# Qualitative Prediction Model for Dynamic Behavior of Ballasted Tracks
## 4 Results and Discussion  
### Assessment of the Track Parameters Using the Proposed Qualitative Analysis  
The effect of the (vertical) track support stiffness and the influence of variations to the rail pad and ballast stiffness on the track support stiffness and associated track responses were then investigated. Some of the values used in the numerical simulations were obtained from previous numerical and experimental studies [1; 2; 9; 17; 18; 23; 41; 45; 49]. Overall, the track response was affected by several parameters, including the rail pad and ballast stiffness, rail surface roughness, and condition of track components; the parameters found to have the greatest impact on the entire ballasted track system and track response were the rail pad and ballast stiffness. These parameters can be estimated using simple field tests, including rail and sleeper displacement and dynamic wheel load tests, and should thus be regularly tested, maintained, and adjusted, as the rail support stiffness directly affects the track displacement, rail support pressure, and track support stiffness. Figure 4: Variations in the dynamic response of the ballasted test track as a function of train velocity: (a) dynamic wheel load; (b) rail displacement; (c) sleeper displacement; (d) rail bending stress. In the graph on the right side of Figure 5 (flat graph), the figures in the white box on top of the white line represent the boundary values of each area. The rail support stiffness (\(k_{d}\)) increased with increasing rail pad stiffness (\(k_{pud}\)) and ballast stiffness (\(k_{b}\)), as is clear from the top-view of Figure 5a presented in Figure 5b. At a constant rail pad stiffness of \(k_{pud}\) = 400 kN/mm, which is the normal thermoplastic polyurethane (TPU) pad used in conventional ballasted railway lines in Korea, the rail support stiffness increased with increasing ballast stiffness. Due to the L-shaped nature of the iso-\(k_{d}\) curves in the \(k_{pud}\)-\(k_{b}\) graph, at \(k_{b}\) > 200 kN/mm, the rail support stiffness was more affected by the rail pad stiffness than the ballast stiffness. For example, at \(k_{b}\) = 300 kN/mm, the rail support stiffness at \(k_{pad}\) = 600 kN/mm was approximately 1.2 times greater than that at \(k_{pad}\) = 400 kN/mm. The track support stiffness also increased with increasing rail pad stiffness and ballast stiffness, as demonstrated in Figure 6. At a constant rail pad stiffness of \(k_{pad}\) = 400 kN/mm, the track support stiffness increased with increasing ballast stiffness. Due to the L-shaped nature of the isotrack support stiffness curves in the \(k_{pad}\)-\(k_{b}\) graph, at \(k_{b}\) > 300 kN/mm, the track support stiffness was more affected by the rail pad stiffness than the ballast stiffness. For example, at \(k_{b}\) = 300 kN/mm, the track support stiffness at \(k_{pad}\) = 600 kN/mm was approximately 1.4 times greater than that at \(k_{pad}\) = 400 kN/mm. Moreover, at \(k_{b}\) = 200 kN/mm, the track support stiffness was similar to the rail pad stiffness at both 400 and 600 kN/mm. Figure 5: Example dynamic rail support stiffness as a function of rail pad and ballast stiffness: (**a**) 3D map; (**b**) 2D map. Figure 6: Example dynamic track support stiffness as a function of rail pad and ballast stiffness: (**a**) 3D map; (**b**) 2D map. The characteristic track length decreased with increasing rail pad and ballast stiffness, as shown in Figure 7. At a constant rail pad stiffness of \(k_{pad}=400\) kN/mm, the characteristic track length decreased with increasing ballast stiffness. Therefore, to allow for an increasing rail pad stiffness while ensuring that the characteristic track length is similar to the sleeper spacing, i.e., the Korean conventional ballasted track of 588-625 mm, the ballast stiffness was held between 200 and 300 kN/mm while varying the rail pad stiffness.