Bending, buckling and free vibration analysis of Euler-Bernoulli nanobeams using Eringen's nonlocal integral model via finite element method

In the present study, finite element formulations are derived for static bending, linear buckling and free vibration analysis of nanobeam structures by utilizing the integral form of Eringen nonlocal model. Formulations are developed according to the minimum total potential energy principle by presenting the differentiation operations explicitly. As being distinct from other studies, a non-uniform mesh distribution is proposed for the corresponding analytical expressions of the beam deflections. With this aid, the discontinuous nature of rotation angle, which is encountered at boundaries of the beam, is aimed to be captured. Many numerical examples are solved, and compared with the exact solutions reported in the literature to demonstrate the versatility of the non-local finite element method with the proposed mesh configuration. It is found out that, with the suggested mesh distribution, the number of elements can be decreased dramatically without sacrificing from the accuracy, which consequently leads to a considerable reduction in the computational cost.