## 3 Seismic Response Spectra in Mexico
### Regional Spectrum According to MDOC Sismo CFE 2015  
According to Table 3, the structures of subgroups B1 and A2 are related to buildings with a high degree of safety, without any restriction regarding their built area and height. Therefore, a basic dynamic analysis of the ground is required to classify it into soil types I, II, and III, depending on three variables: the fundamental period of the soil, \(T_{S}\), the average shear wave velocity, \(v_{s}\), and the thickness of the equivalent homogeneous soil layer, \(H_{s}\). With the basic dynamic properties of the soil and the reference spectrum (ER) in the rock layer available, the seismic elastic design spectrum can be obtained by the regional spectrum procedure [1]. The fundamental period of the soil, \(T_{S}\), using an equivalent homogeneous layer is obtained according to the following equation:  
\[T_{s}=\frac{4H_{s}}{v_{s}} \tag{4}\]  
where \(H_{s}\) is the thickness of the equivalent homogeneous layer and \(v_{s}\) is the average shear wave velocity; both were previously obtained from soil studies through physical tests, such as wave dispersion, cross hole, or any equivalent method. If a soil with multiple layers is present, the value of \(H_{s}\) considers an equivalent homogeneous layer according to the following equation:  
\[H_{s}=\sum_{i=1}^{N}h_{i} \tag{5}\]  
where \(h_{i}\) is the thickness of layer \(i\). The average shear wave velocity, \(v_{s}\), is obtained using the most unfavorable criterion of the equations:  
\[\text{Average Velocity}:\ v_{s}=\frac{\sum_{i=1}^{N}v_{i}h_{i}}{H_{s}} \tag{6}\]  
\[\text{Average Slowness}:\ v_{s}=\frac{H_{s}}{\sum_{i=1}^{N}\frac{h_{i}}{v_{i}}} \tag{7}\]  
where \(v_{i}\) is the shear wave velocity of layer \(i\). To calculate the fundamental soil period, \(T_{s}\), with multiple layers, the following equations are used:  
\[T_{s}=\frac{4}{\sqrt{g}}\sqrt{\left(\sum_{i=1}^{M}\frac{h_{i}}{G_{i}}\right) \left(\sum_{i=1}^{M}\gamma_{i}h_{i}\left(w_{i}^{2}+w_{i}w_{i-1}+w_{i-1}^{2} \right)\right)} \tag{8}\]  
\[G_{i}=\frac{\gamma_{i}v_{i}}{g} \tag{9}\]  
where \(g\) is gravity, \(i\) refers to a soil layer, \(h_{i}\) is the thickness of the layer \(i\), \(G_{i}\) is the shear modulus of layer \(i\), \(\gamma_{i}\) is the volumetric weight of layer \(i\), \(v_{i}\) is the shear wave velocity of layer \(i\), and \(M\) is the total number of layers. The dimensionless values of each layer,\(w_{i}\), are considered \(w_{o}=0\) for the basal rock layer and \(w_{M}=1\) for the superficial layer. The dimensionless value, \(w_{m}\), in the intermediate layer is calculated with the following equation:  
\[w_{m}=\frac{\sum_{i=1}^{m}\frac{h_{i}}{\gamma_{i}v_{i}^{2}}}{\sum_{i=1}^{M} \frac{h_{i}}{\gamma_{i}v_{i}^{2}}} \tag{10}\]  
Once the values of \(T_{s}\), \(H_{s}\), and \(v_{s}\) are obtained, the next step is to classify the soil into types I, II, and III according to the stiffness of the equivalent layer, where soil type I refers to a rock layer and soil type III to a soft layer. According to Table 5, the variables \(T_{s}\), \(H_{s}\), and \(v_{s}\) must be combined into three different cases, locating the coordinates of each case on the microzoning graph shown in Figure 4, where the values of \(v_{s}=720\) m/s, \(v_{c}=360\) m/s, \(H_{s}=2\) m, and \(H_{c}=30\) m are the limits between soil types I, II, and III. Finally, the most unfavorable point location is chosen to assign the soil type. Soil type I is a rigid layer with low dynamic amplifications at the site, soil type II is a medium rigid layer that presents intermediate dynamic amplifications, and soil type III is a soft layer with high dynamic amplifications. With the location of the seismic zone related to Figure 2 and the soil type assignment according to Figure 4, the site factor, \(F_{sit}\), and the response factor, \(F_{res}\), can be selected using the values shown in Table 6, where the reference spectrum (ER) and the spectral acceleration in the rock layer, \(a_{0r}\), are provided by PRODISIS.