# Fragility Assessment of Traffic Embarkments Exposed to Earthquake-Induced Liquefaction
## 4 Results and Discussion
### Interdependencies between Possible EDPs  
Although the permanent vertical displacement of an embankment crest has been proposed in the literature ([14]) as an EDP for the derivation of fragility curves, many other damage indicators could more accurately characterize the state of damage for an embankment after a seismic event. For example, since lateral spreading represents the main failure mechanism, traffic infrastructure could be predominantly damaged by lateral movements or by a difference in horizontal displacements either side of an embankment. Despite this well-known observation, no widely acceptable and experimentally supported threshold values related to horizontal displacement were found in the literature. In order to validate the use of vertical displacement at the top midpoint of an embankment (\(u_{y\_mid}\)) as an EDP,  
Figure 10: Influence of density of liquefiable layer on fragility curves (damage state criteria for railway embankments).  
we studied its correlation with three other possible damage state indicators (\(u_{y\_\_\mathit{edge}}\), \(\left|\Delta u_{y}\right|\) and \(\Delta u_{x}\)), as defined by following equations:  
\[u_{y\_\mathit{edge}}=-\mathrm{max}\Big{(}\left|u_{y\_\mathit{right}}\right|, \left|u_{y\_\mathit{left}}\right|\right|) \tag{6}\]  
\[\left|\Delta u_{y}\right|=\ \mathrm{max}\Big{(}\left|u_{y\_\mathit{mid}}-u_{y\_ \mathit{left}}\right|\right|,\left|u_{y\_\mathit{mid}}-u_{y\_\mathit{right}} \right|\Big{)} \tag{7}\]  
\[\Delta u_{x}=u_{x\_\mathit{right}}-u_{x\_\mathit{left}} \tag{8}\]  
where \(u_{y\_\mathit{right}}\), \(u_{y\_\mathit{mid}}\) and \(u_{y\_\mathit{left}}\) are the maximum vertical displacements at the crest right edge, midpoint and left edge, respectively, and \(u_{x\_\mathit{right}}\) and \(u_{x\_\mathit{left}}\) are the maximum horizontal displacements at the crest right and left edges, respectively.  
The correlations between permanent embankment midpoint settlements (\(u_{y\_mid}\)) and the three alternative damage state parameters are presented in Figures 11 and 12. Nearly 8000 data points from the final calculated state of the numerical simulations are contained in these three graphs, including the results for four different embankment heights, four soil profiles, two density states of the liquefiable layer and 30 GMs with various intensities.  
According to the observed scatter and coefficient of determination, \(R^{2}\), in Figures 11 and 12, it can be seen that both \(u_{y\_\mathit{mid}}\) and \(u_{y\_\mathit{edge}}\), and \(u_{y\_\mathit{mid}}\) and \(\Delta u_{x}\) correlate well. These two alternative damage state indicators are therefore well represented by the vertical displacement of the crest midpoint of the embankment. On the other hand, Figure 11b shows that the difference in vertical displacement of the midpoint and edge points (\(\left|\Delta u_{y}\right|\)) responds differently than the settling of the embankment midpoint.  
These results confirm that permanent settling of the midpoint of the crest of an embankment is a relatively good damage state indicator, since it: (i) can represent well some other possible damage state indicators; and (ii) is not sensitive to possible local phenomena at the edges of the embankment. Nevertheless, other representative EDPs should be used if differential settlements are expected to be the driving mechanism behind the damage of an examined embankment.  
Figure 11: Correlation of \(\mathrm{u_{y\_\mathit{mid}}}\) to (**a**) vertical displacements at crest edges and (**b**) absolute differences between edge and middle crest points.  
Figure 12: Correlation of \(\mathrm{u_{y\_mid}}\) to horizontal differential displacements.