The formulation of 3D geometrically exact beam finite elements relies heavily on the interpolation scheme used for the unidimensional variables describing the rotations or the rigid-body motions of the beam cross-sections. Since these rotations and positions belong to the non-commutative Lie groups SO(3) and SE(3), they are not easily amenable to a direct discretisation.
Several schemes to interpolate the rotations have been suggested in the last two decades, each of them with specific advantages and drawbacks. Although the subject is far from settled, it has reached a stage of maturity which makes it possible to clearly identify several properties that an interpolation scheme should preferably display: it ought to (i) preserve orthogonality, (ii) be independent from the iterative process adopted, (iii) be path-independent, (iv) be frame-invariant, (v) be applicable to an arbitrary number of nodes and (vi) lend itself to a computationally implementable linearisation. Until very recently, none of the interpolation schemes for the rotation field in structural finite elements fulfilled all these requirements.
However, the interpolation rule proposed by Buss and Fillmore [
] and Merlini and Morandini [
] seems to gather all the aforementioned desirable properties. This paper discusses the implementation of a (geometrically exact) Reissner-Simo beam element adopting this interpolation scheme for the rotation field.
In addition, a growing trend in the development of geometrically exact beam finite elements involves the so-called helicoidal interpolation, in which the displacement and rotation fields are assumed to be coupled - this designation stems from the fact that a linear helicoidal interpolation between two adjacent nodes is an helix. In view of some mathematical analogies between groups SO(3) and SE(3), it is believed that the interpolation scheme applicable to rotations can be extended to the case of rigidbody motions.