# Analytical Predictions on the Ground Responses Induced by Shallow Tunneling Adjacent to a Pile Group

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## Abstract

**:**

## 1. Introduction

## 2. Mechanical Model and Method Framework

#### 2.1. Establishment of Mechanical Model

_{t}vertical to the tunnel axis is considered. Notably, the pile group and shallow tunnel are in the same plane when y

_{t}$=$ y

_{p}; the pile group is not in the same plane as the tunnel when y

_{t}$\ne $ y

_{p}, which is a three-dimensional problem.

_{t}, y

_{p}are y coordinates of the tunnel section and pile group, respectively. The pile loads in the ground consist of two parts: the pile tip load q and the pile shaft shear load s. For the convenience of calculations, the pile loads are simplified to an equivalent concentrated load P and equivalent linear load f along the longitudinal axis. The horizontal coordinate of the first pile load d

_{k}is adopted to represent the location of the pile group, and l is the pile spacing.

#### 2.2. Method Framework

## 3. Analytical Solutions of Stress for Tunneling Adjacent to Pile Group

#### 3.1. Stress Boundary Conditions

_{0}is the lateral stress coefficient and is equal to v/(1 − v). v is the Poisson’s ratio. σ

_{x}

_{1}and σ

_{z}

_{1}are the ground stresses induced by the pile group, σ

_{s}is the stress boundary condition of the tunnel periphery, and p

_{x}, p

_{z}are the stress boundary functions of tunnel paralleled to the x and z axes, respectively.

#### 3.2. Analytical Solution of Pile Group Loads

#### 3.2.1. Ground Stress Induced by Pile Group Tip Loads

_{t}plane, respectively, induced by equivalent concentrated load P

_{k}at the pile tip.

#### 3.2.2. Ground Stress Induced by Pile Group Shaft Shear Loads

_{t}plane, respectively, induced by equivalent linear load f

_{k}along the pile length.

#### 3.2.3. Stress Superposition

#### 3.3. Analytical Solution of Shallow Tunnel in Greenfield

#### 3.3.1. Stress-Release Function

_{o}are the center and radius of the circular tunnel, respectively. The angle of point Q is expressed as

_{r}denotes the stress release coefficient. The ground displacement caused by the pile group loads has been completed before the tunnel excavation, so this part of the displacement is no longer of concern during the tunnel excavation and is cleared to 0. Then, the release loads are applied in the opposite direction to the initial stresses (Figure 4, Part 1), and the value of release loads are the initial stresses multiplied by the stress release coefficient.

_{r}= 0, the tunnel periphery is in the initial stress state before excavation, or in a no-stress release state while the lining is applied immediately after the excavation, the stress solution equals Equation (12). When s

_{r}= 1, σ in Equation (10) is equal to 0, which means that the tunnel has no lining during excavation and the stress is in a completely released state. When $0\le {s}_{r}\le 1$, Equation (10) reflects the different degrees of stress release.

_{x}and p

_{z}represent the principal stress of point Q and are the projection of the normal stress on the x′ and z′ axes on the plane MN. Then, the process of transforming the normal stress into the principal stress can be expressed by Equation (11). The stress function of the tunnel perimeter is

#### 3.3.2. Part 2a Solution

#### 3.3.3. Part 2b Solution

_{x}and F

_{z}around the tunnel periphery, both of which are equal to the weight of excavated soil minus the weight of the tunnel lining. The “buoyancy effect” will occur when the weight of excavated soil is greater than the lining [15,40], and is incorporated into the method in this study by using the complex potential functions of Equations (17) and (18).

- I.
- Ground surface:

- II.
- Tunnel periphery:

#### 3.4. Stress Solution

## 4. Parametric Analysis

#### 4.1. Stress Analysis

_{r}values are presented in Figure 7, wherein the pile group loads are 4 m to the right side of the tunnel.

_{r}is small, as shown in Figure 7b. With the increase in the value of s

_{r}, the tensile stress zone on the ground surface gradually develops downward as well as beginning to appear on the upper right side of the tunnel, due to the influence of the pile group load (see Figure 7d). The greater the value of s

_{r}, the greater the disturbance of the stratum stress field.

#### 4.2. Analysis of Potential Plastic Zone

#### 4.2.1. Influence of the Pile Group

_{r}= 0.9, h = 12 m, P = 355 kN, f = 235 kN/m, γ = 20 kN/m

^{3}, c = 40 kPa, $\phi $ = 20°. Figure 10 plots the potential plastic zones for different offsets d

_{k}, and the potential plastic zones are merged in Figure 10a. It can be noted from Figure 10b that the potential plastic zone of the tunnel begins separated from the potential plastic zone of the pile group when d

_{k}= 7 m. Furthermore, the degree of separation increases gradually with the increase in the value of d

_{k}, as shown in Figure 10c,d; In Figure 10e, the potential plastic zone of the tunnel and pile group are completely separated when d

_{k}= 10 m; however, there is some influence between them; In Figure 10d, the potential plastic zone of the tunnel is almost not affected by the pile group, and the potential plastic zone of the tunnel is approximately symmetrical when d

_{k}= 12 m.

#### 4.2.2. Influence of Soil Parameters

_{r}, the relative distance d

_{k}between pile group and tunnel, the pile length h, the pile loads P and f, and soil parameters (γ, c, $\phi $). The working conditions match the real working conditions, so based on the established procedures, incorporating engineering parameters will allow us to qualitatively predict the secondary stress field and potential plastic zone. In future related research, a coupling analysis with analytical solutions and the additional effects of adjacent structures will be proposed to consider more complex underground space structures. The reasonable boundary condition is a key factor.

## 5. Conclusions

- (1)
- The stress around the tunnel is greatly affected by a change in the stress release coefficient. When the s
_{r}is large enough, the tensile stress zone appears in the tunnel crown; notably, the tensile stress zone appears at the right of the tunnel crown due to the influence of pile group loads. The greater the pile group tip loads and shaft shear loads, the greater the degree of stratum disturbance. - (2)
- The stratum stress is disturbed in the horizontal range of approximately −5 m~5 m (h = 8 m, l = 2 m), when the pile group is located 1 m above the tunnel crown; the disturbed range of stratum stress increases with the increase in pile length.
- (3)
- Considering the gravity field, the potential plastic zone of the tunnel is butterfly shaped, and the range of lower part is larger than the upper. The potential plastic zones are merged when the horizontal distance between the pile group and the tunnel d
_{k}$\le $ 3r; the potential plastic zone is completely separated when d_{k}$>$ 3r; the potential plastic zone of the tunnel is almost unaffected by the pile group when d_{k}$\ge $ 4r, and the law of the influence of the relative position of the pile group and tunnel on the plastic zone is consistent with the law presented in Xiang et al. (2013). - (4)
- Shear strength parameters have similar influence rules on the potential plastic zone, and the ranges of the potential plastic zone decrease as the c and φ increase; on the contrary, the ranges of the potential plastic zone increase as the volumetric weight increases.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 7.**Contour diagrams of major principal stresses with various s

_{r}: (

**a**) s

_{r}= 0; (

**b**) s

_{r}= 0.3; (

**c**) s

_{r}= 0.7; (

**d**) s

_{r}= 1.

**Figure 8.**Contour diagrams of major and minor principal stresses with various pile group tip loads P and shaft shear loads f when s

_{r}= 0.9: (

**a**,

**b**) P = 150 kN, f = 70 kN/m; (

**c**,

**d**) P = 355 kN, f = 235 kN/m.

**Figure 9.**Contour diagrams of major principal stresses with various pile lengths and relative positions of the pile group and tunnel when s

_{r}= 0.9: (

**a**) the pile group right on top of the tunnel; (

**b**) the pile group distributed on both sides of the tunnel; (

**c**) the depth of the pile group tip close to the tunnel spring line; (

**d**) the depth of the pile group tip deeper than the tunnel spring line.

**Figure 10.**Potential plastic zones for different offsets d

_{k}of the pile group from the tunnel: (

**a**) d

_{k}= 6 m; (

**b**) d

_{k}= 7 m; (

**c**) d

_{k}= 8 m; (

**d**) d

_{k}= 9 m (

**e**) d

_{k}= 10 m; (

**f**) d

_{k}= 12 m.

**Figure 11.**Potential plastic zones for different soil parameters when d

_{1}= 6 m, d

_{2}= 7 m, d

_{3}= 8 m, d

_{4}= 9 m (x coordinate of each pile) and h = 12 m. (

**a**) Different magnitudes of volumetric weight; (

**b**) Different magnitudes of cohesion; (

**c**) Different magnitudes of the angle of internal friction.

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**MDPI and ACS Style**

Guo, C.; Tao, Y.; Kong, F.; Shi, L.; Lu, D.; Du, X. Analytical Predictions on the Ground Responses Induced by Shallow Tunneling Adjacent to a Pile Group. *Mathematics* **2023**, *11*, 1608.
https://doi.org/10.3390/math11071608

**AMA Style**

Guo C, Tao Y, Kong F, Shi L, Lu D, Du X. Analytical Predictions on the Ground Responses Induced by Shallow Tunneling Adjacent to a Pile Group. *Mathematics*. 2023; 11(7):1608.
https://doi.org/10.3390/math11071608

**Chicago/Turabian Style**

Guo, Caixia, Yingying Tao, Fanchao Kong, Leilei Shi, Dechun Lu, and Xiuli Du. 2023. "Analytical Predictions on the Ground Responses Induced by Shallow Tunneling Adjacent to a Pile Group" *Mathematics* 11, no. 7: 1608.
https://doi.org/10.3390/math11071608