## 4 Case Example  
The background of this case example is a shaft to be constructed in North China with an alluvium depth exceeding 800 m. Equation (41) is used to calculate the design thickness \(t\) of the frozen wall at depths of 800 m, 900 m, 1000 m, and 1100 m.  
In this case example [37], the average temperature of the frozen wall is \(-35\ ^{\circ}\)C. The elastic modulus, Poisson's ratio, internal friction angle, dilation angle, and uniaxial compressive strength of the frozen wall are \(E^{\text{II}}=400\) MPa, \(\mu^{\text{II}}=0.2\), \(\varphi=6\), \(\psi=0\), and \(\sigma_{\text{c}}=12\) MPa, respectively. The elastic modulus and Poisson's ratio of the surrounding unfrozen alluvium are \(E^{\text{III}}=150\) MPa and \(\mu^{\text{III}}=0.2\), respectively. The outer radius of the shaft lining \(r_{\text{a}}\) is 5 m, and the stiffness of the outer edge of the shaft lining \(k\) is 2000 MPa/m. The initial horizontal ground pressure is determined by [37,38]  
\[p_{0}=-0.013\ h \tag{44}\]  
where  
(1) When \(\alpha\) is constant, \(t\) increases nonlinearly and rapidly with \(h\), and \(p\) decreases with \(h\). When \(h\) is constant, \(t\) decreases with \(\alpha\), and \(p\) increases with \(\alpha\). (2) With the parameter \(\alpha\) kept constant, the differences between the frozen thickness-depth curves for both radially incomplete unloading and radial complete unloading conditions enlarge with the increase in depth. When the \(t\) curve with \(\alpha=0\) is used as a baseline, \(t\) is approximately 95% (\(\alpha=0.05\)), 90% (\(\alpha=0.1\)), 85% (\(\alpha=0.15\)), and 80% (\(\alpha=0.2\)) of the baseline at depths ranging from 800 m to 1100 m. When \(\alpha\) increases by 0.05, \(t\) decreases by approximately 0.33 m (\(h=800\) m), 0.42 m (\(h=900\) m), 0.54 m (\(h=1000\) m), and 0.68 m (\(h=1100\) m), indicating that the effect of \(\alpha\) on \(t\) increases with \(h\). (3) [PERSON]'s formula is derived based on the model for the frozen wall where the inner edge is completely unloaded in a radial manner, which should be a special case for \(\alpha=0\) in this paper. Compared with results from [PERSON]'s formula, the results when \(\alpha=0\) in this paper should be the same as those when using [PERSON]'s formula. As can be seen from Table 2, there is no difference between the results of the two methods, which verifies the effectiveness and accuracy of the method proposed in this paper. The above analyses show that \(\alpha\) has an important effect on the design thickness of frozen walls, and the effect intensifies as the depth increases. \(\alpha\) is a construction parameter that depends on the excavation height-radius ratio and the extent of the freezing of the soils. The smaller the excavation height-radius ratio, the larger the value of \(\alpha\) should be. Based on field data, \(\alpha\) can be obtained using the results of the interaction force \(p\) between the frozen wall and the shaft lining. Based on to the field data of the freezing shaft sinking of Wanfu Mine [39], the results of \(p\) from the auxiliary shaft at the alluvium depth of 697.5 m are shown in Figure 11, which is of great reference. Prior to the construction of a section shaft lining, earth pressure cell sensors were affixed as evenly as possible to the inner edge of the frozen wall. After the construction of the section shaft lining, the interaction force \(p\) between the frozen wall and the shaft lining increases at a decreasing rate with the progress of construction, and finally tends towards steadiness, as shown in Figure 14. Due to the complexity of the construction, the duration curves of \(p\) in Figure 14 do not overlap ideally with each other as expected, which is a common problem in field surveys [40]. When \(\alpha=0.05\), the difference between \(p\) in Figure 13 at the depth of 800 m and the max steady value (sensor NO. 6) in Figure 14 is no more than 0.4 Mpa, and when \(\alpha=0.15\), the difference between \(p\) in Figure 11 at the depth of 800 m and the min steady value (sensor NO.