## 2 Mathematical Model for Bridge System Reliability
### Cable Strength Degradation Modeling  
It is known that the cables in a cable-supported bridge are prone to damage due to corrosion and fatigue. The average service time of a bridge cable is usually 20 years. Thus, in the reliability and safety evaluation, cable strength degradation should be taken into consideration. In general, cable damage can be considered in a parallel-series model accounting for the effect of cable length and number of cables [25]. In addition, in order to incorporate the cable strength degradation due to the fatigue-corrosion effect, cable strength can be derived from the strength of a wire using a modified Weibull distribution, which can be written as [26]:  
\[F_{Z}(z)=1-\exp\left[-\lambda\left(\frac{z}{u}\right)^{k}\right] \tag{4}\]  
where \(z\) is the strength of a short wire, \(u\) and \(k\) are the distribution parameters in a conventional Weibull function, \(\lambda=L/\!L_{0}\) is a scale parameter representing the fatigue-corrosion effect, \(L_{0}\) is the length of a wire specimen, and \(IL_{0}\) is the correlation length of the wire. A higher \(\lambda\) indicates a severer damage to the cable and a smaller correlation length. To model the probability distribution of a parallel-series system, a cable consisted of \(n\) wires was considered. [PERSON] et al. [27] provided the probability distribution of a cable based on the aforementioned Weibull distribution. The mean and standard deviation are:  
\[E(n)=nx_{0}(1-F_{Z}(x_{0}))+c_{n} \tag{5}\]  
\[D(n)=x_{0}[nF_{Z}(x_{0})(1-F_{Z}(x_{0}))]^{1/2}, \tag{6}\]  
where \(c_{n}=0.966 an^{1/3},a^{3}=\frac{f_{Z}^{2}(x_{0})x_{0}^{4}}{(2f_{Z}(x_{0})+x_{0 }/(x_{0}))},x_{0}={\left[l\frac{L_{0}}{LR}\right]}^{1/k}\omega_{u},\sigma_{u}\) is the ultimate stress of a wire, and \(f_{Z}(x_{0})\) is a probability density function following Weibull distribution. To model the cable strength degradation model, an experimental study on the strength of corroded cables with a service time of 20 years was considered [28]. Figure 1 presents the probability density function (PDF) of the cable strength of three types of short wires, including a new wire, an uncorroded wire, and a corroded wire. It can be observed that the mean strength value of the corroded wire was the lowest. In addition, the shape of the distribution changed significantly due to the corrosion-fatigue effect. According to the research conducted by [PERSON] et al. [16], the degradation curves of a cable can be obtained by:  
\[y_{1}(t)=-1.5\times 10^{-5}t^{2}-3.2\times 10^{-3}t+0.998 \tag{7}\]  
\[y_{2}(t)=-4.7\times 10^{-4}t^{2}-2.4\times 10^{-3}t+0.996 \tag{8}\]  
where \(t\) is the service time in years, and \(y_{1}\) and \(y_{2}\) are the degradation functions for the uncorroded and corroded cables, respectively. It can be concluded that the mean cable strength of the uncorroded and corroded cables decreased significantly during the service lifetime. In addition, the deviation of the probability was significantly affected by the corrosion effect. The cable strength of a cable with a number of long wires is usually modeled as an equivalent normal distribution function using Equations (5) and (6), as suggested by [PERSON] et al. [27]. Considering a cable with parallel steel wires of \(L_{b}=232\) m, n = 243, and \(\sigma_{u}=1766\) MPa, the equivalent normalized PDFs of the corroded, uncorroded and new cables are shown in Figure 2. It can be observed that the standard deviation exhibited a slight unexpected decrease. This phenomenon might due to the equivalent normalization in the series-parallel system of the cable. Figure 1: Comparison of the PDFs of the short wire strength accounting for fatigue damage and corrosion. By combining the PDFs of different wires (as shown in Figure 1), the time-varying PDF of the cable strength can be estimated. Therefore, based on Figure 2 and Equations (7) and (8), the time-varying PDFs were obtained (as shown in Figure 3). As it can be observed in Figure 3, the probabilistic characteristics of the corroded and uncorroded cables were totally different.