Table Of ContentInternational Union of Theoretical
and Applied Mechanics
Buckling
of Structures
Symposium Cambridge/USA
June 17-21, 1974
Editor
Bernard Budiansky
Springer-Verlag
Berlin Heidelberg New York 1976
Professor Bernard Budiansky
Harvard University
Cambridge, Massachusetts, U.S.A.
With 214 Figures
ISBN 978-3-642-50994-0 ISBN 978-3-642-50992-6 (eBook)
DOl 10.1007/978-3-642-50992-6
Library of Congress Cataloging in Publication Data: Symposium on Buckling of Structures, Harvard
University. 1974. Buckling of structures. At head of title: International Union of Theoretical and
Applied Mechanics. Bibliography: p. Inclndes index. 1. Jlnckling (Mechanics) - Congresses. 2. Struc
tural stabilit~-- Congresses. 1. Jlndiansky, Beruard. II. International Union of Theoretical and App
lied Mechanics. III. Title. TA656.2.S95. 1974. 624'.176. 75-31726.
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Preface
This volume contains the written texts of the papers presented at a
Symposium on Buckling of Structures held at Harvard University in
June 1974. This symposium, one of several on various topics sponsored
annually by the International Union of Theoretical and Applied Me
chanics (IUTAM), was organized by a Scientific Committee consisting of
B. Budiansky (Chairman), A. H. Chilver, W. T. Koiter, and A. S. Vol'
mir. Participation was by invitation of the Scientific Committee, and
specific lecturers were invited to speak in the areas of experimental
research, buckling and post-buckling calculations, post-buckling mode
interaction, plasticity and creep effects, dynamic buckling, stochastic
problems, and design. A total of 29 lectures were delivered, including
a general opening lecture by Professor Koiter, and there were 93 reg
istered participants from 16 different countries.
Financial support for the symposium was provided by IUTAM,
in the form of partial travel support for a number of participants, and
also by the National Science Foundation, the National Aeronautics
and Space Administration, and the Air Force Office of Scientific Re
search, for additional travel support and administrative expenses.
Meeting facilities and services were efficiently provided by the Science
Center of Harvard University, and administrative support was gen
erously provided by the Division of Engineering and Applied Physics
of Harvard University. The scientific chairman enjoyed the invaluable
assistance of his colleagues Professors J. W. Hutchinson and J. L. San
ders in making local arrangements for the symposium; and, finally, the
dedicated secretarial and administrative services provided by Marion
Remillard and Parian Temple, without whom nothing would have
worked, are most gratefully acknowledged.
Cambridge, Mass., October 1975
Bernard Budiansky
Participants
(Authors denoted by asterisks)
Akkas, N. Ankara, Turkey
*Almroth, B. O. Palo Alto, California, U.S.A.
*A mazigo, J. C. Troy, New York, U.S.A.
*A nderson, M. S. Hampton, Virginia, U.S.A.
*Arbocz, J. Pasadena, California, U.S.A.
*A ugusti, G. Florence, Italy
*Babcock, Jr., Ch. D. Pasadena, California, U.S.A.
Basdekas, N. Arlington, Virginia, U.S.A.
Batdorf, S. B. Los Angeles, California, U.S.A.
Bauld, N. R. Clemson, South Carolina, U.S.A.
Besseling, J. F. Delft, Netherlands
Billington, D. P. Princeton, New Jersey, U.S.A.
Blaauwendraad, J. Gouda, Netherlands
Brown, E. H. London, England
Brush, D. O. Davis, California, U.S.A.
*Budiansky, B. Cambridge, Massachusetts, U.S.A.
Bufler, H. Stuttgart, Germany
Bushnell, D. Palo Alto, California, U.S.A.
Calladine, C. R. Cambridge, England
Ceradini, G. Rome, Italy
*Chilver, A. H. Cranfield, England
Como, M. Cosenza, Italy
Danielson, D. A. Charlottesville, Virginia, U.S.A.
Dickie, J. F. Manchester, England
Dill, E. H. Seattle, Washington, U.S.A.
*Duszek, M. K. Warsaw, Poland
Dym, C. L. Cambridge, Massachusetts, U.S.A.
Ebner, H. Aachen, Germany
*Esslinger, M. Braunschweig, Germany
*Gallagher, R. H. Ithaca, New York, U.S.A.
Galletly, G. D. Liverpool, England
*Hansen, H. R. Oslo, Norway
*Hayman, B. Leicester, England
Hedgepeth, J. M. Santa Barbara, California, U.S.A.
Herrmann, G. Stanford, California, U.S.A.
*Hoff, N. J. Stanford, California, U.S.A.
Huang, N. C. Notre Dame, Indiana, U.S.A.
Huseyin, K. vVaterloo, Ontario, Canada
*Hutchinson, J. W. Cambridge, Massachusetts, U.S.A.
*Johns, K. C. Sherbrooke, Quebec, Canada
Participants V
Jones, N. Cambridge, Massachusetts, U.S.A.
Kalnins, A. Bethlehem, Pennsylvania, U.S.A.
Kempner, J. Brooklyn, New York, U.S.A.
Khot, N. S. Dayton, Ohio, U.S.A.
Knets,1. Riga, U.S.S.R.
*Koiter, W. T. Delft, Netherlands
*Kozarov, 1\'I. lVL Sofia, Bulgaria
*Leckie, F. A. Leicester, England
*Leipholz, H. H. E. \Vaterloo, Ontario, Canada
*Leonard, R. VY. Hampton, Virginia, U.S.A.
Libove, C. Syracuse, New York, U.S.A.
Lukasiewicz, S. \Varsaw, Poland
McIvor, 1. K. Ann Arbor, Michigan, U.S.A.
*Massonnet, Ch. Liege, Belgium
Masur, E. F. Chicago, Illinois, U.S.A.
Nachbar, W. La Jolla, California, U.S.A.
*Needleman, A. Cambridge, Massachusetts, U.S.A.
*van der Neut, A. Delft, Netherlands
Niordson, F. 1. Lyngby, Denmark
Ohtsubo, H. Tokyo, Japan
*Pedersen, P. T. Lyngby, Denmark
Pian, T. H. H. Cambridge, Massachusetts, U.S.A.
*Pignataro, M. Rome, Italy
Plaut, R. H. Providence, Rhode Island. U.S.A.
Pomerantz, J. Arlington, Virginia, U.S.A.
Rehfield, L. W. Atlanta, Georgia, U.S.A.
Reissner, E. La Jolla, California, U.S.A.
Rhodes, J. Glasgow, Scotland
Riks, E. Amsterdam, Netherlands
Roorda, J. \Vaterioo, Ontario, Canada
*Roren, E. l\'1. Q. Oslo, Norway
Samuelson, L. A. Bromma, Sweden
Sanders, J. L., Jr. Cambridge, Massachusetts, U.S.A.
Seide, P. Los Angeles, California, U.S.A.
*Sewell, M. J. Reading, England
Simitses, G. J. Atlanta, Georgia, U.S.A.
Simmonds, J. G. Charlottesville, Virginia, U.S.A.
*Singer, J. Haifa, Israel
Stein, M. Hampton, Virginia, U.S.A.
Supple, W. J. Guildford, England
*Tennyson, R. C. Downsview, Ontario, Canada
*Thompson, J. M. T. London, England
Thurston, G. A. Denver, Colorado, U.S.A.
Tillman, S. C. Manchester, England
*Tulk, J. D. London, England
*Tvergaard, Y. Lyngby, Denmark
Uetani, K. Kyoto, Japan
Valid, R. Chatillon, France
de Veubeke, B. F. Liege, Belgium
Wierzbicki, T. \Varsaw, Poland
*Williams, F. W. Birmingham, England
*Yamaki, N. Sendai, Japan
Zyczkowski, ill. Krakow, Poland
Contents
I. Opening Lecture
W. T. Koiter: Current Trends in the Theory of Buckling . . . . . . . . 1
II. Buckling and Post-Buckling Calculations
F. W. Williams, W. H. Wittrick, R. J. Plank: Critical Buckling
Loads of some Prismatic Plate Assemblies. . . . . . . . . . . . . . . . .. 17
P. T. Pedersen: On the Collapse Load of Cylindrical Shells ..... 27
R. H. Gallagher: Finite Element Representations for Thin Shell
Instability Analysis ..................................... 40
B. O. Almroth, E. Meller, F. A. Brogan: Computer Solutions for
Static and Dynamic Buckling of Shells .................... 52
III. Plasticity and Creep
N. J. Hoff: Theory and Experiment III the Creep Buckling of
Plates and Shells ...................................... 67
R. B. Rikards, G. A. Teters: Nonsymmetric Creep Buckling of
Cylindrical Shells under Axial Compression and External Pres-
sure 78
F. A. Leckie, B. Hayman: Creep Instability of Thick Shell Struc-
tures .................................................. 86
M. J. Sewell: A Plastic Flow Rule at a Yield Corner 95
J. W. Hutchinson, B. Budiansky: Analytical and Numerical
Study of the Effects of Initial Imperfections on the Inelastic
Buckling of a Cruciform Column .......................... 98
Contents VII
M. K. Duszek: Stability Analysis of Rigid Plastic Structures at
the Yield-Point Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 100
I V. Mode Interaction
A. van der Neut: Mode Interaction with Stiffened Panels 117
W. T. Koiter, M. Pignataro: An Alternative Approach to the In
teraction between Local and Overall Buckling in Stiffened Panels 133
J. M. T. Thompson, J. D. Tulk, A. C. Walker: An Experimental
Study of Imperfection-Sensitivity in the Interactive Buckling
of Stiffened Plates ...................................... 14H
V. Tvergaard, A. Needleman: Mode Interaction in an Eccentric-
ally Stiffened Elastic-Plastic Panel under Compression ...... 160
V. Stochastics .
J. C. Amazigo: Buckling of Stochastically Imperfect Structures.. 172
G. Augusti, A. Baratta: Reliability of Slender Columns: Compari-
son of Different Approximations .......................... 183
K. C. Johns: Some Statistical Aspects of Coupled Buckling Struc-
tures .................................................. 199
VI. Dynnmics
H. H. E. Leipholz: Some ltemarks on Liapunov Stability of Elas-
tic Dynamical Systems .................................. 208
M. Kozarov, M. Kishkilov: An Investigation of the Stability and
Vibration of a Hyperbolic Shell of one Sheet under Axisymmet-
ric Loading ............................................ 217
VII. Experiments
J. Singer, A. Rosen: The Influence of Boundary Conditions on
the Buckling of Stiffened Cylindrical Shells .... . . . . . . . . . . . .. 227
R. C. Tennyson: The Effect of Shape Imperfections and Stiffening
on the Buckling of Circular Cylinders ....................... 251
M. Esslinger, B. Geier: Calculated Postbuckling Loads as Lower
Limits for the Buckling Loads of Thin-Walled Circular Cylinders 274
VIII Contents
,J. Arbocz, Ch. D. Babcock, Jr.: Prediction of Buckling Loads
Based on Experimentally Measured Initial Imperfections. . . .. 291
N. Yamaki: Experiments on the Postbuckling Behavior of Circu-
lar Cylindrical Shells under Torsion ........................ 312
V III. Design
A. H. Chilver: Design Philosophy in Structural Stability 331
R. W. Leonard, M. S. Anderson, \V. L. Heard, Jr.: Design of a
Mars Entry "Aeroshell" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 346
R. Maquoi, C. Massonnet: Interaction between Local Plate Buck-
ling and Overall Buckling in Thin-Walled Compression Mem-
bers-Theories and Experiments .......................... 365
E. M. Q. Roren, H. R. Hansen: Buckling Design in Ship Struc-
tures .................................................. 383
Current Trends in the Theory of Buckling
W. T. Koiter
Technische Hogeschool, Delft, .Netherlands
Abstract
Some current trends in buckling theory are reviewed. Particular attention is
given to multi-mode bifurcation buckling, the associated imperfection-sensitivity
and the correlation of experimental evidence with theory.
1. Introduction
The task of a speaker in an opening lecture at a meeting of selected
experts in a particular field of science is hardly to be envied. In most
aspects of his topic the audience comprises members with far more
specialized expertise than the lecturer has at his command, and the
eaRY way out, a retreat into generalities, is not likely to be welcomed.
Having accepted the challenge, however, I shall try and review some
CUl'rent trends in the theory of buckling as I see them. I had the good
fortune to read the abstracts of your papers which our chairman kindly
made available to me, and this may reduce to some extent the inevit
able personal bias in my talk.
The simplest possible model of a column, depicted in Fig. 1 a,
consists of a rigid bar supported on an elastic hinge. This simple model
is employed in several elementary text-books, including the author's
[28] which, incidentally, enjoys sales comparable to a volume of poetry.
The perfect column, with zero eccentricity of the applied load (e =
'XL = 0), loses its stability and buckles by bifurcation at the critical
load 1\ = DIL. The post-buckling behaviour depends on nonlinearities
in the spring characteristic. Various possible load-deflection curves are
shown in Fig. 1 b, and the relationship between the load and the overall
shortening is exhibited in° F ig. 1 c. In cases of an unstable descending
post-buckling path (ex =1= or ex = 0, (3 < -1/6) the column is sensitive
to geometric imperfections or load eccentricity. The associated drop in
critical load is shown in Fig. 1 d, and the buckling load FI at the bi
furcation point is replaced by the snap load at a limit point. In the
absence of a descending post-buckling path of the perfect column, how
ever, no loss of stability occurs in the column with imperfections or
2 W. T. Koiter
a load eccentricity. The presence of the buckling load PI of the perfect
column appears here only in the form of a marked increase of the
deflection when the load P approaches and passes the value PI"
Post-buckling equilibrium paths and the influence of imperfections
F
IF
i
I
I
i
L
a
F F F
CJ. = 0 , f3 < -116 a=O,{3>-1I6 e
b
F F*IF,
CI.. = 0 ,/3> -116
F.
I .......
\ .............. e&= 0, f:l < -116
" a=O, f3 < -116 ,n=312
....... a;>! 0
l(lX>O,n=2
,1/L
c d
Fig. 1. a) Simplest column model eccentricity e = uL, spiral spring,
+
ltf = D [0 - or.02 ,803]; b) Load-deflection curves; c) Load-shortening curve for
perfect column. d) Imperfection-sensitivity (1 - F*IF1)n = const· lui FIFr
