# Review of the Main Cable Shape Control of the Suspension Bridge
## 2 The Development of the Shape-Finding Calculation Theory for Main Cables
### Finite Element Method (FEM)
Through the application of this approach, the whole bridge structure was nonlinearly analyzed, avoiding the complexities and mistakes of the traditional method's assumptions on boundary conditions, virtual restrictions, and the creation of equilibrium equations for a single cable system. [PERSON] et al. [48] established a shape-finding approach for a suspension bridge main cable based on a nonlinear FEM in the context of the Eulerian description. Firstly, the control differential equation of the main cable in 3D space was constructed by analyzing the cable as a whole, and then following the introduction of the 1D linear form function, the Newton iteration approach was implemented to evaluate the main cable shape. It can be seen that the optimization of the FEM based on the catenary element mostly focuses on the shape-finding of the space cable. Even though complex nonlinear equations are transformed into linearized equations in shape-finding calculations, iterative calculations are still required in the process of solving the element stiffness matrix, due to the coupling of equations between the elements. Some new computational methods were proposed to improve the efficiency of iterative computation, such as the penalty factor search algorithm and whole mechanical analysis method. The refined analysis  
It is worth mentioning that in all cases, the main cable should be tangential to the saddle, and the tangent point is continually varying. The saddle directly restricts the deformation of the main cable. The above research did not consider this problem. A flexible iterative approach taking into account the slip effect between the main cable and the saddle was developed by [PERSON] et al. [49]. Firstly, the nonlinear equation was linearized by the first-order Taylor expansion, and the numerical approach was exploited to derive the catenary element's tangent stiffness matrix. The Newton-Raphson method was implemented to evaluate the finished main cable shape, free cable shape, and saddle pre-deviation under dead load. It is crucial to realistically simulate the saddle in the process of solving the main cable shape via FEM. The simplified approach exploited by the majority of researchers involved simulating the tangent point and eccentricity of the saddle with multiple rod units and rigid arms. [PERSON] et al. [50] constructed a multi-nodesaddle element composed of five-rod elements to consider the movement of the saddle and the change of the tangent position. As demonstrated in Figure 5a, the saddle is composed of rod elements 1\(\sim\)5; 6 and 7 are the tower top, and the main cable's tangent points on the saddle are nodes 1 and 2. The saddle pushing process is simulated by translating the coordinates of nodes 1\(\sim\)6. [PERSON] [51] employed four beam elements to simulate the saddle and pushing process. According to Figure 5b, the vertical and horizontal forces produced by the main cable's tension are transmitted to the tower top by employing the straight bar elements with large axial stiffness between points 1 and 6 and between points 6 and 7, respectively. The main cable's tangent points on the saddle are 2 and 3. The simulation approach is the same as that provided in Ref. [50], and the calculation's precision is not particularly good. The main cable's stress-free length in the suspension section cannot be modified according to changes in the tangent point's position in relation to the saddle. To solve the above problem, a more accurate simulation methodology [52] for the saddle was developed, as demonstrated in Figure 6. To simulate the saddle body, the rigid beam elements were radially arranged along the saddle's upper surface. A changing range for the tangent point between the main saddle and the main cable was established in the encryption area. The saddle top in the encryption area was linked to the main cable using compression-only connections, which can simulate the separating state and how their tangent point positions vary at each stage.