The analysis of the stability problem of plate subjected to inplane compressive load is important due to the relatively poor capacity of plates in resisting compressive forces compared to tensile forces. It is also significant due to the nonlinear, sudden nature of buckling failures. This study presents the elastic buckling analysis of SSCF and SSSS rectangular thin plates using the single finite Fourier sine integral transform method. The considered plate problems are (i) rectangular thin plate simply supported on two opposite edges, clamped on one edge and free on the fourth edge; (ii) rectangular thin plate simply supported on all edges. The plates are subject to uniaxial uniform compressive loads on the two simply supported edges. The governing domain equation is a fourth order partial differential equation (PDE). The problem solved is a boundary value problem (BVP) since the domain PDE is subject to the boundary conditions at the four edges. The single finite sine transform adopted automatically satisfies the Dirichlet boundary conditions along the simply supported edges. The transform converts the BVP to an integral equation, which simplifies upon use of the linearity properties and integration by parts to a system of homogeneous ordinary differential equations (ODEs) in terms of the transform of the unknown buckling deflection. The general solution of the system of ODEs is obtained using trial function methods. Enforcement of boundary conditions along the y = 0, and y = b edges (for the SSCF and SSSS plates considered) results in a system of four sets of homogeneous equations in terms of the integration constants. The characteristic buckling equation in each case considered is found for nontrivial solutions as a transcendental equation, whose roots are used to obtain the buckling loads for various values of the aspect ratio and for any buckling modes. In each considered case, the obtained buckling equation is exact and identical with exact expressions previously obtained in the literature using other solution methods. The buckling loads obtained by the present method are validated by the observed agreement with results obtained by previous researchers who used other methods.

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 Galerkin-Vlasov 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.

This paper presents the elastic buckling analysis of thin rectangular plates using the Galerkin-Kantorovich method. The plate has two opposite simply supported edges x = 0 and x = a, and two clamped edges y = 0 and y = b where the origin is a corner and the uniform compressive load is applied at the simply supported edges. Mathematically, the problem is a boundary value problem (BVP) represented by a domain differential equation and boundary conditions. The buckling deflection function is assumed as an infinite series of an unknown function of y coordinate (G(y)) and a known function (of the x coordinate) that satisfies all the Dirichlet boundary conditions along the simply supported edges. The Galerkin-Kantorovich formulation of the BVP is an integral equation that further simplified to a homogeneous fourth order ordinary differential equation (ODE) in the unknown function G(y). The general solution of the ODE was achieved using trial function methods. The enforcement of the boundary conditions along the clamped edges resulted in a system of homogeneous equations in terms of the four integration constants. The condition for nontrivial solution is used to obtain the characteristic buckling equation as a transcendental equation which is solved for the elastic buckling load for each buckling mode using computer software based iteration methods. The critical elastic buckling load is found to correspond to the first buckling mode. The characteristic elastic buckling equation obtained is found to be the exact buckling equation for the problem and is identical with previously obtained equations. The elastic buckling loads obtained agree with the previous results from the literature.

This paper presents the generalized integral transform method for solving flexural and elastic stability problems of rectangular thin plates clamped along /2 yb = and simply supported along remaining boundaries (x = 0, x = a) (CSCS plate). The considered plate is homogeneous, isotropic and carrying uniformly distributed transversely applied loading causing bending. Also studied, is a plate subject to (i) biaxial (ii) uniaxial uniform compressive load. The method uses the eigenfunctions of vibrating thin beams of equivalent span and support conditions in constructing the basis functions for the plate deflection and the integral kernel function. The transform is applied to the governing domain equation, converting the problem to integral equations for both cases of bending and elastic buckling. The integral equation reduces to algebraic problems for the bending problem, and algebraic eigenvalue problem for the elastic buckling problem. The deflections are obtained as double infinite series with rapidly convergent properties. Bending moments expressions are double series with infinite terms which are rapidly convergent. Maximum deflections and bending moments values occur at the plate centre in agreement with symmetry. The present results gave double series solutions with good convergent properties in closed form for bending problems. The resulting bending solutions were exact. Solving the resulting eigenvalue equation gave closed analytical equation for the buckling loads. Buckling loads are computed for the cases of biaxial and uniaxial uniform compression of square thin plates using one term approximations. The buckling load obtained for one term approximation of the eigenfunction gave results that are 12.23% greater than the exact solution. The use of more terms in the eigenfunction expansion could give more acceptable results for the eigenvalue problem of buckling of CSCS plates.