# Fatigue Crack Calculation of Steel Structure Based on the Improved McEvily Model
## 2 Improved McEvily Model
### Improvement of Model Parameters  
The McEvilly model mainly explains the problem of fatigue crack growth rate through the effective range of macro crack stress ratio, maximum stress intensity factor, maximum stress intensity factor at the opening level, and stress intensity factor at the threshold level [12]. The McEvily model can be expressed as:\[\left. \begin{array}{l}\frac{da}{dN}=AM^{B}\\ M=K_{\max}(1-R)-\left(1-e^{-ka}\right)\left(K_{op\max}-RK_{\max}\right)-\Delta K _{effth}\\ K_{\max}=\sqrt{\pi r_{e}\left(\sec\frac{\pi\sigma_{\max}}{2\
u_{\rm v}}+1\right) }\left(1+Y(a)\sqrt{\frac{a}{2r_{e}}}\right)\sigma_{\max}\end{array} \right\}, \tag{2}\]  
where \(K_{\max}\) is the maximum stress intensity factor; R is the stress ratio (\(\sigma_{\max}/\sigma_{\max}\)); \(K_{op\max}\) is the maximum stress intensity factor at the opening level for a macroscopic crack; \(k\) is a material parameter; \(a\) is crack length; \(\Delta K_{effth}\) is the effective range of the stress intensity factor at the threshold level; \(r_{e}\) is an empirical material constant of the inherent flaw length of the order of 1 \(\mu\)m; \(\sigma_{\rm v}\) is the virtual strength of the material; \(Y(a)\) is a geometrical factor in calculating the stress intensity factors when under crack length \(a\); \(A\) and \(B\) are material constants, which are the control factors of fatigue crack growth and need to be obtained by fitting the test results. The fatigue cracks that appear in orthotropic steel decks in the actual engineering bridge are difficult to obtain the relevant parameters through the field test method, so the McEvily model is not ideally used in actual engineering. Therefore, the parameters that are \(K_{\max}\), \(k\), \(K_{op\max}\), and \(\Delta K_{effth}\) in the McEvily model were improved by using reasonable assumptions and empirical formulas to improve. First, the simplified calculation of \(K_{\max}\) is carried out, according to the theory of linear elastic fracture mechanics is obtained  
\[\left. \begin{array}{l}\Delta K=K_{\max}-K_{\min}\\ R=\frac{K_{\max}}{K_{\max}}\end{array} \right\}. \tag{3}\]  
Thus, \(K_{\max}\) can be determined by \(\Delta K\) and \(R\), namely  
\[K_{\max}=\frac{\Delta K}{1-R}. \tag{4}\]  
In orthotropic steel bridge decks, the U-rib weld is mainly subjected to the stress along the bridge direction, and the crack of the weld can be regarded as a surface crack subject to the far-field uniform tensile stress [17]. In order to facilitate the analysis of cracks, the cracks assumed to appear in the U-rib but weld are open cracks. Open cracks are the most common and most likely to cause fracture damage [18]. In order to describe the fatigue crack growth process of the U-rib butt weld, the semi-elliptical surface crack model of the finite thickness plate was employed. Figure 1 shows the semi-elliptical crack surface crack model of the finite thickness plate. According to the linear elastic fracture mechanics, the stress intensity factor of the surface crack in Figure 1 reaches the maximum value at the tip Q [19; 20]; its calculation can be expressed as follows:  
\[\Delta K=M_{1}M_{2}\frac{\sigma\sqrt{\pi a}}{E(z)}, \tag{5}\]  
where \(M_{1}\) is the free surface coefficient; \(M_{2}\) is the finite plate thickness correction factor; \(\sigma\) is the stress range; \(E(z)\) is the second type of complete elliptic integrals in calculating the stress intensity factor. The above  
where \(a\) is the crack depth; \(c\) is the crack width; \(w\) is the plate thickness corresponding to the U-rib, and W is the geometric function. During the crack propagation process of Q345 qD steel, to simplify the calculation, we supposed \(a/c\) = 0.1 is its constant while fatigue crack growth, and then \(M_{1}\) is 1.11, \(z\) is 0.995, and \(E(z)\) is 1.01. The range of the stress intensity factor could be rewritten as:  
\[\Delta K=1.11\times(\frac{2w}{\pi a}\tan\frac{\pi a}{2w})^{1/2}\frac{\sigma \sqrt{\pi a}}{1.01} \tag{7}\]  
Then make reasonable assumptions about material parameter \(k\).