Effect of Rock Mass Disturbance on Stability of 3D Hoek–Brown Slope and Charts
Abstract
:1. Introduction
2. The Generalized Hoek–Brown Failure Criterion and Its Applicability
2.1. The Generalized Hoek–Brown Failure Criterion
2.2. The Equivalent Mohr–Coulomb Strength Parameters Method
3. The Kinematic Approach of Limit Analysis
4. Slope Stability Analysis Using Limit Analysis
4.1. Failure Mechanism of a 3D Slope
4.2. FoS Solution of Slope
5. Comparison
5.1. Comparison of FoS
5.2. Validity of Index SR
6. Results and Discussion
6.1. Parametric Analysis of the Disturbance Factor D on Slope Stability
6.2. Design Charts for 3D Rock Slopes
6.3. The Slope Angle Weighting Factor fβ_3D for 3D Slopes
7. Case Study
7.1. A Rock Slope of an Open Pit Mine at Baskoyak Anatolia
7.2. A Rock Slope in Kisrakdere Coal Open Pit Mine in Western Turkey
8. Conclusions
- Validities of the present study and index SR on estimating the FoS solutions of a 3D slope in disturbed rock masses are verified that for a slope with given B/H; slope angle β; and Hoek–Brown strength parameters GSI, mi, and D, the FoS of slope is still only related to index SR.
- A parametric analysis is conducted to investigate the effects of 3D character and rock mass disturbance on slope stability. It is shown that B/H and the rock mass disturbance factor D have significant influences on slope stability and should be considered in stability analyses of slopes in rock masses.
- A series of stability charts are presented and modified equations to determine the slope angle weighting factor fβ_3D considering the 3D character of slope are presented to provide a convenient and straightforward way to estimate FoS solutions of 3D slopes in disturbed rock masses.
- A case study is conducted to apply the presented stability charts to practical cases. The results indicated that the present FoS solutions obtained using the stability charts in conjunction with the slope angle weighting factor fβ_3D are in good agreement with the analytical solutions. The validity of the present stability charts and the equations to estimate the slope angle weighting factors fβ_3D are verified.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Limit Analysis—Lower Bound | SLIDE-Limit Equilibrium Using Equivalent Mohr–Coulomb Parameters | Limit Analysis | |||||||
---|---|---|---|---|---|---|---|---|---|
Nonlinear HB | Equations (5)–(7) | Equations (5), (6) and (9) | Equations (5), (6) and (10) | Present 3D Solutions (B→∞) | |||||
β/° | GSI | mi | SR | FoS0 | FoS1 | FoS2 | FoS3 | FoS4 | FoS |
30 | 100 | 5 | 0.070 | 1.0 | 1.014 | 0.988 | - | 1.0 | 0.9872 |
30 | 100 | 15 | 0.026 | 1.0 | 1.020 | 0.999 | - | 1.024 | 1.0243 |
30 | 100 | 25 | 0.016 | 1.0 | 1.023 | 1.003 | - | 1.036 | 1.0431 |
30 | 100 | 35 | 0.011 | 1.0 | 1.024 | 1.007 | - | 1.044 | 1.0327 |
30 | 70 | 5 | 0.218 | 1.0 | 1.018 | 0.985 | - | 1.011 | 1.0101 |
30 | 70 | 15 | 0.075 | 1.0 | 1.023 | 0.996 | - | 1.028 | 1.0202 |
30 | 70 | 25 | 0.045 | 1.0 | 1.024 | 1.004 | - | 1.035 | 1.0272 |
30 | 70 | 35 | 0.032 | 1.0 | 1.025 | 1.010 | - | 1.040 | 1.0306 |
30 | 50 | 5 | 0.461 | 1.0 | 1.020 | 0.993 | - | 1.014 | 1.0101 |
30 | 50 | 15 | 0.153 | 1.0 | 1.024 | 1.003 | - | 1.026 | 1.0171 |
30 | 50 | 25 | 0.091 | 1.0 | 1.025 | 1.024 | - | 1.032 | 1.0233 |
30 | 50 | 35 | 0.065 | 1.0 | 1.026 | 1.008 | - | 1.036 | 1.0302 |
30 | 30 | 5 | 1.057 | 1.0 | 1.022 | 1.001 | - | 1.012 | 1.0101 |
30 | 30 | 15 | 0.323 | 1.0 | 1.026 | 1.003 | - | 1.026 | 1.0162 |
30 | 30 | 25 | 0.185 | 1.0 | 1.026 | 1.005 | - | 1.031 | 1.0213 |
30 | 30 | 35 | 0.129 | 1.0 | 1.027 | 1.004 | - | 1.035 | 1.0278 |
30 | 10 | 5 | 4.363 | 1.0 | 1.023 | 1.002 | - | 1.006 | 1.0101 |
30 | 10 | 15 | 0.943 | 1.0 | 1.025 | 1.007 | - | 1.023 | 1.0168 |
30 | 10 | 25 | 0.460 | 1.0 | 1.026 | 0.996 | - | 1.033 | 1.0254 |
30 | 10 | 35 | 0.286 | 1.0 | 1.026 | 1.004 | - | 1.040 | 1.0316 |
45 | 100 | 5 | 0.135 | 1.0 | 1.000 | 1.008 | 1.022 | 1.027 | 0.9857 |
45 | 100 | 15 | 0.058 | 1.0 | 1.005 | 1.041 | 1.003 | 1.086 | 0.9886 |
45 | 100 | 25 | 0.036 | 1.0 | 1.012 | 1.047 | 1.003 | 1.110 | 0.9900 |
45 | 100 | 35 | 0.026 | 1.0 | 1.015 | 1.060 | 1.005 | 1.126 | 0.9900 |
45 | 70 | 5 | 0.469 | 1.0 | 1.001 | 1.038 | 1.001 | 1.055 | 0.9840 |
45 | 70 | 15 | 0.176 | 1.0 | 1.012 | 1.080 | 1.002 | 1.098 | 0.9900 |
45 | 70 | 25 | 0.108 | 1.0 | 1.017 | 1.060 | 1.007 | 1.113 | 0.9900 |
45 | 70 | 35 | 0.077 | 1.0 | 1.019 | 1.061 | 1.009 | 1.123 | 0.9900 |
45 | 50 | 5 | 1.046 | 1.0 | 1.004 | 1.045 | 1.001 | 1.063 | 0.9853 |
45 | 50 | 15 | 0.369 | 1.0 | 1.009 | 1.065 | 1.004 | 1.098 | 0.9900 |
45 | 50 | 25 | 0.222 | 1.0 | 1.020 | 1.066 | 1.010 | 1.110 | 0.9900 |
45 | 50 | 35 | 0.158 | 1.0 | 1.021 | 1.044 | 1.011 | 1.118 | 0.9900 |
45 | 30 | 5 | 2.593 | 1.0 | 1.011 | 1.066 | 0.999 | 1.060 | 0.9869 |
45 | 30 | 15 | 0.829 | 1.0 | 1.018 | 1.070 | 1.007 | 1.094 | 0.9900 |
45 | 30 | 25 | 0.480 | 1.0 | 1.021 | 1.074 | 1.010 | 1.110 | 0.9900 |
45 | 30 | 35 | 0.334 | 1.0 | 1.024 | 1.085 | 1.011 | 1.118 | 1.0101 |
45 | 10 | 5 | 13.585 | 1.0 | 1.014 | 1.087 | 1.000 | 1.039 | 0.9847 |
45 | 10 | 15 | 3.155 | 1.0 | 1.023 | 1.106 | 1.005 | 1.080 | 0.9900 |
45 | 10 | 25 | 1.552 | 1.0 | 1.023 | 1.107 | 1.009 | 1.103 | 0.9900 |
45 | 10 | 35 | 0.969 | 1.0 | 1.026 | 1.079 | 1.010 | 1.115 | 0.9900 |
60 | 100 | 5 | 0.232 | 1.0 | 1.001 | 1.033 | 1.043 | - | 0.9822 |
60 | 100 | 15 | 0.130 | 1.0 | 1.004 | 1.114 | 1.026 | - | 1.0101 |
60 | 100 | 25 | 0.088 | 1.0 | 1.004 | 1.146 | 1.035 | - | 1.0155 |
60 | 100 | 35 | 0.066 | 1.0 | 1.004 | 1.141 | 1.040 | - | 1.0241 |
60 | 70 | 5 | 0.946 | 1.0 | 1.013 | 1.059 | 1.024 | - | 0.9891 |
60 | 70 | 15 | 0.435 | 1.0 | 1.004 | 1.143 | 1.033 | - | 1.0153 |
60 | 70 | 25 | 0.276 | 1.0 | 1.004 | 1.161 | 1.043 | - | 1.0252 |
60 | 70 | 35 | 0.20 | 1.0 | 1.005 | 1.183 | 1.047 | - | 1.0284 |
60 | 50 | 5 | 2.337 | 1.0 | 1.005 | 1.124 | 1.026 | - | 0.9900 |
60 | 50 | 15 | 0.953 | 1.0 | 1.004 | 1.171 | 1.036 | - | 1.0201 |
60 | 50 | 25 | 0.584 | 1.0 | 1.008 | 1.176 | 1.046 | - | 1.0267 |
60 | 50 | 35 | 0.419 | 1.0 | 1.009 | 1.172 | 1.049 | - | 1.0302 |
60 | 30 | 5 | 6.439 | 1.0 | 1.009 | 1.150 | 1.023 | - | 1.0101 |
60 | 30 | 15 | 2.317 | 1.0 | 1.009 | 1.197 | 1.044 | - | 1.0239 |
60 | 30 | 25 | 1.356 | 1.0 | 1.010 | 1.201 | 1.049 | - | 1.0294 |
60 | 30 | 35 | 0.945 | 1.0 | 1.011 | 1.230 | 1.051 | - | 1.0319 |
60 | 10 | 5 | 38.926 | 1.0 | 1.004 | 1.183 | 1.013 | - | 0.9900 |
60 | 10 | 15 | 11.734 | 1.0 | 1.013 | 1.257 | 1.048 | - | 1.0288 |
60 | 10 | 25 | 5.928 | 1.0 | 1.017 | 1.261 | 1.054 | - | 1.0366 |
60 | 10 | 35 | 3.729 | 1.0 | 1.018 | 1.258 | 1.059 | - | 1.0403 |
Input Parameters | Case 1 | Case 2 | Case 3 |
---|---|---|---|
GSI | 30 | 30 | 30 |
mi | 8 | 8 | 8 |
β/° | 60 | 60 | 60 |
σci/kPa | 20 | 25 | 250 |
γ/(kN/m3) | 23 | 28.75 | 23.96 |
H/m | 25 | 25 | 300 |
SR (σci/γH) | 34.783 | 34.783 | 34.783 |
FoS (D = 0) | |||
Bishop simplified | 2.026 | 2.026 | 2.026 |
Janbu simplified | 1.934 | 1.934 | 1.934 |
Spencer | 2.032 | 2.032 | 2.032 |
Morgenstern–Price | 2.027 | 2.027 | 2.027 |
Phase2 8.0 (FEM) | 2.000 | 2.040 | 2.030 |
Present 3D solution (B→∞) | |||
D = 0 | 2.0230 | 2.0230 | 2.0229 |
D = 0.3 | 1.7066 | 1.7066 | 1.7066 |
D = 0.7 | 1.2477 | 1.2477 | 1.2477 |
D = 1.0 | 0.8416 | 0.8416 | 0.8416 |
H/m | β/° | γ/ (kN/m3) | σci/ kPa | GSI | mi | D | fβ | ||||
---|---|---|---|---|---|---|---|---|---|---|---|
Shen et al. [11] | Sun et al. [12] | Present solution | |||||||||
B/H =0.7 | B/H =1.0 | B/H →∞ | |||||||||
184 | 55 | 27 | 153 | 47 | 9 | 0.9 | 0.793 | 0.817 | 0.825 | 0.827 | 0.828 |
140 | 34 | 26 | 50 | 28 | 8 | 0.7 | 1.259 | 1.264 | 1.439 | 1.389 | 1.290 |
220 | 45 | 27 | 65 | 44 | 17 | 0.8 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
135 | 65 | 27 | 172 | 58 | 9 | 0.9 | 0.637 | 0.753 | 0.734 | 0.732 | 0.733 |
70 | 50 | 27 | 29 | 41 | 7 | 0.8 | 0.885 | 0.901 | 0.905 | 0.903 | 0.905 |
110 | 45 | 26.5 | 50 | 25 | 10 | 0.7 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
270 | 45 | 27 | 109 | 39 | 18 | 0.9 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
170 | 55 | 30 | 104 | 48 | 7 | 0.7 | 0.793 | 0.814 | 0.830 | 0.828 | 0.831 |
60 | 60 | 27 | 65 | 44 | 13 | 1.0 | 0.711 | 0.710 | 0.754 | 0.752 | 0.751 |
35 | 67 | 27 | 109 | 28 | 12 | 1.0 | 0.609 | 0.571 | 0.664 | 0.659 | 0.651 |
63 | 35 | 27 | 109 | 28 | 12 | 1.0 | 1.232 | 1.252 | 1.361 | 1.326 | 1.256 |
70 | 49 | 27 | 3 | 49 | 25 | 1.0 | 0.905 | 0.941 | 0.921 | 0.920 | 0.922 |
58 | 50 | 27 | 5 | 55 | 22 | 1.0 | 0.885 | 0.922 | 0.900 | 0.902 | 0.904 |
60 | 48 | 27 | 5 | 54 | 22 | 1.0 | 0.925 | 0.944 | 0.937 | 0.939 | 0.940 |
60 | 52 | 27 | 5 | 56 | 22 | 1.0 | 0.847 | 0.930 | 0.865 | 0.868 | 0.869 |
40 | 71 | 27 | 50 | 33 | 14 | 1.0 | 0.558 | 0.538 | 0.621 | 0.614 | 0.602 |
110 | 50 | 27 | 50 | 25 | 14 | 1.0 | 0.885 | 0.903 | 0.899 | 0.900 | 0.901 |
41 | 50 | 27 | 3 | 46 | 24 | 1.0 | 0.885 | 0.899 | 0.900 | 0.902 | 0.904 |
41 | 55 | 27 | 3 | 49 | 24 | 1.0 | 0.793 | 0.848 | 0.820 | 0.819 | 0.820 |
46 | 55 | 27 | 3 | 50 | 24 | 1.0 | 0.793 | 0.810 | 0.820 | 0.819 | 0.821 |
57 | 49 | 27 | 3 | 48 | 24 | 1.0 | 0.905 | 0.909 | 0.921 | 0.921 | 0.922 |
57 | 37 | 27 | 3 | 48 | 24 | 1.0 | 1.179 | 1.185 | 1.374 | 1.328 | 1.221 |
57 | 40 | 27 | 3 | 48 | 24 | 1.0 | 1.103 | 1.130 | 1.288 | 1.246 | 1.154 |
57 | 42 | 27 | 3 | 48 | 24 | 1.0 | 1.056 | 1.083 | 1.235 | 1.198 | 1.113 |
27 | 45 | 25 | 0.75 | 100 | 10 | 0 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
50 | 60 | 23 | 10 | 30 | 8 | 1.0 | 0.711 | 0.722 | 0.746 | 0.746 | 0.745 |
50 | 45 | 27 | 13.5 | 30 | 5 | 0.7 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
25 | 45 | 27 | 5.4 | 20 | 20 | 0.7 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
5 | 30 | 27 | 2.7 | 10 | 5 | 0.5 | 1.375 | 1.438 | 1.615 | 1.551 | 1.433 |
25 | 75 | 25 | 0.625 | 80 | 15 | 0.3 | 0.511 | 0.525 | 0.605 | 0.591 | 0.577 |
250 | 60 | 23 | 46 | 50 | 35 | 1.0 | 0.711 | 0.652 | 0.746 | 0.745 | 0.743 |
B/H | Regression Equations—fβ_B/H | Fitting Degree—R2 |
---|---|---|
0.7 | 35.7 β−0.9091, 30° ≤ β < 45° | 0.9992 |
62.2 β−1.082, 45° < β ≤ 90° | 0.9995 | |
1.0 | 28.52 β−0.8569, 30° ≤ β < 45° | 0.9997 |
2.26 e−0.01849β, 45° < β ≤ 90° | 0.9993 | |
1.2 | 26.04 β−0.8355, 30° ≤ β < 45° | 0.9998 |
2.301 e−0.01882β, 45° < β ≤ 90° | 0.9996 | |
1.5 | 23.91 β−0.816, 30° ≤ β < 45° | 0.9994 |
2.342 e−0.01914β, 45° < β ≤ 90° | 0.9998 | |
2.0 | 22.38 β−0.8012, 30° ≤ β < 45° | 0.9998 |
2.382 e−0.01945β, 45° < β ≤ 90° | 0.9998 | |
5.0 | 19.85 β−0.7745, 30° ≤ β < 45° | 0.9999 |
2.442 e−0.01994β, 45° < β ≤ 90° | 0.9996 |
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Xu, J.; Wang, X.; Xie, P.; Wang, R.; Du, D. Effect of Rock Mass Disturbance on Stability of 3D Hoek–Brown Slope and Charts. Buildings 2024, 14, 114. https://doi.org/10.3390/buildings14010114
Xu J, Wang X, Xie P, Wang R, Du D. Effect of Rock Mass Disturbance on Stability of 3D Hoek–Brown Slope and Charts. Buildings. 2024; 14(1):114. https://doi.org/10.3390/buildings14010114
Chicago/Turabian StyleXu, Jingshu, Xinrui Wang, Pengfei Xie, Ruotong Wang, and Dianchun Du. 2024. "Effect of Rock Mass Disturbance on Stability of 3D Hoek–Brown Slope and Charts" Buildings 14, no. 1: 114. https://doi.org/10.3390/buildings14010114