## 3 Seismic Response Spectra in Mexico
### Regional Spectrum According to MDOC Sismo CFE 2015
\begin{table}
\begin{tabular}{c c c c} \hline
**Acceleration** & **Soil Type I** & **Soil Type II** & **Soil Type III** \\ \hline \(a_{0}\)\(\left(\text{cm}/\text{s}^{2}\right)\) & \(32\leq a_{0}\leq 490\) & \(80\leq a_{0}\leq 690\) & \(94\leq a_{0}\leq 752\) \\ \(c\)\(\left(\text{cm}/\text{s}^{2}\right)\) & \(80\leq c\leq 1225\) & \(320\leq c\leq 2000\) & \(390\leq c\leq 2256\) \\ \hline \end{tabular}  
* The seismic elastic design spectrum is obtained with the following equations: \[S_{a}\frac{\text{cm}}{\text{s}^{2}}=\begin{cases}a_{0}+(c\beta-a_{0})\frac{T_{c }}{T_{a}}&for\ T_{e}<T_{a}\\ c\beta&for\ T_{a}\leq T_{e}<T_{b}\\ c\beta\left(\frac{T_{b}}{T_{c}}\right)^{r}&for\ T_{b}\leq T_{e}<T_{c} \\ c\beta\left(\frac{T_{b}}{T_{c}}\right)^{r}p_{c}\left(\frac{T_{c}}{T_{c}} \right)^{2}&for\ T_{e}\geq T_{c}\end{cases}\] (14) 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.  
\end{table} 
Table 7: Limits for the spectral accelerations \(a_{0}\) and \(c\), reproduced with permission from [PERSON], Comision Federal de Electricidad Manual de Diseno de Obras Civiles; published by Comision Federal de Electricidad (CFE), 2015 [1]. The damping factor \(\beta\) is obtained with the following equation:  
\[\beta= \begin{cases}\left(\frac{0.05}{\xi_{e}}\right)^{0.45}&for\ \ T_{e}<T_{c}\\ \left(\frac{0.05}{\xi_{e}}\right)^{0.45\left(\frac{T_{c}}{T_{e}}\right)}&for\ \ T_{e}\geq T_{c}\end{cases}  \tag{14}\]  
where \(\xi_{e}\) is the structural damping ratio. According to Equation (14), if \(\xi_{e}=0.05\) is used, then the calculated damping factor is \(\beta=1\), proving that the seismic elastic design spectrum has a default damping radio value of 5%. The \(p_{c}\) factor is obtained with the following equation:  
\[p_{c}=k+(1-k)\left(\frac{T_{c}}{T_{e}}\right)^{2} \tag{15}\]  
where \(k\) is a spectral decay parameter for acceleration values after period \(T_{c}\). The parameters \(T_{a}\), \(T_{b}\), \(T_{c}\), \(k\), and \(r\) in previous equations are obtained by using the values shown in Table 8, according to the seismic zone and the soil type. The seismic elastic design spectrum using the regional spectrum procedure can be modified by applying different return periods, \(T_{r}\), and structural damping ratios, \(\xi_{e}\). PRODISIS allows us to apply a specific return period, \(T_{r}\), providing the return period reference  
Figure 5: Seismic elastic design spectrum for structures of importance B1 and A2. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Zone** & **Soil Type** & \(T_{a}\) **(s)** & \(T_{b}\) **(s)** & \(T_{c}\) **(s)** & \(k\) & \(r\) \\ \hline  & I & 0.1 & 0.6 & 2.0 & 1.5 & 0.5 \\ A & II & 0.2 & 1.4 & 2.0 & 1.0 & 2/3 \\  & III & 0.3 & 2.0 & 2.0 & 0.5 & 1.0 \\  & I & 0.1 & 0.6 & 2.0 & 1.5 & 0.5 \\ B & II & 0.2 & 1.4 & 2.0 & 1.0 & 2/3 \\  & III & 0.3 & 2.0 & 2.0 & 0.5 & 1.0 \\  & I & 0.1 & 0.6 & 2.0 & 1.5 & 0.5 \\ C & II & 0.2 & 1.4 & 2.0 & 1.0 & 2/3 \\  & III & 0.2 & 2.0 & 2.0 & 0.5 & 1.0 \\  & I & 0.1 & 0.6 & 2.0 & 1.5 & 0.5 \\ D & II & 0.1 & 1.4 & 2.0 & 1.0 & 2/3 \\  & III & 0.1 & 2.0 & 2.0 & 0.5 & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Values of periods and exponents for the elastic design spectrum, reproduced with permission from [PERSON], Comision Federal de Electricidad Manual de Disénéo de Obras Civiles; published by Comision Federal de Electricidad (CFE), 2015 [1]. spectrum EPR, with values used as initial inputs to apply on the procedure described instead of the reference spectrum ER inputs. To obtain the seismic elastic design spectrum with vertical load, the spectral acceleration values are obtained using the following equation:  
\[S_{av}= \begin{cases}1.4\;F_{d}S_{a}&for\;T_{v}<0.05\;\text{s}\\ 1.4\;F_{d}S_{a}\Big{(}\frac{0.05}{T_{v}}\Big{)}^{2/3}&for\;T_{v}\geq 0.05\; \text{s}\end{cases}