COMPARISON OF NATURAL FREQUENCIES OF LAMINATES BY 3-D THEORY,

doi:10.1006/jsvi.1999.2453, available online at http://www.idealibrary.com on

COMPARISON OF NATURAL FREQUENCIES OF LAMINATES BY 3-D THEORY,

PART I: RECTANGULAR PLATES

C. C. CHAO

Department of Power Mechanical Engineering, National ¹sing Hua ;niversity, Hsinchu, ¹aiwan 300, R.O.C.

AND

YEONG-CHYUANCHERN

Department of Mechanical Engineering, National Chin >i Institute of ¹echnology,

¹aichung, ¹aiwan 411, R.O.C.

(Received 19 August 1998, and in ,nal form 11 June 1999)

The three-dimensional theory of laminated plates and shells has been developed by Chao et al. [10}13, 62, 63] with many applications to impact and shock modal analyses. In this research, a complete survey of the literature is made on the free vibration natural frequencies of simply supported rectangular plates. Various boundary conditions are composed of "xed pin, hinge-roller, and sliding pin supported edges. The lowest frequencies are obtained in the present study in comparison with those in earlier studies as a result of the close natural state reached in keeping with the three-dimensional boundary and interlaminar continuity conditions via a 3-D augmented energy variational approach.

( 2000 Academic Press

1. INTRODUCTION

The mathematical theory of elasticity and vibration problems in engineering were comprehensively discussed by Love [1], and Timoshenko [2], respectively, in the 1920s. It was noted that assumptions of the classical thin plate theory overestimated the structural sti!ness, and hence the natural frequencies. Reissner [3], and Mindlin and Medick [4] considered the e!ect of transverse shear on the bending of isotropic elastic plates, leading to the development of the "rst order, and higher order shear deformation theories.

Since the advent of composites featuring high sti!ness, high strength and light weight, vibration of anisotropic laminated plates has drawn the attention of many researchers. Exact solutions for bending, vibration and buckling of simply supported thick orthotropic and cross-ply laminated rectangular plates were obtained by Srinivas et al. [5], and Srinivas and Rao [6] in 1970.

A three-dimensional solution was found by Noor and Burton [7] for the

0022-460X/00/100985#23 $35.00/0 ( 2000 Academic Press

antisymmetrically laminated anisotropic plates. In view of ever increasing application to general laminated structures in engineering, theories and a number of numerical solution methods have been developed for the "rst order approximation in preliminary design. Assessments of computational models for multi-layered anisotropic, and sandwich plates were published by Noor and Burton [8], and Noor et al. [9] respectively.

Three-dimensional semianalytical solutions have been developed by Chao et al.

on the basis of local 3-D stress equilibrium with many applications to impact and shock modal analysis of laminated plates and curved panels [10}13, 62, 63].

A complete survey of the literature on engineering vibration analysis of laminated plates is presented in Table 1. The studies are classi"ed as theory, material property and numerical methods.

Basic theories of plates and shells can be found in four categories, i.e., (i) classical thin plate theory known as CPT, (ii) "rst order shear deformation theory known as FSDT, (iii) higher order shear deformation theory known as HSDT, and (iv) theory of three-dimensional elasticity.

In the classical plate theory [1, 2, 14}20, 61], the transverse shear e!ects are neglected according to the Kirhho! assumption, and the structural sti!ness and natural frequencies are overestimated.

It was Reissner [3], who "rst considered the e!ect of transverse shear on the bending of elastic plates, that led to the development of the "rst order shear deformation theory FSDT. However, the e!ects of cross-sectional warping is ignored resulting in an unrealistic linear variation of the transverse shear stress

TABLE1

Classi,cation of references

Theory Material property

CPT : 1}2, 14}20, 61 Isotropic : 1}4, 14, 16, 19, 20, 22, 26, 27,

FSDT : 3, 21}30 29, 32, 41}43, 52, 58}60

HSDT : 4, 31}52 Composite : 5}13, 15, 17, 18, 21}26, 28}63 3}D : 5}13, 53}60, 62, 63 Sandwich : 9, 36, 40, 55

Numerical methods

Di!erential quadrature : 18, 30 Finite di!erence : 53

Finite element : 22}24, 26, 31, 33, 36}41, 45, 46, 50}52, 54}56, 59, 60 Finite layer : 58

Finite strip : 27}29, 47 Galerkin technique : 21, 25, 34, 42, 57

Hamilton principle : 3, 4, 10}13, 31, 32, 35, 43, 44, 48, 49, 62, 63 Newton}Raphson : 5, 6

Rayleigh}Ritz : 2, 14, 15}17, 19, 20, 59

Assessment : 8, 9

through thickness of the laminate, and the use of shear correction coe$cients is required [3, 21}30].

The higher order shear deformation theory HSDT was mainly based on a two-dimensional approach by incorporating higher order modes of transverse cross-sectional deformation [4, 31}52]. It began with the work of Mindlin and Medick [4] for isotropic plates. A more reasonable parabolic variation of transverse shear stress/strain through thickness can be obtained with no need for the assumed shear correction coe$cients. The major drawback of the conventional HSDT lies in that it is unable to satisfy the interlaminar continuity from layer to layer and stress equilibrium over the lateral surfaces without regard to the transverse normal stress, which is of special importance in treating the contact and impact problems. Recent development has led to a three-dimensional model in which the six stress/strain components are fully obtainable throughout the laminated plate.

Recently, thick laminate construction has stimulated the interest in use of three-dimensional theory for predictions of structural response and stresses. The 3-D theories [5}13, 53}60, 62, 63] include 3-D exact analysis, 3-D "nite element method, 3-D "nite layer method, 3-D layerwise theory, and the 3-D elasticity theory. The engineering vibration problem has rarely been solvable in exact form of 3-D elasticity for laminated plates and shells, except for a few special cases such as cross-ply by Srinivas et al. [5, 6], and antisymmetrical angle-ply by Noor and Burton [7, 8]. The present study is devoted to the more general case for three-dimensional analysis.

In this research, a thorough analysis and survey of moderately thick or thin plates made up of symmetric or antisymmetric, cross-ply or angle-ply lay-ups is carried out in accordance with the three-dimensional elasticity theory in comparison with earlier studies. Lowest natural frequencies are obtained by taking the three-dimensional boundary and interlaminar continuity conditions into account as the physical requirements of natural state as shown in equations (1)}(5). To facilitate the comparison, several types of plate materials are treated in the present study. The isotropic/metallic plates are discussed

"rst with di!erent length to thickness ratios and in-plane aspect ratios. The rest are concerned with anisotropic laminated composite plates consisting of high strength/modulus aragonite or glass, carbon, boron reinforcing "bers embedded in high-performance matrix. In view of the numerous publications in this "eld, discussions are con"ned to simply supported plates due to the limited scope of this paper.

2. THEORETICAL FORMULATION

Consider a K layered plate of in-plane dimensions a, b and thickness h with simple supports. In the treatment of the various problems of interest, it may pertain to any one of the following three types of boundary conditions, in which local stresses and displacements are concerned rather than the global stress resultants and stress couples in the conventional plate theories.

Figure 1. Schematic of a laminated plate with simple supports.

2.1. THREE-DIMENSIONAL BOUNDARY AND INTERLAMINAR CONDITIONS

The conventional edge boundary conditions are modi"ed in the essence of three-dimensional elasticity in terms of local displacements and stresses for the various support con"gurations for the 3-D boundary conditions as shown in equations (1)}(4). In the present study of free vibration, the entire laminated plate is considered surface traction free over both lateral surfaces. Both the natural and geometrical edge conditions are justi"ed by admissible displacement functions exactly everywhere over all four edges for cross-ply laminations, while speci"ed geometric edge conditions are justi"ed for angle-ply and other laminations. Three types of simply supported edge boundary conditions are treated.

¸ateral surface traction free conditions:

z"0: F(0)1 "pxz"0, F(0)2 "pyz"0, F(0)3 "pzz"0,

z"h: F(K)1 "pxz"0, F(K)2 "pyz"0, F(K)3 "pzz"0. (1) S1 ,xed pin supported edges:

x"0, a: z"0, pxx"u"v"w"0, zO0, pxx"v"w"0, y"0, b: z"0, pyy"u"v"w"0, zO0, pyy"u"w"0. (2) S2 hinge-roller supported edges:

x"0, a: pxx"v"w"0, y"0, b: pyy"u"w"0. (3) S3 sliding pin supported edges:

z"0, x"0, a: pxy"u"w"0, y"0, b: pyx"v"w"0. (4) The surface conditions are labelled as F(0)i and F(K)i for transverse normal and shear stresses free at the bottom and top surfaces respectively. Pasternak or Winkler mode elastic foundation may be incorporated into the surface condition if required.

Interlaminar continuity: Since individual displacement "elds are assumed for each layer of the laminate, interlaminar continuity of layer displacements in

addition to transverse stresses must be satis"ed at each interface between adjacent layers.

F(k)1 "p(k)xz<sup>`</sup>!p(k`1)xz ^~"0, F(k)4 "u(k)<sup>`</sup>!u(k`1)^~"0, F(k)2 "p(k)yz<sup>`</sup>!p(k`1)yz ^~"0, F(k)5 "v(k)<sup>`</sup>!v(k`1)^~"0, F(k)3 "p(k)zz<sup>`</sup>!p(k`1)zz ^~"0, F(k)6 "w(k)<sup>`</sup>!w(k`1)^~"0,

k"1, 2,2, K!1 (5)

where, for simplicity, the interlaminar conditions are denoted as F(k)i with subscripts 1, 2, 3 for the transverse stresses pxz, pyz, pzz and 4, 5, 6 for layer displacements u, v, w. The superscripts#and!denote the upper and lower surfaces of the respective layers. Layers are numbered from the bottom upwards.

2.2. THREE-DIMENSIONAL DISPLACEMENT FIELDS

Three-dimensional displacement "elds are assumed according to the various edge boundary conditions as above for each layer in terms of double Fourier series of x, y for the in-plane co-ordinates and polynomials in z to proper higher orders for the out-of-plane co-ordinate, i.e.,

uk(x, y, z, t)" +

j,m,n[;jmnZj(z);m(x);n(y)]k, vk(x, y, z, t)" +

j,m,n[<jmnZj(z)<m(x)<n(y)]k, wk(x, y, z, t)" +

j,m,n[=jmnZj(z)=m(x)=n(y)]k. (6) S1 ,xed pin displacement ,eld:

uk(x, y, z, t)"+J 1

+M 0

+N

1 ;jmnzj cos xm sin yn, vk(x, y, z, t)"+J

1 +M

1 +N

0 <jmnzj sin xm cos yn, wk(x, y, z, t)"+J

0 +M

1 +N

1 =jmnzj sin xm sin yn, (7) where xm"mnx/a, yn"nny/b.

S2 hinge-roller displacement ,eld:

uk(x, y, z, t)"+J 0

+M 0

+N

1 ;jmnzj cos xm sin yn, vk(x, y, z, t)"J

+ 0

+M 1

+N

0 <jmnzj sin xm cosyn, wk(x, y, z, t)"+J

0 +M

1 +N

1 =jmnzj sin xm sin yn. (8)

S3 sliding pin displacement ,eld:

uk(x, y, z, t)"+M 1

+N

0 ;0mn sin xm cos yn#+J 1

+M 0

+N

1 ;jmnzj cos xm sin yn, vk(x, y, z, t)"+M

0 +N

1 <0mn cos xm sin yn#+J 1

+M 1

+N

0 <jmnzj sin xm cos yn, wk(x, y, z, t)"+J

0 +M

1 +N

1 =jmnzj sin xm sin yn. (9)

2.3. THREE-DIMENSIONAL ENERGY VARIATIONAL APPROACH

Strain components in a layer: In accordance with the three-dimensional consistent higher order theory of plates and shells [7}10], the small strains are expressed in terms of the displacements of the kth layer.

exx"Lu

Lx, eyy"Lv

Ly, ezz"Lw Lz, cyz"Lw

Ly#Lv

Lz, cxz"Lw Lx#Lu

Lz, cxy"Lu Ly#Lv

Lx. (10)

Stress components in a layer: The three-dimensional stresses in the plates are obtained using the anisotropic constitutive law of composites for any layer. The 3-D mechanical properties must be known to perform the three-dimensional elasticity analysis. Since most of the numerical examples in the literature are incomplete in 3-D properties, the transverse the Poisson ratiolp23 can be calculated from equation (13) in reference to Philippidis [61] and the transverse shear modulus is obtained as G23"E2/[2(1#lp23)] in the y}z plane.

i g g j g g k

pxxpyy pzzpyz pxzpxy e g g f g g h

"

CM 11 CM12 CM13 0 0 CM 16 CM 12 CM22 CM23 0 0 CM 26 CM 13 CM23 CM33 0 0 CM 36

0 0 0 CM 44 CM45 0

0 0 0 CM 45 CM55 0

CM 16 CM26 CM36 0 0 CM 66 i g g j g g k

exxeyy ezzcyz cxzcxy e g g f g g h

. (11)

Energy formulation: The generalized equations of motion are derived by means of the strain energy, kinetic energy, and work done by non-conservative forces via a three-dimensional augmented energy variational approach subject to the surface conditions and interlaminar continuity by using Lagrange multipliers.

<"+K

k/1

P

G

H

dx dy dz, i, j"x, y, z,

¹"+K

k/1

P

G

H

dx dy dz,

SC" + k/0,K

+3

i/1

P

IC"K~1+ k/1

+6

i/1

P

P"<!¹!=nc#SC#IC. (12)

¸agrange multipliers: The Lagrange multipliers are assumed according to the various corresponding stress and displacement "eld functions. Using the S1 and S2 models, the six types of Lagrange multipliers are expanded in Fourier series as follows:

j1"+

m,nKxz,mnpxz(xm)pxz(yn)"+

m,nKxz,mn(cos xm sinyn#sinxm cosyn), j2"+

m,nKyz,mnpyz(xm)pyz(yn)"+

m,nKyz,mn(cos xm sinyn#sinxm cosyn), j3"+

m,nKzz,mnpzz(xm)pzz(yn)"+

m,nKzz,mn(cos xm cosyn#sinxm sinyn), j4"+

m,nKu,mn cosxm sinyn, j5"+

m,nKv,mn sinxm cosyn, j6"+

m,nKw,mn sinxm sinyn. (13)

Modi,ed ¸agrange1s equations: The three-dimensional displacements can be partitioned into the lower and higher order parts denoted by vectors ;namely, _land ;h,

M;NT"M;lD ;hNT

M;lNT"M;jmn, <jmn, =jmnNT, j"1, 2,2, J!2,

M;hNT"M;jmn, <jmn, =jmnNT, j"J!1, J. (14) Using the lateral surface and interlaminar constraint conditions as above, the six degrees of freedom of the higher order part can be eliminated in each layer for each Fourier series component. A system of modi"ed Lagrange's equations of motion is obtained via energy variation with respect to the generalized displacements and Lagrange multipliers.

LP

Lji"0N[¸hj]M;hN"![¸lj]M;^lN, LP

L;i"0N[M]M;G N#[K]M;N#[¸Tj]MK]"MPN, (15)

where [¸j] is a matrix representing the surface and interlaminar continuity rela- tionship with [¸^lj] and [¸hj] as submatrices through partition. [M] and [K] are the mass and sti!ness matrices of the system, which can be converted to reduced forms by use of the lower order displacements alone.MPN is the equivalent external forcing term and will vanish to zero vector in free vibration.

Assuming simple harmonic motion, the following eigenvalue problem is derived:

[KM ] M;lN"u2[MM]M;lN, (16) whereu is the natural frequency of the free vibration.

3. RESULTS AND DISCUSSION

By use of the present three-dimensional theory, a general survey is made on free vibration of the various simply supported rectangular plates. Numerical results are tabulated in comparison with the literature in the order of isotropic plates, cross-ply, angle-ply and quasi-isotropic hybrid laminated composites. Di!erent displacement "elds are used for di!erent boundary conditions as the case applies.

Table 2 shows classi"cation of the 3-D boundary conditions, to which each of the references in the literature survey and tables in the present study pertains. Basically, the concepts of constant or averaged transverse shear for the FSDT, and parabolic transverse shear distribution for the HSDT are inconsistent with real physics. These theories are unable to account for the three-dimensional boundary conditions of no lateral surface traction in free vibration, and interface continuity of displacements and transverse stresses as per Newton's third law. The present three-dimensional elasticity theory of laminated plates is rigorous in that all of these conditions are taken into consideration by leaving the higher order displacement coe$cients to be determined through an energy variational approach in pursuit of a natural state for minimum total potential energy. As a result, natural frequencies obtained

TABLE2

Classi,cation of boundary conditions and displacement ,elds

References in literature survey Present Class Boundary condition Displacement "elds Table nos.

S1 18, 23, 24, 59 5, 10, 12

S2 5, 6, 9}13, 18, 23}25, 28, 30}37, 39, 5, 6, 9}13, 25, 28, 30}32, 34, 35, 3}9, 11}13 40, 42}44, 48, 49, 51, 52, 57}60 37, 39, 42}44, 48, 49, 53, 57, 58

S3 7, 8, 18, 21}23, 32, 33, 40, 49, 50, 52 7, 8, 21,32, 49 5, 7, 9, 11}16 Unk. 1}4, 14}17, 19, 20, 26, 27, 29, 38, 1}4, 14}20, 22}24, 26, 27, 29,

41, 45}47, 53}56 33, 36, 38, 40, 41, 45}47, 50}52, 54}56, 59, 60

in the present study are the lowest among all results in the literature. Only a few exceptions are encountered, in which, an * mark will be noted with an explanation.

3.1. CONVERGENCE AND ACCURACY

At "rst, convergence studies were carried out for the isotropic, and cross-ply, and angle-ply antisymmetric and symmetric laminated plates. Accuracy was also veri"ed by checking with Srinivas' exact solutions in close agreement in Tables 3 and 4.

Table 3 shows the normalized fundamental frequencies for thin and thick square plates with the present S2 displacement "elds by changing the order of the polynomial function Zj(z). In the higher order shear deformation theory, the zj usually varies from order 2 to 5. The thicker the plate, the higher is the order of the transverse co-ordinate term zj required. In the present theory, polynomials to order 3 were employed for laminates of moderate thickness, and order 4 for thick plates where h/a*0)1. The "rst part shows the present fundamental frequencies with fast convergence and accurate results as compared to those in earlier studies.

Srinivas, Joga Rao and Rao's vibration analysis of isotropic plate was an exact elasticity solution [5]. Leissa [14] reconsidered the problem with the classical thin plate theory. Farsa et al. [18] conducted the vibration studies of laminated rectangular plates by the di!erential quadrature method. Noor [53] solved the free vibration problem using the 3-D elasticity theory with higher order "nite di!erence schemes. Criterion for convergence on the Fourier series part is whether the assumed functions has attained an adequate set of the series. The second part of Table 3 shows the necessary condition that each frequency tends to a certain limit about the cross-ply laminate in steady, and increasingly smaller changes as the values for m and n are gradually increased. Results of the present S2 symmetric cross-ply and angle-ply thin plates at a/h"20 are always lower as compared with those of Bowlus et al. [25], in which an FSDT-based Galerkin technique was used for determining the natural frequencies and mode shapes.

3.2. ISOTROPIC PLATES

Table 4 shows the normalized frequencies of moderately thick isotropic square plates withl"0)3 at a/h"10 in comparison with Srinivas et al. [5], Reddy [22], Huang and Dasgupta [59], Meimaris and Day [60], and Shankara and Iyengar [52]. The present results are in good agreement with the exact solution of Srinivas et al. In reference [52], a C3 "nite element model based on HSDT was used without considering C1 continuity of the inter-element slope, and a high-low#uctuation in their frequencies was indicated by an *.

The "rst eight frequenciesX"ua2(o/D)1@2 of isotropic (l"0)3) rectangular and square plates are compared in Table 5. Firstly, in consideration of varying aspect ratios, frequencies of the present S1 and S2 thin (a/h"20) rectangular plates are the lowest when compared to those of Leissa [14], Liew et al. [16], Zhou [19],

TABLE3

Convergence and accuracy of normalized frequenciesX for square plates Fundamental freq. by changing order of polynomial zj

Plates S2 Isotropic [0/902]i6 [0/902/0]iiiS [45/!45]ii

a/h 10 1000 5 283 5 100

Present j Xa Xb Xc u, Hz Xd Xd

m"n"8 2 9)4937 19)7391 3)3499 59)7509 8)6138 14)6109 3 9)3153 19)7389 3)3495 59)7370 8)2932 14)6043 4 9)3150 19)7389 3)3495 59)7370 8)2873 14)6041 5 9)3150 19)7389 3)3495 59)7370 8)2872 14)6041

Reference [5] 9)3150 * * * * *

Reference [14] * 19)7392 * * * *

Reference [53] * * 3)4250 * * *

Reference [18] * * * 59)7500 * *

First few modes by changing terms of Fourier series,Xd

a/h"20 m, n X1 S2 ( j"3)X3 X5 X1 Reference [25]X3 X5

[0/90]iiS 2,2 11)749 36)716 * 11)758 36)866 *

4,4 11)749 36)716 42)488 11)758 36)866 42)573

6,6 11)749 36)716 42)488 11)758 36)866 42)573

8,8 11)749 36)716 42)488 11)758 36)866 42)573

S2 ( j"3) Reference [25]

[45/!45]iiS 2,2 14)619 36)075 * 14)699 36)164 *

4,4 14)320 34)849 55)788 14)418 35)444 57)883

6,6 14)177 34)594 55)051 14)283 34)734 55)082

8,8 14)087 34)450 54)825 14)205 34)613 54)856

Material property and notations

Material E1

E2 G12

E2 G23

E2 l12 l23 Frequency

Isotropic 11 ** ** 0)30)3 0)30)3 Xa"100u(oAh/G)1@2Xb"ua2(oA/D)1@2 Composite iiiiii 101511)48 0)60)4290)278 0)500)3430)27 0)250)400)28 0)250)4560)28 Xc"10uh(om/E2)1@2Xd"ua2(om/E2h2)1@2u in Hertz

For material iii: a"b"12 in, E2"2)7 Mpsi, o"1)92]10~4 lb s2 in~4.

Geannakakes [20], and Cheung and Kong [29]. Secondly, in considering varying length to thickness ratios, the frequencies of the present S3 square plates also compare well with those of Chen and Yang [26] and Mizusawa [27]. Speci"c displacement "elds were used as required by the boundary conditions.

TABLE4

Normalized frequencies Xmn"ua2(o/Eh2)1@2 of an isotropic square plate

m, n 1,1A 1,2 0,1 2,2 1,3 1,1S 2,3

Reference [5] 5)7769 13)8050 * 21)2143 25)8699 27)5537 32)4915

Reference [22] 5)793 14)081 * * 27)545 * 35)050

Reference [59] 5)785 13)871 19)483 21)300 26)420 27)662S 32)930

Reference [60] 5)778 13)63 * * 25)26 * 31)08

Reference [52] 5)7712* 13)7904* 19)4838 21)1580* 25)8980 27)5545 32)4340*

S2 5)7769 13)8050 19)4833 21)2145 25)8697 27)5536 32)4916

TABLE5

Normalized frequencies of ,rst eight modes for isotropic plates

X1 X2 X3 X4 X5 X6 X7 X8

a/b Reference For rectangular plates, a/h"20

2/5 [14] 11)448 16)186 24)081 35)135 41)057 45)795 49)384 53)691 [20] 11)448 16)186 24)082 35)147 41)056 45)795 51)357 53)691 S1 11)436 16)175 24)074 35)122 41)038 45)774 49)330 53)669 S2 11)391 16)071 23)829 34)604 40)334 44)899 48)311 52)467 2/3 [14] 14)256 27)415 43)864 49)348 57)024 78)956 80)054 93)213 [20] 14)256 27)415 43)864 49)350 57)024 78)958 80)089 93)218 S1 14)244 27)402 43)845 49)334 56)999 78)923 80)015 93)173 S2 14)167 27)089 43)041 48)310 55)647 76)362 77)389 89)636 1 [14] 19)739 49)348 49)348 78)956 98)696 98)696 128)305 128)305

[16] 19)74 49)35 49)35 79)03 99)25 99)25 * *

[19] 19)739 49)365 49)365 78)979 98)973 98)973 128)534 128)534 [20] 19)739 49)348 49)348 78)956 98)701 98)701 128)309 128)309 [29] 19)74 49)36 49)38 78)98 98)80 99)28 128)410 128)790 S1 19)731 49)331 49)331 78)923 98)653 98)653 128)231 128)231 S2 19)569 48)310 48)310 76)362 94)702 94)702 121)703 121)703

a/h For square plates

10 [26] 19)064 45)489 45)489 69)816 85)147 * * *

[27] 19)058 45)448 45)448 69)717 84)926 84)926 * * S3 17)468 40)099 40)099 60)901 74)090 74)090 93)060 93)060 100 [27] 19)732 49)303 49)303 78)841 98)512 98)512 * *

S3 19)701 49)115 49)115 78)365 97)778 97)778 126)770 126)770

3.3. CROSS-PLY PLATES

¹hickness e+ect: For the varying length}thickness ratios, fundamental natural frequencies of antisymmetric and symmetric cross-ply graphite "ber reinforced laminates are presented in Table 6. Srinivas et al. [5] analyzed the problem in an

TABLE6

Fundamental frequencies X"ua2(o/E2h2)1@2 for cross-ply very thick, moderatelythick, and thin square plates

Reference a/h"2 5 10 20 25 50 100

[0/90] [5] 4)935 8)518 10)333 11)036 11)131 11)263 11)297 [32] 5)699 9)010 10)449 10)968 11)037 11)132 11)156 [42] 4)810 8)388 10)270 11)016 11)118 11)230 11)296

[36] * 8)702 10)415 11)060 * 11)202 11)208

[44] * 9)807 10)568 11)105 * 11)275 11)300

[41] 5)718 9)092 10)576 11)114 11)186 11)293 11)311 [43] 4)939 8)521 10)335 11)036 11)132 11)263 11)297 S2 4)953 8)527 10)335 11)037 11)132 11)262 11)296

S3 4)730 7)567 8)834 9)527 9)708 10)224 10)800

[0/90]S [32] 5)576 10)989 15)270 17)668 18)050 18)606 18)755 [42] 5)923 10)673 15)066 17)535 18)054 18)670 18)835

[44] * 10)263 14)702 17)483 * 18)641 18)828

[41] 6)002 11)772 15)945 18)000 18)308 18)745 18)860 S2 5)164 10)232 14)696 17)481 17)948 18)640 18)825 S3 5)238 9)866 12)790 14)355 14)730 16)054 17)562 E1/E2"40, E3/E2"1, G12/E2"0)6, G13"G12, G23/E2"0)5, l12"l13"0)25, lp23"0)646.

exact elasticity solution. The higher order displacement "eld hypothesis was employed by Reddy and Phan [32] in vibration studies of isotropic, orthotropic and laminated plates. An individual-layer HSDT was used by Cho et al. [42].

Kant and Mallikarjuna [36] developed a higher order theory with C3 "nite element formulation. Shiau and Wu [41] used a high precision higher order triangular element. Nosier et al. [43] employed a layerwise theory. Hamilton's principle was used by Hadian and Nayfeh [44] in a third order shear-deformation plate theory. The lowest frequencies are obtained from the S3 solution in the present study.

Moderately thick orthotropic plate: Aragonite square plates of moderate thickness a/h"10 were studied by Srinivas and Rao [6] in an exact solution. The analyses of Reddy [31], Fan and Ye [57], Cho et al. [42], and Tessler et al. [49] are also listed along with the present S2 method in Table 7. In Reference [49], pre-assumed shear correction coe$cients iz0"0)907, iz1"0)816 caused a few slightly lower frequencies as indicated by an *. Via the present S2 approach, the normalized frequencies of various modes are all in good agreement with the exact analysis.

E+ect of orthotropy-moderately thick to thin: The fundamental natural frequencies of free vibration X"ua2(o/E2h2)1@2 of antisymmetric cross-ply graphite/epoxy thick and thin laminated plates are presented in Table 8. Owen and Li [24] performed a re"ned transverse vibration and buckling analysis using a "nite element displacement method. Ochoa and Reddy [39] also analyzed this topic by

"nite element methods. Argyris et al. [45] used a three-node triangular element in

TABLE7

Natural frequencies of a simply supported aragonite plate,X"uh(o/C11)1@2 Reference m, n I-A I-S II-S II-A III-A III-S IV-S V-S

[6] 1, 1 0)04742 0)21697 0)39405 1)3077 1)6530 2)2722 2)5479 3)2636

[31] 0)04740 * * 1)3086 1)6550 * * *

[57] 0)04751 0)21700 0)39405 * * * * *

[42] 0)0474 0)2170 0)3941 1)3081 1)6536 * * *

[49] 0)0474 0)2170 0)3941 1)3078 1)6530 2)2879 * *

S2 0)04742 0)21697 0)39405 1)3077 1)6530 2)2722 2)5479 3)2636 [6] 2, 1 0)11880 0)35150 0)67278 1)4205 1)6805 2)2537 2)6264 3)2760

[31] 0)11897 * * 1)4216 1)6827 * * *

[42] 0)1188 0)3515 0)6728 1)4208 1)6812 * * *

S2 0)11880 0)35150 0)67278 1)4205 1)6805 2)2537 2)6264 3)2760 [6] 2, 2 0)16942 0)43382 0)78796 1)4316 1)7509 2)2455 2)6334 3)3179

[31] 0)16950 * * 1)4323 1)7562 * * *

[42] 0)1694 0)4338 0)7880 1)4319 1)7523 * * *

S2 0)16942 0)43382 0)78795 1)4316 1)7509 2)2455 2)6334 3)3178 [6] 3, 3 0)33200 0)65043 1)1814 1)5737 1)9289 2)2274 2)7457 3)4085

[31] 0)33260 * * 1)5744 1)9395 * * *

[42] 0)3319 0)6505 1)1815 1)5741 1)9221 * * *

[49] 0)3309* 0)6503* 1)1813* 1)5737 1)9296 2)2918 * * S2 0)33200 0)65043 1)1814 1)5737 1)9289 2)2273 2)7457 3)4085

TABLE8

E+ect of orthotropy on the fundamental frequencies of antisymmetric cross-ply square plates

2 layers 4 layers 10 layers

E1/E2 Reference a/h"10 100 10 100 10 100

10 [24] 7)8699 8)1477 9)5385 10)0934 9)9648 10)5744 [45] 7)7644 8)1090 *9)3764 10)0490 9)8664 10)5293 S2 7)7343 8)0815 9)3888 10)0108 9)8409 10)4852 40 [24] 10)5001 11)3202 14)7357 17)3038 15)8024 18)6394 [39] 10)6100 11)5380 14)8830 17)4930 15)7930 18)8210 [45] 10)3619 11)2890 *14)3459 17)2632 15)6800 18)6014 S2 10)3129 11)2580 14)4778 17)2255 15)6563 18)5207 E3/E2"1, G12/E2"0)6, G13"G12, G23/E2"0)5, l12"l13"0)25, lp23 calculated as per reference [61]

non-linear free vibration with a 0)1}1% lower frequencies for four-layered moderately thick laminated plates of a/h"10.

E+ect of orthotropy-thick laminates: The e!ects of number of layers and degree of orthotropy of the individual layer on the normalized fundamental frequencies are

TABLE9

E+ect of orthotropy on the fundamental frequenciesand symmetric cross-ply square plates [0/902]X"uh(o/E2)1@2 of antisymmetric

E1/E2 3 10 20 40 3 10 20 40

Reference 2 layers Reference 3 layers

[53] 0)25031 0)27938 0)30698 0)34250 [53] 0)26474 0)32841 0)38241 0)43006 [33] 0)24868 0)27955 0)31284 0)36348 [33] 0)26278 0)33192 0)38268 0)43415 [24] 0)25601 0)28712 0)31558 0)35182 [24] 0)26948 0)33917 0)38979 0)43951 [37] 0)24868 0)27955 0)31284 0)36348 [34] 0)26223 0)32692 0)36923 0)40878 [35] 0)24128 0)27769 0)30525 0)34072 [35] 0)25560 0)32586 0)36898 0)40923 [54] 0)24929 0)27821 0)30563 0)34076 [55] 0)26461 0)32451 0)37717 0)42558 [56] 0)25032 0)27939 0)30862 0)34757 [56] 0)26280 0)32675 0)37031 0)41044 [38] 0)24931 0)27822 0)30566 0)34114 [40] 0)26126 0)32528 0)37253 0)41520 [40] 0)24782 0)27764 0)30737 0)34810 [48] 0)26357 0)33342 0)38457 0)43510 [48] 0)25174 0)28129 0)31011 0)34860 [46] 0)264 0)339 0)393 0)447

[46] 0)248 0)282 0)317 0)369 [58] * * * 0)42666

[47] 0)24934 0)27897 0)30586 0)34909 [30] * 0)33117 0)38150 0)43247

[58] * * * 0)33758 S2 0)26225 0)32689 0)36888 0)40965

S2 0)24842 0)27548 0)30424 0)34096 S3 0)22910 0)28510 0)32926 0)37558

S3 0)20003 0)23574 0)26796 0)30968 9 layers

10 layers [53] 0)26640 0)34432 0)40547 0)46679 [53] 0)26583 0)34250 0)40337 0)46498 [33] 0)26384 0)34169 0)40334 0)46580 [24] 0)26916 0)34527 0)40526 0)46590 [24] 0)26971 0)34708 0)40746 0)46803 [37] 0)26337 0)34050 0)40270 0)46692 [34] 0)26375 0)34079 0)40138 0)46260 [35] 0)26308 0)33917 0)39969 0)46120 [35] 0)26356 0)34013 0)39995 0)46009 [40] 0)26331 0)33989 0)40069 0)46295 [40] 0)26298 0)34035 0)40107 0)46222 [48] 0)26329 0)33974 0)40075 0)46285 [48] 0)26390 0)34169 0)40310 0)46510 [46] 0)264 0)344 0)408 0)472 [46] 0)264 0)347 0)410 0)474

S2 0)26402 0)33982 0)40027 0)46103 [30] * 0)34098 0)40217 0)46397 S3 0)20737 0)27258 0)33030 0)39700 S2 0)26456 0)34149 0)40113 0)46082 (lp23 0)55575 0)62409 0)63873 0)64606) S3 0)24700 0)31416 0)36988 0)43122 E3/E2"1, G12/E2"0)6, G13"G12, G23/E2"0)5, l12"l13"0)25, lp23 calculated as per reference [61].

compared with those in the literature in Table 9. Thick square plates of a/h"5 multi-layered antisymmetric and symmetric cross-ply were analyzed with the material properties typical of high performance "brous composites. The ratios of moduli E1/E2 varied from 3 to 40, number of layers between 2 and 10, and the transverse the Poisson ratioA brief review is made on the literature as follows. Noor [53] solved the freelp23 is calculated as per reference [61].

vibration problem using the 3-D elasticity theory with higher order "nite di!erence schemes. Putcha and Reddy [33] used the mixed element based on a re"ned plate theory to analyze anisotropic plates. Owen and Li [24] studied vibration and stability of laminated plates by the "nite element displacement method. Khdeir and Librescu [34] applied the higher order plate theory to analyze cross-ply laminated plates. Ren and Owen [35] studied the vibration and buckling problem based on

Hamilton's principle. Analytical and "nite element solutions of the classical, "rst order, and third order laminate theories were developed by Reddy and Khdeir [37]

to study the buckling and free-vibration behavior of laminates. Jing and Liao [54], and Tseng and Chou [38] developed a partial hybrid element for the vibration of thick laminated composite plates. Rao and Meyer-Piening [55] used a hybrid-stress "nite element to perform the vibration analysis of FRP faced sandwich plates. Chen and Jiang [56] developed a three-dimensional mixed "nite element method for the dynamic failure analysis. Re"ned theories of "ber- reinforced laminated composites and sandwiches were discussed by Mallikarjuna and Kant [40]. Wang and Lin [47] published a "nite strip method based on the higher order plate theory for determining the natural frequencies of laminated plates. He and Ma [48] used a re"ned shear deformation theory to study the vibration of laminated plates. Ghosh and Dey [46] analyzed this using a simple

"nite element based on higher order theory. Kong and Cheung [58] discussed a "nite layer method on free vibration. Bert and Malik [30] analyzed laminated composite structures using the di!erential quadrature numerical method based on the "rst order shear deformation theory with a shear correction factorn2/12. For a thick plate of up to a/h"5, it is unlikely for the transverse displacement to vary through thickness as regular plates. A single term to the zeroth order of =jmnzj is preferred for the displacement "eld of w. On the other hand, it is more likely to deform in the manner of in-plane shear of S3 rather than bending extension}compression of the S2 displacement model. In Table 9, lower frequencies are also shown for the S3 displacement approach of the present theory.Cross-ply of various composites: As for a solution method by using Fourier series, Leissa and Narita [15] performed a vibration study for symmetric cross-ply laminated plates based on the Ritz method. Taking the length}thickness ratio 50 and number of layers from 1 to 15 plies (1L}15L) for composites of E-glass/Ep, Boron/Ep, and Graphite/Ep, the present S1 theoretical predictions compare well with Leissa's thin plate solution. The "rst few frequencies of the symmetric cross-ply square plates are presented in Table 10 as the lowest for all halfwave numbers.

3.4. ANGLE-PLY PLATES

E+ect of thickness and aspect ratio: To demonstrate the e!ects of thickness on the natural frequencies of the angle-ply laminated plates, the present S2, and S3 solutions are compared to those of Bowlus et al. [25]. The "rst and "fth mode frequencies of the [$45]S square plates are shown in Table 11 with "xed m, n"6 and the length to thickness ratios a/h varying from 5 to 50. For the angle-ply laminations, much lower frequencies are provided by the displacement model in the S3 edge condition, especially for the case of higher modes.To examine the combined e!ects of thicknesses and aspect ratios, the "rst mode frequencies of the present theory is listed in Table 12 for the symmetric four-layer angle-ply rectangular graphite/epoxy plates in comparison with Akhras et al. [28], in which a shear-deformable "nite strip was developed in the static and vibration analyses of composite laminates based on FSDT with shear correction factor56.

TABLE10

Normalized frequencies of ,rst few modes for cross-ply laminated square thin plates, X"ua2(o/D

3 )1@2

E-glass/Ep: X1,1 X1,2 X2,1 X2,2 X1,3 X2,3 X3,1 X3,2

[15] 1L 15)193 33)296 44)416 60)770 64)525 90)289 93)661 109)07 5L 15)193 35)894 42)344 60)770 71)569 94)504 88)395 105)44 15L 15)193 38)138 40)334 60)770 77)514 98)240 83)232 101)97 S1 1L 14)861 32)543 43)329 59)355 62)965 88)143 91)124 106)227

5L 14)862 35)063 41)330 59)367 69)779 92)217 86)090 102)787 15L 14)869 37)275 39)278 59)645 75)321 95)709 80)942 99)533

Boron/Ep: X1,1 X1,2 X1,3 X2,1 X2,2 X2,3 X3,1 X3,2

[15] 1L 11)039 17)364 30)905 40)371 44)157 53)269 89)663 92)701 5L 11)039 24)037 49)281 36)790 44)157 63)520 81)425 86)002 15L 11)039 28)866 61)571 33)138 44)157 71)705 72)136 79)306 S1 1L 10)778 16)957 30)162 39)213 42)904 51)785 86)388 89)325 5L 10)782 23)445 47)915 35)793 42)969 61)724 78)343 83)170 15L 10)803 28)156 59)723 32)259 42)931 69)590 69)882 76)854

Gr)/Ep: X1,1 X1,2 X1,3 X2,1 X2,2 X1,4 X2,3 X3,1

[15] 1L 11)290 17)132 28)692 40)740 45)159 45)783 54)082 90)055 5L 11)290 24)035 48)362 37)089 45)159 83)230 64)470 81)205 15L 11)290 28)990 61)156 33)359 45)159 106)740 72)766 72)063 S1 1L 10)727 16)286 27)243 38)565 42)773 43)395 51)237 84)819 5L 10)729 22)794 45)698 35)058 42)712 78)267 60)888 76)221 15L 10)729 27)563 57)738 31)677 42)818 100)098 68)607 67)799 E-glass/Ep: E1/E2"2)45, G12/E2"0)48, G23/E2"0)342, l12"0)23, lp23"0)462, E3"E2.

Boron/Ep: 11 0)34 0)346 0)21 0)444 G13"G12.

(Gr/Ep): 15)4 0)79 0)299 0)30 0)675 l13"l12.

TABLE11

Fundamental and ,fth mode frequencies of [$45]S square plates with varyingthickness, X"ua2(o/E2h2)1@2

Reference a/h 5 10 15 20 25 30 35 40 50

[25] X1 9)57 12)77 13)84 14)28 14)51 14)63 14)71 14)75 14)87 S2 9)46 12)63 13)71 14)18 14)41 14)54 14)63 14)68 14)75 S3 9)13 12)06 13)10 13)58 13)85 14)02 14)14 14)23 14)36 [25] X5 25)51 41)69 50)39 55)08 57)77 59)43 60)49 61)23 62)13 S2 22)11 34)71 43)63 55)05 57)71 59)35 60)41 61)14 62)03 S3 15)93 29)63 40)62 44)97 47)78 49)82 51)40 52)68 54)66 E1/E2"15, E3/E2"1, G12/E2"0)4286, G13"G12, G23/E2"0)3429, l12"l13"0)4, lp23"0)458.

TABLE12

Fundamental frequenciesvarying thickness and aspect ratiosX"ua2(o/E2h2)1@2 of [$45]S rectangular plates with

Reference [28] (b/a) S2 (b/a)

a/h 1 2 3 4 5 1 2 3 4 5

10 12)716 7)849 6)704 6)276 6)073 12)588 7)775 6)650 6)232 6)034 20 14)074 8)346 6)928 6)568 6)342 13)924 8)268 6)968 6)520 6)300 50 14)551 8)505 7)155 6)658 6)425 14)396 8)428 7)098 6)611 6)383 100 14)623 8)529 7)171 6)671 6)437 14)468 8)452 7)115 6)624 6)395

S1 (b/a) S3 (b/a)

10 12)594 7)778 6)652 6)233 6)035 12)160 6)836 5)499 4)908 4)588 20 13)927 8)269 6)967 6)521 6)300 13)472 7)624 6)172 5)571 5)257 50 14)396 8)428 7)098 6)611 6)383 14)145 8)216 6)837 6)296 6)024 100 14)468 8)451 7)114 6)624 6)395 14)382 8)391 7)040 6)532 6)286 E1/E2"14, E3/E2"1, G12/E2"0)533, G13"G12, G23/E2"0)323, l12"l13"0)3, lp23"0)521.

TABLE13

Fundamental frequencieswith varying aspect and thickness ratiosX"ua2(o/E2h2)1@2 of [$45/$45] rectangular plates

a/b 0)2 0)6 1)0 1)6 2)0 0)2 0)6 1)0 1)6 2)0

Reference a/h"10 a/h"20

[21] 8)66 12)82 18)46 27)95 34)87 9)30 14)45 21)87 35)56 46)26 [22] 8)72 12)97 18)61 27)74 34)25 9)48 14)90 22)58 36)25 46)79 [46] 4)93 12)65 18)06 27)18 31)28 9)52 14)72 22)19 35)89 46)45 [52] 8)55 12)56 17)79 26)99 33)55 9)30 14)39 21)68 35)04 45)41 S2 8)38 11)40 15)64 22)73 28)01 9)57 14)38 21)12 32)96 42)02 S3 5)22 8)36 13)65 19)65 24)81 5)94 9)41 15)60 23)02 29)57

a/h"30 a/h"50

[21] 9)44 14)84 22)74 37)82 49)98 9)51 15)04 23)24 39)17 52)29 [22] 9)67 15)39 23)68 38)94 51)13 9)82 15)69 24)34 40)65 53)99 [46] 9)72 15)22 23)28 38)59 50)89 9)84 15)50 23)91 40)24 53)68 [52] 9)49 14)84 22)69 37)59 49)55 9)62 15)12 23)30 39)19 52)25 S3 6)58 10)22 16)67 24)61 31)64 7)66 11)77 18)64 27)29 34)78 E1/E2"40, E3/E2"1, G12/E2"0)6, G13"G12, G23/E2"0)5, l12"l13"l23"0)25.

Combined e+ects of aspect and thickness ratios: To show the e!ects of aspect and thickness ratios, vibration of the [$45$45] skewsymmetric angle-ply laminated rectangular plates is treaded with the S3 sliding pin supported boundary condition.

Results of S3 hinge-roller support displacement "eld are also listed for further comparison. Fundamental frequencies are compared in Table 13, with varying

a/h, a/b ratios, among many authors. Bert and Chen [21] provided a closed-form solution to the problem by way of the classical thin plate theory with shear deformation taken into account. Ghosh and Dey [46] employed a simple "nite element based on higher order theory to analyze free vibration of the laminated plates. Shankara and Iyengar [52] applied a C3 "nite element model based on HSDT to the free vibration of composite plates. Cross references are made to Reddy [22], who used an FSDT quarter and half plate "nite element. Remarkably, lower frequencies are always obtained in the present study except for the case of Ghosh and Dey [46] at a/b"0)2 and a/h"10 in Table 13, of which the frequency was even lower.

Fiber orientation: To examine the e!ect of "ber orientation, the "rst, third, and

"fth mode natural frequencies of symmetrical four-layer angle-ply laminated thin square plates are presented in Table 14 with three materials: E-glass/epoxy, boron/epoxy, and graphite/epoxy. The present S3 theoretical results compare well with all varied ply angles in Leissa and Narita [15], and Chow et al. [17], in which the transverse vibration problems were studied by the Rayleigh-Ritz method.

Quasi-isotropic hybrid: The e!ects of thickness ratio and "ber orientation on fundamental frequencies are presented in Table 15. The present results compare well with the three-dimensional elasticity solution of Noor and Burton [7] for 10-layered angle-ply and 16-layered quasi-isotropic hybrid laminates. Fiber orientation for the quasi-isotropic hybrid laminates is [45/!45/0/90/45/

!45/0/90]2. The top four and bottom four layers are made of graphite-epoxy material, and the middle eight layers are made of glass-epoxy material. (*values are tabulated asX]100 for h/a"0)01)

3.5. GENERAL LAMINATION

In general lamination schemes, the fundamental natural frequencies of laminated plates are shown in Table 16 for varied thicknesses ratios. Lower frequencies are

TABLE14

E+ect of ,ber orientation on ,rst few mode frequencies of symmetrical angle-ply thin square plates with a/h"50,X"ua2(o/D

3 )1@2

[$h]S Reference E-glass/Ep Boron/Ep Gr/Ep

h X1 X3 X5 X1 X3 X5 X1 X3 X5

03 [15] 15)19 44)42 64)53 11)04 30)91 44)16 11)29 28)69 45)16 [17] 15)19 44)52 64)55 11)04 30)92 44)18 11)30 28)70 45)18 S3 14)18 38)37 57)59 9)55 25)81 31)92 10)06 23)89 33)58 303 [15] 16)02 42)62 71)68 12)83 36)62 52)13 12)66 36)67 51)84 [17] 15)94 42)52 71)45 12)78 36)36 51)59 12)56 36)40 51)23 S3 15)22 40)18 66)71 12)42 33)63 46)40 12)42 34)25 47)33 453 [15] 16)29 41)63 77)56 13)46 34)94 57)59 13)17 34)76 57)61 [17] 16)17 41)52 77)33 13)39 34)55 56)84 13)12 34)36 56)85 S3 15)49 40)16 71)00 13)13 33)59 49)98 13)05 33)82 50)96

TABLE15

E+ect of thickness ratio and ,ber orientation on fundamental frequencies of angle-ply and quasi-isotropic hybrid square laminates,X"uh(o/E

3 )1@2

h h/a 0)01* 0)05 0)10 0)15 0)20 0)25 0)30

P1 03 S3 0)1082 0)0233 0)0453 0)1760 0)2855 0)4068 0)5359 153 [7] 0)1328 0)0320 0)1162 0)2304 0)3588 0)4934 0)6307 S3 0)1112 0)0235 0)0842 0)1743 0)2850 0)4097 0)5429 303 [7] 0)1510 0)0362 0)1296 0)2532 0)3889 0)5286 0)6692 S3 0)1274 0)0260 0)0928 0)1915 0)3128 0)4489 0)5930 453 [7] 0)1595 0)0381 0)1351 0)2617 0)3993 0)5400 0)6810 S3 0)1409 0)0285 0)1014 0)2082 0)3379 0)4810 0)6220 P2 [7] 0)1354 0)0329 0)1217 0)2463 0)3893 0)5405 0)6946 S3 0)1229 0)0240 0)0872 0)1824 0)3029 0)4438 0)5965 Properties EL/E0 ET/E0 GLT/E0 GTT/E0 lLT lTT o/o0 Stacking

P1 : 15 1 0)50 0)35 0)30 0)49 1)0 [h/!h/2]10

P2 : Gr/Ep 11)49 1)14 0)56 0)28 0)38 0)49 0)846 H:45/!45/0/90 Gl/Ep 4)46 1 0)566 0)395 0)415 0)49 1)0 [HGr/HGl/HGl/HGr]

TABLE16

Fundamental frequenciesX"ua2(o/E2h2)1@2 for square plates with di+erent lamina-tion schemes

Reference [51] S3

a/h 4 10 100 4 10 100

[0] 7)739 12)465 15)193 6)827 9)520 13)166

[0/30/0] 7)573 12)380 15)353 6)896 10)009 14)254

[0/45/0] 7)413 12)213 15)400 6)821 9)917 14)362

[0/60/0] 7)258 12)005 15)340 6)633 9)611 14)075

[0/90/0] 7)123 11)758 15)177 6)342 9)102 13)436

[0/90] 6)809 8)951 9)690 5)565 6)948 9)423

[0/90]2 7)557 11)845 14)025 5)912 8)040 11)423

[0/$30/0] 7)606 12)447 15)401 6)511 8)951 11)659

[0/$45/0] 7)411 12)272 15)441 6)397 8)865 11)894

[0/$60/0] 7)197 11)930 15)251 6)174 8)608 11)727

obtained in all cases in the present theory of S3 displacement models as compared to Maiti and Sinha [51], in which HSDT with a third order, six degrees of freedom per node "nite element was employed.

4. CONCLUSION

1. In the treatment of free vibration of composite laminated thick and thin plates, a complete survey of the literature and comparisons of natural

frequencies have been performed according to the present three-dimensional theory. Lowest frequencies are obtained with few exceptions via a three-dimensional augmented energy variational approach leading to the natural state.

2. Unlike the traditional theories of laminated plates and shells, the present three-dimensional semi-analytical solutions are based on the theory of elasticity. The three-dimensional boundary conditions and interlaminar continuity of layer displacements and transverse stresses are satis"ed by use of the assumed admissible displacement "elds and Lagrange's multipliers.

3. Systematic three-dimensional displacement functions have been developed for a variety of edge boundary conditions such as the S1 "xed pin, S2 hinge-roller, and S3 sliding pin supported displacement "elds, in keeping with physical reality and mathematical requirements.

4. Judging from the lowest natural frequencies, it is noted that the S2-type displacement functions are most suitable for use with the cross-ply laminates in bending extension}compression, and S3-type displacement functions for angle-plies in in-plane shear, due to ease of normal and tangential movements along the edges respectively.

ACKNOWLEDGMENTS

This work was supported by the National Science Council of Republic of China through grant NSC86-2212-E-007-007.

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