Free vibration of moderately thick FGM plates using the dynamic stiffness method and the Wittrick-Williams algorithm

The dynamic stiffness method (DSM) for free vibration analysis of functionally graded moderately thick plates based on power and sigmoid laws is developed. The functionally graded plates analysed are identified as P-FG and S-FG plates, describing the power and sigmoid laws. The effective material properties of the P-FG and S-FG plates are assumed to vary through the thickness of the plate. First-order shear deformation theory and Hamilton’s principle are used to derive the governing differential equations of motion. The Levy-type solution is sought to derive an efficient analytical formulation for plates with simply supported boundary conditions along two opposite edges. By imposing the boundary conditions for displacements and forces, the dynamic stiffness matrix is developed. Then the Wittrick–Williams (W-W) algorithm is applied to compute the natural frequencies and mode shapes of P-FG and S-FG plates. The influence of the material gradient index, aspect ratio, thickness to length ratio and boundary conditions on the natural frequencies and mode shapes of P-FG and S-FG plates are investigated in detail. Comparing these two grading laws i.e. power and sigmoid grading laws is important, as they yield different stress and stiffness distributions, which directly impact vibration characteristics, stress concentrations, and manufacturability of functionally graded components. To establish the accuracy and validity of results from the present theory, comparative studies utilising existing literature are performed, which show excellent agreement. The results given, can be used as bench-mark solutions when validating finite element and other approximate theories.