# Calculation of Limit Support Pressure for EPB Shield Tunnel Face in Water-Rich Sand

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## Abstract

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## 1. Introduction

## 2. Numerical Simulation of the Distribution Rules of the Seepage Field Near the Tunnel Face

#### 2.1. Overview of Calculation of a Hydraulic Gradient around the Tunnel Face

_{x}, k

_{y}, and k

_{z}are the seepage coefficients in x, y, and z directions. Additionally, h(x,y,z) is the total hydraulic head field.

_{F}is the piezometric head in the working chamber for the case of a closed shield, and △h is the difference between the elevation of the water table h

_{0}and h

_{F}. The constants a and b can be determined by curve fitting of Equations (2) and (3) to the numerical results. The specific values of the two parameters under a different ratio of cover depth to diameter of the tunnel should be determined by curve fittings of Equations (2) and (3) to the numerical results and are given in Reference [14]. Compared with the numerical simulations, the approximate analytical solutions proposed by Perazzelli et al. [14] are proved to provide relatively accurate solutions of a hydraulic gradient for the rectangular tunnel [8,23].

_{x}, f

_{y}, f

_{z}} are equal to the gradient of the numerically computed hydraulic head h(x,y,z), as shown in Equations (4)–(6).

_{cov}and i

_{cro}represent the hydraulic gradient in the cover layer and crossed layer, respectively.

#### 2.2. Finite Element Modeling

_{0}between the water surface and the tunnel floor is 100 m. The symmetry of the model is adopted for analysis. Shield tunneling is a progressive process. Considering that the research focus of the distribution rules of the seepage field near the shield tunnel face, the numerical simulation in this paper adopts the one-time excavation to a certain distance (30 m). Boundary conditions of the model are set as follows. The upper surface of the model is fixed pressure, γ

_{w}(h

_{0}-C), the surroundings boundary, bottom boundary, and tunnel liner are impervious boundaries, and the tunnel face is the fixed hydraulic head, h

_{tunnel face}= 0.

#### 2.3. Seepage Field Analysis

#### 2.4. New Approximate Analytical Solutions of Total Hydraulic Head Ahead of the Tunnel Face for the Circular Tunnel

_{0}, φ’

_{0}) and are dependent on ground geometric characteristics (i.e., cover depth and radius of the tunnel) and head elevation (h

_{w}), which provides a possibility for establishing new 3D formula of hydraulic head field without fitting with the numerical simulations. Figure 7 shows the geometric diagram of the underwater tunnel.

_{0}and h

_{F}denote the elevation of the water table and hydraulic head on the tunnel face, respectively. Additionally, r represents the radius of the tunnel.

## 3. Limit Equilibrium Model of the Tunnel Face Stability Considering the Effect of Seepage Forces

#### 3.1. Outline

_{T}is applied on the tunnel face to sustain stability. The effective weight of the soil is γ’, the cohesion of the soil is c’, the friction angle of the soil is φ’, and the permeability of the soil is k

_{0}. As shown in Figure 8, the wedge is acted upon by the volume forces and the surface forces. The volume forces include its weight and seepage force, while the surface forces include the support force of the slurry on the tunnel face, the resultant normal forces, and shear forces along the failure surfaces ade, bcf, and abfe as well as the resultant vertical force of the prism at the interface defc.

#### 3.2. Mechanical Analysis of the Prism

_{cov}can be calculated with the Equations (6) and (8).

_{cov}, the formula i

_{cov}is divided into N segments along the depth direction and i

_{cov}is assumed to be constant in each segment i

_{cov,i+1}($i=0,\text{}1,\text{}2,\text{}\dots ,N-1$). Then Equation (9) can be written as shown below.

#### 3.3. Mechanical Analysis of the Wedge

_{s}is the support force, N is the normal force acting on the surface of the wedge, and G is the gravity of the wedge.

_{s}on the wedge is then obtained:

_{x}in the wedge block are shown below.

#### 3.4. Calculation of Limit Support Pressure

_{cr.}is determined by iteratively maximizing the necessary support force.

_{1}, f

_{2}, f

_{3}, f

_{4}, and f

_{5}are:

## 4. Comparison of the Limit Support Pressures with the Existing Approaches

_{0}= C + D). The cohesion and friction angle of soil are 0 kPa and 35°. The dry and submerged gravities of the soil are 15.2 kN/m

^{3}and 5.4 kN/m

^{3}. As shown in Figure 11, the calculated limit support pressure increases linearly with the water table. Moreover, the results from this paper were between the results from Perazzelli et al. [14] (the highest solutions) and from Lee et al. [1] (the lowest solutions).

## 5. Sensitivity Analysis of Model Parameters on the Limit Support Pressures

#### 5.1. Influence of the Variables of the Hydraulic Head on Limit Support Pressures

_{T}/(γ’D) with a different effective friction angle of the soil φ’. The tunnel diameter D is 10 m and the cover depth C is 10 m. Hydraulic head h

_{0}are 20 m, 40 m, 60 m, 80 m, and 100 m. The effective friction angle of soil are 15°, 20°, 25°, 30°, and 35°. Submerged unit weight of the soil is 10 kN/m

^{3}.

#### 5.2. Influence of the Variables of the C/D on Limit Support Pressures

_{T}/(γ’D) with a different effective friction angle of the soil φ’. The tunnel diameter D is 10 m, and the cover depths C are 10, 20, 30, 40, and 50 m. Hydraulic head h

_{0}is 100 m. The effective friction angle of soil are 15°, 20°, 25°, 30°, and 35°. Submerged unit weight of the soil is 10 kN/m

^{3}.

## 6. Conclusions

- (1)
- The distribution law of total hydraulic head field along the horizontal distance at the axis of the shield tunnel is analyzed. The results show that the distribution of total hydraulic head at the axis of the shield tunnel face along the horizontal distance is a “negative exponential” function. The distribution law of total hydraulic head along the depth direction of the shield tunnel in front of the tunnel face on the vertical symmetrical surface is analyzed. The results indicate that the distribution of total hydraulic head along the depth is nonlinear when the tunnel is close to the tunnel face.
- (2)
- The comparative results of the horizontal and vertical hydraulic head field with the numerical simulations in this paper and existing approximate analytical solutions demonstrate accuracy of the formula proposed in this paper.
- (3)
- Comparisons of the results of limit support pressure obtained from the theoretical analysis in this paper and the existing approaches show that the failure mechanism proposed in this paper could provide relatively satisfactory results for the limit support pressures applied to the tunnel face.
- (4)
- When the buried depth ratio is the same, the normalized effective limit support pressures of the shield tunnel face increases linearly with the rise of the hydraulic head. When the hydraulic head is the same, the normalized effective limit support pressures of the shield tunnel face decreases nonlinearly with the increase of the buried depth ratio. Moreover, the higher the effective friction angle of the soil, the lower the normalized effective limit support pressures will be.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 8.**Comparison of the total hydraulic head obtained from approximate analytical solutions (this paper and Reference [14]) and numerical simulation for C/D = 1.

**Figure 9.**Comparison of the total hydraulic head obtained from approximate analytical solutions (this paper and Reference [14]) and numerical simulation for C/D = 5.

References | Hydraulic Gradient | Parameter Specification |
---|---|---|

Liu et al. [22] | $\{\begin{array}{l}{i}_{\mathrm{cov}}=\frac{{h}_{0}-{h}_{F}}{C}\\ {i}_{cro}=\frac{{h}_{0}-{h}_{F}}{\sqrt{CD}}\end{array}$ | h_{0}—undisturbed hydraulic headh _{F}—hydraulic head on the tunnel faceC—depth of cover D—diameter of the tunnel face |

Perazzelli et al. [14] | $\{\begin{array}{l}{i}_{\mathrm{cov}}=\frac{\left({h}_{0}-{h}_{F}\right)a}{H}{e}^{-b\frac{x}{H}+a\left(1-\frac{z}{H}\right)}\\ {i}_{cro}=\frac{\left({h}_{0}-{h}_{F}\right)b}{H}{e}^{-b\frac{x}{H}}\end{array}$ | a,b—constants determined by curve fitting to the numerical results H—height of the tunnel face x,z—vertical and horizontal coordinates |

Lei [23] | $\{\begin{array}{l}{i}_{\mathrm{cov}}=\frac{\left({h}_{0}-{h}_{F}\right)a}{r}{e}^{-b\frac{x}{r}+a\left(2-\frac{z}{r}\right)+c}\\ {i}_{cro}=\frac{\left({h}_{0}-{h}_{F}\right)b}{r}{e}^{-b\frac{x}{r}}\end{array}$ | a,b,c—constants determined by curve fitting to the numerical results r—radius of the tunnel face |

Liu et al. [20] | $\{\begin{array}{l}{i}_{\mathrm{cov}}=\frac{{h}_{0}-{h}_{F}}{{d}_{1}+{d}_{2}}\\ {i}_{cro}=\frac{\left({h}_{0}-{h}_{F}\right)}{\lambda}{e}^{-\frac{z}{\lambda}}\\ \lambda =\sqrt{\frac{{k}_{0}D\left({d}_{1}+{d}_{2}\right)}{{k}_{1}}}\end{array}$ | d_{1},d_{2}—the thicknesses of the cover layersk _{0}, k_{1}—the permeabilities in the crossed layer and cover layer |

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**MDPI and ACS Style**

Wang, L.; Han, K.; Xie, T.; Luo, J.
Calculation of Limit Support Pressure for EPB Shield Tunnel Face in Water-Rich Sand. *Symmetry* **2019**, *11*, 1102.
https://doi.org/10.3390/sym11091102

**AMA Style**

Wang L, Han K, Xie T, Luo J.
Calculation of Limit Support Pressure for EPB Shield Tunnel Face in Water-Rich Sand. *Symmetry*. 2019; 11(9):1102.
https://doi.org/10.3390/sym11091102

**Chicago/Turabian Style**

Wang, Lin, Kaihang Han, Tingwei Xie, and Jianjun Luo.
2019. "Calculation of Limit Support Pressure for EPB Shield Tunnel Face in Water-Rich Sand" *Symmetry* 11, no. 9: 1102.
https://doi.org/10.3390/sym11091102