Ritz Variational Method for the Exact Bending Analysis of Clamped Mindlin Plates

Abstract

The exact bending analysis of clamped Mindlin plates have been formulated in this work using Ritz variational method (RVM). The formulation used first principles approach to derive the Ritz variational functional (RVF) as the sum of the strain energy and the potential of the applied load. Transverse shear deformations were catered for in determining the strain energy of the bending plate, making the resulting RVF applicable to thick plates. The RVF is expressed in terms of two rotational displacements and a transverse displacement. Shape functions that satisfy the clamped boundaries were used to express the three unknown displacements in terms of their generalized displacement parameters. This reduced the RVF to a functional dependent on the generalized displacement parameters. The principle of minimization of the RVF was used to determine the equations of equilibrium which were solved to find the generalized displacement parameters. Thus the deflections and bending moments were obtained in analytical form as a rapidly convergent series. The deflections at the center of the plate and bending moments at the middle of the clamped edges were identical to previous study that used a modified Navier series solution technique and closely in agreement with other previous studies that used higher order shear deformation plate theories. The solution obtained is exact within the Mindlin theory framework used because both the boundary conditions and the domain equations are satisfied.