## 5 Comparison of Calculation and Reality
### Assumptions and Calculation  
Primarily, the distinction between static and dynamic bedding approaches is delineated in the context of the portal frame bridge, exemplified by a sample calculation.  
According to the documentation, the soil stiffness is given by \(E_{s}=80\)-\(140\)\(\text{MN}/\text{m}^{2}\). As a first approximation, the static bedding is calculated using the formula \(k_{s}\approx\frac{E_{s}}{f_{\text{shape}}\,b}\), as described in [53]. The shape factor \(f_{\text{shape}}\) depends on the length-to-width ratio of the footing. In the specific scenario considered, characterised by an average stiffness and an estimated length-to-width ratio, the static bedding is determined according to Equation (35).  
\[k_{s}\approx\frac{110\text{ MN}/\text{m}^{2}}{1.4\times 5.5\text{ m}}=14.29\text{ MN}/\text{m}^{3}\] (35) with \[l=b_{T}=15.34\text{ m};\,b=b_{F}=5.5\text{ m};\,f_{\text{shape}}=1.4\]  
Figure 12: Dynamic characteristics of the investigated frame bridges compared to the data collection of the _DIBRST_[52].  
However, in the dynamic bedding calculation, neglecting the \"trench\" and \"wall\" effects, the equivalent spring stiffness is determined using Equation (15).  
\[K_{0,z} =\frac{2\ G\ a}{1\ -\ \
u}\left(0.73+1.54\left(\frac{b}{a}\right)^{0.75 }\right)\cdot I_{Trench}\cdot I_{Wall}=8108.8\ \text{MN/m}\] \[I_{Trench} =I_{Wall}=1\] with \[a=b_{T}/2\ =7.67\ \text{m},\,b=b_{F}/2\ =2.75\ \text{m},\,D_{1}=1.43\ \text{m},\] \[D_{2}=6.5\ \text{m},\,A_{B}=84.37\ \text{m}^{2};\,G=256.33\ \text{MPa};\] \[\rho_{B}=2000\ \text{kg/m}^{3};\,V_{5}=358\ \text{m/s}\ (\text{MASW});\,v=0.3\ (\text{assumption})\]  
Considering the foundation area \(A_{B}\), the equivalent bedding is determined to be 96.11 \(\text{MN/m}^{3}\), an increase of approximately 6.7 times compared to the static approach. The density and shear wave velocity values are based on the MASW method report [54]. Furthermore, if the lateral bedding effects are considered, the vertical spring stiffness, including \(I_{trench,1-2}\) and \(I_{wall,1-2}\), is calculated as \(K_{0,z,emb}=12\),585.9 \(\text{MN/m}\).  
\[I_{Trench,1-2} =1.0+\frac{1}{21}\frac{1.43+6.5}{2.75}\left(1+\frac{4}{3}\times \frac{2.75}{7.67}\right)=1.2\] \[I_{Wall,1-2} =1+0.19\left(\frac{(1.43+6.5)\times 15.34}{5.5\times 13.34} \right)^{2/3}=1.29\]  
The ageing effects of concrete are accounted for in the material approach, resulting in an increase in [PERSON]'s modulus. As required by the Model Code 2010 [55], an increase of 19% is observed. When applied to the frequency/stiffness relationship, this enhancement in [PERSON]'s modulus results in the corresponding stiffening of the eigenfrequency by approximately 4.4%. While analogous increases in [PERSON]'s modulus are often reported in the literature, these cases are often associated with \"dynamic stiffness\" rather than concrete ageing [14; 56]. In contrast, research by [PERSON] asserts that stiffening effects attributed to dynamic loading have only a minor effect on concrete stiffness, with concrete ageing being the main contributor [57].  
The numerical model represents the bridge structure by both C3D8 and C3D10 elements. The backfill (C3D8) is rigidly connected to the bridge. The caps and the trough of the slab track, which are characterised by continuous reinforcement in the transverse direction, are assumed to have a stiffness of 20%. In addition, a material damping of 1.5% is attributed to the subsoil.