VoI-Based Optimization of Structural Assessment for Spatially Degrading RC Structures

: Before implementing a bridge monitoring strategy, a bridge manager would like to know the return on investment. Moreover, in order to spend the available budget as efﬁciently as possible, the monitoring strategy should be optimized, i.e., the type of measurements but also the time and locations at which these are performed. For this purpose, the Value of Information (VoI) can be used. The VoI represents an estimate of the beneﬁt that can be gained from a monitoring strategy before it is actually implemented. By comparing the VoI of different alternative strategies, the one with the highest VoI can be selected. As such, the VoI is a tool for objective decision-making. The calculation of the VoI is based on pre-posterior analyses, including Bayesian updating of model parameters based on yet unknown monitoring outcomes. When calculating the VoI for an actual case, some challenges arise. First, the calculation of the VoI requires a number of assumptions on different input parameters. Second, the VoI is computed by evaluating life-cycle costs for different possible outcomes of the monitoring strategy, leading to a high computational cost. However, for practical implementations, results are preferably available within an acceptable time span and are robust with respect to the chosen input parameters. In this work, the implementation of the VoI approach for optimization of monitoring strategies is investigated by a problem statement in a case study where a reinforced concrete girder bridge is considered. To perform this optimization, the VoI for different monitoring strategies is compared. The calculation time required for the Bayesian updating of the model parameters based on the available data is limited by using Maximum A Posteriori (MAP) estimates to approximate the posterior distribution. The VoI can be used both to optimize a monitoring strategy or for comparison of different strategies. To limit the number of required (computationally expensive) evaluations of the VoI, optimization of the monitoring strategy itself can be simpliﬁed by determining the optimal sensor locations beforehand, based on a different metric than the VoI. For this purpose, the information entropy is used, which expresses the difference between the prior and posterior uncertainty of the model parameters. Finally, the sensitivity of the VoI to different input parameters is investigated.


Introduction
In many countries, bridges are reaching the end of their lifetime. In order to optimize bridge maintenance under the existing budget constraints, condition-based maintenance can be performed instead of predictive-based maintenance (e.g., [1,2]). Based on obtained sitespecific data, prediction models of the structure can be updated by application of Bayesian methods [3]. Nevertheless, monitoring the condition of the bridge also comes at a certain cost. To investigate whether this monitoring is worth its cost, the corresponding Value of Information (VoI) can be calculated. The VoI is the difference in expected service-life cost for the prior maintenance strategy and the expected service-life cost when accounting for data provided by a proposed monitoring strategy [4]. The VoI can be used as a metric to optimize the monitoring strategy, i.e., the type of measurements performed, the time of measuring and the location of the sensors. Moreover, by calculating the VoI for different monitoring strategies, the best strategy can be chosen as the one with the highest VoI.
The calculation of the VoI is based on a pre-posterior analysis where a Bayesian framework is used to update the model parameters based on yet unknown monitoring results. The costs in the calculation of the VoI correspond to the life-cycle costs, including costs for maintenance actions and repairs, costs for monitoring and inspection, and failure costs. For all these costs, relevant values should be used as input in the VoI analysis. Moreover, in a pre-posterior analysis, prior distributions of the model parameters are updated to posterior distributions based on yet unknown monitoring data. These prior distributions to be used in the analysis should also be defined. Assigning prior distributions to the model parameters and estimating the different costs required in the analysis can be a difficult task. Investigation is needed to determine whether these parameters influence the resulting VoI and whether a robust result is obtained or not. The relevant input parameters are pointed out in this work and their influence on the VoI is investigated.
The calculation of the VoI can be computationally challenging. It requires the calculation of two expected values of the life-cycle cost, i.e., the prior and the posterior cost. To evaluate the posterior costs, the expected value of the life-cycle costs is evaluated by taking the average over all possible monitoring outcomes. In this analysis, for the different monitoring outcomes, posterior distributions of the model parameters and the corresponding updated probability of failure should be calculated. Calculating this posterior distribution is often based on sampling based methods, where many model evaluations are required.
Applications of the VoI analysis can be found in several works. For example, in [5] the methodology is applied to steel structures and the VoI is calculated based on partially observable Markov decision processes. The influence of the time to damage, availability and inaccuracy of the data, reaction time, repair costs, and discount factor is investigated for application on a wind turbine structure under continuous damage monitoring. Also in [6], the VoI is explained in a clear way and its benefit of a rational way of making decisions is illustrated. In [7], transition probabilities between different damage states and probabilities of monitoring outcomes are assumed together with repair costs for application in a Bayesian network for the example of a wind farm. Also in [8], the VoI is determined for the application of structural health monitoring to a wind farm. Finally, in [9], assumptions are made on utilities and prior probabilities of bridge states.
In general, all the referred works have contributed greatly to the introduction of the VoI as an objective tool in decision-making. However, it is not always clear how sensitive the computed VoI is to different choices for the relevant input parameters. The literature is especially lacking applications for the optimization of monitoring strategies in the case of reinforced concrete structures. Some publications can be found where the influence of some assumptions on the input variables is investigated. However, this is often very limited and applied to simplified conceptual models of structures. In [10], the effects of different system properties on the VoI are demonstrated. In addition, Reference [11] illustrates how the decision problem is influenced by the assumed probabilistic models, i.e., the type of probability distribution, the level of uncertainty, the choice of degradation law, the quantity and quality of information, and the probabilistic dependencies between components of systems. The influence of the measurement accuracy on the VoI is investigated in [12]. In [12], the challenges in the selection of suitable monitoring thresholds for action alternatives are also pointed out. This concerns both defining the threshold values for monitoring outcomes triggering an action and the choice among different actions. In [13], the influence of the threshold on the annual probability of component fatigue failure is investigated, and in [14], the influence of the inspection time and monitoring plan is studied, together with the influence of the number of monitoring years. In [15], the computational burden of the VoI is pointed out as one of the main challenges in real world applications. From this work and others, it can be concluded that computational cost is an important bottleneck in the calculation of the VoI since calculations over the whole service life of the bridge are required. In [16], the VoI analysis is applied to a reinforced concrete structure to determine the cost effectiveness of measuring the chloride content as an alternative to visual inspections. The work provides a valuable basis for the application of VoI analysis to reinforced concrete structures affected by chloride ingress. However, some issues for future work are mentioned, such as the fact that actual costs should be known and might influence the results, no particular case study is considered in the work, discrete inspection outcomes are assumed (depassivation is detected or not detected), and the accuracy of the measurements might influence the results.
In this work, the practical use of the VoI as a tool for optimization of a monitoring strategy is investigated, particularly focusing on reinforced concrete structures. Section 2 provides a short recapitulation of the calculation of the VoI, and Section 3 focuses on some issues in the practical implementation. In Section 4, a problem statement is introduced in the form of a reinforced concrete (RC) girder bridge subjected to corrosion for which the monitoring strategy should be optimized. In Section 5, the sensitivity of the VoI to initial assumptions on the input values is investigated. Finally, Section 6 examines the influence of using expected values for the costs in the calculation of the VoI.

Calculation of the VoI
The calculation of the VoI for a spatially degrading reinforced concrete structure is explained in [17]. A generalization of the framework from [17] is given in Figure 1. It shows how the VoI depends on the expected prior (E[C pr ]) and posterior (E[C post ]) costs. To calculate these expected values, a series of steps are required. First, the structure needs to be defined together with all relevant input parameters. Since the evaluation of the VoI requires the calculation of service life costs, the service life t SL and different costs C need to be defined. The evaluation of the VoI takes place in a Bayesian framework. Hence, probabilistic analysis is required and stochastic distributions are assigned to the relevant model parameters. By application of Bayesian analysis, these prior distributions of the model parameters f are updated to posterior distributions f" based on (yet unknown) monitoring data. Hence, the prior distributions of the model parameters need to be defined at the start of the VoI analysis.
The prior cost is the life-cycle cost accounting for no monitoring, eventually taking into account a standard inspection strategy (i.e., for example regular visual inspections, indicated with subscript 'st'). In the posterior cost, the monitoring strategy for which the VoI is calculated is taken into account. Hence, this monitoring strategy should also be defined: what type of measurements, when and where to measure, and the corresponding costs. In both the prior and posterior analysis, different decision alternatives a should be considered, which can be based on reliability thresholds or monitoring outcomes. To calculate an expected value of the costs, all possible inspection and/or monitoring outcomes should be sampled, inducing the samples y. These contain both the outcomes from the reference scenario (prior analysis)(y st ) and those of the additional monitoring strategy in the posterior analysis (y add ). For each set of monitoring outcomes, the life-cycle cost C T (y) is calculated. This is the minimum of the life-cycle costs for the different possible action alternatives in the maintenance strategy of the structure, i.e., min(C T (t SL |a,y)). The costs for each of these action alternatives are evaluated by calculating failure probabilities at every time step between the current time t 0 and the service life t SL . The corresponding failure costs C F are accounted for, depending on the cumulative probability of failure p F (t). At times where measurements are obtained, the relevant distributions are updated, i.e., at t insp/meas , prior distributions of the model parameters f are updated to posterior distributions f . The costs of monitoring/inspections C I are also accounted for. When maintenance actions are performed at t action , model parameters are changed accordingly, and the relevant costs C A are accounted for. This framework allows accounting for different types of data gathered from the monitoring strategy, possibly at different locations along the structure. Reference [18] illustrates how strain data obtained from proof-loading and modal data obtained from dynamic tests can be used to infer the state of corrosion of the structure for a reinforcedconcrete bridge. These data can be incorporated into the VoI framework. This enables the optimization of a monitoring strategy itself or choosing the most optimal strategy among different possible strategies.
As pointed out above, different (modeling) assumptions are required in the evaluation of the VoI such as costs, prior distributions, etc. Moreover, since the calculation of the VoI involves Bayesian updating and the consideration of the entire service life, calculating the VoI can be computationally expensive. Using the VoI both for optimizing a monitoring strategy itself and for choosing the most optimal strategy among a set of possible strategies requires many evaluations of the VoI. Hence, some approximations might be required to limit both the number of required evaluations of the VoI and the computational time of a single VoI evaluation. For example, to reduce the number of evaluations of the VoI, part of the optimization of the monitoring strategy can be performed outside the VoI analysis. Optimal sensor positions can be determined beforehand based on optimal sensor placement algorithms as for example in [19][20][21]. Similar methods are applied in this work to derive the optimal sensor positions for updating the corrosion parameters in the degradation model of RC structures. In the following sections, a balance is made between accuracy and computational time, and the influence of different choices on input parameters is investigated.

Approximating the Posterior Distribution
In Section 2 and Figure 1, it is indicated that the prior distributions f of the model parameters are updated to posterior distributions f " based on the available monitoring data. For this purpose, Markov Chain Monte Carlo (MCMC) sampling is generally used. However, the application of MCMC requires a considerable computational effort. When the posterior distribution has to be derived for different possible monitoring outcomes, as is the case when calculating the VoI, this can lead to a very high computational burden. To overcome this issue, approximate methods can be used to estimate the posterior distributions. Since for the calculation of the VoI the sign of the resulting VoI and the relative difference between different monitoring alternatives are most important, an approximation that leads to an error that is small compared to this difference in VoI is justified.
According to [22,23], when a large amount of data is available, the posterior Probability Density Function (PDF) can be asymptotically approximated by a normal distribution, centered around the Maximum A Posteriori (MAP) point and with a covariance matrixΣ po . For a set of parameters θ to be estimated, with prior probability f (θ), the MAP estimate is given by Equation (1) [22,23].
Here, L(ϑ|y) is the likelihood function accounting for data y. When this likelihood function is Gaussian, the first term F ML corresponds to a generalized least squares objective function. The term F MAPr corresponds to a regularization term based on the available prior information.
The approximate posterior covariance matrixΣ po is computed as the inverse Hessian of the MAP objective function F MAP , evaluated at the MAP point according to Equation (2).
In many optimization algorithms, this Hessian is computed as a by-product in the solution of the optimization problem to solve for the MAP point.

Optimization of Sensor Locations Using Information Entropy
The VoI can be used either to optimize a monitoring strategy itself or to choose between different alternative strategies. In the optimization of a monitoring strategy, sensor locations could also be considered as an optimization parameter, resulting in the calculation of the VoI for different sensor configurations. Nevertheless, this requires many calculations of the VoI and hence a very large computational effort. This can be resolved by determining optimized sensor locations for parameter estimation beforehand by minimizing a metric called the information entropy. This information entropy expresses the decrease in uncertainty from the prior distribution of the model parameters to the posterior distribution after updating based on the considered data. The latter depends on the available data y and the sensor configuration L. It is necessary to choose the optimal sensor locations such that the best possible information on the model parameters θ is extracted from the data [24]. By optimizing the sensor locations beforehand, only the number of sensors should be taken along in the optimization procedure of the monitoring strategy, and the required number of evaluations of the VoI can be reduced.
For a large number of data, an asymptotic approximation of the information entropy is given by Equation (3) [19].
Here, θ 0 represents the optimal value of the parameter set θ that minimizes the measure of fit between the measured and modeled data. N θ is the number of model parameters, and the matrix Q(L; θ 0 ) is known as the Fisher information matrix and contains information on the posterior uncertainty of the parameters θ based on the data from all sensor positions in L. For the prior estimate, nominal values can be used for θ 0 which are representative for the system (e.g., the prior mean values). The Fisher information matrix Q(L; θ 0 ) is given by Equation (4).
where ∇ θ x k represents the sensitivities of the measured data type to the model parameters, ∑ is the prediction error covariance matrix, and N is the number of sampled data. The optimal sensor locations can be found by solving a discrete-valued optimization problem. The sensors should be placed such that the resulting measurement data are most informative about the model parameters. Hence, the optimal sensor location L opt is the one that minimizes the information entropy. This minimization is constrained over the set of N p measurable degrees of freedom. Two heuristic sequential sensor placement (SSP) algorithms exist in the literature: the forward and the backward SSP [19]. In the forward algorithm, the sensor that results in the highest reduction in information entropy is added in each iteration. Hence, at each iteration, the sensor configuration is selected with the minimum information entropy. The backward algorithm works in the reverse order, starting from sensors at all possible locations and each time removing the sensor that results in the smallest increase in information entropy.
In this work, the sensitivities required in the evaluation of the Fisher information matrix are calculated based on the method of Nelson [25], as for example in [21]. Furthermore, the method is here applied in relation to a finite element model. The sensitivities of the measurement data to the corrosion parameters are derived and the optimal sensor positions are determined based on the forward SSP algorithm, which is a greedy search algorithm.

Optimization of Time of Monitoring
In addition to choosing the locations of the sensors, the time at which monitoring is performed also needs to be defined. Very often, time bounds where monitoring is useful can already be defined beforehand. For example, when it is the aim of monitoring or inspection to collect data that allow assessing the corrosion degree of a reinforced-concrete structure, these measurements are only useful when it is likely that corrosion has initiated. For this reason, the initial time range following construction is not considered in the greedy search for optimizing the time of monitoring. To optimize the time of monitoring, a number of steps are followed to arrive at a maximum value for the VoI. No minimization algorithm is used in this work, but a greedy search is applied in order to detect local optima, similar to the optimization of the sensor locations. In the first step, the posterior cost is calculated for different monitoring times and for one possible monitoring strategy. Then, the time step leading to the lowest posterior cost is chosen. At this time step, gradually, the number of sensors is increased or decreased, until the minimum posterior cost is obtained. For a given number of sensors, the sensor locations are chosen based on the optimal sensor placement as explained in Section 3.2. These optimal sensor locations are time independent [26].

Problem Description and Probabilistic Models
In this case study, a reinforced-concrete (RC) girder bridge is considered as described in [18,27,28] and illustrated in Figure 2. For more information on the bridge geometry, the reader is referred to [18,27,28]. The bridge is subjected to chloride-induced corrosion, which leads to a reduced resistance over time. The probability of failure P f (t) and the corresponding reliability index β(t) are quantified based on the bending limit state given by Equation (5). The probability of failure can be used to evaluate the cumulative probability of failure p F (t) mentioned in Figure 1. Based on this cumulative probability of failure, the cost of failure at time t (C F (t)) can be calculated ( Figure 1).
In Equation (5), A s (t) is the reinforcement area at time t, which changes over time due to corrosion (see Section 4.2), and P represents the load effects due to permanent loads, which are among others a function of the concrete density ρ c (see Table 1). For the meaning of the other variables in the limit state function and their probability distributions, the reader is referred to Table 1. The characteristic values of the load effects relate to a traffic load according to the Eurocode [29]. The reference period for the probabilistic load models and hence for the resulting probability of failure is equal to one year.

Time Dependent Degradation
The bridge is subjected to chloride-induced corrosion, leading to a reduction of its resistance over time. The use of a chloride-ingress model in the assessment of an existing structure is for example illustrated in [33]. The reduction in resistance of the RC structure due to chloride-induced corrosion depends on the reduction of the reinforcement area over time, A s (t), as given by Equation (6).
where r 0 is the initial reinforcement radius, V corr the corrosion rate, T i the initiation period, and α p a factor accounting for pitting. The initiation period depends on the concrete cover c, the surface chloride concentration C s , the critical chloride concentration C cr , and the diffusion coefficient D according to Equation (7). It should be pointed out that if the concrete is previously cracked, chloride ingress might be faster, resulting in a lower initiation period [34]. This is not considered in this work, since the bridge is assumed uncracked.
Corrosion parameters are modeled in a probabilistic way. Since corrosion can have a spatial variation along the structure (see for example [35]), some of the corrosion variables are modeled by random fields. The corrosion rate V corr , is modeled by means of a 2D lognormal random field, with mean 0.0116 mm/year and a coefficient of variation of 0.2 [28]. The correlation model of the underlying Gaussian random field is a squared exponential correlation model with a correlation length of 2 m along the length of the bridge and a correlation length of 5.8 m along the width of the bridge. For the diffusion coefficient D, a similar random field is applied, with mean value 129 mm 2 /year and a coefficient of variation of 0.10. The chloride concentration at the surface C s is modeled by a scalar random variable, assuming the whole bridge is subjected to identical exposure conditions. A lognormal distribution with mean 0.1% and a coefficient of variation of 0.1 [28] are chosen. To account for the spatial variation of the properties that are modeled by random fields, the bridge is discretized in 50 elements, i.e., each of the five girders is divided into 10 elements of equal length. It should be pointed out that these elements do not necessarily correspond to the finite elements of the finite element model. The elements of the discretization of the random field can be larger than the elements of the finite element model and can each consist of a number of smaller finite elements.
The corrosion degree α at a time instant t depends on the remaining reinforcement area at time t, A s (t), and the initial reinforcement area A s,0 according to Equation (8).

Monitoring Strategies
The required service life of the bridge is equal to 100 years. The standard inspection strategy consists of visual inspections every 5 years. Diagnostic load testing (e.g., measuring strains under a proof-load applied to the bridge) or dynamic tests [36,37] can provide information on the real deterioration state of the bridge [18]. Hence, the question arises whether performing these tests on the bridge would be worth their costs, or whether it would be better to perform only the regular maintenance schedule. The different investigated monitoring strategies are: 1.
Dynamic tests using accelerometers from which natural frequencies and mode shapes can be derived; 2.
Dynamic tests with optic fibers from which natural frequencies and strain mode shapes can be derived; 3.
Static proof-loading tests during which strains are measured.
Other monitoring strategies (e.g., [38]) can also be considered. They can be included in a similar way.
The considered strategies are referred to as strategy 1, strategy 2, and strategy 3, respectively. For strategy 1, the considered data vector contains the natural frequencies of the first four modes and the corresponding mode shapes at multiple sensor locations. The number of sensor locations is varied in the following analyses. The frequencies and mode shapes are accounted for in the Bayesian updating procedure according to the likelihood function given by Equation (9).
This expression is based on the assumption of a normally distributed measurement error and modeling error, both with zero mean and covariance matrixes Σ D and Σ G , respectively. F ML is the maximum likelihood function and is given by Equation (10) [39].
where N m is the number of modes considered, i.e., N m = 4 in this case. In Equation (10), it is implicitly assumed that suitable mode shape scaling has been performed. The error on the frequencies σ λ is initially assumed 0.001, and the error on the norm of the mode shapes σ φ is assumed 0.01. For strategy 2, the considered data vector contains the natural frequencies of the first four modes of the bridge and the corresponding strain mode shapes at top and bottom fiber of the girders. The frequencies are accounted for in the likelihood in a similar way as in Equation (10). The strain mode shapes are accounted for in a similar way as for the static strains (cfr. infra). The error on the frequencies is the same as for strategy 1, and the error on the strain mode shapes is assumed 0.5 µε.
For strategy 3, static strains are assumed to be measured under proof-loading, and the data vector contains these strains measured at different locations along the girders, at both top and bottom fiber. The maximum likelihood function F ML is given by Equation (11).
where ε j are the measured strains, ε j (θ M ) the modeled strains, and σ ε the measurement error when measuring static strains under proof-loading. The error on the strains σ ε is initially assumed to equal 0.2 µε.
In the following, the measurement locations of accelerations, as well as static and dynamic strains, are assumed to be in the middle of the elements in which the structure is discretized. The abovementioned monitoring strategies can be performed at different time instants, with a varying number of sensors.

Repair Strategy
In the maintenance strategy, repair is assumed to be performed once a reliability threshold β repair is reached. At this time of repair, the reliability index is considered to return to its initial value. When the posterior corrosion degree has a mean value larger than the prior mean at the time step under consideration, the planned repair is assumed to be performed in such a way that the corrosion rate and/or initiation period of the repaired structure is reduced compared to the initial structure (for example, by the use of better repair mortars). Furthermore, repair can also be performed immediately after monitoring, before the planned repair based on the reliability index. This repair is only performed if the posterior mean of the corrosion degree following monitoring is larger than a critical value for the corrosion degree α cr .

Costs
For the calculation of the life-cycle costs, different cost values are required: costs of maintenance/repair actions, costs of the monitoring strategy itself, and costs of failure. Although the importance of life-cycle cost analysis is generally acknowledged [40,41], data on these costs might be difficult to obtain. Even though some information on the costs can be available, some of them might still be hard to estimate.
As a first estimate for the monitoring costs, cost values for the three considered monitoring strategies are summarized in Table 2. The number of sensors for strategy 1 refers to the number of uniaxial accelerometers used. The corresponding costs are the costs for a modal analysis, which include the costs for instrumentation of the bridge and the data processing. The number of sensors for strategy 2 refers to the number of elements (in the discretization of the random fields) equipped with a sensor of an optic fiber, both at lower and upper fiber of the girder. The optic fibers used have a length equal to the length of the girder and are applied each time to the top fiber and bottom fiber of the girder. Each of these fibers has 10 sensors along its length, measuring the strains. These 10 sensors correspond to the 10 elements in which the girders are discretized. For strategy 3, the number of sensors refers to the number of discrete strain sensors applied on the bridge. To estimate the costs of failure and repair, in [42], it is stated that these costs could be monetized and presented as a percentage of the total bridge value C BV , which is given by C BV = f B ·C 0 , with C 0 the structural/construction costs defined as the initial bridge value per square meter of bridge deck, and f B a factor for multiplication of the bridge value due to its importance in the network. The basic structural costs C 0 can be based on [43], where a value of 1200 euro per square meter of bridge deck is suggested. The importance of the bridge in the road network f B is evaluated based on:
Each of these variables have grade 1 to 5 and f B = 1 + 1/5[0.25(G RC + G AADT + G DD ) + 0.125(G LS + G TL )]. Values for these variables can be found in [44]. For the bridge under consideration, a structural/construction cost of C 0 = €126,672 is estimated and a factor f B equal to 1/5[0.25(2 + 2 + 5) + 0.125(2 + 2)] = 1.55. This construction cost C 0 only relates to the superstructure of the bridge and is also in line with global cost estimates provided by the local road authorities in Flanders [45].
According to [46], a cost of failure of the bridge equal to 5 times the total bridge value is assumed, resulting in a cost of failure of €981,708.
The influence of different assumptions on the repair costs is investigated in Section 5.2.1. The considered cost models are also summarized there.

Approximation of the Posterior Distribution by MAP Estimates
As pointed out in Section 3.1, approximate methods can be used to estimate the posterior distributions of the model parameters instead of the computationally expensive MCMC sampling. Before using these approximations of the posterior distribution of the model parameters in the VoI calculations, it should be investigated whether the approximation is accurate enough. For the considered bridge, the MAP estimate of the posterior distribution of the diffusion coefficient and the corrosion rate is compared with the posterior distribution found when applying MCMC for the three monitoring strategies. For monitoring strategy 2 implemented at 35 years, the resulting posterior distributions are visualized in Figures 3 and 4. Here, it can be seen that the approximation is quite good when considering the mean and uncertainty of the distribution.  The influence of the use of approximate posterior distributions based on MAP estimates instead of MCMC sampling on the resulting VoI is also investigated. When the VoI is calculated accounting for monitoring strategy 2 implemented at t = 35 years, the difference in posterior costs found based on the MAP estimates and MCMC calculations is equal to €844.88 or 0.4% of the total bridge value. In addition, other results were compared, and it was found that, for this specific case, the difference in VoI by applying MAP estimates instead of full MCMC does not influence the decisions made based on the resulting VoI, and the MAP estimates can be used to speed up the calculations of the VoI. Whether an error on the VoI is acceptable depends on the order of magnitude of the error and the order of magnitude of differences between the VoI's of the different considered monitoring strategies. Hence, before using the approximate posterior distributions, a few checks should be made on the performance of the approximate method by evaluating the posterior distribution both with the approximate method and with MCMC simulations. In addition, the influence on the VoI should be checked at least once. This example shows how the computational demand can possibly be limited since the error on the VoI by using approximate distributions can be small. However, it is advised to at least check this once for individual cases.

Determination of Optimal Sensor Positions
In the case study under investigation, three monitoring strategies are considered, as summarized in Section 4.3. In the following, the optimal sensor locations are calculated for monitoring strategy 1. The optimal sensor locations for updating of the corrosion variables based on the mode shapes are given in Figure 5. The horizontal axis on this figure represents the total number of sensors placed on the structure, and the vertical axis represents the location along the length of the structure. The different girders are represented by the different markers. Hence, when x sensors are placed on the structure, the markers corresponding to that value on the x-axis represent the locations of the sensors, i.e., on which girder and the position along the girder. For example, in case one sensor is placed on the bridge, x = 1 should be considered. A vertical line through x = 1 crosses a single purple pentagon at y = 1.5 m. The purple pentagons correspond to sensors placed on girder 5. Therefore, in case one sensor is placed on the structure, it should be located at girder 5 at 1.5 m from the support. Similarly, when placing five sensors on the structure (x = 5), a vertical line at x = 5 crosses the markers of girders 1-5, all with a y-coordinate of about 1.5 m. Hence, when five sensors are placed on the structure, each girder should be equipped with one sensor, each at about 1.5 m from the support. A similar procedure can be repeated to determine optimized sensor locations for strategies 2 and 3.

Optimization of the Time of Monitoring
To determine its optimal value, the time of monitoring is increased in steps of five years between the time at which the prior reliability index starts to decrease and the prior time of repair. Hence, the first times of monitoring considered are 15,20,25,30,35,40, and 45 years since the time of construction. The minimum posterior costs for measuring strain mode shapes (strategy 2) are found at 35, 40, and 45 years. Hence, the range [34 years, 46 years] years is also considered, with step sizes of 1 year. The lower and upper boundaries of the interval are taken one year apart from the previously identified optimal times of monitoring to verify whether there is no increase in VoI just before or after these times. The results are visualized in Figure 6 for two cases. The first case assumes repair model 7 (see further) and a reliability threshold of β repair = 3. The monitoring strategy is strategy 2 with strain mode shapes measured at 10 elements (case 1). The second case gives the VoI as a function of time for repair model 1 (see further) with reliability threshold β repair = 3 and monitoring strategy 2 with strain mode shapes measured at 50 elements (case 2). In the next step, for monitoring at 35 years or monitoring at 44 years and case 1 (since for these times of inspection the largest VoI was found for case 1), the number of sensors is increased. These results are given in Figure 7 (label 'Strategy 2'). Here, it can be seen that under these specific assumptions, applying a larger number of sensors is not worth the extra cost. Moreover, the behavior is almost the same for the two times of monitoring. Only when sensors are placed at locations chosen such that sensor locations are present at all elements in which the structure is discretized and hence all elements of the random fields, a noticeable difference in VoI is found between the two times of monitoring.  Figure 7 (labeled 'Strategy 1'). Here, the largest VoI is found for the lowest number of sensors at t = 44 years, but the influence of the number of sensors on the VoI is very small. The VoI is also larger than for strategy 2, when monitoring is performed at 44 years but smaller for monitoring at 35 years.
Finally, in strategy 3, static strain data is considered, respectively, at 20, 35, and 50 locations. Here, the number of sensors is also based on advice from experts. The results are given in Figure 7 (labeled 'Strategy 3'). Under the current assumptions, for monitoring at t = 44 years, the influence of the number of sensors on the VoI is limited, and the VoI is lower than for the strategy 1. For monitoring at t = 35 years, a larger VoI is found for the lowest number of sensors, and the VoI is larger than for the strategy 1.

Type of Monitoring Technique and Accuracy
The posterior distributions of the corrosion parameters when Bayesian updating is performed based on the monitoring data depend on the uncertainty of the technique applied, since this is an important parameter in the likelihood function used in the Bayesian updating (see Equations (9)-(11)). For some techniques, this might be hard to estimate accurately. Hence, in this section the influence of this measurement uncertainty is investigated. The error on the measured static strain σ ε is varied from 0.2 to 1.95 µε [47][48][49], the error of the strain mode shape from 0.5 to 1.95 µε [47][48][49], the error on the frequency σ λ from 0.001 to 0.01 [50,51], and the error on the norm of the mode shape from σ φ 0.001 to 0.01 [50,51]. The influence on the VoI is visualized in Figure 8. It can be seen that there can be a substantial influence of the measurement error on the VoI. For this specific case and under the assumed cost values, the VoI increases with decreasing measurement error. The influence on the VoI is largest for the error on the mode shapes. The influence of the error on the frequencies is the second largest, followed by the error on the static strains and the error on the strain mode shapes, respectively. Changing these error models can also change the decision on the most optimal monitoring technique.

Influence of the Cost of Repair
The cost of repair can be described as f rep ·C BV , where f rep , for example, depends on the reliability index at the time of repair or on the damage extent. For the repair costs, seven different models are used in this work, based on the literature. The model proposed by [42] is considered as cost model 1. There, f rep is a function of the reliability index, as provided in Table 3. Table 3. Values for f rep based on [42].

Damage Level [43] 1-Minor Damage 2-Slight Damage 3-Medium Damage 4-High Damage 5-Demolition Eminent
1.00 8. 40 29.15 70.50 140 The other six models for the cost of repair are based on [52], where the cost of repair is given as a function of the ratio of the maximum load that can be sustained by the damaged structure to the service load requirement for the design specifications. In this work, this is slightly adapted to come to the cost models given in Table 4, by replacing the calculation of the load or resistance by the reliability index. In Table 4, n el is the total number of elements in which the structure is discretized, and n d is the number of damaged elements, which are the elements with a posterior mean corrosion degree larger than the a priori expected corrosion degree. Table 4. Cost models based on [52] adapted in order to account for reliability index directly or for the number of damaged elements. The different cost models are visualized in Figure 9. It should be pointed out that the considered models can be subdivided into three categories. The first model (from Table 3) can reach values larger than 100% for f rep , whereas the other models are restricted between 0 and 100%. Moreover, models 2-4 still depend on the reliability index, whereas models 5-6 depend on the number of damaged elements. The influence of each of these cost models on the VoI is investigated. The VoI is calculated for a critical reliability of β repair = 3 and for strain mode shapes measured at 10 elements (strategy 2, indicated with 'Strategy 2' in the following figures) or accelerations measured at 20 locations (strategy 1, indicated with 'Strategy 1' in the following figures) or static strains measured at 20 locations (strategy 3, indicated with 'Strategy 3' in the following figures). The models for the cost of repair are numbered 1-7, and the VoI for the different cost models is visualized in Figure 10 for the three mentioned monitoring strategies. Here, it can be seen that the model used for the cost of repair has a large influence on the VoI. The absolute difference can reach up to more than €20,000, which is 10% of the total bridge value. Hence, inaccurate estimates of the costs of repair might lead to a large over-or under-estimation of the VoI. The coefficient of variation of the resulting VoI's equals 0.35 for the strain mode shapes, 0.41 for the mode shapes, and 0.54 for the static strain data. The monitoring strategy with the highest VoI is in all cases related to the extraction of modal data from acceleration measurements. Figure 10. Influence of the cost model for repair on the VoI for the three considered monitoring strategies (for the strategies, reference is made to Table 2).

Cost
When considering the three categories of models, it can be seen that the third category (models 5-7) provides a larger VoI. This can be ascribed to the fact that a low number of damaged elements can be present, resulting in a low repair cost. However, when these damaged elements become critical, a few damaged elements can already result in a large influence on the reliability index, leading to a larger f rep according to the first four models. When comparing model 1 with models 2-4, it can be seen that the difference is relatively small since the low reliability levels for which f rep becomes larger than 1 (i.e., larger than 100%) in model 1 are not reached.

Influence of the Cost of Failure
Three different values are assumed for the cost of failure: 2C BV , 5C BV , and 10C BV . The results are given in Figure 11 for three different monitoring strategies: measuring accelerations at 50 locations (Strategy 1), measuring strain mode shapes at all 50 elements (Strategy 2) and measuring static strains at 50 locations (Strategy 3). If the cost of failure increases, the prior cost increases together with the posterior cost but still results in a larger VoI. Figure 11. Influence of the cost of failure on the VoI for the three considered monitoring strategies (for the strategies, reference is made to Table 2).

Prior Distribution of the Corrosion Parameters
The diffusion coefficient can be difficult to estimate when one is not sure about the properties of the concrete in the existing structure and when no monitoring data are available. Furthermore, the diffusion coefficient varies in time, and hence the prior estimate at a certain time instant might differ from the actual value. When changing the mean of the diffusion coefficient from 64.5 to 258 mm 2 /year, the VoI can vary with an amount of €34,000, as illustrated in Table 5. While the monitoring strategy with the highest VoI remains the same, the difference in VoI between two techniques is not always of the same order of magnitude. Hence, when taking into account the uncertainty on the estimates of the prior and posterior costs (see Section 6), the optimal strategy might change. When the Coefficient of Variation (COV) of the diffusion coefficient is varied, the influence of the VoI is also given in Table 5. Here, the influence of the VoI is of the order of magnitude of €35,000, which is 18% of the total bridge value for a variation in COV of the diffusion coefficient from 0.05 to 0.2. Moreover, the influence differs depending on the monitoring technique considered. The VoI remains the largest when mode shapes are available (strategy 1), but for the lowest COV the difference with the static strains (strategy 3) is limited. Furthermore, when comparing the strain mode shapes (strategy 2) with the static strains, another technique might be chosen depending on the assumed COV for the diffusion coefficient. For the corrosion rate, different parameters are available in literature for different exposure classes. Some of these models are given in Table 6. When there is doubt on which model to use, the impact on the VoI can be large, ranging from a large positive to a large negative VoI, as can be seen in Table 5. Moreover, the optimal monitoring strategy might change, with large differences in VoI between the different strategies. The surface chloride concentration is also a difficult parameter to estimate. The initial value is once taken half and once double the value originally considered, and it is found that the influence on the VoI is very large and can range from a negative VoI to a large positive VoI, as can be seen in Table 5. Moreover, depending on the value of mean of the chloride concentration at the surface of the concrete (µ Cs ), the difference in VoI between two techniques might change. When changing the COV of the surface chloride concentration, the influence on the VoI is also illustrated in Table 5. Again, the optimal monitoring strategy might change depending on the prior assumptions.

Scatter or Uncertainty on the Posterior Costs
The calculation of the VoI is based on estimated values of the costs, averaged over different possible monitoring outcomes. This results in a distribution for the costs. The expected value of this distribution is then calculated and used in the determination of the VoI, as also illustrated in the flowchart in Figure 1. Nevertheless, besides the expected value, the scatter on the calculated costs could also be accounted for.
For the prior costs, a coefficient of variation of about 0.25 is found for all seven different repair strategies. For the posterior costs, some differences on this COV are found, as given in Table 7. Furthermore, when increasing the number of locations where the strain mode shapes are measured (strategy 2), the uncertainty on the posterior cost decreases from a COV of 0.012 to a COV of 0.006. For the static strains (strategy 3), the uncertainty even reduces from a COV of 0.09 to 0.02 when increasing the number of sensors from 20 to 50. Under these specific assumptions, the uncertainty on the costs is also much larger for the mode shapes (strategy 1) than for the other monitoring data. Hence, data with a larger (measurement and/or model) error leads to a larger posterior uncertainty. This is also illustrated in Figure 12, where the mean value and uncertainty of the posterior cost is given for the different measurement errors given in Table 8. Besides the results visualized in Figure 12, it is also found that, when the prior COV of the corrosion parameters is increased, the COV of the costs also increases.  Table 2). Since the posterior uncertainty of the costs also depends on the monitoring strategy and the accompanying (measurement and/or model) errors, it might be necessary to not only take into account the expected values of the costs in the calculation of the VoI but also to include the accompanying uncertainties. A method to account for the uncertainties on the costs can be based on robust design optimization, where uncertainties on the objective function f (x) are accounted for by changing the minimization according to Equation (12).
where W is a weighting factor within the range [0, 1], and (µ 0 f (x) , σ 0 f (x) ) is the utopia point corresponding to W = 0 or W = 1. By inserting this weighting factor W in the equation, a trade-off is found between performance and robustness.
In the case of the optimization of monitoring strategies, the objective function equals the VoI. This VoI should be maximized, which corresponds to a minimization of the posterior cost, since the prior cost is the same for all monitoring strategies to be compared. Instead of minimizing the expected value of the posterior cost, one could also consider Wµ post + (1 − W)σ post to account for a balance between expected value and the uncertainty on it.
As an example, when applying the latter formula to estimate the posterior costs following the strain mode shapes at different monitoring times (strategy 2), the results are given in Figure 13. Here, it can be seen that the general pattern of the posterior cost as a function of time is conserved but flattens out as W decreases, and hence, more weight is given to the uncertainty than to the mean value. However, the monitoring time with the lowest cost is the same for every value of W. A similar behavior is observed for the costs in function of the number of sensors and for the other monitoring strategies. Taking into account this weighting factor W can alter the optimal solution. For W = 0, the lowest posterior cost is found for strain mode shapes (strategy 2) measured at 40 elements at 35 years. For W = 0.1, the lowest posterior cost is found for strain mode shapes measured at 20 elements at 35 years, and for W = 0.2 to W = 0.4, measuring strain mode shapes at 10 elements is more optimal. For W = 0.5 to W = 1, the optimal solution would be to measure the accelerations at 44 years with 20 sensors. Hence, different values of W could be considered depending on the weight the decision maker wants to give to the expected value and to the uncertainty. This choice might also influence the choice on the most optimal monitoring strategy.

Conclusions
In the literature, the VoI is presented as an important tool for the optimization of monitoring strategies of structures. Nevertheless, this comes with some computational challenges. In this work, some solutions are proposed to reduce the computational time. One of these solutions is the use of approximate posterior distributions based on MAP estimates instead of the computationally expensive MCMC sampling to calculate the posterior distributions. Another solution is performing part of the optimization of the monitoring strategy outside the VoI calculations. The locations of the sensors can be determined beforehand by applying an optimal sensor placement algorithm. As such, given the number of sensors, the optimal locations of these sensors are fixed. This leads to a reduced number of evaluations of the VoI.
Another aspect lacking in the literature is the investigation of the influence of input parameters. Very often, the benefit of the VoI is illustrated by one specific example, where all parameters are assumed to be fixed. If an investigation on the influence of input parameters is performed, this is very often done for simplified conceptual structural models. In a practical case, it might be difficult to decide on the values of these input parameters, especially for complex structures. From the investigation performed in this work, it is concluded that the choice of the input parameters can have a significant impact on the VoI and the optimal monitoring strategy. Hence, caution is required when interpreting results of a VoI analysis, and the results of a VoI analysis should always be evaluated from a critical mindset. It might be beneficial to do the extra effort and calculate the VoI for different scenarios to be sure that the final conclusion (for example with respect to the choice of the monitoring strategy) remains valid. Furthermore, the following conclusions are found in relation to the investigated structure: • A lower measurement error leads to a larger VoI. Changing the error models can also affect the decision on the most optimal monitoring strategy.

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Changing the cost of repair has a large influence on the VoI. • A higher cost of failure leads to a higher prior and posterior cost but also results in a larger VoI.

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Changing the mean and/or COV of the diffusion coefficient has a large influence on the VoI. Moreover, the influence differs depending on the monitoring strategy considered.

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Changing the parameters of the prior distribution for the corrosion rate can have a large impact on the VoI, ranging from a large positive to a large negative VoI. Moreover, the optimal monitoring strategy might change, with large differences in VoI between the different strategies. • When changing the distribution (mean value or standard deviation) of the surface chloride concentration, the influence on the VoI is very large and can range from a negative VoI to a large positive VoI.
In the last section, a possible way to also account for the uncertainties on the costs in the calculation of the VoI is investigated. Here, a weight factor is used to account for this uncertainty on the costs. Depending on this weight factor, the most optimal monitoring strategy might change.
Even though the VoI approach requires a rather high computational cost, these costs are generally very small compared to the costs of the actual monitoring system or the costs of failure of critical infrastructure. The VoI approach seems particularly useful for the management of large and critical bridges, e.g., in terms of traffic flow. For such bridges, large costs can be involved in the closure and/or failure of the bridge, and it can be beneficial to investigate beforehand whether investing in a monitoring strategy is beneficial or whether one can better stick to an a priori determined maintenance schedule.