## 3 Numerical Example  
As shown in Figure 2, an FEM of the single tower cable-stayed bridge named \"Guandu Bridge\" is provided as an example to verify the proposed damage identification method. The main physical parameters of this bridge are listed in Table 1. The relations between the element numbers and the node numbers of this bridge are listed in Table 2.  
Figure 1: The flowchart of the whole technique. Figure 2: A single tower cable-stayed bridge. Two static loading modes with different loading positions were used to obtain the displacement data. The first static loading mode applied two vertical forces on the 67 and 70 DOFs. The second static loading mode applied two vertical forces on the 112 and 115 DOFs. Three damage conditions were simulated in this work. Damage condition 1 assumed a single damage appearing in cable element 31 (long cable) with a stiffness loss of 15%. Damage condition 2 assumed cable element 15 (short cable) was damaged with a stiffness reduction of 15%. Damage condition 3 assumed that cable elements 10 and 30 were both damaged at the same time with a stiffness reduction in both of 20%. For the first damage case, the proposed damage indicator vector \(\gamma\) is shown in Figure 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Elements** &
\begin{tabular}{c} **Elastic** \\ **Modulus** \\ \end{tabular} & **Density** & **Cross Sectional Area** & **Moment of Inertia** \\ \hline Cable & 200 GPa & 7800 kg/m\({}^{3}\) & 4.657 \(\times\) 10\({}^{-3}\) m\({}^{2}\) & / \\ Pylon & 200 GPa & 7800 kg/m\({}^{3}\) & 0.31 m\({}^{2}\) & 0.437 \\ Box girder & 200 GPa & 7800 kg/m\({}^{3}\) & 0.75 m\({}^{2}\) & 1.26 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Physical parameters of the single tower cable-stayed bridge. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Cable Element Number** & **Corresponding Nodes** & **Corresponding DOFs** \\
**(From Left to Right)** & **Corresponding Nodes** & **Corresponding DOFs** \\ \hline
1 & 1, 17 & 1, 2, 3, 49, 50 \\
2 & 2, 18 & 4, 5, 6, 51, 52 \\
3 & 3, 19 & 7, 8, 9, 53, 54, 55 \\
4 & 4, 20 & 10, 11, 12,  
From Figure 3, DOFs 4, 5, 6, and 137 are the identified damaged DOFs, since their corresponding indicator coefficients were noticeably larger than the other coefficients. According to Table 1, cable 31 was the associated component for these damaged DOFs. By using Equation (19), the damage extent of cable 31 was obtained as \(\alpha_{31}=0.15\). For the second damage case, the calculated damage indicator vector is shown in Figure 4. It is clear from Figure 4 that DOFs 43, 44, 45, and 89 are the identified damaged DOFs, for their indicator coefficients were considerably larger than the other coefficients. According to Table 1, cable element 15 was the corresponding damaged element. The damage extent was calculated using Equation (19) to be \(\alpha_{15}=0.15\). For the third damage case, the calculated damage indicator vector is shown in Figure 5. It is clear from Figure 5 that DOFs 7, 8, 9, 28, 29, 30, 74, and 134 are the identified damaged DOFs, for their indicator coefficients were obviously larger than the other coefficients. According to Table 1, cable elements 10 and 30 were the corresponding damaged components. For this multiple-damage case, the damage extents were calculated using Equation (20) to be \(\alpha_{10}=0.2\) and \(\alpha_{30}=0.2\) for cables 10 and 30, respectively. Figure 4: Damage indicator vector when short cable 15 was damaged. Figure 5: Damage indicator vector when cables 10 and 30 were damaged. Figure 3: Damage indicator vector when long cable 31 was damaged. From the above results, it was shown that the proposed method had good identification accuracy for both short and long cable damage cases. The proposed method successfully determined the damaged cable location and extent, even if the static load was unknown. Next, a comparative study is carried out on the calculation accuracy between the proposed method and the displacement sensitivity method.