## 5 Case Study
### Derivation of Fragility Functions  
The cloud analysis [39] requires defining a best-fit curve between an intensity measure (IM) and an engineering demand parameter (EDP) in the logarithmic space.  
The regression algorithm was implemented in the toolkit, where excessively high displacements are treated differently through censored regression analysis [51]. This regression method is based on the fact that numerical analyses can achieve convergence for displacement levels that are incompatible with structural stability. This mainly occurs due to limitations in numerical modeling and can introduce biases into the best-fit curve. Censored regression analysis requires defining a threshold (i.e., maximum displacement or acceleration), beyond which data points are assumed to be affected by numerical errors. This threshold is defined by a censoring factor, which is defined as the ratio between the maximum allowable EDP and the threshold for the last considered damage state (e.g., complete damage). In this case, a censoring factor of 1.5 was used, meaning that EDPs 1.5 times above the threshold for the last damage state will be treated differently [23; 24].  
As explicit consideration of building-to-building variability was not accounted for (i.e., only one capacity curve was provided per building class), it waspossible to increase uncertainty to account for this source of variability. In the VMTK, this adjustment wasmade by adding to the dispersion of variability from record to record (_orrec-to-rec_) [24]. In other words, two records with the exact same IM can result in different structural responses due to the contribution of building-to-building variability (_cbld-to-bld_). For simplicity's sake, we chose to directly increase the standard deviation around the best-fit curve by a factor of 0.30, as described in Equation (1) [23].  
\[\sigma_{total}=\sqrt{\left(\sigma_{rec-to-rec}\right)^{2}+\left(\sigma_{bld- 0-bld}\right)^{2}} \tag{1}\]  
Figure 9: EDP distributions vs. IML for RM2/M medium-height masonry housing (4 to 7 floors). (**a**) Low seismic design code level, (**b**) moderate seismic design code level, and (**c**) high seismic design code level.  
The resulting fragility functions are defined using cumulative log-normal functions, as shown in Figures 10 and 11 for the aforementioned structure classes.  
The fragility functions derived from low- and medium-rise masonry houses werecompared for different seismic design levels. Table 7 provides a summary of the probability of reaching or exceeding various damage states for low-rise houses (RM2/L) at an expected value of 0.50 g of PGA and for medium-rise houses (RM2/M) at an expected value of 0.5 g of SA (T = 0.5 s). For example, it can be observed that the probability of damage decreased- for low-rise houses (RM2/L) with moderate and high levels of seismic design, especially for damage states DS3 and DS4, and the probability of damage waszero for medium-rise houses (RM2/M) in damage state DS4 for moderate and high levels of seismic design.  
Figure 11: Fragility functions for medium-height masonry structures (RM2/M) with spectral accelerations of SA (0.5 s). (**a**) Low seismic design code level, (**b**) moderate seismic design code level, and (**c**) high seismic design code level.  
Figure 10: Fragility functions for low-height masonry structures (RM2/L) with PGA spectral accelerations. (**a**) Low seismic design code level, (**b**) moderate seismic design code level, and (**c**) high seismic design code level.