## 4 Detrimental Effects of SSI
### Case Studies  
Figure 6a,b are the dynamic responses of the Hollywood Storage Building for an assumed SSI system and an assumed fixed-base structure, respectively. The excitation, as shown at the top of the figure, is the acceleration record obtained at the ground floor of the building in the NS direction (a component of the horizontal direction) during the Northridge earthquake in 1994, and it is normalized to an acceleration of a peak value of 0.1 g. The record can be found on the USGS website [[https://strongmotioncenter.org/](https://strongmotioncenter.org/)]([https://strongmotioncenter.org/](https://strongmotioncenter.org/)) (accessed on 28 January 2019). It is also assumed that the excitation is a train of SV waves coming from the bedrock with an incident angle \(\theta\) = 90\({}^{\circ}\). The acceleration responses of both the SSI system and the fixed-base structure are shown in the time history \(\tilde{u}_{\rm b}(t)\) (left column) and the response spectrum of the time history \(R\)S(\(\tilde{u}_{\rm b}\)) (right column). The responses in the time history in Figure 6 were gathered according to the following steps: First, the time history of an earthquake acceleration was converted into the frequency domain at 4096 + 1 equally spaced points from 0 to 50 Hz using Fourier transformation to acquire the spectrum of earthquake acceleration. The corresponding values on the spectrum of the structure response \(u_{\rm b}\) were calculated at these frequency points as well. Thereafter, the spectrum of the structure response was multiplied by the spectrum of the earthquake acceleration at the corresponding frequency points to produce the amplified spectrum of the structure. Finally, the response in the frequency domain was converted to the time domain using inverse fast Fourier transformation. In the time history graph (left column), for the time window 0-40 s, the value denoted by an asterisk \"*\" is the maximum value during the time history, and the value presented in the bottom right corner is the mean value. In the response spectrum graph, the value denoted by an asterisk shows the same. It can be noticed that the maximum value of the SSI system (0.33 g) is 57% larger than that of the fixed-base oscillator (0.21 g), and the mean value of the SSI system (4.33 \(\times\) 10\({}^{-2}\) g) is 48% larger than that of the fixed-base oscillator (2.92 \(\times\) 10\({}^{-2}\) g). Also, the maximum value in the response spectrum of the SSI system (0.96 g) is 25% larger than that of the fixed-base oscillator (0.80 g). We list these three values (the maximum and the mean values in the time history, and the maximum value in the response spectrum) for the four buildings to 42 earthquake events in Table 3 (XOZ plane) and Table 4 (YOZ plane) for both the SSI system and the fixed-base oscillator. The excitations are the acceleration records from the ground floor of each building for each earthquake event ([[https://strongmotioncenter.org/](https://strongmotioncenter.org/)]([https://strongmotioncenter.org/](https://strongmotioncenter.org/)), accessed on 28 January 2019). \begin{table}
\begin{tabular}{c c c c c} \hline  & **S-Wave Velocity** & **Thickness** & **Mass Density** & **Damping Ratio** \\  & **(m/s)** & **(m)** & **(kg/m\({}^{3}\))** & **Damping Ratio** \\ \hline
6 & 762.0 & 16.16 & 1846.9 & 2\% \\ \hline Bedrock & 944.8 & \(\infty\) & 1846.9 & 2\% \\ \hline  & \multicolumn{4}{c}{Sherman Oaks Building} \\ \hline
1 & 160 & 2.7 & 2040.8 & 2\% \\ \hline
2 & 205 & 6.7 & 2040.8 & 2\% \\ \hline
3 & 260 & 13.7 & 2040.8 & 2\% \\ \hline
4 & 330 & 23.8 & 2040.8 & 2\% \\ \hline Bedrock & 514 & \(\infty\) & 2040.8 & 2\% \\ \hline  & \multicolumn{4}{c}{Walnut Creek Building} \\ \hline
1 & 336 & 5.0 & 1846.9 & 2\% \\ \hline Bedrock & 434 & \(\infty\) & 1846.9 & 2\% \\ \hline \end{tabular}
\end{table} 
Table 2: \(Cont.\)