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Rock Mechanics and Rock Engineering

, Volume 52, Issue 11, pp 4565–4588 | Cite as

Coupling Analysis for Rock Mass Supported with CMC or CFC Rockbolts Based on Viscoelastic Method

  • Gang WangEmail author
  • Wei Han
  • Yujing Jiang
  • Hengjie Luan
  • Ke Wang
Original Paper

Abstract

Rockbolts have been widely used in rock reinforcement for high-stress conditions in mining and civil engineering. However, the interaction mechanism between the rockbolt and the rock mass is still unclear. To fully understand the coupling mechanism of a rock mass supported with rockbolts, this article studied the coupling effect and the time-dependent behavior of a rock mass supported with continuously mechanically coupled (CMC) or continuously frictionally coupled (CFC) rockbolts. The elastic solutions of the interaction model were obtained in the coupled state. In addition, viscoelastic analytical solutions were used to describe the rheological properties of the coupling model, and the solutions were acquired by setting the constitutive models of the rockbolt and rock mass to a one-dimensional Kelvin model and a three-dimensional Maxwell model based on the material properties. According to the proposed coupling model, the rock mass stress and displacement fields, and the rockbolt axial force strongly depend on the relative deformation modulus of the rock mass and rockbolt. In addition, a lower viscosity coefficient of the rockbolt or rock mass produces a larger rock mass displacement. Moreover, as the relative deformation modulus increases, the distance to the neutral point beyond the rockbolt head increases. Furthermore, the position of the neutral point is independent of time.

Keywords

Analytical solutions Time-dependent behavior CMC or CFC rockbolt Viscoelasticity 

List of Symbols

Ab

Cross-sectional area of the rockbolt

K

Bulk modulus of the rock mass

ρ

Radial coordinate

r

Tunnel radius

d

Rockbolt diameter

σ0

Initial rock mass stress

\(u_{{\rho_{0} }}\)

Rock mass displacement under the initial rock mass stress

\(\varepsilon_{{\theta_{0} }}\)

Rock mass tangential strain under the initial rock mass stress

\(u_{{\rho_{2} }}^{\prime }\)

Change of displacement in the unreinforced zone

\(\varepsilon_{{\rho_{2} }}^{\prime }\)

Change of radial strain in the unreinforced zone

\(\sigma_{{\theta_{2} }}^{\prime }\)

Change of tangential strain in the unreinforced zone

\(\sigma_{{\rho_{1} }}\)

Radial stress in the reinforced zone

\(\sigma_{{\theta_{1} }}\)

Tangential stress in the reinforced zone

\(\varepsilon_{{\rho_{1} }}\)

Radial strain in the reinforced zone

\(\varepsilon_{{\theta_{1} }}\)

Tangential strain in the reinforced zone

\(u_{{\rho_{1} }}\)

Rock mass displacement in the reinforced zone

ε

Axial strain of the rockbolt

Sθ

Rockbolt spacing in the tangential direction

t

Time

Gr

Shear modulus of the rock mass

σij

Stress tensor

TN

Axial force of the rockbolt in the neutral point

T1

Rockbolt axial force in front of the neutral point

\(\overline{{P_{{{\text{a}}K}} }}^{\prime } (s), \, \overline{{Q_{{{\text{a}}K}} }}^{\prime } (s)\)

Operator function of the rockbolt viscoelastic constitutive model after Laplace transformation

τa

Additional shear stress beyond the neutral point

\(\tau_{{{\text{B}}_{ 1} }}\)

Shear stress before the neutral point

pk,qk

Constant parameters of the rockbolt material

Er

Deformation modulus of the rock mass

Eb

Deformation modulus of the rockbolt

Θ

Angular coordinate

R

Radius of the reinforced zone

L

The length of the rockbolt

μr

Poisson’s ratio of the rock mass

\(\varepsilon_{{\rho_{0} }}\)

Rock mass radial strain under the initial rock mass stress

\(u_{{\rho_{1} }}^{\prime }\)

Change of displacement in the reinforced zone

\(\varepsilon_{{\rho_{1} }}^{\prime }\)

Change of radial strain in the reinforced zone

\(\varepsilon_{{\theta_{1} }}^{\prime }\)

Change of tangential strain in the reinforced zone

c

Distance from the concentrated force to the rockbolt end

\(\sigma_{{\rho_{2} }}\)

Radial stress in the unreinforced zone

\(\sigma_{{\theta_{2} }}\)

Tangential stress in the unreinforced zone

\(\varepsilon_{{\rho_{2} }}\)

Radial strain in the unreinforced zone

\(\varepsilon_{{\theta_{2} }}\)

Tangential strain in the unreinforced zone

\(u_{{\rho_{2} }}\)

Radial displacement in the unreinforced zone

\(\varepsilon_{{\rho_{0} }}\)

Initial strain of the rock mass under the initial stress

Sz

Rockbolt spacing in the longitudinal direction

σb

Axial stress of the rockbolt

u

Displacement in the semi-infinite plane under the Mindlin solution

εij

Strain tensor

ρn

The position of the neutral point

T2

Rockbolt axial force beyond the neutral point

\(\begin{aligned} \overline{P}^{\prime } (s),\;\;\overline{Q}^{\prime } (s) \hfill \\ \overline{P}^{\prime \prime } (s),\;\;\overline{Q}^{\prime \prime } (s) \hfill \\ \end{aligned}\)

Operator function of the rock mass viscoelastic constitutive model after Laplace transformation

τb

Shear stress caused by the rock mass deformation

\(\tau_{{{\text{B}}_{ 2} }}\)

Shear stress beyond the neutral point

D

Differential operator

Notes

Acknowledgements

This study was supported by the National Natural Science Foundation of China (no. 51479108) and the Taishan Scholar Talent Team Support Plan for Advantaged & Unique Discipline Areas.

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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  • Gang Wang
    • 1
    • 2
    Email author
  • Wei Han
    • 1
  • Yujing Jiang
    • 2
  • Hengjie Luan
    • 2
  • Ke Wang
    • 1
  1. 1.Shandong Provincial Key Laboratory of Civil Engineering Disaster Prevention and MitigationShandong University of Science and TechnologyQingdaoChina
  2. 2.State Key Laboratory of Mining Disaster Prevention and Control Co-founded by Shandong Province and the Ministry of Science and TechnologyShandong University of Science and TechnologyQingdaoChina

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