# Review of the Main Cable Shape Control of the Suspension Bridge
## 2 The Development of the Shape-Finding Calculation Theory for Main Cables
### Analytical Methods  
If the weight of the cable is uniformly distributed across its span, the cable will align parabolically and will have a constant horizontal component force [18] (i.e., the parabolic theory of shape-finding of the main cable). The aforementioned theory ignores the cable's weight distribution along the cable curve, which is only applicable to bridges with a small span. To overcome the limitations of the parabolic theory, several studies were carried out by numerous academics, and then the segmental linear method [19], the segmental parabolic method [20], and the segmental catenary method (SCM) [21] were proposed. Additionally, the above methods have been summarized in Ref. [21], and the comparison accuracy and application scope of the above calculation methods for the bridge completion state of the main cable were presented through examples. 1. The optimization of calculation methods  
The spatial cable suspension bridge and three-tower suspension bridge make the traditional calculation method face new challenges. Therefore, some scholars have proposed a new calculation method based on the traditional SCM to analyze the shape-finding problem of space cables and multi-span main cables. Consider the problem where the coupling effect between the space cable and the boom makes the main cable form-finding complex and difficult to converge. [PERSON] et al. [22] employed the least-squares method with Marquardt correction to solve the linear calculation of spatial cable plane suspension bridges. This method was easy to converge and had high precision, but it was only applicable to the case where the hanger is only inclined in the transverse direction of the bridge. To solve the above problem, [PERSON] et al. [23] also derived the bridge alignment analytical formula of the spatial special-shaped main cable, considering the coupling effect of the spatial main cable and hanger, which can consider the case where the hanger is inclined in both the longitudinal and transverse directions. In addition, considering the particularity of the main cable form finding of the three-tower suspension bridge, a three-tower space cable shape-finding algorithm with an unequal main span on the basis of the SCM was developed by [PERSON] et al. [17]. As shown in the calculation flowchart of Figure 1, the main span, secondary main span, side span, and anchor span were solved and analyzed in turn. The control equations equal to the number of basic unknowns were established using the equilibrium conditions of geometric compatibility and force, and the appropriate values of basic unknowns were evaluated. It can be concluded from the above literature that the current analysis objects of main cable shape-finding were mainly concentrated on unconventional spatial cables and multi-tower suspension bridges, but the analysis method was still based on the traditional SCM. Additionally, based on the SCM method, an analytical calculation method that captured the coupling effect of the main cable-hanger-stiffening beam system was established by [PERSON] et al., [7] which can calculate the mechanical properties of the whole structure and the mechanical properties of the system. Figure 1: The main cable’s shape-finding flowchart. the correlation of each component. Due to the poor convergence performance and the low computational efficiency of the SCM, some scholars have proposed a new method for the main cable shape-finding. The suspension bridge's main cable shape was analyzed by [PERSON] et al. [24] by implementing a slope climbing method. The position and slope of the main cable's lowest point were then determined using the unified linear equations for each cable section. The deformation compatibility condition was employed to construct the corresponding equations, and the variation law of the main cable force's horizontal component force was then employed to solve them. [PERSON] et al.