Nonlinear Vibration Analysis of Functionally Graded Beam on Winkler-Pasternak Foundation under Mechanical and Thermal Loading Via Homotopy Analysis Method

In this paper, the homotopy analysis method is used to nonlinear free vibration analysis of a mechanical and thermal loaded functionally graded beam on nonlinear elastic foundation. At first, the governing partial differential equation of the problem has been derived based on the Euler-Bernoulli theory and the Von-Karman strain-displacement relationship. Then, it was reduced to a nonlinear ordinary differential equation via the Galerkin method. The homotopy analysis method which has high accuracy was implemented in order to obtain a closed form solution and study the problem parametrically. The accuracy of the proposed method is verified by those available in literatures. The numerical results demonstrate that proposed method yields a very rapid convergence of the solution as well as low computational effort. Finally, the effects of different parameters such as amplitude, linear and nonlinear elastic foundation, thermal and mechanical loads and boundary conditions were investigated on the beam vibration and their results are presented for future work.