Two-Dimensional Solution for Coupled Thermoelasticity of Functionally Graded Beams Using Semi-Analytical Finite Element Method

This article presents the two-dimensional solution for a simply supported beam made of functionally graded material (FGM) subjected to arbitrary time-dependent lateral thermal shock loads by the semi-analytical finite element method. The method of finite Fourier series is combined with the Galerkin finite element method to solve the two-dimensional coupled thermoelasticity equations. The series solution is assumed along the length of the beam and finite element procedure is adopted across the thickness of the beam such that the two-dimensional character of the solution is preserved. The FGM beam is assumed to be graded across the thickness direction. The material properties across the thickness direction follow the volume fraction of the constitutive materials in power law form. The equation of motion and the conventional coupled energy equation are simultaneously solved to obtain the displacement components and temperature distribution in the beam. The C-1-continuous shape functions are used in the Galerkin finite element method. The Laplace transform technique is used to transform the governing equations into the space domain, where the Galerkin finite element is employed to obtain the solution in the space domain. The inverse of the Laplace transform is performed numerically to obtain the final solution in the real time domain. Finally, the results are validated with the known data reported in the literature.