Vibration of nonlinear graduation of nano-Timoshenko beam considering the neutral axis position

This paper presents a methodology to investigate the free vibration of nonlinear material graduations of a nanobeam based on the nonlocal Timoshenko theory using finite element method. The non-classical nonlocal Eringen nanobeam model considers the length scale parameter, which captures the small scale effect. The material properties of the functionally graded (FG) nanobeam are assumed to vary nonlinearly in the thickness direction. The location of neutral axis is determined and the kinematic fit conditions of the nanobeam are proposed according to super convergent Timoshenko beam. A numerical solution of the equation of motion is obtained through the finite element method (FEM). The developed methodology is applied to detect the free vibration response of different nano-Timoshenko beams with different boundary conditions, material exponents, and nonlocality parameters. The obtained numerical results are reflected the significant effect of neutral axis position, material distribution profile, and the nonlocality parameter on the fundamental frequencies of nano-Timoshenko beams.