Timoshenko nanobeams are investigated by using a nonlocal Eringen-like constitutive law described by two material length-scale parameters. A nonlocal variational treatment, based on a new definition of the Helmholtz free energy is proposed. The ensuing system of ordinary differential equations of nonlocal elastic equilibrium is consistently obtained with the corresponding boundary conditions. New closed form solutions of nonlocal Timoshenko nanobeams are established and comparisons with gradient and local formulations are performed. Finally, two examples are illustrated to assess nonlocality effects.
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