## 3 Description of FE Model and Verification
### Modeling Method
The embankment region of the analysis model is assumed as a normal ballasted track in which the typical values for peak longitudinal and lateral resistance (20 kN/m/track and 18 kN/m/track, respectively) with 2 mm of elastic limit were applied. For the vertical direction, the elastic behavior was assumed and a vertical stiffness of 98.9 N/mm/sleeper was used considering the typical stiffnesses of the ballast (=131.4 N/mm/sleeper) and the subgrade (=400 N/mm/sleeper). In this study, the bridge was modeled with a classic beam element. The dimensions for the section of the open-deck steel plate girder bridge are shown in Figure 8a and Table 3. Note that, in Korea, since the span length of an open-deck steel plate girder bridge varies between 9 m and 30 m, bridge sections for three different span lengths (9 m, 15.2 m, and 30 m) were selected from the existing bridge. A typical 3D FE model is depicted in Figure 8b. Using the 3D analysis model, the section properties for the equivalent beam model are shown in Figure 7 and were derived using unit load analysis. The derived section properties, including axial rigidity (_EA_), \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Stiffness** & **Value** & **References** & **Remark** \\ \hline \(K_{x,f}\)(N/m) & 1.4 \(\times\) 10\({}^{7}\) & KRS TR 0014-20R [14] & Perfect elasto-plastic behavior with 7 kN of peak resistance \\ \(K_{y,f}\)(N/m) & 2.6 \(\times\) 10\({}^{7}\) & [PERSON] et al. [15] & Elastic behavior \\ \(K_{z,f}\)(N/m) & 8.0 \(\times\) 10\({}^{7}\) & [PERSON] et al. [16] & Elastic behavior \\ \(R_{x,f}\)(N\({}_{y,f}\)) and \(R_{z,f}\) & 6.0 \(\times\) 10\({}^{4}\) & [PERSON] et al. [15] & Elastic behavior \\ (N\(\cdot\)m/rad) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Rail fastener properties. Figure 8: Open-deck steel plate girder bridge: (**a**) section dimensions and (**b**) 3D modeling. in-plane and out-of-plane bending rigidity (\(EI_{y}\) and \(EI_{z}\)), and torsional rigidity (\(G\)), are summarized in Table 4.  
To invoke the buckling of a CWR track in the numerical simulation, the initial imperfection of the rail has to be included in the FE model. In this study, the shape of the imperfection is assumed to be a half-sine curve, as shown in Figure 9, with a maximum magnitude, \(e_{R}\), of 17 mm for a rail length, \(L_{R}\), of 10 m. Note that the maximum magnitude of the imperfection was set based on rail maintenance practices in Korea [17], where the maximum misalignment in a normal track is provided as 17 mm for a rail length of 10 m. Thus, the initial imperfection of the rail, \(\delta_{R}\), can be expressed as:  
\[\delta_{R}\ =\ e_{R}\sin\left(\frac{\pi x}{L_{R}}\right)\sin\left(\frac{\pi x }{L_{R}}\right) \tag{3}\]  
In general, the open-deck steel plate girder bridge has a low lateral bending stiffness compared with other types of railway bridges, and a considerable lateral vibration can occur when a train passes through. Since the track is directly attached to the girder and the lateral displacement of the girder can amplify the initial imperfection of the rail, the lateral displacement occurring through the lateral vibration can reduce the buckling stress of the CWR track. For this reason, the lateral displacement of the girder is considered in the analysis. The lateral displacement of the girder, \(\delta_{G}\), is defined as:  
\[\delta_{G}\ =\ e_{G}\sin\left(\frac{\pi x}{L_{G}}\right) \tag{4}\]  
where \(L_{G}\) is the length of the girder. \(e_{G}\) is the maximum lateral displacement of the girder at the mid-span. \(e_{G}\) of 0, \(L_{G}/1500\), and \(L_{G}/1000\) are considered in the analysis. The non-linear analysis is conducted by increasing the temperature of the rail. Then, the axial stress of the CWR can be obtained by converting the rail temperature to the stress (refer to Equation (1)). In order to simulate the snap-through buckling curve of a CWR track, the Riks method was applied.