## 3 Measured Temperature Field of Thin-Walled Concrete
### Seasonal Changes in Vertical Gradient Temperature  
The vertical gradient temperature was flexible across both time and space. As shown in Figure 6, the gradient temperature was unevenly distributed along the thickness, and this distribution pattern also changed over time. However, in the field of civil engineering, extreme patterns are more important than the overall changes in gradient temperature. Generally, extreme patterns in the temperature gradient include two patterns with opposite effects. The first one is the maximum positive pattern (\(T_{\text{g}}^{+}\)), which usually occurs during the day, and the second one is the maximum negative pattern (\(T_{\text{g}}^{-}\)), which usually occurs at night. Both of them could be identified by the following interpretations.  
1. \(T_{\text{g}}^{+}\) is the gradient temperature pattern when the temperature difference between the top and bottom elevations is the maximum positive value within a day. For example, the gradient temperature pattern at 16:00 in Figure 6 corresponds to the largest value of 7.26 \({}^{\circ}\)C within a day (the difference between point \(a1\) and point \(b1\)).
2. \(T_{\text{g}}^{-}\) is the gradient temperature pattern when the temperature difference between the top and bottom elevations is the maximum negative value within a day. For example, the gradient temperature pattern at 6:00 in Figure 6 corresponds to the smallest value of \(-1.63\)\({}^{\circ}\)C within a day (the difference between point \(a2\) and point \(b2\)).  
According to the statistics of the \(T_{\text{g}}^{+}\) and \(T_{\text{g}}^{-}\) by season, the average variation and extreme range within 24 h by season are shown in Figure 7. Whether from the perspective of the values within or the width of the distribution range, the gradient temperature in summer was the strongest and weakest in winter. Moreover, it was found that, for all seasons, both the \(T_{\text{g}}^{+}\) and \(T_{\text{g}}^{-}\) could be expressed by an exponential function, as shown in Equation (2), where \(m\) is the corresponding gradient temperature value at the bottom elevation (\(T_{\text{gb}}\)), while \(n\) is the difference between the gradient temperature at the top (\(T_{\text{gt}}\)) and \(T_{\text{gb}}\), \(a\) is a parameter that needs to be determined, and \(z\) is the distance (m) from the top elevation.  
\[\left\{\begin{array}{l}T_{\text{g}}^{+(-)}=m+ne^{-az}\\ m=T_{\text{gb}}\\ n=T_{\text{gt}}-T_{\text{gb}}\end{array}\right. \tag{2}\]  
Figure 6: Flexible gradient temperature (average values during summer 2022).  
Multiple data fittings were conducted on the average values of \(T_{\text{g}}^{+}\) and \(T_{\text{g}}^{-}\), and it was found that \(a=11.5\) is suitable for both extreme gradient temperature distributions, shown as the dashed lines in Figure 7. Moreover, \(m\) and \(n\) in different seasons are shown in Table 2. For \(T_{\text{g}}^{+}\), \(m\) was always negative but \(n\) was the opposite, and there was a correlation between \(m\) and \(n\) so that \(m\) increased when \(n\) decreased. For \(T_{\text{g}}^{-}\), \(n\) was always negative but \(m\) did not present a regular change pattern. In general, the most significant \(T_{\text{g}}^{+}\) and \(T_{\text{g}}^{-}\) values occurred in the summer and winter, respectively. In addition, the \(T_{\text{g}}^{-}\) in winter was \(-0.26\) times the \(T_{\text{g}}^{+}\) in summer, which is \(86.7\%\) and close to the value of \(-0.3\) recommended by AASHTO LRFD [28].