## 2 Numerical Model
### Wheel-Rail Contact Model  
Several wheel-rail contact theories are used to estimate normal and tangential contact forces in numerical analysis procedures [23; 24]. Hertzian Contact Theory for normal forces and Kalker's Fastsim for tangential forces are the most common and easy-to-adopt contact theories. However, they are mostly used during MBS modelling. In FEM, penalty-based contact models are mostly preferred. Selection of a contact model amongst these contact theories is related to the complexity of the numerical  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Element** & **Type** & **Number of Elements** & **Property** & **Value** \\ \hline \multirow{3}{*}{Rail} & & & \(\rho\)1  & 7800 \\  & Beam & 1980 & E2  & 210 \\  & & & PR3  & 0.3 \\ \hline \multirow{3}{*}{Sleeper} & & & \(\rho\)1  & 2500 \\  & Beam & 3070 & E2  & 38 \\  & & & PR3  & 0.2 \\ \hline \multirow{3}{*}{Rail Pad} & & & \(\rho\)1  & 7800 \\  & Beam & 1980 & E2  & 210 \\  & & & PR3  & 0.3 \\ \hline \multirow{3}{*}{Sleeper} & & & \(\rho\)1  & 7800 \\  & Beam & 1980 & E2  & 210 \\  & & & PR3  & 0.3 \\ \hline \multirow{3}{*}{Sleeper} & & & \(\rho\)1  & 2500 \\  & Beam & 3070 & E2  & 38 \\  & & & PR3  & 0.2 \\ \hline \multirow{3}{*}{Rail Pad} & & Spring & 245 & k4  & 1300 \\ \hline \multirow{3}{*}{Ballast} & & & \(\rho\)4  & 45 \\  & & & \(\rho\)5  & 45 \\  & & & \(\rho\)5  & 32 \\ \hline \multirow{3}{*}{Primary Suspension} & & & \(\rho\)1  & 2500 \\  & Beam & 3070 & E2  & 38 \\  & & & \(\rho\)6  & 0.2 \\ \hline \multirow{3}{*}{Rail Pad} & & & \(\rho\)1  & 7800 \\  & Beam & 1980 & E2  & 210 \\  & Beam & 2070 & E2  & 0.3 \\ \hline \multirow{3}{*}{Sleeper} & & & \(\rho\)1  & 7800 \\model. The model here is relatively simple, which is built with simple 2D beam elements. Furthermore, only the vertical contact forces are considered. More importantly, general behaviors such as sleeper behavior under dynamic loading are of interest rather than wheel-rail interface in this study. As a consequence, simple Hertzian contact theory is applied here to calculate contact forces. The formulation of the contact force is:  
\[F=k\cdot\left(\delta_{w}-\delta_{r}-\delta_{ir}\right)\,,\]  
where \(k\) is the Hertzian spring constant and \(\delta_{w},\delta_{r},\delta_{ir}\) are the displacement values of the wheel, rail and track surface roughness at the location, respectively. The formulation of Hertzian contact stiffness is given in [6] for railway applications as:  
\[k=\sqrt[3]{\frac{3E^{2}Q\sqrt[3]{R_{v\textit{thevel}}.R_{v\textit{thevel}}}}{2 \left(1-v^{2}\right)^{2}}},\]  
where \(E\) is modulus of elasticity, \(v\) is Poisson's ratio, \(Q\) is static wheel load, \(R_{v\textit{thevel}}\) is wheel radius and \(R_{\textit{rail}}\) is rail profile radius. Based on this formulation, contact stiffness is calculated as \(1.4\times 10^{6}\) N/mm.  
The track surface roughness, sometimes called as \"irregularity\", in the formula represents track irregularities (vertical rail levels), which affect the dynamic train-track interaction. In order to simulate realistic conditions, the track irregularities obtained by a track measurement car in [37] are used in this simulation. The vertical measurements of a 30 km track are divided into small sections, the length of which are the same as the 48 m turnout model. Afterwards, the sections are compared with respect to the mean values and standard deviations. The section with the highest standard deviation and mean value is selected for the worst case scenario. The selected irregularity is presented in Figure 2. The profile of track irregularities