## 3 Numerical Modelling of the SSI System  
In the scope of the research at hand, the model tested within the TRISEE project was modelled using the finite element analysis software SAP2000 v21.0.2 [36]. The physical  
Figure 1: Experimental setup (m) [12]. Figure 2: Load application Phase II—low density sand (m). model of the structure was modelled in 3D using shell finite elements representing the foundation slab and frame elements representing the column. The loading curve recorded during the experiment was applied at the top of the column as a horizontal cyclic loading. Multilinear plastic links were used to simulate the soil compliance in the vertical direction. The foundation model was supported by 25 identical link elements. Vertical stiffness of the soil was calculated according to [PERSON] [37] and [PERSON] et al. [38] which are presented in Equations (1)-(3). According to this proposal, the axial stiffness of the spring is a function of the soil's shear modulus (\(G\)), Poisson ratio (\(\
u\)), and geometry of the foundation strip (\(L\),\(B\)) where L is half of the foundation strip length and \(B\) is half of the foundation strip width. \[k_{x}=k_{y}-\frac{0.2\cdot G\cdot L}{0.75-\
u}\cdot\left(1-\frac{B}{L}\right) \tag{1}\]  
\[k_{y}=\frac{2\cdot G\cdot L}{2-\
u}\cdot\left[2+2.5\cdot\left(\frac{B}{L} \right)^{0.85}\right] \tag{2}\]  
\[k_{z}=\frac{2\cdot G\cdot L}{1-\
u}\cdot\left[0.73+1.54\cdot\left(\frac{B}{L} \right)^{0.75}\right] \tag{3}\]  
Multilinear links require information regarding the force-deformation backbone curve as well as the hysteresis type for the soil. The backbone curve was determined according to [PERSON] [39] briefly presented in Figure 3 and Equations (4)-(7). A step-by-step detailed procedure for the backbone curve calculation in sand can be found in [40]. The damping ratio of the soil could have been included into the numerical model within the vertical link elements, but it was neglected, since it did not have a significant impact on results, such as settlement or tilting of the foundation. In the particular case, the inclusion of damping did not to have a significant impact on overall results for the soil-structure system since it was loaded with relatively slow cycles of loading. \[p_{ult}=\overline{A_{s}}\cdot p_{s} \tag{4}\]  
\[p_{m}=\overline{B_{s}}\cdot p_{s} \tag{5}\]  
\[p=k_{py}\cdot z\cdot y_{k} \tag{6}\]  
\[p_{st}=\gamma\cdot z\cdot\left[\frac{k_{0}\cdot z\ tan\varphi\ sin\beta}{\tan (\beta-\varphi)\cdot cos\alpha}+\frac{tan\beta}{\tan(\beta-\varphi)}\cdot(b+z \cdot tan\beta\cdot tan\alpha)+k_{0}\cdot z\cdot tan\beta\cdot(tan\varphi \cdot sin\beta-tan\alpha)-k_{a}\cdot b\right] \tag{7}\]  
To simulate the hysteretic behavior of the soil-foundation system the Takeda model [41] was chosen from the available hysteretic models in the software. This model was primarily developed for the purpose of modelling the response analysis of reinforced concrete structures, yet due to similarities in the shape of the curve to the response of the sand, it can be also used for soil modelling. In recent years it has been successfully implemented to model the nonlinear response of soils [42]. The compression part of the hysteresis loop is  
Figure 3: Backbone curve determination. modelled by force-deformation (p-y) curve (Figure 2:0) while the tension part is neglected. It is important to stress out that use of simplified modelling approaches leads to certain limitations, therefore, initial imperfections of the soil-structure system are not included in the numerical model as well as saturation of the foundation soil. Gap elements and horizontal linear springs are assigned along the edges of the model. As the foundation model is symmetrical, both, the stiffness in x and y direction have the same properties (Figure 4). The model was loaded with three different time-dependent functions. Two functions simulated gravitational loading in the vertical direction and one function simulated seismic loading in the horizontal direction.