Nonlinear dynamic and stability of a beam resting on the nonlinear elastic foundation under thermal effect based on the finite strain theory

Abstract

Nonlinear free vibration and thermal buckling of a beam resting on nonlinear are investigated in this study. In the nonlinear kinematic relations, the finite strain theory is used with Euler-Bernoulli and Hamilton's principle is employed to derive the nonlinear governing of motion. The Galerkin's method is applied to simplify the governing nonlinear partially differential equation to the nonlinear ordinary differential equation. In addition, He's variational method is employed to obtain an analytical expression for the nonlinear natural frequency and thermal buckling temperature. In this study, a comparison between the finite strain theory and the von Kármán Theory is presented and the results shows that the finite strain theory gives more accurate results than the von Kármán Theory for nonlinear natural frequency and thermal buckling temperature. In the numerical results, the effect of different parameters such as linear and nonlinear coefficients of the elastic foundation, boundary conditions and the amplitude of the oscillation on the thermal buckling temperature and the nonlinear natural frequency investigated.