Geometrically exact isogeometric Bernoulli-Euler beam based on the Frenet-Serret frame

A novel geometrically exact model of the spatially curved Bernoulli-Euler beam is developed. The formulation utilizes the Frenet-Serret frame as the reference for updating the orientation of a cross section. The weak form is consistently derived and linearized, including the contributions from kinematic constraints and configuration-dependent load. The nonlinear terms with respect to the cross-sectional coordinates are strictly considered, and the obtained constitutive model is scrutinized. The main features of the formulation are invariance with respect to the rigid-body motion, path-independence, and improved accuracy for strongly curved beams. A new reduced beam model is conceived as a special case, by omitting the rotational DOF. Although rotation-free, the reduced model includes the part of the torsional stiffness that is related to the torsion of the beam axis. This allows simulation of examples where the angle between material axes and Frenet-Serret frame is small. The applicability of the obtained isogeometric finite element is verified via a set of standard academic benchmark examples. The formulation is able to accurately model strongly curved Bernoulli-Euler beams that have well-defined Frenet-Serret frames.