## 4 Analysis of Main Cable Curve of Suspension Bridge
### Calculation Accuracy  
The IPSO method in Section 3 was introduced in the calculation of Equations (16)-(19). In the iteration, the spacing between hangers, the natural load of the bridge, and other initial parameters were taken as known quantities, and the completed coordinate value of the main cable was taken as the unknown quantity. The learning factors were all set as 2, the maximum iteration step number was 5000, \(\omega_{\text{max}}=0.9\), \(\omega_{\text{min}}=0.4\), and the convergence criterion was \(f(\mathbf{x})<1.0\times 10^{-6}\). Due to focus of the paper, only the results of the control points on the main span of the bridge in the completed state are listed in Table 1, where the results are the average values after 50 calculation steps performed by the software MATLAB. (1)  
Results compared with N-R iterative method  
In practical engineering, designers often use the N-R method to find the shape of the main cable of a suspension bridge. Therefore, in this section, the traditional N-R method is used to iteratively solve Equations (16)-(19); the method was presented in detail by [PERSON] and [PERSON] (2013) [31]. A comparison of the main cable curves between the proposed IPSO method and N-R iterative method is shown in Figure 6, and the differences are presented in Figure 7. The maximum difference between the two results is less than 1 cm, which can be regarded as meeting the engineering demand. In order to verify the calculation accuracy of the IPSO method further, based on practical engineering, the three additional cases listed in Table 2 were compared for their rise-span ratios, hunger forces, and area of main cable. In Table 2, \(F_{design}\) stands for the design force of the hanger. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Point Number** & **Position** & **\(\mathbf{x(m)}\)** & \(\mathbf{z(m)}\) \\ \hline
1 & South side of the top of the tower & 0.0 & 211.625 \\
2 & 1/16 point of main span (1) & 72.5 & 181.204 \\
3 & 1/8 point of main span (1) & 145.0 & 154.931 \\
4 & 1/16 point of main span (2) & 217.5 & 132.809 \\
5 & 1/4 point of main span (1) & 290.0 & 114.751 \\
6 & 1/16 point of main span (3) & 362.5 & 100.737 \\
7 & 1/8 point of main span (2) & 435.0 & 90.737 \\
8 & 1/16 point of main span (4) & 507.5 & 84.742 \\
9 & Mid-span of main span & 580.0 & 82.732 \\
10 & 1/16 point of main span (5) & 652.5 & 84.736 \\
11 & 1/8 point of main span (3) & 725.0 & 90.737 \\
12 & 1/16 point of main span (6) & 797.5 & 100.736 \\
13 & 1/4 point of main span (2) & 870.0 & 114.751 \\
14 & 1/16 point of main span (7) & 942.5 & 132.807 \\
15 & 1/8 point of main span (4) & 1015.0 & 154.930 \\
16 & 1/16 point of main span (8) & 1087.5 & 181.200 \\
17 & North side of the top of the tower & 1160.0 & 211.625 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Curve coordinates of the main cable bridge by the IPSO method. Figure 5: Layout of the Yangtze River Bridge. It can be seen from Table 3 that the results obtained by the IPSO and N-R iteration methods are basically consistent. From Figure 7 and Table 3, under these four cases, the maximum calculated difference between the IPSO method, and the N-R iteration method is 0.008%, and the difference ranges from \(-\)0.008% to 0.003%. It can be considered that the results produced by the IPSO method are basically consistent with those of the N-R iteration method. (2) Results compared with FEM \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case Number** & **Rise–Span Ratio** & **Area of Main Cable** & **Hanger Force** \\ \hline O (Practical engineering) & \(1/9\) & 0.321 m\({}^{2}\) & \(F_{design}\) \\ I & \(1/11\) & 0.321 m\({}^{2}\) & \(F_{design}\) \\ II & \(1/9\) & 0.321 m\({}^{2}\) & \(1.1\times F_{design}\) \\ III & \(1/9\) & 0.350 m\({}^{2}\) & \(F_{design}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Four cases. Figure 6: Shape-finding calculation results by the IPSO method and the N-R iterative method. Figure 7: Difference between main cable curves by the IPSO method and the N-R iterative method.