Study On The Behavior Of Box Girder Bridge 4w5e2t

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Study on the Behavior of Box Girder Bridge Thesis · January 2010 DOI: 10.13140/RG.2.1.2747.6641

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STUDY AND BEHAVIOUR OF BOX GIRDER BRIDGE A Project Report Submitted to

Nagarjuna University In Partial fulfillment of the Requirements for the Award of the Degree of

BACHELOR OF TECHNOLOGY with specialization in

CIVIL ENGINEERING Submitted By: J.S.KALYANA RAMA (Y06CE050) V.R.RAGHAVA SUDHIR V.SAMPATH KUMAR (Y06CE039) (Y06CE044) V.VICKRANTH (Y06CE060) Under the Guidance of

V.RAMESH, Asst. Professor & Special Thanks to

N.R.K.MURTHY, HEAD OF THE DEPARTMENT

DEPARTMENT OF CIVIL ENGINEERING V.R.SIDDHARTHA ENGINEERING COLLEGE KANURU, VIJAYAWADA-520007 -I-

APRIL -2010 STUDY AND BEHAVIOUR OF BOX GIRDER BRIDGE

DEPARTMENT OF CIVIL ENGINEERING V.R.SIDDHARTHA ENGINEERING COLLEGE KANURU, VIJAYAWADA-520007

CERTIFICATE This is to certify that the project report entitled “STUDY AND BEHAVIOUR OF BOX GIRDER BRIDGE” is the bona fide work done by J.S.KALYANA RAMA

Y06CE050

V.R.RAGHAVA SUDHIR

Y06CE039

V.SAMPATH KUMAR

Y06CE044

V.VICKRANTH

Y06CE060

Under guidance and supervision of V.RAMESH, Asst.Professor, submitted in partial fulfillment of the requirements for the award of the Degree of Bachelor of Technology, in Civil Engineering by the Acharya Nagarjuna University

(V.RAMESH)

(Dr. N.R.K. MURTHY)

GUIDE:

HEAD OF THE DEPARTMENT

Date

Date:

- II -

ACKNOWLEDGEMENTS We take this opportunity first to express our deep sense of gratitude and gratefulness to our project guide, V.RAMESH, Asst.Professor, Department of Civil Engineering for his expert guidance, constant encouragement and during all phases of our work.

We would also like to thank N.R.K.MURTHY, Professor, Department of Civil Engineering, D Y NARASIMHA RAO, Senior Engineer, Bridges and B SRIKANTH, Design Engineer, S.C.R Secunderabad for their valuable suggestions and encouragement in the successful completion of this Report.

We would also like to thank Dr. N.R.K. MURTHY, Professor and Head, Department of Civil Engineering for his cooperation in providing facilities for the successful completion of this Report.

We would also like to thank Dr.K.MOHAN RAO, Principal,V.R.SIDDHARTHA ENGINEERING COLLEGE for providing the state of the art facilities in the college. We also take this opportunity to thank everyone who helped either directly or indirectly in bringing out the project report to the final form.

PROJECT ASSOCIATES: J.S.KALYANA RAMA

(Y06CE050)

V.R.RAGHAVA SUDHIR

(Y06CE039)

V.SAMPATH KUMAR

(Y06CE044)

V.VICKRANTH

- III -

(Y06CE060)

ABSTRACT “When tension flanges of longitudinal girders are connected together, the resulting structure is called a box girder bridge”. The behavior of box girder section for a general case of an eccentric load has been studied and presented its studies in chapter 2. An encoming review of literature has been made regarding construction and a summary of general specifications with reference to IRC:18 have been discussed in chapter 3. Box girders can be universally applied from the point of view of load carrying, to their indifference as to whether the bending moments are positive or negative and to their torsional stiffness; from the point of view of economy. An ongoing work has been taken as a case study for the present work. Analysis principles for torsion and distortion effects are applied to the section selected, and found satisfactory. Correspondingly, the problem has been analyzed and designed for flexure and shear by giving due considerations for torsional and distortional effects as a precautionary measure.

- IV -

TABLE OF CONTENTS CERTIFICATE ACKNOWLEDGEMENTS ABSTRACT CONTENTS 1.

PAGE.NO

INTRODUCTION TO BOX GIRDER BRIDGES

2.

3.

1

Introduction

2

Historical development

3

Evolution

4

Advantages

5

Disadvantages

5

Specifications

6

BEHAVIOUR OF BOX GIRDER

7

Flexure

9

Torsion

10

Distortion

16

Warping of Cross section

18

Shear lag

19

Diaphragms

22

CONSTRUCTION AND GENERAL ARRANGEMENT

24

General Arrangement

25

Cast-in-situ Construction

26

Construction of Multi-cell

-V-

Box Girder 4.

29

ANALYSIS OF BOX GIRDER BRIDGE

32

(CASE STUDY) Torsional Analysis

33

Distortional Analysis Beam On Elastic Foundation 5.

35

DESIGN OF BOX GIRDER

39

Description

41

Span arrangement

41

Prestress

41

Design Data

42

Sectional properties

44

Bending Moment And Shear Force Calculations

45

Tabulation of Bending Moment And Shear Forces

49

Prestressing forces and other losses Calculations

51

Prestress in service condition

54

Shear force calculations

54

Design for shear

57

Design of Elastomeric Bearing

58

Deflection Calculations

62

Elongation Statement

63

- VI -

Check for ultimate moment of Resistance

64

Design of Deck Slab

65

Design of cantilever deck slab beyond end diaphragms

68

Design of cantilever deck slab below Footpath

69

Provision of untensioned mild steel Reinforcement

71

Design of Intermediate Diaphragms

73

End diaphragms

76

Design of End block

77

6.

CASE STUDY PICTURES

79

7.

CONCLUSIONS

86

CONCLUSION AND FUTURE WORK 8.

87

REFERENCES

88

- VII -

CHAPTER 1

INTRODUCTION

-1-

Introduction The continuing expansion of highway network throughout the world is largely the result of great increase in traffic, population and extensive growth of metropolitan urban areas. This expansion has lead to many changes in the use and development of various kinds of bridges. The bridge type is related to providing maximum efficiency of use of material and construction technique, for particular span, and applications. As Span increases, dead load is an important increasing factor. To reduce the dead load, unnecessary material, which is not utilized to its full capacity, is removed out of section, this Results in the shape of box girder or cellular structures, depending upon whether the shear deformations can be neglected or not. Span range is more for box bridge girder as compare to T-beam Girder Bridge resulting in comparatively lesser number of piers for the same valley width and hence results in economy. A box girder is formed when two web plates are ed by a common flange at both the top and the bottom. The closed cell which is formed has a much greater torsional stiffness and strength than an open section and it is this feature which is the usual reason for choosing a box girder configuration. Box girders are rarely used in buildings (box columns are sometimes used but these are axially loaded rather than in loaded in bending). They may be used in special circumstances, such as when loads are carried eccentrically to the beam axis “When tension flanges of longitudinal girders are connected together, the resulting structure is called a box girder bridge”. Box girders can be universally applied from the point of view of load carrying, to their indifference as to whether the bending moments are positive or negative and to their torsional stiffness; from the point of view of economy.

-2-

1.1

Historical development and description: The first box girder cross section possessed deck slabs that cantilevered out only slightly from the box portion shown in figs a to e. With the prestressed concrete the length of cantilever could be increased. The high form work costs caused a reduction in the number of cells fig (f, g, h). In order to reduce the construction loads to minimum possible extent or to require only one longitudinal girder in working states even with multiple traffic lanes. It was only with the development of high strength prestressing steel that it became possible to span longer distances. The first prestressed concrete bridges, most of I-cross sections were built towards the end of the 1920’s.The great breakthrough was achieved only after 1945. “THE SCLAYN” bridge over the river Maas, which was built by Magnel in 1948, was the first continuous prestressed concrete box-girder bridge with 2 spans of 62.70m. In following years the ratio of wages to material costs climbed sharply. This thereby shifted the emphasis of development of construction method. The box girder cross-section evolved structurally from the hollow cell-deck bridge or T-beam Bridge. The widening of the compression zone that began as a structural requirement at the central piers was in the extended throughout the entire length of bridge because of advantages transverse load-carrying characteristics.

Fig:1-1

-3-

1.2

Evolution: The spanning of bridges started with simple slabs. As the spans increased, the design depth of slab is also increased. It is known that material near centre of gravity contributes very little for flexure and hence can be removed. This leads to beam and slab systems. The reinforcement in bottom bulb of beam provided capacity for tensile forces and top slab concrete, the capacity to resist the compression. They formed a couple to resist flexure. As the width of slab is increased more number of longitudinal girders are required resulting in reduction of stiffness of beams in transverse direction and relatively high transverse curvature. The webs of beams get opened out spreading radially from top slab. Under high transverse bending these will no longer be in their original position. To keep it in their original position the bulbs at bottom should be tied together which in-turn leads to evolution of box girder. Long spans with wider decks and eccentric loading on cross-section will suffer in curvature in longitudinal and transverse direction causing heavy distortion of cross-section. Hence the bridges should have high torsional rigidity in order to resist the distortion of cross-section deck to a minimum. Accordingly box girders are more suitable for larger spans and wider decks, box girders are to be suitable cross-section. They are elegant and slender. Economy and aesthetics further lead to evolution of cantilevers in top flanges and inclined webs in external cells of box girder. The dimension of cell could be controlled by prestressing. As the span and width increases the beams and bottom slabs are to be tied to keep the geometry which in turn leads to evolution box girder. Any eccentric load will cause high torsional stresses which will be counter acted by the box section. The analysis of such sections are more complicated due combination of flexure, shear, torsion, distortion. But it is more efficient cross-section. It is used for larger spans with wide crosssection. It can be used for spans up to 150m depending upon the construction methods. Cantilever method of construction is preferred most. -4-

1.3

Advantages Associated with Box Girders: 

In recent years, single or multicell reinforced concrete box Girder

Bridge have been proposed and widely used as economic aesthetic solution for the over crossings, under crossings, grade separation structures and viaducts found in modern highway system. 

The very large Torsional rigidity of the box girder‘s closed cellular

section provides structures beneath is more aesthetically pleasing than open-web type system. 

In case of long span bridges, large width of deck is available to

accommodate prestressing cables at bottom flange level. 

Interiors of box girder bridges can be used to accommodate service

such as gas pipes, water mains etc. 

For large spans, bottom flange could be used as another deck

accommodates traffic also. 

The maintenance of box girder is easier in interior space is directly

accessible without use of scaffolding. 

Alternatively space is hermetically sealed and enclosed air may be

dried to provide a non-corrosive atmosphere. 

It has high structural efficiency which minimizes the prestessing

force required to resist a given bending moment, and its great Torsional strength with the capacity this gives to re-centre eccentric live loads, minimizing the prestress required to carry them. 1.4

Disadvantages: One of the main disadvantages of box decks is that they are difficult to cast in-situ due to the inaccessibility of the bottom slab and the need to extract the internal shutter. Either the box has to be designed so that the entire cross section may be cast in one continuous pour, or the cross section has to be cast in stages.

-5-

1.5

Specifications: It can cover a range of spans from 25 m up to the largest nonsuspended concrete decks built; of the order of 300 m. Single box girders may also carry decks up to 30 m wide. For the longer span beams, beyond about 50 m, they are practically the only feasible deck section. Below 30m precast beams or voided slab decks are more suitable while above 50ma single cell box arrangement is usually more economic. Single cell box-girder cast-in-situ are used for spans form 40m to 270m.The box arrangement is done in order to give aesthetic appearance where the web of box will act as a slender appearance when combined with a slim parapet profile. Single box arrangements are efficient for both the longitudinal and transverse designs, and they produce an economic solution for mot medium and long span structures. This type of deck is constructed span-by-span, using full-height scaffolding or trusses, or as balanced cantilever using form travelers. This could be particularly important for medium length bridges with spans between 40m and 55m. Such spans are too long for twin rib type decks, and too short for cast-insitu balanced cantilever construction of box girders, while a total length of box section deck of less than about 1,000 m does not justify setting up a precast segmental facility. Haunches: The uprights have to carry the same bending moment as the haunch, but with the benefit of a compression force due to the weight of the roof. Thus they may be slightly thinner than the haunches. Haunches are always economical. They provide the twin benefits of attracting moment away from mid-span and then providing a greater lever arm to resist this moment economically. Even very short haunches are valuable in reducing the hogging reinforcement.

-6-

CHAPTER 2

BEHAVIOUR OF BOX GIRDER BRIDGES

-7-

Fig:2-1

-8-

A general loading on a box girder, such as shown in fig 2-1 for single cell box, has components which bend, twist, and deform the cross section. Thin walled closed section girders are so stiff and strong in torsion that the designer might assume, after computations based on the elemental torsional theory, that the torsional component of loading in fig 2-1(c). has negligible influence on box girder response. If the torsional component of the loading is applied as shears on the plate elements that are in proportion to St. Venant torsion shear flows, fig 2-1 (e), the section is twisted without deformation of the cross section. The resulting longitudinal warping stresses are small, and no transverse flexural distortion stresses are induced. However, if the torsional loading is applied as shown in fig 2-1 (c), there are also forces acting on the plate elements fig 2-1 (f), which tend to deform the cross section. As indicated in fig 2-2 the movements of the plate elements of the cross section cause distortion stresses in the transverse direction and warping stresses in the longitudinal direction.

2.1

FLEXURE:

Fig:2-2

-9-

A vehicle load, placed on the upper flange of box girder can occupy any position, transverse as well as longitudinal. This load is transferred transversely by flexure of deck to the webs of box girder. For understanding the various stresses generated, initially consider that the webs of box girder are not allowed to deflect. The structure resembles a portal frame. The flexure of deck would induce transverse bending stresses in the webs, and consequently in the bottom flanges of the girder. Any vehicle load can thus be replaced by the forces at the intersections of deck and web as shown in fig 2-3. Now the s under the web are allowed to yield. This results in deflection of web and consequently redistribution of forces among web and flanges. Distortion of cross section occurs as a result of the fact that m1 and m2 are not equal resulting in sway of frame, due to eccentrically placed load. The section of box tries to resist this distortion, resulting in the transverse stresses. These stresses are called distortional transverse stresses. The distortion of cross section is not uniform along the span, either due to non uniform loading or due to presence of diaphragms or due to both. However the compatibility of displacements must be satisfied along the longitudinal edges of plate forming the box, which implies that these plates must bend individually in their own plane, thus inducing longitudinal warping displacements. Any restraint to these displacements causes stresses. These stresses are called longitudinal warping stresses and are in addition to longitudinal bending stresses. 2.2

TORSION: The main reason for box section being more efficient is that for eccentrically placed live loads on the deck slabs, the distribution of longitudinal flexural stresses across the section remains more or less identical to that produced by symmetrical transverse loading. In other words, the high torsional strength of the box section makes it very suitable for long span bridges. Investigations have shown that the box girders subjected to torsion undergo deformation or distortion of the section, giving rise to transverse as well as longitudinal stresses. These stresses cannot be predicted by the - 10 -

conventional theories of bending and torsion. One line of approach to the analysis of box girders subjected to torsion is based on the study of THIN WALLED BEAM THEORY. The major assumptions are: a)

Plate action by bending in the longitudinal direction for all plates

forming the cross section, namely webs, slabs is negligible. b) or

Longitudinal stresses vary linearly between the longitudinal ts, the

meeting

points

of

the

plates

section.

Fig: 2-3

- 11 -

forming

the

cross

The kerb, footpath, parapet, and wearing coat generally form the superimposed dead loads acting on the effective section which is responsible for carrying all loads safely and transmitting them to the substructure. Because of symmetry, the self weight of the effective section and the superimposed dead loads do not create any torsional effects. However the non-symmetrical live loads which consist of concentrated wheel loads from vehicles on any part of carriage way and the equivalent uniformly distributed load on one of the footpaths can subject the box girder to torsion.

Fig: 2-4

If the deck slab is considered to be resting on non deflecting s at A and B in fig 2-3(b), the vertical reactions and the moments created by the live loads at these points can be computed. The effects of moments at this stage are treated as separately since they cause only local transverse flexure fig 2-5 and can be evaluated by considering a slice of unit length from the box girder. The effect of superimposed and dead loads should also be taken into in such evaluations.

- 12 -

Fig: 2-5

Coming to the vertical reactions, let equal and opposite vertical forces be applied at A and B. In studying the longitudinal and transverse effects, it should be noted that finally all longitudinal effects have to be superimposed separately on the one hand, and transverse effects on the other. The vertical forces are denoted by P1 and P2 in fig 2-6. As shown, (a) = (b) +(c). Since (c) = (d) + (e), it is evident that (a) = (b) + (d) + (e). Now (b) and (d) are symmetrical loads and, as in the case of superimposed dead loads and self weight, do not create any torsional effects. Let the sum of all these symmetrical loads be denoted by Q, Q, acting at A and B fig. The loads Q, Q cause simple longitudinal flexure only and the structural effects caused are illustrated in fig 2-4(a). The loads P, P cause torsional effects in the box girder, and they are shown in fig b, c. The internal forces generated to counteract P, P are shown in fig 2-7.

- 13 -

Fig: 2-6

Fig: 2-7 In ‘rigid body rotation’ or ‘pure torsion’ effects, the section merely twists or rotates causing St.Venant shear stresses and associated warping stresses which can be evaluated by the elemental theory of torsion as applied to closed sections of thin walled . It may be emphasized that due to very high stiffness in ‘pure torsion’, the box girder will twist very little, and that the webs will remain almost vertical in their original unloaded position. Also the associated longitudinal stresses due to warping restraint when present are negligible as compared to those induced by the longitudinal flexure due to forces Q, Q. The theoretical behavior of a thin-walled box section subject to pure torsion is well known. For a single cell box, the torque is resisted by a shear flow which acts around the walls of the box. This shear flow (force/unit length) is constant around the box and is given by q = T/2A, where T is the torque and A is the area enclosed by the box. The shear flow produces shear stresses and strains in the walls and gives rise to a

Or, Where J is the torsion constant. However, pure torsion of a thin walled section will also produce a warping of the cross-section, Of course, for a simple uniform box section subject to pure torsion, warping is unrestrained and does not

- 14 -

give rise to any secondary stresses. But if, for example, a box is ed and torsionally restrained at both ends and then subjected to applied torque in the middle, warping is fully restrained in the middle by virtue of symmetry and torsional warping stresses are generated. Similar restraint occurs in continuous box sections which are torsionally restrained at intermediate s. This restraint of warping gives rise to longitudinal warping stresses and associated shear stresses in the same manner as bending effects in each wall of the box. The shear stresses effectively modify slightly the uniformity of the shear stress calculated by pure torsion theory, usually reducing the stress near corners and increasing it in mid. Because maximum combined effects usually occur at the corners, it is conservative to ignore the warping shear stresses and use the simple uniform distribution. The longitudinal effects are, on the other hand greatest at the corners. They need to be taken into when considering the occurrence of yield stresses in service and the stress range under fatigue loading. But since the longitudinal stresses do not actually participate in the carrying of the torsion, the occurrence of yield at the corners and the consequent relief of some or all of these warping stresses would not reduce the torsional resistance

Fig 2-8 Warping of rectangular box subjected to pure torsion. If torsional loading is applied, there are forces acting on the plate of elements, which tend to deform the cross section. The movements of the plate elements of the cross section cause distortion stresses in transverse direction and warping stresses in longitudinal direction.

- 15 -

2.3

DISTORTION:

Fig 2-9: Distortional effects

When torsion is applied directly around the perimeter of a box section, by forces exactly equal to the shear flow in each of the sides of the box, there is no tendency for the cross section to change its shape. Torsion can be applied in this manner if, at the position where the force couple is applied, a diaphragm or stiff frame is provided to ensure that the section remains square and that torque is in fact fed into the box walls as a shear flow around the perimeter. Provision of such diaphragms or frames is practical, and indeed necessary, at s and at positions where heavy point loads are introduced. But such restraint can only be provided at discrete positions. When the load is distributed along the beam, or when point loads can occur anywhere along the beam such as concentrated axle loads from vehicles, the distortional effects must be carried by other means. The distortional forces shown are tending to increase the length of one diagonal and shorten the other. This tendency is resisted in two ways,

- 16 -

by in-plane bending of each of the wall of the box and by out-of-plane bending, is illustrated in Figure.

Fig 2-10 Distortional displacements in box girder.

In general the distortional behavior depends on interaction between the two sorts of bending. The behavior has been demonstrated to be analogous to that of a beam on an elastic foundation (BEF), and this analogy is frequently used to evaluate the distortional effects. If the only resistance to transverse distortional bending is provided by out-of-plane bending of the flange plates there were no intermediate restraints to distortion, the distortional deflections in most situations would be significant and would affect the global behavior. For this reason it is usual to provide intermediate cross-frames or diaphragms; consideration of distortional displacements and stresses can then be limited to the lengths between cross-frames. The distortion of section is not same throughout the span. It may be completely nil or non-existent at points where diaphragms are provided, simply because distortion at such points is physically not possible. The warping stresses produced by distortion are different from those induced by the restraint to warping in pure torsion which is encountered in elementary theory of torsion. The compatibility of displacements must be satisfied along the longitudinal edges of the plate forming the box, which implies that these plates must bend individually in

- 17 -

their own plane, thus inducing longitudinal warping displacements. Any restraint to this displacement causes stresses. These stresses are called longitudinal warping stresses and are in addition to longitudinal bending stresses. A general loading on a box girder such as for a single cell box, has components, which bend twice and deform the cross section. Using the principles of super position, the effects of each section could be analyzed independently and results superimposed. Distortional stresses also occur under flexural component, due to poisson effect and the beam reductance of the flange in multi cellular box, the symmetrical component also gives rise to distortion stresses and it is significant percentage of total stresses. With increase in number of cells, the proportion of transverse distortional stresses also increase. How ever for a single cell box the procedure of considering only the distortional component of loading for evaluation of distortional stresses in adequate for practical purposes. The concrete boxes in general have sufficient distortional stiffness to limit the warping stresses to small fraction of the bending stresses, without internal diaphragms. But for steel boxes either internal diaphragms or stiffer transverse frames are necessary to prevent buckling of flanges as well as of webs and in most cases these will be sufficient to limit the deformation of the cross section. Sloping of the webs of box girder increase distortional stiffness and hence transverse load distribution is improved. If section is fully triangulated, the transverse distortional bending stresses are eliminated. This form could be particularly advantageous for multicell steel boxes. Therefore distortion of box girder depends on arrangement of load transversely, shape of the box girder, number of cells and their arrangement, type of bridge such as concrete or steel, distortional stiffness provided by internal diaphragms and transverse bracings provided to check buckling of webs and flanges.

2.4

WARPING OF CROSS SECTION: Warping is an out of plane on the points of cross section, arising due to torsional loading. Initially considering a box beam whose - 18 -

cross section cannot distort because of the existence of rigid transverse diaphragms all along the span. These diaphragms are assumed to restrict longitudinal displacements of cross sections except at midspan where, by symmetry the cross section remains plane. The longitudinal displacements are called torsional warping displacements and are associated with shear deformations in the planes of flanges and webs. In further stage assume that transverse diaphragms other than those at s are removed so that the cross section can distort. (Fig). It results in additional twisting of cross section under torsional loading. The additional vertical deflection of each web also increases the out of plane displacements of the cross sections. These additional warping displacements are called distortional warping displacements/ Thus concrete box beams with no intermediate diaphragms when subjected to torsional loading, undergo warping displacements composing of two components viz, torsional and distortional warping displacements. Both these give rise to longitudinal normal stresses i.e. warping stresses whenever warping is constrained. Distortion of cross section is the main source of warping stresses in concrete box girders, when distortion is mainly resisted by transverse bending strength of the walls and not by diaphragms. 2.5

SHEAR LAG: In a box girder a large shear flow is normally transmitted from vertical webs to horizontal flanges, causes in plane shear deformation of flange plates, the consequence of which is that the longitudinal displacements in central portion of flange plate lag behind those behind those near the web, where as the bending theory predicts equal displacements which thus produces out of plane warping of an initially planar cross section resulting in the “SHEAR LAG". Another form of warping which arises when a box beam is subjected to bending without torsion, as with symmetrical loading is known as “SHEAR LAG IN BENDING”. Shear lag can also arise in torsion when one end of box beam is restrained against warping and a torsional load is applied from the - 19 -

other end fig 2-11. The restraint against warping induces longitudinal stresses in the region of built-in-end and shear stresses in this area are redistributed as a result which is an effect of shear deformation sometimes called as shear lag. Shear distribution is not uniform across the flange being more at edges and less at the centre fig 2-13.

Fig:2-11 In a box beam with wide, thin flanges shear strains may be sufficient to cause the central longitudinal displacements to lag behind at the edges of the flange causing a redistribution of bending stresses shown in fig 2-12. This phenomenon is termed as “STRESS DIFFUSION”. The shear lag that causes increase of bending stresses near the web in a wide flange of girder is known as positive shear lag. Whereas the shear lag, that results in reduction of bending stresses near the web and increases away from flange is called negative shear lag fig 2-12. When a cantilever box girder is subjected to uniform load, positive as well as negative shear lag is produced. However it should be pointed out that positive shear lag is differed from negative shear lag in shear deformations at various points across the girder. At a distance away from the fixed end in a cantilever box girder say half of the span; the fixity of slab is gradually diminished, as is the intensity of shear. From the compatibility of deformation, the negative shear lag yields. Although positive shear lag may occur under both point as well as uniform loading, negative shear lag occur only under uniform load.

- 20 -

Fig:2-12 It may be concluded that the appearance of the negative shear lag in cantilever box girder is due to the boundary conditions and the type of loading applied. These are respectively external and internal causes producing negative shear lag effect. Negative shear lag is also dependent upon ratio of span to width of slab. The smaller the ratio, the more severe are the effects of positive and negative shear lag.

Fig:2-13

The more important consideration regarding shear lag is that it increases the deflections of box girder. The shear lag effect increases with the width of the box and so it is particularly important for modern bridge designs which often feature wide single cell box cross sections. The

- 21 -

shear lag effect becomes more pronounced with an increase in the ratio of box width to the span length, which typically occurs in the side spans of bridge girders. The no uniformity of the longitudinal stress distribution is particularly pronounced in the vicinity of large concentrated loads. Aside from its adverse effects on transverse stress distribution it also alters the longitudinal bending moment and shear force distributions in redundant structural systems. Finally, the effect of shear lag on shear stress distribution in the flange of the box, as compared to the prediction of bending theory is also appreciable. A typical situation in which large stress redistributions are caused by creep is the development of a negative bending moment over the when two adjacent spans are initially erected as separate simply ed beams and are subsequently made continuous over the . In the absence of creep, the bending moment over the due to own weight remains zero, and thus the negative bending moment which develops is entirely caused by creep.

Fig 2-14 Effect of shear lag on distribution of stresses at the of a box girder

2.6

DIAPHRAGMS: Advantage of closed section is realized only when distortion of cross section is restricted. Distortion could be checked by two ways: First by improving the bending stiffness of web and flanges by appropriate reinforcement, so as additional stresses generated due to restraint to distortion are within safe limits. The Second alternative to check distortion

- 22 -

may be to provide diaphragms as shear walls at the section where it is to be checked. These diaphragms distribute the differential shears of web to flanges also by bending in plate ad by shear forces in diaphragm. The introduction of diaphragms into box girders will have two effects on transverse moments in slabs: 1)

If the diaphragm spacing is approximately equal to transverse

spacing of webs, transverse bending moments may be reduced as a result of two way slab action of diaphragm . 2)

The moments caused by differential deflection will be eliminated

over the region influenced by diaphragms. By the provision of diaphragms, transverse bending stresses caused by the moments, resulting from differential deflection of top and bottom slabs are eliminated. Proper spacing of diaphragms can be determined by the use of beam on elastic foundation concept to effectively control differential deflection. The use of diaphragms at s which are definite locations of concentrated loading significantly diminishes the differential deflections near the s and should always be provided. As far as possible interior diaphragms are avoided as they not only result in additional load but also disrupt and delay the casting cycle resulting in overall delay in construction. In general interior diaphragms would be needed for the box section, which has light webs and ed by relatively stiff slabs. Such a form of cross section is not appropriate for concrete box girders, although prestressing is done externally this type of cross section is not justified. Diaphragms which are stiff out of their planes, when provided at the s, restrain warping in continuous spans, resulting in stresses. These stresses add to longitudinal bending stresses. As conditions of maximum torque do not generally coincide with conditions of maximum bending, and the warping stresses, if they occur, may not therefore increase bending stresses to unacceptable values

- 23 -

CHAPTER 3

CONSTRUCTION AND GENERAL ARANGEMENT OF BOX GIRDER

- 24 -

3.1

GENERAL ARRANGEMENT: The deck arrangement is similar to a voided slab, but with the voids occupying a larger proportion of deck area and usually being rectangular in section. The outer webs are often sloped and side cantilevers made longer to improve the appearance. The web thickness is governed by the shear requirements, but they must be wide enough to provide space for reinforcement and concrete to be placed around prestressing ducts. This usually requires a minimum web thickness of 300mm, but may be wider if larger tendons are used. The deck slab size is governed by web spacing and live load carried and is typically between 150mm and 200mm being sufficient. Transverse diaphragms are provided across the full width of the box at each of the locations. The diaphragms provide rigidity to the box assist in transferring the loads in the webs to the s. Intermediate diaphragms are often placed at ¼ or 1/3 points along the span to stiffen up the box and to help distribute the loading between the webs. Access into box cells is achieved through soffit access holes of a minimum of 600mm diameter, and is located near the abutments. Similar sized holes are provided through each of diaphragms and webs, as required to give access into each section of deck. Small drainage holes, typically 50mm diameter, are provided through bottom slab at the low point in each section of deck to ensure that water cannot collect inside box cells. Concreting and construction restraints dictate a minimum deck depth of 1200mm; although for reasonable inspection and maintenance access a depth of at least 1800mm is needed. With an optimum span –to-depth ratio of between 18:1 and 25:1 the preferred span lengths are usually greater than 30m. Multi-strand tendons are used following a draped profile, and are located in the bottom of the webs in the mid span and at the top of webs over the s. For decks with a overall length less than 80m and fully cast before applying prestress, the tendons would usually extend over the full deck length and be anchored on the end diaphragms. Longer decks are cast in stages on span-by-span basis, with the prestress tendons - 25 -

anchored on the webs at the construction t. The tendons are then continued into next stage of deck by using couplers.

CODAL PROVISIONS COARSE AGGREGATES IRC:18 recommends the nominal size of coarse aggregate shall

usually be

restricted to 10mm less than the minimum clear distance between the individual cables or un-tensioned steel reinforcement or 10mm less than the minim um clear cover to un-tensioned steel reinforcement, whichever is less. A nominal size of 20mm coarse aggregate is used for pre-stressed concrete work. FINE AGGREG Fine aggregates shall confirm to clause302.3.3 of IRC21 WATER Water used for mixing hall be as per clause302.4 of IRC21 CONCRETE Concrete shall be used in accordance with clause 302.6 of IRC21 3.2

CAST- IN SITU CONSTRUCTION OF BOX GIRDER A) Casting the cross section in one pour B) Casting the cross section in stages A) Casting the cross section in one pour:

Fig:3-1 Wide bottom slab cast through trunking

- 26 -

Fig:3-2 Narrow bottom slab with concrete cast down webs There are two approaches to cast a box section in one pour. The bottom slab may be cast first with the help of trucking ing through temporary holes left in the soffit form of top slab. This requires laborers to spread and vibrate the concrete, generally possible for decks that are at least two meters deep. The casting of webs must follow closely, so that cold ts are avoided. The fluidity of the concrete needs to be designed such that the concrete will not slump out of the webs. This is assisted if there is a strip of top shutter to bottom slab about 500mm wide along web. This method of construction is most suitable for boxes with relatively narrow bottom flanges. The compaction of bottom slab concrete needs to be effected by external vibrates, which impels the use of steel shutters. The concrete may be cast down both webs , with inspection holes in the shutter that allow air to be expelled and the complete filling bottom slab to be confirmed. Alternatively concrete may be cast down first with the second web being cast only when concrete appears at its base, demonstrating that the bottom slab is full. The concrete mix design is critical and full-scale trials representing both the geometry of the cross section and density of reinforcement and prestress cables are essential. However the section is cast, the core shutter must be dismantled and removed through a hole in the top slab, or made collapsible so it may be withdrawn longitudinally through the pier diaphragm. Despite these difficulties, casting the section in one pour is under-used. The recent development of self-compacting concrete could revolutionize the construction of decks in this manner. This could be particularly important for medium length bridges with spans between 40 m and 55 m. Such spans are too long for twin rib type decks, and too short - 27 -

for cast-in-situ balanced cantilever construction of box girders, while a total length of box section deck of less than about 1,000 m does not justify setting up a precast segmental facility. Currently, it is this type of bridge that is least favorable for concrete and where steel composite construction is found to be competitive. B) Casting the cross section in stages

Fig:3-3 Alternative positions of construction t The most common method of building box decks in situ is to cast the cross section in stages. Either, the bottom slab is cast first with the webs and top slab cast in a second phase, or the webs and bottom slab constitute the first phase, completed by the top slab. When the bottom slab is cast first, the construction t is usually located just above the slab, giving a kicker for the web formwork, position 1 in Figure. A t in this location has several disadvantages. Alternatively, the t may be in the bottom slab close to the webs, or at the beginning of the haunches, position 2. The advantages of locating the t in the bottom slab are that it does not cross prestressing tendons or heavy reinforcement; it is protected from the weather and is also less prominent visually. The main disadvantage is that the slab only constitutes a small proportion of the total concrete to be cast, leaving a much larger second pour. The t may be located at the top of the web, just below the top slab, position 3. This retains many of the disadvantages of position 1, namely that the construction t is crossed by prestressing ducts at a shallow angle, and it is difficult to prepare for the next pour due to the presence of the web reinforcement. In addition, most of the difficulty of casting the bottom slab has been re-introduced. The advantages are

- 28 -

that the t is less prominent visually and is protected from the weather by the side cantilever, the quantity of concrete in each pour is similar and less of the shutter is trapped inside the box. Casting a cross section in phases causes the second phase to crack due to restraint by the hardened concrete of the first phase. Although the section may be reinforced to limit the width of the cracks, it is not desirable for a prestressed concrete deck to be cracked under permanent loads. Eliminating cracks altogether would require very expensive measures such as cooling the second phase concrete to limit the rise in temperature during setting or adopting crack sealing ixtures

3.3

CONSTRUCTION OF IN-SITU MULTI CELL BOX GIRDER Most in situ multi-cell box girders are cast on full height scaffolding built up from the ground. Where good access exists this form of construction provides flexibility in the construction sequence and deck layout. Obstructions under the deck, such as live loads, railways or small rivers, are overcome by spanning with temporary works to the false work. After erecting the scaffolding the formwork is placed to the required shape and profile. Timber formwork, consisting of a plywood facing ed by timber studding, Steel forms are used when long lengths of decks are to be cast in stages and the shutters are used many times. With timber forms it is easier to have squarer angled corners and flat faces while steel forms are able to incorporate curved and sides. Casting the deck section in several stages simplifies the formwork. This also makes the concreting operations much simpler and easier to control. The bottom slab, outer webs and diaphragms are cast first, followed by the inner webs and top slab soon after. The time delay between castings should be kept to a minimum to reduce any early thermal and differential shrinkage effects. It is preferable to cast the outer webs with the bottom slab so that the construction t is at the top of web and hidden in the corner with top slab. A construction t between the bottom slab and the webs is difficult to hide on the concrete surface and, although this is not - 29 -

important for the inner webs, it marks the appearance on the outer webs. The form work for the inner webs and top slab is ed off the bottom slab concrete, simplifying the overall arrangement. With the formwork in position, the next activity is the fixing of the reinforcement, prestressing ducts and anchorages. Short shutters are being installed along the bottom of the webs to form kickers when the webs are cast in next stage. Without the inner web and top slab formwork in place the access for placing, compacting and finishing the concrete in the bottom slab is improved. The subsequent concreting of inner webs and top slab is done from above the deck without needing access to the void. At this stage the deck is still fully ed by the false work which remains in place until the concreting is completed and the tendons installed. Either permanent formwork s or removable table forms are used between the webs to the wet deck slab concrete. The removal of formwork from inside the voids, after the deck is completed, requires it to be broken down into small sections are ed out through the access holes in the diaphragms and bottom slabs alternatively, a larger temporary access hole is left in the top slab at one end of the deck slab which is concrete after the rest of the formwork has been removed. Longer girder bridge decks, extending over several spans, are usually cast in sections on a span-by-span basis. This has several benefits including reducing the size of concrete pours to a more manageable quantity, optimizing the length of pre stress tendons and permitting the maximum re –use of false work and formwork. The first section cast is a complete span plus part of the adjacent spans to give short cantilevers. This moves the construction ts away from the highly stressed region at the pier and helps to balance the deck in temporary and permanent situations. To optimize the overall moment distribution the construction t is placed between the ¼ or 1/3 points of span. Subsequent sections of deck extend from the construction t over the next pier with a short cantilever, as before .This process is continued until the end of deck is reached - 30 -

During concreting of deck slab the level and finishing of the top surface has to be carefully controlled. On smaller decks this is achieved by placing leveling timbers on the reinforcement and screeding the concrete to the top of these. For larger areas of slab a finishing machine is used to assist accurately leveling of top surface. When the concrete has attained the required strength the pre stress tendon are installed and stressed. The deck tends to lift up along its span and reduce the load on the false work as the pre stress applied. The false work is removed after sufficient tendons have been stressed to carry dead load of deck.

- 31 -

CHAPTER 4

ANALYSIS OF BOX GIRDER BRIDGE:

- 32 -

4.1

TORSIONAL ANALYSIS FOR MULTI CELL BOX GIRDER

Multi cell closed section: The analysis of a single cell closed section can be extended to multi cell section. Fig …? Indicated a two-cell closed section with mean areas of cross sections A1 and A 2. The thickness of the cell walls is considered to be uniform. The length ABCD (s1) may be assumed to be of thickness t1, DEFA (s2) of thickness t2 and DA (s3) of thickness t3; the corresponding stresses in the segments are denoted by τ 1, τ 2, and τ 3. The relative magnitudes of the stresses can be determined by applying the equilibrium condition of the longitudinal forces (due to complementary shear stresses) at D. Considering

sections 1, 2, 3 indicated in

figure, equilibrium of the longitudinal forces yield τ1 t1 = τ 1 t2 + τ 3t3 The torque applied is resisted by both the cells; Appling Eqn xy1t1 = xy2 t2 = q T = § τ t r ds = τ1t1 [§ r ds] + τ2t2 [§ r ds] = 2 [τ1t1A1 + τ2t2A2] It may be noted that the contour integral is computed around the first and second cells successively. From the compatibility condition, the angle of twist of the two cells should be equal. The angle of twist of the beam can be determined from the energy principles. The strain energy of the first cell can be related to the work done by the torque component of the cell. We have U = § [ τ2 / (2G) ] l t ds = [ ql/(2G) ] § τds = [ ql/(2G) ] [τ1s1 + τ3s3]

eq.:1

W = [Tiθ/2] = [qA1θ]

eq.:2

Fig:4-1.: torsion in multi cell closed section - 33 -

From the above equation, we obtain θ = [ l/2A1G] [τ1s1 + τ3s3]

eq.:3

From the analysis of the second cell, we obtain θ = [ l/2A2G] [τ2s2 - τ3s3]

eq.:4

It should be noted in eqn.4 that the integral along segment AD is performed in the direction opposite to the assumed shear flow, hence the negative sign, The third equation, that is required to the shear stresses, can be obtained by equating the values of θ from eqn.:1 and 2. In case the thickness of cell walls vary, eqn: 3 and 4 should be modified suitably. This feature is explained in the illustrated examples that follow. The above analysis can be extended to a beam of any number of cells. It may be noted that the shear flow through the intermediate web depends upon the relative dimensions of the ading cell walls. If the cells are of the same dimensions, shear flow through the intermediate webs will be zero; the beam behaves as if comprising one large cell without intermediate webs. The intermediate webs, however, restrict the bending deformations of the beam.

Application to Case Study:

Fig 4-2 Area of hollow portion = 5.48 m2 Torsion on the cross section is given by (assuming CLASS AA TRACKED) = 2835 KN-m Because of symmetry of box section at the middle τ3 = 0

- 34 -

Applying equation of shear flow since t1 = t2 τ1 = τ2 We know that T τ1

= 2 [τ1t1A1 + τ2t2A2]

=0.587 MPa

Shear stress in the bottom flange = τbf =0.43 MPa Φ = [ l/2A1G] § τ ds Therefore deflection of box girder = 0.054 X 10-3 rad/m Hence from the above obtained deflection it can be proved that the box is an ideal shape in of torsional stiffness for any bridge spanning from 20m to 300m.

4.2 DISTORTIONAL ANALYSIS FOR MULTI CELL BOX GIRDER 4.2.1 BEAM ON ELASTIC FOUNDATION: BEF procedure of analysis of a box girder is simplified analytical procedure that s for the important characteristics of their behavior and gives the designer an overall view of the interaction between loading, proportions and response. This procedure could for deformation of the cross section for the effect of rigid or deformable interior diaphragms, longitudinally and transversely stiffened plate elements, non prismatic sections, continuity over intermediate s, and pro arbitrary end conditions. The BEF analogy provides a convenient guide to the diaphragm spacing and stiffness required to keep distortion stresses small in order to avoid fatigue problems from torsional loads that may complete reverse in service. Further the BEF analogy lends itself to graphical presentation of the effects of design parameters on bridge response. In addition the analogy contributes to the understanding of behavior required for good design practice. ANALOGY: A mathematical analogy exists between distortional behavior of a rectangle single cell section box beam and the flexural behavior of a beam on elastic foundation. The following arguments were presented so far.

- 35 -

a) The transverse flexural moments induced by distortional action are proportional to the BEF deflections fig 4-2.

Fig: 4-3 b) Longitudinal warping moments induced by distortional action are proportional to the BEF moments fig 4-3.

Fig 4-4

c) Concentrated torsional moments represented by the force system P, P correspond to concentrated loads on the BEF. Similarly distributed torsional moments correspond to distributed loads on BEF fig 4-4.

- 36 -

Fig: 4-5

d) The resistance to transverse distortion of the box girder section as generated in the top and bottom slabs fig corresponds to the foundation modulus of BEF: also the resistance to longitudinal warping corresponds to the BEF moment of inertia fig 4-5.

Fig: 4-6 e) A diaphragm which is infinitely stiff in its own plane but completely flexible for out-of-plane bending corresponds to a simple non-deflecting in the BEF; similarly, a diaphragm which is addition completely rigid for out-of-plane bending corresponds to a fixed in the BEF. PROCEDURE: Step 1: Resistance of cross section to deformation which-corresponds to the BEF foundation modulus ‘k’ is developed.

- 37 -

For calculating ‘k’ the deflection ‘s1’, of the loaded nodes of the cell, for unit uniform torsional load is computed by considering a unit length of the cell made statically determinate by cutting the bottom plate at its midpoint. For calculation of ‘s1’, only St. Venant torsion is considered with no work of shears in planes of plates. Step 2: Resistance due to restraint of warping-which corresponds to BEF moment of inertia ‘Ib’ evaluated by considering amount of torsional load equilibrated by restraint of warping. The distributed torsional ‘p’ equilibrated restraint of warping is computed using a virtual displacement ‘dw’ and the work of ‘p’ and ‘q’ (shears due to St. Venant torsion). Now, Pw = E*Ib*dx4/dw4. Composition of this equation with expression for ‘pw’ developed value of ‘Ib’ can be calculated. Step 3: Properties of diaphragms- The effect of diaphragm flexibility on box girder behavior is determined by evaluating the corresponding flexibility for the analogous BEF. For diaphragms stiffness is defined as the magnitude of concentrated vertical torsional load, acting at the top of each web, which would deform the diaphragms to produce unit deflection ‘w’ for the cross section itself acting as a linkage. Step 4: Thus an analogous BEF is prepared related to box girder. A number of solutions for beams on elastic foundation are available. Thus knowing response of corresponding BEF, response of box girder to distortion can be found.

- 38 -

CHAPTER 5

DESIGN OF BOX GIRDER BRIDGE

- 39 -

LOADS AND FORCES Loads, forces and their combination as per IRC6-1966 are used Ultimate load is as per clause12 ofIRC18 PERMISSIBLE STRESSES The compressive stress developed due loading mentioned in clause5.2 of IRC18 shall not exceed 0.5 f cj which shall not be greater than 20mpa, where fcj is concrete strength at that time subjected to a maximum value of characteristic compressive strength of concrete(fck). The permissible stress in concrete at service stage shall not exceed 0.33fck. No tensile stress is allowed at service. SECTIONAL PROPERTIES The thickness of web shall not be less than d/36 plus twice the clear cover to the reinforcement plus diameter of duct hole where‘d’ is overall depth of box girder measured from the top of the deck slab to the bottom of the soffit or 200mm plus the diameter of the duct holes, whichever is greater as per clause 9.3.2 of IRC 18. The web thickness shall not less than 350 mm in case of pre-stressing . Minimum dimensions for haunches are provided as per IRC18 clause9.3.2.5. Minimum thickness of diaphragms shall not be less than minimum web thickness. The thickness of bottom flange of box girder shall not be less than 1/20 th of clear web spacing at the junction with bottom flange or 200mm whichever is more.

- 40 -

The minimum thickness of the deck slab with cantilever tips shall be of 200mm. The minimum clear height inside the box girder shall be 1.5m to facilitate inspection 5.1

DESCRIPTION The Superstructure consists of two cell PSC box girder duly connected by three intermediate diaphragms and two end diaphragms. The deck width of box girder is 11.05m. The depth of box girder is 2.0m. Thickness of deck slab is 220 mm. In the design of box girder carriage way width of 7.5m is being taken for which 1 lane of 70R or Two lanes of Class A (live load taken from SERC tables) whichever governs is being considered. Single stage prestressing is envisaged when concrete attains a strength of M35 or at 14 days after casting of box girder which ever is later. The prestressing in box girder is done using 10 nos. of 12T13 cables. The cables are provided with 75ID sheathing. All the cables are stressed from both ends.

5.2

5.3

SPAN ARRANGEMENT a)

Clear span (square)

=

19.500 m

b)

Center to center of Bearings (Effective span )

=

19.500 m

c)

Total length of box girder

=

20.400 m

d)

Over all length of deck slab

=

20.960 m

PRESTRESS It is proposed to use 10 Nos of 12T13 cables. Each cable consists of 12.7 mm dia. 7-ply class 2 Low relaxation strands as per IS : 14268 and has an area of 98.7 mm2.

- 41 -

The properties as given by the manufacturer are as follows. a)

Area of each 12T 13 cable

=

1184.4 mm2

b)

Ultimate tensile strength

=

18960 kg/cm2

c)

Breaking strength of cable

=

2245.62 KN

d)

Anchorage slip at stressing

=

6mm

e)

All cables are used with bright metal sheathing of 75mm ID = 0.25 K = 0.0046/m

f)

Allowable force in 12T 13 cables at stressing end before anchorage = 1718 KN

5.4

DESIGN DATA Effective Span ( C/c of bearings)

=

19.500 m

Length of girder

=

20.400 m

Length of deck slab at top

=

20.960 m

Carriage way width

=

7.500 m

Width of footpath

=

1.500 m

Width of Kerb

=

0.325 m

Depth of kerb

=

0.275 m

Height of parapet wall

=

1.800 m

Thickness of parapet wall

=

0.275 m avg.

Thickness of wearing coat

=

0.075 m

Thickness of Vertical ribs at mid span

=

0.300 m

- 42 -

Thickness of vertical ribs at

=

0.450 m

Thickness of intermediate diaphragm

=

0.300 m

Thickness of end diaphragm

=

0.450 m

Width of box girder at mid span

=

6.600 m

Width of box girder at

=

6.750 m

Width of deck slab

=

11.050 m

Depth of box girder

=

2.000 m

Thickness of deck slab

=

0.220 m

Thickness of cantilever slab at

=

0.350 m

Thickness of cantilever slab at tip

=

0.220 m

Thickness of soffit slab at mid span

=

0.300 m

Thickness of soffit slab at

=

0.450 m

Haunches at top at mid span

=

0.30 m

=

0.150 m

=

0.30 m

=

0.150 m

=

0.15 m

=

0.075 m

Unit weight of concrete VRCC

=

25.00 KN/ m3

Unit weight of concrete VCC

=

24.00 KN/ m3

Bottom haunches at mid span

Top haunches at

- 43 -

5.5

SECTIONAL PROPERTIES CROSS SECTION PROPERTIES AT MIDSPAN

Fig:5-1 Yb

=

1.121 m

Yt

=

0.879 m

Lna

=

3.781 m4

Zb

=

3.373 m3

Zt

=

4.301 m3

GROSS SECTION PROPERTIES AT

Fig:5-2 Yb

=

1.032 m

- 44 -

5.6

Lna

=

4.253 m4

Zb

=

4.121 m3

Zt

=

4.394 m3

Yt

=

0.968 m

BENDING MOMENT AND SHEAR FORCE CALCULATIONS BENDING MOMENT DUE TO SELF WEIGHT OF BOX GIRDER BM at mid span

=

7975.016KN-m

SF at mid span

=

0.01KN

BM at 3/8 span

=

7456.63KN-m

SF at 3/8 span

=

401.94KN

BM at 1/4 span

=

6015.13KN-m

SF at 1/4 span

=

827.17KN

BM at 1/8 span

=

5254.41KN-m

SF at 1/8 span

=

962.91KN

ADDITIONAL LOAD DUE TO PROJECTION OF SLAB BEHIND THE GIRDER :

Fig:5-3

- 45 -

Reaction at

=

40.11KN

Reaction from both sides

=

80.22KN

BENDING MOMENT DUE TO SUPERIMPOSED DEAD LOAD ON BOX GIRDER a)

Footpath slab

=

7.50

KN/m

b)

Kerbs

=

8.94

KN /m

c)

Parapet wall

=

24.8

KN /m

d)

Wearing coat

=

13.5

KN /m

=

54.7

KN /m

Fig:5-4 BENDING MOMENT DUE TO LIVE LOAD ON FOOTPATH

Fig:5-5

Live load on footpath

=

- 46 -

150 KN/m

BENDING MOMENT DUE TO LIVE LOAD ON DECK SLAB BM DUE TO LIVE LOAD Girder Span 19.5 0 21 0 22.5 0 24 0 25.5 0 27 0 28.5 0 30 0 31.5 0 33 0 34.5 0 36 0 38.5 0 40 0

.125L

.25L

.375L

.5L

147 163 179 195 211 227 243 260 276 292 309 324 346 368

244 271 298 327 355 383 411 438 466 494 522 549 576 603

303 338 373 408 443 478 513 547 581 616 651 687 722 758

325.3 363 401 438 476 513 550 588 626 663 701 738 776 814

0.125L 147.00

0.25L 244.00

0.375L 303.00

0.5 L 325.30

0.25 L 287.06

0.375 L 356.47

0.5 L 382.71

BM Due to Live Load (with Impact and Torsion (10%) 0.125 L 0.25L 0.375 L 0 190.24 315.76 392.12

0.5 L 420.98

19.5

0

BM Due to Live Load (with Impact ) 0.125L 0 172.94

SHEAR FORCE DUE TO LIVE LOAD Girder 0.125L Span 19.5 73.724 61.224 21 75.6 63.1 22.5 77.227 64.727 24 78.65 66.15 25.5 79.906 67.406 27 81.022 68.522 28.5 82.021 69.521 30 82.92 70.42 31.5 83.733 71.233 33 84.473 71.973 34.5 85.148 72.648 36 85.767 73.267

0.25L

0.375L

0.5L

48.724 50.6 52.227 53.65 54.906 56.022 57.022 57.92 58.92 59.473 60.148 60.767

36.75 38.33 39.73 41.15 42.41 43.52 44.52 45.42 46.23 46.87 47.65 48.27

25.22 26.71 27.99 29.12 30.11 31.02 32.02 32.92 33.73 34.47 35.15 35.77

- 47 -

Span 19.500

73.72

0.125 L 61.22

0.25L 48.72

0.375 L 36.75

0.5 L 25.22

SHEAR FORCEDUE TO LIVE LOAD (WITH IMPACT) Span 0.125 L 0.25L 0.375 L 19.500 86.73 72.03 57.32 43.23

0.5 L 29.67

SHEAR FORCE DUE TO LIVE LOAD ( WITH IMPACT) & TORSION (10%) Span 0.125 L 0.25L 0.375 L 0.5 L 19.500 95.41 79.23 63.05 47.55 32.64

5.7

TABULATION OF BENDING MOMENTS AND SHEAR FORCES AT VARIOUS SECTIONS Tabulation of Bending moments at various sections (KN-m)

Girder

/ 0.125

Section Self

0.25L

0.375L

0.50L

0.625L

0.75L

0.875L

7456.6

7975.0

7456.6

6015.1

5254.4

L 0.00

5254.4 6015.4

weight

- 48 -

0.00

SIDL

0.00

1131.1 1943.8

2431.4

2594.0

2431.4

1943.8

1131.1

0.00

Live

0.00

1902.4 3157.6

3921.2

4209.8

3921.2

3157.6

1902.4

0.00

0.00

310.2

666.9

711.5

666.9

533.1

310.2

0.00

0.00

8598.1 11649.7 14476.1 15490.2 14476.1 11649.7 8598.1

Load (duly increasing 10%

for

torsional effect) Footpath

533.1

load Total Load

Tabulation of Shear force at Various sections (KN)

- 49 -

0.00

Girder

/ 0.125

Section

0.25L

0.375L 0.50L 0.625L 0.75L

0.875L

L

Self

1949.9

962.9

827.2

401.9

0.00

401.9

827.2

982.9

1949.9

SIDL

557.8

400.0

266.7

133.3

0.00

133.3

266.7

400.0

557.8

Live

954.1

792.3

630.5

475.5

326.4

475.5

630.5

792.3

954.1

153.0

109.7

73.1

36.6

0.00

36.6

73.1

109.7

153.0

3614.8

2264.9 1797.5 1047.4

326.4

1047.4

1797.5 2264.9

weight

Load (duly increasing 10%

for

torsional effect) Footpath load Total Load

5.8

PRESTRESSING FORCES AND OTHER LOSSES CALCULATION ARRANGEMENT OF CABLES AT MID SPAN

- 50 -

3614.8

Fig:5-7 ARRANEMENT OF CABLES AT ANCHORAGE

Fig:5-8 C.G. of Cables: Cables 1 2 3

No. of Cables 3 3 4



1.339 0.938 0.206 0.765 Total Horizontal forces:Cables 1 2 3

420.66 413.49 581.75 1415.90

1/8 Span

¼ span

3/8 span

Mid span

0.874 0.481 0.120 0.455

0.539 0.206 0.120 0.271

0.337 0.120 0.120 0.185

0.270 0.120 0.120 0.165

1/8 Span 432.23 429.72 601.73 1463.31

¼ span 445.27 448.40 608.73 1502.40

3/8 span 457.14 464.09 616.10 1537.33

Mid span 467.71 467.46 623.47 1558.64

Prestressing Details : Details



1/8 span - 51 -

¼ span

3/8 span

Mid span

Prestressing Force (KN) Yb…m C.G. of Cables …m Ecentricity m Area m2 P/A …. KN/m2 Zt …m3 Zb …. m3 Pe / Zt… KN/m2 Pe / Zb…KN/m2 P/A Pe/Zt….KN/m2 P/A +pe/Zb….KN/m2 Prestress at C.G. of cables

14159.0

14633.1

15024.0

15373.3

15586.4

1.032 0.765 0.267 7.566 1871.4 4.394 4.121 859.6 916.4 1011.8

1.082 0.455 0.627 6.965 2100.9 4.353 3.789 2107.4 2420.9 -6.5

1.121 0.271 0.850 6.212 2418.5 4.301 3.373 2967.5 3784.5 -549.0

1.121 0.185 0.936 6.212 2474.7 4.301 3.373 3344.7 4265.6 -870

1.121 0.165 0.956 6.212 2509.0 4.301 3.373 3464.1 4417.8 -955.1

2787.8

4521.7

6202.9

6740.3

6926.7

210.83

349.24

528.67

603.58

627.65

BM due to self Wt (KN-m)

0.00

5254.4

6015.1

7456.6

7975.0

Md/Zt … KN/m2

0.00

1207.2

1398.4

1733.5

1854.0

Md/Zb …. KN/m2

0.00

1386.7

1783.4

2210.8

2364.5

Rresultant Stress at TG (KN/m2)

1011.8

1206.6

849.4

863.5

898.9

Resultant stress at BG ( KN/m2)

2787.8

3135.0

4419.5

4529.5

4562.3

Stress at CG of Cables KN/m2

2108.3

2695.3

3935.1

4190.1

4260.1

No of Strands

=

120

Average Stress in concrete at c.g of

=

3.37 N/mm2

Average force in Cables

=

14955.2 KN

Average Stress in Steel

=

0.665961UTS

Days at Stressing

=

14

Grade of Concrete

=

40 N/mm2

EC

=

29421.78 N/mm2

- 52 -

=

195000 N/mm2

=

11.18 N/mm2

Residual shrinkage

=

0.00025

Shrinkage Loss

=

48.75 N/mm2

ES Losses Due to Elastic shortening Due to Shrinkage

Relaxation Loss Initial Stress 0.5 0.6 0.7 0.8 0.67 0.68 Long Term relaxation loss

% Relaxation 0 1.25 2.5 4.5 2.07

12.5 12.5 20

77.07 N/mm2

Creep loss Creep Strain / 10 Concrete maturity MPA

5.9

75

0.00056

-1E-05

80

0.00051

-0.000007

90

0.00044

-0.000004

100

0.0004

86.56

0.00046

Creep Loss

30.52 N/mm2

Total Loss

198.81 N/mm2 including 20% long term loss

% of Loss

16.05 including 20% long term loss

STRESS IN SERVICE CONDITION Details



1/8 span

¼ span

Prestress at TG after loss KN/m2

849.4

-5.5

-460.9

- 53 -

3/8 span Mid span -730.4 -801.8

Prestress at BG after loss(KN/m2) Md/Zt….KN/m2 Md/Zb … KN/m2 Stress at TG after loss (KN/m2) Stress at BG after loss (t/m2) BM (SIDL + FPLL +LL) (KN.m) Zt… m3 Zb … m3 SIDL + FPLL + LL Stress at top KN/m2 SIDL + FPLL + LL Stress at Bottom KN/m2 Resultant stress at TG (KN/m2) Resultant Strress at BG (KN/m2)

5.10

2340.4

3796.0

5207.4

5658.5

5815.0

0.00 0.00 849.4

1207.2 1386.7 1201.7

1398.4 1783.4 937.5

1733.5 2210.8 1003.1

1854.0 2364.5 1052.2

2340.4 0.00

2409.3 3343.7

3424.0 5634.6

3447.7 7019.5

3450.6 7515.2

4.39 4.12 0.00

4.25 3.79 768.2

4.30 3.37 1309.9

4.30 3.37 1631.9

4.30 3.37 1747.1

0.00

882.5

1670.6

2081.2

2228.1

849.4

1969.9

2247.4

2635.0

2799.3

2340.4

1526.8

1753.4

1366.5

1222.4

SHEAR FORCE CALCULATIONS Shear Forces in Kilo Newton 1/8 span ¼ span Self weight of PSC Box SIDL Liveload

1949.92 557.84 1107.1

962.9 400 902.0

827.2 266.67 703.7

3/8 span Mid span 401.9 0.00 133.34 0.00 512.1 326.4

Partial Safety factor @ ultimate limit state Self weight of PSC Box

1.5

SIDL

2

Liveload

2.5 Shear forces in Kilo Newton@ Ultimate load condition

Self weight of PSC Box SIDL Live load Total



1/8 span

¼ span

2924.9 1115.68 2767.7 6808.2

1444.4 800 2255.1 4499.4

1240.8 533.34 1759.2 3533.3

Bending Moment in KN.m 1/8 span ¼ span Self weight of PSC Box

0.00

5254.4 - 54 -

6015.1

3/8 span Mid span 602.9 0.00 266.68 0 1280.3 816.0 2149.9 816.0

3/8 span Mid span 7456.6 7975.0

SIDL Live load Total

0 0 0

1131.14 2212.6 8598.1

1943.82 2431.43 3690.8 4588.0 11649.7 14476.1

2593.97 4921.2 15490.2

Partial Safety factor @ Ultimate State Self weight of PSC Box 1.5 SIDL 2 Liveload 2.5 Bending Moment in KN.m @ ultimate State 1/8 span ¼ span 3/8 Mid span span Self weight of PSC Box 0.00 7881.6 9022.7 11184.9 11962.5 SIDL 0 2262.28 3887.64 4862.86 5187.94 Live load 0 5531.5 9226.9 11470.1 12303.0 Total 0 15675.4 22137.3 27517.9 29453.5 Vertical Component of Prestressing Force(KN) after Friction and Slip losses Cable No. 1/8 span ¼ span 3/8 span Mid span 1 860.4 714.3 490.6 251.8 0 2 856.2 651.9 332.1 0.00 0 3 373.0 0.00 0.00 0.00 0 Total 2089.5 1366.2 822.7 251.8 0 % of Loss = 160.4979 Vertical Component of Prestressing Force (KN) after all Losses 1/8 span ¼ span 3/8 span Mid span 1754.2 1147.0 690.6 211.4 0 Total Vertical Component of Prestressing Force (KN) Partial Safety Factor(1.00) 1/8 span ¼ span 3/8 span Mid span 1754.2 1147.0 690.6 211.4 0 C.G. of Cables (mm)

1/8 span

0.765 0.455 Check for max. Shear force / Shear Resistance Details 1/8 span 1. Ultimate shear force 5054.1 3352.5 2. Overall depth – cm 200.00 200.00 3. C.G. of all cables = cm 76.526 45.465 4. Effective depth = cm 123.47 154.54 5. Eff. Width of web – cm 120 100.31 (b-2/3 of dia of duct)

- 55 -

¼ span 0.271 ¼ span 2842.7 200.00 27.139 172.86 75

3/8 span Mid span 0.185 0.165 3/8 span 1938.5 200.00 18.514 181.49 75

Mid span 816.0 200.00 16.500 183.50 75

6. max. permissible stress Kg / cm2 7. Shear resistance (KN)

47

47

47

47

47

6963.9 >Vu

7285.9 >Vu

6093.3 >Vu

6397.4 >Vu

6468.4 >Vu

Shear resistance of section as cracked : Vcr = 0.037 bd fck + MCR / M x V MCR = [ 0.37 fck + fpt ] x I / Y Details Vu = Ultimkate shear (Kilo Newton) B = Breadth of rib in mm D = effective depth in mm M = Ultimate moment Fpt = N/mm<x yfc Where (Yfc = 0.87) 0.37 fck (N/ mm^2 ) I/Y = Zb mm Mcr (KNm x 10-1) KN m Mcr / Mu x V (KN) 0.037 bd fck (N) x 10-4 Total Vcr (KN) Vco (Kilo Newton) Shear resistance to be considered (KN)

6808.2

1/8 span 4499.4

¼ span 3533.3

3/8 span 2149.9

Mid span 816.0

1200.00

1003.13

750.00

750.00

750.00

1234.74

1545.35

1728.61

1814.86

1835.00

0.00

1567.54

2213.73

2751.79

2945.35

2.04

3.30

4.53

4.92

5.06

2.34 4.12E+09 18035.0

2.34 3.79E+09 21380.4

2.34 3.37E+09 23173.3

2.34 3.37E+09 24497.0

2.34 3.37E+09 24956.4

-

6137.0 36.28

3698.7 30.34

1913.8 31.85

691.4 32.21

5346.4 5346.4
6499.7 4640.3 4640.3
4002.1 3712.4 3712.4
2232.4 3388.7 2232.4
1013.5 3225.7 1013.5
Computation of f & fpt after losses (Kg /cm2)

Prestress at C.G. of Cables Kg/ cm2 F Kg / cm2 after losses

Prestress at bottom of PSC BOX Kg / cm2 F Kg / cm2 after losses

21.08

1/8 span 34.92

¼ span 52.87

3/8 span 60.36

Mid span 62.76

17.70

29.32

44.38

50.67

52.69

27.88

1/8 span 45.22

¼ span 62.03

3/8 span 67.40

Mid span 69.27

23.40

37.96

52.07

56.58

58.15

Shear resistance of section as uncracked :

- 56 -

Vco = 0.67 bh (ft^< + f. Ft) Details 1. b = breadth of rib in 2. h = overall depth in 3. f : N/mm2 4. ft = B/mm2 5. Vco = (in Killo Neuton) 6. Vertical component of prestressing Total

5.11

Ft = 0.24 Fck =

1.52 N/mm2

1200 2000.00 1.77 1.52 3592.2 1754.2

1/8 span 1003.125 2000.00 2.93 1.52 3493.4 1147.0

¼ span 750 2000.00 4.44 1.52 3021.8 690.6

3/8 span 750 2000.00 5.07 1.52 3177.3 211.4

Mid span 750 2000.00 5.27 1.52 3225.7 0.00

5346.4

4640.3

3712.4

3388.7

3225.7

DESIGN FOR SHEAR Shear Reinforcement : Minimum Shear Reinforcement

SV = 0.87 Asv f y / 0.4 b

Design Shear Reinforcement SV = 0.87 fyv Asv dt / V-Vc +0.4bdt Fyv = 415 N/mm>

Vc(N) Vu (N) Reinforcement Provide diameter No of legs Asv (mm2) B dt Spacing (mm) Spacing to be adopted (mm)

5.12

1.78E+06 1.68 E + 06 Min 10 4 314 400.00 1234.74 708.5606 200

1/8 span 1.55 E +06 1.12E+06 Min 10 4 314 334.38 1545.35 847.6239 200

DESIGN OF ELASTOMERIC BEARING

- 57 -

¼ span 1.09E+06

3/8 span 6.81E+05

Mid span 3.38E+05

9.48+05 Min 10 2 157 250.00 1728.61 566.8485 200

6.46E+05 Min 10 2 157 250.00 1814.86 566.8485 200

2.72E+05 Min 10 2 157 250.00 1835.00 566.8485 200

Fig:5-9 Overall dimension of bearing

=

350 x 500

1) Size of Bearing (a x b)

=

338 x 488

2) Size of elastomer

=

10 mm

3) thickness of M.S. Plates (t)

=

3 mm

4) Top & Bottom cover

=

5 mm

5) Side cover

=

6 mm

6) Total number of elastomers

=

4 Nos.

7) Number of steel plates

=

5 Nos.

9) Total height of bearing

=

65 mm

Pc (DL or slowly applied vertical loads )

=

835.9 KN

Ps (LL or quickly applied vertical loads )

=

369.0 KN

Hc (slow acting horizontal forces)

=

36.1 KN

Hs (quick acting horizontal forces)

=

33.3

KN

Uc (Horizontal (shear) movement due to Hc)

=

Us (Horizontal (shear) movement due to Hs)

=

8) Total thickness of elastomer pad = 50 mm

(Rotation under effect of slow acting loads) (Rotation under effect of quick acting loads) G ( Static shear modulus of elastomer )

=

1

N/mm2

G ( Dynamic shear modulus of elastomer)

=

1.8

N/mm2

Shrinkage strain

=

0.00025

Creep strain

=

0.000464051

Temperature strain for 30 deg change

=

0.00036

- 58 -

Total Strain

=

0.001074051

Half the span

=

10.2 m

Total Horizontal moment due to creep shrinkage & temperature =10.96 mm Longitudinal force

=

200 KN

Dispersion

=

0%

Net Longitudinal force after dispersion

=

200 KN

No. of bearings

=

6

=

7.30

1)

Max and min pressures Max = (Pc + Ps) / a x b

Max pressure should be less than 10N / mm2 O.K Min = Ps/a x b

=

5.07

Minimum pressure should be greater than 2 N/mm2 O.K 2.

Shape factor

(a)

S = (axb) / (2(a+b) x hi)

=

9.98 >6<12

O.K (b)

a/10 338/10

33.8 mm

a/5 338/5

67.6 mm

h=

50 mm h should be greater than a/10 and less than a/5 O.K

(c)

a

=

338mm

b = 488 mm

a should be greater than b/2 1>b/2 O.K

3)

Assumed total horizontal moment due to creep shrinkage and temperature =

10.96 mm

Hc = slow acting horizontal force

=

36.1 KN

Longitudinal force

=

200 KN

Dispersion

=

0%

Longitudinal force on bearing

=

33.3 KN

=

0.20 N/mm2

Quick applied loads Shear stress

- 59 -

Shear strain /50

=

5.61 mm

& slow applied loads

=

16.57mm

Distortion limit in shear

=

70% of the height of bearing

=

0.3314

Total movement due to quick applied loads

4)

No slip condition

1

=

0.22 (for D.L)

1

=

0.18 (DL + LL)

Resisting forces 1

x DL

=

182.6 KN

Hc = 3.61

2

x (DL + LL) =

219.5 KN

Hs = 3.33

Resisting force > Hc Resisting Force > Hc + Hs

5)

Deflection at mid span

=

5 wL4

= 5 ML2

384 EI

48 EI

E = 31622.777 N/mm2 I = 3.781 M4 M = 15490.2 KN.m L = 19.5 m Deflection

=

5.13 mm

Due to dead loads

=

3.50 mm

Due to live loads

=

1.63 mm

Tan c

=

0.0004 radians

Tan s

=

0.0002 radians

(a) for D.L. condition ei

> tan c

a/6 compression of individual layer N = 5.07

- 60 -

ei =

hi x N

= 0.1225788 mm

4 GS2 + 3N Compression of

4

ei

0.0087

=

Layers = 0.49 mm

a/6 (b)

For (DL + LL) condition ei

tan c + 1.5 tan s

a/6 N

=

7.30

ei

=

0.174

Compression of ei

=

4 layers

=

0.695 mm

=

0.0007 radians

=

1.27 N / mm2

0.01233

a/6 tan c + 1.5 tan s

6)

Check for total shear stress

(a)

Shear stress due to compressive load =

1.5 (Pc + 1.5 Ps) S x

axb

b)

Shear stress due to horizontal loads =

c)

Shear stress due to rotation G a2(tan c + 1.5 tan s)

0.42 N/mm2

=

0.08 N/mm2

=

1.77 N/mm2

=

2.41

2 x hi x h Total Shear stress <5 N/mm2 7)

Suitability of steel plates = 2 x (t1 + t2) (Pc + 1.5 Ps) a x b x s 3 mm plates is

5.13

O.K

DEFLECTION CALCULATIONS Maximum Deflection under a UDL causing a maximum moment of ‘M’

- 61 -

=

5 x ML2 48 EI L

=

19.5 m

E

=

29421.78 N/mm2

I

=

3.781 M4

=

7975.0

Moment due to self weight Moment due to SIDL

=

2594.0

KN.m KN.m

Moment due to Live Load

=

4209.8

KN.m

Moment due to Foot path Load

=

711.5

KN.m

Total Downward deflection =

5.515

mm

Upward deflection due to prestress Prerstressing Force at Midspan

=

1.5586.4

Losses

=

16.05 %

Prestressing Force after Losses

=

13084.8

KN

Eccentricity

=

0.956

m

Upward deflection due to Prestress= 4.45

5.14

KN

m

Net Downward deflection

=

1.061 mm

SPAN / 1500

=

13.00

mm

ELONGATION STATEMENT E = 2.00 E +06 Kg / cm2 Area of cable = 11.844 Cm2

- 62 -

Cable

Refer

Force

Averag

Len

Lengt

Elon

Total

Grp

ence

(KN)

e

gth

h

gatio

Elong Retrac Elonga

(mm)

n(m

ation(

tion

tion(m

m)

mm)

(mm)

m)

75.70

11.44

64.26

Point

Force( KN)

1

A1

Slip &

Net

1717.9 2

A

B

C

D

1717.9

1717.9

A1

2

2

A

1709.8

1713.8

AB

73

96

1559.0

1634.4

29

51

1559.0

750

5.44

1020.

7.39

70 BC

9113.

62.88

09 CD

0

29

Cable

Refere Force Averag

Len

Lengt

Elon

Total

Slip &

Net

Grp

nce

gth

h

gatio

Elong

Retrac

Elonga

(mm)

n(m

ation(

tion

tion(m

m)

mm)

(mm)

m)

11.44

63.67

(KN)

Point

e Force( KN)

2

A1

1717. 9

A

B

1717. 1717.9

A1

9

A

1709. 1713.9

AB

9 C

1576. 1643.2

1558. 1567.4

5.44

1021.

7.39

21 BC

6 D

750

6546.

45.41

45 CD

2550

2

- 63 -

16.87 75.11

Cable

Refere Force

Grp

nce

(KN)

Point

Averag

Len

Lengt

Elon

Total

Slip &

Net

e

gth

h

gatio

Elong

Retrac

Elonga

(mm)

n(m

ation(

tion

tion(m

m)

mm)

(mm)

m)

53.95 75.75

11.44

64.31

Force( KN)

3

A1

1717. 9

A

1717.

1717.9

9 B

A1

1710.

1714.0

AB

1002.

7.25

05

1673.

1691.5

BC

0 D

5.44

A

0 C

750

1275.

9.11

87

1614.

1643.6

CD

7775

2

5.15

CHECK FOR ULTIMATE MOMENT OF RESISTANCE Bending Moment in KN.m & Ultimate State 1/8 span Self weight of PSC 0.00 7881.6 Box SIDL 0.00 2262.3 Live Load 0 5531.48 Total 0.00 15675.4 Failure by Yield of steel

=

¼ span 9022.7

3/8 span 11184.9

Mid span 11962.5

3887.6 9226.94 22137.3

4862.9 11470.12 27517.9

5187.9 12303.04 29453.5

0.9 x db x As x fp

As

=

Area of High Tensile Steel

db

=

Depth of beam from compression edge to cg of Steel Tendons

Fp

=

Ultimate tensile Strength of Steel

Mult

=

Ultimate Moment of Resistance

As (cm2) Fp (kg / cm2) db(cm) Mult (KN.m)

118.44 18960 123.47 24954.8

1/8 span 118.44 18960 1545.4 31232.5

¼ span 118.44 18960 1728.6 34936.2

3/8 span 118.44 18960 1814.9 36679.4

Failure by Crushing of Concrete = 0.176bd2fck +2/3x0.8x(Bf-b)x(db-t/s)tfck

- 64 -

Mid span 118.44 18960 1835.0 37086.5

B(cm) 120 Db(cm) 123.47 0.176bdb2fck(KN.m) 12879.617 Bf(cm) 750 T(cm) 22 2/3x0.8x(Bf-b)x(db- 9976.87 t/2)tfck Mult 11264.83

5.16

1/8 span 100.3125 154.54 16864.899 750 22 13130.05

¼ span 75 172.86 15777.073 750 22 15383.24

3/8 span 75 181.49 17390.8083 750 22 16202.98

Mid span 75 183.50 17778.948 750 22 16394.40

14816.54 >Mu

16960.95 >Mu

17942.06 >Mu

18172.29 >Mu

DESIGN OF DECK SLAB Grade of Concrete Grade of steel

= =

M

40

415

Modular Ratio m

=

10.0

Permissible Stress in concrete =

13.37

Permissible stress in steel

=

200.00

K = 1/(1+ (st / m cbc)

=

0.401

J = 1-k/3

=

0.866

Design of

=

2.321

Dead load intensity

=

7.30 KN/m2

N/mm2 N/mm2

Q = ½ x cbc x k x j

Design consideration From SERC tables, para 2.2 of introduction, it may be adequate in most of the cases if mean of the span bending moments for simply ed and fixed edges cases is taken for design loading with the moments of the latter case, as recommended in ONORMB4202 size

=

3.00 x 5.00 m (from chart –81)

Fro Span moments, average of span moments of fixed edges and simply ed edge is considered For edge moments, edge moments of fixed case is considered Referring to key chart No. 81 of SERC tables : Impact coefficient for 70-R Loading is

=

25.0 %

Impact coefficient for Class AA loading

=

25.0 %

Impact coefficient for Class A Loading

=

50.00 %

- 65 -

Type loading

of

Mxc 70-R 1675.380 With impact 2094.225 Class AA 2101.190 With Impact 2626.488 Class A 1043.690 With impact 1565.535 D.L 1.0 3295.50 KN/m2 D.L 0.730 2405.72 KN/m2

Simply Myc 3137.660 3922,075 3750.850 4688.563 1696.560 2544.840 7719.80

Mxc 1091.400 1364.250 1284.190 1605.238 709.800 1064.700 1192.10

Myc 1654.300 2067.875 2086.600 2608.250 1109.800 1664.700 3479.50

Mxe -2640.480 -3300.600 -2935.350 -3669.188 -1737.460 -2606.190 -5142.40

Mye -3010.660 -3763.325 -3540.250 -4425.313 -1861.480 -2792.220 - 7118.70

5635.45

870.23

2540.04

-3753.95

-5196.65

From the above table class AA Loading will govern the design

Design span moment due to Live Load : Mxc

=

2115.863

Kg – m/m

Myc

=

3648.406

Kg – m/m

Mxe

=

-3669.188

Kg – m/m

=

-4425.313

Kg – m/m

Mye Design Moments due to dead load : Mxc

=

163.797

Kg – m/m

Myc

=

408.774

Kg – m/m

Mxe

=

-375.395

Kg – m/m

=

-519.665

Kg – m/m

Mxe

=

2279.66

Kg – m/m

Myc

=

4057.18

Kg – m/m

Mxe

=

-4044.58

Kg – m/m

Mye

=

-4944.98

Kg – m/m

Mxc

=

2279.7

Myc

=

4057.2

Mxe

=

4044.6

Mye Total design moment

ABSOLUTE VALUES

- 66 -

Mye

=

4945

Maximum Value

=

4945

D

=

144.57 mm < 172.00 mm

Reinforcement required at various critical sections are as follows.

1)

Mid Span

a)

Shorter span :

Effective depth available at mid span

=

172.00 mm

Ast for Myc

=

1336.0 mm2

Spacing required

=

150.42

Using 16 mm dia @ 200 mm c/c area of steel

=

1004.8 mm2

Using 16 mm dia @ 200 mm c/c area of steel

=

1004.8 mm2

Total Steel provided

=

2009.6 mm2

=

750.69 mm2

Spacing required

=

150.58 mm

Using 12 mm dia @ 150 mm c/c area of steel

=

753.6 mm2

Using 12 mm dia @ 150 mm c/c area of steel

=

753.6 mm2

Total Steel provided

=

1507.2 mm2

Effective Depth

=

156.00 mm

Ast for Mye

=

1795.4 mm2

Spacing required

=

111.93 mm

Using 16 mm dia @ 200 mm c/c area of steel

=

1004.8 mm2

Using 16 mm dia bar spacing required

Hence provide 16 mm dia @ 200.0 mm c/c Hence, provide 16 mm dia @ 200.0 mm c/c

b) Longer span : Ast for Mxc Using 12 mm dia bar spacing required

Hence, provide 12 mm dia @ 150.0 mm c/c Hence, provide 12 mm dia @ 150.0 mm c/c 2) a) Shorter span :

Using 16 mm dia bar spacing required

- 67 -

Using 16 mm dia @ 200 mm c/c area of steel

=

1004.8 mm2

Total steel provided

=

2009.6 mm2

=

1468.5 mm2

Spacing required

=

136.85 mm

Using 12 mm dia @ 150 mm c/c area of steel

=

753.6 mm2

Using 12 mm dia @ 150 mm c/c area of steel

=

753.6 mm2

Total steel provided

1507.2 mm2

Hence, provide 16 mm dia 200.0 mm c/c Hence, provide 16 mm dia @ 200.0 mm c/c b) Longer span : Ast for Mxe Using 16 mm dia bar spacing required

=

Hence, provide 12 mm dia @ 150.0 mm c/c Hence, provide 12 mm dia @ 150.00 mm c/c

5.17

DESIGN OF CANTILEVER DECK SLAB BEYOND END DIAPHRAGMS:

Fig:5-10

Cantilever portion of deck slab beyond the outer face of end diaphragms =0.505 m Bending moment at face of cantilever : 1)

Dead load bending moment Self weight of deck slab

=

7.125

Weight of wearing coat

=

0.18

- 68 -

2)

=

8.925KN/m2

Bending moment

=

1.14

Live load bending moment with impact

=

4.438 KN-m

Effective width of deck slab for this load

=

1.20a + bw

=

1.436 m

Liveload bending moment per meter width

=

30.91 KN-m

Total design bending moment

=

32.05 KN-m

Effective depth required

=

117.51 mm <

KN-m

300.00 mm Area of steel required

=

616.8 mm2

=

183.26 mm

Using 12 mm dia bar spacing required Spacing required Hence, provide 12 mm dia @ 150.0 mm c/c Hence, provide 12 mm dia @ 150.0 mm c/c

5.18

DESIGN OF CANTILEVER DECK SLAB BELOW FOOT PATH

Fig:5-11

- 69 -

Cantilever portion of deck slab beyond the outer face of end diaphragms =

2.225 m

Self weight of footpath slab

=

7.65625

KN/m

Bending moment

=

6.70

KN-m

Bending moment at face of cantilever : 1)

Footpath Slab :

2)

Railing :

a)

Weight of railing

=

11.69 KN/m

Bending moment

=

24.40 KN-m

Weight of railing rectangular part

=

1.13

KN-m

bending moment

=

2.26

KN-m

Weight of railing triangular part

=

0.22

KN/m

Bending moment

=

0.40

KN-m

Total railing bending moment

=

27.06 KN-m

Self weight of deck slab

=

15.853125

Weight of wearing coat

=

1.8

=

17.65 KN/m2

C.G. of Section

=

1.028 m

Bending moment

=

18.14 KN-m

b)

c)

3)

4)

Dead load bending moment

Live load bending moment : footpath live load P P

=

Live load in Kg/m2

L

=

Effective span of the girder

=

W

=

Width of footwayin m =

1.525 m

1

=

500.00 Kg/m2

5.64

p 5.

= {p1-260 +(4800/L} x {16.5-W)/15}

=

19.500 m

KN-m

Liveload : The specified minimum clearance for class – AA and Class – 70 R is 1.20 m, where as the span of cantilever is only 0.575 excludiing kerbs. Hence class – AA and class – 70R vehicles will not operate in this portion and only class – A loading is considered for design of cantilever slab.

Effective width of deck slab for this load

- 70 -

=

1.20a +bw

=

0.490 m

Live per meter width

=

174.5 KN/m

Bending moment

=

13.09 KN-m

Total design bending moment

=

70.63 KN-m

Effective depth required

=

174.44mm < 300mm

Area of steel required

=

147.84 mm

=

147.84 mm

Using 16 mm dia bar spacing required Spacing required Hence, provide 16 mm dia @ 200.0 mm c/c + provide 16 mm dia @ 200.0 mm c/c

Distribution Steel : Distribution steel shall be designed to resist the moment of = 0.30 LL moment + 0.20 DL moment = 1.038 t-m Area of steel required

=

348.43 mm2

=

225.29 mm

=

0.22429

Using 10 mm dia bar spacing required Spacing required

Hence, Provide 10 mm dia @ 200.0 mm c/c At top and bottom % of steel

5.19

PROVISION OF UN TENSIONED MILD STEEL REINFORCEMENT : Vertical Reinforcement : (CI : 15.2 of IRC 18-2000) : Minimum vertical reinforcement required as per clause 15.2 is

0.18 % of

sectional area Running Section 1) Sectional area of rib in plan

=

300000.0 mm2

Sectional area of vertical reinforcement required

=

540.0 mm2

Area of steel provided 2-L 10 mm dia @

2)

150.0 c/c

Sectional Area of rib at end - 71 -

=

1047.613 mm2

=

600000.0 mm2

Sectional area of vertical reinforcement required

1080.00 mm2

=

Area of steel provided = 2-L 10 mm dia @ 150.0 c/c X 2-L 10 mm dia @ 150.0 c/c +

2095.23 mm2

=

Longitudinal reinforcement : (CI : 15.3 of Irc 18-2000) : Minimum longitudinal reinforcement required as per clause 15.3 is 0.15 % of sectional area 1)

Sectional area of vertical rib

=

444000.0 mm2

Sectional area of vertical reinforcement required

=

666.00 mm2

No. of bars required using 10mm dia bars

=

8.48 Nos.

Hence provide 10 mm dia bars 9 Nos over for both faces of vertical ribs in longer direction in middle. Hence provide 10 mm dia bars 18 nos. over for both faces of vertical ribs in longer direction at ends. LONGITUDINAL REINFORCEMENT IN SOFIT SLAB : (CI:15.4 of IRC 18-2000) : Minimum longitudinal reinforcement for sofit slab required as per clause 15.4 0.18 % of sectional area and it is distributed in at top and bottom. At mid span : Sectional area of soffit slab

=

300000.0

mm2

Area of reinforcement required

=

540.00

mm2

Using 10 mm dia spacing required

=

290.74

mm

Hence provide 10 mm dia @ 200 mm c/c both top and bottom At : Sectional area of soffit slab

=

450000.0

mm2

Area of reinforcement required

=

810.00

mm2

Using 10 mm dia spacing required

=

193.83

mm

Hence provide 10 mm dia @ 100 mm c/c both top and bottom

Minimum transverse reinforcement in soffit slab required as per clause 15.4 is 0.3% of sectional area and it is distributed in at top and bottom Sectional area of soffit slab

=

300000.0

mm2

Sectional area of vertical reinforcement required

=

900.00

mm2

- 72 -

Using 10 mm dia spacing required

=

174.44

mm

Hence provide 10 mm dia bars @ 150 mm c/c across the soffit slab both top and bottom At : Sectional Area of soffit slab

=

450000.0

mm2

Area of reinforcement required

=

1350.0

mm2

Using 10 mm dia spacing required

=

116.30 mm

Hence provide 10 mm dia @ 100 mm c/c both top and bottom

5.20

DESIGN OF DIAPHRAGMS / CROSS GIRDERS : Diaphragms (i.e. Cross Girders) in a multi girder, T- beam and deck slab types of Superstructure are provided to distribute the unsymetrically placed live loads on the deck. These cross girders / diaphragms are transversely stiff and are therefore very effective in distributing the live loads on the deck to several longitudinal girders in the system. In the Multi – girder bridge deck the diaphragms are intended to distribute the live loads to various main girders and are thus subjected to both positive and negative bending moments. The cross girders / diaphragms add to the stiffness of the deck system against torsional buckling of the individual beam . The cross girders / diaphragms are therefore designed for the direct loads that they i.e., 1. Self weight of deck slab 2. Wheel loads coming directly over the diaphragms forms the most critical load for the diaphragms. INTERMEDIATE DIAPHRAGMS : Two (2) end diaphragms and three intermediate diaphragms are provided at mid span. The deck slab is designed as ed on all four sides, and hence the self weight of deck slab transferred to the diaphragm, is triangular. This load can however be converted in to an equivalent UDL (Ref. Reynolds Hand Book) Equivalent UDL due to self Weight of deck slab

=

WL/3

W = Self weight of Slab

=

7.3 KN/m2

Load per meter width

=

24.33 KN/m

If no opening was left in diaphragm, then there would have been no bending of diaphragms. The diaphragms. The diaphragms now bends, over the opening of 1200 mm wide and, at this section the diaphragms has a depth of 220 + 365 = 585

- 73 -

mm. It therefore means that the diaphragms of 585 mm depth is spanning over 1200 mm, as a fixed beam. Width of opening / span

=

1.20 m

Depth above opening

=

585 mm

Bending moment due to slab load = WL2 /12

=

2.92 KN/m

Self weight of cross beam / diaphragm

=

4.4

BM due to self weight of diaphragm

=

0.05 KN-m

Hence total dead load moment

=

3.45

KN/m

KN-m

EFFECT OF LIVE LOAD “Class AA” (tracked) load placed directly over the diaphragm gives the maximum reaction and hence causes the maximum bending moment.

Fig:5-12

Load Transferred to diaphragm

=

285.38

KN

Therefore, with 25% impact

=

356.73

KN

As the width of opening is 1200 mm, only one track will come over opening and when the track is placed exactly in center (Across the width) maximum moments are obtained. Co-efficient CAB for FEM

=

0.125 x (1-a2/3) x W

=

3.714

=

44.568

MOMENT Therefore, BM at A

= CAB x L

- 74 -

KN-m

(-ve bending moment near ) Total moment at A

=

48.01

KN-m

Depth required

=

262.60

mm

Overall depth available

=

585.00

mm

Effective depth

=

535.00

mm

Reinforcement required

=

521.78

mm2

Using 20 mm dia bars. No of bars required =

1.66 Nos.

Provide 4 Nos. TOR 20 mm bars at top as shown in the drawing

SPAN MOMENT :

Fig:5-13 Mc

=

69.12 KN-m

Net bending moment at mid span

=

69.12 KN-m

Dead load moment in span

=

3.45 KN-m

Net design BM in span

=

72.57 KN-m

Reinforcement required

=

788.59 mm2

Using 20 mm dia bars. No of bars required = 2.51 Nos. Provide 3 Nos. TOR 20 mm bars at bottom as shown in the drawing

DESIGN OF SHEAR : Shear Force (Live load)

=

17.84

Dead load / Self weight shear force

=

2.63

Total Design shear force

=

181.00 KN

- 75 -

KN

Shear stress

= V/bxd

=

1.21 < 2.50Mpa

Maximum permissible shear stress for M40 grade of concrete is 2.50 Mpa as per IRC 21-2000 table –12A Shear Resistance of Concrete assumed as 20% Using 12 mm dia 2 legged stirrups spacing required Sv

=

156.08

mm

Provide 12 mm dia @ 150 mm c/c

5.21

END DIAPHRAGM The end diaphragm has a thickness of 450 mm. The load transferred from “ class AA” tracked Vehicle on end diaphragm will be much less, as compared to the intermediate diaphragm, however same reinforcement will be provided in the end diaphragms also.

Design for end diaphragm for lifting condition: In the event of replacement of Elastomeric bearings, it becomes necessary to lift the superstructure from its s and accordingly suitable locations for placement of Hydraulic jacks for lifting the superstructure from over the bed blocks are marked in the sketch below. This arrangement for lifting of superstructure induces bending moment and shear forces in the end diaphragms of the PSC superstructure for which suitable reinforcement will be provided as calculated below.

Fig:5-14

- 76 -

Total Weight of half span of superstructure =

2507.8 KN

Dead load reaction per bearing

=

C/c Distance between bearing and jack position =

0.625 m

Maximum hogging moment

522.45 KN-m

=

835.92 KN

Depth required

=

816.85 mm

Overall depth available

=

2000.00

Effective depth

=

1950.00 mm

Reinforcement required

=

1483.52 mm2

Using 20 mm dia bars. No of bars required Provide 6 Nos, TOR 20 mm bars at top as shown in the drawing.

SHEAR Maximum shear force ( cantilever span)

=

835.92 KN

Shear Stress

=

0.95 < 2.50 Mpa

= V/bxd

Maximum permissible shear stress for M40 grade of concrete is 2.50 Mpa as per IRC 21-2000 table –12A. Assuring 20% shear resistance due to concrete Using 12 mm dia 4 legged stirrups spacing required Sv

=

227.10 mm

Provide 12 mm dia @ 200 mm c/c

5.22

DESIGN OF END BLOCK BEARING STRESS BEHIND ANCHORAGES : (Clause 7.3) The maximum allowable bearing stress, immediately behind the anchorages is given by Fb

=

0.48 fcj  (A2/A1) or 0.8 fcj whichever is smaller

A1

=

bearing area of anchorage ( 27 x 27)

A2

=

Area of concrete within member without overlapping (40 x 40)

fcj

=

Concrete strength at the time of stressing

Where

The strength of concrete at the time of stressing shall not be less than

- 77 -

= 34600.0 KN/ m2 fb = Permissible bearing stress behind anchorage

= 0.48 Fcj  (A2/A1) = 24604.4 KN/m2

or 0.80 Fcj

= 27680 KN/m2

Maximum force in the cable after blocking

= 145.00

Cable force after instantaneous losses

= 0.00

(Elastic shortening, relaxation of steel, and seating of anchorage) = 145.00 Therefore, Bearinig pressure immediately behind the anchorages = 19890.3 KN/m2 < 24604 KN/m2

- 78 -

CHAPTER 6

CASE STUDY PICTURES

- 79 -

Fig: 6-1 STAGING OF PROPOSED BOX GIRDER BRIDGE OF SPAN 21mts

Fig: 6-2 FRONT ELEVATION OF BOX GIRDER - 80 -

Fig: 6-3 END DIAPHRAGM

Fig: 6-4 ELASTOMERIC BEARING

- 81 -

Fig: 6-5 SPAN SHOWING SOFFIT SLAB IN BEARING LEVEL

Fig: 6-6 PIER, BED BLOCK & BOXGIRDER

- 82 -

Fig: 6-7INTERMEDIATE DIAPHRAGM

Fig: 6-8 PRESTRESSING CABLE IN LONGITUDINAL GIRDER

- 83 -

Fig: 6-9 PRESTRESSING CABLE IN SOFFIT SLAB

Fig: 6-10 HYDRAULIC JACK OUTER DIAMETER

- 84 -

Fig: 6-11 CABLE LOCATED IN JACK

Fig: 6-12 POST-TENSSIONING CABLE BY HYDRAULIC JACK

- 85 -

CHAPTER 7

CONCLUSION

- 86 -

CONCLUSION AND FUTURE WORK An approximate method adopted in the box girder bridge design has been taken as a case study. It revealed that the final stresses in concrete at transfer and service load stages very nominal when compared with allowable stresses. But the factor of safety when checked against ultimate loads is only 2.0, and the box section selected when checked for torsional and distortional effects is found satisfactory. A further study is sought in these respects for optimum usage of material strengths using software packages available is suggested.

- 87 -

CHAPTER 8 REFERENCES:  Strength

of

Materials-

A

Practical

Approach

by

D.S.Prakash Rao.  Bridge Superstructures by N.Rajagopalan.  Concrete Box Girder Bridges by Jorg Schlaich and Hartmut Scheef.  The Design of Prestressed Concrete Bridges by Robert Benaim.  M.C.Tandon “Box Girders subjected to Torsion”, Indian Concrete Journal, February 1976.  Koll Brunner C.F. and Busler K ,”Torsion in structure” 1969 edition  Design of Bridges By N.Krishnaraju  IRC codes 5,6,18,21,22,78,83(part II)  IS CODES 456-2000, 1343

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