Designer`s Guide to EN 1992-2

DESIGNERS' EUROCODES an ICG) initiative

DESIGNERS' GUIDES TO THE EUROCODES

DESIGNERS' GUIDE TO EN I992.2 EUROCODE 2: DESIGN OF CONCRETE STRUCTURES

PART 2: CONCRETE BRIDGES

Eurocode Designers' Guide Series to EN

DÊsrgtefs' Guide lY.

1990. Eurocode: Bosis

of

Holickf. 0 7277 301| 8. Published 2002.

DesigneÆ' Guide

Sttucturol Desrbn. H. Gulvanessian, J.-A, Calgaro and

to EN 1994-l-1. Eurocode 4: Design of Composrre

Genero/ Ru/es ond Rules

for Buildings. R-

P.

Steel

ond Concrete Structurci port

Johnson and D, Anderson. 0 7277

3l5l

l.l:

3. Published 2004.

Designers' Cuide to EN 1997-1. Eurocode 7: Geotechnical DesÈn Generoi Ru/es. R. Frank, C. Bauduin, R. Driscoll, lv1. Kawadas, N. Krebs Ovesen, T. Orr and B. Schuppener. 0 7277 3 154 8- Published 2004.

-

Desrgners'Guide to EN 1993-l-1. Eurocode 3: Design ofSree/ Sruau rcs. General Rules ond Ru/es for Buildings. L. Gardner and D. Nethercot. O 7277 3163 7. Published 2004. Designers' GuriCe to EN I 992-l- I ond EN 1992- I-2. Eurocode 2: Design of Concrete Svuctures. Generol Rules ond Rules for Buildings ond SttucturolFire Design A,W Beeby and R- S. Narayanan.0 7277 3105 X. published

2005. Designers' Guride to EN 1998-l ond EN 1998-5. Eurocode 8: Design of Struaures for Eafthquoke Resiston e. Generol Ru/eg SeÈmic Actiong Design Ru/es fot Buildings, Foundotjons ond Re:oining 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-l -l . Eurocode 5: Design of Timber Stru ctures. Common Rules ond Buildings. C. ll'êÊem.0 7277 3 162 9- Forthcoming: 2007 (provisional).

Des{nerc' Guide to EN I99l-4. Eurocode Forthcoming: 2007 (provisional).

!:

Actions on Srructures.

Designers'Cude to EN 1996. Eurocode 6: Port Forthcoming: 2007 (provisionâl).

I.l:

Design

fot

Rules ond

Wind Aajons. N, Cook 0 7277 3lS2 l.

of Mosonry

Sttuctures- J.

Morron.

O

7277 3155 6,

Desrgners' Gui,Ce to EN l99l-l-2, 1992-l-2, 1993-l-2 and ËN 19941-2. Eurocode I: Aajons on Structures. Eurocode 3: Des(n of Stee/ structures- Eurocode 4: Design of Composite Stee/ ond Conqete Sùuctures. Firc Engineering (Actions on Steel and Composite Structures). Y. Wang, C. Bailey, T. Lennon and D. Moore.

0 7277 3157 2. Forrhcoming: 2007 (provisional). Desrgner' Guide to EN 1993-2, Eurocode 3: Desqn ofsteei Structûres, Bridges. C. R. Hendy and C. J. Mu rphy. 0 7277 3l60 2- Forthcoming. 2007 (provisional). Designers'Guide ta EN I99l-2, | 991^l -l, f99!-!-3 ond 199!-l-5 to !-7. Eutocode /j Acr,bns on Srrucrures. Trafic Loods and Other Actions on Eridges. J--A. Calgaro, M. Tschumi, H. Gulvanessian and N. Shetty.

0 7277 3156 4. Forthcoming; 2007 (provisional).

Desrgnels'6uide to EN l99l-l-l,EN l99l-l-3ond !991-l-5to l-7. Eutocode I: Actions on Structurcs. Geûetul Ru/es ond Aaions on Bui/dings (not Wind), H. Gulvanessian, J.-A. Calgaro, P. Formichi and G. Harding.

0 7277 3158 0. Forthcoming 2007 (provisional).

EN 1994-2. Eurocode 4: Design of Composite Steel ond Concrcte Stuctures. port 2: Generol ond Rules for Btidges. C- R. Hendy and R. P. johnson. 0 7277 3l6t 0. Published 2006.

Des€ners' Gur'de to Rules

www.eurocodes.co.uk

DESIGNERS'GUIDES TO THE EUROCODES

DESIGNERS' GUIDE TO EN I992.2 EUROCODE 2: DESIGN OF CONCRETE STRUCTURES

PART 2: CONCRETE BRIDGES

C. R. HENDY and D. A. SMITH

thomastelford

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103

First pubtishcd 2007

Eurocodes Expcrt

Structural [ùrocodes offer the oppoitunity of harmonized design standards for the European construction market ând the rest of the world. To achieve thisJ the constrùction industry needs to becorne acquainted nith the Eurocodes so thât the maximum advantâge can be taken of these opportunities Eurocodcs Expert is a new ICE and Thornas Telford initiative set up to assist in creating a greater awareness of thc impâct and implementation of the Euiocodes within the UK consûuction industry Eurocodes Expert provides a range of products ard se ices to âid and support the transition to Eurocodes. For comprehcnsive and useful information on the adoption of the Eurocodes and their implcmentation process pleas,€ visit our website or email eurocodes@thomastelford,corn

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(O The authors and Thomas

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2007

All rights, including translation, resened. Except as pemitted by

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Preface Aims and objectives of this guide The principal aim ofthis book is to provide the user wilh guidance on the intelpretation and use of EN 1992-2 and to prcscnt worked examples. It covers topics that will be encountered in typical concrete bridge designs and explains the relationship bctween EN 1992-2 antl Lhe other Eurocodes. EN 1992-2 is not a'stand âlone'document ând refers extensiveiy to other Eurocodes. Its format is based on EN 1992-l-1 and generally follows the same clause r.rumhering lt identifies which parts of EN 1992-1-1 are relevant for bridge design and adds fut'ther clauses that are speciflc to bridges. It is therefbre uot useful to produce guidancc on EN 1992-2 in isolation and so this guicle covers material in EN 1992-1-l rvbich will need to be used in bridgc design.

This book also provides background infbrmation and relcrences Eurocode 2 to understand the origin and objeclivcs of its provisions.

to

enable users

of

Layout of this guide EN 1992-2 has â fôreword, l3 sections and l7 annexes. This guidc has an introduction which corresponds to th€ foreword ofEN 1992-2, Chaptcrs I to 10, $'hich correspond to Sections I to 10 of thc Eurocode and Annexes A to Q which âgain correspontl to Annexes A to Q of the Eurocode. The guide generally follows the sectjon numbers and first sub-headings in EN 1992-2 so that guidance can be sought on the code on â section by section basis. The guide also Ibllows thc format of EN 1992-2 to lower levels of sub-heading in cases $'here this can conveniently be done and where thcril is sullicient material to rnerit this. The need to use several Eurocode parts can initially make it â dâunting tâsk to locate information in the order required for a real design, In some places, therefore, additional sub-scctions are included in this guide to pull together rclcvanL dcsign rules for individual elements, such as pile caps. Additional sub-sections are identif,ed as such in the sub-section heading.

The following parts of rhe Eurocodc arc intended to be used in conjunction with Eurocode 2:

EN 1990: Basis of structural design EN 1991: Actions on structures EN 1997: Geotechnical design EN 1998: I)csign of structures for earthquake resistânce hENs: Construction products relevant for concrete structures EN 13670: Execution (construclion) of concrete stmctures will gcnerally be required for a typical concrete bridge design, but discussion on them is generally beyond the scope of this guide.

These documents

DESIGNERS' GUIDE

TO EN I992.2

In this guide. references to Eurocodc 2 are made bv using the abbreviation 'EC2' for EN 1992, so EN 1992-1-l is relerred to as EC2-1-1. Where clausc numbers are relerred to in thc lext, they are prefixed by thc number of the relevant pârt of EC2. Hence:

. . . .

2-2i clause 6.3.2(6) neâns slause 6.3.2, paragraph (6),

of EC2-2 2-1-lr'clause 6.2.5(1) means clause 6.2.5, paragraph (1)- of EC2-l-1 2-2lExpression (7-22) means equation (7.22) in EC2-2 2- l- 1i Expression (7.8) means equâtion (7.8) in EC2-1-1.

Note that, unlike in other guides in this series, eyen clauses in EN 1992-2 itself are pretxed wilh '2-2'. There âre so many references to other parts of Eurocode 2 required that to do otherwise would be confusing.

Where additional cquations are provided in the guide- they are numbered sequentially within each sub-section of a main seotion so that, fbr exâmple, the third additional expression within sub-section 6.1 would be referenccd equation (D6.1-3). Additional Iigures and lables fbllow the same system. For example, the second additional ligurc in scction 6.4 would be referenced Figure 6.4-2,

Acknowledgements Chris Hendy would like to thânk his rvife, Wendy, and two boys. Peter Edrvin Hendy and Matthew Philip Hcndy, 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 his son, Williâm Thomas Smith. and the continued support of Brian and Rosâlind Ruflell-Ward from the very beginning. Both authors would also like to thank their empJoyer, Atkins, lor providing both facilities and time lbr the production of this guide, They also wish to thank Dr Paul .Iackson and Dr Steve DenLon for their helpful commi:nts on the guide.

Chrit Hendy David A. Smith

Contents Preface

Aims and objectives of this guide Layout of this guide vi

Acknowledgements Introduction

7

Additional information spccific to EN 1992-2 Chapter

1.

Generâl

1.1.

3

Scope

1-1.1. Scope of Eurocode ? 1.1.2. Scope of Part 2 of Eulocode

1.2. Normativereferences 1.3. Assumptions 1.4. Distinction between principles 1.5. Definitions 1.6. Slmbols Chapter 2.

3

2

4 4 4

and application rules

5 5 5

Basis of design

7

2.1. Requirements 2.2. Principles oflimit state design 2.3. Basic variables 2.4. Verification by the partial factor method

'7

2.4.1. General 2.4.2. Design r alues 2.4.3. Combinations ol

Concrete

3-1.1. 3.1.2. 3.1.3. 3.1.4. 3.1.5.

1

9

actions

Materials

3.l.

1

9

2.5- Design assisted by testing 2.6. Supplementaryrequirementsforfoundations Chapter 3.

2

Gcncral Strength Elastic deformation Creep and shrinkage Concrete stress strain relation for non-linear structural analysls

I

l0 l0

ll 11

ll t4 t4

l9

DESIGNERS' GUIDE

TO EN I992-2

3-1.6. 3.1,7. 3.1.8J.1.9.

Dcsign conpressive and tensile strengths Stress-strâin relations for the design ol sections Flexural teusile strength Confined concrcte

Reinlbrcing steel

3.2.1. 3,2.2. 3.2.3. 3.2.4. 3.2.5, 3.2.6. 3.2.7. 3.3.

4.

5.

23

Strength

23

Ducrilily

2\

Welding Fatigue Design assumptions

25 25

Prestressing steel 3.3.1. Gencral 3.3.2. Properties 3.3.3. Strength 3.3.4. Ductilitl' characterislics

.J 25

26 27 21 28

28 29 29 29

Prestressing devises

Durability and coycr to reinforcement

31

4.1. General 4.2. Environmentalconditions 4.3. Requirements lbr durability 4.4. Methods of vcrilication

3t

4,4.1. Chapter

22 23 23 23

3.4.1. Anchorages and couplers 3.4.2- External non-bonded Lendons Chapter

2l

General Properties

3.3.5. Fatigue 3.3.6- Design assurnptions 3.4.

20

32 35

36 36

Concrete cover

Structural analysis 5.l. Genelal 5.2. Gcomerric imperfections 5.2.1. General (additional sub-secLion) 5.2.2. Arches (additional sub-section) 5.3 Idealization of the s|Iuoture 5.3,1 Structurâl models for overall analysis

5.3.2- Geon.retric data

5.4. Linear elastic analysis 5.5. Lincar elastic analysis with limited 5.6. Plastic analysrs 5,6.1. 5.6.2.

redistribution

General Plastio analysis 1br beams. Ii'arnes and slabs

5.6..1. RorirtioncaFaciL) 5.6-4. Strut-and-tie models 5.7, NonJinear 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

5,8.

39 39

40

40 43

44 44 44 48 49 52 52 52 53 56 58 58 60 61

stâtes

Analysis of second-order cffects with axial load 5.8.1. Delinitions and introduction to second-order effects 5-8.2. Gcneral 5.8.3. Simplified critcrja for second-order effects

62

62 63

64

CONTENTS

5.8.4. Creep 5.8,5. Melhods of analysis 5-8.6. General method second-order non-linear analysis 5.8.7, Secontl-ordcr analysis based on nominal stiflness 5.8.8. Method based on nominal curvatute 5.8.9. Biaxial bending

5.9, Lateral instability

ol'slender beams 5.10. Prcstressed members and structurcs 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-tensrorung 5.10.5. lmmediate losses of prestl€ss for post-tensioning 5.10.6. Time-dependent Iosses 5.10.?. Consideration of prestress in the analysis 5.10.8. EIIects of prestressing at the ullimate limit statc 5.10.9. Effects of prestressing aL the serviceability and fatigue

limit

statcs 5.11. Analysis for some pârticular structural members

Châpter 6.

stâtes

Ultimate limit

6.1.

69 10 70

7l 76 80 80 ti1

8l 82 83

84 85

90 95 96 98 104

105

lbrce sub-section)

ULS bending with or without axial

105

105 6.1.1. 105 sub-section) 6.1.2. ll8 (additional sub-section) beâms 6.1.3. Prestressed concrete l2l (additional sub-section) Reinforced concret€ columns 6-1.4. (additional prestress with failure ofmcmbers 6.1.5. Brittle

Gcneral (âddirionâl Reinforced concrete bcams (âdditional

t26

sub-scction)

6.2.

Shear

131

6,2.1. General vcrillcation procedure rulcs 6.2.2. Members not requiring design shear reinfbrcement 6.2.3. Membels lequirilg dcsign shear reinlbrcement 6.2,4. Shear betu,een web and flanges o[ T-sections 6-2.5. Shear ât the interfâce betwecn concrete cast at different

132

6.2.6. 6.2-?. 6.3.

times Shcar and transverse bending Shear in precast concrete and composlte constructon (additional sub-section)

Generâl Design procedure Warying torsion

6,4.

1.

reinforccment

6.4.5. Punching

sheâr resistance

t'72 5

l'7 6

t7'7

179

ol slabs and bases with shear

reinforcement

6.5.

160

1'7

6.4.2. Load distribution and basic control pcrirneter 6.4.3. Punching shear calculation 6.4.4. Punching shear resislance of slabs and bases without

Pile caps (additional sub-section) Design with strut-and-ties models 6,5.1. General

6.4.6.

158 160

t7s

Generâl

shear

154

\71 sub-section)

Punching 6.4-

140

166 166 167

Torsion

6.3.1. 6.3.2. 6.3.3. 6.3.4. Torsion in slabs (additional

133

183

185 193 193

IX

DESIGNERS' GUIDE

TO EN I992.2

6.5,2. Stmts 6.5.3. Ties 6.5.4- Nodes 6_6.

Anchorage and laps

6.1

Partially loaded areas

.

6.8.

193

195

r96 201 201 208

Fatigue

6.8.1. 6.8.2. 6.8.3. 6.8.4.

Verification conditions Internal lbrces and stresscs for fatigue veriflcâtion

208

CombinaLion of actions Velification proccdure for reinforcing and prestressing

209

stecl

209 210 212

6.8.5. Veriûcation using damage equivâlent stress range 6.8-6. Other verificâtion methods 6.8.7, Verification of concrete under compression or shear 6.9.

Chapter 7.

Membrane elements

215

Serrjceâbility limit stâtes 7-

1.

225

General

225 226 230

7.2. StrcssliDiitation 7.3. Crack control 7.3.1, 7-3.2. 7.3.3. 7.3.4.

7.4. 7-5. Chapter 8.

General consideratiors Minimum areas of reinforcerrent Control of cracking without direcr calculation Control ofcrack widths by direct calculation Deflection control Early thermal cracking (additional sub-section)

Detailing of reinforcement and prestuessing st€el

243 243

246

248 248 249 251

Anchorage of links and shear reinforcement Anchorage by weldcd bars Laps and mechanical coupJers

8.8. Additional rules for large 8.9. Bundled bars

234

246 247 247

8.4.1. General 8.4.2. Ultimate bond stres,, 8.4.3. Basic anchorage length 8.4.4. Design anchoraee length

8.7.1. General 8.7.2. Laps 8.7.3, Lâp length 8.7.4. Transverse reinforcement in the lap zone 8.7.5. Laps ofwelded mesh fabrics made of ribbed 8.7.6. Welding (additional sub-section)

230 232

245

8,1. General 8.2. Spacing of bars 8.3- Permissible mandrel diameters 1br bent bars 8.4. Anchorage of longitudinal reinforcement

8.5. 8.68.7.

208

251 252

252 252

253 wires

diameter bars

8.10. Prestressing tendôns 8,10.1. Tendon layouts 8.10.2, Anchorage of pre-tensioned tendons 8.10.3. Anchorage zônes o I post-tensioned members 8.10,4. Anchorages and couplers for prcstressing tendons 8-10.5- Deviators

l)l 25',7

251 258 258 258

2s9 262 211

272

CONTENTS

Chapter 9.

Detailing of members and particular rules 9.1. General

n5

9.2.

275 275

Beâms

9.2.1. LongitudiDal reinlorcement 9-2.2. Shear reinfôrcement 9.2.3. Tot'sion reinfbrcement 9.2.4. Surface teinl'orcemenr 9.2.5. Indirect supports

9.3. 9.4, 9.5.

Chapter

10.

10.8. 10.9.

Chapter

11.

2'79

219

9.3.1. Flexural rcinforcement 9.3.2- Shear reinforcement

281

Columns 9.5.1. General

282 282 282 282

9.5.2. Longitudinal reinforcement 9-5.3. Trânsverse reinforcement

283 283

Walls

284

Deep bcams

Foundâtions

284 285

Regions with discontinuity in geometry or acLion

2ti ti

Flat slabs

10.3.2. Prestressing steel

.

219

281

10.4. Not used in EN 1992-2 10.5. Structural analysis

r0.'7

278

Solid slabs

Additional rules for precâst concretc elem€nts and shucturcs 10.1. General 10.2- Basis of design, fundâmental requiremenls 10.3. Materials 10.3.1. Concrctc

10.6.

2'7 5

289 289 289

290 290 290

290 290

10.5.1. General 10.5.2. Losses ol- prcstress

290

Not used in EN 1992-2 Not used in EN 1992-2 Not uscd in EN 1992-2 Particular rules lbr design and detailing 10.9. L Restraining moments in slâbs 10.9.2. Wall to floor connections 10.9.3. Floor systems 10.9-4. Connections and supports for precast elements

29l

10.9.5. Bearings 10,9.6. Pocket foundations

292 293

Lightweight aggregate concrete strùctures 11.1. General

291

291 291

29]' 29'l 291 291 291

29s

1.3-2. Elastic deformation

295 296 296 296 296

11,3.3. Creep and shrinkagc

291

I1.3.4- Stress strain relâtions for non-linear structulal analysis

298

1.3.5 Design compressive and tsnsile strengths I 1.3.6. Stress sLrain relations for the design of sections

298

I1.2. Basis of design 1L3. Mâterials I1.3.1. Concrete 1

1

1.3.7. Confined concrete 11.4. Durabiliiy and cover lo rcinforcement 1

298

298 298

xl

DESIGNERS' GUIDE

TO EN I992.2

1.5. Structural analysis 1.6. Ultimate limit stâtes 11.7. Serviceability linit states

xtl

1

298

1

298

I1.8- Delailing of reinfôraement gcneral I1.9. Detailing of members and particular rules

302

Chapter 12,

Plain and lightly reinforced concrete structures

303

Chapter 13.

Design for thc execùtion stâges 13.1. Gencral 13.2. Actions during execution 13.3. Verification criteria 13.3.1. Ultimate limit stâte 13.3,2. Serviceability limit states

307 107

309

Annex A.

Modification of partial factors for mâteriâls (informative)

311

Annex B.

Creep and shrinkagc strâin (informatiye)

313

Annex C.

Reinforcement properties (normatiye)

316

Annex D.

Detailed câlculâtioo method for pr€stressing steel relâxâtion losses (informati ve)

3t7

Annex E.

Indicatiye strertgth classes for durability (informative)

322

Annex F.

Tension reinforcemeot expressions for in-plâne stress conditions

302 302

308 309 309

(informative)

324

Annex G.

Soil-structure interâction

325

Annex H,

Not ased in EN 1992-2

Annex I.

Analysis of flat slabs (informative)

326

Annex J.

Detailing rules for particulâr situations (informatiye)

327

Annex K.

Structural effects of time-deperdent behaviour (informative)

33r

Annex L.

Concrete shell elements (informative)

344

Annex M.

Shcar and transverse bending (informative)

346

Annex N.

Damage equivalent strcsses for fatigue verification (informatve)

356

Annex O.

Typical bridge disconfinuity regions (informative)

362

Annex P.

Safety format for nonJinear analysis (informative)

363

Annex Q.

Control of shear cracks within webs (informalive)

3@

References

369

Index

371

lntroduction The provisions of EN 1992-2 are preceded by a forcword, most of whjch is common to all Eurocodes. Tbis Foreword conlains clauses on:

. . . . . .

the background to the Eurocode programme the status and teld of application of the Eurocodes national standards implementing Eurocodes links between Eurocodes and harmonized iechnical specif,cations for products

additional information specific to EN 1992-2 Nalional Annex for EN 1992-2.

Guidance on the common text is provided in the introduotion to the Designers' Gui.le to EN 1990 Eurocotle: Basis of Structural Design,l and only background information relevant tô users of EN 1992-2 is given here. It is the responsibility of each nâtionâl standards body to implement each Eurocode part as â nâtional standald, This will con.rprise, without âny alterations, the full text of thc Eurocode and its annexcs as published by the EuropËan Committee for Standardization (CEN, lrom its title in French). This will usually be preceded by a National Title Page and â Nâtional Foreword, and may be followed by a Nâtional Annex. Each Eurocodc recognizes the right of nationâl regulatory authorities to cletermine values related to sâfety matters. Values, classes or methods to be chôsen or determined at national level are ref'erred to as nationally dctermined pârâmeters (NDPs)- Clauses of EN 1992-2 in whioh thesc occur are listed in the Foreword. NDPs are also indicated by notes immediately after relevant clauses- These Notes givc recommended values. It is expected thât most of the MÊmbcr Stâtes of CEN will specify the recommended values, as their use was assumed jn the mâny calibration studies made during drafTing. Reconmended values ate used in this guide, as the Nâtional Annex for the UK was not available ât the time of writing. Comments are made regarding the likely values to be adopted where different. Each National Annex will give or cross-refer to the NDPs to be used in the relevant country. Otherwise the Natiônâl Annex may contain only the followiug:2

. .

decisions on the use of informative annexes, and references

to non-contradictorv complem€ntarv information to assist the user to appll'

the Eurocode, The set ôfEurocodes will supersede the British bridge code, BS 5400, which is required (as

condition of BSI's membcrship of CEN) to be withdrawn by early 2010, ing national standard'. a

as

it is a 'conflict-

DESIGNERS' GUIDE

TO EN I992-2

Additional information specific to EN 1992-2 The information specitc to EN 1992-2 emphasizes that this standard is to be used with other Eurocodes, The standartl includes many cross-references to EN 1992-l-1 and does not itself reproduce material which appears in other parts ôfEN 1992. Where a clause or paragraph in EN 1992-2 modifies one in EN 1992- I -1, the clause or paragraph number used is renumbered by adding 100 to it. For example, ifparagraph (3) ofa clause in EN 1992-l-l is modified in EN 1992-2. it becomes paragraph (103), This guide is intended to be self-contained for the clesign of concrete bridges and therefore provides commentary ôn other parts of EN 1992 as necessary,

The Iorev,orcl lists the clauses of EN 1992-2 in rvhich National choice is pennitted. Eisewhere, there are cross-references to clauscs with NDps in other codes. Otherwise. the Normative rules in the codc must be tbllowed, if the design is to be 'in accordance with the Eurocodes'. In EN 1992-2, Sections I to 13 (actually I l3 becausc cJause 13 does not exist in EN 1992-l - l) are Normative. Of its l7 annexes- onl), its Annex C is'Nôrmative', as alternative approaches may be used in other cases. (Arguably Annex C. wl.fch detnes the properties of reinforcement suitable lbr use with Eurocodes. should not be in Eurocorle 2 as it relates to materiâl which is contained in product standards.) A Nationâl Annex may make Informative provisions Normative in the country concerned, and is itsell Normative in thât côuntry but not elscwhere. The 'non-conlradictory complimentarf infomation' referred to above could include, l'or example, reference to a document based on provisions of BS 5400 on mattcrs not treated in the Eurocodes. Each country can do this, so some âspects of the design of a bridge will continue to depend on whefe it is to be built-

CHAPTER

I

General This châpter is concerned with the general aspects of EN 1992-2, Eurocode 2: Design ol (:oncrete Structure,\. Part 2: Concrete Bridges. Thc material described in this chapter is covered in section 1 of EN 1992-2 in the followine clauses:

. . . . . .

Scope

Normative references Assumptions

Distinction between principles and application rules Definitions Symbols

l.l.

l.l.l.

Clause 1.1 Clause I .2 Clause 1.3 Clause 1.4 Clause 1.5 Clsuse I .6

Scope Scope of Eurocode 2

The scope of EN 1992 is outlined in 2-2/clause 1.1.1 by reference to 2- l- li clause l, l.l. lt is to bc uscd with EN 19q0, Eurocode: Rasîs o-f Strtu:tural Desiga, which is the head document of the Eurocode suite and has an Annex A2,'Application for bridges'. 2-IJ lchuse 1.1'I(2) emphasizes that the Eurocodes are concerned with structural behaviour and that other requirements, e.g. thermâl and acoustic insulâtiôn, âre not considered. The basis for verification ofsalèty and serviceability is the partial factor method. EN 1990

recommends values for load factors and givcs various possibilities for conbinations of âctions. The values and choice of combinations are to be set by thc Nationâl Annex for the country in which the structure is to be constructed. 2-l-l lclause I.l.I (3)P slaLcs that thc following parts of the Eurocode are intcndcd to be used in conjunction with Eurocode 2:

EN 1990: EN l99l:

F\ lqql: EN 1998: hENs: ENl3670:

2-l-l/clouse r

.r. t (2)

2-l-l/clause

t.t.t

(3)P

Basis of structural design Actions on structurrs

Geotcchnicr I tlesign Design of structures fôr eârthquâke resistance Construction products relevant for concrete s[ructures

Execution(construction)ol'concretestructutes

will often be required ftir a lypical concrete bridge design. but discussion on them is generally beyond the scope of this guide. They supplemenL the normative refèrence standards given in 2-2,/clause 1.2. The Eurocodes are conccrned with design and not execution, but minimum standards of workmanship and material specification are required to ensure that the design assumptions are valid, For this reason, 2-l-liclause l.l,l(3)P includes the European standards for concrete products and for the execution of concrete structnres- 2-1-llclause 1.1.1(4)P lists the other pârts ofEC2, These documeuts

2-l-l/clouse

I.t.t(4)P

DESIGNERS' GUIDE

TO EN I992.2

One standard curiously not referenced by EN 1992-2 is EN 15050: Precut Concrete Bridge Elenlents. At lhe trme of rvriting, this document was available only in draft lbr comment, but its scope and content made it felevânt tô precast concrete bridge design. At the time of the review of prEN 15050: 2004, its contents were a rnixture of the following:

. . . . . .

definitions relevant to precast concrete bddges inlbnnative design guidance on items not covered in EN 1992 (e.g, fbr sheal keys) cross-reference to design requirements in EN 1992 (c.g. for longitudinal shear) informâtive guidance duplicating or contradicting normative guidânce in EN 1992-2 (e.g. effective widths for shear lag) cross-reference to EN 13369: Common rules for preutst conuete products requirements for inspection and testing of the finished product.

Comment was nlade that EN 15050 should not contradict or duplicate design requirements in EN 1992. If this is achicvecl in the final version. there will be lirtle Normative in it for the designer to follow, but there mây remain some guidancc on Lopics not covered by EN 1992.

l.1.2, Scope of Part 2 of Eurocode 2 EC2-2 covels structurâl design ofconcreLc bridges. lls format is based on EN 1992-l-l and generally follows the same clause numbering as discussed in the Introduction to this guide. It identifies which pârts of EN 1992-l-1 are relevant for bridge design and which parts need 2-l-l/clouse t.r

.2(4)P

modification. It also adds provisions which are specitc to bridges. clause 1.1.2(4)P states that plain round reinforcement is not covered.

lmportantly,2-lJl

1.2. Normative references References arc given only to other Europeân stândards, all of which are intended to be used as a package.

Formâlly, the Standards ofthe International Organization for Standardization (ISO) apply only if given ân EN ISO designation. National standards for design and for products do not apply if they conflict with a relevant EN stândard. As Eurocodes may not cross-refer to national standards, replacement of national standards for products by EN or ISO standards is in progress, with â time-scâle similar to that for the Eurocodes. During the period of changeover to Eurocodes and EN standards it is possible that an EN referred to, or its national annex, mây not be complete- Designers who then seek guidance Iiom national standards should take âccount of differences between the design philosophies and salèty lâctors in the two sets of documents. Of the material and product standards referred to in 2-l-llclause 1.2, Eurocode 2 relies most heavily on EN 206-1 (for the specification. perfomance. production and compliance criteria for concrete)" EN 10080 (technical delivery conditions and speciflcation of weldable, ribbed reinforcing steel for the reinfblcement of concrete) ând EN 10138 (for the speciflcation and general requirements for preslressing steels). Further reference to and guidance on the use of thcse standards can be found in section 3. which discusses materials.

1.3. Assumptions It is assumed in using EC2-2 that the provisions ôf EN

1990 rvill be followed. In addition, ECl2-2 identifles the following assumptions, some of which reiterate those in EN 1990:

. . . .

Structures are designed by appropriately qualified and experienced personnel and are constructed by personncl with appropriate skill and experience. Thc construction materiâls and products are used as spccified in Eurocodc 2 or in the relevant mâterial or product specitications. Adequate supervision and quality control is provided in làctories, il) plants and on site, The structure will be adequately mâintained and used in accordancc with the design brief.

CHAPTER

.

The requirements for construction and workmanship given in EN 13670 are complied with.

EC2-2 should not be used for tbe dcsign of bridges thât will be executed to specifications othcr than EN 13670 without â careful comparison ofthe respective tolerânce and workmanship requirements. Slender elements in particular arc sensitive to construction tolerances ln

their design.

1.4. Distinction between principles and apPlication rules Reference has to be rnade to EN 1990 for thc distinction between 'Prinoiples' and 'Application Rules'. Essenlially, Principles comprise general stâtem€nts and requirements that musl bc followed and Application Rules are rules that comply with these Principles. Thcre may' however, be other ways to oonply with thc Principles and these methods may be substituted ifit is showu that thcy are at least equivâlent to the Application Rules with respect to safcty, serviceability and dr.rrability. This, holvever, presents thc problem that such a design could not then be deemed to con.rply wholJy with the Eurôcodes. Principlcs are required by EN 1990 to be marked with a 'P' adjâcent to the paragraph number. In addition" Principles can also generally be identifled by the usc of'shall' within a clause, while'should' and'may'are generally used lbr Application Rules, but this is not

completely cônsistent.

1.5. Definitions Refercncc is made to the definitions given in clauses 1.5 of EN 1990 and further bridgespecific definitions are provided. There are sùme signilicant differences in the use of language compared to British codes'l-hese arose frôm the use of English as the base language for the drafting process, and the resulting need to inprove precision of meaning and to facilitate translâtion into other Europcan languages. In pârticulâr:

. .

'action'means a load andlor an imposed delbrnation; 'âction effect' and 'eflect of action' have thc same meaning: any deformation or internal force or moment Lhat results from ân âction.

Actions are furLhcr subdivided intô permânent actions, G (such as dead loads, shrinkage and creep), variable âctions, Q (such as traflic loads, wind loads and temperature loads), and âccidental actions. l. Prestressing, P, is lrcated as a permanent action in most situations. The Eurocodes denote characteristic values of any parametcr with a suliix 'k'. Design

values are denoted with a suflix'd' and include appropriate partial factors- It should be noted that this practice is dillerent from current UK practice in concrete design, whcre material partial factors âte usually included in fbrmulae lo ensure the).'are not forgôtten. It is therefore extremely important to use thc correct parâmeters, duly noting the sullix. to ensure that the maLcrial partial factôrs are included when appropdate.

1.6. Symbols The symbols in the Eurocodcs are all based on ISO standard 3898: 1987.' Each code has its own list, applicable within thât code. Some symbols have more than one mcaning, the particular mearing being staled in the clause. There ate a few important changes from previous practice in the UK. For example, an:c -r axis is along a member and subscripts âre used extensively to distiuguish charactcdstic values from design values. The use of upper-case subscripts for 1 factors for materials implies that the values given allow for two tlpes of lrncertainty, i.e. in the properties of the material and in the rcsistânce model used.

I. GENEML

CHAPÏER 2

Basis of design This châpter discusses the basis of design as covered in section 2 following clauses:

. . . . . . 2.

of F\ l99l-2 in Lhc

Roquirenents Principles of limit state design

Clause 2.1

Basic variables

Clause 2.J

VcrificaLion by lhc partial factor melhod Design assisted by testing Supplementary requirernents lbr fbundations

C lau,se

l.

Chuse 2.2 2.4

Clause 2.5 Clause 2.6

Requirements

2-1-llclause 2.1,1nrakes referelrce to EN 1990 for the basic principles and requirements for 2-l-l/clouse 2.1.1 the design process for concrete bddges. This includes the limit states and combination of actions to consider, together with the lequired per'foruance of the bridge at each limit state- These basic pcrformancc rcquircmcnts arc decmed to be met if the bridge is designed using actions in accordance with EN 1991, combinâtion of actions and load factors at the various limit states in accordance with EN 199û, and the resistances, durability and serviceability provisions of EN 1992. 2-1-l lclause 2,1.3 tefers to EN 1990 tbr rules on design working life, durability and quality 2-I-l/clause 2.L3

managcmcnt for bridges. Design working life predominantly âffects calculatiols on fâtigue ând durability requirements, such âs concrete cover. The lalLcr is discussed in set:tion 4 of this guide. Permanent bridges have an indicative dcsign life of 100 years in EN 1990. For political reasons, iL is likel), rhat rhc UK will adopt a design life of 120 years in the Nâtional Annex to EN 1990 for permanent bridges for consistency with previôus national design standards.

2.2. Principles of limit state design limit sLatc design arc seL out in sccLion 3 ofEN 1990. They are not specific to the design of concrete bridges and are discussed in reference L

The principles of

2.3. Basic variables Actions to consider

2-l-l lclause 2.3.L1(,1.) refers to EN 1991 for acLions to consider in design and also refers to EN 1997 for actions arising from soil ând wâter pressures. Actions not covcred by either of these sources may be included in a Project Specification.

2-l-l/clouse 2.3.t.

r(t)

DESIGNERS' GUIDE

TO EN I992.2

2-1-1/clause 2.3.1.2 and 2-1-l/clause 2.3.1.3 cover thermâl effects ând differential settlements respectively, which are 'indirect' actions. These are essentially imposed deformations

2-l-l/clouse 2.3. r .2(t )

2-l-l/clouse 2.3. r .3(2)

2-l-l/clouse 2.3.t.2(2)

2-l-l/clouse 2.3.t.3(3)

râther than imposed forces. The effects ôf imposed deformations must âlsô âlwâys be checked at the serviceability limit state so as to limit deflections and cracking 2-1-11 clause 2.3.1,2(1) inC, 2-I-llclause 2.3.1.J(2) refer. Indirect actions can usually be ignored Ibr ultimate limit states (excluding fatigue), since yielding of overstressed areas will shed lhe locked-in lbrces generâted by imposcd deformalion. However, a certain amount of ductility and plastic rotation capâcity is required to shed these actions and this is noted itt 2-I-Ilclause 2.3.1,2(2) an C30/37

>

slab

geometry

reduce

by

class

I

by

I

class

I

class

by

I

by by

lnctêâseclass by 2

2

reduce clâss

I

reduce class by I Reduce class by I

Reduce class by I

Reduce class by I

Reduce class

I

I

Reducê clâss

lncrease class by

reduce class

Increâse

XD3/XS2/XS3

2 clâss by 2 > c40/s0 > c40l50 Teduce reouce class by I class by I Rêduce Reduce clâss by I class by I

c30i37 > c35/4s

reduce class by I

Reduce

Reduce class

2

XD2/XSl

>

c40/50 > c4siss by

I

Reduce class

by

I

(position of not affected by construction process) Speciâl

quality

control

Reduce class

by

Reduce by I

Reduce class

by

Reduce class

by

I

by

I

ensured

I The stength class and water/cement ràtio are considered to be related vâlues. The relâtionship i3 subject to a national côde. A special composition (type of cement, ùc value, fine fillers) with the intent to produce low permeability may be 2 ThÊ limic may be reduced by onê strêngth .lass if air encrainment

of more rhân 4%

is applied.

reinforcement, as might occur where fixed length links are used which constrain the posiiion of a layer of reinforcement in one face of the section with respcct to that in the other lace. The recommended valucs for cnrin {lùf are given in Tables 4.4N and 4.5N in EC2-1-l lbr reinfbrcing and prestressing steel respectively, and are reproduced here as Tables 4.4-2 and 4.4-3. They can be amended in the National Annex. Where in situ concrete is placcd against other concrete clements, such as at construction joints, 2-2lclause 4,4.1.2( 109) âllows the minimum concrete cover to reinforcement to be reduced- Thc recommended reduction is to the value required for bond. provided that the concrete clâss is ât leâst C25i30, the exposure time of the temporary concrete surface to

2-Uclouse 4.4. r .2(t 09)

an outdoor environment is less than 28 days, and the intelface is roughcned. This recommendation can be modilied in thc National Annex.

Tâble 4.4-2. Minimum cover requirements, c.".d,., for durability (reinforcing steel) Environmèntal requirements for c.'n.6u. (mm) Exposure class (from

Structurel clâss

sl

t0

s2

s4

t0 t0 t0

s5

t5

s6

20

s3

2-l-llTable 4.1)

xcl

xc2lxc3

t0 t0 t0

t0

t5

t5

20

70

25

t5

25

l0

35

40

45

20

30

35

40

45

50

40

45

50

XD I/XsI

xD2txsz

XD3/XS3

20

25

30

30

35

40

30

37

DESIGNERS' GUIDE

TO EN I992-2

Table 4.4-3. Minimum .over requirements, cmin_du' for durability (prestressing steel) Environmental requirements for

amin,dur

(mm) Exposure class (from 2-l-l/Table 4.1)

xcl

Structural class

sl

s4

t0 t0 t0 t0

s5

t5

s6

20

t5 15 20 25 30 15

xcuxc3

XD I/XS

25 30 35 40 45 50

20 25

30 35

40 45

I

xD2txsT

XD3/XS3 40

30 35

40

40 45

45

45 50

50

55

50

55

60

55

60

65

2-l-l/clouse 4.4. t .2(t

t)

2-l-l/clouse 4.4.r.2(t 3) 2-2/clouse

4.4.t.2(t t4)

2-I-l lclaase 4.4.1,2(11) requires further increases to the minimum covers for exposed agglegate ûnishes, rvhile 2-I-l ldause 4,4.1.2(13) gives requirements where the concrete is

subject to abl'asion. Specific requirements are given rn 2-2lclause 4.4.L2(II4) and 2-21 chuse 4.4.1.2(115J to cover bare concrete decks of road bridges and concrete exposed to abrasion by ice or solid transportalion in running watcr-

2-Uclouse

4.4.t.2(t I5) 2-l-l/clouse 4.4.

t.3(t )P

2-l-l/clause 4.4.t.3(3)

2-l-l/clause 4.4.t.3(4)

in design for deviotion The actual cover to be specified on the drawings, c,,o1n, has to include a ftrrther allowance for deviation (Ac6"u) according to 2-l-lldause 4.4.1.3(1)P such that c.o. : c*;n + Ar:a"u. The value of Aca"" for buildings and hridgcs ma1, be dcfincd in the National Annex and is recommended by ECz to be taken as llmm. 2-l-llclaase 4.4.1.3(3) allows the recommended value of Àr:6"u lo be reduced in situations where accurate measurements of cover achieved can be tâken and non-conforming elements rejected, such as is the câse for 4.4.1 .3. Allowonce

manul-aclure of precast units for erample. Such modif,cations can again be given in the National Anncx- 2-1-llclause 1.4,1.3(4) gives further requirements where concrctc is cast against uneven surfaces, such as directly onLo the ground, or where there are surfâce feâtures which locally reduce cover (such as ribs). The fbrmer will typically cover bases cast on blinding or bored piles cast djrcctly against soil.

54- From Tal

r . .

lor l0Gyear design 1ile. add 2 for concrete grade in exciss of C35/45, deducL

t

for men

Therefore fina1 structural class is 54, From Table 4,4N structural -' -''-'- class ''-: "i4:" c11;030, caioj,' -

of EN t992-l-l tor XC3

25 mm

From bond rxrngiderations.

rn,,n 6

=

20 mm (Lhe ba

r size):

cln6-. max{rn,,n,;.;cqin.a,, -t Ac*,,.1 Acnu..., Ara,,r,n: l0mm} - max{20;25 r 0 0 .0: 10} = 25 -lrr The recommended value of Arde, = l0 mm so: rnom

38

=

cm,n

" Ac6"'

:

25

- l0 -

35 mm

and

CHAPTER 5

Structural analysis This chapter discusses strtrctural analysis as cotered in section following clauscs:

. . . . . . . . . . .

5

oI EN

1992-2

in

rhe

General

Clause 5.1

Geometricimperfèctions Idealization of the strucrure

Ck

Linear elastic anall'sis Lincar elastic analysis with limited redistribution Plastic analysis NonJinear analysis Analysis of second-order elTects tvith axial load Lateral instâbility of slender beams PresLressed members and structures Analysis for some particular structural members

5.l.

se

5.2

Clause 5.3 Clause 5.4 Clause 5.5 Cltruse 5.6 C lause 5 .7 Clause 5.8 Clause 5.9 Clalrse 5 -10 Clause 5.1l

General

2-1-I lclause 5.1.1 ( /) P is a reminder that a global analysis ma1' not côyer all relevant structural effects or the true behaviour so that separate local analysis may also be necessary. A tlpical exanple of this situation includes the grillage analysis of a beam and slab deck

where the lôngitudinaL grillage n]embers have been placed along the a)ds of the main beams only, thus not modelling the effecLs of local loads on tlle slab. The Note to 2-1-liclause 5.1.1(l)P mâkes an imFortant observation regarding the use of shell finite element models: that the application rules given in EN 1992 generally relate to Lhc rcsistance of entire cross-sections to intcrnal forces and lnonlents. bul these stress resultânts âre not determined directly from shell flnite element models- In such cases, either the stresses determined can be integratcd over the cross-section to detemine stress resultants for use with the member application rules or individual elements must be designed directly lbr their stress frelds. 2-1-l lclause 5.1,1f3l refers to Annex F for â method of designing clcncnLs subject to in-plane stress fields only. Annex LL provides a method for dealing with elements also subjected to out of plane forces and moments- The use ôf bôth these annexes can be conservative bccause they do not make allowance for redistribution across a cross-section as is implicit in many of the member rulcs in scction 6 of EN 1992-2. 2-I-llclaase 5.1.7(21 gives other instances where local analysis may be needed. These relaLe to situations where the âssumptions ol beamlike behaviour are not valid and planc sections do not rcmain plane. Examples include those listed in the clause plus any situation where a discontinuity in geomctry occurs, such as at holes in a cross-section. These local analyses can often be carried out using strut-and-tie ana)ysis in accordance with section 6.5 of EN 1992-2. Some situations are covered lully or partially by the application rules

2-l-l/clouse

s.t.t(t)P

2-l-l/clouse

5.t.t(3)

2-l-l/clouse

5.t.t(2)

DESIGNERS' GUIDE

TO EN I992-2

EN 1992- For example, load application in the vicinity of bearings is partially covered by the shear enhancement rules in 2-l-l/clause 6.2, while the design of the bearing zone itselfis parLially covcrcd by 2-l-17clause 6.7. The design of post-tensioned anchorage zones is partially covered by 2-2/Annex J. As a general principle, given in 2-7-1/clause 5.1,I ( 4) P, 'appropriate' idealizations ofboth the geometry and the behaviour of the structure have to be made to suit the partjcular design verification being perlbrmed, Clauses 5.2 and 5.3.2 are relevanL to geometry (covering imperfections and efleclive span). Behaviour relates to the choice of model and section properties. For example, â skew flât slâb could be safely modelled for ultimate limit stâtes elsewhere in

2-l-l/clouse

s.t.t

(4)P

using a torsionless grillage, but this woukl be inappropriatc for serviceability limit state crack width checks; Lhc modcl would fail to predict tôp crâcking in the obtuse qornersSection properties, uith respect to the choice of cracked or un-cracked behaviour, are discussed ir section 5.4 of this guide. Clausc 5.1.2 also covers sheat lag, which affects section stiffness.

2-l-l/clouse

s.t.1(s) 2-l-l/clouse s.t.r(6)P 2-2/clouse

s.t.r(r08) 2-l-l/clouse

5.t.r(7)

2-l-l/douse 5. t .2(t )P 2- I- I /douse

5.t.2(3) to (s) 2-2/clouse

5.t.3(t0t)P

2-1-llclause 5.1,1(5) alnd 2-1-llclause 5.1.1(6)P require each stage of construction to be considered in design as this nray affect the final distribution of internal effects. Timedependent effects, such âs fedistribution of moments due to creep in bridges built itt stagcs, also need to be realistically modelled. 2-27Annex KK addresses the specific case of creep redistribution, which provides some 'recognized design methods' as relerrcd to in 2-2lclause 5.1.1( 108 ). 2-l-I lclause 5.1 ,1( | gives a general stalcment of the types of shuctural analysis that may be used, which include:

. r . .

linear elastic analysis with and rvithout redislribution plastic analysis strut-and-tie modelling (a special câse of plastic analysis) non-linear analysis.

Guidance on when and how tô use these analysis methods is given in the scctions ofthis guide corrcsponding to the relevant sections in EC2. Sôil-structure interaction is a special case of the application of 2- l- l/clause 5- 1 .1(4)P and is covered by 2-1-llclsuse 5.1.2(1)P. It should be considered where jt significantly affects the analysis (as would usually be the qase for in tegral bridge design). 2-1-l lclause 5.1.2 ( 3 ) to ( 5 ) also specifically mention the need to consider the interaction between piles in analysis u'here thev are spaccd centre to centre at less Lhan three times the pile diarneter, 2-2lclause 5.1.3(101)P essentially requires all possible combinations of actions and load positions to be considered, such that the nost critical design situation is identiûed. This has been common practice in bridge dcsign and often necessitates the use of influcnce

surfaces, The Note to the clause allows a National Annex to specify simplified load arrangcments lo minimize the number of arrangements to consider. Its inclusion was driven by the buildings community, where such simplifications are rnadc in current UK practice. The equivalent note in EC2-1-l therefore makcs recommendâtions fbr buildings but none are given for bridges jn Ê.C2-2. The comments made on load factors under clause 2.4.3 are also relevant to the determinâtion of load combinations. 2- I - l/clouse 5.1.4 2-I-l ldaase 5,1.4 requires that second-ordcr cllects should be considered in bridge tlesign and these are much more formally addressed than previously was the case in UK prâctice. DcLailed discussion on analysis for second-order eflccts and when they can be neglected is nresented in section 5.8.

5.2. Geometric imperfections 5.2,l. General (additional sub-section) The term 'geometric iûrped'ections' is used to describe the departures frorn the exact centrelinc, sctting out dimensions specified on drawings that occur duling construction. This is inevitâble as all construction work can onlv be executed to certain tolerances. 40

CHAPTER

5. STRUCTURAL ANALYSIS

2-I-1 lclaase 5.2 ( 1) P requires these irrperl'ections to be considered in analysis. Thc term does not apply to tolerances on cross-scction dimensions. which are accounted for separately in the material factors, but does apply to load position. 2-l-l/clâuse 6.1(4) gives minimum requirements for the latter. Geonetric inperfèctions can apply both to overall structure geo-

metry and locâlly tô members. Geometric imperlèctions can give rise to additional moments from the eccentricity ofaxial loads generated. Thcy are therefore particularly important to consider when a bridge or its elements are sensitive tô second-order effects. 2-1-l lclause 5.2(2)P, however, requires imperfections to be considered for ultimate limit stâtes even when second-order effects can be ignored in accordance with 2-l-liclause 5,8.2(6). Fol short bridge elements, the additional moments caused by imperfections will oftcn be negligibJe and the effects of imperfections could Lhen be ignored in such cases with experience. Imperfections need not be considercd for serviceability limit statcs (2J-l lclause 5.2(3)). 2-2lclause 5.2(104) stûes that the values of imperfections used within 2-2,/clause 5.2 assume that workmanshin is in accordance with deviation class I in EN 13670. If other levels of rvorkmanship are to be used during construction, lhcn the imperfections used in design should be modified accordingly. In general, imperfections can be modelled as either bows or angular dcpartures in members. EC2 generalll' uses angular dcpartures as a simplification. but sinusoidâl bows wjll often be slightly more critical and better reflect the elastic critical buckling mode shâpe. For this reason, sinusoidal imperfections have to be used in the design of ârches to 2-2r'clause 5.2(106). The type of imperfection relevant for member design will depend on the modc o[ buckling. An overall lean to a pier will suffice where buckling is in a swal' mode, which EC2 describes as 'unbraced' conditions in seqtion 5.8. This is becâuse the nron-rents generated b,v tlie imperfection will add to those frôm the additional deflections under load. An overall lean would not, however. sulTice for buckling within a member when its ends are held in position, which EC2 describcs as 'braced' conditions, In this latter casc, a lean of the entire column alone would not induce any rnomenls the "\'ithin column length. It would. however, induce forces in the positional restraints, sô ân imperfection of ihis type would be relevânt for the design of the restraints. A local eccentricity within the member is thcrefore required lor buckling of braced members. This illustraLcs the need to choose the type ofimperfection carefully depending on the effcct being investigated. Further discussion on 'braced' and 'unbraced' cônditions is given in section 5.8 of this guide and Fig. 5.2-l illuslratc' thc dilTerencc. A basic lean imperfection, d1, is defined for bridgcs ft 2-2ltlause 5.2(105l as follows: d1

:

2-2i (s.101)

dno6

2-l-l/clouse 5.2(t )P

2-

l- I /dause s.2(2)P

2-I-l/clouse s.2(3) 2-2/clouse 5.2( t 04)

2-2/douse

s.2(t 0s)

where: 0D

(lh

I

is the basic value of angular departurc is the reduction factor fôr height with ua 2ltA; aa is the length or hcight being oonsidered in metres

:

!

|

(a)Unbraced

Fig. 5.2-1. Effect of geometric imperfecrions in isolated members

4l

DESIGNERS' GUIDE

TO EN I992-2

The lower linril for a6 given in EC2-l-l was removed in EC-'2-2 to avoid excessive imperfèctions in tâll bridge piers- The value of d6 is a nationally determined pârâmeter but the recommended value is I /200^ which is the sârne âs previously used in Model Code 90.ô 2-l-l/clouse 5.2(7) 2-1-llclaase 5.2(7.) allows imperfections in isolaled membcrs to be taken into âccôunt either by modelling them directly in the structurâl system ôr by replâcing them with ôquivaIent forces. Thc latter is a useful alternative, as the same n.rodel can be used to apply different imperfections, but the disadvantagc is rhat the axial forces in members must ûtst be known belbre the equivalent forces can be calcr.rlated. This can become an iterative procedure. These âlternatives are illusLrated in Fig. 5.2-1 for the trvo simple cases ofa pin-ended strut and â

cantilever. They are:

(a) Application of ân eccentricity, sj 2-l-1,/clause 5.2(7) gives the follorving fomula for the imperfection eccenLricity: e;

-

where

2-t-t t(s.2)

0lo12 16

is tbc effective length.

For the unbraced cantilever in Fig. 5.2-l(a). the angle ol lean from 2-2l(5.101) leads directly to the top eccentricity of ei - 011 - 0lsl2, when /0 : 21 (noting that iç > 2/ for cântilever piers wiLh real foundations as discussed in section 5.8.3 of this guide). For the braced pin-ended pier in Fig. 5.2- l(b). partially reproducing 2-1- 1i Fig, 5.1(a2), the ecccnlricjty is shown to be applied predominantly as an end eccentricity. This is not in keeping with the general philosophy ol applying imperfections as angular deviâtions. An altemative, therefbre, is to apply the imperlectiôn for the pin-ended câse as a kink over the half wavelcngth of buckling, based on two angular deviations, dt, as shown in Fig. 5.2-2. This is then consistent with the equivalent forse system shown in Fig. 5.2-l(b). It is also the basis ofthe additional guidance given in EC2-2 fbr arched bridges where a deviatron a : 0rl/2 has Lo be attributed to the lowest symmetric modes as discussed below. This melhod of application is slightl) less conserrative. 2-l-l iExpression (5.2) can be misleading lbr effective lengths less than the height of the member, âs the eccentricity c1 should rcally apply over the hall wavelength of buckling, /n. This interpretation is shown in Fig. 5.2-3 for a pier rigidly built in at each end. It leads to the same peak imperfection as for the pin-ended case, despite th€ iact the effective length lbr the built in case is half that of the pinned case- This illustrates the need to be guided by the buckling mode shape when choosing inperfections.

(b) Application of a transverse force, f{, in the position that giyes maximum moment The following formulae for the imperfection forces to apply are given in 2-l-l/clause 5.2(7): H; = H;

QtN

lbr unbraced members

(see

Fig. 5.2-1(a))

2- 1-

- 20tN for blaced members (see Fig. 5.2-1(b))

2-l - 1(5.3b)

wbere N is tl.re axial load.

I

I

et=0

2

Fig. 5,2-2. Alternative imperfection for pin-ended stTut as ân angular deviâtion

42

l/(s.3a)

CHAPTER 5. STRUCTURAL ANALYSIS

lo

=

I

e

=2e

=

0ll2

112

I

Sinusoidal

impedection

Angular imp€rTection

Fig. 5.2-3. lmperfection for pier buik in ar borh ends

Where the imperfections are applied geometrically as a these forces arc direotly equivalent to the imperfection.

kink or lean

as discussed above,

5.2.2. Arches (additional sub-section) 2-2lclause 5,2(106) covers imperfections for archcs for buckling in plane and out of plane.

2-2/clouse

s.2(t06) lnflone buckling For in-plane buckling cases where â symmetric buckling modc is c tical, for example fiom arch spreâding, a sinusoidal in.rperf'ection of a - 0û12 hâs to be applied as shown in Fig. 5.2-4. This magnitude is derived by idealizing the actual buckling modc as a kink made up from angular deviations, 91, despite the clause's recommendâtion thât the imperfèction be distributed sinusoidally for arch cases as discussed above. Wherc an arch does not spreâd signiflcantly, the lowcst mode of buckling is usually

anti-symmetric, as shown in F-ig. 5.2-5. In this case. the mode shape, and thus imperfection, can be idealized as a saw toolh using the san.re basic anguJar dcviation in conjunction with the reduced lengtlt Ll2 relevânt tô the buckling môde. The imperfection therelbre becomes a - AlLf 4. Once again, EC2 requires the iÛrperfection to be distributed sinusoidally.

Out-of-plone buckling

For out-of-planc buckling, the same shape of imperfection as in Fig- 5.2-4 is suitable but in the hodzontal plâne.

Kink imperfection

Sinusoidal imperfeclion

Fig. 5.2-4, lmperfection for in-plane buckling with spreading foundarions

43

DESIGNERS' GUIDE

TO EN I992-2

Kink impedection

Sinusoidal impertection

Fig. 5.2-5. lmperfection for in-plane buckling with 'rigid' foundations

5.3 ldealization of the structure 2-l-l/clouse s.3. r(t )P

2-l-l/clouse 5.3. r (3), (4), (s)

ond (7)

5,3.1 Structural models for overall analysis 2-1-llclause 5,3,1(1)P lists typical elements comprising a structure and stâtes that rules are given in EC2 to covcr thc dcsign ol these various elements. While detailing rules are provided by element type in section 9 of EN 1992-2, rules lbl resistances arc generally presented by resislance type rather than by element typc in section 6. For example, 2-21clausc 6.1 covers the design of sections in general to cômbinâtions of bending and axial force. These rules içply equally to beams, slabs and columns. 2-I-I lclause 5.3.1( 3 ), ( 4), ( 5 ) and (7) provide definitions of beams, deep beiLms, slabs and columns. The definitions given are self-explanâtory and are otien useful in delining the detailing and analysis requirements lbr the particular element. For cxample, the distinction n.rade between a beam and deep bcam is useful in determining the appropriate verification method and detailing rules. A beam can be checked fol bending, shear and torsion using Z-27clause 6.1 to 6.3, while rleep beams are more appropriately treated using the strut-and-tie rules of 2-2iclause 6.5. No distinction is, however, made between beams with axial force and columns in EC2 in cross-section resistanc€ design. A distinction, however, rematns necessary whcn sclccting tbe most appropriate dctailing rules from section 9. Sometimes it may be appropriate tô treât pârts ûf â beâm, such as a flange in a box girder, as a wall ôr column for detailing purposes! as discussed in section 9.5 of this guide.

5.3.2. Geometric data 5.3.2.L Effeaive width of flonges (oll limit stotes) In widc liangcs, in-plane shear flexibility leads to a non-uniform distribution of bending stress across the flange width. This effect is known as shear lag. The sftess in the flange adjacent to the web is conscqucnLly found to be greâter than expected tiom section analysis with gross cross-sections, while the stress in the flange remote from thir web is lower than expecled. This shear lag also leads to an apparcnt loss ol stiffness of â section in bending. The determination of thc actual djstribution of stress is a complex problen which can, in theory, be determined by finite element ânalysis (with appropriate choice of elements) if realistic behaviour oI reinforcen.rent and concrete can bc modelled. For un-cracked concrete. the behaviour is relatively simple but becomes considerably more complex with cracking

of the concrete and yielding of the longitudinal 2-l-l/clouse s.3.2. t (t )P

44

rcinforcernent, which both help to

redistribute the stress across the cross-section. The ability of the transverse reinforcement to distribute the fbrces is also relevant. 2-IJ lclause 5.3.2.1(1,lP accounts for both thc loss ofstillness and localized increase in flange stresses by the use ofan effective width offlange, which is less than the actual available flange width. The effective flange width concept is artifioial but, when used with engineering bending theory,leads to uniform stresscs across the whole reduced flange width that are equivalent to the

CHAPTER 5. STRUCTURAL ANALYSIS

peâk values adjacent to the wcbs in the 'true' situation. It lbllows from the above that if finite elemenL modelling offlanges is performed using appropriate anâlysis elements, shear lag will be taken into âccount automatically (the accuracy depending on the matedal properties specilied in analysis as discussed above) so an effective flange need not be used. Thc rules for effective width ma1' be used for flanges in members other than just 'T' beams as suggested by 2-l-l/clause 5.3.2.1(l)P; box girders provide an obvious addition. The physical flange width is unlikely to be reduced for many typical bridges, such as precast beam and slab decks where the beams are placed close together. The effect of shear lag is greatest in locâtions of high shear where lhe force in the flânges is changing rapidly Conscquently, eflèctive widths at picr sections will be smallel than those for the span regions. 2-l-1,/clause 5.3.2-1(1)P notes that, in addition to the considerations discussed above, effective width is a function of type of loading and span (which affect th€ distribution of shear along the beam). These are characterized by the distance between points of zero

2-l - l/clouse bending moment. 2-l-llclause 5.3.2.1(2) and (31, together with 2-1-tiFigs 5.2 and 5.3 flange 5.3.2.1(2) ond (3) (not reproduced). allow cllcctive widths to be calculated as a function of the actual width and the distance, 10, between points of zero bending moment in the mâin beam adjacent to the location considerecl. This lcngth actually depends on the load case being considered and the approximations given are intended to save the designer from having to dctcrmine actual values of /0 for each load case. Thc totâl effective wjdth acting with a web, ô"p, is given as follorvs:

à"n:Iô"n.i +à, 55 MPa

(D5.6-6) (D5.6-7)

These are again more onerous than corresponding liD.rits in

2-l-l/clouse 5.6.3(3)

EC2-l-l

due

to the greater

potential dcpth of bridge beams. The plastic rotâtion capacity has to be compared with the actuâl rotation at the hinge implied fronr the global analysis. 2-1-I lclaase 5.6.3(3) statcs only that the rotation should be calculated using design values for materials. One possibility is tô use elâstic anâlysis to deten ne the fraction of Lhe load, a, at which the first plastic hinge forms (that is when the moment resistance is just exceeded at the hinge location) and Lhen to appl), Lhe rcmaining load increments to a model with the plastic hinge modelled as a hinge without any moment rigidity. This determination of the plastic rotation. ds, is shown in Fig. 5.6-4. The question thcn arises as to what stiffness to use lor the âs yet unyielded pârts of the bridge in Fig. 5.6-4 for a uniformly distributed load. If gross elastic cross-section properties are used, bascd on the design valie of the qoncrete, Young's modulus É'"6 4o'/1s with

-

^tce.:1.2 (fron 2-1-1,/clausc 5.8.6(3)), this will overestimate the 1eâl stiffness ând therefore undercstimate rotation at the hinge. A rcasonable approximation might be to use fully cracked properties, again based on the design value of the concrete, Young's modulus Failure of steelbelore concGte (small xu/d) bêfore ste€l (lârss

Simultâneous lailure of concrete and sleel

Reinforcemenl

Fig. 5.6-3. Possible srrain diagrams at failure

54

.dd)

CHAPTER 5. STRUCTURAL ANALYSIS

(aJ l,'lomenls under unilorm load beJorc and

Defleciions under (1

afterlirst hinge formation

- r)W

{b) Dellections ailer firsl hinge lormâtjon

Fig. 5.6-4. Rotation at plastic hinge for two-span bfidge beam wilh uniformly distributed loâd, W

E* - E"./16

and steel modulus 8". This could still overestimate stiffness in the n.rosl highly to lbl.ln, but would undcrcstimate stiffness everywhere else. To minimize âny un-conservatism in this respect, it is desirable to base stressed arcas where another hinge was about

the section bending resistance here on the reinforcement diaglam with thc llat yield plâteâu in 2-l-liFig- 3.8. The 'real' behaviour can only be obtained through nonlinear analysis considering the 'real' mâtefial behâviour. 5.6.3.2. Check of rotntion capocrty when negleding imposed deformotions at the ultimote limit stote (odditional sub-seaion) A similar caLculation of rotation capacity and plastic hinge rotation has

to be mâde when

elastic analysis is used, but the effects of imposed deformations âre to be ignored at lhe ultimate limit staLe as discussed in section 2.3 of this guide. The rotation caused by, for example, settlement cân be checked on the basis ol the angular chânge produced il the setllenent is applied to a model with a hinge ât the locâtion where rotation capacity is being checked in a similar way tù that in Fig. 5.6-4. In general (1or differential temperature or dill'erential shrinkâge for example), the plastic rotations can be obtained from the'free' displacemcnts as shown in Fig. 5.6-5, or by again applying the free curvatures to a model with hinged supports, In verifying the rotation capacity in this wây, it should be recognized that even the usc of elastic global analysis may pJace some demands on rotation capacily see section 5.4 ofthis guide. The entire plastic rotation câpâcity may not therefore be available for the above check. so it is advisable to leave some margin betu,een plastic lotation oapaciLy and plastic rotation due to the imposed delormation. Generally. the proportion of tot:rl rotâtion capacity required lbr this check will be small and adequate by inspection. It has, in any case. been common practice in tl.re UK to ignore in.rposed detbrmations at thc ultimate limit state without an explicit check of rotaLion capacity. EC2, however, demands more caulion, particularly as there are no restrictions on designing reinlbrced concrete bcams with over-reinforced behaviour.

Fig. 5.6"5. Plastic rotations caused by imposed culatures such

as

from differentiâl temperature

55

DESIGNERS' GUIDE

TO EN I992-2

If the limits in (D5.6-l) or (D5.6-2) are met, the effect of imposed defbrmations could automâtically be neglcctcd. Alternatively, 2-27'Expression (5.10a) or 2-2lExpression (5.10b) could be used where it wâs only necessary to shed 15% of the moment (or other limit as specified in thc National Annex). 5.6.4. Strut-and-tie models Anal),sis with strut-ând-tie modeLs is a special case of the application of the lower-bound theorem of plasticity. Strut-and-tie models have not been commonly used by UK engineers, mainly because ôf the lack of codified guidancc. Whcn such modcls have been used, as for example in tl.re design of diâphragms in box girder bridges, there has not been a consistent appfoâch used for the chcck of the strength of compression struts and nùdcs. EC2 now provides guidance on these limits, but it is far from a complete guide in itself ând reference can uscfully be made lo texts such âs reference 8 for more background. There will often be difûculties in applying specific rules from EC2 for compression limits in nodes and sLruts, and engineering judgement will still be needed. However, the use of strut-and-tie modelling is still very valuable. even when used simply to detemine the locations and quantities of reinforcement. 2-1- I iclause 5.6.4 gives general guidânce on the use of strut-and-tie models, Strut-and-tie models are intended to be uscd in areas of rlon-linear strain distribution unless rules are given elsewhere in EC2. Such exceptions include beams with short shear span which are covered in section 6.2. In this particular case, the use of the strut-and-tie rules in prelèrenc€ to the test-based shear rules would lead to a very conscrvative shear rcsistance based on concrete crushing.

2-l-l/clouse

Typical examples where non-linear strain distribution occurs are areas where thcre are concentrated loads, corners, openings or other discontinuities. These areâs are often called 'D-regions', rvhere'D' stands lbl discontinuity^ detail or disturbance. Outside these areas, whele the strain distribution again becomes linear, stresses nray be derived from traditional beam or tluss theory depending on whether the concrete is cracked or uncrâcked respectively. Thesc areas are known as 'B-r'egions', where 'B' stands lor Bernoulli or beam. EN 1992 also refers to them as 'contiûuil),' regions- Some typical l)-regions, together with their approximate extent, are shown in Fig. 5.6-6. Strut-ând-tie models are best developed by following the flow of clasLic force from the B-region boundaries as shown in Fig. 5.6-6, or from other boundary conditiôns, such as support reactions, when the entire system is a D-region (such as a deep beam). 2-1-llclause 5.6.4(1,1 indicates that strut-and-tic modelling

s.6.4(t )

can be used

2-l-l/clause s.6.4(3)

2-l-l/clquse 5.6.4(s)

2-l-l/clause 5.6.4(2)

56

lbr both continuity and discontinuity regions.

Strut-and-tic modelling makes use of the lower-bound theorem of plasticity which states that any distribution of stresses used to resist a given applied loading is safe, âs long âs equilibrium is satisfied throughout and âll parts ôf the structure have stresses less than 'yield'. Equilibrium is a fundamental requirernent of 2-1-l lclause 5.6,4(j).ln reality, concrete has limited ductility so it will not always be safe to design any arbitrary force system using this philosophy. Since concrete pemrits limited plastic deformations, tbc force system has to be chosen in such a way as Lo not exceed the deformation limit anywhere before the assumed state of stress is reached in the rest of the structure. In practice, this is best âchieved by aligning struts and ties to follow the internal forces predicted by an un-cracked elastic analysis. To this end. it may sometimes be advisable to flrst model lhe region rvith finite elenlents to establish the flow of elastic forces before constructing the strut-and-tie model. This is the basis of 2-1-I lclause 5.6.4(5). The advantage ol closely lbllowing the elâstic behaviour in choosing a model is that the same analysis cân then be uscd for both ultimate and scrviccabi]ity limit stâtes. 2-1-1/ clause 5.6.4(2) requires orientation of thÊ struts in accordance with elâstic theory. The stress limit to use at the serviceability limit state lbr clack control may be chosen according to bar size or bâr spacing, as discussed in section 7 of this guidcFailure Lo follow the elastic flow of force and overly relying on the lower-bound theorem oan result in resistance being ovcrestimaLed. This can occur for example in plâin concrete

CHAPTER 5, STRUCTURAL ANALYSIS

É!

r-:--T-

lK"'::--i I -^111 | lj 'i momentl -|

J'

-aarl

Lcros,ns

I

H

tr (a)

*"1

(b)

t I

ffi=o*n." Fig. 5.6.6. Examples of strut-and-tie models and extenr of D-regions

under a ooncentrated verticâl load if the load is appJied to a small width. Neglecting the transverse tensions generaled, as shown in Fig. 6.5-1, by assuming that the load docs not splead across the section, can actually lcad to an overestimate of rcsistance. This pârticulâr casc is discussed at length in section 6.7 of this guide. Similar problems can arise if the âssumed stmt angles dcpart significantly liom the elastic traiectories, as discussed in section 6,5.2. Expericnce, however, shows that it is not alwa!,s necessâry to rigidly fbllorv the elastic flow

of force at the ultimate limit state. The most obvious example is lhc truss model for reinfbrced concrete shear design, which permits considcrable departure of both reinforcement and compression strut directions liom the principal stfess directions of 45' lo thc vertical at the neutral axis. Generally, it is desirable to follow Lhc lines of force from elastic analysis, unless experience shows thât it is unnecessary to do so. Often there appears to be a choioe of model even when the elastic load paths have been lbllowed. In selecting the best sôlution, it should be bornc in mind thât the loâds in the real structure will try to follow the paths involving least force and defbnr.ration. Sincc reinforcement is much more dcformable Lhan the stiff concret€ struts. the hest model will minimize the number and length of the ties. An optimization criterion is given in reference 8. This requires tbc minimization of the internal strain energy, ! F;Z;e,n;, where:

: force in strut or tie I : length of strut or tie i ctrli mcan slrain in slrut or Iie / l"r

Zi

The terms for the concrete struts Çan usually be ignored as their strains are usually much smaller than those of the ties. As â guide to constructing models, in highly stressed node

57

DESIGNERS' GUIDE

TO EN I992-2

regions (e.9. near concentratcd loads) thc compression struts and ties should form an angle of about 60" and not less than 45'. Detailing of nodes is also inportant and guidance is given in sectjon 6.5-4. This guidance also applies to the design oflôcâl areas subject tô côncentrâted lôads even if the design is not performcd using strut-and-tie analysis,

5.7. Non-lineâr analysis 2-l-l/clouse

s.7(t)

2-l-l/clouse 5.7(4)P

2-

l- I/dause

5.7(2)

2- I-l /douse 5.7(3)

Non-linear analysis of concrete bridges, in the contexl o[ 2-1-l lclause 5.7fl), âccôunts for the nonlineâr nature of the matedal properties. This includes the effects of cracking in concrete and non-linearity in the material stless strain curves. The analysis thcn ensures both equilibrium and compatibility of defcctions using these matedal properties. This represents the most realistic representâtion of structural behaviour, provided that the material propcrtics assumed are realistic. Such an analysis may thcn be used at SLS and ULS. Nonlinear analysis may also model non-linearit1, in suucturâl response due to the changing geometry caused by deflections, This is irrportant in the design of elements where second-order effects arc signitcant, as discussed in section 5.8 of this guide. It is fbr second-order calculation for slender members, such as bridge piers, that non-linear analysis is often particularly beneficial, as the simplified alternatives are usually quite conservative. This is discussed in section 5.8.62J-l lclause 5.7(4)P requires the analysis to be pedbrmed using 'realistio' values ofstructural stiflnesses and a mcthod which takes account of unccrlainties in thc resistanse model. The most 'realistic' values âre the meàn propefiies as these âre the prûperties expected to be found in the real slructure. lf mean strergths and stifinesses are used in analysis, rather than dcsign values, there is ân âpparent incompatibility bctween local section design and overall analysis. The rationale often given fbr this is thât materiâl factors are required to account for bad workmanship. It is unlikell' rhat such a severe drop in quality would affcct the matedals in large pârts of the structure. It is more likely that this would be localized and, as such, would not signitcantly allcct global behaviour, but would alïect the ability of local sections to resist the internal effects derived from the global behaviour. This is noted by 2J-l lclause 5.7(2), which requires local cdtical sections to be checked for inelâstic behaviour. For thc buckling ofcolumns. however, it could be argued that even a rclatively small area of'design mâteriâl' ât the critical section could significantly increâse deflections and hence thc moments at the critical section. Care and experience is therefore needed in selecting appropriate mâterial properties in diflerent situations. 2-1-llclause 5.7(3) generâllJ," permits load histories to be ignored for structures subjected predominantly to static load and all the aclions in a combination may then be applied by increasing their values simultaneously.

5.7.l. Method for ultimate limit states (additional sub-section) lbrulti-

2-2/clouse

2-2lclause 5.7(105) makes a proposal for thc properties to use in nonlinear analysis

5.7(t 0s)

mate limit states and provides a sâfety formât. The proposed method, which may be amended in the National Annex, essentially uses mean propcrtics for steel and a reference strength fbr concrete of 0.84/"ç. The material stress strain responses are derived by using the strengths given below in conjunction with the non-linear concrcle stress-strain relationship given in 2-l-1,/clause 3.1.5 (Fig. 3.2), the reinforcement stress strain relationship for curve A in 2-l-liclause 3.2.7 (Frg.3.8) and the prestressing steel stress .strain relationship for curve A in 2-1-llclausc 3.3.6 (Fig. 3-10):

2-l-1/Fig. 3.2 fbr concrete: 2-1- 1/Fig. 3.8

lor reinforcemcnt:

t.t !!

/l-

is replaced uy

dr

is replaced by I -1{.1

%

7"1

icdr is replaced by 1.lkdr.

2-l-l,/Fig. 3.10 for prestressing stccl:

58

lr1 is replaced by l.llrp

CHAPTER 5. STRUCTURAL ANALYSIS

These modifications are necessary to mâke the mâteriâl characteristics compâtible with the

verilication format given. which uses a single value of mâterial saf'ety factor, 1b,, to cover concrete, reinfblcement ând prestressing stecl. This can be seen as followsFor reinforcemcnt lailure, the strength used above corresponds approximately to â mean value and is taken asl,,'' - 1 .lf,ç, and since the desrgn ultimate strength isf,1 : I,r/1.l5 the cquivalent matedal fàctor to use withdn : 1. 1ôr is 1cy = L l x l.I5 - 1.27.

For

the reference strength for analysis is tâken âs /l:

concrcLc failure,

1.1

x

(1.15/ l.5y;k : 0.843ir, and since the /eslpr ultimate strength is/;d : l;k/ 1.5 the equivalent material factor tû use withl : 0.843Ik is a/o, - 0.843 x 1.5 = 1.27. The concrele reference strength is thercfore not a mean strength but one that is necessar)' to give the sâme matefiâl lactor for concrete and steclThc above concrete strengths do not includc the factor o"", which is needed for the reslstânce. To adopt the same global safèty factor âs above and include &"" in the resistance, it nust also be included in the analysis. This inplies that ./1," should be replaced by l.l (ir,/1)rr"" /lç in 2-l-l/Fig. 3.2, contrary to the above. This change appeared to have been agreed by the EC2-2 Project Team on severâl occasions but failed to be made in the final text. Although not stated, lurther modification is required to the stress-strâin curves where there is signiflcant creep- The stress strain curve of 2-l-l/Fig. 3.2 for concrete (and Figs 3.3 and 3.4) is l'ol short-term loading. Creep will therefore have the effect of making the respônse to long-term actions Ûrore flexible. This cân conservâtively be accounted fbr by multiplying all strâin vâlues in the concrete stress strain diagram by a factor (l + (rcf), where {"1 is the effective creep ratio discussed in section 5-8.4 of this guide. This has the ellect of stretching the stress strain curve along the strain âxis, If the analysis is performed until the ultimatc strcngth is reached at one location (based on the âbove material properties) such that the mâximum combinâtion of actions reached is 4"6 = rr;,1(1'6G +7qQ), where 1GG +1aO is the applied design combir.ration of aclions and ag4 is thc loàd factor on these design actions reached in the analysis, then the sâfety verilication on load would bc:

^ 1..V + 1t,U



^."r.i"u""

=

H., -

r

Y-li.

,-r-ri(s2i)

rcquircd

0.125 < 0.2 as rcquircd 6.125

t o* rtrluwo.r'.d

4L' o,!ë

-

2-l-l ^5.24,

example 5.8-2)

0.088 (for p >

0,002)

2-r-l,(5.22)

The concrete modulus

î11,:,;,1.*Ï;-îîilî'"

",lffilïïl-llJïï:] l:=

ing load

*,Éi::., ..ri.-.,.

iïiq

-

ll l;llîi

l

72eo3kN

75

DESIGNERS' GUIDE

TO EN I992-2

1366 kN

Fig. 5.8-9. Pier loading for Worked example 5.8-3

5.8,8. Method based on nominal curvature 2-l-l/clouse 5.8.8. t ( r)

2-l-l/douse s.8.8.2(t )

The method of2-l-1i clause 5.8.8 is based on similar theory to the slender column method in BS 5400 Part 4'' in Lhat an estimate of the maxirrurr possible curyature is used to calculate the second-ôrder momenl 2-l -l lclaase 5.8,8,1(1) notes thal thc method is primarily intended lbr use with members thât cân be isolâted frôm the rest of the bridge, whose boundary conditions can be represented by an effective length applied to the membcr. The

first-order moment, including that from initial imperfections, is added tô the moment from the additional naximum deflection according to the expr€ssion in 2-I-llclause 5,8.8.2(1). (This differs from thc method in US 5400 where initial imperfections are not considered.)

My6-

Mns4+ M2

2- l

-

1/(s.31)

wherc:

ozd M2 M

is thc first-order noment. including the efTect of imperf'ections is the estimatcd (nominal) second-order moment

The additional second-ordcr momcnl is givcn as follows: M2

-

Ns4e2

2-1-1(5.33)

M2 is determined by calculating r:2 from Lhe cstimated curvalurc at failurc, l//, according to the fornrula, e,= (.1lr)li/c. c depends on the distribution of curvâtufe in the column, The definition of r dillers from c6 used in 2-l-ticlause 5.8,7 as it depends on the shape of tlle

76

CHAPTER

5. STRUCTURAL ANALYSIS

totâl curvâture, not just lhe curvature from first-order moment- For sinusoidal curvâture, c: I and for conslant curvature, c - 8 as discusse 0.5 for /11 >

60 MPa

2-2l(6.10.bN)

This factor is discussed l'urther below. is a fâctor uscd to take account ôf comprcssion in the shear ârea ând its value may be given in the National Annex. 2-2,/clausc 6.2.3(103) recommends the

following values: 1.0

for non-prestressed structures

(l +a"o/,{a)

for 0 < o.o < 0.25Id

2-2l(6.1 1.aN)

1.25 foL 0.25À < o-- < 0,51..,

2.5(l

-

o"ol

f")

for 0.5/".1 < o"o

2-2l(6.11.bN)

I

1.016

2-2l(6.11.cN)

u'here a"o is the mean compressive stress (tneâsured positive) in the coucrete. This lactôr is discussed lurther below.

t42

CHAPTER

6, ULTIMATE LIMIT

STATES

Thus substituting for o. in equation (D6.2-5) gives Lhc cxpression in 2-l-liclause 6.2 3(4): 2-

l- 1/(6.14)

This exprcssion cllectively gives the maximum shear tesistance of a section belore lailure occurs due to clushing ofthe concrete struts and is therefore designâted tr/s,i.,nu, in lhe codeFor practical ranges of axial load in prestressed members, the recommcnded values for n.* gcnerally result in increasing the maximum shear stress liniit by up to 25%. If this is considered together with the recommended beneflt from r,lwhere the links are not fuJly stressed, the ûraximum shear limiL can bc around twice that permitted by BS 5400 Part 4' and potentially unsafe. Iiinclined links are used at 45', tr/p1..,, can approach four times the equivalcnt limit in BS 5400 Part 4! .t"o, z1 and inclined links are discussed further belorv. For bcams with vertical links. the âuthors âle unawar€ of test results which suggesl the additional increase for crushing resistance given by û"* is unsafe on its own. It is nôt, however, supported by lhe rulcs for mcmbrane elements in 2-2lclausc 6.109(103), where compression reduces the maximum resistânce to shear, The physical model behind i,1 is also hard to understand as it promotes the design of over-reinforced behaviour in shear, which may invalidate the plastic assumptions behind the truss model which relies on t'otation ol the rveb compression diagonals. The justilicatiou probably relates to limiting the tensile enhancing the strut compression limit, as discussed in soction 6-5.2 of this guide. In conjunction with the use of the upper valucs of o.*, the EN 1992-2 results may become unsafè, particularly where the webs are designed to be very slender becausc of thc high permissible stresses. Slender webs (with high height to thickness ratio) may exhibit significant second-order out-of-plane bending effects which would lead to tàilure at shear forces less than the values based on uniform crushing. There is limited test evidcncc herc and the UK Nâtional Annex thereforc imposes an upper limit on sheâr resistânce based on web slenderness to safeguard against this. z1 is also reduced. For beams with inclined links, no test results are available to check the high predicted valucs of tr/p4,.,u"- One result is available in refèrence 12 but this failed prematurely below the loâd expected from EN 1992-2. It is clear, thereforc, that the reconmcnded values in EN 1992-2 should be used rvith greât caution. The UK National Annex therelbre reduces both r". (to 1.0) and 11 where inclined links are used. Where webs carry signilicant transverse bending in addition to shear, these high shear crushing resistances mây nôt be achievable due to the interaction of the compression lields strains acting skew

to the struts and thcrcby

section 6.2,6 rel'ers. Notwithstanding the above, the examples in this guide use the EN 1992-2 recomuended values for o", and rr1, but with an upper lilnit of 1.0 imposed ôn.ù",. For vertical shear reinforcen.renl, a:90" so cot.r::0 and 2- l - I /Expression (6.14)

simpliûcs to:

rna.,,,r, ôc$..'!v--l

.

/Ld

cold

I

---:------------ Lr.sa$Z/'tl.J /

;i

d\

uano: -/ "

giving the expression in 2-2lclause 6.2.3(103): y'na.-u.

a"nb*zu1f"6

(cotd+tand)

2-21(6.e)

Additionâl longitudinal reinforcement and the shill method The design of the longitudinal reinforcement in the region cracked in flexure, or where shear reinforcement is provided, is aflected by the shear design of members. 2-2lclaute 6.2.3(107) requires additional longitudinal rcinforcement to be provided in certain regions over and abovc that required for the flexural design, âs discussed below. The free bodl' ABD in Fig. 6.2-5 shows a portion of a smeared truss within a membet, ilrcludinq both horizontal and vertical forces. As before. the compressive strut is chosen to

2-2/clquse 6.2.3(r 07)

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DESIGNERS' GUIDE

TO EN I992-2

.1,I

.t... ...1 i...

I

i. 1..

'r.l

l

1...

Fig. 6.2-5. Free body diagram for truss analogy àÇt

at an angle

I to the horizontal and th€ shear reinfôrcement is inclined at an angle legs ol shcar reinforcement crossing the plâne A-D is

of a. Again, the number of -F

0)/r

sin d sin û.

"sin(0 First, consider vertical equilibrium: I/

- I F,-, where Fve,' :

,F

sin o- Thus

. | - f stno-:sinfd n) Êsin(A | .l) ssmf./sinû lsind

(D6.2-6)

Second. consider holizontal equilibrium: C

:

Z+

I.Fh.,,

z sin(d o) _ _ Fz sin(d + ,,) / J /'COSO - l | -:COl J Sln t^-sln a .r sln {/

(

uhere fhor'

:

F cos

q. Thus (D6 2-7)

{}

Finally- consider moment equilibriun about point A:

to' d r \- rn.,,, ,/_ : Therefore, substituting for I F"c., from equation (D6.2-6) and M.

-

T= .-

F.",, 1 \/_.*tr

t

(D6.2-7) gives:

+ (t) z cot 0 Fz sin(P + r'r) +-cot(lrsind Jslno which simplifies to: Fz srû10

I

F'6o;, from equation

z

I

o) ^ : +d coLa) îl,r sln -tcorpt Substituting f liom equatiôn (D6.2-6) into equation f'--2 rin{d

M -I:

(D6 2-8) (D6.2-8) gives:

M:TzrvlGoto+cotq) and rearranging gives: Tz

:

Bùl M

M

-

-

v

icotë

- Vicotu:

M

-

Vz corl

+

rl(cotg

cot cr)

Vzcot0 is the bending moment al section CD (McD) therefore:

Tz: Mco+f

r:Ysp

(cotd

-

coto)

+t(pôtg _ corâ)

(D6.2-e)

hense thc longitudinal reinforcement at section CD should be designed for the force from the bending moment at CD, M.o/;, plus V l2(,cot? cota). This is equivalent to designing for

an ellective bending moment of:

M : Mco + where a,

144

va

côIo) : M(,D + 7(.cot? -

:0.5:(cot9

cot n).

Vûj

(D6.2-10)

CHAPTER 6, ULTIMATE LIMIT STATES

Equation (l)6.2-9) is the hasis of the design tensile force given in 2-2/clause 6.2.3(107): Tensile force

=

Muo z

+

A4o with

AF,6

:0.5t

sa(cotÉl

-

cota)

2-21(6.r8)

For âny section, Mr 1f z I LF.T should not be taken as greater than MEa,-,*/2, as implied by the shift method below, where Ms6.."* is thc maximum mornent along the beam (for either the sagging or hogging zones considered). 2-l-1,/clause 9.2.1-3(.2) allows an alternative nethod based on equation (D6.2-10). This is lo shift lhc dcsign moment envelope horizontally by a distance .?L - 0-52(cot f, - cot r-v) as

shown in 2-l-liFig. 9.2, which effectively introduces the additional moment I/c1 in equation (D6,2- l0), The longitudinal reinforcement is designed to lhis new effective moment envelope. This is equivalent lo designing thc rcinlorcement at a sectiôn to resist only the real bending moment at thât section, but tô then conti[ue this reinforcement beyond thât section by the

a further shift may be required as discusscd under the comments on 2-l-liclause 6.2.4(7). For members without sbcar reinforcement, the value of a1 should be taken as the efIèctive depth, d, at thc section considered, as indicated in 2-l-l/Fig.6.3. distance, ar. For flanged beams,

Behaviour for loads applied close to supports

Wherc a load is applied close to a support, speciflcally wirhin a distance of 2d from the support,2-l-Ilclause 6.2,3(8) permits the contribution of that loâd to producing the shear force l/pa to be reduced by a factor, É -a,l2d.'lhis is the same factor as for members without shear reinforcement in 2-1-1iclâuse 6.2.2(6). The shear force calculated in this way must satisfy the following condition:

Zft

!

6.2.3(8)

2-r-t l(6.19)

,4.*fr.6 sin cr

1.,,[.6

2-[- I /douse

of thc shcar rcinl'orcement crossing the inclined shear crack between load and support. (Note thât this deflnilion of l,* difi'ers from that elsewhere,) For loads close to supports, there clearly cannot be a free choice of strut angle within lhe range allowed in 2-l-1/clause 6.2.3(2). Only the reinforcement within the central 0.754u should be tâken into âccount as shown in Fig. 6.2-6. This limitation is made because tests by Asin'' indicated that the links adjacent to both load and suppofi do not fully yield. The shear crushing limit according to 2-l-\l'clause 6.2.2(6) should be checked without âpplying the factor 3 : a,l2d. This crushing limit is independent of strut angle. This concept ofconsidering only the Iinks between the load and thÊ support works only for single loads. Where there âre other loads contributing to the shear applied further than 2d from the support, the shear design for these loads should be treated in accordance with the remainder of thc clauses in section 6.2.3 of EC2. It would be illogical to constrâin the location fol link provision for these loads according to the location of the load nearest the support. The link requirements from the two systems should then be added. The shear crushing limit should simila y be performed by superposing lhe results from the two systems. For most bridge applications, unless there is onc or more very heavy loads acting within a distancc 2rl from the support which contributes a significant proportion of the total shear, it will usually be sumcient to determine shear reinforcement solely on the basis ofthe variable strut inclination n.rethod in 2-1-l/clause 6-2.3(3).

where

is the resistance

Fig. 6.2-6. Shear reinforcement in short

spans

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DESIGNERS' GUIDE

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TO EN I992-2

CHAPTER 6. ULTIMATE LIMIT STATES

(b) Alternâtively, cônsider links inclined

aL an angl.e

of45'.

and a srrut angle of 45".

Thus from 2- l- l,Expression 16.lzl): tsna,.,*

-

n ,6*z11 /1.(cotd ' cot n)/(l

_ t.0 r =

-

cot'É)

(cor45 x 400 r. t388.1 < 0.516 x 21.3 .-

-cor45) ,, 1^ l0 ,r 'l'i :;:;iùai'x

6675.6 kN

ional Annex may This is a vcry largc increarc in crushing resisla resislance nce and the UK Nat National reqtrict it as ,s.liccrrsr,ed telt above. ahove restrict discussed in the main tert Realranging 2-l-I/Expression (6.|3) for reinforcemenl design gives

r'*., , -;(coto+col")s"r: /.*a iin c +

1:: ;s -

(cot 0

cot o )

lhcrefirre

,t

ll Aû w l0l

+tffi:r're4mm2/mm ..-'. .e4a:,]'.i4! and using the same two legs ol [69 bars ar above (,4.w - 402 mm' increascd from 200 mm ro 4Q2l Llq4 - 288 mm. say 2?5 mm. i.e. links (inclined al 45") a{ 2?5 mm centres.

ir'tËJHî

"'11,ïi,_ï -

.

rer melre width.

Shear at points of contràflexure Neâr tô points ofconffaflexure, a problem arises in determinirg the value ofshear resistance since the strength is dependent on the lever ar.In and area of longitudinal tensile reinfbrccment. It is therefore always necessary to check the maximum shear force cocxistent with a sagging moment and the maximurn shear'lblce coÊxistent with a hogging moment when designing shear reinforcemenl. 6.2.3.2. Prestressed concrete members

Ellective web rvidths Prestressing ducts can reduce the shear crushing resistance of prcstressed members. In calculation, the web width should therefore bcl reduccd to allow for the presence of ducts2-1-l lclsuse 6.2.3(6) deflnes values for such nominal web thicknesses as l'ollows:

.

- b. 0.5ta

l- I /clause 6.2.3(6)

For grouted ducts with a diameter, $ > b,18:.

ô,,,"-,

2-

2-l-r(6.r6)

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DESIGNERS' GUIDE

TO EN I992-2

Fig. 6,2-7. Strut-and-tie model for flow of web compression strut around

a

oucr

wherc d is the outel' diameter of the duct and ! ra is detcrmined aL thc most unfavourable level where multiple ducts occur at the same level.

. For grouted metâl .

ducts with a diameter, ô < b*18, no reduction in web width is

required. Fof non-grouted ducts, unbonded tendons and grouted plastic ducts:

b*,*. =

ô. t.zla

2-t-ri6.r7)

The 1.2 factor in lhe âbove expression js used to account for the splitting of the concrete struts due to transverse tension. It may be reduced to 1.0 if'adequâte trânsverse reinfbrcement' is provided. This may be derived from a strut-and-tie model like that in

Frg.6.2-1. The reinforcement would typically need to be formed as closed links to provide âdequate anchorage. The above expressions penalize the use of plastic ducts compâred to steel ducts. Although Lhe use of plastic ducts is ôften prefcrabJc for durability reasons, the nominal web width expressions favour the use of steel ducts as the webs can be mâde thinner. This is sometimes â significant consideration for long-span post-tensioned bridges where the webs are sized on the shear crusbing limit to keep deâd weight to â minimum- The reason for the cônservâtive view of the Eurocode is that, even if completely filled with grout, the duct itself must still be stiflenough to Lransfer the force from the inclined comFression struts through its walls rather than around the duct as in Fig. 6.2-1 . ECZ assumes that plastic ducts âre not stiff enough to

be able to achieve this. This has not becn verified by tests and the authors are aware of unpublished tests by a prestressing supplier which indicate virtualll' no difference between lhe behaviour of grouted plastic and stecl ducts. Sections cracked in flexure

The definition of crâcked in flexure is discusscd in section 6.2.2.2. The resistances of prestressed beams lhat are cracked in flexure are conservatively predicted by 2-2/clause 6.2.2(10l), but it will be rare for such beams to not requile shear reinlbrcement. Once shear reinforccment is required, the conctete contribution to resistance is lost and the truss trodel of section 6.2.3 ol EC2 must be used. The design procedure is essentially the same as for reinforced concrete, discussed âbove, except that crushing resistance /p6'nu" is influenced by thc axial stress from prestress through the fâctôr o.o in 2-27'clause 6.2.3{103). discussed in section 6.2.3.1 above. Since prestress is treated in EC2 as an âpplied force (2-l-l/clause 6-2.1(3) refers), any truss models developed for shear design must include the effects of prestress forces applied at anchoragcs and arising from tendon curvature. The inclined componenL of prestress oftcn relieves the shear l'rom other imposed loads. For loads applied âfter prestressing ând

t4a

CHAPTER 6. ULTIMATE LIMIT STATES

contributing to the allowable lbrce in the tensile chords. This is thÊ basis of 2-2lclause 6.2.3(1t7) which allows bonded prestrcssing to bc considercd in calculation of the additional longi tudinal tensile fbrce required for shear. (This means thât the fbrce need not nccessarily be providcd by additjonal longitudinal reinforcement.) In so doing, the stress increase in bonded tendons should be limited so that the total stress in â tendon does not exceed iLs design strength. The eilects of unbonded tendons should be trealed as applied forces acting on thc bcanr (bascd on Lhe prestressing force after losscs), although stress increases from overall structure deflection may be included in calculating these fbrces where thcy have been determilled, Any additional tensile fbrce provided by unbondcd tendôns mây losses, bonded tendons may be treâted in the sâme way âs ordinary reinforcement,

2-Uclouse

6.2.3(107)

be considercd whcn applying 2-2/F.xpression (6.1Ii).

at different levels (draped or un-drapcd) together use in thc shear equâtions is not obvious. The simplification of ,- : 0.9rl in 2-1-liclause 6.2.3( l) is restricted to reinforccd concrete sections without axial force and might not be conservative in othf]r situations. The simplest approach is to dclcrminc thc z value from Lhe analysis lor the cross-section bending resistance, which usually precedes the design for shear. This is equivalent lo hasing the lever arm on the centroid ol the steel force in the tensile zone. Thc additional tensile fbrce of 2-2iclause 6.2.3(107) can thcn bc providcd by thc tcnsilc resistance that is spate alter carlying the design bending moment. The additional longitudinal fbrce should be proportioned between the tenclons and the reinforcement in the ratio of their respective forces lion the bending analysis so as to maintain the value of z used in the flexural design. lf thcrc is no spare resistance. th€n additional longitudinal reinlbrcement can be provided in the tensile

For members with prestressing tendo

s

wilh un-tensioned reinfbrcement, the lever artrr, ;, to

face-

The above is essentiâlly equivâlent to the method proposed in the Notc to 2-2l'clause 6.2.3(107). where the shear strength of the merrber is calculated by considering the superposition ol two different truss models with dillerent geometry and two augles of concrete

strut to account for the leinlbrcement and draped prestressing lendons. This method is subjcct to variation in thc National Annex and is illustrated in 2-2/Fig. 6.102N. Using two angles of concrete strut leads to difficulties in interpreting the rules for l/p,1,û1n*, both in terms of the appropriate strut angle to use and the value of;. 2-2-llclause 6,2.3(107) therefore recommends using a weighted mean value for t?. A weighting dccordrng to the longitudinal force in each system leâds to the sâme result. as discussed above. A further simplification, commonly used in prestress design, is to bâse the bendiug ald shear design ôn the prestressing tendons alone- The un-tensioned reitrlorucment can then be taken into account in providing the additional longitudinal reiniorcement. Since the centroid of the longitudinaL reinforcement is usually at a greâter effective d€plh than that of the prestlessing tendons, this is conservative. The approach of considering only the tcndons for thc llcxural design is olter used Lo simplily analysis, so that the reinfbrcement provided can be used fôr the tôrsional design and. in the case ofbox girders, lhc local flexural desisn of deck slabs.

6.2-5"

t49

DESIGNERS' GUIDE

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::a|451

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CHAPTER

6. ULTIMATE LIMIT

STATES

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DESIGNERS' GUIDE

TO EN I992"2

CHAPTER

6. ULTIIYATE LIIYIT

STATES

6.2. 3. 3. Segmental constfuction In segmental construction where there is no bondcd prestressing in the tension chord, it is possible fbr the joints betwcc[ segments to open once the decompression moment of the

section has been reached. The flnal ultimate moment may not be signilicantly higher than the decompression moment. This depends primarily on:

. .

in tbe prestressing tendons can incrcase as a result of the overaLl slructural deflections (see section 5.10.8); whether the lcver arm between tensile forcc and concrete collDrcssioD can increasc significantly after decompression. This depends on the section geometry. For solid rcclangular cross-sections, the incrcasc in lever arm can be large, approaching a limit of twice that at decompression for low prestrcss ftrrce. For a box girdcr wilh vcry Lhin whether the force

webs, the increase in lever arm

nay be very srnall as there

is

little âyâilable web compres-

sion concrete.

The depth to $bich the joint opens is governed by the depth of the tcxural conrprcssion block. /r,"1, ând this in turn depends on the âbôve. This is shown in Fig. 6.2-9. 2-2lcluuse 6.2.3(109) requires that the prestressing force be assumed to rernain constant alTer decompression, unless a detailed ânâlysis. such as lhat refeued lo abovc, has bccn donc. Clearly, if prestress force increases have been considered in the flexural design. the same must be done here to avoid apparent flcxural firilule. The opening of the joint intloduces a reduced depth througb which the web sheff compression struts can pass. Two checks ale necessary for a givcn con.rpression depth. /r-,1. Crushing ofthe web strùts is

l-l/clouse 6.2.3(t09)

checked using 2-2iExpression (6.103) which rearrangcd is: it..6à" i,

1.1

(D6.2-11)

'EJ 'Gotd + tandl

To avoid fâilure local to the joint, shear reinforcement should be provided in the leduced length, /lrcd cot d, adjacent to the joint as shown in Fig. 6.2-9 according to 2-2,/Expressior.r (6,104):

,r", :

vr.t h,.,r

Jy.,r

2-2i6.104)

coT 0

If either ol these checks cannot be satisfied, ir,."6 should be increased prestressing force and thc check repeated,

by ilroreasing the

]* /'

L] Fig.6.2-9.

+l

Effect of ioint opening in precasr construction without bonded prestressing

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DESIGNERS' GUIDE

TO EN I992-2

6.2.4. Shear betwêen web and flanges of T-sections Longitudinal shear Z- I -

l/clouse

6.2.4(r)

2-l-l lclwse 6.2.4( 1l allows longitrLdinal shear in flanges to be checked using a truss model. The check covers the crushing resistânce of the concrete struts and the tensile strength of Lhe transverse reinforcement. Clause 6.2-4 applies to planes through the thickness of the flange. It need not be applied to planes through the web at the web flange junction. If a construction joint has been made between web and flangc, however, the similar provisions of 2-1'1iclause 6.2.5 should bc chccked. Despite the reference to T-sections in the title of the clause, the provisions also apply to other sections, such as boxes, on the basis of the shear per web.

For compâtibility with the web shear design and the design of additional longitudinal reinlbrcement for shear, the flange fbrces should theoretically be determined considering thc samc Lruss model. A typical truss model for the end of an end span is shown in Fig. 6.2-10. This approach would differ from previous UK practice whcre the longitudinal shear rvas determined from elastic cross-section analysis using beam theory and the transverse reinforcement placed accordingly, following the envelope of vertical shear force. It can be seen ftom Fig. 6.2-10 that tlle tlansverse reinforcemcnt predicted by a truss ûrodel of an 'l' beam does not follow the shear force envelope, but is displaced along thc beam from the locâtion of peak shear. 2-2/clouse

6.2.4(t 03)

To avoid the need to draw out truss models lor cvcry loading situâtion, the pragmatic simplification is made tn 2-2lclause 6.2.4fl/r3./ thât the âverage force iucrease per metre mây be calculated over a length A-r, which should not be taken greatcr than half the distance between the point of contraflexure and thc point of maximurn moment in each hog and sag zone. This allows a certain amount of averaging out of the reinforccment from that which would be produced by a detailed truss model. Howcver, where there are point loads, thc lcngth À-r should not bc takcn as greater thân the distance between thlr point loads to avoid significantly underestimâting the rate of change of llange force. Since bridges are usually subject to significant point loads from vehicle axles, it appears that this simplification u ill not usually he appropriate. 2-2,/clause 6.2.4(103) offers a further simplified method by which the longitudinal shear is dctermincd dircctly from the vcrlical shear per web, I's6, in the same way as described in section 6.2.5 below. The expression for shear flolv given between wcb and flange, I/s1/;, is only correct where the flexural neutral axis lics at the web-flange junction, In general, the shear flow between web and flange can be taken as BV;,1f z as discusscd in section 6.2,5 below, where É in this case is the ratio of force jn lhc cffective flange to that in the whole

',,

2-l-I/clouse 6.2.4(6)

u111hçf

cot$

to the

2-1-ll(6.21)

2- l- I /clause 6.2.4(4) limits the angle of spread, d;, to betwcen 26.5' ând 45" lbr compression flanges and 38.6'' to 45" for tension flangcs, unless more detailed modelling, such as a nonlinear finite element ânâlysis thât can consider cracking of concrctc, is used. When the shear stress is less than 40olu (a nationally determined pârâmeter) of the design tensile stress. ,f,6, 2-I-l lclause 6.2.4(rt,) permits the concrete alone to carry the longitudinâl shear and no additional transverse reinforcement is requircd (other than minimum reinforcement). For a concrctc wilh cylindcr strength 40 MPa, this gives a lirniting stress of0.67 MPa. For greater shear stress, the côncrete's resistance is lost completely as jn the main verlical

shear design. It should be noted that interface shear should still be checked according to 2-1-I,/clause 6.2.5 rvhere thel'e are construction joints. Diffèrent concrctc contributions âre calculated depending on the degrec of roughcning of the interfâce. Fol a surface prepared by cxposing aggregate, clâssed âs'rough' in 2-1-1/clause 6.2.5(2), the concrete contribution is 45% ofthe design tensile stress, /",,1, which is a greater proportion than allowed here. EN 1992-2 does not specify the distribution of the required transverse rcjnforcement between the uppel and lower la,vers in the slab. It was a rcquirement of eârly drafts of

t56

CHAPTER 6. ULTIMATE LIMIT STATES

0.5 0

Fig. 6.2- 13. Total transverse reinforcemen! requirements for shear and transverse bending

EN 19S2-2 that the transverse l'einforcement provided should havc the same centre of resistânce âs the longitudinal force in thc slab. This requirement was removedJ presunably because ii has been common practice tô consider th€ shear rcsistance to be the sum of the

from the trvo layers. It should be noLcd that âpplication of Annex MM of EN 1992-2 would necessitate provision of transvetse leinfbrcemcnt rvith the same centre

resisLanccs

of resistance as the longitudinal force in the slab, On the tension flange, the angle of spread ot thc compression strut into the flange requires the longitudinal tension reinlorccment to be required 'eaflier' than expected from considerations of the web truss alone, as shown in Fig. 6.2-10- 2-I-l lclause 6,2.4( 7) requires ilccount to be taken ofthis in determining the curtùilment ofthe longitudinai reinforcemeni. This may be actrieved by inLroducing a further shift ofe cotdf in the shift method described in sectlon 6.2- 3.1 of this guide. {:. is the distance of the longitudinal bar considered lrom the ed ge of thc

2-l-l/clouse 6.2.4(7)

web plus a distance ô,,/4. Flanges with combined longitudinal shear and transverse bending Flanges fbrming deck slabs will atso usually be subjected to transverse bending from dead and livc loads- The flange reinforcement needs to bc checked for its ability tô carry both in-plane shear and any ttansverse bending. A simplified mle fbr combining the reinforcemcnt reqùirements fron.r shcar and bending is given in 2-2lclause 6.2'4(105). This requires the

amount of transverse reinforcement to be lh€ greater of that required for longitudinal shear alone (case (a)) and half Lhat required for longitudinal shear plus that required for transverse bending (casc (b)). This rule is illustrated in Fig. 6.2-13, in u4rich '4..0'6 is the total reinforcement required, and subscripts s and b reler to the reinl'orcement t'equired for shear and bending, respecLively. The reinforcement provided should not be less than the mirinr um requirements of 2- I - I iclause 9.2. L Once again, EN 1992-2 does not spccify the disftibution of the required transverse reinforcement between lhc upper and lorver layers in the slab. particularly wiLh respect to the component due to sheâr. Clearly the reinforcemcnt must ât least bc placed so as to resist the transverse bending lromenL alone. The designer is left to decide where he ùr she places the centre of gravity of the reinfôrcement needed for shear' The intention of the drafters rvas to distdbute the reinlorcenent as shown in Fig.6.2-14 for the two cases (a) and (b) above, Use of 2-2/Annex MM would suggest that these combination rulcs are optin.ristic and indeed they âre not allowed for rveb design. Previous UK prâctice in BS 5400 Part 4'has, however, been to ignore any interaction in flangcs2-2iclause 6.2.4(105) also rcquires that the interaction betwecn bending and longitudinâl shear is chccked in the zone which is in compression under the transversc bending Compression from strut-and-tie action in thc flange resisting shear will add to that from lransverse bending and can lead to crushing failure. It has previor.rsly been UK practice to jgnore this interâction but it can be unsafe to do so. To allow fbr the interaction, a simplified approach is given, whereby the depth of flange required fbr compression in Lransverse bending is deducted from rf when calculatiug thc crushing resistancc. This approach is conservati\€ because the directions of thc compressive stresses from bending and [rom sheâr do not coincide. I1 th(r llange is inâdequate v'hen chccked according to the above simplited approach, then the sandwich n.roclel ol2-2,/Annex MM can be used to check thc

2-2/clouse

6.2.4(r 0s)

t57

DESIGNERS' GUIDE

TO EN I992-2

Compression face:

ffi-'.'' il".'t--)-

uz

aæ

Case (â)

Fig. 6.2-

|

4.

#

oo

Case {b)

Placemem ôf transverse reinforcement for cases (a) and (b)

compressions in the two outer lâyers as discussed in Annex M of this guide- Thc reinforcement should also then be dcrived liom 2-21Annex MM, but this will generally exceed that from the simple rules above. It was not intended that the longitudinul compressive stress from the main beam bcnding

moment bc considered when determining the depth to bc ignored in shear calculation according to 2-2iclause 6.2.4(105). It would, however, be consistent with thc cr"* tem.r in 2-2,i Expression (6.9) rvherc axial force is high. As a result, for.flanges with high compressivc axial stress, 2-2l'Annex MM may not lead to a mote economic check of concrete crushing as thc shear stress does interact with compression from both longitudinal and transvsrse bending see discussion on 2-27Annex MM in this guide.

6.2.5. Shear at the interface between concr€te cast at difierent times Shear stresses âcross construction joints between concrete elenents cast at different times must be checkcd to ensure that the two concrete components act fully compositely, The bending and shear designs of such members are based on this âssumprion. 2-l-l/clause 6.2,5 deals specificalll' with this interfâce shear requirement, which must be considered in 2-

I-

I/clouse

6.2.5(t)

addilion tô the requirements of 2-2iclause 6.2.4. 2-L-llclnuse 6.2.5(/) specilies thât the intcrlaces should be checked to ensurc thàl cfti I r]a6;, where t'g61 is thc design value of shear strcss in the interlàc€ and oa61 is the design shear resistance at the interfacc. The applied design value ol shear stress is given by: u51;

= pV6f

zb1

2-1-11(6.24)

where:

Ii V1,a ô;

is thc ratio of the longitudinal force in the new concrctc area and the total Iongitudinal force either in the compression or the tension zone, both calculated for the section considered is the total vertical shear forcc for the section ir lhe lever arm of the composite section is the width ofthe intcrface shear plane (2-l-liFig. 6.8 gives examples)

2- 1- I /Expression (6.24) is intended to be used assurning all loads arc carried on the composite section. which is compatible with the design approach lbr ultimate flexure. The basic shear stress for design at the inlerface is related to the maximum longitudinal shear stress aL the junction between compression and tension zones given by tr/6,1/:à;. This lbllows from consideration of equilibrium of the forces in cither the tension or compression zone. The instantancous force is given by Msa/z so the change in force per unit length along the beam (the shear flow), assuming that the lever arm, z, remains constânt, is given by:

(D6.2-16)

The shcar stress is then obtained by dividing by the thickness ât the interfâce to give: ,uo

158

V",

- ,Ë

(D6.2-t1)

CHAPTER 6. ULTIMATE LIMIT STATES

Fig. 6.2- 15. Determination of d

If the shear plane checkcd lies within either the compression or tension zoDes' the sheâr stress from equation (D6.2-17) may be reduoed by the factor 3 abovc- It will always be conservâtive to take p:1.0 For flanged beams, much of the force is contained in the

flanges so coustruction joints at the underside of flange will typically have,llt 1.0. In oLher cases, É cân be obtained from the flexural design as shown in Fig. 6.2-15' tion thc forces F1 and F2 (which arc shown with the flange in compression). The qucstion arises as to whât to use lbr the lever arm, :. Strictly, the value of: should reflect the stress block in the beam for the loading considered. In reâlity, this would be time-consuming lo achieve and it $'ill generally be reasonable to use the same vahre ls obtained from the ultimâte bending resistance analysis, as shown in Fig. 6.2-15 For cracked sections, the use of thc ultimate bending resistance lever arm will slightly overestimale thc actual lever ârm ât lower bending trroments. However, for other than very heavily reinforced sections, this difference in lever arm will be small and is compensated fbr by also basing thc value of d on the stress block for ultimate resistance. The above represents a difference to previous UK practice, where the shear stress distdbution was based on the elastic dislribution on an uncracked section regardless offlexural stress distribution. If a section remained uncracked in bending and elastic analysis was used to determine the lever arm. theu the shcar stress determined liom 2- I - l/Expression (6 24) lvould be the same as that from the ânâlysis for the uncracked shear resistance in 2-1-1i clause 6.2-2.

A further point is that if the lcver arm used is not taken to be the sâme as that lor the calculaljons on flexural shear, it would then be possible to find that the maximum shear stress âccording to this clause was exceedcd, while the check against /p6.n,n* fbr flexuraL shear u'as satisfied.

The design sheâr resistance at the interface is based on the CEB Model Code provisions and is given in 2-l-l,iclause 6.2.5(1) as: r.,p6;

:

cjl,6

* pa, *

p/r,1(p sin a

*

cos

o) <

0.5u;f"6

906

2-l-l(6.25)

rvhere c and /r are faclors u'hich depend ôn the roughn€ss of the interlace Recommended values in the absence of results hom tests ale given in 2-I-llclause 6.2.5(2 ) . OIheT factors are defined in 2-1- I i cLause 6.2.5(1). The first lerm in 2-l - l,/Expression (6.25) relates to bond between the surfâces and any mechanical interlock provided by indcnting the surfaces, the second relates to friction across the interface under the action of compressive siress, on, and thc third term relâtes to the mechanical resistance ofreinforcement crossing the interfâce. The reinforcement provided for sheat in accordance with clauses 6.2.1 Io 6.2.4 may be considered in Lhe reinforcement ratio p. The reinforcement docs not need to be taken as that provided in addition to that nccded for ordinary shear. In order to allow the practical placcment of reinfôrcement across the interface in bands of

decreasing longiLudinaL sprcing. 2-1-llckruse 6.2,5(3) allows a stepped distribution of rcinforcement to be used by averaging the shear stress over a given lcngth of the member corresponcling to the length of band chosen. This is illustrated in Worked example 6.2-8.

2-l'l/clouse 6.2.5(2)

2-l-l/clouse 6.2.s(3)

t59

DESIGNERS' GUIDE

TO EN I992-2

No guidance

is given as to by how much the local shear stress may exceed the locâl câlÇulated

A rcasonable approach would be to allow the shear stress to exceed the resistance locally by l0%, providcd that the tôtâl resistance within the band wâs equal to or grcater than the total longitudinal shear in the same length_ This would be consistent with the design of shear conn€ctors in EN 1994-2. Where interfàce shear is checked under dynamic or latigue loads,2-2lclause 6.2.5(I0S) requires that the roughness coefficjent values, c, are tâken as zero lo account for potential detcrioration of the concrete component of resistance across the interface under cyclic reslstance.

2-2/clause

6.2.s(t0s)

loading.

6.2.6. Shear and transverse bending In wehs, particularly those of box girders, transverse bending moments can leâd to reductions in the maximum permissiblc coexistent sheâr force because the compressive stress llclds from sheâr and from transverse bending have to be combined. The two stress ûelds do not, hùwever, fully add because they âct ât different angles. In the UK it has been common lo design reinforcement in wcbs for the combined âction of transverse bending and shear, but not to chcck the concrete itself for the combined effect. The lower crushing limit in shear used in the UK made this a reasonable approximation, but it is potentially unsafe if a less conservative (and more realistic) crushing strength is used2-2,iclause 6.2.106 formally requires consideration olthe above shear moment interaction, but ifthe web shear force according to clause 6.2 is less than 20% ot tr/qd ..x or the transvefse

moment is less than 10% of M11,1,-,,*, the interaction does not need to be considered- Mp6.,n"" is defined as the maximum web resistance to trânsverse bcnding. The subscript ,max' might suggest that this is the maximum obtainable bending resistance if the web wcre to be heavily over-reinforced. It was, however, intended to be thc actual web bending resistânce in the absence of shcar, even though thc former could be considered more relevant fot the crushing check. These criteria are unlikely to be satisfie (t-1, + 0.10o*)

(é.4'7

)

where:

k

: | + \/ (200 /d) (with d in millimetres)

P1: J P1t 14' < pty,

pt

d

0.02

relate to the bonded tension steel in the y- and z-directions respectively. The valucs should be calculated as mean values taking into accoLrnt â slab width equal to the loaded area or column width plus 3d each side should be taken as the averagc cffcctive depth obtained in each orthogônal direction from 2- l-1i Expression (6.32)

- (o", + o",) l2 (in MPa, positive for compression) o"u, o"" arc thc normal concrete stresses from lôngitudinal fbrces in the critical section in o.o

the y- ând z-directions respectively (in MPa, comprcssion taken as positive)

Other terms have lhe same definitions as in 2-l-liclause 6.2.2. The values of Cp4.", u-1n and ÀL may be given in the Natior.ral Annex. EC2 recolr.rmends values of 0,l8/i," and 0.10 (compared to 0.15 for flcxural shear) for these respectively for punching shcar- The value of ln,i,., is recommended tô be the same as for flexural shear. In general. the punching shear resistance for a slab shoukl bc assessed for the basic control section discussed in section 6.4.2 using 2-1-1,i Expression (6.47) and at the pedmeter ofthe loaded area against rp6.-u* defined in 2-1-1/clause 6.4.5(3).2-1-llclause 6.4.4(2). however, requires thât the punching shear resistance of column bases be verifled at control pc meters witllin the 2d periphery of thc column and the lowest value taken. This is because the angle of the punching cone mây be steeper in this situation due to the favourable reaction fiom the soil. A check of punchiug on the basic perimeter ignoring the relieving forcc from the soil would be conscrvativc. C)n any given perimetet, and in the abscnce of transmitted moûr€ot, ugd is as follows: uv1

:

with whcrc

Vvl

,,",1

I/s,1,."d

2-r

f ud

:

i/sa

-

2-

l- I /dause 6.4.4(2)

-1(6.49)

2-l-r(6.48)

À l/ru

trs4 is the column load and À Zs1 is the net upu/ard lbrce

considered (i.e, upward pressule

within the control perimeter

liom the soil calculalcd excluding the self-weight of the

base).

For steeper sheâr planes. an enhancement ofthe basic resistancc to 2-1-1/Expression (6.47) is applicable, whereupon the resistance becomes: trp,1

:

cp6,"/e(1oop t.T.o)t'',

2j

2

r,,n,,

r4

where a is the clistancc from thc periphery of the column to the control

2-l-1(6.50)

pc

meter considered

This formulation for shear enhancement. where the rcsistance is enhanced, is at odds with that for flexural shear. whcre thc shcar itself is reduced. This is discussed in section 6.2.2.1 of this suide.

t79

DESIGNERS' GUIDE

TO EN I992-2

For most bases. column axial load will be accompanied by some bending moment (due to moment fixity at deck lcvel or horizontal forces âpplied to the column top through bearings)In such cases. an increase in the dcsign shear stress to account lbr uneven shear distribution around the control perimeter is necessary- 2- 1- l,/Expression (6.51) is providcd to do this and iL is equivalent to 2- 1- 1/Expression (6.38) fol cases whele there is no reâcting sôil pressure: /ea..".r l, ,- Mro ^. "J ' Ed ,d l' -" v.d*d Iu) 2- I -

2- l

-l (6.51)

1/Expression (6.51) contains ân equiyâlent term to thc tcrm 1.

lvfEd

Vu

U

I

Wt

in

2- 1- l lExprcssion (6.39), but with tr/s6 replaced by I/p6 ,.6. It would be logical to allow Msd to sûrilarly be replaced wilh a reduced value allowing for the soil pressure, but this is consenatively (and probably unintentionally) not done in 2- 1- I /Expression (6.51). ut and Wr are also replaced by I and tr42 which relate to the actual pcrimeter being checked.

iË*

-,'

:

I80

glYeg

tt tt

It | l, .: ---1

500

I I I

CHAPTER

6, ULTIMATE LIMIT

STATES

t8l

DESIGNERS' GUIDE

TO EN I992-2

-- 2.41 \{Pa

(

-,,, tberefore..okay : i.-:.'lr:.1- l.:'..., :.:.....::,1::::':::.:::i The loundation should. also be checked for flexural shear. From 2-l-l-'clause 6.2.1(8). planes nearcr than d need nol be checked where there is approxJmarely umform load. so rhe resistance will be checked f1 a sgctlgn;q1{ àiletlCéri irqry.,i* iqi9ûl:qiq t:::. t:

t82

un

r

CHAPTER 6. ULTIMATE LIMIT STATES

6.4.5. Punching shear resistance of slabs and bases with shear reinforcement Where ts,1 exceeds the vâlue of

'L'R,l.c

fôr the perimeter considered, usually the basic pcrimeter

a1, punching shear

reinforcenent is required in accordance with 2-1-11clause 6.4.5 and three zones are required to be chcckcd:

. . .

the zone immediately adjaccnt to thc loaded area (against the she crushing limii); the zone in rvhich the shear reinforcement is plâced; thc zonc outside the shear reinfbrcenent. 2-l-1,/clause 6.4.5(4) i[cludes a delinition for an outer perimeter where the concrete resistancc alone is sullcient for the punching shear and shear reinforcement hâs to be provided within this zone as discussed bclow.

Where shear reinfolcement is required, the following equâtion is defined in 2-l-I lclause 1 ) lo calculate the punching shear resistance of slabs or column bases:

6,4.5(

r.,p6..,

:

0.?5,p4."

+ ls

(4),r,"f,".r(*)'- "

2-l-l/clouse 6.4.5(t)

2-l-1(6.52)

where:

1,," s, .Â.a,"r

a

is the area of one perimeter of sheâr reinforcement around the column or loaded area accortling to 2-l-liFig. 6.22 is the râdiâl spacing ofperimeters of shear reinforcement is the effective design strength of the punching shear reinlorcement allowing for anchoragc cffrcicncy : 250 | 0.25d ./y.a (in MPa) with d taken, as befbre, as -< the âverâge effective depth (in millimetres) is the angle between the shear reinfbrcement and the planc of the slab

2-1-l iExpression (6.52) diflèrs fionl the fornulae for flexural shear in that a concrete terrn ls added to a shear reinforc€ment term- However, 2-l -1,/Expression (6.52) does not fully combine the concrete resistance and the link resistance h the 2d pedmeter. Thc reasons for this âre cntircly tcst-based. The 0.75 factor t)n the conorete term represents a reduced concrete contribution as one might expect when reinforcement is yielding with the associated in.rplicd concrete oracking and defomation. The use of L5d rather than 2,1is also needcd for adequâte câlibration with test results and reflects observations that shear reinforcement at the ends of shear planes is less eff'ective. It doesn't imply â steeper failure plane. The reduced shear reinforcement strcngth- Âwd.ci. is a furlher anchorage elTiciency factor affecting shaUower slabs. The above expression has been presented assuming a constânt area of shear reinforccment

area Fig. 6.4-4 refers. Irr bridges, reinl'orcement is not usually placed like this, but rather on an orthogonâl rectangular grid, coinciding with horizontal reinforcement ârrangements. This is a necessity where moving loads are to be catered for. Such arrangements inevitably lead to thc area of reinforcement on each perimeter moving away liom the loaded

increasing on successive perimeters away from the loaded area. One solution is Lo apply 2- I - I iExpression (6.52) by considering only the reinfbrcement bars lhat are located âs.in Fig. 6-a-a(h) and ignoring olhcr rein[orceruent providcd between the arms of the crucilbrm shape (which would increase the reinfôrcement area on successive perimeters mor ing away from the loaded area). There would be no need to reduce the effective concrete perimeter as indicatcd in Fig. 6.4-a(b) if additional reinforcemenl wcre so placed to reduce the circumferential spacing. An âlternâtive, less conservative, approach is given later. The control perimeter at u,hich shear reinforcement is not required (aor,, or r.iu,,, in "1 Fig. 6.4-4) is defined in 2-1-l lclause 6.4.5(4, as the perimeter lvhete the concrete resistance alone is suflicient to resist the aDûlied shear stress: ,ôur.ei

,, vxa , : P ?'Rd,c4

with oq6." from

2- 1-

2-

l- l /douse 6.4.s(4)

2- 1-1i (6.54)

I,/Expression (6.47).

The outermost perimeter ofshear reinforcement should be plâced at a distance not grcater than t/ : l.5d (which may be varied in the National Annex) withir this oùter perimetel (as illustrated in Fig,6.4-4) to ensure an inclined failure plane cânnot develop within this

t83

DESIGNERS' GUIDE

TO EN I992-2

(oJ-------'-

t2

-- .-'\â (a) Perlmeter

(b) Peimeter uo(l êr where circumJerential spacing exceeds 2d

uour

Fig.6.4-4. Control perimeters

adlacent

to

loaded area wirh shear reinforcement

pefllnetef ôver a radial distance of 2d without passing through â set of shear reinforcement legs. If the perimeter ,our or aoùr cf is less than 3.0d from the face of the loaded area, shear reinforcemenl should, however, still be provided out tô ât leâst â perimeter 1.5d from the lace of the loaded area such that the required resistance accordhg to 2- 1- I /Expression (6.52) can be achieved. According to 2-1-liFig. 9.10, thc innermùst perimeter ol shear reinfôrcement should not be placed nearer thân 0.3d to the face of the loaded area. This is similar to the reâson for pulting reinforcement in the middle 0.75a, of a shear span, as discusscd under clause 6.2.3(8). The radial spacing ol reinforcement should also not exceed 0.75d in accordance with 2-l-llclause 9,4.3(li. Because of the difficulties with matching availablc reinforcement

tù perimcters for rcal reinforcement layouts and conti:ol perimeter shapes, an alternâtive âpproâch is proposed here rvhich allows successive control perimeters to be checked if necessary. In general, shear failure is deemed to occul over a radial distance of 2/- Consequcntly, lo enhance resistance, shear reinforcement of area 11*" should be placed within ân âreâ enclosed between the control perimetef chosen and one 2d inside it. To correspond to the 1.5d in 2- I - l,/Expression (6.52), it is desirable to consider ônly the reinforcement within a radial band of 1.5d. To comply with the need to consider only reirforcement further thân 0.3d liorn the loaded pcrimeter, only reinforcement further than 0-3d from the inner pcrimcter should be considered. Consequently, only reinforcement fïrther than 0.2d inside the control perimeter should be included. These two limits arc consistent with the fact thât rein[orcement at each end ol' a failure plane is unlikely to be fully effective. This reinforcement zône is shown in Fig.6,4-5.

Reinforcemenl ) Isw to include for check on perimeter !i

u (between

Fig. 6.4-5. Reinforcement

t84

lA,*

to

include in check

to

4

equation (D6-4-3)

and uout

CHAPTER 6. ULTIIYATE LII4IT STATES

I[ the above method is fbllowed, successive perimeters, a1, between the basic control pedmeter al 2d a\d. the perimeter a.,u, are checked to ensure that the reinforcement in each 2d zone âbove satisfies: (LrE6

\- .r

-

0.75r,p1")rr;,.1

(D6.4-3)

It will be noted thât if the abovc is applied to the control perinreter àL 2d, the same total reinforcement requirement as in 2-1- l i Expression (6.52) is produced. If it is âpplied ât the perimeter uoùt, some reinforcement requil€ment will still be predicted because of the 0.75 fàctor on up6.. in equation (D6.4-3). This is unfortunate, but as long as reinforcement is rletailed so that it is stopped no further than l.5d inside the perimeter &oui âs required by 2-1-1/clause 6.4.5(4), some reinforcement will be available for this check. Maximum punching shcar stress 2-1-I lchuse 6.4.5(3) reguires the shear stfess at âny section to be less than t1,1."". This check is equally applicable to scctions with or without shear reinforccmsnt, but is only likely to be critical in slabs with shear reinforcement. Clearly, the most critical s€ction to

check is the column perimeter

2-I-l/clouse 6.4.5(3)

or the perimeter of the loaded area (and 2-l-llclause

6.4.5(3) includes specific &0 vâlues for the special cases close to slab edges or corners). ?.,Rd nrâx may be given in the National Annex but the recommended value in 2-l-liclause 6.4.5(3) is ïrd.max :0.5r,/11; the same as lor flerural shear design in 2-1-1/clause 6.2.2(6).

6.4.6. Pile caps (additional sub-section) EN 1992-2 provides no specifio guidance for checking punching shear in pile caps. The general lules can be applied tô pile caps where thc cdgcs of the piles are located further than 2d fron.r the pier face, but this situation is rare in prâctice. In other cases, a lot of interpretâtion is required. Thc revised thinking on shear enhancement in Lhc final drafting of ECl2, discussed in section 6.2.2.1, hâs made matters more complicated. Some suggestions are made below and in Worked exan.rple 6,4-2. Where pile edges are closer to the piers than 2r1, somc ofthc shear force will be transmitted direcLly into the support by way of a strutting action. The basic punching perimeter cannoL be constructed without encompassing a part of the support perimctcr (a) in Fig. 6.4-6. 2-l-l,/clause 6.4.2(2) r€quires reduced perimeters to be checked, such thât the suppolt reaction is excluded, suggesting aperineter like that oftype (b) in Fig.6.4-6.2-1-l1clause 6.4.3(7) does nol âllôw âny enhancement to be taken on such perimeters. On lhc onc hand, it would not be r€asonable to enhance resistance for support proximity on the whole of such a punching perimeter without making reductions to the ellectiveness of other parts oj Lhe pcrimeter, as the shear will be unevenly distributed around that perimeter. On the other hand, some degree of cnhancement must take place because the fâilure surfàce is, at least locall1,. steepened

by the

presence

(d) Flexural shear plane actoss câp

of

Lhe support and sorne

o_o

load can strut directly into the

(b) Reduced ædmeter

(a) 2d perimeter

(c) Flexural shear plane

Fig. 6.4-6. Corner pile wirhin 2d of a cotumn

base

t85

DESIGNERS' GUIDE

TO EN I992-2

support. In the limit. if rhe pile is very close to the pier face. the load r.ill transfer straight between support and pile in con.rpression and the very short perimeter oftype (b) may underestimate resistance if enhancement is not permitted. The abovc problems sterl from the subjective distinction between 'punching' and'flexural' shear. It is arguable that the above situation is more of a flexural shear problem. The following procedure is thelelbre proposed:

.

.

First, a làilure plane of type (d) in Fig. 6.4-6, extending the full width of the pile cap, should be chccked for flexural shear. A method considering shear cnhancement of the concrete resistânce over the sections where reinforcement crosses the pile head as in Worked example 6.4-2 is recommended. Second, axial loads lrom corner piles can be checkerl for punching at the pile lâce to check the maximum shear stress ând âgâinst the minimum resistance liom: (i) punching at a 2d perimeter (without support enhancement) ignoring thc presence of the support, ând; (ii) a diagonal flexural shear plane at the edge of the pile of type (c) in Fig. 6.4-6, which rvâs the approach uscd in US 5400 Part 4.' A method considering shear enhancement of the concrete resistânce over the pile head as in Worked example 6.4-2 is rccommer-rded.

Chcck (i) will generally be less critical than check (ii). For all of the checks ahovc. where support proximity is included to enhânce part of the shear resistance, it is suggested here thât, in keeping with current UK practice, a, is taken to be the distance between the face of a column or wall and the nearcr edge of the piles plus 20% of the pile diameter. This approach is pulsued in Worked exarnple 6.4-2.

' r

l

: :::::::,II

:

:::

,"

: ::: :: ::: ::: l.:: : :::: :;I;

,..11Y,j Yl

!&o=uoo' L+;-Ir

L-d

I

L-jd

S€ction

Fis.6.4-7.*.,*..."0.::iu,o"#

t86

(Alldlmen$lons in mm) (Àrdrmensronsinmm)

CHAPTER

6. ULTII4ATE LIMIT

STATES

t87

DESIGNER5' GUIDE

TO EN I992-2

nul*19*

t88

,"f'n: 'n:il

CHAPTER 6. ULTIIYATE LIMIT STATES

"

:*

shear

tï: **'

,r, o,.***'

"n."r

s'Lress

rimi'i:

I.,Jt-:i ffi':':;Ï:ï''um

r.r"r,

the corner pilc is checked

,,"_r-,]l"r,*"0 in Fig. 6.4-9 below):

ffiî

*Et{È-H;Ï -

=

Fig. 6.4-9.

'

r,"**,.n*.

H, ,^;:l il*

*-Hffi.t']" '*

-

Or". u..or, .oîn"". "'""'

:,":,:'""

-,tï,'

"rt

hogonar directions; thercfore'4,

:

3272 mm) im

î,"T

t89

DESIGNERS' GUIDE

TO EN I992.2

",-. "^"*.i1

-:l':-:i-ïijï #i*mnit'

ïii] -''o "*- ""'

the contribution from ;0"'t, n"r. laken as the

x t0'- ?14.8 *t{_ ;"-":-: , --'-1500:"li'"î* t760**, "* 434 s singo ^,

..' .::

t90

*

:"

" ""hancins

the concrete resisrauæ:

parts of the section onlv' the enhanced pârts only.

CHAPTER 6. ULTII.IATE LIÈ4IT STATES

Required area per leg is rherefore 1760127 -65 mmz 1i.e. use

l0$

legs)

(4) Punching shear on a ?d perjmerer. as illustratcd in Fig. 6.4- I I ' is checked ignoring enhancement fior ptoximity to lhc column as required by 2-l-l/clâuse 6.4.3(7). lTfis is the oniy punching shear plane explicitly covered in EC2 ) The perimeter is not. however- reduced to exclude lhe ootumn load as required in 2-l-llclâuse 6.4 2(2). as this is effectively checkcd io (3) abôve by way oi a flexural shear plane. .

,ï"tf-

-

r------+.-1;-ldz

-

dt)

CHAPTER 6. ULTIMATE LIMIT STATES

The valuc o[ Fq4, should be reduced in accordance wjtb 2-I -I lclaase 6.7 ( 3 ) if the load is not l.n or ifhigh shear forces exist. Ifthe load is not uniform, the bearing pressure check could be based on Lhe peak pressure. No guidance is given on lhe effects ofshear, but shear force could reasonably be ignored ifit is less than 10% ofthe vertical force, which is cônsistent with 2-l-11clause 10.9.4.3 which deals with precast elements, For'higher

uniformly distributed on the area

2-l-l/clouse 6 7(3)

shcar, the veçtor resultânt, i1r, of the shear force, F1, and the vertical forcc, F", could be used in the bearing check. as recommended in Mo,ltl CoJt 90.Û according to:

,'TH

(D6.7-

l)

This shcar foroe would have to be tied into the surrounding strucLure by tie reinforcement ât the loaded face. It may be tempting to assumc no distribution ofload and set lco equal to 1"1, whereupon thc bearing resistance becomes l"p4u : A"rt-f"a. Many UK designers have in the past effectively taken this as the lin.fting pressure where bursting reinforcement has not been provided. However, a check ofreinforccment is strictly still required for bursLing as discusscd below sincc the transverse spl€ad of loâd in un-reinforced concrete leads to crâcking when the concrete's tensile strength is reached, and this can give risÊ to premature failure at stresses less than ./;,1. No guidânce is given in EC2 on bearing prcssure in the absense of âny suitably placcd reinforcement, The bearing pressure cor.rld safely be limited to dR,r.max:0 6(l - /;k/250)I1/'i; as discussed in section 6,5 of this guide, or the teûsilc resistance oi lhe concrete could be considered to increase resistance âs discussed below. For piers with geometry such that the load has to spread in one direction only, it is likely thal the ninimum perimeter reinforcement would give a reasonable bursting resistance and hence a limiting bearing pressure in excess of0.6(1 - f"yl250)J"y11.. The rules for nodes, as discussed in scction 6.5.4 of this guide, may also apply in cascs other than this simple column case and is illustrâted in both Worked examples 6.5-l and 6.7-1.

Bursting Thc tcnsile forces generated by the trânsverse spread ol load can be resisted as shown in the strut-ând-tie model in Fig. 6.7-1. The depth over which stlesses become unifbrm can be taken equal to the dimension ô, which in this case is the width of the s€ction, or twice the distance from centrc of load to a free edge in the direction considered for eccenûic loads. The strut-ând-tie model shown produces a tension force as fbllows:

t h,n

2-l-1(6.58)

where 'F is the applied vertical forcc. This tension needs to be calculated for bolh transvcrsc directions and reinfbrcement detailed accordingly. Where the load spreâds out from an applied load but tapers back into another node without sp.reading to th€ lull cross-section in between, the tension in 2- 1- 1,/Expression (6.58) should be replaced by thc slightly modiûed expression for a 'full discontinuity' as given in 2- I - l/Expression (6.59) in section 6.5 of this guide, Worked example 6.7'1 illustrates the use of this expression. EC2 gives no guidance where there is no (or insuflicienL) reinforcement tô resist this tie force- Motlel Corlc 906 pcrmits the force to be resisted by the concrete tensile resistance. For the case shown in Fig. 6.7- l, this would lead to a concrete tensile resistance of:

T^* :0.6b ' L' ha

(D6.7-2)

and a Iimiting bcuring reaction of: 2.4b2.

L. J.d

b-a

(D6.7-3)

where L is the length of the loaded ârea perpendicular to the side a, 0.6ô is the height of the t€nsile zone in Fig. 6.7- I and f",6 is the design tensile strength oi the concrete. The limifing

203

DESIGNERS' GUIDE

TO EN I992-2

______________ o.ssf"kr"

Appfoximately 0.6(1

-

U2sô)f.kt^t.

bla

Fig. 6.7-3. Allowable bearing stress for 40l'4Pa unreinforced concrete allowing for ænsile strenSth bearing prcssurc thcn becomes:

(D6.7-4) Equatiôn (D6.7-4) needs to be applied in both perpendicular directions and thc lowest resistanc€ taken. Figure 6.7-3 shorvs equation (D6.7-4) plotted for a concrete with 40 MPa cylinder strength which has ,{.,0 - L42 MPa (inctuding act : 0.85, âs this is a case of sustaincd comprrssion). The minimum strength, based on transvene cracking, occurs at a//r - 2.0 where the predictetl allowable stress is very close to oa6.,,,,* :0.6(l - ik/250)/"1/1 lbr a strut with transverse tension in accordance with 2-1-l iclause 6.5.2, The real failure load observed in tests on tlis conliguration is usually greater than the derived value based on crâcking. Equation (D6.7-4) can also be applied to individual bulging compression struts belween nodes, âs discussed in section 6.5.2 of this guide, where limits on the allorvable compression are suggested. lt would not be appropriate to use this method of allowing for the tensile strength ofthe concrete where the concrcLc is cxpected to be cracked from other effects, such as flexure. ln this case, the bearing stless should be limited to aq,1.,n"" = 0.6(1 - Ik/250):f"r/l- as discussed above, for cascs whcrc therc is no appropriate reinforcement, Where the load is eccenffic to the supporting afea. lurther strut-and-tic idealization would hc ncccssary to dislribute the stresses Lo their values remote from the loaded end, using the methods discussed in section 6.5 of this guide. Similarly, alternative strut-and-tie solutions will have to be developed where the section remote from the applied load is nôt the sâme as at the loaded end. This might, for cxample, occur in hollow piers made solid at thc rop only as in Fig. 6.7-4. In this case, the loâd has to spread out to the pier walls lor equilibrium so reinforccmcnl must be provided at the tic location shown. The bearing resistance may

Fig. 6.7-4. Sru!-and-lie system for hollow pier with solid top

204

CHAPTER

6. ULTIMATE LII4IT

then effectively be govenrcd by tbc compression limit for individual bulging compression struls. as discussed in section 6-5. if they are nôt themselves reiuforccd transversely, 01' bJ the nodes thenselves. Mole complex geometries, such as those in Worked examples 6-5.1 and 6.7-1. will generally rcquirc a chcck ol the struts and nodes. 2-2lclwse 6.7(105,) mâkes reference to EN 1992 Annex J lbr further guidance on bddge bearing areas. 2-2iclause J-104 coniirms that 2-l-1r'clauses 6.5 and 6.7 are lelevant lo the design o[ bcaring arcas and adds some requirements on edge distances and high strength concrete. These are discussed in Annex J of this euide.

.irtgs, .The load

STATES

2-2/clouse

6.7(r0s)

in each beari

strength of The overall strul-ând-tie idealizarjon is shown in Fig. 6.7-5.

srrçrgthr ôf 30 !4Ba.anr1.iÏe

l

Bearing pressure at node I

Node I is a CCT node accortiing to 2-l-l clausc 6.5.4(4Xb) with timiting stre rnano* - kù'J"a- 085r rl- 10/250) r 0.85 x J0/1.5 -l?.72M,Pa. This couLr i*i"ii"A tv ià",^ to l+ ln'lpa in accordance with 2-l-l'clause 6.5.4(51 as t}re angle between srrut-and-tie is greater than 55". Thc partially loaded area rulei; cannot be directly h*. due ro the presence of Tie I passing through the node, generating tensilc

:11:d strÊss.

:

Thc applic.rl srrcss ar lhc bearing surface .11.5.{ 106/i1200''800) - Il.98MPa < I2.?2 Mi'a so OK. The sftess at rh; node edge Ineeling sLrul A must also be checkcd.

The applied stress

-.

. , '|

'4

.;

.,lnu

7

-^" i,.i... '

: i

. ll:l::....,'..'

,

so OK

205

DESIGNERS' GUIDE

TO EN I992-2

e x dllectron:

-l

{1.8

-

l

0.8r 8

Y

I

t.5

t.60 MN

-

<

206

.1

-u

CHAPTER

.

6. ULTIIYATE LIMIT

STATES

iri.âI]ôi{ lol

-

-

-d

PLAN

Îe 1 -

11 no.

tl

.8 m

25t)

72G.,Boommlffi

trffiL I *'= i

--l

'L

\,'';-.",

I 1..*

r ][*

Êrr E or^.^-,,^- and -^-]-resulùng - .,.,^^ reinforcement -^i^r^-----^r (excluding 9-?/^nnav 2-2lAnnexJJl 2-2lAnnex dispelsal ,--.u,*hd ditpergal Flg. l6.7-5. :inforcement {-^from *-u load ,,.^-'^., Pier -*.. cross-s€ction requirements) for Worked example 6.7-l

207

DESIGNERS' GUIDE

TO EN I992-2

6.8. Fatigue This section covers the rules given in sectiôn 6.8 of EC2-2. Guidance on damage equivalent stress calculation is given in the comnentary on 2-2,/Annex NN.

2-2/clause 6.8. | (t 02)

6.8.l. Verification conditions Throughout the lifc of a bridge, constant roâd or rail trallic loading will produce largc nunbers of repetitive loading cycles in bridge elements. Both steel (reinforcing and presffessing) and concrete components which are sub.jccted lo large numbers of repetitive loading cycles can become susceptible to fatigue damage. As a consequence, 2-2lclaase 6.8.1(102) requires that fatigue asscssment is undertaken lbr structures and structural components which are sub.iected to regulâf load cycles- Somc cxccptions, where fatigue verilication is generally unnecessary, are provided in the note to thât clâuse âs follorvs:

. . . . . . .

Footbridges, cxcept those components very sensitive to wind action. The ll.rost con.rmon cause of wind-induced laLigue is vortex shedding. EN 1991-1-4 covers wind-indused fatigue.

Buried arch and frane structures with â minimum earth cover of l.0m (road bridges) or l.5m (railway bridges). This assumes a certain amount of arching of the soil, which suggests that span should âlso be relevant. Foundâtions. Piers iurd columns not rigidly connected to bridge superstructures. 'Rigid' in this context is intendcd to refer to moment connection as pinned connections will not usually lead to cycles of signiûcant livc. load stress range. Retaining walls of embankments for roads ard railways. Abutments which are not rigidly connected to bridge superstructures (with the exception

of the slabs of hollow abutments). Prestressing and reinforcing steel in regions where, under the frequent combination of actions ând Pç (presumably Pç.;"1), only compressive stresses occur at the extrcme concrete tbres. This is because the sirain and hence stress range in the steel is typically smalL while the collcrete remâins in compression.

The National Annex may give other rules.

6.8.2. Internal forces and stresses for fatigue verification 2-l-l/clquse 6.8.2(r)P I- l /douse 6.8.2(2)P 2-

2-l-l/clouse 6.8.2(3)

2-1-l lclause 6,8,2(1)P requifes stresses to be calculated assuming cracked concrete sections, neglecting the tensile strength of the concfete. Shear lag should be taken into account wherc relevant (2-l-1,/clause 5.3.2.1 relers). 2-1-llclause 6.8.2(2)P additionally requires the eflect of different bond bchaviour of prestressing ând reinforcing steel to be taken into account in the calculation of reinforcemenl stress. This results in an increase in sttess in the reinfofcing steel lrom that calculated using a crâcked elastic cross-section analysis by a factor, 4, given in 2-1- 1i Expression (6.64). 2-IJ lclause 6.8,2(3) requires fatiguc vcrification to be undertaken for the design of shear reinforcement, which is a new check for UK prâctice. Steel forces are calculated from the truss analogy using a compressive strut angle ol d1",, For fatigue câlculation, it is impôrtânt to use a realistic estimat€ of the stress range- It is thcrefore appropriate that this angle is taken greater thân that assumed fol the ultimâte limit stâte design (within the angular limits of 2-l-llclause 6.2.3(2), since the latter is the angle at the ultimare limit state âfter a certain âmount of plastic redistribution has taken place to reduce the stress in the links and to use them optimally. As a result, dfiL rnay be taken as: tandio,

: y\alO < t-O

2-l- li(6.6s)

where 0 is the angle of concrete compression struts to thc bcam axis assumed in the ultimate limit rtate sheal'design. For shear reinforcement inclined at an angle.r to the horizontal, the

208

CHAPTER 6. ULTIMATE LIMIT STATES

'lvlean' stress level

lrom non-cyclic loads Compression

Fig. 6.8-

|

.

Stress ranges

for reinforcement fatigue verification

caused by sâme cyclic action at differenl

mean stress levels

steel lbrce can be determined by rearranging 2-

/,.;(cot where À

/

I-

l7'Expression (6.13). thus:

AI/.s i cot o) sin n

(D6.8-1)

d6,,

is the rhear force range

6.8.3. Combination of actions Thc calculalion of thc strcss rangcs for latiguc vcrilication to ECz requires the appJied load to be divided into non-cyclic and fatigue-inducing cyclic action effects. The basic combination of the non-cyclic load is defined by Expressions (6.66) and (6.67) of EN 1992- l-l and is equivalent to the deflnition ofthe lrequent combination for the serviceability limit state. The cyclic action is then combined with the unfavourable non-cyclic action to determine the strcss ranges 2-l-li Expressions (6.68) and (6.69) rcfer. The non-cyclic âction gives a meân stress level upon which the cyclic part of the âction effect is superimposed, as illustrated in Fig. 6.8- I for reinforcement. Meân stress is important as it determines whether the sign of the stress in an element reverses in the course of a cycle of loading. In Fig. 6.8-1, the reinforcement stress range fbr a given cyclic action is less fbr the smaller tensile mean stress as part of the cyclic loading then causes compression in lhc concrete, which reduces the stress in the reinforcement for that part of the cycle.

6.8.4. Verification procedure for reinforcing and prestressing steel The number of cycles to fatigue failure of a steel compônent is a function ol the stress that each loading cycle induces in the component aud the type of component. Since the relationship of stress range (Ao) to the number of cycles to lailure (N) is exponential, thc relationship is normally plotted graphically in the form of a logAo logN curve. These types of curve are comraonly reïèrred to as S N curves.

2-I-l lclause 6.8.4(I

)

a1lows the damage produced by cycles

of a single stress range of N curves for reinforcing

amplitude Âo to be determined by using the corresponding S

2-l-l/clouse 6.e.4(t)

and prestressing steel. The lbrm of these curves is illustrated in Fig. 6.8-2 for reinlbrcement; the diagram for prestressing steel is simiLar, using 0.1 % proôf stress in place of yicld stress,

rog lyk

N,

tog N

Fig, 6.8-2. Characteristic fatigue strength curve (S-N curve) for reinforcing steel

209

DESIGNERS' GUIDE

TO EN I992.2

Recotnmended values dcfining the appropriate S N curve geometry for the steel component

under consideration are given in 2-1-l/Tables 6.3N and 6.4N for reinforcenent and prestressing stecl rcspectively. The recommended pârameters therein may be modified in the

National Annex. 2-l-liclause 6.8.4(l) and also 2-1-l7clause 2.4.2.3()\ require a partial factor. 1p 6",, to bc applied to âll fatigue loads when calculating thÊ stress range. The vâlue for 1F râr is deflned in thc National Annex and is recommendcd by tsC2 to be Laken as 1.0. The resisting stress range at N" cyclcs, 4o1"1. given in 2-l-liTables 6.3N and 6.4N- also has to be divided by Lhe natelial partial safety factor 16.,. The recommended value for 1s rrr from 2-l-lT,clause 2-l-1,/clouse 6.8.4(2)

2.4.1.411; is 1.15. In real fatigue assessnent situations fbr concrete bridge design, there will be more than one sfrcss range actlng on the steel elemcnt throughout its de sign life. 2-I-l lclaase 6.8.4(2 ) allows multiple amplitudes to be treâted by using a linear cumulative damagc calculation, known as

the Palmeren-Miner sunrmation:

,*:Iffi.'o

2-r-t l(6.70)

where:

r(Ào;)

is the applied nunber of oycles lbr a stress range of Ao, N(Ao1) is the resisting number of cycles for a strcss rangc of Ao;, i.e. the number of

loading cycles to làtigue failure For most bridges, thc above is a complex calculation becâuse the stress in eâch component usually varies due to the random passage of vehicles from a spectrurn. Details on a road or râil bridge could be assessed using the âbove procedure if the loading regime is known at design. This includes the weight and number of every type of vehicle thât will use each lane or track of the bridge throughoul its design life. and the correlation between loading in each lane or track. ln the majority of câses this would require lengthy calculations. As an alternative to the use of2-1-liExpression (6.70).2-l-liclause 6,8.5 allows the use of simplified faLigue Load Models 3 and 71. from EN 1991-2. lor road and rail bridges rcspcctively, in order to reduce Lhe complexity of the latigue assessment calculâtiôn- It is âssumed that the fictitious vehicle/train aloDe causes the fatigue damage. The calculat€d stress from the vehicle is then adiusted by fâctors to give a single stress range which. for N* cycles, causes the sâme dâmage as the actual tlalïic dudng the bridge's lifetime. This is called the 'damagÇ equivalent stress' and is discussed in scction 6-8.5. 2-l-l/clouse 6.8.4(3)

2-l-l/clouse 6.8.4(s)

2-Uclouse 6.8.4(r 07)

2-I-llclause 6.8.4(-l) requires that. where prestressing or reinforcing steel is exposed to latigue loads, the calculâted stresses shall not cxceed the design yield strength of the steel as EC2 does not cover cyclic plasticity. 2-1-l lclause 6.8.4(5) relates to assessment of existing structures, which is strictly outside the scope of EC2, so its inclusion is curious. [ts rclercnce to corrosion is not explicit about either the degree of corrosion or its nature (e.g. general or pitting), so a single vâlue of stress cxponent to cover âll situations is dubious. Nevertheless, it was not intended that any such allowancç for corrosion be made in new design. 2-2lclause 6.8.4(107,) permits no latigue check to be conducted lbr external and unbonded tendons lying rvithin the depth ôf the concrete scction. This is because the strain, and hence stress, vâriâtiôn undcr scrvice loads is srnall in such tendons. Consideration should be given to fatigue in external tendons which arc outside the depth of the structure (such as in €xtradosed bridges) as the fluctuation in stress might be more signilicant here. This situation is coverecl by EN l99l-l-11-

2-

6.8.5. Verification using damage equivalent stress range

2-l-l/clouse

In the damage equivalent strcss range nethod, described by 2-l-llclause 6.8.5(1) àîd 2-I-11 clause 6.8.5(2). the real operational loading is represcnted by N" cycles of an equivalent single amplitude stress range, Ao,,..,(N-), which câuses the same damage as the actual

l- l /douse 6.8.5(t ) 6.8.5(2)

2r0

CHAPTER 6. ULTIMATE LIMIT STATES

trâmc during the bridge's liferime. This stress range may be calculated for reinfbrcing or prestressing steel using 2-2iAnnex NN, as illustrated in Worked example 6.8-1.2-1-l/ clause 6.8.5(3) contains a verification formula for reinforcing steel, prcsLressing steel and snlicins devices: 1p,6",

Ao.

(

"0,

N- )

whcrc:

.

2-1-t l(6

2-l-l/clouse 6.8.s(3)

7r)

-.*P

Ao""o,(N-)

is the appropriate damage equivalent stress range (converted ro

Aoq.1(N")

from 2-2iAnnex NN is the resisting stress range limil at ,\'' cycles from the approprialc S N curves given in 2-1-1/Tables 6.3N or 6.4N

N- cycles)

2-1-llExpression (6.?1) does not coyer concrete fatigue vetification. 2-2,/Annex NN3,2 provides a damage equivalent verillcation for concrete in railway bridges, but there is no similar verilication for highway bridges. For highway bridges, concrete can be verified using the methods in 2-2iclause 6.8.7, as illustrated il] Wolked example 6.8-2.

2tl

DESIGNERS' GUIDE

TO EN I992-2

6.8.6. Other verification methods

2- I- I/clouse

6.e.6(t)

2t2

2-1-liclause 6.8.6(l) and (2) give alternative rules for fatiguc vcrilications ofrcinforcing and prestressing steel componcnts. These methods are intended as an âlternâtive to checking fatigue rcsjstance using 2-1-11'clauses 6.8.4 or 6.8.5. 2-1-I lclause 6.8.6( /) allows the lhtigue pelformance of reinforcement or prestressing steel to be deemed satisfactory ilthe stress range under the frequent cyclic load combined witll the

CHAPTER 6. ULTIMATE LIMIT STATES

basic combination is less thân Àl for unwelded reinforcement or Àr for rvelded reinforcenlent. The values ofkl and A: mây be given in the Nâtional Annex and ECl2 recommends Laking valucs of 70 MPa and 35 MPa respectively. The meaning of 'fi'equent cyclic loading' is not

it

implies a calculation based on the fatjgue load models in EN l99l-2. Assumûrg this to be the case, it will usually be preferable to perfonn a damage equivalcnt stress calculation using 2-2iAnnex NN as this also uses the latigue load models of EN lSSl-2 rnd uill lerLd lo a more economic anrwcr. 2-1-l lclause 6.8.6(2) allows the stress range altelnatively to be calculated directly hom the frequent load combination to avoid thc need Lo calculate slrcss ranges from fatigue load rlodels or directly from traffic dâta. However, the recommended allorvable stress ranges abovc would mean that elellents would rarely pass such a check. Where welded joints or splicing dcviccs arc uscd in prcstressed concrete construction, 2-1-llclaase 6.8.6(37 requires thât no tension exists in the côncrete section within 200mm of thc pres|Iessing tendons or reinforcing steel under the Irequent load combination when a reduction factor of k3 is applicd to thc meal valuc ol thc prestressing folce. The value of / 50MPa, where:

tion (compression measured as positive) is the minimum compressive stress LLnder the ftequent lôad combination at Lhe same fibre where oc.mar occurs. o. -1n should be taken as 0 ifnegalive (in tension) is thc concrele design fatigue compressive strength defincd in the code âs:

./"a.r",

-

6.8.6(3)

is the maximum compressive stress at a libre under the frequent load combina-

dc.min

.fa,a,

2-l-l/clouse

fât

but limited to 0.9 for /;k <

dc.max

6.8.6(2)

6.8.7(2)

the non-cyclic loading used for the static design:

9i,r{ < q.5 + 0.45 "-.-i"

2-l-l/clouse

k

r

A*(t r')./,, (

t

-

2-2i6.76)

-&)

where:

k]

is a coeilicient delined in the National Annex and is recommendcd by EC2 to be taken as 0.85

,3""(fu) is the coeflicient for côncrete strength at first cyclic loading from 2-l-l/clause 3.1.2(6)

l0

is the âge of the concrete in days upon flrst cyclic loading, i.e. the age

a

t which live

load is first applied

-4a

is the dcsign comprcssive strength ofconcrete. A value for o". of 1.0 is intended to be used here in conjunction with À1 : 0.85, asÀrperformsa similar luncLion of

accounling for sustained loading

For concrcte road bridgcs, this allerlative concrele faLiguc verification is unlikely to govern design, other thân possibly for very short spans where the majority of thc concrete

213

DESIGNERS' GUIDE

2-l-l/clouse 6.8.7(3)

TO EN I992-2

stress is produced by live load. It will therefbre generally be appropdate Lo usc this simplified check. No gr.ridance is given on thc calculaLion ol the concrete stresses: ignoring concrete in tcnsion will be a conservative assumption. 2-I-llclquse 6.8.7(3J permits thc abovc simplified verification ofconcrete to be applied to the compression stl'uts of members subjected to shear and requiring shear reinforcenent. Since the comprcssion struts have transverse tension passing through lhcm (see discnssions in section 6.5 of this guide), /161o, has to be reduced by the factôr r,, defined in 2-l-liclause 6.2.2(6), and rhe verification becomes:

o.;'",,*

a

u|at,t:tt

0,5

*

0.45

o"_.."'n

(D6.8-2)

u.lctt.tat

The stresses o".-n* and o.,min cân be câlculated for reinforced concrete beams, with shear

reinforcement inclincd aL an angle obtained by re-arranging

2- l -

|i

a to the horizontal, from the lollowing

expression

Expression (6.14):

/ l+eolld \ ô*,: \cot 4 + cot o/ /Fd

(D6.8-3)

I/66 is thc rclevant shear force under the liequent load combination ancl the other symbols

2-l-l/douse 6.e.7H

are defined in 2-1-1,/clause 6.2-3. Thc concrcte stress increases with reducing strut ângle d so. in this case. it is conservative to base 0 on its ULS value in the above calculation rather than the larger angle rllç", from 2- l - l /Expression (6.65)For members subjected to shear but not requiring shear reinforcement. 2-1-1/ clause 6,8.7(4) provides the iollowing expressions for assuming satisfactory fàtigue resistance in shear:

ro, -fn ''" , 0, /' '' ' '*l rLd,rno l/Ro,,

but limited to 0.9 for or-

fo,

t/-. 'Ed'"' 'fd,m'\

.1"1

o.s

-

o.+s

{f0"

"

l

2-r-11(6.'78)

l/Rd."

< 50 MPa or 0.8 for /"1 > r' t/-. I

50

MPa:

'#,*

- r-) ,,!-"'ro: lrRd.

2-1,11(6.7e)

'Rd.

where:

/s..* /s6.n;,

y'ra."

of thc maximum applied shear force under the frequent load combination is thc design value ol the minimum applied shear force under thc frequent load combination in the cross-section, where tr/Ed,,','âx ôccurs is the design value

is the design shear resistance lrom 2-2,/Expression (6.2.a)

:s!tess).1-.,

!!G:'!ru'l) : 4,t

2t4

ln .'.

.

CHAPTER 6. ULTIMATE LIMIT STATES

Since the maximum and minimum fihrc stresses under the frcquenr load combinarion have becn calcLilated. 2- l-llExpression (6.77) must be satisfred:

,.-*. '4a

i'' {o.s-o+s;" Io.o rol.7* < 5o MPa .i.i . i...:

: ,:,.

wherC; from 2-2lFxnrcssion (6.76t:

-

1.,r.,",

Érl*(ro)/.a (l

-

...-.tt

f,kl250) with

À1

-

I ,,:.-:

0.85 (recorn-

mended r alue.1. It is conservatively assumed here rhat construcùon tralÏc uscs the hridç from an age of r*. r, first cyclic loadiug is r0 7 dâys. Cenerally, Joad would not be applied

:

Irdlâ;tn"

From 2-1-llclause 3.1.2(6): s

i*{r)

- cxp(r{i /tÛill

- 0.25 for rapid hardening, normal slrcngth cemcnt and from 2-l-llclause 3.1.ff I ): 1^.' ctuf,y/t, - 1.0 x 35/1.5 = 23.11 MP.t

ltaking o."

=

1.0

for

fa

rigue as discussed in the main texlJ-

Thcreforc

L.(7) =

exp(0.25

" tt t/QW)t - o.t'tts

and /"6.1;1

-

0.85 x 0.7788

x

23.31

{

(l

35/250)

- ll,lMPa

The final checks are therefore as follows:

(l)

Sagging section: o,

(2)

^"*/J,,t.t,,

-

4.0/11.3=o..ro

This is < 0.9 and < 0.5 + 0.45.'cnin/Â,fâr - 0.5 +0.45 r I.0/13.3 = 0.51 as required. Thcrcfore. the saggjng sectiol rt mid-span has adequate fatigue re.;istancc. Hogging secdon:

o,..n / f,a rut: a 5/I 3 3 - tr'34

Tfus

is

J 0.9 and S 0.5 - 0.45o.6"/ /"1rar - 0.5 r 0.15 .

O.7

/13.3 0.52 as

required. Therefore. the hogging section at an intermediate suppofl bas adeguate fatigue rcsistance.

6.9. Membrane elements A

problem encountcrcd when using lineiu elastic finite element techniques to analyse concrete bridges is that the results produccd are usually in the form of stresses, while the code resista[ce rules are presented in terms of stress resultants, such as shear folce and bending moment. 'fhis applies to the rules for bending and shear gir,'en in sections 6.1 and 6.2 respectively of EC2. The rules for mcmbranc eler'ncnts presented in 2-2lclause 6.109(10l) provide a way of designing directly from the stresses produced by a trvodiurensional linear elastio finite element nrodel. The sign convention for stresses in 2-2i' clause 6-109 is shorvn in Fig. 6.9-1. The rr.rles âre âlso intended fol use with elements under out-of-plane bending and torsion iD coniunction with the sanrlwich model of Annex LL. Annex MM gives speciûc recommendations lor thc dcsign of box girder webs in shear and transverse bcncliug. It should be noted thât the use of these mc'rnbrane rules rvherc other member resistânce formulae coukl be used (such as the sheâr model in section 6.2) will generally lead to a

2-2/douse 6.

t0e(t 0t)

2t5

DESIGNERS' GUIDE

TO EN I992-2

ta",

1

| '*"

l-

]

rf; t

Fig, 6.9-1, Sign convention used in membrane rules

2-2,/clouse

6.t09(r02)

lower calculated resistance. This is because the membrane rules do not consider plastic redistribution within the cross-section or allow for the beneficial results of physical testing used specifically to derive the other member resistance rulcs. To design the reinforcement and check the concrete compressive stresses,2-Z/c/czre 6.109( 102 ) requies a lower-bound solution, based on the lower-bound thcorem ofplasticity, to be used.2-2,/Annex F gives equations for designing the reinforcement. Unfbfiunately, the sign convention for direct stress in Annex F differs from that in 2-2,/clause 6.109 and Fig. 6.9-l (as compression is taken as posilive in Annex F but negative in 2-2/clause 6,109). The general equations (F.8) to (F.10) of Annex F âre therefole reproduced below wilh a n.rodification to make them comDatible with Fie. 6.9-l:

: pyo"y : o.,1 : p,crs"

<

p",/ya,^

(D6.9-l)

ltir,yl tan0 a op1, ! l46,rl(tang+ cotd)

pyfya-y

(D6.9-2)

116r"yl

cot

I+

"s,r,

=

rcdrna,(

(D6.e-3)

p.o, and'/ya are the reinforcenent ratio, reinforcement slress and reinforcement

2-2/clouse

6.t09(t03)

2t6

design

yield stress in each direction respectively. Él is the angle of the assumed plastic compression field to thc x-axis. o",1..u, is the maxirlum compressive stress in thc concrete stress field. Significant limitations of Annex F and the above expressions are that the reinibrcement must be aligned with the X and Y directions in the analysis model (although it is possible to rotâte the output sLrcss field to align with lhe rcinforcement using Mohr's circle) and the reinlbrcement must not be skew. In the lâtter case, eilher the expressions need to be modificd or analogous lower-bound methods used. A modified sel of equations for skew reinforcement is presented beneath Worked example 6-9-l . Where it is required to design reinfbrcement lbr moment fields in slabs with or without skew reinfbrcement and without a net in plâne axial force or shcar forcc, mcthods srLch as those in references 19, 20 and 2l qould be used. plovided a check on the concrete compression lield discussed below can be included as neccssary. 2-2lAnnex F gives versions ofthc above, by way ofits expressions (F.2) to (F.7), which are optimized to mirimize the reinforcement provision (taking tan0: I if thc greatest compressive stress is less in magnitude than the shear stress or tan9: l"oa*r/osa,] othenvise). Urfortunately, these optimized equâtiotrs are often not valid according to the rulcs in 2-2/ clâuse 6.109, as the optimized angle may lie outsjdc the allowable limits below. Since concrete hâs limited ductility, it is not possible to indiscriminately apply the lowerbound theorerr of plasticity and therefore limits are set rn 2-2lclause 6,109(103) on the deviation of the assumed plastic compression fiekl direction at angle â. lrom the angle É"1 ol the un-cracked elastic principal compressive stess direction. (This is similar to thc qr.ralitative requiremcnts for strut-an

20 M Pa

CHAPTER 6- ULTIMATE LIMIT STATES

2-2/clouse F. t

.J

(t 04)

JJ

The correct solution is thcn yiclticd. This example is only intcnded tô highlighl tbe prohlcms thaL can occ.ur b5 sticling rigidly lo t}e elaslic compression ang.lc. which might occur where spreadsheels are uscd lo aulomâte calculations for example. The cxamplc of zcro rhear stress mây seent arÛficial. bul a similâr procedurc of departing lronl the ehstic angle is required iI thc above example is repeated with a small lalue c'f shear rtrcss. (5) tiniatial tension (repeated with tension in tbe Y direction) Assume {he lollowing stresses are obtainetl; o1a,

-

0 MPa

ï:-:ii

221

DESIGNERS' GUIDE

TO EN I992-2

.. ..x.

x

i:;;:;; 'l::-..:.

//-ro./4\ i ',r À\r | , *âr

""",=l

\ vl,.\ r; 61 =

:

t oz-z.u

| \\,/ I l-.>o

-Ai

V-*.

:

\\\

I

1,.*

--iî---=--t/ I

rF4{,

\*zt':\,

:':::l: :::.:: :l I

llt1

,""*V-'*o, '. ---l

'K_/

,;;;;*;;*;;*;;;',, (c)

*

(d)

:.

..i:r,trl,:l

Skew reinforcement

It

is possible to derive similar expressions to equatioos (D6.9-l) to (D6,9-3) for cases with skew reinfolcemenl. Such cxpressions have been presented by Bcrlagnoli, G., Carbone, V.L, Giordano, L. and Mancini, G. Unfortunately, their expressions as presented ât the C.I- Pren.rier Congress on 1 2 July 2003 in Milan entitled '2nd Intcmational Speciality Conference on the Conceptuâl Approach to Structural Design' contâined â typographical mistake and cânnot therefore be referenced. They are reproduced below with moditcâtions to côrrect the error and to mâke the notation compatible with EN 1992. The sign convention is shown in Fig. 6.9-3.

** .

I

".,,

W Fig. 6.9-3. Sign convenlion for membrane rules with skew reinforcemenr

222

CHAPTER 6. ULTIMATE LIMIT STATES

Po"' P'o ' .rcd

:

rFdx sin d

cos,6

oËdv cos d

sin(É o66" sin d sin o

oEdx

-

iEdxy

sin B +

-.rlcos(',

TEdxy

cos(0 + B)

* oEdv cos d cos.l + lEo*u sin(t/ + cr) co,ç0 41co.(o ,1

târ0

(D6.9-4)

31

- pâc,s.cos(d o)AT+p,o,rsin19 61-Il!

(D6,9-5) (D6.9-6)

In using the above equations, it is vital that the sign convention for the angles and stresses in Fig. 6.9-3 (anti-clockwise positive) is observed and that the directior of the plastic compression Iield, 0, is taken to be ir Lhe same X, Y quadraut as thc angle of the principal compressive stress in the un-cracked state, d"l, The former is not required in equâtions (D6.9-l) to (D6.9-3), where d is alrvays taken as positive because of Lhe mod sign introduced on the shear stress Lcrms.

223

CHAPTER 7

Serviceability limit states This chapter deals with lhe design at service lir.nit states of rnembers as covered in section 7

of EN 1992-2 in the followins clauses:

. . . .

7.I

General

C leuse

Stress limitation

Clause 7.2

Crack control Deflection control

( ldust

/ -J

Clause 7.4

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

7. 1. General EN 1992-2section 7 covers only the three serviceability limit states relaLing to clâuse 7.2 to 7.4 above. 2-l-llclause 7.1(1)P notes that other serviceability limit stâtes 'may be of importance'. EN 1990/A2.4 is relevant in this respect. It covers partial factors, seryiceability criteria, design situations, comfbrt criteria, detbrmations of railway bridges and criteria for

2-

l-

l/clouse

7.t(t)P

the safety of rail tralllc- Most of its provisions are qualitative but some recommended values are given in various Nôtes, âs guidance for National Annexes. EN 1990 is of general relevance. From clause 6,5.3 of EN 1990, the rÊlevant cômbinâtion of actions for serr.iceability limit states is 'normally' either the characteristic, frequent, or quasipernanent combination. These are all used in EC2-2 and the gcneral forms of these

combinations, togethcr with examples of use, are given in Table 7.1, but refèrence 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. âctions that need not be considered together), recommended sombination factors and partial safcty factors for bridge design are also specified in Annex A2 of EN 1990. Sectjon 2 of this guide gives i-urther commentary on the basis of design and lhe use of pârliâl factors and combinâtions of actions. The general expressions in Table 7.1 have been simplilied assuming that partial fâctors of 1.0 are used throughout lor all actions at the scrviccability limit stâte, âs recommended in Annex A2 of EN 1990, but they may be varied in the Nâtional Annex. Appropriate methods of global analysis for determining design action effects âre 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). Elâstic globâl analysis is most commonly used and it is not normally then necessary to considsr the effects of cracking within it section 5.4 refèrs. 2-I-llcluse 7.1(2) permtts an un-cracked concrete crôss-section to be assuned for stress ând deflection calculation provided thât the flexurâl tensile stress undcr the relevant qombination of actions considered does not exceed .Ât."n. .,[t."n may be tâken as either l;rn

2-l-l/clquse 7.

r(2)

DESIGNERS' GUIDE

TO EN I992-2

Table

7.l.

Combinations of actiôns for serviceability limit states

Combination

General expression

Characteristic

Combination of acrions wirh a fixed (small) probâbiliry of being exceeded durint normal opêration within the structure's design life, e.E. combinâtion eppropriate to checks on stress in reinforcement as it is undesirable for inelâstic deformation of reinforcemenr ro occur ar âny time during the service life.

lG1; +P*Q4r *Ido.,Qt,

Frequent

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

tGkJ +

P

*

4r.rQr.,r

*I4,,,Q0,,

Quâsi-

Combination of actions expected !o be exceeded approximately 50% ofthe time, i.e. a rime-bâsed mean. For example, combination appropriate to crack width checks in reinforced concrete members on the basis that durability is influenced by âverage crâckwidths, nor the worst crack width ever experienced-

I

P

*

ry'2

* E û2.,Q1.,

Permanent

Gç,;

+

|Qp.|

or./"t .n but should be consistent with the vah.re used in the calculation of minimum tension reinfbrcement (see section 7.3)- For the purpose of calculating crack widths and tension stiffening effects, /1,. should be used

7.2. Stress limitation in bridges are limited to ensure that under normal condiLions of use, assumptions made in design models (e.g. linear-elastic behaviour) remâin valid, ând to âvoid deterioration such as the spalling olconcrcte or excessive cracking leading to a reduction ofdurability. For persistent design situations, it is usual to check stresses soon aftcr thc opening of the bridge to traffic, rvhen little crccp has occurred, and also at a later time when cleep and shdnkage are SLrcsses

substantially complete. This aliects the loss of FrcsLress in preôtressed structurils and the modular râtio for strcss and crack width calculation in reinforced concrete structures, It may be necessary to include part of the long-tijrm shrinkage effects in the first check, because up to half of the long-term shdnkage can occur in the fust 3 months âfter the end of curing of the concrete. Calculation of an effective concrete modulus allowing for creep is discussed below. 2-

l- I /dause

7.2(t )P

2-2/douse

7.2(t02)

2-I-llclnase 7.2(1)P regui.res compressive stresses in the concrete to be limited to avoid longitudinal cracking, micro-cracking or exccssivc creep. The trst two can lead to a reduction ofdurabifity. 2-2lclause 7.2(102) add,resses longitudinâl cmcking by requiring the stress level undel the chârâcteristic combination of actions to not exceed a limiting value of t1llp (for areas with exposure classes of XD, XF or XS), where fr1 is a nationalJy dctermined parameter with recômmended value of 0.6. The clause identifies thât the limit cân be increased where specific measures ârc tâken, such as increasing the cover to reinforcement (from the minimum values discussed in section 4) or by providing confinement by transverse reinforcement. The improvement fiôm confining rcinforcement is quantified as an increase in allowable stress of l0oÂ, but this may be varied in the Nâtionâl Annex. The design ofthis reinforcement is not covered hy EC2-2, but ihe strut-and-lie rules in 2-27clause 6.5 and discussions in section 6.5 of this guide are relevant. Such reinforcen.rent would need to operate at low stresses to have any significant effcct in limiting the rvidth of compression-induced cracks. Micro-cracking typically begins to develop in concrete where the compressive stress exceeds approximately 70% of the cômpressive strength. Given the limits above to control longitudiflal cracking, no further criteria are given to control micro-clacking.

CHAPTER 7. SERVICEABILITY LIMIT STATES

Fig.7.l.

Notarion for a rectangular beam

2-1-l lclause 7,2( 3 ) addresses nonlinear creep âs covered by 2-2/clause 3.I .4(4). It requires nor.r-linear creep to be considered where the stress under the quasi-permanent combination of' âctions exceeds À1jly, where È2 is a nationally determined parameter with recommended value of0.45. 2-2iclause 3.1.4(4) gives the same limiting stress, but it is not subject to nâtional variation in that clause so must bc deemcd [o takc prccedence. 2-I-llclause 7,2(4)P rcqtrnes stresses in reinforcement and prestressing stcel to be limited

2-l-l/clause

to ensurc inelastic defomations of the steel are avoided under serviceability loads, which

7.2(4)P

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-l ldause 2.2/5) requires that th(] tcnsilc strcss in reinforcement under the characteristic combination of actions does not exceed k3/r1. Where the stress is caused by imposed deformations, lhe tensile stress should not exceed kaf,ç. although it will be rare for tensile stress to exist solely from imposed deformations. The mean value of stress in prestressing tendons should not exqled lr5fo;. The values of ,t3, Àa and À5 are nâtionâlly determined parameters and arc recommended lo bc taken as 0.8, 1.0 and 0.75 respectively. The higher stress Umit for reinforcement tension under indirect âctiôns reflects the ability for stresses to be shed upon concrete cracking. The lbllowing method can be used to determine stresses in cracked reinforced concrete beams and slabs. The concrcLc modulus to usc for scction analysis depends on tlle ratio olpermanent (long{erm) actions to variable (short-term) actions, The short-term modulus is 8"., and thc long-tcrm modulus is E" l(,1 +,!). The effective conffete modulus for a combination of long-term ând shôrt-term actions can be taken as: (Mqp + M.r ).E'"-

M$+

(1

+ ô)Mqp

7.2(3)

2-l-l/douse

2-l-l/clouse 7.2(s)

(D?-l)

where Mr, is the moment due to short-term actions and Mqe is the monlent from quasip€rmanent actions. The neutrâl axis depth and steel strain can be derjvcd from a cracked elastic anaLysis, assuming plane soctions remain plane. For a rectangulâr beam, from Fig. 7.1: Strains

.,

d-d,.

d. '"

(D7-2)

Forces

Ft; SO

ls4es - 0.5b4€.[

(D7-3)

,eff

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

.

1:

:-

A.E, + V (,4.2,)'. +2bA.E,E'"îd D

Lc clf

(D7-4)

277

DESIGNERS' GUIDE

TO EN I992-2

Fig. 7.2. Cracked section transformed to sreel units

The second môment of arca of

Lhe

cracked section. in sleel units. is derived frol.r the cross-

section in Fig. 7.2:

r- "ld A

d,):

+_:+bd.,

(D7-5)

The elastic section moduli are:

z.- I ld, Steel: ;, : I l(.d d.) Concretc:

(D7-6) (D1-7)

f'or a given noment, MEd, the ( oDcrefe: o -

stresses arc thcrcforc:

-'j-j: ---:l::: ts

Mna qiêê!. -

(D7-8) (D7-e)

Thc strains are:

^

L oncrclc:

__ a^: M"^

1

(D7-10)

-'- I SteelrÉ:M", t;

(D7- I

l)

Thc above may also be applied to flanged beams where either the neutral axis remains in the compression flange (when ô is the flange lvidth) or remains in the web when the flange is wholly in tension (whereupon â is the web width). The procedure for checking stresses is illustrâted in Worked example 7.1. In pârticulâr', this illustrâtes the treatment ol' creep on modular ratio.

Wôrked example

7. | :

Reinforced concrete deck slab

A reinfo rced concete deck sla b. 150 nrd ibiôk aryi wit[itisi Ci s/+j iori.crerçi ii iutûigtÈd, : momËnt:ôf :9tl.I41t:: *4af ,ifie c!Êràçtôriitiia. aôinbiûàtùd n'f : .to a transverse,saggûrg jÏs actions at SLS. momenr comprises | 5Yo permanent adions from self-ueighr and superimposed dead load and 85Yu transienL acrions iiom tratTic. The ullinute design requires a

228

CHAPTER 7. SERVICEABILITY LII.4IT STATES

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TO EN I992-2

Treatment of differential temperature in stress calculation For bridge beams, a calculation of the stresses induced by nôn-uniforin temperature distributions needs to be considered. Strictly, these stresses (which include primary sellequilibriâting stresses and secondary stresses due to rcstrajnt of deflection) need to be included in the stress checks discussed âbôve. For crâcked sections, the analysis to determine

self-equilibriating stresses is complicated and highly iterative. However, since cracking results in a reduction in stiffness of the section, cracking ofa section will lead to â substantial rclaxaLion of the stresses induced by temperature. It is therefore generally satisfactory lo ignore temperâture-induced self-cquilibriating stresses in cracked sections ând to consider only lhe secoldary efïects. The neglect of tcmperature-induced self-equilibriating stresses applies also to crack width calculâtion for reinforced concretc scctions in 2-2iclause 7-3, where temperature is also included in the quasi-permanent combinâtion for bridge members. For prestressed members where dccompression is being checked (and therefore sections are expected to be substantiâll.v uncracked), both primary and secondary effects should, however, be included in the calculation of stresses, This approach is in accordance with pr€vious UK practice and the UK's National Annex.

7.3. Crack control The consistency in use of symbols in this section of EN 1992-2 and EN 1992-1-1 is poor, with the same symbols sometimes changing delinition liom sub-clause to sub-clausc. Care is therelbre needed tô use the correc[ dcfinition in the relevant clause.

2-l-!/clouse 7.3. r (t )P

2-l-l/clouse 7.3. r (2)

2-l-l/clause 7.3. t (3)

2-l-l/clouse 7.3. t (4) 2-Uclause 7.3. t (t 0s)

l1ô

7.3.l. General considerations 2-I-llclause 7.3.1(I)P sLàtcs thal cracking shall bo limitcd to lhc cxtent that it should not impair the proper functioning or durabilitl' of the structure, or cause its appearance to be unacceptable- 2-l-l lclause 7.3,1(2) is. however, a rerrinder that cracking is inevitable in reinïbrced concrete bridges subjected to bending, shear, torsion or tension. Cracking may arisc from the result of either direct loading, from trafic actions, for example, or restraint of imposed deformations, such as sbrinkage or tcmperature movcmcnts. The rules in EC2 cover thc control of cracking from these causes and are discussed in detail in this section. Section 7.5 below discusscs early thermal cracking. Additionally, cracks may arise from other causes such as plastic shrinkage. corrosion of reinforcement or expansive chemical reactions (such as alkali silica reaction). 2-l-llclause 7.3,1(3) notes that the control of such cracks is beyond the scopc of EC2, even though thcy could be vcry large if they occur. 2-1-l lclause 7.3.1(4) and 2-2lclause 7.3.1(105) both essentially require the design crack width to be chosen such thaL cracking does not impair the functioning of the structure. Cracking norn.rally 'impairs' the function of the structure by either helping to initiate reinforcement corrosion or by spoiling its appearance. The relationsl.rip beLw€en cracking

CHAPTER 7. SERVICEABILITY LII4IT STATES

and corrosion in reinfbrced concrete has been cxlensively researched. Thc alkalinitl' of fresh concrete protccts reinforcernent fiom coftosion. This protcction can be destroyed, however, by carbonation or ingress of chlorides. Cracks can lead to an acceleration of both of these Frocesscs by providing a path

for carbon dioxidc and chloride ions to the reinforcement. The size of the cracks also has an influence on the time to initiation of reinfbrcement corrosion. Noticeable cracking in strucLures causes concel'n to the public ancl it is therefore prudent to limil crack widths to a size thât is not readily noticcable Tlre above considerations have led to the crack width limitâtions specilied in 2-2'lttble 7.101N, which is subject to \,âriâtion in a National Annex. 2-27clause 7.3-1(10-5) notcs thàt although complianc€ with the crack width calculation mcthods and adherence to these

liniting crâck widths should guarar.rtee adequate perfonnance, the calculated crack widths themsclves should not be considered as real values. For reinforccd concrete. the crack rvidth check is recommended to be perfbrmed under the quasi-permanent load cômbination. This ellectively excludes tralllc lor highway bridges when the recommended value of ?12 = 0 from Annex A2 of EN 1990 is r.rsed. The quasi-permanent combjnation does, however, include temperature. In ohecking crack widths in reinforced concrete members, only the secondary ellècts of temperâture differcnce need to be co[sidered as discussed in section 7.2 above- For bonded prestressed members, however, the self-equilibriating stresses should also be included in decompression checks. Prestressing steels are much more sensitive to damage lrom corrosion than normal reinforcement, mostly duc Lo their smaller diametef and higher level of stress undel which they normally operâte. It is thereforc widely accepted that it is necessary to l.rave more onerous rulcs for protection of prestressed concrete mcmbcrs against corrosion. This is reflected in stricter crack control criteria fbr prestressed membefs with bonded tendons in 2-2iTable 7.l0lN, It also specifles requircmer.rts for decompression checks for prcstrcsscd members wiLh bonded tendons and defines under which relevant combination of âctions the decompression check is required. For XC2, XC3 and XC4 environments, it is the quasi-permanent conbination while for XD and XS classes, iL is the frequent combination. Metlbers with only unbondcd tendons are tl'eated in the sâme way as reinlbrced concrete

mcmbers 2-2l clause 7.3.1 (6). In order to safeguard bonded tendons from corrosion. it would be logical for two-way

2-2/clouse 7.3. t (6)

spanning elements with prestressing in one direction only, such as a deck slab in a prestressed concrete box girder, to also have stricter crack critcria in thc direction transverse to the prestressing. This is not, however, explicitly required by 2-2iTable 7.101N and was not required in plevior.ls UK codesThe decompression linit check requites that no tensilc stresses oscur in any concrete

within a certâin distance, rccommended to be l00mm. of the tendon or its duct. This cnsures that there is no direct crack path lo the tendon for contaminants. Thc 100 mm requlrement is ûol a cover requirement. It simply means that if the cover is less than 100mm. it must all be in compression. Lessef covels may be acceptable, providing the minimum requirenents of z-2lclâuse 4 are met. Conversely, tensilc stressr:s are permitted in the cover as long as the concrete within 100mm (or amended value in the National Annex) of the tendons or ducts is in compression. Il, in checking decompression, the extreme fibre is found to be cracked, the check of decompression at the specified distânce liom the tendons becomes iterative. Additionally, although not stated in 2-2/Table 7.101N, if decompression is not checked at the surfaoe for XD and XS cnvironmental clâsses, a crack width check should also be performed if untensioned reinforcement is present. It is therefore simpler and conservative to check decompression at the surface of the member. If a crack $'idth check is peformed, the criterion for reinforced concrete in Table 7.101N can be adopted. Stress checks in a pre-tensioned beam are illustrated in Worked example 5,10-l in section 5.10 of this guide. In deep beams and clcments with geometricâl discontinuities, where strut-ând-tie analysis it is still necessâry to check crack widths.2-1-llclause 7,3,1(8) allows Lhc bar forces thus dctcrmined to be used to calculâte reinforcement stresses tô verify crack widths in accordânce u.ith thc rcnainder of 2-l-llclause 7.3. 2-l-llclause 7.3.1(9) iÏr is required,

2-l-l/clouse 7.3. t (8)

2-l-l/clouse 7.3.r (e)

231

DESIGNERS' GUIDE

2-2/douse 7.3.

r

(t r0)

TO EN I992.2

general permits either a direct calculation of crâck widths using 2- l-l/clause 7-3.4 or a check of allowable reinforcement stress for a given crack widtb in accordance with 2-l-l/clause 7.3.3. The latter is simpier as many of the parametets needed in 2-l-l/clause 7.3.4 relate to beam geometry. Where strut-and-tie modelling is used to verify crack widths in this way, the results will only bc rcprescntativc if Lhe strut-and-tie model is based on the elâstic stress trâjectories in the uncrâcked stâte. This is discussed in section 6.5.1 and is noted in 2-1-liclause 7.3.1(8). 2-2lchuse 7.3.1 ( I l0l suggests that 'in some cases it mây be necessâry to check and control shear cracking in webs'. These'cases'âre not delined and 2-2/Annex QQ, which is referenced ftrr further information, is equally vague other than to imply that a check is most relevant lbr prestressed members, perhaps partly because the longitudinal web coûrpression rcduces the tensile strength ofthe concret€ in the direction of maximum principal tensile stress. Previous UK design standards have not required a verification ôf crâcking due to shear in webs, but the sheal design lor reinforced concrete members at ULS differs in EC2 in two wâys. First, highcr crushing resistances are possible which mcans greater forces need to be carried by the links if the web concrete is fully stressed. Second, shear design was previously based on a

truss model with web compression struts fixed at 45". Since tbc Eurocode permits the compression struls lo rolate to flatter angles, fewer links might be provided using EC2 in solne cases to môbilize a given shear fôrce, thus creating greater link stresses at the seNiceability limit state.

7.3.2. Minimum areas of reinforcement In deriving the expressions for the calculâtion ofcrack widths and spacings in section 7.3.3, â fundamental assumptjon is that the reinforcement remains elastic. If the reinforcement yields, deformation rvill become concentrâted ât the crack where yielding is occurring, and this will inevitabl.v invalidate the formulae,

For a section suhjecrcd Lo uniform tcnsion, the force necessâry for the member to crack is N". : 1""/.,-, where N", is the crâcking load, ,1" is the area ofconcrete in Lension and /1,- is the mean tensile strength ofthe concrete. The strength of lhc rcinforcement is lr/r1. To ensure thât distributed cracking develops, the steel must nôt yield when the lirst crack forms hence: (D7-12)

4.1,u > A"l"t'" 2-2,/clouse

7.3.2(r02)

Equation (D7- 12) needs to bc modilied for stress distributiôns other than uniform tension. 2-2lclause 7,3.2(102) introduces a variable, À., to account for different types ôf stress distribution whicl.r has the effect of reducing thc rcinforcement requirement when the tensile stress reduces through the section depth. A furthel'fàctor, fr, is included to allorv for the influence of internal self-equilibriating stresses which arise where the strain varies nonJinearly through the member depth. Common sources of nonlinear strain variation are shdnkage (where the outer concrete shrinks nole rapidly than the interior concrete) and temperaturc djflerence (where the outer concrete heats up or cools more rapidly than the interior concrete). The self-equilibriating stresses that are produced can increase the tension at thc outer fibre, Lhus leading to cracking occurring at a lorver load than expected. This in turn means that less reinforcement is necessâry to carry the fbrce at cracking and thus to cnsure distributed cracking occurs. The lactor /r thercforc reduces the feinforcement necessary where sell-cquilibrjating stresscs can occur. These stresses ate tnore pronounced 1br deeper members and thus k is smaller for deeper members. Tbc minimum required reinlorcement area is thus given as: A. n,,,o.

-

2-2lQ.t)

k"kf6."1sA "1

where:

A"t

232

is the required minimum area of reinforcing steel within the tensile zone is the area ofconcrete within the tensile zone. The tensile zone should be taken as that part of lhe concretc sectioD which is calsulated to be in tension just belore the fbrmation of the first crâck

CHAPTER 7. SERVICEABILITY LIMIT STATES

crs

is the absolute value of the maximum stress permitled in the reinforcement immediately âfter formation of thc {irst crack. This will generally need to be taken as the value to satisfy the mâximum bar size or bar spacing requirements of 2-l-1/Tahtes 7.2N or 7.3N respectively

It"r

is the mean value oftensile strength ofthe concrete eflective ât the time when the cracks are expected to first occur, i.e. if,,n or lower (/",-(l)) ifcracking is expectcd earlier than 28 days. lu maDy cases. where the dominant imposed deformations result from dissipation of the heat of hydration, cracking mây occul within 3 to 5

fron

casting. However, 2-2lclause 7.3.2(105) requires that /;1,,,(t is not 2.9 MPa, which corresponds to the mean 28-dây tensile strength of grade C30/37 côncrete. The use of mean tensile strength (rather than the more conservative uDner characteristic va.lue which was used in Modet Corte 906) has, in nart, bcen used t,r produuc simiLar minimum reinfolcement provisions to those obtâined with previons European practice is a cocilcicnt allowing for the eflèct of non-uniforn self-equilibriating stresses and should be taken as: 1.0 for webs with /2 < 300mm or flanges with widths less than 300mm 0.65 for webs with h ) 800 mm or flanges with widths grcater than 800 mm Intermediate values should be interpolated. A value of À of 1.0 can always days

taken as less than

k

2-2/clouse

7.3.2(t05)

con5crvalivcl) be uied

Æ"

is a coefficient allowing

for the nature of thc

stress

distribution within the sectioù

immediately prior to cracking and the change of the lever arrrr, calculaLed from 2-2lExpression (7.2) or 2-2iExpression (7.3) for webs and flanges of flanged beams respectively. It depends on the mean direct stress (whether tensile or comprcssive) acling on the part of the cross-section being checked. It is equal to 0.4 for rectangular beams without axial force. Prestressing has the effect of reducing À" for webs by way of 2-2iExpression (7.2), while clirect Lcnsion increases its valuc. A value of 1.0 is always conservative. It should be noted that 2-2lExpression (?-2) contains a term, k1, which differs in delinition and value from ihat in 2-l - li Expression (7.2) ând ânother in 2-l-l7clause 7.3.4 For flangetl beams. such as T-beams or box girders, 2-2iclause 7,3.2(102) requires that the minirnum reinforcement provision is determined for each individual part ofthe section (webs and flanges. for example), and 2-21Fig. 7.l0l identifies how the section should be sub-divided for this purposc: Lhe web height, l], is taken to extend over the full heighL of the member. The sub-division used clearly affects the calculation of somc of thc terms in 2-2i Expressions (7.2) and (7.3), particularly concrete area and mean concrete stress. Despite the apparenl similarity betrveen 2-2iQ.l) and the minimum reinforcement r€qirements in 2-l-1i9.2.1.1, the former is associated with timiting crack widths while the lattcr is associated with preventing steel yield upon cracking ol the cross-scction. Both checks must therefore be performed2-1-llcl use 7,.1.2(3/ allows any bonded tendons located within the eflective tension area to contrjbutc to the area of minimurl reinforcement required to control cracking, provided they âre within 150 mm ol the surfacc to be checkecl. 2-2iExpression (7.1) then becomes: ,4,,.n1,o"

{ {1 lnAo, -

k.kf"1,"1çA"1

2-l-l/clouse 7.3.2(3)

(D7-13)

lo

is the area of bonded pre- or post-lensioned tendons within the clTective tensile area, l",.ip (discussed under clause 7.3.4), and Àon is the stress increâse in the tendons from the statc of zero strain of the concrctc aL lhe same level (i.e. the increase in strcss in the tendons after

decompression of the côncrete at the level of the tendons). {1 is the âdjusted ratio of bondstrcngth taking into account the diflèrent diameters of prestressing and rcinlorcing steel,

is the ratio of hond strenglh of prestressing and reinforcing stcel, given in 2-l- 1/clause

6.8.2

155

DESIGNERS' GUIDE

TO EN I992-2

d,

is the largest bâr diameter oJ'reinl'orcing steel

,y',, is the equivalent diameter ofthe tendon in accordance with 2-l-l/clause 6.8.2. lfonly prestressing stecl is used to control cracking, {' - uf 2- I- I /douse

7.3.2(4)

2-I-llclnuse 7.3.2(4) allows minimum reinlorccmcnt to bc omjttcd rvhere, in prestressed concrete members, the stress ât the most tensile fibre is limited Îi] â nationally determined value. recommendcd to be Ârefl, under the characteristic combination of actions and the characteristic vâlue of prestress. This does not remove the need to consider the provision of rcinforcement to control early thennal cracking prior to application of thc prestlessing.

7.3.3. Control of cracking without direct calculation 2-2/douse 7.3.3(t 0 | )

2-l-l/clouse 7.3.3(2)

The basis of the crack width calculation n.rethod in EN 1992 is presenled in section 7.3.4. 2-2lclause 7.3,3(I0I), however, allows 'simplified methods'to be used for the control of

cracking without direct calculation and. undesirably fbr pan-Europeal consistency, allows the National Annex to specify a mcthod- The rccommcnded method is thât given in 2-1-l,iclause 7.3.3. In this method,2-1-llclause 7.3.3(2) requiles the reinlbrcement stress tô be determined from a cracked section analysis (see Worked example ?.1) under the relevant combinâtion of âctions (see section 7.1 of this guide). The relevant effective concrete modulus for long-tenn and short-term loading should also be used. It is assumcd that minimum reinforcement according to 2- 1-1/clause 7.3.2 wilJ be provided. An âdvântâge of rhis simplified approach is that mâny of the diffculties of interpretation of parameter definition involved in direct oalculations to 2-21clause 7.3.4 for non-rectangular crosssections (such âs for circulâr sections, discussed below) can be avoided. For cracks caused nainly by direct actions (i.e. iniposed forces and moments), cracks may be côntrolled by limiting reinl'orcemenl stresses to the values in either 2-1-llTable 7.2N or 2-l-liTable 7.3N. It is not necessâry to satisfy both. The former sets limits on reinforcement stress based on bar diameter and the latter based on bar spacing. For cracks caused mâinly by restraint (for example, due to shrinkage or temperâture), only Table 7.2N can be used; cracks have to be controlled by lirniting the bar size to match the calculated reinforcement stress immediately after cracking. Tables 7.2N and 7.lN of EN 1992-1-l were produced lrom pârâmetric studies carried out using the crâck width calculalion formulae in 2-l-1,/clause 7.3.4, discusscd below. They were based on reinforced concrete rectângular sections (r".:0.5r) in pure bending (ft2 :0.5, È":0.4) with high bond bars (ir1 :0.8) and C30/37 concrete (f;r.eff : 2.9 MPa). The cover to the centroid ôf the main reinforcement was assumed to t:e o.lh (h d :0,1h],.

-

The valucs in brackets above refer to the assumptions given in Note I of 2-l-l/Table 7.2N. (r", and /r are dc6ned in 2-l-l/clause 1.3.3(2), k1 and t2 are defined jn 2-1-I,/clause 7.3.4(3) and k" is defined in 2-2/clause 7.3.2(102).) Correction for other geometries can be

made. as discussed below. The use ofthese tables for bridges rvas criticized by some countries becâuse they hâve been derived for menbers with covers morË typical of those lbund in buildings (specifically 25 mm), whereas bridge covers are typically much greater. Cover is a significant contdbutor to crack spacing and hence crack width, as can be seen in section 7.3.4 and Worked example 7.3. This potentiâlly leads to grcater calculated crack widths for bridges- This criticism was onc reâson l-or the allowance of nâtionâl choice in the câlculation method to be employed,

Detailed arguments over the parameters to use in crack width calculation, however, tcnd to attribute a gfeater implied accuracy to thc crack width calculation than is really justified. Of greaLcr significance is the load combinâtion used tô câlculâte the crack widths, as discussed in section 7.3. 1 abovc- There is a strong argunent that adequate durability is achieved by specifying adequate cover and by limiting reinforcement stresses to sensible values below yield. The former is achievcd through compliance with 2-2i'clause 4 and the lalter by following thc rcinfbrcement stress limits in 2-l-llclause 7.3.3(2) and 2-l-1/clause 7.2(5). For members with geometry, loading or concrete strength other than as in the assumptions above, the maximum bar diameLers in 2-l-llTable 7.2N strictly nced to bc modilied. The

234

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7. SERVICEABILITY LIMIT

STATES

fbllowing twô equations arc given: d,

:

0i

k"h",. (l"r"i/2.q);;; !;1

tbr sections at least parLly in compressron

h.

o. o,(J,, aç12.9);;;:r - lor:,eclion5 er)mplelely in ô l/? - al

tension

2-1-1(7.6N) 2-r

-l(7.7N)

.J

where:

o. ql lr 1", d

is the adjusted bar diameter is the maximum bar sizc givcn in 2-l-llTable 7,2N is the orerall depth of rhe secLion is the depth of the tensile zone immediâtely prior to crâcking, considering the characteristic values ofprestress and axial forces under the quasi-pernanent combination of

actions is here defined as Lhc c feotive depth to the centroid o l the outer hyer of reinlbrcement. This is not intended to be a general dcfinition ofd, but rather a clarification that ifthe whole scction depth is in tension, the effective depth shôuld be measured to the centl'oid of the steel in one facc and not to the centroid of the sLeel in both faccs.

Thc latter could lead to a centroid ât mid-depth of the member. Generally, d is the depth to the ccntroid of the area of the reinforcement in tension, as clarified in 2-l-l/Fig.7.la), where there are Lwo layers ofbars While 2-l-1/clause 7.3.3(2) states that this adjustment of bâr diametÊr 'should' be made. earlier drafts of EN 1992-1-1 and. Model Cotle 90.6 stated that this adjustment 'may' be made. It should be noted that the adjustment can be either beneficial or detrimental to limiting stress. depending on circunstance, so the distinction betweer 'may'and 'should' is importanl. The lormer implies that beneût can be taken from the adjustment, ignoring it in other cases, while the latter implies that it should be considered also where it is nore onerous to do so. Whether ôr nôt this level of sophistication is merited is a mâtter for debate, as mentioncd above. It is noted tlat no such adjustment to bal diameter is made in crack checks in EN 1994-2, where thc same two tables of limiting bar stress appear. In general, thc whole issue of adjusting bar diameters can be avoided by using Table 7.3N to determine limiting stresscs. For a 0.3mm crack width and bar spacing up to 200n.rm. it will be advantageous to use 2-1-1,iTable 7.3N for all bar diameters from l6mm upwards. The adjustment to bar diameter is not practical when strut-and-tie modelling is used to determine reinforcement stresses as the terms all relate to beam behaviour. I)irect use of EN 1992-1-l Table 7.2N or 7.3N would probably be reasonable in such cases. Differences in cover cannol be accommodated by the adjustments of 2- I - l,/Expression (7.6N) and 2-l-liExpression (7.7N) which, ironicaily, is thc most significant fâctor in determining crack widths when using the direct câlculâtion method of 2-l-l/clause 7.3.4. Calculations on reinforced concrcle slabs and rectangulal beams indicate that the simplc method of calculation based on the use of EN 1992-l-l Table 7-2N and 7.3N renain conservâtive relâtive to the more accurate calculation method of 2-l-liclause 7.3.4 for covers up to about 35mm. For greâter covers, Lhe simple method becomes increasingly lrn-conseNativc by comparison. However, considering the limited âccuracy of both crack width calculâtion methods and Lhe benefits to durability associated rvith the provision of grcaler covers as discussed âbove, the simpLified mcthod probabty remains acceptable for greâter sovers- There is certainly no stated limit to its use in EN 1992 based on a n]a-{lltum cover. Where there is a mirture ofplestressing steel ând un-tensioned reinforcement, the prestress can conselvatively be treâted as an external force applied to the cross-section (ignoring the stress inilrease in the tendons aiter cracking) and the stress determined in the reinlbrcement, ignoring concrete in tension as usual. Tbe reinforcement stress derived can then bc compared against thc tabulated limits. For pre-tensioned beams with relatively little untensioned

reinforcement, where crack control is

to be provided mâinly by the

bonded tendons

235

DESIGNERS' GUIDE

TO EN I992-2

themselves, the Note to 2-l-lic)ause ?,3.3(2) permits Tables 7,2N and 7.3N to b{r used with the steel stress taken as the total str'€ss in the tendons aflcr cracking, minus the initial prestrcss aftcr losscs. This is approximatcly equal to the stress increase in thc tendons alleI decornpression at the level of the tendons.

with depth grtater than 1000mm and main reinforcement concentrated in proportion only a smal) of the depth. 2-I-llclause 7.3.3/3.) requiles additional minimum reinfbrcement to be evenly distributed over the side faces of lhc bcams in the tension zone to control cracking. [t is norn]al in bridgc dcsign to distribute steel âround the perimeter of sections tô control early thermal cracking and reinforcement for this purpose should generally be sumcient to meet the requirements of this ciausc; scction 7.5 refers. 2-I-I ldause 7.3.3 ( 4) is a rcmindcr that there is a pârticular risk of large cracks occurring in sections rlhere there are sudden changes of stress such as aL changes of sectiôn, near concentratcd loads. where bars are curtailed or at areas of high bond stresses such as at the end For

2-l-l/clouse 7.3.3(3)

2-I-l/clouse 7.3.3(4)

bean.rs

oflaps. While sudden chânges ofsection should normally be avoided (by introducing tapers), conpliance rvith the reinfbrcement detailing rules givcn in clauses 8 and 9, together with the crâck control rules of clause 7, should normally give sâtisfâctory perfornance.

236

CHAPTER

7. SERVICEABILITY LIMIT

STATES

7.3.4. Control of crack widths by direct calculation The basis of the clack width calculalion to EN 1992 is presented here. first considering the sin.rplified casc of a rcinforced concrete prism in tension as iu Fig. 7.3. The member will first crack when the tensile strength of the weakest section is reached. Cracking leads to a local redistribution of stresses âdjacenl to the crack as indicated in Fig. 7.3 by the strain distributions. At Lhc crack, the entire tensile force is carried by the reinforcement, Moving away from the crack, tensile stress is transferred from lhc reinforcement by bond to the surrounding concrete and, thercforc, at some distance Z" from the ct'ack, thc disftibution of stress is unaltered from thât before the crack formed. At this location, the strain in concrete and reinforcement is equal and lhe slrcss in the concrete is just belorv its tensile strength. The redistribution of stress local to the crâck results in an extension of the member which is taken up in the crack, causing it to open- This also leads to a reducliùn in the member stillness. With increasing tension, a second crack will lbnn at the ncxt weâkest section. This will not be within â distance I- ol the first crack due to the reduction in concrete stresses in that region assooiated wiLh thc first crack, With further increase in lcnsion, more cracks will dcvelop until the mâximum crack spacing anywhere is 22"- No further cracks will then

Fig. 7.3. Strains adjacent to a crack

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DESIGNERS' GUIDE

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|-

l/clouse

7.3.4(t )

TO EN I992-2

form but furthcr loading will cause the exisLing cracks to widen. This is called 'stabilized' cracking. The member stiflness $'ill continue to reduce, tending towards Lhat of the fully cracked section, considering reinforcement alone in the tension zone. The crack width formulae in EC2 are based on the discussions above. The crack width stems ftom the dillèrence in extension of the concrete ând steel over a length equal to the crack spacing. Thc crack width, Lr., is thus given in 2-l-llclause 7.1.4(1) as: tr'1 :.1,,"u*(e". - a"-) 2-l - li(?.8) where:

lr* r,ln,*

is the characteristic crack width is lhe rraximum crack :\f'auing

€cm

strain of the teinforcement in the length sr.n'nx undôr the relevant combination of loads, including the effect of imposed deformatioas and taking into account the effects of tension stiflèning. Only the additional tensile strain beyond zero strain in the concrete is considered is the mean strâin in the concrete in the length s,.-,,, between ctâcks

És. is the mean

These terms are discussed in detail below. Crock spocing

From the above discussion, the minimum crack spacing is Z" and the maximum crâck

spacing is 2I". The average crack spacing, s-, is therefore sornewhere belween these two values. Figure 7,3 illusûates that Z" and thus s- depend on the rate at which stress can be transferred l'rom the reinforcement to the concrete. Assuming a constant bond stress, r, along the length Ze, the strcss at a distance Z" from a crack will rcach Lhe tensile strength

of the concrete when:

rltôLc - Ac.l

(D7- l4)

I

where '4. is the concrete area and .,f, is the tensile strength of the concrete, Introducing the reinforcen.rent ratio p: n!'l4A.into equation (D7-14) gives:

(D7-15) The mean crack spacing cân then bc expressed as: s,",

-

0.25krô/p

(D7-16)

where Àl is a constant which tâkes account of the bond properties of the reinforcement in the concretej4r/Î, and the diference betrveen minimum and mean crack spacing. Equation (D7- l6) docs not Iit test dâtâ well, so an additional term for reinforcement cover, c. needs to be included in the exprcssion for crack spacing:

s^

:

kc + 0.25kél p

(D7-17'

The need fbr the reinforcement cover term probably ariscs because, although equation (D7-14) assumes that the tensile stress is coûstant in the area.4", the concrete sûess will actually be greatest adjacent to the bar and will reduce with distanc€ from it. This reduces the cracking load of the area lc. Equation (D7-17) applies to concfete sections in pure tension and a further factor is thefefore necessary to allow for other oases where the stress distribution varics through the depth of the member- It iri also necessary to define an effective reinforcement ratio, pp since the "ff appropriate concrete ârea is not that of the whole member but rather must be related to the actuâl tension zone. This leads to:

s, : 2-l-l/clouse 7 7 417l

kc + 0.25k1k20.3ô ô >0.36 6 -0.96 6 .:.' .:.n !.:

:

6 :

:ii ijii i ii: i-li ,, rr Æ4 i,:riTitlTi]]TTr: .........iiÏ',

--lmr rri-r-irr-r---

,++itffi

:: tl

r

nus ror worst ca3e ( I on frars. ârransement (a))i /0.,n,"

> max{0.3 x

1.4

x

>

r.o x

I l51

tî|l;.-.", (a) /o

"iï=ïï:

Ï

.5; 15

x

25; 200}

-

4M nm

o.

; I

:' :'i' I :' I :' lt

-fï; uemm

"l:ï

ll,.'..,-.'

256

CHAPTER

8. DETAILING OF REINFORCEMENT AND

PRESTRESSING STEEL

8.7.5. Laps of welded mesh fabrics madê of ribbed wires The use of welded mesh fabric reinforcement is permitted by EC2 but is nol commôn in bridge design due to its reduced fatigue performance. Of the two options for lapping given in 2-l-1,/clause 8.7.5.1, intermcshing or laycring, intermeshing will be most contmon in bridge design as it has to be used where fatigue loads are expected. The requirements for lapping intermeshed bars are tl.re same as for bars, excepl that the transverse reinlbrcement on the mesh cânnot be counted in the calculation of crr, but no further transvcrse reinforcement need be provided.

8.7.6. Welding (additional sub-section) Welding requirernents are not mentioned in section 8 of EC2-l-1. The requirements for welding two bars togcther arc givcn by EC2 in 2-1-1/clause 3.2.5.

8.8. Additional rules for large diameter bars t or

cascs where largc diamctcr bars arc uscd, tests have shown

thât the splitting forces and

dowel âction are significantly greater than for smaller diameter bars. 2J-I lclausc 8.8(1) defines large diameter bars as those with a diameter greater than drurg". The value of o1,.r" is â nâtionâlly determined pârâmeter and EC2-1-1 recommends a value of 32mm- This will be increased to 40 mm in the UK National Annex to be consistent with current practice, where bars of 40 mm diameter are often used in abutments, cellular decks, piles and pile caps for erample. This will mean that the provisions of 2-l-l/clause 8.8 should rarely apply. The rest of 2-l-liclause 8.8 defines additional detailing rules that must be satislied when using bars of a large diameter. The rules have been developed to account for lhe increâsed splitting iblces and dowel action and have been based on the rccommendâtions given in Model Cotle 90.'Thc additioual dctailing rules covcr the following:

. . .

2-l-l/douse 8.8(t )

crack control and surface reinforcement reouiremenls: additional trânsverse reinforcement at straight anchorages; laps.

Where large diameter bars are used, 2-1-l lclause 8.8(2) requires thât crack control is achieved by either direct calculation or by incorporating additional surface rcinl'orcement (see Annex J of this guide and 2-1-1i'clause 8.8(8)). Direct calculation will normally be chosen as it will often be impractical to provide additional surface reinforcemcnt (e.g. in piles).

2-I-l ldaase 8.8(4) requires lhat largc diameter bars are generally not lapped unlcss

sections are at leâst 1m thick or where the stress in the bars is not greater thân 80% of the design ultimate strength. Where straight anchorages are used for large diâmeter bars, 2-1-l lclause 8.8/5J requires links, in addition to those required for shear, to be provided as conûning reinforcement in

2-l-l/clouse 8.8(2)

2-

l- l /dause 8.8(4)

2-l-l/clause 8.8(s)

257

DESIGNERS' GUIDE

2-l-l/clouse 8.8(6)

TO EN I992-2

zones where ffansverse compression is not present, This additionâl reinforcement is defined tn 2-l-llclause 8,8(6) as being not less Lhan Lhc following:

A"n:0.25A,nt in

the direction parallel to the tension fâce

2-r-1(8.12)

4""

2- 1- li (8.l3) - 0.254"n2 in the direction perpendicular to the tension face rvhere l" is the cross-sectional ârea of a single anchored bar, z1 is lhe nunber of layers

anchored ât the same point in the section considered and n2 is the number of bafs anchored in each Jayer. This additional transverse rcinforccnent is required to be unilbrmly distributed along the anchorage zone with bars ât centres nôt exceeding five times the size ofthe longitudinal reinfbrcement. EC2 does not indicate how many legs of links are required per longitudinal bar, but, Model Corlc 906 suggests that 2 lcgs of a stirrup can surround 3 bars per layer ât most.

8.9. Bundled bars 2-Udouse

8.9.r(t0t)

Generally, the rules for individual bals also apply for bundled bars as long as the âdditiônal EC2 detailing requirements for bundlcd bars are met. 2-2lclause 8.9.1(101) requires that bundled bars arc of the same type ând grade, but may be of different sizes providing the

ratio of diameters does not excccd

2-l-l/clouse 8.e. | (2)

8. 10. Prestressing 8.10.1, Tendon layouts

2-l-l/clouse 8. t 0.

1.7.

For design purposes, the bundle of bars should be trcated as a single bar having the sarne area and centre of gravity as the bundle, but the spacing ând cover requirements should be applied to the outer edge of the bundle- For same size bars. the equivalent diameter, on, is g|en n 2-I-l lclaase 8.9.1(2) as ô": (b\E; I 55mm, where n6 is the number of bars jn the bundle. The number of bars in a bundle should be limited to no more thân 4 for verticâl bars in compression (where bond conditions have heen shown to be besl) or at laps, or lo 3 for all other cases. This is to ensure the bond and ânchorâge chârâcteristics ôf bundled bars do not stray too far fiom test cascs against which the empirical fbrmulae have been calibrated. Where two touching bars are positioned above one another in regious classified as having good bond conditions, the bars need not be treâted as bundled. 2-1-17'clauses 8.9.2 and 8.9-3 cover anchorage and laps respectively. These give requirements for staggering bars within the bundle themselves in some situations.

tendons

EC2 gives rules fbr the spâcing of post-tensioning ducts and pre-tensioned strands. These spâcings are intended to ensure thât plâcing ând compacting ol the concrete can be carried out satisfâctodly and that sumoient bond is available between the concrete and the tendons, as required by 2-l-l lclwse 8.10.1.1 (l )P-

t.r (t)P Pre-tensioDed tendons

2-l-[/clouse 8.t

0.

Minimr.rm clear horizontal and vertical spâcings ôf individual pre-tensioned strands are shown in Fig, 8.10-1, where d, is the rnaxinun aggregate size. Other layouts, including bundling, can be used, but only if test results demonstrate satisfacLory ultimatc behaviour and tlrat placement and compaction ofthe concrete is possible 2-l-llclause 8.10.1.2(1).

t.2(t) Post-tensioning ducts

2-I-l/clouse e. r0.

t

.3(t )P

2-l-l lclause 8.10.1.3(1)P requircs ducts to be positioned such that concrete cân be placed without damaging the ducts ând thât the concrete can rcsist thc forces imposed by terdons in curved ducts. Minimum clear spacing between ducts, bâsed on the former requirement, are shown in Fig. 8.10- l. These duct spacings are, however. unlikely to be adequate to comply with the lalter requirement \ryhere the tendôn profiles are tightly curved and thcre is a tendency fbr the concrete between ducLs to split under the bursting stresses generated by the Tâdial force, as shown nr Fig. 8.10-2(a). In this situation, eithcr thc spacings should be

255

CHAPTER

8. DETAILING OF REINFORCEMENT AND

PRESTRESSING STEEL

=4n5 >a

>50 mm

_ol =,iÏl >40mm

I

o a

o-a o_ o_

1

-

Î:ff

o'

r-i:ii.'

ê4u mm

>20

(b) SlÊnds

(a) Ducrs

Fig.8.l0-1.

al- ---L'20

Minimum clear spacing beween ducts and between pre-tensioned strands

increased or the concrete should be reinlbrced between tlle ducts. Curved ducts may also require an increase in cover perpendicular to the plane of the bend. Recommended duct spacings and covers for different duct diâmeters and bend radii were given in BS 5400 Part 4,' and these are reproduced as Tables 8-10-1 and 8.10-2. Tendons will sometimes âlsô need tying back into the main body of the section if the bend is such that the duct pulls against the cover. This is common where tÊndons arc placed in webs curved in plan or flanges curved in clcvation and a suitablc reinforcement detail is shown in

Fig. 8.10-2(b). 2-1-1 lclause 8.10.1.J (2) states that ducts for post-tensioned mcmbers should not normally be bundled except in the case of a pair of ducts placed vertically one above the othcr. Particular care should be taken if this is done in a thin deck slab, such as rnay occur in balanced

cantilever construction, as the transvcrse shear resislance of the slab can be significanlly reduced at the duct positions, pârticulâdy while ducts are un-grouted.

8.10.2. Anchorage of pre-tensioned tendons 8.10.2.1. General

The bond strength applicable to the design of anchoring of pre-tensioned tendôns depends on the type ol loading. Thc highest values are applicablc to the initiâl transmission length, 1or, since the tendons thicken against the concrete in the trânsmission zone at transfer âs the str€ss in them reduces. Lower values are applicable lor calculation of the anchorage length, 16, at ultimate limit states where the force in the tendon increases and the tendon diameter rcduces, thus shrinking away from the concrete. These dilTcrent bond values are reflected in the calculation of iniLial transmission length and ultimâte limit state anchorage length, as shown in Fig. 8.10-4.

(b)

(a)

8, | 0-2. (a) Splitting of concrete betvveen ducts due resisting pull-out

Fig,

to

curyature forces and (b) reinforcement

2-l-l/clquse 8.t 0. t.3(2)

DESIGNERS' GUIDE

TO EN I992-2

Table 8.10-1. Minimum distânce between cenrrelines of ducts in Dlâne of curvature. in mm Duct internal diameter (mm)

19 30 40 Radius

s0

80

of

curvature ôf duct (m) 2

4

90

t00

0

t20

t30 t40

t50

t60

t70

Tendon force (kN)

796 387 960 t337 t920 2640 3360 4320 5t83 60t9 7200 8640 9424 I

t0 t40 150 485 700 960 s5 70 t75 245 350 480 r0 38 60 t20 t65 23s 320 4 t0 6

90 80

t0

t75 240 140 t95 t60

t25 t00

t2 t4 t6 I8

140

t3200

Radii not normâlly

785

940

525

630 470

used

3t5

730 870 545 655 440 525 365 435

t75

270

3

t60

235 210 200

275 245 220

305

395

245 205

3t5

t80

20

t0336 248

l5

22

375 330

290

t045

785 855 630 685 525 570 450 490 39s 430 350 380

r5

265

3

240

285 3 t0 265 285 260 280

24 26

345

940 750 625 535 470 420 375 340

3t5 300

8t5 680 585 510 455

800

4t0

480 435 400 370 345 340

370 340 320

30

745 600

32 34 36

40

38 60 80 t00 t20 t40

Note l. The têndon force shown

is

t60

180 200 220

240

the maximum normâlly availâble for the given size of duct, (Taken

260

280

as 80% of

300

the charàcterhtic strength of the

Note 2. Values less than 2 x duct internal diameter are not included.

2-l-l/clouse

8.t0.2.2(t)

8.10.2.2. Transfer of prestress The transmission length at transfer is determined lrorn 2-1-l lclause 8.10.2.2(1) âssuming a constant bond stress Âùr, where: .fnpr

:

rlprrh

"f"'dQ)

2-l-li(8.15)

with:

\pr 4r : ird(4

2.7 tbr indented wires or 3.2 for 3- and 7-wire strands 1.0 for good bond conditions (as defined in EC2-1-1 Fig. 8.2) or 0.7 otherwise is the concrete design tensile strcngth at time of release

2-|-l/clause

The basic value of transmission length is given in 2-1-llclause 8.10.2.2(2) by:

8.r0.2.2(2)

l$ :

olo2(bo

pno

l

Jtlpl

2-r-r(8.r6)

wherc;

cit : 1.0 for gradual release or 1.25 for sudden release a2 - 0.25 for circular tendons or 0.19 fbr 3- and 7-wire strands is the nominal diamctcr of tendon o .'pn 0 is Lh(j tendon stress just after release 2- l-l /clouse

8.t0.2.2(3)

The design value of the transmission length should be tâket from 2-1-l lclausc 8.10.2.2( j ) as eilher /n1t : 0.8/pr or lpo: 7.2lpr. whichever is most adverse for the check being carried

out. The shorter length,

/o,1, will usuâlly be used for checking stresses at transfer at beam ends, as the increasing dead load sagging moments away fiom the supports help to

260

CHAPTER

8. DETAILING OF REINFORCEMENT AND

Table 8.10.2, Minimum cover to ducts perpendicular to

plâne

PRESTRESSING STEEL

of curvâture, in mm

Duct internal diameter (mm)

19 30 40 Radius

ôf

curvature of duct (m) 2

t00

s0

0

120

t40

r30

t50

r60

t70

Tendon force (kN)

296 387 960 t337 t920 2640 3360 4320 sr83 6019 7200 8640 9424 t0336 11248

50

4

55 50

rss 220 70 r00

6

50 65

t0

50

I

445

t45

205

65 60

t6

55

t8

50

350 220

L65 t65

t25 95 |5 85 t00 75 90 70 8s 65 80 65 75 60 70 55 70 55 65 50 65

90

55

20 22

24 28 30

8s 80 80 75 75 70 70 65 65 60 60

55 34

55 55

50

50

50

50 50

50

s0

Nole: The tendôn fôrce shown is the maximum normally avâilable

420

265 t85 t40 t25

t50 t20 0 t00 |5 95 |0 90 t05

60 60

38

13200

Radii not normally

320

t7 t4

40

lf

r00 95

90 85 85

80

80 75 75

70 70

used

310 375 220 270 t65 205 t45 t65 t30 t50 t25 t40 I t5 r35 ll0 125 t05 120 r00 | 15 i00 I l0 95 105 90 t05 90 100 85 I00 85 95 80 90 80 90

for rhe given size of duct. (Tàken

460 330 250 200

360

395

215

240

27s t85 t75

t70 t60 t50 t45 t40 t30

165 155

t50 t45 t35 r30

t2s t20 t20

t30 t25 r20

5

0 0

l15 I t5 I t0

t05 t00

as 80% of

the

300 200

t90 t80

t70 t60 r55

t50 r45

t,rc t35 | 30

t25 t25

t20

ch

260 215 205 t90 r80 t75 t65 160 155 r50 145 t40 r40 t35 t30

3t5 260 225

2l5 205

t95 t85 180

t70 t65 160

t55 t50 t50 t45

acteristic strength of the

prevent overstress in tension aL thc beam top and in conpression aL the bottom under the elïcctq nf nrecfrê

Natior.ral Antex. 2-1-llclaase 9,2.1,1(l

0,00Ilbd

)

recornmends the

2-1- l (9.1N)

/vk

where:

./rr",,

.lu* ô1 d

is the mean tcnsile strength of the concrete is the charâcteristic yield strength of the reinforcement is the mean width of the tension zone (excluding the compression flange for nonrectangular sections) is the effective depth to the tension reinforcement

This requirement is derived from Lhe development ofan expression to ensure the reinfôrcement does not yield as soon âs crâcking in the côncrete occurs. Considering a rectangular beam, the cracking moment is given by:

M.,

-

/;ûbh2

16

where ô and I are the breadth and ôverâll depth of the concrete beâtn respectively. If the beam is reinforced wiLh an area of steel ls, of yield strength ,{,r at ar.r effective depth r/, the ultimate moment of resistance is given by:

M,,

-

JtyA"z

where z is the lever arm. For the cracked strcngth of thc scction to exceed the meân un-cracked strength at f,rst crâcking, M,, > M",. This requires JykA,z > J..^bh2 f 6 whic,h teârrânges to:

f,h bh) o Jrk z

, -_;---,t as

Introducing the effecLive depth. d. gives;

^" i*(:)'i" A typical value of l/d might be l.l. A conservâtive vâlue of d/z would be 1.25, corresponding to a heavily reinlorced section with z : 0.80d, which is unlikely to arise with minimum reinforcement in â rectangular section, but could arise with other cross-section shapes, such as

T beams. These valu€s l€ad to:

As> o2s,Ëbd 1/Expression (9.1N) above is ofthe same form as this expression lor a rectangular beam, but the 0.25 Iâctor has been replaced by 0.26, which makes some âllowance for other section geometries- The rclatively small increase for other geon.retries reflects the conservative value ot df z already assumed âbove. Elements contâining less thân the minimum area of reinforcement,,4",.1". should be considered as un-reinforced and designed in accordance with 2-2/clause 12 as required hy 2-l-lldaase 9.2.1.1(2). In addition to the minimum reinforcement requirements detailed above, EC2-2 requires that all elements in bridgc design should contain a minimum quantily of reinforcemÊnt to control cracking as discussed in section 7.3. Fôr prestressed concrete sections, additional rules to prevent brittle lailure are given in clause 6,1 and discussed in section 6.1-5 ol'this guide. Similarly, 2-l-Ilclause 9.2.1.1(4) requires the designer to cheÇk thât the ultimate bending resistance of members with permanently un-bonded or external prestressing tendons exceeds the flexural cracking monent by a recomnended factor of 1 .15, In order to ease the placing and compacting of concrete and to prevent cracking from excessive internal resLraint to concrete shlinkase. reinforcement in beams should be 2- 1-

2-l-l/clouse 9.2.r.r(2)

2-l-l/clouse

9.2.t.|(4)

276

CHAPTER

9. DETAILING OF

MEIYBERS

AND PARTICULAR RULES

limited to a maximum value. ,4. n u.. Thc value of 1.,n"" for beams, outside lap locations, can be specified in the National Annex aud 2-l-Ilclaure 9.2,t,1(.1) recommends the yâlue of 0.04,4., where ,{" is tbe gross cross-sectional concrete areâ of the element considered. 1.2. Other detoiling orrongements 9.2.1 .2 (/) requires that for monolithic construction, even where sinple supports have been assumetl in the design, support scctions ofbean-ts should bc rcinforced to provide a minimum hogging moment resistance. This should be a proportion, pr, of the maximum sagging moment in thc span. The value of 31 may be provided in the National Annex and EC2 recommends tâking â vâlue of 0.15. This relatively small minimum design moment could require considerable redistribution of moment to occur and it is likely that the National Annex rvill specify a larger value. For bridges, however, where plasLic analysis is generally not permitted, it will not usually be acceptablc to treat a monolithic connection as a simple support. The results ofan elastic analysis, with ôr without any limited redistribution allowed in clause 5.5, are then likcly to determine minimum hogging momcnt requiremcû1s- Crack width checks u'ould âdditionally have to bc performed using the results of the unmodifled elastic analysis. For flanged beams. 2-1-l lclause 9.2.1 .2(2l requires the total area of tcnsion reinforcement required lo bc spread over the effective width of the flange, although pârt of the reinfbrcement may be concentrated ovcr thc web width. This is presun.rably to limit cracking across the width of the llange at SLS (including from early thcrmal cracking), even though such â distribution might not be necessary at ULS. Where the design of a section has included the contribution of any longitudinal compression reinlorcement in the resistânce câlculation, 2-l-I lclause 9.2.1.2(3) requires the reinforcement to be ellcctively held in place by tlansverse reintbrcement. This is intended to prevent buckling of the bârs out through thc cover. Thc spacing of this trânsyerse reinforcement shoqld be no greatel' than 15 times the diâmeter of the compression bar. The minimum size of the transversc rcinforcement is not giveu, bul jt is recommended hcrc that the requirenents of 2-1-1,/clâuse 9.5-3 for columns be followed. 2-l-llclause 9.2.I.2(3) âlso does not dcfrne specific requirements for how the transverse reinftrrcement should enclose the compressiôn reinforcemcnL- Thc requirenrcnts for columns in 2-1-1t' clâuse 9.5.3, as discussed in section 9.5.3 of this guide, could also be applied to colnpression flanges of beams.

2- l -

l/clouse

e.2.t.t(3)

9.2.

2-I-I lclausc

9.2.

2-l-l/douse e.2.

t.2(t )

2-l-l/clouse 9.2.t.2(2)

2-l-l/clouse 9.2.t .2(3)

1.3. Curtoilment of longîtudinol tension reinforcement

EC2 allows longitudinâl bars to be curLailcd beyond the point which they are no longer required for design, provided suflicient feinforcement remains to adequately resist the envelope of tensile forces acting at all sections (including satisfying minimum requircmcnts). Bars should extend at least an anchorâge length beyond this point. The contribution to the sectiôn resistance ol bars within their anchorage length mây be taken into account assuning a linear variation of force (as indicatcd in 2-l-liFig. 9.2-2 and allowcd by 2-I-l lduuse 9.2.1.3(3)) or r.r.ray conservatively be ignôred, In determining the point bcyond which the reinfbrcement is no longer required, appropriale allowance lnust be given to the requiremcnis to providc additional tensile fôrce for shear design. This lcads to the 'shift lrrethod' desuibed in 2-l-1/clause 9.2,1.3 and discussed in section 6.2.1.1 ôfthis guidc. It shoukl be noted that a further shift is required for posi[ion within wide flanges as discussed in section 6-2.4. 2-l-llclause 9.2.L3(4) also specilies an increâsed ânchorage length requirenent for bent-up bars which contribute to the shear rcsislance of a section. This ensures the bar can reach its design shength at the point it is required âs sheât reinforcement.

2-l-l/clouse 9.2.t.3(3)

2-l-l/clouse e.2.t.3(4)

9.2.1.4. Anchoroge of bottom reinforcement ot on end support For end supports, where litLle or no end fixity is assumed in the design, the designer should provide at least a propoftion ofthe reinforccmcnl provided in thc span. This proportion,732, is recommended by EC2 to be taken as 0,25. This longitudinal teintblcement should not be

277

DESIGNERS' GUIDE

TO EN I992-2

lbr the shear design together with any axial lorce and must be adequately anchored heyond thc face of thc support. less than that required

9.2. | .5. Anchoroge of bottom reinforcement ot intemediote supports For intermediâte supports, EC2 allorvs the minimum bottom reinforcement calculated in accordance with 2-l-l/clause 9,2.1.4 to be curtailed after extending at least l0 bâr diameters (for straight bars) into thc supporl mcasured from the support fâce. For bars with bends or hooks, EC2 requires the bâr to be anchored â length at least equal to the mandrel diameter beyond the support face, to ensure the bend of the bar docs not begin until well within the support. Where sagging moments can develop, the bottom reinforcement should obviously

be continuous through intermediate supports, using lapped bars ifneccssary- Considerations

of early thermal cracking may also lead to the need for continuity of the longitudinal reinforcement.

9.2.2. Shear reinforcement 2-2/douse 9.2.2(t 0 r )

2-l-l/clouse 9.2.2(3)

2-l-l/clouse 9.2.2(4)

2-l-l/clouse e.2.2(s)

2-2lclause 9.2.2(107) requires shear reinforcement to form an angle, a, of belween 45' and

90' to the longitudinal axis of the structural member. In mosL bridge applications, sh€ar reinforcement is provided in the form oflinks enclosing the longitudinal tensile reinforcement and anchored in accordance with 2-l-l/clause 8.5. The inside of link bends should be provided with a longitudinal bar of at least the same diamctcr as the link as discussed in section 8.5. 2-1-l lclnuse 9.2.2(3) allcrws links to be formed by lapping legs near lhe surface of webs (it is oflen convenient to form links fiom two U-bars, for example) only if the links are not requircd to resist torsion. It has, howevcr, been common practice in the UK to form outer links in bridges in this wây and they have performed adequately as torsion links. 2-2,/clause 9.2.2(101) allows shear reinforoement to comprise a combination of links and bent-up bars, but a certain minimum propofiion of the required shear reinforcement must be provided in the form of links. This minirrum proportion, 193, is recommended by 2-1-llclaase 9,2,2(4) to be taken as 0.50 bul it may be varied in the Nâtional Annex. The restriction is largely due to the lack of test dâta for combinations ol 4/2, the column is classed as a wall. For hollow columns, where b > 4h, rto guidance is given.

282

CHAPTER 9. DETAILING OF MEMBERS AND PARTICULAR RULES

9.5.2. Longitudinal reinforcement A minimum percentagc of longitudinal reinlbrcement is required in columns to câter for unintended eccentricities and to control creep deformations. Under long-term application of service loading conditions, load is transferred fron.r the concretc to thc rcinforcement because the côncrete crccps and shrinks. If the area of reinfôrcement in a column is small, the reinforcement may yield undcr long-term service loads. The recommended minimum area requirements in 2-l-Ilclause 9.5.2(2) therclore accounts for the design

2-l-l/clouse

axial compressive force, as wcll as the column gross cross-sectionâl ârea such that:

ls -,n

: 0.l0NEd/ld

> 0.0021"

2- l

-

9.s.2(2)

1(9.12N)

where:

y'y'sa is the nraximum design axial compression force "/ya is the dcsign yield strength of lhe reinforcement

A.

is the gross cross-sectional concrete area

The National Annex may also specily minimum bar sizes to be used for longitr.rdinal rcinlorcement in columns (recommendcd by 2-I-llclause 9.5.2(1) to be 8mm) and maximum areas of reinforcement, The maxirnum areas have been chosen pârtly from practical considerâtions of placing and compacting the concrete and partly to pl'€vent cracking lrom excessive internâl restrâint to concrele shrinkage caused by thc reinllorcemcnt. 2-l-I lclause 9.5.2(3) rccommends a maximum reinforcement content ôf 0.041. outside lâp locations, increasing to 0.081" at laps. For columns with a polygonal cross-section, 2-I-l ldause 9.5.2(1) reqnires at least one longitudinal bâr to be placed at each corner with a minimum of four bars used in circular columns.

2-l-l/clause e.5.2(t

)

2-l-l/clouse e.s.2(3)

2-l-l/clause 9.5.2(4)

lt

is recommended, however, that, as in current UK practice (and as recommended it Model Code 906 ), a minimum of six bars is used for circular columns in order tô ensure the stability of the reinforcement cage prior to casting.

9,5.3, Transverse reinforcement 'Transvelse reinforcement' generally refers to links, loops or spirals enclosing the longitudinal reinforcement- The purpose of transverse reinforcemenL in columns is to provide adequate shear lesistânce and to secure longitudinal compressiôn bars against br.rckling out through thc concrete cover. Although not explicitll, stated in EN 1992, it is recomnendetl here that minimum links are provided lbr columns using the samÊ requirements as those for beams. To ensure the stâbility of the reinforcement cages in columns prior to castil4, 2-2lclause 9.5.3(I0l ) recommends the minimum diameter of the transverse rcinforc€ment to be the greatel of 6 mm or one-quarter the size of the maximum diâmeter of the longitudinal bars. The maximum spacing of transverse reinforcement in columns, -scl.mar. may be given in the Nationaf Annex, but 2-1-llclause 9.5.3(3) recommends Laking a value of the leâst of the

2-2/clouse 9.5.3(t 0 | )

. . .

l/clause 9.s.3(3)

2-

I-

2-

l- l/douse

fbllowing: 20 times the minimum diarneter of the longitudinal bars; the lesser dimension of the column: 400 mm.

2-1-l lclause 9,5,1(4) further recommends that these maximum spacing dimensions are reduced by multiplying them by a factor ôf 0.6 in column sections close to â beâm or slâb, or near lapped joints in tbe longitudinal reinfbrcement where a minimum of three bars evenly placed along the lap length is required. 2-IJ lclause 9.5.3 (6) requires that every longitudinâl bâr or bundle ofbars placed in a corner sl.rould be held by transverse reinforccment. In addition. no bar within a compression zone should be furthcr Lhan 150 mm fiom â 'restrâined' bar. There is, however, no deflnition pro-

vided of what constitutes â restraincd bar.

9.s.3(4)

2-

l- I /dause e.5.3(6)

It could be interpreted as requiring all

compression bars in an outer layer to be rvithin 150 mm of a bar held in place by â link, with links passing around every alternalr bar. For box sections with wide flanges, this would

283

DESIGNERS' GUIDE

TO EN I992-2

require additional link reinforcement in the flanges in addition to web links over the depth ofthe section. This interpretalion was the onc uscd in BS 5400 Part 4' fol'bars contributing to the section resistânce. It is not practicâl to provide this detail in all situations - links in flanges is an obvious example. Where this detailing cannot be achieved, it is recomtlended herc that transveme bars should still be provided on the outside of the longitudinal reinforcement (rvhich 2- l-l /clause 9.6.3 for walls describes as 'horizonlal' rcinforcement râther than transverse reinforcement), but Lhe longitudinal compression bars in an outel layer should not then be included in the resistance calculation- This detailing problem does not arise in circular columns with perimeter links as these will be suiTicient to rcstrain the cômpression bars. It should bc noled that lhe inclusjon of reinforcement in compression zones is implicit in other expressioûs elser.here in the code (such as the noninal sliffness method for ânalysis of slender columns in 2-l-1/clause 5.8.7.2 and Lhe interactiôn formula tbr biaxial bending of columns in 2-1-liclause 5.8-9). For stiffness calculation (as in 2-l-1,/cLause 5.8.7.2), it would be reasonable to use the code fomulae without cnclosing evety other compression bar by a link. For sLrcngth calculation (as in 2-l-1,/clause 5.8.9), it is imporlanl that compression bars are propedy held b1' links as above,

9.6. Walls 2-l-l/clause e.6. t

(r)

2-1-l lclause 9,6.1( 1) defines a wall as having a length to thjckness ratio of at least 4. Where a wall is subjcctcd Lo prcdominanlJy ouL-of-plane bending, clause 9,3 for slabs applies- The âmount of reinforcement in a wall and appropriate detailing for it may be determined lrom a strut-and-tie nodel. Leaf piers may fall into the 'wall' category. 2-1-1,/clause 9.6,3 and 2-1-1,/clause 9.6.4 deal

rvith 'horizontal' and 'transverse' reinfbrcement requircmcnts respectively, Horizontal rcinftrrccnrert lies paralleJ Lo the long lace of the wall while transverse reinforcement passes through the width of the wall in the form of links, It is reconrmcnded here that the requirements lbl columns in clause 9.5 should also be mcl, which can be more onerous, e

.g. minimum vcr[ical rcinforccmcnl.

Considerations of eârly thermal cracking may give rise to greatcr reinforcement requirements. The UK National Annex is likely to givc guidance here.

9.7. Deep beams A deep beam is formally detned in EN 1992 as a member whose span is less than 3 times its overall section depth, In bridge design, this rvill mosl frcquently âpply to transverse diaphragms in hox girdcrs or bcLwccn bridge beams- Strut-ând-tie modelling, as discussed in section 6.5, rvill be the normal method of design and all Lhc rules on anchorage of reilforoement at nodes and limiting concrele prcssure wiJl apply-

2-l-l/clouse e.7(t) 2-2/clouse e.7(r 02)

284

The main reinforccmcnt dctcrmincd for a deep beâm from â strut-and-tie analysis may not require any surface leinfolcement. A recommended minimum rcinforcement ratio of 0.17o (but not less than t5Ommrim) for cach fase and each orthogonal direction is therefore required to be placed near each face in accordance with )-1-l lcla se 9.7(1). Thrs amoùnl can, however, be varied by the National Annex. From 2-2lclause 9.7(102), bar centrcs should not exceed the lesser of 300 mm or the web thickness (which n.ray also be varied by the National Annex). This reinlbrcement is intendecl to control cracking from effects not dircctly modelled in Lhe struland-tie analysis, such as transverse tension liou bulging of cornpression struts as discussed in section 6.5, and surtàce strains rcsulting from tensile ties withjn the concrcte section. (The nominal rcinforcement is nôt intended to be suflicienL to fully restrâin the tensile forces perpendicular to a bulging compression strut and thus increase its allowable compressive stress. If this is required, reinforcement should be cxplicitJy designed for this purpose in acoordance with 2-l-liclause 6.5.) For deep beams, it is likely thât reinforcement to control eatly thermal cracking will exceed thc abovc minimum requirements. The UK National Annex is likely to give guidance herc.

CHAPTER 9. DETAILING OF MEMBERS AND PARTICULAR RULES

9.8. Foundations 2-l-1iclâuse 9.8 gives additional detâiling guidance in section 9.8 for the following types of foundations: pile caps, footings, tie beams and borecl piles- PiLe caps and footings are discussed below. Pile .ops

2-l-I,/clause 9.8.1 gives additional requirements lor the design ôf reinforc€d concrcte pile caps, including:

. . .

Reinforcement in a pile cap should be detern.rined by using either strut-and-tie or flexural mcthods. The distânce from the edge of the pile to the edge of the pile cap should be sulicient to enable the tie forces in the pile cap to be properly anchored. The main lensile rcinl'orccmcnt to rcsist the action effects should be concentrated above the tops of the piles. This is a harsh requirement. Where flexural design is appropriate, it has been co[rmon practice

in the UK to adopt a unil'orm distribution of reinforcement

across the pile cap unless the pile spacings exceeded three pile diamcters. Even rvhere strut-and-tie analysis is used, it may be possible to consider reinforcement outside the pile width, provided transvcrsc rcinforccment is able to distribute the forccs as shown

in Fig.9.8-1. the concentraied reinforcement placed above the pilc tops is ât least equal to thir nrininum rcinforccmcnt rcquirements of the full sectiot,2-2ldause 9.8'1(103) allows evenly distributed bars along the bottom surface of a rl]embet to be omitted. This is not recommended here for the clesign ol bridgc pile caps in order to control early thermal cracking in such areas. Note the provision in 2-l-llclause 9.8.1(3) for buildings, which allows the side fàces and tops ofpile caps to be un-rcinforced where there is no risk of tension dcvcloping, is omittcd in EN 1992-2 for similar reasons. Welded bars are allowed tô provide anchorage to the tensile reinforcementThe corlplession caused by the support reaction from thc pile may be assurnetl to spread at 45' ftom Lhc edge ofthe pile (2-1-1iFig. 9.ll) and mây be taken inlo acgount when

. If

. . .

câlculâting ânchorage lengths. The tops of the piJes should extend a minimum of

50

2-2/douse r(t 03)

9.8.

mm into the pile cap.

Footings

2-1-liclausc 9-8.2 gives additional guidance on the design of column and wall footings including:

. . .

The rlain tensile reinlorcement should be providcd with bars of a minimum diameter, r/',,,;n, with recommended value of 8 mm. The main reinfbrcement of cifcular lbotings may be orthogonal and concentrated in lhe mjddle halfolthe footing (110%). If adopted, the remainder ofthe footing should be considered âs plâin côncrete. The anal.vsis should include checks for any tensile stresses resulting on the upper surl-acc ol thc footing, and lhesc should be adequatel.v reinforced.

Transverse lension resisled by reirlorcement

Tie-backlorce

Fig. 9.t-

l.

Spread of load from â pile to âdjecent ties

285

DESIGNERS' GUIDE

2-l-l/clouse e.e.2.2(t )

2-l-l/clouse e.e.2.2(2)

TO EN I992-2

2-1-liclause 9.8.2.2 descdbes â strut-ând-tie method (Fig. 9.8-2) for calculating the forces along the length ofthe tension reinforcement in order to determine anchorage requirements. This modcl is necessary to account for the effects of inclincd cracks', as no ted, in 2-1-l lclaase 9.8.2,2( I ). Anchorage is particularly important at the edges of the footing in determining whether or not the main tension rcinforcement requires bends, hooks or laps onto side face reinforcemcnt. The tensile force to be anchored at any distance jç fi'om the edge of the base is given in 2-1-I lclause 9.8.2.2rl2) as follows:

F,

:

Rz,lzi

2-1-1(9.13)

where:

4 R l{ra .:" zi

is the force to be anchored at a distance x from the edge ofthe fboting is the resultant fbrce from the ground pressure within disLance.x is the vertical forcc corresponding to the total ground pressure between sections A and B (illustrated in Fig.9.8-2) is the external lever arm (the distance between À and NB,1) is the internal lever arm (the distance between the reinlbrcement and the horizontal

F.

is the compressive force corresponding to the mâximum tensile force, ,f. -,,*

force,,Q)

2-l-l/dause 9.e.2.2(j)

The lever arms, ze ând zj, can bii readily determined from considerations of the conrptession zones from Nsl and 4 respectively, I)tt 2-l-I lclause 9,8.2.2(3) g:.es simplifications where z" may be calculated assuming e :0.15b and ?i is taken as 0.9d. In practice, the value of z1 will aheâdy be known from Lhe ULS bending analysis and this simpliflcâtiôn is unnecessary. The value of À;d, however, depends on the distânce between A and B, and, since B is not initially known, the process of linding a compatible value of e is iterative.

2-l-l/clouse 9.8.2.2(4)

2-l-l/clouse e.8.2.2(s)

Where the available anchorage length, denoted /6 in Fig. 9.8-2, is not sumcient to ânchor thc force 4 at the distânce x,2-l-Ilclause 9.8,2,2(4) allows additional anchorage to be provided by bending the bars or providing suitâble end anchorage devices. Theoretically, the anchorage of the bar should be checked at all values of ,r. For straight bars, the minimum value of r would be the most critical in determining anchorage requirementsThis is because the diagonal compression strut is flatter for small x, and a greater proportion of the lcngth x is concrete cover. If x is taken less than the cover then the reinfùrcement cleally cannût operatc at all. 2-I-llclause 9.8.2.2(5) therefore recommends using â minimum value of À/2 âs a practical simplification. For other types of anchorage, such as bends or mcchanical devices, higher values ôf r may be more critical, since doubling the distance x, for example, will not double the available anchorâge fesistânce. The calculation procedure is illustrated in Worked examplc 9-8-l -

Fig.9.8-2. Model for

296

calculating required anchorage forces in tensile reinforcement

CHAPTER

9. DETAILING OF

IYEMBERS

AND PARTICULAR

RULES

. .l:: .::, :: ::::::::::i ;lt:: :. : ::: i

t-

j

;".",

.

::::

i

I ',:

::. '

287

DESIGNERS' GUIDE

TO EN I992-2

9.9. Regions with discontinuity in geometry or action The definition of a 'D-region', together with typical examples of such regions, is given in section 6.5.1 of this guide. D-regions have to be designed using strut-ând-tie models (see section 6.5), unless specific rules are given elsewhere in EC2. (Such exceptions include beams with short shear span which are covered in section 6.2.)

288

CHAPTER

IO

Additional rules for precast concrete elements and structures This chapter deals rvith Lhe dcsign ol'precast concrete elements and structures as covered in section l0 ofEN 1992-2 in the followins clauses:

. . . . .

General Basis of design, fundamental requirements

Materiâls

Structural analysis Particular rulcs for design and detailing

Clause l0.l Clause 10.2 Clûuse 10.3

Claute 10.5 Clause 10.9

Comment on EN 15050: Precast Concrete Bridge Elements and its applicability to design are nade in section 1.1.1 of this guide.

10.

I.

General

of LC2-2 apply to bridges made partly or entirely of precast concrete, and are supplementary tô the design recommendations discussed in the previous chapters and detailed in the corresponding sections ofEC2-2. EC2-2 addresses the additional rules for the use ofprecast concrcte elements in relation to eâch ofthe main general clauses in sections 1 to 9. The heâdings in section l0 are numbered 10, followed by the number ofthe corresponding main section, For example, rules supplementary tô 'Section 3 - Materials' are given under clausc'10-3 Materials'- The sub-sections in this guide follow the same format. Tl.re design rules in sccLion 10

2-l-llclause 10.1.1 provides def,nitions of terms relating to precâst concrete design. 'Transient situatjon' is perhaps most relevant to bridge design, Examples of trânsient situations include de-moulding, trânsportâtion, storage, erection and assembly. Transient design situations during transportation and storagc are particularly imporLant in the

2-l-l/clouse 10.1.l

design ofprctensioned beams, which can experience moment reversal in a transient situation

but not in the permanent situation.

10.2. Basis of design, fundamental requirements In the design and detailing ofprecast concrete elements and structures, 2-1-1 lclause 10.2(1)P requires the designer to specifically consider connections and joints between eLements, temporary and permânent bearings and trânsient situations as above. 2-1-llclaase 10'2(2) recuires dvnamic effects in transient situations to b{] taken into account wherc relevânt

2-l-l/clouse t0.2(t)P 2-l-l/clouse t0.2(2)

DESIGNERS' GUIDE

TO EN I992-2

Dynamic actiols are particularly likely to arisc during erection when lifting and landing beams, but the clause is not intended to apply to âccidental situations such as a dropped beam. No specific guidance is given in EC2-2 on dynzrmic load câlculation, other than to pennit representation of dynamic effects by mâgnitcâtion of static elTccts by an appropriate fâctor- Factors of0.8 or 1.2 would bc reasonable in such calculations, depending on whether the static effects were l'avourable or unfavourable for the elTccLs beine checked. These factors rverc recommende d n Motlel Cotle 90.6 An analogous rule for amplification of static effects is given in clause 3.2.6 ofEN 1993-l

-11

fbr the dynamic elÏects of the case of â sudden cable failure on a cable-supported structure, but this represents ân extremc câse. It should bc noted that in situ construction may also have transient situations which neerl to be checked and these are covered by 2-2iclause I 13. Some of the recommendations in 2-2lclause 113 also apply to prccast concrete.

10.3. Materials 10.3.l. Concrete The rules in this seclion are mainly concerned with the ellccts of heat curing on the rate of gain of compressive strength and the creep and shrinkage prôperties ol concrete. Where heat curing is applied, the 'maturity function' of 2-1-liAnnex B expression (8,l0) is used to produce a fictitious older age of loading which leads to reduced creep when used in conjunction with the other form ulae in 2-l-liAnnex B. The time ât which the creep effect is calculated shoulcl also be similarly adjusted according to expression (8.10). Exprôssion (8.10) can also be used in coniunction with expression (3.2) to calculate an accelerated gain of compressive strength-

2-l-l/clouse t0.3.2

10,3.2. Prestressing steel 2-I-I lclause 10.3.2 requires thc accelerated relaxation of prestressing steel to be considered where heat curing ol the concrete is undertaken. An expression is given to calculate an equivalent time which may then be used in the stândard relaxation equations of 2-1-1/ clause 3.3.2(7).

10.4. Not used in EN 1992-2 |

0.5. Structural analysis

10.5.l. General 2-2/douse t

0.s. t (t)P

The structural analysis of precast bridges usuâlly involves staged constructiùn, which is covered by 2-2,/clause l13. 2-2lclause 10.5,1(1)P gives further requirements for analysis of precast rnembers. In particular, the designer must specitcally consider:

. . .

2-2/dause r0.s.t (2)

development of composite action (e.g. superstructures comprising pretensioned beams

with in

siLu deck slabs);

behaviour ofcomrections (e.g. in situ stitch joints between pr€cast deck panels); tùlerances on geometry and position that may âffect loâd distribution (e.g. box beam Ianded on Lwin bearings ât each end is susceptible to possible uneven sharing of bearing reactions dus to the torsiônâl stiflness ôf the box).

2-2lclause 10.5.1(2,) allows benetcial horizontal restrainL caused by friction to be used in design, providing the element is not in a seisn.ric zone and the possibility ofsignificant impact loading is eliminatcd both of these could temporarily eliminate the compressive reactions

that led to the frictional restraint. This is particula y âpplicable to bearings but applies equally to precast and in situ construction. Further restrictions are that friction must not provide the sole means of attaining structural stâbilitv and that it should not be relied

290

CHAPTER IO. ADDITIONAL RULES FOR PRECAST CONCRETE ELEMENTS AND STRUCTURES

upon if the element geometry and bearing arrangements cân leâd to irreversible sliding trânslâtiôns. In the âbsence of a mechanical fixed bearing, the latter could oscur between superstructure and substructure under cycles of temperature cxpansion ând contraction, cau:'ing allernale :'ticking and sliding.

10.5.2. Losses of prestress 2-1-llcluse 10.5.2(1) gives a method for calculâting

losses

in tendons lrom temperâtule

dilïerences dudng the heat curing of precast concrete elelllents.

2-l-l/clouse t 0.5.2(t )

10.6,7,8. Not used in EN 1992-2 10.9. Particular rules for design and detailing 10.9.1. Restraining moments in slabs The rules in tl.ris section are of little relevance

Lo bridscs and are

nôt discussed here.

10,9,2. Wall to floor connections Thc rulcs in this scction arc of liLtlc relcvancc to bridees and âre not discussed hcre-

10.9.3. Floor systems The rules in this section are of little relevance to bridges and ale not discussed here.

10.9.4. Connections and supports for precast êlements Generol

The detailing of connections for precast concrete elements js a critical aspect of their design. Materials used for connections must be stable and durable for the lifetimc of the structure and must possess adequate strength. Th€ principles rn 2-1-llclaaset 10.9.4.1 and 10.9.4.2 seek to ensure this. EC2-1-1 gives specific rules for the detailing ofconnections transmitting compressive forces, shear forces, bending moments and tensile forces as discussed in the tbllowing sections. lt also covers the detailing ofhàlfioints and the ânchorage ofreinforcement at supports.

2-l-l/clouses 10.9.4.1 ond t

0.9.4.2

Connections tonsmittin g ..nmprcssive fofces

In the design ofconnections transnitting compressive forces,2-I-I lclause 10.9.4.3(1) allows shear forces to be ignored if they are less than 1OYo of the compressive force. Where greater shear forces exist, the vector resultânt force could be used in the check of bearing pressure, as suggested in section 6.7 of this guide. 2-I-llclause 10.9.4.3(2) reminds the designer thât âppropriâte rreasures should be taken

2-

to prevent relative rnovements that might disrupt bedding materials between elements during sctting. This applics also to stjtchjoints betwcen precast elements. The prevention ofrelative

t

movement may require restrictions to adjâcent construction activities or provision of temporary clamping systems. Bearing areas should be reinforced to rÊsist transverse tensile stresses in adjacent elements, as discussed in sections 6.5 and 6.7 of this guide. The rules in clause 6.7 generâlly dictate the nraxinrun.r achievable bearing pressure. 2-1-llclause 10,9,4.3(3), however, restdcts bearing pressures to 0-3/16 for dry connections without bedding mortar. Additionally, concrete faces with rubber bearings are susceptible to splitting caused by trânsverse expansion olthe rubber and, 2-1-llclause 10.9.4.3(5) gir.es a method of calculating the required surface reinïbrcem€nt.

t

l- l /douse 0.9.4.3(t)

2-l-l/clouse 0.9.4.3(2)

2-l -l /douse | 0.9.4.3(3)

l- I /dause t0.9.4.3(s)

2-

Connections tronsmitt)ng sheor forces

Interfâce sheâr ât construction joints between two concrete el€ments (for example, â precast beam and in situ deck slab) should be checked in accordance with clause 6.2.5.

291

DESIGNERS' GUIDE

TO EN I992-2

Fig. | 0.9-

l,

Connection of precast units by overlâpping reinforcement loops

Conneaions tronsmitfing bending moments or tensile forces

2-[-l/clouse t

0.9.4.5(

t)

2-l-l/clquse t0.e.4.s(2)

For connections transmitling bending rnonents and tensile forces,2-l-llclause 10.9,4,5(1) requires reinforcement to be continuous across the conncction and be adcquately anchored into the adjacent concrete Êlements. 2-I-llclause 10.9.4.5/?) suggests methôds of âchieving continuity. The methods listed, together with some comments on their use, are as follows:

. . . .

. .

lapping ofbars requires a large in situ p1ug, so is often not a suitable option for prccast deck panels landed on beams, for exâmple where limited connection width is available; grouting reinforcemcnt bars into holes requires accurate setting out and placem€nt; overlapping reinforcement loops (sec Fig. 10.9-1) usel ul for minimizing the size oI the in situ joint, but physical testing may be needed to demonstrâte âdequate serviceâbility pefÏofmance; welding ofreinforcement bârs ôr steel plates - useful for minimizing the size of the in situ joint, but bars will require fatigue checks in accordance with 2-1-liclause 6.8.4 and the fatigue verif,cations are morc onerous ior weldecl reinforcement than un-welded reinforcement: prestressing - as in precast segmental box girder construction. Although the use ofprestressing in this manner can eliminate the need lor in situ concrete plugs, the joint mai, require glue to seal the interfàce; couplers - threaded types of coupler are usually il1lpossible to use to join bars in infill bays while mechanically bolted couplers are usually of larger diameter, requiring gIeâter concrete cover.

Holf joints 2-l-l/clquse t

0.9.4.6(t )

2-I-llclause 10.9.4,6(1) provides two alternative strut-ând-tie models for the design of half joints which may be used either scparalely or combined. It is common to usc prestressing tendôns to provide the ties fbr the hâlf-joint nib, pârticularly for those ties angled across the corner from nib to main body of the member. This helps to limit crack sizes in an area which is hard to inspect and ensures the ties âre adequately anshored- The lack of provision for inspection and maintenance and the difliculties of excluding contaminants, such as de-icing salts, dictate that halfjoints should not generally be used for bridge applications. There havc been many examples in the UK of half-joint details which have been âdversely aflècted by comosion.

2-l-l/clause |

0.9.4.7(t )

Anchoroge of rcinfor.ement ot supports 2-1-I lclause 10.9.4,7(1) reminds designers that reinforcement in supported and supporting members sbould bc deLailed to provide adequate ancholage, allovr'ing for construction and

setting-out tolerânces.

2-l-l/douse r0.9.s.2(2)

10.9,5, Bearings 2-l-l/clause 10.9.5 covers the detailing of bcaring areas. 2-1-llclause 10,9,5.2(2) gives recommendations for sizing of bearings based on the following allowable bearing pressures:

. .

0.4,4a for dry connections, i.e. those without bedding mortar: the design strength of the bcdtling material for all other cases but lildted to a maximum

of 0.85/la.

292

CHAPTER IO. ADDITIONAL RULES FOR PRECAST CONCRETE ELEIYËNTS AND STRUCTURES

These pressures are only intended to be used tô determine minimum bearing dimensions. ReinforccmcnL in the bearing areas must still be determined in accordance with clause

2-l-1 6.5. 2-l-liTable 10.2 gives absolute minimum bcaring lengths. These are clearly intended for building structures and bridge bearing dimensions will always be significantly greater. dictated by limitation of bearing stress as above.

10.9.6. Pocket foundations 2-1-l,/clause 10.9.6 covers pocket fbundations which are capable of transferring vertical âctions, bending moments and horizontal shear forces fron.r columns to foundations. For bddges, precast pockets are sometimes used for pile to pile cap connections and pile cap to column connecLions, with in situ concrete used to form a plug between elcmentsFor connections where shcar kcys are provided, 2-1-l lclause 10.9.6.2 allows the connection to be designed as monolithic, but a check on the shear connection is still required. ln parlicular. only if adequate interfàce shear resistânce is provided under bending and axial force (see 2-1-l,iclausc 6.2-5) can punching shear be checked assuming a monolithic colul l,/pocket interface. Lap lengths need to be increased due to the distance belwecn lapping bars in adjacent elenents. as shown in Fig. 10.7(a) of EC2- I - I . Where there are no shear keys and the interlace between precâst elements is snooth. 2-l-l lclause 10.9.6.3(1) allorvs fbrce and moment trânsfel to be achieved by compressive reactiôns on the sides ol the pockcL (through the in situ plug concrete) ând corresponding frictional fbrces, as shown in Fig. 10.7(b) of EC2-l-1- This behaviour is very much rhat ol a dowcl in a socket, rather than of a monolithic connection. 2-1-llclause 10.9.6.3(3) therefore requires the reinforcemcnt jn column and pocket Lo be detailed âccoldingly for the forces acting on them individually. In parti lâr, care is needed with checking shear in the element in thc pocket as high sl.rear forces can be generated in producing the lixity momenl.

2-l-l/clouse t0.9.6.2

2-l-l/clouse |

0.9.6.3(t )

2-l-[/clouse |

0.9.6.3(3)

293

CHAPTER II

Lightweight aggregate concrete structures This chapter dea.ls with the design oflightweight aggregate concrete structures as covered in section l1 of L.N 1992-2 in the followins clauses:

. . . . . . . . .

I.l

General

Clause I

Basis of design

Cluu:ta I1.2

Materials DurabiUty and cover to reinforcement Structurâl ânalysis

Clause I1.3

Ultimate limit states

Claure I I .4 Clause I 1.5 Clause I 1.6

Sen'iceability limit states

Detailing of reinfôrcament - general DcLailing of men.rbers and particular rules

Clause I L8 Clause I L9

No comments are made on the'Additional rules for prccast concrete elements and strucLurcs' in clause 11.10 as it makes no modilications to the rules ofZ-l-liclause 10.

I

l.l.

General

The design rccommenclations discussed in the previous chapters and detailed in the colresponding sections ôf EC2-2 have becn dcveloped for concrete made from normalweighl aggregates. As naturally occurring aggregates become less abundant and increasingly expensive, manufacturcd aggregates are increasingly used and mosL manufactured aggregates are lightweight. The use of lightwcighr aggregatc concretes (LWAC) also has obvious advantages where it is desirable to reduce dead loads, such as in long spans that are dead load dominated. Lightweight aggregate concrete has been used throughout the wodd âlthough less so in the UK, particularly in britlge constluction. There is extensive test data verifying the properlics of lightweight aggregate concrete and thc implications its use has on the design verifications oi concrete structures. Section I I addresses these implicâtiôns on the use of the main general sections for normal-wcight aggregate concrete. All the clauses given in sections 1 to 10 and 12 of EC2 are generally applicable to lightweight aggregate concrete unless they are substituted by special clauses given in section I L The headings in section 1l are numbered 1 l, followed by the number of the corresponding main scction thal it modifies, e.g. section 3 of EC2-2 is 'Materials' so I 1.3 is similarly called 'Materials' and makes specific material requirements for lightrveight âggregate concrete2-1-llclause 11.1.1t'J) clarifies thât section l1 does not apply to air-entrain€d concrete or lightweight concretc wiLh àfl open structure. 2-l-llclause 11.1.1(4) dcfincs lightweight

2-l-l/clouse

H.t.t(3) 2-

l- I /douse

rt.t.t(4)

DËSIGNERS' GUIDE

TO EN I992-2

aggregâte concrete as concrctc having a closed sLructurc with an oven-dry density ofno more than 2200 kgimr, made with â propôrtiôn ôf ârtificiâl or natulâl lightrveight aggregates with a parlicle dcnsity of less lhan 2000kg/m'.

| 1.2. Basis of design 2-2,/clause 2 is vaLid for lightweight aggregate concrete without modificatjons. This includes

the material làctors for concrete.

11.3. Materials

2-l-l/clouse

il.3.t (t) 2-l-l/clouse t t.3.r (z)

2-l-l/clouse t t.3.r (3)

I 1.3. |. Concrete The strcngth classes of lightrveight aggregate concl€te are designated by adopting the symbol LC in place of C f

0.5n"

ln addition to the main reinforcement provided at the top of the corbel (with a totâl area of 1,,-^6), vertical links are required where the shear force exceeds the concrete shear

ANNEX I. DETAILING

RULES FOR PARTICULAR SITUATIONS

Fig. J- l. Corbel strut-and-tie model

resistânce according to 2-l-1,/clause 6.2.2. 2-l-llclause J.J(-l) expresscs this latter condition as aEd > tr/Rd c, but it would be reasonable to interpret this âs /JFEI > I/R,l.c in accordance with 2-l-l1clause 6-2.2(6). rvhere'1 is the reduction làctor to allow lbr. shear enhâncement. Where link reinforcement is required, its provision clcarly no longer relates to the strut-and-tie model of Fig. J-1. 2-l-liclause J.3(3) requires the link force provided to be a minimum of 50% of the applied vertical force. This shoulil be viewed as ân âbsolute minimum requirement. with the link area determined more

2-l-l/clouse 1.3(3)

generally in accordance with 2-1-l/clause 6.2.3(8). The check of the compression strut can agâin effectivcly be made by checking that Lhe shcar is limitcd such that Fea < 0.5b*duJ"a in accordance with 2- I - l,/Expression (6.5).

For all sizes of corbel, the anchorage of the main reinforcement pfojecting into the supporting membcr should be checked for adequacy in accordance u,ith the rules in sections 8 and 9 of EN 1992-2.

14. Partially loaded areas

j4.

|

.

Bearing zones of bridges

The dcsign for allowable bearing pressurc and reinforcernent design is covered in sections 6.5 and 6.7 ôf this gûde. 2-2lclause J..ft4.1 gives somc additional requiren.rents as follows:

.

the minimum distance between the edge of a loaded arca and thi: edgc of thc scction should not be less thân 50mm or less than li6 of the coresponding dimension of the loaded area:

2-2/clouse

J.t04.t

Fig. J-2. Sliding wedge mechanism under concentrated load

329

DESIGNERS' GUIDE

TO EN I992-2

for concrete grades in excess of C55/67, Id needs to be rcplaced by 10.46f:i.'l0 + 0.lik)]Id. This formula doesn't actually reducc the value of fo1 until the concrete gLrde exceeds C60/75: an additional sliding rvedge mechanism, illustrated in Fig, J-2. also needs to be checked.

The 'failure' plane is deflned by d: 30" and an amount of reinforcement given by > -F1.1,/2 must be uniformly distributed over the height lx. The provided reinforce'4. [a ment should be suitably detailed and anchored in accordance with 2-2/clâuse 9, which will normallv necessitate the use of closed links.

J4.2. Anchorage zones of post-tensioned members 2-2/clouse

1.t04.2

330

2-2lclause J.104.2 gives rules supplementary to those given section 8-10.3 of this euide.

for anchor zones of post-lcnsioned members which are in 2-1-11clause 8.10.3. These provisions are discussed in

ANNEX K

Structural effects of time dependent behaviour (informative) Annex KK oi EN 1992-2 is mainly concerncd with the redistribution of internal actions and stresses that occur in bridges built in stâges. This includes, for example, box girder bridges built span by span with striking of formwork at eâch stâge and precast composite members in un-propped constluction where a dcck slab is cast ailer ercction of the mdn precast beans. This section of the guide does nôt fôllow the clause headings of EN 1992 Anncx KK and is structured as follows: General considerations General method Simplified methods

Application to pre-tensioned composite members

Kl.

Section Kl Section K2 Section K3 Section K4

General considerations

Creep will tend to cause action effects built up from staged cônstructiôn to redistribute towards the aqLion effècts thât would have becn produced had the structure been constructed

monolithically all at the same time. The effect of creep redistribution is illustrated in Fig. K-1 for the dead load moments in

a

three-span bridge. The bridge is built span-by-span in the stages shown and the deatl load

molnent creeps from its built-up distribution towârds that for monolithic construction. In prcstressed members, the Frestress sccondary ûroments are similarly redistributed. Similâr creep redistribution also occurs in simply supported pre-tensioned bcams whicb are subsequently made continuous. For prestressed members (whether post-teûsioned or pre-tensioned) this redistribution of moments ând stresscs is particularly important for the serviceability limit state design, as it can lead to unacceptâble cracking and serviceability sftesses if not considered properly 2-2lclause KK.2(101.1 refers. Consideration ofthc redistribution is olten less important at Lhc ultimate linit state and can be ignored where there is suficient rotation capacity available to shed the restraint moments, unless âny ofthe bridge members are susceptible to significant second-order effects - the Note to 2-2iclause KK.2(101) refers. Providing the concrete stress under quasi-permanent loads does not exceed 0.45[k(r, linear creep behaviour may be assumed where the creep strain varies linearly with the crecp 2-2lclause KK.2(102). Where the concrete stress exceeds 0.451"L(r), non-linear creep hâs to be considered, whereupon the creep strain vades exponcntially \À'ith sLrcss-

2-2/clouse

KK2(t

0

t

)

2-2/clause KK.2(

l02)

DESIGNERS' GUIDE

TO EN I992-2

l\4omenls resulling f rom the conslruclion sequence

Fig. K-

|

,

lvlomenls after creep redistribution

Typical redistribution of dead load moments due

Moments assuming b dge built in one go

to

creep

n.ray need to be considered for pre-tensioned beams which are stressed to high loads at â young age. This is discussed in section 3.1.4 of this guide. Annex KK of EC2-2 does not cover the effects of differential shrinkage. This is covered in K4.2 of this guidc. Five methods ofconsidering the effects ofcreep redistribuLion are presented in EN 1992-2:

Non-linear creep

(a) (b) (c) (d) (e)

gcncral stcp-by-stcp analysis 2-2lclause KK.3; differential version of the above 2-2iclause KK.4; applioation of the theorems of visco-elasticity - 2-2iclause KK.5; coefficient of ageing method - 2-2iclause KK.6; simplilied method based on coellicient of ageing method - 2-21clause KK.7.

Generally, onll' method (e) will need to be explicitly considered by designers as this will usually predict long-term eflects adequalcly and is the only method that lends itself to simple hand calculation. Where it is important to pr€dict losses and deflectiôns ât intermediate stages of construction, as rvould be the case in balanced cantilever construction, jt will usually be necessary to use proprietary software which is likely to usc variatiôns on method (â). Onll' methods (a) and (e) âre theltfore considered bclow and their use in the context of both composite and non-composite beams is discussed.

K2. General method Thc gcncral melhod invùlves a slcp by step calculation of the strain according to 2-21 Expression (KK.101), which will typically be perfbrmed by iterativc computer analysis. Such a method will also include losses due to the primary effects of creep and shrinkage, as well as duc to creep redistribution cffccts:

'c(rl

au o" ...

E,t,n1-

'rt)

The application ofthis

où _+ f__L-- {,..t,,)oo,,.,I -e 2- \EJ'J F...{rRli ^o'' _É.,(r./.t 2_2.(KK.tgl)

E,\281-

fomula to unrestrained concrete (ignoring shrinkage strain) under

K-2. ln general for real members, the concrete stress will not change in discrete steps as shown in Fig. K-2, as the stress itself will conslantly vary rvith thc changing strâin due to the presence of extemâl restrâint or internal restraint (from reinforcement or prestressing). A computer analysis is therefore required to split the analysis into a scrjcs of time steps such that the effects ol a series of additive externally applied axial forces is illustrated in Fig.

ANNEX K. STRUCTUML

EFFECTS OF TII4E DEPENDENT BEHAVIOUR

ï

E"(bl

1

t

Fig, K-2. Creep strain accordingto 2-2l(KK.l0l) for unrestrained concrete with several load incrêments the constantly varying stress can be transforrncd into smâll discrete stcps similar to those in

Fig. K-2. A dillèrential version of this procedure is provided in 2-21clause KK.4. The cllccLs of prestressing steel relaxation also need to be considered. Computer methods rvill normally also include the effects oj relaxation loss in a way similar to that discussed in Annex D.

K3. Simplified methods A sirrple n.rethod for calculating the long-term creep redisLribution elTècts is given in ?-2/ dause KK.7 bâsed on the ageing coelTcienl mcLhod- The basic proccdure is to calculatc the distribution of moments (or othcr internal actions) built up by fully modelling thc construction sequence and to then recalculate the distribution of moments assuming thât the structure wâs built in one go with all dead loads, superimposed dead load and prestressing applied to the final slrucLurc. The actual long-tenn [lomcnts âccounting for creep redistributjon can then be found by an interpolation bctween these two scts of moments according to 2-2,/Expression (KK.ll9), without the need for calculation by a time-step method. 2-2lExpression (KK.l19) also applies to stresses:

2-2/clouse KK7

2-2(KK.l l9) The redistribulion of actions due to creep is therefbre:

^s

: (s. so)

ri (o]s,

lo)

-

c;(r", ro)

I + 1rl(co, r")

(DK-r)

where: ,SO

s"

Ploc,loJ

ôQ.,t()

are the inlernal actions obtâined from the construction sequence build-up are the internal actions obtained assuming that the whole sLructure is built in one go and then all the pcrmanent load applied to it. The phrase 'construcLed on centering' is used in EN 1992-2 to convey this idea of the structure belng

constructed in its enlirety on continuous supports, with the supports or 'centering' being released only after the entire structure is compleled- Strictly. the word'centering' is normally uscd to describe temporary suppôrts to arches is tl.re final creep coefficient for a concrete age /0 at time of first loading is the creep coeficient fot a concrele agc l0 at time offirst loading up to an age /", whele t. is the concrete agc at which the structural system is changed, such

333

DESIGNERS' GUIDE

TO EN I992-2

as closure

of a stitch between adjacent spans. It is therefore the amount of

creep lactor'used up'before thc structural system is changed. Where the structural svstem is changed in this way a numbcr of times, a representative averâge age should be used based on the average age at rvhich each stage is connected

,r(ca,

t")

to the next. For example, if each stagc takes 30 days to construct (including shipping of falsework) and is then immediately connected to the previous span, Lhen /c : l0 : 30 days woulcl be a reasonable approximation as the prevlous stage concrete would be ôlder than 30 days, but some ofthe cufrent stâge concrete would be younger is the flnal creep coefficient for a concrere age /c at time offirst loâding

The creep coefficients can be calculated as discussed in section 3.1,4 of this guide. The ageing coeflicient 1 can be thought of as representing the rcduction in creep, and thcrefore increase in stiffness, for restraint of deformations occurring âfter the time tc. It cân be taken as 0.8, which is a good representâtive value for most construction, but in reality it va es with age at time ol loading and cr.eep factor. The redistribution in equation (DK-l) can be thought ofas follows. If the initiâl moments ln â structure are M0,i, resulting from load applied ât time t0, Lhen the free creep culyature fronr tlrese moments occurring alter a time l" will be M11.;lEI(g(,.:n, tl - d(1c, t0)). If the structure wet'e to then bc fully restrained everywherc (a purely theoretical rather thân practical situâtion) at time /c, restraint moments would be developed. The effective young's

modulus fbr this load case would be É/(l +Xé(oo,r")) and hence the fully restrained moments developed would be Mo,i(O(*, ru) - AQc,rù)/(1+ Id(oo, r")). These restraint monents reFresent the ledistribution moments lbr this particular change of shuctural system- Equation (DK-l) predicts the sâme result- In Lhis case. So: Mo.t, S" is obtâined assuming the structure to be in its final condition (i.e. fully restrained everywhere in this case) prior to initial loading. This leads to 56 - M.,;:0 and thus equation (DK-1) gives redistribution moments ol' Mj,;(ci,(cc,ln) - (t(t",tt)))ll + XA(ca, r.)). For non-prestressed concrete bridges. there is no problem with the interpretation of 2-2/ Expression (KK.l l9) and the 'S' terms contain only the dead load and superimposed dead load. The internal actions are then redistributed as in Fig. K-1. For prestressed bridges, however, the magnitude of the prestressing lbrce itself changes during the life of the bridge so the value to use in the interpolâtion needs careful consideration. The secondary eflects of prestress are themselves altcred by creep (due to the loss of prestrcssing force) even without chânging the structural system. There are several interpretâtions possible for the prestressing force to use in the interpolat1on. This is a result ol the lirct that the method is not exact. Four interDretâtions (others are possible) include:

(a) The prcslress force after all short-term and time-dependent long-temr losses is considered in deriving Sc, but the prestress force including only whâtever lossÊs that have

(b)

334

occurred up to 'closing' the structure is considered in dcriving Sp In each case, redisffibution eflects are ignored in calculaLing S" and S0 (othef thân the small change in secondary ell-ects of prestress caused by the loss of prestressing force), This appears to be the literal inleryretation of the definitions of Sc and 56 given in KK.TiExpression (101). This is not the intended interpretation, however, as Soo then does not include all the loss of prestressing force and the effects on rcducing the primary presffess moments. This is because the final stress state is ellectively then obtained i'rom an interpolation between values ofS" which include all losses oIprestress, and values ofS0 which include only the immediate lôsses and a small fraction of the long-term time-dependent losses. This approach is therefbre inappropriate as it underestimates losses and overestimates the actual final prestress force. The prestress force, including âll short-term and time-dependent long-term lossss, is used to determine both 5'" and Sn. The long-term prestrsss losses in each case can be determined from the concrete stresscs after application of the initial prestressing force, including the immediate losses. In eâch case, redistribution eflects are ignored (other

ANNEX K. STRUCTURAL

(c)

EFFECTS OF TIME DEPENDENT BEHAVIOUR

than the small change in prestress secoudary moment resulting from the loss ofprestressing force). S- from 2-2lExprcssion (KK.119) then includes all the long-term loss of prestress as well and can be tâken to represent the fina1 internal actions including all the long-term losses. The prestless lbrce including only short-term losses (i.e. not considering tine-dependent long-lcrm losses) is used to determine both S. and .ln- The ternl

. d(/cr/0) (s.. - s.)ç'(T,/ul 1+ lplco rcl

from equation (DK-l) is then calculated, which represents

the redistribution eflects only. The lông-lerm loss of prestress is then subsequently calculatcd from the built-r.rp internal actions Sn and these built-up internalactions modified accordingly to allow lbr long-Lcrnl losses of prestress. This lcads to a set of long-term internal actions derived from followins the construction sequence but without consider-

ing redistribution. say .!0,-. The term

(S. "rf

tlerived above

Hfi#P

is

added to Sn.- to give thc final set of internâl actions allowing for both creep redistibu-

tion and all long-term

losses of prestress, i.e.

S*

:

So,-

ï

(""

-

ttî*0i#1::i") "r)

(d) As (c) but rhe prestress force used to calculate (,sc so) 4ra;ffi*Gf) i, tt.tut immediately ât the time of the change of structural system, e.g. connection ol adjacent spans. 56 is derived considering the build-up of stresses and all long-term losses of prestrcss.

-

These approaches are all approximate but method (d) often gives results closest to the results of a time-step analysis. This approach is summarized below under the heading 'Application to post-tcnsioned construction'. Only melhod (a) above is completely

ilapprop

ate to use. Given the inherent uncertainty in creep calculations, this interpolation approach ls nornally of satisfactory accuracy for predicting long-term redistribution effects Where there âre mâny stages of construction with concretes of dillerent ages before a struclure is made statically indeterminate (such as occurs in balanced cântilever construction), or where there âre mâny changes to the slrucLural system (such as occurs in span-by-span construction), one difficully is deciding â representative single value for the creep coefficients d(cc,ro) ci,(r", 16) (which is the creep reI aining after the structural system has been modified) and qi(co, r"). It will usualJy he adequâte to select an averâge residual creep value for thc slructure- In môst câses for post-tensioned structures, Lhe value of ,tfiLr/- .,\/r r. \ v--: ,-\ , ""'.'u' *ill typically be betueen 0.ô5 rntl 0.8 1+ig,lo(,r./

A simpler version of 2-2/Expression (KK.l l9) often used (and which was cffectively used in BS 5400 Part 4') is:

S,.:50 * (s. -so)(1 e ")

(DK-2)

Equation (DK-2) has only the one creep fàctor which cân be taken âs ô = o(oc, ro) O(/", /0), but it does not contain the ageing coefficient. lL is therefore less accurate thân 2-2i Expression (KK.I 19). A value for $ of between l -5 and 2 0 will be representâtive in most cases for posL-tensioned bridges. with the above expression, the redistribution of actions due to creep is therefore:

As: (S" Su)(l

e

')

(DK-3)

Equations (DK-2) and (DK-3) tend lo ovcr-predist the amount of redistribution that will occur, pafiicularly in structures where the concrele is quite old ât the time of modifying the structurâl system (such as closing a structure with â stitch).

335

DESIGNERS' GUIDE

TO EN I992-2

Application to post-tensioned construction For post-lensioned beams,

â suggested

approximate proocrlure, following EC2-2 formulae, is

as fbllows:

(l) (2) (3)

Build up thc internal actions for the bridge fiom the construction sequence considering only immediate losses see 2-2iclause 5.10.5Calculate long-term losses in prestressing fbrce using 2-2iclause 5.10.6, based on the cônorete stlesses obtained fiom above. Detcrmine the chânge in primaly ând secondary prestress moments from this loss of ptestress.

(a) Modify the internal actions in (l) by the ellects of the losses in

(3) to givc long-ternr

effects excluding creep rcdistribution.

(5) Modify the interral actions in (4) by adding the effects of rcdistribution according to cquation (DK-l) with S0 and Â, bâsed on thc loss at the time of modifying the structural system.

It

336

is always possible to use a conputer time-step analysis as an alLernative.

ANNEX K. STRUCTURAL

EFFÊCTS OF TIME DEPENDENT BEHAVIOUR

K4. Application to pre-tensioned composite members K4.1. Differential creep in composite beams Whcn a deck slab is cast on a pre-te[sioned beam, the change in cross-section afLer thc beam has been prestressed will gencratc rcstraint stresses, as the loaded pre-tensioned beâm tries to deflect turther with creep and this is resisted by the deck slab- The suesses will redistribute from the built-up values towards those obtained if the dead load and prestress were applied to the flnal cômposite section. Evcn if thc bridge is not built in stâges span-wise, redistribution will take place internal to eâch cross-section.

337

DESIGNERS' GUIDE

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This redistribution is illustrated in Fig. K-4 for a simply supported beam. It illustràtes that the internal actions ând stresses should bc calculated for thc as-built case and monolithic case

and then interpolation between them performed using either equâtion (DK-1) or equation (DK-3) to determine thc rcdistribution effects. As discussed above, equatjon (DK-3) may overestinate the magnitude of redistribution. The prestress forces to use should be those immediately after casting the deck slab, so an assumption has to be made with respecr to this timing and thus how much creep has therefore already occurred. Determination of the crccp fâctors to use in equation (DK-l) or equation (DK-3) also requires estimation of this timing. Worked example K-2 illustrates the calculation. lf the sinply supported pre-tensioned beam is subsequently made continuous, which is lypically achicved by connecting adjacent simply supported spans together, the creep defornation is further restrained by thc modilicd strucLural system and the support reactions are modified. This leads to the development of rnoments ât the supports due to th€ redistribution of dead load rnonents and prestress. These continuity moments develop in the same way as discussed for post-tensioned beams due to the changed support conditions. Usually, sagging moments develop at the supports and additionâl reinforcement is required across the jôint to prevent excessive cracking. This problem can be treatcd by interpolation bctween the as-built case (l) and monolithic câse (2) (with âll loâds âpplied to the composite section) âs in Fig. K-4, but case (2) will also contain the secondary effects of prestress due to the continuitv and the dead load moments will also be modified by côntinuity. Either equation (DK-l) or equation (DK-3) can again be used. This is shown in Fig. K-5. Alternatively, the internal lockcd-in slresses qan first be determined exactly as in Fig. K-4 without considering the bridge to be mâde côntinuous. Restrâint of the creep fiom the

\[ Pnmary Preslress

load

Dead {beam and deck slab)

n Total

(1) Dead load (excluding SDL)and prestress applied to lhe precast beam

\V l\ * l\\ / t\l |\

v

t_l

Primary preslress

Dead load (beam and deck slab)

Totâl

(2) Dead load (excluding SDL) and preslress applied lo the composite beam Diffêrenc€

(2)-

(1)

(monolithic-as.buiit) slress due to creep redishibulion

(3) Stress lrom creep redistibution lrom (1) towards (2)

Fig, K-4. Redistribution of stresses in simply supported composite beam due to creep

ANNEX K. STRUCTURAL

i' "'__:" "i

EFFECTS OF TIME DEPENDENT BEHAVIOUR

tJ \ V tI * t/ | l.l\ -r;\ \ /r l\ I

=

L.il_..llll___:..1_i!\,ttl

P mary prestress

Dead

load

I

Tolal

(Deâm ano

deck slab) (1) Dead load (excluding SDL) and preslress applied to lhe precast bearn

n 1

\

[

u

[

\

Primary Dead loâd Change in pre$ress momenl dead load ignonng momenl due continuity locontinuily \ __Y---/

Secondary

Total

prestress

(2) Dead load (excluding SDL) and prcsiress applied to the composite beam

Ditlerence

(2)-

(monolithic

-

(1) as-built)

Siress due lo creep rêdistibution

+++++++ +

++++*+ (3) Slress from creep redisiribution irom (1) lowards (2)

Fig. K-5. Redistribution of stresses due to creep in simply supported composite beam subsequently madê continuous

to the devslopmcnt of support moments. These restraint momcnts are linear between supports and the crept value at the support can be found using equâtion (DK-l) or (DK-3). [n this case, Sç - 0 at the supports and S" should consider the secondary ellects of prestress ând the dead load support hogging moment on the continuous beam only, as other effects are included in the initial step. In either case, the stress from these redistribution cffects should be added to the built-up stresses for the beam includins all losses. prestress and dead load leads

.-.'..'' 339

DESIGNERS' GUIDE

TO EN I992-2

I - f49 x l0l mm2 z*, - i+.lt x lo6 mm l 2,.^ .rop = 46.96 x 106 mml Eccentricitv of strands from"'""' neurral " axis =

142

mm.

;:":.:-"^-:'*'" Zb.t

2,""

-

*

x l0omml irs.e x tobmml 114.9

Eccenrriiitj of strands from neuLral ar.is:306 mm-

Botrom fibrc srress from prestress

*:::.

rop nbre

1"-

*".-i a # :

Borlom fibre stress from dead load Top tibre

*r.r, t *

lir- ilif - l5,27Mpa

-

dead load

-

:

w..I l;ya #k:

,nrïO-4.1.1

I:

II

I : : I : : : ll :]

i:

::l

:l

-4.30 MPa

MPa

': ree :inr

itr

)s me 101

/1 r\l

,:,'lAi.o.-,;i ':: l::l i:., ' '--: r.:

340

IOn:

r

t'{

\: /

....

:0 .9.4 MPf

ANNEX K. STRUCTUML

EFFECTS OF TIME DEPENDENT BEHAVIOUR

;Ëh:. ,

-l J

.13 It

I '::::."1:::'::r \t'

iroo l.-

rl

't-

.- -...t.-r-, :: :: i :.l: :

fig.

l(,.6;

L!

:

a

,

Iqr

monolil

..t

:ii.!;

i!1

çârnpô

akèd

I

;,r

K4.2. Differential shrinkage EN 1992-2 does not explicitly cover the calculation of the effect of dilïercntial shrinkage. but it must be allowecl for at the seryiceability limit state in composite beams whele the residual shrinkage strain of deck slab and precast beam differ after the beam is tnadc composite. Usually the precast bean will have undergone a significant parl of its totâl shrinkage strain by the time the deck slab is cast, so thc deck slab will shrink by a relatively grcater amount, This relative shrinkage will compress the top of the precast beam (causing axial lorce and sagging momcnt in iL) whiJe generating tension in the deck slab itsell. The differential shrinkage strâin cân be estimated from 2-2i clause 3.1.4 if Lhe approximâte age of the beam at casting is known so that: a,11p:a.1,,66(oc) (e.r,,r*-(co) .,h,b",,-(lL))

(DK-4)

where e"1.1,"","(co) is the total shrinkage strain of the precast beam, êsrr sLat(æ) is the total shrinkage strain ol the slab and e,5s"n,',(r1) is the shrinkage strain of the precasL beam âfler casting the slab. Il Lhc bcam was fully restrained and the shrinkage occurrcd instantâneously, the restraint axial force in the slâb would be .P"1, : e66..É1,66- However, as the shrinkage strain occurs slowly, this shrinkage force is modilied by creep so ttrat ihe fully rcstrained force is actually

more realistically estimated from: F,6

-

ra,xÀ1q.,6

/l -e i;;r

o\

(DK-5)

Once again, it is not simple to estimàte a single creep râtio to apply in this case covering boLh slab and precast beam. Given the uncertainties involved in the calculation as a whole, a value of2.0 will generalll' suffice- This restrained force can be sepârated into a rcstraint axiâl

force and moment acting on the whole cross-section, together with a locked-in self equilibriating stress as shown in Fig. K-7. The axial stress and bending stress can be determined from the axiâl force in equâtiôn (DK-5) acting on the cômpositÊ section. If the bridge is statically determinate, then the restraint rnomcnt component can be releâsed withôut generating any secondary moments. Similarly, if the deck has no restraint

341

DESIGNERS' GUIDE

TO EN I992-2

--l

+J

\t

\

V

l\ ++

+++++++ + ++++r-++

I

Fullrestraint

I

x

\

Self-equilibrialing

Axial

Fig. K-7. Components of differential shrinkage resrrâined stress

to contraction, then the axial component can similarly be

released

without generating

any rcslraining tension. If lhe bridge is statically indeterminate, however. the release of the resûâint momenl component will generate secondary moments. The determination of secondary moments is illustrated in Fig. K-8 for a simplc Lwo-span beam of constant cross-section- In Fig. K-8, the restraint moment is hogging, so to releâse this moment a sagging moment ofequal magnilude must be applied to the beam in the analysis model. A calculation of diffèrential shrinkâge eflects is given in Worked example 5.10-3.

K4.3. Summary procedure for pre-tensioned composite beams For pre-tensioned composite beams, a suggcsted approximate procedure using formulae in EC2-2 is as follows:

(l)

Build up the internal actions for the beams frôm the construction sequence considering only immediâte losses sce 2-2/clause 5.10.4.

(2) Calculate long-term built-up

losses

in prestressing force using 2-2/clause 5.10-6 with the above

stresses-

(3) Determine

the change in presl.rcss moments from lhis loss of prestress. ThaL part of thc loss occurring prior to casting the slâb should be used to reduc€ the prestressing fôrce on the precâst bearn alone. The part ofthe loss occurring afler the slab has been cast can be

(4)

estimated by applying the remaining loss of lorce as a series of tensile forces to the composite section along the line of the prestress centroid. (If the strands are straight then a single force equal and opposite at each of the beam will suflice.) It is quite common, however, to apply all the loss to the precâst beâm as it will generâlly mâke Iittle difference - see Worked example 5.10-3 in section 5.10 of this guide. Modify the internâl âctions in (l) by the effects ofthe losses in (3).

m

Fully €slrained momenls

Resulting secondâry momenls

Fig. K-8. Secondary moments from differential shrinkage

342

ANNEX K. STRUCTURAL

(5)

EFFECTS OF TIME DEPENDENT BEHAVIOUR

Modily the internal actions in (4) by adding the effects of redistribution âccording to equation (DK-l) with S0 ând ,Sc based on the prestressing forces ât the time ofconstructing lhe slab, as described in sectiou K4.l of this guide-

(6) Add the effects of differential shrinkase.

343

ANNEX

L

Concrete shell elements (informative) EN 1992-2 Annex LL, in coujunction with 2-2,/Annex F and 2-2/clause 6.109 (invoked by paragraph (112) of EN 1992-2 Annex LL), gives a method of designing concrete elements subject to in-p1anc axial and shear fbrces, together with out-of-plane rnoments and shear forces. It is most likely that Lhis annex will be used when such a stress ûeld has been determined from a finite element model, âlthough the equâtlons âre presented in terns of stress resultânts (force or moment per metre) rather tlan stresses. The annex can also be used rvhere sinpler analysis has been performed, as illustrated in the discussiôns on Annex MM in this guide, covering the design of bor girder webs in bending and shear, A sandwich model is empJoyed to convert out-of-plane noments and lwisting moments iûLo stress resultants acting in a single plane in each of the outer layers of the sandwich. The outer layers also carry the direct and shear stresses acting in the plane of the element. The core section of the sandwich is used only to carry shear forccs which are transverse tô the elemcnt's thickness. Paragraphs (109) and (110) require the principal transverse shear force in the core to be verified separately by means of the beam rules 1'or shcar in 2-2,/ clause 6.2. Expression (LL.123) in paragraph (110) recommends that the leinforcement ratio for this check bc based on the strength ofthe reinforcemert layers resolved in the direction ot the principal sheâr, v.hereas previous practice (e.g. reference 6) has based the reinforcement ratio on the stiflness in this directiorl. As the latter is more established, it is recommended here that Expression (LL-123) is modifrcd in linc with this assumption such that: Pt

:

tLl

P\ Cos O0

+ pl srn

dn

(DL-1)

Having determined lhe stresses in each outer layer, the rnembrane rules in 2-2/clause 6.109 and 2-2/Annex F can be used to design orthogonal reinlorcement and check concrete stress fields. Skew reinforcement is not covered in EC2, but section 6.9 of this guide provides some guidance in such cascs. lt should be noted that the use ofthe sandwich model and membranc rules in design do not make any allowance for plastic redislribulion across a cross-sectlon and so can bc very conservative. Consequently. it is always better to use the member resistance rules in the main body of EC2, wherc applicable, as such rcdistribution is implicit within thosc rules. The flrst part ofAnnex LL deals with checking rvhether or not the elenrent will crack under the loading considered. If the element is prcdicted to be un-crackcd, the only check required is thât the principal compressive stress is below a"..1"ç/1i or a higher linrit based on.,f1," where there is triaxial compression. Tbc formula prescntcd in paragraph (107) for checking cracking is a general triaxial expression (based on principal stesses o1, o2 and a3), It is only one of severâl existing possible crack prediction lbnr.rulae. More background on this subject

ANNEX L. CONCRETE SHELL

ÊLEMENTS

can bc found in reference 29. It is inappropriate that mean values of concrete strengths should be used jn this vedfication, âs implicit in Expression (LL.10l), as thc calculations relate to ultimâte limit statc strength; design values of concrete strength rvould be more appropriate. It is therelbre recommendcd here that /* and ir- should be replaced by /11 and ;/",6 rcspectivell, in Expressions (LL.l0l) and (LL.ll2). A simpler altemative for biaxial stress states only would be to use the cracking verification ofAnnex QQ, but replacing the characteristic tensile strength with the design tensile strength in Explession (QQ,l0l). The sandwich model and equations are lairly sclf-explanatory and arc much simplified when the reinforcement in each direction in a layer is assumed to hâve the sâme covet. This assumption ûray not be appropriate in very thin elements, The use of the equations is not discussed further here, but the special case of shear and transverse bending in beam webs, with some further simplilications, is investigated in detail in Annex MM of this guide. Use of the sandwich model is complicated slightly where reinforcement is not centred on its respective sandwich layer. It may sometimcs be necessary to choosc layer thicknesses such that this occurs wbere the in-plane compressive stress fleld is high. Annex LL gives a method to account for this eccentricity in its paragraph (l 15) and this is again discussed in Annex M of this guide. A further use of Annex LL would be for the design of slabs subjected principally to transverse loading. In previous UK prâctice, such cases would have bccn dcsigned using the Wood Armer equationsle'2o or the more general capâcity fleld equations.2r The combined use of Amex F, 2-21clause 6.109 and Annex LL to design slab reinforcement does not necessarily lead to conflict with these approaches. The reinforcement produced is

usually the same, ôther than minor differences due

to

assumptions

for lever

arms,

Houever, 2-2/clause 6,109 sometimes limits the use of solutions from the Wood Armer equatrons ôr the more general capacity field equations through its limitation of e ê.r = 15'. It âlso requires a check of the plastic compression field, which refèrences 19, 20 and 2l do not require. Despite neglect of these requirements, the Wood Armer equationsle'20 have. however, successfully been used in the past.

345

ANNEX M

Shear and transverse bending

(informative)

M

l.

Sandwich model

Although the maximum allowâble shear stress, determined by orushing of the web concrete within the diagonal compression struts, is generally significantly higher in EN 1992 than previously used in the UK, there will be occâsions when this higher limit cannol be mobilized. In webs of box girders, transverse bending moments can lead lo significant reductions in the maximum permissible coexistent shear force because the compressive stress fie1ds from shear and from trânsverse bending have to be combined. The stresses from the two fields are not, however, sin.rply additive because they act at different angles. In the UK, it has been common practice to design reinforcement in webs for the combined action oftransverse bending and shear, but not to check the concrete itself for the combined effect, The lower limit in shear used in the UK made this a reasonable âpproximation, but it is potentially unsafe

if a less conservative (and more realistic)

cr-ushing strength is Bsed.

2-2/clause 6.2.106 formally requires consideration ofthe above shear momenl interaction, but if rhe web shear force is less than 20% of tr/p,1.n1u" or the transverse môment is less than 107" of the maximum transverse moment resistance then the interaction does not need to be considered- These criteria are unlikely to be satisted for box girders, but the allowance fol coexisting moment will ollen be su{icient to negate the nced for a check of webs in t}pical beam and slab bridges. Where the interaction has to be considered, Annex MM can be used. 2-2iAnnex MM uses the rules for membrane elements in 2-2,/clause 6.109 and a sandwich model (based on Annex LL) to idealize the web as two separate outer layers subject to inplane forces only. Such câlculâtiôns are potentially lengthy because the longitudinal direct stress varies over the height of the beam and the vertical direct stresses vary through the

2-2/MM(t0t)

thickness, making the angle of the elastic principal compressive stress vary everywhere. Also, the membrane rules of 2-Z/clause 6.109 apply to the design of plates with a general stress field ohtained from finite element (FE) analysis. The check of transverse bending and shear in webs is usually going to be done without reference to ân FE analysis, Consequently, some simplifications are made ln 2-2lMM(101) to fâcilitate the web design as follows:

.

The shear per unit height rnay be considered as having constant value along Â.y in

Fig. M-l:

uea

:

VealAy.

The intent is to permit the shear stress to be taken as constant on all sides of the element. The use of o;4 as a shear flow is unfortunate as it is used as â sheâr stress in the rest of EC2. There is also no guidancc given on the length Ay used lbr averaging

ANNEX IY. SHEAR AND TRANSVERSE BENDING

Fig. M-

l.

Conversion of stress resultants into layer stresses

the shear stress. When the design fbr shear is based on overall member behaviour, the slrear flow could sensibly be taken as uE6 = Vs7fz, where z is the lever arm according to clause 6.2 and tr/s is the she in one web. This makes the shear stress compâtible with thât assumed in the main shear section.

.

The transverse bending moment per unit length should be considered as having constant value along Ly: mEd MEdf Lx

:

This can be a conservative simplification, elTectively making the transverse môment unifôrm over the whole height of the web, based on the greatest moment-

.

Longitudinal axial force (e.g. from prestressing) cân be given a constant valu€ in

Pe,a :

Prd

A/:

lAf

Again, no guidance is giyen ôn limits for the lcngth Ay. A reasonahle interpretation is to consider only the uniform axial component of the prestress, i.e. the stress ât the centroid of the section providing the centroid lies in the web. This is consistent with the approach used in 2-2/clause 6.2.3 for the determinâtiôn of ocp.

.

The transverse shcar within the web, due to the vâriâtion of the corresponding bending moment, cân be neglected in Ay. The transverse shear corresponding to variations in the transverse moment are, in any case, usually fairly small.

One final simplification, which is not explicitly mentioned in EC2-2, but is reâsonâble âs it is compatible with the member shear design rules, is to ignore the effects of the main beam momcnt on longitudinal fofces in the web. With the above assumptions, it is possible to use a sandwich model âs shown in Fig. M-l and the membrane rules of 2-2/clause 6.109 and Annex F to design the reinforcement- Thc designer is free to decide on the thicknesses of the layers. Expressions (MM.101) to

(MM.106) âre given tn 2-2lMM(102) to determine stresses in the layers and they are reproduced as equations (DM1-l) to (DMl-6) below- Thc notation and sign convention Ibr opa, have been amcnded in both the equations below and in Fig. M-l to give compatibility with clause 6.109 and Annex LL in EC2-2. lf Annex M M is used as presented in EC2-2, it is important to note that its sign convention is not consistent with other pârts of

2-2/MM(t 02)

8C7.2: TEdxyl

:

?/Ëd

(2b*

b*-tt - tt -

tùt j

(DM-l)

347

DESIGNERS' GUIDE

TO EN I992-2

b*-tt

,Ldryr

lI

\:/w

l)]'1

mF.d

(,5*

(tt + tù12)t)

(6"

(,t

m.F.d

(2b*

+

t2)12)tz

b*-tt - t1 -t2)t1 ô'" t1

)À ,t, \zuw

Fig. M-2. Internal equilibrium with reinforcement eccentric to layer centre

348

(DM-3) (DM-4) (DM-5)

(DM-6) , \, 't/.2 One remaining problem is thc determination of det (defined in 2-2lclause 6.109) which variÊs through the thickness of the element in the un-crackecl elastic state. The choice of location to determine the un-cracked flexural strcss is therefore critical. It is recommended here, somewhât arbitrarily, that d.] be câlculâted from the stl'€sses determined a/ier the element has been split into the lal'ers of the sandwich. This appcars to be \1'hât is required in Annex MM from the ordcr of thc paragraphs, but it dôes not then strictly relate to the initial principal compression angle in the un-cracked elastic section The simplcst applicalion of Lhe rules can bc done rvhere the layer thicknesses are based on twice the cover sô that the reinforcement is ât the centre of the layer. This will usually then lead to an'un-used'gap between the layers, which reduces the naximum shear resistance, In bridges, the wcbs will often be highly strcssed in shear, so use ôf the full width of web thickness will often be necessary. Consequently. it will olTen be necessary to make the layer thicknesses total either the full width of the web or a significant fraction of it. This, however, leads to the addcd problem that the derived layer forces act eccentdcally to the actual reinlbrcement forces. In this câse, the procedure is to perform Lhc concrete veriflcâtion ignoring this cccentriqity and lhen to make a correction in the calculation of the reinforc€ment forces in accordance with 2-2lMM(103). Formulae for this correction are given in Annex LL as equations (LL,l49) and (LL,l50). They are derivcd here as equations (DM7) and (DM-8) wiLh notation again changed to suit Fig. M-l. The stless resultants in each layer are shown in Fig. M-2. riEd represenLs the forse from the sandwich model acting at the centrc ofthe layer and zi6 rcpresents the force to be used in the reinforcement design so as to maintain equilibrium. dEd\2

2-2/tûM(r 03)

:/Ed,

I

(DM-2)

ANNEX IY. SHEAR AND TRANSVERSE BENDING

Taking momcnts about the reinforcement in layer 2 gives:

(b"

/ t' a', .,\ t 11.'(l /r., t1) .,\ - b; ô'.t,ia. . ,eor(À" -i J ./

\-

Thus:

'; o;)*^.(?-u;) ^"(.u- (â. b,, b5)

(DM-7)

From force equilibrium:

ni;2:

nyar

*.,h0:

-

(DM-8)

tÊar

The above equations also allow diflerent covers to the two faces which (LL 149) and (LL.150) do not. Using the reinforcement design formulae and axes convention in section 6.9 of this guide, the reinforcement rcquirements ni" for the 1' direction in terms of stress resultant are from equations (DM-7) and (DM-8):

", (r" with n,"r n]r2

:

:

', ,,) , ,'(! ,!) (1,* b\

lrs,1,,1 J tan d1

nrul

*

r?.r,2

nJyr

+

(DM-e)

b'r)

xEdyr and r?syr

:

flEdx,'2l

tan 0: +

n Edy2

(DM-10)

Similar equations can be produced lbr the x direction but noting that ts,l"rllcotâ1 * tp4*1 ând n.,2 : ls,lru2icotÉzi np.a'2.

n,^1

-

The use of these equations and the sandwich model is illustraled in Worked example M-1 It illustrates the process of making the correction where rcinforcement is not centred on the layers. It shôws little difl'erence in lhe rcinforcement produced from models with )ayers gleâter than trvice the cover and equal to twice the cover- This will generally be the case wherc Layers are equal, but it is especially important to make thc correction wherc lhe layers are diffèrent sizes lo avoid violâtion of equilibrium between overall web s ess resultants and intctnal actions in concrete and reinforccment. The application of the membrane rules to cases of shear and transverse bending will often leâd to tensile l'orces in the longitudinal direction. For shear acting âlone, the longitudinâl force produced is tr/p6 cot d ând is the same force as predicted in the shear model of section 6.2i there it is shared between tension and compression chords, increasing the tension by 0.5trlp,1cold and reducing the compression by 0.5lll.1cotd In âpplying the membrane rules to webs in shear and transverse bending, it is reasonable to distributc the reinforcement (or forces) betwccn chords in the same way, rather than providing continuous longitudinal reinforsement up the webs. This is the stbject of 2-2lMM(104).

2-UMM(t04)

M2. Alternative approach based on longitudinal shear rules 2-2lMM(105) allows an alternative simpler method for combining the effecls from transverse bending and shear in the absence of axial force, based on the rules for longitudinal shear in 2-2/clause 6.2.4. The deptb of the compression zone required lbr transverse bending, /r,., can first be determined ând then this width discounted for the purposes of calculating the mâximum permissible shear lorce, ,/Rd.,nâx, as shown in Fig. M-3. For concrete verification, this is conservative âs the compressive stress telds for bcnding and shear do not ast at the same angle and therefore add vectorially rather than algebraically. Figure M-3 implies that the centroid of the shear resistance in this simple model is displaced from the centre of the web and therefore the reinforcement for shear would be slightly eccentric to maintain this same centroid of loading. There is no requirement tô consider this effect in Lhe longitudinâl shear rules in Z-2/clause 6.2.4 In the membrane

2-2/MM(t05)

349

DESIGNERS' GUIDE

TO EN I992-2

Compression zone

Bending compr€ssion

\/\./ \

nEd

'/ Fig. M-3. Simplified combination of bending and shear in webs rules, this efect is minimized by having a different compression ângle on each face of the web, with a flatter truss on thc side in fleKurâl tension. The resulting shear steel requirement can then be added to that lbr bending on the tension face. The longitudinal shear rules in 2-27 clause 6.2,4 do not rcquire full combination ofbending and sheâr fôrces in the reinforcement design. Something less than full addition will also be achieved by application of the method in Annex MM, but it will not give as much reduction as âllowed in 2-2,/cJause 6.2.4. Worked example M-l illustrates this. For web design, 2-2,rAnnex MM(105) therefore requires i.ull combination of the reinforcemcnt for shear and bending. The drafters of EC2-2 have specifically intended the rules for flânges with longitudinal shear and transverse bending to nol be used for the dcsign of webs with axial force_ This is because the longitudinal shear rules ignore the presence ôf âny axial compression that is present. The membrane rules of2-2/clause 6-109 give a reduction in shear crushing resistance as soon as any axial load is applied. By contrast, the web shcar rules of2-l-l/clause 6.2.1(3) allow an enhoncemezt of concrete crushing strength in shear for average web compressive stress up to 60% of the design cylinder strength by way of the recommended value of n"*. It is only beyond this vâlue of compression that the shear resistance is actually reduced. In normal prestressed beams the axial force will not be this hieh. in which case it would generally appear reasonable to ignore rhe txial lôad and Lo uù the modilied longitudinâl shear rules above for checking web crushing in combined sheâr ând transverse bending. This would, however, deparL from the recommendations of Annex MM, It is worth noting that various clauses in ECZ can be applied to shear in an element and none are fully consistent:

(l) (2) (3) (4)

The web design rules for shear in 2-l-l/clâuse 6-2 have a reduced crushing strength under very high axial compression by wa), of the factor o",u. The resistance is, however, allowed lo be enhanced in the presence of lorv and moderate axial stress. A similar reduction is made to the allowable torsional shear stress in flanges under very high flange compression by way of a"*. Flanges in longitudinal shcar, however, ale not reduced in strength by high compression as there is no ,1cw term in the relevant formula in 2'zlclause 6.2.4. The mernbrane rules of 2-2,rclause 6-109 implicitly reduce maximum shear strength as

soôn as any axial load is prcsent. Stricter limits are also imposed on the direction

of

the compression struts. These differences are, however, inevitable as each rule for a particular situatiôn (such as a web in shear, a plate Êlement in shear, a flange in longitudinal shear) have only been veriûed in that particular specific application. Generally, the member resistânce rules make allowance for plastic redistdbution across the cross-section on the basis of test results. The more generalized rnembrane rules do not.

350

ANNEX M. SHEAR AND TRANSVERSE BENDING

i4rÈii.' : t

::

;:tl

35

|

DESIGNERS' GUIDE

TO EN I992-2

ÏÏ; J"':::Ï: i;' ;*r":**': *t., Ï;:"' "*"

ruling ,h. ,.*ïor""nl."' ,. ::ï'i ,, ur'o.r,un ,,,=nr* o, *,

ffi

',

"

,r.",

"'="

:"r

*F*fiï**

t'ï

:,,ï,:;;.*'

:sign srrength of 435 MPa gives a steel area

12

4q-x x loo 100

:

2.85

mo:/mm

'*'';,.fr-1 ,--lK-./' / /1 \ \ l/l\ .r

/

\ | \i //

î-.; # 'J*ffi ',

J5,Z

--.1.

ANNEX M. SHEAR AND TRANSVERSE BENDING

:_i

:../

,)

353

DESIGNERS' GUIDE

TO EN I992-2

.:--.

t - lJ:o 4J5

= 2.94nrrn2lmm

The reinforcemeni is essenrraUy tbe same as from Lhe first sandwich arising because of the diflerenr angle used for the comprt $tT1e1ccs ireld). but the concrete srrcss iield is now acceptable. A check in accor

354

ANNEX M. SHEAR AND TRANSVERSE BENDING

,,.-.',1

| ùêèdÈô. .:

:.:i. :6

-

If tfi

1;;;.îiiit

i;i;i;;: i.,; . i:1; ;,l.l. .:,l.t..

:.t

;:::::,,: ....::. i;;,;t;,;:l:... ::.::: ,.:

355

ANNEX N

Damage equivalent stresses for fatigue verification (informative)

N

l.

General

Annex NN of EN 1992-2 is used in conjunction with 2-2,iclause 6.8.5 and gives a simplified procedure to calculâte the damage equivalent stresses for fâtigue veriflcâtion of reinforcement and prestressing steel in concrete road and railway bridge decks, It is based on fatigue loâd models given in EN l99l-2. Although the annex is infùrmative, there is no

other simple alternâtive for the fatigue verification of reinforcement ând prestressing steel. as discussed in section 6.8.4 of this guide. Workcd example 6.8-1 illustrates the damage equivalent stress qalculation for a road bridge. Annex NN also provides a damage equivalent stress method for the verification of concrete in railway bridges, but not for road bridees.

N2. Road bridges

2-2/clsuse

NN.z.r(r0r)

The damage equivalenl stress method for road bridges is based on fatigue Load Model 3, de6ned in EN l99l-2 clause 4.6.4. This model is illustrated in Fig. N-1. The weight of each axle is equal to 120kN. Where required by ËN 1991-2 and its National Annex, two vehicles in the same lane should be considered. For calculation of the damage equivalent stress rangcs for steel verification, 2-2lclause NN.2.1(IU) requires the axle loads of the làtigue model to be multiplied by the following factors:

. .

l.?5 for verification at intermediate support locations; 1.40 for veriflcation at all other locations,

All wheel conlaci surfaces 0,4 m x 0.4 m

Fig. N-1. Fatigue Load Model

3

ANNEX N. DAMAGE EQUIVALENT

STRESSES

FOR FATIGUE VERIFICATION

The modified vehicle is lhen moved âoross lhe bridge along eâch notional lane and the for each cycle of stress fluctuation2-2lclause NN,2.1(102) gives the following expression for calculating the dârnage equivalent stress range for steel verification: sLrcss rangc detcrmined

Ao","uu

: Ao,',p.À,

2-2/dause NN.2.t (t 02)

2-2(NN l01)

where:

Ao,

).

e"

is lhc strcss range causcd by faLiguc Load Model 3, modited as above and applied in accordance with clause 4.6.4 ofEN l99l-2, assuming the load combination given in 2-l-llclause 6,8.3 is a factor to calculate the damagc cquivalent stress rânge fiorn the strcss range caused by the modified fatigue load model

The factor, À,, includes the influences of span, annual tlaflic volume. service life, multiple lares, traffic type and surlace roughncss, and is calculated from: À.

:

2-2l(NN. r02)

t'iu1À.',1À',2À,,3À.,a

Each of these factors is discussed in turn below. {rn, is a damage equivalent impâct fâctor influenced by road surlacc roughness. The factor is defined in Annex B of EN l99l-2, together with recommendations for selecting the

appropriate value, as follows:

. dn,: 1.2 for surfaces of good roughness (recommended for new and mâintâined roadway layers such as asphalt); . drr, - 1.4 for surl'aces ol medium roughness (recommended for old and unmaintâined roadway layers).

In addition to the above factors, where the section under consideration is within a distance 6.0n.r lrom an expansion ioint, a further impacl factor should be applied to the whole loading, as defined in Annex B of EN 1991-2. This clause rcfers to EN 1991-2 Fig.4.7 (reproduced as Fig. N-2). The À. 1 factor is obtained from 2-2lFig. NN.l (for the intermediate support area) or 2-2,/ Fig. NN.2 (for the span and local elements) âs appropriate. For n.rain longitudinal reinlbrcement in continuous beams, it would be reasonable to use 2-21Fig. NN.1 for a length of 15o/o of the span each side ol an intermediate support and to use 2-2/Fig. NN.2 elsewhere. This is the âpproach in EN 1993-2. The design of shear reinforcement is bascd on 2-2lFig. NN,2. À.,1 accounts for the critical length ofthe influence line or surface and the shape ofthe_^S N curve, so its value depends on the type ofelement under consideration. In ENV 1992-2," the sarre graphs were provided but the horizontal axis related to span length. The latter is often appropriate, but not always. For example, for continuous beams in hogging bending at intermediâte supports, there âre two positive lobes to the influence line from each adjacent span, each causing a cyole of stress variation. The length of thc span is therefbre approPriate. The same is true for shear. For reaction, however the greâtest positive lobe length of the

of

È

1.3

.E

=E

t .9

'l 1.0

Distance from expansion joinl (m)

Fig. N-2. Additional amplilication factor for proximity to expansion ioints

357

TO EN I992-2

DESIGNERS' GUIDE

influence line, defining a cycle of stress variation, covcrs two spans, so the total length of the two spâns is morc appropriate in this case. EN 1993-2 uses the above considerâtions to delermine the base length to usc in câlculation and its clause 9.5,2(2) can be used for guidânce in determining the appropfiâte length here. The À,,r correction factor takes account of the annual traffic volume ând traffic type (i.e. weight). It is oalculated liom the following expression:

\".2:Qx

2-2(NN.103)

71{"r,/2,0

u.here:

is a factor tâken from 2-21Table NN.l. It accounts for the damage done by the actual mix of traffic rveights compared to that done by lhe fatigue load model. p

0

need not, howevcr, be detennined fiom traffic spectra (âs the equivâlent pârâmeter in EN 1993-2 needs to be) as it may be deternined from 2-2i Table NN.1 for a given

'ûâmc typc', as delined in Note 3 of given there âre not easy to interpret:

. . .

EN 1991-2 clause 4.6.5(1). The definitions

'long distânce' means hundrcds of kilometres;

'mcdiun distance'meâns 50 to l00km; 'locâl tramc'means distances less than 50km. 'Long distance' will typically apply to motorways and trunk roads, The use of the other cases will need to be agreed with the Client k2 is the slope of the appropriate S N curve fbr the element under cônsideration Nobs is fhe number of lories per year (in millions) in accordance with Table 4.5 of EN 1991-2, reproduced here as Table N-l (these yâlues mây be modified by the National Annex) The

Às.3

Às.r

:

làctor takes account of the required service life:

k.-

2-2(NN.104)

?NYeârs/ 100

where ly'yean is the design life of the bridge (in years). The À".a correction fâctor takes account of the influence of tra{ic fron adjacent lanes. Since most bridge decks are âble to distribute load transversely, clements will also pick up fatigue loading from vehicles passing in lanes rernote from those directly above the element- Às,4 is calculated fiom the following:

.

"''t:

",8N"* \/ N"l"J

2-2(NN.105)

where:

No6r; is the uumber of lorries expccted on lane / per year 1{.6.,1 is the numbcr of lorries expected on the slow lane per year This expression is very approximale and does not contain any direcl measure of the relalive influence of trafic in each lane, unlike the equivalent parametet in EN 1993-2.

Table N-

l.

Indicative number of heavy vehicles expecred per year per slow lane

Trafic categories (

l)

with two or more lanes per direction with flow rates of lorries Roads and motorways with medium flow rates ôf lorries l"lain roâds wirh low flow rârcs of lorries Local roads with low flow rates of lorries Roads and motorlvays

No6, per year per slow lane

2.0

x

106

high

(2) (3) (4)

358

0.5 x 106 0.125 x 106 0.05 x 106

ANNEX N. DAMAGE EQUIVALENT

STRESSES

FOR FATIGUE VERIFICATION

N3. Railway bridges The damage equivalent stress method for fatigue verification of railway bridges is split inro two secLions; one for reinforcing and prestressing steel and one for concrele clements. The key âspects of both are discussed below.

N3.1. Reinforcing and prestressing steel The darnagc equivalent stress method for reinforcement and prestressing steel in railwây bridges follows a similar format to that for road bridgcs, buL is based ôn using the normal rail traflic models (Load Model 7l and Load Model SW/O) delined in EN 1991-2, but excluding Lhc o* factor defined therein. The characteristic (static) values for Load Model 71 consist of â lôâd train of four 250 kN axlcs and uniformly disrributed loads of 80 kN/m. The geonretry of this load model is illustrâted in Fig. N-3. In âddirion. Ibr continuous span bridges (including single spans witl integral abutments). the eflects lrom Load Model SW/0 should also be considered if worse than Load Modcl 71. The characteristic (stâtic) values for Load Model SW/O consist of two unitbrmly distributed loâds of 133 kN/n.r with thc geometry illustrated in Fig. N-4- Refercnce should be made to EN 1991-2 and its relevant Nâtional Annex fbr full details of the load models including transverse eccentricity of vertical loads. For strucLures with rnultiple fiacks, 2-2lclause NN,J,I(101) requires thc relevant load model to be applied to a maximum of two tracks to dcLcrmine the steel stress range. The following expression for calculating the damage equivalent slress range for steel verification 15

Slven:

Aa",*u:À.xçLrAo.,71

2-2i

NN.l0ô)

whele:

4a,.71 is the sLrcss range caused by Load Model 7l or SW/O as above- It should be câlculated in the load combination given in 2-1-l/clause 6.8.3 is a dynan.ric factor which enhances the static loâd effects obtained from the above E load modeLs. This factor is defined in EN 1991-2 clause 6.4.5 (as iD) and should be taken as either Q2 for carefully mâintâined track or Or for track with standard maintenancc- Using the forn.rulae deflned in EN l99l-2,.T'2 lies within the rânge of 1.0 to 1.67 and iD3 lies within the range of 1 .0 to 2.0 À. is a correction factor to calculate the damage equivalent stress range from the stress range caused by above load models Thc correclion faotor, Àr, includes the influences of span, annual traffic volumc, service life and multiple tracks and is calculated from:

À,:

2-2(NN.l07)

À.'.1À'.2À,.1À.,a

4x

Qvk = 250 kN €ach

Unlimited 0.8m 3x1.6m 0-8m lJnlimited Fig. N-3. Load Model 7l and charâcrerisric (saric) values for verrical loads q* -

q"k = 133 kN/m

133 kN/m

15.0

m

5.3

m

15.0 m

Fig. N-4. Load Model 5W/0 and charactêristic (static) values for verticâl loads

2-2/douse

NN.3./(r0r)

DESIGNERS' GUIDE

TO EN I992-2

The correction lactùr À..r takes into account the influence ofthe length ofthe influence line or surlâce and trafûc mix. The values of À,.1 for standard or heavy traffic mixes (defined in EN l99l-2) are given in 2-2lTable NN.2 and 2-2(NN.108). The values given for mixed traflic correspond tô the combination of trains givcn in Annex ts of EN 1991-2. No values for a light traffic mix are provided. In such circumstânces, either À, 1 values can be based on the stândard tralllc mix or calculations on the basis of the actual trafic speclra can be undertaken.

The

À".2

correction fàctor is used to include the influence of the ânnual tralTic volume: Vol

2-2(NN.109)

25;7û where:

Vol k2

is the volume oftraffic (tonnes per year per track) is the slope ofthe appropriate S N curve for the element under consideration

The

2-Uclouse NN.3. / (r 06)

À,.1 coffection factor is uscd to include the influence of the serviqe life and is identical to that for roâd bridges above. The Àr.4 correction factor is used to account for the influence of loading lrom more than one track, The expression given rn 2-2lclause NN,3.1(106) includes the effects of stress rangcs from load on tlacks other thân frôm thât which produces the greatest stress range. It is therefore more rational than the corresponding expression for road bridges, which considers only the trâffic flow in adjacent lanes.

N3.2. Concrete subie€ted to compression

2-2/clause

NN.3.2(t0 t)

The damage equivâlent stress method for concrete under compression in railway bridges is similar to thaL lor reinforcement and preslressi[g steel âbove; the damage equivalent shesses are based on the stress ranges obtained from analysis using Load Model 71. 2-2lclause NN,3.2(101) stâtes that adequate fatigue resistance may be assumed for concrete elements in compression if the follotving expression is satisfied:

I_F."d nr\,e'lu ! /

l+Y:rO

2-2(NN.1l2)

1/il - Requ -

where:

F. -

4crl,max.equ

tcrl.rrin.eou

-

"cd.mrn.equ

- rsd J!cd.l-zil

lcd,max,equ

ocd.nax.eou

rs'J t

./cd,Lat

where:

%d /la6t

is Lhe padial factor for modelling uncertâinty, defined in clause 6.3,2 ofEN 1990. Values are likely Lo be in the range of 1.0 to 1.15 and are set in the National Annex to EN 1990 (1a is not defined in EC2-2 other than in 2-2lclause 5.7) is lhe design fatigue compressive strength ofconcrete from 2-2/clause 6.8,7

and o",1,-;,."u,, are the upper and lower stresses ôf the dâmâge equivalent stress spectrum with 10Ô number of cycles. These upper and lower stresses should be calculated ocd,max.equ

from the following eKptessions: dcd.max.cqu acd,min,eqù

Jbu

- a"0",- f À.(4",."'l :4",0"''' * À"(o"0...

- o".p.r"n) o.,.i".tr)

2-2l(NN.l13)

ANNEX N. DAMAGE EQUIVALENÎ

STRESSES

FOR FATIGUE VERIFICATION

whÊre:

dc,perm is the compressive corlcrete stress caused by the characteristic combinâtion of permânent actions (excluding the effects of the fatigue load models) o".."*.71 is the maximum compressive concrete stress caused by the chatacteristic

combination of actions including Load Model 71, and the âppropriate dynamic factor, /, from EN 1991-2 (as discussed above) o".n,;n.71 is the minimum compressive concrete sûess caused by the characteristic combinaLion of actions including Load Model 71, and the âppropriate dynanic factor, / À" is a correction factor to câlculate the upper ând lower stresses of the damage equivalent stress spectrum from the stresses caused by the fatigue load model

The )c correction lactor covers the same cllccts that À" covers for steel veriûcations, but additionally includes the influence of permanent compressive stress by way of the factor .\",n. The factors Àc,1, lc,r,r and Àc,4 are equivalent to À",1 to À"a for steel verifications.

361

ANNEX O

Typical bridge discontinuity regions (informative) Annex OO of EN 1992-2 provides guidance on the design of diaphragms for box girders with:

. . .

twin hearings; single bearings; or u monolithic connection to a pier.

Guidance is also given on the design of diaphragms

in

decks

with Double Tee cross-

sections.

The inclusion ofthis material was made at a vcry late stage in the drafting ofEN 1992-2. It differs in style and content to the rest of the document in thât it prôvides nô Principles or Application Rules, but rather gives guidance on suitable strut-and-tie idealizations that can be used in design- As such, the matcrial requires no furthcr commentary here other than to note that other idealizations are possible and mây sometimes be dictat€d by constraints on the pôsitioning of rcinforcement and,/or prestressing steel. The layouts provided san, however, be taken to be examples of good practice.

ANNEX

P

Safety format for non-linear analysis (informative) The provisions of Annex PP of EN 1992-2 are discussed in section 5-7 of this guide.

ANNEX Q

Control of shear cracks within webs (informative) 2-2lclatse7.3.1(ll0) statcs that 'in some cases it may be necessary to check and control shear cracking in webs'and reference is made to 2-27Annex QQ. Some suggestions lbr when and why such a check might b€ appropriâte are made in section 7.3.1 of this guide. 2-2lLnnex QQ gives a procedure for checking shear cracking, although it is prefaced by the statem€nt that 'at present, the prediction of shear cracking in webs is accompanied by large model uncertainty'. It also states that the check is particularly necessary for prestressed members. This is probably more because of the perceived need to keep a tighter control on crack widths where prestressing tendons are involved, rather than due to any inherent vulnerability of prestrcsscd sections lo web shear cracking (although thinner webs are often used in prestressed sections, which will be more likely to crack). Worked example Q-l shows the benefits ofprestressing in reducing web crack widths due to shear. In the procedure, the larger principal tensile stress in the web (o1) is compâred to the cracking strength of the côncrete, allowing for â biâxial stress state, given as:

(, o.s+

Jctk,0.05

2-2(QQ.l01)

where o3 is the larger compressive principâl stress (compression taken as positive) but not greater than 0.6/11. Where the greâtest principal tensile stress o1 < /1,6 (in this check, tension is positive), the web is deemed to be un-cracked and no check of crack width is required. Minimum longitudinal reinl'orcement should, however, be provided in accordance with 2-2/clause 7.3.2. Where o1 > .Âb, EC2 stâtes thât cracking in the web should be controlled either by the method of 2-2/clause 7.3.3 or the direct calculation method of 2-27lclause 7.3.4. In both cases, it is necessary tô take account of the deviation ângle between principal tensile stress and reinforcement directions. Since the reinforcement directions do not, in general, align with the direction of principal tensilô stress and the reinforcement is likely to be different in the two orthogonâl directions, there is difficulty in deciding whât unique bar diameter and spacing to usc in the simple method of 2-2/clausc 7.3.3. If the method of 2-2/clause 7.3.4 is used, the eflect of deviation angle between principâl tensile stress ând reinforcement directions on crack spacing. ùn,ix, can be calculated using 2- 1- I /Expression (7-15):

/ cosÉ

sind

\Jr,max.y

Jr.rnar,.

I

2-t-t l(7 .t5)

The eflect of the deviation ângle between principal tensile stress and reinforcemenl directions on calculalion of reinforcement stress is not given in EN 1992- One approach is to calculate the sffess in the reinforcement, ns, by dividing the stress in the concrete in the

ANNEX Q. CONTROL OF SHEAR CRACKS WITHIN WEBS

direction of the principal tensile stress by an effective reinforcement ratio, p], whcre pL - XII , p; cosa a; and p; is the reinforcement ratio in layer I at an angle or; to the direction of principâl tensile stress. This reinforcemenL ratio is bascd on equivalent stiflness in the principal stress dircction, which was the approach used in BS 5400 Part 4.'o" is then used in 2- I - li Expression (7.9) to calculate the strain for use in the crack calculation formula of 2- l- l,/Expression (7.8). In applying 2-l-llExpression (7.9), rr,."n could be taken as pl but, in view of the uncefiainty of the eflècts ot tension stiflening whère the reinforccment direction

is skew to the cracks, it is advisable to ignore the tension stiffening term and take

€.. €,^:

o,lE".

The direction of the principal tensile stress in the un-cracked condition wiJl vary over the height of the web due to the variation oi- both shear stress and flcxural stress. In determining whethÊr or not the section is cracked, all such points should be checked. However, a similar lreatment aftcr cracking would also require crack checks at all locations throughout the depth of the beam. To avoid this situation, and since the subject of the calculation is 'crackirg due to shear". one sirnplilication for doubly flanged beams might be to ignorc flcxural stresscs whcn checking web crack widths and to take the shear slrcss as uniform ovet a depth z, where: is an appropriate lever arm, such as the ULS flexural lever arrrr. This latter assumption is not conseNative for shear stress, but is a reasonable approximation given the uncertainties of thc mclhod. The axial force component of any prestressing should also be considered in the crack width câlcultrtiôn. Thc ncglecl of the flexural stresses above can partly be justified for doubly flanged beams by the fâct that, on cracking, flexural tensile stresses will shed to the flanges where the main flexural reinfbrcement is provided. Despite the shedding ofstress, longitudinal strain will still be produced in thc wcb in order for the flange reinforcement to take up the load. While strictly this strain will âdd to thât calculated from the shear stlesses" it lrray be reasonable lo ignore this as the flexural crack widths must still be checked separately. Care with this approach should, however. be exercised wiLh T bcams when the stem is in tension. Since. in this case, there is no discrete tension flânge containing the main flexural rcinforcement, the flexural forces in thc wcb cannot be shed on cracking. In such cases, it is more conservâtive to consider the full stress field in checking web cracking.

365

DESIGNERS' GUIDE

TO EN I992-2

-tseo example Fig. Q-1. Web relnforcemem for Worked exampE G v-

|I

From 2-l -l /Et

trc

.tr.mar.y

=

-k1k.kaçf

pr.,x..,

' l.{

r,.o'*., Jr.ûÀi,z

-r -- À3c*/