## 2 Case Studies
### Structural Models  
The structure (Figures 1 and 2) represents a low-rise stiff two-story (floor height: 3.2 m, total height: 6.4 m) reinforced concrete building with a 4 \(\times\) 3 columns scheme (4 along the \(x\)-direction and 8 m spaced and 3 along \(y\)-direction and 6 m spaced, cross Area: 0.16 m\({}^{2}\)), designed in accordance with Eurocode 2 and 8 [33; 34]. The hypothesis at the base of the structural model is that because of the presence of the isolation, the building may be assumed to be capacity designed being able to respond in the elastic range [35]. Therefore, the structure has been modelled with Elastic Beam Column elements (characteristics in Table 2). The masses calculated with the seismic combination following [34] were concentrated at each floor and included the structural components (such as beams, columns, slabs, balconies, stairs). The floors were modelled as rigid by applying the RigidDiaphragm [36], and thus the hypothesis of shear type behaviour was assumed (plan and vertical regularity has been assumed). A superficial foundation (26 m \(\times\) 14 m \(\times\) 0.5 m, 15,264 nodes and 11,076 elements, Figure 2) has been simulated with a rigid concrete slab that may be considered typical for low-rise buildings with limited bearing pressures (180 kPa) at the base level, (see [37]). The slab was modeled with the Pressure Independent Multiyield (PIMY, [36; 38], properties in Table 3) and was designed by considering the maximum eccentricity (the ratio between the overturning bending moment and the vertical axial forces) calculated with the minimum vertical loads (gravity and seismic loads) and maximum bending moments (due to the seismic loads). The PIMY is based on the [PERSON] multi-surface kinematic plasticity model with an associate non-linear deviatoric part of the flow rule that is independent from the volumetric part (non-associate and linear-elastic). The backbone curves are defined by hyperbolic relations and described by two parameters (the strain shear modulus and ultimate shear strength). The interface between the structure and the foundation was modeled in order to represent the rocking component (rotations and over turning moments) that is fundamental to reproduce the SSI effects. The nodes of the last 50 cm of the columns were connected with horizontal rigid beam-column links (see [36]) to model the continuity of the link slab-column. EqualDOF constraint (see [36]) were used to connect the points (one belonging to the rigid link and the second to the foundation) in order to have the same displacements between the structure and the slab nodes. Zerolength elements (see [36]) were used to model the interface between the surrounding soil (named S soil) and the foundation by reproducing the uplifts and gapping mechanisms and non-linear effects (i.e., tilts and settlements) due to SSI. This assumption guaranteed to maintain the dynamic equilibrium at each step and thus resolving the convergence problems. The soil around the foundation (soil S) represents the infill weak soil and has been modeled with the Pressure Independent Multiyield (PIMY, [36; 38]). The considered properties are shown in Table 3. \begin{table}
\begin{tabular}{c c} \hline \hline
**Property** & \\ \hline Mass density & 24.0 kN/m\({}^{3}\) \\ Young Modulus & \(3.5\times 10^{7}\) kN/m\({}^{2}\) \\ Shear Modulus & \(1.73\times 10^{7}\) kN/m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Structural properties. Figure 1: 3D view of the numerical model (Model B1, B2 and B3). Soil S: Infill soil (Yellow); Soil F: Foundation soil (Blue). Figure 2: Schematic view of the structure and the foundation (Model B1, B2 and B3).