# A Monte Carlo Approach to Estimate the Stability of Soil–Rock Slopes Considering the Non-Uniformity of Materials

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## Abstract

**:**

## 1. Introduction

## 2. The Monte Carlo Method for Soil–Rock Slopes

#### 2.1. Generation of Independent Random Variables

#### 2.2. Stability Analysis of the Soil–Rock Slope by the Monte Carlo Method

#### 2.2.1. Stability Analysis Process for the Soil–Rock Slope

#### 2.2.2. Analysis Model of the Soil–Rock Slope

## 3. Strength Measurement of Soil–Rock Mixtures

#### 3.1. Acquisition of Samples

#### 3.2. Strength Measurement of Soil–Rock Mixtures

#### 3.2.1. Testing Process

#### 3.2.2. Testing Scheme

#### 3.3. Testing Results

## 4. Comparison with Liu’s Method

## 5. Slope Stability Analysis, Considering Rock Content

## 6. Discussion About Stability Estimation of the Soil–Rock Slope

#### 6.1. Discussion About Characteristic Parameters of the Soil–Rock Slope

#### 6.2. Discussion on the Applicability of Analytical Methods

- (1)
- In the process of sampling for stability analysis of soil–rock slopes, it is advised to survey in the main affecting area first, and then survey in the secondary affecting area. The definition of affecting area is shown in Figure 19, that L is the basic length [3,16]. In this process, L is the horizontal length of plastic belt to slope top in homogeneous soil slope models, and L is 4 m in the article.
- (2)
- When the rock size is 0.1~0.4 Ls (Ls is the width of slope) in the main affecting area, it is advised to use analysis models with the Monte Carlo method, considering the strengthening effect of rocks. When the soil–rock slope is more easily affected by rainfall, traffic load, and other factors, it is advised to use analysis models with the Monte Carlo method, considering the scatter strength of soil–rock mixtures. When the soil–rock interface is weak, it is advised to use analysis models with the Monte Carlo method, considering the weakening effects of soil–rock interface area.

## 7. Conclusions

- (1)
- The Monte Carlo method and its application in stability analysis of soil–rock slopes are introduced in detail.
- (2)
- Based on the strength of soil–rock mixtures arising from the in situ samples and remade samples, the scatter characteristic of cohesion and internal friction angle for soil–rock mixtures were proved, and it can be concluded that the cohesion reduces with the increase of rock content, and the internal friction angle increases with the increase of rock content.
- (3)
- Through comparing with the failed slope and Liu’s method for analyzing the stability of soil–rock slopes, it was proved that analyzing the soil–rock slope with the Monte Carlo method is reasonable and economical.
- (4)
- The stability of the soil–rock slope, considering strengthening effect of rocks to slope and different rock contents, was studied, and the results show that with the increase of rock content, the stability of the soil–rock slope reduces first and then increases, and the minimum value was achieved at 20% rock content. When the rock content is over 30%, the soil–rock slope achieves the largest safety factor, when considering the strengthening effect of rocks, and achieves the smallest safety factor when considering the scatter strength of soil–rock mixtures.
- (5)
- Based on a large number of examples and existing documents, the investigation area for the optimum parameter characteristics of soil–rock mixture in slope stability analysis is proposed. Taking these into account, the chosen models were put forward for different situations, to study which situations are suitable for analysis models, considering the weak soil–rock interface area and the strengthening effect of rocks.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Examples of soil–rock slopes: (

**a**) natural soil–rock slope [12]; (

**b**) manually filled soil–rock slope.

**Figure 6.**In situ sampling diagram: (

**a**) in situ drilling pictures; and (

**b**) samples from in situ drilling.

**Figure 9.**Cohesion results of soil–rock mixtures: (

**a**) cohesion testing results of soil–rock mixtures; (

**b**) cohesion results of Kalender et al. [8].

**Figure 11.**The stability analysis results of the soil–rock slope with Liu’s method: (

**a**) calculation results of C-1; (

**b**) calculation results of C-2; (

**c**) calculation results of C-3; (

**d**) calculation results of C-4; (

**e**) calculation results of C-5; and (

**f**) calculation results of C-6.

**Figure 12.**Results of soil–rock slope analysis of the D-1 group obtained by the Monte Carlo method: (

**a**) cohesion distribution picture; (

**b**) internal friction angle distribution picture; and (

**c**) slope plastic belt picture.

**Figure 13.**Results of soil–rock slope analysis of the D-2 group obtained by the Monte Carlo method: (

**a**) cohesion distribution picture; (

**b**) internal friction angle distribution picture; and (

**c**) slope plastic belt picture.

**Figure 15.**Statistical results of safety factors of D group obtained by the Monte Carlo method. (

**a**) Results of group D-1; (

**b**) results of group D-2.

Specimen | Physical Parameters | ||||
---|---|---|---|---|---|

clay | Specific Gravity | Liquid Limit | Plastic Limit | Plasticity Index | Optimized Water Content |

2.75 | 41.30% | 19.20% | 22.10% | 16% | |

rock | Point pressure strength | crushing index | Density | ||

1234 N | 23% | 2.6 g/cm^{3} |

In Situ Samples | Remade Samples | ||
---|---|---|---|

Number | Rock Content | Number | Rock Content |

I-1 | 18.4 | R-0-1/2/3/4/5 | 0% |

I-2 | 18.7 | R-1-1/2/3/4/5 | 10% |

I-3 | 20.3 | R-2-1/2/3/4/5 | 20% |

I-4 | 25.2 | R-3-1/2/3/4/5 | 30% |

I-5 | 25.9 | R-4-1/2/3/4/5 | 40% |

I-6 | 32.9 | R-5-1/2/3/4/5 | 50% |

I-7 | 33.7 | R-6-1/2/3/4/5 | 60% |

Number | VBP | Time (s) | Safety Factor | Number | VBP | Time (s) | Safety Factor |
---|---|---|---|---|---|---|---|

C-1 | 22.50% | 163 | 1.172 | C-4 | 22.40% | 140 | 1.238 |

C-2 | 22.00% | 135 | 1.25 | C-5 | 23.10% | 149 | 1.266 |

C-3 | 22.50% | 128 | 1.164 | C-6 | 22.80% | 146 | 1.227 |

Number | Cohesion (kPa) | Internal Friction Angle (°) | Standard Deviation of Cohesion (kPa) | Standard Deviation of Internal Friction Angle (°) | Time (s) | Repeated Times | Safety Factor | Standard Deviation of Safety Factor |
---|---|---|---|---|---|---|---|---|

D-0 | 7.3 | 26.4 | 0 | 0 | 47s | 1 | 0.929 | 0 |

D-1 | 7.3 | 26.4 | 0.9 | 1.05 | 953 | 50 | 0.937 | 0.00959 |

D-2 | 7.3 | 26.4 | 0.9 | 1.05 | 965 | 50 | 0.965 | 0.01131 |

Number | MBP | Cohesion (kPa) | Internal Friction Angle (°) | Safety Factor |
---|---|---|---|---|

W-0 | 0% | 14.1 | 21.5 | 1.019 |

W-1 | 10% | 11.5 | 23.3 | 0.992 |

W-2 | 20% | 8.6 | 24 | 0.898 |

W-3 | 30% | 6.2 | 28.7 | 0.953 |

W-4 | 40% | 4.0 | 33 | 0.973 |

W-5 | 50% | 3.1 | 36.4 | 1.031 |

W-6 | 60% | 2.4 | 39 | 1.063 |

Number | MBP | Cohesion (kPa) | Internal Friction Angle (°) | Standard Deviation of Cohesion (kPa) | Standard Deviation of Internal Friction Angle (°) | Average Safety Factor | Standard Deviation |
---|---|---|---|---|---|---|---|

T-0 | 0% | 14.1 | 21.5 | 0.3 | 0.3 | 1.019 | 0.00312 |

T-1 | 10% | 11.5 | 23.3 | 0.6 | 0.6 | 0.9914 | 0.00461 |

T-2 | 20% | 8.6 | 24 | 0.8 | 0.9 | 0.8999 | 0.00597 |

T-3 | 30% | 6.2 | 28.7 | 1.0 | 1.2 | 0.9479 | 0.01262 |

T-4 | 40% | 4.0 | 33 | 1.2 | 1.5 | 0.9678 | 0.02470 |

T-5 | 50% | 3.1 | 36.4 | 1.4 | 1.8 | 1.02 | 0.03866 |

T-6 | 60% | 2.4 | 39 | 1.6 | 2.1 | 1.053 | 0.04056 |

Number | MBP | Cohesion (kPa) | Internal Friction Angle (°) | Standard Deviation of Cohesion (kPa) | Standard Deviation of Internal Friction Angle (°) | Probability of Rock | Average Safety Factor | Standard Deviation |
---|---|---|---|---|---|---|---|---|

R-0 | 0% | 14.1 | 21.5 | 0.3 | 0.3 | 0% | 1.019 | 0.00312 |

R-1 | 10% | 11.5 | 23.3 | 0.6 | 0.6 | 10% | 0.9968 | 0.00391 |

R-2 | 20% | 8.6 | 24 | 0.8 | 0.9 | 20% | 0.9022 | 0.00666 |

R-3 | 30% | 6.2 | 28.7 | 1.0 | 1.2 | 30% | 0.9623 | 0.00934 |

R-4 | 40% | 4.0 | 33 | 1.2 | 1.5 | 40% | 0.9912 | 0.02525 |

R-5 | 50% | 3.1 | 36.4 | 1.4 | 1.8 | 50% | 1.034 | 0.04049 |

R-6 | 60% | 2.4 | 39 | 1.6 | 2.1 | 60% | 1.076 | 0.04573 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, A.; Huang, X.; Li, N.; Jiang, P.; Wang, W.
A Monte Carlo Approach to Estimate the Stability of Soil–Rock Slopes Considering the Non-Uniformity of Materials. *Symmetry* **2020**, *12*, 590.
https://doi.org/10.3390/sym12040590

**AMA Style**

Zhou A, Huang X, Li N, Jiang P, Wang W.
A Monte Carlo Approach to Estimate the Stability of Soil–Rock Slopes Considering the Non-Uniformity of Materials. *Symmetry*. 2020; 12(4):590.
https://doi.org/10.3390/sym12040590

**Chicago/Turabian Style**

Zhou, Aizhao, Xianwen Huang, Na Li, Pengming Jiang, and Wei Wang.
2020. "A Monte Carlo Approach to Estimate the Stability of Soil–Rock Slopes Considering the Non-Uniformity of Materials" *Symmetry* 12, no. 4: 590.
https://doi.org/10.3390/sym12040590