## 3 Numerical Analysis
### Single-Span Bridge
Similar to the third mode, the fourth mode comes into a resonance condition with the loading frequency (see Figure 6b) and thus contributes significantly to the response. Therefore, the change in bearing stiffness can significantly affect the dynamic response of the bridge even its impact on the modal frequencies is small because it leads to a change in the predominant mode of vibration. A similar behavior can also observed for the bearing stiffness value of \(k_{v}=1\times 10^{6}\) kN/m. Figure 9 shows that the behavior of the bridge is dominated by the frequencies 15 Hz and 36 Hz for \(k_{v}=1\times 10^{6}\) kN/m. These correspond to the third and fourth vibration modes of the bridge, respectively, and are very close to the dominant loading frequencies as indicated by the loading function for a speed of 80 km/h; see Figure 6b. When the bearing stiffness value is reduced further to \(k_{v}=1\times 10^{5}\) kN/m and \(1\times 10^{4}\) kN/m, the dynamic response of the bridge continues to be dominated by third mode of the bridge, which has a frequency between 7 and 8 Hz, while the influence of the fourth mode diminishes. The results summarized above indicate the sensitivity of the dynamic response and the dominant mode to the bearing stiffness. Furthermore, the bearing stiffness has a significant influence on the location of the maximum accelerations. While the maximum acceleration is closer to the mid-span for the case of \(k_{v}=1\times 10^{15}\) kN/m, its location shifts towards the ends of the bridge as the bearing stiffness gets lowered. Here, it should be noted that some sudden jumps were observed at the ends of the bridge for \(1\times 10^{8}\) kN/m \(\leqslant k_{v}\leqslant 1\times 10^{10}\) kN/m. A possible explanation for these sudden jumps can be the singularities associated with the Finite Element Method at the boundaries. Since the bridge is modeled as a stand-alone structure without the front and back approaches, the train movement starts and ends abruptly at the two supports which may lead the inaccuracies at the edges depicted in Figure 8. Although the acceleration values at the support for \(1\times 10^{8}\) kN/m \(\leqslant k_{v}\leqslant 1\times 10^{10}\) kN/m may not be exact, it is clear from Figure 8 that the location of the maximum accelerations shifts towards the edges for \(k_{v}\leqslant 1\times 10^{9}\) kN/m. This shift in the location of the maximum acceleration can be explained by the impact of the bearing stiffness on the mode shape of the bridge as it modifies the boundary conditions of the bridge. As an example, the first and third mode shapes for different bearing stiffness  
Figure 9: Fourier Amplitude Spectrum of the acceleration time history response at the mid-span for a train speed of 80 km/h. Different colors correspond to different bearing stiffness values. values are plotted in Figure 10. While the first mode shape remains identical for the two cases, there is a significant difference in the third mode shape, particularly at the bearing locations for \(k_{v}=1\times 10^{10}\) kN/m and \(k_{v}=1\times 10^{7}\) kN/m. The modal displacements at the bearing location increase even further when the bearing stiffness is reduced further. Recalling that the third mode begins to dominate the dynamic response of the bridge for \(k_{v}\leqslant 1\times 10^{7}\) kN/m, the increase in the modal displacement close to the abutments for these bearing stiffness values for the third mode can be indicated as the reason why the location of the absolute maximum acceleration on the bridge shifts from the mid-span towards the ends of the bridge with a decrease in the bearing stiffness.