  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.
