# Time-Dependent Behavior of a Circular Symmetrical Tunnel Supported with Rockbolts

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{0}, the viscosity parameters η

_{1}and η

_{2}of the three-dimensional Burgers model as well as the pre-tension T

_{0}on reinforcement effect were analyzed. According to the proposed model, the smaller η

_{2}is or the larger the pre-tension T

_{0}is, the more effective the support effect. However, when the pre-tension is too large, the support effect is no longer significantly enhanced. In addition, the early reinforcement effect is controlled by the first creep stage in the Burgers model while the ultimate support effect is mainly influenced by the viscosity coefficient of the second creep stage in the Burgers model. This research can provide an important theoretical reference to guide the parameter design of rockbolt reinforcement engineering in a circular symmetrical tunnel.

## 1. Introduction

## 2. Coupling Mechanical Model of the Rockbolt and Rock Mass System

#### 2.1. Principle of the Coupling Model

#### 2.2. Elastic Solutions

_{0}and the reinforcement force P

_{ρ}are uniformly distributed on the excavation surface and the interface between the reinforced zone and the original zone; (c) the problem is axisymmetric, and the lateral pressure coefficient K

_{a}= 1; and (d) the deformation is small.

_{ρ}, σ

_{θ}and u

_{ρ}are respectively the radial stress, tangential stress and displacement; E is Young’s modulus; μ is Poisson’s ratio; ρ is the radial coordinate; A, B, and C are coefficients.

_{ρ}

_{1}, σ

_{θ}

_{1}and u

_{ρ}

_{1}are respectively the radial stress, tangential stress, and displacement in the reinforced zone and A

_{1}and C

_{1}are the pending coefficients.

_{0}and pressure on the inside of a circular hole. A mechanical analysis diagram is shown in Figure 4a. The general forms expressions of the stress and displacement in the original zone (ρ > R) are as follows:

_{ρ}

_{2}, σ

_{θ}

_{2}and u

_{ρ}

_{2}are respectively the radial stress, tangential stress, and displacement in the original zone, and ${A}^{\prime}$ and ${C}^{\prime}$ are coefficients.

_{0}is the support force in tunnel opening; P

_{ρ}is the reinforcement force; L is the length of the free part of the rockbolt; $\u25b3L$ is the deformation of the rockbolt in the axial direction, $\u25b3x$ is the pre-tension length of the rockbolt; S

_{z}is the rockbolt spacing in the longitudinal direction; S

_{θ}is the rockbolt spacing in the tangential direction; ${u}_{\rho 1(\rho =R)}^{0}$ and ${u}_{\rho 1(\rho =r)}^{0}$ are the displacements of the original state without support; A

_{b}is the area of the rockbolt cross section; and E

_{b}is the elastic modulus of the rockbolt.

#### 2.3. Viscoelastic Solutions

#### 2.3.1. Rheological Model of Rockbolt and Definition of Operator Function

_{b}(D) and P

_{b}(D) are the operator functions of the one-dimensional Kelvin rheological model. The space parameter transformation based on the Laplace transform is

_{b}is the rockbolt’s viscosity coefficient, s is the variable obtained from the Laplace transformation.

#### 2.3.2. Rheological Model of Rock Mass and Definition of Operator Function

_{ij}is the partial stress tensor; e

_{ij}is the partial strain tensor; σ

_{ij}is the stress tensor; ${\epsilon}_{ij}$ is the strain tensor; and ${Q}^{\prime}(D)$, ${P}^{\prime}(D)$, ${Q}^{\u2033}(D)$ and ${P}^{\u2033}(D)$ are the operator functions of the rock mass viscoelastic constitutive model.

_{1}is the viscous coefficient of the first creep stage; η

_{2}is the viscous coefficient of the second creep stage; G

_{0}is the elastic shear modulus of the rock mass; G

_{1}is the corresponding viscoelastic shear modulus of the rock mass; and K is the elastic bulk modulus of the rock mass.

#### 2.4. Analytical Solutions in the Laplace Space

## 3. Analysis of the Analytical Solution Using Engineering Parameters

#### 3.1. Axial Force Changes over Time

_{1}and σ

_{0}on the axial force evolve over time. The results indicate that η

_{1}affects not only the axial force initial value but also the rockbolt rheological state. When η

_{1}= 3.0 × 10

^{10}and 5.0 × 10

^{10}Pa·s, the axial force decreases to a stable value over time, the rockbolt rheological state is stress relaxation. When η

_{1}= 7.0 × 10

^{10}Pa·s, the axial force increases to a stable value over time, the rheological state of the rockbolt is stable creep. The axial force will converge to a stable value regardless of whether the rheological state of a rockbolt is creep or stress relaxation; thus η

_{1}does not affect the stable value of the axial force and only influences the early support effect. When η

_{1}and η

_{2}remain unchanged, a larger σ

_{0}will generate a larger initial force and higher stability. Figure 6b indicates that the influences of η

_{2}and σ

_{0}on the axial force evolve over time. The results indicate that η

_{2}affects not only the initial value and stable value of axial force but also the rheological state of the rockbolt; thus, η

_{2}affects not only the early reinforced effect but also the final support effect. When η

_{2}= 1.0 × 10

^{11}Pa·s, 1.5 × 10

^{11}Pa·s, and 2.0 × 10

^{11}Pa·s, the axial force decreases to a stable value over time; thus, the rheological state of the rockbolt is stress relaxation. A smaller value of η

_{2}produces a larger initial value and convergent value of the axial force. When η

_{2}= 3.0 × 10

^{11}Pa·s, the axial force increases to a stable value over time, and the convergent value is less than η

_{2}= 1.0 × 10

^{11}Pa·s, 1.5 × 10

^{11}Pa·s, and 2.0 × 10

^{11}Pa·s; thus, the rheological state of the rockbolt is creep. As shown in Figure 6c, the influence of η

_{b}on axial force evolves over time. When η

_{b}= 1.0 × 10

^{20}Pa·s and 3.0 × 10

^{20}Pa·s, the axial force decreases to a stable value over time, the rockbolt rheological state is stress relaxation. When η

_{b}= 5.0 × 10

^{20}Pa·s, the axial force is gradually reduced to zero and then increases to a stable value over time; the stable value is the same as when η

_{b}= 1.0 × 10

^{20}Pa·s and 3.0 × 10

^{20}Pa·s. When η

_{b}= 6.0 × 10

^{20}Pa·s, the axial force of the rockbolt increases to a stable value over time; thus, the rockbolt rheological state is creep, and the initial value is the same as when η

_{1}= 3.0 × 10

^{10}Pa·s. Figure 6d reveals that the influence of T

_{0}on the axial force evolves over time; larger values of T

_{0}produce a greater absolute value of initial axial force. Thus, the larger is T

_{0}, the better the support effect. When T

_{0}= 4.0 × 10

^{4}N, 5.0 × 10

^{4}N, and 5.5 × 10

^{4}N, the axial force absolute value gradually decreases over time and converges to a fixed value. A larger value of T

_{0}yields a higher fix value. But, when the T

_{0}is too large, the fix value will not increase significantly. At this time, the rockbolt support to the rock mass is in the stage of relaxation.

#### 3.2. Radial Stress Changes over Time

_{1}, η

_{2}and σ

_{0}influenced the radial stress. Figure 7a shows the jump value fitting curve between the reinforced and the original zone when the time is zero. When η

_{2}is fixed, the fitting curve is ${y}_{1}=-4334.5\times x+49929$. ${R}^{2}=0.9198$. When η

_{1}is fixed, the fitting curve is ${y}_{2}=-1160.3\times x+27924$. ${R}^{2}=0.9513$. As shown in Figure 7a, η

_{1}and η

_{2}are negatively correlated with the radial stress jump value, and the slope of y

_{1}is greater than the slope of y

_{2}, which indicates that the effect of η

_{1}on the radial stress jump value is larger than the effect of η

_{2}. The radial stress jumps at the junction of the reinforced zone and the original zone due to the support of the rockbolt. Figure 7b shows the radial stress fitting curve in the excavation surface when the time is zero; when η

_{2}is fixed, the fitting curve is ${y}_{3}=-19850\times x+214942$. ${R}^{2}=0.936$. When η

_{1}is fixed, the fitting curve is ${y}_{4}=-3148.1\times x+109336$. ${R}^{2}=0.9678$. Note that η

_{1}and η

_{2}are negatively correlated with the radial stress in excavation surface. The slope of y

_{3}is greater than y

_{4}, which indicates that the effect of η

_{1}on the excavation surface radial stress is also larger than the effect of η

_{2}. Therefore, the magnitude of the radial stress in the excavation surface of the tunnel is equivalent to the supporting force of the rock mass in the coupling model, and the greater the support force is, the better the support effect. The smaller η

_{1}or η

_{2}is, the better the support effect in the excavation moment. Figure 7c shows the evolution of the radial stress of the reinforced zone monitoring point ρ = 4 m over time (influence of η

_{1}and σ

_{0}). The absolute value of radial stress gradually decreases over time because the stress of the rock mass is gradually released over time. When the parameters η

_{1}and η

_{2}are constant, a larger σ

_{0}yields a greater absolute value of the initial and convergence values. When the parameter η

_{2}is fixed, a larger η

_{1}produces a smaller absolute value of the initial radial stress. The radial stress converges to a fixed value over time, and the fixed value is independent of η

_{1}. A smaller η

_{1}produces a larger change rate. Figure 7d shows the evolution of the radial stress over time for the reinforced zone monitoring point ρ = 4 m (influence of η

_{2}and σ

_{0}). The absolute value of radial stress gradually decreases over time; when the parameters η

_{1}and η

_{2}are fixed, a greater value of σ

_{0}yields a higher absolute value of initial radial stress and convergence value. When η

_{1}is constant, a smaller η

_{2}produces a larger absolute value of the initial and convergence values. Thus, the smaller is the η

_{2}, the better is the support effect. η

_{2}does not have a significant effect on the rate of change, which remains fairly constant.

#### 3.3. Tangential Stress Changes over Time

_{2}on the tangential stress in the reinforced zone over time. The tangential stress gradually decreases over time and then converges to a stable value. When η

_{1}is fixed, a larger η

_{2}generates a larger initial and stable values of the tangential stress. Figure 8b shows the influence of η

_{1}on the reinforced zone tangential stress over time. The tangential stress also decreases over time and finally converges to a stable value. The constant value is independent of η

_{1}, and a smaller η

_{1}yields a smaller initial value but a larger change rate. Figure 8c shows the influence of η

_{2}on the tangential stress of the original zone over time: η

_{1}influences not only the initial value but also the stable value; a smaller η

_{2}yields greater initial and stable values of the tangential stress. Figure 8d shows the influence of η

_{1}on the tangential stress in the original zone over time. The tangential stress also decreases over time, and η

_{1}influences only the initial value. The tangential stress finally converges to a certain value. The constant value is independent of η

_{1}, and a smaller η

_{1}produces not only a greater initial value but also a larger change rate.

#### 3.4. Displacement Evolution over Time

_{2}is fixed, a smaller η

_{1}causes a larger displacement at any position. If η

_{1}is fixed, a smaller η

_{2}yields a larger displacement. Figure 9b shows the displacement when t = 4 d along the radial distance. The amount of displacement significantly changes from the excavation moment. Figure 9c shows the displacement evolution rule of ρ = 7 m over time, the displacement gradually increases and exponentially converges to a fixed value. A smaller η

_{1}produces larger absolute convergence values. A smaller η

_{2}yields larger absolute convergence values. Therefore, important support measures should be taken when η

_{1}or η

_{2}is small.

## 4. Conclusions

- (1)
- The axial force of DMFC rockbolts are positively correlated with the support force at the excavation face in a tunnel, and the greater the convergence value of the axial force is, the better the support effect. In addition, the greater the pre-tension of rockbolt, the better the reinforcement effect, however, when the pre-tension is too large, the rock bolt support effect will not increase significantly.
- (2)
- The η
_{1}of the three-dimensional Burgers model influences the early support effect, η_{2}of three-dimensional Burgers model affects both the early and the ultimate reinforcement effect. In addition, there is a significant negative correlation between rock mass displacement and η_{1}or η_{2}. Therefore, important support measures should be taken when η_{1}or η_{2}is small. - (3)
- In this paper, the interaction model elastic solutions were solved based on the distributed force model. However, the axial force of the rockbolt is similar to the concentrated force to rock mass (Bobet 2006), Hence, a more suitable theoretical model should be explored to solve the coupling model in future.
- (4)
- Continuously Mechanically Coupled (CMC) rockbolts applications are more extensive than DMFC rockbolts, the reasonable and simplified method for the theoretical model of CMC rockbolts can be further studied based on this model, which lays a foundation of the preliminary research for solving the theoretical model of CMC rockbolts.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

A_{b} | Area of the rockbolt cross-section |

L | Length of rockbolt free part |

ρ | Radial coordinate |

r | Radius of tunnel |

T | Axial force of the rockbolt bolt |

P_{0} | Support force in tunnel opening |

k | Stiff of the support system |

σ_{ρ}_{1} | Radial stress in the reinforced zone |

σ_{θ}_{1} | Tangential stress in the reinforced zone |

u_{ρ}_{1} | Displacement in the reinforced zone |

σ_{ρ}_{1s} | Radial stress of the reinforced zone in Laplace |

σ_{θ}_{1s} | Tangential stress of the reinforced zone in Laplace |

u_{ρ}_{1s} | Displacement of the reinforced zone in Laplace |

S_{θ} | Rockbolt spacing in the tangential direction |

η_{b} | Viscosity coefficient of the rock bolt |

η_{2} | Viscosity coefficient of the second creep stage |

Δx | Pre-tension length of the rockbolt |

G_{1} | Viscoelastic shear modulus of rock mass |

t | Time |

S_{ij} | Partial stress tensor |

${Q}_{b}(D),{P}_{b}(D)$ | Operator functions of the Kelvin model |

$\begin{array}{l}{P}^{\prime}(D),{P}^{\u2033}(D)\\ {Q}^{\prime}(D),{Q}^{\u2033}(D)\end{array}$ | Operator function of rock mass viscoelastic constitutive model |

E_{r} | Deformation modulus of the rock mass |

E_{b} | Deformation modulus of the rockbolt |

θ | Circumferential angle |

σ_{0} | Initial stress |

μ | Poisson’s ratio of rock mass |

P_{ρ} | Reinforcement force |

R | Radius of the reinforced rock zone |

σ_{ρ}_{2} | Radial stress in the original zone |

σ_{θ}_{2} | Tangential stress in the original zone |

u_{ρ}_{2} | Displacement in the original zone |

σ_{ρ}_{2s} | Radial stress of the original zone in Laplace |

σ_{θ}_{2s} | Tangential stress of the original zone in Laplace |

u_{ρ}_{2s} | Displacement of the original zone in Laplace |

S_{z} | Rockbolt spacing in the longitudinal direction |

η_{1} | Viscosity coefficient of the first creep stage |

ΔL | Deformation of rockbolt in the axial direction |

G_{0} | Elastic shear modulus of the rock mass |

K | Elastic bulk modulus of the rock mass |

σ_{ij} | Stress tensor |

e_{ij} | Partial strain tensor |

${\overline{{P}_{bK}}}^{\prime}(s),{\overline{{Q}_{bK}}}^{\prime}(s)$ | Operator functions of the Kelvin model in Laplace space |

$\begin{array}{l}{\overline{P}}^{\prime}(s),{\overline{Q}}^{\prime}(s)\\ {\overline{P}}^{\u2033}(s),{\overline{Q}}^{\u2033}(s)\end{array}$ | Operator function of rock mass viscoelastic constitutive model after Laplace transformation |

## Appendix A

## References

- Kovári, K. History of the sprayed concrete lining method—Part I. Milestones up to the 1960s. Tunn. Undergr. Space Technol.
**1992**, 18, 57–69. [Google Scholar] [CrossRef] - Cai, Y.; Tetsuro, E.; Jiang, Y.J. A rock bolt and rock mass interaction model. Int. J. Rock Mech. Min. Sci.
**2004**, 41, 1055–1067. [Google Scholar] [CrossRef] - Sun, J. Rock rheological mechanics and its advance in engineering applications. Chin. J. Rock Mech. Eng.
**2007**, 26, 1081–1106. [Google Scholar] - Phienwej, N.; Thakur, P.K.; Cording, E.J. Time-dependent response of tunnels considering creep effect. Int. J. Geomech.
**2007**, 7, 296–306. [Google Scholar] [CrossRef] - Goodman, R. Introduction to Rock Mechanics; Wiley: New York, NY, USA, 1989. [Google Scholar]
- Nomikos, P.; Rahmannejad, R.; Sofianos, A. Supported axisymmetric tunnels within linear viscoelastic Burgers rocks. Rock Mech. Rock Eng.
**2011**, 44, 553–564. [Google Scholar] [CrossRef] - Ladanyi, B.; Gill, D.E. Tunnel lining design in creeping rocks. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1985**, 22, A17. [Google Scholar] [CrossRef] - Sulem, J.; Panet, M.; Guenot, A. An analytical solution for time-dependent displacements in circular tunnel. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1987**, 24, 155–164. [Google Scholar] [CrossRef] - Ladanyi, B. Time-dependent response of rock around tunnels. In Comprehensive Rock Engineering; Hudson, J., Ed.; Pergamon: London, UK, 1993; Volume 2, pp. 77–112. [Google Scholar]
- Panet, M. Understanding deformations in tunnels. In Comprehensive Rock Engineering; Hudson, J., Ed.; Pergamon: London, UK, 1993; Volume 1, pp. 663–690. [Google Scholar]
- Ghaboussi, J.; Gioda, G. On the time-dependent effects in advancing tunnels. Int. J. Numer. Methods Geomech.
**1977**, 1, 249–269. [Google Scholar] [CrossRef] - Gioda, G. A finite element solution of non-linear creep problems in rock. Int. J. Rock Mech. Min. Sci. Geomech. Abstr.
**1981**, 18, 35–46. [Google Scholar] [CrossRef] - Peila, D.; Oreste, P.; Rabajuli, G.; Trabucco, E. The pre-tunnel method, a new Italian technology for full-face tunnel excavation: A numerical approach to design. Tunn. Undergr. Space Technol.
**1995**, 10, 367–374. [Google Scholar] [CrossRef] - Li, J.J.; Zheng, B.L.; Xu, C.Y. Numerical analyses of creep behavior for prestressed anchor rods. Chin. Q. Mech.
**2007**, 28, 124–128. [Google Scholar] - Wang, G.; Liu, C.Z.; Jiang, Y.J.; Wu, X.Z.; Wang, S.G. Rheological Model of DMFC Rockbolt and Rockmass in a Circular Tunnel. Rock Mech. Rock Eng.
**2015**, 48, 2319–2357. [Google Scholar] [CrossRef] - Sharifzadeh, M.; Tarifard, A.; Moridi, M.A. Time-dependent behavior of tunnel lining in weak rock mass based on displacement back analysis method. Tunn. Undergr. Space Technol.
**2013**, 38, 348–356. [Google Scholar] [CrossRef] - Labiouse, V. Ground response curves for rock excavations supported by ungrouted tensioned rockbolts. Rock Mech. Rock Eng.
**1996**, 29, 19–38. [Google Scholar] [CrossRef] - Bobet, A. A simple method for analysis of point anchored rockbolts in circular tunnels in elastic ground. Rock Mech. Rock Eng.
**2006**, 39, 315–338. [Google Scholar] [CrossRef] - Carranza-Torres, C.; Fairhurst, C. Application of the convergence–confinement method of tunnel design to rock masses that satisfy the Hoek-Brown failure criterion. Tunn. Undergr. Space Technol.
**2000**, 15, 187–213. [Google Scholar] [CrossRef] - Bobet, A.; Einstein, H.H. Tunnel reinforcement with rockbolts. Tunn. Undergr. Space Technol.
**2011**, 26, 100–123. [Google Scholar] [CrossRef] - Wang, Z.Y.; Li, Y.P. Rock Rheology Theory and Numerical Simulation; Science Press: Beijing, China, 2008. [Google Scholar]

**Figure 2.**Three main analytical models of DMFC rockbolts: (

**a**) distributed force model; (

**b**) point load model; (

**c**) equivalent material model.

**Figure 4.**Elasticity analysis mechanics model: (

**a**) force analysis in original zone; (

**b**) force analysis in reinforced zone.

**Figure 6.**Evolution of the axial force over time: (

**a**) influences of η

_{1}and σ

_{0}on the axial force over time; (

**b**) influence of η

_{2}on the bolt axial force over time; (

**c**) influence of η

_{b}on the axial force over time; and (

**d**) influence of T

_{0}on the axial force over time.

**Figure 7.**Mechanical states in cases 1–7: (

**a**) influence of η

_{1}and η

_{2}on the radial stress jump value and its fitting; (

**b**) influence of η

_{1}and η

_{2}on the excavation surface radial stress and its fitting. (

**c**,

**d**) reinforced zone monitoring points ρ = 4 m (influences η

_{1}, σ

_{0}, and η

_{2}, σ

_{0}).

**Figure 8.**Evolution of the tangential stress over time: (

**a**,

**b**) reinforced zone monitoring point ρ = 6 m; (

**c**,

**d**) original zone monitoring point ρ = 10 m.

**Figure 9.**Deformation law of the rock mass: (

**a**) displacement law of the excavation moment along the radial distance; (

**b**) displacement law of t = 4d along the radial distance; (

**c**) displacement evolution curve of ρ = 7 m.

Parameters | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | Case 7 | Case 8 |
---|---|---|---|---|---|---|---|---|

R/m | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 |

r/m | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |

A_{b}/10^{−4} m^{2} | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |

E_{b}/10^{11} Pa | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 | 2.0 |

L/m | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 |

K/10^{9} Pa | 2.2 | 2.2 | 2.2 | 2.2 | 2.2 | 2.2 | 2.2 | 2.2 |

G_{0}/10^{9} Pa | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 | 1.5 |

G_{1}/10^{10} Pa | 6.0 | 6.0 | 6.0 | 6.0 | 6.0 | 6.0 | 6.0 | 6.0 |

η_{b}/10^{20} Pa·s | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 | 3.0 |

η_{1}/10^{10} Pa·s | 5.0 | 3.0 | 7.0 | 5.0 | 5.0 | 5.0 | 5.0 | 5.0 |

η_{2}/10^{11} Pa·s | 2.0 | 2.0 | 2.0 | 4.0 | 6.0 | 2.0 | 2.0 | 2.0 |

T_{0}/10^{4} N | 4.0 | 4.0 | 4.0 | 4.0 | 4.0 | 5.0 | 5.5 | 4.0 |

σ_{0}/10^{6} Pa | −2.0 | −2.0 | −2.0 | −2.0 | −2.0 | −2.0 | −2.0 | −3.0 |

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## Share and Cite

**MDPI and ACS Style**

Han, W.; Wang, G.; Liu, C.; Luan, H.; Wang, K.
Time-Dependent Behavior of a Circular Symmetrical Tunnel Supported with Rockbolts. *Symmetry* **2018**, *10*, 381.
https://doi.org/10.3390/sym10090381

**AMA Style**

Han W, Wang G, Liu C, Luan H, Wang K.
Time-Dependent Behavior of a Circular Symmetrical Tunnel Supported with Rockbolts. *Symmetry*. 2018; 10(9):381.
https://doi.org/10.3390/sym10090381

**Chicago/Turabian Style**

Han, Wei, Gang Wang, Chuanzheng Liu, Hengjie Luan, and Ke Wang.
2018. "Time-Dependent Behavior of a Circular Symmetrical Tunnel Supported with Rockbolts" *Symmetry* 10, no. 9: 381.
https://doi.org/10.3390/sym10090381