## 2 Mapping Relationship Between Rail Deformation and Bridge Deformation
### Wavelength Range of Vertical Deformation of 200-600 m Cable-Stayed Bridge  
The influence of the cable-stay stiffness ratio on the wavelength characteristic of vertical deformation in large-span, cable-stayed bridges under complex loads is examined. Two contrasting scenarios are explored: the strong beam and the weak cable, and the weak beam with the strong cable. To evaluate the vertical deformation under these conditions, adjustments are made to the diameter of the tension cables, resulting in cable-beam stiffness ratios of 100 and 0.001. The deformation of the long-span, cable-stayed bridge is a typical non-stationary signal. And the conventional signal decomposition method is difficult to use in effectively decomposing the long-wave deformation. Employing the Butterworth bandpass filtering principle, the vertical deformation is decomposed into wavelengths ranging from 25 m to 1200 m at intervals of 50 m. The minimum and maximum wavelengths are identified by intercepting those with 1% and 99% of the energy share in the bridge deformation. The resulting wavelength range of vertical deformation in long-span, cable-stayed bridges is presented in Table 5.  
From the table, it becomes apparent that, for the cable-stayed bridge sharing the same span arrangement, the stiffness of both the cable and the beam significantly influences the minimum and maximum wavelengths. Specifically, the weaker stiffness of the beam results in the minimum values for both minimum and maximum wavelengths, while the stronger stiffness of the beam leads to the maximum values. The substantial difference in the range of minimum and maximum wavelengths across these extremes underscores the profound impact of the cable-stayed beam stiffness on the linear wavelength characteristics of the cable-stayed bridge.  
Furthermore, an increase in the number of auxiliary piers contributes to a gradual decrease in the minimum wavelength and the maximum wavelength. While the maximum wavelength remains relatively constant. For instance, in the case of a main span 200 m cable-stayed bridge, the minimum wavelength and the maximum wavelength are 21.38 m and 1268.29 m. In the case of a main span 400 m cable-stayed bridge, these values are 27 m and 1268.81 m. Finally, for a main span 600 m cable-stayed bridge, the minimum wavelength and the maximum wavelength are 38.8 m and 1270.74 m.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **Wavelength** & \multicolumn{2}{c}{**Minimum**} & \multicolumn{2}{c}{**Maximum**} \\
**The Span of the** & **Type** & \multicolumn{2}{c}{**Wavelength/m**} & \multicolumn{2}{c}{**Wavelength/m**} \\ \cline{2-6}
**Main Span/m** & **Cable-Beam** & **0.001** & **100** & **0.001** & **100** \\ \hline  & Three-span & 63.48 & 170.62 & 692.88 & 1268.29 \\
200 & Five-span & 31.74 & 151.37 & 756.29 & 1261.91 \\  & Seven-span & 21.38 & 122.31 & 599.79 & 1260.02 \\ \hline  & Three-span & 38.64 & 304.22 & 1127.19 & 1268.81 \\
400 & Five-span & 28.57 & 266.57 & 1053.87 & 1263.92 \\  & Seven-span & 27.00 & 242.58 & 948.30 & 1262.22 \\ \hline  & Three-span & 61.50 & 419.12 & 1237.67 & 1270.37 \\
600 & Five-span & 38.80 & 370.69 & 972.44 & 1270.74 \\  & Seven-span & 39.87 & 345.21 & 904.67 & 1270.12 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Minimum and maximum wavelengths of deformation of 200 m cable-stayed bridges.