Buckling of cylindrical shells with random imperfections

BUCKLING O F CYLINDRICAL

SHELLS

WITH RANDOM IMPER FEGTIONS

T h e s i s by Rena S c h e r F e r s h t

In P a r t i a l Fulfillment of the R e q u i r e m e n t s F o r the D e g r e e of

Doctor of Philosophy

California Institute of Technology Pasadena, California

ACKNOWLEDGMENT

The author wishes t o e x p r e s s h e r gratitude and s i n c e r e

appreciation to D r , E, E. S e c h l e r f o r h i s help, advice and encourage- m e n t which enabled h e r t o c a r r y out this study. The advice and

suggestions of D r , C. D. Babcock a r e a l s o v e r y m u c h appreciated.

This opportunity is a l s o taken to thank D r , T , K , Caughey f o r m o s t

helpful d i s c u s s ions.

P a r t i c u l a r appreciation is e x p r e s s e d to my husband, D r , S. No F e r s h t , to whom this work is dedicated. His patience, understanding

and encouragement w e r e a s o u r c e of inspiration during my e n t i r e

engineering studies.

This study was supported in p a r t by the National Aeronautics

and Space Administration under R e s e a r c h G r a n t NsG 18-59 and t h i s

ABSTRACT

The buckling stability analysis of long cylindrical shells with

random imperfections subjected to axial load i s t r e a t e d using two

different approaches. The f i r s t study i s based on a Eyapunov method

which enables one t o establish sufficient conditions for buckling

stability of a long cylindrical s h e l l with axisyrnmetric random i m p e r -

fections. A perturbed s y s t e m of equations in the neighborhood of the prebuckling solution i s investigated, By reducing the problem to a

s y s t e m of integral equations, it is observed that the stability boundary

value problem of a long s h e l l i s s i m i l a r t o that s f a dynamical system with random p a r a m e t r i c excitations,

Initial imperfections were a s seurmed to have Gaussian dis t r i -

bution and a n exponential cosine c o r r e l a t i o n function, The c r i t i c a l

load was obtained a s a function of the root m e a n s q u a r e of the imperfections. Results obtained a r e qualitatively s i m i l a r to those

of Koiter for a periodic imperfection (Ref. 1).

The second p a r t i s based on the approximate method of

truncated hierarchy. The prebuckling s t a t e of equilibrium f o r

a s y m m e t r i c imperfections i s found by a successive substitution

technique, A homogeneous variational s y s t e m of equations i s s e t up

in o r d e r to examine the existence of bifurcation in the neighborhood

of the equilibrium state. These l a s t equations involve random

p a r a m e t r i c t e r m s , The truncated h i e r a r c h y method i s applied and

correlation functions associated with asymmetric imperfections are

examined numerically. Qualitatively the results obtained are as

T A B L E O F CONTENTS

P a r t

-

P a g e

I INTRODUCTION 1

11 ALMOST SURE STABILITY O F LONG CYLINDRICAL SHELLS WITH AXISYMMETRIC RANDOM

IMPERFECTIONS

6

1. P r e l i m i n a r i e s

6

2. B a s i c Equations 10

3 . Methad of Solution 11

4. Derivation of Stability Condition 5. NurnericaP E x a m p l e

6.

Concluding R e m a r k s

I11 APPROXIMATE STABILITY ANALYSIS O F LONG

CYLINDRICAL SHELLS WITH ASYMMETRIC

RANDOM IMPERFECTIONS

1. P r e b u c k l i n g Equilibrium with A s y m m e t r i c ICmperfections

2 , V a r i a t i o n a l Equations and Stability Analysis

3 , P a r t i c u l a r C a s e s

i. A x i s y m m e t r i c I m p e r f e c t i o n s

ii. A s y m m e t r i c I m p e r f e c t i o n s

4. Concluding R e m a r k s REFERENCES

LIST O F SYMBOLS

Constant defined on P a g e 1 3

Constant defined on P a g e 13

M a t r i x defined on P a g e 17

M a t r i x defined on P a g e

6

M a t r i x defined on P a g e 1 0

€h3/ l 2 ( 1 - ~ 2 )

E

Young' s modulus

F'

S t r e s s function

-

f=-

P a r a m e t r i c coefficient m a t r i x

- f ~

S c a l a r function defined on P a g e 1 0

+(X,QI)

,

4,

tr,%

,

,%,(r,

*1)

T r a n s f e r functions

3a(t,q)

)

h(3j

q)

1 ht(389),h2tsJq)

G*

G r e e n ' s function defined om P a g e 13

G L

Constant m a t r i x defined on P a g e 1 0

W e

G r e e n ' s function defined on Page 1 3

T r a n s f o r m e d t r a n s f e r fumctions

c,(a,p),

H ( ~ , B ) ~ . H ~ ( ~ , P ) ,

H ~ ( ~ , P B )

&k

Shell thickness

I

Unit m a t r i x

LIST O F FIGURES

F i g u r e

1 T y p i c a l P o w e r S p e c t r u m C u r v e s

P a g e 7 2

2

S t a b i l i t y B o u n d a r y . f o r a C y l i n d r i c a l S h e l l

with A x i s y r n m e t r i c I m p e r f e c t i o n s 7 3

3 Buckling S t r e n g t h Dependence f o r D i f f e r e n t

P o w e r S p e c t r u m P a r a m e t e r s 74

4 Buckling S t r e n g t h Dependence f o r D i f f e r e n t 8(,

LJST O F SYMBOLS (continued)

k

Non-dimensional c i r c u m f e r e n t i a l wave number

Constant defined on P a g e 21

Constant defined on P a g e 2 1

Constant defined on P a g e 21 Constant defined on P a g e 21

Membrane s t r e s s r e s u l t a n t s

Axial wave n u m b e r

T r a n s f o r m a t i o n m a t r i x defined on P a g e 17 H e r m i t i a n m a t r i x defined on P a g e 9

G r e e n s s function defined on P a g e 1 8

C o r r e l a t i o n functions

Radius of the s h e l l

P h a s e function defined on P a g e 40

Power s p e c t r u m functions

Non-dimensional r a d i a l deflection of buckled mode

\%

H e r m i t i a n m a t r i x defined on P a g e

9

W

Radial deflection of s h e l l

rn

Initial r a d i a l deflection of s h e l l (imperfection)

bQd # G

) G

Non-dimensional r a d i a l deflections

,

w,

8 Axial coordinate

LIST O F SYMBOLS (continued)

Y

C i r c u m f e r e n t i a l coordinate

Independent v a r i a b l e s in the t r a n s f o r m e d domain

c

,r*

Functions defined on P a g e 1 5

$

(4 D i r a c ' s d e l t a function

E

j &w P a r a m e t e r s of the power s p e c t r u m function

5

( A ] Function defined on P a g e 2 3

'7,

1

qco

Eigenvalues of

Bi

8

I P a r a m e t e r s of the power s p e c t r u m function

65'

T r a n s f o r m a t i o n m a t r i x defined on P a g e 21

K j

K I

Kt

Root m e a n s q u a r e s of initial imperfections

)a

s./5&

Diagonal m a t r i x defined on P a g e 17

Eigenvalues of

d%

Eigenvalues of

V

P o i s s o n f s s a t i s

Applied a x i a l s t r e s s

LIST O F SYMBOLS (continued)

Non-dimensional s t r e s s function associated with

buckled mode

I, INTRODUCTION

In the l a s t t h r e e decades it h a s been recognized that s m a l l

g e o m e t r i c a l imperfections a r e the m a j o r c a u s e f o r the reduction in

the buckling s t r e n g t h of c y l i n d r i c a l s h e l l s , subjected to a x i a l loads.

P a r t i c u l a r analytical studies of the problem, using approximate

techniques and considering s i m p l e periodic modes of imperfections,

have been c a r r i e d out by Moiter (Refs. 1, 2 ) , Donne11 and Wan (Ref. 3 ) , Hutchinson (Ref, 4), Budiansky and Hutchinson (Ref. 5))

Babcock and S e c h l e r (Ref.

6 )

and o t h e r s . F e w a t t e m p t s have been

m a d e to study p r o b l e m s a s s o c i a t e d with local imperfections, a l m o s t

p e r i o d i c and s t a t i o n a r y random imperfections, In o t h e r words, the

s t u d i e s that have been c a r r i e d out s o f a r a r e r e l a t e d to i d e a l c a s e s

and give qualitative insight to the problem.

In the s e a r c h f o r a m o r e r e a l i s t i c d e s c r i p t i o n of the geometry

of i m p e r f e c t i o n s , i t was suggested by Bolotin (Ref. 7 ) that the i m p e r - fection function should b e considered as a random v a r i a b l e , B y using s t a t i s t i c a l techniques b a s e d on probability distributions and their

t r a n s f o r m a t i o n s one could evaluate the probabilities f o r buckling

f a i l u r e . This outlined p r o c e d u r e is p e r h a p s too g e n e r a l and becomes

i m p r a c t i c a l a s the number of random v a r i a b l e s i n c r e a s e s ,

The f i r s t a t t e m p t to s e l e c t a l e s s g e n e r a l c l a s s of random

imperfections, a s s u m i n g s t a t i o n a r i t y and ergodicity, has been made

imperfections. In a r e c e n t work by Amazigo (Ref.

9),

the problem of buckling of long cylindrical s h e l l s under axial load h a s been solved

f o r the c a s e of a x i s y m m e t r i c initial imperfections. The approximate

technique of truncated h i e r a r c h i e s h a s been utilized in this solution.

In both s t u d i e s , a n exponential c o s i n e c o r r e l a t i o n function f o r the

imperfections has been examined. It should b e noted that the solution

techniques in t h e s e two studies w e r e based on the a s s u m p t i o n s that

t h e initial imperfections w e r e s m a l l .

In the p r e s e n t work two different techniques have been used.

The f i r s t p a r t c o n s i s t s of a stability analysis which is based upon

Lyapunovss d i r e c t method, and h a s been utilized f o r the axisyrnmetric

s t a t e of imperfections. No attempt h a s been m a d e t o extend i t to a

m o r e g e n e r a l s t a t e s f imperfections, although i t is felt that this c a n

a l s o b e achieved. The a n a l y s i s is b a s e d on a study by Caughey and

Gray (Ref. 10) f o r dynamical s y s t e m s with s t a t i o n a r y random

p a r a m e t r i c excitations.

Considering the problem of long cylindrical s h e l l s , a p a r -

t i c u l a r c l a s s of random imperfections, which is of p r a c t i c a l

significance, is the s t a t i o n a r y s t a t e of imperfections with r e s p e c t to

the axial variable, By expanding the imperfection function in F o u r i e r s e r i e s i n the c i r c u m f e r e n t i a l direction, one can s e t up the problem

considering the F o u r i e r coefficients as the random v a r i a b l e s . T h e s e coefficients a r e a s s u m e d to b e s t a t i o n a r y with r e s p e c t to the axial

independent v a r i a b l e and m a y b e c r o s s c o r r e l a t e d . In addition i t is

a s s u m e d that the joint probability distribution f o r t h e s e coefficients

random variables satisfy the ergodic property.

By considering the perturbation equations of the prebuckling

solution it i s possible to obtain a l i n e a r s y s t e m of ordinary differ-

ential equations with constant and random p a r a m e t r i c coefficients.

By disregarding the t e r m s with p a r a m e t r i c coefficients the s y s t e m i s

reduced to a stable one a s long a s the load i s below the c l a s s i c a l

buckling load.

When the p a r a m e t r i c coefficients a r e included by reducing the

problem into a s e t of integral equations i t was observed that, with

proper modifications, the stability analysis i s s i m i l a r to that of a

dynamic s y s tem where the axial variable replaces the time variable,

As soon a s this p a r t of the analysis i s established, the application of

the Lyapunov technique becomes straightforward.

Lyapunov's method yields sufficient conditions f o r stability,

but it often occurs that this technique leads to extremely conservative

conditions. One of the m a j o r problems with Eyapunovq s method i s

that of determining the proper m a t r i x inequalities in o r d e r to derive

s h a r p e r stability conditions. This p a r t of the problem has been

handled with particular c a r e , yet i t i s felt that this p a r t i s s t i l l open,

a s in dynamical s y s t e m s , to improvement,

The p r e s e n t method of stability has been tested numerically

f o r the particular c a s e s f a x i s y m m e t r i c random imperfections, By

considering a Gaussian distribution and a n exponential cosine

correlation function, the c r i t i c a l load was obtained a s a function of the root mean s q u a r e of the imperfections, The curves obtained a r e

s p e c t r u m function coincides with the frequency of the c r i t i c a l l i n e a r

buckling mode.

Finally one should point out that the p r e s e n t study is perhaps

only the f i r s t s t e p in this direction. By using the s a m e technique,

sufficient conditions f o r stability of cylindrical s h e l l s , subjected to

o t h e r types of loads, a s well a s d e t e r m i n i s t i c , a l m o s t periodic s t a t e s

of imperfections, can b e obtained.

The second p a r t of the p r e s e n t work i s b a s e d on the approx-

i m a t e method of truncated h i e r a r c h y , A prebuckling approximate

solution is obtained by using the method of s u c c e s s i v e substitutions,

which is valid under the r e s t r i c t i o n that the root m e a n s q u a r e of the imperfections is s m a l l compared to the s h e l l thickness. Once this

p a r t of the problem is solved one c a n t u r n to the stability analysis.

In o r d e r to verify the existence of a second solution in the neighbor-

hood of the prebuckling equilibrium s t a t e , a v a r i a t i o n a l homogeneous s y s t e m of equations is s e t up.

In

o t h e r words t h e s e equations will

enable one to examine the existence of bifurcation. Assuming that

the initial imperfections a r e s m a l l , the method of truncated h i e r a r c h y

can be applied following (Refs. 11, 12,

91,

As a r e s u l t one obtains a s y s t e m of integro-differential equations f o r the p r o p e r c o r r e l a t i o n

functions, This problem is f u r t h e r reduced by applying double

F o u r i e r t r a n s f o r m s which l e a d s t o a s y s t e m of homogeneous

equations f o r the p r o p e r power s p e c t r u m functions. The condition

f o r existence of a non-trivial solution yields the d e s i r e d relation f o r egistence of bifurcation. Naturally the lowest load and the associated

Exponential cosine correlation functions a r e examined

numerically for combinations of a s y m m e t r i c and axisymmetric modes

of imperfections. The correlation function p a r a m e t e r s a r e selected

carefully in o r d e r to justify the applicability of the numerical results

obtained. This l a s t argument naturally i s based on physical intuition

r a t h e r than on experimental evidence. In a work by Arbocz and

Babcock (Ref. 13) imperfections have been m e a s u r e d by e l e c t r i c a l

m e a n s ; however, the record was too s h o r t and therefore reliable

c o r r e l a t i o n functions could not be established. Although the m e a s u r e d

r e s u l t s a r e p r e c i s e and carefully obtained, the number of c r o s s

sections of the cylinder for which imperfections were m e a s u r e d i s

not sufficient f o r data reduction in o r d e r to s e t up numerically the

s t a t i s t i c a l p r o p e r t i e s s f the imperfections. This, for the time being,

l e a v e s only the possibility of examining known c s r r e l a t i o n functions

f o r testing the theory. As mentioned before, the p a r a m e t e r s in these

functions a r e selected on the b a s i s s f intuition which really relies on

speculations

.

It i s hoped that, in the future, the p r e s e n t m e a s u r e m e n t techniques will be improved considerably, and perhaps new means

f o r the m e a s u r e m e n t sf imperfections will be found, Then the

11. ALMOST SURE STABILITY O F LONG CYLINDRICAL SHELLS WITH AXISYMMETRIC RANDOM IMPERFECTIONS

1. P r e l i m i n a r i e s

The p r e s e n t study t r e a t s the stability of a boundary value problem. In general Lyapunovls second method t r e a t s asymptotic

stability of dynamic s y s t e m s , in other words it i s related to initial

value problems. In o r d e r to r e l a t e the boundary value problem to an

equivalent dynamic s y s tern in a steady s t a t e response o r a stationary

response in a s t a t i s t i c a l s e n s e Pet u s investigate the following system of equations.

where

%

i s a n N-column vector with the components x.

x 9

-

i

=

1 2 N Itj i s a constant N x N m a t r i x and

FQsj

i s an N x N m a t r i x whose nonzero elements a r e stochastic p r o c e s s e s :

It is assumed that the m a t r i x

8

h a s a t l e a s t one s q u a r e root

A

_,

the eig,envaPues of which a r e distinct and have negative r e a l parts.

with the conditions a t infinity

The solution of (1.3) as

5

-r

+

as

can be obtained from the

equation

F u r t h e r m o r e , a s

5

-

-og the solution can be obtained from

Equations (1.5) and (1.4) can be combined to one equation a s

follows

The stability is defined in the s e n s e that a s

₃

-9 the

l a t e r a l deflection of the shell tends to zero. This i s known a s

asymptotic stability, and the t e r m almost s u r e stability is associated

with it, One can therefore s t a t e that conditions (1.4) can be met i f

and only if there i s a m a t r i x P%

,

the eigenvalues of which have

negative r e a l p a r t s , This l a s t condition together with (1.4) a s s u r e s

stable solutions a s

1%

o r by considering (1.7) and

Turning now to (1.1) and assuming that f o r

(3)

=

0

this s y s t e m i s stable, let it a l s o be assumed that the elements of

hf)

_,

fi*

( l )

,

satisfy the following properties,

a. The p r o c e s s e s a r e continuous in -00 (

5

00

b e

The p r o c e s s e s a r e s t r i c t l y stationary, c, The p r o c e s s e s satisfy an ergodic property,

guaranteeing the equality of the a v e r a g e s with r e s p e c t to

5

and the ensemble a v e r a g e s ,

On the b a s i s of the assumptions with r e s p e c t t o

A

and the boundary conditions a t

5

=

4

48

,

_{one can construct a Green's}

function m a t r i x associated with (1.3) o r (1.7)

Equations ( 1 , l ) can therefore be converted into a s y s t e m of integral equations of the form,

By observation one r e a l i z e s that equations (1.9) can be obtained f r o m the s y s t e m of equations

be reduced to the f o r m

where a proper condition a t X,(o)

= &

can be selected.

F r o m this point, the analysis will follow Caughey and Gray

(Ref. 10). If

A

i s a stability matrix, there exists a Hermitian positive definite m a t r i x

V

,

such that (Ref. 14)

where

A*

=

A'

0

A Hermitian m a t r i x

Q(x)

can be formed a s follows

where

vi

and

v-t

a r e positive definite Hermitian m a t r i c e s obtained a s follows : Since is a positive definite Hermitian m a t r i x there exists a n orthogonal transformation

@

such that

V

p o s s e s s e s a u n i q u e s q u a r e root

~f

a l s o

Now, l e t

I

Q(g)II

be the norm of

Q(F)

;

if

E

s u r e l y stable in the large.

-

In the particular c a s e that

Fkx)

may be written in the form

where

Gi

a r e constant m a t r i c e s and

fi(r)

a r e s c a l a r functions

of and

M

<

N~

,

i t is possible to have a s h a r p e r condition of M

stability. If

I

fP'I,

E{fJT;)

1

exists and i s l e s s than

&-I

/

then equation (1.11) is almost s u r e l y stable in the large, where

(

q")I,

i s the numerically l a r g e s t eigenvalue of the

m a t r i x

2, Basic Equations

Let a point on the cylindrical surface of radius

R

be

specified by its axial and circumferential coordinates x and 'j

Due to the presence of imperfections each point is radially displaced

from the cylindrical surface by Q(%) It is assumed that

In the absence of s u r f a c e loads, the equations expressing equilibrium

in the

x

and

y

direction for a shallow shell involve only the

equations a r e satisfied by introducing the s t r e s s function

_f=

(%,YJ

,

F L Y

Ny

=

5 x a

NEY

=

-

F;xy

Let

W(X,Y)

(positive inwards) be the radial displacement of the shell. In the c a s e of axisymmetric imperfections the functions F(x,)r) and

W(%,Y)

satisfy the following two nonlinear equations,

where is Young's modulus, V Poissons s ratio,

k

is the shell1 thickness,

E

k

=

membrane rigidity,

~ r p 3

=

a ( , - J )

= bending rigidity

Equation 42,1) i s the compatibility equation in membrane s t r a i n s and (2,2) i s the radial equilibrium equation.

3 . Method of Solution

Equations (2. l ), (2.2) admit, f o r a n axisymmetrie: imperfect cylindrical s h e l l under axial compression, a n axisynnmetric

w h e r e

8*

is the, axial compressive s t r e s s . Substituting (3.1 ) into

(2.1 ) and ( 2 . 2 ) yields

T h e s e equations can be simplified by reducing them to a nondirnen-

s i o n a l form, Let

Introducing these relations into ( 3 , 2 ) and ( 3 . 3 ) yields

where

Go

(

5-

7)

and

He(

5

-?)

a r e the Green's functions associated with the homogeneous p a r t of (3.5). These functions a r e

where

This solution remains finite a s long a s 6

1

Considering the c a s e of stationary and ergodic random imperfections with z e r o mean, the autocorrelation functions a r e defined a s follows

is the expectation of the function

f

Now

This i s the d e s i r e d relation f o r the l i n e a r p a r t of the solution. Following Koiter (Ref. 1 ), the nonlinear equations (2.1 ), ( 2 . 2 ) may admit a n a s y m m e t r i c solution adjacent t o the s y m m e t r i c one which is specified by

W,(X~Y)

and

( E , ( x ~ ~ )

.

Hence

considering

and taking into account that the deviation f r o m the axisymmetric configuration i s infinitesimal one may l i n e a r i z e the equations with r e s p e c t to

WI

('1 91 and

9,

(%

,

Y ) The compatibility condition and the e q u i l i b r i w equation t h e r e f o r e a r e

Introducing the l a s t expressions into ( 3 . 1 3 ) and ( 3 . 1 4 ) yields

As before, these equations can be reduced to a nondirnensional form using ( 3 , 4 ) and the relations,

which is

where

I-',

(5

1 9

equations ( 3 . 1 8) and (3.14) yield

g,''

=

g ~ ,

+

x3

=

k2x2

+

X,

X;

=

k2x3

-

( h ' ~ ~ +

x4)

+

1 ; ~ )

x 2

where p r i m e denotes differentiation with r e s p e c t to

5

In

m a t r i x notation (3,201

m a y

be w r i t t e n as

4 . Derivation of Stability Condition

F o r the following analysis, the autocorrelation function

a,

(XI

will be assumed a s an exponential cosine function

R

=

K z

e-'Ml

&so5

(4.1)

and

fie)

will be assumed to have

a Gaussian distribution.

Obviously

K

r e p r e s e n t s the root m e a n s q u a r e of the imperfections.

F o r a function

f

with Gaussian distribution

In the c a s e of the cylindrical shell,

and

and

00

where

To complete the stability analysis established in Section 1, one has to find the Hermitian matrix

V

such that

A * V + V A =

- 1

where

A

i s formulated a s shown in

(3.24),

Then one has to.find the transformation matrix

@

such that

where/(*; a r e the eigenvalues of

V

Let

G,

and

C1

be the following m a t r i c e s

and

Let

qy'

be the eigenvalues of

Bl

and

q y

the eigenvalues of A+

Ba

The stability condition for the cylindrical shell will then be

Now from (4.3), (4.41, (4,6) and (4. 7)# it is easily s e e n that

and

where Go and

$Ite

a r e constants.

Introducing these relations into (4.

B

1 ) yields

5. Numerical Example

In o r d e r to evaluate the stability boundary determined in equation (4.12) a specific numerical example has been c a r r i e d out. The following p a r a m e t e r s were used in the calculation,

%/h

=

800

Lp

=

0.3

=

0.2

8

=

1 . 0

The data

(&,@)

for the correlation function of the initial imperfections w e r e selected s o that the peak of the power spectrum. would be in the neighborhood of the peak of the response k e r n e l for We&) This will a s s u r e consideration of the m o s t c r i t i c a l situation, The

numerical evaluation determines the following relation.

The shell will r e m a i n stable a s long as 4

5 h 1

The calculation was c a r r i e d out varying the wave number

k

i n the vicinity of

k =

The stability boundary obtained i s shown in F i g u r e 2 , This r e s u l t is s i m i l a r qualitatively t o the deter- ministic c a s e s associated with sinusoidal imperfections.

6.

Concluding R e m a r k s

The stability condition i s only a sufficient c r i t e r i a f o r the

stability of the shell. The buckling p r o b l e m is s t i l l open f o r s h a r p e r

conditions, n e v e r t h e l e s s the p r e s e n t condition does not r e q u i r e any

f u r t h e r assumptions with r e s p e c t to

>

o r the power s p e c t r u m

functions of the initial imperfections.

It should b e pointed out that, f o r c e r t a i n p a r t i c u l a r c a s e s ,

a s h a r p e r stability condition can be obtained by m e a n s of o t h e r

techniques. F o r example, where the load is c l o s e to the l i n e a r

c r i t i c a l one and the power s p e c t r u m function of the imperfections

v a r i e s slowly in the vicinity of the a x i s y m m e t r i c a ? respolgse

function, the Case c a n be solved in a simplified m a n n e r , considering

a n a r r o w band f i l t e r technique. The stability condition obtained will

III. APPROXIMATE STABILITY ANALYSIS OF LONG CYLINDRICAL

SHELLS WITH ASYMMETRIC RANDOM IMPERFECTIONS

1. Prebuckling Equilibrium with Asymmetric Imperfections

This p a r t of the work i s based on the approximate method of

truncated h i e r a r c h i e s which has been used before f o r the axis y r n - m e t r i c c a s e by Amazigo (Ref.

9).

No attempt will be made to study the validity of this technique, however comparisons between r e s u l t s

obtained by truncated h i e r a r c h i e s and those obtained by other

approximate techniques, such a s perturbation techniques f o r

particular cases, are in good numerical agreement. In o r d e r to

a s s u r e justification for adopting the truncated h i e r a r c h y method a s

used in the following, one should a s s u m e that the root m e a n square

of the imperfections i s s m a l l compared to the shell thickness. The

l a s t assumption s e e m s to be r a t h e r r e s t r i c t i v e , nevertheless the

c a s e s which f a l l into this c l a s s a r e of g r e a t p r a c t i c a l significance,

Considering again the nondimens ionall equations of an

imperfect cylindrical shell subjected to axial load,

Consider a solution of equation ( 3 . 1 ) of the f o r m

The functions

t(f

,?)

and

W$,

7)

satisfy the following s y s t e m of l i n e a r equatians

One r e a l i z e s that these t r a n s f e r functions exist for a l l

_,

as long a s

A

I

and, in particular, for

=

o

and

Introducing (1.2) into (1.1) and neglecting higher o r d e r t e r m s

in

.

4t

( ~ ~ 7 )

and WI ($,

7)

a s well a s multiplications of sub-zero

At this point consider the following useful identity

The solution of ( I , ? ) can be formally written in the form

-

1

w,,~

tr-

r,

,

1-13

Q,I,

c s - 3 , ~ -

.zd

d ~ ,

where the double Fourier transforms of

3

kg(fJ7)

.

F r o m the expressions f o r

Hl(d,p)

and

C2(r(,II)

one can observe

that, for

e=

0 these expressions have a singularity a t 0 ( = 0 ,,

A question a r i s e s a s to the existence of the f o r m a l solutions s e t up

in f

1.9).

The answer to this question l i e s in (1.8)- This nonlinear differential operation enables one t o utilize only second derivatives

Hence, the use of only second derivatives of the t r a n s f e r functions

will remove the singularities a t

p

= O and d m o

.

This completes the solution to the second o r d e r of approx-

imation. A higher o r d e r of approximation can be achieved by

proceeding further with the s u c c e s s i v e substitutions, which will not

be sought h e r e .

2 . Variational Equations and Stability Analysis

With the assumption that a solution of equations (1.1) can be

found to a satisfactory o r d e r of approximation, one can consider the

stability problem by seeking the possibility of admittance of a second

solution. In other words, a variational equation will be s e t up in the neighborhood of the existing solution to verify the existence of

bifurcation.

Returning to equations (1. P ) and, assuming that they admit a

second solution specified by

_{" ( 3 , ~}

₎

and

9

(5)

r)

.

the deviation of which f r o m the b a s i c solution i s small, one may l i n e a r i z e the

equations with r e s p e c t to

M t $ l q )

and

?(Spy)

.,

The compatibility

c - .

-w h e r e w = W I + W l and

+=+,,++,

*

where p r i m e denotes differentiation with r e s p e c t to the f i r s t

argurnent in the function and dot denotes differentiation with respect to the second .argument in the function.

Let the following correlation functions now be defined

where the double integrals associated with the expectation a r e taken over the repeated variables.

At this point one should also consider the proper a p p r o x - imation for truncation in the technique to be used. The correlation discard approximation in the p r o c e s s of closing the hierarchy in a typical c a s e i s

"(Y+PJ

'I+

5 )

E ~ I + P #

r

q.5,)

~ ( ' S + / U . ~ J

%IT,)

u l 1 ~ 1 )

p e r t u r b a t i o n solutions in s i m p l e d e t e r m i n i s t i c buckling problems.

F o r reasonably s m a l l imperfections the a g r e e m e n t between the two

techniques is good and t h e r e f o r e this technique will b e adopted f o r

the problem of s h e l l s f o r the c a s e of s m a l l imperfections a s

compared to the s h e l l thickness.

Equations ( 2 , 2 ) and ( 2 . 3 ) c a n be w r i t t e n f o r the point

( r * g

,*r+

?)

,

in which c a s e a l l differentiations a r e applied with

r e s p e c t to the p r o p e r a r g u m e n t s . Doing so, and multiplying equation

( 2 . 2 ) (written f o r the point

( ' 1 ' 3 ,

q*?)

) by

q(3,

7)

,

taking the

expectation of t h e r e s u l t and using the c o r r e l a t i o n d i s c a r d approx-

(2.7)

Considering equation (2.2) a g a i n a t the point

(5+5

1

9'3)

multiplying by

~

(

J

Y)

1

and taking the expectation of the r e s u l t i n g

The s a m e procedure a s described for equation ( 2 , 2 ) w i l l be

and

To obtain the expressions f o r

all(%

q

;,acr,

5 )

a ,

(31q;

P ,

31

% ( j A j r + , ~ >

and

R ~ ( S I ~ > P I S )

one has to multiply equations ( 2 , Z ) and ( 2 , 3 ) written out a t the point

r-59%)

~ e q )

by

u ( ~ ~ r l ) a ( 8 + / ~ ~

" 1 5 ) take the expectation with r e s p e c t to

3

and

,

u s e the correlation

*

-

-

i

4

-q,-

s,

4,

d?,

and

(2.14)

and

+

4

't(51j9,)

K;

_y%,q%)+

-Feo(tt

jq@)h:(tsj

Q?*)

-

2 J # * ( ~ # J vt9 Q v (53,qa)

+

a:'($-

r,,

?-q,)j"(z,,~)

One can observe that the four equations obtained consist of two identical s e t s of equations for different s e t s of correlation funetions. It will therefore be s d f i c i e n t to concentrate in the

Upon applying double F o u r i e r transform using proper convolution relations, the following equations a r e obtained

where

Equation (2.26) i s an implicit relation between )r

,

o(

and

[.1

.

Naturally the lowest value of

3

i s to be sought.

Formally,by m i n i m i z i n g w i t h r e s p e c t to o( and f4

,

w i l l r e s u l t two m o r e relations which will uniquely determine the minimum value of a s well a s the corresponding d. and

P

at which i t will occur.

Since

1,

4 )

,

I&(&)

P)

and

Is(&,

P)

involve

h

in a n implicit f o r m in a r a t h e r complex fashion, the treatment of the solution f r o m h e r e on will be numerical r a t h e r than proceeding with cumbersome analytical relations. Finally one should point out that by minimizing with r e s p e c t to

d

and

P

one commits himself to a p a r t i c u l a r solution of the variational equation r e p r e - sented by a double simple harmonica1 mode, Since the variational equations a r e introduced in o r d e r to investigate the existence of a second solution in the neighborhood of a given solution, and since one i s concerned with double continuous s p e c t r u m s , i t is possible to choose any a r b i t r a r y non-trivial s p e c t r u m mode. Naturally the double periodic s p e c t r u m modes a r e associated with the lowest value of $B *

3 . P a r t i c u l a r Cases

P r i o r to considering numerical examples of particular e a s e s , one should investigate the power s p e c t r u m function

_SG

(dl

P)

of the imperfections. This function i s dependent on the two arguments

-60-

Considering f i r s t the argument

[j

,

representing a c i r c u l a r modified frequency in the circumferential direction, i t is obvious that this variable i s d i s c r e t e , and can only be of the form

F o r 2)

1 ,

p

can be considered a s a continuous variable for all practical numerical computations, this will be followed only by insignificant numerical e r r o r s . The axially

s y m m e t r i c mode of imperfections i s naturally the c a s e of

_{k s o}

*

Hence, since any general s t a t e of imperfections can be expanded i n F o u r i e r s e r i e s in the circumferential direction i n the f o r m

where

and, since the concept of correlation function can be utilized not

The l a s t expression leads to some interesting conclusions; for c a s e s where the phase

Sm(8)

i s constant (independent of

3

), the

correlationfunction

Rse(p,s)

takes the particular f o r m

which physically means that these a r e no "torsional imperfections" present in the shell. On the other hand, for the m o r e general case where the phase 5% i s a function of

$

,

sttorsional imper- fectionss' a r e present in the shell and the correlation function

R g

(.u)

3 )

takes a m o r e complicated form.

independent of

5

.

F u r t h e r m o r e , exponential cosine correlation functions f o r k ( 8 ) a r e examined numerically. The l a s t

functions a r e often used in control s y s t e m s , and s e e m to be acceptable f o r stationary s t a t e s of random imperfections (Ref. 8,

9).

However, the selection of p a r a m e t e r s in this p a r t i c u l a r correlation function i s of m a j o r significance, since imperfections m a y occur in a certain range of frequencies. P a r t i c u l a r attention should be paid to those modes of the power s p e c t r u m which a r e m o s t effective in reducing

the buckling strength yet r e m a i n reasonably practical.

Since, i n the following the power s p e c t r u m

5%

(d, (3) will be needed, one should note that the double F o u r i e r t r a n s f o r m of

expression (39 2 ) for constant

₉

_,

_yields

where

S4(d)

m

=

0, I , I J

.-

.

i s the F o u r i e r t r a n s f o r m of the correlation function

which in the following analysis will be taken in the f o r m

Typical power spectrum curves a r e presented i n F i g u r e 1 . At this point l e t us turn to particular c a s e s *

f i l

Axisvmmetric Imperfections

F i r s t , considering the c a s e of axisymmetric imperfections

for which

Introducing these expressions into (2.26) yields the final d e s i r e d form

for the c h a r a c t e r i s t i c equation, i n the axisymmetric case. A com- parison between the equation obtained and the one obtained by

Amazigo (Ref,

9) r e v e a l s

a slight difference. However, a numerical comparison between the r e s u l t s obtained f r o m the two equations i s

almost in perfect agreement.

The integrals w e r e evaluated numerically by m e a n s of

Simpson's quadrature which has the f e a t u r e of selecting the proper

s i z e of integration subintervaPs according to the d e s i r e d number of

significant figures. Minimization of

h

a s o( and

f3

vary

continuously has been c a r r i e d out numerically, With the assumption

that the number of waves in the circumferential direction i s l a r g e

enough t o justify continuous variation of f%

,

which not always was

the case, the final minimum values have been obtained.

F r o m the expression for the power spectrum (3.6) one

-

r e a l i z e s that, f o r the axisymmetric case, the mode shape of

Sse(d)

i s dependent on two p a r a m e t e r s , namely,

1,

Qe, The c a s e

1E,= 0 and i s the deterministic c a s e of a s i m p l e

periodic imperfection. The p r e s e n t analysis h a s been examined

numerically, in addition to the deterministic case, for Qe 1 and

various

&

between O and

ii

These r e s u l t s a r e presented

-65-

imperfections i t was expected that the c h a r a c t e r i s t i c equation w i l l be

reduced to the one obtained in (Ref.

9).

It turned out that a slightly different c h a r a c t e r i s t i c equation was obtained. Nevertheless,

applying the present numerical integration technique for the integral

t e r m s in both c a s e s revealed that quantitatively the r e s u l t s obtained

from both c h a r a c t e r i s t i c equations w e r e identical. However, these

numerical r e s u l t s were not i n agreement with those obtained in

(Ref. 9). As explained in this reference the integrals have been

evaluated numerically using calculus of residues. Since this

technique has not been given in detail, i t was impossible to investigate

further the cause for the discrepancy and no further comments can be

made,

(ii) Asymmetric Imperfections

Another c a s e of practical significance i s the one where, in

addition to the axisymmetric mode of random imperfection, a n

a s y m m e t r i c mode of random imperfection i s present, Let the power

spectrum for this c a s e be

Introducing this power spectrum into

(2,251

yields

I,

(.c,

p,

=

I;

(6%

p)

+

-

a""

-67-

Introducing e x p r e s s i o n s (3.8) p r o p e r l y into (3.11) and the resulting

e x p r e s s i o n s into ( 2 . 2 6 ) yields the f i n a l d e s i r e d f o r m f o r the c h a r a c - t e r i s t i c equation f o r this p a r t i c u l a r a s y m m e t r i c c a s e , where the

instability mode o c c u r s a t

p

=

k

.

A

,

in this c a s e is minimized with r e s p e c t to

d

.

The i n t e g r a l s (3.10) o r (3.11) w e r e a l s o evaluated nurner-

ically using S i m p s o n ' s rule. Minimization of

h

has been

c a r r i e d out n u m e r i c a l l y , The v a r i o u s r e s u l t s a r e p r e s e n t e d in

F i g u r e s 4 and 5. The family of c u r v e s obtained a r e qualitatively a s anticipated. Quantitatively t h e s e r e s u l t s a r e valid f o r s m a l l i m p e r

-

fections which, in m o s t p r a c t i c a l applications, a r e the c a s e . As the

imperfections b e c o m e in magnitude of the o r d e r of the s h e l l thickness

the p r e s e n t approach is no longer valid and o t h e r techniques will have

to be sought, This is naturally coupled a l s o into questions of validity

of the equations s f the s h e l l and r e m a i n s f o r future investigations.

P e r h a p s s f a l l known techniques f o r stability analysis i t s e e m s

that the Lyapunov approach is the m o s t powerful tool f o r establishing

sufficient conditions to s u c h questions, This will involve the appli-

- 6 9 -

4. Concluding Remarks

The method of truncated h i e r a r c h y proved to be a powerful

tool in the stability analysis of the cylindrical shell with s m a l l

imperfections. Although this technique i s limited to a narrow class

of imperfections, i t i s this c l a s s which i s of m a j o r concern in

engineering applications. Any attempt to adopt this technique for

m o d e r a t e imperfections will be followed by cumber s o m e computations

associated with higher h i e r a r c h i e s . P e r h a p s a m o r e difficult task

would be to justify the r e s u l t s obtained. This naturally suggests the

examination of other techniques which a r e not based on a d i r e c t s ~ l u t i o n of the eqaations, yet r a t h e r investigate the p r o p e r t i e s of the

solutions. As pointed out before, the kyapunov analysis i s one way

t o approach this problem.

In conclusion, the analysis presented in P a r t 111 and in

particular the c h a r a c t e r i s t i c equation obtained, a r e sufficient to

establish the buckling load in practical applications f o r cylindrical

shells with stationary random imperfections. The p a r t i c u l a r

numerical c a s e s considered in this work a r e only the f i r s t s t e p in

investigating numerically the nature of the problem where a m o r e

complicated s t a t e of imperfections i s concerned, Finally the present

study can easily be extended to the buckling problem where, in

REFERENCES

1. Koiter, W. T., "On the Stability of Elastic Equilibrium", Ph.D. Thesis, Delft, H, J. P a r i s , Amsterdam, 1945.

2. Koiter, W, T., "The Effect of Axisymmetric h p e r f e c t i o n s on the Buckling of Cylindrical Shells under Axial Compression", Koninkl. Nederl. Akademie van Wetenschappen, Amsterdam, S e r i e s B, ,66 (1963)

No.

5.

3, Donnell, L. H. and Wan,

C.

C., "Effect of Emperfections on Buckling of Thin Cylinders and Columns under Axial Compression", J. of Appl, Mech., March 1950,

4. Hutchinson, J. -W., "Axial Buckling of P r e s s u r i z e d Imperfect Cylindrical Shellsse, AEAA Journal, Vol, 3, No, 8, August

5.

Budiansky, B

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and Hutchins on, J. W., "Dynamic Buckling of Imperfection Sensitive S t r u c t u r e s

",

App. PAech, P r o c . of the 11th Intern. Congress, Munich, 1964.

6. Babcock, C. D. and Sechler, E. E., "The Effect of Initial

Imperfections on the Buckling S t r e s s of Cylindrical Shells

",

NASA TN D-2005, 1963,

7, Bolotin, V. V., B'Statistical Methods in the Nonlinear Theory of E l a s t i c Shells

",

NASA TT F-85,

P

962 (originally written in Russian i n 1956)-

9. Amazigo, J . C., "Buckling under Axial Compression of Long Cylindrical Shells with Random Axis y m m e t r i c Imperfections

",

Harvard Rep. SM-20, November 1967.

10. Caughey, T, K. and Gray, A. W., "On the Almost S u r e Stability of Linear Dynarnic Sys tems with Stochastic Coefficients ' I ,

J. of Appl. Mech., June 1965.

11. Keller, J. B., "Stochastic Equations and Wave Propagation in Random Media", P r o c . on Sy-mposia in Applied Mathematics, XVI, Rhode Island: American Math. S c , , p. 145-170, 1964. 12, Richardson, J . M., e'The Application of Truncated Hierarchy

Techniques in the Solution of a Stochastic Linear Differential Equation'@, Proc. on Symposia in Applied Mathematics, XIV, Rhode Island: American Math. Sc., p, 298-302, 1964,

13. Arbocz, J. .and Babcock,

C.

D,

,

@'Experimental Investigation of the Effect of General h p e r f e c t i o n s on the Buckling of

Cylindrical Shells Galcit Rep. SM 48-7, F e b r u a r y 1 968. 14, Gantrnacher,

F,

R . , "The Theory of Matrices", 1959.

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W.

and Battin, Re H., "Randorn P r o c e s s e s i n Auto- matic Controls', McGraw-Hill, 1956,

1 6. Hahn,

W e

,

I'Theory and Applications of Liapunovs s Direct Methodte, Prentice-Hall, 1

963.

P

7. Taus sky,

8,

,

esA R e m a r k on a Theorem of Liapunove', J , Math. Analysis and Appl. 2, p, 105-PO?, 1961.

8

11.

0

2.0

3.0

4.0

5.0

Frequsnqy,

a