## 4 Numerical Model of Structural Components and Details
The element has five degrees of freedom (DOF) considering that the rotation (torsion) around its local vertical barycentric axis (X) was not included in the model. The other translational and rotational stiffnesses were calculated based on the reference literature for the \"elastomeric bearings\" (UNI EN 1337-3, 2005), of which a summary is reported in Table 5. In the real structure, all the prestressed beams are placed directly on the reinforced neoprene supports positioned on the top of the piles. Under the hypothesis made to consider only one beam element per carriageway, this criticality was resolved with the use of \"rigid links\", attributing the \"master\" property to the node on the girders, and \"slave\" to the four support points on the stack. It was decided to insert non-structural elements, such as highway barriers and connecting crosspieces of beams, only as nodal point loads [57]. In addition to the latter and the structural self-weights, the load of the flooring was modelled as a linear load applied to the  
Figure 10: (**a**) Illustration of the beam element model of the bridge deck. (**b**) Plan view of the deck finite element model. (**c**) Perspective view for deck finite element model. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Kv** (vertical stiffness along X-axes) & 1,533,877 & N/mm \\ \hline
**Kh** (horizontal stiffness, X and Y plane) & 2857 & N/mm \\ \hline
**K\(\alpha\),x** (bending stiffness about X-X axes) & 19.8 \(\times\) 10\({}^{9}\) & Nmm \\ \hline
**K\(\alpha\),y** (bending stiffness about Y-Y axes) & 35.2 \(\times\) 10\({}^{10}\) & Nmm \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Stiffness parameters of neoprene supports. deck beams. Overall, the unloaded bridge model (\"scenario A\") with only the permanent loads and loads of the non-structural element applied, appears as shown in Figure 11.  
To summarize, the RC sections of the bridge (i.e., piles, piers, pre-stressed beams, and deck) were modelled solely with beam finite elements. The neoprene bearing elements of the beams on the piers were modelled with \"general-link\" elements in MidasGen, where the stiffness matrix elements which were calculated with reference to the simplified UNI 1337-3 method were manually inserted. The deck was modelled with a single beam element, and the section imported from SPC is shown in Figure 10a. To simulate the actual support of the prestressed RC beams on their respective neoprene bearings, \"rigid-link\" elements were used to account for the vertical eccentricity. To obtain the vibration modes of the complex bridge structure, an eigenvalue analysis was performed. To do this, as a preliminary step, an estimation of the stiffness of the piles in the cracked phase was taken into account. Method 1 \"of effective inertia\" proposed in Eurocode 8 [58] was applied, which is briefly reported below:  
\[J_{eff}=0.08\cdot J_{un}+J_{CR}\]  
\[J_{CR}=\frac{M_{y}}{\left(E_{c}\cdot X_{y}\right)}\]  
where \(J_{un}\) is the moment of inertia of the gross section of the uncracked pier, \(J_{CR}\), is the moment of inertia of the cracked section at the yield point of the tensile reinforcement; \(M_{y}\) and \(\chi_{y}\) are the yield moment and curvature of the section, respectively; and \(E_{c}\) is the elastic modulus of concrete. Therefore, the critical stiffness will depend on the moment-curvature diagrams of the piles (and consequently also on the static axial load of the single pile). From these considerations, an average reduction of the non-cracked inertia of about 80% was applied to the piles, which was managed in MidasGen through the combined use of the \"stiffness scale factor\" command and the \"boundary change assignment\". The first was used to create the case of modified inertias with the cracked phase, whereas the latter was used to separately manage the static analysis (conducted with the original elastic stiffnesses) and the eigenvalue analysis and all subsequent descending considerations on the seismic response (conducted for the cracked phase).