^{1}

^{2}

^{2}

^{1}

^{2}

^{1}

^{2}

Stress and deformation around circular tunnel are crucial for optimizing the support system and evaluating the tunnel stability. The damage zone induced by blasting or mechanical excavation can dramatically influence the support design and methods because the self-weight of broken rock mass at the roof of the tunnel can exert a high pressure on the support system, leading to the support system instability due to the overload. This paper presents a new closed-form solution for analyzing the stress and deformation of deep circular tunnel excavated in elastic-brittle rock mass with the consideration of the rock gravity and damage zone by using the unified strength criterion. A new modified equilibrium equation in the fracture zone is used to determine the stress and the radius of fracture zone. The correctness of the solution is also verified by comparison with the numerical simulation results. The results illustrate that the rock gravity, damage zone radius, and intermediate principal stress have an extremely important influence on the ground response. The tunnel surface convergence and damage zone radius with the consideration of the gravity are obviously larger than those without consideration of the gravity. The rock gravity effect under the high intermediate principal stress gradually weakens, illustrating that the intermediate principal stress is beneficial to tunnel stability. Large deformation instability of the tunnel is dependent on the extension of damage zone. The larger the radius of damage zone, the larger both fracture range and tunnel surface deformation. The proposed solution in this study is novel and can be used to assess the ground convergence for different scenarios and to optimize the support system during the early design stage of the tunnel.

During excavation of rock, the extent of damage zone varies with the tunnel shape, excavation method, and rock mass properties. The distribution of damage greatly impacts the ground response and support design. This damaged zone can be considered a rock failure zone in which the rock mass properties are seriously weakened due to the disturbance caused by the excavation. The extent of the damage zone near the tunnel surface in blasted tunnels is far more significant owing to the impact wave and stress redistribution [

Many researchers have mainly concentrated on determining the range of excavation- or blast-induced damage zones beyond the perimeter of tunnels excavated in strain-softened, elastic-brittle-plastic, and elastoplastic rock masses [

The gravitational forces inside the damage zone have a significant influence on the response characteristic of the rock mass and existing support system. Furthermore, the relation between support pressure and tunnel convergence at the roof is unusual compared with that at the side wall because the gravity forces do not exist at the side wall. Several researchers have analyzed the impact of rock gravity on the ground reaction of circular tunnels excavated in M-C, H-B, or GHB rock mass [

Both intermediate principal stress and gravitational force exert considerable influence on the equilibrium and should be evaluated using the governing equations for accurately predicting the stresses and deformation around the tunnel. Combining the impact of the damage zone, rock gravity, and intermediate principal stress is crucial in the estimation of roof convergence and optimization of the support system but has not been investigated yet. In this paper, the modified equilibrium equation is used in the fracture zone in conjunction with consideration of the gravity effect and accounts for the intermediate principal stress and the damage zone. This analytical solution is beneficial for a more direct understanding of the response behaviour of the damaged zone.

The USC takes into account the effect of all the stress components acting on an orthogonal octahedron based on a twin-shear element model [

In equations (

As shown in Figure

Limiting loci of USC in deviatoric plane.

In Figure

Deep tunnel subjected to a hydrostatic stress field with gravity effect and cylindrical BIDZ: (a)

Simplified mechanical behaviour of rock material in this study for both plastic zone and damaged zone. (a) Stress-strain relationship; (b) flow rule.

The axial stress

The failure parameters of damaged rock mass are denoted by subscript “

In deformation analysis of a circular tunnel, an appropriate plastic potential function needs to be determined according to the incremental theory of plasticity. Because the plastic strain increments depend on the plastic potential function, when the plastic potential function is consistent with the yield criterion, the function is called the associated flow rule; otherwise, it is called the nonassociated flow rule. It is believed, in general, that plastic deformation satisfies the nonassociated flow rule [

Most previous studies regarded

The rock weight of the fracture zone (i.e., plastic and damaged zones) at the tunnel roof can apply higher additional pressure on the support system and significantly affect tunnel stability. Hence, the rock weight of the fracture zone should be considered in the equilibrium equation. Detournay [

The rock mass remains elastic outside the fracture zone, as shown in Figure

For Case 1,

In Case 2, the stresses and deformation of the tunnel in the elastic zone can be obtained only by substituting

By considering the blast-induced damage effect in the same way, substituting equation (

The closed solution is acquired by substituting

From the above equation, the radius of the BIDZ (

The approximate solution for

The total circumferential strain

Assuming that the total strain in the fracture zone consists of a plastic strain component and an elastic strain component, the total strain in the fracture zone can then be expressed as follows:

By adopting Hooke’s law, the elastic strain components

Substituting equations (

Substituting equations (

Upon substituting equation

As shown in equation (

The tunnel roof convergence

The solution for a circular opening consists of a series of simple formulas, which obviate the need to solve a complex differential equation. The solution takes into consideration rock gravity and the BIDZ, which can produce a relatively accurate result. Hence, the new solution has a wider engineering application.

Zhang et al. [

Cheng [

The geometry and rock parameters by Pan et al. [

Geometry and rock parameters | Value |
---|---|

In situ stress, | 30 |

Tunnel radius, | 10 |

Rock gravity, ^{3}) | 0 |

Internal pressure, | 0 |

Dilatancy angle, | 30 |

Initial elastic modulus, | 2 |

Initial cohesive, | 6 |

Initial friction angle, | 30 |

Initial Poisson’s ratio ( | 0.3 |

Damage elastic modulus, | 2 |

Damage cohesive, | 6 |

Damage friction angle, | 30 |

Damage Poisson’s ratio ( | 0.3 |

Comparison of stress distribution under (a)

A 3 m radius transportation roadway of Quandian coal mine in China is buried at about 650 m, with 15 MPa average in situ stress. The surrounding rock is mainly sandy mudstone. The special rock parameters of the roadway are shown in Table

The geometry and rock parameters used in this study.

Geometry and rock parameters | Value |
---|---|

In situ stress, | 15 |

Tunnel radius, | 3 |

Rock gravity, ^{3)} | 27.5 |

Dilatancy angle, | 0 |

Initial elastic modulus, | 2.2 |

Initial cohesive, | 5 |

Initial friction angle, | 30 |

Initial Poisson’s ratio ( | 0.25 |

Residual elastic modulus, | 1.6 |

Residual cohesive, | 3.6 |

Residual friction angle, | 24 |

Residual Poisson’s ratio ( | 0.25 |

Damage elastic modulus, | 0.44 |

Damage cohesive, | 1.0 |

Damage friction angle, | 6 |

Damage Poisson’s ratio ( | 0.25 |

Case 2 is used here to investigate the effect of intermediate principal stress and gravity on the ground response curve (GRC). Figure

GRC at different gravity condition and intermediate principal stress. (a) Ignoring the gravity effect. (b) Considering the gravity effect.

As shown in Table

Roof surface displacement of the tunnel under various gravity conditions.

Parameters | Conditions | |||||
---|---|---|---|---|---|---|

0.0 | 0.25 | 0.5 | 0.75 | 1.0 | ||

Gravity included | 1.088 | 0.507 | 0.305 | 0.210 | 0.157 | |

Gravity ignored | 0.914 | 0.458 | 0.284 | 0.199 | 0.151 |

Considering the gravity condition, the roof surface displacement of the tunnel increases by 19.04% for

With increasing

Figure

Evolution law of damage zone at various gravity conditions and intermediate principal stress.

Damage zone radius of the tunnel under various gravity conditions.

Parameters | Conditions | |||||
---|---|---|---|---|---|---|

0.0 | 0.25 | 0.5 | 0.75 | 1.0 | ||

Gravity included | 3.726 | 2.614 | 2.096 | 1.803 | 1.616 | |

Gravity ignored | 3.427 | 2.496 | 2.034 | 1.765 | 1.591 |

For the damage zone, as the internal pressure decreases gradually, the dimensionless value

For

The roof rock gravity has an essential influence on the radius of damage zone. Considering the gravity effect, the dimensionless value of

The stress and deformation of the tunnel are dependent on the damage zone range. In this work, Case 1 is used to explore the effect of damage zone radius on the stress and deformation of the tunnel while ignoring the gravity effect (

Hoop stress and radial stress of tunnel under various damage ranges (

Figure

Displacement change of tunnel under various damage ranges (

A new elastic-plastic solution of deep circular tunnel excavated in elastic-brittle plastic rock mass is presented by using unified strength criterion. The new solution considers the effect of intermediate principal stress, rock gravity, and damage zone on ground response. Associated and nonassociated flow rules are also considered in the solution. The correctness of the solution is also verified by comparison with the numerical simulation results. This study demonstrates that the intermediate principal stress, rock gravity, and damage zone can dramatically influence the stress and deformation of surrounding rock. Some interesting meaningful conclusions are summarized as follows.

The intermediate principal stress has a significant influence on the tunnel convergence, damage zone radius, and critical pressure at the interface of elastic-damage zone. With the increasing intermediate principal stress, the tunnel convergence, the radius of damage zone, and critical pressure decrease gradually under various gravity conditions, indicating that the intermediate principal stress is beneficial to the tunnel stability.

The surface displacement and damage zone radius of the tunnel with the consideration of the gravity are obviously larger than those with ignoring the gravity, indicating that the rock gravity can accelerate roof instability of the tunnel. Furthermore, the rock gravity effect under the high intermediate principal stress gradually weakens, illustrating again that the intermediate principal stress is crucial for the tunnel stability.

Large deformation instability of the tunnel is dependent on the extension of damage zone. Originally, both fracture range and tunnel convergence are in a lower level before damage zone. Once the rock mass enters the damage zone, both fracture range and tunnel convergence sharply and nonlinearly increase. The larger the radius of damage zone, the larger both fracture range and tunnel surface deformation. Hence, the support method design of the tunnel should fully consider the effect of the range of BIDZ on the ground response.

The intermediate principal stress, rock gravity, and damage zone can significantly influence the ground response. Hence, it is suggested to perform parametric studies during the early design stage of the tunnel to evaluate the ground convergence for different scenarios.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (nos. 51874277 and 51874278) and the Natural Science Foundation of Jiangsu Province of China (no. BK20181357).