Computational Mechanics (1994) 13, 295-314
Computational
Mechanics
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-Mindlinfirst-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, quasiconforming 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 ReissnerMindlin 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 delamination, rotations and shear deformations are important. For this reason, the in-plane firstderivatives 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 nonlinear 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 ' x 2 )
(1)
B,-Z. Huang et al.: A quasi-conforming triangular laminated composite shell element
297
~3 I x3
U2
X2
X1
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 middlesurface 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'ap + 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
i~
1 u
~
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 middlesurface 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 =
-hi2
\x3j
=
~
-h/2
(6,7)
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.
298
Computational Mechanics 13 (1994)
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 Q26I ~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,
eo=[Ell,e22,2e123 r,
M=[Mll,
M22, M t 2 ] r
/r
r
" ~ i j k).
,W'(k)(k,')2,
t (X~k_1))2),132__(X~k_
(Aij, i, D,j) = Z rl
3 X~k-1,"11--([X32,
k
1))2))
(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;
H : ~ k2H44 klkEH4']
Lklk2H4s k2Hss 3
(11)
kl, kz are the shear correction factors and
Hij = V [')(k)[x3
3
(i, j -- 4, 5).
,~.qj ', (k) - - X ( k - 1 ) )
where
k
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+ Ua;10-1k)J;k= 0,
(~ = 1, 2
k, I = 1, 2, 3).
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 0-33may be taken as 1 :h/a:h2/a2 (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~;l0-tk = tr,k(1 + O(e, (h/a)c~)).
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. (ll)
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 "quasiconforming" 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~( a~,
5{A(e)+a:[89176
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 conditionaltwo-field variational problem
6~( S
m\
{A(t) - f ~
.O,,,
~ t~
S~,~
/
89
(m=l,2,...,N)
(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]
g2,,,
dV= 0
(m = 1,2,...,N),
(17)
&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- SamS t~
u=u~
(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)
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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
h3
- gE(w,~ + b~uu);e + (we + b~u~,);a - 7a3;e - 7e3;a] - ~cae) dA
12am
+ h ~ 9
+ b~u~-O~-Ta3)dA
(21)
=0.
A,~
Furthermore, e,ao and ~caacan be divided into two parts, respectively as
_
9
(I)
K~=~c'
K"
(22, 23)
where
9 apo are the components of the linear strain tensor, x'aS is the total curvature changes of the
9
middle-surface,
9(a~)o--- 1 3( u 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
E*aP {2(ua;~ + u~;a- 2bapw) --9 w
+ b"u~,)(w,~ + b~uu)] - 89
dA + ~ 9 .a~ {~[(w,a
,
A,.
A,.
'~
i~ (w,a + bU~u.);~ + (w,~ + b~u.);~] - ~c'~} dA = 0
) dA = 0
(28)
(29)
Am
~b*a{(w., + b~u.) - Oa} dA = O,
(30)'
Am
where the notation 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 32~3in 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.
B.-Z. Huang et al.: A quasi-conformingtriangular laminated composite shell element
301
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
where )~ is the load factor.
Substituting Eqs. (22, 23) into Eq. (10), we have
(32)
= [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 Am
+ 2NIx)r(3u)E~ol')(u", u) + 2N")r e~ol1)(u", 6u)} dA
(36)
3 r/rK=(r/', r/")I/= E ~ {2N(x ')r( 3u, u')elo11)(u', u) + 2N (1a)r(3u, u")e{o1')(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,(o1I~.
(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
Ko~ = Ko + KI(0, r/i) + K2(r//, r/i)-
r/iJJ
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
302
Computational Mechanics 13 (1994)
1
bll--
R1
,
1
b2z-
R2
,
(43)
ba2=O
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(o1), 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 strainenergy) 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, atb, ~t~ and the nodal
values qS of y, respectively as:
(K'tt]
o
0
o
0
L o L1
L2
Lt
Lz
o[
o
0
[2~c'12j
0
{~)1t : I LO
(~2
L1
0
L2
0
~= t~13t =Igo 0
LY23
0
Lo
4=
o LoL1L2
0
o
[Lo
0
0
0
0
0
0
0
0
L0
0
L1
0 2 1 {~q};
ml
0
L2
0 2 1 {qS}
0
L1
0
(44)
in which L~(i = 0, 1, 2) are the area co-ordinates of the element and
(45)
( i = 0 , 1 , 2 i n order).
qS=[y/t3,Y iz3]r
From the expression of Y, we obtain
we"= ~
~23,2
(46)
[ = ~ C q
[Y13,2 -{- Y23,1)
in which A is the area of the element and
CS
r1 l X2 -- X22
2
2 --0 X12
1 0 1 X22 --0 XO
1 0 1 XO
t --X~
X1
=
0
X2 -- X 1
X0 -- X2
t
2
2
1
1
2
2
t
1
X12--X 1 X 1 --X 2 XO - X2 X2 -- X0 X 1 --X 0 Xo
where x ~i (i = O, 1, 2; ~ = 1, 2) are the coordinates of node (i),
x3t 0
xz
~
)
1
RI
21
~ xl
2
~,x~)
" ' ' /1 cx!.x?~
( I, 1)
Fig. 2. Geometry of the element
(47)
B.-Z. H u a n g et al.: A quasi-conforming triangular laminated composite shell element
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 W t," WI1
" , WI2
" , ~ i1 3 , 7 2 i3 - 1"l T
I1 = [ U li, U2,
(i = 0, 1, 2 in order)
(48)
is divided into two parts, q and q~ where
q = [uia, U2'W
, , ' W , iI ' w i,2 qT
(49)
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 O(a)'
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
m
o(t)+ 6nctB~(ol)+ 6C(ol)TBK+ 6KTDIr + 6~.rH7} dA,
(51)
Am
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~}
(52)
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 ~ p C p ;
A
K b ~ cbT(Ab) - T[)bcb
[2H
K s=CSTL s CS; K"-- A
A
12
H
nJ
2H H
H
2H
(53)
where A is the area of the element and the matrices 1) p, 1) bp, 1) b, ~b~, ~sp, i~s are given in Appendix A2.
Likewise, we obtain
6r/rK1 t/= ~ {~qr(KS + K + K~T + Ksb + Kobr)q6qTKS'q' + 6q~rKO~Tq}
TM
(54)
?71
c~~/TK2~/= ~ {6qTKSq}
m
in which
KS = CST(A.) - rDscs;
K w = cgT(A o)- TI~.,,Cp; Kb = CgT(Ag)- Ti~gbCb
K gs = CsT(M) - TI3s~C~; K s ___CgT(M)- ri~sCg
(:55)
Computational Mechanics 13 (1994)
304
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
erda------ f ~ dx3 o r ( Y o d a a n d g t l u r i l 9 9 2 )
-h/ 2
a~3=-
~
x3
j" _~/~(m)dx3
(56,57)
m= l xam_l
where x m3 and x m3- 1 are the x3-coordinates of the top and bottom surfaces of the ruth-layer
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. F r o m 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
~7~3=h j" ~3
, dx 3 = k ~_7~3.
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 ~ = IO "~ol3dx
,
3 = 1.5k~V,aX 3 1 - 3 \ h J _J"
(61)
B.-Z. Huang et al.: A quasi-conforming triangular laminated composite shell element
305
On considering the influence of warping on in-plane stresses, instead of Eq. (1), we obtain
(62)
/A~ = Ua(X 1, X 2) -- X3t~t "~ Us~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.
Ul = [g44(~)713 -}- 84s(~)~23] 2'
~
h
u2 = [E5'*(()7~3 + $55(~)723] ~
(65)
in which
,
,
}
S-44_T1.5{ k ,on 424 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
1.5fk2n
o
~
;
S5s= h [ 2 55 f S55(1- ~2)d~ + k~k2H45 0f
=
l"5Jk
h ~ lk2H45 o~
2
$55(1 - ~2)d~ -t- klH44
oJ'S,5(1 - ~2)d~
}
}
(66)
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
Y12
0.3
=
G12 = 6 4 1 0 M P a ;
E 2 = 13000MPa
Gla =6410MPa;
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
306
Computational Mechanics 13 (1994)
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_~
\2'2//
w(3a
a~
\2-'2J
M,,
(a~,~a)
2x2
4x4
6x6
Jones(1975)
0.8519
0.8895
0.8958
0.9014
0.6104
0.6389
0.6439
0.6483
63.17
61.26
60.87
60.61
(a a)
2.5
M22 2 ' 2
M1,\2
(3a,a_~
2/
[a 3a'~
M22~,~)
51.86
50.19
49.63
48.56
73.38
71.36
70.96
70.69
59.19
57.28
56.71
55.63
6
E
i
t
Wmax
2.0
l
5
M~ax
1.5
%
M~max
o~
o 3
u3
= % 1.0
~~
~=
5
a
0
2
.
.
.
.
.
.
.
.
i
10
.
.
.
.
.
.
.
.
i
1000
100
10000
a/h
0.8
.
f
.
.
I
,
I
2x2
0.71
0.6
I
I
I
I
4x4
r
6x6
Mesh
0.5
0.z,
~ == 0.3
0.2
0.1
0
0.1
4
b
,
,
~
10
,
.
i
100
G~3 / G12
,
,
r
1000
D
,
,
i
10000
,
100000
vs.a/h,
Figs. 4a, b and 5. 4a Deflection at the center
b deflection at the center
5 Percentage
e r r o r s of ]41max, M m1a1x and M m22
ax
vs. G~3/G12.
B.-Z. Huang et al.: A quasi-conforming triangular laminated composite shell element
307
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 = 2 5 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 + E 2 ( I + 2v13))]/[12a4qo(1 - vl2v21)].
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
2 2'2/
Present
Pagano et
Present
Pagano et
Present
Pagano et
Present
Pagano et
al. (1972)
al. (1972)
al. (1972)
al. (1972)
4
4
10
10
20
20
100
100
0.763
0.720
0.558
0.559
0.541
0.543
0.539
0.539
822
812
~i3
823
0.0452
0.0467
0.0275
0.0276
0.0230
0.0233
0.0212
0.0214
0.317
0.219
0.313
0.301
0.317
0.328
0.307
0.339
0.339
0.292
0.217
0.196
0.178
0.156
0.163
0.139
O'12
(00 )
813
0"23
0.0341
0.0328
0.0236
0.0233
0.0218
0.0218
0.0211
0.0213
0.305
0.223
0.263
0.247
0.256
0.255
0.248
0.259
0.319
0.225
0.252
0.226
0.234
0.221
0.224
0.219
\2'2 4/
0.783
0.663
0.407
0.401
0.310
0.308
0.272
0.271
4.63
4.49
1.700
1.709
1.150
1.189
0.972
1.008
Table 3. Maximum stresses and deflections in 9-ply laminates
a/h
0"11
(aah
2 2 2/
Present
Pagano et al. (1972)
Present
Pagano et al. (1972)
Present
Pagano et al. (1972)
Present
Pagano et al. (1972)
4
4
10
10
20
20
100
100
0.736
0.684
0.556
0.55I
0.539
0.541
0.537
0.539
0"22
\2 2'4/
0.733
0.628
0.489
0.477
0.445
0.444
0.430
0.431
4.08
4.078
1.480
1.512
1.090
1.129
0.969
1.005
308
C o m p u t a t i o n a l Mechanics 13 (1994)
Table 4. E r r o r in in-plane stresses in c o m p a r i s o n to the 3D-solutions ( P a g a n o et al. 1972) (~o)
Ply
a/h
Warping
correction
611
622
612
a/h
611
622
612
With
Without
4
4
6.0
-41.4
18.1
-13.0
- 3.2
-32.1
10
10
-0.2
-t2.7
1.5
-4.0
-0,4
-8.7
With
Without
20
20
-0.4
- 3.9
0.6
- 1.0
- 1.3
- 3.9
100
I00
<0.1
<0.1
0.4
0.4
-0.9
-0.9
With
Without
4
4
7.6
- 23.2
16.7
- 22.9
4.0
- 35.7
10
10
0.9
- 5.6
2.5
- 5.5
1.3
- 7.7
With
Without
20
20
0.4
-2.0
0.2
- 1.8
< O.1
- 2.3
100
100
- 0.4
-0.4
- 0.2
-0.2
- 0.9
-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
~
t=
,2j
I
I
o
90
6
a = 16 in (406 rnm)
h = 0.8 in (20.3 ram)
d = 2 in (50.8 ram)
-I
a
I
~,,-X 1
I
l
I
[
I
Present
"~--
0
-~<--
90
' + ......
9-~..... 90
0
Rybicki, Hopper (1973)
-c~.-~---
0
90
Tang (I 979)
Nishioka, Atluri (1982)
~.,... .....
"- 2
-2
.4]
0
I
I
I
I
r
I
I
I
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
B.-Z. Huang et al.: A quasi-conforming triangular laminated composite shell element
309
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. G o o d 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'~-%-..
I
a
a = 355.6 mm
R = 381 mm
d = 50.8 mm
h = 2.272 mm
i
3x8
6x8
160
900
I
f
800
700
I
1
I 600
I
lZ,0
- Stanley (1985)
[] Present 6x8
~7 Present 3x8
~
t
120
~V
'
/
-ff 5oo
~
A
100
"-.
,2.272
_L-
>I
T
I
~oo
z
~ ABAQUS 8x8
\
z~
$4R5
300
~50.8~-
$9R5
"~-,
80
_~_,~4- ~--4
200
I00
O0
9
6O
,,
I
1
1
I
50
100
150
200
Arc position (rnrn)
.i
250
I
I
300
350
0
10
,
~
~
)
2
4
6
8
A (turn2)
10
-
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
310
Computational Mechanics 13 (1994)
12
10
6x6
-----4X4
.
.
.
.
.
.
2x2
j
} Present
/J ,
, / , "~
- - - - - S h i n et al. (1993)
i
/~,-"
8
13
6
fi.
-,,~.+
/>1"
_>_l ;2L?
I
0.5
I
1.0
I
I
1.5
wmax/t
2.0
9
l
2.5
3.0
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)
E2=1890Psi(130MPa)
G12 -- 930 Psi(64.2 MPa)
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
B.-Z. Huang et al.: A quasi-conforming triangular laminated composite shell element
311
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
(Ap)- 1 =
I/A, (Ab)-l=
[i~
C
, (Ag ) - l =
0
where, I =
1
0
,C=~
1
1
b_i
3
--1
A ,
3R1
( c j - c.) A,
24R 1
( b j - bin) A
24R 1
O,
ci
2'
A
3R 2'
(cj - %) A,
24R 2
(bj - bin) A
24R 2
Ci
bi
3' 2'
bJ =x2-x2J
m' c ~ = - ( x J - x ~ )
-
0,
2'
C p = [ M oP, M I P, M 2 P] ,9 M/v=
i0
0,
0,
(i,j,m)=(O, 1,2inorder)
cb:
[
c~(1,1)
...
cb0,15~]
Cb(4,1)
---
c~(7,1)
...
Cb(4, 15) / = [M S + S ~ o , N ~ + S ~ I , Q ~ + Sbo2]
Cb(7, 15)_1
Cb(2,1)
...
cb(2,1s)]
c~(5, 1)
Cb(8, 1)
...
...
cb(s, is)/= tQb, + S o,n b,+ <I,N
Cb(3,1)
Cb(6,1)
Cb(9,1)
...
...-.
cb(3,15)- 1
Cb(8, 15)A
Cb(9,15)J
+ S121
0
Computational Mechanics 13 (1994)
312
bi
3~ '
O,
O,
c~
3R2'
ci
bi
3R 1'
3R 2'
-
M b=
i
-
89
l ( x 2 ~j-- y2
2,
Cj
1
cj
bi
6R 1 '
6R 2 '
-
~2(2Y~-
c,,(5X~_ y2),
~(5y/~_X
i5
(2X~ i _ y2 ),
~
89
C~JXmi L i ,
'
Cm
6R a '
2",
mi
12"
bkbi
bk
6Rt
Ck
CkCi
6R2
4A
Ck
bk
(bkci + Gbi)
6R~
6R 2
4A
bm
lij
--
,
Yij
=
Cm
-lij
-'
--
"'
~2 (5Y~ _ X~,)
bk
bk
24A
24A
Ck
c~ (c~ - c 2
24A
")'
--(b~%-bjcj)
24A
c~ (bj~j- b,.cm) + ~4A (c) - c~),
24A
s, = [ ('4 -
+
-
]
i, j, m = 0, 1, 2 (in order)
k=0,1,2
C o 9
[
C.(1,
1)
C~
1)
"'"
9..
C~
C~
C~
1)
I
15)-1 = [M{, + S~)o, N~) + S~) 1 , Q~ + S~)2 ]
15)_1
9..
C o ( 2 , 1 5 ) - ] = [Q~ + S ~ o , M ~
Cg(5, 1)
9 ..
c,(5,15)d
t0(3,1)
9..
9..
Co(3, 15):] = [N~ + S ~ o , Q ~ + S ~ , M ~
C q 6 , 15)A
2
Cg(6, 1)
2)
bj
4 Xm ' Y~
where
Xij
X~)
--(bjcj-b~%),
4A
O,
-
12"
1(X2 _ Y2i) , c J ( s X ~ . _ r 2 ) ,
bm
6R 2 '
r~
cj_j ( 2 y 2 _ X2mi)
"'
4
6R 2 '
~2 (X2 + 4 Y/~)+ 12cJ(Xmi2+ 4Yg,)
~ XijYiJ
~XijY~j,
71 X .~iY m i ,
O,
(2X/2j - Y~),
Cm
Yij,
2X~j
l_(y~_ x2),
6R 1 '
Sb :
ki
y2 ), c: (7x2 + G) + c+<7x:
~
2
- b,.
-
__ X m~
2 +
1
y/j,
2Xij
6R2,
O~
~ X~j Y~/ + cj
-~X,,,i Y,ni,
-- l ( X m i Vmi -- X i j Yij),
O,
O~
b~ xijYij +bJ x,,iY,~i
4
b1
6R1'
N b=
i
b,.
bj 2 + 4Xmi),
2
-(~(Y,j2 + 4X,)2 + ~-~(Y,,,
gm~- x,jY~j),
+S~t,N~
+S~2]
+ S~2]
bk (b.,c,.- b~cj) + Ck (b2" - b~)
24A
2~A
-
313
B.-Z. Huang et al.: A quasi-conforming triangular laminated composite shell element
6RI'
Mq=
!
O,
O,
fdbi, -~-o (b,.cm-bjcfl,
A7
6R 2 '
N~=
I
Q~=
3b,,,
12R1
0,
0,
A
6R 2 '
A ,
12R 1
0,
Sg =
0,
ki
0,
1 e
1
J
{o(Cm--c]),
(bmCm--bjcj
~ ci'
_ A _ L _ ,
6 '
~bmc~,
3
20
c2
m
30'
. .3 .by,.
1
b mcm
30
c2
30'
~0 1
1
]
bjc
- ~88(bj - bm)t .
C~(cj - c,,),
6'
b2 1
1 bycy,
20
3
Cj ,
20
48
_ c~
30j
1
2o
__ _ _ Cra,
0,
A
12R 2'
(b2-b 2)
48
Ck
-~(bj-bm)J
Appendix A2
I)V=A,
I D~tI, D12I,
I)b= D12I , D22I,
D16I, D26I,
15.(u,,u,,)= IN'q, NI=I l
LN12I, N22I_J'
D161 ]
D261
;
I)bP = gi93
1 B, l)bs = 5193D,
i
I~ sp = B,
f)ss
= D,
D661
f)W(u") = 89
I)gb(u")=L/~21I ' ~22I ' /326IJ '
15I(u', u") = 89162162
+
1,
o,
whereI93= 0, 0, 0, 1,
0, 0, 0, 0,
(fi(U/t) ~ ~ 1(II"),
IN
0
q~(n")Aq~(u'))
o, o, o, o,
1, 1, 0, 0,
0, 0, 1, 1,
, i6 2=
q~2(u")l T ~ : [/~11, B12,
1, 1, 1, 0,
0, 0, 0, 1, 1,
~1~I=
=L 0, 4~2(u"), 4~,(u")]'
/~21, /~==, /~26A q~r(u")B'
'~, N 22, N'23 r = A {e{o')(u '') + 2~?(oiL)(n'll ,,) -q- E~2)(Utt) } --~ Bx(n").
References
Ashwell, D. G.; Sabir, A. B. (1972): A new cylindrical shell finite element based on simple independent strain functions. Int. J.
Mech. Sci. 14, 171-183
Atluri, S. N. (1975): On hybrid finite element models in solid mechanics. In: Vishnevetsky, R. (ed.): Advances in computer
methods for partial differentials equations. AICA, Rutgers University, pp. 346-356
Atluri, S. N.; Murakawa, H. (1977): On hybrid finite element models in nonlinear solid mechanics. In: Berger, P. G. et al. (eds.):
Finite elements in nonlinear mechanics. Tapir, Norway, 3-44
314
Computational Mechanics 13 (1994)
Atluri, S. N.; Galtagher, R. H.; Zienkiewicz, O. C. eds. (1983): Hybrid and mixed finite element methods. Wiley, New York
Atluri, S. N.; Tong, P.; Murakawa, H. (1983): Recent studies on hybrid and mixed finite element methods in mechanics.
In: Atluri, S. N. et al. (eds.): Hybrid and mixed finite element methods. Wiley, New York, 51-71
Chow, T. S. (1971): On the propagation of flexural waves in an orthotropic laminated plate and its response to an impulsive
load. Journal of Composite Materials 5, 306-319
Hibbit, Karlsson and Sorensen Inc. (1990): ABAQUS User manual, Version 4.7
Hu, H. C. (1955): On some variational principles in the theory of elasticity and the theory of plasticity. Scientia Sinica, 4, 33-54
Jones, R. M. (1975): Mechanics of Composite Materials, New York, McGraw-Hill.
Kant, T.; Pandya, B. N. (1988): A simple finite element formulation of a higher-order theory for unsymmetrically laminated
composite plates. Computers Struct. 9, 215-246
Koiter, W. T. (1967): General equations of elastic stability for thin shells. In: Muster, D. (ed.): Proc. of Symposium on the theory
of shells in honor Lloyd H. Donnell. Univ. of Houston, 187-227
Lo, K. H.; Christensen, R. M.; Wu, E. M. (1977)A high-order theory of plate deformation, Part 1: Homogeneous plates, Part
2: Laminated plates. J. Appl, Mech. 44, 663-676
Levinson, M. (1980): An accurate simple theory of the statics and dynamics of elastic plates. Mechanics Research communications,
7, 343-350
Lu, H.-X.; Liu, Y.-X. (1981): The quasi-conforming element in FEM and its application in hyperbolic shell element. J. Dalian
Institute of Technology 20, 75-87 (in Chinese)
Mindlin, R. D. (1951): Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl. Mech. 18,
Trans. ASME 73, 31-38
Nishioka, T.; Atluri, S. N. (1982): Stress analysis of holes in angle-ply laminates: an efficient assumed stress "special-holeelement" approach and a simple estimation method. Comput. Struct. 15, 135-147
Noor, A. K. (1983): Multified (mixed and hybrid) finite element models. State-of-the-art surveys on finite element technology.
ASME, New York, 127-162
Noor, A. K.; Atluri, S. N. (1987): Advances and tends in computational structural mechanics. AIAA J. 25, 7, 976-995
Noor, A. K.; Burton, W. S. (1989): Assessment of shear deformation theories for multilayered composite plates. Appl. Mech.
Review 42, 1 12
Pagano, J. N. (1970): Exact solutions for rectangular bidirectional composite and sandwich plates. J. Composite Materials
4, 20-34
Pagano, J. N.; Hatfield, S. J. (1972): Elastic behavior of multilayered bidirectional composites. AIAA J. 10, 931-933
Pian, T. H. H. (1964): Derivation of element stiffness matrices by assumed stress distribution. AIAA J. 2, 1333-1336
Reddy, J. N. (1984): A simple higher-order theory of laminated composite plates. J. Appl. Mech. 51,745-752
Reissner, E. (1945): The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12, 69-77
Reissner, E.; Stavsky, Y. (1961): Bending and stretching of certain types of heterogeneous aeolotropic elastic plates. J. Appl.
Mech. 28, 402-408
Riks, E. (1972): The application of Newton's method to the problem of elastic stability. J. Appl. Mech. 39, 1060 1066
Rybicki, E. F.; Hopper, A. T. (1973): Analytical investigations of stress concentrations due to holes in fiber reinforced plastic
laminated plates: three-dimensional model. AFML-TR-73-100, Battelle Columbus Labs
Shin, D. K.; Griffin Jr, O. H.; Gurdal, Z. (1993): Postbuckling response of laminated plates under uniaxial compression. Int. J.
Non-Linear Mechanics 28, 95-115
Stanley, G. M. (1985): Continuum-Based Shell Elements. PhD. Dissertation, Dept. of Mechanical Engineering, Stanford Univ.
Tang, L-M.; Liu, Y-X.; Chen, W-J. (1980): Quasi-conforming elements for finite element analysis. J. Dalian Institute of
Technology 19, 135-147
Tang, S. (1979): Variational approach to edge stresses of circular cutouts in composities. AIAA Paper 79-0802, 20th AIAA/
ASME/ASCE/AHS. SMD Conf., 326 332. St. Louis, MO
Washizu, K. (1955): On the variational principles of elasticity and plasticity. Report 25-18, Cont. Nsori-07833, M.1.T.
Whitney, J. M. (1973): Shear correction factors for orthotropic laminates under static loads. J. Appl. Mech. Trans. ASME 40,
302-304
Xue, W.-M.; Karlovitz, L. A.; Atluri, S. N. (1985): On the existence and stability conditions for mixed hybrid finite element
solutions based on Reissner's variational principle. Int. J. of Solids and Structures 21, 97-116
Yoda, T.; Atluri, S. N. (1992): Postbuckling analysis of stiffened laminated composite panels, using a higher-order shear
deformation theory. Comput. Mech. 9, 390 404
Zhang, J. D.; Atluri, S. N. (1988): Postbuckling analysis of shallow shells by the field-boundary-element method. Int. J. Num.
Meth. Eng. 26, 571-587
Zienkiewicz, O. C. (1971): The finite element method. Mc-Graw Hill
Communicated by S. N. Atluri, May 17, 1993