Estimation of Shear Strength Parameters Considering Joint Roughness: A Stability Case Analysis of Bedding Rock Slopes in an Open-Pit Mine

Abstract

This study aims to fully explore the influence of rock joint roughness in the slope stability analysis of an open mine. Therefore, based on the least squares parameter estimation method, a generalized shear strength parameter coupling roughness, which is used to evaluate a slope design, was established by a matching formula reflecting the relationship between the roughness coefficient and the shear strength parameter. Firstly, to verify and calibrate the model, physical direct shear tests were conducted on serrated rock joints, and the effect of roughness was analyzed qualitatively. The results show that an increase in tooth height increases the internal friction angle but decreases cohesion. Therefore, the rock joint direct shear test numerical model by PFC-2D was established and compared with the physical test curve, effectively verifying its correctness. The relationship between cohesion c and internal friction angle φ with roughness coefficient JRC was further quantified, and the new model was then fitted to characterize the relationship between roughness and shear strength parameters. This method is applied to the design optimization of an open pit mine slope. The new parameters are input to GeoSMA-3D, which is used to search for the key block of a bedding slope on an open-pit mine, to optimize the design of the slope.

1. Introduction

A slope is a typical engineering model, and it is susceptible to more serious geological hazards due to complex geological effects and engineering disturbances. Represented by mountainous transportation infrastructure and large open-pit mines, the excavation process often faces some stability problems. It is generally believed that compared with anticline slopes, bedding slopes are an important risk source for slope instability [1]; when the dip angle of the rock layer is less than 15° at the foot, a landslide is very likely to occur [2]. Therefore, landslides are also important objects of study for slope design and the disaster monitoring of large open-pit mines. The study of damage patterns and the mechanisms of bedding rocky slopes began with the major accidents of reservoir arch dams that occurred in France during the 1950s. Muler [3] focused on the influence of rock joints, and he indicated the effective characteristics of slope stability. Goodman et al. [4] introduced the idea of material mechanics into the damage mechanism of bedding slopes. On the basis of studies on the basic instability mode and damage mechanism [5,6,7], the damage mode and instability of rocky slopes under dynamic responses are revealed [8,9,10]. On this basis, numerical research with discrete elements [11], block theory [12], discontinuous deformation analysis methods [13], and other new methods have achieved fruitful research results in slope stability. In addition, along with the development of reliability theory, reliability analysis methods have received general attention based on multiple damage modes [14,15]. The broad application of intelligent algorithms and deep learning also provides an effective means for slope stability analysis [16,17].

The above-mentioned studies show that the mechanical characteristics of the rock mass are the controlling factors of damage [18,19], especially shear mechanical properties. Research on shear mechanical properties is parametrized by the experimental method, which considers engineering applications and analysis accuracy. Therefore, introducing φ and c to explain the shear mechanism is widely recognized and applied in engineering, which was written into the code [20]. In fact, the angle of internal friction is concerned with the geometry of rock joints, while cohesion reflects material properties, which collectively describe the main internal factors affecting the shear strength. Thus, shear properties are transformed into the estimation of shear strength parameters.

To review the current research, most scholars have evaluated slope stability by estimating shear strength parameters, and a series of estimation methods has been established [21,22,23]. Scholars have conducted a series of studies and obtained a variety of methods for obtaining shear strength parameters. For instance, physical tests are used in engineering applicability, especially for single-shape rock masses and insignificant variations of rock joints. In addition, numerical simulation tests are useful, and they require data and cannot carry out a large number of repetitive physical tests, whereas their calibration and testing model need to be studied thoroughly. Furthermore, the engineering analogy method is applied to the case of untestable data, and this method does not lead to large errors. Moreover, the theoretical analysis method is used in parameter estimation, but its accuracy combined with its poor engineering applicability is not recognized. The parameter inversion method for monitoring projects is highly accurate, and it can achieve local estimation. In spite of the intelligent algorithm method with higher accuracy, its analysis is unclear, so it also depends on large amounts of engineering specimens and has weak engineering applicability. Currently, scholars have produced many innovations for the estimation of rock joint shear strength parameters, but the essence of these estimations is the optimization of data processing, which always does not consider the influence of the rough surface morphology on shear strength parameters and ignores the influence of that in different locations.

However, for the problem of the estimation of shear strength, rock joint roughness is an important parameter. Methods for analyzing the data and carrying out quantitative evaluations for rock joints have been a focus of research in the field of rock mechanics. Barton [24] established the classical JRC-JCS model by conducting a study of 300 natural rock joints, which established the basic paradigm of strength theory. As for the strength theory research path with respect to characterizing the roughness of rock joints to establish the friction angle, some scholars have carried out long-term research on this topic. In this case, they have achieved substantial progress and defined a range of roughness coefficients, among which are the most commonly used, such as the root mean square of slope Z₂ [25], structural function SF [26], and profile line roughness coefficient R_p [27]. In addition, from two-dimensional to three-dimensional, some scholars have defined three-dimensional roughness [28,29,30], which further promoted the development of strength theory. In contrast, the existing theories could not be unified. On the other hand, along with the accurate strength theory, the inapplicability of engineering and balancing accuracy is still an important issue.

Considering the science and engineering applicability, the parameter estimation method is further refined and modified in this study. The generalized shear strength parameter is proposed, and the modified formula with full consideration of roughness is fitted. Shear strength can be accurately obtained by shear tests, and the International Society of Rock Mechanics has established the relevant standards for static and dynamic shear tests. Some scholars have further enriched the standards’ applicability by improving testing machines and introducing different influencing factors into the shear test. However, for actual engineering needs, there are many limitations to obtaining the shear strength, especially for a rock joint, because its complex and variable surface morphology makes obtaining a global shear strength impossible using simple sampling tests. Therefore, the shear test is only used as a standard value to verify the correctness of other theories. To overcome the limitations of the shear test, simulation methods based on numerical computation have become a focus of research. Scholars have used different methods to simulate the shear test process so that the shear strength can be obtained with a large number of repetitions, and the particle flow method represented by PFC has achieved substantial progress in simulations [31,32,33,34].

In this study, physical and numerical test on the direct shear is conducted. Relying on the bedding slope of an open-pit mine, the study of rock joint shear strength parameters is carried out, and the generalized shear strength parameter, internal friction angle, and cohesion are determined with different roughness. The influence of the roughness on the shear strength parameters is also summarized. The research results are applied to the slope stability analysis using GeoSMA-3D. As a result, it is meaningful to analyze the stability and optimize the slope again for the bedding slope using the generalized shear strength parameter.

2. Engineering Background and Generalized Shear Strength Parameter

The open-pit mine is located in an inter-river block-type mountainous area with denuded low to medium mountainous terrain. The terrain is high in altitude in the northwest and low in altitude in the southeast, with elevation ranging from 1650 to 1875 m. The mine area is excavated, and it is located at 1875 m above sea level. The mountain is flanked by a river valley, and the mine area was strongly cut by developing ditches, forming a striped ridge with a slope of 30° to 35°. The valleys are mostly “V” shaped. The location and topography of the mine site are shown in Figure 1.

The fracture system in the mine area includes north–west (F2 and F14), east–west (F10 and F15), north–west–west (F11 and F17), north–north–west (F12 and F13), and north–south (F4 and F9) fractures, among which north–west, north–east, and north–south fractures have a controlling effect on the mine area, and north–south fractures are the reproduction of basement fractures in the meridional region.

The fractures also have important hydrogeological characteristics except for mineral control and important mineralization significance. Most river valleys and gullies in the area are developed along the fractures, and groundwater (springs) is often exposed on the side of the river valleys and the top of the gullies. Generally, the fracture zone along the tensional fracture is conducive to the storage of fracture water, while the compressional and compressional torsional fracture upper plate is mostly developed with orthogonal or oblique secondary faults and a dense fracture zone with good open areas. In contrast, the lower plate is mostly developed with feathery shear fractures with poor open areas.

2.1. Research Region

The main sandstone area is located at the east slope of the open-pit section in this study, controlled by F13, F14, and F15 faults, among which the east–west F15 fault has little influence on slope stability while the F13 and F14 faults have greater influence, especially the F14 fault. The natural stability slope of the mountain on the east bank of the river is about 55°. Therefore, the value of the slope angle of the east slope is 50–55° in the ground survey report.

The sampled statistics in this survey report that the friction coefficients are all less than 7 (JRC). In fact, due to the engineering applicability, the slope design of this open-pit mine did not fully consider the frictional properties between the rock joints, ignoring the self-stabilizing ability. In order to consider the roughness comprehensively, to analyze the stability of the east slope of the mine, and to optimize the slope design, this study focuses on a typical area of the slope, as shown in Figure 2. This area is a three-step slope with rich joints and fissures, which are disturbed by engineering, near the outer edge of the slope, and the section’s form is a morphological mutation.

2.2. Generalized Shear Strength Parameter

The shear strength parameter is a basic indicator of the rock joint, and it is widely used in the parameter setting of various analysis methods. In fact, in the analysis of stability problems of rock mass under multi-field conditions, more practical conditions, such as seepage, earthquake, etc., need to be considered, which requires more parameters. This brings a series of parameter analyses and equation establishment. If the shear strength parameters can be further optimized for different conditions so that they can reflect the relevant properties, it will be greatly simplified. In this study, a generalized shear strength parameter was proposed, which is defined as a strength parameter. It can reflect a variety of properties of the rock joints based on the shear strength parameter, which makes some practice on the rough properties.

3. Direct Shear Test of the Jagged Sandstone Joints

For the sandstone slope studied in this paper, taking into account the applicability of the project and the influence of friction characteristics on their shear resistance, the quantitative relationship between the friction coefficient and shear strength parameters applicable to this project is established, and the rough friction characteristics are involved with shear strength parameters. It should be emphasized that physical tests are required to obtain fundamental data for the determination of shear strength parameters, and the rough rock joint is generally prepared by manual carving. Although the accuracy of the 20 standard contour lines determined by Barton roughness values can be guaranteed by a rock engraving machine [35], systematic errors and the accuracy of the specimen’s preparation are difficult to estimate because of two transformations from the digital model to the physical model and then to the digital model in the operation process. In addition, the value preservation of the two-dimensional profile to three-dimensional spreading for roughness values still needs further investigation.

In order to avoid this error, this paper uses PFC simulations. The accuracy of the numerical model needs to be calibrated and checked by physical tests before the simulation so that indoor direct shear tests on the rock joints of the rock can be carried out first.

3.1. Preparation of Specimens

Considering that the direct shear test is intended for calibration and for verifying the accuracy of the PFC model, it is necessary to quickly produce accurate specimens only based on the variation of roughness. In order to avoid errors in the preparation of complex rock specimens, jagged joint specimens were designed for the study, and the rocks were taken from sandstone during engineering processes.

Firstly, the specimens were processed into 100 mm × 100 mm × 100 mm cubes, which should be properly polished after cutting. The size is recommended by ISRM. The centerline was determined in any plane and marked, while jagged rough contour lines were drawn along the center plane on the two opposite sides of the specimen according to the set centerline (Figure 3). The center line divides two sections along the marked line and the contour line of the rock joints, which ensures that it is jagged.

Via the above process, three groups of twelve rock joints specimen were formed, each of which was divided into two sections that were named according to the height of their sawtooth as follows.

RA − X_u/d

Among them, R denotes roughness; A is the serial number in ascending order according to the serrated height, with three assigned values 2, 4, and 5; X is the serial number of the specimen, with four values 1–4; u/d represents the upper and lower sections, respectively.

The specimen after machining accuracy calibration is shown in Figure 4.

3.2. UCS and Direct Shear Test

In this paper, we test the peak shear strength value together with its normal stress as the base data for obtaining shear strength parameters. The peak shear strength is obtained from a physical direct shear test with a maximum vertical force of 500 kN and a horizontal force of 1000 kN (Figure 5a). The data acquisition system can automatically obtain direct shear test data and draw the stress–strain curve. The external displacement sensor can be used to monitor the deformation in all directions during the shear process.

In order to prevent the specimen from being damaged by compression before shearing due to excessive normal stress during the test, the applied normal stress value should be less than 40% of the uniaxial compressive strength of the rock material. It is necessary to measure their uniaxial compressive strength using standard cylindrical specimens before the test [36]. In this study, uniaxial compression tests were conducted using a rock UCS testing machine (Figure 5b). Three standard cylindrical sandstone specimens with a height of 100 mm and a bottom diameter of 50 mm were prepared. The size is recommended by ISRM. Their uniaxial compression curves were obtained by the rock UCS tester, where the peak strength was 31.7 MPa, 27.3 MPa, and 27.3 MPa. Thus, the uniaxial compressive strength of the sandstone material was determined to be 28.8 MPa.

In summary, the normal stress for the straight shear test should satisfy the following conditions.

σ_n < 0.4 σ_c = 0.4 × 28.8 MPa = 11.5 MPa

(1)

3.3. Test Procedure and Results

In this study, twelve groups of rock specimens were tested in three rounds under different compressive forces.

Among them, the determination of normal force should be set according to the compressive strength. Referring to Equation (1), the normal stresses are R2 (2, 4, 6, and 8 MPa), R4 (3, 4, 5, and 6 MPa), and R5 (2.5, 3.5, 4.5, 5.5, and MPa) successively. The test requires subtracting the self-weight of the upper plate of the specimen, the spacer, and the roller plate. The normal and tangential indenters are in contact with the specimen’s surface by 30% displacement loading, and the pre-normal force is at a loading rate of 0.5 kN/s. The shear test would be initiated after the value is stabilized.

The peak shear strength was determined from the displacement–load curve and plotted as a stress–strain curve for the calibration of parameters in the numerical simulation. The normal stress and peak shear strength are recorded in

Table 1. Summary of direct shear test data of rock joints.

Serial Number	Normal Stress (σn)/MPa	Peak Shear Strength (τ)/MPa
R2-1	2	3.683
R2-2	4	6.000
R2-3	6	8.109
R2-4	8	11.217
R4-1	3	4.638
R4-2	4	5.907
R4-3	5	6.512
R4-4	6	7.821
R5-1	2.5	4.457
R5-2	3.5	4.962
R5-3	4.5	7.008
R5-4	5.5	7.080

.

3.4. Estimation of Shear Strength Parameters and Analysis

The least squares method was used for a few sample data and weak dispersion, and the analysis is discussed in detail. Using four sets of σ_n-τ data for each rock joint, a linear fit is performed in Table 1, and the equations are as follows.

$$\begin{array}{l}\tau =A{\sigma }_{\mathrm{n}}+B\\ A=\tan \left(\phi \right)\\ B=\mathrm{c}\end{array}\}$$

(2)

As shown in Figure 6, their corresponding shear strength parameters are calculated separately and summarized in

Table 2. Summary of parameters of R2, R4, and R5 joints.

	R2	R4	R5
Tooth height/mm	2	4	5
Cohesion (c)/MPa	1.0745	1.6502	1.9108
Internal friction angle (φ)/°	51.0	45.4	44.8

. The internal friction angle needs to be further back-calculated according to regression coefficient A.

According to Table 2, it can be observed that shear strength parameters change with the change in tooth height; specifically, the internal friction angle increases with the increase in tooth height, and cohesion decreases with the increase in tooth height.

4. Numerical Estimation of Generalized Shear Strength Parameters of Rock Joints Considering the Roughness

The influence law of roughness on shear strength parameters was studied by conducting physical tests. On this basis, the effect of roughness on the shear strength parameters of real rock joints can be further discussed for engineering research. The standard profile contour line of the rough rock joint proposed by Barton has been generally recognized by scholars, and it has been applied in different quantities during engineering operations. Therefore, Barton’s standard JRC as the roughness definition value has good engineering applicability for this project. In this study, the numerical model of the direct shear test is constructed by using PFC-2D discrete elements to obtain the stress–displacement curve’s evolution and to compare it with the curve derived from the shear test to obtain microscopic parameters which can characterize the rock as well as the micro-parameter of rock joints. The numerical model is consistent in size with the physical test specimens. The size of the specimen of the straight shear test is 100 × 100 mm, and the specimen of the UCS test is 50 mm in diameter and 100 mm in height.

4.1. Establishment of Numerical Model of the Direct Shear Test on Sandstone Joints

In order to effectively and precisely control the relative change in roughness, the physical experiment carried out is based on the idea of preparing a rough rock joint by expanding the two-dimensional contour line to three dimensions (Figure 7). Therefore, in the process of establishing the numerical model, the same idea of forming a rough rock joint by expanding Barton’s standard contour line to three dimensions is adopted. The use of the PFC-2D numerical model can be well adapted to such a method, which is recognized by many scholars. In addition, the advantage of using the 2D model is the ability to use all of Barton’s theories about his standard contour lines in a reasonable manner, since these theories are specific to 2D contour lines.

4.2. Rough Joints Setting

Barton established ten two-dimensional roughness contour lines and defined the JRC values (Figure 8) via a large number of rock joint direct shear tests based on engineering rock masses, which became a widely accepted roughness characterization coefficient. In this study, the standard contour lines with JRC = 0.4, 2.8, 5.8, 6.7, and 9.5 were selected. The literature gives the point coordinate values of Barton’s standard contour lines, which can accurately plot the corresponding contour lines (Figure 9). This will greatly improve the accuracy of the numerical model. The rough contour lines are drawn in CAD using point coordinates and added to the numerical model.

The starting positions of the contour lines are adjusted to the same height, which ensures that the shearing process is for the shearing of the rock joints not for the rock mass. After that, the coordinates are rotated based on the starting level on the basis of obtaining a contour line accurately. Additionally, the new one is imported into the model to establish a two-dimensional PFC numerical model for the direct shearing test.

4.3. Construction of a Direct Shear Test Numerical Model and Parameter Calibration

The direct shear numerical model was constructed by PFC discrete elements, and the steps in model construction were mainly divided into six steps: sample formation, precompression, setting cementation, setting rock joint, and setting normal stress and loading.

The direct shear numerical model was constructed. Microscopic parameters were calibrated by trial uniaxial compression processes and direct shear tests (Figure 10), in which the direct shear model was used as a smooth joint model, and the relevant microscopic parameters are shown in

Table 3. Microscopic parameters of numerical simulation of sandstone uniaxial compression.

Parameter Type	Microscopic Parameters	Value
Basic physical parameters of particles	Density (kg/m3)	2350
	Minimum particle radius (mm)	0.58
	Maximum particle radius (mm)	0.754
	porosity	0.15
	Modulus of linear part (GPa)	2.2
	Stiffness ratio of the linear part	1.2
	Friction coefficient	0.5
Linear parallel bonding microscopic parameters	The modulus of the bonding part	2.2
	Stiffness ratio of bonding part	1.2
	Mean value of normal intensity (MPa)	18
	The variance of normal strength (MPa)	3.6
	The mean value of tangential strength (MPa)	18
	Variance of tangential strength (MPa)	3.6
	Angle of internal friction (°)	30

and

Table 4. Microscopic parameters of smooth joint model.

Parameter Type	Microscopic Parameters	Value
Microscopic parameters of smooth joint model	Normal stiffness (N/m)	5 × 109
	Shear stiffness (N/m)	5 × 109
	Friction coefficient	0.75

.

4.4. Numerical Simulation Results Analysis

4.4.1. Numerical Model Verification Based on UCS Test and Direct Shear Test

As shown in Figure 11a, an oblique damage interface, below which rock breaking occurs, as shown by the red arrows, is the same in physical testing and numerical simulation.

As shown in Figure 11b, the overall performance of jagged bumps is a smoothed, red circle, and there is a local concentration of broken belt, a blue circle.

In addition, the simulation of the peak strength is critical in this study. Numerical tests were carried out using the R2-2 rock joints, and the shear stress and shear displacement curves are shown in Figure 11c, respectively. It can be observed that the peak shear strengths of numerical tests can conform well to physical tests, and their basic trends remain consistent. Both the damage model and the strength prediction show that the PFC model we built has good applicability.

4.4.2. Estimation of Shear Strength Parameters Based on Numerical Simulation Results

The numerical model for direct shear tests was used to apply a standard contour line with JRC values of 0.4, 2.8, 5.8, 6.7, and 9.5 as rough rock joints. Direct shear numerical tests were carried out at normal pressures of 2, 4, 6, 8, and 10 MPa to determine the corresponding peak shear strengths, which are summarized in

Table 5. Summary of numerical test data of rough rock joints.

JRC	σn/MPa	τ/MPa	JRC	τ/MPa	JRC	τ/MPa	JRC	τ/MPa	JRC	τ/MPa
0.4	2	2.8	2.8	2.59	5.8	2.87	6.7	3.43	9.5	3.99
	4	5.2		5.81		6.28		6.42		6.52
	6	6.14		7.32		7.43		7.86		8.42
	8	7.07		7.6		8.65		8.92		9.54
	10	8.75		8.8		8.99		9.35		10.62

.

The data from Table 5 were analyzed using least squares estimation (Figure 12). It was observed from the τ-σ_n scatter plot that the peak shear strength corresponding to a normal stress of 10 MPa exhibited a more significant dispersion. With JRC values of 2.8, 5.8, and 6.7, the results deviated from the point group; therefore, they are excluded from parameter estimation.

Based on the shear strength parameter estimates in

Table 6. Summary of shear strength parameters of JRC0.4~JRC9.5 rock joints.

	JRC = 0.4	JRC = 2.8	JRC = 5.8	JRC = 6.7	JRC = 9.5
Cohesion c/MPa	1.861	1.695	1.685	1.5927	1.4800
Internal friction angle $\phi$/°	34.5	39.6	42.8	44.8	45.4

, it was observed that the cohesion of the rock joint decreased with an increase in JRC, while the internal friction angle increased with an increase in JRC. This trend is consistent with the results from physical tests.

4.5. Generalized Shear Strength Parameter and Empirical Formula between Roughness and Shear Strength Parameters

The scatter plot (Figure 13) is plotted with JRC as the horizontal coordinate, and cohesion c and the internal friction angle were plotted as vertical coordinates. It was observed that they have a good linear relationship, so a linear fit was performed to obtain empirical equations for the generalized shear strength parameter values that were applicable to the subject of this study.

$$\begin{array}{c}\phi =1.2175\mathrm{JRC}+35.284\\ c=-0.0383\mathrm{JRC}+1.8558\end{array}\}$$

(3)

5. Optimization of the Slope’s Design

To recalibrate the internal friction angle and cohesion of joints, the empirical equations relating shear strength parameters and roughness are utilized. The key blocks of the slope model, both before and after the parameter correction, were searched using GeoSMA-3D software. The software takes into account the frictional resistance between joints to resist the sliding force, which allows for a thorough search of key blocks. Finally, a comparison is made between the key block information before and after the recalibration process.

This study used the geotechnical structure and model analysis system (GeoSMA-3D) developed by the team to analyze the stability of the study area. GeoSMA-3D is based on block theory and discontinuous deformation theory (DDA). The program adopts C++ as its programming language, and it uses OpenGL technology for graphic editing and visualization.

To determine the dominant rock joint, the nodal data collected from the engineering survey report are clustered and analyzed. This information is then imported into GeoSMA-3D, which generates the trace model. The program automatically calculates the internal rock joint intersection line (Figure 14).

The software parameter definition interface employs the friction coefficient extracted from the engineering survey report to define the physical parameters of the rock joint both before and after the correction of shear strength parameters via Equation (3), which is utilized for the retrieval of key block calculations (Figure 15).

Figure 14 shows the external morphology of the slope, but it does not reveal the key blocks inside it. Therefore, it is necessary to calculate the data of the key blocks to conduct a thorough assessment of the slope’s stability before and after the parameter correction [37].

Table 7. Key block data before parameter correction.

Serial Number	Volume	Number of Surfaces	Number of Sliding Surfaces	Safety Factor
01	0.3862	4	1	0.878
02	3.9627	5	2	0.557
03	5.7823	5	1	0.356
04	0.6234	4	1	0.719
05	0.3791	4	2	0.885
06	0.8245	4	1	0.639
07	7.3241	5	2	0.159
08	0.3258	4	1	0.889

and

Table 8. Key block data after parameter correction.

Serial Number	Volume	Number of Surfaces	Number of Sliding Surfaces	Safety Factor
01	0.1367	4	2	0.905
02	0.2398	4	1	0.893
03	5.1873	5	1	0.218
04	0.0038	4	1	0.934
05	3.5831	5	2	0.612
06	0.3847	4	1	0.875

present the calculated data, demonstrating that the number of key blocks was reduced from 8 to 6. Furthermore, the volume of the blocks decreased by 51.2%, leading to an improvement in the safety factor. As the slope’s stability has some redundancy, it may be feasible to increase the slope’s dimensions based on the original survey and design.

6. Discussion

6.1. Accuracy Testing of Rock Joints Specimen

While modern industry can guarantee the precision of preparing serrated rock joints, this may not be the case for the sandstone specimen examined in this experiment. The materials used in this study were sourced from an engineering site, which makes them susceptible to rock joint damage during processing and transportation. Additionally, the height differences between the serrations are small. Consequently, performing accuracy tests during specimen preparation is crucial in order to select specimens that satisfy the predetermined test criteria, which need to digitalize the measurement of roughness [38].

This research utilizes a 3D structural light digital scanning method and point cloud model digitization processing technology to scan the morphological features of rock joints specimen in the laboratory and obtain their 3D point cloud data. These data are then used to compare the accuracy of a specimen’s preparation. The 3D digital scanning equipment (Figure 16), which operates on the VisenTOP 2021R platform, is equipped with fully automatic stitching capability and an overall error control module. This allows multiple point clouds to be stitched together without any cumulative error or delamination issues.

Rock joint comparisons require morphology scanning and the precision detection of all 12 rock joints in sequence. In fact, considering the particularity of the rock joint specimens, the regularly serrated rock joints are mass-produced by a unified mold, and the twelve rock joints are divided into three groups, each of which has the same morphological state and high system stability. Therefore, the precision monitoring process can be further simplified by randomly selecting one specimen from each group of rock joint specimens and by further randomly selecting the upper or lower plate of each specimen for sampling inspection to ensure its precision detection effect. In this study, R2-1u, R4-1d, and R5-3u were randomly selected for testing.

After scanning, the 3D point cloud model contains four sides, except for the joint ones. To prepare the model for calibration, redundant planes are removed using plane fitting. Only the jagged rock joints required for calibration are retained, and the model is further denoised. Additionally, the large number of overlapping data points generated during this process, as well as the sparsity of different specimen spacing and point clouds, requires optimization and refinement [39]. The acquired and processed digital model of the rock joint is imported into the software, and two profile contour lines are intercepted along the vertical direction to calculate its sawtooth height. Furthermore, two contour lines are randomly intercepted at any position along the parallel sawtooth’s direction to determine the standard deviation of its coordinate values. The former is used to check whether the sawtooth’s height meets the design requirements, while the latter is used to check the sawtooth’s flatness.

Figure 17a shows that R2-1u, R4-1d, and R5-3u rock joints in the 3D model have differences of 2 mm, 4 mm, and 5 mm, respectively, between their highest and lowest points. However, due to the limitations of point cloud processing and coordinate alignment, the lowest point depicted in the color map of the figure does not accurately represent the true lowest point of the sawtooth. Therefore, the difference can only provide a rough estimate of its height range. To obtain more precise measurements, further analyses via cross-sectional profiling are required.

Figure 17b shows the interception position of each rock joint line that requires analysis. The coordinates of the profile lines are derived, and the standard deviation of the high point, low point, and difference values from the predetermined tooth height for the A-A and B-B profile lines are calculated. Similarly, the standard deviation of the elevation of the 1-1 and 2-2 profile lines is also calculated.

Figure 18 provides an example of the A-A profile line of R2-1u, where the high (red) and low (blue) points of each tooth are identified, and the tooth height is calculated by aligning the principle to the left. Three sets of data, the high-point coordinates, low-point coordinates, and tooth height coordinates are obtained, and their standard deviations are calculated.

Table 9. Summary of detection index data of each profile line.

Joints	Indicators	Profile Line		Indicators	Profile Line
		A-A	B-B		1-1	2-2
R2-1u	Standard deviation of high point	0.004238	0.003856	Standard deviation of elevation	0.00511	0.00429
	Standard deviation of low point	0.005627	0.004036
	Standard deviation of the difference from the predetermined tooth height	0.008011	0.007924
R4-1d	Standard deviation of high point	0.003578	0.003502	Standard deviation of elevation	0.00504	0.00552
	Standard deviation of low point	0.003426	0.002440
	Standard deviation of the difference from the predetermined tooth height	0.006049	0.005697
R5-3u	Standard deviation of high point	0.004201	0.004241	Standard deviation of elevation	0.00436	0.00389
	Standard deviation of low point	0.004529	0.004601
	Standard deviation of the difference from the predetermined tooth height	0.007260	0.007632

indicates that the index error of each profile line is within the 10⁻³ orders of range, confirming that the specimen’s preparation accuracy satisfies the predetermined requirements.

6.2. Determination of Parameter Estimation Methods

There are several parametric methods for estimating shear strength parameters via direct shear tests on rock joints. However, since the test data for the engineering cases studied in this study exhibit limited specimens and non-discrete characteristics, it is necessary to compare different methods for estimating shear strength parameters based on physical tests and to analyze the accuracy of various data processing models. Thus, this study uses four sets of data from specific engineering cases to estimate the shear strength parameters of cohesion (c) and internal friction angle (φ) via the least squares method, point group center method, and stochastic fuzzy mathematics method. The corresponding estimation results for the four sets of data are presented in

Table 10. Summary of shear strength parameters from groups 1-4 estimated by 3 methods.

	Method	c/MPa	$\mathit{\phi }$/°
1	Least squares method	0.103	32.4
	Point group analysis	0.016	33.3
	Random fuzzy mathematics method	0.031	35.7
2	Least squares method	0.103	32.4
	Point group analysis	0.014	31.2
	Random fuzzy mathematics method	0.016	29.8
3	Least squares method	0.084	26.5
	Point group analysis	0.137	20.4
	Random fuzzy mathematics method	0.099	23.8
4	Least squares method	0.053	28.1
	Point group analysis	0.037	29.6
	Random fuzzy mathematics method	0.053	28.1

. Considering the computational complexity, engineering applicability, and stability of parameter estimation, the least squares method is selected for parameter estimations.

6.3. Further Research

Uniaxial compression tests of rock need to be carried out first before conducting direct shear tests of rock joints. Yet, in many cases, it is difficult to carry out physical tests, for which the estimation of uniaxial compressive strength for different rocks is an important research direction [40].

In this paper, generalized shear strength parameters considering roughness are established for sandstone anastomosing non-filled joints, but rock joints under engineering conditions are often non-anastomosing [41] and filled [42], and how to establish generalized shear strength parameters for joints with different conditions using the method in this paper needs further research.

7. Conclusions

This study investigated the stability of a bedding slope in an open-pit mine and focused on estimating the shear strength parameters of rock joints. It used the least square method to conduct indoor direct shear tests and PFC-2D numerical simulation. Then, it analyzed the impact of roughness on the shear strength parameters of rock joints, both qualitatively and quantitatively. The study summarized empirical formulas for roughness coefficients and shear strength parameters. Furthermore, it applied them to optimize the slope design.

(1) Analysis, physical testing, and numerical simulations of the shear strength parameters at various tooth heights indicated that the internal friction angle increased with tooth height while the cohesive force decreased as tooth height increased.

(2) This study focuses on the influence of rock joint roughness on shear strength parameters by conducting numerical simulation using PFC-2D, which is calibrated and validated by physical tests. As a result, a generalized shear strength parameter of rock joint, sandstone from the project in Xinjiang, that couples the roughness and an equation for it were provided, and they were derived to account for the impact of roughness on shear strength parameters. The study also circumvented the issue of accuracy loss in replicating Barton’s standard contour lines in physical testing.

(3) GeoSMA-3D was used to analyze the slope stability of the dependent project, and the research region-established empirical formula was utilized to calibrate the shear strength parameters of the joints. The key block retrieval results indicated that the slope’s design was highly safe, and the slope angle can be appropriately increased.