## 5 Fatigue Crack Propagation Analysis
### Effects of Track Irregularity and Train Speed  
Track irregularities are regarded as the most important excitation of the TTB system [24]. Five different classes of track irregularity, i.e., class 2 to class 6 are considered to investigate their influence on the fatigue crack propagation life. Using Equation (4) and the parameters reported in Table 2, four additional sets of track irregularity samples corresponding to class 2, 3, 4, and 6 were generated to perform multi-scale coupled analysis of the TTB system. Figure 14 shows the fatigue crack propagation curves resulting from the five different classes of track irregularity under \(a_{0}=0.1\) mm. Generally, the crack propagation life rises with the decrease in severity of track irregularities. If the track irregularity class is improved by one grade to class 6 for the Baihe Bridge, the fatigue life of the bracket analyzed will be extended to the year 2082 as shown in Table 4. It is also found in the table that the number of stress cycles \(\sum n_{i}\) in a stress block increases with the increase in degree of track irregularities, whereas a similar tendency is not present for the equivalent stress range \(\Delta S_{\text{eq}}\). For purposes of illustration, maximum vertical, lateral, and torsional wheel-rail forces obtained from the five different classes of track and perfect track (without considering irregularities) are reported in Table 5, corresponding to the first wheelset in the locomotive of the freight train running at 60 km/h. A significant increase can be observed in the wheel-rail interactions in all directions with the deterioration of track condition. These observations suggest that track conditions have considerable effects on the fatigue life of structural details in the bridge deck system. Since dynamic responses of the TTB system to the excitation of track irregularities can change with vehicle and bridge parameters, accurate quantification of the abovementioned effects calls for specific analysis of the TTB system. Figure 12: The photo of the cracked bracket. Figure 13: Crack growth rate d\(a\)/d\(N\) versus relative crack length \(a\)/\(a_{\text{c}}\). In the present analysis, operating speeds of the passenger train and freight train respectively ranging from 50 to 110 km/h and 40 to 100 km/h with the step of 10 km/h are taken into account. A 10 km/h speed gap between the passenger and freight train is still adopted as discussed earlier. Therefore, there are overall seven speed combinations considered as shown in Table 6. Then, the fatigue stresses are computed with multi-scale dynamic analysis of the coupled TTB system subjected to the excitation of class 5 track irregularities. Crack propagation curves corresponding to the different operating speeds were obtained as shown in Figure 15. Great differences in fatigue life in terms of year can be observed. When the passenger and freight train speeds are respectively 50 km/h and 40 km/h, it takes 224 years for the crack in the stringer-to-floor-beam bracket considered to propagate from 0.1 mm to 85 mm. Yet the associated time becomes 8 years if the speed combination is (70, 60) km/h. Additionally, The crack propagation curves obtained from speed combinations (80, 70) km/h and (110, 100) km/h almost overlap completely, since the equivalent stress range of the (80, 70) km/h combination is smaller while the stress cycle number in a stress block is larger compared with their counterparts of the (110, 100) km/h combination as reported in Table 6. One can observe in the table that train speeds can significantly affect the fatigue load effects (i.e., \(\Delta\)Seq and