## 3 SSI Modeling of Selected Buildings
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Shear Wave Velocities (Mean \(V_{s30}\))** & **Stiff** & **Moderate** & **Soft** \\ \hline Horizontal displacement (x) (m/s) & 720 & 285 & 180 \\ Horizontal displacement (y) (m/s) & 900 & 360 & 224 \\ Vertical displacement (z) (m/s) & 720 & 285 & 180 \\ Rotation about \(x\)-axis (xx) (m/s) & 1020 & 405 & 255 \\ Rotation about \(y\)-axis (yy) (m/s) & 1080 & 430 & 270 \\ Torsion about \(z\)-axis (zz) (m/s) & 1020 & 405 & 255 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mean shear wave velocities for considered soils types. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Stiff** & **Moderate** & **Soft** \\ \hline Mass density (\(\rho_{s}\)) & 22 & 20 & 18 \\ (\(\text{K}\)/m\({}^{3}\)) & 0.25 & 0.33 & 0.42 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Mass densities (\(\rho_{s}\)) and [PERSON]’s ratio of soil types. Figure 2: Typical lateral deformation of single degree of freedom system (SDOF) model according to (a) fixed base, (b) SSI system. Modeling of buildings according to the substructure method is essentially performed in two parts. In the first part, the motion of the massless foundation system is calculated without the presence of the superstructure. The second part consists of the application of motion to the structure where soil properties are simulated by a group of equivalent springs and dashpots [21; 22]. Equivalent soil properties of the foundation-soil interface are represented with springs and dashpots. The stiffness and damping of these springs and dashpots were calculated according to equations proposed by [PERSON] and [PERSON] [18]. In the present study, stiffer springs (\(k_{2}\), \(k_{3}\), \(k_{4}\)), which were calculated as a function of interior springs (\(k_{x}\)\({}^{\text{i}}\)) and the stiffness intensity modifiers (\(\text{R}_{\text{k,xx}}\)-\(\text{R}_{\text{k,yy}}\)) along the foundation edges (\(\text{R}_{\text{e}}\)L and \(\text{R}_{\text{e}}\)B), are used to prevent the underestimation of rotational stiffness (Figure 3). Consequently, the total stiffness of the foundation was determined. The symbols \"L\" and \"B\" represent the dimension of the foundation system and they are taken as half of the actual dimensions. \(\text{R}_{\text{e}}\) is the foundation end length ratio which can be taken between 0.3 and 0.5. Damping intensities of dashpots (Figure 3, \(\text{c}_{x}\)\({}^{\text{i}}\)) along the foundation edges were reduced by a damping intensity modifier (\(\text{R}_{\text{c,xx}}\)-\(\text{R}_{\text{c,yy}}\)) to not overestimate the foundation rotational damping (\(c_{2}\), \(c_{3}\), \(c_{4}\)). A schematic illustration of link and dashpot assignments to the foundation-soil interface is shown in Figure 3  
Foundation systems of the selected buildings have shallow foundation, and the foundation dimensions were determined according to design projects of the buildings. Both of the base slab averaging and the embedment effects were taken into account by considering soil shear wave velocities, embedment depth of the foundation, and the natural vibration frequencies of structures. The results  
Figure 3: Calculation and representation of stiffness and damping expressions for the foundation springs added to the foundation systems. indicated that the ground motion reductions (H\({}_{\text{u}}\)) caused by these effects can be neglected for selected buildings since they can be considered as low- and mid-rise buildings and they have no basement. The values calculated for the embedment effects (\(\omega\)D/V\({}_{\text{s}}\)) for all of buildings are lower than 0.006 (Figure 4, left). As a result, the calculated ground motion reduction factors (H\({}_{\text{u}}\)) are very close to 1 which means that a significant reduction is not needed. A similar situation is also valid for the base slab averaging effects (Figure 4, right). Consequently, obtained results indicate that selected ground motions can directly be used for the nonlinear time history analysis of buildings according to both cases (Figure 4).