## 5 Analysis of Structural Performance by Probability-Based Approach
As far as the probability of failure in the case of jointly distributed variables is concerned, it must be calculated by integrating the generic JPDF, \(f_{Z_{x}Z_{Y}}(Z_{X},Z_{Y})\), where \(Z_{X}\) denotes _IDI\({}_{x}\)_ or \(d_{FPS_{X}}\) whereas \(Z_{Y}\) denotes _IDI\({}_{Y}\)_ or \(d_{FPS_{Y}}\). Figures 21-28 illustrate mainly the level curves of the joint PDFs, together with a certain limit state area, considering the cases with and without masonry infills and for both fixed-based and base-isolated buildings. For example, regarding the superstructure, only results related to the 4 th story and the 2 nd story are presented. \begin{table}
\begin{tabular}{l c c c c} \hline \hline
**Limit State** & **IDI [\%]** & **IDI [\%]** & **P\({}_{\textbf{f}}\) in 50 Years [\(-\)]** & **P\({}_{\textbf{f}}\) in 1 Year [\(-\)]** \\  & **for FB Structure** & **for BI Structure** & **P\({}_{\textbf{f}}\) in 50 Years [\(-\)]** & **P\({}_{\textbf{f}}\) in 1 Year [\(-\)]** \\ \hline _Fully operational limit state LS1_ & 0.50 & 0.33 & 5.0 \(\times\) 10\({}^{-1}\) & 1.4 \(\times\) 10\({}^{-2}\) \\ \hline _Operational limit state LS2_ & 1.00 & 0.67 & 1.6 \(\times\) 10\({}^{-1}\) & 3.5 \(\times\) 10\({}^{-3}\) \\ \hline _Life safety limit state LS3_ & 1.50 & 1.00 & 2.2 \(\times\) 10\({}^{-2}\) & 4.5 \(\times\) 10\({}^{-4}\) \\ \hline _Near-Collapse limit state LS4_ & 2.00 & 1.33 & 1.5 \(\times\) 10\({}^{-3}\) & 3.0 \(\times\) 10\({}^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Limit states thresholds in terms of IDI (FB and BI structure) [12; 28]. Figure 18: Mono-variate log-normal distribution (PDFs (**a**) and CDFs (**b**)) related to the in-plane relative displacement with respect to the ground of the isolation level oriented in \(X\) and \(Y\) direction (without masonry infills). Reprinted/adapted with permission from [26]. Figure 19: Probabilities of exceedance with mono-variate assumption in logarithmic scale: fixed-base model **(a)**; base-isolated model (b); isolator devices for base-isolated model **(c)**; (without masonry infills). Reprinted/adapted with permission from [26]. Figure 20: Probabilities of exceedance with mono-variate assumption in logarithmic scale: fixed-base model **(a)**; base-isolated model **(b)**; isolator devices for base-isolated model **(c)**; (with masonry infills). We can observe a much higher probability of exceedance when the parameters _IDIs_ and \(d_{FPS}\) are considered in both the planar directions of the building. This is mainly due to the good degree of correlation that always exists between the abovementioned parameters. Figures 25-28 show how the presence of the infills leads to lower exceeding probabilities as well as a lower dispersion of the spatial response of the structure along its height, especially for the fixed-base structure. Figure 23: Three-dimensional view of the joint PDF for the isolation story **(a)** and level curves (**b**) with a generic displacement threshold (without masonry infills). Figure 21: Level curves of the joint PDF for the 4 th story with a generic limit state: fixed-base model **(a)**; base-isolated model **(b)**; (without masonry infills). Figure 22: Level curves of the joint PDF for the 2 nd story with a generic limit state: fixed-base model **(a)**; base-isolated model **(b)** (without masonry infills). In conclusion, in the case of retrofitting intervention using FP devices, the mentioned above results translate into a reduced damage level for fully operational (LS1) and operational (LS2) limit states, enhancing the recovery of functionality after seismic events. Moreover, with reference to the life safety (LS3) and near-collapse (LS4) limit states, the interaction between the FP isolation system and the presence of an infill wall reduces the probability of exceeding the related limit state thresholds as well as the potential direct (i.e., casualties, recovery, or rebuild of the structure) and indirect (i.e., loss of service of the building related to its function for the community) costs.