Numerical Investigation for the Effect of Joint Persistence on Rock Slope Stability Using a Lattice Spring-Based Synthetic Rock Mass Model

Abstract

This study underscores the profound influence of rock joints, both persistent and non-persistent with rock bridges, on the stability and behavior of rock masses—a critical consideration for sustainable engineering and natural structures, especially in rock slope stability. Leveraging the lattice spring-based synthetic rock mass (LS-SRM) modeling approach, this research aims to understand the impact of persistent and non-persistent joint parameters on rock slope stability. The Slope Model, a Synthetic Rock Mass (SRM) approach-based code, is used to investigate the joint parameters such as dip angle, spacing, rock bridge length, and trace overlapping. The results show that the mobilizing zones in slopes with non-persistent joints were smaller and shallower compared to slopes with fully persistent joints. The joint dip angle was found to heavily influence the failure mode in rock slopes with non-coplanar rock bridges. Shallow joint dip angles led to tensile failures, whereas steeper joint dip angles resulted in shear-tensile failures. Slopes with wider joint spacings exhibited deeper failure zones and a higher factor of safety, while longer rock bridge lengths enhanced slope stability and led to lower failure zones. The overlapping of joint traces has no apparent impact on slope stability and failure mechanism. This comprehensive analysis contributes valuable insights into sustainable rock engineering practices and the design of resilient structures in natural environments.

1. Introduction

Rocks are distinct from most other engineering materials in that they have discontinuities and inherent defects, which lead their structure to be heterogeneous, anisotropic, and discontinuous. Because of their different response under various loading conditions, the fundamental knowledge and approaches used for other engineering materials are often insufficient to understand rock mass behavior. This understanding is imperative for sustainable engineering practices, as it informs strategies to address the distinct challenges posed by the unique characteristics of rocks, contributing to the development of resilient solutions.

Rock mass strength is mainly controlled by discontinuities such as fractures, joints, faults, and beddings. The mechanical and geometrical properties of discontinuities, which represent the weakest component of the rock mass, dominate the stability of natural and engineered rock slopes.

Several empirical geotechnical classification systems are available for characterizing rock mass to take into account the degree of jointing of this heterogeneous media. The Rock Quality Designation (RQD) [1], Rock Mass Rating (RMR) [2], Rock Mass Quality (Q) [3], Rock Mass Index (RMi) [4], and Geological Strength Index (GSI) [5] are the most suitable systems for estimating the rock mass strength. These classification systems utilize empirical relationships to estimate the mechanical properties of large volumes of rock mass from laboratory test results on small-scale samples [6].

Numerical techniques, including continuum, discontinuum, and hybrid methods, have significantly enhanced our understanding of rock slope failure mechanisms over the last few decades. The discontinuum method is highly suitable for simulating the propagation and coalescence of cracks, large deformation, and highly fractured rock mass in the context of slope stability problems [7]. The discrete element approach (DEM) has been adopted to investigate the common failure types, including planar, wedge, toppling, and combined failures [8]. Planar and bi-planar failures were analyzed using a three-dimensional lattice spring code [9,10]. Lian et al. [11] utilized a Distinct Lattice Spring Model (DLSM) to study the toppling failure in a jointed rock slope, and the wedge failure mechanism was evaluated using the DLSM method by Zhao [12]. Havaej et al. [9] utilized a Slope Model (Itasca) to investigate non-daylighting wedges and flexural toppling failures.

Rock bridges, also known as locked sections, comprise a portion of intact rock that separates joint surfaces. The presence of rock bridges along a discontinuity plane can significantly impact the strength of the rock mass. Camones et al. [13] and Yang et al. [14] studied the mechanism of step-path failure in slopes, which form partially along discontinuities and partially through intact rock bridges. Diederichs [15] illustrated the significance of rock bridges by implementing slope support members and comparing those with artificial slope reinforcement systems (e.g., cables or bolts). Al-E’Bayat et al. [16] reported that rock slopes with non-persistent discontinuities have a higher factor of safety and significantly smaller mobilized zones than slopes with fully persistent discontinuities. In addition to considering rock bridges, various joint parameters play significant roles in slope stability, including the mobilized joint roughness coefficient (JRC_m) and its impact on post-peak shear behavior, as demonstrated by Deiminiat et al. [17], who utilized a reduced-order model (ROM) to investigate how the mechanical properties of the rock mass impact the stability of the surrounding rock mass.

The uncertainty associated with the characterization of rock joints imposes constraints on the reliability of predictions regarding the behavior of the rock mass. Researchers commonly employ probabilistic evaluation methods such as the point estimate, the Monte-Carlo, and the response surface that assess the structural safety level by integrating the uncertainty factors inherent in the rock mass [18,19]. When utilized in conjunction with deterministic methods, these approaches significantly improve slope stability calculations, effectively overcoming the limitations of conventional methods.

Most slope stability research efforts are generally based on fully persistent discontinuities or studying the effect of a limited range of non-persistent joint parameters on slope stability. Therefore, the authors believe that the presented work offers a comprehensive study to quantify the impact of persistent and non-persistent joints with various parameters on the stability of rock slopes.

This study presents the lattice spring-based synthetic rock mass (LS-SRM) modeling approach to perform a parametric study of persistent and non-persistent joint parameters on rock slope stability. The Slope Model, a Synthetic Rock Mass (SRM) approach-based code, is used to investigate the effect of various joint parameters (Figure 1). The failure mechanisms in the jointed slopes were also analyzed as a scope of this work. The strength reduction method (SRM) is adopted to estimate the factor of safety for various scenarios of the joint geometry using the lattice-spring-based synthetic rock mass (LS-SRM) modeling approach. Figure 1 presents the non-persistent joint parameters taken into account in this research. These parameters are as follows:

• joint spacing (d);
• joint dip angle (β);
• rock bridge length (RB_L);
• overlapping joint percentages;
• joint segment length (L_j).

2. Persistent and Non-Persistent Discontinuities

Many experimental and numerical studies have been performed on pre-cracked small-scale specimens to further understand the crack initiation, propagation, and coalescence process [20,21,22,23,24,25]. Figure 2 shows simple illustrations of a slope with a single persistent joint and a non-persistent joint in two dimensions. A planar failure takes place once the driving force exceeds the joint shearing strength in the first case. In this case, the factor of safety (FS) can be defined by the following:

$$FS=\frac{C_{0}A+Wcos\theta tan\phi }{Wsin\theta }$$

(1)

where θ is the slope surface angle, and W is the block weight. φ, C₀, and A are the joint friction angle, joint cohesion, and joint area under the block, respectively. In the second case (Figure 2b), the failure occurs along the joint and the rock bridge between joint segments, where the φ and c of the sliding surface were replaced with equivalent parameters C_eq and φ_eq, as expressed in Equations (2) and (3), respectively. C_eq and φ_eq can be estimated by using Jennings’ [26] equations:

$$c_{eq}=\left(1-k\right)c_r+k{c}_j$$

(2)

$$\mathit{tan}{\phi }_{eq}=\left(1-k\right)\mathit{tan}{\phi }_r+k\mathit{tan}{\phi }_j$$

(3)

where φ_r and φ_j are the intact rock and joint friction angles, respectively. c_r denotes the intact rock cohesion, c_j is joint cohesion, and k is the coefficient of continuity along the failure pathway. The coefficient of continuity is defined as:

$$k=\frac{\sum L_j}{\sum L_j+\sum R{B}_L}$$

(4)

where L_j is the joint length, and RB_L is the rock bridge length.

Previous studies have shown that considering single persistent discontinuities along the slope cannot adequately capture the interaction between multiple pre-existing joints and crack propagation through rock bridges, leading to slope failure [27,28]. As demonstrated in Figure 3, failures often result from the interaction of multiple discontinuity surfaces rather than along a single planar surface. During the shearing along pre-existing joints, the initiation and propagation of tensile microcracks in rock bridges can also take place, resulting in a cross-over fracture and a continuous stepped failure surface (step-path failure).

Given the absence of a current framework detailing failure mechanisms in jointed rock slope stability, a parametric investigation becomes imperative to enhance understanding. This study aims to enrich insights by examining the interplay between persistent and non-persistent joints and their impact on slope stability.

3. Methodology

3.1. Slope Model Code

This study used Slope Model software (V3.0), a synthetic rock mass (SRM) approach-based code, to simulate the rock slopes [30]. It is noteworthy to mention that the SRM approach was initially implemented with Itasca’s codes in Particle Flow Code (PFC3D), based on the bonded particle model (BPM), as well as Universal Distinct Element Code (UDEC), based on the bonded block model (BBM). Pierce et al. [31] used the SRM approach to describe discontinuum modeling and quantify the mechanical properties of rock mass in a mass mining project.

The SRM approach has been effectively used for large and fractured rock slopes that successfully simulate the explicitly introduced rock bridges and failure response [32,33,34]. Unlike the BPM approach, which simulates the intact rock with linked spherical particles [35], the SRM uses massless springs, as shown in Figure 4. The contacts in the Slope Model code can be simulated either using parallel bonds or flat-joint contacts. In the parallel bonds model, the particles can roll relative to each other without any resistance once the bond breaks, producing an unrealistic compressive to tensile strength ratio (σ_c/σ_t) of the intact rock. On the other hand, in the flat-joint contact model, each spring is divided into sub-springs, allowing resistance to rotate through normal forces in the sub-springs even after the springs are broken. Thus, this provides more accurate simulations of intact rock’s uniaxial compressive and tensile strength responses. Figure 4A illustrates the configuration of three sub-contacts, the minimum number of sub-contacts that can be used in the flat joint model, around the circumference of the contact disk [35]. The Slope Model code employs an explicit solution scheme, and all nodes in the model are simulated by solving three translations and three rotation equations of motion. Further details about the mechanical formulation of the LS-SRM method can be found in the User’s Guide and Tutorial [36] and references therein.

Unlike the discrete element approaches (DEM), LS-SRM is effective in computational time because it is formulated in a minor strain that does not require detection and updating of the contact. The LS-SRM can also simulate step-path failures by breaking a lattice spring in the normal or shear direction, resulting in micro-crack coalescence that forms a failure surface along the crack propagation direction. According to Sainsbury et al. [37], the slope failures result from crack propagation and coalescence associated with pre-existing discontinuities. As the SRM model can simulate both pre-existing discontinuities as well as the deformation of the intact rock, it is believed that the SRM is a suitable method for modeling the failure of rock slopes.

Despite the promising features of the LS-SRM, this methodology has two main limitations. A primary constraint lies in the user’s requirement to choose between parallel bonds, resulting in a markedly higher σ_c/σ_t, or flat joint contacts that induce an unrealistically low rock matrix Poisson’s ratio. Additionally, the LS-SRM lacks the capability to visualize lattice detachment.

3.2. Numerical Model Set-Up

A parametric study was conducted using the Slope Model code on different slope scenarios having various d, β, L, and overlapping joint parameters to understand and quantify the impact of persistent and non-persistent discontinuities on rock slope stability. Seven groups of slope models were generated and described in Section 4.

The configuration of the slope model, designed to represent a large-scale rock slope, is illustrated in Figure 5. The slope model was generated with the following dimensions: 150 m in length, 125 m in height, 5 m in width, vertical slope height of 100 m, and having a 70° overall slope face angle. The slope geometry was kept the same for all models. The free boundary condition was applied to the upper face of the slope, and all other boundaries (the left, right, and bottom) were fixed in all degrees of freedom. Eleven monitoring points were located on the slope face with 10 m vertical intervals. As slope boundaries tend to have local failures, placing history points on the slope face may measure extraneous displacement and velocity values. For this reason, the history points were located 20 cm deep into the slope face.

Since the LS-SRM modeling approach is efficient in terms of computational time, a high resolution (lattice spring length) of 50 cm was used, which is relatively small compared to the overall size of the slope geometry. A total number of 3,525,171 lattice springs and 715,197 nodes were created to simulate the base case slope without any joints.

Micro-mechanical properties control the material behavior of the Slope Model. Since these lattice parameters (micro-properties) cannot be measured in the laboratory, they are often determined by employing a calibration process to match the results of conducted laboratory tests. The procedure involved an iterative variation of the micro-mechanical properties, which is presented in the following section. The procedure of the LS-SRM numerical method to estimate the stability of rock slope is illustrated in the flowchart (Figure 6).

3.3. The Calibration of Micro-Scale Parameters

The parametric analysis in this study was carried out on limestone slopes with material properties obtained from [38]. The experimental mechanical properties of limestone are shown in

Table 1. Mechanical properties of limestone [38].

Intact Rock Properties	Unit	Value
Density	kg/m3	2125
Young’s modulus	GPa	1.27
Poisson’s ratio	-	0.22
Compressive strength	MPa	29.05
Tensile strength	MPa	2.9
Friction	degree	26
Cohesion	MPa	7.3

. Model calibration in the LS-SRM numerical method, in which micro-mechanical properties govern model response, is the significant step to match the macro-mechanical model parameters and experimentally determined laboratory-scale parameters. The primary aim of the calibration process is to calculate a set of model parameters that can accurately represent the macro properties of intact rock. Figure 7 summarizes the calibration process of the mechanical properties of intact rock and rock mass.

In order to calibrate the mechanical properties of limestone, a series of confined and unconfined compression test simulations were performed on lab-scale specimens with dimensions of 50 mm × 50 mm × 100 mm and a lattice resolution of 2 mm. According to Bastola and Cai [39], the loading rate does not significantly affect the macro-mechanical properties of the model. In order to maintain the quasi-static equilibrium, the tests were conducted by applying a constant vertical displacement rate (velocity) of 0.01 m/s at the top of the specimen [40]. Force-displacement monitoring points were assigned for stress and strain calculations.

The accurate input of micro-mechanical properties plays a crucial role in adequately simulating the rock mass response. Achieving an error margin of less than 10% between numerical simulations and experimental results is imperative for the reliability of the simulation. A trial-and-error approach, as presented in the flowchart in Figure 8, can be arduous and time-consuming. In an effort to make this process more efficient, the present study employs an approach by utilizing the equations proposed by Al-E’Bayat et al. [41]. These equations have been demonstrated to provide estimations of micro parameters with an error margin of less than 10%, ensuring that the simulated values closely align with the experimental data while significantly reducing the challenges associated with the trial-and-error methodology. Using established equations contributes to our numerical simulation efficiency and the precision of the micro-mechanical property input process.

Macro-scale UCS and Young’s modulus values were estimated from the unconfined compressive test simulations by plotting stress vs. axial strain curves with the data collected from the monitoring points. Triaxial compressive strength (TCS) simulations were conducted in three different confinement levels. The macro peak friction angle of the flat joint was estimated from the TCS test simulations with an axial loading rate of 0.01 m/s and of 2, 4, and 6 MPa confinement levels. In LS-SRM, calibration was not required for the macro-scale tensile strength value as it always corresponded to the micro-scale tensile parameter.

Table 2. Micro and macro mechanical properties of limestone.

Intact Rock Properties	Unit	Micro	Macro
Young’s modulus	GPa	0.18	1.26
Uniaxial compressive strength	MPa	21.4	26.50
Tensile strength	MPa	2.9	2.9
Peak friction of flat joint	degree	40	27.4
Residual friction of flat joint	degree	0	0
Radius multiplier	-	0.9	0.9

presents the micro-mechanical properties, input for LS-SRM models, and corresponding calibrated macro-mechanical properties of limestone.

It is well known that the mechanical properties of the rock mass are scale-dependent. Lajtai [42] illustrated that rock specimens on a small scale have higher strength than larger ones because the dimensions of the laboratory specimen restrict the number and size of pre-existing discontinuities. Bastola and Cai [39] performed a scale-effect study on specimens under an unconfined compressive test using the LS-SRM approach and concluded that the strength and deformation modulus of the rock mass decreases with the increase of the rock mass scale. Therefore, the downscaling for the calibrated macro-mechanical properties of the lab-scale parameters was required to simulate the large-scale mechanical response of the rock slope. Several empirical relations are available to estimate the rock mass strength and deformation modulus. Most of these relations are based on the rock mass classification systems such as RMR, RQD, Q index, MRMR, and GSI. In this study, Ramamurthy’s [43] empirical equations (Equations (5) and (6)) were utilized to estimate the strength and deformation modulus of rock mass based on RMR, where it was assumed as 83 for limestone in this study.

$${\sigma }_m={\sigma }_i{exp}^{\left(RMR-100\right)∕2}$$

(5)

$$E_m=E_i{exp}^{\left(RMR-100\right)∕17.4}$$

(6)

where, E_i and E_m are deformation moduli (GPa) for intact rock and rock mass, respectively, σ_i and σ_m are the compressive strength (MPa) for intact rock and rock mass, respectively. From the above empirical relations, E_m and σ_m were estimated as 0.478 GPa and 14.72 MPa, respectively.

In the last part of the calibration studies, the deformation modulus and rock mass strength were calibrated using large-scale confined and unconfined compressive specimens. The dimension of the large specimen is 20 m × 20 m × 50 m, which has 198,512 nodes and 1,005,106 springs with a lattice size of 50 cm, used in the slope scale calibration. The estimated deformation modulus and compressive strength values of the rock mass from empirical and numerical approaches are listed in

Table 3. Empirical and numerical mechanical properties of limestone.

Rock Mass Properties	Unit	Empirical	Numerical	Error (%)
Young’s modulus	GPa	0.040	0.043	7
Uniaxial compressive strength	MPa	14.69	14.63	0

.

3.4. Strength Reduction Method

In slope stability studies, the limit analysis (upper-bound and lower-bound solutions), limit equilibrium method (LEM), and strength reduction method (SRM) are the three most common techniques utilized to determine the factor of safety (FS) of slopes. Zienkiewicz et al. [44] established the strength reduction method, which has gained wide acceptance in assessing slope stability. Numerous studies have adopted the strength reduction method to evaluate the stability of rock slopes [45,46,47,48,49]. In this study, the strength reduction method was adopted to calculate the FS of slopes by applying a strength reduction factor (SRF) to reduce the rock mass strength that induces instability. The SRF can be regarded as the factor equivalent to the FS in the LEM analyses. The following are the common formulas of the strength reduction method that were used to determine the critical SRF for slope:

$${UCS}^{new}=\frac{1}{SRF}\times UCS$$

(7)

$${\phi }^{new}=arctan\left(\frac{1}{SRF}\times tan\phi \right)$$

(8)

$${\sigma }^{new}=\frac{1}{SRF}\times {\sigma }_t$$

(9)

where SRF is the strength reduction factor, UCS, φ, and σ_t are rock mass UCS, friction angle, and tensile strength, respectively. UCS_new, φ_new, and σ_new are the reduced UCS, friction angle, and tensile strength, respectively.

4. Parametric Studies

Estimating the safety factor (FS) and identifying the failure zones are two challenging tasks in slope stability analysis. In this work, parametric studies were conducted to quantify the impact of rock bridge and joint persistency on both the FS and failure mechanisms of slopes having various joint parameters and rock bridge distributions.

This section presents the groups of slope models performed for parametric studies. The slope geometry, spring lattice resolution, and rock mass properties were kept constant in all models, but they differed in the distribution of joints. Joints were generated using CAD software (Rhino V7) and imported into the Slope Model as a discrete fracture network (DFN). Identical joint strength properties were assumed in all models with a 20° friction angle, zero tensile strength, and cohesion.

Table 4. The range of joint parameters.

Persistent Joint Parameters		Values
Group A	Base case	No joint
Group B	Joint spacing (m)	2, 4, 6, 8, & 10
Group C	Joint dip angle (°)	10, 20, 30, …, & 90
Non-Persistent Joint Parameters		Values
Group D	Joint spacing (m)	2, 4, 6, 8, & 10
Group E	Joint dip angle (°)	10, 20, 30, …, & 90
Group F	Rock bridge length (m)	1, 3, 5, 7, & 9
Group G	Joint overlapping (%)	0, 50, & 100

presents seven groups of slope models, the variable joint parameter of each group, and the range of their values. The x-velocity plots, x-velocity charts at monitoring points, number of microcracks, and unit volume of failure zones were recorded for each slope to estimate the factor of safety (FS) and failure mechanism of slopes. The initiation velocity indicated in all x-velocity plots is 1 × 10⁻⁵ m/s to identify the failure zone.

The slope model having no joints was assumed as the base case of the parametric study. The factor of safety of each slope model (FS_s) was compared with the FS of the base case (FS_b) by using a term of FS_b% as the reduction percentage (FS_b% = (FS_b − FS_s)/(FS_b) × 100). The percentage of FS_b reduction was calculated to quantify the effect of joint parameters on the stability of the rock slopes.

The sensitivity analysis of fully persistent joints was performed on the joint spacing and joint dip angle. For Group B, five slopes were generated with varying joint spacing of 2, 4, 6, 8, and 10 m, with dip angle of 50°. Figure 9B shows a slope with a joint spacing of 8 m as an example for this group. In the models designed to analyze joint dip angle (Group C), the range of joint dip was varied from 10 to 90° with an interval of 10°, and the 6 m joint spacing was kept constant. See Figure 9C as an example of a slope with a β of 30°.

The non-persistent joint parameters that were investigated in this study comprised joint spacing of 2 to 10 m with an increment of 2 m, joint dip angle of 10 to 90° with an interval of 10°, rock bridge length of 1 to 9 m with an increment of 2 m, and an overlapping joint percentage of 0, 50, and 100% as illustrated in Figure 9D–G. The other parameters in each group were kept constant at the middle value. For example, in group D, the rock bridge length is 7 m, and the joint dip angle is 50°.

For the geometric consistency of non-persistent joints, joints were dispersed in a limited area of 2 m beyond the slope face and 2 m below the top of a slope. The joint trace length is 10 m for all slope models except group G (varying in the overlapping percentage), where the joint trace, spacing, and rock bridge length of 8 m were utilized to achieve the desired overlapping percentages.

5. Results and Discussion

The safety factor of the base case slope was initially determined to be used as a reference point for the other slope models in this study. The strength reduction approach was used to estimate the FS of the slope models. In the SRM, the strength parameters for both the rock mass and joints, such as compressive strength, tensile strength, and friction angle, were gradually reduced at 0.05 SRF increments until achieving an unstable case. Figure 10A illustrates that the slope was stable at an SRF of 5.50, where there was no mobilized zone observed in the x-velocity plot of a slope having almost zero x-velocities. On the contrary, Figure 10B shows a circular failure with a significant mobilization zone observed at the lower side of the slope face. Also, the accelerating x-velocity indicates the slope is unstable at an SRF of 5.55. These two models imply that the FS of the base slope is between 5.50 and 5.55. In all models, the upper SRF bracket was set to 5.55 and gradually reduced for the rock mass and joints with a 0.05 increment until a stable case was reached.

Model geometries and rock mass properties were chosen to be realistic and suitable for parametric analysis. It was decided that a simulated base model would be appropriate for the following stages of parametric analysis. The following parts of the parametric study present both current FS values and the percent reduction on the base FS value for all models together, allowing for the intuitive quantification of the effect of joint parameters on model stability.

5.1. The Effect of Joint Spacing for Fully Persistent and Non-Persistent Conditions

5.1.1. Slopes Having Fully Persistent Joints

The slope models in Group B cover the effect of joint spacing on the stability of slope models with fully persistent joints dipping at 50° out of the slope. Five different joint spacings (2 m, 4 m, 6 m, 8 m, and 10 m) were analyzed in this part. Figure 11 illustrates the contours of x-displacement plots for slopes at an SRF of 1.00 and indicates that all slopes in this group are unstable and have factors of safety (FS) lower than 1.00 with FS_b reduction percentages of more than 90%.

In rock slope stability analysis, it is widely observed that planar failures occur in slopes with steep and fully persistent joints. This agrees with the slope failure mechanisms demonstrated in Figure 11. The recorded few microcracks indicated that no brittle failure in the rock mass occurred. In other words, the models did not observe microcrack formation since all of the failures were caused by existing fully persistent joints. The failure occurred on the deepest daylight joint in all slopes of this group, independent of joint spacing; hence, identical unstable block volumes were observed in all slope models. It was concluded that the joint dip angle was the main factor affecting the volume of the unstable block.

5.1.2. Slopes Having Non-Persistent Joints

The results of group D, varying in joint spacing, show that whereas all slopes are stable, the stability increases with the increasing joint spacing, see Figure 12. These results agree with Hoek and Bray’s [50] conclusion that the strength and stiffness of rock mass decrease as the number of joints increases. The x-displacement contours (Figure 13) demonstrated that the failure mechanism in all slope models is a step-path failure along the non-coplanar pre-existing joint. According to Figure 12, 2 m and 4 m joint spacing reduced the FS value by 65% compared to the base case. This reduction was found at around 55% for all other joint spacing cases.

The x-displacement contours in Figure 13 demonstrate the failure mechanisms for slopes at the first unstable condition. The presented slopes failed at different SRFs due to the joint spacing impact; this comparison aims to illustrate the failure pattern of slope onset as it fails with a step-path failure. The failure occurred along the closest daylight joint trace to the slope toe as in the fully persistent slopes in Group B. The step-path failure in Group D models began to slip at the lower part of the slope from the block located on the lowest daylight joint trace, whereas the upper part was not yet formed. Then the failure takes place at the upper rock bridges, which failed one by one from the bottom up.

Though the decrease in joint spacings decreased the stability of slopes, the mobilized zone in slopes with a joint spacing of 2 m at the first unstable condition (SRF = 1.95) is shallower and smaller than in the slope with 10 m joint spacing at the failure point (SRF = 2.65). Slopes with larger joint spacing had higher FS, but a larger mobilized zone was produced at the onset of failure. This larger mobilized zone was attributed to the large block interlocks along the joint trace in slopes with higher joint spacing. On the other hand, the failure in slopes with smaller joint spacing produced small-scale unstable zones that could be relatively easier to control than the mobilized zones noted in slopes with largely spaced joints at the first unstable condition.

The slope models of group D are simulated at the same SRF of 2.90 to demonstrate the effect of non-persistent joint spacing on the failure mechanism. According to x-displacement plots, shown in Figure 14, the failure initiates at the lowest joint trace near the slope toe. A tensile-shear step-path failure occurred on all models: tensile failure through non-coplanar rock bridges and shear failure through coplanar rock bridges.

Figure 15 shows the size of unstable zones, in terms of the unit volume, in slopes at the same SRF. The maximum failure zone occurred in the slope with the tightest joint spacing (2 m) because the increase in the joint spacing led to a decrease in rock mass strength, thus decreasing the stability of the entire slope. In addition, the rock mass strength reductions (at high SRF) and the gravity increment at the bottom of the slope caused a higher tensile stress concentration, allowing the wing cracks to initiate at the joint tips that were resulting in more microcracks growing in slopes with tight joint spacing than in the slopes with wider joint spacing, as shown in Figure 15.

5.2. The Effect of Joint Dip Angle for Fully Persistent and Non-Persistent Conditions

5.2.1. Slopes Having Fully Persistent Joints

Figure 16 shows the FS results for group C, varying in persistent joint dip angle. The FSb reduction percentage represents the FS’s ratio for each slope compared to the base case model (FS is 5.55). The results indicate that the FS decreased with increased joint dip angles for slopes with unfavorably oriented joints (10–60°). The estimated FS was smaller than 1.00 for joints dipping between 20 and 60°, while an FS value of 1.15 was estimated for the shallowest joint dip case (10°). On the other hand, slopes with favorably oriented joints (70, 80, and 90°) were stable, with FS higher than 1.0. In these cases, FSb reduction is less than 30%.

The x-displacement contours of slopes at an SRF of 4.55 (Figure 16) display the failure mechanism. A planar failure was noted in slopes with unfavorably oriented joints (10–40°), combined failure (planar and circular) occurred in slopes with steeply dipping joints (50–80°), and the failure pattern in the slope with a joint dip angle of 90° is the combined circular and toppling.

The unit volume of mobilized zones at an SRF of 4.55 decreased as the joint dip angle increased. The largest mobilized zone was noted on the slope with a joint dip angle of 10°, and the vertically jointed slope has a smaller mobilized zone, as illustrated in Figure 17. Also, the microcrack number increased as the joint dip angle decreased. In other words, microcracks can initiate and propagate faster in slopes that have larger mobilized volumes. The failure in these slopes occurs in two distinctive stages:

(i)

the blocks near the slope face are separated and slide along pre-existing joints;

(ii)

the subsequent blocks farther from the slope face are failed and slide along pre-existing joints, as shown in the slope with a joint dip angle of 10° in Figure 18.

In these models, the failure occurred partially along pre-existing joints (shear failure) and partially through the rock mass (tensile failure), resulting in a high number of microcracks.

5.2.2. Slopes Having Non-Persistent Joints

The slope models in group E show the impact of the non-persistent joint dip angle on the slope stability. Figure 19 illustrates that most slopes are stable and have an FS greater than 1.00. In the emphasis on FS values, the more stable slopes were noted in steeply dipping non-persistent joints. In addition, the presence of non-persistent joints does not significantly reduce the FSb% in slopes with steep joint dip angles (50–90°), in which the FSb% is less than 20%.

The x-displacement plots in Figure 20 indicate the failure mechanism of the slope models at an SRF of 5.50. A combination of failure along joint traces and coplanar and non-coplanar rock bridges generated a tensile-shear step-path failure in slopes with shallowly dipping joints (≤50°). In slopes with steep joint dip angles of 60 and 70°, the failure mechanism was evaluated as a planar failure, partially along the pre-existing joint and partially through coplanar rock bridges. A combined circular–toppling failure occurred in slopes with almost vertically dipped joints of 80 and 90°.

Figure 21 illustrates the crack initiation and step-path failure mechanism with a 50° joint dip angle at an SRF of 2.55. In LS-SRM, the step-path failure is a progression of a mixed tensile-shear failure through rock bridges that occur one by one from the bottom up under the action of gravity, similar to the failure mechanism in the PFC models reported by Huang et al. [51]. Since the tensile stress concentration was distributed under the gravity force, the first wing cracks began to form at the inner and outer of most of the joints’ tips, especially the joint at the lower part of the slope. At the lower part of the slope, tensile cracks were propagated in a non-coplanar rock bridge toward the closer joint tips. After that, a secondary shear crack was initiated at the tip of the joint and propagated in a coplanar rock bridge toward the first wing crack, which caused a mixed tensile-shear failure of the rock bridge. As a result of step-path slip initially formed at the lower part of the slope, higher tensile stress concentrations formed in the upper part of the slope and caused tensile (crown) cracks initiated from the joint tip perpendicular to the joint slope and propagated to the surface of the slope. Due to gravity, the upper blocks began to move and slip, resulting in step-path failure.

The unit volume of failure zones in Figure 22 shows that the slopes with unfavorably oriented non-persistent joints produce significantly larger failure zones than ones with favorably oriented joints. The number of cracks was significantly higher on all slopes as the rock mass strength was reduced, where lattice springs at the bottom of the slope failed under the gravitational force, forming wing cracks [51].

5.3. The Effect of the Rock Bridge Length and Overlapping Echelon

Zhang [52] reported that the dimensions of rock bridges are estimated using the joint spacing, number of joint sets, and joint persistence. Consequently, decreases in the joint persistence increase the size of the rock bridge, enhancing the strength of the jointed rock mass. The results of Group F, shown in Figure 23, which included slope models with varying rock bridge lengths, demonstrated that the increases in RBL slightly increased the slope stability. The joint persistence of 67% was found to be a critical limit; no apparent impact of joint persistence was observed higher than this value.

Rock bridge length does not make significant differences in the FS values. The failure mechanism of the slope at an SRF of 3.00, which was displayed through x-displacement contours (Figure 24), is notably different. A tensile-shear step-path failure appeared in slopes with a rock bridge length of 1, 3, and 5 m, whereas a shallow surface and tensile step-path failure occurred in slopes with higher rock bridge length. The unit volume of failure zones decreased as the rock bridge increased. Thus, the macrocrack number decreased when the rock bridge increased, as shown in Figure 25.

Finally, the slope models in group G, designed to analyze the impact of overlapping non-coplanar joint traces on the stability of the slope, were investigated. The results demonstrated that these slope models had a factor of safety of around 3.50 and an FSb reduction of 36%. Figure 26 illustrates that the failure mechanism is nearly identical for all slopes at an SRF of 4.10.

Fully and 50% overlapped joints had almost identical unit volumes of failure zones. However, in the non-overlapping joint slope, a notably smaller failure zone was observed. Accordingly, it can be concluded that the progression of failure was more difficult in the crack tip propagation of non-overlapped joints, and therefore, a smaller mobilized zone was obtained.

5.4. A Comparison of the Slopes Having Persistent and Non-Persistent Joints

The strength of the intact rock bridge dominated the deformability and strength of the non-persistent jointed rock mass. The existence of a rock bridge, which was significantly more robust than the rock mass, enhanced the stability of rock slopes. The results indicated that the factor of safety of the slope models with non-persistent joints is higher than in fully persistent joints. In addition, most non-persistent jointed slopes had an FSb reduction of less than 60%, whereas persistent jointed slopes had an FSb reduction of more than 90%, as shown in Figure 27.

Furthermore, the unit volumes of failure zones for the first unstable conditions and the failure mechanisms were entirely different in slopes with persistent and non-persistent joints. For instance, Figure 28 illustrates the mobilized zones with fully persistent and non-persistent jointed slopes with a 20° joint dip angle at an SRF of 2.70. It is clearly shown that the fully persistent jointed slope had a significantly larger failure zone than the non-persistent jointed slope. A planar-shear failure occurred along the pre-existent persistent slope at a low SRF (Figure 27A). Conversely, a tensile step-path failure was limited around the slope face where the shear failure occurred along pre-existent joints, and non-coplanar rock bridges failed in tension.

Slope models with favorably oriented joints had the highest FS in both persistent and non-persistent slope models in Groups C and E. Furthermore, the failure mechanism on these slopes was convergent, where a circular-shaped rock mass failure was presented.

6. Conclusions

The presence of discontinuities affects both the strength and deformability of rock mass. In this paper, a parametric study on persistent and non-persistent joints was performed to quantify the influence of joints and rock bridge parameters on the stability of rock slopes. Seven groups of slope models varying in their geometries of joint configurations were simulated using the LS-SRM approach. The effect of jointing was studied based on the base case of a slope with no joints, which had the highest safety factor of 5.50. The results show that fully persistent jointed slopes significantly impact the FS more than non-persistent joints, confirming the importance of considering joint persistence in the slope stability analysis. The mobilized zones in slopes were entirely different in persistent and non-persistent cases; smaller and limited zones were noted in slopes with non-persistent joints. Rock slopes with fully persistent joints resulted in larger failure zones than slopes with non-persistent jointing, especially in slopes with joint dip angles between 10 and 40°.

Four failure mechanisms were observed in the slope models: (1) planar failure in slopes with daylight fully persistent joints, (2) combined toppling-circular failures in slopes with favorable persistent and non-persistent joints, (3) shear-tensile step-path failures in coplanar and non-coplanar rock bridges in slopes with non-persistent joints dipping 10 and 50° out the slope face, and (4) step-path failures in a pattern of tensile failure through non-coplanar rock bridges in slopes with higher rock bridge length.

The number of microcracks increased with an increased depth and size of mobilized failure zones. Also, the first wing cracks, formed due to the tensile stress concentration under the gravity force, were significantly higher in slopes with a higher SRF because of the weakness of rock mass.

The increase in joint spacing and rock bridge length increased the size of the rock bridge, enhancing the stability of rock slopes. Even though the overlapping of joint traces altered the geometry of the rock bridge, no variations in the factor of safety or failure mechanisms were observed in slope models with various overlapping percentages, and smaller mobilized zones were observed in the non-overlapped joints case.