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.

On modal analysis of axially functionally graded material beam under hygrothermal effect

“…58 used the GDQ method to investigate the small-amplitude vibrations of FG beams. Paul and Das 59 presented a mathematical model to investigate the modal characteristics of the FGM beam. The authors employed the Ritz method to obtain the solutions for hinged-hinged (H-H) and clamped-clamped (C-C) beams.…”

“…Eroglu 61 carried out a free frequency analysis of FG circular beams under thermal effect. It has been found from literature 52–61 that the modal behavior of the FGM beam was carried out under thermal effect only, where mechanical properties were varied smoothly along the thickness of the beam with power law distribution.…”

“…It has been revealed from the literature 20–32,47–68 that the mechanical properties are varying smoothly along the thickness of the beam. To the author's best knowledge, modal analysis of FGM beam under hygrothermal effect where material properties are graded continuously along the length of the beam has not been reported in the literature before.…”