Free Vibration Analysis of Pre-deformed Bi-directionally Graded Beam Supported on Nonlinear Elastic Foundation

The present article considers a bi-directional functionally graded (BDFG) beam on nonlinear elastic foundation to investigate nonlinear free vibration behaviour. A power-law distribution function is considered to describe material gradation in both the longitudinal and transverse directions. The beam foundation is considered as a nonlinear elastic one that consists of a cubic nonlinear layer and a shear layer. The beam kinematics is modelled following the principle of Timoshenko beam theory. A whole domain semi analytical formulation which considers von Kármán type geometric nonlinearity, is adopted to derive the detailed nonlinear equation sets. The solution is obtained by using an iterative numerical method. The results are furnished to study the influences of material gradation indices of BDFG beam, as well as, the influences of various stiffness parameters of nonlinear foundation, on the natural frequencies of the system. In a few cases, linear and nonlinear mode shape plots are obtained.