Analytical solutions for coupled tension-bending of nanobeam-columns considering nonlocal size effects

Abstract

Considering nanoscale size effects and based on the nonlocal elastic stress theory, this paper presents an analytical analysis with closed-form solutions for coupled tension and bending of nanobeam-columns subject to transverse loads and an axial force. Unlike previous studies where many nanomechanical models do not yield analytical solutions and hence the size effects are studied by molecular dynamics simulation or other numerical means, the nonlocal nanoscale effects at molecular level unavailable in classical mechanics are investigated analytically here and first-known closed-form analytical solutions are presented. New higher-order differential governing equations in both transverse and axial directions and the corresponding higher-order nonlocal boundary conditions for bending and tension of nanobeam-columns are derived based on the variational principle approach. Closed-form analytical solutions for deformation and tension are presented and their physical significance examined. Examples conclude that the nonlocal stress tends to significantly increase the stiffness of a nanobeam-column, which are contradictory to the current belief using the nonlocal analysis that structural stiffness should be reduced but consistent with many non-nonlocal analyses including the strain gradient and couple stress theories that stiffness should be enhanced. The relationship between bending deflection, axial tension and nanoscale is presented, and also its significance on stiffness enhancement with respect to the design of nanoelectromechanical systems.