Theoretical Investigation to the Effect of Bolt Reinforcement on Tunnel Viscoelastic Behavior

  • Kui Wu
  • Zhushan ShaoEmail author
  • Chenglong Li
  • Su Qin
Research Article - Civil Engineering


Rock bolts are widely applied in undergrounding excavation for strengthening rocks internally. For better understanding the effect of bolt reinforcement, the mechanical behavior of a circular bolt-liner combined supported tunnel is investigated using analytical method in this paper. Viscoelastic mechanical model of bolted rocks is established, and analytical solution for displacement and pressure in the rock/liner interface is provided, accounting for the installation delay of liner. Results show that the variations of displacement and pressure can be generally divided into three stages, rapid increasing stage, slow increasing stage and steady stage. Bolt parameter plays an important role on the evolution time of displacement and pressure. It will take shorter time for displacement and pressure to reach a steady state in the case with greater bolt parameter. In addition, installing rock bolts are able to effectively reduce displacement and pressure and there exists an exponential function relationship between displacement (pressure) and bolt parameter. The displacement and pressure decrease in a high rate with bolt parameter under the relatively low class. But, this decrease trend becomes flatter if bolt parameters reaching a certain value. There should be an appropriate bolt parameter for a certain tunnel project, taking into account mechanical parameters of rocks, load-bearing capacity of liner, or allowed deformation of design.


Tunnel Rock bolt Viscoelasticity Deformation Pressure 

List of Symbols

sij, eij

Tensors of the stress and strain deviators

\( s_{ij}^{\text{M}} ,\;e_{ij}^{\text{M}} \)

Tensors of the stress and strain deviators in Maxwell model

\( s_{ij}^{\text{K}} ,\;e_{ij}^{\text{K}} \)

Tensors of the stress and strain deviators in Kelvin model

\( G^{\text{M}} ,\,\eta^{\text{M}} \)

Spring constant and viscosity coefficient of dashpot in Maxwell model

\( G^{\text{K}} ,\,\eta^{\text{K}} \)

Spring constant and viscosity coefficient of dashpot in Kelvin model

\( G^{\text{S}} \)

Spring constant representing performance of rock bolts

\( \sigma_{ij} ,\,\varepsilon_{ij} \)

Stress and strain tensors

\( \sigma_{kk} ,\,\varepsilon_{kk} \)

Stress and strain subjected to Einstein summation convention

\( \delta_{ij} \)

Kronecker delta

\( \sigma_{\text{r}} ,\,\sigma_{\theta } ,\sigma_{z} \)

Radial, tangential and axial stress in cylindrical coordinate system

\( \varepsilon_{\text{r}} ,\,\varepsilon_{\theta } ,\varepsilon_{z} \)

Radial, tangential and axial strain in cylindrical coordinate system

\( \sigma_{\text{mean}} ,\,\varepsilon_{\text{mean}} \)

Mean stress and strain

\( \Delta \sigma_{\text{r}} ,\,\Delta \varepsilon_{\text{r}} \)

Radial deviatoric stress and strain in polar coordinate system

\( \Delta \sigma_{\theta } ,\,\Delta \varepsilon_{\theta } \)

Tangential deviatoric stress and strain in polar coordinate system

\( p_{0} ,\,p\left( t \right) \)

Initial ground stress and pressure on liner/reaction force of liner

\( r,\,\theta \)

Polar coordinates


Radius of the tunnel

\( u\left( {r,t} \right),\,u_{\text{R}} \left( t \right) \)

Radial displacement and radial displacement at tunnel wall

\( K_{\text{S}} \)

Liner stiffness


Liner thickness


Young’s modulus and Poisson’s ratio of the liner


Installation time of liner



This research work is supported by the National Natural Science Foundation of China (No. 11872287), the Found of Shaanxi Key Research and Development Program (No. 2019ZDLGY01-10).


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Copyright information

© King Fahd University of Petroleum & Minerals 2019

Authors and Affiliations

  1. 1.School of Civil EngineeringXi’an University of Architecture and TechnologyXi’anChina

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