# Corrosion Fatigue Assessment of Bridge Cables Based on Equivalent Initial Flaw Size Model
## 4 Corrosion Fatigue Lifespan Assessment of Parallel Wire Cable
### Cable Life Evaluation Model Considering Corrosion Fatigue Effect
From a conservative perspective, in the lifespan assessment, it is assumed that the wire enters corrosion of the bare wire,  
Figure 11: Parallel wire cables corrosion transfer schematic. and pitting pit growth starts immediately after the protective coating fails. The growth of pitting pits is determined based on the deformation of the pitting factor in Equation (18). \[a_{p}(t)=\phi_{\max}a_{uc}(t) \tag{18}\]  
where \(a_{p}(t)\) represents the maximum pitting depth as a function of time and \(a_{uc}(t)\) represents the uniform corrosion depth as a function of time. The pitting factor \(\varphi_{\max}\) is used to evaluate the probability distribution relationship of the maximum pitting depth over time. The Gumbel distribution model for the pitting factor of cable high-tensile steel wires was developed by [PERSON] et al. [44] based on the chunked sample extreme value model as shown in Equation (19):  
\[F_{\phi_{\max}}(x)=\exp\left(-\exp\left(-\frac{(x-\beta_{x})}{\alpha_{x}} \right)\right)\] (19a) with coefficients given by \[\alpha_{x}=0.954,\beta_{x}=0.905\ln(\frac{A_{\kappa}}{A_{0}})+4.078 \tag{19b}\]  
\[A_{\kappa}=\pi DL_{x},\kappa\geq 3 \tag{19c}\]  
where \(A_{0}\) is the surface area of the steel wire with a length of 21 mm and a diameter of 7 mm, \(\alpha_{x}\) and \(\beta_{x}\) denote the range scale and positional parameters of the surface area \(A_{\kappa}\), \(A_{\kappa}\) is the chunking area and \(L_{\kappa}\) is the chunking length. When the diameter is 7 mm, \(A_{0}\) = 329.7 mm\({}^{2}\), \(\alpha_{x}\) = 0.954, \(\beta_{x}\) = 4.078. The series of steel wires used had a diameter of 5 mm, the bare steel quality is the same as that of the literature [44], and the chunking method is used to analyze the probability of corrosion pits according to the surface area of the chunks so that the above model can still be applied. During the growth process of pitting pits, the pits are simplified into hemispherical shapes, and the transformation criterion is used as a criterion for converting pitting pits into cracks. The transformation criteria are given by the following:  
\[\left(\Delta K\right)_{pit}\geq\Delta K_{th} \tag{20}\]  
Figure 11.0  
Figure 11.1  
Figure 11.2  
where \(K_{t}\) is the stress concentration coefficient, \(\Delta\sigma\) is the stress amplitude, \(\varphi\) is determined to be \(\pi/2\), \(c\) is half the width of the pit, and \(a\) is the pit depth. Based on the transformation criterion, the corrosion fatigue lifespan \(N_{CF}\) can be divided into two stages: crack initiation lifespan \(N_{n}\) (i.e., the lifespan from corrosion pit growth to crack transformation) and crack propagation lifespan \(N_{p}\). Equation (24) is used to consider the effect of corrosion-fatigue coupling effect on crack propagation lifespan \(N_{p}\). Figure 11.3  
\(C_{cor}(N)\) is the correction factor of the corrosion fatigue crack growth rate, which is related to the material, environment and load condition. It should be noted that a large number of tests are needed to determine this parameter. At present, there is no correction parameter suitable for the corrosion fatigue crack growth rate of high-strength steel wire. The stress intensity factor (\(K_{IC}\)) reaching the fracture toughness value (\(\sigma_{\text{max}}\)) is considered as the judgment standard for steel wire fracture failure. Figure 11.4  
Figure 11.5  
Figure 11.6  
where is the mass loss rate and \(D_{0}\) is the diameter of the bare steel wire.