A quasi-conforming triangular laminated …-Z. Huang et al.: A quasi-conforming triangular laminated composite shell element ~3 I x3 297 X 1 U2 X 2 Fig. 1. Laminated shell where a

Computational Mechanics (1994) 13, 295-314 C o m p u t a t i o n a l M e c h a n i c s �9 Springer-Verlag 1994

A quasi-conforming triangular laminated composite shell element based on a refined first-order theory

Bao-Zong Huang*, Vijay B. Shenoy**, S. N. Atluri*** Computational Modeling Center, Georgia Institute of Technology, Atlanta, GA 30332-0356, USA

Abstract. A "quasi-conforming" triangular laminated shell element based on a refined first-order shear deformation theory is presented. The Hu-Washizu variational principle, involving strain and displacement fields as variables, with stresses being considered as Lagrange multipliers, is used to develop the laminate composite shell element. Both strains and displacements are discretized in the element, while displacements alone are discretized at the boundary. The inter-element C 1 continuity is satisfied a posteriori in a weak form. Due to the importance of rotations and shear deformation in the geometrically non-linear analyses of shells, 7 degrees of freedom per node are chosen, viz. three displacements, two first-derivatives in the in-plane directions of the out-of-plane displacement, and two transverse shear strains at each node. To consider the effect of transverse shear deformation on the global behavior of the laminated composite shell, the Reissner-Mindlin first-order theory, with shear correction factors of Chow and Whitney, is adopted. The transverse shear stresses are obtained through the integration of the 3-D equilibrium equations; and the warping induced by transverse shear is considered in the calculation of the in-plane stresses to improve their accuracy. Numerical examples show that the element has good convergence properties and leads to highly accurate stresses.

1 Introduction

With the increasing application of fiber reinforced composites in engineering structures, finite element analysis of laminated composite plates and shells under various loading has received a great deal of attention in recent literature. Hitherto, three dimensional finite element models have not been widely used in the analysis of complex composite shell/plate structures, due to their intensive computational requirements; and thus the 2-D models are considered to be the economic alternatives.

In order to gain accuracy and efficiency, many multifield 2-D element models have been formulated and developed during the past twenty years (Pian 1964; Atluri et al. 1983; also see the review article by Noor and Atluri 1987). In multifield elements, the displacements, stresses, and/or strain fields are considered as independent variables. The displacement compatibility and/or traction reciprocity conditions are relaxed and satisfied in weak forms. Besides a few three-field mixed elements, the majority multifield elements developed are the two-field ones; hybrid stress elements (Atluri and Murakawa 1977; Atluri et al. 1983; Noor 1983; Atluri 1985; Xue et al. 1985) and quasi-conforming elements (Ashwell and Sabir 1972; Tang et al. 1980). Multifield elements are based on the Hu-Washizu variational principle or its modified formulations for finite elements (Atluri et al. 1983). When the strains are eliminated through constitutive equations, the stress equilibrium equations in elements and some a priori conditions of traction reciprocity or displacements continuity along the inter-element boundary are satisfied, various hybrid stress elements result. On the other hand, when the stresses are eliminated through the constitutive equations and some weak forms of the displacement compatibility conditions are satisfied posteriorily, quasi- conforming elements are obtained. In general the formulation of the multifietd elements is much

* Visiting Professor (On leave from Northeastern University, Shenyang, P.R. China) ** Graduate Research Assistant *** Institute Professor & Regents' Professor of Engineering

296 Computational Mechanics 13 (1994)

simpler and more convenient than the displacement elements with C 1 continuity, and the accuracy of the multifield elements, especially for the stress field, is higher than in the common displacement elements which are based on C O continuity. In the analysis of a laminated plate with a cut-out, hybrid stress elements gave very good accuracy (Nishioka and Atluri 1982).

For laminated composite plate/shell elements, it is usually necessary to consider transverse shear deformation, when the ratio of the thickness and the in-plane dimensions of the shell is not very small, or when the stress gradient is large. Since Reissner (1945) and Mindlin (1951) presented the first-order shear deformation theory for isotropic plates, many investigations concerning the shear deformation of laminated composite plates have been performed. Reissner and Stavsky (1961) generalized the first-order shear deformation theory to laminated plates. Chow (1971) and Whitney (1973) obtained the shear correction factor introduced by Reissner (1945) for the case of symmetric and asymmetric laminated plates. The correction factors based on the 3-dimensional model improves the accuracy of solutions for the first-order theory significantly. A comparison with the three dimensional results for the bending of plates under transverse loads shows that within a certain range of span to thickness ratio, the first order theory is a very close approximation. Besides the first-order theory, a series of higher-order shear deformation theories has been presented (Lo et al. 1977; Levinson 1980; Reddy 1984; Kant and Pandya 1988; Noor and Burton 1988; Yoda and Atluri 1992). In higher-order theories more complex variations through the thickness, of the in-plane and transverse displacements, are considered to improve the Reissner- Mindlin assumptions.

In this paper, the Hu-Washizu variational principle in which stresses are considered as the Lagrange multipliers, is applied to construct a quasi-conforming triangular composite laminate shell element. The compatibility condition is treated as an additional constraint and is satisfied in a weak form; thus higher accuracy of the elements in comparison with non-conforming elements is expected. Since the mid-surface strains and curvature strains of the element are discretized directly, the accuracy of the stress field is comparable to that of the displacement field. In geometrically non-linear problems with large local deformations, such as postbuckling and de- lamination, rotations and shear deformations are important. For this reason, the in-plane first- derivatives and of the out-of-plane displacement and the transverse shear strains are chosen as nodal degrees of freedom. This is beneficial for the application of the element in non-linear analyses.

The quasi-conforming shell element is based on a refined first-order shear deformation theory. The shear correction factors presented by Chow (1971) and Whitney (1973) are used. In the calculation of the transverse shear stresses, the 3-D equilibrium equations are integrated through the thickness. Warping resulting from transverse shear stresses is considered to improve the in-plane stresses.

The finite element formulation based on incremental iterative method is presented. All non- linear terms in the equilibrium equations are retained, in order to gain higher accuracy and faster convergence in calculation of the complex postbuckling paths. The accuracy and efficiency of the element are ascertained through some linear and non-linear numerical examples. The results are seen to be highly satisfactory.

2 Basic relations

Consider a laminated composite shell as shown in Fig. 1. Assuming that each lamina is of an orthotropic material, with the principal directions being perpendicular to the x3-axis, we have, according to the Reissner-Mindlin displacement model (Reissner 1945; Mindlin 1951) and shallow shell theory (Koiter 1967), the following the basic relations for a laminated composite shell:

2.1 Reissner-Mindlin displacement model

The displacements u at an arbitrary point in the shell is assumed to be

U = / ~ t a ~ -1-/Aaa3; ffi~t ~u~(x1, X2) -- X30~(X1, X2); /~3 = W(x l 'x2) (1)

B,-Z. Huang et al.: A quasi-conforming triangular laminated composite shell element

~3 I x3

297

X 1

U2

X 2

Fig. 1. Laminated shell

where a n and a 3 a r e respectively the base vectors tangent to undeformed middle-surface and unit vector normal to the undeformed middle-surface; u~, w are the displacements at the middle-surface and 0, are the angles which characterize the rotations of the normal to the undeformed middle- surface to its final deformed state.

2.2 Strain-displacement relations

The above displacement field gives rise to the following strain field

e = (e,po + x3~c~a)a'a p + E~3a~a 3 (2)

where e is the Green strain tensor, %~o and K~r are the strains at the middle-surface, and bending curvature strains respectively, %3 are the transverse shear strains; and

1 u i~ 1 ~ = - ~(0,~ + 0B;~)

= - + - ( 3 )

1 E~3 = 27~3

wherein ( );~ denotes the covariant derivative with respect to the middle-surface coordinate x~; q~ are the rotations that characterize the transformation of the normal to the undeformed middle- surface to the deformed middle-surface and are related to u, and w through

(a, = w , + bUuu, (4)

and ]?~3 are the equivalent transverse shear strains

~t3 = q~t - - '0~ ; (5)

b~a denotes the curvature tensor of the undeformed middle-surface.

2.3 Stress resultants and stress couples

The stress resultants and stress couples are defined as

h/2

\M~,ej = \ x 3 j = ~ (6,7) -h i2 -h /2

where a ~a and 0 "~3 a r e the components of the 2"a-Piola-Kirchhoff stress tensor. In Eqs. 6 and 7 the effect due to initial curvature is ignored, since the t e r m s x3b~# are very small for thin shells.


2.4 Constitutive equations

The constitutive equations of the k-th layer may be written as

{0-11 / [ 011 Q12 Q161 (Ell 1 ~ O'13 ff04-4 lk~2E131 0"22~ : / Q12 Q22 Q26 I ~E22 ( ; ~0.23} k ~45 = l_045 055 ~2e23 )k 0-1 j [Q 6 Q:6 Q66/ tzq:)

(8, 9)

where Qi~ are the transformed material constants in the reference surface co-ordinates, and can be obtained from the usual transformation relations (Jones 1975) using the given engineering constants E 1, E 2, v12, G12, G13, G23 and the fiber orientation angles in each lamina. Substitution of Eq. (8) into Eq. (6) leads to

in which N = [ N 1 1 , N 2 2 , N12] r, M = [ M l l , M22, Mt2] r

e o=[El l , e22 ,2e123 r, /r r

(Aij, i, D,j) = Z rl k). 3 " ~ i j ,W'(k)- X~k- 1,"11--([X32,t (k,')2, (X~k_ 1))2), 132__(X~k_ 1))2)) k

(i,j,k= 1,2,6).

Under the equivalence of transverse shear strain energy, using the 3-D linear equations of equilibrium, the relations between the shear stress resultants Q~ and the equivalent shear strains Y~3 were obtained by Chow (1971) and Whitney (1973) as

Q --- Hy; Q = [Q1, Q2]r ~ = [~)13, ~)23]T;

where kl, kz are the shear correction factors and

Hij = V [')(k)[x3 3 (i, j -- 4, 5). ,~.qj ', (k) - - X ( k - 1 ) ) k

H : ~ k2H44 klkEH4'] (11) Lklk2H4s k2Hss 3

For geometrically nonlinear problems, in the absence of body forces, the equations of equilibrium read

+ Vu)) = 0 (12)

where I is the unit tensor. The first two equations in Eq. (12) may be expressed in component form as

(o_ak + U a 0-1k) = 0, (~ = 1, 2 k, I = 1, 2, 3). ;1 J;k For the shallow shells with moderate rotations, the orders of magnitude of u~p and u~3 are same as those of the in-plane strains and rotations, denoted by E and q$, respectively. The ratio of the orders of magnitudes 0fthe stresses a ~', a ~3 and 0 -33 may be taken as 1 :h/a:h2/a 2 (Here a = min (L, x / ~ ) ; L and R are the orders of magnitudes of the dimensions and curvatures of the middle surface respectively). This analysis leads to

0-~k + U ~ 0-tk = tr,k(1 + O(e, (h/a)c~)). ;l Thus the first two equations in Eq. (12) (i.e. except the equation in x 3 direction) may be linearized as

0"~ + O'0t3;3 =0, (13)

and were used by Chow (1971) and Whitney (1973) to calculate the factors kl and k 2. Thus Eq. ( l l )

B.-Z. Huang et al.: A quasi-conforming triangular laminated composite shell element 299

is adopted as the relationship between the stress resultants Q" and the shear strains 7,3, for shallow shells with moderate rotations.

3 Quasi-conforming condition based on the Hu-Washizu variational principle

The basis of multifield elements is the generalized variational principle for a solid continuum, modified for a descretized finite-element assembly (Atluri et al. 1983). Various quasi-conforming finite elements can be formulated based on the Hu-Washizu principle. Instead of the exact compatibility conditions which are used for displacement-based elements with C 1 continuity (in plate/shell problems), some approximate compatibility conditions in weak form (or "quasi- conforming" conditions) are satisfied in "quasi-conforming" elements. The weak form of the compatibility conditions in the non-linear plate/shell problems can be obtained from the Hu-Washizu principle.

If the inter-element C O continuity is satisfied a priori, the Hu-Washizu principle can be written simply as the sum of the respective integrals for each of the finite elements (Atluri et al. 1983), as:

a~( 5{A(e)+a:[�89176 a~, s,,~ 5 t ~ s,~, S n ' a ' ( u - u~ = 0

(14)

where a and E are respectively the second Piola-Kirchhoff stress tensor and Green strain tensor, S.,. and S,,, are the segments of the boundary of element m where displacements and tractions are prescribed respectively. If n = u ~ on S,,., and o- is considered as a Lagrange multiplier, Eq. (14) may be cast into the following conditional two-field variational problem

6~( S {A(t) - f ~ ~ t ~ � 8 9 ( m = l , 2 , . . . , N ) m \ .O,,, S~,~ /

(15, 16)

for all element strain and displacement fields, where N is the number of elements. The constraint condition in Eq. (16) may now be satisfied in a weak form if the independently assumed strain fields c are expressed in terms of the element displacement fields through the element-level "quasi-conforming" condition:

J" &r:[�89 + uV + Vu-uV) - e] dV= 0 (m = 1,2,. . . ,N), (17) g2,,,

&r is an independent test-function for each element. When the element strain fields g are expressed in terms of element displacement fields through Eq. (17), the two-field problem reduces to a one-field variational problem:

6~( ~a. .{A(e(n))-f~ dV- S t~ u=u~ Sam

(18)

The strain energy of the laminated composite shell element can be expressed in the form

A(6)dV= �89 ~ (N~r o + M't~Go + Q'y~)dA, (19)

where A m is the area of the element. Since 6a in Eq. (17) is a continuously differentiable, but otherwise arbitrary, test function that

varies in the three co-ordinates x i of the shell space, we assume a specific form for such a test function, denoted by e*, possessing a linear variation in the thickness direction x 3. We thus have

e* = (E *'~ + x3x*~a)a,aa + r (20)

300 Computational Mechanics 13 (1994

Using Eq. (20) in Eq. (17) and integrating through x 3, we obtain

ts~, ~;~ + u~,~) - 2 b ~ w + (w , + b2u.)(w,r + b"~u.) - ea~o] } dA Am

h 3 - gE(w,~ + b~uu);e + (we + b~u~,);a - 7a3;e - 7e3;a] - ~cae) dA

12am

+ h ~ �9 + b ~ u ~ - O ~ - T a 3 ) d A =0 . A,~

(21)

Furthermore, e,ao and ~caa can be divided into two parts, respectively as

_ ( I ) �9 K~=~c' K" (22, 23)

where �9 are the components of the linear strain tensor, x' is the total curvature changes of the �9 apo aS

middle-surface,

- - 1 u �9 (a~)o - 3( a.p, + up;a) - bapw', ~c'ar = �89 + b"u,),~ + (w,p + bUu~ u);,], (24, 25)

and d~) ~ and K"~p denote the non-linear terms of the in-plane strain tensor, and curvature change due to transverse shear alone, respectively:

t . . . . t ~ (26, 27) E(2)apo = 2-~])a (~fl ' K p - - 2(~ a3;fl -}- ~fi3;c~ )

If Eq. (27) are satisfied a priori, sufficient conditions for the satisfaction of Eq. (21) follow

2bapw) w �9 {~[(w,a + b"u~,)(w,~ + - �89 ) dA = E*aP {2(ua;~ + u~;a- --�9 dA + ~ .a~ , b~uu)] 0 A , . A , .

'~ i~ (w,a + bU~u.);~ + (w,~ + b~u.);~] - ~c'~} dA = 0 Am

~b*a {(w., + b~u.) - Oa} dA = O, Am

(28) (29)

(30)'

where the notat ion E *a3 is replaced by ~b *a. When Eq. (30) is satisfied for a complete family of test functions or the areas A,~ are infinitesimal, we have

(w ,a + - (pa = O.

Thus Eq. (28) reduces approximately to

1 _ 2bar _ r(1) ~ dA = 0; (31) Am

when Eq. (30) is satisfied for a cut-off family of test functions (including a constant one) and the areas A,, are sufficiently small.

The Eqs. (31), (29) and (30), as the "quasi-conforming" conditions, are used in Sect. 5 to determine the undetermined parameters of �9 ~c' o, ~b~ in terms of the displacement parameters, which are subsequently substituted into Eq. ~f8). ap

4 Incremental equilibrium equations

In the quasi-conforming element, the linear part of strains E(~.) o, the total curvature change of the middle surface ~c'r and rotations (Pa in Eq. (18) are descretxPzed directly and the undetermined parameters in them are expressed in terms of the displacement field, viz. the generalized displacements at the nodes. In addition, the transverse shear strains 32~3 in Eq. (18) are descretized by means of their values at the nodes. Based on a Lagrangean description, using Eq. (18) we now formulate the incremental equilibrium equations in terms of the generalized displacements.


Let (2/, u/) denote a point (i) on the equilibrium path, (2i + l, ui + 1) is a required point (i + t)

ai+ 1 = Ui -~- Aui

•i+ 1 = )~i "q- A2i (32)

where )~ is the load factor. Substituting Eqs. (22, 23) into Eq. (10), we have

= [M~,) j + [M~2)j (3331

where

Let q and At/denote the generalized nodal displacements corresponding to the displacements u and Au. We define the matrices K0, K1 and K2 as follows

~r/TKor/-= ~ f {N(1)T(u)~?(ol)(~n) + M(1)T(u)K(~u) + Qr(n))'(au)) dA (35) ra A m

~r/TKl(r/', r/")r/= ~ ~ {2(N(')T(u") + 2N(10T(n',u ") + N(2)T(u"))K(J 1)(U, ~U) m A m

+ 2NIx)r(3u)E~ol')(u", u) + 2N") r e~ol 1)(u", 6u)} dA (36)

3 r/rK=(r/', r/")I/= E ~ { 2N(x ')r( 3u, u')elo 11)( u', u) + 2N (1 a)r(3u, u")e{o 1 ')(n', aft)} dA (37) m Am

in which u' and u" are allowed to assume 0, u~ and Aui, similarly I/' and !/" may assume 0, r/~ and Ar/i, and the symmetric bilinear forms do ~ a) and N a 1) are defined by

= 1 , , ,

~[(jb 1 (n)(])1 ( n ) , (]~2(u')l~D2(utt), (~1 (n ' ) l#2(u ' ) -{- (]~1 (u")(jD2 (u ' )] T

N(1 *~ = A,(o 1 I~. (38)

Using the above definitions in equations Eq. (18) and Eq. (19), retaining all the non-linear terms, we obtain the equilibrium equations

[K o + K~(0, r/,) + K2(r/i, r/i ) + �89 (r/,, Ar/~) + Kz(r/,, Ar/,) + �88 Ar/,)] Ar/, = A2,F (39)

where F is the reference load vector. The iterative form of Eq. (39) for the r-th iteration of the i-th step follows:

KTAAr/7 = AA27F -- ART (40)

in which

AAr/. ". = A-r+ 1 , l , - A'r/~; AA27= A2'+1 - A 2 7 , (41)

and

K 7 = Ko, + K 1 (r/,, ar/;) + 2Kz(r/,, ar/;) + Kz(Ar/;, A r/7)

1 r/iJJ r/i--

Ko~ = Ko + KI(0, r/i) + K2(r//, r/i)- (42)

5 Discretization of strains and displacements

In this section and in the following, we assume that x ~ is the principal co-ordinate system of the shell (Fig. 2). Thus the curvature tensor of the mid-surface of the shell is


1 1 b l l - - , b 2 z - , ba2=O (43)

R1 R2 where R 1 and R 2 are the principal radii of curvature. In the quasi-conforming shell element the shape functions of Tang et al. (1980) and Lu et al. (1981) are used for the discretization of the linear system strain E(o 1), the total curvature change x', the rotations ~b of the mid-surface and displacements. The equivalent transverse shear deformation y are considered as both strains (in shear strain- energy) and generalized displacements (in bending strain energy). The fields c ~) ' o , x , 4 ~ a n d y a r e assumed over each element in terms of the undetermined parameters a~ p, at b, ~t ~ and the nodal values qS of y, respectively as:

(K' t t ] [ L o Lt Lz 0 0 0 0

o [ o o L o L 1 L 2 o [2~c'12 j 0 0 0 0 0 0 L o

0 0

o o

L1 L2

{~)1 t : I LO L1 L2 0 0 0 2 1 {~q}; 4 = (~2 0 0 L 0 L 1

~ = t ~ 1 3 t = I g o 0 0 ml 0 L2 0 2 1 {qS } LY23 Lo 0 L1 0

in which L~(i = 0, 1, 2) are the area co-ordinates of the element and

qS=[y/ t3 ,Y iz3]r ( i = 0 , 1 , 2 i n order).

From the expression of Y, we obtain

we" = ~ ~23,2 [ = ~ C q

[Y13,2 -{- Y23,1)

t - - X ~ X 1

X o

in which A is the area of the element and

r l 2 0 2 2 0 2 2 CS 1 X2 -- X2 1 1 X2 --0 XO 1 1 XO --0 X1 = 0 X 2 -- X 1 X 0 -- X 2

t 2 2 1 1 2 2 t 1 X12--X 1 X 1 --X 2 X O - X 2 X 2 -- X 0 X 1 --X 0

where x ~ (i = O, 1, 2; ~ = 1, 2) are the coordinates of node (i), i

(44)

(45)

(46)

(47)

x3t 0 z 2 x ~,x~)

~ ) " ' ' / cx!.x?~ 1 21 1 ( I, 1)

RI ~ xl Fig. 2. Geometry of the element

B.-Z. H u a n g et al.: A quas i -conforming t r iangular l amina ted composi te shell e lement 303

The displacement field in the element and on the boundary are discretized separately. Linear interpolation is used for the in-plane displacement u~ in the element and fi, on the boundary, while cubic shape functions (Zienkiewicz 1971) are used to discretize the out-of-plane displacement w in the element and # on the boundary. The normal derivative of the out-of-plane displacement #,, on the boundary is interpolated linearly and the derivative of the out-of-plane displacement along the boundary ~ is assumed to be same as that of w along the boundary.

Each node of the element has seven generalized displacements as degrees of freedom. The generalized displacement vector of the element

i i " " " i i "l T (i = 0, 1, 2 in order) (48) I1 = [ U l , U2, W t, WI1 , WI2 , ~ 13 ,723-1

is divided into two parts, q and q~ where

q = [uia, , , i w i qT (49) U2'W ' W , I ' ,2

and q" is an in Eq. (44). Now the test functions E*, K'* and ~* are chosen to have same form as the trial functions e (a)

O '

K' and q~ respectively except for the undetermined parameters in them. On integration of Eqs. (31), (29) and (30), in which the terms containing u~;p and w,p are integrated by parts in order to express them in terms of u~, fi~ and w, ~ , , #,s respectively, we obtain the undetermined parameters ~P, and ~0 in terms of q:

~v = (AV) - ICPq; ~b = (Ab) - XCbq; ~s = (AS) - 1CSq. (5131)

The matrices (A') - ~, (A ~)- t, (M)- ~, C p, C b and C s are given in Appendix A1.

6 Stiffness matrices K0, K1 and K2

Substituting the constitutive relations Eqs. (34) and (11) into the definitions Eq. (35), we obtain

6t/rKoT/= ~ ~ {6r o(t) + 6nctB~(ol) + 6C(ol)TBK + 6KTDIr + 6~.rH7} dA, (51) m A m

On substitution of Eqs. (23), (44), (46) and (50), we have

611TKo~I = ~ {6qr(K p + K bv + K bpT + Kb)q + 6qr(K b~ + K~'r)q ~ + 6q~r(Kb~T + KSp)q rtl

+ 6qSr(K~ + Kh)q ~}

in which

K v = Cp'r(Ap)-TI)VCp; Kb~ ,= Cbr(Ab)-rl~bvCp;

KbS= cbr(Ab)-rl~b~,C.; K~p = c~r l I~pCp; A

(52)

where A is the area of the element and the matrices 1) p, 1) bp, 1) b, ~b~, ~sp, i~s are given in Appen- dix A2.

Likewise, we obtain

6r/rK1 t/= ~ {~qr(KS + K TM + K ~T + K sb + Kobr)q6qTKS'q' + 6q~rKO~Tq} (54) ?71

c~ ~/TK2 ~/= ~ {6qTKSq} m

in which

KS = CST(A .) - rDscs;

K gs = CsT(M) - TI3s~C~;

K w = cgT(A o)- TI~.,,Cp; K b = CgT(A g)- Ti~gbCb

K s ___ CgT(M)- ri~sCg (:55)

K b ~ cbT(Ab) - T [ )bcb

[2H H n J K s=CSTL s CS; K"- - A 2H H

A 12 H 2H

(53)


where the matrices Ilg, ~)w, oYb, ~)gs and I) f are dependent on the solutions of the i-th step and r-th iteration of I /and At/. The modified Newton-Raphson method and arc length approach (Riks 1972; Zhang and Atluri 1988) are used to solve Eq. (40). At bifurcation points on paths, the asymptotic postbuckling solutions are taken as the initial values of iteration, so that any assumed imperfections are rendered unnecessary.

7 Calculation of stresses based on a refined first-order theory

As mentioned above, the equivalent transverse shear strains 7,3 are defined essentially by means of transverse shear strain energy and shear stress resultant Q~, so that �89 is equal to the density of shear strain-energy obtained from the three-dimensional model. Therefore the transverse shear strain-energy based on the first-order theory is quite accurate. This is one of the reasons that the theory gives satisfactory results within a certain range. But ?'~a are neither the real nor the average transverse shear strains (as shown in the following). Thus ~,3 can not be used to calculate a "3 through the constitutive relations, in the first order theory. In fact this theory is not characterized by a uniform distribution of transverse shear strains and stresses. Just as in engineering beam theory, the natural approach to the calculation of ~3, is the use of the 3-D equilibrium equations, simplified for moderate rotations,

h/2 k x3 erda------ f ~ dx3 o r ( Y o d a a n d g t l u r i l 9 9 2 ) a ~ 3 = - ~ j" _~/~(m)dx3 (56,57)

-h/ 2 m= l xam_ l

where x 3 and x 3 are the x3-coordinates of the top and bot tom surfaces of the ruth-layer m m - 1

respectively. It is well known that the transverse shear deformation leads to the warping of the cross section

and changes of the shear deformation along the mid surface causes curvature change (See Eq. (27)). Since re" does not lead to bending deformation it is deducted from the total curvature change. In the case of large shear deformation r" can significantly affect the in-plane stresses of laminated composite plates and shells. It seems that the inaccuracy of r " is one of the main sources of error in the first-order theory. In order to improve in-plane stresses, it is necessary to account for warping. due to transverse shear stresses.

First, consider an orthotropic plate. According to the equilibrium conditions (Eq. (56)), the shear stresses e~a have a parabolic distribution through the thickness of the plate. From the definition of Q~, ~r ~3 can be written as follows

~ 3 = 1.5 1 - . (58)

Using constitutive equations (Eq. (9)) of orthotropic plate under cylindrical bending in x%direction we obtain the real transverse shear strain

~3 hQ u [_

where i -- 4, 5 when ~ = 1, 2 respectively. Substituting Q~ = k2H,7~3 into Eq. (59) (here and in the following _ = ~; no sum on _), we have

the average transverse ~hear strain

1 h/2 ~3 dx = ~_7~3. ~7~3=h j" , 3 k 2 (60)

-hi2

It is seen that the equivalent shear strains are not equal to the average shear strains. Using a parabolic distribution of o -'a, we obtain the wraping of cross-section due to transverse shear

u ~ = I , 3 1 3 \ h J _J" "~ol3dx = 1.5k~V,aX 3 - (61) O


On considering the influence of warping on in-plane stresses, instead of Eq. (1), we obtain

/A~ = Ua(X 1, X 2) -- X3t~t "~ U s (62) ~t"

The displacements of Eq. (62) can be used for calculation of in-plane stresses. For composite laminates, 3-dimensional analysis shows that the distribution of transverse

shear stresses through thickness is close to parabolic even when the span-to-thickness ratio equals 4 (Pagano 1970). Therefore we can assume that equation Eq. (58) is valid for laminated plates and shells. Then shear strains ~* in the k-th layer are given by

(7~3 LS45 $55 k LS45 Ss5 ~= Q4s Qss

The warping of the cross-section in the x 1 and x 2 directions can be obtained

, h. ~ h Ul = [g44(~)713 -}- 84s(~)~23] 2' u2 = [E5'*(()7~3 + $55(~)723] ~ (65)

in which

{ , , } S-44_ T1.5 k , n 4 4 o 2 IS4,,(l_~2)d~_k_k,k2H45SS45(l_~2)dr

S,~s = ktk2H45 ~ $44(1 - ~2) d~ + k2H55 1 S,5(1 - ~2)d~ o o

1.5 fk2n ~ ; } S5s= h [ 2 55 f S55(1- ~2)d~ + k~k2H45 f 0

l"5Jk } = 2 J'S,5(1 ~2)d~ (66) h ~ lk2H45 o ~ $55(1 - ~2)d~ -t- klH44 o -

where ( = 2xa/h. Using Eq. (62), we obtain the improved in-plane stresses.

8 Numerical examples

To evaluate the effectiveness of the element and the refined first-order theory, some benchmark problems are solved. Example 1 is the bending of a simply supported plate under uniform pressure, to check for locking and to assess the convergence of the element. In example 2, the bending of a simply supported plate under sinusoidal pressure is investigated to ascertain the effectiveness of the new method in the calculation of stresses; this is a bench-mark often used for the comparison of various 2-D theories with the exact 3-D results. The behavior of the element under the conditions of high stress gradients, such as high stress concentration near cut-outs, is investigated in examples 3 and 4. In example 5 the postbuckling of a plate is considered to assess the robustness of the element in non-linear analysis.

( t ) Simply supported plate under uniform pressure. A cross-ply (90/0/90/0/0/90/0/90)s laminated square plate, all edges simply supported, under uniform pressure qo = 0.1 MPa is considered. The geometry is shown in Fig. 3. The following material properties are assumed:

E 1 = 135000MPa G12 = 6 4 1 0 M P a ; E 2 = 13000MPa Gla =6410MPa;

Y12 = 0.3 G23 = 4340 MPa.

A quarter of the plate is modeled using a 4 x 4 mesh. The maximum deflections of the plate for various ratios of a/t and (Gt3, G23)/G12 , are obtained and plotted in Fig. 4. It is seen that the element is non-locking. To ascertain the convergence of the element, various mesh sizes ranging from 2 x 2 to 6 x 6 are used. The maximum deflections and bending moments thus obtained are


f/ q __/f / / . . . . 100 mm

~. = 2.272 mm Fig. 3. Simply supported plate under uniform pressure

Table 1. Solutions for example l with different mesh sizes

Mesh w{a_ a_~ w(3a a~ (a a) (a a) (3a,a_~ [a 3a'~

\ 2 ' 2 / / \ 2 - ' 2 J M,, ~,~ M22 2 '2 M 1 , \ 2 2 / M22~,~)

2 x 2 0.8519 0.6104 63.17 4 x 4 0.8895 0.6389 61.26 6 x 6 0.8958 0.6439 60.87 Jones(1975) 0.9014 0.6483 60.61

51.86 73.38 59.19 50.19 71.36 57.28 49.63 70.96 56.71 48.56 70.69 55.63

2.5

2.0

l 1.5 %

=% 1.0

~ ~ 5

0 a 10

0.8

0.71 0.6

I 0.5

0.z,

~ == 0.3

0.2

0.1

0 0.1

4 b

. . . . . . . . i

100 . . . . . . . . i

1000 a / h

. f . . ,

, , ~ , . i

10 100

G~3 / G12

, , r

1000 D

10000

, , i ,

10000 100000

o~ ~ = o u3

6 E i t

Wmax

5

M~ax M~max

3

2

I

I I I I r

2x2 4x4 6x6

Mesh

Figs. 4a, b and 5. 4a Deflection at the center vs. a/h, b deflection at the center vs. G~3/G12. 5 Percentage

M m a x M m a x e r r o r s of ]41ma x, 1 1 and 22


tabulated in Table 1 and plotted in Fig. 5. It is seen that the element has good convergence property and accuracy. It is especially noted that the present element has same accuracy for displacements and stresses. This, evidently, is an important advantage compared to the displacement based element as the stress are usually of more interest.

(2) Simply supported plate under sinusoidal pressure. To check the accuracy of the element in analyzing thick laminated plates, examples of Pagano and Hatfield (1972) are solved. In these examples a 4 x 4 mesh is used over a quarter of the simply supported plate under transverse sinusoidal loading (see Fig. 3). The material properties of the laminae are taken as follows.

E1 =25 x 106 Psi(172600MPa) G 1 2 = 0 . 5 X 106 Psi(3450MPa)

E 2 = 10 6 Psi(6900 MPa) Gt 3 = 0.5 x 106 Psi(3450 MPa);

v12 = vl3 =0.25 G23 =0.2 x 106 Psi(1380MPa).

The shear correction factors kl and k 2 are calculated according to Whitney (1971). The warping displacements u~, given by Eq. (64) are used in the calculation of the in-plane stresses. The transverse shear stresses are obtained through Eq. (57). The results are tabulated in Table 2 in which solution of 3-dimensional elasticity (Pagano and Hatfield 1972) are included. The error in the inplane stresses with and without the warping correction is presented in Table 4. The results are expressed in terms of the following non-dimensional deflections and stresses:

~ = ~r~l~hZ/(qoa2), 6c~3 = a~3h/(qoa);

v~ = wrc'*h3[4Glz + (E 1 + E2(I + 2v13) ) ] / [12a4qo(1 - v l2v21)] .

From Tables 2-3 it is seen that solutions based on the refined first order theory are satisfactorily close to the 3-D elasticity solution. For quite small a/h ratios (a/h = 4) the accuracy of deflections

Table 2. Maximum stresses and deflections in 3-ply laminates

a/h 8il 822 812 ~i3 823

2 2 ' 2 / \ 2 ' 2 4 /

Present 4 0.763 0.783 0.0452 0.317 0.339 4.63 Pagano et al. (1972) 4 0.720 0.663 0.0467 0.219 0.292 4.49 Present 10 0.558 0.407 0.0275 0.313 0.217 1.700 Pagano et al. (1972) 10 0.559 0.401 0.0276 0.301 0.196 1.709 Present 20 0.541 0.310 0.0230 0.317 0.178 1.150 Pagano et al. (1972) 20 0.543 0.308 0.0233 0.328 0.156 1.189 Present 100 0.539 0.272 0.0212 0.307 0.163 0.972 Pagano et al. (1972) 100 0.539 0.271 0.0214 0.339 0.139 1.008

Table 3. Maximum stresses and deflections in 9-ply laminates

a/h 0"11 0"22 O'12 813 0"23

(aah (00 ) 2 2 2 / \ 2 2 ' 4 /

Present 4 0.736 0.733 0.0341 0.305 0.319 4.08 Pagano et al. (1972) 4 0.684 0.628 0.0328 0.223 0.225 4.078 Present 10 0.556 0.489 0.0236 0.263 0.252 1.480 Pagano et al. (1972) 10 0.55I 0.477 0.0233 0.247 0.226 1.512 Present 20 0.539 0.445 0.0218 0.256 0.234 1.090 Pagano et al. (1972) 20 0.541 0.444 0.0218 0.255 0.221 1.129 Present 100 0.537 0.430 0.0211 0.248 0.224 0.969 Pagano et al. (1972) 100 0.539 0.431 0.0213 0.259 0.219 1.005

308 Computa t ional Mechanics 13 (1994)

Table 4. Error in in-plane stresses in compar ison to the 3D-solutions (Pagano et al. 1972) (~o)

Ply Warping a/h 611 622 612 a/h 611 622 612 correction

With 4 6.0 18.1 - 3.2 10 - 0 . 2 1.5 - 0 , 4 Without 4 - 4 1 . 4 - 1 3 . 0 -32 .1 10 - t 2 . 7 - 4 . 0 - 8 . 7

With 20 - 0 . 4 0.6 - 1.3 100 <0.1 0.4 - 0 . 9 Without 20 - 3.9 - 1.0 - 3.9 I00 <0.1 0.4 - 0 . 9

With 4 7.6 16.7 4.0 10 0.9 2.5 1.3 Without 4 - 23.2 - 22.9 - 35.7 10 - 5.6 - 5.5 - 7.7

With 20 0.4 0.2 < O. 1 100 - 0.4 - 0.2 - 0.9 Without 20 - 2 . 0 - 1.8 - 2.3 100 - 0 . 4 - 0 . 2 - 0 . 9

is still satisfactory but the error in stresses becomes larger. The comparison brought out by Table 4 shows that the in-plane stresses are improved significantly due to warping correction.

(3) Laminated plate with a hole under uniaxial tension. The problem of a hole a in four layer (90/0/0/90) laminated plate, discussed by Nishioka and Atluri (1982), Tang (1979) and Rybicki and Hopper (1973) is solved by means of the present element. The geometry and mesh is shown in Fig.

x2 i A

~ ~ , , - X 1

t= a -I

a = 16 in (406 rnm) h = 0.8 in (20.3 ram) d = 2 in (50.8 ram)

6

"- 2

,2j I I I I l I [ I

o Present

90 " ~ - - 0

Nishioka, Atluri (1982) -~<-- 90 ' + ...... 0 �9 -~ ..... 90 Rybicki, Hopper (1973)

-c~.- 0 -~--- 90 Tang (I 979)

~., . . . .....

-2

.4] I I I I r I I I

0 10 20 30 ~0 50 60 ~ 80

0.,-----~-

90 Figs. 6-7 . 6 Plate with a hole subjected to uniaxial tension, and mesh. 7 Circumferential stress in 0 ~ and 90 ~ layers


6. The material properties are assumed as follows

E 1 = 25 x 106 Psi(172600 MPa); E2 = 3 x 106 Psi(20700 MPa)

G12 = G13 = G23 =0.5 x 106psi(3450MPa) v12 =0.336.

The uniaxial tension in the xl-direction is expressed by the average stress through o- o the thickness. The numerical results of the stresses in the circumferential direction along boundary of hole are given in Fig. 7. Good agreement between the present results and those of 3-dimensional analysis (Nishioka and Atluri 1982; Tang 1976) can be seen especially in the regions of high stresses. It is to be noted that it is impossible to satisfy exactly the free stress condition for each lamina on the free edge in a 2-dimensional model.

(4) Cylindrical laminated panel with a central hole under compression. Compression of a cylindrical symmetric ply laminate (45/-45/90/0/0/90/-45/45)s panel with a central hole is considered. See Fig. 8 for the geometry. The material properties of the panel are same as those in example 1 .3 x 8 and 6 x 8 meshes are used in the calculations. The distribution of the stress resultant N along the x2-axis is shown in Fig. 9 where the result obtained by ABAQUS (Hibbitt, Karlsson and Sorensen Inc. 1990) with 8 x 8 mesh and $4R5, $9R5 elements and Stanley (1985) are given as well. It is seen that a rapid convergence of the stress resultant with high stress concentration is attained with less degrees of freedom. The results of the 3 x 8 mesh show that the present element is not shape-sensitive. The influence of stiffeners along the boundary of the hole on the linear buckling loads is given in Fig. 10, where A is the area of cross-section of the stiffener, with 6:3:I height/width/thickness ratio. It is seen that even a quite weak rib can significantly increase the buckling load. But this effect decreases with increasing stiffness of the rib.

x2 ~F'~-%-.. a

a = 355.6 mm

R = 381 mm

d = 50.8 mm

h = 2.272 mm

3x8

I 6x8

i 160

900

800

700

I 600

-ff 5oo ~oo

z 300

200

I00

f

I 1

/

I

$4R5

I ~ ABAQUS 8x8 lZ,0

I - Stanley (1985)

[] Present 6x8 t 120 ~ ~7 Present 3x8 ~V

' ~ A ,2 .272 100 "-. _L-

\ ~- >I ~50.8- T z ~ $ 9 R 5

" ~ - , _ ~ _ , ~ 4 - ~ - - 4 80

6O

O0 ,, I 1 1 I .i I I , ~ ~ ) 50 100 150 200 250 300 350 0 2 4 6 8 10

9 Arc position (rnrn) 10 A (turn 2) -

Figs. 8-10. 8 Cylindrical panel under compression and meshes. 9 Stress resultant N 11 along x 1 = 0. 10 Critical load of the panel with a stiffened cut-out


12

10

i 8

6 13

6x6 j - - - - - 4 X 4 } Present / J , . . . . . . 2x2 , / , "~ - - - - - S h i n et al. (1993) / ~ , - "

-,,~.+ />1" _>_l ;2L? f i .

I I I I �9 l

0.5 1.0 1.5 2.0 2.5 3.0 wmax/t

Fig. 11. Center deflection of square panel under compression

(5) Post-buckling of a laminated panel. As a nonlinear example post-buckling of a (_+ 30)24s laminated square plate with all edges simply supported and restrained x2-normal edges under uniaxial compression is considered. The length and thickness of the plate are 20 in. (508 mm) and 0.12 in. (3.05 ram) respectively. The material properties are taken as:

E 1 = 19010 Psi(1310 MPa) G12 -- 930 Psi(64.2 MPa)

E2=1890Psi(130MPa) Gla=Gz3=93000Psi(6420MPa); v12=0.38.

The results of the present finite element with 2 x 2, 4 x 4 and 6 x 6 meshes given in Fig. 11 are in good agreement with Shin's (1993).

9 Limitations of the present model and proposed remedy

In the 2-D model, the boundary conditions at the free edges cannot be satisfied for each layer, and thus it is impossible to obtain the inter-laminar stresses by using this model, albeit the use of 3-D equilibrium equations (Eq. (56)). In order to obtain the stresses at the free edges, a full 3-D analysis may be essential. This will be addressed in a following paper.

10 Concluding remarks

A quasi-conforming triangular laminated composite shell element with 7-degrees of freedom per node is presented. The results of the linear and non-linear numerical examples show good accuracy and convergence of the element in the analysis of large transverse shear deformations, stress concentration near cut-outs, and large deflection of laminated plate/shell structures. The direct discretization of the strains and the a posteriori satisfaction of the inter-element compatibility by means of the Hu-Washizu variational principle, increases the accuracy of the quasi-conforming elements especially in the computation of the stress fields. In the calculation of the in-plane stresses, the differentiation of the displacement field is rendered unnecessary, so that the accuracy of the stresses and displacements are comparable; sometimes the stresses are even more accurate than the displacements, as shown in the examples. This is of benefit for stress analysis especially in problems with high stress concentration such as near a hole. In addition good accuracy of the in-plane stresses is required for the calculation of the transverse shear stress according to the 3-D equilibrium equations. The element is based on a refined first order theory. For the application to the analyses of problems involving large shear deformations and rotations, both the in-plane first-derivatives of the out-of-plane displacement and transverse shear strains are taken as generalized displacements at the nodes. In the first order theory based on the Reissner-Midlin model, the transverse shear energy which results from 3-D analysis is accounted for by means of the shear


correction factors. But the equivalent shear strains are neither the real nor the average shear strains and cannot be used to calculate the shear stresses a ~3 directly, a ~a can be obtained from the in-plane stresses using equilibrium equations as done in engineering beam theory. The results of example 2 show that if plates are not very thick (for example a/h > 10), the accuracy of a ~a is satisfactory. When transverse shear deformation is large, warping of the cross sections leads to considerable error for in-plane stresses based on first order theory. Accounting for the effect of warping, with the assumption of parabolic distribution of transverse shear stresses, the accuracy of the in-plane stresses can be improved significantly. When a/h < 10 in-plane stresses with warping correction are in good agreement with that of the 3-D model, with an error lower than 3%. It is seen from examples 3-5 that the element can be used efficiently for problems with high stress gradients and non-linearity. To follow post-buckling path at a bifurcation point the asymptotic solution based on Koiter's theory is used as the first iteration result so that the need of initial imperfections is obviated.

Acknowledgements

This research is supported under a grant to the Center of Excellence for Computational Modeling of Aircraft Structures, from the Federal Aviation Administration. This support, and the encouragement of Drs. Bill Wall and Larry Neff are thankfully acknowledged. The first author also wishes to express his thanks to the National Natural Science Foundation of China for its support.

Appendix A1

( A p ) - 1 = I/A, [i~ i0 (A b)- l= C , (A g) - l= 0

where, I = 1 , C = ~ 1 3 -

0 1 --1

b_i

2 '

P P P �9 C p = [ M o , M I , M 2 ] , M/v= O,

Ci 3'

bJ =x2-x2J m' c ~ = - ( x J - x ~ ) (i,j,m)=(O, 1,2inorder) cb:

0, A , ( c j - c.) A, ( b j - bin) A 3R1 24R 1 24R 1

c i A (cj - %) A, (b j - bin) A 2 ' 3R 2' 24R 2

bi 0, 0, 2'

[ c ~ ( 1 , 1 ) . . .

Cb(4,1) ---

c~(7,1) . . .

Cb(2,1) ...

c~(5, 1) ... Cb(8, 1) ...

Cb(3,1) ... Cb(6,1) ..-

Cb(9,1) .-.

c b 0 , 1 5 ~ ]

Cb(4, 15) / = [M S +S~o ,N ~ + S ~ I , Q ~ + Sbo2] Cb(7, 15)_1

cb(2,1s)] cb(s, i s ) / = tQb, + S o,n b, + <I,N + S 121 Cb(8, 15)A

c b ( 3 , 1 5 ) - 1

Cb(9,15)J

24R 2

0

312

M b = i

- bi 3 ~ ' O,

c~ O,

3R2'

ci bi 3R 1' 3R 2'

�89 gm~ - x,jY~j),

- - l ( X m i Vmi -- X i j Yij),

l ( x 2 -- __ X 2 + y2 ), 2, ~j y2 m~

b,. 2 2 bj 2 2 -(~(Y,j + 4X,) + ~-~(Y,,, + 4Xmi),

+ -~X,,,i Y, ni, ~ X~j Y~/ cj

c: (7x2 + G) + c+ <7x: r ~

N b = i

- b1 6R1'

O~

cj

6R 1 '

1 y/j, O, 2Xi j

Cj 1 Y i j ,

6R2, - 2 X ~ j

bi l_(y~_ x2), 6R 2 ' 2

~ (2X/2j - Y~), ~ XijYiJ -

C m ~XijY~j , ~ 2 ( 2 Y ~ - X~)

c , , (5X~_ y2), ~ ( 5 y / ~ _ X 2) i5

- b, .

6R 1 '

O~

Cm

6 R a '

O, 1 Ymi , 7 X .~i

~ � 8 9 ' 6R 2 '

bm 1(X2 _ Y2i) , 6R 2 ' 2" , mi

(2X~ i _ y2 ), bj Xm ' 4 Y~

C~J Xmi L i , c j_j ( 2 y 2 _ X2mi) 4 12" " '

r 2 ) , ~2 (5Y~ _ X~,) c J ( s X ~ . _ 12" " '

S b : ki

- bk

6Rt

bkbi

4A

C k CkCi O,

6R2 4A

Ck bk (bkci + Gbi)

6R~ 6R 2 4A

bk - - ( b j c j - b ~ % ) , 24A

c~ (c~ - c 2 24A ")'

c~ (bj~j- b,.cm) + ~4A (c) - c~), 24A

w h e r e

bm Cm - - , Y i j = - - - - ' Xij li j lij

i, j , m = 0, 1, 2 (in o rder ) k = 0 , 1 , 2

C o �9

s , = [ ( ' 4 - + - ]

[ C . ( 1 , 1 )

C~ 1)

C~ 1)

Cg(5, 1)

I t 0 ( 3 , 1 )

Cg(6, 1)

"'" C~ 15)-1 = [M{, + S~) o, N~) + S~) 1 , Q~ + S~) 2 ] �9 .. C~ 15)_1

�9 .. C o ( 2 , 1 5 ) - ] = [Q~ + S ~ o , M ~ + S ~ t , N ~ + S ~ 2 ] �9 . . c , ( 5 , 1 5 ) d

�9 .. Co(3, 15):] = [N~ + S ~ o , Q ~ + S ~ , M ~ + S~2] �9 .. C q 6 , 15)A 2

Computational Mechanics 13 (1994)

b~ xijYij +bJ x,,iY,~i -

4

~2 (X2 + 4 Y/~) + 12cJ (Xmi2 + 4Yg,)

bk

24A

Ck - - ( b ~ % - b j c j ) 24A

bk (b.,c,.- b~cj) + Ck (b2 " - b~) 24A 2~A

B.-Z. Huang et al.: A quasi-conforming triangular laminated composite shell element

M q = !

I _ A _ L _ ,

N~= 12R1 0, 0, A

6R 2 '

Q~=

12R 2'

Sg = 0, 6 ' 48

ki 0, _ c~ C~(cj - c,,),

6 ' 48

6RI' O, fdbi, -~-o (b,.cm-bjcfl, (b2-b 2)

O, A 7 1 e 1 J 6R 2 ' ~ ci' {o(Cm--c]), (bmCm--bjcj

3b, , , 1 b2 1 2o ~bmc~, 30j 3 c 2 1

__ _ _ Cra, m b mcm 20 30'

A , . . . . 3 by, 1 bycy, ~0 1 12R 1 0, 20 30 0, A 3 c2 1 ]

C j , b j c 20 30'

- ~88 (bj - bm)t . Ck -~(bj-bm)J

313

A p p e n d i x A 2

I D~tI, D12I, I )V=A, I )b= D12I , D22I,

D16I, D26I,

15.(u,, u,,)= IN'q, NI=I l L N12I, N22I_J '

D161 ]

D261 ;

D661

I)bP 1 l)bs = i I~ sp = B, = gi93 B, 5193D,

f)W(u") = �89

I)gb(u")=L/~21I ' ~22I ' /326IJ '

15I(u', u") = �89162162 + q~(n")Aq~(u'))

1, o, o, o, o, o, whereI93= 0, 0, 0, 1, 1, 1, 0, 0, , i6 2= 1, 1, 1, 0,

0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1,

(fi (U/t) ~ ~ 1 (II"), 0 q~2 (u")l T ~ : [/~11, B12, ~1~I= =L 0, 4~2(u"), 4~,(u")]' /~21, /~==, /~26A q~r(u")B'

I N '~, N 22, N'23 r = A {e{o')(u '') + 2~?(oi L)(n'll ,,) -q- E~2)(Utt) } --~ Bx(n").

f)ss = D ,

1,

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Communicated by S. N. Atluri, May 17, 1993

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