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Article

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

by
Aftab Ur Rahman
1,
Guangcheng Zhang
1,*,
Salman A. AlQahtani
2,*,
Hammad Tariq Janjuhah
3,
Irshad Hussain
4,*,
Habib Ur Rehman
5 and
Liaqat Ali Shah
6
1
Faculty of Engineering, China University of Geosciences, Wuhan 430074, China
2
Department of Computer Engineering, College of Computer and Information Sciences, King Saud University, Riyadh 11543, Saudi Arabia
3
Department of Geology, Shaheed Benazir Bhutto University, Sheringal, Dir Upper 18050, Pakistan
4
Faculty of Electrical and Computer Engineering, University of Engineering and Technology Peshawar, Peshawar 25000, Pakistan
5
Gems & Jewellery Centre of Excellence, University of Engineering & Technology, Peshawar 25120, Pakistan
6
School of Architecture and Civil Engineering, Loughborough University Epinal Way, Loughborough LE11 3TU, UK
*
Authors to whom correspondence should be addressed.
Water 2023, 15(10), 1924; https://doi.org/10.3390/w15101924
Submission received: 19 March 2023 / Revised: 4 May 2023 / Accepted: 7 May 2023 / Published: 19 May 2023
(This article belongs to the Special Issue Safety Evaluation of Dam and Geotechnical Engineering)
Browse Figures
Versions Notes

Abstract

:
The purpose of this study is to identify the leading causes of slope instability along a local highway in Anhui, People’s Republic of China. As part of the east expansion project, the mountain range will be excavated to create a two-way, nearly 30 m wide highway. The site’s topography consists of a hill with palm-shaped faces carved from limestone running along its sides. The geological characteristics and slope stability of the research area highlight the possibility of slope failure along both sides of the roadway. Slope stability analysis was performed in order to determine the failure mechanism and create a stable slope. Initial slope characterization and shear properties of the rock were determined by means of fieldwork and laboratory analysis. By causing wedging failure and toppling collapse, the bedding joints and discontinuity orientations increase instability, as determined by a kinematic analysis performed with DIP.6 software. The Limit Equilibrium Method (LEM) of analysis is presented in the software Slide 6.020 to illustrate the instability of the slope. The unstable condition of the slopes was determined using empirical methods that were validated and enhanced by limit equilibrium analysis.

1. Introduction

Analysis of rock slope stability is required to plan safe excavations of slopes such as road cuts, mines, and railways to evaluate the stability of natural slopes. Especially in steep and mountainous regions, rock slopes are one of the most critical concerns in road construction. Distinct rock mass conditions, different discontinuities, shear zones, faults, thrusts, an unprofessional method of slope cutting, excessive rainfall, seismic activity, and tectonic activities make slopes more susceptible to instability problems [1,2,3,4]. Nevertheless, the most frequent types of slope failure are planar, wedge, rock topples, and rockfalls [5]. A complete geological analysis of the slope necessitates a detailed survey and research to assist the engineer in designing the appropriate support system and mitigating slope failures. A well-designed slope improves slope stability and safety, reduces expenses, increases mine life, and minimizes the stripping ratio [3,6].
Due to advancements in geology, we can understand such events, eventually enabling us to predict them so that the damage can be reduced. Modern technology and advanced techniques regarding slope stability have improved in recent years, but rock slope stability is still challenging for engineers. This is mainly due to the use of primitive techniques and the complexity of the rock slope considering the discontinuities influenced by joints, bedding, folds, faults, etc. Geological characteristics of rock mass include location and the number of joint sets, joint spacing, joint orientations, joint material, slope geometry, slope material strength, and seepage [7,8,9,10]. A rock slope can fail due to one or a combination of the different failure mechanisms. When the presence of pre-existing discontinuities dictates the instability, the failure will be in the form of plane sliding, wedge sliding, or toppling [11,12].
Many researchers around the globe have used various approaches for slope stability evaluation. Unlike deterministic analysis, which provides a single measure of the factor of safety (FoS), the kinematic analysis and limit equilibrium method can determine the probability of failure [8,13,14,15]. Several earlier studies have adopted probabilistic approaches to assess the stability of slopes. Probabilistic analysis is a technique for modeling the uncertainty and variation in parameters. The deterministic method minimizes the safety factor to determine the critical failure surface. In contrast, the probabilistic method allows measuring slope reliability based on the probability of failure and reliability index. The deterministic method is characterized by the high uncertainty of the variables considered in the analysis [16,17,18]. As a result, the probabilistic method allows for a new perspective on risk and reliability beyond the scope of traditional deterministic methods [13,18]. The factor of safety is normally expressed as a random variable. It can be substituted by the probability of failure in the assessment of slope stability. In contrast, the sensitivity analysis utilizes the maximum and minimum values of the parameter to determine the slope’s critical condition. Therefore, the relationships among the parameters and their effects on FoS can be evaluated [19,20].
With the development of computer technology, the finite element method has recently been a concern at home and abroad. When the slope reaches the failure state through the finite element strength reduction, the displacement on the sliding surface will change suddenly and produce a large and unlimited plastic flow. Discontinuities play a key contributing factor in slope stability. In addition, rock type, joint properties, geometry, water condition, and failure-plane characteristics are the key parameters that affect slope instability [9,14,16,21,22]. Moreover, assessing rock slopes is challenging to select the appropriate parameters for the stability assessment. Therefore, combining field data with a laboratory analysis approach leads to suitable consideration of the appropriate parameters [23].
Kinematics analysis is the most important method to investigate the slope instability and failure mechanism associated with landsides. The height of the slope also plays a role in the mode of failure; as the height increases, complex failures may occur. Kinematic analysis, the Limit Equilibrium Method (LEM), and numerical modeling are modern techniques widely used for slope stability analysis. To perform kinematic analysis and understand the failure mode, discontinuity orientation, joint sets, and bedding planes are key parameters that can be considered. Kinematic analysis is a very useful method to understand types of failure by using discontinuities and joint orientations [22,24,25]. Various techniques, including kinematic analysis, limit equilibrium analysis, rock mass classification system (SSPC system), and probabilistic analysis, among others, are available for the investigation of rock slope stability [26,27,28,29]. Kinematic analysis using the stereographic projection method is typically carried out before performing a detailed study in nearly all slope stability analyses [30,31,32]. Kinematic analysis is a geometric method that employs angular interactions between discontinuity planes to determine the possibility and failure types in a jointed rock mass [33].
In engineering analysis, numerous limit equilibrium techniques have been developed to analyze and design slopes in both soil mechanics and rock mechanics. It is founded on the idea of safety factors. To compute the global failure analysis, the factor of safety, and the failure surface, limit equilibrium analyses were carried out. Several approaches, including LEM, were used to assess the slope stability for the safety factor. The discrepancies in the benefits and limitations of each strategy must be emphasized. The LEM technique basically segments a soil/rock mass above the slip’s surface into a finite number of slices that can be segmented either vertically or horizontally [34,35]. Despite the widespread usage of traditional 2D modeling, it is generally believed that 2D slope stability analysis is inadequate when compared to 3D [20,36]. However, many researchers have conducted 2D slope stability analysis [8,9,13]. When using the limit equilibrium approach to calculate the forces pushing the rock mass and the opposing forces, the ratio of opposing forces to driving forces at equilibrium is known as the factor of safety (FOS) [11,37,38]. Numerous controlling parameters, including slope geometry, failure plane features, water forces, and external triggering factors, will have an impact on the FOS. Slope height has an inverse relationship with the FOS. Shear strains will increase as slope height (h) increases; as a result, FOS will decline [39,40].
The current study investigated the possible mode and potential of failure along the left and right slopes of a highway. Kinematic analysis tools were used to evaluate the potential failure of rock masses moving along geologic structures on the slope face. The analysis was conducted through DIPS software using graphical stereographic projection to examine the failure potential in both locations. In this study, the main factors affecting the mode of instabilities observed in the field are discontinuity orientation and bedding planes. The geometry of the slope, joints, and bedding plane was considered for the possible design and mode of failure.

Location and Geology of the Study Area

The study area is located in northeastern China in a moderately mountainous area (longitude 30.928194702590133, latitude 117.88073690223622) with an elevation of 35 m in Tongling city, Anhui province, P.R. China, as shown in Figure 1.
The province has different climates in the north and south, associated with geography. The seasons are more pronounced and more moderate towards the north. In January, temperatures are often between 1 °C and 3 °C, whereas in July, they are typically 27 °C or higher (Anhui Meteorological Department). The vegetation on the mountain is not developed, the overall terrain in the upper middle is high, and the two ends are low. The hillsides slope toward the inlet and outlet ends. The natural slope is 18°–26°. The Tongling area is a tectonically active zone with a low thickness of about 32 km. The Tongling district is located in the center of the Middle-Lower Yangtze River Block (MLYB) [41,43]. The dominant lithologies in this region are Silurian Middle Triassic marine sedimentary rocks. NE-trending fold structures are present and are accompanied by a series of NE-, NNE- NW-, and NNW-trending fault systems that control the emplacement of Mesozoic intrusions. The structures in this area include the NE-trending Qingshan anticline [42,44,45]. The sedimentary rocks exposed in the area range from Middle-Upper Silurian to Lower Triassic in age, but no Lower to Middle Devonian rocks are recognized. The main host rocks of the deposit are sandstone, limestone, and dolomitic limestone [46].

2. Materials and Methods

2.1. Field Investigation

The east extension project plan is to excavate the mountain body as a two-way lane, and the width of the road is nearly 30 m. The site terrain consists of a hill with limestone palm faces along both sides and a spherical peak. The planned route joins Tongjing Middle Road and Qifeng Road at their intersection in the west, crosses the northern foot of Qingshan Mountain in the east, travels past the deserted quarrying pond on the mountain’s eastern side, and finally joins Tongjing East Road. The total length of the road is about 2.2 km. It is 18.9~25.6 m from west to east. The project extends east along Tongjing Middle Road. The excavation section of the low mountain is about 600 m, the highest point of the terrain is 144 m, the relative height difference of the mountain is 130 m, and the north side cutting slope is 110 m. The 70 m south slope is shown in Figure 1b. Field investigation shows that the rock formation and structural surface in the northern foothills of Qingshan are well outcropped, the rock formation level is flat, and the joint surface is developed. The local rock mass is severely cut.

2.2. Laboratory Testing and Geotechnical Assessment

In order to assess the rock mass metrics and geotechnical features, fresh bulk samples were extracted and transported to the rock mechanics lab at the China University of Geosciences Wuhan. The cylindrical samples were prepared with dimensions of 100 mm in length and 50 mm in diameter. Both sides of the cylinders were flattened to get a uniform outcome and characteristics across all samples. All samples were drilled perpendicular to the depositional layers in a pattern in which no apparent fractures were observed (Figure 2a,b). All samples were dipped in the water tank before being put in a vacuum saturator to absorb more water and become thoroughly saturated (Figure 2d). Each sample mass was evaluated every 24 h until it reached complete saturation. Throughout every measurement, the saturated mass of each specimen remained unchanged. Therefore, reaching a completely saturated condition in rock is challenging. After calculating the saturated mass, samples were baked for 24 h at 120 °C (Figure 2c). To assess water absorption, the difference between dry and wet mass is divided by the rock dry mass, according to the Chinese rock mass testing standard. The calculated data for physical properties are listed in Table 1.
Before the triaxial test, cylindrical samples were subjected to a drying and wetting procedure. All specimens were triaxially tested to determine the result (Figure 2e). Each sample was placed in a triaxial cell and subjected to 0.20 KN/s of steady vertical pressure until failure (see Figure 2f). ISRM-recommended laboratory testing methods were followed [40]. Figure 2e depicts an RMT-150 rock mechanics testing machine used to perform triaxial tests. These experiments were done at the Chinese Academy of Sciences, Rock and Soil Mechanics Laboratory in Wuhan, China.

2.3. Strength Parameters

According to the Mohr-Coulomb technique, one of the most crucial factors affecting the stability of rock slopes is the shear strength of the discontinuities, which may be computed using the cohesion and friction angle parameters [47]. After metamorphism, the rock is a marbleized limestone and dolomite unit with fine grain structure, a grey-black hue, and thin layers. For limestone, porosity, unit weight, and water absorption were estimated. The test was conducted in the rock mechanics lab at the China University of Geosciences in Wuhan. To assess water absorption, the difference between dry and wet mass is divided by the rock dry mass, according to Chinese rock mass testing standard. The interpretation of stress-strain characteristics (maximum (σ1), minimum (σ3) 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

Kinematic analysis was conducted to highlight the possibility for several types of rock slope failures that arise due to adversely oriented discontinuities [5]. The rock slope moments and likely failure direction were calculated and anticipated using kinematic analysis along both slopes. Three frequent failure modes (planar, wedge, and toppling) that result from adversely oriented discontinuities and bedding planes were examined [11]. The required parameters obtained from the field through geological survey and geotechnical investigation are summarized.
In order to evaluate the slope kinematics, the discontinuity parameters acquired from the scan-line survey that was conducted for each of the analyzed slopes were entered into the DIPS program, which is shown in Figure 3 [11,24,48]. The failure potential and cause, impacted by the numerous slope angles, exiting discontinuities, and bedding planes, were calculated using an arbitrary slope dip angle of 40° to 70°. For this purpose, the slope was modeled to the right and left banks with different slope angle.
To compute the global failure analysis, the factor of safety, and the failure surface, limit equilibrium analysis was carried out. A Slide 6.020 numerical model was utilized to simulate how the material would react to stress, strain, and shearing as the road was built. To better understand the mechanism of weak rock instability brought on by highway excavation, two-dimensional cutting slope models were created. The slopes have an excavation ratio of 130 m long and 130 m high. The model still requires the creation of borders, mesh generation, boundary conditions, addition of traction, field stress, normal condition, seismic condition, and simulation at the end. These studies employed the Bishop, Janbu, and Spencer four-slice methodologies.

3. Results

Plane failure, wedge failure, and toppling failure are the three basic types of failure that occur along rock discontinuities in hard rock slopes [5]. This study sought to determine how geological structures affected the frequency of rock failure along a road in a hilly area with several geological characteristics crossing it. Geological and geotechnical approaches were used to investigate the issue. Kinematic analysis (Stereo-net plots) is one of the geotechnical techniques to determine the failure mechanism (wedge, planar, and toppling analysis) and critical slope failure.
Three main failure mechanisms were examined on both sides of the road using the orientation data and kinematic analysis generated by Dips 6.0 software. We tabulated the rock discontinuity data collected from the sloped site and grouped the data sets based on their similarities. The kinematic analysis findings and the actual field situation agree rather well. Field measurements and observations revealed a joint-controlled toppling and wedge-type failure. The data also show that failure is more critical and likely to occur along the left bank than the right. The result shows that when the slope angle is low, the critical intersection points in the critical zone are lower, while with an increase in slope dip angle, the number of critical intersections also increases (Table 3). The kinematic study of the left bank makes it evident that the toppling mode has been tested with a higher failure probability than the planar and wedge failure.
Overall, from the right bank, it is clear that three forms of failure display rising failure possibilities with increasing dip slope angle, with toppling and wedge failure exhibiting higher failure probabilities than planar (Figure 4a). On the other hand, it can be seen from the left bank that toppling failure and wedge failure both exhibit a pattern of increasing the likelihood of failure and are more problematic, although toppling failure is more prevalent than wedge failure. Additionally, planar failure exhibits very low odds (Figure 4b).
The toppling mode was tested with a higher failure probability than the planar and wedge failure, as shown by the kinematic analysis of the left bank (Figure 5a). According to the results, the likelihood of a planar failing is slightly higher than 1% when the slope dip angle is 40°, but it increases to 1.6% when the angle is 5°. The number of critical failures is 2 (Figure 5a). Also, when the slope dip angle is 55°, 60°, and 70°, failure chances rise to 2.5%, 2.9%, and 5%, respectively. The chance of wedge failure increases to 10% at a slope angle of 55°, as shown in Figure 5d, from 5% at 45° (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. In total, 34 joints have a dip angle of 80°, whereas 63 joints dipped with an angle greater than 70° (Figure 2a). The toppling failure is the rotation of a block or stone column located on a sloping surface, around a point [49]. As demonstrated in Figure 4a, toppling failure has a higher failure probability. Failure probabilities are just below 3% when the slope dip angle is 40°, but they rise somewhat with slope dip angle and can reach up to 4.5% when the dip angle is 55° (Figure 5g,h).
According to the findings from the right bank, the likelihood of a planar failing is 0% at a slope angle of 40°, while the same percentages are repeated at a slope angle of 60°. When the slope dip angle is 65° and 70°, the failure chances increase slightly with values of 1%, which is very low (Figure 4b). The right bank wedge failure study yields nearly identical results as the left bank. Although the odds of failure were lower for a modest dip angle, they rose as the slope dip increased (Figure 4b). The outcome shows that the slope dip angle of 45° had a very low failure chance of 2% (Figure 6d). According to the right bank kinematic analysis results, toppling failure probability is somewhat more than 5% at a slope dip angle of 40°, but gradually increases 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).
To simulate the impact of road building and geological characteristics concerning rock mass displacement, the Finite Element Method (Slide 6.020 software) was utilized. Because of the highly jointed and varied joint spacing of the rocks along the research area road, different-sized blocks can develop. Using a Brunton compass, more than 150 bedding joints and discontinuities with dip/dip directions were measured at each location. The results show that the calculated factor of safety in normal conditions for the Bishop, Janbu, and Spencer method is 1.128, 1.105, and 1.126, respectively (Figure 7a–c). This result of the left bank indicates that the slope is stable when the slope angle is 60° or less, but the slope becomes vulnerable to failure when the overall slope angle increases to 65° and 70°. However, the slope becomes more unstable when the ground acceleration is applied. Along with this increasing slope angle, the FoS value decreases, which ultimately increases the failure probability (Figure 9).

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

The limit equilibrium method (LEM) was used to determine the factor of safety (FoS) and evaluate the surface failure along both sides of the highway slopes. The primary method used with the Slide 6.020 application to assess stability is this analytical methodology for failure surfaces and factor of safety in normal and seismic conditions is shown in Figure 8. For a variety of limit equilibrium approaches, including the simplified Bishop method, the simplified Janbu method, and the simplified Spencer method, the FoS values using the Mohr-Coulomb criteria were computed. Using this technique, forces that lead to rock mass instability, as well as resistant forces are examined, and the resistance to driving force ratio (FoS) is calculated [11,24,42,45]. In a 2-dimensional analysis, a slice that is one unit thick is thought to represent a typical piece of the slope. Previous studies have demonstrated that the limit equilibrium approach provides acceptable results for engineering applications [5,26].

4.2.1. Left Slope

In a deterministic investigation of the left bank slope using LEM under typical circumstances, several FoS values were obtained. Two rock slope stability scenarios were examined: one under typical conditions and the other under peak ground acceleration (PGA). The results show that the calculated factor of safety in normal conditions for the Bishop, Janbu, and Spencer methods is 1.128, 1.105, and 1.126, respectively (Figure 7a–c). This result of the left bank indicates that the slope is stable when the slope angle is 60° or less, but the slope becomes vulnerable to failure when the overall slope angle increases to 65° and 70°. [52] revealed that movement of the anti-dip slope can be influenced by the slope angle when it increased from 60° to 70° while the dip angle of the strata is kept fixed. According to [50], the prior collapse at the quarry was mostly caused by steep slope angles. The Bishop and Spencer technique values are close to 1.7, whereas the Janbu FoS value with its lowest slope angle of 45° is slightly less than 1.6. Higher safety factor values appear with a decreasing slope angle, indicating a reduced chance of failure (Figure 9a). When the slope angle is kept lower, according to LEM analysis of these methods, the left bank is stable. When the slope angle rises to 60° or 70°, however, the slope becomes highly unstable and has a low value of FoS.
However, for the seismic condition with ground acceleration of 0.25, the values of safety for Bishop, Janbu, and Spencer decrease by 0.759, 0.715, and 0.758, respectively (Figure 7d–f). In addition to this, a small change in slope angle can also change the factor of safety, which ultimately affects the slope stability. The FoS of the slope with various slope angles and a peak ground acceleration (PGA) of 0.25 are shown in Figure 9b. The lowest FoS values are 0.715 for the Janbu approach, and 0.759 and 0.758 for the Bishop and Spencer methods, respectively. These results lead to slope vulnerability, which makes it more likely to collapse at any time under these circumstances. In contrast, it can be seen that, due to a decrease in slope angle with PGA of 0.25, the FoS value rises. The FoS value for Janbu, with the lowest slope angle of 45°, is slightly higher than 1, while the values for the Bishop and Spencer methods are somewhat higher than 1.1. This indicates that failure probability increases, and the slope is more unstable (Figure 9b).
However, for the seismic condition with ground acceleration of 0.25, the values of safety for Bishop, Janbu, and Spencer decrease by 0.759, 0.715, and 0.758, respectively (Figure 7d–f). In addition, a small change in slope angle can also change the factor of safety, which ultimately affects the slope stability. The FoS of the slope with various slope angles and a peak ground acceleration (PGA) of 0.25 are shown in Figure 9b. The lowest FoS values are 0.715 for the Janbu approach, and 0.759 and 0.758 for the Bishop and Spencer methods, respectively. These results indicate the slope vulnerability, which makes it more likely to collapse at any time under these circumstances. In contrast, it can be seen that, due to a decrease in slope angle with PGA of 0.25, the FoS value rises. The FoS value for Janbu, with the lowest slope angle of 45°, is slightly higher than 1, while the values for the Bishop and Spencer methods are somewhat higher than 1.1. This indicates that failure probability increases and the slope is more unstable (Figure 9b).

4.2.2. Right Slope

The LEM analysis for right bank resulted in very low factor of safety (FoS) values. In normal conditions, the FoS values for the Bishop and Spencer methods are 1.179 and 1.168, respectively (Figure 8a,c), whereas the FoS for the Janbu method is 1.092, which is comparatively very low (Figure 8b). The results demonstrate that according to the Bishop and Spencer methods, the right slope is stable, while the Spencer method indicates that it is fairly stable when the slope angle is 60°. From Figure 9c, it can be seen that when the slope dip angle decreases, FoS rises. When the slope angle is kept at 45° and 50°, then the values of FoS are 1.3 for all three methods. This indicates that the rock slope is stable when the dip angle is kept lower. However, the slope becomes extremely unstable and has a low value of FoS as the slope angle increases to 60° or 70°.
The computed safety factor for Bishop, Janbu, and Spencer methods decreases with 0.25 ground acceleration in seismic circumstances. The right bank is unstable and at risk of failure at any slope dip angle, while the Bishop and Janbu methods calculate the FoS as 1 when the slope dip angle is 45°; however, the slope is not stable and can fail at any time. Figure 8b shows that the lowest value of FoS for the Janbu method is 0.706, which is very low. On the other hand, the FoS values for the Bishop and Spencer methods are 0.808 and 0.183, which are comparatively higher than the Janbu result, but lower than 1 which indicates a very unstable condition of the slope (Figure 8a,c). In seismic conditions, the FoS of the slope with various slope dip angles and a peak ground acceleration (PGA) of 0.25 is shown in Figure 9d. In this condition, it is observed that the slope is still unstable when the slope angle decreases and shows chances of failure. In addition, the FoS value is slightly less than 0.7 and 0.6 when the slope dip angle increases up to 60° and 70° with PGA of 0.25. This value indicates that the slope becomes very unstable and the chances of failure are higher.

5. Conclusions

The purpose of this study was to investigate potential failure and mode along the left and right slopes of an embankment. Kinematic analysis techniques were used to evaluate the potential for failure of rock masses moving along geologic structures on the slope face. The geometry of the bedding plane, joints, and slope were considered for both design and failure scenarios. The failure potential at both sites was investigated utilizing the graphical stereographic projection of the DIPS software. The two main factors in this study that affect the type of instabilities seen in the field are bedding planes and discontinuity orientation. The results are drawn from the field visit and the slope stability calculations conducted using kinematic and finite element methods. Based on kinematic analysis, the rock slope is stable when the slope angle is lower than 60°, while the failure probabilities are enhanced with the rise in slope dip angle up to 70°.
In the same way, wedging and toppling failures exist even when there is no expected planar collapse. According to the analysis, on the right and left sides, toppling failure has a higher probability than wedge failure. On the other hand, the Limit Equilibrium Method (LEM) output shows that the slope is critical to failure with a large slope dip angle and stable when the slope dip angle is reduced. In normal conditions, the slope is stable on both sides with factors of safety (FoS) more than 1. When the peak ground acceleration (PGA) reaches 0.25, the slope is extremely unstable, and the factor of safety (FoS) is less than 1.

Author Contributions

Conceptualization, L.A.S.; Data curation, H.U.R.; Formal analysis, H.T.J. and L.A.S.; Funding acquisition, S.A.A. and I.H.; Investigation, A.U.R., G.Z. and H.U.R.; Methodology, A.U.R., G.Z., H.T.J. and H.U.R.; Project administration, G.Z., S.A.A., I.H. and L.A.S.; Resources, S.A.A. and I.H.; Software, A.U.R., S.A.A., H.T.J. and I.H.; Supervision, G.Z., S.A.A. and I.H.; Validation, G.Z. and H.T.J.; Visualization, A.U.R. and I.H.; Writing—original draft, A.U.R.; Writing—review & editing, A.U.R. and G.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Research Supporting Project Number (RSPD2023R585), King Saud University, Riyadh, Saudi Arabia.

Data Availability Statement

The dataset and the samples for this study are available from the corresponding author upon reasonable request.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Maps of the study area. (a) Geological map of Tongling district, Anhui province, modified after [41,42]; (b) cross-sectional area of the highway to be constructed.
Figure 1. Maps of the study area. (a) Geological map of Tongling district, Anhui province, modified after [41,42]; (b) cross-sectional area of the highway to be constructed.
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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 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.
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Figure 3. Quantitative data of discontinuities and bedding joints: (a) dip (b) dip direction.
Figure 3. Quantitative data of discontinuities and bedding joints: (a) dip (b) dip direction.
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Figure 4. Failure probabilities of different failure mechanism on the (a) left bank and (b) right bank.
Figure 4. Failure probabilities of different failure mechanism on the (a) left bank and (b) right bank.
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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 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°.
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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 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°.
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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 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.
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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 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.
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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.
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.
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Table
Table 1. Physical Properties of Limestone.
Table 1. Physical Properties of Limestone.
SampleWater Absorption (%)Error (±)Porosity (%)Error (±)Unit Weight (γ)Error (±)
Limestone dry0.120.0420.360.1122.710.010
Dolomite wet0.230.0200.660.0552.820.011
Table
Table 2. Shear Strength Parameters of Limestone.
Table 2. Shear Strength Parameters of Limestone.
Serial No:Unit Weight (kN/m3)Cohesion (kPa)Internal Friction Angle (°)Deformation Modulus (MPa)Poisson Ratio (MPa)
Strong, weathered limestoneNatural26.83503825,0000.28
Saturated27.13003523,0000.31
Table
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).
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).
SiteFailure Type45°50°55°60°65°70°TCI
LBP234568239
W593907126715711967248828,426
T21902366250026672989577328,426
RBP000000239
W59978011131509191528,42628,426
T4846287259151059124728,426
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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

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