## 4 Discussion
### Suggestion for Design and Maintenance  
In the design of fixed-end arch bridges, it is necessary to determine vertical deformation due to temperature change and, thus also, to ensure that the bridge span lengths will not coincide with the critical wavelengths for vehicle vibration. However, it is not easy to predict track-geometrical errors due to temperature change, because the rail levels at the construction stage cannot be precisely known, and, moreover, the support conditions and input temperatures in the numerical models are not exactly the same as those in the real structure.  
Therefore, at the maintenance stage or commissioning stage, it is important to readjust the rail levels within an appropriate range. To do this, first, the rail profiles should be measured for a certain range of temperature, and then the peak track irregularities at the maximum temperature (during the daytime in summer) and at the minimum temperature (at nighttime in winter) can easily be predicted, as was done in the present study. This information will facilitate determination of the extent of rail-height adjustment. The rail profiles after readjustment are given as  
\[y_{r}(x,\ T)=y_{ri}(x,\ T)+\Delta y_{r}(x) \tag{7}\]  
where \(y_{ri}(x,\ T)\) is the reference rail level at temperature \(T\), \(y_{r}(x,\ T)\) is the rail level after adjustment, and \(\Delta y_{r}(x)\) is the extent of the rail height readjustment (i.e., shim thickness) at \(x\). From the measurement data, the reference rail level \(y_{ri}(x,\ T)\) at a specific temperature can be obtained, and the readjusted rail level should be determined to meet the following requirements:  
\[a_{peak,s}=v_{30m,s}\left(\frac{a_{peak}}{v_{30m}}\right)\leq a_{all} \tag{8}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Source** & **Major Mode Direction** & **Natural Frequencies (Hz)** \\ \hline \multirow{4}{*}{Numerical model} & Transverse & 1.24, 1.42, 2.06, 3.05, 4.07 \\ \cline{2-3}  & Longitudinal & 3.46, 11.84 \\ \cline{2-3}  & Vertical & 4.81, 6.71, 7.24, 8.33, 9.20 \\ \hline Measured deck acceleration from test run & Vertical & 4.4, 8.9 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Natural frequencies of test bridge.  
\[a_{peak,w}=v_{30m,w}\left(\frac{a_{peak}}{v_{30m}}\right)\ \leq a_{all} \tag{9}\]  
where \(a_{peak,s}\) and \(a_{peak,w}\) are the peak accelerations of the train body in summer and winter, respectively, \(v_{30m,s}\) and \(v_{30m,w}\) are the peak 30-m-chord track irregularities obtained based on the readjusted rail profiles \(y_{r}(x,\ T)\) in summer and winter, respectively, \(a_{peak}/v_{30m}\) is the peak train body acceleration per unit 30-m-chord track irregularity, and \(a_{all}\) is the allowable limit of train body acceleration. To meet the requirements in both summer and winter, the extent of the rail height readjustment \(\Delta y_{r}(x)\) should be determined so that the waveform of the 30-m-chord track irregularities in summer and winter are almost symmetrical with respect to the rail profiles at neutral temperature; i.e., \(v_{30m,s}\approx v_{30m,w}\). The relationship between track irregularity and train body acceleration, as shown in Figures 13 and 17, can be obtained from the running test at different speeds, or alternatively, it can be predicted by an appropriate numerical dynamic analysis model, which should first be validated based on the test data.