# VoI-Based Optimization of Structural Assessment for Spatially Degrading RC Structures
## 5 Sensitivity of the Vol to Initial Assumptions
### Costs  
#### 5.2.1 Influence of the Cost of Repair  
The cost of repair can be described as \(f_{rep}\cdot C_{BV}\), 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.  
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. Figure 8: Influence of the measurement error on the VoI. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Damage Level [43]**} & **1—Minor Damage** & **2—Slight Damage** & **3—Medium Damage** & **4—High Damage** & **5—Demolition Eminent** \\ \hline Reliability index [-] & \(\beta>3.82\) & \(3.3<\beta\leq 3.82\) & \(3.0<\beta\leq 3.3\) & \(2.3<\beta\leq 3.03\) & \(\beta\leq 2.3\) \\ \(f_{rep}\) [\%] & \(1.00\) & \(8.40\) & \(29.15\) & \(70.50\) & \(140\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values for \(f_{rep}\) based on [42]. 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 \(\beta_{repair}=3\) 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 \(\copy{20,000}\), 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. \begin{table}
\begin{tabular}{c c} \hline \hline
**Cost Model** & \(f_{rep}\) [-] \\ \hline
2 & \(1-\frac{\beta}{38}\) \\
3 & \(1-\left(\frac{\beta}{38}\right)^{2}\) \\
4 & \(\frac{1+\cos\left(\frac{\beta}{38}\pi\right)}{1-\frac{\beta}{46}}\) \\
6 & \(1-\left(\frac{\beta_{0}-\beta_{1}}{\beta_{0}}\right)^{2}\) \\
7 & \(\frac{1+\cos\left(\frac{\beta_{0}-\beta_{1}}{\beta_{0}}\pi\right)}{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Cost models based on [52] adapted in order to account for reliability index directly or for the number of damaged elements. Figure 9: Different assumed cost models.