Nonlinear thermal stability and vibration of pre/post-buckled temperature- and microstructure-dependent functionally graded beams resting on elastic foundation

Abstract In this study, buckling and post-buckling analysis and small amplitude vibrations in the pre/post-buckling regimes of functionally graded beams resting on a nonlinear elastic foundation and subjected to in-plane thermal loads are investigated. The Timoshenko beam theory with the von Karman nonlinearity and the microstructural length scale based on the modified couple stress theory are used to derive the governing nonlinear equilibrium equations. Various types of boundary conditions are considered. Thermo-mechanical properties of the FGM beams are assumed to be functions of both temperature and thickness. The solution is determined in two different regimes. A static phase with large amplitude response and a dynamic regime near the static one with small amplitudes are considered. In order to discretize the motion equations in geometrical domain of both regimes, generalized differential quadrature method (GDQM) is used. The resulting system of nonlinear algebraic equations are solved iteratively using Newton’s method. Numerical results indicate that, depending on the boundary conditions and the type of thermal load, the response of the structure may be of unique stable path or the bifurcation-type in static regime. Also, free vibration of a beam subjected to in-plane thermal load may show zero frequency magnitude at a certain temperature, which specifies the existence of bifurcation-type of instability. Influences of nonlinear elastic foundation parameters, thermal load type, different types of boundary conditions, and microstructural length scale on equilibrium paths, critical buckling load, and fundamental frequencies are studied.