Structural Safety, 6 (1989) 115-127 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

115

RANDOM RESPONSE OF ANTISYMMETRIC ANGLE-PLY LAMINATED PLATES *

M.P. Singh, A.A. Khdeir, G.O. Maldonado and J.N. Reddy

Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061 (U.S.A.)

Key words: composite plates; laminated plates; random vibration; random response; thick plates; shear deformation; first-order theory; random excitation; random field.

ABSTRACT

The random response of an antisymmetric angle-ply composite plate subjected to random lateral load on its surface is obtained The first-order transverse shear deformation theory, involving five coupled partial differential equations, is used. To evaluate the effect of shear deformation, anisotropy and other problem parameters on the response, the numerical results obtained with this theory are compared with the results obtained by the classical plate theory. Significant differences are observed in the two sets of results obtained by the two theories especially for thick plates, thus demonstrating the importance of shear deformations and need for utilization of plate theories which properly account for such deformations.

1. INTRODUCTION

In high-performance systems, composite plates made with several layers of high-strength lamina are often used. Depending upon the orientation of the fibers in the various layers, the composite plates are often classified as cross-ply or angle-ply laminates with symmetric, antisymmetric or asymmetric layouts. Cross-ply laminates are those with laminae oriented at 0 o and 90 o to the laminate (or plate) axes, whereas angle-ply laminates are made of laminae with orientations other than 0 ° and 90 °. Such plates usually exhibit significant anisotropy. The response of these plates is also significantly affected by the transverse shear deformation, because of their low transverse moduli compared to the in-plane moduli. As a result, the classical plate theory based on Kirchhoff's hypothesis fails to predict the response of such composite plates correctly. To incorporate these effects, therefore, the first and higher order theories have been proposed by Mindlin [1], Yang et al. [2], Reddy [3], etc.

* Paper presented at an International Symposium on Methods of Stochastic Mechanics and Applications, Urbana, IL, U.S.A., October 31-November 1, 1988.

0167-4730/89/$03.50 © 1989 Elsevier Science Publishers B.V.

116

In this paper, we study the random dynamic response characteristics of antisymmetric angle-ply laminates. Such a laminate is obtained when the principal directions of orthotropy of an even number of layers are oriented alternately at angles + 0 and - 0 with respect to the x-axis of the laminate. The dynamic response characteristics of such plates have been examined by several investigators in the past. Whitney and Pagano [4], Bert and Chen [5], and Khdeir [6] have studied the free vibration characteristics and Khdeir and Reddy [7] have examined the dynamic response under arbitrary loading on the plates. Random vibrations of the orthotropic, sandwich and composite plates has also been studied by several investigators. To mention a few, for example, Kulkarni et al. [8], Elishakoff [9], Witt and Soboczyk [10], Mei and Wentz [11], Chonan [12], Mei and Prasad [13], Witt [14], Cederbaum et al. [15].

Here in this paper, we use an exact modal analysis approach to obtain the random dynamic response of simply supported antisymmetric angle-ply plates, subjected to lateral loads distrib- uted randomly in the spatial and time domains. Both the first-order shear deformation theory (FSDT) and classical plate theory (CPT) have been used in the analysis. In the first-order theory, the response is governed by a set of five coupled partial differential equations whereas in the classical theory it is governed by three coupled partial differential equations. Several sets of numerical results have been obtained with these two theories and compared to evaluate the effect of the shear deformation on the response.

2. ANALYTICAL FORMULATION

The equations of motion of an antisymmetric angle-ply laminate based on the first-order shear deformation theory have been developed by Yang et al. [2] and Reddy [3]. These equations are:

02u 02u . ~ 2 U ~ 2~t/x ~ 21py 02~y + +(Al z+A66)~-~ fy+2916~)xOy +916 ~x 2 . . . . + B 2 6 - - (la) I1/i = Al10x 2 A66 0y 2 0y 2

, 02U ~2U ~2U 02~x n ~ 2~tJx A66 ~x 2 A220y2 0x 2 1526 0y2 I1iJ = ( Alz + A66 ) O---~fy + - - + - - + B 1 6 - - + - -

0 2 w O 2 w ~-~--~- ~-~--~- Ilff~=A55~x2 +A44--+A553y 2 +A44 +q(x, y, t)

~2U ~W ~2~ x I3~x 02U 02.___~U + 9 2 6 - - --A55--~- x + D l l - - = 2B16 Ox3y + B16 ~X 2 0y 2 0X 2

~21~y + 2B26 3xOy (lb)

(lc)

] ~2~x 02~Y 02~Y A44+y (le) + (D12 + D66J Ox3y + D660x ------5- + D22 0---~- -

where (u, v, w) are the displacements of a point (x, y, 0) along the x, y and z-directions,

~2u + _ O2_.___uu + 2B26220 3w I3fy = B160x--- ~ ~260y 2 3xOy A44 -~y

02~* (ld) +D66 ~2~___~ _ A55G + (D12 + 066) OxOy 3y2

117

respectively; (q'x, I~y) are the rotations of a transverse normal about the y and x-axes, respectively; a dot over a variable indicates differentiation with respect to time; q(x, y, t) is the transverse pressure on the plate; and/1 and 13 are the mass inertias defined as

(11, 13)= fh/2 (1, zE)p (m) dz (2) .I - h / 2

in which t0 (m) is the mass density of the mth layer, and h is the thickness of the plate. Coefficients A~j, B;j, and Dij are the laminate stiffnesses defined as:

(Aij, B~j, D~j)=fh/2(1, z, z2)Q}~ )dz i, j = 1 , 2 , 6 (3a) ¢ - h / 2

A,j=K2£~I2Q~ 7) dz, i, j = 4 , 5 (3b)

In eqn. (3b) K~. are the transverse shear correction factors, and Q}~) are the plane-stress-re- duced stiffness coefficients of the m th layer.

For a simply supported plate, these equations are associated with the following boundary

-4-B16~xX q - B26~yY = 0

q-D11~x x -k" D12~y y = 0

conditions. A t x = 0 a n d x = a : lu w

N6 =A66 -[- ~X

M1 = B16 5Sy +

At y = 0 a n d y = b :

{ o = w = +( = U6 = + ao + O+v +

M2 = B26 "~y ~x D12 ~ D22 0

(4)

(5)

The solution of eqns. ( la)- ( ld) subject to the boundary conditions (4) and (5), in which the time and spatial dependence can be separated, is possible. The time dependence will, of course, be harmonic if no damping is assumed. Such a separable solution, satisfying the boundary condition of eqns. (4) and (5) is of the following form:

u(x, y, ¢)= E m , n ~ l

v(x, y, t)= m , r t 2 1

w(x, y, t ) = (6) rn,~l ~x(x, y, t ) = m 1 ~y(X, y, t) Y'~

m , n = l

Umn(t)flmn( x, Y)

Vm.(t)f2m.(X, Y)

Wmn(t)f3mn(X, Y)

Xm.(t)fzm.(X, Y)

Ym.(t)fam.(x, Y)

1 1 8

where

I f1 .... (x, y) = sin ax cos fly

f3m.(x, y) sin ax sin fly

o~ = m ¢r/a fl = n ~/b .

f2m, (X, y) : COS aX sin fly

(7)

Substituting eqn. (6) in eqn. (1), multiplying by appropriate fjm,(X, y) functions and integrating over the domain of the plate, one obtains a set of five coupled second-order differential equations of the following form:

[M]{2"} + [ K ] ( z } = {F(t)} (8)

in which vector { z } is defined as:

{ Z } V : ( U m . , Vm., Win., Xm., Ym.) (9)

For transverse loading q(x, y, t), only the third element of the vector { F(t)} is non zero and is defined as

4L70" F3(t ) = ~ q(x, y, t)f3m,(X, y) dx dy (10)

The matrices [M] and [K] are the analogues to the conventional mass and stiffness matrices. They are symmetrical and functions of rn and n. Matrix [M] is also diagonal. Some formulations have been developed which lead to unsymmetric matrices [15]. The variational formulation [3] of the governing equations, however, avoids such undesirable characteristics.

For each m and n eqn. (8) can be solved by any standard technique. If desired, viscous damping can also be included to account for the dissipation of energy in the plate. If the damping is assumed to be of proportional type, normal mode approach can be used. For other forms of damping, the state-vector approach with complex modes can be used. Here in this formulation, for analytical convenience, we will assume classical damping with a constant modal damping ratio in all the modes. Utilizing the eigenvectors of the following eigenvalue problem,

03 2 (11) rn n ,l

eqn. (8) can be decoupled. In eqn. (11), 02ran j2 is an eigenvalue, and ~m.j is the corresponding eigenvector. To decouple eqn. (8), we utilize the following standard transformation,

{z} = [Om.]{ p} (12)

where [~mn] is 5 × 5 modal matrix of eigenvectors { qSj,~n } and { p} is the vector of principal coordinates. After substituting eqn. (12) in (8), we premultiply by d0m,; and then use the orthogonal properties of the eigenvectors to obtain the following decoupled equations for the principal coordinates,

2 p. , . /+ 2fijO:m,up.,,, ] + o:~,j = • .S(3)F3(t) (13)

where the modal damping ratio/31, has now been introduced to include the viscous dissipation of energy. Also q~m.j(3) is the third element of the eigenvector {~m.;}" It is assumed that the eigenvectors { 0r..j } are orthonormalized with respect to the mass matrix in eqn. (8).

119

With the Fourier-Stieltjes integral representation of the lateral loading as: oo

q(x, y, t)= f_ e -i~t ds(x, y, to) oo

(14)

the solution of eqn. (13), for zero initial conditions can be written as:

4dPmnj(3) foafob Pm. j ( f ) - ab f3m"(x' y) e-i'~tHm.j(~o, t) ds(x, y, ~o) (15)

where Hm.j(~0, t) is the transient frequency response function of eqn. (13) for the unit harmonic force e -'~'. For large values of t, it approaches the steady-state frequency response function, defined as:

1 Hmnj( (a)' t )t--*c~ = Hmnj( °° ) = ( Oamn j 2 _ _ 0 ) 2 __ 2ifljOam.,Oa)

(16)

We will now consider the steady-state situation where the load has been acting for a long time on the plate. The lateral deflection of the plate at any point (x 1, Yl) can now be written as

5 a.,Ob f~ Hm.j W(Xl, Yl, t)= E f3mn(Xl, Yl) Y'. Ym.j[ [ f3mn( X, Y) (~o) ds(x, y, ,o) do~ dx dy

m,n j = l " 0

(17)

where

2 (3)/ab (18) Ymn j = 4ep~,. j

Similar expressions can also be written for other response quantities such as fibre stresses or strains in a lamina by substituting appropriate modal response functions in place of f3m.(Xl, Yl) in eqn. (17).

3. RESPONSE COVARIANCE

For a random lateral loading on the plate, one can also obtain the random characteristics of the response [16]. We will assume that the lateral loading is a temporally stationary random field, the random variations of which about its mean value can be characterized by a spectral density function in the frequency domain. One can then show that the stationary cross-correlation function of the displacement values at two points on the plate can be written as:

oo oo 5

Rww(Xl, Yl; x2, Y2; tl, t2 )= £ £f3mn(Xl, Yl)f31p(X2, Y2) £ "~mnj'Ylpk m,n l,p j , k = l

foafobfoafo b X famn(X', y')f3lp(X", y " ) d x ' dy' dx" dy"

f × ~(x ' , y'; x", y"; ~o)e"°(t~-t')H.,.j(co)Ht~k(~o)dto (19) oo

where an asterisk as a superscript denotes a complex conjugate and ~ ( x ' , y ' ; x ' , y" , ~0) is the

120

cross-spectral density function of the loadings at points (x' , y ' ) and (x", y" ) of the plate. This is related to ds(x, y, 0)) as:

E[ds(x' , y', c%) d s * ( x " , y" , 0)2) ] = (I)(x ', yt; X", y" , 0)1)~(0)1- 0)2) do), do) 2 (20)

Again, expressions similar to eqn. (19) can be obtained for the correlation functions of other response quantities by a suitable replacement of functions f3m,(Xl, Yl) in the first set of summations in eqn. (19).

Equation (19) has been utilized to obtain the standard deviation of displacement at the center of a square plate for different parameters of the random pressure field. To demonstrate the effect of the shear deformation and rotatory inertia, the results obtained by (FSDT) are also compared with the results obtained by the CPT. Based on Kirchhoff's hypothesis, the governing equations of motion for the CPT are as follows:

03W 03W 3B16 - - 9 2 6 - - (21a) Ox20y Oy 3

11fft=A11 02u - 02__uu+(A12+A66) 02u 0x ---'-~" q'- A66 0y 2 OxOy

. 02U 02U _+_A2202__.~U 03w 3926 03w 11i5 = (A12 + A66 ) ~ -'1- A66 0x------ ~" 3y2 916 3x----- S - 3xSy----- 2

_ 03U 03U 03U - - + - - + 9 1 6 - - +3B26 926 0y 3 0X 3 3xSy

I {02# 02#] 33u I l l 0 - 31 0x 2 -4- 0Y 2 ] = 3B16 0x--~y

(21b)

- D l l 04w 2(D12 + 2D66 ) 04W 04--W + q (21C) OX 4 Ox20y ~ D22 0y4

With the following boundary conditions at the simply supported edges: At x = 0 a n d a:

02---'W = 0 _ _ __ 926 0y2

02"----~WW = 0 _ _ __ D12 0y2

u=w=O

( OU OU -~X 02Wox 2 N6 = A66 77.. q- - B16

(0U 0O ) 02W M1 = B16 ~ q- ~ - Dll 0x 2

(22)

At y = 0 a n d b:

v = w = N 6 = 0

M2 = B 2 6 ( ~ + -~x0V J -- D 1 2 - - _ O2w 32w (23) 3x 2 D22 3Y 2 - 0

The solution of eqns. (21) in which time and spatial dependence can be separated will also be of the same form as in eqns. (6). Of course, only the first three of eqns. (6) are necessary. For each set of (m, n) values, one obtains equations similar to eqns. (8), but of order 3. These equations can again be decoupled by the eigenvectors of their associated eigenvalue problem. The subsequent response equations are identical to the ones formulated earlier. Response covariance is still given by eqn. (19) with the exception that the maximum value of the indices ( j , k) is only 3 instead of 5.

121

4. NUMERICAL RESULTS

Several sets of response results are reported for a square plate of 20" × 20", analyzed by the FSDT and CPT to show the effect of the shear deformation. All laminae have the same material properties, but their angular orientations are arranged such that an antisymmetric arrangement is obtained. Such an arrangement will necessarily consist of even laminae in the plate. The modulus of elasticity in the direction perpendicular to the fibers is taken as E 2 = 1.0 × 1 0 6 psi, whereas for the modulus in the fiber direction several values are considered, with E1/E 2 ratio varying between 5 and 60. Similarly several values of the side-to-thickness ratio, a/h, have been considered. In most cases, the value of shear correction factor, K ~ = K~5 has been taken to be 5/6. However, since this factor is not exactly equal to 5 /6 (a value most commonly used), the results with shear factor values varying between 0.7 and 0.9 have also been obtained. Further- more, the effect of varying the number of layers as well as fiber orientation on the response has also been examined. Other material properties are taken as follows: Poisson's ratio, ~'12 = 0.25; mass density, to = 0.0012 lb s 2 / i n 4 ; shear moduli, G12 = G13 = 0 . 6 E 2 and G23 = 0 . 5 E 2.

The temporal correlation has been assumed to be exponential, and independent of the spatial distribution. For the spatial correlation, three different models: (1) perfectly correlated, (2) delta correlated, and (3) exponentially correlated, have been considered. These three different models of the cross-correlation function Rqq = E[q(x' , y ' , t l )q(x", y" , t2) ] are as follows:

(a) Perfectly correlated model:

R q q = q o e - a l t 2 - q l

(b) Delta correlated model:

(24)

Rqq = qoS(X "' - x ' ) 8 ( y " - y ' ) e -~1,2-,,I (25)

(c) Exponentially correlated model:

R q q = qo e-axlX"-x'l-aylY"-y'l-a[t2-tl[ (26)

The parameters a, a,, and ay are the parameters of the model. Large values of these parameters imply only very localized correlations. A wide range of values have been considered for the temporal correlation parameter a. The parameters a x and ay were assumed to be equal but two different values, 0.1 and 0.3, were used. The pressure intensity parameter has been taken as 500 psi. The response results are, of course, linearly related to this value. In all cases, the results reported are for the standard deviation of the deflection at the center of the plate. For various correlation models, this standard derivation is obtained from eqn. (19), by setting t I ----- t 2

and x 1 =Yl = X2 =Y2 = a/2, as follows:

oO oo

1 1 z q°a E Ef3m,(½a, ~a)f3tp(~a, ½a)l,,,L e 0 w ~ m,n I,p

5

f ~ . do~ X j,k=lE %,,.jY#,k -ooH""J(~°)Ht" k(*°) (e2 + a2) (27)

122

0.30

0.25

" - ' 0 . 2 0 z * * * * * FIRST ORDER THEORY o ~ , • " , CLASSICAL THEORY

~ 0 . 1 5

IlC ,,( CI

~ 0.10

0 .05

-a . . . . . . , . . . . . . . . . . . . . . . . . . ;"1, . . . . . ;?, ; ' " ; ' ~ ,

o.oo 1o 1o-' lo - ' 1 ~_ ~ ' ~ ; . . . . . . _

TIME CORRELATION PARAMETER

Fig. 1. Deflection standard deviation obtained by the FSDT and CPT versus temporal correlation parameter c~. Number of layers = 4, a/h = 10, E~/E 2 = 40, 0 = 45 o, % = % = 0.3.

in which lml for the three spatial correlation models is defined as:

(a) Perfectly correlated load model:

Imt= 1 4aZ/'rtZml' if m, l = 1, 3, 5 . . . . 0, otherwise

(b) Delta correlated load model:

=t½a' ifm=l Ira/

0, otherwise

(c) Exponentially correlated load model:

i , . % a { 1 1 } - - - - 2 + 0 ( 2 ) -~- (~ml-[- - - 2 (0(x +

where 0£m and a I are defined in eqn. (7).

,rr2ml 2 + e - ( ~ a ) [ ( - 1 ) r e + l + ( - 1 ) '+1]

a 2 2 2 (0£2 ..~ 0£m)( 0£x "31- 0£2)

(28)

(29)

(30)

The expressions for J,p are similar, except that a is replaced by b, % is replaced by 0(y and am and a t are replaced by ft, and tip, respectively. The latter parameters are defined in eqn. (7).

Figure 1 shows the standard deviation of the deflection obtained by the FSDT and CPT for different values of the temporal correlation parameter. The both spatial correlation parameter values were taken to be 0.3. The other parameters of the problem are also indicated in the title. It is rather interesting to note that a high value of 0£ --- 10000 produced the highest response. Figure 2 shows the percent difference (error) in the responses obtained by the two theories (FSDT and CPT). This error is defined as the ratio, expressed in percent, of the difference of the responses obtained by the two theories divided by the result of FSDT. For the length-to-thickness ratio, a/h = 10, this difference is quite large. In Fig. 3, the errors in results obtained by the CPT, a vis-a-vis the results obtained by the FSDT, are plotted versus the a/h ratio for plates with 2, 4,

123

44

a: 40

32

TIME CORRELATION PARAMETER

Fig. 2. Pe r cen t e r ro r ve r su s t e m p o r a l co r r e l a t i on p a r a m e t e r a . N u m b e r o f l ayers = 4, a / h = 10, E 1 / E 2 = 40, O = 45 o,

a x = a y = 0.3.

10 and 20 layers. Three graphs show that for higher a/h ratios, this error, of course, becomes small, but not insignificant even for large values. This demonstrates the effect of the shear

a deformation. Figure 3 also shows the results for plates with ~ = 4. Of course, plates with such

small ratios may not be found in practice. But if a fine mesh is used in a finite element analysis,

8 0 ¸

20 ¸

0

~ 6O ̧ k d

Z ~ J

t~ 40 ¸

2.,w

0 ~ i l l , , = l l ,

100.

f-lO,20

, , l i l l J l l , , l J i i l l ,

0 10 20 30 4.0 LENGTH/THICKNESS RATIO

Fig. 3. Pe r cen t e r ro r in the p l a t e s w i t h 2, 4, 10 a n d 20 layers v e r s u s l e n g t h - t o - t h i c k n e s s ra t io , ct = 10000 , E 1 / E 2 = 40,

O = 4 5 o , a x = Oly = 0.3

124

4 3 .

42

41

n,."

0 40 r e "

W

~ 3 9 W

ht a. 3 8

37 ̧

36 ̧

35 0.60

\

\ \

\

\

0.65 0.70 0.75 0.80 0.85 0.90 SHEAR FACTOR

F i g . 4. Percent error versus shear correction parameter. Number of layers = 4, a = 1 0 0 0 0 , E]/E 2 = 40, 0 = 45 ° ,

a x = gty = 0 . 3 .

the localized small a/h ratios of the finite elements may have some significance in the calculation of local response.

Figure 4 shows the error when the shear correction factor is changed. Low shear correction factor values are associated with higher differences (thus larger error) in the results obtained by the two theories.

1.6

~0.8 rm

fY

C~ Z

toO.4

~ ~ FIRST OF DER THI;ORY _ _ _ CLASSIC# L THEOF Y

0.0 0 I 0 20 30 40 50 60

El/E2 RATIO

Fig . 5. Deflection standard deviation obtained by the FSDT and CPT versus E ] / E 2 ratio. Number of layers = 4,

a/h=lO, / 9 = 4 5 ° , Otx=Oty=0.3.

50.

125

40.

n," O

z z 30. bJ

F- Z W

n~ ',' 20 O_

10 ¸

0 0

/ /

/ /

/ /

/ /

10 20 30 40 50 60 E1/E2 RATIO

F ig . 6. Percent error versus E ] / E 2 ratio. N u m b e r of layers = 4, a/h = 10, 0 = 45 o a = 10000 , (x x = ay = 0.3.

Figure 5 demonstrates the effect of E1/E 2 ratio on the deflection response obtained by the two theories. The corresponding difference in the results obtained by the two theories, plotted as error, is shown in Fig. 6. The error is seen to increase with the E 1 / E 2 ratio in Fig. 6.

Figures 7 and 8, respectively, show the variation in standard deviation of the deflection and the corresponding difference between the results obtained by the two theories as the ply angle is changed. The deflection is seen to decrease with an increase in the ply angle. The error in the

0.40

0,35

' ' 0 . 3 0 Z O

~ 0 . 2 5

z 0.20

0.15

\

"• = = • = . IRST ORD[ R THEOR~ • , , , ' , ;LASSICAL I'HEORY

-. . . .

0.10 0 10 20 30 40 50

PLY-ANGLE

Fig. 7. Deflection standard deviation obtained by the FSDT and CPT versus ply-angle. N u m b e r of layers = 4,

a/h = 1 0 , E 1 / E 2 = 40, a = 1 0 0 0 0 , a x = ay = 0.3.

126

39,

38 .:

rv 37.:

; 385

o. 35-:

34 -:

/ \ /

f / J

33--" 0 10 20 30 40 50

PLY-ANGLE

Fig. 8. Percent error versus ply angle. Number of layers = 4, a / h = 10, E 1 / E 2 = 40, a = 10000, a x = a v = 0.3.

results obtained by the CPT first decreases as the angle is increased till a value of about 18 °, but thereafter it is seen to increase again. The deflection and error values are not shown for angles greater than 45 o, as the curves in Figs. 7 and 8 are symmetrical about the 45 o degree line.

Results similar to those described above were also obtained for other values of the spatial correlation parameters as well as perfectly correlated and delta correlated pressure fields. They are not shown here for lack of space. But, qualitatively these results were very similar to the ones presented above. Also the results for other response quantities and the level crossing rates and peak response characteristics, though not obtained here, will show similar characteristics.

5. CONCLUDING REMARKS

The results of the parametric study presented here clearly demonstrate that the transverse shear deformation plays an important role in the dynamic response of laminated plates. The error in the response calculated by the Kirchhoff's (i.e., classical) plate theory can be significant, even in moderately thick composite plates. Thus, utilization of at least the first-order shear deformation plate theory in vibration analyses of these plates is advocated.

REFERENCES

1 R.D. Mindlin, Influence of rotary inertia and shear on flexural motion of isotropic elastic plate, J. Appl. Mech, 18 (1951) 31-38.

2 P.C. Yang, C.H. Norris and Y. Stavsky, Elastic wave propagation in heterogeneous plates, Int. J. Solids Struct., 2 (1985) 665-684.

3 J.N. Reddy, Energy and Variational Methods in Applied Mechanics, Wiley, New York, 1984. 4 J.M. Whitney and N.J. Pagano, Shear deformation in heterogeneous anisotropic plates, J. Appl. Mech., 37 (1970)

1031-1036.

127

5 C.W. Bert and T.L.C. Chen, Effect of shear deformation on vibration of antisymmetric angle-ply laminated rectangular plates, Int. J. Solids Struct., 14 (1978) 465-473.

6 A.A. Khdeir, Free vibration of antisymmetric angle-ply laminated plates including various boundary conditions, J. Sound Vib., 122(2) (1988) 377-388.

7 A.A. Khdeir and J.N. Reddy, Dynamic response of antisymmetric angle-ply laminated plates subjected to arbitrary loading, J. Sound Vib., 126(3) (1988) 437-445.

8 A.M. Kulkarni, J.R. Banerjee and P.K. Sinha, Response of randomly excited orthotropic sandwich plates, J. Sound Vib., 41(2) (1975) 197-205.

9 I. Elishakoff, Random vibrations of orthotropic plates clamped or simply supported all round, Acta Mech., 28 (1977) 165-176.

10 M. Witt and K. Soboczyk, Dynamic response of laminated plates to random loading, Int. J. Solids Struct., 16 (1980) 231-238.

11 C. Mei and K.R. Wentz, Large-amplitude random response of angle-ply laminated composite plates, AIAA J., 20(10) (1982) 1450-1458.

12 S. Chonan, Random vibration of a prestressed, orthotropic, thick rectangular plate on a generalized flexible foundation, J. Acoust. Soc. Am., 78(2) (1985) 598-604.

13 C. Mei and C.B. Prasad, Effects of large deflection and transverse shear on response of rectangular symmetric composite laminates subjected to acoustic excitation, AIAA Dynamics Specialists Conference, Monterey, CA, 1987, pp. 809-826.

14 M. Witt, Dynamic response and reliability of laminated plates to random loading, in: Refined Dynamical Theories of Beams, Plates and Shells and Their Applications, Springer-Verlag, Berlin, 1987, pp. 274-285.

15 G. Cederbaum, L. Librescu and I. Elishakoff, Random vibration of laminated plates modeled within a high-order shear deformation theory, J. Acoust. Soc. Am., 84(2) (1988) 660-666.

16 Y.K. Lin, Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, 1967.