# 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  
Based on the structural design documents, the elastic modulus and support length are the variables used to update the numerical models. Since concrete has a compressive strength of 40 MPa, the initial value of the elastic modulus was calculated to be 35 GPa. The initial value of the support length was set as the value obtained by subtracting the elastic bearings from the total span length. These initial values do not affect the updated final results even if they are set as arbitrary values because they are used to shorten the numerical analysis updating process. Tables 5-7 show the process of performing numerical model updating by using the dynamic properties estimated from the ambient vibration test of each target bridge as target values. Numerical model updating was performed until the relative error of Equation (7) was 1% or less. Consequently, the vertical and torsional modes estimated from the measured data differed by less than 5% from those of the updated numerical model. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Support Length**} & \multicolumn{2}{c}{**Effective Length of Mode**} & \multirow{2}{*}{**Natural Frequency (Hz)**} & \multirow{2}{*}{**Elastic**} & \multicolumn{2}{c}{**Effective Length of Mode**} & \multirow{2}{*}{**Natural Frequency (Hz)**} \\ \cline{2-2} \cline{5-10}  & & **Shape (m)** & & & & **Modulus of** & & **Shape (m)** & \\ \cline{2-2} \cline{5-10}  & **Vertical** & **Torsional** & **Vertical** & **Torsional** & **Concrete** & **Vertical** & **Torsional** & **Vertical** & **Torsiona** \\ \cline{2-2} \cline{5-10}  & **(4.96 Hz)** & **(25.64 Hz)** & **(4.96 Hz)** & **(25.64 Hz)** & **(GPa)** & **(4.96 Hz)** & **(25.64 Hz)** & **(4.96 Hz)** & **(25.64 Hz)** \\ \hline
40 & 42.985 & 40.995 & 4.314 & 19.686 & 40 & 39.775 & 38.587 & 4.918 & 20.273 \\
39.2 & 40.846 & 39.343 & 4.625 & 19.844 & 42 & 39.773 & 38.583 & 5.033 & 20.726 \\
38.4 & 39.776 & 38.590 & 4.829 & 19.927 & 44 & 39.772 & 38.880 & 5.145 & 21.169 \\
37.6 & 38.894 & 37.984 & 5.019 & 19.997 & 46 & 39.770 & 38.577 & 5.255 & 21.603 \\
36.8 & 38.066 & 37.397 & 5.208 & 20.068 & 48 & 39.769 & 38.574 & 5.363 & 22.028 \\
36 & 37.268 & 36.816 & 5.400 & 20.138 & 50 & 39.768 & 38.571 & 5.469 & 22.445 \\ constant & \multicolumn{2}{c}{elastic modulus of concrete: 35 GPa} & \multicolumn{2}{c}{constant} & \multicolumn{2}{c}{support length: 38.4 m} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Sensitivity analysis results for the numerical model of the Imgi 2 nd Bridge. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Parameter**} & \multicolumn{4}{c}{**Dynamic Property**} \\ \cline{2-9} \cline{5-9}
**Number of** & \multirow{2}{*}{**Support Length (m)**} & **Elastic** & \multirow{2}{*}{**Number of**} & \multicolumn{2}{c}{**Natural Frequency (Hz)**} & \multicolumn{2}{c}{**Effective Span Length (m)**} \\ \cline{5-9}  & & **Modulus** & & **Result** & **Vertical** & **Torsional** & **Vertical** & **Torsional** \\ \cline{5-9}  & & **(GPa)** & & & & & & \\ \hline initial & 38.5 & 35.00 & 1 & 4.32 & 21.21 & 39.63 & 38.29 \\
2 & 39.4 & 46.91 & 2 & 4.68 & 24.26 & 41.17 & 38.93 \\
3 & 39.0 & 52.57 & 3 & 5.07 & 25.63 & 40.38 & 38.50 \\
4 & 39.2 & 50.55 & 4 & 4.93 & 25.15 & 40.56 & 36.68 \\
5 & 39.4 & 51.42 & 5 & 4.90 & 25.34 & 40.98 & 37.39 \\
6 & 39.4 & 52.77 & 6 & 4.96 & 25.65 & 40.98 & 37.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Numerical model updating process of the Imgi 2 nd Bridge. Since the traveling speeds of high-speed trains differ from each target bridge, each traveling speed must be applied in order to accurately compare the measured and analyzed signals. The traveling speeds of high-speed trains were estimated by using the beating frequency included in the measured acceleration response.