# Geotechnical Assessment of Rock Slope Stability Using Kinematic and Limit Equilibrium Analysis for Safety Evaluation

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## Abstract

**:**

## 1. Introduction

#### Location and Geology of the Study Area

## 2. Materials and Methods

#### 2.1. Field Investigation

#### 2.2. Laboratory Testing and Geotechnical Assessment

#### 2.3. Strength Parameters

_{1}), minimum (σ

_{3}) normal stresses in MPa, and elastic modulus Εi (GPa)) and shear strength parameters (cohesion (c), internal friction angle (φ), and Poisson ratio) were based on the difference between the dry and saturated states and the confining stress (Table 2).

#### 2.4. Failure Mechanisms and Slope Stability

## 3. Results

## 4. Discussion

#### 4.1. Kinematic Analysis

#### 4.1.1. Left Slope

- (a)
- Planar failure: It is clear from the kinematic analysis of the left bank that the toppling mode tested with a greater failure probability than the planar and wedge failure (Figure 5a). The results shows that when the slope dip angle is 40°, planar failure is slightly more than 1%, but with an increase in 5°, the failure chances reach 1.6%. The number of critical failures is 2 (Figure 5a). Also, when the slope dip angle is 55°, 60°, and 70°, failure probability increases to 2.5%, 2.9%, and 5%, respectively. The key critical zone for planar failure is located on the left slope, as seen in the highlighted red area. The danger of the developing planar slide is represented by intersections in the crucial zone (Figure 5b,c). The condition that must be met for a plane to fail is that the slope-face dip must be larger than the slide-plane dip, and both must be higher than the slide-surface friction angle, i.e., ψf > ψp > ϕ [12]. According to the pole points, none of the major joint sets on the slopes is critical for planar slope failure. As a result of the discontinuities of pole points not being inside the planar daylight envelope, planar failure does not occur on the slopes of the examined area (Figure 5a–c). Because the dip angle of the majority of joints is lower and extremely small, the pole of the junction point of the joints falls within the friction cone or safe region. The number of critical failures likewise rises as the slope dip angle increases (Table 3). Thus, the kinematic analysis indicates that planar failure chances are very low, approaching zero.
- (b)
- Wedge failure: The failure probability of a wedge is 5% at a slope angle of 45° as shown in Figure 5d, and increases to 10% at a slope angle of 55° (Figure 5e). The failure probability is highest (17%) when the slope’s peak orientation has a dip angle of 70° (Figure 5f). The majority of the joints have a dip angle of 20° or more, as shown by the data; moreover, 34 joints have a dip angle of 80°, whereas 63 joints dipped with an angle greater than 70° (Figure 2a). This indicates that wedge failure is more likely to occur along the left bank. The number of critical failures also increases as the slope dip angle increases even though more joints are in the unsafe zone, which raises the probability of wedge failure (Table 3). According to [50], the prior landslide in construction is mostly caused by steep slope angles. Additionally, [51] reported that most of the intersection points of the joints fall in the safe zone because of the low dip angle, which decreases the probability of wedge failure. Azimuth-based discontinuities with NE and SE orientations produce intersection points in the critical zone that lead to wedge failure. The number of crucial locations likewise grows as the dip slope angle rises. Therefore, the probability of wedge failure increases and rises up to 11% when the slope dip angle is 70° (Figure 5f).
- (c)
- Toppling failure: The toppling failure is the rotation of a block or stone column located on a sloping surface, around a point [49]. From Figure 4a, it can be seen that toppling failure shows a higher failure probability. When the slope dip angle is 40°, failure chances are slightly less than 3%, while a small increase in slope dip angle increases the failure chances up to 4.5% when the dip angle is 55° (Figure 5g,h). There is a lower chance of a toppling failure than a wedge failure (Figure 4a). Failure probabilities spike dramatically at a slope dip angle of 70°, where they reach 17%, indicating a dangerously high risk for toppling failure (Figure 5i). By observing the pole orientation of joints, one can determine the most vulnerable areas that could topple. Ning et al. [52] found that the change in slope angle and stratum dip angle could impact the stability of the slope. From Figure 5a, it can be seen that 109 discontinuities are oriented in the NW and SW where the joints and slope dip in opposite directions. This suggests that toppling failure shows a higher probability. From Figure 5g, it can be seen that with a slope angle of 45°, critical failure is comparatively lower, but when the slope angle increases, the critical intersection points increase. With slope angles of 60° and 70°, the toppling failure chances are high and more critical. All results of the critical interactions are listed in Table 3.

#### 4.1.2. Right Slope

- (a)
- Planar failure: Results from the right bank show that when the slope angle is 40°–60°, planar failure chances are 0%. When the slope dip angle is 65° and 70°, the failure chances increase slightly to 1%, which is very low (Figure 4b). The failure probability is very low when the slope dip is lower than the joint dip. On the other hand, there will be no chance for planar failure if the pole points of the joints are not in the critical zone [51,53]. The number of critical failures is 0 when the slope dip angle is kept lower or higher. All results are shown in Table 3. The failure plane created by the friction angle circle and the daylight envelope demonstrates that any junction point is susceptible to failure. According to the results of quantitative data on dip direction, several joints are orientated in the same direction as the slope. As a result, no joint is in danger of failing from the right bank (Figure 6a–c). The azimuth of most discontinuities is oriented in NW and NE where the criteria for planar failure are not met. Additionally, when the slope angle is 45°, the failure chance is zero, but with an increase in slope dip angle up to 70°, the critical intersection points and failure chances are zero and 1%, respectively. This indicates that an increase in slope dip angle has no effect on planar failure in this condition (Figure 6a,c).
- (b)
- Wedge failure: The right bank wedge failure study yields nearly identical results as the left bank. A low dip angle has lower odds, but as the slope dip increases, the likelihood of failure increases (Figure 4b). From the result, it can be seen that the slope dip angle of 45° has a failure probability of 2%, which is very low (Figure 6d). Additionally, the number of intersection points in the critical intersection zone is comparatively lower (Table 3). Additionally, the failure probability exhibits a regular increase with increasing slope dip angle. From the quantitative analysis of discontinuities, it can be seen that 67 discontinuities are oriented in a SW direction. In contrast, 94 are oriented NW (Figure 2b). The kinematic measurements show that wedge-type failure is controlled by azimuth and slope dip and dip direction. In the critical zone, azimuth-based discontinuities with NW and SW orientations result in intersection spots that cause wedge failure. The failure probability becomes very high when the slope angle is 65° and 70°; then, failure chances rise to 7% and 8%, respectively (Figure 6e,f). This indicates that as crucial intersection sites rise, they depend on the slope dip angle, which raises the chance of failure.
- (c)
- Toppling failure: From the results, it can be seen that toppling failure is more crucial in the right bank at any dip angle of the slope. When rock masses have a dominating discontinuity set (often bedding or foliation) with a strike almost parallel to the sloping surface and inward dip, rock slopes are more likely to topple [43]. Landslides mainly occur on dip slopes and sporadically happen elsewhere [14]. In addition, it is possible to decrease the slope angle because doing so lowers the weight of the material, enhancing the slopes’ stability [1]. From the kinematic analysis result of the right bank, it can be seen that toppling failure probabilities are slightly more than 5% when the slope dip angle is 40°, but they gradually increase as the slope dip angle increases, obtaining a maximum of 5.5% at a dip angle of 55°. At 65° and 70°, toppling exhibits the greatest values of 8.5% and 9.5%, respectively (Figure 4b). From Figure 3a,b, 54 discontinuities are oriented NE and SE, where the joints and slope dip in the opposing directions. In addition, most of the discontinuities dip with a high dip angle. This implies that toppling failure is a possibility. In comparison, when the slope angle is 45° and the failure chance is 5.5%, then crucial junction locations are relatively lower (Figure 6g). Critical interaction sites and failure probability both rise as the slope dip angle rises (Table 3). When the slope dip angle rises up to high value of 65° and 70°, then failure probability is slightly more than 8.5% and 9%. This result concurs with kinematic analysis, which indicates that NE–SW dipping discontinuities have a high potential for toppling failure. Moreover, the number of critical points also increases with the increase in slope dip angle (Figure 6h,i). Based on the azimuth and slope geometry of the right bank, it is evident that high failure probability of toppling exists with a high dip slope angle. In this condition, the slope is unstable and more likely to fail.

#### 4.2. Limit Equilibrium Analysis

#### 4.2.1. Left Slope

#### 4.2.2. Right Slope

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Different materials and instruments. (

**a**) Core sample extracted from core drilling machine. (

**b**) All prepared core samples from limestone and dolomite. (

**c**) Oven for dying the saturated samples. (

**d**) Vacuum saturator. (

**e**) RMT-150. (

**f**) Triaxial cell.

**Figure 4.**Failure probabilities of different failure mechanism on the (

**a**) left bank and (

**b**) right bank.

**Figure 5.**Dip analysis for the left bank of the highway: planar failure at (

**a**) 45°, (

**b**) 55°, and (

**c**) 70°; wedge failure at (

**d**) 45°, (

**e**) 55°, and (

**f**) 70°; and toppling failure at (

**g**) 45°, (

**h**) 55°, and (

**i**) 70°.

**Figure 6.**Dip analysis for the right bank of the highway: planar failure at (

**a**) 45°, (

**b**) 65°, and (

**c**) 70°; wedge failure at (

**d**) 45°, (

**e**) 65°, and (

**f**) 70°; and toppling failure at (

**g**) 45°, (

**h**) 65°, and (

**i**) 70°.

**Figure 7.**Slope stability analysis for the left bank of the highway: (

**a**) Bishop method under normal conditions, (

**b**) Janbu method under normal conditions, (

**c**) Spencer method under normal conditions, (

**d**) Bishop method under seismic conditions, (

**e**) Janbu method under seismic conditions, (

**f**) Spencer method under seismic conditions.

**Figure 8.**Slope stability analysis for the right bank of the highway. (

**a**) Bishop method under normal conditions, (

**b**) Janbu method under normal conditions, (

**c**) Spencer method under normal conditions, (

**d**) Bishop method under seismic conditions, (

**e**) Janbu method under seismic conditions, (

**f**) Spencer method under seismic conditions.

**Figure 9.**Factors of safety for different slope angles. (

**a**) Left bank slope under normal conditions, (

**b**) left bank slope under seismic conditions, (

**c**) right bank slope under normal conditions, (

**d**) right bank slope under seismic conditions.

Sample | Water Absorption (%) | Error (±) | Porosity (%) | Error (±) | Unit Weight (γ) | Error (±) |
---|---|---|---|---|---|---|

Limestone dry | 0.12 | 0.042 | 0.36 | 0.112 | 2.71 | 0.010 |

Dolomite wet | 0.23 | 0.020 | 0.66 | 0.055 | 2.82 | 0.011 |

Serial No: | Unit Weight (kN/m^{3}) | Cohesion (kPa) | Internal Friction Angle (°) | Deformation Modulus (MPa) | Poisson Ratio (MPa) | |
---|---|---|---|---|---|---|

Strong, weathered limestone | Natural | 26.8 | 350 | 38 | 25,000 | 0.28 |

Saturated | 27.1 | 300 | 35 | 23,000 | 0.31 |

**Table 3.**Summary of critical interaction points, left bank (LB), right bank (RB), planar (P), wedge (W), toppling (T), critical intersection (CI), total critical interaction (TCI).

Site | Failure Type | 45° | 50° | 55° | 60° | 65° | 70° | TCI |
---|---|---|---|---|---|---|---|---|

LB | P | 2 | 3 | 4 | 5 | 6 | 8 | 239 |

W | 593 | 907 | 1267 | 1571 | 1967 | 2488 | 28,426 | |

T | 2190 | 2366 | 2500 | 2667 | 2989 | 5773 | 28,426 | |

RB | P | 0 | 0 | 0 | 0 | 0 | 0 | 239 |

W | 599 | 780 | 1113 | 1509 | 1915 | 28,426 | 28,426 | |

T | 484 | 628 | 725 | 915 | 1059 | 1247 | 28,426 |

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## Share and Cite

**MDPI and ACS Style**

Rahman, A.U.; Zhang, G.; A. AlQahtani, S.; Janjuhah, H.T.; Hussain, I.; Rehman, H.U.; Shah, L.A.
Geotechnical Assessment of Rock Slope Stability Using Kinematic and Limit Equilibrium Analysis for Safety Evaluation. *Water* **2023**, *15*, 1924.
https://doi.org/10.3390/w15101924

**AMA Style**

Rahman AU, Zhang G, A. AlQahtani S, Janjuhah HT, Hussain I, Rehman HU, Shah LA.
Geotechnical Assessment of Rock Slope Stability Using Kinematic and Limit Equilibrium Analysis for Safety Evaluation. *Water*. 2023; 15(10):1924.
https://doi.org/10.3390/w15101924

**Chicago/Turabian Style**

Rahman, Aftab Ur, Guangcheng Zhang, Salman A. AlQahtani, Hammad Tariq Janjuhah, Irshad Hussain, Habib Ur Rehman, and Liaqat Ali Shah.
2023. "Geotechnical Assessment of Rock Slope Stability Using Kinematic and Limit Equilibrium Analysis for Safety Evaluation" *Water* 15, no. 10: 1924.
https://doi.org/10.3390/w15101924