## 5 Analysis of Structural Performance by Probability-Based Approach  
Starting from the outcomes of the 3D NL dynamic simulations, the horizontal relative displacements of the isolators \(d_{\text{FPS}}\) and _IDIs_ have been the starting point to analyze the structural performance of both fixed-base and base-isolated systems in probabilistic terms. In particular, it has been assumed that both \(d_{\text{FPS}}\) and _IDIs_ are lognormally distributed [12], with mean value \(\mu\) and standard deviation \(\sigma\). By performing a statistical inference analysis, the hypothesis of lognormal distribution has passed the test with a significance level of 5%. The parameters of the probabilistic distribution have been computed adopting the maximum likelihood method according to [48]. In Figures 16 and 17, the mono-variate lognormal distributions are illustrated by presenting either the results in terms of probability density functions (PDFs) or cumulative density functions (CDFs) and by making a comparison between the fixed-base and base-isolated structures, each of which considers once the absence of the masonry infills and once including those elements. Table 4 reports the limit state (LS) thresholds according to [28] in terms of IDI for both the structural systems (i.e., fixed-base and base-isolated) associated with an acceptable limit for the probability of exceedance in 50 years and 1 year. Figure 16a,b reports, for both fixed-base and base-isolated buildings (without masonry infills), the mono-variate lognormal probability distribution functions obtained from the _IDIs_ and considering different stories, while Figure 17a,b reports the related cumulative distribution functions. Figure 16: Mono-variate log-normal distribution (PDFs) related to the _IDIs_ oriented in \(X\) and \(Y\) direction: fixed-base model (**a**), base-isolated model (**b**); (without masonry infills). Reprinted/adapted with permission from [26]. Figure 17: Mono-variate log-normal distribution (CDFs) related to the _IDIs_ oriented in \(X\) and \(Y\) direction: fixed-base model (**a**), base-isolated model (**b**); (without masonry infills). Reprinted/adapted with permission from [26]. In particular, Figure 17a,b illustrates, respectively, the mono-variate lognormal probabilistic and cumulative distribution functions of the \(d_{FPS}\) including both the \(X\) and \(Y\) directions, for the base-isolated building (without masonry infills). First of all, it is shown that the horizontal displacement and the interstory drift index in the X direction are considerably larger than in the other direction, being less stiff in that direction. In addition, it is demonstrated that the retrofit allows a reduction in terms of the probability of exceedance \(P_{f}\) of the defined limit state. The exceedance probabilities for the isolation level and for both the fixed-base and base-isolated buildings (without masonry infills and considering the different stories) are shown in Figure 18a-c. It is shown that the isolation technique is able to decrease the _IDIs_ and, thus, to significantly reduce the probability of exceedance if compared to the non-isolated structure. The previous conclusion is generally true with and without considering the presence of masonry infills, as shown in Figures 19 and 20. As for the effects of the masonry infills, they can be highlighted as they affect the order of magnitude of the _IDIs_ as well as the related exceeding probabilities. Furthermore, the tri-dimensional response of either the fixed-base structure or the base-isolated structure can be performed by evaluating the degree of correlation between the abovementioned parameters _IDIs_ and \(d_{FPS}\) along the two directions \(X\) and \(Y\) of the planar scheme of the structure. Then, the joint log-normal distributions have been computed according to [12].