Free-edge buckling of heterogeneous cylindrical shells in axial compression

Int. J. mech. Sci. Pergamon Press. 1969. Vol. 11, pp. 217-223. Printed in Great Britain

FREE-EDGE BUCKLING OF HETEROGENEOUS CYLINDRICAL SHELLS IN AXIAL COMPRESSION Y . STAVSKY a n d S. F R I E D L A N D Department of Mechanics, Technion-lsrael Institute of Technology, Haifa, Israel (Received 30 September 1968) S u m m a r y ~ A solution is given to the problem of axisymmetric, free-edge buckling of semi-infinite, heterogeneous, orthotropic, cylindrical shells, subject to axial compression. The resulting stability condition is shown to depend on the roots of the characteristic equation in such a way t h a t only their elementary symmetric polynomials appear. Consequently, the stability condition is directly expressed in terms of the coefficients of the characteristic equation which need not be solved. I t is shown t h a t p, which is the ratio between the free-edge buckling force and the critical axial compression for a simply supported shell, strongly depends on a certain elastic parameter, expressing the effect of shell heterogeneity on its stability. I f for a homogeneous isotropic shell p is one-half, for a composite shell it can vary, theoretically, between zero and one. Numerical results indicate t h a t for a specific circular cylindrical shell composed of two orthotropic layers p is 0.16; however, when the location of these layers is reversed p goes up to 0.84. NOTATION a

a~ A1, A2 A~j

A,*, B~ B* c, d

D~j D~ ei

E Eij

/ g, h h -- h l + h ~ i,j, k k m

I, m 2

M~ N~ N~m, N~S P

t

x

radius of reference surface of shell coefficients in equation (17) constants in equations (11), (23) elastic area defined in equation (4) modified extensional rigidity defined in equation (3) elastic statical moment defined in equation (4) extensional-flexural coupled rigidity defined in equation (3) coefficients defined in equations (8), (9) respectively elastic moment of inertia defined in equation (4) modified flexural rigidity defined in equation (3) elementary syrametric polynomials defined in equations (20) Young's modulus elastic stiffness modulus function defined in equation (17) polynomials defined in equation (21) shell thickness, sum of distances to bounding surfaces subscripts in equations (20) composite shell parameter defined in equation (2) constants defined in equations (14), (15) respectively axial bending moment axial force axial buckling loads defined in equation (26), (1) respectively unknown in equation (10) amount of translation in equation (28) axial co-ordinate slope angle of shell generator Kronccker's delta 217

218

Y. STAvsxY and S. FRIEDLAND radial co-ordinate, taken positive inward v Poisson's ratio non-dimensional axial co-ordinate p ratio defined by NeF/N~s 1. I N T R O D U C T I O N

IN HIS v o n K £ r m £ n A n n i v e r s a r y Lecture, H o f f 1 showed t h a t the a x i s y m m e t r i c buckling load, for a free-edge, semi-infinite, h o m o g e n e o u s cylindrical shell in axial compression, is o n e - h a l f t h e classical critical v a l u e for s i m p l y s u p p o r t e d shells. This exciting new result indicated t h a t revised b o u n d a r y conditions m a y h a v e significant effects on t h e b e h a v i o r of cylindrical shells. F u r t h e r contributions were m a i n l y m a d e b y H o f f a n d his associates a n d t h e s t a t e of t h e a r t is b e s t described in a v o n K £ r m £ n Memorial L e c t u r e g i v e n b y Hoff. ~ T h e analysis of heterogeneous s t r u c t u r a l e l e m e n t s gained m u c h interest in recent years. As indicated b y S t a v s k y a n d Hoff, 3 in a concise treatise on "Mechanics of Composite S t r u c t u r e s " , good progress is n o t e d in equilibrium p r o b l e m s of n o n - h o m o g e n e o u s shells, whereas r e l a t i v e l y little w o r k has b e e n done on eigenvalue p r o b l e m s of such systems. A x i s y m m e t r i c buckling load for heterogeneous, o r t h o t r o p i c cylindrical shells in axial compression was given b y Tasi 4 following a D o n n e l l - t y p e a p p r o x i m a t i o n a n d b y S t a v s k y a n d F r i e d l a n d 5 using a R e i s s n e r - t y p e t h e o r y . F u r t h e r studies on general buckling b e h a v i o r of such shells are r e p o r t e d in ref. 5, following a L o v e - S o u t h w e l l - t y p e t h e o r y . I n t h e p r e s e n t p a p e r H o f f ' s 1 free-edge, a x i s y m m e t r i c buckling p r o b l e m is solved for the case of a semi-infinite, o r t h o t r o p i c cylindrical shell, heterogeneous in t h e thickness direction. 2. A X I S Y M M E T R I C B U C K L I N G - - S S 3 BOUNDARY CONDITIONS Axisymmetric buckling of heterogeneous, orthotropie cylindrical shells of finite length in axial compression, subject to classical simple support conditions, designated by Hoff 2 as SS3 conditions, was treated by Tasi 4 using a Donnell-type approximation and by Stavsky and Friedland 5 following a Reissner-type theory. It was shown in ref. 5 that a Donnell-type solution is a fairly good approximation to the critical compression N~s when certain inequalities hold, with the result

a.N~S -= -

A 2 [4(k) + B~'e]

( 1)

where k = Aoo D ~ + B ~

(2)

and a is the radius of the reference surface of the circular cylindrical shell. The subscripts and 0 denote the axial and circumferential directions, respectively. The A*, B* and D* terms are determined by the following matrix relations: A* = A -1,

B* = - A - 1 B ,

D* = D - B A - - 1 B

(3)

whereas the elements of the unstarred matrices are defined by integrals of the form (A,~,B,,,D,,) =

f_+;/(l,~,~,)E.d~

(4)

The stiffness moduli E~j of the heterogeneous shell are assumed to be specified functions of the thickness co-ordinate ~, taken positive in the inward direction. Shell total thickness is denoted by h, being equal to the sum of h 1 and h 2.

Free-edge buckling of heterogeneous cylindrical shells in axial compression

219

N o t e t h a t for a homogeneous isotropic shell Eh s D~e = Dee = 12(1-v2) '

B~e = 0,

1 A~a = E--h

(5)

and t h e classical buckling load becomes Eh s ~/[3(1 - v")]

aNe° = - aN° =

(6)

b y specialization of e q u a t i o n (1). 3. F R E E - E D G E

BUCKLING

I n order to solve I~off's 1 free-edge eigenvalue p r o b l e m for t h e case of composite cylindrical shells we first s t a t e t h e g o v e r n i n g differential e q u a t i o n t h a t accounts for axial prestress as established b y S t a v s k y and F r i e d l a n d :5 f~rV+2cf~+d2f~ = 0

(7)

where fi is t h e slope of t h e shell g e n e r a t o r and a prime denotes differentiation w i t h respect to t h e non-dimensional axial co-ordinate. ~¢ = x/a T h e expressions for t h e coefficients c and d are 2c = - a { 2 B * e + [ A * o ( a - B ~ o ) +Aee* Boe * ] Ne} k

(8)

a 2

d 2 = 7c- ( l + A ~ e N e )

(9)

which are also non-dimensional. L e t t h e solution of e q u a t i o n (7) be t a k e n in t h e f o r m f~ = e~e, t h e n the characteristic e q u a t i o n is of the form p4 + 2cp2 + d 2 = 0 (10) As ~ -~ oo t h e slope f~ m u s t be b o u n d e d and the solution of e q u a t i o n (7) reduces to fl = A 1 e - ~ l e + A s e-~ff where R e (Pl, P2) > 0. The two b o u n d a r y conditions at t h e free edge, ~ = 0, M e = 0;

(11)

Me. e = a N e f l

(12)

l e a d to two homogeneous equations for A1 and A s, h a v i n g a non-trivial solution w h e n t h e i r d e t e r m i n a n t vanishes. I t is n o t e d t h a t for a heterogeneous shell M e is a f u n c t i o n of fl' as well as of f~" : M e = mlfl~'+m2fl" (13) where t h e expressions for m 1 and m s are

B?e k

(14)

m I ~-- A ~ 0 a S

ms=

I B oe-- A oo D ee + B oe N e{A o o [ aA~oa{ • 2

*

$

*

*

* * B* B Co) + A ee 0el}

(15)

I n case t h e shell is s y m m e t r i c a l l y layered, or w h e n it is homogeneous, t h e B* t e r m s v a n i s h and c o n s e q u e n t l y m 1 is zero a n d t h e expression for m s simplifies to Dee/a. The chaxacteristic determinmat is of t h e f o r m : m 1101a A-m s 101

m 1 io~ + ms Ps

ralPh+rasPS--aN e

mlp~+msp~-aN

= 0

(16)

e

which includes H o f f ' s 1 buckling condition (27) as a special ease. I n s t e a d of e v a l u a t i n g t h e roots of t h e characteristic e q u a t i o n (10), which m a y be quite i n c o n v e n i e n t in t h e case of heterogeneous shells, use is m a d e of t h e elementary symmetric polynomial8 of t h e roots.

220

Y. STAVSKYand S. FRIEDL-~-ND 4. E L E M E N T A l l Y

SYMMETRIC

FUNCTIONS

Let Pl, P2 . . . . . p , be the n roots of the algebraic equation f(x) = x"+alxn-l+...+an_

1x+a, = 0

(17)

Then f ( x ) = (x--P1) (x--p2).,. ( x - p , )

= 0

(18)

E q u a t i n g these two expressions we have an identity which is true for all values of x, we find t h a t the coefficient at of x n-~ in equation (17) is given by ai = ( - 1)/ei

(19)

where the elementary symmetric polynomials e i of the roots Pl ..... p , are of the form e~=~Pi,

%=~PiPJ, i
%=

~

i
P~P~Pk . . . . .

e~=Pl...P,

(20)

The elementary symmetric polynomials (20) derive much of their importance from the fundamental theorem of symmetric polynomials which is stated without proof: An y symmetric polynomial of the n arguments Pl ..... p~ can be expressed as a polynomial in the n elementary symmetric polynomials e1, ..., e,. For a proof see, for example, Turnbull's 6 book and for other details one might wish to consult Birklaoff and MacLane's ~ text. I t is recalled that a polynomial g(Pl . . . . . Pn) is called symmetric if it is invariant under any interchange among its n arguments, Consequently, in order to evaluate the polynomial g(p~ . . . . . p~) it is not necessary to solve for the roots of equation (17) but rather to use the above cited theorem, together with relation (19), to get g(Pl . . . . . p~) = h(% . . . . . a~) (21) Let us now consider the application of these ideas to the problem at hand of free-edge buckling of composite cylindrical shells. The characteristic determinant (16) results in a stability equation of the form (P2 --Pl) {PlP~ [ml~ Pl P] + m l mz(Pl + P~) + m]] + a N ~ [m~(p~ + p] + Pl P2) + m2]} = 0

(22)

The case Pl = P~ must be excluded as the expression (11) for f~would then take the form fi = A 1 e - ~ + A~ ~e -~,~

(23)

which will not lead to a solution of interest. I f PxCP~ the buckling condition is obtained when the expression appearing in the curly brackets of equation (22) vanishes, thus constituting a fifth-degree equation for the eigenvalue/Vs. This expression is the g polynomial of our problem whereas the h polynomial enables us to write the stability equation in terms of the coefficients of the characteristic equation (16) as follows: d(m~ d 2- 2m~ m~ c + m~) + andiron( - 2c + d) + m~] = 0

(24)

The elementary symmetric polynomials for the present shell problem are et = O,

~ e~ = --(p~ +p~),

5. F R E E - E D G E

e~ = O,

~ ~ e~ = p~p~

BUCKLING

=

d~

(25)

LOAD

The free-edge buckling load N~F is obtained from the stability equation (24). Noting t h a t B~e and B~$ are of the order of the shell thickness h, and that the non-dimensional quantities B~o/a, B~$/a, A~$ Bo$/aAoa,* * A~$Ne are negligible with respect to unity, the solution of the fifth-degree equation (24) for NtF simplifies to the expression: D* aN~y = -- "-'e~

4k

(26)

-~-1' o
(2v)

The ratio p of N~F to N~s takes the form

P=~F

Free-edge buckling of heterogeneous cylindrical shells in axial compression

221

reducing to Hoff's 1 result, p = ½, as B~'~ vanishes for a homogeneous isotropie shell. As the extensional-flexural coupled rigidity B*$ vanishes also for homogeneous orthotropic shells and for symmetrically layered isotropic or orthotropic shells, Hoff's solution p = ½ is thus shown to hold for such systems. I t is observed that both Nes and NeF, see equations (1) and (26) respectively, depend on the second power of B~$ whereas only N~s, and consequently the ratio p, depend on the first power of B~. The significance of this state of affairs will become clear after some invarianee properties of B ~ are established. 6. I N V A R I A N C E

PROPERTIES

OF A*,B*,D*

I n order to consider some invarianee properties of the elements of the matrices A*, B* and D* let us first treat the transformation laws for the basic elastic matrices A, B, D, as defined by the integrals of (4). 6.1. Translation of thickness co-ordinate The cylindrical reference surface, ~ = 0, was taken in equation (4) to be located at distances hi and h2, respectively, from the outer and inner surfaces of the shell. Let us choose a new thickness co-ordinate ~ related to the old one by the following transformation equation : = ~-t (2s) where t is the amount of translation of the origin of co-ordinates ~ = 0. Then the new elements of A , / } , b are related to the original ones by transformation laws of the form f +hj-t

^

A,j = J_a,_,E,~g)d~ = A;j

(29)

B'~ = J-a,-, Ei,(~) ~d~ = B , , - tA,~

(30)

DiJ = ~+a,-~ E,j(~)~2d~ -- D,~+t2A,,-2tB,j

(31)

J -hl-t

Considering the matrix relations (3) and using equations (29) to (31) the following invarianee and transformation laws are obtained: ~

= A*

(32)

/}* = B * + t~i~

(33)

/5" = D~

(34)

where ~,j, the Kroneeker delta, is defined to have the value one if i = j, zero ff i ¢ j . I t is noted that the A~ and D~ terms are invariant under a transformation of translation, whereas for the B* matrix the same is true only for the off-diagonal elements. The diagonal elements B* are changed by the amount of translation t. As the stability problem should be formulated by invariant quantities, it now becomes clear, in view of relation (33), that particularly equations (1), (24) and (26) are really independent of the location of the origin of co-ordinates. 6.2. Reflection of thickness co-ordinate Let us consider a transformation of reflection as defined by = -~

(35)

Then we find that the unst~rred and starred matrices A, B, D transform according to the following laws A,~ = A,j, A 5 = A~ (36)

~,~=--B,~,

/}~=--B~

(37)

b,~ = Dij,

D~ = D~

(38)

222

Y. STAVSEY and S. FRIEDLA~D

The important result is that B*j changes sign under a transformation of reflection. Such a transformation can be achieved by assuming a suitable thickness variation of the elastic properties of the shell. As an example consider a cylindrical shell composed of two different orthotropic layers, then B~$ that appears linearly in equations (1} and (27) changes sign when the order of the layers is reversed and strongly affects the results for buckling loads. Interestingly, the free-edge critical load N$~, in view of equation (26), is not affected by a layer reversal. 7. N U M E R I C A L

RESULTS

Numerical results were obtained, by using a digital computer, for various combinations of geometric and elastic properties of two-layer circular cylindrical shells. The most interesting results were obtained for the following composite shells : Ca~e I h 1 = 2"0cm,

h ~ = 1.0cm,

a = 300cm

(39)

The elastic stiffness moduli for the outer and internal layers are given, respectively, by the following matrices : [E]t = 10"

2 0

5 0

0 2

kg/cm 2, [El2 = l0 s

[0 0] 4 0

2 0

0 4

kg/cm 2

(40)

The quantities of interest in equations (1), (26) and (27) are A ~ = 1.5×10 -7cm/kg,

B$] = + 0 . 6 c m ,

D~] = 2.85x106kgem,

~/ki= 0-887cm (41)

N~s=-6.611×lO4kg/cm,

N~=-l'O71×lO4kg/cm,

p1--0"1625

(42)

Case I I Interchange the subscripts 1 and 2 in equations (39) and (40) to get a layer reversal. The following results are obtained : A ~ x = 1.5×10 -Tcm/kg,

B~ I---0.6cm,

D~] I = 2.85×10 skgcm,

~]k[i-- 0.887cm (43)

N ~ = - 1-275 × 104 kg/cm,

N~F = N ~ = -- 1.071 × 10' kg/cm, N I INn = 5"18

pIi = 0.8375

(44) (45)

8. C O N C L U S I O N S T h e results o b t a i n e d are r e m a r k a b l e for t h e following p r o p e r t i e s w h i c h s t e m f r o m t h e h e t e r o g e n e i t y o f t h e shells u n d e r c o n s i d e r a t i o n : (1) T h e e x t e n s i o n a l - f l e x u r a l c o u p l e d r i g i d i t y B~'~ s u b s t a n t i a l l y affects t h e * * and b u c k l i n g loads 1V~s a n d N~F as its s e c o n d p o w e r is o f t h e o r d e r o f AeoD~# o f k. (2) A s u i t a b l e l a y e r a r r a n g e m e n t m a y considerably i m p r o v e t h e b u c k l i n g l o a d N~s, o v e r five t i m e s in t h e e x a m p l e cited. (3) A l a y e r r e v e r s a l leaves u n a l t e r e d t h e free-edge a x i s y m m e t r i c b u c k l i n g l o a d N~F. (4) T h e effect o f free-edge b o u n d a r y c o n d i t i o n s o n t h e r e d u c t i o n r a t i o o f t h e s i m p l y s u p p o r t e d critical c o m p r e s s i o n is q u i t e c o n t r o l l a b l e b y t h e t h i c k n e s s v a r i a t i o n o f t h e shell elastic m o d u l i . I n t h e c o n s i d e r e d e x a m p l e s p w a s s h o w n t o be as low as 0.16 a n d as h i g h as 0.84.

Free-edge buckling of heterogeneous cylindrical shells in axial compression

223

An extension of the present axisymmetric theory to the case of general buckling, with a view towards recent results by Nachbar and Heft, s is forthcoming. AcIcnowl~bjements--Preparation of this paper was supported b y the Gerard Swope Foundation and b y the Technion Research Funds. The authors wish to t h a n k Mr. I Smolash for his help in the numerical examples, carried out at the Technion's Computation Center. REFERENCES 1. N. J. HOFF, Prec. Syrup. Distinguished Lecturers in Honor of Dr. Theodore yon Kdrmdn on his 80th Anniversary, p. 1. Institute of the Aerospace Sciences, New York (1962). 2. N. J. HOYF, Israel J. Technol. 4, 1 (1966). 3. Y. STAVSKY and N. J. HOYF, Mechanics of composite structures. I n Composite Engineering Laminates (edited b y A. G. H. DIETZ), chapter 1. MIT Press, Cambridge, Mass. (1969). 4. J. TAsI, A I A A Jnl 4, 1058 (1966). 5. Y. STAVSKYand S. F~mDLAN~, T D M Rep. 67-9, Department of Mechanics, TechnionIsrael I n s t i t u t e of Technology (1967). Also in Israel J. Technol. 1 (1969). 6. H. W. TURNBU'LL, Theory of Equations, chapter V. Oliver & Boyd, Edinburgh (1946). 7. G. BmKHOFF and S. M~cL~-~, A Survey of Modern Algebra, p. 146. MacMillan, New York (1953). 8. W. NACmaAR and N. J. HOFF, Q. appl. Math. 20, 267 (1962).