Keywords: boundary element method, meshless, buckling, analog equation method, elastic plates, radial basis functions, thin plate splines.

Buckling or elastic instability of plates is of great practical importance. Analytical solutions of a plate buckling have been found in the simplest cases of support and loading conditions. Numerical methods such as the finite element method have been used by many researchers to investigate the buckling problems. More recently, the boundary element method (BEM) has proved to be a powerful solver for the plate buckling problems. In the early development of the BEM [1,2,3,4,5], domain integrals containing domain curvature terms arise in the formulations which usually result in a combined boundary-domain solution of the problem. Afterwards, the effort to overcome this limitation has been made by using the concept of the dual reciprocity method. Elzein and Syngellakis [6] have proposed two models for the plate deflection. The accuracy of the results depends on the order of the Fourier approximation or the choice of deflection functions. Lin et al. [7] have proposed the boundary integral formulation which was in terms of three displacements of the plate's middle surface and the planar loading and, or displacements applied at the boundary of the plate. A Fourier series was also applied in their formulations. Recently, Neranzaki and Katsikadelis [8] have proposed an analog equation method (AEM) for buckling of plates with variable thickness. The AEM has been successfully employed to solve the plate buckling problems without the limitations on the plate geometry, thickness variation and the support and loading conditions. However, there are still domain integrals in the formulations.

In this paper a BEM-based meshless method is developed for buckling analysis of elastic plates. The presented method was achieved using the concept of the analog equation method (AEM). According to this method the original eigenvalue problem for a governing differential equation of buckling is replaced by an equivalent problem for plate bending problem subjected to an "appropriate" fictitious load under the same boundary conditions. The domain integral caused by the fictitious load is approximated by using thin plate splines (TPSs) as a radial basis function series and then converted also to a line integral on the boundary using a technique based on the BEM. The formulation has no limitations on plate shape, and transverse and inplane boundary conditions. The eigenmodes of the actual problem are obtained from the known integral representation of the solution for the classical plate bending problem, which is derived using the fundamental solution of the biharmonic equation. The following conclusions can be drawn from this study:

1. As the method is boundary-only, it has all the advantages of the pure BEM, i.e. the discretization and integration are performed only on the boundary.
2. The known fundamental solution of the biharmonic equation is employed to derive the integral representation of the solution. Thus, the kernels of the boundary integral equations are conveniently established and evaluated.
3. Accurate numerical results for the critical load factors and the corresponding mode shapes are obtained. No special computational techniques are required for solving plates having free boundaries
4. The concept of the analog equation in conjunction with radial-basis-functions approximation of the fictitious sources renders the BEM a versatile computational method for solving difficult engineering problems. It depends only on the order of the differential equation and not specific differential operator which governs the problem under consideration.

1

J.A. Costa, C.A. Brebbia, "Elastic buckling of plate using the boundary element method", in "Boundary Elements VII", Springer-Verlag, Berlin, 4-29-4-42, 1985.

2

G. Benzine, A. Cimetiere, J.P. Gelbert, "Unilateral buckling of thin elastic plates by boundary integral equation method", Int J Numerical Methods in Engineering, 21, 2189-2199, 1985. doi:10.1002/nme.1620211206

3

G.D. Manolis, D.E. Beskos, M.F. Pineros, "Beam and plate stability by boundary elements", Computers and Structures, 22(6), 917-923, 1986. doi:10.1016/0045-7949(86)90152-5

4

M. Tanaka, K. Miyazaki, "A boundary element method for elastic buckling analysis of assembled plate structures", Computational Mechanics, 3, 49-57, 1988. doi:10.1007/BF00280751

5

S. Syngellakis, A. Elzein, "Plate buckling loads by the boundary element method", Int J Numerical Methods in Engineering, 37, 1763-1778, 1994. doi:10.1002/nme.1620371008

6

A. Elzein, S. Syngellakis, "Dual reciprocity in boundary element formulations of the plate buckling problem", Engineering Analysis with Boundary Elements, 9, 175-184, 1992. doi:10.1016/0955-7997(92)90059-G

7

J. Lin, R.C. Duffield, H.R. Shih, "Buckling analysis of elastic plates by boundary element method", Engineering Analysis with Boundary Elements, 23, 131-137, 1999. doi:10.1016/S0955-7997(98)00074-5

8

M.S. Nerantzaki, J.T. Katsikadelis, "Buckling of plates with variable thickness-an analog equation solution", Engineering Analysis with Boundary Elements, 18, 149-154, 1996. doi:10.1016/S0955-7997(96)00045-8

9

S.P. Timoshenko, J.M. Gere, "Theory of elastic stability", McGraw-Hill, New York, 1961.

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