Three-Dimensional Analysis of the Free Vibration of Thick Rectangular Plates With Depressions, Grooves or Cut-Outs

A solution is presented for the free vibration of very thick rectangular plates with depressions, grooves or cut-outs using three-dimensional elasticity equations in Cartesian coordinates. Simple algebraic polynomials which satisfy the boundary conditions of the plate are used as trial functions in a Ritz approach. The plate is modelled as a parallelepiped, and the inclusions are treated quite straightforwardly by subtracting the contribution to the strain and kinetic energy expressions of the volume removed, before minimizing the functional. The approach is demonstrated by considering a number of square thick plate cases, including a plate with a cylindrical groove, a shallow depression or a cylindrical cut-out.