# VoI-Based Optimization of Structural Assessment for Spatially Degrading RC Structures
## 2 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 \(f^{\prime}\) are updated to posterior distributions \(f^{\prime\prime}\) 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}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\! 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 reinforced-concrete bridge. These data can be incorporated into the Vol 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 Vol such as costs, prior distributions, etc. Moreover, since the calculation of the Vol involves Bayesian updating and the consideration of the entire service life, calculating the Vol can be computationally expensive. Using the Vol both for optimizing a monitoring strategy itself and for choosing the most optimal strategy among a set of possible strategies  
Figure 1: General framework for the calculation of the Vol. 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.