Buckling and Post-Buckling: Four Lectures in Experimental, Numerical and Theoretical Solid Mechanics Based on Talks Given at the CISM-Meeting Held in Udine, Italy, September 29–October 3, 1985

252 Pages·1987·3.356 MB·English

by  Johann Arbocz, Michel Potier-Ferry, Josef Singer, Viggo Tvergaard (auth.)

Lecture Notes
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288
.J Arbocz M. Potier-Ferry
.J Singer .V Tvergaard
Buckling and Post-Buckling
Four Lectures in Experimental, Numerical
and Theoretical Solid Mechanics
Based on Talks Given at the CISM-Meeting
Held in Udine, Italy, September 29-October 3, 1985
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Berlin Heidelberg NewYork London Paris Tokyo
Authors
Johann Arbocz
Department of Aerospace Engineering, Delft University of Technology
Delft, The Netherlands
Michel Potier-Ferry
Laboratoire de Physique et M~canique des Mat~riaux, Universit~ de Metz
lie du Saulcy, F-57045 Metz, France
Josef Singer
Department of Aeronautical Engineering, Technion, Israel Institute of Technology
Haifa 32000, Israel
Viggo Tvergaard
Department of Solid Mechanics, The Technical University of Denmark
Lyngby, Denmark
ISBN 3-540-18312-4 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-18312-4 Springer-Verlag NewYork Berlin Heidelberg
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PREFACE
This book contains four lectures delivered at the International Centre for
Mechanical Sciences, UDINE, Italy, in 1985.
Slender and thin-walled structures are frequently used in aeronautics and also
in mechanical, nuclear and civil engineering. The computation of their practical
carrying capacity requires a nonlinear analysis. For many shells in the elastic
or elastoplastic range, geometric nonlinearities are needed to calculate the
reduction of the critical load due, for instance, to imperfections. These lectures
aim to give a comprehensive account for the theoretical, numerical and experimen-
tal methods which are useful to analyse the buckling and post-buckling behavior
of structures.
The book is intended for postgraduate students, researchers and engineers who
are interested in this field.
The first lecture by M. POTIER-FERRY is devoted to the elastic post-buckling
theory and some more or less simple applications. The ideas of recent mathematical
approaches like bifurcation, catastrophe and singularity theories are included in
this lecture, but without advanced mathematics. J. ARBOCZ is an aeronautical
engineer. In his paper, he discusses the influence of imperfections and boundary
conditions on the maximum load of stiffened shells of revolution. By coupling
expertise and simple numerical methods, he sets examples how to design real
structures. The state of the art in plastic buckling is presented in the lecture
by V. TVERGAARD. It essentially contains bifurcation criteria for plates and
shells, asymptotic and finite-element analyses of plastic post-buckling and
discussions of the influence of the constitutive laws. In spite of tremendous
progress,~numerical and mechanical models are not sufficiently reliable, especial-
ly to predict a very sensitive phenomenon like buckling. The last lecture by J.
SINGER is an excellent guide to the experimental methods used in testing com-
pressed structures.
We express our sincere thanks to the authorities and staff of the CISM for
their efficiency in the organization of the course.
W.T. KOITER M. POTIER-FERRY
V
TABLE OF CONTENTS
M. Potier-Ferry
FOUNDATIONS OF ELASTIC POSTBUCKLING THEORY ........................ I
.I Introduction ................................................... 2
2. Simple Models ....... ........................................... 3
2.1 Amplitude equation ......................................... 3
2.2 An elementary buckling model ............................... 5
2.3 Beam on foundation ......................................... 9
3. Classical Postbuckling Theory .................................. 12
3.1 Energy criterion. Neutral stability ........................ 14
3.2 Lyapunov and Schmidt reduction ............................. 16
3.3 First case : Bifurcation from a fundamental path ........... 20
3.4 Alternative method ......................................... 25
3.5 Second case : Perturbation of a singular equilibrium ....... 28
3.6 Symmetric structure. Symmetry breaking ..................... 33
3.7 Imperfection sensitivity ................................... 39
3.8 Classifications of singularities ........................... 42
4. Applications ................................................... 44
4.1A lumped arch .............................................. 45
4.2 Examples with a restabilization of the fundamental path .... 48
4.3 Bifurcation from a nearly double buckling mode ............. 50
4.4 Imperfection sensitivity close to a nearly double
buckling mode .............................................. 53
5. Mathematical Foundations ....................................... 59
5.1Lyapunov and Schmidt reduction (continued) ................. 59
5.2 Catastrophe theory : progress or digression ................ 64
5.3 On the energy criterion .................................... 67
6. Structures with a Large Aspect Ratio ........................... 69
6.1 Limitations of classical postbuckling theory ............... 69
6.2 Landau-Ginzburg equation ................................... 71
6.3 Amplitude modulation of buckling patterns .................. 73
6.4 Tendency to localization 75
6.5 Wavenumber selection ....................................... 77
References ........................................................ 79
IV
J. Arbocz
POST-BUCKLING BEHAVIOUR OF STRUCTURES
NUMERICAL TECHNIQUES FOR MORE COMPLICATED STRUCTURES
.I Introduction ..... , .......................................... 84
2. Mathematical Formulation of the Stability Problem ........... 85
3. Classic Linearized Small Deflection Theory for a Perfect Shell 87
4. Effect of Axisymmetric Imperfection ......................... 91
5. Effect of Asymmetric Imperfection ........................... 95
6. Effect of Axisymmetric and Asymmetric Imperfections
(2-Modes Solution) .......................................... 99
6.1 For a perfect shell ~I = ~3 = 0 ........... A ....... A ..... 102
6'2 For a shell with axisymmetric imperfectio~ ~I ~ 0~ ~3 = 0 103
6.3 For a shell with asymmetric imperfection ~I = 0, ~3 ~ 0 106
6.4 For a shel with both axisymmetric and asymmetric
imperfection~ ~I ~ O, ~3 ~ 0 ............................ 106
7. Effect of General Imperfections - Multimode Solution ........ 107
7.1 Newton's method of quasilinearization .~ .................. 107
7.2 Reduction to a set of algebraic equations ............... 111
7.3 Solution using increments in end-shortening ............. 112
7.4 Mode selection .......................................... 113
7.5 Correlation studies ..................................... 115
Effect of Boundary Conditions ............................... 118
.
8.1 Reduction to an equivalent set of ordinary differential
equations ............................................... 119
8.2 Effeot of boundary conditions for isotropic shells ...... 122
8.3 Effect of boundary conditions for stringer stiffened
shells .................................................. 125
9. Stochastic Stability Analysis ................................. 130
10. Concluding Remarks .......................................... 135
References .................................................. 136
Appendix A .................................................. 139
Appendix B .................................................. 140
Appendix C .................................................. 141
Appendix D .................................................. 142
IIV
V. Tvergaard
EFFECT OF PLASTICITY ON POST-BUCKLING BEHAVIOUR
I. Introduction ................................................. 144
2. Basic Equations for Elastic-Plastic Plates or Shells ........ 144
3. Bifurcation Analysis ........................................ 147
3.1 Solids with a smooth yield surface and normality ........ 147
3.2 Solids with non-normality ............................. .. 149
3.3 Solids that form a vertex on the yield surface .......... 149
4. Initial Post-Bifurcation Behaviour .......................... 152
4.1Bost-bifurcation theory ................................. 152
4.2 Results for integrally stiffened panels ................. 157
5. Hypoelastic Estimates of Imperfection-Sensitivity ........... 161
6. Numerical Analyses of Plastic Buckling ...................... 166
7. Localization of Buckling Patterns ........................... 171
8. Effect of Material Rate-Sensitivity ......................... 177
References ..................................................... 180
J. Singer
EXPERIMENTAL TECHNIQUES AND COMPARISON WITH THEORETICAL RESULTS
I. Introduction ............................................... 681
2. Motivation for Experiments ................................. 186
3. Elements of a Simple Buckling Test - a Column Under
Axial Compression .......................................... 981
Buckling and Postbuckling of Columns ....................... 191
.
5. Modeling - Theory and Practice ............................. 202
6. Buckling and Postbuckling of Plates ........................ 207
7. Buckling and Postbuckling of Shells ........................ 220
8. Initial Imperfections in Shells ........................... 231
9. Boundary Conditions and Loading for Shells ................. 237
10. Nondestructive Testing for Shells .......................... 238
FOUNDATIONS OF ELASTIC POSTBUCKLING THEORY
by Michel POTIER-FERRY
Laboratoire de Physique et M~canique des Mat~riaux
Universit~ de METZ, lie du Saulcy
57045 - METZ, France
ABSTRACT
In this paper we present the elastic postbuckling theory, but we intro-
duce two modifications with respect to its classical statement. First,
we ~ account for the influence of symmetries. Second, the singularities
will be classified according to their robustness, what was introduced by
Catastrophe Theory. Various recent progresses are discussed ; mathemati-
cal foundations of the energy criterion, buckling of structures with a
large aspect ratio, contribution of catastrophe theory ...
.I INTRODUCTION
The long story of postbuckling !heory began with a study of the fle-
xible compressed beam by .L Euler /20/ in 1744. The research in the field
of elastic stability became important only a century later, when the
extensive use of steel permitted the introduction of slender structures,
especially in civil engineering. The practical interest of a nonlinear
theory was recognized in the 1930's and the general postbuckling theory
/317,
of elastic structures was enunciated by Koiter in his thesis 1945.
He established that imperfections may give rise to significant reductions
of the critical load, which explained the discrepancy between linear
theories and experiments on shells. The development of aeronautical and
aerospace programs motivated a considerable amount of research in this
field during the 1960's. From that time, the research activity is remai-
ned important and new application~ appeared such as off-shore structures
/48/ /62/, nuclear engineering /3/ ...
Developments subsequent to Koiter's thesis had not fundamentally modi-
fied the theory itself. The British school (Sewell !5Z/, Thompson-Hunt
/60/, Huseyin /25/...) has considered _ylniam _ discrete sxstems. The Har-
vard's school (Budiansky - Hutchinson /11/, Budiansky /9/ ...) relies
rather on the ~rinciple of virtual work than on the potential energy,
rather on a priori set expansions than on the elimination of "passive
variables".
In another respect, bifurcation and stability theory has been the
subject of many mathematical studies. Since the pioneer works of Poincar~
(1885), Lyapunov (1892) and Schmidt (1908), mathematical analysis has
played a great part in bifurcation theory of continuous systems, the basic
tool being the Inverse Function Theorem. Although postbuckling and bifur-
cation theories have progressed in a mutual ignorance, the two points of
view often are similar, mathematical analysis providing quite rigorous
justifications to the asymptotic computations of postbuckling theory±
Accounts of these_researches_ can be found in Vainberg - Trenogin L6Z/,
Keller - Antman ~28/, Sattinger ~49/, Iooss- Joseph ~2!/. More recent
tools of analysis, such as Sobolev spaces or semi-groups, have contributed
to a better understanding of the dynamical foundations of the energy cri-
terion (Ball L~/, Potier-Ferry ~4!/).Catastrophe_theory (Thom /58/)and
singularity theory (Golubitsky and Schaeffer /22/)put forward a more qua-
litative approach than that of the analysts. These theories are issued
from topology and differential geometry. Perhaps the most interesting
idea of catastrophe theory is to classify the singularities according to
their "robustness", what we shall simply include in postbuckling theory.
One can also modify the statement of postbuckling theory in a simple way,
by accounting for the symmetries of the considered structures._ _Recently,
this gave rise to a number of mathematical works (Sattinger /51/, Golubit-
sky and Schaeffer /52/ /53/).
Roughly, we shall follow Koiter's original statement. As we have just
explained, some recent ideas shall be introduced in the classical theory,
but we shall avoid the advanced mathematical techniques that are often
used to present those ideas. In addition, the contribution of mathematics
to postbuckling will be discussed in a separate chapter. Practical appli-
cations are beyond the scope of this short introduction. We shall only
discuss simple models in order to present or to illustrate the basic no-
tions. Especially, we shall give at least one example of each type of
singular point that appears in our classifications. The paper is closed
with a new postbuckling analysis for structures such that postbuckling
patterns have a cellular shape with a large number of cells. Initially± _
the latter methods had been developed for problems of fluid mechanics/68/
The thermodynamical foundations of the energy criterion are well
understood, the main contributions being those of Duhem /17/ and Ericksen
L~9/. This will not be presented here and we refer to recent accountsL33/
!Z/. We neither discuss non-conservative systems /4/ /16/ /55/ that may
lead to dynamic instabilities ~2Z/. A complete bibliography should __-ulcni
de thousands of references. For this_pur~o~e,_we refer ~o_textbooks ~8/
/25/ /48/ /60/ /63/, review papers /9/ /26/ /29/ /33/ /64/and symposium
proceedings ~IO/ ~38/ ~4!/ ~5~/.
.2 SI~LE MODELS
The most typical feature of instability theory is that its fundamen-
tal characteristics can be found in very simple models. Moreover, any
complicated structural system is equivalent, in some sense, to one of
these simple models, at least in the neighbourhood of a critical state.
We present here two simple models and the slightly more intricate example
of a beam on a Winckler foundation, that will be useful for illustrating
the general theory.
2.1 Amplitude equation
To explain the instability phenomena in fluid mechanics, one consi-
ders usually the so-called Landau equation /18/, which can be written as
da 3
dt - %( - %c ) a -- C 3 a (2-I)
or as
da 2
dt - %( - %c ) a - C 2 a (2-2)
The function a(t) is the amplitude of the disturbance, which implies that
the spatial shape of this disturbance does not vary significantly. The
control parameter is denoted by % : in fluid mechanics, it may be the
Reynolds number or the Rayleigh number ... In this course, it will be a
loading parameter. The critical value of % is designated by % .
C
Look at the e~uigibrium paths in the a - % plane for Eqs (2-I) or
(2-2). One finds a ~c~tnemadnu~ solution a I )%( = 0 and a secondary branch
of solutions aii(%), which is given by :
2
aii = %( - %c)/C3 or aii = %( - %c)/C2 (2-3)
i
These two branches intersect at the point (a,%) = (O,%c) , that is called
bifurcation point. The secondary branch (2-3) is called bifurcating
branch. The stability of an equilibrium solution is governed by the linea
rized equation. One obtains, for the fundamental solution,
as
t--d = (% - %c )a (2-4)
and for the solution (2-3),
a = all + b
db db
dt - 2(% - %c)b or t--d = - (% - %c )b (2-5)
By comparison of (2-4) and (2-5), one concludes that the secondary solu-
tions are stable only for % ~reater than %c, i.e. when the fundamental
solution is unstable. Remark also that the bifurcation ~oint coincides
with the point where the fundamental solution los~s its stability. The
appellation "p~nc~ip£e of exchange of stab£f~6~y" is used to designate
the two latter properties, but the terminology is rather inappropriate
except with Eq. (2-2).
The equilibrium paths are pictured in Figure ,1 where the dashed
curves represent unstable solutions. One distinguishes three cases accor-
ding to the equation (2-I) or (2-2) and to the sign of the constant C .3
The bifurcation points of Figures !-a, |-b, I-c are called respectively
stable- s ymm~tric , u~ table- s ymmetric and as ymmetric .
I
I
I
f
7
f
f
a
Fi~. ,al Stable - Sym, Fig. ,b| Unstable - Fig. ,c| Asymmetric
metric bifurcation Symmetric bifurcation bifurcation
Fig. ,I Amplitude - parameter curves for various
bifurcation points