## 4 Seismic Response Spectra in the USA
### Regional Spectrum According to ASCE/SEI 7-16  
To obtain the seismic elastic design spectrum, we must know the location of the site and the physical properties of the soil as initial inputs. With the geographical location of the site on the U.S. seismic hazard map, it is possible to obtain the spectral accelerations in rock for a risk-targeted maximum considered earthquake, \(MCE_{R}\), using the parameter \(S_{s}\) for a short structural period of \(T_{e}=0.2\) s and the parameter \(S_{1}\) for periods of \(T_{e}=1.0\) s.  
With the soil properties available, the site can be classified between classes A and F. If the soil properties are not available, ASCE/SEI 7-16 recommends applying class D, unless a study is required due to the nature of the structure or the soil physical conditions can be classified as site class E or F [2]. First, the spectral accelerations parameters, \(S_{S}\) and \(S_{1}\), related to an earthquake, \(MCE_{R}\), need to be modified by using the site class coefficients to obtain the modified response spectral acceleration parameter, \(S_{MS}\), for short structural periods of 0.2 s and \(S_{M1}\) for periods of 1.0 s, according to the following equations:  
\[S_{MS}=F_{a}S_{S} \tag{28}\]  
\[S_{M1}=F_{v}S_{1} \tag{29}\]  
where \(S_{S}\) and \(S_{1}\) are the spectral response acceleration parameters obtained from the \(MCE_{R}\) at short and long periods, \(F_{a}\) is the site coefficient for short periods, and \(F_{v}\) is the site coefficient for long periods. The coefficients \(F_{a}\) and \(F_{v}\) are obtained using different tables provided in chapter 11 of ASCE/SEI 7-16, depending on the site class and the mapped spectral acceleration parameters [2]. The design spectral acceleration parameters, \(S_{DS}\) for short periods and \(S_{D1}\) for long periods, are determined via the following equations:  
\[S_{DS}=\frac{2}{3}S_{MS} \tag{30}\]  
\[S_{D1}=\frac{2}{3}S_{M1} \tag{31}\]  
where the \(S_{MS}\) and \(S_{M1}\) values, multiplied by the factor of \(2/3\), indirectly modify the \(MCE_{R}\) from a return period of \(T_{r}=4975\) years (with a 1% probability of failure in 50 years) to an \(MCE\) with a return period of \(T_{r}=475\) years (with a 10% probability of failure in 50 years). To obtain the seismic elastic design spectrum with a 5% of structural damping ratio, the following equations are used:  
\[S_{a}(g)=\left\{ \begin{array}{cc}S_{DS}\Big{(}0.4+0.6\frac{T}{T_{0}}\Big{)}& 0<T<T_{0}\\ S_{DS}&T_{0}\leq T\leq T_{S}\\ \frac{S_{D1}}{T}&T_{S}<T\leq T_{L}\\ \frac{S_{D1}T_{L}}{T^{2}}&T>T_{L}\end{array} \right. \tag{32}\]Figure 16 shows the spectral acceleration according to the previous equations with the location of parameters \(S_{DS}\) and \(S_{D1}\). According to the graph, \(T\) is the structural period in seconds; \(T_{0}\) and \(T_{S}\) are the periods used as limits between the maximum acceleration value, obtained with the following equations:  
\[T_{0}=0.2\frac{S_{D1}}{D_{DS}} \tag{33}\]  
\[T_{S}=\frac{S_{D1}}{S_{DS}} \tag{34}\]  
\(T_{L}\) is the long transition period, which is assigned depending on the geographical location of the site. Since \(T_{L}\geq 4\) s, its application is related to structures with seismic isolators or damping systems, where the calculated structural periods usually have high values. PGA is the peak ground acceleration value at period \(T=0\) s. The seismic elastic design spectrum obtained with the previous equations corresponds to a \(MCE\) with a return period of \(T_{r}=475\) years. If a design spectrum for a \(MCE_{R}\) with a return period of \(T_{r}=4975\) years is required, the \(S_{a}(g)\) values in Equation (32) must be multiplied by 1.5. Additionally, to consider the importance factor, \(l_{e}\), all \(S_{a}(g)\) values must be multiplied by the factor provided according to the risk category. If a structural damping ratio value other than 5% is required, ASCE/SEI 41-17 presents the modified spectral accelerations, \(S_{a}(g)\), by applying the provided damping value according to the following equations [19]:  
\[S_{a}(g)=