# The Inversion Method Applied to the Stress Field around a Deeply Buried Tunnel Based on Surface Strain

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## Abstract

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## 1. Introduction

^{™}, resulting in the improved SMARS algorithm. The reliability of the algorithm has been validated through a three-dimensional numerical example, and successfully applied to the stress field measurement of the −930 drift for Macheng Iron Mine, Hebei Province, China.

## 2. Stress Field Inversion Method and Verification of Its Accuracy

#### 2.1. Measurement of Surface Strain on Drift Walls

#### 2.2. Inversion Process

- The geometric dimensions of the tunnel and rock mechanical parameters can be determined through geological surveys and rock mechanics experiments. Based on this information, a finite element model containing the tunnel is established in ABAQUS. The user subroutine SIGINI is adopted to apply the initial stress field X;$$X=\left\{{\sigma}_{x},{\sigma}_{y},{\sigma}_{z},{\tau}_{xy},{\tau}_{yz},{\tau}_{xz}\right\}$$
- Reasonable ranges S of parameter values for the substitute equilibrium stress subroutine in the algorithm’s main program, implemented in MATLAB™, can be defined. Within these ranges, multiple discrete arrays are generated. These arrays X
_{j}can serve as random inputs for the parameters of the stress field subroutine;$${X}_{j}=\left\{{\sigma}_{jx},{\sigma}_{jy},{\sigma}_{jz},{\tau}_{jxy},{\tau}_{jyz},{\tau}_{jxz}\right\}{X}_{j}\in S,S\in \left({x}_{j\mathrm{min}},{x}_{j\mathrm{max}}\right)$$ - The stress field is calculated for each array using ABAQUS. During the calculation process, the overall equilibrium of the stress field is first established, followed by the excavation of the tunnel. Finally, the calculated strains f
^{cm}(Xj) corresponding to each array at the measurement points are obtained; - The strains at the limited measurement points are extracted for each array, and error analysis is conducted using an error analysis subroutine and the on-site measured strain value f
^{a}^{m}(Xj). The objective function J_{i}for error analysis is as follows;$${J}_{i}=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}\parallel}{f}_{i}^{am}({X}_{j})-{f}_{i}^{cm}({X}_{j})\parallel $$ - Based on the results of the error analysis, whether or not the minimum required engineering accuracy is met can be determined. If the error meets the requirements, the program terminates. Otherwise, the parameters are mutated, new random arrays X
_{j}are generated, and Steps 3 and 4 are repeated until the minimum error tolerance is satisfied. Finally, the optimal solution X_{opt}is derived:$${X}_{opt}=\left\{{\sigma}_{x}{}^{opt},{\sigma}_{y}{}^{opt},{\sigma}_{z}{}^{opt},{\tau}_{xy}{}^{opt},{\tau}_{yz}{}^{opt},{\tau}_{xz}{}^{opt}\right\}$$

#### 2.3. Verification of the Accuracy of the Method

_{x}= 19.8 MPa, σ

_{y}= 18.0 MPa, σ

_{z}= 12.6 MPa, σ

_{xy}= 5.4 MPa, σ

_{yz}= 3.6 MPa, and σ

_{zx}= 7.2 MPa. The target values of the measurement points for inversion were obtained through simple finite element calculations (Table 1); ε

_{x}, ε

_{y}, ε

_{z}, γ

_{xy}, γ

_{yz}, and γ

_{xz}represent the elastic normal strains and elastic shear strains at the measurement points.

_{y}, the along-tunnel (z-direction) elastic strain ε

_{z}, and the surface shear strain γ

_{yz}at Points A to D. Additionally, the horizontal (x-direction) elastic strain ε

_{x}, the along-tunnel (z-direction) elastic strain ε

_{z}, and the surface shear strain γ

_{xz}at point E can be measured. Therefore, during the inversion process, this research only considers the elastic strains ε

_{y}, ε

_{z}, and γ

_{yz}at Points A to D, as well as the elastic strains ε

_{x}, ε

_{z}, and γ

_{xz}at Point E. The error is defined as follows:

_{x}= 19.30 MPa, σ

_{y}= 17.69 MPa, σ

_{z}= 12.28 MPa, σ

_{xy}= 5.45 MPa, σ

_{yz}= 3.76 MPa, and σ

_{zx}= 7.34 MPa. From Table 3, the maximum error in the measured strains occurs at Point A. This can be attributed to the small reference strain at Point A (−20.31 με). The maximum error in the strains at the remaining points is 3.34%, which is observed at Point E. In summary, based on Figure 7 and Table 2 and Table 3, except for the strain in the z-direction at Point A, the inversion errors at the other points are less than 5%. This level of accuracy fully satisfies practical engineering requirements, indicating that the proposed method can be successfully applied in practice.

## 3. Engineering Applications

#### 3.1. Rock Mechanics Parameters

#### 3.2. Measurement and Processing of Surface Strain

#### 3.3. Inversion Model

#### 3.4. Inversion of Stress Field

## 4. Distribution of the Stresses around the Tunnel

## 5. Conclusions

- The utilization of the equilibrium stress subroutine can avoid errors caused by artificial stress or displacement boundary conditions. Secondly, by directly inverting the stress field based on strain measurements, it circumvents the secondary errors introduced by calculating stresses based on elastic principles. Additionally, by considering the excavation-induced plastic damage to the surrounding rock, severe distortion of the results arising from use of the governing constitutive model can be prevented.
- Based on accuracy verification through examples and field applications in the Ma Cheng iron mine, the proposed stress inversion method can achieve an accuracy of over 90% in practical applications. Particularly for deeply buried rock masses, this method proves to be a simple and cost-effective approach.
- According to the in situ stress measurement method proposed in this study, the maximum principal stress, intermediate principal stress, and minimum principal stress in the rock at the Ma Cheng iron mine were calculated as 38.78 MPa, 25.68 MPa, and 11.63 MPa, respectively. The ratio between the maximum and minimum principal stresses is 3.33:1, and the maximum principal stress is 1.43 times greater than the intermediate principal stress. The concentration of stress is evident in the left-hand side of the tunnel arch. These findings are consistent with the observed collapse phenomena.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 10.**The strain measurement process: (

**a**) polishing; (

**b**) attaching strain gauges; (

**c**) wire bonding and waterproofing; (

**d**) drilling and data collection; (

**e**) post-drilling inspection for waterproofing; (

**f**) core extraction.

Measurement Points | ${\mathit{\epsilon}}_{\mathit{x}}\mathit{(}\mathit{\mu}\mathit{\epsilon}\mathit{)}$ | ${\mathit{\epsilon}}_{\mathit{y}}\mathit{(}\mathit{\mu}\mathit{\epsilon}\mathit{)}$ | ${\mathit{\epsilon}}_{\mathit{z}}\mathit{\left(}\mathit{\mu}\mathit{\epsilon}\mathit{\right)}$ | ${\mathit{\gamma}}_{\mathit{xy}}\mathit{\left(}\mathit{\mu}\mathit{\epsilon}\mathit{\right)}$ | ${\mathit{\gamma}}_{\mathit{yz}}\mathit{\left(}\mathit{\mu}\mathit{\epsilon}\mathit{\right)}$ | ${\mathit{\gamma}}_{\mathit{xz}}\mathit{\left(}\mathit{\mu}\mathit{\epsilon}\mathit{\right)}$ |
---|---|---|---|---|---|---|

A | 97.37 | −263.86 | −20.31 | 61.28 | −204.97 | 68.75 |

B | 93.85 | −249.94 | −67.14 | −76.26 | −223.13 | 58.03 |

C | 78.86 | −204.86 | −68.03 | 9.83 | −228.99 | 78.62 |

D | 94.72 | −327.85 | −15.71 | −135.78 | −264.40 | 84.46 |

E | −182.64 | 68.44 | −49.09 | −11.36 | −14.86 | −11.01 |

${\mathit{\sigma}}_{\mathit{x}}$ | ${\mathit{\sigma}}_{\mathit{y}}$ | ${\mathit{\sigma}}_{\mathit{z}}$ | ${\mathit{\tau}}_{\mathit{x}\mathit{y}}$ | ${\mathit{\tau}}_{\mathit{y}\mathit{z}}$ | ${\mathit{\tau}}_{\mathit{x}\mathit{z}}$ | |
---|---|---|---|---|---|---|

Target value | 19.80 | 18.00 | 12.60 | 5.40 | 3.60 | 7.20 |

Inversion value | 19.30 | 17.69 | 12.28 | 5.45 | 3.76 | 7.34 |

Error | 2.53% | 1.72% | 2.54% | 0.93% | 4.44% | 1.94% |

Measurement Point | ${\mathit{\epsilon}}_{\mathit{x}}$ | ${\mathit{\epsilon}}_{\mathit{y}}$ | ${\mathit{\epsilon}}_{\mathit{z}}$ | ${\mathit{\gamma}}_{\mathit{x}\mathit{y}}$ | ${\mathit{\gamma}}_{\mathit{y}\mathit{z}}$ | |
---|---|---|---|---|---|---|

A | Inversion value (με) | - | −260.08 | −18.76 | - | −208.22 |

Error (%) | - | 1.43 | 7.63 | - | 1.59 | |

B | Inversion value (με) | - | −247.17 | −66.52 | - | −228.03 |

Error (%) | - | 1.11 | 0.92 | - | 2.20 | |

C | Inversion value (με) | - | −201.75 | −67.22 | - | −233.08 |

Error (%) | - | 1.52 | 1.19 | - | 1.79 | |

D | Inversion value (με) | - | −323.61 | −15.37 | - | −270.68 |

Error (%) | - | 1.29 | 2.18 | - | 2.38 | |

E | Inversion value (με) | −176.88 | - | −47.45 | −11.54 | - |

Error (%) | 3.15 | - | 3.34 | 1.58 | - |

${\mathit{\sigma}}_{\mathit{x}}$ | ${\mathit{\sigma}}_{\mathit{y}}$ | ${\mathit{\sigma}}_{\mathit{z}}$ | ${\mathit{\tau}}_{\mathit{x}\mathit{y}}$ | ${\mathit{\tau}}_{\mathit{y}\mathit{z}}$ | ${\mathit{\tau}}_{\mathit{x}\mathit{z}}$ | |
---|---|---|---|---|---|---|

Target value (MPa) | 19.80 | 18.00 | 12.60 | 5.40 | 3.60 | 7.20 |

Inversion value (MPa) | 18.44 | 16.84 | 11.39 | 4.73 | 4.12 | 7.11 |

Error | 6.87% | 6.44% | 9.60% | 12.41% | 14.44% | 1.25% |

Density (kg/m ^{3}) | Compressive Strength (MPa) | Tensile Strength (MPa) | Poisson’s Ratio | Young’s Modulus (GPa) | Friction Angle (°) | Cohesion (MPa) | |
---|---|---|---|---|---|---|---|

Average value | 2530 | 101.15 | 8.39 | 0.26 | 72.47 | 45 | 20 |

Standard deviation | 0.01 | 16.73 | 0.73 | 0.03 | 8.57 | 6.58 | 3.62 |

Measurement Point | ${\mathit{\epsilon}}_{\mathit{x}}(\mathit{\mu}\mathit{\epsilon})$ | ${\mathit{\epsilon}}_{\mathit{y}}(\mathit{\mu}\mathit{\epsilon})$ | ${\mathit{\epsilon}}_{\mathit{z}}(\mathit{\mu}\mathit{\epsilon})$ | ${\mathit{\gamma}}_{\mathit{x}\mathit{y}}(\mathit{\mu}\mathit{\epsilon})$ | ${\mathit{\gamma}}_{\mathit{y}\mathit{z}}(\mathit{\mu}\mathit{\epsilon})$ |
---|---|---|---|---|---|

A | - | −80 | −16 | - | −83 |

B | - | −60 | 18 | - | −75 |

C | −5.2 | −219 | - | −125 | - |

D | 13 | −370 | - | −42 | - |

E | 11 | −168 | - | −25 | - |

F | 14 | −124 | - | −108 | - |

Measurement Point | ${\mathit{\epsilon}}_{\mathit{x}}$ | ${\mathit{\epsilon}}_{\mathit{y}}$ | ${\mathit{\epsilon}}_{\mathit{z}}$ | ${\mathit{\gamma}}_{\mathit{x}\mathit{y}}$ | ${\mathit{\gamma}}_{\mathit{y}\mathit{z}}$ | |
---|---|---|---|---|---|---|

A | Inversion value (με) | - | 78.95 | −16.74 | - | −80.62 |

Error (%) | - | 1.31 | 4.59 | - | 2.87 | |

B | Inversion value (με) | - | −58.41 | 18.60 | - | −75.56 |

Error (%) | - | 2.66 | 3.34 | - | 0.74 | |

C | Inversion value (με) | 0.76 | 216.69 | - | −125.87 | - |

Error (%) | - | 1.05 | - | 0.7 | - | |

D | Inversion value (με) | 7.91 | −372.88 | - | −38.75 | - |

Error (%) | - | 0.78 | - | 7.74 | - | |

E | Inversion value (με) | 11.63 | −166.29 | - | −24.81 | - |

Error (%) | - | 1.02 | - | 0.76 | - | |

F | Inversion value (με) | 14 | −129.91 | - | −107.93 | - |

Error (%) | - | 1.05 | - | 0.06 | - |

${\mathit{\sigma}}_{\mathit{x}}$ | ${\mathit{\sigma}}_{\mathit{y}}$ | ${\mathit{\sigma}}_{\mathit{z}}$ | ${\mathit{\tau}}_{\mathit{x}\mathit{y}}$ | ${\mathit{\tau}}_{\mathit{x}\mathit{z}}$ | ${\mathit{\tau}}_{\mathit{y}\mathit{z}}$ | |
---|---|---|---|---|---|---|

Inverted Values (MPa) | 34.720 | 25.380 | 15.982 | −3.400 | −6.954 | 6.065 |

Stress Value (Mpa) | Cosine Value with Respect to the x-Axis | Cosine Value with Respect to the y-Axis | Cosine Value with Respect to the z-Axis | |
---|---|---|---|---|

Maximum principal stress | 38.78 | 0.85 | −0.39 | 0.35 |

Second principal stress | 25.68 | 0.47 | 0.84 | −0.28 |

Minimum principal stress | 11.63 | −0.17 | 0.41 | 0.90 |

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**MDPI and ACS Style**

Yan, X.; Hao, Q.; Yang, R.; Peng, J.; Zhang, F.; Tan, S.
The Inversion Method Applied to the Stress Field around a Deeply Buried Tunnel Based on Surface Strain. *Appl. Sci.* **2023**, *13*, 12507.
https://doi.org/10.3390/app132212507

**AMA Style**

Yan X, Hao Q, Yang R, Peng J, Zhang F, Tan S.
The Inversion Method Applied to the Stress Field around a Deeply Buried Tunnel Based on Surface Strain. *Applied Sciences*. 2023; 13(22):12507.
https://doi.org/10.3390/app132212507

**Chicago/Turabian Style**

Yan, Xiaobing, Qiqi Hao, Rui Yang, Jianyu Peng, Fengpeng Zhang, and Sanyuan Tan.
2023. "The Inversion Method Applied to the Stress Field around a Deeply Buried Tunnel Based on Surface Strain" *Applied Sciences* 13, no. 22: 12507.
https://doi.org/10.3390/app132212507