Vibrations of thick isotropic plates with higher order shear and normal deformable Plate theories

We use a higher order shear and normal deformable plate theory of Batra and Vidoli and the finite element method to analyze free vibrations and stress distribution in a thick isotropic and homogeneous plate. The transverse shear and the transverse normal stresses and strains in the plate are considered and traction boundary conditions on the top and the bottom surfaces of the plate are exactly satisfied. All components of the stress tensor are computed from equations of the plate theory. Equations governing deformations of the plate involve second-order spatial derivatives of generalized displacements with respect to in-plane coordinates. Thus triangular or quadrilateral elements with Lagrange basis functions can be employed to find their numerical solution. Results have been computed for rectangular plates of aspect ratios varying from 4 to 20 and with all edges either simply supported or clamped, or two opposite edges clamped and the other two free. Computed frequencies, mode shapes, and through the thickness distribution of stresses for a simply supported plate are found to match very well with the corresponding analytical solutions. Advantages of the present approach include the use of Lagrange shape functions, satisfaction of traction boundary conditions on the top and the bottom surfaces and the use of the plate theory equations for accurate determination of transverse stresses. The order of the plate theory to be used depends upon several factors including the aspect ratio of the plate.