# Fragility Analysis Based on Damaged Bridges during the 2021 Flood in Germany
## 2 Materials and Methods
### Statistics of Population  
Information about the construction date for 184 bridges out of 250 was obtained. According to the age distribution and the number of failures per each structural typology, beams and arches were the most affected types, with a total of 125 and 82 bridges, respectively. Interestingly, there was a high number of newly built beam bridges, mainly footbridges. From the analysis of the post-disaster evidence, many of these structures collapsed due to unexpected water activity, often in combination with driftwood blockage, which increased the horizontal thrust on the decks, as seen in Figure 6a,b.  
Then, the bridge inspection records antecedent to the flood were analyzed, comparing those in the present database to all bridges which had been federally managed. Structures that were present in both databases were deleted from the federal database. From this analysis, a statistically significant worse average condition rating was found among damaged arch bridges compared to their counterparts in the federal database. There were no similar differences for the other bridge typologies. The data also allowed us to make the same comparison by filtering the federal database to analyze only bridges over rivers in the states of NRW and RLP. To this end, the Kruskal-Wallis (K-W) test was performed on the two databases. A K-W test was used as an equivalent to ANOVA but for non-parametric data. The factor was the bridge typology, accounting for the following categories: a) beam and box girder bridges and b) arch bridges. Before the K-W test, the data were tested against [PERSON]'s assumption, resulting in a rejection of the null hypothesis with a confidence level at \(\alpha\) 0.05 [69]. A K-W test was performed, resulting in significance at a \(p\)-value of 0.0001. To shed light on individual subgroups, a nonparametric post hoc test was used, with the \(p\)-value corrected according to [PERSON]'s assumption [70].  
The results showed the greatest difference in condition ratings among beam bridges, with an adjusted \(p\)-value of 0.0010, while among arch bridges it was 0.0043. On average, the condition rating of the damaged bridges measured before the flood was statistically worse compared to that of the undamaged population. These differences obviously increase if the rating assigned to the damaged bridges after the flood is used. The test was significant as reported, but not all the undamaged structures were subjected to the same hazard magnitude. To better explain this point, a correlation between the damage level and an intensity measure representative of the hydraulic force on bridges was searched for. The triggering mechanisms were selected based on evidence from surveys and damage reports. Then, the predominance of one mechanism over the other (scour over hydraulic force) was found and was highlighted in both Figures 7 and 8, with respect to deck typology (beam vs. arch bridges) and weight (masonry and concrete decks, i.e., heavy, vs. steel and wooden ones, i.e., lightweight). Then, the FEMA P-58 method was used to draw fragility curves starting from these correlations [36].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**D1** & **D2** & **D3** & **D4** \\ \hline
1.0–1.9 & 2.0–2.9 & 3.0–3.4 & 3.5–4.0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Equivalence of bridge condition ratings to a qualitative damage level, based on the structural safety parameters.