# Qualitative Prediction Model for Dynamic Behavior of Ballasted Tracks
## 4 Results and Discussion
### Assessment of the Dynamic Track Responses Using Qualitative Analysis  
The resulting variations in dynamic track response plotted against the variation in the rail and ballast stiffness at a vehicle speed of 120 km/h are shown in Figures 9-12. The sleeper reaction force may directly affect the track displacement, rail support pressure, and track deterioration, and thus was determined to depend upon the track support stiffness, rail displacement, and rail bending stiffness. The sleeper reaction force increased with increasing rail pad and ballast stiffness, as shown in Figure 9. At a constant rail pad stiffness of \(k_{pad}\) = 400 kN/mm, increasing the ballast stiffness caused the sleeper reaction force to increase. Furthermore, the sleeper reaction force was more affected by the ballast stiffness than the rail pad stiffness. For example, at \(k_{pad}\) = 400 kN/mm, the sleeper reaction force at a of \(k_{b}\) = 300 kN/mm was approximately 1.1 times greater than that at \(k_{b}\) = 200 kN/mm. Generally, sleeper reaction forces account for 60% of the dynamic wheel load [1; 2; 4; 13]. However, in that case, the majority of the sleeper reaction force could be ensuring that 60% of the dynamic wheel load at the ballast stiffness of 400 kN/mm. Figure 10: Example dynamic rail displacement as a function of rail pad and ballast stiffness (**a**) 3D map; (**b**) 2D map. Figure 9: Example dynamic sleeper reaction force as a function of rail pad and ballast stiffness (**a**) 3D map; (**b**) 2D map. The relationship between the rail displacement, which directly affects the track displacement, passenger comfort, and track deterioration, and the track support stiffness, characteristic track length, dynamic wheel load, and rail bending stiffness was then investigated. The rail displacement was found to decrease with increasing rail pad and ballast stiffness, as shown in Figure 10, although the ballast stiffness did affect the rail displacement more strongly than the rail pad stiffness. In a classic ballasted track, low-average rail displacement of approximately 0.3-0.4 mm under a wheel load of 100 kN corresponds to a subgrade modulus of 0.3-0.5 N/mm\({}^{3}\)[4]. Under Korean and German track regulation, the rail displacement must remain within the very narrow range of 1.2-1.5 mm for high-speed railway (HSR) lines. However, the rail displacement of in-service ballasted tracks depends on the track components and ballast condition. To ensure that the rail displacement did not exceed the Korean regulatory limits during normal operation (0.4-0.5 mm), the ballast stiffness was held constant at 200-220 kN/mm while varying the rail pad stiffness from 400 to 600 kN/mm. At the constant rail pad stiffness of 400 kN/mm, increasing the ballast stiffness also caused the rail displacement to increase. The relationship between the rail bending stress, which directly affects the track displacement and safety, and the track support stiffness, characteristic track length, dynamic wheel load, and rail bending stiffness, was then investigated. As a result, the rail bending stress decreased with increasing rail pad  
Figure 11: Example dynamic rail bending stress as a function of rail pad and ballast stiffness: (**a**) 3D map; (**b**) 2D map. Figure 12: Example dynamic rail support pressure as a function of rail pad and ballast stiffness: (**a**) 3D map; (**b**) 2D map. and ballast stiffness, as shown in Figure 11, although the ballast stiffness affected the rail bending stress more strongly than the rail pad stiffness. Korean and German regulations require that the rail bending stress of the CWR for HSR lines remain lower than 92 and 70 MPa, respectively; however, others have recommended that the rail bending stress of a ballasted track on an HSR remain below 60 MPa [4]. However, in that case, most rail bending stress could be ensuring that of the Korean and German regulation regardless of increasing the ballast and rail pad stiffness.