# Accuracy of Two-Dimensional Limit Equilibrium Methods in Predicting Stability of Homogenous Road-Cut Slopes

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Brief Literature Review on Limit Equilibrium Methods

#### 1.2. Brief Discussion on Literature

## 2. Materials and Methods

#### 2.1. Limit Equilibrium

#### 2.1.1. Ordinary Method

#### 2.1.2. Bishop’s Simplified

#### 2.1.3. Janbu’s Simplified

#### 2.1.4. Janbu’s Generalised

#### 2.1.5. Corps of Engineers

#### 2.1.6. Morgenstern-Price

#### 2.1.7. Spencer’s Method

#### 2.1.8. SLIDES Computer Code Procedures

#### 2.2. Limit Analysis

#### 2.2.1. Governing Equations

#### 2.2.2. Lower Bound Principle

#### 2.2.3. Upper Bound Principle

#### 2.2.4. Bounds

#### 2.2.5. Optum G2 Computer Code Procedures

## 3. Results

#### 3.1. Stability Analysis of a Homogenous Road Embankment Slope

#### 3.1.1. Road Embankment Slope with Loose Sand

^{3}, and other input parameters as shown in Table 1, were taken into consideration. Both rigorous lower and upper bounds limit analysis and limit equilibrium methods were used to perform the analysis through separate numerical codes and were implemented as stated by the methodology section. The results of the limit analysis have shown that the slope is expected to be unstable considering the effect of an increase in strength of material with a depth of slope. Furthermore, the lower bound solution estimated the SRF of 0.6588 while the upper bound solution was about 0.666 SRF. On the other hand, the limit equilibrium method solutions were ranging from 0.718 FOS to 0.776 FOS. For all the cases considered (see Figure 6 and Figure 7), it has been denoted that the exact solutions are bracketed within 8 to 17% of error accuracy. This implies that among the limit equilibrium methods, none of them were able to provide the exact solutions as compared to the rigorous lower or upper bound solutions; however, the LEMs solutions were denoted to be closely related to those of the upper bound solutions. Indeed, this type of result has been demonstrated by previous scholars such as those of Yu et al. [40]. Meanwhile, the Corp of Engineer number two was found to produce the highest accuracy error in predicting the stability number. Though several scholars assumed that the LEMs produce similar results with the limit analysis based on the predicted SRF and FOS number without considering the calculation of error accuracy or benchmarking the two methods, the outcome of this analysis demonstrated different views within those studies, such as Renani and Martin [1]. One may denote that, previous studies such as those of Renani and Martin did not consider exact solutions but closely related solutions and by so doing the author came to the conclusion that the solutions of LEMs and rigorous upper bound solutions are similar.

#### 3.1.2. Road Embankment Slope with Medium Sand

^{3}, and other input parameters as shown in Table 1, were taken into consideration. Both rigorous lower and upper bounds limit analysis and limit equilibrium methods were used to perform the analysis. The results of the limit analysis have shown that the slope is expected to be unstable. The lower bound solution estimated the SRF of 0.7782 while the upper bound solution was about O.8068 SRF. On the other hand, the limit equilibrium method solutions (FOS) ranged from 0.834 to 0.908. For all the cases considered (see Figure 8 and Figure 10), it was denoted that the exact solutions are bracketed within 3 to 13% of error accuracy. This implies the limit equilibrium method is still considered to produce reasonable solutions in medium sand; however, the Janbu and Ordinary methods were found to produce very small accuracy errors ranging between 3 and 5%, which is still within the acceptable error accuracy.

#### 3.1.3. Road Embankment Slope with Dense Sand

^{3}, and other input parameters as shown in Table 1, were taken into consideration. Both rigorous lower and upper bounds limit analysis and limit equilibrium methods were used to perform the analysis through separate numerical codes were implemented as stated by the methodology section. The results of the limit analysis have shown that the slope is expected to be unstable considering the effect of the increase in strength of material with a depth of slope.

#### 3.1.4. Road Embankment Slope with Soft Clay

^{0}was selected as a case study shown in Figure 14, Figure 15 and Figure 16. In the slope, a soft clay with a friction angle of 18

^{0}, the cohesion of 9 kPa, and unit weight of 19 kN/m

^{3}, and other input parameters as shown in Table 1, were taken into consideration. Both rigorous lower and upper bounds limit analysis and limit equilibrium methods were used to perform the analysis. The results of the limit analysis have shown that the slope is expected to be unstable using the rigorous lower and upper bound solutions with the SRF of 0.8257 and 0.8079 for upper and lower, respectively. Meanwhile, the limit equilibrium method solutions estimated a stable slope with FOS ranging from 1.155 to 1.274 (see the simulation results in Figure 14, Figure 15 and Figure 16).

#### 3.1.5. Road Embankment Slope with Firm Clay

^{3}, and other input parameters as shown in Table 1, were taken into consideration. Both rigorous lower and upper bounds limit analysis and limit equilibrium methods were used to perform the analysis through separate numerical codes and were implemented as stated by the methodology section. The results of the limit analysis have shown that the slope is expected to be unstable considering the effect of the increase in strength of material with a depth of slope. Furthermore, the lower bound solution estimated the SRF of 0.6588 while the upper bound solution was about 0.666 SRF. On the other hand, the limit equilibrium method solutions ranged from 1.350 to 1.497. For all the cases considered (see Figure 17 and Figure 19), it has been denoted that the exact solutions are bracketed within 17 to 29% of error accuracy. This implies among the limit equilibrium method, none of them were able to provide the exact solutions as compared to the rigorous lower or upper bound solutions.

#### 3.1.6. Road Embankment Slope with Stiff Clay

^{3}, and other input parameters as shown in Table 1, were taken into consideration. Both rigorous lower and upper bounds limit analysis and limit equilibrium methods were used to perform the analysis through separate numerical codes and (SLIDEs and Optum G2) were implemented. The results of the Limit Analysis have shown that the slope is expected to be stable. Furthermore, the lower bound solutions estimated the SRF of 1.655 while the upper bound solutions were about 1.689 SRF, while the limit equilibrium method solutions ranged from 1.717 FOS to 1.932 FOS. For all the cases considered (see Figure 20 and Figure 22), it has been denoted that the exact solutions are bracketed within 5 to 14% of error accuracy. This implies among the limit equilibrium method, none of them were able to provide the exact solutions as compared to the rigorous lower or upper bound solutions; however, the LEMs solutions were denoted to be closely related to those of the upper bound solutions.

#### 3.2. Accuracy Classification Chart of Limit Equilibrium Method in Predicting the Stability of the Homogenous Slope

## 4. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

FOS | Factor of Safety |

SRF | Strength Reduction Factor |

LEM | Limit Equilibrium Methods |

FEM | Finite Element Method |

FED | Finite Difference Method |

OM | Ordinary Method |

BSM | Bishop’s Simplified Method |

JSM | Janbu’s Simplified Method |

JGM | Janbu’s Generalized Method |

CEM | Corps of Engineers Method |

MPM | Morgenstern–Price Method |

SM | Spencer’s Method |

## References

- Renani, H.R.; Martin, C.D. Factor of safety of strain-softening slopes. J. Rock Mech. Geotech. Eng.
**2020**, 12, 473–483. [Google Scholar] [CrossRef] - Hoek, E.; Bray, J.W. Rock Slope Engineering; Institution of Mining and Metallurgy: London, UK, 1981. [Google Scholar]
- Morgenstern, N.R.; Price, V.E. The analysis of the stability of general slip surfaces. Geotechnique
**1965**, 15, 79–93. [Google Scholar] [CrossRef] - Fredlund, D.G.; Krahn, J. Comparison of slope stability methods of analysis. Can. Geotech. J.
**1977**, 14, 429–439. [Google Scholar] [CrossRef] - Zhou, X.P.; Cheng, H. Analysis of stability of three-dimensional slopes using the rigorous limit equilibrium method. Eng. Geol.
**2013**, 160, 21–33. [Google Scholar] [CrossRef] - Duncan, J.M.; Wright, S.G. The accuracy of equilibrium methods of slope stability analysis. Eng. Geol.
**1980**, 16, 5–17. [Google Scholar] [CrossRef] - Nash, D. Chapter 2: A Comparative Review of Limit Equilibrium Methods of Stability Analysis. In Slope Stability; Anderson, M.G., Richards, K.S., Eds.; John Wiley & Sons, Inc.: New York, NY, USA, 1987. [Google Scholar]
- Jiang, Q.; Zhou, C. A rigorous method for three-dimensional asymmetrical slope stability analysis. Can. Geotech. J.
**2018**, 55, 495–513. [Google Scholar] [CrossRef] - Chen, R.H.; Chameau, J.L. Three-dimensional limit equilibrium analysis of slopes. Geotechnique
**1982**, 33, 31–40. [Google Scholar] [CrossRef] - Lam, L.; Fredlund, D.G. A general limit equilibrium model for three-dimensional slope stability analysis. Can. Geotech. J.
**1993**, 30, 905–919. [Google Scholar] [CrossRef] - Yin, H. A three-dimensional rigorous method for stability analysis of landslides. Eng. Geol.
**2012**, 145–146, 30–40. [Google Scholar] - Jiang, Q.; Zhou, C. A rigorous solution for the stability of polyhedral rock blocks. Comput. Geotech.
**2017**, 90, 190–201. [Google Scholar] [CrossRef] - Lu, R.; Wei, W.; Shang, K.; Jing, X. Stability Analysis of Jointed Rock Slope by Strength Reduction Technique considering Ubiquitous Joint Model. Adv. Civ. Eng.
**2020**, 2020, 1–13. [Google Scholar] [CrossRef] - Bishop, A.W. The use of the Slip Circle in the Stability Analysis of Slopes. Geotechnique
**1955**, 5, 7–17. [Google Scholar] [CrossRef] - Janbu, N. Slope Stability Computations. Soil Mechanics and Foundation Engineering Report; Technical University of Norway: Trondheim, Norway, 1968. [Google Scholar]
- Spencer, E. A method of analysis of the stability of embankments assuming parallel interslice forces. Geotechnique
**1967**, 17, 11–26. [Google Scholar] [CrossRef] - U.S. Army Corps of Engineers. Engineering and Design—Stability of Earth and Rockfill Dams. Engineer Manual EM 1110-2-1902; Department of the Army, Corps of Engineers: Washington, DC, USA, 1970. [Google Scholar]
- Lysmer, J. Limit analysis of plane problems in soil mechanics. J. Soil Mech. Found. Div.
**1970**, 96, 1131–1334. [Google Scholar] [CrossRef] - Anderheggen, E.; Knopfel, H. Finite element limit analysis using linear programming. Int. J. Solids Struct.
**1972**, 8, 1413–1431. [Google Scholar] [CrossRef] - Bottero, A.; Negre, R.; Pastor, J.; Turgeman, S. Finite element method and limit analysis theory for soil mechanics problems. Comput. Methods Appl. Mech. Eng.
**1980**, 22, 131–149. [Google Scholar] [CrossRef] - Sloan, S.W. Lower bound limit analysis using finite elements and linear programming. Int. J. Numer. Anal. Methods Geomech.
**1988**, 12, 61–77. [Google Scholar] [CrossRef] - Sloan, S.W. Upper bound limit analysis using finite elements and linear programming. Int. J. Numer. Anal. Methods Geomech.
**1989**, 13, 263–282. [Google Scholar] [CrossRef] - Sloan, S.W. Limit analysis in geotechnical engineering. Modem Developments in Geomechanics; Haberfield, C.M., Ed.; Monash University: Melbourne, Australia, 1995; pp. 167–199. [Google Scholar]
- Li, C.; Sun, C.; Li, C.; Zheng, H. Lower bound limit analysis by quadrilateral elements. J. Comput. Appl. Math.
**2017**, 315, 319–326. [Google Scholar] [CrossRef] - Sloan, S.W.; Kleeman, P.W. Upper bound limit analysis using discontinuous velocity fields. Comput. Methods Appl. Mech. Eng.
**1995**, 127, 293–314. [Google Scholar] [CrossRef] - Yalçin, Y. Integrated Limit Equilibrium Method for Slope Stability Analysis. Master’s Thesis, The Graduate School of Natural and Applied Sciences of Middle East Technical University, Ankara, Turkey, 2018; p. 8. [Google Scholar]
- Abramson, L.W.; Lee, T.S.; Sharma, S.; Boyce, G.M. Slope Stability and Stabilization Methods; John Wiley & Sons Inc.: Hoboken, NJ, USA, 2002; p. 712. [Google Scholar]
- Duncan, J.M. State of the art: Limit equilibrium and finite-element analysis of slopes. J. Geotech. Eng.
**1996**, 122, 577–596. [Google Scholar] [CrossRef] - Sengani, F.; Mulenga, F. Application of Limit Equilibrium Analysis and Numerical Modeling in a Case of Slope Instability. Sustainability
**2020**, 12, 8870. [Google Scholar] [CrossRef] - Sengani, F.; Mulenga, F. Influence of rainfall intensity on the stability of unsaturated soil slope: Case Study of R523 road in Thulamela Municipality, Limpopo Province, South Africa. Appl. Sci.
**2020**, 10, 8824. [Google Scholar] [CrossRef] - Matsui, T.; San, K.-C. Finite element slope stability analysis by shear strength reduction technique. Soils Found.
**1992**, 32, 59–70. [Google Scholar] [CrossRef] [Green Version] - Griffiths, D.V.; Lane, P.A. Slope stability analysis by finite elements. Geotechnique
**1999**, 49, 387–403. [Google Scholar] [CrossRef] - Dawson, E.M.; Roth, W.H.; Drescher, A. Slope stability analysis by strength reduction. Geotechnique
**1999**, 49, 835–840. [Google Scholar] [CrossRef] - Zheng, H.; Liu, D.F.; Li, C.G. Slope stability analysis based on elasto-plastic finite element method. Int. J. Numer. Methods Eng.
**2005**, 64, 1871–1888. [Google Scholar] [CrossRef] - Griffiths, D.V.; Marquez, R.M. Three-dimensional slope stability analysis by elasto-plastic finite elements. Geotechnique
**2007**, 57, 537–546. [Google Scholar] [CrossRef] [Green Version] - Cheng, Y.M.; Lansivaara, T.; Wei, W.B. Two-dimensional slope stability analysis by limit equilibrium and strength reduction methods. Comput. Geotech.
**2007**, 34, 137–150. [Google Scholar] [CrossRef] - Schneider-Muntau, B.; Medicus, W.G.; Fellin, W. Strength reduction method in Barodesy. Comput. Geotech.
**2018**, 95, 57–67. [Google Scholar] [CrossRef] - Zienkiewicz, O.C.; Humpheson, C.; Lewis, R.W. Associated and non-associated visco-plasticity and plasticity in soil mechanics. Geotechnique
**1975**, 25, 671–689. [Google Scholar] [CrossRef] - Optum, G.; Theory of the Model. Optum Computational Engineering; Denmark. 2019. Available online: https://optumce.com/wp-content/uploads/2016/05/Theory.pdf (accessed on 28 January 2022).
- Yu, H.S.; Salgado, R.; Sloan, S.W.; Kim, J.M. Limit analysis versus limit equilibrium for slope stability. J. Geotech. Geoenvironmental Eng.
**1998**, 124, 1–11. [Google Scholar] [CrossRef] - Bjerrum, L. Theoretical and Experimental Investigations on the Shear Strength of Soils. PhD Thesis, ETH Zurich, Zurich, Switzerland, 1954. [Google Scholar]
- Skempton, A.W. Long-term stability of clay slopes. Geotechnique
**1964**, 14, 77–102. [Google Scholar] [CrossRef] [Green Version] - Hettler, A.; Vardoulakis, I. Behaviour of dry sand tested in a large triaxial apparatus. Geotechnique
**1984**, 34, 183–197. [Google Scholar] [CrossRef]

**Figure 1.**Elements used for lower bound limit analysis [24].

**Figure 2.**Elements used for upper bound limit analysis [18].

**Figure 3.**Free-body diagram of a generic slip surface (

**a**) overall diagram; (

**b**) vertical slice diagram [25].

**Figure 4.**Distribution of Stability number (SRF and FOS) in various homogenous slope materials with limit equilibrium benchmarked with limit analysis.

**Figure 5.**Predicted stability number using limit analysis method of strength reduction factor in loose sand (

**a**) the SRF of the upper bound solutions of limit analysis; (

**b**) the SRF of the lower bound solutions of the limit analysis.

**Figure 6.**Limit equilibrium stability number of loose sand slope produced using (

**a**) Ordinary method; (

**b**) Bishop Simplified method; (

**c**) Janbu Simplified method; (

**d**) Janbu Corrected method.

**Figure 7.**Limit equilibrium stability number of loose sand slope produced using (

**a**) Spencer method; (

**b**) Corp of Engineering one; (

**c**) Corp of Engineering two; (

**d**) Morgenstern–Price method.

**Figure 8.**Predicted stability number using limit analysis method of strength reduction factor in medium sand (

**a**) the SRF of the upper bound solutions of limit analysis (

**b**) the SRF of the lower bound solutions of the limit analysis.

**Figure 9.**Limit equilibrium stability number of medium sand slope produced using (

**a**) Ordinary method; (

**b**) Bishop Simplified method; (

**c**) Janbu Simplified method; (

**d**) Janbu Corrected method.

**Figure 10.**Limit equilibrium stability number of medium sand slope produced using (

**a**) Spencer method, (

**b**) Corp of Engineering one (

**c**) Corp of Engineering two (

**d**) Morgenstern–Price method.

**Figure 11.**Predicted stability number using limit analysis method of strength reduction factor in dense sand (

**a**) the SRF of the upper bound solutions of limit analysis; (

**b**) the SRF of the lower bound solutions of the limit analysis.

**Figure 12.**Limit Equilibrium stability number of dense sand slope produced using (

**a**) Ordinary method; (

**b**) Bishop Simplified method; (

**c**) Janbu Simplified method; (

**d**) Janbu Corrected method.

**Figure 13.**Limit Equilibrium stability number of dense sand slope produced using (

**a**) Spencer method; (

**b**) Corp of Engineering one (

**c**) Corp of Engineering two; (

**d**) Morgenstern–Price method.

**Figure 14.**Predicted stability number using limit analysis method of strength reduction factor in soft clay (

**a**) the SRF of the upper bound solutions of limit analysis; (

**b**) the SRF of the lower bound solutions of the limit analysis.

**Figure 15.**Limit Equilibrium stability number of soft clay slope produced using (

**a**) Ordinary method; (

**b**) Bishop Simplified method; (

**c**) Janbu Simplified method; (

**d**) Janbu Corrected method.

**Figure 16.**Limit Equilibrium stability number of soft clay slope produced using (

**a**) Spencer method; (

**b**) Corp of Engineering one; (

**c**) Corp of Engineering two; (

**d**) Morgenstern–Price method.

**Figure 17.**Predicted stability number using limit analysis method of strength reduction factor in firm clay; (

**a**) the SRF of the upper bound solutions of limit analysis; (

**b**) the SRF of the lower bound solutions of the limit analysis.

**Figure 18.**Limit equilibrium stability number of firm clay slope produced using (

**a**) Ordinary method; (

**b**) Bishop simplified method; (

**c**) Janbu Simplified method; (

**d**) Janbu Corrected method.

**Figure 19.**Limit equilibrium stability number of firm clay slope produced using (

**a**) Spencer method; (

**b**) Corp of Engineering one; (

**c**) Corp of Engineering two; (

**d**) Morgenstern–Price method.

**Figure 20.**Predicted stability number using limit analysis method of strength reduction factor in stiff clay; (

**a**) the SRF of the upper bound solutions of limit analysis; (

**b**) the SRF of the lower bound solutions of the limit analysis.

**Figure 21.**Limit equilibrium stability number of stiff clay slope produced using (

**a**) Ordinary method; (

**b**) Bishop Simplified method; (

**c**) Janbu Simplified method; (

**d**) Janbu Corrected method.

**Figure 22.**Limit equilibrium stability number of stiff clay slope produced using (

**a**) Spencer method; (

**b**) Corp of Engineering one; (

**c**) Corp of Engineering two; (

**d**) Morgenstern–Price Method.

Material Type | Loose Sand | Medium Sand | Dense Sand | Soft Clay | Firm Clay | Stiff Clay | Dimensions of the Model | ||
---|---|---|---|---|---|---|---|---|---|

Stiffness | E (MPa) | 30 | 30 | 30 | 30 | 30 | 30 | Rise (m) | 10 |

ν (-) | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | 0.25 | Depth (m) | 4 | |

Strength | c (kPa) | 03 | 04 | 06 | 09 | 12 | 20 | Left (m) | 10 |

ϕ (°) | 13 | 15 | 17 | 18 | 20 | 22 | Run (m) | 20 | |

Tension cut-off | kt (kPa) | 0 | 0 | 0 | 0 | 0 | 0 | Right (m) | 15 |

ϕt (°) | 90 | 90 | 90 | 90 | 90 | 90 | Slope degree | 27° | |

Unit Weights | γdry (kN/m^{3}) | 14 | 16 | 18 | 19 | 20 | 21 | ||

γsat (kN/m^{3}) | 19 | 20 | 21 | 19 | 20 | 21 | |||

Initial Conditions | K0 (-) | 0.5 | 0.43 | 0.36 | 0.69 | 0.66 | 0.63 | ||

σ0 (kPa) | 0 | 0 | 0 | 0 | 0 | 0 | |||

Hydraulic Model | Kx (m/day) | 1 × 10^{−5} | 1 × 10^{−5} | 1 × 10^{−5} | 1 × 10^{−5} | 1 × 10^{−5} | 1 × 10^{−5} | ||

Ky (m/day) | 1 × 10^{−5} | 1 × 10^{−5} | 1 × 10^{−5} | 1 × 10^{−5} | 1 × 10^{−5} | 1 × 10^{−5} | |||

h (m) | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |

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**MDPI and ACS Style**

Sengani, F.; Allopi, D.
Accuracy of Two-Dimensional Limit Equilibrium Methods in Predicting Stability of Homogenous Road-Cut Slopes. *Sustainability* **2022**, *14*, 3872.
https://doi.org/10.3390/su14073872

**AMA Style**

Sengani F, Allopi D.
Accuracy of Two-Dimensional Limit Equilibrium Methods in Predicting Stability of Homogenous Road-Cut Slopes. *Sustainability*. 2022; 14(7):3872.
https://doi.org/10.3390/su14073872

**Chicago/Turabian Style**

Sengani, Fhatuwani, and Dhiren Allopi.
2022. "Accuracy of Two-Dimensional Limit Equilibrium Methods in Predicting Stability of Homogenous Road-Cut Slopes" *Sustainability* 14, no. 7: 3872.
https://doi.org/10.3390/su14073872