## 4 Results
### Calculation and Analysis Model of Frost-Heave Force
#### 4.2.3. Considering the Frost-Heave Force of Circular Wedge Hollowing under Drainage Conditions  
When there is a groundwater drainage channel in the cavity, part of the groundwater is extruded and discharged during the frost-heave process of the water body in the cavity, and then the drainage channel is frozen and the remaining water body freezes and heaves to generate frost-heave force. The volume of discharged water is mainly related to the location of the drainage channel and the shape of the cavity. The frost-heave force when there is a drainage channel was analyzed using a circular wedge-shaped cavity. As shown in Figure 14, the height of the emptying cavity was \(\zeta_{0}\), and the drainage channel was located in the range of 0-\(\zeta_{0}\). The water body froze from the bottom of the cavity. As the freezing front moved upward, the drainage channel was frozen, and the water body in the cavity could not be drained, resulting in the generation of frost-heave force. When selecting the instantaneous state, the heights of the freezing front and the excretion channel are the same, and when they are located at \(\zeta_{0}-\beta_{0}\), the excretion channel freezes. When there is an excretion channel, the frost-heave force in the cavity is P\({}^{-}\). Let  
\[P^{-}=\beta\cdot P \tag{17}\]  
where \(\beta\) is the discharge coefficient. When the water body in the cavity causes frost heaving, the ratio of the frost-heave force when there is water discharge to the frost-heave force when there is no discharge can be determined according to the relative position of the discharge channel and the void. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Surrounding Rock Class** & **I** & **II** & **III** & **IV** & **V** \\ \hline Elastic resistance coefficient & 1800–2800 & 1200–1800 & 500–1200 & 200–500 & 100–200 \\ \hline \end{tabular}
\end{table} 
Table 2: Elastic resistance coefficient of surrounding rocks at all levels. When the water body below the drainage channel is frozen, the volume of the water body that will generate frost-heave force during the subsequent freezing process is  
\[V_{i}=\alpha V_{v-} \tag{18}\]  
\[V_{v-}=z_{1}r_{0}^{2}\varphi_{0}-z_{1}r_{0}\sqrt{2r_{0}(\zeta_{0}-\beta_{0})- \left(\zeta_{0}-\beta_{0}\right)^{2}} \tag{19}\]  
\[\varphi_{0}=\arcsin\frac{\sqrt{2r_{0}(\zeta_{0}-\beta_{0})-\left(\zeta_{0}- \beta_{0}\right)^{2}}}{r_{0}} \tag{20}\]  
\[z_{1}=\frac{\zeta_{0}-\beta_{0}}{\zeta_{0}}z_{0} \tag{21}\]  
At this time, the drainage channel has frozen. According to the calculation results of Equations (3)-(13), the frost-heave force generated by the remaining water body is  
\[\frac{P^{-}=(K_{i}+K_{l})\times}{\frac{\alpha\left(z_{1}r_{0}^{2}\varphi_{0}- z_{1}r_{0}\sqrt{2r_{0}(\zeta_{0}-\beta_{0})-\left(\zeta_{0}-\beta_{0} \right)^{2}}\right)}{2\left(z_{1}^{2}+(\zeta_{0}-\beta_{0})^{2}\right)-\frac{ 2}{3}\frac{\left(\frac{2}{1}+(\zeta_{0}-\beta_{0})^{2}\right)^{2}}{2r_{0}( \zeta_{0}-\beta_{0})-\left(\zeta_{0}-\beta_{0}\right)^{2}}+\frac{4}{3}z_{1}r_ {0}\varphi_{0}\frac{K_{r}+K_{l}}{K_{l}+K_{l}}}} \tag{22}\]  
Comprehensive Equations (16), (17) and (22) show that  
\[\beta=\frac{P^{-}}{P^{-}}=\frac{\left(z_{1}r_{0}^{2}\varphi_{0}- z_{1}r_{0}\sqrt{2r_{0}(\zeta_{0}-\beta_{0})-\left(\zeta_{0}-\beta_{0}\right)^{2}} \right)}{\left(z_{0}r_{0}^{2}\varphi_{0}^{2}-z_{0}r_{0}\sqrt{2r_{0}\zeta_{0}- \zeta_{0}^{2}}\right)}\times \tag{23}\] \[\frac{\left(2(z_{0}^{2}+\zeta_{0})^{2}\right)-\frac{2}{3}\frac{ \left(\frac{2}{1}+z_{0}^{2}\varphi_{0}^{2}\right)^{2}}{2r_{0}\zeta_{0}-\zeta _{0}^{2}}+\frac{4}{3}z_{0}r_{0}\theta_{0}\frac{K_{r}+K_{l}}{K_{l}+K_{l}}}{ \left(2(z_{1}^{2}+(\zeta_{0}-\beta_{0})^{2})-\frac{2}{3}\frac{\left(z_{1}^{2} +(\zeta_{0}-\beta_{0})^{2}\right)^{2}}{2r_{0}(\zeta_{0}-\beta_{0})-\left( \zeta_{0}-\beta_{0}\right)^{2}}+\frac{4}{3}z_{1}r_{0}\varphi_{0}\frac{K_{r}+K_ {l}}{K_{l}+K_{l}}}\right)}\]  
when \(\beta_{0}=0\)--that is, when the volume of water discharged during the freezing process is \(0\)--the frost-heave force \(P=P^{-}\).