## 2 Numerical Modeling Methodology and Theoretical Framework
### Bridge Description and Assumptions  
The structure analyzed in this study is a three-span prestressed concrete bridge constructed using the balanced cantilever method. The bridge configuration consists of two end spans of 100 m each and a central span of 200 m, creating a 100-200-100 m arrangement. The total height from the foundation to the deck level is 60 m, with the piers themselves being 49 m tall while the height of the deck in this position is equal to 11 m (i.e., 60 m = 49 m + 11 m).  
The foundation system consists of isolated footings, with dimensions of \(15\times 15\) m in plan and 2.5 m in height, designed for soil type C according to Eurocode 8. The shear wave velocity is equal to 300 m/s. This soil classification indicates relatively good ground conditions, characterized by deposits of dense sand and gravel. Figure 5 depicts the geometry of the prestressed bridge under consideration.  
The piers feature a twin-column design, where each theoretical pier consists of two independent columns of 49 m in height. Each column has a composite cross-section consisting of semicircular ends with a diameter of 2 m and a rectangular middle section. The rectangular portion varies from 2 \(\times\) 1 m at the top to 2 \(\times\) 2 m at the base, resulting in overall external dimensions of 2 \(\times\) 3 m near the pier cap and 2 \(\times\) 4 m at the base of each column. Figure 6 shows the column section of the bridge piers. The longitudinal and transverse reinforcement ratios of each column are 1.2% and 2.0%, respectively. The gravity load from the superstructure (deck of the bridge) leads to the column's axial force of 42,464 kN, while the max. axial load for the most detrimental loading combination is 58,897 kN. In these cases, the nominal axial load is 0.21 Agfc and 0.29 Agfc, respectively, where Ag is the gross cross-sectional area and fc is the specified concrete compressive strength (50 MPa).  
Figure 5: Geometry of prestressed concrete bridge.  
Figure 6: Bridge pier sections.  
The superstructure features a box girder cross-section with a deck width of 10 m. The section depth varies along the bridge length, measuring 5 m at the abutments and at the middle of the central span, and increasing to 11 m directly above the piers. This variation optimizes the structure's response to bending moments while maintaining efficiency in material usage.  
The assumed values for springs and dampers were computed using the Mulliken and Karabalis theory [36] for soil type C according to Eurocode 8 [37], with the given 15 \(\times\) 15 \(\times\) 2.5 m isolated footings. The vertical stiffness (Kz) is 1.12 \(\times\) 10\({}^{6}\) kN/m, horizontal stiffness (Kx, Ky) is 0.856 \(\times\) 10\({}^{6}\) kN/m, rocking stiffness (Krx, Ky) is 10.3 \(\times\) 10\({}^{6}\) kNm/rad, and torsional stiffness (Ktz) is 7.87 \(\times\) 10\({}^{6}\) kNm/rad. The corresponding damping coefficients are vertical (Cz) 5.47 \(\times\) 10\({}^{3}\) kNs/m, horizontal (Cx, Cy) 4.14 \(\times\) 10\({}^{3}\) kNs/m, rocking (Crx, Cry) 59.3 \(\times\) 10\({}^{3}\) kNms/rad, and torsional (Ctz) 43.5 \(\times\) 10\({}^{3}\) kNms/rad.