Table Of Content

Concrete Structures Stresses and Deformations Third Edition Also available from Spon Press Abnormal Loading on Structures Design of Offshore Concrete Experimental and Numerical Modelling Structures F. K. Garas, K. S. Virdi, R. Matthews & I. Holand, E. Jersin & O. T. Gudmestad J. L. Clarke Dynamic Loading and Design of Autogenous Shrinkage of Structures Concrete A. J. Kappos Edited by E. Tazawa Earthquake Resistant Concrete Structures Bridge Deck Behaviour G. G. Penelis & A. J. Kappos 3rd Edition E. C. Hambly Global Structural Analysis of Buildings Bridge Loads K. A. Zalka An International Perspective C. O’Connor & P. Shaw Introduction to Eurocode 2 Design of Concrete Structures Circular Storage Tanks and Silos D. Beckett & A. Alexandrou 2nd Edition A. Ghali Monitoring and Assessment of Structures Concrete Ground Floors G. Armer N. Williamson Structural Analysis A Unified Classical and Matrix Concrete Masonry Designer’s Approach Handbook A. Ghali & A. M. Neville 2nd Edition J. J. Roberts, A. K. Tovey & A. Fried Structural Defects Reference Manual for Low-rise Buildings Design Aids for Eurocode 2 M. F. Atkinson Design of Concrete Structures Edited by The Concrete Societies of The Wind Loading of Structures UK, The Netherlands and Germany J. D. Holmes Concrete Structures Stresses and Deformations Third Edition A. Ghali Professor, The University of Calgary Canada R. Favre Professor, Swiss Federal Institute of Technology (EPFL) Lausanne, Switzerland M. Elbadry Associate Professor, The University of Calgary Canada London and New York First published 1986 by E & FN Spon Second edition first published 1994 Third edition first published 2002 by Spon Press 11 New Fetter Lane, London EC4P 4EE Simultaneously published in the USA and Canada by Spon Press 29 West 35th Street, New York, NY 10001 This edition published in the Taylor & Francis e-Library, 2006. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” Spon Press is an imprint of the Taylor & Francis Group © 1986, 1994 A. Ghali and R. Favre © 2002 A. Ghali, R. Favre and M. Elbadry The right of A. Ghali, R Favre and M. Elbadry to be identified as the Authors of this Work has been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this book has been requested ISBN 0-203-98752-7 Master e-book ISBN ISBN 0–415–24721–7(Print Edition) Contents Preface to the third edition xiv Acknowledgements xvi Note xvii The SI system of units and British equivalents xviii Notation xx 1 Creep and shrinkage of concrete and relaxation of steel 1 1.1 Introduction 1 1.2 Creep of concrete 2 1.3 Shrinkage of concrete 4 1.4 Relaxation of prestressed steel 5 1.5 Reduced relaxation 7 1.6 Creep superposition 8 1.7 The aging coefficient χ: definition 10 1.8 Equation for the aging coefficient χ 11 1.9 Relaxation of concrete 12 1.10 Step-by-step calculation of the relaxation function for concrete 14 1.11 Age-adjusted elasticity modulus 17 1.11.1 Transformed section 17 1.11.2 Age-adjusted flexibility and stiffness 18 1.12 General 18 2 Stress and strain of uncracked sections 20 2.1 Introduction 20 2.2 Sign convention 22 2.3 Strain, stress and curvature in composite and homogeneous cross-sections 22 2.3.1 Basic equations 25 vi Contents 2.4 Strain and stress due to non-linear temperature variation 27 Example 2.1 Rectangular section with parabolic temperature variation 29 2.5 Time-dependent stress and strain in a composite section 30 2.5.1 Instantaneous stress and strain at age t 31 0 2.5.2 Changes in stress and strain during the period t to t 33 0 Example 2.2 Post-tensioned section 37 Example 2.3 Pre-tensioned section 43 Example 2.4 Composite section: steel and post- tensioned concrete 44 Example 2.5 Composite section: pre-tensioned and cast-in-situ parts 49 2.6 Summary of analysis of time-dependent strain and stress 57 2.7 Examples worked out in British units 61 Example 2.6 Stresses and strains in a pre-tensioned section 61 Example 2.7 Bridge section: steel box and post-tensioned slab 64 2.8 General 68 3 Special cases of uncracked sections and calculation of displacements 69 3.1 Introduction 70 3.2 Prestress loss in a section with one layer of reinforcement 70 3.2.1 Changes in strain, in curvature and in stress due to creep, shrinkage and relaxation 74 Example 3.1 Post-tensioned section without non- prestressed steel 75 3.3 Effects of presence of non-prestressed steel 78 3.4 Reinforced concrete section without prestress: effects of creep and shrinkage 79 Example 3.2 Section subjected to uniform shrinkage 81 Example 3.3 Section subjected to normal force and moment 83 3.5 Approximate equations for axial strain and curvature due to creep 85 3.6 Graphs for rectangular sections 85 3.7 Multi-stage prestressing 87 3.8 Calculation of displacements 88 3.8.1 Unit load theory 89 3.8.2 Method of elastic weights 89 Contents vii Example 3.4 Simple beam: derivation of equations for displacements 92 Example 3.5 Simplified calculations of displacements 93 3.9 Example worked out in British units 95 Example 3.6 Parametric study 95 3.10 General 98 4 Time-dependent internal forces in uncracked structures: analysis by the force method 100 4.1 Introduction 101 4.2 The force method 103 4.3 Analysis of time-dependent changes of internal forces by the force method 105 Example 4.1 Shrinkage effect on a portal frame 108 Example 4.2 Continuous prestressed beam constructed in two stages 109 Example 4.3 Three-span continuous beam composed of precast elements 113 Example 4.4 Post-tensioned continuous beam 116 4.4 Movement of supports of continuous structures 121 Example 4.5 Two-span continuous beam: settlement of central support 125 4.5 Accounting for the reinforcement 128 Example 4.6 Three-span precast post-tensioned bridge 128 4.6 Step-by-step analysis by the force method 136 4.7 Example worked out in British units 141 Example 4.7 Two-span bridge: steel box and post-tensioned deck 141 4.8 General 144 5 Time-dependent internal forces in uncracked structures: analysis by the displacement method 146 5.1 Introduction 146 5.2 The displacement method 147 5.3 Time-dependent changes in fixed-end forces in a homogeneous member 149 Example 5.1 Cantilever: restraint of creep displacements 152 5.4 Analysis of time-dependent changes in internal forces in continuous structures 153 viii Contents 5.5 Continuous composite structures 154 5.6 Time-dependent changes in the fixed-end forces in a composite member 156 5.7 Artificial restraining forces 158 Example 5.2 Steel bridge frame with concrete deck: effects of shrinkage 160 Example 5.3 Composite frame: effects of creep 164 5.8 Step-by-step analysis by the displacement method 172 5.9 General 175 6 Analysis of time-dependent internal forces with conventional computer programs 176 6.1 Introduction 177 6.2 Assumptions and limitations 177 6.3 Problem statement 179 6.4 Computer programs 179 6.5 Two computer runs 184 6.6 Equivalent temperature parameters 186 6.7 Multi-stage loading 188 6.8 Examples 188 Example 6.1 Propped cantilever 188 Example 6.2 Cantilever construction method 192 Example 6.3 Cable-stayed shed 193 Example 6.4 Composite space truss 197 Example 6.5 Prestressed portal frame 201 6.9 General 205 7 Stress and strain of cracked sections 207 7.1 Introduction 208 7.2 Basic assumptions 209 7.3 Sign convention 209 7.4 Instantaneous stress and strain 210 7.4.1 Remarks on determination of neutral axis position 213 7.4.2 Neutral axis position in a T or rectangular fully cracked section 214 7.4.3 Graphs and tables for the properties of transformed fully cracked rectangular and T sections 216 Example 7.1 Cracked T section subjected to bending moment 234 Example 7.2 Cracked T section subjected to M and N 236 Contents ix 7.5 Effects of creep and shrinkage on a reinforced concrete section without prestress 237 7.5.1 Approximate equation for the change in curvature due to creep in a reinforced concrete section subjected to bending 243 Example 7.3 Cracked T section: creep and shrinkage effects 243 7.6 Partial prestressed sections 246 Example 7.4 Pre-tensioned tie before and after cracking 250 Example 7.5 Pre-tensioned section in flexure: live-load cracking 254 7.7 Flow chart 249 7.8 Example worked out in British units 260 Example 7.6 The section of Example 2.6: live-load cracking 260 7.9 General 262 8 Displacements of cracked members 264 8.1 Introduction 265 8.2 Basic assumptions 266 8.3 Strain due to axial tension 266 Example 8.1 Mean axial strain in a tie 271 8.4 Curvature due to bending 271 8.4.1 Provisions of codes 274 Example 8.2 Rectangular section subjected to bending moment 275 8.5 Curvature due to a bending moment combined with an axial force 276 Example 8.3 Rectangular section subjected to M and N 278 8.5.1 Effect of load history 280 8.6 Summary and idealized model for calculation of deformations of cracked members subjected to N and/ or M 281 8.6.1 Note on crack width calculation 284 8.7 Time-dependent deformations of cracked members 284 Example 8.4 Non-prestressed simple beam: variation of curvature over span 285 Example 8.5 Pre-tensioned simple beam: variation of curvature over span 290 8.8 Shear deformations 293