## 4 Computational and Mathematical Aspects
### Fragility Curves  
One of the problems the engineer must face and resolve at the later stages of this integrated approach involves the quantitative vulnerability assessment of the building in its current state (damaged or not), as well as its projected behavior once it has been \"modified\" after interventions. In other words, a method is necessary to assess the seismic vulnerability of the existing structure as well as to assess the intervention scenarios and rank them according to the reduction they induce to the seismic vulnerability, thus leading to the selection of the optimal intervention scenario. One of the most important tools is considered to be fragility analysis, which provides a measure of the safety margin of the structural system above specified structural performance/hazard levels. Several methodologies for performing fragility analysis have been proposed in the past (used to assess the behavior of structural systems). Simplified methodologies for fragility evaluation were proposed by [PERSON] et al. (1980) [198], [PERSON] et al. (1997) [199], [PERSON] et al. (2007) [200, 201], [PERSON] et al. (2010) [202], [PERSON] and [PERSON] (2012) [203] and [PERSON] et al. (2015) [204]. A detailed state-of-the-art report can be found in HAZUS 2003 [205]. A very interesting 3D homogenized FE limit analysis software package was produced by [PERSON] and [PERSON] (2011) [4] for the fragility curve evaluation of existing masonry churches. It is a two-step approach. In the first phase, homogenized masonry strength domains are obtained using a simplified kinematic procedure applied on a three-leaf unitary cell. In the second phase, homogenized domains are implemented at a structural level on a plate and cell kinematic FE software. The procedure is tested on two real scale existing churches located in Italy and compared to results provided by means of a conventional elastic approach performed by means of a standard commercial software. The global behavior at failure and the overall strength of the buildings are assessed through Monte Carlo simulations varying both masonry mechanical properties and direction of the equivalent seismic load. Recently, [PERSON], [PERSON], [PERSON] and [PERSON] 2018 [206], based on the fact that school facilities in Iran, in particular masonry schools, have shown poor performance during past earthquakes and can be identified as one the country's most vulnerable infrastructures to earthquakes, proposed a method to perform index-based damage assessment for brick masonry schools located in the province of Yazd, the central region of Iran, using a comprehensive database of school buildings. The database occurred from field survey forms applied for each observed school, which collected the features and damages of each structure. The results of a vulnerability index method developed in Iran were employed as input data to obtain empirical fragility curves for the school inventory. The Macroseismic model [207] and GNDT II level method [208] were two empirical methods combined in this procedure. Finally, the procedure was verified using damage survey data obtained after recent earthquakes (1990 Marjil-Rudbar earthquake and 2003 Bam earthquake) that occurred in Iran. Evaluating seismic fragility information curves for structural systems involves: (a) information on structural capacity; and (b) information on the seismic hazard. Since both aforementioned contributing factors are uncertain to a large extent, the fragility evaluation cannot be carried out in a deterministic manner. A probabilistic approach, instead, needs to be utilized in the cases in which the structural response is evaluated and compared against \"limit states\", that is limiting values of response quantities correlated to structural damage (see Table 4). Fragility, as shown in Figure 15, is the probability of the structural damage to reach or exceed a certain damage threshold di (performance level) under a given earthquake level (Peak Ground Acceleration (PGA)). It generally increases as the earthquake intensity level increases.