## 3 Numerical Analysis  
To investigate the impact of the bearing stiffness on the dynamic response of railway bridges, numerical analyses were conducted on three separate bridges, one single-span and two multi-span, under train loading. The single-span bridge used is a 50 m long railway bridge. [PERSON]'s Modulus of concrete was taken as 32 GPa, and the moment of inertia of the cross-section about the main bending axis is 16.89 m\({}^{4}\). The mass per unit length is assumed to be 22.9 t/m including the track bed and the ballast and modeled as lumped mass at each node. Damping is modeled as 2% Rayleigh damping anchored at the first and third mode frequencies. The bridge is discretized at equal distances of 0.025 m. The horizontal movement in the supports is restrained considering their negligible effect while the ends of the bridge are free to rotate. Elastic springs with a stiffness of \(k_{v}\), which varied between \(1\times 10^{4}\) kN/m and \(1\times 10^{15}\) kN/m, is used to simulate the vertical behavior of elastomeric bearings. Here, it should be noted that, for most of the commercially available elastomeric bearings, the stiffness varies between \(1\times 10^{5}\) kN/m and \(1\times 10^{7}\) kN/m, but a wider range was used in this study to consider possible variation in the bearing stiffness due to factors such as aging and deterioration. Stiffer values were considered to simulate the fully restrained case, which is commonly used in literature as well as to account for possible stiffening in bearings with aging. Two- and three-span bridges were created from the single span bridge by introducing elastomeric bearings at the column locations. The cross-sectional properties and the length of the bridge was kept constant. Figure 4 depicts the overview of the two- and three-span bridges generated from the single-span bridge. The multi-span bridges used in the study represent a very common bridge type, where elastomeric bearings are placed at the top of the columns while the bridge deck is continuous throughout the entire bridge length. Introducing the middle supports serves two main purposes: (i) the frequency of the bridge is altered significantly, and thus the dynamic analysis is repeated for a new set of bridge frequencies paving the way to evaluate the effect of the bridge frequency and its interplay with the loading frequency on the dynamic response of the bridge, and (ii) the difference in the behavior of the end and middle spans can be evaluated. As the train model, without losing generality, the ICE-2 train is selected as a representative of modern passenger trains. The configuration and the axle distances for a typical ICE-2 train are depicted in Figure 5. The axle load, suspension stiffness and suspension damping used in the numerical analysis are 19.995 t, 4800 kN/m and 109 kNs/m, respectively. It should be noted that, although only one type of train, ICE-2, is used in this study, the properties of the ICE-2 train are similar to other trains such as TGV [40], and the results presented in this article can be generalized for modern passenger trains. For each bridge and bearing stiffness configuration, dynamic analyses were repeated for speeds of 50 km/h, 80 km/h and 130 km/h. Figure 6 shows the Fourier Amplitude Spectrum (FAS) of the loading function of the ICE-2 train for the relevant speeds. Figure 3: Overview of the numerical model. The track irregularities are not considered in the study because the main focus of the article is to evaluate the impact of the bearing stiffness on the frequency content and bending behavior of the bridge. Although the amplitudes of vibrations are used in the article to identify the resonance conditions, prediction of the acceleration amplitudes is out of the scope of this article. For this reason, only the vibrations that are created by the bending behavior of the bridge are considered and not by other sources such as track irregularities. Figure 4: Overview of the multispan bridges. Figure 5: Axle distance and configuration for a typical ICE-2 X [40].