# Fatigue Crack Calculation of Steel Structure Based on the Improved McEvily Model
## 3 The Verification of Engineering Feasibility
### Comparison of Crack Fatigue Damage  
The nominal stress method is one of the most commonly used methods to evaluate fatigue cracks, and the calculated results of this method are relatively conservative. In  
Figure 10: Crack depth growth curve. Figure 9: Traffic flow predicted by Grey Theory. order to compare the calculation results of the improved [PERSON] model with the nominal stress method, the fatigue crack damage value index was used in this section. The nominal stress method based on Eurocode 3 was introduced, and the \(S-N\) curve of Eurocode 3 is shown in Figure 11.  
For the nominal stress method, Eurocode 3 provides the fatigue strength curves based on:  
\[\Delta\sigma_{R}^{3}N_{R}=\Delta\sigma_{\mathcal{C}}^{3}\cdot 2\times 10^{6}=K_{ \mathcal{C}}\left(N\leq 5\times 10^{6}\right), \tag{20}\]  
\[\Delta\sigma_{R}^{5}N_{R}=\Delta\sigma_{D}^{5}\cdot 5\times 10^{6}=K_{D}\, \left(5\times 10^{6}\leq N\leq 10^{8}\right), \tag{21}\]  
where \(\Delta\sigma_{R}\) is the stress range; \(N_{R}\) is the corresponding life expressed as the number of cycles related to \(\Delta\sigma_{R}\); \(\Delta\sigma_{\mathcal{C}}\) is the fatigue strength when fatigue life is \(2\times 10^{6}\) cycles. \(\Delta\sigma_{D}\) is the fatigue strength when fatigue life is \(5\times 10^{6}\) cycles; \(K_{\mathcal{C}}\) and \(K_{D}\) are fatigue strength coefficients. According to Eurocode 3, \(K_{\mathcal{C}}\) and \(K_{D}\) of the rib-to-rib welded details are \(7.16\times 10^{11}\) and \(1.90\times 10^{15}\), respectively. Therefore, the fatigue damage value of the stress range histogram can be defined as  
\[D=\sum_{S_{i}\geq\Delta\sigma_{D}}\frac{n_{i}S_{i}^{3}}{K_{\mathcal{C}}}+\sum _{S_{j}\leq\Delta\sigma_{D}}\frac{n_{j}S_{j}^{5}}{K_{D}}, \tag{22}\]  
where \(n_{i}\) is the number of stress range \(S_{i}\), which is bigger than \(\Delta\sigma_{D}\); and \(n_{j}\) is the number of stress range \(S_{j}\). In order to observe the fatigue damage value caused by each stress range, the fatigue damage histogram of strain gauge SG15 can be obtained through Figure 8 and Equation (22); the results are shown in Figure 12. Based on the actual annual traffic growth rate of 10.56%, the paper forecasted the fatigue damage value of SG15 from 2010 to 2040, and the annual fatigue damage value D\({}_{total}\) is  
\[D_{total}=\frac{x_{t}\sigma_{e}^{5}}{K_{D}}=365\times X_{d}\times(1+z)^{t} \frac{\sigma_{e}^{5}}{K_{D}}. \tag{23}\]  
Figure 11: \(S-N\) curve of Eurocode 3. The prediction results of the nominal stress method are shown in Figure 13.  
Based on the theory of linear elastic fracture mechanics, the prediction of fatigue crack depth from 2010 to 2040 can be converted into annual fatigue damage according to Equations (24) and (25). The expression of annual fatigue damage value is  
Figure 9:0  
Figure 9:1  
The annual fatigue damage value of the improved McEvily model and Paris model in 2010-2040 is shown in Figure 13, and the initial fatigue damage value is 0. From Figure 13, it can be seen that the fatigue damage value calculated by the improved McEvily model is similar to the nominal stress method. Since the improved McEvily model considers more engineering factors, the growth rate is greater than that calculated by the nominal stress method. The Paris model is significantly less than the nominal stress method and the improved McEvily model in the calculated results, which cannot accurately evaluate the fatigue crack life. Because the Paris model has limited in predicting the fatigue life of small cracks, resulting in smaller calculated results. It shows that the accuracy of the improved McEvily model in the calculated results is greater than that of the Paris model and can be well adapted to practical engineering. The improved McEvily model can preliminarily  
Figure 9:2 Fatigue damage histogram of gauge SG15. Figure 9:3 Fatigue damage prediction results. evaluate the fatigue crack life, providing convenience for bridge managers and ensuring the safety of orthotropic steel bridge decks.