## 3 Results and Discussion
### Effect of the Frost Heave Ratio  
The frost heave ratio range of rock masses under different frost heave grades is shown in Table 2. The frost heave ratio is determined by referring to the method of [PERSON] et al. [33], who proposed a method to value the frost heave ratio of the rock mass in the cold region tunnel and carried out the corresponding classification of frost heave susceptibility (5 grades: non-frost heave susceptible rocks, weakly frost heave susceptible rocks, frost heave susceptible rocks, strongly frost heave susceptible rocks, and extremely strong frost heave susceptible rocks). In this study, four working conditions with frost heave ratios of 0.25%, 0.5%, 1.0%, and 2.0% are designed to calculate the frost heaving force of the lining; the values of the frost heave ratio cover a range from weakly frost heave-susceptible rocks to extremely strong frost heave-susceptible rocks, and the minimum temperature is \(-10\)\({}^{\circ}\)C. Figure 10 shows the distribution of frost heave pressure under different frost heave ratios. The frost heave pressure from the vault to the sidewall gradually increases, the frost heave pressure from the sidewall to the vault bottom gradually decreases, and the frost heave pressure from the inverted arch to the spring line is the maximum. As the frost heave ratio increases, the frost heave pressure around the lining increases accordingly. The maximum frost heaving force is located in the sidewall section, in descending order of the frost heave ratio, which is 0.261 MPa, 0.305 MPa, 0.391 MPa, and 0.565 MPa. The largest increase in the frost heaving force occurs on the sidewall, i.e., when the frost heave ratio increases from 0.25% to 2.0%, the frost heaving force increases by 116%. The frost heave ratio of the surrounding rock has a significant effect on the frost heaving force.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Non-Frost Heave** & **Weakly Frost Heave** & **Frost Heave** & \
\begin{tabular}{c} **Strongly Frost** \\ **Heave** \\ \end{tabular} &
\begin{tabular}{c} **Extremely Strong** \\ **Frost Heave** \\ \end{tabular} \\ \hline \(\varepsilon_{v}\leq\) 0.13 & 0.13 \(<\varepsilon_{v}\leq\) 0.47 & 0.47 \(<\varepsilon_{v}\leq\) 0.80 & 0.80 \(<\varepsilon_{v}\leq\) 1.6 & 1.6 \(<\varepsilon_{v}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Frost heave ratio \(\varepsilon_{v}\) (%) of rock masses under different frost heave grades [33].  
Figure 10: Distribution of frost heave pressure under different frost heave ratios (MPa).  
Figure 11 shows the distributions of the maximum principal stresses of the initial lining under the four frost heave ratios. Figure 10 shows that when the frost heave ratio is 0.25%, the stresses are all negative, and the lining is completely under compression; when the frost heave ratios are 0.5%, 1.0%, and 2.0%, tensile stresses appears inside the lining, with maximum values of 0.05 MPa, 0.17 MPa, and 0.54 MPa, respectively, and the maximum tensile stress occurs at the spring line. The possible reason is as follows. The radius of curvature of the arch is relatively large; under the action of the frost heaving force of the surrounding rock, the backward bending of the inverted arch occurs to a certain extent, resulting in local tension at the spring line. Considering that the tensile strength of concrete is low, the strength of the arch can be improved by adding reinforcement or laying an insulation layer at the spring line [34].