# Fatigue Crack Calculation of Steel Structure Based on the Improved McEvily Model
## 3 The Verification of Engineering Feasibility
### Calculation of Fatigue Crack Growth  
The annual fatigue crack depth is the key index to calculate the fatigue crack damage value by the fracture mechanics method. Firstly, the number of load cycles needs to be analyzed according to the monitoring data, the strain\(-\)time curve in Figure 7d can be transformed into the stress\(-\)time curve by multiplying the elastic modulus, and the stress range was extracted from the stress-time curve using rain\(-\)flow counting method [32], the stress range \(\sigma_{i}\) was from 0 to 35 MPa, as shown in Figure 8. According to [PERSON]'s damage accumulation theory [33], the daily effective stress range \(\sigma_{e}\) under the variable amplitude load is expressed as  
\[\sigma_{e}=\left(\sum\frac{n_{i}}{N_{a}}\sigma_{i}^{m}\right)^{1/m}, \tag{16}\]  
where \(n_{i}\) is the number of cycles of the stress range \(\sigma_{i}\); \(N_{a}\) is the number of cycles of all the stress ranges; \(m\) is related to material properties. Figure 7: Signal of gauge SG15. (**a**) Original strain–time curve; (**b**) Strain\(-\)time curve affected by temperature; (**c**) Strain–time curve caused by random ambient excitations; (**d**) Strain–time curve caused by vehicle loads. Due to the effective daily stress range, \(\sigma_{e}\) is a random variable, the mean value of the effective daily stress range is represented as  
\[E(\sigma)=\int_{0}^{+\infty}\sigma_{e}f(\sigma)d\sigma_{e}, \tag{17}\]  
where \(f(\sigma)\) is the probability density of the daily effective stress range. Secondly, the cumulative damage induced by repetitive vehicle load actions is the primary source of steel bridge fatigue damage. Thus, the growth of traffic flow needs to be considered in calculating the fatigue crack depth. The subjects of this study were a significant amount of field data acquired from the suspension bridge's health monitoring system between 2010 and 2018. Based on the known traffic flow from 2010 to 2015 years, the gray theory [34] is used to predict the traffic flow from 2016 to 2040 years. The prediction model of the annual traffic flow of the suspension bridge is:  
\[\hat{x}^{(1)}(t+1)=4223.976143\times e^{3.018}-3909.976143\] \[\hat{x}^{(0)}(t+1)=\hat{x}^{(1)}(t+1)-\hat{x}^{(1)}(t) \tag{18}\]  
where \(\hat{x}^{(1)}(t)\) is the cumulative annual traffic flow from 2010 to \(t\). \(\hat{x}^{(0)}(t+1)\) is the annual traffic flow in year \((t+1)\). Figure 9 shows the traffic flow from 2016 to 2040 predicted by the gray theory. The maximum error of the comparison between the actual traffic flow and the predicted traffic flow in 2016\(-\)2018 is not more than 5%, indicating that the accuracy of the predicted traffic flow can be verified, which can be seen in Figure 9. The data curve fitting of the predicted traffic flow shows that the annual traffic flow increases exponentially at a 10.56% growth rate. Then, the number of stress cycles of vehicle load within one year \(X(t)\) is  
\[X(t)=365\times X_{d}\times(1+z)^{t}, \tag{19}\]  
where \(X_{d}\) is the mean of the number of daily stress cycles corresponding to the mean of the daily effective stress ranges \(E(\sigma)\); \(z\) is the annual traffic growth rate. Figure 8: Stress range cycle number of gauge SG15. Finally, based on the stress range cycle number histogram and the predicted annual traffic flow, the study used the Markov chain [35] to analyze and obtain the annual fatigue crack depth. Figure 10 showed crack depth increment predicted by the improved McEvily model and the Paris model, in which \(C\) and \(m\) used in Paris model are \(7.158\times 10^{-11}\) and \(3\), respectively, according to the Eurocode 3 [36]. From Figure 10, it can be seen that the crack depth increases exponentially with the increase in annual traffic flow. The crack growth predicted by the two models increases slowly in the initial stage and rapidly in the later stage, which is consistent with the phenomenon observed in practical engineering.