Free vibration behavior of tapered functionally graded material beam in thermal environment considering geometric non-linearity, shear deformability and temperature-dependent thermal conductivity

Abstract

An improved mathematical model is presented to investigate the free vibration behavior of post-buckled tapered functionally graded material beam, subjected to uniform temperature rise and steady-state heat conduction. The material properties including the thermal conductivity are considered to be temperature-dependent and an iterative algorithm for solving temperature-dependent steady-state heat conduction equation is presented to get the correct temperature profile. The initial static post-buckling problem is formulated using minimum potential energy principle and the subsequent free vibration problem is formulated using Hamilton’s principle by employing the tangent stiffness of the post-buckled configuration. The solution of the governing equations is obtained using Ritz method. Following Timoshenko beam theory, a geometrically non-linear mathematical model is developed by employing the non-linear strain–displacement relationships for both normal and shear strains. The study is carried out for both hinged–hinged and clamped–clamped beams. Non-dimensional load–frequency behaviors are presented for different gradation indices, taperness parameters, and length–thickness ratios. Static post-buckling equilibrium path for clamped–clamped beams is also presented. The significant effects of shear non-linearity and temperature-dependent thermal conductivity on dynamics of tapered functionally graded material beam are shown in the paper.