An intrinsic beam model based on a helicoidal approximation—Part I: Formulation

The focus of the present work is directed towards the one‐dimensional non‐linear analysis of space‐curved and twisted beams undergoing large displacements and finite rotations. According to Cosserat's model, a beam is intended here as a continuum generated by the rigid motion of a cross‐section along a curve. The novelty of the proposed methodology is implied in the assumption that the reference line of the beam, both in the undeformed and in the deformed configurations, is a helicoid in space. This formulation radically departs from the classical beam models based on polynomial interpolations of the independent fields. In fact, within the framework of helicoidal geometry, a powerful description of space‐curved beam kinematics with large displacements and finite rotations can be developed. Furthermore, the beam model developed herein can be proved to enjoy several invariance properties that make it particularly attractive for non‐linear analysis. The formulation discussed in this work adheres to a mixed variational principle, the independent fields being the generalized displacements and the sectional stress resultants. In a companion paper, we address the linearization of the governing weak form, the finite element implementation of the proposed methodology with a novel incremental treatment of finite rotations, and the development of a helicoidal model for rigid body dynamics using finite elements in time.