The effect of initial axisymmetric shape imperfections on the buckling behaviour of circular cylindrical shells under axial compression

THE EFFECT OF INITIAL AXISYMMETRIC SHAPE IMPERFECTIONS ON

THE BUCKLING BEHAVIOUR pF CIRCULAR CYLINDRICAL SHELLS UNDER AXIAL COMPRESSION by

•

THE EFFECT OF

I

NTTIAL AX

T

SYMMETRIC

SHArE IMPERFECTIONS ON THE BUCKLING

BEHAYIOUR OF CIRCULAR CYLINDRICAL

SHELLS UNDER AXrAL COMPRESSION

by

D, B, Muggeridge

Manuscript received December,

1969,

ACKNOWLEDGEMENT

It is a pleasure to thank Dr. G. N. Patterson, Director of the Institute for Aerospace Studies, for providing the opportunity of carrying out this study. Special thanks are due to Dr. R. C. Tennyson for his help and guidance throughout the course of the investigation.

I should like to record the help of Mr. J. D. Bradbury, of the Institute's Technical Staff, in the fabrication of my shells.

Acknowledgement is due to Dr. J. W. Hutchinson of Harvard University for the exact model calculations. The author wishes to thank a fellow student, Mr. R. D. Caswell,for a numerical evaluation of Koiter's theory and for some test data.

The assistance of Mrs. B. Waddell in preparing the manuscript and the help of Mr. R. Magid in preparing the figures is sincerely appre ci ated.

Finally I should like to thank my wife. Her encouragement and help throughout the work proved invaluable.

I am indebted to the National Research Council of Canada for fellowship support during the last three years.

This project was supported by the National Research Council of Canada (NRC Grant No. A-2783) and the National Aeronautics and Space Administration (NASA Grant No. NGR 52-026-011).

SUMMARY

Twenty-six accurately made, homogeneous, isotropie cylindrical shells of circular cross-section (100 ~

R

i

t

~ 280, L/R

z

2.8) have been tested in axial compression and found to buckle completely elastically within 10% of predicted loads. Of these shells, twenty-four have had initial axisymmetric imperfections of known properties. All shells were made fr om photoelastic plastic using a spin-casting technique and parti -cular attention was given to the measurement of their median surfac,e profiles •

Koiter's theory for uniform axisymmetric imperfections has been extended to include arbitrary values of wave number and nine shells have provided good agreement with this theory for a wide range of imperfection amplitudes and wavelengths. Five shells containing groups ofaxisymmetric imperfections having different frequencies and constant amplitudes have

been tested and it was found that buckling loads can be predicted by Koiter 's theory when three wavelengths of the same critical frequency exist. H ow-ever, it.would appear that for design purposes the buckling load reduction was essentially governed by the presence of a single wavelength of the critical frequency.

A series of seven shells containing alocal axisymmetric inward dimple have been tested. A broad range of imperfection amplitudes and

wavelengths was investigated and the experimental results were in good agre e-ment with theoretical predictions of Hutchinson and of Amazigo and Budiansky.

Three shells containing 'random' axisymmetric imperfections of small standard deviation have been tested and buckling loads were found in

TABLE OF CONTENTS

NOTATION 1. INTRODUCTION

2. BASIC EQUATIONS AND THEORETICAL MODELS 2.1

2.2

2.3

2.4

Thin Wall Shell Equations for Axisymmetric Impe.rfect Circular Cylinders

Theory for Uniform Axisymmetric Imperfections 2.2.1 Koiter's Extended Theory

2.2.2 Exact Model Formulation

Theory for an Arbitrary, Localized Axisymmetric Imperfection

Asymptotic Theory for Random Axisymmetric Imperfections 3. EXPERIMENTAL TECHNIQUE

3.1 Construction of Circular Cylindrical Shells with Initial Axisymmetric Imperfections 3.2

3.3

3.4

3.5

Shell Mounting Method

Measurement of Median Surface Profiles and Average Shell Wall Thickness

Test Procedure

Photoelastic Method of Stress Analysis and Photographic Technique

4.

DISCUSSION OF TEST RESULTS

4.1

Comparison of Test Data with Theory for

Uniform Axisymmetric Imperfection Distributions

4.2

4.3

Comparison of Test Data with Theory for a Local Axisymmetric Dimple Imperfection Comparison of Test Data with Theory for Random Axisymmetric Imperfections

5.

CONCLUSIONS REFERENCES

APPENDIX A: Derivation of Thin Wall Shell Equations for Axisymmetric Imperfect Circular Cylinders APPENDIX B: Prebuckling Displacements and

Stress Distribution Page 1 2 2 4

4

8

10 12 13 13

16

17

18

20 20 20 23

26

27

29

APPENDIX C: Derivation of S(l) in Equation 2.4.2 APPENDIX D: Construction of a Template with a

Cos~ne Function Imperfection

TABLES FIGURES

•

'

.

"

b c D E f F G~( w- w) h K 1, x 1, x cr L m,n

-N

N , N , N

_x Y xy

NX' N

y '

N

xy P NOTATION* ~

4p

+ 1

4

4p

+ 1

-[3(1-

v

2

)

Et3

12(1-V

2 ) flexural rigidity of shell

modulus of elasticity Airy stress function

2 c .1' E

i

3

one-sided power spectral density function

t/t

L/m

=

77R 2p , axial half-wave length [12(1_V2

f

1_/

4

1 7T(Rt) 2'

shell length

number of half-waves and waves in the axial and circumferential directions respectively

bending and twisting moments per unit distance in median surface of shell

compressive force per unit length of the edge of the shell membrane stress resultants

2 2 2 2cL N 2cL N 2cL N

-3

x'

-3

y' ~ xy Et Et Et respectively mnR 2L

q o R

S(l)

t t u, v, w·

u,

V, W

w

(x)

x,y,z

-x x, y

z

the classical axisymmetric bucking mode wave number

shell radius measured to the median surface two-sided power spectral density ofaxisymmetric imperfections evaluated at the critical frequency shell wall thickness

average shell wall thickness

displacements measured in the axial, circumferential radial directions respectively

uit, vit,

w/t ,

respectively strain energy

potentialof external loads deviation of median surface

axial, circumferential, and radial coordinates respecti vely x/L, y/L L2(1_v2_)1/2

Rt

respectively ....

..

Greek a: ~ ~' )' 5 5' 6. v p (J cr (J e T Symbols

iso clinic parameter

nL/R

tjtx

=

1/2K

cr

(t/R) [12(l_v2) ]

-1

/2

peak axisymmetric imperfection amplitude retardation -00 w t ix e

ciX

root mean square of initial imperfection,

5

/2

t

rms (J

_e_ ) respectively

(J

ct

maximum deviation of median surface/average shell wall thickness

Poisson's ratio (

t

)1/2

P Rc ' nondimensional axial wave number

Et , classical compressive buckling stress for a Rc perfect circular cylindrical shell

critical cylinder buckling stress observed

experimentally

cylinder buckling stress as computed from exact model formulation

theoretical buckling stress

1/2

-( t ) nondimensional circumferential wave number

1\

Rc '

Sub scripts avg cr e o rms T shear stress

deviation fr om the axisymmetric equilibrium configuration

ö

4

/öx

4

+

2

ö

4

/öx

2 Öy2

+

ö

4/Öy4

average classical critical

refers to exact model theory prebuckling quantities

root mean square theoretical

When sub scripts follow a comma, they indicate partial differentiation of the principal variable with respect to the subscripts shown.

•

I, INTRODUCTION

The basic load-carrying member of many aerospace structures is a thin walled shell, As such, the buckling behaviour of a shell is an important design consideration, In the last few years consistent use of

, , 1 2

the large-deflection Karman-Donnell equations ' has generally led to good agreement with experimental data obtained from near-perfect test

3-7

specimens Remaining discrepancies between theory and experiment are gene rally considered to be due to the effects of initial imperfections and the presence of edge constraints. Excellent reviews of the literature can be found in Refs. 8~lO.

Theoretical analyses 11-17 of the effect of edge conditions have shown that, except in the free edge case, a substantial reduction in the buckling load occurs only when the conditions of simple support and of zero tangential displacement are applied. Gorman 18 and Tennyson 19 show test data in good agreement with the theory of Ref, 17. Although clamped edge constraints and eccentricity of loading may be important in some practical applications, they are of minor importance in tests performed on near~perfect

specimens under laboratory conditions Consequently, the presence of initial geometrie imperfections still appears to be the major reason for the dis-crepancy between theory and experiment,

The effe cts of imperfections on the buckling load of circular cylindrical shells were first investigated by Donnell and Wan in Ref, 20,

14 2l~24

Other theoretical analyses' have clearly shown that relatively small

imperfection amplitudes can drastically reduce the critical load of the shell, Despite the substantial theory available, little experiment al data 7,25,26 exists describing the effects of specific imperfections in shape in

reducing the statie buckling load, Consequently, it was of particular interest to determine the buckling load reduction caused by an initial axisrmmetric imperfection in shape defined by a simple trigonometrie function 21,14, In Koiter's 'special theory' 14, which was formulated

neglecting end constraints, he verified the asymptotic relationship presented earlier in his 'general theory' 21 for the limiting case of the axisymmetric imperfection amplitude approaching zero. Moreover, Koiter was áble to ~ive an equation which predicted the effects of finite imperfecttons on the

buckling load for wavelengths equal to the classical axisymmetric buckling mode of a perfect shell. Using Koiter's theory 14, it was possible to extend his analysis for arbitrary values of imperfection wavelength and amplitude, the results of which are discussed in Section 4.1. In order to test Koi ter' s extended theory ~ photoelastiC" plastic circular c;ylilldrical shells containing an axisymmetric imperfection in shape were manufactured as described in Section 3.1 and tested under axial compression. For com-parison purposes, numerical buckling load calculations were performed by Hutchinson 27 based on an exact analytical model in which the effects of a clamped end constraint and the specific geometrical configuration of the cylinder were taken into account. These results were first presented in

Ref. 28. It was also of interest to determine the extent to which the critical wavelength would dominate buckling in a shell containing a group of sine

waves of varying frequency but constant amplitude. The results of this investigation are al~o described in Section 4.1.

Theoretical analyses 23,27,29 of the effect of alocal axisymmetric dimple have shown load reductions only slightly less degrading than

.those caused by uniform axisymmetric imperfections. An earlier experiment al investigation of a similar problem involving a loc al imperfection is described in Ref. 25·. Hence, a series of test shells were made to obtain a comparison with the above theoretical model~. These results are discussed in Section

4.2 and Ref. 30.

In practical shell configurations, imperfection distributions are more likely to be random in nature than uniform. Consequently, it was necessary to determine if the power spectral density of the imperfections evaluated at the critical frequency would govern the buckling behaviour of a circular cylinder. The results obtained from tests on shells containing 'random' axisymmetric imperfections were compared to the asymptotic theory of Amazigo

3l~

A discussion of these results is contained in Section 4.3 and Ref. 35.

2. BASIC EQUATIONS AND_ THEORETICAL MODELS

2.1 Thin Wall Shell Equations for Axisymmetric Imperfect Circular Cylinders The effect of initial imperfections on the buckling load of

theory of elastic stability 21. It assumes that the initial imperfections are small, resulting in a linearization of the problem with respect to the imperfection parameter. The theory is most easily evaluated for the case of imperfections in the shape of the axisymmetric buckling mode of the perfect circular cylindrical shell.

A right-handed coordinate system is established with the origin on the median surface • Axial and circumerential coordinates x and y are taken with a positive outward normal. The median surface is assumed to deviate only slightly from a cylinder of radius

R.

This condition is expressed by the relations

I ;;

(x)

1«

R

I

_;'x

I

_«

_{1 2.1.1}

In addition, the radius of curvature in the meridional plane of the shell is specified to be at least the same order of magnitude as a cylinder of radi us R as expressed by

R

I

;'xx

I

~

0 (1) _2.1.2

Three equilibrium equations together with boundary conditions may be obtained by variation of the total potential as outlined in Appendix A. The equilibrium equations are the weIl known

Kármàn -

Donnell equations.

N

x'x

N y'y

4

~ D\l w = N (w, x xx + N , xyy

o

+ N , 0 xyx + w, ) + N (;, + w, ) xx y yy yy + 2N (;, + w, ) -

!

N xy xy xy R Y

An Airy stress function f is introduced such that

N

x f, yy

N

Y

=

f

'xx

N xy f 'xy

2.1.3

2.1.4

The first two equilibrium equations of 2.1.3 may be replaced by the compatibility equation

1 _'ij4f 2

₊

2w,

=

w,

w,

w,

w Et xy xx yy xy 'xy 2,1.5 1

- w,

w,

-

w, W

+-w

xx yy yy 'xx R 'xx

Since the analysis is restricted to axisymmetric initial imperfections, the equilibrium and compatibility equations ms\y be rewrittenas

4

,..,

D'ij w

=

f, _yy (w, _xx + w, ) + f, w, xx xx yy 2f, w, ""

1:

f, xy xy R xx w2 - w, w - w, w, 'xy xx 'yy xx yy 1 + - w

R

'xx

These two equations for.m the basis of all theoretical models considered in this investigation,

In addition to the equilibrium eq~ations, the variational approach yie1ds the fo11owing natural boundary condition at x

=

~ L/2

f,

=

yy -f _'xy

=

M ;:;. x H =. N

= ..

N x N xy

=

o

-D (w , + VW, ) =. 0 xx yy f, (w_{L +;L) -} f,

yy

Ä X xy

2.2 Theory for Unifor.m Axisymmetric ,I1D.l'erfeètiéns-2,2,1 Koiter~s Extepded ~eprr

2,1,8

14

Koiter showed that the equilibrium and compatibility equations, a1though non1inear, admit a simp1e axisymmetric so1ution for the case of unifor.m axia1 compression

f(x,y) W(x,y)

1

- - 2"

w* (X) 2.2.1

Substituting these quantities into equations 2.1.6 and 2.1.7 we get the following two linear differential equations

*

1 Et f 'xxxx 1 R

*

.

W , _xx = 0 n.~

+!

f

t-

+

at

.vw'xxxx R 'xx W'xx +

at

is now specified as ;(x)

= -

~t cos(2px/R)

*

VI = 0 'xx 2.2.2 2.2.3 2.2.4

a suitable particular solution of Eqs, 2.2.2 and 2.2.3. may be found. If one wants to consider the effect of clamped edge constraints. the stress distribution must be modified near the end of the shell. Here the disturbanee dies out exponentially ~ith increasing distanee from the ends. Hence Koiter's solution can be considered valid except in two boundary

l~ers near the ends if the shell is sufficiently long.

Equilibrium in the axisymmetric configaration. described by Eqs. 2.2.2 and 2.2.3 is called neutral if the nonlinear equations admit a second solution which is infinitesimally close to the axisymmetric

solution. Hence the particular solution is substituted into the general equations (Eqs. 2.1.6 and 2.1.7) which are then linearized with respect to the deviation from the axisymmetric equilibrium configuration.

The compatibility equation (EQ. 2.1.7) then becomes

[-w,

+

4c~ bP~,

cos(2px/R)]

and the equation of equilibrium (Eq. 2.1.7) takes the form

1;74

4c v w + -R

[ct>,

xx

-+

~

t 2 4clJ, bp ~ , cos (2px/R) yy

w'xx

+

~~

(b-1) W'yy cos(2px/R)

=

0 2.2.6

'l'he equilibrium is neutral if and only if these equations have a non ... vanishing solution that satisfies the boundary conditions. Buckling of the axisymmetric shape occurs at the smallest eigenvalue of the load parameter

".r

By using a Galerkin procedure32, an upper bound for the

critical load parameter m~ be obtained,

A buckling mode is assumed to be gi ven by

00

w(x,y)

=

t

I

C_j cos [(2j-l)

~

J

cos(~

) 2.2'1

.1=1

and is substituted into the compatibility equation (Eq.~·.2,5). The resulting solution is substituted into the equation of equilibrium

(Eq. 2,2.6) together with equation 2.2.7 'l'he equation obtained is

00 2

I I

A. C. h cos[(2j-l) px/R]

=

0

J J+ j= -1 h= -2

2.2.-8

'l'he application of Galerkin' s method results, as a first approximation ,14 in an equation whose explicit form is

2 2 2 4 2 2 -2 2 (p +

T)

+ 4p

(p

+

T)

-

4r.p 2

4

2 2 -2 2q.l T [b-1 +

Bb

p ( p + T

J ]

. 2 2

4 4

2 2 -2 + 16 (c~) b p

T [

.

(p

+

T)

+

(9p +T)

2 2- 2

]

=0 2.2·9 Ol

'!t

Using the following definitions, P = Kq

p2=

2K~

Eq, 2,2,9 can be re-written in the form

where

Al

= -512 K6 Q2

A

2 64K 4 Q4 + 1024

~

+ 128K4 BQ2 + 128 2

4

Q2 = c~ T K

A3

= - 16 K Q - 256 K B - 8 Q B -~4 6 ~22 16 c~ T K Q B 222 + 512 Q4B2 + _16K4_B2 2 4 2 ₊

64(c~)2K4Q2

_T4_B2_H

A4

= 64c~T K B Q = 2K2 + T 2 B = 16K4 + 1 H

=

.!....+

1. ₂ 2 (18K2

+

T2) Q 2,2,10 2

6

c~ T K B 2,2,11

Thus, for gi ven values of K and ~, a minimum value of " can be determined from Eq, 2,2,10 by varying the circumferential wave number parameter T. This analysis is limited by the conditions on the wave numbers·(m,n must be large) and by the ini tial buckling mode assumed by Koiter 14, It should be noted that the analysis assumes a shell of constant thickness,

Almroth 23 modified Koiter's analysis such that incremental dis-placements vere represented by

00 w(x,y)

=

t

I

j=l

c.

cos J [(2j-1) ~] mR cos

(~)

2,2,12 The modified formulation was programmed for the digital computer and m was varied to minimize the critical 10ad, The results so obtained vere in good agreement vith results from a finite difference analysis for

reasonably long shells, The modified formulation resulted in a

5%

decrease in buckling 10ad (for ~

=

0,1) from that predicted by Koiter,

2.2.2 Exact Model Formulation

Due to the technique used to fabricate the photoelastic plastic test cylinders, the imperfection profile was cut only on the"inside wall of the shell, thus resulting in axisymmetric thickness and median surface profiles (refer to Fig. 1) given respectively by

t(x.)

=

t

(1 + 5 _cos 2 5 t cos 7TX

r

_x 7T x )

r

_x 2.2.13

Hence, it was necessary to determine the extent to which the thickness variation in combination with the prebuckling deformations due to the

clamped end constraint would reduce the buckling load estimates based on Koiter's extended theory. The following non~dimensional fOrmlof the equilibrium and compatibility equations were derivéd by

Hutchinson27~

[h

3

(W,XX

+

vW, yy)],xx

+

h

3

(W'yyyy*"'W'xxyy)

+

2(1-V)

(h

3 W'Xyy)'X

+

2~

'Z

F,XX

= 2

c [F,yy tw,xx

+

W'xx)

+

F,XX

'

W,yy

- 2

F,XY W'Xy

1

~

(F

'yyyy -

17

F

'xxyy)

+

[~(F

'XX

-17

F ,yy) ] 'XX

+ 2(1+V)

[~

F,XYY

],x -

2~

Z

W'xx

'

2c[-W,yy (W'xx

+

W,xx)

+

(W'xy)2 ]

2.2.14

2.2.15

When h

=

a constant. Eqs. 2.2,14 and 2,2,15 reduce to those used by Koiter 14 t viz: Eqs. 2.1.6 and 2.1.7.

The prebuckHng axisymmetric s oluti on can be obtained from Eqs, 2.2.14 and 2.2.15, noting that F ü' _yy

=

.... N

2.2.16

(

~

FO'XX)

'XX

-

2J3

Z wo'xx + v N

(~),XX;"

0 _2.2.17

For the clamped end constraint appropriate to the cylinder configuration

being studied, the following boundary conditions are applicable

v

=w W = 0 at X 0,1 0 0 o'x U 0 at X = 0 0 U = U (N) at X 1 0 0

Hence Eq, 2.2,17 yields

F

=

-v N

+ 2~

Z h W

o 'XX 0

which, when substituted into Eq, 2,2,16 give~

(h3W

0~XX),XX

+ 2c N (Wo'XX + w'XX) + 12 Z2 h W -

2~

Z v N

=

0 o 2.2.18 2.2.19 2,2,20

The solution of Eqs. 2.2,19 and 2,2,20 yiel~the prebuckling values F ,

o

Wo necessary for the complete buckling solution of Eqs, 2,2,14 and

2,2.15, It is assumed that the form of F and W which will satisfy the

equilibrium and compatibility equations at the inception of buckling is

given by where W = W + E W o 1 F=Fo+ EF 1 W 1 = W1(X) cos ~y F₁ = F1(X) cos ~y 2,2,21 2.2.22

Substitution of Eqs. 2.2.21 and 2.2.22 into Eqs. 2.2.14 and 2.2.15 and retaining only first order terms in E gives

-2 (I-V) t32 (h

3

_{Wl ,x) 'X} + 2.[3 Z Il,xx + 2 c N Wl,XX = 2c

[_13

2 J'l(Wo,XX + W,XX) -

t3~1

'ó'XX

J

~

(13

4 FI + v

13

2 1l'1,XX) + [

È

('1 ,XX + v

13

2 1_{1 )}

J

'XX - 2 (1 + v) t32

[~

'l,X ],X - 2./3 Z Wl,XX 2.2.23 2.2.24

with the appropriate boundary conditions at the shells ends defined by a:~I"'.bc

=

0

at 2,2,25

Equations 2.2.19, 2~2,20, 2,2,23 and 2.2.24 were solved

numerically by Hutchinson 27 using a modification of Potter's method 33 2.3 Theory for an ArM trag L_Localized AxisYJlI!Iletric Imperfection

Amazigo and Budiansky29 obtained a general formula for the buckling load of an infinitely long cylindrical shell under axial compression containing an arbitrary, localized axisymmetric imperfection. Their formula is an asymptotic one which is valid for sufficiently small imperfections in much the same way as is Koiterls simple formula for a sinusoidal axisymmetric imperfection 14.21,

The initial axisymmetric imperfection w in the middle surface of a cylinder of constant thickness tenters into the buckling formula through

-Joo

w ix

6=

where

X

=

7TX/.i _x and .ix _cr cr

is the half wavelength of the classical axisymmetric buckling mode. The asymptotic formula of Ref. 29 for the ratio of the buckling load to the classical bucking load is

3/2

[1 -f..] =

2.3.2

Of primary interest here is a I'cosinell _{dimple defined by'}

5 [1 + cos

r]

Ix I S.i w=- _{2 x} x = 0 lxI

>

.i x

so that.i is the half wavelength of the dimple, In this case,

x

~= sin(

mf)

(-t5 ) (3,2_1

2.3.3 ,

2.3.4

where I?' _. ==.i /.i _{x xcr} 1/2K. When (3'

=

1 (K

=

0

.5), E

q

.

2.3.4 reduces to

I ~ I

=

7T 5/2t 2.3.5

The maximum reduction in the buckling load, for a given imperfection amplitude - thickness ratio 5/t • occurs when (3'~ 0.8 with ~ about five percent larger than Eq. 2.3.5.

Numerical calculations were carried out by Hutchinson 27,3°, for finite length shells with the eosine dimple defined by Eq. 2.3.3 symmetrically located with respect to the ends of the shell as shown in Fig. 2. The calculation procedure. which has been described in Section

2.2.2, takes into account end effects and nonlinear prebuckling deformations. The prebuckling problem and the reduced eigenvalue problem were both obtained

.A ,

without approximation from a Karman~Donnellrtype shell theory. Resulting ordinary differential equations.were cast in finite difference form

and solved by a Gaussian elimination method usually referred to as

Potter's method. Clamped end conditions were chosen since they pertained most closely to those in the testing program.

Almroth 23 investigated alocal axisymmetric dimple imperfection given by

.w

=

2

5 cos ( nx)/ (2 ~) 0

<

x

<

~

x - x

Eq. 2.3.6 may be transformed, by use of the trigonometrie

( r)

1/2 cos

(2~)

=

+ to become 5 (1 + cos w

₌

2 0 cos 2

r-)

x

x -

J

0< x < ~ x ~ _x < x

=s

L/2 2.3.6 identity 2.3.7

This is identieal to Eq. 2,3.3 except for the specifieation of an inward dimple, Almroth noted th at an imperfection of this type, for all values of ~ , is slightly less harmful than a periodie imperfection with the

x

ampl itude 5/2.

2.4

Amazigo's Asymptotic Theory for Random Axisrmmetrie Imperfections

Amazigo 31 assumed that the cylinder imperfections were homogenous random functions of the axial coordinate and hence could be eharaeterized by their mean and autocorrelation. He analyzed the buekling behaviour of infinitety long axisymmetric imperfect shells under axial eompression on

.

'

)

the basis of a modified truncated hierarchy methode The Karman - Donnell equations form the starting point of this analysis. These equations were linearized in the usual manner to investigate the buckling phenomenon.

On the basis of the above. Amazigo derived the following asymptotic buckling load relation

2

2/7

"

=

1 - [

~97T_C

_ _ S(l)J

where 8(1) is the power spectral density of the imperfection distribution evaluated at the critical frequency. In order to estimate 8(1). Fourier series analysis was employed which yielded the following relationship between the complex Fourier series coefficient and the power spectral

density function employed by Amazigo (refer to Appendix C)

2.4.2 8(1)

-Ic 1

2 q L n 0

In the above approximation, the fundamental frequency was taken to be the reciprocal of the shell length.

3. Experimental Technique

3.1 Construction of Circular Cylindrical 8hells with Initial ,Axisymmetric ,Imperfections

All test cylinders were made from a liquid photoelastic epoxy plastic, spun cast in an acrylic tube. The plastic used consisted of 100 parts of Hysol XC9c4l9 resin to 29 parts of Hysol 3561 hardener. A room temperature curing time of at least five hours was required. All shells were cast under similar conditions of humi di ty and temperature • A casting form was manufactured from a piece of acrylic tube by cutting and facing the tube to the appropriate length. Plastic end-plates were machined and O-rings installed to give a snug fit in the ends of the tube. Holes were made in each end - plate so that the liquid epoxy plastic could be poured into the form without removing it from the rotation rig (see Fig. 3). Four holes, 900 apart and diametrically opposed

longitudinally were tapped in both end - plates. Bolts were screwed into these holes with appropriate washers to statically mass balance the form.

A

transparent tube was used to ensure that complete wetting of the inside wall was accomplished before relying on centrifUgal force to drive the liquld epoxy around the perimeter. Due to the transparent nature of the tube, a check could be made for trapped air bubbles. This generally caused na problem because the high rotational ve10cities produced sufficient

Due to irregularities in the wall thickness and internal diameter of the acrylic form, mass balancing was necessary to prevent vibration of the apparatus during spinning. Any such vibration will produce thickness variations in the circumferential direction during the casting process. To ensure a circular cross~section in which to cast the photoelastic shells, a succession of Hysol plastic liner shells were first cast in the acrylic form. Af ter each liner was cast, a change in mass balancing was required. Finally, when the inside of the farm was circular, the addirttLon of another liner shell did not change the mass balance. When this stage was reached a liquid releasing agent, Hysol 4368, was wiped on the inside wall to prevent bonding of the shell to the last liner.

Longitudinal thickness variations were kept to a minimum by carefUl alignment of the ground steel rad on which the form .was attached. Thickness tolerances of + 2 1/2% were achieved both circumferentially and longi tudinally on the two geometrically 'near-perfect' reference shells. Shells having

known geometric imperfections had tolerances wi thin .: 5% (of the gi ven profile) •

The fabrication of the shells containing axisymmetric imper-fections was accomplished in two stages. Initially, a geometrically

'near-perfect' shell was cast. This was left in the farm until the shell's inner wall was machined to a prescribed profile. The profiles were established by aluminum templates which were made by the apparatus shown in Fig.

4.

This consisted of a Slo-Syn HS 50 D viscously damped stepping motor controlled by a Slo-Syn ST 1800 BV adjustabIe speed drive. This enabled the motor speed to be accurately varied from 4 to about 400 steps per second (1.80 + 3% per step). A circular cam with its centre of rotation offset to provide the appropriate amplitudes of the imperfection wasclamped on the shaft of the motor to maintain contact wi th the spring loaded cutting taal. The motor - cutting taal assemhly was then mounted on the taal post of the lathe and a round bar of 2024-T4 Aluminum was mounted in the lathe

chuck.

Various feed and speed combinations of the lathe were used, in conjunction with a particular speed of the motor driving the

cutting tooI, to establish the wavelength of the imperfection. Once all speed combinations were determined, the profile was cut into the aluminum bar. Amplitudes of the imperfections were carefully measured with a

micrometer and the bar was milled until only a central flat remained (see Fig.

4).

The stylus of the hydraulic tracing-tool apparatus ran along the edge of the finished template (see Fig. 5),

The offset cam in the configuration des cri bed above should

produce a perfect eosine function (refer to Appendix D), This was indeed found tQ be the ease for all test series of shells except the one eontaining ax1symmetric inward dimples, Here the imperfeetion cut on the outside wall deviated slightly from a pure eosine function due to same warping of the shell wall,

A slightly more complicated procedure was employed to pr0duce the 1mperfection distributions required for shells AD

6 - 8.

This shell series 'ealled for an imperfection specified by the sum of three eosine functions, In order to achieve this, the desired profile was plotted and an almost identical fit to it was obtained by using several eosine

functions of different amplitudes and frequencies superimposed with either a 00 or 1800 phase relationship, Very high frequency components were replaced by some small flat sections, These sections reintroduced some high frequency terms of small amplitudes when a Fourier series analysis was performed on the resulting profile as discussed in Section 4,3,

In order to machine the profile into the shell wall, the form and shell were clamped to the face plate of the lathe, Extreme care was used to prevent run-out of the form at the free end, A ring with equally spaeed grub serews was used to true up this unsupported end to within ,OOI in, Templates made by the proeess outlined above were used in conjunction with a hydraulic tracer~tool apparatus to cut the desired profile (see Fig, 5),

Before the imperfeetion profiles were made, each shell was bored out to the appropriate starting thickness,

All shells were subsequently removed from the form by means of a machined plug and hydraulic press (see Fig,

6).

A preliminary survey of thiekness variations was made with a dial gauge (eapable of reading to 0,0001 in. ) opposing a ground steel boss which was attaehed to a rigid frame. These measurements were very carefully made because they served as reference starting points for the median surface

traces later obtained for eight generators (equidistant around the circum-ference) ,

3.2 Shell Mounting Method

2024-T4 Aluminum end-plates were machined with parallel faces and a clamping groove to accomodate the ends of the shell (see Fig.

7).

One end-plate was placed in the retaining ring of the measuring apparatus (see Fig.

8)

and carefully aligned. Each shell was fitted snugly around the inside diameter of the groove of both end-plates. Hysol epoxy plastic, the same as used in the manufacture of the shells, was poured into the groove which was usually about 0.5 in. deep and 3/8 in, wide. Af ter the plastic had cured in one end-plate, the shell was inverted and the pro-cedure repeated for the other end-plate, It was established, by profiles obtained from the measuring apparatus, that the groove bottom. was

machined sUfficiently parallel to the face of the end-plates to guarantee less than 0.002 in. deviation between end points of the shell. This was extremely fortunate as it eliminated many trial and error runs which would have been required to establish perpendicularity of the shell wall with

respect to it~ end-plates.

All test shells, except those containing an axisymmetric dimple, were ready for detailed measurement of their thickness and media.n. surface profiles af ter mounting had been accomplished. In the case of the shells containing an axisymmetric dimple, a further step was required before they

•

could be measured. This consisted of ~eturning the shell (with end-plates installed) to the lathe for cutting of the desired profile on the outside wall. The original template was used in the hydraulic tracing apparatus and anormal tool bit substituted for the internal boring bar and bit. Due to the thinness of the shell wall, a supporting ring was installed inside the shell while the profile was being cut on the outside surface. These comments do not apply to Shell D7 as it had a straight outside wall and hence was characterized by a local th~ckness variation rather than an axisymmetric dimple imperfection.

Carbaloy tool-bits were used throughout the machining of the shell wall because of the superior cutting characteristics, Surprisingly enough, it was found that these tool-bits needed to be reground af ter each cut. Omi tting this step led to a dull cutting edge of the tool ... bi t which induced it to chatter. Any chatter of the tool bit resulted in a rough surface of the shell.

3.3 Measurement of Median Surface and Average Shell Wall Thickness Profiles The classical buckling load is obtained from the equation

P

d

= 2 7TE t2

IC

which contains thickness as a squared quant i ty. Hence i t is important to accurately determine an average

thickness for each test shell. The present investigation also called for an assessment of the degree of perfection of the shell wall profile. Both considerations were satisfied. by the construction of a profile measuring apparatus which is shown in Fig.

8.

It consisted of a rigid frame

that could be easily levelled and a motor driven turntable to which the shells were clamped. Profiles of the median surface were required in the axial direction. Thus, a height gauge was driven by a Slo-Syn motor by means of a lead screw, Two contact probes, Schaevitz PC - 107a linear variabIe differential transformers, were mounted on the height gauge so that they opposed one another, Although the use of contact probes in connection with thin walled shells is not an ideal measuring system, it was feIt that the spring constants (trimmed to less than one gram of force exerted on the shell surface) could be used with confidence in resolving

-4

displacements of the order of 10 in. These probes were driven by Atcotran 6101 Edemodulators. A schematic of the associated circui try and

readout is shown in Fig's. 9 and 10. Eight generators were measured for each shell. At the start of each traverse, the difference of the probe sig-nals was set to zero so that a reference could be established at points where the absolute thickness was known from micrometer measurements. The integrator, initially set to zero, was set to run as the probes started to traverse the shell. The time needed for the complete traverse was measured with a stop watch. Median surface profiles were obtained on - line and a value of average thickness (to be subtracted from the starting reference thickness) was obtained at the end of each run. When the profiles were re-drawn, the difference in starting thickness at various circumferential stations was taken into account. Although the circuitry did not provide for any indication of possible lean of the shell with respect to its end-plates, independent traverses of boththe inner and outer surfaces indicated overall leans of less th en 0.0003 in The circuitry, as shown in Fig. 10, was used for all shells except those containing an axisymmetric dimple.

Here the slope of the output - deflection curve for the outside probe was reversed and the difference amplifier was replaced by one in an 'adder' configuration. Although not shown, an additional operational amplifier manifold was used to obtain root mean square values of imperfection amplitudes. It was, of course, necessary to machine a hole in one end plate of each shell in order to insert the inside probe (see Fig.

8).

3.4

Test Procedure

Each shell was loaded in a Tinius Olsen electrically driven four-screw rigid testing machine of 60,000 lb. capacity. The loading range

(600 lb.) appropriate to the requirements of the experiment was calibrated with aMorehouse compression ring dynamometer of 1000 lb. capacity. This instrument was accurate to 1/10 of 1% of range at the calibrated load. Special care was taken to meet requirements concerning the loading of the dynamometer, which specified a surface of RockweIl C hardness 50 to 55 for the lower boss to bear on, and a cold rolled steel pad between the upper boss and the loading surface. These considerations ensured axial loading of the ~namom~ter. A calibration chart was used to compensate all readings for the ambient temperature.

Two geometrically 'near-perfect' shells were constructed to provide a reference basis for the subsequent imperfect shell buckling results. The reference shells were used to obtair. a compression modulus of elasticity from end-shortening data and at the same time, to determine the effect of the clamped end constrain~s. Each shell was placed in the testing

machine (see Fig. 11) and the loading platten was lowered to within 0.01 in. of the top end-plate of the shell. FeeIer gauges were used to check that the adjustable loading platten was parallel to the top end-plate. Changes in the alignment were accomplished by means of three levelling screws which controlled the inclination ofthe platten to within ± 0.001 in.

The elastic modulus as shown in Fig. 12 varied by

±7%.

This was obtained from the data of four dial gauges mounted 900 apart. A variation of only 2% in the elastic modulus was obtained by adjusting the levelling screws so that each dial gauge indicated almost exactly the same amount of travel af ter a predetermined load was applied. This latter procedure demon-strated that the loading was extremely uniform. Any small misalignment of the loading platten resulted in a small region of nonlinearity in the

load-elongation curve. This was circumvented by applying a pre-laad of 25 lbs. Far comparison purposes, a test piece of solid Hysol bar was machined and its modulus of elasticity established under quasi-statie

loading. The value obtained, 4.00 x 105 psi, (see Fig. 13) agreed within 2% of the value obtained by end shortening measurements on the reference shells. It was thus decided to use the modulus obtained from the cylinder compressian tests. Uniformity af strain during loading was also abserved in the isachromatic patterns which remained uniform in colour around the shell at any given load. Af ter buckling, the symmetry af the post-buckling

o 0

4

0

pattern and the 0 ,90 and

5

isoclinics for each. buckle around toe shell also attested to the uniformity of the strain distribution.

The optical sensitivity factor

(K'

factor) and Poisson's ratio were based on previously established data abtained far Hysol plastic under identical canditions. This was judged adequate as this laboratory has had an eight year association with the Hysol plastic in its present mode af

1. t· 3, 4, 19, 28, 30, 35 app lca 1.on

5

-1

All tests were made wi th an applied strain rate

S

3 x 10 - sec up to buckling. Prebuckling deformations were recorded by means af a non-contact displacement transducer. This was a

MTI

Model KD-45 Fotonic Sensor capable of resolving displacements of 10-5 in. (see Fig. 14). Radial dis-placements at a value of (J / (J

=

0.22 are shown as a function of shell

cr

length in Fig. 15. Values of pbuckling and post-buckling loads were re-corded far each shell.

Calour photographs and slides were obtained for most shells inthe pre-buckled and post-buckled configurations. Pictures were taken viewing the shell through aplane reflection-type polariscope with and without a full wave plate. These photographs are discussed in the next sectian and are shown in Fig. 17.

Experimental and geometric properties of each shell are summarized in Tables I, 11, 111 and IV. A discussion of the test results is cantained in Section

4.

It should be noted that because of the elastic behaviour of the shells, as many as 20 to 40 tests were conducted on each shell. The data was thus obtained throughout many runs. The repeatability af this data was established by the constancy of buckling laads, pictures ofisochromatics and isoclinics and axial end-deflection versus machine load curves.

3.5

Photoelastic Method of Stress Analysis and Photographic Technigue.

Phötoelastic analysis was used only to a small extent in this investigation, primarily te determine the effect of end constraints and as a qualitative tooI in assessing general stress distributions. However, the property of the photoelastic epoxyplastic shells to buckle elastically constituted

a

fundamental basis of the entire test programme.

In order to analyze the shells photoelastically it is assumed that the wall thickness is very small compared to the other dimensions of the shell. Hence a two-dimensional stress analysis is employed. The prin-cipal stress difference at any point on the shell was obtained by using a reflection-type polariscope.

Following the work of Gorman 18 and Tennyson 19 , it was of interest to determine the ratio of maximum shear stress resultant to maximum shear stress res~ltant with edge effects neglected. This is shown in Fig.

16

for a value of cr}J

=

0.22. I t can be seen that a region of roughly 0.75 in.

cr

at each end of the shell is affected by the clamped end constraint . A dis-cussion of the pertinent equations and techniques involved is given in

App~ndix B.

To obtain statie photographs of the isoclinic and isochromatic patterns, two types of film were used. Kodacolour - X with an ASA of

80 was used to obtain prints and Kodak High Speed Ektachrome with an ASA of 125 was used to obtain slides which were subsequently processed to yield prints. Exposure times of the order of 1/2 second resulted in accurate

colour reproductions of the stress patterns (see Fig. 17 and Sections 4.1

- 4.3).

In some cases, it was necessary to insert a full wave plate into the field of view to increase the fringe order by unity. Pictures of the post-buckled isochromatics and isoclinics were obtained as quickly as possible so as not to subject the shells to appreciable creep.

4.

DISCUSSION OF TEST RE~ULTS

4.1 Comparison of Test Data With Theory for Uniform Axisymmetric Imperfection Distributions.

Eq. 2.2.10 was solved for minimum values of À by varying the

circumferential wave number parameter T. This was done for specific values of K aild

IJ.

Fig' s 18 and 19 show the solutions obtained for

appropriate for the shells tested). Results were generated for a wide range of the axial wave number parameter K. The envelope defining the lower bound of minimum À (approached fr om above) versus ~ for all

values of K is shown in Fig. 20, which lies somewhat lower than Koiter's results 14 for K

=

0.5. Only as

~ ~O

(i.e.

the~asymptotic

solution ) does K

=

0.5 yield the minimum buckling load as demonstrated in Fig. 21. It is quite clear from the analysis that the critical imperfection wavelength leading to a minimum buckling load increases for increasing values o~ the imperfection amplitude.

Nine shells, Al- 9, (refer to Table I) containing uniform axisymmetric imperfections were tested to compare results with Koiter's extended theory. Two geometrically'near-perfect' shells provided reference data. The appropriate shell geometry for a reference shell and median sur-face profiles of shell No. R2 are shown in Fig. 22. Geometrical properties of shells containing uniform axisymmetric imperfections are shown in Fig. 1. The median surface profiles for shells A 4 - A

9

are shown in Fig's. 23 -25.

Fig. 17 shows, in colour, the prebuckling i~ochromatics for

shells A 2 and A 3 and also the corresponding postbuckled isochromatics and isoclinics. It should be noted that Table I indicates one and two tiers of buckles for shells A 3 and A 2 respectively. However, Fig. 17 shows the

reverse. Both shells had almost interchangeable equilibrium states (at the same load) of one tier of six buckles or two tiers of seven buckles. Only the slightest change in alignment of the top platten would cause the shift in the postbuckled configuration. Data given in Table I is thought to be more representative than that shown in Fig. 17 as it lists modes which were finally established as repeatable.

A comparison of the shell buckling data with Koiter's extended theory and the results of Hutchinson's exact model analysis is contained in Figs. 26 and 27 respectively. In each case, the experimental data is

within 10% of the predicted values. The remarkable feature of the com~

parison between Koiter's model and the exact formulation is that for the range of imperfection amplitude and axial wave number parameters considered, no significant difference exists between the calculated critical loads. Hence, it appears that the load reduction due to initial shape imperfections

is the dominant factor completely overwhelming the effects of thickness variation and end constraint. It. should be noted that the buckling loads obtained for the geometrically 'near-perfect' shells were predicted

exactly by Hutchinson's model.

To determine the effect of imperfection amplitude on the critical buckling load, four shells were tested having nominally the same axial wave

number parameter (K

=

0.594) and varying values of im~erfection

avg.

amplitude _to - thickness ratio. The results are contained in Fig. 28 and a comparison is made with both Koiter's extended theory and the exact

model calcula~ions. One geometrically 'near-perfect' shell is included

for reference purposes. It was also of interest to deterrnine if a critical axisyrnmetric imperfection wavelength existed which would yield a minimum buckling load for a given value of imperfection amplitude. Five shells yere tested having nominally the same imperfection amplitude parameter

(

~

=

0.047) and varying values of the axial wave number K, As shown in Fig. 29, a critical value of Kdoes indeed exist at which a minimum buckling load is observed, consistent with the results of Koiter's extended theory.

·The e~act model results are also shown in Fig. 29 for comparison purposes. It is noted that the difference between Koiter's extended theory and

experiment is small, except as K -7 l.O. In this latter case, the ini tial

buckling mode assumed by Koiter is only an approximation and a more appropriate comparison to the experimental data is afforded by the exact model. Although no shells were tested having the wave number parameter K -7 0, i t would be expected that the exact model would again agree more

closely with experiment. In this range, Koiter's analysis is limited by the conditions on the wave numbers (m, n must be large). Due to the pre-ceding discussion of the validity of Koiter's theory as K -7 0, and 1,

it w~s considered appropriate to test a series of shells having a wave number close to the critical value and hence well away from the above suspect ranges. This was done as previously described for a nominal wave number parameter K

=

0.594, the results of which are shown in Fig. 28.

A series of shells, ADl -

5

(refer to Table 11), containing groups ofaxisyrnmetric imperfections was tested to investigate the effect of having imperfections of various wavelengths in the same shell. A

constant imperfection amplitude-to-thickness ratio nominally equal to

schematic representation of the median surface of the shells tested in this series and the actual median surface profiles of shells ADl - 5 are shown in Figs. 31 - 33. As can be seen, three shells had identical distributions of imperfections except that the critical wavelength, i.e. an imperfection wavelength equal to the classical axisymmetric buckling mode, (in terms of the wave number used in this paper, this cor-responds to K

=

0.5; actually a value of K

=

0.468 was used) was placed at various axial locations in the shell w~ll. It can be seen from the results obtained (Fig. 34) that wavelength location had only a small effect on the buckling loads. However, it was noted that the postbuckled configuration of one shell was biased to the end which contained wave numbers leading to the lowest equilibrium states (see Fig. 17). The prebuckling isochromatics for this particular shell are also shown in Fig. 17. Subsequent tests were conducted with four and six critical half waves located at the centre of the shell between varying numbers of half waves of different frequencies. The latter case resulted in a buckling load equal to that predicted for an equivalent shell containing a uniform axisymmetric distribution of imperfections.

It would appear that ex cept for small differences, for design

purposes the buckling load reduction was essentially governed by the presence of the cri tical frequency.

4.2 Comparison of Test Data with Theory for a Local Axisymmetric Dimple Imperfection

Predictions based on the asymptotic formula, Eq. 2.3.2}and the numerical calculations are compared in Fig. 35 for K

=

0.5. Only the absolute value of the imperfection amplitude enters into the asymptotic formula and thus the curve shown holds for inward and outward dimples. On the other hand, the numerical results bring out some difference between the buckling load for a bulge (curve A) and a pinch (curve B) with the latter having the more degrading effect. Agreement between the asymptotic formula

and the numerical resultsfor the inward dimple is quite reasonable. The length parameter, Z, for these shells w~s taken as 300 so that for all practical purposes, the buckling load was independent for values of Z in this range and greater.

All numerical calculations were made with 120 finite difofèrence stations over the half length of the shell and isolated results were checked using 250 stätions.

From Fig. 35 it can be seen that localized dimples in constant thickness shells have an effect which is somewhat less severe than a sinusoidal axisymmetric imperfection. Almroth 23 has made the same obser-vation on th~ basis of his studies of the effe cts ofaxisymmetric imper-fections on the buckling load of cylindrical shells.

Fischer 13 consideredan initial axisymmetric dimple imperfection of the form

o

<x

<

l

!lil

=

-

-

x

l

_x

< x :::

L/2

o

which he evaluated for two values of land positive values of 5(outward x

bulge). Ris results indicated that an imperfection of the above form is

considerably less critical than a periodic imperfection of the same amplitude. This is, of course, consistent with the numerical results pTeviously

described.

Effects of local thickness variations are illustrated by the results presented in Fig.

36.

These results were also obtained numerically in the manner described in Section 2.3 and detailed further in Section 2.2.2. For both the lowest and uppermost curves)the thickness variation is given by 6t

=

5 (1 + cos

]!)

x

=

0

I xl

<

I xl>

l x

l

x 4.2.1 with either the inner or outer wall unpert~ed as shown. Thus, this variation gives rise to exactly the same initial middle surface variation as described by Eq. 2.3.3. It is not surprising that a shell which has both an inward middle surface varîation and a reduction in its thickness buckles at a load considerably below a constant thickness shell with an

as indicated by the uppermost curve, results in a small reduction in the buckling load if

Bit

is sufficiently small.

Two shells, Dl and D7 (refer to Table 111) were tested under axial compression to verify the theoretical predictions. Shell Dl had a local inward axisymmetric dimple centered at mid~length and was (nominally) of constant thickness. Shell D7 had a thickness variation given by Eq. Eq. 4.2.1.

The average shell wall thickness

t*

,

measured over the dimple area defined by Ixl~x ' was used as the reference thickness in calculating the buckling load of Shell Dl. Good agreement with the numerical calculations was observed for this shell. However, Shell D7 buckled at a load 10% higher than was predicted. It should be noted that the values Qf the wave number parameter K and Poisson's ratio v was somewhat different to those used for the numerical calculations. The difference in v would account for about 3% of the nominal 10% discrepaney between the data and theory and the difference in K would also tend to reduce the discrepancy.

Six shells, Dl -

6

(refer to Table 111), each containing a loeal inward axisymmetric dimple imperfection centered at mid-length were tested under axial compression. Typical median surfaee profiles are compared with eosine functions (of appropriate amplitudes and frequencies) generated by a Wavetek function generator. The measured profiles show good agreement with the assumed geometry (see Fig. 37). Complete median surface profiles

are shown in Figs. 38 - 40.

Typical prebuckling isochromatics and postbuckled isoclinics and isochromatics as seen through a reflection - type polariscope are contained

in Fig. 17 for shell D5. Local stress concentrations around the dimple are evident in the prebuckled shell configuration resulting from local bending deformation. A similar effect is generated by the clamped edge constraints.

A comparison of shell buckling data for shells with inward dimples and theoretical predictions is made in Fig. 41. The theoretieal curves have been obtained using K

=

0.473 whieh is the average of the

corresponding values for the three test specimens (data for a perfect shell was included for reference purposes). As alluded to previously, the average wall thickness

t*,

which was measured only over the dimple area defined

by Ixl<

l

,

was used as the reference thickness in calculating the

-

x

buckling loads. Justification of this step rests on the argument that bucklipg is a localized phenomenon governed by the dimpled region. In any case, the discrepancy between the average shell wall thickness

t

and the loc al average

t*

was less than five percent in every shell but one as

listed in rable 111. All the test specimens fall in the range in which the numerical predictions are essentially independent of the shell length. The agreement between test and theory, particularly with the numerical results is very good.

Fig.

42

displays plots of buckling loads for constant thickness shells ~ith inward dimples over a range of values of the imperfection wave number parameter

K

for a constant value of the imperfection - thickness

-.

6

ratio

5/t

The value of

5/t

=

0.3 3

was chosen as the average of the values for the three experiment

al

points plotted. I t is clear that the critical wavelength occurs for

K

~

0.6

and the associated buckling load is only a few percent below the predicted value for K

=

0.5. The asymptotic formula shows the same general trend as the numerical results.

4.3

Comparison of Test Data with Theory for a Random Distribution of Axisymmetric Imperfections.

Three shells,

AD 6 - 8

(refer to Table

IV),

containing 'random' axisymmetric imperfections were tested to determine if the power spectral density of imperfection components, evaluated at the criti~al frequency

(i. e.

Tr/

l , correspondi,ng to the classical axisymmetric buckling mode x

cr

wavelength), would govern the buckling behaviour of a circular cylindrical shell 31

Originally the imperfection profiles for the above three shells were ~pecified by the sum of three eosine functions, each having different

frequencies and amplitudes. One of these frequencies was determined to be critical for each particular shell. Due to the machining operation, certain frequencies were introduced and altbough the profile was hardly random, in a statiStical sense, the profile certainly had more than three Fourier compohents. Profiles of ·the median surface of the three shells in this series are shown in Figs.

43, 46

and

49.

The complex Fourier series co-efficients of imperfections, for the above shells, are plotted in the

frequency domain in Fig's.

44, 47

and

50.

The corresponding estimated power spectral densities of imperfections are shown in Figs.

45, 48

and

51.

From Fig. 52, experiment al agreement with the asymptotic theory of Amazigo 31 is

~remely

good. It is concluded that for the range of test data obtained, the cri tical frequency component does indeed govern buckling of a circular cylindrical shell as predicted by Amazigo.

The test data mentioned above was compared with the theory presented in Ref.

34.

The author of this paper pointed out that the

stability condition employed in the analysis was only a sufficient'criteria for the shell. It was further noted that when the load is close to 'the linear critical one and the power spectral density function of the im-perfection varies slowly in the vicinity of the axisymmetric response

function, the stability condition obtained will be sharper than that obtained by~apunov's method. Hence it is not surprising that the stability

boundary provides a conservative estimate of the buckling loads (see Fig. 53). It should be noted that the theory of Ref.

34

assumes that imperfections are continuous, stationary and ergodic.

5.

CONCLUSIONS

The spin - casting method outlined in Section 3.1 was found to yield geometrically 'near-perfect' cylindrical shells. These cylinders, with clamped ends, when subjected to axial compression buckled elastically within 1% of the load predicted by Hutchinson 27. The repeatability of the buckling process allowed as many as forty tests to be carried out on one specimen. As aresult, each test series employed only a few cylinders.

The manufacture ofaxisymmetric imperfections inthe test shells . as mentioned in Section 3.1, was carried out with sufficient accuracy to obtain buckling loadswithin 10% of all relevant theory.

It was shown that for shells containing uniform axisymmetric

imperfections, a critical ~avelength did exist for a particular imperfection amplitude and that buckling loads were reduced by increasing values of

imperfection amplitude for a particular wave number, as predicted by

Koiter

14.

It was further shown that as the number of critical imperfection wavelengths was increased to three, in a shell containing groups

the observed buckling load corresponded to that obtained for a shell containing a uniform distribution ofaxisymmetric imperfections of the critical frequency with the same amplitude.

For shells containing an inward axisymmetric dimple, it was shown that for a given value of imperfection amplitude, a critical dimple wave-length exists for which the buckling load is a minimum, at least for the Wave form studied. As expected, for a given wavelength, buckling loads decreased rapidly with increasing values of imperfection amplitude. Both trends were predicted by the numerical calculations of Hutchinson 30 and the asymptotic formula of Amazigo, et. al. 29. In general, for the axi-symmetrie dimple profile considered, buckling load reductions are not as severe as compared to a uniform distribution over the entire cylinder length.

Shells containing 'random' axisymmetric distributions of imperfec-tionshave buckling loads that are essentially governed by the power spectral density of the imperfections evaluated at the critical frequency, at least for the range of test data obtained. This is in agreement wi th the asymptotic theory of Amazigo 31 and the work of Fersht 34.

It would appear that the last series of shells yielded data which tends to validate the general random imperfection approach to the determina-tion of buckling loads of practical shell structures. This method should provide a systematic approach to the determination of the stability of thin walled shells. It would of course be necessary to extend the analysis to include asymmetrie distributions of imperfections.

An

experimental

program to com~liment this theoretical extension should prove to be extremely worthwhile for practical design purposes.