Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates

Abstract In the present work, finite element formulations for nonlocal elastic (i) Euler–Bernoulli beam and (ii) Kirchoff plate have been reported. Nonlocal differential elasticity theory is considered. Galerkin finite element technique has been employed. For both nanobeams and nanoplates weak forms of governing equations are derived and energy functionals are obtained. Present finite element results for bending, vibration and buckling for nonlocal beam with four classical boundary conditions are computed. These results are in good agreement with those reported in the literature. Further, bending, vibration and buckling analyses are carried out for stepped nanobeam. Furthermore, using present finite element bending, vibration and buckling analyses for nonlocal nanoplate are carried out. Present formulation will be useful for structural analyses of nanostructures with complex geometry, material property, loading and boundary conditions.