## 4 Modeling and Results with Discussion
### States without Settlement of the Three Cases  
[PERSON] et al. [40] analyzed the FE models with bonded contacts between track elements, as it expressed the first case. Thus, a comparison between different cases would show the characteristics and benefits of each of these cases. The three cases have different results of stresses and deformation values, and the results were compared in the figures as follows. Figure 11 depicts the values of the wheel/rail pressure for the three cases. It can be seen that the results of the contact pressure are identical, with no difference after the starting zone. The values of contact pressure are converged even after changing the mesh size at the contact of wheel and rail. Figure 12a shows the railhead equivalent stresses. It shows that the third case has the highest stresses compared with the other cases, which show nearly the same results. The highest average value of stresses in the railhead belongs to the third case, which equals 159.1 MPa. However, the highest average values of stresses for the first and second cases are 154.2 MPa and 154.8 MPa, respectively. Away from the starting zone, the maximum vertical deflection of the railhead for the three cases ranged from 2.2 mm to 2.4 mm, as shown in Figure 12b. By comparing these results with [PERSON] and [PERSON] [41], who analyzed 1.9 mm of vertical rail deflection, it can be concluded that the results of the 2D model are reasonable. [PERSON] and [PERSON] [41] used an axle load of 20 tons, 600 mm of Sleeper spacing, and a running speed of 100 km/h on rail 60E1 (UIC60). It can be seen in Figure 12b that there is a slight difference between the results of the three cases, with a 0.1 mm difference of deflection values. Figure 11: Wheel/rail contact pressure. Figure 12: Railroad: (**a**) stresses; (**b**) deformation. Figure 13a illustrates the stress values of the 11 th tie plate, and describes the maximum values of the stresses. The maximum stresses belong to the first case because of the effect of bonded contact. The average stresses for the first, second, and third cases are 39.5 Mpa, 27.2 Mpa, and 33.2 Mpa, respectively. Thus, the second case has the lowest stresses in the 11 th tie plate. The maximum deformation value of the 11 th tie plate, as illustrated in Figure 13b, appears in the third case. Stresses increase gradually in the first case of the 11 th sleeper, reaching the highest value of all cases, i.e., 0.66 MPa, as observed in Figure 14a. Figure 13: The 11 th tie plate: (**a**) stresses; (**b**) deformation. The average stresses for the first, second, and third cases are 0.52 MPa, 0.37 MPa, and 0.4 MPa, respectively. Thus, the second case has the lowest stresses for the 11 th sleeper. However, the minimum deformation values of the 11 th sleeper are registered in the first case, as shown in Figure 14b. The maximum deformation value of the third case equals 2.31 mm. It increased by about 10% of the maximum deformation value of the first case. In Figure 14b, it can be seen that the maximum deformation of the 11 th sleeper occurred when the load affects it directly. Deformation values around the distance of 1100 mm decreased due to the effect of track elasticity, which is presented in the Zimmermann method, while the rail tends to rise and pulls sleepers up with it. On the other hand, deformation values increase at a distance of 600 mm because of track elasticity and the impact of superstructure weight. The stress values of the subballast layer in the three cases show a slight difference, and the subgrade layer has the same status, as shown in Figure 15. The average stresses in the subballast layer are 0.095 MPa, 0.096 MPa, and 0.1 MPa for the first, second, and third cases, respectively. The average stresses in the subgrade layer are 0.047 MPa, 0.049 MPa, and 0.055 MPa for the first, second, and third cases, respectively. However, for the ballast  
Figure 14: The 11 th sleeper: (a) stresses; (b) deformation.