**4. Numerical models**

#### **4.1 Buildings characteristics**

The identification and selection of structural systems to be studied typical of an area is initially carried out. In this case, five buildings of a very common typology in Mexico City were selected, which has been very vulnerable to the two earthquakes of September 19 (1985 and 2017), and other seismic events.

With the purpose of studying existing buildings constructed before 1985 in Mexico City, the five medium-rise buildings damaged by the 2017 earthquake were analyzed (**Figure 5**). These buildings were built on reinforced concrete columns and reticular waffle slabs, four of them are in the Benito Juárez Sector and one more in the Cuauhtémoc Sector. All buildings are for housing use and are classified within group B according to the México City Building Code [8]. **Table 1** summarizes the main characteristics of the five buildings, including the EME-98 classification of the damage degree assigned after the 2017 earthquake.

Once the structures to be studied have been modeled, the main parameters to be considered are defined. The main function of this variation is to obtain a standard deviation from the potential capabilities of each of the analyzed models and consider them within their vulnerability assessment. The definition of the damage thresholds was made according to **Table 2**.

These buildings have long rectangular plants as shown in **Figure 3**, as can be seen, with the exception of the SE2–7 N building, the structures exceed or are equal to the value of 2.5, which is a limit value suggested in all structural recommendations because when exceeded, it has been observed that the structural response can be amplified and the torsional effects can also be increased, in addition, all these structures have no girders and the slab have relatively low depths, which generates flexible


#### **Table 1.**

*General characteristics of the selected buildings [25].*


**Table 2.** *Definition of damage threshold.* frames away from being shear systems. Another factor that increases the vulnerability of these buildings is the unfavorable orientation of the columns with their minor axis on the short side.

#### **4.2 Capacity curves**

For all buildings studied, a nonlinear static analysis (Pushover) was performed using the SAP program to define the capacity curves of each model. The structural configuration and considered loads of buildings are defined in Ref. [25]. The different capacity curves of the models analyzed are presented in part left of **Figure 5**. According to the results, it is possible to note that the model of the structure SE2–10 N is the one that provides the least resistant capacity since its last capacity is presented when a basal shear of Vu = 25.6 Ton occurs, with a roof displacement of 27 cm, while the strongest structure corresponds to the model SE2–7 N, where its basal shear is Vu = 60.31 Ton, with a roof displacement of 31 cm.

To obtain the ductility, μ, of the system it is necessary to represent the capacity curve in its bilinear form, which is obtained by defining the yield point and the ultimate capacity point of the structure. The procedure used in this study corresponds to that proposed by FEMA 356 [14], which is based on matching the energy dissipated by the structure defined by the area under the actual curve with the dissipated energy of the idealized curve. **Table 2** presents some of the most important points that define the capacity curves obtained in their bilinear form.

#### **4.3 Capacity spectra**

To compare the seismic demand with the capacity of the structure, it is necessary to transform the result of pushover to another curve that relates the spectral displacement *Sd* with the spectral acceleration *Sa*. This transformation is known as the capacity spectrum and develops by applying the dynamic characteristics of the fundamental mode. The capacity spectrum is determined using the following equations:

$$\text{Sd}\_{\circ} = \frac{Dt\_{\circ}}{\chi\_{M}\varrho\_{t1}} \tag{2}$$

$$\text{Sa}\_{j} = \frac{V\_{j}}{\mathbf{M}\_{T}\alpha} \tag{3}$$

where:

Dtj = displacement of each point of the capacity curve [cm].

Vj = shear of each point of the capacity curve [Ton].

γ<sup>M</sup> = participation factor of the first mode.

α = effective mass coefficient of the basal shear of the first mode.

φt1 = maximum top amplitude of the structure associated with the first mode.

MT = total mass of the structure.

The capacity spectrum is defined by two main points: the ultimate capacity point (UC) and the yield capacity point (YC), which are expressed as follows:

$$\text{YC}\left[A\_{\text{y}} = \text{S}\_{\text{ay}} = \frac{\text{C}\_{\text{s}} \text{SR}}{a\_{1}}; \text{Dy} = \text{S}\_{\text{dy}}\right] \tag{4}$$

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

**Figure 5.**

*Right: Structural plants of the buildings studied. SE2–5 N, SE2–7 N, SE2–8 N, SE2–10 N, SE2–12 N. Left: The corresponding capacity curves of the studied buildings.*

$$\text{UC}\begin{bmatrix} \mathcal{A}\_u = \mathcal{S}\_{au} = \mathcal{\lambda} \mathcal{A}\_\mathcal{y}; D\_u = \mathcal{S}\_{du} \end{bmatrix} \tag{5}$$

where *Cs* represents the design seismic coefficient. This value is calculated using Eq. (4), and an approximate overstrength of the buildings analyzed is obtained.

#### **Figure 6.**

*Capacity spectrum, derived from the capacity curves of Figure 5 for the five selected buildings.*

#### **Figure 7.**

*Mean capacity spectra for buildings with short and long periods; damage thresholds are indicated in each case (DD2 to DD5).*

As part of the methodology used in this work, the buildings analyzed were divided into two groups. Firstly, based on the characteristics and properties of each one of these structures, the dispersion of the capacity curves is greater among the buildings with five to seven levels and those from eight to 12 levels (**Figure 6**). Secondly, it is desired to reduce the uncertainty that exists due to this difference since there are only five buildings analyzed. For this reason, two mean capacity curves have been calculated, one for buildings with short periods and another for buildings with periods longer than 1 sec (**Figure 7**). This division by periods is also useful and of great interest since it allows classifying the type of buildings that were most vulnerable to the earthquake of September 19, 2017. Capacity spectra of **Figure 7** show the bilinear representation. From these capacity curves in their bilinear form, the thresholds were defined, of damage for each type (**Table 2**).
