A branching solution for the local buckling of a circumferentially cracked cylindrical shell

Int. J. mech. 8¢i. Pergamon Press. 1974. Vol. 16, pp. 689-697. Printed in Great Britain

A BRANCHING SOLUTION FOR THE LOCAL BUCKLING OF A CIRCUMFERENTIALLY CRACKED CYLINDRICAL SHELL M. S. EL NASCHIE D e p a r t m e n t of Civil and Municipal Engineering, University College, London (Received 30 August, 1973, and in reviaed form 1 February 1974)

S u m m a r y - - T h e paper presents an initial post buckling analysis for the rotationally symmetric branching of a cylindrical shell with a free edge following the linear eigenvalue analysis of Ashwell. The shell is supposed to be infinitely long and a closed form solution is obtained for the initial curvature of the post buckling path. This curvature is seen to be negative and more severe t h a n t h a t of the corresponding classical solution. The solution is shown to be relevant to the buckling of a cracked concrete shell such as a cooling tower. NOTATION t shell wall thickness l length of the shell or the strut r shell radius E modulus of elasticity v Poisson ratio -K = [3(1 - v2) (r"/t2)] i a = gi~]2; f l = g ~ e Wink[or constant s = x/r non-dlmensional co-ordinate x co-ordinate axis in the length direction ¢ angular co-ordinate i half wave number a~ generalized co-ordinates W, w = W/r lateral displacement p c critical axial pressure /~c critical axial load PcCl classical eigenvalue V potential energy A diagonal potential energy a perturbation parameter A n Ic initial curvature (.2.) = ~_~ ;

¢iJ... = f2

(),

d() ds

(sum of trigonometrical functions) dx

Ic evaluated a t the critical point INTRODUCTION

ONLY a f e w s u b j e c t s i n t h e t h e o r y o f e l a s t i c i t y h a v e a t t r a c t e d so m u c h r e s e a r c h work as shell buckling with its notorious discrepancy between theoretical and experimental results. After many unsuccessful explanation attempts KKrman and Tsien z in the U.S.A. and, about the same time, Koiter ~ in Holland showed 689 48

690

M.S. EL NASCHIE

for the first time that an explanation is possible s within a non-linear theory. K£rman and Tsien introduced what is now known as the minimal load carrying capacity approach. This method dominated for many years and most of the researchers 3 aimed to get more accurate results b y using the Kkrman method b u t Pfliiger in 1962 discovered the possibility of a negative minimum load. Therefore, he introduced a new buckling criterion and obtained some interesting new results. 5, ~ Some time later, Hoff et al. ~ showed, b y employing a Yoshimura displacement pattern, that the minimum load for the cylindrical shell under compression tends to zero when the number of Fourier terms tend to infinity so that the criterion of a minimal load carrying capacity cannot be used in any universal sense. Koiter's pioneering work, on the other hand, remained, unfortunately, relatively unknown till 1963. 8 This work was part of his Doctoral thesis in which he focused attention on the initial post buckling and related the afterbuckling behaviour to the stability or instability of the bifurcation point itself, obtaining an exact solution in the asymptotic sense and showing that the main reason for the perplexing behaviour of shell buckling is to be found in the non-linear interaction of many buckling modes which are harmless when taken separately. His method is also capable of taking initial deviations into account and it is the severe imperfection sensitivity predicted b y Koiter's theory which causes the high reduction of the buckling load. Koiter's theory is now widely accepted. Apart from these non-linear studies some attempts were made aiming to obtain a smaller bifurcation load and the first of this kind seems to be that of Kuranishi 9 for the cylindrical shell in compression. His reduction was only 4 per cent, b u t about 1961 a very interesting new result for the real bifurcation loads of cylindrical shells was published b y Hoff 1° and subsequently, and apparently independently, b y Ohira 11 in Japan. These results were half of the classical one and in later papers the reduction was even greater. 12,13 This classical bifurcation for these so-called relaxed boundary conditions was then essentially improved b y many other authors. ~4-17 It is interesting t~ note that bifurcation loads, which are half of the classical one, were obtained for the first time in 1927 b y Chwalla TM for the elastically supported strut and that similar results for plates and cylinders were also obtained b y AshwelP 9 in 1951 and the priority should therefore go to Chwalla and Ashwell. The present paper applies general branching theory to the solution of Chwalla and Ashwell for the axisymmetric buckling of the free edge of a very long circular cylindrical shell. Within the framework of the general theory of discrete systems, s°, 21 closed and exact expressions are obtained for the buckling load and initial post buckling behaviour and these are compared with those of the corresponding classical results. The eigenvalue of the problem is in agreement with the previous results of haft of the classical value and surprisingly the initial curvature is negative and more severe than that of the corresponding classical problem, see Figs. 2 and 3. Although a free edge m a y at first seem to be a rather abstract condition, we

Solution for the local buckling of a cireumferentially cracked cylindrical shell

691

would draw attention to the fact that the solution could relate to the buckling of a compressed concrete cylinder containing a circumferential crack as shown in Fig. 1. The work thus relates to the study of degenerating systems by Hayman 24 and Chilver, and would seem to warrant further careful study.

f

t t t FzG. 1. Buckling of a compressed concrete cylinder containing circumferential crack.

~~

• I

_.L2 8'P~

~0m

FIG. 2. A comparison between the initial post buckling path of the simply supported A~iaymmotrically buckled cylinder and the simply supported free strut. (Schematic diagram.)

692

M . S . EL NASCHIE

P

~'cl

2~'~ rz) u

(l[ ,

r

)

.

c

J.

~

P= l p;~--2(O'6 r2)a

2

.L 2

a

FIG. 3. A c o m p a r i s o n b e t w e e n t h e a x i s y m m e t r i c a l post buckling p a t h of t h e s i m p l y s u p p o r t e d a n d t h e free edge cylinder. (Schematic diagram.)

2. T H E

STRUT

ON AN

ELASTIC

FOUNDATION

Because of t h e similarity of t h e g o v e r n i n g e q u a t i o n it is c o n v e n i e n t to consider first t h e analogous p r o b l e m of t h e s t r u t on an elastic foundation. F o r this case t h e p o t e n t i a l functional is g i v e n for a trivial f u n d a m e n t a l p a t h b y ~1

v=v,+v~+v.+v,+ .... v~ = v~ . . . . o, v~ =

I'll

Jo(½EFW: + W W ~ - ½t~W ~) dx,

v~ = f~ (½EF~= W~--~ W')dx, where x is t h e co-ordinate axis in t h e l e n g t h direction, W is t h e lateral d i s p l a c e m e n t c o m p o n e n t , F is t h e m o m e n t of inertia, l is t h e l e n g t h o f t h e strut, c is t h e W i n k l o r constant, /~ is t h e axial load, E is t h e m o d u l u s of elasticity a n d (...) = d( )/dx. I n t r o d u c i n g t h e non-dimensional co-ordinate s = x / r , changing t h e differentiation for d x to ds a n d i n t r o d u c i n g t h e n e w dimensionless q u a n t i t y w = W / r (where r is some characteristic dimension) we can write

v, =

w

,_

P

½EF-Tr,o , - g ~ , )

,.\

.

~ a~.

Solution for the local buckling of a circumferentially cracked cylindrical shell 3. S H E L L

693

ANALOGY

Using the well-known analogy between the strut on a n elastic foundation and the axisymmetrical cylinder2~ we can replace

E

ta

F --> ~-~ r d¢,

(1 --122) '

C -~ ~ r d¢

and

P -->P r d¢,

where t is the shell wall thickness, r is the shell radius, ~ is the angular co-ordinate and P is the axial pressure. The following potential functional for the cylindrical shell is thus obtained

--

f:" f: ({

12'aw.,41Er___rt2 2

2 ( 1 - v 2) 12 r ~ w ' ~ -

V4 =

w "4 r d s d ¢

and writing now 12(1-v ~)~=

4K 4

and integrating with respect to ¢ we get

2 + ½Etw 2_ ½Pw 2/2wr ds,

w V~ = f i ( "1- ~Et

~ = f : (1 4-K iwEt .~w,_~. Pw")2"firds. Using the calculus of variations the following differential equation is generated from

w'" + a/E4K4w" + 4K 4 w = 0, where a/E = P/Et.

We note t h a t this is identical to the eigenvaluo equation obtained by Hoff 1° using the equilibrium method for a cylindrical shell. 4. I N I T I A L

POST BUCKLING OF A SIMPLY SUPPORTED SHELL UNDER AXIAL PRESSURE

The simply supported condition t h a t we shall employ is the classical one which corresponds to Hoff's SS3 as defined in ref. 23. Starting with the potential functional

V=

f:

E

t8

(1 (1-vt) ~.~(~r,+ ~7, W,+ ~r W'+P(½W'- }W4)) 2rrr dx.

We can equate this to the transformed W function of Thompson ~° since the fundamental path is here truly trivial. Expanding the solution in the following Fourier series co

. "fix

W = ~ a ~ s i n z -T , i we get

oo i"fi

i"fix

W = ~a, Tcos T,

/i'fi\ 2 . Prrx = zi

a , t - r ) sm T "

694

M.S. EL NASCHIE

Inserting this in the potential energy and engaging the d u m m y suffix summation convention we obtain the quadratic form

-

['

W , = 2~rr ~ ( l _ v , ) 1 2 \ l* ] ~Ptja~a~+~r~ ~ t j a i a ~ - P

]

2

i~saia~ "

Regarding now the orthogonality conditions of ¢9,~and ~li,j we see that the quadratic form is strictly diagonal so t h a t the transformation W -> A is rendered trivial. We can then write

As=

E

2~rr/[1

ta / i , \ ' .

lEt

.

t

l* ]

P/irr~* .1

and thus pc=

rt a /irr~ ~

E

Introducing, as usual, the so-called characteristic length parameter we come to an algebraic equation for the wave n u m b e r i 141

1

How for a thin shell 1 can be neglected compared with i as is usually done in shell buckling so t h a t i~

l J2

a n d thus we get the m i n i m u m critical load

E

/

t*

t*

\

Et*

in agreement with the well-known classical solution) Reversing the shell analogy we obtain the smallest critical load of the strut on an elastic foundation P g ~ = ~ 4(eEl) in agreement with Chwalla. ~s Since the potential functional does not include third order terms the initial slope of the post buckling p a t h is rendered zero. Now for the initial curvature we evaluate the fourth-order terms and obtain A , = [-12(1 -v*)E 12,T]ta[irr]*-q ) m , - -oP ~ ¢ x m ] 2~rra~, where

fl /

irr ~*.

¢,,,2 = J0 t e ° s 7

fl/

3 • ,,,,

=

=

i~" \ 3 / .

L~cosz~)

i~- \ 3

~sm-T~ J ,= = g .

and thus Allll = _

l

Et*

./iTr\ 4_

(x-v') 4a l(T) ~ "

We still need to evaluate the coefficient Ai~ [c

d t~A,~

o

d

(z-PF'?

~"

a n d following the general theory 2°, 31 the initial curvature for the shell is All no

Ann

=

1 Et

- 3All [o = --~ ~-~.

To obtain the results of the strut we need to change Et/r* to C to get All

f¢ = - ~ c .

Solution for the local b u c k l i n g of a circumferentially cracked cylindrical shell

695

Now T h o m p s o n a n d H u n t S*give t h e initial c u r v a t u r e for t h e s t r u t o n a n elastic f o u n d a t i o n as

Anlc

c___[/~' 7 = E F \~r!

B(~/c)" = ¼(is- 3~/);

and

B=EF,

a n d w r i t i n g for t h e infinitely long s t r u t

w e get A l l I¢ = - ½c

which confirms the p r e s e n t result. 5. T H E

EIGENVALUE

OF THE

FREE

EDGE

CYLINDER

W e c a n write t h e eigenvector of t h e axisynmaetrieal free edge b r a n c h i n g in the following c o n v e n i e n t form 1°, 19 w/a 2e-~'(~/(3) cosf~s--sinf{s), =

where s = x/r, c~ = kA/2, [J = K ~]t a n d a is a n a r b i t r a r y c o n s t a n t , a n d we will n o w proceed to d e t e r m i n a t e t h e associated eigenvalue. Considering some trigonometrical identities we h a v e w S = 4a s e-S~'(2 + cos 2fls - ~/(3) sin 213s), w 'S = 4a s e ~ ' ( a s ( 2 + cos 2f~s- ~/(3) sin 2fls) +fl2(2 - c o s 2fls + 4(3) sin 2fls) + 2 ~ ( s i n 2f~s+ ~/(3) cos 2~s)),

etc.

I n s e r t i n g this in t h e p o t e n t i a l f u n c t i o n a l ~ , e v a l u a t i n g the following i m p r o p e r integrals : o - ~ " sin 2~s ds = ~ K 4t,

fe-~'d.= K-'42 and noting that aS=KS/2,

f ~ S = K St,

etc.

we o b t a i n t h e following diagonal q u a d r a t i c form from ~ , n o t i n g t h a t since t h e problem is a one degree of freedom isometric one, the V -~ W -~ A t r a n s f o r m a t i o n s are trivial. S°, $1

A S = L ( E t - -3-,~] 2- a 2 + E t - -3-~~/ 2- a s - P 3 ~ ( 2 ) K s a ) , where L is a c o n s t a n t = (2/K) 2¢rr, K = [3(1 - v S) (rS/t~)] t. Setting A S = 0 we get t 12)c

~

1 Et

_ - -

2 K'

.~_

1

Et

2 ~/[3(1 -- vs)] (r/t) "

Thus 1 Et z P~ = ~413(i_v3)]r

i n a g r e e m e n t w i t h Ashwell2 F o r t h e s t r u t we get po = 4(EFc)

in a g r e e m e n t w i t h Chwalla. .8

1

=

c

~Pc,

696

M . S . EL NASC~IE 6. T H E

INITIAL FOR

POST BUCKLING THE FREE EDGE

ANALYSIS

W e c o m e n o w t o o u r o r i g i n a l m a i n t a s k w h i c h is t o e x a m i n e t h e s t a b i l i t y of t h e bifurcation point. F o l l o w i n g t h e g e n e r a l t h e o r y of d i s c r e t e s y s t e m s t h e i n i t i a l c u r v a t u r e A n Ic is g i v e n by

Allll c

A l l Ic = - 3Ai----~ a n d we s t a r t b y e v a l u a t i n g t h e f o u r t h - o r d e r t e r m s

Al111 lc ---- A_~_~c4 !

=_ f2,l Et

.2 ,, P

,x2,n.rdso4t

I n s e r t i n g t h e c r i t i c a l e i g e n v e c t o r in A x m ]¢ a n d e v a l u a t i n g t h e i m p r o p e r i n t e g r a l s : e -a== cos 2fls d s = - 7K '

°e -*~` sin 2fl= d s

= ~l

JI

Io°e -4"` sin 4fls ds = ~ ' J l f : e-'~" ds

4_~ 2 4K '

=

we g e t w ~*w ' z d s =

a*16K ~742

and /13

80 \

f : w"a ds = a a l 6K a (-~-~q- ~---~2) . T h u s we o b t a i n Al111 Ic ---- [2EtK9"~-

13

80

and with 1 Et 2 K ~,

we g e t

Al111

_

IC =

153

--EtK ~

2rrr4h

W e n e e d o n l y t o e v a l u a t e t h e s e c o n d o r d e r coefficient A n I~ = ~

.c

1 o~ Wt2

= --~f£ -~21rrds

[c2!.

= -- 12 ~/(2) g 2~rr. T h e i n i t i a l c u r v a t u r e is t h u s

All

Ec =

-

E t 153

21 a n d e q u a t i n g t h e p e r t u r b a t i o n p a r a m e t e r t o t h e edge d i s p l a c e m e n t we o b t a i n 153 1 ,,~ - E t 0.6 21 (2~(3) r) ~ = r ~ ' w h i c h is m o r e s e v e r e t h a n t h e s i m p l y s u p p o r t e d classical one. AI--'~]c =

--Et

Solution for the local buckling of a circumferentiaUy cracked cylindrical shell

697

CONCLUSION T h e e x a c t a s y m p t o t i c solution for t h e initial p o s t b u c k l i n g p a t h o f t h e free edge circular cylindrical shell w i t h d a m p e d oscillations as t h e e i g e n v e c t o r d e m o n s t r a t e s t h a t this k i n d o f b r a n c h i n g n o t o n l y exhibits a n eigenvalue o f h a f t t h e classical one b u t also includes a higher u n s t a b l e initial p o s t b u c k l i n g b e h a v i o u r c o m p a r e d w i t h t h e c o r r e s p o n d i n g classical p r o b l e m . T h e d a m p e d oscillation is n o t a b l e o n l y in a region s -- ½ for t h e r e l a t i v e l y t h i c k shell o f r/t -- 100. This is s t r o n g l y r e d u c e d for t h i n shells o f r/t -- 1000 to o n l y s = 0.08 a n d , thus, t h e a s s u m p t i o n o f a v e r y long shell is justified a posteriori. F r o m these results we can n o w e x p e c t t h e a s y m m e t r i c a l case for which t h e e i g e n v a l u e is k n o w n , to be o n l y 36 per cent o f t h e classical one, t o h a v e a similar b e h a v i o u r a n d t h u s t o possess a higher i m p e r f e c t i o n sensitivity. W e m i g h t n o t e t h a t while o u r s y m m e t r i c p a t t e r n could be e x p e c t e d in t h e local buckling of a c r a c k e d cylinder, it is h o w e v e r n o t clear w h e t h e r t h e a s y m m e t r i c p a t t e r n could be a c c o m m o d a t e d in this w a y . Acknowledgement--The author is indebted to Dr. J. M. T. Thompson for many helpful discussions as well as suggestions concerning the presentation of this paper.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

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