## 2 Reinforcement Steel Corrosion Model
### Corrosion Time and Rate of Reinforcement  
Existing research results show that chloride erosion is the main cause of reinforcement corrosion. The initial corrosion of reinforcement is marked by a critical chloride concentration on the surface of the reinforcement. Additionally, the diffusion process of chloride conforms to [PERSON]'s second diffusion law. It can be concluded that the initial time \(T_{i}\) of reinforcement corrosion can be expressed as follows:  
\[T_{i}=\frac{C^{2}}{D_{C}}\bigg{[}erf\bigg{(}\frac{C_{0}-C_{cr}}{C_{0}}\bigg{)} \bigg{]}^{-2}, \tag{1}\]  
where \(erf\) is the error function; \(C\) is the concrete cover thickness; \(D_{C}\) is the chloride diffusion coefficient; \(C_{cr}\) is the critical chloride concentration; \(C_{0}\) is the mass concentration of chloride on the concrete surface; \(C\), \(D_{C}\), \(C_{cr}\), and \(C_{0}\) are all random variables; the probability distribution types and characteristic parameters of such variables can be obtained by experimental and theoretical model calculation [13; 38]; and the initial corrosion time of reinforcement \(T_{i}\) is also a random variable according to Equation (1).  
The corrosion of reinforcement is an electrochemical reaction process, and there is a quantitative relationship between the corrosion rate of reinforcement and the corrosion current density at time \(t\). [PERSON] provided calculation equations of the corrosion current density and corrosion rate of reinforcement at time \(t\) after the beginning of corrosion [39; 40], which can be expressed as:  
\[i_{c}(t)=i_{c0}\times 0.85(t-T_{i})^{-0.29}, \tag{2}\]  
\[r(t)=0.0116i_{c}(t), \tag{3}\]  
where \(i_{c0}\) is the current density at the initiation of reinforcement corrosion, expressed as \(i_{c0}=\left[37.8(1-w/c)^{-1.64}\right]/C\), which is related to water/cement ratio w/c and concrete cover thickness \(C\). As the water/cement ratio of bridge concrete is difficult to be directly obtained by experimental methods, the relationship between the water/cement ratio and strength of structural concrete in the specification for mix proportion design of ordinary concrete in China is innovatively introduced in this paper [37], and the water/cement ratio of concrete can be indirectly calculated by Equation (4),  
\[w/c=\frac{a_{a}f_{b}}{f_{cu,k}+\alpha_{a}\alpha_{b}f_{b}}, \tag{4}\]  
where \(f_{cu,k}\) is measured compressive strength of concrete; \(f_{b}\) is the 28d compressive strength of cement mortar, and the value is 32.5 MPa; and \(\alpha_{a}\) and \(\alpha_{b}\) are regression coefficients, which are adopted as 0.53 and 0.2, respectively, in this paper.  
By substituting Equation (4) into Equation (3), the relationship between the corrosion rate of reinforcement, the concrete strength, and the concrete cover thickness can be obtained, which is expressed as:  
\[r(t)=0.3727\bigg{(}\frac{f_{cu,k}+3.445}{f_{cu,k}-13.78}\bigg{)}^{1.64}\frac{ (t-T_{i})^{-0.29}}{C}. \tag{5}\]  
Equation (5), used to calculate the corrosion rate, is derived based on Equation (2) to Equation (4). Reinforcement corrosion is related to chloride diffusion, and concrete strength and concrete cover thickness are essential influencing factors for chloride diffusion. The corrosion rate decreases with the concrete strength and concrete cover thickness increasing [39].