# Numerical Assessment of the Seismic Vulnerability of Bridges within the Italian Road Network
## 3 Fragility Curves
### Limit States  
The considered Limit States are reported in Tables 3-7, with some comments to briefly discuss the choices. This is necessary because there is an intrinsic difficulty in the association between physical damage and response thresholds [49]. Shear and chord rotation criteria refer only to inelastic elements. Shear failure is a brittle type of failure that takes place suddenly when the shear capacity is exceeded by the demand. The failure of a pier is a critical event in terms of bridge safety, while failures in transverse beams or connecting elements of the piers refer to localized problems. Nevertheless, their effects must be carefully considered as they can have an impact on the global response of the structure. For chord rotation, three response thresholds have been defined. The CLS refers to the computation of an ultimate capacity \(\theta_{u}\), the LLS involves the calculation of a fraction (3/4) of \(\theta_{u}\), while the DLS requires the definition of a yielding capacity \(\theta_{y}\). The response of bearing elements has been checked through the definition of specific criteria in the analyses. In general, bearings fail due to an excessive displacement demand. The Limit States specified for bearings are slightly different for elastomeric and PTFE-steel bearings. Different approaches have been proposed to assess the behavior of the entire structure [50; 51; 52]. In this work, two different Limit States, Damage and Collapse, have been considered to describe the global response of the bridge. While the Damage Limit State refers to a condition of service for the structure, the Collapse Limit State is an indicator of critical conditions in terms of safety and is reached when there are failures in piers or bearings. It has been assumed that to reach the Collapse Limit State, it was necessary to exceed the Damage Limit State. It has to be pointed out that the definition of the Collapse Limit State does not include the real collapse mechanism of the structure, since it is not possible to capture it with the regular Finite Element models used for the structural analysis, but it is instead a conventional definition employed in the assessment of the case studies. The criteria listed in Table 7 are alternative possibilities and reaching just one of them is enough to consider the corresponding Limit State as being exceeded. \begin{table}
\begin{tabular}{c c} \hline \hline
**Limit States** & **Criteria** \\ \hline
50\% bearing height & Reaching of horizontal displacement corresponding to 50\% of device height in at least one bearing \\ \hline
100\% bearing height & Reaching of horizontal displacement corresponding to 100\% of device height in at least one bearing \\ \hline
150\% bearing height & Reaching of horizontal displacement corresponding to 150\% of device height in at least one bearing \\ \hline
200\% bearing height & Reaching of horizontal displacement corresponding to 200\% of device height in at least one bearing \\ \hline \end{tabular}
\end{table} 
Table 6: PTFE-steel bearings Limit States. \begin{table}
\begin{tabular}{c c} \hline \hline
**Limit States** & **Criteria** \\ \hline
**Chord rotation DLS/LLS/CLS (piers)** & Reaching of chord rotation capacity, for the considered LS, in at least one pier \\ \hline Chord rotation DLS/LLS/CLS (transverse elements of frame & Reaching of chord rotation capacity, for the considered LS, in at least one transverse element of frame piers) & \\ \hline Chord rotation DLS/LLS/CLS (Connections between pier & Reaching of chord rotation capacity, for the considered LS, in at least one connection between pier elements \\ \hline \end{tabular}
\end{table} 
Table 4: Chord rotation Limit States.