Geometrically non-linear large planar deformations of shear deformable beams: derivation,approximation,and numerical validation using Finite Element analysis

Abstract

In this paper, we present derivations for the dynamic equations of motion of a shear deformable beam undergoing geometrically non-linear large deformations in a single plane. In the static case, these governing equations are given by six coupled, non-linear ordinary differential equations of first order which we present in the Lagrangian frame of reference and henceforth refer to as the Brunk–Herbrich–Weckner (BHW) beam theory. We confirm analytically that under pure bending an initially straight beam deforms into a circle, a known bench-mark solution. Next, we confirm numerically that our results agree well with the commercial Finite-Element code Abaqus for several example problems with increasing complexity, including the effect of axial preload as well as varying property domains along the beam. Abaqus is regarded as the state-of-the-art for non-linear structural analysis within the aerospace industry. Next, we show that the well-known Timoshenko–Ehrenfest and Euler–Bernoulli beam theory can be obtained from the general Brunk-Herbrich-Weckner beam theory in the limit of small deformations. Finally, we introduce the Iterative Projection Method which improves the results of the linearized Timoshenko–Ehrenfest beam theory without the computational cost associated with the fully non-linear analysis for all numerical test cases considered.