## 2 Resistance Capacity and Performance Degradation Analysis
### Determination of the Degradation Initiation Time of the PSC-1 Girder  
The performance of a reinforced concrete structure often degrades because of corrosion of the reinforcing bars. Chloride attack is a major factor in the corrosion of reinforcing bars embedded in concrete. [PERSON] et al. [8] proposed a chloride diffusion model based on a diffusion equation derived from [PERSON]'s second law that considers the time dependence of the chloride diffusion coefficient (\(\overline{D}\)) and the dependence of the diffusion coefficient correction factor \((f(w_{c}))\) on the width of cracks in the concrete. In this study, the initiation time for the performance of the structure to degrade was calculated by using the improved chloride diffusion equations given below:  
\[C(x,t)=C_{0}\Bigg{|}1-\text{erf}\left(\frac{x}{2\sqrt{\overline{D}\cdot f(w_{ c})\cdot t}}\right)\Bigg{|}, \tag{8}\]  
\[\overline{D}=\left\{\begin{array}{c}\frac{D_{0}}{1-m}\left(\frac{t_{0}}{t} \right)^{m}t<t_{R}\\ D_{0}[1+\frac{t_{R}}{t}\big{(}\frac{m}{1-m}\big{)}]\big{(}\frac{t_{0}}{t_{R}} \big{)}^{m}t\geq t_{R}\end{array}\right., \tag{9}\]  
\[f(w_{c})=31.61w_{c}^{2}+4.73w_{c}+1\ (w\geq 0.1\ \text{mm}), \tag{10}\]  
where \(\overline{D}\) is the diffusion coefficient dependent on time, \(f(w_{c})\) is the crack effect function, \(w_{c}\) (mm) is the crack width, \(D_{0}\) is the diffusion coefficient at \(t_{0}\), \(m\) is constant depending on the mix proportion, \(t_{0}\) is the reference time, and \(t_{R}\) is the time at which the diffusion coefficient is assumed to be constant.  
When the diffused chloride reaches a certain concentration level at the depth of the rebar, the rebar corrodes. The chloride concentration that causes corrosion of the steel reinforcement is called the chloride threshold level (\(C_{\text{lim}}\)). The point at which the resistance capacity degrades can be predicted by calculating the time \(t\) at which the chloride concentration in the cover thickness of the concrete \(x\) exceeds the chloride threshold value (\(C_{\text{lim}}\)). Thus, the initiation time for the performance degradation of a reinforced concrete structure by steel corrosion can be determined by using the chloride concentration around the steel reinforcement in the concrete:  
\[C(x,t)>C_{\text{lim}}. \tag{11}\]  
The carbonation depth from the surface of the concrete due to the presence of carbon dioxide in the atmosphere can be expressed as Equation (12), which is dependent on time [9]. The progression of internal carbonation is proportional to the square root of the total elapsed time. Based on this equation, the estimated time (\(t\)) when the carbonation depth exceeds its limit in the concrete can be calculated by using the degradation initial time of the PSC-1 girder caused by carbonation (Equation (13)). The carbonation rate coefficient (\(\alpha_{d}\)) signifies the progress rate of carbonation and depends on the material properties of the concrete and the environmental conditions that the concrete is subjected to. The work by [PERSON] [10] was used to calculate the degradation initiation time according to the appropriate carbonation rate coefficient for the given material properties and environmental conditions (Table 2):  
\[y_{p}=\alpha_{d}\sqrt{t}, \tag{12}\]  
\[y_{\text{lim}}<y_{p}, \tag{13}\]  
where \(\alpha_{d}\) (cm/\(\sqrt{t}\)) is the carbonation rate coefficient, \(t\) (year) is the service life, \(y_{p}\) (mm) is the predicted carbonation depth, and \(y_{\text{lim}}\) (mm) is the limit depth.