Table Of ContentPLASTIC DESIGN OF FRAMES
2
APPLICATIONS
PLASTIC DESIGN OF
FRAMES
JACQUES HEYMAN
ReaderinEngineering
UniversityofCambridge
2. APPLICATIONS
UCAMBRIDGE
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© Cambridge University Press 1971
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permission of Cambridge University Press.
First published 1971
First paperback edition 2008
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isbn 978-0-521-07984-6 Hardback
isbn 978-0-521-73087-7 Paperback
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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
