#### Abstract

A slope is a complex engineering geological body that contains many uncertainties. In the present study, the high rock slope of Duimenshan in Gejiu, China is taken as the research object, and the Monte Carlo method is used to perform the analyses. Firstly, based on a large number of rock mass discontinuities, the Fuzzy C-Means (FCM) clustering algorithm is used to determine the dominant discontinuities. Secondly, the affecting parameters of the rock mass such as cohesion, internal friction angle, horizontal seismic acceleration coefficient, and water-filled depth coefficient are taken as random variables, and the failure probability of the slope and the failure index corresponding to different slope heights and slopes are obtained as the output parameter. The obtained results show that the target reliability index of the slope in a certain range of the slope height is far from the safety value, indicating that the slope is unstable, and the combination of slope height and slope angle affects the reliability of the slope. More specifically, as the slope height and angle increase, the number of samples with a stability coefficient of less than 1 increases, thereby increasing the failure probability of the slope indicating that the slope is stable. It is found that unless the slope height is small, it is the main factor of stability. However, when the slope height exceeds 33 m, the effect of the slope inclination on the stability increases gradually.

#### 1. Introduction

Slope Engineering is a complex engineering geological body that deals with many complex problems such as uncertainties of physical and mechanical parameters of rock and soil mass and the space-time variability, the complexity, diversity, and randomness of external factors such as rainstorm [1], earthquake [2], groundwater [3], load [4], artificial blasting [5], and karstification [6]. Accordingly, stability analysis of slope engineering is a very challenging task. Studies show that relying only on a single index or analysis method may lead to inaccurate results that cannot be used in engineering applications [7, 8].

The rigid body limit equilibrium method, which is mainly used in slope stability quantitative analysis, is based on the factor of safety method and cannot objectively consider uncertainty and variability of rock mass parameters. In this method, ignoring the effect of these factors may lead to inaccurate and inconsistent results, thereby increasing potential security risks. Therefore, it is of significant importance to introduce the reliability analysis method into slope engineering [9–11]. Compared with nonphysical assumptions and theories in mathematical analysis methods, numerous physical parameters, including the structural characteristics of the slope, parameters of the rock mass discontinuity, rainfall, earthquake, and groundwater are regarded as random variables, and then the random variables are expressed by rational functions. Consequently, this method can be effectively applied to solve the internal uncertainty and randomness of rock and soil slope. Accordingly, the limit equilibrium method can be applied to obtain more comprehensive and objective results.

Currently, the common slope reliability analysis methods are the Monte Carlo method [12–17], response surface method (RSM) [18–20], Latin hypercube sampling (LHS) multidimensional stratified sampling method [21–24], and first-order second-moment method (FOSM) [25–28]. In this regard, the Monte Carlo method is widely used in financial engineering, biomedicine, economics, computational physics, and geotechnical engineering. Since the convergence rate of this method is not restricted by the dimension of random variables and the function complexity, it can be applied to quickly and accurately solve the problem and determine the simulation error. It can be seen that Monte Carlo method has many advantages, and it can be seen from previous studies that applied research has made more achievements. But in the field of slope engineering, how to introduce the information of structural plane and other random variables into the application of this method more accurately will become an important link whether the analysis method is reliable or not; few studies have done sourcing circumstances analysis of random variables. Therefore, in this study, the high rock slope in the construction site of the heavy metal pollution treatment demonstration zone in the Gejiu area of Yunnan Province was selected as the research object, and the dominant structural planes of the slope were determined by using the Fuzzy C-Means (FCM) clustering algorithm, and then the reliability of the high rock slope was analyzed by using the Monte Carlo method; it provides the basis for the permanent treatment of the slope and the safety guarantee for the construction and operation of the demonstration test area.

#### 2. Geological Condition of the Slope

##### 2.1. Topography

The landform of the slope area belongs to the karst fault block of high and middle mountains with scarce vegetation and up-steep and down-gentle characteristics. The natural slope at the top of the excavation slope is in the range of 5°-10°. The slope is steeper near the ridge, and the maximum slope is 35°. The slope is located in the west section of the north side of the proposed site. The total length and the maximum height in the middle and two sides of the excavated slope are 460 m, 50 m, and 5-10 m, respectively. Moreover, the excavation slope is in the range of 80°-85°.

##### 2.2. Formation Lithology

Based on the engineering exploration, the main strata lithology distributed in the study area are Quaternary Plant Layer (), quaternary slope eluvium () that contains gravelly silty clay, and underlying bedrock strata () that contain limestone of the Middle Triassic Gejiu formation. According to the distribution of rock mass, borehole data, and geological data of the slope, the engineering geological rock group of the slope can be divided into the following four main categories and two subcategories: (1) strong weathered limestone with relatively weak rock groups; (1.1) strong weathered medium-thick limestone with relatively weak rock groups; (1.2) strong weathered thin limestone with relatively weak rock groups; (2) thin bedded limestone hard rock group; (3) medium-thick bedded limestone hard rock group; and (4) middle-thick bedded limestone, which is harder than hard rock groups. Moreover, Figures 1 and 2 illustrate that the slope can be divided into five areas according to the engineering geological rock group. Table 1 shows the shear strength of each rock group.

##### 2.3. Geological Structures

The study area is located in the south of Gejiu. The main regional affecting faults include the Red River deep fault in the south and the east-west Kaofang fault in the west, which is 1.5 km away. Other regional faults are relatively far from the study area and can be ignored. Figure 3 shows that there is a normal fault (f1) in the southeast section of the slope with an attitude of 60°∠60°. The strike of the fault is nearly perpendicular to the strike of the slope. Moreover, the traction structure is developed in the rock strata of the two sides of the fault, the rock mass is broken, and the fault fracture zone is obvious.

Figure 2 shows the elevation diagram of the slope indicating that the whole direction of strata is to the northwest, the direction of tilt is to the southeast, and the slope structure is of oblique type. The dip angle of rock strata varies greatly, which may be attributed to faults and local compressions. In the northwest section of the slope, the bed attitude is 40°-45°∠70°-120°. The attitude of the rock layer in the middle section of the slope is relatively stable, and the bed attitude is 40°-45°∠115°-145°. The southeast section of the slope is located in the footwall of the fault, and the traction deformation of the local rock strata changes the slope structure to a consequent type slope, which is detrimental to the slope stability.

##### 2.4. Hydrogeological Conditions

According to the difference between strata and lithology and the form, space, and hydraulic characteristics of the groundwater, the groundwater in the study area can be mainly divided into two categories, including pore water and karst water. It is worth noting that there is no underground water in the borehole and no underground spring in the vicinity.

#### 3. Determination of Dominant Structural Plane Based on the Fuzzy C-Means (FCM) Clustering Algorithm

According to the performed field survey, the high rock slope is 480 m long. The detailed survey line method is used to investigate the structural plane of rock mass with a length of 13.00 m and 164 joints. The main characteristics of the rock mass are presented in Table 2.

Dunn [29] proposed the Fuzzy C-Means (FCM) algorithm as a clustering scheme based on an objective function. The core idea of this algorithm is to find the minimum objective function by updating the clustering center and membership function. This algorithm has a simple design and wide applications that are widely used as an effective fuzzy cluster analysis method.

The main steps of the FCM algorithm can be summarized as follows: define and normalize the parameters of the structural plane in the computer program, define the initial conditions and set an appropriate number of clusters K in the range of 2~9, and set the convergence precision, the number of iterations , and the fuzzy weighting index . Then, compare the difference of cluster centers under different grouping conditions to check whether the number of iterations has reached or . The values of the validity test of silhouette under different grouping are analyzed to determine the best grouping number and cluster center. Figure 4 illustrates the flow diagram of the FCM algorithm.

Figures 5 and 6 show the plane diagram of the occurrence distribution of the structural plane and the density diagram of the structural plane poles of the slope in the original state. It should be indicated that it is an enormous challenge to classify and analyze these poles using conventional methods. Based on the mathematical statistics function of the FCM algorithm, a cluster analysis is carried out on the five parameters of 164 groups of structural plane parameters. These parameters are dip angle, direction of tilt, ductility, opening degree, and roughness. The number of clusters is 2~9, and the total number of iterations is set to 800.

The cluster validity test value (Table 3) and the cluster validity grouping effect diagram of 3~5 groups (Figure 7) were obtained through iterative calculations. It is observed that when the dominant structure plane is divided into 4 and 5 groups, a negative number of appears, which is not consistent with the interval of the objective function. Moreover, when there are 3 groups, reach the minimum value and negative numbers disappear. Accordingly, the optimal grouping number of dominant discontinuities in the rock mass is 3 groups.

**(a) Three-group classification**

**(b) Four-group classification**

**(c) Five-group classification**

After determining the optimal grouping of dominant discontinuities using the FCM clustering algorithm, the final data in a rock mass can be obtained according to the radian transformation of the data of the cluster center (Table 4). Then DIPS software is utilized to perform the FCM algorithm, and the obtained contour of the structural plane pole is presented in Figure 8. Compared with Figure 6, the distribution of the structural plane of rock mass after clustering analysis is more clear, and the desired result is achieved.

#### 4. Determination of Objective Reliability of the Slope

The objective reliability (also called reliability design) and the acceptable risk of slope design can meet the requirements of slope engineering design. So far, no uniform criterion of slope reliability has been proposed in this regard. To resolve this shortcoming, the calibration method and the analogy method are applied to determine the target reliability index. It is intended to refer to the current standards for reliability indicators, analyze the key points, and refer to the reliable design codes such as construction slope, railway, highway, water conservancy, and hydropower slope. The calibration method not only inherits the current design standard but also fully reflects the long-term practical experience of experts in engineering construction. Therefore, this method has been widely adopted worldwide.

##### 4.1. Slope Service Life

According to “The unified standard for reliability design of engineering structures” (GB50153-2008), designing the service life is the basic prerequisite to satisfy the reliability of engineering structures. Therefore, it is necessary to determine the design life (service life) before determining the target reliability of slope engineering.

According to the “The technical code for Building Slope Engineering” (GB50513-2013), the design service life of a building slope should not be less than the design service life of a protected building (structure). Based on the safety grade table (Table 5) and the design life table (Table 6) of “Building structures specified in the unified standard for reliability design of building structures” (GB50068-2018), the proposed buildings in the slope project (Table 7) and the service life of the buildings in this area are determined to be 50 years.

Table 8 shows the service life table of slope engineering indicating that the service life of slope should be higher than 50 years.

##### 4.2. Slope Safety Grade

The classification of slope safety is mainly affected by the slope height and hazard degree, which can be obtained according to “Unified Standard for Reliability Design of Engineering Structures” (GB50153-2008), “Technical Code for Building Slope Engineering” (GB50513-2008), and “Slope Design Code for Water Resources and Hydropower Projects” (SL386-2007). In this regard, the obtained results are presented in Tables 9 and 10.

##### 4.3. Target Reliability of the Slope

Currently, there is no unified standard for the reliability design of slope engineering. However, the studied slope is of building type slope. Therefore, according to “Unified Standard for Reliability Design of Building Structures” (GB50068-2018), “Unified Standard for Reliability Design of Engineering Structures” (GB50153-2008), and the conventional standards about reliability in railway, highway, water conservancy, hydropower, and port engineering, an inductive analysis can be carried out [30]. In this regard, the obtained results are presented in Tables 11 and 12.

Since the studied slope is a high rock slope with a height of more than 30 m and referred to item 3.2.5 of “Unified Standard for Reliability Design of Engineering Structures” (GB50153-2008) stipulates that “the value of reliability index should be 0.5 for every grade difference of safety level”; the target reliability index of the slope is determined in Table 13.

#### 5. Reliability Analysis of Slope Based on MCS

Considering uncertainty and complexity of affecting factors on the stability of slope engineering, the Monte Carlo simulation (MCS) method is applied to analyze the reliability of slope engineering. In this method, the influence of many uncertain factors is considered in the calculations. The slope reliability which determines whether the slope function can meet the design and use requirements within a certain period of time, that the larger the reliability index, the lower the slope failure probability.

The basic idea of the applied reliability analysis method can be summarized as follows: based on the Monte Carlo method, the slope height , slope angle , cohesion , internal friction angle , and the horizontal seismic acceleration coefficient of rock slope are taken as random variables, and then the stability coefficient corresponding to random samples is calculated by solving the problem in MATLAB environment. Further statistical analysis of the number of failure samples in random samples (i.e., the number of samples with ) is carried out, and then the target failure probability of slope and the statistical characteristics of rock mass parameters are calculated. Finally, the slope reliability is evaluated [31–33].

##### 5.1. Specific Implementation Steps

Figure 9 shows the flowchart of the reliability analysis of the rock slope using the Monte Carlo method.

Similar to Hoek [34] and Wang et al. [35], it is assumed that the rock slope is composed of a single unstable block. In other words, the failure mode is simplified as a planar two-dimensional rock mass sliding failure. Figure 10 shows the slope calculation model, and the mathematical expression is presented.
where , , , and denote the water pressure on the sliding surface, unit width area of the sliding surface, water pressure on the tensile crack surface, and the dead weight of the sliding body, respectively. Moreover, and are horizontal seismic acceleration coefficients and slip surface dip angles, respectively. The gravity of the rock mass and water is 27 kN/m^{3} and 10 kN/m^{3}, respectively, and the dip angle of the sliding surface is 45°. Moreover, is the filling depth of fissure water, which is the product of and .

The slope is 75~85°, the height of the slope is 5~50 m, and the dip angle of the sliding surface is about 45°. Since cracks are mainly concentrated in the range of 25~50 m slope height, the simulation is focused on this range. The discrete random variables of rock mass parameters with a normal distribution include the crack depth , cohesion , internal friction angle , and horizontal seismic acceleration coefficient , while the water filling depth coefficient in the fracture follows a truncated exponential distribution. In this regard, 26 and 11 possible values are considered when the slope height and the slope angle are dispersed at 1 m and 1° equal intervals, respectively. In total, 286 cases are considered in the present study.

Based on the determination results of rock mass parameters, statistical analysis is made, and the distribution of random variable strength parameters of the slope is determined. Table 14 shows the horizontal seismic acceleration coefficient varying from 0 to 0.16, and the depth coefficient of water filling in cracks ranges from 0 to 1.

##### 5.2. Statistical Analysis

Since the slope reliability index is in one-to-one correspondence with failure probability, the slope target failure probability and the minimum sample number required for simulation are set to and , respectively, to ensure the calculation accuracy. The simple statistical analysis of the random variables yields the rectangular statistical Figures 11–14.

Figures 11–14 reveal that among the random variables, the number of samples of slope height and slope angle in each interval is constant, obeys a discrete and uniform distribution, and the number of samples in equal intervals is and , respectively. However, the distribution of two random variables, internal friction angle and cohesion , roughly accords with the standard normal distribution curve indicating that these variables have a normal distribution in the range of 16~37°and 120~240 kPa. Analysis results can be summarized as follows: (1)Overall reliability analysis of the slope:

After demonstrating the rationality of the input random variables, the slope stability is calculated and analyzed in MATLAB environment. In order to improve the calculation accuracy, the slope stability was calculated and statistically analyzed three times, and the obtained results are presented in Table 15 and Figure 15

Figure 15 shows that the stability coefficients are mostly concentrated in the interval of 0.6~2, which reflects the rationality of the input random variables and the stability of the performed calculations. Moreover, Table 15 reveals that the values of and the corresponding failure probability of three groups of slope failure samples are very close, and the average value of the overall failure probability of the slope within the range of 25~50 m is 0.1477, which is much higher than the design failure probability of the slope (). It is concluded that the failure probability of the slope functional structure is extremely high. To sum up, as far as the slope is concerned, the target reliability requirements of the first-class building slope cannot be achieved, and the slope is unstable so it is necessary to take timely treatment measures. (2)Reliability analysis under different combinations of slope heights and angles:

The overall failure probability of slope is obtained after the failure probability of slope corresponding to different combinations of slope height and slope angle is statistically analyzed. Figure 16 shows the statistical chart of stability safety coefficient corresponding to different combinations of slope height and slope angle . It is found that when the stability coefficient in a certain combination state is greater than 1, the slope is stable. Moreover, Figure 16 indicates that as the slope height and slope angle increase, more samples with a stability coefficient of less than 1 are achieved and the probability of slope failure increases, and the slope becomes unstable.

In order to reflect a specific failure probability value under 286 combinations of slope heights and angles, the failure probability value in a specific case was obtained through the calculation formula in literature [31–33] and the results are shown in Figure 17. It is observed that when the slope height is about 25 m, variations of the slope angle slightly affect the failure probability of the slope, indicating that until the slope height is small, it is the main influencing factor of stability. Moreover, when the slope height exceeds 33 m, the influence of the slope inclination on the stability gradually increases. The failure probability values of 249 combinations are obtained within the slope height range of 25~50 m. It is found that less than 87% of the combinations with different slope heights and angles are in the state of functional structure failure.

#### 6. Conclusions

Based on the MCS method and the principle of limit equilibrium analysis, the cohesion , internal friction angle , horizontal seismic acceleration coefficient , and water-filling depth coefficient in rock mass parameters are considered random variables, and the overall failure probability of slope and failure indices corresponding to different slope heights and slopes are calculated numerically. Accordingly, it is found that the target reliability index of the slope is far from the safe value within a certain range of slope height, and the slope is in a state of instability with a high probability. Aiming at performing timely measures, the main achievements can be summarized as follows: (1)FCM-based clustering algorithm has a promising efficiency in analyzing a large number of random rock mass structural planes and can obtain ideal grouping results. Accordingly, this algorithm can provide a solid foundation for subsequent slope stability analysis and can be applied to determine the dominant structural planes of multiparameter rock mass structural planes(2)The overall design service life of the slope has a high probability of functional failure, indicating that the overall slope is in an unstable state. Therefore, it is necessary to treat the slope in time. To this end, the Monte Carlo method is introduced to calculate the overall failure probability of the slope. The obtained results show that the overall failure probability of the slope is 0.1477, which is much higher than the target failure probability of the slope(3)The combination of slope height and slope angle affects slope reliability. More specifically, as slope height and slope angle increase, more samples with a stability coefficient of less than 1 appear, and the failure probability of the slope increases, indicating that the slope is unstable. Until the slope height is small, it is the main influencing factor. However, when the slope height exceeds 33 m, the influence of the slope inclination on the stability gradually increases and 87% of the combinations with different slope heights and angles fail

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare no conflict of interest, financial or otherwise.

#### Authors’ Contributions

A. Fayou and Wan-cheng Pan have contributed equally to this work.

#### Acknowledgments

The authors acknowledge the National Natural Science Foundation of China (Grant: 42267020), the Project of High-quality Course Construction for Graduate Students of Yunnan Province in 2020(numerical simulation of geological engineering), and the Key Research and Development Program of Yunnan Province in 2022(202203 AC100003).