On Timoshenko's iterative method in buckling of nanobeams

Buckling of slender nanobeams is addressed by an iterative procedure. The nonlocal stress gradient theory of elasticity is exploited to account for scale effects. The governing problem is shown to be mathematically represented by a higher-order differential formulation with respect to the locally elastic case. In order to avoid expensive computational approaches, an iterative solution strategy is proposed. Such a procedure is the generalization of a method of successive approximations due to Timoshenko, here extended to the field of nonlocal gradient mechanics. Validation of the proposed methodology is carried out and critical buckling loadings are assessed for case-studies of current interest in Nano-Engineering. The present approach provides a computationally effective tool for the assessment of buckling loadings of nanobeams.