**Open Access**-
**Authors :**Ola Ragb , M. S. Matbuly -
**Paper ID :**IJERTV8IS120057 -
**Volume & Issue :**Volume 08, Issue 12 (December 2019) -
**Published (First Online):**11-12-2019 -
**ISSN (Online) :**2278-0181 -
**Publisher Name :**IJERT -
**License:**This work is licensed under a Creative Commons Attribution 4.0 International License

#### Efficient Quadrature Solution for Composite Plates with Variable Thickness Resting on Non-Uniform and Nonlinear Elastic Foundation

Ola Ragb*, M. S. Matbuly

Department of Physics and Engineering Mathematics, Faculty of Engineering, Zagazig University,

P.O. 44519, Zagazig, Egypt

AbstractTwo Different schemes are examined for vibration analysis of composite plate with variable thickness resting on non-uniform and nonlinear elastic foundation problems. On the basis of first order transverse shear theory a basic equation of vibration is derived. Investigations are made over non-uniform and nonlinear Winkler and uniform Pasternak foundation model. Examined schemes are based on discrete singular convolution and moving least square differential quadrature method. Also, the obtained nonlinear algebraic system is solved by using iterative quadrature technique. This problem is solved for different boundary conditions, different shear correction factor and varying thickness in one and two directions. Numerical analysis is applied to investigate influence of different computational characteristics on convergence and accuracy of the obtained results. The obtained results agreed with the previous analytical and numerical ones. Further a parametric study is introduced to explore the influence of elastic and geometric characteristics of the vibrated plate, on results.

KeywordsComposite; Vibration; Shear Correction Factor; Non-uniform Elastic Foundation; Nonlinear Winkler Foundation; Variable Thickness; Discrete Singular Convolution; Moving Least Square.

INTRODUCTION

Non-uniform elastic plates with varying thickness are commonly used in ship and offshore structures, pavement of roads, footing of buildings and bases of machines. Therefore, the vibration analysis of like plates is of great importance for practical design and structural components.

Depending upon the requirement, durability and reliability, materials are being developed so that these may provide better strength, efficiency and economy. Therefore a study of character and behavior of these plates is required so that the full potential of these plates may be used. So, these plates have been studied analytically and numerically. In most cases, their closed form solutions are extremely difficult to establish. But, for some special cases of variable thickness of rectangular plate, investigations have been made and solutions have been obtained [1-4].The most commonly used numerical methods for such applications are Spline function approximation technique [5-6], Rayleigh-Ritz

method [7-8], Mesh-free method[9], Mixed boundary node method [10], Finite strip method [11] and Finite element method [12], have been widely applied for the plate resting on uniform elastic foundation. Most of these studies are computationally expensive.

In seeking a more efficient numerical method is differential quadrature method (DQM) which requires fewer grid points yet achieves acceptable accuracy [13-15]. By applying DQM, the free vibration problem of plate is translated into the eigenvalue problem. According to the selection of basis functions and influence domain for each point, there are more than versions of DQM. Discrete singular convolution differential quadrature method (DSCDQM) [16-18], Moving least square differential quadrature method (MLSDQM) [19-21] are the most reliable versions. DSCDQ method is that they exhibit exponential convergence of spectral methods while having the flexibility of local methods for complex boundary conditions. MLSDQ exploits the merits of both the DQ and meshless method.

The main aim of present work to apply two different schemes,( DSCDQM and MLSDQM) to solve vibration problems of composite plates with varying thickness. These plates are resting on non-uniform and nonlinear Winkler and uniform Pasternak elastic foundation. Based on a transverse shear theory, the governing equations of the problem are formulated. The unknown field quantities and their derivatives are approximated using DQ approximations. Then the obtained nonlinear algebraic system is solved by using iterative quadrature technique. The program of MATLAB is designed to solve the reduced eign-value problem. Numerical analysis is implemented to investigate convergence and efficiency of each scheme. Accuracy of the obtained results is compared with the existing previous results. Further a parametric study is introduced to investigate the influence of elastic and geometric characteristics on natural frequencies and mode shapes.

FORMULATION OF THE PROBLEM

Consider a composite consisting of n plates interfacialy bonded with variable thickness h(x,y) and resting on non-uniform and nonlinear elastic foundation of Winkler and uniform Pasternak type , as shown in Fig.(1).

D (x , y ) E h 3 (x , y ) / [12(1 2 )]

is the flexural rigidity

of the plate. G, E and v are shear modulus, Youngs modulus and Poisson's ratio of the plate. k is the shear correction factor [25-27].

Assuming harmonic behavior of the problem, the field quantities can be written as:

x (x , y ,t ) x e i t ,

Fig.1 Composite Plate with Variable Thickness resting on Non-uniform and Nonlinear Winkler- Pasternak Foundation.

y (x , y ,t ) y e i t ,

w (x , y ,t ) W e i t

(5)

Each plate occupies (ai 1 x ai , 0 y b, i 1,n) ,

Where is the natural frequency of the plate and

where a and b are width and length of the composite.

Based on a first-order shear deformation theory, the equations of motion for each plate can be written as [20, 22]:

i 1 .

x , y ,W are the amplitudes for x , y , and w ,

2 (1 ) 2 (1 ) 2 y

respectively.

D (x , y ) x x

x 2 2 y 2 2 x y

(1)

Substituting from Eq.(5) into (1-4), One can reduce the

kGh (x , y ) w

2

x I1 x ,

problem to:

2 2 2

x

t 2

x

(1 ) x

(1 ) y

D(x, y)

2 2 x2

2 y2 2 xy

y

y

D (x , y )

(1 ) y

(1 ) 2x

(6)

y 2

2 x 2

2 x y

(2)

W 2

kGh(x, y) x

x

I1x ,

y 1

y 1

kGh (x , y ) w

y

I

2 y

,

t 2

2

(1 ) 2

(1 ) 2

2w

2w

D(x, y) y y

x

kGh (x , y )

x

y

y2 2 x2

2 xy

x 2

y 2 x

y

(3)

(7)

3 2w 2w 2w

kGh(x, y) W

2 I ,

2 0

2 0

K1 (x )w

K 3w

K 2 x 2

I

y

t 2

y y 1 y

Where Io and I1 are mass moments of inertias [23]:

h /2

2W

kGh(x, y)

2W

x

y 3

K1(x)W K3W

0 x2

y2 x

y

(8)

I0 , I1

(1 , z2 ) dz,

(4)

h0 /2

2W

K2 x2

2W

y2

2 I0W

is the plate mass density. K

1 (x ) K1

(1 x / a)

is the non-uniform Winkler foundation stiffness which linearly varying along x-direction. is variation parameter of stiffness of foundaion. K2 is shear modulus of foundation

Boundary conditions can be expressed as follows:

Clamped edge:

reaction.

K 3 is the non-linear Winkler foundation. t: time.

W 0,

nx y

ny x

0,

x (x , y ,t ),y (x , y ,t ),w (x , y ,t ) are normal strain

n n 0

(9)

rotations and transverse deflection [24].

The thickness variation function of plate is

x x y y

h(x , y ) h0 (1 x / a)(1 y / b). h0 is the constant

reference thickness value and , are variation parameter of thickness.

Simply Supporting of the first kind: SS1

(x ,b )

W 0,

W (x ,b

i ) W (x ,bi ),

D ( x i

x

(x ,b )

(x ,b )

(x ,b )

(n 2 n 2 ) x

(1 )n n

x

y i ) D ( x i y i ),

x y x

x y y

1

y

x (x ,bi )

x y

y (x ,bi )

(13)

x y

x y

( n 2 n 2 ) y

y

(1 )nx ny

y 0,

x

(10)

2

1

( y

(x ,b )

x )

(x ,b )

1

D ((n 2 n 2 ) x 2n n

x

( x i

2 y

y i ), (i 1, n )

x

2 x y y

x y x

(n 2 n 2 ) y x y x

2n n y ) 0

x y y

SOLUTION OF THE PROBLEM

Two different differential quadrature techniques are employed to reduce the governing equations into nonlinear eigenvalue problem, as follows [16-21]:

Simply Supporting of the second kind: SS2

W 0, nx y ny x 0,

Discrete Singular Convolution Differential Quadrature Method (DSCDQM)

In this technique, regularized Shannon kernel (RSK) may be used as a shape function such that the unknown u(x)

(n 2 n 2 ) x (1 )n n x

(11)

and its derivatives can be approximated over a narrow

x y

( n 2 n 2 )

x

y

x y

(1 )n n

y

x 0

bandwidth

( x x M , x x M ) as [16-18]:

2

x y x y

M sin[ (x x ) / h ]

( (x i x j ) )

y x

u (x )

i j x e

2 2

u (x

), (14)

i (x x ) / h j

Free edge:

j M i j x

(i N , N ),

kG h (n

W n

W n

n

) 0,

x x

y y

x x y y

where hx is the step size, 2M+1 is the effective

(n 2 n 2 ) x

(1 )n n

x

computational band width, is regularization parameter,

x y

x y

x y

( n 2 n 2 )

x

y

y

x y

(1 )nx ny

y

y 0,

x

(12)

= r hx and r is a computational parameter.

Derivatives of u, can be approximated as a weighted linear sum of ui , (i=-N,N) as [16-18]:

1 D ((n 2 n 2 ) x

2n n

x

u M x

2 x y y

x y x

x x x

i j M

Cij u (x j ),

(15)

2 2 y

y

y

(nx n ) x

y

2nx ny y ) 0

2u

x 2

x x

M

ij j

ij j

C xx u (x ),

i N , N ,

nx and ny are the directional cosines at a point on the

Where

i j M

boundary edge.

Along the interface between ith plate and (i+1) th one, the

(1)i j

hx (

e

(i j )2

2 2 ) ,

i j

continuity boundary conditions can be described as:

hx (i j )

C

C

x

ij

0

0

,

i j

(16)

i j 1

2 (i j )2

x by minimizing the following weighted quadratic for

( 2(1)

hx 2 (i j )2

xx

1 )e

2

hx ( 2 2 ) ,

j

N 1

h 2

C ij

1 2

(a) (x – xi )(u

i 1

(xi ) ui )

(20)

1

1

i j

2

3hx 2

N

(x – xi )(PT (xi )a(x) ui )2

i 1

Similarly, one can approximateu y ,u yy and calculated

C

C

, C

, C

y yy

ij ij .

Moving Least Squares Differential Quadrature Method (MLSDQM)

Where (x-xi) is a positive weight function defined over the influence domain, (i, i=1,N)

The stationary value of (a) with respect to a(x) leads to a linear equation, such as:

In this technique, the influence domain, (i, i=1: N), for each node is determined as shown in Fig. (2). Over each

A(x)a(x) B(x)u

(21)

influence domain, the nodal unknowns can be approximated as [19-21]:

from which

a(x) A1 (x)B(x)u , (22)

u (xi ) u h

N

1

1

(xi ) j (xi ) u (x j ),(i 1, N ),

(18)

where

T

j 1

A(x) P(xi ) i (x)P

(xi )

1

1

N

i i

i i

n

n

i 1

(x)P(x )PT

(xi ),

u=u1 u2

u T ,

B(x)=P(x) (x)

1 (x)P(x1)

2 (x)P(x2 )

n (x)P(xn )

.

Fig.2 Domain discretization for moving least

On suitable substitution from Eq. (22) into (19), uh(x) can then be expressed as:

u h (x) PT (x)a(x)

1

1

i

i

N

squares differential quadrature.

PT (x)A1 (x)B

xu=i (x)ui

(23)

Where the shape function j (xi

) can be obtained using

i 1

MLS approximation as follows:

Where the nodal shape function:

Let

m

i

i

u h (x) Pi (x)ai (x) PT

(x)a(x),

(19)

i (x) = PT (x)A1 (x) B

x

(24)

i 1

W here a(x) a (x),a (x), ,a

(x) T is a vector of

Determination of the shape function i (x) and its partial

1 2 m

unknown coefficients.

derivatives can be simplified as follows [19-21]:

i x PT xA1 xBi x T xBi x

(25)

PT (x) p1 (x), p2 (x), , pm (x) is a complete set

of monomial basis. m is the number of basis terms. The coefficients aj (x),( j 1, m ) , can be obtained at any point

Since A(x) is a symmetric matrix, then obtained through

Ax x P x

x

can be

(26)

Therefore, the problem of determination of the shape function is reduced to solution of Eq.(26). This Equation can be solved using LU decomposition and back-substitution, which requires fewer computations than the inversion of

A(x). Further, the first and second order partial derivatives of

W i 0,

n i n i 0, n i

n

n

y y

y y

j (xi ) can be determined as follows [19-21]:

x y y x x x

Differentiate Eq. (26) with respect to L, K, (L, K =X,Y) such as:

i 0, (i 1, N )

(34)

A(x),L (x) P,L (x) A,L (x) (x),

Simply supporting of the first kind: SS1

(L X ,Y )

(27)

W i 0,

N (n 2 n 2 ) c x (1 )n n c y j

x y ij x y ij x

0,

A(x),LK (x) P,LK (x) A,LK (x)(x)

j 1 ( n 2 n 2 )c y

(1 )n n c x j

(35)

(28)

x y ij x y ij y

A,L (x),K (x) A,K (x),L (x), (L, K X ,Y )

N (n 2 n 2 ) c y 2n n c x j

1 D ij

x y ij x y ij x

0,

2 j 1 (n 2 n 2 ) c x 2n n c y j

The first and second order partial derivatives of the shape function can be described as:

x y ij x y ij y

(i 1, N )

(x ) c L (x ) T (x )B (x )

Simply supporting of the second kind: SS2

j ,L i j i j ,L i j i

T (x )B (x

), (L x , y ),

(29)

W i 0,

n i n i

0,

j i j ,L i

x y y x

N (n 2 n 2 )c x (1 )n n c y j

(x ) c LK (x ) T

(x )B (x )

x y ij x y ij x

0

j ,LK i j i j ,LK i j i

j 1 ( n 2 n 2 )c y (1 )n n c x j

(36)

T (x )B (x ) T (x )B (x ) T

(x )B (x ),

x y ij x y ij y

i j ,LK i j ,L i j ,K i j ,K i j ,L i

(L, K x , y )

(30)

Free edge:

(i 1, N )

On suitable substitution from Eqs.(14-30) into (6-8), the problem can be reduced to the following nonlinear eigenvalue problem:

N

N

x ij y ij x x y y

x ij y ij x x y y

kG h ij ((n c x n c y )W j ) kG h ij (n i n i ) 0,

j 1

N (n 2 n 2 )c x (1 )n n c y j

kGhij c xW j D ij c xx 1 c yy kGh ij j

x y ij x y ij x

0,

N ij

ij

2 ij

x

j 1 ( n 2 n 2 )c y (1 )n n c x j

(37)

x y ij x y ij y

j 1 D ij 1 c x c y j

(31)

N (n 2 n 2 )c y 2n n c x j

2 ik kj y

1 D ij x y ij x y ij x

0

2

2 2 x y j

2 I j , (i , k 1, N )

j 1 (nx ny )cij 2nx ny cij y

1 x (i 1, N )

kGhij c yW

j D ij 1 c x c y

j

N ij

2 ik

kj x

NUMERICAL RESULTS

j 1 D ij c yy

1 c xx

kGhij j

(32)

This section presents numerical results that

ij

2 ij

y

demonstrate convergence and efficiency of each one of

the proposed schemes for vibration analysis of non-

y

y

2 I1 j , (i , k

N

N

kGhij c xx

1, N )

c yy W

j c x j c y j

uniform and nonlinear Winkler and uniform Pasternak foundation of composite plate. This plate varying with thickness. For all results, the boundary conditions (34-37) are augmented in the governing equations (31-33). Then

j 1

ij ij

ij x

ij y

the obtained nonlinear algebraic system is solved by

K c xx

c yy

W j K iW

j K

(W 2 )W j

(33)

using

u

iterative quadrature

ar f

technique

[34]. Thee

2 ij ij 1 3

comp tational ch acteristics o each schem are adapted

2 I 0W j , (i 1, N )

to reach accurate results with error of order 10-10. The obtained frequencies are normalized such as:

The boundary conditions (9-13) can also be approximated using DQMs as:

Clamped edge:

( I (20 ) where 0 is the fundamental

frequency of isotropic squared plate.

For DSCDQ scheme based on regularized Shannon kernel (RSK), the problem is also solved over a uniform grids ranging from 7*7 to 19*19. The bandwidth 2M+1

ranges from 5 to 15 and the regularization parameter = r hx ranges from 1.5hx to 3 hx , where hx =1/N-1.

Table (1) shows convergence of the obtained

fundamental frequency to the exact and numerical ones [28-31] over grid size 13*13, bandwidth 11 and regulization parameter = 2.82 hx.

TABLE. 1 Comparison between the fundamental frequency due to DSCDQM-RSK with the bandwidth 2M+1

fundamental frequency

Number of grid points

9

11

13

15

17

regularization parameter

Band width

DSCDQM- RSK

=1.8 hx

2M+1 =7

2.03

2.46

2.74

2.89

2.9545

2M+1 =9

2.23

2.69

2.82

2.93

2.9736

2M+1 =11

2.58

2.88

2.91

2.95

2.9828

2M+1 =13

2.79

2.92

2.96

2.99

2.9991

=2.4 hx

2M+1 =7

2.29

2.64

2.83

2.92

2.9683

2M+1 =9

2.52

2.72

2.88

2.96

2.9743

2M+1 =11

2.61

2.91

2.94

2.98

2.9915

2M+1 =13

2.86

2.95

2.99

2.99

2.9976

=2.62 hx

2M+1 =7

2.58

2.72

2.89

2.96

2.9697

2M+1 =9

2.70

2.85

2.94

2.99

2.9953

2M+1 =11

2.76

2.94

2.99

3.01

3.0224

2M+1 =13

2.89

2.98

3.01

3.02

3.0213

=2.82 hx

2M+1 =7

2.86

2.92

2.98

2.99

3.0026

2M+1 =9

2.89

2.98

2.99

3.005

3.0141

2M+1 =11

2.94

2.99

3.02

3.022

3.0215

2M+1 =13

2.97

3.01

3.02

3.022

3.0215

Exact results [28]

3.0215

Ritz method[29]

3.0214

Radial basis[30]

3.0216

Element free Galerkin[31] (15×15)

3.0225

fundamental frequency

Number of grid points

9

11

13

15

17

regularization parameter

Band width

DSCDQM- RSK

=1.8 hx

2M+1 =7

2.03

2.46

2.74

2.89

2.9545

2M+1 =9

2.23

2.69

2.82

2.93

2.9736

2M+1 =11

2.58

2.88

2.91

2.95

2.9828

2M+1 =13

2.79

2.92

2.96

2.99

2.9991

=2.4 hx

2M+1 =7

2.29

2.64

2.83

2.92

29683

2M+1 =9

2.52

2.72

2.88

2.96

2.9743

2M+1 =11

2.61

2.91

2.94

2.98

2.9915

2M+1 =13

2.86

2.95

2.99

2.99

2.9976

=2.62 hx

2M+1 =7

2.58

2.72

2.89

2.96

2.9697

2M+1 =9

2.70

2.85

2.94

2.99

2.9953

2M+1 =11

2.76

2.94

2.99

3.01

3.0224

2M+1 =13

2.89

2.98

3.01

3.02

3.0213

=2.82 hx

2M+1 =7

2.86

2.92

2.98

2.99

3.0026

2M+1 =9

2.89

2.98

2.99

3.005

3.0141

2M+1 =11

2.94

2.99

3.02

3.022

3.0215

2M+1 =13

2.97

3.01

3.02

3.022

3.0215

Exact results [28]

3.0215

Ritz method[29]

3.0214

Radial basis[30]

3.0216

Element free Galerkin[31] (15×15)

3.0225

,regularization parameter , the grid points N and the previous results for a regular discretized isotropic simply supported squared plate:( h0/a=0.01,v=0.3, k=5/6,K1=500,=0,=0,=0, K2=0, K3=0).

For MLSDQ scheme, circular influence domain, (i, i=1,N), is considered as shown in Fig.2. Gaussian weight function is employed such as [19-21]:

size 11*11 , completeness order 4 and raduis of support domain dmax = 5 .

TABLE.2 Comparison between the fundamental frequency due to MLSDQM with the radius of support domain dmax completeness order Nc , the grid points N and the previous results for a regular discretized isotropic simply supported squared plate: (h/a=0.01,v=0.3, k=5/6, K1=100,=0,=0,=0, K2=0, K3=0).

fundamental frequency

Number of grid points

7

9

11

13

completeness order

Radius of support domain

MLSDQM

Nc=2

dmax =5

2.64

2.516

2.446

2.4509

dmax =6

2.31

2.295

2.272

2.2612

dmax =7

2.26

2.254

2.251

2.2491

dmax =8

2.25

2.247

2.246

2.2436

Nc =3

dmax =5

2.25

2.245

2.245

2.2441

dmax =6

2.25

2.243

2.243

2.2421

dmax =7

2.24

2.242

2.242

2.2414

dmax =8

2.24

2.241

2.241

2.2413

Nc =4

dmax =5

2.24

2.241

2.2413

2.2413

dmax =6

2.24

2.241

2.2413

2.2413

dmax =7

2.24

2.241

2.2413

2.2413

dmax =8

2.24

2.241

2.2413

2.2413

Nc =5

dmax =5

2.74

2.244

2.2413

2.2413

dmax =6

2.25

2.249

2.2413

2.2413

dmax =7

2.26

2.241

2.2413

2.2413

dmax =8

2.26

2.241

2.2413

2.2413

Exact results [28]

2.2413

Ritz method[29]

2.2413

Radial basis[30]

2.2414

Element free Galerkin[31] (15×15)

2.2427

Table (3) shows that the best value of Shear factor correction is to be taken (5 /6-v). Tables (3,4) show that the fundamental frequency decrease with increasing Poisson ratio and Shear factor correction. Also ,the value of shear factor correction helps to achieve more accurate results.

TABLE.3 Comparison between the fundamental frequency, Poisson , shear factor correction K and the previous results

exp((d

i

/ c )2 ) exp((r / c )2 )

Poisson ratio v

Discrete Green function [33]

=

= ( + ( ))

(( ))

= ( + ( ))

( ( ))

=

=

0.15

8.1424

8.1638

8.1639

8.164

8.163

0.23

8.1110

8.1110

8.1112

8.1112

8.111

8.111

0.3

8.0640

8.04586

8.0460

8.046

8.046

0.5

7.8090

7.7430

7.7432

7.743

7.744

Poisson ratio v

Discrete Green function [33]

=

= ( + ( ))

(( ))

= ( + ( ))

( ( ))

=

=

0.15

8.1424

8.1638

8.1639

8.164

8.163

0.23

8.1110

8.1110

8.1112

8.1112

8.111

8.111

0.3

8.0640

8.04586

8.0460

8.046

8.046

0.5

7.8090

7.7430

7.7432

7.743

7.744

2

di r

(38)

for CCCS squared plate: (h0/a=0.01, K1=0,=0.5,=0.5,=0, K =0 , K =0,E /E =2.45).

w i (x )

0

0

1 exp((r / c ) )

di r

2 3 1 2

Where di is the distance from a nodal point xi

to a field one

x located in the influence domain of xi. r is the radius of support domain and c is the dilation parameter. In the present work, the dilation parameter is selected such as: c=r/4.

The scheme is employed with varying completeness order Nc ranges from 2 to 5 and the raduis of support domain dmax = r/ hx ranges from 4 to 9. Also, the problem is also solved over a uniform grids ranging from 5*5 to 19*19. Table (2) shows convergence of the obtained fundamental frequency to the exact and numerical ones [28-31] over grid

Boundary coditions

natural frequency

1

2

3

Execution time (sec)

variable thickness

results

SSSS

=0

DSCDQM- RSK

4.902

7.253

8.374

1.780280

MLSDQM

4.902

7.253

8.374

1.839144

Element free Galerkin [31]

4.900

7.256

8.382

Discrete Green function [33]

4.902

7.253

8.374

=0.4

DSCDQM- RSK

5.360

7.928

9.150

1.889903

MLSDQM

5.360

7.928

9.150

1.918841

Element free Galerkin [31]

5.356

7.931

9.159

Discrete Green function [33]

5.360

7.928

9.150

=0.8

DSCDQM- RSK

5.770

8.525

9.831

1.900178

MLSDQM

5.770

8.525

9.831

1.928867

Element free Galerkin [31]

5.772

8.529

9.845

Discrete Green function [33]

5.770

8.525

9.831

CCCC

=0

DSCDQM- RSK

6.780

8.953

10.29

1.632990

MLSDQM

6.780

8.953

10.29

1.646533

Element free Galerkin [31]

6.748

8.905

10.24

Discrete Green function [33]

6.780

8.953

10.29

=0.4

DSCDQM- RSK

7.402

9.770

11.23

1.716354

MLSDQM

7.402

9.770

11.23

1.726170

Element free Galerkin [31]

7.371

9.723

11.18

Discrete Green function [33]

7.402

9.770

11.23

=0.8

DSCDQM- RSK

7.945

10.48

12.05

1.731625

MLSDQM

7.945

10.48

12.05

1.742609

Element free Galerkin [31]

7.915

10.43

12.01

Discrete Green function [33]

7.945

10.48

12.05

Boundary coditions

natural frequency

1

2

3

Execution time (sec)

variable thickness

results

SSSS

=0

DSCDQM- RSK

4.902

7.253

8.374

1.780280

MLSDQM

4.902

7.253

8.374

1.839144

Element free Galerkin [31]

4.900

7.256

8.382

Discrete Green function [33]

4.902

7.253

8.374

=0.4

DSCDQM- RSK

5.360

7.928

9.150

1.889903

MLSDQM

5.360

7.928

9.150

1.918841

Element free Galerkin [31]

5.356

7.931

9.159

Discrete Green function [33]

5.360

7.928

9.150

=0.8

DSCDQM- RSK

5.770

8.525

9.831

1.900178

MLSDQM

5.770

8.525

9.831

1.928867

Element free Galerkin [31]

5.772

8.529

9.845

Discrete Green function [33]

5.770

8.525

9.831

CCCC

=0

DSCDQM- RSK

6.780

8.953

10.29

1.632990

MLSDQM

6.780

8.953

10.29

1.646533

Element free Galerkin [31]

6.748

8.905

10.24

Discrete Green function [33]

6.780

8.953

10.29

=0.4

DSCDQM- RSK

7.402

9.770

11.23

1.716354

MLSDQM

7.402

9.770

11.23

1.726170

Element free Galerkin [31]

7.371

9.723

11.18

Discrete Green function [33]

7.402

9.770

11.23

=0.8

DSCDQM- RSK

7.945

10.48

12.05

1.731625

MLSDQM

7.945

10.48

12.05

1.742609

Element free Galerkin [31]

7.915

10.43

12.01

Discrete Green function [33]

7.945

10.48

12.05

TABLE.4 Comparison between the fundamental frequency, Poisson , shear facor correction K and different boundary condition for squared plate: (h /a=0.01, K =100,=0,=0,=0,

0 1

K2=50 , K3=0,E1/E2=2.45).

Poisson ratio v

shear factor

correction

=

SSSS

CCCC

CSSS

CCSC

0.15

0.8547

3.9382

5.20912

4.2312

4.87028

0.23

0.8666

3.9207

5.18596

4.2123

4.84863

0.3

0.8772

3.8935

5.1501

4.1831

4.8151

0.5

0.9091

3.7429

4.95105

4.0213

4.62899

Tables (5, 6) show that execution time of DSCDQM-RSK scheme is less than that of MLSDQM. Therefore, it is more efficient than MLSDQM for vibration analysis of variable thickness, non-uniform and nonlinear elastically supported composite plate.

TABLE. 5 Comparison between the obtained natural frequencies and the previous results for isotropic clamped plate: (h0/a=0.015,v=0.15 ,=0,=0,=0, K2=0,K3=0).

Subgrade reaction

K1=1390.2

K1=2780.4

Execution time (sec)

Results

2

3

1

2

3

MLSDQM (11×11),

Nc=4 and dmax =5

5.245

8.316

8.316

6.463

9.132

9.132

1.676119

DSCDQM- RSK(13×13)

=2.82hx and M=5

5.245

8.316

8.316

6.463

9.132

9.132

1.656792

Mixed finite element [32]

5.245

8.316

8.316

6.463

9.132

9.132

Element free Galerkin[31]

5.267

8.391

8.392

6.477

9.202

9.203

Ritz method[29 ]

5.259

8.432

8.432

6.460

9.248

9.248

Radial basis[30 ]

5.244

8.313

8.313

6.463

9.130

9.130

Also, for different boundary conditions and varying thickness in one and two directions, tables (6,7) insist that DSCDQM-RSK scheme is the best choice for such problem.

TABLE. 6 Comparison between the natural frequency due to DSCDQM-RSK and MLSDQM with the variable thickness in one direction ,different boundary conditions and the previous results for a squared plate: (h/a=0.01,E1/E2=2.45 v=0.23, K1=0, =0, K2=0,K3=0).

TABLE. 7 Comparison between the natural frequency due to DSCDQM-RSK and MLSDQM with the variable thickness , in two direction ,different boundary conditions and the previous results for a squared plate: (h0/a=0.01,E1/E2=2.45, v=0.23, K1=0, K2=0, K3=0).

Boundary coditions

natural frequency

variable thickness

results

SSSS

-0.5

-0.5

DSCDQM- RSK

3.635

5.335

6.087

MLSDQM

3.635

5.335

6.087

Element free Galerkin [31]

3.633

5.3460

6.0957

Discrete Green function [33]

3.635

5.335

6.087

-0.5

0.5

DSCDQM- RSK

4.704

6.937

7.966

MLSDQM

4.704

6.937

7.966

Element free Galerkin [31]

4.707

6.9422

7.9751

Discrete Green function [33]

4.704

6.937

7.966

0.5

-0.5

DSCDQM- RSK

4.708

6.933

7.904

MLSDQM

4.708

6.933

7.904

Element free Galerkin [31]

4.708

6.9420

7.9146

Discrete Green function [33]

4.708

6.933

7.904

0.5

0.5

DSCDQM- RSK

6.086

9.022

10.350

MLSDQM

6.086

9.022

10.350

Element free Galerkin [31]

6.099

9.013

10.359

Discrete Green function [33]

6.086

9.022

10.350

CCCC

-0.5

-0.5

DSCDQM- RSK

4.955

6.548

7.440

MLSDQM

4.955

6.548

7.440

Element free Galerkin [31]

4.936

6.532

7.4167

Discrete Green function [33]

4.955

6.548

7.440

-0.5

0.5

DSCDQM- RSK

6.453

8.510

9.748

MLSDQM

6.453

8.510

9.748

Element free Galerkin [31]

6.427

8.481

9.715

Discrete Green function [33]

6.453

8.510

9.748

0.5

-0.5

DSCDQM- RSK

6.447

8.525

9.671

MLSDQM

6.447

8.525

9.671

Element free Galerkin [31]

6.420

8.501

9.639

Discrete Green function [33]

6.45

8.525

9.671

0.5

0.5

DSCDQM- RSK

8.390

11.076

12.666

MLSDQM

8.390

11.076

12.666

Element free Galerkin [31]

8.36

11.040

12.632

Discrete Gree function [33]

8.390

11.076

12.666

Boundary coditions

natural frequency

variable thickness

results

SSSS

-0.5

-0.5

DSCDQM- RSK

3.635

5.335

6.087

MLSDQM

3.635

5.335

6.087

Element free Galerkin [31]

3.633

5.3460

6.0957

Discrete Green function [33]

3.635

5.335

6.087

-0.5

0.5

DSCDQM- RSK

4.704

6.937

7.966

MLSDQM

4.704

6.937

7.966

Element free Galerkin [31]

4.707

6.9422

7.9751

Discrete Green function [33]

4.704

6.937

7.966

0.5

-0.5

DSCDQM- RSK

4.708

6.933

7.904

MLSDQM

4.708

6.933

7.904

Element free Galerkin [31]

4.708

6.9420

7.9146

Discrete Green function [33]

4.708

6.933

7.904

0.5

0.5

DSCDQM- RSK

6.086

9.022

10.350

MLSDQM

6.086

9.022

10.350

Element free Galerkin [31]

6.099

9.013

10.359

Discrete Green function [33]

6.086

9.022

10.350

CCCC

-0.5

-0.5

DSCDQM- RSK

4.955

6.548

7.440

MLSDQM

4.955

6.548

7.440

Element free Galerkin [31]

4.936

6.532

7.4167

Discrete Green function [33]

4.955

6.548

7.440

-0.5

0.5

DSCDQM- RSK

6.453

8.510

9.748

MLSDQM

6.453

8.510

9.748

Element free Galerkin [31]

6.427

8.481

9.715

Discrete Green function [33]

6.453

8.510

9.748

0.5

-0.5

DSCDQM- RSK

6.447

8.525

9.671

MLSDQM

6.447

8.525

9.671

Element free Galerkin [31]

6.420

8.501

9.639

Discrete Green function [33]

6.45

8.525

9.671

0.5

0.5

DSCDQM- RSK

8.390

11.076

12.666

MLSDQM

8.390

11.076

12.666

Element free Galerkin [31]

8.36

11.040

12.632

Discrete Green function [33]

8.390

11.076

12.666

Furthermore, a parametric study is introduced to investigate the influence of elastic and geometric characteristics of the composite on the values of natural frequencies.

Figures (3-15) show that the natural frequencies decrease with increasing variable thickness (, ), Young's modulus gradation ratio and thickness ratio. As well as, Figs. (3- 10,12-13) show that the natural frequencies are increased with increasing variation parameter of foundation stiffness , Winkler and Pasternak foundation, shear modulus gradation ratio and aspect ratio a/b . The case of (E1=E2=E3=E4=E5, G1=G2=G3=G4=G5 and p=p=p=h4=p) is a limiting case of this study which was previously solved in [28] and [29-33]. Figures(12-15) show the first three mode shapes of the vibration waves along the interface.Furthermore, Figs.(14-

show that the amplitudes of W increase with increasing linear and nonlinear elastic foundation parameters.Also, these figures show that the natural frequencies do not affect significantly by nonlinear elastic foundation parameter k3.

Normalized fundamental frequency

Normalized fundamental frequency

28

26 =1000

24

22 =500

20

18

16 =100

14

12

10 =0

8

6

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Thickness variable

Normalized fundamental frequency

Normalized fundamental frequency

28

26 =1000

24

22 =500

20

18

16 =100

14

12

10 =0

8

6

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Thickness variable

FIG. 3 Variation of the normalized fundamental frequency withthickness variable (, ) and non-uniform Winkler foundationfor squared simply supported plates. A: =0, B: =0.5(K1=1000, K2=0,K3=0).

Normalized fundamental frequency

Normalized fundamental frequency

Normalized fundamental frequency

Normalized fundamental frequency

44 42

42 40

=1000

40 38

38

38

=500

36

36 =100 34

34

32

32 =0 30

30

28

28

=1000

=500

=100

=0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Thikness variable

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Thikness variable

Normalized fundamental frequency

Normalized fundamental frequency

42

40 =1000

38 =500

36 =100

34

FIG. 5 Variation of the normalized fundamental frequency with thickness variable (, )

,non-uniform ,nonlinear Winkler and Pasternak foundation for squared simply supported plates. A: =0, B: =0.5 (K1=1000, K2=10, K3=50)

Normalized fundamental frequency

Normalized fundamental frequency

32 =0 18

30

28 15

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

B Thikness variable

12

=-0.5,-0.5

9 =0.5,0.5

FIG. 4 Variation of the normalized fundamental frequency with thickness variable (, ) and non-uniform Winkler and uniform Pasternak foundation for squared simply

supported plates. A: =0, B: =0.5 (K1=1000, K2=10,K3=0)

Normalized fundamental frequency

Normalized fundamental frequency

44

=-0.5,0.5

=0.5,0.5

6

3

0 2000 4000 6000 8000 10000

A

42

=1000

40

38 =500

36 =100

34

32 =0

30

28

27

Normalized fundamental frequency

Normalized fundamental fequency

24

21

18 =-0.5,-0.5

15 =0.5,0.5

12 =-0.5,0.5

=0.5,0.5

9

6

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0

Thikness variable

3

0 2000 4000 6000 8000 10000

FIG. 6 Variation of the normalized fundamental frequency with thickness variable (, ) and non-uniform Winkler foundation for squared simply supported plates . A: K1=10, B: K1=100 ( K2=0,K3=0)

Normalized fundamental frequency

Normalized fundamental frequency

30

28

26

24 =-0.5,-0.5

22 =0.5,0.5

=-0.5,0.5

20

=0.5,0.5

18

16

0 2000 4000 6000 8000 10000

51

Normalized fundamental frequency

Normalized fundamental frequency

48

45

42

39 =-0.5,-0.5

36 =0.5,0.5

33

30 =-0.5,0.5

27 =0.5,0.5

24

21

18

0 2000 4000 6000 8000 10000

Normalized fundamental frequency

Normalized fundamental frequency

51

48

45

42

39 =-0.5,-0.5

36 =0.5,0.5

33

30 =-0.5,0.5

27 =0.5,0.5

24

21

18

FIG. 8 Variation of the normalized fundamental frequency with thickness variable (, ) and non-uniform, nonlinear Winkler and Pasternak foundation for squared simply supported plates. A: K1=10, B: K1=100 ( K2=10,K3=100)

60

54

Natural frequencies

Natural frequencies

48

42

0 2000 4000

B

6000 8000 10000

36

30 E3=E2=0.2E1

E5=0.2E4

FIG. 7 Variation of the normalized fundamental frequency 24

E3=E2=E1=E4=E5=1

E4=E5=2E1

with thickness variable (, ) and non-uniform 18

E =E =E =3E

Winkler and Pasternak foundation for squared

12 E3=0.3E2 E =E =E =5E

1 4 5 2

simply supported plates. A: K1=10, B: K1=100 ( K2=10,K3=0)

3 2 1 4

6

0

Normalized fundamental frequency

Normalized fundamental frequency

30

28

26

24 =-0.5,-0.5

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

A Location along the interface.

60

54

Natural frequencies

Natural frequencies

48

42

22 =0.5,0.5

=-0.5,0.5

20

=0.5,0.5

36

30 G1=G4=G5=1.5G2

24

G4=G5=2G1

G5=1.75G4 G3=G2=G1=G4=G5=1

18

16

0 2000 4000 6000 8000 10000

A

18 G3=0.75G2

12

6

0

G3=G2=G1=0.5G4 G3=G2=0.25G1

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

B Location along the interface.

FIG. 9 Variation of the natural frequencies with Young's and Shear modulus gradation ratio of a squared simply supported composite (K1=100, K2=10,K3=50, h0/a=0.01,=1000, =-0.5, =-0.5, v1=v2=v3=v4=v5).

250

225

Natural frequencies

Natural frequencies

200

175

150

125

a1=a4=a5=3a2

a5=4a4

48

Natural frequencies

Natural frequencies

42

36

h =h =0.5h

100 a =a =2a

3 2 1

30

p=p=p=.7h4

h =h =h =h =h =1

4 5 1

75

p=.6p

24

3 2 1 4 5

50

3 2

3 2

a =(2/3)a

25 a =a =0.5a

a =a =a =.75a

a3=a2=a1=a4=a5=1

18 h =h =h =3h

h4=p=2p

3 2 1

0

3 2 1 4

1 4 5 2

p=4h4

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

A Location along the interface.

130

120

110

Natural frequencies

Natural frequencies

100

90

12

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

B Location along the interface.

FIG. 11 Variation of the natural frequencies with thickness of a squared elastically supported composite. A: Simply supported plates, B: Clamped plates

80

70 a1=a4=a5=3a2

60

50

40

30

a5=4a4

a4=a5=2a1

0.100

(K1=1000, K2=1000, K3=1000, =1000, =-0.5, =-0.5;E1=E2=E3=E4=E5, G1=G2=G3=G4=G5, v1=v2=v3=v4=v5).

20 a3=(2/3)a2

a =a =a =a =a =1

10 a3=a2=0.5a1 a3=a2=a1=.75a4

0

3 2 1 4 5

Lateral mode shape: w

Lateral mode shape: w

0.075

Fundamental mode

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

B Location along the interface.

FIG. 10 Variation of the natural frequencies with aspect ratio (a/b) and thickness (h/a) for Clamped composite plates. A: h0/a=0.01, B: h0/a=0.1 (K1=500, K2=100, K3=100,=1000, =-0.5, =- 0.5;E1=E2=E3=E4=E5, G1=G2=G3=G4=G5,

v1=v2=v3=v4=v5).

0.050

0.025

0.000

-0.025

Second mode Third mode

50 A

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Time along the interface: (x, y=0)

45

Natural frequencies

Natural frequencies

40

35 h =h =0.5h

0.25

Lateral mode shape: w

Lateral mode shape: w

0.20

Fundamental mode

3 2 1

30

p=p=p=.7h4

h =h =h =h =h =1

0.15

25 p=.6p

3 2 1 4 5

20

p=h4=p=3p

15

10

h4=p=2p

p=4h4

0.10

0.05

0.00

Second mode

Third mode

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5

Location along the interface.

-0.05

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Time along the interface: (x, y=0)

FIG. 12 Variation of the lateral mode shapes with time and non-uniform Winkler foundation for squared clamped plate at =0, =0,K2=0,K3=0, =1000. A: K1=10, B: K1=100

0.150

Lateral mode shape: w

Lateral mode shape: w

0.125

0.100

0.075

0.050

0.025

0.000

-0.025

-0.050

Fundamental mode

Second mode Third mode

0.440

0.385

Lateral mode shape: w

Lateral mode shape: w

0.330

0.275

0.220

0.165

0.110

0.055

0.000

Fundamental mode

Second mode Third mode

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Time along the interface: (x, y=0)

0.385

Fundamental mode

-0.055

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Time along the interface: (x, y=0)

FIG. 14 Variation of the lateral mode shapes with time and non-uniform and nonlinear Winkler

Lateral mode shape: w

Lateral mode shape: w

0.330

0.275

0.220

0.165

0.110

0.055

0.000

-0.055

Second mode Third mode

0.200

0.175

Lateral mode shape: w

Lateral mode shape: w

0.150

0.125

foundation for squared clamped plate at =-0.5, =0.5,K2=0, K3=100, =1000. A: K1=10, B: K1=500

Fundamental mode

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

B Time along the interface: (x, y=0)

FIG. 13 Variation of the lateral mode shapes with time and non-uniform Winkler and Pasternak foundation for squared clamped plate at =-0.5, =0.5,K2=100, K3=0, =1000. A: K1=10, B: K1=500

0.100

0.075

0.050

0.025

0.000

-0.025

-0.050

Second mode Third mode

0.150

0.125

Fundamental mode

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

A Time along the interface: (x, y=0)

0.585

Lateral mode shape: w

Lateral mode shape: w

0.100

0.075

0.050

0.025

0.000

-0.025

-0.050

Second mode Third mode

0.520

Lateral mode shape: w

Lateral mode shape: w

0.455

0.390

0.325

0.260

0.195

0.130

0.065

0.000

-0.065

Fundamental mode

Second mode Third mode

A

0.150

Lateral mode shape: w

Lateral mode shape: w

0.125

0.100

0.075

0.0500.025

0.000

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Time along the interface: (x, y=0)

Fundamental mode

Second mode Third mode

-0.130

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

B Time along the interface: (x, y=0)

FIG. 15 Variation of the lateral mode shapes with time and non-uniform and nonlinear Winkler and Pasternak foundation for squared clamped plate at =-0.5, =0.5,K2=100, K3=100, =1000. A: K1=10, B: K1=500

-0.025

-0.050

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Time along the interface: (x, y=0)

CONCLUSION

Different Quadrature schemes have been successfully applied for vibration analysis of composite plate with variable thickness resting on non-uniform and nonlinear elastic foundation. Iterative quadrature technique is used to solve the nonlinear algebraic system. A matlab program is designed for each scheme such that the maximum error (comparing with the previous exact results) is 1010 . Also, Execution time for each scheme, is determined. It is concluded that discrete singular convolution differential quadrature method based on regularized Shannon kernel (DSCDQM-RSK) with grid size 13*13 , bandwidth 2M+1

11 and regulization parameter = 2.82 hx leads to best accurate efficient results for the concerned problem. shear factor correction (K=5/6-v )is considered for such problem. Based on this scheme, a parametric study is introduced to investigate the influence of elastic and geometric characteristics of the vibrated plate, on results. It is aimed that these results may be useful for design purposes of engineering fields.

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