Beam Bending Solutions Based on Nonlocal Timoshenko Beam Theory

This paper is concerned with the bending problem of micro- and nanobeams based on the Eringen nonlocal elasticity theory and Timoshenko beam theory. In the former theory, the small-scale effect is taken into consideration while the effect of transverse shear deformation is accounted for in the latter theory. The governing equations and the boundary conditions are derived using the principle of virtual work. General solutions for the deflection, rotation, and stress resultants are presented for transversely loaded beams. In addition, specialized bending solutions are given for beams with various end conditions. These solutions account for a better representation of the bending behavior of short, stubby, micro- and nanobeams where the small-scale effect and transverse shear deformation are significant. Considering particular loading and boundary conditions, the effects of small-scale and shear deformation on the bending results may be observed because of the analytical forms of the solutions.