# Corrosion Fatigue Assessment of Bridge Cables Based on Equivalent Initial Flaw Size Model
## 3 Corrosion Fatigue Performance of Steel Wire
### Fatigue Lifespan Evaluation of Steel Wire  
The tested steel wire used here had a length of 300 mm, a nominal tensile strength of 1670 MPa (measured average tensile strength was 1935 MPa), a sample diameter of 5.4 mm, and a diameter of 5.25-5.27 mm after removing the coating. This study used 5.26 mm for calculations. Furthermore, the two-dimensional dimensions of corrosion fatigue sample fracture pits were extracted, and the influence of these pits on fatigue life was analyzed. The two-dimensional pit dimensions mainly consisted of the pit depth (\(a_{p}\)) and width (\(c_{p}\)), which were measured as shown in Figure 4. The three-point circle method was used to process the corroded cross-section. The three blue dots in Figure 4 are on the edge of the corroded cross-section and the red dot is the center of the circle determined by blue dots. Multiple lines were drawn from the center of the circle to the inner wall boundary of the pit. The point where the shortest distance from the line intersects the pit was considered the deepest point of the pit. The lines were extended to the outer surface of the steel wire, and the extension distance represented the pit depth. Additionally, the straight-line distance between points on both sides of the two-dimensional pit was considered the pit width. The measured pit parameters are listed in Table 1.  
Figure 3: Calculation procedure of EIFS. The measured pit parameters are substituted into Equations (1)-(10) in Section 2.2, and the calculation conditions and procedures of the EIFS and lifespan of each stage can conducted according to Figure 3. If the fatigue lifespan \(N_{F}\) of the corroded steel wire is equal to the test value \(N_{E}\), the initial defect depth of the corroded steel wire can be calculated. The calculation results are shown in the Table 1. The corrosion has two main aspects of impact on the fatigue lifespan: (1) The depth and width of the corrosion pit, as well as the size of the fracture area, directly affect the initiation of cracks. Deeper and wider corrosion pits, as well as larger fracture areas, can lead to a shorter crack initiation lifespan. (2) The size of the fracture area resulting from corrosion affects the rate of crack propagation. A larger fracture area can lead to faster crack growth and, consequently, a shorter crack propagation lifespan. Generally, corrosion affects the fatigue lifespan by influencing both the initiation and propagation stages of crack growth. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Extension**} & \multirow{2}{*}{ \begin{tabular}{c} **Depth of** \\ **Erosion** \\ **Pit \(a_{p}\)/mm** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Depth of** \\ **Erosion** \\ **Pit \(c_{p}\)/mm** \\ \end{tabular}  } & \multirow{2}{*}{ \begin{tabular}{c} **Fracture Area** \\ **S\({}_{coor}\)/mm\({}^{2}\)** \\ \end{tabular}  } & \multirow{2}{*}{\ \begin{tabular}{c} **\(a_{0}\)** \\ **/mm** \\ \end{tabular}  } & \multirow{2}{*}{
\begin{tabular}{c} **\(a_{sc}\)**Since the steel wire specimens have the same mass loss rate, the average corrosion depth \(D_{corr}\) can be considered constant, and the fractures of the steel wire are primarily influenced by individual large corrosion pits. Therefore, the corrosion pit depth and width can be considered as the two main factors determining the size of the fracture area. The depth-to-diameter ratio of the corrosion pits remains approximately constant across different depths. The relationship between EIFS and the variation in corrosion pit depth is illustrated in Figure 5.  
The regression fitting curves for equivalent initial crack depth \(a_{0}\) and corrosion pit depth for single and multiple fatigue source extension conditions are given in the following equation:  
\[ \begin{cases}a_{0}=0.000386+0.0156a_{p}^{0.938},\text{ for single fatigue source}\\ a_{0}=0.000443+0.0106a_{p}^{0.431},\text{ for multi fatigue source}\end{cases}