# Wheel Flats in the Dynamic Behavior of Ballasted and Slab Railway Tracks
## 3 Numerical Study
### Results and Discussion
#### 3.3.2 Slab Track  
Figure 10 shows the receptance of the slab track. From this figure, the most relevant frequencies of the slab track, associated with the vertical deformation, were 167, 589, and 1373 Hz, higher values than those identified for the ballasted track, which revealed that the slab track had greater inertia than the ballasted one. Furthermore, the magnitude of those resonance frequencies was smaller when compared to the values of the ballasted track, confirming the greater rigidity of the slab track. By analyzing the effects of the wheel flats, the variation of the dynamic load along the slab track due to a wheel flat with a 2 mm depth for the ICE 2 train was included in the study in order to forecast the eventual dynamic behavior of a train at that speed. Vehicle speeds of 25, 160, 220, 300, and 360 km/h are presented in Figure 11.The lower speed of 25 km/h was numerically tested for two main reasons: (i) to highlight the impact not only of the wheel flat, but also of the vehicle speed; (ii) to contribute to the definition of a \"safe vehicle speed\" in the case of the existence of wheel flats; this \"safe speed\" was not expected  
Figure 10: Receptance of the slab track. Figure 9: Dynamic amplification factor for the force on the fastening system: perfect wheel vs. flat wheel (\(d=2\) mm), ballasted track. to induce severe dynamic contact loads on the track. The higher speed (360 km/h) was unrealistic for the ICE 2 train, but it was included in the study in order to forecast the eventual dynamic behavior of a train at that speed. Figure 11 shows that:  
* The existence of a wheel flat with a 2 mm depth, for vehicle speeds higher than 25 km/h, led to loss of contact between the wheel and the rail immediately before and after the occurrence of a high-impact load;
* The magnitudes of the impact loads \(I_{1}\) and \(I_{2}\), for the speed of 300 km/h, were respectively 952.9 kN and 394.7 kN, which were 9.7- and 4.0-times the static load. From this figure, the dynamic impact load on the track increased with the increase of the vehicle speed; the position of the second impact load from the first impact load increased with the increase of the vehicle speed. Figure 11: Variation of the dynamic load along the slab track for various vehicle speeds due to a wheel flat (\(d=2\) mm). (**a**) V = 25, 160, and 220 km/h; (**b**) V = 300 and 360 km/h. The obtained results showed also that the impact load due to a wheel flat with a 2 mm depth was higher on the slab track than on the ballasted one (see Tables 3 and 4), because, as is known, the overall track stiffness is higher in the case of a slab track than in the case of a ballasted track, which contributes to the high wheel-rail impact load. The evolution of the dynamic load amplification factor with the train speed is presented in Figure 12, for the cases of a perfect wheel and a flat wheel flat with a \(d=2\) mm depth. By observing this figure, the dynamic load amplification factor was approximately constant for speeds between 40 and 80 km/h, presenting a slight decrease for a vehicle speed of 120 km/h. These results were similar to those obtained for ballasted tracks by [8; 12; 40]: the wheel with a flat running at a relatively low speed (between 30 and 40 km/h) may induce higher dynamic loads on the track than those obtained at higher train speeds. From 120 to 360 km/h, the dynamic load amplification factor increased linearly at a high rate. In this research, the influence of the flat depth on the dynamic response of the slab track was also investigated. In Table 4, the dynamic load obtained for different flat depths is indicated for a vehicle speed of 360 km/h. The peak load for a flat depth of 2 mm was approximately four-times higher than that obtained with a flat depth of 0.5 mm. The results also showed that on the slab track, there was a loss of contact between the wheel and the rail when the depth of the flats exceeded 0.5 mm. For the ballasted track, this phenomena only occurred for flat depths greater than 1 mm.