Large amplitude free vibration analysis of functionally graded beams using an homogenisation procedure

The purpose of the present paper is to show that the problem of geometrically non linear free vibrations of functionally graded (FG) beams with immovable ends can be reduced to that of isotropic homogeneous beams with effective bending stiffness and axial stiffness parameters. The material properties of the functionally graded composites examined are assumed to be graded in the thickness direction and estimated through the rule of mixture. The theoretical model is based on the Euler-Bernouilli beam theory and the Von Karman geometrical nonlinearity assumptions. An homogenization procedure is developed using the governing axial equation of the beam in which the axial inertia and damping are ignored. Hamilton's principle is applied and a multimode approach is derived to calculate the fundamental nonlinear frequency parameters, which are found to be in a good agreement with the published results. The non-dimensional curvatures associated to the nonlinear fundamental mode are also given in the case of clamped-clamped FG beams.