Efficient Reliability-Based Inspection Planning for Deteriorating Bridges Using Extrapolation Approaches

Abstract

In this study, inspection planning of deteriorating bridges is optimized to determine the inspection application times and methods based on various objectives. These objectives can be formulated by considering the probabilistic structural performance and service life after inspection and maintenance. Probabilistic structural performance and service life prediction are generally based on the probability of failure (or reliability). However, there are difficulties associated with optimizing inspection planning when a low probability of failure is estimated. In this study, we address inspection planning using extrapolation approaches to efficiently compute a low probability of failure. The inspection planning method proposed in this study determines the inspection application times for a given inspection method. We investigated the applicability of direct Monte Carlo simulation (MCS), subset simulation, and two extrapolation approaches (i.e., kernel density estimation (KDE) and KDE combined with generalized Pareto distribution (GPD)) for inspection planning. The probability of failure for optimum inspection planning was based on the damage detection-based state function and extended service life-based state function. These state functions were formulated by considering damage propagation, damage detection by inspections, and service life extensions by maintenance. Illustrative applications to general examples and an existing bridge are provided to investigate the effects of approaches for computing the failure probability on the accuracy and variation of the optimum inspection application times. Finally, the most appropriate approach for optimum inspection planning was determined considering the accuracy and reliability of the solution, computational efficiency, and the applicability of the probabilistic optimization process. The presented investigations revealed that KDE is more appropriate than MCS and the combination of KDE and GPD for optimum inspection planning.

1. Introduction

Bridges are inspected periodically to ensure their safety, as their performance continuously deteriorates owing to various environmental stressors and loading effects [1,2,3]. Depending on the accuracy and reliability of inspection outcomes, effective decisions can be made with respect to inspection schedules, in-depth inspections, type of maintenance, and replacement [4]. To increase the accuracy and reliability of inspection outcomes, the quality of inspection methods and the number of inspections should be improved and increased, respectively. However, such improvements and increases incur additional costs. Therefore, inspection planning is essential to optimize the inspection application times and methods [5,6]. Over the last few decades, approaches have been investigated for the development of optimum inspection planning with various probabilistic objectives [2,5,7]. These probabilistic objectives are generally based on the damage detection time, probability of failure after inspection and maintenance, value of information obtained from inspections, extended service life with inspection and maintenance, life-cycle cost, and risk, among other factors [8,9,10,11,12,13,14,15,16,17].

Several difficulties are encountered in probabilistic optimum inspection planning. The objectives of optimum inspection planning should be formulated based on the uncertainties associated with damage propagation, damage detection, structural performance assessment and prediction, and estimations of the life-cycle cost and risk. Accordingly, the objectives can be expressed in complex forms using various probabilistic parameters and conditions. Solving a single- or multiobjective optimization problem consisting of probabilistic parameters and conditions is time-consuming [15,18]. The computational time required for optimum inspection planning can be reduced by applying an efficient optimization algorithm and efficient computation of objective values in the optimization algorithm.

Genetic algorithms (GAs) are generally adopted when various conditions are considered in a single- or multiobjective optimization problem and the objectives are expressed in non-closed forms, as GAs can address discrete, continuous, and non-differentiable objectives [19]. An increase in the number of objectives increases the computational cost and reduces the accuracy and reliability of solutions. Continuous efforts have been made over the last few decades to develop more advanced GAs that can improve the accuracy, reliability, and computational efficiency of solutions [20,21,22,23,24,25].

Formulating the objectives for optimum inspection planning requires the computation of time-dependent probability of failure (or reliability). When the state function used to compute the probability of failure is formulated based on various probabilistic conditions, is not in a closed form, and the computed probability of failure is relatively low, so a long computational time is required to obtain an accurate probability of failure. In this paper, we present extrapolation approaches to efficiently address the difficulties associated with computing the probability of failure (or reliability) for optimum inspection planning. The optimum inspection planning method proposed in this study produces the optimum inspection application times for a given inspection method. The use of reliability for optimum inspection planning was also reviewed. The representative difficulties associated with computing the probability of failure for optimum inspection planning are described. Direct Monte Carlo simulation (MCS), subset simulation, and two extrapolation approaches (kernel density estimation (KDE) and a combination KDE and generalized Pareto distribution (GPD)) were considered candidate approaches to compute the probability of failure for optimum inspection planning. These approaches were applied to three general examples and an existing bridge to investigate the accuracy, computational efficiency, reliability, and applicability of the probabilistic optimization process. The damage detection-based and extended service life-based state functions, which were formulated considering damage propagation and detection after inspections and service-life extensions after maintenance, were used to compute the probability of failure for optimum inspection planning. By applying the proposed method to an existing bridge, the effects of the approaches (MCS, KDE, and the combination of KDE and GPD) on the accuracy and reliability of the optimum inspection planning were investigated.

2. Use of Reliability for Optimum Inspection Planning

Reliability (or probability of failure) is one of the most representative probabilistic performance indicators used to design new structures, as well as to evaluate the performance and manage the service life of deteriorating bridges [26,27,28,29,30]. Other representative probabilistic performance indicators, including redundancy, vulnerability, robustness, and risk, have also been formulated based on the probability of failure and reliability [2,6,31].

2.1. General Concept of Probability of Failure

Reliability (p_s) is generally defined as the probability that a structure will survive for a specific time interval. It is the complement of probability of failure (p_f) (i.e., p_s = 1 − p_f) [26,32]. The reliability (p_s) and probability of failure (p_f) can be expressed as:

$$p_s={\displaystyle \underset{g\left(\mathbf{X}\right)>0}{\int }{f}_{\mathbf{X}}\left(\mathbf{x}\right)d\mathbf{x}}$$

(1a)

$$p_f=1-p_s={\displaystyle \underset{g\left(\mathbf{X}\right)<0}{\int }{f}_{\mathbf{X}}\left(\mathbf{x}\right)d\mathbf{x}}$$

(1b)

where f_X(x) is the joint probability density function (PDF) of random variables (X); and g(X) is the state function; and g(X) > 0 and g(X) < 0 indicate the safe and failure states, respectively. Figure 1 shows the relationship between the state function value, its PDF, and cumulative distribution function (CDF). The probability of failure (p_f) is equal to the area under the PDF of g(X), which is upper-bounded by the vertical axis located at g(X) = 0 (see Figure 1a). In addition, p_f is equal to the CDF value when g(X) = 0, as shown in Figure 1b. Generally, the state function (g(X)) is formulated as the difference between the resistance and the load effects of a structural component. The time-dependent state function (g(X, t)) can be expressed by considering the changes in external and internal forces, associated stresses and strains, and the degree of damage (e.g., crack size and corrosion depth) over time [2]. Thus, time-dependent p_f and p_s can be computed for a structural component.

Furthermore, the assessment and prediction of the p_f and p_s of a deteriorating structural system require (a) formulation of the state functions of the individual components involved in the system, (b) identification of critical deterioration mechanisms affecting the state function of the components, (c) appropriate system modeling (e.g., series, parallel, and series-parallel systems), and (d) estimation of the correlation among the safety margins of individual components. El Hajj Chehade and Younes [33] and Shittu et al. [34] reviewed the software programs used to compute the p_f and p_s of structural systems.

2.2. Computational Procedure for Reliability-Based Optimum Inspection Planning

Inspection planning for deteriorating structures can be optimized based on the assessment and prediction of reliability and probability of failure [2,6]. The associated computational procedure is illustrated in Figure 2. This procedure consists of three main phases: (a) prediction of damage propagation and service life considering inspection and maintenance interventions, (b) formulation of the objectives for reliability-based inspection planning, and (c) formulation and computation for single-and multiobjective optimization. These main phases require the establishment of prediction models associated with damage propagation and loading conditions, as well as estimation of the effects of inspections and maintenance on damage propagation and structural reliability. Optimization can be based on several objectives, such as maximization of the reliability index (or reliability) for a predefined period, minimization of the probability of failure for a predefined period, minimization of the total life-cycle cost, minimization of the damage detection-based probability of failure, and maximization of the service life-based safety margin. These objectives are formulated by considering the effects of inspection and maintenance on the service-life extensions and improvement of structural reliability. To select and apply the best solution for multiobjective optimization (i.e., the best optimum inspection and maintenance planning), decision making should be performed.

3. Difficulties in Computing Reliability for Optimum Inspection Planning

If the state function is explicitly formulated, the reliability (p_s) and probability of failure (p_f) can be computed using several methods, including the MCS method [35], the importance sampling (IS) method [36], the Latin hypercube sampling (LHS) method [37], the first-order reliability method (FORM) [38], the second-order reliability method (SORM) [39], and the directional simulation method [40]. Software programs based on these methods can effectively compute the probabilities of p_s and p_f using a closed-form definition of the state function.

When the probabilities of p_s and p_f are used to formulate the objectives and solve the optimization problem for the inspection planning of deteriorating structures, several difficulties may be encountered. For example, (a) the state function may be in a non-closed form, (b) the formulation of the state function considers multiple probabilistic conditions, (c) a low probability of failure (p_f) can affect the optimum inspection planning, and (d) a high computational cost is required to compute p_f for the optimization process based on non-closed forms of the state function and objective functions. The state function used in reliability-based optimum inspection planning can be formulated considering probabilistic damage propagation prediction, probability of damage detection, probability of maintenance application, the effect of inspection and maintenance on service life extension, and reliability improvement. In this case, the state function can be formulated in a non-closed form through preprocessing tasks, as indicated in Figure 2. The accuracy of optimum inspection planning depends on the accuracy of the computed p_f. Furthermore, the cost for computing the p_f of a structure formulated by a non-closed state function is higher than that of a structure formulated by a closed-form state function. Additional computational costs are required when an optimization problem formulated with the non-closed state function and objective function is solved using GAs.

4. Possible Methods for Computing Probability of Failure for Inspection Planning

Efficient reliability-based optimum inspection planning for deteriorating bridges should address the difficulties of formulating with non-closed state functions and objective functions, low probability of failure, and high computational cost, as mentioned previously. To efficiently address these difficulties, direct MCS, subset simulation, and extrapolation methods based on the fitting distribution for the values of the state function can be adopted.

4.1. Direct Monte Carlo Simulation

Using the direct MSC method, the probability of failure (p_f,M) can be computed as [26,41]:

$$p_{f,M}=\frac{1}{N_s}{\displaystyle \sum _{i=1}^{N_s}{I}_f\left({\mathbf{x}}_i\right)}$$

(2)

where N_s is the number of samples, I_f(x_i) is the indicator function of the random variable associated with the i^th sample, and I_f(x_i) is equal to one for g(X_i) < 0 and equal to zero otherwise. The direct MSC is a general and simple method for solving high-dimensional reliability problems. However, p_f,M cannot be obtained when the number of samples (N_s) is less than 1/p_f,M. Furthermore, additional samples and computational costs are required during the computation of p_f,M. Therefore, it is inefficient in computing a low probability of failure.

4.2. Subset Simulation

The subset simulation developed by Au and Beck (2001) is an adapted MSC to estimate a low probability efficiently, wherein a low probability of failure is considered a product of a sequence of conditional probabilities of intermediate failure events. Using subset simulation, a low probability of failure (p_f,s) is computed as:

$$p_{f,S}=P\left(F\right)=P\left(F_1\right){\displaystyle \prod _{j=2}^{N_m}P\left(F_j|F_{j-1}\right)}$$

(3)

where F_j = j^th intermediate failure event, F_Nm = original failure event F, N_m = number of intermediate failure events, and P(F_j) = probability of occurrence of F_j. F₁ is the probability of failure estimated using Equation (2) when the random variables (X_i) are independent and identically distributed according to PDF f_X(x). The conditional probability ($P\left(F_j|F_{j-1}\right)$) is expressed as:

$$P\left(F_j|F_{j-1}\right)={\displaystyle \underset{\mathbf{X}}{\int }{I}_{f,j}\left(\mathbf{x}\right)f_{\mathbf{X}}\left(\mathbf{x}|F_{j-1}\right)d\mathbf{x}}$$

(4)

where I_f,j (x) is the indicator function of random variables (X) for the j^th intermediate failure event. The conditional PDF $f_{\mathbf{X}}\left(\mathbf{x}|F_{j-1}\right)$ in Equation (4) is defined as:

$$f_{\mathbf{X}}\left(\mathbf{x}|F_{j-1}\right)=f_{\mathbf{X}}\left(\mathbf{x}\right)I_{f,j-1}\left(\mathbf{x}\right)/P\left(F_{j-1}\right)$$

(5)

where f_X(x) is the joint PDF of random variables (X). Generating samples using Equation (5) and the acceptance–rejection method [42] is difficult because the acceptance probability is proportional to 1/P(F_j−1), and the generated samples are likely to be rejected before obtaining appropriate samples [42]. Therefore, based on Markov Chain Monte Carlo (MCMC) algorithms, the conditional probability ($P\left(F_j|F_{j-1}\right)$) given in Equation (4) can be computed as:

$$P\left(F_j|F_{j-1}\right)=\frac{1}{N_s}{\displaystyle \sum _{i=1}^{N_s}{I}_{f,j}\left({\mathbf{x}}_i\right)}$$

(6)

where X_i is represented by the conditional PDF $f_{\mathbf{X}}\left(\mathbf{x}|F_{j-1}\right)$ for i = 1, 2, …, N_s and is generated using MCMC algorithms. To improve the efficiency and accuracy of subset simulation, several MCMC algorithms, such as the component-wise Metropolis–Hasting algorithm [43] and the conditional sampling method [44], can be adopted. Detailed theoretical background on subset simulation method and its modifications for efficiency improvement and general applications can be found in [41,43,45,46,47,48], among others.

4.3. Kernel Density Estimation

KDE, a nonparametric method to estimate the PDF of a random variable, is useful when the data cannot be represented using a specific distribution function [49,50,51,52]. KDE has been adapted as an extrapolation approach to efficiently solve probabilistic problems [53]. Based on KDE, the PDF f_X(x) and CDF F_X(x) of a random variable (X) are expressed as [49,54]:

$$f_X\left(x\right)=\frac{1}{N_{s}b}{\displaystyle \sum _{i=1}^{N_s}K\left(\frac{x-x_i}{b}\right)}$$

(7a)

$$F_X\left(x\right)=\frac{1}{N_s}{\displaystyle \sum _{i=1}^{N_s}G\left(\frac{x-x_i}{b}\right)}$$

(7b)

where b is the bandwidth, x_i is a random sample, and K(⋅) is the kernel smoothing function. G(x) is expressed as:

$$G\left(x\right)={\displaystyle {\int }_{-\infty }^{x}K\left(y\right)dy}$$

(8)

The smoothness of the kernel PDF is affected by the bandwidth (b) and kernel smoothing function (K(⋅)). A kernel smoothing function (K(⋅)) must satisfy the following three conditions: (a) K(⋅) is symmetric, (b) ${\displaystyle \int K\left(x\right)dx}=1$, and (c) $\underset{x\to \infty }{lim}K\left(x\right)=\underset{x\to -\infty }{lim}K\left(x\right)=1$. Uniform, Gaussian (i.e., normal), triangular, and Epanechnikov functions can be used as kernel smoothing functions (K(⋅)). The Gaussian-based kernel smoothing function (K(⋅)) is the most widely used function for KDE [55]. When a Gaussian kernel function is adopted for K(⋅), the optimal bandwidth (b) required to minimize the mean integrated square error is selected as b = {4/(3N_s)}^0.2⋅σ, where σ is the standard deviation of the random samples [56]. The reflection method, which is the boundary correction method for KDE, can be used when bounded random samples are considered. Accordingly, the PDF f_X(x) (Equation (7a)) and CDF F_X(x) (see Equation (7b)) were modified as [54]:

$$f_X\left(x\right)=\frac{1}{N_{s}b}{\displaystyle \sum _{i=1}^{N_s}\left[K\left(\frac{x+x_i-2L_l}{b}\right)+K\left(\frac{x-x_i}{b}\right)+K\left(\frac{x+x_i-2L_u}{b}\right)\right]}\mathrm{for}{L}_l\le x\le L_u$$

(9a)

$$\begin{array}{ll}{F}_X\left(x\right)=& \frac{1}{N_s}{\displaystyle \sum _{i=1}^{N_s}\left[G\left(\frac{x+x_i-2L_l}{b}\right)+G\left(\frac{x-x_i}{b}\right)+G\left(\frac{x+x_i-2L_u}{b}\right)\right]}\\ & -\frac{1}{N_s}{\displaystyle \sum _{i=1}^{N_s}\left[G\left(\frac{x_i-L_l}{b}\right)+G\left(\frac{L_l-x_i}{b}\right)+G\left(\frac{L_l+x_i-2L_u}{b}\right)\right]}\end{array}\mathrm{for}{L}_l\le x\le L_u$$

(9b)

where L_l and L_u are the lower and upper limits of a random sample, respectively. To improve the approximation at the tail of the KDE, an adaptive KDE was investigated by [52,53,57,58,59] among others, in which an adaptive bandwidth is used in different regions of the sample space instead of a fixed bandwidth.

4.4. Kernel Density Estimation with Pareto Tails

To fit the sample data accurately by smoothing the tails of the PDF, the combined KDE and generalized Pareto distribution (GPD) can be used as an extrapolation approach. The GPD is fitted for the tails because applying the KDE to the tails is difficult when the number of samples corresponding to the tails is insufficient. KDE can be applied to middle sample data. GPD is expressed as [60,61,62]:

$$f_X\left(x\right)=\frac{1}{\alpha }{\left(1+\delta \frac{x-\rho }{\alpha }\right)}^{-1-1/\delta }\mathrm{for}\mathsf{\delta }\ne 0$$

(10a)

$$f_X\left(x\right)=\frac{1}{\alpha }\exp \left(\frac{x-\rho }{\alpha }\right)\mathrm{for}\mathsf{\delta }=0$$

(10b)

where α = scale parameter, ρ = threshold parameter, and δ = shape parameter. In Equation (10a), the range of random variable X should satisfy x > ρ for δ > 0 and ρ < x < ρ − α/δ for δ < 0. In Equation (10b), the random variable X should be larger than ρ. Exponentially decreasing distribution tails can be represented using a GPD with δ = 0. When the parameters ρ and δ are equal to zero, the GPD is equivalent to the potential distribution. GPD represents the sample data associated with the upper and lower tails of the random variable. The upper and lower tails correspond to random variables larger than the predefined upper threshold and smaller than the predefined lower threshold, respectively. The sample data located between the predefined lower and upper thresholds are represented using KDE.

5. Application to General Examples

To compare the probability of failure (p_f) and reliability index (β) computed using the four methods (i.e., MSC, subset simulation, KDE, and the combination of KDE and GPD), three general examples (i.e., examples I, II, and III) were investigated. The state functions and associated random variables in examples I, II, and III are listed in

Table 1. Probability of failure (p_f) and reliability index (β) when Monte Carlo simulation (MCS), subset simulation, kernel density estimation (KDE), and the combination of KDE and generalized Pareto distribution (GPD) are applied for examples I, II and III.

Example	State Function	Probability of Failure (pf)/Reliability Index (β)	MCS *	Subset Simulation	KDE	Combined KDE and GPD
I	$g\left(\mathbf{X}\right)=X_1+2$ X1 ~ N(0, 1)	pf	0.0227	0.0229	0.0229	0.0229
		β	2.00	2.00	2.00	2.00
II	$g\left(\mathbf{X}\right)=\frac{\left(X_1+X_2\right)}{\sqrt{2}}+3$ X1 ~ N(0, 1) X2 ~ N(0, 1)	pf	0.0014	0.0014	0.0014	0.0015
		β	2.99	2.99	2.99	2.97
III	$g\left(\mathbf{X}\right)=X_1·X_2+7$ X1 ~ N(0, 1) X2 ~ N(0, 1)	pf	1.43 × 10−4	1.33 × 10−4	1.43 × 10−4	1.28 × 10−4
		β	3.63	3.65	3.63	3.66

* Number of samples for MCS = 10⁶.

. These examples are based on the state function without considering structural systems. The state functions of the three examples are represented by simple polynomial functions consisting of one or two random variables, as shown in Table 1. The random variables used in these examples are independent and follow a normal distribution with a mean and standard deviation of 0 and 1.0, respectively. The state function values in examples I, II, and III were computed with 10⁶ samples generated using the MCS, and p_f was obtained using Equation (2). Subset simulation with the state function and random variables defined in Table 1 provided p_f. KDE and the combination KDE and GPD based on the state function values generated using MCS provided p_f and β. In this study, a boundary method was applied for KDE, in which only sample values of the state function less than zero were considered because the purpose of the application was to efficiently compute p_f. The upper and lower thresholds were predefined as the top 1% and bottom 1% of the state function values, respectively, to formulate the combination of KDE and GPD.

A comparison between the p_f and β computed using the four methods (MSC, subset simulation, KDE, and the combination of KDE and GPD) for the three examples is shown in Table 1. A comparison between the PDFs of the state function values obtained using MCS, KDE, the combination of KDE and GPD, subset simulation and the exact PDF for Example I is shown in Figure 3. The exact PDF shown in Figure 3 is a normal distribution with a mean and standard deviation of 2 and 1, respectively, and the exact β is equal to 2 because the state function g(X) = X₁ + 2, where X₁ is normally distributed, as indicated in Table 1. The area under the PDF of g(X) upper-bounded by the vertical axis located at g(X) = 0 indicates p_f. A detailed representation of this area (i.e., detail A) is shown in Figure 3b. The exact value of p_f (or β) and its approximate values derived using MCS, subset simulation, KDE, and the combination of KDE and GPD do not differ significantly. Furthermore, the PDF obtained from MCS has many fluctuations because the state function values are estimated with the sample data generated from MCS; the histogram for the state function values can be produced and converted into the PDF by dividing the sample size and bin size into the frequency of the histogram. The degree of fluctuations depends on the bin size of the histogram. An increase in the bin size of the histogram reduces the degree of fluctuations of the PDF but cannot accurately represent the sample data.

The probability of failure (p_f) and reliability index (β) computed using the four methods (MSC, subset simulation, KDE, and the combination of KDE and GPD) for examples II and III can also be found in Table 1. The exact reliability index (β) for example II was equal to 3.0 because the state function values were normally distributed with a mean of 3 and standard deviation of 1.0 (see the state function (g(X)) and random variables defined in example II in Table 1). The reliability index (β) estimated using the four methods was approximately equal to 3.0. In addition, the p_f and β computed using the four methods were not significantly different in example III, as shown in Table 1.

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(T) = t_cr − t_det

(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]:

$$P_{ins}=1-\Phi \left(\frac{\ln \left(a\right)-\lambda }{\varsigma }\right)$$

(12)

where a is the crack size, and λ and ζ 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 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. Probability of failure (p_f) and reliability index (β) associated with the damage detection time-based state function when MCS, KDE, and the combination of KDE and GPD are applied.

Inspection Application Times	Probability of Failure pf/Reliability Index β	MCS *	KDE	Combined KDE and GPD
tinsp1 = 10 years	pf	0.0034	0.0034	0.0037
	β	2.70	2.70	2.68
tinsp1 = 7 years tinsp2 = 14 years	pf	3.40 × 10−5	3.58 × 10−5	5.13 × 10−5
	β	3.98	3.97	3.88
tinsp1 = 6 years tinsp2 = 12 years tinsp3 = 18 years	pf	2.00 × 10−6	1.84 × 10−6	3.00 × 10−6
	β	4.61	4.63	4.53

* Number of samples for MCS = 10⁶.

presents the probability of failure (p_f) and the reliability index (β) 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 Figure 5b and Figure 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. The CDFs obtained when inspections were applied at 6, 12, and 18 years are shown in Figure 7b. The p_f and β associated with the CDFs in Figure 7 are presented in Table 2. As shown in Figure 7 and Table 2, p_f estimated using the combination of KDE and GPD method was larger than those computed using MCS and KDE. β estimated using MCS was similar to that estimated using KDE. The difference between the CDFs at the state function value equal to zero (i.e., g = 0) led to differences between the values of p_f (or β) based on MCS, KDE, and the combination of KDE and GPD. As shown in Figure 6 and Figure 7, KDE appropriately represents the CDF based on MCS. As a result, there is no significant difference between the values of p_f (or β) based MCS and KDE.

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 extension, reliability can be estimated using the extended service life-based state function (g(T)) as [2,64]:

$$g\left(\mathbf{T}\right)=t_{life,ex}-t_{tg}$$

(13)

where t_life,ex = extended service life, and t_tg = 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(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 (β) 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. Probability of failure (p_f) and reliability index (β) associated with the extended service life-based state function when MCS, KDE, and the combination of KDE and GPD are applied.

Inspection Application Times	Probability of Failure pf/Reliability Index β	MCS *	KDE	Combined KDE and GPD
tinsp1 = 10 years	pf	0.0038	0.0038	0.0053
	β	2.67	2.67	2.56
tinsp1 = 7 years tinsp2 = 14 years	pf	1.30 × 10−4	1.30 × 10−4	1.89 × 10−4
	β	3.65	3.65	3.56
tinsp1 = 6 years tinsp2 = 12 years tinsp3 = 18 years	pf	2.40 × 10−5	2.40 × 10−5	1.64 × 10−6
	β	4.07	4.07	4.65

* Number of samples for MCS = 10⁶.

. p_f and β 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 Figure 8 and Figure 9, respectively. When the inspection was applied at 10 years, the βs 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_fs (or β) based on MCS and KDE were equal, as indicated in Table 3.

The PDFs and CDFs of the state function values when inspections were applied at 7 and 14 years are shown in Figure 9. As shown in Figure 9b, two peaks near -6 years and -14 years were represented using MSC and KDE. However, the PDF obtained using the combination of KDE and GPD had an exponentially decreasing left tail without any peaks. As shown in Figure 9c and Table 3, among MCS, KDE, and the combination of KDE and GPD, the combination of KDE and GPD produced the largest p_f. The PDFs and CDFs of the state function values under three inspections at 6, 12, and 18 years are shown in Figure 10. In addition, the p_f computed using the combination of KDE and GPD was the smallest (see Figure 10 and Table 3). As shown in Figure 8c and Figure 9c, p_f (or β) values based on MCS and KDE were the same, as the CDF based on MCS is appropriately represented by the CDF based on KDE.

8. Application for Optimum Inspection Planning

Optimum inspection planning in this study indicates the process of determining the best inspection application times to minimize the probability of failure (p_f). In general, the optimization problem is formulated with design variables, objectives, constraints, and given conditions. Multiple objectives can be considered individually or simultaneously in an optimization problem for inspection planning [2]. In this application, a single-objective optimization was adopted and formulated as:

Find the inspection application times $t_{\mathrm{insp}}=\left\{t_{insp,1},t_{insp,2},\dots ,t_{insp,N_i}\right\}$

To minimize the probability of failure p_f

Such that 1 year ≤ t_insp,i − t_insp,i−1 ≤ 10 years

Given number of inspections N_i and inspection method (i.e., ECI)

Here, t_insp indicates design variables (i.e., the inspection application times (t_insp,i)). The objective is to minimize p_f. As a constraint, the time interval between inspection application times ranges from 1 year to 10 years. The total number of inspections (N_i) and the inspection methods are given. The probability of failure (p_f) is based on two state functions (i.e., damage detection time-based state function and extended service life-based state function). To investigate the effect of the approaches used for computing the failure probability on the optimization, MCS, KDE, and the combination of KDE and GPD methods were used. The information and assumptions used to formulate the optimization problem are based on the bridge application presented in Section 6 and Section 7. To solve the optimization problem, the genetic algorithm in MATLAB R2022a [65] with 100 generations was adopted because the GA can provide reliable solutions after sufficient generations, regardless of the continuity or differentiability of the objective function [19].

8.1. Optimum Inspection Planning with Damage Detection Time-Based State Function

Inspection planning can be optimized to minimize the probability of failure associated with the damage detection time-based state function (see Equation (11)). The solutions (i.e., optimum inspection application time (t_insp,1) and the associated objective values of p_f) obtained when N_i = 1 are illustrated in Figure 11. Figure 11a–c show the 100 solutions obtained using MCS, KDE, and the combination of KDE and GPD. These solutions were obtained through 100 computations to investigate the effect of a p_f-estimating method on the dispersion of the optimum t_insp,1 and p_f. Each solution was represented by a polyline connecting t_insp,1 and p_f on the vertical axes. As shown in Figure 11a, one of the solutions using MCS indicates the optimum inspection application time t_insp,1 = 7.025 years and the associated p_f = 3.90 × 10⁻⁵. The optimum inspection application times t_insp,1 were differed slightly across solutions, and the largest and smallest t_insp,1 values in the 100 solutions were 7.036 years and 6.985 years, respectively. However, p_f remained near 3.90 × 10⁻⁵. When KDE was used to compute p_f, the obtained optimum inspection application time was 7.013 years for all 100 computations (see Figure 11b). The t_insp,1 value obtained with 100 computations using the combination of KDE and GPD was 7.157 years (see Figure 11c). As shown in Figure 11, (a) a more reliable optimum inspection plan can be obtained when KDE and the combination of KDE and GPD methods are used; (b) the objective values (i.e., p_f) of the optimization are almost non-dispersed when MCS, KDE, and the combination of KDE and GPD are used; and (c) the objective values based on MCS and KDE are almost the same (i.e., p_f = 3.90 × 10⁻⁵ for MCS and p_f = 3.97 × 10⁻⁵ for KDE). p_f obtained using MCS cannot be less than 0.1 × 10⁻⁵, as the sample size used for MCS was 10⁶.

8.2. Optimum Inspection Planning with Extended Service Life-Based State Function

The optimum inspection application times and associated values of p_f are shown in Figure 12 and Figure 13. The objective of the optimization problem is to minimize the probability of failure (p_f) formulated with the extended service life-based state function (see Equation (13)). The number of inspections (N_i) is one and two in Figure 12 and Figure 13, respectively. The 100 solutions obtained using MCS, KDE, and the combination of KDE and GPD are illustrated in parallel coordinates in Figure 12a–c (or Figure 13a–c), respectively.

When MCS was used to compute the probability of failure (p_f) considering one inspection, the optimum inspection application time (t_insp,1) was equal to 10 years (see Figure 12a). If the probability of failure (p_f) is estimated using KDE, the inspection application time of t_insp,1 = 10 years should be applied to minimize p_f (see Figure 12b). Figure 12c indicates that one of the optimum solutions was t_insp,1 = 9.93 years. As shown in Figure 12a–c, (a) the solutions were not dispersed, and (b) the constraint for the optimization problem (i.e., 1 year ≤ t_insp,i − t_insp,i−1 ≤ 10 years) was active because the 100 solutions shown in Figure 12a,b had the same optimum inspection application time of 10 years.

Figure 13 shows the optimum inspection application times and corresponding probability of failure (p_f) when two inspections are considered to formulate the optimization problem. The optimum inspection application times (t_insp,1 and t_insp,2) obtained using MCS were 7.033 and 13.714 years, respectively (see Figure 13a). When KDE was used to compute p_f, t_insp,1 was 7.033 years, and t_insp,2 was 13.718 years (see Figure 13b). With the use of the combination of KDE and GPD, t_insp,1 was 8.606 years, and t_insp,2 was 17.171 years (see Figure 13c). As shown in Figure 13, (a) the optimum inspection application times obtained using MCS are similar to those obtained using KDE; (b) the optimum inspection application times (t_insp,1 and t_insp,2) computed using MCS and KDE were less dispersed than those computed using the combination of KDE and GPD; and (c) the objective values based on MCS are more dispersed than those based on KDE.

The dispersions of the optimum inspection times are affected by the probability of failure based on the extended service life-based state function (see Equation (13)). MCS computes the probability of failure directly (see Equation (2)). However, the probability of failure using KDE and the combination of KDE and GPD is obtained through extrapolation with the sample data generated from MCS. Therefore, KDE and GPD can provide a more stable probability of failure than MCS during the optimization process using GA.

9. Discussion

The accuracy, reliability, and computational efficiency of KDE and the combination of KDE and GPD are affected by various factors, such as the kernel smoothing function; the bandwidth and predefined upper and lower thresholds affect the results of KDE and the combination of KDE and GPD, respectively. Their effects on accuracy, reliability, and computational efficiency should be further investigated to optimize these factors. Furthermore, advanced extrapolation approaches have been developed recently to efficiently and accurately address a low probability of failure [52,66]. The decision-making process for determining the most appropriate approach among the candidate advanced extrapolation approaches must be generalized considering their accuracy, reliability, applicability, and computational efficiency for practical applications.

10. Conclusions

In this study, we proposed an efficient reliability-based optimal inspection planning method for deteriorating bridges using extrapolation approaches, whereby the optimum inspection application times are determined by minimizing the probability of failure. Extrapolation approaches are used to compute the probability of failure for optimum inspection planning. Applications of MCS, subset simulation, and two extrapolation approaches (i.e., KDE and the combination of KDE and GPD) were presented with three general examples. The effects of the approaches (in computing failure probabilities, inspection application times, and the number of inspections) on the probability of failure were investigated using an existing structure. Furthermore, the applications of MCS and two extrapolation approaches for optimum inspection planning of an existing fatigue-sensitive steel girder bridge were presented. Based on this investigation, the following conclusions can be drawn.

• Probabilistic optimum inspection planning for deteriorating bridges generally requires the estimation of the probability of failure. When a state function is formulated considering multiple probabilistic conditions, it is crucial to select an appropriate approach to compute the probability of failure. In this study, MCS, KDE, and a combination of KDE and GPD were considered as candidate approaches.
• When the number of design variables (i.e., inspection application times) in the inspection planning optimization problem is equal to one and the probability of failure can be reliably obtained using MCS, the optimum inspection application time obtained using MCS is almost the same as that computed using KDE. However, if the number of design variables is two and the probability of failure is low, the solutions after 100 computations with MCS are more dispersed than those with KDE.
• When MCS is used, the accuracy of the computed failure probability considerably depends on the number of samples. A large number of samples is required to obtain an accurate and low probability of failure. Because the computational cost increases with the number of samples, MCS may be inefficient. Although KDE and the combination of KDE and GPD are performed with the samples generated using MCS, the associated computational time is negligible, and these two extrapolation approaches can efficiently compute a low probability of failure.
• The most appropriate approach for reliability-based optimum inspection planning should be selected based on the accuracy and reliability of the solution, the computational efficiency, and the applicability of the probabilistic optimization process. KDE and the combination of KDE and GPD can be easily applied using various programming languages and existing codes. The presented applications demonstrate that the probability of failure computed using KDE is more accurate and reliable than that estimated using the combination of KDE and GPD.