## 5 Conclusions  
In this study, aiming to find an efficient solution to calculating the main cable curve of a double-tower three-span steel-box girder suspension bridge, we established a nonlinear solution equation based on the theory of the segmented catenary method and the equations were formulated based on the IPSO method to obtain the node coordinates of the main cable curve. The results were compared with the traditional N-R iteration method and FEM. The following main conclusions were made:  
1. As for the calculation accuracy, the proposed IPSO method shows similar accuracy when solving the multi-segment catenary nonlinear equations compared with the traditional N-R iteration method and FEM.
2. The N-R iteration method might take a long time and have a strong dependence on the initial value. Ensuring an equivalent solution accuracy, the proposed method based on the IPSO method shows a faster convergence speed. Moreover, when the case is more complicated, the proposed method more easily converges.
3. In the current work, the curve-finding problem is only a two-dimensional problem. This is because, in most three-dimensional suspension bridges, the main cable itself is  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{2}{c}{**Computing Time (s)**} \\ \cline{2-3}  & **IPSO Method** & **N-R Iteration Method** \\ \hline O & 35 & 50 \\ I & 37 & 53 \\ II & 32 & 60 \\ III & 34 & 49 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Computing time between the IPSO and N-R iteration methods for four cases.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Initial Horizontal Force**} & \multicolumn{2}{c}{**Computing Time (s)**} \\ \cline{2-3}  & **IPSO Method** & **N-R Iteration Method** \\ \hline \(1.5\times H_{design}\) & Yes & Yes \\ \(2.0\times H_{design}\) & Yes & Yes \\ \(5.0\times H_{design}\) & Yes & No \\ \(10.0\times H_{design}\) & Yes & No \\ \hline \hline \end{tabular}
\end{table}
Table 5: Convergence between the IPSO and N-R iteration methods for different hanger forces.  
a two-dimensional curve, which can be easily transformed into a two-dimensional situation through coordinate transformation. Therefore, two-dimensional problems are statistically and comparatively analyzed to evaluate the convergence of IPSO. However, with the rapid development of bridges with long spans, there are many other engineering problems that are three-dimensional problems and cannot be simplified. In the future, statistical analyses and mathematical research on bridge engineering based on the IPSO method will be considered.  
Conceptualization, Z.W.; methodology, [PERSON] and X.S.; software, [PERSON]; validation, M.S. and M.L.; formal analysis, [PERSON]; investigation, P.H.; writing--original draft preparation, [PERSON] and M.S.; writing--review and editing, [PERSON] and M.S.; visualization, X.S. and J.W.; supervision, X.S.; project administration, Z.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China (Grant No. 52108464) and the Professional Construction Support Plan of Sichuan Agricultural University (Grant No. 2021997791). The authors are grateful to the members of this paper for providing invaluable information and suggestions for this study.  
The authors declare no conflict of interest.