## 5 Vertical Deformation of the Composite Tied-Arch  
The variation in the static level gauge readings of the right assembly set with the air temperature during a 15-day non-operation period from 10 October 2024 to 25 October 2024 is shown in Figure 9. It can be seen that the reading (i.e., liquid level) of the mid-span gauge rises with the increase in the air temperature, while those of the gauges at both the upstream and downstream ends drop.  
For each side of the composite tied-arch, the vertical deformation is the measurement of the static level gauge at the mid-span minus the average of the measurements of the gauges at both ends of the span, and the vertical deformation of the whole composite tied-arch (designated as \(\Delta V\)) is taken as the average of such calculated values of both sides. Due to the viscosity of the liquid in the long connecting tube between two adjacent static level gauges, the liquid level fluctuation in the gauge chamber may not be able to capture the instant deformation of the composite tied-arch. Therefore, in the monitored data analysis, the daily average measurement of each static level gauge is used, and then the daily average vertical deformation of the composite tied-arch (hereinafter referred to as \(\Delta V_{\text{dirr}}\)) can be calculated. The daily average air temperature can also be achieved from thermometer readings, and, by subtracting the temperature at the static level gauge initialization (readings reset to 0), the daily average air temperature change (hereinafter referred to as \(\Delta T_{\text{dirr}}\)) is obtained. Thus, the relationship between \(\Delta V_{\text{dirr}}\) and \(\Delta T_{\text{dirr}}\) during the non-operation period from 1 December 2023 to 31 December 2024 is illustrated in Figure 10.  
Figure 9: Variation in level gauge readings with air temperature \(T\) from 10 October 2024 to 25 October 2024.  
It can be seen from Figure 10 that, under the non-operation status, the correlation between \(\Delta V_{dav}\) and \(\Delta T_{dav}\) is very poor, which indicates that no definitive statistical relationship can be established between the two parameters. However, there is a general trend that the monitored \(\Delta V_{dav}\) data points are, to some extent, evenly distributed about the \(\Delta T_{dav}\) axis, and the magnitudes essentially fall into a narrow range of \(-1\) mm-\(1\) mm.  
During the aqueduct operation from 9 March 2024 to 2 May 2024, the variation in the gauge readings of the right assembly set with the water level inside the trough is shown in Figure 11. It can be found that, under the water conveyance status, besides the regular fluctuation in temperature, as seen in Figure 9, the readings of the mid-span gauge deviate from the general trend under empty trough conditions and notably rise above those of the gauges at the upstream and downstream ends.  
To perform a multiple linear regression, taking \(\Delta T_{dav}\) (\"C) and the daily average water level (hereinafter referred to as \(W_{dav}\), m) as two independent variables and \(\Delta V_{dav}\) (mm) as the dependent variable, the optimal fitting function can be expressed as follows:  
\[\Delta V_{dav}=-0.7955+0.0504\times\Delta T_{dav}+1.916\times W_{dav} \tag{2}\]  
The correlation between the monitored and regression values of \(\Delta V_{dav}\) during aqueduct operation from 9 March 2024 to 2 May 2024 is illustrated in Figure 12, which has an R-square of 0.8166 and an RMSE of 0.28 mm, indicating that the regression function (2) can provide rational predictions for the vertical deformation of the composite tied-arch during operation. It is also found that the t-statistic for \(\Delta T_{dav}\) (5.34) is lower than that for \(W_{dav}\) (11.68), indicating that \(W_{dav}\) is statistically significant.  
Figure 11: Variation of gauge readings with water level during operation.  
Figure 12: Correlation between monitored and regression values of \(\Delta V_{dav}\) during operation.