## 4 Determination of Initial Crack Size and Critical Crack Size  
Chinese code GB/T 18590-2001 \"Corrosion of metals and alloys-Evaluation of pitting corrosion\" [31] states that radiography, one of the in situ measurement methods, can be used to measure the depth of the corrosion pits on real bridges. Due to the short service time of the bridge, the actual measurement data are temporarily difficult to obtain. Therefore, from the perspective of scientific research, the assumed initial crack size, initial shape, and other crack information are used. The thickness of the plate N6', where the structural stress is concentrated at the end of the cable 28, is 24 mm, and the initial crack sizes \(a_{0}\) are selected \begin{table}
\begin{tabular}{c c c} \hline \hline
**Fatigue Detail** & \
\begin{tabular}{c} **Location of Stress** \\ **Concentration at the Bearing** \\ **Structure of Cable 1** \\ \end{tabular} &
\begin{tabular}{c} **Location of Stress** \\ **Concentration at the Bearing** \\ **Structure of Cable 28** \\ \end{tabular} \\ \hline Fatigue damage & 7.69 \(\times 10^{-15}\) & 4.04 \(\times 10^{-13}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fatigue damage caused by a single pass by fatigue-load calculation model II. Figure 7: Time history of stress of load-bearing structures at the ends of cable 1 and cable 28.  
as 1.00 mm, 2.00 mm, and 3.00 mm to study the influence of the corrosion pits and the initial pit size on the fatigue life of the weathering-steel anchor box. The critical crack size can either be determined according to the applicability principle to obtain \(a_{max}\) or be determined according to the fracture mechanics criterion, usually the commonly used K criterion, to obtain \(a_{f}\), and then the smaller of the values of \(a_{max}\) and \(a_{f}\) is taken [32]. For the load-bearing structure at the end of cable 28, the applicability principle assumes that once the crack penetrates the steel plate and the critical crack size is reached, the component will fail and the calculation will be terminated. Therefore, \(a_{max}\) is taken as the N6' plate thickness of 24 mm. The technical conditions of the weathering steel Q355 NHD used in weathering-steel anchor boxes meets the requirements of Chinese code GB/T 4171-2008 \"Atmospheric corrosion resisting structural steel\" [33], and the impact energy at the temperature of \(-20\) degrees is not less than 34 J. According to the conversion formula of impact energy and fracture toughness recommended in Appendix J of British code BS 7910 [34], the fracture toughness of Q355 NHD can be obtained as:  
\[K_{C}=\left[\left(12\sqrt{C_{v}}-20\right)\left(\frac{25}{B}\right)^{0.25} \right]+20 \tag{4}\]  
where \(K_{C}\) is fracture toughness, in MPa\(\cdot\)m\({}^{1/2}\); \(C_{v}\) is the impact energy at a specific temperature, in J; \(B\) is the thickness of the material for which the fracture toughness needs to be evaluated, in mm. Therefore, the fracture toughness of the Q355 NHD steel plate at the service temperature is not less than 70.5 MPa\(\cdot\)m\({}^{1/2}\). According to the commonly used \(K\) criterion, the critical crack length \(a_{f}\) can be obtained according to Equation (5) as [35]:  
\[a_{f}=\frac{Q\left(\frac{K_{C}}{Y\sigma_{max}}\right)^{2}}{\pi} \tag{5}\]  
where \(a_{f}\) is the critical crack length; \(Y\) is the geometric correction factor; \(\sigma_{max}\) is the maximum value of the sum of constant and live load stress; and \(Q\) is the shape factor of the semi-elliptical crack, which can be calculated by Equation (6) [35]:  
\[Q=1+1.464\left(\frac{a}{c}\right)^{1.65}\ \left(\frac{a}{c}\leq 1\right) \tag{6}\]  
The variable amplitude load at the end of cable 28 is mainly the load in the direction of the cable force, while the load amplitude in other directions is much smaller, and therefore, the critical crack obtained by the mode I opening-mode crack is taken as \(a_{f}\).