## 3 Results and Discussion
### Effect of the Number of Days with Negative Temperatures  
To investigate the effects of the duration of negative temperature on the temperature field distribution around the lining, in this study, amplitude modulation is performed according to Equation (9) to obtain a temperature change function with a minimum temperature of \(-10\)\({}^{\circ}\)C. The coldest day of this function is used as the cut-off point to advance or delay the coldest day, the temperature amplitude remains unchanged, and the angular frequency is varied to generate five working conditions (\(\Delta\)T = \(\pm\)15 d, \(\pm\)30 d, and the original function). The positive sign indicates that the coldest day is delayed, the period of the temperature function becomes larger, and the number of days with negative temperatures decreases in that year; the negative sign indicates that the coldest day is advanced, the period decreases, and the number of days with negative temperatures increases in that year. The frost heave ratio is the same as that in Section 3.1. Figure 8 shows the freezing depth of the tunnel vault and sidewall at 365 d under different negative temperature days. The freezing depth of the vault and sidewall decreases with increasing \(\Delta\)T, and the freezing depth becomes smaller when the coldest day is delayed, which is due to the fact that the temperature is approaching the minimum and has not yet reached the minimum. The freezing depth becomes greater when the coldest day advances. As the number of negative temperature days increases, the coldest day is not the day when the freezing depth is the greatest; the thermal response of the surrounding rock has a certain lag, and the peak freezing depth lags behind the minimum temperature.  
Figure 7: Distribution of frost heave pressure under different minimum temperatures (MPa).  
Figure 9 shows the distribution of frost heave pressure under different numbers of negative temperature days, and the distribution characteristics are similar to those in Section 3.1. The distribution characteristics of the frost heaving force under different conditions are basically the same, with the frost heaving force from the sidewall to the spring line being the greatest and the frost heaving force being the least at the arch bottom. The maximum frost heaving force under each working condition in descending order of \(\Delta\)T (+30 d, +15 d, 0 d, -15 d, and -30 d) is 0.239 MPa, 0.247 MPa, 0.252 MPa, 0.256 MPa, and 0.251 MPa, respectively, and the absolute maximum increment in the frost heaving force is 0.013 MPa. The advance or delay of the coldest day (i.e., the change in angular frequency) has an insignificant effect on the frost heaving force. The time accumulation effect is not considered because the angular frequency adjustment would destroy the interannual periodicity of the temperature data, resulting in significant differences in the temperature between a certain day and the same day a few years later, which is inconsistent with the general pattern.  
Figure 8: Freezing depth of the tunnel vault and sidewall at 365 d under different numbers of negative temperature days.  
Figure 9: Distribution of frost heave pressure under different negative temperature days (MPa).