Application of Galerkin’s Variational Method to the Bending Analysis of Rectangular Mindlin Plate with Clamped Edges

The Mindlin plate theory (MiPT) was formulated in this work for bending analysis under distributed load using equilibrium approach. The formulation was presented for isotropic, homogeneous, linear elastic materials and for rectangular plates under distributed load. It used a small displacement assumption that made the formulation use linear elasticity theory and limit the application to exclude nonlinearity of material behavior. The resulting formulation is a system of three coupled partial differential equations in terms of two unknown rotational displacements and one unknown transverse displacement w. The system of equations was then formulated for uniformly loaded plates using Galerkin method as Galerkin functionals (GF) which are Galerkin integral equations (GIEs) in terms of the unknown displacements. The displacements and w are expressed using linear combinations of their respective amplitudes and shape functions that satisfy the displacement and moment conditions at the clamped boundaries. The orthogonality properties of the displacement shape function adopted in this work make the integration problems involved in evaluating the GIEs easier. Consequently, the GIEs are expressed as systems of nonhomogeneous algebraic equations that are more readily solved. The Cramer’s rule for solving nonhomogeneous algebraic equation is implemented to obtain the displacement amplitudes. The resulting displacements and bending moments are thus computed for various ratios of dimension, a, to plate thickness, h, (a/h) varying from a/h = 3, 5, 10. It was found that the present results are identical to previous results that used Ritz variational method and Navier’s method for uncoupled Mindlin equations.