Table Of ContentDesign of Prestressed
Concrete to Eurocode 2
Second Edition
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 33487-2742
© 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 U.S. Government works
Printed on acid-free paper
Version Date: 20161209
International Standard Book Number-13: 978-1-4665-7310-9 (Hardback)
International Standard Book Number-13: 978-1-3153-8952-3 (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 U.S. 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.
copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc.
(CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit 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 Cataloging‑in‑Publication Data
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Preface xv
Authors xix
Acknowledgements xxi
Notation and sign convention xxiii
1 Basic concepts 1
1.1 Introduction 1
1.2 Methods of prestressing 4
1.2.1 Pretensioned concrete 4
1.2.2 Post-tensioned concrete 5
1.2.3 Other methods of prestressing 6
1.3 Transverse forces induced by draped tendons 7
1.4 Calculation of elastic stresses 10
1.4.1 Combined load approach 10
1.4.2 Internal couple concept 12
1.4.3 Load balancing approach 13
1.4.4 Introductory example 13
1.4.4.1 Combined load approach 14
1.4.4.2 Internal couple concept 15
1.4.4.3 Load balancing approach 15
1.5 Introduction to structural behaviour:
Initial to ultimate loads 16
2 Design procedures and applied actions 21
2.1 Limit states design philosophy 21
2.2 Structural modelling and analysis 23
2.2.1 Structural modelling 23
2.2.2 Structural analysis 24
v
vi Contents
2.3 Actions and combinations of actions 26
2.3.1 General 26
2.3.2 Load combinations for the strength limit states 29
2.3.3 Load combinations for the stability
or equilibrium limit states 31
2.3.4 Load combinations for the
serviceability limit states 32
2.4 Design for the strength limit states 33
2.4.1 General 33
2.4.2 Partial factors for materials 33
2.5 Design for the serviceability limit states 34
2.5.1 General 34
2.5.2 Deflection limits 35
2.5.3 Vibration control 37
2.5.4 Crack width limits 37
2.5.5 Partial factors for materials 38
2.6 Design for durability 38
2.7 Design for fire resistance 40
2.8 Design for robustness 43
References 44
3 Prestressing systems 47
3.1 Introduction 47
3.2 Types of prestressing steel 47
3.3 Pretensioning 49
3.4 Post-tensioning 51
3.5 Bonded and unbonded post-tensioned construction 58
3.6 Circular prestressing 59
3.7 External prestressing 60
4 Material properties 63
4.1 Introduction 63
4.2 Concrete 63
4.2.1 Composition of concrete 64
4.2.2 Strength of concrete 65
4.2.3 Strength specifications in Eurocode 2 68
4.2.3.1 Compressive strength 68
4.2.3.2 Tensile strength 69
4.2.3.3 Design compressive and tensile strengths 70
Contents vii
4.2.3.4 C ompressive stress–strain curves
for concrete for non-linear
structural analysis 72
4.2.4 Deformation of concrete 73
4.2.4.1 Discussion 73
4.2.4.2 Instantaneous strain 74
4.2.4.3 Creep strain 76
4.2.4.4 Shrinkage strain 81
4.2.5 Deformational characteristics
specified in Eurocode 2 82
4.2.5.1 Introduction 82
4.2.5.2 Modulus of elasticity 83
4.2.5.3 Creep coefficient 84
4.2.5.4 Shrinkage strain 86
4.2.5.5 Thermal expansion 87
4.3 Steel reinforcement 87
4.3.1 General 87
4.3.2 Specification in Eurocode 2 88
4.3.2.1 Strength and ductility 88
4.3.2.2 Elastic modulus 89
4.3.2.3 Stress–strain curves:
Design assumptions 90
4.3.2.4 Coefficient of thermal
expansion and density 91
4.4 Steel used for prestressing 91
4.4.1 General 91
4.4.2 Specification in Eurocode 2 94
4.4.2.1 Strength and ductility 94
4.4.2.2 Elastic modulus 94
4.4.2.3 Stress–strain curve 96
4.4.2.4 Steel relaxation 96
References 98
5 Design for serviceability 101
5.1 Introduction 101
5.2 Concrete stresses at transfer and under full service loads 102
5.3 Maximum jacking force 105
5.4 Determination of prestress and
eccentricity in flexural members 106
5.4.1 Satisfaction of stress limits 106
5.4.2 Load balancing 114
viii Contents
5.5 Cable profiles 116
5.6 Short-term analysis of uncracked cross-sections 118
5.6.1 General 118
5.6.2 Short-term cross-sectional analysis 120
5.7 Time-dependent analysis of uncracked cross-sections 136
5.7.1 Introduction 136
5.7.2 The age-adjusted effective modulus method 136
5.7.3 Long-term analysis of an uncracked
cross-section subjected to combined axial
force and bending using AEMM 138
5.7.4 Discussion 156
5.8 Short-term analysis of cracked cross-sections 158
5.8.1 General 158
5.8.2 Assumptions 160
5.8.3 Analysis 160
5.9 Time-dependent analysis of cracked cross-sections 170
5.9.1 Simplifying assumption 170
5.9.2 Long-term analysis of a cracked cross-section
subjected to combined axial force and
bending using the AEMM 170
5.10 Losses of prestress 175
5.10.1 Definitions 175
5.10.2 Immediate losses 176
5.10.2.1 Elastic deformation losses 176
5.10.2.2 Friction in the jack and anchorage 177
5.10.2.3 Friction along the tendon 177
5.10.2.4 Anchorage losses 179
5.10.2.5 Other causes of immediate losses 180
5.10.3 Time-dependent losses of prestress 181
5.10.3.1 Discussion 181
5.10.3.2 Simplified method specified
in EN 1992-1-1:2004 182
5.10.3.3 Alternative simplified method 183
5.11 Deflection calculations 187
5.11.1 General 187
5.11.2 Short-term moment–curvature
relationship and tension stiffening 190
5.11.3 Short-term deflection 195
5.11.4 Long-term deflection 200
5.11.4.1 Creep-induced curvature 201
5.11.4.2 Shrinkage-induced curvature 202
Contents ix
5.12 Crack control 208
5.12.1 Minimum reinforcement 208
5.12.2 Control of cracking without direct calculation 211
5.12.3 Calculation of crack widths 213
5.12.4 Crack control for restrained shrinkage
and temperature effects 215
5.12.5 Crack control at openings and discontinuities 216
References 216
6 Flexural resistance 219
6.1 Introduction 219
6.2 Flexural behaviour at overloads 219
6.3 Design flexural resistance 222
6.3.1 Assumptions 222
6.3.2 Idealised compressive stress blocks for concrete 223
6.3.3 Prestressed steel strain components
(for bonded tendons) 226
6.3.4 Determination of M for a singly
Rd
reinforced section with bonded tendons 228
6.3.5 D etermination of M for sections containing
Rd
non-prestressed reinforcement and
bonded tendons 232
6.3.6 Members with unbonded tendons 239
6.4 Design calculations 241
6.4.1 Discussion 241
6.4.2 C alculation of additional non-prestressed
tensile reinforcement 242
6.4.3 Design of a doubly reinforced cross-section 245
6.5 Flanged sections 248
6.6 Ductility and robustness of prestressed concrete beams 254
6.6.1 Introductory remarks 254
6.6.2 Calculation of hinge rotations 257
6.6.3 Q uantifying ductility and
robustness of beams and slabs 257
References 260
7 Design resistance in shear and torsion 261
7.1 Introduction 261
7.2 Shear in beams 261
7.2.1 Inclined cracking 261
