Designers Guide to EN1992-2 Eurocode 2

DESIGNERS’ GUIDES TO THE EUROCODES

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

PART 2: CONCRETE BRIDGES

Eurocode Designers’ Guide SeriesDesigners’ Guide to EN 1990. Eurocode: Basis of Structural Design. H. Gulvanessian, J.-A. Calgaro andM. 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 Rulesand Rules for Buildings and Structural Fire Design. A.W. Beeby and R. S. Narayanan. 0 7277 3105 X. Published2005.

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 andBuildings. 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. FireEngineering (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. GeneralRules 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 1992-2EUROCODE 2: DESIGN OFCONCRETE STRUCTURES

PART 2: CONCRETE BRIDGES

C. R. HENDY and D. A. SMITH

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Preface

Aims and objectives of this guideThe principal aim of this book is to provide the user with guidance on the interpretation anduse of EN 1992-2 and to present worked examples. It covers topics that will be encounteredin typical concrete bridge designs and explains the relationship between EN 1992-2 and theother Eurocodes.

EN 1992-2 is not a ‘stand alone’ document and refers extensively to other Eurocodes. Itsformat is based on EN 1992-1-1 and generally follows the same clause numbering. Itidentifies which parts of EN 1992-1-1 are relevant for bridge design and adds furtherclauses that are specific to bridges. It is therefore not useful to produce guidance onEN 1992-2 in isolation and so this guide covers material in EN 1992-1-1 which will needto be used in bridge design.

This book also provides background information and references to enable users ofEurocode 2 to understand the origin and objectives of its provisions.

Layout of this guideEN 1992-2 has a foreword, 13 sections and 17 annexes. This guide has an introduction whichcorresponds to the foreword of EN 1992-2, Chapters 1 to 10, which correspond to Sections 1to 10 of the Eurocode and Annexes A to Q which again correspond to Annexes A to Q of theEurocode.

The guide generally follows the section numbers and first sub-headings in EN 1992-2 so thatguidance can be sought on the code on a section by section basis. The guide also follows theformat of EN 1992-2 to lower levels of sub-heading in cases where this can conveniently bedone and where there is sufficient material to merit this. The need to use several Eurocodeparts can initially make it a daunting task to locate information in the order required for areal design. In some places, therefore, additional sub-sections are included in this guide topull together relevant design rules for individual elements, such as pile caps. Additionalsub-sections are identified as such in the sub-section heading.

The following parts of the Eurocode are intended to be used in conjunction withEurocode 2:

EN 1990: Basis of structural designEN 1991: Actions on structuresEN 1997: Geotechnical designEN 1998: Design of structures for earthquake resistancehENs: Construction products relevant for concrete structuresEN 13670: Execution (construction) of concrete structures

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

In this guide, references to Eurocode 2 are made by using the abbreviation ‘EC2’ forEN 1992, so EN 1992-1-1 is referred to as EC2-1-1. Where clause numbers are referred toin 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 prefixedwith ‘2-2’. There are so many references to other parts of Eurocode 2 required that to dootherwise would be confusing.

Where additional equations are provided in the guide, they are numbered sequentiallywithin each sub-section of a main section so that, for example, the third additional expres-sion within sub-section 6.1 would be referenced equation (D6.1-3). Additional figures andtables follow the same system. For example, the second additional figure in section 6.4would be referenced Figure 6.4-2.

AcknowledgementsChris Hendy would like to thank his wife, Wendy, and two boys, Peter Edwin Hendy andMatthew Philip Hendy, for their patience and tolerance of his pleas to finish ‘just onemore section’.

David Smith would like to thank his wife, Emma, for her limitless patience during prepara-tion of this guide. He also acknowledges the continued support of Brian and RosalindRuffell-Ward from the very beginning.

Both authors would also like to thank their employer, Atkins, for providing both facilitiesand time for the production of this guide. They also wish to thank Dr Paul Jackson and DrSteve Denton for their helpful comments on the guide.

Chris HendyDavid A. Smith

DESIGNERS’ GUIDE TO EN 1992-2

vi

Contents

Preface v

Aims and objectives of this guide vLayout of this guide vAcknowledgements vi

Introduction 1

Additional information specific to EN 1992-2 2

Chapter 1. General 3

1.1. Scope 31.1.1. Scope of Eurocode 2 31.1.2. Scope of Part 2 of Eurocode 2 4

1.2. Normative references 41.3. Assumptions 41.4. Distinction between principles and application rules 51.5. Definitions 51.6. Symbols 5

Chapter 2. Basis of design 7

2.1. Requirements 72.2. Principles of limit state design 72.3. Basic variables 72.4. Verification by the partial factor method 9

2.4.1. General 92.4.2. Design values 92.4.3. Combinations of actions 9

2.5. Design assisted by testing 102.6. Supplementary requirements for foundations 10

Chapter 3. Materials 11

3.1. Concrete 113.1.1. General 113.1.2. Strength 113.1.3. Elastic deformation 143.1.4. Creep and shrinkage 143.1.5. Concrete stress–strain relation for non-linear structural

analysis 19

3.1.6. Design compressive and tensile strengths 203.1.7. Stress–strain relations for the design of sections 213.1.8. Flexural tensile strength 223.1.9. Confined concrete 23

3.2. Reinforcing steel 233.2.1. General 233.2.2. Properties 233.2.3. Strength 233.2.4. Ductility 243.2.5. Welding 253.2.6. Fatigue 253.2.7. Design assumptions 25

3.3. Prestressing steel 253.3.1. General 253.3.2. Properties 263.3.3. Strength 273.3.4. Ductility characteristics 273.3.5. Fatigue 283.3.6. Design assumptions 28

3.4. Prestressing devices 293.4.1. Anchorages and couplers 293.4.2. External non-bonded tendons 29

Chapter 4. Durability and cover to reinforcement 31

4.1. General 314.2. Environmental conditions 324.3. Requirements for durability 354.4. Methods of verification 36

4.4.1. Concrete cover 36

Chapter 5. Structural analysis 39

5.1. General 395.2. Geometric imperfections 40

5.2.1. General (additional sub-section) 405.2.2. Arches (additional sub-section) 43

5.3 Idealization of the structure 445.3.1 Structural models for overall analysis 445.3.2. Geometric data 44

5.4. Linear elastic analysis 485.5. Linear elastic analysis with limited redistribution 495.6. Plastic analysis 52

5.6.1. General 525.6.2. Plastic analysis for beams, frames and slabs 525.6.3. Rotation capacity 535.6.4. Strut-and-tie models 56

5.7. Non-linear analysis 585.7.1. Method for ultimate limit states 585.7.2. Scalar combinations 605.7.3. Vector combinations 615.7.4. Method for serviceability limit states 62

5.8. Analysis of second-order effects with axial load 625.8.1. Definitions and introduction to second-order effects 625.8.2. General 635.8.3. Simplified criteria for second-order effects 64


viii

5.8.4. Creep 695.8.5. Methods of analysis 705.8.6. General method – second-order non-linear analysis 705.8.7. Second-order analysis based on nominal stiffness 715.8.8. Method based on nominal curvature 765.8.9. Biaxial bending 80

5.9. Lateral instability of slender beams 805.10. Prestressed members and structures 81

5.10.1. General 815.10.2. Prestressing force during tensioning 825.10.3. Prestress force 835.10.4. Immediate losses of prestress for pre-tensioning 845.10.5. Immediate losses of prestress for post-tensioning 855.10.6. Time-dependent losses 905.10.7. Consideration of prestress in the analysis 955.10.8. Effects of prestressing at the ultimate limit state 965.10.9. Effects of prestressing at the serviceability and fatigue

limit states 985.11. Analysis for some particular structural members 104

Chapter 6. Ultimate limit states 105

6.1. ULS bending with or without axial force 1056.1.1. General (additional sub-section) 1056.1.2. Reinforced concrete beams (additional sub-section) 1056.1.3. Prestressed concrete beams (additional sub-section) 1186.1.4. Reinforced concrete columns (additional sub-section) 1216.1.5. Brittle failure of members with prestress (additional

sub-section) 1266.2. Shear 131

6.2.1. General verification procedure rules 1326.2.2. Members not requiring design shear reinforcement 1336.2.3. Members requiring design shear reinforcement 1406.2.4. Shear between web and flanges of T-sections 1546.2.5. Shear at the interface between concrete cast at different

times 1586.2.6. Shear and transverse bending 1606.2.7. Shear in precast concrete and composite construction

(additional sub-section) 1606.3. Torsion 166

6.3.1. General 1666.3.2. Design procedure 1676.3.3. Warping torsion 1716.3.4. Torsion in slabs (additional sub-section) 172

6.4. Punching 1756.4.1. General 1756.4.2. Load distribution and basic control perimeter 1766.4.3. Punching shear calculation 1776.4.4. Punching shear resistance of slabs and bases without

shear reinforcement 1796.4.5. Punching shear resistance of slabs and bases with shear

reinforcement 1836.4.6. Pile caps (additional sub-section) 185

6.5. Design with strut-and-ties models 1936.5.1. General 193

ix

CONTENTS

6.5.2. Struts 1936.5.3. Ties 1956.5.4. Nodes 196

6.6. Anchorage and laps 2016.7. Partially loaded areas 2016.8. Fatigue 208

6.8.1. Verification conditions 2086.8.2. Internal forces and stresses for fatigue verification 2086.8.3. Combination of actions 2096.8.4. Verification procedure for reinforcing and prestressing

steel 2096.8.5. Verification using damage equivalent stress range 2106.8.6. Other verification methods 2126.8.7. Verification of concrete under compression or shear 213

6.9. Membrane elements 215

Chapter 7. Serviceability limit states 225

7.1. General 2257.2. Stress limitation 2267.3. Crack control 230

7.3.1. General considerations 2307.3.2. Minimum areas of reinforcement 2327.3.3. Control of cracking without direct calculation 2347.3.4. Control of crack widths by direct calculation 237

7.4. Deflection control 2437.5. Early thermal cracking (additional sub-section) 243

Chapter 8. Detailing of reinforcement and prestressing steel 245

8.1. General 2458.2. Spacing of bars 2468.3. Permissible mandrel diameters for bent bars 2468.4. Anchorage of longitudinal reinforcement 247

8.4.1. General 2478.4.2. Ultimate bond stress 2488.4.3. Basic anchorage length 2488.4.4. Design anchorage length 249

8.5. Anchorage of links and shear reinforcement 2518.6. Anchorage by welded bars 2518.7. Laps and mechanical couplers 252

8.7.1. General 2528.7.2. Laps 2528.7.3. Lap length 2538.7.4. Transverse reinforcement in the lap zone 2548.7.5. Laps of welded mesh fabrics made of ribbed wires 2578.7.6. Welding (additional sub-section) 257

8.8. Additional rules for large diameter bars 2578.9. Bundled bars 2588.10. Prestressing tendons 258

8.10.1. Tendon layouts 2588.10.2. Anchorage of pre-tensioned tendons 2598.10.3. Anchorage zones of post-tensioned members 2628.10.4. Anchorages and couplers for prestressing tendons 2718.10.5. Deviators 272


x

Chapter 9. Detailing of members and particular rules 275

9.1. General 2759.2. Beams 275

9.2.1. Longitudinal reinforcement 2759.2.2. Shear reinforcement 2789.2.3. Torsion reinforcement 2799.2.4. Surface reinforcement 2799.2.5. Indirect supports 279

9.3. Solid slabs 2819.3.1. Flexural reinforcement 2819.3.2. Shear reinforcement 282

9.4. Flat slabs 2829.5. Columns 282

9.5.1. General 2829.5.2. Longitudinal reinforcement 2839.5.3. Transverse reinforcement 283

9.6. Walls 2849.7. Deep beams 2849.8. Foundations 2859.9. Regions with discontinuity in geometry or action 288

Chapter 10. Additional rules for precast concrete elements and structures 289

10.1. General 28910.2. Basis of design, fundamental requirements 28910.3. Materials 290

10.3.1. Concrete 29010.3.2. Prestressing steel 290

10.4. Not used in EN 1992-2 29010.5. Structural analysis 290

10.5.1. General 29010.5.2. Losses of prestress 291

10.6. Not used in EN 1992-2 29110.7. Not used in EN 1992-2 29110.8. Not used in EN 1992-2 29110.9. Particular rules for design and detailing 291

10.9.1. Restraining moments in slabs 29110.9.2. Wall to floor connections 29110.9.3. Floor systems 29110.9.4. Connections and supports for precast elements 29110.9.5. Bearings 29210.9.6. Pocket foundations 293

Chapter 11. Lightweight aggregate concrete structures 295

11.1. General 29511.2. Basis of design 29611.3. Materials 296

11.3.1. Concrete 29611.3.2. Elastic deformation 29611.3.3. Creep and shrinkage 29711.3.4. Stress strain relations for non-linear structural analysis 29811.3.5 Design compressive and tensile strengths 29811.3.6. Stress strain relations for the design of sections 29811.3.7. Confined concrete 298

11.4. Durability and cover to reinforcement 298

CONTENTS

xi

11.5. Structural analysis 29811.6. Ultimate limit states 29811.7. Serviceability limit states 30211.8. Detailing of reinforcement – general 30211.9. Detailing of members and particular rules 302

Chapter 12. Plain and lightly reinforced concrete structures 303

Chapter 13. Design for the execution stages 307

13.1. General 30713.2. Actions during execution 30813.3. Verification criteria 309

13.3.1. Ultimate limit state 30913.3.2. Serviceability limit states 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


xii

CHAPTER 7

Serviceability limit states

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

. General Clause 7.1

. Stress limitation Clause 7.2

. Crack control Clause 7.3

. Deflection control Clause 7.4

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

7.1. GeneralEN 1992-2 section 7 covers only the three serviceability limit states relating to clause 7.2 to7.4 above. 2-1-1/clause 7.1(1)P notes that other serviceability limit states ‘may be ofimportance’. EN 1990/A2.4 is relevant in this respect. It covers partial factors, serviceabilitycriteria, design situations, comfort criteria, deformations of railway bridges and criteria forthe safety of rail traffic. Most of its provisions are qualitative but some recommended valuesare 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 combinationof actions for serviceability limit states is ‘normally’ either the characteristic, frequent, orquasi-permanent combination. These are all used in EC2-2 and the general forms of thesecombinations, together with examples of use, are given in Table 7.1, but reference tosection 2 and Annex A2 of EN 1990 is recommended for a detailed explanation of theexpressions and terms. Specific rules for the combinations of actions (e.g. actions thatneed not be considered together), recommended combination factors and partial safetyfactors for bridge design are also specified in Annex A2 of EN 1990. Section 2 of thisguide gives further commentary on the basis of design and the use of partial factors andcombinations of actions.

The general expressions in Table 7.1 have been simplified assuming that partial factors of1.0 are used throughout for all actions at the serviceability limit state, as recommended inAnnex A2 of EN 1990, but they may be varied in the National Annex.

Appropriate methods of global analysis for determining design action effects are discussedin detail in section 5. For serviceability limit state verification, the global analysis may beeither elastic without redistribution (clause 5.4) or non-linear (clause 5.7). Elastic globalanalysis is most commonly used and it is not normally then necessary to consider theeffects 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 forstress and deflection calculation provided that the flexural tensile stress under the relevantcombination of actions considered does not exceed fct;eff. fct;eff may be taken as either fctm

2-1-1/clause7.1(1)P

2-1-1/clause7.1(2)

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

7.2. Stress limitationStresses in bridges are limited to ensure that under normal conditions of use, assumptionsmade in design models (e.g. linear-elastic behaviour) remain valid, and to avoid deteriorationsuch as the spalling of concrete or excessive cracking leading to a reduction of durability. Forpersistent design situations, it is usual to check stresses soon after the opening of the bridge totraffic, when little creep has occurred, and also at a later time when creep and shrinkage aresubstantially complete. This affects the loss of prestress in prestressed structures and themodular ratio for stress and crack width calculation in reinforced concrete structures. Itmay 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 endof curing of the concrete. Calculation of an effective concrete modulus allowing for creepis discussed below.

2-1-1/clause 7.2(1)P requires compressive stresses in the concrete to be limited to avoidlongitudinal cracking, micro-cracking or excessive creep. The first two can lead to a reductionof durability. 2-2/clause 7.2(102) addresses longitudinal cracking by requiring the stress levelunder the characteristic combination of actions to not exceed a limiting value of k1 fck (forareas with exposure classes of XD, XF or XS), where k1 is a nationally determined parameterwith recommended value of 0.6. The clause identifies that the limit can be increased wherespecific measures are taken, such as increasing the cover to reinforcement (from theminimum values discussed in section 4) or by providing confinement by transverse reinforce-ment. The improvement from confining reinforcement is quantified as an increase in allowablestress of 10%, but this may be varied in the National Annex. The design of this reinforcementis not covered by EC2-2, but the strut-and-tie rules in 2-2/clause 6.5 and discussions in section6.5 of this guide are relevant. Such reinforcement would need to operate at low stresses to haveany significant effect in limiting the width of compression-induced cracks.

Micro-cracking typically begins to develop in concrete where the compressive stressexceeds approximately 70% of the compressive strength. Given the limits above to controllongitudinal cracking, no further criteria are given to control micro-cracking.

2-1-1/clause7.2(1)P2-2/clause7.2(102)

Table 7.1. Combinations of actions for serviceability limit states

Combination Notes General expression

Characteristic Combination of actions with a fixed (small) probabilityof being exceeded during normal operation within thestructure’s design life, e.g. combination appropriate tochecks on stress in reinforcement as it is undesirablefor inelastic deformation of reinforcement to occur atany time during the service life.

PGk;j þ P þ Qk;1 þ

P 0;iQk;i

Frequent Combination of actions with a fixed probability of beingexceeded during a reference period of a few weeks,e.g. combination used for checks of cracking anddecompression in prestressed bridges with bondedtendons.

PGk;j þ P þ 1;1Qk;1 þ

P 1;iQk;i

Quasi-permanent

Combination of actions expected to be exceededapproximately 50% of the time, i.e. a time-based mean.For example, combination appropriate to crack widthchecks in reinforced concrete members on the basisthat durability is influenced by average crack widths, notthe worst crack width ever experienced.

PGk;j þ P þ 2;1Qk;1 þ

P 2;iQk;i

226


2-1-1/clause 7.2(3) addresses non-linear creep as covered by 2-2/clause 3.1.4(4). It requiresnon-linear creep to be considered where the stress under the quasi-permanent combination ofactions exceeds k2 fck, where k2 is a nationally determined parameter with recommendedvalue of 0.45. 2-2/clause 3.1.4(4) gives the same limiting stress, but it is not subject to nationalvariation 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 limitedto ensure inelastic deformations of the steel are avoided under serviceability loads, whichcould result in excessive concrete crack widths and invalidate the assumptions on which thecalculations within EC2 for cracking and deflections are based. 2-1-1/clause 7.2(5) requiresthat the tensile stress in reinforcement under the characteristic combination of actions doesnot exceed k3 fyk. Where the stress is caused by imposed deformations, the tensile stressshould not exceed k4 fyk, although it will be rare for tensile stress to exist solely fromimposed deformations. The mean value of stress in prestressing tendons should not exceedk5 fpk. The values of k3, k4 and k5 are nationally determined parameters and are recommendedto be taken as 0.8, 1.0 and 0.75 respectively. The higher stress limit for reinforcement tensionunder 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 concretebeams and slabs. The concrete modulus to use for section analysis depends on the ratio ofpermanent (long-term) actions to variable (short-term) actions. The short-term modulus isEcm and the long-term modulus is Ecm=ð1þ �Þ. The effective concrete modulus for acombination of long-term and short-term actions can be taken as:

Ec;eff ¼ðMqp þMstÞEcm

Mst þ ð1þ �ÞMqp

(D7-1)

where Mst is the moment due to short-term actions and Mqp is the moment from quasi-permanent actions. The neutral axis depth and steel strain can be derived from a crackedelastic analysis, assuming plane sections remain plane. For a rectangular beam, from Fig. 7.1:

Strains

"s ¼d � dcdc

"c (D7-2)

Forces

Fs ¼ Fc

so

AsEs"s ¼ 0:5bdc"cEc;eff (D7-3)

Putting equation (D7-2) into equation (D7-3) gives:

dc ¼�AsEs þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðAsEsÞ2 þ 2bAsEsEc;effd

qbEc;eff

(D7-4)

2-1-1/clause7.2(3)

2-1-1/clause7.2(4)P

2-1-1/clause7.2(5)

d

dc

εc

εsAs

d – dc

Stresses Strains

b

Fig. 7.1. Notation for a rectangular beam

227

CHAPTER 7. SERVICEABILITY LIMIT STATES

Index

Page numbers in italics denote figures

accelerated relaxation, 290

actions, 9�10, 288, 308�309actions combinations, 209, 226additional longitudinal reinforcement, 143�145aggregate concrete structures, 295�302allowable maximum strain, 107alternative methods, 349�350amplification factors, 357

analysis, 95�97, 326, 363see also structural analysis

anchorage, 29, 245, 247�251, 247�248, 251, 272bottom reinforcement, 277�8force calculation model, 286prestressing, 87, 89�90, 271pretensioned tendons, 259�262, 262support reinforcement, 292tension, 261�262, 286ultimate limit state, 201, 261�262zones, 262�270, 262, 265�267, 270, 330

angular imperfections, 43application rules, 5

arches, 43assumptions, 4�5, 25, 28�29autogenous shrinkage, 18

axial forces, 78, 304axial loads, 62�80, 123, 125�126

‘balanced’ sections, 78bars, 246�247, 250�258, 254bases, 179�185, 184�185beams, 275�281

bearings, 47composite, 337�339, 338�339creep, 337�339, 338�339doubly reinforced rectangular, 112�114, 113effective spans, 47, 48end stress transfer, 99�100idealization for space frame analysis, 172K values, 251lateral instability, 80�81midspan stress transfer, 100

monolithic with supports, 47

plastic analysis, 52, 53precast pretensioned, 160�164rectangular, notation, 227

shear, 136, 137, 278�279slender, 80�81time-dependent losses, 91with uniform loads, 51

see also slabsbearings, 47, 204, 205�207, 292�293, 329�330bending, 105�131, 299, 304, 350biaxial bending, 80, 125�126biaxial compression, 218biaxial shear, 218

bilinear diagrams, 21bonds, 96, 99�103, 100, 102, 245, 250bottom reinforcement, 277�278box girders, 280�281, 280, 362discontinuity regions, 362effective flange widths, 46�47, 47losses, 87, 94�95post-tensioned, 129�131, 139�140, 148�151shear cracks, 365�367, 366tendon profiles, 151

three span, losses, 87time-dependent losses, 94�95torsion, 172�173without tendon drape, 149�151worked examples, 87�90, 94�95, 172�173,

280�281, 280braced columns, 74braced members, 63bridge piers see piersbrittle failure, 126�131buckling, 43, 63, 65, 305bundled bars, 258bursting, 203�205, 204, 262�263

cables, 126�127cantilevering piers, 67, 70, 75�76, 79�80carbonation corrosion, 34

CCC nodes, 197CCT nodes, 198

centreline distances, 260chemical attack corrosion, 35chloride corrosion, 34

CIRIA Guide 1 method, 264, 265classes, 24, 34�35, 37, 322�323classification, 37

climatic actions, 309close to slab edges/corners, 176close to supports, 135�136, 145coefficients of friction, 86

columns, 185, 282�284see also piers

combinations, 157�158, 169�170, 350composite beams, 161, 337�341, 338�39, 341composite construction, 93, 160�165composite deck slabs, 162�164composite members, 342�343compressive strengths, 11�12, 20�21, 298,

360�361compressive stresses, 115concrete, 11�23cast at different times, 158�160confined, 298

cover verification methods, 36�38creep, 14�17, 333damage, 322

deformation characteristics, 13elastic deformation, 14fatigue verification, 213�215flexural tensile strengths, 22�23heat curing effects, 290indicative strength classes, 322

lightweight aggregate structures, 295�302shear, 213�215shell elements, 344�345shrinkage, 14, 18�19strengths, 11�14, 20�21, 296, 360�361stress characteristics, 13stress limitation, 82�83stress/strain relationships, 19, 19�20, 21�22worked examples, 16�17

contraflexure points, 147

control, cracks, 230�242, 237, 240, 364�367control perimeters, 176�177, 176, 184corbels, 328�329, 329corner piles, 185, 189, 191

corrosion, 32�35, 126�127, 322couplers, 29, 252�257, 271cover, 31�38, 261, 298crackscables, 126�127checks, 236�237, 241�242control, 230�242, 364�367early thermal, 243flexure, 140, 148�149prediction, 240�241sections, 228strains, 237, 240

widths, 237�242, 237, 240worked examples, 236�237, 241�242

creep, 69�70, 313�315, 332, 333�341, 333,338�339, 341

concrete, 14�17, 297, 313�314, 333time-dependent losses, 92�94worked examples, 336�337, 339�341, 341

cross sections see sections

crushing, 169�170, 201�203, 202CTT nodes, 198�199curvature forces, 259curvature methods, 76�80

damage, 210�212, 322, 356�361dead load moments, 332

deck slabs, 228�230, 300�301cover example, 38crack control, 236�237, 241�242shear, 134ultimate limit states, 110�111voided reinforced concrete, 146�147

deep beams, 284definitions, 5, 155, 167, 265deflection, 70, 243deformation, 13, 297

design, 20�21, 298anchorage lengths, 249�251assumptions, 25, 28�29basis, 7�10, 289�290, 296execution stages, 307�309particular rules, 291

shear reinforcement, 133�153, 140�141, 145,153

torsion, 167�171detailed calculation method, 317�321detailing rules, 245�273, 275�288, 291�293,

302, 327�330deviation allowances, 38

deviators, 272�273diaphragms, 199�201, 200�201, 280�281, 280,

362

differential creep, 337�339, 338�339differential shrinkage, 102, 341�342, 342discontinuity, 195�196, 196, 288, 362dispersion, 267doubly reinforced concrete beams/slabs,

112�114, 113D-regions, 57, 288

see also strut-and-tie modeldrying shrinkage, 18ductility, 24, 27�28ducts, 147�148, 148, 258�261, 259durability, 31�38, 298, 322�323

early thermal cracking, 243EC2-2 Annex J method, 263�264eccentricity, 42, 304, 348

edge sliding reinforcement, 206�207effective lengths, 65�67, 65effective thickness, 169

372


effective widths, 44�46, 47, 147�148elastic deformation, 14, 85�86, 296�297end moments, 74end supports, 277�278environmental conditions, 32�35equivalent time method, 318�321, 318examples see worked examplesexecution stages, 307�309expansion joints, 357exposure, 32�35, 37external non-bonded tendons, 29external tendons, 29, 98

external vs. internal post-tensioning, 95�96

failure, 54, 123, 126�127, 248, 304fatigue, 25, 28, 98�99, 208�215, 209, 356�361

worked examples, 211�212, 214�215finite element models, 344

flanged reinforced beams, 115�117, 115, 117flanges, 44�47, 47, 107, 156, 157�158flat slabs, 104, 282, 326

flexural reinforcement, 281flexural shear, 187, 189flexural tensile strengths, 22�23floor systems, 291

footings, 285�286see also pad footings

force calculation model, 286

foundations, 10, 285�288, 285, 287, 293frames, 52, 328free body diagrams, 144

freeze/thaw corrosion, 34�35friction, 86�87, 89

general method, 332�333geometric data, 8, 44�48geometric imperfections, 40�44, 41�44, 74geometry, discontinuity regions, 288

half joints, 292heat curing, 290

heavy vehicles, 358high-strength concrete, 313�314hollow piers, 204

‘I’ beams, 154idealization, 170, 172imperfections see geometric imperfections

imposed curvature, 55indicative strength classes, 322�323indirect supports, 279�280in-plane buckling, 43, 43in-plane stress conditions, 324in situ concrete structures, 311

in situ deck slabs, 160�164interaction diagrams, 122intermediate supports, 278

internal vs. external post-tensioning, 95�96isolated members, 41, 64�67, 65ISO standard 3898: 1997 symbols, 5

joints, 153, 357

K values, anchorage, 249, 251

laps, 201, 252�257, 252, 254, 256large diameter bars, 257�258lateral instability, 80�81layer centres, 348

layer stresses, 347layouts, tendons, 258�259leaf piers, 205�207, 207lightly reinforced structures, 303�305lightweight aggregate concrete structures,

295�302analysis, 298

cover, 298deformation, 297design, 296, 298

elastic deformation, 296�297materials, 296�298non-linear analysis, 298

shear/torsion, 299shrinkage, 297stress, 297�298strut-and-tie models, 300

ultimate limit states, 298�300worked examples, 300�301

limited redistribution, 49�51limiting slenderness checks, 68�69limit states, 7, 58�60, 62, 82�83, 98�103see also ultimate limit states

linear elastic analysis, 48�51, 51links, 251loads

close to slab edges/corners, 176close to supports, 145dispersal, 207distribution, 176�177, 176, 202factors, 125models, 359spread, 285

longitudinal reinforcement, 143�145, 275�278,283

anchorage, 247�251, 247�248, 251, 272longitudinal shear, 154�158, 349�350, 355longitudinal tension reinforcement, 277lossesanchorage, 87

elastic deformation, 85�86post-tensioning, 85�90prestress, 84�90, 89, 291, 317�321, 318pretensioning, 84�85steel relaxation, 317�321, 318worked examples, 87�90

low-relaxation prestressing tendons, 318�321

mandrels, 246�247materials, 8, 10�29, 290, 296�298, 303�304partial factor modification, 311�312

maximum moments, 42�43

373

INDEX

maximum punching shear stress, 185M beams, 162�165, 173�175, 175members, 140�153, 140�141, 145detailing, 275�288, 302not requiring design shear reinforcement,

133�140in tension, 198�199

membrane elements, 215�223, 216, 222membrane rules, 216, 218�222, 222minimum cover, 36�38minimum reinforcement, 127�128minimum shear reinforcement, 137

modification, partial factors, 311�312Mohr’s circles, 222, 352�353moments, 42�43, 70, 74, 332, 342monitoring facility provision, 128

nodes, 196�201, 197�198nominal curvature methods, 76�80nominal stiffness, 71�76non-linear analysis, 19, 19, 58�62, 61, 70�71,

298, 363normative references, 4notation, rectangular beams, 227

open sections, 170�171out-of-plane buckling, 43overlapping, 292

pad footings, 180�182, 180, 287�288, 287parabolic-rectangular diagrams, 21

parameters, creep/shrinkage, 314Part 2 scope, 4partial factor method, 9�10partial factor modification, 311�312partially loaded areas, 201�207, 202, 204, 207,

300, 329�330partial sheared truss models, 141

particular rules, 275�288, 291, 302particular situations, 327�330piers, 43, 63, 68�69, 68, 207worked examples, 16�17, 17see also cantilevering piers; columns

pile caps, 185�193, 186�193, 285piles see corner pilespin-ended struts, 42placement of reinforcement, 158plain reinforced concrete structures, 303�305plastic analysis, 52�58, 53, 55plastic hinges, 55pocket foundations, 293

post-tensioning, 85�90box girders, 129�131, 139�140, 148�153cables, 272�273construction, 336ducts, 258�259members, 262�270, 265, 270, 330

precast concrete, 153, 160�165, 289�293, 312prestress dispersion, 267prestressed beams, 118�131, 119

prestressed members, 137, 140, 147�153, 148prestressed steel, 25�29, 28prestressed structures, 81�103bonded tendons, 99�103, 100, 102�103creep, 92�94fatigue limit states, 98�99limit states, 97�99losses, 84�95, 89, 291post-tensioning, 95�96prestress considerations, 95�97prestressing forces, 82�84primary/secondary effects, 96�97serviceability limit states, 98�103unbonded prestress, 96worked examples, 99�103, 100, 102�103

prestressing devices, 29prestressing forces, 82�90prestressing steel, 209�210, 245�273, 290,

317�321, 318, 359�360prestressing systems, 32�35, 38prestress transfer, 260�261pretensioning, 84�85beams, 131, 160�164composite members, 337�339, 342�343concrete M beams, 162�165, 173�175, 175strands, 259tendons, 258�262, 262

primary prestress effects, 96�97, 96primary regularization prism, 263�264,

264�265principles, 5, 7, 40

properties of materials, 8, 23, 26�27, 316provision of monitoring facilities, 128proximity to expansion joints, 357

punching, 175�193control perimeters, 176�177, 176, 184corner piles, 189, 191load distribution, 176�177, 176pile caps, 185�193shear, 177�179, 185, 192�193, 300shear resistance, 179�185, 184worked examples, 180�182, 180, 186�193,

186�193

railway bridges, 359�361rectangular beam notation, 227rectangular stress blocks, 117redistribution, 332, 338�339, 339�341regions with discontinuity, 288, 362reinforced concrete, 105�117, 108, 121�126,

122�123, 133�147, 303�305see also lightly reinforced . . .

reinforcement, 222�223, 222, 245�273,247�248, 251, 270, 272, 277, 300�2

corrosion potentials, 32�35cover, 31�38, 298crack control, 232�234durability, 31�38eccentric, 348edge sliding, 206�207

374


fatigue verification, 209foundations, 287�288, 287in-plane stress conditions, 324loop overlapping, 292minimum, 37, 127�128, 137normative properties, 316pad footings, 287�288, 287piers, 68, 207

properties, 316punching, 184shear, 133�153, 251surfaces, 327�328suspension, 280tension, 277, 324torsion, 167�169, 167transverse, 254�255, 254webs, 366worked examples, 255�257, 256, 300�301

reinforcing steel, 23�25, 24�25, 209�210, 209,359�360

relaxation, 27, 92, 94, 317�321, 318resultant conversions, stress, 347return periods, 309ribbed wires, 257road bridges, 211�212, 214�215, 356�358rotation capacity, 53�56, 54�55rules see detailing rules

safety format, 61�62, 363St Venant torsion, 166sandwich model, 346�349, 347�348, 353�355scalar combinations, 60�61, 61scope, Eurocode 2, 3�4sea water corrosion, 34

secondary moments, 342secondary prestress effects, 96�97, 96second-order analysis, 62�80

biaxial bending, 80

braced columns, 74creep, 69�70definitions, 62�63methods, 70�71, 76�80nominal stiffness, 71�76simplified criteria, 64�69soil-structure interaction, 64stiffness, 63�64, 71�76unbraced columns, 73

sections, 21�22, 78, 154�158, 169, 170, 170, 298see also prestressed sections

segmental construction, 153, 153, 171serviceability limit states, 225�243

actions combinations, 226crack control, 230�243deck slabs, 228�230, 302execution stages, 309prestressed structures, 98�103reinforced concrete deck slabs, 228�230stress limitation, 226�230worked examples, 228�230

shear, 131�165

alternative methods, 349�350beam enhancement comparisons, 136

bending, 160, 346�355, 347�348, 350between web/flanges, 154�158composite construction, 160�165concrete cast at different times, 158�160contraflexure points, 147cracking, 364�367, 366deck slabs, 134flexure failures, 137flow calculation definitions, 155lightweight structures, 299

members not requiring reinforcement,133�140

members requiring reinforcement, 140�153post-tensioned box girders with tendon drape,

151�153precast concrete, 160�165reinforced concrete, 133�140, 304�305reinforcement, 145, 190, 192�193, 251,

278�279, 282tension, 137�139torsion, 169�170total transverse reinforcement, 157T-sections, 154�158verification procedure rules, 132�133wall analysis, 104worked examples

bending, 351�355box girders, 139�140, 148�153deck slabs, 134, 146�147flexure, 139�140tendon drape, 151�153without tendon drape, 149�151

see also longitudinal shear; punchingshear resistance, 179�185, 184shear stress, 161, 185shell elements, 344�345shift method, 143�145short spans, 145shrinkage, 313�315concrete, 14, 18�19differential, 102, 341�342, 342lightweight structures, 297

time-dependent losses, 92, 94sign convention, 216simplified criteria, 64�69simplified methods, 333�336simplified rectangular diagrams, 21singly reinforced concrete, 107�112, 108sinusoidal imperfections, 43

skew reinforcement, 222�223, 222slabsedges/corners, 176

K values, 251plastic analysis, 52punching, 176, 179�185, 184shear resistance, 179�185, 184shear stress distribution, 161torsion, 172�175, 175

375

INDEX

slabs (continued )see also beams; flat slabs; solid slabs

slenderness, 64�67, 65, 68�69, 80�81, 305see also lightweight . . .

sliding wedge mechanism, 329

SLS see serviceability limit statessoil structure interactions, 64, 325solid sections, 169�170, 281�282space frame analysis, 172spacing, 246, 259spalling, 201�203, 202, 262, 265spans, beams, 47, 48

square sections, 170, 170staged construction, 93static values see characteristic values

steel, 25�9, 28, 228, 359�360see also prestressing steel

stiffness, 63�64, 71�76straincompatibility, 106�107, 118, 122�123cracks, 237, 240�241, 240creep, 313�315, 333distributions, 117external tendons, 98

strength

classes, 296, 322�323concrete, 11�14curve, 209

prestressed steel, 27reinforcing steel, 23�24see also individual strengths

stressanchorage zones, 262block idealization comparisons, 22

concentrated loads, 202concrete characteristics, 13damage equivalent, 356�361differential shrinkage, 102

distances, uniformity, 261fatigue verification, 356�361flanged beams, 117

lightweight structures, 297�298limitation, 226�230Mohr’s circles, 222

non-linear structural analysis, 298prestressed steel, 28redistribution, creep, 338�339reinforced concrete deck slabs, 228�230reinforcement fatigue verification, 209resultant conversions, 347section design, 298

serviceability limit states, 226�230strain relations, 298temperature differentials, 230

transfer in beams, 99�100, 100transverse, 262

stress–strain diagrams, 24�25, 28stress–strain profiles, 119stress–strain relationships, 19, 19�20, 21�22structural analysis, 39�104

geometric imperfections, 40�44, 41�44global, 39, 45�46lateral instability, 80�81lightweight structures, 298linear elastic analysis, 48�51local, 39�40non-linear analysis, 58�62, 298plastic analysis, 52�58post-tensioning, 95�96precast structures, 290�291prestressed members/structures, 81�103principles, 40

reinforced structures, 303�304second-order effects analysis, 62�80structure idealization, 44�48time-dependent behaviour, 331�343, 332worked examples, 67�69, 75�76, 79�80

structural classification, 37

structure idealization, 44�48strut-and-tie models, 56�58, 57, 195�196, 196anchorage zones, 266

corbels, 329ducts, 148hollow piers, 204lightweight structures, 300

nodes, 196�201, 197�198struts, 193�195ultimate limit states, 193�201worked examples, 199�201, 200�201

strutstensile stress, 194�195transverse stress, 193�195see also pin-ended struts; strut-and-tie models

summary procedure, 342�343supports, 135�136, 145, 187, 291�292surface reinforcement, 279, 327�328suspension reinforcement, 280symbols, 5

temperature differentials, 230tendons, 149�151, 151external non-bonded, 29low-relaxation prestressing, 318�321post-tensioned, 86

prestressing, 258�273, 291strain, 98vertical profiles, 87

tensile forces, 82�83, 261�262tensile reinforcement, 286, 324tensile strength, 12�14, 20�21testing, 10

thin walled sections, 170three-span bridges, 87, 336�337ties, 195�196, 196time-dependent behaviour, 90�95, 91, 331�343,

332time-dependent losses, 91, 92�95torsioncombined shear/torsion, 169�170definitions, 167

376


design procedure, 167�171, 170lightweight structures, 299

M beam bridges, 173�175, 175open sections, 170punching, 175�193reinforced structures, 167�169, 167, 279,

304�305St Venant torsion, 166

segmental construction, 171slabs, 172�175, 175ultimate limit state, 166�175warping torsion, 171�172worked examples, 172�175pad footings, 180�182, 180pile caps, 186�193, 186�193slabs, 173�175, 175

total transverse reinforcement, 157transfer, prestress, 260�261transverse bending, 157�158, 157, 160, 346�355,

347�348transverse forces, 42�43transverse reinforcement, 158, 254�255, 254,

283�284transverse stress, 193�195, 262truss models, 140�141, 144, 154T-sections, 154�158two-span beams, 51

ULS see ultimate limit statesultimate bond strength, 248ultimate limit states (ULS), 105�223

anchorage, 201, 261�262beams, 105�117, 108, 112�117, 113bending, 105�131, 122�123brittle failure, 126�131columns, 121�126, 122�123deck slabs, 110�111doubly reinforced rectangular beams,

112�114, 113execution stages, 309fatigue, 208�215flanged beams, 115�117laps, 201lightweight structures, 298�300membrane elements, 215�223methods, 58�60partially loaded areas, 201�207, 202, 204prestressed concrete beams, 118�121prestressed structures, 97�98shear, 131�165singly reinforced concrete beams/slabs,

107�112, 108strain compatibility, 106�107strut-and-tie models, 193�201torsion, 166�175voided reinforced concrete slabs, 111�112worked examples, 110�111, 113�117,

129�131unbonded prestress, 96unbonded tendons, 121

unbraced columns, 73unbraced members, 63

un-cracked in flexure, 137�140uniaxial bending, 123uniaxial compression, 218�220uniaxial shear, 218�220uniaxial tension, 220�222uniformity, stresses, 261

uniform loads, 51unreinforced concrete, 204unrestrained concrete, 333

valuesdrying shrinkage, 18material factors, 10

vertical loads, 359variables, basic, 7�8vector combinations, 61�62vehicle expected numbers, 358verificationexecution stage criteria, 309

fatigue, 208�215, 356�361methods, 36�38partial factor method, 9�10procedure rules, 132�133

vertical profiles, 87voided reinforced concrete slabs, 111�112,

146�147

walls, 284, 291, 304warping torsion, 166, 171�172websbending/shear combinations, 350box girders, 280�281, 280control of shear cracks, 364�367, 366diaphragm region, 280�281, 280longitudinal shear rules, 355reinforcement, 365�367, 366widths, 147�148

welding, 25, 251, 257worked examples

anchorage zones, 267�270, 270bonded tendons, 99�103, 100, 102�103box girders, 87�90, 94�95, 139�140,

280�281, 280, 365�366bridge piers, 16�17, 17brittle failure, 129�131cantilevered piers, 67, 75�76, 79�80composite beams/slabs, 162�164, 339�341,

341crack control, 236�237, 241�242creep, 339�341, 341damage equivalent stress range, 211�212deck slabs, 38, 255�257, 256diaphragm design, 199�201, 200�201doubly reinforced concrete slabs, 113�114effective flange widths, 46�47, 47equivalent time method, 318�321fatigue, 211�212, 214�215flanged beams, 115�117

377

INDEX

worked examples (continued )foundations, 287�288, 287lap lengths, 255�257, 256leaf piers, 205�207, 207lightweight structures, 300�301linear elastic analysis, 51, 51losses, 87�90low-relaxation prestressing tendons, 318�321membrane elements, 218�222, 222partially loaded areas, 205�207, 207post-tensioned members, 139�140, 267�270,

270

prestressed beams, 118�121, 129�131prestressed steel, 27prestressed structures, 99�103, 100, 102�103pretensioned concrete M beams, 162�165punching, 180�182, 180, 186�193, 186�193reinforced concrete

columns, 124�125slabs, 110�112, 228�230, 241�242

relaxation losses, 27, 318�321

sandwich model, 353�355serviceability limit states, 228�230shearcracking, 365�366post-tensioned box girders, 149�153reinforced concrete, 134�135transverse bending combinations,

351�355structural analysis, 67�69, 75�76, 79�80structure idealization, 46�47strut-and-tie models, 199�201, 200�201three-span bridges, 336�337time-dependent losses, 94�95torsion, 172�175transverse bending, 351�355ultimate limit states, 110�112, 115�117voided reinforced concrete slabs, 111�112,

146�147web reinforcement, 366

zones for shear reinforcement, 190

378


Documents

Designers Guide to EN1992-2 Eurocode 2