Table Of ContentCISM COURSES AND LECTURES
Series Editors:
The Rectors of CISM
Sandor Kaliszky- Budapest
Mahir Sayir -Zurich
Wilhelm Schneider -Wien
The Secretary General of CISM
Giovanni Bianchi -Milan
Executive Editor
Carlo Tasso -Udine
The series presents lecture f.lOtes, monographs, edited works and
proceedings in the field of Mechanics, Engineering, Computer Science
and Applied Mathematics.
Purpose of the series is to make known in the international scientific
and technical community results obtained in some of the activities
organized by CISM, the International Centre for Mechanical Sciences.
INTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
COURSES AND LECTURES -No. 379
COUPLED INSTABILITIES
IN METAL STRUCTURES
THEORETICAL AND DESIGN ASPECTS
EDITED BY
JACQUES RONDAL
UNIVERSITY OF LIEGE
~ Springer-Verlag Wien GmbH
Le spese di stampa di questo volume sono in parte coperte da
contributi del Consiglio Nazionale delle Ricerche.
This volume contains 244 illustrations
This work is subject to copyright.
All rights are reserved,
whether the whole or part of the material is concemed
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or similar means, and storage in data banks.
© 1998 by Springer-Verlag Wien
Origina11y published by Springer-VerJag Wien New York in 1998
SPIN 10678627
In order to make this volume available as economically and as
rapidly as possible the authors' typescripts have been
reproduced in their original forms. This method unfortunately
has its typographical limitations but it is hoped that they in no
way distract the reader.
ISBN 978-3-211-82917-2 ISBN 978-3-7091-2510-6 (eBook)
DOI 10.1007/978-3-7091-2510-6
PREFACE
One of the main trends in current developments in the theory of constructional
steelwork is to study the performance of structural elements and systems in the
way they actually behave, i.e. without unnecessary idealizations and
simplifications. The behaviour of up-to-date steel structures (which, with the
view to enhance their competitiveness with regard to structural systems made of
other materials, become more and more thin-walled) is rather complex, with
several parallel phenomena always influencing their performance and limit
states.
The aforesaid is valid in particular as far as the buckling phenomena of the
systems are concerned. Very rarely are their limit states governed by one
stability phenomenon: practically always it is the interaction of several
phenomena of this kind which determine the ultimate strength (and
consequently, also the design) of the system under study.
Although plenty of research has to date been conducted in the field of stability
problems of constructional steelworks, this having also been reflected in the
advanced schools organized by CISM, a lot of work still remains to be done as
far as the coupling of their effects is concerned (see for example the field of
interaction between global and local buckling of steel columns and girders,
interaction between buckling phenomena and fatigue in the plate elements of
steel bridges, etc.). Also the education ofyoung generations of researchers and
practising engineers in this field is largely behind the level desired.
To fill up the gap and to improve the current situation in these directions were
the main objectives of the ''Advanced School on Coupled Instabilities in Metal
Structures - Theoretical and Design Aspects" organized at the International
Centre for Mechanical Sciences in Udine, September 30- October4, 1996. This
monograph is a revised version of the lecture notes. The lectures were
organized in a way leading progressively, from the mathematical and physical
basic theories to the design aspects through numerical and semi-empirical
approaches of the problems.
Part I, by G. Augusti, from the University of Roma La Sapienza, Italy, with the
collaboration of V. Sepe and A. Paolone, gives an historical introduction to
compound and coupled buckling.
Part II, by M. Pignataro, also from the University of Roma La Sapienza, Italy,
deals with the theoretical aspects of stability, bifurcation and postbuckling
analysis.
Part Ill, by V. Gioncu, from the Politehnica University of Timisoara, Romania,
introduces the phenomenological and mathematical modelling of coupled
instabilities.
Part IV, by J.M. Davies, from the University of Manchester, England, is
dedicated to the application of the generalized beam theory to coupled instability
problems.
Part V, by G.H. Hancock, from the University of Sydney, Australia, is devoted to
the use of finite strips in the non-linear analysis and also the distorsional
buckling analysis of thin-walled structural members.
Part VI, by D. Dub ina, from the Politehnica University of Timisoara, Romania,
gives a semiempirical approach of the interactive buckling analysis of
thin-walled steel members based on the load erosion method.
And, finally, part VII, by J. Rondal, from the University of Liege, Belgium gives
a generalization of the notions given in previous parts, to a general metal
material. This part deals also with the optimal design of structures exhibiting
coupled instabilities.
The editor wishes to thank warmly these six colleagues for the excellence of the
work performed during the preparation of this advanced school, which is well
reflected in this monograph.
Special thanks are also due to the CISM Rector, Prof S. Kaliszky, the CISM
Secretary General, Prof G. Bianchi, the Editor of the Series, Prof C. Tasso,
and to all the CISM Staff in Udine.
Jacques Ronda/
CONTENTS
Page
Preface
Part I- An Introduction to Compound and Coupled Buckling
and Dynamic Bifurcations
by G. Augusti, V. Sepe and A. Paolone ............................................................................................................................ 1
Part II - Stability, Bifurcation and Postbuckling Analysis
by M. Pignataro ............................................................................................................................................................................................... 29
Part III -Phenomenological and Mathematical Modelling of
Coupled Instabilities
by V. Gioncu ....................................................................................................................................................................................................... 85
Part IV- Generalised Beam Theory (GBT) for
Coupled Instability Problems
by J.M. Davies ................................................................................................................................................................................................ 151
Part V -Finite Strip Buckling and Nonlinear Analyses and
Distortional Buckling Analysis of Thin-Walled Structural Members
by G.J. Hancock .......................................................................................................................................................................................... 225
Part VI- Interactive Buckling Analysis of Thin-Walled Cold-Formed Steel
Members via Critical Load Erosion Theory
by D. Dub ina .................................................................................................................................................................................................... 291
Part VII - Buckling and Interactive Buckling of Metal Columns, Optimum
Design Under Stability Constraints and Code Aspects
by J. Randal.. ...................................................................................................................................................................................................... 345
PART I
AN INTRODUCTION TO COMPOUND AND COUPLED BUCKLING AND
DYNAMIC BIFURCATIONS
G. Augusti
University ofRoma "La Sapienza", Roma, Italy
with the collaboration of
V. Sepe and A. Paolone
ABSTRACT
This Part I is a general introduction to the subject matter of the Volume, spanning over some
of the main « theoretical aspects ».
The flrst Chapter presents the simple strut models and equations that allowed to discover and
describe for the flrst time, about 30 years ago, the phenomenon of compound buckling in a
static framework. In the second Chapter, the most significant of these models is used to
illustrate the probabilistic characteristics of the load capacity of an imperfection-sensitive
structure.
In Chapter 3, instability is tackled in a dynamic context. Mechanical systems subject to non
conservative actions and containing two real control parameters are considered. Two simple
models are used to show double Hopf and Hopf -divergence bifurcation. Post-critical
solutions are discussed in terms of non-linear coupling between respectively two dynamic
modes or one dynamic and one buckling mode.
2 G. Augusti, V. Sepe and A. Paolone
1. THE DISCOVERY OF COMPOUND BUCKLING-A RECOLLECTION
(G. Augusti)
1.1 Introduction
I have always been convinced that simple models with few degrees of freedom (just the
number necessary to allow the phenomenon under consideration) can be of great help in
understanding the essential of many mechanical phenomena. Accordingly, a simple but
instructive introduction to compound buckling can be to recollect how I first came across
that phenomenon more than 30 years ago, while I was preparing my Ph.D. Thesis at the
University of Cambridge [1] (the First Part of that Thesis was later translated into Ref.[2]).
~ - UNSTABLE.
0
0
0
?
z 8
6
0·5
p - rK
e.-
STABLE
0·1
-1·0 -O·S
Fig. 1.1: First one-degree-of-freedom strut model: load-deformation curves
I had investigated the static buckling and post-buckling behaviour of a number of one
degree-of-freedom strut models, including the two classical models that show respectively
stable and unstable buckling (Figs.1.1 and 1.2). Then, I tried and extended these two
models into two-degrees-of-freedom models, i.e. models again constituted by a rigid
"limb" with different elastic restraints, but supported at the bottom by a spherical, rather
than cylindrical, hinge and thus able to rotate about any axis in the x-y plane (see Fig.I.3,
where the relevant symbols are defined 1 ).
1 The Figures in this Chapter 1 reproduce exactly figures from Ref.[ I].
Introduction to Compound and Coupled Buckling 3
Fig.1.2: Second one-degree-of-freedom strut model: load-deformation curves
Let me say soon that the 2-d-o-f
extension of the second 1-d-o-f model
(i.e. a rigid strut restrained at the top
end by two horizontal elastic springs,
D 0 E S respectively parallel to the x- and y
axis) did not unveil any interesting or
unexpected phenomenon: the post
buckling paths in the z-x and z-y
OA • L planes duplicated the path of the 1-d
081c AO: x11::lsln~ o-f model (Fig.l.4 ). Of course, two
OD'•AB•YA•lsmil buckling loads2 of this model were
OE =AA' • ZA • Leos 0. found: (1) PE = kxL and (2) PE2 = kyL,
kx and ky being the stiffness of the
spring parallel to x andy respectively.
X
Fig. 1.3: 2-d-o-f strut models: definitions
2 In Refs.[1][2]. and in this text, the term buckling is identified with bifurcation of (static) equilibrium;
collapse with a maximum on a load-deformation curve. Often, in the literature, collapse loads are also called
buckling loads.
