## 4 Longitudinal Deformation of the Composite Tied-Arch  
### Evaluation of the Effects of Air Temperature and Water Load  
The typical variation in the LVDT readings of the expansion joints at abutments on both sides of the composite tie arch with the air temperature (hereinafter referred to as \(T\)) during a 15-day non-operation (trough empty) period from 15 November 2024 to 30 November 2024 is shown in Figure 4. It can be seen that the LVDT readings at the downstream abutment expansion joints on both sides essentially exhibit the same magnitude and variation pattern, while those at the two upstream abutments basically maintain around a near zero status due to the fixed basin-type rubber bearings at those locations. It can also be seen that the LVDT readings at the downstream abutment expansion joints are very sensitive to the air temperature, with daily maximum and minimum readings appearing almost concurrently with the daily highest and lowest air temperatures, respectively.  
Figure 4: Variation in LVDT readings with air temperature \(T\) from 2024/11/15 to 2024/11/30.  
Figure 3: Layout of monitoring devices at the upstream abutment. (3) Joint meters (LVDT), (4) strain gauges, (5) thermometer, (6) thrust block, (7) fixed basin-type rubber bearing (downstream end: sliding basin-type rubber bearing).  
For each side of the composite tied-arch, the longitudinal deformation (designated as \(\Delta L\)) is the sum of the readings of the LVDTs at upstream and downstream abutment joints, and the \(\Delta L\) of the whole composite tied-arch is taken as the average of both sides.  
The LVDTs are initialized to 0 right after installation at a temperature of \(-3\)\({}^{\circ}\)C; so, the daily extreme air temperature change (hereinafter referred to as \(\Delta T_{dex}\)) is taken as the daily highest or lowest (if lower than \(-3\)\({}^{\circ}\)C) temperature minus \(-3\)\({}^{\circ}\)C, and the daily extreme \(\Delta L\) (hereinafter referred as \(\Delta L_{dex}\)) is calculated using the daily maximum or minimum readings of LVDTs with respect to daily highest or lowest temperatures. The relationship between \(\Delta L_{dex}\) and corresponding \(\Delta T_{dex}\) during the non-operation period from 1 December 2023 to 31 December 2024 is plotted in Figure 5, which exhibits a good correlation with a R\({}^{2}\) of 0.9738 and a RMSE of 1.33 mm. The regression also shows a \(\pm 1\)\({}^{\circ}\)C air temperature change tends to cause a longitudinal deformation of \(\pm 0.51\) mm for the composite tied-arch.  
During the aqueduct operation from 9 March 2024 to 2 May 2024, the variation in \(\Delta L_{dex}\) with \(\Delta T_{dex}\) and the water level occurring at the same time as \(\Delta L_{dex}\) (hereinafter referred to as \(W_{dex}\)) is shown in Figure 6. It can be seen that even under water conveyance, the longitudinal deformation of the composite tied-arch still closely follows the air temperature changes.  
Figure 5: Correlation between \(\Delta L_{dex}\) and \(\Delta T_{dex}\) during non-operation period.  
To perform a multiple linear regression, taking \(\Delta T_{dex}\) (\({}^{\circ}\)C) and \(W_{dex}\) (m) as two independent variables and \(\Delta L_{dex}\) (mm) as the dependent variable, the optimal fitting function can be expressed as follows:  
\[\Delta L_{dex}=-2.063+0.5669\times\Delta T_{dex}+2.874\times W_{dex} \tag{1}\]  
The relationship between the monitored and regression values of \(\Delta L_{dex}\) during the aqueduct operation from 9 March 2024 to 2 May 2024 is shown in Figure 7, which has a good correlation with an R-square of 0.9434 and a RMSE of 0.81 mm. It is also found that the t-statistic for \(\Delta T_{dex}\) (25.11) is higher than that for \(W_{dex}\) (6.36), indicating that \(\Delta T_{dex}\) is statistically significant.