141336426 bridge deck analysis

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Bridge Deck Analysis

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This book is dedicated to Orlaith, Sadhbh and Ailbhe,and to Margaret

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Bridge Deck AnalysisEugene J.O’Brien and Damien L.Keogh

Department of Civil Engineering,University College Dublin, Ireland

Chapter 4 written in collaboration with the authors by

Barry M.LehaneDepartment of Civil, Structural and Environmental

Engineering, Trinity College Dublin, Ireland

London and New York

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First published 1999by E & FN Spon

11 New Fetter Lane, London EC4P 4EE

Simultaneously published in the USA and Canadaby Routledge

29 West 35th Street, New York, NY 10001

E & FN Spon is an imprint of the Taylor & Francis GroupThis edition published in the Taylor & Francis e-Library, 2005.

To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousandsof eBooks please go to www.eBookstore.tandf.co.uk.

© 1999 Eugene J.O’Brien and Damien L.Keogh

Cover photograph: Killarney Road Bridge, courtesy of Roughan and O’Donovan,Consulting Engineers

All rights reserved. No part of this book may be reprinted or reproducedor utilised in any form or by any electronic, mechanical, or other means,now known or hereafter invented, including photocopying and recording,or in any information storage or retrieval system, without permission in

writing from the publishers.

The publisher makes no representation, express or implied, with regard tothe accuracy of the information contained in this book and cannot accept anylegal responsibility or liability for any errors or omissions that may be made.

British Library Cataloguing in Publication DataA catalogue record for this book is available from the British Library

Library of Congress Cataloging in Publication DataO’Brien, Eugene J., 1958–

Bridge deck analysis/Eugene J.O’Brien and Damien L.Keogh.p. cm.

Includes index.ISBN 0-419-22500-5

1. Bridges-Floors. 2. Structural analysis (Engineering)I.Keogh, Damien L., 1969–. II. Title.

TG325.6.027 1999624’.253–dc21 98–48511

CIP

ISBN 0-203-98414-5 Master e-book ISBN

ISBN 0-419-22500-5 (Print Edition)

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Contents

Preface viii

Acknowledgements x

Chapter 1 Introduction 1

1.1 Introduction 1

1.2 Factors affecting structural form 1

1.3 Cross-sections 2

1.4 Bridge elevations 8

1.5 Articulation 26

1.6 Bearings 29

1.7 Joints 32

1.8 Bridge aesthetics 34

Chapter 2 Bridge loading 40

2.1 Introduction 40

2.2 Dead and superimposed dead loading 42

2.3 Imposed traffic loading 43

2.4 Thermal loading 46

2.5 Impact loading 51

2.6 Dynamic effects 52

2.7 Prestress loading 54

Chapter 3 Introduction to bridge analysis 67

3.1 Introduction 67

3.2 Moment distribution 67

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3.3 Differential settlement of supports 75

3.4 Thermal expansion and contraction 78

3.5 Differential temperature effects 89

3.6 Prestress 104

3.7 Application of moment distribution to grillages 111

Chapter 4 Integral bridges 121

4.1 Introduction 121

4.2 Contraction of bridge deck 128

4.3 Conventional spring model for deck expansion 133

4.4 Modelling expansion with an equivalent spring at deck level 137

4.5 Run-on slab 145

4.6 Time-dependent effects in composite integral bridges 147

Chapter 5 Slab bridge decks—behaviour and modelling 151


5.2 Thin-plate theory 151

5.3 Grillage analysis of slab decks 169

5.4 Planar finite-element analysis of slab decks 185

5.5 Wood and Armer equations 191

Chapter 6 Application of planar grillage and finite-element methods 200


6.2 Simple isotropic slabs 200

6.3 Edge cantilevers and edge stiffening 203

6.4 Voided slab bridge decks 211

6.5 Beam and slab bridges 218

6.6 Cellular bridges 228

6.7 Skew and curved bridge decks 236

Chapter 7 Three-dimensional modelling of bridge decks 240


7.2 Shear lag and neutral axis location 240

7.3 Effective flange width 242

7.4 Three-dimensional analysis 244

7.5 Upstand grillage modelling 245

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7.6 Upstand finite-element modelling 252

7.7 Prestress loads in three-dimensional models 260

AppendixA

Reactions and bending moment diagrams due to applied load 263

AppendixB

Stiffness of structural members and associated bending moment diagrams 265

AppendixC

Location of centroid of a section 267

AppendixD

Derivation of shear area for grillage member representing cell with flangeand web distortion

269

References 272

Index 274

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Preface

Twenty-five years ago, fairly complex skew, prestressed concrete bridge decks could beanalysed with a fair degree of accuracy—but only by using manual methods. The famousRusch and Hergenroder influence surface charts, translated from the German by the Cementand Concrete Association, gave surfaces for various stress and aspect ratios up to a 45° skew.Full analysis of a bridge deck involved, amongst other techniques, the use of planimeters onthe way to calculating volumes under the influence surface, in turn leading to the calculationof mx, my and mxy moments. The method was tedious, somewhat approximate and could oftentake weeks. Indeed, if an error arose early on in the calculations, many days could be spent inre-analysing. Now, it is possible to change a dozen variables and a computer program willrecalculate stresses and reactions in seconds.

There is still a need, however, perhaps more so now than in the past, for a bridge engineerto understand how a bridge deck responds to various combinations of load and to be able todecide if the ‘answer’ (output) is sensible. To be confident of this, an understanding of thebehaviour of non-symmetrical, eccentrically loaded, irregularly supported structures isessential.

This book fulfils just that role. Written by two engineers who have, between them,experience of almost all aspects of modern bridge design and analysis, it includes chapters onevery aspect of bridge deck analysis that a practising bridge engineer is ever likely to need.Written in clear, unambiguous English, copiously and carefully illustrated, it represents yearsof scholarship and research presented in a lucid and understandable style which should makeeven the more complex theory understandable to all engineers.

In many aspects, the book contains either a novel approach to design or entirely newmethods. It covers construction in some detail, with sections on bearings, joints and aestheticsnot commonly found in bridge analysis books, loading (with prestress treated as a special caseof loading) and details of a unique graphical approach to moment distribution—a powerfultool in engendering an understanding of fundamental structural behaviour. This is particularlyuseful for

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checking the output of computer analyses. Other chapters deal comprehensively withintegral bridges (with a major geotechnical input from Dr Barry Lehane) and the increasingacceptance of FE methods of analysis, although the merits of grillage methods are not ignored.

All in all, this must prove the standard work on bridge deck analysis for decades to come.Professor S.H.Perry

Civil, Structural and Environmental EngineeringTrinity College Dublin

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Acknowledgements

We would like to thank Dr A.Ghali most sincerely for major contributions to some of theearlier chapters. He gave most generously of his time with the sole objective of getting it right.The initial writing effort was greatly facilitated for both authors through the support ofProfessor S.H.Perry and Trinity College Dublin.

A sabbatical stay in Slovenia for the first author made the initial drafting of many chapterspossible. This would not have been feasible without the enthusiasm of AlešŽnidaričof theSlovenian National Building and Civil Engineering Institute and the support of the Universityof Ljubljana. The stay in Slovenia was greatly enhanced and enriched by Alenka Žnidarič.

The support of Roughan and O’Donovan Consulting Engineers, where both authors wereemployed for a time, is much appreciated. Special thanks is due to Joe O’Donovan forproviding some of the photographs in the text, including the cover illustration. Ancon CCLare also acknowledged for providing a number of illustrations. The assistance of Chris Davisand Michael Barron of Mott McDonald with Chapter 2 is gratefully acknowledged. Theauthors of STRAP (ATIR software, Tel Aviv) and NIKE3D (Lawrence Livermore NationalLaboratories, USA) are thanked for the use of their programs.

Disclaimer

This publication presents many advanced techniques, some of which are novel and have notbeen exposed to the rigours of time. The material represents the opinions of the authors, andshould be treated as such. Readers should use their own judgement as to the validity of theinformation and its applicability to particular situations and check the references beforerelying on them. Sound engineering judgement should be the final arbiter in all stages of thedesign process. Despite the best efforts of all concerned, typographical or editorial errors mayoccur, and readers are encouraged to bring errors of substance to our attention. The publisherand authors disclaim any liability, in whole or in part, arising from information contained inthis publication.

Chapter 1Introduction

1.1 Introduction

A number of terms are illustrated in Fig. 1.1 which are commonly used in bridge engineering.In this figure, all parts of the bridge over the bearings are referred to as superstructure whilethe substructure includes all parts below. The main body of the bridge superstructure is knownas the deck and can consist of a main part and cantilevers as illustrated. The deck spanslongitudinally, which is the direction of span, and transversely, which is perpendicular to it.There may be upstands or downstands at the ends of the cantilever for aesthetic purposes andto support the parapet which is built to retain the vehicles on the bridge.

Bridge decks are frequently supported on bearings which transmit the loads to abutments atthe ends or to piers or walls elsewhere. Joints may be present to facilitate expansion orcontraction of the deck at the ends or in the interior.

1.2 Factors affecting structural form

In recent years, it has been established that a significant portion of the world’s bridges are notperforming as they should. In some cases, bridges are carrying significantly more traffic loadthan originally intended. However, in many others, the problem is one of durability—thewidespread use of de-icing salt on roads has resulted in the ingress of chlorides into concrete.This is often associated with joints that are leaking or with details that have resulted inchloride-contaminated water dripping onto substructures. Problems have also been reportedwith post-tensioned concrete bridges in which inadequate grouting of the ducts has lead tocorrosion of the tendons. The new awareness of the need to design durable bridges has led todramatic changes of attitude towards bridge design. There is now a significant move awayfrom bridges that are easy to design towards bridges that will require little maintenance. Thebridges that were easy to design were usually determinate,

Fig. 1.1 Portion of bridge illustrating bridge engineering terms

e.g. simply supported spans and cantilevers. However, such structural forms have many jointswhich are prone to leakage and also have many bearings which require replacement manytimes over the lifetime of the bridge. The move now is towards bridges which are highlyindeterminate and which have few joints or bearings.

The structural forms of bridges are closely interlinked with the methods of construction.The methods of construction in turn are often dictated by the particular conditions on site. Forexample, when a bridge is to be located over an inaccessible place, such as a railway yard or adeep valley, the construction must be carried out without support from below. Thisimmediately limits the structural forms to those that can be constructed in this way.

The method of construction also influences the distributions of moment and force in abridge. For example, in some bridges, steel beams carry the self weight of the deck whilecomposite steel and in-situ concrete carry the imposed traffic loading. Various alternativestructural bridge forms and methods of construction are presented in the following sections.

1.3 Cross-sections

1.3.1 Solid rectangular

The solid rectangular section, illustrated in Fig. 1.2, is not a very efficient structural form asthe second moment of area of a rectangle is relatively small. Such a bridge is generallyconstructed of reinforced concrete (particularly for the shorter spans) or prestressed concrete.Due to the inefficiency of this structural form, the stresses

Fig. 1.2 In-situ solid rectangular section: (a) without cantilevers; (b) with cantilevers

induced by the self weight of the concrete can become excessive. However, the shutteringcosts for a bridge with a flat soffit are relatively low and the reinforcement is generally simple.As a result, this form of cross-section is often the most cost-effective for shorter spans (up toabout 20 m). As can be seen in Fig. 1.2, bridges can be constructed with or without cantilevers.Comparing bridges of the same width, such as illustrated in Figs. 1.2(a) and (b), it can be seenthat the bridge with cantilevers has less weight, without much reduction in the second momentof area. However, what is often the more important advantage of cantilevers is the aestheticone, which is discussed in Section 1.8.

Solid rectangular sections can be constructed simply from in-situ concrete as illustrated inFig. 1.2. Such construction is clearly more economical when support from below the bridge isreadily available. When this is not the case, e.g. over railway lines or deep waterways, arectangular section can be constructed using precast pretensioned inverted-T-sections asillustrated in Fig. 1.3. Holes are cast at frequent intervals along the length of such beams tofacilitate the threading through of transverse bottom reinforcement. In-situ reinforced concreteis then poured over the precast beams to form the complete section. With this form ofconstruction, the precast beams must be designed to carry their self weight plus the weight ofthe

Fig. 1.3 Precast and in-situ solid rectangular section

Fig. 1.4 Voided slab section with cantilevers

(initially wet) in-situ concrete. The complete rectangular section is available to carry otherloading.

1.3.2 Voided rectangular

For spans in excess of about 20 m, solid rectangular sections become increasingly less cost-effective due to their low second moment of area to weight ratio. For the span range of 20–30m, it is common practice in some countries to use in-situ concrete with polystyrene ‘voids’ asillustrated in Fig. 1.4. These decks can be constructed from ordinary reinforced concrete orcan be post-tensioned. Including voids in a bridge deck increases the cost for a givenstructural depth because it adds to the complexity of the reinforcement, particularly thatdesigned to resist transverse bending. However, it reduces considerably the self weight andthe area of concrete to be prestressed without significantly affecting the second moment ofarea. The shuttering costs are also less than for in-situ concrete T-sections which are describedbelow. Hence it is, in some cases, the preferred solution, particularly when the designerwishes to minimise the structural depth. It is essential in such construction to ensure thatsufficient stays are provided to keep the voids in place when the concrete is poured and toprevent uplift due to flotation. This problem is not so much one of steel straps failing as ofgrooves being cut in the polystyrene by the straps. Concerns have been expressed aboutvoided-slab construction over the lack of inspectability of the concrete on the inside of thevoid and there are many countries where this form is virtually unknown.

It is common practice to treat voided slabs as solid slabs for the purposes of analysisprovided that the void diameter is less than 60% of the total depth. Regardless of thediameter-to-depth ratio, the voids must be accounted for when considering the design to resisttransverse bending. Guidance is given on the analysis of this type of deck in Chapter 6.

1.3.3 T-section

The T-section is commonly used for spans in the range 20–40 m as an alternative to voided-slab construction. However, the T-section is a less efficient structural form as it tends to havemore material close to the neutral axis of the bridge than a voided slab. As a result, the sectiontends to be deeper for a given span. In-situ T-section decks, illustrated in Fig. 1.5, are moreexpensive in terms of shuttering

Fig. 1.5 In-situ concrete T-sections: (a) single web such as might be used for a pedestrian bridge;(b) multiple webs such as would be used for wider decks

costs than voided slabs but have a major advantage in that all of the bridge deck is totallyinspectable.

Over less accessible places, precast concrete or steel forms of T-section, as illustrated inFig. 1.6, are favoured. These consist of pretensioned prestressed concrete or steel beamsplaced in position along the length of the span. An in-situ concrete slab, supported onpermanent shuttering, spans transversely between the beams while acting as flanges to thebeams longitudinally.

1.3.4 Box sections

For spans in excess of 40 m, it becomes economical to use ‘cellular’ or ‘box’ sections asillustrated in Fig. 1.7. These have a higher second moment of area

Fig. 1.6 T-sections: (a) composite steel and concrete; (b) composite precast Y-beam and in-situconcrete

Fig. 1.7 Box sections: (a) single cell; (b) multi-cellular

per unit weight than voided slab or T-sections. However, they are only considered economicalat higher spans as it is only then that the structural depth becomes sufficiently great (about 2m) for personnel to enter the void to recover the shuttering and, when the bridge is in service,to inspect the inside of the void.

Fig. 1.8 Composite precast and in-situ box section

Box sections can be constructed of in-situ or precast concrete or can be composite with aprecast pre-tensioned U-section and an in-situ concrete slab as illustrated in Fig. 1.8.

1.3.5 Older concepts

Many variations of the above structural forms have been used in the past and are evident inexisting bridge stocks. For example, in the past, it was common practice to construct T-section decks using precast ‘M-beams’ (Fig. 1.9). These have wider bottom flanges than theprecast ‘Y-beams’ (Fig. 1.6(b)) used more commonly today. A disadvantage of the M-sectionis that it is difficult to compact the concrete properly at the top surface of the wide bottomflange. In the past, M-sections were often placed side by side with the bottom flanges withinmillimetres of each other. The analysis of this type of bridge is similar to that of any T-sectionbridge.

It was also common practice in the past to build bridges of ‘pseudo-box’ construction asillustrated in Fig. 1.10. These were constructed of M-beams with insitu concrete near thebottom to form a void. The bottom in-situ concrete was reinforced transversely by threadingbars through holes cast in the M-beams. The section is more efficient than a T-section as moreconcrete is located away from the centroid. However, if water leaks into the voids, corrosionproblems can result and,

Fig. 1.9 Precast M-beam

Fig. 1.10 Pseudo-box section

due to the nature of this structural form, assessment and repair is difficult. The structuralbehaviour of the pseudo-box section is similar to that of a small multi-cellular box section.

Another form of construction used widely in the past is the ‘shear key’ deck, illustrated inFig. 1.11(a). This consists of precast concrete slab strips joined using longitudinal strips of in-situ concrete. The latter ‘shear keys’ are assumed to be capable of transferring shear force butnot transverse bending moment as they have no transverse reinforcement. Thus the transversedeformation is assumed to be as illustrated in Fig. 1.11(b), i.e. rotation is assumed to occur atthe joints between precast units. Shear key decks were popular for railway bridge constructionas the railway line could be reopened even before the in-situ concrete was placed. However,they are no longer popular due to concerns about the durability of the in-situ joints.

1.4 Bridge elevations

The cross-sections described above can be used in many different forms of bridge. Many ofthe alternative bridge elevations and their methods of construction are described in thefollowing sections.

Fig. 1.11 Shear-key deck: (a) section through small portion of deck; (b) assumed transversedeformation

1.4.1 Simply supported beam/slab

The simplest form of bridge is the single-span beam or slab which is simply supported at itsends. This form, illustrated in Fig. 1.12, is widely used when the bridge crosses a minor roador small river. In such cases, the span is relatively small and multiple spans are infeasibleand/or unnecessary. The simply supported bridge is relatively simple to analyse and toconstruct but is disadvantaged by having bearings and joints at both ends. The cross-section isoften solid rectangular but can be of any of the forms presented above.

1.4.2 Series of simply supported beams/slabs

When a bridge crossing is too wide for an economical single span, it is possible to constructwhat is in effect a series of simply supported bridges, one after the other, as illustrated in Fig.1.13. Like single-span bridges, this form is relatively simple to analyse and construct. It isparticularly favoured on poor soils where differential settlements of supports are anticipated.It also has the advantage that, if constructed using in-situ concrete, the concrete pours aremoderately sized. In addition, there is less disruption to any traffic that may be below as onlyone span needs to be closed at any one time. However, there are a great many joints andbearings with the result that a series of simply supported beams/slabs is no longer favoured inpractice. Continuous beams/slabs, as illustrated in Fig. 1.14, have significantly fewer jointsand bearings. A further disadvantage of simply supported beam/slabs in comparison tocontinuous ones is that the maximum bending moment in the former is significantly greaterthan that in the latter. For example, the bending moment diagrams due to a uniformlydistributed loading of intensity ω(kN/m) are illustrated in Fig. 1.15. It can be seen that themaximum moment in the simply supported case is significantly greater (about 25%) than thatin the continuous case. The implication of this is that the bridge deck needs to becorrespondingly deeper.

Fig. 1.12 Simply supported beam or slab

Fig. 1.13 Series of simply supported beam/slabs

Fig. 1.14 Continuous beam or slab

Fig. 1.15 Bending moment diagrams due to uniform loading of intensity ω: (a) three simplysupported spans of length l; (b) one three-span continuous beam with span lengths l

1.4.3 Continuous beam/slab with full propping during constructionAs stated above, continuous beam/slab construction has significant advantages over simplysupported spans in that there are fewer joints and bearings and the applied bending momentsare less. For bridges of moderate total length, the concrete can be poured in-situ in one pour.This completely removes the need for any joints. However, as the total bridge length becomeslarge, the amount of concrete that needs to be cast in one pour can become excessive. Thistends to increase cost as the construction becomes more of a batch process than a continuousone.

1.4.4 Partially continuous beam/slab

When support from below during construction is expensive or infeasible, it is possible to useprecast concrete or steel beams to construct a partially continuous bridge. Precast concrete orsteel beams are placed initially in a series of simply supported spans. In-situ concrete is thenused to make the finished bridge continuous over intermediate joints. Two forms of partiallycontinuous bridge are possible. In the form illustrated in Fig. 1.16, the in-situ concrete is castto the full depth of the bridge over all supports to form what is known as a diaphragm beam.Elsewhere the cross-section is similar to that illustrated in Fig. 1.6. In the alternative form ofpartially continuous bridge, illustrated in Fig. 1.17, continuity over intermediate supports isprovided only by the slab. Thus the in-situ slab alone is required to resist the completehogging moment at the intermediate supports. This is possible due to the fact that members oflow structural stiffness (second moment of area) tend to attract low bending moment. The slabat the support in this form of construction is particularly flexible and tends to attract arelatively low bending moment. There is concern among some designers about the integrity ofsuch a joint as it must undergo significant rotation during the service life of the bridge.Further,

Fig. 1.16 Partially continuous bridge with full-depth diaphragm at intermediate supports: (a)elevation; (b) plan view from below

Fig. 1.17 Partially continuous bridge with continuity provided only by the slab at intermediatesupports

Fig. 1.18 Joint detail at intermediate support of partially-continuous bridge of the typeillustrated in Fig. 1.17

as the main bridge beams rotate at their ends, the joint must move longitudinally toaccommodate this rotation as illustrated in Fig. 1.18.

In partially continuous bridges, the precast concrete or steel beams carry all the self weightof the bridge which generates a bending moment diagram such as that illustrated in Fig.1.19(a) for a two-span bridge. By the time the imposed traffic loading is applied, the bridge iscontinuous and the resulting bending moment diagram is as illustrated in Fig. 1.19(b). Thetotal bending moment diagram will be a combination of that due to self weight and otherloading. Unfortunately, due to creep, self weight continues to cause deformation in the bridgeafter it has been made continuous. At this stage it is resisted by a continuous rather than asimply supported beam/slab and it generates a distribution of bending moment more like thatof Fig. 1.19(b) than Fig. 1.19(a). This introduces a complexity into the analysis compoundedby a great difficulty in making accurate predictions of creep effects.

Fig. 1.19 Typical distribution of bending moment in two-span partially-continuous bridge: (a)bending moment due to self weight; (b) bending moment due to loading applied afterbridge has been made continuous

The great advantage of partially continuous construction is in the removal of all intermediatejoints while satisfying the requirement of construction without support from below. Themethod is also of a continuous rather than a batch form as the precast beams can beconstructed at a steady pace, starting even before work has commenced on site. Constructionon site is fast, resulting in minimum disruption to any existing traffic passing under the bridge.A significant disadvantage is that, while intermediate joints have been removed, intermediatebearings are still present with their associated maintenance implications. Particularly for theform illustrated in Fig. 1.17, two bearings are necessary at each intermediate support.

1.4.5 Continuous beam/slab—span-by-span constructionFor construction of particularly long bridges when access from below is expensive orinfeasible, in-situ construction, one span at a time, can be a viable option. This is achievedusing temporary formwork supported on the bridge piers as illustrated in Fig. 1.20(a).Proprietary post-tensioning couplers, such as illustrated in Fig. 1.21, can be used to achievecontinuity of prestressing across construction joints. In this form of construction, the pointwhere one concrete pour meets the next is designed to transmit bending moment and shearforce and is not intended to accommodate movements due to thermal and creep effects. Thejoint may sometimes be located at the quarter-span position as illustrated in Fig. 1.20(b),where bending moments and shear forces are relatively small.

In particularly long continuous beam/slabs, an intermediate joint may become necessary torelieve stresses due to expansion/contraction. It has been said that joints should be providedevery 100 m at least. However, this figure is constantly being revised upwards as theproblems of bridge joints in service receive ever more attention.

Fig. 1.20 Temporary support system for span-by-span construction: (a) joint over intermediatepier; (b) joint at quarter span

Fig. 1.21 Post-tensioning coupler to transmit prestress forces across a construction joint(photograph courtesy of Ancon CCL)

1.4.6 Continuous beam/slab—balanced cantilever constructionWhen the area under a bridge is inaccessible and spans are in excess of about 40 m, it is ofteneconomical to construct bridges by the balanced cantilever method. At spans of this length,precast beams are not generally available to span the complete length at once. The cross-section is generally of the box type constructed either of in-situ concrete or precast segmentsof relatively short length (4–5 m longitudinally). This form of bridge is generally made ofpost-tensioned prestressed concrete.

The sequence of construction is illustrated in Fig. 1.22. An intermediate pier is cast firstand a small part of the bridge deck (Fig. 1.22(a)). This is prevented from rotation either by arigid connection between pier and deck or by construction of a temporary prop or propsconnecting the deck to the foundation as illustrated. However, either method is only capableof resisting a relatively small out-of-balance moment so it is necessary to have approximatelyequal lengths of cantilever on each side at all times during construction. Segments of deck arethen added to the base segment, either alternately on opposing sides or simultaneously in pairs,one on each side. The segments are supported by a ‘travelling form’ connected to the existingbridge (Fig. 1.22(b)) until such time as they can be permanently posttensioned into place asillustrated in Fig. 1.22(c). Ducts are placed in all segments when they are first cast, inanticipation of the need to post-tension future segments at later stages of construction.Segments can be cast in-situ or precast; in the case of

the latter, there is typically a ‘shear key’ as illustrated in Fig. 1.22(d) to provide a positivemethod of transferring shear between segments. Moment is transferred by the concrete incompression and by the post-tensioning tendons. While epoxy resin is commonly used to joinsegments, it does not normally serve any structural purpose.

Fig. 1.22 Balanced cantilever construction: (a) elevation of base segment and pier; (b) temporarysupport of segments; (c) sectional elevation showing tendon; (d) precast segment

Segments are added on alternate sides until they reach an abutment or another cantilevercoming from the other side of the span. When cantilevers meet at mid-span, a ‘stitchingsegment’ is cast to make the bridge continuous as illustrated in Fig. 1.23. Post-tensioningtendons are placed in the bottom flange and webs by means of ‘blisters’, illustrated in Fig.1.24, to resist the sagging moment that will exist in the finished structure due to applied trafficloading.

The bending moment in a balanced cantilever bridge is entirely hogging while the bridgeremains in the form of a cantilever. Thus, the moment due to self weight during constructionis such as illustrated in Fig. 1.25(a). After the casting of the stitching segments andcompletion of construction, the bridge forms a continuous beam and the imposed serviceloading generates a distribution of moment, such as illustrated in Fig. 1.25(b). This form ofbridge is quite inefficient as parts of it must be designed to resist a significant range ofmoments from large hogging to large

Fig. 1.23 Casting of stitching segment

Fig. 1.24 Blisters and tendon in the bottom flange (sectional elevation)

sagging. Nevertheless, it is frequently the most economical alternative for construction overdeep valleys when propping from below is expensive.

The analysis of balanced cantilever bridges is complicated by a creep effect similar to thatfor partially continuous beams. This is caused by a tendency for the distribution of momentdue to self weight to change in the long term from the form illustrated in Fig. 1.25(a) towardsa form approaching that illustrated in Fig. 1.25(b). This results from creep deformationswhich are still taking place after the bridge has been made continuous.

1.4.7 Continuous beam/slab—push-launch construction

For spans in excess of about 60 m, ‘incremental-launch’ or ‘push-launch’ becomes a viablealternative to balanced cantilever as a method of construction. In pushlaunch construction, along segment is cast behind the bridge abutment as illustrated in Fig. 1.26(a). Hydraulic jacksare then used to ‘push’ this segment out into the first span to make way for the casting ofanother segment behind it (Fig. 1.26(b)). This process is continued until the complete bridgehas been constructed behind the abutment and pushed into place. When the deck is beingpushed over intermediate supports, temporary sliding bearings are used to minimise frictionforces.

Fig. 1.25 Distributions of bending moment in balanced cantilever bridge: (a) due to self weightduring construction; (b) due to imposed loading after completion of construction

Fig. 1.26 Push-launch construction: (a) casting of the first segment; (b) pushing of the partiallyconstructed bridge over first span

The method has a considerable advantage of access. All of the bridge is constructed in thesame place which is easily accessible to construction personnel and plant. A significantdisadvantage stems from the distribution of bending moment generated temporarily duringconstruction. Parts of the deck must be designed for significant hog moment duringconstruction as illustrated in Fig. 1.27(a). These same parts may be subjected to sag momentin the completed bridge as illustrated in Fig. 1.27(b). The effect is greater than in balancedcantilever construction as the cantilever length is the complete span length (as opposed to halfthe span length for the balanced cantilevers). This doubling of cantilever length has the effectof quadrupling the moment due to self weight during construction. Bridges designed for push-launch construction, like those designed for balanced cantilever construction, must bedesigned for the creep effect and are subject to the associated complexity and uncertainty indesign.

1.4.8 Arch bridges

For larger spans (in excess of about 50 m), the arch form is particularly effective. However,arches generate a significant horizontal thrust, as illustrated in Fig. 1.28(a), and are only aviable solution if it can be accommodated. This can be achieved if the bridge is located on aparticularly sound foundation (such as rock). If this is not the case, an arch is still a possibilityif it is tied such as illustrated in Fig. 1.28(b). In a tied arch, the horizontal thrust is taken bythe tie. Some engineers design bridges in an arch form for aesthetic reasons but articulate thebridge like a

Fig. 1.27 Distributions of bending moment in push-launch bridge: (a) due to self weight duringconstruction; (b) due to imposed loading after completion of construction

Fig. 1.28 Arch bridges: (a) conventional form with deck over the arch; (b) tied arch with deck atbase of arch

simply supported beam, as illustrated in Fig. 1.29. This is perfectly feasible but, as the bridgehas no means by which to resist the horizontal thrust, it behaves structurally as a simplysupported beam. While traditional masonry arches were designed to be completely incompression, modern concrete or steel arches have no such restriction and can be designed toresist bending as well as the axial compression generated by the arch form.

Concrete arches are particularly effective as concrete is very strong in compression. Thearch action causes the self weight to generate a compression which has all the advantages ofprestress but none of the disadvantages of cost or durability associated with tendons. Thus theself weight generates a distribution of stress which is, in fact, beneficial and assists in theresistance of stresses due to imposed loading. Other advantages of arches are that they areaesthetically pleasing in the right environment, the structural depth can be very small andlarge clear spans can readily be accommodated. For example, while a continuous beam/slabcrossing a 60 m motorway would normally be divided into two or four spans, an arch canreadily span such a distance in one clear span creating an openness under the bridge thatwould not otherwise be possible. An additional major advantage is that arches require nobearings as it is possible to cast the deck integrally into the substructures. As can be seen inFig. 1.30, movements due to thermal expansion/contraction and creep/shrinkage do generatesome stresses but these are not as significant as those in the frame form of constructiondiscussed below.

The principal disadvantage of concrete arches, other than the problem of accommodatingthe horizontal thrust, is the fact that the curved form results in shuttering which is moreexpensive than would otherwise be the case. If arches are located over inaccessible areas,considerable temporary propping is required to support the structure during construction.

Fig. 1.29 Simply supported beam bridge in the shape of an arch

Fig. 1.30 Deflected shape of arch subjected to thermal contraction

1.4.9 Frame/box culvert (integral bridge)

Frame or box bridges, such as illustrated in Fig. 1.31, are more effective at resisting appliedvertical loading than simply supported or continuous beams/slabs. This is because themaximum bending moment tends to be less, as can be seen from the examples of Fig. 1.32.However, accommodating movements due to temperature changes or creep/shrinkage can bea problem and, until recently, it was not considered feasible to design frame bridges of anygreat length (about 20 m was considered maximum). The effects of deck shortening relative tothe supports is to induce bending in the whole frame as illustrated in Fig. 1.33. If some of thisshortening is due to creep or shrinkage, there is the usual complexity and uncertaintyassociated with such calculations. A further complexity in the analysis of frame bridges is that,unless the transverse width is relatively small, the structural behaviour is three-dimensional.Continuous slab bridges on the other hand, can be analysed using two-dimensional models.

The minimal maintenance requirement of frame/box culvert bridges is their greatestadvantage. There are no joints or bearings as the deck is integral with the piers and abutments.Given the great upsurge of interest in maintenance and

Fig. 1.31 Frame/box culvert bridges: (a) box culvert; (b) three-span frame

Fig. 1.32 Typical distributions of bending moment: (a) simply supported spans; (b) continuousbeams; (c) frames/box culverts

Fig. 1.33 Effect of thermal contraction of deck in frame bridge: (a) deflected shape; (b)distribution of bending moment

durability in recent years, this lack of maintenance has resulted in an explosion in the numbersof bridges of this form. Ever longer spans are being achieved. It is now considered thatbridges of this type of 100 m and longer are possible. There are two implications for longerframe-type bridges, both relating to longitudinal movements. If the supports are fully fixedagainst translation, deck movements in such bridges will generate enormous stresses. Thisproblem has been overcome by allowing the supports to slide as illustrated in Fig. 1.34. If thebridge is supported

Fig. 1.34 Sliding support and run-on slab in frame bridge

on piles, the axes of the piles are orientated so as to provide minimum resistance tolongitudinal movement. The second implication of longer frame bridges is that the bridgemoves relative to the surrounding ground. To overcome this, engineers specify ‘run-on’ slabsas illustrated in the figure which span over loose fill that is intended to allow the abutments tomove. The run-on slab can rotate relative to the bridge deck but there is no relative translation.Thus, at the ends of the run-on slabs, a joint is required to facilitate translational movements.Such a joint is remote from the main bridge structure and, if it does leak, will not lead todeterioration of the bridge itself.

A precast variation of the frame/box culvert bridge has become particularly popular inrecent years. Precast pretensioned concrete beams have a good record of durability and do notsuffer from the problems associated with grouted post-tensioning tendons. These can be usedin combination with in-situ concrete to form a frame bridge as illustrated in Fig. 1.35. Cross-sections are typically of the form illustrated in Fig. 1.6(b). There are a number of variations ofthis form of construction which are considered further in Chapter 4.

Fig. 1.35 Composite precast and in-situ concrete frame bridge

1.4.10 Beams/slabs with drop-in span

For ease of construction and of analysis, some older bridges were constructed of precastconcrete with drop-in spans. A typical example is illustrated in Fig. 1.36. This bridge isdeterminate as the central ‘drop-in’ part is simply supported. The side spans are simplysupported with cantilevers to which point loads from the drop-in span are applied at their ends.The form has the disadvantage of having joints and bearings at the ends of the drop-in span aswell as at the extremities of the bridge itself. However, it can readily be constructed overinaccessible areas. The drop-in span, in particular, can be placed in position very quickly overa road or railway requiring a minimum closure time. Thus, it is still popular in some countriesfor pedestrian bridges over roads.

The joint and bearing detail at the ends of the drop-in span in this form of construction isparticularly important. In older bridges of the type, two ‘halving joints’, as illustrated in Fig.1.37(a), were used. This detail is particularly problematic as access to inspect or replace thebearings is extremely difficult. A more convenient alternative, which provides access, isillustrated in Fig. 1.37(b).

Fig. 1.36 Beam bridge with drop-in span

Fig. 1.37 Halving joint at end of drop-in span: (a) traditional detail (no access); (b) alternativedetail with access

Fig. 1.38 Reinforcement detail in halving joint

However, regardless of which alternative is chosen, halving joints frequently cause difficultyfor a number of reasons:

• Even for pedestrian bridges in which de-icing salts are not used, the joints tend to leak,which promotes corrosion of the halving joint reinforcement.

• There are very high tensile and shear stresses at a point where the structural depth isrelatively small.

• As can be seen in Fig. 1.38, there can be difficulty finding space to provide sufficientreinforcement to resist all of the types of structural action that take place in the halvingjoint.

1.4.11 Cable-stayed bridgesCable-stayed construction, illustrated in Fig. 1.39, becomes feasible when the total bridgelength is in excess of about 150 m and is particularly economical for lengths in the 200–400 mrange. The maximum main span achievable is increasing all the time; the current limit is ofthe order of 1000 m. The concept of cable-stayed bridges is simple. The cables are onlyrequired to take tension and they provide support to the deck at frequent intervals. The deckcan then be designed as a continuous beam with spring supports. An analysis complication isintroduced by sag in the longer cables which has the effect of making the stiffness of thesupport provided non-linear. It is also generally necessary to carry out a dynamic analysis forbridges of such slenderness. For spans of moderate length, the cross-sections of cable-stayedbridges are often composite with steel beams and concrete slabs; for the longest spans, steelbox section decks are used to reduce the bridge self weight.

Fig. 1.39 Cable-stayed bridge

The economy of the cable-stayed form stems from its ease of construction over inaccessibleplaces. It lends itself readily to staged construction with the cables being added as required tosupport successively placed segments of the deck. As for balanced cantilever bridges,segments are placed successively on alternate sides of the pylon.

1.4.12 Suspension bridges

The very longest bridges in the world, up to about 2000 m span, are of the suspension typeillustrated in Fig. 1.40. In suspension bridges, the main cables are in catenary and the deckhangs from them applying a substantially uniform loading. They are more expensive toconstruct than cable-stayed bridges as they are not particularly suited to staged constructionand the initial placing of the cables in position is onerous. Further, it is sometimes difficult tocater for the horizontal forces generated at the ends of the cables. For these reasons, cable-stayed construction is generally favoured except for the very longest bridges.

1.5 Articulation

Bridge design is often a compromise between the maintenance implications of providingjoints and bearings and the reduction in stresses which results from the accommodation ofdeck movements. While the present trend is to provide ever fewer joints and bearings, theproblems of creep, shrinkage and thermal movement are still very real and no one form ofconstruction is the best for all situations.

The articulation of a bridge is the scheme for accommodating movements due to creep,shrinkage and thermal effects while keeping the structure stable. While this clearly does notapply to bridges without joints or bearings, it is a necessary consideration for those which do.Horizontal forces are caused by braking and traction of vehicles, wind and accidental impactforces from errant vehicles. Thus, the bridge must have the capacity to resist some relativelysmall forces while accommodating movements.

Fig. 1.40 Suspension bridge

In-situ concrete bridges are generally supported on a finite number of bearings. The bearingsusually allow free rotation but may or may not allow horizontal translation. They aregenerally of one of the following three types:

1. fixed—no horizontal translation allowed;2. free sliding—fully free to move horizontally;3. guided sliding—free to move horizontally in one direction only.

In many bridges, a combination of the three types of bearing is provided. Two of the simplestforms of articulation are illustrated in Figs. 1.41(a) and (b) where the arrows indicate thedirection in which movements are allowed. For both bridges, A is a fixed bearing allowing nohorizontal movement. To make the structure stable in the horizontal plane, guided slidingbearings are provided at C and, in the case of the two-span bridge, also at E. These bearingsare designed to resist horizontal forces such as the impact force due to an excessively highvehicle attempting to pass under the bridge. At the same time they accommodate longitudinalmovements, such as those due to temperature changes. Free sliding bearings are providedelsewhere to accommodate transverse movements. When bridges are not very wide (less thanabout 5 m), it may be possible to articulate ignoring transverse movements such as illustratedin Fig. 1.41(c).

Fig. 1.41 Plan views showing articulation of typical bridges: (a) simply supported slab; (b) two-span skewed slab; (c) two-span bridge of small width

When bridges are not straight in plan, the orientation of movements tends to radiate outwardsfrom the fixed bearing. This can be seen in the simple example illustrated in Fig. 1.42(a).Creep, shrinkage or thermal movement results in a predominantly longitudinal effect whichcauses AB to shorten by δ1 to AB'. Similarly, BC shortens by δ2 to BC'. However, as B hasmoved to B', C' must move a corresponding distance to C″. If the strain is the same in AB andBC, the net result is a movement along a line joining the fixed point, A to C. Further, themagnitude of the movement |CC″|, is proportional to the radial distance from the fixed point,|AC|. The orientation of bearings which accommodate this movement is illustrated in Fig.1.42(b). Similarly for the curved bridge illustrated in plan in Fig. 1.42(c), the movementswould be accommodated by the arrangement of bearings illustrated in Fig. 1.42(d).

Bearings are generally incapable of resisting an upward ‘uplift’ force. Further, ifunanticipated net uplift occurs, dust and other contaminants are likely to get into the bearing,considerably shortening its life. Uplift can occur at the acute corners of skewed bridges suchas B and E in Fig. 1.41(b). Uplift can also occur due to applied

Fig. 1.42 Plan views showing articulation of crooked and curved bridges: (a) movement ofcrooked bridge; (b) articulation to accommodate movement; (c) movement of curvedbridge; (d) articulation to accommodate movement

Fig. 1.43 Uplift of bearings due to traffic loading

Fig. 1.44 Uplift of bearing due to transverse bending caused by differential thermal effects

loading in right bridges if the span lengths are significantly different, as illustrated in Fig. 1.43.However, even with no skew and typical span lengths, differential thermal effects can causetransverse bending which can result in uplift as illustrated in Fig. 1.44. If this occurs, not onlyis there a risk of deterioration in the central bearing but, as it is not taking any load, the twoouter bearings must be designed to resist all of the load which renders the central bearingredundant. Such a situation can be prevented by ensuring that the reaction at the centralbearing due to permanent loading exceeds the uplift force due to temperature. If this is notpossible, it is better to provide two bearings only.

1.6 Bearings

There are many types of bearings and the choice of which type to use depends on the forcesand movements to be accommodated and on the maintenance implications. Only a limitednumber of the more commonly used types are described here. Further details of these andothers are given by Lee (1994).

1.6.1 Sliding bearingsHorizontal translational movements can be accommodated using two surfaces which are incontact but which have the capability to slide relative to one another.

Fig. 1.45 Guided sliding bearing (photograph courtesy of Ancon CCL)

This is possible due to the availability of a material with a high durability and a very lowcoefficient of friction, namely polytetrafluoroethylene (PTFE). Sliding bearings todaygenerally consist of a stainless steel plate sliding on a PTFE-coated surface. They can takemany forms and are often used in combination with other forms of bearing. In somecombinations, rotation is facilitated through some other mechanism and plane sliding surfacesare used which allow translation only. In other cases, the sliding surfaces are spherical andallow rotation; this form is also referred to as the spherical bearing. When translation is to beallowed in one direction only, guides are used such as illustrated in Fig. 1.45.

Sliding bearings offer a frictional resistance to movement which is approximatelyproportional to the vertical force. Some bearings are lubricated, resulting in a reducedcoefficient of friction. However, it is common in such systems for the lubricant to be squeezedout after a number of years, at which time the coefficient returns to the unlubricated value.Whether or not sliding bearings are lubricated, it has been suggested that they be treated aswearing parts that eventually need to be replaced.

1.6.2 Pot bearingsPot bearings, such as illustrated in Fig. 1.46, consist of a metal cylinder containing anelastomer to which the force is applied by means of a metal piston. They are frequently usedfor motorway bridges of moderate span. The elastomer effectively acts as a retained fluid andfacilitates some rotation while preventing translation. Thus, pot bearings by themselves arecommonly used at the point of fixity. They are also used in combination with plane slidingsurfaces to provide free sliding

Fig. 1.46 Pot bearing

bearings. By incorporating guides (Fig. 1.45), such a combination can also be used to form aguided sliding bearing.

1.6.3 Elastomeric bearings

When the forces to be resisted are not very high, e.g. when bearings are provided under eachbeam in precast construction, elastomeric bearings can be a very economical alternative tosliding or pot bearings. They are made from rubber and can be in a single layer (for relativelylow loading) or in multiple layers separated by metal plates. Elastomeric bearingsaccommodate rotation by deflecting more on one side than the other (Fig. 1.47(a)) andtranslation by a shearing deformation (Fig. 1.47(b)). They are considered to be quite durableexcept in highly corrosive environments and require little maintenance.

Fig. 1.47 Elastomeric bearing: (a) rotation; (b) translation

1.7 Joints

While bearings in bridges can frequently be eliminated, movements will always occur withthe result that joints will always be needed. Even in integral construction, the movement mustbe accommodated at the end of the run-on slab. However, the number of movement jointsbeing used in bridge construction is decreasing with the philosophy that all of the associatedmaintenance implications should be concentrated into as few joints as possible. Joints arenotoriously problematic, particularly in road bridges, and frequently leak, allowing salt-contaminated water to wash over the substructures.

1.7.1 Buried joint

For movements of less than 10–20 mm, joints buried beneath road surfacing are possible and,if designed well, can result in a minimum maintenance solution. A typical arrangement isillustrated in Fig. 1.48. The material used to span the joint is important; for larger gaps, it isdifficult to find a suitable material which carries the impact loading due to traffic across thegap while facilitating the necessary movement.

1.7.2 Asphaltic plug joint

The asphaltic plug joint is similar to the buried joint in that the gap is protected by roadsurfacing. However, in this case the road surfacing over the joint consists of a speciallyformulated flexible bitumen, as illustrated in Fig. 1.49. This form has been successfully usedfor movements of up to 40 mm and is inexpensive to install or replace.

Fig. 1.48 Buried joint (after Lee (1994))

Fig. 1.49 Asphaltic plug joint (after Lee (1994))

1.7.3 Nosing joint

Very popular in the 1960s and 1970s, the nosing joint, illustrated in Fig. 1.50, is no longerfavoured in many countries. It can accommodate movements of similar magnitude to theasphaltic plug joint but has a reputation for frequent failure and leakage. The nosings todayare made up of cementitious or polyurethane binders instead of the epoxy mortars popular inthe 1970s which were often found to deteriorate prematurely.

Fig. 1.50 Nosing joint (after Lee (1994))

1.8 Bridge aesthetics

The art of bridge aesthetics is a subjective one with each designer having his/her own stronglyheld opinions. However, there is generally some common ground, particularly on whatconstitutes an aesthetically displeasing bridge. Certain bridge proportions in particular, lookbetter than others and attention to this can substantially improve the appearance of thestructure. The aesthetics of the more common shorter-span bridges are considered in thissection. Further details on these and longer-span bridge aesthetics can be found in theexcellent book on the subject by Leonhardt (1984).

Some aspects of aesthetics are common to most bridges. It is generally agreed that theupstand and parapet are important and that they should be carried through from the bridge tocorresponding upstands and parapets in the abutment wing walls as illustrated in Fig. 1.51.This serves to give a sense of continuity between the bridge and its setting as the eye canfollow the line of the bridge from one end to the other. The sun tends to shine directly onupstands while the main deck tends to remain in shadow (Fig. 1.52). This effect can be useful,particularly if the designer wishes to draw attention away from an excessively deep main deck.The effect can be emphasised by casting the upstand in a whiter concrete or by casting theouter surface at an angle to the vertical as illustrated in Fig. 1.53. The depth of the upstandand the main deck relative to the span is a critical issue as will be seen in the followingsections.

Fig. 1.51 Continuity of upstand and parapet (photograph courtesy of Roughan and O’DonovanConsulting Engineers, Dublin)

Fig. 1.52 Shading of main deck relative to upstand (photograph courtesy of Roughan andO’Donovan Consulting Engineers, Dublin)

Fig. 1.53 Section through upstand

1.8.1 Single-span beam/slab/frame bridges of constant depthFor very short-span bridges or culverts, the shape of the opening has a significant influence onthe aesthetics. The abutment wing walls also play an important role as can be seen in theexample of Fig. 1.54. In this example, the shape of the opening is square (span equals height)and the abutment wing walls are large triangular

Fig. 1.54 Square opening with alternative span/upstand and span/main deck depth ratios: (a) 10and 5 with brick wing walls; (b) 20 and 5; (c) 20 and 10; (d) 10 and 5

blocks. For such a bridge the main deck can be constructed of the same material (e.g.concrete) as the abutment walls. However, it may be difficult to get a good finish with in-situconcrete and, if aesthetics are important, it may be better to clad the wing walls in masonry asillustrated in Fig. 1.54(a) while leaving the main deck and upstand in concrete.

For a square opening, a relatively deep main deck is often recommended such as one-fifthof the span. However, this clearly is a matter of opinion and also depends on the relativedepths of the main deck and the upstand. Three alternatives are illustrated in Fig. 1.54. Atypical solution is illustrated in Fig. 1.54(b) with a span/upstand depth ratio of 20 and aspan/main deck depth ratio of 5. Ratios of 20

Fig. 1.55 Rectangular opening with small wing walls: (a) slender deck and deep upstand; (b)deep deck and slender upstand

and 10 are illustrated in Fig. 1.54(c) for upstand and main deck respectively, while ratios of10 and 5 are illustrated in Fig. 1.54(d) and (a).

For a 2×1 rectangular opening with wing walls of similar size, a much more slender deck isdesirable; span/upstand depth ratios of 20 and a span/main deck depth ratio of 10 is oftenrecommended. For rectangular openings with less pronounced wing walls, an even moreslender deck is favoured. Typical ratios are illustrated in Fig. 1.55(a) with a span/upstanddepth ratio of 40 and a span/main deck depth ratio of 20. The heavier looking alternativeillustrated in Fig. 1.55(b) has ratios of 60 and 10. It can be seen that the upstand appears toothin and/or the deck too deep. Leonhardt points out that scale is important as well asproportion. This is illustrated in Fig. 1.56(a), where people and traffic are close to thestructure which is large relative to their size. (In this structure, a parapet wall is integral withthe upstand making it look deeper than necessary.) A structure with similar proportions looksmuch better in Fig. 1.56(b) as it is smaller and is more likely to be viewed from a distance.

1.8.2 Multiple spans

The relative span lengths in multi-span bridges have a significant effect on the appearance.For aesthetic reasons, it is common practice in three-span construction to have the centre spangreater than the side spans, typically by 25–35% as illustrated in Fig. 1.57. This can beconvenient as the principal obstruction to be spanned is often in the central part of the bridge.When the ground level is lower at the centre, as illustrated in the figure, this proportioningalso tends to bring the relative dimensions of the rectangular openings closer, which has agood aesthetic effect. The bridge illustrated is probably typical with a main span/upstanddepth

Fig. 1.56 The influence of scale on appearance: (a) large structure near the viewer looks heavy;(b) small structure remote from the viewer looks better than in(a)

Fig. 1.57 Three-span bridge with good proportions

Fig. 1.58 Variable depth bridges: (a) straight haunches; (b) curved alignment achieved using twocurves of differing radius; (c) curved haunches

ratio of 40 and a span/deck depth ratio of 20. As for single-span bridges, the upstand iscontinuous from end to end, effectively tying the bridge together. An open parapet is alsoused in the bridge of Fig. 1.57 to increase the apparent slenderness of the bridge.

Varying the depth of bridges allows the depth to be increased at points of maximummoment. This greatly complicates the detailing but makes for an efficient light structure andtends to look very well. When a road or rail alignment is straight, straight haunches arepossible as illustrated in Fig. 1.58(a), where the depth is increased at the points of maximum(hogging) moment. Straight haunches are considerably cheaper than curved ones, both interms of shuttering and reinforcement details. However, they are not as aesthetically pleasingas a curved profile, illustrated in Figs. 1.58(b) and (c). When alignments are curved, curveddecks are strongly favoured over straight ones.

Chapter 2Bridge loading

2.1 Introduction

For bridges, it is often necessary to consider phenomena which would normally be ignored inbuildings. For example, effects such as differential settlement of supports frequently need tobe considered by bridge designers while generally being ignored by designers of buildingstructures. These and other more common forms of bridge loading are considered in thischapter. The various types of loading which need to be considered are summarised in Table2.1. Some of these are treated in greater detail in the following sections as indicated in thethird column of the table. Other types of loading which may occur but which are notconsidered here are the effects of shrinkage and creep, exceptional loads (such as snow) andconstruction loads. Another source of loading is earth pressure on substructures. This isconsidered in Chapter 4 in the context of integral bridges. Three codes of practice are referredto in this chapter, namely, the British Department of Transport standard BD37/88 (1988), thedraft Eurocode EC1 (1995) and the American standard AASHTO (1995).

Dead and superimposed dead loads consist of permanent gravity forces due to structuralelements and other permanent items such as parapets and road surfacing. Imposed trafficloads consist of those forces induced by road or rail vehicles on the bridge. The predominanteffect is the vertical gravity loading including the effect of impact. However, horizontalloading due to braking/traction and centrifugal effects in curved bridges must also beconsidered. Where footpaths or cycle tracks have been provided, the gravity loading due topedestrians/cyclists can be significant.

Thermal changes can have significant effects, particularly in frame and arch bridges. Boththe British standard and the AASHTO treatments of temperature are somewhat tedious in thatdifferent load ‘combinations’ must be considered. For example, the AASHTO standardspecifies one combination which includes the effects of temperature, wind and imposed trafficloading. An alternative, which

Table 2.1 Summary of bridge loads

Load type Description Section1. Dead Gravity loading due to structural parts of bridge 2.2

2. Superimposed dead Gravity loading due to non-structural parts of bridge 2.2

3. Imposed traffic Loading due to road or rail vehicles 2.3

4. Pedestrian and cycletrack

Gravity loading due to non-vehicular traffic –

5. Thermal Uniform and differential changes in temperature 2.4

6. Differential settlement Relative settlement of supporting foundations –

7. Impact Impact loading due to collision with errant vehicles 2.5

8. Dynamic effects Effect of bridge vibration 2.6

9. Wind Horizontal loading due to wind on parapets, vehicles and thebridge itself

–

10. Prestress Effect of prestress on indeterminate bridges 2.7

must also be considered, excludes some thermal and wind effects but includes a higher trafficloading. The calculation is complicated by the use of different factors of safety and thespecification of different design limits for the different combinations. For example, theservice stresses permitted in prestressed concrete bridges are higher for the combinations inBD37/88 which include temperature than for combinations which do not. The draft Eurocodetreats temperature in a manner similar to other load types and applies the same method ofcombining loads as is used throughout EC1.

Differential settlement of supports can induce significant bending in continuous beam orslab bridges, as will be demonstrated in Chapter 3. The draft Eurocode on GeotechnicalDesign, EC7 (1994), recommends that the process of soil/structure interaction be taken intoconsideration for accurate analysis of problems of this type, i.e. it is recommended that acombined model of the bridge structure and the supporting soil be used to determine thestresses induced by settlement. No geotechnical guidance is given in either BD37/88 orAASHTO on how bridges should be analysed to determine the effect of this phenomenon.

The loading due to impact from collisions with errant vehicles can be quite significant forsome bridge elements. The load specified in the UK has increased dramatically in recent years.Similarly high levels of impact loading are in use in many European national standards, inAASHTO and in the draft Eurocode.

Vibration is generally only significant in particularly slender bridges. In practice, thisusually only includes pedestrian bridges and long-span road and rail bridges, where thenatural frequency of the bridge is at a level which can be excited by traffic or wind. Inpedestrian bridges, it should be ensured that the natural frequency of the bridge is not close tothat of walking or jogging pedestrians.

In addition to its ability to induce vibration in bridges, wind can induce static horizontalforces on bridges. The critical load case generally occurs when a train of high vehicles arepresent on the bridge resulting in a large vertical projected area. Wind tends not to be criticalfor typical road bridges that are relatively wide but can be significant in elevated railwayviaducts when the vertical projected surface area is large relative to the bridge width. Both theBritish and the American standards specify a simple conservative design wind loadingintensity which can be safely used in most cases. More accurate (and complex) methods arealso specified for cases where wind has a significant effect.

Prestress is not a load as such but a means by which applied loads are resisted. However, inindeterminate bridges it is necessary to analyse to determine the effect of prestress so it isoften convenient to treat prestress as a form of loading. The methods used are very similar tothose used to determine the effects of temperature changes.

2.2 Dead and superimposed dead loading

For general and building structures, dead or permanent loading is the gravity loading due tothe structure and other items permanently attached to it. In BD37/88, there is a subdivision ofthis into dead loading and superimposed dead loading. The former is the gravity loading of allstructural elements. It is simply calculated as the product of volume and material density. Forprestressed concrete bridges, it is important to remember that an overestimate of the dead loadcan result in excessive stresses due to prestress. Thus dead load should be estimated asaccurately as possible rather than simply rounded up.

Superimposed dead load is the gravity load of non-structural parts of the bridge. Such itemsare long term but might be changed during the lifetime of the structure. An example ofsuperimposed dead load is the weight of the parapet. There is clearly always going to be aparapet so it is a permanent source of loading. However, it is probable in many cases that theparapet will need to be replaced during the life of the bridge and the new parapet could easilybe heavier than the original one. Because of such uncertainty, superimposed dead load tendsto be assigned higher factors of safety than dead load.

The most notable item of superimposed dead load is the road pavement or surfacing. It isnot unusual for road pavements to get progressively thicker over a number of years as eachnew surfacing is simply laid on top of the one before it. Thus, such superimposed deadloading is particularly prone to increases during the bridge lifetime. For this reason, aparticularly high load factor is applied to road pavement.

Bridges are unusual among structures in that a high proportion of the total loading isattributable to dead and superimposed dead load. This is particularly true of long-span bridges.In such cases, steel or aluminium decks can become economically viable due to their highstrength-to-weight ratio. For shorter spans, concrete or composite steel beams with concreteslabs are the usual materials. In some cases, lightweight concrete has been successfully usedin order to reduce the dead load.

2.3 Imposed traffic loading

Bridge traffic can be vehicular, rail or pedestrian/cycle or indeed any combination of these.Vehicular and rail traffic are considered in subsections below. While pedestrian/cycle trafficloading on bridges is not difficult to calculate, its importance should not be underestimated.Bridge codes commonly specify a basic intensity for pedestrian loading (e.g. 5 kN/m2 in thedraft Eurocode and the British standard and 4 kN/m2 in the American code). When astructural element supports both pedestrian and traffic loading, a reduced intensity is allowedby some codes to reflect the reduced probability of both traffic and pedestrian loadingreaching extreme values simultaneously. Most codes allow a reduction for long footpaths.

2.3.1 Imposed loading due to road trafficWhile some truck-weighing campaigns have been carried out in the past, there has been ascarcity of good unbiased data on road traffic loading until recent years. Bridge traffic loadingis often governed by trucks whose weights are substantially in excess of the legal maximum.In the past, sampling was carried out by taking trucks from the traffic stream and weighingthem statically on weighbridges. There are two problems with this as a means of collectingstatistics on truck weights. In the first place, the quantity of data collected is relatively smallbut, more importantly, there tends to be a bias as drivers of illegally overloaded trucks quicklylearn that weighing is taking place and take steps to avoid that point on the road.

In recent years the situation has improved considerably with the advent of weigh-in-motion(WIM) technology which allows all trucks passing a sensor to be weighed while they travel atfull highway speed. WIM technology has resulted in a great increase in the availability oftruck weight statistics and codes of practice are being revised to reflect the new data.

Bridge traffic loading is applied to notional lanes which are independent of the actual lanesdelineated on the road. In the Eurocode, the road width is divided into a number of notionallanes, each 3 m wide. The outstanding road width between kerbs, after removing these lanes,is known as the ‘remaining area’. The AASHTO code also specifies notional lanes of fixedwidth. The British Standard on the other hand (for carriageway widths in excess of 5 m)allows the lane width to vary within bands in order to get an integer number of lanes withouthaving any remaining area.

The AASHTO code specifies a traffic lane loading which consists of a knife-edge load plusa uniformly distributed lane loading. Alternatively, a truck of specified dimensions and axleweights must be considered. A dynamic factor is applied to the truck to allow for theincreased stresses which result from the sudden arrival of a speeding vehicle on a bridge. Ingeneral, the imposed traffic loading specified by AASHTO is considerably less onerous thanthat specified by both BD37/88 and the Eurocode.

BD37/88 and the draft Eurocode specify two types of traffic loading, ‘normal’ and‘abnormal’. Normal traffic loading or Highway A (HA) represents an extreme

combination of overloaded trucks of normal dimensions. This could be a traffic jam involvinga convoy of very heavy trucks as would tend to govern for a long bridge. On the other hand, itcould be a chance occurrence of two overloaded moving trucks near the centre of a shortbridge at the same time, Particularly on roads with rough surfaces, there can be a considerabledynamic component of truck loading which is deemed to be included in the specified normalload. Eurocode normal loading consists of uniform loading and a tandem of four wheels ineach lane as illustrated in Fig. 2.1(a). In addition, there is uniform loading in the remainingarea. While there are a number of factors which can vary between road classes and betweencountries, the standard combination is a load intensity of 9 kN/m2 in Lane No. 1 and 2.5kN/m2 elsewhere. The four wheels of the tandems together weigh 600 kN, 400 kN and 200kN for Lanes 1, 2 and 3, respectively. In the British standard, ‘full’ HA lane loading consistsof a uniform loading whose intensity varies with the loaded length and a ‘knife edge’concentrated loading of 120 kN. For bridges with many notional lanes, a number ofpossibilities must be considered, a typical one being full HA in Lanes 1 and 2 combined with60% of full HA in the other lanes as illustrated in Fig. 2.1(b). The AASHTO code allowssimilar reductions in lane loading for multi-lane bridges to account for the reduced probabilityof extreme loading in many lanes simultaneously.

The possibility of abnormal or Highway B (HB) loading must also be considered in Britishand Eurocode designs. This consists of an exceptionally heavy vehicle of the type which isonly allowed to travel under licence from the road/bridge authority. Different countries havedifferent classes of abnormal vehicle for which bridges must be designed. A large number ofalternative abnormal vehicle classifications are specified in the draft Eurocode from whichindividual countries can select combinations for which roads of specified classes are to bedesigned. In BD37/88, only one abnormal vehicle is specified but it may have a length of 9.6,14.6, 19.6, 24.6, or 29.6 m. Illustrated in Fig. 2.2, the vehicle is known as the Highway B orHB vehicle. It is scaled in gross ‘units’ of 40 kN so that a minor road bridge can be designed,for example, to take 25 units (a 1000 kN vehicle) while a highway bridge can be designed for45 units (a 1800 kN vehicle).

Combinations of normal traffic and an abnormal vehicle must be considered in bridgedesign. While there are exceptions, the abnormal load in BD37/88 is

Fig. 2.1 ‘Normal’ road traffic loading: (a) Eurocode normal loading; (b) British standard HAloading

Fig. 2.2 British standard abnormal (HB) vehicle consisting of 16 wheel loads of F=2.5 kN perunit

generally taken to replace the normal loading throughout the length of the vehicle and for adistance of 25 m before and after it. Normal load is placed throughout the remainder of thelane and in the other lanes.

2.3.2 Imposed loading due to rail traffic

The modelling of railway loading is considerably less onerous than that of road traffic loadingas the transverse location of the load is specified. This follows from the fact that the train cangenerally be assumed to remain on the tracks. However, there are some aspects of trafficloading that are specific to railway bridges which must be considered.

The weights of railway carriages can be much better controlled than those of road vehicleswith the result that different load models are possible depending on the railway line on whichthe bridge is located. However, bridges throughout a rail network are generally designed forthe same normal load model. The standard Eurocode normal load model consists of fourvertical point loads at 1.6 m intervals of magnitude 250 kN each and uniform loading ofintensity 80 kN/m both before and after them. In addition, the Eurocode provides for analternative abnormal load model. In BD37/88, the normal load model, known as RailwayUpper (RU), is similar in format. On passenger transit ‘light rail’ systems, less onerous loadmodels can be applied. A standard light rail load model, Railway Lower (RL), is specified inthe British standard. However, less stringent models have been used for the design of bridgeson some light rail networks.

The static loads specified for the design of railway bridges must be increased to takeaccount of the dynamic effect of carriages arriving suddenly on the bridge. This factor is afunction of the permissible train speed and of the natural frequency of the bridge. Railwaytracks on grade are generally laid on ballast. On bridges, tracks can be laid on a concrete‘track slab’ or the bridge can be designed to carry ballast and the track laid on this. There aretwo disadvantages to the use of track slabs. When used, an additional vertical dynamic load isinduced by the change from the relatively ‘soft’ ballast support to the relatively hard trackslab. This effect can be minimised by incorporating transition zones at the ends of the bridgewith ballast of reducing depth. The other disadvantage to the use of track slabs depends on themethod used to maintain and replace ballast. If this is done using automatic

equipment, a considerable delay can be caused by the need to remove the equipment at thestart of the bridge and to reinstall it at the end.

Another aspect of loading specific to railway bridges is the rocking effect. It is assumed fordesign purposes that more than half of the load (about 55%) can be applied to one rail whilethe remainder (about 45%) is applied to the other. This can generate torsion in the bridge.

Horizontal loading due to braking and traction is more important in railway bridges than inroad bridges as the complete train can brake or accelerate at once. While it is possible in roadbridges for all vehicles to brake at once, it is statistically much less likely. Longitudinalhorizontal loading in bridges can affect the design of bearings and can generate bendingmoment in substructures and throughout frame bridges.

2.4 Thermal loading

There are two thermal effects which can induce stresses in bridges. The first is a uniformtemperature change which results in an axial expansion or contraction. If restrained, such as inan arch or a frame bridge, this can generate significant axial force, bending moment and shear.The second effect is that due to differential changes in temperature. If the top of a beam heatsup relative to the bottom, it tends to bend; if it is restrained from doing so, bending momentand shear force are generated.

Uniform changes in temperature result from periods of hot or cold weather in which theentire depth of the deck undergoes an increase or decrease in temperature. Both the draftEurocode and the British standard specify contour plots of maximum and minimum ambienttemperature which can be used to determine the range of temperature for a particular bridgesite. The difference between ambient temperature and the effective temperature within abridge depends on the thickness of surfacing and on the form of construction (whether solidslab, beam and slab, etc.). The American approach is much simpler. In ‘moderate’ climates,metal bridges must be designed for temperatures in the range −18 °C to 49 °C and concretebridges for temperatures in the range−12 °C to 27 °C. Different figures are specified for‘cold’ climates.

It is important in bridge construction to establish a baseline for the calculation of uniformtemperature effects, i.e. the temperature of the bridge at the time of construction. It is possibleto control this baseline by specifying the permissible range of temperature in the structure atthe time of completion of the structural form. Completion of the structural form could be theprocess of setting the bearings or the making of a frame bridge integral. In concrete bridges,high early temperatures can result from the hydration of cement, particularly for concrete withhigh cement contents. Resulting stresses in the period after construction will tend to berelieved by creep although little reliable guidance is available on how this might be allowedfor in design. Unlike in-situ concrete bridges, those made from precast concrete or steel willhave temperatures closer to ambient during construction. The AASHTO code specifies abaseline temperature equal to the mean ambient in the day preceding completion of the bridge.The British Standard and the draft Eurocode specify no baseline.

As is discussed in Chapter 4, integral bridges undergo repeated expansions and contractionsdue to daily or seasonal temperature fluctuations. After some time, this causes the backfillbehind the abutments to compact to an equilibrium density. In such cases, the baselinetemperature is clearly a mean temperature which relates to the density of the adjacent soil.

In addition to uniform changes in temperature, bridges are subjected to differentialtemperature changes on a daily basis, such as in the morning when the sun shines on the topof the bridge heating it up faster than the interior. The reverse effect tends to take place in theevening when the deck is warm in the middle but is cooling down at the top and bottomsurfaces. Two distributions of differential temperature are specified in some codes, onecorresponding to the heating-up period and one corresponding to the cooling-down period.These distributions can be resolved into axial, bending and residual effects as will beillustrated in the following examples. As for uniform changes in temperature, the baselinetemperature distribution is important, i.e. that distribution which exists when the structuralmaterial first sets. However, no such distribution is typically specified in codes, theimplication being that the distributions specified represent the differences between thebaseline and the expected extremes. Transverse temperature differences can occur when oneface of a superstructure is subjected to direct sun while the opposite side is in the shade. Thiseffect can be particularly significant when the depth of the superstructure is great.

Cracking of reinforced concrete members reduces the effective cross-sectional area andsecond moment of area. If cracking is ignored, the magnitude of the resulting thermal stressescan be significantly overestimated.

The effects of both uniform and differential temperature changes can be determined usingthe method of ‘equivalent loads’. A distribution of stress is calculated corresponding to thespecified change in temperature. This is resolved into axial, bending and residual distributionsas will be illustrated in the following examples. The corresponding forces and moments arethen readily calculated. Methods of analysing to determine the effects of the equivalent loadsare described in Chapter 3.

Example 2.1: Differential temperature I

The bridge beam illustrated in Fig. 2.3 is subjected to the differential increase in temperatureshown. It is required to determine the effects of the temperature change if it is simplysupported on one fixed and one sliding bearing. The coefficient of thermal expansion is12×10−6 and the modulus of elasticity is 35000 N/mm2.

The applied temperature distribution is converted into the equivalent stress distribution ofFig. 2.4(a) by multiplying by the coefficient of thermal expansion and the modulus ofelasticity. There is an ‘equivalent’ axial force and bending moment associated with anydistribution of temperature. The equivalent axial force can readily be calculated as the sum ofproducts of stress and area:

Fig. 2.3 Beam subject to differential temperature change

Fig. 2.4 Components of imposed stress distribution: (a) total distribution; (b) axial component;(c) bending component; (d) residual stress distribution

This corresponds to a uniform axial stress of 579600/(600× 1200)=0.81 N/mm2 as illustratedin Fig. 2.4(b). However, this beam is supported on a sliding bearing at one end and istherefore free to expand. Thus, there is in fact no axial stress but a strain of magnitude0.81/35000=23×10−6.

The equivalent bending moment is found by taking moments about the centroid (positivesag):

The corresponding extreme fibre stresses are:

as illustrated in Fig. 2.4(c). As the beam is simply supported, it is free to rotate and there is infact no such stress. Instead, a strain distribution is generated which varies linearly in the range±1.11/35 000=±32×10−6. The difference between the applied stress distribution and that whichresults in axial and bending strains is trapped in the section and is known as the residual stressdistribution, illustrated in Fig. 2.4(d). It is found simply by subtracting Figs. 2.4(b) and (c)from 2.4(a).

Example 2.2: Differential temperature II

For the beam and slab bridge illustrated in Fig. 2.5(a), the equivalent axial force, bendingmoment and residual stresses are required due to the differential temperature increases shownin Fig. 2.5(b). The coefficient of thermal expansion isαand the modulus of elasticity is E.

Fig. 2.5 Beam and slab bridge subject to differential temperature: (a) cross-section; (b) imposeddistribution of temperature

Table 2.2 Calculation of force

Block Details Force

a 3αE (2.4×0.15)= 1.080αE

b 1.890αE

c 0.150αE

d 0.100αE

Total force= 3.220αE

Fig. 2.6 Division of section into blocks: (a) cross-section; (b) corresponding imposed stressdistribution

By summing moments of area, the centroid of the bridge is found to be, below thetop fibre. The bridge is split into two halves, each of area, 0.70 m2 and second moment ofarea, 0.064 86 m4. The temperature distribution is converted into a stress distribution in Fig.2.6 and divided into rectangular and triangular blocks. The total tensile force per half is thenfound by summing the products of stress and area for each block as shown in Table 2.2.

The total force of 3.22αE corresponds to an axial tension of 3.22αE/0.70= 4.60αE.Similarly moment is calculated as the sum of products of stress, area and distance from thecentroid as outlined in Table 2.3 (positive sag). The total moment of −0.718αE corresponds tostresses (positive tension) of:

Table 2.3 Calculation of moment

Block Details Momenta −0.262αE

b −0.506αE

c −0.012αE

d 0.062αE

Total moment= −0.718αE

Fig. 2.7 Resolution of stress distribution into axial, bending and residual components: (a) totaldistribution; (b) axial component; (c) bending component; (d) residual stressdistribution

Hence the applied stress distribution can be resolved as illustrated in Fig. 2.7. The residualdistribution is found by subtracting the distributions of Figs. 2.7(b) and (c) from the applieddistribution of Fig. 2.7(a).

2.5 Impact loading

Most bridge analysis is based on static linear elastic principles. However, the collision of avehicle with a bridge is highly non-linear. To overcome the resulting complications, codes ofpractice often greatly simplify the procedure by specifying equivalent static forces. Thissection considers the basis on which these forces are derived.

The simple case illustrated in Fig. 2.8 is considered first. An undeformable sphere of mass,m, travelling at a velocity, v, collides with a spring of stiffness, K. The kinetic energy of thesphere is:

(2.1)

On impact, this is converted into strain energy in the spring. A static force, Peq, which causesa deflection, Δ, generates a strain energy of:

(2.2)

Fig. 2.8 Impact of undeformable sphere with spring

Hence the equivalent static force is:

(2.3)

For a spring of stiffness, K, a force Peq generates a deflection:

(2.4)

Substituting for Δ in equation (2.3) gives an alternative expression for Peq:

(2.5)

Substituting for Ek in this equation gives the equivalent force in terms of mass and velocity:

(2.6)

While this is a very simple case, it can be used as a basis for determining equivalent staticforces. The mechanics of a collision between a vehicle and a structure are quite complex.Further, a small difference in the impact location or the impact angle can result in asubstantial change in the effect. For these reasons, the situation is simplified by treating thevehicle as undeformable and the structural element as a spring. It follows from theseassumptions that a vehicle with kinetic energy, Ek, will generate the equivalent force given byequation (2.5) on the outer surface of a structural element. A table of design static forces isspecified in the draft Eurocode based on the expected masses and velocities of trucks on roadsof various class. Similar equivalent static loadings are specified in the AASHTO standard andin BD37/88.

On bridge piers, the draft Eurocode specifies that the impact force due to a truck be appliedat a specified height above the road surface. An impact force is also specified for a derailedtrain colliding with a pier. On bridges over road carriageways, there is a possibility that truckspassing underneath will collide with the bridge deck. However, because only the top of thevehicle is likely to impact on the bridge, a substantial reduction factor applies. It is notnecessary, in the draft Eurocode, to consider collision of trains with bridge decks overhead.

2.6 Dynamic effects

Vibration can be a problem in slender bridges where the natural frequency is at a level whichcan be excited by wind or traffic. Such a possibility can be investigated by means of adynamic analysis. In dynamics, mass has a significant effect on the response of the structureto a given load and computer models must incorporate a representation of mass as well asstiffness. This is frequently done by ‘lumping’ the distributed mass of a bridge at a finitenumber of nodes. For example, the simply supported beam bridge of Fig. 2.9(a) could berepresented by the lumped mass

model of Fig. 2.9(b). All structures have a number of natural frequencies at which they tend tovibrate. If the bridge of Fig. 2.9 is excited, by wind, say, it may vibrate at one of thesefrequencies, as illustrated in Fig. 2.10. The shape of the structure during such vibration isknown as the mode shape.

In the simplest form of dynamic analysis, the source of excitation of the bridge is notconsidered and only the natural frequencies and mode shapes are determined. If it can beshown that the natural frequencies of the bridge are not close to the frequency of all expectedsources of excitation, there may be no need for further dynamic analysis. Common forms ofexcitation are truck vibration, wind, and jogging or walking pedestrians. Even when thefrequencies are not close, as would often be the case for a road bridge excited by traffic, asuddenly applied load generates significantly more stress than a statically applied one.However, the equivalent static loads specified in codes of practice take account of thisphenomenon and incorporate a ‘dynamic amplification’ factor. Dynamic amplification can bedefined as the ratio of the actual stress to that due to the corresponding static load.

If the excitation frequency is close to one of the natural frequencies of the bridge, furtheranalysis is required to determine the dynamic amplification in what is known as a ‘forcedvibration’ analysis. In such an analysis, the interaction of the

Fig. 2.9 Idealisation of beam for dynamic analysis: (a) original beam; (b) lumped mass model

Fig. 2.10 Mode shapes of simply supported beam: (a) typical first mode shape; (b) typical secondmode shape

applied loading and the bridge is taken into account. In the case of road traffic, the appliedloading is a truck or trucks of considerable mass, vibrating on their own tyres and suspensions.Furthermore, the trucks are moving so the location of their masses are changing with time.Such an analysis is currently only possible with specialist computer programs whichincorporate the complexities of truck rocking and bouncing motions and the variations intruck dynamic characteristics which may be expected in typical traffic.

In Section 2.5, design for the impact of vehicles colliding with bridges was discussed. Forsuch cases, the draft Eurocode specifies an equivalent static force. As an alternative, the codeallows for the carrying out of a dynamic analysis. In such cases, the equivalent force isassumed to increase from zero to its full value over a very short time (measured inmilliseconds) and to maintain a constant value for a further short time (of the order of 200 ms).Such a loading can readily be specified in a computer model and the maximum distribution ofstress determined.

2.7 Prestress loading

While prestress is not in fact a loading as much as a means of resisting load, it is oftenconvenient to treat it as a loading for analysis purposes. Like temperature, prestress can behandled using the method of equivalent loads. Such a method is only necessary in the case ofindeterminate bridges. However, even for simply supported slab or beam-and-slab bridges, itis often necessary to analyse to determine the degree to which prestressing of one memberaffects others. Whether the bridge consists of beams or a slab, equivalent loadings can befound for individual tendons. The combined effect of a number of tendons can then be foundby simply combining the loadings.

Examples of analysis using equivalent prestress loads are given in Chapter 3. In this section,methods will be given for the calculation of their magnitudes. For a qualitative understandingof the effects of prestress, the concept of linear transformation is also introduced.

2.7.1 Equivalent loads and linear transformationThe equivalent loading due to prestress can generally be found by simple equilibrium offorces. For example, for the externally prestressed bridge illustrated in Fig. 2.11(a),equilibrium of vertical forces gives an upward force at B of:

As the angle, θ, is generally small, this can be approximated as:

(2.7)

It also follows from the small angle that the horizontal force is P cosθ ≈P. Finally, as theforces are eccentric to the centroid at the ends, there are concentrated

Fig. 2.11 Prestressed concrete beam with external post-tensioning: (a) elevation showing tendon;(b) equivalent loading due to prestress

moments there of magnitude (Pcosθ)e2≈Pe2. Hence the total equivalent loading due toprestress is as illustrated in Fig. 2.11(b). It can be shown that the equivalent loading due toprestress is always self-equilibrating.

A parabolically profiled prestressing tendon generates a uniform loading which again canbe quantified using equilibrium of vertical forces. A small segment

Fig. 2.12 Segment of parabolically profiled tendon: (a) elevation; (b) equivalent loading

of such a profile is illustrated in Fig. 2.12(a). At point 1, there is an upward verticalcomponent of the prestress force of:

(2.8)

As the angles are small:

(2.9)

where x1 is the X coordinate at point 1. This force is upwards when the slope is positive.Similarly the vertical component of force at 2 is:

(2.10)

where F2 is downwards when the slope is positive. The intensity of uniform loading on thissegment is:

(2.11)

The equivalent loads on the segment are illustrated in Fig. 2.12(b).

Example 2.3: Parabolic profile

The beam illustrated in Fig. 2.13 is prestressed using a single parabolic tendon set outaccording to the equation:

(2.12)

where s is referred to as the sag in the tendon over length l as indicated in the figure. It is

required to determine the equivalent loading due to prestress.

Fig. 2.13 Beam with parabolic tendon profile: (a) elevation; (b) equivalent loading due toprestress

Differentiating equation (2.12) gives:

(2.13)

As θA is small:

For a positive slope, the equivalent point load at A would be upwards and of magnitudeP(eB−eA−4s)/l. However, in this case, the slope is negative and the force is downwards ofmagnitude P(−eB+eA+4s)/l.

The slope at B is calculated similarly:

As B is on the right-hand side, this force is downwards when positive. Hence, the equivalentpoint loads are as illustrated in Fig. 2.13(b). The intensity of uniform loading is given byequation (2.11) where the second derivative is found by differentiating equation (2.13):

(2.14)

This too is illustrated in the figure.

Example 2.3 illustrates the fact that the intensity of equivalent uniform loading due to aparabolic tendon profile is independent of the end eccentricities. A profile such as thatillustrated in Fig. 2.13(a) can be adjusted by changing the end eccentricities, eA and eB whilekeeping the sag, s, unchanged. Such an adjustment is known as a linear transformation andwill have no effect on the intensity of equivalent uniform loading as can be seen fromequation (2.14). This phenomenon is particularly useful for understanding the effect ofprestressing in continuous beams with profiles that vary parabolically in each span.

Example 2.4: Qualitative profile design

A prestressed concrete slab bridge is to be reinforced with 10 post-tensioned tendons. Thepreliminary profile for the tendons, illustrated in Fig. 2.14(a), results in insufficientcompressive stress in the top fibres of the bridge at B. It is required to determine anamendment to the profile to increase the stress at this point without increasing the prestressforce.

In a determinate structure, stress at the top fibre can be increased by moving theprestressing tendon upwards to increase the eccentricity locally. This increase in tendoneccentricity, e, increases the (sagging) moment due to prestress, Pe, which increases thecompressive stress at the top fibre. However, in an indeterminate structure, the response of astructure to such changes is not so readily predictable. In the structure of Fig. 2.14, increasingthe eccentricity locally at B without changing the sags, as illustrated in Fig. 2.14(b), does littleto increase the compressive stress at the top fibre at that point. This is because the eccentricityat B has been increased without increasing the tendon sag in the spans. As was seen above,the equivalent uniform loading due to prestress is a function only of the sag and is, in fact,unaffected by eccentricity at the ends of the span. Thus, the change only results in adjustmentsto the equivalent point loads at A and B and to the equivalent loading near B. As these forcesare at or near supports, they do not significantly affect the distribution of bending momentinduced by prestress. A more appropriate revision is illustrated in Fig. 2.14(c) where theprofile is lowered in AB and BC while maintaining its position at the support points. This hasthe effect of increasing the tendon sag which increases the intensity of equivalent uniformloading. Such a uniform upward loading in a two-span beam generates sagging moment at theinterior support which has the desired effect of increasing the top-fibre stress there.

Fig. 2.14 Adjustment of tendon profile: (a) original profile; (b) raising of profile at B by lineartransformation; (c) lowering of profile in AB and BC to increase sag

Most prestressing tendons are made up of a series of lines and parabolas and the equivalentloading consists of a series of point forces and segments of uniform loading. This can be seenin the following example.

Example 2.5: Tendon with constant prestress force

A three-span bridge is post-tensioned using a five-parabola symmetrical profile, half of whichis illustrated in Fig. 2.15(a). It is required to determine the equivalent loading due to prestressassuming that the prestress force is constant throughout the length of the bridge.

The intensities of loading are found from equation (2.11). For the first parabola:

Fig. 2.15 Tendon profile for Example 2.5: (a) partial elevation showing segments of parabola; (b)equivalent loading due to prestress

Similarly, the intensities of loading in the second and third parabolas are respectively:

and

The point load at the end support is the vertical component of the prestress force.Differentiating the equation for the parabola gives the slope, from which the force is found tobe:

All of the equivalent loads due to prestress are illustrated in Fig. 2.15(b). Verifying thatthese forces are in equilibrium can be a useful check on the computations.

Note that in selecting the profile, it has been ensured that the parabolas are tangent to oneanother at the points where they meet. This is necessary to ensure that the tendon does notgenerate concentrated forces at these points.

2.7.2 Prestress losses

In practical post-tensioned construction, prestress forces are not constant through the length ofbridges because of friction losses. This is illustrated in Fig. 2.16(a)

Fig. 2.16 Equivalent loading due to varying prestress force: (a) segment of beam and tendon; (b)equivalent loading (Pav=(P1+P2)/2)

where the forces at points 1 and 2 are different. However, the difference between prestressforces at adjacent points is generally not very large. Therefore, a sensible approach to thederivation of equivalent prestress loading is to start by substituting the average prestress forcefor P in equations (2.9)–(2.11). The resulting loading is illustrated in Fig. 2.16(b). It will beseen in Example 2.6 that this equivalent loading satisfies equilibrium of forces and moments.

The use of equivalent loads which do not satisfy equilibrium can result in significant errorsin the calculated distribution of prestress moment. A useful method of checking the equivalentloads is to apply them in the analysis of a determinate beam. In such a case, the moment dueto the equivalent loading should be equal to the product of prestress force and eccentricity atall points.

Example 2.6: Tendon with varying prestress force

The post-tensioning tendon of Example 2.5 is subject to friction losses which result in theprestress forces presented in Fig. 2.17. The eccentricities given in this figure have beencalculated from the equations for each parabola given in Example 2.5. It is required todetermine the equivalent loading due to prestress taking account of the loss of force. Thebridge is post-tensioned from both ends with the result that the prestressing forces varysymmetrically about the centre.

Fig. 2.17 Tendon profile showing varying prestress force (in kN) and eccentricity (in m)

With reference to Example 2.5 but using average prestress forces, the equivalent intensities ofuniform loading are:

In addition, point loads must be applied at the end of each segment in accordance with Fig.2.16(b). In segment AB, the equation for the parabola is:

At A, x=0, the slope is −0.1322 and the upward force is:

Fig. 2.18 Equivalent loading due to prestress: (a) loading on each segment; (b) total

the minus sign indicating that the force is actually downwards. At B, the slope of the profileis:

giving a downward force at the right end of magnitude:

The corresponding point load components for the other segments of parabola are calculatedsimilarly and are presented, together with the other equivalent uniform loads, in Fig. 2.18(a).It can be verified that the forces and moments on each segment are in equilibrium. The forcesand moments at the ends of each segment are summed and the result is illustrated in Fig.2.18(b).

2.7.3 Non-prismatic bridgesThe eccentricity of a prestressing tendon is measured relative to the section centroid. In non-prismatic bridge decks, the location of this centroid varies along the length of the bridge. Thisclearly affects the eccentricity and hence the moment due to prestress.

A segment of beam with a curved centroid is illustrated in Fig. 2.19(a). In such a beam, theprestress forces are resolved parallel and perpendicular to the centroid and the eccentricity ismeasured in a direction perpendicular to it. The resulting equivalent loading is illustrated inFig. 2.19(b) where s is distance along the centroid.

Fig. 2.19 Equivalent loading due to variation in location of centroid: (a)segment of beam andtendon; (b) equivalent loading

Example 2.9: Equivalent loading due to change in geometry

The beam illustrated in Fig. 2.20 has a non-prismatic section; the centroid changes depthlinearly between A and B and between B and C. It is prestressed with a tendon following asingle parabolic profile from A to C. In addition, there are friction losses of 12% which varylinearly between A and C (friction losses generally do not vary linearly but this is a widelyaccepted approximation). It is required to determine the equivalent loading due to prestress.

The beam is divided into just two segments, AB and BC. The definition of the parabola isindependent of the section geometry. With reference to Example 2.3, it is defined by anequation of the same form as equation (2.12), i.e.:

Fig. 2.20 Elevation of beam and tendon profile

If the eccentricity is approximated as the vertical distance, it can be found as the differencebetween y and the line representing the centroid. Hence for segment AB:

Similarly for segment BC, the eccentricity is given by:

Differentiating the equation for segment AB gives:

Similarly, for BC, the derivatives are:

Fig. 2.21 Equivalent loading: (a) loading on each segment; (b) total

Differentiating again gives, for both segments:

The average values for prestress force in segments AB and BC are 0.97P and 0.91Prespectively, where P is the jacking force. The resulting equivalent loading due to prestress isillustrated for each segment in Fig. 2.21(a). The forces are combined in Fig. 2.21(b).

Chapter 3Introduction to bridge analysis

3.1 Introduction

Two approaches to bridge analysis are presented in this chapter, moment distribution and themethod of equivalent loads. Moment distribution is a convenient hand method that can beused in many cases. It is, of course, not practical in most situations to analyse bridges by hand.However, a knowledge of such methods is extremely useful for developing a completeunderstanding of the nature of bridge behaviour under load. Moment distribution has beenselected as there is a physical action corresponding to each stage of the calculation whichmakes it easier to develop a qualitative understanding of the phenomena. The method is alsouseful for checking computer output as it provides approximations of increasing accuracythroughout the analysis process.

In addition to moment distribution, the method of equivalent loads is presented as a meansof analysing for the effects of ‘indirect actions’, i.e. actions other than forces that can inducestress in a bridge. The method consists of determining loads which have the same effect onthe structure as the indirect action. Analysis for the equivalent loads can be carried out byconventional computer methods or by moment distribution.

3.2 Moment distribution

Moment distribution can be used to check computer output and to develop insight into thebehaviour of a great range of bridge types subjected to many different types of action. In thissection, the method is illustrated using some simple examples.

The approach to moment distribution used in this book is a little different in its presentationto that used traditionally. The process of releasing joints, familiar to most engineers, isperformed not by adding numbers in a table but rather by adding bending moment diagrams.This may be slower to perform in practice but provides a much clearer explanation of theprocess and is less prone to error.

The analysis procedure consists of four steps. These are presented in the left-hand columnof Table 3.1 and are illustrated using the example presented in the right-hand column.

Table 3.1 Moment distribution

General Example

The 3 members are isolated by applying fixities at B and C as shown:

Step 1: All members of thestructure are isolated fromone another by applying anumber of fixities. Thefixities are numbered andthe direction of each isdefined. The bendingmoment diagram (BMD)due to the applied loadingon the resulting ‘fixed’structure is sketched.Appendix A gives theBMDs for members with arange of end conditions.

This fixed structure is equivalent to:

The resulting bending moment diagram (BMD) is found (with reference toAppendix A):

General ExampleUnit rotation at B induces a BMD of (refer to Appendix B):Step 2: The BMDs due

to application of unitdisplacements at eachof the fixities arefound. Appendix Bgives the BMDs for awide range of suchdisplacements. TheseBMDs are thennormalised to give aunit value at each pointof momentdiscontinuity.

The total discontinuity at B is 3EI/l+4EI/(1.25l)=6.2EI/l. Dividing the BMDby this gives the normalised version, boxed below, i.e. a BMD with a unitdiscontinuity at B which results from some applied rotation at B:

The corresponding BMD for rotation at C is found similarly:

General ExampleThe moment just left of B in the fixed BMD (1) of 0.125wl2 is lessthan that just right of B (0.130wl2) by 0.005wl2. This discontinuity isremoved by adding the normalised BMD corresponding to rotation atB (2), factored by 1.005wl2. The resulting BMD is:

Step 3: In the fixed BMD(Step 1), there is generally alack of equilibrium of bendingmoment at the fixing points,often characterised bydiscontinuities in the BMD. Inthis step, such discontinuitiesare successively removed byapplying ‘rotations’. This isperformed simply by addingor subtracting the normalisedbending moment diagrams,scaled in each case by theappropriate discontinuity.

The discontinuity at C is now (0.131–0.125)wl2=0.006wl2. This isremoved by subtracting the normalised BMD corresponding torotation at C (3), factored by 0.006wl2 (the BMD is subtracted, asadding it would increase the discontinuity). The resulting BMD is:

Step 4: The process describedin Step 3 is repeated until aBMD is arrived at in whichequilibrium is satisfiedeverywhere. This is the finalsolution.

The correction of the discontinuity at B had the effect of increasing thediscontinuity at C. Similarly, the correction of the discontinuity at Chad the effect of reintroducing a discontinuity at B. Hence, to get anexact answer, the process of adding normalised BMDs, factored by thediscontinuities, must be repeated until no discontinuity remains. Forthis particular example, the first iteration has resulted in a BMD whichis sufficiently accurate for most practical purposes.

Example 3.1: Continuous beam using symmetry

Concepts of symmetry can be used to great effect when analysing by moment distribution. Aswill be demonstrated in this example, it is possible to isolate members from each other by thesimultaneous application of a pair or pairs of equal and opposite fixities.

The beam of Table 3.1 is analysed again, this time using symmetry. The beam is fixedsimultaneously at B and C as illustrated in Fig. 3.1(a). The arrows indicate the directions ofpositive rotation for Step 2. The fixed bending moment diagram (BMD) (Step 1) is unaffectedby the symmetric system of fixities but, in Step 2, two equal and opposite rotations must beapplied simultaneously at B and C, as illustrated in Fig. 3.1(b). The resulting BMD (from ) isillustrated in Fig. 3.1(c) and the normalised version, which gives a unit discontinuity ofmoment at B and C, is illustrated in Fig. 3.1(d) (boxed). The discontinuities at B and C in thefixed BMD are, as before, 0.005wl2. Hence, Step 3 consists of removing these twodiscontinuities (simultaneously) by adding the BMD of Fig. 3.1(d), scaled by 0.005wl2. Theresulting BMD is illustrated in Fig. 3.2. As there are no further discontinuities, no iteration isrequired for this example and the BMD of Fig. 3.2 is, in fact, exact.

Fig. 3.1 Moment distribution using symmetry: (a) symmetrical system of fixities; (b) unitrotation simultaneously at B and C; (c) BMD due to unit rotation; (d) normalisedBMD

Fig. 3.2 Final BMD for three-span beam

Example 3.2: Box culvert

The application of moment distribution to a two-dimensional frame type of structure isdemonstrated using the box culvert illustrated in Fig. 3.3. For simplicity, this culvert isassumed to be supported at two discrete points under the walls and to have constant flexuralrigidity throughout.

Step 1: The members are isolated by applying fixities at A, B, C, and D as illustrated in Fig.3.4(a). Symmetry is exploited by simultaneously fixing A and B and simultaneously fixing Cand D as shown. The fixed BMD is, from , as illustrated in Fig. 3.4(b).

Step 2: Applying unit rotation simultaneously at A and B (Fig. 3.4(c)) results in the BMDillustrated in Fig. 3.4(d). When normalised, this becomes the BMD of Fig. 3.4(e). Thenormalised BMD due to rotation at C and D is found similarly and is as illustrated in Fig.3.4(f).

Step 3: The discontinuity at A and B in the fixed BMD (Fig. 3.4(b)) is Pl/8. This iscorrected by applying the BMD of Fig. 3.4(e), factored by Pl/8, which gives the BMD of Fig.3.5(a). The discontinuity now present at C and D is 0.4(Pl/8). This is corrected by adding theBMD of Fig. 3.4(f), factored by that amount. The resulting BMD is illustrated in Fig. 3.5(b).

Fig. 3.3 Box culvert example

Fig. 3.4 Analysis of box culvert (a) system of fixities; (b) fixed BMD; (c) moments required toinduce unit rotation at A and B; (d) BMD associated with unit rotation at A and B; (e)normalised BMD for rotation at A and B; (f) normalised BMD for rotation at C and D

Fig. 3.5 BMD after successive corrections: (a) after correction of discontinuity at A and B; (b)after correction of discontinuity at C and D; (c) a after second correction at A and B;(d) after second correction at C and D

Step 4: The correction at C and D has reintroduced a discontinuity at A and B of 0.16(Pl/8).Adding the BMD of Fig. 3.4(e), factored by this amount, gives the BMD of Fig. 3.5(c).Finally, the BMD of Fig. 3.4(f), factored by 0.064(Pl/8), is added to give the BMD of Fig.3.5(d). The discontinuity now existing at A and B is considered to be sufficiently small for thepurposes of this example and the BMD of Fig. 3.5(d) is deemed to be the final solution.

This box culvert of Example 3.2 was assumed to be supported at two discrete points. A moretypical situation would be that of continuous support from granular material throughout thelength of the base and side walls. A more realistic finite-element (FE) model taking accountof these effects and assuming typical soil properties is illustrated in Fig. 3.6(a) and theresulting BMD in Fig. 3.6(b). A higher hogging moment (0.825(Pl/8)) is found with acorresponding reduced sagging moment. Earth pressure on a structure of this type generatesan additional distribution of moment. The interaction of bridges with the surrounding soil isconsidered further in Chapter 4.

Fig. 3.6 Finite-element model of box culvert and surrounding soil: (a) finite-element mesh; (b)resulting BMD

3.3 Differential settlement of supports

There is considerable research and development activity currently taking place in the field ofsoil/structure interaction. Clearly soil deforms under the vertical forces applied through bridgepiers and abutments. If the deformation is not uniform, distributions of bending moment andshear are induced in the deck. To accurately analyse for this effect, the structure and thesurrounding soil may be represented using non-linear computer models. However, as theeffect is often not very significant, many structural engineers treat the soil as a spring or aseries of springs in the numerical model. The disadvantage of this is that differentialsettlement is more often caused by a relatively weak patch of soil under one support ratherthan by a non-uniform distribution of applied loads. Thus, an alternative approach, frequentlyadopted by bridge engineers, is to assume that a foundation support settles by a specifiedamount, Δ, relative to the others and to determine the effects of this on the structure. The following example serves to demonstrate the effect of a differential settlement on acontinuous beam bridge.

Example 3.3: Differential settlement by moment distribution

The continuous beam illustrated in Fig. 3.7 is subjected to a settlement at B of Δ relative tothe other supports. The resulting BMD is required given that the beam has uniform flexuralrigidity, EI. The system of fixities cannot be symmetrical as the ‘loading’ is not symmetrical.Hence, the beam is fixed as illustrated in Fig. 3.8(a).

Step 1: Referring to (4th and 5th BMDs), the fixed BMD is as illustrated in Fig. 3.8. Thediscontinuity of moment at B is 1.5EI∆/l2. By coincidence, the discontinuity at C is of thesame magnitude.

Step 2: Applying a unit rotation as illustrated in Fig. 3.9(a) results in the BMD illustrated inFig. 3.9(b). When normalised, this becomes the BMD of Fig. 3.9(c). Similarly, a rotation at Cresults in the normalised BMD of Fig. 3.9(d).

Fig. 3.7 Three-span beam example: (a) geometry; (b) imposed support settlement

Fig. 3.8 First step in analysis of three-span beam: (a) system of fixities; (b) fixed BMD

Fig. 3.9 Effect of rotations at points of fixity: (a) unit rotation at B; (b) BMD associated with unitrotation at B; (c) normalised version of BMD associated with rotation at B; (d)normalised BMD associated with rotation at C

Fig. 3.10 BMD after successive corrections: (a) after correction at1; (b) after correction at 2; (c)after second corrections at1 and 2

Step 3: The discontinuity at B in Fig. 3.8(b) is 1.5 EI∆/l2, Adding the BMD of Fig. 3.9(c),factored by this amount, gives the BMD of Fig. 3.10(a). This correction at B has the effect ofincreasing the discontinuity at C to 1.8 EI∆/l2. This is corrected by adding the BMD of Fig.3.9(d) factored by this amount to give the BMD of Fig. 3.10(b).

Step 4: One further iteration gives the BMD of Fig. 3.10(c) which is deemed to be ofsufficient accuracy.

Differential settlement has the effect of generating sagging moment at the support whichsettles. This is important as supports in continuous beams are generally subjected to hoggingmoment and are often not designed to resist significant sag. It is interesting to note twoadditional things about the final BMD illustrated in Fig. 3.10(c), which are typical ofdifferential settlement:

1. The moment at the support which settles is proportional to the second moment of area, I,divided by the square of the span length, l. It is usual to size a bridge by selecting a depthwhich is proportional to span length (i.e. depth=l/k for some constant, k). As the secondmoment of area is proportional to the cube of

the depth, the maximum moment due to differential settlement is roughly proportional to(l/k)3/l2=l/k3, i.e. moment is proportional to span length. The implication of this is that, for agiven settlement Δ, the induced moment is more critical for bridges with long spans than for those with short ones. Further, the span/depth ratio is particularly important; a modestincrease in slenderness can considerably reduce the moment due to differential settlement.It might be expected that for longer spans, the differential settlement should be larger as thesupports are further apart and soil conditions are more likely to be different. However, inpractice, values are often specified which are independent of span length.

2. Unlike BMDs due to applied forces, the distribution of moment due to differentialsettlement is proportional to the elastic modulus. This is particularly significant for concretebridges where considerable creep occurs. A widely accepted approximate way to model theeffect of creep is to reduce the elastic modulus. As moment is proportional to this modulus,it follows that creep has the effect of reducing the moment due to differential settlementover time.

This beneficial effect of the creep in concrete is countered by the fact that themagnitude of the differential settlement itself often increases with time due to time-dependent behaviour in the supporting soil. However, if the specified settlement isdeemed to include such time-dependent effects, it is reasonable to anticipate somereduction in moment due to concrete creep.

3.4 Thermal expansion and contraction

As discussed in Chapter 2, there are two thermal effects for which bridge analysis is required,namely, axial expansion/contraction and differential changes in temperature through the depthof the bridge deck. In this section, analysis for the effects of axial expansion/contraction dueto temperature changes is considered.

If a beam is on a sliding bearing as illustrated in Fig. 3.11(a) and the temperature is reducedby ΔT, it will contract freely. A (negative) strain will occur of magnitudeα(∆T) where αis thecoefficient of thermal expansion (strain per unit change in temperature). The beam thencontracts by α(∆T)l where l is its length. However, no stresses are generated as no restraint isoffered to the contraction. As there is no stress, there can be no tendency to crack. If, on theother hand, the beam is fixed at both ends as illustrated in Fig. 3.11(b), and its temperature isreduced by ΔT, then there will be no strain. There cannot be any strain as the beam is totallyrestrained against contraction. This total restraint generates a stress of magnitude Eα(∆T),where E is the elastic modulus. The stress is manifested in a tendency to crack.

Fig. 3.11 Extreme restraint conditions For axial temperature: (a) free; (b) fully fixed

The most common case requiring analysis is the one in between the two extreme casesdescribed above, where a beam is partially restrained. This happens for example in archbridges where contraction is accommodated through bending in the arch (Fig. 1.30). It alsohappens in frame bridges where the piers offer some resistance to expansion or contraction ofthe deck.

Example 3.4: Restrained axial expansion by moment distribution

For the bridge illustrated in Fig. 3.12(a), it is required to find the bending moment, shear forceand axial force diagrams due to an increase in deck temperature of ΔT.

Fig. 3.12 Frame subjected to axial change in temperature: (a) original geometry; (b) deformedshape after expansion of deck

Fig. 3.13 First step in analysis of frame: (a) fixing system; (b) fixed axial force diagram

The deck is supported on a bearing at B which prevents relative translation between it and thesupporting pier but allows relative rotation. Thus, a thermal expansion tends to bend the pieras illustrated in Fig. 3.12(b). As the pier is fully fixed at its base, its resistance to bendingrestrains the expansion a little and generates a small compressive stress in the deck between Aand B. In addition, bending moment is generated in the pier, BD.

Fig. 3.14 Effect of translation at fixing point: (a) forces required to induce unit translation; (b)associated axial force diagram; (c) associated shear force diagram; (d) free bodydiagram showing lack of equilibrium of forces at B; (e) normalised free body diagram

Step 1: The substructure and superstructure of the bridge are isolated from one another by theimposition of a translational fixity at B as illustrated in Fig. 3.13(a). (While a rotational fixityat this point is also possible, such a fixity is not necessary to isolate the members in this case.)There is no bending moment or shear force in the fixed structure. However, the stress in AB isα(∆T)E, whereαis the coefficient of thermal expansion and E is the elastic modulus. Thecorresponding force is α(∆T)E(area)=6000α(∆T)EI/h2. Hence the axial force diagram is asillustrated in Fig. 3.13(b).

Step 2: To apply a unit translation at B requires a force to compress AB ofE(area)/(length)=1500EI/h3 as illustrated in Fig. 3.14(a). In addition, a force is required tobend BD (, 4th case) of 3EI/h3 giving a total required force at B of 1503EI/h3. The associatedaxial force and shear force diagrams are illustrated in Figs. 3.14(b) and (c). The requiredexternal force at B can be seen in the free-body diagram of Fig. 3.14(d). The lack of forceequilibrium in this diagram corresponds to the moment discontinuity in the BMDs of theusual moment distribution problems. The normalised version of Fig. 3.14(d), corresponding tounit discontinuity of force at B, is illustrated in Fig. 3.14(e).

Fig. 3.15 Results of analysis: (a) free body diagram with restored equilibrium at B; (b) axialforce diagram; (c) shear force diagram; (d) bending moment diagram

Step 3: There is a lack of force equilibrium in the fixed structure at B (Fig. 3.13(b)) as there isno axial force in BC and no shear force in the pier to correspond to the axial force in AB. Thissituation is corrected at B by subtracting the forces illustrated in Fig. 3.14(e) factored by6000α(∆T)EI/h2, i.e. by adding an axial tension in AB of 0.998×6000α(∆T)EI/h2 and a shearforce in BC of 0.002×6000α(∆T)EI/h2 to Fig. 3.13(b). The result is illustrated in Fig. 3.15(a).As there is no further force discontinuity, this is the final free body diagram. The final axialforce and shear force diagrams are illustrated in Figs. 3.15(b) and (c). As the shear forceacross the pin at B is 12α(∆T)EI/h2, the bending moment in BD varies from zero at this pointto a maximum of 12α(∆T)EI/h at D as illustrated in Fig. 3.15(d).

There are some points of interest about axial temperature effects apparent from this simpleexample. Most noteworthy is the effect of the relative values of deck area and pier secondmoment of area. The area of the deck is typically numerically much larger than the secondmoment of area of the pier with the result that the restraint to deck expansion is relativelysmall. Hence the rise in temperature results in a lot of strain and in very little stress in thedeck. It is also of interest to note that, as for differential settlement, the moments and forcesdue to changes in temperature are proportional to the elastic modulus. This means that suchstresses, if sustained in a concrete structure, may be relieved by the effect of creep.Substantial temperature changes occur on a short-term basis during which the effects of creepdo not have a significant ameliorating effect. However, in-situ concrete bridges generatesignificant quantities of heat while setting and consequently have their initial set when theconcrete is warm. The sustained stresses generated by the subsequent contraction of theconcrete as it cools can be relieved substantially by creep.

Example 3.5: Thermal contraction in frame bridge by moment distribution

The frame structure illustrated in Fig. 3.16 is integral having no internal bearings or joints. Asa result, thermal contraction or expansion induces bending moment as well as axial force andshear. It is subjected to a uniform reduction in temperature through the depth of the deck(ABC) of 20°C and no change in temperature elsewhere. The resulting distribution of bendingmoment is required given that the coefficient of thermal expansion is 12×10−6. The relativeflexural rigidities are given on the figure and the area of the deck is 500I0/l2.

Fig. 3.16 Integral frame of Example 3.5

Fig. 3.17 First step in analysis of frame: (a) system of fixities; (b) fixed axial force diagram; (c)free body diagrams showing shear and axial forces in fixed frame

Step 1: Due to symmetry, there is no tendency for point B to rotate and this point can beconsidered fixed without applying a fixity. However, as points A and C will tend to rotate aswell as translate, two fixities are needed at each, one translational and one rotational asillustrated in Fig. 3.17(a). Due to symmetry, the fixities at A and C are taken to be equal andopposite as illustrated.

In the fixed structure, the attempt to contract generates a tensile stress in ABC of(12×10−6)(20°)E and an axial force reaction at each end of (12×10−6)(20°)(500EI0/l2)−0.12EI0/l2. No distribution of bending moment is present in the fixed structure but the axialforce diagram is as illustrated in Fig. 3.17(b). As there is axial force in ABC but nocorresponding shear force in AD or CF, there is a lack of force equilibrium at A and C asillustrated in Fig. 3.17(c).

Step 2: Applying a unit rotation in Direction 1 (Fig. 3.17(a)) requires the moments andforces illustrated in Fig. 3.18(a) and generates the BMD illustrated in Fig. 3.18(b). Thenormalised version is found by dividing by 7.2EI0/l and is illustrated in Fig. 3.18(c). As therotation is applied while fixing against translation, no axial forces are generated in themembers. However, there is a shear force just below A and C which is unmatched by an axialforce in AB or BC. Hence, there is a lack of force equilibrium at A and C which, whendivided by 7.2EI0/l, is as illustrated in Fig. 3.18(d).

Applying a unit translation in Direction 2 (while preventing rotation) requires the momentsand forces illustrated in Fig. 3.19(a) and generates the BMD illustrated in

Fig. 3.18 Effect of rotation at A and C: (a) moments and forces required to induce unit rotation;(b) BMD associated with unit rotation; (c) normalised BMD associated with rotation;(d) normalised shear and axial forces associated with rotation

Fig. 3.19(b). In addition, a distribution of axial force is generated which is illustrated in Fig.3.19(c). It can be seen in Fig. 3.19(d) that there is a shear force just below A which is notmatched by the axial force to its right. The discontinuity or lack of equilibrium at A is538.4EI0/l3. Normalising with respect to this value gives Figs. 3.20(a) and (b). The normalisedlack of joint equilibrium is illustrated in Fig. 3.20(c).

Step 3: The lack of force equilibrium in the fixed structure illustrated in Fig. 3.17(c) iscorrected by factoring Fig. 3.20(a) by 0.12EI0/l2 and, as there is no moment induced in thefixed structure, adding it to a BMD of zero. In addition, the joint forces of Fig. 3.20(c)

must be factored by 0.12EI0/l2 and added to those of Fig. 3.17(c). The results are illustrated inFig. 3.21. There is a discontinuity in the BMD (or lack of moment equilibrium) at A and Cevident in Fig. 3.21(a). This is corrected by scaling Figs. 3.18(c) and (d) by 0.00214EI0/l andadding them to Figs. 3.21(a) and (b) respectively. The results are illustrated in Fig. 3.22.

Step 4: The removal of the moment discontinuity reintroduces a lack of force equilibriumwhich is evident in Fig. 3.22(b) of magnitude, 0.00285EI0/l2. Figures 3.20(a) and (c) are

Fig. 3.19 Effect of translation at A and C: (a) forces and moments required to induce unittranslation; (b) BMD associated with unit translation; (c) axial force diagramassociated with unit translation; (d) free body diagram at joint A showing lack ofequilibrium

Fig. 3.20 Normalised effect of translation at A and C: (a) normalised BMD; (b) normalised axialforce diagram; (c) free body diagram showing unit discontinuity of forces

Fig. 3.21 Effect of correcting for lack of force equilibrium: (a) corrected BMD; (b) corrected freebody diagram

Fig. 3.22 Effect of correcting for discontinuity in BMD: (a) corrected BMD; (b) corrected freebody diagram

Fig. 3.23 Results of analysis for effects of thermal contraction: (a) BMD; (b) free body diagram

scaled by this amount and added to Figs. 3.22(a) and (b). The resulting moment discontinuityis corrected by factoring and adding Figs. 3.18(c) and (d). This leads to Fig. 3.23 where thelack of force equilibrium is deemed to be sufficiently small. Figure 3.23(a) is thereforeadopted as the final BMD.

Example 3.5 serves to illustrate the effect of a moment connection between the bridge deckand the piers. In such a case, thermal movement is resisted by bending in both the piers andthe deck. To some extent this alters the resistance to contraction or expansion. However, amore important effect of the moment connection is the bending moment induced in the deckby thermal movement. This can become a significant factor in bridge deck design.

3.4.1 Equivalent loads method

The method of equivalent loads is a method by which a thermal expansion/contractionproblem can be converted into a regular analysis problem. While the method may not at firstseem to be any simpler to apply than the procedure used above, it is particularly useful when acomputer is available to carry out the analysis but the program does not cater directly fortemperature effects.

Example 3.6: Introduction to equivalent loads method

The equivalent loads method will first be applied to the simple problem of the partiallyrestrained beam illustrated in Fig. 3.24 which is subjected to an axial increase in temperatureof ΔT. The expansion is partially restrained by a spring of stiffness AE/(2l) where A is cross-sectional area and E is the elastic modulus of the beam. The equivalent loads method consistsof three stages as follows.

Stage A—Calculate the equivalent loads and the associated stresses: The loading is foundwhich would generate the same strain in an unrestrained member as the distribution oftemperature. An axial expansion can be generated in an unrestrained beam by applying anaxial force, F0, where:

where αis the coefficient of thermal expansion. However, temperature on an unrestrainedmember generates strain but not stress. The equivalent force, on the other hand, will generateboth, even on an unrestrained beam. Therefore, it is necessary to identify the ‘associatedstresses’, i.e. that distribution of stress which is inadvertently introduced into the structure bythe equivalent loads. In Stage C, this distribution of stress must be subtracted to determine thestresses generated indirectly by the change in temperature. The equivalent loads for thisexample are illustrated in Fig. 3.25(a) and the associated stress distribution in Fig. 3.25(b).

Stage B—Analyse for the effects of the equivalent loads: The beam is analysed for theloading illustrated in Fig. 3.25(a). Normally this stage would be done by computer but it

Fig. 3.24 Beam on rollers with partial (spring) restraint

Fig. 3.25 Analysis by equivalent loads method: (a) equivalent loads; (b) associated stressdistribution; (c) equilibrium of forces at spring; (d) stress distribution due totemperature change

is trivial for this simple example. It is well known that, when a load is applied to two springs,it is resisted in proportion to their stiffnesses. In this case, the beam acts as a spring ofstiffness AE/l. Hence the force is taken in the ratio 1:2 as illustrated in Fig. 3.25(c). Thedistribution of stress due to application of F0 is an axial tension throughout the beam ofmagnitude 2F0/(3A) as illustrated in Fig. 3.25(d).

Stage C—Subtract the associated stresses: The distribution of associated stresses issubtracted from the stresses generated by the equivalent loads. For this example, this consistsof subtracting the axial stress distribution of Fig. 3.25(b) from that of Fig. 3.25(d). The resultis an axial compression of F0/(3A) throughout the beam. This is the final result and is whatone would expect from a thermal expansion in a partially restrained beam; strain is generatedbut also some compressive stress.

3.5 Differential temperature effects

When the sun shines on the top of a bridge, the top tends to increase in temperature faster thanthe bottom. Thus, a differential temperature distribution develops which tends to cause thebridge to bend. If a linear distribution of this type is applied to a simply supported single-spanbeam, the bending takes place freely and the beam curves upwards as the top expands relativeto the bottom. This corresponds to the case of a beam on rollers subjected to an axial increasein temperature in that strains take

place but not stress. If such a differential temperature distribution is applied to a beam inwhich the ends are fixed against rotation, the free bending is prevented from taking place andthe situation is one of stress but no strain. In multi-span beams and slabs, partial restraintagainst bending is present as will be seen in the following examples.

Example 3.7: Differential temperature in two-span beam

The two-span beam illustrated in Fig. 3.26 is subjected to a change of temperature which isnon-uniform through its depth. The temperature change varies linearly from an increase of 5°at the top to a decrease of 5° at the bottom. The centroid of the beam is at mid-height, theelastic modulus is E and the second moment of area is I. It is required to determine the BMDdue to the temperature change given that the coefficient of thermal expansion is α.

The BMD will be determined using the method of equivalent loads.Stage A—Calculate the equivalent loads and the associated stresses: The temperature

change would generate a distribution of strain varying from 5αat the top to −5αat the bottomof an unrestrained beam, where αis the coefficient of thermal expansion. Consider thefamiliar flexure formula:

where M is moment, R is radius of curvature, σis stress and y is distance from the centroid.

where εis strain. The ratio 1/R is known as the curvature, κ. In this case, the change intemperature generates a curvature of:

The corresponding equivalent moment is:

Fig. 3.26 Beam of Example 3.7 and applied distribution of temperature

Fig. 3.27 Application of equivalent loads method: (a) equivalent loads; (b) associated BMD

Temperature on an unrestrained structure generates strain and curvature but not bendingmoment or stress. The equivalent moment on the other hand will generate both curvature andbending moment, even on unrestrained beams. Therefore, it is necessary to identify the‘associated BMD’, i.e. that distribution of moment which is inadvertently introduced into thestructure by the equivalent loading. The equivalent loads and associated BMD are illustratedin Figs. 3.27(a) and (b) respectively.

Fig. 3.28 Stages in equivalent loads method: (a) applied equivalent loads; (b) BMD due toapplication of equivalent loads; (c) BMD after subtraction of associated BMD

Stage B—Analyse for the effects of the equivalent loads: Analysis of a symmetrical two-spanbeam is trivial because, due to symmetry, the central support point, B, does not rotate. Hence,it is effectively fixed as illustrated in Fig. 3.28(a) and the solution can be determined directlyfrom Appendix B. The BMD due to the applied equivalent loading is as illustrated in Fig.3.28(b).

Stage C—Subtract the associated stresses: Subtracting the associated BMD of Fig. 3.27(b)from Fig. 3.28(b) gives the final result illustrated in Fig. 3.28(c).

Example 3.8: Differential temperature change in continuous beam

The three-span beam illustrated in Fig. 3.29 is subjected to an increase in temperature whichvaries linearly from a maximum of 20° at the top to 10° at the bottom. The depth of the beamis h and the centroid is at mid-depth, the elastic modulus is E and the second moment of areais I. It is required to determine the BMD due to the temperature increase given that thecoefficient of thermal expansion is α.

The temperature distribution is first converted into a strain distribution by multiplying bythe coefficient of thermal expansion, a. The distribution is then resolved into two components,axial strain and bending strain, as illustrated in Fig. 3.30. As the beam is free to expand, theaxial component will result in a free expansion, i.e. a strain but no stress. The bendingcomponent will result in some moment but not as much as would occur if the beam weretotally prevented from bending. The BMD will be determined using the method of equivalentloads.

Stage A—Calculate the equivalent loads and the associated stresses: In this example, thecurvature is, from Fig. 3.30:

Fig. 3.29 Differential temperature example

Fig. 3.30 Resolution of applied change in strain into axial and bending components

Hence, the equivalent moment becomes:

Thus, the equivalent loads and associated BMD are as illustrated in Fig. 3.31.

Fig. 3.31 Application of equivalent loads method: (a) equivalent loads; (b) associated BMD

Stage B—Analyse for the effects of the equivalent loads: The frame is analysed for theloading of Fig. 3.31(a). Normally this stage would be done by computer but it will be doneusing moment distribution for this simple example.

Fig. 3.32 First step in analysis by moment distribution: (a) system of fixities; (b) applied loadingon fixed structure; (c) BMD in fixed structure

Step 1: The beam is fixed at B and C in order to isolate the three spans. As the ‘loading’ issymmetrical, the fixities at B and C are equal and opposite as indicated in Fig. 3.32(a). Theapplied loading on the fixed structure is illustrated in Fig. 3.32(b) and the resulting BMD(Appendix B) in Fig. 3.32(c).

Fig. 3.33 Effect of rotation at fixing points: (a) moments required to induce unit rotation; (b)BMD associated with unit rotation; (c) normalised BMD associated with rotation

Fig. 3.34 Completion of equivalent loads method: (a) BMD due to analysis by momentdistribution; (b) BMD after subtraction of associated BMD

Step 2: Unit rotation at B and C requires the application of the moments illustrated in Fig.3.33(a) and generates the BMD illustrated in Fig. 3.33(b). The normalised version isillustrated in Fig. 3.33(c).

Step 3: The discontinuity of moment at B and C evident in Fig. 3.32(c) is corrected byadding Fig. 3.33(c) factored by 5EIα/h. The result is illustrated in Fig. 3.34(a).

Step 4: As no discontinuity now exists, no further iteration is required.Stage C—Subtract the associated stresses: Subtracting the associated BMD of Fig. 3.31(b)

from Fig. 3.34(a) gives the final result illustrated in Fig. 3.34(b). This is the BMD due to thedifferential temperature increase.

Example 3.9: Bridge diaphragm

The bridge diaphragm illustrated in Figs. 3.35(a) and (b) is subjected to the differentialincrease in temperature shown in Fig. 3.35(c). It is required to determ if there will be uplift atB c due to combined temperature and dead load. The upward reaction from the bearing due tothe dead load is 300 kN, the coefficient of thermal expa is 12×10−6 and the modulus ofelasticity is 35 000 N/mm2.

The cross-section and temperature distribution for this examp le are identical to those ofExample 2.1 (Chapter 2). Referring to that example, the equi ivalent loading is a force of 580kN and a moment of 160 kNm of which only the mome is of relevance.

Fig. 3.35 Bridge diaphragm example: (a) plan of geometry; (b) section through diaphragm; (c)applied temperature distribution

Fig. 3.36 Analysis to determine effect of imposed differential temperature: (a) equivalentloading; (b) associated BMD; (c) results of analysis; (d) final BMD

Fig. 3.37 Free body diagram for diaphragm beam

To determine the reaction due to this moment, the structure is analysed for the loadingillustrated in Fig. 3.36(a). The associated BMD is illustrated in Fig. 3.36(b).

By symmetry, Point B does not rotate and is effectively fixed. Hence (as in Example 3.7)the BMD due to applied loading is as illustrated in Fig. 3.36(c). Subtracting the associatedBMD gives the final BMD illustrated in Fig. 3.36(d). The reactions at A and C can be foundfrom the free body diagram illustrated in Fig. 3.37:

Hence the reaction at B is 80+80=160 kN. As the reaction due to dead load exceeds thisvalue, there is no uplift of this bearing due to the differential temperature change.

Example 3.10: Differential temperature in bridge of non-rectangular section

The beam-and-slab bridge whose section and temperature loading is described in Example 2.2consists of two 10 m spans. It is required to determine the maximum stresses due to thedifferential temperature change.

In Example 2.2, it was established that the equivalent moment due to the temperaturechange is −0.718αE for half of the bridge. Using the method of equivalent loads:

Stage A: The equivalent loads are illustrated in Fig. 3.38(a) and the associated BMD in Fig.3.38(b).

Stage B: Analysis by computer or by hand gives the BMD illustrated in Fig. 3.38(c).Stage C: Subtracting the associated BMD of Fig. 3.38(b) from Fig. 3.38(c) gives the final

distribution of moment due to restrained bending illustrated in Fig. 3.38(d).

Fig. 3.38 Analysis to determine effect of differential temperature change: (a) equivalent loading;(b) associated BMD; (c) results of analysis; (d) final BMD

Thus, a sagging bending moment is induced over the central support, B, of 1.077αE whichgives stresses (tension positive) of −5.29αE and 11.32αE at the top and bottom fibresrespectively. It was established in Example 2.2 that the residual stresses are−5.38αE and−5.45αE (restraint to expansion induces compression at the extreme fibres). Hence, the totalstress at the top fibre is −5.29αE−5.38αE=−10.67αE. At the bottom fibre the total stress is11.32αE−5.45αE=5.87αE.

Example 3.11: Variable section bridge

Figure 3.39(a) shows the elevation of a pedestrian bridge while Figs. 3.39(b), (c) and (d) showsections through it. The deck is subjected to the differential decreases in temperature shownin the figure. The bridge is first restrained when its temperature is somewhere between 5°Cand 25°C and the minimum temperature attained during its design life is −15°C. It is requiredto determine the equivalent loading and the associated stress distributions given a coefficientof thermal expansion, α=12×10−6/°C and a modulus of elasticity, E=35×106 kN/m2.

Fig. 3.39 Pedestrian bridge: (a) elevation; (b) section A—A and corresponding imposedtemperature distribution; (c) section B—B and corresponding imposed temperaturedistribution; (d) section C—C

By summing moments of area it is found that the centroids are 0.5 m and 1.033 m below thetop fibre for the solid and hollow sections respectively (Figs. 3.39(b)–(d)). Summing productsof stress and area in Fig. 3.39(b) gives the equivalent force (positive tension) on the solidsection due to the differential temperature distribution:

The corresponding equivalent moment (positive sag) is:

Fig. 3.40 Model of pedestrian bridge: (a) geometry showing differences in level of centroids; (b)equivalent loading; (c) associated axial force diagram; (d) associated BMD

In the hollow section, the equivalent force due to the differential temperature distribution is:

and the equivalent moment is:

The maximum axial decrease in temperature is (25−(−15))=40°C and the correspondingstress is 40αE. For the solid section of Figs. 3.39(b) and (d), the area is 2.6×1=2.6 m2 givingan equivalent force of:

For the hollow section, the area is 2.64 m2 and the equivalent force is:

A model which allows for the difference in the level of the centroids is illustrated in Fig.3.40(a). Note that the short vertical members at b and c could be assumed to have effectivelyinfinite stiffness. However, using members with very large stiffnesses can generate numericalinstability in a computer model. Therefore, a second moment of area several times as large asthe maximum used elsewhere in the model (e.g. ten times) generally provides sufficientaccuracy without causing such problems.

Noting that the axial effects apply to all members while the differential temperaturedistributions only apply to the deck (abcd), the equivalent loads are illustrated in Fig. 3.40(b).The associated axial force and bending moment diagrams are illustrated in Figs. 3.40(c) and(d), respectively. The bending moment and axial force distributions due to the temperaturedecreases can be found by analysing for the equivalent loading illustrated in Fig. 3.40(b) andsubtracting the associated distributions of Figs. 3.40(c) and (d) from the results.

3.5.1 Temperature effects in three dimensions

When the temperature of a particle of material in a bridge is increased, the particle tends toexpand in all three directions. Similarly, when a differential distribution of temperature isapplied through the depth of a bridge slab, it tends to bend about both axes. If there is restraintto either or both rotations, bending moment results about both axes as will be illustrated in thefollowing example.

Example 3.12: Differential temperatureThe slab bridge of Fig. 3.41 is articulated as shown in Fig. 3.41(a) to allow axial expansion

in both the X and Y directions. However, for rotation, the bridge is two-span

Fig. 3.41 Slab bridge of Example 3.12: (a) plan showing directions of allowable movement atbearings; (b) section A-A; (c) imposed temperature distribution in deck (section 1−1);(d) imposed temperature distribution in cantilever (section 2–2)

longitudinally and is therefore not able to bend freely. Further, there are three bearingstransversely at the ends so that it is not able to bend freely transversely either. The deck andcantilevers are subjected to the differential temperature increases illustrated in Figs. 3.41(c)and (d) respectively. It is required to determine the equivalent loading and the associatedBMD due to this temperature change. The coefficient of thermal expansion is 9×10−6/°C andthe modulus of elasticity is 32×106 kN/m2.

The specified temperature distributions are different in the cantilevers and the main deck ofthis bridge. However, for longitudinal bending, the bridge will tend to act as one unit andbending will take place about the centroid. The location of this centroid is:

below the top surface. The bridge deck is divided into parts as illustrated in Fig. 3.42corresponding to the different parts of the temperature distribution and the temperature

Fig. 3.42 Cross-section with associated distribution of imposed stress: (a) deck; (b) cantilevers

changes are converted into stresses. Taking moments about the centroid gives a longitudinalbending moment per metre on the main deck of:

The corresponding bending moment per metre on the cantilever is:

These equivalent longitudinal moments are illustrated in Fig. 3.43.The transverse direction is different from the longitudinal in that the cross-section is

rectangular everywhere. In the cantilever region, bending is about the centroid of the

Fig. 3.43 Equivalent loading due to temperature

Fig. 3.44 Resolution of imposed stress in cantilever into axial and bending components

cantilever. The applied stress distribution is resolved into axial and bending components asillustrated in Fig. 3.44. The axial expansion is unrestrained while the bending stressdistribution generates a moment of:

Fig. 3.45 Associated BMDs: (a) plan showing section locations; (b) section B–B; (c) section A–A;(d) section C–sC

In the main deck, the differential distribution is applied to a 0.8 m deep rectangular sectiongiving a moment about the centroid of:

As M3 is applied to the outside of the cantilever, only (M4−M3) needs to be applied at thedeck/cantilever interface as illustrated in Fig. 3.43. As these applied moments generatedistributions of longitudinal and transverse moment, there are two associated BMDs asillustrated in Fig. 3.45. As for the previous example, the problem is completed by analysingthe slab (by computer) and subtracting the associated BMDs from the solution.

3.6 Prestress

The effects of prestress in bridges are similar to the effects of temperature and the sameanalysis techniques can be used for both. However, there is one important distinction. Anunrestrained change in temperature results in a change in strain only and no change in stress.Prestress, on the other hand, results in changes of both stress and strain. For example, if abeam rests on a sliding bearing at one end, it can undergo axial changes in temperaturewithout incurring any axial stress. However, prestressing that beam does (as is the objective)induce a distribution of stress. When the movements due to prestressing are unrestrained, thestress distributions are easily calculated and analysis is not generally required. However, thereare many bridge forms where the effects of prestress are restrained to some degree or otherand where analysis is necessary.

Example 3.13: Frame subject to axial prestress by moment distribution

The frame of Fig. 3.16, reproduced here as Fig. 3.46(a), is subjected to a prestressing forcealong the centroid of the deck, ABC, of magnitude, P. It is required to determine the netprestress force in the deck and the resulting BMD. The frame is analysed by momentdistribution.

Step 1: The system of fixities used in Example 3.5 is used again here as illustrated in Fig.3.46(b). The BMD due to applied ‘loading’ on the fixed structure is zero everywhere as theprestress forces are applied at fixing points.

Step 2: The effects of inducing rotations or translations at the fixing points are the same asfor Example 3.5. The normalised versions are presented here in Figs. 3.47(a) and (b) (unitdiscontinuity in moment) and in Figs. 3.47(c) and (d) (unit discontinuity in force).

Fig. 3.46 Frame subjected to prestress force: (a) geometry and loading; (b) system of fixities

Fig. 3.47 Effect of displacements at fixing points: (a) normalised BMD due to rotation; (b)normalised forces due to rotation; (c) normalised BMD due to translation; (d)normalised forces due to translation

Fig. 3.48 Effect of prestress force: (a) BMD after correction for force equilibrium; (b) internalforces after correction for force equilibrium; (c) BMD after correction for momentequilibrium; (d) internal forces after correction for moment equilibrium

Step 3: The translational fixity is released first to apply the prestress force. This consistssimply of factoring Figs. 3.47(c) and (d) by P. It can be seen in the results, illustrated in Figs.3.48(a) and (b), that equilibrium of forces at A and C is then satisfied. The discontinuity ofmoment which results is removed by factoring Figs. 3.47(a) and (b) by 0.0178Pl and addingto give Figs. 3.48(c) and (d).

Step 4: As force equilibrium in Fig. 3.48(d) is satisfied to a reasonable degree of accuracy,no further iteration is deemed necessary.

Example 3.13 serves to illustrate the ‘loss’ of prestress force that occurs in a frame due to therestraint offered by the piers. In this example, about 5% of the applied force is lost as shearforce in the piers. It is also of importance to note the bending moment that is inadvertentlyinduced by the prestress. Interestingly, this bending moment is independent of the elasticmodulus and is therefore unaffected by creep. In a concrete frame, a prestressed deck willcontinue to shorten with time due to creep. However, the bending stresses induced by thisshortening are also relieved by creep with the result that creep has little net effect on thebending moment due to prestress.

Example 3.14: Analysis for eccentric prestressing

The beam illustrated in Fig. 3.49 is prestressed with a straight tendon at an eccentricity, e,from the centroid with a prestress force, P. It is required to determine the induceddistributions of axial force and bending moment.

The method of equivalent loads is applicable to prestress just as it is to temperature. Theonly difference is that, as prestress generates stress as well as strain, it is not appropriate todeduct the associated stresses from the analysis results as was necessary in temperatureanalysis. In this example, the prestress force is applied at an eccentricity to the centroid. Thisis equivalent to applying a moment alongside the force as illustrated in Fig. 3.50(a). The axialforce diagram is clearly as illustrated in Fig. 3.50(b). To determine the bending momentdiagram, however, is not so straightforward as the beam is not free to lift off the supports at Band C. The analysis to determine the BMD will be carried using moment distribution.

Fig. 3.49 Beam subjected to eccentric prestress force

Fig. 3.50 First stage in equivalent loads method: (a) equivalent loads; (b) axial force diagramdue to prestress

Step 1: The beam is fixed as illustrated in Fig. 3.51(a). The BMD in the fixed structure due tothe equivalent loading is as illustrated in Fig. 3.51(b).

Step 2: The moments required to induce unit rotation at B and C are illustrated in Fig.3.52(a), the resulting BMD in Fig. 3.52(b) and the normalised BMD in Fig. 3.52(c).

Step 3: The discontinuity of bending moment evident in Fig. 3.51(b) is removed byfactoring Fig. 3.52(c) by Pe/2 and adding. The result is illustrated in Fig. 3.53. As there is nofurther discontinuity, this is the final BMD due to prestress.

Fig. 3.51 First step in analysis by moment distribution: (a) system of fixities; (b) fixed BMD

Fig. 3.52 Effect of rotation of fixing points: (a) moments required to induce unit rotation; (b)BMD associated with unit rotation; (c) normalised BMD

Fig. 3.53 Final BMD due to eccentric prestress force

It is interesting to note from Example 3.14 that the effect of the tendon below the centroid isto generate sagging moment in the central span. In a simply supported beam, a tendon belowthe centroid generates hogging moment.

Example 3.15: Profiled tendons

In most post-tensioned bridges the tendons are profiled using a combination of straightportions and parabolic curves. For preliminary design purposes, the actual profiles aresometimes approximated by ignoring the transition curves over the internal supports asillustrated in Fig. 3.54. For this beam, it is required to find the BMD due to a prestress force,P.

A parabolic profile generates a uniform loading, the intensity of which can be determinedby considering equilibrium of forces at the ends of the parabola. (This was covered in greaterdetail in Chapter 2.) For the parabola in Span AB, the slope is found by differentiating theequation as follows:

Fig. 3.54 Beam with profiled prestressing tendon

Fig. 3.55 Equivalent loading due to profiled tendon: (a) equivalent forces in span AB; (b)summary of all equivalent forces on beam

Fig. 3.56 Equivalent loads method: (a) system of fixities for analysis by moment distribution; (b)equivalent loads and BMDs due to prestress in fixed structure; (c) BMD aftercorrection for discontinuity in BMD

At A, x=0 and the slope becomes −0.08. From Fig. 3.55(a) it can be seen that the verticalcomponent of the prestressing force at A is P sin θ1≈P tanθ1=0.08P. Similarly, at x=l, theslope is 0.12 and the vertical component of prestress is 0.12P. Hence, equilibrium of verticalforces requires a uniform loading of intensity:

In BC, the vertical components can be found similarly. They are both equal to 0.1P and theintensity of loading is, coincidentally, wBC=0.2P/l. In CD, the intensity is, by symmetry,wCD=wAB=0.2P/l. Thus, the complete equivalent loading due to prestress is as illustrated inFig. 3.55(b). The beam is analysed for this loading using moment distribution.

Step 1: The symmetrical system of fixities is illustrated in Fig. 3.56(a) and the associatedBMD (Appendix A) is given in Fig. 3.56(b).

Step 2: The BMD associated with simultaneous rotations at B and C is identical to thatderived for Example 3.14 and illustrated in Fig. 3.52(c).

Step 3: To remove the moment discontinuity of 0.00833Pl in Fig. 3.56(b), Fig. 3.52(c) isfactored by this amount and added to Fig. 3.56(b). The result is illustrated in Fig. 3.56(c). Asthere is no further discontinuity, this is the final BMD due to prestress in this beam.

Example 3.15 serves to illustrate that the effect of profiled prestressing tendons can be quitesimilar to the effect of self weight in that it applies a uniform loading throughout the beam.The obvious difference is that typical prestress loading is in the opposite direction to loadingdue to self weight.

3.7 Application of moment distribution to grillages

A great many bridges are analysed by computer using the grillage analogy. In this method,described in detail in Chapters 5 and 6, the continuous bridge slab is represented by a mesh ofdiscrete beams. In most practical grillages, significant vertical translational displacementsoccur at the joints. As a result, moment distribution is applicable to the analysis of grillagesbut is tedious to apply for most examples. In this chapter, only those grillages are consideredin which there is no such joint displacement.

Example 3.16:Torsion due to vertical loading

When bridges are curved or crooked in plan, vertical loading induces torsion as will bedemonstrated in this example, which is illustrated in Fig. 3.57(a). This bridge is long andnarrow so it can be idealised by two beam members as illustrated in Fig. 3.57(b). Thetorsional rigidity, GJ, is 1.5 times the flexural rigidity, EI. The bridge is subjected to uniformvertical loading of intensity, w.

Step 1: The two members are isolated from each other by the fixing of point B. As there isa support there already, it is not necessary to provide a vertical translational fixity.

Fig. 3.57 Analysis of crooked bridge: (a) plan view of geometry; (b) plan view of idealisation; (c)plan view of system of fixities; (d) elevation of applied loading and resulting BMD inAB while fixed; (e) plan view of fixed BMD

However, two rotational fixities are required in orthogonal directions as illustrated in Fig.3.57(c) (rotation about two axes). The double headed arrows indicate rotational fixities wherethe positive direction is clockwise when looking in the direction of the arrow. Vertical loadingon AB in the fixed structure is applied to a beam which is fixed at one end. From Appendix A,the bending moment diagram is as illustrated (in elevation) in Fig. 3.57(d). In plan, the BMDfor the two beams are illustrated in Fig. 3.57(e). However, the BMDs for each of these beamsis about the axis of that beam so the discontinuity of moment at B is not apparent from thediagram. At B, there is a transition between bending moment and torsion in the members.

The internal bending moment at the left end of BC is wl2/8 as illustrated in Fig. 3.58(a). Inorder to compare this moment to that just left of B, it needs to be resolved into componentsparallel and perpendicular to AB. This is done in Fig. 3.58(b) and it can be seen that there is adiscontinuity of moment at B of (1−1/√2)wl2/8=0.293wl2/8. In addition, there is adiscontinuity in torsion of wl2/(8√2).

Step 2: The second step in moment distribution is to find the bending moment and torsiondiagrams due to unit rotation at each of the points of fixity. A unit rotation is first applied

Fig. 3.58 Plan views showing internal moment: (a) end moment in BC; (b) resolution of endmoment in BC parallel and perpendicular to AB

in Direction 1 (Fig. 3.57(c)). Unit rotation at the end of AB results in the deformed shape andBMD illustrated in elevation in Fig. 3.59(a). To determine the effect on member BC, it isnecessary to resolve the rotation into components parallel and perpendicular to that memberas illustrated in Fig. 3.59(b). Hence the BMD due to unit rotation at B is as illustrated in Fig.3.59(c). As there is no resistance to twisting at C, the application of a twist of 1/√2 at B doesnot generate any torsion in BC. The discontinuity of moment at B in the BMD of Fig. 3.59(c)can be seen when the moments are resolved in Fig. 3.59(d). It is (3+3/2)EI/l=9EI/2l; thenormalised version of Fig. 3.59(d), normalised for moment, is illustrated in Fig. 3.59(e).

Applying a unit rotation in Direction 2 (Fig. 3.57(c)) generates no bending but a torsion ofGJ/l in AB. In BC, the rotation at the joint must be resolved into components as illustrated inFig. 3.60(a) and it can be seen that it generates no torsion and the BMD illustrated in Fig.3.60(b). The discontinuity of moment at the joint can be seen by resolving the internalmoments and torsions in Fig. 3.60(c). The discontinuity parallel to Direction 2 at B isGJ/l+3EI/2l=3EI/l; dividing by this value gives the normalised version illustrated in Fig.3.60(d).

Step 3: In the third step, the discontinuities in the fixed bending moment and torsiondiagrams are removed by scaling and adding the diagrams derived in Step 2. The

Fig. 3.59 Effect of rotation in direction 1 at B: (a) elevation of AB showing imposed unit rotationand associated BMD; (b) resolution of rotation parallel and perpendicular to BC; (c)BMD due to unit rotation; (d) free body diagram showing lack of equilibrium ofmoments at B; (e) normalised free body diagram

discontinuity in the fixed BMD of Fig. 3.58(b) is (1−1/√2)wl2/8=0.293wl2/8. Adding the BMDof Fig. 3.59(e) scaled by minus this value gives the moments and torsions illustrated in Fig.3.61(a). While the discontinuity parallel to Direction 1 (Fig. 3.57(c)) has now been removedat B, there is still a discontinuity parallel to Direction 2 of 0.805wl2/8. This is removed byadding the diagram of Fig. 3.60(d), scaled by minus this value to give the diagram illustratedin Fig. 3.61(b). The new discontinuity now introduced parallel to Direction 1 is removed byadding a diagram proportional to Fig. 3.59(e) to give Fig. 3.61(c) and the discontinuity in thatis removed by adding a diagram parallel to Fig. 3.60(d) to give the diagram illustrated in Fig.3.61(d). The corresponding bending moment and torsion diagrams are illustrated in Figs.3.61(e) and (f).

Fig. 3.60 Effect of rotation in direction 2 at B: (a) resolution of rotations parallel andperpendicular to BC; (b) BMD due to unit rotation; (c) free body diagram showinglack of equilibrium of moments at B; (d) normalised free body diagram

Fig. 3.61 Successive corrections to internal moments and torsions: (a) after correction ofmoments in direction 1; (b) after correction of moments in direction 2; (c) after secondcorrection in direction 1; (d) after second correction in direction 2; (e) final BMD; (f)final torsion diagram

Step 4: To get an exact answer, the process must be continued until no discontinuity remains.

Example 3.17: Torsion due to skew supports

The skewed bridge illustrated in Figs. 3.62(a) and (b) is long relative to its width and can bemodelled using a single longitudinal member and a pair of outriggers at the ends as

Fig. 3.62 Long skewed bridge: (a) plan view; (b) cross-section; (c) plan view of idealisation

illustrated in Fig. 3.62(c). This deck has flexural rigidities of (EI)ABC= (EI)DEF=0.8, (EI)BE=3.6and torsional rigidities of (GJ)ABC=(GJ)DEF=2.0, (GJ)BE=4.0. It is subjected to verticaluniform loading of intensity w.

Step 1: To isolate the members from one another, total fixity must be imposed at B and E,i.e. it is necessary to fix against vertical translation and against rotation about both axes. Thisis represented diagrammatically in Fig. 3.63(a). The circles in this figure indicate translationalfixities with a direction of positive upwards. The symmetry of the system is exploitedrecognising that the three fixities at B are identical to the corresponding fixities at E.

Fig. 3.63 First step in analysis by moment distribution: (a) complete system of fixities with tworotations and one translation; (b) simplified system of fixities; (c) fixed BMD; (d) freebody diagram showing lack of moment equilibrium in fixed structure

The system of fixities illustrated is adequate, but unnecessary in practice. As the two bearingsare relatively close together (i.e. 12<<l1), the vertical deflection at B and E will be relativelysmall and can be neglected. By the same token, the rotation in Direction 2 will be small, as toapply a unit rotation there would require a moment that is very large. Therefore it is sufficientto fix the bridge as illustrated in Fig. 3.63(b). In the fixed structure, BE is fixed at each endand the BMD (Appendix A) is as illustrated in Fig. 3.63(c). There is no torsion in the bridgein its fixed state. The discontinuity of moment parallel to the direction of fixity is found byresolving the moment reaction of wl2/12 parallel to the direction of fixity as illustrated in Fig.3.63(d). The discontinuity is (wl2/12) cosθas there is a moment of this amount to the right ofB and zero moment/ torsion on the other side of it.

Step 2: The second step consists of applying a rotation at B and E. Applying a unit rotationto ABC and DEF about their own axes requires no moment as no torsional resistance wouldbe offered by the bearings. However, to rotate member BE in Direction 1 requires it to berotated and twisted. This can be seen by resolving the unit rotation into directions parallel andperpendicular to the member as illustrated in Fig. 3.64(a). The required twist of one endrelative to the other is 2 sinθ; the torsion required to generate such a twist (Appendix B) is (2sin θ)(GJ)BE/l1=8 sin θ/l1. The member must also undergo bending in order to rotate at eachend through cos θ. The elevation showing the

Fig. 3.64 Effect of rotations at B and E: (a) plan showing resolution of rotations into componentsparallel and perpendicular to BE; (b) elevation showing moments required to inducerotations in BE; (c) BMD associated with unit rotations at B and E

required deflected shape is illustrated in Fig. 3.64(b); the corresponding BMD (Appendix B,last BMD) is illustrated in Fig. 3.64(c). Thus, to apply a unit rotation in the direction of fixitygenerates this BMD plus a distribution of constant torsion throughout member BE ofmagnitude 8 sin θ/l1 .

The internal moments and torsions at B and E are illustrated in Fig. 3.65(a). Resolvingparallel to the direction of fixity gives the discontinuity of moment corresponding to unitrotation:

Normalising with respect to this discontinuity, results in the internal moments and torsionsillustrated in Fig. 3.65(b).

Step 3: As stated in Step 1 and illustrated in Fig. 3.63(d), the discontinuity in momentparallel to Direction 1 in the fixed structure is (wl2/12) cosθ. This is removed by adding Fig.3.65(b) scaled by that amount. The resulting diagram is illustrated in Fig. 3.66(a).

Fig. 3.65 Internal moments associated with rotations at B and E: (a) free body diagram withresolution of moments parallel and perpendicular to direction of fixity; (b) normalisedfree body diagram

Fig. 3.66 Corrected internal moments: (a) free body diagram showing moments after correctionfor discontinuity at B and E; (b) plan showing final BMD; (c) plan showing finaltorsion diagram

There is no discontinuity remaining in this diagram as the torsion and moment componentsare in equilibrium. Hence, no further distribution of moment is required. The final momentand torsion diagrams are illustrated in Figs. 3.66(b) and (c) respectively.

It can be seen in Example 3.17 that the skew supports have the effect of introducing a smallhogging moment at the ends of the bridge. If the skew, θ, were zero, the bridge would ineffect be simply supported and the moment would be positive everywhere (sagging). On theother hand, if the skew were very large, the end hogging moment would be correspondinglylarge approaching a maximum of wl2/12. The skew also has the effect of introducing asignificant distribution of torsion into the bridge.

Chapter 4Integral bridges

4.1 Introduction

Integral bridges are those where the superstructure and substructures are continuous orintegral with each other. While the concept is well established, many bridges built in the1960s and 1970s were articulated with expansion joints and bearings to separate thesuperstructure from the substructure and the surrounding soil. In the 1980s and 1990s, manyof these required rehabilitation due to serviceability problems associated with the joints. As aresult, integral construction has recently received a great deal of attention and this form islikely to become much more widespread in the future. In the UK in particular, designers arenow required to consider the use of the integral form for most shorter bridges (up to 60 mspan and 30° skew).

4.1.1 Integral construction

There are many variations on the basic integral bridge. In the bridge of Fig. 4.1(a), the deck iscomposed of separate precast beams in each span. While in the past such a deck might havehad a joint over the central support, a more durable form of construction is to make itcontinuous over the support using in-situ concrete, as illustrated. A bridge is shown in Fig.4.1(b) in which the deck is continuous over the internal support and integral with theabutments at the ends. Figure 4.1 (c) illustrates another variation; this bridge is integral withboth the abutments and the intermediate pier.

While there are considerable durability advantages in removing joints and bearings, theirremoval does affect the bridge behaviour. Specifically, expansion and contraction of the deckis restrained with the result that additional stresses are induced which must be resisted by thebridge structure. The most obvious cause of expansion or contraction in bridges of all forms istemperature change but other

Fig. 4.1 Integral bridges: (a) precast beams made integral over the interior support; (b) deckcontinuous over interior support and integral with abutments; (c) deck integral withabutments and pier

causes exist, such as shrinkage in concrete bridges. In prestressed concrete decks, elasticshortening and creep also occur. A simple integral bridge is illustrated in Fig. 4.2(a). If thebases of the abutments are not free to slide, deck contraction induces the deformed shapeillustrated in Fig. 4.2(b) and the bending moment diagram of Fig. 4.2(c). Partial slidingrestraint at the bases of the abutments results in the deformed shape of Fig. 4.2(d) and abending moment diagram which is similar in shape to that of Fig. 4.2(c), but of a differentmagnitude.

Time-dependent contractions in concrete bridge decks induce bending moments in integralbridges. While the magnitude of creep contraction is time dependent, creep also has the effectof relieving the induced bending moments over time. The net effect of this is that momentsinduced by creep contraction are small. Shrinkage strain increases with time but the resultingmoments are also reduced by creep.

Elastic shortening occurs in post-tensioned prestressed concrete decks during theapplication of prestress. If the deck is integral with the supports at the time of stressing,bending moments are induced. On the other hand, many integral bridges are constructed fromprecast pretensioned beams and the bridge is not made integral until after the pretensioningprocess is complete. In such cases, no bending moments are induced by the elastic shortening.

Temperature changes are another major source of deck expansion and contraction.Temperature can be viewed as having a seasonal and hence long-term component as well as adaily or short-term component.

The resistance of an integral bridge to movement of any type depends largely on the formof construction of the substructures. Three alternative forms are illustrated in Fig. 4.3. In eachcase, a run-on slab is shown behind the abutment. These are commonly placed over thetransition zone between the bridge and the

Fig. 4.2 Frame bridge subject to contraction: (a) geometry; (b) deformed shape if bases arerestrained against sliding; (c) bending moment diagram if bases are restrained againstsliding; (d) deformed shape if bases are partially restrained against sliding

adjacent soil which generally consists of granular backfill material. Figures 4.3(a) and (b)show two bridges which are integral with high supporting abutments and piled foundations. Insuch a case, a reduction in lateral restraint can be achieved by using driven H-section pileswith their weaker axes orientated appropriately. An alternative form of integral construction isone in which abutments sit on strip foundations like the small bank seat abutment illustratedin Fig. 4.3(c). Minimising the sliding resistance at the base of these foundations helps toreduce the lateral restraint. Care should be taken in the design to ensure that bank seats havesufficient weight to avoid uplift from applied loads in other spans.

Fig. 4.3 Ends of integral bridges: (a) deep vertical abutment; (b) deep inclined abutment; (c)bank seat abutment

4.1.2 Lateral earth pressures on abutmentsThe lateral earth pressures (σh) that the abutments of integral bridges should be designed forare those that take place during the maximum expansion of the bridge deck combined withany additional surcharge. The expansion has the effect of pushing the abutment laterally intothe backfill. The resulting earth pressures developed on the abutment are dependent on thestiffness and strength of the backfill and on the amount of movement of the abutment.

The maximum lateral earth pressure that can be sustained by the backfill is termed thepassive pressure (σhp) which, for dry backfill at a depth z and no surcharge at ground level, isgiven by the expression:

(4.1)

where Kp is the coefficient of passive pressure andγsoil is the unit weight of the backfill. Thecoefficient Kp may be estimated from Fig. 4.4 for a given angle of internal friction of thebackfill and a given ratio, where δa is the angle of interface friction between theabutment and backfill.

One design approach would be to use equation (4.1) directly to determine the maximumlateral pressure distribution on the abutment. This approach, however, is generally overlyconservative as abutment movements are usually significantly less than those required togenerate passive pressures. The preferred approach is one

Fig. 4.4 Coefficients of passive earth pressure (horizontal component) for horizontal retainedsurface (after Caquot and Kersiel (1948))

involving an appropriate soil/structure interaction analysis which takes due account of thestiffness of the soil. Such an approach is described later in this chapter.

A third (and commonly used) approach relates the pressure distribution on the abutment tothe degree of mobilisation of its maximum (or passive) lateral capacity. This method is basedon experimental observations which indicate that movements to develop full passive pressurestypically correspond to an abutment rotation equal to one-tenth of the retained height of soil,Hret/10, or to a wall translation of Hret/20. It follows that lateral pressures may be relatedapproximately to the average displacement of the abutment over the retained height (δav).Expressions for σh emerging from this rationale are given below; these are in keeping with thegeneral guidelines set out in BA42/96 (1996).

(4.2)

where

(4.3)

and

(4.4)

(4.5)

It will be seen later that the actual thermal expansion in integral bridge decks is closelycomparable to that which occurs in a similar unrestrained deck (as the

restraint offered by typical abutments and backfill is relatively small). Therefore, for a bridgedeck of length L which experiences an increase in temperature of ΔT, δav may be calculatedas:

(4.6)

(4.7)

where αis the coefficient of thermal expansion of the deck.Implicit in equations (4.6) and (4.7) is the assumption that a bank seat experiences a lateral

translation while a deeper abutment bends and rotates about a point just below the groundlevel on its inner face. For the latter case, therefore, it is reasonable to assume that horizontalstress acting on both sides of the abutment are given by equation (4.5) when the depth exceedsapproximately 1.2 Hret.

Example 4.1: Determination of design abutment earth pressures

A 50 m long integral bridge has deep wall abutments which retain 6 m of well compactedgranular fill. The peak angle of friction of the fill is 45° and its dry density is 1900 kg/m3.The design extreme event for the determination of maximum abutment pressures is a 40°increase in temperature. Assume αfor the deck is 12×10−6 per °C and

From equation (4.7):

Figure 4.4 indicates that Kp=17.5 for and . As Hret=6 m and δav<Hret/20,equation (4.3) gives:

The unit weight of the soil (γsoil) is

Therefore for z<6 m (Hret):

4.1.3 Stiffness of soil

The longitudinal expansion of integral bridge decks is resisted not just by the abutmentsupports but also by the backfill soil behind the abutments and the natural/imported soilbeneath them. For most cases, it is necessary to quantify

the restraint provided by the soil. This can only be achieved with a knowledge of theappropriate soil stiffness parameters. Clearly, a higher soil stiffness will lead to higher axialforces and bending moments in the deck due to its longitudinal expansion or contraction. Thedesign stiffness used for the calculation of such forces and moments should therefore be amaximum credible value.

The stress-strain relationship for soil is non-linear at strains in excess of about 0.000 05(50×10−6) and it is therefore common to refer to a secant modulus defined as the ratio of stressto current strain. The value of the secant modulus at a given strain for a typical cohesionlesssoil (such as the granular type generally used for backfill) depends primarily on its density (orvoid ratio), the level of confining stress and the loading history. A typical approximaterelationship has been proposed by Lehane et al. (1996):

(4.8)

where Es is the secant Young’s modulus in kN/m2, e is the void ratio of the soil, p' is the meanconfining stress less the pore water pressure in the soil, patm is the atmospheric pressure (100kN/m2), used as a reference stress and γis the shear strain which is taken to lie within therange 50×10−6 to 0.01.

The degree of compaction of backfill on site is often specified in terms of the dry density,ρd, which is related to the void ratio, e, by the expression:

(4.9)

where Gs is the specific gravity of the soil particles (typically 2.65) andρw is the

Fig. 4.5 Secant Young’s modulus for granular soil (assuming Gs=2.65) (after Lehane et al.(1996))

density of water. Thus, specification of the dry density effectively dictates the void ratio, e.The secant Young’s modulus, derived using equation (4.8), is plotted in Fig. 4.5 for a range

of in-situ dry densities (ρd), mean confining stresses (p'), and shear strains (γ). Equation (4.8)or Fig. 4.5 can be used to estimate the secant Young’s modulus for cohesionless soil.Guidance on appropriate values for ρd, p' andγfor specific cases is given in subsequentsections.

4.2 Contraction of bridge deck

There is generally a lesser height of soil in front of bridge abutments than behind them. As aresult, the resistance provided by such soil to the contraction of a bridge deck is usually small.This means that, in an analysis to determine the effects of elastic shortening, creep, and/orshrinkage, the principal uncertainty relates to the resistance to movement at the bases of thepiers and abutments.

4.2.1 Contraction of bridge fully fixed at the supports

The case is first considered of an integral bridge in which no translational movement canoccur at the base of the abutments. These conditions are applicable if the abutmentfoundations are cast in very dense soil or rock. However, an analysis of this type is often usedas a first step to determine a limit on the stresses induced by deck contraction when thesupports are partially fixed.

Fig. 4.6 Contraction of frame rigidly fixed at supports: (a) geometry; (b) bending momentdiagram from example 3.5

The bridge illustrated in Fig. 4.6(a) was considered in Chapter 3 for an axial contraction dueto temperature of 20° in the deck (ABC) (Example 3.5). In that case, the bridge was fullyrestrained at the base of each abutment and pier. However, some movement of the deck waspossible through bending in the abutments. If the ends of the deck were fully prevented fromcontracting, the decrease in temperature would generate a large tensile force in the deck andthere would be no contraction. However, the resistance of the abutments to movement wasconsiderably less than the axial stiffness of the deck (Fig. 3.20(c) shows that only 7% of thepotential force is applied to the abutments). Thus, there was a much greater tendency for thetemperature decrease to cause the abutments to bend than to cause an axial stress in the deck.The end result for that example was a relatively small axial tension in the deck, only 5% ofthe potential level, and a relatively large contraction. The axial contraction induced bending inthe abutments and, due to the integral nature of the bridge, bending in the deck also. Thecomplete bending moment diagram is illustrated in Fig. 4.6(b).

4.2.2 Contraction of bridge on flexible supportsMost bridges are constructed on supports which have some degree of flexibility. Abutmentsand piers are generally either supported on foundations bearing directly on the ground belowor on pile caps underlain by piles. Quantification of the pile resistance is beyond the scope ofthis text and interested readers are referred to books such as that of Tomlinson (1994).

Strip foundations or pile caps are commonly founded at around 0.5–1.0 m below theground level on the inside of the abutment as illustrated in Fig. 4.7. It is this small depth ofsoil, together with sliding resistance at the base of the pad, that resists bridge contraction. Thesoil around the strip foundation can be idealised by a number of linear elastic springs.Expressions for the stiffness of such springs have been deduced here from relationshipsprovided by Dobry and Gazetas (1986) for an elastic soil. Design spring stiffnesses on theinside of the abutment for a strip foundation of width B, embedded to a depth of between 0.5m and 1.0 m below the ground level are given in equation (4.10):

(4.10)

where kvert, khori and krot are the stiffnesses per metre length of strip foundation for vertical,horizontal and rotational displacement respectively. Conservative, upper bound estimates ofthe secant Young’s modulus of elasticity, Es, may be calculated using equation (4.8) assuminga p' value equivalent to the foundation bearing pressure and a shear strain (γ) of 0.001.

Fig. 4.7 End of integral bridge showing shallow depth of soil on inside: (a) bank seat; (b) deepabutment

Example 4.2: Contraction for shallow strip foundation

The bridge illustrated in Fig. 4.8 is subjected to a shrinkage strain of 200×10−6. It is requiredto determine the distribution of bending moment and axial force generated in the deck giventhat the Young’s modulus for the concrete is 30×106 kN/m2. The foundation is assumed to beworking under a bearing pressure of 300 kN/m2 and the breadth of the strip foundation is 2.5m. The degree of compaction has been controlled by specifying a dry density of backfill, ρd,of 1900 kg/m3.

Inverting equation (4.9) and assuming Gs=2.65, gives a void ratio of:

Fig. 4.8 Bridge of Example 4.2: (a) elevation; (b) detail at abutment

Substituting in equation (4.8) then gives:

Equation (4.10) then gives spring stiffnesses per metre run for the supports of:

The equivalent load for a shrinkage strain of 200×10−6 is the product of the strain, themodulus of elasticity of concrete and the cross-sectional area (per metre run):

The equivalent loads and the associated axial force diagram are illustrated in Fig. 4.9. Theframe was analysed using a standard analysis package which gave the deflected shape, axialforce and bending moment diagrams illustrated in Fig. 4.10(a)–(c). Subtracting the associatedaxial force diagram gives the actual distribution of axial force generated by the shrinkage,illustrated in Fig. 4.10(d). No adjustment is necessary for the deflected shape or bendingmoment diagram.

Fig. 4.9 Computer model for bridge of Example 4.2: (a) equivalent loading and springs; (b)associated axial force diagram

Fig. 4.10 Analysis results: (a) deflected shape; (b) axial force diagram from computer analysis;(c) bending moment diagram; (d) corrected axial force diagram

Example 4.2 is interesting in that it gives an indication of the magnitude of bending momentsand axial forces that can be generated by a restrained shrinkage. Out of a total potentialshortening of 6 mm (200×10−6×30000 mm) at each end, 5.7 mm is predicted to actually occur.However the restraint which prevents the remaining 0.3 mm does generate distributions ofstress in the frame. The axial tension is relatively small at 337 kN corresponding to a stress inthe deck of less than 0.3 N/mm2. However, the bending moment at the ends are moresignificant at 568 kNm. Assuming uncracked conditions, this corresponds to a maximumflexural stress of 2.4 N/mm2.

4.3 Conventional spring model for deck expansion

Soil generally provides considerably more resistance to deck expansion than contraction asabutments are generally backfilled up to the level of the underside of the run-on slab (Fig. 4.7).Thus, the stresses generated by an increase in deck temperature, for example, will be affectedsignificantly by the properties of the soil behind the abutments.

The selection of a suitable soil stiffness value (Es) is essential for appropriate modelling ofthe backfill. Some notable features have been observed from experimental studies bySpringman et al. (1996), and others:

1. Cyclic variations in temperature (and associated expansions and contractions of the deck)cause the backfill to compact and, with time, to tend to an equilibrium density compatiblewith the strain amplitude that it is regularly subjected to. There is some evidence to suggestthat the granular backfill at this stage will have increased in density by a maximum of about20% from its as-placed density for loose fills and by a maximum of about 10% for wellcompacted fills.

2. The horizontal stresses acting on an abutment following cyclic expansions and contractionsof the deck remain approximately constant to depths of up to 6 m and typically havemagnitudes of between 25 kN/m2 and 50 kN/m2 (depending on the type of compactionplant used). This observation suggests that the use of a constant soil stiffness value withdepth (for a given strain) is reasonably realistic.

3. The stiffness of the soil is influenced by the shear strain in the backfill. The maximumshear strain induced in the backfill as the deck pushes out the abutment a distance δisapproximately 2δ/H, where H is the height of the retained fill. To adopt a single soilstiffness value, an average shear strain must be assumed. The average shear strain in thebackfill must be less than 2δ/H and could conservatively be assumed as about 2δ/3H. Non-linear elastic finite-element analyses by Springman et al. (1996) support the validity of thisassumption.

The conventional spring model represents the backfill soil and soil beneath the abutment by aseries of spring supports. Such a model is imperfect as it does not allow for shear transferwithin the soil as there is no interaction between the

springs. It does, however, have the advantage of simplicity and is considered here because itremains a popular approach among bridge engineers.

An approximate expression, assuming linear elasticity, has been developed for thehorizontal spring stiffness per square metre, of the backfill behind an abutment of depth H andtransverse length, L:

(4.11)

The application of equation (4.11) is illustrated in the following example.

Example 4.3: Conventional spring model

The culvert illustrated in Fig. 4.11 is subjected to an increase in temperature of 20 °C. Theresulting distribution of bending moment is required given that the culvert is made fromconcrete with an elastic modulus of 28×106 kN/m2 and a coefficient of thermal expansion of12×10−6 per °C. The dry density of the backfill,ρd, has been specified as 1600 kg/m3. Thepiles are assumed to provide insignificant lateral restraint to the deck.

It is assumed that the density of the backfill reaches an equilibrium value 20% in excess ofthat specified, i.e.:

Inverting equation (4.9) and assuming Gs=2.65, gives a void ratio of:

Fig. 4.11 Culvert of Example 4.3

To estimate the average shear strain induced in the backfill, the expansion of the culvert isestimated as its unrestrained value, i.e. the product of the temperature increase, the coefficientof thermal expansion and the distance of the abutment from the stationary point (the centre ofthe culvert):

In accordance with Note 3 above, the average shear strain in the affected backfill is then:

On the basis of Note 2, a horizontal stress of p'=50 kN/m2 is assumed. Then equation (4.8)gives:

The horizontal spring stiffness is then given by equation (4.11):

The model for a 1 m strip of the frame is then as illustrated in Fig. 4.12(a). The equivalentloading is:

and the associated distribution of axial force is illustrated in Fig. 4.12(b). The bending

Fig. 4.12 Computer model for culvert of Example 4.3: (a) springs and equivalent loads; (b)associated axial force diagram

Fig. 4.13 Bending moment diagram for Example 4.3

moment diagram was found from a computer analysis and is illustrated in Fig. 4.13. As therewas no associated distribution of bending moment, this is the final distribution of moment dueto the expansion. The moment in the abutments can be seen to change sign through its lengthdue to the flexible nature of the horizontal support. The deflection found from the computeranalysis was 1.19 mm. As this is similar in magnitude to the deflection of 1.20 mm assumedin the estimation of shear strain, iteration was not considered necessary.

4.4 Modelling expansion with an equivalent spring at deck level

An alternative to the conventional spring model is presented here which has a number ofadvantages over the traditional approach. This technique consists of modelling both theabutment and the surrounding soil with an equivalent lateral and rotational spring at decklevel. The approach used to derive the spring constants represented the soil as a completemesh of finite elements rather than a series of springs and is therefore considered theoreticallymore sound than the conventional spring model, described in Section 4.3. This method doesnot, however, provide details concerning the distribution of moment in the abutment or thepressure distribution in the soil.

4.4.1 Development of general expressionLehane (1999) determined the forces and moments associated with lateral displacement androtation of the top of an abutment with retained backfill, i.e. the forces and momentsassociated with passive movements which occur as a consequence of deck expansion. Heconducted a series of finite-element analyses which involved the application at the top of theabutment of (i) a horizontal displacement δwith zero rotation, and (ii) a rotation θwith zerohorizontal displacement (Fig. 4.14).

The purpose of the analyses was to provide credible upper bound estimates of soilresistance. It was therefore assumed conservatively that the soil had limitless compressive andtensile strength (e.g. no passive failure or abutment lifting were allowed) and that no slipbetween the abutment and the soil occurred (e.g. base sliding or slip on the abutment stemwere not permitted). However, given that relatively small movements are required to reducepressures to their minimum (active) values on the inner face of the abutment, the analysesassumed that any soil present on this side did not contribute to the resistance.

Fig. 4.14 Stiffness components at top of abutment: (a) unit translation; (b) unit rotation

It was found that the flexural rigidity of the abutment (EIa) and the ratio, r, defined as:

(4.12)

were the most important factors controlling the magnitudes of the lateral force (Fh) andmoment (M) at the top of the abutment (Fig. 4.14). The values of Fh and M were also seen toincrease systematically as the base width (B) increased and its height (H) reduced.

Best-fit expressions were obtained for Fh and M for the range of parameter values given inTable 4.1. They are given here in matrix form:

(4.13)

where f1 and f2 are functions of the ratio, H/B which are given by equation (4.14) for r>0.05m−3. All values in this stiffness matrix can be reduced by 15% if friction between theabutment and soil is considered negligible.

(4.14)

Fig. 4.15 End part of frame bridge showing locations and directions of fixity

Table 4.1 Range of parameters used in derivation of equation (4.13)

Parameter Allowable rangeEs(kN/m2) 10000−500000

EIa(kNm2/m) 1.0×104−2.5×106

r=Es/EIa(m−3) >0.05

H (m) (Fig. 4.14) 1.5−12

B (m) (Fig. 4.14) 0.5–3.5

For the range of parameters listed in Table 4.1, equation (4.13) was found to predict values ofFh and M to within 10% of the values given by the finite element analyses.

When a frame bridge with an abutment height of H is fixed rigidly at the supports and thesystem of fixities illustrated in Fig. 4.15 is used, the stiffness matrix, [K], in the absence ofsoil, is:

(4.15)

where Ad, Ld and Id are the cross-sectional area, span length and second moment of area of thedeck respectively. When the bridge is embedded in soil and this is taken into account, theterms involving Ia and H are replaced with terms from equation (4.13) with the result thatequation (4.15) becomes:

(4.16)

A comparison of equations (4.15) and (4.16) shows that the influence of soil can be taken intoaccount by analysing a model of a form similar to that illustrated in Fig. 4.15. This couldreadily be achieved in computer analysis programs by allowing the appropriate stiffness termsto be changed in the program to those given in equation (4.16). Alternatively, it is possible toallow for soil in a conventional structural analysis program through the use of an equivalentabutment second moment of area and height and the addition of a horizontal (translational)spring at X. Equating the K22 (second row, second column) terms in equations (4.15) and(4.16), gives:

(4.17)

where Heq and Ieq are the equivalent abutment height and second moment of area respectively.Similarly, equating the K12 (and K21) terms gives:

(4.18)

Equations (4.17) and (4.18) can be simultaneously satisfied by selecting an equivalentabutment second moment of area equal to:

(4.19)

The equivalent abutment height is then:

(4.20)

To make the first terms (K11) equal requires a further adjustment which can be achieved bythe addition of a linear horizontal spring at X of stiffness:

(4.21)

4.4.2 Expansion of frames with deep abutments

The equivalent single-spring model can be simplified for the case of deep abutments. Forvalues of (H/B) in excess of 10, the parameters f1 and f2 approach their minimum values of0.33 and 0.40 respectively. As a result, the equivalent abutment second moment of area can beset equal to the actual second moment of area without great loss of accuracy:

(4.22)

Substituting for f2 in equation (4.20) gives an equivalent height of:

(4.23)

Finally, substituting for f1 and f2 in equation (4.21) gives a spring stiffness of:

(4.24)

These equations can be used to estimate the properties of an equivalent frame for an integralbridge with deep abutments.

Example 4.4: Equivalent single-spring model for frame with deep abutments

The equivalent single-spring model is used to determine the maximum moment in the culvertillustrated in Fig. 4.11 due to a temperature increase of 20°. The concrete has an elasticmodulus of 28×106 kN/m2 and a coefficient of thermal expansion of 12×10−6 per °C. The drydensity of the backfill has been specified as 1600 kg/m3.

The elastic modulus of the soil is found as for Example 4.3 to be:

and the second moment of area of a 1 m strip of the abutment is:

The ratio defined by equation (4.12) is then:

The equivalent height of abutment is then, from equation (4.23):

The stiffness of the single spring on each side is given by equation (4.24):

The equivalent frame and loading are illustrated in Fig. 4.16. The magnitude of theequivalent loads, as for Example 4.3, is:

Fig. 4.16 Computer model for bridge of Example 4.4

Fig. 4.17 Bending moment diagram from computer analysis of bridge of Example 4.4

The associated axial force diagram is as illustrated in Fig. 4.12(b). The model was analysedusing a standard computer program and the resulting bending moment diagram is illustrated inFig. 4.17. It is important to remember that the distribution of moment in the abutment is notrealistic; the true shape of this distribution will be similar to that given in Fig. 4.13. However,the magnitude of moment in the deck, 87 kNm, is likely to be more reliable than the valuefound in Example 4.3.

4.4.3 Expansion of bank seat abutmentsEquations (4.13) and (4.14) imply that an abutment provides a greater resistance to deckexpansion if it has a lesser depth of embedment (H). This implication arises because of theassumption that the soil is an elastic material with infinite strength and that no sliding alongthe abutment base can take place. The reality, of course, is that shallow abutments are morelikely to slide than deep ones and will therefore offer less restraint to deck expansion thanequations (4.13) and (4.14) would suggest.

The influence of a limited soil strength on the resistance offered by a bank seat is illustratedin Fig. 4.18. In Figure 4.18(b), predictions from finite-element analyses are presented of ahorizontal force/deflection relationship. When the soil is linear elastic and infinitely strong,the function is, of course, linear. On the other hand, when the soil is treated as an elasticperfectly plastic material, with a finite strength defined by its friction angle, , thedeflections per unit load can be seen to be significantly greater. Similar results can be shownfor moment/rotation functions and for force/rotation and moment/deflection functions.

In the example of Fig. 4.18, it can be seen that the effective lateral stiffness for a movementat the top of the abutment of 10 mm is only about half that of the purely elastic case. Effectiverotational stiffnesses at this lateral movement are about 75% of the purely elastic case.

It is not possible to generalise the observations made from calculations such as thosesummarised in this figure other than to say that the restraint provided by bank seats will beless than that predicted by equation (4.13). It is therefore recommended that this equation beused in preliminary analysis and that a finite-

Fig. 4.18 Finite-element analysis results for bank seat abutment (Es= 100000 kN/m2, Ec=30×106

kN/m2, foundation bearing pressure= 200 kN/m2, soil friction angle, ): (a)section through bank seat; (b) horizontal force/displacement relationship

element soil/structure analysis incorporating a realistic constitutive model for the soil isperformed if the effects of deck expansion have a significant influence on the final bridgedesign.

Example 4.5: Equivalent single-spring model of bank seatThe equivalent single-spring model is used to determine the maximum moment in the

culvert illustrated in Fig. 4.19 due to a temperature increase of 20°C. The concrete has anelastic modulus of 28×106 kN/m2 and a coefficient of thermal expansion of 12×10−6/°C. Thedry density of the backfill has been specified as 1600 kg/m3.

The elastic modulus for the soil is found in the same manner as for Example 4.3 but usingthe smaller abutment height,

and the second moment of area of a 1 m strip of the abutment is Ia=0.018 m4. The ratiodefined by equation (4.12) is then, r=0.31.

Fig. 4.19 Bridge of Example 4.5

For this example, the ratio of embedment depth to foundation breadth, H/B, is 2.5/3=0.83. Theparameters, f1 and f2 are calculated from equation (4.14):

The equivalent height is then calculated directly from equation (4.20):

The equivalent abutment second moment of area is given by equation (4.19):

Finally, the spring stiffness is, from equation (4.21):

Fig. 4.20 Computer model for bridge of Example 4.5

Fig. 4.21 Bending moment diagram from computer analysis of bridge of Example 4.5

The equivalent frame and loading are illustrated in Fig. 4.20. The magnitude of the equivalentloads, as for Examples 4.3 and 4.4, is:

This model was analysed and the bending moment diagram is illustrated in Fig. 4.21. Themaximum magnitude of moment in the deck due to the expansion is 114 kNm. It is clear fromFig. 4.18 that this result is quite conservative.

4.5 Run-on slab

It has been seen in this chapter that soil provides some restraint against deck movement inintegral bridges but that most of the movement still takes place. On a road bridge, this must beaccommodated if premature deterioration of the pavement is to be avoided. This is achievedin many cases by the installation of a run-on slab as illustrated in Fig. 4.22. The effect of sucha slab is to allow relative rotation between the deck and the run-on slab while preventingrelative translation. Preventing relative vertical translation significantly improves therideability for vehicles travelling over the bridge. Preventing relative horizontal translation isnot so simple. Clearly, the bridge still expands and contracts relative to the surrounding soiland the incorporation of a run-on slab does not prevent this. In effect, it transfers the relativehorizontal movement from the end of the deck to the end of

Fig. 4.22 Run-on slab

the run-on slab. This approach is widely adopted as the failure of a joint at the end of a run-onslab is a minor maintenance problem whereas a leaking joint at the end of a deck can result indeterioration of the bridge itself.

Run-on slabs are designed to span the settlement troughs that develop behind the abutmentsof integral bridges. An asphaltic plug joint positioned at the juncture between the run-on slaband the bridge approach road is commonly used to facilitate horizontal movements.

Settlement troughs arise because of the tendency for cohesionless backfill, whatever itsdensity, to contract and increase in density in response to cyclic straining. Such straining isimposed on the backfill by the abutment which moves in response to thermal movements ofthe deck. Analytical prediction of the shapes and magnitudes of settlement troughs is,however, not commonly attempted by bridge designers. This is because existing modelswhich attempt to simulate the soil’s response to a complex history of cyclic straining are veryapproximate, difficult to use and require measurement of a large range of representativegeotechnical parameters from cyclic laboratory tests.

Settlement profiles may be approximated as having a triangular shape varying from amaximum settlement (δmax) at the abutment to zero at a distance Lt from it. It has been shownby Springman et al. (1996) that, after many cycles of imposed lateral movement δ, δmax variesbetween about 10δand 20δin well-compacted fill for both deep abutments and bank seats.

The assessment of the required length of the run-on slab relies on observations of measuredbehaviour and engineering judgement. Both analytical and model test studies have shown thatthe surface settlement trough tends to an equilibrium profile after a large number of cyclicabutment movements of the same magnitude. Much larger settlements occur in initially loosebackfills where considerable volumetric contractions take place before an ‘equilibrium’density is attained. The extent of the settlement trough is also controlled by the amount ofbackfill subjected to cyclic abutment movements and therefore, for a given movement of thetop of the abutment, could be assumed to vary approximately with the height of the retainedfill (H).

Table 4.2 Approximate upper limits on expected trough lengths

Granular fill

Well compacted Loosely compactedDeep abutments 0.6H 1.4H

Bank seats 0.9H 2.1H

These observations and those taken during centrifuge model tests by Springman et al. (1996)suggest that the length of the trough (Lt) is unlikely to exceed the limits given in Table 4.2.

As an example, the length of run-on slab required for the bridge of Example 4.5 (Fig. 4.19)is calculated. As the backfill is loosely compacted (density= 1600 kg/m3) and the abutmentsare not deep, a maximum trough length of 2.1H can be assumed from Table 4.2. Hence, therun-on slab should have a length of at least 2.1(2.5)=5.25 m.

4.6 Time-dependent effects in composite integral bridges

Many integral bridges are constructed using a combination of precast prestressed beams andin-situ concrete such as illustrated in Fig. 4.23. When the in-situ concrete is cast, the precastbeams are simply supported and the self weight of the bridge induces a sagging moment, asillustrated in Fig. 4.24(a). When the in-situ concrete

Fig. 4.23 Composite integral bridge made from precast and in-situ concrete: (a) elevation; (b)section A—A

subsequently sets, the bridge acts as a frame and imposed traffic loading generates saggingnear the centres of the spans and hogging over the supports (Fig. 4.24(b)). The net result issubstantial sagging near the centres of the spans and some hogging over the supports (Fig.4.24(c)).

Non-prestressed reinforcement is generally provided at the top of the deck over thesupports to resist the hogging moment as illustrated in Fig. 4.25 and it is often necessary toprovide great quantities of closely spaced bars to prevent excessive cracking. Further, currentUK practice is to design to ensure no tensile stress whatsoever in the prestressed beams. Thiscan be quite difficult at points such as A in the figure as these same pretensioned beams mustbe designed to resist substantial sagging moment near mid-span. The resultant prestress forceis therefore designed to be below the centroid near mid-span (Fig. 4.25) to ensure a hoggingprestress moment. Near the supports, the hogging prestress moment combines with hoggingdue to applied loading, making it very difficult to prevent tension in the beams. The problemcan be countered by the debonding of strands near the ends to prevent the prestress force fromacting there. However, this can be quite uneconomical in its use of prestressing strand.

Fig. 4.24 Bending moment diagrams due to short-term loading: (a) due to self weight; (b) due toimposed traffic loading; (c) due to self weight plus traffic loading ((a) plus (b))

Fig. 4.25 Detail near support of composite integral bridge

All of the above effects occur in the short term, i.e. the period immediately following theconstruction of the bridge. In the long term, the distributions of bending moment change dueto creep in the prestressed beams. The equivalent loading due to prestressing strands belowthe beams’ centroid is illustrated in

Fig. 4.26 Equivalent loading due to a prestress force, P, at a mean eccentricity, e: (a) completeprestress force applied at ends; (b) debonding near ends of beam

Fig. 4.27 Effects of prestress on composite integral frame: (a) equivalent prestress loading andbending moment diagram at time of transfer of prestress; (b) equivalent prestressloading and bending moment diagram due to creep strains after frame is madeintegral; (c) total bending moment diagram due to prestress

Fig. 4.28 Detail at support showing points where long-term cracking is likely to occur

Fig. 4.26(a). If some strands are debonded, the equivalent moments at the ends are less but afurther increment of equivalent moment is applied at the points where debonding ceases (Fig.4.26(b)). When prestress is first applied below the centroid, the beams hog upwards asillustrated in Fig. 4.27(a). As they are simply supported, such curvature is unrestrained so itresults in instantaneous strain and a moment which is the simple product of prestress forceand eccentricity. Due to creep, these hogging strains increase with time. When the bridge ismade integral, further curvature is resisted and the resulting distribution of moment is asillustrated in Fig. 4.27(b). The long-term result is a distribution of prestress moment such asthat illustrated in Fig. 4.27(c). This phenomenon is particularly significant if the bridge ismade integral when the precast concrete is young as this causes most of the creep strain tooccur when it is in the integral form. It can result in cracking at the bottom of the deck overthe supports as illustrated in Fig. 4.28, particularly at the interface between the precast and in-situ concretes.

Clark and Sugie (1997) carried out a parametric study of the time-dependent effects incomposite integral bridges. They suggest that there is little point in trying to determine thedistribution of bending moment that develops in the long term as there are fewcreep/shrinkage computer models that give consistently reliable results. In a study ofcontinuous bridges made integral at the interior supports, they calculated the maximum long-term sagging moment for beams made integral when between 21 and 100 days old. Theypropose the assumption of a sagging moment of 750 kNm (per beam) for spans in the 20–36m range where the beams are 1100 mm deep or greater. For smaller beams, they suggestdesigning for a moment of 600 kNm.

Chapter 5Slab bridge decks—behaviour and modelling

5.1 Introduction

The development of a closed-form solution for bridge slabs under the action of applied load isachievable for a limited number of cases, but is generally impractical. Fortunately, slabs canreadily be idealised using one of a number of well-proven methods and analysed usingstructural analysis programs. To understand the basis of such programs and their limitations, itis necessary to first consider the theory of bending of plates.

5.2 Thin-plate theory

Slabs used in the construction of bridge decks are generally thin relative to their span lengths.Such slabs can be assumed to behave like thin plates which can be thought of as the two-dimensional equivalent of beams. Thick plates correspond to deep beams and are notconsidered here. Thin plates get their strength from bending, in a similar way to beams,except that bending takes place in two mutually perpendicular directions in the plane of theplate.

5.2.1 Orthotropic and isotropic platesA material in which the behaviour in each direction is independent of the others is referred toas anisotropic. A subset of anisotropic materials are orthotropic materials in which thebehaviour varies in mutually perpendicular directions (X and Y) only. Orthotropy representsthe most general material behaviour usually considered for bridge decks. A further subset oforthotropic materials are isotropic

materials in which the behaviour in all directions is the same. Although this type of material israrely found in bridge construction, isotropic plate theory can be used with reasonableaccuracy for the analysis of many bridges.

A materially (or naturally) orthotropic plate is composed of a homogeneous material whichhas different elastic properties in two orthogonal directions, but the same geometric properties.This implies that the plate has a uniform thickness and hence the same second moment of areain both directions but different moduli of elasticity. Such a plate might be constructed of amaterial where the microstructure is orientated in two mutually perpendicular directions, suchas timber. This type of plate is not typical of that found in bridge decks but is frequently usedas an approximation of actual conditions.

Many bridge slabs possess different second moments of area in two directions, such asreinforced concrete slabs with significantly different amounts of reinforcement in the twodirections or voided slabs. These types of slab are referred to as geometrically (or technically)orthotropic. In the following sections, the theory of materially orthotropic thin plates isdeveloped. While the theory is strictly only applicable to cases of material orthotropy, it iscommon practise to extend it to include geometric orthotropy. Thus, equations are derivedassuming the plate to have a uniform depth but they are subsequently extended to decks whichhave different second moments of area in orthogonal directions.

5.2.2 Bending of materially orthotropic thin platesFigure 5.1 shows a portion of a thin plate in the X−Y plane. The origin of the axis system is atmid-depth in the plate, at which point, z=0. Figure 5.2 shows a small segment of plate withdimensions δx×δy and a cube of material in that segment a distance z above the origin whichhas a height of, δz. In this figure the thickness of the plate is taken to be d. When a load isapplied, the cube both moves and distorts. Considering initially the X−Z plane, the points a, b,c and d shown in Fig. 5.2 move to a',b', c' and d' as illustrated in Fig. 5.3. The displacement ofpoint a in the X direction is denoted u. Considering point b, a distance δx from a, thedisplacement at that point in the X direction, will be u plus the change in u over the distanceδx, i.e.:

Hence the length of a'b' projected onto the X axis is:

Fig. 5.1 Portion of thin plate and co-ordinate axis system

Fig. 5.2 Segment of thin plate and elemental cube of material

Fig. 5.3 Distortion of cube of material in X- Z plane

By definition, the strain in the X direction is:

(5.1)

Similarly, if v and w are the displacements in the Y and Z directions respectively, it can beshown that:

(5.2)

and:

(5.3)

The shear strain in the X−Z plane is defined as the change in the angle, cab from the original90°, i.e. the difference between c'a'b' and cab. As can be seen in Fig. 5.3, there are twocomponents, α, andβ. Referring to the figure:

As ∂u/∂x is small, this reduces to:

The other component of strain can be found similarly to be:

Hence the shear strain is:

(5.4)

Similarly the shear strains in the X−Y and Y−Z planes are respectively:

(5.5)

(5.6)

In thin-plate theory, a number of assumptions are made to simplify the mathematics involved.The first of these assumptions is that there is no strain in the Z direction, i.e.:

(5.7)

This implies that w is independent of z, or that w is a function of x and y only. Figure 5.4illustrates the implications of this assumption. The physical meaning is

Fig. 5.4 Segment of plate showing uniformity of distortion in Z direction

that there is no compression or extension of the bridge slab in a direction perpendicular to itsplane. In other words, the depth of the slab remains unchanged throughout, and all pointsdeflect vertically by exactly the same amount as the points directly above and below them.Clearly this is a simplification but the strains in the Z direction are generally so small that theyhave negligible effect on the overall behaviour of the bridge slab.

The second assumption which is made is that the deflection of the plate is caused bybending alone and that shear distortion makes no significant contribution, i.e.:

(5.8)

(5.9)

The consequences of this are shown in Fig. 5.5 where it can be seen that the 90° angle of cabis preserved in the distorted c'a'b '. This assumption is again a simplification of the truebehaviour, but is justified by the fact that, bridge slabs being relatively thin, their behaviour isdominated by bending rather than shear deformation. Notwithstanding this, concrete bridgeslabs do not have great shear strength, and although shear strains are small, a means fordetermining shear stresses will be required. Such a method is presented later in this section.

Fig. 5.5 Segment of plate in X−Z plane showing assumed lack of shear distortion

Rearranging equation (5.8) gives:

As w is independent of z, this implies:

(5.10)

where C is a constant of integration. As the origin is located at the centre of the plate andbending is assumed to occur about that point, there is no displacement in either the X or Ydirections at z=0. Hence, at z=0, u and v are both zero. Substituting this into equation (5.10)implies that the constant C is zero giving:

(5.11)

By rearranging equation (5.9), a similar expression can be derived for v:

(5.12)

Substituting equations (5.11) and (5.12) into equations (5.1) and (5.3) respectively gives:

(5.13)

(5.14)

Similarly equation (5.5) gives:

(5.15)

In the flexural theory of beams, the curvature is defined as:

where κis the curvature and R is the radius of curvature. In thin-plate theory, the equationsare similar, but there are now curvatures in the X, Y and XY directions which are given by:

(5.16)

(5.17)

(5.18)

Substituting equations (5.16)–(5.18) into equations (5.13)–(5.15) respectively then gives:

(5.19)

(5.20)

(5.21)

Examination of equation (5.19) shows that strain in the X direction is a linear function of z, asκx=∂2w/∂x2 is independent of z. Equation (5.20) shows that the same applies to the strain in theY direction. From this, it follows that plane sections remain plane, as is generally assumed inbeam theory. This is generally a reasonable assumption for slab bridge decks, but some casesdo exist where this is not so. Such cases are discussed further in Chapter 7.

5.2.3 Stress in materially orthotropic thin plates

In the previous section, expressions were established for the various strains in a thin plate.Expressions are now developed for the corresponding stresses. Figure 5.6 (a) shows a one-dimensional bar subjected to a tensile force. The only significant strain in this system is in adirection parallel to the axis of the bar. This strain, ε, is related to the stress,σ, and modulus ofelasticity, E, by:

In the three-dimensional case, strains in the other two directions become significant, as isindicated in Fig. 5.6(b). By defining the X axis as the direction of the applied force, the strainin that direction is given by:

(5.22)

Fig. 5.6 Distortion in one- and three-dimensions: (a) one-dimensional bar; (b) three-dimensionalbody showing the effect of stress in the axial direction on strains in the orthogonaldirections

where Ex, Ey and Ez are the moduli of elasticity in the X, Y and Z directions respectively, andνx, νy and νz are the corresponding Poisson’s ratios. Equation (5.22) assumes that the plate ismade of a homogenous material and that the elastic constants (Ex, νx, etc.) are independent ofeach other, as is appropriate for the materially orthotropic (or anisotropic) case.

For a thin plate in bending, the stress in the Z direction is small and the Poisson’s ratio isgenerally small for bridge deck materials. Consequently the last term of equation (5.22) canbe ignored. An expression for strain in the X direction for the case of an orthotropic materialwith the elastic constants varying in the X and Y directions is then given by:

and likewise the strain in the Y direction is given by:

In matrix format this becomes:

and by rearranging and inverting the matrix we get:

which yields expressions for the stresses as follows:

(5.23)

(5.24)

The shear modulus, Gxy, is defined as the ratio of shear stress, to shear strain, γxy, whichgives:

(5.25)

Substituting equations (5.19)–(5.21) into equations (5.23)–(5.25) respectively givesexpressions for the stresses in terms of curvature:

(5.26)

(5.27)

(5.28)

5.2.4 Moments in materially orthotropic thin plates

Figure 5.7 shows a small cube taken from a thin plate with the associated normal stressesσx,σy andσz and shear stresses. It is well established that, to satisfy

Fig. 5.7 Elemental cube of material showing normal and shear stresses

equilibrium, pairs of shear stresses must be equal as follows:

(5.29)

Considering the normal stresses first, Fig. 5.8(a) shows a vertical line of cubes (such as that ofFig. 5.7) through the depth of the plate in the X−Z plane. Each of these cubes is subjected to anormal stress in the X direction as indicated in the figure. When there are no in-plane forces ina bridge deck, the sum of the forces in these cubes is zero. As each cube is of the same surfacearea, it follows that:

However, there is a bending moment caused by these stresses. The term mx is used torepresent the moment per unit breadth due to theσx stresses, summed through the depth of thedeck. Figure 5.8(b) shows the depths of the cubes δz and their distances from the origin, z1, z2,z3, etc. Each cube has a width perpendicular to the page of δy (not shown in the figure). Theforces F1, F2, F3, etc., due to each of the stresses are also shown. The ith cube contributes acomponent of hogging bending moment of magnitude (σxiδzδy)zi. Taking sagging moment aspositive and summing over the depth of the plate gives:

(5.30)

Substituting equation (5.26) into equation (5.30) gives:

which gives:

(5.31)

Fig. 5.8 Vertical line of elemental cubes through the depth of a plate: (a) stresses on each cube;(b) forces on the cubes and distances from the origin

Applying a similar method it can be shown that the stress σy causes a moment per unit breadthmy which is given by:

(5.32)

The second moment of area per unit breadth of the plate, i is defined by:

(5.33)

Therefore equations (5.31) and (5.32) can be rewritten in terms of the second moment of areaas follows:

(5.34)

(5.35)

It is important to remember that mx is the moment per unit breadth on a face perpendicular tothe X axis and not about the X axis, i.e. in a reinforced concrete deck it is the moment whichwould be resisted by reinforcement parallel to the X axis. Likewise, my is the moment per unitbreadth on a face perpendicular to the Y axis.

Referring to Fig. 5.7, it can be seen that the shear stresses result in forces parallel to the Yaxis which will also cause a moment. The moment per unit breadth due to is termed mxy.Figure 5.9 shows a number of cubes through the depth of the plate in the Y−Z plane. The shearforce on the face of each cube is given by:

and the moment per unit breadth due to this force is given by:

Taking anti-clockwise as positive on the +X face, the total moment per unit breadth due tois given by:

(5.36)


Fig. 5.9 Stack of elemental cubes in the Y−Z plane showing shear stresses

which gives:

(5.37)

Similarly the moment per unit length, myx, caused by (on the Y face) can be shown to be:

(5.38)

(5.39)

However, as indicated in equation (5.29), equilibrium requires and to be equal andcomparison of equations (5.36) and (5.38) yields:

(5.40)

It follows from the definition of curvature (equation (5.18)) that the two twisting curvaturesare the same:

(5.41)

so there is no contradiction between equations (5.37) and (5.39). These equations can berewritten as:

(5.42)

Fig. 5.10 Bending and twisting moments in a plate: (a) segment of plate and directions ofmoments; (b) associated distortions

where j is known as the torsional constant and is given by:

(5.43)

The moment mxy (=myx) is often referred to as a twisting moment and is distinct from thenormal moments mx and my. Figure 5.10(a) shows the direction in which each of thesemoments acts while Fig. 5.10(b) shows the type of deformation associated with each of them.

5.2.5 Shear in thin plates

Vertical shear forces occur in bridge decks due to the shear stresses, andillustrated in Fig. 5.7. Unlike beams, there are two shear forces at each point, one for eachdirection (X and Y). Defining qx and qy as the downward shear forces per unit breadth on thepositive X and Y faces respectively then gives:

(5.44)

and:

(5.45)

It was assumed earlier (equations (5.8) and (5.9)) that shear deformations in the plate werenegligible. This is a reasonable assumption as shear deformation is generally small in bridgeslabs relative to bending deformation. However, shear stresses, while numerically small, canbe significant, particularly in concrete slabs which are quite weak in shear. In the simpleflexural theory of beams, the same phenomenon exists and an expression is found fromequilibrium of forces on a segment. Figure 5.11 shows a segment of a beam of length dx inbending. The moment and shear force at the left end are M and Q respectively and at the rightend are M+dM and Q+dQ respectively. Taking moments about the left hand end gives:

Rearranging and ignoring the term, dQdx which is relatively small, gives an expression for theshear force Q:

(5.46)

i.e. the shear force is the derivative of the moment. In thin-plate theory, a similar expressioncan be derived.

Fig. 5.11 Equilibrium of small segment of beam

A small element from the plate of base dimensions dx×dy is shown in Fig. 5.12, with varyingbending moment and shear force. The terms qx and qy refer to shear forces per unit breadthwhile mx, my and mxy refer to moments per unit breadth. This is different from the beamexample above where Q and M referred to total shear force and total moment.

Taking moments about the line a–b (Fig. 5.12) gives:

where Fz is the body force acting on the segment of slab (for example, gravity). Dividingacross by dx dy gives:

where fz is the body force per unit area. The second and third terms of this equation representvery small quantities and can be ignored giving:

(5.47)

By taking moments about the line b–c (Fig. 5.12), an equation for qy can be derived in asimilar manner:

(5.48)

It can be seen that the expressions for the shear forces per unit breadth (equations (5.47) and(5.48)) are of a similar form to that for a beam (equation (5.46)) except for the addition of thelast term involving the derivative of mxy or myx.

Fig. 5.12 Equilibrium of small segment of slab

5.3 Grillage analysis of slab decks

The idea of grillage analysis has been around for some time but the method only becamepractical with the increased availability of computers in the 1960s. Although computationalpower has increased many-fold since then, the method is still widely used for bridge deckanalysis. Some of the benefits that have been quoted are that grillage analysis is inexpensiveand easy to use and comprehend. These benefits traditionally favoured the method over finite-element analysis which was typically only used for the most complex problems. In today’senvironment of inexpensive, high-powered computers coupled with elaborate analysisprograms and user-friendly graphical interfaces, the finite-element method has begun toreplace the grillage method in many instances, even for more straightforward bridge decks.That said, the grillage method has proved to be a versatile tool for the analysis of manybridges and benefits from numerous favourable comparisons with experiments such as thoseof West (1973).

The plane grillage method involves the modelling of a bridge slab as a skeletal structuremade up of a mesh of beams lying in one plane. Fig. 5.13(a) shows a simple slab bridge decksupported on a number of discrete bearings at each end and Fig. 5.13(b) shows an equivalentgrillage mesh. Each grillage member represents a portion of the slab, with the longitudinalbeams representing the longitudinal

Fig. 5.13 Grillage idealisation of a slab: (a) original slab; (b) corresponding grillage mesh

stiffness of that part of the slab and the transverse grillage members representing thetransverse stiffness. In this way, the total stiffness of any portion of the slab is represented bytwo grillage members. The grillage mesh and individual beam properties are chosen withreference to the part of the slab which they represent. The aim is that deflections, momentsand shears be identical in both the slab and the grillage model. As the grillage is only anapproximation, this will never be achieved exactly. Clearly different levels of accuracy areacceptable for different applications. For example, a crude representation might be sufficientat the preliminary design stages.

5.3.1 Similitude between grillage and bridge slabIt is necessary to achieve correspondence or similitude between the grillage model and thecorresponding bridge slab. A point p is illustrated in Fig. 5.13 corresponding to the junction oflongitudinal beams b1 and b2 and transverse beams b3 and b4. Figure 5.14 shows an enlargedview of the junction along with the forces and moments acting on beams b1 and b3 in thegrillage. The forces and moments have not been shown on beams b2 and b4 for clarity.

The moments at the ends of beams b1 and b2 adjacent to p in the grillage give a measure ofthe moment mx in the slab while the moments at the ends of beams b3 and

Fig. 5.14 Segment of grillage mesh showing forces and moments on members b1 and b3

b4 give a measure of the moment my. The moments in the grillage members are total momentswhile those which are required in the slab are moments per unit breadth. Therefore, it isnecessary to divide the grillage member moments by the breadth of slab represented by each.This breadth is indicated in Fig. 5.13 as sx and sy for the longitudinal and transverse beamsrespectively. Unfortunately, in the grillage, the moments at the ends of beams b1 and b2adjacent to p are generally not equal, nor are those in beams b3 and b4. For a fine grillagemesh, the difference is generally small, and it is sufficiently accurate to take the averagemoment at the ends of the beams meeting at the junction. The magnitude of this difference isoften used as a check on the accuracy of the grillage, but it should be borne in mind that asmall inequality does not necessarily mean an accurate grillage, as other factors may beinvolved.

The moments per unit breadth in the slab at point p are therefore obtained from the grillageusing the following equations, with reference to Figs. 5.13 and 5.14:

or:

(5.49)

Similarly:

(5.50)

The moments at any other point in the slab can be found in a similar way. If the point is not atthe intersection of longitudinal and transverse grillage members, it is necessary to interpolatebetween adjacent beams. Care should be taken while doing this, especially if a coarse grillagemesh is used. Some computer programs carry out this interpolation automatically, in whichcase it is necessary to confirm that the program has interpolated the results in a sensiblemanner. It is often more convenient to start by considering the locations at which momentswill be required and to formulate the grillage mesh in such a way as to avoid the need forinterpolation between beams.

The twisting moments per unit breadth in the slab, mxy and myx, are found from the torquesin the grillage members in a similar manner. These moments at point p (again with referenceto Figs. 5.13 and 5.14) are given by:

(5.51)

and:

(5.52)

Equation (5.40) stated that mxy and myx are equal for materially orthotropic plates, but thetorques in grillage members b1 and b2 will not necessarily be equal to the torques in b3 andb4. Therefore the twisting moment in the slab is arrived at by averaging the torques per unitbreadth in all four beams meeting at the point p. This may be quite unsatisfactory, as largevariations of torque may exist between the longitudinal and transverse beams, particularly fororthotropic plates with significantly different flexural stiffnesses in the two directions. Thesituation can be improved by choosing torsion constants for the longitudinal and transversebeams which promote similar levels of torque per unit breadth in both. This technique isdiscussed further in the next section.

The shear forces per unit breadth in the slab, qx and qy, are found from the shear forces inthe grillage members in a similar manner to the moments. At point p, (Figs. 5.13 and 5.14)these are given by:

(5.53)

and:

(5.54)

Equations 5.47 and 5.48 gave expressions for the shear forces per unit breadth in the slab.Examining, for example, the shear force Vb1 in Fig. 5.14, it can be seen that this shear forcewill be equal to the derivative of the moment Mb1 with respect to x as this beam will complywith equation (5.46). This accounts for the first term of equation (5.47), but there is noaccount taken in the grillage analysis of the second term, namely, the derivative of myx withrespect to y. This could be calculated in the grillage by finding the derivative of the torques inb3 and b4 with respect to y. However, unless myx is particularly large, this is not normallydone as the resulting inaccuracy in the shear forces tends to be small.

5.3.2 Grillage member properties—isotropic slabsA grillage member in bending behaves according to the well-known flexure formula:

(5.55)

where M is the moment, I the second moment of area, E the modulus of elasticity and R theradius of curvature. By substituting the curvature 1/R withκand rearranging, the moment perunit breadth, m is found:

(5.56)

where i is the second moment of area per unit breadth.

Equation (5.34) gives an expression for the moment per unit breadth in the X direction inthe slab. For an isotropic slab, there is only one value for E andν. Substituting E for Ex and νfor νx andνy in that equation gives:

As Poisson’s ratio, v, is relatively small in bridge slabs (approximately 0.2 for concrete), it iscommon practice to ignore the second term in this equation, giving:

A further simplification is made by equating the term below the line to unity. This can bejustified by the fact that Poisson’s ratio is small. Further, if this approximation is applied toboth mx and my, they are both affected by the same amount. As it is the relative values ofstiffness that affect the calculated bending moments and shear forces, such an adjustment hasvery little effect on the final results. The moment/curvature relationship then becomes:

(5.57)

To achieve similitude of moments between a slab and the corresponding grillage, the stiffnessterms of equations (5.56) and (5.57) must be equated. This can clearly be achieved byadopting the same elastic modulus and second moment of area per unit breadth in the grillageas that of the slab.

A grillage member in torsion behaves according to the well known equation:

(5.58)

where is the angle of twist, T is the torque, l is the length of the beam, G is the shearmodulus and J is the torsion constant (St. Venant constant). Figure 5.15 shows a portion of abeam of length δx in torsion. The displacement in the Z direction is given by w and the angleof twist over the length δx is given by:

Hence:

(5.59)

Fig. 5.15 Segment of beam subjected to torsion


(5.60)

Applying equation (5.58) to the beam of Fig. 5.15 gives:

(5.61)


(5.62)

This can be rewritten in terms of torque per unit breadth, t:

(5.63)

where jgril is the torsion constant per unit breadth in the grillage member.Equation (5.42) gives an expression for the twisting moment per unit breadth in the bridge

slab:

(5.64)

To achieve similitude of moments, mxy, in the slab and torques, t, in the grillage members, thestiffness terms of equations (5.63) and (5.64) must be equated. This can clearly be achievedby adopting the same shear modulus and torsion constant in the grillage member as is in theslab.

Equation (5.43) gives an expression for the torsion constant of the slab. Equating this to jgril

gives:

(5.65)

where d is the slab depth. Equation (5.65) ensures that the grillage members in both directionswill have the same torsional constant per unit breadth. However, they will not necessarilyhave the same total torsional constant as they may represent different breadths of slab if thegrillage member spacing in the longitudinal and transverse directions differ. The torsionconstant for the grillage member can alternatively be expressed in terms of the slab secondmoment of area:

(5.66)

Although equations (5.65) and (5.66) are based on the grillage member having the same shearmodulus as the slab, it will not generally be necessary to specify Gxy for the grillage model.The behaviour of a grillage member is essentially one dimensional and consequently its shearmodulus can be derived from the elastic modulus and Poisson’s ratio directly using the well-known relationship:

(5.67)

Typically, this is carried out automatically by the grillage program.The preceding derivation of grillage member torsional properties is applicable to thin plates

of rectangular cross-section where equation (5.43) for the torsional constant is valid. Torsionin beams is complicated by torsional warping (in all but circular sections) and formulas havebeen developed to determine an equivalent torsional constant for non-rectangular sectionssuch that equation (5.58) can be applied.

For rectangular beams with depth d and a breadth of greater than 10d, the torsional constantmay be approximated with:

(5.68)

It can be seen that equation (5.68) predicts a torsion constant for the beam which is twice thatpredicted by equation (5.66) for isotropic slabs. The reason for this lies in the definition oftorsion in a beam and of moment mxy in a slab. Figure 5.16 shows a portion of a beam ofbreadth b and depth d in torsion. The shear stresses set up in the beam are shown, in both thehorizontal and vertical directions. The torque in the beam results from both of theseshear stresses and is given by:

(5.69)

In the slab, equation (5.36) shows that the moment mxy is arrived at by summing only theshear stresses in the horizontal direction (i.e. only). Consequently the torsion constant for agrillage member representing a portion of an isotropic slab is only half that of a regular beam(or a grillage member representing a regular beam). In the slab, the shear stresses in thevertical direction are accounted for by the shear force per unit breadth, qx as illustrated inFig. 5.17. It has been recommended that the edge grillage members be placed at 0.3 times theslab depth from the edge so as to coincide with the resultant of the shear stresses. The verticalshear stresses are accounted for in the grillage in the same manner by the shear forces qy inthe transverse beams.

5.3.3 Grillage member properties—geometrically orthotropic slabsEquation (5.34), reproduced here, applies to materially orthotropic slabs:

However, most bridges have the same modulus of elasticity, E, for both directions. Further,many bridges are geometrically orthotropic, i.e. they have different second moments of areaper unit breadth in the orthogonal directions. It is common practise to use the equationsdeveloped for materially orthotropic thin plates to represent geometrically orthotropic bridges.This is achieved in a grillage by basing the second moment of area per unit breadth of thegrillage members in the X direction on that of the slab in that direction. Similarly, in the Ydirection, the second moments of area per unit breadth for the grillage and the slab areequated.

Fig. 5.16 Beam subjected to torsion showing resulting shear stresses

Equation (5.40) stated that the two twisting moments at a point in a materially orthotropic slabare equal to each other. Further, as stated in equation (5.41), the two twisting curvatures arethe same. If it is assumed that the same conditions hold for geometrically orthotropic slabs,i.e.:

and:

There is no facility in a grillage model to ensure that the two curvatures at a point are equal.However, in a fine grillage mesh, curvatures in the orthogonal directions at a point will beapproximately equal. Then, if the same shear modulus and torsional constant are used in thetwo directions, it follows from equation (5.42),

Fig. 5.17 Slab with vertical shear stresses and corresponding grillage members with shear forcesper unit breadth

reproduced and adapted here as equation (5.70), that the twisting moments are equal:

(5.70)

Hambly (1991) recommends using such a single torsional constant for both orthogonaldirections:

(5.71)

It can be seen that this equation is consistent with equation (5.66) for an isotropic slab. Theshear modulus for a slab made from one material, G, is a function of the elastic modulus, E,and Poisson’s ratio, ν. It is generally calculated internally in computer programs usingequation (5.67).

5.3.4 Computer implementation of grillages

There are many computer programs commercially available which are capable of

analysing grillages. These programs are generally based on the same theory, that of thestiffness method, with some variations from program to program.

The computer implementation of a plane grillage consists of defining a mesh ofinterconnected beams lying in one plane. The points at which these beams are connected arereferred to as nodes. Each node has the capability to deflect vertically out-of-plane or to rotateabout each axis of the plane. There is no facility for the nodes to deflect in either of the in-plane directions or to rotate about an axis perpendicular to the plane. The nodes are thereforesaid to have three degrees of freedom, two rotations and one translation. Consequently, in-plane axial forces are not modelled by the grillage. This inhibits the calculation of in-planeeffects such as axial thermal expansion or contraction or in-plane prestressing. Such effectsare normally determined separately (often by hand due to their simplicity) and added toresults from the grillage, according to the principle of superposition.

Some grillage programs allow, or require, the definition of a cross-sectional area for thebeams. This may be used to define the bridge self weight. In such cases, care should be takento ensure that the self weight is not applied twice by applying it to both the longitudinal andtransverse beams. Some programs also use the cross-sectional area definition to model sheardeformation. Even though the thin plate behaviour considered in Section 5.2 assumed thatthere was no shear deformation, some grillage programs do allow for shear deformation. Thisis generally achieved by defining a cross-sectional area and a shear factor, the product ofwhich gives the shear area. While shear deformation is generally not very significant intypical bridges, it should improve the accuracy of the results if it is allowed for in thecomputer model. Some programs which allow the modelling of shear deformation will onlygive results of shear stresses when this option is invoked.

Grillage programs model the supports to the bridge slab as restraints at various nodes. Ittherefore makes sense, when formulating the grillage, to locate nodes at the centres of thebearings or supports. These nodal supports may be rigid, allowing no displacement or rotationin either of the two directions, or may allow one or more of these degrees of freedom. Mostgrillage programs will allow the use of spring supports, and the imposition of specific supportsettlements. These facilities may be used to model the soil/structure interaction as discussed inChapter 4.

5.3.5 Sources of inaccuracy in grillage models

It should always be borne in mind that the grillage analogy is only an approximation of thereal bridge slab. Where the grillage is formulated without regard to the nature of the bridgeslab, this approximation may be quite inaccurate, but when used correctly it will accuratelypredict the true behaviour. However, even if due care is taken, some inherent inaccuraciesexist in the grillage, a number of which are described here.

It has been pointed out that the moments in two longitudinal or two transverse grillagemembers meeting end to end at a node will not necessarily be equal. The discontinuitybetween moments will be balanced by a discontinuity of torques in the beams in the oppositedirection to preserve moment equilibrium at

the node. Where only three beams meet at a node, such as where two longitudinal beamsalong the edge of a grillage meet only one transverse beam, this discontinuity will beexaggerated. This is illustrated in Fig. 5.18 where it can be seen that the torque T in thetransverse beam, having no other transverse beam to balance it, corresponds to thediscontinuity between the moments Mb1 and Mb2 in the longitudinal beams. The requiredmoment is arrived at by averaging the moments on either side of the node. The magnitude ofthese discontinuities can be reduced by choosing a finer grillage mesh. The same phenomenoncauses discontinuities in torques and shears, which should be treated in the same manner. Aswas mentioned earlier, excessively large discontinuities in moments, torques or shearsindicate a grillage mesh which is too coarse, and requires the addition of more beams. Theopposite of this is not necessarily true, as other factors may also have an effect.

Equation (5.34) gave an expression for moment per unit breadth, mx, in the slab. Thisexpression involved terms accounting for the curvature in the X and Y directions. Whenderiving the properties of a grillage member parallel to the X axis, the effect of curvature inthe Y direction was ignored (see equation (5.57)). A similar simplification was made for my.As a result of this, the curvatures in the grillage members in one direction do not effect themoments in the beams in the other direction in the same manner as they do in the bridge slab.This potential inconsistency is reduced by the low Poisson’s ratio of bridge slab materialswhich limits the influence of curvatures in one direction on moments in the orthogonaldirection.

Equation (5.40) stated that the moments mxy and myx are equal in a slab, as are thecorresponding curvatures in the two directions. There is no mathematical or physical principlein the grillage to make this so. Torsions per unit breadth of similar magnitude in bothdirections in a grillage can be promoted by choosing the same torsional constant per unitbreadth for the longitudinal and transverse beams. However, significant differences canremain.

Fig. 5.18 Distribution of bending moment in a segment of grillage mesh showing discontinuity inmoment (Tb3=Mb1−Mb2)

Equations (5.47) and (5.48) provide expressions for the shear forces per unit breadth, qx andqy. The first of these equates the shear force per unit breadth qx to the sum of two derivatives:

In the grillage, the shear force in a longitudinal or transverse beam will simply be thederivative of the moment in that beam with respect to X or Y, whichever direction the beamlies in. There is no account taken of the derivative of the twisting moments, mxy or myx.Fortunately, except for bridges with high skew, the magnitude of these moments is generallyrelatively small.

5.3.6 Shear force near point supports

There is a particular problem in using grillage models to determine the intensity of shear force(shear force per unit breadth) near a discrete bearing. When bridges are supported at discreteintervals, there are sharp concentrations of shear intensity near each support. Each grillagemember represents a strip of slab with the result that a point support at a node in a grillagemodel has an effective finite breadth. It follows that, if the grillage mesh density increases, theeffective breadth decreases and the calculated concentration of shear adjacent to the supportincreases. This direct relationship between mesh density and the calculated maximum shearintensity means that, if reasonably accurate results are to be obtained, the grillage memberspacing has to be fixed near the support so that it gives the correct result.

O’Brien (1997) found that the grillage member spacing had a much reduced influence onthe results for shear at distances of more than a deck depth from the support. If it wereassumed that shear enhancement was sufficient to cater for local concentrations of shear neara support, then grillage member spacing would assume a much reduced importance. Thus thedesigner would design for the shear force calculated at a deck depth from the support. Greatershear forces at points closer to the support would be ignored on the basis that load would becarried by direct compression rather than shear mechanisms.

5.3.7 Recommendations for grillage modelling

It is difficult to make specific recommendations on the use of a technique such as grillagemodelling, which is applicable to such a wide variety of structural forms. However, somegeneral recommendations are valid for most grillage models. These should not be viewed asabsolute, and should be used in the context of good engineering judgement. Some morespecific recommendations, such as those relating to voided or skewed bridge decks, are givenin Chapter 6. It will be seen from the recommendations given here that the traditional need foreconomy in the

numbers of grillage members no longer applies, as the computational power available totoday’s engineers is well in excess of that available when earlier recommendations were made.Nonetheless, there is no advantage in providing excessive numbers of grillage members as theamount of output data will be excessive, and, beyond a certain point, no additional accuracywill be achieved.

1. Longitudinal grillage members should be provided along lines of strength in the bridge slab,should these exist. Lines of strength may consist of concentrations of reinforcement,location of prestressing tendons, or precast beams in beam-and-slab bridges.

2. Where possible, grillage members should be located such that nodes coincide with thelocations of supports to the bridge slab. The procedure of moving nodes locally to coincidewith supports, illustrated in Fig. 5.19(a), should be avoided if possible, as this may result inskewed members which complicate the interpretation of results.

3. There is little point in having longitudinal beams too closely spaced. Spacing will often bedictated by the location of supports or lines of strength in the bridge slab. A reasonablespacing of longitudinal beams is between one and three times the slab depth. However,significantly greater spacings are often possible without great loss of accuracy, particularlyin wide bridge slabs.

4. Transverse beams should have a spacing which is similar to that of the longitudinal beams.Often this spacing will be greater than that of the longitudinal beams, as the magnitude ofmoment in the transverse beams is generally relatively small. A choice of between one andthree times the longitudinal spacing would be reasonable. The transverse grillage members

Fig. 5.19 Alternative grillage meshes near point supports: (a) local adjustment to mesh nearsupports to maintain constant spacing of members elsewhere; (b) non-constant meshspacing

should also be chosen to coincide with lines of transverse strength in the bridge slab,should they exist, such as heavily reinforced diaphragms above bridge piers.

5. If the spacing of grillage members is in doubt, a check can be performed by comparing theoutput of a grillage with that from a more refined grillage, i.e. one with more longitudinaland transverse beams at a closer spacing. For bending moment results, increasing the meshdensity tends (up to a point) to increase the accuracy.

6. It has been recommended by Hambly (1991) that the row of longitudinal beams at eachedge of the grillage should be located in a distance of 0.3d from the edge of the slab, whered is the slab depth. The objective is to locate these beams close to the resultant of thevertical shear stresses, , in the bridge slab as illustrated in Fig. 5.17. It has also beenrecommended that, when determining the torsional constant of these longitudinal grillagemembers, the breadth of slab outside 0.3d should be ignored. The second moments of areaof these beams are calculated using the full breadth of slab in the normal way. The validityof this recommendation has been confirmed by the authors through comparisons of grillageanalysis results with those of elaborate three-dimensional finite-element models. Careshould be taken, however, that this recommendation does not result in supports beingplaced in the wrong locations. Figure 5.20 illustrates an example where a member iscorrectly placed more than 0.3d from the end, so that the span length between supports inthe grillage and the bridge slab are the same.

7. Supports to the grillage should be chosen to closely resemble those of the bridge slab. Thismay involve, for example, the use of elastic springs to

Fig. 5.20 Segment of grillage mesh showing longitudinal members 0.3d from the edge except forthe end transverse members

simulate deformable bearings or ground conditions as discussed in Chapter 4.8. Beyond a deck depth from the face of the support, reasonable accuracy can be achieved

with most sensible member spacings. Closer to the support, where shear enhancementoccurs, grillage analysis is much less reliable.

5.4 Planar finite-element analysis of slab decks

The finite-element (FE) method was pioneered in the mid 1950s for use mainly in theaeronautical industry. Originally it was used for in-plane analysis of structures but it was soonextended to the problem of plate bending by Zienkiewicz and Cheung (1964). Muchdevelopment has taken place since this pioneering work and many texts now exist which givea comprehensive description of the method (see, for example, that of Zienkiewicz and Taylor(1989)). Finite-element analysis is relatively easy to use and comprehend and, when appliedcorrectly, is at least as accurate as, and often more accurate than, the grillage method.

Finite-element analysis is well known to bridge designers, some of whom consider it to bethe most general and accurate method available for bridge deck analysis while others view itwith a degree of scepticism. The authors have used the method extensively for the analysis ofbridge decks and have found it to be an excellent analysis tool in many cases. This said, thescepticism expressed by some bridge designers is quite often well founded as the perceivedaccuracy of the method often overshadows the importance of using it correctly. There is a riskthat inexperienced users will attempt to analyse complex bridges without understanding thetrue nature and behaviour of the structure. A useful method of gaining familiarisation with aspecific FE program is to begin by analysing simple structures, the behaviour of which isknown, and then to progress to more complex structures.

When applied to the analysis of slab bridge decks, the FE method involves the modelling ofa continuous bridge slab as a finite number of discrete segments of slab or ‘elements’. All ofthe elements generally lie in the one plane and are interconnected at a finite number of pointsknown as nodes. The most common types of element used are quadrilateral in shape althoughtriangular elements are sometimes also necessary. Some elements do not model in-planedistortion and consequently the nodes have only three degrees of freedom, namely out-of-plane translation, and rotation about both in-plane axes. No particular problem arises fromusing elements which allow in-plane deformations in addition to out-of-plane bending, but thesupport arrangement chosen for the model must be such that the model is restrained from freebody motion in either of the in-plane directions or rotation in that plane. Such analyses areonly necessary if it is specifically required to model in-plane effects, such as axial prestress.Sometimes it is more convenient to carry out an FE analysis with out-of-plane deformationonly and to add the in-plane effect of prestress afterwards (which may often be determined byhand). Finiteelement models in which the elements are not all located in the one plane can beused to model bridge decks which exhibit significant three-dimensional behaviour. Some ofthese types of model are discussed in Chapter 7.

5.4.1 Similitude between finite-element model and bridge slab

The moments per unit breadth, mx, my and mxy, are output directly by FE programs. These aregenerally given at the element centres and/or corners. Many programs provide the ability todetermine these values at any arbitrary point using interpolation. If this facility is used, acheck is useful to ensure that the values given are consistent with those at the neighbouringnodes. Equations (5.34) and (5.35) give expressions for the moments mx and my in a thin plate.Each of these expressions involves terms relating to the curvature in both the X and Ydirections. The finite elements will behave according to these equations, and unlike a grillageanalysis, will account for the effect of curvature in one direction on the stiffness in the otherdirection. This is a significant advantage of the FE method over the grillage approach.Equation (5.42) gave an expression for the moments mxy and myx in a thin materiallyorthotropic plate. The finite elements will satisfy this equation, and the problem inherent ingrillage modelling of torques per unit breadth not being equal in orthogonal directions doesnot arise. Finally, equations (5.47) and (5.48) give expressions for the shear force per unitbreadth in a thin plate. These expressions involve derivatives of the direct moment mx (or my)and the twisting moment myx (or mxy). It was shown above that a grillage model does not takeaccount of the derivative of the twisting moment. In FE analysis, shear force per unit breadthcan be calculated, although not all programs offer this facility. The twisting moment term canreadily be accounted for, although in some programs it may not be. Where the twistingmoments are significant, it is advisable to determine whether or not shear forces are calculatedcorrectly using equations (5.47) and (5.48).

5.4.2 Properties of finite elements

The types of finite element considered here are those used for the modelling of slab bridgedecks. These are plate elements which can model out-of-plane bending, in-plane distortion ora combination of both of these. The material properties of the elements are defined in relationto the material properties of the bridge slab. When materially orthotropic finite elements areused, five elastic constants, Ex, Ey, Gxy, νx, and νy, typically need to be specified. Someprograms assume a value for Gxy based on the values input for the other four elastic constants.If this is the case, the validity of this relationship should be checked for the particular plateunder consideration.

Isotropic bridge slabs

In the case of bridges which are idealised as isotropic plates, only two elastic constants needto be defined for the finite elements, E and ν. The shear modulus, G, is determined by theprogram from these constants directly according to equation (5.67). As the element is ofconstant depth, the second moment of area per unit breadth is given by equation (5.33):

In a typical program, the user simply specifies the element depth as:

(5.72)

Geometrically orthotropic bridge slabs

Geometrically orthotropic bridge decks are frequently modelled using materially orthotropicfinite elements. In such cases, ix≠iy, but only one depth can be specified. This problem can beovercome by determining an equivalent plate depth and altering the moduli of elasticity of theelement to allow for the differences in second moments of area.

Equation (5.34) gives an expression for the moment, mx, which will be satisfied by amaterially orthotropic finite element:

where and ielem are the element elastic modulus and second moment of area per unitbreadth respectively. Equation (5.35) gives a similar expression for my. In most geometricallyorthotropic bridge slabs, there is only one modulus of elasticity, Eslab, for both directions, butthere are two second moments of area per unit breadth, and . However, similitudebetween the finite element and the bridge slab can be achieved by keeping the products ofelastic modulus and second moment of area equal:

(5.73)

(5.74)

The modulus of elasticity of the element in the X direction may be chosen arbitrarily to beequal to the modulus of elasticity of the bridge slab, i.e.:

(5.75)

Substituting this into equations (5.73) and (5.74) gives:

(5.76)

and

(5.77)

The equivalent element depth can be calculated from equation (5.72).

For a materially orthotropic slab, the moment/curvature relationship for the twistingmoment, mxy, is given by equation (5.42). An approximate expression for the constant, Gxy,has been suggested by Troitsky (1967):

(5.78)

For a geometrically orthotropic slab with a single modulus of elasticity and Poisson’s ratio, asimilar expression can be determined by substituting from equations (5.73) and (5.74) to give:

(5.79)

To be consistent with the equations for and ielem derived above, the modulus of elasticityof the element in the X direction is taken to be equal to the modulus of elasticity of the bridgeslab. Then, equation (5.76) applies and equation (5.79) becomes:

(5.80)

Equation (5.78) was derived by assuming an average value of the elastic moduli in the twodirections and an average Poisson’s ratio. Consequently the accuracy of this and equation(5.80) diminishes as the variation in the elastic properties in the two directions increases. Insuch cases the shear modulus may need to be reduced. It was reported by Troitsky (1967),from the results of analysis and experimentation on steel orthotropic bridge decks, that theshear modulus given by the above expression may need to be reduced by a factor of between0.5 and 0.3. The lower value of 0.3 was reported to come from an extreme case where theflexural stiffness in the two directions varied by a factor of 20.

To determine if the influence of the shear modulus on the analysis is significant, the authorswould suggest analysing the orthotropic plate using a value predicted by equation (5.80) andanalysing again using a shear modulus of half this value. As an alternative, the orthotropicnature of the plate might be better handled using a combination of elements and beammembers or a three-dimensional model. These types of model are discussed further inChapters 6 and 7.

Instead of arbitrarily equating the modulus of elasticity of the finite element in the Xdirection to the corresponding modulus of the slab, the moduli in the Y direction could beequated. This would lead to alternative expressions to the above. Alternatively, an arbitrarydepth of finite element could be chosen (say, a depth that would result in a second moment ofarea equal to the average of the second moments of area of the bridge slab in the twodirections) and expressions determined for the corresponding values of the moduli ofelasticity of the element.

The expressions given above relate to bridge slabs with the same modulus of elasticity in bothdirections, but can easily be modified where this is not the case.

5.4.3 Recommendations for finite-element analysis

There are many commercially available computer programs for FE analysis of bridge decks.Quite often the same program can be used for grillage and FE analysis which saves the userhaving to become familiar with two separate programs. The implementation of the FE modelis carried out in a similar manner to a grillage and many of the comments in Section 5.3 apply.One variation between the two methods is that the FE model may allow for in-planedeformations and consequently the nodes will often have five or six degrees of freedom. Thistype of model is useful where in-plane effects (such as axial prestress) are to be considered.

As with grillage modelling, it is difficult to make specific recommendations relating to FEmodelling of bridge slabs but some general guidelines are given here. Once again these shouldnot be viewed as absolute. In contrast to grillage modelling, it may become necessary to limitthe number of elements, as some programs may not be able to deal with excessive numbers.In general, more elements tend to result in greater accuracy although this is by no meansguaranteed. Many engineers use denser meshes of elements in those parts of a bridge wherebending moment changes rapidly such as near an interior support. However, it is often moreconvenient if a consistent mesh density is used throughout a bridge.

Unlike the grillage method, the finite element response to applied loading is based on anassumed displacement function. This function may be applicable to elements of a certainshape only, and quite often the program will allow the user to define elements which do notconform to this shape. Considering, for example, quadrilateral elements with nodes at the fourcorners, a typical program may be able to deal with elements of the type shown in Fig. 5.21(a)but may give an inaccurate representation for the elements shown in Fig. 5.21(b). Morespecific recommendations are given below and further guidance, applicable to voided andskewed bridge decks, is given in Chapter 6.

1. Regularly shaped finite elements should be used where possible. These should tend towardssquares in the case of quadrilateral elements and towards equilateral triangles in the case oftriangles. Obviously, considerable deviation from these shapes may be permissible and thedocumentation provided with the program should be consulted for specificrecommendations. In the absence of information to the contrary, two rules commonlyapplied to quadrilateral elements are that the ratios of the perpendicular lengths of the sidesshould not exceed about 2:1 and that no two sides should have an internal angle greaterthan about 135°.

2. Mesh discontinuities should be avoided. These may occur when attempting to refine themesh such as in Fig. 5.22(a) where elements (1) and (2) are connected to each other at pointP but are not connected to element (4). Some elements have mid-side nodes so that it ispossible for example to have

Fig. 5.21 Possible shapes of quadrilateral finite elements: (a) generally good shapes; (b)potentially problematical shapes

Fig. 5.22 Meshes of finite elements at transition between coarse and dense mesh: (a) potentiallyproblematic arrangement; (b) good arrangement

elements (3) and (4) connected to the mid-side node of element (1) at Q. A mesh isshown in Fig. 5.22(b) where mid-side nodes are not needed and all elements areconnected.

3. The spacing of elements in the longitudinal and transverse directions should be similar.This will be complied with if the first recommendation is adhered to.

4. There is little point in using too many elements as an excessive number slows the runningof the program and may not result in significantly greater accuracy. If mesh density is inquestion, it is useful to compare the output of a model with the chosen mesh density to thatof a model with a greater density. Similar results from both would suggest that the meshwas sufficiently dense.

5. Elements should be located so that nodes coincide with the bearing locations. This isgenerally easily achieved.

6. Supports to the finite-element model should be chosen to closely resemble those of thebridge slab. This may involve, for example, the use of elastic springs to simulatedeformable bearings or ground conditions as discussed in Chapter 4.

7. Shear forces near points of support in finite-element models tend to be unrealistically largeand should be treated with scepticism. However, results at more than a deck depth awayfrom the support have been found in many cases to be reasonably accurate (O’Brien et al.1997).

5.5 Wood and Armer equations

Much of this chapter has been concerned with methods of analysis of slab bridges. The resultsof such analyses give three components of bending moment at each point, mx, my and mxy.This section addresses the design problem of how the engineer should calculate the momentcapacity required to resist such moments. As bending moment is a vector, the threecomponents can be combined using vector addition in a manner similar to the concept ofMohr’s circle of stresses. Resultant moments can be calculated at any angle of orientation andcan, if excessive, result in yield of the slab at any such angle.

A small segment of slab is illustrated in Fig. 5.23(a) and the possibility is considered offailure on a face, AB, at an angle of θto the Y axis. The length of the face AB is l and, as canbe seen in the figure, the projected lengths on the X and Y axes are l sin θand l cos θrespectively. The moment per unit length on the X face is mx so the moment on BC is mxl cosθ. The corresponding moment on AC is myl sin θ. These moments are illustrated in Fig.5.23(b) using double headed arrows to denote bending moment, where the moment is aboutthe axis of the arrow. The twisting moments per unit length, mxy and myx, are also illustrated inthis figure.

The vectors representing the moments are resolved to determine the moments on the faceAB. For convenience, a second axis system, N–T, is introduced where N is normal to the faceAB and T is parallel (tangential) to it. The direct moment per unit length on AB is denoted mnand the twisting moment per unit length is denoted mnt. All vectors are resolved parallel andperpendicular to AB in Fig. 5.24.

Fig. 5.23 Segment of slab: (a) geometry; (b) applied bending and twisting moments

Considering components parallel to AB first:

(5.81)

Considering components perpendicular to AB gives:

(5.82)

The components of moment on a face perpendicular to AB are considered in Fig. 5.25 whereresolution of components gives:

(5.83)

and:

(5.84)

Fig. 5.24 Resolution of moments on a segment of slab parallel and perpendicular to AB

Fig. 5.25 Resolution of moments on a face perpendicular to AB

A comparison of equations (5.82) and (5.84) verifies that mnt and mtn are equal. Equations(5.81)–(5.84) can be used to resolve all components of moment on a small segment of plateinto a new axis system as illustrated in Fig. 5.26.

In an orthotropic steel plate, moment capacity is generally provided in the two orthogonaldirections. In a concrete slab, ordinary or prestressing reinforcement is provided in twodirections, which are not necessarily orthogonal. In this section, only orthogonal systems ofreinforcement are considered; similar equations for non-orthogonal systems are given byClark (1983). Furthermore, only the case in which mn is positive is considered here. The casewhen mn is negative is also treated by Clark.

Fig. 5.26 Transformation of applied moments to an alternative co-ordinate system: (a) momentsin X-Ysystem; (b) moments in N-T system

An orthogonal system of reinforcement provides moment capacity in two perpendiculardirections which are taken here to be parallel to the co-ordinate axes. Hence, the momentcapacities per unit length can be expressed as and as illustrated in Fig. 5.27. This figureis different from Fig. 5.26(a) in that there are no twisting moment terms; no capacity to resisttwisting moment is assumed to be provided. Equation (5.81) gives the moment on a face at anangle θto the Y axis. A corresponding equation can readily be derived for the momentcapacity. Leaving out the mxy term in equation (5.81) leads to:

(5.85)

Fig. 5.27 Segment of slab illustrating the moment capacities provided

While no capacity to resist twisting moment is explicitly provided, capacity can be shown toexist on face AB (Fig. 5.23(a)) by considering equation (5.82) which gives:

(5.86)

Similarly, from equation (5.83):

(5.87)

To prevent failure on face AB of Fig. 5.23, the moment capacity must exceed the appliedmoment. As only the case for which mn is positive is being considered, this becomes:

Substituting from equations (5.81) and (5.85) gives:

Dividing the equation by cos2 θgives:

This can be expressed as:

where

(5.88)

and

The function, f(k), is the excess moment capacity for the angleθ, i.e. the amount by which themoment capacity exceeds the applied moment for that angle. To prevent failure of the slab, itis clearly necessary that this function exceeds zero for all values of θ. The most critical anglewill be that for which f(k) is a minimum. This minimum value is found by differentiating thefunction and equating to zero, that is:

As k=tanθ, differentiating with respect toθgives:

which is never zero. Hence the minimum value for f (k) occurs when:

(5.89)

where is a critical value for k. For this to be a minimum excess moment capacity rather thana maximum, the second derivative of f (k) must be positive, i.e.:

(5.90)

Taking equations (5.89) and (5.90) together, it can be seen that and mxy must be of the samesign. This fact will be shown to be of significance later in the derivation.

Example 5.1: Moment capacity check

At a point in a bridge slab, the moments per unit length due to applied loads have been foundto be, mx=190, my=80 and mxy=20. It is required to determine if it is sufficient to providemoment capacities of, and

Equation (5.89) is used to determine the angle for which the excess moment capacity isminimum:

i.e. the critical angle is 29.7°. The minimum excess capacity is then found by substitution inequation (5.88):

As the excess capacity is negative, the slab will fail for this value of θ.

When new bridges are being designed, the moment capacities are not generally known inadvance and the problem is one of selecting sufficiently large values for and . It can beseen from equation (5.89) that effectively dictates the value for for a particular set ofmoments, i.e. choosing amounts to choosing . Thus the designer’s problem can beviewed as one of choosing a suitable value for provided that equation (5.89) is satisfied, i.e.choosing such that:

(5.91)

It is, of course, also necessary to have a positive excess moment capacity. The minimumrequired excess moment capacity is:

Substituting from equation (5.91), this becomes:

It was established earlier (by comparing equations (5.89) and (5.90)) that and mxy were ofthe same sign. Hence, their product is positive giving:

(5.92)

Similarly, equation (5.91) becomes:

(5.93)

Any value for can be selected by the designer and these equations used to determine theminimum required moment capacities. The cost of providing moment capacity in the two co-ordinate directions may not necessarily be equal as a bridge may, for example, be prestressedin one direction and reinforced with ordinary reinforcement in the other. In general, the costof providing moment capacity at a point may be taken to be proportional to:

The value for which results in minimum cost is found by differentiating:

(5.94)

This can be used to find an economical value for in equations (5.92) and (5.93). If the costof providing moment capacity is the same in both directions, thenρ=1 and equations (5.92)and (5.93) become:

(5.95)

(5.96)

These are known as the Wood and Armer equations (Wood, 1968).

Example 5.2: Wood and Armer equations II

At a point in a bridge slab the moments per unit length due to applied loads have been foundto be, mx=190, my=80 and mxy=20. It is required to determine economical moment capacitiesgiven that providing costs twice that of providing .

As cost is proportional to the constant, ρ, is 0.5 and the minimum cost value for thecritical angle is defined by:

i.e.

Equations (5.92) and (5.93) then give the required moment capacities:

Chapter 6Application of planar grillage and finite-element

methods

6.1 Introduction

In Chapter 5, the behaviour of bridge slabs is considered. Two methods of analysis areintroduced, grillage and finite-element methods, both of which consist of members lying inone plane only. In this chapter, both of these planar methods of analysis are used to model arange of bridge forms. Planar methods are among the most popular methods currentlyavailable for the analysis of slab bridges. They can, with adaptation, be applied to manydifferent types of slab as will be demonstrated. Further, their basis is well understood and theresults are considered to be of acceptable accuracy for most bridges.

In Chapter 7, more complex non-planar methods of analysis are considered. For certainbridges, non-planar models are considerably more accurate than planar models. However,they can also be considerably more complex and can take much longer to set up. For thisreason, planar grillage and finite-element models are at present the method of choice of agreat many bridge designers for most bridge slabs.

6.2 Simple isotropic slabs

When bridge slabs are truly planar, it is a simple matter to prepare a computer modelfollowing the guidelines specified in Chapter 5. This will be demonstrated in the followingexamples.

Example 6.1: Grillage model of two-span right slab

A two-span bridge deck is illustrated in Fig. 6.1. It is to be constructed of prestressed concreteand is to have a uniform rectangular cross-section of 0.8 m depth. The deck is supported onfour bearings at either end and on two bearings at the centre as illustrated in the figure. Acombination of fixed, free-sliding and guided-sliding bearings is used so that the bridge canexpand or contract freely in all directions in plane. It is required to design a grillage mesh toaccurately represent the deck given that the concrete has a modulus of elasticity of 35×106

kN/m2.Figure 6.2 (a) shows a convenient grillage mesh for this bridge deck. The longitudinal

members have been placed along the lines of the bearings, with an additional line at the centreof the deck. As recommended in Section 5.3, a row of longitudinal members has been placedat a distance of 0.3 times the depth from the edge of the slab. The transverse members havebeen placed at a spacing of 1.5 m which gives a ratio of transverse to longitudinal spacing ofbetween 1.2 and 1.5. The end rows of transverse members are taken through the centres of thebearings.

Fig. 6.1 Plan view of two-span bridge

Fig. 6.2 Grillage mesh for bridge of Fig. 6.1: (a) plan; (b) section

Table 6.1 Grillage member properties for Example 6.1

Second moment of area (m4) Torsion constant (m4)Longitudinal members

R1, R9 0.0371 0.0537

R2, R8 0.0483 0.0964

R3, R7 0.0470 0.0938

R4, R6 0.0491 0.0981

R5 0.0470 0.0938

Transverse Members

End members 0.0534 0.0862

All intermediate members 0.0640 0.1280

Figure 6.2 (b) shows a cross-section of the slab with the grillage members superimposed. Thisis used to determine the breadth of slab attributable to each longitudinal grillage member. Itcan be seen that this breadth is taken to be from midway between adjacent members on eitherside. The bridge slab is assumed to be isotropic and the second moments of area per unitbreadth are taken to be equal to those of the slab:

The torsion constants per unit breadth are calculated according to equation (5.66):

The second moments of area and torsion constants of the grillage members are thendetermined by multiplying these values by the relevant breadth of each member as given inFig. 6.2 (b). These values are presented for all of the grillage members in Table 6.1. Thelongitudinal members have been grouped by row as R1 to R9 and the transverse membershave been grouped as end members and all intermediate members as illustrated in Fig. 6.2.For the transverse end members, the breadth is 1.5/2+0.5 as the slab extends 0.5 m past thecentre of the bearing. However, in keeping with recommendation number 6 of Section 5.3.7,this is reduced by 0.3d=0.24 m for the calculation of the torsion constant. Similarly, whendetermining the value of the torsion constant of the longitudinal members in rows R1 and R9,a reduced breadth of (0.87−0.24)=0.63 m was used.

Example 6.2: Finite-element model of two-span right slab

A planar finite-element model is required for the bridge deck of Example 6.1 and Fig. 6.1.

Figure 6.3 shows a convenient finite-element mesh. The breadths of the elements arechosen such that nodes coincide with the locations of the supports. The two rows of

Fig. 6.3 Finite-element mesh for bridge of Fig. 6.1

elements at each edge of the model could be replaced with one row of 1.5 m breadth, but theextra number of elements in the model chosen is not considered to be excessive. The length ofthe elements along the span of the bridge was chosen as 1.2 m which is equal to the breadth ofthe widest element. This is a somewhat arbitrary choice, and had the length been taken as,say, equal to the average breadth of the elements, a similar degree of accuracy could beexpected. As this is an isotropic bridge slab, the only geometric property which has to beassigned to the elements is their depths. All of the elements are assigned a depth of 0.8 mwhich is equal to the actual depth of the bridge slab. As for Example 6.1, the elastic modulusis taken to be that of the slab, E=35×106 kN/m2.

6.3 Edge cantilevers and edge stiffening

Slab bridge decks often include a portion of reduced depth at their edges known as an edgecantilever. This type of construction is chosen partly for its reduced self weight and partly forits slender appearance (see Section 1.8). Cross-sections of typical slab decks with edgecantilevers are illustrated in Fig. 6.4. Upstands or downstands, such as those illustrated in Figs.6.4(c) and (d), are often included at the edges of the slab, either to stiffen the edge, to carry aprotective railing, or simply for aesthetic reasons. These are frequently important aestheticallyand, in the case of concrete bridges, may be precast to ensure a good quality of finish. In suchcases, the upstand may not be integral with the bridge deck and can simply be considered asan additional load on it. If they are made integral with the deck, then the increased stiffnesswhich they provide generally needs to be considered. It is not necessarily conservative toignore the additional stiffness provided by them.

The effect of an edge cantilever or an integral upstand/downstand is to change the stiffnessof the bridge deck. In slab bridges, the appropriate stiffness is determined by first finding theneutral axis location for the complete deck. The properties of each part are then calculatedabout this axis. In some bridge decks, finding the location of the neutral axis may not bestraightforward. Figure 6.5 shows the cross-section of a deck with a long slender edgecantilever with an upstand at its edge. In such a case, the neutral axis will not remain straightas the upstand tries to bend about its own axis, causing the bridge neutral axis to rise. Bridgedecks of this type are discussed further in Chapter 7. Only decks where the neutral axisremains substantially straight are considered here. These will be similar to those illustrated inFig. 6.4, where the edge cantilever is relatively short or stocky

Fig. 6.4 Typical cross-sections of slab decks showing cantilevers and upstands

Fig. 6.5 Cross-section of slab deck with slender cantilever and upstand

or where the upstand is not excessively stiff. The neutral axis is then taken to be straightacross the complete deck and to pass through its centroid.

Example 6.3: Grillage analysis of slab with edge cantilever

The cross-section of a prestressed concrete bridge slab with edge cantilevers is illustrated inFig. 6.6. The bridge deck, which has a constant cross-section through its length, spans 20 mand is simply supported on three bearings at each end as indicated in the figure. It is requiredto design a suitable mesh of grillage members to model the structure.

The first task is to determine the location of the deck neutral axis which is taken to bestraight and to pass through the centroid. This can be determined by hand or by using one ofmany computer programs available for such purposes. In this case, the neutral axis is found tobe 563 mm below the top of the bridge deck. Details of a general approach to this calculationare given in Appendix C.

Fig. 6.6 Bridge deck of Example 6.3 (dimensions in mm): (a) section; (b) plan

The cross-section is divided into a number of segments, each of which is represented by a rowof grillage members. Figure 6.7(a) shows the divisions chosen and the corresponding grillagemembers. The spacings of longitudinal grillage members is given in Fig. 6.7(b). The reasonsfor this particular arrangement are as follows:

• Each edge cantilever is modelled with two separate rows of members so that the reduceddepth towards the edge can be allowed for.

• The outermost row of grillage members, Row R1, is placed at a distance of 90 mm from theedge of the cantilever. This distance corresponds to 0.3 times the average depth ofcantilever. This is in keeping with recommendation number 6 of Section 5.3.7.

• The second row of grillage members from the edge, R2, is located at the centre of theportion of cantilever which it represents.

• The third row of members from the edge, R3, is placed at a distance of 0.3 times the depthof the deck (0.3×1200=360 mm) from the midpoint of the sloping edge of the main deck.The location from which this distance is taken is somewhat arbitrary, but that chosen hereseems reasonable.

• The fourth row, R4, and middle row, R7, of grillage members are located to coincide withthe supports to the bridge deck. Note that row R4 is not exactly at the centre of the portionit represents.

• Two rows of grillage members, R5 and R6 (and R8 and R9), are chosen between thesupports. In each case, these members represent a portion of bridge slab of breadth 1000mm and they are located at the centre of that portion.

Fig. 6.7 Grillage model (dimensions in mm): (a) cross-section showing grillage members andcorresponding segments of deck; (b) schematic of cross-section showing spacingbetween members; (c) plan of mesh

Figure 6.7 (c) illustrates a plan of the grillage mesh with dimensions in mm. Twenty one rowsof transverse members with a spacing of 1000 mm were chosen. This is a very dense meshhaving a spacing less than the slab depth. However, it gives a good longitudinal to transversespacing ratio, between 1:1 and 1:1.27. Due to the variation in depth between rows R2 and R3,the transverse members between these rows have been modelled as two separate memberswith a row of nodes where they join.

For this example, each row of longitudinal grillage members is considered separately. Thesecond moment of area about the centroid (of the bridge) of each portion of deck isdetermined. The second moment of area relative to the centroid of the bridge is always greaterthan (or equal to) that relative to the centroid of the individual portion

of deck. For example the second moment of area of row R7 is given by:

All of the longitudinal grillage member second moments of area are presented in Table 6.2.The transverse members are divided into two groups. The first group are those in the

cantilever portion, running from the edge as far as the row of nodes indicated in Fig. 6.7.These are labelled Tc in Fig. 6.7(c). The second group are those in the main portion of thedeck and account for all of the other transverse members. These are labelled Tm in the figure.The second moment of area of the transverse members in the cantilever, Tc, are taken abouttheir own centroids as they will bend (transversely) about their own centroids. The depth ofthese members is taken as the average depth of the cantilever, i.e. 300 mm. The secondmoment of area per unit breadth of these members is therefore:

The second moment of area of the transverse grillage members in the main part of the deck,Tm, are also calculated about their own centroids as it is about these that they will bend. Thesecond moment of area per unit breadth of these members is therefore:

The second moment of area of the transverse members is then found by multiplying thesevalues by the breadth of the members (which for this example is 1 m). The results arepresented in Table 6.2.

The torsion constants for the members are determined in accordance with equation (5.71)as this is an orthotropic deck :

Table 6.2 Grillage member properties for Example 6.3

Second moment of area (m4) Torsion constant (m4)

Longitudinal members

R1, R13 0.029 0.010

R2, R12 0.034 0.013

R3, R11 0.110 0.143

R4, R10 0.131 0.261

R5, R6, R7, R8, R9 0.146 0.290

Transverse members

Tc—End members 0.002 0.019

Tc—Intermediate members 0.002 0.021

Tm—End members 0.144 0.178

Tm—Intermediate members 0.144 0.278

where and are the second moments of area per unit breadth in the X and Y directionsrespectively. To apply this equation, the X direction is arbitrarily chosen as the longitudinaldirection. Considering the longitudinal members in row R1 and the transverse members Tc,the second moment of area per unit breadth of the longitudinal members (with reference toTable 6.2) is given by:

The second moment of area per unit breadth of the transverse members is 0.002 m3. Hence,the torsion constant per unit breadth of the longitudinal members, R1, and the transversemembers, Tc, is given by:

Considering next the longitudinal members in row R2 and the transverse members Tc, thesecond moment of area per unit breadth of the longitudinal members (with reference to Table6.2) is given by:

Therefore the torsion constant per unit breadth of the longitudinal members, R2, and thetransverse members, Tc, is given by:

This gives a value for the torsion constant per unit breadth for each of the longitudinalmembers R1 and R2 but there are two distinct values for the transverse members Tc. At thisstage, an approximation is made by taking an average value for the torsion constant per unitbreadth of the transverse members. In doing this, the condition of Section 5.3.3 is not satisfiedwhich required that the torques per unit breadth in the grillage members in the longitudinaland transverse directions be of the same magnitude. However, as the two distinct values arevery close, the average value is considered acceptable. The torsion constant per unit breadthof the transverse grillage members, Tc, is therefore:

Considering the longitudinal members in row R3 and the transverse members Tm, thesecond moment of area per unit breadth of the longitudinal members (with reference to Table6.2) is given by:

The second moment of area per unit breadth of the transverse members is 0.144 m3 andtherefore the torsion constant per unit breadth of the longitudinal members, R3, and thetransverse members, Tm, is given by:

This value is adopted for the longitudinal members in row R3. The other longitudinalmembers, R4 to R10, have the same second moment of area per unit breadth (with referenceto Table 6.2) which is:

Hence, the torsion constant per unit breadth of the longitudinal members, R4 to R10, andthe transverse members, Tm, is given by:

This value is adopted for longitudinal members R4 to R10. The average of the two values istaken for the transverse members Tm:

The torsion constant for each grillage member is then arrived at by multiplying the torsionconstant per unit breadth by the breadth of slab represented by that member. For the endtransverse members, Tm, the breadth is reduced by 0.3×1.2=0.36 m. For the longitudinalmembers in rows R3 and R11, the breadth is reduced by 0.3×0.9=0.27 m. For the endtransverse members, Tc, and the longitudinal members in rows R1 and R13, the breadth isreduced by 0.3×0.3=0.09 m. These values are given in Table 6.2. It can be seen that bysplitting the transverse members running between rows R2 and R3 (and R11 and R12) intotwo separate transverse members, the need to average two dissimilar values of torsionconstant was avoided.

Example 6.4: Finite-element analysis of slab with edge cantilever

It is required to prepare a finite-element model for the bridge deck of Example 6.3 and Fig.6.6.

The cross-section of Fig. 6.6(a) is divided into a number of segments in a similar manner tothe grillage model. As the nodes form the boundaries of the elements and the location of thesupports must coincide with nodes, the division of the deck for the finite-element modelvaries somewhat from that of the grillage. Figure 6.8(a) shows the division of the deck andFig. 6.8(b) shows a cross-section through the finite-element model. The depths of theelements have not been drawn to scale in this figure. Figure 6.8(c) shows a plan of the finiteelement model with rows of elements labelled r1 to r14. The length of the elements (in thelongitudinal direction) is taken as 1000 mm. This results in 20 elements in each of the 14longitudinal rows.

Fig. 6.8 Finite-element model (dimensions in mm): (a) cross-section showing division of deck intoelements; (b) schematic of cross-section showing breadths of elements; (c) plan ofelement mesh

The X axis is again chosen to be in the longitudinal direction and the Y axis to beperpendicular to this. The second moments of area per unit breadth, and , aredetermined for each portion of the bridge deck. In the X direction, these are calculated aboutthe centroid of the bridge which was seen in Example 6.3 to be located 563 mm below the topsurface. In the Y direction, the second moment of area per unit breadth of each portion isdetermined about its own centroid as it is about this that transverse bending occurs. In the caseof the elements representing the edge cantilevers (rows r1, r2, r13 and r14), the transversestiffness is based on the average depth of that portion of cantilever. In the case of the elementsin row r3, it is difficult to determine the transverse stiffness as the depth varies significantly.A depth of 1000 mm is chosen as this seems to be a reasonable compromise and it is felt thatthe problem does not warrant an in-depth analysis. The second moments of area per unitbreadth for each row of elements are given in Table 6.3.

Table 6.3 Finite-element properties for Example 6.4

Finite element row number (m3) (m3) delem (m)

r1, r14 0.0490 0.0013 Ec 0.838 0 .027 Ec 0. 068 Ec

r2, r13 0.0561 0.0036 Ec 0.876 0 .064 Ec 0. 106 Ec

r3, r12 0.1138 0.0833 Ec 1.109 0.732 Ec 0. 356 Ec

r4, r11 0.1456 0.1440 Ec 1.204 0 .989 Ec 0. 414 Ec

r5 r6, r7, r8, r9, r10 0.1456 0.1440 Ec 1.204 0.989 Ec 0. 414 Ec

The bridge deck is geometrically orthotropic, as the second moments of area vary in twoorthogonal directions. In the finite-element program, it is modelled as materially orthotropicwith a single value for element depth. The variation of second moment of area in the twodirections is allowed for by specifying two different elastic moduli. Arbitrarily choosing theelastic modulus in the X direction, , to be equal to the elastic modulus of the concrete, Ec,then the equivalent depth, delem, to be used for the finite elements is found by equating thesecond moments of area of the element and the slab (equation (5.76)):

Equation (5.77) then gives an expression for the elastic modulus in the Y direction, , interms of the elastic modulus of the concrete, Ec:

The elastic moduli in the two directions and the equivalent depths of each row of elementsare given in Table 6.3.

The shear modulus, , is calculated using equation (5.80) by substituting values for thePoisson’s ratio, the elastic modulus and the second moments of area per unit breadth.Assuming a Poisson’s ratio of 0.2 for concrete, values of were arrived at for each row ofelements. These values are also given in Table 6.3.

6.4 Voided slab bridge decks

Longitudinal voids are often incorporated into concrete slab bridge decks to reduce their selfweight while maintaining a relatively large second moment of area. These are created byplacing void formers, usually made from polystyrene, within the formwork before casting theconcrete. Figure 6.9 shows a cross-section through a typical voided slab bridge deck withtapered edges. It is common practice to discontinue the voids over the supports which has theeffect of creating solid diaphragm beams there.

When the void diameter is less than about 60% of the slab depth, it is common practice tomodel the voided slab using the same methods as are used for

Fig. 6.9 Cross-section through voided slab bridge

solid slab decks. On the other hand, when the void diameter exceeds about 60%, thebehaviour becomes more ‘cellular’. Cellular decks are characterised by the distortionalbehaviour illustrated in Fig. 6.10 which can be modelled using a variation of the conventionalgrillage or FE methods known as ‘shear flexible’ grillage or FE. Even if the voids are large, avoided slab deck is less likely to distort than the box girder section of Fig. 6.10 and, withoutspecific guidance, such a shear flexible model would be difficult to implement. Bakht et al.(1981) reviewed many methods of analysing voided slab bridges. They propose that,regardless of the size of the voids, such slabs can be analysed using the same techniques asthose used for solid slab decks but with modified member properties.

The first step in the modelling of a voided slab deck is to determine the location of theneutral axis. This is generally taken to be at a constant depth transversely and to pass throughthe centroid of the deck. If the bridge deck has edge cantilevers or if the voids are not locatedat the centre of the deck, then the position of the centroid may not be at mid-depth and shouldbe calculated in the usual way. For planar grillage or finite-element models, the properties ofeach part of the deck are then calculated relative to the neutral axis of the complete deck.

Determination of the longitudinal second moment of area per unit breadth of a voided slab,, is straightforward. The stiffness of the voided portion is simply subtracted from the

stiffness of the solid slab. Determination of the transverse second moment of area and thetorsional rigidity are not so simple. For the

Fig. 6.10 Characteristic behaviour of cellular bridge deck: (a) original geometry; (b) deformedshape showing characteristic cell distortion

transverse second moment of area, Bakht et al. (1981) recommend using the method of Elliottwhich gives this quantity in terms of the depth of the slab, d, and the diameter of the voids, dv

(Fig. 6.11):

(6.1)

Equation (6.1) does not take into account the spacing of the voids as the authors maintainedthat this was not a significant factor. Clearly this equation is only applicable to slabs with asensible void spacing. A slab where the voids were spaced three to four times the slab depthapart would have a transverse rigidity in excess of that predicted by equation (6.1). Thisequation assumes that the centre of the voids and the deck centroid (for transverse bending)are located at mid-depth. This is quite often a reasonable assumption when consideringtransverse bending.

When the void diameter to slab depth ratio is 0.6 or less, the transverse stiffness can beapproximated as being equal to the longitudinal stiffness. Examination of equation (6.1)shows that the presence of the voids reduces the transverse stiffness by only 12% for a ratio of0.6.

Fig. 6.11 Cross-section through segment of voided slab bridge

Table 6.4 Ratio of torsional stiffness of voided slab, iv-slab, to that of solid slab, islab (from Bakht etal.1981)

0.9 0.8 0.7 0.6 0.50.90 0.45 0.48 0.51 0.56 0.62

0.85 0.55 0.58 0.61 0.64 0.69

0.80 0.64 0.66 0.68 0.71 0.75

0.75 0.70 0.72 0.74 0.77 0.80

0.70 0.76 0.78 0.79 0.82 0.84

0.65 0.81 0.82 0.84 0.86 0.88

0.60 0.85 0.86 0.87 0.89 0.90

For the torsional stiffness of voided slabs per unit depth, jv-slab, Bakht et al. (1981)recommend using the method of Ward and Cassell. This gives the values presented here inTable 6.4 for the ratio of torsional stiffness of the voided slab jv-slab, to that of a solid slab ofthe same depth, jslab. For a grillage model, jslab can be determined from equations (5.65) or(5.71) and Table 6.4 can then be used to determine jv-slab. It was suggested that the valuesgiven in Table 6.4 are only applicable to internal voids in an infinitely wide slab becausethose at the edges possess much lower torsional rigidities. However, Bakht et al. conclude that,in most practical cases, reduction of the torsional rigidity for the edge voids is not warrantedas voided slab bridge decks are usually tapered at their edges or have substantial edge beams.

Example 6.5: Grillage model of voided slab bridge

Figure 6.12 shows the cross-section of a prestressed concrete bridge deck which incorporatescircular voids along its len gth. The deck spans 24 m between the centres of supports and issupported on four bearings at either end as illustrated in the figure. The voids stop short ateach end forming solid diaphragm beams 1 m wide over the supports. Thus the total bridge is25 m long consisting of 23 m of voided section and two 1 m diaphragms . The neutral axispasses through the centroid of the deck which is located at mid-depth as the voids are locatedthere. The layout and member properties are required for a grillage model.

Fig. 6.12 Cross-section through bridge of Examples 6.5 and 6.6

Fig. 6.13 Grillage mesh for bridge of Example 6.5

Fig. 6.14 Segment of voided slab

Figure 6.13 shows a suitable grillage mesh. The longitudinal members are located midwaybetween voids, with the exception of the outer row on each side where they are locatedmidway between the edge of the outermost void and the edge of the deck. It is not consideredappropriate to locate these grillage members at 0.3 times the depth of the slab from the edgeas this location is within the void. By using this arrangement, the supports coincide with thelocations of nodes in the grillage mesh. The transverse grillage members are located in 17rows, 1.5 m apart.

As the void diameters are in excess of 60% of the slab depth, the slab is treated as anorthotropic plate and the properties of the longitudinal and transverse members aredetermined separately. The longitudinal direction is taken to be the X direction. The internallongitudinal grillage members represent the portion of deck illustrated in Fig. 6.14. Thesecond moment of area of this member is found by subtracting the second moment of area ofthe circle from that of the rectangle, i.e.:

The edge longitudinal grillage member represents a portion of deck equal to exactly halfthat of the internal members with the result that its second moment of area is given by:

The second moments of area of the internal transverse members are determined usingequation (6.1):

Hence, for the internal transverse members, the second moment of area is:

For the 1m wide end diaphragms, the second moment of area is simply:

As the diaphragm is only 1 m wide and the transverse members are spaced at 1.5 m, thenext row of transverse members, adjacent to the diaphragm, will be 1.75 m wide and will havea second moment of area of:

The torsion constant for the grillage members is found from Table 6.4. Both the ratio dv/svand dv/d are 0.67. Interpolating in the table gives a ratio for the torsion constants per unitbreadth of:

Taking equation (5.65) to calculate the torsion constant per unit breadth for a solid slabthen gives:

The torsion constants for both the longitudinal and transverse members in the voided slabare then found by multiplying this value by their respective breadths. The torsion constant perunit breadth for the diaphragm is given by equation (5.71):

Example 6.6: Finite-element model of voided slab bridge

A finite-element model is required for the 25 m long voided slab deck of Example 6.5 andFig. 6.12.

For convenience, a mesh consisting largely of 1.2 m square elements is chosen, asillustrated in Fig. 6.15. At the ends, two transverse rows of elements, each 0.5 m wide, areused to represent the diaphragm. The transverse rows of elements adjacent to the diaphragmsat each end are 1.3 m wide in order to make up the correct total length.

Each longitudinal row of elements represents a strip of the deck from midway between onevoid to midway between the next. The second moment of area per unit

Fig. 6.15 Finite element mesh for bridge of Example 6.6

breadth in the longitudinal direction can be found by considering a 1.2 m wide strip of thedeck. The total second moment of area of this strip is again calculated by subtracting thesecond moment of area of the void from that of the equivalent rectangular section:

Hence, the second moment of area per unit breadth is:

For the transverse direction, equation (6.1) gives:

The slab is geometrically orthotropic, as the second moments of area (rather than themoduli of elasticity) are different for the longitudinal and transverse directions. To model thisas a materially orthotropic plate, it is necessary to calculate a single equivalent value for slabdepth, de. Selecting the modulus of elasticity in the X direction, Ex, equal to the modulus forthe concrete, then equation (5.76) implies a depth of element of:

Equation (5.77) gives an expression for the elastic modulus in the Y direction:

where Ev-slab is the modulus of elasticity of the concrete in the voided slab. The shear modulusis calculated from equation (5.80):

Taking a Poisson’s ratio of 0.2, this gives:

The diaphragm beams are solid so the corresponding elements are 1.2 m thick and havemoduli of elasticity in both directions equal to that of the concrete. The shear modulus for thediaphragms is given by equation (5.67).

6.5 Beam and slab bridges

Beam and slab decks are used for a wide variety of modern bridges. They differ from slabbridge decks in that a large portion of their stiffness is concentrated in discrete beams whichrun in the longitudinal direction. Load sharing between the beams may be provided by a topslab or by a combination of a top slab and a number of transverse diaphragm beams. Beamand slab bridges are generally suitable for similar span lengths as slab bridges but are oftenchosen in preference because of their ability to be easily erected over inaccessible areas suchas deep valleys or live roads or railways.

Beam and slab decks may be formed in a number of ways, the most obvious being thecasting of an in-situ concrete slab on steel or precast concrete beams as shown in Fig. 6.16 (a)and (b). Many other methods exist, such as steel beams with a composite steel and concreteslab, a precast concrete slab or even a completely in-situ beam and slab as illustrated in Fig.6.16 (c).

During construction, the beams generally act alone and must be capable of carrying theirself weight, the weight of the slab and any construction loads present. On completion, thestructural action of these decks is considered to be two-dimensional. Therefore they can beanalysed by similar methods to those proposed for slab decks in the preceding sections. Themain load-carrying component of a beam and slab deck is the longitudinal spanning beams.The slab acts to transmit applied loads to the beams by spanning transversely between them.In addition to this, the slab provides a means for load sharing between longitudinal beams.The extent of this load sharing is largely dependent on the stiffness of the slab. Consequently,it is important that the slab be idealised correctly in the model as, for example, an overly stiffslab may lead to a prediction of load sharing between adjacent beams which does not occur inreality. This phenomenon is indicated in Fig. 6.17.

Transverse diaphragm beams can be used to provide additional load sharing betweenlongitudinal beams. Wide diaphragms also serve to improve the shear

Fig. 6.16 Forms of beam and slab construction: (a) in-situ slab on steel beams; (b) in-situ slab onprecast concrete beams; (c) in-situ beam and slab

Fig. 6.17 Load sharing in beam and slab decks: (a) thin slab—little load sharing; (b) thick slab—increased load sharing

capacity by extending the portion of the bridge near a support which is solid. In precastconcrete beam construction, continuity between adjacent spans may be provided by the slabalone, but quite often, a diaphragm beam is constructed over intermediate supports to provideadditional continuity.

6.5.1 Grillage modelling

Grillage modelling of beam and slab decks generally follows the same procedures as for slabdecks. The obvious exception is that grillage beams should normally be

Fig. 6.18 Effective flange width of diaphragm beam: (a) plan at end; (b) section through L-beam

positioned at the location of the longitudinal beams. This generally complies with the need tolocate beams at the supports as, in beam and slab construction, supports are normallyprovided directly beneath the beams. It is possible to use one grillage member to representtwo or more actual beams but this complicates the calculation of properties and interpretationof the results with little saving in analysis time in most cases.

The properties of the longitudinal grillage members are determined from the properties ofthe actual beams and the portion of slab above them. Unlike slab decks, the section propertiesfor beam and slab decks are generally calculated about the centroid of this composite section,not about the centroid of the whole bridge. This approach is justified on the basis that, due tothe low stiffness of the slab, there will be a much greater variation in the depth of the neutralaxis than in slab bridges.

Transverse grillage members should clearly be placed at the location of all diaphragmbeams. The slab will act as a flange to such beams making them T- or L-section in shape.Hambly (1991) suggests an effective flange breadth of bw+0.3s for L-sections as illustrated inFig. 6.18, where s is the spacing between beams.

In addition, transverse members are required to represent the transverse stiffness of the slab.For slab decks, Section 5.3.7 stated that transverse member spacing should be between oneand three times the longitudinal member spacing. This spacing is also recommended for beamand slab bridges although greater spacings are possible without significant loss of accuracy.The properties of the transverse grillage members should be derived from the properties of therelevant diaphragm beam or slab as appropriate, each acting about its own axis.

Example 6.7: Grillage model of beam and slab bridge

Figure 6.19 shows the cross-section of a beam and slab bridge deck consisting of a cast in-situslab on precast concrete Y-beams. Each precast beam is supported on a bearing at each endand the deck has a single span of 20 m (centre to centre of bearings). Solid diaphragm beams,1 m wide, are provided at each end and no additional transverse beams are located betweenthese. The elastic modulus of the precast beams is 34 kN/mm2 and that of the in-situ slab is 31kN/mm2. A grillage model of the beam and slab deck is required.

Fig. 6.19 Beam and slab bridge deck: (a) cross-section; (b) detailed dimensions of Y-beam

The modular ratio for the in-situ and precast concrete is:

The procedure adopted is to assign a modulus of elasticity of 34 kN/mm2 to all of thegrillage members (except for the end diaphragms), but to factor the stiffness of the slab by thismodular ratio.

The section properties of the precast beam are generally given by the manufacturer; in thiscase, the properties are:

Area = 0.374 m2

Second moment of area = 0.0265 m2

Height of centroid above soffit = 0.347 m

The torsion constant is generally not given and must be determined by the analyst. Figure6.19 (b) shows the exact dimensions of the precast beam. For the purposes of determining thetorsion constant, the beam cross-section is approximated as two rectangles as illustrated inFig. 6.20. The torsion constant of a cross-section made up of rectangles is commonlyestimated by calculating the torsion constants of the individual rectangles and summing. Thetorsion constant, J, for a rectangular section according to Ghali and Neville (1997) is:

(6.2)

Fig. 6.20 Equivalent section made up of rectangles for determination of torsion constant

where b is the length of the longer side and a is the length of the shorter side. Applying thisequation to the rectangles of Fig. 6.20 gives a torsion constant for the Y-beam of:

The constant can be found more exactly by applying Prandtl’s membrane analogy asdescribed by Timoshenko and Goodier (1970). A finite-difference technique was used todetermine the constant in this case and a value was found as follows:

The simplified method can be seen to be accurate to within 7% for this section. Figure 6.21shows a suitable grillage layout for this bridge deck. A longitudinal grillage member ispositioned at the location of each Y-beam. Transverse members are positioned at each end tomodel the diaphragms. Additional transverse beams are located at 2 m centres between theseto represent the transverse stiffness of the slab. This gives a transverse to longitudinal memberspacing ratio of 2:1 which is acceptable. ‘Dummy’ longitudinal members with nominalstiffness are provided at the edges and transverse members are continued past the ends of theedge Y-beams to connect to them. This is a convenient method for applying loads such asthose due to parapet railings. Some grillage programs allow the definition of ‘dummy’ beams.If this is not the case, then these beams should be assigned very small section propertiesrelative to those used elsewhere in the grillage (say, 0.5%).

Supports are located at the ends of each longitudinal beam (other than the dummy beams).As the grillage model is planar, consideration need not be given to in-plane horizontalmovements at this stage.

For the interior longitudinal members, the second moment of area is the sum of the secondmoment of area of the Y-beam plus the 1 m width of slab above it, both

Fig. 6.21 Plan view of grillage mesh

taken about the common centroidal axis of the section. The stiffness of the slab is reduced byfactoring it by the modular ratio. Hence the equivalent area of the combined section is:

The section centroid is found by summing moments of area about the soffit:

where yb is the distance of the centroid above the soffit. Hence:

The second moment of area of the combined section is:

The torsion constant is taken as the sum of the torsion constants of the Y-beam and theslab. The torsion constant of the slab is determined using equation (5.65). Hence:

Each edge longitudinal member is similar to the interior members except for a 0.2×0.3 m2

upstand. This raises the centroid above that for the interior members. Summing moments ofarea about the soffit gives:

Fig. 6.22 Section through end diaphragm beam

Hence the second moment of area of the edge section is:

For the transverse members, the properties are determined in the usual manner. For thesecond moment of area:

The torsion constant is:

The slab acts as a flange to the diaphragm beams. The recommended flange breadth is thesum of the web breadth plus 0.3 times the beam spacing:

Hence the centroid is:

above the soffit. For the slab bending about its own axis, the row of transverse membersadjacent to the diaphragm accounts for the slab up to 1 m from the centre of the diaphragm asillustrated in Fig. 6.22.

This leaves 0.5 m of slab to be accounted for in the diaphragm stiffness, 0.2 m of which isdeemed to be bending about its own axis. The second moment of area is thus:

The torsion constant is calculated allowing for 0.5 m of flange from equations (6.2) and(5.65):

The modulus of elasticity for in-situ concrete is used for the diaphragm beams.

If the web width at the top of the longitudinal beams in a beam and slab deck is large relativeto their spacing, then the slab can inadvertently be modelled as having an excessively longtransverse span. Figure 6.23(a) shows a deck consisting of a concrete slab on precast concreteU-beams. Figure 6.23(b) shows a grillage model with longitudinal grillage beams for the U-beams and transverse beams spanning between them representing the slab. It can be seen fromthis that the span of the slab in the model is too long. This would lead to an excessivelyflexible slab which in turn would lead to the incorrect modelling of load sharing between theU-beams. One possible solution to this is shown in Fig. 6.23(c), where the transverse grillagemembers have been subdivided to include much stiffer portions at their ends.

6.5.2 Finite-element modelling

In finite-element modelling of beam and slab decks, a combined model is generally usedwhich represents the slab with finite elements and the beams with grillage

Fig. 6.23 Transverse modelling of decks with wide flanges: (a) in-situ slab on precast concrete U-beams; (b) conventional grillage model where slab has excessive transverse span; (c)improved grillage model

members. This is generally straightforward to implement and follows the recommendationsmade for slab bridge decks. Care should be taken when determining the properties of thefinite elements representing the slab. One of two approaches can be taken. In the firstapproach, the slab is modelled using isotropic elements which are assigned a thickness equalto the depth of the actual slab. They are also assigned the elastic properties of the slab. Thelongitudinal grillage members are then assigned the stiffnesses of the combined beam andassociated portion of slab minus those already provided through the finite elements. In thesecond approach, the slab is modelled using orthotropic finite elements with the truetransverse and longitudinal properties applied in both directions. The beams are then modelledby grillage members with the properties of the actual beams excluding the contribution of theslab.

Example 6.8: Finite-element model of beam and slab bridge

A finite-element model is required for the beam and slab bridge of Example 6.7 and Fig. 6.19.Figure 6.24 shows a suitable finite-element mesh incorporating grillage members

longitudinally. Grillage members are used for each of the Y-beams and for each of the enddiaphragms. The finite elements continue to the edge of the deck resulting in a row ofelements 0.5 m wide at each side. An element length of 1 m in the longitudinal directionresults in a maximum element aspect ratio of 1:2 which is considered to be acceptable.Supports are provided at the ends of each longitudinal grillage member.

The finite elements are assigned a thickness of 0.16 m which is equal to the depth of theslab. They are assigned a modulus of elasticity and a Poisson’s ratio equal to those of theconcrete in the slab.

For the longitudinal grillage members, the properties of the combined Y-beam and the 1 mwidth of slab above it are determined relative to the centroidal axis of the combined section.The stiffness of the slab which has already been applied through the finite element issubtracted. The modulus of elasticity and Poisson’s ratio for the beams are used for thesemembers. From Example 6.7, the second moment of area of the combined section is:

Fig. 6.24 Combined finite-element and grillage mesh

The second moment of area of the 0.16 m thick finite elements is then subtracted to give thesecond moment of area to be used for the grillage member:

The torsion constant for the combined section was arrived at in Example 6.7 by adding theindividual torsion constants of the Y-beam and slab. As the slab is represented by theelements, the torsion constant to be assigned to the grillage members is simply that of the Y-beam:

In Example 6.7, the second moment of area for the end diaphragms in the grillage modelwas calculated as (refer to Fig. 6.22):

For the finite-element model, the elements are present up to the centre of the diaphragm torepresent the transverse stiffness of the slab about its own axis. Hence, the stiffness of the slabbending about its own axis is not required and a small component of stiffness inadvertentlycontributed by the elements must be subtracted:

The torsion constant is that of a rectangular section less the portion inadvertently addedthrough the elements. From equations (6.2) and (5.65):

6.5.3 Transverse behaviour of beam and slab bridges

The top slab in a beam and slab bridge is often designed transversely as a one-way spanningslab supported by the longitudinal beams (Fig. 6.25). However, such an approach results in agreat quantity of reinforcement and has been shown to be

Fig. 6.25 Detail of section in beam and slab deck

quite conservative. The beams have a considerable lateral stiffness and have the effect ofconfining the slab. The result is that load is transferred from the slab to the beams by archingaction rather than bending action alone. In two reported cases (Bakht and Jaeger, 1997),Canadian bridges have been built without any transverse slab reinforcement but using steelstraps to guarantee confinement. In these cases, the slab depth to beam spacing ratios were1:12 and 1:13.5. To account for observed arching action, the Ontario Highway Bridge DesignCode (OHBDC, 1992) allows for the provision of much less reinforcement than would befound by an assumption of bending.

6.6 Cellular bridges

Cellular bridge decks are formed by incorporating large voids within the depth of the slab.The most common type are box girder decks, with single or multiple rectangular cells. Voidedslab bridges, with large diameter circular voids, can also be considered to be of a cellular form.However, as was discussed in Section 6.4, alternative methods are available for their analysiswhich are generally more convenient. Figure 6.26 shows a number of commonly used cellulardeck forms.

There are four principal forms of structural behaviour associated with cellular bridges. Thefirst two of these are longitudinal and transverse bending, as illustrated in Figs. 6.27(a) and(b). The third form of behaviour is twisting, as indicated in Fig. 6.27(c). The fourth form,which characterises cellular structures, is transverse cell distortion, as indicated in Fig. 6.27(d).This distortion is caused by the localised bending of the webs and flanges of the individualcells. The behaviour is similar to that observed in Vierendeel girders.

It is the transverse distortional behaviour that makes the analysis of cellular decks differentfrom other forms. The principal factors affecting the distortion are the dimensions of the cellsrelative to the deck depth, the stiffness of the individual webs and flanges, and the extent (ifany) of transverse bracing to the cells. Clearly the provision of transverse diaphragms alongthe span of a cellular deck will significantly reduce the degree of transverse distortion.

Fig. 6.26 Sections through alternative cellular bridge decks

Fig. 6.27 Behaviour of cellular decks: (a) longitudinal bending; (b) transverse bending; (c)twisting; (d) transverse distortion

6.6.1 Grillage modelling

Grillage modelling of cellular bridge decks can be achieved by use of what is commonlyreferred to as a ‘shear flexible’ grillage. In this method, the deck is idealised as a grillage ofbeam members in the usual manner, except that the transverse members are given a reducedshear area such that they experience a shear distortion equal to the actual transverse distortionof the cells in the bridge deck. Clearly such a method requires a grillage program whichmodels shear deformation as well as bending and which allows for the specification of a shear(or ‘reduced’) area for the members independently of the other section properties. The methodis illustrated below by means of an example.

Figure 6.28(a) shows a single cell of width l of a cellular bridge deck under the action(transversely) of a vertical load P. If it is assumed for now that the webs are stiff and thattransverse distortion is caused by bending of the flanges only, then the distorted shape of thecell is as shown in Fig. 6.28(b). If the flanges are of equal thickness, then the load acting oneach can be taken as P/2.

The vertical deflection due to the bending of a beam of length l, fixed against rotation atboth ends and subjected to a vertical force P/2, is:

(6.3)

where I is the second moment of area and E is the modulus of elasticity. The flanges of a cellwill act as beams transversely with a second moment of area per unit breadth equal to d3/12,where d is their thickness. Hence, from equation (6.3), the deflection due to flange distortionis:

(6.4)

Fig. 6.28 Distortion of single cell with stiff webs: (a) applied loading; (b) distorted shape

The total deflection in a cantilever of length l, subjected to a vertical load per unit breadth ofP at its free end is:

(6.5)

where G is the shear modulus and as is the shear area of the section per unit breadth. Thesecond term is the deflection due to shear deformation which, for most structures, is smallrelative to the deflection due to bending. By equating the shear deformation in a transversegrillage member to the bending deformation of the cell flanges in the bridge, an expression forthe required shear area per unit breadth of a shear flexible grillage member is found:

(6.6)

In this example, it was assumed that transverse distortion was caused by the distortion of thecell flanges only. In practice, the webs of cellular decks are also flexible and consequentlythey too contribute to the overall transverse distortion.

Figure 6.29 shows a single cell of a cellular bridge deck with a constant web thickness butdifferent upper and lower flange thicknesses. Assuming points of contraflexure at mid-heightand equating the deflection of this cell to the shear deformation of a grillage member gives amore exact and general expression for shear area per unit breadth:

(6.7)

Details of the derivation of this formula are given in Appendix D. For cellular decks of othershapes, it has been suggested by others that a plane frame analysis be carried out to determinethe equivalent shear area of the transverse grillage members. However, this may be difficult tocarry out accurately in practice due to such factors as cracking in concrete sections.

Fig. 6.29 Cross-section through cellular deck showing dimensions of cell

The second moments of area of the longitudinal members in a shear flexible grillage aredetermined in the same way as for slab decks. As for slab bridges, the neutral axis of thebridge deck is first determined and the second moment of area of the portion of deckrepresented by each longitudinal grillage member is determined about that axis. For thetransverse members, the second moment of area of the top and bottom flanges is calculatedabout an axis at the bridge mid-depth as illustrated in Fig. 6.30.

(6.8)

The first term in equation (6.8) is generally small relative to the second and is often ignored.The torsion constants of the longitudinal and transverse grillage members are based on the

portion of section represented by the members. As mentioned previously, the torsion constantfor a thin rectangular section twisting about its own axis may be approximated by bd3/3,where b is the breadth and d the thickness. Such an equation is valid when the shear flows areopposing through the depth of the section as illustrated in Fig. 6.31(a). For a portion of boxsection, this is not the

Fig. 6.30 Longitudinal section through deck for transverse bending

Fig. 6.31 Shear stresses due to torsion: (a) rectangular section; (b) portion of box section withcantilever

case as illustrated in Fig. 6.31(b) except in the edge cantilevers. The torsion constant for athin-walled box section is given by:

(6.9)

where a is the area enclosed by the centre line of the wall, li, is an increment of length and diis the thickness of that increment. Applying equation (6.9) to the single cell of Fig. 6.29would give:

However, the contribution of the webs is accounted for through the shear forces in thelongitudinal beams and should not be accounted for again here. A formula suggested byHambly (1991) halves the constant and removes the web term:

(6.10)

Example 6.9: Shear flexible grillage model of a cellular bridge deck

Figure 6.32 illustrates a two-span, three-cell bridge deck with edge cantilevers. There are 2 mthick solid diaphragms at the end and central supports. It is assumed that the deck iscontinuously supported transversely at each support. A grillage model is required.

Figure 6.33 shows a convenient grillage mesh. Four longitudinal members are chosen, oneat the centre of each web. The two edge members represent the portion of deck from the edgeto halfway between the first and second webs (Fig. 6.32). The two internal members representthe portion of deck from halfway between the first and second webs to the centre. Transversegrillage members are located at the ends and at the central support to represent the transversediaphragms. Additional transverse members are placed at 2 m centres giving a longitudinal totransverse member spacing ratio of 1:1.11.

The first step in determining the grillage member properties is to find the neutral axis of thedeck which is assumed to pass through the centroid. By summing moments of area about anypoint in the section, the centroid can be shown to be located at 0.65 m

Fig. 6.32 Cellular bridge of Example 6.9 (dimensions in m): (a) cross-section; (b) longitudinalsection

Fig. 6.33 Plan view of grillage mesh

above the soffit. The second moments of area for the longitudinal members about this axis arethen determined. For the edge longitudinal members:

For the internal longitudinal members:

For the transverse members, the second moment of area per unit breadth, itrans is given byequation (6.8):

The breadth of the transverse members is 2 m, giving:

The torsion constant per cell is given by equation (6.10):

This gives a torsion constant for the interior longitudinal members of 0.36 m4. The edgemembers only represent half a cell and the contribution of the cantilever is added:

The torsion constant per unit breadth for the transverse members is taken to be equal to thatof the longitudinal members:

The shear area per unit breadth of the transverse grillage members is given by equation(6.7):

For concrete, a Poisson’s ratio of 0.2 is assumed. Then equation (5.67) gives:

which results in a shear area of:

The breadth of the transverse members is 2 m, giving:

For the longitudinal members, the shear area is taken as the area of the webs, a commonapproximation for I-sections, giving:

The end and central diaphragm beams are 1.2 m deep by 2 m wide. The second moment ofarea of the grillage members representing these is therefore:

The torsion constant for the diaphragms is determined using equation (6.2):

The shear area of the transverse diaphragm is taken as the actual shear area as nosignificant transverse distortion is assumed to take place. For a rectangular section, the sheararea can be shown to equal 83.3% of the actual area. Hence:

6.7 Skew and curved bridge decks

Many bridge decks incorporate some degree of skew and others are curved in plan. A grillageor finite-element model can be formulated for such decks based on the recommendationsgiven in earlier sections along with some additional considerations given here.

Significant skew in bridge decks leads to a non-uniform distribution of reactions betweensupports. Care is needed in modelling the support system in such cases as any flexibility willcause a redistribution of reactions. The greatest reactions will tend to occur at obtuse cornersin skew decks and the smallest reactions at acute corners. In highly skewed decks, uplift canoccur at acute corners which is generally to be avoided. Large reactions at obtuse corners leadto high shear forces which can also be difficult to design for. A high degree of twistingaccompanied by large torsional moments (mxy) are also associated with skew decks. As aresult, in reinforced concrete, the Wood and Armer equations can dictate a requirement fortop reinforcement near supports where hogging would not normally be expected.

6.7.1 Grillage modellingA suitable grillage model of a skew deck will depend largely on the angle of skew, the spanlength and the width of the deck. An important consideration is to place the grillage membersin the directions of principal strength. Figure 6.34(a) shows a long narrow bridge deck with ahigh degree of skew and Fig. 6.34(b) shows a suitable grillage layout. This deck will tend tospan in the skew direction so the longitudinal grillage members are aligned in that direction.The transverse grillage members should generally be oriented perpendicular to thelongitudinal members. An exception to this is in concrete decks where the transversereinforcement is not

perpendicular to the longitudinal reinforcement. In such cases, it is generally more appropriateto orientate the transverse members parallel to the transverse reinforcement as illustrated inthe alternative grillage layout of Fig. 6.34(c).

Figure 6.35(a) shows a short, wide bridge deck with a small angle of skew and Fig. 6.35(b)shows a suitable grillage layout. This deck will tend to span perpendicular to the supportsrather than along the skew direction. Consequently, the longitudinal grillage members areorientated in this direction. Once again, the transverse grillage members are orientatedperpendicular to the longitudinal members. Care should be taken with the edge grillagemembers which generally will have to be orientated in the skew direction. If significant edgebeams or stiffening is provided to the bridge deck, then this should be allowed for whenassigning the properties of the edge beams in the grillage. Bridge decks which fall betweenthe extremes of Figs. 6.34 and 6.35 will require a greater amount of judgement by the analystin choosing a suitable grillage layout.

Curved decks pose no particular problem for grillage modelling. Some analysis programswill allow the use of curved beams, but straight beams will be sufficiently accurate if thegrillage mesh is fine enough. Figure 6.36 shows a suitable grillage mesh for a curved bridgedeck. The longitudinal members, although straight, follow the curved layout closely due to thefineness of the mesh. The

Fig. 6.34 Long, narrow, highly skewed bridge deck: (a) plan view; (b) grillage layout; (c)alternative grillage layout

Fig. 6.35 Short, wide bridge deck with small skew: (a) plan view; (b) grillage layout

Fig. 6.36 Grillage layout for curved bridge deck

transverse members radiate from the centre of the curve. In this way, they are approximatelyperpendicular to the longitudinal members.

6.7.2 Finite-element modelling

Finite-element modelling of skew or curved decks should be carried out according to therecommendations for right decks. Generally, no special consideration need be given todirections of strength as the elements are two-dimensional and will model the two-dimensional behaviour of the skew or curved slab. This is an advantage that the finite-elementmethod has over the grillage method, especially for

Fig. 6.37 Alternative finite-element meshes: (a) skewed quadrilateral finite elements; (b)alternative triangular elements

inexperienced users who might not have the expertise to formulate a suitable grillage model.Skewed quadrilateral elements, as illustrated in Fig. 6.37(a), can give results which are just

as accurate as those for rectangular elements and they are very easy to implement. However,highly skewed quadrilaterals may result in round-off errors due to calculations involvingsmall angles. In such cases, triangular elements, as illustrated in Fig. 6.37(b), may be moreeffective.

Chapter 7Three-dimensional modelling of bridge decks

7.1 Introduction

In Chapter 6, the analysis of bridge decks using planar models is discussed. Bridge decks withedge cantilevers are considered but it is stipulated that only those with short cantileversshould be analysed by the methods proposed. In this chapter, the problems associated withbridge decks such as those with wide edge cantilevers are discussed. The theoreticalbackground is reviewed and a number of solutions are suggested including three-dimensionalmethods of analysis.

7.2 Shear lag and neutral axis location

When a bridge deck flexes, longitudinal bending stresses are set up. These are distributedtransversely from one part of the deck to adjacent parts by interface shear stresses. Thus,when the bending moment in a flanged beam varies from one point to another, interfacestresses are generated as illustrated in Fig. 7.1. When flanges or cantilevers are wide andslender, the edges do not receive the same amount of axial stress as those near the centre ofthe bridge. This phenomenon is known as ‘shear lag’ as it is associated with interface shearand is characterised by the lagging behind of axial stresses at the edges of cantilevers. Theextent of the reduction of stress is dependent on both the geometric shape of the bridge deckand the nature of the applied loading.

Figure 7.2(a) shows a bridge deck with the edge cantilevers separated from the main part ofthe deck. If a load were applied to the deck in this condition, each part would bend about itsown centroid, independently of the rest. In this condition, the bridge deck has a non-continuous neutral axis as indicated in the figure. If the bridge deck is now rejoined, acommon centroid can be found and the entire bridge is often assumed to bend about a neutralaxis passing through this point. This common neutral axis can be seen in Fig. 7.2(b). As therejoined bridge bends, the

remote edges of the cantilevers, due to shear lag, do not experience the same amount of axialstress as the main part of the deck, as can be seen in Fig. 7.2(c). The effect of bending is notfelt to the same extent in the edges of the cantilevers as it is elsewhere. This is because theedges of the cantilevers tend to bend about their own

Fig. 7.1 Interface shear stresses in flanged beam subject to bending

Fig. 7.2 Transverse variation in neutral axis location: (a) if cantilevers and main deck were freeto act independently; (b) commonly assumed straight neutral axis; (c) variation inlongitudinal stress at top of deck; (d) actual neutral axis location

centroidal axes. Obviously they are not free to do this, but this tendency causes the overallbridge deck neutral axis to move towards the centroid of the cantilevers at the edges. Such anon-uniform neutral axis is illustrated in Fig. 7.2(d).

There is a strong link between shear lag and neutral axis location. It could be said that thevariation in the neutral axis location in a bridge deck is caused by shear lag or that shear lag iscaused by the tendency of each part of the bridge deck to bend about its own neutral axis. Athree-dimensional analysis can automatically account for shear lag as it allows for variationsin neutral axis location directly.

7.3 Effective flange width

In the design of bridge decks, a two-dimensional analysis, as described in Chapter 6, is oftenused which does not take account of shear lag. It is possible to overcome this problem byassuming an ‘effective flange width’ for the edge cantilevers, as illustrated in Fig. 7.3. Themethod uses a notional width of cantilever in the grillage or finite element model which has auniform stress distribution equal in magnitude to the maximum stress in the actual cantilever.Hence, a two-dimensional model with an effective flange width, analysed with no allowancefor shear lag, can be used to determine the maximum stress in the cantilever.

The correct effective flange width to be used for the cantilever is largely dependent on theratio of the actual cantilever width to the length between points of zero moment (points ofcontraflexure), as it is from these points that longitudinal stresses begin to spread out into thecantilevers. The effective flange width is also dependent on the form of the applied loading.Hambly (1991) presents a chart for the determination of effective flange widths for beamssubjected to distributed and concentrated loads. The chart, reproduced here as Fig. 7.4, relatesthe ratio of effective flange width, be, and actual flange width, b, to the ratio of actual flange

Fig. 7.3 Actual and calculated distributions of longitudinal bending stress at top of flanged deck

Fig. 7.4 Effective flange width for different loadings (solid line), and common approximations(dashed line)

width, b, and length between points of contraflexure, L. Also shown in the figure are thepopular approximations for this relationship:

and:

which can be seen to be reasonably accurate for relatively wide flanges.

Figure 7.5(a) shows the cross-section of a bridge deck with edge cantilevers. The cantileversare 2.4 m wide and the deck has a single simply supported span of 20 m.

Example 7.1: Effective flange width

Fig. 7.5 Cross-section of bridge deck of Example 7.1: (a) showing actual cantilever widths; (b)showing effective flange width

As the span is simply supported, the length between points of contraflexure, L, is equal to thespan length in this case. Hence, the ratio of the cantilever width to this length, b/L, is2.4/20=0.12. From Fig. 7.4 the ratios of be/b are 0.93 and 0.67 for the uniformly and pointloaded cases respectively. This results in effective flange widths of 2.23 m and 1.61 mrespectively. Figure 7.5(b) shows this effective flange width for one of these load cases. Aconstant stress is assumed in the modelled portion of the cantilever and that part of it outsidethe effective flange width is ignored.

This example highlights the limitations of the effective flange width method as the nature ofthe loading causes a substantial variation in the effective flange width.

7.4 Three-dimensional analysis

The use of two-dimensional analysis methods with effective flange widths is approximate atbest and does not address the issue of upstands which are often provided at the edges ofbridge cantilevers. When the effects of shear lag are significant, some form of three-dimensional model is necessary to achieve an accurate representation of the behaviour of thestructure.

One such technique is three-dimensional finite-element analysis using solid ‘brick’ typeelements. Figure 7.6 shows such a model of a portion of bridge deck with edge cantilevers.The benefit of this type of model is that it can be used to describe the geometry of highlycomplex bridge decks very accurately. Inclusion of voids, a cellular structure or transversediaphragms pose no particular problems. In addition to this, the model automatically allowsfor any variations in the location of the neutral axis and hence allows for shear lag in edgecantilevers. Unfortunately, the use of such models is currently limited mainly to research andhighly specialised

Fig. 7.6 Portion of bridge deck modelled with solid brick elements

applications due to excessive run times and computer storage requirements and due to ashortage of user-friendly software, particularly for post-processing of the large quantities ofoutput data generated. The authors have used this type of model extensively to develop andtest a number of simplified three-dimensional models which are suitable for everyday bridgedesign. Some of these simplified models are discussed in the following sections.

7.5 Upstand grillage modelling

In Chapter 6, grillage modelling is applied to bridge decks including those with edgecantilevers. That type of analysis is referred to as planar grillage as all of the grillage membersare located in one plane. It is only suitable for bridge decks where the neutral axis remainssubstantially straight across the deck and is coincident with the centroidal axis of the bridge.When this is not the case, a three-dimensional technique, such as upstand grillage modelling,is required. The upstand grillage analogy is a direct extension of the planar grillage analogy,but involves the modelling of each part of the bridge deck as a separate plane grillage locatedat the centroid of the portion of bridge deck which it represents. The plane grillage meshes arethen connected using rigid vertical members. Figure 7.7 shows an upstand grillage model fora bridge deck with edge cantilevers. In this, the edge cantilevers are modelled with grillagemembers which are located at the centroid of the cantilevers while the main part of the deck ismodelled with grillage members located at the centroid of that part.

The properties of each part of the deck are determined relative to its own centroid.Consequently, there is no need to make an assumption as to the location of the overall bridgeneutral axis. There is also no need to assume an effective flange width to allow for shear lageffects. As the model is three-dimensional, it will automatically determine the location of theneutral axis, be it straight or varying, for each load case considered. Consequently, shear lagwhere it exists, will be accounted for automatically.

Although the upstand grillage seems to be a relatively simple and powerful model,difficulties arise when in-plane effects are considered. Unlike the plane

Fig. 7.7 Upstand grillage model

grillage, the three-dimensional nature of the model causes in-plane displacements in thegrillage mesh. This results in a requirement to specify the cross-sectional areas of the grillagemembers as well as the second moments of area (about both axes) and the torsion constants,However, the real problem is the occurrence of local in-plane distortions of the grillagemembers, as illustrated in Fig. 7.8, which are clearly inconsistent with the behaviour of thebridge deck. Such behaviour in the model can be avoided in one of two ways. The memberscan be given very large in-plane second moments of area, or the nodes at the ends of themembers can be restrained against in-plane rotation. Both of these measures will have similareffects, and it may even be prudent to adopt both.

Restraining in-plane rotations in the model may have adverse effects in some cases. If partof the actual bridge deck deforms in-plane, as tends to occur at the ends of edge cantilevers,the imposition of rotational restraints will prevent this behaviour from occurring in the modelwhich may significantly affect the accuracy of the results. Figure 7.9(a) shows the cross-section of a 24.8 m single-span bridge deck with wide edge cantilevers. This bridge wasanalysed under the action of a constant longitudinal bending moment using a planar grillagemodel and an upstand grillage model (Keogh and O’Brien, 1996). To test the accuracy of bothmodels, a three-dimensional finite-element analysis using solid ‘brick’ type elements, similarto that shown in Fig. 7.6, was also carried out. Figure 7.9(b) shows an exaggerated plan viewof the deflected shape of the three-dimensional brick finite-element model (only one-half ofthe model is shown as it is symmetrical). The in-plane distortion seen at the end of thecantilevers is made up of both in-plane shear distortion and in-plane bending. It is the in-planebending component which is not modelled by an upstand grillage with in-plane rotationalrestraints. Figure 7.10(a) shows the longitudinal bending stress predicted along the top of thisbridge deck at mid-span by the three-dimensional brick finite-element model, the planegrillage model and the upstand grillage model. Only half of the width is shown and the cross-section is included for reference. Figure 7.10(b) shows the corresponding quantities at span.The upstand grillage predicts almost the same stress as the elaborate three-dimensional brickfinite-element model at mid-span while the plane grillage predicts a higher stress in thecantilever and a lower stress elsewhere. Assuming the elaborate model

Fig. 7.8 In-plane distortion of members in upstand grillage model

Fig. 7.9 In-plane deformation in cantilevers of deck: (a) cross-section; (b) plan view of deflectedshape (half)

to be accurate, the benefits of the upstand grillage can be seen at this location. However, thisis not the case at span where the upstand grillage in fact makes a poorer prediction of stressin the cantilever than the plane grillage. This inaccuracy in the upstand grillage is attributableto the use of inappropriate rotational restraints at the ends of the cantilevers. Unfortunately,the complete removal of the rotational restraints resulted in the behaviour illustrated in Fig.7.8 which caused inaccuracies elsewhere in the upstand grillage model. One solution is toremove the rotational restraints only where the in-plane bending actually occurs but thismethod requires a degree of knowledge regarding the behaviour of the deck, which may notbe available prior to analysis. Clearly this is not a satisfactory approach for many bridge decks.

Fig. 7.10 Calculated longitudinal bending stress on top surface of deck: (a) at mid-span; (b) atspan

Example 7.2: Upstand grillage model

Figure 7.11(a) shows the cross-section of a bridge deck with edge cantilevers. The deck is 25m long with a single, simply supported span between bearings of 24 m and is supported alongthe entire width of the main part of the deck at each end. An upstand grillage model isrequired.

Fig. 7.11 Upstand grillage model of Example 7.2 (dimensions in mm): (a) cross-section; (b) cross-section with grillage members superimposed; (c) plan view of grillage mesh

Figure 7.11(b) shows the cross-section with a suitable upstand grillage model superimposed;the portion of bridge deck associated with each grillage member is indicated by the brokenlines. Figure 7.11(c) shows a plan of the upstand grillage mesh. The members representing theedge cantilevers are located at the centroid of the cantilevers which is 0.2 m below the top.The grillage members representing the main part of the deck are located at the centroid of thatpart which is at 0.6 m from the top. This gives vertical members with a length of 0.4 m. Thegrillage members representing the cantilevers and the outermost members in the main part ofthe deck have been positioned at 0.3 times the depth of the side (at that location) from theedge in accordance with the recommendations of Section 5.3.7. Seventeen rows of transversemembers are provided at a constant spacing of 1.5 m.

The properties of the members in the upstand grillage model are easily determined. The Xdirection is arbitrarily chosen to be parallel to the span of the bridge. Assuming the main deckslab to be isotropic, the second moment of area per unit breadth is calculated

as for a beam:

The torsion constant per unit breadth for longitudinal and transverse members is calculatedaccording to equation (5.65):

The second moments of area and torsion constants for the grillage members are thendetermined by multiplying these values by the relevant breadths of the members shown in Fig.7.11. For the longitudinal members in the main deck, this gives a second moment of area of:

and a torsion constant of:

except for the edge member in the main deck where the torsion constant is:

The area of the longitudinal members is also required and is given by:

For the transverse members other than those at the ends of the deck, the second moment ofarea is:

The torsion constant is:

and the area is:

At the ends, the bridge extends 0.5 m past the centres of the bearings, giving memberbreadths of 1.25 m, resulting in a second moment of area of:

a torsion constant of:

and an area of:

For the edge cantilevers, the bridge slab is again assumed to be isotropic and the secondmoment of area per unit breadth is calculated according to the simple beam formula:

and the torsion constant per unit breadth is calculated according to equation (5.65):

The second moments of area and torsion constants for the grillage members are once againdetermined by multiplying these values by the relevant breadths of the members. For thelongitudinal cantilever members this gives a second moment of area of:

The torsion constant is based on the breadth excluding the portion outside 0.3 times thedepth (at that location) which gives:

The area of the longitudinal cantilever members is given by:

For the transverse cantilever members, other than those at the ends of the deck, the secondmoment of area is:

the torsion constant is:

Table 7.1 Upstand grillage member properties for Example 7.2

I (m4) J (m4) A (m2)

Longitudinal members

Cantilever 0.0042 0.0073 0.32

Main deck (interior) 0.173 0.346 1.44

Main deck (edge) 0.173 0.276 1.44

Transverse members

Cantilever (interior) 0.0080 0.0161 0.60

Main deck (interior) 0.216 0.432 1.80Cantilever (ends) 0.0066 0.0121 0.50

Main deck (ends) 0.180 0.256 1.50

and the area is:

At the ends, the member breadths are less than those of the internal members, so onceagain, the member properties are less. The grillage member properties are given in Table 7.1.

A row of nodes is located at the junction of the edge cantilever and the main part of thedeck (Fig. 7.1(b)) so that the transverse members on the cantilever side can be given theproperties of the cantilever and those on the other side can be given the properties of the mainpart of the deck. The vertical members are given very large properties so that they will notbend or deform. The values chosen are dependent on the computer and software used asexcessively large values may result in round-off errors. In the authors’ experience, a secondmoment of area and torsion constant of between 100 and 1000 times the largest values in themodel is usually appropriate. Thus, for this example, the second moment of area could be 22m4 (100×0.216) and the torsion constant 43 m4 (100×0.432). This approach may need to beverified for particular computers and software. A useful way of achieving this is to increasethe member properties in successive runs until just before the program becomes unstable dueto round-off errors. Some programs may have the facility to assign ‘rigid’ properties tomembers. If this is available then it should be used for the vertical members.

It is of importance that no longitudinal grillage member be located at the top of the verticalmembers. The longitudinal member at the bottom is sufficient and the specification ofmembers at one level only simplifies the determination of their properties and theinterpretation of results.

When interpreting the results of an upstand grillage model, it is important to realise that themoments are not comparable to those in a planar grillage, as bending in the upstand model isnot about the bridge neutral axis. However, the difference is accounted for by the presence ofaxial forces which the bridge must be designed to resist.

7.6 Upstand finite-element modelling

Upstand finite-element (FE) modelling is an extension of plane FE modelling in the same waythat upstand grillage modelling is an extension of plane grillage modelling. The upstand FEmodel consists of a number of planes of plate finite elements connected together by rigidvertical members. It is generally more convenient, although not essential, that vertical beammembers are used rather than vertical elements. Figure 7.12 shows an upstand FE model for abridge deck with edge cantilevers. The cantilevers are idealised as finite elements located atthe level of the centroids of the actual cantilevers while the main part of the deck is idealisedusing finite elements located at the centroid of that part. The finite-element meshes on eachplane are connected by rigid vertical grillage members.

In a series of tests, the authors have found the upstand FE method to be very suitable formodelling bridge decks with wide edge cantilevers. It benefits from being three-dimensionalwhile being relatively simple to use. Most significantly, it does not suffer from the problemsof modelling in-plane behaviour associated with upstand grillages. This is largely due to thewell proven ability of finite-elements to model in-plane behaviour.

Fig. 7.12 Upstand finite-element model

Fig. 7.13 Calculated longitudinal bending stresses at span on top surface of deck

The bridge deck of Fig. 7.9 was analysed by the authors using an upstand FE model (O’Brienand Keogh, 1998). Figure 7.13 shows the longitudinal stresses predicted along the top surfaceof the deck at of the span in the same format as that used in Fig. 7.10. The distributionspredicted by the elaborate three-dimensional brick FE model and the plane and upstandgrillage models described previously are also shown. It can be seen that the upstand FE modelpredicts an almost identical stress distribution to the elaborate three-dimensional brick FEmodel. Similar results were found at midspan and for all other cases considered.

The interpretation of results from upstand FE models is not comparable to those fromplanar FE models (as is the case for upstand and planar grillage models).

Example 7.3: Upstand finite-element model

Figure 7.14(a) shows the cross-section of a bridge deck with wide edge cantilevers. The deckis continuous over two spans of 24.8 m and is supported along the entire width of the mainpart of the deck at each support location. An upstand FE model is required.

Fig. 7.14 Upstand finite-element model of Example 7.3: (a) cross-section (dimensions in mm); (b)finite-element mesh

Figure 7.14(b) shows a three-dimensional view of a suitable upstand finite-element mesh. Allof the elements are 1.2 m wide and 1.24 m long (in the span direction). The elementsrepresenting the edge cantilevers are located at the centroid of the cantilevers which is 0.2 mbelow the top of the bridge deck. Those representing the main part of the deck are located atthe centroid of that part which is 0.6 m from the top of the deck. This results in verticalmembers with a length of 0.4 m.

The main part of the deck and the edge cantilevers are both taken to be isotropic andconsequently the only properties associated with the elements (other than their materialproperties) are their depths. The elements in the main part of the deck are given a depth of 1.2m and those in the edge cantilevers a depth of 0.4 m.

This model was analysed by the authors under the action of self weight. A plane FE model(in accordance with the recommendations of Chapter 5) and a three-dimensional FE modelusing solid ‘brick’ type elements were also analysed. Figure 7.15(a) shows the longitudinalstress distribution at the top of the bridge deck along the centreline of the deck as predicted byeach of the models. As the model is symmetrical about the central support, only one span isshown in the figure. This stress distribution follows the expected pattern with zero stress atthe ends, maximum compressive stress close to span, zero stress close to span andmaximum tensile stress above the central support. The three-dimensional brick FE andupstand FE models predict a very similar stress at all locations and the plane FE model is inreasonable agreement. Figure 7.15(b) illustrates the corresponding distribution along a line0.6 m in from the edge of the cantilever. The three-dimensional brick FE and upstand FEmodels once again predict very similar stress at all locations but the plane FE model is in pooragreement with these. The plane FE model predicts a significantly greater stress at both themid-span and central support locations. This is caused by the inability of the planar model toallow for the rising neutral

Fig. 7.15 Longitudinal bending stress at top fibre for bridge of Example 7.3: (a) at centre; (b) 0.6m in from edge of cantilever

axis in the edge cantilever. Alternatively this can be viewed as the inability of the planarmodel to allow for shear lag. This example shows the benefits of three-dimensional modellingover planar modelling for bridge decks of this type.

7.6.1 Upstand finite-element modelling of voided slab bridge decks

The three-dimensional nature of upstand FE modelling requires the specification of thecorrect area for the elements as well as the correct second moment of area, as the stiffness ofeach part of the deck is made up of a combination of both of these. Most FE programs onlyallow the specification of a depth for the finite elements which does not

allow the independent specification of area and second moment of area. This is sufficientwhen dealing with solid slabs, but causes problems when dealing with voided slabs.

Modelling of voided slabs by the plane FE method is discussed in Section 6.4. Whenconsidering the longitudinal direction, the depth of the finite elements is determined byequating the second moment of area of the voided slab to that of an equivalent depth of solidslab. As the voids are generally located close to mid-depth of the slab, the equivalent depth ofthe elements will generally be quite close to (but smaller than) the actual depth of the voidedslab. In other words, the presence of the voids does not greatly affect the longitudinal secondmoment of area of the deck. This is not the case when considering the cross-sectional areawhich is greatly reduced by the presence of the voids. Therefore, a finite element with a depthchosen by considering the second moment of area of the voided slab will have an excessivearea. As stiffness in the upstand FE model is made up of a combination of both the secondmoment of area and the cross-sectional area of the elements, this will result in an overly stiffmodel. A solution to this problem is to reduce the area of the elements. In theory, this couldbe done by incorporating additional grillage members into the model with a negative area andzero second moment of area. Clearly a member with negative area has no physical meaningand, quite sensibly, most computer programs will not allow this. A more feasible alternative isto choose the depth of the finite elements so that they have the correct area and then to addadditional grillage members to make up the shortfall in second moment of area. Theadditional grillage members should have zero area. They should also have zero in-planesecond moment of area as the in-plane behaviour is still modelled by the finite elements.

Example 7.4: Upstand FE model of voided slab

Figure 7.16(a) shows the cross-section of a voided slab bridge deck with wide edgecantilevers. The deck is simply supported with a 24 m span and is supported continuouslyacross its breadth at each end. An upstand FE model is required.

Figure 7.16(b) shows the cross-section of a suitable upstand FE model for this bridge deck.A choice of 20, 1.2 m long elements in the longitudinal direction would be appropriate for thismodel. The length of the rigid vertical members is equal to the distance between the centroidof the cantilevers and that of the main part of the deck. In this case the vertical members are0.35 m long.

The X direction is chosen as the longitudinal direction. For the elements in the main part ofthe deck, each element represents a portion of deck 1.2 m wide with one void. The secondmoment of area of this is:

and the area is:

Fig. 7.16 Upstand finite-element model of Example 7.4 (dimensions in mm): (a) cross-section; (b)section through finite-element model

Equating this to an equivalent solid element with the same area gives an equivalent elementdepth, deq, of:

The second moment of area of this equivalent solid element, Ieq, is:

This gives a shortfall in second moment of area which has to be made up by additionalgrillage members. The second moment of area of these additional members, , is:

To incorporate the additional members in the model, each finite element in the main part ofthe deck is replaced by four elements and four grillage members as illustrated in Fig. 7.17.These elements have the same equivalent depth of 0.879 m and the longitudinal grillagemembers have second moments of area of 0.093 m4.

The required transverse second moment of area per unit breadth is given by equation (6.1):

Fig. 7.17 Replacement of plate element: (a) original element; (b) corresponding combination ofelements and grillage members

Hence, the required additional second moment of area which is provided by the transversegrillage members is:

Fig. 7.18 Upstand finite-element model with additional grillage members (half)

The edge cantilevers are modelled as finite elements with a depth of 0.5 m which is equal tothe actual depth of the cantilever. Rigid, or very stiff vertical grillage members are specified at0.6 m intervals to join the meshes on the different planes. The elements used for this exampleonly had nodes at the corners with the result that they could only be joined to the verticalmembers at their corners. Therefore, the originally proposed 1.2×1.2 elements in thecantilever were replaced with four 0.6×0.6 elements to give nodes at 0.6 m intervals. The finalupstand FE model with grillage members shown as dark lines is illustrated in Fig. 7.18. Onlyone-half of the model is shown as it is symmetrical.

7.6.2 Upstand finite-element modelling of other bridge types

It is possible to extend the principles of upstand FE analysis to types of bridge other than solidand voided slabs, provided care is taken to ensure that good similitude exists between themodel and the actual structure. Figure 7.19(a) shows a beam and slab bridge. Each beam inthis bridge will act compositely with the slab above it and they are normally assumed to bendabout their own centroid rather than that of the bridge as a whole. However, this is clearly anapproximation as the exact location of the neutral axis will depend on the flange widths andthe relative stiffnesses of the members. In such cases where the location of the neutral axis isunclear, an upstand FE analysis can be used to represent the behaviour more accurately thanthe alternative planar models.

The slab can be represented in the model using finite elements located at its centroid ofequal depth to it. The properties of the remaining parts of the deck are then calculated, eachabout its own centroid, and are represented by grillage members at the levels of thosecentroids as illustrated in Fig. 7.19(b). The horizontal members at different levels are joinedby stiff vertical members. This approach has the advantage of simplicity as there is a directcorrespondence between each member and

Fig. 7.19 Upstand finite-element model of beam and slab bridge: (a) cross-section; (b) sectionthrough upstand finite-element model

Fig. 7.20 Plate finite-element model of cellular bridge: (a) original bridge; (b) finite-elementmodel

a part of the structure. However, the interpretation of the output can be tedious. The calculatedmoment for each beam member is only applicable to bending about its own centroid. Ifreinforcement is to resist the stresses in a beam and the adjacent elements, then the totalmoment will have to be calculated taking account of the axial forces in the beam and theelements and the distance between them.

Figure 7.20(a) shows a cellular bridge deck and Fig. 7.20(b) shows a suitable model basedon a variation of the upstand FE analogy. This model, as well as dealing with a varyingneutral axis, has the advantage of automatically allowing for transverse cell distortion asdiscussed in Section 6.6. Transverse diaphragms could also be incorporated into this modelwith ease. Care should be taken with such a model to ensure that sufficient numbers ofelements are provided through the depth of the webs, to correctly model longitudinal bendingthere. Unfortunately, the number of elements required to achieve this is very large and this,combined with the tedium of interpreting the results, often rules out its use.

7.7 Prestress loads in three-dimensional models

When analysing for the effects of prestress in bridge decks, it is usual to uncouple the in-planeand out-of-plane behaviours. The in-plane behaviour is governed by the distribution of axialstress in the bridge deck and is often determined by a hand calculation. The out-of-planebehaviour is affected by the vertical components of tendon force and by the moments inducedby tendon eccentricity. These effects are generally dealt with by calculating the equivalentloading due to prestress (Chapter 2) which is often based on an assumed neutral axis location.The bridge deck is then analysed to determine the effects of the equivalent loading. Thestresses determined

from this analysis are combined with the in-plane axial stresses to obtain the overall effect ofthe applied prestressing forces.

When using a three-dimensional model, such as the upstand grillage or upstand FE methods,the equivalent loading due to prestress can be applied in a three-dimensional manner. Many ofthe complications involved in determining equivalent loads due to prestress can be avoided inthis way. There is no uncertainty concerning the location of the neutral axis about whicheccentricity of prestress must be calculated. There are also advantages to be gained in theinterpretation of results, because they can be related directly to the design without the need todistinguish between primary and secondary effects. This method is often simpler toimplement as there is no need to uncouple the in-plane and out-of-plane behaviours. In thethree-dimensional approach, the prestress forces are applied directly to the model at thecorrect vertical location by means of stiff vertical grillage members. It follows that thecalculation of moments due to cable eccentricity are not dependent on any assumed neutralaxis location.

The sources of error in a traditional planar model, with the equivalent loading calculated inthe normal manner, are two fold. Firstly, as discussed in previous sections, the inability of theplanar model to allow for the variation in neutral axis location may cause inaccuracies in thecalculated response to equivalent loading. However, there is an additional error, as themagnitude of the equivalent loading is itself dependent on the eccentricity of prestress and istherefore affected by the neutral axis location. It should be mentioned that, as the neutral axislocation is load dependent, the location which is applicable to, say, self weight may not beapplicable to prestressing.

Figure 7.21(a) shows a portion of a bridge deck with an edge cantilever. The location of theneutral axis is indicated in the figure, but it is unknown at this stage. The deck is subjected toa prestress force, P, at a distance, h, below mid-depth of the main part of the deck. Thisprestress force has an unknown eccentricity, e, which is also indicated in the figure. Figure7.21 (b) shows the equivalent portion of an upstand FE model. The prestress force is applieddirectly to the model through a rigid vertical member of length h. The eccentricity of thisforce is once again e but a knowledge of the magnitude of the eccentricity is not necessary.The model is subjected to an axial force which generates a moment of:

To avoid the necessity of adding a large number of vertical grillage members to the model, theprestress force can alternatively be applied at the level of the elements along with anadditional moment to allow for the difference in level between the true point of applicationand the element. Figure 7.21 (c) shows this alternative model. The additional moment is theproduct of the prestress force and the distance, h. The equivalence of Figs. 7.21(a) and (c) canbe seen by considering the applied moment. In the latter, the applied moment is:

which is equal to the applied moment of the former. In this way, the independence of theprestress loading from the neutral axis location is retained but the necessity for a large numberof vertical members is avoided.

Fig. 7.21 Portion of prestressed concrete deck: (a) original deck; (b) upstand finite-elementmodel with vertical member at point of application of prestress; (c) alternativeupstand finite-element model

The authors have found this direct method of representing the effects of prestress to be themost accurate of many methods tested when compared to results from elaborate three-dimensional finite-element analyses with brick type elements. In particular, upstand FEanalyses with equivalent loading calculated in the traditional way (as described in Chapter 2)did not always give accurate results.

Appendix AReactions and bending moment diagrams due to

applied load

Appendix BStiffness of structural members and associated

bending moment diagrams

Appendix CLocation of centroid of section

The centroid, , of any section can be found from the co-ordinates of the perimeter pointsusing the formula:

(C.1)

where xi and y i are the co-ordinates of point i and n is the number of co-ordinate points. Forthe purposes of this calculation, point n+1 is defined as equal to point 1. For the section of Fig.6.6, the co-ordinates are taken from the figure starting at the top left corner and specifyingonly half the section (which will have the same centroid as the full section). The terms ofequation (C.1) are given in Table C.1 where Top and Bottom refer to the numerator anddenominator respectively of the fraction specified in the equation.

Table C.1 Evaluation of equation (C.1)

xi yi (xi−x i+1) yi+y i+1 Top Bottom0 1200 −5500 4320000 2400 −23.76×109 −39.60×106

5500 1200 0 1440000 1200 0 0

5500 0 4000 0 0 0 0

1500 0 300 640000 800 0.19×109 0.72×106

1200 800 1200 2440000 1800 2.93×109 6.48×106

0 1000 0 3640000 2200 0 0

0 1200 0 4320000 2400 0 0

Sum= −20.64×109 −32.40×106

The y coordinate of the centroid is then:

The same answer can be found by dividing the section into rectangles and triangles andsumming moments of area about any common point.

Appendix DDerivation of shear area for grillage member

representing cell with flange and web distortion

The transverse shear force half way across the cell will be distributed between the flanges inproportion to their stiffness. Hence, the shear force in the top flange will be:

Fig. D.1 Cell with flange and web distortion: (a) assumed distortion; (b) segment of cell betweenpoints of contraflexure

where V is the total shear force and it and ib are the second moments of area per unit breadthof the top and bottom flanges respectively. This force is illustrated in Fig. D.1 for a segmentof cell between points of contraflexure. Hence the total moment at the top of the web is:

The rotation of the web due to this moment is:

where h is the bridge depth (centre to centre of flanges) and iw is the web second moment ofarea per unit breadth. The total deflection in the top flange results from this rotation plusbending in the flange itself:

Similarly the deflection in the bottom flange can be shown to be:

The mean deflection is:

Equating this to the shear deformation in a grillage member gives:

If the second moments of area per unit breadth are expressed in terms of the flange and webdepths , this becomes equation (6.7):

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OHBDC (1992) Ontario Highway Bridge Design Code, Ministry of Transportation of Ontario,Downsview, Ontario, Canada.

Springman, S.M., Norrish, A.R.M. and Ng, C.W.W.W. (1996) Cyclic Loading of Sand Behind IntegralBridge Abutments, UK Highways Agency, TRL Report 146, London.

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Foundation, Cleveland, Ohio.West, R. (1973) C&CA/CIRIA Recommendations on the Use of Grillage Analysis for Slab and

Pseudo-slab Bridge Decks, Cement and Concrete Association, London.Wood, R.H. (1968) The reinforcement of slabs in accordance with a pre-determined field of moments,

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Index

Page numbers appearing in bold refer to major entries

AASHTO 40Aesthetics 34–9Aluminium deck 42Analysis, introduction to 67–120Anisotropic 151, 160Application of planar grillage and finite element methods 200–39Arch 18–20Articulation 26–9Asphaltic plug joint 32–3

Balanced cantilever 14–17BD37/88 40Beam and slab bridge 183, 218–28

arching action of slab 228finite element modelling 225–7grillage modelling 219–25transverse behaviour 227–8upstand finite element model 259

Bearing 29–31, 169, 180, 182, 185elastomeric 31pot 30–1sliding 17, 29–30spherical 30see also Supports

Bending moments due to applied loading 263–4, 265–6Blister 16Box culvert 21–3, 72–4Box girder 212, 228Box section 5–7Bridge bashing see Loading, impactBuried joint 32

Cable-stayed 25–6Cantilever 3

balanced 14–17Cellular bridge 212, 228–36

grillage modelling 230–6three-dimensional finite element model 260transverse cell distortion 228, 229, 231, 260, 269–71

Cellular section see Cross-section, boxCentroid, location of 267–8Collision loads see Loading, impactComposite 25, 42, 147–50Computer implementation of grillages 179–80Concrete, lightweight 42Continuous beam/slab 10, 13–18Contraction of integral bridges 128–33Coupler, post-tensioning 14

Creep 12, 17, 18, 28, 78, 82, 147–50Cross-section 2–8

box 5–7older concepts 7−8solid rectangular 2–4T- 4–5voided rectangular 4

Culvert, box 21–3, 72–4Curved bridge 236–9

finite element modelling 238–9grillage modelling 236–8

Dead loading 40, 42Density, dry 127Diaphragm 10, 95, 184, 211, 218–19, 220, 228Differential settlement 9, 40, 41, 75–7Differential temperature 47–51, 89–104Downstand 203Drop-in span 24–5Dry density 127Durability 1Dynamic amplification 53Dynamic effects 52–4

Earth pressure 124–6Edge cantilever 203–11, 237, 240, 244–6, 252Edge stiffening 203–11, 237Effective flange width 242–4, 245Elastomeric bearing 31Elevations 8–26Equivalent loading due to prestress 54–66, 260–1Equivalent loading due to temperature/ thermal effects 47, 49, 88Equivalent loads method 67, 88–9, 90, 92, 107Eurocode 40Expansion of integral bridge see Integral bridge, expansion

FEA see Finite element analysisFinite element analysis (FEA)

application of planar 200–39beam and slab bridge 225–7brick elements 244, 246, 253, 262mesh 189–91properties of elements 186–9recommendations for modelling 189–91similitude with bridge slab 186slab bridges 185–91see also Upstand finite element modelling

Foundation, shallow strip 130Forced vibration 53Frame bridge 21–3Frequency see Loading, dynamic

Geometrically orthotropic 152, 177–9, 187–9, 211, 217Grillage

accuracy 171analysis of slabs 169–85application of moment distribution 111–20application of planar 200–39beam and slab bridges 219–25cellular bridges 230–6computer implementation 179–80member properties 173–9mesh 169–71, 178, 180–2, 184recommendations for modelling 182–5

shear flexible 212, 230similitude with bridge slab 171–3sources of inaccuracy 180–2U-beams, modelling of 225see also Upstand grillage modelling

Halving joint 24

Impact loading 41, 51–2, 54Imposed traffic loading 40, 43–5Inaccuracy, sources of see Grillage, sources of inaccuracyIncremental launch 17In-plane effects 162, 180, 185, 189, 245–7, 252, 260–1Integral bridge 21–3, 121–50

contraction 128–33cracking over supports 147–50expansion 137–45

bank seats 142–5deep abutments 140–2

time-dependent effects 147–50Interface shear stress 240Inverted T 3

Isotropic 151–2, 173–7, 179, 186, 200–3

Joint 13, 32–3asphaltic plug 32–3buried 32construction 13halving 24nosing 33

Key, shear 15

Lane, notional 43Launch, incremental 17Lightweight concrete 42Linear transformation 54–8Loading 40–66

abnormal traffic 44–5cycle track 40, 43dead 40, 42dynamic 41, 45, 52–4equivalent due to prestress 54–66, 260–1equivalent due to thermal effects 47, 49, 88HA 43–4HB 44–5horizontal 40, 46impact 41, 51–2, 54normal traffic 43–4pedestrian 40, 41, 43prestress 42, 54–66, 104–11rail traffic 45–6road traffic 43–5superimposed dead 40, 42thermal 40, 46–51

differential 47, 89–104uniform 46, 78–89

traffic 40, 43–5wind 42

Losses, prestress 60–3, 107

Materially orthotropic 152–67, 173, 177–8, 186–8, 211, 217M-beam 7Mesh see Finite element analysis, mesh; Grillage, meshModulus

secant 127shear 161, 176, 178, 186, 188, 231

Momentcapacity see Wood and Armer equationsdistribution 67–120in orthotropic plates 161–7twisting see 166–7, 172–3, 176, 178–9, 182, 186, 188, 191–9, 236

Movement, accommodation of 26

Natural frequency see Loading, dynamicNeutral axis 203–4, 212, 220, 232, 240–2, 244–5, 252, 259–61Nosing joint 33Notional lane 43

Orthotropic 151–2, 160, 173, 188, 193geometrically 152, 177–9, 187–9, 211, 217materially 152–67, 173, 177–8, 186–8, 211, 217

Parapet 34, 40, 42Partially continuous beam/slab 10–13Passive earth pressure 124Pavement 42Pier 184Poisson’s ratio 160, 174, 176, 179, 181, 188Pot bearing 30–1Prandtl’s membrane analogy 222Precast beam 183Pressure, earth 124–6

passive 124Prestress

loading 54–66loading in three-dimensional models 260–2losses 60–3, 107parabolic profile 56–8qualitative profile design 58–9tendon 183

Pseudo-box construction 7Push-launch construction 17–18

Rail traffic loading 45–6Reactions due to applied loading 263–4Recommendations

for finite element analysis 189–91for grillage modelling 182–5

Rectangular section see Cross-section, solid rectangularRemaining area 43Rigid vertical members 245, 252, 259, 261Road traffic loading 43–5Run-on slab 23, 122, 145–7

Secant modulus 127Section see Cross-sectionSegment, stitching 16Series of simply supported beams/slabs 9Settlement, differential 9, 40, 41, 75–7Settlement trough 146Shallow strip foundation 130Shear

area 180, 230–1, 269–71close to point support 182, 191distortion/deformation 156–7, 180, 230–1, 246enhancement 182, 185flexible grillage 212, 230force, from grillage 173key 15key deck 8lag 240–2, 244–5modulus 161, 176, 178, 186, 188, 231strain 155–6, 161strength of concrete 156in thin plates 167–9

Simply supported beam/slab 9, 24Skew deck 116–20, 236–9

finite element modelling 238–9grillage modelling 236–8

Slabbridge decks, behaviour and modelling 151–99run-on 23, 122, 145–7

Sliding bearing 17, 29–30Soil stiffness 126–8Soil/structure interaction 41, 74, 125, 180Span-by-span construction 13Span/depth ratios 36Spring

model (of soil) 133–6, 137stiffness (for soil) 130supports 180, 184, 191

Steel deck 42Stiffness of structural members 265–6Stitching segment 16Stress in orthotropic plates 159–61Strip foundation 130Structural form, factors affecting 1–2Superimposed dead loading 40, 42

Supports 180, 183–5, 189, 191, 220see also Bearing; Shear, close to point support

Suspension bridge 26Symmetry 71

T-section 4–5Temperature, differential 47–51, 89–104Temperature loading see Loading, thermalTerms 1Thermal loading see Loading, thermalThin plate theory 151–69Three-dimensional modelling of bridge decks 240–62Torsion

constant 167, 173–9, 181, 184, 209, 221–2, 232–3see also Prandtl’s membrane analogy

moment distribution 111–20Traffic loading 40, 43–5Transformation, linear 54–8Twisting moment 166–7, 172–3, 176, 178–9, 182, 186, 188, 191–9, 236

U-beam, grillage modelling of 225

Uplift 28Upstand 34, 203, 244Upstand finite element modelling 252–60, 261

of other bridge types 259–60of voided slabs 255–9

Upstand grillage modelling 245–52, 253, 261

Vibration see Loading, dynamicVoided slab 4, 152, 211–18, 228

torsional stiffness 214

WIM 43Wing wall 35Wood and Armer equations 191–9, 236

Y-beam 7

141336426 bridge deck analysis

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