###### Abstract  
The technology for freezing shaft sinking is widely used for shafts to pass through deep, unstable alluvia with the continuous exploitation of mineral resources. Due to the technique using the sectional excavation and shaft lining construction adopted in deep alluvia, the radial stress at the inner edge of a frozen wall is incompletely unloaded. In this paper, a mechanical model was established for a frozen wall with its inner edge radially incompletely unloaded. A parameter, \(\alpha\), expressing the degree of being unloaded was introduced, and then a new method of designing and calculating the thickness of the frozen wall was proposed. The range of parameter \(\alpha\) was estimated based on the frozen wall-shaft lining interaction forces from field data from a given project. The results indicate that the range of \(\alpha\) can be chosen to be from 0.05 to 0.15 in deep alluvia. The design thickness of the frozen wall can be reduced by at least 5% for the frozen wall with the inner edge radially incompletely unloaded. The design thickness is significantly influenced by the strength and elastic modulus of the frozen soil and the elastic modulus of the surrounding unfrozen alluvium. The design and calculation method of frozen wall thickness can provide new ideas for guiding the design of frozen walls in deep alluvia. f +
Footnote †: journal: Applied  
significant increase in the technical difficulty of freezing shaft sinking. Therefore, how to ensure the safety of freezing shaft sinking while taking into account the economy can provide new ideas for guiding the design thickness of frozen walls in deep alluvialvia. Theories for designing the frozen wall thickness have developed with the widespread use of freezing shaft sinking in alluvialvia. The earliest formula [12] can be traced back to the Lame formula based on the elastic single-layer, thick-walled cylinder model to guide the freezing shaft sinking process in topsoil at a depth of approximately 120 m or less. As the Lame formula was demonstrated to be too conservative, the Domke formula [13] was presented based on an elastoplastic single-layer, thick-walled cylinder model with the limitation that the plastic zone outer radius is smaller than the radius at which freezing pipes are set. This formula was widely used in practical freezing shaft sinking projects at depths of approximately 120 m to 300 m. Considering the frozen soil as a kind of friction material, the Klein formula [14] improved the Domke formula by replacing the Tresca yield criterion with the Mohr-Coulomb yield criterion. Removing [PERSON]'s limitation on the outer radius of the frozen wall plastic zone, [PERSON] [15] proposed a fully plastic single-layer, thick-walled cylinder model formula. Recently, there has been a surge in the utilization of unloading models to enhance the analysis of frozen walls [16; 17; 18; 19; 20]. Because the Domke formula has large errors when the modulus ratio of the frozen wall to its surrounding unfrozen alluvium is less than 10, [PERSON] [17] started with the geostress unloading mechanism and presented a formula that comprehensively considered the excavation unloading effect, the interaction between the frozen wall and the surrounding unfrozen alluvium, and [PERSON]'s limitation on the outer radius of the frozen wall plastic zone. This formula was successfully applied in an alluvium depth range of 400 m to 700 m. The results showed that the radial stress at the outer boundary of the frozen wall following excavation was evidently lower than the initial geostress, leading to a decrease in the thickness of the frozen wall by 10% to 36% in comparison to the results derived from the Domke formula. The geotechnical support characteristic curves suggest that the stability of the underground structure is highly influenced by the support effect. The movement of the inner boundary of the frozen wall is closely linked to the stability of the shaft wall.