Solving Clamped Kirchhoff Plate Bending Problems Using Superposition of Sinusoidal and Polynomial Basis Functions in the Ritz Variational Method

Thin plate bending analysis is an important research subject due to the extensive use of plates in the different fields of engineering and the need for accurate solutions. This article uses the Ritz variational method and a superposition of trigonometric and polynomial basis functions to solve the Kirchhoff-Love plate bending problems (KLPBPs). The unknown displacement function in the Ritz variational functional (RVF) to be minimized is sought as linear combinations of basis functions Fm(x) and Gn(y) that are found by superposing sine series and third degree polynomial functions with the polynomial parameters determined such that all boundary conditions of deformation and force are satisfied. The displacement is thus expressed in terms of unknown displacement parameters Amn which are found upon minimization of RVF with respect to Amn. The minimization process gave a matrix stiffness equation in Amn with the stiffness matrix and force matrix found from Fm(x) and Gn(y) and their derivatives. The algebraic equation is solved, and the deflection and bending moments obtained. The problems considered were clamped (CCCC) plates under uniform and hydrostatic distribution of loads and plates with opposite edges clamped, the rest simply supported (CSCS) under uniformly distributed loading. Comparison of the solutions by Generalized Integral transform method, Levy-Nadai series method, and symplectic eigenfunction superposition confirms that the present results are accurate.