## 3 Sequential Interaction Analysis Scheme  
A separate analysis, which involves examining the effects of temperature and train load separately and then simply summing the results for evaluation, underestimates the resulting displacement [24]. Therefore, a separate analysis is not appropriate for evaluating the relative displacements of QHT segments due to track-bridge interactions. As shown in Figure 4, a separate analysis overestimates the longitudinal resistance when the train load is applied after the temperature load. Previous studies [13; 21] used the separate analysis scheme to evaluate the safety of anchor blocks, as it handled applied forces conservatively. However, this is not the case for displacement evaluation. Before anchors are installed, evaluating the longitudinal displacement is important, and the separate analysis is not appropriate in this regard. A sequential scheme can accurately evaluate the longitudinal forces and displacements due to interaction. In this study, therefore, a sequential analysis was applied to consistently analyze the longitudinal behavior of a QHT on the railway bridge. Equations (1) and (2) show the corresponding maximum and minimum longitudinal resistances calculated by a sequential analysis when the loading state changes from having only a temperature load to having a train load additionally applied. The maximum and minimum longitudinal displacements generated in the elastic zone are shown in Equations (3) and (4), respectively. \[\mathrm{F_{max}^{i+1}}=-(\mathrm{F_{i}}-\mathrm{F_{u}})+\mathrm{F_{max}^{i}}- \mathrm{F_{i}} \tag{1}\]  
\[\mathrm{F_{min}^{i+1}}=-(\mathrm{F_{i}}-\mathrm{F_{u}})+\mathrm{F_{min}^{i}}- \mathrm{F_{i}} \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & & **Maximum** & **Maximum** \\
**Case** & **Friction** & **Limit** & **Longitudinal** & **Longitudinal** \\
**Coefficient** & **Displacement, u\({}_{\mathbf{0}}\)** & **Resistance, F\({}_{\mathbf{max}}\)** & **Resistance, F\({}_{\mathbf{max}}\)** \\
**(mm)** & **(km)** & **(km/m) Without** & **(km/m) With Train Load** & **Load** \\ \hline Rail fastener spring & - & 0.5 & 40.0 & 60.0 \\
0.7 & 0.05 & 24.9 & 80.9 \\ Friction spring & 0.5 & 0.05 & 17.8 & 57.8 \\
0.3 & 0.05 & 10.7 & 34.7 \\
0.1 & 0.05 & 3.6 & 11.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Bilinear properties of fastener and friction springs. Figure 4: Longitudinal resistance determined through separate and sequential analyses. \[\mathrm{u}_{\mathrm{max}}^{\mathrm{i+1}}=\left(-\frac{1}{\alpha}+1\right)\mathrm{u} _{0}+\left(\mathrm{u}_{\mathrm{max}}^{\mathrm{i}}-\mathrm{u}_{\mathrm{i}} \right)/\alpha \tag{3}\]  
\[\mathrm{u}_{\mathrm{min}}^{\mathrm{i+1}}=\left(\frac{1}{\alpha}-1\right) \mathrm{u}_{0}+\left(\mathrm{u}_{\mathrm{min}}^{\mathrm{i}}-\mathrm{u}_{ \mathrm{i}}\right)/\alpha \tag{4}\]  
where \(\mathrm{F}_{\mathrm{I}}\) is the longitudinal resistance with train load, \(\mathrm{F}_{\mathrm{u}}\) is the longitudinal resistance without train-load, and \(\alpha\) denotes their ratio (i.e., \(\mathrm{F}_{\mathrm{I}}/\mathrm{F}_{\mathrm{u}}\)). As interaction analysis models used in previous studies [25; 26] comprise the rail and bridge, spring elements were introduced only for a fastener spring. This study expands the model to the rail-track-bridge system that considers the rails, QHT, and bridge. A full sequential analysis should include both the rail fastener spring between the rail and the track and the friction spring between the track and the bridge. Figure 5 shows the example of the nonlinear behavior of the longitudinal resistance spring. Spring type A is for when the train load is applied while the longitudinal resistance spring is within the elastic range due to the temperature load; spring type B is for when the train load is applied when a longitudinal slip has occurred and the longitudinal resistance reaches its maximum limit owing to a temperature change.