Application of Beam Functions for Static Calculation of Bending Rods and Rectangular Plates

Abstract

In this paper, the deflections of a rod and a rectangular plate with any types of boundary conditions are proposed to be represented as a series and a double series of eigenfunctions of the differential equation of bending vibrations of a beam or plate with the corresponding boundary conditions. Then, using the Ritz method, the functional of the total energy of rod bending and bending and torsion of the plate and the external load acting on them is determined. For a rod with any types of boundary conditions, by differentiating the functional of the total energy, an exact solution for deflections in the form of a rapidly converging series is obtained. In this case, previously published results by S.P. Timoshenko and E.S. Sorokina are used. For a rectangular plate, using the orthogonality property of eigenfunctions and their second derivatives, a quadratic functional of the indefinite coefficients at the eigenfunctions is calculated. Differentiating the functional with respect to each of the unknown coefficients forms an infinite system of linear algebraic equations, the solution of which by the truncation method allows us to find the deflections of the plate. Further, the forces in the plate are found using known methods. Two examples of calculating a rectangular plate with four supported edges and a plate with two supported edges are given. The proposed approach is simple, universal and allows calculating rectangular plates with any types of boundary conditions on the contour for an arbitrary external load. The article provides a table of eigenvalues and forms for calculating rectangular plates.