Three-dimensional vibration behavior of bi-directional functionally graded curved parallelepipeds using spectral Tchebychev approach

This paper presents the application of the three-dimensional spectral-Tchebychev technique to accurately predict the vibration behavior of bi-directional functionally graded material curved parallelepipeds including geometries such as beams, thin/thick plates, and solids. In this study, the material distribution within the domain of the structure is obtained using bi-directional Mori-Tanaka method. To derive the boundary value problem governing the dynamics of functionally graded curved parallelepipeds, three-dimensional elasticity equations are used together with extended Hamilton's principle. Numerical solution of the integral boundary value problem is performed using the three-dimensional spectral Tchebychev approach. To validate and assess the performance of the presented solution approach, a number of case studies are conducted. In each case study, the non-dimensional natural frequencies and mode shapes are calculated and compared to those found using a finite element solution approach. Furthermore, computational time of the simulation is measured in each case. It is shown that the presented solution technique enables accurate prediction of vibration behavior of bi-directional functionally graded curved parallelepipeds as precise as a finite elements method, but for a fraction of the computational cost.