A Fully Nonlinear Beam Model of Bernoulli–Euler Type

Abstract

This work presents a geometrically exact Bernoulli–Euler rod model. In contrast to Pimenta (1993b), Pimenta and Yojo (1993), Pimenta (1996), Pimenta and Campello (2001), where the hypothesis considered was Timoshenko’s, this approach is based on the Bernoulli–Euler theory for rods, so that transversal shear deformation is not accounted for. Energetically conjugated cross-sectional stresses and strains are defined. The fact that both the first Piola–Kirchhoff stress tensor and the deformation gradient appear again as primary variables is also appealing. A straight reference configuration is assumed for the rod, but, in the same way, as in Pimenta (1996), Pimenta and Campello (2009), initially curved rods can be accomplished, if one regards the initial configuration as a stress-free deformed state from the straight position. Consequently, the use of convective non-Cartesian coordinate systems is not necessary, and only components on orthogonal frames are employed. A cross section is considered to undergo a rigid body motion and parameterization of the rotation field is done by the rotation tensor with the Rodrigues formula that makes the updating of the rotational variables very simple. This parametrization can be seen in Pimenta et al. (2008), Campello et al. (2011). A simple formula for the incremental Rodrigues parameters in function of the displacements derivative and the torsion angle is also settled down. A 2-node finite element with Cubic Hermitian interpolation for the displacements, together with a linear approximation for the torsion angle, is displayed within the usual Finite Element Method, leading to adequate $C_{1}$ continuity.