# Method of Calculating the Vertical Displacement and Additional Stress of Existing Tunnels under the Influence of Grouting Rings of New Tunnels

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## Abstract

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## 1. Introduction

## 2. Proposed Calculation Method

#### 2.1. Method of Solving Additional Stress on Existing Tunnels at Any Angle

#### 2.1.1. Establishment of a Mechanical Model

_{0}), and the radius of the tunnel (R). The influencing factors involved in the excavation process include the additional thrust (q) of the cutter head, the friction resistance (f) of the shield shell, the additional grouting force (p), and soil loss characteristics. The relevant parameters of the existing tunnel are the buried depth of the axis (h) and the radius of the tunnel (R

_{s}). In this work, we considered the excavation surface position and the tunnel crossing angle (α) to be variable and improved the deficiencies of the calculation method presented in ref [16] to make our calculation method more widely applicable.

#### 2.1.2. Solution for Additional Stress Caused by Shield Tunneling

_{1}along the z-axis and load component p

_{2}along the y- axis.

_{x}along the x-axis, force σ

_{y}along the y-axis, or force σ

_{z}along the z-axis, stress components σ

_{x-z}, σ

_{y-z}, and σ

_{z-z}of any point $(x,y,z)$ in the soil in the vertical z direction can be defined as:

_{z-q}, σ

_{z-f}, σ

_{z-p1}, and σ

_{z-p2}can be obtained for stresses q, f, p

_{1}, and p

_{2}at any point $(x,y,z)$ along the z-axis. Due to the limited space in this paper, the specific integration and derivation process is not repeated.

_{z-s}on the axis of the existing tunnel caused by soil loss, this paper refers to the method described in ref [18], and we obtain:

- $B=\frac{4H[H+d-\sqrt{{(H+d)}^{2}-\eta (x){(R+d)}^{2}}]}{R\eta (x)(R+d)}$,
- $\delta =\frac{1}{2}-\frac{1}{\pi}\mathrm{arcsin}[\frac{2d}{R(1+\sqrt{1-\eta (x)})}]$,
- $\lambda =\frac{1}{4}-\frac{2(1-\sqrt{1-\eta (x)})}{\pi \eta (x)}[\mathrm{arcsin}(\frac{d}{R\sqrt{1-\eta (x)}})+\sqrt{1-{(\frac{d}{R\sqrt{1-\eta (x)}})}^{2}}-1]$,
- $\eta (x)=\frac{{\eta}_{\mathrm{s}}}{2}[1-\frac{x}{\sqrt{{x}^{2}+{H}^{2}}}]$.

_{s}is the percentage of soil loss in the tunnel excavation, and η(x) represents the soil loss rate as a function of the x-axis; B, δ, and λ are intermediate calculation variables; d stands for the distance from the focal point of the soil movement to the tunnel center, and ${U}_{\mathit{\text{z-s}}}$ is the vertical displacement of soil at point $(x,y,z)$; k is the foundation bed coefficient.

_{z-k}) can be expressed in:

#### 2.2. Analysis of the Principle of Circumferential Grouting and Its Influence on the Surroundings

#### 2.2.1. Analysis of the Principle of Circumferential Grouting

_{1}. Due to the injection of a large amount of slurry, the voids in the original native body are further filled and compacted. With the continuous increase in the grouting volume, part of the soil in the area to be grouted is squeezed out in the radial direction due to extrusion after the soil voids in the grouting area are filled. When the grouting ring is solidified and stabilized, the extruded soil adheres to the surface of the grouting ring to form an edge expansion zone, and the grouting ring forms grouting and solidification with a thickness of t

_{2}. In summary, the influence of the grouting behind the wall on the external soil deformation can be reflected in the volume growth of the grouting affected zone. The volume of the space to be grouted before grouting is V

_{1}, and it increases to V

_{2}after grouting; thus, the volume expansion rate (Q) can be expressed through:

_{inj}) and the efficiency of grouting (ξ), which can be expressed as:

_{1}) can be reasonably designed according to the construction conditions and equipment. The determination of the volume of injected grout (V

_{inj}) only needs to substitute the efficiency of grouting (ξ), the optimal volume expansion rate, and the volume of the space to be grouted (V

_{1}) into Equation (10); the volume expansion rate (Q) is a parameter that reflects the impact of the grouting ring on the surrounding environment, and we provide a method to estimate its reasonable range of values in the following sections.

#### 2.2.2. Establishment of the Grouting Ring Expansion Model and Calculation of Its Influence on the Surroundings

_{z}stands for the width coefficient of the soil settlement trough, $\phi $ is the friction angle in the soil, ${i}_{z}={i}_{0}{\left(1-\frac{z}{h}\right)}^{0.3}$, and i

_{0}is the width coefficient of the ground settlement trough, which can be obtained according to the method of Knothe [19], that is, ${i}_{0}=\frac{h}{\sqrt{2\pi}\mathrm{tan}\left({45}^{\circ}-\frac{\phi}{2}\right)}$.

_{1}+ L

_{2}, where L

_{1}and L

_{2}represent the length of the back section and the length of the front section respectively. Also, the buried depth of the axis of the new tunnel and the radius of the new tunnel are represented by H and R respectively. Any calculation unit in the grouting ring is taken as $\mathrm{d}V=\mathrm{d}\xi \mathrm{d}\zeta \mathrm{d}\eta $, and the buried depth of the calculation unit is equal to η; the thickness of the grouting ring is t

_{1}. Due to the influence of the grouting, part of the soil in the original grouting zone is extruded to form an edge expansion zone. The thickness of the edge expansion zone is represented by Δt, and the final thickness of the slurry– soil mixture is t

_{2}. The volume expansion rate is indicated by Q. Based on the stochastic medium theory and the research results of Qi Jingjing et al. [14], the vertical deformation (U

_{z-u}) of and the additional stress (σ

_{z-u}) on the surrounding soil caused by the grouting can be obtained by integrating the grouting ring and the edge expansion zone as follows:

#### 2.3. Calculation of the Vertical Displacement of Existing Tunnels

_{z}) acting on the axis of the existing tunnel after taking into account the influence of the grouting ring is expressed by:

_{t}is the ring width of the segment, and l stands for the length variable along the axis of the existing tunnel.

## 3. Analysis and Reliability Verification of an Engineering Case

#### 3.1. Example Conditions

#### 3.2. Theoretical Calculation Results

#### 3.3. Reliability Verification

_{0}= 40 m, and the volume expansion rate of the grouting ring is 1.58%. Figure 8 illustrates the comparison of the settlement distribution curves of the existing upline. According to the results, first, the settlement curve of the tunnel obtained from the proposed calculation method is roughly in the form of a normal distribution, and the measured settlement data also present a distribution of “large settlement values in the middle and small settlement values at both ends”. Thus, the overall distribution of the calculated settlement is similar to that of the measured settlement. Second, the measured and the calculated maximum tunnel settlement in the crossing center are 3.2 and 3.3 mm respectively, which denotes a difference of only 0.1 mm, thereby fulfilling the accuracy requirement. Third, the range of the settlement of the tunnel figured out by the proposed calculation method is relatively close to the measured range and is approximately symmetrically distributed about the crossing center.

## 4. Influence of a Single Factor on Existing Tunnels

#### 4.1. Volume Expansion Rate

_{1}= L

_{2}= 15 m. The volume expansion rate is also set at 0, 1, 2, and 3%, and the other relevant parameters remain unchanged.

#### 4.2. Length of Grouting Rings

_{1}= L

_{2}= 0.5 L.

#### 4.3. Tunnel Crossing Angle

_{1}and L

_{2}are set at 15 m, and Q is equal to 1%. The tunnel crossing angles are set at 15, 30, 45, 60, and 90°, and the other relevant parameters remain unchanged.

_{0}= 40 m, the additional stress near the negative direction is slightly larger than that in the positive direction. Fourth, the settlement of the existing tunnel increases with the decrease in the tunnel crossing angle. Indeed, when α is set at 90, 60, 45, 30, and 15°, the maximum settlement of the existing tunnel across the center is 5.35, 6.13, 7.26, 9.34, and 13.14 mm respectively. Fifth, as α declines, the influence range of the settlement of the existing tunnels gradually extends.

## 5. Conclusions

- The results of this theoretical calculation method are in good agreement with the measured data. The proposed method can be used to calculate the vertical displacement of the existing tunnel caused by a new tunnel crossing, under the influence of the grouting rings.
- Installing grouting rings on new tunnels can effectively reduce the disturbance to existing tunnels caused by tunnel crossing and can significantly decrease the additional stress on and the settlement of the existing tunnels.
- When the volume expansion of the grouted volume is within a certain range, increasing Q can effectively reduce the additional stress on and the settlement of the existing tunnel. When the volume expansion rate is too large, the impact of the grouting ring on the existing tunnel exceeds that of the tunnel excavation, and the variation in the additional stress on the existing tunnel shifts to a vertical upward trend; thus, the tunnel is bulged.
- Properly increasing the length of the grouting ring can decrease the additional stress on and the settlement of the existing tunnel, but the effect gradually lessens. As the tunnel crossing angle decreases, the range of the settlement and the settlement value of the existing tunnel gradually rise.

_{0}= 0 and substitute it into the calculation method in this paper. The vertical displacement value of the existing tunnel at the crossing center can be obtained as ω

_{0}< 0 (a negative number indicates settlement). Third, take a new volume expansion rate Q

_{1}= Q

_{0}+ △k (△k is the calculation accuracy, which can be determined artificially), and by substituting Q

_{1}into the calculation method in this paper, the new vertical displacement value can be obtained as ω

_{1}. Fourth, if ω

_{1}> 0, it means that the best volume expansion rate is between Q

_{0}and Q

_{1}. If ω

_{1}< 0, you need to take a new volume expansion rate Q

_{2}= Q

_{1}+ △k, and by substituting Q

_{2}into the calculation method in this paper, the new vertical displacement value can be obtained as ω

_{2}. Fifth, if ω

_{2}> 0, it means that the best volume expansion rate is between Q

_{1}and Q

_{2}. If ω

_{2}< 0, you need to continue the above process, take Q

_{i}

_{+1}= Q

_{i}+ △k (i = 2,3,4,...,n). Substituting Q

_{i}

_{+1}into the calculation method in this paper until ω

_{i}

_{+1}> 0 is obtained, then the calculation can be stopped, and the optimal volume expansion rate of the grouting ring is between Q

_{i}and Q

_{i+1}.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 10.**The comparison of the additional stress of the existing tunnel at different volume expansion rates.

**Figure 11.**The comparison of the vertical displacement of the existing tunnel at different volume expansion rates.

**Figure 12.**The comparison of the additional stress on the existing tunnel at different lengths of the grouting ring.

**Figure 13.**The comparison of the vertical displacement of the existing tunnel at different lengths of the grouting ring.

**Figure 14.**The comparison of the additional stress on the existing tunnel at different tunnel crossing angles.

**Figure 15.**The comparison of the vertical displacement of the existing tunnel at different tunnel crossing angles.

Parameters | New Tunnel | |
---|---|---|

H | 17.6 m | |

R | 3.35 m | |

η_{s} | 2% | |

q | 45 kPa | |

f | 110 kPa | |

p | 120 kPa | |

d | 2.68 m | |

E_{s} | 10.47 MPa | |

μ | 0.27 | |

φ | 28° | |

Parameters | Existing tunnels | |

Upline | Downline | |

h | 11.0 m | 11.1 m |

α | 62° | 61° |

L | L_{1} = 18.25 mL _{2} = 31.25 m | L_{1} = 31.25 mL _{2} = 18.25 m |

R_{s} | 3.1 m | |

D_{t} | 1.5 m | |

E_{t}I_{t} (tunnel equivalent bending stiffness) | 1.1 × 10^{8} kN·m^{2} | |

k_{s} (shear stiffness between rings) | 7.45 × 10^{5} kN/m | |

k_{t} (tensile stiffness between rings) | 1.94 × 10^{6} kN/m | |

J (coefficient of rotation effect of rigid body) | 0.3 |

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**MDPI and ACS Style**

Qi, Y.; Wei, G.; Xie, Y.
Method of Calculating the Vertical Displacement and Additional Stress of Existing Tunnels under the Influence of Grouting Rings of New Tunnels. *Symmetry* **2020**, *12*, 1623.
https://doi.org/10.3390/sym12101623

**AMA Style**

Qi Y, Wei G, Xie Y.
Method of Calculating the Vertical Displacement and Additional Stress of Existing Tunnels under the Influence of Grouting Rings of New Tunnels. *Symmetry*. 2020; 12(10):1623.
https://doi.org/10.3390/sym12101623

**Chicago/Turabian Style**

Qi, Yongjie, Gang Wei, and Yu Xie.
2020. "Method of Calculating the Vertical Displacement and Additional Stress of Existing Tunnels under the Influence of Grouting Rings of New Tunnels" *Symmetry* 12, no. 10: 1623.
https://doi.org/10.3390/sym12101623