## 3 Damage Indexes Effectiveness
### MSA Results  
In this study, the results of the MSA are used to calculate the values of the considered damage indexes and to assess their effectiveness and reliability compared to various damage scenarios and performance levels, selected according to the DPC-ReLUIS project [28; 29] and Eurocode 8 [27]: Usability Preventing Damage (UPD), Life Safety (LS), Collapse Prevention (CP) and Global Collapse (GC). Table 9 lists the limit states and the corresponding structural conditions. Figure 4a provides an example of the results obtained from the analysis as diagrams showing the damage index values as a function of the spectral acceleration. The complete graphical representation is reported in Appendix A for different building configurations. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case Study** & **Building Fundamental Period (T\({}_{1}\))** & **Selected Conditioning Period (T\({}_{j}\))** \\ \hline Existing building (BF) & 1.55 & 1.50 \\ \hline Existing building (IF) & 0.27 & 0.15 \\ \hline Code-conforming wall building (BF) & 0.62 & 0.50 \\ \hline Code-conforming wall building (IF) & 0.17 & 0.15 \\ \hline Code-conforming MRF building (BF) & 0.56 & 0.50 \\ \hline Code-conforming MRF building (IF) & 0.18 & 0.15 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Conditioning periods chosen for each configuration of the building. \begin{table}
\begin{tabular}{c c} \hline \hline
**Limit State/Performance Level** & **Description** \\ \hline \multirow{6}{*}{Usability Preventing Damage (UPD)} & Defined according to Eurocode 8: it aims to withstand a seismic action having a larger probability of occurrence than the design seismic action, without the occurrence of damage and the associated limitations of use, the costs of which would be disproportionately high in comparison with the costs of the structure itself. \\ \cline{2-3}  & Defined according to Eurocode 8: it aims to protect occupants from serious injury and to allow safe evacuation. Although significant structural and non-structural damage may occur, the critical load-bearing elements of the building must remain intact to prevent complete structural failure. Herein considered when the inter-story drift ratio reaches a value of 2\%. \\ \hline \multirow{6}{*}{Collapse Prevention (CP)} & Defined according to Eurocode 8: it represents the stage at which the structure is severely damaged, with large permanent drifts, but retains its vertical load-bearing capacity and most of the ancillary components, if any, have collapsed. Herein considered when the top displacement reaches the value corresponding to a 20\% degradation of the maximum base shear obtained from a non-linear static, pushover, analysis. \\ \cline{2-3}  & Defined according to Eurocode 8: it represents the stage at which the structure could be considered to have collapsed. \\ \cline{2-3}  & Herein considered when the top displacement reaches the value corresponding to a 50\% degradation of the maximum base shear obtained from a non-linear static, pushover, analysis. \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Considered limit states and performance levels. In Figure 4a, the solid blue line and the blue circles show the median value of the damage indexes calculated over the 10 IM levels. This provides a representation of the central tendency of the damage indexes. The light blue shaded area shows the interquartile range (25 th to 75 th percentile), which illustrates the variability of the index, with wider ranges reflecting greater uncertainty. The vertically shaded areas correspond to the defined performance levels: UPD (green), LS (orange), CP (red) and GC (dark red). The width of these areas represents the interquartile range of the performance levels while the dashed vertical line represents the median value. This representation highlights the damage index value dispersion within the same intensity level and offers a clear view of the variability and robustness of the indicators.