Keywords: thin quadrilateral plate, conformal mapping, extended Kantorovich method, bending analysis.

The extended Kantorovich method (EKM) has been extensively used to obtain approximate solutions for various two-dimensional elasticity problems including bending, vibration and buckling analyses of plates [1,2,3,4,5,6,7]. Among the advantages of the method, one can refer to the rapid convergence and highly accurate results together with less computational effort and the possibility of obtaining approximate closed form solutions for the governing partial differential equations. In addition, the EKM, in comparison with other weighted residual methods, has an attractive feature in the sense that the initial estimate for the solution has no effect upon the final results. Furthermore, in the EKM a set of ordinary differential equations (ODEs) should be solved iteratively instead of solving more complicated partial differential equations. However, it should be noted that application of the EKM in the literature is limited to rectangular plates. Various two-dimensional elasticity analyses including bending [2,3,4], vibration [5,6] and buckling [7] of rectangular isotropic and orthotropic plates with either constant or variable thickness can be found in the literature.

The attention of this article is focused on the applicability of the EKM to obtain solutions for plates with a more general shape. As the first step, bending of a fully clamped, curve-sided quadrilateral thin plate is considered. To do so, the physical domain of the problem is mapped to a proper rectangular domain using conformal mapping. The new version of the governing partial differential equation and proper boundary conditions in the mapped coordinate system is derived. This equation consists of a forth order PDE with variable coefficients. Based on the EKM, assuming a separable function for displacement of the plate together with Galerkin weighted residual technique converts the governing equation into two separate forth order ordinary differential equations in terms of u and v. Finally, the ODEs obtained are solved using the boundary conditions of the plate in an iterative way until convergence of the results is achieved.

An error analysis is carried out to examine rapid convergence and accuracy of the method by substitution of the obtained solutions to the original governing equation after every iteration. It is shown that the method offers very fast convergence as two to three iterations were enough to obtain final results. Results of this study also revealed that the method provides highly accurate predictions for a general shape quadrilateral thin plate. Apart from the error analysis, the accuracy of the results for both deflection and stress components at various points of the plate were examined using finite element analysis. Due to lack of results for this type of plate in the literature, a finite element analysis was also carried out using the commercial finite element code ANSYS. Comparisons of the results show close agreement with those of the finite element analysis. It is also found that the method is computationally efficient as minimum computational time needed to obtain final results with desired accuracy. Furthermore, results of this study were found to be quite encouraging to apply the method to more complicated engineering problems in order to benefit the various aforementioned advantages of the method.

1

A.D. Kerr, "An extension of the Kantorovich method", Quart Appl. Math., 26, 219-229, 1968.

2

A.D. Kerr, and Alexander H, "An application of the extended Kantorovich method to the stress analysis of a clamped rectangular plate", Acta Mech., 6, 180-196, 1968. doi:10.1007/BF01170382

3

M.M. Aghdam, M. Shakeri and S.J. Fariborz, "Solution to the Reissner plate with clamped edges", ASCE J Eng Mech 122 679-682, 1996. doi:10.1061/(ASCE)0733-9399(1996)122:7(679)

4

M.M. Aghdam and S.R. Falahatgar, "Bending analysis of thick laminated plates using extended Kantorovich method", Composite Structures, 62, 279-283, 2003. doi:10.1016/j.compstruct.2003.09.026

5

M. Dalaei and A.D. Kerr, "Natural vibration analysis of clamped rectangular orthotropic plates", J. Sounds & Vibration, 189, 399-406, 1996. doi:10.1006/jsvi.1996.0026

6

I. Shufrin and M. Eisenberger, "Vibration of shear deformable plates with variable thickness - first-order and higher-order analyses", Journal of Sound and Vibration, 290, 465-489, 2006. doi:10.1016/j.jsv.2005.04.003

7

V. Ungbhakorn and P. Singhatanadgid, "Buckling analysis of symmetrically laminated composite plates by the extended Kantorovich method", Composite Structures, 73, 120-128, 2006. doi:10.1016/j.compstruct.2005.02.007

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