Transverse vibration of symmetrically laminated elastically restrained plates of variable thickness

Ashour International Islamic University Malaysia, Department of Science in Engineering, 53100 Kuala Lumpur, Malaysia e-mail: ashour@iiu.edu.my Key words: Finite strip method, vibration,

Computational Mechanics July 7-12, 2002, Vienna, Austria Eds.: H.A Mang, F.G Rammerstorfer,

J Eberhardsteiner

Transverse Vibration of Symmetrically Laminated

Elastically Restrained Plates of Variable Thickness

Ahmed S Ashour

International Islamic University Malaysia, Department of Science in Engineering, 53100 Kuala Lumpur,

Malaysia e-mail: ashour@iiu.edu.my

Key words: Finite strip method, vibration, symmetrically laminated, elastic restrained

Abstract

The natural frequencies of symmetrically laminated plates of variable thickness are analyzed using the finite strip transition matrix technique In this paper, the natural frequencies of such plates are determined for edges with elastic restrained against rotation, transition or both

A successive conjunction of the classical finite strip method and the transition matrix method is applied to develop a new modification of the finite strip method to reduce the complexity of the problem The displacement function is expressed as the product of a basic trigonometric series function in the longitudinal direction and an unknown function that has to be determined in the other direction Using the new transition matrix, after necessary simplification and the satisfaction of the boundary conditions, yields a set of simultaneous equations that leads to the characteristic matrix of vibration

Numerical results for different combinations of elastic rotational or transitional edges have been presented and compared with those available from other methods in the literature Also, the effect of the tapered ratio and the aspect ratio on the natural frequencies of the plates is presented The good agreement with other methods demonstrates the validity and the reliability of the proposed method

1 Introduction

Laminated plates are widely used in many engineering application In some applications, the designer has to construct variable thickness plates to save material or to meet certain criteria The vibration of uniform plates with elastic restrained boundary conditions has been investigated by many authors [1-8] On the other hand, the vibration of elastically restrained plates of variable thickness has been studied by relatively fewer authors

Kobayashi and Sonoda [9] presented an exact method for analyzing the free vibration and buckling of isotropic plate with linearly tapered thickness in one direction with two opposite edges are simply supported and the other two edges are elastically restrained against rotation Grossi and Bahat [10] used Rayleigh-Ritz method with the boundary characteristics orthogonal polynomials as shape functions and Rayleigh Schmidt method to find the natural frequencies of isotropic tapered rectangular plates with edges restrained against rotation and translation Gutierrez and Laura [11] used the differential quadrature method to determine the fundamental frequencies of rectangular plates with linearly varying thickness and non-uniform boundary conditions Filipich et al [12] used the Galerkin method to obtain an approximate solution to the vibration of isotropic rectangular plates of variable thickness with two opposite edges simply supported and very general boundary conditions on the other two edges

Rais-Rohani and Marcellier [13] presented approximate analytical solutions for the free vibration and buckling of rectangular anisotropic plates as well as rectangular sandwich plates with edge restrained against rotation

To the best of the author’s knowledge there is no publication available in the open literature on this problem The main objective of this paper is to determine the natural frequencies of cross play symmetrically laminated plates of variable thickness subjected to elastically restrained boundary conditions against both rotation and translation in the variable thickness direction and any combination

of clamped or simply supported boundary conditions in the other direction

2 Governing Equations of Elastic Restrained plates

Consider a cross play symmetrically laminated rectangular plate of variable thickness h(y), length a, width b, density ρ and with elastic restrained boundary conditions at y = 0 and y = b as shown in Fig

1 The governing equation can be written as:

3 22

0

where W is the flexural displacement,

3 3

ij ij

o

h

h

= are the bending rigidities, h is the plate height at y o

= 0,

/ 2 ( ) 2

1 / 2

, 1, 2,6

h N

k

ij ij

k h

= −

( )k

ij

Q are the material constants of the kth lamina , N no of layers, z kis the distance from the midplane of the plate to the bottom of the kth lamina and Q ij( )k are the plane stress reduced

stiffness coefficient of the lamina k given by

Fig 1: Laminated plate of variable thickness

yy yx xx xx

xy

xy yx yx xy xy yx

E

HereE xx, E yy are the longitudinal and transverse plate moduli, respectively and G xy is the in plane shear modulus, and νxy and νyx are the Poisson’s ratios

2.1 Boundary Conditions

The considered boundary conditions along the y-direction are elastically restrained against both rotation and translation Aty=0, the boundary condition for this case are,

0

∂ ν

∂

whereD3=D12+2D66, the suffixes “0, b” means the rigidities are calculated at y = 0 and y = b respectively The boundary conditions on the other elastically restrained end (y = b) are

b

∂ ν

∂

where suffixes “b, 0” means rigidities are calculated at y = 0, y = b respectively The boundary conditions at the other two edges x = 0 and x = a can be any combination of the classical boundary

conditions (simple, clamped, free)

2.2 Method of solution and the eigen value problem

Assuming a solution of the form

1

( ) ( )

M

m m m

=

where Xm (x) are the beam functions that satisfy the boundary conditions at x = 0 and x = a The governing equation can be transformed into 4N number of first order differential equations in terms of

the normalized coordinates which can be solved as in [14] In many papers, e.g [1, 2], the boundary

conditions are solved approximately by neglecting some of the terms in Eqs (4-7) In this paper, the elastically restrained boundary conditions are solved exactly Using Eq (8), the boundary conditions

at the normalized coordinates η=0are

0

3

3

1 22

2

i jj T

Y

0

2

1

1

M

j xy ij i

i

i ij R

Y

ν

where 0

0

22

0

T

D

bT

0

22 0

R

D bR

φ = and at η=1

3

3

1 22

2

M

D

Y

2

1

1

b

M

i

i ij R

Y

ν

where 22b

b

T

b

D

bT

φ = and 22b

b

R b

D bR

φ =

In the next sections, the symbol S-C-S-ER (for example) means that the edges x = 0, y = 0, x = a, y = b are simply supported, clamped, simply supported and elastic restrained respectively A linearly tapered plate is used to illustrate the above technique with the following non-dimensional variable thickness ( ) 1hη = +δη, where δ is the taper ratio

3 Convergence and Comparison Investigation

3.1 Convergence analysis

Since no solution exists for the above problem, one has to carry out several convergence studies First,

a convergence investigation is carried out to examine the effect of number of terms of the power series

M used in the solution The results are shown in table 1 for uniform three-layer cross-ply symmetrically laminated plates The material properties used in this case are given

byE11/E22=40,G12/E22=0.6, ν12=0.25, and the frequency parameter λ used is given

(E /E )/

λ λ= β Form Table 1, it is very clear that the method converge very rapidly Other convergence studies are carried out for linearly tapered plate but they are not shown here due to the limitation of the space

Table 1: Convergence investigation of frequency parameters λ for three-ply (0,90,0) fully clamped

plates (CCCC)

1

3.2 Comparison analysis

Table 2 shows the frequency parameters for clamped unidirectional laminated uniform plates compared with some available results in the literature The material properties used in this case are given by E11/E22=15.4, G12/E22 =.79, ν12=.30, the results agree very well with other results in the literature

Table 2: Comparison of frequency parameters of five-layer laminated plate (CCCC)

1

4 Numerical Results

The frequency parameter λ for uniform laminated plates subject to classical boundary conditions are presented and compared with results from ref [16] in table 3 It can be seen that the results agree very well with ref [16] A parametric investigation is carried out to study the effect of elastic restrained boundary conditions on the frequency parameters for uniform and non-uniform laminated plates The results are shown in Fig 2a and Fig 2b for uniform plates and in Figs 3-5 for non uniform plates First, we consider(S-ER-S-ER) uniform laminated plates Figure 2 shows the frequency parameter surfaces of the fundamental (a) and the second modes (b) surfaces versus the elastic restrained against rotation coefficient φr and against translation coefficient φt for three layer laminated plate (90, 0,90) Table 3: The frequencies parameters of three-ply uniform laminated plates (0,90,0) for different

combinations of boundary conditions with some comparison

Boundary Condition

1

t r

O

O

r

t

Fig 2: The frequency parameters l for laminated plate with edges elastically restrained against both

rotation and translation (a) Fundamental mode (b) Second mode

7

8

9

10

1.0E-08 1.0E-06 1.0E-04 1.0E-02 1.0E+00 1.0E+02

Elastic Restrained Against Rotation )r

δ = 0.2

δ = 0.4

(a)

8

9

10

11

12

13

14

15

Elastic Restrained Against Rotation )

r

δ = 0.2

δ = 0.4

(b) Fig 3: The Frequency parameters λ of the fundamental (a) and the second mode (b) versus the elastic restrained against rotation coefficient φr with different taper thickness ration δ for S-ER-S-ER

plate (Φt = 0.000001)

14

15

16

17

18

Elastic Restrained Against Rotation )r

δ = 0

δ = 0.2

δ = 0.4

(a)

15

16

17

18

19

20

21

22

Elastic Restrained Against Rotation )r

δ = 0

δ = 0.2

δ = 0.4

(b) Fig 4: The Frequency parameters λ of the fundamental (a) and the second mode (b) versus the elastic restrained against rotation coefficient φr with different taper thickness ration δ for C-ER-C-ER plate

(Φt = 0.000001)

Fig 4: The fundamental frequency parameters λ versus the elastic restrained against rotation coefficient φr with different taper thickness ration δ for C-ER-C-ER plate (Φt = 0.1)

The relation between the frequency parameter λ and the elastic restrained against rotation coefficient

φr while the elastic restrained against translation Φt is kept equal 0.000001, almost zero, ( for different taper ratios δ = 0., 0.2 and 0.4) is shown in Fig 3 for S-ER-S-ER plate and in Fig 4 for C-ER-C-ER plate In both figures, the fundamental mode is presented in (a) while the second mode is presented in (b) In Figure 5, the fundamental frequency parameters is plotted versus the elastic restrained against translation coefficient φr while the coefficient of elastic restrained against translation is considered equal (Φt = 0.1) It can be observed that in the case of S-ER-S-ER, the elastic restrained against rotation has more effect on the fundamental frequency than in the case of C-ER-C-ER In all cases, the effect is significant only in the range (.001 <Φt< 1) Also, it can be seen that the second mode is affected more than the first mode From Fig 5, it can be seen that φr has no effect on the fundamental frequency for uniform plate and has a slight effect for taper plates Finally, the fundamental frequency increases as the taper ratio increases as it can be seen for Fig 3a, 4a and 5 While it decrease as the taper ratio increases, figs 3b and 4b for the second mode

5 Conclusion

In this paper, the finite strip transition matrix is used to investigate the vibration of laminated plates of variable thickness The lamination used is limited to cross-ply The boundary conditions considered

in the papers are any combination of elastic restrained against translation, rotation or both in the variable direction and any combination of elastic or clamped boundary condition in the other side The effect of the elastic restrained boundary conditions on the vibration frequency parameters are investigated and presented in tabulated and graphic forms

The results presented in this paper for elastically restrained laminated plated can be considered as new

in the literature and can be useful for designers and engineers

13

14

15

16

17

Elastic Restrained Against Rotation )r

r O

δ = 0

δ = 0.2

δ = 0.4

Acknowledgements

This research was supported by the Research Center, International Islamic University Malaysia

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