A semi analytical approach for large amplitude free vibration and buckling of nonlocal FG beams resting on elastic foundation

In this paper, an attempt is made to obtain a closed form solution for both natural frequency and buckling load of nonlocal FG beams resting on nonlinear elastic foundation. Implementing Eringen’s nonlocal elasticity theory, the effect of nonlocality is introduced into the Euler–Bernoulli beam theory to obtain the nonlinear governing partial differential equation. Application of the Galerkin technique to the governing equation leads to a nonlinear ODE in the time-domain. Finally, natural frequency of the FG nano beam is obtained using He’s variational method. It is shown that considering the nonlocal effects decreases the buckling load as well as natural frequency. Results also reveal that effects of nonlocal parameters on fully clamped beams are more than other types of boundary conditions. Moreover, it is shown that the effect of nonlocality decreases by increasing length of the beam.