# A Co-Rotational Meshfree Method for the Geometrically Nonlinear Analysis of Structures
## 2 Theoretical Formulation
### Geometry and Kinematics of Three-Dimensional Beam
#### 2.2.2 Kinematics of Deformed Beam  
From the kinematic assumptions made in the present paper, the deformations of the beam can be described by the displacements and rotation of the beam centroid axis. As shown in Figure 1, the position vector of an arbitrary point \(P\) on the beam centroidal axis at the undeformed and deformed configuration can be expressed as follows:  
\[\mathbf{r}_{P}(S)=X_{\text{o}}(S)\,\mathbf{e}_{X}+Y_{\text{o}}(S)\,\mathbf{e}_{Y}+Z_{\text{ o}}(S)\,\mathbf{e}_{Z} \tag{1}\]  
And  
\[\mathbf{r}_{P}\text{ }(\text{s})=X(\text{s})\,\mathbf{e}_{X}+Y(\text{s})\,\mathbf{e}_{Y}+Z( \text{s})\,\mathbf{e}_{Z}, \tag{2}\]  
where \(\mathbf{e}_{X}\), \(\mathbf{e}_{Y}\), and \(\mathbf{e}_{Z}\) are the unit basis vector of the global coordinates; \(S\) and \(s\) are the arc length of beam centroidal axis in undeformed and deformed configuration, respectively. Moreover, \(X_{\text{o}}(S)\), \(Y_{\text{o}}(S)\), \(Z_{\text{o}}(S)\) and \(X(\text{s})\), \(Y(\text{s})\), \(Z(\text{s})\) are the coordinate components of the position vector associated with undeformed and deformed configuration. From the geometric relationship between the global and co-rotational local coordinates at deformed configuration, Equation (3) can be given as follows:  
\[\mathbf{e}_{z}(\text{s})=\mathbf{d}(\text{s})/\mathbf{d}\text{s}. \tag{3}\]  
The relationship between the global and co-rotational local coordinate associated with each nodal point by collocation point method on the deformed beam axis can be represented by the rotation matrix \(\mathbf{R}(\text{s})\):  
\[\mathbf{e}_{xyz}(\text{s})=\mathbf{R}(\text{s})\,\mathbf{e}_{XYZ}, \tag{4}\]  
where \(\mathbf{R}\) is the rotation matrix defined according to the Rodrigues formula [35], as follows:  
\[\mathbf{R}=I+\frac{\sin\theta}{\vartheta}\Theta+\frac{1-\cos\theta}{\vartheta^{2} }\Theta^{2}\]  
\[\vartheta=\left[ \begin{array}{c}\varphi\\ \chi\\ \psi\end{array} \right],\,\mathbf{\Theta}=\left[ \begin{array}{ccc}0&-\psi&\chi\\ \psi&0&-\varphi\\ -\chi&\varphi&0\end{array} \right],\,\Theta^{2}=\Theta\Theta=\left[ \begin{array} []{ccc}-(\chi^{2}+\psi^{2})&\chi\varphi&\varphi\psi\\ \chi\varphi&-(\psi^{2}+\varphi^{2})&\psi\chi\\ \varphi\psi&\psi\chi&-(\varphi^{2}+\chi^{2})\end{array} \right]. \tag{5}\]  
The rotation \(\vartheta\) is a vector that lies on the axis of rotation, with its length equal to the angle of rotation. The rotation vector \(\vartheta\) can be written as a column matrix composed of components \(\varphi\), \(\chi\), and \(\psi\) in the fixed global coordinates. Furthermore, \(\Theta\) is a skew-symmetrical matrix composed from the rotation vector. Figure 1: Coordinate systems of the co-rotational beam. For the convenience of later derivation, the curvature vector is defined to represent the bending and twisting of the beam in terms of the co-rotational local coordinate as follows:  
\[\mathbf{\omega}=\omega_{x}\,\mathbf{e}_{x}+\omega_{y}\,\mathbf{e}_{y}+\omega_{z}\,\mathbf{e}_{z}, \tag{6}\]  
where \(\omega_{x}\), \(\omega_{y}\), and \(\omega_{z}\) are curvatures with respect to the \(x\), \(y\), and \(z\) axes of the co-rotational coordinates, respectively. Hence, the change rates of unit vectors with respect to the chord length of beam axis at deformed configuration can be expressed as follows:  
\[\mathbf{e}^{\prime}\mathbf{xyz}=\mathbf{K}\,\mathbf{e}_{\mathbf{xyz}} \tag{7}\]  
where \(\mathbf{K}=\left[ \begin{array}{ccc}0&\omega_{z}&-\omega_{y}\\ -\omega_{z}&0&\omega_{x}\\ \omega_{y}&-\omega_{x}&0\end{array} \right]\) is the curvature matrix, and \(\left[ \begin{array}{c}e_{x}\\ e_{y}\\ e_{z}\end{array} \right]\) is the basis vectors of the co-rotational local coordinates. From Equations (4) and (7), the curvature matrix can be represented in terms of the rotation matrix as follows:  
\[\mathbf{K}=\mathbf{R}^{\prime}\mathbf{R}^{\mathbf{T}} \tag{8}\]  
Then, substituting Equation (5) into Equation (8) will yield the curvature of the beam at the deformed configuration as follows:  
\[