# A New Method for Finding the Shape of the Main Cable in the Special Cable Plane
## 2 Force Analysis of Spatial Special-Shaped Cable Plane
### Force Transfer Equation of Space Suspender  
The suspender is an important part of the space shaped cable plane, and the clear force transmission process of the suspender is the key to finding the shape of the main cable of the space shaped cable plane. Based on the force transfer equation of the inclined plane suspender, the recurrence relationship between the force of the main cable section on the special plane of the space cable and the force of the main cable end is proposed (Equation (2)). By analyzing the force transfer equation of the suspender, it can be found that the transverse bridge component \(H_{xi}\) in different sections of the main cable changes significantly under the action of the suspender transverse bridge component \(P_{xi}\). Due to the existence of the transverse bridge force component of the suspender, it is necessary to calculate the transverse bridge force component in the middle of the span during the calculation of the finished bridge alignment of the spatial special-shaped cable plane, which increases the calculation difficulty of finding the main cable shape of the spatial special-shaped cable plane. \[ \begin{cases}H_{xi}&=H_{xi-1}-P_{xi}=H_{x0}-\frac{n}{\sum\limits_{i=1}^{n}P_{ xi}}\\ H_{zi}&=H_{zi-1}-P_{zi}=H_{20}-\frac{n}{\sum\limits_{i=1}^{n}P_{zi}}\\ V_{i}&=V_{i-1}-g_{m}L_{mi-1}-P_{vi}-g_{n}L_{ni}\\ &=V_{0}-g_{m}\sum\limits_{i=1}^{m-1}L_{mi}-g_{m}\sum\limits_{i=1}^{n}L_{ni}- \sum\limits_{i=1}^{n}P_{vi}\end{cases}  \tag{2}\]  
where \(g_{m}\) is the dead weight of the sling; \(L_{ni}\) is the unstressed length of the ith hanger; and \(P_{xi}\), \(P_{zi}\) and \(P_{vi}\) are the along the bridge, the transverse bridge, and the vertical components  
Figure 1: Calculation Diagram of Main Cable of Single Cable Plane Curved Suspension Bridge [11]. of the suspender of section \(i\), respectively, at the lower end. The specific meanings of other letters are the same as those above. **3. New Calculation Model**  
A numerical calculation model is the key to solve the problem. [PERSON] [10] proposed the calculation model of the space main cable (Equation (3)) based on the analysis of the stress of the space main cable. The calculation model of the space main cable proposed by [PERSON] is:  
\[\left\{ \begin{array}{l}f_{1}(H_{x0},H_{z0},V_{0}):ey1=\sum\limits_{i=1}^{n}h_ {i}-\Delta y1\\ f_{2}(H_{x0},H_{z0},V_{0}):ey2=\sum\limits_{i=1}^{m}h_{i}-\Delta y2\\ f_{3}(H_{x0},H_{z0},V_{0}):ez4=\sum\limits_{i=1}^{m}z_{i}-\Delta z4\end{array} \right. \tag{3}\]  
At present, there is a lack of efficient numerical calculation models for the shape-finding calculation of the main cable in the space special-shaped cable plane. On the basis of [PERSON], this paper proposes a calculation model (Equation (4)) suitable for finding the shape of the main cable of the spatial special-shaped suspension bridge, which takes into account the influence of the calculation error of the midspan transverse direction. The new calculation model considers both the vertical rise span ratio and the transverse rise span ratio of the spatial main cable, which is more consistent with the actual situation of the spatial special-shaped cable plane. The calculation model of the space main cable proposed in this paper is:  
\[\left\{ \begin{array}{l}f_{1}(H_{x0},H_{z0},V_{0}):ey1=\sum\limits_{i=1}^{n}h _{i}-\Delta y1\\ f_{2}(H_{x0},H_{z0},V_{0}):ey2=\sum\limits_{i=1}^{m}h_{i}-\Delta y2\\ f_{3}(H_{x0},H_{z0},V_{0}):ez3=\sum\limits_{i=1}^{n}z_{i}-\Delta z3\\ f_{4}(H_{x0},H_{z0},V_{0}):ez4=\sum\limits_{i=1}^{m}z_{i}-\Delta z4\end{array} \right. \tag{4}\]  
where \(ey1\) is the vertical calculation error at the right end of the main cable, \(ey2\) is the vertical calculation error in the middle of the main cable span, \(ez3\) is the calculation error in the transverse direction of the right end of the main cable and \(ez4\) is the transverse calculation error in the middle of the main cable span.