## 2 Development of the Numerical Model
### Modeling of the High-Speed Train and Slab Track  
As a commonly operated vehicle on high-speed railway networks in China, the China Railway High-speed (CRH) 380 Electric Multiple Unit (EMU) train is simulated in this model. The vehicle consists of a one-car body, two bogies, four wheelsets, and two stage-suspension systems, as illustrated in Figure 1. The car body, bogies, and wheelsets are simplified as the rigid-bodies with shell and beam elements. These multi-rigid-bodies are connected by the springs and dashpots. As the vertical vibration is the primary excitation to the infrastructures, the vertical degrees of freedom (DOF) of the vehicle are considered in this model. The vehicle has a total of 10 DOF including the vertical and pitch motion of the car body (\(Z_{c}\), \(\beta_{c}\)), the vertical and pitch motion of the bogies (\(Z_{bi}\), \(\beta_{bi}\)\(i=1,2\)), and the vertical motion of the wheelsets (\(Z_{bi}\)\(i=1,\ldots,4\)).  
The China Railway Track System (CRTS) II slab track is adopted in the model. It consists of rail, rail pads, concrete slab, a cement asphalt (CA) mortar layer, and concrete base [38]. The rail is simulated as the Euler beam, which is supported by the discrete springs and dashpots to represent the rail pads. The concrete slab, CA mortar, and concrete base are simulated as solid elements.  
The contact between wheel and rail is simulated based on the Hertz contact theory in LS-DYNA [37; 39]. The following equation is used to calculate the dynamic wheel-rail contact force:  
\[F=K_{H}\times(Z_{w}-Z_{r}-\delta) \tag{1}\]  
where \(K_{H}\) is the vertical stiffness of the wheel-rail contact spring, \(K_{H}=1.325\times 10^{9}\) N/m in this study [37]; \(Z_{w}\) is the vertical displacement of the wheel; \(Z_{r}\) is the vertical displacement of the rail; and \(\delta\) is the track irregularity. The Germany high-speed low disturbance irregularity is used to excite the wheel-rail contact [37]. The power spectrum density (PSD) function of the track irregularity is calculated as follows:  
\[S_{\rm p}(\Omega)=\frac{A_{\rm p}\Omega_{\rm c}^{2}}{\left(\Omega^{2}+\Omega_ {r}^{2}\right)\left(\Omega^{2}+\Omega_{\rm c}^{2}\right)} \tag{2}\]  
where \(S_{\rm p}(\Omega)\) is the vertical power spectral density; \(A_{\rm p}\) is the roughness constant (\(A_{\rm v}=4.032\times 10^{-7}\) m\({}^{2}\). Rad/m); \(\Omega_{\rm c}\) and \(\Omega_{r}\) are the cutoff frequency (\(\Omega_{\rm c}=0.8246\) rad/m, \(\Omega_{r}=0.0206\) rad/m); and \(\Omega\) is the spatial frequency of the irregularity. The PSD function can be transformed into vertical irregularities along the longitudinal distance of the track by means of a time-frequency transformation technique, as shown in Figure 2.  
In LS-DYNA, the built-in keywords: *Rail_Track and *Rail_Train are used to define the contact parameters such as the stiffness of the wheel-rail contact spring, the track irregularity, and so on.  
The material properties of the CRH380 EMU Train and CRTS II slab track are shown in Table 1. Since most previous studies adopted the static material properties of a slab track despite the fact that the actual loads from high-speed trains onto slab tracks are dynamic excitation [40; 41], the dynamic  
Figure 1: Simulation of the vehicle.  
material properties of the CRTS II slab track are used in this model in order to obtain a more realistic vibration response. The stiffness of rail pads is determined by the dynamic value [37] and the moduli of the concrete slab's elasticity, CA mortar, and concrete base are considered as the strain-rate dependent values.