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Design of Rigid Pavement
Elements of a Typical Rigid Pavement y A typical rigid pavement has three elements :
i) Subgrade; ii) Sub‐base; iii) Concrete slab y Subgrade is the in situ soil over which the pavement structure is ed. y Stiffness of the subgrade is measured by modulus of subgrade reaction (K). y K is determined with the assumption that the slab is resting on dense fluid and thus the reactive pressure of soil on pavement is linearly proportional to the deflection of the slab. y Value of K is widely dependant upon the soil type, soil density, and moisture content. y K is determined by plate bearing test.
Elements of a Typical Rigid Pavement
(contd)
y Sub‐base is the layer of selected granular materials placed
on the subgrade soil and immediately below the concrete pavement y It is provided for the following purposes yTo provide an uniform and reasonable firm pavement . yTo prevent mud pumping. yTo provide levelling course on undulated and distorted subgrade. yTo act as capillary cut off. y It is not a part of the rigid pavement structure as it is not provided to impart strength to the pavement structure.
Elements of a Typical Rigid Pavement
(contd)
y Construction of sub‐base is generally done by yGranular material like natural gravel, crushed slag,
crushed concrete, brick metal, laterite, soil aggregate etc. yGranular construction like WBM or WMM yStabilized soil ySemi rigid material like Lime clay Puzzolana Concrete, Lime Flyash Concrete, Dry Lean Concrete . y Concrete Slab is designed on the basis of flexural strength of concrete. y Due to repeated application of flexural stresses by the traffic loads, progressive fatigue damage takes place in the cement concrete slab in the form of gradual development of micro‐cracks.
Elements of a Typical Rigid Pavement
(contd)
y The ration between flexural stress due to the load and the
flexural strength of concrete is termed as the Stress Ratio (SR). If SR < 0.45 the concrete is expected to sustain infinite number of repetition. y Various properties of concrete as recommended for use as rigid pavement are y Flexural strength: 45 kg/cm2; y Modulus of Elasticity: 3 x 105 kg/cm2; y Poisson's ratio: 0.15; y Coefficient of thermal expansion: 10 x 10‐6 /⁰C.
Types of Rigid Pavement y There are four types of Rigid Pavement y ted and unreinforced concrete pavement y ted and reinforced concrete pavement y Continuously reinforced concrete pavement y Prestressed Concrete Pavement y Most of the rigid pavements in India are ted and
unreinforced concrete pavement. The necessary IRC design guidelines are y IRC: 58– 2002 (Guideline for the Design of Plain ted Rigid Pavements for Highways) y IRC: 15‐ 2002 (Standard Specifications and Code of Practice for Construction of Concrete Roads) y IRC: SP: 62‐ 2007
Stresses in Rigid Pavement y Stresses in concrete pavement are produced due to
following reasons y Applied wheel load y Changes in temperature y Changes in moisture content y Volumetric changes in soil subgrade y In design of rigid pavement stresses due to applied wheel load and changes in temperature are considered. y Since the nature of the stresses due to changes in moisture content is reverse that of stresses due to changes in temperature, it is not considered in thickness design. y Stresses due to volumetric changes of subgrade soil is taken care by properly selected sub‐base course.
Westergaard Analysis y H. M. Westergaard is considered to be pioneer person in
rigid pavement design. y The basic assumptions in Westergaard (1925) analysis for computation of stresses are i. Concrete slab acts as a homogenous, isotropic, and elastic solid in equilibrium. ii. The reaction of subgrade are vertical only and they are proportional to the deflection of the slab. This reaction of subgrade per unit area at any given point is equal to a constant K multiplied by the deflection at that point. iii. The thickness of the slab is uniform. iv. The load at the interior and at the corner of the slab is distributed uniformly over a circular area of .
Westergaard Analysis
(contd)
v. For corner loading the circumference of the area of
is tangential to the edge of the slab. vi. For the load at the edge of the slab is uniformly distributed over a semi‐circular area . The diameter of the semi‐circle is with the edge of the slab. y Critical stress locations y Interior: This is the position within the slab which is at any place remote from all the edges. y Edge: This is the position of the slab which is situated in the edge, remote from the corners. y Corner: This is the position which is situated at the bisector of the corner angle.
Westergaard Analysis
(contd)
Interior Edge
Corner
Stresses due to Wheel Load y Under the wheel load the interior and the edge of the slab
y y
y y y
behaves like a simple ed beam having tension at the bottom. Under the action of wheel load corner may behave as a cantilever specially when the slab is casted by . The maximum tensile stress may be found at corner as this location is considered as discontinuous from all the directions. As the edge is discontinuous in one direction this location may encounter lesser stress than the corner. Loads applied at the longitudinal edge can produce more stress than that at the transverse edge. Least stress is occurred at the interior as this position of the slab is continuous in all directions.
Stresses due to Wheel Load
(contd)
y Computation of stress at edge location y The original equations of Westergaard has been
modified by several researchers. y As per IRC the stresses due to wheel load may be determined by the software IITRIGID developed at IIT Kharagpur. y The stresses at edge may also be computed by the following equation as modified by Teller & Sutherland
P h
l b
σ le = (0.529 2 )(1+ 0.54μ)[4 log10 ( ) + log10 (b) − 0.4048]
Stresses due to Wheel Load
(contd)
y σle = Wheel Load Stress at Edge Region (kg/cm2) y P = Design Wheel Load (kg) or ½ of Single Axle Load (kg) or y y y y y y y
¼ of Tandem Axle Load (kg) h = Pavement thickness (cm) μ = Poisson's Ratio E = Modulus of Elasticity of Concrete (kg/cm2) k = Modulus of Subgrade Reaction (kg/cm3) l = Radius of Relative Stiffness (cm) b = Equivalent Radius of Resisting Section (cm) a = Radius of Load Area (cm)
Stresses due to Wheel Load
(contd)
y Relative stiffness of slab to sub‐grade y A certain degree of resistance to slab deflection is
offered by the sub‐grade. y The sub‐grade deformation is same as the slab deflection. Hence the slab deflection is direct measurement of the magnitude of the sub‐grade pressure. y The resistance to deformation depends on the stiffness of the ing medium as well as on the flexural stiffness the slab. y This pressure deformation characteristics of rigid pavement lead Westergaard to define the term radius of relative stiffness (l). l in cm is given by Eh 3 l=4 12 (1 − μ 2 ) k
y Equivalent Radius of Resisting Section y The wheel load concentrates on a small area of the
pavement y The area of the pavement that is effective in resisting the bending moment due to that load may be more than tyre imprint area. y The maximum bending moment occurs under the loaded area and acts radial in all directions. y The area of the pavement that is effective in resisting the bending moment due to a wheel load is known as Equivalent Radius of Resisting Section or also as Radius of Equivalent Distribution of Pressure. b = a =
a ≥ 1 . 724 h
for 1 .6 a
2
+ h
2
− 0 . 675 h
for
a < 1 . 724 h
Stresses due to Wheel Load
(contd)
y Computation of Stress at Corner Location y Wheel load stress at corner region is obtained as per
Westergaard’s analysis modified by Kelley
3P σ lc = 2 h
⎡ ⎛ a 2 ⎞1 .2 ⎤ ⎟ ⎥ ⎢1 − ⎜ ⎜ ⎢ ⎝ l ⎟⎠ ⎥ ⎣ ⎦
y σlc = Wheel Load Stress at Corner Region (kg/cm2)
Stresses due to Temperature Variation y Stresses are induced in the slab due to variation of
temperature y The temperature variation may be of two types y daily variation resulting in a temperature gradient across the thickness of the slab, and y seasonal variation resulting in uniform change in the slab temperature. y The former results in warping stresses and the later in frictional stresses.
Stresses due to Temperature Variation
(contd)
y Temperature Warping Stresses y Cement concrete pavement undergoes a daily cyclic
y y
y
y
change of temperature as thermal conductivity of concrete is low. The top surface of the pavement becomes hotter than bottom during day time and cooler during night. In the daytime thus the top surface of the pavement expands more than that in the bottom. This results the slab to warp upwards (top convex). The restraint offered to this warping tendency by self‐ weight and the dowel bars of the pavement induces stresses in the pavement. This is known as warping stress.
Stresses due to Temperature Variation
(contd)
y Flexural tensile stress will be generated at the bottom y
y
y y
surface during day time. Conversely, in the night the slab warp downward (top concave). Flexural tensile stress will be generated at the top surface. As the restraint offered to warping at any section of the slab is a function of weight of the slab upto the section, the corner has very little of such restraint for slabs without dowel bars and is free to warp. Thus warping stress is negligible. The interior can offer maximum restraint to warp and has maximum warping stress. The equations for warping stresses are available due to Westergaard.
Stresses due to Temperature Variation
(contd)
y The critical combination of stress indicates that most critical
location is the edge. y The equations for the warping stress at the edge as recommended by IRC is obtained as per Westergaard’s analysis using Bradbury’s coefficient.
σ twe =
Eαt C 2
y σtwe = Temperature Warping Stress at Edge Region (kg/cm2) y E = Modulus of Elasticity of Concrete (kg/cm2) y α = Coefficient of thermal expansion of concrete y t = temperature difference between top and bottom of slab y C = Bradbury’s coefficient depends on L/l or W/l of slab
Stresses due to Temperature Variation
(contd)
y Temperature Friction Stresses y Uniform seasonal temperature variation cause the slab
expands and contracts in the longitudinal direction. y This expansion and contraction of the slab is prevented by the friction between the slab and the subgrade. Stresses are thus set up in the slab. L/2 σcAc
σ c × h × B × 100 = B × ∴σ
c
=
L h × ×W ×f 2 100
WLf 2 × 10000
Stresses due to Temperature Variation
(contd)
B = Slab width (m) h = slab thickness (m) L = Length of the slab (m) σtfe = Temperature friction stress in concrete (kg/cm2) W = Unit weight of concrete in (kg/cm2) f = Coefficient of friction between concrete and subgrade y The temperature friction stress is taken care in rigid pavement by providing ts in plain ted pavement or by reinforcement in reinforced concrete pavement
Critical Combination of Stresses y Combination of flexural stresses due to wheel load and that
y y
y
y
to temperature warping provides the critical stress for design of rigid pavement. Maximum combined stress at the three critical locations will occur when these two stresses are additive. Warping stresses at three locations decrease in the order of interior, edge and corner whereas the wheel load stresses decrease in the order of corner, edge and interior. Therefore, critical stress condition is reached at edge location where neither wheel load stress nor the warping stress is minimum. Since at night due to warping the corner may behave as cantilever it is recommended to check the wheel load stress at corner.
ts in Rigid Pavement y The rigid pavement slab is deliberately divided into blocks of
appropriate sizes in order to take care the effects of temperature friction stress or stresses due to moisture variation. y These deliberate planes of weaknesses in the slab are known as ts. A good t should have the following functional requirements: y Must be waterproof [proper sealing to be provided] y Riding quality should not be deteriorated y Should not make any structural weakness [for example staggered ts should be avoided]
ts in Rigid Pavement
(contd)
y Classification of the ts according to location in the
pavement y Longitudinal ts y Transverse ts Transverse ts Longitudinal ts
ts in Rigid Pavement y Classification of ts according to Forms y Dummy t y Butt t y Tongue and Groove t y ts with Clear Gap y Classification of ts according to Function y Expansion t y Contraction t y Longitudinal t y Construction t
(contd)
ts in Rigid Pavement
(contd)
Expansion t Dowel Bar [Fully Bonded part]
Sealer
Expansion Cap with Cotton Waste at the Back
t+12
t
Filler
75mm
Dowel Bar [Bitumen Painted part]
t+6
Schematic Drawing of Expansion t with Dowel Bar
ts in Rigid Pavement
(contd)
y The pavement slab tends to expand when the temperature y y y
y y
rises above that at which the pavement was laid. Expansion of the slab is prevented by friction between the slab and the subgrade. Compressive Stress is thus set up and this may try to buckle or blow up the slab . In order to prevent this stress, Expansion ts in the transverse direction of the pavement are provided to allow space for expansion of the slab. The t is formed by maintaining a gap of about 20 to 25 mm between two slabs. The gap is filled up by a non‐extruding compressive filler material.
ts in Rigid Pavement
(contd)
y A sealing compound is provided on the top of the filler
material to prevent entry of water and dust. y To ensure transfer of load between the two slabs on each side of the t dowel bars are provided. y Dowel bars are usually mild steel round bars of short length. y Half length is bonded into concrete on one side of the t and the other half is painted by bitumen in order to prevent bonding with concrete. y A metal cap with cotton waste at the back is provided at the painted half end of the dowel bar. This ensures free movement of the slab during expansion.
ts in Rigid Pavement
(contd)
y Dowel bars not only permits the expansion of the slabs but
also holds the slab ends on each side of the t as nearly as possible. y Deflection of one slab under load is resisted by the other slab which, in turn is caused to deflect and thus carry a portion of the load imposed upon the first slab. y The spacing of expansion t may vary from twenty meters to a few hundred meters.
ts in Rigid Pavement
(contd)
Contraction t Sealer
Contraction t with Butt t Dowel Bar Sealer
Contraction t with Dummy t
ts in Rigid Pavement
(contd)
y Stresses are also generated in the concrete pavement slab
due to contraction of concrete when the temperature is reduced with respect to that during laying. y Contraction ts are thus provided to reduce tensile stress due to contraction or shrinkage of concrete. y There are two types of contraction t: y Dummy t: In this type no t is made in reality. Only a small groove is cut on top of the slab for a depth of ¼ to 1/3 of the thickness of the slab. If stress becomes more than that the slab can withstand, a crack may develop at the location of the grove as this is the weakest plane in the slab. Simple dummy t may not contain any dowel bar. If dowel bar is not provided the load transfer is ensured by particle interlocking.
ts in Rigid Pavement
(contd)
y Butt t: In case of a butt t two slabs abut each
other. Therefore, a clear plane of separation will exist in this t. Dowel bars may or may not be provided. y In case of dummy or butt t good sealing material is provided at the top of the t in order to prevent entry of water and dust inside the t. y Spacing of ts varies with thickness of the slab and also with the existence of reinforcement. For slab of thickness upto 250mm ts maximum spacing may be 4.5m whereas upto 350mm thick pavement, maximum spacing will be 5.0m.
ts in Rigid Pavement
(contd)
Longitudinal t y Longitudinal ts are necessary in the concrete slab for the pavement having more than 4.5m wide. y Longitudinal t prevents longitudinal cracking. y Mild steel bars known as Tie bars are provided across the longitudinal t to hold the t tightly closer and to keep both the slabs at the same level. y Tie bars are not provided to act as load transfer device. y Both the ends of the tie bars are fully bonded in the concrete. y Longitudinal ts may be butt type or keyed [Tongue and Groove] type.
ts in Rigid Pavement
(contd)
Construction t y Construction ts are provided in the transverse
direction whenever the placing of concrete is suspended for more than 30 minutes. y As far as possible construction t should coincide with either expansion t or contraction t. y If a construction t is provided at the location of any contraction t it should be of butt type with dowel bar. y If a separate construction t is needed, it should be provided within the middle third of two contraction ts.
ts in Rigid Pavement
(contd)
Arrangement of ts y Staggered t y When transverse ts are staggered with respect to the longitudinal t, sympathetic cracks may occur. y These cracks often occur in the line with the t in the other side of the transverse crack.
Sympathetic cracks
ts in Rigid Pavement
(contd)
y Skew and Acute Angle t y Use of skew t increases the risk of cracking at the
acute angle corners. y At the acute angles the stresses become very excessive. y At the time of warping the acute angles become completely uned and cause more stresses than that would occur in right angle corner.
Design of Rigid Pavements y Step 1: Stipulate design values for various parameters y Step 2 : Decide type and spacing between ts. y Step 3 : Select a trial design thickness of the pavement
slab. y Step 4 : Compute the repetitions of axle loads of different magnitudes during design period y Step 5 : Calculate stresses due to single and tandem axle loads and determine cumulative fatigue damage (CFD). y Step 6 : If CFD is more than 1.0, select a higher thickness and repeat the procedure from step 4.
Design of Rigid Pavements
(contd)
y Step 7 : Compute the temperature stress at edge. If sum of
the temperature stress and the flexural stress due to highest wheel load is greater than modulus of rapture select a higher thickness and repeat the procedure from step 4. y Step 8 : Design the pavement thickness on the basis of corner stress, if no dowel bar is provided and no load transfer is possible due to lack of aggregate interlocking.
Design of Rigid Pavements
(contd)
Design Example y Design a cement concrete pavement for a two lane two way National Highway in Karnataka State. The initial total two way traffic is 3000 commercial vehicles per day. The other deign parameters are: y Flexural strength of cement concrete: 45 kg/cm2. y Effective modulus of subgrade reaction of the DLC sub‐ base: 8 kg/cm3. y Spacing of contraction ts: 4.5 m. y Width of slab: 3.5m y Tyre pressure: 8 kg/cm2. y Rate of traffic increase: 7.5%. Axle load spectrum obtained from axle load survey is given below
Design of Rigid Pavements Single Axle Load Axle load (tonnes) 19‐21 17‐19 15‐17 13‐15 11‐13 09‐11 Less than 9 Total
% of axle loads 0.6 1.5 4.8 10.8 22.0 23.3 30.0 93.0
(contd)
Tandem Axle Load Axle load (tonnes) 34‐38 30‐34 26‐30 22‐26 18‐22 14‐18 Less than 14 Total
Design of Rigid Pavements
% of axle loads 0.3 0.3 0.6 1.8 1.5 0.5 2.0 7.0
(contd)
Design Traffic y Present traffic = 3000 cvpd; Design life (assumed) = 20 years; y Cumulative repetition in 20 years = 47, 418, 626 cv y N = 365 A [(1+r)n – 1]/r where A is the initial number of axles in
the year when road is operational ; r is the rate of annual growth of traffic; n is the design life. y Design traffic = 25% of repetition of commercial vehicles
= 11,854, 657 cv y Total repetitions of single axles and tandem axles are
Design of Rigid Pavements Single Axle Load Load (tonnes) Expected repitions 20 71127 18 177820 16 569023 14 1280303 12 2608024 10 2762135 Less than 10 3556397
(contd)
Tandem Axle Load Load (tonnes) Expected repitions 34‐38 35564 30‐34 35564 26‐30 71128 22‐26 213384 18‐22 177820 14‐18 59273 Less than 14 237093
Design of Rigid Pavements
(contd)
Thickness Design y Trial thickness = 32 cm; Load safety factor =1.2 y To take care unpredicted heavy truck loads the magnitude of axle loads should be multiplied by load safety factor (LSF). y National highways and other roads where there will be uninterrupted traffic flow and high volumes of truck traffic: 1.2 y Lesser important roads with lesser incidence of truck traffic : 1.1 y Residential and other local streets : 1.0
Design of Rigid Pavements Axle load AL x 1.2 Stress tonnes (kg/cm2) (AL)
Stress Ratio
(contd)
Expected Fatigue life Fatigue life Repetition (N) consumed (n) (n/N)
Single axle 20
24.0
25.19
0.56
71127
94.1 x 103
0.76
18
21.6
22.98
0.51
177820
4.85 x 103
0.37
16
19.2
20.73
0.46
569023
14.33 x 104
0.04
14
16.8
18.45
0.41
128030
Infinity
0.00
36
43.2
20.07
0.45
35560
62.8x 106
0.0006
32
38.4
18.4
0.40
35560
Infinity
0.00
Cumulative fatigue life consumed
1.1705 > 1
Tandem axle
Design of Rigid Pavements
(contd)
y Relation between fatigue life (N) and Stress ratio (SR) y N = unlimited for SR, 0.45 y N= [4.2577/ (SR ‐0.4325)]1.324 0.45≤SR≥0.55 y Log10 N = (0.9718 – SR) / 0.0.828 for SR > 0.55 y Since CFD fro thickness of 32 cm>1, increase thickness y Take next trial thickness = 33cm y Repeat the steps from 4 y Cumulative fatigue life consumed for thickness of 33 cm =
0.47 y Highest stress load stress for thickness of 33 cm = 24.10 kg/ cm2
Design of Rigid Pavements
(contd)
Temperature Warping stress y For E= 3 x 105 kg/cm2, α
=
10 x 10‐6 /⁰C, t= 21⁰ [slab thickness 33 cm in Karnataka state], K= kg/cm3, L= 4.5m,], Temperature Warping stress = 17.3 kg/ cm2 y Total of temperature warping stress and the highest axle load stress = 17.3 + 24.1 = 41.4 kg/ cm2 < 45 kg/ cm2
Elements of a Typical Rigid Pavement y A typical rigid pavement has three elements :
i) Subgrade; ii) Sub‐base; iii) Concrete slab y Subgrade is the in situ soil over which the pavement structure is ed. y Stiffness of the subgrade is measured by modulus of subgrade reaction (K). y K is determined with the assumption that the slab is resting on dense fluid and thus the reactive pressure of soil on pavement is linearly proportional to the deflection of the slab. y Value of K is widely dependant upon the soil type, soil density, and moisture content. y K is determined by plate bearing test.
Elements of a Typical Rigid Pavement
(contd)
y Sub‐base is the layer of selected granular materials placed
on the subgrade soil and immediately below the concrete pavement y It is provided for the following purposes yTo provide an uniform and reasonable firm pavement . yTo prevent mud pumping. yTo provide levelling course on undulated and distorted subgrade. yTo act as capillary cut off. y It is not a part of the rigid pavement structure as it is not provided to impart strength to the pavement structure.
Elements of a Typical Rigid Pavement
(contd)
y Construction of sub‐base is generally done by yGranular material like natural gravel, crushed slag,
crushed concrete, brick metal, laterite, soil aggregate etc. yGranular construction like WBM or WMM yStabilized soil ySemi rigid material like Lime clay Puzzolana Concrete, Lime Flyash Concrete, Dry Lean Concrete . y Concrete Slab is designed on the basis of flexural strength of concrete. y Due to repeated application of flexural stresses by the traffic loads, progressive fatigue damage takes place in the cement concrete slab in the form of gradual development of micro‐cracks.
Elements of a Typical Rigid Pavement
(contd)
y The ration between flexural stress due to the load and the
flexural strength of concrete is termed as the Stress Ratio (SR). If SR < 0.45 the concrete is expected to sustain infinite number of repetition. y Various properties of concrete as recommended for use as rigid pavement are y Flexural strength: 45 kg/cm2; y Modulus of Elasticity: 3 x 105 kg/cm2; y Poisson's ratio: 0.15; y Coefficient of thermal expansion: 10 x 10‐6 /⁰C.
Types of Rigid Pavement y There are four types of Rigid Pavement y ted and unreinforced concrete pavement y ted and reinforced concrete pavement y Continuously reinforced concrete pavement y Prestressed Concrete Pavement y Most of the rigid pavements in India are ted and
unreinforced concrete pavement. The necessary IRC design guidelines are y IRC: 58– 2002 (Guideline for the Design of Plain ted Rigid Pavements for Highways) y IRC: 15‐ 2002 (Standard Specifications and Code of Practice for Construction of Concrete Roads) y IRC: SP: 62‐ 2007
Stresses in Rigid Pavement y Stresses in concrete pavement are produced due to
following reasons y Applied wheel load y Changes in temperature y Changes in moisture content y Volumetric changes in soil subgrade y In design of rigid pavement stresses due to applied wheel load and changes in temperature are considered. y Since the nature of the stresses due to changes in moisture content is reverse that of stresses due to changes in temperature, it is not considered in thickness design. y Stresses due to volumetric changes of subgrade soil is taken care by properly selected sub‐base course.
Westergaard Analysis y H. M. Westergaard is considered to be pioneer person in
rigid pavement design. y The basic assumptions in Westergaard (1925) analysis for computation of stresses are i. Concrete slab acts as a homogenous, isotropic, and elastic solid in equilibrium. ii. The reaction of subgrade are vertical only and they are proportional to the deflection of the slab. This reaction of subgrade per unit area at any given point is equal to a constant K multiplied by the deflection at that point. iii. The thickness of the slab is uniform. iv. The load at the interior and at the corner of the slab is distributed uniformly over a circular area of .
Westergaard Analysis
(contd)
v. For corner loading the circumference of the area of
is tangential to the edge of the slab. vi. For the load at the edge of the slab is uniformly distributed over a semi‐circular area . The diameter of the semi‐circle is with the edge of the slab. y Critical stress locations y Interior: This is the position within the slab which is at any place remote from all the edges. y Edge: This is the position of the slab which is situated in the edge, remote from the corners. y Corner: This is the position which is situated at the bisector of the corner angle.
Westergaard Analysis
(contd)
Interior Edge
Corner
Stresses due to Wheel Load y Under the wheel load the interior and the edge of the slab
y y
y y y
behaves like a simple ed beam having tension at the bottom. Under the action of wheel load corner may behave as a cantilever specially when the slab is casted by . The maximum tensile stress may be found at corner as this location is considered as discontinuous from all the directions. As the edge is discontinuous in one direction this location may encounter lesser stress than the corner. Loads applied at the longitudinal edge can produce more stress than that at the transverse edge. Least stress is occurred at the interior as this position of the slab is continuous in all directions.
Stresses due to Wheel Load
(contd)
y Computation of stress at edge location y The original equations of Westergaard has been
modified by several researchers. y As per IRC the stresses due to wheel load may be determined by the software IITRIGID developed at IIT Kharagpur. y The stresses at edge may also be computed by the following equation as modified by Teller & Sutherland
P h
l b
σ le = (0.529 2 )(1+ 0.54μ)[4 log10 ( ) + log10 (b) − 0.4048]
Stresses due to Wheel Load
(contd)
y σle = Wheel Load Stress at Edge Region (kg/cm2) y P = Design Wheel Load (kg) or ½ of Single Axle Load (kg) or y y y y y y y
¼ of Tandem Axle Load (kg) h = Pavement thickness (cm) μ = Poisson's Ratio E = Modulus of Elasticity of Concrete (kg/cm2) k = Modulus of Subgrade Reaction (kg/cm3) l = Radius of Relative Stiffness (cm) b = Equivalent Radius of Resisting Section (cm) a = Radius of Load Area (cm)
Stresses due to Wheel Load
(contd)
y Relative stiffness of slab to sub‐grade y A certain degree of resistance to slab deflection is
offered by the sub‐grade. y The sub‐grade deformation is same as the slab deflection. Hence the slab deflection is direct measurement of the magnitude of the sub‐grade pressure. y The resistance to deformation depends on the stiffness of the ing medium as well as on the flexural stiffness the slab. y This pressure deformation characteristics of rigid pavement lead Westergaard to define the term radius of relative stiffness (l). l in cm is given by Eh 3 l=4 12 (1 − μ 2 ) k
y Equivalent Radius of Resisting Section y The wheel load concentrates on a small area of the
pavement y The area of the pavement that is effective in resisting the bending moment due to that load may be more than tyre imprint area. y The maximum bending moment occurs under the loaded area and acts radial in all directions. y The area of the pavement that is effective in resisting the bending moment due to a wheel load is known as Equivalent Radius of Resisting Section or also as Radius of Equivalent Distribution of Pressure. b = a =
a ≥ 1 . 724 h
for 1 .6 a
2
+ h
2
− 0 . 675 h
for
a < 1 . 724 h
Stresses due to Wheel Load
(contd)
y Computation of Stress at Corner Location y Wheel load stress at corner region is obtained as per
Westergaard’s analysis modified by Kelley
3P σ lc = 2 h
⎡ ⎛ a 2 ⎞1 .2 ⎤ ⎟ ⎥ ⎢1 − ⎜ ⎜ ⎢ ⎝ l ⎟⎠ ⎥ ⎣ ⎦
y σlc = Wheel Load Stress at Corner Region (kg/cm2)
Stresses due to Temperature Variation y Stresses are induced in the slab due to variation of
temperature y The temperature variation may be of two types y daily variation resulting in a temperature gradient across the thickness of the slab, and y seasonal variation resulting in uniform change in the slab temperature. y The former results in warping stresses and the later in frictional stresses.
Stresses due to Temperature Variation
(contd)
y Temperature Warping Stresses y Cement concrete pavement undergoes a daily cyclic
y y
y
y
change of temperature as thermal conductivity of concrete is low. The top surface of the pavement becomes hotter than bottom during day time and cooler during night. In the daytime thus the top surface of the pavement expands more than that in the bottom. This results the slab to warp upwards (top convex). The restraint offered to this warping tendency by self‐ weight and the dowel bars of the pavement induces stresses in the pavement. This is known as warping stress.
Stresses due to Temperature Variation
(contd)
y Flexural tensile stress will be generated at the bottom y
y
y y
surface during day time. Conversely, in the night the slab warp downward (top concave). Flexural tensile stress will be generated at the top surface. As the restraint offered to warping at any section of the slab is a function of weight of the slab upto the section, the corner has very little of such restraint for slabs without dowel bars and is free to warp. Thus warping stress is negligible. The interior can offer maximum restraint to warp and has maximum warping stress. The equations for warping stresses are available due to Westergaard.
Stresses due to Temperature Variation
(contd)
y The critical combination of stress indicates that most critical
location is the edge. y The equations for the warping stress at the edge as recommended by IRC is obtained as per Westergaard’s analysis using Bradbury’s coefficient.
σ twe =
Eαt C 2
y σtwe = Temperature Warping Stress at Edge Region (kg/cm2) y E = Modulus of Elasticity of Concrete (kg/cm2) y α = Coefficient of thermal expansion of concrete y t = temperature difference between top and bottom of slab y C = Bradbury’s coefficient depends on L/l or W/l of slab
Stresses due to Temperature Variation
(contd)
y Temperature Friction Stresses y Uniform seasonal temperature variation cause the slab
expands and contracts in the longitudinal direction. y This expansion and contraction of the slab is prevented by the friction between the slab and the subgrade. Stresses are thus set up in the slab. L/2 σcAc
σ c × h × B × 100 = B × ∴σ
c
=
L h × ×W ×f 2 100
WLf 2 × 10000
Stresses due to Temperature Variation
(contd)
B = Slab width (m) h = slab thickness (m) L = Length of the slab (m) σtfe = Temperature friction stress in concrete (kg/cm2) W = Unit weight of concrete in (kg/cm2) f = Coefficient of friction between concrete and subgrade y The temperature friction stress is taken care in rigid pavement by providing ts in plain ted pavement or by reinforcement in reinforced concrete pavement
Critical Combination of Stresses y Combination of flexural stresses due to wheel load and that
y y
y
y
to temperature warping provides the critical stress for design of rigid pavement. Maximum combined stress at the three critical locations will occur when these two stresses are additive. Warping stresses at three locations decrease in the order of interior, edge and corner whereas the wheel load stresses decrease in the order of corner, edge and interior. Therefore, critical stress condition is reached at edge location where neither wheel load stress nor the warping stress is minimum. Since at night due to warping the corner may behave as cantilever it is recommended to check the wheel load stress at corner.
ts in Rigid Pavement y The rigid pavement slab is deliberately divided into blocks of
appropriate sizes in order to take care the effects of temperature friction stress or stresses due to moisture variation. y These deliberate planes of weaknesses in the slab are known as ts. A good t should have the following functional requirements: y Must be waterproof [proper sealing to be provided] y Riding quality should not be deteriorated y Should not make any structural weakness [for example staggered ts should be avoided]
ts in Rigid Pavement
(contd)
y Classification of the ts according to location in the
pavement y Longitudinal ts y Transverse ts Transverse ts Longitudinal ts
ts in Rigid Pavement y Classification of ts according to Forms y Dummy t y Butt t y Tongue and Groove t y ts with Clear Gap y Classification of ts according to Function y Expansion t y Contraction t y Longitudinal t y Construction t
(contd)
ts in Rigid Pavement
(contd)
Expansion t Dowel Bar [Fully Bonded part]
Sealer
Expansion Cap with Cotton Waste at the Back
t+12
t
Filler
75mm
Dowel Bar [Bitumen Painted part]
t+6
Schematic Drawing of Expansion t with Dowel Bar
ts in Rigid Pavement
(contd)
y The pavement slab tends to expand when the temperature y y y
y y
rises above that at which the pavement was laid. Expansion of the slab is prevented by friction between the slab and the subgrade. Compressive Stress is thus set up and this may try to buckle or blow up the slab . In order to prevent this stress, Expansion ts in the transverse direction of the pavement are provided to allow space for expansion of the slab. The t is formed by maintaining a gap of about 20 to 25 mm between two slabs. The gap is filled up by a non‐extruding compressive filler material.
ts in Rigid Pavement
(contd)
y A sealing compound is provided on the top of the filler
material to prevent entry of water and dust. y To ensure transfer of load between the two slabs on each side of the t dowel bars are provided. y Dowel bars are usually mild steel round bars of short length. y Half length is bonded into concrete on one side of the t and the other half is painted by bitumen in order to prevent bonding with concrete. y A metal cap with cotton waste at the back is provided at the painted half end of the dowel bar. This ensures free movement of the slab during expansion.
ts in Rigid Pavement
(contd)
y Dowel bars not only permits the expansion of the slabs but
also holds the slab ends on each side of the t as nearly as possible. y Deflection of one slab under load is resisted by the other slab which, in turn is caused to deflect and thus carry a portion of the load imposed upon the first slab. y The spacing of expansion t may vary from twenty meters to a few hundred meters.
ts in Rigid Pavement
(contd)
Contraction t Sealer
Contraction t with Butt t Dowel Bar Sealer
Contraction t with Dummy t
ts in Rigid Pavement
(contd)
y Stresses are also generated in the concrete pavement slab
due to contraction of concrete when the temperature is reduced with respect to that during laying. y Contraction ts are thus provided to reduce tensile stress due to contraction or shrinkage of concrete. y There are two types of contraction t: y Dummy t: In this type no t is made in reality. Only a small groove is cut on top of the slab for a depth of ¼ to 1/3 of the thickness of the slab. If stress becomes more than that the slab can withstand, a crack may develop at the location of the grove as this is the weakest plane in the slab. Simple dummy t may not contain any dowel bar. If dowel bar is not provided the load transfer is ensured by particle interlocking.
ts in Rigid Pavement
(contd)
y Butt t: In case of a butt t two slabs abut each
other. Therefore, a clear plane of separation will exist in this t. Dowel bars may or may not be provided. y In case of dummy or butt t good sealing material is provided at the top of the t in order to prevent entry of water and dust inside the t. y Spacing of ts varies with thickness of the slab and also with the existence of reinforcement. For slab of thickness upto 250mm ts maximum spacing may be 4.5m whereas upto 350mm thick pavement, maximum spacing will be 5.0m.
ts in Rigid Pavement
(contd)
Longitudinal t y Longitudinal ts are necessary in the concrete slab for the pavement having more than 4.5m wide. y Longitudinal t prevents longitudinal cracking. y Mild steel bars known as Tie bars are provided across the longitudinal t to hold the t tightly closer and to keep both the slabs at the same level. y Tie bars are not provided to act as load transfer device. y Both the ends of the tie bars are fully bonded in the concrete. y Longitudinal ts may be butt type or keyed [Tongue and Groove] type.
ts in Rigid Pavement
(contd)
Construction t y Construction ts are provided in the transverse
direction whenever the placing of concrete is suspended for more than 30 minutes. y As far as possible construction t should coincide with either expansion t or contraction t. y If a construction t is provided at the location of any contraction t it should be of butt type with dowel bar. y If a separate construction t is needed, it should be provided within the middle third of two contraction ts.
ts in Rigid Pavement
(contd)
Arrangement of ts y Staggered t y When transverse ts are staggered with respect to the longitudinal t, sympathetic cracks may occur. y These cracks often occur in the line with the t in the other side of the transverse crack.
Sympathetic cracks
ts in Rigid Pavement
(contd)
y Skew and Acute Angle t y Use of skew t increases the risk of cracking at the
acute angle corners. y At the acute angles the stresses become very excessive. y At the time of warping the acute angles become completely uned and cause more stresses than that would occur in right angle corner.
Design of Rigid Pavements y Step 1: Stipulate design values for various parameters y Step 2 : Decide type and spacing between ts. y Step 3 : Select a trial design thickness of the pavement
slab. y Step 4 : Compute the repetitions of axle loads of different magnitudes during design period y Step 5 : Calculate stresses due to single and tandem axle loads and determine cumulative fatigue damage (CFD). y Step 6 : If CFD is more than 1.0, select a higher thickness and repeat the procedure from step 4.
Design of Rigid Pavements
(contd)
y Step 7 : Compute the temperature stress at edge. If sum of
the temperature stress and the flexural stress due to highest wheel load is greater than modulus of rapture select a higher thickness and repeat the procedure from step 4. y Step 8 : Design the pavement thickness on the basis of corner stress, if no dowel bar is provided and no load transfer is possible due to lack of aggregate interlocking.
Design of Rigid Pavements
(contd)
Design Example y Design a cement concrete pavement for a two lane two way National Highway in Karnataka State. The initial total two way traffic is 3000 commercial vehicles per day. The other deign parameters are: y Flexural strength of cement concrete: 45 kg/cm2. y Effective modulus of subgrade reaction of the DLC sub‐ base: 8 kg/cm3. y Spacing of contraction ts: 4.5 m. y Width of slab: 3.5m y Tyre pressure: 8 kg/cm2. y Rate of traffic increase: 7.5%. Axle load spectrum obtained from axle load survey is given below
Design of Rigid Pavements Single Axle Load Axle load (tonnes) 19‐21 17‐19 15‐17 13‐15 11‐13 09‐11 Less than 9 Total
% of axle loads 0.6 1.5 4.8 10.8 22.0 23.3 30.0 93.0
(contd)
Tandem Axle Load Axle load (tonnes) 34‐38 30‐34 26‐30 22‐26 18‐22 14‐18 Less than 14 Total
Design of Rigid Pavements
% of axle loads 0.3 0.3 0.6 1.8 1.5 0.5 2.0 7.0
(contd)
Design Traffic y Present traffic = 3000 cvpd; Design life (assumed) = 20 years; y Cumulative repetition in 20 years = 47, 418, 626 cv y N = 365 A [(1+r)n – 1]/r where A is the initial number of axles in
the year when road is operational ; r is the rate of annual growth of traffic; n is the design life. y Design traffic = 25% of repetition of commercial vehicles
= 11,854, 657 cv y Total repetitions of single axles and tandem axles are
Design of Rigid Pavements Single Axle Load Load (tonnes) Expected repitions 20 71127 18 177820 16 569023 14 1280303 12 2608024 10 2762135 Less than 10 3556397
(contd)
Tandem Axle Load Load (tonnes) Expected repitions 34‐38 35564 30‐34 35564 26‐30 71128 22‐26 213384 18‐22 177820 14‐18 59273 Less than 14 237093
Design of Rigid Pavements
(contd)
Thickness Design y Trial thickness = 32 cm; Load safety factor =1.2 y To take care unpredicted heavy truck loads the magnitude of axle loads should be multiplied by load safety factor (LSF). y National highways and other roads where there will be uninterrupted traffic flow and high volumes of truck traffic: 1.2 y Lesser important roads with lesser incidence of truck traffic : 1.1 y Residential and other local streets : 1.0
Design of Rigid Pavements Axle load AL x 1.2 Stress tonnes (kg/cm2) (AL)
Stress Ratio
(contd)
Expected Fatigue life Fatigue life Repetition (N) consumed (n) (n/N)
Single axle 20
24.0
25.19
0.56
71127
94.1 x 103
0.76
18
21.6
22.98
0.51
177820
4.85 x 103
0.37
16
19.2
20.73
0.46
569023
14.33 x 104
0.04
14
16.8
18.45
0.41
128030
Infinity
0.00
36
43.2
20.07
0.45
35560
62.8x 106
0.0006
32
38.4
18.4
0.40
35560
Infinity
0.00
Cumulative fatigue life consumed
1.1705 > 1
Tandem axle
Design of Rigid Pavements
(contd)
y Relation between fatigue life (N) and Stress ratio (SR) y N = unlimited for SR, 0.45 y N= [4.2577/ (SR ‐0.4325)]1.324 0.45≤SR≥0.55 y Log10 N = (0.9718 – SR) / 0.0.828 for SR > 0.55 y Since CFD fro thickness of 32 cm>1, increase thickness y Take next trial thickness = 33cm y Repeat the steps from 4 y Cumulative fatigue life consumed for thickness of 33 cm =
0.47 y Highest stress load stress for thickness of 33 cm = 24.10 kg/ cm2
Design of Rigid Pavements
(contd)
Temperature Warping stress y For E= 3 x 105 kg/cm2, α
=
10 x 10‐6 /⁰C, t= 21⁰ [slab thickness 33 cm in Karnataka state], K= kg/cm3, L= 4.5m,], Temperature Warping stress = 17.3 kg/ cm2 y Total of temperature warping stress and the highest axle load stress = 17.3 + 24.1 = 41.4 kg/ cm2 < 45 kg/ cm2
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