## 4 Seismic Response Spectra in the USA
### Initial Inputs for the Elastic Design Spectrum According to ASCE/SEI 7-16  
To obtain the seismic elastic design spectrum, we need to have the importance of the structure, the spectral accelerations parameters, \(S_{S}\) and \(S_{1}\), according to the seismic hazard map related to an earthquake, \(MCE_{R}\), and the soil physical properties, used to perform the dynamic amplifications due to the site effects. Figure 15: Seismic hazard map for the USA according to the USGS [17]. The risk category classification applied to each structure depends of the use of the structure, where a probability of failure by \(MCE_{R}\) and a structural importance factor, \(I_{e}\), are assigned [2]. Risk category I is related to common structures with a probability of failure by \(MCE_{R}\) of 10% in 50 years and an importance factor of 1.0; risk category III is focused on the energy sector, such as hydroelectric and nuclear plants, where their failure can have negative economic impacts and lead to environmental catastrophes, using a probability of failure by \(MCE_{R}\) of 5% in 50 years and an importance factor of 1.25; risk category IV is related to the same structures of category III with a very high risk, using a probability of failure of 2.5% and an importance factor of 1.5; finally, risk category II is related to all other structures except of the previous risk categories, using the same probability of failure and importance factor assigned in category I. To consider the dynamic amplification of the ground due to the site effects, a physical soil study must be performed on the field using available methods, such as cross hole or the equivalent, to obtain the dynamic properties of the layers and assign a site class. As in Figure 11, previously shown as an example, the physical properties of each layer, \(i\), are required, such as the thickness, \(d_{i}\), the shear wave velocity, \(v_{si}\), the volumetric weights, \(\rho_{i}\), the damping ratio, \(\xi_{i}\), the number of blows of the standard penetration resistance, \(N_{i}\), and the shear stress, \(s_{ui}\). A minimum soil thickness \(H=30\) m is required to obtain the dynamic properties [2]. According to ASCE/SEI 7-16, the soil can be classified from site class A to F, where class A is very stiff rock and class F very soft soil. The variables used to classify the soil are the average shear wave velocity \(\overline{v}_{s}\), the average standard penetration resistance \(\overline{N}\) and the average undrained shear stress \(\overline{s}_{u}\). Site class A and B are related to rock soils where \(\overline{v}_{s}\) is used; site class C and D are related to dense and stiff soils, using one of the different values of \(\overline{v}_{s}\), \(\overline{N}\) and \(\overline{s}_{u}\); site class E and F are related to soft soils where different mechanical and dynamic properties are required. The values of \(\overline{v}_{s}\), \(\overline{N}\) and \(\overline{s}_{u}\) required to classify the site are located in chapter 20 of ASCE/SEI 7-16 [2]. Therefore, for a layered soil, as shown in Figure 11, the average shear wave velocity, \(v_{s}\), is obtained according to the following equation:  
\[\overline{v}_{s}=\frac{\sum_{i=1}^{n}d_{i}}{\sum_{i=1}^{n}\frac{d_{i}}{ \overline{v}_{si}}} \tag{22}\]  
where subscript \(i\) refers to a layer, \(d_{i}\) is the thickness of layer \(i\), \(v_{si}\) is the shear wave velocity of layer \(i\), and \(n\) is the number of layers. To obtain the average standard penetration resistance for cohesionless soils, \(\overline{N}\), the following equation is used:  
\[\overline{N}=\frac{\sum_{i=1}^{n}d_{i}}{\sum_{i=1}^{n}\frac{d_{i}}{\overline{ N}_{i}}} \tag{23}\]  
where \(N_{i}\) is the standard penetration resistance for layer \(i\); the values \(N_{i}\) and \(d_{i}\) are related to cohesionless soil, cohesive soil, and rock layers.