# VoI-Based Optimization of Structural Assessment for Spatially Degrading RC Structures
## 4 Case Study: A Simply Supported RC Girder Bridge
### 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).  
\[A_{s}(t)=(r_{0}-x(t))^{2}.\pi\cdot n_{r}\;with\;x(t)=V_{corr}\cdot\alpha_{p} \cdot(t-T_{i}) \tag{6}\]  
where \(r_{0}\) is the initial reinforcement radius, \(V_{corr}\) the corrosion rate, \(T_{i}\) the initiation period, and \(\alpha_{p}\) 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.  
\[T_{i}=\frac{1}{4D}\frac{c2}{\left(erf^{-1}\left(1-\frac{C_{cr}}{C_{s}}\right) \right)^{2}} \tag{7}\]  
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 \(\alpha\) 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).  
\[\alpha(t)=\frac{A_{s,0}-A_{s}(t)}{A_{s,0}} \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Variable** & **Name** & **Distr.** & **Mean** & **COV** & **Ref.** \\ \hline \(f_{y}\) [MPa] & Yield stress of reinforcement & LN & 550 & 0.02 & [30] \\ \(h\) [mm] & Height of the beam + slab & Det. & 790 & / & [28] \\ \(a\) [mm] & Concrete cover & Det. & 69 & / & [28] \\ \(h_{f}\) [mm] & Height of the slab & Det. & 190 & / & [28] \\ \(f_{c}\) [MPa] & Concrete compressive strength & N & 25.9 & 0.15 & [28] \\ \(b_{f}\) [mm] & Slab flange width & Det. & 2500 & / & [28] \\ \(K_{R}\) [-] & Resistance model error & LN & 1.05 & 0.105 & [30] \\ \(\rho_{c}\) [kg/m\({}^{3}\)] & Concrete density & N & 2500 & 75 & [31] \\ \(K_{E}\) [-] & Load effect model error & LN & 1 & 0.1 & [30] \\ \(Q\) [kNm] & Variable load effect & GU & 0.728Q\({}_{k}\) & 0.146Q\({}_{k}\) & [32] \\ \hline \hline \end{tabular}
\end{table}
Table 1: Probabilistic distributions for the evaluation of the probability of failure.