Free vibration and bending of functionally graded beams resting on elastic foundation

In this paper, free vibration and static bending analysis of functionally graded (FG) beams resting on Winkler foundation are investigated within Euler-Bernoulli beam theory and Timoshenko beam theory. Material properties of the beam change in the thickness direction according to power-law distributions. The governing equations and the related boundary conditions are derived using the principal of the minimum total potential energy. The Navier-type solution is used for simply-supported boundary conditions, and exact formulas are proposed for the static deflections and the fundamental frequencies. In order to establish the accuracy of the present formulation and results, the natural frequencies are obtained, and compared with the published results available in the literature. Good agreement is observed. Numerical results are presented to investigate the influences the different material distributions, foundation stiffness, and shear deformation on the bending and free vibration behavior of FG beams.