Limit Analysis of Progressive Asymmetrical Collapse Failure of Tunnels in Inclined Rock Stratum
Abstract
:1. Introduction
2. Methods
2.1. Theoretical Basis
2.1.1. Upper Bound Theorem of Limit Analysis
2.1.2. Nonlinear Hoek–Brown Yield Criterion
2.2. Progressive Asymmetrical Collapse Mechanism in Inclined Rock Stratum
3. Upper Bound Analysis of Progressive Asymmetrical Collapsing Block
3.1. Upper Bound Analysis of Rectangular Tunnels
3.2. Upper Bound Analysis of Circular Tunnels
4. Results and Discussion
4.1. Rationale Verification
4.1.1. Rectangular Tunnels
4.1.2. Circular Tunnels
4.2. Sensitivity Analysis
4.2.1. Cases with Rectangular Tunnels
4.2.2. Cases with Circular Tunnels
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Appendix A. Derivation of the Total Rate of Energy Dissipation
Appendix B. Calculation of the Range and Total Weight of the Collapsing Block
References
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A1 | B1 | σc1 /MPa | γ1 /kN·m−3 | L1 /m | L2 /m | L3 /m | L4 /m | h1 /m | h2 /m | h3 /m | W1 /103 kN |
---|---|---|---|---|---|---|---|---|---|---|---|
0.1 | 0.8 | 8 | 20 | 1.88 | 0.37 | 1.92 | 0.44 | 6.65 | 6.75 | 6.51 | 0.34 |
0.2 | 0.8 | 8 | 20 | 3.72 | 0.69 | 3.85 | 0.92 | 6.71 | 6.94 | 6.43 | 0.67 |
0.3 | 0.8 | 8 | 20 | 5.53 | 0.98 | 5.78 | 1.49 | 6.76 | 7.14 | 6.32 | 1.01 |
0.4 | 0.8 | 8 | 20 | 7.33 | 1.35 | 7.70 | 2.44 | 6.82 | 7.43 | 6.16 | 1.38 |
0.5 | 0.8 | 8 | 20 | 9.16 | 2.23 | 9.98 | 5.41 | 6.98 | 8.23 | 5.63 | 2.11 |
0.4 | 0.76 | 8 | 20 | 8.80 | 2.42 | 9.23 | 4.49 | 7.01 | 8.42 | 5.79 | 1.94 |
0.4 | 0.77 | 8 | 20 | 8.40 | 2.05 | 8.83 | 3.88 | 6.95 | 8.06 | 5.90 | 1.76 |
0.4 | 0.78 | 8 | 20 | 8.03 | 1.75 | 8.44 | 3.33 | 6.90 | 7.78 | 6.00 | 1.61 |
0.4 | 0.79 | 8 | 20 | 7.67 | 1.53 | 8.07 | 2.85 | 6.86 | 7.58 | 6.09 | 1.48 |
0.4 | 0.8 | 7 | 20 | 6.02 | 0.17 | 6.07 | 0.29 | 6.62 | 6.65 | 6.54 | 0.92 |
0.4 | 0.8 | 8.5 | 20 | 7.85 | 1.86 | 8.22 | 3.18 | 6.91 | 7.88 | 6.03 | 1.58 |
0.4 | 0.8 | 9 | 20 | 8.32 | 2.32 | 8.63 | 3.77 | 7.00 | 8.32 | 5.92 | 1.76 |
0.4 | 0.8 | 10 | 20 | 9.10 | 3.12 | 9.29 | 4.71 | 7.14 | 9.16 | 5.76 | 2.10 |
0.4 | 0.8 | 8 | 19 | 7.84 | 1.85 | 8.31 | 3.32 | 6.91 | 7.87 | 6.00 | 1.59 |
0.4 | 0.8 | 8 | 21 | 6.86 | 0.91 | 7.10 | 1.59 | 6.75 | 7.09 | 6.31 | 1.22 |
0.4 | 0.8 | 8 | 22 | 6.43 | 0.53 | 6.55 | 0.86 | 6.68 | 6.84 | 6.43 | 1.11 |
0.4 | 0.8 | 8 | 23 | 6.07 | 0.22 | 6.10 | 0.33 | 6.63 | 6.67 | 6.53 | 1.04 |
A2 | B2 | σc2 /MPa | γ2 /kN·m−3 | L1 /m | L2 /m | L3 /m | L4 /m | h1 /m | h2 /m | h3 /m | W1 /103 kN |
---|---|---|---|---|---|---|---|---|---|---|---|
0.28 | 0.7 | 10 | 21 | 7.32 | 1.43 | 7.67 | 2.64 | 6.84 | 7.57 | 6.12 | 1.40 |
0.32 | 0.7 | 10 | 21 | 7.33 | 1.30 | 7.73 | 2.33 | 6.82 | 7.34 | 6.18 | 1.37 |
0.34 | 0.7 | 10 | 21 | 7.34 | 1.27 | 7.76 | 2.27 | 6.81 | 7.28 | 6.19 | 1.36 |
0.36 | 0.7 | 10 | 21 | 7.34 | 1.26 | 7.78 | 2.23 | 6.81 | 7.23 | 6.19 | 1.36 |
0.3 | 0.4 | 10 | 21 | 7.41 | 1.26 | 8.44 | 2.81 | 6.81 | 6.81 | 6.09 | 1.41 |
0.3 | 0.5 | 10 | 21 | 7.40 | 1.25 | 8.29 | 2.64 | 6.81 | 6.84 | 6.12 | 1.40 |
0.3 | 0.6 | 10 | 21 | 7.38 | 1.24 | 8.01 | 2.34 | 6.80 | 6.98 | 6.17 | 1.37 |
0.3 | 0.65 | 10 | 21 | 7.36 | 1.24 | 7.86 | 2.26 | 6.81 | 7.13 | 6.19 | 1.36 |
0.3 | 0.7 | 8 | 21 | 6.98 | 1.03 | 7.04 | 1.56 | 6.77 | 7.23 | 6.31 | 1.21 |
0.3 | 0.7 | 9 | 21 | 7.13 | 1.17 | 7.31 | 1.91 | 6.79 | 7.31 | 6.25 | 1.28 |
0.3 | 0.7 | 11 | 21 | 7.57 | 1.58 | 8.28 | 3.22 | 6.87 | 7.60 | 6.02 | 1.53 |
0.3 | 0.7 | 12 | 21 | 7.89 | 1.89 | 9.05 | 4.28 | 6.92 | 7.83 | 5.83 | 1.76 |
0.3 | 0.7 | 10 | 22 | 7.32 | 1.35 | 7.68 | 2.45 | 6.83 | 7.45 | 6.16 | 1.43 |
0.3 | 0.7 | 10 | 23 | 7.31 | 1.36 | 7.66 | 2.45 | 6.83 | 7.46 | 6.16 | 1.48 |
0.3 | 0.7 | 10 | 24 | 7.30 | 1.36 | 7.63 | 2.45 | 6.83 | 7.48 | 6.15 | 1.53 |
0.3 | 0.7 | 10 | 25 | 7.29 | 1.37 | 7.61 | 2.46 | 6.83 | 7.50 | 6.15 | 1.58 |
A1 | B1 | σc1 /MPa | γ1 /kN·m−3 | L1 /m | L2 /m | L3 /m | L4 /m | h1 /m | h2 /m | h3 /m | W2 /103 kN |
---|---|---|---|---|---|---|---|---|---|---|---|
0.10 | 0.8 | 18 | 18 | 2.23 | 1.20 | 1.91 | 0.87 | 3.38 | 4.96 | 3.35 | 0.71 |
0.11 | 0.8 | 18 | 18 | 2.36 | 1.22 | 2.04 | 0.90 | 3.41 | 5.03 | 3.40 | 0.74 |
0.12 | 0.8 | 18 | 18 | 2.50 | 1.23 | 2.24 | 0.96 | 3.45 | 5.11 | 3.48 | 0.79 |
0.13 | 0.8 | 18 | 18 | 2.59 | 1.23 | 2.42 | 1.02 | 3.48 | 5.16 | 3.56 | 0.83 |
0.14 | 0.8 | 18 | 18 | 2.74 | 1.25 | 2.62 | 1.09 | 3.52 | 5.25 | 3.65 | 0.87 |
0.10 | 0.72 | 18 | 18 | 2.88 | 1.29 | 2.88 | 1.23 | 3.57 | 5.43 | 3.77 | 0.94 |
0.10 | 0.74 | 18 | 18 | 2.71 | 1.26 | 2.55 | 1.08 | 3.52 | 5.29 | 3.61 | 0.86 |
0.10 | 0.76 | 18 | 18 | 2.55 | 1.24 | 2.32 | 0.99 | 3.47 | 5.17 | 3.51 | 0.81 |
0.10 | 0.78 | 18 | 18 | 2.40 | 1.22 | 2.10 | 0.92 | 3.43 | 5.07 | 3.42 | 0.76 |
0.10 | 0.8 | 17 | 18 | 1.97 | 0.99 | 1.52 | 0.55 | 3.29 | 4.56 | 3.26 | 0.57 |
0.10 | 0.8 | 19 | 18 | 2.03 | 1.04 | 1.61 | 0.63 | 3.31 | 4.66 | 3.28 | 0.60 |
0.10 | 0.8 | 20 | 18 | 2.11 | 1.10 | 1.71 | 0.70 | 3.34 | 4.76 | 3.30 | 0.64 |
0.10 | 0.8 | 21 | 18 | 2.18 | 1.15 | 1.81 | 0.78 | 3.36 | 4.87 | 3.33 | 0.67 |
0.10 | 0.8 | 18 | 17 | 2.29 | 1.23 | 1.98 | 0.92 | 3.40 | 5.02 | 3.37 | 0.72 |
0.10 | 0.8 | 18 | 19 | 2.23 | 1.19 | 1.85 | 0.82 | 3.38 | 4.95 | 3.33 | 0.70 |
0.10 | 0.8 | 18 | 20 | 2.20 | 1.18 | 1.79 | 0.78 | 3.37 | 4.91 | 3.32 | 0.69 |
0.10 | 0.8 | 18 | 21 | 2.18 | 1.16 | 1.74 | 0.74 | 3.36 | 4.88 | 3.31 | 0.68 |
A2 | B2 | σc2 /MPa | γ2 /kN·m−3 | L1 /m | L2 /m | L3 /m | L4 /m | h1 /m | h2 /m | h3 /m | W2 /103 kN |
---|---|---|---|---|---|---|---|---|---|---|---|
0.2 | 0.9 | 10 | 25 | 1.90 | 1.08 | 1.61 | 0.76 | 3.29 | 6.83 | 3.26 | 0.66 |
0.3 | 0.9 | 10 | 25 | 2.14 | 1.14 | 1.80 | 0.81 | 3.35 | 5.84 | 3.32 | 0.69 |
0.4 | 0.9 | 10 | 25 | 2.25 | 1.19 | 1.87 | 0.85 | 3.38 | 5.34 | 3.34 | 0.71 |
0.6 | 0.9 | 10 | 25 | 2.29 | 1.23 | 1.93 | 0.88 | 3.40 | 4.76 | 3.36 | 0.71 |
0.5 | 0.5 | 10 | 25 | 2.41 | 1.34 | 2.02 | 0.96 | 3.45 | 3.65 | 3.38 | 0.724 |
0.5 | 0.6 | 10 | 25 | 2.41 | 1.33 | 2.01 | 0.96 | 3.45 | 3.72 | 3.38 | 0.723 |
0.5 | 0.7 | 10 | 25 | 2.38 | 1.31 | 2.00 | 0.94 | 3.44 | 3.91 | 3.37 | 0.723 |
0.5 | 0.8 | 10 | 25 | 2.33 | 1.27 | 1.97 | 0.92 | 3.42 | 4.32 | 3.37 | 0.720 |
0.5 | 0.9 | 21 | 25 | 2.01 | 0.99 | 1.69 | 0.66 | 3.30 | 4.64 | 3.30 | 0.61 |
0.5 | 0.9 | 22 | 25 | 2.07 | 1.04 | 1.73 | 0.70 | 3.32 | 4.71 | 3.31 | 0.63 |
0.5 | 0.9 | 23 | 25 | 2.13 | 1.09 | 1.78 | 0.75 | 3.34 | 4.79 | 3.32 | 0.66 |
0.5 | 0.9 | 24 | 25 | 2.19 | 1.15 | 1.84 | 0.80 | 3.36 | 4.88 | 3.34 | 0.68 |
0.5 | 0.9 | 10 | 21 | 2.52 | 1.44 | 2.17 | 1.11 | 3.50 | 5.41 | 3.42 | 0.715 |
0.5 | 0.9 | 10 | 22 | 2.45 | 1.38 | 2.09 | 1.04 | 3.46 | 5.29 | 3.40 | 0.714 |
0.5 | 0.9 | 10 | 23 | 2.38 | 1.32 | 2.02 | 0.97 | 3.44 | 5.18 | 3.38 | 0.713 |
0.5 | 0.9 | 10 | 24 | 2.32 | 1.26 | 1.96 | 0.92 | 3.41 | 5.08 | 3.36 | 0.712 |
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Lyu, C.; Zeng, Z.; Dong, Y. Limit Analysis of Progressive Asymmetrical Collapse Failure of Tunnels in Inclined Rock Stratum. Symmetry 2019, 11, 904. https://doi.org/10.3390/sym11070904
Lyu C, Zeng Z, Dong Y. Limit Analysis of Progressive Asymmetrical Collapse Failure of Tunnels in Inclined Rock Stratum. Symmetry. 2019; 11(7):904. https://doi.org/10.3390/sym11070904
Chicago/Turabian StyleLyu, Cheng, Zhengqiang Zeng, and Yucang Dong. 2019. "Limit Analysis of Progressive Asymmetrical Collapse Failure of Tunnels in Inclined Rock Stratum" Symmetry 11, no. 7: 904. https://doi.org/10.3390/sym11070904