Free vibration analysis of two directional functionally graded beams using a third order shear deformation theory

This paper presents the free vibration behavior of two directional functionally graded beams subjected to various sets of boundary conditions which are simply supported (SS), clamped-simply supported (CS), clamped-clamped (CC) and clamped-free (CF) by employing a third order shear deformation theory. The material properties of the beam vary exponentially in both directions. In order to investigate the free vibration response, the equations of motion are derived by means of Lagrange equations. The axial, transverse deflections and rotation of the cross sections are expressed in polynomial forms including auxiliary functions which are used to satisfy the boundary conditions. The verification and convergence studies are performed by using computed results from a previous study which is based on the Timoshenko beam formulation. The results for extensive studies are provided to understand the influences of the different gradient indexes, various aspect ratios and boundary conditions on the free vibration responses of the two directional functionally graded beams.