# 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)  
With the continuous development of computer applications in engineering calculations, a nonlinear FEM based on the strict mathematical formula derivation and finite displacement theory was produced. The cable element and tangent stiffness matrix are the basis of the FEM. At present, there are three main approaches to simulate cable elements: rod element [35], multi-node curved cable element [36], and two-node catenary element [37; 38]. The rod element method divides the cable into many hinged rod elements to simulate the curvy shape of the cable. This approach not only has a low calculation accuracy, but it also cannot capture the actual cable shape well. Due to a large number of nodes, the program processing of the multi-node curved element method is multifaceted, and obtaining the explicit stiffness matrix is challenging. The catenary element approach, which has few nodes and can simulate any stress level and length of the suspension cable with high accuracy, can be utilized to assess the explicit stiffness matrix. As a result, this element has emerged as the top option for the shape-finding calculations of the main cable of suspension bridges [39]. Since the catenary element was first used to simulate the suspension cable catenary, many investigators have improved and refined the catenary approach, and have exploited that in the analysis of the suspension bridge's cable. (1) The optimization of calculation methods  
The tangential stiffness matrix of the catenary cable element was derived from the exact analytical formula to obtain the cable shape and node force. The geometric state and internal force state of the whole cable could be determined using the known stress-free length and node coordinates, thus simplifying the shape-finding analysis of the cable structure. Figure 4: Flow chart for shape-finding, considering saddles. The target configurations under dead loads (TCUD) methodology originally defined by [PERSON] et al. [40] is a classical method for shape-finding of the main cable of a suspension bridge, which uses the target configuration of the structure under dead loads to solve the linear equilibrium equation of the cable element, with the node coordinates and the length of the strain-free element as unknown quantities. Some scholars have solved the problem where the determination of the main cable shape of the self-anchored suspension bridge is affected by the axial deformation of the bridge tower and the beam by improving the TCUD method [16; 41], and later it has been extended to solve the main cable shape of the 3D suspension bridge [42]. [PERSON] et al. [43] established the equilibrium equation of each segment force of the main cable and comprehended the 3D cable's shape-finding by employing the coordinate and matrix iteration approach. Furthermore, formulations of geometrically nonlinear total Lagrangian-based FEM [44] that can be applied to 3D and 2D cables were displayed. A representative application revealed that this approach was successful in both static and transient analyses. Some scholars have also conducted a series of investigations to enhance computational efficiency. A nonlinear FEM and interaction matrix-based analysis methodology was developed by [PERSON] et al. [45], which was able to reasonably predict suspension bridge shapes conveniently and efficiently. For the main cable system's nonlinear analysis under the static action, [PERSON] et al. [46] established a catenary element. Based on the accurate analytical expression of the elastic catenary, the element's internal forces in each direction, as well as the tangent stiffness matrix, were explicitly determined. The penalty factor search algorithm was also introduced to improve on the calculation accuracy and the convergence rate. Based on the elastic catenary theory, a novel method for determining the main cable's shape of a suspension bridge was developed by [PERSON] et al. [47].