## 6 Application to an Existing Bridge with Damage Detection Time-Based State Function  
The state function for optimum inspection planning can be formulated based on the damage detection time [2; 5; 6; 13]. The damage detection time-based state function (g(T)) is expressed as:  
\[g(\textbf{T})=t_{cr}-t_{det} \tag{11}\]  
where \(t_{cr}\) is the time at which the damage reaches the critical state, and \(t_{det}\) is the damage detection time. The formulation of \(t_{cr}\) was based on probabilistic damage propagation prediction. \(t_{det}\) was formulated using an event tree. An event tree represents all possible outcomes and their probabilities. In this case, the outcome was the damage detection time, and its probability was estimated by considering the effect of the degree of damage and the quality and application time of an inspection method for damage detection under uncertainty. Therefore, \(t_{cr}\) and \(t_{det}\) could be treated as the random variables. The detailed formulations of \(t_{cr}\) and \(t_{det}\) are described by [5; 13]. This application is based on an existing steel girder bridge, the I-64 Bridge over the Kanawha River in West Virginia. This bridge was constructed in 1974. The illustrative example is based on a fatigue critical detail, the gap between the bottom flange of the exterior girder and the end of the transverse connecting plate of the exterior girder, as shown in Figure 4. The detailed fatigue crack propagation is described by [18]. Herein, a nondestructive eddy current inspection (ECI) was applied to detect fatigue crack damage. The associated probability of damage detection is expressed as [6]:  
\[P_{ins}=1-\Phi\left(\frac{\ln(a)-\lambda}{\xi}\right) \tag{12}\]  
where \(a\) is the crack size, and \(\lambda\) and \(\zeta\) are the location and scale parameters. These parameters for ECI are \(-0.968\) and \(-0.571\), respectively [63]. The probability of damage detection (\(P_{ins}\)) was used to formulate \(t_{det}\). The damage detection time-based state function (g(T)) was formulated considering multiple probabilistic conditions; however, the state function (g(T)) expressed in Equation (11) is theoretical. The damage detection time (\(t_{det}\)) is influenced by the time  
Figure 4: Fatigue critical detail for illustrative application (adapted from [41]). of inspection, the number of inspections, quality of inspection, and damage propagation. State function \(g\)(T) cannot be expressed in a single closed form. Furthermore, the computational cost for estimating the probability of failure and reliability index should be minimized. Therefore, subset simulation, which computes multiple conditional probabilities using MCMC for intermediate failure events and requires a high computational cost, may be inappropriate for computing the probability of failure based on the damage detection time-based state function. Table 2 presents the probability of failure (\(p_{f}\)) and the reliability index (\(\beta\)) when MCS, KDE, and the combination KDE and GPD were applied. The entire PDF of the state function values when the inspection was applied at 10 years is illustrated in Figure 5a. Detail B in Figure 5a is illustrated in Figure 5b. Figure 6a shows the entire CDF corresponding to the PDF shown in Figure 5a. The probability of failure (\(p_{f}\)) is the CDF value when the state function value is equal to zero. Detail C in Figure 6b shows the differences between the CDFs obtained using MCS, KDE, and the combination of KDE and GPD in detail. A slight difference exists between the PDFs (or CDFs) provided by MCS, KDE, and the combination of KDE and GPD (see Figures 5b and 6b). The probability of failure (\(p_{f}\)) estimated using MCS and KDE was 0.0034. \(p_{f}\) computed using the combination of KDE and GPD was 0.0037, which was larger than those computed using MCS and KDE, as shown in Table 2. The CDFs of the state function values when inspections were applied at 7 years and 14 years obtained using MCS, KDE, and the combination of KDE and GPD are shown in Figure 7a.