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This paper presents the Galerkin-Vlasov variational method for the elastic buckling analysis of SSCF and SSSS rectangular plates. The thin plate problems studied are: (i) simply supported along two opposite sides x = 0, and x = a, clamped along the third side y = 0, and free along the fourth side y = b; (ii) simply supported along the four sides x = 0, x = a, y = 0 and y = b. In each case the edges x = 0 and x = a are subjected to uniform compressive load. Mathematically, the considered stability problem is a Boundary Value Problem (BVP) expressed by a domain fourth order partial differential equation (PDE) whose general solution should satisfy all the boundary conditions determined by the edge support conditions. By the GalerkinVlasov method, the unknown deflection shape function is chosen as the product of the eigenfunctions of a vibrating thin beam of identical span in the x direction and an unknown function of y(Gn(y)). The Galerkin-Vlasov variational integral equation is simplified using the Leibnitz rule, integration by parts and the orthogonality properties of the eigenfunctions of simply supported thin beams to a system of fourth order ordinary differential equations (ODEs). The general solution of the system of ODEs is obtained using trial function methods in terms of hyperbolic and trigonometric functions. The imposition of boundary conditions is used in each of the two cases to find the characteristic buckling equation. The buckling equation is obtained in each case as a transcendental equation, which is solved to obtain the eigenvalues from which the buckling loads are determined. The results obtained in each case for the buckling equation are identical to previous results obtained by other scholars who used classical methods and energy minimization methods. The results obtained for the buckling loads are in agreement with previously obtained solutions in the literature. The results obtained in each presented case in this study are found to be exact because exact shape functions were used in the x direction and the general solution was obtained for the domain PDE at every point in the plate domain. In addition, the solution obtained was made to satisfy all the boundary conditions at all the edges of the plate.

Galerkin-Vlasov Variational Method for the Elastic Buckling Analysis of SSCF and SSSS Rectangular Plates