Vibration of shear deformable plates with variable thickness — first-order and higher-order analyses

This work presents accurate numerical calculations of the natural frequencies for elastic rectangular plates of variable thickness with various combinations of boundary conditions. The thickness variation in one or two directions of the plate is taken in polynomial form. The first-order shear deformation plate theory of Mindlin and the higher-order shear deformation plate theory of Reddy have been applied to the plate analysis. The governing equations and the boundary conditions are derived using the dynamic version of the principle of minimum of the Lagrangian function. The solution is obtained by the extended Kantorovich method. This approach is combined with the exact element method for the vibration analysis of members with variable flexural rigidity, which provides for the derivation of the exact dynamic stiffness matrix of varying cross-sections strips. The large number of numerical examples demonstrates the applicability and versatility of the present method. The results obtained by both shear deformation theories are compared with those obtained by the classical thin plate theory and with published results.