A New Semi-Analytical Method for Elasto-Plastic Analysis of a Deep Circular Tunnel Reinforced by Fully Grouted Passive Bolts
Abstract
:Featured Application
Abstract
1. Introduction
2. Definition of the Problem
2.1. Basic Assumptions
2.2. Tunnel Face Effect during Tunneling
2.3. Equation of Equilibrium
2.4. Failure Criterions
2.4.1. Mohr–Coulomb Criterion
2.4.2. Generalized Hoek–Brown Criterion
2.5. Plastic Potential Equation
2.6. Equations of Strains and Displacements
2.6.1. In an Elastic Region
2.6.2. In a Plastic Region
2.7. Interaction between the Rock Mass and the Passive Bolts
3. Stress–Strain Analysis of the Rock Mass and Passive Bolts Using the Finite Difference Method
3.1. Stress–Strain Analysis in an Unbolted Region
3.2. Stress–Strain Analysis in a Bolted Region
3.2.1. Stress–Strain Analysis in an Elastic Region
3.2.2. Stress–Strain Analysis in a Plastic Region
3.2.3. Behaviors of the Passive Bolts
4. Verification
4.1. Comparison with the Results from the Commercial Numerical Software
4.1.1. Establishment of the Numerical Model
4.1.2. Mohr–Coulomb Criterion
4.1.3. Generalized Hoek–Brown Criterion
4.2. Comparison with the In Situ Test Results from the Kielder Experimental Tunnel
5. Discussion
5.1. Comparison of the Solutions That Considered or Neglected Decoupling at the Bolt–Rock Interface
5.2. Effect of the End Plate and Shear Stiffness at the Bolt–Rock Interface
5.3. Effect of Bolt Length and Density
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A. Stress–Strain Analysis in an Unbolted Tunnel
Appendix A.1. Stress–Strain Analysis in an Unbolted Elastic Region
Appendix A.2. Stress–Strain Analysis in an Unbolted Elastic Region
Appendix B. Calculation Procedure
- (1)
- Input parameters: R, p0, β, p, E, μ, ψ, Eb, lb, ds, Ab, lz, ω, Ks, Kep, cs, φs, Δr, cp, cr, φp, φr, σci,p, σci,r, mb,p, mb,r, sp, sr, ap, and ar.
- (2)
- (3)
- Assume an initial value of σr(R+lb) = (σr1(R+lb)+σr2(R+lb))/2, where σr1(R+lb) = 0 and σr2(R+lb) = p0.
- (4)
- Calculate ur(R+lb) and σθ(R+lb) by replacing p and R in the equations in Appendix A with σr(R+lb) and R+lb, respectively.
- (5)
- Assume an initial value of = (+)/2, where = 0 and= 10.
- (6)
- Calculate τs(0) and Fn(0) using Equations (32) and (33), respectively.
- (1)
- Verify the elastoplastic state of the rock mass in the ith annulus using Equation (4) (M–C rock mass) or Equation (5) (H–B rock mass).
- (2)
- Calculate σr(i) using Equation (24) (elastic state), or Equation (27) (plastic state, M–C rock mass) or Equation (29) (plastic state, H–B rock mass).
- (3)
- Calculate σθ(i) using Equation (25) (elastic state), or Equation (28) (plastic state, M–C rock mass) or Equation (30) (plastic state, H–B rock mass).
- (4)
- Calculate ur(i) using Equation (26) (elastic state) or Equation (31) (plastic state).
- (5)
- Calculate τb(i) and Fn(i) using Equations (32) and (33), respectively.
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Rock Mass | Passive Bolts | ||
---|---|---|---|
Symbol (Unit) | Value | Symbol (Unit) | Value |
R (m) | 3 | Eb (GPa) | 210 |
p0 (MPa) | 1 | lb (m) | 3 |
E (GPa) | 0.5 | ds (mm) | 25 |
μ | 0.2 | Ab (mm2) | 491 |
cp (MPa) | 0.1 | lz (m) | 1 |
cr (MPa) | 0.1 | ω (°) | 10 |
φp (°) | 30 | Ks (MPa) | 10 |
φr (°) | 30 | Kep (MN/m) | 0 |
ψ (°) | 0 | β | 0.3 |
p (MPa) | 0 | ||
cs (kPa) | ∞ | ||
φs (°) | 0 |
Symbol (Unit) | Value | Symbol (Unit) | Value |
---|---|---|---|
R (m) | 3 | mb,r | 2 |
p0 (MPa) | 15 | sp | 4 × 10−3 |
E (GPa) | 5.7 | sr | 4 × 10−3 |
μ | 0.25 | ap | 0.55 |
σci,p (MPa) | 30 | ar | 0.55 |
σci,r (MPa) | 30 | ψ (°) | 15 |
mb,p | 2 |
Rock Mass | Passive Bolts | ||
---|---|---|---|
Symbol (Unit) | Value | Symbol (Unit) | Value |
R (m) | 1.6 | Eb (GPa) | 210 |
p0 (MPa) | 2.6 | lb (m) | 1.8 |
E (GPa) | 5 | ds (mm) | 25 |
μ | 0.25 | Ab (mm2) | 491 |
σci,p (MPa) | 37 | lz (m) | 0.9 |
σci,r (MPa) | 37 | ω (°) | 32 |
mb,p | 0.1 | Ks (MPa) | 70 |
mb,r | 0.05 | Kep (MN/m) | 30 |
sp | 8 × 10−5 | β | 0.3 |
sr | 1 × 10−5 | p (MPa) | 0 |
ap | 0.5 | cs (kPa) | ∞ |
ar | 0.5 | φs (°) | 0 |
ψ (°) | 0 |
Symbol (Unit) | Value | Symbol (Unit) | Value |
---|---|---|---|
R (m) | 3 | mb,r | 0.615 |
p0 (MPa) | 5 | sp | 7.3 × 10−4 |
E (GPa) | 2.57 | sr | 1.7 × 10−4 |
μ | 0.25 | ap | 0.516 |
σci,p (MPa) | 30 | ar | 0.538 |
σci,r (MPa) | 24 | ψ (°) | 0 |
mb,p | 0.981 |
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Wang, M.; Zhang, X.; Tong, J.; Yi, W.; Wang, Z.; Liu, D. A New Semi-Analytical Method for Elasto-Plastic Analysis of a Deep Circular Tunnel Reinforced by Fully Grouted Passive Bolts. Appl. Sci. 2020, 10, 4402. https://doi.org/10.3390/app10124402
Wang M, Zhang X, Tong J, Yi W, Wang Z, Liu D. A New Semi-Analytical Method for Elasto-Plastic Analysis of a Deep Circular Tunnel Reinforced by Fully Grouted Passive Bolts. Applied Sciences. 2020; 10(12):4402. https://doi.org/10.3390/app10124402
Chicago/Turabian StyleWang, Mingnian, Xiao Zhang, Jianjun Tong, Wenhao Yi, Zhilong Wang, and Dagang Liu. 2020. "A New Semi-Analytical Method for Elasto-Plastic Analysis of a Deep Circular Tunnel Reinforced by Fully Grouted Passive Bolts" Applied Sciences 10, no. 12: 4402. https://doi.org/10.3390/app10124402
APA StyleWang, M., Zhang, X., Tong, J., Yi, W., Wang, Z., & Liu, D. (2020). A New Semi-Analytical Method for Elasto-Plastic Analysis of a Deep Circular Tunnel Reinforced by Fully Grouted Passive Bolts. Applied Sciences, 10(12), 4402. https://doi.org/10.3390/app10124402