Table Of ContentProblems ofNonlinear Deformation
PROBLEMSOF
NONLINEAR
DEFORMATION
The Continuation Method
Applied to Nonlinear
Problems in Solid Mechanics
by
E. I. GRIGOLY UK
Auto-Mechanical Institute, Moscow, U.S.S.R.
and
V.1. SHALASHILIN
Aviation Institute, Moscow, U.S.S.R.
"
SPRINGER-SCIENCE+BUSINESS MEDIA, B.V
Library of Congress Cataloging-in-Publication Data
Gr 1 90 l,-uk. Eduard Ivanov 1 ch.
lPrcblemy n811n81n090 aefo:-mlrovanl1a. Engl1shj
PrOblems of nonllnear deformat lan the contlnuatlon method
applled ta nonl1near problems 1n solid mechanlcs E.:. Grlgo1yuk
and V.I. Shalashll1n.
p. cm.
Translat lan of Problemy ne11n81nogo deformlrovanl1â.
Includes blbllographlcal references ana lndex.
ISBN 978-94-010-5681-6 ISBN 978-94-01 1-3776-8 (eBook)
DOI 10.1007/978-94-011-3776-8
1. Contlnuatlon methods. 2. Deformatlons (Mechanlcs)
3. NonJlnear theorl'2S. !. Shalashll1n. V. I. (VladJmlr Ivanovlchl
II. Title.
QA377.G7513 1991
531 .38--dc20 90-45736
ISBN 978-94-010-5681-6
Translated [rom Problemy Nelineynogo Deformirovaniya
(Nauka, Moscow, 1988) by M. P. Konyaeva
Printed on acid-free paper
All Rights Reserved
© 1991 Springer Science+Business Media Dordrecht
Originally published by Klu"er Academic Publishers in 1991
Softcover reprint ofthe hardcover Ist edition 1991
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner.
Contents
Preface VII
Introduction
B.1. Two Forms of the Method of Continuation of the Solution with
Respect to a Parameter 6
B.2. The Problem of Choosing the Continuation Parameter and Its
Relation to the Behaviour of the Solution in the Neighbourhood
ofSingular Points 12
Chapter 1. Generalized Forms ofthe Continuation Method 20
1.1. Generalized Forms ofContinuous Continuation ofthe Solution 20
1.2. Generalized Forms ofDiscrete Continuation ofthe Solution 31
1.3. Examples ofApplying Different Formsofthe Continuation Method 41
1.4. Optimum and Near-Optimum Continuation Parameters 52
1.5. Formsofthe Continuation Method with PartialOptimizationofthe
Continuation Parameter 60
Chapter 2. Continuation ofthe Solution Near Singular Points 65
2.1. Classification ofSingular Points 65
2.2. The Simplest Form ofBifurcation Equations 70
2.3. The Simplest Case ofBranching (rank(fO) = m- 1) 77
2.4. The Case ofBranching When rank(fO) = m- 2 79
Chapter 3. The Continuation Method for Nonlinear Boundary
Value Problems 86
3.1. Continuous Continuation ofthe Solution in Nonlinear One-
Dimensional Boundary Value Problems 87
3.2. DiscreteContinuationoftheSolutionin NonlinearOne-Dimensional
Boundary Value Problems 91
3.3. The Discrete Orthogonal Shooting Method 96
v
VI Contents
3.4. Algorithms for Continuous and Discrete Continuation ofthe
Solution with Respect to a Parameter for Nonlinear One-
Dimensional Boundary Value Problems 104
Chapter 4. Large Deflections of Arches and Shells ofRevolution 114
4.1. Large Elastic Deflections ofPlane Arches in Their Plane 114
4.2. Stability ofan Inextensible Circular Arch under Uniform Pressure 121
4.3. Algorithms for the Method ofContinuation of the Solution with
Respect to a Parameter for Large Deflectionsofa Circular Arch 125
4.4. Large Deflections ofa Circular Arch Interacting with a Rigid
Half-Plane 134
4.5. Equations for Large Axisymmetric DeflectionsofShells ofRevolution 143
4.6. Toroidal Shell ofCircular Section under Uniform External Pressure 150
Chapter 5. Eigenvalue Problems for Plates and Shells 160
5.1. General Formulation of the Continuation Method In Eigenvalue
P~~ms 100
5.2. Natural Vibrations ofa Parallelogram Membrane 167'
5.3. Natural Vibrations ofa Trapezoidal Membrane 177
5.4. Eigenvalue Problems for Homogeneous and Sandwich Plates and
Spherical Panels ofParallelogram and Trapezoidal Form in Plan.
Membrane Analogy 181
5.5. Solution for a Parallelogram Membrane by the Perturbation Method 183
Appendix I. A Survey ofLiterature on the Use ofthe Continu
ation Method for Nonlinear Problems in the
Mechanics ofDeformable Solids 193
1.1. General Formulation ofthe Continuation Method 193
1.2. Continuation ofthe Solution in the Neighbourhood ofSingular
Points and the Problem ofChoosing the Continuation Parameter 196
1.3. Different Forms ofthe Continuation Method 199
1.4. Application to Geometrically Nonlinear Systems 203
1.5. The Use ofthe Continuation Method in Conjunction with the
Finite Element Method 207
1.6. The Continuation Method in Physically Nonlinear Problems 211
1.7. A Comparison ofthe Different Formsofthe Continuation Method 212
Appendix II. A BriefSummary ofthe Notation and Basic Defi-
nitions in the Algebra ofVector Spaces 215
Bibliography 220
Author's index 257
Preface
Interestin nonlinear problemsin mechanics hasbeen revived and intensified by the
capacityofdigitalcomputers. Consequently, aquestion offundamental importance
is the development ofsolution procedures which can be applied to a large class of
problems. Nonlinear problems with a parameter constitute one such class. An
important aspect of these problems is, as a rule, a question of the variation of
the solution when the parameter is varied. Hence, the method ofcontinuing the
solutionwith respect to aparameterisa natural and, to acertain degree, universal
tool for analysis. This book includes details ofpractical problems and the results
of applying this method to a certain class of nonlinear problems in the field of
deformable solid mechanics.
In the Introduction, two forms ofthe method are presented, namely continu
ous continuation, based on the integration of a Cauchy problem with respect to
a parameter using explicit schemes, and discrete continuation, implementing step
wise processes with respect to a parameter with the iterative improvement ofthe
solution at each step. Difficulties which arise in continuing the solution in the
neighbourhood ofsingular points are discussed and the problem of choosing the
continuation parameter is formulated.
In Chapter I, generalized forms of the method are constructed which provide
a unified process for continuing the solution at regular and limit points ofthe set
ofsolutions. It is shown that the problem ofchoosing the parameter is connected
with the solution oflinearized systems ofequations by the traditional elimination
method and that this problem does not arise ifthe orthogonalization procedure is
used instead. It is also shown how the continuation process can be constructed
so that the linearized systems will be best conditioned, and how the optimum (in
this sense) continuation parameter can be chosen. Also, examples are considered
illustratingthe application ofthe method tomodel problemssuch as ashallowarch
and a three-bar truss.
Chapter 2 is devoted to analyzing the behaviour of the solution in the neigh
bourhood of singular points by expanding the solution in a Taylor series of the
generalized parameter in the neighbourhood ofsingular points. Thesimplest form
vii
Vlll Preface
ofthe bifurcation equation is derived, and thesimplest caseofbranchingis consid
ered when it happens in a two-dimensional subspace ofthe variableand parameter
space. The analysis ofbranchingin a three-dimensional space is also carried out.
In Chapter3, nonlinear boundary value problemsfor ordinarydifferentialequa
tions are considered. The use oftheshootingmethod for thesolution oflinearized
boundary value problems enables one to map the functional set ofsolutions of a
nonlinear problem onto a curve in a vector space of small dimension, which in
turn permits a generalization of the methods discussed in Chapter I to nonlinear
boundary value problems.
In Chapter 4, the methods ofChapter 3are used for constructing solutions in
large deflection problems for arches and in large axisymmetric deflection problems
for shellsofrevolution. In particular, the results ofnumerical analysisare given for
circular arches and circular toroidal panels.
Chapter 5 presents an example ofapplying the method ofcontinuation of the
solution with respect to a parameter to stability and vibration problems for plates
and shells ofnoncanonical shape in plan, whose deviation from the canonical form
(rectangle, circle, etc.) is determined by a certain parameter. The problem is
illustrated by considering vibrations of parallelogram or trapezoidal membranes.
By using the membraneanalogy, the results areextended tovibration and stability
problemsforflat andshallowsphericalpanels. The mostcommonly used method in
this kind ofproblemis the perturbation method. Acomparison is made, therefore,
between the perturbation method and the method ofcontinuing the solution with
respect to a parameter.
Appendix I provides a literature survey ofinvestigations in which the contin
uation method has been directly used in the solution of nonlinear problems for
deformable solid mechanics as well as listing publications where the stepwise pro
cesses have been used for constructing solutions which are related to one or other
forms ofthis method.
Inchoosingthedegreeofgenerality inpresentingthe material, theauthorshave
restricted themselves to that which requires the reader's knowledgeofmathematics
to be at the level offirst year graduate student. Some additional information on
the algebra ofvector spaces, a necessary prerequisite, isgiven in Appendix II.
The book summarizes the authors' publicationsfrom 1976. The actual writing
ofthe book has been allocated as follows: V.1. Shalashilin has written Chapters I,
3, and 5, the remaining chapters were written by the authors together.
The authors are grateful to Prof. 1.1. Vorovich, Corresponding Member of
the Academy of Sciences of the U.S.S.R., and to Prof. A.S. Kravchuk for taking
the trouble to read the manuscript; their comments have been included in the
final version of the book and have undoubtedly contributed to its improvement.
The authors are also thankful to A.Ya. Sorodin for his help in the numerical
computations.
Introduction
Many problems in the mechanics ofdeformable solids reduce or can be reduced to
thesolutionofsystemsofnonlinearalgebraic, transcendental,differentialorintegral
equationscontaining an explicit parameter. Theseare problemssuch asstatic non
linear deformation, stability, optimization, and others. A parameter appearing in
these nonlinearequationsmay bea load parameter, a temperaturefield parameter,
a geometric or astructural parameter, etc.
Historically, thefirst problemofthiskind originated withJ. Bernoulli,1.Euler
and J.L. Lagrange who studied the problem of deformation of slender bars (the
problem of elastica). This is an example of a geometrically nonlinear problem
which reduces to the boundary value problemfor a nonlinear differentialequation.
The difficulties arising in geometrically nonlinearsystems are apparent even in
the analysis of the deformation process for the simplest two-bar system, namely
the Mises truss, shown in Fig. B.1. Theequilibriumequation for thejoint Aofthe
truss is ofthe form
P
N= ---, (a)
2cosa
where N are the compressiveforces on the bars, and a is the angle betweenthebar
and the vertical. Ifa in the undeformed state is ao, theshorteningofthe bars is
ill= _a a_. (b)
sinao sina
Providingthe deformationshorteningofthebarsis within Hooke'slaw, the relation
between the forces N and the shorteningofthe bars is linear
o
ill= Nl (c)
EF'
where F and10 are, respectively, thecross-sectionalareaand theundeformedlength
ofthe bar, and E is the tensileelasticmodulusofthe material. Forthe Mises truss,
the most useful information about its properties is provided by a diagram relating
2 Introduction
Figure B.I Figure B.2
the force P to the vertical displacement v of thejoint A as shown in Fig. B.2. It
can be derived by adding to equations (a)-(c) the relation between the angle0' and
the displacement v
V = a-t-g"-0-'--=-t-g-"0'':0" (d)
tg0" tg0'0
Solvingall these equations, it is not difficult to obtain the relation P(v) in analytic
form
However, the analytic form of the solution can only be obtained in exceptional
cases. Ifimperfect bars are considered in this problem and their buckling is taken
into account, the analytic solution becomes impossible. In this case relation (c) is
replaced by
Nlo 11"2 (2 2)
Lli = EF +41 W - Wo ,
0
(e)
Wo
W = 1-N/N*'
Hereit isassumed that the initialshapeofimperfect bars isa half-waveofthesine
curve
. 1I"X
Yo(x) = WOSIll-,
10
where x and yare the coordinatesalong the barand alongthenormal toit, respec
tively, Wo is the amplitude ofthe initial imperfection. The deformed shape of the
bar is also considered as a half-waveofa sine curve
Y(X) = WSI.ll T1Ic"X;"'
Introduction 3
[1'-.-----,,~---..-..::._r_::_
, U
Figure B.3 Figure B.4
N is the criticalforce for the bar. (Relations(e) are derived and discussedin detail
in Section 1.3.)
The diagramP(v) becomes considerably more complicated and takes the form
shownin Fig. B.3. The curve 0-2-3-4-6correspondsto thedeformationofthe truss
with perfectlystraight bars, without takingintoaccount their lossofstability. The
curve0-1-3-5-6shows thedeformationofthesametrussbut withthelossofstability
ofthe bars taken into account (portion 1-3-5). The curve 0-7-3-8-6 demonstrates
the deformation ofthe truss with imperfect bars.
The equations (a), (b), (d), (e) represent a system offive algebraic and tran
scendentalequationswithsixunknownquantities,P,N,.6.1,Q,v,w. Systemsofsuch
equations cannot, as a rule, be solved analytically, so they are solved numerically.
In this case, a series of values ofone of the unknowns are prescribed, taking it as
a parameter, and the remaining unknowns are found. As is seen even from this
simple example, it is preferable to choose the displacement vasa parameter since
thesystemhas threesolutionsfor certain valuesofP. Thelatter circumstanceadds
considerably to the complexity ofthe numerical solution process.
Stillgreater difficulties arise in thesolutionofproblemsdealing with the defor
mationofsystemssuchasshallowarches andshallowshells. In particular,asshown
in (147],for ashallowarch (Fig. B.4) the relation betweentheload qand thethrust
N is ofthe form ofaloop-shaped curve (Fig. B.5). The equation relating qand N
is given in Section 1.3.
~"
'. \ \ \. I
Figure B.5 Figure B.6
