Postbuckling analysis of Timoshenko nanobeams including surface stress effect

Abstract

A modified continuum model is developed to predict the postbuckling deflection of nanobeams incorporating the effect of surface stress. To have this problem in view, the classical Timoshenko beam theory in conjunction with the Gurtin–Murdoch elasticity theory is utilized to propose non-classical beam model taking surface stress effect into account. The geometrical nonlinearity is considered in the analysis using the von Karman assumption. By employing the principle of virtual work, the size-dependent governing differential equations and related boundary conditions are derived. On the basis of the shifted Chebyshev–Gauss–Lobatto grid points, the generalized differential quadrature (GDQ) method is adopted as an accurate, simple and computational efficient numerical solution to discretize the non-classical governing differential equations along with various end supports. Selected numerical results are worked out to demonstrate the nonlinear equilibrium paths of the postbuckling behavior of nanobeams corresponding to different values of beam thickness, buckling mode number, surface elastic constants, and various types of boundary conditions.