Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECTURES No. 238
STABILITY OF
ELASTIC STRUCTURES
EDITED BY
H. LEIPHOLZ
UNIVERSITY OF WATERLOO
SPRINGER-VERLAG WIEN GMBH
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©1978 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien-New York in 1978
ISBN 978-3-211-81473-4 ISBN 978-3-7091-2975-3 (eBook)
DOI 10.1007/978-3-7091-2975-3
PREFACE
This monograph is based on the notes of a lecture series on Stability
of Elastic Structures given at the International Centre for Mechanical Sciences
(CISM) In Udine, Italy, from October 4 to October 13, 1976 by Professors
H. H. E. Leipholz, K. Huseyin, both from the University of Waterloo, Ontario,
Canada, and M. Zyczkowski from the Technical University of Cracow, Poland.
The main objective of the lecture series was to report on the developments in
the domain of the stability of elastic structures-which have occurred during
the last three decades, and to make the audience familiar with a number of
new problems and methods for their solution as these are typical for the
modern perception of stability.
Actually, there were new problems, posed to structural engineers in
the age of nuclear technology and spacemechanics, that enforced the revision
of the classical theory of stability of structures. The existence of
deflection depending, so called follower forces had to be acknowledged, and
it had to be admitted that the transition of a struc~ure from a stable to an.
unstable state is essentially a dynamic process. Only in exceptional cases,
which, however, were those mostly considered in Civil Engineering practice in
the past, a static approach to stability problems is possible. In general,
only the dynamic approach is powerful enough to reveal the stability behaviour
of a structure to the full extent. This is the case, because the flutter
phenomenon is not only restricted to structures subjected to time depending
loading but may also be observed for structures subjected to follower loads
which are time independent. In the light of these facts, it became clear
that attempts had to be made to view the stability of elastic structures in
the broader context of dynamical systems. As a consequence, methods developed
in other areas of mechanics and engineering science had to be made suitable
i
II
for an application to structures. This was specifically true for Liapunov's
Second Method which is largely applied in Mechanical Engineering and Control
Theory for the treatment of stability problems of discrete systems. However,
structures are essentially continuous systems (if not approximately discretized).
Therefore, Liapunov's method had to be reformulated and extended for an
application to continuous systems. This specific subject will be treated
thoroughly in this treatise.
The monograph consists of three parts. Each part is self-contained,
but deep-rooted connections between the various parts can easily be detected.
The first part deals with basic concepts, definitions, and criteria. The
stability problem is mathematically formulated, and methods for the solution
are presented and discussed. The second part emphasizes the multiple
parameter aspect of stability. Already by using the dynamic approach, two
parameters, namely frequency and load, are involved. But, there may be
several load parameters instead of a single one and in addition imperfection
parameters may be contained in the problems concerned. As a consequence, the
investigations are carried out in an n-dimensional space. Moreover, the
algebraic method is being used, assuming that the problems are posed in terms
of lumped-mass systems rather than in terms of continuous systems. The third
part deals with some special and practical aspects of non-conservative systems
subjected to follower forces which are of interest to the designer. Influence
of damping, optimal design of columns and shells, and the investigation of
the postbuckling behaviour of imperfect structures may be mentioned as some of
the topics considered in that part of the monograph.
It is my pleasure to thank CISM for the support and hospitality which
the three authors received during their stay at Udine. We owe special thanks
to Professor W. Olszak, who suggested the course on Stability of Elastic
III
Structures and encouraged us to write this monograph. It is dedicated to all
those devoted to problems of mechanics in connection with structural engineering
and over all to those interested in that fascinating and practically so
important subject of stability.
Waterloo, Cracow, Spring 1978
H. H. E. Leipholz
K. Huseyin
M. Zyczkowski
CONTENTS
PART I by H. H. E. Leipholz, University of Waterloo
1. Introduction 1
1.1 Basic concepts 1
1.2 Differential geometric aspects 4
1.3 Stability definitions, topological aspects 6
2. The Mathematical Formulation of the Stability Problem 9
2.1 Equation of motion 10
2.2 Variational equation, the fundamental problem 22
3. Approaches to the Solution 36
3.1 Liapunov's approach 36
3.2 Energy approach 61
3.3 Modal approach, descretization, algebraization 65
4. Conclusion 87
Figures 89
References 97
PART II by K. Huseyin, University of Waterloo
Introduction 100
1. Stability of Gradient Systems 100
1.1 Introductory remarks 100
1.2 Classification of critical conditions 102
1.3 Equilibrium surface 106
1.4 Stability boundary 109
1. 5 Geometry of the equilibrium surface and connec.tion with the
catastrophe theory 112
1. 6 Examples 116
2. Stability of Autonomous Systems 121
2.1 Definitions 121
2.2 Classification of systems 126
2.3 Cons·ervative systems 128
2.4 Pseudo-conservative systems 134
2.5 Gyroscopic systems 1~ 1
2.6 Circulatory systems 156
Figures 161
References 179
VI
PART III by M. Zyczkowski, Technical University of Cracow, Poland
1. Influence of the Behaviour of Loading on Its Critical Value 181
1.1 Introduction 181
1.2 Derivation of the equation detennining the exact value of the
critical force 183
1. 3 Analysis of the stability curve 189
Figures 198
References 200
2. Influence of Simultaneous Internal and External Damping on the
Stability of Non-Conservative Systems 201
2.1 Introduction 201
2.2 Analysis of Ziegler's model 201
2.3 Particular cases 206
Figures 207
References 209
3. Interaction Curves in Non-Conservative Problems of Elastic
Stability 211
3.1 Introduction 211
3.2 Assumptions 212
3.3 Static criterion of stability 213
3.4 Energy method 215
Figures 222
References 224
4. Optimal Design of Elastic Columns Subject to the General
Conservative and Non-Conservative Behaviour of Loading 225
4.1 Introduction 225
4.2 Statement of the problem in the general conservative case 225
4.3 General considerations 229
4.4 Particular solutions 233
4.5 Optimization of an elastic bar compressed by antitangential
forces 236
Figures 238
References 241
VII
s. Investigation of Postbuckling Behaviour of Imperfect
Cylindrical Shells by Means of Generalized Power Series 243
5.1 Introduction 243
5.2 Non-linear problem of stability of a circular cylindrical
shell subject to hydrostatic loading 244
5.3 Elimination of the parameter d for moderate-length shells 247
5.4 Pressure in terms of the deflection amplitude 251
5.5 Upper critical pressure 252
5.6 Lower critical pressure 258
5.7 Summary of results written in physical quantities 263
References 265
6. Optimal Design of Shells with Respect to Their Elastic
Stability 268
6.1 Introduction 268
6.2 Parametric optimal design of a spherical panel 269
6.3 Optimal design of a cylindrical shell under pure bending 272
ass
Figures
References 290
PART I
by H.H.E. Leipholz
University of Waterloo
Ontario Canada
1. Introduction
Concepts and quantities used in stability theory are to a large
extent not invariant. They are chosen and defined according to the
particular intent of the researcher and the purpose of his investigation. In
many cases, practical aspects of the problems involved dictate the point of
view to be adopted for the approach to stability. Therefore, a stability
theory as such does, strictly speaking, not exist. It is necessary,
before starting any stability considerations, to define clearly the
stability concepts to be used in order to avoid misunderstanding and
confusion. According to the chosen concepts and definitions, the specific
stability theory for a specifit situation is then developed. We shall
proceed according to these guidelines in the following.
1.1 Basic concepts
The behaviour of an elastic system is described, by one or more
characteristics which exhibit at the onset of instability a certain
property suitable for the formulation of a stability criterion. In a
general state, the system possesses a degree of stability which is the
norm of the perturbation (magnitude) necessary to drive the system to the
stability boundary and slightly beyond it. In addition, the system
depends on a variety of parameters, e.g. load parameters, structural
parameters and possibly the time, which may act as a parameter by, for
example, effecting the elastic properties of the system through aging.
The parameters control the system's behaviour as the degree of stability
depends on the parameter values. At critical values, the degree of stability
