## 2 Numerical Modeling Methodology and Theoretical Framework
### Methodology Validation Through Experimental Comparisons  
To verify the proposed finite element methodology's use in analyzing prestressed bridges, three benchmark examples were thoroughly examined. In the first example, a direct tension test was carried out to validate the model's ability to capture the complete stress-strain relationship. As illustrated in Figure 1, the concrete damage model successfully reproduced the full tensile stress-strain curve, including the post-peak softening branch, which is traditionally challenging to capture using conventional elastoplastic models. Thecomputational results demonstrate very good agreement with the experimental data of [PERSON] and [PERSON] [39], particularly in the representation of the degradation of material properties and the progressive damage evolution during loading.  
The second verification example focuses on cyclic compression loading, where the model's ability to simulate repeated loading-unloading scenarios is evaluated. Figure 2 presents the cyclic compressive stress-strain relationship, clearly demonstrating that the damage model accurately captures not only the softening behavior shown in the experimental test of [PERSON] et al. [40] but also the unloading and reloading paths. The model's performance is particularly noteworthy in its representation of the degradation in stiffness and the accumulation of permanent deformations during cyclic loading, which are crucial aspects of the seismic analysis of bridge structures.  
The third and final validation example examines a bridge pier subjected to seismic excitation, specifically analyzing the response during the first 15 s of the 1995 Kobe earthquake. The pier geometry and reinforcement details are presented in Figure 3, and were taken from [PERSON] and [PERSON] [41].  
Figure 1: Direct uniaxial tension: experimental test [39] vs. proposed model.  
Figure 2: Uniaxial cyclic compression: experimental test [40] vs. proposed model.  
The time-history displacement plot in Figure 4 provides a critical comparison between the experimental shake table test results and the proposed numerical model. The analysis focuses on the horizontal displacements at the pier head in both the X and Y directions during the first 15 s of the 1995 Kobe earthquake.  
The numerical model demonstrates remarkable precision in reproducing the experimental displacement patterns. By closely tracking peak displacement amplitudes in both horizontal directions, the analysis reveals the model's exceptional ability to capture the complex dynamic response of the bridge pier. The time-history plot not only illustrates the peak displacements but also effectively represents the residual deformations, which are crucial for understanding the structural integrity under seismic loading.  
The very good correlation between the experimental data and numerical predictions validates the proposed finite element methodology. This validation is particularly significant in demonstrating the model's ability to simulate the complicated nonlinear dynamic responses of bridge structures subjected to seismic excitation.  
Figure 4: Time–history of horizontal displacements of the head of the pier: experimental test [41] vs. proposed model.  
Figure 3: Bridge pier geometry based on the work presented in Ref. [41] (in millimeters).