# Fatigue Crack Calculation of Steel Structure Based on the Improved McEvily Model
## 2 Improved McEvily Model
### Improvement of Model Parameters
The material parameter \(k\) reflects the change of the crack closure effect with the crack length. With the gradual increase in the material parameter \(k\), the crack growth rate decrease, and the crack length will also decrease [21]. It is obvious to know that the material parameter \(k\) has a certain effect on the crack growth rate. According to the research of [PERSON] [22], when calculating surface cracks, the value of \(k\) is expressed as a function of crack length \(a\), and the expression of the value of \(k\) is as follows:  
\[k=\left\{ \begin{array}{ll}\exp(10.6236-0.0096228a)&a\leq 200\ \upmu\\ 6000&a>200\ \upmu\end{array} \right.. \tag{8}\]  
Some studies have investigated that the reasonable lower limit of the average initial crack depth of welding details is 0.1 mm, much larger than 200 \(\upmu\)m [23; 24; 25]. Hence, the paper takes the average value of the initial crack \(a_{0}\) as 0.1 mm, and the material parameter \(k\) is set to 6000. Finally, the parameters of \(\Delta K_{\textit{effth}}\) and \(K_{\textit{opmax}}\) are simplified. Based on the concept of crack closure, through the research of [PERSON] [26], the \(K_{\textit{opmax}}\) can be defined with  
\[\Delta K_{\textit{opmax}}=K_{\textit{maxth}}-\Delta K_{\textit{effth}}, \tag{9}\]  
where \(K_{\textit{maxth}}\) is the maximum stress intensity factor at the threshold level. In order to obtain the empirical formula of \(\Delta K_{\textit{effth}}\), the parameter of \(\Delta K_{\textit{th}}\) is introduced, which is the range of the stress intensity factor range at the threshold. According to the average value of typical specimens used in the fatigue crack growth test, the ratio of \(\Delta K_{\textit{eff}}/\Delta K\) could be determined. The correlation between \(\Delta K_{\textit{eff}}\) and \(R\)[27] can be reflected as follows:  
\[\Delta K_{\textit{eff}}/\Delta K=f_{\textit{eff}}=\left\{ \begin{array}{ll}0. 52+0.42R+0.06R^{2}&R\geq 0\\ (0.52-0.1R)/(1-R)&-2\leq R<0\end{array} \right. \tag{10}\]  
Figure 1: The theoretical model of surface cracks. where \(\Delta K_{eff}\) is the range of the effective stress intensity factor. Therefore, the ratio of \(f_{eff}\) could be obtained according to Equation (10). A quantitative equation of crack growth rate was proposed by [PERSON] et al. [28], which can reflect the crack growth rate under different \(R\) by the crack growth rate of \(R\) is 0, and \(\Delta K_{th}\) can be inferred from \(\Delta K_{th0}\). \[\Delta K_{th}/\Delta K_{th0}=f_{th}=\left\{ \begin{array}{cl}(1-R)^{0.5}&-5 \leq R<0\\ (1-R)^{0.3}&0\leq R<0.5\\ (1.05-1.4R+0.6R^{2})^{0.3}&0.5\leq R<1\end{array} \right., \tag{11}\]  
where \(\Delta K_{th0}\) is the threshold stress intensity factor range when \(R\) is 0. According to the Equations (9)-(11), \(\Delta K_{effh}\) and \(K_{opmax}\) could be rewritten as:  
\[\Delta K_{effh}=f_{eff}f_{th}\Delta K_{th0}, \tag{12}\]  
\[\Delta K_{opmax}=K_{\text{max}th}-\Delta K_{effh}=[1/(1-R)-f_{eff}]f_{th} \Delta K_{th0}, \tag{13}\]  
when the above parameters are determined through assumptions and empirical formulas, the [PERSON] model can be modified as follows:  
\[\left. \begin{array}{l}\frac{da}{dN}=AM^{B}\\ M=[1+\frac{R}{1-R}(1-e^{-ka})]\Delta K-\frac{1-e^{-ka}}{1-R}f_{th}\Delta K_{ th0}-e^{-ka}f_{eff}f_{th}\Delta K_{th0}\\ \Delta K=1.11\times(\frac{2w}{\pi a}\tan\frac{\pi a}{2w})^{1/2}\frac{\Delta \varphi\sqrt{\pi a}}{1.101}\end{array} \right\}. \tag{14}\]