## 3 Seismic Response Spectra in Mexico
### Regional Spectrum According to MDOC Sismo CFE 2015
Figure 4: Soil microzoning using thickness, \(H_{s}\), and average shear wave velocity, \(v_{s}\), adapted with permission from [PERSON], Comisión Federal de Electricidad Manual de Diséno de Obras Civiles; published by Comision Federal de Electricidad (CFE), 2015 [1]. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Layer Data to Combine 1** & **Thickness \(H_{s}\) to Obtain Velocity \(v_{s}\) to Obtain** \\ \hline
1 & \(H_{s}\)\(y\)\(v_{s}\) & \(H_{s}\) & \(v_{s}\) \\
2 & \(T_{s}\)\(y\)\(v_{s}\) & \(H_{s}=v_{s}T_{s}/4\) & \(v_{s}\) \\
3 & \(H_{s}\)\(y\)\(T_{s}\) & \(H_{s}\) & \(v_{s}=4H_{s}/T_{s}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Combination values of the homogeneous layer for soil microzoning, adapted with permission from [PERSON], Comisión Federal de Electricidad Manual de Diséno de Obras Civiles; published by Comision Federal de Electricidad (CFE), 2015 [1]. With the seismic zone and soil dynamic factors available, the seismic elastic design spectrum can be obtained. First, it is required to obtain the maximum accelerations, \(a_{0}\) and \(c\), by applying the following equations:  
\[a_{0}=F_{sit}a_{0r} \tag{11}\]  
\[c=F_{res}a_{0} \tag{12}\]  
where \(a_{0}\) is the maximum acceleration on the ground and \(c\) is the maximum spectral acceleration. Due to the empirical origin of the dynamic amplification factors shown in Table 6, a few restrictions related to the spectral accelerations values are added to avoid any oversizing result, using the limits shown in Table 7.  
The seismic elastic design spectrum is obtained with the following equations:  
\[S_{a}\frac{\text{cm}}{\text{s}^{2}}= \begin{cases}a_{0}+(c\beta-a_{0})\frac{T_{c }}{T_{a}}&for\ T_{e}<T_{a}\\ c\beta&for\ T_{a}\leq T_{e}<T_{b}\\ c\beta\left(\frac{T_{b}}{T_{c}}\right)^{r}&for\ T_{b}\leq T_{e}<T_{c}\\ c\beta\left(\frac{T_{b}}{T_{c}}\right)^{r}p_{c}\left(\frac{T_{c}}{T_{c}}\right) ^{2}&for\ T_{e}\geq T_{c}\end{cases}  \tag{13}\]  
where \(S_{a}\) is the spectral acceleration, \(\beta\) is the structural damping factor, \(r\) is a spectral decay parameter, \(T\) is the vibration period of the structure, and \(T_{a}\), \(T_{b}\), and \(T_{c}\) are the limits of the spectral acceleration graph. The results are shown in Figure 5. \begin{table}
\begin{tabular}{c c c c} \hline
**Zone** & **Soil Type I 1** & **Soil Type II** & **Soil Type III** \\ \hline A & \(F_{sit}=1.0\) & \(F_{sit}=2.6\) & \(F_{sit}=3.0\) \\ \(F_{res}=PRODISIS\) & \(F_{res}=3.8\) & \(F_{res}=4.2\) \\ B & \(F_{sit}=1.0\) & \(F_{sit}=2.6-0.2\left(\frac{4a_{0}-50}{50}\right)\) & \(F_{sit}=3.0-0.3\left(\frac{4a_{0}-50}{50}\right)\) \\ \(F_{res}=PRODISIS\) & \(F_{res}=3.8-0.2\left(\frac{4a_{0}-50}{50}\right)\) & \(F_{res}=4.2-0.3\left(\frac{4a_{0}-50}{50}\right)\) \\ C & \(F_{sit}=1.0\) & \(F_{sit}=2.4-0.3\left(\frac{4a_{0}-100}{100}\right)\) & \(F_{sit}=2.7-0.4\left(\frac{a_{0}-100}{100}\right)\) \\ \(F_{res}=PRODISIS\) & \(F_{res}=3.6-0.2\left(\frac{4a_{0}-100}{100}\right)\) & \(F_{res}=3.9-0.3\left(\frac{4a_{0}-100}{100}\right)\) \\ D & \(F_{sit}=1.0\) & \(F_{sit}=2.1-0.5\left(\frac{4a_{0}-200}{200}\right)\) & \(F_{sit}=2.3-0.6\left(\frac{4a_{0}-200}{200}\right)\) \\ \(F_{res}=PRODISIS\) & \(F_{res}=3.4-0.5\left(\frac{4a_{0}-200}{200}\right)\) & \(F_{res}=3.6-0.6\left(\frac{4a_{0}-200}{200}\right)\) \\ \hline \end{tabular}  
* The response factor, \(F_{res}\), for soil type I soil requires the geographic location using the seismic hazard map. Therefore, PRODISIS is required to obtain the value [4].  
\end{table} 
Table 6: \(F_{sit}\) and \(F_{res}\) factors for dynamic amplification factors at the site, reproduced with permission from [PERSON], Comision Federal de Electricidad Manual de Diseno de Obras Civiles; published by Comision Federal de Electricidad (CFE), 2015 [1].