## 3 Results  
### Structural Analysis Results  
The eigenvalue analysis results indicated that the natural frequency of the first bending mode for the reference model was 7.94 Hz. The natural frequencies for each stiffness reduction model are summarized in Table 3. The stiffness reduction rates for each model can be calculated from the obtained natural frequencies using Equation (2). The stiffness reduction rates ranged from 0% for the reference model to a maximum of 54.19%. The deflection values for each stiffness reduction model are listed in Table 4, showing a gradual increase in deflection due to the reduction in stiffness. The deflection response ratios for each stiffness reduction model were obtained from these deflection values using Equation (1).  
The correlation between the deflection response ratios and stiffness reduction rates was evaluated using the deflection response ratios and stiffness reduction rates listed in Tables 3 and 4, as shown in Figure 14. In addition, regression analysis was performed based on the values for each model, and the regression equations obtained for each load case are listed in Table 5. The coefficient of determination (\(R^{2}\)) for each load case was greater than 0.997, indicating a very high correlation between the deflection response ratio and stiffness reduction rate. Accordingly, the field measurement data were used to analyze and evaluate the bridge damage.  
Figure 13: Loading tests applied to the target bridge.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multirow{2}{*}{**Classification**} & \multirow{2}{*}{**1.00 \(\mathbf{h}\)**} & \multicolumn{8}{c}{**Damage Model**} \\ \cline{3-10}  & & & **0.95 \(\mathbf{h}\)** & **0.90 \(\mathbf{h}\)** & **0.85 \(\mathbf{h}\)** & **0.80 \(\mathbf{h}\)** & **0.70 \(\mathbf{h}\)** & **0.60 \(\mathbf{h}\)** \\ \hline LC1 & Deflection (mm) & 0.42 & 0.46 & 0.51 & 0.57 & 0.63 & 0.81 & 1.06 \\ LC3 & Deflection response ratio (\%) & 1.00 & 1.10 & 1.22 & 1.36 & 1.52 & 1.94 & 2.53 \\ \hline LC2 & Deflection (mm) & 0.34 & 0.38 & 0.42 & 0.47 & 0.53 & 0.68 & 0.89 \\  & Deflection response ratio (\%) & 1.00 & 1.11 & 1.23 & 1.37 & 1.54 & 1.98 & 2.61 \\ \hline LC4 & Deflection (mm) & 0.56 & 0.62 & 0.68 & 0.75 & 0.83 & 1.05 & 1.37 \\ LC6 & Deflection response ratio (\%) & 1.00 & 1.10 & 1.21 & 1.33 & 1.48 & 1.87 & 2.43 \\ \hline LC5 & Deflection (mm) & 0.48 & 0.52 & 0.58 & 0.64 & 0.71 & 0.90 & 1.18 \\  & Deflection response ratio (\%) & 1.00 & 1.10 & 1.21 & 1.34 & 1.50 & 1.90 & 2.48 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Deflection response ratio obtained from the structural analysis results.  
Figure 14: Correlation between deflection response ratio and stiffness reduction rate from structural analysis for (**a**) LC1, LC3; (**b**) LC2; (**c**) LC4, LC6; (**d**) LC5.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Classification**} & \multirow{2}{*}{**1.00 \(\mathbf{h}\)**} & \multicolumn{8}{c}{**Damage Model**} \\ \cline{3-10}  & & & **0.95 \(\mathbf{h}\)** & **0.90 \(\mathbf{h}\)** & **0.85 \(\mathbf{h}\)** & **0.80 \(\mathbf{h}\)** & **0.70 \(\mathbf{h}\)** & **0.60 \(\mathbf{h}\)** \\ \hline Natural frequency, Hz & 7.94 & 7.63 & 7.31 & 6.98 & 6.66 & 6.02 & 5.38 \\ Stiffness reduction rate (\%) & 0.00 & 7.85 & 15.42 & 22.69 & 29.63 & 42.60 & 54.19 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Natural frequencies and stiffness reduction rates obtained from the structural analysis results.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Classification** & **Regression Analysis Equation** & \(\mathbf{R^{2}}\) \\ \hline LC1, LC3 & \(y=0.0004x^{2}+0.0047x+1.0247\) & 0.9973 \\ LC2 & \(y=0.0004x^{2}+0.0047x+1.0258\) & 0.9973 \\ LC4, LC6 & \(y=0.0004x^{2}+0.0041x+1.0258\) & 0.9972 \\ LC5 & \(y=0.0004x^{2}+0.0042x+1.0239\) & 0.9973 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Regression analysis equation for each load case.