A new trigonometric shear deformation plate theory for free vibration analysis of FGM plates with two-directional variable thickness

Abstract

The present research aims to propose a new trigonometric shear deformation plate theory (TSDPT) for free vibration analysis of functionally graded plates with two-directional variable thickness. Trigonometric functions are utilized to determine the distribution of transverse shear stress in the new TSDPT. The thickness variation follows three distinct patterns: linear, concave, and convex. The governing equations of the composite plates are derived via the new theory and the Hamilton's principle. Then, via the Galerkin's method, the current study establishes a solution for determining natural frequencies of variable-thickness FGM plates. The obtained outcomes of FGM plates are calculated by using the MATLAB program. To confirm the accuracy of the current computational model, we conducted comparisons between the obtained results and data previously reported in the open literature. The results obtained from the new theory are compared not only for the case of a plate made of isotropic material but also in comparison to a plate made of functionally graded material. In both scenarios, the proposed theory showcases superior performance when juxtaposed against classical plate theory (CPT), first-order shear deformation plate theory (FSDPT), and other shear deformation plate theories, displaying significantly reduced error rates. Furthermore, graphical representations of the influences of three-parameter stiffness of Kerr foundation, boundary conditions, geometrical parameters, two different FGMs, FGM index, and mode numbers on the vibrational behaviors of FGM plates are provided.