# A Simplified Method for Effective Calculation of 3D Slope Reliability

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## Abstract

**:**

## 1. Introduction

## 2. Three-Dimensional Limit Equilibrium Method Based on the Modification of Normal Stresses over Slip Surface

#### 2.1. Fundamental Assumptions

#### 2.2. Three-Dimensional Limit Equilibrium Equations

${D}_{i}={\displaystyle \iint {\sigma}_{0}{\xi}_{i}{{s}^{\prime}}_{y}\mathrm{d}x\mathrm{d}y\left(i=1,2,3\right)}$ | $E=-{\displaystyle \iint {\sigma}_{0}{\xi}_{0}{{s}^{\prime}}_{y}\mathrm{d}x\mathrm{d}y}$ |

${A}_{1i}=-{\displaystyle \iint {\sigma}_{0}{\xi}_{i}{{s}^{\prime}}_{x}\mathrm{d}x\mathrm{d}y(i=1,2,3)}$ | ${A}_{14}={\displaystyle \iint {\sigma}_{0}{\xi}_{0}{{s}^{\prime}}_{x}\mathrm{d}x\mathrm{d}y}$ |

${A}_{2i}={\displaystyle \iint {\sigma}_{0}{\xi}_{i}\mathrm{d}x\mathrm{d}y}(i=1,2,3)$ | ${A}_{24}={\displaystyle \iint w\mathrm{d}x\mathrm{d}y}-{\displaystyle \iint {\sigma}_{0}{\xi}_{0}\mathrm{d}x\mathrm{d}y}$ |

${A}_{3i}={\displaystyle \iint {\sigma}_{0}{\xi}_{i}\left(x+s\cdot {{s}^{\prime}}_{x}\right)\mathrm{d}x\mathrm{d}y(i=1,2,3)}$ | ${A}_{34}={\displaystyle \iint \left[w\cdot x-{\sigma}_{0}{\xi}_{0}\left(x+s\cdot {{s}^{\prime}}_{x}\right)\right]\text{}\mathrm{d}x\mathrm{d}y}$ |

${B}_{1i}={\displaystyle \iint \rho \psi {\sigma}_{0}{\xi}_{i}\mathrm{d}x\mathrm{d}y}(i=1,2,3)$ | ${B}_{14}=-{\displaystyle \iint \rho \left({c}_{u}+\psi {\sigma}_{0}{\xi}_{0}\right)\text{}\mathrm{d}x\mathrm{d}y}$ |

${B}_{2i}={\displaystyle \iint \rho \psi {\sigma}_{0}{\xi}_{i}{{s}^{\prime}}_{x}\mathrm{d}x\mathrm{d}y}(i=1,2,3)$ | ${B}_{24}=-{\displaystyle \iint \rho \left({c}_{u}+\psi {\sigma}_{0}{\xi}_{0}\right){{s}^{\prime}}_{x}\mathrm{d}x\mathrm{d}y}$ |

${B}_{3i}={\displaystyle \iint \rho \psi {\sigma}_{0}{\xi}_{i}\left(x\cdot {{s}^{\prime}}_{x}-s\right)\mathrm{d}x\mathrm{d}y(i=1,2,3)}$ | ${B}_{34}=-{\displaystyle \iint \rho \left(x\cdot {{s}^{\prime}}_{x}-s\right)\left({c}_{u}+\psi {\sigma}_{0}{\xi}_{0}\right)\text{}\mathrm{d}x\mathrm{d}y}$ |

${G}_{1}=-{\displaystyle \iint w\mathrm{d}x\mathrm{d}y}$ | ${G}_{2}=0$ |

${G}_{3}=-{\displaystyle \iint w\cdot {z}_{c}\mathrm{d}x\mathrm{d}y}$ |

## 3. Simplified Method of 3D Slope Reliability Calculation

#### 3.1. Critical Horizontal Acceleration Coefficient ${K}_{c}$

#### 3.2. Limit State Performance Function

#### 3.3. Three-Dimensional Slope Reliability Calculation Method

## 4. Numerical Examples

#### 4.1. Example 1

#### 4.2. Example 2

## 5. Conclusions

- (1)
- This method has the advantages of simple calculation, no iterative convergence problem, and high calculation efficiency. Combined with the SS method, it can fully reflect the advantages of high accuracy and efficiency.
- (2)
- By changing ${K}_{c0}$ to a value greater than zero, this method can conveniently calculate 3D slope reliability under seismic loads without large-scale modification of the calculation program.
- (3)
- In the case of a long slope, the results of 2D reliability calculation do not necessarily underestimate the stability of the slope, so it is necessary to carry out 3D slope reliability analysis.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Three-dimensional slope, soil column, and forces. (

**a**) Three-dimensional slope and sliding body. (

**b**) Soil column and forces.

**Figure 4.**The sliding surface position and range of Example 1. (

**a**) Three-dimensional slip surface at ${K}_{c0}=0$. (

**b**) Three-dimensional slip surface at ${K}_{c0}=0.1$.

**Figure 6.**The slip surface position and range of the three-layer slope in Example 2. (

**a**) Three-dimensional sphere slip surface. (

**b**) Three-dimensional ellipsoid slip surface.

**Figure 7.**Relationship curves between reliability results and slope length. (

**a**) The curve of failure probability with slope length. (

**b**) The curve of reliability index with slope length.

$\mathbf{Cohesion}\text{}\mathit{c}$ (kPa) | $\mathbf{Friction}\text{}\mathbf{Angle}\text{}\mathit{\phi}$ (°) | $\mathbf{Unit}\text{}\mathbf{Weight}\text{}\mathit{\gamma}$ (kN/m^{3}) | ||
---|---|---|---|---|

${\mu}_{c}$ | ${\sigma}_{c}$ | ${\mu}_{\varphi}$ | ${\sigma}_{\varphi}$ | |

41.65 | 8.00 | 15.00 | 3.00 | 18.82 |

Performance Functions | Reliability Methods | $\mathbf{Horizontal}\text{}\mathbf{Seismic}\text{}\mathbf{Coefficient}\text{}{\mathit{K}}_{\mathit{c}0}$ | $\mathbf{Sampling}\text{}\mathbf{Number}\text{}\mathit{N}$ | $\mathit{\beta}$ | ${\mathit{P}}_{\mathit{f}}$ | Slip Surface Shape | $\mathbf{Computation}\text{}\mathbf{Time}\text{}\mathit{t}$ (s) |
---|---|---|---|---|---|---|---|

${F}_{s}$ method | MCS | 0 | 40,000 | 2.6376 | 0.0042 | 3D S1 | 21.70 |

${K}_{c}$ method | MCS | 0 | 40,000 | 2.6606 | 0.0039 | 3D S1 | 12.45 |

${F}_{s}$ method | SS | 0 | 40,000 | 2.6780 | 0.0037 | 3D S1 | 6.80 |

${K}_{c}$ method | SS | 0 | 40,000 | 2.6438 | 0.0041 | 3D S1 | 5.45 |

${F}_{s}$ method | FOSM | 0 | 2.6596 | 0.0039 | 3D S1 | 0.30 | |

${K}_{c}$ method | FOSM | 0 | 2.6937 | 0.0035 | 3D S1 | 0.13 | |

Bishop | GA + FORM | 0 | 2.6100 | 0.0045 [8] | 3D S1 | >3.8 h | |

Bishop (GeoStudio) | MCS | 0 | 40,000 | 2.2422 | 0.0120 | 2D circular S1 | |

${K}_{c}$ method | MCS | 0.1 | 40,000 | 1.8317 | 0.0335 | 3D S2 | 12.98 |

${K}_{c}$ method | SS | 0.1 | 40,000 | 1.8633 | 0.0312 | 3D S2 | 5.33 |

${K}_{c}$ method | FOSM | 0.1 | 1.8432 | 0.0326 | 3D S2 | 0.14 | |

Bishop (GeoStudio) | MCS | 0.1 | 40,000 | 1.1870 | 0.1176 | 2D circular S2 |

Soil Layer | $\mathbf{Cohesion}\text{}\mathit{c}$ (kPa) | $\mathbf{Friction}\text{}\mathbf{Angle}\text{}\mathit{\phi}$ (°) | $\mathbf{Unit}\text{}\mathbf{Weight}\text{}\mathit{\gamma}$ (kN/m^{3}) | ||
---|---|---|---|---|---|

${\mathit{\mu}}_{\mathit{c}}$ | ${\mathit{\sigma}}_{\mathit{c}}$ | ${\mathit{\mu}}_{\mathit{\varphi}}$ | ${\mathit{\sigma}}_{\mathit{\varphi}}$ | ||

Layer 1 | 0 | 0 | 38 | 0.1 | 19.5 |

Layer 2 | 5.3 | 0.53 | 23 | 4.6 | 19.5 |

Layer 3 | 7.2 | 1.44 | 20 | 4.0 | 19.5 |

Performance Functions | Reliability Methods | $\mathbf{Sampling}\text{}\mathbf{Number}\text{}\mathit{N}$ | $\mathit{\beta}$ | ${\mathit{P}}_{\mathit{f}}$ | Slip Surface Shape | $\mathbf{Computation}\text{}\mathbf{Time}\text{}\mathit{t}$ (s) |
---|---|---|---|---|---|---|

${F}_{s}$ method | MCS | 100,000 | 2.8576 | 0.0021 | 3D sphere | 58.98 |

${K}_{c}$ method | MCS | 100,000 | 2.8894 | 0.0019 | 3D sphere | 31.48 |

${F}_{s}$ method | SS | 100,000 | 2.8682 | 0.0021 | 3D sphere | 18.23 |

${K}_{c}$ method | SS | 100,000 | 2.8707 | 0.0020 | 3D sphere | 16.89 |

${F}_{s}$ method | FOSM | 2.7411 | 0.0031 | 3D sphere | 0.80 | |

${K}_{c}$ method | FOSM | 2.7152 | 0.0033 | 3D sphere | 0.44 | |

Bishop | GA + FORM | 2.8900 | 0.0019 [8] | 3D sphere | ~7 h | |

${K}_{c}$ method | MCS | 100,000 | 2.5414 | 0.0055 | 3D ellipsoid ($\eta $ = 7) | 163.59 |

${K}_{c}$ method | SS | 100,000 | 2.5360 | 0.0056 | 3D ellipsoid ($\eta $ = 7) | 56.92 |

${K}_{c}$ method | FOSM | 2.4290 | 0.0076 | 3D ellipsoid ($\eta $ = 7) | 0.61 | |

Bishop (GeoStudio) | MCS | 100,000 | 2.5465 | 0.0054 | 2D circular |

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**MDPI and ACS Style**

Chen, J.; Zhu, D.; Zhu, Y.
A Simplified Method for Effective Calculation of 3D Slope Reliability. *Water* **2023**, *15*, 3139.
https://doi.org/10.3390/w15173139

**AMA Style**

Chen J, Zhu D, Zhu Y.
A Simplified Method for Effective Calculation of 3D Slope Reliability. *Water*. 2023; 15(17):3139.
https://doi.org/10.3390/w15173139

**Chicago/Turabian Style**

Chen, Juxiang, Dayong Zhu, and Yalin Zhu.
2023. "A Simplified Method for Effective Calculation of 3D Slope Reliability" *Water* 15, no. 17: 3139.
https://doi.org/10.3390/w15173139