# Numerical Study of the Seismic Response of an Instrumented Building with Underground Stories
## 3 Numerical Model  
### Description of the Numerical Models  
The finite element model was created using the structural drawings of the structure [38], the soil mechanics report [39], and the state of completion at the time of instrumentation. The low-intensity of the seismic events (maximum recorded acceleration was 0.0112 g) allowed for a linear elastic model, considering the concrete elasticity modulus obtained from Equation (4), given in Article 8.5.1 of the ACI318S-08 [35]:\[E_{c}=4700\sqrt{f_{c}^{2}}\ (\text{MPa}) \tag{4}\]  
The discretization of the slab elements was made so that the mesh nodes were coincident with the ones of both columns and walls, using an average square size of 1 m. Vertical elements, such as columns and walls, were divided into three parts for each story. Even though the structure did not present visible cracks or damage after the Maule earthquake, the models incorporated a stiffness reduction using a factor \(I_{g}\) affecting [PERSON]'s modulus of concrete: \(E_{c}I_{g}\). This further made it possible to verify the consideration of cracked sections for a low-intensity earthquake analysis.  
As the building was uninhabited during the data collection period, only the deadweight load of the reinforced concrete was considered with a unit weight of \(w_{c}=24.5\ \text{kN/m}^{3}\).  
For comparison purposes, the foundation modeling options were:  
1. Embedded on the first floor, see Figure 2a;
2. Embedded at the lowest underground level, see Figure 2b;
3. Embedded at the lowest underground level, with horizontal ballast, see Figure 2c;
4. Foundation slab supported on vertical springs, with horizontal ballast, see Figure 2d.  
Each model that incorporated the soil-structure interaction considered the soil through equivalent springs, with stiffness constants given according to the ballast coefficient. The springs used were considered in two ways: (1) only in compression behavior and (2) compression and tension behavior. The latter, although formally incorrect, allowed an evaluation of the simplification of the model (because it requires only a linear-elastic analysis). Linear-elastic springs were used for all models, and the only nonlinearity  
Figure 2: Model of the building: (**a**) embedded at the base of the first story; (**b**) embedded at the lowest underground level and without lateral ballast; (**c**) embedded at the lowest underground level with horizontal ballast; (**d**) using horizontal and vertical ballast in the soil-structure interface.  
considered was the only compression behavior already described, trying to keep the model as simple as possible. As is described in the next section, the ground motion records used had very low intensities in terms of maximum accelerations, so the soil behavior is expected to remain in the linear-elastic range, making the assumptions of this paragraph suitable for analysis purposes.  
The soil ballast coefficients were obtained from the soil mechanics report [39], considering [PERSON]'s [37] formulas (Equations (1) and (2)), and the foundation width and length values B = 33.6 m and L = 44.8 m, which yielded a vertical soil ballast coefficient \(k_{v}\) = 12.43 kN/mm\({}^{3}\).  
Horizontal ballast coefficient \(k_{h}\), where necessary, was modeled in three ways: (1) as a constant horizontal ballast equivalent to one-half the value of the vertical ballast \(k_{v2}\); (2) as a linear distribution of horizontal ballast applied in the center of the vertical sections of the perimeter walls (using the approach given by Equation (3)); and (3), calculated using the method proposed by [PERSON] [40] and revised by [PERSON] and [PERSON] [41], which depends on the maximum horizontal acceleration considered. However, given that the horizontal ballast for each model--using the maximum acceleration of each seismic event--did not vary substantially (variance < 0.00X%), the average horizontal ballast from the 11 seismic events analyzed was used for all the models where necessary. The characteristics of the models analyzed in this work are summarized in Table 1.