**5. Fragility curves and damage probability matrices**

Fragility curves are a graphical representation of the cumulative distribution function of the probability of reaching or exceeding a specific state of damage, given a *Probabilistic Seismic Vulnerability and Loss Assessment of the Buildings in Mexico City DOI: http://dx.doi.org/10.5772/intechopen.109761*


#### **Table 3.**

*Damage degree and damage index in the five buildings.*

structural response, to a given seismic action. The HAZUS methodology [11] defines fragility curves and a lognormal probability distribution defined by the following equation.

$$P[ED \ge ED\_i] = \Phi\left[\frac{\mathbf{1}}{\beta\_{ED}} \ln\left(\frac{\mathbf{S}d}{\mathbf{S}d\_{ED}}\right)\right] \tag{6}$$

where *SdED* is the average spectral displacement for which the probability of exceedance is 50%; *βED* is the standard deviation of the natural logarithm of spectral displacement, *Φ* is the cumulative standard normal distribution function, and *Sd* is the spectral displacement.

In this methodology, each fragility curve is defined by the average spectral displacement value corresponding to the threshold of each damage state defined in **Table 2**. The standard deviation was calculated using two methods and adjusted to the mean of both. The first method obtains the standard deviation using the different actual capacity curves, while the second method adjusts the fragility curves with a discrete probability distribution (**Table 3**).

#### **5.1 Fragility curves for buildings constructed before 1985**

The fragility curves calculated for the first type of buildings, that is, between five and seven stories (with vibration periods between 0.60 and 0.86 s), are presented on the left side of **Figure 8**. In addition, the values of the spectral displacements Sd corresponding to the hazard spectrum for a return period Tr = 100 years in Zone IIIb, are included in each curve (see **Figure 4**). **Table 3** indicates the average damage factors and the central damage factor, FDC, for each case.

#### **5.2 Fragility curves for buildings constructed after 1985**

In order to include buildings built with recent building regulations, theoretical models of buildings were included from the design spectra of the 2004 Complementary Technical Standards for Design by Sism (GDF, 2004). **Figure 9** shows the

#### **Figure 8.**

*Fragility curves that are indicated by dashed lines the Sd displacements for each building according to its period calculated assuming a return period Tr = 100 years.*

#### **Figure 9.**

*Capacity spectra obtained from NTCS2004 [8] design spectra for Zone IIIb in Mexico City and using an overstrength factor of 1.5*.

capacity spectra determined for three ductility reduction factors, Q = 2, Q = 3, and Q = 4, for structures built in Zone IIIb, and an overstrength factor of 1.5. These spectra are the basis for determining the fragility curves in **Figure 10**, following the same procedure used earlier. Fragility curves were calculated for six structural models with periods between 1.0 s and 2.0 s, with intervals of 0.2 s.

**Figure 4b** shows the displacement response spectra for 5% of critical damping at eleven stations in Mexico City's zone IIIb of September 19, 2017, which were calculated for elastic models and for models with ductility equal to two (Q-2). They are also indicated in the same **Figure 4**, the average or average response spectra and the spectrum corresponding to the 84% percentile, for all cases. It is important to note that there is little dispersion between the spectra of the different stations for displacement spectra, compared to acceleration response spectra.

In order to calibrate the fragility curves in **Figure 10**, the spectral displacement, Sd, for the return period Tr = 100 years (**Figure 4a**) associated with each natural period, is included in each of the six boxes. It can see that, in general, there is very good correlation with the behavior of the buildings of this group (built after 1985), observed during the 19 September 2017 earthquake.

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

**Figure 10.**

*Fragility curves for buildings using a coefficient of Q = 2 and with fundamental periods of 1.0, 1.2, 1.4, 1.6, 1.8, and 2.0 s.*

### **6. Conclusions**

In the first part of this article, a probabilistic seismic hazard analysis (PSHA) was carried out, which allowed the definition of seismic hazard curves as well as uniform hazard spectra for each of the six seismic zones of Mexico City; in this analysis, the seismogenic origin of the earthquakes was considered, that is, interplate or intraslab origin.

In the second part, the fragility curves that are the basis for establishing the damage index of buildings for a specific scenario were defined. Two groups of fragility curves were defined, the first for buildings built before 1985 and the second for buildings built after 1985. In the first case, nonlinear static analyzes (Pushover) of five buildings, which were performed to define the capacity curves and later the capacity spectra elements, were necessary to estimate the fragility curves. In the second case, the capacity spectra were directly defined from the design spectra of the Mexico City Building Regulations.

For the study of buildings built before 1985, five buildings were selected as representatives of this group, which have a structural system of columns of reinforced concrete with a waffle slab, and were damaged by the earthquake of September 19, 2017.

With the purpose to assign the vulnerability of residential buildings in Mexico City, the fragility curves of buildings built before 1985 were defined, which have a structural system of reinforced concrete columns with a lightened flat slab and which were damaged by the earthquake of September 19, 2017.

The findings of the probabilistic seismic hazard and probabilistic vulnerability analysis resulted in the following conclusions:

1.Uniform hazard spectra, UHS, were computed for return periods of 50, 100, 475, 975, and 2475 years. Specific attenuation relationships were developed for

Mexico City that directly includes local effects; in addition, to complement the study, classical predictive functions were used.


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