Ritz Method Application to Bending, Buckling and Vibration Analyses of Timoshenko Beams via Nonlocal Elasticity

Bending, buckling and vibration behaviors of nonlocal Timoshenko beams are investigated in this research using a variational approach. At first, the governing equations of the nonlocal Timoshenko beams are obtained, and then the weak form of these equations is outlined in this paper. The Ritz technique is selected to investigate the behavior of nonlocal beams with arbitrary boundary conditions along them. To find the equilibrium equations of bending, buckling, and vibration of these structures, an analytical procedure is followed. In order to verify the proposed formulation, the results for the nonlocal Timoshenko beams with four classical boundary conditions are computed and compared wherever possible. Since the Ritz technique can efficiently model the nano-sized structures with arbitrary boundary conditions, two types of beams with general boundary conditions are selected, and new results are obtained.