## 2 Modelling of Soil-Structure Interaction
### Fundamentals of Soil-Structure Interaction  
The excitation of the bridge superstructure causes displacements at the abutments, which usually results in a phase-shifted ground reaction and thus in radiation damping. The quotient of the ground reaction \(R\) and displacement \(x\) is both material- and frequency-dependent and is called dynamic stiffness or the impedance function \(K\). Due to the frequency dependence, the impedance function appears as a complex function by transforming and substituting (not shown here) [16; 17].  
\[K =\frac{R(t)}{x(t)} \tag{1}\] \[...\] \[=K_{0}\cdot(k_{z}+i\Omega c_{z}) \tag{2}\]  
The constant \(K_{0}\) in Equation (1) is equivalent to the quasi-static stiffness (\(a_{0}=0\)), which is calculated as a function of the geometry and the soil stiffness. Solutions for the quasi-static stiffness are given in the literature, e.g., according to [PERSON] for circular foundations [18] as a function of the shear modulus \(G\), the radius \(r\) and Poisson's ratio \(\
u\):  
\[K_{0,K}=\frac{4\ G\ r}{1\ -\ \
u} \tag{3}\]  
Through the substitution and rearrangement of Equations (5) and (6) into the fundamental equations of linear dynamics (omitted here), the computation of the radiation damping \(\zeta_{A}\) associated with the embedded foundation is achieved (Equation (4)). This is accomplished by determining the ratio between the imaginary (3) and real (\(\Re\)) components of the impedance function.  
\[\zeta_{A}=\frac{\Im(a_{\circ})}{2\cdot\Re(a_{\circ})} \tag{4}\]  
Figure 1 shows the frequency-dependent components (\(k_{z}\), \(c_{z}\)) of the impedance function of a massless rigid rectangular foundation in a trench on a homogeneous elastic half-space. Additionally, comparative reference data from the literature, neglecting the trench, are included [17; 19; 20; 21].  
\[k_{z} =\Re\left(\frac{F_{(\omega)}}{x_{(\omega)}}\right)*\frac{1}{K_{0}} \tag{5}\] \[c_{z} =\Im\left(\frac{F_{(\omega)}}{x_{(\omega)}}\right)*\frac{1}{a_{0} \ K_{0}}\] (6) with \[x_{(\omega)} =\text{complex displacement}\]  
Figure 1: (**a**) A massless rigid square foundation under harmonic excitation in a trench on a homogeneous elastic half-space, (**b**) the system sketch, (**c**) the impedance function for \(\
u=0.3\) and data according to the literature [17; 19; 20; 21].  
In the following discussion, we will refer to the frequency-dependent component of the function as the impedance function, which is a slight modification of the definition in Equation (1). Figure 1c illustrates the impedance function with respect to the dimensionless frequency \(a_{0}\). Using \(a_{0}\) facilitates a straightforward plot of the impedance function with changes in the shear wave velocity \(V_{S}\) for varying foundation geometries. In the case of rectangular foundations, the equivalent radius \(r_{0}\) can be assumed.  
\[a_{0} =\frac{\omega r_{0}}{V_{S}} \tag{7}\] \[r_{0} =\sqrt{\frac{4\;a\;b}{\pi}}\] (8) \[\text{with}\;\omega =2\pi f \tag{9}\]  
The shown impedance function is based on the analysis of the hybrid model described in Section 2.2. The presented relief (trench) corresponds to the BEM model outlined in Section 2.3. Due to the centred position of the hybrid solution in between the reference data in Figure 1, the relief only causes a minor modification of the impedance function. For \(a_{0}<2\), the impedance function undergoes minimal variations (\(k_{z}\approx 1\)). Consequently, this observation has led to the widespread application of frequency-independent methods for the impedance function in simple geometric configurations.  
The literature extensively covers the calculation of impedance functions, their derivation and their approximation for simple foundation geometries [17; 19; 20; 21; 22; 23; 24; 25; 26]. In contrast, the solution for complex geometries, considering soil profiles and interactions between adjacent foundations, is typically addressed by numerical methods implemented in computer programs.