Designers Guide to EN1992-2 Eurocode 2

DESIGNERS’ GUIDES TO THE EUROCODES

DESIGNERS’ GUIDE TO EN 1992-2 EUROCODE 2: DESIGN OF CONCRETE STRUCTURES PART 2: CONCRETE BRIDGES

Eurocode Designers’ Guide Series Designers’ Guide to EN 1990. Eurocode: Basis of Structural Design. H. Gulvanessian, J.-A. Calgaro and M. Holicky´. 0 7277 3011 8. Published 2002. Designers’ Guide to EN 1994-1-1. Eurocode 4: Design of Composite Steel and Concrete Structures. Part 1.1: General Rules and Rules for Buildings. R. P. Johnson and D. Anderson. 0 7277 3151 3. Published 2004. Designers’ Guide to EN 1997-1. Eurocode 7: Geotechnical Design – General Rules. R. Frank, C. Bauduin, R. Driscoll, M. Kavvadas, N. Krebs Ovesen, T. Orr and B. Schuppener. 0 7277 3154 8. Published 2004. Designers’ Guide to EN 1993-1-1. Eurocode 3: Design of Steel Structures. General Rules and Rules for Buildings. L. Gardner and D. Nethercot. 0 7277 3163 7. Published 2004. Designers’ Guide to EN 1992-1-1 and EN 1992-1-2. Eurocode 2: Design of Concrete Structures. General Rules and Rules for Buildings and Structural Fire Design. A.W. Beeby and R. S. Narayanan. 0 7277 3105 X. Published 2005. Designers’ Guide to EN 1998-1 and EN 1998-5. Eurocode 8: Design of Structures for Earthquake Resistance. General Rules, Seismic Actions, Design Rules for Buildings, Foundations and Retaining Structures. M. Fardis, E. Carvalho, A. Elnashai, E. Faccioli, P. Pinto and A. Plumier. 0 7277 3348 6. Published 2005. Designers’ Guide to EN 1995-1-1. Eurocode 5: Design of Timber Structures. Common Rules and for Rules and Buildings. C. Mettem. 0 7277 3162 9. Forthcoming: 2007 (provisional). Designers’ Guide to EN 1991-4. Eurocode 1: Actions on Structures. Wind Actions. N. Cook. 0 7277 3152 1. Forthcoming: 2007 (provisional). Designers’ Guide to EN 1996. Eurocode 6: Part 1.1: Design of Masonry Structures. J. Morton. 0 7277 3155 6. Forthcoming: 2007 (provisional). Designers’ Guide to EN 1991-1-2, 1992-1-2, 1993-1-2 and EN 1994-1-2. Eurocode 1: Actions on Structures. Eurocode 3: Design of Steel Structures. Eurocode 4: Design of Composite Steel and Concrete Structures. Fire Engineering (Actions on Steel and Composite Structures). Y. Wang, C. Bailey, T. Lennon and D. Moore. 0 7277 3157 2. Forthcoming: 2007 (provisional). Designers’ Guide to EN 1993-2. Eurocode 3: Design of Steel Structures. Bridges. C. R. Hendy and C. J. Murphy. 0 7277 3160 2. Forthcoming: 2007 (provisional). Designers’ Guide to EN 1991-2, 1991-1-1, 1991-1-3 and 1991-1-5 to 1-7. Eurocode 1: Actions on Structures. Traffic Loads and Other Actions on Bridges. J.-A. Calgaro, M. Tschumi, H. Gulvanessian and N. Shetty. 0 7277 3156 4. Forthcoming: 2007 (provisional). Designers’ Guide to EN 1991-1-1, EN 1991-1-3 and 1991-1-5 to 1-7. Eurocode 1: Actions on Structures. General Rules and Actions on Buildings (not Wind). H. Gulvanessian, J.-A. Calgaro, P. Formichi and G. Harding. 0 7277 3158 0. Forthcoming: 2007 (provisional). Designers’ Guide to EN 1994-2. Eurocode 2: Design of Concrete Structures. Bridges. D. Smith and C. Hendy. 0 7277 3159 9. Forthcoming: 2007 (provisional).

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DESIGNERS’ GUIDES TO THE EUROCODES

DESIGNERS’ GUIDE TO EN 1992-2 EUROCODE 2: DESIGN OF CONCRETE STRUCTURES PART 2: CONCRETE BRIDGES

C. R. HENDY and D. A. SMITH

Published by Thomas Telford Publishing, Thomas Telford Ltd, 1 Heron Quay, London E14 4JD URL: http://www.thomastelford.com

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First published 2007

Eurocodes Expert Structural Eurocodes offer the opportunity of harmonized design standards for the European construction market and the rest of the world. To achieve this, the construction industry needs to become acquainted with the Eurocodes so that the maximum advantage can be taken of these opportunities Eurocodes Expert is a new ICE and Thomas Telford initiative set up to assist in creating a greater awareness of the impact and implementation of the Eurocodes within the UK construction industry Eurocodes Expert provides a range of products and services to aid and support the transition to Eurocodes. For comprehensive and useful information on the adoption of the Eurocodes and their implementation process please visit our website or email [email protected]

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# The authors and Thomas Telford Limited 2007

All rights, including translation, reserved. Except as permitted by the Copyright, Designs and Patents Act 1988, no part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying or otherwise, without the prior written permission of the Publishing Director, Thomas Telford Publishing, Thomas Telford Ltd, 1 Heron Quay, London E14 4JD. This book is published on the understanding that the authors are solely responsible for the statements made and opinions expressed in it and that its publication does not necessarily imply that such statements and/or opinions are or reflect the views or opinions of the publishers. While every effort has been made to ensure that the statements made and the opinions expressed in this publication provide a safe and accurate guide, no liability or responsibility can be accepted in this respect by the authors or publishers.

Typeset by Academic þ Technical, Bristol Printed and bound in Great Britain by MPG Books, Bodmin

Preface Aims and objectives of this guide The principal aim of this book is to provide the user with guidance on the interpretation and use of EN 1992-2 and to present worked examples. It covers topics that will be encountered in typical concrete bridge designs and explains the relationship between EN 1992-2 and the other Eurocodes. EN 1992-2 is not a ‘stand alone’ document and refers extensively to other Eurocodes. Its format is based on EN 1992-1-1 and generally follows the same clause numbering. It identifies which parts of EN 1992-1-1 are relevant for bridge design and adds further clauses that are specific to bridges. It is therefore not useful to produce guidance on EN 1992-2 in isolation and so this guide covers material in EN 1992-1-1 which will need to be used in bridge design. This book also provides background information and references to enable users of Eurocode 2 to understand the origin and objectives of its provisions.

Layout of this guide EN 1992-2 has a foreword, 13 sections and 17 annexes. This guide has an introduction which corresponds to the foreword of EN 1992-2, Chapters 1 to 10, which correspond to Sections 1 to 10 of the Eurocode and Annexes A to Q which again correspond to Annexes A to Q of the Eurocode. The guide generally follows the section numbers and first sub-headings in EN 1992-2 so that guidance can be sought on the code on a section by section basis. The guide also follows the format of EN 1992-2 to lower levels of sub-heading in cases where this can conveniently be done and where there is sufficient material to merit this. The need to use several Eurocode parts can initially make it a daunting task to locate information in the order required for a real design. In some places, therefore, additional sub-sections are included in this guide to pull together relevant design rules for individual elements, such as pile caps. Additional sub-sections are identified as such in the sub-section heading. The following parts of the Eurocode are intended to be used in conjunction with Eurocode 2: EN 1990: EN 1991: EN 1997: EN 1998: hENs: EN 13670:

Basis of structural design Actions on structures Geotechnical design Design of structures for earthquake resistance Construction products relevant for concrete structures Execution (construction) of concrete structures

These documents will generally be required for a typical concrete bridge design, but discussion on them is generally beyond the scope of this guide.

DESIGNERS’ GUIDE TO EN 1992-2

In this guide, references to Eurocode 2 are made by using the abbreviation ‘EC2’ for EN 1992, so EN 1992-1-1 is referred to as EC2-1-1. Where clause numbers are referred to in the text, they are prefixed by the number of the relevant part of EC2. Hence: . . . .

2-2/clause 6.3.2(6) means clause 6.3.2, paragraph (6), of EC2-2 2-1-1/clause 6.2.5(1) means clause 6.2.5, paragraph (1), of EC2-1-1 2-2/Expression (7.22) means equation (7.22) in EC2-2 2-1-1/Expression (7.8) means equation (7.8) in EC2-1-1.

Note that, unlike in other guides in this series, even clauses in EN 1992-2 itself are prefixed with ‘2-2’. There are so many references to other parts of Eurocode 2 required that to do otherwise would be confusing. Where additional equations are provided in the guide, they are numbered sequentially within each sub-section of a main section so that, for example, the third additional expression within sub-section 6.1 would be referenced equation (D6.1-3). Additional figures and tables follow the same system. For example, the second additional figure in section 6.4 would be referenced Figure 6.4-2.

Acknowledgements Chris Hendy would like to thank his wife, Wendy, and two boys, Peter Edwin Hendy and Matthew Philip Hendy, for their patience and tolerance of his pleas to finish ‘just one more section’. David Smith would like to thank his wife, Emma, for her limitless patience during preparation of this guide. He also acknowledges the continued support of Brian and Rosalind Ruffell-Ward from the very beginning. Both authors would also like to thank their employer, Atkins, for providing both facilities and time for the production of this guide. They also wish to thank Dr Paul Jackson and Dr Steve Denton for their helpful comments on the guide. Chris Hendy David A. Smith

vi

Contents Preface Aims and objectives of this guide Layout of this guide Acknowledgements

v v v vi

Additional information specific to EN 1992-2

1 2

Chapter 1.

General 1.1. Scope 1.1.1. Scope of Eurocode 2 1.1.2. Scope of Part 2 of Eurocode 2 1.2. Normative references 1.3. Assumptions 1.4. Distinction between principles and application rules 1.5. Definitions 1.6. Symbols

3 3 3 4 4 4 5 5 5

Chapter 2.

Basis of design 2.1. Requirements 2.2. Principles of limit state design 2.3. Basic variables 2.4. Verification by the partial factor method 2.4.1. General 2.4.2. Design values 2.4.3. Combinations of actions 2.5. Design assisted by testing 2.6. Supplementary requirements for foundations

7 7 7 7 9 9 9 9 10 10

Chapter 3.

Materials 3.1. Concrete 3.1.1. General 3.1.2. Strength 3.1.3. Elastic deformation 3.1.4. Creep and shrinkage 3.1.5. Concrete stress–strain relation for non-linear structural analysis

11 11 11 11 14 14

Introduction

19


viii

3.1.6. Design compressive and tensile strengths 3.1.7. Stress–strain relations for the design of sections 3.1.8. Flexural tensile strength 3.1.9. Confined concrete 3.2. Reinforcing steel 3.2.1. General 3.2.2. Properties 3.2.3. Strength 3.2.4. Ductility 3.2.5. Welding 3.2.6. Fatigue 3.2.7. Design assumptions 3.3. Prestressing steel 3.3.1. General 3.3.2. Properties 3.3.3. Strength 3.3.4. Ductility characteristics 3.3.5. Fatigue 3.3.6. Design assumptions 3.4. Prestressing devices 3.4.1. Anchorages and couplers 3.4.2. External non-bonded tendons

20 21 22 23 23 23 23 23 24 25 25 25 25 25 26 27 27 28 28 29 29 29

Chapter 4.

Durability and cover to reinforcement 4.1. General 4.2. Environmental conditions 4.3. Requirements for durability 4.4. Methods of verification 4.4.1. Concrete cover

31 31 32 35 36 36

Chapter 5.

Structural analysis 5.1. General 5.2. Geometric imperfections 5.2.1. General (additional sub-section) 5.2.2. Arches (additional sub-section) 5.3 Idealization of the structure 5.3.1 Structural models for overall analysis 5.3.2. Geometric data 5.4. Linear elastic analysis 5.5. Linear elastic analysis with limited redistribution 5.6. Plastic analysis 5.6.1. General 5.6.2. Plastic analysis for beams, frames and slabs 5.6.3. Rotation capacity 5.6.4. Strut-and-tie models 5.7. Non-linear analysis 5.7.1. Method for ultimate limit states 5.7.2. Scalar combinations 5.7.3. Vector combinations 5.7.4. Method for serviceability limit states 5.8. Analysis of second-order effects with axial load 5.8.1. Definitions and introduction to second-order effects 5.8.2. General 5.8.3. Simplified criteria for second-order effects

39 39 40 40 43 44 44 44 48 49 52 52 52 53 56 58 58 60 61 62 62 62 63 64

CONTENTS

5.8.4. Creep 5.8.5. Methods of analysis 5.8.6. General method – second-order non-linear analysis 5.8.7. Second-order analysis based on nominal stiffness 5.8.8. Method based on nominal curvature 5.8.9. Biaxial bending 5.9. Lateral instability of slender beams 5.10. Prestressed members and structures 5.10.1. General 5.10.2. Prestressing force during tensioning 5.10.3. Prestress force 5.10.4. Immediate losses of prestress for pre-tensioning 5.10.5. Immediate losses of prestress for post-tensioning 5.10.6. Time-dependent losses 5.10.7. Consideration of prestress in the analysis 5.10.8. Effects of prestressing at the ultimate limit state 5.10.9. Effects of prestressing at the serviceability and fatigue limit states 5.11. Analysis for some particular structural members

Chapter 6.

Ultimate limit states 6.1. ULS bending with or without axial force 6.1.1. General (additional sub-section) 6.1.2. Reinforced concrete beams (additional sub-section) 6.1.3. Prestressed concrete beams (additional sub-section) 6.1.4. Reinforced concrete columns (additional sub-section) 6.1.5. Brittle failure of members with prestress (additional sub-section) 6.2. Shear 6.2.1. General verification procedure rules 6.2.2. Members not requiring design shear reinforcement 6.2.3. Members requiring design shear reinforcement 6.2.4. Shear between web and flanges of T-sections 6.2.5. Shear at the interface between concrete cast at different times 6.2.6. Shear and transverse bending 6.2.7. Shear in precast concrete and composite construction (additional sub-section) 6.3. Torsion 6.3.1. General 6.3.2. Design procedure 6.3.3. Warping torsion 6.3.4. Torsion in slabs (additional sub-section) 6.4. Punching 6.4.1. General 6.4.2. Load distribution and basic control perimeter 6.4.3. Punching shear calculation 6.4.4. Punching shear resistance of slabs and bases without shear reinforcement 6.4.5. Punching shear resistance of slabs and bases with shear reinforcement 6.4.6. Pile caps (additional sub-section) 6.5. Design with strut-and-ties models 6.5.1. General

69 70 70 71 76 80 80 81 81 82 83 84 85 90 95 96 98 104

105 105 105 105 118 121 126 131 132 133 140 154 158 160 160 166 166 167 171 172 175 175 176 177 179 183 185 193 193

ix


6.6. 6.7. 6.8.

6.9.

x

6.5.2. Struts 6.5.3. Ties 6.5.4. Nodes Anchorage and laps Partially loaded areas Fatigue 6.8.1. Verification conditions 6.8.2. Internal forces and stresses for fatigue verification 6.8.3. Combination of actions 6.8.4. Verification procedure for reinforcing and prestressing steel 6.8.5. Verification using damage equivalent stress range 6.8.6. Other verification methods 6.8.7. Verification of concrete under compression or shear Membrane elements

193 195 196 201 201 208 208 208 209 209 210 212 213 215

Chapter 7.

Serviceability limit states 7.1. General 7.2. Stress limitation 7.3. Crack control 7.3.1. General considerations 7.3.2. Minimum areas of reinforcement 7.3.3. Control of cracking without direct calculation 7.3.4. Control of crack widths by direct calculation 7.4. Deflection control 7.5. Early thermal cracking (additional sub-section)

225 225 226 230 230 232 234 237 243 243

Chapter 8.

Detailing of reinforcement and prestressing steel 8.1. General 8.2. Spacing of bars 8.3. Permissible mandrel diameters for bent bars 8.4. Anchorage of longitudinal reinforcement 8.4.1. General 8.4.2. Ultimate bond stress 8.4.3. Basic anchorage length 8.4.4. Design anchorage length 8.5. Anchorage of links and shear reinforcement 8.6. Anchorage by welded bars 8.7. Laps and mechanical couplers 8.7.1. General 8.7.2. Laps 8.7.3. Lap length 8.7.4. Transverse reinforcement in the lap zone 8.7.5. Laps of welded mesh fabrics made of ribbed wires 8.7.6. Welding (additional sub-section) 8.8. Additional rules for large diameter bars 8.9. Bundled bars 8.10. Prestressing tendons 8.10.1. Tendon layouts 8.10.2. Anchorage of pre-tensioned tendons 8.10.3. Anchorage zones of post-tensioned members 8.10.4. Anchorages and couplers for prestressing tendons 8.10.5. Deviators

245 245 246 246 247 247 248 248 249 251 251 252 252 252 253 254 257 257 257 258 258 258 259 262 271 272

CONTENTS

Chapter 9.

Detailing of members and particular rules 9.1. General 9.2. Beams 9.2.1. Longitudinal reinforcement 9.2.2. Shear reinforcement 9.2.3. Torsion reinforcement 9.2.4. Surface reinforcement 9.2.5. Indirect supports 9.3. Solid slabs 9.3.1. Flexural reinforcement 9.3.2. Shear reinforcement 9.4. Flat slabs 9.5. Columns 9.5.1. General 9.5.2. Longitudinal reinforcement 9.5.3. Transverse reinforcement 9.6. Walls 9.7. Deep beams 9.8. Foundations 9.9. Regions with discontinuity in geometry or action

275 275 275 275 278 279 279 279 281 281 282 282 282 282 283 283 284 284 285 288

Chapter 10.

Additional rules for precast concrete elements and structures 10.1. General 10.2. Basis of design, fundamental requirements 10.3. Materials 10.3.1. Concrete 10.3.2. Prestressing steel 10.4. Not used in EN 1992-2 10.5. Structural analysis 10.5.1. General 10.5.2. Losses of prestress 10.6. Not used in EN 1992-2 10.7. Not used in EN 1992-2 10.8. Not used in EN 1992-2 10.9. Particular rules for design and detailing 10.9.1. Restraining moments in slabs 10.9.2. Wall to floor connections 10.9.3. Floor systems 10.9.4. Connections and supports for precast elements 10.9.5. Bearings 10.9.6. Pocket foundations

289 289 289 290 290 290 290 290 290 291 291 291 291 291 291 291 291 291 292 293

Chapter 11.

Lightweight aggregate concrete structures 11.1. General 11.2. Basis of design 11.3. Materials 11.3.1. Concrete 11.3.2. Elastic deformation 11.3.3. Creep and shrinkage 11.3.4. Stress strain relations for non-linear structural analysis 11.3.5 Design compressive and tensile strengths 11.3.6. Stress strain relations for the design of sections 11.3.7. Confined concrete 11.4. Durability and cover to reinforcement

295 295 296 296 296 296 297 298 298 298 298 298

xi


11.5. 11.6. 11.7. 11.8. 11.9.

xii

Structural analysis Ultimate limit states Serviceability limit states Detailing of reinforcement – general Detailing of members and particular rules

298 298 302 302 302

Chapter 12.

Plain and lightly reinforced concrete structures

303

Chapter 13.

Design for the execution stages 13.1. General 13.2. Actions during execution 13.3. Verification criteria 13.3.1. Ultimate limit state 13.3.2. Serviceability limit states

307 307 308 309 309 309

Annex A.

Modification of partial factors for materials (informative)

311

Annex B.

Creep and shrinkage strain (informative)

313

Annex C.

Reinforcement properties (normative)

316

Annex D.

Detailed calculation method for prestressing steel relaxation losses (informative)

317

Annex E.

Indicative strength classes for durability (informative)

322

Annex F.

Tension reinforcement expressions for in-plane stress conditions (informative)

324

Annex G.

Soil-structure interaction

325

Annex H.

Not used in EN 1992-2

Annex I.

Analysis of flat slabs (informative)

326

Annex J.

Detailing rules for particular situations (informative)

327

Annex K.

Structural effects of time-dependent behaviour (informative)

331

Annex L.

Concrete shell elements (informative)

344

Annex M.

Shear and transverse bending (informative)

346

Annex N.

Damage equivalent stresses for fatigue verification (informative)

356

Annex O.

Typical bridge discontinuity regions (informative)

362

Annex P.

Safety format for non-linear analysis (informative)

363

Annex Q.

Control of shear cracks within webs (informative)

364

References

369

Index

371

CHAPTER 7

Serviceability limit states This chapter deals with the design at service limit states of members as covered in section 7 of EN 1992-2 in the following clauses: . . . .

General Stress limitation Crack control Deflection control

Clause 7.1 Clause 7.2 Clause 7.3 Clause 7.4

An additional section 7.5 is included to discuss early thermal cracking.

7.1. General EN 1992-2 section 7 covers only the three serviceability limit states relating to clause 7.2 to 7.4 above. 2-1-1/clause 7.1(1)P notes that other serviceability limit states ‘may be of importance’. EN 1990/A2.4 is relevant in this respect. It covers partial factors, serviceability criteria, design situations, comfort criteria, deformations of railway bridges and criteria for the safety of rail traffic. Most of its provisions are qualitative but some recommended values are given in various Notes, as guidance for National Annexes. EN 1990 is of general relevance. From clause 6.5.3 of EN 1990, the relevant combination of actions for serviceability limit states is ‘normally’ either the characteristic, frequent, or quasi-permanent combination. These are all used in EC2-2 and the general forms of these combinations, together with examples of use, are given in Table 7.1, but reference to section 2 and Annex A2 of EN 1990 is recommended for a detailed explanation of the expressions and terms. Specific rules for the combinations of actions (e.g. actions that need not be considered together), recommended combination factors and partial safety factors for bridge design are also specified in Annex A2 of EN 1990. Section 2 of this guide gives further commentary on the basis of design and the use of partial factors and combinations of actions. The general expressions in Table 7.1 have been simplified assuming that partial factors of 1.0 are used throughout for all actions at the serviceability limit state, as recommended in Annex A2 of EN 1990, but they may be varied in the National Annex. Appropriate methods of global analysis for determining design action effects are discussed in detail in section 5. For serviceability limit state verification, the global analysis may be either elastic without redistribution (clause 5.4) or non-linear (clause 5.7). Elastic global analysis is most commonly used and it is not normally then necessary to consider the effects of cracking within it – section 5.4 refers. 2-1-1/clause 7.1(2) permits an un-cracked concrete cross-section to be assumed for stress and deflection calculation provided that the flexural tensile stress under the relevant combination of actions considered does not exceed fct;eff . fct;eff may be taken as either fctm

2-1-1/clause 7.1(1)P

2-1-1/clause 7.1(2)


Table 7.1. Combinations of actions for serviceability limit states Combination

Notes

Characteristic

Combination of actions with a fixed (small) probability of being exceeded during normal operation within the structure’s design life, e.g. combination appropriate to checks on stress in reinforcement as it is undesirable for inelastic deformation of reinforcement to occur at any time during the service life.

Frequent

Combination of actions with a fixed probability of being exceeded during a reference period of a few weeks, e.g. combination used for checks of cracking and decompression in prestressed bridges with bonded tendons.

Quasipermanent

Combination of actions expected to be exceeded approximately 50% of the time, i.e. a time-based mean. For example, combination appropriate to crack width checks in reinforced concrete members on the basis that durability is influenced by average crack widths, not the worst crack width ever experienced.

General expression P

Gk;j þ P þ Qk;1 þ

P

Gk;j þ P þ

1;1 Qk;1

þ

P

P

Gk;j þ P þ

2;1 Qk;1

þ

P

P

0;i Qk;i

1;i Qk;i

2;i Qk;i

or fctm;fl but should be consistent with the value used in the calculation of minimum tension reinforcement (see section 7.3). For the purpose of calculating crack widths and tension stiffening effects, fctm should be used.

7.2. Stress limitation

2-1-1/clause 7.2(1)P 2-2/clause 7.2(102)

226

Stresses in bridges are limited to ensure that under normal conditions of use, assumptions made in design models (e.g. linear-elastic behaviour) remain valid, and to avoid deterioration such as the spalling of concrete or excessive cracking leading to a reduction of durability. For persistent design situations, it is usual to check stresses soon after the opening of the bridge to traffic, when little creep has occurred, and also at a later time when creep and shrinkage are substantially complete. This affects the loss of prestress in prestressed structures and the modular ratio for stress and crack width calculation in reinforced concrete structures. It may be necessary to include part of the long-term shrinkage effects in the first check, because up to half of the long-term shrinkage can occur in the first 3 months after the end of curing of the concrete. Calculation of an effective concrete modulus allowing for creep is discussed below. 2-1-1/clause 7.2(1)P requires compressive stresses in the concrete to be limited to avoid longitudinal cracking, micro-cracking or excessive creep. The first two can lead to a reduction of durability. 2-2/clause 7.2(102) addresses longitudinal cracking by requiring the stress level under the characteristic combination of actions to not exceed a limiting value of k1 fck (for areas with exposure classes of XD, XF or XS), where k1 is a nationally determined parameter with recommended value of 0.6. The clause identifies that the limit can be increased where specific measures are taken, such as increasing the cover to reinforcement (from the minimum values discussed in section 4) or by providing confinement by transverse reinforcement. The improvement from confining reinforcement is quantified as an increase in allowable stress of 10%, but this may be varied in the National Annex. The design of this reinforcement is not covered by EC2-2, but the strut-and-tie rules in 2-2/clause 6.5 and discussions in section 6.5 of this guide are relevant. Such reinforcement would need to operate at low stresses to have any significant effect in limiting the width of compression-induced cracks. Micro-cracking typically begins to develop in concrete where the compressive stress exceeds approximately 70% of the compressive strength. Given the limits above to control longitudinal cracking, no further criteria are given to control micro-cracking.

CHAPTER 7. SERVICEABILITY LIMIT STATES

b

εc dc

d d – dc

εs

As

Stresses

Strains

Fig. 7.1. Notation for a rectangular beam

2-1-1/clause 7.2(3) addresses non-linear creep as covered by 2-2/clause 3.1.4(4). It requires non-linear creep to be considered where the stress under the quasi-permanent combination of actions exceeds k2 fck , where k2 is a nationally determined parameter with recommended value of 0.45. 2-2/clause 3.1.4(4) gives the same limiting stress, but it is not subject to national variation in that clause so must be deemed to take precedence. 2-1-1/clause 7.2(4)P requires stresses in reinforcement and prestressing steel to be limited to ensure inelastic deformations of the steel are avoided under serviceability loads, which could result in excessive concrete crack widths and invalidate the assumptions on which the calculations within EC2 for cracking and deflections are based. 2-1-1/clause 7.2(5) requires that the tensile stress in reinforcement under the characteristic combination of actions does not exceed k3 fyk . Where the stress is caused by imposed deformations, the tensile stress should not exceed k4 fyk , although it will be rare for tensile stress to exist solely from imposed deformations. The mean value of stress in prestressing tendons should not exceed k5 fpk . The values of k3 , k4 and k5 are nationally determined parameters and are recommended to be taken as 0.8, 1.0 and 0.75 respectively. The higher stress limit for reinforcement tension under indirect actions reflects the ability for stresses to be shed upon concrete cracking. The following method can be used to determine stresses in cracked reinforced concrete beams and slabs. The concrete modulus to use for section analysis depends on the ratio of permanent (long-term) actions to variable (short-term) actions. The short-term modulus is Ecm and the long-term modulus is Ecm =ð1 þ Þ. The effective concrete modulus for a combination of long-term and short-term actions can be taken as: Ec;eff ¼

ðMqp þ Mst ÞEcm Mst þ ð1 þ ÞMqp

2-1-1/clause 7.2(3)

2-1-1/clause 7.2(4)P 2-1-1/clause 7.2(5)

(D7-1)

where Mst is the moment due to short-term actions and Mqp is the moment from quasipermanent actions. The neutral axis depth and steel strain can be derived from a cracked elastic analysis, assuming plane sections remain plane. For a rectangular beam, from Fig. 7.1: Strains "s ¼

d dc "c dc

(D7-2)

Forces Fs ¼ Fc so As Es "s ¼ 0:5bdc "c Ec;eff Putting equation (D7-2) into equation (D7-3) gives: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi As Es þ ðAs Es Þ2 þ 2bAs Es Ec;eff d dc ¼ bEc;eff

(D7-3)

(D7-4)

227

Index Page numbers in italics denote figures accelerated relaxation, 290 actions, 9ÿ10, 288, 308ÿ309 actions combinations, 209, 226 additional longitudinal reinforcement, 143ÿ145 aggregate concrete structures, 295ÿ302 allowable maximum strain, 107 alternative methods, 349ÿ350 amplification factors, 357 analysis, 95ÿ97, 326, 363 see also structural analysis anchorage, 29, 245, 247ÿ251, 247ÿ248, 251, 272 bottom reinforcement, 277ÿ8 force calculation model, 286 prestressing, 87, 89ÿ90, 271 pretensioned tendons, 259ÿ262, 262 support reinforcement, 292 tension, 261ÿ262, 286 ultimate limit state, 201, 261ÿ262 zones, 262ÿ270, 262, 265ÿ267, 270, 330 angular imperfections, 43 application rules, 5 arches, 43 assumptions, 4ÿ5, 25, 28ÿ29 autogenous shrinkage, 18 axial forces, 78, 304 axial loads, 62ÿ80, 123, 125ÿ126 ‘balanced’ sections, 78 bars, 246ÿ247, 250ÿ258, 254 bases, 179ÿ185, 184ÿ185 beams, 275ÿ281 bearings, 47 composite, 337ÿ339, 338ÿ339 creep, 337ÿ339, 338ÿ339 doubly reinforced rectangular, 112ÿ114, 113 effective spans, 47, 48 end stress transfer, 99ÿ100 idealization for space frame analysis, 172 K values, 251 lateral instability, 80ÿ81 midspan stress transfer, 100

monolithic with supports, 47 plastic analysis, 52, 53 precast pretensioned, 160ÿ164 rectangular, notation, 227 shear, 136, 137, 278ÿ279 slender, 80ÿ81 time-dependent losses, 91 with uniform loads, 51 see also slabs bearings, 47, 204, 205ÿ207, 292ÿ293, 329ÿ330 bending, 105ÿ131, 299, 304, 350 biaxial bending, 80, 125ÿ126 biaxial compression, 218 biaxial shear, 218 bilinear diagrams, 21 bonds, 96, 99ÿ103, 100, 102, 245, 250 bottom reinforcement, 277ÿ278 box girders, 280ÿ281, 280, 362 discontinuity regions, 362 effective flange widths, 46ÿ47, 47 losses, 87, 94ÿ95 post-tensioned, 129ÿ131, 139ÿ140, 148ÿ151 shear cracks, 365ÿ367, 366 tendon profiles, 151 three span, losses, 87 time-dependent losses, 94ÿ95 torsion, 172ÿ173 without tendon drape, 149ÿ151 worked examples, 87ÿ90, 94ÿ95, 172ÿ173, 280ÿ281, 280 braced columns, 74 braced members, 63 bridge piers see piers brittle failure, 126ÿ131 buckling, 43, 63, 65, 305 bundled bars, 258 bursting, 203ÿ205, 204, 262ÿ263 cables, 126ÿ127 cantilevering piers, 67, 70, 75ÿ76, 79ÿ80 carbonation corrosion, 34


CCC nodes, 197 CCT nodes, 198 centreline distances, 260 chemical attack corrosion, 35 chloride corrosion, 34 CIRIA Guide 1 method, 264, 265 classes, 24, 34ÿ35, 37, 322ÿ323 classification, 37 climatic actions, 309 close to slab edges/corners, 176 close to supports, 135ÿ136, 145 coefficients of friction, 86 columns, 185, 282ÿ284 see also piers combinations, 157ÿ158, 169ÿ170, 350 composite beams, 161, 337ÿ341, 338ÿ39, 341 composite construction, 93, 160ÿ165 composite deck slabs, 162ÿ164 composite members, 342ÿ343 compressive strengths, 11ÿ12, 20ÿ21, 298, 360ÿ361 compressive stresses, 115 concrete, 11ÿ23 cast at different times, 158ÿ160 confined, 298 cover verification methods, 36ÿ38 creep, 14ÿ17, 333 damage, 322 deformation characteristics, 13 elastic deformation, 14 fatigue verification, 213ÿ215 flexural tensile strengths, 22ÿ23 heat curing effects, 290 indicative strength classes, 322 lightweight aggregate structures, 295ÿ302 shear, 213ÿ215 shell elements, 344ÿ345 shrinkage, 14, 18ÿ19 strengths, 11ÿ14, 20ÿ21, 296, 360ÿ361 stress characteristics, 13 stress limitation, 82ÿ83 stress/strain relationships, 19, 19ÿ20, 21ÿ22 worked examples, 16ÿ17 contraflexure points, 147 control, cracks, 230ÿ242, 237, 240, 364ÿ367 control perimeters, 176ÿ177, 176, 184 corbels, 328ÿ329, 329 corner piles, 185, 189, 191 corrosion, 32ÿ35, 126ÿ127, 322 couplers, 29, 252ÿ257, 271 cover, 31ÿ38, 261, 298 cracks cables, 126ÿ127 checks, 236ÿ237, 241ÿ242 control, 230ÿ242, 364ÿ367 early thermal, 243 flexure, 140, 148ÿ149 prediction, 240ÿ241 sections, 228 strains, 237, 240

372

widths, 237ÿ242, 237, 240 worked examples, 236ÿ237, 241ÿ242 creep, 69ÿ70, 313ÿ315, 332, 333ÿ341, 333, 338ÿ339, 341 concrete, 14ÿ17, 297, 313ÿ314, 333 time-dependent losses, 92ÿ94 worked examples, 336ÿ337, 339ÿ341, 341 cross sections see sections crushing, 169ÿ170, 201ÿ203, 202 CTT nodes, 198ÿ199 curvature forces, 259 curvature methods, 76ÿ80 damage, 210ÿ212, 322, 356ÿ361 dead load moments, 332 deck slabs, 228ÿ230, 300ÿ301 cover example, 38 crack control, 236ÿ237, 241ÿ242 shear, 134 ultimate limit states, 110ÿ111 voided reinforced concrete, 146ÿ147 deep beams, 284 definitions, 5, 155, 167, 265 deflection, 70, 243 deformation, 13, 297 design, 20ÿ21, 298 anchorage lengths, 249ÿ251 assumptions, 25, 28ÿ29 basis, 7ÿ10, 289ÿ290, 296 execution stages, 307ÿ309 particular rules, 291 shear reinforcement, 133ÿ153, 140ÿ141, 145, 153 torsion, 167ÿ171 detailed calculation method, 317ÿ321 detailing rules, 245ÿ273, 275ÿ288, 291ÿ293, 302, 327ÿ330 deviation allowances, 38 deviators, 272ÿ273 diaphragms, 199ÿ201, 200ÿ201, 280ÿ281, 280, 362 differential creep, 337ÿ339, 338ÿ339 differential shrinkage, 102, 341ÿ342, 342 discontinuity, 195ÿ196, 196, 288, 362 dispersion, 267 doubly reinforced concrete beams/slabs, 112ÿ114, 113 D-regions, 57, 288 see also strut-and-tie model drying shrinkage, 18 ductility, 24, 27ÿ28 ducts, 147ÿ148, 148, 258ÿ261, 259 durability, 31ÿ38, 298, 322ÿ323 early thermal cracking, 243 EC2-2 Annex J method, 263ÿ264 eccentricity, 42, 304, 348 edge sliding reinforcement, 206ÿ207 effective lengths, 65ÿ67, 65 effective thickness, 169

INDEX

effective widths, 44ÿ46, 47, 147ÿ148 elastic deformation, 14, 85ÿ86, 296ÿ297 end moments, 74 end supports, 277ÿ278 environmental conditions, 32ÿ35 equivalent time method, 318ÿ321, 318 examples see worked examples execution stages, 307ÿ309 expansion joints, 357 exposure, 32ÿ35, 37 external non-bonded tendons, 29 external tendons, 29, 98 external vs. internal post-tensioning, 95ÿ96 failure, 54, 123, 126ÿ127, 248, 304 fatigue, 25, 28, 98ÿ99, 208ÿ215, 209, 356ÿ361 worked examples, 211ÿ212, 214ÿ215 finite element models, 344 flanged reinforced beams, 115ÿ117, 115, 117 flanges, 44ÿ47, 47, 107, 156, 157ÿ158 flat slabs, 104, 282, 326 flexural reinforcement, 281 flexural shear, 187, 189 flexural tensile strengths, 22ÿ23 floor systems, 291 footings, 285ÿ286 see also pad footings force calculation model, 286 foundations, 10, 285ÿ288, 285, 287, 293 frames, 52, 328 free body diagrams, 144 freeze/thaw corrosion, 34ÿ35 friction, 86ÿ87, 89 general method, 332ÿ333 geometric data, 8, 44ÿ48 geometric imperfections, 40ÿ44, 41ÿ44, 74 geometry, discontinuity regions, 288 half joints, 292 heat curing, 290 heavy vehicles, 358 high-strength concrete, 313ÿ314 hollow piers, 204 ‘I’ beams, 154 idealization, 170, 172 imperfections see geometric imperfections imposed curvature, 55 indicative strength classes, 322ÿ323 indirect supports, 279ÿ280 in-plane buckling, 43, 43 in-plane stress conditions, 324 in situ concrete structures, 311 in situ deck slabs, 160ÿ164 interaction diagrams, 122 intermediate supports, 278 internal vs. external post-tensioning, 95ÿ96 isolated members, 41, 64ÿ67, 65 ISO standard 3898: 1997 symbols, 5

joints, 153, 357 K values, anchorage, 249, 251 laps, 201, 252ÿ257, 252, 254, 256 large diameter bars, 257ÿ258 lateral instability, 80ÿ81 layer centres, 348 layer stresses, 347 layouts, tendons, 258ÿ259 leaf piers, 205ÿ207, 207 lightly reinforced structures, 303ÿ305 lightweight aggregate concrete structures, 295ÿ302 analysis, 298 cover, 298 deformation, 297 design, 296, 298 elastic deformation, 296ÿ297 materials, 296ÿ298 non-linear analysis, 298 shear/torsion, 299 shrinkage, 297 stress, 297ÿ298 strut-and-tie models, 300 ultimate limit states, 298ÿ300 worked examples, 300ÿ301 limited redistribution, 49ÿ51 limiting slenderness checks, 68ÿ69 limit states, 7, 58ÿ60, 62, 82ÿ83, 98ÿ103 see also ultimate limit states linear elastic analysis, 48ÿ51, 51 links, 251 loads close to slab edges/corners, 176 close to supports, 145 dispersal, 207 distribution, 176ÿ177, 176, 202 factors, 125 models, 359 spread, 285 longitudinal reinforcement, 143ÿ145, 275ÿ278, 283 anchorage, 247ÿ251, 247ÿ248, 251, 272 longitudinal shear, 154ÿ158, 349ÿ350, 355 longitudinal tension reinforcement, 277 losses anchorage, 87 elastic deformation, 85ÿ86 post-tensioning, 85ÿ90 prestress, 84ÿ90, 89, 291, 317ÿ321, 318 pretensioning, 84ÿ85 steel relaxation, 317ÿ321, 318 worked examples, 87ÿ90 low-relaxation prestressing tendons, 318ÿ321 mandrels, 246ÿ247 materials, 8, 10ÿ29, 290, 296ÿ298, 303ÿ304 partial factor modification, 311ÿ312 maximum moments, 42ÿ43

373


maximum punching shear stress, 185 M beams, 162ÿ165, 173ÿ175, 175 members, 140ÿ153, 140ÿ141, 145 detailing, 275ÿ288, 302 not requiring design shear reinforcement, 133ÿ140 in tension, 198ÿ199 membrane elements, 215ÿ223, 216, 222 membrane rules, 216, 218ÿ222, 222 minimum cover, 36ÿ38 minimum reinforcement, 127ÿ128 minimum shear reinforcement, 137 modification, partial factors, 311ÿ312 Mohr’s circles, 222, 352ÿ353 moments, 42ÿ43, 70, 74, 332, 342 monitoring facility provision, 128 nodes, 196ÿ201, 197ÿ198 nominal curvature methods, 76ÿ80 nominal stiffness, 71ÿ76 non-linear analysis, 19, 19, 58ÿ62, 61, 70ÿ71, 298, 363 normative references, 4 notation, rectangular beams, 227 open sections, 170ÿ171 out-of-plane buckling, 43 overlapping, 292 pad footings, 180ÿ182, 180, 287ÿ288, 287 parabolic-rectangular diagrams, 21 parameters, creep/shrinkage, 314 Part 2 scope, 4 partial factor method, 9ÿ10 partial factor modification, 311ÿ312 partially loaded areas, 201ÿ207, 202, 204, 207, 300, 329ÿ330 partial sheared truss models, 141 particular rules, 275ÿ288, 291, 302 particular situations, 327ÿ330 piers, 43, 63, 68ÿ69, 68, 207 worked examples, 16ÿ17, 17 see also cantilevering piers; columns pile caps, 185ÿ193, 186ÿ193, 285 piles see corner piles pin-ended struts, 42 placement of reinforcement, 158 plain reinforced concrete structures, 303ÿ305 plastic analysis, 52ÿ58, 53, 55 plastic hinges, 55 pocket foundations, 293 post-tensioning, 85ÿ90 box girders, 129ÿ131, 139ÿ140, 148ÿ153 cables, 272ÿ273 construction, 336 ducts, 258ÿ259 members, 262ÿ270, 265, 270, 330 precast concrete, 153, 160ÿ165, 289ÿ293, 312 prestress dispersion, 267 prestressed beams, 118ÿ131, 119

374

prestressed members, 137, 140, 147ÿ153, 148 prestressed steel, 25ÿ29, 28 prestressed structures, 81ÿ103 bonded tendons, 99ÿ103, 100, 102ÿ103 creep, 92ÿ94 fatigue limit states, 98ÿ99 limit states, 97ÿ99 losses, 84ÿ95, 89, 291 post-tensioning, 95ÿ96 prestress considerations, 95ÿ97 prestressing forces, 82ÿ84 primary/secondary effects, 96ÿ97 serviceability limit states, 98ÿ103 unbonded prestress, 96 worked examples, 99ÿ103, 100, 102ÿ103 prestressing devices, 29 prestressing forces, 82ÿ90 prestressing steel, 209ÿ210, 245ÿ273, 290, 317ÿ321, 318, 359ÿ360 prestressing systems, 32ÿ35, 38 prestress transfer, 260ÿ261 pretensioning, 84ÿ85 beams, 131, 160ÿ164 composite members, 337ÿ339, 342ÿ343 concrete M beams, 162ÿ165, 173ÿ175, 175 strands, 259 tendons, 258ÿ262, 262 primary prestress effects, 96ÿ97, 96 primary regularization prism, 263ÿ264, 264ÿ265 principles, 5, 7, 40 properties of materials, 8, 23, 26ÿ27, 316 provision of monitoring facilities, 128 proximity to expansion joints, 357 punching, 175ÿ193 control perimeters, 176ÿ177, 176, 184 corner piles, 189, 191 load distribution, 176ÿ177, 176 pile caps, 185ÿ193 shear, 177ÿ179, 185, 192ÿ193, 300 shear resistance, 179ÿ185, 184 worked examples, 180ÿ182, 180, 186ÿ193, 186ÿ193 railway bridges, 359ÿ361 rectangular beam notation, 227 rectangular stress blocks, 117 redistribution, 332, 338ÿ339, 339ÿ341 regions with discontinuity, 288, 362 reinforced concrete, 105ÿ117, 108, 121ÿ126, 122ÿ123, 133ÿ147, 303ÿ305 see also lightly reinforced . . . reinforcement, 222ÿ223, 222, 245ÿ273, 247ÿ248, 251, 270, 272, 277, 300ÿ2 corrosion potentials, 32ÿ35 cover, 31ÿ38, 298 crack control, 232ÿ234 durability, 31ÿ38 eccentric, 348 edge sliding, 206ÿ207

INDEX

fatigue verification, 209 foundations, 287ÿ288, 287 in-plane stress conditions, 324 loop overlapping, 292 minimum, 37, 127ÿ128, 137 normative properties, 316 pad footings, 287ÿ288, 287 piers, 68, 207 properties, 316 punching, 184 shear, 133ÿ153, 251 surfaces, 327ÿ328 suspension, 280 tension, 277, 324 torsion, 167ÿ169, 167 transverse, 254ÿ255, 254 webs, 366 worked examples, 255ÿ257, 256, 300ÿ301 reinforcing steel, 23ÿ25, 24ÿ25, 209ÿ210, 209, 359ÿ360 relaxation, 27, 92, 94, 317ÿ321, 318 resultant conversions, stress, 347 return periods, 309 ribbed wires, 257 road bridges, 211ÿ212, 214ÿ215, 356ÿ358 rotation capacity, 53ÿ56, 54ÿ55 rules see detailing rules safety format, 61ÿ62, 363 St Venant torsion, 166 sandwich model, 346ÿ349, 347ÿ348, 353ÿ355 scalar combinations, 60ÿ61, 61 scope, Eurocode 2, 3ÿ4 sea water corrosion, 34 secondary moments, 342 secondary prestress effects, 96ÿ97, 96 second-order analysis, 62ÿ80 biaxial bending, 80 braced columns, 74 creep, 69ÿ70 definitions, 62ÿ63 methods, 70ÿ71, 76ÿ80 nominal stiffness, 71ÿ76 simplified criteria, 64ÿ69 soil-structure interaction, 64 stiffness, 63ÿ64, 71ÿ76 unbraced columns, 73 sections, 21ÿ22, 78, 154ÿ158, 169, 170, 170, 298 see also prestressed sections segmental construction, 153, 153, 171 serviceability limit states, 225ÿ243 actions combinations, 226 crack control, 230ÿ243 deck slabs, 228ÿ230, 302 execution stages, 309 prestressed structures, 98ÿ103 reinforced concrete deck slabs, 228ÿ230 stress limitation, 226ÿ230 worked examples, 228ÿ230 shear, 131ÿ165

alternative methods, 349ÿ350 beam enhancement comparisons, 136 bending, 160, 346ÿ355, 347ÿ348, 350 between web/flanges, 154ÿ158 composite construction, 160ÿ165 concrete cast at different times, 158ÿ160 contraflexure points, 147 cracking, 364ÿ367, 366 deck slabs, 134 flexure failures, 137 flow calculation definitions, 155 lightweight structures, 299 members not requiring reinforcement, 133ÿ140 members requiring reinforcement, 140ÿ153 post-tensioned box girders with tendon drape, 151ÿ153 precast concrete, 160ÿ165 reinforced concrete, 133ÿ140, 304ÿ305 reinforcement, 145, 190, 192ÿ193, 251, 278ÿ279, 282 tension, 137ÿ139 torsion, 169ÿ170 total transverse reinforcement, 157 T-sections, 154ÿ158 verification procedure rules, 132ÿ133 wall analysis, 104 worked examples bending, 351ÿ355 box girders, 139ÿ140, 148ÿ153 deck slabs, 134, 146ÿ147 flexure, 139ÿ140 tendon drape, 151ÿ153 without tendon drape, 149ÿ151 see also longitudinal shear; punching shear resistance, 179ÿ185, 184 shear stress, 161, 185 shell elements, 344ÿ345 shift method, 143ÿ145 short spans, 145 shrinkage, 313ÿ315 concrete, 14, 18ÿ19 differential, 102, 341ÿ342, 342 lightweight structures, 297 time-dependent losses, 92, 94 sign convention, 216 simplified criteria, 64ÿ69 simplified methods, 333ÿ336 simplified rectangular diagrams, 21 singly reinforced concrete, 107ÿ112, 108 sinusoidal imperfections, 43 skew reinforcement, 222ÿ223, 222 slabs edges/corners, 176 K values, 251 plastic analysis, 52 punching, 176, 179ÿ185, 184 shear resistance, 179ÿ185, 184 shear stress distribution, 161 torsion, 172ÿ175, 175

375


slabs (continued ) see also beams; flat slabs; solid slabs slenderness, 64ÿ67, 65, 68ÿ69, 80ÿ81, 305 see also lightweight . . . sliding wedge mechanism, 329 SLS see serviceability limit states soil structure interactions, 64, 325 solid sections, 169ÿ170, 281ÿ282 space frame analysis, 172 spacing, 246, 259 spalling, 201ÿ203, 202, 262, 265 spans, beams, 47, 48 square sections, 170, 170 staged construction, 93 static values see characteristic values steel, 25ÿ9, 28, 228, 359ÿ360 see also prestressing steel stiffness, 63ÿ64, 71ÿ76 strain compatibility, 106ÿ107, 118, 122ÿ123 cracks, 237, 240ÿ241, 240 creep, 313ÿ315, 333 distributions, 117 external tendons, 98 strength classes, 296, 322ÿ323 concrete, 11ÿ14 curve, 209 prestressed steel, 27 reinforcing steel, 23ÿ24 see also individual strengths stress anchorage zones, 262 block idealization comparisons, 22 concentrated loads, 202 concrete characteristics, 13 damage equivalent, 356ÿ361 differential shrinkage, 102 distances, uniformity, 261 fatigue verification, 356ÿ361 flanged beams, 117 lightweight structures, 297ÿ298 limitation, 226ÿ230 Mohr’s circles, 222 non-linear structural analysis, 298 prestressed steel, 28 redistribution, creep, 338ÿ339 reinforced concrete deck slabs, 228ÿ230 reinforcement fatigue verification, 209 resultant conversions, 347 section design, 298 serviceability limit states, 226ÿ230 strain relations, 298 temperature differentials, 230 transfer in beams, 99ÿ100, 100 transverse, 262 stress–strain diagrams, 24ÿ25, 28 stress–strain profiles, 119 stress–strain relationships, 19, 19ÿ20, 21ÿ22 structural analysis, 39ÿ104

376

geometric imperfections, 40ÿ44, 41ÿ44 global, 39, 45ÿ46 lateral instability, 80ÿ81 lightweight structures, 298 linear elastic analysis, 48ÿ51 local, 39ÿ40 non-linear analysis, 58ÿ62, 298 plastic analysis, 52ÿ58 post-tensioning, 95ÿ96 precast structures, 290ÿ291 prestressed members/structures, 81ÿ103 principles, 40 reinforced structures, 303ÿ304 second-order effects analysis, 62ÿ80 structure idealization, 44ÿ48 time-dependent behaviour, 331ÿ343, 332 worked examples, 67ÿ69, 75ÿ76, 79ÿ80 structural classification, 37 structure idealization, 44ÿ48 strut-and-tie models, 56ÿ58, 57, 195ÿ196, 196 anchorage zones, 266 corbels, 329 ducts, 148 hollow piers, 204 lightweight structures, 300 nodes, 196ÿ201, 197ÿ198 struts, 193ÿ195 ultimate limit states, 193ÿ201 worked examples, 199ÿ201, 200ÿ201 struts tensile stress, 194ÿ195 transverse stress, 193ÿ195 see also pin-ended struts; strut-and-tie models summary procedure, 342ÿ343 supports, 135ÿ136, 145, 187, 291ÿ292 surface reinforcement, 279, 327ÿ328 suspension reinforcement, 280 symbols, 5 temperature differentials, 230 tendons, 149ÿ151, 151 external non-bonded, 29 low-relaxation prestressing, 318ÿ321 post-tensioned, 86 prestressing, 258ÿ273, 291 strain, 98 vertical profiles, 87 tensile forces, 82ÿ83, 261ÿ262 tensile reinforcement, 286, 324 tensile strength, 12ÿ14, 20ÿ21 testing, 10 thin walled sections, 170 three-span bridges, 87, 336ÿ337 ties, 195ÿ196, 196 time-dependent behaviour, 90ÿ95, 91, 331ÿ343, 332 time-dependent losses, 91, 92ÿ95 torsion combined shear/torsion, 169ÿ170 definitions, 167

INDEX

design procedure, 167ÿ171, 170 lightweight structures, 299 M beam bridges, 173ÿ175, 175 open sections, 170 punching, 175ÿ193 reinforced structures, 167ÿ169, 167, 279, 304ÿ305 St Venant torsion, 166 segmental construction, 171 slabs, 172ÿ175, 175 ultimate limit state, 166ÿ175 warping torsion, 171ÿ172 worked examples, 172ÿ175 pad footings, 180ÿ182, 180 pile caps, 186ÿ193, 186ÿ193 slabs, 173ÿ175, 175 total transverse reinforcement, 157 transfer, prestress, 260ÿ261 transverse bending, 157ÿ158, 157, 160, 346ÿ355, 347ÿ348 transverse forces, 42ÿ43 transverse reinforcement, 158, 254ÿ255, 254, 283ÿ284 transverse stress, 193ÿ195, 262 truss models, 140ÿ141, 144, 154 T-sections, 154ÿ158 two-span beams, 51 ULS see ultimate limit states ultimate bond strength, 248 ultimate limit states (ULS), 105ÿ223 anchorage, 201, 261ÿ262 beams, 105ÿ117, 108, 112ÿ117, 113 bending, 105ÿ131, 122ÿ123 brittle failure, 126ÿ131 columns, 121ÿ126, 122ÿ123 deck slabs, 110ÿ111 doubly reinforced rectangular beams, 112ÿ114, 113 execution stages, 309 fatigue, 208ÿ215 flanged beams, 115ÿ117 laps, 201 lightweight structures, 298ÿ300 membrane elements, 215ÿ223 methods, 58ÿ60 partially loaded areas, 201ÿ207, 202, 204 prestressed concrete beams, 118ÿ121 prestressed structures, 97ÿ98 shear, 131ÿ165 singly reinforced concrete beams/slabs, 107ÿ112, 108 strain compatibility, 106ÿ107 strut-and-tie models, 193ÿ201 torsion, 166ÿ175 voided reinforced concrete slabs, 111ÿ112 worked examples, 110ÿ111, 113ÿ117, 129ÿ131 unbonded prestress, 96 unbonded tendons, 121

unbraced columns, 73 unbraced members, 63 un-cracked in flexure, 137ÿ140 uniaxial bending, 123 uniaxial compression, 218ÿ220 uniaxial shear, 218ÿ220 uniaxial tension, 220ÿ222 uniformity, stresses, 261 uniform loads, 51 unreinforced concrete, 204 unrestrained concrete, 333 values drying shrinkage, 18 material factors, 10 vertical loads, 359 variables, basic, 7ÿ8 vector combinations, 61ÿ62 vehicle expected numbers, 358 verification execution stage criteria, 309 fatigue, 208ÿ215, 356ÿ361 methods, 36ÿ38 partial factor method, 9ÿ10 procedure rules, 132ÿ133 vertical profiles, 87 voided reinforced concrete slabs, 111ÿ112, 146ÿ147 walls, 284, 291, 304 warping torsion, 166, 171ÿ172 webs bending/shear combinations, 350 box girders, 280ÿ281, 280 control of shear cracks, 364ÿ367, 366 diaphragm region, 280ÿ281, 280 longitudinal shear rules, 355 reinforcement, 365ÿ367, 366 widths, 147ÿ148 welding, 25, 251, 257 worked examples anchorage zones, 267ÿ270, 270 bonded tendons, 99ÿ103, 100, 102ÿ103 box girders, 87ÿ90, 94ÿ95, 139ÿ140, 280ÿ281, 280, 365ÿ366 bridge piers, 16ÿ17, 17 brittle failure, 129ÿ131 cantilevered piers, 67, 75ÿ76, 79ÿ80 composite beams/slabs, 162ÿ164, 339ÿ341, 341 crack control, 236ÿ237, 241ÿ242 creep, 339ÿ341, 341 damage equivalent stress range, 211ÿ212 deck slabs, 38, 255ÿ257, 256 diaphragm design, 199ÿ201, 200ÿ201 doubly reinforced concrete slabs, 113ÿ114 effective flange widths, 46ÿ47, 47 equivalent time method, 318ÿ321 fatigue, 211ÿ212, 214ÿ215 flanged beams, 115ÿ117

377


worked examples (continued ) foundations, 287ÿ288, 287 lap lengths, 255ÿ257, 256 leaf piers, 205ÿ207, 207 lightweight structures, 300ÿ301 linear elastic analysis, 51, 51 losses, 87ÿ90 low-relaxation prestressing tendons, 318ÿ321 membrane elements, 218ÿ222, 222 partially loaded areas, 205ÿ207, 207 post-tensioned members, 139ÿ140, 267ÿ270, 270 prestressed beams, 118ÿ121, 129ÿ131 prestressed steel, 27 prestressed structures, 99ÿ103, 100, 102ÿ103 pretensioned concrete M beams, 162ÿ165 punching, 180ÿ182, 180, 186ÿ193, 186ÿ193 reinforced concrete columns, 124ÿ125 slabs, 110ÿ112, 228ÿ230, 241ÿ242 relaxation losses, 27, 318ÿ321

378

sandwich model, 353ÿ355 serviceability limit states, 228ÿ230 shear cracking, 365ÿ366 post-tensioned box girders, 149ÿ153 reinforced concrete, 134ÿ135 transverse bending combinations, 351ÿ355 structural analysis, 67ÿ69, 75ÿ76, 79ÿ80 structure idealization, 46ÿ47 strut-and-tie models, 199ÿ201, 200ÿ201 three-span bridges, 336ÿ337 time-dependent losses, 94ÿ95 torsion, 172ÿ175 transverse bending, 351ÿ355 ultimate limit states, 110ÿ112, 115ÿ117 voided reinforced concrete slabs, 111ÿ112, 146ÿ147 web reinforcement, 366 zones for shear reinforcement, 190

Designers Guide to EN1992-2 Eurocode 2

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