Postbuckling analysis of Timoshenko nanobeams including surface stress effect R. Ansari a , V. Mohammadi a , M. Faghih Shojaei a , R. Gholami b,⇑ , S. Sahmani a a Department of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran b Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, P.O. Box 1616, Lahijan, Iran article info Article history: Received 17 May 2013 Received in revised form 12 October 2013 Accepted 14 October 2013 Available online 15 November 2013 Keywords: Nanobeams Surface stress Postbuckling Size effect Generalized differential quadrature method abstract A modified continuum model is developed to predict the postbuckling deflection of nano- beams 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 uti- lized 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 sup- ports. 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. Ó 2013 Elsevier Ltd. All rights reserved. 1. Introduction During the last three centuries, buckling characteristics of different elastic structures have been studied. In a straight beam, which is subjected to an axial compressive load, the buckling phenomenon may occur when the applied compressive load exceeds a critical value that can be led to turn the beam into an unstable condition and then it will buckle into one of several stable curves (Magnucka-Blandzi & Magnucki, 2011). However, the beam is still able to carry considerable load be- yond buckling, namely as postbuckling response. Various investigations have been carried out to analyze the postbuckling behavior of beam structures. The thermal buckling and postbuckling behaviors of a composite beam with embedded shape memory were investigated analytically by Lee and Choi (1999). Zhang and Taheri (2003) simulated the dynamic pulsebuckling and postbuckling re- sponses of fiber-reinforced plastic laminated beams, having initial geometric imperfections, subjected to axial impact. Abu-Salih and Elata (2005) analyzed the postbuckling characteristics of an infinite beam that is bonded to a linear elastic foundation and they assumed that the beam is subjected to an internal compressive stress. The nonlinear large deflec- tion-small strain analysis and postbuckling behavior of Timoshenko beam-columns of symmetrical cross-section with semi-rigid connectors subjected to conservative and non-conservative loads were presented by Aristizabal-Ochoa (2007). Machado (2008) studied nonlinear buckling and postbuckling of thin-walled composite beams, considering shear 0020-7225/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijengsci.2013.10.002 ⇑ Corresponding author. Tel./fax: +98 141 2222906. E-mail address: gholami_r@liau.ac.ir (R. Gholami). International Journal of Engineering Science 75 (2014) 1–10 Contents lists available at ScienceDirect International Journal of Engineering Science journal homepage: www.elsevier.com/locate/ijengsci