Bending and Buckling of Timoshenko Nano-Beams in Stress-Driven Approach

A modified total potential energy functional is derived for stress-driven non-local model of Timoshenko beam subject to transverse load and/or critical axial load. The modified functional includes expressions representing the constitutive boundary conditions, which are a peculiarity of the adopted stress-driven approach. The Euler equations of the modified functional are the governing equations of the stress-driven non-local model. Instead of solving directly the Euler equations, approximate solutions are searched by imposing the stationary condition of the modified functional through the Ritz method. In order to validate the method, the proposed numerical solutions are compared with closed-form expressions, in load cases where closed-form solutions are available. Finally, the proposed numerical method is used for determining the buckling load of non-local Timoshenko beam.