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

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.


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][2][3][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][14][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][17][18], Moving least square differential quadrature method (MLSDQM) [19][20][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.

II. 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).
where a and b are width and length of the composite.
Assuming harmonic behavior of the problem, the field quantities can be written as: ( , , ) , Where ω is the natural frequency of the plate and 1 i =− .
Substituting from Eq.(5) into (1-4), One can reduce the problem to: Boundary conditions can be expressed as follows: •  (10) • Simply Supporting of the second kind: SS2 22 22 0, 0, • Free edge: and xy nn are the directional cosines at a point on the boundary edge.
Along the interface between ith plate and (i+1) th one, the continuity boundary conditions can be described as: where x h is the step size, 2M+1 is the effective computational band width, σ is regularization parameter, σ = 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][17][18]: Similarly, one can approximate y u , yy u and calculated y ij In this technique, the influence domain, (Ωi, i=1: N), for each node is determined as shown in Fig. (2). Over each influence domain, the nodal unknowns can be approximated as [19][20][21]: Where the shape function () ji  x can be obtained using MLS approximation as follows: Where ϖ(x-xi) is a positive weight function defined over the influence domain, (Ωi, i=1,N) The stationary value of Π(a) with respect to a(x) leads to a linear equation, such as: On suitable substitution from Eq. (22) into (19), u h (x) can then be expressed as: Where the nodal shape function: Determination of the shape function () i  x and its partial derivatives can be simplified as follows [19][20][21]: 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 , , , The first and second order partial derivatives of the shape function can be described as: (29) , , , On suitable substitution from Eqs. (14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30) into (6)(7)(8), the problem can be reduced to the following nonlinear eigenvalue problem: The boundary conditions (9)(10)(11)(12)(13) can also be approximated using DQMs as: • Clamped edge: • Simply supporting of the first kind: SS1 • Simply supporting of the second kind: SS2 IV.

NUMERICAL RESULTS
This section presents numerical results that demonstrate convergence and efficiency of each one of the proposed schemes for vibration analysis of nonuniform and nonlinear Winkler and uniform Pasternak foundation of composite plate. This plate varying with thickness. For all results, the boundary conditions (34)(35)(36)(37) are augmented in the governing equations (31)(32)(33). Then the obtained nonlinear algebraic system is solved by using iterative quadrature technique [34]. The computational characteristics of each scheme are adapted to reach accurate results with error of order ≤10 -10 . The obtained frequencies ω are normalized such as: 100 ranges from 5 to 15 and the regularization parameter σ = r hx ranges from 1.5hx to 3 hx , where hx =1/N-1.  exp( ( / ) ) exp( ( / ) ) () 1 exp( ( / ) ) 0 Where i d 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][29][30][31] over grid size ≥11*11 , completeness order ≥4 and raduis of support domain dmax = 5 .  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.  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. 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.  Furthermore, a parametric study is introduced to investigate the influence of elastic and geometric characteristics of the composite on the values of natural frequencies.

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V. 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 10 10 −  . 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 σ = 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.