On Calculation of Bending of a Thin Orthotropic Plate Using Legendre and Chebyshev Polynomials of the First Kind

The problem of bending of a thin orthotropic rectangular plate clamped at the edges is considered in the paper. The solution is obtained using the Legendre and Chebyshev polynomials of the first kind. The function that approximates the solution of the biharmonic equation for an orthotropic plate is presented in the form of a double series expansion in these polynomials. Matrix transformations and properties of the Legendre and Chebyshev polynomials are also used. Roots of these polynomials are used as collocation points, and boundary value problem is reduced to a system of linear algebraic equations with respect to coefficients of the expansion. The problem of bending of a plate caused by the action of a distributed transverse load of constant intensity that corresponds to hydrostatic pressure is considered. This boundary value problem has analytical solution. The results of calculations for various ratios of the lengths of sides of the plate are presented. The values of deviation of solutions constructed using Legendre and Chebyshev polynomials from the analytical solution of the problem are presented in terms of the infinite norm and the finite norm in the space of square-integrable functions.