Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory

In this paper, static bending and buckling of a functionally graded (FG) nanobeam are examined based on the nonlocal Timoshenko and Euler–Bernoulli beam theory. This non-classical (nonlocal) nanobeam model incorporates the length scale parameter (nonlocal parameter) which can capture the small scale effect. The material properties of the FG nanobeam are assumed to vary in the thickness direction. The governing equations and the related boundary conditions are derived using the principal of the minimum total potential energy. The Navier-type solution is developed for simply-supported boundary conditions, and exact formulas are proposed for the deflections and the buckling load. The effects of nonlocal parameter, aspect ratio, various material compositions on the static and stability responses of the FG nanobeam are discussed. Some illustrative examples are also presented to verify the present formulation and solutions. Good agreement is observed. The results show that the new nonlocal beam model produces larger deflection and smaller buckling load than the classical (local) beam model.