# VoI-Based Optimization of Structural Assessment for Spatially Degrading RC Structures
## 4 Case Study: A Simply Supported RC Girder Bridge
### 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). \[L\sim\left(\det(\mathbf{\Sigma}_{D}+\mathbf{\Sigma}_{G})\right)^{-1/2}\exp \left(-\frac{1}{2}F_{ML}\right) \tag{9}\]  
This expression is based on the assumption of a normally distributed measurement error and modeling error, both with zero mean and covariance matrixes \(\mathbf{\Sigma}_{D}\) and \(\mathbf{\Sigma}_{G}\), respectively. \(F_{ML}\) is the maximum likelihood function and is given by Equation (10) [39]. \[F_{ML}=\sum_{r=1}^{N_{m}}\frac{\left(\overline{\lambda}_{r}-\lambda_{r}( \boldsymbol{\theta}_{M})\right)^{2}}{\sigma_{\lambda}^{2}\overline{\lambda}_{ r}^{2}}+\sum_{r=1}^{N_{m}}\frac{\left\|\overline{\boldsymbol{\Phi}}_{r}- \boldsymbol{\phi}_{r}(\boldsymbol{\theta}_{M})^{2}\right\|}{\sigma_{\phi}^{2} \left\|\overline{\boldsymbol{\Phi}}_{r}\right\|^{2}} \tag{10}\]  
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 \(\sigma_{\lambda}\) is initially assumed 0.001, and the error on the norm of the mode shapes \(\sigma_{\phi}\) 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 \(\mu\varepsilon\). 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). \[F_{ML}=\sum_{j=1}^{N}\frac{\left(\overline{\varepsilon}_{top,j}-\varepsilon_{ top,j}(\boldsymbol{\theta}_{M})\right)^{2}}{\sigma_{\varepsilon}^{2}}+\sum_{j=1}^{N} \frac{\left(\overline{\varepsilon}_{bottom,j}-\varepsilon_{bottom,j}( \boldsymbol{\theta}_{M})\right)^{2}}{\sigma_{\varepsilon}^{2}} \tag{11}\]  
where \(\overline{\varepsilon}_{j}\) are the measured strains, \(\varepsilon_{j}(\boldsymbol{\theta}_{M})\) the modeled strains, and \(\sigma_{\varepsilon}\) the measurement error when measuring static strains under proof-loading. The error on the strains \(\sigma_{\varepsilon}\) is initially assumed to equal 0.2 \(\mu\varepsilon\).