Time-Dependent Solutions for Lined Circular Tunnels Considering Rockbolts Reinforcement and Face Advancement Effects
Publication: International Journal of Geomechanics
Volume 21, Issue 10
Abstract
During deep excavation in weak rock masses, the time-dependent deformation of the rock and longitudinal advancement process have significant influences on the tunnel performance. Accordingly, a combined support system consisting of rockbolts and linings is commonly used in such tunnels to ensure their stability. However, there are no well-established solutions for quickly and accurately evaluating the safety of those support systems. To address this problem, mechanical model of a circular tunnel with a rockbolt–lining combined system is analytically built in this study. Then mathematical formulas for the stresses and displacements around the tunnel are derived, considering the time-dependent behaviors of the rocks and tunnel face advancement effect. Furthermore, the analytical formula proposed in this study is compared with existing solutions and numerical results obtained using the finite-element software Abaqus, and a good agreement is achieved. Finally, a parametric investigation is conducted regarding the influences on the tunnel’s mechanical performance. The results show that the installation time (t0) and distribution density of rockbolts (SθSz) and the installation time of the lining (t1) have a significant influence on the deformation of the surrounding rock. Although the installation times of bolt and lining are positively related to the deformation suppression effect of the surrounding rock, they are negatively related to the stress of the bolt and lining. It is found that the closer the installation time of bolt and lining, the greater the stress of lining; otherwise, the greater the stress of bolts. Furthermore, the initial release coefficient (m) and the ratio of tunnel excavation rate to influence radius can influence the deformation rate of surrounding rock. Finally, the study proposes a new method for quickly evaluating the viscoelastic deformation of a tunnel.
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Acknowledgments
This research was supported by the National Natural Science Foundation of China (Nos. 11872287, 51908431) and the Found of Shaanxi Key Research and Development Program (No. 2019ZDLGY01-10).
Notation
The following symbols are used in this paper:
- Ab
- section area of rockbolts (m2);
- E
- elastic modulus (MPa);
- Eb
- elastic modulus of rockbolts (MPa);
- G
- shear modulus (MPa);
- Gk
- shear modulus of Kelvin model (MPa);
- Gm
- shear modulus of Maxwell model (MPa);
- J(s)
- Laplace transform of J(t);
- J(t)
- flexibility modulus (MPa−1);
- K
- bulk modulus (MPa);
- KE
- lining stiffness (MPa);
- Kb
- rockbolt stiffness (MPa/m);
- L
- length of rockbolts (m);
- m
- initial stress release coefficient;
- P0
- initial ground stress (MPa);
- Pa
- fictitious support pressure (MPa);
- PT1
- support stress by the bolt at the inner boundary of the reinforcement area (MPa);
- PT2
- support stress by bolt at outer boundary of the reinforcement area (MPa);
- operator functions of rock viscoelastic model in Laplace space;
- R
- outer radius of the bolt-reinforced zone (m);
- RL
- influence radius of excavation face;
- r
- distance from tunnel center (m);
- r1
- tunnel radius (m);
- r2
- inner radius of the lining (m);
- T(s)
- axial force in Laplace space;
- T2
- axial force of rockbolts (kN);
- (t)
- stress release coefficient varying with time;
- t
- time (days/years);
- t0
- installation time of rockbolts (days/months);
- t1
- installation time of lining (days/months);
- u
- radial displacement of rock;
- v
- excavation speed (m/days or m/months);
- X
- distance from the face (m)
- Laplace Transform of *;
- α
- rockbolt tangential spacing radius;
- δ(τ)
- impact function;
- ηk
- Newtonian viscosity coefficient of Kelvin model (MPa · days or MPa · month);
- ηm
- Newtonian viscosity coefficient of Maxwell model (MPa · days or MPa · month);
- λ(X)
- stress release coefficient varying with distance;
- μ
- Poisson’s ratio;
- σr
- radial stress (MPa);
- σ1
- stress of rock mass at the inner boundary of unreinforced area (MPa); and
- σθ
- tangential stresses (MPa).
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International Journal of Geomechanics
Volume 21 • Issue 10 • October 2021
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Aug 18, 2020
Accepted: May 2, 2021
Published online: Jul 28, 2021
Published in print: Oct 1, 2021
Discussion open until: Dec 28, 2021
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