Large Amplitude Free Vibrations of FGM Beams on Nonlinear Elastic Foundation in Thermal Field Based on Neutral/Mid-plane Formulations

Abstract

The present research deals with the small and large amplitude free vibrations of thick functionally graded material (FGM) beams. Beams are resting on an elastic foundation which consists of the Winkler layer, shear layer and nonlinear springs. The effect of uniform temperature elevation is also included into the formulation. A third-order shear deformation beam theory is adopted as the basic assumption of the formulation which satisfies the traction-free boundary conditions on the top and bottom surfaces. Using two different formulations, i.e. the mid-surface-based formulation and the neutral surface-based formulation, the governing equations of motion are established. For the case of beams with immovable edges in in-plane direction and simply supported in flexure, the governing equations are given in a new presentation. These equations are solved using the two-step perturbation technique. Closed-form expressions are provided to obtain the small and large amplitude natural frequencies of the FGM beam resting on nonlinear elastic foundation and operating in thermal environment. Numerical results are given to discuss the effects of thermal environment, elastic foundation parameters, different formulations, and the geometrical parameters.