# An Improved Analytical Algorithm on Main Cable System of Suspension Bridge
## 5 Conclusions  
* It is theoretically proved that there is no solution for calculating the main cable system in main span or side span under certain loading conditions.
* By introducing the search algorithm and penalty factor, a numerical analysis method was improved to overcome the problem of no solution under certain loading conditions, and to develop the segmental catenary theory.
* The necessity and effectiveness of the improved analytical method were described by the theoretical calculation results and numerical examples. The program using proposed method has been successfully applied in shape finding during design and configuration control during construction of main cable system for suspension bridges in China.  
Conceptualization, methodology, software and writing, C.L., J.H. and H.K.; Investigation and validation, C.D. and H.L.; Resources, supervision and revision, C.L., Z.Z. and Y.L.  
This research was funded by [National Natural Science Foundation of China] grant number [51778069 and 51308070], [National Basic Research Program of China] grant number [973 Program, No. 2015 CB057702], [Key Discipline Fund Project of Civil Engineering of Changsha University of Sciences and Technology] grant number [13 ZDX04, 13 KA04].  
The authors acknowledge funding from the National Natural Science Foundation of China (Nos. 51778069, 51308070), National Basic Research Program of China (973 Program, No. 2015 CB057702), Key Discipline Fund Project of Civil Engineering of Changsha University of Sciences and Technology (13 ZDXK04, 13 KA04). We thank the reviewers and the editor for the valuable comments and suggestions that helped us improve the manuscript.  
The authors declare no conflicts of interest.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Node**} & \multicolumn{4}{c}{\(x\)} & \multicolumn{4}{c}{\(y\)} \\ \cline{2-7}  & **Method 1** & **Method 2** & **Method 3** & **Method 1** & **Method 2** & **Method 3** \\ \hline
**1** & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 \\
**2** & 30.995 & 30.996 & 30.995 & \(-9.641\) & \(-9.638\) & \(-9.640\) \\
**3** & 61.389 & 61.391 & 61.389 & \(-18.595\) & \(-18.589\) & \(-18.595\) \\
**4** & 91.357 & 91.356 & 91.357 & \(-26.942\) & \(-26.945\) & \(-26.942\) \\
**5** & 121.075 & 121.075 & 121.075 & \(-34.748\) & \(-34.748\) & \(-34.748\) \\
**6** & 151.276 & 151.276 & 151.276 & \(-30.242\) & \(-30.243\) & \(-30.242\) \\
**7** & 181.404 & 181.404 & 181.404 & \(-25.275\) & \(-25.277\) & \(-25.275\) \\
**8** & 211.641 & 211.641 & 211.641 & \(-19.818\) & \(-19.817\) & \(-19.818\) \\
**9** & 242.166 & 242.166 & 242.166 & \(-13.825\) & \(-13.824\) & \(-13.825\) \\
**10** & 273.159 & 273.159 & 273.159 & \(-7.241\) & \(-7.240\) & \(-7.241\) \\
**11** & 304.800 & 304.800 & 304.800 & 0.000 & 0.000 & 0.000 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Node coordinates of cable segments under applied load (unit: m).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Element**} & \multicolumn{3}{c}{**Tension Force**} \\ \cline{2-4}  & **Method 1** & **Method 2** & **Method 3** \\ \hline
1 & 94.41 & 94.40 & 94.40 \\
2 & 93.98 & 94.00 & 94.00 \\
3 & 93.58 & 93.60 & 93.70 \\
4 & 93.21 & 93.20 & 93.30 \\
5 & 91.15 & 91.20 & 91.20 \\
6 & 91.37 & 91.40 & 91.40 \\
7 & 91.61 & 91.60 & 91.70 \\
8 & 91.87 & 91.90 & 91.90 \\
9 & 92.16 & 92.20 & 92.20 \\
10 & 92.48 & 92.50 & 92.50 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Tension force of cable segments under applied load (unit: kN).