## 3 Verification by a Two-Dimensional Finite Element Model
### Extreme Damage Cases and Identification Results  
Eighteen extreme damage scenarios are designated in the FEM, and all damage cases are attributed to cable failure. Table 1 lists all the damage conditions investigated in the FEM. The damage of one single hanger and the simultaneous damage of two hangers were simulated. The damage condition can be divided into two categories. DC1-DC12 is the first category, which belongs to the one single hanger damaged at a time; DC13-DC18 is the second category, considering the damage of two hangers simultaneously. In the finite element model, the complete damage of each hanger (that is, the cable force reduction ratio is 100%) is simulated by deleting the damaged hanger. The deflection difference between the complete damage of each hanger and the healthy state can be obtained from the model. As a result, the deflection difference influence matrix can be obtained, and the result is plotted in Figure 4. It can be seen from Figure 4 that when each hanger is completely damaged separately, the deflection difference vector will reach a clear peak at the damaged hanger. When the damage occurs in the symmetrical position, the deflection difference vector is also symmetrical. In the FEM, the damage degree of the hanger is simulated by changing the cross-sectional area of the hanger. The deflection difference vector at the anchorage point between the hanger and the tie-beam under each damage condition is put forward. Then, the deflection difference vector and the influence matrix of the deflection difference are brought into Equation (9). Under each damage condition, the proportion vector of cable force reduction of each hanger can be obtained. The results are plotted in Figures 5 and 6. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Damage Type** & **Damage Case No.** & **Damage Hanger No.** & **Damage Degree** \\ \hline \multirow{4}{*}{**Single hanger**} & DC 1–3 & N2 & 10\% 20\% 30\% \\  & DC 4–6 & N3 & 10\% 20\% 30\% \\  & DC 7–9 & N4 & 10\% 20\% 30\% \\  & DC 10–12 & N5 & 10\% 20\% 30\% \\  & DC 13–15 & N2 and N4 & 10\% 20\% 30\% \\  & DC 16–18 & N3 and N7 & 10\% 20\% 30\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Eighteen damage conditions simulated by FEM. Figure 4: Deflection change of each anchorage point when N1–N9 is wholly damaged. Figure 3: Diagram of the two-dimensional FEM. From Figure 5, it can be seen that when a single hanger is damaged, the recognition result is perfect. As can be seen from the figure, the reduction ratio of cable force of the damaged hanger is proportional to the degree of damage. The reduction of cable force only occurs in the damaged hanger. Therefore, the damaged hanger of Figure 5a-d can be quickly and easily located, which is N2, N3, S4N4, and N5, respectively. The ratio of cable force reduction and quantitative identification results are shown in Table 2. From Table 2, it can be seen that when a single hanger is damaged, the identification result of the cable force reduction ratio is completely consistent with the actual reduction ratio. However, the reduction ratio of cable force is less than the degree of damage, so this method can only relatively quantify the degree of damage. Figure 5: Identification results for DC1–DC12: (**a**) the preset damage hanger is N2; (**b**) the preset damage hanger is N3; (**c**) the preset damage hanger is N4; (**d**) the preset damage hanger is N5. Figure 6: Identification results for DC13–DC18. Identification results for DC1–DC12: (**a**) the preset damage hangers are N2 and N4; (**b**) the preset damage hangers are N3 and N7. The identification results of simultaneous damage of two hangers are shown in Figure 6. As can be seen from Figure 6a,b, the hangers with reduced cable force are N2 and N4, and N3 and N7, respectively. Therefore, the damaged hangers can be identified as N2 and N4, and N3 and N7, respectively.