A Homogenization Procedure for Nonlinear Free Vibration Analysis of Functionally Graded Beams Resting on Nonlinear Elastic Foundations

Abstract

The present work deals with a homogenization procedure (HP), which is developed to reduce the problem of geometrically nonlinear free vibrations of functionally graded beams (FGB) resting on elastic nonlinear foundation with immovable ends 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 Kármán geometrical nonlinearity assumptions. 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.

A homogenization procedure for geometrically non-linear free vibration analysis of functionally graded circular plates involving the coupling between transverse and in-plane displacements

“…The differential quadrature method, which has been successfully used in solving boundary value problems, has also been extended to solve initial value problems of plates and was used to discretize the time domain [21,22]. Zerkane et al [23] used a homogenization procedure to nonlinear free vibration analysis of functionally graded beams resting on nonlinear elastic foundations. Sepahi et al [24] investigated the effects of three-parameter of elastic foundations and thermo-mechanical loading on the axisymmetric large deflection response of a simply supported annular FG plate based on the first-order shear deformation theory (FSDT) in conjunction with nonlinear von Karman assumptions.…”