Analogies between nonlocal and local Bernoulli-Euler nanobeams

Small-scale effects in nanobeams are effectively described by the Eringen model of nonlocal elasticity. The nonlocal elastostatic problem of Bernoulli-Euler nanobeams is here formulated in variational terms by recognizing that the nonlocality effect is equivalent to a bending curvature distortion prescribed on a corresponding local nanobeam, subjected to the same kinematic boundary constraints and applied loads. The conditions to be imposed for the kinematic integrability of the bending curvature field are also provided to evaluate the bending moment solution in statically indeterminate nonlocal nanobeams. Since the curvature distortion describing the nonlocality effect is kinematically integrable in statically determinate structures, bending moments do not exhibit small-scale effects in non-redundant nanobeams. The equivalence method illustrated in the present paper is resorted to for solving the nonlocal elastostatic problem of nanobeams under constant transversal load distributions.