Table Of ContentSTUDIES IN APPLIED MECHANICS
5. Mechanics of Structured Media, Parts A and Β (Selvadurai, Editor)
6. Mechanics of Material Behavior (Dvorak and Shield, Editors)
7. Mechanics of Granular Materials: New Models and Constitutive Relations (Jenkins and
Satake, Editors)
8. Probabilistic Approach to Mechanisms (Sandler)
9. Methods of Functional Analysis for Application in Solid Mechanics (Mason)
10. Boundary Integral Equation Methods in Eigenvalue Problems of Elastodynamics and Thin
Plates (Kitahara)
11. Mechanics of Material Interfaces (Selvadurai and Voyiadjis, Editors)
12. Local Effects in the Analysis of Structures (Ladeveze, Editor)
13. Ordinary Differential Equations (Kurzweil)
14. Random Vibration - Status and Recent Developments. The Stephen Harry Crandall
Festschrift (Elishakoff and Lyon, Editors)
15. Computational Methods for Predicting Material Processing Defects (Predeleanu, Editor)
16. Developments in Engineering Mechanics (Selvadurai, Editor)
17. The Mechanics of Vibrations of Cylindrical Shells (Markus)
18. Theory of Plasticity and Limit Design of Plates (Sobotka)
19. Buckling of Structures-Theory and Experiment. The Josef Singer Anniversary Volume
(Elishakoff, Babcock, Arbocz and Libai, Editors)
20. Micromechanics of Granular Materials (Satake and Jenkins, Editors)
21. Plasticity. Theory and Engineering Applications (Kaliszky)
22. Stability in the Dynamics of Metal Cutting (Chiriacescu)
23. Stress Analysis by Boundary Element Methods (Balas, Sladek and Sladek)
24. Advances in the Theory of Plates and Shells (Voyiadjis and Karamanlidis, Editors)
25. Convex Models of Uncertainty in Applied Mechanics (Ben-Haim and Elishakoff)
26. Strength of Structural Elements (Zyczkowski, Editor)
27. Mechanics (Skalmierski)
28. Foundations of Mechanics (Zorski, Editor)
29. Mechanics of Composite Materials-A Unified Micromechanical Approach (Aboudi)
30. Vibrations and Waves (Kaliski)
31. Advances in Micromechanics of Granular Materials (Shen, Satake, Mehrabadi, Chang and
Campbell, Editors)
32. New Advances in Computational Structural Mechanics (Ladeveze and Zienkiewicz, Editors)
33. Numerical Methods for Problems in Infinite Domains (Givoli)
34. Damage in Composite Materials (Voyiadjis, Editor)
35. Mechanics of Materials and Structures (Voyiadjis, Bank and Jacobs, Editors)
36. Advanced Theories of Hypoid Gears (Wang and Ghosh)
37A. Constitutive Equations for Engineering Materials
Volume 1: Elasticity and Modeling (Chen and Saleeb)
37B. Constitutive Equations for Engineering Materials
Volume 2: Plasticity and Modeling (Chen)
38. Problems of Technological Plasticity (Druyanov and Nepershin)
39. Probabilistic and Convex Modelling of Acoustically Excited
Structures (Elishakoff, Lin and Zhu)
40. Stability of Structures by Finite Element Methods
(Waszczyszyn, Cichon and Radwanska)
General Advisory Editor to this Series:
Professor Isaac Elishakoff, Center for Applied Stochastics Research, Department of Mechanical
Engineering, Florida Atlantic University, Boca Raton, FL, U.S.A.
STUDIES IN APPLIED MECHANICS 40
Stability of Structures
by Finite Element
Methods
Zenon Waszczyszyn
Czestaw Cichon
Maria Radwanska
Institute of Computer Methods in Civil Engineering
Cracow University of Technology
Cracow, Poland
ELSEVIER
Amsterdam- Lausanne-New York-Oxford-Shannon-Tokyo 1994
ELSEVIER SCIENCE B.V.
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© 1994 Elsevier Science B.V. All rights reserved.
No part of this publication may be reproduced, stored in a retrieval system, ortransmitted, in any
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No responsibility is assumed bythepublisherfor any injury and/or damageto persons or property
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This book is printed on acid-free paper.
Printed in The Netherlands
To
Our Families
vii
PREFACE
The presented book is a result of the authors' and their co-workers' research and
teaching in the field of advanced problems of structural mechanics. Stability analysis
of structures belongs to such problems because of the complex models and
computational methods needed for the analysis.
In the mid 70-ies a group of research-staff and teachers of the Department of
Civil Engineering and Computer Center of the Cracow University of Technology
started with a Seminar on Finite Element Methods. It was the beginning of joint
effort which produced many papers, computer programs, monographs and a book
in the end. The book is a more comprehensive form, but similar contents were
published in Polish1. Ad hoc translations of parts of this book have been well
appreciated by foreign students at the Cracow University of Technology (CUT).
They have also been useful in cooperation of CUT with European universities.
It is difficult to mention all our Colleagues who contributed to the creation of
the background to the book. But we have to express our deep thanks to the Authors
of the computer codes ANKA and MANKA, Authors of procedures for finite
elements and related numerical examples. That is why we thank very much Doctors
A.Mlodzianowski, E.Pabisek, D.Zbos, Mr. K.Rozycki and Mr. M.Wojtasiewicz as
the main Authors of the codes which were used to compute the majority of
numerical examples included into the book. We are grateful to our Colleagues who
contributed in writing of selected Sections of the book: Mr. A.Winnicki (Sections
5.2, 8.2), Dr. J.Pieczara (Sections 6.7, 9.5), Mr. J.Pamin (Sections 6.2, 6.6),
Mrs.M.Janus (Sections 5.5, 9.4, 9.6). We would like to express our thanks also to
Professor M.Wieczorek and Dr. J.Czmochowski for their permission to include
examples taken from their papers. The book could not be written without careful
proof-reading of the English manuscript by Mrs, E.Han-Wiercinska and word-proces
sing under WordPerfect v.5.1 by Mrs.K.Nuckowska and Miss B.Kielb as well as
by Mr. R.Przybylski. The graphics and figures were made by Mrs.M.Bacewicz and
Mr. A.Szewczyk. We would also like to thank to Professor M.Dacko and Professor
A.Gomulinski for their valuable remarks to the Polish book 1.
1 Z.Waszczyszyn, Cz.Cichon, M.Radwanska: Finite Element Method in Stability of
Structures (in Polish), Arkady, Warszawa 1990.
viii
The basic stages of the work on this book were carried out at the Institute of
Structural Mechanics of CUT in the years 1988-90 in the framework of the Polish
government-supported research program CPBP 02.01 on Mechanics of Materials,
Structures and Technological Processes and continued in the years 1991-93 due to
the support of the Committee of Scientific Research - Grant Nr 30401 91 01 on
Nonlinear Analysis of Structural Stability.
The financial support of the TEMPUS Program JEP-0438, coordinated by
Professor G.Guerlement from Faculte Polytechnique de Mons, Belgium, enabled us
to cover the expenses related to the English translation and word-processing of the
book.
The generous support of all the above persons and institutions is gratefully
acknowledged.
Finally the authors would like to express their deep thanks to Professor
D.Elishakoff. It took place in Delft, The Netherlands, in August 1992 that he
suggested to Z.Waszczyszyn starting with preparation of this book in English.
Without such an encouragement the book would exist only in a Polish, much
simpler version.
Cracow, August 1994 Zenon Waszczyszyn
Czeslaw Cichon
Maria Radwaaska
1
CHAPTER 1
INTRODUCTION
1.1. Theory of Stability and Finite Element Method
The theory of stability of structures is an important part of structural mechanics.
It is a basis for rational design of new structures and examination of the safety of
existing structures. The design corresponds in fact to the synthesis, i.e. parameters
of a structure are calculated for known input data and known response of the cf Fig.
1.1. The examination deals with the analysis of structures, i.e. for given parameters
of the structure the relation between excitation and response of the structure is
analyzed.
ANALYSIS
((llooaaddss)) ιι ((ddiissppll..))
((ssttrruuccttuurree))
input ttrraannssffoorrmmaattoorr oouuttppuutt
DESIGN
(SYNTHESIS)
Fig. 1.1 Analysis and design of structures
If during the design process the parameters of structures obey given constraints
and certain criteria are fulfilled, then the optimum design takes place.
Design problems are usually much more difficult than those problems associated
with the analysis of structures. That is why the design is generally changed into a
sequence of analysis problems with iterative improvement of structural parameters.
Both design and analysis problems are solved by means of computer methods.
Such an effective method is the Finite Elements Method (FEM).
In the book only the analysis problems in the sense of the definition given
above are considered. Optimum design problems with instability of structures taken
into account were discussed by Gajewski and Zyczkowski [1].
2 INTRODUCTION
The theory of stability deals with critical loads and deformation of structures
which are associated with sudden (quantitative) changes of the structure state
(Skalmierski [2]). In what follows the stability of motion and stability of equilibrium
state will be considered.
The actual configuration has to be taken into account in the theory of stability
of structures. This means that stability equations are actually nonlinear. They can
be linearized in case of linear buckling theory. Such a theory enables us to come
to a number of results in the frame of the so-called classical theory of structural
buckling.
The theory of stability of structures started more than 200 years ago if we
accept the paper by Euler published in 1759 on buckling of columns as the first
one. The 'classical' period of the theory reaches as far back as 30-40's of our
century. At that time both experiments and theoretical investigations on the
instability of shells revealed defects of the linear theory. Efforts on clearing up the
discrepancies between the experiments and theoretical predictions for buckling of
cylindrical shells significantly influenced the development of nonlinear theories of
the analysis of structures (Grigolyuk [3], Pignataro, Rizzi and Luongo [5]).
The beginning of the modern theory of structural stability can be related to the
Doctoral Thesis by Koiter. The Thesis was defended in Delft (The Netherlands) in
1945 but it became more commonly known after the English translation [4] was
made in 1967. The Thesis initiated theoretical investigations of postbuckling behavior
of structures in the vicinity of critical states. It started with the analysis of sensitivity
of structures to initial imperfections.
The consideration of the influence of imperfections on the response of structures
is of primary importance in the stability analysis. The next factor that has to be
considered is the behavior of loads during the deformation process of structures. It
is well known that the critical state under spatially fixed action of loads (dead loads)
can be significantly different than under follower loads. The first case corresponds
to the so-called conservative (potential) systems whereas the latter one to
nonconservative systems. In the first case a 'statical' approach can successfully be
applied to the stability analysis. In the case of nonconservative systems vibrations
of structures have to be considered to compute critical parameters of buckling loads.
The theory of structural stability needs complex models of structures. It makes
the problem of stability much more complicated, even at the level of linear analysis,
than those from linear statics or dynamics of structures. That is why in the
precomputer period only buckling of simple structures or with a small number of
degrees of freedom were analyzed.
1.1. THEORY OF STABILITY AND FINITE ELEMENT METHOD 3
The arrival of computers and development of computational methods enable us
to deal with complex problems of structural stability. The development in this field
has also been strongly stimulated by the needs of engineering applications, especially
by those related to aeronautics, naval, nuclear, offshore and - of course - related
to aerospace structures. Realization of engineers' expectations appears to be possible
due to the development of computer methods and first of all due to FEM.
The papers from the early 60's, initiated by Gallagher, Argyris, Zienkiewicz and
others in USA and Europe conduced to the initial stress matrix. This fact opened
the door to the application of methods of linear algebra to the analysis of buckling
problems. The formulation of incremental models and development of iterative pro
cedures has initiated the nonlinear analysis of structures since the beginning of 70-
ties (Oden [6], Bathe [7], Kleiber [8]).
In such a way the computer has started to influence the formulation of models
as well as methods of analysis. This concerns also the domination of FEM in the
analysis of structural stability.
In the scope of the linear analysis FEM enables us first of all to analyze
systems with a large number of DOF. The method has also been applied success
fully to the analysis of nonconservative systems.
The basic importance of FEM lies in the analysis of structures with geometrical
orland material nonlinearities. In this way the postcritical behavior as well as
buckling analysis of inelastic structures can be carried out.
In the first place FEM gets ahead due to the unification of structural models,
algorithms and software. The analysis at the level of the structure is the same for
all types of structures. They are distinguished at the finite element level where
appropriate matrices are to be formulated individually. At this level and at the point
level the mechanics of structures (more precisely: continuum mechanics and strength
of materials) is applied. Computer sciences and numerical methods dominate at
higher level of the analysis.
In the book mainly the stability of equilibrium states will be considered. This
can be treated as a special case of the stability of motion but due to such limitation
more efficient FEM procedures can be applied to the analysis. Another reason is
that the authors' experience is related rather to civil engineering than mechanical
engineering structures.
The problems of structural instability should be considered in the frame of a
more general approach of the probabilistic or stochastic processes analysis
(Elishakoff [9]), as a part of the theory of safety of structures (Murzewski [10]).
This is caused by random imperfections of structures, features of materials and load
data.
4 INTRODUCTION
The above approach leads to a very complicated analysis. This concerns the
formulation of models, computational methods and moreover lack of statistics for
data preparation. That is why the deterministic approach is generally preferred by
engineers and scientists and why this approach will also be used in the book.
The literature devoted to the deterministic analysis of structure stability is quite
rich. In the book [1] over 220 books on stability of structures were listed. A great
number of books in Russian deserves notice. An interesting Polish contribution is
that by Zyczkowski [11].
The literature associated with the application of FEM to the structural stability
analysis is very extensive. First of all it concerns papers in scientific journals and
conference proceedings, then chapters in books on FEM - cf Noor [12].
Stability of structures, and especially application of FEM to the buckling analy
sis, is also being extensively developed in Poland. That is why the book is based
also on Polish achievements. Besides the extended chapters in books (Kleiber [13],
Zyczkowski [11], Gomulinski and Witkowski [14], Waszczyszyn [15]) also a great
number of Doctoral Theses and conference contributions were devoted to the
nonlinear analysis and stability of structures by means of FEM. Also proceedings
[16,17] related the advanced courses on stability of structures should be mentioned
here. They will be quoted in appropriate Points of the book.
1.2. Aims, Assumptions and Contents of the Book
The application of FEM to the structural stability analysis is considered in the
book. The formulation of FEM models, suitable not only for engineering
applications but also for the description of the phenomena associated with instability
of structures is presented herein. The efficiency of models and methods will be
discussed from the viewpoints of linear and nonlinear analyses. The numerical
results enable us not only to evaluate the accuracy of FEM models and methods
applied to the analysis but they also should turn our attention to new possibilities
of the numerical analysis.
The scope of the book is limited to the deterministic analysis of stability of
equilibrium states. It is related to problems typical for civil engineering. That is why
we extended the chapters associated with the linear buckling analysis.
The book is addressed to research-workers as well as to postgraduate and
doctoral students. Separate parts of the book were provided for students at the
Cracow University of Technology on M.Eng. and Doctoral Courses. That is why
