Buckling analysis of Euler-Bernoulli beams using Eringen's two-phase nonlocal model

The inconsistency of Eringen's nonlocal differential model, as applied to investigate nanostructures, has recently triggered the study of nonlocal integral models. In this paper we adopt Eringen's two-phase nonlocal integral model to carry out an analytical study on the buckling problem of Euler-Bernoulli beams. By using a reduction method rigorously proved in the previous work, the resulting integro-differential equation for the problem is firstly reduced to a fourth order differential equation with mixed boundary conditions. Exact characteristic equations are then obtained for four types of boundary conditions. Further, after some detailed asymptotic analysis, asymptotic solutions for the critical buckling loads are obtained, which are shown to have a good agreement with the numerical solutions. The analytical solutions show clearly that the nonlocal effect reduces the buckling loads. It is also found that the effect could be first-order or second order depending on the boundary conditions.