## 3 Description of the Assumed Numerical Model  
In order to identify the data assumed in the analytical model, a numerical model was used--developed at the Chair of Roads, Railways and Traffic Engineering, Faculty of Civil Engineering, Cracow University of Technology. The analyses were performed in Autodesk Simulation Multiphysics 2013 calculation software and--as an auxiliary--in Excel software. The following type of analysis was performed: MES (Mechanical Event Simulation) with Nonlinear Material Models, using elements of the model--type 2-D (2D strain state). To develop numerical models of track with a given structure, the following types of materials were used:  
1. Elastic (isotropic) materials: rails, sleepers, selected parts of fastening system \(\bullet\) Density--\(\varrho\) (kg/m\({}^{3}\)); \(\bullet\) Modulus of elasticity--E (N/mm\({}^{2}\)); \(\bullet\) Poisson's Ratio--\(\
u\) (\(-\)); \(\bullet\) Shear modulus of elasticity--G (N/mm\({}^{2}\)). Table 1 presents the assumed data of the numerical model for elastic materials. \begin{table}
\begin{tabular}{c c c} \hline
**UIC 60 Rails** & **Sleepers** & **Fastening System–Rail Pad** \\ \hline steel & concrete & steel \\ \hline e—7872 kg/m\({}^{3}\) & \(\varrho\)—2351 kg/m\({}^{3}\) & \(\varrho\)—7872 kg/m\({}^{3}\) \\ E—200,000 N/mm\({}^{2}\) & E—31,026,408 N/mm\({}^{2}\) & E—200,000 N/mm\({}^{2}\) \\ v—0.29 & \(\
u\)—0.15 & v—0.29 \\ G—80,000 N/mm\({}^{2}\) & G—13,489 N/mm\({}^{2}\) & G—80,000 N/mm\({}^{2}\) \\ \hline \end{tabular} (2) Elastic-plastic materials with reinforcement (two-line material) (von Mises with Isotropic Hardening): ballast and fastening system elements.  
\(\bullet\) Mass density—\(\varrho\) kg/m\({}^{3}\);  
\(\bullet\) Modulus of elasticity—E (N/mm\({}^{2}\));  
\(\bullet\) Poisson’s Ratio--\(\
u\) (\(-\));  
\(\bullet\) Yield strength—\(\sigma_{\rm Y}\) (N/mm\({}^{2}\));  
\(\bullet\) Strain hardening modulus—\(\epsilon_{\rm T}\) (N/mm\({}^{2}\)).  
\end{table} 
Table 1: Assumed data of the numerical model for elastic materials. Figure 2: Equilibrium path with the lateral resistance of the ballast determined to be r\({}_{0}\) = 7 kN/m and different type rail-sleeper structure bending stiffness. Table 2 presents the assumed data of the numerical model for elastic-plastic materials. Resistances of the ballast and of the fastening system both in the lateral and longitudinal directions are elastic in a very small range. The values of the parameters describing a given material in the model have been assumed in such a way that even with forcing small displacement, the results correspond to the shapes of the charts of experiments. The critical value in which material characteristics change is variable and depends on numerous factors. The range of the elastic zone of ballast response in lateral and longitudinal directions typically is 5 \(\div\) 7 mm. For the fastening system, the critical value of elastic response for longitudinal resistance is approximately 0.4 \(\div\) 0.6 mm. The selection of the above material parameters was based on the charts of longitudinal resistance, lateral resistance, and torsional resistance of the fastening system, and the ballast resistance. Rails were modelled as beams, determining the cross-section, moments of inertia, and Young's modulus. Fastening system parameters were selected in such a way that the characteristics of the longitudinal resistances and rotation moment of the model corresponded to the charts of elastic-plastic characteristics obtained as a result of experimental studies selected in literature review. The assumed longitudinal resistance and rotation characteristic of fastening system (Figure 3) of the fastening system were described with the use of a chart assumed on the basis of [5]. In order to obtain more accurate results, contact surfaces between the rails and the fastening system (e.g., anchor and steel plate) were used in the model Figure 4 presents the model and the details of the contact surface between the rail and the fastening system element.