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

Abstract

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.

Adding kinematic constraints to purely differential dynamics

“…In fact, the orientation matrix cannot be directly interpolated. However, one can exploit the helicoidal strain assumption [39], mutuated from the screw theory used to describe rigid body dynamics, and interpolate the orientation as…”

Introducing the Logarithmic finite element method: a geometrically exact planar Bernoulli beam element

“…the positions and orientations of the cross-sections of the beam along the neutral axis, with elements of the Lie group SE (3), and linearly interpolating on the associated Lie algebra, the formulation allows for a coupling of the translational and rotational components of the deformation. An earlier formulation that implicitly uses an interpolation of the neutral axis of a beam as well as of the orientation of its cross-section on a the Lie algebra se(3), instead of polynomial shape functions, has been presented by Borri and Bottasso [7]. Selig and Ding explicitly introduce Lie groups and Lie algebras in their exposition of a screw-theoretic formulation for a planar beam [26].…”

A variational formulation for finite elasticity with independent rotation and Biot-axial fields

“…As concerns the Biot-axial, it should actually be null in this case of isotropic material in the absence of body couples. Due to the numerical approximation, values of the Biot-axial of the order of 10 ÿ 6 7 In the expression of d G2 ; and L are tensors entering the calculus of rotations parametrized with a rotation vector, see for instance Borri and Bottasso (1994), Appendix II. Tensor L is also a function of the current residual of the equation of rotational equilibrium, hence it depends on the stress and deformation and on the acting couple.…”

“…The variational principles of the polar continuum will be drawn joining the outline followed by Borri (1994) in analytical dynamics. Consider the equilibrium Eqs.…”