## 3 Calculation of Cable Saddle Position
### Determination of Cable Saddle Position  
For a section of the main cable unit placed on the cable saddle, when it is in equilibrium, the left node is \(I\) and the right node is \(J\). Its position can be uniquely determined by a specific value of fixed-point \(Q\) and independent variable \(Z\). When an initial position of the main cable unit is given; that is, an initial independent variable \(Z\) is given, the left node is \(I^{\prime}\), and the right node is \(I^{\prime}\), as shown in Figure 7. The horizontal and vertical deviations of node \(I^{\prime}\) and \(I\) are \(r_{1}\) and \(r_{2}\), respectively, and the horizontal and vertical deviations of nodes \(J^{\prime}\) and \(J\) are \(r_{3}\) and \(r_{4}\), respectively. In order to facilitate an iterative solution, the deviation \(r_{1}\)-\(r_{4}\) is composed of deviation vector \(r\). Since point \(Q\) is a fixed point, when the positions of point \(I\) and point \(J\) change, the tangent points \(B\) and \(C\) of the suspension cable and cable saddle will also change. According to the tangent point coordinates and friction relations expressed by the fixed point \(Q\) of the cable saddle [Equations (6)-(12)] and the analytic formulas of the horizontal and vertical projection length of the elastic catenary [Equations (13) and (14)], the components of the deviation vector \(r\) is obtained. \[r_{1}=x_{I}-\{R\sin\bigl{(}Z_{2}-\theta_{Q}\bigr{)}+p_{1}.\biggl{[}\frac{1}{q} \ln\bigl{(}\frac{t_{B}-p_{2}}{t_{I}+u_{2}}\Bigr{)}-\frac{L_{1}}{EA}\bigr{]}\} \tag{15}\]  
\[r_{2}=y_{I}-[R\cos\bigl{(}Z_{2}-\theta_{Q}\bigr{)}-\frac{t_{I}-t_{B}}{q}.( \frac{t_{I}+t_{B}}{2 EA}+1)] \tag{16}\]  
Figure 6: Catenary cable calculation model diagram. Figure 7: Schematic diagram of cable saddle force and deviation. \[r_{3}=x_{I}-\{R\sin(Z_{3}+\theta_{Q})\}+p_{3}[\frac{1}{q}.\ln\left(\frac{t_{C}-p_{4} }{t_{I}+u_{4}}\right)-\frac{L_{2}}{EA}\right)] \tag{17}\]  
\[r_{4}=y_{I}-\{R\cos{(Z_{3}+\theta_{Q})}-\frac{t_{I}-(t_{C}+f)}{q}.[\frac{t_{I}+( t_{C}+f)}{2 EA}+1]\} \tag{18}\]  
From the horizontal force of the main cable and context relation  
\[t_{B}=Z_{1} \tag{19}\]  
\[t_{c}=Z_{1}\frac{\cos{(Z_{2}-\theta_{Q})}}{\cos{(Z_{3}+\theta_{Q})}}+f \tag{20}\]  
\(u_{1}\), \(u_{2}\), \(u_{3}\) and \(u_{4}\) are the corresponding node forces of node \(I\) and \(J\), respectively. \(p_{1}\), \(p_{2}\), \(p_{3}\) and \(p_{4}\) are the internal forces corresponding to tangent points \(B\) and \(C\), respectively. \(t_{B}\) is the main cable tension at the left tangent point \(B\); \(t_{c}\) is the main cable tension at the right tangent point \(C\). \(x_{i}\) and \(y_{i}\) are the horizontal and vertical coordinates of node \(I\), respectively; \(x_{j}\) and \(y_{j}\) are the horizontal and vertical coordinates of node \(J\), respectively; \(L_{1}\) is the unstressed length between \(I\) and \(B\) of the main cable of the left span (i.e., \(Z_{4}\)); \(L_{2}\) is the unstressed length between \(C\) and \(J\) of main cable of right span. According to [PERSON]'s iterative method, a small increment in the independent variable \(\Delta\textbf{Z}\) produces a small increment in the deviation \(\Delta\textbf{r}\), then \(\Delta\textbf{r}\) can be approximately expressed by the following formula  
\[\Delta\textbf{r}\approx j\Delta\textbf{Z} \tag{21}\]  
\(J\) is the Jacobian matrix of order 4. The element \(j_{ij}=\partial r_{i}/\partial Z_{j}\), \(I\), \(J\) take 1, 2, 3, 4, respectively. The increment in the independent variable that reduces the deviation can be expressed as  
\[r_{1}=x_{I}-\{R\sin\bigl{(}Z_{2}-\theta_{Q}\bigr{)}+p_{1}.\biggl{[}\frac{1}{q} \ln\bigl{(}\frac{t_{B}-p_{2}}{t_{I}+u_{2}}\Bigr{)}-\frac{L_{1}}{EA}\bigr{]}\} \tag{15}\]0  
Using the [PERSON] iterative method, until the deviation is less than the expected limit, the cable saddle element can be considered to be in equilibrium. The tangent stiffness matrix of the cable saddle element is derived.