KANTOROVICH VARIATIONAL METHOD FOR THE ELASTIC BUCKLING ANALYSIS OF KIRCHHOFF PLATES WITH TWO OPPOSITE SIMPLY SUPPORTED EDGES

Abstract

We present the Kantorovich variational method for the elastic buckling solutions of Kirchhoff plates with two opposite simply supported edges. Uniform compressive loading is applied on the two simply supported edges, x = 0 and x = a. Three cases of edge support considered along the y = 0 and y = b edges are (i) both are clamped (ii) both are simply supported, and (iii) edge y = 0 is simply supported while edge y = b is free. The Kantorovich method which is based on seeking to minimize the total potential energy functional P with respect to the unknown deflection w(x, y) assumes the deflection in variable separable form as an infinite series of the product of an unknown function f_m(y) and a known sinusoidal function that satisfies the Dirichlet boundary conditions along the simply supported edges. Euler-Lagrange differential equation is used to obtain the fourth order ordinary differential equation which f_m(y) must satisfy to minimize P. The general solution for f_m(y) is obtained as a combination of hyperbolic and trigonometric functions with four unknown integration constants. The conditions for nontrivial solutions are applied to find the characteristic stability equations in each considered case. Exact solutions are obtained for the elastic buckling equations which are identical with previously obtained solutions in the literature. The elastic buckling equations are solved by iteration methods and by closed form solution methods for the Kirchhoff plate to obtain the elastic buckling loads expressed in terms of the elastic buckling load coefficients. It was found that the method yields exact elastic buckling load and exact elastic buckling load coefficient for each boundary condition considered in this study.