Homogenization procedure for the free and the steady state periodic forced response of laminated composite beams resting on non-linear foundation

Abstract

In the present paper, a theoretical model based on Hamilton's principle and spectral analysis is used to study the non-linear free vibration of laminated composite beams (LCB) resting on nonlinear Pasternak elastic Foundation with immovable ends. A homogenization procedure is performed to reduce the problem to that of isotropic homogeneous beams with effective bending stiffness and axial stiffness. Iterative form solutions are presented to determine the influence of the foundation parameters on the nonlinear frequency to the linear frequency ratio of the LCB. Then, a multi-mode approach is developed to study the nonlinear steady state periodic forced response of the beam. The results are given for various foundation stiffness parameters, showing that an increase in the value of linear elastic foundation stiffness or in modulus of Pasternak foundation leads to a decrease in the nonlinear to linear frequency ratio, but this ratio enhances with an increase in nonlinear elastic foundation stiffness. On the other hand, the beam is subjected to a concentrated force, an explicit formulation is used for 102 Merrimi El Bekkaye et al. calculation of the non-linear steady state periodic forced response of beams.