## 5 Fatigue Crack Propagation Analysis
### Fatigue Life of the Bracket  
As discussed in the previous section, fatigue cracks are likely to initiate and propagate in the high-stress area of the stringer-to-floor-beam bracket. Since stress responses of the bracket are within the elastic range, based on the LEFM theory, Paris law characterizing the crack propagation relationship is adopted as [36]  
\[\frac{\mathrm{d}a}{\mathrm{d}N}=C(\Delta K)^{m} \tag{5}\]  
where \(a\) is the crack length; \(N\) is the number of stress cycles; \(C\) and \(m\) are the material constants; and \(\Delta K\) is the stress intensity factor range. Equation (5) defines the relationship between the crack growth rate \(\mathrm{d}a/\mathrm{d}N\) and the stress intensity factor \(K\), which can be expressed as:  
\[K=Y\cdot\sigma\cdot\sqrt{\pi a} \tag{6}\]  
where \(Y\) is the dimensionless factor, which is associated with the crack geometry and applied loads; and \(\sigma\) is the applied stress. Accordingly, the stress intensity factor range \(\Delta K\) can be written as:  
\[\Delta K=Y\cdot\Delta\sigma\cdot\sqrt{\pi a} \tag{7}\]  
Figure 10: Time-histories of the stresses in the high-stress area of the stringer-to-floor-beam bracket: (**a**) passenger train, 50 km/h; (**b**) freight train, 40 km/h; (**c**) passenger train, 70 km/h; and (**d**) freight train, 60 km/h. where \(\Delta\sigma\) is the constant-amplitude stress range. Combining Equations (5) and (7), fatigue life in terms of stress cycle count \(N\) associated with the stress range \(\Delta\sigma\) can be obtained as:  
\[N=\int_{a_{0}}^{a_{c}}\frac{\mathrm{d}a}{C(\Delta K)^{m}}=\int_{a_{0}}^{a_{c}} \frac{\mathrm{d}a}{C\big{(}Y\Delta\sigma\sqrt{\pi a}\big{)}^{m}} \tag{8}\]  
where \(a_{0}\) and \(a_{c}\) are the initial and critical crack length, respectively. If the dimensionless factor \(Y\) is a constant, through the integration operation of Equation (8), an explicit expression of fatigue life \(N\) can be obtained as:  
\[N=\frac{2}{(2-m)\cdot C\big{(}Y\Delta\sigma\sqrt{\pi}\big{)}^{m}}\Big{(}a_{c} ^{\frac{2-m}{2}}-a_{0}^{\frac{2-m}{2}}\Big{)}\;m\;\
eq 2 \tag{9}\]  
It should be noted that Equations (8) and (9) only provide fatigue life estimation related to the crack propagation. Since micro-cracks generally exist when the bridge is completed and put into service, fatigue life associated with the crack initiation may be conservatively ignored. In addition, the calculation of fatigue life \(N\) and stress intensity factor \(K\) requires knowledge of the dimensionless factor \(Y\). As discussed earlier, fatigue cracks tend to initiate from the upper border of the vertical plate of the stringer-to-floor-beam bracket (Figure 3). Therefore, an empirical formula of \(Y\) for single edge crack is adopted in this study as [37]  
\[Y=\sqrt{\frac{2b}{\pi a}\tan\frac{\pi a}{2b}}\frac{0.752+2.02\big{(}\frac{a}{b} \big{)}+0.37\big{(}1-\sin\frac{\pi a}{2b}\big{)}^{3}}{\cos\big{(}\frac{\pi a}{2 b}\big{)}} \tag{10}\]  
where \(b\) is the plate width. Equation (10) indicates that \(Y\) is the function of the crack length and plate dimension. As a result, the closed-form expression of fatigue life \(N\), i.e., Equation (9), may become unavailable due to the difficulty in the integration of Equation (8). In the present analysis, a numerical integration procedure is developed to obtain the fatigue crack propagation life. As shown in Figure 10, the Baihe Bridge always sustains variable-amplitude stresses subjected to the moving passenger and freight trains. For the sake of the application of Equation (8), the variable-amplitude stress ranges associated with the critical stringer-to-floor-beam bracket are converted into a constant-amplitude stress range, i.e., the equivalent stress range \(\Delta S_{\text{eq}}\) expressed as [38]  
\[\Delta S_{\text{eq}}=\left[\sum\frac{n_{i}}{\sum n_{i}}.(\Delta\sigma_{i})^{3 }\right]^{1/3} \tag{11}\]  
where \(n_{i}\) is the number of applied stress cycles corresponding to the stress range \(\Delta\sigma_{i}\); \(\sum n_{i}\) is the total number of stress cycles in a stress block.