## 5 Aerodynamic Loads and Safety Analysis
### Safety Analysis
#### 5.2.2. Wheel Load Reduction Rate  
When the vertical contact force of a wheel exceeded its original static wheel load, the wheel lifted, and the wheel unloading accident was likely to happen. The corresponding safety coefficient WRR [39] is defined as:  
\[\text{WRR}=\Delta P/\overline{P} \tag{29}\]  
where \(\Delta P\) represents the reduction amount of wheel load in one wheel, which is \(+\) for reducing and \(-\) for the increase of the vertical contact force. \(\overline{P}\) represents the load that the total train static self-weight equally distributes on each wheel.  
Figure 20 demonstrates the time-history of WRR of the head carriage wheelsets during the train entering tunnel under the conditions of train speed \(=250\) km/h and crosswind velocity \(=15\) m/s.  
Figure 19: Contrast of global maximum derailment coefficients conditioned train speed \(250\) km/h, crosswind velocity \(25\) m/s: (**a**) head, (**b**) middle, and (**c**) rear carriages.  
Figure 20: Time-history of head carriage WRR conditioned train speed 250 km/h, crosswind velocity 15 m/s, (**a**) 1 st wheelset, (**b**) 2 nd wheelset, (**c**) 3 rd wheelset, (**d**) 4 th wheelset (open air (OA); entering entrance (E); tunnel (T)).  
Figure 20a-d shows that the carriage deflected to the leeward side, the vertical contact force decreased in the windward side, and WRR stabilized at a higher level in the windward side when the train ran in the crosswind environment completely. Meanwhile, the variation amplitude of the WRR in leeward side wheel was remarkably larger than that of the opposite side because the wheel tread on the leeward side had close contact with the rail top. When the train was in the process of entry, the carriage turned aside towards the windward side due to the sudden disappearance of wind excitation. As shown in Figure 20a, the WRR in the windward side decreased to \(-2.63\) (the lowest value among the four wheelsets), whereas that in the leeward side increased to \(0.64\) instantaneously because of self-weight. Then, the WRR in the windward side rose to its global maximum of \(1.0\), indicating that the wheel had been detached from the rail. Finally, the WRRs on both sides rapidly decreased when the train entered the tunnel due to the buffering effect of the suspension system.  
Similar to DC, the variation amplitude of WRR of the first and third wheels in the windward side was larger than those of the other wheels and occurred instantaneously when the train entered tunnel. Figure 21 presents the comparison of all the wheels' maximum WRR values amongst the head, middle and rear carriages during the two processes (i.e., only in open air and entering tunnel) under conditions of train speed = 250 km/h and crosswind velocity = 25 m/s. A safety risk was likely to be observed in the head carriage based on the discussion of the DCs and WRRs.  
Figure 21: Contrast of global maximum wheel reduction rate conditioned train speed 250 km/h, crosswind velocity 25 m/s: (**a**) head, (**b**) middle, and (**c**) rear carriages.  
To verify the effectiveness of the track irregularity spectrum used in the present study, Figure 22 shows the time-history of the WRR in the first wheelset of the head carriage during the train entering the tunnel at a speed of 250 km/h (no crosswind) under the track spectrum condition in this paper and the corresponding result under the condition of the Chinese track spectrum [40], respectively. It is shown from Figure 22 that the WRRs of wheels on both sides fluctuated at the same level due to the absence of crosswind. The maximum WRR value, shown in Figure 22a,b, was 0.51 and 0.17, respectively. Both were significantly smaller than the corresponding result shown in Figure 20a. Therefore, the track irregularity spectrum adopted in the present study is effective and conservative.