# Using the Morgenstern–Price Method and Cloud Theory to Invert the Shear Strength Index of Tailings Dams and Reveal the Coupling Deformation and Failure Law under Extreme Rainfall

^{*}

## Abstract

**:**

## 1. Introduction

^{3}, which affected the office buildings, markets, and some residential buildings of the mining area about 500 m downstream, resulting in 277 deaths, 4 missing, and 33 injured, with direct economic losses of CNY 96.192 million [9]. In the red mud storage yard of the Ajkai alumina plant in southwest Hungary, 1,000,000 m

^{3}of red mud leaked into at least 7 villages, resulting in 4 deaths, 3 missing, and more than 150 injured. Finally, red mud began to flow into the Danube River, and the spread of red mud along the river triggered panic in many European countries [9]. On 5 November 2015, the dam of the Fundão tailings pond in Minas Gerais, Brazil, collapsed. 45 million cubic meters of toxic sludge flowed into nearby rivers and oceans. The accident had a devastating impact on the local river basin. A total of 19 people died, and about 700 were homeless. The water source for hundreds of thousands of people has been polluted, and a large number of wild animals have died [10]. Therefore, the deformation and destruction of the tailings dam caused by rainfall seriously threaten the sustainable operation of the tailings pond and the life and property safety of the surrounding residents and staff [9,11,12]. Thus, this article further explores tailings dam stability and accident disaster control.

## 2. Methodology

#### 2.1. Engineering Background

#### 2.1.1. Engineering Geological Conditions of the Tailings Stacking Field

#### 2.1.2. Judgement of the Unstable State of the Tailings Dam

#### 2.2. Morgenstern–Price Method

_{u}is the ratio of the pore water pressure in the soil to the self-weight of the soil.

_{r}to x

_{r}+ Δx intervals so that the normal inter-slice force F can be obtained from F

_{1}to Fn one by one, and then the tangential inter-slice force X of each block can be obtained.

_{0}and F

_{n}of the whole landslide mass.

#### 2.3. Cloud Theory

#### 2.4. Rainfall Conditions

_{S}/C

_{V}) is 1.92. Then, the hourly rainfall can be calculated using Formula (15):

#### 2.5. Theoretical Model Construction

_{m}is water capacity, k

_{s}is saturated permeability, µ is fluid dynamic viscosity, k

_{r}is relative permeability, ρ is fluid density, g is gravity acceleration, D is location head, and Q

_{m}is the mass source.

_{v}is the volume strain of the medium, and β is the Boit coefficient. The second item on the right in Formula (17) is the effect of deformation of soil particles on pore water seepage. This is the coupling term of stress and seepage. It is also the effect of stress on pore pressure.

_{a}and u

_{w}are air pressure and water pressure, respectively; (u

_{a}–u

_{w}) is matrix suction; χ is the matrix suction coefficient, which is approximately equal to saturation.

_{ij}is the stress tensor, σ’

_{ij}is the effective stress tensor, F

_{i}is the volume force, D

_{ijkl}is the elastic tensor, p is the pore water pressure, δ

_{ij}is the Kronecher sign, ε

_{ij}is the strain tensor, and u

_{i}is the solid displacement. The above describes the influence of seepage on the internal stress state of saturated-unsaturated soil due to the change in pore water pressure on the effective stress between soil particles.

_{w}is the volumetric specific humidity; K

_{w}is the hydraulic conductivity; ɑ, m, and l are model fitting parameters; θs is the saturated water content; θr is the residual water content; and S

_{e}is the effective saturation. The modeling steps of the tailings dam fluid structure coupling model under extreme rainfall conditions are shown in Figure 6.

## 3. Results and Discussion

#### 3.1. Sensitivity Analysis of Shear Strength Parameters for Tailings Dams

_{S}are cohesion C and internal friction angle φ. Therefore, the values of cohesion C and internal friction angle φ are changed to reflect the change in safety factor F

_{S}. Nine data points are used to conduct sensitivity analysis on the safety factor F

_{S}. In the analysis method, the value of fixed cohesion C remains unchanged, while the value of internal friction angle φ is fine-tuned to analyze its influence on the minimum safety factor Fs. Then, the value of the fixed internal friction angle φ remains unchanged, the value of cohesion C is fine-tuned, and its influence on the minimum safety factor Fs is analyzed. The fitting is shown in Figure 7.

^{0.05486C}, and the relationship between the internal friction angle φ and the safety factor Fs is Fs = 0.97086e

^{0.01412φ}. According to Figure 7, cohesion C and internal friction angle φ are positively related to the safety factor Fs of the tailings dam. However, the change in cohesion C leads to more changes in the safety factor Fs, so the influence of cohesion C on the safety factor Fs of the tailings dam is greater.

#### 3.2. Theoretical Tailings Dam Cohesion Parameter Inversion Based on the Cloud Model

#### 3.2.1. Select Parameter Range

_{max}− Ex

_{min}) = 0.266. When 0 < He < En/3, the certainty of the cloud model is uncertain and distributed in a normal distribution [55]. According to the observation of the discrete cloud map, it is more appropriate to choose He = 0.0266. The forward cloud corresponding to the qualitative concepts of these five intervals is shown in Figure 9.

#### 3.2.2. Selection of Training Samples and Calculation of Safety Factor

_{1C}, Ex

_{2C}, Ex

_{3C}, Ex

_{4C}, and Ex

_{5C}. See Table 1, Table 2, Table 3, Table 4 and Table 5 for the calculation results of the safety factor Fs of cohesion C.

#### 3.2.3. Uncertain Cloud Reasoning

- (1)
- If only one group of uncertainties is greater than 0, the output is directly generated by the inverse cloud generator;
- (2)
- If there are 2 degrees of certainty of confirmation (K
_{i}and K_{i+}_{1}) greater than 0, when using these 2 degrees of certainty to activate the corresponding rules consequent, select the 2 cloud droplets generated and cover these 2 cloud droplets with a virtual cloud. Then, the output method of cohesion C is Formula (24):$$C=\frac{{x}_{1}\sqrt{-2ln{k}_{2}}+{x}_{2}\sqrt{-2ln{k}_{1}}}{\sqrt{-2ln{k}_{1}}+\sqrt{-2ln{k}_{2}}}$$ - (3)
- When more than three uncertainties are greater than 0, the expectation Ex of the virtual cloud is generated directly by the inverse cloud generator, and the value of Ex is the output value of cohesion C.

#### 3.2.4. Verification of the Inversion Method of Strength Parameters

#### 3.3. Analysis of the Deformation Destruction Characteristics of a Tail Mine Dam under Extreme Rainfall Conditions

^{2}.

^{−4}m, the maximum cell growth rate was 1.1, and the curvature factor was 0.2.

^{2}. At 0.05 h, the plastic strain area of the tailings dam suddenly increases, and a new plastic strain area appears inside with a total area of 0.2626 m

^{2}and an irregular shape. After 1 h, the area of plastic strain is 0.2781 m

^{2}. The extreme rainfall intensity is greater than the seepage capacity of the tailings dam. The dam surface produces runoff, and the rainwater that fails to seep will gather at the dam toe. As a result, the footing on the outside of the tailings dam is gradually eroding. The increase in water content at the dam toe will generate pore water pressure. The rock and soil mass at the dam toe is humidified and softened and gradually develops into the tailings dam, creating a priority channel for the subsequent rainwater infiltration into the tailings dam, resulting in the formation of a free face at the dam toe and causing collapse and failure.

## 4. Conclusions

- (1)
- The correlation between cohesion C and safety factor Fs is significant. The safety factor of a tailings dam is obtained by the Morgenstern–Price method, and the specific cohesion parameters are inversed by using cloud theory within the corresponding cohesion C range. The final calculation result is 8.6901 kPa, which overcomes the problem that the fuzziness and randomness of the quantitative cohesion value are transferred to the qualitative concept of the safety factor;
- (2)
- The characteristics of coupling deformation and failure under extreme rainfall conditions are as follows: the plastic deformation area gradually develops on the inside of the tailings dam after dampness and softening, and the area gradually expands. The dam toe and abutment area have obvious displacement, and the whole displacement field gradually transfers from the accumulative tailings to the tailings dam with the rainfall, which intensifies the deformation and damage of the tailings dam. The seepage of rainwater and the hydrodynamic force generated by runoff drive the deformation and failure of tailings dams, and the deformation and failure of tailings dams provide a dominant transport path for rainwater seepage;
- (3)
- Under extreme rainfall conditions, the dam toe and abutment are high-risk areas. They should be taken as the target areas for priority prevention and control. In actual projects, measures such as the drainage or covering of the dam surface should be taken to avoid damage to the rainwater acceleration tailings dam.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Burritt, R.L.; Christ, K.L. Water risk in mining: Analysis of the Samarco dam failure. J. Clean. Prod.
**2018**, 178, 196–205. [Google Scholar] [CrossRef] - Dong, L.; Deng, S.; Wang, F. Some developments and new insights for environmental sustainability and disaster control of tailings dam. J. Clean. Prod.
**2020**, 269, 122270. [Google Scholar] [CrossRef] - Hancock, G. A method for assessing the long-term integrity of tailings dams. Sci. Total Environ.
**2021**, 779, 146083. [Google Scholar] [CrossRef] [PubMed] - Pacheco, F.; Oliveira, M.; Oliveira, M.; Libanio, M.; Valle, J.F.; Melo, S.M.; Pissarra, T.; Melo, M.; Valera, C.; Fernandes, L. Water security threats and challenges following the rupture of large tailings dams. Sci. Total Environ.
**2022**, 834, 155285. [Google Scholar] [CrossRef] [PubMed] - Buch, A.; Niemeyer, J.; Marques, E.; Silva-Filho, E. Ecological risk assessment of trace metals in soils affected by mine tailings. J. Hazard Mater.
**2021**, 403, 123852. [Google Scholar] [CrossRef] - Clarkson, L.; Williams, D. An Overview of Conventional Tailings Dam Geotechnical Failure Mechanisms. Min. Met. Explor.
**2021**, 38, 1305–1328. [Google Scholar] [CrossRef] - Piciullo, L.; Storrøsten, E.; Liu, Z.; Nadim, F.; Lacasse, S. A new look at the statistics of tailings dam failures. Eng. Geol.
**2022**, 303, 106657. [Google Scholar] [CrossRef] - Owen, J.; Kemp, D.; Lèbre, É.; Svobodova, K.; Pérez, M.G. Catastrophic tailings dam failures and disaster risk disclosure. Int. J. Disaster Risk Reduct.
**2020**, 42, 101361. [Google Scholar] [CrossRef] - Rico, M.; Benito, G.; Diez-Herrero, A. Floods from tailings dam failures. J. Hazard Mater.
**2008**, 154, 79–87. [Google Scholar] [CrossRef] - Hatje, V.; Pedreira, R.; Rezende, C.; Schettini, C.; Souza, G.; Marin, D.; Hackspacher, P. The environmental impacts of one of the largest tailing dam failures worldwide. Sci. Rep.
**2017**, 7, 10706. [Google Scholar] [CrossRef] - Dong, L.; Tong, X.; Li, X.; Zhou, J.; Wang, S.; Liu, B. Some developments and new insights of environmental problems and deep mining strategy for cleaner production in mines. J. Clean. Prod.
**2019**, 210, 1562–1578. [Google Scholar] [CrossRef] - Čarman, M.; Jemec Auflič, M.; Komac, M. Landslides at a uranium mill tailing deposit site Boršt (Slovenia) detected by radar interferometry. Landslides
**2014**, 11, 527–536. [Google Scholar] [CrossRef] - Jia, Y.; Xu, B.; Chi, S.; Xiang, B.; Xiao, D.; Zhou, Y. Joint back analysis of the creep deformation and wetting deformation parameters of soil used in the Guanyinyan composite dam. Comput. Geotech.
**2018**, 96, 167–177. [Google Scholar] [CrossRef] - Yu, X.; Gong, B.; Tang, C. Study of the slope deformation characteristics and landslide mechanisms under alternating excavation and rainfall disturbance. Bull. Eng. Geol. Environ.
**2021**, 80, 7171–7191. [Google Scholar] [CrossRef] - Yu, Y.; Zhang, B.; Yuan, H. An intelligent displacement back-analysis method for earth-rockfill dams. Comput. Geotech.
**2007**, 34, 423–434. [Google Scholar] [CrossRef] - Lin, X.; Zhang, L.; Yang, Z.; Li, P.; Li, T. Inversion analysis of the shear strength parameters for a high loess slope in the limit state. J. Mt. Sci.
**2020**, 18, 252–264. [Google Scholar] [CrossRef] - Daftaribesheli, A.; Ataei, M.; Sereshki, F. Assessment of rock slope stability using the Fuzzy Slope Mass Rating (FSMR) system. Appl. Soft. Comput.
**2011**, 11, 4465–4473. [Google Scholar] [CrossRef] - Luo, Z.; Xie, C.; Jia, N.; Yang, B.; Cheng, G. Safe roof thickness and span of stope under complex filling body. J. Cent. South. Univ.
**2013**, 20, 3641–3647. [Google Scholar] [CrossRef] - Xie, C.; Jia, N.; He, L. Study on the Instability Mechanism and Grouting Reinforcement Repair of Large-Scale Underground Stopes. Adv. Civ. Eng.
**2020**, 10, 8832012. [Google Scholar] [CrossRef] - Kang, F.; Xu, B.; Li, J.; Zhao, S. Slope stability evaluation using Gaussian processes with various covariance functions. Appl. Soft. Comput.
**2017**, 60, 387–396. [Google Scholar] [CrossRef] - Li, D.; Liu, C.; Gan, W. A new cognitive model: Cloud model. Int. J. Intell. Syst.
**2009**, 24, 357–375. [Google Scholar] [CrossRef] - Wang, G.; Xu, C.; Li, D. Generic normal cloud model. Inform. Sci.
**2014**, 280, 1–15. [Google Scholar] [CrossRef] - Wang, M.; Wang, X.; Liu, Q.; Shen, F.; Jin, J. A novel multi-dimensional cloud model coupled with connection numbers theory for evaluation of slope stability. Appl. Math. Model.
**2020**, 77, 426–438. [Google Scholar] [CrossRef] - Wang, Y.; Yin, X.; Jing, H.; Liu, R.; Su, H. A novel cloud model for risk analysis of water inrush in karst tunnels. Environ. Earth Sci.
**2016**, 75, 1–3. [Google Scholar] [CrossRef] - Yao, X.; Deng, H.; Zhang, T.; Qin, Y. Multistage fuzzy comprehensive evaluation of landslide hazards based on a cloud model. PLoS ONE
**2019**, 14, e0224312. [Google Scholar] [CrossRef] - He, X.; Wang, J.; Feng, J.; Yan, Z.; Miao, S.; Zhang, Y.; Xia, J. Observational and modeling study of interactions between urban heat island and heatwave in Beijing. J. Clean. Prod.
**2020**, 247, 119169. [Google Scholar] [CrossRef] - Naveendrakumar, G.; Vithanage, M.; Kwon, H.; Chandrasekara, S.; Iqbal, M.; Pathmarajah, S.; Fernando, W.; Obeysekera, J. South Asian perspective on temperature and rainfall extremes: A review. Atmos. Res.
**2019**, 225, 110–120. [Google Scholar] [CrossRef] - Zipser, E.; Liu, C. Extreme Convection vs. Extreme Rainfall: A Global View. Curr. Clim. Change Rep.
**2022**, 7, 121–130. [Google Scholar] [CrossRef] - Du, C.; Liang, L.; Yi, F.; Niu, B. Effects of Geosynthetic Reinforcement on Tailings Accumulation Dams. Water
**2021**, 13. [Google Scholar] [CrossRef] - Kang, C.; Chen, S.; Chan, D.; Tfwala, S. Numerical modeling of large-scale dam breach experiment. Landslides
**2020**, 17, 2737–2754. [Google Scholar] [CrossRef] - Gonzalez, F.; Raval, S.; Taplin, R.; Timms, W.; Hitch, M. Evaluation of Impact of Potential Extreme Rainfall Events on Mining in Peru. Nat. Resour. Res.
**2018**, 28, 393–408. [Google Scholar] [CrossRef] - Nazrien, N.J.; Mohd, T.A.; Razali, I.; Abd, R.N.; Wan, M.W.; Karim, O.A.; Mat, D.S.; Awang, S.; Mohd, M. The Effect of Extreme Rainfall Events on Riverbank Slope Behaviour. Front. Env. Sci.
**2022**, 10, 176. [Google Scholar] [CrossRef] - Zhao, N.; Hu, B.; Yi, Q.; Yao, W.; Ma, C. The Coupling Effect of Rainfall and Reservoir Water Level Decline on the Baijiabao Landslide in the Three Gorges Reservoir Area, China. Geofluids
**2017**, 12, 3724867. [Google Scholar] [CrossRef] - Yeh, H.; Tsai, Y. Effect of Variations in Long-Duration Rainfall Intensity on Unsaturated Slope Stability. Water
**2018**, 10, 479. [Google Scholar] [CrossRef] - Gui, R.; He, G. The Effects of Internal Erosion on the Physical and Mechanical Properties of Tailings under Heavy Rainfall Infiltration. Appl. Sci.
**2021**, 11, 9496. [Google Scholar] [CrossRef] - Yang, Y.; Cai, R.; Zhang, G.; Su, S.; Liu, W. Evaluating and analyzing the stability of loess slope using intermittent rainfall and various rainfall patterns. Arab. J. Geosci.
**2022**, 15, 218. [Google Scholar] [CrossRef] - Jiang, X.; Wörman, A.; Chen, P.; Huang, Q.; Chen, H. Mechanism of the progressive failure of non-cohesive natural dam slopes. Geomorphology
**2020**, 363, 107198. [Google Scholar] [CrossRef] - Hu, W.; Xin, C.; Li, Y.; Zheng, Y.; van Asch, T.; McSaveney, M. Instrumented flume tests on the failure and fluidization of tailings dams induced by rainfall infiltration. Eng. Geol.
**2021**, 294, 106401. [Google Scholar] [CrossRef] - Xu, J.; Ueda, K.; Uzuoka, R. Evaluation of failure of slopes with shaking-induced cracks in response to rainfall. Landslides
**2021**, 19, 119–136. [Google Scholar] [CrossRef] - Tian, S.; Dai, X.; Wang, G.; Lu, Y.; Chen, J. Formation and evolution characteristics of dam breach and tailings flow from dam failure: An experimental study. Nat. Hazards
**2021**, 107, 1621–1638. [Google Scholar] [CrossRef] - Zhong, Q.; Chen, S.; Mei, S.; Cao, W. Numerical simulation of landslide dam breaching due to overtopping. Landslides
**2017**, 15, 1183–1192. [Google Scholar] [CrossRef] - Xie, C.; Nguyen, H.; Choi, Y.; Jahed, A.D. Optimized functional linked neural network for predicting diaphragm wall deflection induced by braced excavations in clays. Geosci. Front.
**2022**, 13, 101313. [Google Scholar] [CrossRef] - Qiu, X.; Li, J.; Jiang, H.; Ou, J.; Ma, J. Evolution of the Transient Saturated Zone and Stability Analysis of Slopes under Rainfall Conditions. KSCE J. Civ. Eng.
**2022**, 26, 1618–1631. [Google Scholar] [CrossRef] - Khan, K.; Wang, C.; Khan, M.; Liang, Z.; Li, S.; Li, B. Influence of rainfall infiltration on the stability of unsaturated coal gangue accumulated slope. J. Mt. Sci.
**2021**, 18, 1696–1709. [Google Scholar] [CrossRef] - Zhou, C.; Ai, D.; Huang, W.; Xu, H.; Ma, L.; Chen, L.; Wang, L. Emergency Survey and Stability Analysis of a Rainfall-Induced Soil-Rock Mixture Landslide at Chongqing City, China. Front. Earth Sci.
**2021**, 9, 774200. [Google Scholar] [CrossRef] - Wu, K.; Chen, N.; Hu, G.; Han, Z.; Ni, H.; Rahman, M. Failure mechanism of the Yaoba loess landslide on 5 March 2020: The early-spring dry spell in Southwest China. Landslides
**2021**, 18, 3183–3195. [Google Scholar] [CrossRef] - Li, S.; Yuan, L.; Yang, H.; An, H.; Wang, G. Tailings dam safety monitoring and early warning based on spatial evolution process of mud-sand flow. Saf. Sci.
**2020**, 124, 104579. [Google Scholar] [CrossRef] - Xie, C.; Nguyen, H.; Bui, X.; Nguyen, V.; Zhou, J. Predicting roof displacement of roadways in underground coal mines using adaptive neuro-fuzzy inference system optimized by various physics-based optimization algorithms. J. Rock Mech. Geotech.
**2021**, 13, 1452–1465. [Google Scholar] [CrossRef] - Sun, G.; Cheng, S.; Jiang, W.; Zheng, H. A global procedure for stability analysis of slopes based on the Morgenstern-Price assumption and its applications. Comput. Geotech.
**2016**, 80, 97–106. [Google Scholar] [CrossRef] - Ouyang, W.; Liu, S.; Yang, Y. An improved morgenstern-price method using gaussian quadrature. Comput. Geotech.
**2022**, 148, 104754. [Google Scholar] [CrossRef] - Liu, X.; Zhang, M.; Sun, Z.; Zhang, H.; Zhang, Y. Comprehensive evaluation of loess collapsibility of oil and gas pipeline based on cloud theory. Sci. Rep.
**2021**, 11, 15422. [Google Scholar] [CrossRef] - Cao, W.; Deng, J.; Yang, Y.; Zeng, Y.; Liu, L. Water Carrying Capacity Evaluation Method Based on Cloud Model Theory and an Evidential Reasoning Approach. Mathematics
**2022**, 10, 266. [Google Scholar] [CrossRef] - Zhang, H.; Wang, T.; Ding, Z.; Zhang, X.; Han, L. Uncertainty analysis of impact factors of eco-environmental vulnerability based on cloud theory. Ecol. Indic.
**2020**, 110, 105864. [Google Scholar] [CrossRef] - Wu, D.; Deng, T.; Duan, W.; Zhang, W. A coupled thermal-hydraulic-mechanical application for assessment of slope stability. Soils Found.
**2019**, 59, 2220–2237. [Google Scholar] [CrossRef] - Wang, T.; Li, S.; Xu, Z.; Hu, J.; Pan, D.; Xue, Y. Risk assessment of water inrush in karst tunnels excavation based on normal cloud model. Bull. Eng. Geol. Environ.
**2018**, 78, 3783–3798. [Google Scholar] [CrossRef] - Cao, J.; Xie, C.; Hou, Z. Spatiotemporal distribution patterns and risk characteristics of heavy metal pollutants in the soil of lead–zinc mines. Environ. Sci. Eur.
**2022**, 34, 1–14. [Google Scholar] [CrossRef] - Zou, J.; Zhang, R.; Zhou, F.; Zhang, X. Hazardous area reconstruction and law analysis of coal spontaneous combustion and gas coupling disasters in goaf based on DEM-CFD. ACS. Omega
**2023**, 8, 2685–2697. [Google Scholar] [CrossRef] - Cao, J.; Xie, C.; Hou, Z. Transport patterns and numerical simulation of heavy metal pollutants in soils of lead-zinc ore mines. J. Mt. Sci.-Engl.
**2021**, 18, 2345–2356. [Google Scholar] [CrossRef]

**Figure 2.**Landslide location of the tailings dam. Yellow frame: dam surface under unstable state. Red frame: shallow landslide area on the dam surface.

**Figure 3.**Morgenstern–Price method theory (

**a**) Model of an arbitrarily shaped soil slope (

**b**) Force analysis diagram for any differential soil strip. dW—weight of a soil strip; dN′—effective normal reaction force at the bottom of the soil strip; dT—frictional resistance of the soil strip; F′, F′ + dF′—horizontal effective normal strip force on both sides of the soil strip; X, X + dX—force between tangential strips at both sides of the soil strip; V, V + dV—pore water stress acting on both sides of the soil strip; dV

_{S}—pore water stress acting on the bottom of the soil strip.

**Figure 7.**Sensitivity analysis of the change in shear strength parameters on the safety factor Fs. (

**a**) Cohesion C; (

**b**) internal friction angle φ.

**Figure 8.**The minimum safety factor and sliding surface of tailings dams under different cohesion C: (

**a**) Fs = 1.000; (

**b**) Fs = 1.050; (

**c**) Fs = 1.103.

**Figure 9.**Forward cloud map of each interval: (

**a**) Ex = 7.8, En = 0.266, He = 0.0266; (

**b**) Ex = 8.2, En = 0.266, He = 0.0266; (

**c**) Ex = 8.6, En = 0.266, He = 0.0266; (

**d**) Ex = 9, En = 0.266, He = 0.0266; (

**e**) Ex = 9.4, En = 0.266, He = 0.0266.

**Figure 13.**Equivalent plastic strain of the tailings dam: (

**a**) T = 0.01 h; (

**b**) T = 0.05 h; (

**c**) T = 1 h.

**Figure 14.**Equivalent plastic strain area cloud pictures of the tailings dam: (

**a**) T = 0.01 h; (

**b**) T = 0.05 h; (

**c**) T = 1 h.

**Figure 15.**Displacement cloud pictures of the tailings dam: (

**a**) T = 0.01 h; (

**b**) T = 0.05 h; (

**c**) T = 0.5 h; (

**d**) T = 1 h.

**Figure 16.**Monitoring point M1 and M2 displacement. (Ⅰ: Unstable change phase of displacement change. Ⅱ: Stable growth phase of displacement change. A: Maximum value in unstable change stage of displacement. B: Minimum value in unstable change stage of displacement).

Cohesion (kPa) | Fs | |
---|---|---|

1_{C} | 7.50131 | 0.943 |

8.30150 | 1.027 | |

7.70046 | 0.976 | |

7.40259 | 0.939 | |

7.89953 | 0.979 | |

8.00075 | 1.006 | |

8.20211 | 1.023 | |

7.60060 | 0.952 | |

Ex_{1C} | 7.8 | 1.000 |

Cohesion (kPa) | Fs | |
---|---|---|

2_{C} | 8.00480 | 1.006 |

8.10081 | 1.021 | |

8.30078 | 1.030 | |

8.40121 | 1.035 | |

7.79991 | 0.998 | |

7.90093 | 0.979 | |

8.50072 | 1.043 | |

8.60047 | 1.050 | |

Ex_{2C} | 8.2 | 1.023 |

Cohesion (kPa) | Fs | |
---|---|---|

3_{C} | 8.40168 | 1.035 |

8.50064 | 1.043 | |

8.70093 | 1.061 | |

8.80129 | 1.070 | |

8.20208 | 1.023 | |

8.30044 | 1.030 | |

9.00298 | 1.084 | |

8.90135 | 1.079 | |

Ex_{3C} | 8.6 | 1.05 |

Cohesion (kPa) | Fs | |
---|---|---|

4_{C} | 8.80021 | 1.070 |

8.90051 | 1.079 | |

9.10094 | 1.094 | |

9.20022 | 1.104 | |

8.60354 | 1.050 | |

8.70189 | 1.061 | |

9.30214 | 1.109 | |

9.40796 | 1.113 | |

Ex_{4C} | 9 | 1.066 |

Cohesion (kPa) | Fs | |
---|---|---|

5_{C} | 9.20095 | 1.104 |

9.30010 | 1.109 | |

9.50246 | 1.122 | |

9.60023 | 1.136 | |

9.00350 | 1.091 | |

9.10623 | 1.094 | |

9.70047 | 1.144 | |

9.80063 | 1.152 | |

Ex_{5C} | 9.4 | 1.103 |

1_{C} | 2_{C} | 3_{C} | 4_{C} | 5_{C} | ||
---|---|---|---|---|---|---|

Fs | Ex | 1.000 | 1.023 | 1.050 | 1.066 | 1.103 |

En | 0.252 | 0.0194 | 0.0217 | 0.020 | 0.202 | |

He | 0.0030693 | 0.0023693 | 0.0025607 | 0.0025000 | 0.0025006 |

Cloud Representation of Qualitative Concept of Rule Antecedents of Fs | Cloud Representation of Qualitative Concept of Rule Consequent of C |
---|---|

PRE_{A}_{C1} = PRE (1.000, 0.0252, 0.0030693) | POST_{B}_{C1} = POST (7.8, 0.266, 0.0266) |

PRE_{A}_{C2} = PRE (1.023, 0.0194, 0.0023693) | POST_{B}_{C2} = POST (8.2, 0.266, 0.0266) |

PRE_{A}_{C3} = PRE (1.050, 0.0217, 0.0025607) | POST_{B}_{C3} = POST (8.6, 0.266, 0.0266) |

PRE_{A}_{C4} = PRE (1.066, 0.0200, 0.0025000) | POST_{B}_{C4} = POST (9.0, 0.266, 0.0266) |

PRE_{A}_{C5} = PRE (1.103, 0.0202, 0.0025006) | POST_{B}_{C5} = POST (9.4, 0.266, 0.0266) |

Material | Bulk Density ρ (kg/m ^{3}) | Young’s Modulus E (Pa) | Poisson’s Ratio μ | Cohesion C (kPa) | Internal Friction Angle φ (°) | Hydraulic Conductivity K (m/s) |
---|---|---|---|---|---|---|

Tailings dam | 2200 | 2.0 × 10^{7} | 0.28 | 9.56 | 23.1 | 4.1 × 10^{−5} |

Tailings | 2820 | 3.5 × 10^{7} | 0.27 | 9.80 | 26.6 | 5.5 × 10^{−5} |

Foundation | 2300 | 3.0 × 10^{7} | 0.30 | 30.00 | 25.0 | 1.5 × 10^{−7} |

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## Share and Cite

**MDPI and ACS Style**

Chen, Z.; Xie, C.; Xiong, G.; Shen, J.; Yang, B. Using the Morgenstern–Price Method and Cloud Theory to Invert the Shear Strength Index of Tailings Dams and Reveal the Coupling Deformation and Failure Law under Extreme Rainfall. *Sustainability* **2023**, *15*, 6106.
https://doi.org/10.3390/su15076106

**AMA Style**

Chen Z, Xie C, Xiong G, Shen J, Yang B. Using the Morgenstern–Price Method and Cloud Theory to Invert the Shear Strength Index of Tailings Dams and Reveal the Coupling Deformation and Failure Law under Extreme Rainfall. *Sustainability*. 2023; 15(7):6106.
https://doi.org/10.3390/su15076106

**Chicago/Turabian Style**

Chen, Ziwei, Chengyu Xie, Guanpeng Xiong, Jinbo Shen, and Baolin Yang. 2023. "Using the Morgenstern–Price Method and Cloud Theory to Invert the Shear Strength Index of Tailings Dams and Reveal the Coupling Deformation and Failure Law under Extreme Rainfall" *Sustainability* 15, no. 7: 6106.
https://doi.org/10.3390/su15076106