## 5 Fatigue Crack Propagation Analysis
### Fatigue Life of the Bracket
Herein, one stress block is defined as the stress time-histories generated by 22 passenger train passages and 28 freight train passages in an entire day. Equation (11) indicates that the fatigue damage caused by the constant-amplitude equivalent stress range \(\Delta S_{\text{eq}}\) is equal to that produced by the variable-amplitude stress ranges \(\Delta\sigma_{i}\). Consequently, the number of stress cycles \(N\) required for the crack propagation from initial length \(a_{0}\) to critical length \(a_{c}\) can be rewritten as:  
\[N=\int_{a_{0}}^{a_{c}}\frac{\mathrm{d}a}{C(Y\Delta S_{\text{eq}}\sqrt{\pi a} )^{m}} \tag{12}\]  
In this study, fatigue stresses in the high-stress area of the stringer-to-floor-beam bracket obtained from the multi-scale vibration analysis of the coupled TTB system associated with both the passenger and freight trains are processed to generate the stress range histograms using the rain-flow counting method. Specifically, element stress results extracted from the bridge FE model corresponding to the high-stress area of the bracket are considered herein. Subsequently, the equivalent stress range \(\Delta S_{\text{eq}}\) is calculated based on the stress range histogram of one stress block. Fatigue life \(N\) is then estimatedby assuming a small crack length increment at each step of iteration and numerically solving the integration shown in Equation (12). Values of the initial crack length can impact the estimation results of fatigue life [39]. These values can be determined by performing an in-situ nondestructive test on the components of the considered bridge with initial flaws. Due to the lack of initial crack size data associated with the investigated bracket, in this study several initial crack length values in the range of 0.1 mm to 0.25 mm recommended by BS 7608 [40] are adopted considering the uncertainty in determination of the initial length. The critical crack length represents a certain crack size beyond which fracture failures or loss of load-carrying capacity will occur to the structural details. Accordingly, this value could be determined by fracture mechanics or assuming a size proportional to the dimensions of structural components, e.g., plate width or thickness. In addition, accurate identification of the two material constants for applying the Paris law in Equation (5), i.e., \(C\) and \(m\), usually requires a fatigue crack growth rate test. These two constants take the values of \(1.26\times 10^{-13}\) and 3.0 respectively, based on the investigation in [PERSON] 1984 [41]. Figure 11 shows the computed crack propagation history of the stringer-to-floor beam bracket from the year 1980 when the Baihe Bridge was opened to railway traffic. Four different initial crack sizes \(a_{0}\) = 0.1 mm, 0.15 mm, 0.2 mm, and 0.25 mm are considered. Additionally, the critical crack length \(a_{\rm c}\) = 85 mm is adopted herein, which represents half of the bracket width. It can be observed in Figure 11 that the crack propagation amount in the loading period 1980-1997 is quite small; however, after 1997, the crack grows quickly because the running speeds of the passenger train and freight train respectively increase to 70 km/h and 60 km/h as discussed earlier. Therefore, train speeds can significantly affect the fatigue cracking of the bracket, and this effect will be further discussed later on. When the initial crack size \(a_{0}\) takes the values of 0.1 mm, 0.15 mm, 0.2 mm, and 0.25 mm, fatigue life in terms of year needed for the crack length to reach 85 mm is calculated as 24.9, 23.5, 22.6, and 21.9 years respectively. It was reported in [PERSON] et al. [19] that a 60 mm crack was detected in the bracket under consideration in 2006 and a round hole was drilled to prevent further development of the crack (Figure 12). Based on the results shown in Figure 11, the year associated with a critical crack size of 60 mm and initial crack size of 0.1 mm is about 2004. Hence, the predicted crack growth rate is slightly greater than the situation in the real world.