Explicit Analysis for the Ground Reaction of a Circular Tunnel Excavated in Anisotropic Stress Fields Based on Hoek–Brown Failure Criterion
Abstract
:1. Introduction
2. Problem Description
2.1. Correlation between Confinement Loss and the Advancing Excavation of Tunnels
2.2. Variations in Stress around a Circular Tunnel
- (1)
- Stresses in the far field (initial in situ stresses, as r → ∞): Before tunnel excavation, the stresses within the rock masses are in equilibrium and mirror the lithostatic stresses, remaining constant in the surrounding medium of the tunnel. Thus, the initial anisotropic stress within the rock masses can be illustrated as depicted in Figure 3 and represented by the following equation:
- (2)
- Stresses in the near field (final stresses or boundary stresses at the tunnel periphery, R ≤ r < ∞): Upon tunnel excavation, the stresses at any given point within the rock masses surrounding the tunnel undergo alteration. These altered stresses can be determined using the Kirsch solutions, outlined as follows:
3. Derivation of Stress/Displacement Equations in the Plastic Region
3.1. Derivation of the Confinement Loss in Elastic Limit with the Non-Linear Failure Criteria
3.2. Derivation of the Plastic Radius
3.3. Derivation of Stress in the Plastic Region
3.4. Derivation of Displacement in the Plastic Region
- (1)
- Homogeneous solution ():
- (2)
- Particular solution ():
4. Utilization of an Incremental Procedure for the Analytical Solution
Calculation Steps in the Incremental Procedure Method
- (1)
- Input data: This relates to the data of the initial in situ stress, geometry of the tunnel, material properties, and unsupported distance.
- (2)
- To estimate the confinement loss λz at a certain distance z from the tunnel working face, one can use a given value of λz as a chosen effect of the unsupported distance of tunnel excavation. Therefore, λz can be determined from the Equation (1).
- (3)
- Dividing the confinement loss λz by n segments, the incremental step λ can be expressed as
- (4)
- Calculating each step value of λ as
- (5)
- Attaining the final value
- (6)
- According to Equation (23), estimate the confinement loss in the elastic limit situation (λe).
- (7)
- If , it means that the stress state is in the elastic region and that the radial and tangential stresses/displacements can be calculated with Equations (11)–(13).
- (8)
- If , it means that the stress state is in the plastic region and that the plastic radius Rp can be calculated with Equation (29). Once one obtains this value Rp, the procedure automatically substitutes into Equations (31), (32) and (52) for the radial and tangential stresses, and the radial displacement, respectively.
- (9)
- Recording the calculated data relates to the representation of the distribution of stresses/displacements (), (), and () on the cross-section of the tunnel and () at the intrados of the tunnel.
- (10)
- When i < n − 1, repeat steps (4) through (10).
- (11)
- When i = n − 1, the process is not repeated, and the data from each step is recorded.
- (12)
- Drawing the distribution of stresses/displacements at the intrados and on the cross-section of the tunnel.
5. Comparison of Results between the Findings of This Study and Published Data
5.1. Stress/Displacement in the Elastic Region
- (1)
- When λ = 0 (), it indicates that the dimensionless radial stress (σr/σv) and tangential stress (σθ/σv) used in the calculation were, respectively, 1 and 0.67 in the initial anisotropic stress state as shown the horizontal line on the right side in Figure 9a;
- (2)
- When 0 < λ < 1.0 (), it describes that the stresses were also in the elastic region, the radial and tangential stresses increased with the increase of the confinement loss, and both stresses were separated along the horizontal axis (r/R);
- (3)
5.2. Stress/Displacement in the Plastic Region
- (1)
- When λ = 0 (), it indicates that the stresses were in the initial stress state as shown in the horizontal line on the right side in Figures; the radial stress (σh/σv) and tangential stress (σθ/σv) used in the calculation were 0.67 and 1.0, respectively.
- (2)
- When 0 (), it describes that the stresses were in the elastic region, and the radial and tangential stresses increased with the increase of the confinement loss, and both stresses were separated along the horizontal axis (r/R).
- (3)
- When (), the plastic radius appeared, and stresses were at the elastic-plastic interface. The radial stress began to change the curvature, and the tangential stress attained the maximum value.
- (4)
- When (), it indicates that the stresses are in the plastic region, and this leads to both the radial stress and tangential stress being decreased steeply.
- (5)
- Until (), the radial stress becomes zero and the tangential stress is equal to the coefficient proposed by the Hoek–Brown failure criterion.
6. Conclusions
- (1)
- A coherent closed-form analytical solution has been derived for the elastic–perfectly plastic analysis of a circular tunnel within rock masses governed by the Hoek–Brown non-linear failure criterion, considering anisotropic in situ stress.
- (2)
- The agreement between published results and the proposed closed-form solutions using the explicit procedure was excellent within elastic–perfectly plastic media. The percentage error for radial displacement ranged from 0.93% to 1.18% in the elastic region and from 3.28% to 5.51% in the plastic region. Additionally, the percentage error for the radius of the plastic zone ranged from 4.99% to 5.6%.
- (3)
- An incremental approach within the explicit analysis method (EAM) was proposed to model the advancing excavation of the tunnel face. This method calculates stresses and displacements at each step, facilitating the generation of ground reaction curves, stress paths at the intrados of the tunnel, and stress/displacement distributions across tunnel cross-sections.
- (4)
- The fluctuation of stresses during tunnel excavation can primarily be explained by the stress gradient, which represents the disparity between far-field and near-field stresses around the tunnel. This gradient can be computed using incremental assumptions in numerical analysis, with the confinement loss considered as a portion of the stress gradient.
- (5)
- Confinement loss in the elastic limit was determined by the peak strength parameters of the rock masses and initial vertical stress.
- (6)
- The plastic radius was influenced by the peak strength parameters of the rock masses, confinement loss in the elastic limit, and dependency on confinement loss. Increased confinement loss resulted in a larger plastic radius, reducing both radial and tangential stress while increasing radial displacement in the plastic region.
- (7)
- The incremental procedure method proposed herein accommodates the nonlinear failure criterion of the rock masses. It not only serves as a valuable tool for analyzing circular tunnels under isotropic stress conditions but also holds promise for simulating tunnel behavior under anisotropic stress conditions.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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---|---|---|---|---|---|
Parameter | Case I | Case II | Case III | Case IV | Case V |
E (GPa) | 60 | 90 | 40 | 5.5 | 27.6 |
ν | 0.2 | 0.2 | 0.2 | 0.25 | 0.2 |
mi | 10.84 | 16 | 7.5 | 17 | 15 |
GIS | 89 | 90 | 79 | 50.08 | 50.31 |
D | 0 | 0 | 0 | 0 | 0 |
σci (MPa) | 210 | 200 | 300 | 30 | 69 |
Kψ | 0 | 0 | 0 | 0 | 0 |
R (m) | 10.0 | 10.0 | 4.0 | 5.0 | 6.1 |
Published Studies | Radial Displacement, uR (m) | Plastic Zone Radius, Rp (m) | EAM Radial Displacement, uR (mm) (Error * %) | EAM Plastic Zone Radius, Rp (m) (Error * %) |
---|---|---|---|---|
Case I | 0.09 | N/A | 0.0909 (1.0%) | N/A |
Case II | 0.05 | N/A | 0.0505 (0.93%) | N/A |
Case III | 0.11 | N/A | 0.1113 (1.18%) | N/A |
Case IV | 0.24 | 21.5 | 0.2268 (5.51%) | 22.705 (5.60%) |
Case V | 0.038 | 14.3 | 0.0368 (3.28%) | 15.014 (4.99%) |
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Lee, Y.-L.; Chen, C.-S.; Lee, C.-M. Explicit Analysis for the Ground Reaction of a Circular Tunnel Excavated in Anisotropic Stress Fields Based on Hoek–Brown Failure Criterion. Mathematics 2024, 12, 2689. https://doi.org/10.3390/math12172689
Lee Y-L, Chen C-S, Lee C-M. Explicit Analysis for the Ground Reaction of a Circular Tunnel Excavated in Anisotropic Stress Fields Based on Hoek–Brown Failure Criterion. Mathematics. 2024; 12(17):2689. https://doi.org/10.3390/math12172689
Chicago/Turabian StyleLee, Yu-Lin, Chih-Sheng Chen, and Chi-Min Lee. 2024. "Explicit Analysis for the Ground Reaction of a Circular Tunnel Excavated in Anisotropic Stress Fields Based on Hoek–Brown Failure Criterion" Mathematics 12, no. 17: 2689. https://doi.org/10.3390/math12172689