Size effect in the bending of a Timoshenko nanobeam

The size effect should be considered due to the large ratio of surface area to volume when the characteristic length of a beam lies in the nanoscale. The size effect in the bending of a Timoshenko nanobeam is investigated in this paper based on a recently developed elastic theory for nanomaterials, in which only the bulk surface energy density and the surface relaxation parameter are involved as independent parameters to characterize the surface property of nanomaterials. In contrast to the Euler nanobeams and the classical Timoshenko beam, not only the size effect but also the shear deformation effect in Timoshenko nanobeams is included. Closed-form solutions of the deflection and the effective elastic modulus for both a fixed–fixed Timoshenko nanobeam and a cantilevered one are achieved. Comparing to the classical solution of Timoshenko beams, the size effect is obviously significant in Timoshenko nanobeams. The shear deformation effect in nanobeams cannot be neglected in contrast to the solution of Euler–Bernoulli nanobeams when the aspect ratio of a nanobeam is relatively small. Furthermore, the size effect exhibits different influences on the bending behavior of nanobeams with different boundary conditions. A nanobeam with a fixed–fixed boundary would be stiffened, while a cantilevered one is softened by the size effect, compared to the classical solution. All the findings are consistent with the existing experimental measurement. The results in this paper should be very useful for the precision design of nanobeam-based devices.