# Numerical Study of the Seismic Response of an Instrumented Building with Underground Stories
## 3 Numerical Model
### Ground Motion Records  
The records used as ground motion input for the time-history analyses were taken from the accelerometers located at the foundation slab. As the foundation-structure interaction was considered to be registered by the accelerometers, and the foundation slab was not included in the models embedded at the base level (level-3). As a result, models that included vertical springs could not be compared to site records.  
From the 16 installed accelerometers, records from units 1, 2, 3, and 4 were used as input for the numerical models. The records from accelerometers 1 and 2 were used as inputs in the X and Y directions of the building, respectively. On the other hand, the average of the records from accelerometers 3 and 4 (vertical direction) was used as input in the Z-direction.  
All the seismic records were subjected to the same centering and filtering procedure to minimize high and low-frequency noise errors before comparison and/or use as ground motion input. As a first step, each record was centered to zero for ordinate axes with a tolerance of \(10^{-5}\) g, to then use a band-pass filter via Fourier analysis to reduce the low and high frequencies that distort the signal. Equation (9) was used for this last step, with \(F_{min}=0.25\) Hz and \(F_{max}=25\) Hz, typical values used in record filters.  
Figure 4: Normalized seismic shear coefficient for granular soils.  
Figure 3: Horizontal spring model for underground structures.  
\[H(f)=\frac{1}{\sqrt{1+\left(\frac{f^{2}-F_{min}\cdot F_{max}}{f\cdot(F_{max}-F_{ min})}\right)^{2}}}. \tag{9}\]  
Rocking is the deformation of the foundation plane to the horizontal due to differential deformations of the soil or the foundation itself; while this factor is traditionally assumed to be very small, this work evaluated it in some models. Cases of rocking or rolling as the input used the average Z from vertical accelerometers 3 and 4, divided by the X and Y arms to the center of gravity (considering the foundation slab dimensions), to introduce rotation in the corresponding X and Y directions.  
Each model was subjected to a linear dynamic modal analysis considering every seismic event. This used the first 120 modes, which included more than 90% of the translational mass of the building. All models were assumed to have damping of 5% of the critical damping.  
For each analysis, inter-story shears and drifts were obtained and used as representative of the net internal forces and deformations for the comparison between the model results and the empirical measurements. Notwithstanding, models used to compare inter-story shears and drifts did not include cracking, rocking, or vertical ballast. This lattermost case made use of models, including horizontal ballast in the three forms presented in this work, as well as with or without soil-spring traction.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & **Embedded** & **Embedded** & **Vertical** & **Constant** & **Linear** & **Kuesel** & **Ballast in** & **Collression** & **Packing** \\  & **1 st Floor** & **Foundation** & **Ballast** & **Horizontal** & **Ballast** & **Ballast** & **Ballast** & **Ballast** & **Only** & **Rocking** & **Factor** \\ \hline
1 & Yes & & & & & & & & & Yes \\  
\hline \hline \end{tabular}
\end{table}
Table 1: Models analyzed in this study.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Event** & \begin{tabular}{c} **Date and Time** \\ **(CLI)** \\ \end{tabular} & \begin{tabular}{c} **Magnitude** \\ **(Mw or Mb)** \\ \end{tabular} & \begin{tabular}{c} **Mercalli Intensity in** \\ _Vida dd Mar_ \\ \end{tabular} & \begin{tabular}{c} **Hypocenter Depth (km)** \\ \end{tabular} & \
\begin{tabular}{c} **Epicenter Distance from** \\ _Vida del mar (Gam)_ \\ \end{tabular} &
\begin{tabular}{c} **Maximum Input Acceleration** \\ **(g)** \\ \end{tabular} \\ \hline
5 &
\begin{tabular}{c} 04/04/2010 \\ 2323:212 \\ 05/01/20