On the differentiation of the Rodrigues formula and its significance for the vector‐like parameterization of Reissner–Simo beam theory

In this paper we present a systematic way of differentiating, up to the second directional derivative, (i) the Rodrigues formula and (ii) the spin‐rotation vector variation relationship. To achieve this goal, several trigonometric functions are grouped into a family of scalar quantities, which can be expressed in terms of a single power series. These results are then applied to the vector‐like parameterization of Reissner–Simo beam theory, enabling a straightforward derivation and leading to a clearer formulation. In particular, and in contrast with previous formulations, a relatively compact and obviously symmetric form of the tangent operator is obtained. The paper also discusses several relevant issues concerning a beam finite element implementation and concludes with the presentation of a few selected illustrative examples.