Nonlocal finite element analysis and small scale effects of CNTs with Timoshenko beam theory

In the present work, finite element formulations for nonlocal elastic Euler–Bernoulli beam theory and Timoshenko beam theory have been reported. Nonlocal differential elasticity theory is considered. Galerkin finite element technique has been employed. For CNTs, weak forms of governing equations are derived and energy functionals are obtained. With present finite element analysis bending, buckling and vibration for nonlocal beams with clamped–clamped, hinged–hinged, clamped–hinged and clamped–free (C–C, S–S, C–S and C–F, respectively) boundary conditions are computed. These results are in good agreement with those reported in the literature. Further, bending, buckling and vibration analyses are extended to tapered beams. Present formulation will be useful for structural analyses of nanostructures with complex geometries, material properties, loadings and boundary conditions.