## 5 Conclusions  
1. When the non-uniform characteristics were considered, the radial stress varied linearly with the relative radius \(r\), and the circumferential stress of the frozen wall varied approximately in a parabolic shape. In this case, under the elastic limit state, the maximum value of the circumferential stress of the frozen wall appeared at \(r=1.5\). In the elastoplastic state, the maximum circumferential stress appeared at the elastoplastic junction. In the plastic limit state, the maximum circumferential stress appeared at \(r=2.0\);
2. The displacement of the outer and inner edges of the frozen wall increased with the increase in formation depth; the lower the average temperature of the frozen wall was, the smaller the displacement value. The displacement of the outer edge of the frozen wall was always greater than that of the inner edge, but the influence of temperature on the inner edge was greater than that of the outer edge;
3. When the frozen wall was in the elastic state, the displacement caused by frost heave was constant. However, when the frozen wall entered the elastic-plastic state, the displacement of the inner edge of the frozen wall caused by frost heave increased with the increase in the plastic zone, and the displacement of the outer edge of the frozen wall caused by frost heave decreased with the increase in the plastic zone;
4. When the frozen wall entered the plastic state, its bearing capacity increased with the increase in the radius of the plastic zone, but the growth rate decreased gradually. When a part of the frozen wall entered the plastic state, the displacement of the whole frozen wall increased considerably. Therefore, to improve the stability of the frozen wall, its bearing capacity should be enhanced by increasing its thickness or decreasing its average temperature, which would prevent the plastic state of the frozen wall.  
Conceptualization, B.W.; methodology, S.L.; software, Y.C.; validation, C.R., S.Y.; formal analysis, B.W.; investigation, B.W.; resources, B.W.; data curation, S.L.; writing--original draft preparation, B.W.; writing--review and editing, B.W.; visualization, S.L.; supervision, Y.C.; project administration, B.W.; funding acquisition, B.W. All authors have read and agreed to the published version of the manuscript.  
This work was support by the Natural Science Foundation of Anhui Province (Grant No. 2108085 QE251), China Postdoctoral Science Foundation (Grant No. 2021M703621), Anhui Postdoctoral Science Foundation (Grant No. 2022B635), Open Fund grants projects of Engineering Research Center of Underground Mine Construction of Ministry of Education (Grant No. JYBGCZX2022103), Key Project of Natural Science Research in Universities of Anhui Province (KJ2021A0425), Natural Science Foundation of Anhui University of Science and technology (xjzd2020-18), Talent Introduction Project of Anhui University of Science and Technology (13200403), Major Science and Technology Special Project of Anhui Province (202003c08020007), College student innovation and entrepreneurship program (S202210361048). Not applicable.  
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.  
The data used to support the findings of this study are available from the corresponding author upon request.  
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.