Post Buckling

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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


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


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


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


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


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


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


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

35 8


7/14

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


8/14

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


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.,

and

Stoker,

J. J.,

The

nonl inear

boundary

value

problem

of the

buckled plate, American

J.

of

Math. 63, 839-888, 1941.

t8l

Fricdrichs,

K.

O., and

Stoker,

J.

J.,

Buckl ing

of thc

circular plate

beyond

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