Life-cycle management of deteriorating bridge networks with network-level risk bounds and system reliability analysis

Structural Safety 83 (2020) 101911

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Life-cycle management of deteriorating bridge networks with network-level risk bounds and system reliability analysis David Y. Yang, Dan M. Frangopol

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Department of Civil and Environmental Engineering, ATLSS Engineering Research Center, Lehigh University, Bethlehem, PA, USA

ARTICLE INFO

ABSTRACT

Keywords: Life-cycle management Risk bounds System reliability analysis Bridge deterioration Optimization

Structural deterioration poses a substantial threat to the safety, serviceability, and functionality of bridges. Since bridges are connected in transportation networks, their failure can dramatically alter traffic flow, causing immense social consequences such as large-scale traffic delay and additional vehicle operating cost. In this paper, a novel method is proposed for life-cycle management of deteriorating bridge networks. The proposed method aims at minimizing network-level risks associated with deterioration-induced bridge failure. To combat the deficiencies of existing Monte Carlo simulation-based risk assessment methods, the proposed method adopts a non-simulation approach that relies on risk bounds of deteriorating bridge networks. In particular, this method uses system reliability analysis to determine the occurrence probabilities of various bridge failure scenarios in a network. For each failure scenario considered, traffic assignment is conducted to predict the traffic flow in the damaged network and the network-level consequences. The upper and lower bounds of the network-level risk are then formulated, allowing for efficient and accurate risk assessment with only a few traffic assignment operations. Estimated by the average value of its upper and lower bounds, the network-level risk is used as the optimization objective in a metaheuristic search procedure to obtain optimal life-cycle maintenance plans including the maintenance schedule and the investment on each maintenance action. The proposed method excels in its ability to account for the impacts of unlikely, yet high-impact events, as well as its ability to obtain optimal or near-optimal life-cycle maintenance plans that can more effectively reduce risks than those obtained using simulation-based methods.

1. Introduction Deterioration of surface transportation infrastructure is a major challenge of transportation agencies around the world. Deterioration of bridges is especially detrimental due to their cruciality to network functionality, high maintenance cost, and dire consequences of failure. According to the American Society of Civil Engineers [1], the average age of bridges in the United States is 43 years, already passing the middle age for their design service life. Up to 2016, 9.1% or about 56,000 bridges have been rated structurally deficient, while funding required to clear this backlog by 2030 was estimated at over US$2 trillion. Similar situations are prevalent in many other countries [2,3]. This deterioration problem is further compounded by the growing traffic load due to socioeconomic development [4–6] and increasing hazard frequency and intensity due to climate change [7,8]. The structural performance of a bridge deteriorates gradually as a result of environmental stressors (e.g. corrosion) and long-term effects (e.g. fatigue, shrinkage, and creep). Bridge failure induced by structural

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deterioration can impose dire consequences upon bridge owners/ managers (direct consequences) and society at large (indirect consequences) [9]. Therefore, the risk posed by bridge deterioration should be rigorously controlled through rational life-cycle management [10–12]. Different from many other assets, bridges are located in transportation networks, in which they are connected by roadway links. Essentially, bridges in a transportation network (herein referred to as a bridge network) form a spatially distributed system, in which spatial correlation exists with respect to bridge conditions and bridge functionality in the network [13]. Yang and Frangopol [14] showed that the maintenance priorities of deteriorating bridges can be drastically different when ranked at network and project levels. Due to this characteristic of spatially distributed systems, risk assessment and management should be conducted at the bridge network level. Despite the necessity of network-level analysis, risk assessment of bridge networks faces two major challenges: (a) consideration of the correlation of bridge failure events and (b) efficient evaluation of indirect consequences.

Corresponding author. E-mail address: [email protected] (D.M. Frangopol).

https://doi.org/10.1016/j.strusafe.2019.101911 Received 20 June 2019; Received in revised form 26 October 2019; Accepted 19 November 2019 0167-4730/ © 2019 Elsevier Ltd. All rights reserved.

Structural Safety 83 (2020) 101911

D.Y. Yang and D.M. Frangopol

As mentioned previously, bridges in a network are spatially distributed. Adjacent bridges may suffer from similar environmental stressors and traffic loads, leading to correlated failure events [14,15]. Modeling this spatial correlation is conventionally conducted using two approaches: event-based simulation and random field-based approximation. In the first approach, hazards (either deterministic or stochastic) are hypothetically exposed upon a bridge network. Spatial correlation of bridge failure is then obtained by simulating bridge performance under the hazards considering bridge locations, ages, types, and relative distances to one another, among many other factors. This simulation-based technique has been widely used for various types of hazards such as earthquakes [13,16,17], floods [18], and overweight trucks [19]. In the second approach, random field theory is used directly to approximate the spatial correlation of bridge failure [20–22]. Instead of performing Monte Carlo simulation (MCS) for bridges under hazards, this random field-based approximation can directly and efficiently generate failure scenarios of bridges in a network, though this approximation may compromise the actual correlation condition compared to that of event-based simulation. Although the random field models are mainly developed for certain disasters (e.g. due to earthquakes), it is believed that they are also applicable to deteriorating bridge networks under normal traffic conditions [14,15]. Bridge failure as well as the subsequent reconstruction can severely jeopardize the functionality of bridge networks. Consequences of bridge failure can be categorized as direct consequences, imposed upon transportation agencies, and indirect consequences, imposed upon traffic users. Generally, the latter are usually expressed as the sum of traffic delay and extra vehicle operating costs, which can be several orders higher than the direct consequences [23,24]. The difficulty in evaluating indirect consequences arises from the need to model variation in traffic flow due to bridge failure [14]. In order to capture this variation, traffic assignment models that can predict traffic flow in a bridge network are needed. Currently, the few existing studies that do consider traffic flow variation use MCS to generate failure scenarios of bridges and calculate the associated indirect consequences [14,25–28]. However, for deteriorating networks where failure probabilities of individual bridges are much lower than those of bridge networks under disasters, MCS with a small number of samples might overlook those failure scenarios that are unlikely to appear in MCS but can trigger enormous consequences (i.e. “black swans”), leading to underestimated risks. If more samples are used to capture these “black swans”, the corresponding computational cost can easily become prohibitive. To curtail the computational demand, bookkeeping techniques have been employed so that repetitive traffic assignment can be avoided when the same failure scenario of bridges reappears in MCS [14,29]. Other techniques are also available to reduce computational cost by selecting representative failure scenarios [30] and by using performance proxy [30] or computational surrogates [31] for traffic assignment. Nevertheless, the effects of “black swans” cannot be totally eliminated by using these techniques. To avoid the challenge brought by computationally expensive traffic assignment, many life-cycle risk assessment and management studies chose to use simplistic methods to estimate indirect consequences including using traffic flow data in the intact bridge network [23,32–34] and introducing a random variable to represent indirect-to-direct consequence ratio [35,36]. However, these methods ignore traffic flow variation entirely and, thus, cannot be regarded as network-level risk analysis. Once network-level risks are determined, life-cycle management of bridge networks becomes a resource-constrained project scheduling (RCPS) problem that has proved to be non-deterministic polynomialtime-hard (NP-hard) [37]. To obtain optimal maintenance plans, most life-cycle management frameworks rely on metaheuristic optimization techniques such as generic algorithm [38–40], particle swarm optimization [27,41], harmony search [42], and simulated annealing [43],

among others. As a result, the network-level risk needs to be assessed multiple times in the lifespan of bridges as well as for different life-cycle decision alternatives. Hence, the computational challenge previously mentioned become even more amplified. In this paper, a novel method is proposed for life-cycle management of bridge networks based on network-level risks. The proposed nonsimulation method can significantly alleviate the computational demand. The occurrence probabilities of different failure scenarios are determined using the matrix-based system reliability (MSR) method [44–46]. The MSR method used herein can consider spatial correlation of bridge failure arising from correlated structural capacities and demands. Since the probabilities associated with different network failure scenarios can be directly computed with the MSR method, the expected indirect consequences (i.e. indirect risks) can be calculated analytically by a small number of traffic assignment operations without MCS. Different from existing non-simulation methods for indirect consequence evaluation [47], the proposed method uses risk bounds to (a) reduce the number of failure scenarios needed to be considered and (b) take into account the contribution of unlikely, yet high-impact events. Based on the network-level risk, optimization is then conducted using genetic algorithm to determine optimal life-cycle maintenance plans of deteriorating bridge networks. 2. Description of the proposed method 2.1. Risk bounds of deteriorating bridge networks The risk of structural failure in a bridge network is related to (a) direct consequences such as debris removal and reconstruction and (b) indirect consequences such as traffic delay and extra vehicle operating costs imposed upon traffic users. As the latter are usually much higher than the former [23], the proposed method considers primarily risks associated with indirect consequences. However, the method can be easily expanded to include direct consequences as exemplified in Yang and Frangopol [14]. As stated previously, bridge networks are spatially distributed systems for which risk assessment and management should be conducted at the network level. Mathematically, the network-level risk associated with a deteriorating bridge network can be expressed as

Rnet (t ) =

nsc i=1

Lnet (si)· pf (t , si)

(1)

where Rnet (t ) is the network-level risk in year t ; nsc is the total number of failure scenarios of bridges; Lnet (si) is the network-level (indirect) consequences associated with failure scenario i ; pf (t , si ) is the timevariant occurrence probability of failure scenario i in year t . Herein, a bridge network with failed bridges is referred to as a damaged network, while an intact network means that all bridges in the network are undamaged. Following this terminology, a failure scenario of bridges corresponds to a damage state of the network, represented herein by si . Note that the probability of the occurrence of a damaged network is time-variant due to structural deterioration of bridges. If maintenance actions are considered, this probability is also related to the maintenance actions undertaken. In some cases, the network-level consequences can also be time-variant due to socioeconomic development of communities. Nonetheless, to focus on the formulation of the proposed method, it is assumed herein that the consequences are only related to the network damage state. As the number of bridges increases, it is easy to anticipate that the number of failure scenarios (nsc ), or, equivalently, the number of network damage states, grows exponentially. For instance, a simple network with 5 bridges encompasses 25 1 = 31 damage states, while a network with 100 bridges entails 2100 1 1.27 × 1030 damage states. If each network damage state mandates a traffic assignment operation for consequence evaluation, the second case is obviously impossible. Therefore, upper and lower risk bounds are formulated herein to approximate network-level risk defined in Eq. (1). Given an ascendingly 2

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D.Y. Yang and D.M. Frangopol

sorted sequence of failure scenarios based on their numbers of failed bridges, network-level risk bounds can be formulated as Rnet, upp (t ) =

Rnet, low (t ) =

msc i=1

msc i=1

Lnet (si)·pf (t , si) + Lnet (smax )· 1

ps (t )

Lnet (si)·pf (t , si) + Lnet (smin )· 1

ps (t )

msc i=1

msc i=1

pf (t , si )

(2a)

pf (t , si )

(2b)

where Rnet , upp (t ) and Rnet , low (t ) are the upper and lower bounds of the network-level risk in year t , respectively; msc is the number of scenarios in the sorted sequence of failure scenarios; smax and smin are, respectively, the most and least severe failure scenario among the nsc msc scenarios not selected for risk assessment; ps (t ) is the probability of having an intact network in year t . If Braess’ paradox [48] is not considered (as in most well-design networks), the most severe failure scenario can be identified as the one where all bridges failed. Usually in deteriorating bridge networks (in contrast to those under disasters), this scenario is unlikely to occur yet may have a high impact on the functionality of the network. Therefore, by considering this upper bound, the effect of “black swans” can be properly contemplated. On the other hand, the determination of smin is more complex because the least severe scenario is hard to identify without analyzing all the remaining scenarios. Certainly, one can use the least severe scenario among all nsc cases or even use Lnet (smin) = 0 assuming that no bridges fail in the least severe case. Nevertheless, this will result in a risk band too wide to be useful. Herein, it is assumed that the severity of a failure scenario increases with the number of the failed bridges. Based on this assumption, smin can be identified from the msc scenarios. Consider the msc scenarios that include all possible failure scenarios involving up to nfb bridges. In this case, if it is assumed that more failed bridges always indicate more severe consequences, smin can be identified as the scenario with the highest consequences among those scenarios with nfb failed bridges. If a network with 10 bridges is assumed, the first 100 scenarios considered (i.e. msc = 100 ) will include two complete groups involving one (10 scenarios) and two (45 scenarios) failed bridge(s) but not include the complete group involving three failed bridges. Therefore, for nb = 10 and msc = 100 , nfb = 2 and smin is the most severe scenario among the 45 scenarios with two failed bridges. Based on the risk bounds, the network-level risk can be approximated as

Rnet (t ) =

Rnet , upp (t ) + Rnet , low (t ) 2

Fig. 1. Conceptual diagram of the proposed method.

shown in this subsection how system reliability analysis can be used to calculate these probabilities. For the ease of the following description, a simple network with five bridges is considered. The five bridges are referred to as bridges B1 to B5. Without losing any generality, the scenario with two failed bridges (i.e. bridges B1 and B2) is first considered. Accordingly, the occurrence probability of this failure scenario can be written as

pf (s1 2) = Pr[E1 E2 E3 E4 E5] = Pr[E1 E2]

Pr[(E1 E2)

(E3

E4

E5)] (4)

where s1

(3)

2

denotes the damage state of bridge network (or failure

scenario of bridges) where only bridges B1 and B2 failed; Ei and Ei with i = 1, 2, 5 represent the events that bridges B1 to B5 failed or survived, respectively; Ei Ej is equivalent to Ei Ej , representing the joint event of occurrence of Ei and Ej (the operator is omitted whenever no ambiguity arises); Ei Ej is the union event of Ei and Ej . Compared to Eqs. (1)–(3), the time factor in network vulnerability is dropped temporarily to facilitate notation. It can be realized that the two terms in the last part of Eq. (4) can be interpreted as the failure probabilities of a parallel system and a parallel-series system, respectively. The two systems are illustrated in Fig. 2. Many existing system reliability methods [e.g. 47] can be employed to calculate these two probabilities and, thereby, obtain the occurrence probability of the failure scenario under consideration. There are three exceptions that are worth mentioning: (a) when only one bridge failed, Pr[E1 E2] in Eq. (4) requires a component reliability analysis rather than a system one; (b) when only one bridge survived, the last term in Eq. (4) becomes a parallel system reliability problem; (c) when all bridges failed, the scenario probability can be determined directly by conducting reliability analysis for a parallel system. In the system reliability analysis herein, a component is associated with the failure event of a bridge. This component reliability problem can be represented by the following generalized performance function:

where Rnet (t ) is the approximate network-level risk obtained from risk bounds. From Eq. (2), it can be seen that only msc (instead of nsc ) failure scenarios need to be considered. To ensure that the approximate risk is accurate, a convergence study is needed to determine the required number of failure scenarios. It is anticipated that by taking the average of the two bounds, this approximate network-level risk will converge to the precise value in a faster rate than either the upper or the lower bound alone would do. Based on the network-level risk, life-cycle maintenance actions can be planned for bridges in the network. Due to the complexity of RCPS problems, genetic algorithm, a metaheuristic method, is used in the proposed method to decide when and which bridges need to be maintained. Herein, focus is placed on essential maintenance actions that can result in improvement of structural safety such as repair, retrofit, renewal and rehabilitation actions, among others. The conceptual diagram of the proposed method is shown in Fig. 1. The following subsections describe each of the modules shown in Fig. 1. 2.2. Determination of network vulnerability using system reliability analysis In this paper, the probabilities of all damage states of a network are collectively referred to as network vulnerability. Network-level risk assessment requires the determination of network vulnerability. It is

gi = Ri ( i ) 3

S ( i)

(5)

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D.Y. Yang and D.M. Frangopol

herein on the traffic delay. Due to bridge failure, traffic delay of all traffic users in a bridge network can be expressed as [25]:

Tdelay =

where Ri ( i ) is the structural capacity modeled by random variables i ; S ( i ) is the structural demand modeled by random variables i ; the subscript i indicates that the performance function is with respect to bridge Bi . Since nearby bridges of the same type may share similar environmental and mechanical stressors and traffic loads, and for different bridges are correlated [50]. In system reliability analysis, correlated bridge failure (i.e. correlation between gi < 0 and gj < 0 where i j ) can be also considered using first order approximation [51]. Specifically, the correlation coefficient between two bridge failure events can be determined as follows [51]:

=

T i · j

fa' · ta' (f a' )

f · ta (fa ) a A a

(7)

where Tdelay is the traffic delay in time units; a is a network link from the set A of all network links; fa' and fa are the traffic flow on damaged and intact link a , respectively, measured by passenger car units (PCUs) per unit time; ta' (f a' ) and ta (fa ) are the travel time on damaged and intact link a , respectively, depending on link properties (free speed travel time and practical capacity) and traffic flow. If the hourly demands within peak hours are used in analysis, fa' and fa are hourly traffic flow while Tdelay is the cumulative traffic delay in hours experienced by all commuters who embarked on their trips within one hour. The traffic delay in Eq. (7) can be converted to economic losses in monetary units considering the type of vehicles, purpose of trips, and time period of interest (e.g. duration of peak hours) [14]. In this paper, however, indirect consequences are measured straightly by traffic delay Tdelay . The traffic flow volumes on network links can be predicted by traffic assignment [55]. A transportation network is represented by a graph model containing nodes and links. Traffic demands are PCUs that need to travel from an original node to a destination node. Such pairs of origin–destination (OD) nodes are referred to as OD pairs. Herein, static traffic assignment based on fixed traffic demands is performed to predict traffic flow volumes on network links based on user equilibrium (UE) principle [55]. This traffic assignment model assumes that traffic users have perfect information about their travel time and always behave accordingly to minimize it. In addition, traffic demands are rigid meaning that the OD pairs and their associated demands do not vary with travel time. Despite these assumptions, the static traffic assignment model has been found to perform well in the prediction of steadystate traffic flow after a disturbance event such as bridge failure [56]. As a result, it is used herein for network-level risk assessment. The proposed method is also compatible with other traffic assignment models. For instance, stochastic UE can be used when perception uncertainties of traffic users are considered [57]; gravitational models for traffic demand prediction can be integrated when demand variation (e.g. after a hazard) is taken into account [24,56]; dynamic UE or agentbased modeling (ABM) can be applied when queueing behavior and short-term disturbance are investigated, e.g. when immediate traffic delay after a hazard is of interest [59,60]. As the complexity increases, the number of traffic assignment operations that can be performed will become limited. This may pose a computational challenge to the proposed method. Nevertheless, the proposed method can still out-perform simulation-based methods for which even more traffic assignment operations are needed. The UE principle for traffic assignment states that in an equilibrium state, any traffic user cannot shorten his/her travel time by switching to an alternative path to the destination. Mathematically, traffic assignment based on the UE principle can be formulated as a convex optimization problem solved by Frank–Wolfe algorithm with various shortest path subroutines (e.g. Dijkstra’s shortest path algorithm) [55]. More details for solving traffic assignment problems can be found in Yang and Frangopol [14]. For a damaged network, its link properties are affected by the bridge failure scenario. In particular, it is considered that bridge failure can reduce the link capacity and free speed travel time on a link. The reduction of both is computed as [14]:

Fig. 2. System reliability analysis for network vulnerability evaluation.

ij

a A

(6)

where ij is the correlation coefficient between the failure of bridge Bi and that of bridge B j ; i and j are the expanded vector of direction cosines at the design point obtained from first order reliability (FOR) analysis of components [51,52]. With Eq. (6), probabilities associated with different failure scenarios can be determined using system reliability methods considering correlated failure of components. MCS for failure scenarios is therefore no longer necessary. The preceding discussion is valid for most system reliability methods. Among these methods, matrix-based system reliability (MSR) method is used [44–46]. Compared to other methods, the MSR method can reduce the number of system reliability evaluation from two (see Eq. (4)) to only one. This is achieved by directly devising a generalized system that corresponds to the failure scenario of interest. Note that a generalized system is not of the traditional parallel or series type. Instead, it is defined by an event vector [44]. For instance, the probability obtained from Eq. (4) can be determined directly by using a one-hot vector representing event s1 2 [44]. In addition, the MSR method employs common source random variables to simplify the correlation condition obtained from Eq. (6) [45]. As a result, the dimensionality of system reliability problems can be lessened, rendering improved computational efficiency in medium- to high-dimensional problems [46]. 2.3. Indirect consequence evaluation with traffic analysis Performance of a bridge network can be analyzed by network connectivity or traffic flow capacity [30]. Although the former is useful for networks under disasters [19, e.g. 53,54], deteriorating bridge networks should be better analyzed by the latter since failure due to deterioration can rarely escalate to such an extent that jeopardizes network connectivity, especially in an urban or highway network where local detours are abundant. Indirect consequences considered herein arise from variation in traffic flow due to bridge failure, closure, or reconstruction. On the network-level, this traffic flow variation usually indicates that traffic users in the network have to endure traffic delay due to detours and congestion as well as extra vehicle operating cost due to rerouting. When detours are available, cost associated with traffic delay is often higher than that associated with extra vehicle operating cost. Hence, evaluation of indirect consequences is focused

ta' ,0 =

la,0 +

l b Fa b u

fa' , c = fa, c (1 where

4

' ta,0

maxdb )

b Fa

(8a) (8b)

is the free speed travel time on a damaged link (i.e. link with

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D.Y. Yang and D.M. Frangopol

failed bridges); u is the free speed on line a ; Fa is the set of failed bridges on link a ; la,0 is the length of an intact link; lb is the detour length associated with bridge b; fa, c is the capacity of an intact link,

where sum (·) is the summation operator for a vector that adds up all its elements; B is the total budget for maintenance; i (t i, vi, t ) is the annual reliability index of bridge i with maintenance plans (t i, vi) ; T is the threshold annual reliability index; is the minimal interval between maintenance actions; |·| is the cardinality operator to obtain the number of elements in a vector; nm is the maximum number of maintenance actions for a bridge; t is the time in service in years (t is an integer); tL is the time horizon of interest in years; tstart and tend are the first and last year that maintenance actions are planned. Since multiple maintenance actions (but less than nm ) can be applied to each bridge, the vector representing network maintenance time (tm ) contains the vector (ti ) representing maintenance time of each bridge, i.e. tm: ={t i} with i = 1, 2, , nbr ; the network investment vector vm similarly contains the bridge investment vector vi corresponding to ti , i.e. vm: ={vi} with i = 1, 2, , nbr . In the objective function of Eq. (10), the network-level risk is slightly different from that defined in Eq. (1) as its value is not only related to deterioration but also controlled by the decision on maintenance actions. Apart from the resource constraint shown as Constraint (a) in Eq. (10), the optimization problem also considers constraints related to structural safety and maintenance actions. Constraint (b), the safety constraint, states that the minimum annual reliability index of a bridge cannot be lower than a predefined threshold value T . Constraint (c) indicates that the interval between maintenance actions for a bridge should be larger than a certain value (in years). Constraint (d) requires that the first maintenance action should be later than tstart while the last one should be earlier than tend . As stated previously, the optimization problem in Eq. (10) is a NPhard RCPS problem. Various metaheuristic optimization methods have been developed to seek optimal or near-optimal solutions. In this paper, genetic algorithm (GA) is employed to obtain the optimal time and investment for all bridges in a network. GA is a metaheuristic algorithm that mimics the natural evolution process. In GA, candidate solutions go through crossover and/or mutation processes across multiple generations to evolve to the optimal or near-optimal solution. It has been widely used in the life-cycle management of civil and marine structures [e.g. 38–40]. Herein, the problem is different from the many existing GA operations in that both the time and the investment associated with maintenance actions are design variables. The former are integers in the time-horizon of interest while the latter are real numbers representing investments on maintenance actions. Fig. 3 shows the operations in the GA used herein. Each candidate solution is analogized to the chromosome of an individual, while its design variables (i.e. elements in the decision vector d in Eq. (10)) are treated as genomes. Herein, the genomes in a chromosome are divided into two groups representing time and investment of maintenance, respectively (see Fig. 3). The maintenance plan of each bridge is denoted by nm genomes representing the chronologically sorted maintenance time and nm genomes representing the corresponding investments. In each generation, two-point crossover or mutation is executed among all candidate solutions, as illustrated in Fig. 3. A solution that went through crossover or mutation is named as a raw offspring. It should be noted that after crossover and mutation, a raw offspring may no longer be a valid maintenance plan since (a) the maintenance time of a bridge may not be chronologically sorted, (b) the interval between two maintenance actions may become smaller than the minimal interval ; and/or (c) the total maintenance investment may not equal to the budget B . Therefore, upon crossover and mutation, an operation termed herein as CleanPlan is implemented to convert the raw offspring to a legitimate candidate solution (see Fig. 3). For each raw offspring, the maintenance time for each bridge is (a) sorted chronologically and (b) merged to the earliest point-in-time of all maintenance actions if they are spaced less than . In addition, the investments for all bridges are adjusted to have a sum of B based on the proportion of maintenance investments in the raw offspring. Using the GA procedure described herein, the optimal or

beyond which congestion starts to occur; fa' , c is the link capacity associated with a damaged link; db is the link capacity drop due to failure of bridge b . With link properties updated based on Eq. (8), traffic flow in a damaged network ( fa ' in Eq. (7)) can be determined using the static traffic assignment model mentioned previously. Finally, the indirect consequences in terms of traffic delay can be evaluated by Eq. (7). 2.4. Risk-based optimization of Life-cycle maintenance plans Structural capacities of bridges decrease due to deterioration, increasing the network-level risk over time. In order to control this risk, maintenance actions should be planned in the service life of bridges. The effect of a maintenance action is modeled herein as an increase in the structural capacity of a bridge, which is related to the investment on the maintenance action. Specifically, the effect of a maintenance action is expressed as [19]

kR =

·Cm·(1 + r )tm Cbr

(9)

where kR is the relative increase in the structural capacity because of the maintenance action, and it is with respect to the initial structural capacity without deterioration; is the maintenance efficiency factor reflecting the cost-effectiveness of the action; Cm is the investment on the maintenance action in terms of net present value; r is the discount rate of money; tm is the time of the planned maintenance action; Cbr is the initial cost of the bridge. Since the initial cost of a bridge is a sunk cost, its value is used merely as a reference of how substantial the maintenance investment is. Therefore, the time value of Cbr should not be used. The maintenance efficiency factor can only have positive values. If = 1, Eq. (9) indicates that in order to double the structural capacity, the investment on maintenance should be the same as the initial cost. For more efficient maintenance actions, is larger than one, and the efficiency increases with . Normally, the investments on maintenance actions are subjected to budget constraints. Under budget constraints, the objective of life-cycle management is to minimize the life-cycle network-level risk by planning maintenance actions for various bridges at different points-in-time. Mathematically, the optimization problem can be formulated as Given Structural capacity and demand models associated with all bridges; Structural deterioration models associated with all bridges; Effects of maintenance actions; Number of maintenance actions; Find (design variables) Time (tm ) and investment (vm ) for maintenance actions for a bridge network, denoted herein by a decision vector d: ={tm, vm} So that (objective function) The maximum annual risk is minimized in the time horizon of interest: nsc

min max Rnet (d, t ): = d D 1 t tL

Lnet (si)· pf (d, t , si)

Subjected to (constraints) (a) Budget constraint nbr sum (vi) = B i=1 (b) Safety constraint min i (t i, vi , t ) T fori = 1, 2, 1 t tL

(10)

i=1

, nbr

(c) Maintenance constraint 1 ti, j ti, j 1 forj = 2, , |t i| nm andi = 1, 2, (d) Maintenance constraint 2 tstart t i tend fori = 1, 2, , nbr

, nbr

5

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D.Y. Yang and D.M. Frangopol

Fig. 3. Operations of genetic algorithm in the proposed method.

at least near-optimal maintenance plan can be obtained.

Table 1 Failure scenarios of the 5-bridge network.

3. Discussion The proposed method is first applied to a simple bridge network with 4 nodes, 5 links, and 5 bridges [57]. The bridge network is shown in Fig. 4. The traffic demands consist of 4000 PCUs/hour from Node 1 to Node 2 and 4000 PCUs/hour from Node 3 to Node 2. The practical capacities of Links 1 to 5 are 2000, 4000, 6000, 8000, and 10,000 PCUs/hours, respectively. All links have a length of 50 km, and the free speed on links is 50 km/h. The travel time on a link can be estimated based on the link flow and the following travel time function [61]:

ta (fa ) = ta,0 1 +

fa fa, c

(11)

where ta,0 and fa, c are the free speed travel time and link capacity, respectively; = 0.15 and = 4 are used [61]. The five bridges form 25 1 = 31 failure scenarios (damage states of the network) together with an intact network state with no failed bridges. The definitions of failure scenarios are summarized in Table 1. For links with failed bridges, Eq. (8) is used to update their free speed travel times and link

Scenario number

Definition

1 2 ⋮ 5 6 7 ⋮ 15 16 17 ⋮ 25 26 27 ⋮ 30 31

Only bridge B1 fails Only bridge B2 fails ⋮ Only bridge B5 fails Bridges B1 and B2 fail, while others survive Bridges B1 and B3 fail, while others survive ⋮ Bridges B4 and B5 fail, while others survive Bridges B1, B2, and B3 fail, while others survive Bridges B1, B2, and B4 fail, while others survive ⋮ Bridges B3, B4, and B5 fail, while others survive Only Bridge B5 survives Only Bridge B4 survives ⋮ Only Bridge B1 survives All bridges fail

capacities. It is assumed herein that links with failed bridges will loss 50% of their initial capacities (i.e. db = 0.5 in Eq. (8b)). The structural capacities and demands of all five bridges are assumed to be identical and are treated as primary random variables (i.e. R ( ) = R and S ( ) = S ). In this paper, relative values without units are used to describe structural capacities and demands. They are treated as values normalized by the mean value of the corresponding structural demand. Accordingly, the structural demand is assumed to follow a normal distribution with a mean value of 1 and a standard deviation of 0.375 [62]. The structural capacity is also assumed to follow a normal distribution. Its coefficient of variation is assumed to be 0.15 [62]. It should be noted that the statistics used herein are associated with the flexural capacity of RC girder bridges and the load effects from moving vehicles. The failure mode implied is thus the flexural failure under vehicle load, which is usually the dominant ultimate limit state for

Fig. 4. Five-bridge network. 6

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D.Y. Yang and D.M. Frangopol

Fig. 5. Network-level risk and risk-bound associated with different (a) detour lengths (Case A: lb = 0.5km , Case B: lb = 5km , and Case C: lb = 50km ), (b) threshold annual reliability indices (Case A: T = 1.5, Case B: T = 2 , and Case C: T = 2.5), and (c) correlation condition (Case A: R = S = 0.1, Case B: R = S = 0.5, and Case C: R = S = 0.9).

bridges in non-seismic zones. The mean value of this distribution is determined using first order second moment method (FOSM) [63] based on the annual reliability indices of bridges at the time of analysis, shown as follows

µR,0 =

0·

2 R

2 S

+

+ µS

(i.e. T in Eq. (10)), and (c) the correlations among bridge failures. The one-parameter-at-a-time method [68] is used for the sensitivity analysis. In this analysis, the focus is on the effects of different factors on the network-level risk bounds and estimated risk, determined by Eqs. (2) and (3), respectively. Therefore, only the annual risk at a specific point-in-time is calculated, and deterioration is not considered. Moreover, it is assumed that the annual reliability indices of all bridges are identical and equal to the threshold annual reliability index T . Based on this reliability index, the mean values of structural capacities can be determined by Eq. (12). A baseline case, referred to as Case B, is established by specifying values of the three factors analyzed herein. In particular, the detour length in Case B is assumed 5 km. The threshold annual reliability index is set as 2.0. To consider correlation of bridges failure, the correlation coefficient between structural capacities of any two bridges ( R ) is assumed 0.5; the same correlation coefficient is assigned for structural demands associated with different bridges ( S ). For each factor under consideration, two other representative values are used to depict its range. The case with respect to the smaller value of the two is denoted as Case A, whereas the larger one as Case C. Specifically, Cases A and C associated with lb assume detour lengths of 0.5 km and 50 km, respectively; Cases A and C associated with T use 1.5 and 2.5 as threshold annual reliability indices, respectively; Cases A and C associated with correlation use R = S = 0.1 and R = S = 0.9 to consider different correlation cases. Except for the factor under consideration, the other two factors in Cases A and C are the same as those associated with the baseline case (Case B). To compare different cases, risk bounds and the approximate risk are computed with Eqs. (2) and (3), respectively. They are plotted with respect to the increasing number of failure scenarios included in the risk assessment (i.e. msc in Eq. (2)). The risk and risk bound values are normalized by the approximate risk in the scenario where only the failure of Bridge B1 is considered (i.e. msc = 1). Fig. 5 compares the effects of different factors on the accuracy of risk approximation (i.e. Eq. (3)) and the width between risk bounds. Fig. 5(a) shows the risks

(12)

where µR,0 is the mean value of the structural capacity at the time of the analysis; 0 is the annual reliability index at the time of the analysis; R and S are the standard deviations of the structural capacity and demand, respectively, assumed herein to be time-independent; µS is the mean value of the structural demand. Structural deterioration is represented by the time-dependent mean structural capacity shown as follows [64]:

µR (t ) = µR,0

·t

(13)

where µR (t ) is the mean value of the structural capacity after t years of deterioration; is the annual deterioration rate; is the exponent. Different deterioration mechanisms can be reflected by different values of the exponent . For instance, linear ( = 1), parabolic ( = 2 ), and square-root ( = 0.5) functions can be used to model corrosion-induced deterioration, sulfate attack, and diffusion-controlled degradation, respectively [64]. The effects of different exponents on optimal maintenance times are discussed in Yang & Frangopol [65]. Ideally, deterioration models are determined with probabilistic life-cycle analysis [66,67]. Since corrosion poses a substantial threat to bridge safety, the linear model is considered. Nevertheless, the proposed method can accommodate other deterioration models. For the bridge network analyzed herein, it is considered that = 0.05 for all bridges. 3.1. Sensitivity analysis Three factors influencing the network-level risk are investigated in a sensitivity analysis. They are (a) the detour length associated with bridges (i.e. lb in Eq. (8a)), (b) the threshold annual reliability index 7

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and risk bounds associated with different cases of detour lengths. As expected, as the number of considered scenarios increases, the risk bounds become narrower while the approximate risk quickly approaches the precise value obtained using msc = nsc . The detour length does not seem to significantly affect the speed at which the narrowing as well as the convergence occurs. The effects of reliability thresholds are shown in Fig. 5(b). In contrast to Fig. 5(a), it can be seen that the threshold annual reliability index can noticeably influence the speed at which the risk bounds narrow down. As the threshold annual reliability index decreases, more failure scenarios need to be included in the analysis in order to have a narrow bound. Nevertheless, for the considered range of reliability 2.5), the approximate risk can still quickly conthresholds (1.5 T verge to the precise value. Finally, different cases associated with different correlation conditions are compared in Fig. 5(c). It can be noticed that the risk bounds and their narrowing are significantly influenced by the correlation condition. For near-independent case (i.e. Case A with R = S = 0.1), both the approximate risk and the risk bounds converge to the precise value swiftly. However, for the near-fully-correlated case (i.e. Case C with R = S = 0.9 ), risk bounds cannot effectively narrow down even in the last few scenarios where msc is close to nsc . Additionally, the approximate risk does not converge. In fact, although the approximate risk seems to converge from msc = 15 to msc = 25, it increases again to the precise value in the last few scenarios. This is because under high correlation condition where most bridges tend to survive or fail simultaneously, the probabilities from msc = 15 to msc = 25 are actually lower than those scenarios with more failed bridges. Therefore, the lower bounds of risks are prone to be drastically underestimated, leading to underestimated risk values from msc = 15 to msc = 25. This argument is supported by Fig. 6, where occurrence probabilities of different failure scenarios are calculated using the MSR method. To ensure the accuracy of these probabilities obtained from MSR, MCS with 2,000,000 samples are also conducted, the results of which are presented in Fig. 6. Fig. 7 shows the relative variation in the obtained risk value compared to that in the baseline case. All risk values are obtained with msc = nsc = 31 cases. It can be observed that for the ranges under consideration, the reliability thresholds have the most substantial impact on the network-level risk. Even though T is merely lowered by 0.5 from 2, the network-level risk almost tripled. On the other hand, the correlation condition does not affect the obtained network-level risk as dramatically as it does for convergence. Results from the preceding sensitivity analysis have important

Fig. 7. Sensitivity of risk magnitude to detour lengths, threshold annual reliability indices, and correlation.

implications on the application of the proposed method. The fast convergence in cases with a high threshold annual reliability index indicates that the proposed method is efficient when the safety constraint [Constraint (b) in Eq. (10)] is stringent. This usually needs to be supported by a sufficient budget for maintenance. Nevertheless, existing studies on target reliability indices of existing structures can provide a reasonable range of threshold annual reliability indices used in the proposed method. For instance, it is suggested that the annual target reliability index should be 2.3 for existing bridges with redundant spans [69] and higher for nonredundant or single load path bridges [70]. From the preceding sensitivity analysis, it can be realized that the proposed method can perform very well in this range of reliability thresholds. Another implication from the analysis herein is that the proposed method is suitable for bridge networks with low correlation among bridge failure events (i.e. R 0.5 and S 0.5). This relatively low correlation may be reasonable for a large number of highway bridge networks where bridges are widely spread out covering a large region. Even in cases where correlation is high, it is acceptable to slightly underestimate the correlation to ensure computational efficiency. This simplification is backed by the relative insensitivity of the network-level risk to different correlation conditions (see Fig. 7). 3.2. Optimization results For the 5-bridge network, GA is used to obtain the optimal life-cycle management plan. It is assumed herein that the annual reliability indices at the time of the analysis are 4 for all bridges, i.e. 0 = 4 in Eq. (12). Accordingly, the mean values of the initial structural capacities can be determined by Eq. (12). Specifically, µR,0 = 3.66 for all five bridges. Structural deterioration of bridges is quantified using Eq. (13). The other parameters are selected according to the baseline case in the previous sensitivity analysis, i.e. lb = 5km, db = 0.5, T = 2.0 , and R = S = 0.5. The time-horizon of interest is 75 years leading the time of the analysis (i.e. tL = 75). During this period, each bridge may receive at most two essential maintenance actions (i.e. nm = 2 ). The first essential maintenance action should start 10 years later, while the last essential maintenance action should be conducted at least 10 years before the end of analyzing period (i.e. tstart = 10 and tend = 65 ). The total budget for essential maintenance actions (B ) is US$1.5 million. The minimal interval between two essential maintenance actions of a bridge ( ) is 5 years. The maintenance efficiency factor is 2.0. The initial cost Cbr of each bridge is US$ 1 million, and the discount rate of money r is 2%. The GA is first implemented following the procedure described previously considering all failure scenarios (i.e. msc = nsc = 31). 500 candidate solutions are used to form the population in each generation,

Fig. 6. Occurrence probabilities of failure scenarios in Case C of sensitivity analysis of correlation ( R = S = 0.9). 8

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Fig. 8. Optimal maintenance plan for the 5-bridge network (all failure scenarios considered).

and 120 generations are considered to ensure convergence. In each generation, 60% of the population undergoes crossover operation, while the other 40% is subject to mutation (see Fig. 3). For the mutation group, the mutation rate of a genome is 0.1. The obtained optimal lifecycle management plan is shown in Fig. 8 and summarized in Table 2. The corresponding annual reliability profiles, annual network-level risk, and cumulative maintenance investment are presented in Fig. 9. It can be seen from Fig. 9(b) that the maximum annual network-level risk is substantially reduced due to maintenance. Since the number of the included failure scenarios (msc ) can directly affect the computational efficiency and even the feasibility of GA, different values of msc are investigated. In particular, five cases are taken into account, corresponding to different numbers of failed bridges covered in the msc scenarios used. These cases are denoted as Cases I to V. In Case I, the failure scenarios with only one failed bridge are considered, i.e. msc = 5 ; in Case II, scenarios with at most two failed bridges are considered, i.e. msc = 5 + 10 = 15; finally, in Case V, all scenarios (i.e. scenarios with at most five failed bridges) are considered, i.e. msc = 31. To compare the proposed method to the existing simulationbased method, another case, termed herein as Case MC, is also included in the analysis. In this case, the network-level risk is determined using MCS to generate failure scenario samples and their corresponding network-level consequences. 10,000 samples are used in MCS. Fig. 10 shows the convergence of GA for different cases. For all cases, the plots start from the first generation where candidate solutions satisfy all constraints. It can be seen that all cases converge to an optimal (or near-optimal) value. However, Case MC and Case I are prone to under- and overestimating the optimal network-level risk,

Fig. 9. Results associated with the optimal maintenance plan for the 5-bridge network (all failure scenarios considered): (a) time-variant annual reliability indices of bridges, (b) annual network-level risk, and (c) cumulative maintenance investments.

Table 2 Optimal maintenance plan for the 5-bridge network. Bridge

Essential maintenance time (year) I II

Maintenance investment (1000 USD) I II

B1 B2 B3 B4 B5

21 15 12 10 12

158 117 300 287 265

46 39 40 45 29

Fig. 10. GA convergence for different cases.

81 128 30 60 75

respectively. For Case MC, the underestimation is because of the omission of more severe failure scenarios that may not appear in MCS. For Case I, the overestimation stems from the fact that 26 among the overall 31 scenarios are assumed to have the worst-case consequence

9

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Fig. 12. Network model of the Sioux Falls bridge network (24 nodes, 76 links, and 10 bridges).

space (i.e. the obtained life-cycle maintenance plan). This finding indicates that the proposed method may only need to consider the scenarios with one failed bridge in the optimization process. To correct the overestimated risk, more accurate risk assessment can be conducted for the obtained optimal solution by considering more failure scenarios. In contrast to Case I, simulation-based optimization in Case MC cannot even lead to near-optimal solutions. Although it appears that the risk is only slightly underestimated, the real value of the network-level risk associated with the solution in Case MC is significantly higher than those in other cases. This observation demonstrates the superiority of the proposed method over the conventional simulation-based methods. 4. Illustrative example Based on the results from Discussion, the proposed method is employed for life-cycle management of an existing bridge network to demonstrate its application. In this example, the Sioux Falls network is considered. This network, adapted from the transportation network in Sioux Falls, South Dakota, has been widely used to test traffic assignment models [71]. The bridge network consists of 24 nodes, 76 links, and 10 deteriorating bridges located on the links [47]. The spatial distributions of nodes, links, and bridges are presented in Fig. 12. The traffic demands within the network are shown in Fig. 13 based on different origin and destination nodes. The link properties including link capacity, link length, and free speed are summarized in Table A1. The travel time function follows the same form in Eq. (11) with = 0.15 and = 4 . The bridge properties at the time of the analysis are presented in Table 3. To represent the spatial heterogeneity of bridge conditions, the initial annual reliability indices are generated by a normal distribution with a mean value of 4.0 and a standard deviation of 0.2; the deterioration rates of bridges are sampled from a Uniform distribution ranging from 0.02 to 0.08; the detour lengths of bridges are estimated based on the local road layouts as shown on the Google Maps [72]. It should be noted that Bridges B3 and B5 are two river bridges that cannot find local detours. Therefore, their detour lengths are set as 999 km. Numerically, this is equivalent to removing the links where Bridges B3 and B5 are located from the network if any of the bridges fails. Since the bridge network is highly redundant, the loss of these two links will not affect the connectivity within the network. As in the 5bridge network, the link capacities will drop by 50% if failed bridges are present on the link. The initial costs of bridges, a value needed in Eq. (9), are assumed based on the data in a previous study [14] and are summarized in Table 3.

Fig. 11. Comparison of optimal solutions for different cases: (a) cumulative maintenance investments and (b) approximate and precise maximum annual network-level risks (Case MC uses MCS to estimate network-level risk; Case I–V use risk-bounds involving one to five failed bridges to estiamte network-level risks).

(i.e. the consequence associated with 5 failed bridges) to obtain upperbound of the risk. Although the minimized risks in different cases are similar, the associated life-cycle maintenance plans are drastically different, as shown in Fig. 11(a) in terms of cumulative maintenance costs. This observation indicates the optimization problem might have multiple optimal solutions, and/or GA is only able to converge to a near-optimal solution. To appraise the quality of the obtained optimal solutions, the objective function with msc = 31 is used to calculate the precise network-level risks associated with the obtained maintenance plans in different cases. The results are shown in Fig. 11(b). It can be found that solutions in Cases I to V, albeit their difference in the degrees of approximation in risk evaluation, all result in a very similar minimized risk to the precise solution. Interestingly, the overestimation of the objective value in Case I does not compromise the quality of its solution in design variable

10

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Fig. 13. Traffic demands in the Sioux Falls network.

Table 3 Bridge information in the Sioux Falls network. Bridge

Initial annual reliability indexa

Mean structural capacityb

Deterioration ratec

Initial costd (USD)

Detoure (km)

On linkf

B1 B2 B3 B4 B5 B6 B7 B8 B9 B10

3.987 3.950 3.882 4.038 4.093 4.213 4.082 3.921 3.839 3.803

3.642 3.594 3.509 3.710 3.785 3.959 3.770 3.557 3.456 3.413

0.02277 0.06849 0.07236 0.07116 0.02002 0.04846 0.06185 0.02661 0.02925 0.02147

2,528,482 2,289,328 1,605,177 2,155,577 2,446,449 3,584,922 1,097,027 2,204,237 1,094,328 1,689,624

3.219 1.609 999 0.322 999 0.322 0.644 3.219 3.219 0.322

L2 & L5 L4 & L14 L70 & L72 L45 & L57 L41 & L44 L28 & L43 L42 & L71 L66 & L75 L37 & L38 L34 & L40

Note: a Initial annual reliability indices are generated based on a normal distribution with a mean value of 4.0 and a standard deviation of 0.2. b Mean structural capacities are relative values (unitless) back-calculated based on the initial annual reliability indices. c Deterioration rates are generated based on a Uniform distribution ranging from 0.02 to 0.08. d Initial cost data are from Yang and Frangopol [14]. e Detour lengths are estimated based on Google Maps [72]. f Bridges are present in both directions of traffic, so each bridge is on two links (descriptions of links are provided in Table A1).

Table 4 Criteria for determining correlation coefficients of structural demands. Criterion

Description

I II III IV V VI

Two Two Two Two Two Two

bridges bridges bridges bridges bridges bridges

Correlation coefficient of structural demands are are are are are are

connected by highways connected by highway and major local roads connected by major local roads on two local roads that are connected by one node on two parallel local roads that are next to each other connected but are not in the criteria mentioned above

11

0.7 0.5 0.7 0.5 0.5 0.3

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Table 5 Correlation matrix of structural demands of bridges. Bridge

Bridge B1 B2

B1 B2 B3 B4 B5 B6 B7 B8 B9 B10

1

0.5 1

B3

B4

B5

B6

B7

B8

B9

B10

0.3 0.3 1

0.3 0.3 0.3 1

0.3 0.3 0.5 0.5 1

0.5 0.7 0.3 0.5 0.5 1

0.3 0.3 0.5 0.3 0.5 0.3 1

0.7 0.5 0.3 0.3 0.3 0.3 0.3 1

0.7 0.5 0.3 0.3 0.3 0.3 0.3 0.7 1

0.3 0.3 0.3 0.3 0.5 0.5 0.5 0.3 0.3 1

Symmetric

Correlation among structural capacities of bridges are related to many factors such as bridge ages and bridge types. Similar to the 5bridge network, the correlation coefficient between the structural capacities of any two bridges is assumed to be 0.3, i.e. R = 0.3. By contrast, the correlation among structural demands is determined in a more sophisticated manner. The correlation coefficients between structural demands of two bridges are determined based on the criteria summarized in Table 4. The obtained correlation coefficients are provided in Table 5. It should be noted that the correlation coefficient matrix thus obtained may not be positive-definite. In this case, the closest correlation coefficient matrix is sought by Higham’s [73] algorithm [74]. Based on the deterioration rates in Table 3, deterioration is represented by Eq. (13). The time horizon under consideration is 75 years. For life-cycle essential maintenance actions, nm = 2 , tstart = 10 , tend = 65 , = 5, = 2 , and T = 2 are used similar to the 5-bridge network. The total maintenance budget (B ) is $15 million. For GA, 500 candidate solutions are used in a generation, and overall 150 generations are implemented to ensure convergence of solutions. For the evaluation of the objective function, two cases are considered in this example. In the first case (denoted herein as Case X), the failure scenarios with only one failed bridge are included in the objective function (i.e. msc = 10 ). In the second case (denoted herein as Case Y), the failure scenarios with up to three failed bridges are considered (i.e. msc = 175). The network-level consequences associated with Cases X and Y are shown in Fig. 14. Fig. 15 shows the optimal maintenance plans for Cases X and Y, respectively. These two plans are also summarized in Table 6. Similar to the results of the 5-bridge network, the optimal plans in both cases are different. The cumulative maintenance costs in Cases X and Y are presented in Fig. 16(a). This figure indicates that Case X arrives at a

Fig. 15. Optimal maintenance plans for (a) Case X and (b) Case Y.

solution where maintenance actions are more evenly spread out in the time horizon of interest, while Case Y shows two clear phases of spending at the beginning and the end of the time period under consideration. The risk value associated with the solution in Case X is also calculated using msc = 175, as presented in Fig. 16(b). It can be seen that although the objective function with msc = 10 significantly overestimates the network-level risk, the associated optimal solution actually performs quite well compared to that obtained considering more failure scenarios. This finding is consistent with that for the 5-bridge network and indicates the potential computational benefit of the proposed method. For instance, to improve computation efficiency, the optimization can be conducted only considering failure scenarios with one or two failed bridges. The precise magnitude of risk reduction can then be evaluated by calculating the network-level risk considering more failure scenarios.

Fig. 14. Network-level consequences for Cases X and Y. 12

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5. Conclusions

Table 6 Optimal maintenance plan for the Sioux Falls network.

In this paper, a novel method is proposed for life-cycle management of deteriorating bridge networks. The proposed method uses system reliability analysis to obtain the network-level risk bounds. The risk bounds are then employed to approximate the network-level risk, based on which life-cycle management can be conducted using metaheuristic optimization. Compared to conventional simulation-based methods for network-level risk assessment, the proposed method does not resort to Monte Carlo simulation and exhibits noticeable computational advantage. In addition, it is demonstrated that the proposed method is more likely to lead to optimal or near-optimal solutions compared to conventional methods. The proposed method has been systematically investigated through a sensitivity analysis and applied to an existing bridge network. Based on the results, the following conclusions can be drawn:

(a) Case X (only one failed bridge is considered) Bridge Essential maintenance time (year) Maintenance investment (1000 USD) I II I II B1 B2 B3 B4 B5 B6 B7 B8 B9 B10

25 15 10 13 14 21 21 27 19 18

49 48 35 26 59 29 39 40 61 54

518 506 819 717 777 1332 933 738 240 877

773 1285 542 1009 446 1301 726 294 304 863

(b) Case Y (three failed bridges are considered) Bridge Essential maintenance time (year) Maintenance investment (1000 USD) I II I II B1 B2 B3 B4 B5 B6 B7 B8 B9 B10

11 12 11 10 54 10 22 10 22 10

29 17 28 56 – 57 34 26 52 42

375 1467 1507 807 725 1364 275 144 428 1282

1. System reliability methods can be used to calculate the occurrence probabilities of different failure scenarios of bridges in a network. The calculated probabilities can be employed in the formulation of upper and lower bounds of network-level risks. With the risk bounds, only a few traffic assignment operations are needed in order to estimate network-level risks of deteriorating bridge networks. As a result, the proposed method exhibits considerable computation advantage compared to conventional simulation-based methods. 2. Conventional simulation-based methods are not able to consider unlikely, yet high-impact events (“black swans”), leading to underestimated risks. The maintenance plan thus obtained may not be able to effectively lower network-level risks. Without adding any computational cost, the use of risk bounds in risk assessment can take into account these “black swans” and their impacts on networklevel risks. This advantage is especially relevant to deteriorating bridge networks where high-impact events involving multiple failed bridges are prone to be ignored in simulation-based methods due to their low probabilities of occurrence. 3. Various factors influencing risk bounds and the associated approximate risks are investigated. As more failure scenarios of bridges are considered in the risk assessment, the risk bounds narrow down, and the network-level risk approximated by risk bounds approaches its precise value. From the sensitivity analysis, it is found that the threshold annual reliability index and the correlation among bridge failure events can significantly affect the width between upper and lower risk bounds as well as the convergence of the approximate risk. On the other hand, the detour lengths and the threshold annual reliability index have a substantial impact on the magnitude of network-level risks. For the parametric values commonly encountered in practice, the risk approximated by risk bounds can swiftly converge to the precise value using a relatively small number of failure scenarios. 4. Genetic algorithm (GA) can be used to solve the life-cycle management problem. By adding an additional operation to the crude GA, the GA in the proposed method can obtain both optimal maintenance time and optimal maintenance investment. In addition, the number of maintenance actions for a bridge can be variable. It is also found that the proposed method can rely on objective functions involving failure scenarios with only one or two failed bridges. Although the network-level risk is likely to be overestimated using these objective functions, the obtained maintenance plans are still near-optimal. Once these maintenance plans are determined, the corresponding precise values of network-level risks can be efficiently calculated using risk bounds determined with more failure scenarios. 5. The risk-bound method powered by system reliability analysis can also be applied to other risk assessment and management problems where consequence evaluation is more computationally expensive than system reliability analysis. The computation complexity of the

1022 1833 173 1390 – 1039 154 139 684 192

Fig. 16. Comparison of optimal solutions for Cases X and Y: (a) cumulative maintenance investments and (b) approximate and precise maximum annual network-level risks. 13

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proposed method is directly related to the number of system components (i.e. bridges) under consideration. It will affect the computational efficiency of system reliability analysis, the number of failure scenarios that need to be considered, and the design variable space in the optimization process.

U.S. National Science Foundation (Grant CMMI 1537926) and the U.S. Department of Transportation Region 3 University Transportation Center (Grant CIAM-UTC-REG6). The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organizations.

Acknowledgements The authors are grateful for the financial support received from the Appendix

Table A1 Link properties of links in the Sioux Falls network (node numbers and locations are provided in Fig. 12). Link notation

Connection (start node, end node)

Capacity (PCUs/hour)

Free speed (km/h)

Length (km)

L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11 L12 L13 L14 L15 L16 L17 L18 L19 L20 L21 L22 L23 L24 L25 L26 L27 L28 L29 L30 L31 L32 L33 L34 L35 L36 L37 L38 L39 L40 L41 L42 L43 L44 L45 L46 L47 L48 L49 L50 L51 L52 L53 L54 L55 L56 L57

(1, 2) (1, 3) (2, 1) (2, 6) (3, 1) (3, 4) (3, 12) (4, 3) (4, 5) (4, 11) (5, 4) (5, 6) (5, 9) (6, 2) (6, 5) (6, 8) (7, 8) (7, 18) (8, 6) (8, 7) (8, 9) (8, 16) (9, 5) (9, 8) (9, 10) (10, 9) (10, 11) (10, 15) (10, 16) (10, 17) (11, 4) (11, 10) (11, 12) (11, 14) (12, 3) (12, 11) (12, 13) (13, 12) (13, 24) (14, 11) (14, 15) (14, 23) (15, 10) (15, 14) (15, 19) (15, 22) (16, 8) (16, 10) (16, 17) (16, 18) (17, 10) (17, 16) (17, 19) (18, 7) (18, 16) (18, 20) (19, 15)

25,900.2 23,403.5 25,900.2 4958.2 23,403.5 17,110.5 23,403.5 17,110.5 17,782.8 4908.8 17,782.8 4948.0 10,000.0 4958.2 4948.0 4898.6 7841.8 23,403.5 4898.6 7841.8 5050.2 5045.8 10,000.0 5050.2 13,915.8 13,915.8 10,000.0 13,512.0 4854.9 4993.5 4908.8 10,000.0 4908.8 4876.5 23,403.5 4908.8 25,900.2 25,900.2 5091.3 4876.5 5127.5 4924.8 13,512.0 5127.5 14,564.8 9599.2 5045.8 4854.9 5229.9 19,679.9 4993.5 5229.9 4824.0 23,403.5 19,679.9 23,403.5 14,564.8

80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 80 45 80 80 45 45 80 80 45 80 80 80 80 80 45 80 80 80 45 80 80 80 80 80 45 45 45 80 45 80 80 80 80 80 80 45 80 80 80 80 80 80

8.00 5.33 8.00 6.67 5.33 5.33 5.33 5.33 2.67 8.00 2.67 5.33 6.67 6.67 5.33 2.67 2.25 2.67 2.67 2.25 7.50 6.67 6.67 7.50 4.00 4.00 6.67 8.00 5.33 6.00 8.00 6.67 8.00 3.00 5.33 8.00 4.00 4.00 5.33 3.00 3.75 3.00 8.00 3.75 4.00 4.00 6.67 5.33 2.67 4.00 6.00 2.67 2.67 2.67 4.00 5.33 4.00

(continued on next page) 14

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Table A1 (continued) Link notation

Connection (start node, end node)

Capacity (PCUs/hour)

Free speed (km/h)

Length (km)

L58 L59 L60 L61 L62 L63 L64 L65 L66 L67 L68 L69 L70 L71 L72 L73 L74 L75 L76

(19, (19, (20, (20, (20, (20, (21, (21, (21, (22, (22, (22, (22, (23, (23, (23, (24, (24, (24,

4824.0 5002.6 23,403.5 5002.6 5059.9 5075.7 5059.9 5229.9 4885.4 9599.2 5075.7 5229.9 5000.0 4924.8 5000.0 5078.5 5091.3 4885.4 5078.5

80 80 80 80 80 45 80 80 80 80 45 80 45 45 45 45 45 80 45

2.67 5.33 5.33 5.33 8.00 3.75 8.00 2.67 4.00 4.00 3.75 2.67 3.00 3.00 3.00 1.50 3.00 4.00 1.50

17) 20) 18) 19) 21) 22) 20) 22) 24) 15) 20) 21) 23) 14) 22) 24) 13) 21) 23)

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