Rr Solution for Sym Angle-ply - Liew Et Al(1)

Composite Structures 20 (1992) 213-226

Transverse vibration of symmetrically laminated rectangular composite plates

S. T. Chow, K. M. Liew & K. Y. Lain Department of Mechanical & Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511,

Republic of Singapore

A general numerical method for estimation of the vibrational response of symmetrically laminated rectangular composite plates is presented. This is based on the Rayleigh-Ritz method using the admissible 2-D orthogonal polynomials to derive the governing eigenvalue equation. The natural frequencies and mode shapes for the laminated plates are obtained by solving this governing eigenvalue equation. Several test problems are solved to demonstrate the accuracy and flexibility of the proposed method. The present results, where possible, are verified with those available values from the literature. The effects of the material, number of layers and fibre orientation upon the frequencies and mode shapes are discussed. This study may provide valuable information for researchers and engineers in design applications.

1 INTRODUCTION

Due to the common use of composites in in- dustries, especially in the aircraft industry, vibration analysis for composite structural components such as plates becomes an important design procedure. Composites have very high strength- to-weight and stiffness-to-weight ratios when compared to isotropic materials. Aircraft are typical weight-sensitive structures in which composite materials are favourable in addition to being cost-effective.

Several approximate methods have been proposed for free vibration analysis of angle-ply plates involving analysis which is considerably complicated and where exact solutions are out of the question. The Rayleigh-Ritz method was used by Ashton and Anderson j to obtain natural frequencies for laminated rectangular plates using a series expansion of beam mode shape functions. The results obtained, for the plates with laminated plies of orthotropic boron/epoxy composite having fully fixed boundary conditions, were verified with experimental values. Mohan and Kingsbury 2 USed the Galerkin method with the assumed displacement functions in the form of beam eigenfunctions. The results for several single-layer parallel-fibre boron/epoxy rectangular plates are given; however, the few results given for frequencies and nodal patterns were

213

considerably inaccurate. Whitney 3 obtained the solution for free vibration of single-layer high- modulus graphite/epoxy composite plates by using the classical Fourier analysis in which the assumed functions satisfy both the geometrical and natural boundary conditions.

BerP introduced a simple reduction method to study the vibration of single-layer composite plates with various planforms and edge conditions. Fundamental frequencies for rectangular plates with 15 different combinations of edge conditions were published. He also developed an approximate close-form formula for the frequencies of simply supported plates of arbitrary ply-stacking sequence and derived an optimisa- tion procedure based upon it. 5 Reiss and Rama- chandran 6 generated a frequency optimization method for symmetrically laminated plates with arbitrary edge conditions. The method was demonstrated for simply supported four-layer angle-ply plates.

Sivakumaran 7 proposed a numerical method to estimate the natural frequencies for symmetrically laminated rectangular plates having completely free edges. A set of complete power series was used as the admissible function in the Rayleigh- Ritz method to obtain the approximate natural frequencies. The effects of aspect ratios and fibre orientations on the natural frequencies of symmetrically laminated plates are presented.

Composite Structures 0263-8223/92/S05-00 O 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain

214 S. T. Chow, K. M. Liew, K. Y. Lam

Leissa and Narita 8 recently published an accurate and reasonably comprehensive results for the free-vibration frequencies of symmetrically laminated simply supported rectangular plates, especially for lay-ups other than cross-ply, for which no exact solutions are possible. The Rayleigh-Ritz method is used with the double- sine series as the admissible displacement function.

A general numerical method has been developed by Liew 9 to study the transverse vibration of plates. A set of 2-D orthogonal polynomials is used as the admissible displacement function in the Rayleigh-Ritz method to derive the governing eigenvalue equation. The method has been applied to obtain natural frequencies and mode shapes for triangular plates, t° skew plates, II trape- zoidal plates ~2 and regular polygonal plates.13 The authors further extend the method to study the vibrational response of symmetrically laminated rectangular composite plates. In this paper, relatively accurate and reasonably comprehensive results for the free-vibration frequencies of these symmetrically laminated rectangular composite plates are presented for which no exact solutions are possible. The effects of material, number of layers and fibre orientation upon the frequencies may be seen from the extensive results presented. Mode shapes for several representative laminated composite plates are also presented by means of contour plots.

Fig. 1.

t t

Layer coordinates and orientation for laminates.

approximately by using the Rayleigh-Ritz approach with a set of 2-D orthogonal polynomials.

The strain energy due to bending can be expressed as:~

Ill Vm~= ~ [M][K]dxdy (1) R

where the integration is carried out over the entire plate domain R and

[MI=tM,,,M,,,M,,,,] / (2)

[KI = [ K . Kv, K . I (3)

in which [M] is the moment resultants and [K] is the bending curvatures.

The bending curvatures are related to the dis- placements by

2 RAYLEIGH-RITZ EIGENVALUE F O R M U L A T I O N

0 2 W Kx = Ox 2 (4)

Consider a thin, fibre reinforced composite, laminated rectangular plate lying in the x-y plane, and bounded by - a / 2 < x < a / 2 and - b / 2<y< b/2. The plate, with thickness h in the z- direction, consists of n layers of orthotropic plies perfectly bonded together by a matrix material. The reference plane z--0 is considered to be located at the undeformed middle plane as shown in Fig. 1. The fibre direction within a layer is indicated by the angle ft. The moduli of elasticity for layer parallel to the fibres is Et and perpendicular to the fibres is E 2.

In the present study, the layers are so arranged that a mid-plane symmetry exists. By these special symmetrical arrangements, coupling between transverse bending and in plane stretching is avoided. An attempt is made to solve the natural frequencies and mode shapes of these problems

0 2 W Ky-- (5)

0y 2

o2W Kxr = 2 0x ay (6)

For anisotropic materials, the moment resultants are given by

[m] =[D][K] (7)

where [D] is a 3 x 3 symmetric matrix of bending stiffness coefficients.

For symmetric angle-ply laminates, the coefficients of the bending stiffness matrix are given by

t l

1 Z (N,)k(h3-h3-~); c , t - -1 ,2 ,6 (8) Oct=-3 k=l

Vibrational response of composite plates 215

(Nc,)t , are the reduced stiffness of the kth ply which is defined by the elastic constants of the layer and fibre orientation angle ilk" (N.)t` are given by

NII. ffi Q, i. cos4flk

+ 2(Qi2, + 2Q66.) sin2fl, COS2flk

+ 022 Sin'flk (9)

NI2. = (Ql l . + Q22.- 4066,) sin2 fit, cos 2ilk • 4 + Q,2,(sm flk +cos*ilk) (10)

N22. -- Ql 1, sin4flk

+ 2(Qi2, + 2066.) sin2fl, cos2flk 4

+ Q22. cos flk (11)

NI6. = ( Q , , . - QI2, - Q66.)sin flk COS3flk

+ ( Q,2. - 022. + 2 Q66.) Sill3 #k COS #k ( 1 2)

N261--(Q,,,- Q t 2 . - 2Q66.) Sin3#k COS #k

+(012. -- Q22. + 2 Q66,) sin fl~ COS3#k (13)

-- - 2 Q6~,) sm flk cos t . N66, (QII ,+Q=,-2QIe. • 2 • 4 + Q66,(sm flk +COS'ilk) (14)

where

Eq (15) Qi,. 1 - )'l_~.v2,.

v12'E2' (16) Q12, = 1 - lPl2 . 1/21 i

E2t (17) Q22, ---- 1 - v~2. v2~.

Q66.--- GI2. (18)

V21 El.---- VI2 E2. (19)

in which El. and E2. a r e the Young's moduli parallel to and perpendicular to the fibres and v~2. and v2~. are the corresponding Poisson's ratios.

Substituting eqns (2-8)into eqn (1) results in

= ! f f [ [a2W] 2 r0,wa2w V.

[ =" 0 0

re wo w [ +4D26[~p ~ +4D66 d x d y

(20)

The maximum kinetic energy of the plate dur- ing small amplitude vibration is given by.

1 II Tm,~,=~phto 2 W2(x .y)dxdy (21)

R

where/9 is the mass per unit area of plate, h is the thickness and to is the angular frequency of vibration.

The displacement function W(~. r/) may be expressed in terms of 2-D orthogonal polynomials which is given by

W(l~, rl)= Z CqOq(If, rl) (22) q=l

where ~ = x/a, rl =y/b and Cq are the unknown coefficients.

Substituting eqn (22) into eqns (20) and (21) results in

°~'-2 I, " ' a~ 2

I m a ~ Z c .%(~, ,1)

+ 2D~2 q. a - ~ a~ 2

- ] a ~ Z c~%(~,~) x ~'t a~12 [ }2

a 2 ~. c,¢,q(~,,7) + _~3.4: q-~ at/2

4D,6 [ a2 ~" CqdPq(~, rl)

+ a3b O~ 2

o 2

X q-I a~a,7

I m + ~4D26 02 q.t ~" Cq~q(~, r/)

ab 3 OT12

216 S. T. Chow, K. M. Liew, K. Y. Lain

0 2 £ Cq(~q(~, 77)1 X q=l

o~ or/ [ ]2/ 02 Z G%(t, ,1)

+ 4D66 q = ) d t dr/ afar/

(23)

If . , ~ - ~ phto 2 ~, [Cq~q(t, )/)]2 d t dr/ Rq=l (24)

By minimizing the energy functional ( Vm~ - T.,~.) with respect to each coefficient Cq

0 - - ( Vm~,- Tin,,,,)= 0, q = l , 2 .... ,m (25) 0Cq

leads to the governing eigenvalue equation

Y(Ho-3.L~i)C,=O, i , j = l , 2 .... ,m (26)

where the parameter 3. is defined as

3. = ph~2a4 (27) Do

Elh 3 D O = (28)

12(1 - v12v21)

in which

1 Ho='-~o {D1 ,(Fo), + a4D22(Fo)2

+ Ct 2 D I2[ ( FO)3 "t" ( Fij)4] + 2aDl6[(Fij)5 + (Fij) 6] + 2a3Dz6[(Fii)7 + (F0.) s] + 4a2D66(Fij)9}

(29)

L,/=[[ @,(t, r / )O/(~,r / )dtdr/ (30) .I.I R

[H] is a symmetrical matrix, and [L] is a diagonal matrix. Equation (26) forms a set of i x j homogeneous linear simultaneous equatiom expressed in terms of the unknown coefficients Cv For a non-trivial solution, the determinant of the coefficient matrix is set to zero. There are i x j values of 3. that satisfy eqn (26), which are the upper-bound approximations to the exact frequencies.

Several terms on the right-hand side of eqn (29) need to be defined. They are

a a = - (31)

b

(Fij)l = R 0 ~ 2 0 t 2 dtdr / (32)

(F*j)2=ffR a2°'(~J' r/) 0~%(~' r/) d~ d r / o r / 2 Or/2 (33)

(F,,)3=ffR 02¢i(~' r/) O2@/('!J' r/) d t d r / 0 r / 2 O~ 2 (34)

O2(l)'(t' r/) O2(1)/('!J' r/) d td r / a t 2 Or/2 (35)

f f °~¢"(t' r/) o~%.(t, ,7) (F°)5 = R O~ 2 a t Or/ d t dr/ (36)

f f 02~,(t, )1) 02~/(~, r/) (F°)~= R at0r/ at 2 dtdr / (37)

I f o~a',(t, v) a2c'/(t, ,7) (Fij)7 = R Or/" a t Or] dtdr / (38)

ff a2¢'(t' r/) az%(~, ,1) (F//)8= R a t0 r / Or/2 d t dr/ (39)

(Fii)9=fIR 02¢'(t' r/) a2¢/(~' r/) dt d r / 0 t 0)7 Ot 0)7 (40)

3 ADMi&SBLE DISPLACEMENT FUNCTIONS

The displacement function W(t, r/), as given in eqn (22), is expressed in terms of a series of 2-D orthogonal polynomials. The recurrence formula of 2-D orthogonal polynomials involves every other member in the set

a.,.(t, r/)= g,.(t, r/) ~.(t. r/) m - I

E n=l

g,,,,.,,@,,(/~, q); m > 1 (41)


where Ip,,.. is a constant and g.,(~, r/) is a generating function (which will be discussed later). The constant ~p=,. is #oven by

ffRg,,,(¢,r/)Ot(¢,r/)*,,(¢,r/)dCdr/

. (42)

in the series satisfies the orthogonality condition

R if m ~?1

(43)

3.1 Generating functions

The generating function g,.(~,r/) which are obtained empirically can be determined by the following general procedures.

The parameters a l and a2 are determined from the expressions

a l = [,]-mm - l l (44)

a z = ( m - 1 ) - a 2 (45)

If at is an even number then the generating function is given by

g,,,(,e, r/) = ,e", r/", (46)

where a3 is defined as

a2. O<a3<a I (47) Q 3 m y ,

starting functions are constructed to satisfy the geometrical boundary conditions since, they are the only necessary requirement for the admissible functions in the Rayleigh-Ritz method. The geometrical boundary conditions are ~t = 0 for simply supported edges, Ol=0(~t/0~=0(~t/ Or/=0 for clamped edges and no geometrical boundary condition exists for free edges.

For rectangular plates, the starting functions can be expressed as

4

• r/)= I-I x,(¢, ,1) (50) i - I

where X;(~, r/) are the edge functions for the four edges.

By applying the appropriate boundary conditions at the edges, the edge functions for the plate can be obtained.

For the simply supported edges, the edge functions are #oven by

X i ( , , r / ) = I , - a fo redges ,=a (51) - b foredges r/--b

and the edge functions for the clamped edges are #oven by

2

x,(¢, r/)= (r/ for edges ~ = a

(52) for edges r/= b

and the edge functions for the free edges are given by

X,(~, r/)= 1 (53)

On the other hand, if a I is an odd number then the generating function becomes

gm(~, r/)= ~a' r/~' (48)

where a3 now is defined as

(a2- 1) a3-- 2 ; l < a 3 < a l (49)

The symbol R adopted in eqn (44) denotes the greatest integer function.

3.2 Determination of starting functions

The starting functions Ol(~, r/) chosen for the analysis must satisfy the prescribed boundary conditions of the plate. In the present study, the

4 NUMERICAL APPLICATIONS

The geometry of a rectangular laminated composite plate of dimensions a x b, is shown in Fig. 2, with the fibre direction within a layer indicated by the angle ti-

The proposed method is applied to determine the frequency parameters (phwea4/Do)l/2 for several square plates with various numbers of layers in the angle-ply stacking sequences. The material properties for the composite materials relevant in the study are given in Table 1 which adopted from the book by Vinson and Siera- kowski . 14

An example is given here to demonstrate the procedure of forming the 2-D orthogonal polynomials for a plate having the prescribed boun-


dary conditions. For a rectangular plate having the edges simply supported at ~ -- - a, a and clamped at 17= - b, b, the starting function ~ ( ~ , r/) for this plate is

• ,(~,17)=(~+a)(~-a)[(17+b)(7]-b)] ~ (54)

It can be seen that eqn (54) satisfies the required

l

L

Fig. 2.

I_ LI I - Q -I

Geometry of rectangular plate with angle of fibre orientation fl*.

Table 1. Layer material properties for composite plates studied

Materials E,/E 2 G12/E 2 V~2

E-glass/epoxy (E/E) 2.45 0-48 0"23 Boron/epoxy (B/E) 11"0 0.34 0.21 Graphite/epoxy (G/E) 15.4 0.79 0.30

geometrical boundary conditions for this plate

[(~1(~, ?'1)1~- -o ~ [(l)l(~, 7])]~-a ~--" 0 (55)

[~,(~, 7])]~.-b=[~,(~, 7])]~.b=0 (56)

[ 0~,,(~, 7]) 17)]~ =0 (57)

For the purpose of demonstration, the five subsequent terms (~2(~, 7]) to ~6(~, 17)) for this plate are also given:

%(~, 17)= ~' , (~, 7])- ~2.~1(~, 7]) (58)

a,3(~, 17)= 7],:,~(~, 7])- W3 ,~'1(~, 7]) - ~P3.2~2(~, 7]) (59)

,:,,(~, 17)= ~7],t,,(~, 7])- w,.,~',(~, 17) -- lP4.2(~2(~, 7])-- 1/34.31~3(~, 17) (60)

• s(~, 7])= ~ ' 1 ( ~ , 7])- ~5 ,~,(~, 7]) -- 1ff5,2([32(~, 7])-- 1])5,31~3(~, 17)

- ~5.4'tI)4(~, 7]) ( 6 1 )

(I)6(~, 7])-- 712(~1(~, 7])-- lP6.1(~ll(~, 7])

-- lP6.2(~2(~, 7])-- lP6.3(I)3(~, 7])

- ~P6.4~4(~, 7])- ~P6.5~5(~, 17) (62)

The number of terms required depend on the satisfactory convergence of the solution. The recurrence formula given in eqn (41) can be used

Table 2. Convergence patterns of frequency p m m e t m (Ohw2a'lDo) '/2 for five-layer itnimted simply mpperted graphite/ epoxy s q w e p m e ~ with ~ ~ u ~ ~ , - # , ~ , - # , # )

fl* No. of Mode sequence number terms

1 2 3 4 5 6 7 8

0 30 11.30 17.14 29-01 40.81 45.38 47.29 55-63 70.64 36 11.30 17.13 28-90 40.79 45.27 47-01 55.14 69-65 44 11-30 17.13 28-72 40-78 45.18 46.24 54-98 69-64 50 11-30 17-13 28-70 40-77 45.18 46.23 54.98 69.64

30 11.82 19-76 33"29 39.54 47.47 53" 15 61.40 77.39 36 11.82 19.76 33"27 39-53 47.46 53-03 61.32 75-06 44 11.82 19"76 32.93 39-53 47-43 52.75 61-15 74-13 50 11"82 19.76 32"93 39"53 47"42 52"73 61"11 74-08

30 12"98 25"22 36-99 43"01 53"09 67.46 76"94 77"86 36 12"98 25-21 36-98 43.00 53-08 67"16 76"81 77"81 44 12"98 25"21 36"97 42"65 52"83 66"51 75-77 77"67 50 12"98 25"21 36"97 42"65 52-83 66"48 75"76 77"65

30 13"62 28"76 34.71 49-04 60-12 66" 15 75-21 90.83 36 13-61 28"75 34-69 49-01 60"01 66" 13 74"93 90-56 44 13-61 28"75 34"69 49"90 59"27 65-37 74'31 88"87 50 13-61 28"75 34"68 48-90 59"25 65.34 74"28 88"86

15

30

45


Table 3. Convergence pmterm of frequency ptrameters (pkm2a4/Do) '/2 for five-layer laminated fully clamped ip~ehite/epoxy

fl* No. of Mode sequence number terms

1 2 3 4 5 6 7 8

0 30 23.86 29"71 41.77 60-25 62.93 67.45 76.49 91.30 36 23-86 29.71 41.74 60-24 62.93 67.45 76.49 91.28 44 23-86 29.71 41.73 60.24 62.93 67-45 76.46 854)6 50 23-86 29-71 41.73 60.24 62.93 67.45 76.46 85-06

30 23.46 31"48 45.84 60.38 66.52 67.97 81-39 97.42 36 23-46 31"48 45.84 60-38 66.51 67.97 81.38 96.13 44 23.46 31"48 45.80 60.38 66.48 67.96 81.37 92.42 50 23.46 31"48 45.80 60.38 66.47 67.96 81"36 92.42

30 22-72 36"55 54"04 57"23 70-13 83"51 95"32 100-6 36 22-72 36"55 54"02 57"20 70-12 83"48 95"31 100-6 44 22"72 36"54 54"02 57"17 70.09 83"37 95"28 100-4 50 22-72 36-54 54"02 57"17 70-09 83"37 95"27 100"4

30 22"40 41"65 48"33 65-11 77"90 84"22 93"79 109"4 36 22"40 41-65 48"32 65"10 77'87 84"17 93"78 109"4 44 22"40 41"64 48"32 65-09 77-76 84"06 93-59 109"1 50 22"40 41"64 48"32 65"09 77"76 84-06 93"58 109-0

15

30

45

to generate the 2-D orthogonal polynomials up to rn terms where m - - 1, 2, 3,..., and m is a positive integer number.

Convergence studies are carded out for the simply supported and fully clamped graphite/ epoxy plates. Frequency parameters for the five- layer angle-ply of equal thickness laminated square plates with stacking sequence/5, -/5,/5, -/5,/5 are given in Tables 2 and 3. The first eight frequency parameters are tabulated in each case. As expected convergence characteristics are monotonic as more and more terms are added.

For the simply supported laminated square plates, it can be seen that the solutions obtained by using 44-term and 50-term produce no drastic changes regardless of the fibre orientation. For the fully clamped laminated square plates, comparison is made between the 44-term and 50-term solutions. A maximum of 0.1% discrepancy is observed. FOr the present study, it is decided to use 50 terms for all the subsequent numerical calculations.

Table 4 gives the first eight frequency parameters for the simply supported single-layer parallel-fibre square plates, as the fibre orientation angle varies between 0* and 45*. Results are presented for the E-glass/epoxy (E/E), boron/ epoxy (B/E) and graphite/epoxy (G/E) composites described in Table 1. The following observations are obtained from the results. FOr the E/E material, maximum frequency parame-

ters of the first, second, fourth and fifth modes occur at fl=45", the third and seventh modes occur at 15--0", and the sixth and eighth modes occur at fl-- 15", respectively. For the B/E material, maximum frequency parameters of the first, second, fifth, sixth and eighth modes occur at fl=45", the third and seventh modes occur at /5 -- 30* and the fourth mode occurs at/5-- 0". For the G/E material, maximum frequency parameters of the first, second, third, fifth and sixth modes occur at /5--45", the fourth and eighth modes occur at /5--0* and the seventh mode occurs at/5 = 30*. A comparison of the numerical values and the above observations with the published results by Leissa and Narita s reveals excellent agreements for all cases.

The first eight frequency parameters for the fully clamped laminated square plates are given in Table 5. A study on the variation of fibre orientation upon the vibration response is carded out for the plates. For the E/E material, maximum frequency parameters of the second and fifth modes occur at /5--45", the first, third, fourth and seventh modes occur at/5-- 0", and the sixth and eighth modes occur at /5= 15". For the B/E material, maximum frequency parameters of the second, fifth and sixth modes occur at fl-- 45", the first, fourth and eighth modes occur at/5-- 0", and the third and seventh modes occur at f l - 30*. FOr the G/E material, maximum frequency parameters of the second, fifth, sixth and eighth modes

220

Table 4.

S. T. Chow, K. M. Liew, K. Y. Lain

Effect of varying ~ upon the frequency parameters (phmZa4/Dop/z of single-layer simply ~pported square plates

fl* Source of results

Mode sequence number

1 2 3 4 5 6 7 8

(a), E-glass/epoxy 0 Present 15.19 33.31 44.52 60.78 64-55 90.31 93-67 108.7

Leissa & Narita a 15.19 33.30 44.42 60.77 64.53 90-29 93-66 108.6 15 Present 15-35 33.97 43.78 60.59 66.60 91.20 91-37 108.8

Leissa & Narita 15.41 34.04 43.85 60.65 66.70 91.26 91.37 108.8 30 Present 15"60 35.55 42.46 60.75 71.37 85.62 92.61 108.7

Leissa & Narita 15.84 35.65 42.61 60"92 71-70 85.68 92-79 108.7 45 Present 15-98 36.46 41.71 60.93 76.47 79.88 93.07 108.7

Leissa & Narita 16.06 36.58 41.86 61.16 76.78 80.17 93"38 108.8

(b) Boron~epoxy 0 Present 11.04 17"36 30.92 40.37 44.18 51.94 53"32 70-03

Leissa & Narita 11"04 17.36 30.91 40.37 44.16 51.13 53.27 69.46 15 Present 11"23 18.76 32.62 38.46 44-54 51.54 59-75 72-90

Leissa & Narita 11.24 18-92 32.87 38.69 44.96 51.20 59.00 71.19 30 Present 11.79 2t.36 34-88 34-98 47.71 52.58 70"22 71.45

Leissa & Narita 11.86 21.79 35.40 35.89 48.83 53.21 69.27 71.90 45 Present 12.13 22.40 33.24 36.99 52.22 53.97 64.60 73"83

Leissa & Narita 12.27 22-96 34.05 37.09 52.56 54.22 66-08 74-21

(c) Graphite~epoxy 0 Present 11.30 17.13 28.70 40.77 45.18 46-23 54-98 69.64

Leissa & Narita 11.29 17.13 28.69 40.74 45.16 45.78 54.08 68-14 15 Present 11.26 18.16 30.22 38-73 45-04 47.74 58-35 67.47

Leissa & Narita 11.33 18-35 30.62 38.98 45.41 47.16 58"01 66.02 30 Present 11.49 20-29 32.54 34"85 47-03 48.57 67.12 68.61

Leissa & Narita 11.56 20.78 33-61 35.40 48.24 49.25 66.97 67.49 45 Present 11,66 21.34 32-77 33.61 48.92 50.57 66"22 68.77

Leissa & Narita 11.79 21-91 33.65 34-78 50-21 51.92 65.20 69.28

~Ref. 8.

Table 5. Effect of varying ~ upon ti~ frequency parameters (pho~Za4/De) i/z of single-layer fully clamped square plates

~° Mode sequence number

1 2 3 4 5 6 7 8

(a) E-glass~epoxy 0 29.13 50.82 67.29 85-67 87-14 118.6 126.2 137.5

15 28.90 51-37 65"90 84.29 89.82 119"0 122.7 139.8 30 28-49 52.92 62.68 83.22 95.22 114.1 119~6 138.3 45 28.28 54.24 60-53 82-89 101.7 105.8 l 19.9 137.0

(b) Boron~epoxy 0 23-89 30.59 44.97 62.71 66.94 67.14 76"53 93.45

15 22"97 30.91 46"23 59"35 65"36 67"87 78.72 91"81 30 21"16 32"49 49.40 51.58 64"55 69-42 87-44 92.14 45 20"28 34.34 46"53 50"88 69-04 70.22 82.89 92.82

(c) Graphite~epoxy 0 23"86 29-71 41"73 60"24 62-93 67.45 76.46 85"06

15 22"93 29"92 43.16 59.37 62"55 65.54 77"61 84"83 30 20.97 31-38 46"82 51-79 64-04 65-04 85.48 85.57 45 19.97 33.35 46"50 48"40 66-01 68.64 82-42 88.42

occur at fl ffi 45*, the first and fourth modes occur at fl ffi 00, and the third and seventh modes occur at fl = 30 °. No comparison for the present study is possible due to the lack of results from literature.

The variation of the frequency parameters with respect to the fibre orientation for the three-layer

angle-ply simply supported square plates with stacking sequence 3, - f l , 3 are given in Table 6. For the E /E material, maximum frequency parameters of the first, second, fourth, fifth, seventh and eighth modes occur at fl = 45 °, the third mode occurs at fl = 0", and the sixth mode occurs at


Table 6. Effect of varying ~ upon the frequency parameters (pkmaa4/DO In of three4ayer laminated simply supported square O H ~th ~ ~,queo,.'e ~, - p, p)



1 2 3 4 5 6 7 8

(a) E-glass~epoxy 0 Present 15.19 33.31 44.52 60"78 64.55 90.31 93.69 108.7

Leissa & Narita* 15.19 33.30 44.42 60"77 64.53 90.29 93.66 108-6 15 Present 15.37 34-03 43.80 60-80 66.56 91.40 91.51 108.9

Leissa & Narita 15.43 34.09 43.87 60-85 66.67 91.40 91.56 108.9 30 Present 15"86 35.77 42.48 61.27 71.41 85.67 93.60 108-9

Leissa & Narita 15"90 35.86 42.62 61.45 71.71 85.72 93.74 108.9 45 Present 16-08 36-83 41.67 61.65 76.76 79.74 94.40 109.0

Leissa & Narita 16.14 36.93 41.81 61.85 77.04 80-00 94.68 109.0

(b) Boron~epoxy 0 Present 11.04 17.36 30.92 40.37 44.18 51.94 53.32 70-03

Leissa & Narita 11-04 17.36 30-91 40.37 44.16 51.13 53.27 69.46 15 Present 11.27 19-07 33"09 38.65 45.09 52.67 60.14 73.51

I.,cissa & Narita 11.37 19.21 33.32 38"86 45.46 52.14 59.48 72.77 30 Present 12.16 22.42 35.44 37-10 49-14 56.30 72.43 73.14

Leissa & Narita 12.21 22.78 35.86 37.90 50.04 56.70 71.36 73.57 45 Present 12"68 24.05 33"80 39"30 53.30 58.42 65.26 78.46

Leissa & Narita 12.71 24.51 34.44 40.23 54.44 59.40 66.38 78-00

(c) Graphite/epoxy 0 Present 11.30 17.13 28-70 40.77 45.18 46.23 54.98 69"64

Leissa & Narita 11.29 17-13 28%9 40.74 45.16 45-78 54-08 68.14 15 Present 11-43 18.53 30.84 38.92 45.59 48.93 58.99 70-36

Leissa & Narita 11.46 18%9 31-20 39.15 45.91 48-19 58.70 67.84 30 Present 11.86 21.56 35.10 35.44 48.60 52.99 71.22 72.16

Leissa & Narita 11-97 21-97 35.88 36-04 49.60 53.43 70"04 72.35 45 Present 12-27 23-24 33.45 37.48 53.94 55.27 66-93 76.99

Leissa & Narita 12.31 23.72 34.14 38.45 54.10 56-31 66.20 76%3

aRef. 8.

Table 7. Effect of varying ~ upon the frequency parameters (/aha~Za4/D,) vz of three-layer laminated fully clamped square plates with stacking sequence ~B, - ~ , fl)

/~° Mode sequence number

1 2 3 4 5 6 7 8

(a) E-glass~epoxy 0 29.13 50.82 67.29 85.67 87.14 118.6 126-2 137.5

15 28.92 51.43 65.92 84.55 89.76 119.3 122.7 139.9 30 28"55 53"15 62"71 83"83 95.21 114.1 120"7 138% 45 28"38 54%5 60"45 83%5 102-0 105"6 121"4 137"3

(b) Boron~epoxy 0 23"89 30"59 44.97 62"71 66.94 67" 14 76 "53 93"45 15 23-09 31"21 46-70 59"54 65"91 68"70 79"42 93"49 30 21-57 33"62 51"58 52" 15 66.02 73"29 89%4 96"26 45 20-87 36"18 47"11 54"52 71"25 76"18 83"52 97"08

(c) Graphite/epoxy 0 23"86 29.71 41.73 60"24 62.93 67-45 76"46 854)6

15 23"05 30"28 43-79 59.64 63-49 66"10 78"43 86%9 30 21"45 32.76 49"55 52.40 65%8 69.72 88"33 92"20 45 20"67 35.54 47.20 52.77 71.24 73.07 83"89 96"17

/ ~ -15 °. For both the B/E and G/E materials, maximum frequency parameters of the first, second, fifth, sixth and eighth modes occur at flffi45 °, the third and seventh modes occur at

fl ffi 30 ° an d the f o u r t h m o d e o c c u r s a t fl - 0 °. T h e first e ight f r e q u e n c y p a r a m e t e r s o b t a i n e d by the p r e s e n t m e t h o d a re c o m p a r a b l e to t hose va lues p r e s e n t e d b y Le i s sa a n d Nar i t a s wh ich w e r e


obtained by using the Rayleigh-Ritz method with the beam functions.

Table 7 shows the results for the fully clamped three-layer angle-ply square plates. The frequency parameters are given for various fibre orientation angle and the three materials again are considered in the analysis. For the E/E material, maximum frequency parameters of the first, third, fourth and seventh modes occur at fl = 0", the second and fifth modes occur at f l= 45", and the sixth and eighth modes occur at/5 = 15*. For both the B/E and G/E materials, maximum frequency parameters of the first and fourth modes occur at /5 = 0", the second, frith, sixth and eighth modes occur at/5 = 45", and the third and seventh modes occur at/5 = 30*.

The effect of fibre orientation on the vibrational response is examined for the simply supported laminated plates with stacking sequence/5, -/5, -t5, /5 and the numerical results are presented in Table 8 together with the available results from the literature. For the E/E material,

maximum frequency parameters of the first, second, fourth, fifth, seventh and eighth modes occur at/5 = 45", the third mode occurs at t5--0", and the sixth mode occurs at/5 = 15*. For both the B/E and G/E materials, maximum frequency parameters of first, second, fourth, fifth, sixth and eighth modes occur at/5 = 45 °, and the third and seventh modes occur at 15= 30*. The predicted trends concur with the reported results given by Leissa and Narita 8 for the three materials considered.

The first eight frequency parameters for the fully clamped four-layer angle-ply square plates with various fibre orientations are obtained. The numerical results are given in Table 9 for the three composites which were described earlier in Table 1. For the E/E material, maximum frequency parameters of the first, third, fourth and seventh modes occur at/5 = 0", the second and fifth modes occur a t /5= 45", and the sixth and eighth modes occur at /5= 15". For both the B/E and G/E materials, maximum frequency parameters of the

Table 8. Effect of varying ~ upon the frequency parameters (#hm2a4/De)t/z offourAayer laminates simply supported square #ates with stacking sequence (~, - #, - #, #)



1 2 3 4 5 6 7 8

(a) E-glass~epoxy 0 Present 15.19 33.31 44-52 60.78 64.55 90-31 93.69 108.7

Leissa & Narita ~ 15" 19 33-30 44.42 60.77 64-53 90.29 93.66 t 08.6 15 Present 15"40 34.15 43.84 61.23 66.48 91.47 92.13 109" 1

Leissa & Narita 15.47 34.21 43.91 61.28 66.57 91.47 92-19 109.1 30 Present 15.94 36.23 42-52 62.46 71.45 85-79 95.71 109.3

Leissa & Narita 16-02 36.30 42.62 62.57 71.68 85.81 95.79 109.3 45 Present 16.17 37.62 41.52 63.15 77.33 79.40 97-32 109.4

Leissa & Narita 16.29 37.71 41.63 63.29 77.56 79.60 97.53 109.4

(b) Boron~epoxy 0 Present 11.04 17.36 30-92 40.37 44-18 51.94 53.32 70-03

Leissa & Narita 11.04 17.36 30.91 40.37 44-16 51.13 53.27 69.46 15 Present 11.54 19.67 34.01 39.01 46.16 54-81 60-90 76.78

Leissa & Narita 11.61 19.77 34-17 39.19 46-42 53-90 60.47 75.95 30 Present 12.78 24.34 36-36 40.95 51.59 63.26 74.60 75-06

Leissa & Narita 12"83 24.57 36-62 41.53 52.13 63.12 74.82 74-89 45 Present 13.39 27.01 34"55 45.38 56.84 65.62 68.46 85-03

Leissa & Narita 13.46 27.30 34.94 45-98 57.59 66.30 69.01 84.58

(c) Graphite~epoxy 0 Present 11.30 17.13 28-70 40.77 45.18 46.23 54.98 69.64

Leissa & Narita 11-29 17-13 28-69 40.74 45.16 45-78 54-08 68.14 I5 Present 11.67 19.25 32.07 39-28 46~7 51.18 60.23 72.24

l.,¢issa & Narita 11.71 19-36 32.32 39.48 46.89 50-15 60.09 71.39 30 Present 12.56 23.77 36.40 39.62 51.23 60-95 74-95 75.39

& Narita 12.66 24-03 36.67 40-31 51.84 60.88 74-19 75.19 45 Present 13-12 26-53 34.36 44.22 56.85 65.60 66-42 84"31

Leissa & Narita 13.17 26.82 34.76 44.84 57.61 66.33 67.01 83-90

°Ref. 8.


Table 9. Effect of varying ~ upon the frequency parameters (pkoJ2 a' / D.) va of four4ayer laminated ruby clamped square plates with st,,aune seque .ce (p. - p. - p. p )

fl" Mode sequence number

1 2 3 4 5 6 7 8

(a) E-glass~epoxy 0 29.13 50.82 67.29 85.67 87.14 118.6 126"2 137.5

25 28-98 5I'56 65-97 85"11 89.57 119.9 122"8 140.1 30 28.69 53"62 62-74 85-09 95.15 114.3 122"9 139.1 45 28.53 55.56 60.22 85.25 1026 105.2 124.5 137.6

(b) Boron~epoxy 0 23"89 30-59 44"97 62"71 66"94 67"14 76"53 93"45

15 23"32 31"81 47"59 59"90 66"98 70"22 80"83 96"78 30 22"27 35"59 53-11 55"44 68"55 80"32 93"28 98"74 45 21"80 39"43 47-84 61-04 74.89 83"82 87-02 104"5

(c) Graphite/epoxy 0 23.86 29.71 41.73 60.24 62.93 67.45 76.46 85"06

15 23"29 30-99 44-98 60-09 65"23 67.18 80-14 90"11 30 22.24 35.09 53-40 54.21 68.42 77.91 92'76 99"29 45 21"75 39"20 48-07 60-12 75.33 84.43 85"20 104.2

Table 10. Effect of varying ~ upea the frequency l immeters (pkoJza'/Do) 1/2 of five-layer himinated simply supported square #ttes ,~th m d U n e sequeoce te. - p. p. - p. p)



1 2 3 4 5 6 7 8

(a) E-glass~epoxy 0 Present 15.19 33.31 44.52 60-78 64.55 90.31 93"69 108-7

Leissa & Narita 15.19 33-30 44.42 60-77 64.53 90.29 93.66 108.6 15 Present 15.46 34.24 43.88 61.59 66.42 91.52 92"62 109.3

Leissa & Narita 15-50 34.30 43.93 61.62 66.48 91.51 92-67 109"3 30 Present 15"98 36.58 42.53 63.37 71.43 85.86 97.42 109.4

Leissa & Narita 16.10 36"64 42.62 63.45 71.60 85"88 97.44 109.4 45 Present 16"29 38.30 41.32 64-35 77.77 79-09 99-79 109.3

Leissa & Narita 16.40 38.37 41.40 64.41 77.94 79-23 99.95 109.3

(b) Boron~epoxy 0 Present 11.04 17.36 30-92 40.37 44.18 51.94 53.32 70-03

Leissa & Narita 11-04 17.36 30.91 40.37 44.16 51.13 53-27 69.46 15 Present 11.78 20-10 34-68 39.26 46.92 52.34 61.43 74.22

Leissa & Narita 11-89 19-82 33.10 39-71 47.58 51.48 61-09 73"95 30 Present 13-04 25.62 36.91 43.57 53.11 66.14 75.31 77-02

Leissa & Narita 13.10 25.36 37.15 43.13 53.18 65.97 75.87 76.57 45 Present 13-63 29"08 34.81 49.73 59.07 65.26 75.18 89.38

Leissa & Narita 13"70 28"94 34"92 49"29 59"73 65"78 74"56 88"41

(c) Graphite/epoxy 0 Present 11-30 17.13 28.70 40.77 45.18 46.23 54"98 69"64

Leissa & Narita 11.29 17.13 28.69 40.74 45.16 45.78 54-08 68-14 15 Present 11.82 19.76 32.93 39-53 47.42 52.73 61.11 74-08

Leissa & Narita 11.89 19.82 33-10 39.71 47.58 51.48 61"09 73.95 30 Present 12.98 25.21 36-97 42.65 52.83 66.48 75.76 77"65

Leissa & Narita 13-10 25.36 37.15 43.13 53.18 65.97 75.87 76.57 45 Present 13.61 28.75 34-68 48-90 59.25 65.34 74.28 88"86

Leissa & Narita 13.70 28.94 34.92 49.29 59.73 65.78 74.56 88.41

first a n d f o u r t h m o d e s o c c u r a t /3 = 0", the s e c o n d , fifth, s ixth a n d e igh th m o d e s o c c u r at fl ~ 45", an d the th i rd a n d s e v e n t h m o d e s o c c u r a t fl ffi 30*.

The first eight frequency parameters for the

simply supported five-layer angle-ply laminated square plates with stacking sequence, fl, - f l , fl, - fl , f l are obtained by varying the angle of fibre orientation. The results are given in Table 10. A

224 S.T. Chow, K. M. Liew, K. Y. Lain

= 11.30

r@ @l @@1

~ = 1,5.18

,/~z= 1Z13

: z,6.23 /-~,: 5~.9e ff,:sm6~ Fig. 3. Mode shapes of five-layer G/E simply supported

square plate with stacking sequence (0", 0", 0", 0", 0%

' ~1 = 11.82 ,1"~7 : 19.76 ,/')~,'3 : 32.93 J')~l.: 39.53

Hg. 4. Mode shapes of five-layer G/E simply supported square plate with stacking sequence (15", -15", 15",

- 15", 15").

careful examinat ion of the results with various fibre or ientat ion is carr ied ou t for the three materials. For the E / E material, max imum f requency parameters of the first, second, fourth, fifth and

~:12.9a

Fig. 5. Mode shapes of five-layer G/E simply supported square plate with stacking sequence (30 °, -30", 30".

- 30", 30"/.

seventh m o d e s occur at f l - 4 5 * , the third m o d e occurs at fl = 0", the sixth m o d e occurs at 13 = 15*, and the eighth m o d e occurs at 13 = 30*. For both the B / E and G / E materials, max imum frequency parameters of the first, second, fourth, fifth and e ighth m o d e s o c c u r at fl = 45", and the third, sixth and seventh modes occur at fl=30*. The predicted trends are in agreement with the corresponding reported results given by Leissa and Narita s which were obtained by using the 144- term beam functions in the Rayleigh-Ritz method. The mode shapes for these simply supported five-layer angle-ply laminated square plates having fl=0*, 15", 30* and 45* are presented in Figs 3-6, respectively.

Numerical results of the first eight frequency parameters for the fully clamped five-layer laminated square plates are given in Table 11.

Table 11. Effect of varying ~ upon the frequency i~rmaeters (phmZa4/Do) I/z of five-layer Izminzted [uily clamped zqmure

fl* Mode sequence number

1 2 3 4 5 6 7 8

(a) E-glass~epoxy 0 29.13 50-82 67.29 85.67 87.14 118.6 126.2 137.5

15 29.00 51.65 66-01 85.55 89.40 120.5 122.9 140.1 30 28.78 53-98 62.76 86.09 95434 114.4 124.6 139-3 45 28.68 56-34 59.94 86.48 10343 104.9 127.3 137.5

(b) Boron~epoxy 0 23"89 30.59 44.97 62.71 66.94 67.14 76"53 93.45

1.5 23"48 32.22 48.21 60"16 67.74 71"22 81"91 99"16 30 22.72 36"87 53"71 57.95 70"17 85435 95-46 99"61 45 22.38 41.68 48.04 65-64 77.12 83"38 94.78 109.1

(c) Graphite[epoxy 0 23"86 29.71 41"73 60-24 62-93 67-45 76-46 85436

15 23"46 31"48 45"80 60-38 66"47 67"96 81"36 92"42 30 22"72 36-54 54"02 57-17 70"09 83"37 95"27 100-4 45 22-40 41"64 48"32 65-09 77"76 84436 93"58 10943


J~l: 13.61

N Fig. 6. square

: 2675

: 65.3/'

N N :7,28

Mode shapes of five-layer G/E simply supported plate with stacking sequence (45*, - 4 5 * , 45*,

- 45", 45") .

• /~'~: &1.73 J'~'~,: GO.2/'

~ s : 62.93 ,~; : $7./,5 ~ = 7S,/,6 • f~ ' l : 85.06

F'~. 7. Mode shapes of five-layer G/E fully clamped square plate with stacking sequence (0", 0", 0", 0", 0").

:31.k8 : 23./,6 ~ : 1,6.1q ,r'~'~ = B.38

~s:- / '7 /-~7,:67.s6 /g,:.36 /~',:sz/'2 Fig. 8. Mode shapes of five-layer G/E fully clamped square plate with stacking sequence (15", -15", 15",

- 15", 15").

:2272 :36s/' :s/'02 :s717

f~S=70.O9 /'~'~6 = H 3 7 ~,r~7 = 95.27 ¢r'~a = 100.1,

Fig. 9. Mode shapes of five-layer G]E fully clamped square plate with stacking sequence (30", -30*, 30",

- 30", 30").

: 22./,0 =/'1.6/'

0 ff,:.s8

lr~. 10. Mode shapes of five-layer G/E fully clamped square plate with stacking sequence (45", -45", 45",

- 45", 45*).

occur at fl = 45*, and the third, sixth and seventh modes occur at fl--30*. For the G/E material, maximum frequency parameters of the first mode occur at fl = 0", the second, fourth, fifth, sixth and eighth modes occur at/~ = 45", and the third and seventh modes occur at/~-- 30*. Furthermore, the mode shapes of these fully clamped five-layer angle-ply laminated square plates having fl = 0", 15", 30* and 45* are obtained and are presented in Figs 7-10, respectively.

5 CONCLUDING REMARKS

Again the vibrational responses of the three composites with various fibre orientation are carefully examined and the following conclusiom are made. For the E/E material, maximum frequency parameters of the first and third modes occur at fl = O0, the second, fourth, fifth and seventh modes occur at fl = 45*, and the sixth and eighth modes occur at t - 1 5 °. For the B/E material, maximum frequency parameters of the first mode occurs at fl = 00, the second, fourth, fifth and eighth modes

A simple, computationally efficient and relatively accurate approximate method was introduced to study the vibrational behaviour of symmetrically laminated rectangular composite plates. The general governing eigenvalue equation for the plates was derived based on the Rayleigh-Ritz procedure with a set of 2-D orthogonal polynomials by minimizing the energy functional with respect to each unknown coefficient. By solving this governing eigenvalue equation, the natural

226 S.T. Chow, K. M. Liew, K. Y. Lam

frequencies and mode shapes for the laminated composite plates can be obtained.

It is known that the present method gives an upper bound solution to the problem. To demonstrate the rate of convergence and to ensure that the results obtained converge several convergence studies were carried out for some of the selected examples considered. In the present study, results for rectangular plates with simply supported and fully clamped boundary conditions are considered. To verify the present method, several examples with known values were solved and compared. Excellent agreement was obtained from the comparisons. For the other cases considered, the results were also shown to be accurate. Moreover along with the illustrative examples considered, the present method was shown to be able to be applied to a wide range of symmetrically laminated rectangular plates having different combinations of boundary conditions and plate aspect ratios (a/b). The authors believe that the present work is of practical importance since it provides additional useful design information for engineers who work in the area of thin laminated composite plate elements.

REFERENCES

1. Ashton, J. E. & Anderson, J. D., The natural modes of vibration of boron-epoxy plates. Shock & Vibr. Bull., 39 (1969) 81-91.

2. Mohan, D. & Kingsbury, H. D., Free vibrations of general orthotropic plates. J. Acoustical Soc. Amer., 50 ( 1971 ) 266-9.

3. Whitney, J. M., Free vibration of anistropic rectangular plates. J. Acoustical Soc. Amer., 52 ( 1971 ) 448-9.

4. Bert, C. W., Fundamental frequencies of orthotropic plates with various planforms and edge conditions. Shock & Vibr. Bull.,47 (1977) 89-94.

5. Bert, C. W., Optimal design of a composite-material plate to maximize its fundamental frequency. J. Sound & Vibr., 50 (1977) 229-37.

6. Reiss, R. & Ramachandran, S., Maximum frequency design of symmetric angle-ply laminates. Proc. 4th Int. Conf. on Composite Structures, London. 1987, pp. 1.476-7.

7. Sivakumaran, K. S., Natural frequencies of symmetrically laminated rectangular plates with free edges. Com- posite Structures, 7 (1987) 191-204.

8. Leissa, A. W. & Narita, Y., Vibration studies for simply supported symmetrically laminated rectangular plates. Composite Structures, 12 (1989) 113- 32.

9. Liew, K. M., The development of 2-D orthogonal polynomials for vibration of plates. PhD thesis, National University of Singapore, Singapore, 1990.

10. Lam, K. Y., Liew, K. M. & Chow, S. T., Free vibration analysis of isotropic and orthotropic triangular plates. Int. J. Mechanical Sci., 32 (1990) 455-64.

11. Liew, K. M. & Lam, K. Y., Application of two-dimen- sional orthogonal plate functions to flexural vibration of skew plates. J. Sound & I~br., 139 (1990) 241 - 52.

12. Liew, K. M. & Lain, K. Y., A Rayleigh-Ritz approach to transverse vibration of isotropic and anisotropic trape- zoidal plates using orthogonal plate functions, lnt. J. Solids & Structures, 27 ( 1991 ) 189-203.

13. Liew, K. M. & Lain, K. Y., A set of orthogonal plate functions for vibration analysis of regular polygonal plates. ASME Journal of Vibration and Acoustics, 113 (1991) 182-6.

14. Vinson, J. R. & Sierakowskir, L., The Behaviour of Struc- lures Composed of Composite Materials. Martinus Nijhoff Publishers, Dordrecht, The Netherlands. 1986.

Documents

Rr Solution for Sym Angle-ply - Liew Et Al(1)