## 3 Numerical Analysis
### Single-Span Bridge  
Figure 7 presents the natural frequencies of the bridge for different bearing stiffness values. This plot indicates that the vertical motion at the supports can be assumed to be restrained once the vertical stiffness of the bearing exceeds \(1\times 10^{8}\) kN/m as the natural frequencies of the bridge remains unchanged beyond this threshold value. Figure 7 shows that the frequencies of the bridge start to change significantly when \(k_{v}<1\times 10^{7}\) kN/m. Next, elastic response history analyses were conducted to investigate the effect of the bearing stiffness on the dynamic response of the bridge. Figure 8 presents the absolute maximum accelerations computed at nodes discretized at one meter for varying bearing stiffness and train speeds. Maximum acceleration is selected as the main response parameter because it is the main parameter that impacts the ballast stability and, therefore, is arguably the most important criterion for the serviceability of the railway bridges. The maximum accelerations at the mid-span are shown to be very sensitive to the bearing stiffness. Meanwhile, the maximum acceleration values remain virtually constant at the mid-span for \(k_{v}\geqslant 1\times 10^{8}\) kN/m, and they start to increase significantly as the bearing stiffness drops below this threshold value. The variation in the maximum acceleration demands at the mid-span with bearing stiffness can be explained relatively simply with the effect of the latter on the vibration frequencies for \(k_{v}\leqslant 1\times 10^{6}\) kN/m as depicted in Figure 7. For bearing stiffness values under this threshold, the vibration frequencies decrease significantly compared to the virtually rigid cases leading to an increase in the acceleration amplitudes. However, it is difficult to use the same argument to explain the increase in the accelerations for \(k_{v}=1\times 10^{7}\) kN/m. For this bearing stiffness value, the vibration frequencies computed from an eigenvalue analysis remain virtually identical compared to those for \(k_{v}\geqslant 1\times 10^{8}\) kN/m. On the other hand, the maximum accelerations increase significantly compared to the stiffer cases. For example, even at the mid-span, the maximum accelerations reach 0.9 m/s\({}^{2}\) for \(k_{v}=1\times 10^{7}\) kN/m for a train speed of 80 km/h compared to 0.05 m/s\({}^{2}\) for \(k_{v}=1\times 10^{15}\) kN/m. Figure 7: Variation of the bridge natural frequencies with the bearing stiffness. To understand the reason behind this discrepancy, the Fourier Amplitude Spectra (FAS) of the acceleration time histories at the mid-span for different bearing stiffness values were plotted in Figure 9. The FAS for bearing stiffness values of \(k_{\rm p}\geqslant 1\times 10^{8}\) kN/m indicate that the behavior of the bridge is dominated by the first mode at \(f<5\) Hz although some influence of the third mode around 25 Hz is also visible. On the other hand, the third mode becomes the dominant mode for \(k_{\rm p}=1\times 10^{7}\) kN/m with very little energy at frequencies \(f<5\) Hz compared to the higher bearing stiffness values. A closer look at Figure 7 shows that, although the frequencies of the first two modes remain virtually identical when the bearing stiffness is reduced to \(k_{\rm p}=1\times 10^{7}\) kN/m, the third mode frequency drops from 27.5 Hz for \(k_{\rm p}\geqslant 1\times 10^{8}\) kN/m to 25.2 Hz for \(k_{\rm p}=1\times 10^{7}\) kN/m. The Fourier Amplitude Spectrum of the loading for 80 km/h depicted in Figure 6b indicate that the decrease in  
Figure 8: Absolute Maximum Accelerations along the bridge length for different bearing stiffness and train speeds. the third mode frequency for \(k_{v}=1\times 10^{7}\) kN/m leads to a resonance condition between the loading and the bridge at around 25-26 Hz leading to a significant increase in the contribution of the third mode for this bearing stiffness. In addition, Figure 9 indicates that the fourth mode with a frequency around 45 Hz also contributes significantly to the acceleration response.