile provide confidence in the accuracy of these results. The deformed shape of the contiguous pile wall, floors and tie-slab has been determined and predicted deformations are very small with peak total displacements of around 0.6 mm, 0.7 mm and 3 mm respectively.

4

DISCUSSION AND CONCLUSIONS

An understanding of the underlying engineering principles is essential and a review of the literature on the subject is good practice. Reference was made to Matlock & Reese 1960, Elson 1984, Smith & Griffiths 1988. Sensitivity studies were performed to investigate: mesh density, model extent, load application approaches and material property variation. An investigation was made to explore the effect of a reduction in the stiffness of the cement-bentonite mix between the concrete caisson and steel pipe. The reduction of this from 600 MPa to 200 MPa was found to have little effect on the behaviour of the monopile. The analyses did not consider the detailed stages of excavation and construction of the basement (Ng, 2004) so the deformations and loads predicted to develop in the basement could not reliably be taken into . To address this, the prediction of movement and structural forces induced in the basement during this phase were discounted by zeroing displacements at the start of Phase 2 and by external post-processing of the structural forces developed between Phase 3 and Phase 1. Therefore, the structural forces and displacements reported are in addition to the existing structural forces and displacements due to the wall and floor loads and live loads applied to the basement. The results of the ERCAP analyses (Figures 4 to 6) predict that the monopile will deflect laterally between 4–6 mm with bending moments in the range of 5014 kNm to 6642 kNm – (this is without the tie-slab in place).

The ERCAP analyses identified that the worst case crane loads were the in-service combination comprising a horizontal thrust, axial load and overturning moment of 40 kN, −1662 kN and 6242 kN respectively and these were assumed to act at the crane base, which is 10m above ground level. PLAXIS predicted the worst case deflection to be 5.4 mm horizontal (Model 1 – without tie-slab) and this results in a pile head rotation at ground level of 0.000831 radians (which would result in a 29 mm deflection 34.4 m above ground level). The bending moment in the monopile for this case is 6170 kNm which is of the same order of magnitude as that found in the ERCAP analysis. The results of the PLAXIS analysis predict that the monopile will deflect laterally by 0.5–0.8 mm with peak bending moments in the range of 6960 kNm to 6980 kNm and shear forces in the range 922 kN to 928 kN (this is with the tie slab in place). The maximum horizontal wall deflection is of the order of 0.3 mm which is negligible and the peak bending moment of 200 kNm is generated. The maximum compressive axial load developed in the 500 mm thick tie-slab is 1490 kN for the loading condition in Model 3. The maximum tensile axial load developed is 942 kN for the loading condition in Model 2. The structural capacity of the monopile and tie-slab system is adequate under the action of these structural loads (well within the 20 MPa compressive strength of the concrete and the 275 MPa yield stress of the steel). This case study illustrates the importance of identifying the issues associated with the problem before any analysis is carried out and the best practice (Mar, 2002) of performing ing calculations on simplified idealisations of the problem to gain an idea of orders of magnitude of the results and to develop confidence in the numerical analysis predictions. REFERENCES Duncan, J. M. 1996. State of the art: Limit Equilibrium and Finite-Element Analysis of Slopes, Journal of Geotechnical Engineering, ASCE 122, No.7, July, pp. 557–596. Elson, W.K. 1984. Design of laterally loaded piles, CIRIA Report 103. Mar, A. 2002. How To Undertake Finite Element Based Geotechnical Analysis. NAFEMS (The International Association for the Engineering Analysis Community). Matlock, H. and Reese, L.C. 1960. Generalised solutions for laterally loaded piles, Proc. ASCE, J. Soil Mech. Found. Div. Vol 86 (SM5), pp. 63–91. Ng, C.W.W., Simons, N. and Menzies, B. 2004. A Short Course in Soil-Structure Engineering of Deep Foundations, Excavations and Tunnels, Thomas Telford, London. PLAXIS BV, 2008. PLAXIS 3D Foundation. Version 2.2, PLAXIS BV, Delft, the Netherlands. Poulos, H.G. and Davis, E.H. 1980. Pile Foundation Analysis and Design. New York: John Wiley & Sons. Poulos, H.G. 1992. Program ERCAP (Earth Retaining Capacity of Piles) s Manual, Coffey Geosciences Pty Ltd. Smith, I.M. and Griffiths, D.V. 1988. Programming the finite element method (2nd edition), John Wiley, Chichester.

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Influence of excavation and wall geometry on the base stability of excavations in soft clays T. Akhlaghi, H. Norouzi & P. Hamidi Faculty of Civil Engineering, University of Tabriz, Tabriz, Iran

ABSTRACT: Evaluation of safety factor for base stability of excavations is of great importance in the design of excavations in soft soils. The two dimensional base stability of excavations is evaluated using either the traditional limit equilibrium techniques or the finite element methods. In this study, the FEM with shear strength reduction is used to evaluate the 2D base stability of excavations in soft clays. The influences of the width and the depth of the excavation and the embedded depth of the wall on the base stability have been investigated and discussed. The results indicate that the safety factor against base instability increase almost with the ratio of the depth to the width of excavation. Also with increasing the wall embedded depth, the safety factor increases, whereas the increase in the safety factor is smaller in the case of the large clay thickness under the base of excavation than in the case of small clay thickness. 1

INTRODUCTION

The estimations of ground movement and base instability are the most important aspects that must be considered in excavation works, specially excavations in soft clays, to prevent any hazardous effect to nearby buildings. The temporary struts are required to wall safely and the bending moments in the wall should not go beyond the designed capacity and serviceability. In recent years, it is becoming possible to do detailed numerical analyses for the various mechanical behavior of braced excavation by researching and developing the numerical analysis technique such as the finite element method (FEM). Economic demands favor excavations without sheeting and bracing, but safety requirements limit these solutions. The importance of base stability problems has been indicated with increasing excavation scale and depth for underground space development. Evaluation of the safety factor against base instability is important in the design of excavations in soft soils. If the factor of safety is smaller than from acceptable value, expensive ground improvement schemes may become necessary to stabilize the soil below the final excavated level (O’Rourke et al., 1997). Short-term undrained stability often controls the design of excavations in soft clays immediately after construction. In the practical point of view, there are two distinct methods used to perform stability calculations for all excavations: (1) traditional limit equilibrium methods (LEM); and (2) finite element methods. Several limit equilibrium analysis methods are available for evaluating the base stability of excavations; however, the calculated safety factors vary with different methods. Up to the present, limit equilibrium methods has been widely used in practice

Figure 1. Geometry of excavation.

for predicting the safety factor of excavations, and include separate calculations of basal stability based on failure mechanisms proposed by Terzaghi (1943); Bjerrum and Eide (1956); or overall slope stability using circular or noncircular arc mechanisms based on well established methods proposed by Bishop (1955); Spencer (1967); Morgenstern and Price (1967). It is noted that, Terzaghi’s method is only used for shallow or wide excavations with H /B ≤ 1, where B is the width and H is the depth of the excavation (Figure 1). Since The three-dimensional effect is important at the length to width ratio smaller than six, Terzaghi’s equation for base stability can be used to evaluate the three-dimensional base stability of rectangular (or shallow) excavations after modification.

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In this study, the finite element method with reduced shear strength is used to evaluate the 2D base stability of excavations. Numerical results of this paper indicate that the base stability of excavations was significantly influenced by the factors such as: (1) the depth to width ratio, (2) the thickness of the soft soil layer between the excavation base and the stiff stratum, and (3) the depth of walls inserted below the excavation base. Surely, the base stability of excavations in soft clays was also influenced by the stiffness of walls, that were not considered here. 2

FEM WITH STRENGTH REDUCTION TECHNIQUE

The stability problems (such as, slope stability and base stability of excavations in soft soils) are commonly analyzed by a limit equilibrium method. However, such a traditional approach is limited by assumptions concerning the analysis method itself and failure mechanism of the slope. On the other hand, the most difficult problem faced in the utilization of displacement based finite element method in the stability analysis is the calculation of the safety factor. Such a difficulty can be overcome by the introduction of FEM with shear strength reduction technique which was proposed as early as 1975 by Zienkiewicz et al. The FEM with reduced shear strength has been applied to the slope stability analysis in twodimensional situations by Zienkiewicz et al. (1975), Matsui et al. (1992), Ugai, and Griffiths et al. (2001). Recently the reduced shear strength FEM has been successfully applied to evaluate the base stability of circular excavations. The essence of the finite element method with shear strength reduction technique is the reduction of the soil strength parameters until the soil fails. In other words, in the finite element with reduced shear strength, the safety factor is evaluated by the gradual reduction of the shear strength parameters (c, ϕ) of soil inducing the divergence (failure) of the nonlinear analysis. The shear strength equation is given by:

The reduced shear strength parameters cf and ϕf replace the corresponding values of c and ϕ in the above equation to:

Table 1.

Properties of the soft soil.

Property

Value

Su Eu ν γ

35(kPa) 250Su (kPa) 0.45 18(kN/m3 )

The initial value of F is assumed to be sufficiently small so as to produce a nearly elastic problem. Then the value of F is increased step by step until finally a global failure develops (Cai et al., 2001). The load step is controlled by the increment of the safety factor. If convergence cannot be reached after, for example, 1000 iterations, the F value just before this load step is taken to be the unique safety factor Fs . This method is called FEM with shear strength reduction technique. The finite element method with strength reduction technique used in stability analysis offers a number of advantages over traditional limit equilibrium method. For instance, it eliminates the need for a priori assumptions on the shape or location of the failure surface, and can automatically trace the progressive failure from localized areas all the way to the overall shear failure. Furthermore, it is possible for the FEM with reduced shear strength to evaluate the slope stability under a general framework even for the case of base stability of circular excavations. However, the success of reduced shear strength FEM relies strongly on the determination of global instability of soil slopes, i.e. definition of failure. A 2D finite element program is used for the present analysis, and the secant Newton method is used to accelerate the convergence of the modified Newton– Raphson scheme (Sloan, 1983). The isoparametric elements with eight nodes are used to model the soil and wall elements. In order to obtain the precise results it is found that at least six rows of elements must be put below the bottom of the wall, where the soil failure should take place. The horizontal distance from the wall to the outer boundary of the excavation should be not less than at least two times of excavated depth (H ). The vertical distance in the mesh under the base of excavation must be as fine as possible which depends on the thickness of the clay under the base of excavation. The mechanical properties of the soft soil are shown in Table 1. 3

CALCULATION OF BASAL STABILITY USING LEM

and

so the shear strength equation with reduced shear strength can be written as:

For excavations in an homogeneous clay, the stability of the excavation can be most conveniently expressed in of the stability number, Nc = γH /Su , where γ and Su are the average total unit weight and undrained shear strength in the retained soil, respectively. Table 2 shows the various methods for assessing the stability number from limit equilibrium methods. As indicated in the table, there are subtle differences in these basic

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Table 2. Various proposed stability numbers. case

Stability number (Nc )

Terzaghi

Bejrrum and Eide

Notes

√ H γH = Nc1 + 2 B Su γH H = Nc1 + Su d √ H γH Sub = Nc1 + + 2 Suu Suu B

B Base case for homogeneous deep clay Nc1 = 5.7, d ≥ √ 2 B Effect of underlying bearing layer d < √ 2 Two clay layer: Suu & Sub are strengths in retained soil and below excavated level (anisotropic shear strength in clay)

√ H +D D γH +2 = Nc1 + 2 B B Su

Effect of rigid wall embedment

γH = Nc2 Su

Base case for homogeneous deep clay Nc2 = f (H /B)

Eide et al.

D γH = Nc2 + 2α Su B

O’Rourke

π2 My γH = Nc2 + x Su D(D + h)Su

Effect of rigid wall embedment α is the adhesion between interior soil plug and wall My is yield moment of wall, h is the excavation depth below lowest and x is equal to 1/8, 9/32 or 1/2 for free, sliding and fixed end conditions respectively

solutions associated with assumed values of the bearing capacity factor, Nc , the location of the vertical shear surface in the retained soil, and the inclusion of shear tractions along this plane. Simple modifications are widely used to for the proximity of an underlying bearing layer and for contrast in undrained shear strength above and below the excavated grade.The effects of wall embedment are usually computed the approach proposed by Terzaghi (1943), assuming that failure occurs below the base of the wall and is resisted by the weight of the interior soil plug and adhesion acting along the plug–wall interface. A similar approach is used by Eide et al. (1972). Both methods implicitly assume that the wall is rigid (i.e., does not yield). O’Rourke (1993) assumes that wall embedment does not alter the basal failure mechanism in the soil, but does contribute to the stability due to the elastic strain energy stored in flexure. The resulting stability numbers are functions of the yield moment and assumed boundary conditions at the base of the wall. 4

UPPER BOUND LIMIT ANALYSIS FOR BASE STABILITY OF EXCAVATIONS

The upper bound formulation equates the power dissipated in a kinematically issible velocity field with the power expended by the external loads. A kinematically issible velocity field is one which satisfies the (1) compatibility equations; (2) velocity boundary conditions; and (3) the flow rule. Power is dissipated by the plastic yielding of the soil mass, during plastic flow. It is noted that, limit analysis theorems (upper and lower bound theorems) in conjunction with finite element, are very powerful methods for calculations of stability problems in soil mechanic.

If the soil layer of the thickness D below the excavation base acts as a load (γD), and the wall is rigid, the following equations that are summarized in Table 3, could be derived from the energy dissipation assumption. In the following equations that are considered in Table √ 3, dc is called the critical depth and equals to B 2. It is noted that, the value of γH /Su in the upper bound analysis is equivalent to the Nc -value in FE analysis. The equations from upper bound limit analysis are considered here for making a comparison with proposed FEM with strength reduction, and results of this comparison are considered in the next section. 5

DISCUSSIONS ON THE RESULTS

5.1 Influence of depth to the width ratio

H : B

For the case which the thickness of the soft soil layer below the excavation base is less than the critical depth (d ≤ dc ), the results of the FEM with shear strength reduction indicate that the location of stiff layer has influence on the Nc -value. Figure 2 indicates the influence of depth to the width ratio (H /B) on the base stability of excavation, when the thickness of the soft soil layer below the excavation base is bigger than the critical depth (d ≥ dc ), and the embedded depth is equal to zero, i.e. D = 0.0 (case 1 in Table 3). For more comparison the results of Nc -value are evaluated for different values of d/dc and are shown in Figure 2, where Nc is defined by:

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Table 3.

Stability numbers from limit analysis calculations.

Eq. Number case 1 2 3

4 5 6 7 8 9

Stability Number (Nc )

Terzaghi Nc =

γH H = 5.81 + Su d

Notes

Condition

–

Case (1)

1 H – Nc = 6.14 + √ 2 d 3 θ3 is variable and must be chosen Nc = 2 1 + π − θ1 − θ3 + tan θ1 4 to minimize Nc - value 1 H + tan θ3 + √ 2d 2 d cos θ1 = cos θ3 B 1 H 2d 3 θ1 = cos−1 Nc = 2 1 + π − θ1 + tan θ1 + √ 4 B 2 d √ H +D Terzaghi Nc = 5.71 + 2 – B

D = 0.0 and d ≤ dc

Prandtl

H +D B H +D Terzaghi Nc = 5.71 + dn H +D Prandtl Nc = 6.14 + √ 2dn 3 Nc = 2 1 + π − θ1 − θ3 + tan θ1 4 1 H + tan θ3 + √ 2 d Prandtl

Nc = 6.14 +

in which Nc andFs are dimensionless coefficients depending on the geometry of the excavation and the safety factor of base stability of excavation, respectively. Figure 3 presents the Nc -values, for various ratios of depth to the width, from proposed FEM with strength reduction, the method of Bjerrum & Eide which is considered in Table 2 and Equations (1) and (2) from Table 3. In this figure the FEM results for Nc -values are compared with those obtained from the limit equilibrium (Bjerrum & Eide’s method) and upper bound analysis (Equations (1) and (2) from Table 3). From Figure 3 it is obvious that for the shallow or wide excavation with H /B ≤ 1.0, the Nc -value, predicted using the FEM with reduced shear strength, is close to Equation (1) from Table 3. It is noted that for H /B > 1.0, the Nc -value, from proposed FEM is less than Prandtl’s method (Eq. 2 in Table 3). As shown on Figure 3 the FEM computations give less safety factors than those obtained from the upper bound analysis that are considered in Table 3, and for H /B > 1.0 the Nc -value obtined from FEM increases with the ratio of depth to the width. Figure 3 also shows that the differences between the proposed FEM with strength reduction, limit equilibrium (Bjerrum & Eide’s method) and upper bound analysis (Equations (1) and (2) from Table 3) increase with increasing the ratio of depth to the width. 5.2 Influence of soft soil thickness (d/dc ) In this section the effect of the thickness of soft soil layer under the base of excavation is considered. For

Case (2) D ≥ 0.0 and dn > dc

– –

Case (3)

– θ3 is variable and must be chosen to minimize Nc - value 2d cos θ3 cos θ1 = B

Figure 2. In fluence of H /B on Nc -value with different values of d/dc from FEM (D = 0).

the case d ≤ dc , the results of the FEM with strength reduction indicate that the thickness of the clay layer has influence on the Nc -value as shown in Figure 4. This figure presents the influence of the soft soil thickness for H /B = 1.0. As shown in Figure 4, the safety factor decreases with increasing the thickness of clay under the base of excavation. It can be concluded that when d < dc , the presence of the stiff layer close to the base of excavation increases the Nc -value. It is noted that for clay extending to a considerable depth below the base of excavation (d > dc ), the increase of clay depth has not more influence on the Nc -value. In other words, in this case the Nc -value slightly decreases with increasing the clay thickness. Figure 4 also shows

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Figure 5. Influence of wall embedded depth below the excavation base (d = dc , H /B = 1.5). Figure 3. Influence of H /B on Nc -value with different methods (D = 0).

Figure 4. Influence of d/dc on Nc -value (H /B = 1, D = 0).

that the Nc -values predicted using Eq.(1) and Eq.(2) are larger than that of the FEM with shear strength reduction. When d/dc > 0.5, the difference of the Nc -values between the FEM with reduced shear strength, Eqs.(1) and (2) are small as shown in Figure 4. When the thickness of the soft soil layer are equal to the critical depth, the difference of the Nc -values between the FEM with reduced shear strength, Eqs.(1) and (2) is smaller than 1. 5.3

Influence of embedded depth of the wall

The normalized maximum displacement on the base of excavation () can be used for base instability computations. The normalized displacement increases with the reduction factor of the shear strength F, and develops with F trending to the safety factor of the excavation bases. Thus, the safety factor can be determined from the -F curve. Figure 5 illustrates the influence of the embedded depth on the safety factor of excavation base. If we define D/dc , as the embedment ratio, in case that the soft soil depth are equal or smaller

than the critical depth (d ≤ dc ), the results of proposed FEM with strength reduction shows that, the safety factor increase with increasing embedment ratio. That is because the wall precludes from the movement of soil towards the base of excavation, which is the cause of increasing the safety factor against base instability. The influence of embedded depth on F values for clay extending to a considerable depth below the base of excavation (d > dc ), are illustrated in Figure 6. When d > dc , the effect of increasing the embedded depth of the wall will leads to a marginal increase in the safety factor. From Figures. 5 and 6 we can see that, for clay that has smaller thickness (d ≤ dc ) below the base of excavation, by increasing the embedded wall depth, the increase of the safety factor will be bigger than that for clay with larger thickness (d = 2dc ) below the excavation base. It is obvious that, with increasing the wall embedment ratio, the safety factor will increase, whereas the increase of the safety factor in the case of small clay thickness under the base of excavation (d ≤ dc ) is bigger than in the case of large clay thickness (d = 2dc ). It can be seen that, when the embedded depth is shorter, the higher resistance will be taken place against the soil movement. FEM with reduced sear strength shows that displacement of the wall at its end is significantly smaller than that of the adjacent soil displacements. This means that the wall resistance to the soil movement was high. While in the case of large embedded depth the displacement of the wall at its end is almost the same as the adjacent soil, this means that the resistance of the wall to the soil movement was small. It is noted that, the analyses of the effect of the wall stiffness on the safety factor shows that, there are relations between the embedded depth and the wall stiffness. The safety factor increases with increasing the stiffness of the wall, which depends on the embedded depth of the wall. 6

CONCLUSIONS

In this paper, the base stability analysis of excavations in soft clays was evaluated by the 2D finite

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Figure 6. Influence of wall embedded depth below the excavation base (d = 2dc , H /B = 1.0).

element method with shear strength reduction technique. Emphasis has been given to the numerical comparison of the safety factor of base stability obtained by this technique and the upper bound limit analysis method. The influence of the width and the depth of the excavation and the embedded depth of the wall on the base stability are investigated and discussed. The results indicate that the safety factor against base instability increase almost with the ratio of the depth to the width of excavation. Also with increasing the wall embedded depth, the safety factor increases, whereas the increase in the safety factor is smaller in the case of the large clay thickness under the base of excavation than in the case of small clay thickness. It has demonstrated that the finite element method with shear strength reduction technique is an effective method for assessing the safety factor of base stability. It should be pointed out that, the determination of the global base instability of excavations in soft clays in the reduced shear strength FEM analysis should be received much more attention in the future.

Cai, F. & Ugai, K. 2001. Base stability of circular excavation in soft clay estimated by FEM. Proceedings of the Third International Conference on Soft Soil Engineering, HongKong; 305–10. Goh, A.T.C. 1994. Estimating basal-heave stability for braced excavations in soft clay. J Geotech Engrg, ASCE; 120 (8): 1430–6. Griffiths, D.V. 1980. Finite element analyses of walls, footings and slopes. PhD thesis, , UK. Griffiths, D.V. & Lane, P.A. 1999. Slope stability analysis by finite elements. Geotechnique; 49 (3): 387–403. Hata, S., Ohta, H., Yoshida, S., Kitamura, H. & Honda, H. 1985. A deep excavation in soft clay Performance of an anchored diaphragm wall. Fifth Inter. Conf. on Numerical Methods in Geomechanics, Nagoya; 725–730. Matsui, T., San, K.C. 1992. Finite element slope stability analysis by shear strength reduction technique. Soils and Foundations; 32 (1): 59–70. O’Rourke, T.D. & O’Donnell, C.J. 1997. Deep rotational stability of tieback excavations in clay. J. Geotech. Geoenviron. Eng; 123(6), 506–515. Sanematsu, T. & Isobe, T. 1998. Behavior of braced excavations and simulation analysis of eExcavation in soft ground. In: Annual Report, KAJIMA Technical Research Institute, 46. (in Japanese). Sloan, S.W. 1983. Elastoplastic analyses of deep foundation in cohesive soil. Int J Numer Anal Meth Geomech; 7: 385–93. Terzaghi, K. 1943. Theoretical soil mechanics. NewYork: John Wiley. Ugai, K. 1989. A method of calculation of global safety factor of slopes by elasto-plastic FEM. Soils and Foundations; 29 (2): 190–5 (in Japanese). Ugai, K. & Leshchinsky, D. 1995. Three-dimensional limit equilibrium and finite element analyses: a comparison of results. Soils and Foundations; 35 (4): 1–7. Zienkiewicz, O.C., Humpheson, C. & Lewis, R.W. 1975. Associated and non-associated visco-plasticity and plasticity in soil mechanics. Geotechnique; 25 (4): 671–89.

REFERENCES Bjerrum, L. & Eide, O. 1956. Stability of strutted excavations in clay. Geotechnique; 6 (1): 32–47. Cai, F. & Ugai, K. 2000. Numerical analysis of the stability of a slope reinforced with piles. Soils and Foundations; 40 (1): 73–84.

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Numerical modelling of a steel sheet-pile quay wall for the harbour of Ravenna, Italy D. Segato, V.M.E. Fruzzetti, P. Ruggeri, E. Sakellariadi & G. Scarpelli Università Politecnica delle Marche, Italy

ABSTRACT: In this paper, the case of a bulkhead in the Ravenna harbour is presented, where it was possible to monitor the behaviour of a steel sheet pile structure. This monitoring included inclinometer probes in the ground, topographical displacement readings at the top of the bulkhead, vibrating wire load cells on the anchors. On the basis of a good knowledge of the subsoil geotechnical characteristics, both in of the soil stratigraphy and of the constitutive behaviours of the various soils, a finite element numerical model of the problem, both in 2D and 3D, was used to analyze the data from the monitoring and to interpret the observed displacement fields. From this work some conclusions about the selection of the appropriate numerical scheme and of the soil constitutive modelling to better represent the observed behaviour were reached; such conclusions appear useful in the design of these very common marine infrastructures.

1

INTRODUCTION

Despite the large number of applications and high costs involved in the construction of sheet pile bulkheads for harbour facilities, a comprehensive monitoring of the structure behaviour is often lacking, which limits the possibilities to increase our knowledge on the behaviour of such important infrastructures, especially when difficult subsoil conditions are encountered, as is usually the case for harbour environments. On the other hand, the behaviour of a bulkhead for marine applications is often characterized by the simplicity of the boundary conditions, both in of the groundwater regime and of the problem geometry, typically plane, whereas most of the uncertainties for a soil–structure interaction analysis relate to the actual construction and loading histories. In this paper, the case of the so called “ex-Cabot” bulkhead in the Ravenna harbour is presented, where it was possible to monitor the behaviour of a steel sheet pile structure, during construction and subsequent load testing. The growing demand for facilities development within the Ravenna Port area required the construction of new quay walls able to guarantee higher draft levels and the adjustment of older water- ways to allow the transit of larger ships. One recent work is the removal of the ex-Cabot dock bottleneck produced by the progressive widening of Candiano Channel (Figure 1). For this purpose, after a partial demolition of the existing structure, the building of a new quay wall, aligned with the neighbouring docks and able to deepen the water level from −9.50 m to −11.50 m, was planned. The new retaining structure has been constructed using ARBED HZ975A-12/AZ18 sheet pile elements anchored in correspondence to the top concrete beam.

Figure 1. Plan of the Candiano channel at the end of years ’90 and indication of the future channel widening.

Figure 2. Plan view of the quay and location of monitoring instruments.

The design validation process, established by the Port Authority of Ravenna, required monitoring of the structure during the static testing prescribed at the end of construction. Figure 2 shows the monitoring points and the location of measuring devices.

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Figure 3. Cross section of the new ex-Cabot quay wall before the removal of the old wall.

The wall deformations were measured by two inclinometers (IV1-IV2) installed behind the top beam about 1.5 m away from the sheet pile face. At monitoring point 1, where the static tests were to be run, the anchor load was controlled by two vibrating wire load cells (VW) and topographic targets (MT) controlled the top beam displacements. In this paper, the data collected is interpreted and compared to the results of a numerical model. Although the monitoring was not arranged for back analysis purposes, in the present paper the collected data are interpreted and compared to the results of a numerical model in order to if the structural behaviour can be effectively predicted. The main issue is to interpret the complex loading history of the new structure ing for the geometrical evolution of the boundary conditions.

2

PROJECT DESCRIPTION

Figure 3 shows how the sheet piles of the new quay, anchored at 0.5 m above sea level through five-tendon active ground anchors placed at the same spacing of the sheet pile element (1.79 m), extend to a depth of 27.50 m below sea level where a dense silty sand deposit is located. Since the old quay was only partially demolished before the construction of the new wall, in Figure 3 the profile of the ground is represented as expected during the works in the central part of the quay. At the end of the quay the concrete structure was replaced by a breakwater with the side sloping at the same angle. Such a non-horizontal boundary condition determines non-K0 initial conditions throughout the entire quay length. During this preliminary phase, a working surface at +1.5 m above the mean sea level was prepared; after that, the sheet pile wall was constructed, and finally the ground anchors were installed and pulled to the design tension (300 kN).

Figure 4. Time sequence and test setup of the load test stages.

Next a two step excavation was executed: an initial excavation to the depth of −5 m below sea level; • complete excavation to the project depth (−10.50 m below sea level). •

The demolition of the older structure induced a further local deepening of the ground level to 12 m below the mean sea level. After completion of the above phases, two static tests were performed in the area facing the monitoring point 1, involving in the application of a distributed load on the quay. The two tests differ in the shape of the load area and in the amount of surcharge applied; the time sequence of the test program is shown in Figure 4. TEST 1 simulates the maximum expected crane load: a 150 kPa pressure load is applied on a small area. In TEST 2 a smaller pressure (80 kPa) is applied on

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a larger surface in order to charge the active wedge behind the sheet pile. 3

GEOLOGICAL AND GEOTECHNICAL DESCRIPTION OF THE SITE

Several borings, integrated with laboratory and in situ tests, have been carried out in the recent years to define the geological and geotechnical features of the area. With reference to the ex-Cabot site, ground investigation showed homogeneous conditions in the horizontal direction; such homogeneity is consistent with the geological evolution of the area of Ravenna and related to effects of eustatic phenomena in the first 30 m of the ground. Three main layers were identified in the project area; the first 13.50 m from sea level are characterized by a sand layer representing the most recent phase of the still active olocenic regression. From 13.50 m to 26 m below sea level, a layer of soft silty clay with small sandy-silty lenses deposited in marine environment during the extension of the olocenic transgression can be found. Below this layer, dense gray sandy silts and silty sands forming the beginning of the continental sequence are found. From the sea level to the top surface a 1.5 m thick landfill is present. In order to achieve a geotechnical characterization of the mainly incoherent layers cone penetration testing was used. The shear strength of the silty sands determined using the Robertson and Camla (1983) empirical correlation gives a friction angle of 36◦ associated, according to Meyerhof (1956), to loose ÷ medium dense sands. The shear modulus value was assigned by reducing the initial G0 value in relation to the expected deviatoric strain. In order to define the shear modulus G0 , the Robertson (1982) and Rix & Stokoe (1992) correlations have been adopted giving a value of 65 MPa and consequently a 150 MPa tangent stiffness E0 . For flexible retaining walls, deviatoric strains ranging between 10−1 and 3 × 10−2 % (Mair, 1993) are usual, thus a G/G0 ratio of 0.4 can be chosen; the corresponding unloading modulus is 60 MPa. The former value corresponds to a ratio E / qc = 15 ÷ 17 similar to that suggested by Baldi (1989) in a similar strain field (10−1 %). The geotechnical characterization of the sandy silt was basically obtained from laboratory testing. Index properties are described in Table 1. Triaxial compression tests were used to derive strength and deformability parameters. The slope of CSL (Roscoe & Burland; 1968), coinciding in this case with the failure envelope, was equal to 1.25 associated to a critical state angle ϕcs of 31◦ . The Young secant modulus at a stress level similar to that expected in situ ranges between 6 and 7 MPa. 4

Table 1.

NUMERICAL MODEL

In order to interpret the monitored behaviour of the structure during the test, two numerical schemes have

Index properties of clayey soil.

Depth

LL

PL

PI

A

CF

27.8 ÷ 28.3 12.0 ÷ 12.5 13.1 ÷ 13.6 22.5 ÷ 23.0 16.0 ÷ 16.5 21.5 ÷ 22.0 12.0 ÷ 12.5 18.0 ÷ 18.5 16.0 ÷ 16.5

36 51 44 27 43 93 52 42 31

19 24 22 23 23 21 23 23 21

17 27 22 4 20 15 29 19 10

0.59 0.90 1.05 0.50 0.59 1.07 – 1.12 1.00

29 30 21 8 34 14 – 17 10

been implemented using FEM commercial codes: the first in plane strain conditions by means of PLAXISV8 and the second using the three-dimensional 3D Tunnel v1.2 code. Since the monitoring was intended for static testing only, the inclinometer zero reading refers to a date when the excavation depth was already at −5 m below sea level. The 2D scheme was used in order to refine the soil mechanical characterization by comparing results from the numerical analyses with the observed behaviour from the monitoring. Data during the excavation phases (plane strain conditions) required to bring ground level from 5 m to 10.50 m below sea level were used. The 3D scheme was then implemented, based on the soil characterization validated through the plane strain model, and used to predict the load test results at monitoring point 1. 4.1 Model description The 2D and 3D model cross section geometry coincide although a different number of elements was necessary. The mesh spans 40 m upstream and 30 m downstream and vertically extends to 37 m in depth; it is restrained horizontally at the sides and fixed at the base. The 3D model, reproducing a whole bulkhead cell, measures 50 m in the longitudinal direction. 4.2 Construction phases All construction phases as described in Table 2 were reproduced in the analyses, in order to achieve a realistic simulation of the entire load history. 4.3 Material properties Soil behaviour was modelled using the hardening soil (HS) model, an elastic-plastic model with deviatoric isotropic hardening, which can for pre-failure non linearity and for changes in stiffness when the direction of the stress path is reversed (Brinkgreve; 2002). In the model the elastic behaviour depends on the effective stress state through an hyperbolic relationship.

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For the silty sands, referring to Baldi (1989) for NC ref and OC sands, Eref 50 = 12 MPa and Eur = 60 MPa was adopted. Regarding the oedometric modulus, according to Lunne and Christophersen (1983), a value of Eref oed = 14 MPa = 4qc was chosen. Values of power m (Brinkgreve; 2002) were chosen according to the soil type. The values of mechanical parameters for the clayey layer were chosen on the basis of drained and undrained triaxial test results (Figure 5).The unloading ref modulus Eref ur was set equal to three times E50 . The soil model parameters are summarized in Table 3 where the assumed Mohr Coulomb (MC) model and its basic parameters for the landfill material are also given. With reference to the structural materials, the sheet pile wall was modelled as an elastic beam; a Young modulus of 210 GPa, a Poisson coefficient ν equal to 0.15 and a unit weight of 78 kN/m3 were assumed for the steel; Young modulus of 30 GPa, Poisson coefficient ν of 0.2 and unit weight of 25 kN/m3 for the reinforced concrete. The node-to-node steel anchors are 14 m long, at a 20◦ angle. A geotextile element was added at the end of each anchor in order to simulate the grouted anchor foundation. The grout elements simulated with the geotextile are assumed to have a normal stiffness EA, of 6.6 × 105 kN/m; a pre stress load equal to 300 kN per anchor was imposed during the staged construction. In the end, considering the extension of the surface between sheet piling and soil in the longitudinal direction, an interface friction equal to the Table 2.

soil friction angle was assumed (value of the interface coefficient R in Plaxis set to unity). The analyses were developed in of effective stresses assuming a drained behaviour both for non cohesive and cohesive materials.

5

DISCUSSION ON THE MONITORING DATA AND ANALYSIS RESULTS

The first set of analysis was run in plane strain conditions to reproduce the effect of the excavation from 5 m below the mean sea level, corresponding to the zero reading of the IV2 inclinometer, to 10 m below sea level. In Figure 6, measured and computed displacements referred to this stage are compared. The measured displacement profiles (represented by a line connecting the experimental points) are satisfactorily reproduced by both numerical schemes to a

2D and 3D numerical simulation stages.

Description

2D

3D

K0 stress state generation Initial ground profile Sheet pile installation Anchors at the design tension load Excavation to −5 m below s.l. Fill Excavation to −10.5 m below s.l. Testing load 1 (2 areas 5 × 2 m2 150 kPa ) Eliminate testing load 1 Testing load 2 (area 10 × 15 m2 80 kPa) Eliminate testing load 2

Phase 0 Phase A Phase B Phase C Phase D Phase E Phase F – – – –

Phase 0 Phase 1 Phase 2 Phase 3 Phase 4 Phase 5 Phase 6 Phase 7 Phase 8 Phase 9 Phase 10

Table 3.

Depth Model γ Eref 50 Eref oed Eref ur m ϕ’ c’ K0

Figure 5. Stress – strain relation comparison between experimental results and numerical simulation of triaxial CIU for clayey soils.

Soil model properties.

[m] [−] [kN/m3 ] [MPa] [MPa] [MPa] [−] [◦ ] [kPa] [–]

Landfill

Sands

S.Clay

D. Sands

+1.5 ÷ 0 MC 18 10 – – – 26 – 0.560

0 ÷ − 13.5 HS 18 12 14 60 0.5 36 – 0.420

−13.5 ÷ 26 HS 19 6.3 3.1 25 1 31 – 0.485

−26 ÷ 40 HS 19 12 6 40 1 34 – 0.422

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depth of 22 m with a maximum displacement located at −48 m from the top of the inclinometer. The maximum calculated displacement values are about 10 ÷ 15% less than the measured ones. Below 22 m the calculated displacement profiles diverge from the real deformation profile exhibiting a stiffer behaviour. In Figure 7 the displacements measured at point 1 and those computed by the 3D model are compared with reference to the phases involved in the load test (7 and 9), where non plane strain conditions occur; it is worth to note that the 2D model give for the same phases a maximum displacement 9 and 4 times higher respectively. From the 2D model a deeper pressure bulb resulted in comparison with the 3D model, inducing an high stress change in the softer lower clay strata. During TEST 1 (phase 7 in Table 1), the position of the maximum incremental displacement point is not correctly reproduced by the 3D model although a good estimation of the overall deformation is achieved. The former inconsistency can be partially explained by the limited extent of the loading area which is comparable with the distance from the reference vertical and the sheet pile; nevertheless an overestimation of the silty clay compressibility at low stress is still appreciable. TEST 2, which was carried out on a wider load area, is reproduced satisfactorily by the model, concerning both the incremental displacement pattern and the maximum value which appears overestimated of about 15% during the loading phase. The displacement predicted in excess during the tests, develop inside the

lower part of the silty clays which again in this occasion, contrary to phase 6, exhibit a softer response. Outside the load area, the sheet pile is influenced by the induced stress on a wide area (Figure 8). The topographic survey of the beam horizontal displacement (Figure 9) concerning the test phases, appears in agreement with the anchor load measured by the VW1 cell; the trend would confirm an elastic behaviour of the anchor system except for the “softening” recorded far from loading stages. In particular, after the increment due to the phase 5 excavation, the measured load reverted to the pre-stress value. The former load seem in turn to be overestimated by the model (Table 4). The displacement patterns of the topographic targets (MT) allow a further comparison between numerical predictions and measured deformations; Figure 10 shows that the 3D model is able to reproduce the trend of both vertical and horizontal displacement of the

Figure 7. Comparison of computed and measured horizontal displacements of the wall from phase 6. Table 4. loads.

Figure 6. Comparison of computed and measured horizontal displacements of the wall from phase 5 to phase 6.

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Comparison of computed and measured anchor 3D model [kN]

VW1 cell [kN]

Phase no.

Total

Incremental

Total

Incremental

6 7 8 9

342 350 350 429

–

306 323 315 364

– 17 −8 49

8 0 79

concrete dock beam for both the first and the second load tests. 6

Figure 8. Predicted deformed shape of the wall for the entire quay cell from phase 6 to 9.


An assessment of the effects produced by a full scale load test on a sheet pile bulkhead was achieved through F.E.M. analysis which ed for both soil behaviour and three dimensional aspects. The problem was characterized by a complex load geometry which couldn’t be handled in plane strain conditions without heavily over-predicting the final deformed shape. An elastic-plastic model with deviatoric isotropic hardening was used in order to model the wide range of stress and strain induced behind the wall both in extension (excavation phases) and in compression (test phases). Nevertheless, a lack of accuracy is observed in the lower part of the soft silty clay layer, where a stiffer response (compared to the real structure) is predicted by the model under small incremental unloading stress and vice versa a softer behaviour is shown under small compression stress. Such behaviour seems to be linked to the inability of the non linear model to deal with small strain deformability. ACKNOWLEDGEMENTS Authors wish to acknowledge Port Authority of Ravenna and Coop. C.M.C. for their assistantship and for having made available the data from the monitoring.

Figure 9. Comparison of measured anchor load trend and topographic target displacement.

REFERENCES

Figure 10. Comparison of computed and monitored displacements (MT) of the wall from phase 6 to 9.

Baldi, G. & al. 1982. Design Parameters for Sands from T. Proc. 2nd Eur. Symp. on Penetration Test, ESOPT II Amsterdam 425–432. Rotterdam: Balkema, Brinkgreve, R.B.J. 2002. Plaxis v.8, Material models manual, Rotterdam: Balkema. Clayton C.R.I. 1995. “The Standard Penetration Test (SPT): Methods and use” CIRIA Report n◦ 143. Lunne,T. & Christophersen, H. P. 1983. Interpretation of cone penetrometer data for offshore sands. Proceedings of the Offshore Technology Conference, Houston, Texas, Paper No. 4464, 191–192. Meyerhof, G.G., 1956. Penetration tests and bearing capacity of cohesionless soils. Journal of Geotechnical Engineering, ASCE 82, 1–12. Rix, G.J. & Stokoe, K.H.1992. Correlation of initial tangent modulus and cone resistance. Proceedings of the International Symposium on Calibration Chamber Testing, Postdam 1991, 351–362. New York: Elsevier. Robertson, P.K. 1990. Soil Classification using the Cone Penetration Test”. Canadian Geotechnical Journal, 27 (1), 151–158. Robertson, P.K. & Camla, R.G. 1983. Interpretation of cone penetration tests. Part 1: sand. Canadian Geotechnical Journal, 20 (4), 719–733.

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Numerical modelling of spatial ive earth pressure in sand M. Achmus, S. Ghassoun & K. Abdel-Rahman Institute of Soil Mechanics, Foundation Engineering and Waterpower Engineering Leibniz University of Hannover, Hannover,

ABSTRACT: The calculation of earth pressures acting on underground structures is a central problem in soil mechanics. In special cases spatial earth pressure problems also have to be considered. In order to investigate the spatial ive earth pressure acting on rigid walls in sand, numerical simulations with the finite element method (FEM) were carried out. A hypoplastic material law was adopted to for the complex stress-strain behaviour of sand. Different wall geometries with varying spatiality ratios (breadth to height) and different deformation modes of the rigid wall were considered. The results show a high dependency of the earth pressure coefficients on the wall spatiality ratio. For a smooth wall, a good agreement with the approach of the German standard DIN 4085 is obtained. The wall displacement necessary to reach the maximum earth pressure is also dependent on the wall geometry, with the displacement decreasing with increasing spatiality ratio. The analysis of the earth pressure distributions shows that the earth pressure increase due to spatiality occurs almost only in the lower half of the wall.

1

INTRODUCTION

The calculation of ive earth pressures acting on underground structures is a central problem in soil mechanics. The horizontal pressure acting on a fixed rigid wall is termed earth pressure at rest. When the wall is moved towards the soil the pressure increases until it reaches a maximum, which is called ive earth pressure. For a wall with height H and a breadth B much greater than H the two-dimensional (plane strain) solution of the earth pressure problem applies, with the ive earth pressure force (Ep ) in non-cohesive soils to be determined by the following equation:

where γ = unit weight of soil; H , B = height and breadth of the wall; and kp = ive earth pressure coefficient. The earth pressure coefficient in the twodimensional case is dependent on the angle of internal friction of the soil ϕ and on the wall friction angle δ. According to the German standard DIN 4085, Equation (1) is valid for parallel movement of a wall. For other deformation modes, the resulting force is smaller. For a rigid wall rotating around the top the reduction factor can be estimated to about 0.67 and for a rotation around the toe between 0.5 and 0.67 (DIN 4085). The ive earth pressure coefficient for the threedimensional case is known to be greater than for the two-dimensional case. According to an approach in

DIN 4085 based on the investigations of Weißenbach (1961), the increase is dependent on the spatiality ratio B/H and on the angle of internal friction ϕ. Open questions regarding spatial ive earth pressure concern the exact value of the earth pressure coefficient, the wall displacement at peak state and the earth pressure distribution behind the walls. 2

OVERVIEW OF PREVIOUS WORK

Experimental investigations were first done by Weißenbach (1961&1983) on model walls (I-beam) embedded in sand. He developed a semi-empirical formula to calculate the increase of the earth pressures due to spatial conditions. Ovesen (1964) conducted an extensive series of model tests on anchor plates to investigate the 3-D effects. His tests showed that ive earth pressures are higher than those predicted using conventional methods. Brinch Hansen (1966) developed a method for correcting the results of conventional ive pressure theories behind anchor plates to for 3-D effects. He also found that the geometry of the ing structure is an important factor for the analysis of 3-D ive earth pressure. Neuberg (2002) established an experimental model of a retaining wall (I-profile) in order to determine the 3-D ive earth pressure. He also performed numerical simulations using a discrete element model. Based on the results he proposed a mobilization function for the dependence of ive earth pressure on the wall displacement. A number of researchers developed theories for determining spatial ive earth pressure. Blum

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Table 1.

Index properties of Karlsruhe medium sand.

Unit weight of the grains, kN/m3 D10 , mm D60 , mm Uniformity coefficient, Cu Min. void ratio, emin Max. void ratio, emax

26.5 0.240 0.443 1.85 0.53 0.84

Table 2. Input parameters for hypoplastic material law for Karlsruhe Sand. Figure 1. Spatial ive earth pressure increase with spatiality ratio for different friction angles (Benmebarek et. al. 2008).

φc

grain stiffness hs ed0

30.0◦ 5800 MN/m2

(1932) developed a failure mechanism, as in Coulomb’s theory, by creating a three-dimensional flat failure surface. Based on the log spiral method, a spreadsheet was developed by Duncan & Mokwa (2001) to calculate the spatial ive earth pressure. Soubra & Regenass (2000) developed a method to calculate 3-D ive earth pressure coefficients based on the upper-bound method of limit analysis. Jung (2007) used a finite element simulation to investigate the behaviour of I-profile walls, applying an elasto-plastic material model for sand. His numerical results were calibrated with in-situ measurements of a large scale model made at the University of Texas (Briaud, J. & Lim,Y. 1999). With the results, he derived a mobilization function based on subgrade theory. Benmebarek et al. (2008) carried out a numerical study of 3-D ive earth pressure for parallel movement of rigid walls by means of a finite element simulation. They used a linear elastic-ideal plastic constitutive model with Mohr-Coulomb failure criterion and an associated flow rule. Their results were presented in design tables relating different geometrical input parameters and 3-D ive earth pressure coefficients. Figure 1 summarizes the numerical results showing the decay of the spatial ive earth pressure factor (µ = kp3D /kp2D ) by increasing the spatiality ratio (B/H ) until it reaches the 2-D ive earth pressure conditions.

ec0

N

ei0

α

β

0.53 0.84 0.25 1.00 0.13 1.05

a material, taking the current stress state and void ratio into . This constitutive material model has the following advantages: 1) It is able to describe the behaviour of granular materials with one tensorial equation, i.e. there is no need to distinguish elastic or plastic regions. 2) Hypoplasticity describes loading, un- and reloading processes without any additional material information. 3) The material law takes the effects of the stress level and the relative density on the soil behaviour and in particular on the shear strength into . The constitutive equation has eight constants, which are measured in a simple way by standard tests in soil mechanics. The eight constants used in the finite element analysis are listed inTable 2. For a detailed discussion of the mathematical background and physical significance of the input parameters for the constitutive model reference is made to Herle (1997). More details concerning this material law are given in Herle & Gudehus (1999).

4

DESCRIPTION OF THE NUMERICAL MODEL

4.1 Model features 3

MATERIAL BEHAVIOUR

The numerical modelling was done for Karlsruhe medium sand material. The behaviour of Karlsruhe sand is well documented. It consists mainly of sub round quartz grains. The index properties of the sand are given in Table 1. The modelling of the material behaviour of the soil is of course of crucial importance for the quality of the computation results. The computations were executed for each movement mode using the hypoplastic material law developed for sandy soil. Hypoplasticity was first developed at the University of Karlsruhe. Using a tensorial function, it connects stress and strain rate in

For the numerical investigation with the finite element method (FEM), the ABAQUS program was used. The main aspects of the modelling are listed below:

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– Due to symmetry, only half of the model was discretized. The dimensions of the three-dimensional model area were varied in order to fit different wall dimensions (breadth/height). The geometrical model for B = 10 m is shown in Figure 2. It was verified that with the model dimensions used the calculated behaviour of the wall is not influenced by the boundary conditions. – The soil was modelled with 8-noded solid elements (Fig. 3). The interaction behaviour in the boundary

Table 3.

Input parameters for primary stress state.

eo void ratio

no porosity

D relative density

ϕ (◦ ) friction angle

γ (kN/m3 ) unit weight

Ko earth press.

0.55 0.65 0.75

0.355 0.394 0.428

92.0% 56.8% 26.0%

40.0 36.0 33.0

17.1 16.1 15.1

0.46 0.48 0.50

Figure 2. Geometrical model used.

Figure 4. ive earth pressure coefficient as a function of the wall displacement for parallel movement (H = 10 m, B/H = 1). Figure 3. Finite element mesh.

–

–

–

–

surface between wall and soil was modelled using interface elements. In the front surface of the model area twelve different smooth rigid walls were specified (Fig. 2). By moving certain walls, different breadth-to-height values could be easily realized. The earth pressure on the wall elements moved was calculated by integrating the horizontal soil stresses behind the wall elements. Regarding the mode of wall movement, parallel movement, rotation around the toe of the wall and rotation around the top of the wall were examined. For the calculations described here, the coefficient of friction between the soil and the walls was set to zero, i.e. a wall friction angle δp = 0 was considered. Geometrical non-linearity was also implemented to for the effect of the relatively large deformations necessary to mobilize ive earth pressures. Three different initial void ratios (e0 = 0.55, 0.65, 0.75), i.e. three different relative densities, were considered as shown in Table 3.

The simulation process is executed in stages. First, the primary stress state using own weight of the soil medium is generated. Then the required rigid walls are moved gradually under three different basic wall movements as mentioned before. In Table 3 friction angles for peak states ϕ of the sand are also given. In the hypoplastic formulation the shear strength is stress-dependent and is not described by an explicit parameter.Thus, the friction angles given

were derived by numerical simulation of a direct shear test under vertical stresses of 100, 200 and 400 kN/m2 . These values were used for the conventional calculation of earth pressures with regard to Equation (1). In a similar way, the K0 -values given were derived by a numerical simulation of the stress state under the own weight of the soil using the hypoplastic material law.

4.2 Model verification The reference wall examined with the finite element method (FEM) exhibits a height (H ) of 10 m and a breadth (B) of 10 m (B/H = 1). Figure 4 shows the relationship between the ive earth pressure coefficient (kph ) and the normalized displacement (U /H ) under parallel movement. The earth pressure coeffifem cient at peak (kp3D ) ranges from 8.0 for dense sand (e0 = 0.55) to 5.0 for loose sand (e0 = 0.75). The ive earth pressure for plane strain confem ditions (kp2D ) was calculated from the model with B = 260 m (i.e. B/H = 26). The results were compared to the equations given in the German standard DIN 4085. According to this standard, the earth pressure coefficient calculated for plane strain conditions should be multiplied by a correction factor (µDIN ) to calculate the spatial ive earth coefficient as follows:

The obtained results are summarized inTable 4. Evidently, for the investigated case of parallel movement

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Table 4. Comparison of earth pressure coefficients from numerical simulation and from the German standard DIN 4085. Initial void ratio

Friction angle ϕ (◦ )

DIN (kp2D )

0.75 0.65 0.55

33.0 36.0 40.0

3.39 3.85 4.60

µDIN

fem kp2D

fem kp3D

µ = kfem p3D / fem kp2D

1.39 1.44 1.50

3.55 4.00 5.10

5.0 6.0 8.0

1.41 1.50 1.56

Figure 5b. Spatial earth pressure coefficient (µ) as a function of the spatiality ratio (B/H ) for rotation around the top of the wall.

Besides the numerical simulation results, the results of the DIN 4085 approach and – for parallel movement only – of Benmebarek et al. (2008) are also presented. From these figures, the following can be concluded:

Figure 5a. Spatial earth pressure coefficient (µ) as a function of the spatiality ratio (B/H ) for parallel movement.

of a smooth rigid wall the agreement of the numerical results and the DIN 4085 approach is rather good. The coefficients for plane strain conditions obtained from the numerical simulation are slightly higher than the coefficients obtained from Equation (1), which of course is a result of the different assumption regarding the material behaviour. However, the µ factors of both methods coincide very well, with deviations of less than 4%. From the results of Benmebarek et al. (2008) obtained with a simple elasto-plastic material law (see Section 2) larger µ values between 1.65 and 1.90 were reported. 5

PARAMETRIC STUDIES

In the following numerical results regarding the spatial ive earth pressure coefficient, the wall displacement at peak state and earth pressure distribution will be presented, taking different spatiality ratios and wall deformation modes into . 5.1

Spatial ive earth pressure coefficient

The numerical modelling was done for a certain wall height (H ) of 10 m and the wall breadth (B) was varied from 10 m to 260 m. In the Figures 5a to c the dependence of the correction factor µ for spatial ive earth pressure and thus of the ive earth pressure coefficient (kp ) on the spatiality ratio (B/H ) is shown for the three considered wall movement modes.

1) The limiting value of the ive earth pressure depends on the wall breadth, i.e. on the spatiality ratio, for the three different wall movements. With a higher spatiality ratio (n = B/H ), the ive earth pressure coefficient (kph ) decreases until it reaches the standard 2-D ive earth pressure condition (µ = 1). 2) The parallel wall movement gives higher µ values – and thus larger ive earth pressure coefficients – than rotation around the top of the wall and also around the base of the wall. 3) The numerical results presented here match the German standard DIN 4085 very well, whereby the results of Benmebarek et al. (2008) shown in Figure 5a overestimate the spatial ive earth pressure compared to the numerical results. 5.2 Wall displacement at peak state The normalized wall displacement at peak (Up /H ) obtained from the numerical modelling are shown in the following Figures 6a, b for parallel wall movement and wall rotation around the top. For rotation around the toe a maximum ive earth pressure coefficient could not be reached in the numerical simulation, since even with very large displacements no peak stress was reached. For the presentation in Figure 5c the earth pressure at a wall toe displacement of 20% of the wall height was taken. Figure 6a shows that the wall displacement (Up ) required to mobilize the maximum ive earth pressure decreases by increasing the spatiality ratio (B/H ). Also, by increasing the initial void ratio (e0 ), the wall displacement required to mobilize the maximum ive earth pressure increases. Similar results are shown in Figure 6b for wall rotation around the top of the wall, which confirms that the wall displacement at peak state for spatial ive earth pressure is higher than in the 2-D ive earth pressure case.

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Figure 5c. Spatial earth pressure coefficient (µ) as a function of the spatiality ratio (B/H ) for rotation around the toe of the wall.

Figure 7a. ive earth pressure distributions in the centre of the wall for medium dense sand (e0 = 0.65) and parallel wall movement.

Figure 6a. Normalized wall displacement at peak (Up /H ) as a function of the spatiality ratio (B/H ) for parallel movement.

Figure 6b. Normalized wall displacement at peak (Up /H ) as a function of the spatiality ratio (B/H ) for rotation around the top of the wall.

5.3 Earth pressure distribution at peak state The stress distributions on the wall at peak state for different spatiality ratios is given in Figures 7a-c for the centre line of the wall, i.e. the symmetry axis. For parallel movement (Fig. 7a) the earth pressure distribution is almost linear in the upper half of the wall and is not subject to change with spatiality ratio. In the lower part, the earth pressures are different, with a non-linear distribution and a large increase in the lower third particularly for B/H = 1.

Figure 7b. ive earth pressure distributions in the centre of the wall for medium dense sand (e0 = 0.65) and wall rotation around the top.

Regarding the rotation around the top of the wall (Fig. 7b), the earth pressure starts from very small values in the upper half of the wall. Differences between the distributions for different spatiality ratios are limited to the lower third of the wall. The maximum earth pressures at the wall toe are greater than in the case of parallel movement of the wall.

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the breadth to height ratio (B/H ) and only to a minor degree on the relative density of the sand. The wall displacement at peak state was found to decrease with increasing B/H value. Thus, for the mobilization of the larger 3-D ive earth pressures, larger wall deformations than in the 2-D case are necessary. The consideration of stress distributions showed that the stress increase due to spatial conditions occurs almost completely in the lower half of the wall. For forthcoming research, a parametric study of different soil-wall friction angles and different wall heights will be undertaken. REFERENCES

Figure 7c. ive earth pressure distributions in the centre of the wall for medium dense sand (e0 = 0.65) and wall rotation around the toe.

For rotation around the toe of the wall (Fig. 7c), the earth pressure increases up to its maximum value at almost 2/3 of the wall height and then decreases towards the toe of the wall. The differences due to different spatiality ratios occur mainly between 1/2 and 3/4 of the wall height. For the 2-D case, the resultant ive earth pressure force for rotation around the wall top amounts to about 80% of the force for parallel wall movement and for rotation around the toe to about 70%. Thus, the reduction factors recommended in DIN 4085 (see Section 1) lie on the safe side. For 3-D cases, reduction factors between 82 and 86% (rotation around the top) and between 70 and 75% (rotation around the toe) were obtained with the numerical simulations. This means that the effect of the wall deformation mode on the resultant earth pressure is similar in 3-D and 2-D cases. 6


For the investigations presented a numerical model was developed to simulate spatial ive earth pressure problems in sand. The computations were executed using the hypoplastic material model developed for granular materials like sand. It was proved by the numerical results that for the investigated case (smooth wall) the 3-D approach given in the German standard DIN 4085 produces very reasonable results. The spatial ive earth pressure factor µ depends mainly on

ABAQUS 2008. Manual, Version 6.8, Simulia, Providence, RI, USA. Benmebarek, S. et. al. 2008. Numerical evaluation of 3D ive earth pressure coefficients for retaining wall subjected to translation, Computers and Geotechnics (35), 47–60. Blum, H. 1932. Wirtschaftliche Dalbenformen und deren Berechnung, Bautechnik 10(5). Brinch Hansen 1966. Comparison of Methods for stability Analysis, Three-dimensional effect in stability analysis resistance of a rectangular anchor slab.Danish Geotechnical Institute Bulletin, No.21. Briaud, J. & Lim, Y. 1999, Tieback walls in sand: numerical simulations and design implications, Journal of Geotechnical and Geoenvironmental Eng. ASCE. DIN 4085. 2007. Berechnung des Erddrucks, Deutsches Institut für Normung. Beuth Verlag. Duncan, M., Mokwa, R. 2001. ive Earth Pressures – Theories and Tests, ASCE Journal of Geotechnical and Geoenvironmental Engineering, Vol. 127, No. (3), 248–257. Herle, I. 1997. Hypoplastizität und Granulometrie von Korngerüsten, Veröffentlichungen des Instituts für Bodenmechanik und Felsmechanik der Universität Karlsruhe, Heft 142. Herle, I., Gudehus, G., 1999. Determination of a hypoplastic constitutive model from properties of grain assemblies. Mechanics of Cohesive-frictional Materials, 4: 461–486. Neuberg, C. 2002. Ein Verfahren zur Berechnung des räumlichen Erddruckes vor parallel verschobenen Trägern, Veröffentlichungen des Instituts für Geotechnik, Technische Universität Dresden, Heft 11. Ovesen, NK. 1964. Anchor slabs, Calculation methods and model tests, Danish Geotechnical Institute Bulletin, No. 16, 5–39. Soubra, A-H, Regenass, P. 2000. Three-dimensional ive Earth Pressures by Kinematical Approach, ASCE Journal of Geotechnical and Geoenvironmental Engineering, Vol. 126, No. (11), 969–978. Stefan, J. 2007. Nichtlinearer Horizontaler Bettungsmodulansatz für Trägerbohlwände in mitteldicht gelagertem Sand, Veröffentlichungen des Instituts für Bodenmechanik und Grundbau, Technische Universität Kaiserslautern. Weißenbach, A. 1961. Der Erdwiderstand vor schmalen Druckflächen, Mitteilung des Franzius-Instituts für Grund- und Wasserbau, Hannover. Weißenbach, A. 1983. Beitrag zur Ermittlung des Erdwiderstands, Bauingenieur 58, 161–173.

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Practical numerical modelling for very high reinforced earth walls A. Mar, D.M. Tonks & D.A. Gorman Coffey Geotechnics Limited, Manchester, UK

ABSTRACT: The authors have carried out numerical studies in connection with several very high walls, ranging from 19–80 m tall. To date, few such studies have been reported and comparison of analysis results with full scale wall instrumentation data has been infrequent, or where done, it has generally been somewhat lacking in data and specific modelling details. There are however, a few notable cases giving detailed methodologies and comparison of results with well instrumented walls up to about 4 m high. Following similar methodologies for the very high walls considered here; some modelling challenges have been encountered. Novel ways to overcome this have been explored enabling the influence and sensitivity of the walls to a range of key parameters to be investigated. The models have reached the stage where they can thus provide valuable assistance to design and prediction; including an understanding of performance as the walls are constructed and data is obtained. 1

INTRODUCTION

1.1 Background This paper addresses practical numerical modelling for a range of very high, multi-tiered geogrid reinforced earth (RE) walls, the highest being some 80 m tall arranged in up to four tiers. The wall length varies depending on the particular lift; some sections being well over 500 m long. Initial design of the reinforcement grid layout had been carried out by the designers using standard software using the Bautechnik design method. This is a limit equilibrium approach based on the tie back wedge method. The purpose of this work was to provide an independent geotechnical assessment of the RE walls to gain insight into the structural behaviour of the systems and to predict likely movements to address the serviceability limit state (SLS). Modelling runs have been performed using finitedifference based FLAC (Itasca Consulting Group 2001). The deformed shape of the three tier reinforced wall is shown in Figure 1. The facing blocks would be tied mechanically to the geogrid using full strength connectors. This has been idealised as shown in Figure 2. This paper explores what can practically be done in a design office using first order models and this forms the basis of ongoing work using more sophisticated approaches. 2

to investigate the influence of key model parameters with the aim of developing a model and set or sets of parameters that can predict wall behaviour generally in line with observations and in particular, the relatively low wall displacements observed at Dibba Road. Working stress analysis runs using FLAC have been carried out. The analyses assume fully drained, dry conditions. The backfill mesh comprised 3 rows of elements between each geogrid. Stress-dilatancy has been explored in some of the analyses (Rowe 1962, Bolton 1986, Simoni & Houlsby 2006). The backfill soil and geogrid elements reported here are linear elastic-perfectly plastic. Stress redistribution due to local yielding of the backfill has taken place during the staged construction with an active wedge of plastic points developing behind the wall. The geogrid elements are tension only cable elements

SINGLE TIER ANALYSES

2.1 18 m high Dibba Road RE wall The purpose of this study was to carry out a range of sensitivity analyses using ideal plastic Mohr-Coulomb

Figure 1. Deformed mesh of 56 m high wall (Scaled 20x).

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Table 1.

Sensitivity analysis matrix for single tier work.

2 month geogrid stiffness parameters Higher QC geogrid stiffness parameters

φbackfill of 38◦

φbackfill of 50◦

ψbackfill of 0

ψbackfill of 20◦

ψbackfill of 0

ψbackfill of 20◦

ψbackfill of 0

ψbackfill of 20◦

ψbackfill of 0

ψbackfill of 20◦

φbackfill backfill effective angle of shearing resistance. ψbackfill backfill angle of dilation.

Figure 2. Facing block idealisation detail.

and are fully bonded to the adjacent soil zones. More advanced nonlinear work is ongoing. Extensive review of the literature, notably the work of Bathurst et al. (2001) explores the use of more sophisticated nonlinear soil and geogrid models to take of stress-dependency of stiffness but space precludes inclusion in this paper. (Hatami & Bathurst, 2005, 2006a, b) tested the shear strength of the block-block interface and measured a cohesion of 46 kPa and an angle of shearing resistance of 57◦ . A back of wall interface angle of shearing resistance of 27◦ was used in the analysis. This is based on an interface strength of (2/3)*tan(38◦ ) where 38◦ is the lower bound angle of shearing resistance of the backfill. A series of analyses have been performed to assess the sensitivity of the results to changes in the input parameters. More runs were carried out using the input parameters that the model proved to be sensitive to. Initial analysis runs indicated that the analysis results, in particular the magnitude of wall displacements were sensitive to the following parameters: backfill angle of shearing resistance, backfill angle of dilation and geogrid stiffness. An initial sensitivity analysis of backfill stiffness was carries out in which Young’s moduli of 150 MPa and 1000 MPa were considered. The difference between maximum total and horizontal wall displacements was less than 5% and 2% respectively. For this reason further sensitivity analysis of the backfill stiffness was not investigated. For this reason the sensitivity analysis matrix shown in Table 1 excludes backfill stiffness as a variable. As can be seen a total of 8 sensitivity analyses have been carried out with linear elastic-perfectly plastic models. 2.2 Results Figure 3 indicates that increased angles of shearing resistance and dilation of the backfill significantly reduces the predicted maximum wall displacement, with a minimum value of 60 mm comparable to the 50 mm actually measured. This trend was also observed for the 2-month geogrid stiffness however the minimum value was around 100 mm.

Figure 3. Peak resultant displacement predictions with angles of friction and dilation (QC geogrid stiffness).

Figure 4 shows the peak axial forces generated in selected geogrids from the eight sensitivity analyses of Table 1. It can be seen that the peak geogrid forces are sensitive to the backfill angle of shearing resistance and to a lesser extent the dilation angle. However, the results of the investigation appear to show only minor sensitivity to the geogrid stiffness. Figure 5 shows the force distribution along two selected layers of geogrid, at 3 m above the toe of the wall. The magnitude of the angle of dilation appears to influence the pattern of axial force distribution along the geogrid. The analyses with high dilation angle produce considerably smoother distributions. The areas under these curves are analogous to the work done by the geogrid in resisting the active pressures developed behind the wall. The stronger the backfill soil the lower the active pressures developed behind the wall and hence the less work the geogrid has to do to resist lateral movement. This is evidenced by the smaller areas below the curves for the stronger soil.

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Table 2.

Summary of analyses (56 m high RE wall).

No. of element rows between Analysis geogrids 1 2 3 4 5

Backfill/ wallfill stiffness Run-time* E (MPa) (Hours)

3 1000 2 1000 2 150 ‘Super-geogrids’ 150 2 150

48 (Tier 1 only) 15 (Tier 1 only) 97 (3 tiers) 14.5 (3 tiers) 12 (3 tiers with surcharges replacing tier 2 and tier 3)

*PC with a 2.4 GHz processor and 2 Gb RAM.

Table 3.

Figure 4. Peak geogrid force distribution with height above toe of wall (QC geogrid stiffness).

Mesh Sensitivity Studies (Analyses 1–2).

No. of elements between Analysis geogrids

Peak horizontal displ. of T1* (mm)

Peak vertical displ. of T1* (mm)

Peak geogrid force (kN/m)

1 2

10.0 12.0

12.0 12.5

7.0 6.0

3 2

*T1 = Tier 1.

Strength and stiffness properties for the geogrids have been provided by the suppliers. From the single tier analyses it was found that using the higher QC geogrid stiffnesses gave the best predictions to the displacements observed for the Dibba Road wall. A total of 5 analyses have been performed as summarised in Table 2. 3.2 Analyses 1& 2 (Tier 1 only)

Figure 5. Geogrid force distribution 3m above toe (QC geogrid stiffness).

3 THREE TIER ANALYSES 3.1 56 m high RE wall The single tier sensitivity analyses indicated that a linear-elastic perfectly-plastic Mohr-Coulomb soil model could match the supposed low horizontal wall displacements of around 50 mm for the 18 m high Dibba Road wall, by invoking high strengths (φ = 50◦ , ψ = 20◦ ) and stiffnesses for the backfill and geogrid. With reference to Figure 1 the upper, middle and lower tiers are referred to as 3, 2 and 1 respectively. Tiers 3, 2 and 1 possess total geogrid axial capacities of 2.03, 1.95 and 4.41 MN/m respectively.

As this model is approximately 3 times geometrically bigger than the Dibba Road model and due to the closer spacing of the geogrids in the bottom tier, considerably more elements are required thus increasing the computer run-times for the analysis. Therefore, a number of mesh refinements were trialled in order to reduce the number of elements as far as possible without significantly affecting the accuracy of the results. These refinements included reducing the number of rows of elements between each geogrid from three to two. With reference to Table 2, it can be seen that the run time for the coarser mesh of Analysis 2 is more than 3 times quicker than the finer mesh of Analysis 1. For the bottom tier, the difference between peak predicted values of displacements and geogrid forces were found to be within 10%. As it was likely that the run-time would increase non-linearly for the second and third tiers, element tyings were incorporated to enable coarse grids to be used in non-critical areas, i.e. the foundation and un-reinforced backfill regions. This further reduced the computational run-time for tier 1 to 10 hours.

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Table 4. Peak wall displacements for the three tier model (Analysis 3).

Table 5. Peak geogrid forces for the three tier model (Analysis 3).

Horizontal (mm) Vertical (mm) Construction stage Tier 1 Tier 2 Tier 3 Tier 1 Tier 2 Tier 3 Tier 1 Tier 2 Tier 3

17.5 40.0 60.0

– 60.0 90.0

– – 60.0

7.5 10.0 20.0

– 60.0 80.0

– – 80.0

Maximum axial force in geogrid (kN/m) Construction stage

Tier 1

Tier 2

Tier 3

Tier 1 Tier 2 Tier 3

9.0 15.1 18.5

– 23.0 24.2

– – 29.2

As shown in Table 3, the results for the two analyses show that the coarser mesh produces similar displacement and force predictions for tier 1. The coarser mesh is predicting horizontal movements that are 20% higher inferring that more conservative predictions will be obtained compared to using a finer mesh. However the peak axial force mobilised in the geogrid is 15% lower for the coarser mesh and so this must be borne in mind when analysing the results for the three-tier model. 3.3 Analysis 3: (Tiers 1, 2 & 3) This model exhibited analysis difficulties during the construction of the final 2.2 m lift of the upper tier 3. The model therefore falls 2.2 m short of the 56 m height and the 20 kPa traffic surcharge was not yet applied. The problem manifested as excessive distortion of the soil element at the toe of tier 3. Table 4 shows the peak horizontal and vertical displacements of the wall after sequential construction of tier 1, tier 2 and tier 3. The results show that the peak horizontal displacement after completion of tier 1 is 17.5 mm. This peak horizontal displacement in tier 1 then increases to 40 mm after completion of tier 2 and then further increases to 60 mm after construction of tier 3. The peak horizontal displacement of 60 mm was predicted for both tier 2 and tier 3 under the action of their own self-weights. This was expected as the tiers are of similar height and contained approximately the same amount of geogrid reinforcement. The horizontal displacement in tier 2 is considerably higher than for tier 1 as the first tier is much more heavily reinforced (4.4 MN) than the others (both approximately 2.0 MN) as discussed in Section 3.1. The largest horizontal displacement of 90 mm was predicted in tier 2 after completion of tier 3 above it. The movements are greater in tier 2 than in tier 1 even though there is clearly more material and hence selfweight load acting on tier 1. This is again due to the fact that there is considerably more reinforcement in tier 1. Globally, the maximum vertical displacement is predicted to be 80 mm. This is concentrated in the facing block at the base of tier 3. The results indicate that the facing blocks in tier 1 will not settle more than about 20 mm whereas those in tier 2 and tier 3 may settle by 60 mm to 80 mm. These blocks are ‘punching’ into the tiers they are founded on.

Figure 6. Geogrid force vs. height above tier base at end of construction of tier 3 (Analysis 3).

Again, the presence of bearing pads beneath the blocks may reduce this effect and overcome the convergence difficulties experienced for Analysis 3. Table 5 shows the peak axial force generated in the geogrids after successive construction of each tier. The results indicate that the axial forces in the geogrids increase as expected as the height of the wall increases. The geogrid force in tier 2 and tier 3 exceeds that in tier 1 by factors of 2.6 and 3.2 respectively; a result of fewer geogrids in these tiers. Furthermore, the maximum geogrid force of 29.2 kN/m is predicted to occur in tier 3 rather than tier 2 and this is likely to be due to tier 3 being founded on a the less stiff tier 2 compared to the stiffer and more heavily reinforced tier 1. Figure 6 shows the distribution of mobilised axial force in the geogrids with height for each of the three tiers for the completed construction stage. The graphs show that the maximum force in the geogrids for each tier is predicted in the bottom third of the tiers. The maximum force in tier 1 is located nearer the base of the tier than that for tier 2 and tier 3. Table 6 shows the maximum axial force mobilised in the geogrids in each of the three tiers following each construction stage. As stated in Section 3.1, the total axial force that can be resisted by the geogrid in tier 1, tier 2 and tier 3 is 4.41 MN, 1.95 MN and 2.03 MN respectively.

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Table 6. Sum of geogrid axial forces mobilised after each construction stage (Analysis 3).

Table 8. Peak geogrid forces for the ‘super-geogrid’ model (Analysis 4). Maximum mobilised axial force in super-geogrid

Sum of axial forces in geogrids (kN/m) Construction stage

Tier 1

Tier 2

Tier 3

Construction stage

Tier 1

Tier 2

Tier 3


393.0 663.0 800.0

– 532.0 591.0

– – 638.0


37.0 (7.4) 57.6 (11.5) 70.4 (14.1)

– 58.0 (19.3) 64.3 (21.4)

– – 62.0 (20.7)

Table 9. Peak results for tier 1 for (tier 2 + tier 3) surcharge model (Analysis 5).

Table 7. Peak wall displacements for the 3-tier ‘super geogrid’ model (Analysis 4).

Construction stage

Horiz. wall displ. (mm)

Vert.wall displ. (mm)

Geogrid axial force (kN/m)

Tier 1 Tier 2 + Tier 3 surcharge

17.5 35.0

25.0 120.0

9.0 12.0

Horizontal (mm) Vertical (mm) Construction stage Tier 1 Tier 2 Tier 3 Tier 1 Tier 2 Tier 3 Tier 1 Tier 2 Tier 3

20.0 50.0 70.0

– 70.0 90.0

– – 60.0

7.5 10.0 20.0

– 50.0 80.0

– – 100.0

Therefore, the minimum factor of safety (force available/force mobilised) is 3.18 in tier 3 and the overall factor of safety is 4.14.

Analysis 3 it was in tier 3. Intuitively, this seems more plausible due to the fact that tier 2 is being loaded by tier 3. It was noted that distortion of the facing block elements occurred in Analysis 3 and this may have contributed to the prediction of the peak geogrid force in tier 3 for that analysis.

3.4 Analysis 4: ‘Super-geogrid’ model In order to further reduce the computer run-time an analysis was run whereby the tiers were split into approximately 1m high sections with one ‘supergeogrid’ to represent all the geogrids in that section. The stiffness and strength properties of this ‘supergeogrid’were factored up by superposition to represent the combined properties of the other geogrids. Table 7 shows the peak horizontal and vertical displacements for each of the three tiers at each stage of construction. Comparison with Table 4 illustrates consistently slightly higher peak displacements with a similar pattern of movement as Analysis 3 with higher values at the facing owing to the larger spacing of the ‘super-geogrids’. The results of Analysis 4 indicate that modelling of the RE wall by combining groups of grids into one super-geogrid gives similar results to Analysis 3. This approach drastically reduces the computer run-time from 97 hours to 14.5 hours; a factor of almost 7 times quicker. This will be particularly useful for carrying out sensitivity analyses in the future. Table 8 shows the maximum axial force mobilised in the ‘super-geogrids’. The values in brackets represent the maximum force per single geogrid, i.e. the force in the ‘super-geogrid’ divided by the number of single geogrids it represents. Although the axial forces mobilised in the geogrids for Analysis 4 are smaller than those in Analysis 3 (Table 5), the general trend of increasing force as construction proceeds is evident. However the peak geogrid force in Analysis 4 occurs in tier 2 whereas in

3.5 Analysis 5: Surcharge model In order to obtain an idea of the likely outcome of the 3-tier analysis quickly, without explicitly modelling the upper two tiers, a surcharge was applied to the tier 1 model of analysis 2. This would simulate the selfweight load imparted by tier 2 and tier 3 but would not model the stiffness offered by these two tiers. The results are presented in Table 9. The results for this analysis show vertical displacements consistent with Analyses 3 and 4. However, peak horizontal displacements are approximately three times smaller. The horizontal movements are considered to be unrealistically small as the surcharge is not capable of modelling actual redistribution of load that will occur in the actual geogrid-reinforced upper tiers. The forces mobilised in the geogrids are also less as a result of smaller horizontal movements. 4

DISCUSSION AND CONCLUSIONS

4.1 Analysis methods Analyses have been carried out for the 3 stages of the 3-tier, 56 m high RE wall. Following on from the single tier work, higher QC geogrid stiffnesses have been used and high angles of shearing resistance and dilation have been adopted for the backfill material (Table 1). The ‘detailed’ FLAC Analysis 3 (Table 2) is considered the most reliable for present purposes as it does not include the simplifications made in Analyses 4 and 5 such as combining the geogrid layers

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into ‘super-geogrids’, or modelling the upper tier as a surcharge. Analyses 1 and 2 model the first tier only and are therefore not suitable for consideration of the full scale deformations. However, modelling has been very time consuming to run (about 4 days for the full model), precluding practical exploration of sensitivity and parametric studies. The FLAC ‘super-geogrid’ model simplifies the analysis by substituting a single geogrid per approximate 1 m height of RE wall with equivalent strength and stiffness to the actual geogrids it represents. This gives peak horizontal displacements within 5 mm (better than 20%) agreement on the conservative (higher) side. Predicted geogrid forces are lower by typically 4 kN/m or say 20%. These are minor differences in comparison to the other uncertainties in such modelling. The ‘super-geogrid’ approach opens up the opportunity to study the 80 m RE wall. 4.2

Horizontal movements

Analysis 3 predicts that the bottom tier 1 will exhibit peak horizontal wall movement of around 20 mm after construction; 40 mm after completion of tier 2 and 60 mm upon completion of tier 3. The final peak horizontal displacement is predicted to be around 90 mm at approximately mid-height in tier 2. The higher displacement in tier 2 rather than tier 1 is attributed to having far less reinforcement than tier 1 (2.0 MN/m compared to 4.5 MN/m). This suggests that the reinforcement in tier 1 could be substantially reduced. However, this would imply approximately proportionate increases in horizontal movement which could exceed acceptable wall movement (SLS) requirements. Halving the amount of reinforcement in tier 1 might approximately double the wall movements to about 100 mm. Conversely for tier 2, doubling the reinforcement could reduce movement to about 50 mm. 4.3

Geogrid forces

Analysis 3 predicts that the bottom tier 1 geogrids will experience forces up to 18.5 kN/m, very well within the yield strength. The maximum predicted force is in a geogrid placed 1.4 m from the base of the bottom tier. This corresponds to a factor of safety on rupture of 3.5 on the 120 year creep limited strength. This suggests fewer and/or weaker grids could be used, subject to satisfying SLS deflection requirements. There appears to be no significant force mobilised in the geogrids in tier 1 at more than about 12 m from the face therefore the lengths could be reduced. 4.4 Vertical displacements The maximum vertical displacements are predicted to be of the order of 100 mm in the unreinforced backfill material behind the reinforced zones, at around midheight (i.e. middle tier 2), considerably influenced by the soil stiffness. Of more consequence may be

the localised vertical displacements in the vicinity of the wall/facing. The overall picture is of movements of perhaps 100 mm in the reinforced fill behind the centre of each wall tier, not greatly affected by the subsequent lifts. Noting that the Dibba Road 18 m wall is understood to have performed satisfactorily, the vertical strain effects of subsequent tiers appears limited and hence probably manageable, noting also the considerable set-backs in this case. That said, the facing, connection and adjacent area warrants better modelling in detail. 4.5

Conclusions

The studies are giving increasing understanding of practical numerical modelling, parameters and factors influencing the design and performance of high RE tiered walls. The present analyses have reached the point where we can practically model the full proposed 80 m tiered wall. Calibrations can be set against the models to date and a reasonably consistent picture is emerging. Validation depends on field data and test data for the fill materials, but the numerical methods now have the potential to back fit existing factual data (Class C) and monitoring data obtained during construction (Class B predictions/observational method), and hence have increasing value for design and (Class A) prior prediction purposes. REFERENCES Bathurst, R.J., Walters, D.L., Hatami, K. and Allen, T.M. (2001). Full-scale performance testing and numerical modelling of reinforced soil retaining walls. In Proceedings of the 4th International Symposium on Earth Reinforcement, IS Kyushu 2001, Fukuoka, Japan, 14– 16 November 2001. Edited by H.Ochiai, J. Otani, N. Yasufuku and K. Omine. A.A. Balkema, Rotterdam, the Netherlands, Vol.2, pp. 777–799. Bolton, M.D. (1986). The strength and dilatancy of sands. Géotechnique, 36, pp.65–78. Hatami, K. & Bathurst R.J.(2005). Development of a numerical model for the analysis of geosynthetic-reinforced soil segmental walls under working stress conditions. Can. Geotech. J. 42: 1066–1085. Hatami, K. and Bathurst, R.J. (2006a). Numerical Model for Reinforced Soil Segmental Walls under Surcharge Loading. ASCE. Journal of Geotechnical and Geoenvironmental Engineering. Hatami, K. and Bathurst, R.J. (2006b). Parametric analysis of reinforced soil walls with different height and reinforcement stiffness. 8th international Geosynthetics Conference Proceedings. Itasca Consulting Group. 2001. FLAC—Fast Lagrangian Analysis of Continua. Version 4.00. Itasca Consulting Group Inc., Minneapolis, Minn. Rowe, P.W. (1962). The stress-dilatancy relation for static equilibrium of an assembly of particles in . Proc. R. Soc. 269A, 500–527. Simoni, A. and Houlsby, G.T. (2006). The direct shear strength and dilatancy of sand-gravel mixtures. Geotechnical and Geological Engineering Vol.24, pp. 523–549.

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Short term three dimensional back-analysis of the One New Change basement in London R. Fuentes University College of London, UK Arup Geotechnics, UK

A. Pillai & M. Devriendt Arup Geotechnics, UK

ABSTRACT: This paper presents the results of a three dimensional back-analysis in the short term with the aim to validate the soil parameters used in the BRICK model. The comparison has been made between the results of the numerical analysis and the field measurements obtained on site during the works. An additional comparison between detailed modelling and simplified sequences is presented. Finally a LS-DYNA tool to model bearing piles as embedded beams using beam elements is presented. The results of this work have provided increased confidence in the new BRICK model parameters. Furthermore, the conclusions and lessons learnt in using the new system of pile modelling, as well as the construction sequence used, provide valuable information for their application in future projects and research work.

1

INTRODUCTION

The BRICK model for London Clay has been extensively used and published by means of the original parameters presented by Simpson (1992), and later reviewed by Pillai (1996). However, not much work has been presented on the application of the new parameters, or so called ‘Most probable’ parameters derived by Arup (SCOUT, 2007) to deep excavations. This paper presents the back-analysis carried out for short term performance of a deep basement excavated in London Clay. In this work, the authors have focused on the new retaining wall performance in the short term, and more specifically on the eastern side of the excavation where a complete understanding of the construction sequence adopted on site is known. Devriendt et al. (in press) have recently prepared a paper which shows the effect of this basement construction on the adjacent tunnels. The aim of the present paper is to validate and examine the performance of the Most probable BRICK parameters and compare it to the original, or Characteristic parameters, adopting the EUROCODE 7 terminology. This will increase confidence in this set of parameters, and also provide the platform for others to extend their knowledge and research on the behaviour of deep basements in stiff clays in constrained urban environments. 2

SITE DESCRIPTION

The site is located at New Change in the City of London. It covers an entire city block bounded by building

Figure 1. Site location and adjacent tunnels shown indicatively.

on all sides (Fig. 1). It has maximum dimensions of 105 m (north to south) by 150 m (west to east) and covers an approximate area of 12,000 m2 . The existing ground level of the site is approximately +17.5 mOD along Cheapside at the north of the site and the surrounding topography generally falls towards the River Thames in the south. St. Paul’s Cathedral and St. Paul’s Cathedral Choir School are located immediately to the west.

2.1 Existing structures The proposed development involved the demolition of the former Bank of England offices. This was a seven-storey steel framed building. Two levels of basement (basement and sub-basement) were present, with approximate floor levels of +14 mOD and +11 mOD.

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Table 1.

Soil type

Soil stratigraphy and properties. Top of layer γ (mOD) (kN/m3 )

Made Ground 17.5 Brickearth 11.7 Terrace deposits 10.0 London Clay 6.6 Lambeth G Clay −29.0 Lambeth G Sand −42.0 Thanet Sand −48.0

18 18 20 20 20 20 20

E ϕ (GPa) (deg)

Table 2.

K0

10 25 0.58 20 25 0.58 20 36 0.41 Used BRICK model Used BRICK model 125 36 0.41 250 38 0.38

BRICK parameters.

Parameter

Characteristic

Most probable

λ κ ι ν µ βG ,βϕ

0.1 0.02 0.0019 0.2 1.3 4.0

0.1 0.02 0.00175 0.2 1.3 4.0

String length (Gt/Gmax)

KEY: γ represents the bulk density, E is the drained Young’s Modulus, ϕ is the angle of shear resistance, and K0 represents the earth pressure coefficient at rest.

Portion of material (strain,%)

Characteristic

Most probable

0.08 0.17 0.22 0.24 0.16 0.055 0.031 0.027 0.0135 0.0035

3.04E-05 6.08E-05 1.01E-04 1.21E-04 8.20E-04 1.71E-03 3.52E-03 9.69E-03 2.22E-02 6.46E-02

3.00E-05 7.50E-05 1.50E-04 4.00E-04 7.50E-04 1.50E-03 2.50E-03 7.50E-03 2.00E-02 6.00E-02

KEY: Gmax is the maximum shear modulus, Gt is the shear modules at a given strain level, λ is the slope of the isotropic normal compression line and κ is the slope of the isotropic swelling line in εvol – ln p space, ι is a parameter controlling elastic stiffness, ν is the Poisson’s ratio, µ controls string length due to changes in orientation in the π-plane, βG ,βϕ control the amount of initial stiffness and strength gain from overconsolidation respectively.

Figure 2. Groundwater measurements at the Terrace Deposits within the site.

2.2

Nearby tunnels

The twin bored tunnels of the LU (London Underground) Central Line run beneath Cheapside to the north of the site. The tunnels are approximately 3.8 m in diameter and lined with a cast iron segmental lining. Figure 1 shows the approximate position of these tunnels as dotted lines, with respect to the new development. Their presence and construction was modelled as part of this work. Devriendt et al. (in press) have recently published the effects of the basement construction on the tunnels.

3

GROUND AND GROUNDWATER CONDITIONS

Table 1 shows the soil stratigraphy and the ground parameters that were chosen for the analysis. Most of these parameters were all derived from a thorough site investigation. Figure 2 shows the ground water readings of the boreholes located within the site. From this information, a water level of +8.5 mOD was assumed for the site in its current conditions. An under drained

profile was taken for the London Clay and underlying Lambeth Clay using a 60% of a fully hydrostatic profile. A fully hydrostatic profile was then chosen below the clay materials. This assumption was based on the results of nearby sites presented by Simpson et al (1989), as no data was available for the London Clay and underlying soils for this site. 3.1 Brick parameters and initial stresses Simpson (1992) introduced the two-dimensional BRICK model. This is a non-linear model, which was formulated in strain space. The non-linearity of the stiffness is modelled using the analogy of strings attached to bricks being dragged by a man. Simpson has since developed a three dimensional version of this model which has never been published in detail. A detailed explanation of this model has been recently given by Ellison (in prep.). The values of the parameters used in this case are shown in Table 2. These show the characteristic parameters as presented by Pillai (1996), and most probable, as presented in SCOUT (2007). The values of K0 are calculated by the program based on the stress history of the material. An overburden of 206 m was used for this site to model the overconsolidated nature of the London Clay and Lambeth Clay.

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Table 3.

4.2 Construction sequence

Construction sequence in the model.

A cantilever type construction was adopted to reach level +11.0 mOD. Below this level, a top down sequence was followed under the proposed lower ground floor slab. Moling techniques were used to excavate the underlying ground. However, this paper covers only the cantilever stage up to 13/08/08. Seven different construction sequences were adopted during construction of the perimeter wall at the cantilever excavation stage. These consisted of very different arrangements which ranged from the use of hit-and-miss techniques using buttress walls combined with raking props, thrust blocks and / or berms as temporary . Since this paper focuses on the east area only, all these different sequences were not modelled in detail, and only wished in place at Stage 8 (see Table 3). They are believed to have little effect on the behaviour of the east wall, especially far from the corners. On the other hand, a detailed excavation construction sequence was modelled for the east side. Details of this sequence are shown in Table 3.

Stages Description 1 2 3

Initialisation. Apply volume loss to existing tunnel material. Tunnel material excavation and wish in place lining. 4 Wish in place existing building (self-weight of slabs above formation level neglected) without bottom slab to allow heave in the short term. 5 Apply loads from existing building. 6 Long term conditions (up to date) using existing groundwater conditions. 7 Demolish existing building, install new wall in all areas except east side. 8 Install berm in east side. Wish in place rest of temporary works around the site. 9 Excavate berm partially and install new secant wall. 10 Excavate berm down to +11.0 mOD and connect existing wall to new wall – 17/04/08. 11-8q Wish in place cofferdam for Core 2 and excavate inside to +6.6 mOD approx – 23/04/08. 12-9q Excavate in from of NE corner to +110.0 mOD, install access ramp in east side – 28/05/08. 13-10q Excavate inside cofferdam for Core 2 to +3.2 mOD, wish in place cofferdam for Core 3, and dig to +6.6 mOD – 26/06/08. 14-11q Install access ramp in SW corner, excavate inside cofferdam for Core 3 to +3.2 mOD, wish in place bearing piles inside cofferdam, superstructure for Core 2 begins (25% of total weight assumed) – 09/07/08. 15-12q Removal of ramp in centre of east side, wish in place cofferdam for Core 1, excavate inside to +6.6 mOD. Core 3 excavated to formation level. Core 2 reaches top level (assume 90% of total load). – 13/08/08.

4

4.3 FE elements The model comprises of over 500,000 elements. Three different element types were used: (a) solid elements to model the soil, (b) shell elements to model retaining walls, internal walls, core walls and slabs, and (c) beam elements, used to model temporary props and bearing piles (see Section 5.8 of this paper for more details on the modelling of bearing piles). 4.4 Boundary conditions

FE MODEL

The mesh for the problem was built using Hypermesh v. 10, whereas the analysis was carried out using LS-DYNA CEAP software. The latter allows for multiprocessor analysis, which decreases computing times dramatically. An Intel® Xenon® U with 8 X5482 @ 3.20 GHZ processors and 8.00 GB of RAM computer was used, which led to a total computation time of approximately 26 minutes per stage. 4.1 Analysis models Two main analysis have been carried out; one using the characteristic values and the second using the most probable parameters, as shown in Table 2. A further model has been carried out with a less detailed construction sequence, where all the temporary works were ‘wished in place’. The stages of this model are shown in Table 3 using the letter ‘q’. This less detailed model was carried out to show the effects of using the less detailed approach, which is typical of projects when the FE analysis is undertaken at stages when the construction sequence has not yet been totally defined.

The model extended from +17.5 mOD at the top or existing ground level, to −60 mOD or top of the Chalk layer. It was assumed that the Chalk layer represented a fixed boundary. The model is 310 m wide and 490 m long approximately. The following displacement boundary conditions have been applied: (a) the horizontal base of the model was restrained in all directions since at this depth the effects of the excavations and surface surcharges were negligible and restraint was anticipated to have no effect on the behaviour of the system, (b) all of the vertical boundaries were restrained in both horizontal directions but were free to move vertically. These were assumed to be sufficiently far from the excavations to have no effect on the behaviour of the basement itself. 4.5

Soil models

London Clay and Lambeth Group Clay were modelled using the BRICK model, and all the other remaining soils where modelled using the linear elastic perfectly plastic Mohr-Coulomb soil model. 4.6 Tunnel modelling The construction of the tunnel was modelled using a 2% volume loss along its length.

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Figure 3. Typical output using embedded piles showing axial force decreasing with depth.

This is done in LS-DYNA by assuming an isotropical compression of the material using an algorithm similar to that of a thermal compression (LS-DYNA Manual, 2008). The self-weight of the material inside the tunnel needs to be set to zero or a very low value, in order to avoid collapse of the tunnel and error of the program.

Figure 4. Location of ground settlement points and inclinometers in the east side of the site. KEY: SM is settlement monument, I-1 denotes inclinometers and LS denotes levelling studs. Table 4.

4.7

Dimensions of structural elements in the model.

Bearing piles modelling Element

An Arup internally developed routine allows modelling of piles as beam elements connected to solid elements, which represent the soils.

– New secant wall 1 – New secant wall 2 – Cores secant pile walls – Sheet piles for capping beam – Sheet piles elsewhere

4.7.1

Brief description (after LS-DYNA Manual, 2008) The beam elements modelling the piles have no coincident nodes with the solid elements that surround them. The behaviour of the pile-soil interaction is modelled using a stress-displacement relationship. An elastic-perfectly plastic and hyperbolic relationships are offered in the program. defined relationships can also be inserted. In this case an elastic-perfectly plastic relationship was used. Furthermore, the interface pile-soil is also modelled by means of a reduced undrained shear strength or angle of shear resistance, depending if it is an undrained or drained situation. Visualisation of the stresses at the shaft and the base of the piles is achieved by the definition of null beams. Figure 3 shows a typical output obtained for a pile under Core 2 (see Fig. 4) at Stage 15. 4.7.2 Advantages of this approach The proposed approach allows the designer to model the piles separately from the soil giving a much increased flexibility. For example, the costs of remeshing the problem are reduced significantly, should the position of piles change during the design process.

diam/s * (m)/(mm)

t (m)

Emod (kPa)

0.9/1.25 1.2/1.80 0.9/1.25

0.78 1.04 0.78

1.82E+07 1.69E+07 1.82E+07

AU14

0.51

5.43E+06

AU21

0.57

6.22E+06

Element

Dimension in model (m)

– Existing retaining wall – Existing floor slabs – Proposed floor slabs – Proposed raft

0.68 m thick 0.30 m thick 0.4 m thick 1.5 m thick

*Sheet piles were calculated using the Piling Handbook (2005).

As the pile-soil interaction is modelled using a stress-displacement relationship, the data obtained from pile tests can be used to model site specific relationships. A further advantage of this method is the reduction of element numbers, and therefore, computing time, versus a solution where piles are meshed and modelled as solid elements.

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Figure 5a & 5b. Inclinometer I11 readings and FE predictions a) Above: Characteristic, b) Below: Most probable.

4.8 Shell elements and other structural elements The walls were modelled as continuous elements using linear elastic model. The width and stiffness of these elements was calculated such that the modelled values of axial and flexural stiffness (perpendicular to the

Figure 6a & 6b. Inclinometer I12 readings and FE predictions. a) Above: Characteristic, b) Below: Most probable.

wall) were equal to those of the proposed secant pile walls. The Young’s modulus of concrete and steel were taken as 2.8 × 107 MPa and 2.1 × 108 MPa respectively. Table 4 shows the results of these calculations for the different elements used in this project.

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6

CONCLUSIONS

This study has allowed the comparison and verification of different set of parameters of the BRICK soil model against a case study. This provides confidence in the model to be used in future projects. It has also been shown that modelling the actual construction sequence as closely as possible is important to obtain the correct movements of walls and ground. The match with ground movements is not as good. Finally, a method to model piles using beam elements has been shown. This offers great advantages during pre-processing, analysis and post-processing. ACKNOWLEDGMENTS

Figure 7. Comparison between detailed and less detailed construction sequence models.

5

The authors thank Dr Pedro Ferreira and Professor Swain of UCL, and Dr Pellew of RKD Consultants for their , and to all the other partners in the project: Land Securities, Bovis Lend and Lease, Skanska, McGee and Warner Land Surveys. This work is funded by the EPSRC and Arup, and forms part of an Engineering Doctorate degree the first author is undertaking at UCL.

FIELD MEASUREMENTS AND RESULTS

Figure 4 shows the settlement monitoring and inclinometers that were installed on site. Figures 5a and 6a show the readings of inclinometers I11 and I12 respectively at stages of construction as close as possible to those modelled in this study. It can be seen that I12 is showing smaller movements than those in I11, which shows the stiffening effects of the corner. The top 1m of the readings in I11 have been removed from the plots after comparing the unusual readings to the capping beam survey measurements. A maximum settlement of +1.65 mm was measured at point LS37 behind the retaining wall on the east side at Stage 15. The FE results gave a settlement value of −0.38 mm. A potential explanation could be that the readings are close, and possibly smaller, than the actual achievable accuracy of the instrument. Investigating further the causes of this, is outside the scope of this paper. Figures 5 and 6 show that the agreement is better for the model using the ‘Most Probable’ parameters than the one using ‘Characteristic’ parameters for the retaining wall displacements. Figure 7 shows that the less detailed model presents a different deformation mode in the top 10 m. This can have a significant impact on the prediction of ground movements.

REFERENCES Simpson, B. 1992. Thirty-Second Rankine Lecture “Retaining Structures: displacement and design. Geotechnique,42, No. 4, 541–576. Pillai A. 1996. Review of the BRICK model of soil behaviour. MSc dissertation, Imperial College, London. Devriendt, M, Doughty, L, Morrison, P and Pillai, A. (in press) Displacement of cast iron tunnels arising from a deep basement excavation in central London. Geotechnical Engineering. Simpson, B, Blower, T, Draig, R, N, Wilkinson, W, B. 1989. The engineering implications of rising groundwater levels in the deep acquifer beneath London. CIRIA SP69. Piling Handbook. 2005. Arcelor. Ellison, K, Soga, K & Simpson, B. (in prep.) A strain space model for overconsolidated clay with evolving stiffness anisotropy. SCOUT project. 2007. Report on Observational Method under the Framework of Eurocodes. Report no D18. SCOUT. LS DYNA. 2008. CEAP (Civil Engineering Application Program) Manual. Livermore Software Technology Corporation.

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Tunnels and caverns


3D analysis of a micropile umbrella for stabilizing the tunnel face of a NATM tunnel Falko Schmidt TERRASOLUM S.L., Santander, Spain

César Sagaseta University of Cantabria, Santander, Spain

Heinz Konietzky Technical University Bergakademie Freiberg,

ABSTRACT: In this paper, a 3D numerical analysis is presented to investigate the effect of a micropile umbrella on the stability of the tunnel face. Data from an actual NATM tunnel, constructed in Burgos (Spain) in Miocenic sandy clay are taken as a basis. During the development of numerous FLAC3D models, several mechanism and factors were discovered, that influence the tunnel face safety and the behaviour of the micropile umbrella. A sensitivity analysis was performed, evaluating the effect of some of the problem variables: piles inclination (α), length (L), spacing (s) and overlapping (d) and ground cohesive strength (c). The evaluation is performed regarding the influence of micropile parameters on safety factors and the extent of the failure zone around the tunnel face.

1

INTRODUCTION

Excavation under a micropile umbrella (canopy) is a potential alternative for hand-excavated tunnels in difficult ground conditions, for instance soft soil. Micropiles are installed from the periphery of the tunnel face, and extended over a given length. If needed, consecutive umbrellas can be constructed with some overlapping between them. Evaluation of the protective effect of this element is difficult due to the three dimensional character of the problem and the complex soil-structure interaction. This study is based on three dimensional numerical simulations in order to investigate this kind of used commonly in tunnel excavations.

2

MICROPILES

In contrast to common concrete piles, reinforced only to bear bending moments and shear forces, the micropile consists mainly of a high resistance steel tube. The tube is injected with cement in order to achieve proper coupling between surrounding soil and micropile. Many authors categorise the micropiles according to the injection method applied: IGU unique global injection, usually from the tip of the perforation

IR the injection of the micropile can be repeated in order to improve coupling to soil IRS the injection process can be repeated and the zone to be treated can be selected.

3 TUNNEL FACE STABILITY Usually, excluding the tunnel entrance problems, there are two types of stability problems when facing a tunnel construction. One type can be a possible failure affecting an already ed zone of the tunnel. Another type of failure affects the tunnel face, where the excavation and work takes place. The objective of this paper is to investigate the influence of micropile umbrellas on tunnel face stability. Figure 1 shows a typical tunnel face failure: the soil mass collapses at the uned tunnel face area. In this case preliminary in form of micropiles is not installed. This enables mass movement (collapse) and produces a so-called ‘chimney’ - a cylindrical failure body, which might reach the surface and create a sinkhole under unfavourable conditions.

4

FLAC 3D MODELLING

The analysis of tunnel face stability is a three dimensional problem. Although there are several methods available to simulate this phenomenon (e.g. Panet

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Figure 1. Collapse of a tunnel face forming a chimney (displacements in m).

1982), which uses a decreasing inner pressure, there is no exact way to formulate these types of problems in the two dimensional space. Additionally, there is no appropriate way to model the effect of a forerunning micropile umbrella. Therefore, a model in three dimensions is chosen for this investigation. 4.1

Soil softening constitutive model

Additionally to the parameter study on micropiles using the classical Mohr-Coulomb constitutive law for the soil mass, a special strain hardening / softening Mohr-Coulomb model [ITASCA 2005] has been applied, which simulates the soil behaviour at the Fuente Buena tunnel site in a more realistic manner, especially if failure occurs. In order to calibrate the constitutive model, laboratory data are needed. Often, and also in this case, post failure behaviour is rarely investigated by laboratory tests, because only peak values are of interest for many projects. Therefore, laboratory data of a similar soil were used to calibrate the post failure behaviour, which should represent the soil material at the Fuente Buena site. The test data were obtained from the Geotechnical Institute at the Technical University of Freiberg, . The calibration of the softening soil model was performed by the numerical backanalysis of triaxial lab tests with FLAC3D [ITASCA 2005]. Exemplary, Figure 2 illustrates the mesh of the 3-dimensional model and the softening function for the friction angle. The mesh size of the triax model was sized according to the mesh size of the tunnel face in order to avoid zone size dependence of the softening. A detailed simulation of the shearband evolution was not the aim of this modelling. 4.2

Figure 2. Three-dimensional numerical specimen used for parameter calibration and diagram showing the reduction of the friction angle as a function of the accumulated plastic strain.

Modelling of the tunnel

One part of the Fuente Buena tunnel was modelled over a length of 60 metres with a soil overburden of

Figure 3. Three dimensional tunnel model at a certain excavation stage with micropiles and coloured plastified elements.

54 metres; the tunnel geometry and dimensions (cross section approx. 95 m2 ) were according to the construction project and the excavation steps were divided into heading and bench. The excavation was executed in steps of 1 metre within the numerical model. Taking advantage of the symmetry of the tunnel geometry only half of the tunnel was modelled (see Figure 3). Exemplary, Figure 3 shows a plot which indicates plastified zones around the tunnel for a certain excavation stage. All the elements, like micropiles, steel arcs and shotcrete and their interaction with the surrounding soil were applied. Of particular interest are the micropiles: they are rigidly attached to the tunnel liner, fixed and braced by the steel arcs. They interact with the surrounding soil by springs and frictional-cohesive interfaces in axial and normal direction. Figure 4 shows the excavation sequence used in the FLAC3D model: in order to close the ring of the tunnel

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Figure 5. Example with micropiles: factor of safety and corresponding collapsed soil volumes (white colour corresponds to FoS values >1.6).

Figure 4. Illustration of the applied excavation sequence.

immediately, a separation of 20 metres was established between heading and bench excavation. Having modelled the whole tunnel and the excavation process an intensive parameter study was carried out. This analysis was dedicated to micropiles, varying pile spacing (s), overlapping (d), inclination (α) and length (L). The provided project data showed a wide range of soil cohesion and in order to show its influence on the tunnel face stability, this value was changed as well during the parameter study. 5

EVALUATION OF THE NUMERICAL MODELLING RESULTS

As mentioned above, the investigation was carried out in two phases: •

Phase 1: general parameter study using the classical Mohr-Coulomb constitutional material model. • Phase 2: site specific simulations using the soil softening constitutive law

Figure 6. Example without micropiles: factor of safety and collapsed soil volumes(white colour corresponds to FoS values >1.6).

5.1

The following conclusions can be drawn from the Figures mentioned above:

Parametric study with posterior FoS calculations

The objective of this study was to find the optimal micropile configuration for the given soil parameters. Having installed the umbrella, a factor of safety (FoS) analysis was conducted including the determination of the collapsed soil volume.As an example Figures 5 and 6 show the difference in respect to the factor of safety and the corresponding collapsed soil volumes for the same excavation stage by either applying micropiles or not. The procedure to obtain the FoS value is based on c-reduction-technique, reducing progressively these parameters by a factor, and detecting unstable grid point values within the model. An additional algorithm marks every failed model zone with the actual FoS and sums up the failed soil volume.

•

The FoS in the immediate uned tunnel face is increasing from 1.3 to 1.4 by installing micropiles. • The soil volume affected by a potential collapse is much smaller by having micropiles installed.

In order to evaluate the study and to gain tendencies from it, the results were presented in tables containing the collapsed soil volume and the corresponding FoS as a function of the micropile configuration. Table 1 shows only an extract of the parameter study. More information is given by Schmidt (2007), where a total of 39 parameter calculations are documented in detail. From the data obtained by the parameter study, an optimal configuration was chosen for the Fuente Buena tunnel under safety and economical criteria. This configuration was later investigated in more detail

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Table 1. Volume of failed soil mass as a function of micropile umbrella configuration (for soil of 20 kPa cohesion). Umbrella configuration

Failed soil volume [m3 ] at FoS

α[◦ ] s[m] L[m] d[m]

1.4

4 4 4 4 4 4 6 10

0.5 0.5 0.5 0.5 0.3 1.0 0.5 0.5

14 20 14 14 14 14 14 14

1.5

1 403 16e3 5 (**) 73 9e3 8 0 700 5 (*) 0 1.4e3 5 0 580 5 101 772 5 33 8e3 5 252 13e3

No micropiles installed

421 21e3

1.6

1.7

1.8

17e3 17e3 7e3 14e3 8e3 10e3 16e3 15e3

19e3 18e3 18e3 15e3 15e3 16e3 16e3 20e3

19e3 18e3 18e3 15e3 16e3 19e3 17e3 20e3

21e3 22e3 22e3

(*) Optimal configuration for this case. (**) Plastic moment of pile elements results in a lower FoS value.

with the softening soil model adequate for the specific tunnel site conditions. A model without micropile umbrella was calculated as well, in order to evaluate the effect of the canopy. It can be seen that an overlapping of only 1 m gives similar results in comparison to an uned tunnel that means a minimum overlapping of a few meters is recommended anyway. Figure 7 elucidates the influence of each investigated pile parameter on the failed soil mass for a FoS of 1.5. The analysis of the influence of each parameter is obtained by individual variation, leaving the other three parameters unchanged. This sensitivity study shows that reducing the pile inclination and length are positive measures to reduce the failed volume, while reducing the pile segment overlapping favours collapsing. The influence of the pile spacing could not be clarified during this study and needs further investigations. Increasing the pile length and maintaining pile overlapping result in a higher collapsed soil mass, because the piles yield limit is reached and failure occurs.

5.2

Calculation applying the soil softening model

The authors had access to construction site data like convergence and settlement of the tunnel. So it was possible to compare numerical calculation results with measuring data obtained from the Fuente Buena tunnel. For instance, the convergence measurement on site coincided with the values obtained from the numerical model with about 60 mm of closure. Other results, difficult to measure on a tunnel site could be derived from the FLAC3D calculations: Figure 8 shows maximum bending moments of 29 kNm in the pile elements. This helps to dimension micropiles and understand their influence on safety of the tunnel face.

Figure 7. Influence of pile geometry parameters on collapsed soil mass for a FoS equal to 1.5.

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The length of the pile should not be too long in order to avoid high bending moments, which can exceed material resistance and produce failure. The separation of the piles should not permit soil collapse between the pile tubes. As well as the overlapping of the single umbrellas with each other in order to secure a forerunning : it should be chosen in a matter to grant a mutual between the umbrella segments. The use of the soil softening model enabled the calculation of realistic stress and deformation values for the Fuente Buena tunnel including the loading of the micropiles, as shown by comparison with construction site data. REFERENCES

Figure 8. Bending moments [Nm] acting in the micropiles.

6

CONCLUSIONS

The parametric study led to the following practical results: The micropiles should be installed as parallel as possible to the tunnel axis, minimising the pile angle.

ITASCA Consulting Group, Inc. 2005. “Flac 3D – Manual, Second Edition“. Minneapolis, United States of America. Panet, M. y Guenot, A. 1982. “Analysis of convergence behind the face of a tunnel”. Proc. Tunnelling’ 82, Institution of Mining and Metallurgy, London, United Kingdom, p. 197–204. Schmidt, F. 2007. “Tunnel Fuente Buena – The use of micropiles in tunnelling – a numerical approach”. Diploma Thesis. Geotechnical Institute, TU Freiberg, .

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Analysis and design of a two span arch cut & cover structure Sachin Kumar & Tony Suckling Arup Geotechnics, Solihull, UK

Lewis Macdonald Arup, London, UK

Hoe-Chian Yeow Arup Geotechnics, London, UK

ABSTRACT: This paper describes a case study of a finite element analysis undertaken using Plaxis to model a two span arch cut and cover structure with significant backfilling above. The methods used to model the backfilling and the effects of compaction are described. The limitations found of using Plaxis are discussed along with recommendations for the future development of this tool.

1 1.1

INTRODUCTION General

Arup UK was employed to undertake independent design work to check the design of 17 deep cut and cover structures carried out in Australia for a large infrastructure project in Brisbane. The designer was a t venture between two international consultants. The work described in this paper was undertaken as part of this checking process for one particular cut and cover structure called CC210. CC210 is a 400 m long cut and cover structure and carries up to 5 lanes of traffic, which merge and diverge inside the structure. An 80 m long section of CC210 was designed as a twin span cast in-situ concrete arched roof structure. An arched roof structure was chosen as it could efficiently carry the high vertical loads due to the significant depth of backfill over the tunnel. The span of the arches is approximately 20 m and 22 m. Each of the arches has a ring thickness generally of 500 mm. These large span arches were cast in 12m long sections. The base slab, interior and exterior walls of the CC210 structure below the arches are constructed from cast in-situ reinforced concrete. The wall concrete and the slab immediately below the walls are cast directly against the excavated rock surface. The remainder of the slab is cast onto a no fines concrete drainage layer. The permanent walls were laterally ed at the top by the smoke duct tie slab. In the UK there was a high profile collapse in 2005 during backfilling over a single span precast concrete arch structure at Gerrards Cross near London. The risks of working with such structures were therefore very well appreciated by the design and checking team,

and by the contractor. Considering the complexity of the structure, the interaction between the structure and fill, and the interaction with temporary works, the arch structure was analysed in different ways using 2D finite element software Plaxis v9. The different methods ensured that all anticipated effects of compaction of the fill were ed for, such as introducing stress history, locked in horizontal ground stresses and of potential variations to the construction sequence. A structural model was also developed to validate the forces in the various of the structure. Results from the Plaxis analyses will be compared and conclusions drawn on the accuracy of the results. Comment will also be made on the mesh generation facility within Plaxis 2D and the limiting factors encountered during the design. 1.2 Geology The geology of Brisbane is complex with a great diversity of the rocks within a small area. During the late Triassic period (220 million years ago), volcanoes in the region were active. Actual lava flows were rare but instead large volumes of rock were blasted out of the vents as floods of fragments in gas-rich clouds.Around Brisbane itself, violent volcanic eruptions of rhyolitic compositions saw clouds of ash mixed with air and fluids rush down valleys in the hilly terrain. Such clouds consist of a dense mass of semi-liquid pumice and dust close to their melting points with hot gas between acting as a lubricant. Some of the ash flows travelled over 100 km, but when they came to rest, the material was so hot that its own weight compacted and welded it to a very strong rock called welded tuff or ignimbrite. This rock unit is known as the Brisbane

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Figure 3. Typical Cross-section.

than the upper ground water level at +11.5mAHD. Based on this, the artesian water pressure in the Brisbane Tuff was considered in the design with head at +11.5mAHD in addition to the ground water level of +7mAHD for the soil layers above the Brisbane Tuff. 3 ARCH STRUCTURE 3.1 Description of the structure Figure 1. 3D Geological Model.

Figure 2. Geological Cross-section.

Tuff and provided the main founding medium for the CC210 structure. Overlying the Brisbane Tuff here is the Aspley / Tingalpa formation which are shales, siltstones and sandstones formed by the deposition of fine sediments. 2

GROUND MODEL

2.1 Geological cross-section After examining various cross-sections from a 3D ground model, see Figure 1, the geological crosssection adopted for the analysis of the structure is shown in Figure 2. 2.2 Ground water Data from boreholes and shallow wells installed in the area of CC210 indicate ground water level to be at +7mAHD. Some deeper boreholes installed into the Brisbane Tuff indicated sub-artesian ground water conditions within the Brisbane Tuff. The piezometers installed in the Tuff indicated water pressure higher

CC210 is a 42 m wide cut and cover structure with an arched roof consisting of two concrete arches with maximum backfill thickness of 22 m above them, see Figure 3. The arches are fully fixed at the springing points to the tie slab and to the walls of the carriageway section below. The base is fully fixed to the carriageway section walls, which is required for their stability. The form of the arches near to the springing points was sub-optimal resulting in greater bending stress in the arches in this area. To accommodate this, the arches were thickened and more heavily reinforced at the springing points. The thickness of the arch is 500 mm around the crest and then increases gradually to 1000 mm at the springing points. The structure follows a bottom-up construction sequence with the excavation ed by temporary anchored secant piles to the top of the Brisbane Tuff and below using temporary rock bolts. The base slabs were constructed first, then the walls followed by the arches, and finally the tie slab. As a result of this sequence the forces resulting from the arch selfweight are transferred to the walls. The tie slab acts to tie the arch helping it to resist all the imposed loading including the weight of backfill. The backfilling over the arches is divided into two zones, referred to as A and B, as shown in Figure 3. Zone A covers the area from the top of the structure up to 2 m above the arch crown, and beyond that up to ground level is zone B. The permanent external walls will be designed for full hydrostatic water pressure whilst the base slab and internal walls are drained. 4

FINITE ELEMENT ANALYSIS (FE)

4.1 The FE software The FE software Plaxis 2D was used as the tool for the analysis of this structure. This plane strain

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6.

7. 8.

Figure 4. Different aspects to be included in the Plaxis analysis.

problem would be adequately investigated in a 2D analysis instead of a complete 3D model, saving a lot of computational time. Plaxis was chosen because this was the only 2D FE tool common to both the t venture design team in Australia, and to Arup UK. A Plaxis 3D model was also developed to study the interaction of the ends of the arches with another structure but this is not described in this paper.

9. 10. 11.

a) Maximum vertical load + Minimum horizontal load b) Minimum vertical load + Maximum horizontal load c) Unbalanced loading from left d) Unbalanced loading from right.

4.2 Analysis methodology Due to limited available information on the material parameters, the linear elastic-plastic Mohr-Coulomb model was selected to model the ground and backfill materials. After finalising the geological model, various other critical issues needed consideration in the analysis including backfilling and achieving the longterm conditions. Due to so many variables involved in the analysis, it was imperative that none of them should be missed when studying their impact on the arch forces and deformations. To help a chart was developed as shown in Figure 4. A brief explanation of Figure 4 follows; 1. Due to the overlap between the arch plates and soil, the weight of the plate was reduced to for this overlap. 2. The friction slip between the arch and soil was considered by using various values of interface friction factor (Rint). A brief description of this is presented in section 4.4. 3. Different methods of modelling horizontal ground pressure due to compaction needed to be studied. This is discussed in more detail in section 4.5. Ranges of stiffness were also considered. 4. Water pressure was considered to be developed behind the permanent walls, soon after the completion of backfilling. 5. Any low permeability soil modelled as undrained during construction will be changed to drained,

with the appropriate consolidation stage to eliminate the excess porewater pressure developed during construction. There is a wide range of possible water pressure regimes over the arches after backfilling is complete, e.g. seepage through the secant piles, seepage under the toe of the piles, surface water infiltration etc. Therefore, assessment of worst loading on the arches was undertaken assuming different water pressure distributions on the arches. Effects of differential effective stresses were considered. The stiffness of structural elements to be changed to long term properties. The properties of the arch elements for the short term were calculated assuming the full Young’s modulus of concrete (Ec) and were reduced to values corresponding to 70% of Ec for long term conditions to for shrinkage and cracking. The properties of the wall elements were calculated based on the recommendations of CIRIA C5801 for retaining walls. The different potential sequences for de-stressing of the temporary ground anchors was considered in the analysis. Similar to the ground anchors, various sequences of de-stressing of the temporary rock bolts was considered. Unbalanced loading over the structure was applied as below;

12. To consider the effect of rock creep over the structure, a reduction in properties of the Brisbane Tuff was considered in the analysis. 4.3 FE mesh The initial Plaxis model attempted to include all the variations in the ground model, but with the structural elements and with the addition of surcharge to model the compaction pressure Plaxis failed to generate the mesh due to the closely spaced nodes. The model was subsequently refined by simplifying the ground model; after five attempts a satisfactory mesh was successfully achieved having 7724 elements and 63,314 nodes, refer to Figure 5. 4.4 Interface between arch and soil To ensure no water seepage into the structure, a waterproofing membrane was specified around the arches. The presence of the membrane will reduce the interface friction between the fill and arches. To consider this effect in the Plaxis analysis, a simplified model was set up considering a range of values of the interface friction factor from 0 to 0.67.A value of Rint = 0.3

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Figure 5. Final FE mesh, 7724 Elements & 63,314 Nodes.

was then chosen to model slip between the fill and the arches in the final models.

4.5

Backfilling sequence

An important issue to consider in the analysis was the backfilling sequence and the application of compaction pressure. This required a careful examination of the load deformation behaviour and variation in arch forces under different backfilling sequences. As described in section 3.1, the area to be backfilled was divided into two zones, referred to as A and B. The properties of the granular backfill and the compaction equipment were defined for each of the two zones. In zone A, the backfilling layer thickness was specified as a maximum of 300 mm to achieve adequate compaction and the maximum differential on either side of each arch was specified as 600 mm. While in zone B a maximum differential of 1m was specified again with the maximum layer thickness of 300 mm in order to achieve adequate compaction. The horizontal compaction pressure to be applied to zones A and B was calculated based on the method prescribed in CIRIA C5162 , for the assessment of compaction pressure on retaining walls, refer to Figure 6. This method is the procedure recommended in Australian standard AS5100.3 section C8.2. The horizontal compaction pressure to be applied during backfilling was calculated based on Figure 6 and was 10 kPa for zone A and 20 kPa for zone B. In order to reduce the number of stages in Plaxis, it was decided that modelling of backfilling would be in 600 mm thick layers in zone A and 1m in zone B with the corresponding minimum horizontal compaction pressures to be achieved. A study was then needed to assess the value of the vertical surcharge required over each backfill layer to achieve the calculated minimum horizontal compaction pressure and also the effects of different backfilling sequences on the arch forces and deflections. Carrying out this study on the full Plaxis model would have required enormous computational time and hence was not considered practical. Therefore a simple model was produced to carry out this study, refer Figure 7. The model shown in Figure 7 considered only the arches and tie slab as structural units with backfilling

Figure 6. Assessment of compaction pressure on retaining walls – Reproduced from CIRIA C5162 .

Figure 7. Simplified model to study the backfilling sequence.

and vertical surcharge/compaction pressure as applied forces. In the first step, the aim was set to assess the vertical surcharge required to produce the required minimum horizontal compaction pressure. This would have been much easier for any vertical wall but since the arches were curved, and due to ground arching between them, several trials were carried out with different values of surcharge to make sure that the required lateral stress to be applied over the arches was achieved. This assessment was done for each backfilling stage up to the top of zone A. The next step was to assess the limitations on the backfilling sequence over the arches. After many iterations, the following two methods were identified as being critical in of structural stability, and as would be expected the method of backfilling in zone A was found to be the predominant factor influencing the resulting loads in the structure. Method 1: Backfilling between the arches first over the full width with a level difference of one layer thickness on either side and apply vertical surcharge/compaction pressure. The compaction pressure was then removed from inbetween the arches and backfilling carried out on the sides followed by the compaction pressure on the sides. The compaction pressure on the sides was then removed and backfilling carried out back in the middle with application of compaction pressure and so on. Figure 8 shows the stage of backfilling in the middle of the arches.

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Figure 8. Backfilling as per Method 1.

Figure 10. Plot of Steady State Seepage. Figure 9. Backfilling as per Method 2.

Method 2: Backfilling between the arches first over the full width with a level difference of one layer thickness on either side of arches and apply the compaction pressure. With the compaction pressure in the middle of arches present, carry out backfilling on the sides and apply the compaction pressure. Backfill in the middle of the arches with compaction pressure on the sides present, apply compaction pressure in the middle and at same time remove previous compaction pressure present in the middle. The same approach was also followed for the backfilling on the sides. Figure 9 shows the stage of backfilling in the middle of the arches. The first method was found to induce more sway in the arches resulting in higher bending stresses in the third points of the arches during the backfilling in zone A. The second method resulted in greater bending stresses near to the springing points of the arches. Apart from this, both methods indicated final forces generally of a similar magnitude throughout the structure. Both methods considered similar modelling of the backfilling in zone B, which was backfilling in one single layer over the full span of the structure and applying vertical surcharge/compaction pressure in four stages of the same length. This was finalised after studying the impact of different ways of backfilling and compaction in zone B, which was not found to significantly influence the arch forces. 4.6

Output

The output from the Plaxis analysis in of stresses, deformations and structural forces was carefully examined and interpreted to make sure that the structure was behaving as predicted. One of the examples on the checking of the model performance would be to check the behaviour of the Brisbane Tuff under the base slab. Since drainage is provided for the base slab and considering the Brisbane Tuff permeability to be very low, it was important to check the distribution of pore water pressure in the Brisbane Tuff and its stability under that water pressure regime. The distribution of pore water is shown in Figure 10, and Figure 11 shows the plot of principal stresses. These confirmed that due to the high strength of the Brisbane Tuff it will form an inverted arch to sustain the water pressure which could potentially develop in the long term.

Figure 11. Plot of Principal Stresses.

Figure 12. Bending moment envelope of all of the structural elements.

Figure 12 shows typical output of the forces in the structural elements for the long term conditions. 5

STRUCTURAL DESIGN

5.1 Independent forces check Due to the complex soil structure interaction being modelled as a part of the arch design, Plaxis was relied upon for the overall design of the arch. To the structural results, normal ground stresses were extracted from Plaxis and applied to a structural model produced using Oasys GSA. This demonstrated a satisfactory correlation between member forces in GSA and in Plaxis. GSA was then used in the analysis of the seismic and fire load cases.

5.2 Structural design of the arch The precise construction sequence of the arches was considered to be a key element of the design, in particular the process of backfilling above the arches. The arches flex considerably during construction; their shape changing depending on backfill state. Any alteration to the construction sequence from that prescribed would result in stresses that the structure was not designed for, which would be likely to affect the long term stability of the arches.

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CONCLUSIONS

Plaxis is an excellent tool to carry out routine geotechnical analysis and works satisfactory for such work. However, for analysis requiring complex models, like one described in this paper, Plaxis becomes rather difficult to control and time consuming due to the following reasons;

Figure 13. Example of Moment-Axial interaction chart used in arch design. The solid line represents the ultimate strength envelope of the section considered and the discrete points represent co-existent loads from all stages modeled in the Plaxis analysis.

Significant axial loads exist in the arches, meaning the interaction of moment and axial force was important. Due to the depth of the structure, this was also the case in the walls, with significant axial compression loads being developed. As a result, structural capacity was checked on the basis of co-existent loads (Moment-Axial and Shear-Axial) as opposed to the use of envelopes. For each critical stage, the co-existent member forces (M, N & V) were investigated, allowing the structural stability to be verified for all considered load scenarios. Moment-Axial interaction charts were used to check that the arches were not overstressed during any stage of construction or in the long term. The co-existent values from each Plaxis stage for each structural element were extracted and plotted with the section interaction chart. Figure 13 shows one such chart, which shows the ULS capacity of a meter-wide strip of arch near to the springing point. The plot demonstrates the capacity of the arch to be sufficient for all stages.

1. Due to unstructured and relatively difficult to control mesh generation, significant problems were encountered to achieve the satisfactory mesh for this analysis. The authors request that Plaxis enhances the software’s applicability by allowing increased manual control on the mesh generation. An application to import a mesh into Plaxis will be helpful. 2. Extracting the forces from Plaxis is another aspect which needs to be improved. The lead author spent significant time pulling out the forces and displacements of each structural element for the critical stages, exporting them to Microsoft Excel, and then combining them to produce design envelopes. If Plaxis could allow one to choose the appropriate stages, to then export to Excel in one go this would really enhance its features. 3. The automatic stage calculation is a very good upgrade in the version 9.0. However, its effectiveness is impaired due to its requirement to re-calculate the steady state seepage calculations. REFERENCES 1. Gaba, A.R. & Simpson, B & Powrie, W & Bean, D 2003. CIRIA C580 Embedded retaining walls. CIRIA, UK 2. Chapman, T. 2000. CIRIA C516 Modular gravity retaining walls. CIRIA, UK

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Analysis of a bolt-reinforced tunnel face using a homogenized model E. Bourgeois Université Paris-Est, LC-MSRGI,

E. Seyedi Hosseininia University of Tehran, Tehran, Iran

ABSTRACT: This communication deals with the effect of tunnel face reinforcement on the wall convergence and on the loads in the lining, studied by means of a homogenized anisotropic model for reinforced ground. While the tunneling process is generally handled in a plane strain framework (within the so-called convergenceconfinement method), tunnel face reinforcement makes it necessary to take into the three-dimensional nature of the problem. In the case of an isotropic and uniform initial stress state, and of a circular tunnel, analyses can be performed in axisymmetric mode. Within this framework, finite element simulations have been carried out, using the finite element software CESAR-LC, to simulate the process of tunnel excavation and lining construction. Results indicate that reinforcement of the tunnel face reduces tunnel convergence and decreases the compressive forces in the lining. In the last place, it is shown that results obtained with the anisotropic multiphase approach can be approximated with an isotropic model with adjusted parameters, which may be useful for preliminary design.

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2

INTRODUCTION

Reinforcement of tunnel face by bolts is, in the first place, an efficient technique to improve the stability of the ground during construction. It is also seen as a way of decreasing deformations around the tunnel, in order to keep surface settlements within acceptable limits in the case of shallow urban tunnels. The technique is very often used in tunneling engineering. However, there is still no simple and well recognized design method to choose the number, length and diameters of bolts, or the type of bonding between the bolts and the ground, etc. Numerous studies have been undertaken to get a better understanding of the influence of tunnel face reinforcement on the tunnel behavior: numerical analyses have proven that face reinforcement can reduce tunnel face displacements (Kavvadas & Prountzopoulos, 2009) as well as convergence and change the loads on the lining (Chungsik & Hyun-Kang, 2003). Such analyses remain difficult to perform, because of the number of bolts in the face (several tens) and their size. In this context, it can be efficient, to use homogenized approaches to take into the role of the bolts (Bourgeois et al, 2002, Wong et al, 2004, Wong et al, 2006). In this paper, we use such a homogenization procedure to discuss the influence of bolt reinforcement on wall convergence and on compressive forces in the lining, on the basis of numerical simulations of the excavation process of a deep tunnel with circular section.

MULTIPHASE MODEL

2.1 Principles The principle of the homogenization procedures is to replace the heterogeneous composite material made of the association of the ground with the bolts by a homogeneous material having “equivalent” mechanical properties. Recently, de Buhan and Sudret (1999) have introduced a model in which the reinforced ground is replaced by the superposition of two continua in mutual mechanical interaction. In such a framework, a displacement field and stress field is associated with each phase. Phases are connected to each other through an interaction law (Bennis & de Buhan, 2003). An example of application of this approach to tunnel reinforcement by bolts can be found in de Buhan et al (2008). The multiphase approach has been introduced in the finite element code CESAR-LC (Humbert et al, 2005), and used for the analyses presented here, under the assumption that there is a perfect bonding between the ground and reinforcements: within this framework, one has to handle only one displacement field, common to both phases; however, the model still includes two distinct stress fields. The elastic properties of the reinforced ground as a whole are the sum of the elastic properties of the initial ground and of a uniaxial tensor increasing stiffness in the direction of the bolts. Thus, even if the ground is initially isotropic, the reinforced material has anisotropic elastic properties. Much in the

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same way, strength properties of the reinforced ground are improved in an anisotropic way. 2.2

Overview of the general formulation

The “multiphase model” is a generalized homogenization procedure in which the bolt-reinforced ground is represented, not by one single medium, but by the superposition of two continuous media: one, called the “matrix phase” represents the ground, whereas the “reinforcement phase” is the macroscopic counterpart of the bolts network. This leads to the introduction, at the macroscopic scale, of two displacement fields denoted by ξ m for the matrix phase and ξ r for the reinforcement phase. The matrix phase is associated with a Cauchy stress tensor σ m , and the reinforcement phase with a (scalar) density of axial force in the bolts per unit area transverse to the direction of the bolts, denoted by σ r . The momentum balance is expressed for each phase separately as:

where er is the unit vector in the direction of the bolts, and I er denotes the volume density of interaction forces exerted by the reinforcement phase on the matrix phase (Volume forces have been omitted to keep equations simple). Three constitutive laws describe the behavior of the reinforced ground mass: one for the ground, one for the reinforcement phase, and one for the interaction. Since the volume of the bolts is small compared with that of the reinforced ground, it is assumed that the matrix phase has the same behavior as the initial ground. In what follows, we have adopted the usual Mohr-Coulomb constitutive model. The behavior of the reinforcement phase is described by a linear elastic model:

where εr denotes the vertical strain of the reinforcement; E r is the product of the Young’s modulus of fiberglass bolts Eb by the ratio η of the bolts volume over the overall volume of reinforced ground. The density of interaction force between the matrix and the reinforcement phases is described by a onedimensional constitutive law, that can be linear or not. In what follows, we use a simplifying assumption and it is not necessary to describe precisely the constitutive law associated with the interaction. 2.3 The simplified case of perfect bonding We make the additional assumption that there is a perfect bonding between the bolts and the ground, in the sense that the displacement fields of the matrix and the reinforcement are equal: ξ m = ξ r . With this assumption, the numerical implementation of the multiphase model is much simpler, since we can use standard finite

elements, without having to introduce extra nodal degrees of freedom. However, the stresses associated with the “matrix” and the “reinforcement” are computed separately, in order to compute the plastic strains in the ground. 3

NUMERICAL MODEL

In this study, we present simulations of the excavation of a tunnel with a sequential method. The tunnel section is assumed to be circular, with a radius R = 2.5 m, the depth of the tunnel axis is equal to 75 m and the initial stress state is isotropic, so that the coefficient of lateral pressure at rest K0 is equal to 1. It is assumed that variations of the initial in-situ stress field can be neglected: the stress field is homogeneous, and the mean stress is equal to 1.5 MPa. Under these assumptions, the problem can be dealt with in axisymmetric conditions. Simulations do not integrate the introduction of the bolts in the ground, but we assume that there is a preexisting longitudinal reinforced zone along 35 m of the tunnel axis. This assumption simplifies greatly the preparation of data, but can be criticized because the typical length of actual bolts lies in the range between 10 to 25 m. However, it can be expected that the traction forces in the bolts are almost negligible beyond a given distance from the tunnel face, so that the length of bolts taken into in the simulations makes little difference on the final results. This assumption is discussed later. The density is equal to 1 bolt per square meter of tunnel face, which corresponds to a volume ratio of η = 10−3 if the bolt diameter is equal to 35 mm. Each step of excavation consists in excavating the stross over a length of 2.5 m. For practical reasons, the installation of the shotcrete lining is performed only after the excavation of the 2.5 m step is completed. There is therefore a given length of ground left uned behind the tunnel face. In the simulations, for each step of excavation, a 2.5 m-long concrete lining segment with a thickness e = 20 cm is installed to the ground excavated during the previous step. The mesh used is presented in Figure 1.All elements are of quadratic type. Elements far from the tunnel are triangular; elements close to the tunnel are quadrangular. The mesh includes 3100 nodes and 1300 elements. The simulation of the excavation process (deactivation of excavated zones, activation of the lining in sequence) leads to defining 34 different zones. The interest of the multiphase model lies in the fact there is no need to describe each bolt separately. The behaviour of the unreinforced ground is described by the Mohr-Coulomb model with a linear isotropic elasticity. We adopted the following values of the parameters E = 150 MPa ; ν = 0.4; c = 100 kPa ; φ = 32 degrees, ψ = 2 degrees. The behavior of bolts is linear elastic with Eb = 20 GPa (Young’s modulus for fiber-glass bolts). Bolts are parallel to the tunnel axis.

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Figure 2. Generic step of the modeling sequence of tunnel drilling: forces are applied on the boundary of the excavated zone (arrows); a lining segment is activated behind the tunnel face. Figure 1. Mesh used in CESAR-LC.

The lining segments are assumed to behave as a linear elastic material with EC = 5000 MPa and ν = 0.25. The initial (in situ) stresses in the model are introduced and then, excavation steps are carried out. Each simulation step of the construction process includes several elements (Figure 2): – the stiffness of the elements of the zone to be excavated is set to zero; – the excavated ground exerted on the ground that remains a system of forces that must be « removed ». The finite element procedure consists in computing the appropriate nodal forces to take into this « unloading ». On the whole, the remaining ground was subjected to compression forces from the excavated ground ; in the final state, it is subjected to zero surface forces : the excavation process is therefore equivalent to applying tensional forces, shown in figure 2; – activating the lining segment corresponding to the previous excavation step.

Figure 3. Traction force in a bolt placed in the center of the tunnel face.

The excavation process starts from the bottom of the mesh and the tunnel face moves upwards.

excavation steps). The results show clearly that the traction force decreases rapidly ahead of the tunnel face, in such a way that the traction in the bolt is negligible at a distance of 10 m from the tunnel face. In other words, assuming that the ground is reinforced over a distance larger than that of the actual bolts has no significant influence on the results of the simulations.

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4.2 Wall convergence

RESULTS

4.1 Traction forces in the bolts Figure 3 shows the values of the traction forces computed in the bolts placed in the tunnel face (after 10

Figure 4 shows the convergence (i.e. the radial displacement) of the tunnel wall along the axis of the tunnel. The abscissa x = 25 m corresponds to the position of the tunnel face after the completion of the tenth

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Figure 5. Homogeneous homogeneous elements. Figure 4. Comparison of the radial displacement of the wall along the tunnel axis, in the reinforced and unreinforced cases, after completion of 10 excavation steps (tunnel face is located at x = 25 m, the excavated zone being on the left).

excavation step (the section of the tunnel already excavated is on the left, for x < 25 m, and the ground not yet excavated corresponds to x > 25 m). In the first place, it is worth noting that displacements are almost uniform at a distance larger than 5 m behind the tunnel face, showing that the lining is stiff enough to prevent further convergence of the ground. Besides, with the parameters taken for simulations presented here, the radial displacement in the case where the tunnel face is reinforced is about 10% smaller than in the case without bolting (38 mm vs. 42 mm).

4.3

Compressive force in the lining

For a circular section, it is possible to find the compressive force in the lining. Since the lining is a thin ring (R/e > 10), the compressive force N can be calculated by (Panet, 1995):

where: urmean is the mean radial displacement of the lining, Ksn is the normal stiffness of the lining:

Ksf is the flexural stiffness of the ring, given by:

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(two-phase)

and

non-

SIMPLIFIED ANISOTROPIC MODEL FOR THE REINFORCED ZONE

The ground is modeled as an isotropic elastic- perfectly plastic material; however, the reinforced zone overall behavior is anisotropic. In this section, we discuss the possibility to use an isotropic model for the reinforced ground as a whole, with “improved” values of the parameters. The role of reinforcement element in the reinforced ground is to increase the rigidity as well as the strength of the ground in the direction of bolts. In what follows, we propose to for the increase in strength provided by the bolts by replacing the cohesion of the initial unreinforced ground by an “equivalent” increased cohesion, all other parameters remaining unchanged. The improved cohesion is denoted by cH and its value is estimated as follows. Consider a reinforced soil element as shown in Figure 5 in which the reinforcements are placed horizontally. The element is subjected to a mechanical loading defined by major (#2 ) and constant minor (#1 ) principal stresses. Now, it is possible to simply replace it by a homogeneous two-phase element. Since there is a perfect bonding between phases, the equilibrium condition results in:

where σim (i = 1,2) and σ r correspond to local stress components in matrix and reinforcement phases, r respectively. εm i and ε are the local strain components in the same order. Consider the failure criterion of the soil as follows:

Substituting (7) in (8), one gets: The simulations presented above give a compressive force in the lining of 9 MN/m without bolts and 8.1 MN/m in the reinforced case. This shows that tunnel face reinforcement can not only improve tunnel face stability, but also decrease tunnel wall convergence and the compressive force in the lining.

where E r is the stiffness of the reinforcement phase (equal to the product of the modulus of the bolts Eb by

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increased to for the bolts would lead to smaller radial displacements, and would not provide a better agreement. Sensitivity analyses (not detailed here) also show that results are entirely different if the increased homogeneous cohesion is associated with the whole ground mass and not only with the ground ahead of the tunnel face. They also show that the radial displacement depends strongly on the value of the increased cohesion.

6 Figure 6. Comparison of convergences obtained between with an isotropic and an anisotropic model (reinforced tunnel face).

their volume fraction η). On the other hand, if the new soil element with cH is subjected to the same stress conditions defined by #1 and #2 , failure is associated with the following condition:

By comparing equations (8) and (10), the value of the cohesion cH can be assessed as:

The cohesion found depends on the deformability of the reinforcement phase. It can be noted that Equation (11) is similar to the expression stated by Charmetton (2001) if the term Er εr is replaced by ultimate tensile strength of the reinforcement phase. Analyzing the results of the simulation for the nonreinforced tunnel problem, one observes that the axial strain in the vicinity of the tunnel face is about 2.5%. Assuming that the strain would be about 1% in the case of a bolt reinforced face, the value of the improved “equivalent” cohesion cH is approximately equal to 280 kPa. A simulation with this value of the cohesion gives for the axial strain a value of 0.8% which is close to the value of 1% taken into to estimate the equivalent cohesion. In Figure 6, the result of the new isotropic analysis is compared with the simulation carried out with the (anisotropic) multiphase model. As can be seen, the agreement between the results of both models is very satisfactory: the radial displacements obtained with the homogeneous isotropic cohesion are almost equal to those obtained with the multiphase model. In other words, the homogeneous equivalent model, with a modified value of the cohesion and all other parameters (especially stiffness parameters) unchanged, makes it possible to reproduce the increase of stiffness of the ground mass as a whole. It can be expected that a simulation in which the elastic moduli of the reinforced ground were

CONCLUSION

Reinforcement of tunnel faces by bolts is a common practice, but a difficult problem for designers. The difference between the dimensions of the bolts and the area in which the stress state is modified by the excavation, the mechanical interaction between the bolts and the ground, and the three-dimensional nature of the problem make it difficult to build models to analyze the performance of the technique. It is worth mentioning that the role of radial bolts placed in the tunnel wall, in planes perpendicular to the tunnel axis, can be taken into in classic plane strain analyses (using the convergence confinement method), the bolts being seen as an increase in stiffness of the ground surrounding the excavation. In the case of bolts placed in the tunnel face, things are more complex, because the area reinforced by the bolts is eventually excavated and the bolts are destroyed as the tunneling process goes on. The finite element simulations presented here are based on a homogenized approach that makes it possible to overcome the main difficulties of the problem. They are based on several assumptions that can be discussed, but provide a way of overcoming the complexity of the problem. It can be pointed out that the simulations presented here are carried out in axisymmetric conditions, but the model is available to perform fully three-dimensional simulations if necessary (in the case of a non-circular section, or if the initial stress state is not isotropic and homogeneous). Results tend to show that tunnel face reinforcement reduce both the convergence of tunnel wall and the compressive forces in the lining. The decrease is of the order of 10%, which remains moderate, but it must be recalled that many parameters are involved in the analysis (one could for elastic non linearities, or discuss the influence of the length of uned ground behind the tunnel face). From a qualitative point of view, it is interesting to note that a longitudinal increase in stiffness due to bolts results in a decrease of the wall convergence. This is clearly the result of the modification of the three-dimensional stress distribution due to bolts: it seems therefore difficult to take them into in a plane strain analysis (as is usually done using the usual convergence-confinement method). Unlike the reinforcement of the surrounding ground by radial bolts, the use of longitudinal bolts in the tunnel face

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cannot be analyzed without taking into the three dimensional nature of the problem. It is also recalled that the model provides an estimated of the traction forces in the bolts, which may be useful to choose the number and diameter of bolts. In the last place, a simple analysis makes it possible to model the reinforced zone with a classical homogeneous isotropic model, provided that a suitable increased value of cohesion is taken into . The increased cohesion depends on the deformability of the bolts, and requires making an assumption regarding the axial strain in the bolts close to the tunnel face. This assumption has to be based on engineering judgment, empirical knowledge, or numerical analysis. REFERENCES Bennis, M. & de Buhan, P. 2003. A multiphase constitutive model of reinforced soils ing for soil-inclusion interaction behavior, Mathematical and computer modeling 37: 469–475. Bourgeois, E., Garnier, D. & Semblat, J-F. 2002.A 3D homogenized model for the analysis of bolt-reinforced tunnel faces, 5th Int. Conf. on Num. meth. In Geotech. Eng.: 573–578. Charmetton, S. 2001. Renforcement des parois d’un tunnel par des boulons expansifs retour d’expérience et étude numérique, Ph D thesis, Ecole centrale de Lyon. Chungsik, Y. & Hyun-Kang, S. 2003. Deformation behavior of tunnel face reinforced with longitudinal

pipes-laboratory and numerical investigation, Tunneling and unground space technology 18(1): 303–319. de Buhan, P., Bourgeois, E. & Hassen, G. 2008. Numerical simulation of bolt-ed tunnels by means of a multiphase model conceived as an improved homogenization procedure, Int. J. for Num. and Analytical Meth. Geomech. 32 (13): 1597–1615. de Buhan, P. & Sudret, B. 1999. A two-phase elastoplastic model for unidirectionally-reinforced materials, Eur. J. Mech. A/Solids 18: 1995–1012. Humbert, P., Dubouchet, A., Fezans, G., Remaud, D. 2005. CESAR-LC : A computation software package dedicated to civil engineering uses, Bull. des laboratoires des ponts et chaussées, n◦ 256–257,7–37. Kavvadas, M., & Prountzopoulos, G., 2009. 3D Analyses of Tunnel Face Reinforcement using Fibreglass Nails, 2nd Int. Conf. on Computational Methods in Tunnelling, Bochum, 9–11 September 2009, Aedificatio Publishers, 259–266. Panet, M. 1995. Le calcul des tunnels par la méthode convergence-confinement, Presses de l’Ecole Nationale des Ponts et Chaussées, Paris. Wong, H., Trompille, V. & Dias, D. 2004. Extrusion analysis of a bolt-reinforced tunnel face with finite element ground-bolt bond strength, Canadial Geotech. J. 41 (2): 326–341. Wong, H., Subrin, D., & Dias, D. 2006. Convergenceconfinement analysis of a bolt-ed tunnel using homogenization method. Canadian Geotechnical Journal Vol. 43, n◦ 5, pp. 462–483.

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Class A prediction of the effects induced by the Metro C construction on a preexisting building, in Rome F. Buselli, A. Logarzo & S. Miliziano Dipartimento di Ingegneria Strutturale e Geotecnica “Sapienza” Università di Roma, Italia

A. Zechini Roma Metropolitane, Italia

ABSTRACT: This paper reports the main results obtained through a numerical study aimed to predict the effects induced on an old building by tunneling operations for the construction of Metro C line, in Rome. In order to achieve high quality of class A prediction, full 3D finite elements numerical analyses have been carried out. The most important simulated features are: i) the advancement of tunnel front; ii) the pressure for front ; iii) the TBM geometry (weight and conicity of shield); iv) the tail void grouting; v) the building foundations. A simple elasto-plastic model with Mohr-Coulomb strength criterion and tension cut off is used to describe both soil and foundation behaviour. Results of preliminary analyses carried out using different mesh densities and tolerated errors are used to optimize the complete analysis achieving an acceptable compromise between calculation time and accuracy of the results. The study clearly underlines the necessity to incorporate into the model the presence of the building; in fact, both the weight and the stiffness of the building largely influence the solution.

1

INTRODUCTION

It is important to predict settlements due to excavation of tunnels in urban area to determine the effects induced on pre-existing structures. This prediction is very difficult because of the tridimensionality of the problem and due to the large number of factors that influence the interaction between soil-tunnel-surface structures. Using 3D numerical analysis it is possible to reproduce realistically the main features of the excavation process, without those assumptions that are necessary when the problem is approached by plain strain analysis (2D analysis). In detail, a 3D approach allows to simulate easily the main features of the excavation via Tunnel Boring Machine (TBM): the front , the conicity of the shield and the grouting in pressure of the tail void between the excavation profile and the lining. Moreover, in 3D numerical analysis it is possible to introduce 3D building structures into the model without conceptual difficulties. Settlements profile and volume loss obtained performing 3D simulations are simply analysis results which depend on geometry, soil characteristics and modality of excavation. Numerically obtained stress and strain distributions in the building structure can be analyzed to assess the level of damage induced both in term of cracks formations and reduction of structural safety level. The case study presented in this paper is a real case: the interaction between Metro C line and an

old building located near Pigneto Station, in Rome. Two circular cross-section tunnels will be driven by Earth Pressure Balanced Shield (EPB), to minimize as much as possible the excavation-induced settlements. The building is located above one of the tunnels but it interacts with both of them. Full 3D numerical analyses were carried out to obtain a true class A prediction, modeling in detail the most important aspects of the excavation process. Only the foundation slab of the building was simulated. A distributed load was applied on the foundation to take into the weight of the building. Young’s Modulus of the foundation was appropriately increased to represent the global stiffness of the entire building. Soil and building foundation behaviour have been modeled as simply elasto-plastic and the Mohr Coulomb strength criterion was adopted. In this paper, the geotechnical characterization of the site is reported after the descriptions of the building structure and the tunnels. Then, main features of the numerical models are described and main numerical results are related and compared. Three types of finite element analysis were carried out using Plaxis 3D Tunnel code (Plaxis 3D Tunnel, 2007): i) preliminary analyses with a simplified model to study the influence of mesh density and tolerated error on results accuracy and calculation time; the aim of those analyses was to achieve an acceptable compromise between accuracy and calculation time in the complete analyses; ii) greenfield analysis, that involves only the

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Figure 1. Building involved in this study.

construction of both tunnels (the building is not simulated); iii) full analysis to study the interaction between soil-tunnels-building. Finally, the main conclusions of this research are summarized.

2

BUILDING AND TUNNELS

The building under investigation is a 27 meters high civil habitation located near Pigneto Station, in Rome. The building was built around 1930; at the beginning it was a “T” shaped masonry structure of three levels (Fig. 1). In the fifties of last century, after the damages suffered during the Second World War, it was refurbished and four levels were added using reinforced concrete. The foundation of the building is a masonry slab placed two meters below the ground surface. The quality and the maintenance of the building are poor; a large number of cracks were detected. The structure is located just above the tunnel A; the tunnel B is 23 m far away (Fig. 2). The tunnels have a cover of 21 m.The adoptedTunnel Boring Machine has an external diameter, D, of 6.7 m; seven 0.30 m thick precasting reinforced concrete elements compose the lining (the outside diameter is 6.4 m).

3

Figure 2. Plan view of building and tunnels.

GEOTECHNICAL CHARACTERIZATION

Soil profile and hydraulic conditions have been determined on the base of an extensive geotechnical investigation involving laboratory and in-situ tests. The main results of the geotechnical characterization are reported in Buselli (2009). The subsoil consists of a top layer (8 m) of man-made ground (R), overlaying a volcanic soil deposit, characterized by alternation of hard rock layers, lithoid tuffs with variable cementation (T1 and T2), and thin layers of partially weathered tuff (Ta). The thickness of each layer ranges between 1.5 to 3.0 m. The presence of T1 and T2 strata is important because of their high mechanical properties and their location (immediately above the tunnels crown). The

Figure 3. Geotechnical soil profile.

tunnels are mostly bored through the underlying fluvial prevolcanic deposit made up of three main levels, from the top to the bottom: St, a very dense silty sand and clayey silt; Ar, a clayey silt and silty clay; and, at the end, Sg, a sandy gravel layer. The bedrock is at the bottom of Pleistocenic fluvial deposit: it consists of hundred meters of consistent and overconsolidated Pliocenic clay, Apl. Geotechnical cross section is represented in Fig. 3. Groundwater conditions are hydrostatic with the water table located 16 m below the ground level. Therefore, the tunnel will be excavated under the water level. The constitutive model adopted for soil behaviour is elastoplastic with Mohr-Coulomb strength criterion. Due to the adopted excavation modality (TBM, EPB), small strain levels are expected in the soil around the tunnel. Therefore, small strain stiffness moduli determined by dynamic soil testing were selected. According to the relationship proposed by Hardin (1978), Young’s moduli were assumed to increase with depth. Each layer is characterized by a Poisson’s ratio of 0.2 and the horizontal effective stresses are calculated according to the classical equation K0 = 1 − sin ϕ . The physical-mechanical soil parameters adopted in the analyses are summarized in Tab. 1.

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Table 1.

Physical and mechanical parameters of soils.

Soils

γ (kN/m3 )

c (kPa)

ϕ (◦ )

E (MPa)

K0

R Ta T2 T1 St Ar SG

17.5 16 16 16 18 18 20

5 17.5 100 450 25 40 0

33 32 45 59 30 30 35

185–190 298–302 998–1002 5998–6002 184–190 372–378 370–380

0.45 0.47 0.29 0.14 0.5 0.5 0.43

Figure 5. Detail of local refinement of the clusters around the tunnels.

Figure 4. Adopted mesh.

4

Figure 6. Scheme of excavation simulation. (1) TBM plate; (2) Front pressure; (3) Conicity of the shield; (4) Grout injection; (5) Fresh grout; (6) Hard grout.

3D NUMERICAL MODELLING

4.1 Geometry

Table 2.

Finite element mesh for soil is generated using “Plaxis 3D Tunnel v.2” commercial code. The model is 123 m wide, 47 m deep, and the total length in the z direction is 170 m (Fig. 4). In order to minimize the influence of mesh boundaries, according to Franzius & Potts (2005), the longitudinal distance between the tunnel face and the remote vertical boundary are fixed equal to 7 × D and 13 × D, ahead and behind the position of the studied structure building, respectively. The mesh is divided, in tunneling direction, in slices of 2.5 m thick. The deformation analyses do not extend into the deeper layer Apl. Since the bottom boundary is assumed to represent such a stiff ground layer, both horizontal and vertical total fixities have been adopted as displacement restrains. Roller s are applied to all vertical sides. For the non-symmetric analyses it was necessary to mesh the complete problem; a coarse mesh is considered in these analyses with a local refinement of the clusters around the tunnels (Fig. 4 and 5). Tunnel Boring Machine has a 6.7 m diameter and the shield is 10 m long. It is modeled as a four slices long ring of plate element; the weight of the plate is 53 kN/m, representing the full weight of the TBM including equipments for each meter. 4.2 Tunnel construction The construction process is simulated discontinuously removing slices of elements inside the profile of excavation 2.5 m tick for each step, while at the same time the TBM shield advance, activating shell on front and

Plate characteristics.

Plate

E GPa

υ

EA kN/m

EI kN/m2 /m

TBM Lining

210 38

0.33 0.15

1.2 · 108 3.7 · 106

3.8 · 108 6.8 · 105

deactivating shell on tail. A total pressure equal to the active horizontal earth pressure is applied to the front to the soil during excavation. This pressure is about 100 kPa at the crown and increases linearly (15 kPa/m) moving to the invert (Fig. 6). The shield is slightly conical: the tail radius is 10 mm less than the frontal one. In order to simulate a reduction of the tunnel cross section area, the shield has got a truncated cone shape. The continuous homogeneous final lining is switched on from behind the tail of the shield. Both TBM and tunnel lining are modeled as linear elastics, using shell elements with flexural stiffness, EI, and normal stiffness, EA, as reported in Tab. 2. The tail void injection is simulated using an axial pressure applied to the ring cluster between the lining and the excavation profile, opposite to the direction of tunnel advancement (Fig. 6), 50 kPa higher than the maximum value of the pressure at the front. The stage construction process considers the ageing process of the grout: fresh grout is characterized by a low shear modulus and incompressibility (γ = 21 kN/m3 , E = 1 MPa, υ = 0.49), while hard grout is very stiff (γ = 21 kN/m3 , E = 14 GPa, υ = 0.15). Volume elements are activated to fill the tail void (15 cm) assuming linear elastic properties for

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Figure 7. Foundation model.

them. The grout is assumed to be hardened after four slices. Tunnel A is the first one to be realized and, once it is completed, tunnel B advances. 4.3

Building model

Complete analysis studies the influence of building’s stiffness. Only the foundation of the structure is modeled (Fig. 7). Foundation consists of elasto-plastic cluster of elements with high compressive strength (σc = 1200 kPa) and a relatively low strength in tension (σt = 1/20σc ). The foundation is a masonry slab 1 m thick with a Young’s Modulus E = 35 GPa. This modulus was chosen about 100 times greater then the real one to consider approximately the stiffness of the whole structure. Assuming the Mohr-Coulomb strength criterion, the adopted strength parameters are c = 300 kPa, ϕ = 40◦ and σt = 60 kPa. A distributed load of about 10 kPa for each of the seven levels is applied on the foundation to take into the weight of the building. The foundation is connected to the soil by an interface, using a value of 0.5 for the strength reduction factor (the strength of the interface is half of the strength of surrounding soil). The presence of other buildings is simulated by the application of distributed load acting directly at the upper bound of the model. In the first phase of complete interaction analysis the building is “constructed” and the distributed load representing the surrounding buildings is activated; the resulting displacements are not taken into in further steps: displacements are set equal to zero before starting the construction of the tunnels. 5 ANALYSIS OF THE RESULTS 5.1

Preliminary analyses: influence of mesh density and error tolerance

In preliminary analyses, a small portion of complete mesh (the mesh was reduced in longitudinal direction; 123 × 47 × 30 m) has been adopted and only one tunnel was modeled. Geotechnical profile and geometry are the same as in the complete analyses. Fig. 8 shows the maximum vertical displacement and the calculation time obtained varying the density of the mesh (very coarse, coarse, medium, fine, very fine) and the tolerated error (0.1, 0.01, 0.001). The accuracy of the solution increases by increasing the density of the mesh and decreasing the tolerated error, but

Figure 8. Results of preliminary analyses.

Figure 9. Greenfield analysis: settlements profile in the cross section a-a and comparison with Gaussian curve on advancement steps of tunnel A.

the calculation time speedily increases. An acceptable compromise between calculation time and accuracy of the results is obtained adopting a coarse mesh density and a tolerated error equal to 0.01 (see Fig. 8). In fact, the difference between this solution and the most accurate solution obtained adopting a very fine mesh and a value of 0.001 for tolerated error is only about 5%. However, the calculation time reduces drastically. 5.2 Greenfield and complete interaction analyses Fig. 9 shows the evolution of the greenfield settlements profile calculated along the cross section indicated in Fig. 15a, during the construction of the tunnel A, at foundation level. The settlements begin to appear when the front of excavation is about 2 diameters behind the reference section (located at 72.5 m from the edge of the mesh); the maximum value was 3.2 mm at the end of the tunnel excavation. The numerical result matches quite well the Gaussian distribution originally proposed by Peck (1969) and successively adapted by Moh et al. (1996) for calculation of the settlements profile under ground level, selecting for K a value of 0.7 (Fig. 9). The resulting settlements profile is quite flat according to the presence of rock levels T1 and T2 located immediately above the tunnels crown. Similarly, the volume loss, VL , calculated from numerical results, starts to be appreciable when the front is

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Figure 10. Greenfield analysis: volume loss on advancement steps of tunnel A, cross section a-a.

Figure 11. Greenfield analysis versus complete interaction analysis: settlements profile in the cross section b-b at the end of tunneling operations.

Figure 13. Greenfield analysis versus complete interaction analysis: settlements profile in longitudinal direction at the end of tunneling A operations, section c-c.

Figure 14. Greenfield analysis versus complete interaction analysis: settlements profile in longitudinal direction at the end of tunneling B operations, section d-d.

Figure 12. Greenfield analysis versus complete interaction analysis: volume loss on advancement steps of tunnel B, cross section b-b.

about 2D behind the reference section. Then, it regularly increases and reaches its maximum value (0.32%) when the front is about 3D far away from the reference section. Further excavations do not affect the value of VL (Fig. 10). The weight and the stiffness of the building (Figs. 11, 12, 13 and 14) modify the subsidence basin. The presence of the building significantly increases the settlements: the maximum settlement value is about two times higher than the value obtained in greenfield analysis. This result is attributed to: i) the high weight of the building, only partially compensated; ii) the relative position: the building is located just above tunnel A; iii) the building is set entirely inside the sagging area of the greenfield settlements profile (probably the most important). Due to the building’s stiffness, however, the differential settlements and consequently the angular distortions are not appreciable (0.004%). According to classification proposed by Burland (1995), the expected class of damage of the building is “0”.

Figure 15. Considered sections for the representation of numerical analyses results.

Figs. 16 and 17 show normal stresses, N, and bending moments, M, on the lining in the cross section a-a, at the end of construction of tunnel A. The normal stresses obtained in the complete interaction analysis result higher then those obtained in greenfield conditions. In particular, at the crown, N increases of about 75%. Similarly for M, in the presence of the building, the results show an appreciable increment of the maximum values (for crown, sidewalls and invert arch). In this case too, the phenomenon is due to the building’s presence into the model. In fact, the stresses in the soil before the tunnel construction are higher compared to the greenfield conditions and, consequently, the resulting N and M on lining after tunnel construction are higher too.

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Figure 16. Greenfield analysis versus complete interaction analysis: normal forces distribution on lining at the end of tunnel A operation.

Figure 17. Greenfield analysis versus complete interaction analysis: bending moments distribution on lining at the end of tunnel A operation.

6

full 3D finite elements numerical analyses shall be carried out. The analysis shall include the simulation of the of the tunnel face, the conicity of the shield as well as the grouting in pressure of the annulus between lining and excavation profile. Settlements profile and volume loss obtained numerically are simply analysis results depending on geometry, soil characteristics and modality of excavation. The study clearly underlines the necessity to incorporate the presence of the building into the model: both the weight and the stiffness of the building play an important role. For the case under investigation, the presence of the building significantly increases the amount of settlements; this phenomenon is essentially due to the weight of the building and the relative position between the building and tunnel A. The weight of the building produces an increase of normal forces and bending moments on the tunnel lining. Due to the stiffness of building, however, the differential settlements are small and the angular distortions are not appreciable (0.004%). Consequently, the expected class of damage for the building is zero. REFERENCES Burland, J.B. 1995. Assessment of risk of damage to building due to tunnelling and excavation. Invited special lecture to IS-Tokio: 1st Int. Conf. On Earthquake Geotechnical Engineering. Buselli, F. 2009.Analisi dell’interazione tra lo scavo delle gallerie della Metro C in Roma e un edificio a struttura mista nel quartiere Pigneto. Tesi di laurea. Sapienza, università di Roma. Franzius, J.N., Potts, D.M. 2005. Influence of Mesh Geometry on Three-Dimensional Finite-Element Analysis of Tunnel Excavation. International journal of geomechanics, ASCE. Hardin, B.O. 1978. The nature of stress-strain behaviour for soils. State of the Art. Proc. Geotechnical Eng. Division Specialty Conference on Earthquake Engineering and Soil Dynamics, ASCE, Pasadena, California. Moh, Z.C. & Hwang, R. N. 1996. Ground movements around tunnels in soft ground. London, UK. Peck, R.B. 1969. Deep excavations and tunnelling in soft ground. Proc., 7th Int. Conf. Soil Mech., Mexico City, State of the art 3. pp. 225–290. PLAXIS, Plaxis 3D Tunnel Version 2.4, 2007. Finite Element Code for Soil and Rock Analyses.

CONCLUSIONS

In order to obtain realistic results about the effects induced on surface structures by tunneling operations,

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Estimated settlements during the Brescia Metrobus tunnel excavation A. Sanzeni, L. Zinelli & F. Colleselli University of Brescia, DICATA, Brescia, Italy

ABSTRACT: The paper describes ground subsidence and effect on historic buildings induced by the Earth Pressure Balanced (EPB) shield single tunnel construction of the first line of the Brescia Metrobus (Italy, 2005– 2009). The diameter of the shield is 9.15 m, the tunnel is 5.6 km long and excavation was carried out mainly in alluvial gravelly soil deposits. Among the buildings in Brescia the Palazzo della Loggia has been the venue of the city municipality since its construction between the 15th and 16Ith century. The progressive deterioration of the building massive piers and the forthcoming tunnel construction – located 25 m from the building and 20 m below ground level – required the consolidation of the soil beneath the foundations by means of lowpressure grouting and assessment of building settlements induced by the excavation. A number of finite element numerical simulations were carried out on a calibration-purpose model and preliminary results were compared with measured subsidence obtained from tunnel sections previously constructed. Predicted settlements and settlement distribution at the Loggia section were found in good agreement with movements measured during construction. Among the factors affecting subsidence prediction, particular attention was given to the ground loss during tunnel excavation and the presence of loads due to the building foundations.

1

INTRODUCTION

This paper presents the study carried out to predict subsidence induced by the construction of the tunnel for the first underground metropolitan line in the city of Brescia, Italy, the “Metrobus Brescia” project. The project comprises the construction of a lightrailway line which includes surface and under-ground cut-and-cover trunks (3.4 km and 3.8 km respectively) and a 5.9 km single tunnel trunk, for a total extent of 13 km. The tunnel was excavated between 2005 and 2009 with a Tunnel Boring Machine equipped with a shield and earth pressure balance technology (EPBS). The machine is suitable for most soils, can operate below water table and is capable of automatically laying the permanent tunnel lining. The shield diameter is 9.15 m and the permanent lining diameter is 8.85 m, consisting of pre-cast reinforced concrete elements of 0.35 m thickness. The excavation was carried out mainly in alluvial gravelly soil deposits and the soil cover is generally in the range of 17–20 m. Face pressure was approximately equal to 130 kPa at the top and 250 kPa at base, backfill injection of the tunnel lining was 4.5 m3 /m and advance rate was 30 mm/min (average measured values). The tunnel layout lies beneath the historic center of Brescia where a number of buildings of historical and social interest are located (Fig. 1). Among the most important structures, Palazzo della Loggia has been the venue of the city istration since its construction between the 15th and 16th centuries, during the Republic of Venice domination. Figure 2 shows the main East façade of the building, Figure 3 reports a schematic

Figure 1. Layout of the Brescia Metrobus light-railway line.

section showing the position of the tunnel and Figure 4 illustrates the tunnel layout and indicates the control system installed around the building area. The two storey building ground plan is 47 m × 30 m and about 30 m high. The structure is made of bricks and stones in elevation, covered by wooden floors and cross vaults, and the dome is shaped like a trough vault covered with lead sheets (Marini and Riva 2003, Giuriani 2007). The foundations are continuous in the West part and are isolated piers in the East part; the foundation

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Figure 3. Schematic section at Palazzo della Loggia.

Figure 4. Plan view of Palazzo della Loggia and tunnel location; benchmarks around and connected to the building. Figure 2. Palazzo della Loggia East façade.

level is approximately 5.0 m below ground surface. As documented by Giuriani (2007), the building suffers foundation problems due to partial degradation of the short wooden piles driven for soil improvement purposes in the 15th century (Fig. 2). The tunnel axis is located between 23 m and 25 m from the West side of the building and 24.5 m from ground level. A number of other buildings of different sizes, mainly devoted to residential purposes are located near the palace (Figs. 3–4). The progressive deterioration of the building foundations and the forthcoming tunnel construction required the improvement of soil conditions beneath the foundations and assessment of building settlements induced by the excavation; the improvement of the building foundation was accomplished only one month before the tunnel excavation, by means of low-pressure grout injection of the cavities left by the degraded wooden piles (preparatory studies are described by Giuriani 2007). Before the authors’ involvement the ground subsidence induced by the Metrobus tunnel was computed using the classic empirical equation proposed by Peck in 1969; a similar approach has been recently used with acceptable results to estimate subsidence induced by the excavation of Line-1 extension in Milan (Antiga and Chiorboli 2007). In this study however the effect of tunneling was analyzed using a plain-strain finite element model. To predict soil movement and building response, a number of numerical simulations were carried out on

a calibration-purpose model and preliminary results were compared with measured settlements obtained from a number of tunnel sections, previously constructed. The model was subsequently used to predict subsidence induced at Palazzo della Loggia section (Zinelli et al. 2010).

2

SOIL PROFILE AND PARAMETERS

2.1 Site characterization The tunnel excavation was carried out mainly in alluvial gravelly soil deposits. Soil characterization was performed before the authors’ involvement and included a comprehensive preliminary desk study with results of pre-existing investigations, and a site investigation campaign. The latter was carried out in 2003 and 2004 and consisted of 22 borings with execution of Standard Penetration Tests, geophysical down-hole tests, continuous dynamic penetration tests and soil sampling. Figure 5 shows a schematic soil profile obtained by collecting results of bore-holes and SPT tests along the layout of the tunnel near Palazzo della Loggia. From the ground surface (146.3 m a.s.l.) the following layers are encountered:

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– Made ground: mainly cohesionless, medium to loose soil, NSPT = 15–25.The thickness of this layer varies between 1.0 m to 6.0 m along the line layout

Figure 5. Soil profile at Palazzo della Loggia with SPT test results.

and is approximately 5.0 m at the Loggia palace section. – Gravel: well graded cohesionless (45–60% gravel, 20–35% sand, 10–25% silt and clay), medium to dense soil, NSPT = 35–60. Local sandy and clayey soil deposits as well as weak cemented volumes are encountered along the tunnel layout. – Weathered limestone: this geological unit was encountered only at Palazzo della Loggia section at an elevation below 118–119 m a.s.l. (z = 27–28 m). The water table is generally a few meters below the tunnel axis, is located at an elevation 118 m a.s.l. at the Loggia section, and rises above the tunnel axis southward of the city. 2.2 Constitutive model and soil parameters The subsidence induced by tunnel excavation was studied through a number of finite element analyses using the code Plaxis (version 8.6, Delft University) with a plain-strain, 15-node triangular element model. The mechanical behavior of the soil around the excavation was described using the constitutive model Hardening Soil, available in the code library (Shanz et al. 1999). This is an elastic-plastic rate independent model with isotropic hardening and stress-dependent stiffness according to a power law. The shear resistance parameters were determined based on the authors’ experience with local soil deposits and with empirical correlations with results from SPT tests (De Mello 1971, Shioi and Fukuni 1982, Yoshida et al.1988). The soil stiffness was estimated from experimental data obtained from geophysical down-hole tests performed along the tunnel axis, as described by Rampello and Callisto (2003). In the Hardening Soil (H-S) constitutive model the elastic

Figure 6. Small strain shear stiffness obtained from down-hole tests and assumed profile for the alluvial soil deposit.

behavior of granular soils is defined by isotropic elasticity through a stress-dependentYoung’s modulus:

where σ’3 is the minimum principal effective stress, pref = 100kPa is a reference pressure, E ref and m are model parameters. The Young’s modulus E ref has been related to the shear modulus at small strain G0 obtained from down-hole tests. Figure 6 shows experimental values of G0 estimated from measurements of the shear wave velocity Vs obtained from several down-hole tests (SB 23, 7, 11, 13 and, more recently, S5). Although there is some scatter in the experimental data, it is possible to identify a unique profile for the alluvial gravel deposit. ref The parameter E50 , which identifies the secant stiffness modulus at the reference confining pressure pref , was estimated assuming the Poisson’s ratio ν = 0.20–0.25 and introducing a ratio ref E ref /E50 = 10 − 12 in relation to the expected soil shear strains. The mechanical behavior of the superficial layer of made ground and of the altered limestone were described using a perfectly plastic model with MohrCoulomb failure criterion (M-C) and a liner elastic model (L-E) respectively. Table 1 reports soil constitutive models and parameters adopted in the analyses.

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Table 1.

Soil constitutive models and parameters. ref

ϕ’ Soil layer

Soil model

◦

E ref MPa

m –

E50 MPa

Made ground Gravel Weathered rock

M-C H-S L-E

32 36 –

– 750 –

– 0.4 –

25 65 270

3 3.1

EXCAVATION ANALYSIS Evaluation of ground loss

Ground conditions and tunneling method combine to control the ground movements which result from subsurface excavation. To normalize the volume of lost ground with respect to tunnel size, the volume of the settlement basin at the surface, Vs , can be expressed as a percentage of the excavated tunnel volume, Vexc . Before the magnitude of ground movements can be predicted it is necessary to estimate the expected ground loss Vs /Vexc . This estimate is generally based on case history data and should include an engineering appraisal that takes into the adopted tunneling method and site conditions (New and O’Reilly 1991). Alternatively, the movement of soil can be estimated through the determination of the so called gap parameter (Lee et al. 1992), which takes into the ground loss as a function of soil strength and deformation behavior, physical clearance between the excavated diameter and the lining, and workmanship. In the numerical simulations, that were performed using a plain-strain model, the effect of ground loss during excavation and construction was simulated by applying a contraction to the tunnel lining equal to the ratio Vs /Vexc . The numerical value of the applied contraction was estimated with measurement of surface settlements obtained from a number of sections, taken from a trunk of tunnel already constructed, where the soil profile had geotechnical features similar to those encountered nearby Palazzo della Loggia. Figure 7 shows one of the sections (named SCBF 8, located at km7+597) with indication of the soil profile, depth of the water table and tunnel location, while Figure 8 reports vertical settlements taken during tunnel excavation and construction. The most appropriate value of lining contraction was calculated integrating the settlement profiles such as the one reported in Figure 8 and dividing the obtained value by the area of the TBM face. Figure 9 shows the variation of surface vertical settlement along the longitudinal tunnel crown axis with the advance of the TBM machine: the vertical settlement is normalized by the maximum measured value and the machine advance is expressed both in meters and in shield diameters. The reported data show that vertical displacement ahead of the TBM face amounts to 20–22% of the total and starts showing approximately at a distance equal to 2 diameters; ground loss over the shield consists of some other 25–28% of the total and finally,

Figure 7. Calibration section SCBF 8 at km 7 + 597.

Figure 8. Ground surface settlements at section SCBF8.

although backfill grouting of tunnel lining is systematically executed, the settlement ed over the permanent lining is approximately 50% of the total. At the time of the authors’ involvement an estimated ground loss 0.45–0.50% represented the most likely value with frequency of occurrence in the range of 86-90% (the frequency of occurrence of ground loss higher than 0.5% was 10–14%). 3.2 Back analysis and prediction simulations The back analysis simulations of tunneling through the reference sections were performed with a numerical model assuming the soil profile illustrated in Figure 7 and absence of buildings at the ground surface. Since the examined sections were symmetric, only one half of the tunnel section was taken into . The soil behavior was described with constitutive models and parameters reported in Table 1. The lining contraction was applied instantly during one calculation phase – because the permanent lining is constructed immediately after the excavation – and the effect of a

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Figure 10. Comparison of observed and computed vertical settlements at section SCBF8. Figure 9. Ground surface settlements with TBM advance.

higher contraction as a direct consequence of technical problems encountered during excavation was examined. The prediction analysis at Palazzo della Loggia section (Fig. 3) considered the soil profile represented in Figure 5 and soil constitutive models and parameters reported in Table 1. The presence of the palace and other buildings were conservatively taken into by applying distributed loads at the foundation levels (150 kPa for Palazzo della Loggia and 50 kPa for the other buildings), therefore neglecting the stiffness of the structures in elevation.

4

COMPARISON OF OBSERVED AND COMPUTED BEHAVIOR, CONCLUSIONS

In Figure 10 the vertical displacements measured at ground level 33 days after the excavation and construction of section SCBF 8 are compared with those computed in the numerical analysis and with those estimated with the empirical equation proposed by Peck in 1969 assuming parameter K = 0.35, in agreement with the settlement prediction in the Metrobus project. Measurements and computations show a satisfactory agreement, particularly regarding the amount of settlement above the tunnel axis (with maximum settlement not exceeding 13–14 mm). However, the amplitude of the ground surface affected by subsidence is significantly underestimated by Peck’s empirical correlation, due to the numerical value assumed for parameter K, which affects the transverse distance of the point of inflection from the tunnel longitudinal axis, whereas the FEM computed settlement distribution reproduced the real subsidence well. The comparison also demonstrates that the value of applied lining contraction, estimated as the ratio between measured subsidence and the area of the TBM face, Vs /Vexc , is suitable for describing the effects of excavation and construction process on ground movement around the tunnel, although the plain-strain numerical model does not

Figure 11. Comparison of observed and predicted vertical settlements at Palazzo della Loggia.

allow to appreciate three dimensional and time effects as observed in Figure 9. To investigate the effects of ground loss a number of numerical analyses were performed applying values of lining contraction in the range between 0.3% and 1.5%, as this was intended to represent the highest possible figure in case of technical difficulties during tunneling such as a sudden stop, sudden loss of face pressure, uncontrolled machine tilting and overcutting, or interruption of backfill grouting of tunnel lining. The expected vertical settlement above the tunnel axis greatly increases with applied contraction values higher than 0.7%. Similarly, the amplitude of the subsidence basin increases as a result of the increase in volume of soil subjected to high shear strains near the tunnel lining. Figure 11 shows the results obtained from the prediction analysis at Palazzo della Loggia in comparison with selected measurements of vertical settlements

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obtained from the control system in Figure 4. The computed settlement is taken from a horizontal section located at the building foundation level of the numerical model. The measured settlement is obtained from geometric leveling of benchmarks connected to the building and refers mainly to the north and south side of Palazzo della Loggia, plus some other points across the tunnel. The maximum vertical settlement, measured along the West side of the building at points 34 and 31, is less than 2 mm, and decreases to values comparable with the control system error within 10 m from the West side of the building (benchmarks 44 and 924). The resulting angular distortion is estimated as 1/5000. The settlement prediction compares well with measurements and confirms that, due to the distance of the building from the longitudinal axis of the tunnel, the risk of damage is limited, although the group of buildings directly above the tunnel axis is subjected to more severe effects. The vertical settlements measured above the tunnel show the effect of the buildings stiffness in the sagging zone of the subsidence profile. ACKNOWLEDGEMENTS The authors wish to thank Brescia Mobilità S.p.A., Astaldi S.p.A and Stone S.p.A. for providing technical documents of the Metrobus project. REFERENCES Antiga, A. and Chiorboli, M. 2007. L’analisi dei cedimenti nella progettazione di gallerie realizzate con EPB-Shield in terreni incoerenti. Proc. XXIII Convegno Nazionale di Geotecnica, Padova. Associazione Geotecnica Italiana. Patron Editore (ed.): 103–110.

De Mello, C.F.B. 1971. The standard penetration test. Proc. IV Panamerican Conf. of SME, San Juan, Puerto Rico. Vol. 1: 1–86. Giuriani, E. 2007. Il percorso delle indagini e degli studi per gli inerventi sulla struttura del Palazzo della Loggia. Il Palazzo della Loggia di Brescia, Indagini e progetti per la conservazione. Starrylink (ed.), ISBN 978-88-89720: 55–4. Lee, K.M., Kerry Rowe, R. and Lo, K.Y. 1992. Subsidence owing to tunneling. I. Estimating the gap parameter. Canadian Geotechnical Journal vol. 29: 929–940. Marini, A. and Riva, P. 2003. Nonlinear analysis as a diagnostic tool for the strengthening of an old wooden dome. J. of Structural Engineering, ASCE vol. 129, n. 10: 1412–1421. ISSN: 0733-9445. Peck, R.B. 1969. Deep excavation and tunneling in soft ground. Proc. 7th Int. Conf. on Soil Mechanics and Foundation Engineering, Mexico City. State-of-the-Art: 225–290. Rampello, S. and Callisto, L. 2003. Predicted and observed performance of an oil tank founded on soil-cement columns in clayey soils. Soils and Foundations. Japanese Geotechnical Society vol. 43 n. 4: 229–241. Shanz, T., Vermeer, P.A. and Bonnier, P.G. 1999. Formulation and verification of the Hardening-Soil model. R.B.J. Brinkgreve, Beyond 2000 in Computational Geotechnics. Balkema, Rotterdam: 281–290. Shioi, Y. a nd Fukui, J. 1982. Application of N-value to the design of foundations in Japan. Proc. 2nd European Symp. on Penetration Testing (ESOPT2). Amsterdam, 1: 159–164. The Brescia Metrobus Project. Metropolitana leggera automatica di Brescia. http://www.metro.bs.it/. Accessed November 2009. Yoshida, I. and Motonori, K. 1988. Empirical formulas of SPT blow-counts for gravelly soils. Proc. of ISOPT-1. Orlando (USA). Zinelli, L., Sanzeni, A. and Colleselli, F. 2010. Subsidenza indotta da scavi in sotterraneo. Technical Report. University of Brescia, Dipartimento DICATA (in italian).

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Numerical investigation of the face stability of shallow tunnels in sand Ansgar Kirsch∗ ILF Consulting Engineers, Rum/Innsbruck, Austria

ABSTRACT: Tunnels with low cover are often headed using the shield technique. In this context the face stability is an important issue. The tunnel face must be ed in order to minimise settlements on the ground surface in front of the tunnel and to prevent failure of the soil ahead of the face. Still, the mechanisms that occur during face collapse are not completely understood. In this paper a numerical study of the given problem with finite elements will be presented. Different material models, such as a simple elasto-plastic with Mohr-Coulomb failure criterion and a hypoplastic model, were investigated with respect to their ability to model the collapse of a tunnel face. The numerical results were compared to experimental results that were obtained with small-scale experiments at single gravity. The observed necessary pressure and incremental displacements were predicted sufficiently well by both constitutive models. Moreover, the numerical results were compared to the predictions of some chosen theoretical models that are in use in engineering practice. Good agreement was achieved with an upper bound solution by Léca/Dormieux and an empirical equation by Vermeer/Ruse.

1

INTRODUCTION

The face stability of shallow tunnels (cf. Fig. 1) must be guaranteed to minimise settlements at the ground surface and to prevent an uncontrolled collapse of the soil ahead of the tunnel. For this reason, a necessary pressure, pf , must be prescribed for slurry and EPB shield machines, which needs to counteract water and earth pressure with a sufficient safety margin. For the determination of pf theoretical models as well as laboratory investigations and numerical calculations have been published. The theoretical approaches can be subdivided into kinematic approaches with failure mechanisms (e.g. Horn [10] and variations [1, 2, 19], Krause [17], Léca and Dormieux [18] and derivations [22]) and static approaches with issible stress fields (e.g. Léca and Dormieux [18], Atkinson and Potts [3]). Some additional approaches are neither purely kinematic nor purely static (Kolymbas [16], Balthaus [4]). A number of numerical studies of the face stability of shallow tunnels have been performed, mainly with three-dimensional finite element calculations (e.g. Schubert and Schweiger [21], Sterpi and Cividini [23], Mayer et al. [21] and Chaffois et al. [5]). Ruse andVermeer [20, 24] also investigated the necessary pressure for the face of shallow tunnels with finite elements, making use of a linear elastic, perfectly plastic constitutive model with a Mohr-Coulomb failure condition for the soil. From their numerical ∗

Formerly: Division of Geotechnical und Tunnel Engineering, University of Innsbruck, Innsbruck, Austria

Figure 1. Tunnel geometry.

results the authors derived an empirical equation for the determination of pf . A two-dimensional discrete element model was applied by Kamata and Mashimo [11] to analyse face stability, thus modelling the soil as a discontinuous medium. To assess the quality of proposed models for face stability analysis, the author performed two series of small-scale model experiments, which are described briefly in section 2 and in detail in Kirsch [12, 13]. Moreover, a numerical study with finite elements served to study the given problem. A simple elastoplastic model with Mohr-Coulomb failure criterion and a hypoplastic model were investigated with respect

779

Figure 2. Box and model tunnel for the first series of experiments.

Figure 3. Box and model tunnel for the second set of experiments.

to the following task: to predict the displacement pattern and the necessary pressure for face stability analyses. For all calculations the commercial software Abaqus/Standard was used. The performed sandbox experiments offered the opportunity to compare the simulation results of the two material models not only with each other, but also with the results of the laboratory experiments. 2

SETUP OF THE PHYSICAL MODEL

For the sake of clarity, the author’s 1g-experiments, which served as one reference for the numerical study, are briefly desribed. In a first series of experiments the evolution of failure mechanisms in dense and loose sand with different overburden was investigated, making use of Particle Image Velocimetry. The resulting force on the tunnel face was studied in a second series of experiments. The first series of experiments was conducted in a model box (Fig. 2) with inner dimensions 37.2 × 28.0 × 41.0 (width × depth × height in cm). The outer frame was made of steel, bottom and side walls wooden, and the front wall was 1 cm thick hardened glass. Thus, the soil grains adjacent to the glass wall could be observed throughout the test. The problem was modelled in half; therefore, the tunnel was represented by a half-cylinder of perspex, with an inner diameter of 10.0 cm and a wall thickness of 0.4 cm. This model tunnel protruded 7.0 cm into the soil domain, and its axis lay approx. 8.0 cm above the bottom of the sandbox. An aluminium piston was fitted into the tunnel to the soil. In the second series the geometry of the experiment was slightly modified to model the full problem. Therefore, the tunnel was modelled with a hollow aluminium cylinder with an inner diameter of 10 cm and the same wall thickness of 0.4 cm (Fig. 3). The face of the model tunnel was ed by an aluminium disk with a slightly smaller diameter than the inner diameter of the tunnel (Dpiston = 9.8 cm). The piston rod was ed by a linear roller bearing, embedded in the side wall. The rod made with a miniature load cell that was mounted on a sliding carriage on the outside of the side wall. The carriage could be moved by turning a knob (Fig. 4).

Figure 4. Carriage construction with load cell, goniometer and turning knob.

The experiments were performed displacementcontrolled: by incrementally retracting the carriage, and thus the piston, the failure of the tunnel face was triggered. The load cell measured the resulting force exerted by the ground on the piston. All tests were performed with dry quartz sand with grain diameters between 0.1 and 2.0 mm. In a parametric study the cover-to-diameter ratio C/D and the initial density Id were varied.

3

MATERIAL MODELS

The applied constitutive models will briefly be described in the following, together with information about the model calibration. The “loose” sandbox experiments served as reference for the numerical simulations, because shear localisation was expected to play a minor role. Thus, regularisation was avoided.

3.1 Mohr-Coulomb A linear elastic, perfectly plastic material model with a Mohr-Coulomb failure criterion was applied as a simple elasto-plastic model. Despite its simplicity it is still one of the most commonly applied material models in engineering practice. Moreover, it has been shown that it performs well for the calculation of ultimate limit states (e.g. [6]). The Mohr-Coulomb (“MC”) model

780

Table 1. Applied material parameters. Mohr-Coulomb model (MC) E ν 440/825/1190 kPa 0.31

ϕ 34◦

Hypoplastic model (HYPO) hs n ed0 ϕc 32◦ 1000 MPa 0.30 0.42

ec0 0.75

c 0.005 kPa

ei0 0.76

α 0.10

ψ 2◦

β 2.25

requires five parameters:Young’s modulus E and Poisson’s ratio ν describe the material behaviour in the elastic domain. The friction angle ϕ, cohesion c and dilation angle ψ govern the plastic behaviour of the material. The elastoplastic Mohr-Coulomb model is implemented in Abaqus. 3.2 Hypoplasticity To make realistic predictions for the deformation characteristics, the consideration of basic soil behaviour, such as dilatancy, different stiffnesses for loading and unloading, dependence on stress level and others, is essential (e.g. [8]). These requirements are fulfilled by the applied hypoplastic model (“HYPO”): Hypoplastic models are formulated as non-linear tensorial equations of the rate type (evolution equations), i.e. the stress rate is expressed in of strain rate, actual stress state and void ratio. The equations are incrementally non-linear and hold, equally, for loading and unloading. Moreover, the applied hypoplastic formulation takes pressure level and density into . An overview of the derivation and features of hypoplastic models was given by Kolymbas [14, 15]. The author applied the version of von Wolffersdorff [25] for his simulations. Hypoplasticity requires eight input parameters: void ratios at zero stress level, ei0 , ec0 and ed0 , critical friction angle ϕc , granular hardness hs and exponents α, β and n. These parameters are assumed insensitive to pressure or density and can easily be determined from simple laboratory and index tests [9]. Hypoplasticity is not implemented in Abaqus. It can be incorporated via a -defined subroutine (UMAT), though. For the present study, a UMAT by Fellin and Ostermann [7] was used, which provides error-controlled time integration of constitutive models in rate form. 3.3 Calibration Both models were calibrated for Ottendorf-Okrilla sand, a material which was used in the author’s experimental campaign. The material parameters for both models are summarised in Tab. 1; they were quantified with back-calculations of element tests (triaxial, oedometer and index tests). The Mohr-Coulomb parameters were carefully adapted to the low stress levels prevailing in the

sandbox models. Moreover, a Young’s modulus E for unloading was used, because the ground was predominantly relieved of pressure by the failure of the tunnel face. As the stress level varies with overburden, Tab. 1 lists three values for E corresponding to the respective C/D = 0.5 . . . 1.5. As the “loose” experiments were performed with densities Id = 0.27 . . . 0.33 (corresponding to void ratios e between 0.64 and 0.66), the self-weight of the soil was set to γ = 16.0 kN/m3 (e0 = 0.65, Id = 0.3). Further information about calibration is given by Kirsch [12]. 4

SETUP OF THE NUMERICAL MODEL

The simulation of the sandbox tests included a preliminary parametric study for the numerical model: as a first step, three different modelling procedures for the given problem were tested with both, the Mohr-Coulomb and the hypoplastic model. Then, the numerical configuration of the favoured procedure, e.g. time and space discretisation, element type and integration schemes, was varied. Finally, a mesh study served to assess the influence of the finite element mesh on the results of the simulations. The author finally favoured the follwing setup of the numerical model: 4.1 Geometry and boundary conditions The spatial discretisation of the numerical model was chosen in accordance with the performed experiments, obeying recommendations by Ruse [20] (Fig. 5a). The system’s symmetry was ed for. On the symmetry plane displacements in 2direction were prohibited, same as on the back of the model (Fig. 5b). Displacements in 1-direction were restricted on the left and right planes, whereas the bottom boundary was fixed in vertical direction. The tunnel lining was considered rigid and rough. Therefore, the nodes on the tunnel perimeter were fixed in all directions. In agreement with the experimental investigation, the construction process was not modelled, i.e. the tunnel was “wished-in-place”. 4.2 Mesh The preliminary investigation revealed that the mesh size had the biggest influence on the numerical predictions of the necessary pressure ND . The soil was finally modelled with a total of 3829 elements (Fig. 6) with 52395 degrees of freedom. Twenty-noded brick elements with quadratic interpolation functions and reduced integration were used. 4.3 Analysis steps In the laboratory experiments, the failure of the face was triggered by retracting a piston into the model tunnel. Therefore, the piston was modelled as independent part. The piston had linear elastic properties, with a stiffness roughly five orders of magnitude higher

781

Figure 7. Load-displacement curves for a variation of C/D (range of experimental results shaded in grey).

the displacement of the piston in 1-direction was prescribed. In agreement with the laboratory experiments, increments of 0.25 mm were used for each analysis step.

4.4 Evaluation criteria The numerically predicted (dimensionless) pressure ND = pf /(γ D) and the development of p vs. piston displacement s served as main evaluation criterion for the comparison of different model configurations. In addition, the incremental displacements in the plane of symmetry served as a criterion for the comparison of the finite element calculations with the laboratory experiments. 5

Figure 5. Size and boundary conditions of the FE model.

5.1

Figure 6. Finite element mesh.

than the soil stiffness. The diameter of the piston was slightly smaller than the tunnel diameter. Piston and soil interacted via a law, which allowed for separation of the elements. The between piston and soil was assumed frictionless. In the initial analysis step the piston front was aligned with the front of the tunnel. Subsequently,

RESULTS AND INTERPRETATION Results

The performed sandbox experiments with loose sand were simulated, varying the cover-to-diameter ratio between 0.5 and 1.5. The resulting load-displacement curves for a variation of C/D and both material models are shown in Fig. 7. All curves drop to approximately the same residual value of ND ≈ 0.10 after sufficient displacement of the piston. For the chosen input parameters and model configuration, the overburden C/D and the material model only have a marginal influence on the resulting ND (Tab. 2). Plots of incremental displacement for an advance step from 0.25 to 0.50 mm (Fig. 8) reveal a small difference between the Mohr-Coulomb and the hypoplastic model: although both models predict soil movements up to the ground surface, the magnitude of incremental displacements in the failure zone is smaller for the Mohr-Coulomb calculation. The result for the Mohr-Coulomb simulation for C/D = 1.5 was taken at a piston advance of 1.0 mm.

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Table 2.

Computation times and predicted ND values. Mohr-Coulomb

Hypoplasticity

C/D

Comput. time (h)

ND

Comput. time (h)

ND

0.5 1.0 1.5

1.18 2.23 92.41

0.103 0.100 0.099

28.14 37.08 118.71

0.098 0.095 0.098

probably due to the fact that the kinematics of the problem are slightly different, if the soil is allowed to bulge into the tunnel. There is also a good agreement with the theoretical model by Léca and Dormieux [18], which predicts a value of ND = 0.0903. Sterpi and Cividini [23] modelled the problem with a strain softening material model. They found that neglecting strain softening, as with the Mohr-Coulomb model, led to an underestimation of displacements. This statement is in agreement with the obtained patterns of incremental displacements for the two applied models (Fig. 8). For the simulations with hypoplasticity there are no published references. But the coincidence between predictions for ND with both material models is remarkable. The results of both the Mohr-Coulomb and the hypoplastic model are in good quantitative agreement with the measured pressures (shaded in grey in Fig. 7). The numerically obtained ND ≈ 0.10 value is roughly 10% smaller than the mean ND ≈ 0.11 from the laboratory experiments. A reason for this might be that the applied mesh (Fig. 6) is still not fine enough. Still it allowed to perform the parametric study in practicable computation times.

6

CONCLUSION

The numerical study has shown that both models are capable of predicting the necessary pressure sufficiently well. Also the resulting displacement pattern match the experimental observations well, with some advantages for the hypoplastic model. The Mohr-Coulomb model might seem easier to grasp, but the calibration procedure is error-prone: the expected loading history of the soil, its density and the stress level need to be considered correctly. The hypoplastic model, in contrast, has the advantage that a single set of input parameters is sufficient for one type of soil. The effects of density and stress level on the strength of the material are incorporated in the model by means of the state parameter e. But, there is a price for this capacity: the finite element calculations with hypoplasticity lasted about 25 times longer than the Mohr-Coulomb calculations. Figure 8. Plots of incremental displacements for a piston advance from 0.25 to 0.50 mm for C/D = 1.0.

REFERENCES 5.2 Interpretation The obtained load-displacement curves for the MohrCoulomb calculations are in good qualitative agreement with results published by Ruse [20]. In both investigations, no influence of C/D on the necessary pressure was observed. The absolute value for ND is slightly smaller than predicted by Ruse’s empirical formula: for ϕ = 34◦ , ND = 0.1147. Reason for this might be that Ruse triggered the face-collapse load-controlled. This is

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[1] Anagnostou G. and Kovári K. (1992), Ein Beitrag zur Statik der Ortsbrust beim Hydroschildvortrieb, in Symposium ’92, Probleme bei maschinellenTunnelvortrieben?, Gerätehersteller und Anwender berichten. [2] Anagnostou G. and Kovári K. (1996), Face Stability Conditions with Earth-Pressure-Balanced Shields, Tunnelling and Underground SpaceTechnology, 11(2): pp. 165–173. [3] Atkinson J.H. and Potts D.M. (1977), Stability of a shallow circular tunnel in cohesionless soil, Géotechnique, 27(2): pp. 203–215.

[4] Balthaus H. (1988), Standsicherheit der flüssigkeitsgestützten Ortsbrust bei schildvorgetriebenen Tunneln, in Festschrift H. Duddeck, Institut für Statik der Technischen Universität Braunschweig, pp. 477–492, Springer, Berlin. [5] Chaffois S., Laréal P., Monnet J. and Chapeau C. (1988), Study of tunnel face in a gravel site, in Proc. 6th Int. Conf. on Numerical Methods in Geomechanics, Innsbruck (ed. G. Swoboda), vol. 3, pp. 1493–1498. [6] de Borst R. and Vermeer P. (1984), Possibilities and Limitations of Finite Elements for Limit Analysis, Géotechnique, 34(2): pp. 199–210. [7] Fellin W. and Ostermann A. (2002), Consistent tangent operators for constitutive rate equations, Int. J. Numer. Anal. Methods Geomech., 26: pp. 1213–1233. [8] Hejazi Y., Dias D. and Kastner R. (2008), Impact of constitutive models on the numerical analysis of underground constructions, Acta Geotechnica, 3: pp. 251–258. [9] Herle I. (1997), Hypoplastizität und Granulometrie einfacher Korngerüste, No. 142 in Veröffentlichungen des Institutes für Bodenmechanik und Felsmechanik der Universität Karlsruhe. [10] Horn M. (1961), Horizontaler Erddruck auf senkrechte Abschlussflächen vonTunneln, in Landeskonferenz der ungarischen Tiefbauindustrie (German translation by STUVA, Düsseldorf). [11] Kamata H. and Mashimo H. (2003), Centrifuge model test of tunnel face reinforcement by bolting, Tunnelling and Underground Space Technology, 18: pp. 205–212. [12] Kirsch A. (2009), On the face stability of shallow tunnels in sand, No. 16 in Advances in Geotechnical Engineering and Tunnelling, Logos, Berlin. [13] Kirsch A. (2010), Experimental investigation of the face stability of shallow tunnels in sand, Acta Geotechnica, accepted for publication. [14] Kolymbas D. (1977), A rate-dependent constitutive equation for soils, Mech. Res. Comm., 4: pp. 367–372. [15] Kolymbas D. (2000), Introduction to hypoplasticity, No. 1 in Advances in Geotechnical Engineering and Tunnelling, Balkema, Rotterdam.

[16] Kolymbas D. (2005),Tunnelling andTunnelMechanics, Springer, Berlin. [17] Krause T. (1987), Schildvortriebmit flüssigkeitsund erdgestützter Ortsbrust, No. 24 in Mitteilung des Instituts für Grundbau und Bodenmechanik der Technischen Universität Braunschweig. [18] Leca E. and Dormieux L. (1990), Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material, Géotechnique, 40(4): pp. 581–606. [19] Mayer P.M., Hartwig U. and Schwab C. (2003), Standsicherheitsuntersuchungen der Ortsbrust mittles Bruchkörpermodell und FEM, Bautechnik, 80: pp. 452–467. [20] Ruse N.M. (2004), Räumliche Betrachtung der Standsicherheit der Ortsbrust beim Tunnelvortrieb, No. 51 in Mitteilungen des Instituts für Geotechnik der Universität Stuttgart. [21] Schubert P. and Schweiger H.F. (2004), Zur Standsicherheit der Ortsbrust in Lockerböden, in Proc. ISRM regional symposium EUROCK 2004 and 53rd Geomechanics Colloquium, October 7–9, 2004, Salzburg, Austria (ed. W. Schubert), pp. 99–104. [22] Soubra A.H. (2000), Three-dimensional face stability analysis of shallow circular tunnels, in Int. Conf. on Geotechnical and Geological Engineering, Melbourne, Australia, November 19–24, pp. 1–6. [23] Sterpi D. and Cividini A. (2004), A Physical and Numerical Investigation on the Stability of Shallow Tunnels in Strain Softening Media, Rock Mech. and Rock Engng., 37(4): pp. 277–298. [24] Vermeer P.A., Ruse N.M. and Marcher T. (2002), Tunnel heading stability in drained ground, Felsbau, 20(6): pp. 8–18. [25] von Wolffersdorff P.A. (1996), A hypoplastic relation for granular materials with a predefined limit state surface, Mechanics of Cohesive-Frictional Materials, 1: pp. 251–271.

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Numerical modeling of a bolt-reinforced tunnel in a fractured ground E. Seyedi Hosseininia Ferdowsi University of Mashhad, Mashhad, Iran

E. Bourgeois & A. Pouya Université Paris Est, LC-MSRGI,

ABSTRACT: During the drilling of galleries in the deep underground laboratory of ANDRA in Meuse Haute Marne, it has been observed that the excavation process created in the surrounding ground fractures with very specific shapes. A research programme was undertaken to model the mechanical behaviour of the fractured zone and the influence of radial bolts on the ground deformation around the galleries. For simplicity, the tunnel section is assumed to be circular and the problem is analyzed in axisymmetric mode. An original approach was used, that combines two homogenization procedures, to for the role of the fractures and of the bolts. This approach was implemented in the finite element code CESAR-LC. Computations give larger wall displacements if fractures are taken into , and show that the most efficient way to reduce wall convergence is to place bolts perpendicular to the axis of tunnel, regardless of the inclination of fractures.

1

INTRODUCTION

In finite element simulations of tunnelling in rock masses, it is often assumed that the ground behaviour is homogeneous and isotropic. However, discontinuities of the rock mass can induce anisotropic deformability properties. Since it remains difficult to deal with a large number of fractures in a numerical model, it is worth using an “equivalent” anisotropic model for the fractured ground. In practice, the stability of deep underground excavation is improved by means of bolts placed in the tunnel walls. Since bolts are placed in an ordered manner, the reinforced zone can be modelled using a homogenisation approach too. In the present paper, the process of drilling a tunnel through a fractured ground, whose walls are reinforced by bolts, is modelled by a general finite element code. Two homogenization procedures are used to for the role of the fractures and of the bolts. The paper presents a preliminary parametric study of the influence on wall convergence of the orientation of the fractures and of the direction in which bolts are placed. Analyses are carried out in axisymetric condition.

2

MODELLING OF FRACTURED GROUND

During the drilling of galleries in the deep underground laboratory of ANDRA in Meuse Haute Marne (), it was observed that the excavation process resulted in the creation of fractures in the vicinity of the galleries. Fractures show a complex three-dimensional

Figure 1. Geometry of fractures with chevron shape ; (a) vertical plane parallel to the tunnel axis, (b) horizontal plane, (c) vertical plane orthogonal to the axis.

shape, and form a network of discontinuities more or less uniformly distributed along the axis of the tunnel, illustrated on Figure 1 by three plane sections. It is not our purpose to explain the fracture pattern here, but it reflects : – a complex initial behaviour of the ground (including material anisotropy), – the anisotropy of the initial stress field (observations show that the geometry of the fractures depends on the direction of the gallery axis), – and of the excavation technique itself. Unfortunately, it is very difficult to get undisturbed samples to improve the understanding of the rock behaviour. In what follows, an equivalent anisotropic model is used for the fractured zone. It is assumed that the shape of fractures can be simplified as conical, as shown in Figure 2. For the initial intact ground, the elastic–perfectly plastic Drucker – Prager model is applied with a linear and isotropic elasticity. As a preliminary step, it

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Denoting by E and ν the Young modulus and Poisson’s ratio of intact material, respectively, parameters in equation (2) are given by:

Figure 2. Simplified geometry: fractures are replaced by cones (having the same axis as the tunnel).

with

3

Figure 3. (a) Multi-layered system; (b) unit vector n.

is assumed that the fractures change the elastic characteristics of the ground, but the strength remains the same as that of the intact ground. An homogenization procedure makes it possible to replace the fractured discontinuous ground by a continuum with equivalent anisotropic elastic properties. In order to derive the elastic tensor behaviour, the fractured medium is considered in a first step as the superposition of homogeneous layers separated by plane of finite but small thickness ts. For layers perpendicular to the z-axis (Figure 3a), the equivalent compliance matrix of the system is calculated. In a second step, one takes into the local orientation of the fracture by replacing the z-axis by the direction og the unit vector n perpendicular to the plane tangent to the cones in a global coordinate system as shown in Figure 3b. Four parameters mentioned below are added to the existing Drucker – Prager model in order to consider the fractures: – – – –

normal stiffness of fractures, kn tangential stiffness of fractures, kt distance between two successive fractures, D angle of orientation of fractures, which is equal to the angle between the axis of symmetry and the plane tangent to the conical fracture, α.

As a consequence, the stress-strain relationship for the fractured material is given by:

where:

MODELLING OF REINFORCED GROUND

The role of the bolts is taken into by means of the so-called multiphase model introduced for reinforced materials by de Buhan and Sudret (2000). The framework is an extension of classical homogenization methods. In this model, the whole medium constituted by a ground mass and the bolts it contains is represented by two continuous superposed media (or “phases”), one representing the ground and the other standing for inclusions. In other words, there are, in every geometrical point, two material particles in mutual mechanical interaction. Different kinematic fields are associated with each phase, and the model includes a description of the mechanical interaction between them (Bennis and de Buhan, 2003). The overall properties of the equivalent material are elastically and plastically anisotropic. In the present study, the interaction between phases is described as a perfect bonding.

4

CASE STUDY AND NUMERICAL MODEL

The present case study includes the process of drilling a 5-meter diameter gallery with a length of 25 m. The tunnel is drilled in ten successive steps. The ground is isotropic prior to drilling. However, the drilling process generates fractures in the ground around the gallery. The extent of the fractured zone is assumed to be 2 m beyond the wall and 2.5 m ahead of the tunnel face. The tunnel is supposed to be excavated at a depth of 75 m with an isotropic (K0 = 1) initial stress state (σ ◦ = 1.5 MPa). The problem is dealt with in axisymmetric condition The problem is simulated by a “research version” of the finite element code CESAR-LC, in which both homogenization techniques are implemented. The dimension of the mesh used for the problem is 35 m by 55 m. All elements are quadratic. Far from the tunnel, triangular elements are used; however, quadrangular elements are used around the excavated zone.

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Figure 6. Variation of convergence of wall tunnel along the tunnel axis (non-reinforced ground). Figure 4. Modelling sequence of tunnel drilling.

5

RESULTS

The results of analyses are presented in of tunnel convergence (i.e. of radial displacements of the tunnel wall) along the axis. 5.1 Effect of fracturing

Figure 5. Definition of angles α and β.

For each step of drilling, the simulation of construction process consists of (Figure 4): – deactivating the zone to be excavated, – introducing boundary forces representing the forces applied by the excavated zone, – associating the fractured model with the ground in front of the tunnel face, – associating the combination of the fractured model and the multiphase model to the ground next to the wall in the excavated zone. The constitutive law for the ground is described by the Drucker-Prager model with E = 50 MPa, ν = 0.35, φ = 30 degrees, ψ = 20 degrees. For the fractured ground, additional parameters are: kn = 50 MPa/m, kt = 5 MPa/m, D = 0.2 m. For the opening of the cones that represent the fractures, computations were made for six values of the angle α between the vertical plane containing the tunnel axis and the plane tangent to the cones: α = −60, −45, −30,+30,+45 and +55 degrees. Bolts are assumed to be elastic, made of steel (Young’s modulus Eb = 210 GPa), with a diameter of 25 mm. The density of bolts is taken equal to 1 bolt per square meter of tunnel wall. Three values were studied for the angle β between the tunnel axis and bolt direction β = 30, 60, and 90 degrees. The schematic definition of α and β angles is presented in Figure 5.

Figure 6 compares the convergences obtained for a non-reinforced tunnel with different directions of fractures. It is found that displacements are larger in a fractured rock mass than in an intact ground, but it is not possible to establish a simple relationship between the convergence and the fractures inclination. On the other hand, convergences obtained with α = +30◦ and α = −30◦ are obtained very close to those obtained with α = −60◦ and α +55◦ , respectively. It is more obvious especially for distances far from the tunnel face. 5.2 Effect of bolting The calculated tunnel convergence is presented for different bolting directions (β = 30, 60, and 90 degrees) for all cases of fractured ground in Figure 7. It is very interesting to observe that the inclination of fracturing has no effect on the optimized direction of bolting. The more inclined the bolts, the smaller the convergence. In other words, the best way of stabilizing the radial wall displacement is to place the bolts perpendicular to the tunnel axis, regardless the inclination of fractured ground around the tunnel. 6

CONCLUSION

The generation of fractures during tunnelling and the reinforcement of the wall can be modelled using a combination of homogenization procedures. Numerical simulations show that the wall convergence is larger in a fractured ground than in an intact ground and that the best way of reinforcing the tunnel wall is to place the bolts perpendicular to the tunnel axis, regardless the inclination of fractures. In other words,

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Figure 7. Variation of convergence of wall tunnel along the tunnel axis (fractured and reinforced ground).

anisotropy in the ground around the excavated zone has no effect on the pattern of bolting. It is important to recall that in the models used here, anisotropy is only taken into for the elastic part of the behavior of the fractured ground: the influence of fractures on plastic properties will be considered in future studies. Besides, results presented here only reflect numerical analyses and should be confirmed by comparisons with in situ observations.

REFERENCES de Buhan, P. & Sudret, B. 2000. Micropolar multiphase model for materials reinforced by linear inclusions, Eur. J. Mech., A/Solids, 19: 669–689. Bennis, M. & de Buhan, P. 2003. A multiphase constitutive model of reinforced soils ing for soil-inclusion interaction behavior, Mathematical and Computer Modelling, 37(5–6): 469–475.

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On the effects of modelling gap closure and assumed soil behavior on the FE predictions of ground movements induced by tunneling in soft clay C. Miriano Università degli Studi G. Marconi and DISG, Sapienza Università di Roma, Rome, Italy

C. Tamagnini Dipartimento di Ingegneria Civile e Ambientale, Università degli Studi di Perugia, Perugia, Italy

ABSTRACT: In this work, the effects of two key aspects of simplified 2d FE analyses of tunnel excavation in clays – namely, the simulation of TBM tunnel driving and the assumed constitutive model for the soil – are investigated with particular reference to the prediction of ground movements induced by the excavation. The technique proposed for the simulation of the tunnel face advancement can be considered an improvement of the method originally developed by Lee et al (1992). Here, the original methodology is extended to take into the possibility of a non–uniform gap closure around the tunnel lining, giving rise to the ovalization of the originally circular tunnel section. As for the constitutive modelling of the excavated soil, Modified Cam-clay model has been compared with an hypoplastic model for clays, recently developed by Mašín The Lower Market Street tunnel has been reanalyzed, comparing the predictions obtained with the proposed modelling technique and the hypoplastic soil model with those provided by standard approaches. The results of the simulations indicate that both the tunnel ovalization and the use of the hypoplastic model lead to a significant improvement in the predicted ground movements and potential damage on surrounding structures.

1

INTRODUCTION

The FE simulation of tunneling represents an interesting benchmark for the constitutive framework adopted to describe soil behavior. In fact, the imposed stress paths are strongly non–proportional and characterized by significant changes in both magnitude and direction, in the different zones of soil surrounding the excavation (crown, invert and springlines). In order to investigate this aspect of the numerical modelling process, two inelastic soil models – namely, the Modified Cam–clay model (MCC, Roscoe and Burland 1968), and a recently developed hypoplastic model for clays (HMC, Mašín 2005) – have been used to simulate the excavation of a circular tunnel in soft clays. The Lower Market Street tunnel, driven in the S. Francisco Bay mud, for which displacements measured during the construction process have been provided by Kuesel (1972), has been reanalyzed. The MCC model is a classical isotropic hardening elastoplastic model for clays, currently available in the material libraries of most widely used FE codes. The HMC model, developed within the framework of the theory of hypoplasticity, is a non–linear inelastic model for clays characterized by the lack of a large elastic domain, and by the continuous dependence of the tangent stiffness on the direction of the loading path.

The particular choice of the two aforementioned model is motivated by the following reasons: i) both MCC and HMC incorporate in their mathematical formulation the basic concepts of Critical State Mechanics and, for this reason, share a large number of material constants (e.g., the slopes of the CSL and of logarithmic compression curves); ii) they can be calibrated from the same set of conventional laboratory tests; iii) the mathematical structure of the HMC model is sufficiently simple to allow a straightforward implementation in commercial FE codes. A rational approach to the FE simulation of the complex activities taking place around the advancing shield during tunnelling operations should in principle be based on the use of large scale 3d models. Such an approach is nowadays fortunately made possible by current advances in digital computers and numerical algorithms. However, the availability of simplified two–dimensional approaches for modelling the effects of the excavation is still of great practical value in tunnel design. This is particularly true when significant changes in tunnel geometry, soil conditions and groundwater regime are expected to occur along the tunnel axis, thus requiring a frequent update of the numerical model. Moreover, a two–dimensional, plane strain approach is ideally suited for a unified modelling of both short– and long–term effects of the excavation. In fact, the consolidation process, induced in clayey

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soils by the development of excess pore pressures, usually takes place long after the front of the excavation has ed the section under examination, and a full 3d analysis is therefore unnecessary. In this work, the simple, yet effective, numerical technique proposed by Tamagnini et al. (2005) to simulate shield tunnelling under plane strain conditions has been adopted. This technique, based on a modification of the so–called “gap closure” approach of Lee et al. (1992), allows to take into both the effects of the volume loss at the tunnel face and of tunnel section ovalization, by imposing a non–uniform gap closure. The importance of tunnel ovalization on the predicted displacement field has been evaluated by comparing the results of FE simulations run with and without ovalization with the available measurements for the Lower Market Street tunnel. 2 THE MODELS CONSIDERED The classical MCC model is thoroughly discussed in many textbooks, see e.g., Wood (1990), and the reader is referred to them for further details on it. In the present work, we have adopted the standard implementation provided in the FE code ABAQUS Standard (v6.4, ABAQUS 2002). The HMC model has been developed by Mašín (2005), starting from the basic principles of Critical State soil mechanics. Its constitutive equations possess the classical structure of hypoplastic constitutive equations:

Closed form expressions for the two tensors L(σ) and N (σ), as well as for the barotropy function fs (p) and the pyknotropy function fd (p, e) have been obtained by Mašín (2005) adopting the bilogarithmic compression laws of Butterfield (1979), as well as the elliptic yield surface and the CS locus of the MCC model. The details of the constitutive equations are provided by Mašín (2005) and are not repeated here. The HMC model is capable of reproducing such aspects of clay behaviour as pressure– dependent stiffness (barotropy); history–dependent behavior (pyknotropy); existence of a critical state condition; contractant/dilatant behaviour related to stress state and to loading history. To improve the model response at small strain levels and upon cyclic loading conditions, the version of the HMC model used in this work is an extended version, equipped with a tensorial internal variable – the so–called intergranular strain (Niemunis and Herle 1997) – which preserves the memory of the previous deformation history. For the FE simulation of the tunnel excavation, the HMC model has been implemented in the FE code ABAQUS Standard, using an explicit, adaptive stress point algorithm with error control, based on Runge–Kutta explicit schemes of 2nd and 3rd order, respectively. See Miriano (2008) for details.

Figure 1. Virtual temperature change imposed to simulate a non–uniform gap closure along the excavation boundary.

3

SIMULATION OF TBM TUNNEL DRIVING

The technique proposed for the 2d FE simulation of the advancement of the tunnel face can be considered an improvement of the method originally proposed by Lee et al. 1992, based on the concept of “volume loss”. A key point of this approach is the idea of modelling the volume loss at the face and at the tail of the shield by reducing the initial diameter of the excavation, D0 , by a quantity G, called gap parameter. Although simple and physically appealing, this approach is not completely satisfactory, as available experimental evidence suggests that the deformations occurring at the tunnel boundary induce not only a reduction of its size, but also of his shape (tunnel ovalization, see e.g. Sagaseta 1998). Among the factors associated with current shield tunnelling practice which may induce tunnel ovalization, we recall: i) the practice of driving the tunnel with the TBM axis slightly inclined upwards, to prevent the effects of its self weight; ii) the deformation of the tunnel lining due to self–weight, during the erection stage; iii) the non–uniform filling of the tail void during tail grouting – if employed. All the aforementioned effects are likely to produce, for a given amount of volume loss, vertical ground movements at the crown which are larger than the gap parameter G, as defined by Lee et al. 1992. In order to take such effects into , the procedure adopted in this work – after Tamagnini et al. (2005) – allows to prescribe both the volume loss and the degree of ovalization, by imposing non–uniform tangential displacements to the shield elements after their installation. In particular, the closure of the gap around the excavation boundary is simulated by imposing a virtual temperature change to the structural elements representing the shield, for which a fictitious coefficient of thermal expansion is assumed. Tamagnini et al. (2005) have shown that a realistic displacement pattern around the tunnel boundary can be obtained by prescribing a sinusoidal temperature change along the shield perimeter (see figure 1). The amount of tunnel ovalization is quantified by the dimensionless factor β := a/b, where 2a and 2b represent the tunnel dimensions measured after the gap closure in the horizontal and vertical directions, respectively. In practice, due the above mentioned

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Figure 3. FE discretization adopted in the simulations.

5 THE FE MODEL Figure 2. Soil profile at the instrumented section.

construction details affecting the spatial distribution of the displacements due to the gap closure, a value of β > 1 is generally to be expected. Given the total volume loss Vl and the tunnel ovalization after the gap closure, β, the two constants T0 and A defining the virtual temperature change field to be imposed on the shield elements can be easily determined (see Tamagnini et al. 2005). Note the method of Lee et al. 1992 is recovered as a particular case for A = 0.

4 THE LOWER MARKET STREET TUNNEL The constitutive models mentioned in Sect 2 have been used to simulate the excavation of the Lower Market Street tunnel in S. Francisco, using the technique discussed in the previous Sect. 3. The Lower Market Street tunnel is a part of the Bay Area Rapid Transit (BART) system in San Francisco. It is a 5.65 m diameter circular tunnel, excavated in a very soft, normally consolidated clay, the San Francisco Bay Mud. The details of the project are reported in Kuesel 1972. During the excavation, a monitoring program was undertaken to check the ground displacement induced by tunnelling operations. The ground movements reported by Rowe and Kack 1983, relative to one of the instrumented sections, are considered in the following. The soil profile at the instrumented section comprises a 33 m thick layer of San Francisco Bay Mud, underlain by a layer of dense fine sand, 6 m thick, and by a 30 m thick layer of soft silt and clay (see Fig. 2). The bedrock is found at 69 m below the ground surface. At the instrumented section, the tunnel depth at springline is 19 m, with a depth to diameter ratio of 3.4. The groundwater table is located approximately at the ground surface.

As the tunnel was excavated in a clay soil and the coupling between the solid skeleton and pore water is expected to play a major role in the evaluation of the displacement field associated to the excavation process and its evolution with time, the FE analyses have been carried out adopting a fully coupled formulation. An extensive preliminary study has been carried out in order to assess the influence of element size and time steps on the computed solution, with particular reference to the distribution of pore pressure increments in the soil surrounding the excavation. The finite element discretization adopted in all the simulations is composed of a total of 888 elements and 2693 nodes, corresponding to 6338 degrees of freedom (see Fig. 3). All soil layers have been modelled using eight–noded E8RP elements, with biquadratic interpolation for displacements and linear interpolation for pore pressures. Fully drained conditions have been enforced in the dense fine sand layer by fixing the pore pressure degrees of freedom to their initial, hydrostatic values. Two different sets of 16 three–noded B22 beam elements have been used to model the shield and the tunnel lining. The mechanical behavior of S. Francisco Bay Mud has been described using both the MCC and HMC models. For the sand layer, a three–invariant extended Drucker–Prager model has been adopted. In lack of detailed information on the behavior of the deep soft silt and clay layer, it has been assumed as linear elastic. This assumption appears reasonable considering that, given its distance from tunnel boundaries and the nature of stress–paths induced by excavation, plastic deformations are unlikely to occur in this layer. The calibration of all the elastoplastic models, including the MCC for the Bay Mud, has been performed based on available experimental data, as discussed in Tamagnini et al 2005.As for the HMC model, those material constants which share the same physical interpretation with the corresponding constants of the MCC model have been determined in the same way, ensuring consistency between the two sets of parameters. The constants controlling the evolution of

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Table 1.

Stages of the FE simulation of tunnel driving.

Stage #

t (days)

1 2

— 4.5

3 4

2.25 2.25

Table 2.

Description Geostatic loading Excavation of soil elements and activation of the shield Gap closure Removal of shield elements and activation of lining elements

Program of the FE simulations.

Run #

Soil model

Vl (%)

β (–)

r01 r02 r03

MCC HMC HMC

2.46 2.46 2.46

1.0 1.0 1.0125

Figure 4. Ground surface displacements: a) computed vs. measured vertical displacements; b) computed vs. measured normalized vertical displacements.

soil stiffness at small strain levels – which appear in the evolution law of the intergranular strain – have been assumed based on the indications provided by Niemunis and Herle 1997 for a wide range of fine– grained soils. The full calibration process has been validated by comparing the predicted responses of MCC and HMC models along conventional axisymmetric loading paths (Miriano 2008). The FE simulations of the tunnel excavation has been performed in 4 stages, as detailed in Tab. 1. After the first geostatic stage, the soil elements within the tunnel boundary are removed and, at the same time, the shield elements are activated. Subsequently, the process of gap closure is simulated by imposing a suitable temperature variation to the shield elements. Finally, the shield elements are removed and simultaneously replaced by lining elements. Three different simulations are considered in this work, as shown in Tab 2. They have been selected among the complete set of FE simulations in order to highlight the influence of the constitutive model adopted and of the excavation simulation technique on the predicted ground displacement pattern. The coefficient of ovalization β adopted in Run r03 has been obtained by trial and error as the value which provides the best fit with the observed maximum ground surface settlement along the tunnel axis.

6

DISCUSSION OF SELECTED RESULTS

Computed vertical displacements at ground surface for the three different simulations are compared with available measurements (after Rowe and Kack 1983) in Fig. 4a. In the same figure, the results of the FE simulations are also compared with the Gaussian curve obtained as the best fit to the experimental data. By comparing the measured settlement with the results of Runs r01 and r02 – in which β = 1.00

has been assumed – it is apparent that none of the two model captures correctly both the maximum settlement at tunnel centerline and the shape of the settlement through, although the performance of the HMC model is significantly better than that of the classical MCC model. In particular, while for the MCC model the maximum computed settlement at centerline is only 40% of the measured one, the ratio between the computed and measured maximum settlement increases to about 65% for the HMC model. On the contrary, a strikingly good match between predictions and measurements is obtained by using the hypoplastic model and introducing a small amount of tunnel ovalization (β = 1.0125). It is worth noting that the same level of accuracy in the settlement predictions could be obtained with the MCC model, but at the expense of a larger ovalization ratio β, see Miriano (2008). The effects of the constitutive model adopted and of the assumed ovalization ratio β on the shape of the settlement trough can be assessed by normalizing computed displacements with respect to the maximum settlement at tunnel centerline, as shown in Fig. 4b. The use of classical elastoplastic model in connection with the standard procedure of Lee et al (1992) results in a large overestimation of the settlement through width. This problem can be partially solved by resorting to a more appropriate description of soil behavior, but the consideration of a certain amount of tunnel ovalization during gap closure is nonetheless necessary in order to accurately reproduce the observed displacement pattern. The relevance of the above results in of the potential tunnelling hazard evaluation can be assessed by means of the damage charts proposed by Boscardin and Cording (1989), where the damage due to tunnelling operations to existing structures is related to the quantities:

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representing, respectively, the horizontal strain and the angular distortion at the ground surface. In Fig. 5, the values of $h and γvh computed in Runs r02 (HMC, β = 1.0) and r03 (HMC, β = 1.0125) are superimposed to the damage level isolines proposed by Boscardin and Cording (1989). As potential building damage is typically associated to extension longitudinal strains, in the figure only the positive part of $h , defined as:

has been considered. Computed values of |γvh | and $h at ground surface are plotted in the figures as a continuous curve, parameterized by the distance x to the tunnel axis. The dashed curves in the figure define the boundaries of the 5 zones corresponding to different damage categories, as defined by Boscardin and Cording (1989). The damage categories are identified with numbers ranging from (1) – almost no damage – to (5) – severe damage. In both the FE simulations, the points close to the tunnel axis undergo compressive horizontal strains, and therefore, the potential damage is associated with angular distortions only. As the distance x from the tunnel axis increases, distortions tend to decrease in magnitude, but extensional horizontal strains develop at the same time, so that the potential damage level remains more or less constant. Eventually, as the distance to the tunnel axis has increased sufficiently that the effects of the excavations on ground displacements are negligible, the damage levels decrease and the curves representing the results of each FE analyses converge towards the origin of the plot. The comparison between the damage curves corresponding to the two simulations, shown in Figs. 5a and 5b, clearly indicates that the use of the conventional approach of gap closure can lead to a significant underestimation of potential building damage levels. In fact, in Run02, only slight damage is predicted in the zone between 6 and 14 m from the tunnel axis, whereas in Run03, the zone affected by category (2) damage level is much larger (from about 3 m to about 18 m), and damage level (3) can be expected in the zone between 7 and 14 m from the tunnel axis.

7

CONCLUDING REMARKS

In this work, two important aspects in the FE simulation of shield tunnel driving – the constitutive model adopted for the soil and the technique employed to simulate the advancement of the shield in a plane–strain approximation – have been investigated with reference to a particular case–history: the Lower Market Street tunnel in S. Francisco. The results obtained in a series of 2d FE simulations carried out using two different inelastic constitutive models – the Modified Cam–Clay model and

Figure 5. Computed building damage levels, after the damage categories of Boscardin and Cording (1989): a) HMC model with β = 1.00; b) HMC model with β = 1.0125. Labels on the symbols indicate the distance, in m, to the tunnel axis.

the hypoplastic model for clays by Mašín (2005) – have shown that the adoption of an inelastic model, with no elastic domain and capable of providing a better description of the evolution of soil stiffness with changing loading direction, can improve significantly the prediction of soil displacements induced by the excavation at ground surface, which are largely underestimated by the use of classical elastoplastic models. However, a key factor in obtaining a good quantitative agreement between predictions and measurements is the modelling of tunnel ovalization which typically accompanies the volume loss due to the tunnel driving operations. In fact, a strikingly good match between predictions and measurements has been obtained for the Lower Market Street tunnel by using the hypoplastic model and introducing a small amount of tunnel ovalization, while the classical approach of uniform gap reduction by Lee et al. (1992) yields a significant underestimation of the maximum settlement at the tunnel centerline. While other factors linked to specific features of the mechanical behavior of the soil – such as, for example, intrinsic or induced anisotropy (Lee and Rowe 1989) or strain localization (Callari 2004) – can also have some impact on the displacement field around the tunnel and at ground surface, the role played by tunnel ovalization appears to be at least of equal importance in actual predictions. In particular, the modelling of tunnel ovalization appears necessary in order to get a realistic assessment of potential building damage levels. In this respect, it is worth noting that, while the a priori estimation of the proper value of the ovalization ratio is indeed a difficult task, the field data reported by González and Sagaseta (2001) suggest that

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a well defined correlation exists between tunnel ovalization and normalized volume loss which can be used to obtain a first estimate of tunnel ovalization ratio β, see Tamagnini et al. (2005). REFERENCES ABAQUS (2002). Abaqus/Standard v. 6.3, ’s Manual. Hibbit, Karlsson & Sorensen, Inc. Boscardin, M. D. and E. J. Cording (1989). Building response to excavation–induced settlements. J. Geotech. Engng., ASCE 115(1), 1–21. Butterfield, R. A. (1979). A natural compression law for soils. Géotechnique 29(4), 469–480. Callari, C. (2004). Coupled numerical analysis of strain localization induced by shallow tunnels in saturated soils. Comp. & Geotechnics 31(2), 193–207. González, C. and C. Sagaseta (2001). Patterns of soil deformations around tunnels. Application to the extension of Madrid Metro. Comp. & Geotechnics 28, 445–468. Kuesel, T. R. (1972). Soft ground tunnels for the BART project. In Proc. 1st Rapid Excavations Tunnelling Conference, pp. 287–313. Lee, K. M., K. R. Rowe, and K. Y. Lo (1992). Subsidence owing to tunnelling. I. Estimating the gap parameter. Can. Geotech. J. 29, 929–940. Lee, K. M. and R. K. Rowe (1989). Deformations caused by surface loading and tunnelling: The role of elastic anisotropy. Géotechnique 39(1), 125–140.

Mašín, D. (2005). A hypoplastic costitutive model for clay. Int. J. Num. Anal. Meth. Geomech. 29, 311–336. Miriano, C. (2008). Modellazione numerica dei movimenti indotti dallo scavo di gallerie superficiali in terreni a grana fine. Ph. D. thesis, Università degli Studi di Perugia – Dottorato in Ingegneria Civile. Niemunis, A. and I. Herle (1997). Hypoplastic model for cohesionless soils with elastic strain range. Mech. Cohesive–Frictional Materials 2, 279–299. Roscoe, K. H. and J. B. Burland (1968). On the generalised stress–strain behaviour of ‘wet’ clay. In J. Heyman and F. A. Leckie (Eds.), Engineering Plasticity, pp. 535–609. Cambridge Univ. Press, Cambridge. Rowe, R. K. and G. J. Kack (1983). A theoretical examination of the settlements induced by tunnelling: four case histories. Can. Geotech. J. 20, 299–314. Sagaseta, C. (1998). On the role of analytical solutions for the evaluation of soil deformation around tunnels. In A. Cividini (Ed.), Application of Numerical Methods to Geotechnical Problems. Invited Lecture. CISM Courses and Lectures No. 397, pp. 3–24. Tamagnini, C., C. Miriano, E. Sellari, and N. Cipollone (2005). Two-dimensional FE analysis of ground movements induced by shield tunnelling: the role of tunnel ovalization. Rivista Italiana di Geotecnica 1, 11–33. Wood, D. M. (1990). Soil behaviour and critical state soil mechanics. Cambridge University Press.

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Role of numerical modelling in the current practice of tunnel and cavern design for hydroelectric projects C. Vibert, G. Colombet & O.J. Gastebled Coyne et Bellier Consulting Engineers, Tractebel Engineering, Gennevilliers,

ABSTRACT: While engineering expertise, professional recommendations and empiricisms play an important role in current tunnel and cavern design practice, the tremendous increase of computer power and the improvement of the -friendliness of available software have produced a step change which has led to routinely adopting numerical modelling techniques at all stages of the design. This paper reviews the benefits and the limitations in the application of 2D and 3D finite element analysis. The discussion is illustrated with learning points using tunnel and cavern design case studies of past or current hydroelectric projects.

1

INTRODUCTION

2.2 Characterisation of the rock mass

The paper reviews the evolution of design practices in underground works relating to hydropower projects during the last decades, with the apparition of ever more powerful computation means and the elaboration of sophisticated modelling software. The benefits and drawbacks of the different practices are analysed, and conclusions are drawn on the best practice on modelling. Hydropower projects are specific in nature due to the generally high level of rock cover, water pressures, rock mass, as opposed to soil conditions and absence of urban constraints.

2

The characterisation of the different geotechnical units constituting the rock mass is the basic task, for which field investigations and laboratory tests are essential. Investigations allow defining, among other aspects, the nature and strength of the rock matrix, the distribution and the geotechnical characteristics of the discontinuities, the hydrogeological conditions as well as the in-situ stress field through which the excavation will take place. It is obvious that obtaining knowledge of the geotechnical properties of the rock mass is, in many cases, a rather challenging task, since many unexpected local variations in the geotechnical properties can occur, the logical consequence of an often eventful geological past.

MAJOR DESIGN ISSUES

2.1 General overview of underground works in hydropower projects

2.3 Definition of shape and

Hydropower projects involve the design of various types of underground structures, depending on the projects. These are access tunnels, derivation tunnels for temporary deviation of the river, water pressure tunnels, shafts (gate shaft, surge shaft, pressure shafts), and various large caverns, (power houses and transformer halls, etc.). The present paper concerns the structural design of the tunnels, shafts and large caverns excavated in rock, that is to say, the required measures to ensure long-term stability of the works, for which numerical modelling has now become routine work. The design of the works from the hydraulic point of view (leakages, potential of hydrofracturing, etc.), which requires other specific means, is not the subject of this paper.

A shape is to be defined for the works, and the available means identified, which requires, for each category of , the characteristic strength of steel, concrete or other material to be placed. This task is generally easier to achieve, since we deal here with man-made materials, the properties of which can be controlled. The modelling of the is established on this basis.

3

EARLY TIMES, WITHOUT COMPUTATIONAL MEANS

3.1 First step: characterisation of the rock mass A large number of projects have been designed and constructed without the help of computers.The general

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approach adopted in design of underground works, in hydropower projects, included the topics developed below of which the first consists in the description and assessment of the mechanical properties of a rock mass as a whole. The investigations focused on the geology, the geomechanical characteristics (strength, deformability, alterability) of the rock matrix and of the t sets (density, roughness, persistence), in order to estimate shear strength along those ts. Taking into the scale effect could be done using in-situ deformability tests and geophysics. The definition of rock mass parameters as the basis for design included safety margins (not a mean value), to allow taking into local weaknesses, which may trigger instability. This approach is still valid nowadays, and this first step remains of absolute necessity. 3.2

Design of temporary and permanent

was estimated as a structure able to withstand the external loads as defined here above, and to prevent rock wedge instabilities, using stereonets and assumed shear strength of ts. The rock- interaction analysis (or convergence-confinement method) was a key element towards a better evaluation of the rock loads, although application of this method is usually restricted to a round shape excavation within a hydrostatic in-situ stress field (therefore, the method can not be applied to large cavern design, the shape of which is generally far from circular). The method allows for checking the adequacy of the designed for stabilising the excavation, using the equivalent pressure. It was on the basis of the obtained results, but also largely on the basis of experience and engineering judgement of the Designer, that the final design was established. Although the well-known empirical patterns using the rock mass characteristics established by Bieniawski (1974) and Barton (1974) achieved substantial progress towards formalisation of this experience, engineering judgment still remains necessary. 3.4

4

EMERGENCE OF FEM ANALYSIS

4.1 The power of numerical modelling

Definitions of loadings

The rock pressure acting on the of tunnels was often determined by means of empirical assumptions, such as the one by Terzaghi (1946). The design of hydraulic tunnels also requires the consideration of all possible hydrostatic loadings which may occur during construction and operation of the structure (from transitory conditions to emptying). 3.3

Nevertheless, the collection of the necessary data was the subject of particular attention as the foundation stone necessary for any reliable engineering judgement. A tendency by some owners to restrict investigations, due to budget pressure has always been a major hurdle, still of actuality. The analytical calculations undertaken by the designer, by hand, allowed him to obtain a real insight into the site specific problems. It must be said that the rock actually placed, rather than being determined by measurements and calculation, was very often defined on site, being adapted to the encountered geological conditions according to the experience of the Site Engineer.

Comments on this process

When no computer was available, implementation of the here above described approaches required time-consuming calculation, especially if one wanted to perform a parametric analysis, to check the impact of the variation of one of the input data.

With the increase in power of computers, it has become common practice from the 1980’ to proceed with 2D numerical modelling, at least for caverns and other excavation shapes, which could not be satisfactorily covered by the rock- interaction analysis. Finite element calculations (here after referred to as FEM) enabled the analysis of any shape of excavation within a plane, to divide the rock mass according to expected variations of mechanical characteristics, to simulate the different excavation phases and the placing of elements. Nevertheless, the emergence of powerful calculation technique raised new issues about numerical modelling practices. Some of these are discussed here after.

4.2 Rock mass characteristics and failure criteria The FEM modelling requires input data in order to reproduce, as closely as possible, the mechanical properties of the rock mass in which the structures are to be excavated. This is not different from the previous approaches, in the case of elastic calculation (still a recommended first approach). However the definition of a failure criterion for the rock mass implies an elastoplastic approach with the simulation of the limit strength of the rock mass. Such problems were already known by engineers using the rock- interaction analysis methods, where the definition of a failure criterion for the rock (generally of Mohr-Coulomb type) is necessary, but the question becomes more acute as the then available software had not yet reached sufficient maturity. The use of numerical modelling conceals some unexpected traps in this respect, and it is always worth to remind younger colleagues the now wellknown case of the earlier softwares using a DruckerPrager failure criterion, as experienced by two of the authors in 1991. Although requiring a definition of a cohesion and friction angle, this conical-shaped criterion does not fit with the Mohr-Coulomb criterion, and overviewing this important aspect could

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lead to the absence of failure of the material under a bi-compressive state of stress. Such a finding illustrates how the Designer requires an in-depth knowledge of the actual modelling tool, otherwise he may be led to perform erroneous calculation Subsequent definition of the rock-dedicated Hoek & Brown failure criterion (1980) and ted rock model (ubiquitous t) brought significance advance in capturing the real behaviour of the rock, but added one more difficulty in the choice of rock mass parameters. 4.3 Numerical modelling of The use of FEM analysis allowed the direct modelling of the elements, such as anchors, shotcrete and plain concrete, steel ribs, etc. (using different formulations, which are outside of the scope of this paper). However, the requirement of connecting the elements to nodes of the mesh was restricting the number of patterns that could be modelled. The activation of the elements remains a major problem in 2D modelling, as it is necessary to define, empirically, under which conditions and after how much time the is placed after excavation is carried out. The rock- interaction analysis has often been the chosen method of solving the problem. Axisymmetrical FEM modelling of tunnel excavation were able to provide a value for the stress release depending on distance from the face. The stress release ratio is hence still used in 2D modelling to assess the forces to be withstood by the , the internal pressure within the part of the work to be excavated being initially diminished to simulate partial stress release, and then the elements are activated before complete release. Consequently, as in the rock- interaction method, the final stresses within the will depend on the assumed stress release ratio. Quite different practices may coexist depending on the conditions, from the assumed zero stress release at installation (upper bound of forces within the ), through a 0.3 stress release ratio for a placed immediately at the face, to complete stress release (upper bound of convergence displacement). 4.4 Conclusion on early 2D FEM analyses The use of 2D FEM modelling unquestionably led to significant improvement in the design of underground structure, by enabling direct simulation of an excavation of any shape, through the various construction phases and within several rock materials of distinct geomechanical properties. Direct modelling of the has also been made possible. However, the limited capabilities of the first 2D software packages could not allow rapid computations and any change in the mesh or in the material properties was time-consuming and prone to errors due to

Figure 1. Numerical modelling for determination of in situ stresses around the caverns to be excavated.

the cryptic nature of input data files. Furthermore the available softwares were not able to reproduce precisely the failure criterion commonly used in rock mechanics. The Designer had thus to remain vigilant as to the consequences of his assumptions. Modelling of the also implies an assumption about the stress release ratio to be considered when being placed.

5

CURRENT APPLICATION OF 2D NUMERICAL MODELLING

5.1 The additional capabilities of software modelling packages Nowadays, 2D numerical modelling, using different methods, have become very common through a reduction in their cost, significant improvement in their -friendly interfaces and the reduction in computer computation time. Advanced modelling of specific failure criteria has also become possible. Therefore, numerical modelling is now routinely used in the design of underground structures in most hydropower projects. The main benefits of these improvements is the ability to obtain results within a few minutes, for a 2D analysis, thus permitting numerous scenarios through changes in the input data to check the influence on the design of the uncertainty on some factors, such as the characteristics of the rock mass, the knowledge and relevancy of which still remain questionable in many cases. It also enables swift back-analysis on the basis of monitoring results of the deformation in the excavated cavities, to check the relevance of the input data.

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Figure 2. Extent of plastic zones around the caverns, due to the asymmetrical stress field (Midas GTS).

This increased ease in modelling opens the possibility of analysis at different scales around the project under study. One example of the benefits of such software can be illustrated using the recent computations performed using GTS software for the design of adjacent power house and transformer cavern – the latter including the downstream surge shaft in its bottom part – in a project in the Ecuadorian Andes. The caverns are located under steep slopes, thereby producing a strongly asymmetrical stress field. A large-scale model (Figure 1) was used to assess the distribution of stresses according to the topography (some slight readjustments were necessary to mach the stress ratios found in the hydrofracturing tests). The importance of such analysis is illustrated in Figure 2, reproducing the model of the two caverns under a stress field, as determined by the large-scale model. The strong influence of the asymmetrical stress field increased plasticity in the direction of the principal stress, resulting in the lengthening of the anchors (originally assumed from empirical design) within the pillar separating the two caverns. Similar effects were noted on another project, in Nepal, using a different software (FLAC 2D). It is to be highlighted that a significant improvement has been achieved in recent software packages as the modelling of the elements do not need anymore to be mandatorily linked to a node of the mesh.

5.2

Remaining limitations of the method

Despite the great progress achieved, some major points still call for caution in the use of 2D numerical analysis. Apart from the problem in choosing the characteristics of the rock mass and its modelling, the problem of selecting a stress release ratio when placing the s remains. This is an in-built limitation of the 2D modelling approach as it can not reproduce three-dimensional effects. A perfect example of this was witnessed by the authors, and also concerned a group of two caverns. The initial model had been made using the most unfavourable section, along a vertical plane cutting

Figure 3. Extension of plastic zones in 2D model through the gate shaft (Phase 2D).

Figure 4. Model under the same conditions of the horizontal section located at the floor of the transformer cavern; no plastic zone extends within the pillar (Phase 2D).

through one of the two gate shafts located beneath the transformer cavern. Extensive plasticity was found within the pillar separating the two caverns (see Figure 3). However, a simple 2D model on a horizontal cross-section was sufficient to ascertain that plasticity under the invert of the cavern was actually limited to the direct vicinity of the shafts, due to their relatively small size compared to the caverns (see section of Figure 4). Therefore, the extensive originally foreseen, on the basis of the previous model, could be significantly reduced. Subsequent elaboration of a three-dimensional model confirmed this conclusion.

6

3D NUMERICAL MODELLING

6.1 When to use three-dimensional modelling? Three-dimensional numerical modelling is unquestionably a great achievement, since it allows for the reproduction of the real geometry and sequence of excavation of complex underground works.

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Nevertheless, the implementation of a 3D model remains time-consuming with respect to the mesh definition and construction as well as in computational time. Although the computation time will still decrease due to advances in computer technology, the authors’ opinion is that a 3D model is justified where a real three-dimensional effect exists due to the complexity of the geometry, the geology or due to the construction sequence 6.2 About solving the problem of stress release ratio Theoretically, one of the key benefits of 3D modelling is that rock- interaction analysis is duly taken into , and avoids the use of an arbitrarily chosen stress release ratio before the placement of the rock . For instance, the displacements and stresses within a newly excavated stretch of tunnel are immediately deduced from the total release of internal pressure. Thus in the next step, the can therefore be directly applied, without the need to define a stress release ratio. This implies, however, that: 1. the analysis stages of excavation and subsequent activation should be strictly separate, 2. the dimensions of each of the excavated and then ed stretches should follow the real excavation sequence of the work. Condition 2 is especially constraining, and if the considered length of the excavated stretch in the model is greater than the actual one, it shall be small enough to allow neglecting the actual effect generated by the proximity of the excavation front at the different intermediate stages of placement to take place in reality over the considered length. Typically, the simulation of the excavation, and subsequent activation, of a large part of a cavern in only one computing step, cannot model the stress conditions, since it would overview the whole intermediate stages of excavation and placement which occur during the actual construction process of this part of the cavern, thereby leading to overestimated deformations of the ground and underestimated the ing forces. In practice, condition 2 is rarely fulfilled, especially in case of large excavations, since it would imply the consideration of a large number of computation steps and the knowledge of the construction methods that would be chosen; generally not available at the design phase. Therefore, the use of a stress release ratio remains necessary in many cases.

Figure 5. Representation of the excavated tunnel and intake shaft, under the right dam abutment (Midas GTS).

with a future intake shaft that will convey the water to the power plant during operation. Therefore, the sequence of the work, and consequently computational steps, was as follows. 1. Application of in-situ stresses, 2. Excavation of tunnel portal, shaft platform and dam abutment foundation, 3. Excavation of the diversion tunnel with the placement of temporary (taking into a stress release ratio), 4. Excavation of the intake shaft with temporary , down to the tunnel (taking into a stress release ratio), 5. Placement of the tunnel lining, followed by full stress release, 6. Placement of the lining of the shaft and tunnel/shaft junction, followed by full stress release, 7. Construction of the dam abutment and the intake tower, 8. Reservoir infilling and application of hydrostatic pressure wherever it is to act, 9. Simulation of dewatering of the tunnel with full reservoir. The considerations of these different steps in the computation of the model made clear the very different loadings applied to the lining, during the construction and operation, once the reservoir filled. In such a configuration, and under such different load cases, the 3D numerical model is of special importance in the design of the concrete structure at the intersection of shaft and tunnel. Stresses within this structure were calculated, and the reinforcement designed accordingly. A graphical output of results is shown in Figure 6.

6.3 An example of 3D modelling A particularly illustrative example of 3D modelling has been recently performed for a hydropower project, in Laos, shown in figure 5. The design of the river diversion tunnel, located beneath the right abutment of the dam to be constructed, had to consider the intersection

7

CONCLUSIONS

The development of computer technology during the last decades has deeply modified the design approach of underground works in hydropower projects, and has

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Progress has also been achieved in the modelling of rock wedges, although 3D-analysis is of course necessary to fully comprehend the problem. However, it is obvious that using the most improved software will not guarantee pertinent results if the input data are not representative of the actual conditions that are to be met. From this point of view, progress, from the early computer-less times, is not so significant, and the careful recollection of geotechnical data is a mandatory step prior to any calculation. Although software have become extremely powerful and useful tools, engineering judgement, experience, and monitoring during excavation are still required for selection of input data, proper interpretation of results, and therefore, pertinence of the modelling. Figure 6. Representation of forces within the lining of the intersection between the tunnel and the intake shaft (Midas GTS).

brought unquestionable progress in solving, economically; the complex problems faced occurring during construction and operation. In most of the cases, a two-dimensional modelling, combined with engineering judgment is sufficient to gain a satisfactory assessment necessary in dimensioning the stresses for design, furthermore it enables parametric analysis approaches in order to evaluate the weight of the different input data. The implementation of three-dimensional model should be used for solving particularly complex problems or for projects with high financial impacts, unless further reduction in time enables a more rapid implementation of the models.

REFERENCES Terzaghi, K. 1946. Rock defects and Load on Tunnel s. In Rock Tunnelling with Steel , Editors R.V. Proctor and T.White. Youngstown: Commercial and Sharing Co. Bieniawski, Z.T. 1974. Geomechanics classification of Rock Masses and its application in tunnelling. Proc. 3rd ISRM International Congress. Denver Barton, N., Lien, R. & Lunde, J. 1974. Engineering Classification of Rock Masses for the design of Tunnel . Rock Mechanics, Vol. 6, No.4: 189–236 Hoek, E. 2007. Practical Rock Engineering. Available at www.rockscience.com

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Some modeling techniques for deep tunnels in rock with FE-continuum models T. Marcher ILF Consulting Engineers, Rum/Innsbruck, Austria

ABSTRACT: There are many computation methods available to estimate the ground- interaction and the stress-strain behavior of deep tunnel excavations. Various methods are compared with each other in this paper based on representative examples. The advantages of the FE continuum analyses with respect to an appropriate initial stress state are obvious as is the correctness of tunnel lining geometry, the simulation capabilities for all construction stages and elements. Nevertheless, the successful application of elasto-plastic FE continuum analyses for deep seated tunnel excavation simulations require special attention to both the realistic rock stiffness and the rock strength in connection with the depth from a geomechanical point of view. This paper addresses a modeling technique for deep tunnels in rock with FE continuum models which takes into the full overburden and the elasto-plastic stress-strain behavior in the vicinity of the tunnel opening. 1

2

INTRODUCTION

The paper discusses continuum modeling approaches in deep tunnel engineering. Hence, stress induced failure conditions are considered. Discontinuity induced failure conditions, which would rather require a discontinuum model, are not part of the present paper. A brief review is presented in connection with the use of closed-form solutions such as the convergenceconfinement method. There is no clear definition for shallow/ deep tunnels. In shallow tunnels, the potential for failure mechanisms under low stress conditions is prevailing; in deep tunnels, the effects and significance of strength and stiffness are of particular interest due to the higher overburden pressures. The present paper considers conditions for deep tunnels in rock including “pseudosolid” rocks, e.g. the transition from “soft rock” to “hard soil” such as marl or weathered claystone. Squeezing conditions of weak rock and swelling pressures are not considered in this paper. The deformability and strength of the rock mass significantly influences the tunnel behavior thus being an important factor for the design of deep tunnels. The modulus of deformation and the strength properties of rock mass are often obtained using standard test equipment assuming moderate to large strain characteristics. Generally, the dimensions of the tunnel opening are small compared to the overburden rock mass. Such boundary conditions require an enormous numerical model to reproduce the whole mountain region. The present paper focuses on model approaches for deep tunnels in rock with special consideration for in-situ stress conditions, stiffness behavior and strength.

CHARACTERISTICS OF DEEP TUNNELLING

2.1 Magnitude of in-situ rock pressure (initial stress state) The magnitude of the rock pressure is affected by the shape, width and height of the tunnel opening and the excavation scheme. Amongst the most important factors influencing the effective stress state on a tunnel lining are the rock strength and the initial stress state. Other decisive effects are the geological stratification (see Kastner 1962) and the groundwater conditions including the groundwater flow (percolation water) in case of drained tunneling conditions. Full geostatic pressure will generally act only on tunnel portals. With higher overburden depth instead of full geostatic pressure loosening pressure acts on the tunnel lining. It has to be outlined that a prediction of the magnitude of rock pressure (both vertical and horizontal primary stress state) is generally most uncertain. Consequently, the magnitude of the secondary stress state (i.e. influenced by the excavation process) implies a high degree of uncertainty. Measurements of an initial stress field are generally very demanding (see John et al. 2004). The scattering of measurement data makes it difficult to determine characteristic input parameters for calculation models.

2.2 Rock pressure on tunnel lining (secondary stress state) Assuming pseudosolid rock conditions and classical excavation methods (e.g. NATM method) the rock pressure is characterized by the loosing process in the vicinity of the tunnel opening. Practical values

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Figure 1. Protodykanov’s theory for rock pressure evaluation.

for rock pressure evaluations are given in the literature within rather wide ranges, see Kastner 1962 (e.g. Terzaghi, Bierbäumer, Jaky, etc.). Protodykanov’s theory is based on the simple determination of natural arching in the rock (Figure 1). 2.3

Rock mass parameters

2.3.1 Strength The Mohr Coulomb criterion has its origins in soil mechanics. In slopes, excavation pits, foundations and shallow tunnels such a constant strength approach is appropriate as normal stresses do not change significantly. However, rock masses have rather different shear strength characteristics, particularly in applications to design of large slopes or deep tunnels. In such cases the change in normal stresses can be decisive. The solution for such conditions consists of either introducing individual rock layers increasing the Mohr-Coulomb strength properties continuously or introducing a curvilinear Mohr envelope, of which the Hoek Brown criterion is one (see Hoek et al. 2002). Such a non-linear criterion completely elimates the problem of trying to estimate equivalent Mohr Coulomb cohesion and friction parameters. 2.3.2 Stiffness Constant deformation properties are common practice in modeling rock tunnels. Nevertheless, the significance of confining stress on the stiffness properties of rock is generally known. Asef et al. (2002) presents results of literature review combined with laboratory testing, which outlines the significance of confinement on the stiffness of the rock mass for numerical modeling. Based on this paper the impact of the confining stresses on the deformation properties of the rock mass is even more pronounced for weaker rock masses (e.g. Verman et al., 1997).

Figure 2. Concept of soil-structure interaction by the convergence confinement method.

of relevant discontinuities, ground water conditions, primary stress field, etc.. Depending on hard rock or squeezing conditions deep tunnels in rock are characterized by either fracturing induced by stresses and/or discontinuities, progressive failure induced by stresses, or squeezing conditions. 3

COMPUTATIONAL METHODS

3.1 General Numerous methods are available for stress analyses of tunnels, from very simple closed-form solutions to high-sophisticated numerical model approaches. 3.2 Convergence-confinement method Closed-form solutions are still of great importance for conceptual design purposes to understand response effects of tunnel excavations. One of those is the ground/ reaction curve (also defined as convergence-confinement method). Such a method is proposed originally by Fenner (1938), which is an analytical rock- interaction solution based upon the development of a “plastic zone” in the rock mass surrounding a tunnel (see Figure 2). The advantage of such methods is the simplicity of the approach and the time-saving handling, nevertheless assumptions such as circular tunnel and hydrostatic stress field limit these methods mainly to conceptual studies. 3.3 Continuum approaches

2.4

Rock mass behavior

The ground behavior is defined as ground reaction to excavation without consideration of or subdivision of the cross section taking into local influencing factors such as the relative orientation

In continuum approaches the rock mass is treated isotropic, i.e. with equal input data in all directions. A direct determination of rock mass properties on a true scale for tunnel excavation problems is not available. The usual approach is down-scaling the properties of

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the intact rock taking into rock mass ting. The mechanical and hydraulic properties shall be based on data which describe the ground at a scale proportionate to the volume of rock affected by the tunnel structure. Laboratory tests provide data of the selected rock matrix, without effects of discontinuities and other defects. Usually properties for the rock mass are determined by indirect methods. The application of indirect methods considers either an empirical scaling of existing lab- and in-situ data or the use of empirical classification systems or both. Due to the subjective interpretations possible it is necessary to cross check between data obtained from different methods. The rock mass rating by Bieniawski, Z. T. (1974) provides a method of determination of the properties of the rock mass. The concurrent version used is the RMR89 from Bieniawski, Z. T. (1989). The “Q” Rock Mass Rating System originally proposed by Barton et al., (1974) also considers intact rock and rock mass properties, including RQD. The most decisive parameter of the Hoek and Brown Method is the so called Geotechnical Strength Index (GSI). The GSI value is a classification of the rock structure and conditions of discontinuities. The mechanical properties are determined on the basis of an empirical formula by Hoek et al., (2002). A number of computer based numerical methods are available. These numerical methods can be divided into boundary and domain methods. In boundary methods such as the the Boundary Element method (BEM) the mesh needs to be constructed only on the excavation surface and the effect of the infinite or semi-infinite domain is considered without having to revert to mesh truncation. Inside the rock mass no shape functions have to be used and therefore equilibrium conditions are satisfied (see Beer 2003). On the other side domain methods include the finite element (FEM) and finite difference methods where the physical problem is modeled by discretizing the problem region.

geological conditions also play a role. In general, squeezing rock in lower locations, and increasing lateral pressure necessitate an approximately circular cross-section. The examples chosen for this paper are typical cross-sections which were used for the construction of railway and highway tunnels worldwide. To illustrate different geometries the following cross-sections will be discussed hereafter: • • •

4.2 Stress release at face The excavation of a tunnel generally leads to a complex stress state at the tunnel face. At a certain distance behind the tunnel face, where initial has been applied, plane strain conditions can be assumed. However, the distribution of stresses and strains close to the tunnel face is three dimensional. For the design of the measures, the 3-D behavior near the face is to be approximated when using 2-D models. Plane strain models are strongly dependent on the assumed degree of ground stress relief at the time of lining installation. By doing so it is necessary to consider deformations of the ground, which precede the excavation itself, i.e. to approximate the so-called load sharing effect. To evaluate the stress release of the ground ahead of the tunnel face due to the excavation the β-method is used for the numerical analysis. The initial stress pk acting around the tunnel excavation is divided into a part (1-β) · pk that is applied to the uned tunnel ahead of the face and a part β · pk that is applied to the ed tunnel. The factor (1-β) represents the stress release factor (see Figure 3). The stress release factor can be estimated according to Kielbassa et al. (1991). The proportion of the forces that act on the ground ahead of the face and the proportion which act on the combined system of ground and lining depends on: • • • •

3.4 Discontinuum approaches The rock is represented as a discontinuum and the focus lies on characterizing both, the intact rock and the discontinuities (ts and bedding planes). The use of discontinuum theories has been gaining attention in tunnel engineering e.g. through the use of UDEC (ITASCA software), which uses a force-displacement law specifying interactions between the deformable t bounded blocks and Newton’s second law of motion, providing displacements induced within the rock mass.

Cross-section with flat invert arch, Cross-section with deep invert arch, Circular cross-section.

Size of excavation Stiffness of ground Stiffness of the lining Uned length of excavation.

It is to be noted that the calculation of the stress release factor by Kielbassa et al. (1991) is based on the theory of elasticity. In case of fracturing ahead of the face due to a high virgin stress field the stress release factor will be increased based on experience and engineering judgment.

4.3 Finite element model with “heavy layer” 4

FE-CONTINUUM MODELING TECHNIQUES

4.1 Tunnel lining geometry The form and size of the tunnel cross-section generally depends on its intended use and thus on the required serviceability envelope. The groundwater and

For most sections analyzed the tunnel depth is 100 m or more from the surface. Creating a mesh which would encom the whole overburden height would be computationally inefficient. Also in FE-programs the minimum element size depends on the overall dimensions of the problem. If the mesh extent is too large,

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ground to form these arches. This leads to unrealistically high lining loads. To for the arching effect and obtain more realistic lining loads stiffening of the top and side surface layers is required. The “restraining” of the boundaries of the model s for the typical 3-D restraining effects of rock type material when undergoing tunnel excavation. Increasing the E-modulus of the continuum elements of the top and side surface layers (see Figure 4) by a factor of 100 simulates the restraining of the boundaries. 4.4 Finite element model for “high overburden” Due to the observations that the E-modulus of rock mass increases with tunnel depth (Asef at al., 2002) and depth dependency of the E-modulus is more pronounced in weaker rock masses (Verman et al., 1997) weak rock mass conditions combined with high overburden are modeled taking a relation of elasticity modulus to the horizontal stress field into . The approach is based on basic assumptions given in Asef at al. (2002) and Verman et al. (1997) and shown in Figure 5.

Figure 3. Stress Release in 2D FE calculations.

Figure 4. 2-D FE-model with a “Heavy Layer”.

the elements are too coarse which can lead to meshing problems around the tunnel. High overburden can be modeled introducing a stiff and heavy layer at the top of the model replacing further overburden. The specific weight of the layer is calculated according to the resembling overburden. This implies the characteristic virgin stress distribution without further interference of the model. Model sizes can thus be reduced to a manageable size resulting in finer meshes and higher precision of the results. As illustrated in Figure 4 the computational section is chosen of such size as to be able to neglect all secondary influences at the boundaries, the initial boundary conditions of the FE-mesh are as follows: the top face is free to displace, the side surfaces have roller boundaries (horizontal fixities) and the bottom face of the FE-model is fully fixed, the overburden is simulated with a FE-mesh that extends to a maximum height (for example maximum 50 m above the tunnel crown in Figure 4). Overburden weight exceeding the maximum “FE-height” is taken into by using an additional “high overburden layer” of 1 m thickness. The specific weight of this equivalent layer is chosen in such a way that full overburden stress is achieved. Tunneling causes a transfer of the ground load by arching to the sides of the opening. However, cutting off a finite element mesh reduces the space for the

The E-modulus E0 for the rock mass determined on the basis of tests (lab or in-situ tests) and/or empirical classification methods is appropriate for regions with larger strains. In deep tunnelling considerable areas around the tunnel opening behave almost fully reversible or elastic. Figure 5 illustrates an approach with linear increase of stiffness with depth according to formula (1). E0 is used for the rock mass close to the surface and in the vicinity of the tunnel opening. The area of large strain in the vicinity of the tunnel may be defined either by the loosening area according to Protodjaknov (see Figure 1) or simply by the plastic zone using convergence-confinement method (see section 3.2). 5

CASE EXAMPLES

5.1 Example 1 – motorway tunnel in U.S. A new 2-lane roadway tunnel (approx. 1.3 km long) at the Route 1 south of San Francisco is under construction at the moment. The NATM tunnels with a diameter of approx. 10 m are shown in figure 6. In this calculation section the initial stress state is assumed to be given by an overburden of 145 m. The ground is modeled using the constitutive model of Mohr Coulomb with linear elasticity and perfect plasticity. The section is characterized by an investigated shear zone between the 2 tunnel openings. This shear zone has

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Figure 7. Example for FE-model with “High Overburden” model.

5.2 Example 2 – railway tunnel in greece

Figure 5. Depth-dependent stiffness approach for deep tunnels.

In the course of the preliminary design for the New Railway Line from Kalambaka to Igoumenits a 10 km long tunnel (single track line with a diameter of approx. 10 m) has to be considered using NATM. Figure 7 illustrates the section with the highest overburden, the initial stress state is assumed to be given by an overburden of approx. 500 m. Figure 7 illustrates a linear increase of stiffness with depth according to formula (1). E0 is used for the rock mass close to the surface and in the vicinity of the tunnel opening (E0 ≈ 900 ÷ 1000 MN/m2 ). The area of large strain in the vicinity of the tunnel has been defined by the loosening area according to Protodjaknov resulting in approx. 60 m. Outside this area the effects of confining pressure and small strain result in an E-modulus of maximum E1 = 4000 MN/m2 . 5.3

Figure 6. Example for FE-model with “Heavy Layer”.

been assumed to be 2 m with an angle of inclination of 45◦ to horizontal level. A variation of the location of the shear zone has been investigated during an iterative numerical process to analyze the anisotropic deformation behavior. Construction stages are considered according to sequencing. The mesh above the tunnels extends 50 m above the crown and 27.5 m below the invert. The full overburden of 145 m is simulated with application of additional stresses at the top of the FE-model of 95 m * 24 kN/m3 = 2280 kN/m2 .

Example 3 – motorway tunnel in greece

These 2-tube NATM tunnels of the Maliakos Kleidi Motorway project in Greece have an opening width of ∼15 m (2 lanes plus an emergency lane). The tunnels exhibit an overburden of up to 300 m consisting of rock of the Ossa Unit (phyllites and limestones). Between these major rock types “transitional” rock units such as deformed phyllites with intercalations of crystalline limestones are found. To for the arching effect and obtain realistic lining loads stiffening of the top and side surface layers is used according to the description in section 4.3. The “restraining” of the boundaries of the model s for the typical 3-D restraining effects of rock type material when undergoing tunnel excavation. Figure 8 shows a longitudinal section of the tunnel with a maximum overburden of approx. 300 m. Due to the fact that approx. 15 m wide fault zone may influence the primary stress field, FE-analysis are carried out. The result of these calculations is that due to the large variety of stiffness of the different lithological units the stress distribution changes considerably. Due to the alternating sequence of soft and hard rock formations and their inclination the vertical stress state will not only be governed by the height of the overburden and the specific weight of the ground. In the fault

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Figure 8. Example for a FE in-situ stress analyses in longitudinal direction.

empirical design rules and engineering judgment shall always be the starting phase. Due to the large amount of unknowns numerical calculations shall concentrate on the factors, which significantly influence the behavior of deep seating tunnel structures, such as the deformability and the strength of the rock mass. For other factors it is proposed to take results of various empirical methods into and to define characteristic values for the rock mass properties on the basis of a engineering judgment regarding the conclusiveness of results With respect to modeling techniques for deep tunnels the realistic judgment of the in-situ stress fields (taking into arching effects and silo effects of individual geological stratifications and the large variety of stiffness of the different lithological units into ), the high degree of stress arching at the tunnel face and the influence of small strains on the stiffness of the rock mass depending on confining pressure and strain conditions have to be considered. REFERENCES

Figure 9. Example for FE-model of a local “Weakness Zone”.

zone the vertical stress varies approximately between 0.5 and 1.0 MN/m2 while the full overburden (γ · h) leads to approximately 5 MN/m2 . This result is used for the numerical analyses of the cross sections. Cataclasites and disintegrated serpentinized peridotites characterize local weak zones in with peridotite with a height of weakness zone of ≤50 m and an extension of ≤40 m. Such a local transition zone is illustrated in Figure 9 overlain by compact limestone. Such conditions of a local soft pocket allow cutting off the finite element mesh. Figure 9 illustrates a FE-cross section with an overburden of 60 m used for dimensioning excavation & measures in this area.

6

CONCLUSIONS

This paper discusses modeling techniques for deep tunnels in rock with FE-continuum calculations. In summary the following findings can be listed: Rock mechanical analyses of deep tunnels are often characterized by a low level of both input data and understanding. Use of simple empirical or semiempirical methods, analytical solutions combined with

Asef, M. & Reddish, D.J. 2002. The impact of confining stress on the rock mass deformation modulus. Gèotechnique 52, No. 4: 235–241. Barton, N. Lien, R., and Lunde, J. 1974. Engineering Classification of Rock Masses for the Design of Tunnel , Rock Mechanics, Vol. 6, No. 4. Beer G., Dünser C. & Noronha M. 2003. Recent advances in the numerical simulation of tunnel excavation. ISRM 2003–Technology roap for rock mechanics, South African Institute of Mining and Metallurgy. Bieniawski, Z.T. 1974. Geomechanics classification of rock masses and its application in tunneling, Proc. of the 3rd International Congress on Rock Mechanics, Denver, 27–32. Bieniawski, Z.T. 1989. Engineering Rock Mass Classification: Manual. Wiley, New York, 205–219. Hoek, E. 1994. Strength of Rock and Rock Masses. ISRM News Journal, 2 (2), 4–16. Hoek, E., Carranza-Torres, C. and Corkum, B. 2002. HoekBrown criterion—2002 edition. Proc. NARMS-TAC Conference, Toronto, 1, 267–273. Hoek, E.1990. Estimating Mohr-Coulomb friction and cohesion values from the Hoek-Brown failure criterion. International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts, 12 (3), 227–229. John M. Poscher G. 2004. Primärspannungsmessungen: Zurecht oder zu Unrecht ein Stiefkind der Felsmechanik. Band 6, 2. Felsmechanik Kolloquium. Wien. Kastner, H. 1962. Statik des Tunnel- und Stollenbaus. Berlin: Springer. Kielbassa, S. & Duddeck, D. 1991. Stress-Strain Fields at the Tunnelling Face – Three dimensional Analysis for Twodimensional Technical Approach, in: Rock Mechanics and Rock Engineering, 24, 115–132. Verman, M., Singh, B., Viladkar, N., & Jethwa, J.L. 1997. Effect of Tunnel Depth on Modulus of Edformation of Rock Mass. Rock Mech. Rock Eng. 30 (3): 121–127.

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Stress-strain behaviour of a soft-rock pillar acted upon vertical loads F. Federico & S. Screpanti University of Rome “Tor Vergata”, Rome, Italy

G. Rastiello Université Paris-Est, Laboratoire Central des Ponts et Chaussées, Paris,

ABSTRACT: Many abandoned “room and pillar” mines were excavated in weak pyroclastic soils not far from the surface of large areas of Rome (depths ranging from 3 to 15 m). A lot of them collapsed; others appear to be in stable condition, although a large percentage of their structural components shows increasing signs of distress from both morphological and mechanical points of view. The stress-strain behaviour of the pillars under vertical loads (imposed settlements of the ground surface) has been simulated (FEA) starting from the in situ initial conditions due to excavation. The soft rock composing the pillars is mechanically modeled through a Modified Cam-Clay Model, suitable to describe the transition from a brittle-dilatant to a ductile-contractive behaviour. It is shown that the average vertical stress in the horizontal cross section, at the mid-height of the pillar, increases following a non linear trend, from the initial, up to the yielding condition.

1

INTRODUCTION

1.1 Roman “room and pillar” mines Several systems of abandoned room and pillar mines (cavities), at shallow depth, have been found in European cities (Bekendam & Price 1993). In Rome, these cavities (rooms) have been usually excavated at 3 to 15 m depths from the ground surface, in weakly cemented puzzolanas (pillars) under a layer of consistent tuffs (roofs or vaults); the rooms compose an approximately regular grid, at one or more levels. Pillar shape is generally quadrangular. The main causes of limit states (LS) of roofs and pillars are: a) changes of stresses and strains due to additional loads on ground surface, dynamic actions (earthquake, traffic, . . .), local collapse and redistribution of loads on pillars and roofs, increase of extraction ratio; b) weathering, change of water content or chemical actions c) ageing (time effects) (Nova et al. 2003). Pillar failure mechanisms can lead to global (ULSG e.g., crushing, punching, shear failure) and local ultimate limit states (ULSL ; e.g., slabbing) (Lembo Fazio & Ribacchi 1990).

roof, pillar, base layer (Fig. 1) has been numerically analyzed through the ABAQUS code. Starting from the geostatic condition, the room and pillar cavities follow the progressive removal of the soft-rock material (Fig. 1). Similar analyses were made on coal pillars (Murali Mohan et al. 2001). The pillars are loaded by the weight of overlying layers and additional external loads, both acting within an influence (tributary) area At ; therefore, the interaction between two contiguous pillars, due to the irregularities of the system, the stress redistribution after the collapse of some pillars, are neglected. Stability of pillars has been analysed through “virtual” load tests, by imposing assigned displacements of the ground surface, up a conventional ultimate LS (collapse) is attained. This numerical procedure has been applied on 67 pillars characterized by different shapes (circular, squared, rectangular) and sizes (cross section area, height); further numerical simulations (149) concerned square shaped pillars (parameter: height) to better define how the stiffness and shear strength parameters of the composing material influence the pillar response. 2.2 Geometry

2 2.1

FEA PROCEDURE Problem setting

The stress-strain behaviour of pillars, referred to in situ initial conditions, due to excavation, and its evolution following increasing vertical loads, have been analysed (FEA). The geotechnical system composed by the shallow layers beneath the ground surface (g.s.),

The analyses regard shallow soils overlying a thin hardrock layer (a few meters) superposed on a thick soft – rock layer, down to 30.5 m under the ground surface, resembling a typical stratigraphy of soils potentially seat of systems of shallow cavities, in Italian urban areas. The following hypotheses have been assumed: horizontal g.s. and soil and soft rock layers; the cross

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Table 1. Physical and mechanical parameters characterizing involved materials. E MPa

ϕ ν

◦

c’ MPa λ

1 100 0.25 – – 2 1000 0.3 30 1 3 – 0.25 – –

k

– – – – 0.13 3.2 × 10−3

M

pc MPa

– – 1.2 1.3 1.5

– – 3.0 4.5 6.0

pertaining to black roman pozzolana (Cecconi & Viggiani 2001). Each layer is homogeneous; each material exhibits isotropy as concerns the relevant mechanical parameters. The vertical stresses affecting more in-depth layers are not appreciably modified by the excavation. 2.3 Finite element analysis procedure

Figure 1. Typical geometry of the geotechnical system.

section area (Ap ) and the shape of the pillar do not vary along the height of the pillar. The following stratigraphy holds: 1) thick layer (10 m thickness) of covering soils, affected by poor mechanical properties; being generally unknown their plastic mechanical parameters, the linear elastic constitutive model has been chosen to represent their mechanical behaviour; 2) a 2.0 m thickness sound rock layer, composing the roof of the cavities; the mechanical behaviour is modelled through an elastoperfectly plastic constitutive model, coupled with a Mohr-Coulomb failure criterium and associated flow law; 3) soft-rock layer, whose upper part is the seat of excavation activities (thickness: 17.5 m). As concerns its constitutive modelization, this layer has been further sub-divided: a) the behaviour of the material composing the pillars and the layer immediately beneath the pillars, for a global thickness equal to 7 m, is simulated through a Modified Cam Clay constitutive model (implemented as Critical State Clay-Plasticity Model, hardening-softening elastoplasticity) (Leroueil & Vaughan 1990, Cecconi et al. 2002); b) the lower layers exhibit a linear elastic behaviour. The relevant mechanical parameters have been quoted referring to typical values of materials in the roman area (Tab. 1). To define the Cam-Clay model parameters, reference has been made to the values reported in (Federico & Screpanti 2003), obtained through the comparison among theoretical results of “virtual” (FEA) triaxial tests and experimental data

Five FE meshes have been defined. Stresses and strains and the role of cross section area on the pillar strength have been analysed by simulating load tests on 2D and 3D models. The 3D mesh (Fig. 2) takes into the peculiar symmetries referred to the geometry and applied loads (geostatic load and possible uniformly distributed loads over the “g.s.”); it represents only a quarter of the whole system beneath the tributary area; it describes a prismatic part of the subsoil wide 4.93 m; its height is 30.5 m; the roof of the cavity is 12.0 m below the g.s.; the base of the cavity parametrically varies between 15.5 m and 17.5 m below the g.s.; correspondingly, the height hp of the pillars ranges between 3.5 m and 5.5 m. Due to the strain-softening behaviour, FE results could be mesh-dependent. So, preliminary analyses have been developed to evaluate the effect of the FE size on the stress-strain behaviour of the pillar. As a result, the optimal mesh is composed by about 250.000 FE of the first order, hexahedrally shaped, with 8 nodes. Their density increases near the pillar and roof, where high gradients of stresses and strains are expected. Initial conditions and loads - The initial conditions, expressed through the stresses components and the void index (e), have been previously determined under the hypothesis of non-elastic (porous elastic) behaviour for all materials. Excavation phases - The excavation has been simulated by removing prefixed sets of finite elements, ing for the whole height of the pillar, starting from the symmetry axis of the cavity. By this way, although the unrealistic, contemporary excavation of 4 tunnels around the pillar is modelled, it allows to describe the growth of pillars whose geometry progressively varies; moreover, a careful choice of the number and geometry of the elements to be removed, ensures a valuable estimate of the current stresses and strains in the pillars, corresponding to each imposed displacement of the g.s..

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Figure 3. Vectorial representation of the principal stresses evolution in a vertical section crossing the diagonal of a squared cross section pillar (six phases of the “virtual” load test).

Figure 2. 3D FE mesh of a pillar quarter.

“Virtual” load tests - “Virtual” load tests have been simulated by imposing assigned displacements of the g.s., until a conventional ULS (collapse) of the pillar is attained. The maximum allowable displacement (δg.s.,max ) and the number (nmax ) of load increments have been imposed. The numerical code fixes the generic load increment (δg.s.,i ) within the range (10−5 m ≤ δg.s.,i ≤ 2 × 10−2 m). This procedure (imposed displacements) does not cause the divergence of the numerical solution; moreover, it allows to simulate the behaviour of the system after the peak strength is attained. This phase is very interesting because, at the verge of the failure, part of the applied load is transferred to the contiguous pillars, although the collapsing pillar still sustains a not negligible load. On the contrary, the equilibrium condition is no longer satisfied if the external applied loads are not balanced by the cavity-pillar system. 3 3.1

RESULTS OF NUMERICAL ANALYSES Stress-strain evolution

Due to the imposed displacements of the g.s. and the related stresses variation affecting the geotechnical system, the pillar shortens its height and widens its cross section area (orthogonal to the pillar axis); this mechanical behaviour strongly depends on the slenderness of the pillar. In reason of the roof stiffness, the pillar is not uniformly loaded; thus, during the

excavation phases, not uniform stresses and strains occur both in the roof and pillar; the reduced confinement allows the maximum strains affecting the external regions of the pillar. At roof-pillar and base-pillar connections, the shear stresses cause further compressive stresses that, in turn, increase the corresponding confinement and prevent transversal dilatations, whose maximum values are assumed at the pillar mid-height. The principal stresses in the pillar strongly rotate according to the virtual load test phases. Referring to a vertical section along the diagonal of the square cross section of a pillar, whose height is hp = 3.5 m, the vectorial representation of principal stresses (σ1 and σ3 ≡ σ2 ) corresponding to the six phases through which the test develops, is depicted in Figure 3. The corresponding evolution of plastic strains is represented in Figure strain mag√ 4 through the √ plastic p p nitude PEMAG = (2/3 εp: εp ) = (2/3 εij εij ), which is a scalar measure of the accumulated plastic strain (εp being the plastic strains tensor). Initially (displacements are not still imposed at the g.s.) (Figures 3a, 4a), the greatest stresses arise at the corners of the base-pillar and roof-pillar connections. The internal regions, close to the symmetry axis of the pillar, bear small stresses. High shear stresses at the roof-pillar and pillar-base intersections induce appreciable rotations of principal stresses, respect to the external reference system (O,1,2,3). The rotations progressively decrease, for the whole pillar height, in proximity of the symmetry axis,

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Figure 5. Average normal stress σ p vs the average axial strain εp . “Virtual” load tests on squared cross section pillars; height hp = 3.5 m; variable area Ap .

Figure 4. Evolution of the yielding in a vertical cross section of the pillar (square cross section; variable PEMAG; six phases of the “virtual” load test).

along which the maximum principal stresses coincide with the vertical ones. Referring to the horizontal cross section at the pillar mid-height, for each volume element, the greatest principal stress (σ 1 ) approximately coincides with the vertical stress (σ v ) and the horizontal plane can be considered as a principal plane of stress; the intermediate (σ 2 ) and minor (σ 3 ) principal stresses, for symmetry reasons, assume the same value and are directed according to the diagonals of the section. The progressive increase of the imposed displacements on the g.s. initially induces (Figures 3b, 4b) an increase of stresses of the mostly stressed elements, until yielding is attained. A stress redistribution takes place; new plastic strains thus develop, firstly affecting large subhorizontal volumes of materials near the base of the pillar and at the roof-pillar connection (Figs 3c, 4c, 3d, 4d); then, plastic strains affect the external parts of the mid-height pillar too. Further increases of the plastic strains in the internal part of the pillar follow additional g.s. displacements (Figs 3e, 4e, 3f, 4f). These phases of the “virtual” load test are respectively related to the conventional pillar collapse (peak strength) and to a post-peak phase; during this final phase, the most part of applied loads is beared by the internal, better confined volumes of the pillar. Principal stresses within the pillars (Figures 3e, f) tend to align according to a “hourglass” shaped configuration. Results of simulations, particularly the plastic strain evolution in the pillar, appreciably agree with experimental observations described by Martinetti & Ribacchi (1965). On the base of a systematic

examination of in situ damages, cracks and visible local collapses observed in puzzolana pillars and walls, these Authors distinguished five classes qualitatively describing a “damage degree”: 1◦ degree – less damaged pillars; only sub-horizontal cracks, some centimeters wide; 2◦ degree – local collapses, some decimetres wide at the external parts of the roof-pillar connection; 3◦ degree – wedge collapse (thickness some decimetres, height 0.5 – 2 m); 4◦ degree – further wedge or prismatic sub-vertical collapses coupled with visible, large cracks along the whole height of the pillar; 5◦ degree – the pillar assumes a “hourglass” shape. Yielding of large sub-horizontal volumes of materials near the base of the pillar and at the external parts of roof-pillar connection (Fig. 4) may be related to observations reported by (Martinetti & Ribacchi 1965), referring to the first crack degrees. High stresses affect yielded material, due to high confinement deriving from the pillar-roof interaction. Fitting among results of numerical simulations and in situ observations appears more evident if reference is made to phases close to the conventional collapse of the system. Under this condition, the external, less confined parts of the pillar are highly plasticized and bear very small stresses. The peculiar “hourglass” shape configuration assumed by the principal stresses distribution may be related to the similar configuration observed in highly damaged pillars (4◦ , 5◦ damage degree).

3.2 Global response variables The variables describing the global behaviour of the pillar are listed below. σ p is the average vertical stress acting in the horizontal cross section, at the mid-height of the pillar:

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Figure 6. a) Average normal stress σp vs the average axial strain εp . “Virtual” load tests on squared cross section pillars; height hp = 3.5 m. Evolution of yielding of base, mid-height, top cross sections; b) Evolution of the average vertical stress σv on the pillar in the mid-height horizontal cross section of the pillar (six phases of the “virtual” load test).

where σ v = average vertical stress on the pillar deriving from the overburden (weight of soils above the pillar); σ v (δg.s .) = increase of σ v due to the increase of external loads (or imposed displacements δg.s ) and Ap = area of the pillar cross section. εp is the average pillar axial strain:

where hp,i = initial height of the pillar, hp (δg.s .) = shortening of the pillar computed along the symmetry axis. Strains occurring with excavation are referred to the unstrained state (εp = 0) of the system, while stresses induced by excavation are taken in in the following steps of the analysis.

It is worth observing that the change of the mechanical characteristics of the upper soil layers changes the response of the whole geotechnical system; this one, in turn, may induce a global ultimate limit state (ULSG ) in different structural elements (e.g., the roofs). However, these variations do not play any role on the response (σ p , εp ) of a single pillar as well as on the ULSG corresponding to its crushing, because they depend only by the geometry (area, height) and properties of the pillar material. The curve (σ p , εp ) represents a constitutive relationship of a pillar as a whole (“macro-element”), if subjected to vertical stress increments (Fig. 5, square cross section, 7.07 m2 ≤ Ap ≤ 63.42 m2 , constant height hp = 3.5 m). FEA results show that σ p increases following a non linear trend, from the initial value, corrisponding to the

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vertical stress associated with the weight of the covering soils (tributary area) up to the yielding condition. The initial trend is characterized by a progressively increasing slope. The only exception regards the smallest pillar (Ap ≈ 7 m2 ), widely affected by plastic strains arised with excavation: a progressive decrease of the slope is obtained. The smaller pillars (Ap < 19.63 m2 ) exhibit a peak strength σ p,peak , coupled with a brittle post-peak behaviour. If Ap increases (Ap > 28.27 m2 ), a ductile behaviour is observed (Martin & Maybee 2000). In the following, the yielding condition is represented by the stress-strain state for which the psudolinear trend ends and a change of the slope of the curve (σ p , εp ) is observed. Referring to the conventional analytical methods, the yielding stress σ p,max represents the “compressive strength” of the pillar. The curve (σ p , εp ) and the evolution of the PEMAG variable of a pillar (height 3.5 m), for three horizontal sections (base, mid-height, close to the roof) are reported in Figure 6. Plastic strains evolve from both highly stressed upper and lower corners (point 1), starting from the free sides of the upper section (point 2); then, plastic strains occur also at the base of the pillar. Further increases of the axial strain induce the progressive yielding of the mid-height cross section (Point 3) that, at the peak-strength condition, appears fully yielded (point 4). By imposing further g.s. displacements after the peak-strength is attained, plastic strains at the midheight of the pillar increase; at the same time, the internal parts of the upper cross sections yield too. Due to the excavation of the cavities system, initially (absence of applied loads or of ground displacements), the maximum vertical stress (σ v ) affects a circular shaped region whose diameter is approximately equal to 1/10 of the side length, located in proximity of the external corner, along the diagonal. Then, by increasing the g.s. displacements, the loads are progressively borne by the internal, mechanically better confined regions; the maximum stress concentration falls in the centre of the cross section; the contours of the stress levels assume a circular shape too, whose centre is the pillar axis. 4

CONCLUSIONS

The stress-strain behaviour of pillars under vertical loads has been numerically simulated through FEA. The features of the soft rock composing the pillars has been modeled through a Modified CamClay Model, allowing the transition from a brittledilatant to a ductile-contractive behaviour, if the pre-consolidation pressure increases.

The effects on the pillar strength of the cross section area, shape, height, slenderness of the pillar, as well as of the mechanical parameters characterizing the constitutive model, are numerically evaluated. Obtained results show that, for all pillars, σ p (average vertical stress acting in the horizontal cross section, at the mid-height of the pillar) increases following a non linear trend, from the initial, up to the yielding condition. The smallest pillars are widely affected by plastic strains arisen after excavation: they exhibit a peak strength coupled with a brittle post-peak behaviour. If the cross section area increases, a progressive increase of the strength (ductile behaviour) is observed. REFERENCES Bekendam, R. and Price, D. 1993. On the stability of abandoned room and pillar mines in very weak Maasrtichtian calcarenites in the Netherlands. In Anagnostopulos, A. et alii editors, Geotechnical Engineering of Hard Soils, Soft Rocks, vol. 2. Cecconi, M. & Viggiani., G. M. B. 2001. Structural features and mechanical behaviour of a pyroclastic weak rock. Int. J. for Num. Anal. Meth. Geomech., 25(15):1525–1557. Cecconi, M. Tamagnini A., De Simone C., Viggiani G.M.B. 2002. A constitutive model for granular materials with grain crushing and its application to a pyroclastic soil. Int. J. for Num. Anal. Meth. Geomech., 26:1531–1560. Federico, F. & Screpanti, S. (2003). Analytical criteria and numerical procedures for safety analyses of pillars and vaults excavated in pyroclastic rocks. XXII Convegno Nazionale di Geotecnica, Palermo (in Italian). Lembo Fazio, A. & Ribacchi, R. 1990. Problemi di stabilità di scarpate e cavità sotterranee in rocce piroclastiche. M.I.R., vol. II, pp. 1–13. Leroueil, S. and Vaughan, P. 1990. General and congruent effects of structure in natural soils and weak rocks. Geotechnique, 40:467–88. Martin, C. & Maybee, W. 2000. The strength of hardrock pillars. Int. J. of Rock Mech. and Min. Sc., 37(8): 1239–1246. Martinetti, S. and Ribacchi, R. 1965. Osservazioni sul comportamento statico dei pilastri di una cava in sotterraneo di materiali piroclastici. Simp. Probl. Geomin. Sardi, Cagliari. Murali Mohan, G., Sheorey, P., and Kushwaha, A. 2001. Numerical estimation of pillar strength in coal mines. Int. J. of Rock Mech. and Min. Sc., 38(8):1185–1192. Nova, R., Castellanza, R., and Tamagnini, C. 2003. A constitutive model for bonded geomaterials subject to mechanical and/or chemical degradation. Int. J. for Num. Anal. Meth. Geomech., 27(9):705–732.

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Tunnel face stability with groundwater flow P.M. Ströhle & P.A. Vermeer Institute of Geotechnical Engineering, University of Stuttgart,

ABSTRACT: The topic of this paper is the tunnel heading stability with groundwater flow towards the excavation face. The starting point is a formula for the minimum face pressure as considered by other researchers. It is shown that some stability numbers, as used in this formula, need further calibration. In this study, calibration is done by finite element analyses. Pore pressures are solved by a simple steady-state groundwater flow calculation and the minimum face pressure is obtained from a nonlinear elastoplastic failure analysis. Both the size and the fineness of the finite element mesh are studied in full detail, as they condition the accuracy of the finite element calculations.

1

INTRODUCTION AND MOTIVATION

For each individual project, it is a technical challenge to build a tunnel under difficult groundwater conditions. Indeed, a disaster occurred to Marc Isambard Brunel as he built the Thames-Tunnel (1825–1842). In conventional NATM tunneling a lowering of the groundwater or alternatively a soil freezing is essential. In this paper, however, conventional NATM tunneling is not to be discussed; instead the use of a closed shield will be subject of this paper. Closed shields are intended on the one hand to the tunnel face and on the other hand to reduce or stop the flow of groundwater. For this purpose, two types of shield machines can be used, i.e. the earth pressure and the slurry shield. The choice of the machine type depends on the soil parameters. Earth pressure shields are predominantly used in cohesive soils. Conditioning of the soil can significantly increase the fields of application. In general slurry shield machines will be used with cohesionless soils such as fine sand formations up to coarse gravel, see Bielecki. On considering the minimum pressure at the tunnel face, we will extend previous work by Anagnostou & Kovári. They considered a tunnel with groundwater flow towards the face and proposed a particular formula for the minimum face pressure, assuming a wedge-sliding mechanism at failure. Later, Vermeer et al. performed nonlinear finite element analyses to improve the face-pressure formula. However, they did not consider groundwater flow. In this paper we will extend the work by Vermeer et al. to include groundwater flow. 2 ACTUAL STATE OF SCIENCE Anagnostou & Kovári described an analytical formula for the calculation of face stability including the effect

of ground flow. The pressure is expressed as a function of the piezometric height. Anagnostou & Kovári only considered the of the tunnel face by the earth pressure shield, which as mentioned is used in cohesive soils. The failure pressure equation for the tunnel face according to Anagnostou & Kovári depends on following parameters (see equation 1): diameter of the tunnel D, overburden H, piezometric head in the working chamber hF , elevation of the water table h0 , cohesion c , as well as frictional angle ϕ and the submerged unit weight γ (for soil below the water table) and on the dry unit weight γd (for soil above the water table). According to Anagnostou & Kovári the equation for the failure pressure pf

F0 , F1 , F2 and F3 are functions of the overburden H and the diameter D of the tunnel as well as the angle of friction ϕ’ of the soil. h is the difference (h = h0 − hF ) between groundwater level and the water level in the working chamber. For dry soils equation (1) reduces to the first two . In Figure 1 the situation without flow pressure is shown. Vermeer et al. performed a very large number of non-linear finite element calculations and were able to fit computational results by the functions

at least under the conditions that ϕ > 20◦ and H > 2D. For friction angles beyond 25◦ , the above expressions even hold for H > D. From Figure 2 it can be observed that the above F0 -function corresponds well with data by Léca & Dormieux and Krause. In contrast to the soil

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Figure 1. pressure without seepage pressure.

Figure 4. Chart for the dimensionless coefficient F2 after Anagnostou & Kovári.

Figure 2. The soil weight stability number as determined by different methods.

Figure 5. Chart for the dimensionless coefficient F3 after Anagnostou & Kovári.

probably due to their use of a very approximate wedge sliding model. In Figure 4 and Figure 5 the functions F2 and F3 by Anagnostou and Kovári are presented. F2 is the function for the head difference on failure pressure in cohesion less soil and F3 is for the head difference on pressure in cohesive soil. The analyses of these two functions F2 and F3 with the finite element method with the elastic-plastic Mohr-Coulomb constitutive model will be the aim of the authors work. Figure 3. The cohesion stability number F1 according to different models.

3 weight number, the cohesion number can be derived theoretically. In fact, Vermeer & Ruse derived the simple expression F1 from above, which was also verified by use of the finite element method. Again this expression can be compared to findings by other researchers, as also done in Figure 3. Once more Krause’s results are based on shell-shaped failure body, whereas the data by Anagnostou & Kovári are based on the sliding wedge model. On the other hand Anagnostou & Kovári proposed higher values for F0 and F1 , but this is

CALCULATIONS FOR FACE STABILITY WITHOUT GROUNDWATER

For limit load analyses, pre-failure deformations are not of great importance and are assumed to be linearly elastic, as is usual within the elastic-plastic Mohr-Coulomb model being used in this paper. Elastic strains are governed by the elasticity modulus E and Poisson’s ratio υ. The particular values of these input parameters influence load displacement curves as shown in Figure 7, but not the failure pressure pf .

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Figure 7. Typical pressure displacement curve. Table 1. water.

Figure 6. Flow area at collapse.

For this reason they will not get any further attention in this study. In addition to the elasticity modulus and Possions’s ratio, there are three material parameters for the plastic behavior: the effective cohesion c , the effective angle of friction ϕ and the angle of dilatancy ψ. Different dilatancy angles give different collapses mechanism, but they have very little influence on the failure load. For this reason, nearly all our computations were performed for non-dilatant material. As symmetrical tunnel are considered, the collapseload calculations are based on only half circular tunnel which is cut lengthwise along the tunnel axis. Figure 6 shows a typical finite element mesh as used for the calculations. The ground is represented by 10-noded tetrahedral volume elements. The boundary conditions of the finite element mesh are as follows: the ground surface is free to displace, the side surface have roller boundaries and the base is fixed. It is assumed that the distribution of the initial stresses is geostatic according to the rule σh = K 0 σv , where σh is the horizontal effective stress and σv is the vertical effective stress. K0 is the coefficient of lateral earth pressure. Vermeer & Ruse investigated the possible influence of the initial state of stress, by varying the coefficient of lateral earth pressure and found that the K0 -value influences the magnitude of the displacements but not the pressure at failure. The first stage of the calculations is to remove the volume elements inside the tunnel and to activate the shell elements of the lining. This does not disturb the equilibrium as equivalent pressures are applied on the inside of the entire tunnel. To get full equivalence between the initial ing pressure and the initial geostatic stress field, the pressure distribution is not constant but increases with depth. This is obviously significant for very shallow tunnels, but a nearly constant pressure occurs for deep tunnels. The minimum amount of pressure needed to the tunnel is then determined by a stepwise reduction of the ing pressure. A typical pressure-displacement curve is shown in Figure 7, where p is the pressure at the level

Input parameter for the analyses without ground-

saturated unit soil weight cohesion angle of friction angle of dilatancy young’s modulus coefficient of lateral earth pressure

γ = 22 kN/m3 c’= 0 kN/m2 ϕ’= 25–40◦ ψ = 0◦ E = 10.000 kN/m2 K0 = 1.0

Figure 8. The soil weight stability number as determined by Vermeer & Ruse and current calculations.

of the tunnel axis and u the displacement of the corresponding control point at the tunnel face. The control point has to be chosen within the collapsing body; otherwise the load-displacement curve in Figure 7 will come to an almost sudden end and the curve then cannot be used to conclude that failure has been reached. In order to validate the assumptions defined here, first of all calculations without groundwater were carried out and compared with the results from Ruse. The calculations were made using the three dimensional FE-Code Plaxis GiD. The diameter of the tunnel is D = 10 m, the width of the domain is B = 20 m and the length is L = 26 m. The material parameters of the subsoil are shown in Table 1. In Figure 8 the effect of the angle of friction on the normalized failure pressure, i.e. on pf /γD is shown.

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Figure 10. Effect of rear mesh boundary on the normalized pressure. Figure 9. Flow net in front of the tunnel face.

The calculations were carried out and the results were compared with those of Ruse. The difference with Ruse can be explained by the choice of the pressure. Ruse used for his calculation a linear increase of the pressure over the depth at the tunnel face. In the calculations, presented here, a constant pressure over the depth was assumed. Considering the importance of the face distribution and the fact that a trapezoidal distribution is most realistic, it was decided to continue this study with a trapezoidal distribution. For the analyses with groundwater flow Plaxis 3D Tunnel is used.

4

GROUNDWATER FLOW CALCULATIONS

Assuming a horizontal groundwater level all water particles has the same hydraulic head before the tunnel is constructed. The hydraulic head is defined as the elevation head plus pressure head plus velocity head. As soon as a tunnel is constructed below the ground water table, there is a difference in hydraulic head and a groundwater flow towards the tunnel face occurs. Because of the low permeability, the velocity head can be neglected in the given boundary value problem. The groundwater flow causes a seepage pressure on the tunnel face. At the tunnel face holding and driving forces acting on the failure body. The seepage forces on the failure body have a negative effect on the stability of the tunnel face and should be considered in the determination of the failure pressure. The flow net in front of the tunnel face, (see Figure 9) is independent of the coefficient of permeability k in steady state conditions. Only the hydraulic gradient i increases in direction to the tunnel face, because the potential lines are closer in front of the tunnel face (see Höfle et al.). For the calculations the following hydraulic boundary conditions were chosen: the groundwater table was at the ground surface as in Anagnostou & Kovári. Symmetry was also used, that means only one half tunnel was modeled. The symmetry line was taken as an impermeable boundary. In the working chamber atmospheric pressure was described.

Figure 11. Effect of front mesh boundary on the normalized pressure.

The seepage forces in front of the tunnel face are maximum for these boundary conditions. An impermeable tunnel lining was used. In Figure 10 the effect of the rear mesh boundary is shown graphically. For a mesh boundary at only a distance of 0.25D behind the tunnel face the calculated failure pressures for the impermeable or permeable boundary are very different. However at a distance of 4D the failure pressures are the same and remain constant. In Figure 11, the influence of the front mesh boundary can be seen. The mesh independent failure pressure occurs in this case at a distance of 6D from the tunnel face. The distribution of the hydraulic head is shown in Figure 12. The left picture shows the head with a permeable front boundary, the right picture with an impermeable front boundary. This has a clear effect on the groundwater head in front of the tunnel face. In order to neglect this effect, it is recommended to use a length of six times the diameter in front of the tunnel face.

5

PRESSURE CALCULATIONS IN GROUNDWATER

All the failure pressure calculations presented are for drained conditions. Steady state groundwater flow will

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Figure 12. Potential lines for permeable front boundary (left) and impermeable front boundary (right), in each case for a distance of 3D in front of and 4D behind the tunnel face. Figure 14. Effect of the domain size.

Figure 15. Mesh variation.

Figure 13. Geometry variation dependent on the diameter D.

be considered. The parameters for the subsoil are shown in Table 1. In addition a permeability of k = 10−5 m/s for the subsoil is used. This corresponds to a fine sand. Before the effect of the mesh fineness or the shear parameters on the failure pressure can be investigated, a suitable domain size for this problem has to be found. The mesh should have minimum dimensions such that no change in the failure pressure occurs for the same soil parameters and hydraulic boundary conditions. For this reason the size of the domain was varied in the width (variant A), in the depth below the tunnel (variant B), in the length of the tunnel (variant C) and in the length in front of the tunnel face (variant D). For all variations the overburden was 2D, therefore 20 m. In Figure 13 the four variants are illustrated. The outcome of the domain size variation is also shown in Figure 13 by using in variant D a length of 6D in front of the tunnel face. The final domain has a width of 4D, a height of 7D, a length of 6D in front of and 4D behind the tunnel face by using an overburden of 2D. Results of the analyses respect the variations can be obtained from Figure 14, which shows the results for the four calculations (variant A-D). The pressure which is calculated in the final stages for a particular variant is not always the same, as can be seen in Figure 14. This depends on the chronology of the investigation of the variants.

It was started from a domain with a length of 5D, a width of 2D and a height of 4D. Than variant A was investigated, i.e. the width was varied from 1D to 8D. The dashed line in Figure 14 represents variant D and thus the final result for the domain. During these calculations a mesh with the same average element size at the tunnel face and around the tunnel cross section was used. In the area of the failure body the mesh was refined in the longitudinal direction. 6

DISCRETISATION EFFECT

The mesh needs special monitoring. With a finer mesh the accuracy of the results increases, but this result in longer calculation time. Special attention should be paid to the area in front of the tunnel face, because in this space the failure body develops. Therefore, in an area of 10m in front of the tunnel face a finer mesh for the used soil parameters (see Table 1) was used, in which the failure body according to Horn occurs. Different lengths of refinement between 5 m and 15 m were tested. The mesh dependency was investigated according to Figfure 15. The average element size was changed in three different areas: inside the failure body (FB), at the tunnel face (TF) and outside these two areas (SUB). The average element sizes in these areas in longitudinal direction (subscript L) as well as in the cross section (subscript C) were varied. At the end of the first mesh variation cross the tunnel axis, it can be seen, that the effect of the average

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results by Anagnostou & Kovári equation (1) reduces in cohesionless soil to

Figure 16. Final finite element mesh in longitudinal direction (left) and cross the tunnel axis (right).

element size in area SUBC has no significant effect on the failure pressure. Simply the effect of the average element size at the tunnel face (TF) is in this case important for the results. Here, the average element size in FBC was the same as in SUBC . In example, a mesh with an average element length of 1.5 m at the tunnel face (TFC ), achieve the same failure pressure for average element lengths between 4 m and 8 m around the tunnel (SUBC ). In a next step, the mesh (FBC ) was refined in the failure body. Similar to the first mesh variation cross the tunnel axis, the effect of the average element size at the tunnel face is important and a small effect of the average element size in the failure body (FBC ) can be seen. The average element size in longitudinal direction of the tunnel axis plays also an important rule. Different average element sizes in SUBL and FBL was used in the mesh. In FBL the average element size from 0.5 m to 2 m and in SUBL from 2 m to 10 m was analysed. As well as in the mesh variation cross the tunnel axis the influence of the average element size in SUBL is of small importance for the failure pressure. Finally, according to Figure 16 a mesh with an average element size of 10 m in area SUBL and 1m in FBL as well as 10 m in SUBC , 4 m in FBC and 1m in TF will be used for further analyses. This model has a length of 100 m, a height of 70 m and a width of 40 m consists to 10.500 15-noded elements with quadratic interpolation. It is assumed that for this mesh size the mesh dependency can be neglected.

7

COMPARISON TO ANAGNOSTOU & KOVÁRI

In comparison to the results of Anagnostou & Kovári it can be seen that their results, because of the use of the sliding-wedge model, are extremely on the safe side. The necessary pressure by use of equation 1 for a cohesionless soil with a friction angle of ϕ’= 30◦ , a diameter of 10 m, an overburden of H = 2D, a saturated unit soil weight of γ = 22 kN/m3 and a maximum h is nearly 150 kPa by Anagnostou & Kovári. The necessary pressure for the same conditions in current calculations is 110 kPa. Assuming to the

Using the coefficient F0 and F2 of Anagnostou & Kovári pf is 150 kPa. In contrast to this, the ing pressure becomes nearly 140 kPa if F0 after Vermeer et al. and F2 by Anagnostou & Kovári was used. F2 for a soil with a friction angle of ϕ = 30◦ by Anagnostou & Kovári is arround 0.46. One can deduce from the presented calculations, that F2 becomes lower than stated by Anagnostou & Kovári. F2 ranges between 0.3 and 0.4. Both approaches offer nearly the same result, using a cohesion of 70 kPa. In this case no ing pressure is needed. 8

CONCLUSIONS AND OUTLOOK

It has been shown that the face-pressure formula is suitable for tunneling in dry soil. For groundwater flow towards the excavation face, however, the stability number F2 requires further calibration. As yet, this number has been calibrated on the basis of a slidingwedge stability model by Anagnostou & Kovári, which would seem to be inaccurate. For dry soil this has been shown by Vermeer et al. and shown in this study for a case with groundwater flow. Further such FEanalyses are needed to arrive at graphs or mathematical expressions for the stability numbers F2 and F3 . REFERENCES BIELECKI, R. 2009. Schildvortriebe mit Tübbingausbau. Editor: Wissenschaftsstiftung Deutsch-Tschechisches Institut. GbR Veröffentlichungen Unterirdisches Bauen. Hamburg VERMEER, P.A.; RUSE, N.; MACHER T. 2002. Tunnel heading stability in drained ground. Felsbau 20. No. 6. pp. 8–18 ANAGNOSTOU, G.; KOVÁRI, K. 1996. Face Stability Conditions with Earth-Pressure-Balanced Shields. Tunnelling and Underground Space Technology. Vol. 11. No. 2 LÉCA, E.; DORMIEUX, L. 1990. Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material. Geotechnique 40 (4) KRAUSE, T. 1987. Schildvortrieb mit flüssigkeits- und erdgestützter Ortsbrust. Institut für Grundbau und Bodenmechanik. Technische Universität Braunschweig. Dissertation, Heft 24 VERMEER, P.A.; RUSE, N. 2001. Die Stabilität der Tunnel in homogenen Baugrund. Geotechnik 24. No. 3. pp 186–193 RUSE, N. 2004. Räumliche Betrachtung der Standsicherheit der Ortsbrust beim Tunnelvortrieb. Dissertation, Mitteilung 51. Institut für Geotechnik. Universität Stuttgart HÖFLE, R.; FILLIBECK, J.; VOGT, N. 2008. Time dependent deformations during tunnelling and stability of tunnel faces in fine-grained soils under groundwater. Acta Geotechnica 3. pp. 309–316 HORN, M. 1961. Horizontaler Erddruck auf senkrechte Abschlussflächen von Tunnelröhren. In: Landeskonferenz der ungarischen Tiefbauindustrie. Budapest – Übersetzung ins Deutsche durch die STUVA

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Viscoplastic models for the analysis of tunnel reinforcement in squeezing rock conditions G. Barla & D. Debernardi Department of Structural and Geotechnical Engineering, Politecnico di Torino, Italy

D. Sterpi Department of Structural Engineering, Politecnico di Milano, Italy

ABSTRACT: Numerical analyses of tunnels in rock masses which exhibit a time dependent behaviour, such as in severely squeezing conditions, require the use of viscoplastic constitutive laws. Tunnel excavation needs to be modelled with attention paid to the excavation-construction sequence, in order to correctly estimate the interaction between the rock mass and the system. This paper presents the numerical analyses carried out to study the excavation process of the Saint Martin La Porte access adit (Lyon-Turin Base Tunnel), where very severely squeezing conditions were met. In order to face these conditions a novel construction system has been implemented, with performance monitoring being used systematically in order to closely follow the tunnel response. Two different viscoplastic constitutive models (SHELVIP and 3SC) have been adopted. A comparison between numerical results and monitoring data allows one to assess the potential of the two models.

1

INTRODUCTION

In rock masses characterized by a time dependent behaviour tunnelling at great depth is to encounter difficulties whenever the increasing ground deformations lead to large tunnel convergences and high pressure on the s, even in the short term, i.e. in a time span comparable with the excavation advance time (Steiner 1996, Hoek 2001). This is the case of the so called squeezing conditions (Barla 1995), usually met in weak rock masses or during excavation through fault zones (Cristescu & Hunsche 1998, Dusseault & Fordham 1993). Among the approaches proposed and applied to tunnel design (Barla, 2002), numerical analysis is most effective whenever a detailed response of the interaction between rock mass and structures in the short and long term is to be assessed. In order to correctly predict the tunnel convergence and the pressures that the rock mass will exert on the ing structures, an accurate modelling of both the excavation-construction sequence and the mechanical constitutive behaviour of the rock mass is of primary importance. The first requirement is the use of threedimensional or axi-symmetric models and a realistic simulation of the actual excavation sequence and structures installation. Here, the term “structures” refers to the reinforcement measures, installed at the tunnel face and in the heading behind the face, including the primary and the final lining. These aspects turn out to be crucial elements for design, especially

in cases where squeezing effects show up in the short term and last for a long time. The second requirement is particularly true when using elasto-viscoplastic constitutive laws and time dependence is explicitly taken into . The complexity of the problem has often led to the adoption of simplified models where time dependence is ed for by degrading stiffness and shear strength properties with time. However, this approach is very limited in the case of very severely squeezing conditions. With the choice of the constitutive model, the necessity is to calibrate its parameters with reference to the in situ rock mass behaviour. This task might represent the drawback of advanced constitutive models. In fact, in situ creep tests involve operational difficulties and suitable scaling rules, allowing for estimating the rock mass parameters on the basis of laboratory data, have yet to be validated. In some cases, when severely squeezing conditions arise and performance monitoring is adopted systematically, convergence and stress measurements could provide data for the back analysis of the creep phenomenon. In this framework, the case study of the Saint Martin La Porte access adit, along the Lyon-Turin Base Tunnel, is taken as representative of very severe squeezing conditions. The excavation process has been analysed with a two-fold purpose to assess the potential of two different elasto-viscoplastic constitutive models and to get insights into the response of a novel yielding system, especially devised to cope with the encountered conditions.

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Figure 1. Geological profile along the Saint Martin La Porte access adit.

Figure 3. Horseshoe cross section initially adopted showing the large convergences experienced.

Figure 2. Typical geological conditions at chainage 1444 m.

2 THE CASE OF SAINT MARTIN LA PORTE ACCESS ADIT The Saint Martin La Porte access adit is located along the Lyon-Turin Base Tunnel and crosses a Carboniferous formation (marked as hSG in Fig. 1), consisting of black schists (45 ÷ 55%), sandstones (40 ÷ 50%), coal (5%), clay-like shales and cataclastic rocks. The typical geological condition at the tunnel face shows a highly heterogeneous and fractured rock mass, often crossed by faults (Fig. 2). Initially, with a horse-shoe cross section ed by anchors, yielding steel ribs with sliding ts and shotcrete, convergences up to 2 m were observed, with overburden around 300 m (Fig. 3). Having to face higher overburden pressures under the same squeezing conditions, a novel construction method was then implemented, in a near circular cross section, consisting of: – Face reinforcement, including a ring of grouted fibre-glass dowels around the opening perimeter. – Mechanical excavation, carried out in steps of 1 m length, with 8 m long untensioned anchors, 1 m spaced, yielding steel ribs with sliding ts and shotcrete (first phase). – Completion of the excavation, 30 m from the tunnel face, with yielding steel ribs with sliding ts and 9 longitudinal slots fitted with Highly Deformable Concrete (HiDCon) elements (second phase). – Installation of a coffered concrete lining, 1 m thick, approximately 80 m from the tunnel face (third phase).

The deformable concrete elements provide the tunnel with a yielding which allows controlled deformations to take place. The hollow glass particles included in the concrete mixture increase its void fraction and collapse at a given compressive stress, thereby providing the large deformability required. In this case the elements have height 40 cm, length 80 cm and thickness 20 cm. They have been designed to yield up to 50 per cent strain approximately in a ductile manner, while the yield stress has been chosen to be 8.5 MPa. A monitoring system was installed to control the tunnel performance, in particular: – the convergence, with optical targets along the perimeter; – the in depth radial displacements, with multi-point borehole extensometers; – strains and stresses in the concrete lining, with embedded strain cells. The available data exhibit a clear anisotropic timedependent behaviour of the rock mass, slow movements taking place even during standstill, a large extent of the zone of influence of the excavation, and a long lasting tendency to undergo deformations. In this way a back analysis can be performed and the mechanical parameters can be identified.

3 VISCOPLASTIC CONSTITUTIVE MODELS Two constitutive models have been developed in order to describe the squeezing behaviour of the rock mass during tunnel excavation. The first one is referred to as Stress Hardening ELasto VIscous Plastic (SHELVIP) model (Debernardi 2008, Debernardi & Barla 2009). It has been derived from the classical theory of elastoplasticity, which is frequently adopted in design analysis of tunnels, by adding a viscoplastic component based on the Perzyna’s overstress theory (Perzyna 1966). The time-independent plastic strains

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Figure 4. Stress Hardening ELasto VIscous Plastic model (SHELVIP).

Figure 5. 3 Stages Creep rheological model (3SC) and strains developing during primary (dotted line) and secondary/tertiary creep (solid line).

develop only when the stress point reaches the plastic yield surface fp (σ) = 0 which is defined by the Drucker-Prager criterion (Fig. 4). The non associated flow rule and the consistency condition of the classical theory of elasto-plasticity allow the rate of plastic strains to be evaluated. The time-dependent viscoplastic strains develop only if the effective stress state exceeds a viscoplastic yield surface fvp (σ) = 0, which is also defined by the Drucker-Prager criterion and is internal to the plastic yield surface. The rate of viscoplastic strain, which is both deviatoric and volumetric, can be evaluated by using the viscoplastic flow rule of Perzyna and depends on the deviatoric state of overstress referred to the viscoplastic yield surface. Inside the viscoplastic yield surface the behaviour is purely linear elastic. The plastic yield surface is fixed, while the viscoplastic yield surface, which defines the stress threshold of development of viscoplastic strains, can harden according to a stress-based hardening rule. The second model has been devised to for the 3 Stages of Creep (3SC model), namely primary, secondary and tertiary creep, by way of the connection of simple elements into the rheological model shown in Fig. 5 (Gioda & Cividini 1996, Sterpi & Gioda 2009). The first elastic element represents

the time independent reversible response. The viscoelastic Kelvin element s for the deviatoric primary creep, i.e. time-dependent reversible strains which develop with decreasing rate, thus reaching a stable value. The viscoplastic element s for the deviatoric secondary creep, i.e. time-dependent non reversible strains, which develop with constant rate. The secondary creep is onset when the deviatoric component of the state of stress exceeds the limit established by the viscoplastic envelope Fvp (σ) = 0, defined by a non associated Drucker-Prager criterion. The secondary creep rate is therefore governed by the coefficient of plastic viscosity ηvp and by the portion of deviatoric stress exceeding the limit. Finally, the tertiary creep corresponds to an increase of the creep rate and is introduced by a progressive reduction of the coefficient ηvp and a contraction of the viscoplastic envelope, i.e. a reduction of its parameters clim , φlim . This sort of progressive damage is controlled by the cumulated viscoplastic strains. The two models have been tested to their effectiveness in reproducing creep at the laboratory scale. Several triaxial creep tests on rock samples of various nature have been considered to this purpose, and among them also samples of coal taken from the Saint Martin La Porte access adit (Barla et al. 2009). Due to the different mathematical formulation it is rather difficult to compare the two constitutive models on a theoretical basis. The main distinguishing features of the SHELVIP model are: (i) the true yielding limit, which cannot be exceeded by the stress state and induce an instantaneous irreversible strain, (ii) the possibility to accurately describe the load dependency, (iii) the volumetric component of time-dependent strains. For the 3SC model one can mention: (i) the possibility to reproduce the tertiary phase of creep, (ii) the intuitive physical meaning of the time-dependent parameters, which could be obtained on the basis of a controlled procedure (Sterpi & Gioda, 2009).

4

NUMERICAL ANALYSES

Tunnelling through the Carboniferous rock mass at the Saint Martin La Porte access adit has been analysed by using SHELVIP and 3SC, respectively implemented in the Finite Difference Method code FLAC (Itasca 2006) and in the non commercial Finite Element Method code SoSIA (Gioda & Cividini 1996). The tunnel section between chainage 1394 and 1527 m has been chosen for the analyses carried out, where the overburden is approximately 360 m. The initial stress state is assumed to be isotropic and equal to 9.8 MPa. The water table is not present. Axi-symmetric conditions have been adopted in order to reproduce the three-dimensional influence of the tunnel face, which is known to play a significant role in squeezing conditions. The tunnel cross section is assumed to be circular, with an equivalent radius of 6 m. Full face excavation with a constant advance rate

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Table 1. models.

Figure 6. Sketch of the numerical model adopted to simulate the excavation-construction sequence.

of 0.54 m/day is considered. The total size of the mesh is very large in order to minimize the boundary effects that are very significant in the case of large deformations. In addition, the need to represent the various excavation-construction phases, which are quite distant from each other, forces one to adopt a large size model along the tunnel axis. After a series of preliminary back analyses, meant to calibrate the mechanical parameters on the basis of the monitored tunnel convergence only (Barla et al. 2009), the attention has been focused on the modelling of the system and the excavation-construction sequence. The reinforcement ahead of the face has been described by using an equivalent pressure of 0.1 MPa (Fig. 6). Similarly, the first phase has been simulated by using an equivalent radial pressure which is assumed to reach the constant value of 0.1 MPa, 5 m behind the tunnel face, as depicted in Fig. 6. The second phase has been described as an elasto-perfectly plastic ring, installed at a distance of 30 m from the face, with a given stiffness and yield limit. The equivalent stiffness of the ring depends on the stiffness of the single deformable element and on the volume fraction of the 9 elements installed in the ring pertaining to the cross section of interest. The yield limit of the ring is equal to the yield limit of the element. Laboratory compression tests allowed one to evaluate this yield limit to be 8.5 MPa (Barla et al. 2007). As a consequence, for a thickness of 0.2 m and a tunnel radius of 6 m, the second phase at yielding gives a maximum radial pressure equal to 0.283 MPa. In the FDM analysis with the SHELVIP model the second phase has been simulated by using a refined FDM mesh (Fig. 6a). A new linear elastierfectly plastic model has been specially developed to allow the material to yield only in the out of plane direction. In the FEM analysis with the 3SC model this has been introduced by activating specially devised elastic-perfectly plastic “rib” elements

Mechanical parameters of SHELVIP and 3SC

SHELVIP model E ν φ c σt ωp γ m n l ωvp

640 MPa 0.3 26◦ 0.56 MPa 0.10 MPa 0 5.1E-5∗ 2.2∗ 0.18∗ 0.01∗ 0.735∗

3SC model E ν Gve ηve φlim clim ψlim ηvp

640 MPa 0.3 52 MPa 2.3 GPa·d 16.3 ÷ 13.2◦ 0.76 ÷ 0.64 MPa 7.8 ÷ 6.4◦ 18 ÷ 14.4 GPa·d

∗

time in year and pressure in kPa.

(Fig. 6b). These act as out of plane springs, by way of a nodal force applied in the radial direction and equivalent to the radial pressure due to the ring being compressed. Finally the third phase lining has been modelled as a linear elastic ring at a distance of 80 m from the tunnel face. In the FDM analysis it has been introduced by using a refined FDM mesh (Fig. 6a), while in the FEM analysis by using “shell” elements (Fig. 6b). An elastic modulus E = 30 GPa and a Poisson’s ratio of 0.2 have been assumed. In the FEM analysis for each computational step the following simulation stages are considered: excavation of 1 m length by element removal; application of pressure at the face and along the newly excavated tunnel perimeter; activation of a length of 1 m of the second and third phase s, respectively 30 m and 80 m behind the face. The same procedure has been adopted for the FDM analyses, however with the following differences. An excavation step length of 0.5 m has been adopted. The third phase has been activated in steps of 5 m length and the elastic modulus has been incremented gradually to reach the desired value in 28 days, to better reproduce the real construction procedure and the setting of the concrete. The constitutive parameters of the SHELVIP model have been chosen on the basis of the preliminary back analyses above mentioned (Barla et al. 2009), while the parameters of the 3SC model have been assessed with new back analyses taking into the monitored state of stress in the final lining. Table 1 gives a summary of the constitutive parameters for the two models. The different mechanical meaning of the two sets of viscous and plastic parameters does not permit a straightforward comparison

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Figure 7. Computed vs. monitored radial displacements with time, at the section at chainage 1444 m: SHELVIP model (solid line) and 3SC model (dashed line).

between these values. In addition, besides the different formulations of the two constitutive models, various differences exist between the two numerical analyses in of modelling and procedures, as pointed out above. Despite these differences, the validity of the comparison still holds as long as the aim is to investigate the different potential of the two numerical analyses in capturing the overall response of the rock mass and the ing structures to the excavation. This response is checked in of strain and stresses in the short and long term.

5

Figure 8. Computed vs. monitored radial displacements at depth, at chainage 1444 m: SHELVIP model (solid line) and 3SC model (dashed line).

DISCUSSION OF THE RESULTS

In the following Figures 7 to 9 the computed and measured displacements and concrete lining stresses for the sections at chainage 1444 m and at chainage 1457 m respectively are compared. The tunnel convergence versus time (Fig. 7) is well reproduced by the numerical analyses performed with both the two constitutive models SHELVIP and 3SC, notwithstanding the scattering of the monitoring data. The available data point out the heterogeneity and anisotropy of the rock mass, which cannot be represented in a simplified axi-symmetric model. A good agreement between computed and observed radial displacements measured around the tunnel up to a depth of 24 m is also shown in Figure 8, where reference is made to both phases I and II, i.e. respectively 32 and 84 days after installation of the multipoint extensometers. For the second phase , the state of stress in the out of plane direction was proved to be equal to the given yield limit. This value is reached a short time after activation of the . It is noted that rock mass anisotropy affects the stress distribution in the final lining (Fig. 9), with a large scattering in the monitoring data. In this case, the two models show rather different responses: while the stress predicted by SHELVIP rapidly reaches a constant value, corresponding to the average value from

Figure 9. Computed vs. monitored circumferential membrane stress in the concrete liner, at chainage 1457 m: SHELVIP model (solid line) and 3SC model (dashed line).

monitoring, 3SC provides a short time response in agreement with the data, but it overestimates the rate of increasing stress, thus leading to a stable value which is however greater, in the long term, than for SHELVIP. This discrepancy could be due also to the slow setting of the concrete which has not been taken into with the 3SC model.

6

CONCLUSIONS

Two elasto-viscoplastic models have been introduced to study the behaviour of tunnel excavation in severely squeezing rock conditions.

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The first model (SHELVIP) couples the general theory of elasto-plasticity with a time-dependent component. The elasto-plastic behaviour is associated with an external plastic surface that defines the locus of plastic strains development, while the viscous behaviour depends on an internal stress-hardening viscoplastic surface that establishes the onset of viscoplastic strains. The second model (3SC) consists of the connection of basic rheological elements, which reproduce, for different values of the applied deviatoric stress, the three creep stages, namely primary, secondary and tertiary creep. Both models have shown their effectiveness in reproducing the tunnel convergence versus time, the radial displacements around the tunnel and the state of stress in the final lining. The results obtained with the two models have also shown some differences, due to their intrinsic theoretical differences. A final comment can be made on the need for reaching a satisfactory result in simulation of tunnel excavation: an accurate description of the excavationconstruction sequence and the adoption of a suitable elasto visco-plastic constitutive model. The calibration of the parameters pertaining to each model is crucial for obtaining a good numerical prediction of tunnel behaviour based on back analysis and in situ monitoring. REFERENCES Barla, G. 1995. Squeezing rocks in tunnels. ISRM News Journal, Vol.II (3–4), 44–49 Barla, G. 2002. Tunnelling under squeezing rock conditions. In D.Kolymbas (ed), Tunnelling mechanics, Eurosummer School, Innsbruck, Logos Verlag, 169–268

Barla, G., Bonini, M.C. & Debernardi, D. 2007. Modelling of tunnels in squeezing rock. In J.Eberhardsteiner et al. (eds), Euro:Tun 2007, Proc. 1st Int. Conf. Computational Methods in Tunnelling, Vienna: Vienna University of Technology Barla, G., Debernardi, D. & Sterpi, D. 2009. Numerical analysis of tunnel response during excavation in squeezing rock by using two constitutive models. In G.Meschke et al. (eds), Euro:Tun 2009, Proc. 2nd Int. Conf. Computational Methods in Tunnelling, Freiburg: Aedificatio Publishers, Vol.1, 389–396 Cristescu, N.D. & Hunsche, U. 1998. Time effects in rock mechanics. Wiley & Sons. Debernardi, D. 2008. Viscoplastic Behaviour and Design of Tunnels. Ph.D. Thesis, Politecnico di Torino, Department of Structural and Geotechnical Engineering, Italy Debernardi, D. & Barla, G. 2009. New viscoplastic model for design analysis of tunnels in squeezing conditions. Rock Mech. Rock Engng., 42, 259–288 Dusseault, M.B. & Fordham, C.J. 1993. Time-dependent behaviour of rocks. In J.A.Hudson (ed), Comprehensive rock engineering, Pergamon Press, Vol.3, 119–149 Gioda, G. & Cividini, A. 1996. Numerical methods for the analysis of tunnel performance in squeezing rocks. Rock Mech. Rock Engng. 29, 171–193 Hoek, E. 2001. Big tunnels in bad rock (36th Terzaghi Lecture). Int. J. Geotech. Geoenv. Engng. ASCE, 127, 726–740 Itasca (2006), FLAC Fast Lagrangian Analysis of Continua, Version 5.0. Itasca Consulting Group, Minneapolis, USA Perzyna, P. 1966. Fundamental problems in viscoplasticity. Advances in Applied Mechanics, Academic Press, 9, 243– 377 Steiner, W. 1996. Tunnelling in squeezing rocks: Case histories. Rock Mech. Rock Engng., 29, 211–246 Sterpi, D. & Gioda, G. 2009. Visco-plastic behaviour around advancing tunnels in squeezing rock. Rock Mech. Rock Engng., 42, 319–339

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3D FEM analysis of soil improving resin injections underneath a mediaeval tower in Italy M. Gabassi, A. Pasquetto & G. Vinco Uretek, Verona, Italy

F. Mansueto Studio Montaldo & Associati, Genova, Italy

ABSTRACT: In order to stop the settlement process of a mediaeval tower located in Città di Castello (Italy), polyuretanic resins injections were performed in the foundation soil. The deg of the ground improving intervention was made with a 3D finite elements code and an analytical method based on the finite cavity expansion theory (Yu H.S. e Houlsby G.T., 1991), which allows to predict soil parameters changes due to resin expansion in the ground. During job site activity and for a long period after the works were finished the structure has been accurately monitored; the measured data seem to get on well with the one obtained from model analysis. The model creation, starting from the avilable geological data input, was necessary for the understanding of the causes which trigged to settlements. The Safety Factor improvement experienced during the simulation was about 30%.

1 THE CITTÀ DI CASTELLO CIVIC TOWER

2

1.1

2.1 Real time monitoring

Historical overview

The tower, initially built for military purposes, can be dated around the thirteenth century and is the only slim structure, together with the “Campanile Rotondo”, left in the old town Città di Castello. The building has a rectangular shape, dimensions 6,10 times 6,80 m and has a maximum height in the front of 39,80 m. It is divided into seven different levels, four of which were previously used as a prison. The tower, like we see it today, is the result of several collapses and reconstructions occurred over time; this can be gathered from the different wall textures, which interchange themselves along the whole tower height.

1.2

Settlement detection

In March 2007, following an earthquake ed in the area, with a magnitude of 2.2 of the Richter scale, a separation of 4 cm was detected in the purpose made seismic t between the tower and the Bishop’s Palace. By analyzing the data of the cracks monitoring, a differential settlement caused by the earthquake was clearly identified. This settlement strongly increased the before measured leaning of the tower towards the main square. In detail, the leaning grew from 72 to 78 cm, making this way even worse a strain state already close to the limit.

GEOTECHNICAL INVESTIGATION

The real time electronic monitoring was started on October 3rd 2007 and the zero measurement showed a leaning of 74 cm towards the main square and 34 cm towards the contiguous alley. During the next eleven days, a further settlement of 8 mm was ed in both directions.

2.2 Geological survey During October 2003 a geological survey was performed including four deep soundings, ground penetration radar and laboratory tests. The foundation depth from the ground level, varies from 2.3 m, on the sides facing the square (front side) and the alley, to 3.6 m on the side ted to the Bishop’s Palace and the backside. The underground is constituted by a superficial inhomogeneous replenishment layer, which thickness varies from 1.5 to 5.7 m, over a sequence of silty sands and sandy silts layer, followed by a bottom layer of clay and clayey silts at a depth varying from 10.0 to 13.0 m. These kind of soils, characterized by a strong geometric and granulometric as well as geomechanical variability, determine different responses to static and dynamic stress states, worsen by replenishment layers with strong thickness variability due to the ancient old town urbanization.

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Table 1.

OCR values.

S1 C3 (7.7–8.0 m) OCR = σ ’p /σ ’v0 = (179.95/156.91)kPa = 1.147 S1 C4 (11.3–11.5 m) OCR = σ ’p /σ ’v0 = (229.97/225.55)kPa = 1.019 S1 C5 (15.2–15.5 m) OCR = σ ’p /σ ’v0 = (499.99/304.00)kPa = 1.645

The ground water table was detected at a depth of 10 m from the ground level, but is capable of relevant changes depending from the different soils permeability. Also suspended underground water was detected in several spots, coming from water pipes leakages and from the square, following big rainfall events. 2.3 Geotechnical parameters The Consistency Index (IC), varies from 0.738 to 0.950, revealing a solid to plastic consistency of the analyzed soils. These values are proper of groups of inorganic clays with low to medium plasticity, silty and sandy clay and fine silty sands. Sandy soils have a medium-high consistency, whereas clayey soils are characterized by high drained cohesion values (c’) varying from 25 to 30 kPa and oedometric moduli M included between 6.2 and 17.4 MPa meaning a coefficient of volume compressibility mv ranging from 0.16 e 0.06 m2 /MN. From the oedometric tests performed, the consolidation pressure and the over consolidation ratio (OCR) were calculated; the tested samples are all in the range of normal consolidated to poorly overconsolidated soils with some peaks in the clays of the deepest part of the soundings: 3 3.1

GROUND IMPROVEMENT DESIGN Uretek deep injections method®

Due to the need of a low impact technology, which could guarantee low vibrations and small diameter drillings, a polyuretanic resin injections technique was chosen. Uretek Deep Injections® is a very particular technology, consisting of local injections into the soil of a high-pressure expansion resin; which produces a remarkable improvement of the geotechnical properties of the foundation soil. The operation steps are relatively simple and do not require invasive excavations or connection systems to existing and new foundation structures. Small quantities of expanding materials are injected precisely underneath the foundation level into the soil volume were the stress state reaches its peak. In order to avoid the material to flow outside from this volume, the expansion together with the viscosity increase of the resin have to be very quick. Therefore, after having injected the soil to be treated, resin immediately starts to expand. A high expansion pressure of the injection grout is also needed to guarantee a proper compaction of

the soil. It has to be way higher than the stress state induced by the overlaying structures both to allow a certain expansion rate and to avoid higher material consumption. The expansion process, first leads to the compaction of the surrounding soil and then, in case of light overstructures, also to the lift. All the procedure is monitored by electric receivers lighted by a laser emitter and anchored to the building whose foundation is treated. A wide set of laboratory tests have been carried out on the Uretek® resin, named Geoplus® , in order to evaluate its main mechanical properties. Vertical compression with free lateral expansion and vertical expansion in oedometric conditions tests were performed in the geotechnical laboratory of the University of Padova (Favaretti et al. 2004). 3.2 Theoretical view and simulation of the expanding process The expansion process of the resin, locally injected into the soil, can be theoretically studied as a spherical cavity (or cylindrical, if several injections are performed very close each to other, along the same vertical line) expanding in quasi-static conditions. The soil is modelled as a liner elastic-perfectly plastic material with a non-associated Mohr-Coulomb yield criterion and is considered initially subjected to an isotropic state of stress. During the first part of the expansion process, when the internal pressure of the cavity increases, soil shows an elastic behavior, while after reaching a specific value of the internal pressure plastic deformation starts, similarly to the elastic phase, until it reaches the pressure limit (σlim ). It is assumed that as soon as pressure limit is reached, the resin solidifies (Dei Svaldi et al. 2005). The expansion process is theoretically treated adopting analysis at large and small strains, respectively, on the plastic and elastic region (Yu & Houlsby 1991). 3.3 Uretek ground improvement calculation software The analytical model of the expansion process together with the resin expansion law obtained in laboratory, were recently used to develop a software, Uretek S.I.M.S. 1.0, capable to predict the ground improvement index of a soil injected with Geoplus® resin. Uretek S.I.M.S. 1.0 computerizes the above explained model and enables designers to get the improved ground parameters rapidly. To perform a stress-strain analysis this parameters can later on be used to perform a FEM analysis. The quality of the previsions, provided by the analytical model, has been verified on a number of real cases. The reliability of the theoretical previsions increases with the quality of the geotechnical investigation available to designer.

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During first phase injections, due to the expansion of the grout, all voids are filled, the ground is compacted and its stiffness increases. In normal consolidated ground conditions, this leads to the rise of the horizontal stress to values close to the vertical one in a limited volume around the injection point. When the isotropic stress state is reached, the expansion pressure also develops in vertical direction, inducing a surface lifting (Schweiger et al. 2004). The isotropic volume growth is obviously a simplification, because the expansion pressure first develops on the lowest stress plane in homogeneous soil conditions. 3.4 3D FEM analysis The analysis has been performed using a PLAXIS 3D Tunnel software version 1.2 of the Dutch Plaxis b.v. company. In order to model the intervention, some simplifications were adopted and the injections were this way modelled as a volumetric expansion of solid elements. A stiffness increase of both the surrounding as well as the treated soil has been adopted; the isotropic expansion implemented in Uretek S.I.M.S. 1.0 was modelled in the 3D FEM analysis, by forcing the volumetric strain value of the element according to the volume increase calculated with Uretek S.I.M.S. 1.0 (Mansueto et al. 2007). Doing so, an accurate determination of the grout quantities to be injected has been possible. The quick reaction time, as a matter of fact, prevent the material to flow away from the injection point, making this way easier the determination of the injected volumes in a certain soil volume. Considering that the material flows for one meter at the most, the added volume in a sphere of one meter radius around the injection point is equal to the injected quantity times the expansion factor calculated with Uretek S.I.M.S. 1.0 (Pasquetto et al. 2008). Also the soil stiffness increase was taken from the Uretek S.I.M.S. 1.0 output. Figure 1 shows the different foundation levels of the tower: they are higher towards the square (x < 0) and towards the alley (z > 0) as verified in the tests. A stress-strain analysis of the tower for every scheduled injection phase has been performed, simulating the injected volume as an expansion of the soil element located exactly in correspondence of the injection point (x, y and z). The volumetric expansion rate has been assigned to every element, according to the volume of resin to be injected in every injection point and the calculated expansion factor of the resin. The construction of the 3D model, interested 14.310 m3 of soil and required the generation of 8.708 elements, 25.053 nodes and 52.248 stress points internal to the elements. The tower has been modeled in vertical position in the input data. Afterwards, the construction phases have been simulated using intermediate steps, until the final configuration has been reached. The error

Figure 1. 3D FEM model of the tower.

between the modeled tilting and the measured one, lower than 4%, has been evaluated acceptable. The model has been based on the soil stratigraphy, on the precise geometry of the tower and on the scheduled injection phases. The initial condition analysis pointed out that, apart from the rather complex local stratigraphy characterized by the presence of overconsolidated material lenses into much more deformable soils, the different foundation levels determined the tower rotation. As a matter of fact, to a higher foundation level, corresponds a thicker layer of deformable soil, which origins, therefore, a differential settlement and the rotation of the tower. The leaning direction towards the less deeper foundation can be read as a confirmation of this. The FEM analysis clearly evidenced this point. The stress state, in correspondence to the foundation/soil interface, reaches the highest level (700 kPa) underneath the foundation facing the square, exactly were the settlement is the highest. These are the effects of the stress redistribution caused by the tower eccentricity. Figure 2 shows the distribution of the relative shear stresses (meant as the ratio of the existing shear stresses and the resisting ones calculated with a

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Table 2.

OCR values. Parameter γsat KN/m3

E kPa

c kPa

ϕ

Soil Type Replenishment (Silty Clay) Replenishment (Sandy Silt) Replenishment (Sand) Sandy Silt Silty Sand Clay and Clayey Silt

19.5 20.0 18.5 20.0 20.0 21.2

6250 4000 3000 8000 9000 13000

31 30 0 18 18 10

23 28 32 30 30 27

◦

ψ ◦

Constitutive law

– −1 – −3 −2 –

Mohr-Coulomb Mohr-Coulomb Mohr-Coulomb Mohr-Coulomb Mohr-Coulomb Mohr-Coulomb

Figure 2. Relative shear stresses distribution.

Figure 4. Injections points distribution and monitoring points.

Therefore, if the first one is a typical superficial punching failure mechanism, the second one depends from the stress state transferring to deeper soil layers; the two effects are certainly related, depending the second from the first one. 3.5 Executive project Figure 3. Relative shear stress in the center cut of the tower before the injections.

Mohr-Coulomb failure criterion) just underneath the foundations. It has been observed, that where the settlements are the highest, the existing stresses are equal to the resisting ones, meaning that the soil reached a plastic equilibrium condition. This obvious result is important, because proves the correspondence of the analysis performed; the foundation ground reached the full mobilization of the end-bearing capacity. Figure 3 shows an interesting double failure mechanism mobilization. The first one, more superficial, lays just underneath the foundation level and is limited to the first sandy silt soil layer; on the other hand, the second and deeper one, also interests other soil layers under the first one.

Based on the indications come from the FEM analysis, an executive project has been arranged, which has been changed continuously, depending on the reaction of the tower during the different injection phases. During a total of 14 working days, 2.475,5 kg of resin were injected. The amount of injected grout per day has been very different, depending on the real time monitoring data analysis. 4

FIELD AND DESIGN DATA COMPARISON

As mentioned before, during the whole work a real time electronic monitoring was operating. These data have been, afterwards, compared with the settlements calculated with the FEM analysis. 4.1 Expected settlements Figure 5 shows the expected settlements shells for monitoring points A and B, representing two limit

830

Figure 5. Calculated settlements and monitoring data graph.

Figure 7. Safety factor graph.

Figure 8. Settlement/Time graph. Figure 6. Relative shear stress in the center cut of the tower after the injections.

scenarios with zero and full expansion of the resin. The graph also withholds the settlements data, measured on field after each one of the three injection phases. It can be observed that, according to the modeling, little settlements had to be expected, due to a double effect: a lateral soil flow due to the resin injection and expansion first and a ground strain due to the increase of the effective soil stress, also caused by the resin volume expansion, second. Figure 5 shows how little are the differences between the calculated time/settlement curve and the real settlements measured on field after every injection phase.

4.2 Final stress state distribution Referring to relative shear stress (Fig. 2), the FEM analysis clearly shows how the injections strongly reduced this value within the improved ground volume. This reduction is the effect of the soil compaction induced by the resin expansion.

4.3 Safety factor increase The determination of the safety factor, was done using a “c-ϕ reduction” procedure, which foresees a

progressive reduction of the ground parameter values until the soil body collapse is reached. The final result is a movement/reduction factor graph, which represent the safety factor of the structure. Figure 7 shows a comparison of the safety factor before and after the intervention; it can be observed that the injections effect was the raising of the safety factor of about 30%. 4.4 Post intervention monitoring The precision monitoring of three datum points, started on March 25th 2007 and has been necessary for measuring the settlements of the structure before during and after the job site. Figure 8 shows the settlement/time graph, from which clearly appears how the settlement speed rapidly decreases after the injections. Also other electronic devices have been installed on the tower before the intervention, such as three electronic inclinometers with a 10−3 degrees precision and two electronic crack monitors with a 10−2 mm precision. In this case the monitoring had to eventual settlement trends in the short such as in the long period. In order to obtain a significant measurement, also a thermometer has been installed to neglect movements only due to thermal shocks. Analysing the data, it has been observed that during the drilling phase no significant settlement were ed, meaning that the small diameter drills made with hand augers didn’t influence the tower stability.

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On the other hand, during the injection phase, a variation of the cracks opening, such as a tower leaning progress have been observed, confirming this way the results of the FEM analysis. The monitoring is still working and the tower didn’t any further settlements in the last two years. 5

CONCLUSIONS

In this interesting case history, clearly appears how helpful a 3D FEM analysis can be, to take important job site decisions. In this delicate compensation grouting with polyuretanic resin injections, underneath a mediaeval tower, key choices like the injections sequence such as the grout quantities, were taken according to the modeling outputs. At the end of the work a good correspondence between settlements data measured on field and the ones forecasted with the analysis was found, confirming the good quality of the model; also in of bearing capacity increase, a significant rise of the safety factor was observed. The aim of this deg approach was the evaluation of the strain behavior of the tower during the different injection phases, in order to analyze the critical points of the work. To cover the stability problem at hand, also the increasing action of gravity, because of the increasing tilting should be taken into in a leaning instability problem, which wasn’t, however, the purpose of this modeling.

Favaretti, M. Germanino, G. Pasquetto, A. & Vinco, G. 2004. Interventi di consolidamento dei terreni di fondazione di una torre campanaria con iniezioni di resina ad alta pressione d’espansione. In XXII Convegno Nazionale di Geotecnica; Congress proceedings, Palermo, 22–24 October 2004: 357–364. Bologna: Pàtron. Foti, S. & Manassero, M. 2009. Rinforzo e adeguamento delle fondazioni per sollecitazioni statiche e dinamiche. In Risk mitigation and soil improvement and reinforcement; Proc. intern. symp., Torino, 18–19 November 2009. Mansueto, F. Gabassi, M. Pasquetto, A. & Vinco, G. 2007. Modellazione numerica di un intervento di consolidamento del terreno di fondazione di un palazzo storico sito in Rue Joseph de Maistre sulla collina di Monmatre in Parigi realizzato con iniezioni di resina poliuretanica ad alta pressione d’espansione. In XXIII Convegno Nazionale di Geotecnica; Congress proceedings, PadovaAbano Terme, 16–18 May 2007: 277–284. Bologna: Pàtron. Pasquetto, A. Gabassi, M. Vinco, G. & Guerra, C. 2008. Consolidation du sol par injection de résine polyuréthane, afin d’atténuer le gonflement e le retrait des sols argileux. In SEC 2008-Symposium international sécheresse et constructions; Congress proceedings, Marne-La-Valée, 1–3 September 2008: 343–348. Plaxis B.V. 2004. Plaxis 3D Tunnel, Tutorial Manual. Schweiger, H. F. Kummerer, C. Otterbein, R. & Falk, E. 2004. Numerical modelling of settlement compensation by means of fracture grouting. Soils and foundations 44 (1): 71–86. Yu, H.S. & Houlsby, G.T. 1991. Finite cavity expansion in dilatant soils: loading analysis. Géotecnique 41 (2): 173–183.

REFERENCES Dei Svaldi, A. Favaretti, M. Pasquetto, A. & Vinco, G. 2005. Analytical modelling of the soil improvement by injections of high expansion pressure resin. In 6th International Conference on Ground Improvement Techniques; Congress proceedings, Coimbra, 18–19 July 2005: 577–584.

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A numerical study of factors governing the performance of stone columns ing rigid footings on soft clay M.M. Killeen & B.A. McCabe Department of Civil Engineering, National University of Ireland, Galway, Ireland

ABSTRACT: The Vibro Replacement technique is now frequently used as a means of improving the bearing capacity and settlement performance of soft cohesive soils. In this paper, a parametric study using the finite element method is presented which examines the influence of some key variables on the behaviour of small groups of stone columns ing rigid footings. There is great potential to use the finite element method in an applied sense, as analytical approaches have many shortcomings and high quality field data is scarce.

1

INTRODUCTION

The optimum design of stone columns ing small loaded areas (such as pad and strip footings) is arguably the most challenging aspect of stone column design in soft soils. Analytical theories developed to date assume an infinite grid of stone columns subjected to wide-area loading which is implemented mathematically using the unit cell approach (for example, Priebe’s 1995 method for settlement design). Therefore they do not directly capture the behaviour of those columns under footings that are not equally confined on all sides; correction factors are applied within the design for this purpose. In addition, the vertical stress beneath footings decays much more sharply with depth than the stress beneath loaded wide areas, allowing partial-depth treatment to be used. Current design practice relies heavily on empirical methods for partial-depth treatment as analytical theory is much less well developed in this area. Equally, a database of measured field settlement improvement factors in fine soils compiled by McCabe et al. (2009) highlights a dearth of data for strip and pad footings. High quality physical models of footings on soft clay ed by stone columns (i.e. McKelvey et al., 2004, Black et al., 2010) have been informative, although there are obvious difficulties in extrapolating model test findings to field scale, and the proportion of area under each pad that has been replaced with stone in these tests has tended to lie at the high end of what might commonly be used in practice. Publications in which the finite element method has been used to model ground improved with stone columns mostly relate to wide-area loading, using either a unit cell (i.e. Domingues et al., 2007) or 2-D axisymmetric (i.e. Elshazly et al., 2008) approximation. Some 3-D modelling of wide-area loading has also been carried out (i.e. Gab et al., 2008); however, hardly any 3-D modelling of footings has been

published. In this paper PLAXIS 3-D Foundation (Version 2.2) is used to model the behaviour of rigid square pad footings ed by stone columns. The soil profile at the former geotechnical test site at Bothkennar, Scotland, is used as it is representative of many soft soil profiles in Ireland and the UK for which the applicability of stone columns is of growing interest. This paper begins to identify some of the key factors relevant to the design of small groups of stone columns, such as column arrangement, spacing, length, and Young’s modulus of the column material. 2

MODEL OF BOTHKENNAR SOIL PROFILE

Located on the Firth of Forth estuary near Grangemouth, in Scotland, Bothkennar is the former UK test site for soft soil engineering research and as a result is extensively characterised. A weathered crust extends to a depth of 1.5 m and is underlain by 13.0 m of soft uniform Carse clay, deposited under shallow marine or estuarine conditions. 2.1 General soil parameters The clay properties used in the soil model are presented in Table 1 and separated into crust, upper Carse clay and lower Carse clay. A high critical state friction angle (φ ) of 34◦ (attributable to a high proportion of angular silt particles, Allman & Atkinson, 1992) is used for the Carse clay, and a nominal cohesion value of 1 kPa is used for numerical stability. A slightly higher cohesion value of 3 kPa was used for the weathered crust layers. Nash et al. (1992a) report the variation of yield stress ratio, which is equivalent to the overconsolidation ratio (OCR) measured in an oedometer, and in situ lateral earth pressure coefficient (K0 ) with depth, suggesting that the stress state of the Carse clay may have been influenced by erosion of material, a relative drop in sea

833

Table 1.

Depth γ φ ψ c OCR POP K0 1 ref E50 Eref ur pref m

Parameters for Bothkennar soil model.

(m) (kN/m3 ) (◦ ) (◦ ) (kPa) (−) (kPa) (−) (kPa) (kPa) (kPa) (−)

1 ref E50 assumed

Crust

Upper Lower Stone Carse clay Carse clay backfill

0.0–1.5 18.0 34 0 3 1.0 15 1.5 1068 5382 13 1.0

1.5–2.5 16.5 34 0 1 1.0 15 1.0 506 3036 20 1.0

2.5–14.0 16.5 34 0 1 1.5 0 0.75 231 1164 30 1.0

– 19.0 45 15 1 – – 0.3 70000 210000 100 0.3

equal to Eref oed (i.e. Elshazly et al., 2009).

level and fluctuating groundwater levels (summing to a 15 kPa drop in vertical effective stress). In choosing the friction angle of the stone backfill, reference was made to McCabe et al. (2009) who used measured settlement improvement data from the field to suggest that the conventionally- used value of φ = 40◦ may be conservative for columns in soft cohesive soils constructed using the bottom feed system. Subject to adequate workmanship, the value of φ = 45◦ shown in Table 1 should be readily achievable. The angle of dilatancy (ψ) was calculated based on the relationship ψ = φ − 30◦ .

2.2

Hardening soil parameters

The advanced elastic-plastic Hardening Soil (HS) model in PLAXIS 3-D Foundation was chosen to simulate the behaviour of the weathered crust, Carse clay and stone backfill. The HS model is an extension of the hyperbolic model developed by Duncan and Chang (1970). Creep behaviour is not considered in this model. Nash et al. (1992b) report the variation with depth of the initial voids ratio (e0 ), compression index (Cc ) and swelling index (Cs ) of the Carse clay. These parameters were entered into PLAXIS, which uses standard relationships to convert these one-dimensional parameters to three-dimensional quantities for the HS model; Young’s modulus at half the maximum deviator stress (E50 ) and the oedometric modulus (Eoed ) are derived from Cc while the unload-reload Young’s modulus (Eur ) is derived from Cs . The stiffness parameters adopted for the stone backfill are less certain, and thus are subject to parametric study in section 4.3. The reference stiffness (Eref ) is the stiffness corresponding to the confining pressure (pref ); pref is the horizontal ref effective stress in the case of Eref 50 and Eur , and is the vertical effective stress in the case of Eref oed . An important feature of the HS model is its ability to capture the stress dependence of soil stiffness. The parameter ’m’ is used to control the relationship

Figure 1. Validation of soil profile and parameters for a footing on untreated Carse clay.

between soil stiffness (E) and the corresponding confining stresses (p) according to eqn (1).

The reference stiffness moduli were chosen from the aforementioned Nash et al. (1992b) tests at the values of pref quoted in Table 1. This data also indicates that m = 1 is appropriate for the Carse clay. The value of m = 0.3 for the stone backfill is assumed based upon Gab et al. (2008) and others. 3


3.1 Validation of soil model A well-documented field test on a pad footing at the Bothkennar site (no stone column ), see Jardine et al. (1995), was simulated using PLAXIS 3-D Foundation in order to substantiate the use of the parameters in Table 1. The footing, which was 2.2 m square and 0.8 m thick, was loaded to failure over 3 days using kentledge blocks, with loading pauses overnight and whenever settlement rates exceeded 8 mm/h. The Carse clay is modelled as undrained due to the short duration of the load test; concrete is modelled as a linear elastic material (Young’s Modulus Econc = 30 GPa; Poisson’s ratio νconc = 0.15).The loadsettlement response of the footing recorded by Jardine et al. (1995) is shown in Figure 1 together with the PLAXIS 3-D prediction. It is clear that both curves are in good agreement, affirming the selection of the adopted soil profile and material parameters.

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Table 2.

Details of parametric study.

Test name

Ftg size (m)

k (−)

s (m)

F (−)

E50,col (MPa)

A B C D1 D2 D3 E1 E2 E3 F1 F2 F3 G

2×2 3×3 3×3 3×3 3×3 3×3 3×3 3×3 3×3 3×3 3×3 3×3 4×4

4 4 4 4 4 4 5 5 5 9 9 9 16

1.0 1.0 1.5 2.0 2.0 2.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

3.5 8.0 8.0 8.0 8.0 8.0 6.4 6.4 6.4 3.5 3.5 3.5 3.5

70 70 70 70 50 30 70 50 30 70 50 30 70

This approach was also adopted by Guetif et al. (2004) and others. 3.4 Parametric study details Figure 2. Influence of column length, arrangement and drainage analysis type upon settlement performance of stone columns.

3.2 Modelling of drainage PLAXIS 3-D enables the long-term behaviour to be modelled in two ways: (i) drained analysis, using effective stress parameters and (ii) undrained analysis, using effective stress parameters, followed by consolidation. Figure 2 shows that for a 3 m × 3 m footing, loaded to 50 kPa and ed by various configurations of stone columns (depicted in the inset, refer also to Table 2), approaches (i) and (ii) produce quite similar (if not perfectly consistent) results. On this basis, the parametric study in Section 4 is based on a comparison of type (i) analyses only. Type (ii) analyses take longer to perform but these analyses are underway and will be published at a later date. 3.3 Other modelling issues The stone columns in this study have been wishedin-place, i.e. the ground properties have not been modified to reflect changes induced by installation of the columns. Some authors (i.e. Watts et al., 2000, Kirsch, 2008) have reported increases in total stress after column installation, but as noted by McCabe et al. (2009), it is the equalized effective stresses around columns (once pore pressures have dissipated) which influence column performance under load, and these have not been measured. The interaction between the stone column and the surrounding soil is simulated using elastic-plastic interface elements. Owing to the process of column construction, the stone is tightly interlocked with the surrounding soil and it is assumed that a perfect bond (total adhesion) occurs along this interface.

Settlement rather than bearing capacity criteria generally govern the design of stone columns in soft soils. Key variables in the settlement design of stone columns to footings include footing size (B), column length (L), column spacing (s), number of columns (k), column arrangement and stiffness of the column material (E50 ). The various parametric combinations considered in this paper are labelled A-G in Table 2.The footing, which is 600 mm thick, is founded 600mm below ground level. Column diameter is not normally a significant variable in design as the poker is of fixed diameter and the final column size is a function of soil consistency. A diameter of 600 mm is assumed here for soft soils constructed using the instrumented bottom feed system. In Table 2, the quantity F = Af /kAc is referred to as the footprint replacement ratio, and is a measure of the extent to which the soil area under the footing is replaced by stone (Af is the footing area, k is the number of ing columns andAc is the cross-sectional area of each column). Since each footing is square (of width B) and all columns are 600mm diameter, the expression for F can be simplified to:

F is an adaptation for footings of the area replacement ratio (A/Ac , A is the total loaded area) used by Priebe (1995) for unit cells / infinite grids. For each configuration A-G outlined in Table 2, at least 9 different lengths were investigated; covering various extents of partial depth treatment and full depth treatment to 14.5 m below ground level. The effect of varying the E50 value of the column material is assessed for cases D1-3, E1-3 and F1-3. In subsequent plots, the extent of settlement improvement is quantified by means of the settlement

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improvement factor (n), defined as the ratio of the settlement of the footing without treatment to the corresponding settlement with treatment (with the same stress applied to the footing in each case). 4 4.1

PARAMETRIC STUDY Influence of column length and footprint replacement ratio

The type (i) data for the 3 m × 3 m footing shown in Figure 2 is revisited, which shows the influence of column length (L) and footprint replacement ratio (F) upon the predicted value of settlement improvement factor (n) for a column stiffness E50,col = 70 MPa. In all cases, it is clear that n increases with L. Settlement improvement is not observed until columns are installed beyond L/d ≈ 3, as the weathered crust, which extends to 1.5 m, is already competent. The classical Boussinesq solution for vertical stress distribution under a footing would suggest that the stress increment applied to the footing is no longer perceived at L/B = 2, which is equivalent to L/d = 10 in this instance. The PLAXIS output indicates that improvement can still be achieved by constructing columns longer than L = 10d. McKelvey et al. (2004) suggest that while no benefit to bearing capacity was achieved by extending columns beyond L/d = 6, additional benefit to settlement was achieved up to L/d = 10, the maximum length of the partial depth columns. The length effect in this study is most pronounced for the 9 column group (F = 3.5) which also appears to benefit greatly from end bearing (onto a boundary at 14.5 m, modelled as rigid), suggesting that the applied load in this case is being transmitted to great depth. This tendency is also noted in the model test data of Black et al. (2010) with similarly low F values. This would indicate that the benefit of lengthening columns is greatest when they are already closely spaced. However, the F values for the 4 column (F = 8.0) and 5 column (F = 6.4) groups are more representative of practice. Muir Wood et al. (2000) conducted a series of laboratory scale model tests to investigate the influence of column diameter, length and spacing upon the mechanisms of stone column behaviour. The authors observed that as the replacement ratios increased, the columns bulged in the upper zones of the soil layers and transferred the load to greater depth. 4.2 Influence of column position beneath footing The column spacing for a given footprint replacement ratio (or alternatively thought of as the position of the columns in relation to the edge of the footing) is seen in Figure 3 to have a minor influence on the behaviour of the footing. It appears that small benefits can be gained by keeping the columns closer to the footing edge, although these benefits become negligible beyond L/d ≈15. It is well known that the

Figure 3. Effect of column spacing (or position in relation to the edge of the footing) on settlement performance.

stress distribution beneath rigid footings in clay is such that higher stress concentrations develop towards the edges, so columns placed here have the potential to absorb more load and develop improved n values. Also, the stress concentrations that develop decay rapidly with depth and this may explain why the influence of column spacing is restricted to short columns. A finite element study by Wehr (2006) to examine the group behaviour of stone columns demonstrates that shear zone develop at the edges of a pad footing and extend to a depth beneath the centre of the footing. Positioning the columns closer to the edge of the footing, and thus closer to these shear zones, may also explain the enhanced settlement performance. 4.3 Influence of column deformability Figure 4 demonstrates that the stiffness (E50 ) of the stone backfill has an influence on the settlement improvement behaviour of stone column reinforced foundations. A reduction in the stiffness of the column leads to a reduction in settlement improvement, which is similar to findings from finite element modelling on an embankment (Domingues et al., 2007) with columns having A/Ac ≈5. Interestingly, from Figure 4, the effect of column deformability on settlement appears to be much more pronounced for the 9 column group (F = 3.5) than for the two groups with higher F values. 4.4 Column confinement Figure 5 compares three footings sizes, with the number of columns chosen to maintain equal footprint

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Figure 6. Mode of deformation for 4 m × 4 m group G.

Figure 4. Effect of column stiffness upon settlement performance.

bulge outwards from the footing centre and towards the unconfined side; the inner columns appear to bulge less and more uniformly. This behaviour was also observed by McKelvey et al. (2004), who examined the interaction between stone columns beneath strip and pad footings. As the number of columns beneath a footing increases, so does the number of columns with full confinement on all sides and therefore the average settlement performance of a group column will improve. It should be noted that an increased footing size will stress the soil to a greater depth, which should induce more settlement. However, Figure 5 indicates that this effect is more than offset by the positive effects of column confinement.

5

Figure 5. Effect of column confinement upon settlement performance.

replacement ratios (F = 3.5). The settlement performance of the footings improves as the footing size and the number of ing stone columns increases. It is well documented that an isolated column will tend to bulge when loaded. This bulging tends to occur near the ground surface, where the overburden stresses are at their lowest. Figure 6 (which is a diagonal cross section through the 4 m × 4 m group G) highlights that the outer columns beneath the pad footings tend to

CONCLUSIONS

A parametric finite element study with an advanced soil model was carried out to assess the effect of a number of key design variables on the settlement performance of rigid pad footings ed by stone columns. The following conclusions may be drawn, which are specific to a type (i) drained analysis for the ground profile modelled: The PLAXIS 3-D output suggests that settlement performance continues to improve beyond L/d = 10, and this improvement is more pronounced for groups with a low footprint replacement ratio. End bearing is also significant for the F = 3.5 case, but this may be related in part to the assumption of a rigid layer. Columns closer to the footing edge perform better for short column lengths (L/d < 10) than for columns closer to the centre, but the ‘n’ values converge with depth and long stone columns are relatively insensitive to column spacing. The stiffness of the stone backfill has a significant influence on the settlement performance of a footing ed by a large number of stone columns. However, as the number of ing columns reduces, so does the influence of column stiffness. For a given footprint replacement ratio, an increased number of columns ing a footing leads to an increase in the proportion of group columns that have

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full confinement (i.e. behave like a unit cell) resulting in enhanced settlement performance of the footing. It is acknowledged that more definite conclusions from the finite element work are pending upon the outcome of the type (ii) analyses (undrained loading followed by consolidation), and the output from the modelling in general can only be satisfactory validated by full scale field testing. The long term settlement behaviour of footings on soft soils must also consider creep. ACKNOWLEDGEMENTS The authors wish to acknowledge the of the Irish Research Council for Science, Engineering and technology EMBARK Initiative for funding the first author’s research. The of by the Geotechnical Trust Fund (engineers Ireland) is also appreciated. Finally, the authors are extremely grateful for the input of Keller Foundations (UK) to this study. REFERENCES Allman, M. A. & Atkinson, J. H. 1992. Mechanical properties of reconstituted Bothkennar soil. Geotechnique 42(2): 289–301. Black, J.A.., Sivakumar V. and Bell, A.L. (2010) The settlement of a footing of soft clay ed by stone columns, accepted and awaiting publication by Geotechnique. Duncan, J. M. & Chang, C. Y. 1970. Nonlinear analysis of stress and strain in soil. ASCE Journal of the Soil Mechanics and Foundations Division 96: 1629–1653. Domingues, T. S., Borges, J. L. & Cardoso, A. S. 2007. Stone columns in embankments on soft soils. Analysis of the effects of the gravel deformability. 14th European Conference on Soil Mechanics and Geotechnical Engineering. Elshazly, H. A. Hafez, D. H. & Mossaad, M. E. 2008. Reliability of conventional settlement evaluation for circular foundations on stone columns. Geotechnical and Geological Engineering 26(3): 323–334. Gab, M., Schweiger, H. F., Kamrat-Pietraszewska, D. & Karstunen, M. 2008. Proceedings of the 2nd International Workshop on the Geotechnics of Soft Soils, Glasgow, 2008 137–142.

Guetif, Z., Bouassida, M. & Debats, J.M. 2007. Improved soft clay characteristics due to stone column installation. Computers and Geotechnics 34 (2007): 104–111. Hight, D. W., Bond,A. J. & Legge, J. D. 1992. Characterisation of the Bothkennar clay: an overview. Geotechnique 42(2): 303–347. Institution of Civil Engineers. 1992. Bothkennar soft clay test site: characterization and lessons learned. Geotechnique 42(2): 161–378. Jardine, R. J., Lehane, B. M., Smith, P. R. & Gildea, P.A. 1995. Vertical loading experiments on rigid pad foundations at Bothkennar. Geotechnique 45(4): 573–597. Kirsch, F. 2008. Evaluation of ground improvement by groups of vibro stone columns using field measurements and numerical analysis. Proceedings of the 2nd International Workshop on the Geotechnics of Soft Soils, Glasgow, 2008. 241–248. McCabe, B. A., Nimmons, G. J. & Egan, D. 2009. A review of field performance of stone columns in soft soils. Proceedings of ICE Geotechnical Engineering, accepted for publication, May 2009. McKelvey, D. V. Sivakumar, Bell, A.L. & Graham, J. 2004. Modelling vibrated stone columns in soft clay. Proceedings of the Institution of Civil Engineers: Geotechnical Engineering 157(3): 137–149. Muir Wood, D., Hu, W. & Nash. D. F. T. 2000. Group effects in stone column foundations: model tests. Geotechnique 50(6): 689–698. Nash, D. F. T., Powell, J. J. M. & Lloyd, I. M. 1992. Initial investigations of the soft clay site at Bothkennar. Geotechnique 42(2): 163–181. Nash D. F. T., Sills, G. C. & Davison, L. R. 1992. Onedimensional consolidation testing of soft clay from Bothkennar. Geotechnique 42(2): 241–256. Priebe, H.J. 1995. The design of Vibro Replacement, Ground Engineering (Dec), pp 31–37. Watts K. S., Johnson D., Wood L. A. & Saadi, A. 2000. An instrumented trial of vibro ground treatment ing strip foundations in a variable fill. Geotechnique 50(6): 699–708. Wehr, W.C.S. 2006. Stone columns – Group behaviour and influence of footing flexibility. Proceedings of the 6th European Conference on Numerical Methods in Geotechnical Engineering – Numerical Methods in Geotechnical Engineering.

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Calibration and verification of numerical model of ground improved by dynamic replacement S. Kwiecien Silesian University of Technology, Gliwice, Poland

ABSTRACT: Dimensioning of driven stone columns on the basis of the existing methods may raise certain doubts. It is due to underestimation of their load capacity and overestimation of their settlement. The alternative solution could be a computational numerical model of subsoil strengthened with the use of dynamic replacement method, which is presented in this paper.

1

INTRODUCTION

More and more often we are forced to set buildings and engineering structures on weak and strongly deformable soils. Indirect footings, like piles, can be applied in such a situation. However, this type of foundation is often very costly. Direct footings can also be successfully applied, but only after strengthening of the weak subsoil. One of the many methods of subsoil strengthening is a dynamic replacement method, called driven stone columns. Dimensioning of driven columns requires checking of two limit states. The existing methods of calculating the load capacity and settlements cause underestimation of the former (Kwiecien 2007) and overestimation of the latter (Kwiecien 2004). The alternative solution is a numerical analysis with the use of the finite elements method (FEM) and selection of appropriate soil constitutive models. It shall be based on field tests, e.g. broad range of trial load testing of columns. In the paper the author proposes a computational model of subsoil strengthened with stone columns and calibrated in the course of semi-reverse analysis based on a trial load test of a driven column.

2 TRIAL LOAD TEST OF THE STONE COLUMN The tested stone column was formed at a test site in Lubien close to Myslenice (Kwiecien 2007). The subject of the strengthening here were the top layers of aggraded mud of the thickness reaching up to a few meters underlaid by gravels. The column was formed from crushed rock of fraction 0/400 with the use of a tamper weighting 11.5 t and dropped from the height of 13.5 m Trial load test stand (Fig. 1) has been designed for the strength equal to 1.5-times the anticipated load capacity, determined by means of Brauns’ method

Figure 1. Trial load test stand.

(Brauns 1978). A group of ten I-sections I500, anchored with the use of piles of diameter 75 and 150 cm and respectively 15.8 and 10 m, constituted the retaining beam. The trial load test of the column was carried out by means of a constant load steps method. Each load step was maintained until the column’s settlement velocity was lower than 0.05 mm/15 min. The loading was conducted with the use of three hydraulic jacks of the range 0-1300 kN. Three electronic sensors of the range 0-100 mm and accuracy of reading 0.01 mm enabled the measurement of settlements. The final load amounted to q = 1373 kPa (Fig. 2). 3

CALIBRATION AND VERIFICATION OF THE COMPUTATIONAL MODEL OF THE STRENGTHENED SUBSOIL

3.1 Selection of soil constitutive models Consideration of the problem of strengthening cohesive (organic) soils with grainy material (stone column

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Figure 3. Construction of cylindrical unit.

Figure 2. Results of the load plate test of stone column.

aggregate) suggests the use of specific soil models in the computational analysis. In case of grainy low compressible soils forming a column the elastic-perfectly plastic model with the Coulomb – Mohr failure surface can be successfully used. Unfortunately one of the deficiencies of this model, which may cause some numerical complications, is the presence of sharp corners in the yield surface. When the stress path achieves the yield surface at that point, there appears the problem of determination of the appropriate direction of the plastic strain increment vector (Cudny & Binder 2005). The alternative solution may be the use of a modification suggested by Menetrey and Willam (1995) (M-W) with a smooth single-surface approximation of the Coulomb – Mohr yield surface that is free from the singular points. There are four parameters to be determined in this model: modulus of elasticity – E, Poisson’s ratio – ν, internal angle of friction – φ and cohesion – c. The weak, strongly deformable soil surrounding the column, that is to be strengthened, is generally poorly preconsolidated. Plastic strains will dominate in this region. To simulate the specificity of the process of column driving, models ing plastic strengthening and weakening with the porosity changes, i.e. compaction and loosening of soil, have to be used. In the described method the compaction of the surrounding soil is considerably more intensive than in some other technologies, thanks to the strong dynamic consolidation. A high hydraulic gradient appears in the weak surrounding soil as a result of dynamic expansion of great energy after tamping, causing water flow from pores of the fine-grained soft soil to the column. The heterogeneity of compaction during column forming may be simulated by introduction of different material zones in the numerical analysis. The appropriate computational model should be also capable of simulating further compaction and strengthening caused by loading of structure. Critical state models are preferred here and especially the Modified Cam-Clay (MCC) model as the one that is well known and implemented

to most of the FEM programs. The model requires five material constants: slope of the critical state line – M, slope of the normal consolidation line – λ, slope of the swelling line – κ, Poisson’s ratio – υ and specific volume of the critical state line at unit pressure – . 3.2 The essence of calibration and verification The criterion of the computational model adequacy is the accuracy of fitting the “load-settlement” characteristic obtained with the use of the model with the results of the trial load test carried out within the wide range of loading - from zero to 1.5 times the subsoil load capacity according to the Brauns’conception. The calibration of the computational model involves such an estimation of all the nine parameters of the constitutive models used for description of column material and weak subsoil, to ensure the best fit. The process of optimization is realized in the course of semi-reverse analysis, where some of the parameters are estimated in the laboratory tests and treated as given constant data. Due to the complexity of constitutive models, the optimal set of parameters is determined with the use of direct search techniques with multiple FEM analyses simulating the process of the trial load test. Verification of the computational model comes down to a visual or statistical evaluation of discrepancies between experimental data and theoretical predictions with the use of the optimal set of parameters. 3.3 Geometrical model of “stone column – weak subsoil” system The trially loaded stone column was surrounded by a group of columns formed in the net of equilateral triangles of 3 m long sides. Therefore, in the numerical analysis the problem came down to the concept of cylindrical unit cell, in which the equilibrium of individual stone column surrounded by weak soil is considered (Fig. 3). To identify the stone column geometrically, the column was carefully measured in field (Fig. 4), in an excavation dug right after the trial load test had been finished. The soft soil (aggradate mud) samples were

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Figure 6. Results of oedometer tests.

Figure 4. Results of investigation of stone column.

Figure 7. Dermining of CSL slope (M).

Figure 5. Geometrical model of “stone column – weak subsoil” system.

then also taken for the laboratory tests to determine the physical properties as well as parameters of the Modified Cam-Clay constitutive model. The investigated stone column was barrel-shaped along its height and based on the load bearing layer consisting of medium compacted gravel with cobbles. The unit cell modelled in Z_Soil.PC software as axially symmetric and taking into the actual shape of the column has been shown in Figure 5.

3.4 Estimation of soil models’ parameters of strengthened subsoil Estimation of parameters’ values of the soil models used in the analysis was done concurrently. In case of the aggradate mud, represented by MCC model, the parameters were determined on the basis of laboratory triaxial and oedometric tests on undisturbed specimens collected after the trial load test. The fraction size of the crushed rock (0/400) used for forming of columns made the laboratory tests in direct shear box or oedometric apparatus impossible. Field tests (except for trial load test) couldn’t be done neither. Therefore, it was decided that parameters

of the M-W model representing the column material will be estimated on the basis of a reverse analysis, provided the earlier estimation of the aggradate mud’s parameters according to the above mentioned procedure. The M-W model parameters for the gravel underlying the soft soil were determined on the basis of the standard PN-81/B-03020 knowing that the density index ID = 0.5. The parameters of a concrete slab, modelled as linearly elastic, were assessed based on the standard PN-B-03264 as for concrete C12/15. The MCC parameters: λ and κ were estimated in standard oedometric tests conducted in cycles: initial loading (0–200 kPa), unloading (200–12.5 kPa) and secondary loading (12.5–400 kPa) (Fig. 6). They equalled λ = 0.087 and κ = 0.0028. The slope of the critical state line (CSL): M = 1.48, was determine on consolidated specimens in triaxial tests with enabled drainage. The tests were conducted on three specimens subjected to cell pressures 40, 140 and 240 kPa respectively. The experiments were carried out until the specimens achieved 10% of the relative strains. The stress paths p’ – q for the tested specimens of aggradate mud, including CSL, have been shown in Figure 7. The line has been determined based on the first two tests (40 and 140kPa). It turned out that the third test has been conducted for too small range of strains.

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Figure 8. Results of FEM analysis.

Figure 10. Vertical displacement.

Figure 9. Deformed Finite Element Methods mesh.

Knowing the value of the initial voids ratio e0 as well as the slope of the normal consolidation line λ we can determine the value of = ecs + 1, where ecs is determined at unit pressure p = 1 kPa. The Poisson’s ratio was assumed as υ = 0.3 – within the range (υ = 0.23 – 0.45) suggested by Lechowicz and Szymanski (2002) for organic soils. The tested stone column was formed with the use of crashed rock (sandstone) of big fraction – 0/400. Due to the reasons mentioned above, estimation of its parameters was done by means of semi-reverse FEM analysis for the whole “stone column – weak subsoil” system. The analysis carried out with the use of heuristics method (trial and error) was based on the assumption

of employing only the commonly accepted values of the geotechnical parameters. The issue of determining the strength parameters of coarse-grained soils was widely dealt with by Pisarczyk (2000). Internal friction angles obtained in the tests varied from 30 to 60◦ . It depended on the testing method (triaxial or direct shear), material graining, compaction and moisture content. For the material of driven, heavily compacted stone column internal friction angle shall be close to the upper boundary. This optimistic evaluation however has to be slightly lowered due to the presence of a thin film of irrigated binding agent between grains. In case of modulus of elasticity E, as recommended by Pieczyrak (2001), the output value in semi-reverse analysis is the tangent of the slope of the initial experimental curve “load – settlement”. In the analysed case (Fig. 2), within the rage 0–120 kPa, it equals E = 76 MPa. Value of Poisson’s ratio υ was estimated after Jurik (Wilun 2003): υ = 0.2. Also Pieczyrak (2001) employed similar value of Poisson’s ratio in his tests during investigation of the stone column parameters with the use of reverse analysis method. 3.5 Numerical simulation of trial load test As the result of dozens of full numerical analyses, the investigated column’s M-W strength parameters have been determined as: φ = 43.5◦ , c = 5 kPa. The numerical “load – settlement” curve fits best the curve obtained in the field test (Fig. 8). The determinant of

842

Numerical analysis the problem came down to the concept of cylindrical unit cell. The process of calibration of the employed constitutive soil models was based on a semi-reverse analysis. Such a calibration as well as the use of the real shape of the column allowed to obtain very good convergence of the curves referred above. The value of the modified determination coefficient (R2 = 0.9982), very close unity, is the best proof. However, it needs to be noted that this verification refers only to one case/example. Further research and comparisons are inevitable. Author didn’t test influence of boundary conditions on results. It will be next author’s step. REFERENCES

Figure 11. Total vertical stress.

the good fitting was the high value of the modified determination coefficient (R2 = 0.9982). Results of adjustment shown in Figure 8 positively the proposed model. Deformed mesh is presented in Figure 9. Vertical displacement have been shown in Figure 10. Figure 11 shows total vertical stress. 4

FINAL COMMENTS

The trial load of the driven stone column was the basis for creation, calibration and verification of the proposed computational model of the subsoil strengthened with driven stone columns.

Brauns, J. 1978. Initial bearing capacity of stone column and sand piles. Proc. Symp. Soil Reinforcing and Stabilizing Techniques in Engineering Practise, Sydney. Cudny, M. & Binder, K. 2005. Criteria of soil shear strength in geotechnics problems. Marine engineering and Geotechnics, 6: 456–465. Kwiecien, S. 2004. Comparative analysis of calculated and measured settlements of stone columns strengthening weak subsoil., 5th Civil Engineering Departments PhD Students Scientific Conference, 102: 273–282. Kwiecien, S. 2007. Trial load of driven stone column. Field tests results. Scientific Conference on the Occasion of Professor Maciej Gryczmanski’s Seventieth Birthday, 111, Gliwice: 267–274. Lechowicz, Z. & Szymanski, A. 2002. Strains and stability of the embankments on the organic soils. SGGW Publishing house, Warsaw. Menetrey, Ph. & Willam, K.J. 1995. A triaxial failure criterion for concrete and its generalization. ACI Journal 92(3): 311–318. Pieczyrak, J. 2001. Determination of the selected soil models parameters based on trial load tests. Habilitation thesis. Silesian University of Technology, 91, Gliwice. Pisarczyk, S. 2000. Strength of thick-cluster soils from upper Vistula river basins used in the embankments of hydrotechnical structures. Scientific works of Warsaw University of Technology, 32: 5–51. Wilun, Z. 2003. Outline of the Geotechnics. Communication Publishing House, edition. IV, Warsaw.

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Identification and quantification of the mechanical response of soil-wall structures in soft ground improvement X. Liu, Y. Zhao & A. Scarpas Section of Structural Mechanics, Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands

Arian de Bondt Ooms Avenhorn Holding bv, Scharwoude, The Netherlands

ABSTRACT: The extensive surface deposits of clay/peat in the Netherlands cause difficulties in the field of construction and maintenance of infrastructure. One of the main concerns is long term settlement due to the high compressibility and low strength of the soil. Of the various available vertical reinforcing techniques for in-situ ground improvement, reinforcing elements using cast-in piles/walls have received increasing popularity in the recent past. In this investigation, by means of 3D non-linear finite element analyses, the interaction of the components of pavement/soil-wall structures on soft soils and the internal mechanisms that lead to damage have been addressed. The emphasis is placed not on the manifestations of damage but on the actual causes.

1

INTRODUCTION

For a soft ground without sufficient bearing capacity, the available techniques for ground improvement include compaction, stabilization, replacement, consolidation and reinforcement. Although ground reinforcement can be achieved either vertically or horizontally, the vertical reinforcement method is more popular because of easier installation compared to horizontal elements. Of the various available vertical reinforcing techniques for in-situ ground improvement, reinforcing elements using cast-in piles/walls have received increasing popularity in the recent past. The piles/walls can be constructed out of different materials such as stone or gravel, with or without cement, sand mixed with cement, local soil stabilized with cement and others. The selection of a suitable material requires considerations of the existing ground condition, material availability and cost. The piles/walls carry substantially greater proportion of the applied loads with a relatively smaller amount of deformation as compared to the in-situ soft soil deposits. They also help to speed up consolidation process in the soft ground and hence, as a consequence, the post construction settlements of the structure built on them are smaller. The cast-in-place piles/walls in the soft ground together with the cushion form a composite foundation, which s the pavement. For the development of a rational design method for this type of structure, the load transfer from the pavement through the cushion to the pile/walls, for a specific materials/loading combination, needs to be understood

thoroughly. The efficiency of load transfer is determined by the interaction between the soil-walls and the overlying structure. Modern finite element analysis techniques provide a powerful tool for understanding this interaction. According to the literature survey, it becomes apparent that most studies until now have been conducted by using simplified assumptions for the geometry, soil properties and wall/soil interaction mechanisms. As a result, they have only provided limited information about the response of individual components. Because of the great potential of pavement soil-wall structures to increase bearing capacity and to reduce settlement, there is a strong need for a balanced construction and maintenance strategy for this type of structure in an integrated, objective and reliable way. In this paper, the governing equations for the description of the motion of a porous medium and the material constitutive model for describing the nonlinear behaviour of soil material are presented. By means of 3D non-linear finite element analyses, the interaction of the components of pavement/soil-wall structures on soft soils and the internal mechanisms that lead to damage have been addressed. The emphasis is placed not on the manifestations of damage but on the actual causes. The findings of this project will enable the road authorities to set up acceptance criteria for the materials and the components of this type of construction and, also, provide to the manufacturers/suppliers/ construction industry the necessary numerical information for the development of design guides and guidelines.

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2

GOVERNING EQUATIONS FOR POROUS MEDIA

The chosen form of the model yield function is given by:

Geomaterials, in particular soils, show an internal pore structure that consists of a solid phase and a number of fluid phases, e.g. gas, water, oil. The solid phase and the fluid phase in the void space have different motions. Due to these motions and due to the different material properties, there is interaction between these phases. This makes the description of the mechanical behavior of the porous material more difficult. Hence, a theory that can for the behavior of the soil skeleton, the pore fluids, and the interaction between the two, becomes necessary. The governing equations for the description of the motion of porous medium are formulated on the basis of modern mixture theory, see Bowen (1976), Lewis and Schrefler (1998) and Liu (2003). The porous medium is postulated to be a mixture, consisting of two basic continua, solid and water, superimposed in time and space. The balance of momentum equation for the whole two-phase medium is expressed by:

where I1 and J2 are the first and second stress invariants, pa is the atmospheric pressure with units of stress, parameter m controls the nonlinearity of the ultimate surface, Liu et al. (2004a), R represents the triaxial strength in tension, Fs is the function related to the shape of the flow surface in the octahedral plane. The isotropic hardening/softening of the material is described by means of parameter α. Parameter γ is related to the ultimate strength of the material. Parameter n is related to the state of stress at which the material response changes from compaction to dilation. For simulation of the hardening response of the material, parameter α of the yield function in Eq. (4) is expressed as a function of both volumetric and deviatoric hardening components, αV and αD :

where a is the acceleration vector of the solid phase, g is the gravity, ρ = (1 − φ)ρs + φρw is average mass density in which the subscripts s, w indicate solid and water phase, φ represents the porosity. σ represents the total Cauchy stress tensor consisting of the σ effective stress tensor and excess water pressure pw :

in which I is the identity tensor, α˜ is Biot’s constant. The mass balance equation for the solid-water mixture is expressed as:

where D/Dt is the material time derivative operator. vws is water velocity relative to the solid phase. vs is the velocity of the solid phase. φ) Qw = (α˜ − + Kφw , K and Kw are the bulk modulus K of solid and water respectively. Eq (1), (2) and (3) constitute the general governing field equations that can be utilized to simulate the saturated soil in the numerical analyses.

3

CONSTITUTIVE MODEL

To capture the main features of the mechanical behavior of geotechnical materials under complex states of stress, is very important for the type of finite element analysis. In this contribution, the yield function which was proposed originally by Desai (1980) is utilized to simulate nonlinear characteristics of soil material.

v where the ratio ηh = ξv ξ+ξ denotes the contribution of d volumetric hardening to the overall material hardening response. Details of the development of mathematical expressions for αv and αD including the determination of the corresponding hardening parameters are presented in Liu et al. (2004a). Simulation of the material softening phase can be achieved by means of specifying the variation of parameter α, after response degradation initiation, as an increasing function of the monotonically varying equivalent post fracture plastic strain ξpf :

in which ηs = e−κ1 ·ξpf , αu and αR are the values of α corresponding to material ultimate stress response and residual stress state respectively. The parameter κ1 is a material parameter that determines the material degradation rate. 4

NUMERICAL INVESTIGATION OF THE PAVEMENT / SOIL-WALL STRUCTURE RESPONSE

In this investigation, the results of the numerical simulations in Zhao (2009) are presented and analyzed. The finite element analyses are made under a wide range of influencing factors such as soil-wall end-bearing conditions, soil-wall rigidity, friction ratio between the soil-wall and the soil, and load positions. 4.1 Finite element example For the numerical analyses, a 3D mesh represented a pavement ed on cement stabilized soil walls

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Table 1.

Clay modeling parameters.

E (MPa) ν 4.0

λ

kx = ky = kz (m/s)

κ

0.35 0.0765 0.13 9.0 1.0 0.053 1.2e-07 b1

c1

d1

3.05e-04

−74.3

3.0e-06

0.012

Crushed subbase modeling parameters.

E (MPa)

ν

γ

R

n

λ

κ

250.0 a1

0.35 b1

0.075 c1

0.01 d1

5.79

1,0

0.05

2.1e-04

−60.3

3.6e-04

0.03

Table 3.

Elastic material modeling parameters.

Figure 1. Schematic of the finite element mesh.

Asphalt concrete Cement treated mix Stabilized soil wall

is built by using finite element system CAPA-3D (Scarpas & Liu, 2000). The mesh consists of components with specific properties which represent specific construction materials. In the regions between the soilwall structure and the soil, interface elements are implanted to capture the interaction characteristics. The geometric characteristics of the 3D finite element mesh are shown in Figure 1. The depth of the soil-wall varies between 4 m and 7 m, with a wall thickness of 0.35 m. The width between the centers

n

a1

Table 2.

Figure 2. The components of the finite element mesh.

R

γ

E (Mpa)

ν

kx = ky = kz (m/s)

ρ (kg/m3 )

2000

0.35

—

2350

400

0.3

—

1750

100/200

0.3

1.0e-8∼ 1.0e-10

1400

of two adjacent walls is 2 m. Because of symmetry, only quarter of the pavement is simulated. The finite element mesh is composed of 1580, 20-noded brick elements and 76, 16-noded interface elements. The physical meanings of the components of the mesh are denoted in the Figure 2. It needs to be pointed out that the subsoil consists of two layers in the mesh composition. The upper soil layer down to the depth of 7 meter consists of soft clay, while the property of the underlying layer is case dependent. For floating wall, the material stiffness equals to that of the soft clay. In end-bearing cases, the material stiffness is taken as that of the bedrock. The self-weight of the materials is taken into in the analyses. In correspondence with the construction sequence, the staged-construction is utilized to simulate the actual construction process. Table 1 summarizes the model parameters for clay material. The explanation of physical meaning of each parameter can be found in Liu et al. (2004a). The parameters of crushed subbase material are listed in Table 2. The parameters for other materials involved in the simulations are summarized in Table 3. 4.2 Performance analysis The analyses are categorized into two basic groups listed in Table 4: pavements without soil-walls (coded as NW) and pavements with soil-walls. Then, the cases of pavements with soil-walls are further divided into two sub-categories: end-bearing walls (coded as

847

Table 4.

Cases for the parametric analysis. Wall/soil interface properties

Case code NW1 FW1 FW3 FW5 FW7 FW9 FW11 FW13 FW17 FW19 EW1 EW3 EW5 EW7 EW9 EW11

Soil-wall type

Soil-wall stiffness (MPa)

Without soil-wall Floating (7 m) 100 200 Floating (4 m)

100

End Bearing

200 100 200

Adhesion ca (kPa)

Skin friction angle δ(o)

Wheel load location (to the soil-wall)

3.94 4.64 5.51 3.94 4.64 5.51 3.94 5.51 3.94 3.94 4.64 5.51 3.94 4.64 5.51

13.6 17.0 20.4 13.6 17.0 20.4 13.6 20.4 13.6 13.6 17.0 20.4 13.6 17.0 20.4

Directly on the top Directly on the top Directly on the top Directly on the top Directly on the top Directly on the top Directly on the top Directly on the top Directly on the top Directly on the top Directly on the top Directly on the top Directly on the top Directly on the top Directly on the top

Figure 3. Ground settlements during pavement construction.

EW) and floating soil-walls (coded as FW). For each group, two soil-wall stiffness values (100/200 MPa) and three surface roughness parameters are chosen. Furthermore, two different wheel loading locations are simulated. Figure 3 shows the in-time settlement of the ground surface in groups. It can be observed, for each construction phase, the settlement rates of the sub-soil slow down due to the dissipation of the ground water. The influence of the soil-walls to the settlement is apparent. Because the soil-wall transfers the overburden load to the deeper parts of the foundation, the ground surface settlement is substantially reduced. It is also observed that presence of the soil-wall not only minimize the settlements of the pavement, but also increase the consolidation rate of the soil. The reason for this phenomenon is that the soil-wall changes the load transmission mechanism and hence enhances the dissipation of the excess water pressure in the soil. As a result the constructed structure can be completely or

Figure 4. The distribution of excess pore pressure along depth in various construction phases for three cases.

virtually subsidence-free, therefore more sustainable and requires less maintenance. In order to gain more insight into ground water dissipation during construction, the in-time distribution of excess pore water pressure along the depth in the

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Figure 5. Skin friction distribution along the soil-walls.

underlying soil is plotted in Figure 4, for four representative cases: NW1, FW1, FW13 and EW1. At early stages of construction, high excess pore water pressure is generated in the soil for all cases. In the process of time, excess pore water pressure decreases due to the drainage of water. Because the load from the overburden is transferred to the deep underlying soil via the floating soil-wall, it can be seen that, at the foot of the soil-walls (about 7 m and 4 m deep), higher excess pore water pressure is generated. In Figure 5 the negative skin frictions along the soil-walls are presented. Note that in the figure the skin friction distribution is plotted against the original wall. As expected, for end-bearing soil-walls, the skin friction firstly increases up to a certain depth then tends to decrease in magnitude reaching the value zero at the bedrock level. The difference between the endbearing soil walls and floating soil-walls is significant. From the graph it can be seen that the skin friction is negative on the upper portion but becomes positive (i.e. it acts upwards) after a certain depth. The elevation at which this change in sign of the skin friction takes place is known as the neutral depth. It can be seen from Figure 4 that, with other soil and geometric parameters kept same, the increase of soil-wall surface roughness induces increased magnitude of shear. The

Figure 6. Vertical stress distribution along the soil-walls.

present analysis clearly demonstrates the influence of factors such as soil-wall roughness, stiffness, bearing status on the skin friction distribution on the soil-wall. Figure 6 shows the influence of the bearing type on the final vertical stress profile. The vertical stress on the end-bearing soil-walls increases continuously with depth. This increase shows slightly concave curvature because of the mobilized negative skin friction at the wall-soil interface. For the floating soil-walls, due to the presence of a negative and positive friction, the vertical stress decreases after reaching the neutral point. Same as for the settlement, the surface roughness of the soil-wall has a greater influence in the case of floating soil-wall compared to the case of end-bearing soil-wall. Figure 7 shows the stress distribution inside pavement with floating soil-walls. It can be observed that sharp transit from compression to tension of bending stress forms on top of the fixed head of the soil-walls, while elsewhere in the pavement it is nearly bending stress free. High stress concentration zones forms near the fixed head of soil-walls. Obviously it is induced by the transition of the weights to the lower strata via

849

on the bottom of the pavement around the fixed heads of the soil-walls, as shown in σxy distribution, the ing function performed by the soil-walls becomes more significant compared to floating soilwalls. Trajectory of the principle stress is also plotted, to illustrate the location and extent of the arch actions within the pavement. 5

CONCLUSIONS

From the analysis, the ing function of the soilwalls to the pavement is apparent. This improvement of the loading capacity is more prominent with endbearing soil-walls. The magnitude of bending stress in the cement treated layer of pavement is reduced by the presence of soil-walls, due to the fact that the soil-walls distribute the loads to the lower strata. High level of stress concentration forms in the cement treated layer of the pavement around the fixed heads of the soil-walls. This phenomenon becomes more significant with wheel loads applied. These areas are the critical areas that require special attention from a design point of view. The influence of the soil-walls to the settlement is apparent. Because the soil-walls transfer the overburden load to the deeper parts of the foundation, the ground surface settlement is substantially reduced. It is also observed that the presence of soil-wall also increases the consolidation rate of the soil. These effects are more significant with increased soil-wall length, stiffness and surface roughness. The present analysis clearly demonstrates the influence of factors such as soil-wall roughness, stiffness, bearing status on the performance of the structure.

Figure 7. Stress (MPa) distribution for case FW1.

REFERENCES

Figure 8. Stress (MPa) distribution for case EW1.

soil-walls. Judged from the pattern of σxy in the cross section, bending momentum establishes on the bottom of the pavement around the fixed heads of the soil-walls. It can be concluded that, it is caused by the ing function performed by the soil-walls. Contour of the principle stress is also plotted, to illustrate the location and extent of the arch actions within the pavement. Figure 8 shows the stress distribution inside pavement with end-bearing soil-walls. It can be observed that the bending stress σxx is largely reduced. The pavement is nearly bending stress free. Compared to pavement with floating walls, even higher concentration zones of σyy form on top of the fixed head of soil-walls. Stronger bending momentum establishes

Bowen, R.M. 1976. Continuum physics. Academic Press, New York, San Fransisco. London. vol. III. 1–127. Desai, C.S. 1980. A general basis for yield, failure and potential functions in plasticity. International Journal for Numerical and Analytical Methods in Geomechanics 4, 361–375. Lewis, R.W. and Schrefler, B.A. 1998. The finite element method in static and dynamic deformation and consolidation of porous media. 2nd edition, John Wiley, Chichester, U.K. Liu, X. 2003. Numerical modeling of porous media response under static and dynamic load conditions. Ph.D. Thesis, Faculty of Civil Engineering, Delft University of Technology, The Netherlands. Liu, X., Cheng, X.H., Scarpas, A. and Blaauwendraad, J. 2004a. Numerical modelling of non-linear response of soil, Part 1: Constitutive Model. International Journal of Solids and Structures 42(7), 1849–1881. Scarpas, A. and Liu, X. 2000. CAPA-3D finite element system-’s Manual”, Part I, II and III, Section of Structural Mechanics, Delft University of Technology, the Netherlands. Zhao, Y., 2009. Integral pavement/soil-wall structures: a numerical study. Ph.D. Thesis, Faculty of Civil Engineering, Delft University of Technology, The Netherlands.

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Modelling embankments on floating stone columns D. Kamrat-Pietraszewska & M. Karstunen Department of Civil Engineering, University of Strathclyde, Glasgow, UK

ABSTRACT: The paper investigates the quantitative improvement in predicted total settlements of a stone column ed embankment on soft soil through a comprehensive parametric study using 3D finite element analyses. To for the complexity in the behaviour of natural soft deposit, the analyses use the advanced constitutive model S-CLAY1S, which takes for plastic anisotropy and interparticle bonding. Several factors are considered, such as the stiffness and the compaction of the granular material, the diameter and the spacing of columns and the thickness of the deposit.

1 1.1

INTRODUCTION Design of stone columns

As population is increasing and the boundaries of the cities are expanding, there is an increasing need to construct on soils of marginal quality, such as very soft soils. The mechanical properties of soft soils can be improved with ground improvement techniques, such as stone columns. These can be used for all types of civil constructions to reduce the settlement and to increase the bearing capacity of the soil. Stone columns rely on the lateral resistance of the hosting soil, and therefore reliable modelling of surrounding material through appropriate constitutive modelling of the column material and the soft soil is required. There are several proposed design methods for stone columns. The most common methods for bearing capacity consideration follow the ideas by Baumann & Bauer (1974) and Hughes & Withers (1974). For settlement calculations, Priebe’s approach (1995) is extensively used in Europe, but alternatively the method by Baumann & Bauer (1974) can be adopted. All these include a number of simplifying assumptions that limit their applicability. The 3D finite element method (FEM) allows recent developments in constitutive modelling of soft soils to be used in the context of complex soil-structure interaction problems, such as stone column – ed foundations. Whilst 3D finite element analyses are time consuming and require more expertise from the than corresponding 2D analyses, it is a very useful tool in understanding the intricacies of soil-structure interactions. 3D parametric studies can also help in design optimization, as demonstrated in this paper. 1.2

Soft soil and its complexity

There are number of geological processes influencing the stress-strain behaviour of natural soft soils. Anisotropy of soils is produced by sedimentation and

subsequent loading history (Casagrande & Carrillo 1944). The yield characteristics of the natural soil can give an indication of the anisotropy of the material (Graham & Houlsby 1983), which is demonstrated via apparently inclined yield surfaces in stress space. Because most sedimented soils have experienced consolidation under their own self weight, they are inherently anisotropic. Structure, which is described as the combination of the soil fabric and interparticle bonding, is a result of clay mineralogy, depositional environment and post-depositional processes. Bonded soils have often stiffer elastic response than unbonded materials (Graham & Li 1985). Moreover, they exhibit greater peak strength than the equivalent unbonded soil (Leroueil & Vaughan 1990) and have additional resistance to yielding. Destructuration of these bonds as a result of plastic straining may lead to dramatic collapse settlements in high sensitivity clays, such as Canadian or Scandinavian quick clays, but even a moderate sensitivity can have major impact on soft soil response. 1.3 Bothkennar clay The 3D benchmark problem considered in this paper looks at a soft soil deposit improved by floating stone columns underneath embankment fill. This represents a typical application of the stone columns for settlement reduction of earth structures. The parameters for the soft soil used in the simulation have been chosen to represent Bothkennar clay, soft clay from Scotland (UK). The site is situated between the cities of Edinburgh and Glasgow, on the Forth River estuary. The slightly over-consolidated Bothkennar clay, which is overlain by a dry crust, is soft recently deposited marine sediment. It has been strongly influenced by the changes in relative sea level over past 13000 years. Bothkennar clay has a significant organic content, as indicated by loss of ignition, and it is classified as a silty clay.

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Table 1. State parameters and soil constants: Bothkennar clay and dry crust. Material Depth

[m]

Dry crust 0–1

Bothkennar clay 1–30

γ k ν e0 K0 POP OCR

[kN/m3 ] [m/s] [-] [-] [-] [-] [-]

19.00 1 × 10−9 0.20 1.37 0.70 30.00 –

16.50 2.89 × 10−9 0.20 2.00 0.50 – 1.50

where: POP is the pre-overburden pressure. Figure 1. Geometry of the problem and assumed soil profile.

Table 2. Additional soil constants: Bothkennar clay and dry crust. Material

Determination of the yielding characteristics of Bothkennar clay concluded in high degree of anisotropy, see Smith et al. (1992) and McGinty (2006). Moreover, isotropic loading/unloading/reloading tests on vertically and horizontally orientated samples conducted by McGinty (2006) showed evidence of crossanisotropy of elastic behaviour. Oedometer tests done to study interparticle bonding by comparison of natural and reconstituted soil samples show that the yield stress for natural sample is 1.5 times greater than for the reconstituted soil at the same void ratio (Smith et al. 1992). Additionally, triaxial tests conducted on natural samples showed that the breakdown of the bonding is progressive (Clayton et al. 1992, McGinty 2006). 2 2.1

Geometry of the problem

2.2

Constitutive models used to represent soft deposit and embankment fill

[m]

Dry crust 0–1

Bothkennar clay 1–30

M κ λi β µ α0 a b χ0

[-] [-] [-] [-] [-] [-] [-] [-] [-]

1.51 0.02 0.15 1.00 30.00 0.59 9.00 0.20 4.00

1.51 0.02 0.15 1.00 50.00 0.59 9.00 0.20 8.00

where: M is the slope of the critical state line, κ and λi are the gradients of the elastic swelling line and intrinsic normal compression line, α0 is the initial inclination of the yield surface, β and µ govern the rotation of the yield surface, a and b control the rate of destructuration, and χ0 is the amount of bonding.

NUMERICAL MODELLING

The geometry of the embankment considered is shown in Figure 1. Due to the symmetry conditions just half of the geometry is considered in the simulations. The embankment is 2 m high with a gradient of embankment slope of 1:2. The underlying deposit consists from two layers: over-consolidated dry crust and slightly over-consolidated soft Bothkennar clay. The groundwater table is assumed to be located at depth of 1 m. For all simulations the PLAXIS 3D Foundation v.2.2 FE code has been used, taking advantage of 3D modelling, to which the advanced constitutive model S-CLAY1S (Karstunen et al. 2005) has been implemented as -defined model. Mesh sensitivity studies have been done before performing the parametric studies in order to reduce the influence of the mesh on the results of the simulations.

Depth

been chosen. This model s for both plastic anisotropy and destructuration of interparticle bonds using the concept of intrinsic yield surface. The elastic part of the model is isotropic. In the case of Bothkennar clay, ideally one should also for time-dependency; however, this feature of soft soil behaviour is not considered in S-CLAY1S formulation. This would be of great importance if one were trying to attempt to for installation effects. The state parameters and soil constants describing the deposit are shown in Tables 1 and 2 (for determination of the parameter values see Kamrat-Pietraszewska et al. (2008) and McGinty (2006)). The construction of the embankment has been simulated in two stages; two layers of 1 m embankment fill each are placed within 5 days. The elasto-plastic Mohr Coulomb model is used to represent the granular fill and the material is assumed fully drained. 2.3 Constitutive models used to represent column material

To simulate the complexity of natural soft clay the S-CLAY1S model (Karstunen et al. 2005) has

For reference analysis (labelled REF) floating stone columns are assumed to be installed in a square grid

852

Table 3.

Soils constants for stone columns.

Material γ νur E50 ref = Eoed ref Eur ref k c ϕ ψ m

Table 4.

Stone columns 3

[kN/m ] [-] [kN/m2 ] [kN/m2 ] [m/s] [kN/m2 ] [◦ ] [◦ ] [-]

Parameter

19.00 0.30 80000 260000 1.97 × 10−4 0 42 12 0.30

with spacing between the columns SSC equal to 2 m. The diameter of the columns DSC is 0.6 m and the length LSC is 10 m. A representative slice of the full 3D geometry can be modelled with strip having a width equal to the centre-to-centre spacing of the columns. In the FE analyses the stone columns are ‘wished-inplace’ as an undrained process without considering installation effects. In fact, the installation of stone columns on structured soils reduces the amount of bonding (and sometimes also strength of the soil) next to the columns, as well as changes anisotropy and the coefficient of earth pressure at rest, as shown by Castro & Karstunen (in press), in line with the field observations by Guetif et al. (2007) and Kirsch (2006). The Hardening Soil Model is used to model the granular material of the stone columns and the values for material parameters are listed in Table 3. Throughout the parametric studies, the reference modulus E50 ref has been assumed to be the same as the reference oedometer modulus Eoed ref , whereas the reference unloading/reloading modulus Eur ref is assumed to be 3.25 times greater than the E50 ref . 3

NUMERICAL SIMULATIONS

Many researches have tried to identify the optimum stone columns parameters in order to develop a design approach which allows for the maximum improvement performance at the minimum economical costs, see e.g. Ambily & Gandhi (2007), Herle et al. (2007) and Watts et al. (2000). This process-based knowledge (either via experimental model tests or via numerical modelling) can give some recommendations for the industry. In this study, first the influence of the reference moduli and friction angle of the columns is studied. Next, the effect of column diameter and spacing is assessed. Finally, the impact of thickness of the soft deposit on the settlements is investigated. Due to the space limitations this paper focuses on settlement analysis only. 3.1

Numerical simulations: summary.

Influence of the mechanical properties of stone column material

For the construction of stone columns a gravely material is usually used, which meets the specified

Simulation

Eoed ref [kN/m2 ]

ϕ [◦ ]

DSC [m]

SSC [m]

dSS [m]

REF M1 M2 M3 M4 M5 P1 P2 P3 P4 T1 T2

80000 80000 80000 80000 100000 120000 80000 80000 80000 80000 80000 80000

42 40 44 50 42 42 42 42 42 42 42 42

0.6 0.6 0.6 0.6 0.6 0.6 0.7 0.8 0.6 0.6 0.6 0.6

2.0 2.0 2.0 2.0 2.0 2.0 2.0 2.0 1.7 2.3 2.0 2.0

30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0 30.0 20.0 10.0

SC = stone column, SS = soft soil, dSS = thickness of soft soil layer.

requirements regulated by the various standards, see e.g. BS EN 14731:2005. As the column is formed using different types of gravel, the mechanical properties of the stone column will vary depending on the mechanical properties of the material used. Numerical simulations have been performed to asses the influence of the compaction and the stiffness of the stone column material on settlement predictions. The friction angle plays a crucial role in the calculations of the bearing capacity of a soil improved with stone columns and the dilatation, the volume increase of the granular material at yield, has a significant effect on the settlement reduction.As the columns are usually designed to yield, friction and dilatancy angles of granular material will influence the overall behaviour of the system. Thus, a parametric study to investigate the impact of those two factors on the numerical predictions has been carried out. The degree of compaction of the granular material is investigated by varying the angles of friction ϕ and dilatancy ψ only. The relationship between the friction and dilatancy angle is assumed to be:

The most compacted stone column material is assumed to have an angle of friction ϕ of 50◦ and results in relatively high angle of dilatation (ψ = 20◦ ). All performed simulations are listed in Table 4. The importance of the degree of compaction is evident looking at the plot of the settlement reduction ratio sr in relation of the angles of friction and dilatancy of the stone column material, see Figure 2. The settlement reduction ratio sr is determined as the ratio between the settlement with stone columns and settlement in case of no ground improvement. Great reduction of settlement is evident in columns, which allow high concentration of stresses. Recent research indicates that in most cases the conventional design value of angle of friction of 40◦ is far too conservative, as in

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Figure 2. Effect of friction and dilatancy angles on settlement reduction ratio.

large scale shear box tests in dense samples of granular material very high friction angles (above 50◦ ) have been measured at low normal stresses, see Herle et al. (2007). The stiffness of the granular material used for stone column construction varies dependant on the origin of the material and the stiffness of the surrounding soil. The influence of the stone material has been explored using three sets of stiffness parameters for the columns, see Table 4, which represent a typical possible range. The effect was found to be almost negligible (settlement reduction ratios of 0.837, 0.828 and 0.821 for REF, M4 and M5, respectively), so in of design, the friction and dilatancy angles of the columns and the properties of the surrounding soil are more influential than the stiffness of the column material. 3.2

Effect of diameter and spacing of stone columns

The stone columns installed in Europe have usually a diameter of 0.6–1.2 m and a centre-to-centre spacing of about 2 m. In most cases the length of the granular columns does not exceed 15 m, although in extreme situations the stone column length can be in excess of 60 m (Bell 2009). The geometry of the area improved with stone columns has been changed in numerical simulations, by varying column diameter and spacing (see Table 4). The results by the elastic solution (Balaam & Booker 1981) and the solution for plastic columns (Priebe 1995), as well as some laboratory data are compared with the numerical predictions in Figure 3, as a function of the area replacement ratio Ar . The area replacement ratio Ar is the proportion of the total area of the improved soil in which the stone columns have been installed. Experimental data is based on endbearing columns. The advanced model predicts similar

Figure 3. Effect of column diameter: parametric study and comparison with simple solutions and lab data (after Charles & Watts 2002).

settlement reduction ratio for the floating columns than the simple design methods for end bearing columns, demonstrating how conservative the latter are. The evolution of the surface settlement at three chosen points (the centreline, crest and toe of an embankment) as a function of the centre-to-centre spacing is shown in Figure 4. The column spacing has major effect at the centreline of an embankment, but with the distance from the symmetry axis its significance reduces, most likely due to the flexibility of the system. The results for the whole set of parametric studies dealing with the geometry of stone columns is summarised in Figure 5, by plotting the impact of SSC /DSC on the settlement reduction ratio. Additionally, the results for selected field data (all end bearing columns) and the experimental tests conducted on floating stone columns constructed in soft clay in a cylindrical tank by Ambily & Gandhi (2007) have been plotted. Ambily & Gandhi used reconstituted clay and an angle of friction for the stone column of 43◦ . The ratio SSC /DSC has significant influence on the obtained settlement reduction ratio, as seen in Figure 5. One should note that the numerical simulations considered different loading mode at ground level, and different parameters for granular material than used in the experiments, therefore only the general tendencies can be compared. The trends look very similar, and both the numerical simulations and the experiments suggest that there is a certain SSC /DSC ratio, beyond which improvements become marginal. The numerical

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Figure 4. Evolution of the surface settlement with column spacing.

Figure 6. Effect of thickness of soft deposit: a) Total settlements; b) Consolidation time.

Figure 5. Effect of SSC /DSC on settlement reduction factor.

simulations suggest that the optimum ratio is about 2.7. . .2.8, beyond which the rate of increase on the benefit reduces. 3.3

Impact of thickness of soft deposit

been conducted (considering thickness from 10 to 30 m), see Table 4. For the case with 10 m deep deposit, the columns are end bearing. Figure 6a shows that although the thickness influences the total settlement, from practical point of view, the differences are marginal. However, in of consolidation time, the thickness of the deposit is an important factor (Figure 6b). Consequently, although floating columns can be very effective in overall settlement reduction, in a case of a thick deposit, column lengths might need to be extended in order to reduce consolidation time.

4

As the stone columns in practice are installed either as floating inclusions or end-bearing columns, analyses with varying thickness of the soft deposit have

CONCLUSIONS

3D FE simulations of an embankment on floating stone columns have been performed to study the influence

855

of stone columns material and geometry to the predicted settlement. Finally, the effect of the thickness of the deposit was studied. The non-linearity of the stone columns material was modelled with the Hardening Soil model and for the surrounding soil, the advanced elasto-plastic S-CLAY1S model (Karstunen et al. 2005) was used. The parametric studies reveal that the key design parameters are the friction and dilatancy angle of the stone columns and the spacing (SSC ) to diameter (DSC ) ratio. For the latter, the numerical simulations and field data collected from literature suggest typically a value SSC /DSC about 2.7–2.8, as with higher ratios the differences in settlement reduction ratio are rather marginal. The actual stiffness of the column material has little influence on the numerical results. In of settlement reduction, floating columns appear to work as well as end bearing columns. Indeed, the results are very similar to those predicted by simple design methods for end bearing columns. This demonstrates how conservative the simple design methods are. However, as the thickness of the deposit increases, so does the time required for consolidation. Therefore, in deep deposits the length of the columns needs to be optimised to achieve a desired rate of consolidation. In all simulations conducted, the columns have been wished in place. The installation of stone columns changes the structure of the soil and causes increase in excess pore pressures. Provided these excess pore pressures are allowed to dissipate before construction, the installation effects are likely to be very beneficial. ing for installation effects will be a subject for further studies. ACKNOWLEDGEMENTS The research was carried out as part of a “GEOINSTALL” (Modelling Installation Effects in Geotechnical Engineering), ed by the European Community through the programme “Marie Curie Industry-Academia Partnerships and Pathways” (Contract No PIAP-GA-2009-230638). REFERENCES Ambily, A. P. & Gandhi, S. R. 2007. Behavior of stone columns based on experimental and FEM analysis, J. of Geotech. and Geoenviromental Engineering, ASCE, 133(4): 405–415, ASCE. Balaam, N. P. & Booker, J. R. 1981. Analysis of rigid rafts ed by granular piles. Int. J. Numer. Anal. Methods Geomech. 5(4): 379–403. Baumann, V. & Bauer, G. E. A. 1974. The performance of foundations on various soils stabilised by the vibrocompaction method. Can. Geotech. J. 11(4): 509–530. Bell, A. 2009. Private correspondence. BS EN 14731:2005, Execution of special geotechnical worksGround treatment by deep vibration.

Casagrande, A. & Carrillo, N. 1944. Shear failure of anisotropic soils. Journal of the Boston Society of Civil Engineering, Contribution to Soil Mechanics: 1941–1953. Charles, J.A. & Watts, K. S. 1983. Compressibility of soft clay reinforced with granular columns. Proc. 8th European Conf. Soil Mech. Found. Engineering: 347–352, Helsinki. Clayton, C. R. I., Hight, D. W. & Hopper, R. J. 1992. Progressive destructuring of Bothkennar clay: implications for sampling and reconsolidation procedures. Gèotechnique 42(2): 219–240. Craig, W. H. & Al-Khafaji, Z. A. 1997. Reduction of soft clay settlement by compacted sand columns. Proc. 3rd Int. Conf. Ground Improvement Geosystems: 218–231, London. Graham, J. & Houlsby, G.T. 1983. Anisotropic elasticity of a natural clay. Géotechnique 33: 165–180. Guetif, Z., Bouassida, M. & Debats, J. 2007. Improved soft clay characteristics due to stone column installation. Computers and Geotechnics 34: 104–111. Herle, I., Wehr, J. & Arnold, M. (2007). Influence of pressure level and relative density on friction angle of gravel in vibrated stone columns. Mitteilung des Instituts für Grundbau und Bodenmechanik, J. Stahlmann (ed.), published in German, TU Braunschweig, Helft 84: 81–93. Hughes, J. M. & Withers, N. J. 1974. Reinforcing of soft cohesive soils with stone columns. Ground Engineering 7(3): 42–49. Kamrat-Pietraszewska, D., Krenn, H., Sivasithamparam, N. & Karstunen, M. 2008. The influence of anisotropy and destructuration on a circular footing. Proc. 2nd British Geotechnical Association, Int. Conf. on Foundations, ICOF 2008, Dundee, IHS BRE Press, 1527–1536. Karstunen, M., Krenn, H., Wheeler, S. J., Koskinen, M. & Zentar, R. 2005. Effect of anisotropy and destructuration on the behaviour of Murro test embankment. ASCE Int. Journal of Geomechanics 5(2): 87–97. Kirsch, F. 2006. Vibro stone column instllation and its effects on ground improvement. Numerical Modelling of Construction Processes in Geotechnical Engineering for Urban Environment, Th. Triantafyllidis (ed.), Taylor and Francis, London. Leroueil S. & Vaughan, R. R. 1990. The general and congruent effects of structure in natural soils and weak rocks. Gèotechnique 40(3): 467–488. McGinty, K. 2006. The stress-strain behaviour of Bothkennar clay. PhD thesis, Department of Civil Engineering, University of Glasgow, UK. Munkfakh, G. A., Sarkar, S. K. & Castelli, R. J. 1983. Performance of a test embankment founded on stone columns. Proc. Int. Conf. on Advances in Piling and Ground Treatment for Foundations: 259–265, Institution of Civil Engineers, London. Priebe, H. J. 1995. The design of vibro replacement. Ground Engineering 18: 31–37, December. Raju, V. R. 1997. The Behaviour of Very Soft Soils Improved by Vibro Replacement, Ground Improvement Conference, London, 12–64E. Smith, P. R., Jardine, R. J. & Hight, D. W. 1992. The yielding of Bothkennar clay. Gèotechnique 42(2): 257–274. Watts, K. S., Johnson, D., Wood, L. A. & Saadi, A. 2000. An instrumented trial of vibro ground treatment ing strip foundations in a variable fill. Gèotechnique 50(6): 699–708.

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Numerical investigation of the mechanical behaviour of Vibro Replacement stone columns in soft soils T. Meier, E. Nacke & I. Herle Institute of Geotechnical Engineering, Technische Universität Dresden,

W. Wehr Keller Holding GmbH, Offenbach,

ABSTRACT: Vibro Replacement is a ground improvement technique commonly used in practice. The installation of gravel or stone columns leads to an improvement of the mechanical properties of the subsoil in of bearing capacity and more important stiffness, thus settlements can be reduced. For the design of vibro replacement a procedure by Priebe is often employed in practice. It contains some major simplifications and assumptions, e.g. the soil surrounding the column behaves linear-elastically with Poisson’s ratio ν = 1/3 or the earth pressure coefficient after the installation process K = 1.0. By means of comprehensive laboratory tests and a finite element model using advanced hypoplastic constitutive equations, the mechanical behaviour of vibro replacement columns in soft soils was investigated. With the aid of such a model it is possible to analyse the simplifications of Priebe’s theoretical model mentioned above.

1

INTRODUCTION

In this study the mechanical behaviour of so-called unit cells (Fig. 1) consisting of very soft soil and a granular replacement column ed by a rigid underlying stratum are investigated. Considering such unit cells with oedometric boundary conditions corresponds to an “infinite” (large-area) grid of replacement columns. The design of a vibro replacement measure consists of the choice of the spacing, the diameter and the depth of the columns. In the following the spacing is expressed as the area ratio A/Ac , where A is the area of the complete cell (soil plus column) and Ac is the area of the column.

The design method by Priebe (Priebe 1995) is an analytical method based on the cylindrical cavity expansion problem in a linear-elastic half-space. It contains several major simplifications and assumptions: • •

Linear-elastic behaviour of the column and the soil. Design charts are given only for one constant Poisson ratio ν = 1/3 of the soil to be improved. • Uniform radial strains εr = rc /rc0 of the column due to loading (cylindrical cavity expansion). • Earth pressure coefficient of the soft soil K = 1.0 due to the column installation process.

The aim of the presented study is to compare improvement factors n (Eq. 1) obtained from this simple analytical model with those from finite element analyses using two advanced hypoplastic constitutive models.

where s0 – settlement of untreated ground simp – settlement with replacement columns Furthermore, the results of the finite element analyses allow an insight into deformations of the system column/soft soil due to loading. 2 Figure 1. Unit cell.

CONSTITUTIVE MODELS

Two hypoplastic constitutive equations were employed for the mathematical description of the mechanical

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Table 1.

Material parameters of the soft soil.

Table 3.

Material parameters of the rockfill.

ϕc

λ∗

κ∗

N

r

ϕc

hs

n

ed0 /ec0 /ei0

α

β

26.1◦

0.086

0.018

0.9894

0.33

40.7◦

41 GPa

0.24

0.50/0.82/0.93

0.10

1.53

Table 2. Index properties of the soft soil (with ρs –grain density, wL/P –liquid/plastic limit, CC–clay content, IL–ignition loss). ρs

wL

wP

CC

IL

2.752 g/cm3

43.4%

20.9%

50%

3.1%

incremental shear modulus for undrained shearing at the same isotropic stress state with OCR = 1.0. Determined by means of CU triaxial tests. For this numerical study the material parameters (Tab. 1) of a natural clay (CL, index properties cf. Tab. 2) were used.

behaviour of the soft soil to be improved and the granular material of the column. The main advantage of these equations is a clear distinction between material parameters (e.g. critical friction angle) and state variables (stress and density), i. e. for one set of material parameters, which are determined by means of standard laboratory tests on disturbed or remoulded samples, respectively, it is possible to simulate the mechanical behaviour of soils over a wide range of stress states and densities. In the following subsections the two constitutive models are briefly described with respect to the incorporated parameters. 2.1

2.2 Stone column The column material was modeled using a hypoplastic equation for granular soils (Wolffersdorff 1996). It contains eight material parameters: • •

Critical friction angle ϕc (see above). Limit void ratios at zero stress ed0 , ec0 , ei0 : ed0 = emin is a lower bound void ratio of a grain skeleton at zero pressure, ec0 = emax is the void ratio in the critical state at zero pressure. Both, emin and emax are determined through standard index tests (e.g. according to ASTM D4254 and D4253). ei0 is an upper bound void ratio of a simple grain skeleton at vanishing pressure (without macropores). ei0 ≈ 1.15emax and ed0 ≈ 0.6ec0 are simple estimates (Herle 1997) which have been used successfully for more than a decade. • Granulate hardness hs and exponent n: In the case of an isotropic compression of a very loose sample (e0 = emax ) the hypoplastic constitutive equation reduces to the compression law by Bauer

Soft soil

For the description of the soft soil a model proposed by Mašín (Mašín 2005) was used. It contains five material parameters: •

Critical friction angle ϕc : The critical friction angle ϕc determines the resistance of a soil subjected to monotonic shearing in critical state, i.e. when σ˙ = 0 and the volumetric strain rate ε˙ v = tr(˙ε) = 0 hold. Drained or undrained triaxial tests, simple shear or direct shear tests on initially very loose specimens are appropriate for the determination of ϕc . • Compression and swelling index λ∗ and κ∗ of Butterfield’s compression law (Eq. 2) (Butterfield 1979)

where p = −tr(σ)/3 is the effective mean pressure and p0 together with the void ratio e0 define the normal consolidation line. Determined by means of isotropic or oedometric compression tests. In case of swelling λ∗ is replaced κ∗ . • N = ln (1 + e0 ), an expression for the reference void ratio e0 at isotropic effective stress σii = −1 kPa of a normally consolidated sample, (OCR = 1.0) Determined by isotropic or approximated by oedometric compression tests. • Stiffness ratio r = Ki /Gi , where Ki is an incremental bulk modulus for isotropic compression and Gi is an

where a reference pressure, the so-called granulate hardness hs , together with the exponent n govern the compression for an increasing effective mean pressure p . • Exponents α and β: The exponent α controls the peak friction angle of the material, and hence also the dilatancy behaviour. The stiffness of a grain skeleton with e < ec can be adjusted via the exponent β. A comprehensive step-by-step description of the calibration procedure can be found in (Meier 2009). The parameters were calibrated from special laboratory tests with rockfill (d50 ≈ 15 mm, dmax = 40 mm, CU = 2.7, CC = 1.0, ρs = 2.77 g/cm3 ). The resulting parameters are given in Table 3. Figure 2 shows the comparison between oedometric compression tests and the corresponding results obtained by hypoplasticity.

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Figure 2. Oedometric compression tests on rockfill.

Figure 4. Improvement factor n vs. area ratio A/Ac for different Poissons’ ratios ν and constrained moduli Es of the soft soil.

With the aid of this model the influence of the following quantities on the improvement factor was studied: • • •

area ratio A/Ac magnitude of superimposed load initial stress state in of the earth pressure coefficient K = σr /σv (σr – radial and σv – vertical stress) • relative density of the column

4

Figure 3. Layout of the FE model.

RESULTS

4.1 Influence of Poisson’s ratio 3

FINITE ELEMENT MODEL

Figure 3 depicts a layout of the axisymmetric FE model. To allow for the influence of the geostatic stress state, which leads to an increase of stiffness of both materials with depth, a 5 m long column was modeled. It is assumed that the influence of the length of the column on the improvement factor is negligible, as has been shown numerically e.g. by (Borges and Domingues 2009). The analyses were performed with the FE code Tochnog (version 5.0) using evenly spaced first oder crossed triangle elements. The top nodes (“rigid plate”) were constrained to have all the same vertical displacement and the loading was done by a uniformly distributed load linearly increasing with time (mixed boundary conditions). In case of non-linear calculations both spatial and time discretisation can have a strong impact on the calculation results. Hence both were refined until no change in the results due to this refinement could be observed anymore. The initial conditions were: •

Stress State: bulk density of both materials γ = 18 kN/m3 , σv0 = γ · z, σr0 = K · σv0 • Density: Column e0 = 0.6, soft soil OCR = 1.05 (stress dependent void ratio e)

As mentioned earlier, the design charts given by Priebe are based on calculations with Poisson’s ratio ν = 1/3 of the soft soil. To study the influence of this quantity on the improvement factor n = f (A/Ac ) the method was programmed and comparative calculations were carried out. The results are depicted in Figure 4. As can be clearly seen, the influence of ν on the improvement factor n is negligible.

4.2 Influence of the load magnitude Figure 5 shows the influence of the load magnitude on n as obtained from the Priebe method and from the finite element analyses. Both methods show a decrease of n with increasing load. The effect is more pronounced for small area ratios and in general in case of the FE method. For loads p ≤ 125 kPa the Priebe method is conservative compared to the FE results.

4.3

Influence of the stress state

The stress state always plays a key role as it directly influences stiffness and shear resistance of soils. Priebe assumed an earth pressure coefficient K = 1.0 for the soft soil, which is higher than the at-rest value K0 ≈ 1 − sin ϕ (Jaky 1944) for granular and normally

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Figure 5. Improvement factor n vs. area ratio A/Ac for different loads.

Figure 7. Improvement factor n vs. area ratio A/Ac for different initial stress states.

Figure 8. Change in radial stress due to cylindrical cavity expansion (vertical stress kept constant) for different initial stress states.

Figure 6. Distribution of the earth pressure coefficient K along the interface (cross symbols – initial values, boxes – after loading).

consolidated soft soils. This is justified by the construction process, which can be seen as a cylindrical cavity expansion in the ground, where the magnitude of the radial stress is increasing until a limit state is reached eventually. In our FE calculations only the initial stress state can be varied as the evolution of K is governed by the constitutive equations during loading. Figure 6 depicts the distribution of K along the interface between the column and the soil for different initial values and with a superimposed load of 125 kPa. Independently from the initial values K tends to 1 during loading. Figure 7 shows the influence of the initial stress state on the improvement factor.As expected n strongly depends on the initial stress state. The higher the confining pressure on the column from the soft soil, the stiffer the system reacts to loading. The resulting improvement factor after Priebe lies between the FE results for K = 0.5 and K = 1.0 Extra FE simulations of the cylindrical cavity expansion under drained conditions yield K > 1 in the vicinity of the cavity (Fig. 8). Therewith it can be concluded from the numerical results that the assumption K = 1 made

by Priebe is physically sound and conservative for typical loads ≤125 kPa. For loads p ≈ 250 kPa both the Priebe method and the FE analyses yield very similar results and for higher loads the Priebe method overestimates the improvement factor compared with the FE results. 4.4 Influence of the column’s relative density The second key factor influencing stiffness and strength of soils is density. The results presented above were obtained from simulations with dense columns (void ratio e0column = 0.6, i. e. relative density ID = 69%). To judge the effect of the column density on the improvement factor an extra series of calculations with e0column = 0.8, i. e. ID = 6%, was conducted. The results shown in Figure 9 represent a lower bound of the improvement factor and correspond to undensified drainage columns. Figure 10 depicts the ratio nlc /ndc (lc/dc – loose/dense column) as a function of the area ratio. The resulting improvement factors are markedly lower for columns with ID = 6%. In the Priebe method density is taken into using Equation (4) for

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Figure 11. (a) Radial strains εr of the columns vs. depth t (b) contour plot of void ratio (ranging from 0.57 (dark blue) to 0.76 (dark red)) in the unit cell under load. Figure 9. Improvement factor ratio n vs. area ratio A/Ac for insufficiently densified columns.

5 •

•

•

•

Figure 10. Improvement factor ratio nlc /ndc vs. area ratio A/Ac for different loads.

•

the determination of the constrained modulus Es of the column.

where σv is the vertical effective stress and λ = −de/d ln (σ /σref ) is the compression index. In case of the loose columns the Priebe method overestimates the improvement factor compared to the FE results. While for typical loads (p = 125 kPa) the results are only slightly higher, this effect is much more pronounced for higher load magnitudes. 4.5

Deformation behaviour

Figure 11(a) shows the radial strain of the columns after applying a surcharge of 125 kPa for different area ratios. The lateral deformations are clearly concentrated in the upper part of the column. The contour plot of the void ratio in Figure 11(b) reveals a deformation mechanism with shear localization. At the top of the column a cone of very dense material forms and is pushed into the underlying column material which is loosened and displaced downwards and radially.

6

CONCLUSIONS The influence of the Poisson’s ratio ν of the soft soil on the improvement factor determined according to the design method of Priebe is negligible. The improvement factor n depends on the superimposed load. The higher the load, the lower the improvement factor n. The Priebe method clearly underestimates this influence compared to the FE results. For typical loads (p ≤ 125 kPa) the FE calculations yield n = f (A/Ac ), which for small area ratios is up to 50% higher than according to Priebe. Here, using the Priebe method is conservative. The density of the replacement columns has a strong influence on n, which is more pronounced for low area ratios and loads. The deformation mechanism of the column according to the FE analyses is different to Priebe’s assumption of a uniform cylindrical deformation. The numerical calculations revealed shear localization in the upper part of the column, resulting in a cone of very dense material which is pushed into the underlying part of the column.

OUTLOOK

In the actual second phase of this research and development project, model tests in a large oedometer (30 cm in height and diameter) will be conducted to investigate qualitatively the dependence of the achievable column density on the consistency of very soft soils. The FE model will be validated with unit cell model loading tests. Further investigations of the deformation behaviour will be carried out with the validated model and materials parameters of three different natural soft soils. Another main objective will be the numerical examination of the installation process with respect to the transient development of the effective stress state in the vicinity of the replacement column, which essentially influences the stiffness of the system column/soil. Based on the numerical results improvements of the Priebe design procedure may be proposed.

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REFERENCES Borges, J. L. and T. S. Domingues (2009). Embankments on soft soil reinforced with stone columns: Numerical analysis and proposal for a new design method. Geotech. Geol. Eng. 27, 667–679. Butterfield, R. (1979). A natural compression law for soils. Géotechnique 29(4), 469–480. Herle, I. (1997). Hypoplastizität und Granulometrie einfacher Korngerüste. Ph. D. thesis, Veröffentlichungen des Instituts für Bodenmechanik und Felsmechanik/Universität Karlsruhe. Heft 142. Jaky, J. (1944). The coefficient of earth pressure at rest. J. Soc. Hungarian Architects and Eng., 355–358.

Mašín, D. (2005). A hypoplastic constitutive model for clays. Int. J. Numer. Anal. Meth. Geomech. 39, 311–336. Meier, T. (2009). Application of Advanced Constitutive Soil Models for Geotechnical Problems. Ph. D. thesis, Veröffentlichungen des Instituts für Bodenmechanik und Felsmechanik/Universität Karlsruhe. Heft 171. Priebe, H. J. (1995). The design of vibro replacement. Ground Engineering 28(10), 31–37. Wolffersdorff, P. v. (1996). A hypoplastic relation for granular materials with a predefined limit state surface. Mech. Cohes.-Fric. Mater. 1, 251–271.

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Numerical modelling of consolidation around stone columns J. Castro & C. Sagaseta University of Cantabria, Santander, Spain

ABSTRACT: Coupled finite element analyses of the consolidation around stone columns are performed to assess the accuracy of different analytical solutions. The numerical model reproduces the hypotheses and assumptions made in the closed-form solutions. Therefore, a rigid load is applied to a “unit cell”, formed by a fully penetrating column and its surrounding soil, and simple elastic or elasto-plastic soil models are used. The surface settlement, the dissipation of pore pressure and the vertical stress concentration on the column are studied. The important influence of the radial and plastic strains in the column is highlighted. On the other hand, the surrounding soil does not yield for usual conditions, which reasonably justifies the elastic soil behaviour assumed in the analytical solutions. The comparison of the numerical results with the closed-form solutions shows the implications of the assumptions made in each solution.

1

INTRODUCTION

Stone columns, either by the vibro-replacement or vibro-displacement methods, are one of the most common improvement techniques for foundation of embankments or structures on soft soil. A considerable number of stone columns are usually involved in a problem, what implies a complex modelling process. There are mainly five different alternatives: – Unit cell. Only a “unit cell”, i.e. one column and its surrounding soil, is modelled in axial symmetry (e.g. Balaam & Booker 1981). – Plane strain. The cylindrical columns are converted to gravel trenches (e.g. Van Impe & De Beer 1983). It is commonly used under long loads, such as embankments. – Axial symmetry. The cylindrical columns are converted to gravel rings when columns are used under circular loads, such as tanks (e.g. Elshazly et al. 2008). – Homogenization technique. The soil and columns are modelled as a homogeneous soil with improved properties (e.g. Schweiger 1989). – A full 3D modelling using complex numerical models (e.g. Weber et al. 2008). The unit cell concept is generally used by analytical solutions, while the other four options are common in numerical analyses. In this paper, numerical analyses are performed to assess the accuracy of different analytical solutions that study the consolidation process around stone columns (Barron 1948, Han & Ye 2001) and are especially focused on the validation of an analytical solution recently developed by the authors (Castro & Sagaseta 2009).All these solutions are based

Figure 1. Unit cell.

on a unit cell model in axial symmetry, with a rigid load and a fully penetrating column (Figure 1). The study is focused on closed-form solutions because they still form the basis of methods of calculation commonly used in practice, and even in those complex cases where numerical modelling is required, closed-form solutions allow a preliminary study of the problem and are useful for identification of the relative influence of the different parameters. The main features of a stone column treatment, such as the dissipation of excess pore pressures, the settlement reduction and the stress concentration on the columns, are compared with the numerical results.

863

Table 1.

Main features of some analytical solutions.

Barron Column behaviour

Soil behaviour Applied load Study

2

Not included Oedometric Elastic Elasto-plastic Oedometric Elastic Rigid Flexible Consolidation Settlement

•

•

Balaam & Booker

•

Castro & Sagaseta

•

•

•

• • •

•

• •

• •

• ◦

•

• •

REVIEW OF ANALYTICAL SOLUTIONS

As the results of the numerical model will be compared with some analytical solutions, a brief review of those solutions is here presented. Barron (1948) developed an analytical solution to study the radial consolidation around vertical drains and hence it does not consider the column stiffness, which is included by Han & Ye (2001) and Castro & Sagaseta (2009) by means of a modified coefficient of consolidation. Barron’s approach is based on the use of the average value of the excess pore pressure along the radius. As the presented numerical analyses will later illustrate, the exact distribution along the radius may well be different from that assumed by Barron for the first stages of the consolidation process. Soil and column behaviour is usually assumed as elastic in the analytical solutions. Han & Ye (2001) further assume the soil and the column in laterally confined conditions, neglecting the radial displacement of the soil/column interface. Castro & Sagaseta (2009) include the lateral deformation of the column and its possible yielding using the Mohr-Coulomb yielding criterion and a non-associated flow rule for the plastic strains. Castro & Sagaseta (2009) study not only the consolidation process but also the settlement improvement, which for an elastic column agrees with that of Balaam & Booker (1981), who analytically solved the elastic problem of soil and column with simultaneous consideration of the horizontal and vertical components of the deformation. However, Balaam & Booker (1981) solve the consolidation process numerically. A summary of the commented features of each analytical solution is shown in Table 1. 3

Han & Ye

NUMERICAL MODEL

Coupled numerical analyses were performed using the finite element code Plaxis v8.6 (Brinkgreve 2007). The same assumptions and boundary conditions of the closed-form solutions are chosen for the numerical model. Therefore, a unit cell model in axial symmetry (Figure 2) is used.

Figure 2. Numerical model.

The column reaches a rigid substratum because the influence of the column length in floating columns (e.g. Barksdale & Bachus 1983) is beyond the scope of the paper. A rigid load is applied by means of a rigid plate and simple elastic or elasto-plastic soil models are used. For plastic strains, the Mohr-Coulomb yielding criterion and a non-associated flow rule are adopted. Roller boundaries are used but for the upper boundary, where a rigid plate is located and then the horizontal displacements are restricted. For that reason, the results of the upper part (more or less the first meter) are not used to compare with the analytical solutions. The closed-form solutions that are the purpose of the study do not consider a finite permeability of the column or a smear zone, and therefore those aspects are not included in the model. Only one load step is applied in undrained conditions and the excess pore pressures generated in the soil are subsequently assumed to dissipate towards the permeable column. The geometry of the problem and the soil and column properties are chosen to be in typical ranges used

864

in the field. Some of them are varied to perform parametric studies. The specific values are detailed for each case. 4

ELASTIC SOLUTION

This is the simplest case and the soil and column behaviour is assumed as elastic. The results only depend on the problem geometry and the relation between the elastic parameters of soil and column. Consequently, the results are independent of the load level and the stress paths followed. Balaam and Booker (1981) developed a numerical model to study this case and obtained the exact analytical solution of the initial (undrained) and final (drained) states. Their results were used to validate the numerical model and good agreement was reached. Only minor concerns about the undrained soil Poisson’s ratio and the time steps arose. The undrained Poisson’s ratio of the soil was chosen to be 0.499 because a value of 0.495 leads to minor differences in the initial state. The automatic time steps used by the code give a slightly slower consolidation than if the time steps are triggered to be smaller. However, the differences in both cases are not relevant for practical purposes. The numerical analysis allows an accurate evaluation of the total principal stresses and the excess pore pressure (Figure 3). A thorough analysis of these parameters and their evolution with time helps to understand the column behaviour and its interaction with the surrounding soil. Initially, the soil deforms in undrained condition, and hence, with a relatively high stiffness (constant volume). So, the vertical stress is higher on the soil than on the column and the soil pushes the column radially inwards, leading to higher hoop stresses than radial stresses in the soil. During consolidation, excess pore pressures start to be dissipated in the soil close to the column, which gradually reduces the volume of the soil and consequently its apparent stiffness also reduces towards its drained value. This causes a reduction of all the total principal stresses, especially of the total hoop stress. As consolidation progresses, the outer parts follow the same process that has already happened in the soil close to the column. Finally, there are not any excess pore pressures and the column s considerably more vertical stress than the soil because the column is stiffer and pushes the soil radially outwards, with higher radial stresses than hoop stresses in the soil. In Figure 3 (a), the vertical stress on the soil decreases monotonically but this may not happen for other cases. For instance, the vertical stress on the soil close to the column may quickly decrease and part of this decrease is later recovered for lower area replacement ratios (ar < 0.11). The Mandel-Cryer effect in the outer part of the surrounding soil is only observed for those cases where the column s little load. The settlement rate obtained with the numerical model is compared with different analytical solutions

Figure 3. Results of the elastic case: (a) Total vertical stress; (b) Total radial and hoop stresses and (c) Excess pore pressure.

for two usual cases (Figure 4). As it is visible, consolidation under constant load (Barron’s solution) is much slower than in the numerical analysis. On the contrary, the assumption of variable load with lateral confinement (Han & Ye 2001) results in a too fast process, particularly for consolidation rates above 40%. Another drawback is that immediate settlements are not considered. The solution by Castro & Sagaseta (2009), which assumes a variable load and includes the radial displacement of the soil/column interface, shows a better agreement with the numerical results. The

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Figure 5. Initial value condition in Barron’s solution.

Figure 4. Elastic settlement with time.

differences are greater for degrees of consolidation below 40%. However, this also happens with all the other approaches. Balaam & Booker (1981) explain that in Barron’s solution the distribution of the initial excess pore pressures is not uniform, as it should be (Figure 5). That assumption causes some differences that are particularly important in the initial part of the consolidation process. The same comment is also applicable to Han & Ye (2001) and Castro & Sagaseta (2009) because they use Barron’s solution as the reference solution. The differences between the real initial excess pore pressure, which is constant with radius, and the initial excess pore pressure assumed by Barron, which varies exponentially with radius, are very small for high diameter ratios (N > 20), which are common for vertical drains (Figure 5). However, stone columns have lower diameter ratios (N < 5) and then, the effects of Barron’s assumption get visible at the beginning of the consolidation process. The analytical solutions predict different settlement rates because they assume different distributions of the vertical load between soil and column. Figure 6 shows

Figure 6. Stress concentration factor with time.

the stress concentration factor, σzc /σzs , and its variation with time for the same two cases shown in Figure 4.The values predicted by Castro & Sagaseta (2009) agree very well with the numerical solution because lateral deformations are included. On the contrary, if the column is assumed to be laterally confined (Han & Ye 2001), the final SCF is overestimated. Similar results

866

Figure 8. Time-settlement curves for different cases.

Figure 7. Dissipation of pore pressure at different depths.

are obtained for the settlement reduction, as it depends directly on the SCF. The initial and final values of the SCF and the settlement reduction predicted by Castro & Sagaseta (2009) are equal to those of the exact solution by Balaam & Booker (1981) and therefore equal to those of the numerical analysis.

The rough upper boundary in the numerical model and the fact that the analytical model does not consider shear stresses cause only some subtle differences. Because a rigid plate is located on top of the unit cell, the surface settlement is uniform. The variation of the surface settlement with time is plotted in Figure 8. As mentioned above, Castro & Sagaseta (2009) agrees very well with the numerical analyses for degrees of consolidation higher than 40%.

5 YIELDING OF THE COLUMN The simplification of an elastic column can only be considered as a first rough estimation because plastic strains develop in the column even for low loads. Hence, the next step is to include an elastic-perfectly plastic behaviour of the column. As previously mentioned, the Mohr-Coulomb failure criterion and a non-associated flow rule are used for this purpose. An initial geostatic state (K0s , γs , γc ) is considered and the effects of column installation are neglected. As column yielding starts at the surface and it progresses downwards with time, now the results depend on the depth. For the sake of simplicity, the groundwater table is assumed to be at the ground surface. As an example of the numerical analyses that are still in progress, Figure 7 shows the dissipation of the excess pore pressure at different depths. Note that the excess pore pressure varies along the radius but its average value is used. These results confirm that the consolidation slows down when the column yields. As for the elastic case, the numerical results agree reasonably well with Castro & Sagaseta (2009) but for degrees of consolidation below 40%. The results are analyzed at different depths (z = 1, 5, 10 m) to highlight the differences between them. The depth of 1 m is used instead of the values at the surface (z = 0 m) because the upper rough slightly alters the values at the surface. The column yields as the excess pore pressures are dissipated. Plastic points start to appear at the surface for low degrees of consolidation and the whole column is at its active state at the end of consolidation.

6 YIELDING OF THE SURROUNDING SOIL Most of the theoretical analyses (Pulko & Majes 2005; Castro & Sagaseta 2009) justify that plastic strains are limited to the column and therefore the assumption of an elastic behaviour for the surrounding soil is valid. Finite element analyses of the unit cell demonstrate that this is true for usual conditions of the problem (Pulko & Majes 2005) but when the surrounding soil is very soft and the applied load is high, plastic strains appear in the soil close to the column (Tan et al. 2008). A preliminary result of the ongoing numerical analyses is that the differences between an elastic soil and an elastic-plastic one are important for very limiting cases, which are hardly ever issible. So, for distributed loads, a suitable stone column design should not cause a considerable yielding of the soil that surrounds the column, and therefore, an elastic behaviour of the soil is a reasonable assumption of the analytical models.

CONCLUSIONS Coupled finite element analyses of a “unit cell” were performed to study the consolidation around stone columns. The computed values of the most important parameters, such as the settlement reduction, the stress concentration factor and the dissipation of the excess pore pressure, agree very well with an analytical solution recently developed by the authors (Castro &

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Sagaseta 2009) for degrees of consolidation higher than 40%. The discrepancies in the first part of the consolidation process are inherent to the assumptions of Barron’s solution. When the column yields, dissipation of excess pore pressure slows down in a similar way to the analytical solution (Castro & Sagaseta 2009). The elastic behaviour of the soil assumed in most of the analytical solutions is a reasonable hypothesis for common cases. ACKNOWLEDGEMENTS The work presented was part of a research project on stone columns for the Spanish Ministry of Public Works (Ref.: 03-A634). The first author received also a Grant from the Spanish Ministry of Education (Ref.: AP2005-195). REFERENCES Balaam, N.P. & Booker, J.R. 1981. Analysis of rigid rafts ed by granular piles. Int. Journal for Numerical and Analytical Methods in Geomechanics 5: 379–403. Barksdale, R.T. & Bachus, R.C. 1983. Design and construction of stone columns. Report FHWA/RD-83/026. Springfield: Nat. Tech. Information Service. Barron, R.A. 1948. Consolidation of fine-grained soils by drain wells. Transactions ASCE 113: 718–742. Brinkgreve, R.B.J. 2007. Plaxis finite element code for soil and rock analysis, 2D, version 8. Rotterdam: Balkema.

Castro, J. & Sagaseta, C. 2009. Consolidation around stone columns. Influence of column deformation. Int. Journal for Numerical and Analytical Methods in Geomechanics 33: 851–877. Elshazly, H. Elkasabgy, M. & Elleboudy, A. 2008. Effect of inter-column spacing on soil stresses due to vibro-installed stone columns: interesting findings. Geotechnical and Geological Engineering 26: 225–236. Han, J. & Ye, S.L. 2001. A simplified solution for the consolidation rate of stone column reinforced foundations. Journal of Geotechnical and Geoenvironmental Engineering 127(7): 597–603. Pulko, B. & Majes, B. 2005. Simple and accurate prediction of settlements of stone column reinforced soil. 16th Int. Conf. on Soil Mechanics and Foundation Eng.: 1401–1404. Rotterdam: Millpress. Schweiger, H.F. 1989. Finite element analysis of stone column reinforced foundations. PhD Thesis, University of Wales, Swansea. Tan, S.A. Tjahyono, S. & Oo, K.K. 2008. Simplified planestrain modeling of stone-column reinforced ground. Journal of Geotechnical and Geoenvironmental Engineering 134(2): 185–194. Van Impe, W.F. & De Beer, E. 1983. Improvement of settlement behaviour of soft layers by means of stone columns. 8th Int. Conf. on Soil Mechanics and Foundation Eng.: 309–312. Rotterdam: Balkema. Weber, T.M. Springman, S.M. Gäb, M. Racansky, V. & Schweiger, H.F. 2008. Numerical modelling of stone columns in soft clay under an embankment. In Karstunen & Leoni (eds), Geotechnics of Soft Soils-Focus on Ground Improvement: 305–311. London: Taylor & Francis.

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Numerical modeling of inertial soil-inclusion interaction X. Zhang, Ph. Gotteland & P. Foray L3S-R, UJF-INPG-CNRS, Grenoble Universités, Grenoble,

S. Lambert & A. Hatem Keller Fondations Spéciales, KELLER-, Duttlenheim,

ABSTRACT: The ground reinforcement by Mixed Module Columns allows a high increase in bearing capacity and a reduction in settlement. With its high flexibility, the upper part of the Mixed Module Columns (CMM) can absorb most of seismic energy. Numerical modeling was conducted on a shallow foundation lying on a soft clay reinforced by four CMM subjected to static and dynamic horizontal cyclic loading. The numerical results indicate that strength of CMM decreases with increasing height of stone columns. The inertial effect is well demonstrated in the dynamic analyses in comparison to the static ones.

1

INTRODUCTION

Vertical Rigid Inclusions network associated to a granular layer (IR) is widespread in as a means of ground reinforcement for soft soil. Less used but also highly efficient as ground improvement system, Mixed Module Columns (CMM) present significant advantages for the construction in seismic areas. A short Stone Column in the upper part with the length usually less than 2 m is associated to a rigid inclusion in down part of CMM (Bustamante et al. 2006). Both CMM and IR systems allow a high increase in bearing capacity and a reduction in settlement. However, CMM have the advantage that the upper part of the system which is more flexible works as a plastic hinge and less seismic energy is transmitted upwards and downwards to and from the superstructure. Many studies have been carried out on vertically loaded shallow foundations lying on soft soil reinforced by vertical rigid inclusions and on vertically and laterally loaded pile foundations (Chenaf 2006, Georgiadis et al, 1992, Li & Byrne 1992, Remaud 1999, Rosquoët et al, 2007). However, little research has been performed on the CMM reinforcement and the IR systems in seismic areas. In Laboratory 3SR, earlier experimental work has been performed in a large visualization tank (Figure 1) in order to analyse the mechanisms of the CMM and the IR system under horizontal loadings applied to the foundation (Zhang et al, 2010). Two dimensional reduced models were examined in these experiments under both quasi-static and dynamical horizontal cyclic loadings. The reduced physical models consisted of a 20 cm wide and 2 cm thick square footing made from aluminium alloy AU4G, lying on soft clay reinforced by the CMM. CMM were modeled by two stone columns with a rectangular section (20 cm*9 cm) for the upper

Figure 1. Photo of two-dimensional CMM experimental model.

part and two pieces of aluminium plate with a rectangular section (20 cm*0.3 cm) for the lower part (rigid inclusions) instead of four cylindrical columns for both parts in real foundation. Between the upper part and the lower part, two PVC plates with the same rectangular section as the stone columns were installed horizontally in order to the gravel and also to simulate the transition zones in real CMM. It was confirmed that upper part of CMM absorbed more energy than that of the IR system. Horizontal displacement of heads of the rigid inclusions was observed in the dynamical tests due to the inertial effect. While no horizontal displacement of heads of the rigid inclusions was observed during the static tests. With mechanism and behavior clarified by the experiments, this paper presents a numerical study on the CMM system. Numerical modeling was carried out on a four CMM system under a square footing. A dynamical horizontal cyclic loading was applied to

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the square footing with a nominal static vertical loading after the maximum static vertical and horizontal loading had been determined. The length of the upper part of the CMM varied to illustrate its influence on the response of the rigid inclusions. 2 THE COMPUTER PROGRAM The computer program used is FLAC3D (Fast Lagrangian Analysis of Continua in 3 Dimensions). It is a three-dimensional explicit finite difference program for engineering mechanics computation designed by Itasca Consulting Group Inc. It simulates the behavior of three dimensional structures built of soil, rock or other materials that undergo plastic flow when their yield limits are reached. The dynamic analysis option permits to resolve the full equations of motion, using the fully nonlinear method embodied in FLAC3D , rather than the “equivalent-linear” method which is commonly used in earthquake engineering for modeling wave transmission in layered sites and dynamic soil-structure interaction. The fully nonlinear method follows any prescribed nonlinear constitutive relation, and irreversible displacements and other permanent changes are modeled automatically. 3 3.1

NUMERICAL MODELS OF THE FOUR CMM SYSTEM

Figure 2. The configuration and the dimensions of the numerical models (cm).

Figure 3. The grid of the numerical model of the CMM system.

Presentation

The studied foundation system consists of a square footing 2 m wide and 0.5 m thick. It is totally embedded in soil. Four (2*2) CMM are placed in the soil under the footing. The upper part of the CMM is a stone column with 0.9 m diameter and varying length (0.3 m, 1.0 m and 1.5 m). The lower part of the CMM is a rigid inclusion made of plain concrete with 0.34 m diameter, and the length of the rigid inclusion is 5 m. Between the upper and the lower part is an area called “transition zone” which has the same diameter as the stone column and a length of 0.5 m. The transition zone is designed to better transmit vertical loading to the lower part and consists of a mixture of concrete and gravel. The axial distance between the CMM is 1.2 m, so it is observed that the area of the CMM exceeds slightly the square footing. Two soil layers were taken into in the numerical modeling. A soft clay layer and a more resistant gravel layer to obtain the embedment of the rigid inclusions. The dimensions of the different parts of the foundation system are illustrated in Figure 2. 3.2 Numerical modeling Within the numerical models, system composed of footing, stone columns with the “transition zones” and soil media was discretized using predefined 6-node radial cylinder elements and 8-node brick elements (Figure 3). In fact, finer meshes could lead to more

accurate results because they provide a better representation of high-stress gradients. Numerical studies by FLAC3D have been found on piles that the finer meshes hardly improve the results.The mesh employed here is rather coarse to find a compromise between the accuracy and the calculation efficiency. While the rigid inclusions were modeled by threedimensional pile elements and each rigid inclusion was discretized in ten pile elements. In FLAC3D , in addition to providing structural behavior of a beam, both a normal-directed and a shear-directed frictional interaction occur between the pile and the grid. Each pile structural element is defined by its geometry, material and coupling-spring properties. A pile element is assumed to be a straight segment of uniform, bisymmetrical cross-sectional properties lying between two nodes. For the heads of the rigid inclusions, the nodes of the pile element were linked rigidly to the “transition zones” in the three displacement directions (no relative displacement between the grid and the node) and free in the three rotational directions. To form the embedment of rigid inclusions in gravel layer, the links between the pile element nodes and the gravel layer were set rigid in all the degrees of freedom. Behavior of stone columns, soft clay and gravel layer was described by an elastoplastic constitutive model based on the non-associated Mohr-Coulomb criterion. The Mohr-Coulomb model is the simplest elastoplastic constitutive law which presents quite

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Table 1. The material parameters introduced in the numerical models. Young’s Friction Modulus Poisson’s angle Cohesion ◦ MPa ratio kPa Stone columns 60 Clay layer 6 Gravel layer 100 Transition zones 600 Rigid inclusions 5285* Footing 10000

0.3 0.3 0.3 0.3 0.2 0.2

38 0 45

0 20 0

* Edyn = 15855 MPa in the dynamic analyses (Edyn = 3*Estat ).

satisfactory results with not many soil parameters required. Linear elastic model was applied to the footing, “transition zones” and rigid inclusions. The conditions between the foundation and the soil were simulated by interface elements of Mohr Coulomb type. The properties of the interface elements were determined according to the type of soil which they were in with. The same type of interface between the piles and the soil were used via the pile elements. The cohesion and the friction angle of the concrete-clay interface were respectively 0 kPa and 38◦ , while the values were changed to 20 kpa and 0◦ for the concrete-gravel interface. All the material parameters are summed up in Table1. It is noted that the Young’ Modulus of the rigid inclusions is twice as high under dynamic loading than under the static loading (Edyn = 3*Estat ), which is commonly used in geotechnical engineering. The numerical modeling was carried out in three stages. First of all, a vertical loading was applied to the footing until the soil collapsed to obtain the bearing capacity of the foundation. Then, a horizontal loading was applied to the foundation coupled to a nominal vertical loading. The maximum horizontal loading was determined when the footing tended to slide. Finally, a dynamical horizontal cyclic loading was applied with the same vertical loading as in the second stage. In the static analyses, only one half of the soilfoundation system was modeled thanks to the symmetry in order to optimize the calculation duration. While for the dynamic analyses, the whole system was required because of the generation of proper boundary conditions. The lateral boundaries of the main grid were coupled to a free-field grid (Figure 3) by viscous dashpots to simulate a quiet boundary and to ensure wave transmission through the lateral boundaries of the soil mass. The procedure of free-field boundaries used in FLAC3D aims to absorb outward waves originating from the structure. The method involves the execution of free-field calculations in parallel with the main-grid analysis. This procedure uses only the p-waves and s-wave speeds at the side boundary. The damping was considered here using the local damping defined by FLAC3D which is less time-consuming than Rayleigh damping and gives good results in the simple cases, because it is frequency-independent and

Figure 4. Static bearing capacity of the foundation.

needs no estimate of the natural frequency of the system being modeled. The values of damping ratio were 0.05 for the soil and 0.02 for the footing and the rigid inclusions. The dynamic analyses in FLAC3D are very timeconsuming, especially when the contrast of stiffnesses between the materials is large as this case. It is induced by the very small values of critical timestep. The critical timestep is defined such that elastic waves could propagate through the smallest dimension of all the elements and it is given by:

where is the p-wave speed, V is the volume of the f element, and Amax is the maximum face area associated with the element. The min{} function is taken over all zones and includes contributions from the structural and interface modules. A safety factor of 2 is used, because Equation 1 is only an estimate of the critical time step. Hence, the time step used for dynamic procedures is:

Here the value of the critical time step is about 4µs. 4

BEARING CAPACITY OF THE FOUNDATION

In the first calculation stage, in order to investigate the ultimate vertical bearing capacity, a static vertical loading was applied to the footing by means of a very slow continuous displacement. Conventionally, the ultimate vertical loading is defined when the settlement of the foundation reaches 10% of the foundation width, which is 20cm here. From the vertical forcesettlement plots shown in Figure 4, it can be seen that the bearing capacity increases with the decreasing length of the stone columns.

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Figure 6. The horizontal loading-sliding curves. Figure 5. The axial forces in the rigid inclusions (Q = 320 kN).

Figure 5 shows the axial forces in the rigid inclusions when the footing was loaded vertically with a nominal vertical loading determined by the maximum force of the 1.5 m long CMM system with a safety factor of 3, i.e. 320 kN. The length 0 in the figure corresponds to the heads of the rigid inclusions. Because of the symmetry, only one rigid inclusion for each case is studied here. The CMM system with 0.3 m long stone columns has the bearing capacity much higher than that of the two others. The length of 0.3 m long stone columns is so small that their behavior is quite similar to a pile foundation. While for the two others, the stone columns are longer and almost all the settlement takes place inside the soft clay layer and the stone columns around. Due to the different settlements between the stone columns and the soft clay, negative friction occurs on the upper part of the rigid inclusions. In these cases, the axial forces in the rigid inclusions increase till a neutral point, then decrease all along to the base. It was observed that the axial forces in the rigid inclusions decreased with the length increase of the stone columns.

5

HORIZONTAL RESPONSE OF THE CMM SYSTEM

At the second calculation stage, with nominal vertical force defined previously (Q = 320 kN), the footing was subjected to a horizontal loading by means of a very slow continuous displacement. It can be observed from the horizontal load-sliding plots (Figure 6) that the footing did not slide much until reached the shear capacity, but once it reached the shear capacity it started to slide. The shear capacity of the CMM system with 0.3 m long stone columns is much higher than the two others which have almost the same shear capacity. The maximum horizontal displacement before sliding was between 10 mm and 15 mm. The nominal horizontal loading was defined when the displacement

attained 4 mm which corresponded to a horizontal force of about 130 kN. Figure 7 presents internal forces in the rigid inclusions under the nominal vertical and horizontal loadings. The maximum shear forces were obtained at the heads of the rigid inclusions, while the bending moments attained the maximum values about 1m below. It can be seen that all the internal forces have a decreasing tendency when the length of the stone columns in the CMM system increases. It is also observed that the internal forces in the front rigid inclusion are larger than those in the behind one, which agrees well with the pile-group effect. 6

DYNAMIC ANALYSES OF THE CMM SYSTEM

The dynamic analyses were performed under a horizontal cyclic loading with a sinusoidal form. The amplitude of the loading is 4 mm which was determined beforehand by the static horizontal response in order to compare their results. The dynamic loading has a frequency of 1Hz and lasted 10 seconds. The horizontal displacement was applied to the footing by means of the velocity imposed through all the dynamic time steps. During the dynamic loading, after a transition stage, no changes were observed in the response of rigid inclusions from the 3rd cycle to the 10th cycle because the soil-footing-CMM stabilization has been obtained. The envelopes of the internal forces in the rigid inclusions during the dynamic loading are illustrated in Figure 8. Because of the symmetry of the dynamic loading, only one of the rigid inclusions is studied here. Note that there is a phase difference of the responses between the front and the behind rigid inclusions. Due to the inertial effect, the response of the rigid inclusions in the dynamic analyses is more evident than in the static calculation. This agrees well with the observation during the 2D dynamical experiments in the visualization tank (Figure 1). The movement of the heads of the rigid inclusions was quite evident, while it didn’t exist at all in the static ones. That means the

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Figure 7. The response of the rigid inclusions under the static horizontal loading, 7a) Axial forces, 7b) Shear forces, 7c) Bending moments.

rigid inclusions were much more loaded in the dynamical tests, which corresponds to the numerical results. The maximum bending moment here is almost twice larger than the static case.

Figure 8. The envelopes of the internal forces in the rigid inclusions, 8a) Axial forces, 8b) Shear forces, 8c) Bending moments.

Finally, the same tendency with respect to the variation of the length of the stone columns is examined: longer are the stone columns, lower is the strength reponse in rigid inclusions.

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7

CONCLUSIONS

Numerical modeling of ground reinforcement by Mixed Module Columns has been performed following the two-dimensional experimental research work carried out previously. It has been clarified that the length of the stone columns had an important influence on the response of the rigid inclusions. The longer the stone column is, the less the rigid inclusion is loaded. The comparison of the results between the static analysis and the dynamic one showed that there was an important inertial effect on the internal forces of the rigid inclusions. In a further research program, physical models in three dimensions will be built to calibrate the numerical models. REFERENCES Bustamante, M., Blondeau, F. & Aguado, P. 2006. Cahier des charges Colonnes à Module Mixte. KELLER Fondations Spéciales.

Chenaf, N. 2006. Interaction inertielle et interaction cinématique PhD Dissertation, Ecole Centrale de Nantes. Hatem, A., Shahrour, I., Lambert, S. & Alsaleh, H. 2009. Analyse du comportement sismique des sols renforcés par des inclusions rigides et par des colonnes à module mixte. AUGC. Li,Y. & Byrne, P. M. 1992. Lateral pile response to monotonic head loading. Canadian Geotechnical Journal. No. 29, pp. 955–970. Georgiadis, M., Anagnostopoulos, C. & Saflekou, S. 1992. Centrifugal testing of laterally loaded piles. Canadian Geotechnical Journal. No. 29, pp. 208–216. Remaud, D. 1999. Pieux sous charges laterals: étude expérimentale de l’effet de group” PhD Dissertation, Ecole Centrale de Nantes. Rosquoët, F., Thorel, L., Garnier, J. & Canepa, Y. 2007. Lateral cyclic loading of sand-installed piles. Soils & Foundations. Vol. 47, no. 5, pp. 821–832. Zhang, X., Foray, P., Gotteland, Ph., Lambert, S. &Alsaleh, H. 2010. Seismic performance of mixed module columns and rigid inclusions. 7th International Conference on Physical Modelling in Geotechnics (IMG 2010), 8p.

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Performance of geogrid-encased stone columns as a reinforcement of soft ground M. Elsawy, K. Lesny & W. Richwien University of Duisburg-Essen, Essen,

ABSTRACT: Stone columns are a very effective technique for improvement of soft soils particularly under flexible structures. Stone columns are generally used to increase the bearing capacity which depends on the lateral . To avoid dispersion of the stones into the clay and to improve the stone columns as reinforcing elements, geogrids are used as an encasement of the stone columns. In this research the behavior of full scale stone columns in Bremerhaven clay has been analyzed using the FE program Plaxis. The stone columns are loaded under undrained and drained conditions to investigate the effect of varying parameters like the geogrid stiffness and the depth of the encasement on the behavior of the stone columns in short and long term conditions. When using geogrid as encasement for the stone column, an important increase in its bearing capacity as well as a significant reduction in the lateral bulging occurs. More improvement occurs in the behavior of the encased stone columns with increasing encasement stiffness. The bearing capacity of the partially encased stone columns increases with increasing encasement depth. The increase in the bearing capacity in long term is more significant than that in short term conditions under working loads. 1

INTRODUCTION

A lot of areas all over the world, particularly along the rivers and the seas, are covered with thick soft alluvial and marine clay. As increasing developments on these areas recently, a lot of buildings and industry structures are being constructed. Construction on soft natural soil is considered a risk and poses major problems to geotechnical engineers due to its low shear strength and high compressibility. Ground reinforcement by stone columns solves theses problems by providing advantage of reduced settlement and accelerated consolidation process. Another advantage of this method is the simplicity of its construction. The stone columns derive their load carrying capacity from the ive earth pressure resistance developed against the bulging of the column which thereby depending on the shear strength of the surrounding soil. The stone column technique was adopted in the European countries in the early 1960s and thereafter it has been used successfully. Several researches were published in the past three decades which dealt with the stone column technique (Balaam & Booker 1985; Lee & Pande 1998; Wood et al. 2000; Christoulas et al. 2000). Bergado & Long (1994) proved from field measurements and numerical studies that the installation of stone columns in soft soil increases the bearing capacity and accelerate the consolidation.They also reported that stone columns imply more reductions in the total settlement of the soft clay when compared with vertical drains. Ambily & Gandhi (2007) stated that when the column area only loaded, failure occurs by bulging.

Further developments of the stone column technique include the reinforcement of the column using either horizontal layers of reinforcement (Sharma et al. 2004) or encasing the individual stone column by geosynthetics (Nabil 1995; Murugesan & Rajagopal 2006). The geosynthetic encasement leads to more increases in the load bearing capacity of the stone column and reduces its bulging due to the additional confinement from the encasement (Malarvizh & Ilamparuth, 2007). The geosynthetic encasement also prevents the lateral squeezing of stones when the stone column is installed in some extremely soft soil, leading to minimal loss of stones and quicker installation. The published literature on the performance of the encased stone columns is limited especially in the long term conditions. Most of researchers also used the geotextile encasement material in spite of the geogrid encasement has the more stiffness. The scope of this study is to understand the behavior of the stone columns in soft soil and to extract the parameters which play a dominant role in the bearing capacity increase and in the settlement reduction in short and long term conditions. Based on this objective, the influence of the parameters such as the geogrid encasement, the encasement stiffness and the encasement depth is analyzed. 2

FINITE ELEMENT SIMULATIONS

In order to make a realistic modeling of the behavior of the geogrid reinforced stone column-soft soil system, full scale stone columns in Bremerhaven clay

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Table 1. Properties and parameters of the stone column and the soil.

Table 2.

Properties of the geogrid materials.

Parameter

Stone column

Bremerhaven clay

Property

Model conditions γwet (kN/m3 ) E (kN/m2 ) υ(-) λ∗ (-) κ∗ (-) µ∗ (-) c(kN/m2 ) ϕ◦ ψ◦

Mohr coulomb drained 19 55,000 0.3 0 43 10

Soft soil creep undrained and drained 15 0.203 0.025 0,007 5 37,75 0

Mass per unit area (g/m2 ) Axial stiffness, J (kN/m) Aperture size (mm × mm)

Secugrid 20/20 Q1

Secugrid 30/30 Q1

Combigrid 40/40 Q1

155

200

240

400

600

800

33 × 33

32 × 32

31 × 31

stone material which has a bad effect on the drainage of the stone column. The geotextile is arranged in such a way that it would not contribute either to the vertical or the lateral stiffness of the encased stone column. The geogrid encasement is modeled as a linear elastic continuum element. The properties of geogrid materials are tabulated in Table 2.

3

PARAMETRIC STUDY

In order to evaluate the improvement achieved due to the encasement, the following cases have been analyzed. 1. Ordinary stone columns (OSC) installed in soft soil. 2. Geogrid-encased stone columns (ESC) installed in soft soil.

Figure 1. Model of the unit cell (a) Model parts, (b) FEM mesh.

are analyzed. The Mohr Coulomb model is used for the stone column material and the Soft Soil Creep model is used to describe the behavior of the Bremerhaven clay. The finite element program Plaxis 9 has been used for the FE analyses. The properties of the stone column materials and the Bremerhaven clay were adopted from the study of Ambily & Gandhi (2007) and the study of Geduhn (2005), respectively. The properties of these soils are tabulated in Table 1. The stone columns are installed in a square pattern into the soft soil. The “unit cell” analysis has been conducted for a column and the surrounding soft soil using axisymmetric conditions, as illustrated in Figure 1. Half of the model has been used. A medium finite element mesh has been used with 15 nodes triangular elements. Three types of geogrid reinforcement with different stiffness are used as encasement for the stone column, Secugrid 20/20 Q1, Secugrid 30/30 Q1 and Combigrid 40/40 Q1 (Naue GmbH). The last type is a composite of geogrid/nonwoven geotextile. The geotextile is used mainly to prevent the mixing of the clay grains with the

Initially, the analyses have been performed by applying uniform load (q, kPa) on the stone column portion only in undrained and drained condition of the surrounding soil in order to directly assess the influence of the confinement effects due to encasement. Detailed parametric analyses were performed by varying stiffness of the geogrid encasement (J) and depth of the encasement from the top of the column (h). All cases have been idealised through the axisymmetric modelling. The improved performance has been evaluated based on the increase of the bearing capacity and the reduction in the lateral bulging of the stone column. The foundation soil in all cases is assumed to be a 6 m thick Bremerhaven clay layer underlain by rigid hard stratum. In all cases also, the ordinary and the encased stone columns have a diameter of d = 0.6 m and a spacing ratio of S/d = 2. 4


4.1 Effect of Encasement of Stone Column and Influence of Encasement Stiffness (J) 4.1.1 Undrained conditions When the stone column is encased with geogrid materials, a significant increase in the bearing capacity occurs. This is due to the increase of the column confinement with geogrid materials. The encasement materials provide also a stronger lateral by generating radial tension forces. There is a further

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Figure 4. Effect of the encasement stiffness on the tension forces of the encasement. Figure 2. Effect of the encasement stiffness on the bearing capacity of the stone column.

Figure 3. Effect of the encasement stiffness on the lateral bulging of the stone column.

increase in the bearing capacity of the stone column with increasing geogrid stiffness, as shown in Figure 2. The stone column confinement increases with increasing geogrid stiffness which leads to an increase of the overall stiffness of the encased stone columns. The lateral bulging of the stone column and the hoop tension forces in the encasement were calculated at a column load of 180 kPa in undrained conditions. The load of 180 kPa is the minimum failure load of the ordinary stone columns. Figure 3 shows a significant reduction in the lateral bulging of the stone column with encasement. The lateral bulging decreases and the confinement increases with increasing geogrid stiffness. It is observed that in the ordinary stone columns lateral bulging occurs only near the ground surface up to a depth equal to twice the diameter of the stone column. The forces are high within a depth equal to almost twice the diameter of the stone column, as illustrated in Figure 4. Then; the forces reduce gradually with depth to reach zero values below a depth equal to 4 times the column diameter. 4.1.2 Drained conditions When the stone column is encased with geogrid materials and loaded until failure in drained conditions, a significant increase in the bearing capacity occurs.

Figure 5. Effect of the encasement stiffness on the bearing capacity of the stone column.

The bearing capacity of the encased stone column has further increases with increasing geogrid stiffness especially at the higher loads, as shown in Figure 5.The load-settlement relationship of the encased stone column is approximately linear and no yield point occurs. When consolidation occurs, the encasement provides a stronger lateral for the stone column. The confinement of the stone column increases with increasing geogrid stiffness which leads to an increase in the overall stiffness of the encased stone columns, too. The lateral bulging of the stone column and the hoop tension forces in the encasement were calculated at a column load of 300 kPa in drained conditions. The load of 300 kPa is the minimum failure load of the ordinary stone columns. Reductions also occur in the lateral bulging when using encasement, as shown in Figure 6. These are due to the confinement from the encasement of stone column which provides a stronger lateral than that in the ordinary stone column. It is observed that in the ordinary stone columns lateral bulging values along the column occur. The encasement makes the lateral bulging distribution along the column depth more organized due to more stress transfer within lower depths. The lateral bulging of

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Figure 6. Effect of the encasement stiffness on the lateral bulging of the stone column.

Figure 7. Effect of the encasement stiffness on the tension forces of the encasement.

the encased stone column decreases with increasing geogrid stiffness values. Figure 7 shows the radial hoop tension forces developed in the geogrid encasement. The hoop tension forces distribute along the encased stone column. The forces distribution is similar to that for the lateral bulging of the stone column. The tension forces increase with increasing geogrid encasement stiffness. The degree of improvement in the encased stone column-soft soil foundation behavior increases with increasing generation of the tension forces in the encasement material. In comparison with the undrained conditions, the hoop tension forces in drained conditions extend to lower depths due to the stress transfer within downward direction.

4.2

Effect of geogrid encasement depth (h)

The encased stone columns with various encasement depths have been loaded in undrained and drained conditions. The analyses have been carried out using stone columns with a diameter of 0.6 m, a spacing ratio S/d = 2 and a geogrid stiffness of 800 kN/m. The encasement depth to column diameter ratios of h/d = 1, 2, 3, 4, 6, 8 and 10 have been used.

Figure 8. Effect of the encasement depth on the bearing capacity of the stone column.

4.2.1 Undrained condition The bearing capacity of the encased stone columns increases with increasing encasement depth, as shown in Figure 8. The increase in the bearing capacity is more at the higher loads. The highest bearing capacity occurs when the fully encased stone column is used. Loads have been applied on the partially and the fully encased stone column up to a load of 300 kPa which acts as a working load.The encasement beyond a depth equal to twice the diameter of the column doesn’t lead to further improvement in the bearing capacity of the stone columns, as shown in Figure 8. Similar results were stated by Murugesan and Rajagopal (2006). The lateral bulging of the partly and the fully encased stone columns is also investigated. The lateral bulging decreases with increasing depth of the encasement. It is well established from Figure 9 that the bulging of the encased stone column is predominant up to a depth equal to 2–2.5 times the diameter of the column. Hence, the partly encasement of the stone columns with a depth of 3 times the diameter of the column is sufficient to reduce the bulging of the column to minimum values and to provide the required confinement of the column. When the stone columns are reinforced by encasement depth smaller than h/d = 3, larger lateral bulging values of the column occur at the end point of the encasement. Largely differentially lateral displacements are also generated at the encasement end. This phenomenon is clear especially when using encasement depth of h/d = 1. Figure 10 shows the hoop tension forces distribution along the column for various encasement depths. The development of the hoop tension forces looks like that of the column bulging. The distribution of the hoop tension forces is the same for depths larger than h/d = 3. At encasement depths shallower than h/d = 3, there are peak values in the hoop tension forces at the end point of the encasement where there is a largely differentially lateral bulging. The encased stone column with encasement depth of h/d = 1 implies the highest peak value of tension forces at the end point of the encasement, as shown in Figure 10. Because the upper

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Figure 9. Effect of the encasement depth on the lateral bulging of the stone column under a column load of 300 kPa.

Figure 10. Effect of the encasement depth on the hope tension force of the encasement under a column load of 300 kPa.

zone of the stone column is the more loaded zone. Hence, the upper zone needs to be confined. 4.2.2 Drained conditions The partially and the fully encased stone columns have been also studied in drained conditions. When using encasement depth ratio of h/d = 1, the stone columns have a large increase in the bearing capacity. The increase in the bearing capacity of stone columns continues with increasing encasement depth, h/d. The rate of the increase is significant at higher loads, as shown in Figure 11. Loads have been applied on the partially and the fully encased stone columns up to a load of 300 kPa which is a working load. The bearing capacity of the encased column increases also with increasing encasement depth. It was well established that the lateral bulging is distributed along the stone column when it is loaded in drained conditions, as shown in Figure 12. Hence, the encasement is required to a depth that equals the depth of the stone column. The bulging reduces to minimum values in all the column depth when the column is encased completely, h/d = 10. When the stone column is partially encased, its bulging in the encased zone is slightly smaller than that of the full encased column case, while the non-encased zone has so higher values of the column bulging. The non-encased zone in the column starts with a maximum value which generates

Figure 11. Effect of the encasement depth on the bearing capacity of the stone column.

Figure 12. Effect of the encasement depth on the lateral bulging of the stone column under a column load of 300 kPa.

a largely differential bulging at the end point of the encasement. Below the end point of the encasement, the bulging values decrease gradually with depth until it reaches zero at the column base. The shallower the encasement depth is, the higher the bulging values are in the non-encased zone of the stone column. The full encasement induces values of hoop tension forces along the column and the encasement has also the larger tension forces in comparison with undrained conditions. The distribution of the tension forces is similar to that of the stone column bulging. When the stone column is reinforced by partially encasement, the tension forces are implied in the encased part of the column. The tension forces in the partial encasement of the stone column are smaller than those of the full encasement. While the end point of the partial encasement has a peak value of tension force which is so larger than that of the full encasement at the same location, as shown in Figure 13. Because the end point of the partial encasement is free and is subjected to lateral stress from the stone column where there is a largely differentially lateral displacement. The shallower the encasement is, the higher the peak value of the tension force at the end point of the encasement is.

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column implies increases with increasing depth of the encasement. The deeper the encasement of the stone column is, the more the bearing capacity is and the smaller the lateral bulging is. Therefore, full encasement of the stone column leads to the optimum performance of the encased stone columns. REFERENCES

Figure 13. Effect of the encasement depth on the hope tension force of the encasement under a column load of 300 kPa.

5

CONCLUSIONS

The concept of using geogrid encasement to provide a stronger lateral to stone columns installed in weak soil is considered relatively new. In this research, the performance of the fully and the partially encased stone columns with geogrid material was studied. The results obtained from this study showed that, the bulging of the stone column disappears below a depth equal two times the column diameter in undrained conditions. While the bulging implies values along the column in drained conditions due to the stress transfer. The load capacity and the stiffness of the stone column increase by geogrid encasement. When the stone columns are encased, they are confined and the lateral bulging is minimized. The geogrid stiffness plays an important role in enhancing the bearing capacity and the stiffness of the encased column. The stiffer the geogrid is, the higher is the load capacity of the column and the smaller is the lateral bulging. The bearing capacity of the partially encased stone column increases with increasing encasement depth in short and long term conditions. When the work load of 300 kPa is applied on the encased stone column in undrained conditions, the increase of the bearing capacity beyond an encasement depth that equals three times the column diameter is not significant. Therefore, the encasement depth of three times the column diameter is sufficient to minimize the values of the lateral bulging of the stone column. When the work load of 300 kPa is applied in drained conditions, the bearing capacity of the encased stone

Ambily,A.P. & Gandhi, S.G. 2007. Behavior of stone columns based on experimental and FEM analysis. Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 133, No. 4, pp. 405–415. Balaam, N.P. & Booker, I.R. 1985. Effect of stone column yield on settlement of rigid foundations in stabilized Clay. International Journal for Numerical and Analytical Methods in Ceomechanics 9, pp. 331–351. Bergado, D.T. & Long, P.V. 1994. Numerical analysis of embankment on subsiding ground improved by vertical drains and granular piles. Proceedings of the XIII ICSMFE, 1994. New Delhi, India, pp. 1361–1366. Christoulas, ST., Bouckovalas, G. & Giannaros, CH. 2000. An experimental study on model stone columns. Soils and Foundations Journal 40, No. 6, pp. 11–22. Geduhn, M. 2005. Geokunststoffummantelte Vacuumsäulen: Ein Gründungsverfahren für sehr weiche bindige Böden. PhD. Thesis of Duisburg-Essen University, Essen, . Gniel, J. & Bouazza, A. (2009). Numerical modelling of small-scale geogrid encased sand column tests. Geotechnics of Soft Soils-Focus on Ground ImprovementKarstunen & Leoni (eds). Taylor & Francis Group, London. Lee, J.S. & Pande, G.N. 1998. Analysis of stone-column reinforced foundations. International Journal for Numerical and Analytical Methods in Geomechanics 22, pp. 1001–1020. Malarvizhi, S. N. & Ilamparuthi, K. (2007). Comparative study on the behavior of encased stone column and conventional stone column. Soils and Foundations Journal 47, No. 5, 873–885. Murugesan, S. & Rajagopal, S. 2006. Geosynthetic-encased stone columns: numerical evaluation. Geotextiles and Geomembranes Journal 24, pp. 349–358. Nabil, M.A. 1995. Laboratory and analytical investigation of sleeve reinforced stone columns. Ph.D Thesis of Carleton University, Ottawa, Canada. Sharma, R.S., Kumar, B.P. & Nagendra, G. 2004. Compressive load response of granular piles reinforced with geogrids. Canadian Geotechnical Journal 41, pp. 187–192. Wood, D., Hu, W. & Nash, D.F.T. 2000. Group effects in stone column foundations model tests. Géotechnique Journal 50, No. 6, pp. 689–698.

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Offshore geotechnical engineering


A new elasto-plastic spring element for cyclic loading of piles using the p-y-curve concept Ole Hededal & Rasmus Klinkvort Department of Civil Engineering, Technical University of Denmark

ABSTRACT: Modeling the response of large diameter piles subjected to lateral loading is most often done by means of p-y-curves in combination with Winkler beam models. Traditionally the p-y curves are formulated as non-linear (elastic) relations between the lateral movement y and the soil response pressure p in of monotonic loading (until failure) as e.g. prescribed by API (2000). However, the cyclic and dynamic performance is only to a limited degree ed for. Here the elasto-plastic framework is applied allowing definition of unloading-reloading branches, hence enabling modeling of cyclic response. The present model can for effects like pre-consolidation and creation of gaps between pile and soil at reversed loading. Results indicate that the model is able to capture hysteresis during loading with full cycles and model the accumulated displacement observed on piles subjected to “half cycles” as e.g. seen from centrifuge tests carried out. This article presents the theoretical formulations, discusses numerical implementation and finally presents simulations.

1

INTRODUCTION

Modeling the response of large diameter piles subjected to lateral loading is most often done by means of p-y-curves in combination with Winkler beam models. Traditionally, the p-y curves are formulated in of non-linear (elastic) relations between the lateral movement y and the soil response pressure p in of monotonic loading (until failure). These curves were established by back-analysis of a series of tests carried out in the 1950es by Matlock and co-workers. The tests were primarily static, monotonic load tests, but also a few cyclic tests were carried out. Matlock (1970) carried out further cyclic tests on piles in clay that revealed a general reduction of the ultimate capacity for piles subjected cyclic loading compared to monotonic loading. This led to a general reduction of the cyclic ultimate capacity compared to the monotonic ultimate capacity. This reduction or cyclic degradation as it is commonly denoted is incorporated in almost all design codes, e.g. API (2000), as a formal reduction of the ultimate capacity. Still, the models does not directly correlate the reduction to the characteristics of the cyclic loading, i.e. number of cycles, loading amplitude or frequency. Matlock (1970) and later Mayoral et al. (2005) set up a conceptual model for pile-soil interaction from these observations, cf. Figure 1. The model consists of 3 parts. Firstly, a loading phase where the soilpile interaction follows the virgin curve. Secondly, an unloading phase that due to irreversible deformations in the soil will imply the development of a gap between the pile and the soil. Finally, a phase where the pile moves towards the initial position and into the opposite

Figure 1. Typical loading cyclic for a model pile in clay, from Mayoral et al. (2005).

soil face in the cavity created behind the pile during initial loading. In this phase it may be assumed that there exists a drag or friction along the side of piles. Whether or not the gap will develop may depend on the type of soil type. El-Naggar et al. (2005) assumes that the gap will develop for cohesive soils, whereas for cohesionless soils, the soil will cave in and close the gap. Still, centrifuge tests carried out on a pile in

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dry sand indicate that this cave-in effect may not be fully developed, Klinkvort (2009), thus there is probably a need to include the drag effect in a model even for cohesionless soils. Klinkvort (2009). One of the first attempts in formulating p-y-curves that reflected the observed behavior was done by Matlock et al. (1978). Later, Boulanger et al. (1999) proposed an elasto-plastic p-y model based on a two component set-up in which the loading response is handled by a series connection of springs – one spring handling loading (ive failure mode) and another spring handling the unloading-reloading properties of a pile subjected to cyclic loading that is gradually creating a gap behind the pile. Taciroglu et al. (2006) further developed these ideas and proposed a macroelement consisting of three components; leading-face element, rear-face element and drag-element. The two face-elements are formulated in of elasto-plastic springs supplemented with a tension cut-off. The drag element controls the side friction, when the pile is moving inside the cavity during unloading. In the present work, the principles of the abovementioned models are incorporated in a single spring element that can be directly incorporated in a standard finite element code. In the following the elasto-plastic constitutive relations will be presented. Then follows a discussion about the implementation and finally some results from simulations.

2

As mentioned above the flow rule is associated to the yield function, hence rewriting Eqn. (2) by use of Eqn. (3), we find

In case of plastic loading f = 0 the consistency requirement requires the stress point to remain on the yield surface, hence

where the hardening modulus H is the scalar contraction of the partial derivatives of the yield function with respect to α. For isotropic hardening, only a single hardening parameter is needed, i.e. α ≡ α, but since we need to for the development of a gap on the front and on the rear of the pile, respectively, it is necessary to introduce two hardening parameters as is presented in the coming sections. As always the fundamental assumption of common elastic and plastic stress is used, hence

where k is the elastic stiffness. Combining Eqn. (5) and Eqn. (6) yields the definition of the plastic multiplier dλ,

ELASTO-PLASTIC MODEL

A simple one-dimensional elasto-plastic spring is defined. The model is expressed in of the earth resistant force p and the associated displacement u. The standard procedure for development of elastoplastic models are used. First the operator split between elastic and plastic components is assumed.

where due is the elastic part and dup is the plastic part of the total displacement increment du. The plastic displacement component is defined in of the gradient to the plastic potential, i.e.

Here it is used that the displacement increment is associated to the loading direction, hence p · du = 1. This relation is then entered back into Eqn. (6) to produce the elasto-plastic tangent stiffness,

This completes the formal definition of the plasticity model. Remaining is now to define the yield strength as a function of the hardening parameters. 2.1 Yield function

with dλ as the plastic multiplier. The direction of the plastic displacement increment is fixed to the loading plane, implying that the plastic flow potential is by definition associated to the yield surface, i.e. f = g. The simplest yield function may be written as

Following the terminology of Mayoral et al. (2005) and Matlock (1970) we divide the current yield strength into two parts; one relating to the drag contribution and one relating to the earth pressure.

in which pu (α) is the current strength yield strength and α = (α1 , α2 , . . . ) are the hardening parameters (to be defined later).

The first term pu is the drag capacity, which in this version of the model is assumed to be constant. Below this value, the spring is assumed linear elastic with a stiffness k. The second term must for the

drag

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develop when the pile is in with the soil. As long as the pile is sliding in the cavity created by the cyclic motion, the model should behave ideally plastic. Introducing once again the step function we may find

in which the definition of the plastic displacement, Eqn. (4), is utilized. Having established the evolution law, it is finally possible to identify the model specific hardening modulus, H , by revisiting the consistency equation, Eqn. (5). After some manipulation we find that

Figure 2. Schematic drawing of the spring element.

earth pressure when either of the pile faces are in with the soil. If there is no , this term must vanish. This can be achieved by introducing a multiplier to the virgin curve. The obvious candidate is a smooth step function,

The parameter β defines the curvature and the coordinate x is

Note that the arguments αi and xi has been omitted in the formula. Analyzing Eqn. (14), it is noted half of the contributions vanishes if the soil is in with either the front face or the rear face of the pile, since the for the unloaded face S = 0. Likewise this relation ensures that H = 0 in the cavity since S = 0 for all . 3

A typical value for β would be around 1.000.000. The coordinate x thus defines the current position of the pile relative to the soil. If the pile is in with the soil x ≥ 0 and if there is a gap x < 0. Using Eqn. (9) we can write the yield function as

The hardening parameters αi , i = 1, 2 represents either loading of the front or rear face of the pile. The virgin virgin curve pu (α) depends on the soil conditions as e.g. given by API (2000). 2.2

IMPLEMENTATION

The proposed spring element is implemented in an inhouse MATLAB based FE code, Hededal and Krenk (1995). The implementation consists of two parts. Firstly implementation of the spring element using a backward Euler integration scheme for integration of the constitutive relation. Secondly, a Winkler model based on the proposed model has been defined and analyzed using a Newton Raphson based non-linear solver. For this specific application it has been chosen to use the (API 2000) definition of the p-y curves for sand,

Evolution law for hardening parameters

Referring to Figure 2 it is easily seen that the hardening parameter αi is defined as the plastic displacement accumulated during between soil and pile. Physically α is representing the progressive development of the gap. Using the experience from mechanics, it is deemed that a formulation of unloading and reloading in of a displacement criterion (rather than the usual stress based criterion) allows us to keep the formulation simple, even for the discontinuous phase when the pile is moving in the developed cavity. The evolution law for the hardening parameters should thus be defined in such a way that they only

Here pult is the ultimate capacity, A is a strength reduction parameter, k is the subgrade reaction modulus, X is the depth and u is the total lateral displacement. Still, in order to implement this relation into the proposed format, it is necessary to divide the total resistance into a drag contribution and a face loading contribution, i.e.

This is not a trivial task, since the hyperbolic function can not be easily inverted in order to allow

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Figure 4. Overall response on a pile subjected to monotonic loading loading.

Figure 3. API curve versus the elasto-plastic curve. Table 1.

Pile soil properties.

Pile diameter Pile length Load eccentricity Frictional angle Soil density

D L e φ γ

1m 6m 2.5 m 42◦ 16 kN /m3

for a split of elastic and plastic contribution. In the present situation, it has be chosen to use the following approximation,

Figure 5. Overall response on a pile subjected to one-way loading.

Eqn. (17) is a implicit function in α since we have u = α + p/k. This implies that the derivative with respect to α is not trivial. Here we use

as a first order approximation. Comparing the API curve to the prediction of the model, Figure 3, this approximation appears to be acceptable.

4

RESULTS

To demonstrate the ability of the model to capture the pile-soil interaction as observed by Matlock (1970) and Mayoral et al. (2005), three test simulations have been carried out. The material properties used in the three test examples are shown in Table 1. The three tests have been performed with a monotonic or cyclic laterally load applied in the top of the pile.A rather large stiffness has been used for the sand in order to clearly demonstrate the capability of the spring element.

4.1 Example 1 – monotonic loading The spring element presented here is capable of performing cyclic tests. As demonstrated in Figure 3 the elasto-plastic element follows the virgin curve recommended by API (2000). Monotonic tests can therefore also easily be performed with this element. In Figure 4 the result as pile head deflection versus applied laterally load from a monotonic test can be seen. The maximum bearing capacity of the pile is calculated to Pmax = 1122 kN . Using the theory from Hansen (1961), the maximum bearing capacity can be calculated to Pmax = 1152 kN . This results fits very well with the calculation performed in the model. 4.2 Example 2 – one way loading The second example illustrates a pile that is subjected to a load varying from zero and to a given value in the same direction, this is called one-way loading. The maximum load during the cycles is close to the ultimate capacity, so that the accumulation effect is clearly seen. The overall pile response can be seen in Figure 5. This figure shows the pile top deflection versus the

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Figure 6. Spring response on a pile subjected to one-way loading.

applied force. The model simulates a load controlled test with constant load amplitude in a total of ten cycles. It can be seen from Figure 5 that the deflection increases with every cycle. Still, the rate of increase for every cycle is getting smaller and smaller. This shows that the model is able to take for the accumulation of displacement when the model is subjected to one-way loading. The response from one of the springs near the soil surface can be seen in Figure 6. The spring reaches fast the maximum bearing capacity. This is due to the high stiffness. It unloads elastically and then the development of a cavity can be seen. As described in section 2.2, no hardening occurs when the pile is moving in this cavity. It can also be seen that after the first cycle the the spring does not go back to its initial position, but exhibits a permanent deformation. This is due to the accumulation of deflection. The accumulation of deflections occurs due to the development of cavity in several springs and the subsequent redistribution of the force therefore occurs. 4.3 Example 3 – two way loading In this example the pile is subjected to a given load varying between negative and positive values, this is called two-way loading. The overall pile response can be seen in figure Figure 7. The pile is loaded five full cycles. The same maximum force is applied for both direction. It can be seen from Figure 7 that the deflection is getting larger and larger from every load cycle. This is valid for both sides and the increase in deflection is also the same for both sides. This means that the average deflection of the load cycles is constant and equal to zero. It is though interesting that the deflection amplitude increases, hence the secant stiffness will decrease as a consequence of cyclic loading. This effect is extremely important if we are to model the cyclic response of monopile foundations for wind turbine, since the load here is frequency dependent. It should be noted that the number of iterations increases dramatically after the first half cycle when

Figure 7. Overall response on a pile subjected to two-way loading.

Figure 8. Spring response on a pile subjected to two-way loading.

the pile is in a position around the mean deflection. This is due to the development of a cavity in nearly all spring elements. In this position the system have very low stiffness. A simple remedy to this could be to include a small amount of kinematic hardening to the drag-term in a manner as proposed by Hededal and Strandgaard (2008). The response from one of the springs can be seen in Figure 8. It can be seen that a cavity develops as expected. As for the overall pile response, an increase in deflection of the single spring for every load cycle is observed. Also here the average deflection for an overall load cycle is constant and equal to zero. There is no degradation of the springs which can be seen in one-way loading example.

5

DISCUSSION

The cyclic spring presented in this paper is capable of capture physical aspects as seen in tests Matlock (1970), Mayoral et al. (2005) and Klinkvort (2009). Still, improvements are needed. In this section ideas

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which will improve the performance of the spring element and the representation of the physical world. The presented model operates with the same virgin stiffness as un-/reloading stiffness. This could be changed and it must also be expected that a soil not will load and unload with the same stiffness. With a change like this the model will probeable start to accumulate displacements in a smaller loading range. When springs moving in the cavity some sort of hardening should occur. This can also be seen in the Figure 1 by Mayoral et al. (2005). As a side effect an introduction of hardening in the cavity will help the global iterations to converge faster. Other effects which should be incorporated in the future is suction release for clay springs and the fall back of sand particle when dealing with sand springs. 6

CONCLUSION

An elasto-plastic spring element has been defined. The spring element embeds two fundamental features of cyclically loaded piles. It is able to for preloading of the soil by tracing the virgin curve. Secondly, the creation of a gap after reloading, which is undeniably developing in cohesive soils, is ed for by introducing a smoothed step function that keeps track of the current position of the pile-soil interfaces. The element is not only relevant for the quasi-static loading with random time series, but also has a potential in dynamic analysis, where it will provide a physically based hysteretic damping.

Boulanger, R. W., C. J. Curras, B. L. Kutter, D. W. Wilson, and A. Abghari (1999). Seismic soil-pile-structure interaction experiments and analyses. Journal of Geotechnical and Geoenvironmental Engineering 125(9), 750–759. El-Naggar, M. H., M. A. Shayanfar, M. Kimiaei, and A. A. Aghakouchak (2005). Simplified bnwf model for nonlinear seismic response analysis of offshore piles with nonlinear input ground motion analysis. Canadian Geotechnical Journal 42, 365–380. Hansen, J. B. (1961). The ultimate resistance of rigid piles against transversal forces. Danish Geotechnical Institute, Copenhagen, Denmark Bulletin NO. 12, 5–9. Hededal, O. and S. Krenk (1995). FEMLAB – matlab toolbox for the Finite Element Method. Aalborg University. Hededal, O. and T. Strandgaard (2008). A 3d elasto-plastic soil model for lateral buckling analysis. In Proc. 18th International Offshore and Polar Engineering Conference, ISOPE 2008. Klinkvort, R. T. (2009). Laterally loaded piles – centrifuge and numerical modelling. Master’s thesis, Technical University of Denmark. Matlock, H. (1970). Correlations for design of laterally loaded piles in soft clay. In Offshore Technology Conference, pp. 577–594. Matlock, H., H. C. Foo, and L. M. Bryant (1978). Simulation of lateral pile behavior under earthquake motion. In Proc. American Society of Civil Engineers Specialty Conference on Earthquake Engineering and Soil Dynamics,Volume 2, pp. 600–619. Mayoral, J. M., J. M. Pestana, and R. B. Seed (2005). Determination of multidirectional p-y curves for soft clays. Geotechnical Testing Journal of Computational Mechanics Vol. 28, No.3. Taciroglu, E., C. Rha, and J. Wallace (2006). A robust macroelement model for soil-pile interaction under cyclic loads. Journal of Geotechnical and Geoenvironmental Engineering, ASCE 132(10), 1304–1314.

REFERENCES API (2000). American petroleum institute. recommended practice for planning, deg and constructing fixed offshore platforms- working stress design, api recommended practice 2a-wsd (rp2a-wsd), 21st edition, dallas.

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Behaviour of cyclic laterally loaded large diameter monopiles in saturated sand H. Ercan Ta¸san, Frank Rackwitz & Stavros A. Savidis Soil Mechanics and Geotechnical Engineering Division, Berlin Institute of Technology,

ABSTRACT: Monopiles are suitable foundations for offshore wind energy converters. The monopiles are highly laterally loaded due to harsh environmental conditions especially from wind and water waves. Therefore a monopile diameter of up to 7 m in water depth of about 30 m will be necessary to maintain serviceability of the wind energy converter over several years. A three-dimensional finite element model was developed to investigate the behaviour of large diameter piles under cyclic lateral loading taking the interaction between the pile and the surrounding sandy soil into . The material behaviour of sand is described by a hypoplastic model with intergranular strain, suitable to for cyclic behaviour of cohesionless soils. The frictional behaviour in the interface between monopile and soil is modeled by elements. The focus of the investigation is the pore water pressure accumulation close to the monopile. For this purpose a three-dimensional fully coupled two-phase finite element were developed and implemented. The two phase material is assumed to consist of a solid phase, the skeleton, and a fluid phase which fully occupies the pores in the skeleton. The governing equations of the coupled fluid-structure problem are the equations of pore fluid flow and the equilibrium conditions. The effects of pile diameter and cyclic loading on pore water pressure accumulation and soil-pile interaction are discussed on the basis of the numerical results. 1

INTRODUCTION

Increasing importance of sustainable wind energy over the past years led to the planning of offshore wind energy converters in the North Sea and the Baltic Sea as well. The planned constructions require special foundations due to the site and loading conditions. Driven monopiles, i.e. single open ended steel pipe piles with large diameters, are suitable foundations up to 30 m water depth in mostly sandy soil conditions. A design pile diameter of up to 7 m is presumably required to resist the wind and wave loading actions. It is of great importance to consider possible pore pressure accumulation due to highly cyclic loading in the design of monopiles. Current design of monopiles applies the p-y method according to the existing guidelines (API 2000, DNV 2004, GL 2005). The p-y method is based on field measurements with pile diameters d < 1.0 m and load cycles N ≤ 100 (Cox et al. 1974). The straightforward application of these findings to monopiles under offshore highly cyclic loading conditions is still subject to research. In recent years, the behaviour of cyclic lateral loaded offshore pile foundations with focus on pore pressure generation and its consequences have been investigated by means of experimental and numerical analysis (Grabe et al. 2004, Kluge 2007, Ta¸san et al. 2007). The paper focuses on the determination of pore pressure accumulation in the pile-soil interface and

interaction region. Finite element analyses are used to investigate the influence of monopile diameter as well as cyclic loading conditions on the rate of accumulation. These analyses require adequate finite element formulation to for pore pressure generation and effective stresses in the soil. For that purpose a three dimensional two-phase finite element based on the theory of porous media was developed and implemented into an existing finite element code. The finite element fully couples the solid and the fluid phase of the soil and allows for consideration of a nonlinear stress-strain relationship. The latter is described using a hypoplastic constitutive formulation with intergranular strain. It is capable to for the stress and density dependent mechanical behaviour of sands and the influence of cyclic loading as well. The important interface between the pile shaft and the surrounding soil is modelled using a quite simple Coulomb friction model which enables open gaps too. 2

FULLY COUPLED TWO-PHASE MODEL

Two-phase models are used for investigations of geotechnical problems in which the mechanical behaviour of soil is affected significantly by pore fluid. The used two-phase model is based on the theory of porous media and described in detail in Zienkiewicz et al. (1984) and Potts & Zdravković (1999). The principal equations of the model will be recalled in the following.

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Two phase mixture is assumed to consist of a solid phase, the skeleton, and a fluid phase which fully occupies the pores in the skeleton. It is assumed that the solid and fluid constituents can be modelled as incompressible. The problem is formulated in of the absolute displacement of the soil skeleton u and the pore water pressure p. The balance of momentum for the mixture can be written neglecting the accelerations of the relative movement between water and skeleton and considering Terzaghi’s effective stress principle by:

Here, L is the divergence operator, which is formulated in the following vectorial form:

and σ = effective stress vector; ρ = density of the mixture; b = body forces; ς = damping ratio and mT = (1,1,1,0,0,0). The mass balance of fluid phase is formulated as

with k p = permeability matrix, ρw = the density of water and ηw = the dynamic viscosity of water. For isotropic problems the permeability matrix can be written as k p = kp I with unit matrix I. The hydraulic conductivity kd can be derived from the permeability kp by

where g = gravity acceleration. A domain with boundary is filled by the mixture. The boundary conditions to be prescribed for equation (1) and (3) are displacements u˜ on solid displacement boundary u , surface tractions σ˜ on solid traction boundary t , pore water pressure p˜ on fluid pressure boundary p and surface flow q˜ on fluid flux boundary q . The equations (1) and (3) are discretized using standard Galerkin techniques. Thereby the displacement and pore water pressure fields are approximated as

where Nu and Np are the shape functions, u and p are corresponding vectors of unknowns. The following matrix equations are finally determined:

and

with

where B = LNu ; Bp = ∇Np ; and Dt = matrix of tangential moduli, which may be determined from a nonlinear stress-strain relationship. Generalized Newmark method (Bathe 1996) is used for time integration of the coupled equations (6) and (7).

3

HYPOPLASTIC CONSTITUTIVE LAW

Cohesionless soils can be modelled by hypoplastic constitutive law considering the influence of stress level and soil density on the soil behaviour. Stiffness, dilatancy, contractancy and peak friction is followed by the soil state and the deformation direction. Plastic deformations are simulated without using potential or switch functions. A single tensorial equation is used to describe plastic as well as elastic deformations. The basic hypoplastic theory (Kolymbas 1988, von Wolffersdorf 1996) is expanded by Niemunis & Herle (1998) to model realistically the accumulation effects and the hysteretic material behaviour under cyclic loading. An additional state variable, which is the intergranular strain, is introduced to consider the influence of changing direction of deformation on the mechanical behaviour of soil. A total of eight material parameters are required for the basic hypoplastic model. The expanded model with intergranular strain requires five additional parameters (Niemunis & Herle 1998).

4

3-D TWO-PHASE ELEMENT

A 3-D continuum element u20p8 on the basis of two phase model is implemented, where the displacement field is approximated using triquadratic interpolation functions and the pressure field is approximated using trilinear interpolation functions. A high order interpolation of the displacement field is required to overcome unstable tendencies of the finite elements (Zienkiewicz 1986). In the following two numerical examples are presented to the two-phase model and implemented u20p8 element.

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Figure 1. Finite element model, boundary and loading conditions. Table 1.

Material parameters and hydraulic conductivity.

E [kN/m2 ] 40·103

ν 0.0 · · · · 0.48

kd [m/s] 2·10−6

Figure 3. Time histories of settlements.

Figure 4. Numerical results of cyclic undrained triaxial compression test.

pressure is beginning to dissipate and the effective stress is increasing. The simulated time histories of settlements for ν = 0.0, 0.25 and 0.48 are shown in Figure 3. A comparison with the calculated results by Booker (1974) shows a good agreement.

Figure 2. Contours of pore water pressure for ν = 0,25.

4.1 3-D consolidation The problem of the consolidation of a clay layer subjected to surface loads was investigated by Booker (1974). In his investigation it was assumed that the soil consists of an isotropic perfectly elastic skeleton saturated with water. 3-D simulations with u20p8 Elements were performed to compare these finite element results with Booker (1974). A fully water saturated clay layer resting on a rigid base, i.e. fixed boundary conditions, and subject to circular surface loading is modeled and shown in Figure 1. Due to the symmetric loading condition only a quarter of the system is considered. The model surface is assumed to be permeable. The parameters Young’s modulus E, Poisson’s ratio ν and the hydraulic conductivity kd of clay layer are given in Table 1. The contours of pore water pressure at t = 0.5 s and t = 100 s for ν = 0.25 are shown in Figure 2.The results at the beginning of loading demonstrate that the load is carried by pore water, i.e. an excess pore water pressure is developed. The load is with increasing time transferred to solid phase, so that the excess pore water

4.2 Cyclic undrained triaxial test Cyclic undrained triaxial test with dense Hochstetten sand, whose hypoplastic material parameters are given in Niemunis & Herle (1998), is simulated using an u20p8 element. The simulation is performed for an initially isotropic pressure of p = 300 kN/m2 and a stress deviator amplitude of q = 30 kN/m2 . The results of numerical simulation are presented in Figure 4. A comparison with the numerical results by Niemunis & Herle (1998) shows a good agreement. 5

BEHAVIOUR OF MONOPILES DUE TO CYCLIC LATERAL LOADING

5.1 Finite element modelling 3-D finite element simulations are performed to investigate the behaviour of monopiles with large diameter under cyclic lateral, cyclic moment and static vertical loading. The response of monopiles is calculated by a

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Table 2. ◦

Soil parameters for Berlin Sand.

ϕc [ ]

hs [MPa]

n

32.0

3730

0.20

0.41

mR 5.0

mT 2.0

R 1·10−4

ed0

ec0 0.74 βr 0.4

ei0

α

β

0.89

0.14

1.0

χ 6.0

Figure 5. FE mesh of soil-pile system.

steady state analysis. The focus of the investigations is the quantitative determination of pore water pressure accumulation close to the monopile. Because of the symmetry in geometry and load, only a half of soil-pile system is considered, as shown in Figure 5. The monopile with an embedded length l and a diameter d is modelled with 20-node continuum elements. The u20p8 elements are used for modelling the soil. The dimensions of modelled pile-soil system, which are given in Figure 5, are sufficient for monopiles with 4.0 ≤ l/d ≤ 9.0, so that the calculated behaviour of pile is not influenced by the boundaries. The boundary conditions, which imposed on mesh, are fixing of nodes at the bottom of the mesh against displacement in all directions, on the plane of symmetry against displacement normally on that plane and on the periphery of the mesh against displacement in both horizontal directions. Furthermore non permeable conditions are assumed on all boundaries except boundaries on model surface, where pore pressure due to existing water level must be considered. The frictional behaviour in the interface between monopile and soil is modeled by elements with Coloumb friction law. Therefore a wall friction angle of δ = 21◦ is assumed. A linear elastic material behaviour with E = 2.1 · 108 kN/m2 and ν = 0.3 is assumed for the monopile. The material behaviour of sand is described by a hypoplastic model with intergranular strain. The parameters of Berlin Sand, which are given in Table 2, are used in model. The modelled Berlin Sand has a dry density in loosest condition ρd,min = 1.52 g/cm3 and in densest condition ρd,max = 1.88 g/cm3 . Prior to first phase of the simulation a vertical and a horizontal effective stress as initial loading must be defined for soil to determine the required state variables of hypoplastic model. Therefore, as first a calculation under gravity loading is performed with a

Figure 6. Characteristic of cyclic lateral und moment load.

coefficient of the at rest value of earth pressure, k0 , determined by means of Jaky’s equation. The static water pressure is also thereby determined.

5.2 Influence of load cycle number A monopile with diameter d = 7.0 m, embedded length l = 35.0 m and steel pipe wall thickness t = 0.09 m was modelled to study the influence of the number of load cycles on the pore pressure accumulation. The pile is subjected to a cyclic horizontal load with amplitude Hˆ = 2.5 MN and bending moment ampliˆ = 125 MNm. A static vertical load V = 12 MN tude M acts additionally. The frequency of cyclic loads is f = 0.06 Hz. Figure 6 illustrates the cyclic load function of horizontal load as well as bending moment over number of cycles. A water table of +30.0 m over sea bed was chosen in the model. Initial relative density ID = 0.92 and hydraulic conductivity kd = 2.0·10−4 m/s represent the in situ sandy soil conditions. A maximum time step size of 0.2 s is used for simulation a load step. The calculated maximum number of cycles is limited to 5 at the moment because of high computational costs. Contour plots of excess pore pressures are shown in Figure 7 for three different cycle numbers, i.e. N = 1.5, 3.0 and 4.5. In the direction of loading excess pore pressure develops with increasing cycle number as can be seen in Figure 7. Furthermore negative pore pressure occurs opposite to the loading direction. The magnitude of accumulated pore pressure increases with the number of cycles as well as the location of maximum pore pressure simultaneously moves into the direction of the pile base

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Figure 8. Excess pore water pressure in different depths.

Figure 9. Excess pore pressure evolution versus depth for different monopile diameters. Figure 7. Contour plots of excess pore water pressure for N = 1.5, 3.0 and 4.5.

The calculated pore water pressures at the pile-soil interface up to z = 9.6 m depth reach the range of initial vertical effective soil stresses. Figure 8 represents the calculated excess pore pressure over the number of cycles N at the pile-soil interface in depths z = 2.6 m, 6.1 m, 12.0 m and 21.3 m below sea bed.At z = 2.6 m and 6.1 m depth the change of excess pore pressure is almost constant after N = 3.0 cycles, whereas at z = 12.0 m and 21.3 m depth the change of excess pore pressure is almost continuously increasing with cycle number N . The maximum excess pore pressure was calculated after 5 cycles at z = 12.0 m depth. 5.3 Influence of monopile diameter Finite element analyses with cyclic lateral displacement controlled loading of the pile were done to

investigate the influence of pile diameter on the pore pressure evolution. The cyclic displacement amplitude was chosen to be wˆ = d/100 and using the characteristic according to Figure 6 as well as the loading frequency f = 0.06 Hz too. Steel pipe wall thickness t in m was determined according to the following relation (API 2000)

Figure 9 shows calculated maximum excess pore pressure versus depth at the pile-soil interface within the first loading cycle. The location of the maximum excess pore pressure moves in the direction of the pile base with increasing pile diameter. Maximum excess pore pressure arises with the largest monopile diameter d = 8.0 m, as expected before, and in a depth of 18.4 m. The rate of the determined excess pore water pressure to initial vertical effective soil stress up to this depth is more than 80%.

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6

CONCLUSIONS

The results of finite element simulations with fully coupled two-phase element and using the constitutive formulation of hypoplasticity with intergranular strain for sand are offered that due to cyclic loading of monopile with large diameter a pore water pressure accumulation can occur. It’s shown that the accumulated pore water pressure can reach the range of the initial vertical soil stress in the pile-soil interaction region. With increasing number of cycles and increasing diameter, the location of developed maximum pore water pressure moved in direction to pile tip. The pile behaviour is affected by pore water pressure accumulation around the pile. Therefore it’s necessary to consider this accumulation by the design of monopile foundation to ensure the serviceability of offshore wind energy converters. The aim of the numerical, which are presented here, and experimental investigations (Ta¸san et al. 2007) is to propose a practical design method for monopiles with large diameter in which the effect of cyclic lateral loading on the behaviour of pile is considered. ACKNOWLEDGEMENTS The work presented within this paper is the result of a research project, which is funded by the Federal Ministry for the Environment, Nature Conversation and Nuclear Safety. Its is gratefully acknowledged. REFERENCES API 2000. American Petroleum Institute: Recommended practice for planning, deg and constructing fixed offshore platforms – working stress design. Recommend practice design 2A–WSD, Washington D.C. Bathe, K.-J. 1996. Finite element procedures. New Jersey: Prentice Hall. Booker, J. R. 1974. The consolidation of a finite layer subject to surface loading, International Journal of Soils and Structures, 10(9), 1053–1065.

Cox, W. R., Reese, L. C. & Grubbs B. R. (1974). Field testing of laterally loaded piles in sand, Proceedings of the Sixth Offshore Technology Conference, OTC 2079, Houston, 459–472. DNV. 2004. Det NorskeVeritas: Offshore standard DNV–OS– J101, design for offshore wind turbine structures, Norway. GL 2005. Germanischer Lloyd: rules and guidelines, IV – industrial services, Part 2 – guideline for the certification of offshore wind turbines, Hamburg. Grabe J., Dührkop J. & K.-P. Mahutka 2004. Monopilegründungen von Offshore-Windenergieanlagen – Zur Bildung von Porenwasserüberdrücken aus zyklischer Belastung, Bauingenieur 79(9), 418–423. (in German) Kluge K. 2007. Soil liquefaction around offshore pile foundations – scale model investigations. Mitteilung des Instituts für Grundbau und Bodenmechanik Technische Universität Braunschweig, Heft 85. Kolymbas, D. 1988. “Eine konstitutive Theorie für Böden und andere körnige Stoffe”, Veröffentlichung des Institutes für Bodenmechanik und Felsmechanik der Universität Fridericana in Karlsruhe, Heft 109. (in German). Niemunis, A. & Herle, I. 1998. Hypoplastic model for cohesionless soils with elastic strain range, Mechanics of Cohesion-Fractional Materials, 2(4), 279–299. Potts, D. M. & Zdravkoviæ, L. 1999. Finite element analysis in geotechnical engineering. theory. London: Thomas Telford. Ta¸san, H. E., Rackwitz F. & Savidis, S. A. Modellversuche in der geotechnischen Versuchsgrube zur Untersuchung des Tragverhaltens von Offshore-Monopilegründungen. Veröffentlichung des Grundbauinstitutes der Technischen Universität Berlin, Heft Nr. 42, 197–213. (in German). von Wolffersdorff, P.-A. 1996. A hypoplastic relation for granular materials with a predefined limit state surface, Mechanics of Cohesive-Frictional Materials 1(3), 251–271. Zienkiewicz, O. C. & Shiomi, T. 1984. Dynamic behaviour of saturated porous media; the generalized Biot formulation and its numerical solution. International Journal of Numerical Analytical Methods in Geomechanics 8(1): 71–96. Zienkiewicz, O. C. 1986. The patch test for mixed formulations. International Journal for Numerical Methods in Engineering, 23(10), 1873–1883.

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Caisson movement caused by wave slamming—a comparison of ABAQUS and FLAC analyses L. Andersen, H.F. Burcharth & T. Lykke Andersen Department of Civil Engineering, Aalborg University, Aalborg, Denmark

A.H. Augustesen Department of Civil Engineering, Aalborg University, Aalborg, Denmark COWI A/S, Aalborg, Denmark

ABSTRACT: During wave slamming, caisson movement may occur as a combination of sliding along the caisson–foundation interface and local failure in the foundation and seabed. The paper presents a comparison between different techniques applied to the analysis of this movement. Thus, a finite-difference analysis has been performed by means of the commercial code FLAC. Similarly, ABAQUS has been employed for finiteelement analyses based on linear as well as quadratic spatial interpolations, assuming fully drained conditions and utilizing an elastic–plastic model for the rubble foundation and the seabed. The results of the numerical codes are compared to an analytical solution in which the deformation of the subsoil has been disregarded.

1

INTRODUCTION

Wave slamming on a caisson breakwater may lead to movement of the structure. As shown in Figure 1, irreversible deformations may occur in different forms including sliding at the interface between the structure and the foundation, slip failure between the foundation and the seabed, and local collapse of the quarry rock constituting the foundation. Typically, a static design is carried out for breakwater. However, this demands a static pressure distribution equivalent to the dynamic pressure stemming from the wave. As indicated by the pressure distribution sketched in Figure 2 and the example results from measurements of the horizontal force resultant shown Figure 3, defining a proper level of this equivalent static load may not be possible. Thus, shock-like peaks exist in the pressure due to wave slamming, and different sampling frequencies in model tests may lead to very different conclusions regarding the design. Hence, a dynamic model is preferred. Burcharth et al. (2008) proposed a simple model based on the one-dimensional equation of motion. Due to its simplicity, the model can be used for lifetime analysis of a caisson; but it only s for sliding along the interface between the caisson and the rubble foundation. A more realistic modelling of the subsoil can be achieved by utilization of numerical methods. Employing a nonlinear finite-element model, Barquín (1998) included a visco-elastic soil model, and plastic deformation of the foundation and seabed was considered by Burcharth et al. (2009) as well as Kudella & Oumeraci (2009). However, the reliability of such numerical models must be verified

Figure 1. Failure modes for a caisson subjected to wave impact.

before their application in design. Hence, based on the commercial codes ABAQUS and FLAC, finiteelement and finite-difference solutions are compared with focus on their ability to quantify the displacement of a caisson due to horizontal sliding. The numerical models both employ a Lagrangian formulation ing for elastic and plastic material behaviour as well as geometrical nonlinearity. Finally, a comparison is made with the result of the solution proposed by Burcharth et al. (2008).

2

COMPUTATIONAL MODELS

The overall geometry of the vertical breakwater is defined in Figure 4. The wall on top of the caisson has a width of 5 m at the base and 2.5 m at the top. The caisson is assumed to consist of reinforced concrete backfilled with sand. It has been found that the stress levels will not lead to failure in the concrete. Hence, the structure is modelled as a linear elastic material with the properties listed in Figure 4, where E, ν and ρ

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Figure 2. Dynamic part of the hydraulic pressure distribution from wave slamming on a caisson.

Figure 3. Horizontal force resultant from wave slamming on caisson measured in model scale at Aalborg University.

Figure 4. Geometry and material properties of the considered caisson, rubble foundation and seabed.

signify Young’s modulus, Poisson’s ratio and the total mass density, respectively. The foundation and the seabed are assumed to consist of quarry rock and sand, respectively. Linear response is assumed in the elastic regime, and the Mohr-Coulomb criterion is employed to identify stresses at initial yielding and strains during plastic deformation. The material properties are given in Figure 4, where φ and ψ are the angles of internal friction and dilation, respectively, whereas c is the cohesion. The latter has been introduced to increase stability of the computational methods. Finally, an interface exists between the caisson and the foundation. Sliding occurs along this interface in accordance to Coulomb’s friction law. In the present case, a friction coefficient of µ = 0.6 is assumed (see Figure 4) corresponding to an interface friction angle

Figure 5. Simplified distribution of the hydraulic pressure on the caisson during wave slamming.

of 31◦ . Initially, is assumed between the structure and the foundation. However, during the dynamic response slip is allowed to happen in the numerical models, i.e. the tensile strength and stiffness of the interface are both zero. Rayleigh damping is applied, since it is available in ABAQUS as well as FLAC. The damping ratio ζ = 5% is assumed at the frequency 1 Hz and only mass proportional damping is used. Added mass is disregarded, and the water (including the pore fluid in the foundation and subsoil) has not been modelled directly. Instead, a pressure is applied on the caisson surface.The transient part of this pressure is divided into two components as illustrated in Figure 5. The load time histories assumed for the components P1 (t) and P2 (t) are shown in Figure 6. Thus, further to the buoyancy provided by the hydrostatic pressure P0 , a quasi-static pressure, P1 (t), is applied with a uniform distribution on the front side of the caisson and a triangular distribution on its base. The latter part of the load is assumed to act in phase with the load on the front side whereas in reality a short delay occurs due to a finite velocity of the pressure wave travelling through the pore fluid. Finally, wave slamming is modelled by a shock load applied as a pressure, P2 (t), distributed uniformly on the upper half of the front side. The simplified model should be compared to the measured wave load reported in Figure 3 and the pressure distribution illustrated in Figure 2. In the present simulations, the maximum value of P1 (t) is given by Pstatic /P1max = 2, where Pstatic is the value of P1 leading to sliding failure when applied statically in the absence of the shock load P2 . Based on Figure 4, the value Pstatic = 98 kPa is determined, provided a hydrostatic pressure of P0 = 10 kPa at the base of the caisson, i.e. at a water depth of 10 m. In order to study the influence of the peak height as well as the peak width of the shock load, four wave load time histories are analysed and compared. Thus, as listed in Figure 6, the short-term excitation from wave slamming has a duration of t2 = 0.2 or 0.4 s with the peak occurring after t1 = 0.1 or 0.2 s. The peak height is defined by P2max /P1max = 4 or 8. 2.1 ABAQUS models Two finite-element (FE) models are analysed by ABAQUS (Simulia 2009). Plane strain is assumed, utilizing four and eight-node quadrilateral elements with linear and quadratic spatial interpolation of the

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Figure 6. Time history of the two components constituting the simplified wave load on the caisson.

Figure 8. Deformations (scaled 50 times) at ultimate failure in the ABAQUS models using linear (top) and quadratic (bottom) spatial interpolation.

Figure 7. Discretization in the ABAQUS models using linear (top) and quadratic (bottom) spatial interpolation. The dark, medium and light shades of grey indicate concrete, rubble and sand, respectively. The soil domain is 120 m × 40 m.

displacement field, respectively. Figure 7 shows the mesh in the two models. Thus, the node distance is the same, whereas the mesh size is doubled in the second-order elements compared to the linear model. In order to establish the in situ stresses in the structure, foundation and subsoil, gravity is applied incrementally and stepwise. The full mass density is employed and the gravitational acceleration is set to g = 5 m/s2 in order to compensate for uplift on the soil skeleton from the water. This ensures a correct computation of the effective stresses as well as the dynamic response. Full gravity with g = 9.82 m/s2 is applied on the caisson and buoyancy is introduced by a pressure of P0 = 10 kPa as discussed above. The FE models are utilized to identify the failure mode in static loading. Horizontal fixities introduced

at the vertical edges of the soil domain, whereas vertical and horizontal fixities are employed along the bottom of the computational model. As illustrated in Figure 8, static failure occurs as a combination of sliding at the interface and local soil collapse at the heel of the caisson. The local deformations behind the heel are slightly different in the models with linear and quadratic spatial interpolation. However, the ultimate value of P1 is found as 95 kPa in either model, i.e. a value below the capacity related to pure translational sliding, Pstatic . Further, it has found that a reduction of the foundation height by a few metres leads to foundation slip failure at a pressure of the same magnitude. Thus the failure modes shown in Figure 1 occur at similar levels of the load. For this reason, the value Pstatic = 98 kPa is used in the dynamic analysis which is performed by ABAQUS/Standard, i.e. an implicit Lagrangian solution scheme is employed. 2.2 FLAC model A finite-difference method solution is computed by the commercial code FLAC3D (Itasca 2007). A 1 m wide strip is constrained in the y-direction to simulate plane strain. The model employs the discretization used in the linear FE model, i.e. the grid shown in Figure 7. However, the cells are not regarded as elements. Instead, the terminology “zone” is introduced for the space between four adjacent nodes. The volumes and edges of the zones are utilized for application of external forces and boundary conditions as well as evaluation of strains and stresses.

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Figure 9. Vertical normal stresses due to gravity in the FLAC model. The dark and light shades of grey indicate high and low pressure, respectively.

Similarly to the FE model, gravity is applied incrementally. However, since FLAC builds on an explicit dynamic solver, a ramp function is introduced such that the load is increased linearly over time slowly enough to allow static equilibrium between exterior and interior forces to develop. Alternatively, the entire gravitational load is applied at once and time integration is performed until static equilibrium is reached, i.e. the correct stress distribution has been established. In this quasi-static part of the solution, mass scaling is employed to allow the use of larger time increments, leading to a faster solution. Figure 9 shows the distribution of the vertical normal stresses at the end of the gravitational step and after introduction of buoyancy on the caisson, i.e. just before the application of the transient load. The discontinuity of the stresses that can be observed at the interface between the structure and the foundation stems from the hydrostatic pressure applied at the base of the caisson. The ultimate capacity for static loading cannot be identified by application of a forced displacement, since the kinematics of the structure and subsoil at failure are unknown. It has not been possible to determine the ultimate capacity of the caisson for a static load distributed as P1 in Figure 5. However, the main purpose is to find the magnitude of the displacement as well as the deformation mode during wave impact. Thus, only the dynamic analysis has been carried out, based on the explicit solver with the true mass lumped at the nodes.

3


Figure 10 shows the final deformations obtained in the FLAC simulations. A similar deformation is obtained in the ABAQUS simulations and will not be shown. For all combinations of t1 and P2max /P1max , a combination of the three failure modes illustrated in Figure 1 can be observed. Thus, significant plastic deformation occurs under the heel of the caisson due to the localized stresses stemming from the high pressure evolving during wave impact. Further, foundation slip failure clearly occurs at some stage during

Figure 10. Final deformation in the FLAC models. Dark and light shades of grey indicate high and low magnitudes of the strain, respectively. The displacement is scaled by a factor 10.

the response. This is different from the response to static loading where failure occurs as a combination of local soil collapse and sliding. Figure 11 shows the results of the ABAQUS and FLAC models for the different combinations of t1 and P2max /P1max . The horizontal displacements at the top left

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Figure 11. Time history of the horizontal displacements at the base and top corners on the front side of the caisson. The peak of the wave-impact force occurs at the time t1 and the peak height is P1max + P2max . The grey lines indicate the horizontal sliding obtained by the simple analytical solution proposed by Burcharth et al. (2008).

corner (caisson top) and the lower left corner (caisson base) are plotted as functions of time. A comparison is made with the simple analytical solution proposed by Burcharth et al. (2008). Since the analytical solution only concerns the horizontal translation, it provides the same displacement at the top and base of the caisson. A number of observations can be made by inspection of Figure 11. Firstly, significantly different magnitudes of the horizontal displacement are predicted by the finite-difference and finite-element models with FLAC providing the smaller value. Thus, ABAQUS calculates a final displacement that is about a factor two higher than the FLAC model when quadratic spatial interpolation is employed. In the case of linear interpolation, the displacements obtained by the FE model are even greater. On the other hand the difference between the displacements at the caisson top and base is nearly the same in all models, suggesting that the rotation of the caisson is not influenced by the choice of model. The displacements obtained by the numerical models at the base of the caisson should be compared to the analytical solution indicated by the grey shaded line in Figure 11. Evidently, the simple model predicts a displacement that is much smaller than the values provided by ABAQUS for all considered combinations of

the peak duration and magnitude. In most cases, the analytical solution is also far below the displacement given by FLAC. The difference is most pronounced for t1 = 0.1 s and P2max /P1max = 4. Here the analytical approach provides a displacement smaller than 5 mm in contrast to FLAC which predicts a displacement of approximately 2 cm. The difference between the analytical and numerical results becomes less significant with increasing magnitude of the shock load. Especially for t1 = 0.2 s and P2max /P1max = 8 the results of the simple method and FLAC are close to being identical. This may be explained by the fact that all failure modes indicated in Figure 1 are active in the numerical models, whereas only sliding is included in the simple model. Thus, Figure 10 shows that plastic deformations occur at the foundation–subsoil interface for all load histories— even for the relatively short shock load with low magnitude (t1 = 0.1 s and P2max /P1max = 4). Apparently for wave loads with a high peak value, sliding along the structure–foundation interface becomes the dominant mode of deformation and the simple model is more accurate than in the case of low-magnitude forces. Slightly different formulations of the at the structure–foundation interface in ABAQUS and FLAC may cause a significant part of the deviation between

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the results provided by the two codes. In ABAQUS a master–slave definition is employed for the interface with the top of the foundation acting as master surface and the base of the caisson acting as the slave. is then identified when a node from the slave surface lies on or es the element edges on the master surface. Pressure over-closure is penalized by a spring with the stiffness 1010 N/m/m2 acting in the normal direction; but no elastic deformation can occur in the tangential direction. In FLAC, so-called interface elements are applied at the base of the caisson. is established whenever a node from another domain (in this case the foundation) hits these interface elements. Then linear elastic springs counteract penetration in the normal direction and sliding or shear in the tangential direction. In the present analyses, the stiffness 1011 N/m/m2 has been utilized for both stiffnesses. It has been checked that there is no significant change in the results if the interface elements are instead applied on the top of the foundation or if the spring stiffnesses are changed by one order of magnitude. Another explanation for the deviation between the solutions obtained by FLAC and ABAQUS may be a poor degree of convergence. For this reason, a FLAC computation has been carried out with one-and-a-half times as many nodes in each direction. The result is nearly identical to those presented in Figure 11, indicating that the FLAC model is fully converged. On the other hand, an ABAQUS model using linear interpolation and a mesh size that is two thirds of the original mesh size provides the same elastic response but slightly higher displacements at failure. This indicates that the FE solution does not have the same degree of convergence. Furthermore, there is a great influence of the element type (interpolation order) on the outcome. Figure 8 brings some insight into this by clearly demonstrating that the quadratic and linear elements treat at the heel of the caisson in very different manners. 4

CONCLUSIONS

Caisson movements due to shock loads from wave impact have been analysed by the finite-element code ABAQUS and the explicit finite-difference solver FLAC3D under the assumption of plane strain and elastic–plastic response of the foundation and subsoil. It has been found that linear and quadratic interpolation in ABAQUS lead to a significant difference in the total displacements regardless of the load magnitude and load duration. The horizontal displacements achieved by the ABAQUS models are greater than the displacements obtained by FLAC. Both codes

predict more displacement than a simple solution only ing for sliding along the structure–foundation interface, since a combination of foundation slip failure, sliding and local soil deformation under the heel occurs in the refined models. Given that a consistent and fully converged result has been obtained with FLAC, this code is proposed for further analyses. However, a comparison with ABAQUS/Explicit simulations and results based on adaptive mesh refinement within the finite-element analysis may be relevant. The results presented in this paper are based on a model of finite extent. In ABAQUS, semi-infinite elements are available and FLAC allows the use of absorbing boundary conditions for dynamic analysis. In both cases, the idea is to provide boundary conditions that more realistically model the behaviour of unbounded soil. This will be considered in future analyses. Furthermore, it has been assumed that the soil is fully drained and the dynamic pressure on the base of the caisson develops immediately when the wave plunges on the front of the structure. This may not be realistic—in particular not when the foundation and seabed consist of materials with low permeability. Hence, the next step is to include a dynamic pore pressure model in the numerical simulations. Finally, accumulated displacement after a series of wave impacts should be analysed, possibly using a model that s for liquefaction of the seabed.

REFERENCES Barquín, G.G. 1998. Dynamic analysis of a vertical breakwater—extension to the Escombreras Basin Port of Cartagena, Spain. Madrid. Burcharth, H.F., Andersen, L. & Lykke Andersen, T. 2009. Analyses of stability of caisson breakwaters on rubble foundation exposed to impulsive loads. In Smith, J.M. (ed.), Proc. 31st Int. Conf. Coastal Eng., Hamburg, , 31 Aug. – 5 Sep. 2008: 3606–3618. World Scientific. Burcharth, H.F., Lykke Andersen, T. & Meinert, P. 2008. The Importance of Pressure Sampling Frequency in Models for Determination of Critical Wave Loadings on Monolithic Structures. In Proc. COPEDEC VII, Dubai, UAE. Itasca 2006. FLAD3D – Fast Lagrangian Analysis of Continua in 3 Dimensions – ’s Guide. Minneapolis, Minnesota USA: Itasca Consulting Group, Inc. Kudella, M. & Oumeraci, H. 2009. Experimental and numerical study of the response of a sandbed beneath a caisson breakwater subject to cyclic wave load. In Smith, J.M. (ed.), Proc. 31st Int. Conf. Coastal Eng., Hamburg, , 31 Aug. – 5 Sep. 2008: 3619–3631. World Scientific. Simulia 2009. ABAQUS Version 6.9 Documentation. Providence, RI, USA: Dassault Systèmes Simulia Corp.

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Comparison of calculation approaches for monopiles for offshore wind turbines A.H. Augustesen COWI A/S, Aalborg, Denmark Department of Civil Engineering, Aalborg University, Aalborg, Denmark

S.P.H. Sørensen, L.B. Ibsen & L. Andersen Department of Civil Engineering, Aalborg University, Aalborg, Denmark

M. Møller & K.T. Brødbæk COWI A/S, Århus/Aalborg, Denmark

ABSTRACT: Large-diameter (4 to 6 m) monopiles are often used as foundations for offshore wind turbines. The monopiles are subjected to large horizontal forces and overturning moments and they are traditionally designed based on the p-y curve method (Winkler type approach). The p-y curves recommended in offshore design regulations were developed for piles with diameters up to approximately 2.0 m and are based on a very limited number of tests. Hence, the method has not been validated for piles with diameters of 4 to 6 m. This paper aims to assess different calculation approaches for monopiles subjected to quasi-static loading. The methods are: 1) a traditional Winkler model as recommended in the offshore standards; 2) a modified Winkler model in which the initial stiffness of the p-y curves depends on the pile diameter, depth below mudline as well as the internal angle of friction; 3) a three-dimensional continuum model established by means of the commercial program FLAC3D . The approaches are compared based on a monopile used as foundation for a wind turbine at Horns Rev which is located in the North Sea west of Denmark.

1

INTRODUCTION

Large-diameter piles for offshore wind turbines. have diameters around 4–6 m and embedded lengths of 20– 30 m depending on the magnitude of the loads and the soil conditions, i.e. the length-diameter ratio is approximately 5. Monopiles are traditionally designed based on p-y curves, i.e. a Winkler approach, cf. Section 3. For piles in sand, the p-y curves proposed in design regulations such as API (1993) are based on few full-scale measurements on two steel pipe piles (diameter = 0.61 m, wall thickness = 9.5 mm and embedded length = 21 m) leading to a length– diameter ratio equal to 34.4, as discussed by Cox et al. (1974) and Reese et al. (1974). The piles have been tested three and four times within a period of approximately two months after installation. Both cyclic and static loading tests were conducted. Further, the proposed p-y curves have been tested against a database of lateral pile load tests with satisfaction as described by Murchinson and O’Neill (1984). However, they indicate that the assessment of the p-y curves are based on a small database due to the unavailability of appropriately documented full-scale test data.

Briaud et al. (1984) postulate that the soil–pile behaviour is affected by the flexibility of the pile. Criteria for rigid or flexible behaviour have been suggested by various researchers, e.g. Poulus and Hull (1989). According to their recommendations, the piles used for the development of the p-y curves behave as flexible piles. In contrast, the monopile considered in this paper, and generally the monopiles for offshore wind turbines, behave more like rigid piles than flexible ones, i.e. they merely rotate when subjected to large horizontal loads and rocking moments implying a “toe kick”. Hence, the deformation behaviour of the piles and thereby the soil in the case of monopiles for nowadays offshore wind turbines is very different from the conditions from which the p-y curves are derived. These scale effects have not been taken into in the p-y curve formulations currently recommended by API (1993). Commercial finite-element programs such as PLAXIS (2006) and ABAQUS (Simulia 2009) as well as the finite-difference program FLAC3D (Itasca 2007) do not suffer from these shortcomings. Further, much more complicated models for both soil and pile can be employed leading to a more accurate estimation of the load–deformation behaviour of the pile.

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In this paper, results of numerical calculations, conducted by means of FLAC3D , of the load–deflection behaviour of a monopile for an offshore wind turbine are presented and compared to the results obtained by means of a traditional Winkler-type approach employing the currently recommended p-y curves as well as a modified expression of the p-y curves (Sørensen et al. 2010). The three approaches are compared for a real case, i.e. a monopile used as foundation for a wind turbine at Horns Rev located in the Danish sector of the North Sea. Cyclic degradation and partially drained conditions may be relevant. However, for simplicity, drained conditions and a static load scenario, corresponding to the ultimate limit state (ULS), are considered. 2

HORNS REV WIND FARM

The wind turbine considered is part of Horns Rev Offshore Wind Farm, built during 2003 and located in the North Sea west of Esbjerg in Denmark. 2.1

Pile conditions

The steel monopile considered is the foundation for wind turbine 14, which in the following is denoted M14. The outer diameter is 4 m, the length is 31.6 m and the wall thickness WT, and thereby the bending stiffness, Ep Ip , varies along the pile as shown in Figure 1. Ep denotes Young’s modulus, and Ip is the second moment of area around a horizontal axis perpendicular to the pile axis. The monopile has been driven to its final position 31.8 m below the mean sea level leading to an embedded depth of 21.9 m. The pile behaviour is investigated in an ultimate limit state (ULS). It is subjected to the static extreme loads: the horizontal load H = 4.6 MN and the bending moment M = 95 MNm, both acting at seabed level, whereas the vertical load is V = 5.0 MN. Analyses show that the vertical force V has a negligible effect (less than 0.1%) on the deflection pattern as well as the moments in the pile. Therefore, V = 0 is assumed in the following.

Figure 1. Geometry and properties of the pile. Table 1. Soil conditions including average values of the strength and stiffness parameters for each soil layer.

2.2 Soil conditions The soil profile at the location of M14 consists primarily of sand with the stratification and properties summarized in Table 1. Here ϕ is the angle of internal friction and ψ is the dilation angle. The friction angle ϕ is determined from Ts according to the procedure proposed by Schmertmann (1978). Apart from the silt/sand layer with organic material all other layers have relatively high angles of internal friction (36.6◦ < ϕ < 45.4◦ ). Further, it is assumed that ψ = ϕ − 30◦ . The unit weight γ and the submerged unit weight γ of the soil are, except in the layer including the organic material, 20 kN/m3 and 10 kN/m3 , respectively. In the organic sand layer, γ/γ yields 17/7 kN/m3 . The classical Mohr-Coulomb criterion and a linear elastic material model have been combined to describe

Sand Sand Sand(silty) Sand(silty) Sand/silt/Org Sand

Depth [m]

ϕ [◦ ]

Es [MPa]

v

0–4.5 4.5–6.5 6.5–11.9 11.9–14.0 14.0–18.2 18.2→

45.4 40.7 38.0 36.6 27.0 38.7

130 114.3 100 104.5 4.5 168.8

0.28 0.28 0.28 0.28 0.28 0.28

the elasto-plastic material behaviour of the soil in the numerical calculations, cf. Section 4. It is assumed that the stiffness of the soil can be represented by the secant Young’s modulus Es corresponding to an average axial strain of 0.1% as well as Poisson’s ratio ν. According to Lunne et al. (1997), this level of strain is reasonably representative for many well-designed foundations. Es is stress-dependent and it is determined according to

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loading conditions; A = 0.9 for cyclic loading and A = (3.0 − 0.8x/D) ≥ 0.9 for static loading. ∗ With reference to Equation 1 the initial stiffness Epy of the p-y curves is given by

Hence, the initial stiffness is considered independent of the pile properties and linearly dependent on x. Sørensen et al. (2009), Lesny & Wiemann (2006) and Leth et al. (2008) all document that the initial stiffness of the p-y curves is in general significantly overestimated with depth. Hence, Sørensen et al. (2010) propose another formulation for the initial ∗ stiffness Epy :

Figure 2. Winkler approach and definition of p-y curves.

Lengkeek (2003). Apart from the silt/sand layer with organic material, all soil layers have relatively high stiffness (100 MPa < Es < 168.8 MPa). 3 WINKLER MODEL—API METHOD Laterally loaded monopiles used for wind turbine foundations are traditionally designed based on a Winkler approach. In contrast to a Pasternak foundation, the soil has no shear stiffness but only lateral stiffness, represented by nonlinear elastic springs based on semiempirical relations between the soil pressure p acting against the pile wall and the lateral deflection y of the pile, cf. Figure 2. The spring stiffness Epy employed in the Winkler model is provided by the p-y curves as the secant modulus, cf. Figure 2. Generally, Epy increases with depth x and decreases with increasing deflections y. Further, Epy depends on the soil conditions, and in the static loading case, the p-y curves reach a horizontal asymptote corresponding to the capacity of the soil pult . The pile is modelled as a Bernoulli-Euler beam in spite of its relatively small length–diameter ratio. In principle, the Timoshenko beam theory could preferably be applied. However, in practice the BernoulliEuler beam theory suffices since the pile is very stiff, behaving more like a rigid pile than a flexible one. According to API (1993), the p-y relationship for piles in sand is given by

where pult is the ultimate lateral resistance at depth x below the surface, ksand is the initial modulus of the subgrade reaction, dependent on ϕ, y is the lateral deflection, D is the average pile diameter and A is a factor ing for cyclic or static

where xref = 1 m and Dref = 1 m. The depth x and the diameter D should be inserted in meters and the friction angle in radians. The Winkler approach based on the p-y curves proposed by API (1993), cf. Equation 1, will, in the following, be referred to as the API method whereas the Winkler approach based on the p-y curves given by Equation 1 but modified according to the initial stiffness described by Equation 3 is denoted the modified API method. The p-y curves are based on tests on piles located in almost homogeneous soil. The soil profile at Horns Rev is layered. Therefore, the procedure of Georgiadis (1983), which s for layered soil within the framework of the p-y curve method, has been employed. In this study, the governing differential equation for the Winkler approach incorporating the p-y curves has been solved under the auspices of the finiteelement method by introducing appropriate boundary conditions. It turns out that the solution converges if approximately 100 elements are used. 4

FLAC MODEL

A three-dimensional numerical model has been established in FLAC3D (Itasca 2007) which is a three-dimensional, dynamic, explicit finite-difference solver. Due to symmetry, only one half of the pile and the surrounding soil is considered, cf. Figure 3. The model has an outer diameter of 40D = 160 m based on the recommendations by Abbas et al. (2008), and the boundary at the bottom is placed approximately 18 m below the pile toe. The external load in the model is applied as a horizontal force of H = 2.3 MN (=0.5·4.6 MN due to the symmetry) acting at the height h = 20.65 m above seabed level. This combination of the height and horizontal force provides a bending moment of M = 95 MNm at seabed level, corresponding to the design criterion for the prototype, cf. Section 2. For both the soil and pile, zone elements are used to model the geometry. Each zone consists of five

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Figure 3. FLAC3D model. Figure 5. Moments in the pile at maximum loads.

The material behaviour of the soil is modelled using the classical Mohr-Coulomb model. Thus, the soil is simplified as a linear elastic, perfect plastic material. The employed material properties are given in Table 1. The interaction between the monopile and the soil is modelled using a standard FLAC3D interface. The interface is one-sided and is attached to the soil. A linear Coulomb shear-strength criterion is employed for the interface to limit the shear forces acting at the interface nodes. The wall friction angle, δ, is determined according to Equation 4:

Figure 4. Pile deflections at maximum horizontal load.

first-order, constant-rate-of-strain, tetrahedral subelements. The monopile is assumed to be linear elastic steel with the parameters Ep = 210 GPa and ν = 0.3. However, the pile is modelled as a solid cylinder rather than an open tubular pile with an internal soil plug. Young’s modulus is reduced so that the bending stiffness Ep Ip corresponds to that of M14, cf. Figure 1. Thus, for the equivalent solid pile, Ep is within the range 12.3 GPa to 21.0 GPa. Poisson’s ratio is unaltered, since the value for the soil is close to that of steel, cf. Table 1. It should be noted that the shear stiffness of the pile is incorrectly scaled, but it has been found that the shear deformations of the pile have a negligible influence on the overall response of the pile and soil. Similarly to the stiffness, the weight of the solid monopile is adjusted in such way that it corresponds to that of M14. Here, it is assumed that the real, open-ended tubular pile behaves in an unplugged way, assuming that the soil inside the pile is located at seabed level.

Further, the interface elements allow gapping and slipping between the soil and the pile. The finite-difference calculations are executed stepwise. First, the initial stress state is established in the entire model using the submerged unit weight for both the soil elements and the elements that later become the pile. A K0 -procedure, in which it is assumed that K0 = 1 − sin ϕ, is employed to establish the initial horizontal effective stresses. Subsequently the pile is generated by replacing the soil elements that now become the pile with the adjusted strength and stiffness parameters as well as the adjusted weight corresponding to the monopile for M14. Further, the monopile elements are extended above the ground surface in order to realize the loading conditions described above. Between the pile elements and the soil elements, an interface is established to model the pile–soil interaction. The system is brought to equilibrium. Finally, the horizontal load is applied incrementally and new equilibrium states are calculated. Combined damping, which is preferred for uniform motions (Itasca 2007), is introduced to provide a quasi-static solution. Further, different types of grids have been employed to assure convergence, which has been achieved with the grid shown in Figure 3.

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Figure 6. Soil pressures as function of deflection and depth.

5


In Figure 4, the deflections of the monopile for M14 are shown for the case in which the pile is subjected to the static extreme loads presented in Section 2.1. The monopile behaves relatively rigid implying that a “toe kick” occurs; this is especially pronounced when considering the deflection behaviour predicted by means of FLAC3D and the modified API method. The maximum horizontal deflection at seabed level determined by means of the API method is significantly lower compared to the deflections predicted by the modified API method and FLAC3D , cf. Table 2. Below 14 m the deflection pattern estimated by the API method and FLAC3D deviate significantly. FLAC3D estimates, for example, greater horizontal deflections at the pile toe compared to the API method, cf. Table 2. The deviation in deflection pattern may be due to the fact that the stiffness Epy provided by the API method is overestimated at great depths. However, the modified API method gives rise to a deflection pattern similar to the one predicted by FLAC3D but with a much softer response at the lower part of the pile. Since the API method overestimates the stiffness with depth compared to FLAC3D and the modified API method, the depth for zero deflection predicted by the API method is located closer to the seabed, cf. Table 2. The three approaches predict similar distributions of the moment with depth, cf. Figure 5. However, FLAC3D estimates slightly lesser and higher moments at moderate and deep depths, respectively, compared to the API method and the modified API method. The maximum moments determined by the three approaches are almost identical, cf. Table 2. Further, the depths to the maximum moment are 3.4 m and 2.1 m, respectively, with FLAC3D giving rise to the latter value.

Figure 7. Soil pressures as function of deflection and depth. Table 2. Results obtained by means of FLAC3D , the API method and modified API method.

Max. moment [MNm] Depth max. moment [m] Horz. def., seabed [mm] Horz. def., toe [mm] Rotation seabed [°] Depth to zero def. [m]

API

FLAC3D

mod. API

105.4 3.4 26.8 −1.6 0.26 9.9

101.3 2.1 41.8 −4.5 0.31 14

105.4 3.4 46 −13.3 0.35 11.7

The p-y curves at different depths are shown in Figures 6 and 7. In connection with FLAC3D , the soil resistance p is calculated by integration of the interface stresses. Generally, there is a reasonable concordance between the p-y curves estimated by FLAC3D and the modified API method. Except for the depth x = 2.1 m

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the API method has a tendency to overestimate the soil pressures p at a given deflection y, compared to the other two approaches. The deviation may be caused by an overestimation of the stiffness of the sand and/or there are some shortcomings in the method proposed by Georgiadis (1983), in which layered soil profiles are taken into . The pressures, estimated by means of FLAC3D , mobilised at the depth x = 7.4 m are less than the pressures at both x = 2.1 m and x = 3.9 m for a given deflection y. This is due to the lower angle of internal friction of the third layer compared to the first layer, cf. Table 1. When considering the API method the initial stiffness of the p-y curve for x = 7.4 m is slightly higher ∗ than Epy for x = 3.8 m even though the angle of internal friction of the third layer is lower compared to the first layer, i.e. the depth compensate for the differences in ksand and thereby ϕ. This is not the case for the modified API method. However, it should be mentioned that Equation 3 has been calibrated based on angles of internal friction between 30◦ and 40◦ and therefore it may be less reliable for values of ϕ beyond 40◦ .

6

CONCLUSIONS

The behaviour of a monopile used as foundation for a wind turbine at Horns Rev located in the Danish sector of the North Sea has been investigated. The pile is located primarily in sand and it has been subjected to extreme static loads. The paper presents a comparison of the results of 1) numerical calculations conducted by means of the commercial three-dimensional finite-difference code FLAC3D ; 2) a traditional Winkler-type approach employing p-y curves as proposed in current design regulations for offshore wind turbines as well as 3) a modified version of the traditional Winkler-type approach. All approaches indicate that the monopile behaves as a relatively rigid pile, implying that only one point of zero deflection exists. The maximum horizontal deflection at seabed level determined by means of the traditional Winkler-type approach is significantly lower compared to the deflections predicted by the modified Winkler-type approach and FLAC3D . Further, the maximum moments in the pile predicted by the three methods are almost identical. Generally, there is a reasonable concordance between the p-y curves estimated by FLAC3D and the modified Winkler-type approach as function of depth. Compared to the two other methods the traditional Winkler-type method overestimate the stiffness of the soil. Moreover, extreme care should be taken in the estimation of the soil stiffness associated with the constitutive soil models employed in commercial programs such as FLAC3D . Further research is needed to develop new p-y curves for large-diameter piles in sand. This requires the conduction of full-scale tests and numerical analyses based on more advanced constitutive soil models.

REFERENCES Abbas, J.M., Chik, Z.H. & Taha, M.R. 2008. Single pile simulation and analysis subjected to lateral load. Electronic Journal of Geotechnical Engineering, 13E: 1–15. API 1993. RP2A-WSD: Recommended practice for planning, deg and constructing fixed offshore platforms – working stress design, American Petroleum Institute, 20th edition. Briaud, J.L., Smith, T.D. & B.J. Meyer 1984. Using pressuremeter curve to design laterally loaded piles. In Proc. of the 15th Annual Offshore Technology Conference, Houston, Texas, USA, OTC 4501: 495–502. Cox, W.R., Reese, L.C. & Grubbs, B.R. 1974. Field testing of laterally loaded piles in sand. In Proc. of the 6th Annual Offshore Technology Conference, Houston, Texas, US, OTC 2079: 459–472. Georgiadis, M. 1983. Development of p-y curves for layered soils. In Proc. of the Conference on Geotechnical Practice in Offshore Engineering: 536–545. Itasca 2007. FLAD3D – Fast Lagrangian Analysis of Continua in 3 Dimensions – ’s Guide. Minneapolis, Minnesota USA: Itasca Consulting Group, Inc. Lengkeek H.J. 2003. Estimation of sand stiffness parameters from cone resistance. PLAXIS Bulletin No. 13: 15–19. Lesny, K. & Wiemann, J. 2006. Finite-element-modelling of large diameter monopiles for offshore wind energy converters. In Geo Congress 2006, February 26 to March 1, Atlanta, GA, USA. Leth, C.T., Krogsbøll, A. & Hededal, O. 2008. Centrifuge facilities at Technical University of Denmark. NGM2008, Nordisk Geotekniker Møte, Norway. Lunne, T., Robertson, P.K. & Powell, J.J.M. 1997. Cone penetration testing in geotechnical practice. Blackie Academic & Professional, 1997. Murchinson, J.M. & O’Neill, M.W. 1984. Evaluation of p-y relationships in cohesionless soil. In Analysis and Design of Pile Foundations. Proceedings of a Symposium in Conjunction with the ASCE National Convention, ASCE: 174–191. PLAXIS 3D 2006. PLAXIS 3D Foundation - Manual, Brinkgreve, R.B.J. & Broere, W. (edt.), the Netherlands,. Poulus, H. & Hull, T. 1989. The role of analytical geomechanics in foundation engineering. In Foundation Engineering: Current Principles and Practices, 2: 1578–1606. Reese, L.C., Cox, W.R. & F.D. Koop 1974. Analysis of laterally loaded piles in sand. In Proc. of the 6th Annual Offshore Technology Conference, Houston, Texas, USA, OTC 2080: 473–484. Schmertmann, J.H. 1978. Guidelines for cone penetration test, Performance and Design. Report, FHWA-TS-78-209, 145, US Federal Highway istration, Washington, DC. Simulia 2009. ABAQUS Version 6.9 Documentation. Providence, RI, USA: Dassault Systèmes Simulia Corp. Sørensen, S.P.H., Brødbæk, K.T., Møller, M., Augustesen, A.H. & Ibsen, L.B. 2009. Evaluation of the LoadDisplacement Relationships for Large-Diameter Piles in Sand. In Proc. of The Twelfth International Conference on Civil, Structural and Environmental Engineering Computing, September 1 to September 4, Funchal, Madeira, Portugal: paper 244. Sørensen, S.P.H., Ibsen, L.B. & Augustesen, A.H. 2010. Effects of diameter on initial stiffness of p-y curves for large-diameter piles in sand. In Proc. of NUMGE, Trondheim, Norway.

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Effects of diameter on initial stiffness of p-y curves for large-diameter piles in sand S.P.H. Sørensen & L.B. Ibsen Aalborg University, Aalborg, Denmark

A.H. Augustesen COWI, Aalborg, Denmark Aalborg University, Aalborg, Denmark

ABSTRACT: For offshore wind turbines, monopile foundations with diameters of 4–6 m are often employed. The Winkler model approach, where the soil resistance is modelled as uncoupled springs with spring stiffness’ given by p-y curves, is traditionally employed for the design of monopiles. However, this method is developed for slender piles with diameters up to approximately 2.0 m. Hence, the method is not validated for piles with diameters of 4–6 m. The aim of the paper is to extend the p-y curve method to large-diameter non-slender piles in sand by considering the effects of the pile diameter on the soil-pile interaction. Hence, a modified expression for the p-y curves for statically loaded piles in sand is proposed in which the initial slope of the p-y curves depends on the depth below the soil surface, the pile diameter and the internal angle of friction. The evaluation is based on three-dimensional numerical analyses by means of the commercial program FLAC3D incorporating a MohrCoulomb failure criterion. The numerical model is validated with laboratory tests in a pressure tank at Aalborg University.

1

INTRODUCTION

For modern offshore wind turbines several types of foundations exist. The choice of foundation depends on site and loading conditions. An often used foundation concept is monopiles, which are single steel pipe piles driven open-ended. Recently installed monopiles have diameters of 4–6 m and slenderness ratios, L/D, around 5, where L is the embedded length of the pile and D is the pile diameter. The maximum forces acting at the mudline of a foundation for a typical offshore wind turbine is according to Ubilla et al. (2006) in the order of magnitude of 4 MN in horizontal load, 6 MN in vertical load, and 120 MNm in overturning moment. Hereby, offshore wind turbine foundations are highly subjected to lateral loads and bending. When deg laterally loaded offshore monopiles the offshore design regulations, e.g. API (1993) and DNV (1992), recommend the use of the Winkler model approach in which the pile is modelled as a beam on an elastic foundation, cf. Figure 1. The elastic foundation consists of a number of springs with spring stiffness, K, given by means of p-y curves. p-y curves describe the relationship between the soil resistance, p, acting against the pile wall and the lateral pile deflection, y. Several formulations of p-y curves exist depending on the type of soil. The offshore design regulations,

Figure 1. Winkler model approach and definition of p-y curves.

e.g. API (1993) and DNV (1992), recommend the p-y curve given in Equation 1 for piles in sand.

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A is a factor taking the ratio between depth and pile diameter into , pu is the ultimate soil resistance, k is the initial modulus of subgrade reaction and x is the depth measured from the soil surface. The initial modulus of subgrade reaction, k, is determined in of the internal friction angle or the relative density of the sand and governs the initial slope of the p-y curves. The ∗ initial stiffness of the p-y curves, Epy , recommended in the design regulations is given in Equation 2.

Hereby, the initial stiffness is considered independent of the pile properties and linearly dependent of x. The hyperbolic expression, cf. Equation 1, is based on the testing of two identical instrumented piles installed at Mustang Island, cf. Cox et al. (1974). A total of seven load tests were performed on the piles consisting of two static tests and five cyclic tests. Both piles had a diameter of 0.61 m, a slenderness ratio of 34.4 and were installed in similar soil conditions. Due to the limited number of tests, the effects of a large number of parameters on the soil response are still to be clarified. Among these is the pile diameter. Due to serviceability only small rotations at the soil surface are allowed for modern wind turbines. Further, strict demands are set to the total stiffness of wind turbine foundations due to resonance in the serviceability mode. Therefore, the initial stiffness of the p-y curves is of great importance. It seems ques∗ tionable that Epy is independent of the pile diameter. The research within the field gives contradictory conclusions. Terzaghi (1956), Ashford & Juirnarongrit (2003), and Fan & Long (2005) all conclude that the effect of diameter on the initial stiffness of the p-y curves is insignificant. In contrast Carter (1984) and ∗ Ling (1988) suggest a linear relationship between Epy and D. Lesny & Wiemann (2006) postulate that the initial stiffness of the p-y curves is overestimated at large depths when employing a linear variation of initial stiffness with depth. Instead they suggest a parabolic variation of initial stiffness with depth. However, the research is based on a limited number of tests and most researchers consider only flexible piles with slenderness ratios larger than 10, which is rarely the case for modern wind turbine foundations. In the present paper the effect of pile diameter on the initial stiffness of the p-y curves is assessed by means of numerical simulations. At first a numerical model is calibrated to six laboratory tests conducted in a pressure tank in the Laboratory of Foundation at Aalborg University, Denmark. After the calibration the numerical model is extended to simulate large-scale offshore wind turbine foundations. The model is conducted by means of the commercial program FLAC3D .

Figure 2. Three-dimensional mesh employed in the numerical model.

2

NUMERICAL MODELLING OF MONOPILE UNDER STATIC LOADING

A three-dimensional numerical model is constructed in the commercial program FLAC3D which is a dynamic solver incorporating the finite difference method. The Mohr-Coulomb material model is employed in the numerical modelling. 2.1 Construction of numerical model Due to symmetry only half of the pile and half of the surrounding soil are modelled. The mesh employed in the numerical model is shown in Figure 2. In order to reach convergence the mesh is very fine near the pile. The pile is modelled as a solid cylinder in contrast to the hollow pipe piles which are normally employed for offshore wind turbine foundations. Young’s modulus of elasticity for the solid cylinder, Ep , is therefore determined so that the bending stiffness, Ep Ip , of a pile in the numerical model and of a full-scale pile is equivalent. Similarly the density of the pile is determined requiring equivalence in the total weight of the pile. The grid is generated of zone elements. Each zone consists of five 4-noded constant strain-rate subelements of tetrahedral shape. The interface between the soil and the pile is modelled by means of a linear Coulomb shear-strength criterion. The interface allows gapping and slipping between the pile and the soil. The interface is one-sided and is attached to the soil. The model is damped using combined damping which is preferred for uniform motions, cf. FLAC3D 3.1 manual (2006). The horizontal load is applied as a horizontal velocity at the centre nodes at the pile head. A number of calculation steps are prescribed in order to reach the desired pile deflection. To ensure a static behaviour of the pile velocities in the order of 10−6 m/s are employed. For piles with slenderness ratios smaller

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than 10 it is difficult to determine a velocity resulting in a stable simulation. Therefore the horizontal load is applied as a load distributed to all nodes at the pile head. In order to obtain several points on the p-y curves, the load is applied in small load steps. The numerical simulations are executed in steps. At first the initial stresses in the soil is generated using the K0 -procedure. Secondly the pile is installed, the pile parameters are introduced and an equilibrium state is calculated. Here the soil-pile interface is assumed smooth. Thirdly the soil-pile interface is given the correct interface properties and a new equilibrium state is calculated. After reaching equilibrium the horizontal load is applied. 2.2 Computation of soil resistance In order to determine p-y curves from the numerical simulations the soil resistance needs to be determined. The soil resistance is calculated by integration of the interface stresses as described by Fan & Long (2005). 2.3 Calibration of model The numerical model is calibrated to six laboratory tests conducted at the Laboratory of Foundation at Aalborg University, reported by Sørensen et al. (2009). The tests are conducted in a pressure tank in which the pressure can be increased. The test setup enables the possibility of applying an excess effective stress to the soil leading to increased effective stresses. When conducting small-scale tests in sand at 1-g an often introduced source of error is the low stress levels causing the soil parameters and in specific the internal angle of friction to vary strongly with depth. When applying an excess pressure this variation is minimised. Quasi-static tests are conducted on instrumented piles with D = 60 mm and D = 80 mm and a slenderness ratio of L/D = 5. When simulating the laboratory tests, the outer boundaries of the numerical model are set equal to the inner diameter of the pressure tank. The load-displacement relationship for the calibration of the numerical model is shown in Figure 3 for one of the tests. The load-displacement relationships obtained with the numerical model is in good agreement with the laboratory tests. 2.4 Simulation of large-scale piles Large-scale steel pipe piles with varying internal friction angle of 30–40◦ and pile diameter of 1–7 m are simulated. For all simulated piles, embedded lengths of 20 m are employed. The wall friction angle, δ, for the interface between the soil and the pile is determined by Equation 3.

The tangential Young’s modulus of elasticity for the soil, E0 , is varied with the relative density, ID , and the

Figure 3. Calibrated load-displacement relationships for D = 0.08 m and an excess effective stress of 100 kPa.

minor principal stress, σ3 , on basis of Equation 4 in which the output is given in kPa. The equation has been proposed by Ibsen et al. (2009) and is valid for the sand in used in the laboratory tests. σ3ref is the reference minor principal stress given as 100 kPa.

The internal friction angle for the sand used in the laboratory tests can according to Ibsen et al. (2009) be determined by equation (5).

The influence of the minor principle stress is not taken into for the numerical simulations and instead Equation 6 is employed.

In Figure 4 the deflections of the pile is shown for pile diameters of D = 1–7 m. An internal friction angle of 40◦ is employed. The displacements are shown for a pile displacement at the soil surface of 0.024 m. As the wall thickness is held constant for all pile diameters, the bending stiffness varies such that the pile with D = 7 m has the largest bending stiffness and the pile with D = 1 m has the lowest bending stiffness. The pile with D = 7 m behaves very stiff as it deflects almost as a rigid body. In contrast the pile with D = 1 m exhibits a very flexible behaviour. The depth of the point of zero deflection increases for increasing pile diameter as the depth is approximately 8.5 m for D = 1 m and 16.3 m for D = 7 m. Further a significant negative deflection can be observed near the pile toe for the piles with a low slenderness ratio, L/D. Hereby, the slenderness ratio, L/D, has a significant influence on the pile behaviour. An example on the distribution of soil resistance along the pile is shown in Figure 5 for D = 4 m and

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Figure 4. Pile displacement along the pile for pile diameters of D = 1–7 m and ϕtr = 40◦ . For all pile diameters a wall thickness of 0.05 m is employed.

Figure 5. Distribution of soil resistance along the pile a pile diameter of D = 4 m and an internal friction angle of ϕtr = 40◦ . ◦

ϕtr = 40 . The soil resistance is zero in a depth of approximately 14.5 m. This is in good concordance with the pile displacement shown in Figure 4 where the point of zero displacement is located at approximately x = 15.0 m. From the calculated soil resistance along the pile and the measured pile deflection, p-y curves can be ∗ determined. From these the initial stiffness, Epy , can be estimated as the initial slope of the p-y curves, cf. Equation 2. In Figure 6 the initial stiffness along the pile for varying pile diameters of D = 1–7 m and an internal friction angle of ϕtr = 40◦ is shown. The initial stiffness increases non-linearly with depth which is in contrast to the linear dependency proposed by the design regulations, e.g. API (1993) and DNV (1992). Further, the initial stiffness increases for increasing pile diameter. This observation is also in contrast with the design regulations, where only the internal friction angle and the depth below soil surface are taken to affect the initial stiffness. At depths of approximately 13–15 m discontinuities in the initial stiffness can be

∗ Figure 6. Distribution of initial stiffness, Epy , along the pile for D = 1–7 m and an internal friction angle of ϕtr = 40◦ .

∗ Figure 7. Distribution of initial stiffness, Epy , along the pile for varying internal friction angles. The pile diameter is 4 m.

observed.These discontinuities are caused by the small pile deflections at these depths, as the points of zero deflection, are located at these depths, cf. Figure 4. In Figure 7 the initial stiffness of the p-y curves is shown for D = 4 m and varying values of the internal friction angle. The initial stiffness is highly dependent on the internal friction angle such that an increase in the internal friction angle results in an increase in initial stiffness. This observation is in concordance with the design regulations.

3

MODIFIED EXPRESSION FOR THE INITIAL STIFFNESS OF THE P-Y CURVES, E∗PY

Figure 6 and Figure 7 indicate that the initial stiffness depends on the depth below soil surface, the internal friction angle and the pile diameter. A modified ∗ expression for the initial stiffness, Epy , is therefore proposed in Equation 7 in which the initial stiffness depends on the depth below soil surface, the pile diameter and the internal friction angle. The expression

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Table 1. Values of the constants given in Equation 7. a [kN/m2 ]

b [–]

C [–]

d [–]

50000

0.6

0.5

3.6

has been determined on the basis of the numerical simulations employing the least square method.

In Equation 7 b, c and d are dimensionless constants, xref = 1 m, Dref = 1 m and a is a factor specifying the initial stiffness for D = 1 m, x = 1 m and ϕtr = 1 rad. Further, x and D should be inserted in meter and the friction angle in radians. The dependency with depth is proposed to be a nonlinear dependency as proposed by Lesny & Wiemann (2006) with a factor of b = 0.6. On the basis of the numerical simulations the constants a, c and d have been determined. The values of the factors are shown inTable 1. It should be mentioned that other results might be obtained for the factor a if a more advanced material model and a more advanced description of the interface between the pile and the soil are employed. Further, the driving of the pile might also influence a. Figure 8 shows the initial stiffness normalised with respect to ϕ3.6 for varying internal friction angles. The figure shows that when d = 3.6 the proposed expression for the initial stiffness provides a good description of the dependency of the internal friction angle in moderate depths. For D = [1;2;3;5;6;7] similar dependency on the internal friction angle have been observed. Respectively, a curve with a linear (b = 1) and two non-linear (b = 0.6; b = 0.3) variation of the initial stiffness is shown in Figure 8. All curves have been forced trough the average normalised initial stiffness in a depth of 1 m. From the figure it is seen that the linear variation highly overestimates the initial stiffness for large depths. Further, b = 0.3, is a lower limit. For b = 0.6 a good fit is obtained for moderate depths. However a slight overestimation of the initial stiffness is observed for large depths. Figure 9 shows the normalised initial stiffness when employing the proposed expression given in Equation 7. ϕtr = 40◦ and with D = 1–7. The pile bending stiffness has been held constant corresponding to D = 4 m and a wall thickness of 5 cm in order to exclude minor effects from the pile bending stiffness. The figure shows that the proposed expression for the initial stiffness of the p-y curves produces a good fit. Figure 10 and Figure 11 presents a comparison of the initial stiffness obtained from the numerical simulations with Equation 7. Equation 7 produces in general a good fit. However, deviations can be

Figure 8. Normalized initial stiffness with respect to ϕ3.6 for ϕ = 30–40◦ and D = 4 m.

∗ Figure 9. Normalised initial stiffness, Epy , for varying diameter. The internal friction is 40◦ and the magnitude of the pile bending stiffness corresponds to a pile diameter of 4 m and a wall thickness of 5 cm.

observed for large depths. The best fit between Equation 7 and the numerical simulations is obtained for internal friction angles of approximately 40◦ , which is typical for offshore sand. The initial stiffness proposed by the design regulations highly overestimates the initial stiffness in comparison with the numerical simulations.

4

CONCLUSIONS

This paper presents numerical simulations of largescale monopiles for offshore wind turbines. The numerical model is calibrated to small-scale experiments conducted in a pressure tank at Aalborg University and extended to simulate large-scale monopiles. The conclusions that can be drawn are:

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•

Non-slender piles behave as almost rigid objects, and for these piles significant negative deflections near the pile toe take place.

programme “Physical and numerical modelling of monopile for offshore wind turbines”, journal no. 033001/33033-0039. REFERENCES

∗ Figure 10. Comparison of the initial stiffness, Epy , for the nu-merical simulations, Equation 7 and the design regulations. The internal friction angle is 30◦ .

∗ Figure 11. Comparison of the initial stiffness, Epy , for the numerical simulations, Equation 7 and the design regulations. ◦ The internal friction angle is 40 .

•

The initial stiffness of the p-y curves depends on the pile diameter, the internal friction angle and the depth below mudline. A modified expression of the initial stiffness is proposed.

ACKNOWLEDGEMENTS The research has been funded by the Energy Research Programme istered by the Danish Energy Authority. The research is associated with the EFP

API, 1993. Recommended practice for planning, deg, and constructing fixed offshore platforms – Working stress design, API RP2A-WSD, American Petroleum Institute, Washington D.C., 21. edition. Ashford, S.A., & Juirnarongrit, T. 2003. Evaluation of Pile Diameter Effect on Initial Modulus of Subgrade Reaction, Journal of Geotechnical and Geoenvironmental Engineering, 129(3), 234–242. Carter, D.P. 1984. A Non-Linear Soil Model for Predicting Lateral Pile Response, Rep. No. 359, Civil Engineering Dept., Univ. of Auckland, New Zealand. Cox, W.R., Reese, L.C. & Grubbs, B.R. 1974. Field Testing of Laterally Loaded Piles in Sand, Proceedings of the Sixth Annual Offshore Technology Conference, Houston, Texas, 2079. DNV, 1992. Foundations – Classification Notes No. 30.4, Det Norske Veritas, Det Norske Veritas Classification A/S. Fan, C.C. & Long, J.H. 2005. Assessment of existing methods for predicting soil response of laterally loaded piles in sand, Computers and Geotechnics, 32, 274–289. FLAC3D 3.1 manual, 2006. Fast Langrangian Analysis of Continua in 3 Dimensions, Itasca Consulting Group Inc., Minneapolis, Minnesota, USA. Ibsen, L.B., Hanson, M., Hjort, T.H. & Thaarup, M. 2009. MC-Parameter Calibration for Baskarp Sand No. 15, DCE Technical Report No. 62, Department of Civil Engineering, Aalborg University, Denmark. Lesny, K. & Wiemann, J. 2006. Finite-Element-Modelling of Large Diameter Monopiles for Offshore Wind Energy Converters, Geo Congress 2006, February 26 to March 1, Atlanta, Georgia, USA. Ling, L.F. 1988. Back Analysis of Lateral Load Test on Piles, Rep. No. 460, Civil Engineering Dept., Univ. of Auckland, New Zealand. Sørensen, S.P.H., Brødbæk, K.T., Møller, M., Augustesen, A.H. & Ibsen, L.B. 2009. Evaluation of the LoadDisplacement Relationships for Large-Diameter Piles in Sand, Proceedings of The Twelfth International Conference on Civil, Structural and Environmental Engineering Computing, September 1 to September 4, Funchal, Madeira, Portugal, 244. Terzaghi, K. 1956. Evaluation of coefficients of subgrade reaction, Geotechnique, 5(4), 297–326. Ubilla, J., Abdoun, T. & Zimmie, T. 2006. Application of in-flight robot in centrifuge modeling of laterally loaded stiff pile foundations, Physical Modelling in Geotechnics, Taylor & Francis Group, London, 259–264.

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Numerical investigations for the pile foundation of an offshore wind turbine under transient lateral load P. Cuéllar BAM, Federal Institute for Materials Research and Testing, Berlin,

M. Pastor & P. Mira CEDEX, Centro de Estudios y Experimentación de Obras Públicas, Madrid, Spain

J.A. Fernández-Merodo IGME, Instituto Geológico y Minero de España, Madrid, Spain

M. Baeßler & W. Rücker BAM, Federal Institute for Materials Research and Testing, Berlin,

ABSTRACT: Numerical analysis can be useful for the investigation of important aspects of offshore foundation prototypes that otherwise could hardly be studied experimentally, like the evolution of pore-water pressure around the monopile foundation of an offshore wind turbine under extreme loading. A combination of mixed pressuredisplacement formulations along with a constitutive model for sands based on the Generalized Plasticity Theory can replicate accurately the soil behaviour in saturated conditions. However, additional issues must be taken into in order to perform numerical simulations of offshore piles. Some implications of the Babuska-Brezzi restriction, as well as considerations about the pile-soil interface and suitable solution strategies are discussed here. Due to the high cost of the transient analysis, the parallel computation offers a promising perspective, but can be complex and needs to be implemented carefully in order to avoid a performance deterioration. A brief overview on current trends and functional software is given here.

1

INTRODUCTION

Offshore wind energy converters must be able to resist the loads from the environment, in particular those arising from extreme weather conditions. Largediameter monopiles and tripods can be a suitable foundation for such structures but their performance under cyclic and extreme lateral loading has still to be assessed. The foundation of the structure can be particularly affected by the ing of an extreme storm and its stability must be ensured. In saturated soils, cyclic loading within the extreme regime often involves a pore-pressure build up that eventually can lead to liquefaction phenomena and foundation failure. The numerical simulations offer an insight into the fundamental behaviour of the system that is normally not available by means of experimental investigations. For instance, the opening of a gap at the pile-soil interface and the pore pressure evolution in the vicinities of the pile, in particular the possibility of liquefaction around it, are some of the key aspects that can be studied with a numerical model but are hardly traceable in model tests or in-situ conditions.

In the frame of the RAVE (Research at Alpha Ventus) research initiative of the German government for the first German offshore wind-farm, a number of experimental and numerical investigations are being carried out to gain knowledge and ensure the safety of future offshore wind-parks (Quell et al., 2007; Cuéllar et al., 2009). In this paper, some details are given concerning a numerical model for the 3D analysis of the offshore pile foundation and some preliminary results are discussed. The academic finite element code GeHoMadrid has been properly adapted to meet the requirements for this specific problem, and the parameters of the constitutive law for the saturated sand (the Pastor-Zienkiewicz model in the frame of the generalized plasticity) have been calibrated to reproduce the behaviour of the sand at the Alpha Ventus site in the North Sea. Important issues for the transient coupled simulations, such as pore-pressure oscillations, time consumption and selection of an efficient solution strategy, will be addressed in the following sections. Finally, an outlook to the parallelization of the FE code for a high-performance computing is also presented.

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2 2.1

BASIC ASPECTS FOR THE SIMULATIONS

2.2 Numerical model

Mathematical model

In order to achieve a realistic representation of the offshore pile behaviour, the first requisite is to use a suitable mathematical model to describe the main physical processes taking place within the saturated soil, in particular the interaction between the solid skeleton (the soil grains) and the pore fluid (the sea water). The equations of dynamic poroelasticity due to Biot have been extended and modified by Zienkiewicz and coworkers (e.g. in Zienkiewicz & Shiomi, 1984) in order to broaden their scope and ease their implementation within numerical models. The model included in GeHoMadrid is the so-called u-pw formulation described by Zienkiewicz et al. (1999), which expresses the governing equations in of only two variables, namely the displacements u of the solid matrix and the pressure pw of the pore fluid. These equations, which do not consider convective and are based on the assumption of negligible relative accelerations between solid and fluid, can be summarized as follows (Fernandez Merodo, 2001): (i) Balance of linear momentum for the solid-fluid mixture:

where σ is the vectorial form of the effective stress tensor, vector m represents the second-order Kroneckerdelta, ρm is the mass density of the solid-fluid mixture, b is the vector of body forces per unit mass, ü is the acceleration of the solid skeleton and S the vectorial form of the strain operator. (ii) Combination of the equations for fluid mass conservation and fluid linear momentum balance:

where kw is the permeability matrix, Q∗ is the coupled volumetric stiffness of solid grains and fluid, and ρw is the specific weight of the pore fluid. (iii) A suitable constitutive equation for the soil skeleton (see section 2.3 below):

where Dep is the constitutive operator and ε is the strain vector. (iv) and the kinematic relationships between displacements and strains:

Finally, an appropriate set of boundary conditions in of u and pw needs to be introduced in order to define completely the problem under consideration (see, e.g., Mira, 2001).

The continuous partial differential equations presented above can be converted to ordinary differential equations in a discrete form using standard Galerkin techniques (see for instance Zienkiewicz & Taylor, 2000). Introducing two appropriate sets of shape functions Nu and Np for the spatial interpolation of the displacement and pressure fields, and employing a generalized Newmark scheme for the time discretization (GN22 for displacements and GN11 for pore pressures), a non-linear system of equations with discrete variables (in both time and space) can be obtained out of the equations above. Then, for the computation of every time-step t, the non-linear system of equations can be solved iteratively using an appropriate algorithm, typically of the Newton-Raphson type, which results in the following linear system of equations:

where the matrices KT , M , Q, H and C stand for the usual tangent stiffness, mass, coupling, permeability and compressibility respectively, as defined e.g. in (Zienkiewicz et al., 1999), and the constants β1 , β2 and θ are the parameters of the Newmark scheme for time integration. The vectors δ(ü) and δ(p˙ w ) contain the iterative corrections to the variables, while u and p are the residuals. The subscripts between parentheses denote the step of the iterative process, which is to be continued until a suitable tolerance criterion is fulfilled. Further details about the attainment of these equations are out of the scope of this paper and can be found, for instance, in (Pastor & Tamagnini, 2002; Mira, 2001). At this point it is important to note that, although the second set of equations in (5) can be multiplied by a scalar in order to achieve symmetry in the coupling , the overall symmetry of the linear system of equations will ultimately depend on the symmetry of the stiffness matrix KT . This matrix is defined as

and it will, in general, be asymmetric whenever a nonassociative constitutive law is used, as is normally the case for modeling cohesionless soils. The matrix B is the discrete form of the strain operator. As it will be shown later, the lack of symmetry of the linear system of equations carries important consequences for the choice of a solution strategy. 2.3 Constitutive model Along with the mathematical and numerical models described so far, the third main ingredient for the numerical analysis of the foundation’s behaviour is

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the choice of an appropriate constitutive model for the sand. One of the particular features of cohesionless soils is their tendency to contract when they are subject to cyclic loading. In saturated soils, this cyclic densification can lead to an increment of the pore pressure if the permeability of the soil is low or the drainage of pore water is somehow inhibited. Such increments in pore pressure will reduce the effective stress within the soil skeleton and eventually can lead to a total loss of resistance (liquefaction) with potential catastrophic consequences (Pastor et al., 2009). Therefore it is essential that the constitutive model can reproduce the real behaviour of sand. In particular, three key issues need to be considered, namely, a non-associative flow rule, a hardening law not only dependent on the soil density, and the possibility of plastic strains upon unloading. The Generalized Theory of Plasticity, initially proposed by Zienkiewicz and Mroz (1984) and later extended by Pastor and Zienkiewicz (1986) offers a convenient framework for the consideration of such features. In its basic form, it relates the increments of stress (dσ) and strain (dε) as follows:

where the first summand provides the elastic strain through the use of an elastic constitutive tensor Ce , and the second term introduces the plastic strain in dependence of a scalar H (the plastic modulus) and the product of two normalized vectors n and ng . The loading direction n discriminates the stress increments between loading and unloading, and the plastic flow direction ng defines the direction of the plastic strains. In order to achieve irreversible plastic deformations within a closed stress cycle, H and ng need to be defined differently for loading and unloading states, and hence the subscript L or U in eq. (7). The Pastor-Zienkiewicz Mark III model (in the following referred to as PZ model) suitably defines the three directions and the scalar functions in (7) to accurately reproduce the behaviour of sand (see Figure 1). In particular, the definition of the plastic modulus HL , here shown in eq. (8), permits the hierarchical consideration of single aspects of sand behaviour, like for instance the existence of a critical line where all the residual stress states lie or the fact that failure does not necessarily occur when this line is first crossed.

This way, HL incorporates the following ingredients: (i) a pressure dependence through the effective confining stress p , (ii) an isotropic plastic modulus H0 , (iii) a “frictional” factor Hf that limits the possible stress states within the sand, (iv) a volumetric strain hardening function Hv with a dependence on the mobilized stress ratio which makes it zero at the critical state line, (v) a deviatoric strain hardening Hs , which models the material degradation by accumulated strains and permits the crossing of the critical state line without

Figure 1. Experimental and computed results of monotonic undrained triaxial tests with samples of Berliner sand. Test results taken from (Rackwitz, 2003). Solid lines show the predictions with the PZ model.

causing immediate failure, and finally (vi) a discrete memory factor HDM that s for the effects of past events upon cyclic reloading. Additional details can be found in the references mentioned above. Altogether, a set of 12 parameters needs to be determined in order to fully characterize the sand at a given pressure and density. A useful extension to the model introducing a new state parameter has been recently proposed by Manzanal (2008), which permits a unified definition of the material parameters valid for the full range of pressures and densities. The estimation of the model parameters can be done by means of representative tests, like monotonic and cyclic triaxial tests, as described for instance in (Chan, 1988; Manzanal, 2008). 3

SPECIFIC REQUIREMENTS FOR THE SIMULATION OF AN OFFSHORE PILE UNDER LATERAL LOAD

3.1 Pile-soil interface and gapping In order to model adequately the laterally loaded offshore pile, several additional requisites must be met. Apart from the fact that only a full 3D model can be used, since the axial symmetry is no longer valid due to the lateral loading, special care has to be taken when modeling the pile-soil interface. An important feature of laterally loaded piles is the possible formation of a gap behind the pile as the loading progresses, and, therefore, the ability to model a discontinuous interface has to be somehow included. A natural way to do this is to define the interaction between pile and soil as a multi-body problem, as described for instance in (Belytschko et al., 2000), with the introduction of the “master-slave” surface concepts and an efficient detection algorithm.

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Figure 2. Gap formation behind a laterally loaded pile. Pile-soil interface implemented by means of no-tension t elements.

An alternative approach, which can be conveniently implemented into a FE code without major changes, is the use of special interface “t” elements to allow for differential movement (slip and separation) between pile and soil (Potts & Zdravkovic, 1999). The isoparametric t element, described e.g. by Carol and Alonso (1983), can be suitably adapted to reproduce no-tension conditions (i.e. gap opening) by simply defining a non-linear normal stiffness (discontinuous in this case) with different values for tension and compression. However, t elements might present problems of numerical instability and ill-conditioning and should be handled with care (see e.g. Day & Potts, 1994). The use of such elements for a pile interface in cohesive soil is shown exemplarily in Figure 2. 3.2

Element technology to avoid pore pressure instabilities

Due to the high computational cost of the 3D modeling, it appears desirable to use low-order elements like the classical 8-node hexahedron in order to keep the total number of degrees of freedom within affordable margins. However, the use of such elements entails some limitations, as poor bending behaviour or locking phenomena under certain conditions (see for instance Pastor & Tamagnini, 2002). On the other hand, the coupling with pore water pressure brings about an additional restriction, namely the so-called Babuska-Brezzi condition, which states that the order of interpolation for the displacement field must be higher than that for the pressure field if the permeability is very low (nearing undrained incompressible conditions). In such cases, both the permeability and compressibility matrices H and C in equation (5) tend to zero, producing a system of equations in the form of

which will be singular and present pore pressure oscillations unless the number of variables in ξ is greater than in φ or special stabilization techniques are employed (Pastor & Tamagnini, 2002). This leads to the dilemma of either using the “expensive” quadratic interpolation for displacements (e.g. the 20-node hexahedron with 8 nodes for the pressure field h20p8), or resorting to some special techniques, like the stabilized elements from fluid mechanics (see the divergence, fractional-step or alpha methods, e.g. in Brezzi & Pitkaranta, 1984; Pastor et al., 1996) or special implementations of enhanced-strain elements as shown in (Mira et al., 2003). The latter provide the additional advantages inherent to the enhanced-strain elements, namely a better performance in bending and the prevention of locking problems (Mira, 2001). At this point it is important to stress that the term “very low permeability” is a relative one, in particular relative to the element size. Even high permeability values typical of clean sands (around 10−4 m/s) will cause pore pressure instabilities if the element size is big enough, like for instance when modeling real offshore prototypes which might incorporate element dimensions of half a meter or bigger. Therefore, the possibility of pressure oscillations should never be excluded “a priori”, regardless of the soil permeability. In this respect, the use of high-order or special element implementations might be unavoidable for the particular case of real-size offshore foundation prototypes. 3.3

Solution strategies and computational cost

The main drawback of 3D modeling is normally the huge amount of degrees of freedom being considered and the consequent size of the linear system of equations. The assembled matrices can easily include tens or hundreds of thousands of rows and columns, and non-linear transient simulations will usually require that the system of equations be solved repeatedly, often involving several iterations for every time step. Therefore, it is of paramount importance the choice of an appropriate solution strategy. A crude and inefficient method for solving the sparse linear system Ax = b when A is square and nonsingular is to compute the inverse A−1 . This is numerically unstable when A is ill-conditioned and very costly, since normally the inverse of a sparse matrix has no zero entries at all (Davis, 2006). In principle, iterative solvers like the preconditioned conjugate gradient or the Jacobi methods, are adequate choices for large problems and can be programmed easily using Fortran or C (Pastor & Tamagnini, 2002). However, they are only conditionally convergent and in general cannot deal with non-symmetric matrices, which makes them unsuitable for problems with non-associative materials like sand. In order to solve non-symmetric systems of equations, special schemes such as the generalized minimum residual (GMRES) will be required. On the other hand, direct solvers like the Gaussian elimination method are in principle unconditionally

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stable and require relatively few changes to deal with non-symmetric systems, but as the size of the system grows their computational and storage cost can become prohibitive. As shown in (Dongarra et al., 1998), the direct solution of a system of order n requires O(n2 ) storage and O(n3 ) floating-point operations. To alleviate this problem, the sparsity of the matrix can be exploited in order to reduce the storage and number of operations, for instance with special storage schemes (like the skyline or the compressedcolumn) or by additionally performing suitable row and column permutations to reduce the amount of fill-in during factorization. The aim of the sparse algorithms is to perform the solution in a number of operations proportional to O(n) + O(τ), being τ the number of non-zero entries of the matrix. In any case, regardless of the type of solver to be used, the management of data traffic and memory locality can have a decisive impact on the overall efficiency of the algorithms. The performance of a solver can indeed be dominated by the amount of data traffic rather than the number of operations involved (Dongarra et al., 1998). Most computers nowadays include temporary fast-access blocks of memory called caches, where frequently-used data is stored. Since the data flow from the main memory is very expensive (has a much longer access time), an efficient use of the caches can be critical for the performance. This has motivated a restructuring of existing algorithms and new methods that minimize the data movement have arisen. Among these, the BLAS (Basic Linear Algebra Subprograms) constitute a key building block that provides efficient subroutines for basic vectorial and matrix operations. This way, it can be very advantageous to organize the calculations so that the matrices are partitioned into small blocks that fit in the caches and then perform the computations by matrix-matrix operations on the blocks. This constitutes the basis for the so-called supernodal and multifrontal approaches for the LU factorization, which can obtain very high performances and have been described in (Davis, 2006). However, this might not be enough in view of the high cost of the transient calculations proposed here, where, due to the sensitivity of the constitutive model, hundreds of time steps and several iterations per time step might be required for the computation of a single load cycle. In this respect, further reductions in computational time can be achieved by the use of high-performance multi-processor computing. 3.4

parallel processing does not necessarily imply fast and efficient computing. For instance, in a typical FE calculation with tens of thousands degrees of freedom, more than 97% of the computing time can be spent in a single loop for the dot product, which is required for the LU factorization of the linear system, but a direct parallelization of that loop will surely be counterproductive, since the workload of each dot product hardly justifies the associated overhead (i.e. the time spent in creating the threads, synchronizing the jobs and gathering the results at the end). As the function for the dot product is constantly being called, the total overhead will by far overshadow any parallel gains in performance. Therefore, instead of local parallelization of key loops (fine grain parallelization), a more general division of computational tasks should be pursued (coarse grain parallelism). In general, the sparse linear system of equations can show three inherent levels of parallelism: (a) Parallelism at a system level, where the underlying problem (for instance the partial differential equations or the physical structure) can be divided into a set of small subproblems through domain decomposition or substructuring, (b) parallelism at matrix level, where sparsity can lead to simultaneous operations taking place in independent parts of the matrix, and (c) parallelism at submatrix level, where the dense submatrices of the overall sparse system can be treated with the parallel techniques of the dense linear algebra (level 3 BLAS) (Dongarra et al., 1998). Parallelization at the system level has normally the greatest potential for performance gains but is problem-dependent and usually involves major preprocessing tasks in order to achieve an efficient division of the system. A survey on domain decomposition techniques and parallel meshing is given, for instance, in (Chrisochoides, 2005). On the other hand, parallelization at the matrix and submatrix levels can be costly and involve complex algorithms, but in the past years it has been subject of extensive research and now there exists a number of efficient packages that can be used as external libraries for the solution of the linear system (see Davis, 2006, for a comprehensive summary of parallel packages with direct solvers for sparse linear systems). In particular, the non-commercial external libraries SuperLU (Demmel et al., 2009) and MUMPS (Amestoy et al., 2009) do include both sequential and parallel algorithms for an efficient LU factorization of nonsymmetric matrices (supernodal or multifrontal methods) and the subsequent solution of the linear system.

Parallelism

A couple of de facto standards have arisen for the parallel computation with shared-memory systems (multi-processor workstations) and distributed-memory systems (computing clusters), namely the OpenMP and the MPI standards, respectively. They provide a robust framework for parallelization that can be easily incorporated into Fortran and C codes, but require a careful planning of the task division, since

4 A PRACTICAL APPLICATION Monopiles of large diameter (up to 8 meters in diameter) are being studied as feasible solutions for the foundation of offshore wind turbines (Lesny, 2008). In the frame of the RAVE research project, numerical simulations of a laterally loaded monopile with an embedment length of 30 m and an outer diameter

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Figure 3. Excess of pore-water pressure generated around a monopile under sinusoidal lateral loading. Dark and pale contours denote excess and decrease over the hydrostatic pressure respectively.

of 8 m are being undertaken by the authors. Figure 3 shows some preliminary results depicting the excess of pore-water pressure generated around the pile at a cyclic load peak. The calibration of the sand model was done with the triaxial test results presented in (Rackwitz, 2003) for Berliner sand, which is very similar to the typical siliceous sands of the North Sea that can be found off the German coasts. For the preliminary calculations, inertia effects have been neglected and the loading produced by an extreme storm has been introduced as a sinusoidal wave with a frequency of 0.1 Hz and an amplitude of 5 MN applied to the pile-head, 30 m above the seabed. It should be noted that although in this case the cohesionless nature of the sand prevents the development of a gap behind the pile, tangential differential displacements between pile and soil (i.e. slip) do indeed take place at the interface. During a loading peak, maximum pore-water overpressures ranging 35 kPa can be observed taking place at a depth around 4 meters under the soil surface. It can also be observed that the relatively low aspect ratio of the pile (L/D ∼ 4) makes it behave as a short stiff pile, with a displacement pattern similar to that of a rigid body. That implies that under extreme lateral load, the pile base also experiences some lateral displacements, inducing additional pore pressures as well. Further details about the pore-pressure evolution and the danger of soil liquefaction, including a parametrical study of the system, are part of an on-going doctoral project and shall be published soon.

5

CONCLUSIONS

Mixed pressure-displacement formulations provide a comprehensive description of coupled systems that can be suitably implemented into a FE code. However, the quality of the calculations will ultimately depend on the ability of the constitutive model to reproduce

the real behaviour of the soil. A model for sands based on the Generalized Plasticity Theory and calibrated with suitable laboratory tests can replicate accurately the sand behaviour. Nevertheless, additional issues must be taken into in order to perform numerical simulations of offshore piles. Apart from the element technology to deal with the Babuska-Brezzi condition and the provision of an appropriate pile-soil interface model, a suitable solution strategy must be adopted for the transient calculations, where the linear system of equations will have to be solved thousands of times. In this respect, the parallel computation offers a promising perspective, but needs to be implemented carefully in order to avoid a performance deterioration. A brief overview on current trends and functional software has been presented here. Finally, a practical application shows that the numerical model can be useful for the analysis of important aspects of offshore foundation prototypes that otherwise could hardly be investigated.

ACKNOWLEDGEMENTS Special gratitude is due to the research group at the Geotechnical Laboratory of CEDEX in Madrid, as well as to Stavros Savidis, Frank Rackwitz and Ercan Tasan from the Technical University of Berlin for the insightful discussions and kind provision of experimental data. The German Federal Ministry for the Environment (BMU) is gratefully acknowledged for the funding of these investigations. REFERENCES Amestoy, P., Buttari, A., Guermouche, A., L’Excellent, J.-Y. & Ucar, B. 2009. Multifrontal massively parallel solver (MUMPS 4.9.1) s’ guide. CERFACS. Belytschko,T., Liu, W. K. & Moran, B. 2000. Nonlinear Finite Elements for continua and structures. John Wiley & Sons, Ltd. Brezzi, F. & Pitkaranta, J. 1984. On the stabilization of finite element approximations of the Stokes problem. In: Efficient solutions of elliptic problems, notes on numerical fluid mechanics, 10, W. Hakbusch, ed., 11–19. Carol, I. & Alonso, E. E. 1983. A new t element for the analysis of fractured rock. Proc. 5th. Int. Congress on Rock Mech., Melbourne, 147–151. Chan, A. 1988. A unified finite element solution to static and dynamic geomechanics problems, (Thesis), University College of Swansea. Chrisochoides, N. 2005. A survey of parallel mesh generation methods. Brown University. Cuéllar, P., Baeßler, M. & Rücker, W. 2009. Ratcheting convective cells of sand grains around offshore piles under cyclic lateral loads. Granular Matter, 11(6), 379–390. Davis, T. A. 2006. Direct methods for sparse linear systems. Society for Industrial and Applied Mathematics. Day, R. A. & Potts, D. M. 1994. Zero thickness interface elements – numerical stability and application. Int. J. for Numerical and Analytical Methods in Geomechanics, 18, 689–708.

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Demmel, J. W., Gilbert, J. R. & Xiaoye, S. L. 2009. SuperLU s’ guide. Computer Science Division, University of California. Dongarra, J. J., Duff, I. S., Sorensen, D. C. & van der Vorst, H. A. 1998. Numerical linear algebra for high-performance computers. Society for Industrial and Applied Mathematics. Fernandez Merodo, J. A. 2001. Une approche a la modelisation des glissements et des effondrements de terrains: Initiation et propagation, (Thesis), Ecole Centrale Paris. Lesny, K. 2008. Gründung von Offshore-Windenergieanlagen Entscheidungshilfen für Entwurf und Bemessung. Bautechnik, 85(8), 503–511. Manzanal, D. G. 2008. Modelo constitutivo basado en la teoría de la plasticidad generalizada con la incorporación de parámetros de estado para arenas saturadas y no saturadas, (Thesis), Universidad Politecnica de Madrid, ETSIC, Madrid (Spain). Mira, P. 2001. Análisis por elementos finitos de problemas de rotura de geomateriales, (Thesis), Universidad Politecnica de Madrid, ETSIC, Madrid (Spain). Mira, P., Pastor, M., Li, T. & Liu, X. 2003. A new stabilized enhanced strain element with equal order of interpolation for soil consolidation problems. Computer Methods in Applied Mechanics and Engineering, 192(37–38), 4257–4277. Pastor, M., Manzanal, D. G., Fernandez Merodo, J. A., Mira, P., Blanc, T., Drempetic, V., Pastor, M. J., Haddad, B. & Sanchez, M. 2009. Form solids to fluidized soils: Diffuse failure mechanisms in geostructures with applications to fast catastrophic landslides. Granular Matter. 10.1007/s10035-009-0152-4. Pastor, M., Quecedo, M. & Zienkiewicz, O. C. 1996. A mixed displacement-pressure formulation for numerical analysis of plastic failure. Computers & Structures, 62(1), 13–23.

Pastor, M. & Tamagnini, C. (eds.) 2002. Numerical modelling in geomechanics. Revue francaise de génie civil, Hermes Science Publications. Pastor, M. & Zienkiewicz, O. C. 1986. A generalized plasticity, hierarchical model for sand under monotonic and cyclic loading. 2nd Int. Symp. on Numerical Models in Geomechanics, Ghent, Belgium, 131–150. Potts, D. M. & Zdravkovic, L. 1999. Finite element analysis in geotechnical engineering: Theory. Thomas Telford. Quell, P., Knops, M., Heinicke, M., Rettenmeier, A., Kühn, M., Rücker, W., Baeßler, M., Rolfes, R., Haake, G., Hahn, B., et al. 2007. Offshore wind energy research in – RAVE – research at alpha ventus. European Offshore Wind 2007 Conference & Exhibition, December 4–6, 2007, Berlin. Rackwitz, F. 2003. Numerische Untersuchungen zum Tragverhalten von Zugpfählen und Zugpfahlgruppen in Sand auf der Grundlage von Probebelastungen, (Thesis), Veröffentlichungen des Grundbauinstitutes der Technischen Universität Berlin. Zienkiewicz, O. C., Chan, A. H. C., Pastor, M., Schrefler, B. A. & Shiomi, T. 1999. Computational geomechanics, with special reference to earthquake engineering. John Wiley & Sons Ltd. Zienkiewicz, O. C. & Mroz, Z. 1984. Generalized plasticity formulation and applications to geomechanics. In: Mechanics of engineering materials, C. S. Desai & R. H. Gallagher, eds., Wiley, 655–679. Zienkiewicz, O. C. & Shiomi, T. 1984. Dynamic behaviour of saturated porous media: The generalized Biot formulation and its numerical solution. Int. J. for Numerical and Analytical Methods in Geomechanics, 8, 71–96. Zienkiewicz, O. C. & Taylor, R. L. 2000. The Finite Element Method., Butterworth-Heinemann.

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Numerical study of piping limits for installation of large diameter buckets in layered sand L.B. Ibsen Aalborg University, Denmark

C.L. Thilsted Dong Energy Power, Denmark

ABSTRACT: The bucket foundations, often referred as ‘suction caissons’, are large cylindrical structures, typically made in steel. The bucket foundations have the potential to be the cost-effective option for offshore wind turbines, if suction assisted penetration is employed. Suction installation may cause formation of piping channels, which break down the hydraulic seal and prevent further installation. This paper presents a numerical study of failure limits during suction installation in respect to both homogenous and layered soil profile. A numerical flow analysis is performed to determine the hydraulic gradients developing in response to applied suction and the results are presented as simple closed form solutions useful for evaluation of suction thresholds against piping. These close form solutions are compared with large scale model test, performed in a natural seabed at a test site in Frederikshavn, Denmark. These solutions are also valid for penetration studies of other offshore skirted foundations and anchors using suction assisted penetration in homogeneous or layered sand. Due to the complexity of the domain and the governing differential equation, the problem is solved numerically. A numerical solution can be obtained using either finite difference or finite element methods. In this paper, the problem is solved using the commercial finite difference program FLAC3D (Itasca, 2005).

1

INTRODUCTION

This study has been a part of a research project whose aim is to develop a bucket foundation as a foundation for offshore wind turbines. The bucket foundation, often referred to as “suction caisson”, is a large cylindrical structure, typically made of steel, see Figure 1. Depending on the skirt length and diameter, the bucket has a moment resistance equivalent to a monopile, a gravity foundation or in between. The bucket is installed using suction assisted penetration where suction is applied within the caisson after an initial penetration into the seabed caused by self-weight. The suction creates a pressure differential across the caisson lid, effectively increasing the downward force on the caisson while reducing the skirt tip resistance. Suction assisted penetration has the potential to significantly reduce installation costs, since large jack-up, driving and drilling equipment can be avoided. At the time of writing, two bucket foundation have been installed, one at Horns Rev II and the other located in Frederikshavn, Denmark, (Ibsen (2008); however, it is likely that there will be more in the near future. The suction installation technology was originally introduced by Shell (Senepere andAuvergne 1982) and is currently widely used for suction anchor piles and skirted foundations within the oil and gas offshore industry.

The installation of bucket foundation for offshore wind turbines differs for several reasons. Compared to oil and gas jackets, the bucket foundation offers less self-weight to assist penetration and sites for installation are predominantly located at shallow waters, <30 m, which reduces the maximum available suction capacity. The bucket foundation is a large diameter moment resistant structure and its cost efficiency is significantly improved by increasing the ratio of skirt length L over diameter D to approximately L/D ≈ 1 while the wall thickness t is kept at a minimum. The geometric definitions are shown in Figure 2. This paper presents a numerical study of the installation of large diameter thin-walled suction caissons in sand. The objective is to evaluate suction failure limits during installation in respect to piping in both homogeneous and layered sand. Steady-state flow analyses were performed to determine the flow and the hydraulic gradients developed in response to applied suction beneath the caisson lid. The results are presented as simple closed form solutions, valid for a wide range of boundary conditions, and useful for evaluation of suction thresholds against piping in homogeneous or layered sand. These closed form solutions are compared with a large scale field test, installed in a natural seabed at the test site in Frederikshavn, Denmark.

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Figure 3. The critical suction has been achieved and soil failure by piping has occurred. The test was performed with a 4 × 4 m bucket.

Figure 1. Offshore wind turbine installed on a bucket foundation.

Figure 4. T test performed prior to the installation of the buckets.

Figure 2. Definition of dimensions.

2

FIELD TEST DATA

Since installation data from field installation of suction caissons in sand are limited this project has conducted a substantial amount of installation tests on 2 × 2 m and 4 × 4 m buckets which have been performed at the offshore test site in Frederikshavn, Denmark, Ibsen (2008). One of the focus points for these installation tests has been to study the critical suction causing piping. Failure during suction assisted installation occurs when certain thresholds are exceeded. The failure

may be caused by either formation of piping channels or cavitation of pore water. The formation of piping channels occurs when the applied suction increases and causes an upward flow, reducing the effective stresses within the caisson, and eventually liquefying parts of the internal soil matrix. Local piping channels break down the hydraulic seal and prevent further installation, as shown in Figure 3. Three installation tests are studied in this paper. They are all installed in a 13 m × 14 m basin and the T test performed prior to the installation is shown in Figure 4. The applied suction p needed to install the 2 m × 2 m buckets can be seen in Figure 5. In the figure, the normalized suction p/γ D is plotted against the normalized penetration depth h/D where γ is the submerged unit weight of the soil and D is the diameter of the bucket. In the figure 5 it is also seen that installation failure by piping did occur during the installation of bucket 4, at a depth 1.56 m. The piping channels

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Figure 6. Schematic illustration of the axisymmetric flow domain during suction installation.

(r, z, φ) due to the circular geometry of a suction caisson:

Figure 5. The applied suction p needed to install the three buckets. The diameter and shirt length is 2 m × 2 m.

were filled with sand and the outer soil surface leveled, in order to restart the installation. A new failure occurred at a depth of 1.7 m and the test was stopped. Figure 5 shows that the suction needed to install bucket 5 is higher than the suction resulting in piping during the installation of bucket 4. The only difference between the installation tests is the presence of silt layers, see Figure 4. • •

Bucket 2 is installed in a homogeneous sand layer. Bucket 4 is installed where one thin silt layer is present at a depth of 2.7 m. • Bucket 5 is installed in a layered soil profile with thin silt layers at depth of 1.2, 2.4 and 3.5 m. It is assumed that these thin silt layers act as flow boundaries and change the steady-state flow field around the skirt tip as it approaches the layer. The theory is that the presence of these flow boundaries will increase the suction thresholds against piping as it was observed from the installation test with Bucket 5. The influence of the flow boundary is modeled and studied in the following sections.

3

NUMERICAL MODEL

The term (1/r2 )∂2 u/∂φ2 vanishes due to the axissymmetry of the caisson. The differential equation must be solved with appropriate boundary conditions to determine the hydraulic gradient field which arises from the pressure difference, between the ambient seabed water pressure, γw hw + pa and the pore pressure beneath the lid, γw hw + pa + p. pa is the atmospheric pressure. Due to the complexity of the domain and the governing differential equation, the problem is solved numerically. A numerical solution can be obtained using either finite difference or finite element methods. In this paper, the problem is solved using the commercial finite difference program FLAC3D An axisymmetric model was created with a grid consisting of a total of 5,904 zones and an outer boundary located, in the distance, 20R the caisson, as shown in Figure 6. The case where L → ∞ is simulated as L = 20R. The boundary conditions along the caisson skirt, the bottom boundary and the axisymmetric axis are Neumann’s conditions, preventing a flow orthogonal to the boundary. The boundary conditions of the soil surface in the caisson, the free surface and the outer boundary are Dirichlet conditions with prescribed pore pressures. The steady-state flow model computes the exit hydraulic gradient i next to the skirt and that gradient is used to calculate the seepage length s in of the applied suction p as:

The normalized seepage length s/h is a unique function of the relative penetration length h/D.

Excess pore pressure, as a result of suction p inside the bucket, causes a steady-state flow field to evolve in the soil, as shown in Figure 6. This yields a constant influx of water, which must be pumped out to maintain a constant level of suction. Assuming isotropy the seepage problem reduces to the well-known Laplace’s differential equation, ∇ 2 h = 0. It may conveniently be expressed in of pore pressure, u = γw h and cylindrical coordinates

4

NUMERICAL RESULTS

The steady-state flow simulations were conducted for two different cases at various embedment depths 0.1D > h > 1.2D. In the first case, simulations were conducted to investigate bucket installation in homogeneous soil, the results are shown in Figure 7a. The

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Figure 7. The results of the FLAC calculation are plotted as normalized seepage length for exit gradient versus relative penetration. a) Installation in homogenous sand. b) Installation in sand over a flow boundary.

Figure 8. Seepage length for exit gradient versus relative penetration predicted by equation (5), (6) and (7).

second case simulates a bucket installed in sand over a flow boundary, located in the depth L . The results are shown in Figure 7b. 4.1

Installation in homogeneous sand

The following empirical expression is given to approximate the numerical data for the installation in homogeneous sand.

Equation (5) is fitted to two boundaries. For a very small h/D ratio, equation (3) approaches 2.86, a theoretical solution for a sheet-pile wall, suggested by Hansen (1978). For an infinitely long bucket, all the

Figure 9. Normalized critical suction versus relative penetration. The critical suction is calculated with different ratios L /D.

hydraulic head loss occurs inside the bucket with evenly spaced horizontal equipotential lines. Therefore, the normalized length tends to unity. For installation in homogenous sand the internal hydraulic gradients have been investigated by several researchers using finite element programs as Plaxis and SEEP. Senders & Randolph (2009) performed calculations with the finite element program. Plaxis and propose a similar expression for the exit gradient:

For very small h/D ratio equation (4) approaches π, which is a theoretical solution for a sheet-pile wall, suggested by Scott (1963).

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Figure 10. Installation tests analyzed using equation 12 with the flow boundaries interpret from the T tests in Figure 4.

Feld (2001) performed calculations with the finite element program SEEP and proposed that the seepage length could be estimated as:

Figure 8 show that these three different formulations predict similar seepage length for penetrations of practical interest 0.1 ≤ h/D ≤ 1. 4.2

Installation in sand over a flow boundary

The following empirical expression is given to approximate the numerical data for the installation in layered sand:

where (s/h)ref is calculated from equation (3). It is seen that equation (6) approaches equation (3) if the distance to the flow boundary L is large in comparison to the diameter of the bucket D. 5

The exit hydraulic gradient i can also be expressed in of the applied suction p and the seepage length s as:

where γw is the unit weight of water and γ is the submerged unit weight. The critical suction resulting in formation of local piping channels are therefore

By combining equation (6) with equation (9) the critical suction can be expressed as:

Figure 9 shows the critical suction calculated by equation (10) with different ratios L /D. If L /D is large then the critical suction approaches the threshold for penetration in homogeneous sand. It is also seen that the presence of a flow boundary will increase the threshold where critical suction will occur. 6

CRITICAL SUCTION

The formation of local piping channels occurs when the exit hydraulic gradient, next to the caisson wall, exceeds the gravitational force, and thereby reduces the effective stresses to zero. The critical gradient is:

PREDICTION OF FIELD TEST DATA

In Figure 10, the suction needed to install the bucket is plotted against equation (3) and (10). The figure shows that suction close to or higher than critical, predicted by equation (3), can be applied without significant consequences. This is particularly seen in the installation test with bucket 5. It is seen that the suction needed to overcome the resistance during the installation of the bucket 2 never

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violated the critical suction predicted by equation (10) with the flow boundary at 2.7 m. This was not the case in the installation test with bucket 4. At a depth of 1.56 m the applied suction violated the failure criterion predicted by equation (10) and piping channels were formed and observed during the test. At the test with bucket 5 the flow boundary was at a depth of 1.2 m. This increases the suction capacity and the bucket was penetrated with the highest applied suction without any failure occurring. It is shown that these thin silt layers act as flow boundaries and increase the suction thresholds against piping. 7

CONCLUSION

By comparing the numerical studies with the installation tests it is shown that it is the exit gradient next to the skirt which controls when piping will occur. For installation in homogeneous sand, the internal hydraulic gradients have been investigated by several researchers using programs as Plaxis, SEEP and FLAC. These studies have resulted in different formulations, but the empirical expressions predict similar critical suctions for skirt penetrations of practical interest. However, experience from installation of prototype foundations have shown that gradients close to critical, predicted by the expressions for homogenous sand, can be applied without significant consequences. The same was observed in the field test reported in this paper. It is stated that the presence of thin silt layers will act as flow boundaries and increase the suction thresholds against piping.

The influence of the flow boundary was studied in this paper. The results are presented as simple closed form solutions and shown to predict thresholds against piping in homogeneous or layered sand. Future studies have to be performed in order to establish the thresholds against piping when the skirt penetrates through a flow boundary. REFERENCES Erbrich, C. T., Tjelta T. I. (1999) “Installation of bucket foundations and suction caissons in sand: geotechnical performance.” Proc., Offshore Technology Conf., Houston, Texas, Paper OTC 10990. Feld, T (2001). “suction bucket, a new innovative foundation concept applied to offshore wind turbines.” Aalborg university, Aalborg. Hansen, B. (1978). Geoteknik og fundering del II. Laboratoriet for fundering. DTH. (In Danish). Ibsen, L.B (2008). Implementation of a new foundations concept for Offshore Wind farms. Proc. Nordisk Geoteknikermøte nr. 15 NGM 2008, 3–6 September 2008 Sandefjord, Norge, 1–15. Itasca (2005). “FLAC3D – Fast lagrangian analysis of continua: Fluid-Mechanical Interaction”, Itasca Consulting Group Inc., Minneapolis, USA. Scott, R.S. (1963). “Principles of soil mechanics. AddisonWesly Publiching Company, Inc. Senders. M., Randolph M. F.,(2009) “T-Based Method for the Installation of Suction Caissons in Sand” Jour. of Geotechnical and Geoenvironmental Enginnering. Senepere, D., Auvergne, G. A. (1982) “Suction anchor piles – a proven alternative to driving or drilling.” Proc., 14th Offshore Technology Conf., Houston, Texas, 483–493.

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Shallow circular foundations under undrained general combined loading in three-dimensional space B. Bienen Centre for Offshore Foundation Systems, University of Western Australia, Perth, Australia

ABSTRACT: The behaviour of shallow foundations under combined loading has been a topic of geotechnical research for some time. While the modelling framework has moved away from considering ultimate load only to treating combined loading through force-resultant models based on plasticity theory, for undrained loading conditions these have only been developed for vertical, horizontal and moment loading (i.e. the three degrees-offreedom) in two-dimensional space. Although an extension to all six degrees-of-freedom in three-dimensional space has been suggested, to date this has not been validated. As the response is similar in both orthogonal horizontal and moment planes, only information in the torsional plane is required in order to extend the model to six degrees-of-freedom. Therefore, only the interaction of vertical and torsional load was considered in the shallow footing experiments presented here, conducted on over-consolidated clay at 1g. The experimental results were used to validate the proposed extension to the force-resultant footing model for shallow circular footings on clay. To conclude, the footing model, integrated into a finite element structural analysis program, was used to predict the load-displacement behaviour of a mobile offshore drilling rig resting on three shallow spudcan footings on over-consolidated clay soil.

1

INTRODUCTION

As mobile jack-up drilling rigs (Fig. 1) are not custombuilt for a particular site, the current guidelines (SNAME 2002) require a site-specific assessment to be performed for each new location before the platform can be installed. This demonstration of the rig’s capacity to withstand the 50-year return period storm is typically carried out as a push-over analysis. Recently, force-resultant (or macro-element) models based on plasticity theory (Houlsby 2003, Cassidy et al. 2004) have been developed that, if coupled with a structural analysis program, enable integrated simulation of the platform and its foundation-soil interaction to be carried out. To date, similar but separate models exist for undrained (clay) and drained (sand) response, respectively. Originally developed for vertical (V ), horizontal (H ) and in-plane moment (M ) loading (Martin & Houlsby 2001, Houlsby & Cassidy 2002), these models enable analysis in two-dimensional space only. Though Martin (1994) postulated extension of the foundation-soil interaction model for clay to cater for all six degrees-of-freedom in three-dimensional space, to date only the corresponding model representing the non-linear response of a shallow circular footing on sand has been formally extended to six degrees-of-freedom based on experimental evidence (Bienen et al. 2006).To the author’s knowledge, neither experimental nor continuum finite element analysis results have been published based on which an

Figure 1. Jack-up rig (schematic, after Reardon 1986).

expression describing the shape and size of the yield surface in planes including torsion (Q) can be developed for shallow circular footings on clay soil. Though Yun et al. (2009) included a flat circular footing in their study of combined vertical-torsional loading, only the maximum magnitude of sustained torsion of a fully rough footing was provided. This paper presents results from model physical experiments of a flat circular footing and a spudcan on clay under vertical-torsional loading. Based on this evidence, the footing macro-element model is extended to for all six degrees-of-freedoms. Example analyses of a jack-up rig conclude the paper.

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2 2.1

NUMERICAL FORCE-RESULTANT MODEL Background

Following the proposition by Butterfield & Ticof (1979) of utilising ‘interaction diagrams’, a number of models allowing prediction of shallow footing loaddisplacement response on both sand and clay have been formulated within the framework of displacementhardening plasticity (including Nova & Montrasio 1991, Gottardi et al. 1999, Martin & Houlsby 2000, 2001, Byrne & Houlsby 2001, Houlsby & Cassidy 2002, Bienen et al. 2006). These models consist of a yield surface (written directly as a function of the combined loads) which expands according to a hardening law, a description of elastic behaviour within the yield surface and a flow rule to describe the behaviour at yield. The entire footing-soil interaction is encapsulated into a point element. This framework has the advantage that models can be directly implemented into finite element programs to integrate the footing-soil response with the structural response. Notable examples of this have been in predicting the response of entire jack-up platform systems under environmental loading (Thompson 1996, Williams et al. 1998 and Cassidy et al. 2002 for two-dimensional simulation and Bienen & Cassidy 2006, 2009 for three-dimensional simulations on sand). The clay model (Model B) used in this paper but proposed by Martin (1994) and Martin & Houlsby (2001) is part of the ISIS family of plasticity models (Houlsby 2003, Cassidy et al. 2004). It was developed based on an extensive series of experiments on a 125 mm spudcan footing on soft heavily over-consolidated Speswhite kaolin clay (Martin & Houlsby 2000). The tests were carried out on the laboratory floor (at 1g) and comprised VHM loading combinations in-plane. However, Martin (1994) postulated the following expression to describe the yield surface in of all six degrees-of-freedom in three-dimensional space:

where the load components are shown in Figure 2. V0 is the uniaxial vertical capacity at the current penetration, D is the footing diameter in with the soil and h0 , m0 and q0 determine the yield surface size in the horizontal, moment and torsional directions, respectively. The parameter a determined the ellipse eccentricity in the HM plane, β1 , β2 and β12 shape the yield surface in planes including vertical load. For reasons of symmetry there cannot be any cross-coupling within the horizontal or moment

Figure 2. Sign convention (after Butterfield et al. 1997). Table 1.

Kaolin clay properties.

Liquid Limit (LL) Plastic Limit (PL) Specific gravity (Gs) Angle of Internal Friction φ

61% 27% 2.60 23◦

planes, nor can there be coupling torsion to other degrees-of-freedom. 2.2 Yield surface in the vertical-torsional plane The maximum sustainable torsion was derived to be 0.333ADsu for clay (Martin 1994), based on the assumption of mobilising the full soil’s undrained shear strength, su , along the footing-soil interface. This was confirmed by small strain finite element analysis of a flat rough circular footing byYun et al. (2009).This brings the normalised yield surface size in the torsional direction, q0 , to about 0.05 for a surface footing. However, as Martin (1994) pointed out, it is unlikely that a spudcan under horizontal load (shear) will mobilise the undrained shear strength in full, which was confirmed by the experimental results of a model spudcan on over-consolidated clay published in Martin & Houlsby (2000). Similar behaviour is expected under torsional load. This is confirmed by results obtained from physical experiments conducted in this study. The tests with load application in the vertical-torsional plane were carried out at 1g on two different model footings: (1) a flat circular footing 60 mm in diameter and (2) a spudcan 50 mm in diameter. The experimental apparatus used is described in Bienen et al. (2007). The undrained shear strength of the overconsolidated kaolin clay soil (for properties see Table 1) was determined to be about 10 kPa at the depth of the tests. Figure 3 shows results of swipe tests (Tan 1990), in which the footing is penetrated vertically before being twisted at no further penetration. If the elastic stiffness far exceeds the plastic stiffness, the resulting load path closely traces the yield surface (Tan 1990, Martin 1994). The envelope at low vertical load may be traced by unloading the footing vertically before

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Figure 3. Yield surface in the vertical-torsional plane.

Figure 4. Numerical model (SOS_3D).

torsion is applied, noting that the load path will initially fall inside the yield surface. In addition to the experimental results, Figure 3 includes numerical predictions of the swipe tests, which were obtained with the footing macro-model. The yield surface size in the torsional direction, q0 , was assumed to be 0.0275 as suggested by the experimental evidence (corresponding to an interface roughness coefficient of about 0.5). The shape of the yield surface is captured well with β1 = 0.764 and β2 = 0.882 (Eqn. 1) as suggested for the three degree-of-freedom model (Martin & Houlsby 2000), although β1 = β2 = 1.0 would provide a slightly better fit in the torsional plane at low vertical loads.

Table 2.

Leg length Separation of fwd. leg to centerline of aft legs Separation aft legs Young’s modulus (E) Shear modulus (G) Iy,hull beams , I z,hull beams ∗ J hull beams ∗ Ahull beams ∗ Iy,leg beams , Iz,leg beams Jleg beams Aleg beams Footing diameter ∗

3

INTEGRATED ANALYSIS

Soil-structure interaction macro-elements, such as the clay model discussed in this paper, can be incorporated into general finite element programs to allow integrated analyses to be performed. Examples include not only the simulation of jack-up behaviour using in-house software as mentioned above but also applications such as pipeline analysis with commercial software such as Abaqus (Tian et al. 2009), for instance. Integrated analyses are required where the changing footing stiffness significantly influences the global response, as is the case in jack-ups. Decreasing footing stiffness increases the P- effect on such tall flexible structures. The increased overturning load decreases the footing stiffness further. Therefore, the system’s ultimate capacity can only be accurately predicted if this inter-relation is ed for. The software employed for the example integrated analysis presented here is a fluid-structure-soil interaction program developed in-house called SOS_3D (Bienen & Cassidy 2006). The jack-up structure is modeled using beam-column elements. A footing macro-element is attached to each of the three bottom nodes of the structural model. Though no experimental (or field) data were available to benchmark the numerical results for the jack-up system against, a similar comparison for planar conditions only (loading along the axis of symmetry) based on 1g physical experiments showed good agreement (Vlahos 2004).

Jack-up and footing properties. 130 m 45 m 50 m 200 GPa 81 GPa 20 m4 350 m4 10 m2 2 m4 35 m4 5 m2 20 m

Assumed significantly stiffer than the legs.

Table 3.

Footing model parameters.

h0 m0 q0

0.127 0.083 0.028

e1 e2

0.518 1.180

β1 β2

0.764 0.882

Eccentricity (Eqn. 1) a = e1 + e1 (V /V0 ) (V /V0 -1)

3.1 Numerical model The numerical model is shown in Figure 4. Table 2 summarizes the relevant properties, which correspond to a generic field jack-up currently in use. As in the experiments, the spudcan shape was similar in elevation to that used by Martin (1994). The clay soil is assumed to be homogeneous with an undrained shear strength of 50 kPa, submerged unit weight γ = 6 kN/m3 and a rigidity index Ir = 500. As the previously recommended shape factors provide a good fit to the data (Fig. 3), all model parameters are assumed as set out in Cassidy et al. (2004), apart from q0 = 0.0275 (Table 3). Elastic stiffness coefficients are assumed according to Doherty & Deeks (2003). Mimicking the installation procedure employed in the field, the jack-up is preloaded to twice its selfweight, which applies a vertical load of 90 MN on each of the three footings. This corresponds to a bearing pressure of about 286 kPa, which is within the range quoted by Osborne et al. (2006).

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Figure 5. Global load-displacement response.

After preloading, the jack-up is brought back to its self-weight of 135 MN in total. This was followed by a quasi-static push-over for the example analyses presented here, though dynamic fluid-structure-soil analyses can also be performed in SOS_3D. The analyses were carried out for the loading directions shown in the inset of Figure 4, i.e. along the rig’s axis of symmetry and at an angle of 30◦ to it. The former orientation could be simplified and modeled in twodimensional analyses, the latter, however, requires three-dimensional analysis to be performed. 3.2

Numerical model

In the push-over analyses a horizontal load, applied at the jack-up hull was monotonically increased until failure of the system was observed. The global loaddisplacement response of the modeled jack-up is shown in Figure 5. Only resultant magnitudes are provided, i.e. directionality is not indicated. In the current example, the system’s capacity is lower when loaded symmetrically than when loaded at 30◦ to the axis of symmetry. However, this may not always be the case as shown previously for jack-ups on sand (Bienen & Cassidy 2006, 2009). Further, the planned position of the jack-up on site with respect to the dominant loading direction may not allow two-dimensional modelling. The individual footing load paths, shown in Figure 6, shed further light on the global system response. The different orientation of the rig to the push-over load manifests itself in the footing load paths: When loaded symmetrically (0◦ ) the aft spudcans B and C (Fig. 4) experience the same loads, and part of the overturning is resisted by a vertical push-pull mechanism between the forward and aft legs. In the 30◦ loading direction, spudcan A is slightly less heavily loaded while the vertical load on spudcan B reduces even stronger than in the symmetrical orientation during the push-over. The vertical load on spudcan C remains largely unchanged. As the response is shown in of the different loading components, visualization of the yield surface in the respective plane illustrates moment loading as the dominant load component. Torsion of the system is mainly resisted by horizontal footing reactions, torsion of each of the legs (and thus the footings) is low

Figure 6. Footing load paths.

even at a loading angle of 30◦ . However, incorporation of the torsional component into the footing macroelement enables full three-dimensional analyses to be performed. In all loading planes, a distinct change in response is visible upon yield of a footing. This is due to the single yield surface formulation of the model and thus the sharp transition from elastic to elasto-plastic behaviour. The influence of the footing stiffness on the global system stiffness is evident in Figure 5 (note that the structure itself is modeled as elastic). The decreasing rotational footing stiffness during yield is compensated by a stronger vertical pushpull mechanism between the forward and aft footings (Fig. 6). While the yield surface expands on the heavily loaded spudcan(s), the model formulation predicts footing heave and thus a contracting yield surface on the rear spudcan B. Note that the load paths of spudcans A and B travel towards the respective yield surface apex points. Therefore, even though the yield surface associated with spudcan A continuously

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increases in size, the available combined capacity at the current load point (the yield surface cross-section) decreases. The numerical analysis suggests failure of the jackup system not at first yield of the footings, but predicts significant additional capacity. Only when the yield surface corresponding to spudcan A can no longer expand at a sufficiently fast rate to accommodate for the increased combined loading does the jack-up fail in the numerical prediction.

4

CONCLUDING REMARKS

A foundation-soil interaction macro-element for shallow circular footings on clay soil has been extended to cater for all six degrees-of-freedom in threedimensional space. The required evidence for the extension, the interaction in the vertical-torsional plane, was obtained through model experiments. As the footing macro-element predicts not only the combined ultimate capacity, but the entire non-linear loaddisplacement response, integrated analysis allows for the prediction of the jack-up’s behaviour up to ultimate failure of the system. The extended footing macro-element enables simulation of the structure-soil interaction in three dimensions, which is important when the loading direction does not align with the system’s axis of symmetry, for instance. It should be noted that the footing model was developed for shallow circular footings, i.e. the corresponding failure mechanisms extend to the soil surface. On soft soil sites, however, the footings of jack-up rigs penetrate up to three diameters, with soil flowing around the edges of the footing, localizing the failure mechanisms (deep failure mechanism). Work is currently ongoing at the Centre for Offshore Foundation Systems (COFS) to establish a similar footing model that takes into the increased combined load capacity of deeply penetrated spudcan footings.

5 ACKNOWLEDGEMENTS The work described here forms part of the activities of the Centre for Offshore Foundation systems (COFS), established under the Australian Research Council’s Research Centres Program and now ed by Centre of Excellence funding from the State Government of Western Australia. The original code for Model B was written by Dr Chris Martin of Oxford University. The current version of the SOS_3D program uses the ISIS computer code originally written in collaboration between the University of Oxford and The University of Western Australia. Professor Guy Houlsby has been instrumental in its development and his contribution is acknowledged. The author is thankful for the help of Nick Bennett and Brett McKiernan in carrying out the experiments.

REFERENCES Bienen, B., Byrne, B., Houlsby, G.T. & Cassidy, M.J. (2006). Investigating six degree-of-freedom loading of shallow foundations on sand. Géotechnique, Vol. 56, No. 6, pp. 367–379. Bienen, B. & Cassidy, M.J. (2006). Advances in the threedimensional fluid-structure-soil interaction analysis of offshore jack-up structures. Marine Structures, Vol. 19, No. 2–3, pp. 110–140. Bienen, B. & Cassidy, M.J. (2009). Three-dimensional numerical analysis of centrifuge experiments on a model jack-up drilling rig on sand. Canadian Geotechnical Journal, Vol. 46, No. 2, pp. 208–224. Bienen, B., Gaudin, C. & Cassidy, M.J. (2007). Centrifuge tests of shallow footing behaviour on sand under combined vertical-torsional loading. International Journal of Physical Modelling in Geotechnics, Vol. 7, No. 2, pp. 1–21. Butterfield, R. & Ticof, J. (1979). Design parameters for granular soils. Proc. 7th ECSMFE, Brighton, UK, pp. 259–261. Butterfield, R., Houlsby, G.T. & Gottardi, G. (1997). Standardised sign conventions and notation for generally loaded foundations. Géotechnique, Vol. 47, No. 4, pp. 1051–1054; corrigendum Vol. 48, No. 1, p. 157. Byrne, B.W. & Houlsby, G.T. (2001). Observations of footing behaviour on loose carbonate sands. Géotechnique, Vol. 51, No. 5, pp 463–466. Cassidy, M.J., Houlsby, G.T., Hoyle, M. & Marcom, M. (2002). Determining appropriate stiffness levels for spudcan foundations using jack-up case records. Proc. 21st Int. Conf. on Offshore Mechanics and Arctic Engineering (OMAE), Oslo, Norway, OMAE2002-28085. Cassidy, M.J., Martin, C.M. & Houlsby, G.T. (2004). Development and application of force resultant models describing jack-up foundation behaviour. Marine Structures, Vol. 17, pp. 165–193. Doherty, J.P. & Deeks, A.J. (2003). Elastic response of circular footings embedded in a nonhomogeneous half-space. Géotechnique, Vol. 53, No. 8, pp. 703–714. Gottardi, G., Houlsby, G.T. & Butterfield, R. (1999). Plastic response of circular footings on sand under general planar loading. Géotechnique, Vol. 49, No. 4, pp. 453–469. Houlsby, G.T. (2003). Modelling of shallow foundations for offshore structures. International Conference on Foundations (ICOF), Dundee, Scotland, pp. 11–26. Houlsby, G.T. & Cassidy, M.J. (2002). A plasticity model for the behaviour of footings on sand under combined loading. Géotechnique, Vol. 52, No. 2, pp. 117–129. Martin, C.M. (1994). Physical and numerical modelling of offshore foundations under combined loads. DPhil. thesis, University of Oxford. Martin, C.M. & Houlsby, G.T. (2000). Combined loading of spudcan foundations on clay: Laboratory tests. Géotechnique, Vol. 50, No. 4, pp. 325–338. Martin, C.M. & Houlsby, G.T. (2001). Combined loading of spudcan foundations on clay: Numerical modelling. Géotechnique, Vol. 51, No. 8, pp. 687–700. Nova, R. & Montrasio, L. (1991). Settlement of shallow foundations on sand. Géotechnique, Vol. 41, No. 2, pp. 243–256. Osborne, J.J., Pelley, D., Nelson, C., & Hunt, R. (2006). Unpredicted jack-up foundation performance. Proc. JackUp Asia Conference and Exhibition, PetroMin, Singapore. Reardon, M.J. (1986). Review of the geotechnical aspects of jack-up unit operations. Ground Engineering, Vol. 19, No. 7, pp. 21–26.

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SNAME (Society of Naval Architects and Marine Engineers) (2002). Site specific assessment of mobile jack-up units, SNAME Technical and Research Bulletin 5-5A. 1st Ed., 2nd Revision, New Jersey. Tan, F.S.C. (1990). Centrifuge and numerical modelling of conical footings on sand. PhD thesis, University of Cambridge. Tian,Y. & Cassidy, M.J. (2009). The challenge of numerically implementing numerous force–resultant models in the stability analysis of long on-bottom pipelines. Computers and Geotechnics, doi:10.1016/j.compgeo.2009.09.004. Thompson, R.S.G. (1996). Development of non-linear numerical models appropriate for the analysis of jackup units. DPhil. thesis, University of Oxford.

Vlahos, G. (2004). Physical and numerical modelling of a three-legged jack-up structure on clay soil. PhD thesis, University of Western Australia. Williams, M.S., Thompson, R.S.G. & Houlsby, G.T. (1998). Non-linear dynamic analysis of offshore jack-up units. Computers and Structures, Vol. 69, pp. 171–180. Yun, G.J., Maconochie, A., Oliphant, J. & Bransby, F. (2009). Undrained capacity of surface footings subjected to combined V-H-T loading. Proc. 19th Int. Offshore and Polar Engineering Conference (ISOPE), Osaka, Japan, pp. 9–14.

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Undrained ultimate capacity of suction anchors using an advanced constitutive model S. Panayides & M. Rouainia Newcastle University, Newcastle Upon Tyne, UK

ABSTRACT: This paper investigates the ultimate capacity of suction anchor foundations, using an advanced constitutive model. The paper explores the effect of several factors on the undrained load-carrying capacity of the bucket foundations in clay. A parametric study is carried out to assess of the effects of initial structure, the rate of destructuration and the form of the destructuration strain. Illustrative numerical results for an inorganic clay of low sensitivity from the Norrköping region in southern Sweden demonstrate the potential of the constitutive model.

1

INTRODUCTION

The relative inefficiency of piles in resisting lateral forces has led the offshore industry to consider alternative anchorage systems such as suction caissons. Suction caissons can be installed very quickly and precisely at the desired location with less heavy installation equipment and at lower cost. Therefore they are considered as a viable anchorage system in a wide variety of soils ranging from soft clay to dense sands and overconsolidated clays and for a wide variety of structures ranging from floating exploration platforms to permanent production facilities. The development of suction caissons in recent years has seen them used around the world in more than 36 fields in the last decade alone (Andersen, 2002). Suction caissons are large cylindrical shells, with an open bottom and a closed top fitted with valves. The aspect ratio of these piles, defined as the length to diameter ratio, is relatively small when compared with the aspect ratio of conventional piles, typically six or less (Andersen, 2005). Internal stiffeners are usually added, to resist buckling during the installation process, since the caisson walls are relatively thin. They are installed partly by self weight and partly by differential pressure between the surrounding environment and the inside of the skirted foundation. In some cases, dead weights can be applied on the top of the cap to ensure that compressive loads are acting on the suction anchors (Zdravkovic, 2001). Once full penetration has been achieved, the valve is closed. Any vertical movement during service will result in the generation of suction pressure inside the anchor which will mobilize the reverse end-bearing mechanism, as it is described by Finn & Byrne (1972). Foundations for offshore structures, however, experience significant environmental loads from waves, currents and wind giving

rise to lateral loads that could be up to one third of the vertical loads. The direct consequence of that is that resultant loads can be inclined to the vertical. It is well documented that suction anchors are capable of resisting both lateral and axial loads as well as inclined loads. The ultimate capacity of suction anchors has been the focus of many investigations in recent years. Following the work of Hogervorst (1980) Keaveny et al (1994) showed that lowering the load attachment point at mid depth increased the capacity significantly. Suction caisson capacity studies based on upper bound limit analyses from Randolph et al (1998) and a finite element study from Sukumaran et al (1999) indicate that the anchor capacity can be maximised when the load is located at a point which forces the anchor to fail in a translational mode of failure rather than rotational. Murff & Hamilton (1993) presented a three dimensional quasi upper bound formulation for predicting the ultimate capacity of laterally loaded piles. The three dimensional mechanism which they proposed comprised of a conical wedge near the free surface and a flow around zone below the wedge (Randolph & Houlsby, 1984). This paper presents a study of the shot-term pullout capacity of suction caissons in soft clay using an advanced constitutive model. The failure envelope was produced for one reference caisson. The soil was modelled using the Kinematic Hardening Soil Model (KHSM) as proposed by Rouainia & Muir Wood (2000).A parametric analysis was carried out, in which the initial structure size r0 , the destructuration strain rate k and parameter A which controls the relative proportions of distortional and volumetric destructuration were varied. These studies provided a general picture of the effect of each of these parameters on the ultimate capacity of suction caissons.

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2

GEOMETRY

The geometry for the first reference suction anchor foundation adopted for this study is provided in Figure 1. It comprises of a cylindrical suction anchor with closed top with a diameter D of the cylinder of 5 m, while the skirt length L is set to 7.5 m, giving rise to an aspect ratio of L/D = 1.5. A wall thickness of 50 mm (or D/100) was used in all cases. The caisson is embedded with the top cap flush with the surrounding ground level and the load attachment point, or padeye location, at a depth zp along the caisson shaft. Loads are applied at an angle θ from the horizontal. The caisson is considered to be very stiff compared to the soil. The pullout loads are applied on different points on the side of the caisson with at an inclination θ to the horizontal to produce the failure envelopes. The geometry is extended 3 times the length L around to avoid influence of the geometry boundaries. 3


The cylindrical suction anchor was modelled using the Plaxis 3D with 15 node wedge elements. The anchor was modelled by linear elastic wall elements with a high stiffness making them virtually rigid. Since the governing failure mechanisms do not involve the soil plug inside the anchor, this material was modelled as a stiff, elastic material. The mesh with approximately ∼13000 elements and ∼26000 nodes was found to be sufficiently refined in order to minimize the discetization error. For all the FE-models in this study, interface elements along the outside caisson walls have been used with a strength reduction factor of the interface (Rinter) set to 0.65. The interface properties are estimated using the strength reduction factor and the soil properties as follows:

where ϕi and ci are the interface effective friction angle the interface effective cohesion, respectively and Eoed is the constrained modulus of the soil. ϕ and c are the friction angle and effective cohesion of the soil. Since pore water pressures or the installation of the suction anchor is not considered, the phreatic level was placed at the bottom of the geometry. 4

MATERIAL MODEL

The model used in this study was formulated for natural clays within the framework of kinematic hardening with some elements of bounding surface plasticity. It is a rate independent model and it takes into the effects of damage to structure caused by irrecoverable plastic strains, resulting from sampling or geotechnical loading. KHSM is an extension of the well known

Figure 1. Finite Element geometry for the reference suction anchor.

Cam-Clay model. The steady fall of stiffness with strain is controlled by an interpolation function which ensures a smooth advancement of the elastic domain (which is enclosed in a small bubble) towards the bounding surface during loading. A scalar variable r, which is a monotonically decreasing function of the plastic strain, represents the progressive degradation of the material. Accordingly, the following exponential destructuration law, is proposed

where ro denotes the initial structure and k is a parameter which describes the rate of destructuration process with strain. The rate of the destructuration strain εd will be assumed to have the following form

where A is a non-dimensional scaling parameter and p ε˙ q and ε˙ vp are the plastic shear strain and the plastic volumetric strain, respectively. The governing constitutive relations of the KHSM are summarized in the Appendix. 5

CLAY PROPERTIES

The parameters required for the analysis correspond to inorganic clay of low sensitivity from the Norrköping region in southern Sweden. An effective unit weight γ = 10 KN/m3 was used. The model parameters for the soil and the interface were taken from Westerberg, (1995). An over consolidation ratio (OCR) of 1 was adopted for the analyses. The coefficient of lateral earth pressure (K0NC = 1 − sin ϕ ) was taken as 0.5 which corresponds to an effective friction angle ϕ of 30◦ . For the soil-structure interface, an oedometric Young’s modulus (Eoed ) of 1800 KPa and cohesion c of 2.1 KPa were used. These optimized parameters are described as reference parameters and correspond to the KHSM model in all the analyses below.

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Figure 3. Incremental Displacement Vector plot indicating the failure mechanism.

Figure 2. Predicted and measured undrained shear strength for Norrköping clay.

As with all critical state soil models, the undrained shear strength is not an input parameter for KHSM. This issue was resolved, by running undrained triaxial compression tests starting from the insitu stress conditions for several points along the length of the pile. The intersection between the critical state line and the undrained stress path is the failure point, and consequently the corresponding ultimate deviator stress is the difference between the axial and radial total stresses at the end of each test, (Muir Wood, 1990). Hence that point is twice the undrained shear strength of the soil at that depth. The procedure described above, provided the theoretical prediction of the undrained strength profile. The predicted undrained shear strength distribution has a zero intercept at the surface, linearly increasing with depth at a rate of 2.6z. Figure 2 provides a comparison of the predicted TXC strength profile from KHSM with laboratory tests and field vane test as well as the empirical relation for Norrköping clay from Westerberg (1995). As it can be seen, the predicted shear strength distribution lies within the range of values provided by the field and laboratory tests. It is possible to replicate the exact shear strength distribution, by having an initial OCR value that is higher than 1 and decreasing with depth. This process would result in a constant value of the undrained strength with depth, up to the point where it starts varying linearly with depth.

6 6.1


Figure 4. Comparison of KHSM and Bubble models with two analytical methods.

should be noted however that the gradient of the wedge varies with depth. As the wedge approaches the tip of the caissons it tends to curve ing tangentially at the bottom of the caisson. 6.2 Failure envelope comparison Figure 4 shows the comparison of the failure envelope for non-horizontal loadings for the structured model (KHSM), the Bubble model and two analytical methods suggested by Supachawarote (2005) and Senders & Kay (2002). It is evident that the structured model predicts the ultimate load for all loading angles very well. In contrast, the Bubble model, consistently, underestimates the ultimate load by an average of ∼12%. This behaviour is as expected, since the Bubble model was formulated to represent the behaviour of reconstituted material and cannot for added strength the natural clay deposits exhibit.

Failure mechanisms

Figure 3 shows the Incremental Displacement Vector Plots and Failure Mechanism obtained from the analysis corresponding to pure horizontal loading, located at the optimal loading point. A well defined failure surface develops on both the active and ive side of the caisson. As it can be seen the failure extends to the bottom of the caisson, with no flow around zone visible. It

6.3 Parametric analysis The first simulation of the parametric analyses comprised of an investigation of the effect of the measure of initial structure on the ultimate load. All the model parameters values were kept the same as in Table 1, except r0 which controls the size of the structure surface as seen in equation (3).

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Table 1.

Soil parameters for Norrköping Clay.

Material constants

Value

Slope of swelling line κ∗ Slope of normal compression line λ∗ Poisson’s ratio v Critical state stress ratio M Ratio of size of bubble and reference surface R Stiffness interpolation parameter B Stiffness interpolation parameter ψ Destructuration parameter k Destructuration strain parameter A Initial degree of structure r0 Anisotropy of initial structure η0

0.0297 0.252 0.22 1.35 0.145 1.98 1.547 4.16 0.494 1.75 0.5

Figure 6. Load- Displacement curves for the parametric analysis on the destructuration strain rate.

Figure 5. Load- Displacement curves for the parametric analysis on the size of the initial structure.

Increasing r0 by ∼14% of the reference value increases the initial degree of structure so that the load displacement curve, shown in Figure 5 exhibits a stiffer behaviour at ∼0.06 m of pile head displacement. This difference results in an overestimation of the ultimate load by ∼6%. An identical reduction to r0 has the consequence of underestimating the ultimate load by ∼7%. Similarly the deviation from the KHSM curve is initiated at about 0.06 m of displacement. This gives a first indication that the stress state up to that point is situated within the boundaries of elastic domain. Furthermore, the analysis that did not include the effects of destructuration concluded to an ultimate horizontal load of 3070 kN, which gives rise to a deviation of ∼16% from the reference load. It can be said that the model is not very sensitive to the change in value of r0 , however the absence of any degree of structure increases the error significantly. The next model parameter that was investigated was k, which controls the rate of loss of structure with damage strain. Figure 6 depicts the effects on varying this parameter by ∼50% either way. An increase in this parameter results to a more rapid loss of structure and an overall softer behavior as seen in the load displacement curve. This in turn, results in an underestimation of the ultimate load by ∼7%. The softer behaviour

Figure 7. Load- Displacement curves for the parametric analysis on the form of the destructuration strain.

exhibited is as expected, since the reduction of k allows the structure surface to collapse faster towards the reconstituted Cam clay surface. As expected, a decrease of the value of the destructuration parameter reduces the rate at which structure is lost with continuing strain and hence a stiffer response is observed, which results in an overestimation of the ultimate load by ∼8%. The deviations from the load-displacement curves using the KHSM parameters, becomes clear at a pile head displacement of 0.06 m. Parameter A which controls the relative proportions of distortional and volumetric destructuration was then investigated. The two extreme cases have been examined. When setting A to be equal to 1, it is assumed that the destructuration process is entirely distortional. As a result of this process the ultimate load is reduced by ∼7%. The deviation from the reference value of 0.494 used in the KHSM model increases the rate of loss of structure and suggests that the suction caisson loading causes largely distortional deformation. On the other hand, by setting A = 0 and hence presuming

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that destructuration is produced solely by volumetric strain, has the effect of slowing down the process of losing structure and overestimates the ultimate load by ∼18%.As for the previous two investigations, the divergence of the load displacement curves occurs at 0.06 m of pile head displacement. This observation s the hypothesis that until that point of loading, the stress state is situated within the elastic domain. 7


This paper presents the numerical analysis of the undrained ultimate capacity of a suction caisson, in a soft structured clay deposit. Two finite element analyses were conducted in order to compute the ultimate capacity for a variation of load inclinations, to provide the full failure envelope. In the first analysis of the study, the clay deposit was modeled with a constitutive model for reconstituted clays (Bubble model), which cannot take into the destructuration process, whereas, in the second part, the clay deposit was modeled with an advanced constitutive model for natural clays, namely the Kinematic Hardening Structure Model, which can simulate destructuration. Furthermore, a parametric analysis involving the initial measure of structure r0 , the destructuration strain rate k and parameter A illustrated the effect these parameters have on the ultimate capacity corresponding to pure horizontal loading. The main conclusions that can be drawn from the comparison of the two analyses are as follows: •

•

•

•

• •

Both analyses predict similar failure mechanisms, namely a wedge failure on both sides of the caisson with no flow around zone. The model for truly reconstituted soils underestimates the ultimate capacities for all load inclinations by ∼12%. The failure envelope from the KHSM analysis agrees very well with two simple design equations from previous finite element analyses of suction caissons. The parametric analysis on parameter A indicates that the suction caisson loading causes largely distortional deformation. The results from this study show that the reverse end bearing factor of 10.6. The normalized horizontal capacity was found in this study to be 9.6

The above conclusions imply that the KHSM model can predict the undrained ultimate capacity of suction caissons accurately. The work presented in this paper shows that an analysis, which ignores destructuration may underestimate the ultimate capacity of caissons, which could result to uneconomical designs

With p and s are the effective pressure and the deviator stress tensor. K and G are the bulk and shear moduli and κ∗ is the slope of the swelling line. εd and εv are the deviatoric and the volumetric components of the strain tensor ε, respectively. Equation of the reference, bubble and structure surfaces:

where α = (pα I + sα ) denotes the location of the centre of the bubble in the stress space, R represents the ratio of the sizes of the bubble and the structure surface and r represents the destrucruration law which is a decreasing function of the plastic damage strain and pc is the preconsolidation effective stress. Mθ is the slope of the critical state line function of the Lode angle, and η0 is a fixed dimensionless deviatoric tensor Isotropic hardening law:

where λ∗ is the slope of the normal compression line p and ε˙ v is the plastic volumetric strain. Kinematic Hardening law

where µ is a positive scalar of proportionality and σ c is the stress on the structure surface. αˆ = (rpc I , (r − 1)ηo pc ) is the position of the centre of the structure surface. Plastic modulus at current stress:

8 APPENDIX

Where Hc is the plastic modulus at conjugate stress, b is a normalized distance b which vanishes when the bubble and the structure surfaces are in and bmax is its maximum value. ψ and B are two additional material parameters.

Non-linear elastic constitutive laws:

REFERENCES Andersen, K.H., Murff J.D., Randolph M.F.,Clukey E.C., Erbrich C., Jostad H.P., Hansen B., Aubeny C., Sharma P.,

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and Supachawarote C. 2005. Suction anchors for deepwater applications. Int. Symp. on Frontiers in Offshore Geotechnics, ISFOG. Sept. 2005. Perth, Western Australia. Proc. A.A. Balkema Publishers. Andersen L.,Edgers L.and Jostal H.P. 2008 Capacity Analysis of Suction Anchors in Clay by Plaxis 3D Foundation. Plaxis Bulletin, Issue 24. Oct. 2008. Aubeny, C., S. Moon, and J. Murff 2001. Lateral undrained resistance of suction caison anchors. International Journal of Offshore and Polar Engineering 11(2), 95–103. Aubeny, C.P., Moon, S.K., and Murff, J.D. 2001. “Lateral undrained resistance of suction caisson anchors.” International Journal of Offshore and Polar Engineering., 11(2), 95–103. Aubeny, C.P., Han, S.W., and Murff, J.D. 2003. “Inclined load capacity of suction caissons.” International Journal for Numerical and Analytical Methods in Geomechanics., 27(14), 1235–1254. Bransby, M.F., and Randolph, M.F. 1998 “Combined loading of skirted foundations.” Geotechnique., 48(5): 637–655. Finn, W.D.L and Byrne, P.M 1972. “The evaluation of the breakout force for a submerged ocean platform” Proceedingsof international conference: Centrifuge 91, Rotterdam, The Netherlands, A.ABalkema. Hogervorst, J.R 1980 “Field trials with large diameter suction piles” Proceedings Annual offshore technology conference”, Houtson, OTC 3817:217–222. Keaveny, J.V., Hansen S.B., Madshus. C. & Dyvik, R. 1994” Horizontal capacity of large-scale model anchors “Proceedings 13th International conference on soil mechanics and foundation engineering, New Delhi, 2: 677–680. Muir Wood, D. 1990. “Soil behavior and critical state soil mechanics” Cambridge, Cambridge University press. Murff, J.D., and Hamilton, J.M. 1993. “P-Ultimate for undrained analysis of laterally loaded piles.” ASCE, Journal of Geotechnical Engineering., 119(1): 91–10. Randolph, M.F., and Houlsby, G.T. 1984. “The limiting pressure on a circular pile loaded laterally in cohesive soil.” Geotechnique., 34(4): 613–623.

Randolph, M.F., O’Neill, M.P., Stewart, D.P., and Erbrich, E. 1998. “Performance of suction anchors in fine-grained calcareous soils.” Proc. Offshore Tech. Conf., Houston”, OTC 14236. Randolph, M.F., & House A.R. 2001. “Analysis of Suction Caisson Capacity in Clay” Proc. Offshore Tech. Conf., Paper No.8831: 521–52. Rouainia, M. and D. M.Wood 2000. A kinematic hardening constitutive model for natural clays with loss of structure. Geotechnique 50(2): 153–164. Senders, M. and Kay, S. 2002. “Geotechnical Suction Pile Anchor Design in Deep Water Soft Clays”, Conference Deepwater Risers, Moorings and Anchorings, London, UK. Sukumaran, B., McCarron, M.O., Jeanjean, P.& Abouseeda H. 1999 “Efficient finite element techniques for limit analysis of suction caissons under lateral loads” Computer and geotechnics, 24: 89–107. Supachawarote, C., Randoplh, M.F and Gourvenec, S. 2004. “Inclined Pull-Out Capacity of Suction Caissons” Proceedings of the 14th International Society of Offshore and Polar Engineering Conference: 500–506. Westerberg, B. 1995. Lerors mekaniska egenskaper. Licenciate thesis, Luleå University of Technology, Sweden. Zdravkovic, L., Potts, D.M. and Jardine, R.J. 1998. Pull-Out Capacity of Bucket Foundations in Soft Clay”. Proceedings of the International Conference on Offshore Site Investigation and Foundation Behaviour: 301–324. Zdravkovic, L., Potts, D.M. and Jardine, R.J. 2001. “Parametric Study of the Pull-Out Capacity of Bucket Foundations in Soft Clay”, Geotechnique, 51(1): 55–67. Zhao, J., D. Sheng, M. Rouainia, and S. Sloan (2005). Explicit stress integration of complex soil models. Int. J. for Num. and Anal. Methods in Geomech. 29(12), 1209–1229.

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Numerical methods and Eurocode


Embedded cantilever retaining wall ULS design by FEA in accordance with EN 1997-1 A.S. Lees & S. Perdikou Frederick University, Nicosia, Cyprus Geofem Ltd.

ABSTRACT: As numerical methods expand into more areas of geotechnical design it is inevitable that conformance to design codes will become more of an issue. The recent introduction of Eurocode 7 EN 1997 provides an opportunity to develop new modeling methods in accordance with a design code. However, there are many issues to overcome, particularly in ULS design, including one of partial factoring. While a material factoring approach is relatively straightforward to implement using numerical methods because the material properties are input values, factoring geotechnical actions and, in particular, geotechnical resistances, however, is not straightforward because these values are determined by the analysis itself. This paper proposes a method of factoring Ka and Kp through manipulation of ϕ and c values in order to factor geotechnical actions and resistances on an embedded cantilever retaining wall. Credible results are obtained for the example problem studied, particularly for ive resistance factoring, but more study is needed of the methods on other examples. 1

INTRODUCTION

By the time the NUMGE 2010 Conference starts, the Structural Eurocodes will have superseded existing design codes in many of the States of Europe. Eurocode 7 (EC7) (CEN, 2004) covers the geotechnical aspects of the design of buildings and civil engineering works. While the use of numerical methods in geotechnical design offices has increased enormously in recent years, arguably their use would be more widespread if more guidance existed on their use in accordance with design codes and standards. Introduction of EC7 with its common design framework for all geotechnical structure types provides an opportunity to produce such guidance. The Eurocodes provide little guidance on specific design methods. This is particularly the case with EC7 for numerical methods although it does contain some recommendations for their use in assessing deflections in soil-structure interaction problems. The analysis of such serviceability limit state (SLS) problems by numerical analysis has become relatively commonplace but there are also gains to be had from using numerical methods in ultimate limit state (ULS) designs, provided one is prepared to invest in the additional time and expertise that these require. One major advantage is the checking of multiple failure forms which are not pre-determined in the analysis, unlike more traditional design methods which, while robust, may only provide a check on one pre-determined failure mechanism. There are many difficulties to overcome before numerical methods can be used routinely in ULS

design in accordance with EC7. These include understanding the influence of parameters other than those defining the failure criteria, e.g. stiffness, stress ratio, dilation angle, even in the most basic constitutive models. Factoring the Mohr-Coulomb failure criterion in accordance with EC7 should be relatively straightforward, but methods of factoring other criteria, such as a hyperbolic model, are less obvious. A particular difficulty lies in the application of partial factors to geotechnical actions and resistances, since these are not pre-determined values in a numerical analysis. By using the analysis of a ed embedded cantilever retaining wall by the finite element method as an example, this paper describes the application of the partial factors and verification of structural and geotechnical ULSs for all three Design Approaches (DAs). The methods are shown to provide credible results for this particular example.

2

FACTORING FOR ULS VERIFICATION

Bauduin et al. (2005) described methods of implementing ULS partial factors in finite element analyses (FEAs), although no examples were presented. They proposed a material factoring approach (MFA), which is applicable to DA1-Combination 2 (DA1(2)) and DA3 and a load and resistance factoring approach (LRFA), applicable to DA2. All analyses should be conducted using characteristic values of all parameters, with parameters changed stepwise to their design values only to ULSs and obtain design values of structural forces as necessary at each construction

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Figure 1. Strip foundation example.

stage. Following verification of the ULS, the analysis should continue from the previous characteristic state on to the next construction stage. The MFA is straightforward to implement because the material properties are pre-determined parameters. Factoring of external actions is also straightforward, but factoring geotechnical actions and resistances in the LRFA is less so as these values are determined by the analysis itself. Ultimate resistance values are particularly difficult to determine because failure will not normally occur in an FEA of a real problem unless it is forced in some way. For geotechnical structures with external loads only (e.g. shallow foundations), this can be achieved by increasing the applied loads to failure and determining the ultimate resistance value from a load-deflection graph (Bauduin et al. 2005). Both the MFA and LRFA were implemented in an FEA of the strip foundation in clay shown in Figure 1. The LRFA was implemented by increasing a common load factor γA applied to the DA2 design values of all the applied actions on the foundation from 1.0 until failure and plotting the load factor against average vertical deflection of the foundation, as shown in Figure 2. The MFA was implemented firstly by increasing the applied loads to their DA3 design values and then invoking the ϕ − c reduction facility in Plaxis v.9 (Plaxis 2008) which steps up the material factor γM (γϕ and γc equally on tan ϕ and c respectively) until a failure mechanism is detected. It was found informative to plot also the MFA against average vertical deflection of the foundation, as shown in Figure 2. The MFA produces a more definitive failure when plotted (Fig. 2) with a mechanism occurring at γM = 3.0. Due to compression of the soil under the increasing applied load, the LRFA produces a less definite failure. The γA value at failure would correspond with the resistance factor γR used in DA2 (since it is a factor on the design values of the applied loads). Overcoming the difficulty of factoring geotechnical resistance in the strip foundation example was relatively straightforward because only structural loads were the cause of failure. As Bauduin et al. (2005) recognised, in situations where actions from the ground provide resistance (e.g. ive resistance),

Figure 2. Plotting factors to detect failure in strip foundation example.

mobilisation of maximum resistance requires possibly highly inaccurate manipulation of the FEA. In the following retaining wall example, a method of overcoming this difficulty is proposed and implemented in DA2.

3

EMBEDDED CANTILEVER RETAINING WALL EXAMPLE

A design example from Frank et al. (2004) was selected for plane strain FEA, to allow a ready comparison with the published results using more conventional design methods. Figure 3 shows the geometry of the problem: in the FEA the ground anchor was represented by a horizontal bar element connected to the wall at −1.0 m elevation, with spring stiffness (k = EA/L) of 21 kN/mm per m run. The sheet pile wall was modelled with beam elements with flexural stiffness EI = 5 × 104 kNm2 /m, Poisson’s ratio ν = 0.1 and ultimate moment of resistance Mu,d = 613 kNm/m. To model soil-wall friction, elastic perfectly plastic interface elements were used with ϕ and c set to 2/3 the adjacent soil values on both sides of the wall. The soil was represented by 15-node triangular elements, each typically 0.5 m wide around the wall increasing to 10 m at the remote boundaries (50 m distant horizontally and 50 m below ground level). The soil was modelled with a linear elastic-perfectly plastic Mohr-Coulomb model with the properties shown in Table 1. For the calculation of pore pressures, steady-state seepage conditions were established for the groundwater levels shown in Figure 3. The retaining wall and were installed in the first construction stage, followed by excavation to −5.4 m level, both using all characteristic values.

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Figure 4. Development of ive failure due to material strength factoring in DA3 and DA1(2) with d = 4.95 m. Figure 3. Embedded cantilever retaining wall example (Frank et al. 2004). Table 1.

Layer A Layer B

Layer A Layer B

Soil parameters. ϕ deg.

c kPa

ψ deg.

E kPa

ν

35 24

0.1 5

0 0

4 × 104 3 × 104

0.2 0.2

K0

γsat kN/m3

γunsat kN/m3

0.5 0.95

20 20

18 –

Once the d value was determined, a further construction stage was introduced with ϕ and c factored by exactly 1.25 to obtain the design values of structural forces. This avoids the excessively high structural forces that are predicted following ive failure on completion of the ϕ − c reduction. The design wall bending moment profile is shown as the fourth line in Figure 5. The maximum value was 288 kNm/m while the design value of the force was 202 kN/m. 3.2 DA1(1)

k m/s 1 × 10−5 1 × 10−10

ϕ = internal friction angle, c = cohesion, ψ = dilation angle, E = drained stiffness, ν = Poisson’s ratio, K0 = stress ratio, γsat = saturated weight density, γunsat = unsaturated weight density, k = permeability (equal horizontally and vertically).

3.1 DA3 & DA1(2) A design value surcharge of (γQ = 1.3) × 10 kPa = 13 kPa was applied to the ground surface on the retained side. Following this, a ϕ − c reduction was invoked and wall toe horizontal deflection was monitored. Complete analyses were repeated with different d values until ive failure was observed to occur at a material strength factor (γM ) in excess of 1.25. A graph of toe deflection against material strength factor is shown in Figure 4 for the adopted d value of 4.95 m. The graph in Figure 4 highlights the importance of plotting the response of a critical part of the model to changing partial factors, rather than simply applying factors or a ϕ − c reduction and relying on the end result as a lower bound failure state. With a shallower d, a ϕ − c reduction ending with a factor in excess of 1.25 could still have been achieved but clearly, failure initiates before the ϕ − c reduction is complete and this can only be identified by plotting the results in this way.

In DA1, in addition to Combination 2 of partial factors on variable loads and material strength, Combination 1 of partial factors on all loads needs to be checked. The most conservative result of each combination should be used in design. Usually, Combination 2 provides the critical geotechnical failure state while either of the two combinations may produce the higher structural forces. Factors on actions may be applied either directly to actions or to the effects of actions. This is particularly convenient for geotechnical loads where it is difficult to apply factors directly in numerical analyses. It is recommended therefore (Frank et al. 2004) that variable (external) loads are factored by 1.11 (γQ /γG = 1.5/1.35) and design values of structural forces are obtained by factoring them by the combined effect of the variable and permanent actions (i.e. by 1.35). An FEA was performed with a surcharge of 1.11 × 10 kPa = 11.1 kPa and d = 4.9 m. The resulting bending moment in the wall both before and after multiplying by 1.35 (“unfactored” and “unfactored × 1.35”) are shown in Figure 5. The maximum design bending moment was 240 kNm/m and the design force was 168 kN/m. On this occasion, Combination 2 of partial factors produced the critical structural forces, which was also the case using the limit equilibrium method (LEM) of design (Frank et al. 2004). The drawback of this method is that the ULS of ive resistance under the factored load is not checked. Where it is obvious that Combination 2 governs the

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Figure 6. Earth pressure on retained side of wall for DA1(1).

Figure 5. Design value wall bending moment profiles for DA1(1), except where shown.

design, calculation with Combination 1 need not be carried out but, in order to investigate a plausible method for checking Combination 1, the following analysis was also performed. The ϕ values of the soil on the retained side of the wall only were reduced in order to increase the active earth pressure coefficient Ka by a factor of 1.35, with the notion of increasing the earth pressure on the retained side of the wall by the same factor. The appropriate ϕ reductions were estimated from the figures in Annex C of EN 1997-1 (CEN 2004). The ϕ values for Layers A and B became 27.4◦ and 16.3◦ respectively. Assuming the limiting√active pressure is determined from σa = Ka σv − 2c Ka , this necessitated a reduction in c for Layer B to 4.3 kPa in order to reduce σa by a total factor of 1.35. Figure 6 compares the horizontal effective stresses (σh ) on the retained side of the wall calculated by the FEA between the unfactored and reduced ϕ (“factored Ka”) cases.Also plotted are the unfactored values from the FEA multiplied manually by 1.35. Down to an elevation of −7.4 m, where the soil was in an active state, the effect of reducing ϕ was very similar to manual multiplication by 1.35. At lower elevations where the soil was not in an active state, the effect of reducing ϕ was actually to reduce σh . Consequently, the overall resultant horizontal load on the retained side of the wall due to σh was increased by a factor of only 1.17 rather than 1.35. The corresponding wall bending moment profiles are compared in the first three lines in Figure 5. The “unfactored × 1.35” profile was obtained by multiplying the unfactored bending moment profile by 1.35 (i.e. factoring the effect of the action as carried out to

obtain the maximum value 240 kNm/m above). Down to −4.0 m elevation there is also a very close match between the reduced ϕ (“factored Ka”) and manually factored profiles. Below this level there is less reversal of the profile due to the reduced σh on the retained side of the wall near the toe. The maximum bending moment and force values of 243 kNm/m and 170 kN/m respectively match closely with those obtained from manually factoring the effects of the actions above. It appears that the method of reducing ϕ and c values to simulate Ka reduction and hence load factoring of earth pressures on the retained side of the wall worked well in this example wherever the soil was in an active state. However, if the soil has a stress ratio in excess of 1, ϕ reduction will have the effect of reducing horizontal earth pressure. However, it could be argued that since soil strength has been reduced, the overall effect is conservative. This would need further investigation. Incidentally, ive failure of the wall did not occur, suggesting that d = 4.95 m is adequate for the load case modelled. 3.3

DA2

DA2 is the most challenging to implement in retaining wall design using numerical analysis because it involves factoring both geotechnical actions and geotechnical resistances, neither of which can be determined with ease, particularly the latter. It is straightforward to implement by LEM where the actions and resistances are pre-determined; ULSs are checked by moment equilibrium of the factored loads and resistances. Determining moment equilibrium of earth pressure outputs from numerical analysis is fruitless because the pressures are already in equilibrium and the required safety factor on ive resistance can rarely be achieved.

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Figure 7. Development of ive failure due to resistance factoring for DA2 with d = 5.8 m.

Attempting artificially to force ive resistance failure (in a similar fashion to the strip foundation example above) creates highly unrepresentative conditions in the model and is not recommended. Rather than attempting to factor the mobilised ive resistance, the most credible results were obtained by factoring the available ive resistance. This was achieved in a similar fashion to the Ka factoring described for DA1(1) above. Using the figures in EN 1997-1 Annex C (CEN 2004), ϕ = 18.3◦ for Layer B on the excavated side only of the wall was estimated in order to reduce the ive earth pressure coefficient Kp by the resistance factor γR = 1.4. A c correction to 4.2 kPa was also necessary to √ ensure the limiting ive pressure (σp = Kp σv + 2c Kp ) was factored by 1.4. In the FEA, a surcharge of 11.1 kPa was applied and the Ka value increased by a factor of 1.35 to produce the effect of factoring the horizontal effective stress on the retained side, as described for DA1(1) above. The quickest means of determining d was found to be initially reducing the Kp value by, in equal steps to, a factor of 1.6 and plotting the resulting horizontal deflection of the toe of the wall. ive failure was determined to occur with a resistance factor in excess of 1.4 for a d value of 5.8 m, as shown in Figure 7. The analysis was then repeated with d = 5.8 m but Kp unfactored in order to obtain the structural forces both with Ka factored and without Ka factored but with structural forces multiplied by 1.35 for comparison, as carried out for DA1(1). The resulting bending moment plots are shown in Figure 8. Again, there is a close match between the profiles for the two factoring methods down to the maximum bending moment value at −4.5 m elevation but with less reversal of the “factored Ka” profile below this level due to the reduced σh near the toe on the retained side. There was close agreement between the maximum design values of the structural forces: the “factored Ka” method gave maximum bending moment and force values of

Figure 8. Design value wall bending moment profiles for DA2.

225 kNm/m and 164 kN/m respectively while the manually factored values were 227 kNm/m and 161 kN/m respectively. The effect of factoring Kp on σh on the excavated side of the wall is shown in Figure 9. Where the soil was in a ive state down to −9.3 m elevation with characteristic “unfactored” values, the “factored Kp” σh matches very closely with unfactored σh /1.4. Below this level, the “factored Kp” σh is higher because the reduced ϕ value has forced the wall to mobilise the available ive resistance at lower elevations. This higher σh is less concerning than the reduced σh on the retained side in DA1(1) and DA2 because the available ive resistance has been factored and the increased σh merely provides confirmation that with factored ive resistance, the wall is approaching ive failure. Relatively, the d values obtained in each DA compare well with those obtained by LEM (Frank et al. 2004). In these FEAs, the d = 5.8 m in DA2 is 17% bigger than d = 4.95 m in DA1(2) and DA3. Using LEM, d = 7.89 m in DA2 is 19% bigger than d = 6.62 m in DA1(2) and DA3. As expected, LEM produces significantly more conservative d values than FEA, but the similar relative differences give some credibility to the FEA methods employed, in this example at least. An alternative method of ing the ULS, as described by Frank et al. (2004) when using a spring model for this problem, is to apply the load and resistance factors together on the ive resistance, i.e. 1.35 × 1.4 = 1.89. This was applied in a further FEA where ϕ and c were reduced stepwise to 12.9◦ and 3.6 kPa and lower respectively in order to reduce the

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4

Figure 9. Earth pressure on excavated side of wall for DA2.

Figure 10. Development of ive failure due to combined load and resistance factor on ive resistance for DA2.

available ive resistance by a factor of approximately 1.89 and above. The Ka value was also factored as described above. The horizontal deflection of the wall toe was plotted against the approximate ive resistance factor and a resistance factor in excess of 1.89, as shown in Figure 10, was obtained with d = 6.4 m. This value is approaching the LEM value for DA3 and is considered an overly conservative result for an FEA.

CONCLUSIONS

For the particular embedded cantilever wall example studied, credible ULS designs were achieved using FEA following the three different Design Approaches of EN 1997-1. The relative differences between the embedment depths determined showed consistency with the LEMderived values. DA3 and DA1(2) are the most straightforward to implement in FEA because they involve a material factoring approach. A means of factoring the geotechnical action on the retained side of the retaining wall by indirectly factoring Ka values through manipulation of ϕ and c of soils on the retained side was successful wherever active pressures were already mobilised with characteristic values. Where stress ratios exceeded 1 at lower elevations, the effect of reducing ϕ was to reduce σh . Since the soil strength had been reduced, the overall effect of the method may be conservative but this needs more investigation. For DA2, where the ive resistance must be factored, the most credible results were obtained by factoring the available resistance by reducing the Kp value through manipulation of ϕ and c of soils on the excavated side of the wall only. These conclusions are valid for the particular example studied only. A considerable amount of study is needed to test these methods or others on the full range of geotechnical structures and variables. REFERENCES Bauduin, C., Bakker, K.J. & Frank, R. 2005. Use of finite element methods in geotechnical ultimate limit state design. Proc. XVI ICSMGE, Osaka: 2775–2779. CEN. 2004. EN 1997-1 Eurocode 7: Geotechnical design – Part 1: General rules. CEN/TC 250/SC7. Brussels: European Committee for Standardization. Frank, R., Bauduin, C., Driscoll, R., Kavvaddas, M., Krebs Ovesen, N., Orr, T. & Schuppener, B. 2004. Designer’s guide to EN 1997-1: Geotechnical design – General rules. London: Thomas Telford. Plaxis. 2008. Plaxis 2D Version 9.0 Manual. Delft: Plaxis.

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Ultimate Limit State Design to Eurocode 7 using numerical methods C.C. Smith & M. Gilbert Department of Civil and Structural Engineering, University of Sheffield, UK

ABSTRACT: Eurocode 7 has been criticised because the suggested load (or action) factor (Design Approach 1, Design Combination 1) and load/resistance factor (Design Approach 2) methods appear not to readily lend themselves to numerical analysis. However, although these methods appear somewhat less versatile than Design Approach 1, Design Combination 2, this paper describes an approach by which they can be applied in conjunction with numerical analysis methods. Computational limit analysis results obtained using the LimitState:GEO software are used to illustrate the points made, and advantages and disadvantages of the various design approaches are discussed. Ramifications of implementing other features of Eurocode 7 in numerical analysis software are also briefly discussed.

1

2 THE ULTIMATE LIMIT STATE

INTRODUCTION

Eurocode 7 (BSI 2004) exists within the framework of the more general structural Eurocodes and, in contrast with many preceding geotechnical design codes, has been developed to be as general as possible in application. While this results in a number of benefits, such as apparently straightforward integration with general purpose numerical analysis methods, it can result in potential differences in interpretation and application. Assumptions that were implicit in previous practice often need to be stated explicitly in Eurocode 7. Eurocode 7 describes two general approaches for assessing the ultimate limit state (ULS). In broad these involve one in which material properties are factored prior to the design assessment and one in which factors are applied to actions and resistances in the problem prior to the design assessment. The former approach generally presents no problems in application and in integration with general purpose numerical methods. The latter method, however requires greater precision in problem definition, and at first sight is challenging to integrate fully with a general purpose numerical approach. Addressing this challenge is the subject of this paper. Since an ultimate limit state (ULS) design approach links naturally to a limit analysis or limit equilibrium analysis, the problem will be studied in the context of limit analysis. However the broad methodology should be applicable to all ULS analysis methods. In this paper the Discontinuity Layout Optimization (DLO) method (Smith & Gilbert 2007) as implemented in the LimitState:GEO software (LimitState 2009) is utilized to illustrate the application with numerical limit analysis; however the approach is applicable to most generic numerical software packages.

In general any given design is inherently stable and is by implication nowhere near to its ultimate limit state. The principle of a ULS assessment is to drive the system to collapse by some means and assess the difference between the actual and ULS state. This difference can be considered a measure of over-design or, conventionally, Factor of Safety (FoS). This process can be achieved implicitly or explicitly. In many conventional analyses the process it is typically implicit. In a general numerical analysis it must be done explicitly. There are three main ways of driving a system to the ULS: 1. Increase an existing load in the system 2. Reduce the soil strength 3. Impose an additional load in the system In current numerical limit analysis approaches, using optimization techniques to identify the collapse state, increasing an existing load to drive the system to collapse (i.e. 1. in the list above) is inherent to the method. A supplementary load factor, henceforth referred to as an ‘adequacy factor’, λ, is applied to one or more unfavourable loads and the software identifies the magnitude of λ required to achieve collapse. 3

DESIGN TO EUROCODE 7

3.1 Background to Eurocode 7 The Eurocodes bring a unified approach to civil engineering design. Eurocode 7 has undergone significant change since its original ENV draft form to accommodate the differing approaches and design philosophies used across Europe. There are now three Design Approaches (DA) that may be utilized. Normally only

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Table 1.

Eurocode 7 partial factors for STR and GEO.

Parameter Action/action effect

Permanent Variable

Resistance Soil parameters

Unfavourable Favourable Unfavourable Favourable

c tan φ cu

DA1/1

DA1/2

DA2

DA3

1.35 1.0 1.5 0.0

1.0 1.0 1.3 0.0

1.35 1.0 1.5 0.0

1.0/1.35* 1.0 1.3/1.5* 0.0

1.0

1.0

1.1/1.4†

1.0

1.0 1.0 1.0

1.25 1.25 1.4

1.0 1.0 1.0

1.25 1.25 1.4

*DA3 values for actions are given for geotechnical/structural actions. † Factors on resistance depend on resistance type.

one is permitted by the National Application (NA) document of each nation. Each approach has its advantages and disadvantages and are (broadly) summarized as follows:

corresponding design resistance (Rd ) at the ultimate limit state, i.e.:

DA1 This approach requires two design combinations to be considered:

This is straightforward in principle but it is necessary to clarify the definition of an action, an action effect and a resistance. Note that this key Eurocode equation is an inequality; it provides no measure of the degree of over-design.

DA1/1 Factors on actions DA1/2 Factors on material strengths DA2 Factors on actions and resistances DA3 Factors on material strengths In this paper they will be considered in the context of the Eurocode ultimate limit states GEO and STR which refer to failure in the ground and failure in the structure respectively. Since the use of DA1/2 (and also DA3) is generally straightforward, discussion in this paper will focus on the use of action/resistance type approaches DA1/1 and DA2 in numerical models.

3.2 Partial factors and the limit state In Eurocode 7 it is stated that ‘For each geotechnical design situation it shall be verified that no relevant limit state … is exceeded’ [EN 1997-1 2.1(1)P]. To achieve this, the general approach in Eurocode 7 is to apply factors on uncertainties at their source in the calculation, rather than being applied to the whole calculation. Thus factors may be applied to: 1. 2. 3. 4. 5.

actions (F) action effects (E) material properties (X ) resistances (R) geometrical parameters (a)

in order to obtain design values (designated by a subscript ‘d’). It is these design values that are used in any stability calculation. Factors to be used on actions, material properties and resistances are given in Table 1. Factors on geometry parameters are not addressed in this paper. To prevent limit state STR or GEO from occurring, design (i.e. factored) actions or effects of actions (denoted Ed ) must be less than or equal to the

3.3 Actions and resistances 3.3.1 Introduction In Eurocode 7 there appears to be scope for a range of interpretations of the designations of actions and resistances and for the stages at which partial factors are applied to these. This may be seen by examining much of the current literature on Eurocode 7, in which different authors apply factors in different ways. In this paper the following interpretations are proposed for consistent numerical analysis. 3.3.2 ‘Actions’ and ‘Action effects’ The Eurocode defines an action or action effect as a force acting on a structure or on a body of soil or within a soil. In this paper an action will be taken as a quantity whose value is explicitly known prior to the analysis. This generally restricts it to a dead weight or an externally applied load (and will typically include variable actions). It cannot therefore be e.g. an active earth pressure, the calculation of which also involves soil strength. In contrast an action effect will be taken as one that is a function of soil strength and an action, such as an active earth pressure. While an action may be favourable or unfavourable (i.e. promoting or opposing collapse), an action effect is always considered unfavourable. If it is favourable then it is regarded as a resistance. 3.3.3 Resistances Eurocode 7 defines resistance as: ‘capacity of a component, or cross-section of a component of a structure

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to withstand actions without mechanical failure e.g. resistance of the ground, bending resistance, buckling resistance, tensile resistance’ [EN1997-1 1.5.2.7]. It is assumed that this applies equally to the ground. This definition can for example be interpreted to mean: (i) the shear strength of the soil, or (ii) the combined resistance of soil strength and weight (e.g. a ive pressure), or (iii) shear/bending resistance of a structural element. In this paper two types of resistance are distinguished, analogous to the distinction between action and action effect, a structural resistance e.g. plastic moment of resistance of a sheet pile wall and a geotechnical resistance. e.g. a ive earth pressure that also involves self weight.

yielding in the soil adjacent to the wall, and (ii) a specific failure mode (Rankine earth pressure distributions are only strictly applicable to certain wall failure modes). Such implicit assumptions cannot be made in a general numerical analysis. Additionally such assumptions almost always lead to a system which is not in equilibrium. To achieve the same effect with a numerical method a specific failure mode must therefore be explicitly induced. Once the collapse mechanism has been identified, the actions and resistances are then available after the analysis. At this stage partial factors may be applied and equation 1 evaluated.

3.3.4 Commentary In any form of plastic analysis commonly used for ULS type assessments, actions will be known in advance of the analysis. However action effects and geotechnical resistances, being functions of material strengths and actions, are not known in advance of a calculation. Strictly they can only be determined after a collapse analysis has been carried out. In the Eurocode 7 Design Approaches where factors of unity are applied to action effects and resistances this is not an issue since it is not necessary to be able to quantify these in the design assessment. Such design approaches include DA1/2 and DA3 where the only non-unity factors are applied to variable actions and structural actions, both of which are almost always externally applied and thus known in advance. DA1/1 can also fall into this category if the pertinent action effect in equation (1) arises purely from actions, such as in a simple bearing capacity problem. For these approaches, overall stability can simply be assessed by checking if the scenario is stable based on pre-factored parameters. In a numerical limit analysis this corresponds to an adequacy factor λ ≥ 1 on any unfavourable action in the problem. However in Eurocode 7 Design Approaches where non-unit factors are applied to action effects and/or resistances, then it is necessary to be able to quantify these in a design assessment. The means by which these are determined must therefore be considered carefully.

In the preceding section, it was argued that a true Action and Resistance factor approach requires predetermination of the collapse mode in order to identify actions and resistances and thus to enable application of partial factors. Rather than considering a specific collapse mode (such as e.g. wall sliding), it is more correct to consider loss of equilibrium in a specific direction (in the case of wall sliding, this might be horizontal equilibrium). Since equation (1) is cast in of forces rather than energy, it must be evaluated in a specific direction. In order to drive the system to collapse, and simultaneously preserve equilibrium, it is necessary to apply an unfavourable ‘hypothetical’external force (or moment) H parallel to the equilibrium direction to be checked, and to then increase this force until failure occurs. For example, for a horizontal equilibrium check on a retaining wall, H would be applied to the centroid of the wall (to avoid applying additional moments) and in the direction that would induce the expected failure (i.e. in the direction of the active pressure). This is a straightforward calculation for a computational limit analysis method, and ensures equilibrium is preserved at all times while generating appropriate soil failure conditions compatible with the mode of failure being investigated. After the numerical analysis the ULS actions and resistances on the wall are available and can be (post) factored prior to use in a stability assessment (equation 1). H itself is not directly needed and is discarded. The disadvantage of this approach is that the mode of failure must be pre-determined and equation 1 can only be applied in the sense of the imposed failure mode e.g. it would not be theoretically correct to use the values computed for wall sliding to make an assessment against overturning.

4

DETERMINATION OF STABILITY WITH AN ACTION AND RESISTANCE FACTOR APPROACH

4.1 Current practice Consider the ULS design of a retaining wall. This is typically carried out by assuming active and ive Rankine pressures on opposite sides of the wall. These would be designated as an action effect and resistance respectively in this problem. However the values are not strictly quantifiable in advance of a collapse analysis because in assuming these pressure distributions, there is an implicit assumption of: (i)

4.2 Proposed procedure for numerical analysis

5 5.1

SUMMARY OF GENERAL NUMERICAL ANALYSIS APPROACH Problem classification

Eurocode 7 ULS problems can be divided into two broad classes, those amenable to either pre-factoring or post-factoring in relation to the stability analysis.

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5.2

Pre-factoring approach

Design approaches amenable to a pre-factoring approach are those that where the factors on action effects and resistances are 1.0. Actions may have factors which are non-unity applied to them. Note that: 1. This generally includes DA1/2, DA3 and also DA1/1 (where only actions are involved). 2. Designs can be directly analysed using general numerical methods. There is no need to explicitly identify matched actions and resistances, or to test a specific failure mode. 3. The numerical method can be used to automatically identify the critical collapse mechanism 4. Equation (1) will be identically satisfied if the adequacy factor λ applied to any unfavourable action satisfies λ ≥ 1. 5.3

Table 2. Parameters for the problem shown in Fig. 1 (prefactoring approach).

Quantity

Post-factoring approach

Design approaches that require a post-factoring approach are those are those that involve action effects and/or geotechnical resistances, and where the factors on actions and/or resistances are not equal to 1.0. Note that: 1. This generally includes DA2 and DA1/1 (where actions effects are involved). 2. Designs can be analysed indirectly using general numerical methods. Matched actions and resistances must be identified in advance in conjunction with a specific equilibrium direction to be checked. 3. The numerical method can be used to identify the specific collapse mechanism corresponding to a specified equilibrium direction for this case, by applying a disturbing force (or moment as required) in the equilibrium direction. 4. The result of the numerical analysis will allow actions and resistances to be explicitly determined. These values may then be factored as appropriate and stability checked using equation 1. 6 WORKED EXAMPLES 6.1

Figure 1. Simple strip footing on undrained soil, showing typical pattern of slip-lines at collapse. Footing width, B = 2 m, soil undrained shearing resistance cu = 80 kN/m2 , soil unit weight γ = 20 kN/m3 , applied load, V = 500 kN/m, footing weight = 40 kN/m.

Surface footing - DA1/1 pre-factoring approach

6.1.1 Problem statement It is required to carry out a DA1/1 assessment of the vertical bearing stability of the footing shown in Fig. 1. Simple bearing capacity problems such as this do not generally involve action effects and are therefore amenable to the pre-factoring approach in DA1/1 (which factors actions but not resistances). In of actions and resistances, the actions in this problem are clearly the external load on the footing and the weight of the footing itself. The resistance is the resistance of the soil and is experienced by the footing at the soil/footing interface. As part of the assessment, the overdesign factor with respect to applied load will be determined.

Pre-analysis Applied load Footing weight Soil unit weight Adequacy Factor λ on: Numerical analysis Adequacy factor λ λ ≥ 1.0 ⇒ Ed ≤ Rd ?

Partial factor

Type

Unfav. permanent Unfav. permanent Neutral applied load

true ⇒ safe

1.35 1.35 1.0

1.15

In LimitState:GEO this means applying the adequacy factor λ to the applied load, and results in the collapse mechanism shown in Fig. 1. 6.1.2 Numerical stability assessment The values of the parameters used in the pre-factoring analysis are given in Table 2 (unfactored soil properties are shown in Fig. 1), together with results from analysis of the problem in LimitState:GEO using a fine nodal density† . In this problem the soil self weight has no effect (i.e. is Neutral). The result indicates that the design is safe and is overdesigned (over and above the Eurocode factors) by a factor of 1.15 on the applied load. 6.1.3 Analytical check The collapse soil resistance R beneath the footing may be calculated from the Terzaghi bearing capacity equation:

For an undrained failure with no surcharge, Nc = 5.14 and:

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Hence R = 822 kN/m

Table 3. Analytical determination of actions and resistances in the problem shown in Fig. 1 (Actions in kN/m).

Quantity Soil properties cu (kN/m2 ) Actions (F) Applied load Footing weight sum Resistances (R) Base resistance F ≤ R?

Characteristic value

Partial factor

Design value

80

1.0

80

500 40 540

1.35 1.35

675 54 729

822 true

1.0

822 true

a ‘hand’ calculated stability assessment is summarized in Table 3. The over-design factor with respect to the applied load is (822 − 54)/675 = 1.14. The numerical result (in of adequacy, or over-design factor) is thus ≈1% different from the exact theoretical answer.

Figure 2. Simple gravity wall (sliding) problem, showing typical pattern of slip-lines at collapse. Wall height, 4 m, embedment depth, 1 m, wall width, 1.5 m, soil angle of drained shearing resistance φ = 30◦ , soil unit weight γ = 16 kN/m3 , wall vertical faces smooth, wall base angle of drained shearing resistance φ = 30◦ . Wall sits on a rigid base layer. Table 4. (a) Parameters for the problem shown in Fig. 2 (pre-factoring approach).

6.2 Gravity wall sliding – DA2 post-factoring approach

Quantity

6.2.1 Problem characteristics It is required to carry out a DA2 assessment of the horizontal stability of the gravity wall shown in Fig. 2 (i.e. a check against horizontal sliding failure). A DA2 analysis could be carried out by factoring the source actions (i.e. self weight of soil behind the wall) prior to solving. However this is not conventional practice for a load and resistance factor type approach, in which factors are normally applied to the active, ive and base shear resultant forces. The problem thus requires a post-factoring approach and the inclusion of a hypothetical horizontal force H acting in the direction of failure to be checked, as depicted in Fig. 2. The adequacy factor λ is applied to this load H , while the degree of overdesign can be determined by the ratio of resistance to actions. 6.2.2 Numerical stability assessment Pre-factoring settings are given in Table 4(a) (unfactored soil properties are shown in Fig. 2), together with results from the analysis of the problem in LimitState:GEO, using a medium nodal density.† The result indicates that an additional horizontal force H = 64.5 kN/m is required to cause failure. Table 4(b) lists the key action effects and resistances available after the analysis. The data indicates that the factored resistances are greater than the factored action †

LimitState:GEO interconnects nodes laid out across the problem domain with potential slip-lines, from which the critical failure mechanism is then identified. Thus the more nodes that are present, the more accurate the solution is likely to be. Fine and medium nodal densities correspond to targets of 1000 and 500 nodes respectively.

Partial factor

Type

Pre-analysis Retained soil Neutral 1.0 Soil on excavated side Neutral 1.0 Wall weight Neutral 1.0 unit load H Neutral 1.0 Adequacy Factor λ on: load H Numerical analysis Adequacy factor λ 64.5 Post analysis stability check (horizontal equilibrium) Active force Unfav. permanent 1.35 ive force Resistance 1.4 Base friction Resistance 1.1 load H ignore 1.0 b) Post analysis stability check (horizontal equilibrium); all forces in kN/m (taken from output of numerical analysis)

Quantity Action effects (E) Active force Total Resistances (R) ive force Base Friction Total Outcome

Characteristic value

Partial factor

Design value

42.7 42.7

1.35

57.6 57.6

24 83.1 107.1 E≤R

1.4 1.1

17.1 75.6 92.7 Ed ≤ Rd

effects. The wall is over-designed (for sliding only) by a factor of (resistances/actions) 92.7/57.6 = 1.61. 6.2.3 Analytical check In a conventional factored load and resistance approach it is implicitly assumed that the soil is

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yielding either side of the wall. In this case active and ive Rankine pressure distributions are typically taken to act on each side of the wall. This is valid in this case but not necessarily correct in all situations. For a DA2 analysis soil strengths are unfactored and φd = 30◦ . Thus the active force A is given by:

and the ive force P is given by:

which matches the numerical analysis. Since the walls are frictionless, the normal force on the wall is equal to the wall weight (144 kN/m) and the base friction is given by 144 × tan30 = 83.1. Once partial factors are applied, the over-design factor is therefore the same as before, i.e. (24/1.4 + 83.1/1.1)/(42.7 × 1.35) = 1.61. 7

2. Eurocode 7 DA1/2 and DA3 (and DA1/1 in certain cases) can generally straightforwardly be assessed with a numerical model, and have the advantage that the numerical model can be used to identify the most critical collapse mechanism. 3. Eurocode 7 DA1/1 and DA2 present more problems as they require the disturbing action required to cause collapse to be specified in advance of the numerical analysis. The design check also then becomes slightly more involved. This paper has outlined an approach that allows the use of these action/resistance factor methods to be carried out in conjunction with numerical methods, and provides details of simple examples where these approaches have been applied. ACKNOWLEDGEMENTS All numerical analyses described herein were undertaken using LimitState:GEO version 2.0c; see: http://www.limitstate.com/geo REFERENCES

CONCLUSIONS

1. Eurocode 7 provides a new unified design approach for geotechnical design, with a clear and explicit methodology which allows partial factors to be used to for uncertainty at source. The unified approach additionally permits the use of general purpose numerical methods to be applied across a broad range of problem types.

BSI (2004). BS EN 1997-1:2004 Geotechnical design. General rules. LimitState (2009). LimitState:GEO Manual VERSION 2.0 (Sept 3 ed.). LimitState Ltd. Smith, C. C. & M. Gilbert (2007). Application of discontinuity layout optimization to plane plasticity problems. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, 2086, 2461–2484.

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Thomas Benz_ Steinar Nordal-numerical Methods In Geotechnical Engineering _ Proceedings Of The Seventh European Conference On Numerical Methods In Geotechnical Engineering, Trondheim, Norway, 2-4 June 681e5s

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