## 2 Modelling of Soil-Structure Interaction
### Numerical Modelling of Soil-Structure Interaction  
The consideration of the soil-structure interaction (SSI) in the numerical solution of a dynamic problem requires compliance with the Sommerfeld radiation condition (Figure 2a) [27]. This condition states that at the transition to the far field (infinity), only one energy flow in the form of propagating waves is permitted. Due to the energy propagation associated with space and surface waves, there is continuous energy dissipation, quantifiable by the parameter of radiation damping \(\zeta_{A}\).  
Consider the vertical oscillation of a uniformly loaded surface, where the analogue damper is proportional to the primary wave velocity \(V_{P}\). The primary wave velocity is determined by the Lame constant \(\lambda\), the shear modulus \(G\) and the material density \(\rho\). In addition, for spatially limited surfaces, such as foundation blocks, the Lysmer equivalent velocity \(V_{LA}\), which depends on the Poisson ratio \(\
u\), proves beneficial. This is particularly relevant due to the restricted wave propagation (with \(\theta<90^{\circ}\), as shown in Figure 2c)  
Figure 2: (**a**) Illustration of the SSI of a concrete portal frame bridge depending on the numerical solution: (**b**) using the direct method; (**c**) using the substructure method.  
beneath the foundation resulting from the discontinuous loading. The derivation and further explanations can be found in the literature [12; 15; 28].  
\[V_{P} =\sqrt{\frac{\lambda+2G}{\rho}} \tag{10}\] \[V_{S} =\sqrt{\frac{G}{\rho}}\] (11) \[V_{LA} =3.4\frac{V_{S}}{\pi\left(1-\
u\right)} \tag{12}\]  
For the numerical modelling of the soil-structure interaction, two main methods can be used [19]:  
1. Direct Method: The near field is modelled using the FEM and the far field (Sommerfeld radiation condition) is implemented using artificial boundary conditions. (Figure 2b);
2. Substructure Method: The near and far fields are considered separate subsystems and solved independently, e.g., using impedance functions or macro finite elements based on the Boundary Element Method (BEM) [29] (Figure 2c).  
The two methods can be applied and varied regardless of the dimension of the problem. Several subvariations of the numerical modelling of the SSI are known and described in the literature. The methods are overviewed in [12]. Furthermore, both methods can be recombined so that hybrid models can integrate the advantages of both approaches [29]. This way, structures with arbitrarily shaped or wide and flexible foundations, i.e., complex foundation geometries such as partly embedded frames, can be optimally represented. In addition, different structures can interact with each other within a hybrid model, and the structure-soil-structure interaction can be modelled in detail. However, these methods are predominantly situated in an academic context and are not designed for construction practice yet.  
This paper presents a simple numerical Calculation Model. The validation of the model is carried out using the complex hybrid method, as discussed previously [12]. The Calculation Model is based on quasi-static approaches within the substructure method using spring-damper elements, as explained in [15].