# Corrosion Fatigue Assessment of Bridge Cables Based on Equivalent Initial Flaw Size Model
## 4 Corrosion Fatigue Lifespan Assessment of Parallel Wire Cable
### Load Effect on Corrosion Fatigue Lifespan  
The load effect is a crucial factor for the fatigue lifespan of a structure, especially considering the joint effects of corrosion. To measure the effects of corrosion-induced degradation on the bridge load response, [PERSON] [45] proposed fault trees based on the experience of technicians and numerical simulations of different failure scenarios and successfully applied them to a historic steel truss bridge. [PERSON] [46] conducted a dual experimental and analytical study on the fatigue performance of the Quisi and Ferrandet Bridges and investigated the load effect on the corrosion fatigue lifespan. Based on the inspiration of the above research, this study predicts the corrosion fatigue lifespan of the Runyang Suspension Bridge by calculating the response of the actual active load. The rainfall counting method is used to process the stress time spectrum, and the equivalent stress amplitude [47] is introduced to calculate the lifespan. The equivalent force amplitude and the number of cycles can be expressed as follows:  
\[\sigma_{eq}=\left(\frac{\sum\sigma_{eq}n_{i}}{\sum n_{i}}\right)^{\frac{1}{n}} \tag{29}\]  
\[N_{eq}=\sum n_{i} \tag{30}\]  
\[f_{eq}=\frac{t_{eq}}{N_{eq}} \tag{31}\]  
where \(\sigma_{i}\) represents the stress amplitude for the \(i\)th type, \(n_{i}\) represents the number of cycles, and \(m\) represents the slope of the S-N curve, which is assumed to be 3. \(t_{eq}\) represents the time duration for that stress amplitude. Calculations yield an equivalent stress amplitude of 23.65 MPa and 70 cycles for the Runyang Suspension Bridge during the specified time period. For the calculation, data with stress amplitudes below 1 MPa are excluded. The theoretical model proposed in Section 4.1 is adopted. After the coating is completely corroded, the uniform corrosion depth of the bare steel wire is calculated. Assuming the uniform corrosion depth at the current time \(t_{i}\) is \(a_{uc,i}\), the uniform corrosion depth can be represented as follows:  
\[a_{uc,i}=1.79011\cdot\left[\tau(t_{i}-t_{0})\right]^{0.62044} \tag{32}\]  
Equation (32) represents the uniform corrosion depth of the most severely corroded steel wire inside the sheathing. In the equation, \(t\) represents the total time the steel wire has experienced, \(t_{0}\) represents the time for the coating to be depleted, and \(\tau\) represents the conversion coefficient between the actual environment and the experimental environment. Equation (33) is used to investigate the uniform corrosion degree of the remaining cable wires:  
\[a_{uc,i}=R_{j}a_{uc,i} \tag{33}\]  
where \(R_{j}\) represents the corrosion ratio of the \(j\)th wire. Assuming that pitting corrosion starts to occur after the coating is completely corroded, the calculation method for the corrosion depth is as follows: sampling is performed for the pit factor \(\varphi_{\text{max}}\). When the calculation time reaches \(t_{i}\), the maximum pit depth calculated is \(a_{p}(t_{i})\). When the calculation time reaches \(t_{i}\) +1, the maximum pit depth calculated is \(a_{p}(t_{i}\) +1). The calculation formula is as follows:  
\[a_{p}(t_{i+1})=a_{p}(t_{i})+\phi_{\text{max},i+1}[a_{uc}(t_{i+1})-a_{uc}(t_{i})] \tag{34}\]  
The corrosion rate is calculated using the growth rate of pitting pits. \[v_{p}=\frac{a_{p}(t_{i+1})-a_{p}(t_{i})}{t_{i+1}-t_{i}}=\phi_{\text{max},i} \frac{a_{uc}(t_{i+1})-a_{uc}(t_{i})}{t_{i+1}-t_{i}} \tag{35}\]When using the aforementioned method for calculation, the influence of the corrosion-fatigue coupling effect on crack propagation is not considered. \(C_{cor}\) (corrosion factor) is set as 1 for both conditions. The calculation time interval (\(t_{i+1}-t_{i}\)) is set as 1 month. Additionally, considering that the nighttime traffic volume is negligible, the contribution of traffic loads to the fatigue damage of the steel wire can be ignored. Therefore, when calculating the equivalent stress amplitude using the stress spectrum conversion method mentioned above, only the daytime effect is considered.