# Influence Analysis of Traveling Wave Effect on Rail Interaction of Long-Span Suspension Bridge
## 5 Response of Beam-Rail System under Traveling Wave Effect
### Earthquake Response of Bridges
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **The Moment When** & **The Moment When** & **The Moment When** & **The Moment When** \\
**Apparent Wave** & **Point A Reaches the** & **Point B Reaches the** & **Point C Reaches the** & **Point D Reaches the** \\
**Speed (m/s)** & **Maximum Positive** & **Maximum Positive** & **Maximum Positive** & **Maximum Positive** \\  & **Longitudinal** & **Longitudinal** & **Longitudinal** & **Longitudinal** \\  & **Displacement (s)** & **Displacement (s)** & **Displacement (s)** & **Displacement (s)** \\ \hline
200 & 7.52 & 39.24 & 18.52 & 22.08 \\
500 & 9.04 & 12.14 & 13.88 & 16.26 \\
1000 & 8.94 & 9.28 & 19.66 & 4.52 \\
2000 & 8.9 & 7.76 & 19.58 & 6.38 \\
4000 & 8.82 & 3.56 & 3.5 & 3.02 \\
8000 & 8.8 & 3.48 & 3.32 & 9.02 \\ \(\infty\) & 8.8 & 3.12 & 3.12 & 8.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Time for each position to reach the maximum positive longitudinal displacement at different apparent wave velocities. Figure 20: Time course of main bridge end displacement at different visual wave speeds. Table 9 gives the Pearson product moment correlation coefficients of the displacement time course curve of point A beam end and the stress time course curve of point A rail at different apparent wave speeds, and gives the comparison graph of the stress time course curve of point A and the displacement time course curve of the north approaching bridge end (under rail beam at point A) when the apparent wave speed is 200 m/s, as shown in Figure 21. The analysis shows that the displacement time course curve at point A and the stress time course curve at point A are very similar, and the peaks and valleys correspond to each other, so it can be considered that the stress at point A is closely related to the longitudinal displacement of the under-rail beam, that is, the change of stress at point A can be judged by the change of displacement at point A. Generally speaking, the influence of traveling wave effect on the displacement of the beam end cannot be ignored, and the influence of traveling wave effect on the displacement of the main bridge end is much greater than that of the approach bridge end. When the apparent wave velocity is 200 m/s, the absolute maximum displacement of the main bridge end reaches 564 mm, which is 479 mm larger than the absolute maximum displacement of the main bridge end under consistent excitation, but still less than the allowable expansion and contraction of the beam end expansion device, which is 940\(--\)820 mm. Under high apparent wave velocity and consistent excitation, the displacements of the north and south main bridge ends will approximately keep changing synchronously, but under a low apparent wave velocity, the displacements of the north and south main bridge ends will no longer keep changing synchronously. The change in rail stress at point A can be judged by the change in displacement at point A. Figures 22 and 23 show the absolute maximum values of shear force and bending moment at the bottom of pier and main tower under different apparent wave velocities and consistent excitation. Among them, N2 and S2 are the approach piers with fixed bearings, and 3# and 4# are the main towers. Analysis can be seen: the traveling wave effect only on the N2, S2 pier bottom and the main tower bottom force, the rest of the pier bottom force does not change with the change in apparent wave speed, this is because these piers are set up longitudinal movable bearing, no restriction on the longitudinal movement of the superstructure, so N2, S2 pier and the main tower will become the concentration of force parts, apparent wave speed on N2, S2 and the main tower bottom force effect is obvious  
Figure 21: Comparison of stress–time-history curve at point A and displacement time history curve at point A.