Effect of fibre orientation and boundary conditions on the vibration behaviour of orthotropic rhombic plates

Effect of fibre orientation and boundary conditions on the vibration behaviour of orthotropic rhombic plates S.K MALHOTRA, N. GANESANand M.,~ VELUSWAMI (Indian Institute of Technology, India)

The effect of fibre orientation on the frequencies of thin, rhombic, orthotropic plates has been studied using a parallelogrammic, orthotropic plate finite element for various boundary conditions and skew angles. The results indicate that, for a given skew angle and boundary conditions, the fibre orientation of such plates can be chosen to achieve the desired natural frequency. Key words: composite materials; vibration orthotropic plates; fibre orientation; skew angle; boundary conditions; fundamental frequency parameter

NOTATION Ai

constants

x,y

rectangular coordinates

a, b

lengths of sides of parallelogrammic plate element

7

skew angle

D~

equal to Elt 3/12(1 -- v~2 v21)

0

fibre orientation

D2

equal to Ezt 3/12(1 -- v12 v21)

X~

fundamental frequency parameter

p

circular frequency

v~2

major Poisson's ratio

t

plate thickness

vz~

minor Poisson's ratio

U

strain energy

~, ~/ oblique coordinates

{v}

column matrix of nodal displacements

P

w

transverse displacement in the Z-direction

% X slopes Ow/Orh Ow/O~

Study of the vibration behaviour of skew orthotropic plates is of interest because such structures are incorporated in modern high speed aircraft and missiles. Chelladurai et al ~ have carried out limited studies for rectangular/square orthotropic plates for two boundary conditions. Bert 2 has derived a formula for approximate values of the fundamental frequency of orthotropic plates of arbitrary shape and boundary conditions if frequencies of the corresponding isotropic plates are known. The present study is carried out for five boundary conditions for rhombic plates with different skew angles. The influence of fibre orientation

mass per unit area

and boundary conditions on fundamental frequencies is studied. The parallelogrammic plate finite element of Dawe, 3 modified for orthotropic materials, is used for the analysis. Elements of the [D] matrix are obtained from laminated plate theory. 4

FINITE ELEMENTFORMULATION Fig. 1 shows a parallelogrammic plate finite element. It is assumed that the element deforms under load according to the following simple polynomial deflection

0010-4361/88/030127-06 $3.00(~)1988 Butterworth & Co (Publishers) Ltd COMPOSITES. VOLUME 19 . NUMBER 2. MARCH 1988

127

L

2/v

°

•

xE

[D]

Oil

D12 Dl6

D12

D22 D26

D16 D26 D66 N

/4

Dij = ~1

(ai/)k (z k3- Z k =3 l)

3 ~,- = Skew angle

(i = 1 , 2 , 6 ; / = 1,2, 6; N = number of layers) Y Fig. 1

The stiffnesses Qu are as given as:

Parallelogrammic plate element

expression in terms of oblique coordinates ~ and r/:

*/+At 60 = A1 +A 2 ~- + A 3 -~

~2

~-- + A s

(~u = Qll c0s40 + 2(Q12 + 2Q66) sin20cos20

+ Q22sin40

tirt

-ab -

(~12 = (Q11 + Q22 - 4 Q ~ ) sin20 cos20 772 /j3 tj2ri, /jrl2 + A6 b--~- + A,7 a--T- + As --a2 b + A9 --ab2

+ Ql2(sin40 +c0s40) (~22 = Qusin*0 + 2(Q12 + 2 Q ~ ) sin20 cos20

773

"1" Alo V

~j3~/ +A12

"!- All ~

~j~3

+ Qm cos40

ab 3

or

Q16 = (Qu - Qx2 - 2Q6~) sin0 cos30 co

[m] {A}

=

(1)

+ (Ql2 -Q22 + 2Q66) sin30 cos0

where {A} is the column matrix of constants Ai. Using the coordinate transformations:

Q26

=

(Ou - Ql2 - 2Q~s) sin30 cos0

+ (QI2 - Q22 + 2Q¢,6) sin0 cos30 = x + y tan7

(~

= (Qxl +Q22-2Qm-2Q~s) sin20eos20

1"/ = ySCCT

+ Q66 ( sinaO + cos40)

it is possible to express the lateral deflections in terms of the rectangular coordinates x and y. The two oblique slopes are given by: aw = ~ and off

{A} =

[ B -1 ]

{V}

Q22

i)121)21 ) '

Q12

E2 (I --/"12 P21) ' Q66

(1 - v12 v2x)

G12

P21E 1 = uI2E2 The orthogonal and oblique curvatures are related as:

('d2, • .-,

X4)

I a2w ]

Matrix [B-q is the same as that used in the analysis for a rectangular plate element.

=

[C]

~-

[D] [C] d x d y

(3)

02w/ • axayj

where ~"

(a2w

a2w

a2w t

\ax2

aY2

axay]

B

~ ~2w ' ay2 [

The bending strain energy of the element is:

128

-

and

(2)

{V} = ((a)l , ,I/,/1 , Xl,

{C}

V12 E2

E1

QII - (1

where

U

where

aw Z = bff

The constants of the deflection expression (Equation (1)) are evaluated by satisfying the boundary conditions at node points l, 2, 3 and 4 to give:

(4)

_

a2w

1

0

0

sin27 cos2?

1 cos27

2sin7 cos2T

sin 'y cos'),

0

1 cos

0~2

7

a2w a~ 2 a2w a~an

or

{ c } = [G] {Cob}

(5) C O M P O S I T E S . M A R C H 1988

Table 1.

Material properties of typical unidirectional composites 5

Composite

E1 (GPa)

E2 (GPa)

V12

G12 (GPa)

Specific gravity

Glass/epoxy Kevlar/epoxy 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 7.1 7

1.80 1.46 2.00 1.60

Table 2.

Fundamental frequency parameters ( t/~ 1 ) for a clamped square plate (fibre orientation, 0 = 0 °)

Material

Present s o l u t i o n

Reference1

Reference 6

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

39.89 48.97 46.42 52.06

38.54 45.72 43.46 48.34

38.58 45.83 43.54 48.46

Table 3. studied

Definition of boundary conditions

Boundary Plan form and edge condition (BC) condition (skew angle number = 0 °)

Notation

1

CCFC

~

where [k] is the stiffness matrix for the parallelogrammic plate element. The inertia matrix for the parallelogrammic plate element is given as:

{Fi.}=pp=cos~[B-']r

(~

?

~=0

2

~

CCCC

3

,~

CSSC

4

~

CCSC

5

[]

SSSS

- - - - -- Simple support ( S ), free (F)

/////

x [m]T [m] d~dr/) [B-1]

where ~ is proportional to p2, and [Me] is the inertia matrix of the parallelogrammic plate element.

clamped ( C ) ,

The governing equation of vibration in matrix form is:

(Cob} .= [El { A } = [El [B-']

[K] {v} - X [ 3 ( ]

(6)

{v}

Substituting Equation (6) into Equation (5) gives:

(c) = [a] [El [B-'] {v}

a

n" o

f= [E]T [G]T [D] ~ o

[B-']){v}

or

E = ~ [1v ]

T

[k] {v}

(8)

Application of Castigliano's theorem, OU/avi = Fi, yields: { F } = [k]

{v}

COMPOSITES. MARCH 1988

{v} = 0

(11)

where [K] is the assembled stiffness matrix, [At] is the assembled inertia matrix, k is the eigen value and {v} is the eigen vector. Equation (11) is solved using a standard algorithm for obtaining eigen values and eigen vectors.

(7)

The bending strain energy, Equation (3), can be expressed in terms of oblique coordinate quantities as:

x [G] [E]d, dr0

{v} = x [Me] {v} (10)

Also, the oblique curvatures can be obtained by differentiating Equation (1):

U=~- {v} cos~[K ~]r

~=0

(9)

NUMERICAL EXAMPLES AND DISCUSSION

The above finite element formulation is used to study the effect of fibre orientation and boundary conditions on the frequencies of rhombic plates made of glass/ epoxy, Kevlar/epoxy, boron/epoxy and graphite/epoxy, the material properties of which are given in Table 1.5 These properties are based on a fibre volume fraction, Vf, of 0.70 for all four materials and an eight-layered symmetric lay-up (0, -0,/9, - 6 / - 0 , 0, -0, 0). The fundamental frequency parameters ()~ = pt p2L4/X/D1D2) obtained for a clamped square plate (a = b = L) are compared with solutions from References 1 and 6 in Table 2. The values of v/At obtained in this study are slightly higher than those in previous work. ',6 In the study of orthotropic rhombic plates, five boundary conditions (defined in Table 3) and four skew angles 0r/12, rd6, rr/4 and rr/3) are considered. 129

Plots of the fundamental frequency parameter (v/Xa) fibre orientation (8) for the five boundary conditions, four skew angles and four materials are shown in Figs 2-5.

vs

These plots provide useful information for the design of orthotropic skew plates. The fibre angle at which the maximum value of v/X~ occurs for a given boundary condition is different for different skew angles. For example, considering boundary condition 1 (see Table 3): for skew angles of rr/12 and rr/6, v/X~ is maximum at a fibre angle of around 15°; for a skew angle of rr/4, v/X1 is maximum around 30°; while for a skew angle of zr/3,

maximum v/ X~ occurs around 90 °. Also, the nature of the variation of v / X~ with fibre angle for a given boundary condition is different for different skew angles as shown in Table 4. CONCLUSIONS

The following observations may be made from Figs 2-5. 1)

40

For a given boundary condition and skew angle, the variation of the fundamental frequency parameter, X/X1, with fibre orientation, 0, follows the same pattern for all four materials considered. For a given boundary condition, the nature of the variation of v/)h with ~ is dependent on the skew angle of the plate. For a given boundary condition, the value of 0 at which maximum v/X, occurs is different for different skew angles. In general, the value of v/X1 increases with increasing skew angles for all boundary conditions.

30

80

2)

60 o Glass/epoxy o Kevlar/epoxy A Graphite/epoxy Y

50

3) 4)

a

20

o []

Glass / epoxy Kevlar / epoxy

6O x

IO

a

i

i

I

I

i

I~ 40

i

60

20

50 i

i

i

I

I

i

40

3o

b

~

t

I

I

,°l

I

I

^

^

o

-0

~ 60

5O

4O

20

0

- ~

I

I

I

I

- 40

o

~

20 80

60

40

40

I ~

6O e

3O 4O

~2o

~- zo IO

0

15

30

45

60 8(°)

75

90

Fig. 2 Relationship between fundamental frequency parameter (v',kl) and fibre orientation (8) for skew angle of ~r/12: (a) BC 1 (CCFC); (b) BC 2 (CCCC); (c) BC 3 (CSSC); (d) BC 4 (CCSC); and (e) BC 5 (SSSS)

130

0

I 15

I 30

t 45

I 60 #(°)

1 75

I 90

Fig. 3 Relationship between fundamental frequency parameter ( V'~ 1) and fibre orientation (8) for skew angle of ~r/6: (a) BC 1 (CCFC); (b) BC 2 (CCCC); (c) BC 3 (CSSC); (d) BC 4 (CCSC); and (e) BC 5 (SSSS)

COMPOSITES.

MARCH

1988

Table 4. Effect of boundary condition and skew angle on the relationship between v'X1 and 0 for orthotropic skew plates BC

Skew angle =/1 2

7r/6

~r/4

rr/3

CCFC

One maximum around 1 5 ° then decreasing

One maximum around 1 5 ° then decreasing

One maximum around 30 °

Increasing with 0

CCCC

One minimum around 45 °

Increasing with 0

One maximum around 75 °

Increasing steeply with 0

CSSC

Increasing with 0 (slow increase)

Increasing with 0

Increasing with 0

Increasing with 0

CCSC

One maximum around 15 °

One maximum between One maximum around 30 ° and 45 ° 60 °

Increasing with 0

SSSS

One maximum around 60 °

Increasing with 0

Increasing steeply with 0

Increasing with 0

120 a

o D A x

90

Gloss/epoxy Kevlar/ epoxy Graphite/epoxy Boron/epoxy

d

100

50

6O

30

40

b

I

I

I

30

45

I

I

I

60

75

90

e

120 60 I00 4o Be

20

15

@(*)

60

40 100

I

I

I

I

I

I

15

30

45

I

I

I

8O

6O

40

I

I

I

60

75

90

Fig. 4 Relationship between fundamental frequency parameter (v"~l) and fibre orientation (0) for skew angle of 7r/4: (a) BC 1 (CCFC); (b) BC 2 (CCCC); (c) BC 3 (CSSC); (d) BC 4 {CCSC); and (e) BC 5 (SSSS)

e (o)

5)

6)

Table 4 demonstrates how the relationship between v/X1 and 0 is affected as the skew angle is changed from ~-/12 to rr/3 for different boundary conditions. For an orthotropic plate with a given skew angle and boundary conditions, a suitable fibre orientation can be chosen to achieve the desired natural frequency.

C O M P O S I T E S . MARCH 1988

REFERENCES 1 2 3

Chelladurai, T. et al "Effect of fibre orientation on the vibration behaviour of orthotropic rectangular plates" Fibre Sci and Technol 21 (1984) pp 73-81 Bert, C.W. 'Fundamental frequencies of orthotropic plates with various planforms and edge conditions', Shock Vib Bull No 47 (1977) pp 89-94 Dawe, D.J. "Parallelogrammic elements in the solution of

131

a o

14o

d

Gloss/ epoxy 220

120 180 L~>- i o o L~- 140 80 I00 ! I

60

I

I

I

I

I I

5O

b

260

I

I

I

I

170 e

22(2 140 --

[.~- 180 120 -

~

140 ~

IO0 -

I001 I

I

I

I

I

I

80 -

//~

220 50

I

0

15

30

45

I

#

('60)

I

I

75

90

180

I0o

50 0

I 15

I 50

I 4.5

I 60

I 75

I 90

Fig. 5 Relationship between fundamental frequency parameter ( v ' h l ) and fibre orientation (8) for skew angle of 7r/3: (a) BC 1 (CCFC); (b) BC 2 (CCCC); (c) BC 3 (CSSC); (d) BC 4 (CCSC); and (e) Be 5 (SSSS)

0(*)

rhombic cantilever plate problems"J Strain Analysis 1 No 3 (1966) 4 Jones, R.M. 'Mechanics of Composite Materials' (McGraw Hill, 5 6

New York, 1975) Tsai,S.W. et al. 'Introduction to composite materials, vois I and II"AFML Tech Report TR-78-201 (1979) Lekhnitskii,S,(;. 'Anisotropic Plates' (Gordon and Breach

AUTHORS The authors are with the Indian Institute of Technology, Madras 600 036, India. Enquiries should be addressed to Dr Malhotra at the FRP Research Centre.

Science Publishers, London, UK, 1956)

132

COMPOSITES. MARCH 1988