# An Improved Strength Reduction-Based Slope Stability Analysis

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## Abstract

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## 1. Introduction

## 2. Fundamental Theories

#### 2.1. Spectral Element Method

#### 2.2. Strength Reduction Method

## 3. Methodology

## 4. Case Study Models

## 5. Results and Discussion

#### 5.1. Case 1

#### 5.2. Cases 2 and 3

#### 5.3. Safety Factor Determination

- The displacement-based curves were smoother than the gradient-based ones.
- In displacement-based curves, the three plots resulting from three DT values were very close to each other. However, they had meaningful differences in the gradient-based plots.
- The range of SRF in which the system moved from the linear to plastic phase was longer for Cases 2 and 3 than Case 1.
- In all case studies, the safety factor for the system with DT = 30% was the minimum, and for the system with DT = 70%, it was the maximum.
- The safety factor of the homogeneous model (i.e., Case 1) was more than the two other cases (with a weak layer). This is consistent with the physics of the problem.
- Comparing Cases 2 and 3 revealed that the safety factor of the dry model (Case 2) was higher than the partially-wet slope (i.e., Case 3).
- The number of required simulations depended on the LS and pre-defined accuracy.

#### 5.4. Validation

- The SFs obtained by Gharti et al. [14] were approximately equal to the proposed hybrid SEM-SRM with DT of 50%, 10%, and 60% for Case Studies 1, 2, and 3, respectively. This indicates that the inconsistency in SF resulted from the conventional SRM.
- The differences between SFs were equal to 0.02, 0.03, and 0.04 within DT of 30–70% for Case Studies 1, 2, and 3, respectively. This reveals that the influence of DT on SF was numerically significant for multilayer soil slopes. One should note that from the practical point of view, these differences are negligible.
- The number of required analyses to evaluate a particular example that considered both design and precision parameters was about 10 using the proposed method. However, it was a considerable number in other methods (depending on the technique).

## 6. Summary

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Location of index points in a hexahedron element. (

**a**) Nodal points, (

**b**) interpolation points, and (

**c**) Gauss–Legendre–Lobatto points.

**Figure 5.**Schematic convergence to the design trigger (DT) within two iterations; for State 2 in Figure 4. (

**a**) Initial (old) analysis and (

**b**) updated (new) analysis.

**Figure 6.**Generic discretized model of the case studies. (

**a**) Applied boundary conditions, (

**b**) Case 1 and (

**c**) Cases 2 and 3.

**Figure 7.**Determination of the SF based on the maximum displacement and relevant gradient diagrams. (

**a**) Case 1, displacement, (

**b**) Case 1, gradient, (

**c**) Case 2, displacement, (

**d**) Case 2, gradient, (

**e**) Case 3, displacement, and (

**f**) Case 3, gradient.

Methods | Case 1 | Case 2 | Case 3 |
---|---|---|---|

Xing [32] | 2.122 | 1.553 | 1.441 |

Bishop’s modified method (Lam and Fredlund [31]) | - | 1.607 | 1.511 |

Janbu’s simplified method (Lam and Fredlund [31]) | - | 1.558 | 1.481 |

CLARA (Lam and Fredlund [31]) | - | 1.62 | 1.54 |

CLE (Lam and Fredlund [31]) | - | 1.603 | 1.508 |

Chen et al. [30] | 2.262 | 1.717 | - |

Chen et al. [30] | 2.187 | 1.603 | - |

FEM (Griffiths and Marquez [33]) | 2.17 | 1.58 | - |

SEM (Gharti et al. [14]) | 2.18 | 1.57 | 1.49 |

SEM (DT = 30%)—present study | 2.17 | 1.59 | 1.46 |

SEM (DT = 50%)—present study | 2.18 | 1.61 | 1.48 |

SEM (DT = 70%)—present study | 2.19 | 1.62 | 1.5 |

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**MDPI and ACS Style**

Seyed-Kolbadi, S.M.; Sadoghi-Yazdi, J.; Hariri-Ardebili, M.A.
An Improved Strength Reduction-Based Slope Stability Analysis. *Geosciences* **2019**, *9*, 55.
https://doi.org/10.3390/geosciences9010055

**AMA Style**

Seyed-Kolbadi SM, Sadoghi-Yazdi J, Hariri-Ardebili MA.
An Improved Strength Reduction-Based Slope Stability Analysis. *Geosciences*. 2019; 9(1):55.
https://doi.org/10.3390/geosciences9010055

**Chicago/Turabian Style**

Seyed-Kolbadi, S. M., J. Sadoghi-Yazdi, and M. A. Hariri-Ardebili.
2019. "An Improved Strength Reduction-Based Slope Stability Analysis" *Geosciences* 9, no. 1: 55.
https://doi.org/10.3390/geosciences9010055