## 8 Verification of Theoretical Impact Forces  
Derailment simulations were conducted under the assumption that the vehicle details due to unknown abnormal situations of either the track or vehicle. The simulation conditions are shown in Table 7, and the derailment speed is set to 300 km/h, which is the current maximum operating speed of the KTX, as a worst condition. The reason for separating the curve radius into three cases is that the minimum curve radius of Korean high-speed rail is 5000 m and 7000 m in a ballasted track. In addition, since the minimum design curve radius at 300 km/h is 3500 m in a concrete track, the condition for 3500 m was also analyzed to consider the worst-case condition. Since 300 km/h is the maximum operating speed, the impact load generated at speeds below 300 km/h will be lower than the impact force generated at 300 km/h impact. \begin{table}
\begin{tabular}{c c c c c} \hline \multirow{2}{*}{**Curve Radius** **[m]**} & \multicolumn{2}{c}{**Mean Impact Forces [kN]**} & \multicolumn{2}{c}{**Maximum Impact Forces [kN]**} \\ \cline{2-5}  & **Impact Angle \(\theta_{1}\)** & **Impact Angle \(\theta_{2}\)** & **Impact Angle \(\theta_{1}\)** & **Impact Angle \(\theta_{2}\)** \\  & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) & (\({}^{\circ}\)) \\ \hline R400 & 733.49 & 1038.26 & 1571.1 & 2223.96 \\ R1600 & 366.27 & 518.30 & 784.54 & 1110.20 \\ R2400 & 298.97 & 423.03 & 640.39 & 906.13 \\ R3500 & 247.51 & 350.20 & 530.18 & 750.12 \\ R5000 & 207.05 & 292.93 & 443.50 & 627.44 \\ R7000 & 174.96 & 247.52 & 374.77 & 530.18 \\ \hline \end{tabular}
\end{table} 
Table 6: Impact forces considering suspension stiffnesses. Figure 12: Impact loads considering suspension stiffnesses. In the simulations, Cases 1, 3, and 5 represent the conditions in which the vehicle advances along the tangential direction of a curved track after derailment and impacts with the DCP at the impact angle (\(\theta_{1}\)), derived in Equation (3). On the other hand, Cases 2, 4, and 6 consider the additional angles (\(\beta\)) due to the centrifugal force effect and represent the impact angle (\(\theta_{2}\)). The DCP used in the simulations has a curved shape. While additional derailment angles were not considered in Cases 1, 3, and 5, the vehicle was slightly yawed in Cases 2, 4, and 6, as shown in Figure 13, to account for the additional derailment angle caused by the centrifugal force. Since multiple simulation conditions were tested, only one case is presented as a representative in Figure 14. The other conditions were conducted using the same simulation procedure. The presented simulation depicts a scenario where the train derails within the track while traveling at a speed of 300 km/h in a curved track with a radius of 5000 m, with an additional angle caused by centrifugal force, and collides with the DCP. The collision behavior is observed from both the front and rear angles in Figure 14. After colliding with the DCP, the wheels were guided by it and continued moving forward without leaving the track. DCP material was modeled using Mat159, which is a continuous surface cap model. The compression strength of the DCP is 45 MPa, and the elements erode when damage exceeds 0.99 and the maximum principal strain exceeds ERODE-1.0. Ductile damage accumulation occurs when the pressure (P) is compressive, and an energy type term, \(\tau_{c}\), exceeds the damage threshold, \(\tau_{0c}\). Ductile damage accumulation depends upon the total strain components, \(\epsilon_{ij}\), as follows \(\tau_{c}=\sqrt{\frac{1}{2}\sigma_{ij}\epsilon_{ij}}\). The stress components \(\sigma_{ij}\) are the elastoplastic stresses (with kinematic hardening) calculated before application of damage and rate effects [26]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Curve Radius (m)** & **Velocity (km/h)** & \(\beta\) (\({}^{\circ}\)) \\ \hline
1 & 3500 & 300 & 0.893 \\
2 & 3500 & 300 & 0 \\
3 & 5000 & 300 & 0.747 \\
4 & 5 & 0 & 0 \\
6 & 7000 & 300 & 0.631 \\ \hline \hline \end{tabular}
\end{table}