This paper presents a comprehensive engineering method to investigate the failure mechanism of the jointed rock slopes. The field geology survey is first carried out to obtain the slope joint data. A joint network model considering the structural complexity of rock mass is generated in the PFC software. The synthetic rock mass (SRM) approach for simulating the mechanical behavior of jointed rock mass is employed, in which the flat-jointed bonded-particle model (FJM) and smooth joint contact model (SJM) represent intact rock and joints, respectively. Subsequently, the effect of microparameters on macromechanical properties of rock is investigated for parameter calibration. Moreover, the scale effect is analyzed by multiscale numerical tests, and the representative elementary volume (REV) size in the selected research area is found as 16 m × 16 m × 16 m. The microparameters of the SRM model are calibrated to match the mechanical properties of the engineering rock mass. Finally, an engineering case from Shuichang open-pit mine is analyzed and the failure process of the slope during the excavation process from micro- to macroscale is obtained. It has been found that failure occurs at the bottom of the slope and gradually develops upwards. The overall failure of the slope is dominated by the shallow local tension fracture and wedge failure.
The rock mass is a geological body with fractures complexly distributed. These natural discontinuities such as joints and faults are ubiquitous in rock, which have an important impact on the mechanical behaviors of rock material and lead to the anisotropy, heterogeneity, and scale effect [
Considerable research studies have been carried out on the joint network modelling and analysis methods of jointed rock mass [
Particle-based models, which were originally introduced by Cundall and Strack [
The remainder of this paper is organized as follows. Section
The synthetic rock mass (SRM) approach has been implemented in PFC software, where particles are rigid spherical bodies joined by deformable contacts, to solve problems by using the explicit formulation of the distinct element method (DEM). This new technique can be used as a virtual laboratory to conduct numerical experiments to represent in a qualitative and quantitative mechanical behavior of jointed rock mass and has already made it possible to extend the volumes of rock at the scale of 10–100 m containing thousands of nonpersistent joints. In PFC software, the SRM brings together two well-established techniques: bonded-particle model for intact rock deformation and fracture and discrete fracture network (DFN) for joints. That is to say, the SRM model represents the jointed rock mass as an assembly of intact rock and an embedded DFN. The main components of SRM sample are shown in Figure
Synthetic rock mass basic components: (a) intact rock; (b) DFN; (c) SRM.
FJM (flat-jointed bonded-particle model) was proposed based on the distinct element method to simulate the mechanical behavior of intact rock in PFC software by Potyondy [
A typical schematic diagram of flat joint contact.
The smooth joint contact model (SJM) simulates the behavior of a smooth interface, regardless of the local particle contact orientations along the interface [
A smooth joint model (from PFC software manual [
A smooth joint can be envisioned as a set of elastic springs uniformly distributed over a circular cross section, centered at the contact point and oriented parallel with the joint plane. The area of the smooth-joint cross section is given by
When the unbonded SJM model is specified, the incremental force calculation is performed depending on the elastic portion of the displacement increment
If
The natural joints often comprise complex network and dominate the mechanical behaviors of jointed rock mass. Therefore, it is important to obtain the geometrical characteristic representation of complex two or three-dimensional joint network for modelling jointed rock mass. However, it is impossible to directly obtain the in situ joint data deep in the rock. The details of 3D structures of joint network are usually inferred from the observation information of lower dimensional limited exposures like borehole logging, outcrop scanline, or window mapping. A large number of field survey data show that the joints in the rock mass mostly appear in sets, and it is necessary to identify the dominant sets of joints which highly affect the mechanical behavior of rock.
In this paper, a field survey was carried out in the north of Shuichang open-pit iron mine by using borehole logging and scanline methods. A total of 311 sets of joint are measured, and the data show that joints in this area are well developed. In addition, some joints have potential adverse effects on slope stability because the strike and dip direction of joints are almost the same with those of the slope. Statistical analyses are conducted on the data by using rose diagram or stereogram so that joints can be grouped into different sets with their orientations (Figure
Statistical analysis results: (a) contour plot of the joint plane; (b) rose diagram of dip direction.
By using the hierarchical cluster method, the measured data were grouped into four dominant orientations, namely, A, B, C, and D (Table
Range of dominant orientations.
ID | A | B | C | D |
---|---|---|---|---|
Dip direction (°) | 40–70 | 100–170 | 210–240 | 280–340 |
Dip angle (°) | 13–75 | 19–86 | 25–77 | 19–69 |
Number | 56 | 78 | 61 | 53 |
The characteristic of distribution of joints can be described by geometric parameters such as dip direction, dip angle, and spacing. To provide necessary information for the subsequent establishment of joint network model, probability distribution modes of those parameters are studied with orientation characterized by a uniform, normal, or negative exponential distribution, and the histogram and fitting curve are shown in Figure
Histogram and probability density: (a) group A; (b) group B; (c) group C; (d) group D.
The basic parameters and distributions for dominant orientations in Shuichang open-pit iron mine are obtained as shown in Table
The basic parameters and distribution for dominant orientations.
Parameters | ID | ||||
---|---|---|---|---|---|
A | B | C | D | ||
Dip direction (°) | Mean value | 57 | 131 | 249 | 318 |
Standard deviation | 8.39 | 20.3 | 8.58 | 20.76 | |
Distribution law | P11 | P22 | P11 | P11 | |
Dip angle (°) | Mean value | 44 | 54 | 41 | 42 |
Standard deviation | 16.25 | 14.11 | 16.53 | 15.44 | |
Distribution law | P22 | P22 | P22 | P22 | |
Half trace length (m) | Mean value | 1.68 | 2.18 | 1.77 | 1.96 |
Standard deviation | 0.87 | 0.85 | 0.63 | 1.01 | |
Distribution law | P33 | P33 | P33 | P33 | |
Spacing (m) | Mean value | 0.26 | 0.35 | 0.31 | 0.43 |
Standard deviation | 0.11 | 0.18 | 0.14 | 0.29 | |
Distribution law | P33 | P33 | P33 | P33 |
1P1 means uniform; 2P2 means normal; 3P3 means negative exponential.
The shape of the joint in three-dimensional model is regarded as planar discs of radius in PFC. In this case, the probability density function of the disc’s radius is calculated as
The mean radius of joint disc is given as
In a relatively homogeneous area, the volume density is independent of direction and relatively constant and has the following relationship with the average linear density [
Based on formulas (
According to formulas (
The parameters of joint disc.
ID | Mean radius of disc (m) | Linear density of joint (crack/m) | Volume density of joint (crack/m3) |
---|---|---|---|
A | 2.14 | 3.81 | 0.1325 |
B | 2.77 | 2.75 | 0.0612 |
C | 2.25 | 3.22 | 0.1047 |
D | 2.50 | 2.28 | 0.0657 |
The Monte Carlo method is a good tool for joint network modelling, which can approximately solve uncertain problems with a series of random numbers. In this paper, combined with the Monte Carlo method, the built-in language Fish provided by PFC is used to construct a three-dimensional joint network model. The random numbers of joint disc parameters in accordance with corresponding probability distribution are generated in a 10 m × 10 m × 10 m cube space. Based on the OPEN GL technique, the joint network model in PFC is shown in Figure
Stochastic joint network model.
Then, the reliability of this model is tested by comparing the distribution of main joints in the effective area between the two-dimensional section of the model and the window of field survey. Figure
Comparison and verification of joints: (a) actual joints; (b) simulation joints.
The input microparameters in PFC software cannot be measured directly through conventional laboratory tests, which also have highly nonlinear relationships with the parameters such as Young’s modulus, uniaxial compression strength, and Poisson’s ratio in continuum numerical simulation. Therefore, parameter calibration is an essential part of simulation process in PFC to ensure that the microparameters can represent the macromechanical properties of rock well.
Before the calibration process, a series of conventional laboratory tests are carried out on Shuichang iron mine granite to obtain the mechanical parameters. The mechanical test equipment is shown in Figure
Mechanical test equipment: (a) uniaxial and triaxial compression test (TAW-2000); (b) direct shear test (XYJ-1).
Mechanical properties of Shuichang open-pit iron mine granite from laboratory tests.
Material | Property | Value | Property | Value |
---|---|---|---|---|
Rock | Density, | 2640 | Young’s modulus, | 55.23 |
Poisson’s ratio, | 0.183 | Uniaxial compression strength, | 125.71 | |
Cohesion, | 12.35 | Internal friction angle, | 41.51 | |
Joint | Tensile strength, | 6.65 | ||
Cohesion, | 27.83 | Internal friction angle, | 21.06 |
As is known, calibration of the microparameters in PFC is a complicated process because it is impossible to describe the relationship between macro- and microparameters with quantitative mathematical relation. The basic method to determine the microparameters is “trial-and-error.” The specific method is as follows: by keeping other parameters constant, investigate the effect of one single parameter on simulation results, compare the macroparameters from laboratory tests results, and then repeat the above steps time and again until the parameter could be in good agreement with the observed macroscale response.
In this paper, the microparameters of the FJM in PFC, which affect the macromechanical properties of numerical sample, should be determined as shown in Table
Microparameters of FJM.
Microparameters | Description | Unit |
---|---|---|
Minimum particle radius | mm | |
Particle size ratio | — | |
Radius multiplier | — | |
Number of elements in radial direction | — | |
Number of elements in circumferential direction | — | |
Effective modulus of both particle and bond | GPa | |
Normal-to-shear stiffness ratio of both particle and bond | — | |
Friction coefficient | — | |
Bond tensile strength | MPa | |
Bond cohesion strength, | MPa | |
Φ | Friction angle | ° |
Based on previous research, there are six microparameters that have significant effects on the numerical sample in the FJM, i.e.,
Numerical model for rock mechanic test: (a) uniaxial compression test; (b) triaxial compression test; (c) direct tension test.
In the FJM, the calibration process is time-consuming due to no one-to-one correspondence existing between macroparameters and microparameters. For example, the UCS, Poisson’s ratio, and cohesion occur in varying degrees of change with the change of microparameter
Results of range analysis.
Macroparameters | Range value and ranking | Microparameters | |||||
---|---|---|---|---|---|---|---|
Φ | |||||||
15.882 | 18.656 | 21.266 | 23.886 | 107.58 | 13.212 | ||
Ranking | |||||||
56.56 | 18.296 | 16.4 | 8.638 | 9.784 | 17.104 | ||
Ranking | |||||||
0.1001 | 0.29348 | 0.10572 | 0.13274 | 0.14612 | 0.099 | ||
Ranking | |||||||
0.742 | 1.3158 | 0.9906 | 5.598 | 0.8236 | 0.464 | ||
Ranking | |||||||
4.346 | 8.518 | 7.84 | 3.818 | 9.14 | 2.514 | ||
Ranking | |||||||
2.572 | 4.53 | 8.71 | 3.17 | 5.66 | 10.124 | ||
Ranking | Φ |
On the basis of aforementioned analytical results, a new improved trial-and-error method with high efficiency is used to calibrate the FJM. The final microparameters obtained are given in Table
Microparameters of the FJM after calibration.
Microparameter | Φ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Value | 1.0 | 1.66 | 1 | 1 | 3 | 76 | 1.8 | 0.4 | 14 | 135 | 5 |
Then, the microparameters were verified by using the numerical test. Figure
Comparison of uniaxial compressive test from experimental and numerical results.
The comparison of basic mechanical parameters between the experimental and numerical results is shown in Table
Comparison of mechanical parameters between experimental and numerical tests.
Mechanical property | ||||||
---|---|---|---|---|---|---|
Experimental result | 55.23 | 0.183 | 125.71 | 12.35 | 41.51 | 6.65 |
Numerical result | 55.75 | 0.190 | 128.61 | 13.54 | 41.73 | 6.86 |
Error (%) | 0.94 | 3.83 | 2.31 | 9.64 | 0.53 | 3.16 |
The macroparameters of the SJM are usually determined by simulating the direct shear test sample of rock joint. In this paper, an unbonded SJM is selected, that is, the parameter “
Microparameters of the SJM after calibration.
Microparameter | Description | Value |
---|---|---|
Normal stiffness | 2.5 × 107 | |
Shear stiffness | 2.5 × 107 | |
Friction coefficient | 0.5 | |
Dilation angle | 5 | |
Joint bond state | 0 | |
Bonded tensile strength | 0 | |
Bonded cohesion | 0 | |
Bonded friction angle | 0 |
The shear stress-shear displacement curves under different normal stresses from experimental and numerical direct shear tests are shown in Figure
Shear stress-shear displacement curves under different normal stresses: (a) numerical results; (b) experimental results.
Due to the randomness in the distribution of joints, scale effect can substantially influence the rock strength, namely, the strength of a region decreases with increase in region size up to the point at which a representative size is reached. Compared with the engineering rock mass, the laboratory samples are usually small and do not contain systematic joints which affect the rock strength. For this reason, the microparameters of engineering rock mass in the SRM model should be considered for the potential impacts of joints and should be calibrated to match a strength that reflects the size of a typical rock mass, instead of a core sample. Hence, a concept named “representative elementary volume (REV)” is introduced, which is the minimum size with which the rock mechanical properties can be treated as equivalent continuous [
In order to investigate the scale effect and REV size of rock mass, seven SRM models with different sizes were established to conduct numerical uniaxial compression tests. These cubical models with side lengths of 1, 2, 6, 10, 14, 16, and 20 m, respectively, are shown in Figure
Multiscale SRM models.
Figure
Stress-strain curve of the multiscale SRM model.
The results are presented in Figure
Scale effect analysis of different strength parameters: (a) Young’s modulus; (b) UCS.
In this paper, combined with the results from the laboratory tests, the mechanical parameters of engineering rock mass were estimated by using RocLab software based on the Hoek–Brown (H-B) method. The failure envelope curve of engineering rock mass in Shuichang open-pit iron mine is shown in Figures
Normal stress-shear stress curve.
Minor-major principal stress curve.
Subsequently, a series of sample models were established based on the REV size to conduct numerical tests. Making use of the method mentioned in Section
The microparameters of engineering rock mass in the SRM model.
Microparameter | FJM | SJM | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
Φ | ||||||||||
Value | 18 | 1.6 | 0.5 | 0.18 | 0.84 | 6 | 2.5 × 107 | 2.5 × 107 | 0.5 | 5 |
The comparison of basic mechanical parameters between the Hoek–Brown method and numerical results is shown in Table
Comparison of mechanical parameters between the H-B method and numerical tests.
Mechanical property | |||||
---|---|---|---|---|---|
H-B method | 14.09 | 9.153 | 3.513 | 41.73 | 0.154 |
Numerical result | 13.15 | 10.041 | 3.814 | 43.36 | 0.151 |
Error (%) | 6.671 | 9.702 | 8.568 | 3.906 | 1.948 |
The field joint survey was carried out on the north side of the open-pit mine slope, No. II mining area of Shuichang iron mine. Rock mass quality in this area is declined due to the well-developed discontinuous nature of rock such as fault and joint, causing local instability of slope. Records show multiple rock fall and landslips occurred in this area, especially at elevation 44m–116 m. A 3D numerical model is developed using the PFC software in order to investigate the micromechanism of failure of jointed rock slope during excavation process.
The model takes the direction of perpendicular to the stope as the
Jointed rock slope model before excavation of Shuichang open-pit iron mine.
In PFC, particles are rigid spherical bodies with bonded contacts representing intact rock. And those contacts will be broken in shear mode or tension mode when external force exceeds the bonding strength. Therefore, the number, mode, and propagation of crack can be used to analyze the failure evolution. In this paper, the model is excavated by four steps, one bench at a time. Subsequently, the failure mechanism during the excavation process was investigated from micromechanical viewpoint based on PFC software.
Figure
The number of microcracks during excavation.
Figure
Evolutionary process of microcrack.
Diagrams of velocity vector (Figure
Velocity vector diagram.
Displacement vector diagram.
Figure
Failure mode of the slope: (a) simulation result; (b) current situation.
In this paper, the SRM model was established in PFC software to represent the mechanical properties of rock mass from Shuichang open-pit iron mine. Based on the prepared model, the failure mechanism during excavation process was investigated from the micromechanical viewpoint. The conclusions of this study are as follows: statistical analysis method can act as a useful tool to group the dominant orientations of joints and determine the geometric parameters such as dip angle, size, and trace length. The joint distributions of Shuichang slope were obtained by the field survey and quantitatively described in the numerical model. The effects of different microparameters on macromechanical properties of rock are different. Through the orthogonal experiment and variance analysis, the calibration and optimization of the parameters were accomplished. The mechanical properties including stress-strain curve and failure pattern from numerical results were verified with those from experimental results. The existence of joints results in scale effect and anisotropic behavior of rock mass, and these properties tend to gradually weaken with the increasing rock block size. By carrying out numerical tests on multiscale SRM models, the representative elementary volume (REV) in the selected research area was obtained with the size 16 m × 16 m × 16 m. Then, the microparameters of the SRM model were calibrated to match the mechanical parameters of the engineering rock mass. SRM is an effective method to analyze the failure evolution of jointed rock slope. The failure of the slope was dominated by the tensile microcracks between bonded particles. The microcracks primarily occurred at the bottom of the slope and gradually developed upwards. In addition, microcracks were mainly distributed on the shallow part of the slope. After excavation, the wedge occurred in the middle and bottom part of the slope (Appendix) (Tables
Microparameters of orthogonal test.
Plan | Φ | |||||
---|---|---|---|---|---|---|
1 | 20.00 | 2.00 | 0.40 | 12.00 | 110.00 | 15.00 |
2 | 110.00 | 2.00 | 1.00 | 6.00 | 150.00 | 35.00 |
3 | 20.00 | 1.00 | 0.20 | 6.00 | 70.00 | 5.00 |
4 | 40.00 | 3.00 | 1.00 | 14.00 | 110.00 | 5.00 |
5 | 20.00 | 2.50 | 1.00 | 10.00 | 130.00 | 45.00 |
6 | 80.00 | 2.00 | 0.60 | 10.00 | 90.00 | 5.00 |
7 | 110.00 | 2.50 | 0.60 | 14.00 | 70.00 | 15.00 |
8 | 80.00 | 1.00 | 0.40 | 14.00 | 150.00 | 45.00 |
9 | 40.00 | 2.00 | 0.80 | 8.00 | 70.00 | 45.00 |
10 | 60.00 | 3.00 | 0.40 | 10.00 | 70.00 | 35.00 |
11 | 60.00 | 2.00 | 0.20 | 14.00 | 130.00 | 25.00 |
12 | 20.00 | 3.00 | 0.60 | 8.00 | 150.00 | 25.00 |
13 | 80.00 | 3.00 | 0.80 | 6.00 | 130.00 | 15.00 |
14 | 110.00 | 3.00 | 0.20 | 12.00 | 90.00 | 45.00 |
15 | 110.00 | 1.50 | 0.40 | 8.00 | 130.00 | 5.00 |
16 | 80.00 | 1.50 | 1.00 | 12.00 | 70.00 | 25.00 |
17 | 40.00 | 1.50 | 0.20 | 10.00 | 150.00 | 15.00 |
18 | 110.00 | 1.00 | 0.80 | 10.00 | 110.00 | 25.00 |
19 | 60.00 | 2.50 | 0.80 | 12.00 | 150.00 | 5.00 |
20 | 20.00 | 1.50 | 0.80 | 14.00 | 90.00 | 35.00 |
21 | 80.00 | 2.50 | 0.20 | 8.00 | 110.00 | 35.00 |
22 | 60.00 | 1.50 | 0.60 | 6.00 | 110.00 | 45.00 |
23 | 60.00 | 1.00 | 1.00 | 8.00 | 90.00 | 15.00 |
24 | 40.00 | 1.00 | 0.60 | 12.00 | 130.00 | 35.00 |
25 | 40.00 | 2.50 | 0.40 | 6.00 | 90.00 | 25.00 |
Calculated macroparameters of orthogonal test.
Plan | UCS | TS | ||||
---|---|---|---|---|---|---|
1 | 106.88 | 14.25 | 0.2284 | 9.807 | 19.90 | 49.40 |
2 | 200.36 | 72.51 | 0.3986 | 4.96 | 33.54 | 53.10 |
3 | 63.35 | 17.61 | 0.1074 | 5.93 | 19.31 | 29.06 |
4 | 129.14 | 24.78 | 0.3519 | 10.43 | 29.25 | 42.55 |
5 | 155.18 | 12.48 | 0.4105 | 7.75 | 27.45 | 52.43 |
6 | 104.03 | 58.98 | 0.2012 | 8.17 | 25.52 | 39.12 |
7 | 97.28 | 77.64 | 0.1963 | 10.83 | 24.23 | 38.46 |
8 | 172.42 | 70.83 | 0.1108 | 8.42 | 36.36 | 45.22 |
9 | 109.90 | 29.02 | 0.2393 | 6.56 | 17.70 | 56.28 |
10 | 87.02 | 36.01 | 0.3577 | 7.41 | 19.30 | 44.00 |
11 | 109.63 | 40.45 | 0.2712 | 11.42 | 24.05 | 44.16 |
12 | 130.47 | 55.78 | 0.1073 | 5.95 | 25.44 | 49.13 |
13 | 136.94 | 43.27 | 0.5526 | 4.41 | 27.81 | 47.21 |
14 | 96.59 | 61.62 | 0.4252 | 8.86 | 20.63 | 45.47 |
15 | 118.23 | 83.75 | 0.2174 | 6.98 | 27.02 | 42.04 |
16 | 116.38 | 67.68 | 0.1085 | 10.61 | 24.75 | 44.48 |
17 | 108.39 | 28.81 | 0.2592 | 8.789 | 23.30 | 45.19 |
18 | 155.92 | 104.16 | 0.0686 | 9.88 | 37.77 | 39.27 |
19 | 154.22 | 37.83 | 0.3760 | 9.30 | 32.35 | 45.72 |
20 | 133.09 | 16.76 | 0.1200 | 12.03 | 29.36 | 43.67 |
21 | 97.65 | 44.29 | 0.5010 | 6.19 | 21.25 | 44.72 |
22 | 138.49 | 45.49 | 0.2425 | 5.23 | 26.45 | 49.71 |
23 | 130.88 | 58.14 | 0.0563 | 7.86 | 32.75 | 37.74 |
24 | 166.91 | 36.89 | 0.0861 | 9.56 | 37.11 | 43.12 |
25 | 91.87 | 23.91 | 0.4128 | 4.61 | 20.36 | 44.06 |
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This research was funded by the National Natural Science Foundation of China (grant no. 51034001).