# Fragility Analysis Based on Damaged Bridges during the 2021 Flood in Germany
## 2 Materials and Methods
### Fragility Curves Generation  
Fragility curves link the hazard intensity to the damage experienced by structures. Various metrics can be chosen to represent the hazard. In case of floods, it is common to use flow discharge or water elevation, depending on the situation. In the present case, a metric called \(h^{\star\star}\) was used, which is the ratio between the flood height and the bridge deck elevation, as per Figure 9. The symbol \(h^{\star\star}\) was chosen to differentiate it from \(h^{\star}\), called the 'inundation ratio', defined in flume experiment study as \(h^{\star}=(h_{u}-h_{b})/s\), where \(s\) is the thickness of the bridge deck. The metric allowed us to use a single category for various bridge geometries, as the relative height had a good correlation with the recorded damage levels. In other flood phenomena, such as hurricanes, the storm surge can be a good alternative, but in regions such as those of the case under examination, varying bridge clearances pose an issue. Bridges in mountainous environments usually have low clearances, while downstream bridges have higher clearances. Despite this difference, both upstream and downstream bridges sustained the same damage levels. However, in such cases, recorded water elevation and flow discharge were different. Conversely, \(h^{\star\star}\) was comparable; in this  
Figure 8: Percentages of heavy and lightweight bridge decks with respect to the two damage mechanisms. Figure 7: Percentages of arch and beam bridges with respect to the two damage mechanisms. Figure 9: Definition of \(h^{\star\star}\) as fragility intensity measure. way, from a structural point of view, \(h^{\star\star}\) helped to homogenize the population from the hazard point of view. The other considered intensity measure was water velocity, but additional information to distinguish between different soils was not available, resulting in considerable uncertainties. In addition, the simulated velocity had to be considered upstream of the bridge, as local contractions could significantly increase it. The reconstructed peak discharge, as per Figure 2, was useful to investigate many aspects of the flood process. In such a context, [PERSON], [PERSON] and [PERSON] developed an hydraulic model of the Ahr river between Altenahr and Sinzig, studying the effect of houses on the increased volume of water [71]. As mentioned, the water discharge was similar to the 1804s, although only minor damage was observed at the time. The study by [PERSON], [PERSON] and [PERSON] is of particular interest for infrastructure managers, as the increased water height directly affected bridges in the sense that buildings subtracted areas that would have otherwise been occupied by water, as in 1804. This effect increased the water levels, supporting the use of water height to characterize the hazard intensity. Although discharge represented a better intensity measure, the required data to obtain that information were affected by high uncertainty, as the river overtopped bridges causing a pressurized flow underneath many structures. The fragility curves are obtained by means of the following expression:  
\[F_{d}(r)=\Phi\left(\frac{\ln\left(\frac{r_{i}}{\mu_{d}}\right)}{\beta_{d}}\right) \tag{1}\]  
where the parameters of the distribution are obtained by using the maximum likelihood estimation method:  
\[\mu_{d}=\exp\left(\frac{1}{n_{d}}\cdot\sum_{i=1}^{n_{i}}ln\left(r_{i}\right)\right) \tag{2}\]  
\[\beta_{d}=\sqrt{\frac{1}{n_{d}-1}\sum_{i=1}^{n_{i}}\left[\ln\left(\frac{r_{i} }{\mu_{d}}\right)\right]^{2}} \tag{3}\]  
Here, \(F_{d}(r)\) is the fragility estimated at intensity \(r\) for damage state \(d\). The parameters \(\mu\) and \(\beta\) are the mean and standard deviation values of the lognormal cumulative distribution, respectively, and \(n\) is the number of elements or specimens of empirical data. The subscript \(d\) is used to differentiate between damage levels. Two tests were employed to validate the model: the goodness of fit test and a criterion to manage outliers.