## 2 Numerical Modeling Methodology and Theoretical Framework  
### Comprehensive Numerical Modeling Approach and Analytical Framework  
The theoretical framework of this numerical modeling approach is grounded in a holistic multidisciplinary perspective that integrates advanced computational mechanics with damage theory principles. By synthesizing damage mechanics principles with macro-structural response mechanisms, the model establishes a robust theoretical framework that combines fundamental material behavior insights and advanced computational simulation techniques. This approach enables a comprehensive understanding of structural performance by accounting for material characteristics, macroscale structural dynamics, and the complex interplay between material degradation, loading conditions, and structural response.  
More specifically, the analysis of prestressed concrete bridges under combined seismic and train loading requires a sophisticated multi-physics approach to capture the complex dynamic interactions. The numerical framework developed in this study employs a hierarchical finite element model structured to represent the varying cross-sectional properties of prestressed concrete bridge components. The prestressing system modeling incorporates both geometric and material nonlinearity, with particular attention to friction losses at deviator points and stress concentrations at section depth transitions. In this study, the integrated environment of SIEMENS PLM Simcenter 3D [31] was used.  
The concrete material model used the FOM damage model for concrete [32; 33], calibrated for complex bridge geometries. This material model was implemented in SIEMENS PLM Simcenter 3D [31], employing the User-Defined Material (UMAT) subroutine. This approach accounts for the tension stiffening effects and is calibrated for varying section depths while simultaneously incorporating aggregate interlock mechanisms specific to box girder webs and flanges. The model pays attention to the confinement effects in columns, which is especially critical in variable-section designs.  
For the steel rebars, the Menegotto-Pinto model [33; 34] was adopted. This model effectively describes the cyclic stress-strain behavior of reinforcing steel, capturing key phenomena like the Bauschinger effect [35]. Its curved transition between elastic and plastic regions accurately represents the nonlinear response and strain hardening that occur during seismic loading.  
The soil-structure interaction was modeled through specialized interface elements incorporating spring-damper systems derived from Mulliken and Karabalis theory [36], allowing for the accurate representation of foundation behavior under dynamic loading.  
The propagation of seismic waves (traveling waves) is a crucial consideration in the analysis of long-span bridges, as the differential motion of piers can induce significant structural demands. Thus, for bridges with spans of 100 to 200 m and soil with a shear wave velocity of 300 m/s (i.e., soil type C according to ECS [37]), the time delay of the seismic wave's arrival at consecutive piers ranges from 0.33 to 0.67 s. This time difference leads to the asynchronous excitation of supports; this must be incorporated into the dynamic bridge model through proper time delays in the acceleration time histories at support joints.  
The train loading implementation utilizes a moving mass approach calibrated for high-speed rail vehicles. The track structure model incorporates beam elements for rails, discrete spring elements for rail pads, and a continuous support system representing sleepers and ballast layers. This comprehensive approach ensures an accurate representation of the dynamic load transfer from moving trains to the bridge structure [38].