# 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  
When dealing with the load-bearing capacity of parallel wire cables, the bundle-based series-parallel model is a commonly used method. The main idea is to consider the cable as a parallel combination of equi-length wires with each wire being treated as a structure composed of a certain number of wire units connected in series. Friction units are used between adjacent wires to represent micro-motion wear between wires. For parallel wire cables, especially suspension cable systems, the stress on the wires is primarily uniaxial tension, so the friction effects between the wires can be omitted. Corrosion in cable systems is mainly manifested as external corrosion and internal corrosion, with the former having a greater impact. When the sheath cracks, the wires come into contact with the external corrosive environment directly. Therefore, the focus in this paper is primarily on the influence of external corrosion on the cable while neglecting the effects of wear between wires. As shown in Figure 10, due to the spatial arrangement of the wires, the influence of the corrosive environment on the corrosion transmission path of individual wires varies within the cable. Therefore, under a specified stress amplitude, each wire will have a different time to fracture. The cable is considered to have failed when the specified wire breakage rate is reached. The corrosion failure process of a corroded cable system mainly includes the failure of the corrosion protection system and the failure of the wire's load-bearing system. The failure of the corrosion protection system for the wires primarily refers to the corrosion of the coating. The failure of the wire's load-bearing system occurs during the entire stage from the onset of corrosion of the bare wire to its fracture. This includes uniform  
Figure 10: Parallel wire cables whole lifespan calculation model. corrosion and pitting corrosion of the bare wire. After the occurrence of pitting corrosion, the pits undergo three stages: nucleation, growth, and transformation into cracks. The wire fractures when the crack extends to the critical fracture depth. The parallel wire cables corrosion transfer process is shown in Figure 11.  
After simplifying the parallel wire cables to a series-parallel model, stress redistribution occurs in the parallel wire cables when breakage occurs in a single wire. The fatigue stress after wire breakage is calculated using the following equation:  
\[\sigma_{a,i}=\frac{m}{m-i}\sigma_{a} \tag{16}\]  
where \(m\) represents the total number of wires in the cable system. \(i\) represents the total number of wire breaks after experiencing a certain fatigue load. \(\sigma_{a}\) is the stress amplitude in the initial state. \(\sigma_{a,i}\) is the fatigue stress amplitude after wire breakage. In the study conducted by [PERSON] [43], the parallel wire ropes from the Shimen Bridge in Chongqing, with different levels of corrosion and corroded cross-sectional distribution, were investigated. They established a statistical corrosion distribution model for corroded wires. To describe the transmission path of corrosive medium within the rope, the model is adopted. The corrosion rate (\(R\)) is used to evaluate the transmission process:  
\[R=\frac{d_{o}-d_{\min}}{d_{o}-d_{\min,0}} \tag{17}\]  
where \(d_{0}\) represents the nominal diameter of the wire, \(d_{\min}\) represents the minimum diameter of the remaining wires at other positions on the cross-section, and \(d_{\min,0}\) represents the minimum diameter of the reference wire. The expected value of \(R\) in the radial direction is \(E(R_{i})=0.844^{i}\), and in the circumferential direction, it is \(E(R_{i})=0.868^{i}\). After coating corrosion failure, the corrosion products still provide some corrosion protection to the bare wire, and there is a certain incubation period for the initiation of pitting pits after uniform corrosion of the bare wire begins.