Table Of ContentDevelopments in Civil Engineering
Vol. 1 The Dynamics of Explosion and its Use (Henrych)
Vol. 2 The Dynamics of Arches and Frames (Henrych)
Vol. 3 Concrete Strength and Strains (Avrametal.)
Vol. 4 Structural Safety and Reliability (Moan and Shinozuka, Editors)
Vol. 5 Plastics in Material and Structural Engineering (Bares, Editor)
Vol. 6 Autoclaved Aerated Concrete, Moisture and Properties (Wittmann, Editor)
Vol. 7 Fracture Mechanics of Concrete (Wittmann, Editor)
Vol. 8 Manual of Surface Drainage Engineering, Volume II (Kinori and Mevorach)
Vol. 9 Space Structures (Avram and Anastasescu)
Vol. 10 Analysis and Design of Space Frames by the Continuum Method (Kollar and Hegedus)
Vol. 11 Structural Dynamics (Vertes)
Vol. 12 The Selection of Load-Bearing Stuctures for Buildings (Horvath)
Vol. 13 Dynamic Behaviour of Concrete Structures (Tilly, Editor)
Vol. 14 Shells, Membranes and Space Frames (Heki, Editor)
Vol. 15 The Time Factor in Transportation Processes (Tarski)
Vol. 16 Analysis of Dynamic Effects on Engineering Structures (Bata and Plachy)
Vol. 17 Post-Buckling of Elastic Structures (Szabo, Gaspar and Tarnai, Editors)
Vol. 18 Fracture Toughness and Fracture Energy of Concrete (Wittmann, Editor)
Vol.19 Pavement Analysis (Ullidtz)
Vol. 20 Analysis of Skeletal Structural Systems in the Elastic and Elastic-Plastic Range (Borkowski)
Vol. 21 Creep and Shrinkage of Concrete Elements and Structures (Smerda and Kf istek)
Vol. 22 Theory and Calculation of Frame Structures with Stiffening Walls (Pubal)
Vol. 23 Time Effects in Concrete Structures (Gilbert)
Vol. 24 Stresses in Layered Shells of Revolution (Kovafik)
Vol. 25 River Intakes and Diversion Dams (Razvan)
Vol. 26 Analysis of Dimensional Accuracy of Building Structures (Vorlicek and Holicky)
Vol. 27 Reinforced-Concrete Slab-Column Structures (Ajdukiewicz and Starosolski)
Vol. 28 Finite Models and Methods of Dynamics in Structures (Henrych)
Vol. 29 Endurance of Mechanical Structures (Nemec and Drexler)
Vol. 30 Shells of Revolution (Mazurkiewicz and Nagorski)
Vol. 31 Structural Load Modeling and Combination for Performance and Safety Evalution (Wen)
Vol. 32 Advanced Analysis and Design of Plated Structures (Kfistek and Skaloud)
Vol. 33 Regular Lattice Plates and Shells (Sumec)
Vol. 34 Combined Ultrasound Methods of Concrete Testing (Galan)
Vol. 35 Steel-Concrete Structures for Multistorey Buildings (Kozak)
Vol. 36 Analytical Methods in Bin-Load Analysis (Drescher)
Vol. 37 Design of Welded Tubular Connections - Basis and Use of A WS Code Provisions (Marshall)
Vol. 38 Fresh Concrete Properties and Tests (Bartos)
STABILITY, BIFURCATION
AND POSTCRITICAL
BEHAVIOUR OF
ELASTIC STRUCTURES
M. PIGNATARO
N. RIZZI
A. LUONGO
Department of Structural and Geotechnical Engineering,
University of Rome 'La Sapienza\ Italy
ELSEVIER
Amsterdam - London - New York -Tokyo
1991
ELSEVIER SCIENCE PUBLISHERS B.V.
Sara Burgcrhartstraat 25
P.O. Box 211, 1000 AE Amsterdam, The Netherlands
Distributors for the United States and Canada:
ELSEVIER SCIENCE PUBLISHING COMPANY INC.
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New York, NY 10010, U.S.A.
Library of Congress Cataloging-in-Publication Data
ignataro, M. (Marcello)
Stability, bifurcation, and postcritical behaviour of elastic
structures / M. Pignataro, N. P.izzi, A. Luongo.
p. cm. — (Developments in civil engineering ; v. 39)
Includes bibliographical references and index.
ISBN 0-444-88140-9
1. Elastic analysis (Engineering) 2. Bifurcation theory.
I. Rizzi, N. II. Luongo, A. III. Title. IV. Series.
TA653.P54 1991
624. 1 '71—dc20 91-24932
CIP
ISBN 0-444-88140-9
® 1991 Elsevier Science Publishers B.V. All rights reserved.
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or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written
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Printed in the Netherlands
To Professor Warner T. Koiter
vii
PREFACE
The study of the stability of mechanical systems is a fascinating subject
which has stimulated the interest of mathematicians and engineers since the
time of Euler. Apart from his pioneering groundwork, which was limited to
the particular case of the critical and post-critical equilibrium of compressed
rods, all successive literature up to the 1930s attempted solely to determine
the critical loading of mechanical systems, erroneously claiming this to be an
adequate indicator for the evaluation of structural behaviour.
The analytical results obtained, which were based on linear analysis, agreed
within the errors with experimental results for beams and plates, but not for
shells, particularly those of cylindrical form, where the experimental results
showed a very great dispersion.
The work of Marguerre and Trefftz on the post-critical behaviour of plates,
and of von Karman and Tsien on that of cylinders, based on non-linear anal-
ysis, showed how the behaviour of the two structural types was qualitatively
different. Even though these works describe in detail the phenomena of buck-
ling and post-buckling, revealing the inadequacy of the critical load for eval-
uating structural behaviour, the lack of an adequate theory was reflected in
the impossibility of providing indications of a general character, such as for
post-critical behaviour and collapse loads.
Not until 1945 did Koiter, in his doctoral thesis, present a general theory
on the critical and post-critical analysis of elastic structures. As well as con-
stituting a fundamental source for a deeper understanding of buckling, this
also allows us, through the introduction of the concept of "initial imperfec-
tions", to arrive at a quantitative evaluation of the collapse load in a simple
and rational way. It also helps to explain the reason for the wide spread of
experimental results obtained for cylinders.
The work of Koiter, written in Dutch, remained virtually unknown until
1967, when it was translated into English. Meanwhile, towards the end of
the 1950s, a non-linear theory on the post-critical equilibrium of discrete sys-
tems was developing independently in England with the work of Sewell and
Thompson. All the work published in recent years presents either the analy-
sis of continuous systems, following the theory of Koiter, or that of discrete
systems, following the English school. In the present volume the authors
present a treatment, as far as possible organic, of the non-linear behaviour
of discrete and continuous systems, trying to point out the common method-
ology of investigation based on asymptotic analysis. In the light of Koiter's
theory, in fact, it is evident how such a method of investigation, despite its
"local" nature, is quite adequate for providing not only a qualitative but also
a quantitative estimate of the phenomena studied, in this connection better
viii
than costly finite analysis methods are able to do.
The majority of modern texts focus totally or mainly on the construction of
a bifurcated path, whilst in the present volume the authors attach importance
to stability analysis corresponding to the bifurcation which always precedes
the investigation of post-critical behaviour.
As stability is a dynamic concept the authors thought it right to study
it in the wider context of the theory of the stability of equilibrium, accord-
ing to Liapunov. In such a context, and like Koiter, Sewell and Thompson
limiting the study to elastic systems subjected to conservative forces, the dy-
namic criterion is reduced to the total potential energy criterion by means
of the theorems of Lagrange and Koiter for discrete and continuous systems
respectively.
It is necessary to mention that the present volume lacks a systematic
treatment of initial imperfections and of simultaneous modes which have a
marginal importance in the investigation of some simple structural models.
Furthermore, in the review of structural topologies, the study of arches has
been omitted completely; this is to limit the text, keeping it at an elementary
level. In spite of this, the authors hope that they have been able to provide
an exhaustive view of the problems of buckling.
December 1990
M. Pignataro
N. Rizzi
A. Luongo
A CKNO WLEDGEMENTS
The authors wish to thank the publishers Springer Verlag-Vienna, for au-
thorising the reproduction of Figs. 7.86 to 7.40 and 7.45 from the book 'Post-
Buckling Behaviour of Structures' by M. Esslinger and B. Geier, No. 286 of
the series CISM Courses and Lectures, 1975, and Granada Publishing Ltd for
permitting the reproduction of Figs. 7.41 to 7.48 from the book 'Thin-Walled
Structures' by J. Rhodes and A.C. Walker, London 1980, and Figs. 5.1 to 5.5
and 6.1 to 6.4 from the book 'Design for Structural Stability' by P.A. Kirby
and D.A. Nethercot, London 1979.
viii
than costly finite analysis methods are able to do.
The majority of modern texts focus totally or mainly on the construction of
a bifurcated path, whilst in the present volume the authors attach importance
to stability analysis corresponding to the bifurcation which always precedes
the investigation of post-critical behaviour.
As stability is a dynamic concept the authors thought it right to study
it in the wider context of the theory of the stability of equilibrium, accord-
ing to Liapunov. In such a context, and like Koiter, Sewell and Thompson
limiting the study to elastic systems subjected to conservative forces, the dy-
namic criterion is reduced to the total potential energy criterion by means
of the theorems of Lagrange and Koiter for discrete and continuous systems
respectively.
It is necessary to mention that the present volume lacks a systematic
treatment of initial imperfections and of simultaneous modes which have a
marginal importance in the investigation of some simple structural models.
Furthermore, in the review of structural topologies, the study of arches has
been omitted completely; this is to limit the text, keeping it at an elementary
level. In spite of this, the authors hope that they have been able to provide
an exhaustive view of the problems of buckling.
December 1990
M. Pignataro
N. Rizzi
A. Luongo
A CKNO WLEDGEMENTS
The authors wish to thank the publishers Springer Verlag-Vienna, for au-
thorising the reproduction of Figs. 7.86 to 7.40 and 7.45 from the book 'Post-
Buckling Behaviour of Structures' by M. Esslinger and B. Geier, No. 286 of
the series CISM Courses and Lectures, 1975, and Granada Publishing Ltd for
permitting the reproduction of Figs. 7.41 to 7.48 from the book 'Thin-Walled
Structures' by J. Rhodes and A.C. Walker, London 1980, and Figs. 5.1 to 5.5
and 6.1 to 6.4 from the book 'Design for Structural Stability' by P.A. Kirby
and D.A. Nethercot, London 1979.
ix
INTRODUCTION
This volume is intended for students of structural engineering, those study-
ing for a PhD degree and for young researchers interested in problems of
non-linear mechanics, and requires a knowledge of the basics of analysis and
algebra. In the Appendix the authors give a short account of aspects of the
calculus of variations not usually studied in classical courses of analysis, which
is sufficient for the study of this text.
In Chapter 1 the authors discuss the nature of the equilibrium points of
an autonomous system of differential equations, both linear and non-linear,
followed by the concept of stability according to Liapunov and Chetayev.
Then the conditions under which it is possible to analyse a non-linear problem
are discussed making reference to the corresponding linearised problem.
In Chapter 2, after a brief revision of the Lagrange and Hamilton equations
of motion, the authors apply the Liapunov definition of stability to mechan-
ical systems. This is followed by a demonstration of the Lagrange-Dirichlet
theorem and enunciation of the theorems of Liapunov and Chetayev. A dis-
cussion of the criterion of stability of discrete systems follows, providing the
conditions necessary and sufficient for the total potential energy to be positive
definite. In the second part of the chapter some simple mechanical models of
one or two degrees of freedom are examined in detail, with the aim of pre-
senting in an elementary context basic ideas such as primary and secondary
curves of equilibrium, bifurcation and limit points, the energy criterion for
stability and the influence of initial imperfections and simultaneous modes.
In Chapter 3 a local analysis of the properties of points belonging to an
equilibrium path is carried out, determining the conditions which distinguish
a regular equilibrium point from one of bifurcation and from a limit point.
Successively, the use of perturbation analysis in the asymptotic determina-
tion of equilibrium curves through a point is illustrated. The critical points
along a known equilibrium path are then looked for, and local analysis of the
bifurcated path around the bifurcation point is effected. Finally, a system
of two degrees of freedom characterised by a non-linear fundamental path is
analysed and the bifurcation point and the branched curve are determined,
critically comparing the results thus obtained with those corresponding to a
linearisation of the fundamental path.
Chapter 4 extends the results obtained in Chapters 2 and 3 for discrete
systems to continuous systems. Having enunciated the theorem of Koiter
which provides, by means of the potential energy, the sufficient condition
of stability of continuous systems, the authors go on to discuss firstly the
critical condition of equilibrium and then the conditions under which the total
potential energy is positive definite. Finally, an illustration of the construction
X
of bifurcated equilibrium paths by means of perturbation analysis is given.
The last three chapters describe applications of the general theory to par-
ticular structures such as plane frames, thin-walled beams and shells.
In Chapter 5 two models of plane beams are presented, one characterised
by three internal degrees of freedom, two displacement components and a ro-
tation of the section, the other by pure flexural deformation. For both models
the total potential energy is given for a system of beams, up to terms of the
fourth order. For the internally constrained model the energy is modified
by means of additional terms, which take account of the constraint condi-
tions by using Lagrange multipliers. By using the constrained model, various
problems affecting beams subjected to different boundary conditions and ax-
ial loads are successively resolved. Solutions to the post-critical problem are
given for some of these. The Ritz method is often used for determination of
the critical load. Using the same model, the authors examine several types
of simple frames, for which the complete analytical solution is given. At the
end of the Chapter a series of results for complex frames, obtained through a
compatible finite elements technique which makes use of the first beam model,
is presented.
In Chapter 6 Vlasov's theory on thin-walled open cross-sections, based on
the two fundamental hypotheses of the underformability of the section in its
own plane and of the absence of shear deformation in the middle surface of the
beam, is presented. The total potential energy up to the second-order terms
is then given, as this Chapter is for the most part based only on the study of
critical behaviour. The Eulerian equations of the problem are then deduced
for the most general cases of load and boundary conditions. The equations
are successively specialised to the study of a series of classical problems, such
as those of flexural-torsional instability and lateral instability of beams. The
Ritz method of discretisation (for which an application is given) and of com-
patible finite elements are illustrated. Finally, the post-critical behaviour of
thin-walled members under the hypothesis of simultaneous modes is exam-
ined. The analysis is carried out by using the Vlasov model and writing the
total potential energy up to third-order terms.
Chapter 7 concludes the volume with an examination of two-dimensional
structures. After a brief mention of kinematics in general curvilinear coordi-
nates, the authors come to the expression of the total potential energy based
on the simplified kinematic relations of shallow shells, according to Donnell-
Mushtari-Vlasov. The total potential energy of the plate is consequently
obtained through a limit process. Successively, after discussing the stability
of the post-critical behaviour of plates, the authors pass on to the application
of the bifurcation theory to a series of classical problems regarding various
boundary and load conditions. For some problems, to which no exact solu-
tion is known, Ritz's method is used. Mention is also made of the problem
XI
of stiffened plates. In the second part of the Chapter the critical behaviour
of cylinders subjected to radial, axial or hydrostatic pressure, and of spheres
under hydrostatic pressure, is studied. Indications of a qualitative character
only are given for the post-critical behaviour of shells. These are accompained
by a series of diagrams showing experimental results and of photographs of
the models.
Finally, the Appendix reports some essential notions of the calculus of vari-
ations necessary to the understanding of the analytical development reported
in the text.
While not considering the present volume to be a complete work, the
authors nevertheless hope to have stimulated the reader's interest in the study
and further investigation of the subject.
