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A refined mixed shear flexible finite element for the nonlinear analysis of laminated plates
compurm& Sc?wrunl Rinwd in Great Britain.
Vol. 22. No. 4. pp. 529-m.
1986
ao45-7w9/86 0 19136
53.00 c .oa
Perg~mon F’rcsr Ltd.
A REFINED MIXED SHEAR FLEXIBLE FINITE ELEMENT FOR THE NONLINEAR ANALYSIS OF LAMINATED PLATES N. S. PUTCHA* and J. N. RUDDY Department of Engineering Science and Mechanics, Virginia Polytechnic University, VA, U.S. (Received
24 August
Institute and State
1964
shear stresses on the top and bottom faces of the plate. Numerous studies involving the application The two-dimensional analyses of laminated comof the fust-order theory to bending, vibration and posite plates in the past have been based on one of transient analysis can be found in Refs. [8-101. the following two types of plate theories: A variationally consistent higher-order theory (1) the classical lamination theory, which not only accounts for the transverse shear (2) shear deformation theories. deformation but also satisfies the zero transverse In both of the theories it is assumed that the lamshear stress conditions on the top and bottom faces inate is in a state of plane stress, the individual lamof the plate and does not require shear correction ina are linearly elastic, and there is perfect bonding factors was suggested by Reddytll, 121. The disbetween layers. The classical laminate theory, placement field used by Reddy is similar to that of which is an extension of the classical plate theory Levinson[l3] and Murthy[ 141but the last use the (CPT) to laminated plates, ignores the transverse equilibrium equations of the classical plate theory, stress components and models a laminate as an which are inconsistent with the assumed displaceequivalent single layer. The first complete lamiment field. Reddy’s modifications consist of a more nated anisotropic plate theory is attributed to Reissner and Stavsky[I]. The classical laminate theory systematic derivation of displacement field and variationally consistent derivation of the equilibrium is adequate for many engineering problems. Howequations. The refined laminate plate theory preever, laminated plates made of advanced filamendicts a parabolic distribution of transverse shear tary composite materials, such as graphite-epoxy stresses through thickness, contains the same de(whose elastic modulus to shear modulus ratios are pendent variables as in the first-order shear deforvery large), are susceptible to thickness effects because their effective transverse shear moduli are mation theory, and requires no shear correction significantly smaller than the effective elastic mod- coefficients. The conventional variational formulation of the ulus along the fiber direction. These high ratios of classical plate theory as well as the higher-order elastic modulus to shear modulus render the clastheory involves higher order (i.e. second-order) desical laminate theory inadequate for the analysis of composite plates. An adequate theory must account rivatives of the transverse displacement. Therefore, for accurate distribution of transverse shear in the finite-element modelling of such theories one should impose the continuity of not only the transstresses. verse displacement but also its derivatives along the Many theories which account for the transverse element boundary. In other words, a conforming shear and normal stresses are available in the litplate bending element based on displacement forerature (see, for example Refs. [2-53). The Reissz ner-Mindlin-type theories are based on an as- mulation of these theories requires continuity of sumed displacement field that is expanded in terms transverse displacement and their derivatives of the thickness coordinate. A generalization of the across the inter-element boundaries. The construcfirst-order shear deformation plate theory[4, 51 for tion of such an element is algebraically complicated homogeneous isotropic plates to arbitrarily lami- requiring, for example, a quintic polynomial with nated anisotropic plates is due to Yang ef af.[6] and 21 degrees of freedom for a six-node triangular eleWhitney and Pagano[7]. In this theory, the normals ment. Computationally the element requires much to the mid-plane before deformation remain straight storage and computer time. To overcome the stringent continuity requirebut not necessarily normal to the mid-plane after deformation, and consequently, a correction to ments placed by the conventional variational fortransverse stiffnesses is required. The theory also mulation of the classical plate theory, several alternative formulations and associated elements does not satisfy the conditions of zero transverse have been developed (see Ref. [IS] for a review). These include the hybrid finite elements and the * Present address: EngineeringMechanics Research Corporation, Troy, MI, U.S.A. mixed finite elements. The hybrid elements are 1. INTRODUCTION
529
530
N. S. Porctl+and J. N.
based on variational statements that use independent variation of displacements inside the domain of the element and tractions on the boundary of the element. The mixed elements use stationary variational principles, such as the Reissner variational principle or the Hu-Washizu variational principle, to construct independent variations of both displacements and bending moments in a plate. The conventional variational formulation of the firstorder shear deformation theory of plates leads to a CO-element, often referred to as the Mindlin element. Here a mixed finite element model is developed based on the higher-order laminate theory. The mixed finite element consists of 11 degrees-of-freedom (three displacements, two rotations, and six moment resultants) per node. The geometric nonlinearity is incorporated via the von Karman strains. The element is numerically evaluated for accuracy in static analyses of layered anisotropic plates.
REDDY
a% - d.rd?. ) .
a\L,. + ,
a.r
2.2
Constitutive equations
For a lamina of constant thickness h and made of an orthotropic material (i.e. the plate possesses a plane of elastic symmetry parallel to the x-_v plane) the constitutive equations for a layer can be written in the principal material directions as:
2. A REVIEW OF THE THEORY
2. I Kinemutics The elasticity solutions indicate that the transverse shear stresses vary parabolically through the thickness. A displacement field that possibly gives such a parabolic distribution is of the form (see 1111):
UI =
where Q;j are the plane-stress reduced stiffnesses in the material axes of the layer and are given by:
uo + z Jlx 1
u2 =
4
uo + z qJy-
(ZlWW, + w,y)
3
[
1
(2.1)
uj = w.
The von Karman strains associated with the displacement field in (2.1) are:
The stress-strain relations ordinates can be written as
in the laminate
co-
El = E: + Z(Kp + Z2K:)
151 = E; + Z(Ki + Z’K;) 23 = 0
(2.2) {::}
Eq = El + Z’K: Ej = E; + Z2K: 65 = C: + Z(K: + Z’Ki). where
,KI
0
_ --,
ah a-r
=
[$lg:]{z} *
where Qij are the transformed 2.3 Equations
au0+ 1 aY
i! ?,
2 ( ay>
Kq
-
=
aJI,
ay ’
reduced stiffnesses.
of motion
Hamilton’s principle can be used to derive the equations of motion appropriate for the displacement field (2. I) (see [ 161). The equations are given by = Ilii
et =
(2.6)
gV?% -z+
aN2 _ ay
_
+ 1-G _ I
II?, + 74 _ AI 2 )’ 3h’
!!? 4
ay
531
Nonlinearanalysis of laminatedplates
tions-which is compu~tionally and algebraically complicated. Consequently, we resort to the mixed variational formulation of the problem with u, z’,W.&. I&, M, , M:. Mb, P, , P? and Pb as the primary variables. The governing equations for U. v, w’.JIXand Jr, are given in eqn (2.7). To obtain governing equations for the other six primary variabfes (Ml. M2, Mb, PI, P2 and Pbl. we consider the piate constitutive equations
and
where Aij, Bij, etc., are the plate stiffnesses, de-
fined by (A,, Bij, D,, Eij, Fij, Hij) hi2 =
f
_h~
Qijtl, 2, t’, z3*z4,~~1 dZ (i,j = 1, 2. 6) hl?
(Aijt Dijv Fij) =
I
_h,2 Q,(lv 2’. z4) dZ
(i, j = 4, 5). (3.3) Equation (3.1) can be written in the alternative form (to develop the mixed model) as -N $ i}[
I-G’]
I-H’]
=
f
B’
I-L’] I
I
symm.
K2
c’
8 I
[F’l
(a)
M
W,
(cl
Ii} p
(3.4) h/2
=
I_ ~(1, z, z’, z3,z4,z6)de _hi,
iz = tz - ~h,is = z, - &z7,
7, = I~
(2.9
where [G’], [B’] and [f’] are symmetric but [WI, [L.‘] and (C’] may not be symmetric. From eqns (2.71, (3.2) and (3.4), we obtain the governing equations for the mixed formulation (for simplicity the superscripts, primes on gs, hs, etc., (2.10)are not used):
3. MIXED ffHtTE ELEUENT MODEL
If a displacement finite element model (consisting of N, V, w, JI, and I&.as primary variables) of the governing equations in (2.7) is to be developed, we must use interpolation functions that guarantee interelement continuity of not only displacements but also slopes. Apart from the boundary conditions, the terms a2dax2, a2d8y2 and a%da.ray show up in the variational formulation of the problem. So, the finite element model requires a quintic polynomial to represent the interpolation func-
6~: (k,,c’: + g&! + gg3& + (httMt f h,tMr! +
A,,&)
-e
(f,ip,
+
+ Ng,,cP f g& +
h3&&
+
=
rrii 5
6~: Kg& +
+
(b,p,
+
(h3lMI
132Pr
+
b3P6)l.y
..
&Jf,
-
$
+
(/31p,
(3.5)
iJ 0,
+ gze: + g&$
h33M6)
113p6)1~
+
+ &,&!I +
hMd
_
h&
+
f (h,Mt
~3,p2~33p6)1,
+
h32M2
532
N. S. F’u-rcu~and J. N.
&DDY
+ cz2M~ Kg,*cY+ g224+ g2,C)+ (h2lMI + h22M2 6Pz:- (I,&+ 1& t /&) + (cIZM, +
h23M6)
+
(hPI
_ =
I,i;
+
+
M2
+
k3Pb)l.y
..
t2Jlr
-
j$
+
cxMd
-
K;
+ D&)
+
+ (f,zP,
+ FxP2
+
F23P6)
(3.14)
0
(3.6)
A@.?
6P6: -
6~: [(A&
=
(Asset + Dw&l,
+ [(Ad' + D.&f)+ (A&
+ DdC;)l.y
(I,34
+
+
c33M6)
-
K;
=
12,~; +
+
(Fd’,
I,3&) +
+
(c,3M,
+ cZ3MZ
F23P2 + F,,P‘j)
(3.15)
0
- ;{[(D& + Fa 4, + (Da li+ Fw& + [(Dad + F&J + $ (PI, +
2P6*
+
p2,yy)
N6w.y)~
+
(N6W.sr
4
2
(>
-
x z&i,
+ (Lk8 + +
Y, t)
q(X,
I
R {[eqn(35)16u
+ (N1w.x +
= z,a
To obtain the finite-element model of the above governing equations, we construct a variational forFuK;)~,~} mulation of eqns (3.5)-(3.15) over a typical element (see (171):
9
I,(@*,
+
N2w.y).y
+ tii;‘,,) + $
+ [eqn(3.6)16v
+ [eqn(3.7)16w + [eqn(3.8)16$,
4
+ [eqn(3.9)16$,
+ i;$) +
(3.7)
+ teqn(3.1
(3.16)
+ [eqn(3.10)]6MI
l)JfiM~ + [eqn(3. 12)]6M6
+ [eqn(3.13)16P1 + [eqn(3.14)]6P? 64,: MI,
+
+ DJS~$)
(A& + As_& +
-
M6.y
+ 4
(D&
h’
D&
+
+
D4j~a
+ [eqn(3.15)16Ps} dr dy = 0.
FJs~f
After integrating by parts and substituting terpolation for all variables we obtain:
+ Fd) - $ - $ SJI,: Me,
(P,,
+ P4
[[K’] + [If’]] {A’} + [M’] {A’} = {F’}
= &ii + i&
i,Gi;,
(3.8)
(AMa!+
+ Mz, -
4 + - (D&
+ D45K:)
+ FM~:
+ Due:
0 Ddse~
+
h’
A.&
- $
where [K’], [H’], {A’}, [M']and {F’} are the generalized element liinear stiffness matrix, element non-linear stiffness matrix, element displacement vector, element mass matrix and element force vector respectively, and
WI, w , . * . , w, 7 $1, JI: 1 . . . , JI;, &e,
(3.9)
JI.:. $f, . . . . JI;, MI, M;, . . . , M;, M:,
EM,: + -
SM2:-
(h,, E: + hy e;+ h,, b,3M6)
+
+
&t
(b,,M,
cd?
+
+
b23M6) K;
=
hzl $+ +
(c2,P,
+
(b,,M,
cd2
+
+
(3.11)
+
k31p1
+
&t C32p’
(buM, +
+ buMz
c33p6)
(3.12) 1214
+
+
c>,Ma) + (FIIPI
-
Kf
0
b22M2
c23P,)
0
SP,:- (I,,4 + =
h3? &t
hz3 &+ hjj
+ bnM6) =
PI,P:,... . p;. Pi, Pi, . . . ,P$, PA,Pi,. ..P6n](".
0
6M6: - (h,je:+
Kg
Mf,...,M$,M:,M;,...,M:,
(3.10)
(h,zc’j’+
-
b,zMz
c,3p6)
KY = 0
+
-
(c,,p,
(3.17)
{A’}’ = [u,, 142,. . . , u,, VI, v?, . . . , II,,
+ Fd5~jS)
= I23 + i&
the in-
l3,e:)
+
+ FnPz
(CIIMI +
c21M2
+ Fd’d
(3.13)
(3.18) Adding the contributions main, we obtain
of all elements in the do-
HKI + [HII I4 + [Ml I& = {FI,
(3.19)
where [ICI, [HI, {A}, [Ml and {F}are the generalized linear-stiffness matrix, nonlinear stiffness matrix, displacement vector, mass matrix and force vector, respectively. 4. NUMERICAL
RESULTS
Numerical results obtained using the refined mixed element model developed herein are pre-
533
Nonlinear analysis of laminated plates
sented for the linear and nonlinear bending of laminated anisotropic plates. The effects of transverse shear deformation, material anisotropy and coupling between stretching and bending on deflections and stresses are investigated. The finite-element model developed herein is validated by comparing the results with the exact solutions (see [ 11, 121) of the linear higher-order shear deformation plate theory (HSDPT). Once the accuracy of the finite-element model is established, results of the linear and nonlinear analysis of general laminates (i.e. those which do not admit exact solutions) with arbitrary edge conditions are presented. In all the problems considered, the individual layers are taken to be of equal thickness, and in most cases only two sets of material properties are used [these values do not necessarily indicate the actual properties of any material, but they serve to carry parametric studies]:
v-0 ,?i nT=o P,*OE
= 0.5,
FSS-2
U-l
= 0.5, v12 = 0.25.
P,‘O
-I
v=*=r,=w*=P*=O
v=u=yx=H*=P*=O
(aI
(b)
Fig. 1. Boundary conditions for the mixed finite-element model in the full plate analysis of rectangular domains. (a) Cross-ply laminate (FSS-I), (b) anti-symmetric angle-ply laminate (FSS-2) simply supported boundary conditions. Ye
"=W=*x=M*=P2=0
u=*=*,=M*=P2=0 u-0 VI-0
u =o yi p6=o 6
SS-I
y
=
'f u=o
v=o
ly’O
r;O
ns=o
t$=o
P6'0
PgO
s-x H6=pb=o
ty=O s-2
P,'O I
u=*)#= 6'P6'0
Table I. Effect of reduced integration and type of mesh on the accuracy of the solution: (O”/W)
Type of integration Full Mixed Reduced Exact Sol Full Mixed Reduced Exact Sol Full Mixed Reduced Exact Sol Full Mixed Reduced Exact Sol Full Mixed Reduced Exact Sol
w&h’
square plate under UDL (q,, = 1.0); w = -;
Type of mesh
a
0 5;
2
4
5
6.25
IO
n,=o
-
x
The coordinate system and the boundary conditions used for simply supported plates are shown in Fig. 1 for both cross-ply (FSS-1) and angle-ply (FSS-2) laminates. Whenever biaxial symmetry exists in a problem, only a quadrant of the plate is modelled. The boundary conditions for a quadrant of cross-ply @S-l) and angle-ply (SS-2) laminates are shown in Fig. 2. The type of boundary condi-
(4.2)
It is assumed that Gi3 = Glz and v,, = v,~. For the analysis based on the first-order shear deformation plate theory (FSDPT), the shear correction coefficients are taken to be Ki = Ki = 516.
supported
M,=O
model in a quadrant of rectangular laminate. (a) Cross-ply laminate (SS-1) and (b) angle-ply laminate (SS-2) simply supported boundary conditions.
(4.1)
= 0.6,
GdEz
oy=o
(b) (a) Fig. 2. Boundary conditions for the mixed finite-element
G2,1Ez = 0.2, viz = 0.25 Material 11: E,IEz = 40, G,JEz
]gx $ilrl
lI=O W-0
u-y
Material I: E,tEz = 25, G,JE2
v=w=ux=M*=P*=O
W*,‘M2=P*=O
‘I
4 x 4L
2 x 2Q
7.0469 7.1119 7.1289 6.9648
6.9684 6.9705 6.9771 6.9648
3.0792 3.1045 3.1145 3.0706
3.0723 3.0723 3.0769 3.0706
2.5728 2.5990 2.6090 2.5791
2.5813 2.5808 2.5850 2.5791
2.2397 2.2732 2.2834 2.2623
2.2646 2.2640 2.2678 2.2623
1.8409 I .9185 1.9288 1.9173
1.9192 1.9193 1.9218 1.9173
10’
90a4 a 0 5;
12.5
20
SO
100
simply
Material = I Type of mesh 4 x 4L
2 x 2Q
1.7174 1.8364 1.8467 1.8375
1.8384 I .8397 1.8417 1.8375
I .4684 I .7475 I .7576
1.7479 I .7534 I .7548
1.7509
1.7509
1.3261
I .7252 1.7334 1.7348
1.7’70 2s
cross-ply
1.7371 1.7310
1.7310
0.7589 1.6996 1.7098 1.7043
1.6898 1.7068 1.7081
0.2826 I .6927 I .7029 I .6977
I .6789 1.7001 I .7014 1.6977
1.7043
SW
J. N. REDDY
N. S. PVrcmand
tions and the biaxial symmetry of cross-ply and angle-ply laminates are recently discussed by Reddy[ 181.
::
z t:
i
_
Exact elasticity wl~:lon
_._ Exacthigher-order tneary 4. I Linear
analysis
i
In the linear analysis, the nonlinear contributions (i.e. matrix [HI) in eqn (3.19) are set to zero. The finite element model for static analysis becomes
n.a
---
:: * .c 3
(VSDPT)
Exact first-order theory (FSQPT) rder theory (HSQPT)
5.6-
I+ 8 G
Side to thickness ratio, d/h
First, the effect of the reduced-order integration of the stiffness coefficients on the solution is investigated. Table 4.1 contains maximum nondimensionalized center transverse deflections of a simply supported (SS-1) cross-ply (O”190”)square plate under uniform load. Two different meshes (in a quadrant), fi) a 4 x 4 mesh containing four-node linear elements (4 x 4L) and (ii) a 2 x 2 mesh containing nine-node quadratic elements (2 x 2Q) are considered. The effects of full (2 x 2 for linear element and 3 x 3 for quadratic element) integration rule for both bending and shear energy terms, mixed (full integration for bending and reduced integration for shear energy terms) and reduced integration for both bending and shear energy terms are investigated. The equivalent nodal force vector (F} due to external forces is calculated using the full integration. The results are compared with the exact solutions based on the higher-order shear defo~ation plate theory (HSDPT). The type of integration used has considerable effect on the linear element mesh, whereas it has little or no effect when quadratic elements are used. This is due to the fact that the quadratic elements are more flexible and will relax the “locking” effect of shear energy terms in the governing equations.
-.-
Exactelasticity rclution ExactHSOPT
___
Exact FSQPT
I Fig. 4. Effect of length-to-thickness ratio on the in-plane normal stress in a three-layer (O”/W/OO)cross-ply square plate under sinusoidal load.
Since quadratic elements give better accuracy than linear elements and mixed or reduced integration requires less computational time than full integration, a 2 x 2 mesh of quadratic elements with mixed integration is used in all problems discussed, un1es.sotherwise specified. The next example deals with a three-layer, equal-thickness cross-ply (O’/~‘/O’~ square laminate (Material I) subjected to sinusoidal loading. The plate is assumed to be simply-supported (SS1) as shown in Fig. 2. The deflections and stresses are non-d~mensio~a~i~edas follows:
-
Exact elasticity solution
-._ .--
Exact WSOPT Exact FSDPT
\ CPT
0.01 . 0
( la
,
, 20
,
,
,
30
( 40
.
1 50
Stde to thickness ratio. a/h Fig. 3. Effect of length-to-thicknessratio on the center deflectionof a three-layer (O”/~*/Oa)cross-ply square lam-
inate under sinusoidal load.
0
10
20
30
40
50
Side to thickness ratio, a/h
Fig. 5. Effect of length-to-thickness ratio on the transverse shear stress in a three-ply (~*~*/O’~ square plate under sinusoidal load.
Nonlinearanalysisof laminatedplates 0.5
0.5
0.0
- Exactelasticft~~4
0.3
0.3
-__ Exact HSOPT
0.2
o FM
0.1 ;
-_
0.0 -0.1 -0.2 -0.3
0.2
HSWT
Exact FSDPT
0.1 r 0.0 F;
~y,=~yyr(+~)lqO
-O" -0.2 _. 3
q,=~x*(a.;*rh*
-0.4
HaterfaI 1: gi=
-0.5 0.0
535
.06 .03 Shear stress,
.09 ZyL
.12
I
-0.4 10
-0.5
__
a.0
.I5
f -1
‘2
.3
Shear stress.
.4
.5
.ti
TX+
Fig. 6. ~st~bution of the transverse shear stresses thryugh the laminatethickness for a three-layer cross-ply (O*/~*/O’) laminateunder sinusoidalIoad (a/h = IO). where q. denotes the intensity of the load. The stresses are evaluated at Gauss points. In Fig. 3, the maximum deflections at the center of the plate for various side-to-thickness ratios are compared. The deflections predicted by the higherorder theory are in very good agreement with the 3-D elasticity solutions. This is p~icul~ly true for plates which range from medium to thick (5 S. a/h s 20). It can be obsetved that the shearing deformation is more pronounced for decreasing side-tothickness ratios and is almost negligible for a/h 2 50. Therefore, the Cm is adequate for thin (a/h 2: 50) laminates, in which shearing defo~ation is negligible. The mixed finite element results are in excellent agreement with the exact solution of the refined theory. Also, in the range of medium to thick plates (u/h I 20), the higher-order theory yields more accurate results than the ftrst-order theory. In Fig. 4, the nondimension~ inplane normal stresses (ZJ vs side-to-thickness ratios are compared. It is interesting to note that for increasing side-to-thickness ratio, the first-order theory solution converges to the classical theory solution from below, whereas the higher-order solution converges from above, and that the hirer-order theory solution is in very good agreement with the 3D-elasticity solution. The finite element solution is also in very good agreement with the exact solution. Figure 5 shows the effect off side-t~~ickness ratio on the transverse shear stress (23,,) of the plate. Although the first-order theory solution has the same behavior as the higher-order theory and 3-D elasticity solutions, the higher-order theory solution is in a considerable improvement over the first-order theory solution. Again, the finite element solutions are in very good agreement with the exact solution of the higher-order theory. Of course the classical plate theory predicts zero transverse stresses-unless the equilibrium equations of 3-D elasticity theory are used to compute them. Note that the transverse shear stresses predicted by the first-order theory is a constant whereas that predicted by the higher-order theory is quadratic through each lamina of the plate (see Fig. 6). The
higher-order theory results are in good agreement with the 3-D elasticity solutions. Figure 7 contains a plot of the maximum center deflection of a simply supported (SS-2) anti-symmetric angle-ply (45”/-45”/. . .) square laminate (Material I) under sinusoidal Ioading vs side-tothickness ratio. Both two- and eight-layer laminates are considered. The classical theory predicts results which are grossly incorrect. The results show, for a two-layer laminate, some discrepancy between the results of the higher-order theory and the firstorder theory for a/h s 20, and for the six-layer laminate they agree with each other. This is due to the bending-stretching coupling, whose effect is reduced with an increasing number of layers. Again, the finite element results are in very good agreement with the exact solutions. 4.2 nonlinear analysis
In the nonlinear static analysis, eqn (3.19) reduces to
L
t
-
Exact (IsOPT)
o FM (HSDPTB TWO-layer FM
(HSOPT) Eight-layer
0.0 0
10
20 30 40 Stde to thfckness ratio. a/h
i
so
Fig. 7. Effect of number of layers and thickness on the center deflection of 8 (4s”/-450i. . .) square @ate.
536
N. S.Purcmand
where [K]denotes the linear stiffness matrix and [H] denotes the nonlinear (geometric) stiffness matrix. Because [HI depends on the unknown solution {h}, the assembled equations must be solved iteratively until an appropriate convergence criterion is satisfied. In the present study the Picard-type successive iteration scheme is employed. In this scheme, eqn (4.5) is approximated by [[Kl + [H(A71 {A-‘}
= IF‘),
J. N.
REDDY
0.5,
1
. _0.4; z.
-
Experimental
---
Present
_
Higher-order
theory
(HSOPT)
0 _._
[19]
/
CPT [19]
S
/
1
,
/
,
/
/
/
(4.6)
where r is the iteration number. In other words, the nonlinear stiffness matrix for the (r + 1)th iteration is computed using the solution vector from the rth iteration. Such successive iterations are continued until the error in two consecutive iterations satisfies the inequality
O.O#./ a 0.0
’
’
’
1
0.8
0.4
Intensity
’
’
1.2
’ 1.6
of the distributed
’ 2.0
load, q. (psi)
Fig. 9. Experimental correlation of the present mixed element with the center deflection results of Ref. [19]. A simply supported 4-ply symmetric bidirectional square plate under uniformly distributed load.
where 8 is a preassigned value. In the present analin considerable error, which can be attributed to the ysis p is taken to be 1%. At the start of the iteration neglect of the transverse shear strains. In Fig. 9 the procedure, the solution is assumed to be zero so center deflection of a clamped 4-ply symmetric bithat the linear solution is obtained at the end of the directional (o”/90°/90”/O”) square plate (a = b = 12 first iteration. in., h = 0.096 in.) under uniform loading is preIn Figs 8 and 9, the finite element results are sented. The clamped boundary conditions are used compared with the experimental and thin-plate theory results of Zaghloul and Kennedy[l9]. In Fig. 8 (in a quadrant of the plate). The material properties are: El = 1.8282 x 106, El = 1.8315 x 106, Glz = center deflections vs the load parameter are preGr3 = Gzs = 3.125 x 10’ psi, u = 0.23949. Again, sented for a simply supported (5X-l) square plate the finite element results are in good agreement (a = b = 12 in., h = 0.138 in.) under uniform loading. The material properties used are El = 3 x 106, with the experimental results. Next, the effect of number of layers and the lamE2 = 1.28 x 106, Glz = GIJ = Gt, = 0.37 x IO6 ination scheme on the center deflection is investipsi, Y = 0.32. The finite element results are in exgated in Figs. 10 and 11 for two- and eight-layer cellent agreement with the experimental results. anti-symmetric cross-ply and angle-ply laminates, The thin plate theory results (taken from 1191) are
-
-
Experimental
---
Present
ORTHOTROPIC
[19]
Higher-order
PLATE
~_~oa4 ,,(i;)
I
Material a/b = 1. a/h = 10 O.O/'
0.0
’
0.4 Intensity
.
’
0.8
’
’
1.2
of the distributed
I
’
1.6
2.0
load, q,(psl)
O.OO~
50 Load Paramter.
Fig. 8. Experimental correlation of the present mixed element with the center deflection results of Ref. [19]. A simply supported unidirectional square laminate under uniformly distributed load.
P
Fig. 10. Number of layers effect on a clamped cross-ply (0”/90”/. .) square plate (Material I, a/h = 10) under uniformly distributed load.
537
Nonlinear analysis of laminated plates
boundary conditions give lower displacements when compared with simply-supported (SS-I) boundary conditions. 5. SUMMARYAh.D COSCLUSIONS
Load parameter,
P
Fig. I I. Number of layers effect on a clamped angle-ply (45”/-45”/. .) square plate (Material I, a/h = 10) under uniformly distributed load.
respectively. Clamped square plates with (a/h = 10) and Material I properties are used in the analysis. It can be seen that increasing number of layers has the effect of reducing the nonlinearity, both in cross-ply and angle-ply laminates. Also, the angleply plates exhibit less degree of nonlinearity compared with the cross-ply plates. Finally, in Fig. 12, the effect of boundary conditions on the center deflection of a quasi-isotropic square laminate (Material II) is shown for increasing load parameter (F). As expected, the clamped
A mixed shear flexible finite element with relaxed continuity, is developed for the geometrically linear and nonlinear analysis of laminated anisotropic plates. The element is formulated based on a refined higher-order theory which satisfies the zero transverse shear stress boundary conditions on the top and bottom faces of the plate and requires no shear correction coefficients. The mixed finite element developed herein consists of 11 degrees-of-freedom per node which include three displacements, two rotations and six moment resultants. The element is evaluated for its accuracy in the bending of laminated anisotropic rectangular plates with different lamination schemes, loadings and boundary conditions. In comparison to displacement finite element models, the present mixed element has relaxed inter-element continuity, computational simplicity and reduced formulative effort. The advantages of the element include accurate representation of displacements and stresses and the ease with which the model can be applied to nonlinear problems. It is shown that since the element is based on a refined higher-order theory without the use of the shear correction factors, it is less sensitive to reduced order integration on shear energy terms. It is demonstrated by several example problems that the displacements and stresses predicted by the present element are accurate when compared to other experimental and theoretical results, including the three-dimensional elasticity theory. Acknowledgments-The results presented herein were obtained during an investigation supported by NASA Langley Research Center (through Grant NAG-I-459) and the ARMY Research Office (through Grant DAAG29-85-K-0007). The support is gratefully acknowledged. REFERENCES
0
50
100
150
Load parameter.
200
250
p
Fig. 12. Effect of boundary conditions on the center deflection of a quasi-isotropic [0”/45”1-45”/907, square laminate.
1. E. Reissner and Y. Stavsky, Bending and stretching of certain types of heterogeneous aelotropic elastic plates. /. appl. Mech. 28. 40248 (1961). 2. E Reissner, On the theory of bending of elastic plates. J. Murh. Phys. 23, 184-191 (1944). 3. E. Reissner, “The effect of transverse shear deformation on the bending of elastic plates.” J. appl. Mech. 12, A69-A77 (1945). 4. H. Hencky, “Uber die Uerucksichtigung der Schubverzemungen in ebenen Platten,” fng.-Arch. 16 (1947). 5. R. D. Mindlin, “Infiuence of rotatory inertia and shear on flexural motions of isotropic, elastic plates,” 1. appf. Mech. 18, A31 (1951). 6. P. C. Yang, C. H. Norris and Y. Stavsky. “Plastic wave propagation in heterogeneous plates.” INI. 1. Solids Srrucr. 2, 665-648 (1966).
7. J. M. Whitney and N. J. Pagano, “Shear deformation in heterogeneous anisotropic plates.” J. appl. Mech. 37, 1031-1036 (1970).
538
N. S. PurcIu
8. J. N. Reddy. “A penalty plate-bending element for the analysis of laminated anisotropic composite plates.” Inr. 1. numer. Merhs Engng 15(8), 1187-1206 (1980). 9. J. N. Reddy and W. C. Chao, “Non-linear bending of thick rectangular, laminated composite plates.” ht. 1. ‘Van-Linear Mech. 11X3/4).291-301 (1981). 10. J. N. Reddy, “Geometri&y~nonlinear transient analysis of laminated composite plates,” ALU 1. 21(4), 621-629 (1983). I 1. J. N. Reddy, “A simple higher-order theory for laminated composite plates.” J. appl. Mech. 51, 745-752 (1984). 12. J. N. Reddy. “Refined nonlinear theory of plates with
transverse shear deformation.” Inr. J. Solids Srrucr. 20(9/10), 881-896 (1984). 13. M. Levinson, “An accurate, simple theory of the statits and dynamics of elastic plates.” Mech. Res. Commun. 7(6), 343-350 (1980). 14. M. V. V. Murthy, An improved transverse shear de-
and J. N.
15.
16. 17. 18.
REDDY
formation theory for laminated anisotropic plates. NASA Tech. Paper 1903 (1981). J. N. Reddy. Mixed and hybrid finite element analysis of problems in structural mechanics. Res. Rep. No. OU-AMNE-77-3, University of Oklahoma (1977). J. N. Reddy, Energy and Variational Methods in Applied Mechanics. John Wiley, New York (1984). J. N. Reddy, An Inrroducti& to the Finite Element Method. McGraw-Hill. New York (1984). J. N. Reddy, A note on symmetry‘considerations in the transient response of symmetrically laminated anisotropic plates. Int. /. turner. Meths Engng 20, 175181 (1984).
19. S. A. Zaghloul and J. B. Kennedy, “Nonlinear behavior of symmetrically laminated plates.” J. appl. Mech.
42, 234-236,
1975.
20. N. S. Putcha and J. N. Reddy, “A mixed shear flexible finite element based on a refined theory of laminated plates.” Res. Rep. No. VPI-E-84.13, Virginia Polytechnic Institute, Blacksburg (1984).
Vol. 22. No. 4. pp. 529-m.
1986
ao45-7w9/86 0 19136
53.00 c .oa
Perg~mon F’rcsr Ltd.
A REFINED MIXED SHEAR FLEXIBLE FINITE ELEMENT FOR THE NONLINEAR ANALYSIS OF LAMINATED PLATES N. S. PUTCHA* and J. N. RUDDY Department of Engineering Science and Mechanics, Virginia Polytechnic University, VA, U.S. (Received
24 August
Institute and State
1964
shear stresses on the top and bottom faces of the plate. Numerous studies involving the application The two-dimensional analyses of laminated comof the fust-order theory to bending, vibration and posite plates in the past have been based on one of transient analysis can be found in Refs. [8-101. the following two types of plate theories: A variationally consistent higher-order theory (1) the classical lamination theory, which not only accounts for the transverse shear (2) shear deformation theories. deformation but also satisfies the zero transverse In both of the theories it is assumed that the lamshear stress conditions on the top and bottom faces inate is in a state of plane stress, the individual lamof the plate and does not require shear correction ina are linearly elastic, and there is perfect bonding factors was suggested by Reddytll, 121. The disbetween layers. The classical laminate theory, placement field used by Reddy is similar to that of which is an extension of the classical plate theory Levinson[l3] and Murthy[ 141but the last use the (CPT) to laminated plates, ignores the transverse equilibrium equations of the classical plate theory, stress components and models a laminate as an which are inconsistent with the assumed displaceequivalent single layer. The first complete lamiment field. Reddy’s modifications consist of a more nated anisotropic plate theory is attributed to Reissner and Stavsky[I]. The classical laminate theory systematic derivation of displacement field and variationally consistent derivation of the equilibrium is adequate for many engineering problems. Howequations. The refined laminate plate theory preever, laminated plates made of advanced filamendicts a parabolic distribution of transverse shear tary composite materials, such as graphite-epoxy stresses through thickness, contains the same de(whose elastic modulus to shear modulus ratios are pendent variables as in the first-order shear deforvery large), are susceptible to thickness effects because their effective transverse shear moduli are mation theory, and requires no shear correction significantly smaller than the effective elastic mod- coefficients. The conventional variational formulation of the ulus along the fiber direction. These high ratios of classical plate theory as well as the higher-order elastic modulus to shear modulus render the clastheory involves higher order (i.e. second-order) desical laminate theory inadequate for the analysis of composite plates. An adequate theory must account rivatives of the transverse displacement. Therefore, for accurate distribution of transverse shear in the finite-element modelling of such theories one should impose the continuity of not only the transstresses. verse displacement but also its derivatives along the Many theories which account for the transverse element boundary. In other words, a conforming shear and normal stresses are available in the litplate bending element based on displacement forerature (see, for example Refs. [2-53). The Reissz ner-Mindlin-type theories are based on an as- mulation of these theories requires continuity of sumed displacement field that is expanded in terms transverse displacement and their derivatives of the thickness coordinate. A generalization of the across the inter-element boundaries. The construcfirst-order shear deformation plate theory[4, 51 for tion of such an element is algebraically complicated homogeneous isotropic plates to arbitrarily lami- requiring, for example, a quintic polynomial with nated anisotropic plates is due to Yang ef af.[6] and 21 degrees of freedom for a six-node triangular eleWhitney and Pagano[7]. In this theory, the normals ment. Computationally the element requires much to the mid-plane before deformation remain straight storage and computer time. To overcome the stringent continuity requirebut not necessarily normal to the mid-plane after deformation, and consequently, a correction to ments placed by the conventional variational fortransverse stiffnesses is required. The theory also mulation of the classical plate theory, several alternative formulations and associated elements does not satisfy the conditions of zero transverse have been developed (see Ref. [IS] for a review). These include the hybrid finite elements and the * Present address: EngineeringMechanics Research Corporation, Troy, MI, U.S.A. mixed finite elements. The hybrid elements are 1. INTRODUCTION
529
530
N. S. Porctl+and J. N.
based on variational statements that use independent variation of displacements inside the domain of the element and tractions on the boundary of the element. The mixed elements use stationary variational principles, such as the Reissner variational principle or the Hu-Washizu variational principle, to construct independent variations of both displacements and bending moments in a plate. The conventional variational formulation of the firstorder shear deformation theory of plates leads to a CO-element, often referred to as the Mindlin element. Here a mixed finite element model is developed based on the higher-order laminate theory. The mixed finite element consists of 11 degrees-of-freedom (three displacements, two rotations, and six moment resultants) per node. The geometric nonlinearity is incorporated via the von Karman strains. The element is numerically evaluated for accuracy in static analyses of layered anisotropic plates.
REDDY
a% - d.rd?. ) .
a\L,. + ,
a.r
2.2
Constitutive equations
For a lamina of constant thickness h and made of an orthotropic material (i.e. the plate possesses a plane of elastic symmetry parallel to the x-_v plane) the constitutive equations for a layer can be written in the principal material directions as:
2. A REVIEW OF THE THEORY
2. I Kinemutics The elasticity solutions indicate that the transverse shear stresses vary parabolically through the thickness. A displacement field that possibly gives such a parabolic distribution is of the form (see 1111):
UI =
where Q;j are the plane-stress reduced stiffnesses in the material axes of the layer and are given by:
uo + z Jlx 1
u2 =
4
uo + z qJy-
(ZlWW, + w,y)
3
[
1
(2.1)
uj = w.
The von Karman strains associated with the displacement field in (2.1) are:
The stress-strain relations ordinates can be written as
in the laminate
co-
El = E: + Z(Kp + Z2K:)
151 = E; + Z(Ki + Z’K;) 23 = 0
(2.2) {::}
Eq = El + Z’K: Ej = E; + Z2K: 65 = C: + Z(K: + Z’Ki). where
,KI
0
_ --,
ah a-r
=
[$lg:]{z} *
where Qij are the transformed 2.3 Equations
au0+ 1 aY
i! ?,
2 ( ay>
Kq
-
=
aJI,
ay ’
reduced stiffnesses.
of motion
Hamilton’s principle can be used to derive the equations of motion appropriate for the displacement field (2. I) (see [ 161). The equations are given by = Ilii
et =
(2.6)
gV?% -z+
aN2 _ ay
_
+ 1-G _ I
II?, + 74 _ AI 2 )’ 3h’
!!? 4
ay
531
Nonlinearanalysis of laminatedplates
tions-which is compu~tionally and algebraically complicated. Consequently, we resort to the mixed variational formulation of the problem with u, z’,W.&. I&, M, , M:. Mb, P, , P? and Pb as the primary variables. The governing equations for U. v, w’.JIXand Jr, are given in eqn (2.7). To obtain governing equations for the other six primary variabfes (Ml. M2, Mb, PI, P2 and Pbl. we consider the piate constitutive equations
and
where Aij, Bij, etc., are the plate stiffnesses, de-
fined by (A,, Bij, D,, Eij, Fij, Hij) hi2 =
f
_h~
Qijtl, 2, t’, z3*z4,~~1 dZ (i,j = 1, 2. 6) hl?
(Aijt Dijv Fij) =
I
_h,2 Q,(lv 2’. z4) dZ
(i, j = 4, 5). (3.3) Equation (3.1) can be written in the alternative form (to develop the mixed model) as -N $ i}[
I-G’]
I-H’]
=
f
B’
I-L’] I
I
symm.
K2
c’
8 I
[F’l
(a)
M
W,
(cl
Ii} p
(3.4) h/2
=
I_ ~(1, z, z’, z3,z4,z6)de _hi,
iz = tz - ~h,is = z, - &z7,
7, = I~
(2.9
where [G’], [B’] and [f’] are symmetric but [WI, [L.‘] and (C’] may not be symmetric. From eqns (2.71, (3.2) and (3.4), we obtain the governing equations for the mixed formulation (for simplicity the superscripts, primes on gs, hs, etc., (2.10)are not used):
3. MIXED ffHtTE ELEUENT MODEL
If a displacement finite element model (consisting of N, V, w, JI, and I&.as primary variables) of the governing equations in (2.7) is to be developed, we must use interpolation functions that guarantee interelement continuity of not only displacements but also slopes. Apart from the boundary conditions, the terms a2dax2, a2d8y2 and a%da.ray show up in the variational formulation of the problem. So, the finite element model requires a quintic polynomial to represent the interpolation func-
6~: (k,,c’: + g&! + gg3& + (httMt f h,tMr! +
A,,&)
-e
(f,ip,
+
+ Ng,,cP f g& +
h3&&
+
=
rrii 5
6~: Kg& +
+
(b,p,
+
(h3lMI
132Pr
+
b3P6)l.y
..
&Jf,
-
$
+
(/31p,
(3.5)
iJ 0,
+ gze: + g&$
h33M6)
113p6)1~
+
+ &,&!I +
hMd
_
h&
+
f (h,Mt
~3,p2~33p6)1,
+
h32M2
532
N. S. F’u-rcu~and J. N.
&DDY
+ cz2M~ Kg,*cY+ g224+ g2,C)+ (h2lMI + h22M2 6Pz:- (I,&+ 1& t /&) + (cIZM, +
h23M6)
+
(hPI
_ =
I,i;
+
+
M2
+
k3Pb)l.y
..
t2Jlr
-
j$
+
cxMd
-
K;
+ D&)
+
+ (f,zP,
+ FxP2
+
F23P6)
(3.14)
0
(3.6)
A@.?
6P6: -
6~: [(A&
=
(Asset + Dw&l,
+ [(Ad' + D.&f)+ (A&
+ DdC;)l.y
(I,34
+
+
c33M6)
-
K;
=
12,~; +
+
(Fd’,
I,3&) +
+
(c,3M,
+ cZ3MZ
F23P2 + F,,P‘j)
(3.15)
0
- ;{[(D& + Fa 4, + (Da li+ Fw& + [(Dad + F&J + $ (PI, +
2P6*
+
p2,yy)
N6w.y)~
+
(N6W.sr
4
2
(>
-
x z&i,
+ (Lk8 + +
Y, t)
q(X,
I
R {[eqn(35)16u
+ (N1w.x +
= z,a
To obtain the finite-element model of the above governing equations, we construct a variational forFuK;)~,~} mulation of eqns (3.5)-(3.15) over a typical element (see (171):
9
I,(@*,
+
N2w.y).y
+ tii;‘,,) + $
+ [eqn(3.6)16v
+ [eqn(3.7)16w + [eqn(3.8)16$,
4
+ [eqn(3.9)16$,
+ i;$) +
(3.7)
+ teqn(3.1
(3.16)
+ [eqn(3.10)]6MI
l)JfiM~ + [eqn(3. 12)]6M6
+ [eqn(3.13)16P1 + [eqn(3.14)]6P? 64,: MI,
+
+ DJS~$)
(A& + As_& +
-
M6.y
+ 4
(D&
h’
D&
+
+
D4j~a
+ [eqn(3.15)16Ps} dr dy = 0.
FJs~f
After integrating by parts and substituting terpolation for all variables we obtain:
+ Fd) - $ - $ SJI,: Me,
(P,,
+ P4
[[K’] + [If’]] {A’} + [M’] {A’} = {F’}
= &ii + i&
i,Gi;,
(3.8)
(AMa!+
+ Mz, -
4 + - (D&
+ D45K:)
+ FM~:
+ Due:
0 Ddse~
+
h’
A.&
- $
where [K’], [H’], {A’}, [M']and {F’} are the generalized element liinear stiffness matrix, element non-linear stiffness matrix, element displacement vector, element mass matrix and element force vector respectively, and
WI, w , . * . , w, 7 $1, JI: 1 . . . , JI;, &e,
(3.9)
JI.:. $f, . . . . JI;, MI, M;, . . . , M;, M:,
EM,: + -
SM2:-
(h,, E: + hy e;+ h,, b,3M6)
+
+
&t
(b,,M,
cd?
+
+
b23M6) K;
=
hzl $+ +
(c2,P,
+
(b,,M,
cd2
+
+
(3.11)
+
k31p1
+
&t C32p’
(buM, +
+ buMz
c33p6)
(3.12) 1214
+
+
c>,Ma) + (FIIPI
-
Kf
0
b22M2
c23P,)
0
SP,:- (I,,4 + =
h3? &t
hz3 &+ hjj
+ bnM6) =
PI,P:,... . p;. Pi, Pi, . . . ,P$, PA,Pi,. ..P6n](".
0
6M6: - (h,je:+
Kg
Mf,...,M$,M:,M;,...,M:,
(3.10)
(h,zc’j’+
-
b,zMz
c,3p6)
KY = 0
+
-
(c,,p,
(3.17)
{A’}’ = [u,, 142,. . . , u,, VI, v?, . . . , II,,
+ Fd5~jS)
= I23 + i&
the in-
l3,e:)
+
+ FnPz
(CIIMI +
c21M2
+ Fd’d
(3.13)
(3.18) Adding the contributions main, we obtain
of all elements in the do-
HKI + [HII I4 + [Ml I& = {FI,
(3.19)
where [ICI, [HI, {A}, [Ml and {F}are the generalized linear-stiffness matrix, nonlinear stiffness matrix, displacement vector, mass matrix and force vector, respectively. 4. NUMERICAL
RESULTS
Numerical results obtained using the refined mixed element model developed herein are pre-
533
Nonlinear analysis of laminated plates
sented for the linear and nonlinear bending of laminated anisotropic plates. The effects of transverse shear deformation, material anisotropy and coupling between stretching and bending on deflections and stresses are investigated. The finite-element model developed herein is validated by comparing the results with the exact solutions (see [ 11, 121) of the linear higher-order shear deformation plate theory (HSDPT). Once the accuracy of the finite-element model is established, results of the linear and nonlinear analysis of general laminates (i.e. those which do not admit exact solutions) with arbitrary edge conditions are presented. In all the problems considered, the individual layers are taken to be of equal thickness, and in most cases only two sets of material properties are used [these values do not necessarily indicate the actual properties of any material, but they serve to carry parametric studies]:
v-0 ,?i nT=o P,*OE
= 0.5,
FSS-2
U-l
= 0.5, v12 = 0.25.
P,‘O
-I
v=*=r,=w*=P*=O
v=u=yx=H*=P*=O
(aI
(b)
Fig. 1. Boundary conditions for the mixed finite-element model in the full plate analysis of rectangular domains. (a) Cross-ply laminate (FSS-I), (b) anti-symmetric angle-ply laminate (FSS-2) simply supported boundary conditions. Ye
"=W=*x=M*=P2=0
u=*=*,=M*=P2=0 u-0 VI-0
u =o yi p6=o 6
SS-I
y
=
'f u=o
v=o
ly’O
r;O
ns=o
t$=o
P6'0
PgO
s-x H6=pb=o
ty=O s-2
P,'O I
u=*)#= 6'P6'0
Table I. Effect of reduced integration and type of mesh on the accuracy of the solution: (O”/W)
Type of integration Full Mixed Reduced Exact Sol Full Mixed Reduced Exact Sol Full Mixed Reduced Exact Sol Full Mixed Reduced Exact Sol Full Mixed Reduced Exact Sol
w&h’
square plate under UDL (q,, = 1.0); w = -;
Type of mesh
a
0 5;
2
4
5
6.25
IO
n,=o
-
x
The coordinate system and the boundary conditions used for simply supported plates are shown in Fig. 1 for both cross-ply (FSS-1) and angle-ply (FSS-2) laminates. Whenever biaxial symmetry exists in a problem, only a quadrant of the plate is modelled. The boundary conditions for a quadrant of cross-ply @S-l) and angle-ply (SS-2) laminates are shown in Fig. 2. The type of boundary condi-
(4.2)
It is assumed that Gi3 = Glz and v,, = v,~. For the analysis based on the first-order shear deformation plate theory (FSDPT), the shear correction coefficients are taken to be Ki = Ki = 516.
supported
M,=O
model in a quadrant of rectangular laminate. (a) Cross-ply laminate (SS-1) and (b) angle-ply laminate (SS-2) simply supported boundary conditions.
(4.1)
= 0.6,
GdEz
oy=o
(b) (a) Fig. 2. Boundary conditions for the mixed finite-element
G2,1Ez = 0.2, viz = 0.25 Material 11: E,IEz = 40, G,JEz
]gx $ilrl
lI=O W-0
u-y
Material I: E,tEz = 25, G,JE2
v=w=ux=M*=P*=O
W*,‘M2=P*=O
‘I
4 x 4L
2 x 2Q
7.0469 7.1119 7.1289 6.9648
6.9684 6.9705 6.9771 6.9648
3.0792 3.1045 3.1145 3.0706
3.0723 3.0723 3.0769 3.0706
2.5728 2.5990 2.6090 2.5791
2.5813 2.5808 2.5850 2.5791
2.2397 2.2732 2.2834 2.2623
2.2646 2.2640 2.2678 2.2623
1.8409 I .9185 1.9288 1.9173
1.9192 1.9193 1.9218 1.9173
10’
90a4 a 0 5;
12.5
20
SO
100
simply
Material = I Type of mesh 4 x 4L
2 x 2Q
1.7174 1.8364 1.8467 1.8375
1.8384 I .8397 1.8417 1.8375
I .4684 I .7475 I .7576
1.7479 I .7534 I .7548
1.7509
1.7509
1.3261
I .7252 1.7334 1.7348
1.7’70 2s
cross-ply
1.7371 1.7310
1.7310
0.7589 1.6996 1.7098 1.7043
1.6898 1.7068 1.7081
0.2826 I .6927 I .7029 I .6977
I .6789 1.7001 I .7014 1.6977
1.7043
SW
J. N. REDDY
N. S. PVrcmand
tions and the biaxial symmetry of cross-ply and angle-ply laminates are recently discussed by Reddy[ 181.
::
z t:
i
_
Exact elasticity wl~:lon
_._ Exacthigher-order tneary 4. I Linear
analysis
i
In the linear analysis, the nonlinear contributions (i.e. matrix [HI) in eqn (3.19) are set to zero. The finite element model for static analysis becomes
n.a
---
:: * .c 3
(VSDPT)
Exact first-order theory (FSQPT) rder theory (HSQPT)
5.6-
I+ 8 G
Side to thickness ratio, d/h
First, the effect of the reduced-order integration of the stiffness coefficients on the solution is investigated. Table 4.1 contains maximum nondimensionalized center transverse deflections of a simply supported (SS-1) cross-ply (O”190”)square plate under uniform load. Two different meshes (in a quadrant), fi) a 4 x 4 mesh containing four-node linear elements (4 x 4L) and (ii) a 2 x 2 mesh containing nine-node quadratic elements (2 x 2Q) are considered. The effects of full (2 x 2 for linear element and 3 x 3 for quadratic element) integration rule for both bending and shear energy terms, mixed (full integration for bending and reduced integration for shear energy terms) and reduced integration for both bending and shear energy terms are investigated. The equivalent nodal force vector (F} due to external forces is calculated using the full integration. The results are compared with the exact solutions based on the higher-order shear defo~ation plate theory (HSDPT). The type of integration used has considerable effect on the linear element mesh, whereas it has little or no effect when quadratic elements are used. This is due to the fact that the quadratic elements are more flexible and will relax the “locking” effect of shear energy terms in the governing equations.
-.-
Exactelasticity rclution ExactHSOPT
___
Exact FSQPT
I Fig. 4. Effect of length-to-thickness ratio on the in-plane normal stress in a three-layer (O”/W/OO)cross-ply square plate under sinusoidal load.
Since quadratic elements give better accuracy than linear elements and mixed or reduced integration requires less computational time than full integration, a 2 x 2 mesh of quadratic elements with mixed integration is used in all problems discussed, un1es.sotherwise specified. The next example deals with a three-layer, equal-thickness cross-ply (O’/~‘/O’~ square laminate (Material I) subjected to sinusoidal loading. The plate is assumed to be simply-supported (SS1) as shown in Fig. 2. The deflections and stresses are non-d~mensio~a~i~edas follows:
-
Exact elasticity solution
-._ .--
Exact WSOPT Exact FSDPT
\ CPT
0.01 . 0
( la
,
, 20
,
,
,
30
( 40
.
1 50
Stde to thickness ratio. a/h Fig. 3. Effect of length-to-thicknessratio on the center deflectionof a three-layer (O”/~*/Oa)cross-ply square lam-
inate under sinusoidal load.
0
10
20
30
40
50
Side to thickness ratio, a/h
Fig. 5. Effect of length-to-thickness ratio on the transverse shear stress in a three-ply (~*~*/O’~ square plate under sinusoidal load.
Nonlinearanalysisof laminatedplates 0.5
0.5
0.0
- Exactelasticft~~4
0.3
0.3
-__ Exact HSOPT
0.2
o FM
0.1 ;
-_
0.0 -0.1 -0.2 -0.3
0.2
HSWT
Exact FSDPT
0.1 r 0.0 F;
~y,=~yyr(+~)lqO
-O" -0.2 _. 3
q,=~x*(a.;*rh*
-0.4
HaterfaI 1: gi=
-0.5 0.0
535
.06 .03 Shear stress,
.09 ZyL
.12
I
-0.4 10
-0.5
__
a.0
.I5
f -1
‘2
.3
Shear stress.
.4
.5
.ti
TX+
Fig. 6. ~st~bution of the transverse shear stresses thryugh the laminatethickness for a three-layer cross-ply (O*/~*/O’) laminateunder sinusoidalIoad (a/h = IO). where q. denotes the intensity of the load. The stresses are evaluated at Gauss points. In Fig. 3, the maximum deflections at the center of the plate for various side-to-thickness ratios are compared. The deflections predicted by the higherorder theory are in very good agreement with the 3-D elasticity solutions. This is p~icul~ly true for plates which range from medium to thick (5 S. a/h s 20). It can be obsetved that the shearing deformation is more pronounced for decreasing side-tothickness ratios and is almost negligible for a/h 2 50. Therefore, the Cm is adequate for thin (a/h 2: 50) laminates, in which shearing defo~ation is negligible. The mixed finite element results are in excellent agreement with the exact solution of the refined theory. Also, in the range of medium to thick plates (u/h I 20), the higher-order theory yields more accurate results than the ftrst-order theory. In Fig. 4, the nondimension~ inplane normal stresses (ZJ vs side-to-thickness ratios are compared. It is interesting to note that for increasing side-to-thickness ratio, the first-order theory solution converges to the classical theory solution from below, whereas the higher-order solution converges from above, and that the hirer-order theory solution is in very good agreement with the 3D-elasticity solution. The finite element solution is also in very good agreement with the exact solution. Figure 5 shows the effect off side-t~~ickness ratio on the transverse shear stress (23,,) of the plate. Although the first-order theory solution has the same behavior as the higher-order theory and 3-D elasticity solutions, the higher-order theory solution is in a considerable improvement over the first-order theory solution. Again, the finite element solutions are in very good agreement with the exact solution of the higher-order theory. Of course the classical plate theory predicts zero transverse stresses-unless the equilibrium equations of 3-D elasticity theory are used to compute them. Note that the transverse shear stresses predicted by the first-order theory is a constant whereas that predicted by the higher-order theory is quadratic through each lamina of the plate (see Fig. 6). The
higher-order theory results are in good agreement with the 3-D elasticity solutions. Figure 7 contains a plot of the maximum center deflection of a simply supported (SS-2) anti-symmetric angle-ply (45”/-45”/. . .) square laminate (Material I) under sinusoidal Ioading vs side-tothickness ratio. Both two- and eight-layer laminates are considered. The classical theory predicts results which are grossly incorrect. The results show, for a two-layer laminate, some discrepancy between the results of the higher-order theory and the firstorder theory for a/h s 20, and for the six-layer laminate they agree with each other. This is due to the bending-stretching coupling, whose effect is reduced with an increasing number of layers. Again, the finite element results are in very good agreement with the exact solutions. 4.2 nonlinear analysis
In the nonlinear static analysis, eqn (3.19) reduces to
L
t
-
Exact (IsOPT)
o FM (HSDPTB TWO-layer FM
(HSOPT) Eight-layer
0.0 0
10
20 30 40 Stde to thfckness ratio. a/h
i
so
Fig. 7. Effect of number of layers and thickness on the center deflection of 8 (4s”/-450i. . .) square @ate.
536
N. S.Purcmand
where [K]denotes the linear stiffness matrix and [H] denotes the nonlinear (geometric) stiffness matrix. Because [HI depends on the unknown solution {h}, the assembled equations must be solved iteratively until an appropriate convergence criterion is satisfied. In the present study the Picard-type successive iteration scheme is employed. In this scheme, eqn (4.5) is approximated by [[Kl + [H(A71 {A-‘}
= IF‘),
J. N.
REDDY
0.5,
1
. _0.4; z.
-
Experimental
---
Present
_
Higher-order
theory
(HSOPT)
0 _._
[19]
/
CPT [19]
S
/
1
,
/
,
/
/
/
(4.6)
where r is the iteration number. In other words, the nonlinear stiffness matrix for the (r + 1)th iteration is computed using the solution vector from the rth iteration. Such successive iterations are continued until the error in two consecutive iterations satisfies the inequality
O.O#./ a 0.0
’
’
’
1
0.8
0.4
Intensity
’
’
1.2
’ 1.6
of the distributed
’ 2.0
load, q. (psi)
Fig. 9. Experimental correlation of the present mixed element with the center deflection results of Ref. [19]. A simply supported 4-ply symmetric bidirectional square plate under uniformly distributed load.
where 8 is a preassigned value. In the present analin considerable error, which can be attributed to the ysis p is taken to be 1%. At the start of the iteration neglect of the transverse shear strains. In Fig. 9 the procedure, the solution is assumed to be zero so center deflection of a clamped 4-ply symmetric bithat the linear solution is obtained at the end of the directional (o”/90°/90”/O”) square plate (a = b = 12 first iteration. in., h = 0.096 in.) under uniform loading is preIn Figs 8 and 9, the finite element results are sented. The clamped boundary conditions are used compared with the experimental and thin-plate theory results of Zaghloul and Kennedy[l9]. In Fig. 8 (in a quadrant of the plate). The material properties are: El = 1.8282 x 106, El = 1.8315 x 106, Glz = center deflections vs the load parameter are preGr3 = Gzs = 3.125 x 10’ psi, u = 0.23949. Again, sented for a simply supported (5X-l) square plate the finite element results are in good agreement (a = b = 12 in., h = 0.138 in.) under uniform loading. The material properties used are El = 3 x 106, with the experimental results. Next, the effect of number of layers and the lamE2 = 1.28 x 106, Glz = GIJ = Gt, = 0.37 x IO6 ination scheme on the center deflection is investipsi, Y = 0.32. The finite element results are in exgated in Figs. 10 and 11 for two- and eight-layer cellent agreement with the experimental results. anti-symmetric cross-ply and angle-ply laminates, The thin plate theory results (taken from 1191) are
-
-
Experimental
---
Present
ORTHOTROPIC
[19]
Higher-order
PLATE
~_~oa4 ,,(i;)
I
Material a/b = 1. a/h = 10 O.O/'
0.0
’
0.4 Intensity
.
’
0.8
’
’
1.2
of the distributed
I
’
1.6
2.0
load, q,(psl)
O.OO~
50 Load Paramter.
Fig. 8. Experimental correlation of the present mixed element with the center deflection results of Ref. [19]. A simply supported unidirectional square laminate under uniformly distributed load.
P
Fig. 10. Number of layers effect on a clamped cross-ply (0”/90”/. .) square plate (Material I, a/h = 10) under uniformly distributed load.
537
Nonlinear analysis of laminated plates
boundary conditions give lower displacements when compared with simply-supported (SS-I) boundary conditions. 5. SUMMARYAh.D COSCLUSIONS
Load parameter,
P
Fig. I I. Number of layers effect on a clamped angle-ply (45”/-45”/. .) square plate (Material I, a/h = 10) under uniformly distributed load.
respectively. Clamped square plates with (a/h = 10) and Material I properties are used in the analysis. It can be seen that increasing number of layers has the effect of reducing the nonlinearity, both in cross-ply and angle-ply laminates. Also, the angleply plates exhibit less degree of nonlinearity compared with the cross-ply plates. Finally, in Fig. 12, the effect of boundary conditions on the center deflection of a quasi-isotropic square laminate (Material II) is shown for increasing load parameter (F). As expected, the clamped
A mixed shear flexible finite element with relaxed continuity, is developed for the geometrically linear and nonlinear analysis of laminated anisotropic plates. The element is formulated based on a refined higher-order theory which satisfies the zero transverse shear stress boundary conditions on the top and bottom faces of the plate and requires no shear correction coefficients. The mixed finite element developed herein consists of 11 degrees-of-freedom per node which include three displacements, two rotations and six moment resultants. The element is evaluated for its accuracy in the bending of laminated anisotropic rectangular plates with different lamination schemes, loadings and boundary conditions. In comparison to displacement finite element models, the present mixed element has relaxed inter-element continuity, computational simplicity and reduced formulative effort. The advantages of the element include accurate representation of displacements and stresses and the ease with which the model can be applied to nonlinear problems. It is shown that since the element is based on a refined higher-order theory without the use of the shear correction factors, it is less sensitive to reduced order integration on shear energy terms. It is demonstrated by several example problems that the displacements and stresses predicted by the present element are accurate when compared to other experimental and theoretical results, including the three-dimensional elasticity theory. Acknowledgments-The results presented herein were obtained during an investigation supported by NASA Langley Research Center (through Grant NAG-I-459) and the ARMY Research Office (through Grant DAAG29-85-K-0007). The support is gratefully acknowledged. REFERENCES
0
50
100
150
Load parameter.
200
250
p
Fig. 12. Effect of boundary conditions on the center deflection of a quasi-isotropic [0”/45”1-45”/907, square laminate.
1. E. Reissner and Y. Stavsky, Bending and stretching of certain types of heterogeneous aelotropic elastic plates. /. appl. Mech. 28. 40248 (1961). 2. E Reissner, On the theory of bending of elastic plates. J. Murh. Phys. 23, 184-191 (1944). 3. E. Reissner, “The effect of transverse shear deformation on the bending of elastic plates.” J. appl. Mech. 12, A69-A77 (1945). 4. H. Hencky, “Uber die Uerucksichtigung der Schubverzemungen in ebenen Platten,” fng.-Arch. 16 (1947). 5. R. D. Mindlin, “Infiuence of rotatory inertia and shear on flexural motions of isotropic, elastic plates,” 1. appf. Mech. 18, A31 (1951). 6. P. C. Yang, C. H. Norris and Y. Stavsky. “Plastic wave propagation in heterogeneous plates.” INI. 1. Solids Srrucr. 2, 665-648 (1966).
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538
N. S. PurcIu
8. J. N. Reddy. “A penalty plate-bending element for the analysis of laminated anisotropic composite plates.” Inr. 1. numer. Merhs Engng 15(8), 1187-1206 (1980). 9. J. N. Reddy and W. C. Chao, “Non-linear bending of thick rectangular, laminated composite plates.” ht. 1. ‘Van-Linear Mech. 11X3/4).291-301 (1981). 10. J. N. Reddy, “Geometri&y~nonlinear transient analysis of laminated composite plates,” ALU 1. 21(4), 621-629 (1983). I 1. J. N. Reddy, “A simple higher-order theory for laminated composite plates.” J. appl. Mech. 51, 745-752 (1984). 12. J. N. Reddy. “Refined nonlinear theory of plates with
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and J. N.
15.
16. 17. 18.
REDDY
formation theory for laminated anisotropic plates. NASA Tech. Paper 1903 (1981). J. N. Reddy. Mixed and hybrid finite element analysis of problems in structural mechanics. Res. Rep. No. OU-AMNE-77-3, University of Oklahoma (1977). J. N. Reddy, Energy and Variational Methods in Applied Mechanics. John Wiley, New York (1984). J. N. Reddy, An Inrroducti& to the Finite Element Method. McGraw-Hill. New York (1984). J. N. Reddy, A note on symmetry‘considerations in the transient response of symmetrically laminated anisotropic plates. Int. /. turner. Meths Engng 20, 175181 (1984).
19. S. A. Zaghloul and J. B. Kennedy, “Nonlinear behavior of symmetrically laminated plates.” J. appl. Mech.
42, 234-236,
1975.
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