Vibrations of orthotropic square plates having variable thickness (linear variation)

Vibrations of orthotropic square plates having variable thickness (linear variation)

Vibrations of orthotropic square plates having variable thickness (linear variation) S. K. MALHOTRA, N. GANESAN and M. A. VELUSWAMI (Indian Institute ...

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Vibrations of orthotropic square plates having variable thickness (linear variation) S. K. MALHOTRA, N. GANESAN and M. A. VELUSWAMI (Indian Institute of Technology, India) The effect of fibre orientation on the natural frequencies of thin square orthotropic plates having variable thickness is studied using the Rayleigh-Ritz method for various boundary conditions. In the present study plates having linear variation in thickness in one direction ( e i t h e r X o r Y) are studied.

Key words: fibre-reinforced plastics; fibre orientation; thin square orthotropic plates; linear variation; variable thickness

NOMENCLATURE a, b Lengths of sides of rectangular plate (a = b = L in present study) p circular frequency U bending strain energy w transverse displacement in Z-direction x, y rectangular coordinates fl plate thickness tav average thickness of the plate El(tl) 3 D~ equal to 12(1 - v12v21)

D2

equalto

E2(t') 3 12(1 - v12~21)

Ez Young'smodulus along fibre direction E2 Young'smodulus in transverse direction GI2 shearmodulus ~1 0 v12

v21 P

degree of taper fibre orientation.wrt x-axis major Poisson's ratio minor Poisson's ratio mass per unit area

Fibre reinforced plastics (FRP) are finding increasing applications in aerospace structures due to their high strength to weight and high stiffness to weight ratios. Orthotropic plates of varying thickness are used for some aerospace structural components. Determination of vibration behaviour of such components is necessary for their design.

boundary conditions on fundamental frequency are studied for two cases eg, linear variation in thickness along the x-direction and the y-direction.

Chelladurai et al. ~have carried out limited studies using the finite element method for square and rectangular orthotropic plates of constant thickness for two boundary conditions and fundamental frequencies.

A rectangular plate element with coordinate directions is shown in Fig. 1.

Bert 2 has derived a formula for the approximate values of fundamental frequency of orthotropic plates (constant thickness) of arbitrary shape and boundary conditions, if frequencies of the corresponding isotropic plates are known. In the present study orthotropic plates of variable thickness (linear variation) are studied for eight boundary conditions, eg, CCCC, CCSC, CCFC, CCSS, CCFS, CCFF, SSSS and FCFF (see Table i for notations). The effect of fibre orientation and

RA YLEIGH-RITZ METHOD

The bending strain energy of the plate is given by:

a

where,

~_~[ ~2W ~2W ~2W ]

[c]

L ax~ ay~ ~xay j

FDll D12 D16] [D] = ID,2 D22 D261 LD16 D26 D66J

0010-4361/88/011467-06 $3.00(~)1988 Butterworth & Co (Publ ishers) Ltd COMPOSITES. VOLUME 19. NUMBER 6. NOVEMBER 1988

467

Table 1. Notations for planform and edge conditions

Where, N

Dij = ! E (O,j)k

S No. Planformand edgeconditions Notation 1 2

//////('l / ///////////

CCCC

~

CCSC

t

(i = 1,2,6;j = 1,2,6; N = no. of layers) T h e stiffness

Qiiare given as: 3

011 °:4 -I- 2(012 -t- 2066)0C2~ 2 + Q22~ 4

011 =

0s2 (Qll + 022 - 4066)0:2~ 2 + 012 (0:4 + 022 Q l l ~ 4 -t- 2 ( 0 1 2 + 2066)0:2~ 2 + Q220:4 016 (011-- Q12- 2066)0:J~ -t-

(2)

(012 - Q22 + 2066)0:~ 3 026 ~-- (011 - Q12 - 2066)0:~ 3 + (012 - Q22 + 2Q66)0:3~ 0 6 6 = ( 0 1 1 Jr - 2012 - 2066)0:2~ 2 Jr 066(0:464)

/

CCFC

--

where Qll

I/I/11

E1 , Q12 1J12E2 (1 - ~12v21) - ( 1 - v12v21)

Q22 -

E2 (1 -- 1-1121J21) ' 0 6 6 =

ccss "/111/11111

G12

0: = Cos 0, 13 -- Sin 0, and IJ21E 1 = 1)12E2 . The kinetic e n e r g y of the plate is given by: b

t-I

5

134)

Q22

/

3

- zL,)

3k=l

1

CCFS

T= ~ S]~

ptli4~2dxdy

(3)

where, w = 3iv (t = time) 9t Using E q u a t i o n s (1) and (3), U and T are evaluated.

6

'

!l/llllllJ

CCFF

I ?1

ssss FCFF

8

LINEAR VARIATION OF THICKNESS T h e v o l u m e of the plate is kept constant. Thickness of the plate having linear variation of thickness in the x-direction is given by: t1- tar(1-

1"1-Xa)

(4)

(1 - 0.5,1)

( (/////////////)

)Simplesupport(S); clamped(C);(

Thickness of the plate having linear variation of thickness in the y-direction is given by:

)free(F) t1 -

(5) (1 - 0 . 5 0 )

W h e r e ~1 is a constant whose value varies b e t w e e n 0 and 1 (rl -- 0.0 represents constant thickness plate, rl = 1.0 represents m a x i m u m taper), t I av = 1.0 mm, t 1max = 2.0 m m , tlmin - 0.0 m m (ll = 1.0), t I = tav -----1.0 m m (TI = 0.0).

a

b

METHOD OF SOL UTION 0

lY

~-~ x

A series solution satisfying b o u n d a r y conditions is t a k e n as: oo

m-l,2

Fig. 1

468

Rectangular plate e l e m e n t with coordinate directions

oQ

n=l,2

w h e r e (I),,,(x) represents the b e a m function satisfying the plate b o u n d a r y condition in the x-direction and

COMPOSITES. NOVEMBER 1988

~.(y) represents the beam function satisfying the plate boundary condition in the y-direction. °

II

i

=i

In the present study the conventional beam functions are assumed. The assumed displacement function is substituted in the energy equation. Integrations have been carried out by Gaussian Quadrature scheme. The energy equations are varied to obtain the standard eigen value problem. The resulting eigen value problem has been solved by a simultaneous iteration technique.

x

NUMERICAL EXAMPLES AND DISCUSSION The above Rayleigh-Ritz formulation is used to study the effect of fibre orientation and boundary conditions on the frequencies of square orthotropic plates of variable thickness made of graphite/epoxy, the material properties of which are given in Table 2. 4 In order to validate the present formulation, the fundamental frequency parameter (~'1 = ptlp 2L4/ ~'DID2) of clamped square plate for which solutions are available in References (1) and (5) are obtained and are shown in Table 3. Our values are close to these. Two cases of orthotropic square plates having linear variation in thickness are studied in this paper. Tables 4 and 5 give results for eight boundary conditions using n = 4 in Gaussian integration scheme. Tables 6 and 7 give results for two boundary conditions (ie, CCCC and SSSS) using n = 8 in Gaussian integration scheme.

Case I: linear variation in x-direction (graphite/ epoxy plate)

b

The effect of fibre orientation on fundamental frequency parameter x,'kl for various boundary

Fig. 2 Linear variation of thickness in (a) x direction. (b) y direction

Table 2

Material properties of typical unidirectional composites 4

Composite

EI(GPa)

E2(GPa)

1) 12

G12(GPa)

Specific gravity

Fibre vol fraction (Vf)

Glass/epoxy Kev la r/e poxy Boron/epoxy Graphite/epoxy

38.6 76.0 204.0 181.0

8.27 5.50 18.50 10.30

0.26 0.34 0.23 0.28

4.14 2.30 5.59 7.17

1.80 1.46 2.00 1.60

0.5 0.5 0.5 0.5

Table 3 Clamped square plate (constant thickness)

conditions and degree of taper (~1) is analysed. The values of ~kl are tabulated in Table 4.

Material

Fibre ~'kl orientation, (0) Present Ref. 1 Ref. 5 sol.

Glass/epoxy Kevlar/epoxy Boron/epoxy Graphite/epoxy

0° 0° 0° 0°

37.75 44.87 42.94 47.24

Plate thickness = 1.0 m m

COMPOSITES. NOVEMBER 1988

38.54 45.72 43.46 48.34

38.58 45.83 43.54 48.46

BC-1 (CCCC): For ~1 = 0.0, vkl decreases as fibre angle increases from 0 ° to 45 ° becoming minimum at 0 = 45 ° and then it increases for 0 = 450-90 °. For ~1 = 0.2, ~/kl decreases as fibre angle increases from 0 ° to 60 ° becoming minimum at 0 = 60 ° and then it increases for 0 = 600-90 °. For ~ = 0.4 to 1.0, ~X1 is maximum at 0 = 0 ° and decreases continuously as 0 increases from 0 ° to 90 °.

BC-2 (CCSC): For rI = 0.0 to 1.0, ~'kl is maximum at 0 ° and decreases continuously as 0 increases from 0 ° to 90 °.

469

Table 4. epoxy) BC

Linear variation in xdirection (graphite/

~"1 for ~1values

0 (degree)

BC

0.0

0.2

0.4

0.6

0.8

1.0

1 0 (CCCC) 15 30 45 60 75 90

47.24 47.14 46.93 46.83 46.93 47.14 47.24

47.59 47.46 47.20 47.08 47.06 47.21 47.28

48.98 48.78 48.28 47.86 47.58 47.51 47.46

52.17 51.77 50.78 49.66 49.30 48.16 47.81

58.33 57.57 55.61 53.23 50.98 49.28 48.31

69.13 67.73 64.10 59.49 54.90 50.90 48.84

2 0 (CCSC) 15 30 45 60 75 90

46.53 46.18 44.96 42.72 39.24 36.04 34.68

46.89 46.54 45.34 43.46 39.73 36.41 34.74

48.29 47.88 46.57 44.17 40.81 37.29 35.00

51.50 50.93 49.25 46.41 45.32 38.26 35.59

57.69 56.79 54.29 50.47 59.92 40.21 36.67

3 0 (CCFC) 15 30 45 60 75 90

45.08 42.99 37.25 29.49 21.69 16.18 14.22

45.43 43.74 38.22 30.44 22.40 16.54 14.31

46.84 45.56 40.17 32.20 23.66 17.25 14.64

50.06 49.20 43.72 35.66 25.78 18.54 15.41

4 0 (CCSS) 15 30 45 60 75 90

33.73 34.79 36.87 37.17 36.87 34.79 33.72

34.56 35.39 37.00 37.70 36.65 34.56 33.28

36.19 36.72 37.74 38.01 37.11 34.67 32.90

5 0 (CCFS) 15 30 45 60 75 90

31.81 32.65 30.58 25.87 19.74 14.23 11.91

32.79 33.37 30.97 26.01 19.80 14.33 11.93

6 (CCFF)

0 15 30 45 60 75 90

8.35 11.64 12.92 13.80 12.97 10.73 8.33

7 0 (SSSS) 15 30 45 60 75 90 8 (FCFF)

470

0 15 30 45 60 75 90

Table5. epoxy)

Linearvariationin ydire~ion(graphite/

0 (degree)

,J~-i for ~1values 0.0

0.2

0.4

0.6

0.8

1.0

1 0 (CCCC) 15 30 45 60 75 90

47.24 47.14 46.93 46.83 46.93 47.14 47.24

47.28 47.21 47.07 47.11 47.20 47.46 47.59

47.46 47.53 47.60 47.96 48.29 48.78 48.98

47.81 48.23 49.13 49.70 50.80 51.78 52.17

48.32 49.50 52.53 53.38 55.70 57.60 58.33

48.85 51.42 55.59 60.00 64.43 67.85 69.13

68.53 67.01 63.00 57.27 50.36 43.18 38,45

2 0 (CCSC) 15 30 45 60 75 90

46.53 46.18 44.96 42.72 39.24 36.04 34.68

45.86 45.49 44.41 42.01 39.14 36.54 35.47

45.16 44.88 43.73 41.81 39.62 37.77 37.03

44.46 44.47 49.30 42.52 41.22 40.28 39.94

43.76 44.46 44.77 44.82 44.88 45.07 45.18

42.95 44.98 47.76 49.98 52.08 53.75 54.39

56.25 55.74 59.79 40.09 29.26 20.74 16.92

67.05 66.76 59.65 47.80 34.72 24.32 19.59

3 0 (CCFC) 15 30 45 60 75 90

45.08 42.99 37.25 29.49 21.69 16.18 14.22

42.05 40.21 35.33 27.96 20.97 15.99 14.23

38.51 36.98 32.67 26.42 20.38 16.09 14.58

34.56 33.41 29.98 25.10 20.28 16.90 17.03

30.60 30.08 28.05 24.87 21.67 19.43 18.65

27.18 28.04 28.76 28.25 27.23 26.42 16.13

39.16 39.34 39.62 39.23 37.54 34.93 32.69

44.48 44.24 43.54 42.10 39.45 35.74 32.82

53.76 52.95 50.82 47.59 43.07 37.49 33.46

4

30 45 60 75 90

33.73 34.79 36.87 37.17 36.87 34.79 33.72

33.28 34.57 36.77 37.70 36.99 35.38 34.56

32.90 34.55 36.78 38.01 37.74 36.71 36.19

32.71 34.90 37.58 39.26 39.64 39.35 39.16

32.87 36.14 39.59 42.20 43.61 44.26 44.48

33.55 37.96 43.56 48.01 51.09 53.04 53.76

34.54 34.91 32.11 26.77 20.29 14.70 12.13

37.65 37.88 34.61 28.63 21.58 15.57 12.67

43.05 43.29 39.41 32.40 24.21 17.28 13.82

52.37 52.77 47.87 39.07 28.88 20.31 16.03

5 0 (CCFS) 15 30 45 60 75 90

31.81 32.65 30.58 25.87 19.74 14.23 11.91

29.74 30.69 28.96 24.78 19.24 14.28 12.22

27.34 28.44 27.21 23.71 18.92 14.67 12.95

24.83 27.22 25.53 22.94 19.16 17.25 14.49

22.22 24.10 24.65 23.36 20.90 18.65 17.81

20.84 23.76 26.45 27.31 26.72 25.85 25.53

9.19 11.48 13.37 13.88 12.72 10.35 7.95

10.50 12.70 14.24 14.23 12.65 10.02 7.58

12.64 14.78 15.83 15.20 12.97 9.88 7.32

16.43 18.63 19.11 17.53 14.25 10.33 7.48

24.39 26.82 26.37 23.05 17.81 12.37 8.95

6 (CCFF)

0 15 30 45 60 75 90

8.35 11.64 12.92 13.80 12.97 10.73 8.33

7.96 10.36 12.73 13.88 13.53 11.48 9.19

7.58 10.03 12.66 14.24 14.20 12.70 10.50

7.32 9.89 12.99 15.21 15.83 14.78 12.63

7.48 10.35 14.29 17.58 19.14 18.64 16.42

8.95 12.46 18.06 23.37 26.61 26.91 24.38

22.58 24.47 27.59 29.42 27.59 24.46 22.58

22.66 24.58 28.02 29.58 28.40 24.56 22.64

22.96 24.99 28.62 30.25 28.58 24.92 22.88

23.66 25.95 30.00 31.78 29.87 25.73 23.40

25.08 27.88 32.68 34.65 32.19 28.17 24.51

27.81 31.98 37.43 39.29 36.42 29.40 26.08

7 0 (SSSS) 15 30 45 60 75 90

22.58 24.47 27.59 29.42 27.59 24.46 22.58

22.68 24.57 28.01 29.58 28.43 24.57 22.66

22.87 24.92 28.59 30.26 28.61 24.98 22.96

23.37 27.17 29.91 31.80 30.00 25.94 23.66

24.29 27.30 32.40 34.78 32.72 27.88 25.08

25.77 29.68 36.45 39.88 37.62 31.65 27.81

6.48 6.32 5.45 3.99 2.59 1.85 1.72

7.12 7.02 6.16 4.55 2.96 2.11 1.96

7.90 7.92 7.14 5.33 3.48 2.49 2.31

8.92 9.10 8.52 6.51 4.28 3.06 2.85

10.58 10.89 10.61 8.41 5.63 4.04 3.77

14.39 14.63 14.42 11.92 8.24 6.02 5.61

8 (FCFF)

6.48 6.32 5.45 3.99 2.59 1.85 1.72

6.51 6.36 5.49 4.02 2.61 1.86 1.73

6.62 6.49 5.62 4.11 2.67 1.90 1.77

6.84 6.77 5.90 4.33 2.81 2.00 1.86

0

(CCSS) 15

0 15 30 45 60 75 90

7.23 7.29 6.45 4.76 3.10 2.21 2.05

7.93 8.21 7.44 5.56 3.63 2.59 2.39

C O M P O S I T E S . N O V E M B E R 1988

B-C-3 (CCFC):

Table 7.

V a r i a t i o n o f ~,'L1w i t h 0 f o l l o w s t h e s a m e p a t t e r n as B C - 2 f o r all v a l u e s o f Vl. B u t d e c r e a s e i n v a l u e of ~,'kl w i t h 0 is m u c h m o r e for B C - 3 c o m p a r e d to t h a t o f BC-2.

BC

Linear variation in ydirection 0 (degree)

~.1 for q values 0.0

0.2

0.4

0.6

0.8

1.0

(CCCC) 15 30 45 60 75 90

48.55 48.31 47.84 47.61 47.84 48.31 48.55

48.60 48.41 47.79 47.82 48.14 48.68 48.95

48.49 48.52 48.60 48.69 49.36 50.18 50.56

49.19 49.42 50.27 50.68 52.14 53.60 54.21

49.82 50.67 52.27 54.57 57.53 60.14 61.20

50.73 52.42 56.50 61.42 66.94 71.50 73.30

0 7 (SSSS) 15 30 45 60 75 90

23.18 25.04 28.16 29.92 28.16 25.04 23.18

23.22 25.10 28.61 30.16 28.65 25.12 23.25

23.46 25.51 28.90 30.77 28.93 25.58 23.55

24.00 26.34 30.44 32.31 30.58 26.55 24.23

25.08 27.80 32.74 35.22 33.25 28.48 25.66

26.63 32.05 37.05 39.82 38.01 32.57 28.41

BC-4 (CCSS):

F o r ~1 = 0.0, 0.2 a n d 0.4 "q'~'l i n c r e a s e s as 0 i n c r e a s e s f r o m 0 ° to 45 ° b e c o m i n g m a x i m u m at 0 = 45 °, t h e n it d e c r e a s e s for 0 = 4 5 0 - 9 0 °. F o r rl = 0.6 ~'k~becomes m a x i m u m at 0 = 30 ° a n d t h e n it d e c r e a s e for 0 = 3 0 0 - 9 0 °. F o r ~1 = 0.8 a n d 1.0, \'~.1 is m a x i m u m at 0 = 0 ° a n d it falls c o n t i n u o u s l y as 0 i n c r e a s e s f r o m 0 ° to 90 °.

BC-5 (CCFS): F o r q = 0 . 0 - 1 . 0 , ~'~ i n c r e a s e s for 0 = 0 ° to 15 °, b e c o m i n g m a x i m u m at 0 = 15 °, t h e n it d e c r e a s e s as 0 i n c r e a s e s f r o m 15 ° to 90 °.

BC-6 (CCFF):

F o r rl = 0.0 a n d 0.2, \'~'1 i n c r e a s e s for 0 = 0 ° to 45 ° b e c o m i n g m a x i m u m at 0 = 45 ° t h e n it d e c r e a s e s as 0 i n c r e a s e s f r o m 45 ° to 90 °. F o r ~1 = 0.4, 0.6 a n d 0.8, ~'kt i n c r e a s e s for 0 = 0 ° to 30 ° b e c o m i n g m a x i m u m at 0 = 30 °, t h e n it d e c r e a s e s as 0 i n c r e a s e s f r o m 30 ° to 90 °. F o r ~1 = 1.0, ~.~ is m a x i m u m at 0 = 15 °.

BC-7 (SSSS): F o r q = 0.0 to 1.0, v'~.l i n c r e a s e s as 0 i n c r e a s e s f r o m 0 ° to 45 °, b e c o m e s m a x i m u m at 0 = 45 °, t h e n it d e c r e a s e s f o r 0 = 45 ° to 90 °.

BC-8 (FCFF): F o r q = 0 . 0 a n d 0.2, ~:kl d e c r e a s e s as 0 i n c r e a s e s f r o m 0 ° to 90 °. F o r q = 0 . 4 - 1 . 0 , f~.~ i n c r e a s e s f r o m 0 = 0 ° to 15 °, b e c o m i n g m a x i m u m at 0 = 15 °, it t h e n d e c r e a s e s for 0 = 15 ° to 90 °.

Case I1: Linear variation in y-direction: (graphite/ epoxyplate) T h e effect of f i b r e o r i e n t a t i o n o n f u n d a m e n t a l f r e q u e n c y p a r a m e t e r vki for v a r i o u s b o u n d a r y c o n d i t i o n s a n d d e g r e e o f t a p e r (~1) is a n a l y s e d . T h e v a l u e s o f ~ a r e t a b u l a t e d in T a b l e 5.

Table 6. BC

BC-1 (CCCC): F o r r I = 0.0, x'~.l d e c r e a s e s for 0 = 0 ° to 45 °, b e c o m i n g m i n i m u m at 0 = 45 °, it t h e n i n c r e a s e s for 0 = 45 ° to 90 °. F o r 11 -- 0.2, x~.~ b e c o m e s m i n i m u m at 0 = 30 °. F o r ~l 0.4 to 1.0, ~'~1 i n c r e a s e s for 0 = 0 ° to 90 °.

BC-2 (CCSC): F o r 11 = 0.0, F o r 11 = 0.6, m a x i m u m at 90 °. F o r r I =

0.2 a n d 0.4, x'~.l d e c r e a s e s for 0 = 0 ° to 90 °. ~.1 i n c r e a s e s for 0 = 0 ° to 30 ° b e c o m i n g 0 = 30 °, it t h e n d e c r e a s e s for 0 = 30 ° to 0.8 a n d 1.0, ~.1 i n c r e a s e s for 0 = 0 ° to 90 °.

BC-3 (CCFC): F o r ~1 = 0 . 0 to 0.8, V~.l d e c r e a s e s as 0 i n c r e a s e s f r o m 0 ° to 90 °. F o r q = 1.0, ~ . l i n c r e a s e s for 0 = 0 ° to 30 °, t h e n it falls for 0 = 30 ° to 90 °.

BC-4 (CCSS):

BC-5 (CCFS):

~'~.1 for vl values

0.0

0.2

0.4

0.6

0.8

1.0

1 0 (CCCC) 15 30 45 60 75 90

48.55 48.31 47.84 47.61 47.84 48.31 48.55

48.95 48.68 48.14 47.82 47.79 48.41 48.60

50.56 50.18 49.36 48.69 48.60 48.52 48.49

54.21 53.60 52.14 50.68 50.27 49.42 49.19

61.20 60.14 57.53 54.57 52.27 50.67 49.82

73.30 71.50 66.94 61.42 56.50 52.42 50.73

7 0 (SSSS) 15 30 45 60 75 90

23.18 25.04 28.16 29.92 28.16 25.04 23.18

23.25 25.11 28.64 30.15 28.61 25.09 23.22

23.55 25.57 28.93 30.76 28.90 25.50 23.46

24.23 26.55 30.58 32.30 30.44 26.34 24.00

25.66 28.48 33.25 35.22 32.74 27.79 25.08

28.41 32.57 38.01 39.82 37.04 32.05 26.63

COMPOSITES. NOVEMBER 1988

0

F o r ~1 -- 0.0, 0.2 a n d 0.4, ~kl i n c r e a s e s for 0 = 0 ° to 45 °, t h e n it d e c r e a s e s for 0 = 45 ° to 90 °. F o r 11 = 0.6, m a x i m u m ~.~.~o c c u r s at 0 = 60 °. F o r ~1 = 0.8 a n d 1.0, ~'~,1 is m i n i m u m at 0 = 0 ° a n d it i n c r e a s e s as 0 i n c r e a s e s f r o m 0 ° to 90 °.

Linear variation in xdirection 0 (degree)

1

F o r ~1 = 0 . 0 to 0.6, ~'~.~i n c r e a s e s for 0 = 0 ° to 15 °, b e c o m i n g m a x i m u m at 0 = 15 °. It t h e n d e c r e a s e s for 0 = 15 ° to 90 °. F o r ~1 = 0.8, m a x i m u m ~X~ o c c u r s at ~1 = 30 °. F o r ~1 = 1.0 m a x i m u m ~-1 o c c u r s at 0 = 45 °.

BC-6 (CCFF): F o r 11 = 0.0, 0.2 a n d 0.4, ~:kl i n c r e a s e s for 0 = 0 ° to 45 ° a n d t h e n it d e c r e a s e s for 0 = 45 ° to 90 °. F o r 11 = 0.6 a n d 0.8, m a x i m u m ~.1 o c c u r s at 0 = 60 °, a n d for q = 1.0 m a x i m u m x'~.l o c c u r s at 0 = 75 °.

BC- 7 (SSSS): F o r ~1 = 0.0 to 1.0, V~,l i n c r e a s e s for 0 = 0 ° to 45 °, t h e n it d e c r e a s e s for 0 = 45 ° to 90 °.

BC-8 (FCFF): F o r 11 = 0.0 to 0.6, ~kl d e c r e a s e s as 0 i n c r e a s e s f r o m 0 ° to 90 °. F o r I1 -- 0.8 a n d 1.0, m a x i m u m ~.~ o c c u r s at 0 = 15 ° .

471

CONCLUSIONS

REFERENCES

1) For linear variation in thickness along the xdirection, fundamental frequency parameter x~.l increased as ~1(degree of taper) is increases from 0.0 to 1.0 for almost all boundary conditions and fibre orientation. This does not hold true for linear variation in thickness in the y-direction.

1 Chelladurai, T., Shastry, B. P. and Ran, G. V. 'Effect of Fibre orientation on the Vibration Behaviour of Orthotropic Rectangular Plates' Fibre Sci Techno121 (1984) pp 73-81 2 Bert, C. W. 'Fundamental Frequencies of Orthotropic Plates with Various Planforms and Edge Conditions', Shock Vib Bull 47 (1977) pp 89-94 3 Jones, R. M. 'Mechanics of Composiw Materials' (McGraw Hill, New York, 1975) 4 Tsai, S. W. and Hohn, H. T, 'Introduction to Composite Materials', Vols I and II, AFML Tech. Report, TR-78-201, (1979) 5 Lekhnitekii, S. G. 'Anisotropic Plates' (Gordon and Breach Sci. Pub. London, 1956)

2) Fibre angle, at which ~.1 is maximum for a given boundary condition, is also dependent on degree of taper 01). 3) From Tables 4 and 5, it is possible to choose the fibre angle that gives maximum ~.l for a given boundary condition and degree of taper. 4) For both the cases, pattern of variation of ',~'1 with fibre angle is dependent on degree of taper (aq) for most of the boundary conditions.

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A U THORS

The authors are with the Indian Institute of Technology, Madras 600 034, India. Enquires should be addressed to Dr Malhotra at the FRP Research Centre.

COMPOSITES. NOVEMBER 1988