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7/24/2019 Post Buckling
1/14
&ffiffiffiffiffiffi
fiffiffiffi&ffiffiffi%fiffiWffiffiW
Postbuckl ing
heo y
J. W. HUTCHINSON
HARVA RD
U N VE
RSITY,
CAM BR DG
E, MASSACHUSETTS
W.T.
KOITER
TECHNISCHE
HOGESCHOOL,
DELFT.
THE NETHERLANDS
Introduction
A division of elastic stabil i ty theory
and its applications
into
separatecategoriesof
buckling and postbuckling
is
not entirely rational, but the creation of such an artif i-
cial distinction
is
almost
essential o bring
either into
manageableproportions. This point
is
brought
home by
more than 1600 referenceson cylindrical
shell buckling
alone in the
surveyby Grigol'uk
and Kabanov
IRef.
1]
.
Postbuckling
aspects
of
stabil i ty
theory
for flat plates
received
prominence
in the early 1930's, after
Wagner
IRef.
2] had establisheda sound
theoretical
foundation
for the load-carrying capacity of
deeply wrinkled
flat
shear panels.
The
approximate
discussion
by Cox
fo r
plates in uniaxial compression
IRef.
3]
was
followed by
a r igorous analysis by
Marguerre and
Trefftz
[Ref.
4] .
This work was continued
first in Gernrany
IRefs.
5, 6],
and somewhat
later in the
United
States
[Refs.
7-15],
Britain [Refs. 16-18] and The Netherlands IRefs.
1.9-281
Reviews of this work and subsequent
dcvelop-
ments are to be
found in
[Ref.
29]
.
The entirely
different postbuckling
behavior of
th in
shell
struc-
tures came
to l ight in the early
1940's when Karman
and Tsien
IRefs.
30-331
showed
that the
large dis-
crepancies bctween test
and theory
for the buckling
of certain typcs
of th in
shell
structures
was due
to
the highly unstable
postbuckling behavior
of these
structures.
At roughly the
same
time
in war-time
Holland,
Koiter
IRef.
34]
developeda
general heory of
stabil i ty
for
elastic
systems
subject to conservative
oad-
ing which was published as
his doctoral thesis
n
1945.
Karman's and Tsien's papers
spawned
many more in
the following 30 years, and most of them have been
directed to
obtaining
more accurate
results for the
cylindrical shell under axial compression
or to studying
this struct ure under
other
loadings.
Koiter 's work, on the
other hand, attracted relatively
little attention until the
early 1960's when
interest
in
the general heory sprang
up almost simultaneously
in
England
and
in the United
States.
Our
review
will be centered
mainly on the
theory
and
applications that have emerged
from
these two ap-
proaches to postbuckiing theory. A major
stimulus
to
development in this
subject
continues to be generated
by the quest for predictive methods for the buckling of
shell structures, and a sizable portion of this
review
will
concern applications of the t heory to this area,
A
state-of-the-art discussion of shell buckling as of 1958
is
given
in
the excellent
survey
by
Fung
and
Sechler
[Ref.
35],
and that
article
provides a background
for
assessinghe
progresswhich has taken place in this sub-
ject
in
just
over a decade.
Our
lack of
command
of the
Russian
anguage
prevents
us
from
doing
justice
to the
contr ibutions of Soviet scientists n this
field.
Reference
may be made, however,
to
Volmir 's
outstanding reatise
on the
stabil i ty of
deformable
systems
[Ref.
36],
an d
to the
survey
on
cylindrical shell
buckling
[Ref.
1] .
Our discussionwil l not touch on research nto stabil-
i ty of elasticsystemssubject o nonconservativeoadings,
but fortunately
we can
cite
a recent
survey
in Applied
Mechanics
Reuiews by
Herrmann
[Ref.
37]
on this
aspect of
stability
theory,
as well
as a
similar survey by
Nemat-Nasser
IRef.
38]
on thermoelastic
stabil i ty under
general oads.
We also would call attention to another
recent review
article by Thompson
[Ref.
39]
which
does
cover some of the same ground as the present
survey, but the emphasis there is on elastic
systems
characterized
by
a
finite
number
of degrees
f
freedom,
whereas
most of the
applications
we will discuss
will
be
within
the realm
of continuum
theory, We
also omit a
detailed
discussion
of
secondary branching
IRefs.
29 ,
40,
4ll of the initially
stable postbuckling behavior
of
f lat plates, even if this phenomenon is of some im-
portance for
these
and
similar
completely
symmetric
structures
IRef.
42] .
Theory
and Appl ications
The energy
criterion of
stability
for elastic
systems
subject to conservative
loading is
almost
universally
i
353
7/24/2019 Post Buckling
2/14
adopted
by workers in the
field
of
structural stability
[Refs.
43-45]
.
A positive
definite second variation of
the potential energy
about a
static
equil ibr ium
state is
accepted as a
sufficient condition
for
the stability of
that
state. Numerous attempts to
undermine
these
two
pillars of structural
stability
theory
have been made, but
confidence in them remains
undiminished.
With
a
proper
shoring-up of
certain aspe cts of these criteria, they will
undoubtedly continue
to serve as the
foundation
of
elastic
stability theory. An
account of the present status
of the
fundamentals
of stability theory
is
given in
IRef.
46] .
In this review, we
are concerned mainly with the
class of problems most frequently encountered
in the
field
of structural
stability
in which
the
loss
of stability
of one
set of equil ibr ium
states of an
idealized,
or
perfect,
elastic
structure is associatedwith
bifurcation
into another
set of
equil ibr ium
states.
The first
set
is
referred to as the
prebuckling state and the bifurcated
state as the buckled configuration . The
bifurcation
load
of
the perfect
structure
is
commonly called the classical
buckling load
and
is
denoted here by
P-.
This is
by
no
means
the only
circumstance under which a structure
can become unstable. For example a very shallow
clamped
spherical cap subject to a uniform pressure
loading reaches
a maximum,
or
limit, load
at which point
i t
becomes unstable under prescribed pressure
with
no
occurrence
of bifurcation.
However,
due to the
sym-
metry of both the idealized
structure and its loading,
bifurcations frequently do
occur at
load
levels below the
limit load.
The
classical analysis of the
stability
of the prebuck-
ling
state
of the perfect
structure takes the
form
of an
eigenvalue
problem
for
the lowest l oad level for
which
the second
variation of the potential energy is
semi-
definite. The Euler
equations associated
with this
vari-
ational principle are inear
and the eigenmode
or
modes)
associated
with the critical eigenvalue P. is
termed
the
buckling mode
(or modes). While the
classical
analysis
yields
the load
at which a perfect realization of a
struc-
ture wil l
at
least
start to undergo
buckling
deflections, t
gives no indication
of the character of the postbuckling
behavior,
nor do es it give
any
insight into
the way a
non-
perfect realization
wil l behave.
Typical load-deflection
curves characterizinq static
equilibrium
configurations are
shown
in Fig. 1 for the
case of structures which have
a unique buckling mode
associated
with the classical buckling load. In each
of
the three
cases hown the prebuckling
state
ofthe perfect
structure
is
stable
for
P I P,
and
is
unstable
fot P)
P,
where it is
shown as a dotted
curve.
Case I illusrrares a
structure with a stable postbuckling behavior which can
support
loads in excess
of
P,
in the buckled
state.
Th e
behavior
of a slightly imperfect versio n of the
same
structure (or perhaps
for
a
slightly
misaligned
application
of the load) is depicted
by the dashed curve. Case
II
is
an example
of
a
structure which goes into a
stable
or
unstable
postbuckling
behavior depending
on
whether
the load increases
or decreases
ollowine
bifurcation. An
initial
imperfection
is all
that is needed to
prejudice the
deflection
one way
or the other. If an imperfection
in -
duces
a positive
buckling deflection,
the load-deflection
curve of Case II has
a limit load P.,
the buckling load
of
the imperfect
structure, which i s- less
than the
bifurca
tion load
of the
perfect
structure
P..
Case
I
is an ex
ample
of asymmetric
branching behavior,
while Case
II
illustrates
a structure
whose
buckling
behavior is
sym
metric
with respect o
the
buckling deflection
and whose
initial postbrrckling behavior is always unstable under
prescribed
oading
conditions.
The
classical buckling
load is
a reasonably
good
measure
of the load level
at which an imoerfect
realiza
tion
of the
structure begins
to undergo
significant
buck-
ling
deflections
if the
structure has
a
fully
stable
initial
postbuckling
behavior. In
the early
days,
eliance
on the
classical
buckling load
stemmed
from
the fact
that al
confrontations
between
test
and theory
were
for
eithe
columns
or plates,
and both
of these are
stable
in
the
postbuckling
regime.
When testswere
carried out on thin
shell
structures n
the early 1900's,
the
classical heory
became
suspectbecause
hen actuai
buckling loads
were
frequently
found
to
be as little
as one-quarter
of the
classical oad.
Except
for
a very
few
structures,
such as he
column
under axial
compression
[Ref.
47] and the circular r ing
subject
to inward
pressure
[Ref.
43]
,
ir is not
possibl
to obtain
closedform
solutions governingthe
entire
post
buckling
behavior.
Starting with Karman's
and Tsien'
work
on spherical
and
cylindrical shells,
a
large
number
of numerical
calculations
in
the
large-deflection
range
have
been
made.
Each
of these
calculations reDresen
an
attempt to
obtain the
load-deflection
behavio-r
of
the
perfect
structure,
and
in
particular
the minimum
load
the
structure can
support in the
buckled
state
(i.e.,
P*
in Fig.
1). This load
was held
to be
significant as a
pos
sible design oad
on the
grounds hat
the strucrure
could
always support at least this much load and that even
imperfections
would not
reduce the
buckling load
below
this value. This
concept
could not
be useful in
any uni-
versal
ense ince here
are well-known
examples
of struc
tures
with
negative
minimum
postbuckling loads.
An -
other
difficulty with
this proposal is
that the
buckling
process
of
such a structure is
a dynamic
one and the
buckled
structure may end
up deformed
quite
differ-
ently
from
what is
predicted
on the
basis of static
equil ibr ium
considerations
alone. In
any case,efforts to
validate
the minimum
load for
design
purposes
for
cylin-
drical
shell
structures have not
paid off,
as we will dis-
cuss n
the next
major
section of this
article; and this
idea
perhaps
should
be abandoned.
Emphasis in the analysis of imperfection-se nsitive
structures has
shifted to
the maximum load P,
which
can
be
supported
before
buckling is triggered
and to re-
lating P,
to the
magnitude
and
forms
imperfections
actually
take. This
was the
point of view adopted in
the
general nonlinear
theory
of
stability
[Ref.
34]
and
in
subsequent nvestigations
based on
the general heory
IRefs.
49-55]
.
135 4
7/24/2019 Post Buckling
3/14
The initial
postbuckling analysis
in its
simplest
form
yields
an asymptotically
exact relation between
the
load
parameter
P and the buckling deflection
6,
valid
in
the
neighborhood of the bifurcation
point of
the perfect
structure.
This expansion
is in the
form
P=P,( l+a6+b62+...)
(1)
where the coefficients d, b, ...
determine
the initial post-
buckling behavior. A number of important problems
are
characterized
by multiple buckling
modes associated
with the
classical
buckling
load. Two
of
these problems
will be
discussed n the next
section.
For
such
problems,
the
initial
postbuckling analysis
may
be considerably
more complicated,
Asymptotically exact estimates of the buckling
load
of the
imperfect
structure are
obtained by including the
first-order effects
of small
initial deflections. Only
the
component
of the initial deflection in the shape of the
classicalbuckling mod e enters into the resultant formula.
, l
, l
Pl
CASE tr
CASE Itr
Gcn.rol izcd
dofl6ction
FIG. 1.
LOAD.DEFLECTION CUFVES
FOR
SINGLE
Buckl ing dcflccl ion
-
8
BIFURCATION
BEHAVIOR
7- i
135 5
MODE
7/24/2019 Post Buckling
4/14
I f
the
amplitude of
this imperfection component
is
de -
noted by
6-,
then
for Case
II
with
5
)
O,
the general
theory
yields
ment
for
the
effect
of
an imperfection
-
a slightly
offse
load
-
on the buckling
load is remarkably good.
The
first
application
of initial postbuckling
theory
was
to
the
monocoque cyl indrical shell under
axial com
pression
[Refs.
34, 50]
which
wil l be discussed
n the
next section.
The
second
was a study of a
narrow
cylin
drical panel under axial compression
[Ref.
49]
which
displays a transition, depending on
its width,
from
a
stable
postbuckling behavior typical
of a flat plate to the
unstable behavior which characterizes the cylindrica
shell.
Beaty
[Ref.
94]
and Thompson
[Ref.
63]
applie
the general theory to
the axisymmetric buckling
of a
t'
I
(
deg
ees
FIG.
2, COMPARISON
OF THEORY
ON A SIMPLE
TWO-BAR FRAME
to/'.
t
/2
AND EXPERIMENT
(2)
['
+] =
['
]J
=.rsrf,
_P-
o6
PC
P
1__t
: t ' .
'\
-
(ad) ' ' -
where
a is
a
constant which
depends
on propert ies
of the
structure.
For
the symmetric
branching
point of Case
II I
(a= o,b
(
0),
the analogous
formula
is
1
-
P'
-
(a l6- l )
u
Pc
In
both
instances,very small values
of
q6
have
a sizable
effect
on PrlP-
and
it is this feature which accounts
for
the
extreme imperfection-sensitivity
of
many
structures
which have an unstable
postbuckling
behavior.
Interest in
the
initial postbuckling
approach
mush-
roomed in
England
in the
early 1960's
[Refs.
56-93]
.
Some
fine experimentation
is included in this work,
an d
an extensive series
of theoretical
investigations
have
centered around conservative
systems
characterized
y a
finite number of degrees f
freedom. A great deal of
this
work has been carried
out
at
University
College,
London,
by Chilver,
Thompson, Walker
and their students.
Sewell, at the University of Reading, also has concen-
trated
on the postbuckling
behavior
of discretesystems.
He
has particularly
emphasized
the systematics
of the
perturbation
procedure and
has used the
terminology
the
static perturbation
method
in referr ing
to this
method of analysis.
Both Thompson
and
Sewell
have
explored a variety of bifurcation
possibil i t ies
or
such
systems.
Recent studies
have been made with
an
eye to
providing
a
framework for postbuckling calculations
via
finite
element
methods.
Some
of
this work
is
summarized
in
[Ref.
39].
Perhaps
the
best direct
experimental validation
of
the
initial postbuckling approach
was
obtained
in a
series of tests
by Roorda
[Refs.
67, 68].
The simple
two-bar frame shown in Fig. 2 was loaded in a testing
machine which, i n
effect, prescribed
displacement
ather
than dead
load
and
in this way
the postbuckling
equili-
brium states were
recorded.
Experimental points
an d
theoretical predictions
IRefs.
53,
85] shown
as a solid
l ine
for
the load,
P,
versus
buckling
rotation,0,
for
the
perfect
structure
are displayed
in
the upper
half of
the
figure. The
agreement
between
theoty and
experi-
(3)
Effecl of lood
eccenlr ic i l ies
1
356
7/24/2019 Post Buckling
5/14
complete
spherical shell. Their analysis
yielded
the ex-
pected result that
the sphere has an unstable
postbuck-
l ing behavior. However,
it has been
shown
recently
IRef.
55]
that the initial
postbuckling analysis
n
i ts
simplest
form leads
to results that are of little practical
consequence
or
the complete
sphereproblem. We defer
a discussion of this matter
also to the
next
section.
The general theory
IRef.
34]
has been applied to
a
variety
of shell structures by Budiansky
and
Hutchinson
and their students at Harvard University
IRefs.
95-110] .
Included
among them
are toroidal shel l segments
under
various loadings, cylindrical
shells subject to torsion and
spheroidal
shells
oaded
by
external
pressure.
Results
shown in
Fig.
3
due to Budiansky and
Amazigo
IRef.
103] very nicely i l lustrate the outcome
of
such an analysis and its interpretation. In the upper
half of the figure
the classicalbuckling pressurep. in
nondimensional form
is
olotted as a
function
of the
length
param etet
Z
appropriate for either
a simply-
supported
cylinder of
length
L or a segmentof length L
of an infinite
cylinder
reinforced
by
rings
which permit
no lateral
deflection but allow rotation. The initial
post-
,.2.
t
=
(+)
Jt-vz
FIG.
3. COMPARISON
BETWEEN TEST
AND CLASSICAL
THEORY
WITH INITIAL POSTBUCKLING
PREDICTIONS
FOR A
CYLINDRICAL
SHELL UNDER EXTERNAL PRES-
SU
RE
buckling behavior is
symmetric with
respect to the
buckling
amplitude
6,
and
therefore the pressure-
deflection relation takes he
form
where b
is plotted in th e lower half of Fig.
3.
In the
figure, D is the bending
stiffness,
z is the Poisson atio
and t is the shell thickness. n this case, he asymptotic
relationship
between the buckling pressureand
the
im-
perfection s
12
l5
(-b)
l ;
It
n
p
where 6- is the
amplitude of the
component of the im-
perfection in the
shape of the
classical uckling mode.
A
wide range
of test data, col lected
by Dow
IRef.
111],
is
also included in
the
figure.
Measurements
of initial de-
flections
were
not
made n any
of
these
ests,
so
it is not
possible
o make a direct
comparison of test and theory.
On
the other hand, the
coincidence
of the large dis-
crepancy between
test and
classicalpredictions
within
the
Z-range
in
which
b
is
most negative bears out
the
im-
perfection-sensitivity
predicted.
The
complementary
character of an initial postbuck-
ling
analysis and a large
deflection analysis s
brought out
by
studiesof long oval
cylindrical
shellsunder axial com-
pression.
Kempner 's
and Chen's
IRefs.
112-114] numer-
ical results for
the
advanced postbuckling
behavior of
shells
with
sufficiently eccentr ic oval
cross section
showed
that
loads
above
the cla ssical bucklins load
could be
supported. Complete
collapseoccurs onll i when
the
buckling deflections
engulf the high
curvature ends
of the
shell.According
to an
initial
postbuckling
analysis
IRef.
105]
,
initial
buckling
wil l be
ust
about as
sensitive
to imperfections
as
for
the
circular cylindrical
shell as
would indeed
be expected
becauseof the
shallow buck-
l ing
behavior i.r .ach
case.The
composite picture is that
initial
buckiing will
be imperfection-sensitive
but not
catastrophic,
and loads
above the classical alues
should
be
possible. Recent
tests
[Ref.
115] have
confirmed
both
these
eatures.
One aspect
of shell buckling
which has attracted a
great
deal of theoretical
and experimental
attention in
the last few
years
is
the role
which stiffening plays in
strengthening
shell structures.
Some
rather
unexpected
effects
have turned
up. One of the most interesting
wa s
the observation
by
van
der
Neut
[Ref.
116]
over 20 years
ago that
the axial buci
7/24/2019 Post Buckling
6/14
imperfection-sensitive
than
their
unstiffened
counter-
parts,
but the level
of
sensitivity
can vary
widely.
For
example,
while the
classical
buckling load
of an
outside-
stiffened
cylinder
may be much
higher
(a
factor
of two
is
not
untypical) than
an inside-stiffened
one, rhe
im-
perfection-sensitivity
of
the outside-stiffened
shell
may
be much higher
as
well. Only a few experiments
are
available
to put these
predictions to a
test but those
which have
been reported
[Refs.
119, I20l
show pre-
cisely this trend.
An
extensive
series of bucklins tests
on stiffened
shell
structures
has
been underwiy
fot
a
number
of
years
under
the
direction
of
Singer at the Technion in
Israel
[Refs.
1.21.,L22l. This
program has produced
ample
evidence that
the classicalbuckling load is a more
reliable
measure
of actual buckline l oads for
stiffened
shells than
it
is
for
unstiffened ,h l l .. N u rthele ss, the
experimental
scatter and
the
discrepancies
rom
th e
classical
predictions
in these tests
are
indicative
of the
fact
thai
stiffening does
not
eliminate the problem of
imperfection-sensitivity.
An
up-to,date discussion f this
matter is
given
n
IRef.
123]
.
Other recent
work
in
the area of postbuckling theory
includes
an investigation
of the interaction of local
buckling and column failure for
thin-walled
compression
members
[Refs.
124, 125].
This work may help
to
eradicate
the
naive
approach
to optimal design of
struc-
tures l iable
to buckling by
an attempt to
equalize
the
local
and overall
buckling stresses.
There have
been important
advances n both theo-
retical
and
calculational aspects
of shell buckling in t he
last
decade. Buckling
equations have
been proposed
[Refs.
52,
126] which
are exact within the
context
of
first-order
shell theory. Computer
programs are
no w
available
for
accurate
computation of classical buckling
loads for
a wide
classof shells of
revolution
subiect to
axially
symmetric loads
[Refs.
1,27, 1281. These pro-
grams incorporate
effects of nonlinear prebuckling be-
havior
and discrete ings.
When the prebuckling deforma-
t ion
of the perfect
shell is a purely membrane
one with
no
bending, the initial
postbuckling analysis s
generally
simpler than
when bending,
and
nonlinear
prebuckling
effects must
be taken into
account. Most appl icationsof
postbuckling theory
to date have been in
cases
n
which
the prebuckiing response
s exactly
a purely membrane
state or could be reasonably
approximated by one.
Within the last
three
years
there have
been severalap -
plications
of the general theory to problems in
which
i t
is essential
to
include nonlinear
prebuckling effects
IRefs.
104, 109, 110, 1291,
and a generalpurpose
com-
puter
program
for
shellsof
revolution
subject to axisym-
metric loads has been put together IRef. 130] .
Status of
the
Postbuckling
Theory
of the
Cylindrical
Shell
under
Axial Compression
and the
Spherical
Shell
under
Uniform Pressure
The
cylindrical
shell under axial compression
ha s
served as the
prototype in
studies of shell buckling
IRefs.
131-161], but its geometric
simplicity is decep
tive, and in many respects his
structure,
together
with
the sphere subject to pressure, has the most
compli
cated postbuckling
behavior
of all.
In
large part, this
stems
from
the
fact
that a
large
number of buckling
modes
are
associatedwith the classical
uckling
load in
each
of these problems, and
consequently these shel
are
susceptible
to a wide
spectrum of
imperfection
shaoes.
it
is quite possible hat a paper by
Hoff,
Madsenan d
Mayers
[Ref.
152] has put an end to the
quest
for
th e
minimum load which
the
buckled
shell can support.
A
sequence f
large deflection
calculations
of the minimum
postbuckiing load have
been
reported, each
calculatio
more accurate han those which preceded t. Hoff, et al.
give a
convincing argument that a completely accurat
calculation, based on the
procedure which
had been em
ployed in all the previous nvestigations,would
lead
to a
value for
the minimum postbuckling
load
which
would
tend to zero
for
a
vanishing
hickness o
radius ratio.
O
course,
a
shell with a
finite
thickness to
radius ratio
would actually have a nonzero minimum load, but there
now
appears o be generalagreement hat this minimum
is not nearly
as
elevant
as had been thought.
The
roie
of boundary
conditions
was exte nsively ex
plored
in the
1960's and is now fair ly
well
settled. Ac
curate
numerical
calculations
of classical
ucklins load
which
account for nonlincar
prebuckling effects
,'rd
uo.
ious boundary
condit ions have been
made
IRcf.
162]
The
classical load for
cylindrical
shell of moderat
length
which
is clamped
at both ends is
about
93 per
cent of the load
predicted
by the
well-known classica
formula
based
on a
calculation which isnores
end con
ditions
altogether.
So-called *e^k bo,r,ldrry
condition
for
which no tangential
shearstresss exertcd
on the end
of
the shell
educe
he
buckling load
by a
factor
of abou
two
IRefs.
163,164].
More
reccntly
i t has been
show
that relaxed tangential restraint along a small raction of
the edge
has
an almost equally
detr imental effec
IRef.
165].
Near
perfect
cylindrical
shell specimens have
bee
produced
by Tennyson
IRef.
166] and Babcock
and
Sechler
IRef.
144],
and these
shel lsbuckle very
c lose o
the classical
uckling load. High-speed
hotography ha
resolved
the
apparent
discrepancy betwecn
observe
buckling
patterns
and the
classicalmode
shapes
IRefs
167-1711.
Buckle
wavelengths
associatedwith the
col
iapsed
shell are much longer
than the
classical uckling
mode
wavelengths,
but the
wavelengthsassociated
with
deformation
patterns
photographed
ust
after buckling
has
been tr iggered
are in
good agreement
with the pre
dictions
of initial
postbuckling theory.
The
long
cylindrical
shell subject to axial
compres
sion
is
one of the examples
used n
IRef.
34]
to i l iustratc
the general
theory
of elastic
stabil i ty. Boundary condi
tions
are
neglected,
and consideration
is restr icted
to
mode
shapeswhich
are periodic in both
the axial
an d
circumferential
coordinates.
Due to the iarse
number
of
1
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simultaneous
buckling
modes, there are
many possible
static
equil ibr ium branches which emanate
from the bi-
furcation
point
of a perfect cylinder, and all
of them are
unstable.
Remarkably, t he initial curvat ure
of the curve
of load versus end shortening (Pversus
e) is the same
or
all of them and
just
after bifurcation
Effects
of certain
imperfections were
also studied.
Among the
shape
possibil i t ies
represented by all t he
classical
buckling modes, a geometric
imperfection
in
the shape of
the axisymmetric buckling
mode results
n
the
largest relative reduction in the b uckling
load
IRef.
51].
The ratio of the buckling
load
to the
classical
oa d
is related
to the amplitude
of
such
an axisymmetric
imperfection 6
by
where c
=
l,6\-n
and
z
is
Poisson ratio. When the
maximum load
{
is
attained,
the shell
has undergone
lateral deflections which are only
a fraction of the shell
thickness,
and
an imperfection amplitude of only
one-
fifth
a shell thickness causesalmost a 50 oer
cent reduc-
t ion in the buckl ing load.
fq.
(7)
is an asymptotic
formula;
and
like all initial
postbuckling predictions of this type,
it is accurateonly
for
sufficiently small
mperfection amplitudes. n general,
it
is difficult
to assess
he range
of validity of
asymptotic
results. For
this
particular imperfection
shape,
it
is
possible
to obtain
an independent
and more
accurate
estimate
of the P,
uemns
5
relation
which is
basedon a
f in i te
deflection
calculation
IRef.
51].
The
asymptoric
predictions
( q.
7) are plotted
with the more
accurate
results
n
Fig.
4, and two
predictions
compare well over
a
substantial part
of the range of int erest.
The calcula-
t ion
of
IRef.
51] was
repeated
by Almroth
[Ref.
172]
with the aim of get ting better accuracy.He found that
the buckling load prediction
may be somewhat lower
than
those shown
in Fig.
4
if the parameter Z
(Fig.
3)
is very large.
Some
experimental
verification of these buckling
load
calculations
has
been obtained by Tennyson and
Muggeridge
IRef.
173] with tests of
cylindrical shell
specimens with axisymmetric
sinusoidal imperfections
which
were carefully
machined into
them.
Points repre-
senting
he experimental
loads
for five
shells
are plotted
with
two theoretical
curves
in Fig,
5.
The
solid curve
is
the result of the finite
deflection prediction of
IRef.
51 ]
for
the imperfection
wavelength coincident wit h
those
of the
specimens.
The dashed
curve
is a plot of the re-
sults of a numerical caiculation which takes nto account
nonlinear
prebuckling deform ations associated
with
the
clamped end conditions and the thickness variations
present in the t est cylinders
IRef.
173]
.
At extremely
small
imperfection levels, end
conditions dominate.
Once
the
imperfection amplitude is
just
a
few
per cent
of the
shell
hickness, he end effect is negligibleand the
two predictions are virtually identical.
There have been some efforts to translate and adaot
the
initial
postbuckling predictions into a
form
directiy
useful for engincering
design
IRefs.
t29, 1 30, 1611
.#,-1]
(6)
f
1, .12
3c
l6- l
p
l t - - l=r l ' l
;
e)
L
P,J
2lr l l ,
Gcn6rol hcory
Eq.
7)
-a.
E
t
FIG. 4. EFFECT
OF AXISYMMETRIC IMPERFECTIONS IN
THE SHAPE
OF
THE
AXISYMMETRIC BUCKLING MODE
OF A
PERFECT
CYLINDRICAL SHELL UNDER AXIAL
COMPRESSION
o.o2 o.o4 oo6
FIG.
5. COMPARISON
BETWEEN TEST
AND THEORY
FO R
BUCKLING
OF CYLINDRICAL
SHELLS WITH
AXISYM-
METRIC
IMPEBFECTIONS
LOADED
IN
AXIAL COMPRES-
SION
1
35 9
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Reinforcing this trend is
recent
work
by Arbocz and
Babcock
IRefs.
17
4,
17
5]
in which
imperfections
present in a
number of test
specimens
were
carefully
measured
and mapped. Special
techniques have been
used to calculate buckling
loads
for
certain
imperfection
shapes
which are quite different
from the classical uck-
l ing modes or
for
imperfection shapes
which are
only
characterized statistically. Some
of
these will be touched
on in the next section.
Since
Karman's and
Tsien's
[Ref.
301
first large de-
flection calculation for the axisymmetric postbuckling
behavior of
a
complete
sphericai shell under
uniform
pressure,a
fair ly large number of paperson
the
buckling
of
caps
and spheres
haue
been
published.
Many of these
are
referenced n a recent papcr on the subject
IRef.55].
Here,
we wil l on ly mention some
of the salient
eatures
peculiar
to the
sphere
problem which
have recently
come to
light.
Like the cyli ndrical shell under
axial comprcssion,
a
complete
sphcrical shell under
external pressure
ha s
many buckling
modes associatcd
with the
classical
uck-
l ing pressure, and onc of these
is axisymmetric.
Many
unstable equil ibr ium branchcs
emanate
rom
the bifurca-
tion poin t of the perfect shell.
A
shallow
shell analysis
of
mult imode buckl ing of the sphere
IRef.
101] indicates
that this
shell
is highly
sensitive
o both axisymmetric
and nonaxisymmctric
imperfections, again
ust
as
or
th e
cylinder. While this shallow shell
analysis
s not
r igor-
ously applicable to
thc
complete
sphere,
its
relative
simplicity
reveals he mechanicsof
the buckling phenom-
ena of the spherc vcry clearly.
In particular, the shallow
buckling modes are
in
str iking
similarity
with expcri-
mental modes
.eported in
IRef.
1761.
An al tcrnat ive
approach
to
this
problem by a straightforward
pcrturba-
t ion tec lrn ique s detc lopcd
in
IRefs.
177,1781 .
Initial postbuckling analyses
or axisymmetric
buck-
l ing have been
given
by Beaty
[Ref.
94],
Thompson
lRef.63l ,
Walker
[Ref.
80
1,
and most
reccntly by
Koiter
IRcf.
55] . A mult imode analys is in which
the ax isymmetr ic
mode coLlples
with
nonaxisymmetr ic
modes, analogor.rs
o
cylindrical
shell behavior,
also
wa s
given n
IRef.
55] .
Impcrfcct ion-sensi t iv i tywas
found
to
be highe
for
multimode buckling
than
for
axisymrnetr ic
buckl ing.
A most
important d iscovery of
1nef.
55]
was
that
the axisynlmetric
rcsults
just
mentioned,
which were
obtained
by
a
perturbat ion expansionabout
the bi furca-
tion point, havc an
cxtremely
small
range of validity
which
:LctLrally anishes
as the thickncss to
radius
ratio
of thc shell
goes to
zero. The principal
reason
for
this
unusual
l imitation
on the initial postbuckling
results s
thc
occurrence of modes,
associatcdwith
eigenvaiues
which are only vcry sl ightly largcr than the critical
eigenvalr.res,
hosc
amplitudes
become comparable
in
magnitude
to the
amplitudes of the
classical
buckling
modes
outside
of a
very
small
neighborhood
of
the bi,
furcation
point.
For
all practical
purposes, the initiai
postbuckling
analysis n
its standard form
breaks down
under
such
cr'rcumstances.
more
powerful
form
of
asymptotic
expansion,
also
first
detailed in
[Ref.
34]
with an extended range
of validi ty was applied to th
axisymmetric problem
IRef.
55].
The
outcome of th
special
analysis
s that the
spherical
shell is highly im
perfection-sensitive,
more or less to the
same
degreea
the
cylindrical shell under axial
compression,
even
whe
the deformations are
constrained
to be axisymmetri
An approximate
analysis
IRef.
179] and several
ume
ical
calculat ions
Refs.
180-1831
back
up th is conclus io
New ResearchDirections
It seems
safe
to predict that much
of
the
impetus
postbuckling
theory as a developing
subject
wil l
contin
to come
from
questions which
arise
in
the analysis
sheil structures.
Efforts
are
being directed to
interpretin
postbuckling predictions and rendering
them useful
fo
engineering
purposes. If this is to be
accomplishe
further
calculations wil l be needed for more realis
imperfections
than those
which
are readily accomm
dated by the initial
postbuckling analysis. Approach
which incorporate
statistical
descriptions
of
imperfe
tions are also l ikely to receive growing
attention
in th
coming years.
Very reccntly,
Sewell
IRef.
184] has explored vario
ramifications
of the more powerful expansion metho
of
[Ref.
34] (which
was
employed
to get around rh
difficulties in the
sphere problem
referred
to above)
it
applies o conservative
ystemswith
a
finite
number
degrees f
freedom.
Thompson
IRcf.
185] haspropos
a somewhat different variation of the method, also
fo
discrete rather than
continuous systems.
The aim
each
case s to develop a uniformly valid asymptotic e
pansion which gives an extended range of
validity
fo
those rathcr exceptional
problems
n
which the standa
expansion breaks down
outside the
immediate vicini
of the bi furcat iorr
poinc.
Effects
of
localized
dimple
imperfections
on
th
buckling of a bcam on a
nonlinear
elastic
foundatio
have been explored usir ig a variety of techniques, an
asymptotic
formulas
analogous o eqs.
(2) and (3) ha
been uncovered
IRef.
186]
. Appl icat ion of these
spec
techniques has
been made
to study the effcct
of
loc
nonaxisymmetric
imperfections
on buckling
of cylindr
cal
shclls
under
external pressure
IRef.
187] . In
additio
both
theoretical
and exocrimentai
studies have be
made
of cyiindrical and
conical shelisunder axial com
pression in the presence of axisymmetric dimple im
pcrfect ions
IRefs.
188, 1891. With a
continuing
rap
growth of numerical methods
of shell
analysis, t
shou
soon be possible to make
reasonably
accurate
calcu
tions of the buckling l oad of an arbitrary shell structu
in the presence
of almost any
form
of
imperfection. Th
potential of
such
methods has already
been
demonstrat
IRefs.
190-1931
for
problems n which i t is necessary
convert the governirrg
onlinear
partial differential equ
tions to an algcbraic system by spreading a tw
dinensior.ral inite-diffcrcnce
srid
over the shell
midd
surface. No doubt calcullt ionl of t his sort wil l not b
I
360
7/24/2019 Post Buckling
9/14
inexpensive
in
terms
of
computation
t ime
for
many
years to come;
but if
thcy are
carefully
selected,
such
calculations
should be most
usefui for
imperfection-
sensi
iv i ty asscssmcnt.
The dcvelopment
of numer ical
methods for
carry ing
out
the initial
postbuckling
analysis
seems ikely
to pro-
grcss along
the l ines
of both
the
f in i te
difference
pro-
ccdure
IRef.
1301
and
the
f in i te
element
method
IRefs.
80, 194].
An interest ing
scheme
has been
pu t
forward
for
approximately
accounting for
nonlinear
pre-
buckling behavior in an initial postbuckiing analysis n a
way
which exploits
the
computational
advantages
f the
f in i te
element
procedure
IRefs.
195, 196]
.
The idea
be-
hind
the scheme is the treatment
of the
prebuckling
nonlincarity
as a
generalized mperfection.
Preliminary
attempts
have
been
made to
come to
grips
with
somc of the
statistical
aspectsof imperfection-
sensi t iv i ty
IRefs.
197-199].
A rather
complete
analys is
IRcf.
200]
of an infinite
beam
with random
initial de-
flections
resting
on
a
nonlinear
elastic oundation
yields
a
relationship
between the
buckling load
of the
beam and
the root
mean
square
of
the
imperfection
amplitudes
with
an
implicit
dependence
on the
correlatjon
func-
tion for
the imperfection.
Calculations or
the buckling
load of an infinite cylindrical shell under
axiai
compres-
sion with random
axisymmetric imperfections
have been
made
IRefs.
20I,
202]. Here
again, there
emerge
asymptotic formulas
similar
to
eqs.
(2) and
(3) which
now
relate
the buckling
load to a value
of the imperfec-
tion
power
spectral density
at a particular frequency
which corresponds
to the
frequcncy
of
the classical
axisymme
ric mode
IRef.
201]
.
Thcre
is
sti l l a long
wa y
to go
before
results
such as thcsc
wil l be useful
to the
structural cngineer,
but an encouraging irst
stcp
in
this
direct ion has
been taken
IRef.
2031.
Important
extensions
of
postbuckiing theory remain
to be
made to include
buckling
under dynamicaily
ap -
plied loads
and buckling
in the plast ic range.
Dynamic
buckling calculations are not new, but no unifying
theory
is available
which is at all
as comprehcnsive
as
that for
static
conservative oadings.
The initial
post,
buckling
analysishas
been broadened
n an
approximate
fashion
to
inc lude dynamic
effects
IRefs.
95-97],
and
some
simple results
with general implications
have
been found.
Buckling
in
the plastic range is
a phenomenon
that is
not
yet
fully
understood. Nonuniqueness
of incremental
solutions
in plastic
branching
problems
starts at the
lowest
bifurcation load,
usually without loss of
stabil i ty,
and a further
load increase
s required
initial ly in
th e
postbuckling
range
[Rcfs.
204, 205).
A detailed dis-
clrssion of the famous
column
controversy among
Considerc, Engesser,
von Karman
and
Shanley
IRefs.
206-2711
is
given in
[Ref.2121.
ln the
case
ofplates
and
shells he problem
is evcn
more
complicated
because
the physicaliy
unacceptable deformation
theory of
plasticity
yields results in
better
agreement
with exper-
iments
than thc
simplest low
theory.
Even
if
flow theory
is
ccrtainly
more acceptable rom
the physical point
of
view,
i t is
somewhat
doubtful ,
however,
that the
simplest
J
2
theory
would
be
adequare in
bucki ing
problems.
To
some
extent
this
discrepancy
also
has been
explained
by
Onat
and Drucker
IRef.
213]
,
by proper
al lowance for
in i t ia l
imperfections
in thei r
simple example. Even
more
signi f icant,
perhaps,
are Hutchinson's
recent
resul ts
on
the postbuckl ing
behavior
in
the plastic
range
of
struc-
tures
which
are highly
imperfection-sensi t ive
in
the
elastic
domain
[Ref.
2141
.
lt
is
found
in his
examples
(a simple
structural
model
and
a spherical
shel l under
external pressure) that the sensi t iv i ty to imperfections
is equal ly
marked in
the
plastic range,
in
spi re
of the
initial rise
in
the load
at the lowest
bifurcation
point of
the
perfect
structure.
A
simi lar phenomenon
has been
observed
by Leckie
IRef.
215] in
the postcoi lapse
be -
havior
of a r ig id-plastic
sphericai
shel l cap. Hutchinson,s
analysis
also
confi rms
the earl ier f indings
IRef.
213]
that the
discrepancy
betweer.r
the predict ions
of
deforma-
tion theory
and f low
theory
largely
disappears
in the
presence
of
imperfections.
References
[1]
Grigol 'uk,
E.
I., and Kabanov,
V. V.,
Stabi l i ry
of
Cyl in-
drical Shel ls, ( in Russian),SeriesMechanics8, Nauki , Moscow,
L969.
l ,2 l
Wagner, H.,
Ebene blechwandtrager
mit sehr dunnem
stegblcch, Zeitschr.
f.
I:lugtechn.
u.
Motorluftschiffahrt
2O,
200,
227 256
27
9 306 t929 .
L3]
Cox,
H. L.,
Buckl ing
of thin plares
n compression,
Acron. Res.
Comm.,
R. and
M.
7554,1,934.
l,4l
Marguerre,
K., and
Trcfftz, E.,
Ubcr
die tragfahigkcit
eines langsbclasteten
plattenstrei fens
nach.
Uberschrei ten
de r
beul last,
Zei tschr.
fur
Angew.
Math u, Mech.
17. 85-1OO, .937.
t5]
Marguerre,
K.,
Die
mittragcndc
brei te der
gcdruckten
platte,
Lu
ftf
ahr
for
schung
1
4,
121-128, 19
37
t6]
Kromnt,
A., and Marguerre,
K.,
Verhal ten
eines von
schub- und
druckkraften
beansoruchten
olattenstrei fensober-
halb
der
beulgrenzc, Luftfahrtf i rschung
14,
627-639, 1937.
I7l
Friedrichs,
K.
O.,
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