Bending of a clamped thin isotropic plate by the Kantorovich method using special polynomials

Abstract

The problem of bending a thin isotropic rectangular plate clamped on all four sides under the action of a normal load uniformly distributed over its surface is considered. An analytical solution of the boundary value problem for the resolving differential equation with respect to the normal deflection of the plate is obtained by the method of L. V. Kantorovich using special-type polynomials satisfying homogeneous boundary conditions. A feature of these polynomials is the so-called “quasi-orthogonality” property of the first and second derivatives, which leads to the splitting of the system of ordinary differential equations of the L. V. Kantorovich method into separate ordinary differential equations that are easily solved analytically. However, this property of polynomials is only approximately fulfilled. Two solutions are compared: an analytical one under the assumption of “quasi-orthogonality” of the first and second derivatives of polynomials and a numerical-analytical one without this assumption. The stress-strain state in the neighborhoods of corner points has been studied. It is shown that the moments and shear forces tend to zero when approaching the corners of the plate, as well as a double change in the sign of the shear force on the edge of the plate in the neighborhoods of the corner points.