Analysis and modeling the size effect on vibration of functionally graded nanobeams based on nonlocal Timoshenko beam theory

In this study Timoshenko beam theory that applies the size dependent effects in functionally graded material (FGM) beam is discussed. The material properties of FG nanobeams are considered to vary over the thickness based to the power law. The equations of motion according to Eringen nonlocal theory, using Hamilton’s principle are derived and a closed-form solution is presented for vibration behavior of the proposed model. The nonlocal elasticity theory contains a material length scale parameter that can apply the size effect in a FG material. The model is verified by comparing the obtained results with benchmark results available in the literature. In following a parametric study is accompanied to examine the effects of the gradient index, length scale parameter and length-to-thickness ratio on the vibration of FGM nanobeams. It is observed that these parameters are vital in investigation of the free vibration of a FG nanobeam.