# Dynamic Stability Assessment of High-Speed Railway Bridges Using Numerical Model Updating
## 4 Dynamic Stability Assessment for High-Speed Railway Bridges
### Numerical Model Updating and Moving Load Analysis
The speeds obtained were 300 km/h for the Imgi 2 nd Bridge, 275 km/h for the Maeaji Bridge, and 285 km/h for the Hwalchun Bridge. For the moving load analysis, the length, wheel base, and wheel load of a high-speed train passenger car were assumed to be 18.5 m, 15.5 m, and 84,700 kN, respectively. The moving load was applied to the time history analysis of the updated numerical model. It was applied to each node of the rail element over time in the form of a concentrated load, taking into account the speed, wheel base, and wheel load of the high-speed train, and then removed. The damping ratio of 1.66% to 2.88%, which was estimated using the extended Kalman filter in Table 3, was applied to the target bridges. The frequency to apply the low-pass filter was determined to exclude the influence of the beat frequency by the wheel spacing between each passenger car on the interaction between the high-speed train and the bridge. The train has a 3 m distance between the rear wheels of the front passenger car and the front wheels of the rear passenger car. The corresponding frequency is approximately 27 Hz when the train speed is 300 km/h, and 24 Hz when it is 270 km/h. Therefore, 10% to 15% of the corresponding frequency was applied to the low-pass filter for a comparison between the measured response and the analyzed response in the moving load analysis. Figure 8 compares the responses measured at the reference point of each target bridge and the responses analyzed using the numerical models in the time and frequency domains. The analyzed responses were slightly larger than the responses measured in the ambient vibration test; however, they were similar when the noise component was excluded. This indicates that numerical model updating was performed well, and that the numerical analysis based on the updated models can replace field testing of high-speed railway bridges. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Parameter**} & \multicolumn{3}{c}{**Dynamic Property**} \\ \cline{2-7}
**Number of** & \multirow{2}{*}{
\begin{tabular}{c} **Support** \\ **Length (m)** \\ \end{tabular} } & **Elastic** & \multicolumn{2}{c}{**Number of**} & \multicolumn{2}{c}{**Natural Frequency (Hz)**} & \multicolumn{2}{c}{**Effective Span Length (m)**} \\ \cline{3-7}
**Iteration** & **Support** & **Elastic** & \multicolumn{1}{c}{**Modulus**} & **Result** & **Vertical** & **Torsional** & **Vertical** & **Torsional** \\ \cline{3-7}
**Initial** & **24.2** & **35.00** & **1** & **7.70** & **31.74** & **25.57** & **29.23** \\
2 & **23.6** & **51.79** & **2** & **9.54** & **32.00** & **24.56** & **29.66** \\
3 & **23.6** & **51.79** & **3** & **9.44** & **31.64** & **24.39** & **25.99** \\
4 & **23.6** & **43.86** & **4** & **9.44** & **31.64** & **24.43** & **26.58** \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Numerical model updating process of the Maeaji Bridge. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Parameter**} & \multicolumn{3}{c}{**Dynamic Property**} \\ \cline{2-7}
**Number of** & \multirow{2}{*}{
\begin{tabular}{c} **Support** \\ **Length (m)** \\ \end{tabular} } & **Elastic** & \multicolumn{1}{c}{**Number of**} & \multicolumn{2}{c}{**Natural Frequency (Hz)**} & \multicolumn{2}{c}{**Effective Span Length (m)**} \\ \cline{3-7}
**Algorithm** & **Support** & **Elastic** & \multicolumn{1}{c}{**Modulus**} & **Result** & **Vertical** & **Torsional** & **Vertical** & **Torsional** \\ \cline{3-7}
**Initial** & **24.2** & **35.00** & **1** & **7.70** & **31.74** & **25.57** & **29.23** \\
2 & **23.6** & **51.79** & **2** & **9.54** & **32.00** & **24.56** & **29.66** \\
3 & **23.6** & **51.79** & **3** & **9.44** & **31.64** & **24.39** & **25.99** \\
4 & **23.6** & **43.86** & **4** & **9.44** & **31.64** & **24.43** & **26.58** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Numerical model updating process of the Hwalchun Bridge.