Analysis of the Buckling process of circular cylindrical shells under axial compression

IsmE

HOGESCROOL OELFl

IGBOUW~~E

8IBLIOTHErK"",.uYSIS OF THE BUCKLING ffiOCESS OF! CIRCULAR CYLINDRICAL SHELLS UNDER AXIAL COMPRESSION

"Qy

•

ANALYSIS OF THE BUCKLING PROCESS OF CIRCULAR CYLINDRICAL

SHELLS UNDER AXIAL COM}RESSION

by

R. C. Tennyson and S. W. Welles

Manuscript received November

1967

ACKNOWLEDGEMENTS

The authors are grateful to Dr. G. N. Patterson, Director of the Institute for Aerospace Studies, for his interest and encouragement in the re-ported research. We wish to express our appreciation to Dr. R. W. Leonard, Head, Structural Mechanics Branch at NASA ~angley, and to Dr. J. HutchinsJn, Assistant

Professor in the Division of Engineering and Applied Mechanics at Harvard Uni-versity, for their many stim~lating discussions pertaining to this work.

This research was made possib1e through the financial assistance of the Nationa1 Research Counci1 of Canada (NRC Grant No.A 2783) and the

Nationa1 Aeronautics and Space Administration of the United States (NASA Grant No. NGR 52-026-(011»).

'J

SUMMARY

Geometrically 'near-perfect' circular cylindrical photoelastic shells having radius-to-thickness' ratios of, the order 100 "'" 440 have been te.sted in p~re axial compression • . The critical buckling loads were found to agree within 10 "'" 14% of the classical value,. or within a few. percent of the reduced buckling load taking into acdount the clamped end constraint.

High speedphotographs of the buckling process were obtained using two cameras viewing the change in the 450 isoclinics over the entire cylinderls length and over

60%

of the cylinder~s perimeter. A theoretical analysis of the inception of buckling using Koiter's mode shapes has demonstrated that the classical buck11ng mode was observed in the experiments for the first time.

Further investigation of the nonlinear postbuckling mode shapes just af ter initlal buckling has predicted the wave forms observed. It was also determined that the

shallow shell equations used to describe the large-deflection postbuckling be-haviour do not predict isoclinic patterns which are observed in the later stages of buckling. Consequently:i.t is concluded that postb:y.okling load calculations based on these equations are inaccurate beyond the early stages of buckling.

.. :., }' ..

1. 2.

3.

4.

TABLE OF CONTENTS NarATION v INTRODUCTION 1

BASIC EQUATIONS USED IN THE ANALYSIS 1

2.1 Derivation of the Isoclinic Equations 1

2.2 Equilibrium and Compatibility Equations

3

2.3

Solution of Equations of Equilibrium for Pure Axial

Compression

4

2.4

Analysis of the Classical Buckling Mode

45

0 Isoclinics

7

2.5

Analysis of the Nonlinear Postbuckling Mode

45

0

Isoclinics

9

EXFERIMENTAL TECHNIQUE AND RESULTS

3.1

Construction and Testing of Photoelastic Circular Cylindrical Shells

3.2

Experimental Results C ONCWS I ONS REFERENCES TABLE FIGURES

15

15

16

17

18

Al,A2 D E F G L ;'x, ;'y m,n p

Po

R t u,v,w x,y,z

X,Y

Z Greek Symbols

NOTATION

integration

EtS

constants (Eq. (47)) flexural rigidity of shell modulus of elasticity

Airy stress function E 2(1+V) , shear shell length modulus L , nR respectively 'm n

the number of half-waves and waves in the axial and circumfer-ential directions respectively

rmrR

1 1

[12{1 -

v2)J~

(

~

)2,

the critical axisymmetric wave number shell radius, measured to the mid-surface

shell wall thickness

displace~ents measured in the x,y and z directions respectively Cartesian coordinates measured in the axial, circumferential and radial directions respectively

;,

i

';'~

respectively

12 1

Rt {1_v2 ₎₂

v

4 E x' E _Y ~ Tj

e

",

"1 j.l v p cr _cr cr , cr x _Y T xy rp1' rp2

normal strains measured in the x and y directions re'spective1y

ang1e of inc1ination of principa1 s~esses crcr E respective1y Poisson's ratio cr x cr cr

critica1 buck1ing stress

normal stresses measured in the x and y directions respective1y shear stress in the x-y p1ane

P L o

2J2"R

1

P L

( 1 +p ) 2" ,

21\}~

R 1 (l_p)2" respective1y

,

1. INTRODUCTION

During the past five years, several attempts have been made to

determine the initial buckling modes of a circular cylindrical shell under axial compression. The methods usedlby various investigators have included dyna~c3 radial deflection measurements , recording of changes in iao§linic patterns ' and gbservations of reflected light from a shell's surface ' . As noted by Hoff , the classicalor 'checkerboard' asymmetrie wave pattern has never been

observed in experiment even though it clearly exists in theory.

This report describes the results obtained in a research pro-gramme which was initiated to determine the initial buckling modes of a

circu-lar cylinder as it collapsed under axial compressive loading. Photoelastic shells were constructed using the spin-casting technique

7

and the buckling pro-cess was studied by recording the change in the

45

degree isoclinic patterns using high speed cameras. From a theoretical analysis of both the linear

and nonlinear shell equilibrium equations in terms of the isoclinic patterns, it isfel

_B

that conclusive evidence of the classical bucklingmodes as defined by Koiter have been obtained for the first time. Consequently, it has been shown in this report that a circular cylindrical shell sufficiently free of imperfections in shape under axial compression buckles near the reduced

classical value (taking into account the effect of end constraints) in a mode shape predicted by classical theory.

A theoretical analysis of the

48

0 isoclinic patterns using the classical buckling mode shapes given by Koiter is first presented to describe the inception of buckling for a perfect

axial compression. The nonlinear shell

mine subsequent

45

0

isoclinic patterns. with experiment and conclusions made.

2. BASIC EQUATIONS USED IN THE ANALYSIS

circular cylindrical shell under pure

equations are then analysed to deter-The predicted patterns are compared

2.1 Derivation of the Isoclinic Equations

In order to study the buckling mode shapes of a circular

cylindri-cal shell under axial compression, the method of isoclinics was selected. To successfully employ this technique analytically, it is necessary to derive the equation of an isoclinic in terms of the stresses or strains. Since the stress distribution in a circular cylindrical shell element can be approximated by a

plane stress system, it is relatively easy to determine the isoclinic equation. Relationships between the stresses (or strains) and shell displacements for the

median surface must then be obtained. Hence, the problem of defining an

iso-clinic of parameter 8 during the buckling process requires an adequate descrip-tion of the displacement field.

From the theory of photoelasticity9, it is known that an iso-clinic of parameter 8 defines the locus of points in a body subjected to a

plane stress system whose principal stresses ~l _{andrr2 are inclined at the} angle

e

to a set of orthogonaroaxes, x, y (refer to Fig. 1). It is also, known

from the theory of elasticity that principal planes are shearless planes, and

the maximum shear stress occurs on planes inclined at an angle of

45

0 _{to the} principal planes. Bymaking use of the zero shear stress condition and the property of isoclinics, an equation relating the plane stresses (or strains)

For a plane stress system aeting on an element of shell, the shear stress on any set of reetangu~ar axes·x', y' inelined at an angle 8 to the x, y axes ean be written aslO

2 T _{x'y' -}- (~ - ~ ) sin 28 + 2 T cos 28

Y x xy

set of It is possible to ehoose a set ofaxes sueh that TX'y'

= 0

i.e., a

prineipal axes. Thus, for Tx'y' ::: 0, the neeessary tondition on 8 is tan 28 ::: . 2 T ~ x xy

-

~ y ( 1) (2)

On any plane defined by Eq. (2) there will be no shearing stresses, but only normal stresses aeting on the element. These stresses are called

.prineipal normal stresses and the planes on whieh they act are prineipal planes. It may then be eoneluded that Eq. (2) defines an isoclinic of parameter 8 for a

plane stress system.

,

For a linear elastie material, the median surface stresses are related to the strains by Hooke's law;

E ( Ex + v E?j) ~ ::: l_v2 X E ( _{Ey' + Ex)} ~ ::: l_v2 v y

,

_E T ::: G T where G

2(

l+v) xy xy

Henee Eq. (2) ean be re-written in terms of the strains

tan 28 Î'xy

E. - E'

x Y

If 8 450, Eqs. (2) and (6) reduee to ~ ~ 0 if T

r

0

x y xy

and _{::: 0}

if Î'xy

r

0 E. _x E _y

respeetively. Henee Eqs. (7) and (8) define the 450 isoclinics.

(4)

(6)

(7)

,

2.2 Equilibrium and Compatibility Equations

The following equations of equilibrium were first presented in References 11 and 12 for a linear elastic, perfect cylindrical shell. The post-buckling behaviour of the cylinder for moderately large displacements and

small strains is adequately described by these relations*

~ +T

=

0 x'x xy,y T + ~

=

0 xY,x y,y

(10)

Q

~

w =

~

w, + 2 T W, +

~

w, +

~

IR

t x xx xy xy y yy y (n)

The 'corresponding strain-displacement equations used in the study of the postbuckling states are

E _{x u'x} + 2" 1 w, _x 2 (12)

1. w 2 W

E = V +

Y 'y 2 'y R (13)

)'xy = u _'y + v, _x + w, _x w, _y

(14)

The conditions of equilibrium described by Eqs. (9) and

(10)

are

identically satisfied by the introduction of an Airy stress function F(x,y) defined by the relations

cr

=

F

(15)

x 'yy ~

=

F

(16)

y 'xx T F

(17)

xy 'xy

Substituting Eqs. (12) to (17) into Eqs.

(3) (4)

and

(5)

and making ~se of Eqs.

(9) and

(10),

a compatibility equation, is obtained yr4F=E( W 2

'xy w 'xx w 'yy

1

- w )

R

'xx

* Subscripts following a coma indicate differentia~ion with respect to the variables shown.

Equation (11) can also be re-written in terms of the stress function

D-t

I?w

=

F }ol - 2 F, w + 1i' W +

1:

F

'yy 'xx xy 'xy "'xx 'yy R 'xx

An alternative equation of equilibrium can be obtained by combining Eqs.

(18)

and

(19).

If the nonlinear terms are omitted, Donnell's linear equation des-cribing the b~ckling of a cylindrical shell ean be derived

13

in terms of the

radial displacement only. The eomI?atibili~'y eqûation, c,al'). also be reduced to the linear form

~

F + -RE w, = 0

xx

(

\

2.3

Solution of Equatiorn of Equilibrium for Pure Axial Compression

(20)

Th

_{e e aSSlca}

1

" 1 1"

_{1near equa lOrrS} t" _I ₀ f _{equl 1 r1um} "

1" b" 14

_{ean e sa}

b

t" f_{lS 1e} ·

d

by assuming

v

=

0, v u'x = -w

=

a constant

(21)

R

This sol~tion represents the cylindrical form of equilibrium in which the com-pressed shell expands uniformly in the radial direction, neglecting the effects

of end constraint.

Another solution to the classical problem of buekling of cireu-lar cylindrical shells can be obtained

_bY

assuming v == 0 and

sin (m;:)

(22)

u cos cos (m;:)

(23)

w == sin

where the phell is presumed to initially buckle into m half-waves in the axial direction

14 .

In Ref.

14,

i t is shown that if the cylinder is assumed to buckle into many waves along the length in the axisymmetric mode, the corresponding critical buckling wave length is

L

= TT

m

Hence the axisymmetric buckling mode is characterized by

u sin (POX) cos R w == C?S (POX) Sln R where 1

[12(1-v2)]~

(

~

)2

Again, the effects of end constraint are neglected.

(24)

If at the inception of buckling the cylinder is presumed to buckle

into a periodic array, defined by the characteristic wave lengths

tx

and ty,

the displacements~, v and w (as well as the stresses) may be assumed to be

periodic with a wave length

2tx

in the axial direction and a wave length 2ty in the circumferential direction. Hence, the shell at the moment of buckling

is ass~med to subdivide into 2 mn rectangular regions with sides

L

tx

=

-m

77R

n

In each of ~hese regions, the cylinder buckles inwards or outwards~ The regions

in which buckling occurs in the same direction are distributed in a

'checker-board' pattern.

A more general solution of the buckling problem is obtained by

assuming a displacement pattern of the form m71X sin

EX

( 29) u u cos L mn

_1\

ID71X ny (30) v

=

v sin - - cos mn L R m71X _{sin ny} (31) w

=

w sin -mn L R

Because only the radial displacement w(x,y) appears in Do~~ell's equation, it is not necessary to specify the pand v displacement functions in order to

determine the critical buckling load. The solution of Donnell' s equation

us-ing the asymmetrie (or square wave) mode for w(x,y) yields

Equation (32) relationship nL where

S

=

-rrR Equation (33) cr cr

is valid only as long as the wave numbers

2 1

(

m2 +

t

2 )

₌

(12rr~2

)2

m2 L2 1 , Z -RT (1_v 2

)2

can be re-written in the from

p2 _ P P + n2

=

0

0

mand n satisfy the

• where p

=

~R/L and Po is given by Eq. (27). These results were derived on (32)

(33)

(34)

(35)

the assumptions that both pand n are large compared with unity and that boundary

conditions at the shell ends may be ignored. Equation (35 is plotted in Fig. 2.

KOiter

8

,15 has shown that for each

val~e

of n, the general

two properly seleeted non-symmetrie terms. Consequently, for the linear buekling problem, a suffieiently general linear combination of buekling modes is obtained from the following:

(36)

Koiter further showed that the asymmetrie buekling mode ampli-tudes for the radial defleetion funetion ean be related to the wave numbers and the symmetrie mode amplitude by the relations

Pnl wnl

= ::

Pn2 wn2 (39) -4(1-,-À) R p 2 W + 3 ~ n2 w wn2 0 (40) 1 0 0 n nl where w

=

±

~

(À-À ) R (41) 0 _{3 1} o-x 0-cr À = - _{, Àl =}

E

E 1

(~)

= I [3(1_v2_)]2 Equation (41) may be re-written in the form

Wo

~

- 0.42 t (1 -

~~r)

(42)

for v

= 0.49

The amplitudes of the u and v displacement funetions defined in Eqs. (36) and (37) eart be determined in terms of the amplitudes of the radial . displacement by means of the linear eompatibility equation (20), Hooke's law (Eqs. (3), (4) and (5) } and the linearized form of the strain-displaeement relations i.e., E _u'x (43) x w (44) E

=

v, -

-y y R -Y'xy

=

u, y + v, x (45)

As the eircular cylindrical shell is loaded in pure axial

com-pression, classical theory neglects the effects of· end constraint and assumes

the shell expands uniformly. However, although end constraints do not. signi- 16

ficantly reduce the classical bucklingload for a cylinder having clamped ends ,

they do alter the prebuckling radial displacement pattern of the shell. In any case, the deformation is axisymmetric. As aresult, the principal normal

stresses are coplanar with the x and y axes (i.e.,

8

=

0

0_,

₉₀

_{0 ).}

The prebuckling radial displacement function wo(x) has been de-termined16 by solving the axisymmetric form of the equilibrium equation (11)

(46)

The solution of'Eq. (46) for the case of a cylinder in axial compression having

clamped edges with the origin of the coordinate system at the midpoint of the cylinder is where cj) 1 p cr sinh 1 1

2 [(1_P)2 cos cj)l sinh cj)2 - (1+p)2s in cj)lcosh cj)2]

I I . 1 -2 [(1_P)2 sin cj)l 1 (1+p)2 sinh 2cj)2 p L 1 0 _{(1 +p)2} 2;';2 R cr x cr 1

cosbcp2 + (1+p)2 coscj)l sinhcj)2]

I

t

(1 _p)2 sin 2cj)1 P L 1 cj)2 _2;';2 0

_R

(1_p)2 v~ cr .. x w -E 00

Eq"ilation (47) is p10tted in Fig.

3

for p =

0.9.

It is evident that for

x

1f2

<

0.6

w (x) o ~ w 00

=

a constant.

2.4 Analysis of the Classica1 Buck1ing Mode

45

0 Isoclinics

For a given structural configuration, isoclinics próvide a means of determining the inclination of the principa1 stresses independent of the

their magnitude. Consequent1y, if the configuration does not change geometrica1bY the family of isoclinics wi1l not change even if the app1ied 10ading changes in

magni tude • The main point to consider is therefore the proper geometric descri.

Consider the domain of the cylinder suffieiently far removed from the ends (i.e., 0.2

<

x/L

<

0.8 (refer to Fig. 4» that the effect of elamped end constraints ean be negleeted. The total radial deformation funetion ean be written in the following form, making use of Eq.

(38),

w(x,y)=="" w (

00 + W o

Pox cos - - +

R w mn cos

p;

cos

n:

(48)

assuming Pnl;'; Pn2 == p. _{For this particulat. case, wnl} == _{wn2 (refer to Eq. (39».}

Sinee the origin of the eoordinate system is arbitrary, wmn == ~ wnl • Fro.m Eqs. (40) and (41), the asymmetrie wave amplitude ean be determined in terms of the axisymmetrie value

2

w2

P 2 2 2 o 0 W_nl == wn2 == 2 n Hence, 2w _Po 0 w _mn == n (50)

Sinee the elassieal buekling modes oeeurring at the ineeption of buekling are being investigated, the linearized form of the eompatibility

equation (Eq. (20) ) ean be used. Hence, the appropriate form of the stress f'unetion is

r

Pox px ny

F(x,y) == -

U

'

:

F 0 cos R + F mn eos R cos

R

+ (51)

Substituting Eqs. (48) and (51) into Eq. (20) yields, Fo/w_o == - ER/p~

(52)

F

/w

== - ER/p~

mn mn

The equation for the 450 isoclinies in terms of the stresses, Eq.

(7),

ean be re-written in terms of the stress funetion with the aid of Eqs. (15) and (16). Thus,

F, -F, =0

yy xx

(53)

defines the 450 _{isoelinies providing F,xy}

f

_{O. Substituting Eqs. (51) and (52)}

into Eq.

(531

yields, w mn

?

_o 0- R x E

or, substituting Eq. (50) for W

mn one obtains

p x 0- R

o 2 (2 2) px ~ X

eos

R

+ p - n c o s R eos R == woE pon

which defines the 450 isoclinic equation at the inception of buckling.

When p = n, Eq. (54) reduces to

cos p x o - - = R CJR x wE o

Equation (55) defines 450 axisymmetric isoclinic rings separated by a distance

2nR/po (refer to Fig. 5).

When p

f

n, Eq. (54) plots in the form of an oval shaped

iso-clinic pattern, as shown in Fig. 6.

2.5 Analysis of the Nonlinear Postbuckling Mode 450 Isoclinics

The nonlinear form of the compatibility equation used to date in the study of the postbuckling behaviour of cylindrical shells is given by

Eq. (18).

A general mode shape will be assumed having the form

w(X,Y)

= -

W

oo+ wl1 cos H X cos H Y + w20 cos 2 H X

+ w

02 cos 2 H Y + w22 cos 2 H X cos 2 H y.+

+ w

40 cos 4 H X (56)

which corresponds to one of Almroth's17 modes used to determine the postbuckled equilibrium load of the shell. From existing analyses of the postbuckling

con-figuration18 , it has been found that w02

~

O. Although we are studying the

unstable equilibrium states of the cylinder during the buckling process, it

will also be assumed that w02

=

O.

Substituting Eq. (56) into Eq. (18) yields

cos 2 H X

-• cos 2 H Y - cos H X cos

3

H Y

(4

"'w

_n

_w22 2 w₂₀ w_ll H4 8 H 4 _{w40wll )}

.e?

]'j2

+ ]x2

]'j2

+ .ex2 .ey2 cos

3

H X cos H Y

(-40 16

1f'

w 2

8

,,4 ) 8 H4 w2

+ 22 cos 4 H X - 22 cos 4 H Y +

327T4 w40w22

$? $']2

cos

6

7T X cos 2 7T Y

Assume a stress function of the form

F(X,Y) [ F 20 cos 2 7T X + F 02 cos 2 7T Y + F 11 cos 7T X cos 7T Y

+ F 13 cos 7T X cos 3 7T Y + F 31 cos 3 7T X cos 7T Y + F

40 cos 4 7T X

+ F 04 cos 4 7T Y + F 22 cos 2 7T X cos 2 7T Y + F₄₂ cos 4 7T X cos 2 7T Y

+ F 51 cos 5 7T X cos 7T Y + F 62 cos 6 7T X cos 2 7T Y + ()2

X (

11RnY

)2 ]

(58)

Substituting Eq. (58) into the 1eft-hand side of Eq. (57) yie1ds,

\?F/E

1 [167T 4 F 20 E $x4 cos 2 7T X + 16~02 $y4 cos 2 7T Y + _{cos 7T X cos 7T Y +} + + 256 7T4 $y4 16 7T4F42

:ex

4 $y4($~ +

4

$y2)2

Equating eoeffieients of the trigonometrie funetions in Eqs. (57) and (59): gi ves, = E

~l

(

8

w20

ty2 )

32

~2

1 -

i2R

w

2 11 2 E

wll~2

(

(1

+ ~2)2 w20

-~)

4 E

~2wllw2.2

(9 + ~2)2 2 E ~2wll (1 +

9

~2)~

_(w20 + _{2 w22} + _{4 w40 )} (60) E ~2 ~2 F₀₄ = 32 2 E ~2 w 22 w40 (1 - ty2 ) F 22 (1 + ~2)2

87?R

w 4 . F₄₂

=

E ₂₍₁₊₄ ~2w20 _~2)2 w22 F 51

8

E ~2w40 w

_n

(1 + 25 ~2)2 F₆₂

=

2 E ~2w40 _w22 (1 + 9 ~2)2

On substitution of Eq.

(!8)

into the 450 isoclinic equation (53), one obtains 4-n2F

02 _{'àY -} 4-rfF20 _{eos~ ~rX}

+ -n2F 11

(~y2

_-

t;2)

_eos7TX

i>?

cos

_:ëx

2

+ _-rfF

13

G~

-

t~2

)eos7TX cos 37TY + -rfF 31

G? -

t~2)

cos 37TX

+ 16-n2F 04

i?

cos 47TY - 16-n2F 40

tx2

cos 47TX + 4 -rfF 22 ( )

t? -t;2

eos7TY

eos7TY

cos 27TX

+

+ 4

~

F

42

(.e? -

.e~2)

COS47T X cos 27TY + 'Tf-F 51

(.e~2- .e~~)

cos 57TX cos7TY

+ 4

'TfF

62

(t? -

.e~2)

cos 67TX cos 27TY

=

CJ_x (61)

As a fi~st approximation to an unstable postbuckling mode shape, Eq. (56) can be simplified to the form,

w(X,Y)

= -

w

OO + w11 COS7TX cos7TY + w20 cos 27TX (62)

which is similar to Eq. (4S) used to study the inception of buckling. It is reasonab1e to assume that Eq. (62) adequately describes a mode shape; just af ter the initial buckling stage.

Hence, the stress function coefficients (Eqs. (60)) reduce to

(63) E W 11 2 (1 w 20S.ey2

)

F

20

=

32 1J.2 w 2

'?R

11 E 1J.2 w 2 F₀₂

=

₃₂ 11

and the 450 isoclinic equa~ion (61) becomes,

F 11

~ (~

-

.e~2)

COS7TX cos7TY + 'Tf- F 31

G? -

.e~2)

cos 37TX cos7TY

+ cos 27TY - cos 27TX

=

CJ_x (64)

Since we are considering a mode shape not far removed from the classica1 wave, let us assume that (refer to Eq. (50))

Let us impose the classica1 conditions on the wave numbers and assume also that IJ. ~ 1.0. Hence

• .. .... ~.;-.A' :.' • and Eqs.

(p4)

and

(p5)

reduce to 2 F 31 cos 3~X eosrrY ::: and ==

4

18

It is interesting to note that Hoff et al has shown that

Eqo (68) is true for the postbuckling equilibrium state, independent of end

shortening.

(67)

(68)

Under these conditions, the stress function coefficients (Eqs. 63») reduce to E W ll 2 F 20 ::: 32 E F 31 == E w 2 11 32 w 20 (1 - _w l12 w

n

w20 50 32

R2)

~

₀

Substituting Eqs. (69) into Eq. (67) yields

cos ~y

-(1

or upon substitution of Eq. (68) into Eq. (70) one gets

cos

2~Y

- (1 2R 2' ) cos

2~X

- 28

5 cos

3~X cos~Y

:::

w

20Po (69) (70)

rI

(J~tJ\)

p

~!

2 ' \ 0 20 (71) From Ref. 18, the postbuckled equilibrium configuration analysis

yields

2R

::: 1 regardless of the choice of n,

and the stress function coefficient F20 vanishes. However, since we are

examining unstable equilibrium states, let us construct a plot of E~. (71) by

20R

w₂₀

=

~

Po

The right-hand side of Eq. (71) then becomes

2~0

(

CT~tRX,~

)

= E

Hence for high

Rit

ratios (i.e.,

~«

1.0) and

~~tR) ~

~hus

Eq. (71) becomes

\-0.6, E

«

l.o.

cos 2 rr Y - 0.90 cos 2 rr X - 0.32 cos 3 rr X cos rr Y

E

or substituting for cos 2 rr Y

=

2 cos2rrY - 1 we obtain

2 cos2rrY - 0.90 cos2rrX - 0.32 cos 3rrX cosrrY

=

1 + E

~ 1.0

or cos2rrY - 0.45 cos 2rrX - 0.16 cos 3rrX cosrrY - 0.50

=

0

Equation (74) is plotted in Fig. 7. The transition from the linear (initial buckling) to the nonlinear (postbuckling) modes in terms of the

450 isoclinics is shown in Fig. 8.

Since a more general plot of the 450 isoclinics employing Eq. (61) is not possible during the snap-through buckling process because the radial deflection,coefficients are not known, recourse is made to examining Eq. (61) in an analysis of th~ postbuckled equilibrium state of the cylinder. From the paper by Hoff et al lö , the following coefficient values are selected,

corresponding to their Case 2 for cylinder No. 8 (refer to Table 1);

1.980 w 22 - 0.343 1) = t w 40 0.059 1) . = t 0.512

Hence for Case 2, we have

~

=

0.649, 1)

=

0.1099 and

GXE~)

Substituting these values into Eqs. (60) one obtains,

(75)

}

" F₂₀

=

0 F₀₄

=

0.059

a

ly2

_167T2 F 02

=

0.160

alT

F22

=

0.050

alT

41?

4~

(1-1l

2) Fll 0 F42

=

0.033

alT

47?(1-4 112) (76) F13 0.263 a/'~ F₅₁ - 0.069

a

/'y2

7?(9-1l

2) 7? (1-25112) F 31

a/'r

_F 62 0.020

a/'r

- 0.025 7?(1-91l2 ) _{4 7)2(1-91l}2₎ F₄₀

=

0

Thus Eq. (61) can be written as

0.160 cos 2 7TY+ 0.263 cos 7T K cos 3 7T Y - 0.025 cos 3 7T X cos 7T Y,

+ 0.059 cos 4 7TY + 0.050 cos 2 7T X cos 2 7T Y + 0.033 cos 4 7T X cos 2 TrY

0.069 cos 5 Tr X cos 7T Y + 0.020 cos 6 7T X cos 2 7T Y

=

0.054

Equation (77) is plotted in Fig. 9.

3. EXPERIMENTAL TECHNIQUE AND RESULTS

3.1 Construction and Testing of Photoelastic Circular Cylindrical Shells Geometrically 'near-perfect' circular cylindrical shells were manufactured by spin-casting a liquid photoelastic plastic in a rotation

apparatus as shown in Fig. 10. In order to circularize the interior of the casting form, inner.,'liner shells were first constructed. The final photo-elastic plastic she~l was separated from the form by applying axial pressure on the shell wall and pushing it out through one end (refer to Fig. 11).

The inner wallof

.

the cylinde'r was _{. '} then coated wi th a reflecti ve surface in

order to analyse the isoclinic patterns during buckling. Subsequently, the cylinder was fittea with end plates (Fig. 12) and mounted in a compression

machine (Fig. 13). A more detailed description of the fabrication technique

is contained in Ref.

7.

Table 1 summarizes the geometry of several of the shélls tesgeà, each of which had a clamped end constraint corresponding to Almroth'sl case Cl.

~he shells, when tested under pure axial compression were found to buckle within 10 - 14% of the classical buckling load, or within a few

end constraint. The buckling results are shown in Fig. 14 and each test was repeatable as many as twenty times.

Since the primary purpose of this investigation was to analyse the initial buckling modes of a circular cylindrical shell as it collapsed under axial compression, our attention was confined to analysing one shell in detail. The selected cylinder was number

8

(refer to Table 1) which re-presented geometrically one of the most accurate circular cylinders fabricated in the test series to date.

Two 16 mmo high speed cameras (a Fastax and a Hycam) were position-ed such that each camera viewed the entire length of the cylinder and about 30 - 40% of the perimeter, as shown schematically in Fig. 15. High intensity light sources we re employed (quartz iodide lamps and Fastlites ) to illuminate the cylinder walls. Since the method of analysis involved isoclinic patterns, the light had to be plane polarized, which also necessitated the use of infra red filters to prevent damage to the linear polarizers. Linear polarizers were also mounted on each camera lens with the axis of polarization orthogonal to that of each lamp. To maximize the amount of light being transmitted from the cylinder, only 450 _{isoclinics were studied7 . Since light intensities}

were very low and reasonably high framing rates required (about 2000 pps), 16 mm Kodak estar base 2475 high speed camera film was used. With proper developing techniques, an ASA rating as high as 1600 was obtained. In order to synchronize events on the film in both cameras, a single shot electronic

flash unit was triggered shortly aft er the cameras started. The image of the flash permitted one to determine where buckling had first initiated and in which direction (circumf'erentially) i t was propagating. The high speed photo-graphic results' are discussed in the next section.

3.2 Experimental Results

From Figs. 16 and 17, which show simultaneous high speed photo-graphic runs using two cameras arranged according to Fig. 15, buckiing initiates near the top of the cylinder on a side not viewed by the cameras. Subsequently the buckling process begins to propagate circumferentially in both directions, as evidenced by the spreading of the buckling in both films. In each case where the whole buckling process can be observed in a localized region of shell wall, the initial buckling mode isoclinic patterns consist of two oval shapes which rapidly merge into a 'double-diamond' shaped iso-clinic pattern (refer to Figs.

6, 7

and

8).

It is of interest to note that the buckling process and the mode shapes incurred by the local shell wall are always the same, independent of the stage or degree of buckling of the adja-cent shell wall. The final form of the buckles bear no resemblance to the theoretical postbuckling equilibrium mode shown in Fig.

9.

Figure 18 contains the results of a very unique run. In this case, only one camera was used and buckling is again observed to initiate in the upper corner (not directly observed by the camera) of the cylinder. How-ever, as the buckling process progresses in a similar manner to Figs. 16 and 17, suddenly the shell wall (directly in. view of the camera) begins to

collapse in an entirely different mode not observed before. Beginning with frame (2,2), Fig. 18, the initial buckling mode consists'of a seriesof axisymmetric isoclinic rings which rapidly degenerate into 'oval-shaped' patterns (frame (3,2)) and later develop into the 'double-diamond' isoclinic

those observed before. It is the initial stage consisting ofaxisymmetric isoclinic rings, which is predicted by theory (Eq. (54)) for the unique case when p = n, which is of particular interest since theory defines their separa-tion distance as 2nR/Po (refer to Eq. (55)). This quantity can be measured directly from the photographs and compared with theory. From Fig. 19, which contains an enlargement of the initial buckling mode shapes (i.e., the 450 isoclinic patterns corresponding to these initial mode shapes), the first appearance of the rings can be seen along with the subsequent 'oval-shaped' patterns. The measured (average) separation distance between the axisymmetric isoclinic rings is 1.23 in. as compared to the theoretical value of 1.14 in. corresponding to the particular shell's geometry. Hence theory and experiment are in reasonable agreement.

Figure 20 shows the results of a previous investigation3 of the buckling process involving shell number 4 which collapsed into a single tier of buckles. In this case, no axisymmetric rings are observed. However, the standard pattern of two 'oval-shaped' isoclinics merging to form the

'double-diamond' shaped isoclinic pattern is in evidence (frames (l,1}and)C2,1)). These forms agree with the theoretical mode shapes plotted in Figs.

6, 7,

and

8.

Again, however, there is no resemblance of the final postbuckling equili-brium mode shape with theory (refer to Fig.

9).

4. CONCLUSIONS

Evidence has now been obtained for the first time of the classical buckling mode shapes assumed by a circular cylinder at the inception of buck-ling. Contrary to initial expectations, the classical buckling mode is not described by a square wave mode (i.e., an asymmetrie term only) but rather it must be described by Koiter's combination of an axisymmetric term plus an asymmetrie term.

It was further observed that buckling is generally a very localized phenomenon, unaffected by adjacent regions of shell which mayor may not be in a more advanced stage of buckling. The experimental evidence obtained clearly indicates that in order for a region of shell wall to buckle, it must undergo a fundamental geometrical transformation, the mode shapes of which are initially described by Koiter's functions. Although the axisymmetric rings represent a particular case which was observed only when a large segment of the shell wall collapsed simultaneously, the more general oval shaped

isoclinic pattern was usually observed before the presence of the 'double-diamond' sqaped isoclinic pattern, which is characteristic of a nonlinear (large deflection) buckling mode.

It was found that adding more terms to Eq. (62) and subsequently solving the 450 _{isoclinic equation did not produce mode shapes bearing any} resem~lance to later modes observed in the high speed photographs. It is thus conjectured that since the early mode shapes are adequately described by

theory for both the linear and nonlinear ranges, the shallow shell equations themselves inadequately describe the more advanced postbuckling stage. Con-sequently, current theory based on these equations to determine minimum post-buckling loads is not valid beyond the early stages of post-buckling.

1. Ricardo, O.G.S. 2. Tennyson, R.C. 3. ~ennyson, R.C. 4. Evensen, D.A.

5.

Almroth, B.O. Holmes , A.M.C. Brush, D.O. 6. Hoff, N.J.

7.

Tennyson, R.e.

8.

Koi t-er, W. T •

9.

Frocht, M.M. 10. Timoshenko, S. Goodier, J.N. 11. Donnell, L.H. 12. von Karman, T. Tsien, H.S. 13. ponnell, L.H. REFERENCES

A Report on Three Series of Experiments and the Description of a Simplified Model of the Thin Wall Cylinder and Cone Buckling Mechanism. Collected papers on Instability of Shell Structures, NASA TN D-1510 (1962).

A Note on the Classical Buckling Load of Circular ~ylindrical Shells U~der Axial Compression.

AlAA J. Vol. 1, No. 2 (19q3).

Buckling of Circular Cylindrical Shells in Axial fjompression, AlAA J., Vol. 2, No. 7 (1964).

High Speed Photographic Observat ion of the Buckling of Thin Cylinders. J. Exp't. Mech., SESA, Vol. 4, No. 4 (1964).

An

Experimental Study of the Buckling of

Cylinders Under Axial çompression, J. Exp't. Mech., SESA, Vol. 4, No. 9 (1964)..

Thin Shells in Aerospace Structures, J. Astro-nautics and AeroAstro-nautics, AI AA, Vol. 5, No. 2,

(1967).

An

Experimental Investigation of the Buckling of Circular Cylindrical Shells in Axial Com-pression Using the Photoelastic Technique, UTIAS Report No. 102 (1964).

On the Stability of Elastic Equilibrium.

(in Dutch with English summary). Thesis, Delft, H.J. Paris, Amsterdam (1945).

Photoelasticity, Vol. 1 (1941), Vol. 2 (1948),

J.

Wiley and Sons Inc., New York.

Theory of Elas~icity, 2nd edition, McGraw-Hill, Inc., New York (1951).

A New Theory for the Buckling of Thin Cylinders Under Axial Compression and Bending. ASME Trans., Vol. 56 (1934).

The Buckling of Thin Cylindrical Shells Under Axial ~ompression. J. Aeronautical Sciences, Vol. 8, No. 8, (1941).

Stability of Thin-Walled Tubes Under Torsi n. NACA Report 479 (1933).

\ ~ , • 14. Timoshenko, S. 15. Koiter, W.T. 16. Almroth, B.O. 17. Almroth, B.O. 18. Hoff,

N.J.

Madsen, W.A. Mayers, J.

Theory of Elastic Stability. Eng. Soc. Mono-graph, McGraw-Hill Inc., New York, (1936). Elastic Stability and Post-Buckling Behaviour.

Proc. Symp. on Non-Linear Problems, edited by R.E. Langer, Univ. of Wisconsin Press, Madison, Wis. (1963).

Influence of Edge Conditions on the Statrility ofAxially Compressed Cylindrical Shells NASA CR-161 (1965).

Postbuckling Behaviour ofAxially Compressed Circular Cylinders. AIAA Journal, Vol. 1, No.3,

(1963).

The Postbuckling Equilibrium ofAxially Compressed Gircular Cylindrical Shells SUDAER No. 221 (1965).

TABLE 1

PHOTOELASTIC SHELL

DATA

Shell

R

t

R

L

T

R

n

m

No.*

in.

in.

'. _{-' +}

1

4.00

0.022-5%

182

3.50

6

1

2

4.00

0.041±6%

98

2.75

5

1

3

4.15

0.042±3%

99

3.61

4

1

4

4.90

0.042~105%

117

2.45

5

1

5

5.55

0.o168!4%

330

4.13

₉

2

6

8.30

0.0187~4%

443

3.05

8

1

7

5.68

O.0217!3%

262

3.86

8

2

8

5.52

0.0189±1.5% 292

3.11

9

2

* All shells had clamped ends, and were tested in a rigid compression

machine (Ref. 16, Case C-l)

y

ISOCLINIC

FIG. 1 ISOCLINIC OF PARAMETER 8

n

k

ur

1. 5 ""00 1.0

o.

5

o

o

O. 1

0.2

O. 3

0.4

FIG. 3 AXISYMMETRIC PREBUCKLING RADIAL DEFORMATION FOR A

CIRCULAR CYLINDER AXIALLY COMPRESSED WITH CLAMPED

END CONSTRAINTS

x

% ,

R

\IJ'

,

~

,

I

,

I

,

t I

,

\ \

\

"

I-M - - - -

L

---~~I

axisymmetric isoclinic rings

~"R

=

1.0

"0

E FIG. 5 region affected by end constraints i

2.JtR

-

_.,.

0'""

UT"

R

-1

CLASSICAL BUCKLING MODE AXISYMMETRIC ISOCLINIC RINGS (9 = 45 deg. )

x

FIG. 6

ox

~

=

1.0

WóE

GENERAL ISOCLINIC PATTERN CORRESPONDING TO

THE CLASSICAL BUCKLING MODE

(9

= 45 DEG. )

W:

2 0 - _ _ -

20 R

_

1'0

2

~<

'-0

O. 2 O. 4 O. 6 O. 8 1. 0

FIG. 7

ISOCLINIC PATTERN OF A POSTBUCKLING MODE SHAPE

(e

=

45 deg. )

radial deflection

1/

profile ( top v~ie:.w~)~/

_ _

...::t::~~::1~~~~~

__

+-...

_y

axisymmetric ~-+----;""'-+--f--+-+--+--+---:I--+-

___

-rings

FIG. 8

radial deflection profile ( side view)

SUMMARY OF THE 45 DEGREE ISOCLINIC PATTERNS CORRESPONDING TO THE CLASSICAL AND EARLY POSTBUCKLING MODE SHAPES ASSUMED BY A CIRCU-LAR CYLINDRICAL SHELL UNDER AXIAL COMPRESSION

FIG. 9

x

45° ISOCLINICS CORRESPONDING TO THE POST-BUCKLING EQUILffiRIUM CONFIGURATION

Casting

FIG. 11 SEPARA TION OF INNER SHELL FROM

S hel I wit

h R

e f

lee

t

i

v e

Surface

FIG. 12 MACHINED END PLATES WITH AND WITHOUT SHELL ATTACHED

Three Screw

Plate n

5 R- 4

Leveling

Dial

Gages

( ace u rot

e

t

0

0·000

I

in. )

l

Photoelastic

5 heli

(RI

t

=

3 2 6 )

Per

Pel

1.0

-

-

-

- -elossieal-lineortheory

-

-• _ _ -• _ _ _ _ _ _

~lamped~nd~o.!'stroLn-t

(el.c3)

.9

.8

.7

.6

.5

.4

.3

.2

.1

100

FIG. 14 o o o

200

•

•

•

V

0

•

•

upper bound

experimentol

1960

lower bound

of

v

V

V

experi mentol dat a. 1960

experimental data

0.5

<

L/R

<

5.0

Seehier. Bobeoek

Peterson, Dow

Tennyson

300 400 500

1000

LIGHT SCREEN 16 mm F AST AX CAMERA ÛFASTLITE LAMPS

~

HEAT FILTER LINEAR POLARIZER FRONT SURFACE CYLINDER (TOP VIEW) BACK SURFACE

--

-~

----==

V

b

~

QUARTZ IODIDE LAMPS 16 mm HYCAM CAMERA

FRAME 1 2 3 4 5 6 7 8 9 FIG. 16 1 2 3

FRAME 1 2 3 4 5 6 7 8 9 FIG. 17 1 2

FRAME

2 :3 4 5 6 7 8 9 2 3

fRAME 2 4 5 6 7 8 9 2 ₃

FIG. 19

first appearanee ofaxi-symmetrie isoelinic rings

superposition of oval shaped isoclinie patterns

FIG. 19 (continued)

emergenee of the "double -diamond " shaped isoelinie pattern between two axi-symmetrie rings

•

..

FIG. 19 (continued)

emer gence ofaxisymmetric iso clinic rings further down

the cylinder's length

note the growth of the "double - diamond" shaped

FRAME

2

3

4

5

6

7

8

9

2

•