# Seismic Risk Analysis of Existing Link Slab Bridges Using Novel Fragility Functions
## 2 Seismic Reliability Framework
### Seismic Scenario and Hazard  
A typical scenario of medium-high seismicity regions in Italy is considered for the purposes of these applications.  
As an intensity measure, IM, the maximum spectral acceleration component [38], IM = max[S\({}_{\text{a},X}\)(T), S\({}_{\text{a},Y}\)(T)], is chosen for the following two reasons: (i) it allows accounting for the bidirectional features of the seismic input (both the horizontal, X and Y, seismic components are considered), (ii) unlike the Peak Ground Acceleration (PGA) [29], the spectral amplification related to the system dynamic properties can be accounted for through the definition of the vibration period, and an average period between transversal and longitudinal vibration modes is considered in this study. However, several alternative (scalar and vector-valued) IMs are also available but this aspect goes out of the scope of the present study.  
A widely used and validated stochastic ground motion model [39; 40] is adopted to characterize the underlying seismicity and provide a proper IM hazard curve (through the procedure detailed in the work [41]). Moreover, sets of synthetic ground motion samples are generated through the stochastic model in order to obtain a suitable database of earthquakes conditional to different IM levels, according to the analysis setup described in the next subsection. It is worth noting that, using simulated ground motion samples, the typical issues of deficit of high magnitudes real records and the consequent need for scaling procedures (especially at the highest IM levels) are overcome. The only limit of the adopted stochastic model is the incapability to produce pulse-like ground motions, so this type of event is not considered in this work. Further details concerning the seismic scenario and the stochastic model are provided in Appendix A to the present paper.  
The chart of Figure 2 reports the IM hazard curve, i.e., a plot of the Mean Annual Frequency (MAF) of exceeding different intensity spectral acceleration values. Black markers on the curve identify the IM levels at which seismic analysis will be performed. Two horizontal dotted lines are superimposed on the chart to identify the MAF corresponding to two main limit states considered in this study: Damage onset (blue) and Near-Collapse limit states, corresponding to return periods \(\mathrm{T_{R}}\) equal to 100 years and 1950 years, respectively.  
These limits stem from the current Italian seismic code (but the values comply with European Standards as well, e.g., Eurocode 0 [42]) under the assumption of a strategic role played by the bridge during its service life (note that these values are halved for ordinary constructions). The correlation between return periods and IM levels is summarized in Table 1. In order to show the features of the seismic ground motion samples used to perform probabilistic analyses, the charts in Figures 3 and 4 are provided. In Figure 3, the response spectra (single components of the 20 earthquakes, average and percentiles) are provided for two IM levels with a return period close to the reference Limit States; the conditioning on the target IM is highlighted through red markers. In Figure 4, pairs (X and Y) of accelerations selected for the two aforementioned IM levels are shown.  
Figure 3: Response spectra of the ground motion samples conditional to two different IM levels: (**a**) IM 7 (83 years return period) and (**b**) IM 13 (2200 years return period). Red dots identify the conditioning IM value.  
Figure 2: IM hazard curve with IM levels and (damage and near-collapse) limit states highlighted.