Analytical Approach and Shooting Method Solution of Nonlocal Eringen Elasticity Problem of Nanorods

The aim of this paper is to study the postbuckling behaviors of the nanorods combined with the small scale effects. In this paper, buckling of a nanorod column subjected to tip load is investigated. The nanorod is on a clamped support at one end, while the other end is simply-supported subjected to axial compression. At this end, the nanorod is movable only in a horizontal direction. The governing differential equation describing the behavior of the nanorod is developed by the moment–curvature relationship in analogy with the classical Euler–Bernoulli beam theory together with the equilibrium equations, including the effects of nonlocal elasticity as well as corresponding boundary conditions. The numerical shooting methods are derived and employed to solve the differential equations in this problem. The results, including nonlocal elasticity, reveal that nanorods have decreased structural stiffness and show the significant effect of geometrical parameters on the stability of buckled nanorods, emphasizing the importance of accounting for their interaction in the design of nanostructural systems.