# Cable Force Identification of Two Bending Cable Networks with Arbitrary Boundary Conditions
## 4 Application of Cable Force Identification Method
### A Two-Cable Network with Hinged-End Supports  
In this case study, two cables have similar cross-sections and materials but vary in cable lengths and tensions. The cable properties are selected from the cable network of the stay cables on the Fred Hartman Bridge [14]. The flexible cross-tie is located at the position of \(0.400L_{1}\) and \(0.429L_{2}\) from the left-side supports of Cables 1 and 2, respectively. The boundary conditions of the two cables are hinged ends, as shown in Figure 6.  
Step 1: All necessary information on the cable network is listed as follows:  
\[\left\{ \begin{array}{l}T_{1}=1900\text{~{}kN};~{}T_{2}=1596\text{~{}kN}\\ E_{1}=E_{L}=2700\text{~{}kN}\text{~{}m}^{2}\\ \overline{m}_{1}=\overline{m}_{2}=47.9\text{~{}kN}\text{~{}m}\\ K_{c}=4487.5\text{~{}kN}\text{~{}m}^{-1}\end{array} \right. ;\left\{\begin{array}{l} L_{1}=76.55\text{~{}m}\\ L_{2}=67.34\text{~{}m}\end  
Step 2 + 3: The cable network model is created in ABAQUS to obtain measured frequencies at the first 10 modes, as shown in Table 1.  
Step 4: In the following step, the predicted cable forces are calculated using Equation (15). From the observation of many study cases of two-cable networks, we found that the fundamental natural frequency of the cable network is usually closest to the fundamental natural frequencies of the single Cables 1 and 2. Hence, we use the fundamental measured frequency of the cable network to compute the predicted range of cable forces 1 and 2. For higher modes, the mode order is selected by the smallest difference between the cable forces obtained from the fundamental measured frequency and those obtained from the other measured frequencies and the corresponding mode order. For example, the second measured frequency of the cable network, i.e., 2.2564 Hz, is used to calculate the predicted cable force of single Cable 1. If the mode order equal to 1 is selected, the predicted cable force of the single Cable 1 is 5711.79 kN. If the mode order equal to 2 is selected, the predicted cable force of the single Cable 1 is 1410.90 kN. Since the cable force difference of 1974.93 kN and 5711.79 kN is larger than that of 1974.93 kN and 1410.90 kN, the selected mode order is 2. Higher modes follow the same procedure. The predicted value of cable forces is shown in Table 2.  
By applying the _MAD_ technique, the detected ranges of the cable forces are from 1498.83 kN to 2355.9 kN for the first single cable and from 1022.23 kN to 1995.03 kN for the \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Mode Number** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** \\ \hline Measured Frequency (Hz) & 1.3278 & 2.2564 & 2.6883 & 3.2030 & 4.0023 & 4.6050 & 5.4582 & 6.5048 & 6.6917 & 7.2340 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measured frequencies of the cable network with hinged-end support determined by ABAQUS. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Cable Network** & \multicolumn{3}{c}{**Single Cable 1**} & \multicolumn{3}{c}{**Single Cable 2**} \\ \hline
**Mode** & **Measured** & **Mode** & **–** & **Mode** & **–** \\
**Number** & **Frequency (Hz)** & **Number** & **\(\mathbf{T_{1}}\) (kN)** & **Number** & **\(\mathbf{T_{2}}\) (kN)** \\ \hline
1 & 1.3278 & 1 & 1974.93 & 1 & 1525.94 \\
2 & 2.2564 & 2 & 1410.90 & 2 & 1082.39 \\
3 & 2.6883 & 2 & 2010.34 & 2 & 1546.27 \\
4 & 3.2030 & 3 & 1238.92 & 3 & 937.52 \\
5 & 4.0023 & 3 & 1957.38 & 3 & 1493.50 \\
6 & 4.6050 & 4 & 1415.32 & 3 & 1994.30 \\
7 & 5.4582 & 4 & 2017.81 & 4 & 1523.76 \\
8 & 6.5048 & 5 & 1786.58 & 5 & 1323.60 \\
9 & 6.6917 & 5 & 1897.34 & 5 & 1409.32 \\
10 & 7.2340 & 5 & 2236.50 & 5 & 1671.78 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The predicted values of cable forces. Figure 6: Cable network’s configuration with hinged-end supports. second single cable.