# 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
[25] developed a novel coordinate system approach based on the overall mechanical analysis to assess the self-anchored suspension bridge's cable shape, which had a good convergence speed and calculation accuracy. This methodology took the main cable as a whole for stress analysis. It was assumed that the original cable shape would be a straight line; the reaction forces were calculated, and then the iteration process calculations were carried out. After that, a modified force density method [26] was suggested for determining the main cable's shape, and the structural balancing equation was exploited to evaluate the coordinates of the main cable nodes. The method had stable convergence and a high computational efficiency. At present, the shape-finding calculation has high calculation accuracy, which can meet the engineering requirements, and the calculation efficiency attracted more and more attention. Therefore, some scholars have greatly improved on the computational efficiency by introducing intelligent optimization algorithms with the help of the powerful computing power of computers. An improved particle swarm optimization (IPSO) algorithm proposed by [PERSON] et al. [27] can be exploited for the calculation of the main cable shape subjected to various initial values, complex boundaries, and load conditions. The developed methodology was able to solve the difficulty of the initial value setting and the calculation convergence of the traditional Newton-Raphson method. The computational efficiencies of four different methods, namely, genetic algorithm (GA), particle swarm optimization (PSO), GA-PSO, and IPSO, were also compared. Figure 2 presents that IPSO has a higher computational accuracy and efficiency than other ones. [PERSON] et al. [28; 29] also established two efficient shape-finding methods, namely, the explicit analytical iteration method and the enhanced particle swarm optimization (EPSO). The harmony search (HS), GA, PSO, and the proposed EPSO are the four approaches that are being compared in Figure 3 for their computational effectiveness. Commonly, the EPSO is the most effective strategy, saving more than 60% in computing time when compared to the other three approaches. Figure 2: Comparison of the four methods’ convergence rates. The refined analysis  
However, most of the above studies did not consider the influence of the geometric relationship between the saddle and the main cable on the main cable shape. An accurate analysis of the tangent relationship between the saddle and the main cable is necessary for shape-finding. Therefore, a novel approach for computing the suspension bridge's saddle pre-deviation was proposed by [PERSON] et al. [30]. This method modified the stress-free cable length by taking into account the influence of various tangent points, non-contact segments, and cable-saddle friction. Finally, the method's reliability was verified via the model test. By considering the saddle's impact on the length of the main cable, the main cable curve correction algorithm at the saddle was derived by [PERSON] et al. [31] from the perspective of the geometric relationship. To this end, the obtained binary nonlinear equations were analyzed via the Newton-Raphson method. [PERSON] et al. [32] developed a high-precision calculation method for the finished bridge alignment of the pinned cable clip suspension bridge. The main cable and hanger's stress-free length were precisely analyzed using this approach, which considered the tangent contact condition and the friction impact between the main cable and the saddle, as well as the influence of the clamp on the main cable and hanger. By utilizing the dichotomy method, [PERSON] et al. [33] proposed an enhanced algorithm that had great convergence for locating the saddle position of a composite curve. Additionally, a methodology [34] was established for addressing the suspension bridge's target configuration while taking the saddle effect into account.