Eringen’s nonlocal theories of beams accounting for moderate rotations

The primary objective of this paper is two-fold: (a) to formulate the governing equations of the Euler–Bernoulli and Timoshenko beams that account for moderate rotations (more than what is included in the conventional von Kármán strains) and material length scales based on Eringen’s nonlocal differential model, and (b) develop the nonlinear finite element models of the equations. The governing equations of the Euler–Bernoulli and Timoshenko beams are derived using the principle of virtual displacements, wherein the Eringen’s nonlocal differential model and modified von Kármán nonlinear strains are taken into account. Finite element models of the resulting equations are developed, and numerical results are presented for various boundary conditions, showing the effect of the nonlocal parameter on the deflections.