## 2 The Geometrically Exact Beam Model  
### Beam Kinematics  
In this section, a straight beam element with a square cross section was used as an example, and this method can be easily extended to curved beam elements with arbitrary cross sections. A reference configuration \(\mathcal{B}_{0}\) and a current configuration \(\mathcal{B}\) are displayed in Figure 1. For an arbitrary cross-section, the reference point of the local frame is assumed to coincide with the centroid \(O(o_{2},o_{3})\). A unit orthogonal basis system \(\mathbf{A}=\{\mathbf{A}_{1},\ \mathbf{A}_{2},\ \mathbf{A}_{3}\}\) along with the material coordinates \(\{\xi_{1},\ \xi_{2},\ \xi_{3}\}\) is introduced. The axis of the beam is initially along \(\mathbf{A}_{1}\), with the length of beam \(l\) and the arc-length parameter \(S=\xi_{1}\in[0,\ l]\) of the spatial curve. The orientation of the cross-section is described by the basis vectors \(\{\mathbf{A}_{2},\ \mathbf{A}_{3}\}\). The local frame \(\mathbf{a}=\{\mathbf{a}_{1},\ \mathbf{a}_{2},\ \mathbf{a}_{3}\}\) is attached to the cross-section in the current configuration, which is characterized by the time parameter \(t\). For convenience, it is assumed that the material frame \(\mathbf{A}_{i}\) and inertial frame \(\mathbf{e}_{i}\) initially coincide, and thus the relationship between the basis vectors \(\mathbf{A}_{i}\) and \(\mathbf{a}_{i}\) is as follows:  
\[\mathbf{a}_{i}=\mathbf{R}\mathbf{A}_{i}\quad\text{for}\ i=1,2,3 \tag{1}\]  
where the operator \(\mathbf{R}\) is the cross-section orthogonal rotation tensor with respect to inertial frame. In general, the rotation tensor is defined from the exponential map and can be expressed directly by Rodrigues formula [32]:  
\[\mathbf{R}=\exp(\mathbf{\widetilde{\Theta}})=\mathbf{I}_{3\times 3}+a_{1}( \mathbf{\Theta})\mathbf{\widetilde{\Theta}}+a_{2}(\mathbf{\Theta})\mathbf{ \widetilde{\Theta}}\mathbf{\widetilde{\Theta}} \tag{2}\]  
with  
\[a_{1}(\mathbf{\Theta})=\frac{\sin(\left\|\mathbf{\Theta}\right\|)}{\left\| \mathbf{\Theta}\right\|},\quad a_{2}(\mathbf{\Theta})=\frac{1-\cos(\left\| \mathbf{\Theta}\right\|)}{\left\|\mathbf{\Theta}\right\|^{2}} \tag{3}\]  
The notation \(\mathbf{\widetilde{\bullet}}\) indicates that there is an invertible linear map from \(\mathbb{R}^{k}\) to the Lie algebra; and the expression of which depends on the dimension k. The variable \(\mathbf{\widetilde{\Theta}}\) in the \(SO(3)\) framework represents  
Figure 1: Initial configuration and the current configuration.  
the skew-symmetric matrix of the rotational vector \(\mathbf{\Theta}\) and \(\|\mathbf{\Theta}\|\) means the magnitude of the rotational vector. On the contrary, the inverse map of the exponential map is the logarithmic map [33], giving:  
\[\log(\mathbf{R})=\mathbf{\tilde{\Theta}}=\begin{cases}\mathbf{0}_{3}&\text{if }\mathbf{R}= \mathbf{I}_{3\times 3}\\ \frac{\theta}{2\sin(\theta)}(\mathbf{R}-\mathbf{R}^{T})&\text{if }\mathbf{R} \
eq\mathbf{I}_{3\times 3}\end{cases} \tag{4}\]  
with \(\theta=\arccos(\frac{1}{2}(\text{trace}(\mathbf{R})-1))\) and the absolute value \(|\theta|<\pi\).  
An arbitrary point \(\mathrm{p}\) on the cross-section in the current configuration can be expressed by:  
\[\mathbf{x}_{p}(\xi_{1},\xi_{2},\xi_{3})=\mathbf{x}\big{(}\xi_{1})+\mathbf{R}( \xi_{1})\mathbf{y}_{p}(\xi_{2},\xi_{3}) \tag{5}\]  
in which \(\mathbf{y}_{p}=\xi_{2}\mathbf{A}_{2}+\xi_{3}\mathbf{A}_{3}\), and is defined from the reference configuration since the cross-section is assumed to remain undeformed.