# Damage Detection in Beam Bridges Using Quasi-static Displacement Influence Lines
## 3 Numerical Example
### Damage Detection Performance of the Proposed Method  
With the numerical model described in the preceding section, 45 different cases are considered and two types of quasi-static loads are utilized; specifically, 100-kN quasi-static loads are applied in the first 36 cases and 50-kN loads are applied in the other 9 cases. For the first 36 cases, three different damage extents, namely, 5%, 10% and 15%, are implemented and 9 different damage locations are considered for each degree of damage. For the last 9 cases, only an extent of 5% is simulated and the same 9 damage locations are employed. For each case, a total of 9 displacement ILs are obtained and one metric is calculated using the data of those ILs. The damage detection results of all cases are listed in Table 2. A comparison of the metrics among the different cases is shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Item** & \begin{tabular}{c} **Young’s Modulus of** \\ **C50 Concrete** \\ \end{tabular} & \
\begin{tabular}{c} **Bending Moment of** \\ **Inertia** \\ \end{tabular} &
\begin{tabular}{c} **Torsional Moment of** \\ **Inertia** \\ \end{tabular} \\ \hline Value of parameter & \(3.497\times 10^{7}\) kN/m\({}^{2}\) & \(2.395\times 10^{-2}\) m\({}^{4}\) & \(4.353\times 10^{-2}\) m\({}^{4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters used for the finite element model of the simply supported bridge.  
Figure 4: Comparison of the metric values of the different cases.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Damage** & \begin{tabular}{c} **Extent of** \\ **Damage** \\ \end{tabular} & \begin{tabular}{c} **Metric** \\ **Metric** \\ \end{tabular} & \begin{tabular}{c} **Extent of** \\ **Damage** \\ \end{tabular} & \begin{tabular}{c} **Metric** \\ **Metric** \\ \end{tabular} & \begin{tabular}{c} **Extent of** \\ **Damage** \\ \end{tabular} & \begin{tabular}{c} **Metric** \\ **Metric** \\ \end{tabular} & \
\begin{tabular}{c} **Extent of** \\ **Damage** \\ \end{tabular} &
\begin{tabular}{c} **Metric** \\ **Metric** \\ \end{tabular} \\ \hline
3 m & 0\% & \(9.992\times 10^{-16}\) & 15\% & \(4.306\times 10^{-4}\) & 10\% & \(2.701\times 10^{-4}\) & 5\% & \(1.275\times 10^{-4}\) \\
6 m & 0\% & \(1.129\times 10^{-15}\  
As shown in Figure 4, the following results are obtained. The value of the damage feature metric obtained by Equation (12) increases for the same damage location as the extent of damage becomes more serious. However, under the same extent of damage, the values of the damage feature metric change as the damage location varies; the closer the damage location is to the supported end, the greater the value of the damage feature metric obtained. When the damage location is at the middle of the span of the main beam of the bridge, 4 of the displacement ILs are similar, whereas when the damage location is near the supported end, almost all 9 displacement ILs exhibit discrepancies. Consequently, more information in the displacement ILs is affected by the damage when the damage exists at the supported end than when the damage exists in the middle span of the main beam. This discrepancy constitutes the main reason for the observed phenomenon. Furthermore, as shown in Figure 4, the metric values of the last two groups (total 18 cases) are identical, which means that the proposed method is independent of the load conditions before and after damage.  
As discussed above, similar to the traditional method, the effectiveness of the proposed method lies in the change in the displacement ILs before and after damage occurs; however, in contrast to the traditional approach, the proposed damage feature based on the correlation coefficient between different displacement ILs has no relation to the difference in the quasi-static weight of the loading vehicle before and after damage. This characteristic is demonstrated in the numerical example illustrated in Figure 4.