Free vibration of symmetrically laminated plates using a higher-order theory with finite element technique

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGJNFERTNG, VOL. 28, 1875-1889 (1 989)

FREE VIBRATION OF SYMMETRICALLY LAMINATED

FINITE ELEMENT TECHNIQUE PLATES USING A HIGHER-ORDER THEORY WITH

M A L L I K A R J U h A AND T KANT

Department of Cii 11 tnginerring, Indian lnstitutr of Technolog], PoNai. Bombaj 4110 076, lndia

SUMMARY

A C'O finite element formulation of the higher-order theory is used to determine the natural frequencies of isotropic, orthotropic and layered anisotropic composite and sandwich plates. The material properties that are typical of high modulus fibre reinforced composites are used to show the parametric effects ofplate aspect ratio, Icngth-to-thickness ratio, degree of orthotropy, number of layers and lamination angle/scheme. The present theory is based on a higher-order displacement model and the three-dimensional Hooke's laws for plate material. The theory represents a more realistic quadratic variation of the transverse shearing strains and linear variation' of the transverse normal strains through the plate thickness. A special mass matrix diagonalization scheme is adopted which conserves the total mass of the element and includes the effects due to rotary inertia terms. The results.presented should he useful in obtaining better correlation between theory and experiment, and to numerical analysts in vcrifying their results.

INTRODUCTION

The increasing use of composite materials as thick laminates, in aerospace and other industries, has clearly demonstrated the need for the development of new theories to efficiently and accurately predict the behaviour of such structural components. In classical thin plate theory one assumes that normals to the mid-surface during deformation remain straight and normal to it, implying that the transverse shear effects are negligible. As a result, the free vibration frequencies calculated by using the thin plate theory are higher than those obtained by the Mindlin plate theory,' in which transverse shear and rotary inertia effects are included; the deviation increases with increasing mode number. The transverse shear effects are even more pronounced, owing to the low transverse shear modulus relative to the in-plane Young's moduli, in the case of filamentary composite plates. A reliable prediction of the response characteristics of high modulus composite plates requires the use of shear deformable theories.

A number of shear deformable theories have been proposed and some are reviewed in Reference 2. They range from the first such theory by Stavsky3 for laminated isotropic plates, through the theory of Yang et uL4 for laminated anisotropic plates to various effective stiffness theories such as those discussed by Sun and Whitney,' Whitney and Sun's higher-order theory,6 the three- dimensional elasticity theory approach of Srinivas et ~ 1 . ~ 7 and Noor." It has been shown by various investigators5 p 8 that the YNS (Yang-Norris-Stavsky) theory was adequate for predicting gross structural behaviour in the first few flexural modes. Whitney and Pagano" employed the YNS theory to study the cylindrical bending of antisymmetric cross-ply and angle-ply plate strips (see also References 11 and 12). Bert and Chent3 and Reddy14 presented, using the YNS theory, a

0029- 598 1/89/081875-15$07.50 0 1989 by John Wiley & Sons, Ltd.

Received 24 May 1988 Revised 6 December 1988

1876 MAI.1,IKARJIJNA AND T KANT

closed form solution and finite element results respectively for the free vibration of simply supported rectangular plates of antisymmetric angle-ply laminates. In References 15 and 16 vibration of only cross-ply laminated plates was considered.

In recent years, many refined plate theories have been presented to improve the predictions of ~ t a t i c ’ ~ - ’ ~ and d y n a m i ~ ’ ~ - ~ ’ behaviour of laminated plates. The present investigation is concerned with the dcvclopment of a simple Co isoparamctric finite element model based 011 a higher-order theory and its application to the free vibration of symmetric cross-ply and angle-ply laminated composite and sandwich plates.

EQUATIONS O F MOTION

Hamilton’s principlc is employed here to derive the equations of motion. The functional of interest IS

F = r’ ( E - n) dt (1) r l

where ‘1’ is the time, E IS the total kinetic energy of the system and 71 is the potential energy of the system, including both strain energy V and potential of conservative external forces W. Since the primary interest here is in the free vibration analysis, the potential energy due to applied loads Wis zero.

Using d , , d,, . , . , d, as the generalized displacements and assuming that they are independent, the Euler-Lagrange equations then yield the well known Lagrange equations of motion, given as follows:

d dF 2F dt($d,,

=0, r = l , 2 , . . . , R

where R is the total number of degrees of freedom in the system.

of motion in matrix form can be written as When space is discretized with the usual finite element method, the above Lagrange equations

Kd+Md=O (3) where K and M are the global stiffness and mass matrices respectively obtained by the assembly of the corresponding element matrices, d is the nodal displacement vector and d is the second derivative of the displacements of the structure with respect to time.

The above relation (3) is the global discrete equation for free vibration. We now assume a solution for d of the form

(4) where d is the vector of unknown amplitude at the nodes (modal vector), and w is the circular natural frequency of the system. When equation (4) is substituted in relation (3), one gets

d = d ei<Ot

(K - w2M)d = 0 (54 Equation (5a) can be solved, after imposing boundary conditions of the problem, by any standard eigenvalue program. For the purpose of evaluation, relation (5a) is converted into the standard eigenvalue format,

( K - A M ~ = O ; (5b)

FREE ViBKh?IObi OF LAMlNATFD KATFS 1x77

arid a subspace iteration method3’ is used to obtain the eigenvalues 3, and associated eigenvecfors a.

A Cartesian co-ordinate system is chosen in such a way that the x y plane coincides with the midplane of the platc of sides 0, b and of total thickness h (also h , . h,, h,, etc. are the thickness of individual layers in the case of layered platc). The components of displacements are assumed to be as follows:

w(x, y1 z, t ) -=I W{,( Y, y , t ) -i zJM$(x. y, t )

where t is the time, u, u, w are the dtsplacements in the x , y, z directions respectively, wo denotes the transyerse displacements of the midplane, cc/x and i,hY are the middle-surface slopes in the YZ and yz planes respectively. ‘The parameters w;, @, 4; are the higher-order terms in the Taylor‘s series expansion. It may be noted that the expressions (6) for displacement components result in the linear variation of transverse normal strain and quadratic variation of transverse shear strains through the thickness of the plate.

In Co finite element theory, the continuum displacement vector within the element is discretized such that

in which N N i s the number of nodes in an element. N , is the simple isoparametric interpolating (shape) function associated with node i in terms of the normalized co-ordinates 5 and q, and 6, is the generalized displacement vector corresponding to the ith node of an element. The generalized strain E at any point within an element can be expressed by the following relationship^:^^

“

E = C B,8, ( 8 4 I = 1

where

The non-zero elements of the strain-displacement matrix B are given below:

1878 MALLIKARJLINA AND 7. KANT

where

in which

C 1 1 = E l ( 1 -V32V23)/A, '1 2 = E2(vl 2 + '1 3'32)IA

C13=E3(V13 +'1ZVZ3)IA, C22=E2(1 -V31V13)/A

C 2 3 = E 3 ( v 2 3 + 1'13V21)/A, '33 = E3(1 - V12v21)/A (94

C 4 4 = G l Z i C 5 5 = G 2 3 , C 6 6 = G 1 3

and A =( l -v23v32-vl 2v21 -'13'31 -'I ZV23'31 -v13v32v21)

where C = cos 0, S =sin 0. It may be that the above notations Qij and Cij are used here to denote the transformed reduced

stiffnesses with respect to x and y and reduced stiffnesses with respect to directions 1 and 2 respectively (see Figure I) instead of Q i j and Qij as given in Reference 34.

Further, moment and force stress-resultants-per unit length are introduced in the present shear deformation theory after analytical integration through the plate thickness. The elements of the

FREE VIBRATION OF LAMINATED RATES 1879

TYPICAL LAMINA

I1,2,3 ) - LAHINA REFERENCE AXES

_-.-.-.-.-.-.-. LAMINATE MID-PLANE

( X,Y,z )-LAbIINA?E REFERENCE AXES

Figure 1. Laminate geometry with positive set of 1aminaAaminate reference axes, displacement components and fibre orientation

moment vector ‘ M are defined as follows:

(M:, M; , M:J = c (ox, d y , T x y ) z2 dz L = l rL+’ h L

The components of the shear force vector ‘Q’ are defined as follows:

1880 MALLIKARJ UNA AND T. KANT

Substituting equations (9) in equations (10) and integrating with respect to z we obtain the stress-resultants expressed in terms of six generalized displacements as

6=DE

where the moment stress-resultants are

= ? I.= 1

Q11H3

Symmetric

In these relations, L defines the Lth layer, n=number of layers in a laminate and

Hi=-@ ! 2 ; + , - h i ) , i = l , 3 , 5 , 7 (1 Id)

Upon evaluating the matrices B and D as given by equations (8) and (ll), respectively, the element stiffness matrix can be readily computed using the standard relation,

1881 FREE VIBRATION OF LAMINATED RATES

where IJ/ is the determinant of the Jacobian matrix. A diagonal mass matrix that is more sophisticated than a lumped mass matrix is used here, and

is discussed elsewhere.35 Test cases to date show that the acculracy of this form of diagonal mass matrix is excellent.

Thc mass matrix M in equation ( 5 ) is given by

M = NTmN d(Area) ( 1 3 4

N=CNi, N,, . . . I NNL~ ( 1 3b)

Area

where

in which I , and I, are the usual inertias corresponding to wo and 0, (also 0,) degrees of freedom, while I , and 1, are the higher-order inertia terms corresponding to the higher-order degrees of freedom wg and 0: (also 0:) respectively, and these are defined explicitly as follows:

where p I , is the material density of the Lth layer.

the global stiffness and inertia properties by following the standard assembly pr~cedure . ,~ The preceding matrix operations, though performed at element level, are made use of to obtain

DISCUSSION OF NUMERICAL RESULTS

To demonstrate the versatility of the present method, a considerable number of examples, including composite sandwich plates with simply supported and clamped boundary conditions, are investigated. The general configuration of the rectangular plate is shown in Figure 1. A 4 x 4 finite element mesh for the whole plate is used throughout. The selective integration scheme, i.e. 3 x 3 Gauss rule for bending and inertia terms and 2 x 2 Gauss rule for shear terms in the energy expression, was empioyed in the numerical integration. The computer programs PFOST (Program for First Order Shear deformation Theory) and PHOST (Program €or Higher-Order Shear deformation Theory) were developed separately to predict the natural frequencies and modal shapes of composite and sandwich plates. The following boundary conditions were used:

wa = * ~ == &l* o - - $* , -0 -. a tx=O,a ~ ~ = l j i ~ = w ~ = $ ~ = O at y=O, b

Simply supported (SS):

Clamped (CC): wa = +x = t,hy = W E = = I): = 0 at x = O , a and y=O, h

In all the numerical examples, a plate is discretized with nine-noded L,agrangian quadrilateral elements. All the computations were carried out on a CYBER 180,4340 computer in single precision. For the purpose of comparison, some of the evaluations were conducted on sample

1882 MALLIKARJlJNA AND T. KANT

problems for which 3-D elasticity solutions, Mindlin's thick plate theory and finite element results, and classical thin plate theory solutions are available in the open l i t e r a t~ re .~ p9. 14- 2 6 , 36, 37 A s a check on the numerical accuracy of the finite element method, natural frequencies (0) were first obtained for a thick isotropic square plate with a side-to-thickness ratio of 10. In the second example the following orthotropic material properties, typical of aragonite crystals, were used8

CZ2/C1 =0.543103; C3JCl1 =0.530172

C,,/C,, =@23319; C131Cll =0.010776

C2,IC,, =0.098276; C,,/C,, =0.262931

C6,/Cl, =0.159914; C,,/C,, =0.26681

The values of C,, and p are arbitrary because of the non-dimensionalization used (set to unity here). The results are compared with those from a 3-D linear elasticity solution,8 Mindlin's analytical37 and finite element results'". 36 and the classical thin plate theory (CPT) in Tables I and 11. It is seen that the results obtained by using present higher-order theory are very close to those of 3-D elasticity solutions. The CPT overestimates the frequencies and the effect of transverse shear deformation increases with increasing mode numbers. The analytical results of the Mindlin theory3' presented in Table I appear to be better than the finite element results with both the Mindlin theory36 and the present higher-order theory. It should, however, be noted that the finite element method is an approximate one and the present results are obtained with only a 4 x 4 mesh in a full plate. Further refinement of the mesh could yield improved results.

Table I. Comparison of non-dimensional frequencies w= ci~Jph*,G of a square simply supported plate, v = 0.3, ajli = I0

Linear Present 3-D ela- FEM Mindlin's Rock and Hinton3' Classical sticity using thick plate thin plate

rn n theory' HOST theory" Scheme 1 Scheme 2 Scheme 3 ReddyI4 theory

0,0929 0.0930 00931 * 0'0932 ( - 0.32) (-0.21) ( - 4 . 1 1 )

0.2216 02218 0.2217 2 1 0.2226

(-0.44) (-0.38) (-0.41)

0.3379 0.3402 0.3409 2 2 0,3421

(- 1-22) (-0.55) (-0.34)

0,4184 0.4144 0.4387 3 1 0.4171

(t0.31) (-0.66) (+5.16)

0.5152 0.5197 05484 ( - 1.66) (~~ 0.80) ( + 4.66)

3 2 05239

06941 0.682 I 0,7838

( + 0.75) i - 099) ( i- 13.77) 3 3 0.6889

0.7610 07431 0.8779 4 2 0.7511

(f1.31) (-1.06) ( t16.88)

0.0935 ( + 0 3 2 ) 0.22 I6

( - 0.47)

0.3280 (-4.12)

0.3931 ( - 5.77)

0.4707 ( - 10.15)

0.5551 ( - 19.43)

0.9892 ( + 3 1.70)

0-09 3 5 (+ 0.32) 0.2236

( + 0.43)

0.3327 ( - 2.75)

0.4 I73

i + 0.05) 0.5084

(- 2.97)

0.6532 (- 5.1 8)

0.7626 (+ 1.53)

0.0934 (+0.21)

( + 1.97) 0,2270

0.444 1

(+ 6.47) 0-5651

( + 7.86) 0.80 13

(+ 16.31)

0.0963 (+ 3-32)

0.2408 (+8.17)

0.3853 (+ 12.63)

0.4816 ( + 15.45)

06261 (+ 19.50)

0.8686 (+ 25.83)

09632 (+ 28.23)

Values in parentheses give perccntagc errors with respect to the elasticity solution


Table TI Comparison of circular frequencie\ oi=toh(t'p/C, ,). of a simply supported orthotropic square plate (u, h = 10)

FEM Linear 3-D Classical rcsults

elasticity Closed form Present thin plate using jn n theory* solutlon'6 HOST theory FOST

1 1

1 2

2 1

2 2

1 3

3 1

2 3

3 2

1 4

4 1

3 3

2 4

4 2

0.0474

0.1033

0.1188

0.1 694

01888

02180

0.2475

02624

0.2969

0.3319

0.3320

0.3476

0.3706

0.0474

0.1033

0.1 189

0.16%

0.1888

0.2 184

0.2477

0.2629

0.2969

0.3330

0.3326

0.3479

0,3720

0.0474 (0.00) 0.1032

( - 0.09) 0.1 190

( + 0.1 6) 0.1687

(-0.41) 0.1903

( + 0.79) 0.220 1 (t 0.96) 0-2450

( - 1.01) 0.2605

( - 0.72) 0.2855

( - 3.83)

( - 3.43)

( - 2.83)

0.3205

0-3226

0-3410 (- 1.89) 0-3725

(+0.51)

0.0493 (+4.01) 0 1095

( + 6.00) 0.1327 (+ 11.7) 0.1924

( + 13.57) 0.2070 (+ 9.64) 0.2671 + 2 2 5 2 ) 0.2879 + 16.32) 0.3248 t23.78) 0.3371 t 13.54) 0.447 1 + 34.71) 0.41 72 + 25.66) 0.41 52 + 19.45) 0.501 8

( + 35.40)

0-0473 (-0.21) 0.1 03 15 (-0.14) 0.1 190

(+@I61 0.1682

0. I906 ( + 0.95) 0.2205

( +. 1.1 5 ) 0,2432

( - 1.73) 0.2590

(- 1.30)

( ~ 0.7 1)

0.2849 (- 4.04) 0.3116

( - 6-12) 0.3218

f - 3.07) 0-3390

( - 2.47) 0.3798

( + 2.48)

Values in parentheses give percentage errors with respect to the clasticity solution

Rock and H i n t ~ n ~ ~ showed that, of all the available mass lumping techniques, the special mass lumping scheme (Scheme 3), in which mass is lumped in the proportion of the diagonal entries of the consistent mass matrix while at the same time preserving the total mass of the element, gives excellent results. This particular approach is adopted here in the present work. Both Rock and H i n t ~ n ~ ~ and Reddy14 have presented Mindlin finite element results with eight-noded serendipity quadrilateral elements using uniform reduced integration ( i s . 2 x 2 Gauss rules for evaluation of all the terms of K and M). It is seen in Table I that the results from the present formulation are much superior and are close to the 3-D elasticity solutions compared to FEM results using FOST.I4. 36 In Table 11, it is found that the difference in the percentage of error between present HOST and FOST is upto 2.5 per cent for higher modes. For lower modes, the results of both HOST and FOST are close to 3-D elasticity solutions.

In the third example, two sets of dimensionless material properties (typical of high-modulus graphiteeepoxy) are used to study the dynamic behaviour of laminated composite plates. Each

layer is a unidirectional fibre reinforced composite possessing the following engineering elastic cmstants:

"Iz=vz3=Ylj=o~:s The properties for Material-l above are taken from t%gann'H 39 and that for Material-2 from Noor.' It should be noted that these two rnaterials are not transversely so tropic,'. ", 39 but are orthotropic with nine elastic constanis.* These ply elastic constants are used here tn the computations only to illusmte the general nature nf the behaviour of the problems coasidered here. (These constants arc selected for historical reabons h they have been used in many previous studies, see e.g Refererices 5, 6, 17-10, 24, 38 and 39.1 It IS assumed that both materials have p = 1.0 and E , = L , =- 1.0. Table IJI d m v s the effect of fenrrth-eo-thickness ratio on the dimensionless fundarnentai frequencies. The plate is made up . ~ f four-layer, equal thickness, symmetric cross-ply (On/90n/90*/O') laminae with Material-2. 'The 1 esailts are very clwe to the closrd form solutions ofL% high-order theory.2h Of course, FOST m u ! t s are also close to the CFS, since these are first fundamental frequencies. These results clcarly indicate that the effect of thickness on the fundamental frequencies is considerable and in general the inaccuracy IS seen to increase with decreasing ( a h ) ratios of laminates. While in the case of thin laminates the difference between the present and the CPT results is negligible, it i s as high as 93 per cent for a thick laminate with aili = 4. It was demonstrated by Wu and Yinson,4" for example, that the effect ofn/h in reducing the fundamental frequency is much more pronounced for plates of unidirectional advanced composite material than it is for homogeneous, isotropic plates of the same planforrn diniensjons. The explanation for this Is the low ratio of transverse shear moduli to in-plane

rahle Ill Effect of length to thicknebs ratio (uih) on the drrnenaioriless fundamental frequencies, w =(ou' ,k) J p / E , , of wnply supported cross-ply (0 /90 /90L,i0') square

platcs, Material-2, n/h= 10

ajh 4 5 10 20 50 100

Present HOST 9.258 10.740 15.090 17.637 18.669 18.835

FEM results using 9.227 10.736 15.073 17.628 38.679 18.835 (-2.51) (-2-26) (-1.17) (--0.17) (1-0.35) (+0.42)

FOST ( ~ - 2.84) (- 2.30) ( - I "29) (- 0-22) (+ 0.35) (+ 0.42) Closed form 9.497 10.989 15.270 17.668 18.606 18.755 solution2"

plate theory (+88.5) (+65.7) ( t 2 2 . 1 ) (+6.2) (c1.03) (+0.26) Classical 17.907 18.215 18.652 18.767 18.799 18.804

Values in parenthcses give percentage errors with respect to the clmed form solution 26

*The elastic constants listed here do not constitute a symmetnc array and thus cannot conserve energy, 1.e. the whole concept of potential energy is invalid here because vl2/E,', # vI3/E,

Tabl

e JV

. Ef

fect

of

degr

ee o

f or

thot

ropy

of t

he in

divi

dual

lay

ers

on th

e fu

ndam

enta

l fre

quen

cy o

f si

mpl

y su

ppor

ted

squa

re

mul

tilay

ered

com

posi

te p

late

s w

ith a

,& =

5, r0

= w

(ph2

/Ez)

','. M

ater

ial-2

Ell

&

No.

of

laye

rs

Sour

ce

3 I 0

20

30

40

3-D

ela

stic

ity

theo

ryY

Pr

esen

t (E

rror

) 3

FOST

(E

rror

) C

PT (E

rror

) 3-

D e

last

icity

th

eory

' Pr

esen

t (E

rror

) 5

FOST

(E

rror

) C

PT (

Err

or)

3-D

ela

stic

ity

theo

ry'

0.26

474

0.26

1 26

(- 1

.3)

0.29

198

(+ 10

.2)

0.26

587

0.26

124

( - 1.

32)

0262

55 (

- 1.

2)

0.26

255

(-

1.2)

0.

2919

8 (i

9.8

) 0.

2664

0

0.32

841

0,32

528

0.32

5 I9

0.

4126

4 0.

3408

9

0.33

621

0.33

621

0.4 1

264

0.34

4 32

0.38

241

0-41

089

0430

06

-0.9

) 03

7253

(-2.

5)

0.39

884

(-2.

9)

0415

21 (

-3.4

) --

0.98

) 0.

3722

1 (-

2.6

7)

0.39

721

(~ 3

.3)

0.41

501

(- 3

.5)

+ 25.

6)

0.54

043

( +41

3)

0.64

336

(+ 56

.5)

0.73

196

(+ 70

.2)

0.39

792

0.43

140

0.45

374

- 1

.3)

0'39

192

(- 1

.5)

0.42

482

(--

1.5)

0.

4469

5 (-

1.5

) --

1.3

) 0.

3919

2 (-

1.5

) 0.

4245

6 (~

1.5

8)

0.44

628

(- 1

.6)

+21.

0)

0.54

043

(+35

.8)

0.64

336

(750

.9)

0.73

196

(+61

.3)

0-40

547

0,44

2 10

0,46

679

Pres

ent (

Err

or)

0.26

298

(--

1.28

) 0.

3403

5 ( - 1.

1 5)

0.40

1 07 i -

1 Q8)

0.

4375

5, (-

1.

02)

0,46

222

( - 0.

97)

9 FO

ST (E

rror

) 0.

2629

7 (-

1.2

8)

0.34

035

1 -~

1.15

) 0.

4010

7 (

-~

1.

08)

0437

55 (-

1.

02)

0462

22 (

-0.9

7)

CPT

(E

rror

) 0.

2919

8 (+

9.6)

0,

4126

4 (-

+ 19

.8)

0.54

043

(+ 33

.7)

0.64

336

(~4

5.5

2) 0.

73 L9

6 (+

56.8

)

1

?J m

rn

Val

ues

in p

aren

thes

es g

ive

perc

enta

ge e

rror

s w

ith

resp

ect t

o th

e el

astic

ity s

oiur

ion'

1886 MALLIKARJUNA A N D T. KANT

Young's moduli for the advanced composite. The same trend was found in the present investigation. For example, for a square plate with a/h= 10, the HOST prediction of the fundamental frequency is 3.65 per cent lower than the CPT one for a homogeneous, isotropic plate but 23.6 per cent lower for the present cross-ply graphiteeepoxy plate. The effects of number of layers and degree of orthotropy of the individual layers (OC/9Oc/O0/ . . . . / O 0 ) on the dimensionless fundamental frequency (Material-2) are shown in Table IV. The present HOST results are in good agreement with the 3-D elasticity solution. Since the 3-D elasticity results of first fundamental frequencies are being compared with HOST and FOST, the same trend between HOST and FOST is observed here also as in Table 111. The error in the CPT predictions is mainly attributed to the neglect of shear deformation. This is demonstrated by the fact that the error in the predictions of HOST with 3-D elasticity theory did not exceed 3.4 per cent (even for the case of highly orthotropic thick plate). Further mesh refinements could yield still better accuracy.

In the fourth example for the convenience of designers in a wide variety of design situations, the fundamental frequency results are presented here in dimensionless form as a function of plate aspect ratio, length-to-thickness ratio, lamination angle and material orthotropy. A three-layered, symmetric angle-ply ( O / - O / # ) plate of equal thickness was used for the parametric study. From Table V, it is found that, as lamination angle (0) increases, the fundamental frequencies decrease with plate aspect ratio a/b < 1.0 and increase with u/h > 1.0 keeping a = 1. The frequencies increase with increase of aspect ratio, but decrease with increase in length-to-thickness ratio. It is also seen that non-dimensional fundamental frequencies increase with the increase of degree of orthotropy ( E I I E Z ) .

Table V. Effect of plate aspect ratio (alh), length-,to-thickness ratio (czlh), lamination angle ( f l ) on the dimensionless fundamental frequencies, W = w(\/ph2/E2) x 100 of simply supportcd rectangular plate

with stacking sequence as ( f ? / - U / f I )

Material-1 Material-2

ilih aih 0' 0 2 0.5 1 .0 1.5 2.0 0.2 0 5 1 .0 1.5 2.0

30 5 45

60 30

10 45 60

3 0 20 45

60

30 30 45

60

30 40 45

60

30 50 45

60

27.23 1 21.952 16.348

9. I535 6.9203

4.7850

2,6056 1,9025 1,2652

1.1936 0.8640 0.5690

0.6790 0.4899 03215

0.4369 0.31 47 0.2061

29.157 36.220 25,629 36-647 22.1 73 36.220

9.8782 12.709 8.3Y 2 I 12.926

6.98 12 12.709

2.8231 3.7332 2.3539 3.8262 1.9097 3.7332

1.2946 I7263 1.0745 1.7748 08659 1.7263

0.7367 0.9859 0.6105 14.)150 0-4907 0.9859

0.4741 0.6356 0.3926 0.6549 03151 0.6356

46.552 58-51 5 32.985 50.208 65.576 27.21 5 53,430 72.1 76 19.723

17.015 22.173 11.360 18.758 25.629 8.5811 20.338 29.157 56776 5.1763 6,981 0 3.2622 5.8561 8.3919 2-3538 6,4338 9.8780 1.4926

2.4220 3.3085 1.4968 2.7679 4'0457 1.0680 3,051 5 4.8245 0.6706

1.3901 1.9097 0.8519 1.5958 2.3540 0.6054 1.7614 2.8232 0.3787

0.8985 1-2379 0.5483 1.0338 1.5324 0-3888 1.1417 1,8432 0.2428

35.275 43.867 31.589 45.043 26.856 43.867

12.177 15.51 I 10.330 15.904 8,4042 15.5 11

3.5064 4.5679 2.8890 46958 2.292 I 4.5679

1,6101 2.1 129 1.3174 2.1756 1.0386 2.1 129

0-9 167 1.2067 0-748 1 1,2436 0.5885 1.2067

0.5901 0.7779 0.48 I0 0.802 1 0.3779 0.7779

56.522 71.359 61-654 80.550 63-988 85.589

20656 26.856 23.088 31,589 24.775 35.275

6.27 19 8-4040 7.1939 10.330 7.9263 12.176

2,9322 3.9746 3.3950 4-9709 3,7689 5-9801

1,6824 2.2922 1,9558 2.8891 2.1774 3.5065

1.0872 1.4852 1-2665 1.8795 1.4118 2.2913

Tab

le V

T. Ef

fect

of s

hear

rigi

dity

of s

tiff l

ayer

s, b

ound

ary

cond

ition

s and

sid

e-to

-thi

ckne

ss ra

tio

on th

e na

tura

l fre

quen

cies

(o$2n

cycl

esis

cc) o

f a s

even

- la

yer (0"~45'/90-,'core/90"~45-j0')

squa

re (a

ih = 1 .

0) co

mpo

site

san

dwic

h pl

ates

1

P E

Con

side

ring

G,,

and

GI3

of s

tiff

laye

rs

Neg

lect

ing

G,,

and

G,3

of s

tiff

laye

rs

, .,

Sim

ply

supp

orte

d C

lam

ped

Sim

ply

supp

orte

d C

lam

ped

5 u/

h = 100

8

1 473

593

70

70

606

866

134

135

334

356

69

69

385

407

128

128

2 3 3 w >

ulh =

10

aih=

100

alh-

10

aih =

100

alh=

10

u/h=

100

clih

= 10

Mod

al

z no

. H

OST

F

OST

H

OST

FO

ST

HO

ST

FOST

H

OST

FO

ST

HO

ST

FOST

H

OST

FO

ST

HO

ST

FOST

H

OST

FO

ST

8 r

2 774

1203

166

168

856

1399

251

259

518

524

I61

161

547

546

233

232

3 1003

1331

194

196

1081

1512

291

297

617

707

188

189

663

717

271

275

4

1097

1363

268

271

1176

1877

375

388

720

713

257

257

741

728

344

346

6 1321

2005

400

407

1383

2180

552

584

865

820

375

383

876

820

485

484

1376

2172

409

421

1455

2285

566

587

872

827

383

385

896

831

501

519

8 1476

2180

479

492

1556

2408

647

683

885

907

444

450

938

916

560

558

P

4

5 1173

1719

345

358

1245

2128

477

514

731

802

323

321

766

815

416

410

b

rn

3

v)

1888 MALLIKARJUNA AND T. KANT

In the last and fifth example, a seven layer (Oc/4S’/90u/core/900/45”/0”) square composite sandwich plate (u = h = 100 cm) was analysed for two different boundary conditions: simply supported and clamped. The following material properties were used: for face sheets, the assumed ply data based on, Hercules AS1/3501-5 graphite-epoxy prepreg system are E , = 13.08 x lo6 N/cm2, E2=E3=1~06x1O6N/crnZ, GI2=Gl3=0.6x 106N/cm2, GZ3=0.39x 106N/cm2, v,,=v,,=0~28, v,,=O34, p = 15.8 x N-secz/cm4, thickness h,=h45=h90=005h, and the core material is of U.S. commercial aluminium honeycomb (1/4 in cell size, 0.003 in foil) with G,3 =GyZ= 1.772 x lo4 N/cm2, Gl3=Cr,,= 5.206 x lo4 N/cmZ, E,= E,=3.013 x lo5 N/cm2, thickness hG,,,=0.7h. The results obtained by using our present HOST and FOST formulations are presented in Table VI. The effect of transverse shear modulii (G23 and GI3) of stiff layers is more pronounced in thicker plates (low a/h ratio) than it is for thin plates (high u/h ratio). The error between the results of the HOST and FOST increases with increasing mode numbers.

CONCLUSIONS

The results from the higher-order two-dimensional plate theory developed here compare well with three-dimensional elasticity solutions. A Co continuous finite element model of the present higher- order theory governing the behaviour of symmetric, laminated, anisotropic composite and sandwich plates is presented. In contrast to the classical shear deformation theories, the present higher-order theory does not require a shear correction coefficient, owing to more realistic representation of the cross-sectional deformation. The simplifying assumptions made in CPT and FOST are reflected by the high percentage error in the results of thick composite and sandwich plates with highly stiff facings. The increase in number of layers without changing the total thickness increases the fundamental frequency. The effects of plate aspect ratio and transverse shear rigidities of stiff layers on fundamental frequencies are more pronounced in thicker plates than they are for thin plates. It is believed that the improved shear deformation theory presented here is essential for reliable analyses of sandwich type laminated composite plates.

ACKNOWLkDGEMENTS

Partial support of this research by the Aeronautics Research and Development Board, Ministry of Defence, Government of India through its Grant No. Aero/RD-l34/100/84-85/362 is gratefully acknowledged.

The authors are indeed grateful to the reviewer for his extensive constructive comments and suggestions which have been incorporated in this final version of the paper.

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