# Wheel Flats in the Dynamic Behavior of Ballasted and Slab Railway Tracks
## 3 Numerical Study
### Results and Discussion  
#### 3.3.1 Ballasted Track  
In Figure 5, the receptance of the ballasted track is shown. The resonance frequencies that were more relevant for the vertical dynamic behavior of this ballasted track were 75, 230, and 1035 Hz, corresponding respectively to the ballast, rail, and pin-pin deformation. By looking up the magnitude of the receptance values, the flexibility of the track at the rail pad level was evident. Regarding the effects of wheel flats on the ballasted track, the variation of the dynamic load along the ballasted track due to a wheel flat with a 2 mm depth running at 300 km/h is presented in Figure 6.  
The variation of the dynamic load on the track due to the wheel flat obtained in this numerical study was similar to that found by other authors, such as [12, 38, 39]. By Looking at this figure, during the movement of the wheel flat on the track, two impact loads (\(I_{1}\) and \(I_{2}\)) appeared in addition to the occurrence of a loss of contact between the wheel and rail (Figure 6b). At 300 km/h, the impact load \(I_{1}\) was 3.57-times the static load, and the second impact force was 1.97-times that. The magnitude of these impact loads depends significantly on the train speed and strongly influences the track degradation process. Figure 7 shows the evolution of the dynamic load amplification factor with the train speed, for the cases of a perfect wheel and a wheel flat. The dynamic amplification factor (DAF) is defined as the ratio between the dynamic and the static responses, and it is normally used to illustrate the dynamic effect. There are some empirical mathematical expressions that may estimate the DAF usually based on the vehicle speed and on the track's distributed irregularities. However, those expressions are limited, as they do not consider the effect of isolated irregularities of the track or wheel defects such as the flats on the dynamic response of the track. From Figure 7, the amplification factor increased almost four times until 160 km/h; then, the factor was approximately constant up to a speed of 300 km/h; further, there was another increase of the amplification factor from speeds of 300 to 360 km/h. This trend has been observed in the bibliography. Figure 5: Acceptance of the ballasted track. In order to analyze the influence of the depth of the wheel flat on the magnitude of the dynamic force on the track, wheel flats with depths from 0 to 2 mm at intervals of 0.5 mm were analyzed. The flat length \(l\) depends on the flat depth \(d\) as shown in Equation (5). For this calculations, a train speed of 360 km/h was considered, and the results are presented in Table 3. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(F\) & \(d\) & \(d\) & \(d\) & \(d\) & \(d\) \\ (**kN**) & **0 mm** & **0.5 mm** & **1.0 mm** & **1.5 mm** & **2.0 mm** \\ \hline Min. & 91.7 & 23.3 & 0.0 & 0.0 & 0.0 \\ Max. & 107.7 & 164.3 & 238.8 & 314.3 & 433.3 \\ \hline \hline \end{tabular} Min., minimum value; Max., maximum value; \(d\), wheel flat depth.  
\end{table} 
Table 3: Dynamic load on the ballasted track for different wheel flat depths (V = 360 km/h). Figure 6: Variation of the dynamic load along the ballasted track due to a wheel flat running at 300 km/h (\(d=2\) mm). (**a**) Along the track; (**b**) detail. From Table 3, the maximum dynamic force increased with the depth of the flat. For the ballasted track under consideration, there was a loss of contact between the wheel and the rail for flat depths equal to and greater than 1.0 mm. Figure 8 presents the dynamic amplification factor for the track vertical displacement function of the train speed. The results in Figure 8 show that the maximum track displacement, over which a wheel flat ran at a 2 mm depth, increased as expected with the vehicle speed, and for speeds greater than 150 km/h, it was almost double the static displacement. In the case of speeds greater than 80 km/h, the displacement was higher than 2 mm, the recommended value for this type of track in normal operating conditions.