Com~ulers 65 Slrucfures Vol. 18, No. 2, pp. 263-272. 1984 &MS-7949/84 $3.00 + .la

Printed in Great Brihn. Pergamon Press Ltd.

A THREE-DIMENSIONAL NONLINEAR ANALYSIS OF CROSS-PLY RECTANGULAR

COMPOSITE PLATES

T. KUPPUSAMY t and J. N. REDDY~

Virginia Polytechnic Institute and State University, Blacksburg, VA 24861, U.S.A.

(Receioed 20 October 1982; received for publication 17 January 1983)

Abstract-The results of a three-dimensional, geometrically nonlinear, finite-element analysis of the bending of cross-ply laminated anisotropic composite plates are presented. Individual laminae are assumed to be homo- geneous, orthotropic and linearly elastic. A fully three-dimensional isoparametric finite element with eight nodes (i.e. linear element) and 24 degrees of freedom (three displacement components per node) is used to model the laminated plate. The finite element results of the linear analysis are found to agree very well with the exact solutions of cross-ply laminated rectangular plates under sinusiodal loading. The finite element results of the three-dimensional, geometrically nonlinear analysis are compared with those obtained by using a shear deformable, geometrically nonlinear, plate theory. It is found that the deflections predicted by the shear deformable plate theory are in fair agreement with those uredicted by three-dimensional elasticity theory; however stresses were found to be not in good agreement. _

1. INTRODUCTION Composite materials exhibit higher stiffness to weight ratios and increased corrosion resistance compared to isotropic materials. The anisotropic material properties of layered composites can be varied by varying the fibre orientation and stacking sequence. While this feature gives the designer an added degree of flexibility, the stiffness mismatch of the orthotropic layers bonded together with different fiber orientations leads to inter- laminar stresses in the vicinity of free edges. For certain stacking sequences, loading, and boundary conditions interlaminar stresses can be so large that they dictate the design of the structure.

Analyses of layered composite plates can be divided into two groups: (i) analyses based on a laminate plate theory, and (ii) analyses based on a three-dimensional laminate (elasticity) theory. The laminate plate theory is the extension of the classical plate theory (CPT) or the Reissner-Mindlin shear-deformable plate theory (SDT) to layered composite plates. The first lamination theory including bending-stretching coupling is apparently due to Reissner and Stavsky[l]. Yang et al.[2] presented a generalization of the Reissner-Mindlin thick- plate theory for homogeneous, isotropic plates to arbitrarily laminated anisotropic plates. Whitney and Pagano[3] (also see Reddy and Chao[4]) presented closed-form solutions to the theory when applied to certain cross-ply and angle-ply rectangular plates. Reddy [S] presented a finite-element analysis of the lamination theory. A higher-order lamination theory that accounts for a cubic variation (as opposed to linear ini2-51) of the inplane displacements and quadratic variation of the transverse displacement through the thickness is presented by Lo er al.[6], and hybrid-stress finite-element analysis of the theory is presented by Spilker [7].

In laminate plate theory it is assumed that the laminate

tAssistant Professor of Civil Engineering SProfessor of Engineering Science and Mechanics.

is in a state of plane stress (an assumption carried from the classical plate theory) and integrals through the thickness of a laminate are equal to the sum of integrals through the thickness of individual laminae. These assumptions lead to inaccurate prediction of interlaminar stresses at the free edges, although the solution is reason- ably accurate away from free edges. The laminate plate theory is not accurate in a boundary layer region which extends inward from the edge to a distance approximately equal to the laminate thickness,

The fully three-dimensional laminate theory is an extension of the elasticity theory of a three-dimensional solid composed of layers of different material properties. Pipes and Pagano[81 and Pipes[9] used a finite-difference technique to solve the quasi-three-dimensional, linear elasticity equations for laminates (also see Hsu and HerakovichllO]). Lin[ll], Dana[l2] and Dana and Barker[l3] used a cubic, three-dimensional, isoparametric element with 72 degrees of freedom to analyze laminated plates (also see Putcha and ReddyIl41). The numerical results in these studies agree very well with those of Pagano[lS, 161.

The present study is motivated by the lack of finite- element results for three-dimensional, geometrically non- linear analysis of layered anisotropic composite plates. A finite-element formulation of the geometrically non- linear theory of a laminated, three-dimensional, elastic continuum is presented. Numerical results of the linear as well as nonlinear analysis are presented for several cross-ply plate problems. The formulation is validated by comparing the present results of the linear analysis with those of PaganollS, 161. The results of the nonlinear analysis should serve as bench mark results for future investigations.

2. GOVERNING EQUATIONS

Consider a laminate (a) composed of N orthotropic layers with axes of elastic symmetry parallel to the plate axes. The laminate is subjected to normal traction tp=

263

264 T. KUPPUSAMY and J. N. REDDY

q(x$, x3 at its upper surface (i.e. x3 = h/2). The con. stitutive equations for any Iayer are

c,, c,, c,, 0 0 0 c,z c** $3 0 0 0

where & and iii are the components of the stress and strain tensors, respectively, defined in the material- coordinates, and cjii are the material stiffness coefficients. The coefficients, Eli = Cii, are given in terms of the engineering constants by (see Jones f 171)

(3’,, = Ed1 - VZ~V~Z)/A

A=(1 - V~ZV~I- VZV~Z- v,t~,~--2~2tv~,v,‘32)> (2)

where E,, EZ, E, = Young’s moduli in 1, 2 and 3 (material) directions, respectively; vii = Poisson’s ratio for transverse strain in the J-direction when stressed in the i-direction; and GZ3, G13, Glz = Shear moduli in the 2-3, 1-3, and l-2 planes, respectively.

In view of the reciprocal relations

2-s

Ei - Ej’ (3)

there are only nine independent elastic constants for an orthotropic elastic medium.

The constitutive equations (1) when transformed to the plate coordinate system take the form

Cl, c22 c25 0 0 0 622

Cl3 c23 c33 0 0 0

I

‘;2: (4)

Yl3 0 0 0 0 0 c* yzz )

Cl, = C’, , cos4e + 2(C,z + 2C,) sin28 cos% + Cz2 sin%

c,z = (C,, + c,, - 4&,) COS% sin28 + Clz(sin48 t cos%)

2 - *, cl3 = c,3 cos et Cz3 sm-0

CM = ((?.I I -e,, - 2C& sin e c0sae t (C,? - S,, t 2&,) cos 8 sin%

CZt = C’, , sin40 + 2(Clz + 2&) sin*@ C02e t G,, c0s4e

Ct3 = C,, sin20 t& c02e

CB~ = (Cl1 - C,, - 2Q sin% cos 8 +(C,z-C2zt 2&) C0s3esifle

c,, = C,,

Cj6 = (Cl3 - L) sin 0 cos 0 (5)

and 8 is the angle between the positive X,-axis and ff,-axis (measured counter clockwise positive).

The nonlinear kinematic description of an elastic body yields the following equations of equilibrium (in the absence of body forces and moments)

& f

Oij(&ni + 2 ,)I

=O. fi, j = 1,2,3), 16) f

wherein S,i denotes the Kronecker delta symbol, and the summation convention on repeated subscripts is used. The strain-displacement equations of the large-deflection theory of elasticity are given by

1 '$ =? (

au, atij au, au, -- aX,+aX,+ exi axj > .

(7)

To complete the description of the field equations, eqns (4)-(6), and (7) should be adjoined by boundary conditions. At any point of the boundary of the body one should specify one of the following two types of boun- dary conditions:

_ (i) essential (or geomeric) boundary conditions urn = &n

(ii) natural (or dynamic) boundary conditions

1, 3 nj~ii(~~~ f U,+i) = im. (8)

Here ni denotes the jth component of the unit normal to the boundary, t, the* mth component of the boundary traction, and I?,,, and t, denote specified values.

3. ~EL~FO~ATION Here we present a displacement finite-element model

of eqns (3)-(8). To this end we construct the variational formulation of the equations over an arbitrary element Q@’ of the finite-element mesh. We have (see [18, p. 3821)

Here S denotes the variational symbol, and Uij is given in terms of ui via eqns (4) and (7).

The displacements urn are interpolated by expressions of the form,

urn = f: u~*+~ (m = 1,2,3) 00) n=l

where G-(x, y, z) (a! = 1, 2,. . .8) are the trilinear inter- polation functions of the eight-node isoparametric ele- ment in three dimensions, and u,,,~ denotes the value of u, at node a. Substituting eqn (10) into (9), we obtain

K,","u,' t F," =O, (m=1.2,3; (u=l,2 ,,,. 8)

(11)

Three-dimensional nonlinear analysis of cross-ply rectangular composite plates 265

where (no summation is implied on n)

F,” = ,t<, t,,& ds, (j. m, n = 1,2,3). (12)

Every isoparametric finite element fi”) of the finite- element mesh can be generated from the master element via the transformation

x, = Xi&, &?, 57)

=~~,ii”fj&,.6.5) (i= k&3) (13)

where xla are the global coordinates of the element nodes and &( 1, 2, 3) are the local coordinates. Therefore, the integrals in eqn (12) can be transformed to the master element and evaluated nume~~ally using Gauss quadra- ture. The transformation of (~~~/~xi) to (~~~/~~,) is per- formed as follows:

{~]=[Jl-‘($j

3xi 3x1

dx, dxzdxt = (det VI) d& d&d&

where

VI _ 3x3-

For example, consider

x,’ x2’ x3l

XI 2 Xz2 x,= . . . . . . . x,8 XzR x38

8x3 _

(14)

=ii 2,$, F!, (PI. P,, PK)w,w,wK (15)

where PI and W; are the Gauss points and weights, respectively, and the integrand Fgp(ij = 1, 2, 3; (I, p = 1, 2, . .8) is given by

WI being the elements of the inverse of the Jacobian matrix, [J]. This procedure can be implemented on a digital computer, and the element coefficient matrices in eqn (10) can be evaluated numerically. It is well known that when the ratio of side to thickness of the plate is very large (i.e. when the plate is thin, say a/h > 20) one

should use reduced integration to evaluate the shear . . terms (i.e. terms mvolvrng C,,, t_& and C,,). For a trilinear element, 1 x I x 1 Gauss quadrature must be used to evaluate the coefficients of QU, Q55. and 2 x 2 x 2 Gauss rule to evaluate all other terms.

Since the coefficient matrix [K] depends on the un- known solution vector {to, one should employ an itera- tive solution procedure to solve the finite-element equa- tions. Here we use the Picard type iterative technique. which begins with an assumed displacement field (usu- ally, set to zero to obtain the linear solution) to compute [K] at the beginning of the first iteration. In subsequent iterations, solution obtained from the previous iterations is used to compute [K]. The iteration is continued until the solutions obtained in two consecutive iterations differ by a preassigned error margin (say, 10~“). It is more economical to use load incremental methods in con- junction with the iterative technique described above. In other words, for each increment of the load the incre- ment to the nonlinear solution is obtained (see [9]).

4. DISCUSSIONOFTHENUME~CALRESULTS

Results of fhe linear analysis

In order to validate the present formulation and ele; ment, first the linear analysis of a symmetric three-layer square laminate with the x,-direction (of material prin- cipal axes) coinciding with the x,-direction (of the plate axes) in the outer layers and the ffz-direction parallel to the x,-axis in the center layer is performed. The thick- ness of the outer layers is assumed to be one-half of the thickness of the center layer (h, = h3 = h/4, h, = h/2). The loading is assumed to be sinusoidal with respect to the x, - xt plane,

&q(x,, xz)=qocos~cos~ (17)

and the boundary conditions are of the simply supported type which allow normal displacement on the boundary, but prevent tangential displacement. For a quarter plate these imply

at x,=a/2: i12=r(=ir=0

atx,=O: li,=O. i,=O, tA,=O

at xz=b/2: li,=&=i,=O

at x2 = 0: 6, = 0. i, = 0, f: = 0.

(18)

These material of the laminae is assumed to have the following values for the engineering constants

E, = 1.724~ lO’kN/m (25 x 1O”psi)

EZ = E3 = 6.89 x 10” kN/m* (IOh psi)

Cl2 = G,.? = 3.45 x lo6 kN/m* (0.5 x 10” psi) (19)

G23 = 13.78 x 106kN/m~ (0.2 x 106psi)

V,2 = VT, = 52 = 0.25.

The following nondimensionalization is used to present

266 T. KUPPUSAMY and J. N.REDDY

the displacements and stresses:

_ E,h’ fi, _ LU3u~102 u’ = qoasU’, q0a

* (61, dz, C.12) = &&I, 02, UI?L

II

The nondimensional center deflection &(O, 0,O) obtained using the laminate plate theory[4, 51 and the elasticity theory are presented in Table 1. One can conclude from the results that the finite element results obtained by using the reduced integration (R) are in good agreement with the analytical solution of Pagano[ 161. and the shear- deformable plate theory solution[4, 51 differs from the three-dimensional elasticity solution by about 10% for the problem at hand. The classical plate theory (CPT) solution differs from the shear deformable plate theory

Table I. Nondimensionalized center deflectiontin a square cross-ply(0"/90"/0") plate (h, = h3 = h/4, h> = h/?)under sinusoidal loading (E2 = EL El = 25E2. GI: = GI1 = 0.5E2. G!i =0.X2, v,! = I+ = y,1 = 0.25)

Shear Elasticity Solution Deformable Plate Solution[4]

a/h Pagan0 2x2x3 4x4x3 2~2x3 4~4x3 CFS 2x2 4x4 2x2 4x4 CPT cl61 F F R R F F R R

2 5.075 5.052 4,986 5,341 5.051 5.063 5.253 5.109 5.525 5.170 0.431 4 1.937 1.841 1.872 1.983 1.906 1.709 1.722 1.713 1.814 1.734 0.431 10 0.737 0.614 0.694 0.734 0.728 0.663 0.590 0.643 (~.665 0.663 0.431 20 0.513 0.307 0.437 0.496 0.506 0.491 0.324 0.435 0.476 0.487 0.431 50 0.445 0.091 0.124 0.423 0.438 0.441 0.105 0.245 0.420 0.436 0.431

100 0.435 0.027 0.089 0.412 0.479 0.434 0.032 0.103 0.417 0.47R 0.411

t- u (O,O,h); mesh shown is for a quarter plate; boundary conditions are tie samz as those shown in Eq. (18).

Table 2. Comparison of nondimensional stressest for three-layer (0°/900/00) cross-ply square plate under sinusoidal loading (a/b = 1, h, = hj = h/4, hz = h/2)

3LElasticity Theory Plate Theory [S]

t! type Stress P;;;;o 2x2x3 4x4x3 2x2x3 4~4x3 2x2 4x4 2x2 4x4

F F R 4 F F R R

0 1

2 T5

: 6

;i 1

4 :5

a 6

a 1

10 a5

0 6

1 1

20 '5

a 6

a 1

50 15

0 6

a 1

100 y5

r 6

1.388 0.603 0.676 0.133 0.165 0.259 0.318 0.259 0.318

0.153 0.224 0.224 0.179 0.197 0.242 0.286 0.255 0.289

0.086 0.047 0.053 0.025 0.031 0.027 0.033 0.028 0.033

0.720 0.389 0.456 0.185 0.232 0.302 0.377 0.306 0.379

0.219 0.182 0.209 0.138 0.205 0.276 0.330 0.293 0.335

0.047 0.029 0.034 0.021 0.025 0.023 0.029 0.024 0.029

0.599 0.326 0.425 0.268 0.331 0.334 0.451 0.380 0.467

0.301 0.235 0.281 0.147 0.286 0.316 0.389 0.353 0.400

0.028 0.028 0.022 0.015 0.018 0.016 0.022 0.019 0.023

0.543 0.240 0.391 0.298 0.364 0.265 0.437 0.404 0.494

0.328 0.421 0.342 0.277 0.312 0.303 0.395 0.373 0.420

0.023 0.010 0.017 0.013 0.016 0.011 0.018 0.017 0.021

0.539 0.083 0.234 0.308 Cl.376 0.098 0.279 0.413 0.504

0.337 0.755 0.990 0.286 0.321 0.252 0.348 0.380 0.427

0.021 0.003 0.009 0.012 0.015 0.004 0.011 0.016 0.020

0.271 0.025 0.095 0.310 0.378 0.030 0.120 0.414 0.505

0.339 0.877 1.569 0.287 0.274 0.230 0.298 0.381 0.428

0.339 0.001 0.004 0.012 0.015 0.001 0.005 0.016 0.020

tThe stresses in the finite-eleme.nt analysis are computed at the Gauss points.

Three-dimensional nonlinear analysis of cross-ply rectangular composite plates 261

(SDPT) by 35% for side to thickness ratio of a/~ = 10. perties are the same as those given in eqns (If), (18) and The nondimensionalized stresses (C,, trc = ail, &, = (19, respectively. The unite-eIement results are in good

~5,~) for the problem are compared in Table 2. We agreement with the corresponding exact solutions [ 151 observe that for a/h 2 50 the results obtained using the and plate theory solutions [5]. reduced integration are in good agreement with the analytical solutions, and for a/h < 20 the full integration Results of the nonlinear analysis gives better results. First, geometrically nonlinear analysis of isotropic

Nondimensionalized center deflection and stresses of plates is performed and the results are compared with three-layer (hi = h/3) rectangular plates (b/a = 3) of cross- the results available in the literature. A square isotropic ply (O”/so”/O”) construction are presented in Tables 3 and plate (Y = 0.3) subjected to uniformly distributed load is 4. The loading, boundary conditions, and material pro- analyzed using simply supported boundary conditions.

Table 3. Nondimensionalized center deflection in a rectangular (h =3a) cross-ply @"/90"/O")_plate_(h~= hz=k= h/3) under sinusoidal loading (BCI, Ez = E,, Et = 2.FE2, G,? = Glr = 0.5 Ez, Gz3 = 0.2 Ez, VIZ = ~23 = 0.25)

Shear Elasticity Solution lIeformable Plate Solution [5] CPT

a/h Pagan0 2x2x3 4x4x3 2~2x3 4x4x3 CFS 2x2 4x4 2x2 4x4 Solution [15]

2 8.17 7.953 7.831 8.410 7.934 10.11 8.153 7.900 8.600 8.000 0,480 4 2.02 2.702 2.716 2.905 7.764 2.97 2.418 2.376 2.655 2.405 0.480 10 0.919 0.801 0.872 0.915 0.902 0.93 0.748 0.788 0.810 0.804 0.480 20 0.610 0.411 0.540 0.589 0.600 0.60 0,425 0.530 0,560 0.574 0.480 50 0.508 0.137 0.305 0.494 0.513 0.51 0.150 0.327 0.490 0.509 0,480

100 0.503 O.f!42 0.134 0.481 0.501 0.50 0.050 0.153 0.480 0.500 0.480

Table 4. Comparison of nondimensionalized stresses? for three-layer cross-ply (~/W’/OO) square plate under sinusoidal loading (a/b = 1, bj = h/3)

f 30-Elasticity Theory Plate Theory [5]

Stress Pagan0 2x2x3 4x4x3 2x2~3 4x4x3 2x2 4x4 2x2 4x4 Type F F R R F F R R

0 1

2 a5

r 6

Cl 51

0.938 0.550 0.619 0.097 0.122 0.274 0.337 0.273 0.336

0.257 0.267 0.212 0.236 0.124 0.295 0.263 0.290

0.044 0.050 0.025 0.032 0.037 0.046 0.038 0,046

0.309

0.070

a 1 0.755 0,400 0.470 0.162 0.205 0.324 0.406 0.330 0.408

4 a5 0.282 0.221 0.247 0.209 0.244 0.284 0.340 0.303 0,345

a 6

0.051 0.031 0.036 0.021 0.025 0.028 0.034 0.030 0.035

'ii 1

10 ;i5

;I 6

0.590 0.325 0.424 0.244 0.301 0.343 0.464 0.392 0.481

0.357 0.271 0.333 0.301 0.343 0.311 0.382 0.348 0.393

0.029 0.017 0.022 0.014 0.017 0.017 0.023 0.020 0.024

-d 1

20 QS

a 6

0.552 0.233 0.381 0.267 0.327 0.267 0.440 0.408 0.499

0.385 0.426 0.361 0.327 0.369 0.295 0.381 0.360 0.405

0.023 o.nlO 0.016 0.012 0.014 0.011 0.019 0.017 0.021

1 1

50 OS

‘;i 6

0.541 0.080 0.225 0.274 0,335 0.098 0.280 0.414 0,504

0.393 0.757 0.992 0.335 0.377 0.248 0.338 0.363 0.408

0.022 0.003 0.009 0.011 0.013 0.004 0.011 0.017 0.020

;j 1

100 :5

7 6

0.539 0.024 0.092 0.276 0.336 0.030 0.120 0.414 0.505

0.395 0.878 1.570 0.337 0.378 0.229 0.290 0.364 0.409

0.021 0.001 0.004 0.011 0.013 0.001 0,005 0.016 0.020

'The stresses in the finite-element analysis are computed at the Gauss points.

268 T, KUPPUSAMY and f. N. REDDY

Due to the biaxial symmetry, only one quadrant of the plate is modeled with 4 x 4 x 1 mesh of linear elements. Since membrane and bending contributions dominate the stiffness matrix, full integration is used to evaluate the element matrices. The present results for the center deflection and stresses are compared in Figs. 1 and 3 with the nonlinear plate theory (i.e. the von Karman plate theory) solutions available in the literature[20,21]. From the results of the isotropic plate one can conclude that the nonlinearity exhibited by the three-dimensional elasticity theory is relatively smaller than that included in the classical von Karman plate theory[21] but larger than that in the shear deformable plate theory[20].

Next, a symmetric cross-ply (OWWP) square plate under uniformly dist~buted load is analyzed. The boun- dary conditions and material properties are given by eqns (18) and (19), respectively. The three-dimensional elasti- city results for center deflection and stresses are com- pared with the shear deformable plate theory results[20] in Figs. 3-5. The effect of thickness (i.e. ratio of side to thickness) on the deflections and stresses can also be seen from Figs. 3 to 5. It should be noted that the plate theory solutions deviate more from the three-dimen- sional etasticity solutions as the thickness increases. Also, it should be pointed out that the stresses in the two theories are computed at different locations of the plate,

3D-Elasticity theory (4x4~1 mes

Fig. 1. Load-deflection curves for simply supported isotropic (II= 0.3) square plate under uniform load.

8.0

* s2 x.

EZ 4.0

- 3-D Elasticity (4x4~1 mesh)

-.- Levy (CPT)

___ SDPT (4x4 mesh)

- 9 Load parameter , P = 9, 4

Fig. 2. Center normal stress vs the load parameter for simply supported isotropic (v = 0.3) square plate under uniform load (rf, is computed at Y = (0.0164, O.OZ64, 0.0789 in three-dimensional elasticity theory and x = (O.M?S,

0.(16X 0.1) in the plate theory).

Three-dimensional nonlinear analysis of cross-ply rectangular composite plates 269

4.0

3.0

",(O,O,# --

h

- 3-D elasticity (2x2~3 mesh)

SDPT (4x4 mesh) t a/h = 10

-._- 3-D elasticity (2x2~3 mesh) (a/h = 4)

-._._ SOPT (4x4) mesh

Load parameter, P = $-- 2

Fig. 3. Load-deflection curves for cross-ply (O’/~~O’, equal thickness layers) square plates under uniformly dis~buted load (El/E? = 25, Glz = Gtl = 0.5 E, Gz, = 0.2 Ez, Y,? = 0.25.

140

100 -

0 s2 ;=-E- - x E2

80 -

60 -

Load parameter, P = F- 2

Fig. 4. Normal stress vs load parameter for three-layer cross-ply (O”/90V’) square plate under uniform loading (S = a/h = IO).

270 T. KUPPUSAMY and J.N. REDDI

0 100 200 300 400 500

Loading parameter, P = A- E2

Fig. 5. Center normal stress vs load parameter for three-layer cross-ply (O”/90”/O”) square plate under uniform load IS = a/h = 4).

and therefore part of the difference in the two solutions exhibited by the three-dimensional elasticity theory is is attributable to the differences in the location of points less than that of the plate theory for load parameter at which the stresses are calculated. values less than 250.

Finally, a two-layer cross-ply (O”/!W) square plate under uniformly distributed load is analyzed and the plots of 5. CONCLUSIONS center deflection and stresses versus the load parameter are presented in Figs. 6 and 7. Note that the nonlinearity

A finite-element analysis of geometrically nonlinear, three-dimensional theory of laminated plates is presen-

3D-Elasticity (4x4~2 mesh)

SDPT (4x4 mesh) i

3D-Elasticity (4x4~2 mesh) i

SDPT (4x4 mesh)

I

qos4 Load parameter, P = T-

2

Fig. 6. Load-deflection curves for two-layer, cross-ply (0”/90”) square plate under uniformly distributed load (E, =25E2, G,z=G,j=O.SEz. Gx=0.2E2. ~,~=0.25),

Three-dimensional nonlinear analysis of cross-ply rectangular composite plates

1

20 - - 3-O Elasticity -

-.- SDPT 1 OX

0 s2 16 _ ._.. SDPT

0=x X

E2

axzs TX, = E 12 - 2

a

Load parameter, p = %I!? E2

Fig. 7. Normal and shear stresses vs load parameter for two-layer. cross-ply (O”/WO, S = 10) square plate under uniformly distributed load (Cs, is computed at x = (0.0625, 0.0625, 0.1) in SDPT; &,, is computed at x = (0.4736,

0.4736, 0.0894) in three-dimensional and at x = (0.4375. 0.4375. 0.1) in SDPT).

271

ted. It is found that the shear deformable plate theory results are fairly accurate when compared to the three- dimensional elasticity theory for side to thickness ratio of 10. The difference between the two solutions is larger for side to thickness ratio of 4. It is also found that the stresses predicted by the plate theory are in larger error than the deflections when compared to those predicted by the three-dimensional elasticity theory. The reduced integration technique is found to have an effect on the accuracy of the solution: reduced integration is recom- mended for thin plates (a/h > 10) and full integration for thick plates (a/h c lo), especially when geometric non- linearities are included.

Ackno&dgements-The work reported herein is supported by the Structural Mechanics Section of NASA/Lewis and the Mechanics Division of the Office of Naval Research. The support is gratefully acknowledged.

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3.

4.

5.

REFERENCES

E. Reissner and Y. Stavsky, Bending and stretching of cer- tain types of heterogeneous aerolotropic elastic plates. J. Appl. Mech. 28.402-408 (1961). P. C. Yang, C. H. Norris and Y. Stavsky, Elastic wave propagation in heterogeneous plates. Int. J. Solids Structures 2. 665-684 (1%6). J. M. Whitney and N. J. Pagano, Shear deformation in heterogeneous anisotropic plates. J. Appl. Mech. 37, 1031- 1036 (1970). J. N. Reddy and W. C. Chao. A comparison of closed-form and finite element solutions of thick laminated anistropic rectangular plates. Nucl. Engng Design 64, 153-167 (1981). J. N. Reddy, A penalty plate-bending element for the analysis of laminated anisotropic composite plates. Inf. J. Num. Meth. Engng 15. 1187-1206 (1980).

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