PLASTIC DESIGN OF FRAMES 2 APPLICATIONS PLASTIC DESIGN OF FRAMES JACQUES HEYMAN ReaderinEngineering UniversityofCambridge 2. APPLICATIONS UCAMBRIDGE ::: - UNIVERSITYPRESS cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge cb2 8ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521730877 © Cambridge University Press 1971 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1971 First paperback edition 2008 A catalogue record for this publication is available from the British Library isbn 978-0-521-07984-6 Hardback isbn 978-0-521-73087-7 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Information regarding prices, travel timetables, and other factual information given in this work is correct at the time of first printing but Cambridge University Press does not guarantee the accuracy of such information thereafter. CONTENTS PREFACE page vii I THE YIELD SURFACE 1 I .I The definition ofcollapse I 1.2 Characteristics ofthe yield surface 6 1.3 Frame with distributed load 9 1.4 Combined bending and torsion I I Examples 18 2 ELEMENTARY SPACE FRAMES 23 2.1 The right-angle bent 23 2.2 Rectangular grillages 33 Examples 43 3 UNSYMMETRICAL BENDING 45 3.1 Bending ofrectangular section about an inclined axis 45 3.2 The effect ofaxial load 46 3.3 The general unsymmetrical section 48 3.4 The unequal angle 50 Examples 55 4 REINFORCED CONCRETE AND MASONRY 60 4.1 The simple plastic hinge 61 4.2 Bending with axial load 63 4.3 The collapse ofsimple frames 67 4.4 The masonry structure 73 4.5 Reinforced-concrete arches 84- Examples 86 5 ELASTIC-PLASTIC ANALYSIS 88 5.1 Virtual work for elastic-plastic frames 91 5.2 Fixed-ended beam 94- 5.3 Deflexions at collapse 101 5.4 A four-storey frame 108 5.5 The combination ofmechanisms I13 "5.6 The design ofcolumns; preliminary remarks 122 Examples 123 v CONTENTS 6 REPEATED LOADING page 126 6.1 The shakedown theorem 132 6.2 A two-span beam 135 6.3 The combination ofmechanisms 139 6.4 Rectangular portal frames 143 6.5 The relation between Acand As 149 6.6 A two-storey frame 153 6.7 Rolling loads 158 6.8 The significance ofshakedown calculations 161 Examples 164- 7 MINIMUM-WEIGHT DESIGN 167 7.I Dynamic programming 167 7.2 The linear weight function 176 7.3 Foulkes's theorem 183 7.4 Limitations on sections 187 7.5 Upper and lower bounds 191 7.6 Alternative loading combinations 196 7.7 Shakedown loading 199 7.8 Absolute minimum-weight design 202 Examples 215 8 NUMERICAL ANALYSIS 221 8.1 Linear programming 221 8.2 Static collapse analysis 226 8.3 Static collapse under alternative loading combinations 232 8.4 Shakedown analysis 234- 8.5 Minimum-weight design 237 9 MULTI-STOREY FRAMES 242 9.I Design considerations 242 9.2 Braced frames (I) 249 9.3 The load factor 254 9.4 Braced frames (2) 256 9.5 Swayframes 258 10 A DESIGN EXAMPLE 268 10.1 Frame dimensions and loadings 268 10.2 Preliminary design: static collapse 270 10.3 Shakedown analysis ofpreliminary design 283 10.4 Redesign offrame 288 INDEX 291 VI PREFACE The first volume of SirJohn Baker's The Steel Skeleton was published in.1954; volume 2 (Plastic behaviour anddesign), withjoint authorship, waspublishedin1956.Thetwovolumesgiveanaccountofthedevelop mentofsteelworkdesign, thefirst coveringtheperiodupto 1936,when the Recommendations ofthe Steel Structures Research Committee were made, and the second taking the story forward to 1954. The books are thus compounded ofthe historyofstructural design, ofaccounts ofthe importantresearchresults, andoftherelevanttechnicaladvancesinthe theory ofstructures. As soon as volume 2 of The Steel Skeleton had been completed, we wished to construct amore orthodoxtextbook on plastictheory; inthe event, volume 1 (Fundamentals), of Plastic Design of Frames was not published until 1969, again withjoint authorship. The present volume completes the original plan, and covers some topics which were not treated in volume 2 of The Steel Skeleton. The first three chapters deal with the notion of the yield surface in the theory of plasticity; the ideas are developed simply and with reference to the frame rather than the continuum, and are applied to reinforced concrete and masonry in chapter 4. The remaining six chapters of the book return to the problem ofthe plane steel building frame. Inchapter5istreatedthequestionofelastic-plasticanalysis,and in particular the calculation of deflexions. Chapter 6 deals with shake down, andchapter7withminimum-weightdesign; inboththesetopics, some advances have been made since the corresponding chapters in The Steel Skeleton were written. Examples are given at the ends of each of the first seven chapters; as in vol. 1, these are roughly graded from easy to difficult. Chapter 8 discusses methods of numerical analysis, not from the point of view of the construction of detailed programs, but in a way which exposes the analytical skeleton of plastic methods of design. Finally, chapter9discussessomeofthe problemsfacing the designerof multi-storey buildings, and chapter 10 uses many of the techniques presentedinbothvolumesforthesolutionofapracticaldesignproblem. vii PREFACE Dr A. C. Palmer has read very carefully a draft of the manuscript, and has been quick to spot loose expositions and hurried logic; any remaining imprecision in the book is entirely the fault of the author. Similarly, Mr B. D. Threlfall, who has taken a great deal oftime and trouble with the examples, both in the text and at the ends of the chapters,cannotbeheldresponsibleforanyremainingnumericalerrors. VIII 1 THE YIELD SURFACE 1.1 The definition ofcollapse ChapterI ofthepreviousvolumestartedwithageneraldiscussionofthe kind ofbehaviour that might be expected ofa simply-supported mild steel beam, loaded transversely out ofthe elastic into the plastic range. The idealized load-deflexion curve is reproduced in fig. 1.1. As the transverseloadWisincreasedslowlyfromzero,thecentraldeflexion8of the beam atfirst increases elastically and in proportion, so that OA is a straightline.WhentheloadexceedsthevalueatA,someplasticdeforma tionoccurs,withacorrespondinggreaterincreaseindeflexion.Whenthe point B is reached, the deflexion 8increases without limit at a constant load, sayWo. w B C weI---~.---+------~~ o Fig. 1.1 The idealized curve of fig. I.I results from the assumption of an elastic-perfectlyplasticbehaviour ofthe material; that is, itis assumed thatthematerial doesnotstrainhardenasthestrainisincreased. Strain hardeningisinfactsmallforastructuralmildsteel,butwillalwaysoccur eventually; intuitively, the neglect of such increased stresses, at least for relatively small strains, would seem to be 'safe'. It will be assumed I HPD I. THE YIELD SURFACE throughoutthepresentvolumealsothatthematerialiselastic-perfectly plastic.(Thus,ifthetheoryisappliedpracticallytoareinforced-concrete structure, for example, thenthe designermustensurethatthe detailsof the reinforcement are such that any particular plastic hinge does not strain-soften due to crushing of the concrete, before deflexions of the structureasawholebecomelarge.) Similarly, although collapse is characterized by the onset of large deflexions,theassumptionwillcontinuetobemadethatdeflexionsupto andatincipientcollapsearesmall.comparedwiththeoveralldimensions ofa structure. As was remarked in vol. I, this assumption implies that w· o Fig. 1.2 theequationsofequilibriumareunchangedafterdeformation; theshear balance across a storey of a multi-storey frame, for example, is not affectedbythesmallswayofthatstorey.Thusthedeflexions8infig. I.I are supposed to bevery small compared with the span1ofthe simply supportedbeam. The two essential features ofthe load-deflexion curve in fig. I.I are thatadefinitecollapseload%isreachedatpointB,andthatthecollapse load stays constant as the deflexions increase from B. The'flatness' of Be thecollapseportion ofthecurve,alongwhichcollapseoccurs,maybe denotedbydW =0,wheredWstandsforaninfinitesimalvariationinthe valueofthecollapseload.Bycontrast,fig. 1.2illustratesalsothetwocases ofstrain hardening andstrainsoftening; for theformer, dW> 0, while for strainsofteningdW < 0. 2