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A REFINED NONLINEAR THEORY OF PLATES WITH TRANSVERSE SHEAR DEFORMATION

J. N. &DDY Dcpar:ment of Engineering Science and Mechanics, Virginia Polytechnic Institute and

State University, Blacksburg, VA 24061, U.S.A.

(Received 1 August 1983; in revisedfonn 17 November 1983)

M-A higher-order shear deformation theory of plates accounting for the von Karman strains is presented. The theory contains the same dependent unknowns as in the Hcncky-Mindlin type firstorder shear deformation theory and accounts for parabolic distribution of the transverse shear strains through the thickness of the plate. Exact solutions of simply supported plates are obtained using the linear theory and the resulta arc compared with the exact solutions of 3-D &sticity theory, the first order shear dcfonnation theory, and the classical plate theory. The present theory predii the ddlections, stresses, and frquncics more accurately when compared to the 6rstorda theory and the classical plate theory.

INTRODUCTION

First general solutions to the equations of linear elasticity corresponding to thin plates were presented by Cauchy [ l] and Poisson [2] using the methods of series expansion, and by KirchhoffI31 using certain hypothesis. An expansion in powers of the thickness of the plate was used by Goodier[4] to obtain a general solution in terms of a series of biharmonic functions for a plate subjected to edge tractions. It is well known from experimental observations that the Poisson-Kirchhoff theory of plates, in which it is assumed that normals to the midplane before deformation remain straight and normal to the plane after defor- mation, underpredicts deflections and overpredicts natural frequencies. These results are due to the neglect of transverse shear strains in the classical plate theory (CPT).

Refined plate theories, due to Levy [5j, Reissner[6,7], Hencky[8], Mindlin[9], and Kromm [ lo] are improvements of the classical plate theory in that they include the effect of transverse shear deformation (see[ 111). In the Hencky-Mindlin theories the displacements are expanded in powers of the thickness of the plate (see[12-141). Extensions of the Kirchhoff-von Karman theory[l5], a geometrically nonlinear theory associated with the classical plate theory, to refined plate theories were considered by Reissner[l6,17] and Medwadowski[l8]. Extension of the Kromm’s theory to geometrically nonlinear analysis, in the sense of von Karman, is due to Schmidt[l9].

These higher-order theories are cumbersome and computationally more demanding, because, with each additional power of the thickness coordinate, an additional dependent unknown is introduced into the theory. Further, these theories require an arbitrary cor- rection to the transverseshear stiffnesses, and the transverse shear stresses do not satisfy the conditions of zero transverse shear stresses on the top and bottom surfaces of the plate. Of course, the Reissner-Kromm theories satisfy the stress free conditions, but these are based on the stress fields. Thus, need exists for the development of a higher-order shear defor- mation theory that avoids the shear correction factors, and accurately predicts transverse shear stresses. Levinson [20] considered such a plate theory, in which the in-plane displace- ments are expanded as cubic functions of the thickness coordinate. Unfortunately, both Levinson[20] and Schmidt[l9] used variationally inconsistent set of equilibrium equations (they used the equilibrium equations of the classical plate theory), and therefore did not correctly account for all of the strain energy associated with the displacement field.

The present theory accounts for the cubic variation of the in-plane displacements through the plate thickness, the van Karman strains, and transverse shear strains which vanish on the top and bottom faces of the plate. The equations of motion are derived using Hamilton’s principle, and therefore they are consistent with the assumed displacement field. In order to illustrate the accuracy of the present theory, the exact solutions of the linear

881

882 J. N. REDDY

theory are presented for bending and vibration of simply supported, homogeneous, iso- tropic and orthotropic rectangular plates. Comparison of the present solutions with the 3D elasticity solutions shows that the present theory yields more accurate stresses and natural frequencies than the first-order shear deformation theory.

KINEMATICS

We begin with the displacement field in which the displacements along the X- and y-directions are expanded as cubic functions of the thickness coordinate, and the transverse deflection is assumed to be constant through thickness:

h(X, Y9 z,t) = 4x9 Y, 0 + z$,(x, Y, 0 + z2t,(x, Y9 0 + &(x, Y, 0

u2(x, Y, f, 0 = ex, Y, I) + z$y,(x, Y, t) + z’t;<x, Y, 0 + z3C@,y, 2)

a, Y9 0 = 4% Y, 0. (1)

Here U, o, and w denote the displacements of a point (x, y) on the midplane, and I/, and t,G,, are the rotations of normals to midplane about they and x axes, respectively. The functions {,, c,, 6, and c,, will be determined using the condition that the transverse shear stresses, a, = a, and a,, = a, vanish on the plate top and bottom surfaces:

s,(x,Y, *g,r>=o, ,(x., *$)=o (2)

these conditions are equivalent to the requirement that the corresponding strains be zero on these surfaces. We have

au2 au, 4 = 5 + aY - = l)Y + 2z& + 3z25, + 5.

Setting L~(x, y, f h/2, t) and c4(x, y, f h/2, I) to zero, we obtain

The displacement field in eqn (1) becomes

u, =u +z[S,-;($(ll.+$)]

u2=v +z~y-~(;>‘(~y+3] us = w.

(3)

(4)

(9

One should note that, although cubic variation of the in-plane displacements through thickness is accounted, the displacement field in qn (5) contains the same number of dependent variables as in the first-order shear deformation theory. This is an attractive feature from finite-element modeling considerations.

A refined nonlinear theory of p&es with transverse shear deformation

The von Karman strains associated with the displacement field in qn (5) are

883

where

It is interesting to note that the new strain components contain higher-order derivatives of the transverse detbtion.

CONSTITLJTIVE EQUATIONS

For a plate of constant thickness h and made of an orthotropic material (i.e. the plate possesses a plane of elastic symmetry parallel to the x-y plane) the constitutive equations can be written as

where Q, are the plane-stress reduced elastic constants in the material axes of the plate:

Q,,= ” 1 - h2V21 ,Q,,= 1 - vi;‘v 12 21 ,Qu= 1 - 7, 12 21

Qu = G23, Qs = (53, Qei = G12.

(W

EQUATIONS OF MOTION Here we use Haminton’s principle to derive the quations of motion appropriate for

the dispIacement field (5) and constitutive equations (8). The principle can be stated in analytical form as (see Reddy and Rasmussen[21])

J. N. REDDY

(a&, + a&, + a&, + a&, + a&,) dA dcz + b ]

q&v dx dy dr

p [(li,)’ + (iJ2 + (ii,)? dA dz dr

I

=- S[f-I ( N ah+awasw

R

-- +M,Z+P 1 ax ax ax

+k )

)

,[-g$+$)]

ah %W + M astiy 2 7 + ay ay 2T+P2[-g3+t$)]

where the stress resultants N,, M,, P,, Qi and R, are defined by

W,,M,P,)= ” s 0x1, z, 9) dz(i = 1,2,6) -AD

s ” (Q,, 4) = a,& 23 dz

-W

and the inertias I,Ji = 1,2,3,4,5,7) are defined by

h/2 K,12,4,4,I5,&) =

s

~(1, z, z2, z3, z’, z”) dz. -W

(9)

(10)

(11)

A r&cd nonlinear theory of plates with transverse shear deformation 88s

Integrating the expressions in eqn (9) by parts, and collecting the coefficients of SU, 60, SW, S$,, and S$,, we obtain the following equations of motion:

where

& = 12 - $ I,, & = I, - & I,,

r, = I,

The boundary conditions are of the form: specify

un or N,, \

uns or Nms

w or Qn

dW an or Pn 1 on r

where r is the boundary of the midplane f2 of the plate, and

(13)

(14)

886 J. N. &DDY

Qi = Qi -$Ri(i= 1,2)

a - = n, an

and P, and P,,, are defined by expressions analogous to N. and N,,,, respectively. This completes the derivation of the governing equations. An examination of the boundary conditions in qn (14) shows that both $” and awlan are geometric boundary conditions in the present theory. Consequently, one should use interpolation functions that guarantee interelement continuity of slopes in the finite-element modeling of the theory.

The resultants defined in qn (10) can be related to the total strains in qn (6) by the following equations:

=

& D,, 0 .

D, 0

sym. Dbb.

sym-

where A, Bi/, etc. are the plate stiffnesses, defined by

(Ati,D,,Fv,H,,)= J ~~Qy(llz2,1’,z6)dz(i,j= 1,2,6)

(A,,D&)= @+Q&,z2,z4)dz(iJ=4,5) J or

A,= Q, D, = Q&h3/W

FM = Q&h5/80), Hy = Q&h’/448)

A,, = G,h, AS5 = G13h

D ,, = G,(h3/12), D55 = G,,(h3/12)

FM = Gu(h5/80), F,, = G,,(h5/80).

(17)

(18)

A rcfhcd nonlinear theory of plates with transverse shear deformation aal

EXACT SOLUTIONS FOR SIMPLY SUPPORTED RECTANGULAR PLATES

The exact analytical solution of the nonlinear partial differential equations in eqn (12) is an impossible task. Even the linear equations do not allow an exact solution for all geometries and boundary conditions. Here we consider the exact solutions of eqns (12) and (13) for infinitesimal displacement theory of simply supported, rectangular plates. Since the coupling between stretching and bending is zero for the linear theory, we consider only the flexural displacements and natural frequencies. The following “simply-supported” bound- ary conditions are assumed (a and b are the plane-form dimensions and the origin of the coordinate system is taken at the lower left comer of the plate):

w(x, 0) = w(x, b) = w(O,y) = w(u,y) = 0

4(x, 0) = &k b) = P,(O, y) = P&Y) = 0

wx, 0) = K(x, fJ) = M,(O, Y) = M,(u, Y) = 0 (19)

$,(x9 0) = $xk b I= $yKJ, Y) = !byL,(u, Y) = 0.

The resultants of eqn ( 16) can be expressed in terms of the generalixed displacements, for the case of infinitesimal displacements, as

au N2=All- ax+A,afr

8.Y

(20)

888 J. N. RilDDY

Ttie last three ~Uatio~s in eqn (12), for the linear theory, am be expressed in term of the displacements as

( )(

@Y ; d’w +H,* -4 +4,$x

3 x

3h2 a&y &%y” 2

-$[D,,($+~)+F,( -~)(~+~)

+D,,&+t$)+FI( -;)(f$+$)]

+~ss~~+$)+D5s(-;)(~+$)

w4

+&2( -$($+$)+bs($+~)

++($+k)+&( -;)(h+g)]

WI

A refined nonlinear theory of plates with transverse shear deformation

-_ jj *JIy -- at2

WC)

Following the Navier solution procedure, we assume the following solution form that satisfies the boundary conditions in eqn (19),

w= f W, sin ax sin fly eVrw’ m,n-I

$, = f X, cos ax sin /3y e+ ma-I

(22)

$, = i Y, sin QT cos fly e+ nsn-1

where a = mn/a and /3 = m/b, and.o is the frequency of’the natural vibration. Further we assume that the applied transverse load, q, can be expanded in the double-Fourier series as

q = f Q,,,, sin a~ sin /3y. m.n-I

(23)

Substituting eqns (22) and (23) into eqn (21), and collecting the coefficients, we obtain

(24)

the coefficient

890

matrices [c] and [M] are given by

1. N. &DDY

c,, = u*A,, + p2A,., - J (a*D,, + B’Dlo

*[aw,, + 2(H,* + 2&)a*j3’ + jfPHJ

8 Cl2 = aA,, - - aD,, +

h*

2icr)KI +ab2W,2+ %611

c,= Ass + a*D,, + f12De 4 *

-+,s+ s;z

0 4,

- $ (a2FlI + 8*Fd + ( >

T$ *(a2Hll + P2H66)

c2) = -&F,,+F&+ 3h2 ($Hn + Hd]

4 * c33 = A,fa2D,+#12Dn-SD,,+ S;I

0 FM

- $ (a2Fas + f12F2*) + ( >

$ *(j2Hu + a*Hd

M II= &+I7 gj (>

4 (a*+fl*)

M 12 =

Ai,= 4, lu33=r3, A&=0.

For static bending eqn (24) takes the form

and for free vibration we have

(25)

(26)

(27)

A rclincd nonlincar theory of plates with transverse shear deformation 891

where

CII = A MzG13 + %J + z [a4Qll + W% + Q12)a2P2 + B’Qil

= s haGI c12 15 - g b3Q,, + a~2(2G12 + Qdl

~13 = $ WG3 - $ b28W12 + Q12> + B3Qil

cz=~hG13+~ a “” ( 2Q,, + /J2Gd

c23 =g(G,,+Q,,)

c~~=$IzG=+?~~ a 17/r’ ( 2G,2 -t P2Qu)

and Qi, are given by eqn (8b). The static solution is given by eqn (22) with z = 0 and (IV, X, Y) from eqn (26). Note

that for uniformly distributed load Q,,,,, is given by

7,m,n=l,3 ,...

O,m,n ==2,4,... (28)

Bending

NUMERICAL RESULTS

Numerical results are presented in Tables 1 and 2 for homogeneous isotropic (v = 0.3) and orthotropic plates under uniformly distributed transverse load of intensity qo. The

Table I. Comparison of deflections and stresses in isotropic (v = 0.3) plates under uniformly distributed transverse load (m,n = 1,2,. . e ,19)

bl4 - r I

L 2

-

r/h

5

10

1W

CPT

--

5

10

100

CPT

HSDT 0.0535 0.2944

FSDT 0.0536 ';.:;;;I+ .

HSDT 0.0467 0.2690

FSOT 0.0467 ';.;;;;' .

WSDT 0.0444 0.2873 FSDT 0.444 ';.gl

0.0444 0.2673

0.4640 (0.4871) (0.3324)

0.2890 y;;’

.

0.2673 ';.::;:I

0.4690 0.4543 ';.::;;I

0.1947 ';.;;;;I

.

.~.. 0.3926 ‘0.4909'

D.469U 0.4543 '"0.;;;;' ';.:;;;I

0.4909 0.4905 0.4905 (0.4959) 0.4909 (0.4965)

0.2813 0.1946 0.0 0.4909 (0.4965)

HSDT 0.1248 0.6202 0.2927 0.6745 0.5201 0.5615 FSDT 0.1246 0.6100 z:: . 0.2769 0.5451 0.4192 0.6813 %B"E .

HSDT 0.1142 0.6125 0.2769 0.2809 0.6794 0.5230 0.6446 0.5051 FSDT 0.1142 0.6lOD 0.2779 0.2769 0.5451 0.4192 0.6613 0.5240

HSDT 0.1106 0.6100 0.2779 0.2769 0.6813 0.5240 0.6609 FSDT O.llW 0.6100 0.2779 0.2769 0.5451 0.4192 0.6813 L%: .

0.1106 0.61W 0.2779 0.2769 0.0 0.0 0.6813 0.5240

+nu&err in pwcnthesis nrc obtained by n." = 1.3,....29 in the Series Of Eq. (22).

892 J.N. REDDY

Table 2. Comparison of dcikctions and stresses in orthotropic square plates under uniform transverse load (m, n = I, 3,. . . (19)

cllwh, 4’90 $hoW

b/a h/a Exactt HSDPT FSUPT CPT EXdCC HSDPT FSOPT CPT Exrct HSOPT FSOPT CPT

0.05 2

0.10

0.14

0.05

1 0.10

0.14

0.05

0.5 0.10

0.14

21.542 21,542 21.542 21.210 262.67 262.6 262.0 262.2

1,40R.5 1,408.S 1.408.5 1.326 65.975 65.95 65.38 65.55

387.23 387.5 3Rl.B 345.1 33.R62 33.84 33.21 33.44

10.443 10450. 10450. 10250. 144.31 144.3 143.9 144.4

688.57 689.5 689.6 640.7 36.021 36.01 35.62 36.09

191.07 191.6 191.6 166.R 18.346 18.34 11.94 18.41

2.048.7 2051.0 2051.0 19R9.0 40.657 40.67 40.50 40.04

139.08 139.8 139.8 124.3 10.025 lo.n5 9.m 10.21

39.790 40.21 40.23 32.36 5.036 5.068 4.903 5.209

14.048 13.98 11.20 (13.57) 114.00) yy) 6.921 6.958 (6.229) (6.998) I";;;"' 4.8lB 4.944 (4.n27) (4.997) (i.999)

10.873 10.85 8.701 (10.45) (1n.w (ln.88) 5.341 5.382 4.338 (4.657) ;5i;:2) ;5;;;2' 3.731 (2.884) (j.857) (;.R87)

6.243 6.163 4.940 (5.765) ;"(;;r' ~6;;:"' 2.951 I';$" ;ioJ$4) $A:"

(i.186) (i.131) G.219)

II .n

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

- __ *numbers in parenthcsls denote the sheer stress vrlues obtained from the stress cquilibrim cqurtions.

*from Reference 1223

following orthotropic material properties, typical of aragonite crystals (converted from elastic constants given in[22] to engineering constants), are used.

El = 20.83 x lo6 psi, Ez = 10.94 x 106 psi

G12 = 6.10 x lo6 psi, GIJ = 3.71 x 106 psi, G2) = 6.19 x lob psi

VI2 = 0.44, y21 = 0.23. (29)

The elastic constant cl1 used in Table 2 has the value of 23.2 x 106psi. The following nondimensionalized deflections and stresses are tabulated in Table 1:

ui - - - 0 (h2/w2), i=1,2

(h2/qoo2) (30)

-- c4 - a4

(uw).

Two pairs of transverse shear stresses, one obtained from the constitutive equations and the other from equilibrium equations are presented in the tables. In the first-order shear deformation theory, the shear correction factors are assumed to be L,’ = K: = 5/6. The following conclusions can be drawn from the results of Tables 1 and 2:

(1) Even for the isotropic plates the effect of transverse shear deformation is significant. The classical plate theory (CPT) under predicts (for a/b = I) the deflections by 4.9% at a/h = 10 and 170/, at a/h = 5, and stress u, by 0.7% at a/h = 10 and 2.6% at a/h = 5 when compared to the higher-order shear deformation plate theory (HSDPT); see Table 1.

A refined nonlinear theory of plates with transverse shear deformation 893

_._

-*_

HSDPT from constitutlve equations HSDPT from equilibrium equat fans

FSDPT frun constitutfve equations

r h ,s I

.3

.l

0 -. 1 XL

-. 3

-. 5

Fig. 1. Distribution of the tranmnrc shear strw across the thicti of simply supported mctangular plrtcs under uniformly distributad transverse load (orthotropic case).

(2) The first-order shear deformation plate theory (FSDPT) is quite accurate when the transverse de&&ions are concerned. But the stresses are no better than those predicted by CPT; see Tables 1 and 2.

(3) The transverse shear stresses predicted by the constitutive equations (8) of the higher-order theory are the most accurate of the three plate theories when compared to the exact solution of Srinivas and Rao[22]; see Table 2 and Fig. 1. It is interesting to note that FSDPT and CPT give more accurate transverse shear stresses than HSDPT when the stress equilibrium equations of 3-D elasticity theory are used:

etc. (see Appendix 1). (4) The infinite series for deflections converges faster than those for stresses; the

convergence is slower for thick plates than for thin plates; see Table 1. In summary, the higher-order theory yields more accurate distribution of stresses, especially

shear stmsses, when compared to the other plate theories. This feature of the higher-order theory is of umsiderabk interest in the analysis of laminated composite plates, because an accurate prediction of the interlaminar shear stresses enables an accurate determination of the strength and failure of laminates.

Natural vibration The numerical results of the natural vibration of isotropic (Y = 0.3) and orthotropic (see

eqn (29) for material properties) square plates are presented in Tabks 3 and 4, resp. The results are compared with the exact solutions of the thmedimensional elasticity theory[22-24). In Table 4 the first three eigenvalues obtained by the present theory are compared with the exact values, and the values obtained by FSDPT and CPT. From the results presented in Tabks 3 and 4 the following observations can be made:

894 J. N. hDDY

Table 3. Comparison of natural frequencies, B = ~(~), of isotropic (V = 0.3) p&cs

l- -r

--

-

n

1

2

2

3

3

.i

3

4

4

5

5

4

5

-

--- (a/h = IO) -

n

-

1

2

1

3

2

3

4

1

2

4

3

5

5

-

_- b/a = f1

Exact HSDPT FSDPT CPT 1243

b/a = I HSDPl FSDPT

cPT I 111

0.0932

0.226

0.3421

0.4171

0.5239

0.3732 2 (0.3853) 0.4629 1 (0.4816) ! 0.3951 2 [0.6261) 0.7668 2

0.6389

0.7511

0.9268

! 1.0889

4.0931 0.093u

0.2222 0.2219

0.3411 0.3406

0.4158 0.4149

D.5PPi 0.52C6

0.6545 0.6520

0.6862 0.6134

0.7481 0.7446

0.8949 0.8896

0.9230 0.9174

2.0053 0.9984

I.D84? 1.W64

1.1361 1.1268 i

0.704 0.7038 0.7436

I.376 1.373a 1.3729

2.018 2.0141 2.0123

2.431 2.4263 2.4235

2.634 2.6283 2.6250

3.612 3.6013 3.594%

3.RO9 3.1891 3.7818

3.987 3.9148 3.9666

4.535 .4.51>8 4.5089

4890 4.8737 4.8600

5.411 5.3915 5.3754

5.411 5.3915 5.3754

6.449 6.3846 6.3609

----- ____-_- +,,"mbers (,, p4r~,,thqsI~ denote n4turrl frcqucncics ohtdlned by Omitting tho rOt4tWY inerti&

0.7180 10.7224) 1.4273 (1.4448) 2.1281 (il.lGIl) 2.5908 12.648?) i.8247 (2.8895t 3.9575 (4.0935) 4.1822 (4.3343) 4.4062 f4.57Sl) 5.0729 'p;:'

'p;",'

(6:5014) 6.1680

Table 4. Comparison of natural fnquencies, i = 0hhGGh of an orthotropic square plate (a/h 5: 10)

f&act lul HSDPT FSDPT CPT

m n I II* III I 11, III I Ii’ III I

1 I

1 2

2 1

2 2

1 3

3 1

2 3

3 2

1 4

4 I

3 3

2 4

4 2

0.0474 1.3077 1.6530 0.0474 1.30% 1.6550 0.0474 1.3159 I.6646

0.1033 1.3331 1.7160 0.1033 1.3339 1.7209 0.1032 1.3410 1.7305

0.1188 1.4205 1.6805 0.1189 1.4216 1.6827 0.1187 1.4285 1.6921

0.1694 1.4316 1.7509 0.1695 1.4323 1.7562 0.1692 1.4393 1.7655

0.1888 I.3765 1.8115 0.1888 1.3772 1.8210 0.1884 1.3841 1.8305

0.2180 1.5777 1 .I334 0.2184 I.5789 1.7361 0.2178 I.5857 1.7450

0.2475 1.45% 1.8523 0.2477 1.4603 I.8622 0.2469 I A670 1.8714

0.2624 1.5651 1.8195 0.2629 I.5658 I.8255 0.2619 1.5725 1.8341

0.2%9 1.4372 1.9306 0.2%9 I.4379 1.9466 0.2959 I.4445 I.9560

0.3319 1.7179 1.8548 0.3330 1.7186 1.8588 0.3311 1.7265 1.8657

0.3320 1.5737 1.9289 0.3326 I.5744 I.9395 0.3310 1.5812 1.9479

0.3476 1.5068 1.9749 0.3479 1.5076 1.9912 0.3463 1.5141 2.OwJ2

0.3070 1.6940 1.9447 0.3720 1.6947 1.9514 0.36% 1.7022 1.95%

0.0(93 (0.0497) + 0.1095

(0.1120) 0.1327

CO. 1354) 0.1924 (0.1987) 0.2070 (;;2g)

(0:2?79) 0.2879

(0.3029) 0.3248

(0.3418) 0.3371

‘E!) (0:4773) 0.4172

l Pure thick-twist modes + Numbers in parenthesis indicate frequencies obtained by omitting the rotatory inertia.

(1) The classical plate theory overestimates the frequencies. The errors increase with increasing mode numbers.

(2) The frequencies predicted by FSDPT are fairly accurate; the error increases with increasing mode number.

(3) The frequencies predicted by HSDPT are the most accurate of all. (4) The effect of transverse shear deformation increases with increasing mode

number.

A refined nonlinear theory of plates with transverse shear deformation 895

CONCLUSIONS

A refined nonlinear shear deformation theory of flat plates is presented. The theory accounts for (a) zero traction boundary conditions on the top and bottom faces of the plate, (b) cubic variation of in-plane displacements through thickness (hence, a parabolic distribu- tion of transverse shear stresses through thickness), and (c) the von Karman strains. Additional features of the theory are that no shear correction factors are used in the theory, and the resulting equations of motion include the same variables as in the first-order shear deformation theory. Exact solutions for the case of infinitesimal displacements are presented for bending and free vibration of simply supported rectangular plates of isotropic as well as orthotropic materials. The solutions of the higher-order theory are found to be in excellent agreement with the exact solutions of the threedimensional theory of elasticity. The numerical results should serve as references for those who wish to develop a finite-element model of the higher-order theory described herein. Extension of the present theory to laminated anisotropic plates is presented by the author (see[251).

Acktwwfedgement-The support of the research reported here, in parts, by the Air Force Oflice of Scientific Research through grant AFOSR-61-0142 and NASA Langley through grant NAG-I-459 are gratefully acknow- ledged. The author is thankful to Dr. Anthony Amos (AFOSR) and Dr. James Stames, Jr. (NASA) for the encour- agement and support of the work.

:: 3.

4.

8. 9.

IO. II. 12.

13.

14.

IS.

16. 17

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Department, Virginia Polytechnic Institute, Blacksburg, VA (June 1983); also to appear in 1. Appl. Mech.

APPENDIX Transverse sheor stresses from stress equilibrium equarkms

The transvcrsc normal and shear stresses obtained from the stress equilibrium equations for the classical plate theory, the first-order shear deformation theory, and the higher-order theory (for simply supported, orthotropic

J. N. %DDY 8%

mtangular p&s) arc @en below:

(1) Cfossimf plate fdeory

a,=- ~[~-~)rlbJQ,,+(tc,,+~,~~q~msrudnav

hZ1 22

uw=T [ 0~ - b &WG,2 + Q13 + B’QrJ W sin 0~ ~0s PY

when Qe arc @cn by cpn (86), and W dcnotcs the ampiitudc in eqn (22).

(2) Firsr-order shear &formation theory