## 3 Numerical Analysis
### Multi-Span Bridges
Furthermore, the second and third modes contribute much more to the dynamic response for \(k_{v}=1\times 10^{4}\) kN/m and \(k_{v}=1\times 10^{5}\) kN/m. Nonetheless, the first mode dominates the behavior compared to the higher bearing stiffness values. Finally, the numerical analysis was repeated for the three span bridge shown in Figure 4. This bridge has much higher natural vibration frequencies compared to the single- and two-span bridges as depicted in Figure 14. However, similar observations regarding the variation of the natural frequencies can be made as the frequencies tend to remain constant for \(k_{v}\geqslant 1\times 10^{8}\) kN/m while decreasing significantly when the bearing stiffness drops below this value. The change in the frequencies is especially sharp between \(1\times 10^{6}\) kN/m \(\leqslant k_{v}\leqslant 1\times 10^{8}\) kN/m. The variation of maximum accelerations along the length of the bridge shown in Figure 15 is also similar to those observed for single-span and two-span bridges. Firstly, the maximum accelerations increase with a decrease in the bearing stiffness. Furthermore, the location of the maximum acceleration shifts from the middle of the spans towards the ends of the bridge. The accelerations at the middle supports tend to remain to be lower than the other parts of the bridge. Finally, Figure 16 shows the Fourier amplitude spectrum of the accelerations computed at the middle of the second span for the three-span bridge for speeds of 50 km/h and 80 km/h. Comparing the FAS for 50 km/h in Figure 16a and for 80 km/h in Figure 16b reveals the impact of the train speed on the frequency content of the bridge vibrations. For the higher bearing stiffness values, the first mode is dominant for the 50 km/h, while higher modes become more active for lower stiffness values. When the train speed is increased to 80 km/h, the second mode becomes the dominant mode of vibration for higher bearing stiffness values as well. On the other hand, for this train speed and \(k_{v}=1\times 10^{4}\) kN/m, the vibrations are dominated solely by the first mode of vibration. Figure 12: Absolute Maximum Accelerations along the bridge length for different bearing stiffness and train speeds for the two span bridge. Figure 14: Vibration Frequencies–Three span bridge. Figure 13: Fourier Amplitude Spectrum of the accelerations at the middle of the first span for the two-span bridge. Different colors correspond to different bearing stiffness values. Figures 9, 13, and 16 show the complex relationship between the frequency content of the vibrations and the bearing stiffness for bridges with both relatively low (single-span) and high (multi-span) natural frequencies. Thus, variations in the bearing stiffness throughout the lifetime of the structure can lead to significant variations in its dynamic response under train loading. Furthermore, the results also indicate the importance of accurately modeling the bearing stiffness in the evaluation of the bridge under service loading. Considering the uncertainties in the production stage that may lead to deviations from the prescribed bearing stiffness values such as partial loss of contact between the bearing and concrete surfaces through stochastic analysis can be necessary in certain cases where the response parameters are close to the threshold values. Figure 15: Absolute Maximum Accelerations along the bridge length for different bearing stiffness and train speeds for the three span bridge.