## 3 Numerical Models  
In this section, two series of numerical models are considered: with (1) fixed-base and (2) SSI. Several case studies are herein performed by considering several values of the height of the soil layer and the shear wave velocity. In particular, the case studies consider a 2-floor shear-type building (representing a low-rise residential structure) founded on shallow foundations. Sixteen finite-element models were performed in order to assess the mutual interaction between the soil and the structure. The first one consists of the fixed-based (SSI neglected) structure while the other 15 cases are based on soil layers with different characteristics: different depths (10, 20 and 30 m) and stiffness (100 m/s, 200 m/s, 300 m/s, 400 m/s and 500 m/s). The 3D numerical models (soil + foundation + structure, Figures 2-4) were performed with OpenSees PL, [25; 26]. In order to reasonably maintain the computational costs some assumptions were considered. First of all, (1) the foundation slab was modelled as rigid adopting an equivalent linear elastic material with large stiffness, (2) deformations inside the foundations, soil and intermediate nodes along the structure were calculated but not memorized. (3) The floors were considered rigid by introducing rigid diaphragms instead of deformable beams. Figure 2: Three-dimensional (3D) mesh (Model 1). **FIXED model**: the nodes at the base of the structure were restrained in all six directions (longitudinal, transversal and vertical) to fix the structure at the base by neglecting the effects of SSI. This model was used to calculate the periods (0.13 s and 0.04 s) and verify that the first one is representing the dynamic behaviour of the structure (participation mass: 89%). The building is regular in plan (30 m \(\times\) 20 m rectangular base) and regular along its height (3.3 m, each floor, total height: 6.6 m, above ground level). The seismic-resistant system consists of RC columns (three in longitudinal direction, 6 m spaced, and three in transversal direction, 6 m spaced) that were modelled with elastic beam column elements  
Figure 4: Three-dimensional (3D) mesh (Model 3). Figure 3: Three-dimensional (3D) mesh (Model 2). (11 elements for each floor, parameters: Young's modulus 3.5 \(\times\) 10\({}^{7}\) kN/m\({}^{2}\) and shear modulus: 1.73 \(\times\) 10\({}^{7}\) kN/m\({}^{2}\)). The seismic masses were concentrated in correspondence with the connections between the beams and the columns and the floor were tied together to maintain a shear-type behavior, by following the approach already proposed in [27, 28]. **SSI models**: the 15 numerical models were built up to realistically represent non-linear behaviours of the soil consisting of amplitude-dependent amplifications (or de-amplifications), plastic accumulations of ground deformation, and permanent movements (rotations, longitudinal displacements) of the entire system (soil + foundation + structure). Advanced material models were adopted in order to represent the high non-linear behaviour of a weak ground soil. The foundation consists of a 0.50 m-thick RC rigid slab (28.4 \(\times\) 34.4 m) designed by calculating the eccentricity loads (the ratio between the overturning bending moment and the vertical force) in the most detrimental way (minimum vertical loads and maximum bending moments) at foundation level. The bearing pressure of the building is: 195 kPa. EqualDOF [26] were used to link together the nodes at the base of the columns and the ones belonging to the foundation in order to model the slab as a rigid element. The nodes of the interface between the column and the foundation were connected with a horizontal rigid beam-column link to model the continuity of the link slab-column. Two solutions were applied to allow continuity of deformation between the foundation (equivalent concrete material) and the soil.