# An Improved Analytical Algorithm on Main Cable System of Suspension Bridge
## 4 Numerical Examples
### Example 2  
A single span flexible cable that is fixed at both ends subjected to multiple concentrated loads is adopted as an example, as shown in Figure 3, to compare the analytical results from the proposed algorithm with that from other methods. The node coordinates and unstressed lengths of cable segments at initial state are shown in Tables 4 and 5, respectively. In addition, the cross-sectional area of cable is 5.48386 \(\times\) 10\({}^{-4}\) m\({}^{2}\), and the elastic modulus is 13,1473.43 MPa, the weight of unit length cable is 47.02594 kN/m.  
The configuration and tension force of cable at the equilibrium state under applied load were calculated by different methods, including method 1: improved analytical algorithm in present study; method 2: finite element method in Ref. [41]; and, method 3: traditional analytical method in Ref. [42]. The segmental catenary theory is adopted for both method 1 and 3, however, method 1 uses the search algorithm with penalty factor, while method 3 uses traditional Newton-Raphson iteration algorithm. The comparison of cable configuration and internal forces are shown in Tables 6 and 7, respectively. It can be found that the note coordinates and tension force of cable under applied load calculated from improved analytical algorithm agree well with that from method 2 and 3, the maximum difference of node coordinate between method 1 and 2 is 6 mm (\(y\) of node 3) with relative error of 0.03%, and 1 mm (\(y\) of node 2) between method 1 and 3 with relative error of 0.01%; the maximum difference of the cable tension force between method 1 and 2 is 0.05 kN (element 5) with a relative error of 0.06%, and 0.12 kN (element 3) between method 1 and 3 with relative error of 0.1%. In compression of method 2, the initial node coordinates of cable segments are not necessary for the proposed algorithm to calculate the configuration and tension force of cable at equilibrium state. In comparison to method 3, the initial value is not sensitive to solve cable segment equations for the proposed algorithm especially under the conditions of asymmetric and uneven loads, and also the number of iterations is significantly reduced, resulting in faster convergence speed.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Node** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline \(x\) & 0.0 & 30.48 & 60.96 & 91.44 & 121.92 & 152.40 & 182.88 & 213.36 & 243.84 & 274.32 & 304.80 \\ \(y\) & 0.0 & \(-\)11.0642 & \(-\)19.5986 & \(-\)25.6565 & \(-\)29.2760 & \(-\)30.4800 & \(-\)29.2760 & \(-\)25.6565 & \(-\)19.5986 & \(-\)11.0642 & 0.0 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Node coordinates of cable segments at initial state (unit: m).  
Figure 3: A single span cable subjected to multiple concentrated loads.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Element** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline
**Unstressed Length** & 32.4175 & 31.6441 & 31.0683 & 30.6865 & 30.4962 & 30.4962 & 30.6865 & 31.0683 & 31.6441 & 32.4175 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Node coordinates of cable segments at initial state (unit: m).