Design of Prestressed Concrete to Eurocode 2 - Second Edition

Raymond Ian Gilbert Neil Colin Mickleborough Gianluca Ranzi Design of Prestressed Concrete to Eurocode 2 Second Edition Design of Prestressed Concrete to Eurocode 2 Second Edition Taylor Francis Design of Prestressed Concrete to Eurocode 2 Second Edition Raymond Ian Gilbert Neil Colin Mickleborough Gianluca Ranzi CRC Press Taylor Francis Group 6000 Broken Sound Parkway NW Suite 300 Boca Raton FL 334872742 2017 by Raymond Ian Gilbert Neil Colin Mickleborough and Gianluca Ranzi CRC Press is an imprint of Taylor Francis Group an Informa business No claim to original US Government works Printed on acidfree paper Version Date 20161209 International Standard Book Number13 9781466573109 Hardback International Standard Book Number13 9781315389523 eBook This book contains information obtained from authentic and highly regarded sources Reasonable efforts have been made to publish reliable data and information but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint Except as permitted under US Copyright Law no part of this book may be reprinted reproduced transmitted or utilized in any form by any electronic mechanical or other means now known or hereafter invented including photocopying microfilming and recording or in any information storage or retrieval system without written permission from the publishers For permission to photocopy or use material electronically from this work please access www copyrightcom httpwwwcopyrightcom or contact the Copyright Clearance Center Inc CCC 222 Rosewood Drive Danvers MA 01923 9787508400 CCC is a notforprofit orga nization that provides licenses and registration for a variety of users For organizations that have been granted a photocopy license by the CCC a separate system of payment has been arranged Trademark Notice Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe Library of Congress CataloginginPublication Data Visit the Taylor Francis Web site at httpwwwtaylorandfranciscom and the CRC Press Web site at httpwwwcrcpresscom v Contents Preface xv Authors xix Acknowledgements xxi Notation and sign convention xxiii 1 Basic concepts 1 11 Introduction 1 12 Methods of prestressing 4 121 Pretensioned concrete 4 122 Posttensioned concrete 5 123 Other methods of prestressing 6 13 Transverse forces induced by draped tendons 7 14 Calculation of elastic stresses 10 141 Combined load approach 10 142 Internal couple concept 12 143 Load balancing approach 13 144 Introductory example 13 1441 Combined load approach 14 1442 Internal couple concept 15 1443 Load balancing approach 15 15 Introduction to structural behaviour Initial to ultimate loads 16 2 Design procedures and applied actions 21 21 Limit states design philosophy 21 22 Structural modelling and analysis 23 221 Structural modelling 23 222 Structural analysis 24 vi Contents 23 Actions and combinations of actions 26 231 General 26 232 Load combinations for the strength limit states 29 233 Load combinations for the stability or equilibrium limit states 31 234 Load combinations for the serviceability limit states 32 24 Design for the strength limit states 33 241 General 33 242 Partial factors for materials 33 25 Design for the serviceability limit states 34 251 General 34 252 Deflection limits 35 253 Vibration control 37 254 Crack width limits 37 255 Partial factors for materials 38 26 Design for durability 38 27 Design for fire resistance 40 28 Design for robustness 43 References 44 3 Prestressing systems 47 31 Introduction 47 32 Types of prestressing steel 47 33 Pretensioning 49 34 Posttensioning 51 35 Bonded and unbonded posttensioned construction 58 36 Circular prestressing 59 37 External prestressing 60 4 Material properties 63 41 Introduction 63 42 Concrete 63 421 Composition of concrete 64 422 Strength of concrete 65 423 Strength specifications in Eurocode 2 68 4231 Compressive strength 68 4232 Tensile strength 69 4233 Design compressive and tensile strengths 70 Contents vii 4234 Compressive stressstrain curves for concrete for nonlinear structural analysis 72 424 Deformation of concrete 73 4241 Discussion 73 4242 Instantaneous strain 74 4243 Creep strain 76 4244 Shrinkage strain 81 425 Deformational characteristics specified in Eurocode 2 82 4251 Introduction 82 4252 Modulus of elasticity 83 4253 Creep coefficient 84 4254 Shrinkage strain 86 4255 Thermal expansion 87 43 Steel reinforcement 87 431 General 87 432 Specification in Eurocode 2 88 4321 Strength and ductility 88 4322 Elastic modulus 89 4323 Stressstrain curves Design assumptions 90 4324 Coefficient of thermal expansion and density 91 44 Steel used for prestressing 91 441 General 91 442 Specification in Eurocode 2 94 4421 Strength and ductility 94 4422 Elastic modulus 94 4423 Stressstrain curve 96 4424 Steel relaxation 96 References 98 5 Design for serviceability 101 51 Introduction 101 52 Concrete stresses at transfer and under full service loads 102 53 Maximum jacking force 105 54 Determination of prestress and eccentricity in flexural members 106 541 Satisfaction of stress limits 106 542 Load balancing 114 viii Contents 55 Cable profiles 116 56 Shortterm analysis of uncracked crosssections 118 561 General 118 562 Shortterm crosssectional analysis 120 57 Timedependent analysis of uncracked crosssections 136 571 Introduction 136 572 The ageadjusted effective modulus method 136 573 Longterm analysis of an uncracked crosssection subjected to combined axial force and bending using AEMM 138 574 Discussion 156 58 Shortterm analysis of cracked crosssections 158 581 General 158 582 Assumptions 160 583 Analysis 160 59 Timedependent analysis of cracked crosssections 170 591 Simplifying assumption 170 592 Longterm analysis of a cracked crosssection subjected to combined axial force and bending using the AEMM 170 510 Losses of prestress 175 5101 Definitions 175 5102 Immediate losses 176 51021 Elastic deformation losses 176 51022 Friction in the jack and anchorage 177 51023 Friction along the tendon 177 51024 Anchorage losses 179 51025 Other causes of immediate losses 180 5103 Timedependent losses of prestress 181 51031 Discussion 181 51032 Simplified method specified in EN 1992112004 182 51033 Alternative simplified method 183 511 Deflection calculations 187 5111 General 187 5112 Shortterm momentcurvature relationship and tension stiffening 190 5113 Shortterm deflection 195 5114 Longterm deflection 200 51141 Creepinduced curvature 201 51142 Shrinkageinduced curvature 202 Contents ix 512 Crack control 208 5121 Minimum reinforcement 208 5122 Control of cracking without direct calculation 211 5123 Calculation of crack widths 213 5124 Crack control for restrained shrinkage and temperature effects 215 5125 Crack control at openings and discontinuities 216 References 216 6 Flexural resistance 219 61 Introduction 219 62 Flexural behaviour at overloads 219 63 Design flexural resistance 222 631 Assumptions 222 632 Idealised compressive stress blocks for concrete 223 633 Prestressed steel strain components for bonded tendons 226 634 Determination of MRd for a singly reinforced section with bonded tendons 228 635 Determination of MRd for sections containing nonprestressed reinforcement and bonded tendons 232 636 Members with unbonded tendons 239 64 Design calculations 241 641 Discussion 241 642 Calculation of additional nonprestressed tensile reinforcement 242 643 Design of a doubly reinforced crosssection 245 65 Flanged sections 248 66 Ductility and robustness of prestressed concrete beams 254 661 Introductory remarks 254 662 Calculation of hinge rotations 257 663 Quantifying ductility and robustness of beams and slabs 257 References 260 7 Design resistance in shear and torsion 261 71 Introduction 261 72 Shear in beams 261 721 Inclined cracking 261 x Contents 722 Effect of prestress 262 723 Web reinforcement 264 724 Design strength of beams without shear reinforcement 267 725 Design resistance of beams with shear reinforcement 268 726 Summary of design requirements for shear 273 727 The design procedure for shear 275 728 Shear between the web and flange of a Tsection 281 73 Torsion in beams 282 731 Compatibility torsion and equilibrium torsion 282 732 Effects of torsion 284 733 Design provisions for torsion 285 74 Shear in slabs and footings 291 741 Punching shear 291 742 The basic control perimeter 292 743 Shear resistance of critical shear perimeters 294 744 Design for punching shear 296 References 307 8 Anchorage zones 309 81 Introduction 309 82 Pretensioned concrete Force transfer by bond 310 83 Posttensioned concrete anchorage zones 315 831 Introduction 315 832 Methods of analysis 319 8321 Single central anchorage 321 8322 Two symmetrically placed anchorages 322 833 Reinforcement requirements 325 834 Bearing stresses behind anchorages 326 84 Strutandtie modelling 342 841 Introduction 342 842 Concrete struts 343 8421 Types of struts 343 8422 Strength of struts 344 8423 Bursting reinforcement in bottleshaped struts 344 843 Steel ties 346 844 Nodes 346 References 348 Contents xi 9 Composite members 351 91 Types and advantages of composite construction 351 92 Behaviour of composite members 352 93 Stages of loading 354 94 Determination of prestress 357 95 Methods of analysis at service loads 359 951 Introductory remarks 359 952 Shortterm analysis 360 953 Timedependent analysis 362 96 Flexural resistance 392 97 Horizontal shear transfer 392 971 Discussion 392 972 Design provisions for horizontal shear 394 References 398 10 Design procedures for determinate beams 399 101 Introduction 399 102 Types of sections 399 103 Initial trial section 401 1031 Based on serviceability requirements 401 1032 Based on strength requirements 402 104 Design procedures Fullyprestressed beams 404 1041 Beams with varying eccentricity 405 1042 Beams with constant eccentricity 422 105 Design procedures Partiallyprestressed beams 432 Reference 440 11 Statically indeterminate members 441 111 Introduction 441 112 Tendon profiles 443 113 Continuous beams 446 1131 Effects of prestress 446 1132 Determination of secondary effects using virtual work 447 1133 Linear transformation of a tendon profile 453 1134 Analysis using equivalent loads 455 11341 Moment distribution 456 1135 Practical tendon profiles 465 1136 Members with varying crosssectional properties 468 1137 Effects of creep 470 xii Contents 114 Statically indeterminate frames 474 115 Design of continuous beams 478 1151 General 478 1152 Service load range Before cracking 479 1153 Service load range After cracking 482 1154 Overload range and design resistance in bending 483 11541 Behaviour 483 11542 Permissible moment redistribution at the ultimate limit state condition 484 11543 Secondary effects at the ultimate limit state condition 485 1155 Steps in design 486 References 499 12 Twoway slabs Behaviour and design 501 121 Introduction 501 122 Effects of prestress 504 123 Balanced load stage 507 124 Initial sizing of slabs 509 1241 Existing guidelines 509 1242 Serviceability approach for the calculation of slab thickness 510 12421 Slab system factor K 512 1243 Discussion 514 125 Other serviceability considerations 516 1251 Cracking and crack control in prestressed slabs 516 1252 Longterm deflections 517 126 Design approach General 519 127 Oneway slabs 519 128 Twoway edgesupported slabs 520 1281 Load balancing 520 1282 Methods of analysis 522 129 Flat plate slabs 533 1291 Load balancing 533 1292 Behaviour under unbalanced load 535 1293 Frame analysis 537 1294 Direct design method 539 1295 Shear resistance 540 1296 Deflection calculations 541 1297 Yield line analysis of flat plates 555 Contents xiii 1210 Flat slabs with drop panels 559 1211 Bandbeam and slab systems 560 References 561 13 Compression and tension members 563 131 Types of compression members 563 132 Classification and behaviour of compression members 564 133 Crosssection analysis Compression and bending 566 1331 Strength interaction diagram 566 1332 Strength analysis 568 1333 Biaxial bending and compression 579 134 Slenderness effects 580 1341 Background 580 1342 Slenderness criteria 584 1343 Moment magnification method 585 135 Reinforcement requirements for compression members 591 136 Transmission of axial force through a floor system 591 137 Tension members 593 1371 Advantages and applications 593 1372 Behaviour 594 References 600 14 Detailing Members and connections 601 141 Introduction 601 142 Principles of detailing 602 1421 When is steel reinforcement required 602 1422 Objectives of detailing 603 1423 Sources of tension 604 14231 Tension caused by bending and axial tension 604 14232 Tension caused by load reversals 604 14233 Tension caused by shear and torsion 605 14234 Tension near the supports of beams 605 14235 Tension within the supports of beams or slabs 606 14236 Tension within connections 607 14237 Tension at concentrated loads 607 14238 Tension caused by directional changes of internal forces 608 14239 Other common sources of tension 610 xiv Contents 143 Anchorage of deformed bars 610 1431 Introductory remarks 610 1432 Design anchorage length 613 1433 Lapped splices 617 144 Stress development and coupling of tendons 619 145 Detailing of beams 619 1451 Anchorage of longitudinal reinforcement General 619 1452 Maximum and minimum requirements for longitudinal steel 623 1453 Curtailment of longitudinal reinforcement 624 1454 Anchorage of stirrups 625 1455 Detailing of support and loading points 630 146 Detailing of columns and walls 634 1461 General requirements 634 1462 Transverse reinforcement in columns 635 1463 Longitudinal reinforcement in columns 638 1464 Requirements for walls 638 147 Detailing of beamcolumn connections 638 1471 Introduction 638 1472 Knee connections or twomember connections 639 14721 Closing moments 640 14722 Opening moments 640 1473 Exterior threemember connections 642 1474 Interior fourmember connections 645 148 Detailing of corbels 646 149 Joints in structures 647 1491 Introduction 647 1492 Construction joints 648 1493 Control joints contraction joints 649 1494 Shrinkage strips 651 1495 Expansion joints 652 1496 Structural joints 652 References 654 Index 655 xv Preface For the design of prestressed concrete structures a sound understanding of structural behaviour at all stages of loading is essential Also essential is a thorough knowledge of the design criteria specified in the relevant design standard including the rules and requirements and the background to them The aim of this book is to present a detailed description and expla nation of the behaviour of prestressed concrete members and structures both at service loads and at ultimate loads and in doing so provide a comprehensive guide to structural design Much of the text is based on first principles and relies only on the principles of mechanics and the prop erties of concrete and steel with numerous worked examples Where the design requirements are code specific this book refers to the provisions of Eurocode 2 EN 1992112004 and other relevant EN Standards and where possible the notation is the same as in the Eurocode A companion edition in accordance with the requirements of the Australian Standard for Concrete Structures AS 36002009 is also available with the same nota tion as in the Australian Standard The first edition of the book was published over 25 years ago so a com prehensive update and revision is long overdue This edition contains the most uptodate and recent advances in the design of modern prestressed concrete structures as well as the fundamental aspects of prestressed con crete behaviour and design that were well received in the first edition The text is written for senior undergraduate and postgraduate students of civil and structural engineering and also for practising structural engineers It retains the clear and concise explanations and the easytoread style of the first edition Between them the authors have almost 100 years of experience in the teaching research and design of prestressed concrete structures and this book reflects this wealth of experience The scope of the work ranges from an introduction to the fundamentals of prestressed concrete to indepth treatments of the more advanced topics in modern prestressed concrete structures The basic concepts of prestressed xvi Preface concrete are introduced in Chapter 1 and the limit states design philoso phies used in European practice are outlined in Chapter 2 The hardware required to pretension and posttension concrete structures is introduced in Chapter 3 including some construction considerations Material properties relevant to design are presented and discussed in Chapter 4 A comprehen sive treatment of the design of prestressed concrete beams for serviceability is provided in Chapter 5 The instantaneous and timedependent behaviour of crosssections under service loads are discussed in considerable detail and methods for the analysis of both uncracked and cracked crosssections are considered Techniques for determining the section size the magnitude and eccentricity of prestress the losses of prestress and the deflection of members are outlined Each aspect of design is illustrated by numerical examples Chapters 6 and 7 deal with the design of members for strength in bend ing shear and torsion and Chapter 8 covers the design of the anchorage zones in both pretensioned and posttensioned members A guide to the design of composite prestressed concrete beams is provided in Chapter 9 and includes a detailed worked example of the analysis of a composite through girder footbridge Chapter 10 discusses design procedures for stat ically determinate beams Comprehensive selfcontained design examples are provided for fullyprestressed and partially prestressed posttensioned and pretensioned concrete members The analysis and design of statically indeterminate beams and frames is covered in Chapter 11 and provides guidance on the treatment of second ary effects at all stages of loading Chapter 12 provides a detailed discus sion of the analysis and design of twoway slab systems including aspects related to both strength and serviceability Complete design examples are provided for panels of an edgesupported slab and a flat slab The behav iour of axially loaded members is dealt with in Chapter 13 Compression members members subjected to combined bending and compression and prestressed concrete tension members are discussed and design aspects are illustrated by examples Guidelines for successful detailing of the structural elements and connections in prestressed concrete structures are outlined in Chapter 14 As in the first edition the book provides a unique focus on the treatment of serviceability aspects of design Concrete structures are prestressed to improve behaviour at service loads and thereby increase the economi cal range of concrete as a construction material In conventional pre stressed structures the level of prestress and the position of the tendons are usually based on considerations of serviceability Practical methods for accounting for the nonlinear and timedependent effects of cracking creep shrinkage and relaxation are presented in a clear and easyto follow format Preface xvii The authors hope that Design of Prestressed Concrete to Eurocode 2 will be a valuable source of information and a useful guide for students and practitioners of structural design Ian Gilbert Neil Mickleborough Gianluca Ranzi Taylor Francis xix Authors Raymond Ian Gilbert is emeritus professor of civil engineering at the University of New South Wales UNSW and deputy director of the UNSW Centre for Infrastructure Engineering and Safety He has more than 40 years of experience in structural design and is a specialist in the analy sis and design of reinforced and prestressed concrete structures Professor Gilbert has taught successive generations of civil engineering students in Australia on subjects related to structural engineering ranging from statics and structural analysis to the design of reinforced and prestressed concrete structures His research activities are in the field of concrete structures with a particular interest in serviceability Professor Gilbert has published six books including Structural Analysis Principles Methods and Modelling and TimeDependent Behaviour of Concrete Structures which are also published by CRC Press and more than 350 technical papers and reports He was awarded Honorary Life Membership of the Concrete Institute of Australia in 2011 Neil Colin Mickleborough is professor of civil engineering and the direc tor of the Center for Engineering Education Innovation at Hong Kong University of Science and Technology He has been actively involved in the research development and teaching of prestressed and reinforced concrete structural analysis and tall building and bridge design in Australia Asia and the Middle East for the past 30 years He has acted as an expert design consultant on tall buildings and longspan bridge projects in both Dubai and Hong Kong In addition he is a chartered structural engineer and a Fellow of the Hong Kong Institution of Engineers Gianluca Ranzi is professor of civil engineering ARC Future Fellow and director of the Centre for Advanced Structural Engineering at the University of Sydney Gianlucas research interests range from the field of structural engineering with focus on computational mechanics and the service behaviour of composite steelconcrete and concrete structures to architectural science Taylor Francis xxi Acknowledgements The authors acknowledge the support given by their respective institutions and by the following individuals and organisations for their assistance Mr Brian Lim VSL International Limited Hong Kong Mr Brett Gibbons VSL Australia Sydney Taylor Francis xxiii Notation and sign convention All symbols are also defined in the text where they first appear Throughout the book we have assumed that tension is positive and compression is nega tive and that positive bending about a horizontal axis causes tension in the bottom fibres of a crosssection Latin uppercase letters A Crosssectional area or accidental action Ac Crosssectional area of concrete Aceff Effective area of concrete in tension surrounding the tendons with depth hcef equal to the lesser of 25hd hx3 or h2 Act Area of the concrete in the tensile zone just before cracking Ac0 Bearing area Ac1 Largest area of the concrete supporting surface that is geometrically similar to and concentric with Ac0 Ag Gross crosssectional area Ak Area of the ageadjusted transformed section at time tk Ap Crosssectional area of prestressing steel Api Crosssectional area of the prestressing steel at the ith level Apc Crosssectional area of the precast member As Crosssectional area of nonprestressed steel rein forcement or crosssectional area of a single bar being anchored Asi Crosssectional areas of nonprestressed steel reinforce ment at the ith level Asb Area of transverse reinforcement in the end zone of a pretensioned member Equation 86 Asc Crosssectional area of nonprestressed steel reinforce ment in the compressive zone xxiv Notation and sign convention Ast Crosssectional area of nonprestressed transverse steel reinforcement or crosssectional area of nonprestressed reinforcement in the tension zone Asmin The minimum area of bonded longitudinal reinforcement in the tensile zone Equation 149 or minimum area of longitudinal reinforcement in a column Equation 1416 Asw Crosssectional area of the vertical legs of each stirrup or area of the single leg of transverse steel in each wall of the idealised thinwalled section in torsion Aswmax Maximum crosssectional area of shear reinforcement Equations 713 and 714 Aswmin Minimum crosssectional area of shear reinforcement Equation 717 A0 Area of the transformed section at time t0 Bc First moment of the concrete part of the crosssection about the reference axis Bk First moment of the ageadjusted transformed section at time tk B0 First moment of area of the transformed section about the reference axis at time t0 C Strength class of concrete or carryover factor or Celsius D0 Matrix of crosssectional rigidities at time t0 Equation 542 E subscript Effect of actions Ecefft t0 Eceff Effective modulus of concrete at time t for concrete first loaded at t0 Equations 423 and 556 E t t E ceff ceff 0 Ageadjusted effective modulus of concrete at time t for concrete first loaded at t0 Equations 425 and 557 Ecm Secant modulus of elasticity of concrete Ecm0 Secant modulus of elasticity of concrete at time t0 Ep Design value of modulus of elasticity of prestressing steel Epi Design value of modulus of elasticity of the ith level of prestressed steel Es Design value of modulus of elasticity of reinforcing steel Esi Design value of modulus of elasticity of the ith level of nonprestressed steel Fbc Transverse compressive force due to bursting moment in a posttensioned end block Fbt Transverse tensile force due to bursting in a posttensioned end block the bursting force Fc Force carried by the concrete Fcc Compressive force carried by the concrete Fcd Design force carried by the concrete or design compres sive force in a strut Notation and sign convention xxv Fcdf Design force carried by the concrete flange Equation 636 Fcdw Design force carried by the concrete web of a flanged beam Equation 637 Fe0 Ageadjusted creep factor Equation 560 Fpt Tensile force carried by the prestressing steel Fptd Design tensile force carried by the prestressing steel at the ultimate limit state Fs Force carried by nonprestressed steel reinforcement Fsc Force carried by nonprestressed compressive steel reinforcement Fsd Design force carried by nonprestressed steel reinforcement Fst Force carried by nonprestressed tensile steel reinforcement Ft Resultant tensile force carried by the steel reinforcement and tendons Fk Matrix relating applied actions to strain at time tk Equation 5102 F0 Matrix relating applied actions to strain at time t0 Equation 546 G Permanent action Gk Characteristic permanent action I Second moment of area moment of inertia of the crosssection Iav Average second moment of area after cracking Ic Second moment of area of the concrete part of the crosssection about the reference axis Icr Second moment of area of a cracked crosssection Ief Effective second moment of area after cracking Ig Second moment of area of the gross crosssection Ik Second moment of area of the ageadjusted transformed section at time tk Iuncr Second moment of area of the uncracked crosssection I0 Second moment of area of the transformed section about reference axis at time t0 Jtt0 Creep function at time t due to a sustained unit stress first applied at t0 K Slab system factor or factor that accounts for the position of the bars being anchored with respect to the transverse reinforcement Figure 1414 Ldi Length of drawin line adjacent to a liveend anchorage Equation 5150 M Bending moment M Virtual moment Mb Bursting moment in a posttensioned anchorage zone xxvi Notation and sign convention Mc0 Moment resisted by the concrete at time t0 Mcr Cracking moment MEd Design value of the applied internal bending moment MEdx MEdy Design moments in a twoway slab spanning in the x and ydirections respectively Mext0 Externally applied moment about reference axis at time t0 Mextk Externally applied moment about reference axis at time tk MG Moment caused by the permanent loads Mint Internal moment about reference axis Mintk Internal moment about reference axis at time tk Mint0 Internal moment about reference axis at time t0 Mo Total static moment in a twoway flat slab or decom pression moment MQ Moment caused by the live loads MRd Design moment resistance Ms Spalling moment in a posttensioned anchorage zone Msus Moment caused by the sustained loads Msw Moment caused by the selfweight of a member MT Moment caused by total service loads Mu Ultimate moment capacity Mvar Moment caused by variable loads M0 Moment at a crosssection at transfer N Axial force Nck Axial forces resisted by the concrete at time tk Nc0 Axial forces resisted by the concrete at time t0 NEd Design value of the applied axial force tension or compression Next Externally applied axial force Nextk Externally applied axial force at time tk Next0 Externally applied axial force at time t0 Nint Internal axial force Nintk Internal axial force at time tk Nint0 Internal axial force at time t0 Npk Axial force resisted by the prestressing steel at time tk Np0 Axial force resisted by the prestressing steel at time t0 NRd Design axial resistance of a column NRdt Design axial resistance of a tension member Nsk Axial force resisted by the nonprestressed reinforce ment at time tk Ns0 Axial force resisted by the nonprestressed reinforce ment at time t0 P Prestressing force applied axial load in a column Ph Horizontal component of prestressing force Pinit i Initial prestressing force at the ith level of prestressing steel Notation and sign convention xxvii Pj Prestressing force during jacking the jacking force Pmt Effective force in the tendon at time t after the longterm losses Pm0 Initial force in the tendon immediately after transfer after the shortterm losses Px Py Prestressing forces in a slab in the x and ydirections respectively Pv Vertical component of prestressing force P0 Initial force at the active end of the tendon immediately after stressing Q Variable action Qk Characteristic variable action RAk RBk RIk Crosssectional rigidities at time tk Equations 584 585 and 589 RAp RBp RIp Contribution to section rigidities provided by the bonded tendons Equations 5125 through 5127 RAs RBs RIs Contribution to section rigidities provided by the steel reinforcement Equations 5122 through 5124 RA0 RB0 RI0 Crosssectional rigidities at time t0 Equations 535 536 and 539 S First moment of area T Torsional moment TEd Design value of the applied torsional moment TRdc Torsion required to cause first cracking in an otherwise unloaded beam TRdmax Maximum design torsional resistance Equation 734 U Internal work V Shear force Vccd Shear component of compressive force in an inclined compression chord VEd Nett design shear force VRd Design strength in shear VRdc Design shear strength of a beam without shear rein forcement Equations 72 and 74 VRdmax Maximum design shear strength for a beam with shear reinforcement Equation 78 VRds Design strength provided by the yielding shear reinforce ment Equation 77 W External work W1 Elastic energy Figure 620 W2 Plastic energy Figure 620 Z Section modulus of uncracked crosssection Zbtm Bottom fibre section modulus Iybtm Ztop Top fibre section modulus Iytop xxviii Notation and sign convention Latin lowercase letters a a Distance b Overall width of a crosssection or actual flange width in a T or L beam beff Effective width of the flange of a flanged crosssection bw Width of the web on T I or L beams c Concrete cover d Effective depth of a crosssection that is the depth from the extreme compressive fibre to the resultant tensile force in the reinforcement and tendons at the ultimate limit state dn Depth to neutral axis dn0 Depth to neutral axis at time t0 do Depth from the extreme compressive fibre to the centroid of the outermost layer of tensile reinforcement dp Depth from the top fibre of a crosssection to the p restressing steel dpi Depth from the top fibre of a crosssection to the ith level of prestressing steel dref Depth from the top fibre of a crosssection to the refer ence axis ds Depth from the top fibre of a crosssection to the non prestressed steel reinforcement dsi Depth from the top fibre of a crosssection to the ith level of nonprestressed steel reinforcement dx dy Effective depths to the tendons in the orthogonal x and ydirections respectively e Eccentricity of prestress eccentricity of axial load in a column axial deformation fbd Design value of the average ultimate bond stress Equation 145 fc Compressive strength of concrete fcct Compressive stress limits for concrete under full load fcc0 Compressive stress limits for concrete immediately after transfer fcd Design value of the compressive strength of concrete fck Characteristic compressive cylinder strength of concrete at 28 days fcm Mean value of concrete cylinder compressive strength fct Uniaxial tensile strength of concrete fctd Design value of the tensile strength of concrete fctmfl Mean flexural tensile strength of concrete fctk005 Lower characteristic axial tensile strength of concrete fctk095 Upper characteristic axial tensile strength of concrete Notation and sign convention xxix fctm Mean value of axial tensile strength of concrete fctt Tensile stress limits for concrete under full load fct0 Tensile stress limits for concrete immediately after transfer fp Tensile strength of prestressing steel fpk Characteristic tensile strength of prestressing steel fp01k Characteristic 01 proofstress of prestressing steel ft Tensile strength of reinforcement fy Yield strength of reinforcement fyd Design yield strength of reinforcement fyk Characteristic yield strength of reinforcement fywd Design yield strength of shear reinforcement fcp0 Vector of actions to account for unbonded tendons at time t0 Equation 5100 fcrk Vector of actions at time tk that accounts for creep dur ing previous time period Equation 596 fcsk Vector of actions at time tk that accounts for shrinkage during previous time period Equation 597 fpinit Vector of initial prestressing forces Equation 544 fprelk Vector of relaxation forces at time tk Equation 599 h Overall depth of a crosssection he Depth of the symmetric prism hp Dimension of a posttensioning anchorage plate h0 Notional size or hypothetical thickness i Radius of gyration i j k Integers k Coefficient or factor or angular deviation in radiansm or stiffness coefficient kr Shrinkage curvature coefficient l Length or span lbpd Anchorage length required to develop the design stress in a tendon at the ultimate limit state lbrqd Required anchorage length Equation 144 leff Effective span of a slab strip longer of the two effective spans on either side of a column lh Length of plastic hinge ln Clear span as defined in Figure 21 lpt Transmission length lt Transverse span lx ly Longer and shorter orthogonal span lengths respec tively in twoway slabs l0 Distance along a beam between the points of zero moment or effective length of a column or design lap length mp Number of layers of prestressed steel ms Number of layers of nonprestressed reinforcement xxx Notation and sign convention qk Characteristic uniformly distributed variable action rextk Vector of applied actions at time tk Equation 575 rext0 Vector of applied actions at time t0 Equation 541 rintk Vector of internal actions at time tk Equation 576 s Spacing between fitments sf Spacing between transverse reinforcement in a flange Figure 710 slmax Maximum spacing between stirrups or stirrup assem blies measured along the longitudinal axis of the member Equation 1413 st Stirrup spacing required for torsion sv Stirrup spacing required for shear srmax Maximum crack spacing Equation 5201 stmax Maximum transverse spacing of the legs of a stirrup Equation 1414 t Thickness or time t0 The age of concrete at the time of loading u Perimeter of concrete crosssection v Deflection or shear stress vcc Deflection due to creep vcs Deflection due to shrinkage vcx vmx Deflection of the column strip and the middle strip in the xdirection vcy vmy Deflection of the column strip and the middle strip in the ydirection v0 Deflection immediately after transfer vmax Maximum permissible total deflection or maximum final total deflection vmin Minimum shear stress vsus0 Shortterm deflection at transfer caused by the sustained loads vtot Total deflection w Uniformly distributed load or crack width wbal Uniformly distributed balanced load wEd Factored design load at the ultimate limit state wG Uniformly distributed permanent load wk Calculated crack width design crack width wp Distributed transverse load exerted on a member by a draped tendon profile wpx wpy Transverse loads exerted by tendons in the x and ydirections respectively wQ Uniformly distributed live load ws Uniformly distributed service load wsw Uniformly distributed load due to selfweight of the member Notation and sign convention xxxi wu Collapse load wunbal Uniformly distributed unbalanced load wunbalsus Sustained part of the uniformly distributed unbalanced load x Neutral axis depth at the ultimate limit state xyz Coordinates ybtm Distance from the centroidal axis to the bottom fibre of a crosssection yc Distance from reference axis to centroid of the concrete crosssection yn0 Distance from the reference axis to the neutral axis at time t0 ypi ycoordinate of ith level of prestressed steel ysi ycoordinate of ith level of nonprestressed reinforcement ytop Distance from the centroidal axis to the top fibre of a crosssection z Lever arm between internal forces zd Sag or drape of a parabolic tendon in a span Greek lowercase letters α Angle or ratio or index or factor αc Modular ratio Ecm2Ecm1 in a composite member αepi0 Effective modular ratio EpiEceff of the ith layer of prestressing steel at time t0 αesi0 Effective modular ratio EsiEceff of the ith layer of nonprestressed steel at time t0 αepik Ageadjusted effective modular ratio EpiEceff of the ith layer of prestressing steel at time tk αesik Ageadjusted effective modular ratio EsiEceff of the ith layer of nonprestressed steel at time tk αpi0 Modular ratio EpiEcm0 of the ith layer of prestress ing steel at time t0 αsi0 Modular ratio EsiEcm0 of the ith layer of non prestressed steel at time t0 α1 α2 Creep modification factors for cracked and uncracked crosssection Equations 5184 and 5185 respectively or fractions of the span l shown in Figure 1114 β Angle or ratio or coefficient or slope βcct Function describing the development of concrete strength with time Equation 42 βx βy Moment coefficients Table 123 χtt0 Aging coefficient for concrete at time t due to a stress first applied at t0 xxxii Notation and sign convention γ Partial factor γC Partial factor for concrete γG Partial factor for permanent actions G γP Partial factor for actions associated with prestressing P γQ Partial factor for variable actions Q γS Partial factor for reinforcing or prestressing steel Δ Increment or change Δslip Slip of the tendon at an anchorage Equation 5148 ΔPcsr Timedependent loss of prestress due to creep shrinkage and relaxation ΔPel Loss of prestress due to elastic shortening of the member Equation 5146 ΔPdi Loss of prestress due to drawin at the anchorage Equation 5151 ΔPμ Loss of prestress due to friction along the duct Equation 5148 Δtk Time interval tk t0 Δσpcsr Timedependent change of stress in the tendon due to creep shrinkage and relaxation Equation 5152 Δσpc Change in stress in the tendon due to creep Δσpr Change in stress in the tendon due to relaxation Δσps Change in stress in the tendon due to shrinkage Δσp0 Change in stress in the tendon immediately after transfer η Ratio of uniform compressive stress intensity of the idealised rectangular stress block to the design com pressive strength of concrete fcd ε Strain εc Compressive strain in the concrete εca Autogenous shrinkage strain εcc Creep strain component in the concrete εcd Drying shrinkage strain εce Instantaneous strain component in the concrete εcs Shrinkage strain component in the concrete εk Strain at time tk εk Vector of strain at time tk Equations 594 and 5101 εpiinit Initial strain in the ith layer of prestressing steel pro duced by the initial tensile prestressing force Piniti εprelik Tensile creep strain in the ith prestressing tendon at time tk Equation 573 εpe Strain in the prestressing steel caused by the effective prestress Equation 613 εptd Concrete strain at the level of the tendon Equation 614 εpud Strain in the bonded tendon at the design resistance Equation 615 Notation and sign convention xxxiii εr Strain at the level of the reference axis εrk Strain at the level of the reference axis at time tk εr0 Strain at the level of the reference axis at time t0 εsd Design strain in the nonprestressed steel reinforcement εuk Characteristic strain of reinforcement or prestressing steel at maximum load εyk Characteristic yield strain of reinforcement or prestress ing steel ε0 Strain at time t0 ε0 Vector of strain components at time t0 Equations 543 and 545 ϕ Diameter of a reinforcing bar or of a prestressing duct φtt0 Creep coefficient of concrete defining creep between times t and t0 related to elastic deformation at 28 days φt0 Final value of creep coefficient of concrete φpi Creep coefficient of the prestressing steel at time tk κ Curvature κcc κcct Creepinduced curvature Equation 5183 κcr Curvature at first cracking κcs κcst Curvature induced by shrinkage Equation 5187 κef Instantaneous effective curvature on a cracked section κk Longterm curvature at time tk κp Curvature of prestressing tendon κsus Curvature caused by the sustained loads κsus0 Curvature caused by the sustained loads at time t0 κud Design curvature at the ultimate limit state Equation 610 κudmin Minimum design curvature at the ultimate limit state Equation 622 κuncr Curvature on the uncracked crosssection κ0 Initial curvature at time t0 λ Ratio of the depth of the rectangular compressive stress block to the depth of the neutral axis at ultimate limit state ν Poissons ratio θ Angle or sum in radians of the absolute values of succes sive angular deviations of the tendon over the length x or slope θp Angle of inclination of prestressing tendon θs Rotation available at a plastic hinge θv Angle between the axis of the concrete compression strut and the longitudinal axis of the member ρ Reinforcement ratio for the bonded steel As Apbdo ρcw Longitudinal compressive reinforcement ratio related to the web width Ascbw d xxxiv Notation and sign convention ρw Longitudinal reinforcement ratio for the tensile steel related to the web width As Aptbw d or shear rein forcement ratio σ Stress σc Compressive stress in the concrete σcbtm Stress in the concrete at the bottom of a crosssection σc0 σct0 Stress in the concrete at time t0 σck σctk Stress in the concrete at time tk σcp Compressive stress in the concrete from axial load or prestressing σcs Maximum shrinkageinduced tensile stress on the uncracked section Equation 5179 σctop Stress in the concrete at the top of a crosssection in positive bending σcy σcz Normal compressive stresses on the control section in the orthogonal y and zdirections respectively σp Stress in the prestressing steel σpi0 Stress in the ith layer of prestressing steel at time t0 σpik Stress in the ith layer of prestressing steel at time tk σpi Initial stress in the prestressing steel immediately after tensioning σpj Stress in the prestressing steel at the jack before losses σpmax Maximum permissible stress in the prestressing during jacking Equation 51 σpud Design stress in the prestressing steel σp0 Initial stress in the prestressing steel immediately after transfer σs Stress in the nonprestressed steel reinforcement σsik Stress in the ith layer of nonprestressed steel at time tk σsi0 Stress in the ith layer of nonprestressed steel at time t0 σsd Design stress in a steel reinforcement bar σ1 σ2 Principal stresses in concrete Ω A factor that depends on the timedependent loss of prestress in the concrete Equation 5112 ψ ψ0 ψ1 ψ2 Factors defining representative values of variable actions ζ A distribution coefficient that accounts for the moment level and the degree of cracking on the effective moment of inertia Equation 5181 1 Chapter 1 Basic concepts 11 INTRODUCTION For the construction of mankinds infrastructure reinforced concrete is the most widely used structural material It has maintained this position since the end of the nineteenth century and will continue to do so for the foresee able future Because the tensile strength of concrete is low steel bars are embedded in the concrete to carry the internal tensile forces Tensile forces may be caused by imposed loads or deformations or by loadindependent effects such as temperature changes and shrinkage Consider the simple reinforced concrete beam shown in Figure 11a where the external loads cause tension in the bottom of the beam leading to cracking Practical reinforced concrete beams are usually cracked under the daytoday service loads On a cracked section the applied bending moment M is resisted by compression in the concrete above the crack and tension in the bonded reinforcing steel crossing the crack Figure 11b Although the steel reinforcement provides the cracked beam with flex ural strength it prevents neither cracking nor loss of stiffness during crack ing Crack widths are approximately proportional to the strain and hence stress in the reinforcement Steel stresses must therefore be limited to some appropriately low value under inservice conditions in order to avoid exces sively wide cracks In addition large steel strain in a beam is the result of large curvature which in turn is associated with large deflection There is little benefit to be gained therefore by using higher strength steel or concrete since in order to satisfy serviceability requirements the increased capacity afforded by higher strength steel cannot be utilised Prestressed concrete is a particular form of reinforced concrete Prestressing involves the application of an initial compressive load to the structure to reduce or eliminate the internal tensile forces and thereby con trol or eliminate cracking The initial compressive load is imposed and sus tained by highly tensioned steel reinforcement tendons reacting on the concrete With cracking reduced or eliminated a prestressed concrete sec tion is considerably stiffer than the equivalent usually cracked reinforced concrete section Prestressing may also impose internal forces that are of 2 Design of Prestressed Concrete to Eurocode 2 opposite sign to the external loads and may therefore significantly reduce or even eliminate deflection With service load behaviour improved the use of highstrength steel reinforcement and highstrength concrete becomes both economical and structurally efficient As we will see subsequently only steel that can accom modate large initial elastic strains is suitable for prestressing concrete The use of highstrength steel is therefore not only an advantage to prestressed concrete it is a necessity Prestressing results in lighter members longer spans and an increase in the economical range of application of reinforced concrete Consider an unreinforced concrete beam of rectangular section simply supported over a span l and carrying a uniform load w as shown in Figure 12a When the tensile strength of concrete fct is reached in the bottom fibre at midspan cracking and a sudden brittle failure will occur If it is assumed that the concrete possesses zero tensile strength ie fct 0 then no load can be carried and failure will occur at any load greater than zero In this case the collapse load wu is zero An axial compressive force P applied to the beam as shown in Figure 12b induces a uniform com pressive stress of intensity PA on each crosssection For failure to occur the maximum moment caused by the external collapse load wu must now induce an extreme fibre tensile stress equal in magnitude to PA In this case the maximum moment is located at midspan and if linearelastic material behaviour is assumed simple beam theory gives Figure 12b M Z w l Z P A u 2 8 Flexural cracking Reinforcing bars a b M σc σs Fc Fs Figure 11 A reinforced concrete beam a Elevation and section b Freebody diagram stress distribution and resultant forces Fc and Fs Basic concepts 3 based on which the collapse load can be determined as w Z l P A u 8 2 If the prestressing force P is applied at an eccentricity of h6 as shown in Figure 12c the compressive stress caused by P in the bottom fibre at mid span is equal to P A Pe Z P A Ph bh P A 6 6 2 2 and the external load at failure wu must now produce a tensile stress of 2PA in the bottom fibre This can be evaluated as follows Figure 12c M Z w l Z P A u 2 8 2 and rearranging gives w Z l P A u 16 2 fct 0 A bh I bh312 Z bh26 h b wu 0 collapse load w l wu h2 P P l P A wu 8Z l2 Due to wu PA PA MZ MZ Due to P Resultant Concrete stresses PA PeZ 2PA 2PA MZ Due to wu Due to P Resultant Concrete stresses P A wu16Z l2 l P P e h6 wu a b c Figure 12 Effect of prestress on the load carrying capacity of a plain concrete beam a Zero prestress b Axial prestress e 0 c Eccentric prestress e h6 4 Design of Prestressed Concrete to Eurocode 2 By locating the prestressing force at an eccentricity of h6 the load carrying capacity of the unreinforced plain concrete beam is effectively doubled The eccentric prestress induces an internal bending moment Pe which is opposite in sign to the moment caused by the external load An improve ment in behaviour is obtained by using a variable eccentricity of prestress along the member using a draped cable profile If the prestress countermoment Pe is equal and opposite to the load induced moment along the full length of the beam each crosssection is subjected only to axial compression ie each section is subjected to a uni form compressive stress of PA No cracking can occur and if the curvature on each section is zero the beam does not deflect This is known as the balanced load stage 12 METHODS OF PRESTRESSING As mentioned in the previous section prestress is usually imparted to a con crete member by highly tensioned steel reinforcement in the form of wire strand or bar reacting on the concrete The highstrength prestressing steel is most often tensioned using hydraulic jacks The tensioning operation may occur before or after the concrete is cast and accordingly prestressed members are classified as either pretensioned or posttensioned More information on prestressing systems and prestressing hardware is provided in Chapter 3 121 Pretensioned concrete Figure 13 illustrates the procedure for pretensioning a concrete member The prestressing tendons are initially tensioned between fixed abutments and anchored With the formwork in place the concrete is cast around the highly stressed steel tendons and cured When the concrete has reached its required strength the wires are cut or otherwise released from the abut ments As the highly stressed steel attempts to contract it is restrained by the concrete and the concrete is compressed Prestress is imparted to the concrete via bond between the steel and the concrete Pretensioned concrete members are often precast in pretensioning beds that are long enough to accommodate many identical units simultaneously To decrease the construction cycle time steam curing may be employed to facilitate rapid concrete strength gain and the prestress is often transferred to the concrete within 24 hours of casting Because the concrete is usu ally stressed at such an early age elastic shortening of the concrete and Basic concepts 5 subsequent creep strains tend to be high This relatively high time dependent shortening of the concrete causes a significant reduction in the tensile strain in the bonded prestressing steel and a relatively high loss of prestress occurs with time 122 Posttensioned concrete The procedure for posttensioning a concrete member is shown in Figure 14 With the formwork in position the concrete is cast around hollow ducts which are fixed to any desired profile The steel tendons are usually in place unstressed in the ducts during the concrete pour or alternatively may be threaded through the ducts at some later time When the concrete has reached its required strength the tendons are tensioned Tendons may be stressed from one end with the other end anchored or may be stressed from both ends as shown in Figure 14b The tendons are then anchored at each stressing end The concrete is compressed during the stressing operation and the prestress is maintained after the tendons are anchored by bearing of the end anchorage plates onto the concrete The posttensioned tendons also impose a transverse force on the member wherever the direction of the cable changes a b c Figure 13 Pretensioning procedure a Tendons stressed between abutments b Concrete cast and cured c Tendons released and prestress transferred 6 Design of Prestressed Concrete to Eurocode 2 After the tendons have been anchored and no further stressing is required the ducts containing the tendons are often filled with grout under pressure In this way the tendons are bonded to the concrete and are more efficient in controlling cracks and providing ultimate strength Bonded ten dons are also less likely to corrode or lead to safety problems if a tendon is subsequently lost or damaged In some situations however tendons are not grouted for reasons of economy and remain permanently unbonded In this form of construction the tendons are coated with grease and encased in a plastic sleeve Although the contribution of unbonded tendons to the ultimate strength of a beam or slab is only about 75 of that provided by bonded tendons unbonded posttensioned slabs are commonly used in North America and Europe Most insitu prestressed concrete is posttensioned Relatively light and portable hydraulic jacks make onsite posttensioning an attractive proposi tion Posttensioning is also used for segmental construction of largespan bridge girders 123 Other methods of prestressing Prestress may also be imposed on new or existing members using exter nal tendons or such other devices as flat jacks These systems are useful a b c Hollow duct Figure 14 Posttensioning procedure a Concrete cast and cured b Tendons stressed and prestress transferred c Tendons anchored and subsequently grouted Basic concepts 7 for temporary prestressing operations but may be subject to high time dependent losses External prestressing is discussed further in Section 37 13 TRANSVERSE FORCES INDUCED BY DRAPED TENDONS In addition to the longitudinal force P exerted on a prestressed member at the anchorages transverse forces are also exerted on the member wher ever curvature exists in the tendons Consider the simplysupported beam shown in Figure 15a It is prestressed by a cable with a kink at midspan The eccentricity of the cable is zero at each end of the beam and equal to e at midspan as shown The slope of the two straight segments of cable is θ Because θ is small it can be calculated as θ θ θ sin tan e l 2 11 In Figure 15b the forces exerted by the tendon on the concrete are shown At midspan the cable exerts an upward force FP on the concrete equal to the sum of the vertical component of the prestressing force in the tendon on both sides of the kink From statics F P Pe l P sin 2 4 θ 12 e B θ l2 l2 A a C b C P cos θ P P sin θ A FP2Psin θ B P cos θ P P sin θ FP 2P sin θ P P θ c P sin θ l2 Pe Figure 15 Forces and actions exerted by prestress on a beam with a centrally depressed tendon a Elevation b Forces imposed by prestress on concrete c Bending moment diagram due to prestress 8 Design of Prestressed Concrete to Eurocode 2 At each anchorage the cable has a horizontal component of P cos θ which is approximately equal to P for small values of θ and a vertical component equal to P sin θ approximated by 2Pel Under this condition the beam is said to be selfstressed No external reactions are induced at the supports However the beam exhibits a non zero curvature along its length and deflects upward owing to the internal bending moment caused by the prestress As illustrated in Figure 15c the internal bending moment at any section can be calculated from statics and is equal to the product of the prestressing force P and the eccentricity of the tendon at that crosssection If the prestressing cable has a curved profile the cable exerts trans verse forces on the concrete throughout its length Consider the pre stressed beam with the parabolic cable profile shown in Figure 16 With the x and ycoordinate axes in the directions shown the shape of the parabolic cable is y e x l x l 4 2 13 and its slope and curvature are respectively d d y x e l x l 4 1 2 14 and d d 2 p y x e l 2 2 8 κ 15 From Equation 14 the slope of the cable at each anchorage ie when x 0 and x l is θ d d y x e l 4 16 e l2 l2 x P y θ P Figure 16 A simple beam with parabolic tendon profile Basic concepts 9 and provided the tendon slope is small the horizontal and vertical compo nents of the prestressing force at each anchorage may therefore be taken as P and 4Pel respectively Equation 15 indicates that the curvature of the parabolic cable is constant along its length The curvature κp is the angular change in direction of the cable per unit length as illustrated in Figure 17a From the freebody dia gram in Figure 17b for small tendon curvatures the cable exerts an upward transverse force wp Pκp per unit length over the full length of the cable This upward force is an equivalent distributed load along the member and for a parabolic cable with the constant curvature of Equation 15 wp is given by w P Pe l p p κ 8 2 17 With the sign convention adopted in Figure 16 a positive value of wp depicts an upward load If the prestressing force is constant along the beam which is never quite the case in practice wp is uniformly distributed and acts in an upward direction A freebody diagram of the concrete beam showing the forces exerted by the cable is illustrated in Figure 18 The zero reactions induced by the prestress imply that the beam is selfstressed With the maximum eccen tricity usually known Equation 17 may be used to calculate the value of P required to cause an upward force wp that exactly balances a selected FP2Psin κp2 κp κp P P P Fp Unit length a b P Pκp Figure 17 Forces on a curved cable of unit length a Tendon segment of unit length b Triangle of forces 4Pel P e l2 l2 4Pel P wp 8Pel2 Figure 18 Forces exerted on a concrete beam by a tendon with a parabolic profile 10 Design of Prestressed Concrete to Eurocode 2 portion of the external load Under this balanced load the beam exhibits no curvature and is subjected only to the longitudinal compressive force of magnitude P This is the basis of a useful design approach sensibly known as load balancing 14 CALCULATION OF ELASTIC STRESSES The components of stress on a prestressed crosssection caused by the pre stress the selfweight and the external loads are usually calculated using simple beam theory and assuming linearelastic material behaviour In addition the properties of the gross concrete section are usually used in the calculations provided the section is not cracked Indeed these assump tions have already been made in the calculations of the stresses illustrated in Figure 12 Concrete however does not behave in a linearelastic manner At best linearelastic calculations provide only an approximation of the state of stress on a concrete section immediately after the application of the load Creep and shrinkage strains that gradually develop in the concrete usually cause a substantial redistribution of stresses with time particularly on a section containing significant amounts of bonded reinforcement Elastic calculations are useful however in determining for example if tensile stresses occur at service loads and therefore if cracking is likely or if compressive stresses are excessive and large timedependent shortening may be expected Elastic stress calculations may therefore be used to indi cate potential serviceability problems If an elastic calculation indicates that cracking may occur at service loads the cracked section analysis presented subsequently in Section 583 should be used to determine appropriate section properties for use in serviceabil ity calculations A more comprehensive picture of the variation of concrete stresses with time can be obtained using the time analyses described in Sections 57 and 59 to account for the timedependent deformations caused by creep and shrinkage of the concrete In the following sections several different approaches for calculating elastic stresses on an uncracked concrete crosssection are described to pro vide insight into the effects of prestressing Tensile compressive stresses are assumed to be positive negative 141 Combined load approach The stress distributions on a crosssection caused by prestress selfweight and the applied loads may be calculated separately and summed to obtain the combined stress distribution at any particular load stage We will first consider the stresses caused by prestress and ignore all other loads On a crosssection such as that shown in Figure 19 equilibrium requires that Basic concepts 11 the resultant of the concrete stresses is a compressive force that is equal and opposite to the tensile force in the steel tendon and located at the level of the steel ie at an eccentricity e below the centroidal axis This is statically equivalent to an axial compressive force P and a moment Pe located at the centroidal axis as shown The stresses caused by the prestressing force of magnitude P and the hogging ve moment Pe are also shown in Figure 19 The resultant stress induced by the prestress is given by σ P A Pey I 18 where A and I are the area and second moment of area about the centroidal axis of the crosssection respectively and y is the distance from the centroi dal axis positive upwards It is common in elastic stress calculations to ignore the stiffening effect of the reinforcement and to use the properties of the gross crosssection Although this simplification usually results in only small errors it is not encouraged here For crosssections containing significant amounts of bonded steel reinforcement the steel should be included in the determina tion of the properties of the transformed crosssection Section Elevations P e Pe P ytop ytop ybtm e ybtm Centroidal axis Centroidal axis y y Section e Due to P Due to Pe Resultant Stresses due to prestress PA PeyI PeyI PA Figure 19 Concrete stress resultants and stresses caused by prestress 12 Design of Prestressed Concrete to Eurocode 2 The elastic stresses caused by an applied positive moment M on the uncracked crosssection are σ My I 19 and the combined stress distribution due to prestress and the applied moment is shown in Figure 110 and given by σ P A Pey I My I 110 142 Internal couple concept The resultant of the combined stress distribution shown in Figure 110 is a compressive force of magnitude P located at a distance zp above the level of the steel tendon as shown in Figure 111 The compressive force in the concrete and the tensile force in the steel together form a couple with mag nitude equal to the applied bending moment and calculated as M Pzp 111 y yytop e y ybtm Due to prestress Due to moment Combined PA PeyI MyI MyI PA PeyI Centroidal axis Figure 110 Combined concrete stresses P P zp M Pzp Figure 111 Internal couple Basic concepts 13 When the applied moment M 0 the lever arm zp is zero and the resultant concrete compressive force is located at the steel level As M increases the compressive stresses in the top fibres increase and those in the bottom fibres decrease and the location of the resultant compressive force moves upward It is noted that provided the section is uncracked the magnitude of P does not change appreciably as the applied moment increases and as a consequence the lever arm zp is almost directly proportional to the applied moment If the magnitude and position of the resultant of the concrete stresses are known the stress distribution can be readily calculated 143 Load balancing approach In Figure 18 the forces exerted on a prestressed beam by a parabolic ten don with equal end eccentricities are shown and the uniformly distributed transverse load wp is calculated from Equation 17 In Figure 112 all the loads acting on such a beam including the external gravity loads w are shown If w wp the bending moment and shear force on each crosssection caused by the gravity load w are balanced by the equal and opposite values caused by wp With the transverse loads balanced the beam is subjected only to the longitudinal prestress P applied at the anchorage If the anchor age is located at the centroid of the section a uniform stress distribution of intensity PA occurs on each section and the beam does not deflect If w wp the bending moment Munbal caused by the unbalanced load w wp must be calculated and the resultant stress distribution given by Equation 19 must be added to the stresses caused by the axial prestress PA 144 Introductory example The elastic stress distribution at midspan of the simplysupported beam shown in Figure 113 is to be calculated The beam spans 12 m and is posttensioned by a single cable with zero eccentricity at each end and e 250 mm at midspan The prestressing force in the tendon is assumed to be constant along the length of the beam and equal to P 1760 kN w wp wl 2 wpl 2 P l wl 2 wpl 2 P Figure 112 Forces on a concrete beam with a parabolic tendon profile 14 Design of Prestressed Concrete to Eurocode 2 Each of the procedures discussed in the preceding sections is illustrated in the following calculations 1441 Combined load approach The extreme fibre stresses at midspan σtop σbtm due to P Pe and M are calculated separately in the following and then summed At midspan P 1760 kN Pe 1760 250 103 440 kNm and M wl 2 2 8 30 12 8 540 kNm Due to MPa top btm P P A σ σ 1760 10 220 10 8 0 3 3 Due to MPa top top Pe Pey I σ 440 10 485 20 000 10 10 67 6 6 σbtm btm MPa Pey I 440 10 415 20 000 10 9 13 6 6 Due to MPa top top M My I σ 540 10 485 20 000 10 13 10 6 6 σbtm btm MPa My I 540 10 415 20 000 10 11 21 6 6 350 150 y Centroidal axis ytop485 ybtm415 6000 6000 Elevation e 250 30 kNm includes selfweight Parabolic tendon 110 100 430 160 100 450 Section A 220 103 mm2 I 20000 106 mm4 P 1760 kN Figure 113 Beam details Introductory example Notes P is assumed constant on every section all dimensions are in millimetres Basic concepts 15 The corresponding concrete stress distributions and the combined elastic stress distribution on the concrete section at midspan are shown in Figure 114 1442 Internal couple concept From Equation 111 z M P p mm 540 10 1760 10 306 8 6 3 The resultant compressive force on the concrete section is 1760 kN and it is located 3068 250 568 mm above the centroidal axis This is statically equivalent to an axial compressive force of 1760 kN applied at the cen troid plus a moment Munbal 1760 568 103 100 kNm The extreme fibre concrete stresses are therefore σtop unbal top P A M y I 1 760 10 220 10 100 10 485 20 000 1 3 3 6 0 10 43 6 MPa σbtm unbal btm P A M y I 1 760 10 220 10 100 10 415 20 000 1 3 3 6 0 5 92 6 MPa and of course these are identical with the extreme fibre stresses calculated using the combined load approach and shown in Figure 114 1443 Load balancing approach The transverse force imposed on the concrete by the parabolic cable is obtained using Equation 17 as w Pe l p kNm upward 8 8 1 760 10 250 12 000 24 44 2 3 2 1043 1310 80 267 1067 592 1121 1713 913 80 Due to PPe P Pe M P Pe M Figure 114 Component stress distributions in introductory example 16 Design of Prestressed Concrete to Eurocode 2 The unbalanced load is therefore wunbal kNm downward 30 0 24 44 5 56 and the resultant unbalanced moment at midspan is M w l unbal unbal kNm 2 2 8 5 56 12 8 100 This is identical to the moment Munbal calculated using the internal couple concept and as determined previously the elastic stresses at midspan are obtained by adding the PA stresses to those caused by Munbal σtop unbal top MPa P A M y I 10 43 σbtm unbal btm MPa P A M y I 5 92 15 INTRODUCTION TO STRUCTURAL BEHAVIOUR INITIAL TO ULTIMATE LOADS The choice between reinforced and prestressed concrete for the construc tion of a particular structure is essentially one of economics Aesthetics may also influence the choice For relatively shortspan beams and slabs reinforced concrete is usually the most economical alternative As spans increase however reinforced concrete design is more and more controlled by the serviceability requirements Strength and ductility can still be eco nomically achieved but in order to prevent excessive deflection cross sectional dimensions become uneconomically large Excessive deflection is usually the governing limit state For medium to longspan beams and slabs the introduction of pre stress improves both serviceability and economy The optimum level of prestress depends on the span the load history and the serviceability requirements The level of prestress is often selected so that cracking at service loads does not occur However in many situations there is no valid reason why controlled cracking should not be permitted Insisting on enough prestress to eliminate cracking frequently results in unnecessarily high initial prestressing forces and consequently uneconomical designs In addition the high initial prestress often leads to excessively large camber andor axial shortening Members designed to remain uncracked at service loads are commonly termed fully prestressed In building structures there are relatively few situations in which it is necessary to avoid cracking under the full service loads In fact the most Basic concepts 17 economic design often results in significantly less prestress than is required for a fullyprestressed member Frequently such members are designed to remain uncracked under the sustained or permanent load with cracks open ing and closing as the variable live load is applied and removed Prestressed concrete members generally behave satisfactorily in the postcracking load range provided they contain sufficient bonded reinforcement to control the cracks A cracked prestressed concrete section under service loads is signifi cantly stiffer than a cracked reinforced concrete section of similar size and containing similar quantities of bonded reinforcement Members that are designed to crack at the full service load are often called partiallyprestressed The elastic stress calculations presented in the previous section are applicable only if material behaviour is linearelastic and the principle of superposition is valid These conditions may be assumed to apply on a pre stressed section prior to cracking but only immediately after the loads are applied As was mentioned in Section 14 the gradual development of creep and shrinkage strains with time in the concrete can cause a marked redis tribution of stress between the bonded steel and the concrete on the cross section The greater the quantity of bonded reinforcement the greater is the timedependent redistribution of stress This is demonstrated subsequently in Section 573 and discussed in Section 574 For the determination of the longterm stress and strain distributions elastic stress calculations are not meaningful and may be misleading A typical moment versus instantaneous curvature relationship for a pre stressed concrete crosssection is shown in Figure 115 Prior to the application of moment ie when M 0 if the prestressing force P acts at an eccentric ity e from the centroidal axis of the uncracked crosssection the curvature is κ0 PeEcmIuncr corresponding to point A in Figure 115 where Ecm is the elastic modulus of the concrete and Iuncr is the second moment of area of the uncracked crosssection The curvature κ0 is negative because the internal moment caused by prestress is negative Pe When the applied moment M is less than the cracking moment Mcr the section is uncracked and the moment curvature relationship is linear from point A to point B in Figure 115 and κ M PeEcmIuncr κcr It is only in this region ie when M Mcr that elastic stress calculations may be used and then only for shortterm calculations If the external loads are sufficient to cause cracking ie when the extreme fibre stress calculated from elastic analysis exceeds the tensile strength of concrete the shortterm behaviour becomes nonlinear and the principle of superposition is no longer applicable As the applied moment on a cracked prestressed section increases ie as the moment increases above Mcr from point B to point C in Figure 115 the crack height gradually increases from the tension surface towards the compression zone and the size of the uncracked part of the crosssection in compression above the crack decreases This is different to the post cracking behaviour of a nonprestressed reinforced concrete section where at first cracking the crack suddenly propagates deep into the beam and 18 Design of Prestressed Concrete to Eurocode 2 the crack height and the depth of the concrete compression zone remain approximately constant as the applied moment is subsequently varied As the moment on a prestressed concrete section increases further into the overload region approaching point D in Figure 115 the material behav iour becomes increasingly nonlinear Permanent deformation occurs in the bonded prestressing tendons as the stress approaches its ultimate value the nonprestressed conventional reinforcement yields at or near point C where there is a change in direction of the moment curvature graph and the compressive concrete in the ever decreasing region above the crack enters the nonlinear range The external moment is resisted by an internal couple with tension in the reinforcement crossing the crack and compression in the concrete and in any reinforcement in the compressive zone At the ultimate load stage ie when the moment reaches the ultimate resistance Mu at a curvature κu the prestressed section behaves in the same way as a rein forced concrete section except that the stress in the highstrength steel ten don is very much higher than in conventional reinforcement A significant portion of the very high steel stress and strain is due to the initial prestress For modern prestressing steels the initial stress in the tendon immediately after the transfer of prestress is often about 1400 MPa If the same higher strength steel were to be used without being initially prestressed excessive deformation and unacceptably wide cracks may result at only a small frac tion of the ultimate load well below normal service loads Moment M Mu My Mcr B C D Asc Ap Ast A κu κy EcmIuncr McrPe κcr EcmIuncr Pe κ0 Curvature κ EcmIuncr Figure 115 Typical moment versus instantaneous curvature relationship Basic concepts 19 The ultimate strength of a prestressed section depends on the quantity and strength of the steel reinforcement and tendons The level of pre stress however and therefore the quantity of prestressing steel are deter mined from serviceability considerations In order to provide a suitable factor of safety for strength additional conventional reinforcement may be required to supplement the prestressing steel in the tension zone This is particularly so in the case of partiallyprestressed members and may even apply for fullyprestressed construction The avoidance of cracking at service loads and the satisfaction of selected elastic stress limits do not ensure adequate strength Strength must be determined from a rational analysis which accounts for the nonlinear material behaviour of both the steel and the concrete Flexural strength analysis is described and illus trated in Chapter 6 and analyses for shear and torsional strength are presented in Chapter 7 Taylor Francis References 2 Design procedures and applied actions 1 EN 199111 2002 Eurocode 1 Actions on structures Part11 General actions Densities selfweight imposed loads on buildings British Standards Institution London UK 2 EN 199113 2003 Eurocode 1 Actions on structures Part 13 General actions Snow loads British Standards Institution London UK 3 EN 199114 2005 A1 2010 Eurocode 1 Actions on structures Part 14 General actions Wind actions British Standards Institution London UK 4 E N 1990 2002 A1 2005 Eurocode Basis of structural design British Standards Institution London UK 5 EN 199211 2004 Eurocode 2 Design of concrete structures Part 11 General rules and rules for buildings British Standards Institution London UK 6 EN 199112 2002 Eurocode 1 Actions on structures Part12 General actions Actions on structures exposed to fire British Standards Institution London UK 7 E N 199115 2003 Eurocode 1 Actions on structures Part 15 General actions Thermal actions British Standards Institution London UK 8 E N 199116 2005 Eurocode 1 Actions on structures Part 16 General actions Actions during execution British Standards Institution London UK 9 EN 199117 2006 Eurocode 1 Actions on structures Part 17 General actions Accidental actions British Standards Institution London UK 10 EN 199212 2004 Eurocode 2 Design of concrete structures Part 12 General rules Structural fire design British Standards Institution London UK 11 R anzi G and Gilbert RI 2014 Structural Analysis Principles Methods and Modelling Boca Raton FL CRC Press 12 Gilbert RI and Ranzi G 2011 TimeDependent Behaviour of Concrete Structures London UK Spon Press 13 Mayer H and Rüsch H 1967 Building damage caused by deflection of reinforced concrete building components Technical Translation 1412 National Research Council Ottawa Ontario Canada Berlin West Germany 1967 14 ISO 4356 1977 Bases for the design of structures Deformations of buildings at the serviceability limit states International Standards Organisation Geneva Switzerland 15 A S36002009 2009 Australian standard for concrete structures Standards Australia Sydney New South Wales Australia 16 ACI318M14 2014 Building code requirements for structural concrete and commentary American Concrete Institute Detroit MI 17 I rwin AW 1978 Human response to dynamic motion of structures The Structural Engineer 56A9 237244 18 M ickleborough NC and Gilbert RI 1986 Control of concrete floor slab vibration by LD limits Proceedings of the 10th Australasian Conference on the Mechanics of Structures and Materials University of Adelaide Adelaide South Australia Australia 19 S mith AL Hicks SJ and Devine PJ 2009 Design of floors for vibration A new approach Revised edition SCI P354 Steel Construction Institute Berkshire UK 20 Willford MR and Young P 2006 A design guide for footfall induced vibration of structures CCIP016 The Concrete Centre Surrey UK 21 Murray TM Allen DE and Ungar EE 1997 Steel Design Guide Series 11 Floor vibrations due to human activity American Institute of Steel Construction Chicago IL 22 ISO 10137 2007 Bases for design of structures Serviceability of buildings and walkways against vibrations International Standards Organisation Geneva Switzerland 4 Material properties 1 EN 199211 2004 Eurocode 2 Design of concrete structures Part 11 General rules and rules for buildings British Standards Institution London UK 2 EN 1961 1995 Method of testing cement Part 1 Determination of strength British Standards Institution London UK 3 EN 1971 2011 Cement Part 1 Composition specifications and conformity criteria for common cements British Standards Institution London UK 4 EN 9432 2009 Admixtures for concrete mortar and grout British Standards Institution London UK 5 E N 2061 2000 Concrete Specification performance production and conformity British Standards Institution London UK 6 BS 85001 2006 Part 1 Method of specifying and guidance for the specifier British Standards Institution London UK 7 EN 123902 2009 Testing hardened concrete Making and curing specimens for strength tests British Standards Institution London UK 8 EN 123903 2009 Testing hardened concrete Compressive strength of test specimens British Standards Institution London UK 9 EN 123905 2009 Testing hardened concrete Flexural strength of test specimens British Standards Institution London UK 10 EN 123906 2009 Testing hardened concrete Tensile splitting strength of test specimens British Standards Institution London UK 11 BS 1881121 1983 Testing concrete Methods for determination of static modulus of elasticity in compression British Standards Institution London UK 12 E N 10080 2005 Steel for the reinforcement of concrete weldable ribbed reinforcing steel British Standards Institution London UK 13 EN 101381 2005 Prestressing steel Part 1 General requirements 1 EN 101382 Prestressing steels Wire EN 101383 Prestressing steels Strand EN 101384 Prestressing steels Bars British Standards Institution London UK 14 N eville AM 1996 Properties of Concrete 4th edn London UK Wiley 15 M etha PK and Monteiro PJ 2014 Concrete Microstructure Properties and Materials 4th edn New York McGrawHill Education 16 Kupfer HB Hilsdorf HK and Rüsch H 1975 Behaviour of concrete under biaxial stresses ACI Journal 66 656666 17 T asuji ME Slate FO and Nilson AH 1978 Stressstrain response and fracture of concrete in biaxial loading ACI Journal 75 306312 18 D arwin D and Pecknold DA 1977 Nonlinear biaxial stressstrain law for concrete Journal of the Engineering Mechanics Division 103 229241 19 N eville AM 1970 Creep of Concrete Plain Reinforced and Prestressed Amsterdam the Netherlands NorthHolland 20 N eville AM Dilger WH and Brooks JJ 1983 Creep of Plain and Structural Concrete London UK Construction Press 21 G ilbert RI 1988 Time Effects in Concrete Structures Amsterdam the Netherlands Elsevier 22 G ilbert RI and Ranzi G 2011 TimeDependent Behaviour of Concrete Structures London UK Spon Press 426pp 23 G hali A and Favre R 1986 Concrete Structures Stresses and Deformations London UK Chapman and Hall 24 Ghali A Favre R and Eldbadry M 2002 Concrete Structures Stresses and Deformations 3rd edn London UK Spon Press 584pp 25 R üsch H Jungwirth D and Hilsdorf HK 1983 Creep and Shrinkage Their Effect on the Behaviour of Concrete Structures New York SpringerVerlag 284pp 26 A CI 209R92 1992 Prediction of creep shrinkage and temperature effects in concrete structures ACI Committee 209 American Concrete Institute Farmington Hills MI reapproved 2008 27 A CI 2091R05 2005 Report on factors affecting shrinkage and creep of hardened concrete ACI Committee 209 American Concrete Institute Farmington Hills MI 28 A CI Committee 209 2008 Guide for modeling and calculating shrinkage and creep in hardened concrete ACI 2092R08 American Concrete Institute Farmington Hills MI 44pp 29 Trost H 1978 Creep and creep recovery of very old concrete RILEM Colloquium on Creep of Concrete Leeds UK 30 Trost H 1967 Auswirkungen des Superpositionsprinzips auf Kriech und Relaxations Probleme bei Beton und Spannbeton Beton und Stahlbetonbau 62 230238 261269 31 Dilger W and Neville AM 1971 Method of creep analysis of structural members Australasian Conference on Information Security and Privacy 2717 349379 32 B azant ZP April 1972 Prediction of concrete creep effects using ageadjusted effective modulus method ACI Journal 69 212217 33 fib 2013 Fib Model Code for Concrete Structures 2010 Fib International Federation for Structural Concrete Ernst Sohn Lausanne Switzerland 434pp 34 O stergaard L Lange DA Altouabat SA and Stang H 2001 Tensile basic creep of earlyage concrete under constant load Cement and Concrete Research 31 18951899 35 C hu KH and Carreira DJ 1986 Timedependent cyclic deflections in RC beams Journal of Structural Engineering ASCE 1125 943959 36 B azant ZP and Oh BH 1984 Deformation of progressively cracking reinforced concrete beams ACI Journal 813 268278 37 G ilbert RI 2002 Creep and Shrinkage Models for High Strength Concrete Proposals for inclusion in AS3600 Australian Journal of Structural Engineering 42 95106 38 B rooks JJ 2005 30year creep and shrinkage of concrete Magazine of Concrete Research 579 545556 39 L oov RE 1988 A general equation for the steel stress for bonded prestressed concrete members Journal of the Prestressed Concrete Institute 33 108137 5 Design for serviceability 1 EN 199211 2004 Eurocode 2 Design of concrete structures Part 11 General rules and rules for buildings British Standards Institution London UK 2 ACI 31814M 2014 Building code requirements for reinforced concrete American Concrete Institute Detroit MI 3 M agnel G 1954 Prestressed Concrete 3rd edn London UK Concrete Publications Ltd 4 Lin TY 1963 Prestressed Concrete Structures New York Wiley 5 Warner RF and Faulkes KA 1979 Prestressed Concrete Melbourne Victoria Australia Pitman Australia 6 G ilbert RI and Ranzi G 2011 TimeDependent Behaviour of Concrete Structures London UK Spon Press 7 Gilbert RI 1988 Time Effects in Concrete Structures Amsterdam the Netherlands Elsevier 8 Ghali A Favre R and Elbadry M 2002 Concrete Structures Stresses and Deformations 3rd edn London UK Spon Press 9 AS36002009 Australian standard concrete structures Standards Australia Sydney New South Wales Australia 10 Branson DE 1963 Instantaneous and timedependent deflection of simple and continuous reinforced concrete beams Alabama Highway Research Report No 7 Bureau of Public Roads Montgomery AL 11 Bischoff PH 2005 Reevaluation of deflection prediction for concrete beams reinforced with steel and FRP bars Journal of Structural Engineering ASCE 1315 752767 12 G ilbert RI 2001 Deflection calculation and control Australian code amendments and improvements Chapter 4 ACI International SP 203 Code Provisions for Deflection Control in Concrete Structures Farmington Hills MI American Concrete Institute pp 4578 6 Flexural resistance 1 EN 199211 2004 Eurocode 2 Design of concrete structures Part 11 General rules and rules for buildings British Standards Institution London UK 2 ACI 318M14 2014 Building code requirements for structural concrete Detroit MI American Concrete Institute 3 Beeby AW 1999 Safety of structures a new approach to robustness The Structural Engineer 774 1621 0 005 010 015 θ pld mrad 020 025 030 035 040 045 xd 35 30 25 20 15 10 5 C90105 Class C Class B C5060 C5060 C90105 Figure 623 Basic values of allowable rotation θ pld when λ 3 1 7 Design resistance in shear and torsion 1 Ritter W 1899 Die Bauweise Hennebique Construction Methods of Hennebique Zurich Switzerland Schweizerische Bayzeitung 2 Hognestad E 1952 What do we know about diagonal tension and web r einforcement in concrete University of Illinois Engineering Experiment Station Circular Series No 64 Urbana IL 3 EN 199211 2004 Eurocode 2 Design of concrete structures Part 11 General rules and rules for buildings British Standards Institution London UK In each of the eight lines of shear reinforcement perpendicular to the column face as shown in Figure 726 the first vertical bar is located at about 05d eff 90 mm from the column face the outermost bar must be at least 656 mm from the column face The shear reinforcement layout is shown in Figure 727 135 135 135 135 135 90 10 mm shear links 300 600 a b 180 1 1 Figure 727 Sections through slabcolumn connection Example 75 a Section perpendicular to slab edge b Section 11 8 Anchorage zones 1 Hoyer E 1939 Der Stahlsaitenbeton Berlin Germany Elsner 2 BS EN 199211 2004 Eurocode 2 Design of concrete structures Part 11 General rules and rules for buildings British Standards Institution London UK 3 L ogan DR 1997 Acceptance criteria for bond quality of strand for pretensioned concrete applications PCI Journal 422 5290 4 R ose DR and Russell BW 1997 Investigation of standardized tests to measure the bond performance of prestressing strands PCI Journal 424 5680 5 Martin L and Korkosz W 1995 Strength of prestressed members at sections where strands are not fully developed PCI Journal 405 5866 6 MNL1204 PrecastPrestressed Concrete Institute 2005 PCI Design Handbook Precast And Prestressed Concrete 6th edn Chicago IL Precast Prestressed Concrete Institute pp 427429 7 Gilbert RI 2012 Unanticipated bond failure over supporting band beams in grouted posttensioned slab tendons with little or no prestress Bond in Concrete Fourth International Symposium 1720 June Brescia Italy 8 M arshall WT and Mattock AH 1962 Control of horizontal cracking in the ends of pretensioned prestressed concrete girders Journal of the Prestressed Concrete Institute 75 5674 9 G uyon Y 1953 Prestressed Concrete English edn London UK Contractors Record and Municipal Engineering 10 I yengar KTSR 1962 Twodimensional theories of anchorage zone stresses in posttensioned concrete beams Journal of the American Concrete Institute 59 14431446 11 I yengar KTSR and Yogananda CV 1966 A three dimensional stress distribution problem in the end zones of prestressed beams Magazine of Concrete Research 18 7584 12 S argious M 1960 Beitrag zur Ermittlung der Hauptzugspannungen am Endauflager vorgespannter Betonbalken Dissertation Stuttgart Germany Technische Hochschule 13 F oster SJ and Rogowsky DM 1997 Bursting forces in concrete members resulting from inplane concentrated loads Magazine of Concrete Research 49180 231240 14 M agnel G 1954 Prestressed Concrete 3rd edn New York McGrawHill 15 G ergely P and Sozen MA 1967 Design of anchorage zone reinforcement in prestressed concrete beams Journal of the Prestressed Concrete Institute 122 6375 16 W arner RF and Faulkes KA 1979 Prestressed Concrete 1st edn Melbourne Victoria Australia Pitman Australia 17 M arti P 1985 Truss models in detailing Concrete International American Concrete Institute 71 4656 18 M arti P 1985 Basic tools of reinforced concrete beam design Concrete International American Concrete Institute 712 6673 19 S chlaich J Schäfer K and Jennewein M 1987 Towards a consistent design of structural concrete Special Report PCI Journal 323 74150 9 Composite members 1 EN 199211 2004 Eurocode 2 Design of concrete structures Part 11 General rules and rules for buildings British Standards Institution London UK 2 Gilbert RI and Ranzi G 2011 TimeDependent Behaviour of Concrete Structures London UK Spon Press The spacing can be increased further into the span as the shear force V Ed decreases It is important to ensure that these bars are fully anchored on each side of the shear plane If the contact surface were not deliberately roughened but screeded and trowelled c 025 and μ 05 and Equation 961 gives v A Rdi s 0 25 1 2 0 5 0 0405 435 0 5 1 0 0 300 1000 0 3 2 0 725 10 3 A s From Equation 960 A s 2 mm m 1 59 0 32 0 725 10 1752 3 and with 212 mm bars 226 mm 2 cross the shear interface one in each web the required spacing s near each support is s 1000 226 1752 129 mm 11 Statically indeterminate members 1 Ranzi G and Gilbert R I 2015 Structural Analysis Principles Methods and Modelling Boca Raton FL CRC Press Taylor Francis Group 2 Cross H 1930 Analysis of continuous frames by distributing fixedend moments Transactions of the American Society of Civil Engineers 961793 110 3 EN 199211 2004 Eurocode 2 Design of concrete structures Part 11 General rules and rules for buildings British Standards Institution London UK With the effective prestress at C P mt 3337 kN the extreme top fibre stress under the unbalanced loads is σ ctop 3337 10 645 10 2540 10 178 4 10 5 17 14 24 9 0 3 3 6 6 7 MPa With the tensile strength taken as f ctm 35 MPa cracking will occur under the full unbalanced moment The error associated with estimates of the cracking moment based on elastic stress calculation may be significantly large As was discussed in Section 573 and illustrated in Example 55 creep and shrinkage may cause a large redistribution of stress on the crosssection with time particularly when the crosssection contains significant quantities of nonp restressed reinforcement as is the case here If a more accurate estimate of stresses is required a time analysis is recommended see Section 573 A cracked section analysis similar to that outlined in Section 583 is required to calculate the loss of stiffness due to cracking and the increment of tensile steel stress in order to check crack control The maximum inservice moment at C is equal to the sum of the moment caused by the full external service loads and the secondary moment M C 809 25 155 20 926 790 4338 kNm A cracked section analysis reveals that the tensile stress in the nonprestressed top steel at this moment is only 106 MPa which is much less than the increment of 200 MPa specified in EN 199211 3 and given in Table 55 for 25 mm diameter bars if the maximum crack width is to be limited to 03 mm Crack widths should therefore be acceptably small This design example is taken no further here Deflections are unlikely to be excessive but should be checked using the procedures outlined in Section 511 The design for shear and the design of the anchorage zones are in accordance with the discussions in Chapter 10 4 Lin TY and Thornton K 1972 Secondary moments and moment redistribution in continuous prestressed concrete beams Journal of the Prestressed Concrete Institute 171 120 5 M attock AH 1972 Secondary moments and moment redistribution in continuous prestressed concrete beams Discussion of Lin and Thornton 1972 Journal of the Prestressed Concrete Institute 174 8688 6 Nilson AH 1978 Design of Prestressed Concrete New York Wiley 7 Warner RF and Faulkes KA 1983 Overload behaviour and design of continuous prestressed concrete beams Presented at the International Symposium on NonLinearity and Continuity in Prestressed Concrete University of Waterloo Waterloo Ontario Canada 12 Twoway slabs Behaviour and design 1 EN 199211 2004 Eurocode 2 Design of concrete structures Part 11 General rules and rules for buildings British Standards Institution London UK 2 V SL Prestressing Aust Pty Ltd 1988 Slab Systems 2nd edn Sydney New South Wales Australia VSL 3 ACI 31814M 2014 Building code requirements for reinforced concrete Detroit MI American Concrete Institute 4 PostTensioning Institute 1977 Design of posttensioned slabs Glenview IL PostTensioning Institute 5 EN 199212 2004 Eurocode 2 Design of concrete structures Part 12 General rules Structural fire design British Standards Institution London UK 6 G ilbert RI 1985 Deflection control of slabs using allowable span to depth ratios ACI Journal Proceedings 82 6772 Figure 1226 Bandbeam and slab floor system 7 Gilbert RI 1989 Determination of slab thickness in suspended posttensioned floor systems ACI Journal Proceedings 86 602607 8 Gilbert RI 1979a Timedependent behaviour of structural concrete slabs PhD Thesis Sydney New South Wales Australia School of Civil Engineering University of New South Wales 9 Gilbert RI 1979b Timedependent analysis of reinforced and prestressed concrete slabs Proceedings Third International Conference in Australia on Finite Elements Methods Sydney New South Wales Australia University of New South Wales Unisearch Ltd pp 215 230 10 M ickleborough NC and Gilbert RI 1986 Control of concrete floor slab vibration by LD limits Proceedings of the 10th Australian Conference on the Mechanics of Structures and Materials University of Adelaide Adelaide South Australia Australia pp 5156 11 W ood RH 1968 The reinforcement of slabs in accordance with a pre d etermined field of moments Concrete 22 6976 12 A S3600 2009 Australian standard for concrete structures Standards Australia Sydney New South Wales Australia 13 Gilbert RI 1984 Effect of reinforcement distribution on the serviceability of reinforced concrete flat slabs Proceedings of the Ninth Australasian Conference on the Mechanics of Structures and Materials University of Sydney Sydney New South Wales Australia pp 210214 14 Nilson AH and Walters DB 1975 Deflection of twoway floor systems by the equivalent frame method ACI Journal 72 210218 15 Nawy EG and Chakrabarti P 1976 Deflection of prestressed concrete flat plates Journal of the PCI 21 86102 16 J ohansen KW 1962 YieldLine Theory London UK Cement and Concrete Association 17 Johansen KW 1972 YieldLine Formulae for Slabs London UK Cement and Concrete Association 18 R itz P Matt P Tellenbach Ch Schlub P and Aeberhard HU 1981 PostTensioned Concrete in Building Construction PostTensioned Slabs Berne Switzerland Losinger 13 Compression and tension members 1 Ranzi G and Gilbert RI 2015 Structural Analysis Principles Methods and Modelling Boca Raton FL CRC PressTaylor Francis Group 562pp 2 EN 199211 2004 Eurocode 2 Design of concrete structures Part 11 General rules and rules for buildings British Standards Institution London UK 3 Bresler B 1960 Design criteria for reinforced concrete columns under axial load and biaxial bending ACI Journal 57 481490 4 Ospina CE and Alexander SDB 1997 Transmission of high strength concrete column loads through concrete slabs structural engineering Report No 214 Department of Civil Engineering University of Alberta Edmonton Alberta Canada 5 Gilbert RI and Ranzi G 2011 TimeDependent Behaviour of Concrete Structures London UK Spon Press and the elongation caused by N is given by Equation 1348 e N 1 000 10 10 000 120 670 11 670 1 16 71 0 00848 17 14 0 3 00666 5 65 mm The net effect is a shortening of the member by e e e Pcs N mm13 28 5 65 7 63 14 Detailing Members and connections 1 EN 199211 2004 Eurocode 2 Design of concrete structures Part 11 General rules and rules for buildings British Standards Institution London UK 2 Park R and Paulay T 1975 Reinforced concrete structures John Wiley Sons 769pp Chapter 13 The art of detailing 3 Leonhardt F December 1965 Reducing the shear reinforcement in reinforced concrete beams and slabs Magazine of Concrete Research 1753 187198 4 L eonhardt F 1965 Über die Kunst des Bewehrens von Stahlbetontragwerken Beton und Stahlbetonbau 608 181192 609 212220 5 Leonhardt F 1971 Das Bewehren von Stahlbetontragwerken BetonKalender Berlin Germany Wilhelm Ernst Sohn Part II pp 303398 6 Leonhardt F and Teichen KT 1972 DruckStösse von Bewehrungstäben Deutscher Ausschuss für Stahlbeton Bulletin No 222 Wilhelm Ernst Sohn Berlin Germany pp 153 7 L eonhardt F Walther R and Dilger W 1964 Schubversuche an Durchlaufträgern Deutscher Ausschuss für Stahlbeton Heft 163 Berlin Germany Ernst Sohn 8 G oto Y 1971 Cracks formed in concrete around deformed tension bars ACI Journal 684 244251 9 T epfers R 1979 Cracking of concrete cover along anchored deformed reinforcing bars Magazine of Concrete Research 31106 312 10 T epfers R 1982 Lapped tensile reinforcement splices Journal of the Structural Division ASCE 1081 283301 11 Untrauer RE and Henry RL 1965 Influence of normal pressure on bond strength ACI Journal 625 577586 12 Mayfield B Kong FK Bennison A and Triston Davies JCD 1971 Corner joint details in structural lightweight concrete ACI Journal 685 366372 13 S wann RA 1969 Flexural strength of corners of reinforced concrete portal frames Technical Report TRA 434 Cement and Concrete Association London UK 14 F ranz G and Niedenhoff H 1963 The Reinforcement of Brackets and Short Deep Beams Library Translation No 114 London UK Cement and Concrete Association 15 ACI 31814M 2014 Building Code Requirements for Reinforced Concrete Detroit MI American Concrete Institute 16 F intel M 1974 Handbook of Concrete Engineering New York Van Nostrand Reinhold Company