## 3 Parametric Studies  
Several parametric studies have been herein considered by considering the three different DOF with that the entire system (structure + isolation + soil) has been described. The first DOF, representing the structural DOF was modeled with 10 buildings (B1-B10) with increasing stiffness and the same mass and thus different dynamic properties (frequency and fundamental period), in order to represent several conditions of structural deformability. Table 1 displays the structural stiffness, the structural mass, the frequency (\(\omega_{s}\)) and the fundamental period (Ts). It is worth noting that B1-B2 need to be considered representative of particularly deformable cases, to demonstrate that when the structure is deformable, the effects of SSI and BI are not significant. The second DOF represents the isolation system whose characteristics have been varied to study several isolation ratios (n), that is defined as the ratio between the period of the isolation system and the fundamental period of the structure. In particular, n is defined as:  
\[n=\frac{1}{\varepsilon}=\left( \begin{array}{c}\omega_{s}\\ \omega_{b}\end{array} \right)^{2}=\left( \begin{array}{c}T_{b}\\ \frac{T_{s}}{T_{s}}\end{array} \right)^{2} \tag{7}\]  
and realistic values of two, three and four have been assumed herein. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Structure** & **Stiffness (N/m)** & **Mass (kN)** & **Frequency (rad/s)** & **Fundamental Period (s)** \\ \hline B1 & 9.92\(\cdot 10^{6}\) & 2600 & 6.11 & 1.028 \\ B2 & 1.98\(\cdot 10^{7}\) & 2600 & 8.64 & 0.727 \\ B3 & 2.98\(\cdot 10^{7}\) & 2600 & 10.58 & 0.594 \\ B4 & 4.96\(\cdot 10^{7}\) & 2600 & 13.65 & 0.460 \\ B5 & 6.94\(\cdot 10^{7}\) & 2600 & 16.16 & 0.389 \\ B6 & 8.93\(\cdot 10^{7}\) & 2600 & 18.32 & 0.343 \\ B7 & 1.19\(\cdot 10^{8}\) & 2600 & 21.15 & 0.297 \\ B8 & 1.69\(\cdot 10^{8}\) & 2600 & 25.18 & 0.249 \\ B9 & 2.68\(\cdot 10^{8}\) & 2600 & 31.73 & 0.198 \\ B10 & 4.66\(\cdot 10^{8}\) & 2600 & 41.86 & 0.150 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structural properties. The third DOF consists of the soil layer and several values were considered to represent eight typologies of soils. The value of soil layer thickness (H) was taken at 30 m, to follow the Eurocode [31] reference depth at which the average shear wave velocity is calculated. Table 2 shows the properties of the considered soil layers: shear wave velocity (Vs), mass density, shear modulus, Poisson ratio and the typology of the soil, as defined in the Eurocode [31]. Table 3 shows the results of the parametric studies. The lines represent the different structures divided for the eight values of shear wave velocities, the color of the columns represents the isolation ratio (n = 2: red, n = 3: green, n = 4: blue). The columns with Tb contain the values of the isolated period (without SSI), the column with T1 contains the period of the system calculated with Equation (6) and the column \"ratio\" contains the ratio between the period of the system and the isolation period. Such values represent the period elongation due to the presence of the deformable soil. It is worth noting that the maximum elongation of the period occurs with n = 2. If the results are compared for the same n, soil deformability does not affect significantly the elongation for the stiffer soils (Vs bigger than 200 m/s). For Vs = 150 m/s and 130 m/s, ratios between the period of the system and the isolation period increases, with maximum values in correspondence with n = 2 and stiff structures (B9 and B10). The maximum ratio is 2.6061, for Vs = 150 m/s, n = 2 and B10. It is worth noting that for deformable structures (B1-B2) and rigid soil conditions (vs = 1000, 800, 700 m/s), the results are similar because SSI effects are not significant, as in [22]. These outcomes represent the cases where the effects of SSI may be neglected, and they may be considered a validation of the present methodology.