**2. Methodology**

With the purpose of assigning structural vulnerability to the building population of México City, it is necessary to apply an analytical methodology for assessing the seismic response of a class of buildings, which is combined with a damage model in order to derive sets of fragility functions for different levels of seismic behavior. The seismic vulnerability assessment developed in this paper focuses mainly on the HAZUS methodology [10] and RISK-EU [11], which has been adapted for damaged buildings in Mexico City during the September 19, 2017, earthquake.

The methodology developed consists in general of the following steps: (1) seismic hazard assessment for Mexico City, using a probabilistic procedure, for the calculation of the uniform hazard spectra (UHS). (2) Definition of the capacity curves of structural systems, in this case, data from damaged buildings representative of the most vulnerable typology were used for the September 19, 1985 and 2017 earthquake, but analytical models representative of other common typologies of structures built after 1985 were also used; (3) Definition of the damage criteria adopted and development of capacity spectrums; (4) Estimation of the medium capacity curves, which are represented in their bilinear form to define the damage thresholds and thereby build the fragility curves and their damage probability matrices; (5) calculation of overall damage of the structure analytically. The fragility curves resulting from this study were calibrated with the event of September 19, 2017.

### **3. Seismic hazard analysis**

#### **3.1 Probabilistic seismic hazard analysis PSHA**

The seismic hazard for Mexico City was developed with a probabilistic seismic hazard analysis (PSHA). The PSHA use of probabilistic concepts has allowed uncertainties in the location, size, and rate of recurrence of earthquakes and in the variation of ground motion characteristics with earthquake size and location to be explicitly considered in the evaluation of seismic hazards [12]. In this study, the seismic hazard

settings were defined, using uniform hazard spectra (UHS) and seismic parameters. Three ground motion prediction equations (GMPEs) were revised and used, and a disaggregation study was performed. Site effects estimations were included in the prediction equations. UHS was established for both firm and soft soils for Mexico City. Finally, the computed seismic parameters were compared to those for the seismic design guidelines for Mexico City, and we found out that our UHS captured better the seismic settings.

The model characterization for the earthquake occurrence, the seismic sources, the magnitude-recurrence, and the attenuation laws are statistically evaluated (see for example [13, 14]). The probabilistic seismic hazard analysis is characterized by four stages [12]), these stages can be summarized as 1) identification and characterization of seismic sources, historical descriptions, seismic catalogs, isoseismal maps, and instrumental information; 2) Definition of recurrence relationships, it is necessary to ensure that data is homogeneous, independent, and complete (events with M > 5). The recurrence ratio, which specifies the average rate at which an earthquake of some size will be exceeded, is used to characterize the seismicity of each source zone. Gutenberg's law was used. 3) Definition of predictive relationships. 4) development of seismic hazard curves, for which, the uncertainties in earthquake location, earthquake size, and ground motion parameter prediction are combined to obtain the probability that the ground motion parameter will be exceeded during a particular time period.

The equation used to assess the seismic hazard according to the classical methodology is:

$$\lambda(\mathbf{y} > \mathbf{Y}) = \sum\_{i=1}^{N} \lambda\_i(\mathbf{y} > \mathbf{Y}) = \sum\_{i=1}^{N} \mathbf{u}\_i \iiint \mathbf{P}\_i(\mathbf{y} > \mathbf{Y}|\mathbf{m}, \mathbf{r}, \mathbf{e}) \int \mathbf{f}\_{\text{Mi}}(\mathbf{m}) \, \mathbf{f}\_{\text{Ri}}(\mathbf{r}) \, \mathbf{f}\_{\text{ei}}(\mathbf{e}) \, \mathbf{dm} \, \mathbf{dr} \, \mathbf{de} \tag{1}$$

where λ(y > Y) is the annual rate of exceedance of the level of ground motion Y, due to the occurrence of earthquakes in the N seismic sources, which is equal to the sum of the annual rates of exceedance in each source zone λi(y > Y), the same ones that present an annual rate of earthquakes vi; the term Pi[y > Y | m, r, ϵ] gives the probability of conditional exceedance to the trio of variables m, r, and ϵ representing magnitude, distance, and epsilon; fM*i* (*m*)fR*i* (*r*) fϵ*i*(ϵ) are the probability density functions of magnitude, distance, and epsilon [15].

#### **3.2 Hazard curves and UHS for Mexico City**

The identification of the seismotectonic regions is one of the most important stages in the seismic hazard procedure, in order to understand the seismicity of each region that is the characteristics of earthquake occurrence and to identify whether a source, close to the site, is active or not, as well as the definition failure mechanisms. We considered two types of seismic sources capable of producing earthquakes in Central Mexico: faults and areas. Seismic sources are modeled in a seismic hazard assessment with their geometric and recurrence characteristics.

Of the thirteen segments or gaps characteristic of the country, six faults were selected and characterized in this study since the interplate sources that are within a radius of influence of 500 km were analyzed. **Figure 2a** shows the six fault segments *Probabilistic Seismic Vulnerability and Loss Assessment of the Buildings in Mexico City DOI: http://dx.doi.org/10.5772/intechopen.109761*

**Figure 2.**

*(Left) Faults along the Pacific Coast [11]. (Center and Right) Seismic sources type Area affecting Mexico City [12].*

described by Ref. [16] that affect the Central Mexico segment are, Oaxaca Este (OX-E), Oaxaca Central (OX-CI and OX-CII), Oaxaca Oeste (OX-O), Ometepec (OX-M), Acapulco-San Marcos (AC-SM), Guerrero Central (GC), Petatlán (PE), and Michoacán (MI). We have considered an area where earthquakes might occur, with a radius 500 km (red circle in **Figure 2b**), these areas were selected according to the faults described by [17].

Part right and center of **Figure 2** illustrate the 19 seismic sources type-Area parameters, which describe the area where seismic events are presented; these parameters can be consulted in Ref. [18], namely, the focal depth, the activity rate or number of events per year that exceed the minimum magnitude for each source, the parameters of the Richter model, and the estimated minimum and maximum seismic moment magnitudes (Mw).

We use two sets of spectral acceleration attenuation (SA) functions; the first group includes classical functions [19–23] developed with a global database, while for the second group, we use attenuation relations developed by us [18] of strong ground motion recorded in Mexico City with M > 6.0, in both groups we considered interplate and intraslab seismic sources.

For the Hazard assessment the probabilistic procedure described above was used. For the calculation of the hazard curves and the uniform hazard spectra, UHS, we used the EZ-FRISK package [24]. **Figure 3** shows the seismic hazard curves in Zone I, Zone II, and Zone IIIb, as an example of the several sites studied. In **Figure 3**, uniform hazard spectra (UHS) are included for six return periods, namely, 40, 100, 250, 475, 975, and 2475 years, which represent, respectively, 68%, 39.3%, 20% 10%, 5%, and 2% exceedance. Probability of occurring in 50 years. These spectra show the highest spectral acceleration values in Mexico City, ranging from 0.8 g to 1.05 g, indicating a high hazard in Zone III for intraplate and interplate events.

When the shape of the spectral ordinates in sites of Mexico City obtained with our functions are compared with other relationships, we observed significant differences in periods ranging between 1 and 2 s, mainly due to the incorporation of a regression model, which considers local effects, and as seen in **Figure 3**, it provides good estimates of the spectral accelerations. UHS from sites in Zone I, Zone II, and Zone IIIb cover a larger range of periods.

As an example, **Figure 4** present the displacement spectra in Zone IIIb calculated using synthetic accelerograms, which were defined from the UHS assuming a return

**Figure 3.**

*Left: Hazard curves obtained in Zone I, Zone II, and Zone IIIb and Right: the corresponding uniform hazard spectra (UHS) for several return periods.*

#### **Figure 4.**

*Left: Displacement spectra in Zone IIIb calculated from synthetic records for UHS and a return period Tr = 100 years Right: Displacement spectra at stations in Zone IIIb due to September 19, 2017, Earthquake.*

period Tr = 100 years. The right part of **Figure 4** is compared this result with the displacement spectra of stations located in Zone IIIb due to the September 19, 2017, earthquake.

*Probabilistic Seismic Vulnerability and Loss Assessment of the Buildings in Mexico City DOI: http://dx.doi.org/10.5772/intechopen.109761*
