Large amplitude free vibrations of beams made of functionally graded material

Abstract

Large amplitude vibration of uniform slender functionally graded material (FGM) beams is studied by using the Finite Element Method (FEM), based on Euler beam theory. The von-Karman type strain-displacement relations are used to account for the moderately large deflections. Beam with classical boundary conditions such as hinged-hinged, clamped-clamped and hinged-clamped, with axially immovable ends is analyzed. A power law distribution is assumed for the variation of properties through the thickness. The effect of bending-extension coupling arising due to heterogeneity of material through the thickness is included. The governing non-linear equations of motion are obtained by using the principle of virtual work. Iterative eigenvalue analysis has been carried out to solve for the non-linear radian frequency. Hardening type of non-linearity is observed for the boundary conditions considered. For the FGM beams, it is observed that the non-linear frequencies are dependent on the sign of the amplitude of vibration. The effect of non-linearity is observed more for the hinged-hinged beam when compared to the clamped-clamped and the hinged-clamped beams.