A single variable shear deformable nonlocal theory for transversely loaded micro- and nano-scale rectangular beams

In this paper, a simple single variable shear deformable nonlocal theory for bending of micro- and nano-scale rectangular beams is presented. To incorporate small size effects, the theory uses Eringen’s nonlocal differential constitutive relations. The theory has only one fourth-order governing differential equation involving a single unknown variable. The governing equation and the expressions for the bending moment and shear force of the present theory are strikingly similar to those of nonlocal Euler-Bernoulli Beam Theory (EBT) formulated based on Eringen’s nonlocal elasticity theory. The theory assumes that the axial and lateral displacements have bending and shear components such that the bending components do not contribute towards shear force, and the shear components do not contribute towards bending moment. Also, the chosen displacement functions of the theory give rise to a realistic parabolic transverse shear stress distribution across the beam cross-section. Efficacy of the proposed theory is demonstrated through bending of simply supported, cantilever and clamped-clamped micro- and nano-scale beams of rectangular cross-section. The numerical results obtained by using the present theory are compared with those predicted by other nonlocal first-order and higher-order shear deformation beam theories. The results obtained are quite accurate.