# Longitudinal Seismic Response of Continuously Welded Track on Railway Arch Bridges
## 2 Calculation Model for CWR Seismic Response on Deck Arch Bridges
### Numerical Model  
We developed a numerical model of CWR on bridges. The junction piers at both ends are relatively high and pier mass distribution has a great impact on the structural dynamic characteristics, so the actual pier structure should be considered when modelling with beam elements [17]. In particular, secondary dead loads of the bridge deck (including ballast/track slabs, sleepers, rails, fasteners,sidewalk slabs) were simplified into a quality system longitudinally evenly distributed along the beam surface.  
To include the influence of the track on the seismic response of the bridge, a vertical spring was introduced that simulated the vertical stiffness of the track, and an elastic-plastic spring element was used to simulate the longitudinal resistance of the track [19]. The dynamic hysteresis curve of the spring element is shown in Figure 1, in which \(F_{\mathrm{j}}\) is the longitudinal resistance of the track and \(\delta_{\mathrm{j}}\) is the track-bridge relative displacement.  
In addition, outside of the bridge, the track together with 100 m of the embankment was simulated, in order to reduce the influence of the boundary conditions and to accurately simulate the track-bridge interaction. The finite element model is shown in Figure 2. From a practical point of view, the orthogonal damping model-Rayleigh method was adopted [20; 21]. The structural vibration equation under seismic loading can be expressed as  
\[[M]\big{\{}\ddot{x}(t)\big{\}}+[C]\big{\{}\dot{x}(t)\big{\}}+[K]\big{\{}x(t) \big{\}}=-[M][I]\big{\{}\ddot{x}_{\mathrm{g}}(t)\big{\}} \tag{1}\]  
where [M], [C], and [K] refer to the mass matrix, damping matrix, and stiffness matrix of the system respectively; \(\big{\{}\ddot{x}(t)\big{\}}\), \(\big{\{}\dot{x}(t)\big{\}}\), and \(\big{\{}x(t)\big{\}}\) refer to the acceleration, speed, and displacement time interval in relation to the subgrade ground, respectively; \([I]\) refers to the influence matrix; and \(\big{\{}\ddot{x}_{g}(t)\big{\}}\) refers to the seismic wave acceleration time interval within the subgrade ground. The damping matrix can be expressed based on the Rayleigh method as  
\[[C]=\alpha[M]+\beta[K] \tag{2}\]  
where \(\alpha\) and \(\beta\) refer to the mass damping coefficient and stiffness damping coefficient respectively. They can be calculated based on the damping ratios as  
\[\alpha=\frac{2\omega_{\mathrm{i}}\omega_{\mathrm{j}}(\ddot{x}_{\mathrm{i}} \omega_{\mathrm{j}}-\ddot{\zeta}_{\mathrm{j}}\omega_{\mathrm{i}})}{\omega_{ \mathrm{j}}^{2}-\omega_{\mathrm{i}}^{2}} \tag{3}\]  
\[\beta=\frac{2(\ddot{\zeta}_{\mathrm{j}}\omega_{\mathrm{j}}-\ddot{\zeta}_{ \mathrm{i}}\omega_{\mathrm{i}})}{\omega_{\mathrm{j}}^{2}-\omega_{\mathrm{i}}^ {2}} \tag{4}\]  
where \(\omega_{\mathrm{i}}\) and \(\omega_{\mathrm{j}}\) refer to the natural frequencies of the \(i\)th and \(j\)th order of the structure, respectively; \(\xi_{\mathrm{i}}\) and \(\xi_{\mathrm{j}}\) refer to the damping ratios in relation to the \(i\)th and \(j\)th order, respectively. In regards to the test detailed in this paper, the longitudinal first-order natural frequencies of the long and short beams were calculated. When discussing the dynamic response of CWR on bridges, the damping coefficient is usually calculated by selecting the natural vibration frequency of the relevant vibration mode. The damping ratio of a concrete bridge is generally 0.02-0.05, and for a steel bridge it is  
Figure 1: Hysteresis of resistance spring element.  
generally 0.02-0.03. In this paper, the former two-class nature frequency of the system were taken and the damping ratio was taken to be 0.05. The damping coefficients \(\alpha\) and \(\beta\) were taken to be  
\[\alpha=2\frac{0.05\omega_{1}\omega_{2}}{\omega_{1}+\omega_{2}},\ \beta=2\frac{0.05}{ \omega_{1}+\omega_{2}}. \tag{5}\]  
The analysis was based on the Newmark-\(\beta\) method and the integral parameters used for the calculation are \(\gamma\) = 0.5 and \(\beta\) = 0.25 respectively.