Calculation Method of the Design Thickness of a Frozen Wall with Its Inner Edge Radially Incompletely Unloaded

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, α, 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 α 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 α 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.

1. Introduction

The technology for freezing shaft sinking is a special construction technology that utilizes artificial ground freezing methods to form a closed frozen geotechnical curtain (frozen wall) around a supporting structure (a shaft) by temporarily freezing the water in strata to improve the bearing capacity and water-sealing performance of the strata and protect the construction safety of shafts [1,2,3]. It has been gradually popularized and applied due to its outstanding advantages of strong adaptability, high reliability, and good technical and social benefits, and it has become the main construction technology for underground structures in deep water-bearing unstable soil layers and water-rich rock strata [4,5,6,7]. The depth that the shaft is sunk through an alluvium is a landmark indicator to measure the level of technology for freezing shaft sinking [8]. For example, the auxiliary shaft of Wanfu Coal Mine was sunk through alluvium to a depth of 754.96 m (world record) using freezing sinking technology [9]. More than 1100 vertical shafts have been constructed in China using this technology.

The thickness of the frozen wall is closely related to the safety and cost of the freezing shaft sinking project [10,11]. For example, the thickness of the frozen wall in the auxiliary shaft of Wanfu Coal Mine reaches 12.5 m, with an average temperature of −23 °C, which requires a cooling capacity of up to 15,407 kW. Ground pressure and ground temperature increase with the depth that the shaft is sunk through the alluvium, which leads to a 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 alluvia.

Theories for designing the frozen wall thickness have developed with the widespread use of freezing shaft sinking in alluvia. 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 Domke’s limitation on the outer radius of the frozen wall plastic zone, Sanger [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, Yang [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 Domke’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. The existing unloading models do not take into account the effect of displacement and stress unloading at the inner edge of the frozen wall on the design thickness of the frozen wall.

To reduce the displacement of frozen walls during excavations, a technique using short-sectional excavation and shaft lining construction is adopted [21]; it causes interaction between the frozen wall and the shaft, and the radial stress in the inner edge of the frozen wall is incompletely unloaded. With the increasing demand for underground resources, the depth that the shaft is sunk through the alluvium increases, and the excavation height–radius ratio for each section is generally reduced to 1/3~2/3. The characteristics of the inner edge of the frozen wall being radially incompletely unloaded become more obvious, and the deformation of the frozen wall is subject to greater constraints. Therefore, the design process should consider the attributes of being radially incompletely unloaded at the inner boundary of the frozen wall.

In this paper, based on the construction technique using short-sectional excavation and shaft lining, a mechanical model was established for a frozen wall with its inner edge radially incompletely unloaded, comprehensively considering the in situ stress and the frozen wall–shaft lining interaction. Analytical solutions were derived. A parameter α expressing the degree of being unloaded was introduced, and then a new method of designing and calculating values for an elastoplastic frozen wall thickness was developed based on α and the solutions. The influences of the mechanical parameters of the shaft lining, frozen wall, and surrounding unfrozen alluvium on the design thickness of the frozen wall were also discussed. The range of parameter α was estimated based on the frozen wall–shaft lining interaction forces from field data from a given project. A concrete application case is presented at the end of this paper.

2. Mechanical Model and Solution

The process for the method of frozen wall thickness design in this paper is shown in Figure 1.

The schematic for the technique using short-sectional excavation and shaft lining is illustrated in Figure 2. Following the lining of the inner edge of the frozen wall, excavation proceeds downward, causing the working face to move downward, thus creating a new exposed section of the frozen wall. The frozen wall at the depth of the exposed section is selected as the research object. The frozen soil below the exposed section and the shaft wall above it jointly restrict the wall’s deformation. With the gradual sinking of the working face, the exposed section of the frozen wall continues to deform and begins to interact with the shaft. During the interaction between the research object and the shaft, the constraint on the object from the working face gradually decreases until it disappears completely, while the research object–shaft interaction force synchronously increases until it stabilizes. The constraint on the exposed section is approximated to be transformed into the freezing wall–shaft interaction. For the reasons above, on the basis of Yang’s model for the frozen wall where the inner edge is radially incompletely unloaded, with some basic assumptions added, a mechanic model for the frozen wall where the inner edge is radially completely unloaded is ultimately created in this paper.

2.1. Mechanical Model

The mechanical model is shown in Figure 3, where N, I, II, and III represent the shaft lining, the frozen wall plastic zone, the frozen wall elastic zone, and the surrounding unfrozen alluvium, respectively; $r_a$, $r_b$, and $r_c$ are the outer radius of the shaft lining, the excavation, and the outer radii of the frozen wall, respectively; and $t$ is the design thickness of the frozen wall.

The following assumptions are made in the model.

(1) The shaft lining, the frozen wall, and the surrounding unfrozen alluvium are axisymmetric, infinitely long, and thick-walled cylinders, and the displacement of the surrounding unfrozen alluvium at $r$ → ∞ is zero.

(2) The shaft lining and the surrounding unfrozen alluvium are homogeneous, isotropic, linear elastic materials; the frozen wall is an ideal elastoplastic material, and the stresses in the plastic zone satisfy the Mohr–Coulomb yield criterion.

(3) Before and after the formation of the frozen wall, the in situ stresses remain constant.

$${\sigma }_r^0={\sigma }_{\theta }^0=p_0$$

(1)

where $p_0$ is the initial horizontal ground stress, $\sigma$ represents the stress, and the subscripts $r$ and $\theta$ represent the radial and circumferential directions, respectively. The positive and negative signs of the stresses follow the conventions of elasticity theory.

(4) At the instant that the sectional excavation is completed, the inner edge of the frozen wall is displaced in the radial direction and then blocked by the outer edge of the shaft, causing the interaction force between the frozen wall and the shaft to be $p$.

2.2. Initial, Boundary, and Continuity Conditions

The initial states of the frozen wall and the surrounding unfrozen alluvium prior to the excavation are

$${\sigma }_r={\sigma }_{\theta }=p_0$$

(2)

$$u=0$$

(3)

where u denotes the displacement, with a negative notation, following the sign convention of displacement in elasticity theory.

At the interface between the elastic and plastic zones of the frozen wall, the displacement and stress continuity conditions are [22]

$${\left.u^{\mathrm{I}}\right|}_{r=r_p}={\left.u^{\mathrm{I}\mathrm{I}}\right|}_{r=r_p}$$

(4)

$${\left.{\sigma }_r^{\mathrm{I}}\right|}_{r=r_p}={\left.{\sigma }_r^{\mathrm{I}\mathrm{I}}\right|}_{r=r_p}=p_p$$

(5)

$${\left.{\sigma }_{\theta }^{\mathrm{I}}\right|}_{r=r_p}={\left.{\sigma }_{\theta }^{\mathrm{I}\mathrm{I}}\right|}_{r=r_p}$$

(6)

At the interface between the frozen wall and the surrounding unfrozen alluvium, the displacement and stress continuity conditions are

$${\left.u^{\mathrm{I}\mathrm{I}}\right|}_{r=r_c}={\left.u^{\mathrm{I}\mathrm{I}\mathrm{I}}\right|}_{r=r_c}$$

(7)

$${\left.{\sigma }_r^{\mathrm{I}\mathrm{I}}\right|}_{r=r_c}={\left.{\sigma }_r^{\mathrm{I}\mathrm{I}\mathrm{I}}\right|}_{r=r_c}$$

(8)

At the infinity of the surrounding unfrozen alluvium, the stress and displacement boundary conditions are

$${\left.u^{\mathrm{I}\mathrm{I}\mathrm{I}}\right|}_{r\to \infty }=0$$

(9)

$${\left.{\sigma }_r^{\mathrm{I}\mathrm{I}\mathrm{I}}\right|}_{r\to \infty }={\left.{\sigma }_{\theta }^{\mathrm{I}\mathrm{I}\mathrm{I}}\right|}_{r\to \infty }=p_0$$

(10)

2.3. Displacement and Stress of the Frozen Wall Elastic Zone

To facilitate the solution, the mechanical model shown in Figure 3 is split into two basic models as shown in Figure 4 and Figure 5.

The analytical solutions of stress and displacement for the elastic zone of the frozen wall are equal to the solutions for the unloading model shown in Figure 3 superimposed over the initial stress and displacement [23,24].

The unloading amount of the radial stress at the inner edge of the frozen wall elastic zone is

$$\Delta p_p=p_p-p_0$$

(11)

According to the elastic thick-walled cylinder theory [25,26] and combining Equations (7), (10) and (11), the stress and the displacement of the unloading model can be obtained. Based on the aforementioned superposition, analytical solutions of the stress and displacement for the frozen wall elastic zone and the surrounding unfrozen alluvium are obtained as follows:

$${\sigma }_r^{\mathrm{I}\mathrm{I}}=-\Delta p_p\frac{\zeta -\zeta \frac{r_p^2}{r^2}-\frac{r_p^2}{r_c^2}+\frac{r_p^2}{r^2}}{\frac{r_p^2}{r_c^2}-1}+p_0$$

(12)

$${\sigma }_{\theta }^{\mathrm{I}\mathrm{I}}=-\Delta p_p\frac{\zeta +\zeta \frac{r_p^2}{r^2}-\frac{r_p^2}{r_c^2}-\frac{r_p^2}{r^2}}{\frac{r_p^2}{r_c^2}-1}+p_0$$

(13)

$$u^{\mathrm{I}\mathrm{I}}=r\frac{\Delta p_p\left(1+{\mu }^{\mathrm{I}\mathrm{I}}\right)}{E^{\mathrm{I}\mathrm{I}}}\left[1-2{\mu }^{\mathrm{I}\mathrm{I}}-\frac{1-\zeta }{1-\frac{r_p^2}{r_c^2}}\left(1-2{\mu }^{\mathrm{I}\mathrm{I}}+\frac{r_p^2}{r^2}\right)\right]$$

(14)

$${\sigma }_r^{\mathrm{I}\mathrm{I}\mathrm{I}}=\Delta p_p\frac{\zeta r_c^2}{r^2}+p_0$$

(15)

$${\sigma }_{\theta }^{\mathrm{I}\mathrm{I}\mathrm{I}}=-\Delta p_p\zeta \frac{r_c^2}{r^2}+p_0$$

(16)

$$u^{\mathrm{I}\mathrm{I}\mathrm{I}}=-\Delta p_p\zeta \frac{1+{\mu }^{\mathrm{I}\mathrm{I}\mathrm{I}}}{E^{\mathrm{I}\mathrm{I}\mathrm{I}}}\frac{r_c^2}{r}$$

(17)

where $\zeta =\frac{2}{\left[2+M\left(\frac{r_c^2}{r_p^2}-1\right)\right]}$; $M=\left(\frac{E^{\mathrm{I}\mathrm{I}}}{E^{\mathrm{I}\mathrm{I}\mathrm{I}}}-1\right)\frac{1+{\mu }^{\mathrm{I}\mathrm{I}\mathrm{I}}}{1-{\left({\mu }^{\mathrm{I}\mathrm{I}}\right)}^2}+2+\frac{{\mu }^{\mathrm{I}\mathrm{I}\mathrm{I}}-{\mu }^{\mathrm{I}\mathrm{I}}}{1-{\left({\mu }^{\mathrm{I}\mathrm{I}}\right)}^2}$; $E$ and $\mu$ are the elastic modulus and the Poisson’s ratio, respectively; and superscripts II and III represent the frozen wall elastic zone and the surrounding unfrozen alluvium, respectively.

From Equations (13) and (14), the solutions for the circumferential stress and the displacement at the inner edge of the frozen wall elastic zone are

$${\left.{\sigma }_{\theta }^{\mathrm{I}\mathrm{I}}\right|}_{r=r_p}=-\Delta p_p\frac{2\zeta -\frac{r_p^2}{r_c^2}-1}{\frac{r_p^2}{r_c^2}-1}+p_0$$

(18)

$${\left.u^{\mathrm{I}\mathrm{I}}\right|}_{r=r_p}=r_p\frac{\Delta p_p\left(1+{\mu }^{\mathrm{I}\mathrm{I}}\right)}{E^{\mathrm{I}\mathrm{I}}}\left[1-2{\mu }^{\mathrm{I}\mathrm{I}}-\frac{2\left(1-\zeta \right)\left(1-{\mu }^{\mathrm{I}\mathrm{I}}\right)}{1-\frac{r_p^2}{r_c^2}}\right]$$

(19)

2.4. Displacement and Stress of the Frozen Wall Plastic Zone

From the schematic representation of ground–support interaction [27,28] and the intersection of the ground reaction and the support characteristic curves, the radial stress and the displacement at the outer edge of the shaft lining and the radial stress and the displacement at the inner edge of the frozen wall are determined to be

$${\left.{\sigma }_r^{\mathrm{N} }\right|}_{r=r_a}=p$$

(20)

$${\left.{\sigma }_r^{\mathrm{I} }\right|}_{r=r_b}=p$$

(21)

$${\left.{\sigma }_r^{\mathrm{N} }\right|}_{r=r_b}=\frac{p}{k}$$

(22)

$${\left.u^{\mathrm{I}}\right|}_{r=r_b}=r_a-r_b+{\left.u^N\right|}_{r=r_a}$$

(23)

where superscript N represents the shaft lining, k is the stiffness of the shaft’s outer edge, and $p$ represents the interaction force between the frozen wall and the shaft lining.

The Mohr–Coulomb yield criterion describes the relationship between the first and third principal stresses, and the axial stress is generally regarded as the intermediate principal stress [29,30,31]. The relationship between the radial and circumferential stresses in the frozen wall plastic zone is [32,33,34]

$${\sigma }_{\theta }^{\mathrm{I}}=A{\sigma }_r^{\mathrm{I}}-{\sigma }_c$$

(24)

where $A=\frac{1+\mathit{sin}\phi }{1-\mathit{sin}\phi }$, ${\sigma }_c=\frac{2c}{\mathit{tan}\left(45^{°}-\frac{\phi }{2}\right)}$. $\phi$, $c$, and ${\sigma }_c$ are the internal friction angle, cohesion, and uniaxial compressive strength of the frozen soils, respectively. Superscript I represents the frozen wall plastic zone.

The static equilibrium differential equation [35] for the frozen wall plastic zone (see Figure 5) is

$$\frac{d{\sigma }_r}{dr}+\frac{{\sigma }_r-{\sigma }_{\theta }}{r}=0$$

(25)

Substituting Equations (24) and (25) and combining them with Equation (21), solutions of the radial and circumferential stresses for the plastic zone of the frozen wall are

$${\sigma }_r^{\mathrm{I} }=\frac{-{\sigma }_c}{A-1}\left[{\left(\frac{r}{r_b}\right)}^{A-1}-1\right]+p{\left(\frac{r}{r_b}\right)}^{A-1}$$

(26)

$${\sigma }_{\theta }^{\mathrm{I} }=\frac{-A{\sigma }_c}{A-1}\left[{\left(\frac{r}{r_b}\right)}^{A-1}-1\right]+Ap{\left(\frac{r}{r_b}\right)}^{A-1}-{\sigma }_c$$

(27)

From Equations (5) and (26),

$$p_p={\left.{\sigma }_r^{\mathrm{I}}\right|}_{r=r_p}=\frac{-{\sigma }_c}{A-1}\left[{\left(\frac{r_p}{r_b}\right)}^{A-1}-1\right]+p{\left(\frac{r_p}{r_b}\right)}^{A-1}$$

(28)

From Equation (27), the solution of the circumferential stress at the outer edge of the frozen wall plastic zone is

$${\left.{\sigma }_{\theta }^{\mathrm{I}}\right|}_{r=r_p}=\frac{-A{\sigma }_c}{A-1}\left[{\left(\frac{r_p}{r_b}\right)}^{A-1}-1\right]+Ap{\left(\frac{r_p}{r_b}\right)}^{A-1}-{\sigma }_c$$

(29)

Substituting Equations (18) and (29) into Equation (9), the relationship among r_b, r_c, and r_p is obtained:

$${\left(\frac{r_p}{r_b}\right)}^{A-1}=\frac{\frac{p_0}{p\left(A-1\right)-{\sigma }_c}-\frac{1}{A-1}\frac{{\sigma }_c}{p\left(A-1\right)-{\sigma }_c}}{\left(\frac{1}{M}-\frac{1}{2}\right)\frac{r_p^2}{r_c^2}+\frac{A+1}{2\left(A-1\right)}}$$

(30)

If $r_b=r_a$ and $p=0$, the mechanical model in this paper (Figure 3) degenerates to that presented by Yang [17] for the frozen wall where the inner edge is completely unloaded in a radial manner; thus, Equation (30) degenerates to

$${\left(\frac{r_p}{r_a}\right)}^{A-1}=\frac{\frac{1}{A-1}-\frac{p_0}{{\sigma }_c}}{\left(\frac{1}{M}-\frac{1}{2}\right)\frac{r_p^2}{r_c^2}+\frac{A+1}{2\left(A-1\right)}}$$

(31)

Correspondingly, Equation (31) is the same as the formula presented by Yang [17].

Geomaterials do not follow the associated flow rule, and their plastic potential function φ is generally expressed as [36]

$$\varphi =\beta {\sigma }_r-{\sigma }_{\theta }$$

(32)

where $\beta =\frac{1+\mathit{sin}\psi }{1-\mathit{sin}\psi }$, and $\psi$ is the dilation angle of frozen soil, 0 ≤ $\psi$ ≤ $\phi$.

Ignoring the contribution of the elastic deformation and according to plastic potential theory, the plastic strains of the frozen wall satisfy the following relationship:

$${\epsilon }_r^{\mathrm{I}}+\beta {\epsilon }_{\theta }^{\mathrm{I}}=0$$

(33)

where ${\epsilon }_r^{\mathrm{I}}$ and ${\epsilon }_{\theta }^{\mathrm{I}}$ are the radial and circumferential strains, respectively.

The geometric equations for the axisymmetric plane strain are

$${\epsilon }_r=\frac{du}{dr}$$

(34)

$${\epsilon }_{\theta }=\frac{u}{r}$$

(35)

The substitution of Equations (34) and (35) into Equation (33) results in

$$\frac{du}{dr}+\beta \frac{u}{r}=0$$

(36)

Combining Equation (36) with Equations (4) and (19), the analytical solution of the displacement for the frozen wall plastic zone is

$${\mathit{u}}^{\mathbf{I}}=\frac{{\mathit{r}}_{\mathit{p}}}{{\mathit{E}}^{\mathbf{I}\mathbf{I}}}\left(1+{\mathit{\mu }}^{\mathbf{I}\mathbf{I}}\right){\left(\frac{{\mathit{r}}_{\mathit{p}}}{\mathit{r}}\right)}^{\mathit{\beta }}\left\{\frac{-{\mathit{\sigma }}_{\mathit{c}}}{\mathit{A}-1}\left[{\left(\frac{{\mathit{r}}_{\mathit{p}}}{{\mathit{r}}_{\mathit{b}}}\right)}^{\mathit{A}-1}-1\right]+\mathit{p}{\left(\frac{{\mathit{r}}_{\mathit{p}}}{{\mathit{r}}_{\mathit{b}}}\right)}^{\mathit{A}-1}-{\mathit{p}}_0\right\}\frac{\left(\frac{2}{\mathit{M}}-1\right)\left(1-2{\mathit{\mu }}^{\mathbf{I}\mathbf{I}}\right)-\frac{{\mathit{r}}_{\mathit{c}}^2}{{\mathit{r}}_{\mathit{p}}^2}}{\frac{2}{\mathit{M}}-1+\frac{{\mathit{r}}_{\mathit{c}}^2}{{\mathit{r}}_{\mathit{p}}^2}}$$

(37)

3. Design for the Frozen Wall Thickness

3.1. Calculation Method and Formula

For the frozen wall with its inner edge radially incompletely unloaded, the deformation is constrained and the displacement of the inner edge is limited to a part of that with the inner edge radially completely unloaded. To express the degree of being unloaded, parameter α is introduced using the ratio of the displacement of the inner edge of the frozen wall with its inner edge radially incompletely unloaded to that of the inner edge of the same frozen wall with its inner edge radially completely unloaded; see Equation (38):

$$\alpha =1-\frac{{\left.u^{\mathrm{I}}\right|}_{r=r_b}}{{\left.U^{\mathrm{I}}\right|}_{r=r_b}}$$

(38)

where α decreases with the degree of being unloaded, and α = 0 represents that the inner edge of the frozen wall is radially completely unloaded. ${\left.U^{\mathrm{I}}\right|}_{r=r_b}$ is the displacement of the inner edge of the same frozen wall with its inner edge radially completely unloaded.

Substituting r_p = R_p and $p$ = 0 into Equation (37), ${\left.U^{\mathrm{I}}\right|}_{r=r_b}$ is expressed as

$${\left.U^{\mathrm{I}}\right|}_{r=r_b}=\frac{R_p}{E^{\mathrm{I}\mathrm{I}}}\left(1+{\mu }^{\mathrm{I}\mathrm{I}}\right){\left(\frac{R_p}{r_b}\right)}^{\beta }\left\{\frac{-{\sigma }_c}{A-1}\left[{\left(\frac{R_p}{r_b}\right)}^{A-1}-1\right]-p_0\right\}\frac{\left(\frac{2}{M}-1\right)\left(1-2{\mu }^{\mathrm{I}\mathrm{I}}\right)-\frac{r_c^2}{R_p^2}}{\frac{2}{M}-1+\frac{r_c^2}{R_p^2}}$$

(39)

where R_p is the outer radius of the plastic zone of the same frozen wall with its inner edge radially completely unloaded.

Substituting r_p = R_p and p = 0 into Equation (30) R_p can be solved using

$${\left(\frac{R_p}{r_b}\right)}^{A-1}=\frac{-\frac{p_0}{{\sigma }_c}+\frac{1}{A-1}}{\left(\frac{1}{M}-\frac{1}{2}\right)\frac{R_p^2}{r_c^2}+\frac{A+1}{2\left(A-1\right)}}$$

(40)

Following Domke’s limitation imposed on the outer radius of the frozen wall plastic zone [13], $r_p=\sqrt{r_b{r}_c}$ is substituted into Equations (30) and (37). Combining Equation (38) with Equations (38)–(40), the formulas for the design thickness of the frozen wall with its inner edge radially incompletely unloaded are determined as follows:

$$\begin{gathered} {\left(\frac{r_c}{r_b}\right)}^{\frac{A-1}{2}}=\frac{\frac{p_0}{p\left(A-1\right)-{\sigma }_c}-\frac{1}{A-1}\frac{{\sigma }_c}{p\left(A-1\right)-{\sigma }_c}}{\left(\frac{1}{M}-\frac{1}{2}\right)\frac{r_b}{r_c}+\frac{A+1}{2\left(A-1\right)}} \\ \frac{r_c^{\frac{1+\beta }{2}}{r}_b^{\frac{1-\beta }{2}}\left(1+{\mu }^{\mathrm{I}\mathrm{I}}\right)}{E^{\mathrm{I}\mathrm{I}}}\left[\left(\frac{-{\sigma }_c}{A-1}+p\right){\left(\frac{r_c}{r_b}\right)}^{\frac{A-1}{2}}+\frac{{\sigma }_c}{A-1}-p_0\right]\frac{\left(\frac{2}{M}-1\right)\left(1-2{\mu }^{\mathrm{I}\mathrm{I}}\right)-\frac{r_c}{r_b}}{\frac{2}{M}-1+\frac{r_c}{r_b}}=\frac{p}{k}+r_a-r_b \\ =\frac{R_p\left(1-\alpha \right)\left(1+{\mu }^{\mathrm{I}\mathrm{I}}\right)}{E^{\mathrm{I}\mathrm{I}}}{\left(\frac{R_p}{r_b}\right)}^{\beta }\left\{\frac{-{\sigma }_c}{A-1}\left[{\left(\frac{R_p}{r_b}\right)}^{A-1}-1\right]-p_0\right\}\frac{\left(\frac{2}{M}-1\right)\left(1-2{\mu }^{\mathrm{I}\mathrm{I}}\right)-\frac{r_c^2}{R_p^2}}{\frac{2}{M}-1+\frac{r_c^2}{R_p^2}} \\ {\left(\frac{R_p}{r_b}\right)}^{A-1}=\frac{\frac{1}{A-1}-\frac{p_0}{{\sigma }_c}}{\left(\frac{1}{M}-\frac{1}{2}\right)\frac{R_p^2}{r_c^2}+\frac{A+1}{2\left(A-1\right)}} \\ t=r_c-r_b \end{gathered}$$

(41)

r_a has a known value, and parameter α in Equation (41) will be discussed later. The mechanical parameters of the shaft lining, the frozen wall, and the surrounding unfrozen alluvium are substituted into Equation (41) to obtain $t$, $p$, $r_b$, $r_c$, and $R_p$.

If the inner edge is radially completely unloaded, α = 0, $p$ = 0, $R_p=r_p$, and ${\left.{{\left.u^{\mathrm{I}}\right|}_{r=r_b}=U}^{\mathrm{I}}\right|}_{r=r_b}$; thus, Equation (41) degenerates to

$$\begin{gathered} {\left(\frac{r_c}{r_b}\right)}^{\frac{A-1}{2}}=\frac{\frac{p_0}{p\left(A-1\right)-{\sigma }_c}-\frac{1}{A-1}\frac{{\sigma }_c}{p\left(A-1\right)-{\sigma }_c}}{\left(\frac{1}{M}-\frac{1}{2}\right)\frac{r_b}{r_c}+\frac{A+1}{2\left(A-1\right)}} \\ t=r_c-r_b \end{gathered}$$

(42)

Correspondingly, Equation (42) is the same as the formula presented by Yang [17] for the frozen wall where the inner edge is completely unloaded in a radial manner.

3.2. Influences of the Parameters on the Design Thickness of the Frozen Wall

To facilitate the analysis of the influences of mechanical parameters, σ_c, E^II, E^III, k, and t are made dimensionless, as shown in Equation (43):

$$\tilde{t}=\frac{t}{r_a}, {\tilde{\sigma }}_c=\frac{-{\sigma }_c}{p_0},{\tilde{E}}^{\mathrm{I}\mathrm{I}}=-\frac{E^{\mathrm{I}\mathrm{I}}}{p_0},{\tilde{E}}^{\mathrm{I}\mathrm{I}\mathrm{I}}=\frac{-E^{\mathrm{I}\mathrm{I}\mathrm{I}}}{p_0},\tilde{k}=-\frac{k{r}_a}{p_0}$$

(43)

Given typical engineering parameters, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 show that $\tilde{t}$ decreases significantly with α.

Figure 6 shows that $\tilde{t}$ decreases significantly with ${\tilde{\sigma }}_c$. The larger α is, the slower $\tilde{t}$ decreases with ${\tilde{\sigma }}_c$. α has a distinct effect on $\tilde{t}$ while ${\tilde{\sigma }}_c\le 1$, and it has a slight effect while ${\tilde{\sigma }}_c\ge 1.2$. The difference between the curves decreases with ${\tilde{\sigma }}_c$. Figure 6 indicates that the smaller the radial unloading of the frozen wall, the thinner the design thickness, especially when the strength of the wall is relatively low.

Figure 7 and Figure 8 show that $\tilde{t}$ decreases significantly with ${\tilde{E}}^{\mathrm{I}\mathrm{I}\mathrm{I}}$ and increases significantly with ${\tilde{E}}^{\mathrm{I}\mathrm{I}}$. The higher the elastic modulus of the frozen wall, or the lower the elastic modulus of the surrounding unfrozen alluvium, the better for the stability of the frozen wall.

Figure 9 shows that $\tilde{t}$ decreases merely by 0.15% when $\tilde{k}$ varies from the typical value to infinity. Figure 10, Figure 11 and Figure 12 show that $\tilde{t}$ increases slightly with ψ, μ^II, and μ^III, and the maximum increases are 3.5%, −2%, and 7.2%, while α = 0.3, respectively. With the change in these four variables, the curves of the design thickness of the frozen wall are maintained at their respective relatively stable levels.

Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 reveal that the design thickness of the frozen wall varies significantly in sensitivity to various influencing factors. Further, by means of single-factor analysis of variance in mathematics, quantitative data on the sensitivity of freezing wall design thickness to various factors by different values of α are shown in

Table 1. Results of single-factor analysis of variance.

Factor	α				Average
	0	0.1	0.2	0.3
${\tilde{\sigma }}_c$	3.9941	3.2104	2.5281	1.9556	2.9221
${\tilde{E}}^{\mathrm{I}\mathrm{I}\mathrm{I}}$	0.0315	0.0393	0.047	0.0584	0.044
${\tilde{E}}^{\mathrm{I}\mathrm{I}}$	0.0017	0.0030	0.0045	0.0064	0.0039
μIII	0.0014	0.0017	0.0020	0.0024	0.001
ψ	0.000	0.0002	0.000	0.0006	0.0003
μII	0.0001	4.0 × 10−6	3.9 × 10−5	0.0002	0.0001
$\tilde{k}$	9.3 × 10−10	2.6 × 10−6	1.3 × 10−6	2.2 × 10−6	1.5 × 10−6

.

Of the many factors, identifying the dominant ones is crucial, as they have a significant impact on design, construction, and similar simulation tests related to the frozen wall; therefore, they require special attention.

As seen in Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12 and Table 1, the sensitivity of the design thickness of frozen walls to the strength and elastic modulus of the frozen soil, and the elastic modulus of the surrounding unfrozen alluvium, change significantly as the value of α increases. When α is kept constant, if the frozen wall design thickness varies significantly with a factor, the variance associated with this factor in Table 1 is also relatively large, indicating that the frozen wall design thickness has a high sensitivity to this factor. Table 1 shows that the mean variance of frozen wall strength at different values of α clearly exceeds that of other factors. The ranking of these mean variances indicates that the strength and elastic modulus of the frozen soil and the elastic modulus of the surrounding unfrozen alluvium are the main influencing factors. The dilation angle and Poisson’s ratio of frozen soil, Poisson’s ratio of the surrounding unfrozen alluvium, and the stiffness of the shaft’s outer edge are secondary influencing factors.

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 °C. The elastic modulus, Poisson’s ratio, internal friction angle, dilation angle, and uniaxial compressive strength of the frozen wall are E^II = 400 MPa, μ^II = 0.2, φ = 6°, ψ = 0°, and σ_c = 12 MPa, respectively. The elastic modulus and Poisson’s ratio of the surrounding unfrozen alluvium are E^III = 150 MPa and μ^III = 0.2, respectively. The outer radius of the shaft lining r_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]

$$p_0=-0.013 h$$

(44)

where p₀ is the initial horizontal ground stress (MPa) and h is the depth (m).

Figure 13 shows the curves of $t$ and $p$ with $h$ and different values of $\alpha$. Based on Figure 13 and

Table 2. Computational results of frozen wall thickness $t$ (m) with different values of α.

Formula		h/m
		800	900	1000	1100
Yang’s formula [17]		6.85	8.90	11.32	14.21
$\alpha$	0	6.85	8.90	11.32	14.21
	0.05	6.50	8.46	10.75	13.51
	0.1	6.15	8.00	10.20	12.79
	0.15	5.83	7.59	9.65	12.11
	0.2	5.48	7.15	9.10	11.43

, the analyses are as follows:

(1) When $\alpha$ is constant, $t$ increases nonlinearly and rapidly with $h$, and $p$ decreases with $h$. When $h$ is constant, $t$ decreases with α, and $p$ increases with α.

(2) With the parameter α 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% (α = 0.2) of the baseline at depths ranging from 800 m to 1100 m. When α 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 α on $t$ increases with $h$.

(3) Young’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 Yang’s formula, the results when $\alpha$ = 0 in this paper should be the same as those when using Yang’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. α 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 α = 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 α = 0.15, the difference between $p$ in Figure 11 at the depth of 800 m and the min steady value (sensor NO. 2) in Figure 14 is less than 0.3 Mpa, and thus the upper limit and lower limit of α are approximately 0.15 and 0.05, respectively.

According to the calculation results shown in Table 2, when α is chosen as the lower limit (α = 0.05), the calculated design thickness of the frozen wall is approximately 95% of that of the frozen wall with the inner edge radially completely unloaded. When α is chosen as the upper limit (α = 0.15), the calculated design thickness of the frozen wall is approximately 85% of that of frozen wall with the inner edge radially completely unloaded. These results indicate that the design thickness of the frozen wall in deep alluvia can be reduced by at least 5% for the frozen wall with the inner edge radially incompletely unloaded.

5. Conclusions

According to the construction technique using sectional excavation and shaft lining, a mechanical model was established for a frozen wall with its inner edge radially incompletely unloaded, comprehensively considering the in situ stress and the frozen wall–shaft lining interaction. Analytical solutions were derived. A new method of designing and calculating an elastoplastic frozen wall thickness was developed on the solutions. The main conclusions are as follows:

(1) The increase in the design thickness with the depth, based on the new model, is smaller than that based on the mechanical model of radial complete unloading at the inner edge of the frozen wall.

(2) The design thickness of the frozen wall is significantly influenced by the strength and elastic modulus of the frozen soil and the elastic modulus of the surrounding unfrozen alluvium. The dilation angle and Poisson’s ratio of frozen soil, Poisson’s ratio of the surrounding unfrozen alluvium, and the stiffness of the shaft’s outer edge are secondary influencing factors.

(3) The degree of being radially unloaded has an important effect on the design thickness of the frozen wall, and the effect becomes more pronounced with increasing depth. Using the new method of designing and calculating the frozen wall thickness, the design thickness in the case example of this paper can be reduced by at least 5%. This paper provides theoretical support for guiding the design of frozen wall thickness in deep alluvia.