Table Of ContentBifurcation and
Buckling in Structures
Bifurcation and
Buckling in Structures
Kiyohiro Ikeda
Kazuo Murota
First edition published 2022
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
© 2022 Kiyohiro Ikeda and Kazuo Murota
CRC Press is an imprint of Taylor & Francis Group, LLC
Reasonable efforts have been made to publish reliable data and information, but the author and pub-
lisher 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 stor-
age or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, access www.copyright.
com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA
01923, 978-750-8400. For works that are not available on CCC, please contact mpkbookspermis-
[email protected]
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
Names: Ikeda, Kiyohiro, author. | Murota, Kazuo, 1955- author.
Title: Bifurcation and buckling in structures / Kiyohiro Ikeda, Kazuo
Murota.
Description: First edition. | Boca Raton, FL : CRC Press, 2022. | Includes
bibliographical references and index.
Identifiers: LCCN 2021031070 (print) | LCCN 2021031071 (ebook) | ISBN
9780367631611 (hbk) | ISBN 9780367631604 (pbk) | ISBN 9781003112365
(ebk)
Subjects: LCSH: Buckling (Mechanics) | Deformations (Mechanics) |
Bifurcation theory.
Classification: LCC TA656.2 .I43 2022 (print) | LCC TA656.2 (ebook) | DDC
624.1/76--dc23
LC record available at https://lccn.loc.gov/2021031070
LC ebook record available at https://lccn.loc.gov/2021031071
ISBN: 978-0-367-63161-1 (hbk)
ISBN: 978-0-367-63160-4 (pbk)
ISBN: 978-1-003-11236-5 (ebk)
DOI: 10.1201/9781003112365
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
Contents
Preface.......................................................................................................................xi
TheAuthors............................................................................................................xiii
Part I Bifurcation in Structures
Chapter1 IntroductiontoBucklingandBifurcation........................................3
1.1 Summary.................................................................................3
1.2 Whatarebucklingandbifurcation?........................................3
1.3 Mathematicalframework........................................................6
1.4 Snapbucklingofastructure.................................................10
1.5 Bifurcationbucklingofastructure.......................................12
1.6 Problems...............................................................................15
Chapter2 AnalysisofBucklingLoadandMode...........................................17
2.1 Summary...............................................................................17
2.2 Mathematicalframework......................................................17
2.3 Procedureofbucklinganalysis.............................................19
2.4 Jacobianmatrixandcriticalpoint.........................................21
2.5 Totalpotentialenergy...........................................................25
2.5.1 Derivationofgoverningequation............................25
2.5.2 Existenceofapotential............................................27
2.6 Stability.................................................................................28
2.7 Exampleofbucklingloadanalysis.......................................31
2.8 Examplesofbucklingmodeanalysis...................................35
2.8.1 Bar-springsystemwithtwodegreesoffreedom.....35
2.8.2 Bar-springsystemwithmanydegreesoffreedom...37
2.9 Problems...............................................................................39
Chapter3 NumericalAnalysisI:PathTracing..............................................43
3.1 Summary...............................................................................43
3.2 Theoreticalfoundation..........................................................43
3.3 Pathtracingmethods.............................................................46
3.3.1 Loadcontrolmethod................................................46
3.3.2 Displacementcontrolmethod..................................48
3.3.3 Arc-lengthmethod...................................................49
v
vi Contents
3.4 Structuralexample................................................................50
3.5 Problems...............................................................................54
Chapter4 NumericalAnalysisII:BranchSwitching.....................................55
4.1 Summary...............................................................................55
4.2 Theoreticalfoundation..........................................................55
4.2.1 Directionofbifurcatingpath....................................56
4.2.2 Bifurcationanalysisprocedure................................59
4.3 Bar-springsystem.................................................................60
4.3.1 Linear-symmetricspring..........................................60
4.3.2 Nonlinear-asymmetricspring...................................62
4.4 Appendix:Linearsimultaneousequations............................63
4.5 Problems...............................................................................64
Chapter5 BifurcationTheoryI:Basics.........................................................67
5.1 Summary...............................................................................67
5.2 Simpleexampleofbifurcationequation...............................68
5.3 Derivationofbifurcationequation........................................71
5.4 Classificationofsimplecriticalpoints..................................75
5.4.1 Maximalandminimalpointsofload.......................76
5.4.2 Asymmetricbifurcationpoint..................................77
5.4.3 Symmetricbifurcationpoint....................................79
5.5 Directionofbifurcatingpaths...............................................81
5.6 Structuralexamplesofthreekindsofbifurcations...............82
5.6.1 Unstable–symmetricbifurcation..............................84
5.6.2 Stable–symmetricbifurcation..................................85
5.6.3 Asymmetricbifurcation...........................................87
5.7 Problems...............................................................................89
Chapter6 BifurcationTheoryII:SymmetricStructures................................91
6.1 Summary...............................................................................91
6.2 Bifurcationduetoreflectionsymmetry................................91
6.2.1 Proppedcantilever....................................................92
6.2.2 Two-bartrussarchwithbilateralsymmetry............93
6.3 Basicsofgroups....................................................................95
6.3.1 Groupsandsubgroups..............................................96
6.3.2 Dihedralandcyclicgroups......................................96
6.3.3 Trussdomewithregular-triangularsymmetry.........97
6.4 Group-theoreticbifurcationtheory.......................................99
6.4.1 Generaltheory..........................................................99
6.4.2 Bifurcationofaregular-triangularsystem.............101
6.4.3 Trussdomewithregular-hexagonalsymmetry......105
6.5 Studiesofsymmetryandbifurcationofstructures.............107
Contents vii
6.5.1 Symmetryinstructuralmechanics.........................107
6.5.2 Developmentofcatastrophetheory.......................108
6.5.3 Mathematicsonsymmetryandbifurcation............109
6.6 Problems.............................................................................110
Part II Buckling of Structures
Chapter7 MemberBucklingofColumnsandBeams.................................117
7.1 Summary.............................................................................117
7.2 Beam-columnequation.......................................................117
7.2.1 Totalpotentialenergy.............................................118
7.2.2 Derivationofbeam-columnequation....................119
7.3 Beam-columnsubjectedtoaxialcompression...................122
7.3.1 Procedureofbucklinganalysis..............................123
7.3.2 Bucklingundertypicalboundaryconditions.........124
7.3.3 Effectivebucklinglength.......................................127
7.3.4 Cross-sectionalshapeandbucklingstress.............128
7.4 Beam-columnonelasticfoundation...................................131
7.5 Beam-columnsubjectedtoaxialforceanddistributed
load.....................................................................................133
7.6 Initialdeflection..................................................................134
7.7 Inelasticbuckling................................................................136
7.7.1 Elastic-perfectlyplasticbody.................................137
7.7.2 Generalinelasticbody............................................138
7.8 Appendix:Linearordinarydifferentialequations..............139
7.9 Problems.............................................................................140
Chapter8 StructuralBucklingI:Truss........................................................143
8.1 Summary.............................................................................143
8.2 Finitedisplacementanalysis...............................................144
8.2.1 Memberstiffnessequation.....................................144
8.2.2 Structuralequilibriumequation.............................147
8.2.3 Structuralexample.................................................149
8.3 Smalldisplacementanalysis...............................................151
8.3.1 Memberstiffnessequation.....................................151
8.3.2 Structuralstiffnessequation...................................153
8.3.3 Structuralexample.................................................153
8.4 Linearbucklinganalysis.....................................................155
8.4.1 Formulation............................................................155
8.4.2 Structuralexample.................................................157
8.5 Bucklingoftrussmembers.................................................159
8.6 Problems.............................................................................161
viii Contents
Chapter9 StructuralBucklingII:Frame......................................................163
9.1 Summary.............................................................................163
9.2 Stiffnessequationsofbeam-column...................................164
9.2.1 Memberstiffnessequation.....................................164
9.2.2 Structuralstiffnessmatrix......................................167
9.2.3 Structuralexampleofsmall-displacement
analysis...................................................................169
9.3 Linearbucklinganalysis:Introductoryexample................171
9.4 Formulationimplementingaxialdeformation....................173
9.4.1 Memberstiffnessmatrix........................................174
9.4.2 Proceduretoobtainaxialforces.............................177
9.5 Structuralexampleoflinearbucklinganalysis...................177
9.5.1 Definitionofvariables............................................177
9.5.2 Smalldisplacementanalysis..................................179
9.5.3 Linearbucklinganalysis........................................182
9.6 Linearbucklinganalysisintheglobalcoordinates.............185
9.6.1 Globalcoordinatesystem.......................................185
9.6.2 Structuralexample.................................................188
9.7 Problems.............................................................................191
Chapter10 AdvancedTopicsonImperfectSystems......................................195
10.1 Summary.............................................................................195
10.2 Structuralexamplewithimperfection................................195
10.3 Formulationofimperfectionsensitivity.............................197
10.3.1 Maximal/minimalpointofload.............................198
10.3.2 Symmetricbifurcationpoint..................................199
10.3.3 Structuralexample.................................................201
10.4 Worstimperfectionpattern.................................................202
10.4.1 Formulation............................................................203
10.4.2 Structuralexample.................................................203
10.5 Bucklingloadsforrandomimperfections..........................205
10.5.1 Formulation............................................................205
10.5.2 Structuralexamples................................................207
10.6 Imperfectionsensitivityofelastic–plasticplates................208
10.7 Problems.............................................................................211
Chapter11 HistoryofImperfectBuckling.....................................................213
11.1 Summary.............................................................................213
11.2 Initialpost-bucklingbehaviors...........................................213
11.3 Searchforprototypeinitialimperfections..........................214
11.4 Probabilisticscatterofbucklingloads................................215
11.5 Asymptoticmethodandplasticbifurcationofmaterials ...217
11.6 Hilltopbranchingformaterialsandstructures...................218
Contents ix
AppendixA AnswerstoProblems...................................................................229
Index......................................................................................................................259
