## 7 Application to an Existing Bridge with Extended Service Life-Based State Function  
The service life of deteriorating bridges can be optimally managed using reliability (or probability of failure). Considering the effect of inspections and maintenance on service life  
Figure 7: CDFs of damage detection time-based state function values using Monte Carlo simulation, kernel density estimation, and the combination of general Pareto distribution and KDE when inspections are applied at: (**a**) \(t_{imp1}\) = 7 years, \(t_{imp2}\) = 14 years and (**b**) \(t_{insp1}\) = 6 years, \(t_{insp2}\) = 12 years, \(t_{insp3}\) = 18 years. extension, reliability can be estimated using the extended service life-based state function (g(T)) as [2, 64]:  
\[g(\text{T})=t_{life,ex}-t_{lg} \tag{13}\]  
where \(t_{life,ex}\) = extended service life, and \(t_{lg}\) = predefined target service life. The extended service life (\(t_{life,ex}\)) is formulated based on an event tree. The outcome of the event tree is the extended service life. The probability of each outcome incorporates the probability of performing an inspection before reaching the service life, the probability of damage detection, and the probability of applying maintenance. Accordingly, the state function \(g(\text{T})\) cannot be expressed in a single closed form. Therefore, the subset simulation was not adopted in this illustrative application. Details on the formulation of \(t_{life,ex}\) are described in [2, 64]. This application is also based on the fatigue critical detail of an existing bridge shown in Figure 4.  
The probability of failure (\(p_{f}\)) and reliability index (\(\beta\)) based on the extended service life-based state function with one, two, and three inspections were applied at 10 years, 7, and 14 years, as well as 6, 12, and 18 years, respectively, are provided in Table 3. \(p_{f}\) and \(\beta\) were computed using MCS, KDE, and the combination of KDE and GPD. The PDFs and CDFs of the state function values for one-, two-, and three-time inspections are presented in Figures 8 and 9, respectively. When the inspection was applied at 10 years, the \(\beta\)s obtained using MCS, KDE, and the combination of KDE and GPD were 2.67, 2.67, and 2.56, respectively. As shown in Figure 8a,b, the PDFs obtained using MCS and KDE adequately represent the peak located near-10 years. This peak was not observed in the PDF obtained using the combination of KDE and GPD because for the combination of KDE and GPD, the predefined upper and lower thresholds were set as the top 1% and bottom 1% of the state function values, respectively, and the peak near -10 years was represented by a GPD in which the tail decreased exponentially. The CDFs corresponding to the PDFs shown in Figure 8b are shown in Figure 8c. The \(p_{f}\), which is the value of CDF when the state function is zero, estimated using the combination of KDE and GPD was 0.0053. As shown in Figure 8c and Table 3, the \(p_{f}\) under 10-year inspection estimated using the combination of KDE and GPD was the largest among the values obtained using MCS, KDE, and the combination of KDE and GPD. Because the PDFs based on MCS and KDE appropriately represented the peak located near -10 years, the \(p\)/s (or \(\beta\)) based on MCS and KDE were equal, as indicated in Table 3.