# A New Discrete Form of Hoek–Brown Criterion and Its Application to Limit Equilibrium Analysis of Rock Slope Stability

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## Abstract

**:**

## 1. Introduction

## 2. Generalized Hoek–Brown Criterion

#### 2.1. General form in Terms of Minor Principal Stress

#### 2.2. Normalized Form of the GHB Criterion

#### 2.3. Shear–Normal Stress Relation of the GHB Criterion

## 3. New Approximate Formulation of the Mohr Envelope for GHB Criterion

#### 3.1. Approximate Mohr Envelope Based on Taylor Expansion of the Balmer’s Equation

- (i)
- Linear approximation ($n=1$)

- (ii)
- Quadratic approximation ($n=2$)

- (iii)
- Cubic approximation ($n=3$)

#### 3.2. Discussions on the Accuracy of New Formulations of the Mohr Envelope

## 4. Limit Equilibrium Analysis of a Slope in GHB Rock Mass

#### 4.1. Geometry of Rock Slope Models

**Figure 7.**Geometry of slope showing forces of interaction acting on a typical slice: (

**a**) model for plane failure; (

**b**) model for circular failure.

#### 4.2. Modified Bishop Approach for the Assessment of Safety Factor

#### 4.3. Comparison of Safety Factors Based on the Simplified and Modified Bishop Methods

#### 4.4. Assessment of the Critical Value of Surface Load

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Comparison of the proposed approximate Mohr envelopes with those obtained from the numerical (exact) solution: the approximate Mohr envelopes are calculated by using the proposed analytical shear strength formulas (Mohr envelopes) incorporating the approximate solutions for ${N}_{3}$ corresponding to (

**a**) the linear, (

**b**) quadratic and (

**c**) cubic Taylor polynomial approximations of the Balmer’s equation.

**Figure 5.**Percentage errors incurred in the shear strength predictions using the newly formulated approximate Mohr envelopes and Lee and Pietruszcak’s formula [15] for rock masses with GSI values of (

**a**) 20, (

**b**) 40, (

**c**) 60 and (

**d**) 80.

**Figure 6.**Effect of GSI value on the percentage errors of shear strength prediction based on newly formulated approximate Mohr envelopes; (

**a**) $\sigma /{\sigma}_{ci}$ = 0.2, (

**b**) 0.4, (

**c**) 0.6 and (

**d**) 0.8.

**Figure 8.**Factors of safety versus the value of GSI for a slope with embedded vertical tension crack of depth 5 m located at (

**a**) ${x}_{c}$ = 5 m and (

**b**)${x}_{c}$ = 10 m.

**Figure 9.**The circular failure surfaces giving the minimum FS for four different GSI values; slope model with${x}_{c}$ = 10 m and z = 5 m: (

**a**) simplified Bishop method; (

**b**) modified Bishop method.

**Figure 10.**Variation in factor of safety with the location of 5 m deep tension crack for three different values of GSI: simplified vs. modified Bishop approach.

**Figure 11.**The critical value of distributed load versus GSI for two different crack depths, i.e., 5 m and 10 m; planar and circular failure modes: (

**a**) x

_{c}= 5 m and (

**b**) x

_{c}= 10 m.

**Figure 12.**The critical value of distributed load versus GSI for four different crack depths, i.e., 2.5 m, 5.0 m, 7.5 m and 10.0 m; circular failure mode: (

**a**)${x}_{c}$ = 5 m; (

**b**)${x}_{c}$ = 10 m.

**Figure 13.**The geometry of the critical circular failure surfaces for four different values of GSI, i.e., 20, 40, 60 and 80; slope model with a 10 m deep crack: (

**a**)${x}_{c}$ = 5 m; (

**b**)${x}_{c}$ = 10 m.

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**MDPI and ACS Style**

Lee, Y.-K.; Pietruszczak, S.
A New Discrete Form of Hoek–Brown Criterion and Its Application to Limit Equilibrium Analysis of Rock Slope Stability. *Sustainability* **2022**, *14*, 12113.
https://doi.org/10.3390/su141912113

**AMA Style**

Lee Y-K, Pietruszczak S.
A New Discrete Form of Hoek–Brown Criterion and Its Application to Limit Equilibrium Analysis of Rock Slope Stability. *Sustainability*. 2022; 14(19):12113.
https://doi.org/10.3390/su141912113

**Chicago/Turabian Style**

Lee, Youn-Kyou, and S. Pietruszczak.
2022. "A New Discrete Form of Hoek–Brown Criterion and Its Application to Limit Equilibrium Analysis of Rock Slope Stability" *Sustainability* 14, no. 19: 12113.
https://doi.org/10.3390/su141912113