Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction

Abstract

The free and forced vibration of a laminated functionally graded beam of variable thickness under thermally induced initial stresses is studied in this paper within the framework of Timoshenko beam theory. The beam consists of a homogeneous substrate and two inhomogeneous functionally graded layers whose material composition follows a power law distribution in the thickness direction in terms of the volume fractions of the material constituents. Both the axial and rotary inertia of the beam are considered in the present analysis. It is assumed that the beam may be clamped, hinged, or free at its ends and is subjected to one-dimensional steady heat conduction in the thickness direction before undergoing dynamic deformation. To include the effect of temperature change, the initial stress state is determined through a thermo-elastic analysis before the free and forced vibration analyses. The differential quadrature method that makes use of Lagrange interpolation polynomials is employed as a numerical solution tool to solve both the thermo-elastic equilibrium equation and dynamic equation. Numerical results are presented in both tabular and graphical forms for various laminated functionally graded beams, showing that vibration frequencies, mode shapes and dynamic response are significantly influenced by the thickness variation, temperature change, slenderness ratio, volume fraction index, the thickness of the functionally graded layer, and the end support conditions.