## 2 Modelling of Soil-Structure Interaction
### Introduction to Structure-Soil-Structure Interaction  
[PERSON] describes the structure-soil-structure interaction (SSSI) effect at adjacent piles using the resonating mass between the piles and the ratio of the shear wavelength \(\lambda_{S}\) to the pile spacing \(S_{S}\)[19]. Resonant masses are well described for large Poisson's ratios (\(\
u>1/3\)), particularly in the context of single foundations, as explored in [19; 30]. Here, resonant masses continuously reduce the real part of the impedance function \(k_{z}\) with increasing frequency since the resonant masses work/move contrary to the stiffness. In the context of group foundations, this decay is interrupted by the continuously increasing dominance of the shear wave (\(\lambda_{S}\to S_{S}\)) between the piles. Upon reaching a limit criterion \(f_{G}=V_{S}\) / 3\(S_{S}\), the soil oscillates increasingly in antiphase, leading to a significant gain in stiffness. The maximum stiffness is achieved when both piles oscillate exactly out of phase (\(f=V_{S}\) / 2\(S_{S}\) ). An essential difference in the pile group effect compared to portal frame bridge foundations lies in the relatively small diameter of the piles compared to their spacing. Nevertheless, similar relationships described for pile foundations can be observed for the twin foundations of portal frame bridges.  
Further investigations are performed using the hybrid models described in [12]. Figure 3 presents the impedance function (\(k_{z}\), \(c_{z}\)) for both single and twin foundations (abutments), considering spans of \(L=9\) m and \(L=21\) m, respectively. The similarity in characteristics between the impedance function of single foundations and the well established functions associated with rigid, massless single foundations on a flat surface can be observed (Figure 1, comp. [17, 19, 20, 21]).  
Negative values of \(k_{z}\) are due to a phase shift in the ground reaction force, similar to the transition from a stiffness-dominated system to an inertia-dominated system. In this context, the stiffness decreases significantly, allowing the resonant mass, which opposes the stiffness, to become predominant. The 180\({}^{\circ}\) vectorial disparity results in a negative real solution for the impedance function. Detailed representations of the single-mass oscillator and its vectorial equilibrium of forces can be found in the existing literature [31].  
The use of a modified limit criterion \(a_{G,i}\), inspired by [PERSON][19], effectively highlights the group effect (SSSI), as shown in Figure 3. This criterion is derived from the previously discussed effects associated with achieving the maximum stiffness.  
\[a_{G,i} = \frac{2\ \pi\ r_{0}}{L_{l}}*(i-0.5)\text{with}i\in\mathbb{N}\] (13) with \[L_{i}=\text{clear width between the foundation bodies, here, }L_{l}=L-4\ \text{m}\]  
In the region of the \(i\)th limit criterion \(a_{G,i}\), the effect of the resonant mass becomes apparent, causing the impedance function of the twin foundation to oscillate around the solution of a single foundation. The pronounced nonlinearity of the impedance function, which is significantly influenced by wave interference, underlines the complexity inherent in this seemingly simple matter of the (S)SSI.  
When Equation (4) is considered in conjunction with the distinctly nonlinear impedance function for twin foundations (shown in Figure 3), it becomes obvious that the SSSI has a significant effect on the resulting radiation damping of the structure, particularly in the context of a portal frame bridge. This underlines the need for straightforward modelling approaches considering application limits when simulating dynamic soil relationships.  
Simplified approaches to implementing the SSSI in combination with frequency-independent methods are partially available in the literature [32]. However, these cannot yet be practically implemented due to their interaction with time integration methods and currently remain limited to academic contexts only.