Some closed-form solutions for static, buckling, free and forced vibration of functionally graded (FG) nanobeams using nonlocal strain gradient theory

In this paper, static bending, buckling, free and forced vibration of functionally graded (FG) nanobeams are studied within the framework of the recently proposed nonlocal strain gradient theory and the Euler-Bernoulli beam theory. The material properties of nanobeam are presumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fraction of the constituents. The governing equation and the related boundary conditions are derived via the principle of the calculus of variation. In order to eliminate the axial displacement in the formulation, the concept of the neutral surface is adopted. Some analytical solutions are obtained for the static displacement, critical buckling load, free vibration frequencies and the dynamic displacement for the case of the simply-supported end condition. In the dynamic analysis, three different loading cases, a moving load with constant velocity, point and distributed harmonic loads, are considered, and Duhamel’s integration is utilized for obtaining the corresponding dynamic deflections. Several numerical examples are presented in figures and tables in order to examine the effects of the strain gradient and the nonlocal parameters, the gradient index, the excitation frequency and the moving load velocity on the mechanical behavior of FG nanobeam.