Stability of Elastic Structures

343 Pages·2000·8.431 MB·English

by  N. A. Alfutov (auth.)

Foundations of Engineering Mechanics
N.A. Alfutov, Stability of Elastic Structures
Springer-Verlag Berlin Heidelberg GmbH
ONLINE LIBRARY
Engineering
http://www.springer.de/engine/
N. A. Alfutov
Stability
of Elastic Structures
Translated by E. Evseev and Y.B. Balmont
With 128 Figures
, Springer
Series Editors:
V. I. Babitsky, DSc J. Wittenburg
Loughborough University Universitat Karlsruhe (TH)
Department of Mechanical Engineering Institut fur Mechanik
LE11 3TU Loughborough, Leicestershire Kaiserstrafie 12
United Kingdom D-76128 Karlsruhe I Germany
Author:
N.A. Alfutov
Moscow State University of Technology
2-nd Baumanskaya Str. 5
M-1 Department
107005 Moscow I Russia
Translators:
Evgeny Evseev
Professor Vladimir B. Balmont
Ap.34, 50 Skhodnenskaya Street
Library of Congress Cataloging-in-Publication Data
Alfutov, N.A. (Nikolai Anatol'evich)
Stabiltiy of elastic structures I N. A. Alfutov; translated by E. Evseev and V:B. Balmont
(Foundations of engineering mechanics)
Includes bibliographical references and index.
I. Structural stability. 2. Thin-walled structures. 3. Elastic analysis
I. Title. II. Series
TA656.A62 2000
624.1'7--dc21
ISBN 978-3-642-08498-0 ISBN 978-3-540-49098-2 (eBook)
DOI 10.1007/978-3-540-49098-2
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Preface
Improvements of the strength characteristics of traditional structural materi
als and the development of new, high-strength composite materials have en
abled the wide application of light, sophisticated, efficient thin-walled struc
tures in modern mechanical engineering. The role of stability estimations of
these structures has significantly increased, as failure of a thin-walled struc
ture is most often caused by its total stability loss or by the loss of stability
of its elements. The number of publications devoted to general approaches to
and methods of investigation of thin-walled structures, as well as to particular
problems of estimation of the stability of slender columns, column systems,
stiffened plates, sandwich structures, etc., has radically increased.
Different numerical methods for the investigation of structural stabil
ity have been developed. However, in the traditional curricula of most
mechanical-engineering courses, the problems of structural stability have not
been sufficiently reflected. An engineer is often introduced to this analysis
only within a general course on strength of materials. Thus, a decision was
made to write a book which facilitates for an engineer the transition from
general educational courses to reading and understanding the specialist lit
erature on stability of thin-walled load-carrying structures.
During recent decades, in stability theory, probably more than in any
other area of mechanics, contradictory concepts have been introduced, ques
tionable points of view have been discussed, and even incorrect recommenda
tions have been given. It is worth mentioning the so-called low-critical-load
method for structural stability analysis, which was propagated by many spe
cialists and even presented in some reference books. Only recently, thanks
to the efforts of leading scientists, have common approaches to most of the
principal problems of stability theory been developed.
The subject discussed in this book is the stability of thin-walled elastic
systems under static loads. The presentation of these problems is based on
modern approaches to elastic-stability theory. Special attention is paid to the
formulation of elastic-stability criteria, to the statement of column, plate,
and shell stability problems, to the derivation of basic relationships, and to a
discussion of the boundaries of the application of analytic relationships. The
author has tried to avoid arcane, nonstandard problems and elaborate and
unexpected solutions, which bring real pleasure to connoisseurs, but confuse
VI
students and cause bewilderment to some practical engineers. The author
has an apprehension that problems which, though interesting, are limited in
application can divert the reader's attention from the more prosaic but no less
sophisticated general problems of stability theory. The author has tried to
make clear the derivation of each relationship to even an unprepared reader.
From a variety of stability problems of thin-walled structures, some principal
ones have been selected that show specifics of elastic-stability problems. The
author hopes that the reader will easily and deeply understand other known
stability problems, and, what is more important, will learn more quickly how
to state and solve new problems independently.
In order to apply successfully the reference data and the available arsenal
of powerful software, an engineer definitely requires an understanding of the
physics of stability loss of thin-walled structures. He or she also needs to have
a clear idea about the simplifying assumptions and hypotheses, application of
which allows him or her to make mathematical formulations of real problems
and to get reliable results. A presentation of these problems is the main
purpose of this book.
Moscow, August 1999 Nikolai Alfutov
Table of Contents
1. Basic Theory of Elastic Stability. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Equilibrium Paths for Deformed Systems. . . . . . . . . . . . . . . . . . 2
1.2 Stable and Unstable Equilibrium States. . . . . . . . . . . . . . . . . . . 6
1.3 Bifurcation Points, Limit Points, and Critical Loads ........ 12
1.4 Energy Criterion for Bifurcational Stability Loss. . . . . . . . . . .. 15
1.5 Homogeneous Linearized Equations. . . . . . . . . . . . . . . . . . . . . .. 23
1.6 Supercritical Behavior of Elastic Structures . . . . . . . . . . . . . . .. 27
1. 7 Stability of Elastic Structures Under
Combined Loading: Boundary of Stability Region. . . . . . . . . .. 31
1.8 On the Statement of Stability Problems
for Thin-Walled Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37
2. Energy Method
for the Solution of Stability ProbleITls .................... 45
2.1 Principle of Virtual Displacements. . . . . . . . . . . . . . . . . . . . . . .. 45
2.2 Variational Approaches in the Linear Theory
of Elasticity ........................................... 52
2.3 Two Basic Forms of the Energy Criterion
for Bifurcational Stability Loss . . . . . . . . . . . . . . . . . . . . . . . . . .. 56
2.4 Energy Criterion in Bryan Form. . . . . . . . . . . . . . . . . . . . . . . . .. 60
2.5 Energy Criterion in Timoshenko Form .................... 67
2.6 Rayleigh-Ritz Method in Stability Analysis. . . . . . . . . . . . . . .. 72
2.7 The Galerkin Method and its Relationship
to the Rayleigh-Ritz Method ............................ 77
3. Stability of Straight ColuITlns . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85
3.1 Statement of the Problem:
Basic Linearized Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85
3.2 Examples of the Analytic Solution of the Basic Equation . . .. 94
3.3 Columns on Elastic Foundations and Elastic Supports ....... 103
3.4 Stability of Self-Gravitating Column ...................... 110
3.5 Lateral Torsional Beam Buckling ......................... 121
3.6 The Influence of Transverse Shear Strains:
Stability of Sandwich Struts ............................. 129
VIII Table of Contents
3.7 Method of Initial Parameters in Stability Analysis .......... 135
4. Stability of Plates ........................................ 143
4.1 Statement of the Problem:
Basic Initial Relations ................................... 143
4.2 Basic Linearized Equation ............................... 152
4.3 Solution of Basic Equation for a Rectangular Plate ......... 161
4.4 Solution of Basic Equation for a Circular Plate ............. 173
4.5 Approximate Solutions of the Basic Linearized Equation ..... 178
5. Energy Method for the Study of the Stability of Plates ... 189
5.1 Energy Method for the Problem of Plate Bending:
Accounting for Shears. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
5.2 Application of the Energy Criterion in Bryan Form ......... 194
5.3 The Energy Criterion in Timoshenko Form:
Thermoelasticity Problem of Plate Stability ................ 198
5.4 Stability Criterion Statement
via Statically Admissible Initial Internal Forces ............. 203
5.5 Examples of Applications of the Energy Method:
Influence of Transverse Shear ............................ 206
5.6 Stability of Plates under Local Loads ..................... 216
6. Stability of Shells ......................................... 221
6.1 Stability of Circular Ring ................................ 221
6.2 Basic Initial Relations for a Cylindrical Shell ............... 239
6.3 Stability of Cylindrical Shell
Subjected to Axial Compression .......................... 252
6.4 Determination of Critical Value of External Pressure ........ 258
6.5 Stability of Cylindrical Shell
Under Torsion and Transverse Bending .................... 266
6.6 Stability of a Shell Stiffened by Elastic Frames ............. 271
6.7 Determination of Critical Loads
Using the Stability Criterion in Timoshenko Form .......... 278
7. Nonlinear Problems: Stability of Real Bars,
Plates, and Shells ......................................... 287
7.1 Deformation of Compressed Bar After Stability Loss ........ 287
7.2 Supercritical Behavior of Elastic Plates .................... 294
7.3 Nonlinear Approaches in Stability Problems
for Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
7.4 Initial Imperfections in Stability Problems
for Bars and Plates ..................................... 308
7.5 Initial Imperfections in Stability Problems for Shells ........ 315
Table of Contents IX
A. Appendix ................................................. 321
A.l Eigenvalue Problems .................................... 321
A.2 Stationary Values and Extrema of Functions
and Functionals ........................................ 324
References ................................................ 331
Index ..................................................... 335
1. Basic Theory of Elastic Stability
The stability loss of any deformed system is a process that proceeds in
time and, naturally, it should be studied in terms of dynamics. Hence, the
general theorems and methods of stability investigation are based on a dy
namic approach, on a study of the system behavior in time (Bolotin (1963),
Ziegler (1968), Leipholz (1980)). However, in practice a static approach can
be applied for most load-carrying structures. In this case, the equilibrium
conditions at stability loss can be formulated without taking into account
the inertial forces caused by the deformations of the structure.
In this book we shall confine our studies to conservative systems. In this
case a static approach brings us to the same results as does a much more
complicated dynamic approach (Ziegler (1968)). A system is conservative if it
consists of an elastic body and ideal constraints, and is loaded by conservative
forces. Forces are conservative if they have a potential; the work done by such
forces depends only on the initial and final configurations of the body they
are applied to, and does not depend on the path of the transition from one
configuration to another. Constraints are ideal if their reaction forces do not
do any work through any virtual displacement of the system points they are
applied to.
In order to clarify the typical peculiarities of elastic-stability problems, it
is not necessary to analyze complicated mechanical systems. We can do this
within a study of the simplest mechanical systems, which are described by
elementary analytical formulas.
In this chapter we shall introduce and illustrate the basic ideas and main
definitions of the theory of elastic stability (bifurcation point, critical load,
linearized equation, stability region boundary, and energy criterion of stabil
ity) using examples of elastic systems with one or two degrees of freedom,
much as is usually done in the theory of mechanical oscillations. Moreover, in
this chapter we shall consider the restrictions and assumptions that are usu
ally applied when stating and solving the stability problems for thin-walled
load-carrying structures.
N. A. Alfutov, Stability of Elastic Structures
© Springer-Verlag Berlin Heidelberg 2000