## 3 Calculation Model
### Modelling  
The approach described in the studies of [PERSON] and [PERSON] for arbitrarily shaped foundations with embedded configurations was adopted and implemented in the numerical simulations using a special modelling technique [15; 20]. The purpose is to present a simple model characterised by robustness and user-friendliness. Figure 4 shows a schematic representation of the system together with the corresponding spatial coordinates. Subsequently, an exemplary calculation of the vertical (\(z\)) spring-damper elements within the substructure matrix (\(\overline{K_{F,i}}\), \(\overline{C_{F,i}}\)) is presented. Additionally, the dynamic bedding of the horizontal degree of freedom \(x\) is shown, denoted by (\(k_{WDL},c_{WDL}\)). The backfill areas situated between the wing walls and above the foundation (see Figure 4 within \(b_{F}\)) are considered directly in the numerical model. Additional masses  
Figure 4: Calculation Model of a portal frame bridge. (**a**) Problem description; (**b**) system sketch. beyond the foundation slab but within the wing walls are considered by accounting for 40% of the enclosed mass. The wing walls are modelled directly in the numerical model. This approach provides the optimal capture of the stiffening effect of the backfill, particularly for slender abutment walls. Only the adjacent horizontal soil layers are considered using equivalent spring-damper approaches (\(k_{WDL}\), \(d_{WDL}\)). The influence of this horizontal bedding through \"wall\" and \"trench\" effects is discussed in detail in Gazetas [28]. For separate determination of the \"wall\" and \"trench\" effects when calculating the spring and damper elements, it is necessary to distinguish between horizontal \({}_{h}\) and vertical \({}_{v}\) components when selecting the input variables. The horizontal bedding surfaces (\(D_{1}\), \(D_{2}\)) are distinguished in the modelling procedure. \(D_{1}\) is directly incorporated into the substructure matrix, while the effects of \(D_{2}\) are represented as dynamically distributed bedding with stiffness and damping components. This method leads to the determination of the following input parameters:  
**Vertical spring-dashpot elements in \(\overline{K_{F,i}}\) and \(\overline{C_{F,i}}\):**  
\[K_{0,z,emb}=\frac{2\ G\ a}{1\ -\ v}\left(0.73+1.54\left(\frac{b}{a} \right)^{0.75}\right)\cdot I_{v,Trench,i=1}\cdot I_{v,Wall,i=1}\ \text{[N/m]} \tag{15}\] \[I_{v,Trench,i}=1.0+\frac{1}{21}\ \frac{D_{i}}{b}\ \left(1+\frac{4}{3}\ \frac{b}{a}\right)\] (16) \[I_{v,Wall,i}=1+0.19\ \left(\frac{A_{S,i}}{A_{B}}\right)^{2/3} \tag{17}\]  
\[C_{0,z,emb}=\rho_{v}V_{LA,v}A_{B}+\rho_{h}V_{S,h}A_{s,1}\ \text{[N s/m]}\] (18) with \[\rho_{v}=\ \text{density vertical subsoil}\] \[\rho_{h}=\ \text{density horizontal soil}\]  
**Geometry parameters:**  
\[a =\frac{b_{T}}{2} \tag{19}\] \[b =\frac{b_{F}}{2}*\chi_{F}\] (20) \[a >b\] (21) here: \[K_{0,z,emb}=\frac{2\ G\ a}{1\ -\ v}\left(0.73+1.54\left(\frac{b}{a} \right)^{0.75}\right)\cdot I_{v,Trench,i=1}\cdot I_{v,Wall,i=1}\ \text{[N/m]} \tag{15}\]0 is the elongation factor of the effective foundation due to the wing walls and the backfill: \[K_{0,z,emb}=\frac{2\ G\ a}{1\ -\ v}\left(0.73+1.54\left(\frac{b}{a} \right)^{0.75}\right)\cdot I_{v,Trench,i=1}\cdot I_{v,Wall,i=1}\ \text{[N/m]} \tag{15}\]1 (22) \[K_{0,z,emb}=\frac{2\ G\ a}{1\ -\ v}\left(0.73+1.54\left(\frac{b}{a} \right)^{0.75}\right)\cdot I_{v,Trench,i=1}\cdot I_{v,Wall,i=1}\ \text{[N/m]} \tag{15}\]2 (23) \[K_{0,z,emb}=\frac{2\ G\ a}{1\ -\ v}\left(0.73+1.54\left(\frac{b}{a} \right)^{0.75}\right)\cdot I_{v,Trench,i=1}\cdot I_{v,Wall,i=1}\ \text{[N/m]} \tag{15}\]3 (24) \[K_{0,z,emb}=\frac{2\ G\ a}{1\ -\ v}\left(0.73+1.54\left(\frac{b}{a} \right)^{0.75}\right)\cdot I_{v,Trench,i=1}\cdot I_{v,Wall,i=1}\ \text{[N/m]} \tag{15}\]4  
The dynamic bedding parameters (\(k_{WDL},c_{WDL}\)) for the adjacent horizontal soil layer are calculated in a similar fashion using the methods outlined in [15; 20] and implemented with \(x\), \(y\) and \(z\) components.