# Novel Intelligent Approach for Peak Shear Strength Assessment of Rock Joints on the Basis of the Relevance Vector Machine

Authors: Caichu Xia; Man Huang; Xin Qian; Chenjie Hong; Zhanyou Luo; Shigui Du
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

This study mainly establishes a novel intelligent assessment model for peak shear strength of rock joints based on the relevance vector machine (RVM). RVM is a state-of-the-art soft computing technique that has been rarely utilized in joint shear strength assessment. To establish the hybrid intelligent model, three-dimensional scanning tests and direct shear tests on 36 granite joint specimens were conducted. The peak shear strength ratio (τp/σn) is perceived as an explanation of four types of influencing factors, including joint surface roughness, strength of rock material, basic friction angle, and normal stress. In particular, the compressive strength and tensile strength of rock material are first considered together. A total of 36 experimental data were used in this study to train the RVM model to predict the peak shear strength of rock joints. The performance of the RVM model was assessed using the direct shear test data of rock joints collected from previous researches. Four different kinds of kernel functions were adopted to obtain the optimal model. Results show that the proposed model is significantly efficient in predicting the peak shear strength of rock joints. The proposed model is also a promising tool for peak shear strength of rock joints and provides a new research approach to research the mechanical properties of rock joints.


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## Body

## 1. Introduction

The deformation capacity, strength, and stability of fractured rock mass are strongly influenced by the shear behavior of joints. Therefore, the accurate determination of shear strength is the key to design the safety structure in rock mass or rock masses [1]. Over the past decades, empirical [2–9], semi-theoretical [10], and theoretical methods [11] have been proposed to determine the shear strength of rock joints. Various factors, such as rock type, joint surface roughness, joint size, and infilling materials, exhibit a wide variation of joint shear strength [12, 13]. Considering their strong randomness, fuzziness, and uncertainty, the peak shear strength of rock joints has a complex nonlinear relationship with the aforementioned factors [14, 15]. The traditional methods demonstrate strong usability and dependability in practical engineering, but large computations and complex algorithms limit the use of these methods. Thus, the deterministic analysis methods in assessing the joint shear strength remain difficult. In recent years, data mining and artificial intelligence technologies, such as artificial neural network (ANN) and support vector machine (SVM), have been widely used to estimate the mechanical properties of rocks, such as the compressive strength and tensile strength [16–21]. However, ANN has several disadvantages, such as poor generalizing performance, large training data requirement, slow convergence velocity, and over-fitting problem [22–24]. SVM can effectively solve the aforementioned problems and exhibit excellent generalization capability and high regression accuracy [25, 26]. Despite many advantages, SVM suffers from several limitations. The output of an SVM represents decisions rather than posterior probabilities [27]. In addition, the kernel function of SVM must fulfill Mercer’s condition. The relevance vector machine (RVM) is a probabilistic model based on Bayesian theory for regression and classification; it has high generalization capability, sparse model structure, and low computation complexity, avoiding the principal limitations of SVM [28]. Especially for small samples, RVM can map the limited input data into a high-dimensional feature space and finally use several relevance vectors to deal with the dimensionality problems. The primary purpose of this study is to introduce a reliable estimation method of peak shear strength of rock joints on the basis of RVM. To this end, 36 rock joints under different stress were designed and tested in laboratory. The rest of this article is organized as follows: Section 2 presents the theoretical background of RVM. Section 3 introduces the description of the experimental procedure and results. Section 4 describes the establishment of the RVM model. Section 5 presents the comparison and discussion. The final section presents the conclusions.


## 2. RVM Method

RVM is a probabilistic learning model based on the Bayesian framework to solve complex regression and classification problems [28]. It generates a mapping between the targets tnn=1N and the corresponding input vectors xnn=1N.(1)tn=yxn;ω+εn=∑n=1NωnKx,xn+ω0+εn,where N is the total sample number, n is the nth sample number, y is the output variable, ωn is the weight of the nth sample, Kx,xn is the kernel functions, and εn is a mean-zero white noise process with variance σ2.Assuming an independent distribution fortn, the likelihood of the entire dataset can be expressed as follows:(2)ptω,σ2=2πσ2−N/2exp−t−Φω22σ2,where t=t1,t2,…,tNT, ω=ω0,ω1,…,ωNT, and Φxn=1,Kxn,x1,…,Kxn,xNT.Direct maximum-likelihood estimation ofω from equation (2) may suffer from overfitting [29]. Therefore, a zero-mean Gaussian prior distribution over every weight ωi with the diagonal covariance α is adopted and described as follows:(3)pωα=∏i=0NNωi0,αi−1.The posterior distribution for the weights can then be calculated using Bayes’ law based on the Gamma prior distributions [30]:(4)pωt,α,σ2=ptω,σ2pωαptα,σ2.Integrating the weight parameters, the marginal distribution of hyperparametersα and σ2 can be obtained:(5)ptα,σ2=2π−N/2Ωexp−tTΩ−1t2,where Ω=σ2E+ΦA−1ΦT, A=diagα0,α1,…,αN, and E is a unit vector. Taking its partial derivatives to equation (5) by iteration, the values of α and σ2 can be obtained.


## 3. Experimental Procedure and Results

### 3.1. Preparation of Rock Samples

To investigate the effect of wide-range normal stress on peak shear strength, rectangular granite joints were selected for the direct shear tests because of their high strength. Rectangular blocks of rock specimens with dimension of 100 mm × 100 mm × 200 mm were labeled as SJ1–SJ36. Brazilian splitting tests were conducted by using a TSF-600 shear-flow coupled test system for rock joints at Tongji University. The granite joint surface dimensions were approximately 100 mm wide and 200 mm long (Figure1). After testing, no apparent defects were found in the specimens. σc is the uniaxial compressive strength, σt is the tensile strength, E is Young’s modulus, ρ is the rock density, and γ is Poisson’s ratio, and basic friction angle φb was determined using the suggested methods of the International Society for Rock Mechanics (ISRM) [31], and the results are listed in Table 1.Figure 1
Granite block and granite joint specimen.
(a)
(b)Table 1
Physical and mechanical parameters of granite rock.
σ
c (MPa)
σ
t (MPa)
E (GPa)
Ρ (g/cm3)
γ
φ
b
161.0
10.8
45.5
2.64
0.23
33.28°


### 3.2. Surface Topography Measurement

To investigate the influence of the three-dimensional (3D) surface roughness on the mechanical properties of rock joints quantitatively, the joint surface topography of measuring each shear test sample block before using a noncontact direct shear test, 3D stereo-topometric measurement system (positioning accuracy: 20μm) was developed by Xia et al. [32]. Before the estimation of joint surface roughness, the 3D surface topography was reconstructed on the basis of the interpolation of the scanned point clouds. Tatone and Grasselli indicated that the digitized surface topographies of rock joints with sampling interval <0.55 mm using linear interpolation technique can capture microfeatures of surface morphology [33]. In this study, we set the sampling intervals in length (L) and width (W) directions to be 0.3 mm. Taking upper surface topographies of four joint samples as examples, the digitized surface topographies of rock joints (the height magnified to 3 times its normal size) are shown in Figure 2.Figure 2
3D digitized upper surface topographies of rock joints: (a) SJ3, (b) SJ12, (c) SJ21, and (d) SJ30.
(a)
(b)
(c)
(d)


### 3.3. Test Procedure and Results

This test was designed to obtain an extensive peak shear strength model of rock joints. We conduct the direct shear tests on the test samples under nine normal stress levels, namely, 1.0, 2.0, 3.0, 5.0, 8.0, 10.0, 12.0, 15.0, and 20.0 MPa. Four direct tests were performed on the test specimen at each level of normal stress. The values of the peak shear strength for each test specimen are listed in Table2. The rate of shear displacement was set to be 0.5 mm/min, and the maximum displacement was 20 mm which is 10% of the joint length.Table 2
Peak shear strength of test specimens.
Sample name
σ
n (MPa)
τ
p (MPa)
SJ1
1
2.30
SJ2
2
4.55
SJ3
3
4.78
SJ4
5
8.30
SJ5
8
9.63
SJ6
10
12.75
SJ7
12
13.64
SJ8
15
17.28
SJ9
20
21.26
SJ10
1
1.89
SJ11
2
3.64
SJ12
3
7.41
SJ13
5
8.28
SJ14
8
11.15
SJ15
10
13.23
SJ16
12
14.61
SJ17
15
18.07
SJ18
20
19.76
SJ19
1
3.07
SJ20
2
3.65
SJ21
3
5.46
SJ22
5
7.05
SJ23
8
11.92
SJ24
10
12.57
SJ25
12
14.49
SJ26
15
16.15
SJ27
20
20.99
SJ28
1
2.49
SJ29
2
5.21
SJ30
3
5.29
SJ31
5
7.70
SJ32
8
13.18
SJ33
10
12.23
SJ34
12
15.44
SJ35
15
17.74
SJ36
20
24.91


## 3.1. Preparation of Rock Samples

To investigate the effect of wide-range normal stress on peak shear strength, rectangular granite joints were selected for the direct shear tests because of their high strength. Rectangular blocks of rock specimens with dimension of 100 mm × 100 mm × 200 mm were labeled as SJ1–SJ36. Brazilian splitting tests were conducted by using a TSF-600 shear-flow coupled test system for rock joints at Tongji University. The granite joint surface dimensions were approximately 100 mm wide and 200 mm long (Figure1). After testing, no apparent defects were found in the specimens. σc is the uniaxial compressive strength, σt is the tensile strength, E is Young’s modulus, ρ is the rock density, and γ is Poisson’s ratio, and basic friction angle φb was determined using the suggested methods of the International Society for Rock Mechanics (ISRM) [31], and the results are listed in Table 1.Figure 1
Granite block and granite joint specimen.
(a)
(b)Table 1
Physical and mechanical parameters of granite rock.
σ
c (MPa)
σ
t (MPa)
E (GPa)
Ρ (g/cm3)
γ
φ
b
161.0
10.8
45.5
2.64
0.23
33.28°


## 3.2. Surface Topography Measurement

To investigate the influence of the three-dimensional (3D) surface roughness on the mechanical properties of rock joints quantitatively, the joint surface topography of measuring each shear test sample block before using a noncontact direct shear test, 3D stereo-topometric measurement system (positioning accuracy: 20μm) was developed by Xia et al. [32]. Before the estimation of joint surface roughness, the 3D surface topography was reconstructed on the basis of the interpolation of the scanned point clouds. Tatone and Grasselli indicated that the digitized surface topographies of rock joints with sampling interval <0.55 mm using linear interpolation technique can capture microfeatures of surface morphology [33]. In this study, we set the sampling intervals in length (L) and width (W) directions to be 0.3 mm. Taking upper surface topographies of four joint samples as examples, the digitized surface topographies of rock joints (the height magnified to 3 times its normal size) are shown in Figure 2.Figure 2
3D digitized upper surface topographies of rock joints: (a) SJ3, (b) SJ12, (c) SJ21, and (d) SJ30.
(a)
(b)
(c)
(d)


## 3.3. Test Procedure and Results

This test was designed to obtain an extensive peak shear strength model of rock joints. We conduct the direct shear tests on the test samples under nine normal stress levels, namely, 1.0, 2.0, 3.0, 5.0, 8.0, 10.0, 12.0, 15.0, and 20.0 MPa. Four direct tests were performed on the test specimen at each level of normal stress. The values of the peak shear strength for each test specimen are listed in Table2. The rate of shear displacement was set to be 0.5 mm/min, and the maximum displacement was 20 mm which is 10% of the joint length.Table 2
Peak shear strength of test specimens.
Sample name
σ
n (MPa)
τ
p (MPa)
SJ1
1
2.30
SJ2
2
4.55
SJ3
3
4.78
SJ4
5
8.30
SJ5
8
9.63
SJ6
10
12.75
SJ7
12
13.64
SJ8
15
17.28
SJ9
20
21.26
SJ10
1
1.89
SJ11
2
3.64
SJ12
3
7.41
SJ13
5
8.28
SJ14
8
11.15
SJ15
10
13.23
SJ16
12
14.61
SJ17
15
18.07
SJ18
20
19.76
SJ19
1
3.07
SJ20
2
3.65
SJ21
3
5.46
SJ22
5
7.05
SJ23
8
11.92
SJ24
10
12.57
SJ25
12
14.49
SJ26
15
16.15
SJ27
20
20.99
SJ28
1
2.49
SJ29
2
5.21
SJ30
3
5.29
SJ31
5
7.70
SJ32
8
13.18
SJ33
10
12.23
SJ34
12
15.44
SJ35
15
17.74
SJ36
20
24.91


## 4. Establishment of a Peak Shear Strength Assessment Model Based on RVM

### 4.1. Selecting the Influencing Factors

According to the study by Barton and Choubey [34], the joint surface roughness, rock material strength, basic friction angle, and normal stress exhibit a wide variation of shear behavior. Therefore, normal stress, joint surface roughness, rock strength, and basic friction angle were selected in this work to assess the peak shear strength.Various parameters, including empirical, statistical, and fractal, have been proposed to quantify the surface roughness of rock joints. Numerous experimental tests demonstrated that partial contact between the lower and upper joint surface was located in the steepest zones facing the shear direction during the shearing [35–37]. Grasselli et al. [38] established a relationship between total potential contact area ratio Aθ∗ and the corresponding apparent dip angle θ∗, as shown in the following equation:(6)Aθ∗=A0θmax∗−θ∗θmax∗C,where A0 is the maximum potential contact area under apparent dip angle θ∗ at 0°; θmax∗ is the maximum apparent dip angle in the shear direction; and C is a dimensionless roughness parameter characterizing the distribution of the apparent dip angle over the surface, which is calculated using equation (6). Considering the possible range of C∈0,∞, the surface roughness of rock joints can be measured by the roughness metric θmax∗/C+1 [33]. A0 varies significantly in different directions; thus, the 3D roughness metric A0 and θmax∗/C+1 are both included to fully describe the roughness anisotropy. The 3D morphology parameters of the granite joints, namely, A0, θmax∗, and C, were calculated using the measured data from 36 surfaces on the basis of equation (6). The values of A0 and θmax∗/C+1 are listed in Table 3. Figure 3 shows the fitting curves of SJ3, SJ12, SJ21, and SJ36 for calculating the roughness parameter C.Table 3
3D morphology parameters of test specimens.
Sample name
A
0
θ
max
∗
/
C
+
1 (deg)
SJ1
0.479
10.465
SJ2
0.498
12.433
SJ3
0.504
10.139
SJ4
0.510
12.384
SJ5
0.467
10.870
SJ6
0.477
12.718
SJ7
0.542
10.237
SJ8
0.504
10.197
SJ9
0.472
10.208
SJ10
0.502
9.952
SJ11
0.484
11.961
SJ12
0.498
14.253
SJ13
0.528
14.405
SJ14
0.520
11.266
SJ15
0.498
10.411
SJ16
0.508
10.659
SJ17
0.501
11.699
SJ18
0.485
11.298
SJ19
0.496
12.499
SJ20
0.481
10.708
SJ21
0.505
12.808
SJ22
0.532
12.134
SJ23
0.471
14.002
SJ24
0.505
11.191
SJ25
0.499
11.398
SJ26
0.495
10.365
SJ27
0.433
10.083
SJ28
0.493
11.231
SJ29
0.498
13.550
SJ30
0.501
10.829
SJ31
0.512
11.177
SJ32
0.502
11.975
SJ33
0.454
10.467
SJ34
0.479
12.858
SJ35
0.492
9.942
SJ36
0.514
15.303Figure 3
Fitting results of the contact area ratio and apparent dip angle by equation (6): (a) SJ3, (b) SJ12, (c) SJ21, and (d) SJ30.
(a)
(b)
(c)
(d)Joint wall compressive strength (JCS) is commonly used for representing the influence of rock material strength on the shear behavior of rock joints [3, 8, 25, 39–41]. For the fresh rock, JCS can be replaced by the uniaxial compressive strength σc. Based on experimental observation and numerical simulations, several researchers have also found that the failure of asperities in rock joints under direct shear loading was possibly attributed to shear-induced tensile fracture [5, 6, 42–44]. However, the failure mechanism of asperities has not been clearly stated, and controversies still exist. Therefore, uniaxial compressive strength σc and tensile strength σt of the rock are both selected in this study to describe the effect of rock material strength on the peak shear strength of rock joints.Basic friction angleφb is crucial to determine the shear strength of rock joints. Barton suggested a fundamental form of joint shear strength [3]:(7)τp=σntanφb+dn+sn,where σn is the normal stress, dn is the peak dilation angle, and sn is the shear component angle. On the basis of equation (7), various 2D or 3D morphology parameters have been used for representing the sum of peak dilation angle dn and shear component angle sn. No uniform approach for obtaining the basic angle φb is currently available. In this work, laboratory direct shear testing method for two smooth joint surfaces is adopted to obtain the basic angle φb and the extensive shear strength model, as shown in Table 1.


### 4.2. Training and Testing of the RVM Model

In this study, the aforementioned original influencing factors, namely,σn, σc, and σt, can be dimensionless preprocessed as two factors: JCS/σn and σn/σt. Therefore, the five influencing factors, which include JCS/σn, σn/σt, A0, θmax∗/C+1, and φb, form the input vector of the regression model. All these index values can be measured in the laboratory (see Tables 1–3). Mapping the above five influencing factors to the five-dimensional feature space according to the definition of RVM, the appropriate kernel function was chosen to simplify the nonlinear transformation. The linear kernel function was first used due to its parameter-free and rapid calculation as follows:(8)Kx,z=x,z,where x,z are the training dataset of five influencing factors. The results of direct shear tests on 36 rocks in this study were used to cultivate and advance the RVM model. The results of direct shear tests by Grasselli and Egger [5] and Yang et al. [8] were used for testing the RVM model, in which a total of 50 tests were conducted on five different material types of rock joints (granite, sandstone, limestone, marble, and serpentinite). The 3D morphology parameters of 50 groups of rock joints are represented in Table 4. After the analysis was completed, the quality of the regression model was estimated by the average error formula and correlation coefficient [45]:(9)σ¯avg=1n∑i=1nτmea−τestτmea×100%,(10)Fitness_σ¯avg=train_σ¯avg⋅train_n+test_σ¯avg⋅test_ntrain_n+test_n,(11)R2=1−∑i=1nτest−τmea2∑i=1nτest−τmea¯2,(12)Fitness_R2=train_R2⋅train_n+test_R2⋅test_ntrain_n+test_n,where σ¯avg is the average estimation error, n is the sample size, τmea is the measured peak shear strength, τest is the estimated peak shear strength, τmea¯ is the average value of the measured peak shear strength, Fitness_σ¯avg is the comprehensive estimation error of the RVM model, and Fitness_R2 is the correlation coefficient of the train and test model; train_n is the number of the training sample, and test_n is the number of the testing sample. In this work, train_n and test_n are 36 and 50, respectively. The average estimation errors of the RVM model for the training and testing data are 6.46% and 21.12%, respectively. The correlation coefficient of the RVM model for the training and testing data is 0.977 and 0.769, respectively. The comprehensive estimation error Fitness_σ¯avg and correlation coefficient Fitness_R2 of the RVM model for all the data are 14.98% and 0.856, respectively. This finding indicates that the RVM model exhibits good training accuracy and high generalization capacity. Therefore, the RVM model is feasible and effective for estimating the peak shear strength of rock joints and can be used in practical applications.Table 4
Comparison between the measured peak shear strength and the calculated results by three different models.
Sample name
JCS
/
σ
n
σ
n
/
σ
t
θ
max
∗
/
C
+
1 (deg)
A
0
φ
b(deg)
Peak shear strength (MPa)
Measured
Grasselli
Yang
Proposed
G1
81.2
0.26
11.016
0.493
34
5.7
4.6
6.0
4.8
G2
81.2
0.26
12.121
0.498
34
5.6
5.1
6.0
4.8
G4
85.2
0.25
10.031
0.498
34
4.8
4.3
4.3
4.6
G5
166.7
0.13
9.005
0.46
34
2.4
2.3
2.0
2.7
G6
166.7
0.13
10.012
0.477
34
2.9
2.4
2.7
2.7
G7
166.7
0.13
9.939
0.47
34
2.8
2.4
2.6
2.7
G9
166.7
0.13
10.949
0.508
34
3.0
2.6
2.6
2.7
C1
23.3
0.45
9.963
0.491
36
2.2
1.9
2.2
1.7
C2
23.3
0.45
12.048
0.462
36
2.1
2.2
2.6
1.8
C3
6.7
1.55
10.179
0.46
36
5.5
5.5
5.1
4.8
C4
10.2
1.02
11.324
0.508
36
4.6
4.1
4.2
3.3
C5
8.1
1.30
11.821
0.495
36
5.0
5.2
5.1
4.1
C6
24.4
0.43
10.985
0.546
36
2.1
2.1
2.1
1.7
C8
8.1
1.30
11.028
0.555
36
4.9
4.9
4.8
4.1
M1
100.0
0.09
7.143
0.513
37
1.7
1.8
1.7
2.1
M2
50.0
0.19
5.909
0.492
37
2.3
3.2
2.4
3.8
M3
100.0
0.09
5.652
0.471
37
1.2
1.7
1.4
2.1
M4
23.3
0.41
6.689
0.513
37
5.8
6.3
5.7
6.3
M5
33.3
0.28
5.948
0.533
37
4.4
4.6
3.8
4.9
M6
33.3
0.28
6.082
0.45
37
4.3
4.4
4.0
4.9
M7
23.3
0.41
6.001
0.502
37
5.6
6.1
5.8
6.2
M8
22.7
0.42
6.250
0.459
37
6.4
6.1
5.7
6.3
M9
33.3
0.28
5.194
0.494
37
4.5
4.4
3.6
4.9
M10
100.0
0.09
5.683
0.515
37
1.5
1.7
1.4
2.1
M12
47.6
0.09
6.643
0.429
37
3.0
3.6
2.8
3.8
ML1
9.8
1.46
8.000
0.573
37
1.4
1.3
1.4
1.4
ML2
2.4
5.90
6.988
0.505
37
4.5
4.8
3.7
5.1
ML3
4.8
2.99
7.491
0.523
37
2.3
4.7
2.2
2.7
S1
83.3
0.32
13.621
0.504
39
4.3
6.0
8.1
5.2
S2
166.7
0.16
13.787
0.466
39
3.4
1.9
4.1
3.1
Granite (G3)
201.3
0.09
7.075
0.61
34
1.5
1.5
1.5
1.3
Granite (G4)
100.6
0.18
7.535
0.53
34
3.1
2.9
3.0
3.4
Granite (G5)
67.1
0.27
7.551
0.48
34
4.8
4.0
4.4
4.9
Granite (G6)
50.3
0.36
7.982
0.51
34
5.8
5.2
5.8
6.1
Granite (G7)
40.3
0.45
6.952
0.47
34
6.4
5.8
6.4
7.1
Granite (G8)
33.5
0.55
6.902
0.49
34
7.3
6.7
7.5
7.9
Granite (G9)
28.8
0.64
6.661
0.53
34
8.5
7.6
8.3
8.7
Granite (G10)
25.2
0.73
7.959
0.56
34
11.0
8.8
10.5
9.6
Granite (G11)
22.4
0.82
7.784
0.51
34
11.2
9.3
11.2
10.4
Granite (G12)
20.1
0.91
8.842
0.53
34
12.7
10.6
13.3
11.2
Sandstone (S3)
200.0
0.13
8.459
0.51
28
0.7
0.5
0.5
0.4
Sandstone (S4)
100.0
0.25
9.555
0.61
28
0.9
1.0
1.1
1.1
Sandstone (S5)
66.7
0.38
8.797
0.54
28
1.5
1.3
1.5
1.6
Sandstone (S6)
50.0
0.50
8.453
0.49
28
2.0
1.6
1.9
2.0
Sandstone (S7)
40.0
0.63
7.863
0.55
28
2.1
1.9
2.3
2.3
Sandstone (S8)
33.3
0.75
9.150
0.58
28
3.0
2.3
2.9
2.6
Sandstone (S9)
28.6
0.88
8.344
0.5
28
3.3
2.4
3.1
2.9
Sandstone (S10)
25.0
1.00
8.489
0.47
28
3.6
2.6
3.5
3.2
Sandstone (S11)
22.2
1.13
8.116
0.44
28
3.8
2.8
3.8
3.5
Sandstone (S12)
20.0
1.25
9.016
0.51
28
4.7
3.3
4.4
3.7


## 4.1. Selecting the Influencing Factors

According to the study by Barton and Choubey [34], the joint surface roughness, rock material strength, basic friction angle, and normal stress exhibit a wide variation of shear behavior. Therefore, normal stress, joint surface roughness, rock strength, and basic friction angle were selected in this work to assess the peak shear strength.Various parameters, including empirical, statistical, and fractal, have been proposed to quantify the surface roughness of rock joints. Numerous experimental tests demonstrated that partial contact between the lower and upper joint surface was located in the steepest zones facing the shear direction during the shearing [35–37]. Grasselli et al. [38] established a relationship between total potential contact area ratio Aθ∗ and the corresponding apparent dip angle θ∗, as shown in the following equation:(6)Aθ∗=A0θmax∗−θ∗θmax∗C,where A0 is the maximum potential contact area under apparent dip angle θ∗ at 0°; θmax∗ is the maximum apparent dip angle in the shear direction; and C is a dimensionless roughness parameter characterizing the distribution of the apparent dip angle over the surface, which is calculated using equation (6). Considering the possible range of C∈0,∞, the surface roughness of rock joints can be measured by the roughness metric θmax∗/C+1 [33]. A0 varies significantly in different directions; thus, the 3D roughness metric A0 and θmax∗/C+1 are both included to fully describe the roughness anisotropy. The 3D morphology parameters of the granite joints, namely, A0, θmax∗, and C, were calculated using the measured data from 36 surfaces on the basis of equation (6). The values of A0 and θmax∗/C+1 are listed in Table 3. Figure 3 shows the fitting curves of SJ3, SJ12, SJ21, and SJ36 for calculating the roughness parameter C.Table 3
3D morphology parameters of test specimens.
Sample name
A
0
θ
max
∗
/
C
+
1 (deg)
SJ1
0.479
10.465
SJ2
0.498
12.433
SJ3
0.504
10.139
SJ4
0.510
12.384
SJ5
0.467
10.870
SJ6
0.477
12.718
SJ7
0.542
10.237
SJ8
0.504
10.197
SJ9
0.472
10.208
SJ10
0.502
9.952
SJ11
0.484
11.961
SJ12
0.498
14.253
SJ13
0.528
14.405
SJ14
0.520
11.266
SJ15
0.498
10.411
SJ16
0.508
10.659
SJ17
0.501
11.699
SJ18
0.485
11.298
SJ19
0.496
12.499
SJ20
0.481
10.708
SJ21
0.505
12.808
SJ22
0.532
12.134
SJ23
0.471
14.002
SJ24
0.505
11.191
SJ25
0.499
11.398
SJ26
0.495
10.365
SJ27
0.433
10.083
SJ28
0.493
11.231
SJ29
0.498
13.550
SJ30
0.501
10.829
SJ31
0.512
11.177
SJ32
0.502
11.975
SJ33
0.454
10.467
SJ34
0.479
12.858
SJ35
0.492
9.942
SJ36
0.514
15.303Figure 3
Fitting results of the contact area ratio and apparent dip angle by equation (6): (a) SJ3, (b) SJ12, (c) SJ21, and (d) SJ30.
(a)
(b)
(c)
(d)Joint wall compressive strength (JCS) is commonly used for representing the influence of rock material strength on the shear behavior of rock joints [3, 8, 25, 39–41]. For the fresh rock, JCS can be replaced by the uniaxial compressive strength σc. Based on experimental observation and numerical simulations, several researchers have also found that the failure of asperities in rock joints under direct shear loading was possibly attributed to shear-induced tensile fracture [5, 6, 42–44]. However, the failure mechanism of asperities has not been clearly stated, and controversies still exist. Therefore, uniaxial compressive strength σc and tensile strength σt of the rock are both selected in this study to describe the effect of rock material strength on the peak shear strength of rock joints.Basic friction angleφb is crucial to determine the shear strength of rock joints. Barton suggested a fundamental form of joint shear strength [3]:(7)τp=σntanφb+dn+sn,where σn is the normal stress, dn is the peak dilation angle, and sn is the shear component angle. On the basis of equation (7), various 2D or 3D morphology parameters have been used for representing the sum of peak dilation angle dn and shear component angle sn. No uniform approach for obtaining the basic angle φb is currently available. In this work, laboratory direct shear testing method for two smooth joint surfaces is adopted to obtain the basic angle φb and the extensive shear strength model, as shown in Table 1.


## 4.2. Training and Testing of the RVM Model

In this study, the aforementioned original influencing factors, namely,σn, σc, and σt, can be dimensionless preprocessed as two factors: JCS/σn and σn/σt. Therefore, the five influencing factors, which include JCS/σn, σn/σt, A0, θmax∗/C+1, and φb, form the input vector of the regression model. All these index values can be measured in the laboratory (see Tables 1–3). Mapping the above five influencing factors to the five-dimensional feature space according to the definition of RVM, the appropriate kernel function was chosen to simplify the nonlinear transformation. The linear kernel function was first used due to its parameter-free and rapid calculation as follows:(8)Kx,z=x,z,where x,z are the training dataset of five influencing factors. The results of direct shear tests on 36 rocks in this study were used to cultivate and advance the RVM model. The results of direct shear tests by Grasselli and Egger [5] and Yang et al. [8] were used for testing the RVM model, in which a total of 50 tests were conducted on five different material types of rock joints (granite, sandstone, limestone, marble, and serpentinite). The 3D morphology parameters of 50 groups of rock joints are represented in Table 4. After the analysis was completed, the quality of the regression model was estimated by the average error formula and correlation coefficient [45]:(9)σ¯avg=1n∑i=1nτmea−τestτmea×100%,(10)Fitness_σ¯avg=train_σ¯avg⋅train_n+test_σ¯avg⋅test_ntrain_n+test_n,(11)R2=1−∑i=1nτest−τmea2∑i=1nτest−τmea¯2,(12)Fitness_R2=train_R2⋅train_n+test_R2⋅test_ntrain_n+test_n,where σ¯avg is the average estimation error, n is the sample size, τmea is the measured peak shear strength, τest is the estimated peak shear strength, τmea¯ is the average value of the measured peak shear strength, Fitness_σ¯avg is the comprehensive estimation error of the RVM model, and Fitness_R2 is the correlation coefficient of the train and test model; train_n is the number of the training sample, and test_n is the number of the testing sample. In this work, train_n and test_n are 36 and 50, respectively. The average estimation errors of the RVM model for the training and testing data are 6.46% and 21.12%, respectively. The correlation coefficient of the RVM model for the training and testing data is 0.977 and 0.769, respectively. The comprehensive estimation error Fitness_σ¯avg and correlation coefficient Fitness_R2 of the RVM model for all the data are 14.98% and 0.856, respectively. This finding indicates that the RVM model exhibits good training accuracy and high generalization capacity. Therefore, the RVM model is feasible and effective for estimating the peak shear strength of rock joints and can be used in practical applications.Table 4
Comparison between the measured peak shear strength and the calculated results by three different models.
Sample name
JCS
/
σ
n
σ
n
/
σ
t
θ
max
∗
/
C
+
1 (deg)
A
0
φ
b(deg)
Peak shear strength (MPa)
Measured
Grasselli
Yang
Proposed
G1
81.2
0.26
11.016
0.493
34
5.7
4.6
6.0
4.8
G2
81.2
0.26
12.121
0.498
34
5.6
5.1
6.0
4.8
G4
85.2
0.25
10.031
0.498
34
4.8
4.3
4.3
4.6
G5
166.7
0.13
9.005
0.46
34
2.4
2.3
2.0
2.7
G6
166.7
0.13
10.012
0.477
34
2.9
2.4
2.7
2.7
G7
166.7
0.13
9.939
0.47
34
2.8
2.4
2.6
2.7
G9
166.7
0.13
10.949
0.508
34
3.0
2.6
2.6
2.7
C1
23.3
0.45
9.963
0.491
36
2.2
1.9
2.2
1.7
C2
23.3
0.45
12.048
0.462
36
2.1
2.2
2.6
1.8
C3
6.7
1.55
10.179
0.46
36
5.5
5.5
5.1
4.8
C4
10.2
1.02
11.324
0.508
36
4.6
4.1
4.2
3.3
C5
8.1
1.30
11.821
0.495
36
5.0
5.2
5.1
4.1
C6
24.4
0.43
10.985
0.546
36
2.1
2.1
2.1
1.7
C8
8.1
1.30
11.028
0.555
36
4.9
4.9
4.8
4.1
M1
100.0
0.09
7.143
0.513
37
1.7
1.8
1.7
2.1
M2
50.0
0.19
5.909
0.492
37
2.3
3.2
2.4
3.8
M3
100.0
0.09
5.652
0.471
37
1.2
1.7
1.4
2.1
M4
23.3
0.41
6.689
0.513
37
5.8
6.3
5.7
6.3
M5
33.3
0.28
5.948
0.533
37
4.4
4.6
3.8
4.9
M6
33.3
0.28
6.082
0.45
37
4.3
4.4
4.0
4.9
M7
23.3
0.41
6.001
0.502
37
5.6
6.1
5.8
6.2
M8
22.7
0.42
6.250
0.459
37
6.4
6.1
5.7
6.3
M9
33.3
0.28
5.194
0.494
37
4.5
4.4
3.6
4.9
M10
100.0
0.09
5.683
0.515
37
1.5
1.7
1.4
2.1
M12
47.6
0.09
6.643
0.429
37
3.0
3.6
2.8
3.8
ML1
9.8
1.46
8.000
0.573
37
1.4
1.3
1.4
1.4
ML2
2.4
5.90
6.988
0.505
37
4.5
4.8
3.7
5.1
ML3
4.8
2.99
7.491
0.523
37
2.3
4.7
2.2
2.7
S1
83.3
0.32
13.621
0.504
39
4.3
6.0
8.1
5.2
S2
166.7
0.16
13.787
0.466
39
3.4
1.9
4.1
3.1
Granite (G3)
201.3
0.09
7.075
0.61
34
1.5
1.5
1.5
1.3
Granite (G4)
100.6
0.18
7.535
0.53
34
3.1
2.9
3.0
3.4
Granite (G5)
67.1
0.27
7.551
0.48
34
4.8
4.0
4.4
4.9
Granite (G6)
50.3
0.36
7.982
0.51
34
5.8
5.2
5.8
6.1
Granite (G7)
40.3
0.45
6.952
0.47
34
6.4
5.8
6.4
7.1
Granite (G8)
33.5
0.55
6.902
0.49
34
7.3
6.7
7.5
7.9
Granite (G9)
28.8
0.64
6.661
0.53
34
8.5
7.6
8.3
8.7
Granite (G10)
25.2
0.73
7.959
0.56
34
11.0
8.8
10.5
9.6
Granite (G11)
22.4
0.82
7.784
0.51
34
11.2
9.3
11.2
10.4
Granite (G12)
20.1
0.91
8.842
0.53
34
12.7
10.6
13.3
11.2
Sandstone (S3)
200.0
0.13
8.459
0.51
28
0.7
0.5
0.5
0.4
Sandstone (S4)
100.0
0.25
9.555
0.61
28
0.9
1.0
1.1
1.1
Sandstone (S5)
66.7
0.38
8.797
0.54
28
1.5
1.3
1.5
1.6
Sandstone (S6)
50.0
0.50
8.453
0.49
28
2.0
1.6
1.9
2.0
Sandstone (S7)
40.0
0.63
7.863
0.55
28
2.1
1.9
2.3
2.3
Sandstone (S8)
33.3
0.75
9.150
0.58
28
3.0
2.3
2.9
2.6
Sandstone (S9)
28.6
0.88
8.344
0.5
28
3.3
2.4
3.1
2.9
Sandstone (S10)
25.0
1.00
8.489
0.47
28
3.6
2.6
3.5
3.2
Sandstone (S11)
22.2
1.13
8.116
0.44
28
3.8
2.8
3.8
3.5
Sandstone (S12)
20.0
1.25
9.016
0.51
28
4.7
3.3
4.4
3.7


## 5. Comparison and Discussion

### 5.1. RVM with Different Kernel Functions

The inner product of kernel functions is usually substituted for the complex calculation of high-dimensional feature space to deal with the dimensionality problem under high-dimensional feature space. The nonlinear shining mode from the raw feature space to high-dimensional space relays on the type of kernel functions. The approximate kernel function and model parameters have a considerable effect on the model stability and regression precision [46].To further enhance the accuracy of the proposed model, three other common kernel functions were introduced: polynomial kernel function, Gaussian kernel function, and sigmoid kernel function.(13)Polynomial kernel function:Kx,z=a⋅x,z+bc,(14)Gaussian kernel function:Kx,z=exp−x−z22d2,(15)Sigmoid kernel function:Kx,z=tanhk⋅x,z+p,where a, b, c, d, k, and p are kernel function parameters.In the RVM model, the kernel function parameters must be defined. The most relevant papers which include the determination of the optimal kernel function parameters for RVM is lacking. Several different values of kernel function parameters were selected as determining which one provides better regression accuracy. The four kernel function parameters and the accompanying average estimation errors and correlation coefficients are represented in Table5. Synthetically considering the regression performance and the kernel function parameter number, a peak shear strength of RVM model based on a Gaussian kernel function is established. The optimal value of the kernel function parameter d is 52.43.Table 5
Kernel function parameters and average estimation errors.
Kernel function type
Fitness
_
σ
¯
avg (%)
Fitness
_
R
2
Parameter number
Linear
14.98
0.856
0
Polynomial
12.41
0.957
3
Gaussian
12.24
0.956
1
Sigmoid
13.44
0.952
2


### 5.2. Comparison with Existing Criteria

To certify the rationality of the proposed model, the two models developed by Grasselli and Egger [5] and Yang et al. [8] have been chosen for comparison.(16)Grasselli’s model:τp=1+exp−θmax∗σn9A0Cσtσntanφb+θmax∗C1.18cosβ,(17)Yang’s model:τp=σntanφb+θmax∗C0.45e−σn/JCSC0.75.The 3D morphology parameters of the 50 rock joints are represented in Table4. The estimated results using equations (16) and (17), the RVM model results, and the measured peak shear strength are listed in Table 4. The prediction results of the peak shear strength of rock joints are compared with the measured data, as shown in Figures 4 and 5.Figure 4
Measured peak shear strength of rock joints by Grasselli and Egger [5], which was calculated using Grasselli’s model, Yang’s model, and the proposed model.Figure 5
Measured peak shear strength of rock joints by Yang et al. [8], which was calculated using Grasselli’s model, Yang’s model, and the proposed model.According to equation (9), the mean errors of the estimated value results calculated using Grasselli’s model, Yang’s model, and RVM model are 16.3%, 8.9%, and 15.4%, respectively. Based on equation (11), the correlation values of the estimated value results calculated using Grasselli’s model, Yang’s model, and RVM model are 0.889, 0.940, and 0.938, respectively. From the mean error and correlation coefficient analysis, the average estimation error values of the three models are less than 20% and the correlation coefficients are greater than 0.85, which indicate that the three models were quite efficient to be used to predict the shear strength of rock joints. Moreover, the proposed model is more accurate than that of Grasselli’s model and has almost equivalent precision to that of Yang’s model. After analysis of the above average errors, the existent limitations of the three models should be pointed out. In Grasselli’s and Yang’s models, the function forms are complex. According to Grasselli’s model (i.e., equation (14)), the ratio (θmax∗/C) was taken as a measure of joint surface roughness, but C cannot be zero (C=0 for saw-tooth joints). Yang’s model is not fully explained on how to use the two parameters θmax∗ and C in combination, and A0 is not even considered. Furthermore, Grasselli and Egger [5] and Yang et al. [8] deemed that the actual contact area was closely related with σn/σt and σn/σc, respectively. Compared with Grasselli’s and Yang’s models, the proposed model was less complex for large samples and comprehensively considered many factors; in particular, the effect of compressive strength and tensile strength on the peak shear strength of rock joints was first investigated. However, the determination of kernel function parameter needs to be further improved and optimized.In this study, the investigation is limited to the shear behavior of unfilled joints. Influencing factors, such as size of joint (scale effect), sampling interval, joint matching coefficient, and joint filling, should be further investigated. Although the proposed model has limitations, it is believed to be a novel research idea for assessing peak shear strength of rock joints.


## 5.1. RVM with Different Kernel Functions

The inner product of kernel functions is usually substituted for the complex calculation of high-dimensional feature space to deal with the dimensionality problem under high-dimensional feature space. The nonlinear shining mode from the raw feature space to high-dimensional space relays on the type of kernel functions. The approximate kernel function and model parameters have a considerable effect on the model stability and regression precision [46].To further enhance the accuracy of the proposed model, three other common kernel functions were introduced: polynomial kernel function, Gaussian kernel function, and sigmoid kernel function.(13)Polynomial kernel function:Kx,z=a⋅x,z+bc,(14)Gaussian kernel function:Kx,z=exp−x−z22d2,(15)Sigmoid kernel function:Kx,z=tanhk⋅x,z+p,where a, b, c, d, k, and p are kernel function parameters.In the RVM model, the kernel function parameters must be defined. The most relevant papers which include the determination of the optimal kernel function parameters for RVM is lacking. Several different values of kernel function parameters were selected as determining which one provides better regression accuracy. The four kernel function parameters and the accompanying average estimation errors and correlation coefficients are represented in Table5. Synthetically considering the regression performance and the kernel function parameter number, a peak shear strength of RVM model based on a Gaussian kernel function is established. The optimal value of the kernel function parameter d is 52.43.Table 5
Kernel function parameters and average estimation errors.
Kernel function type
Fitness
_
σ
¯
avg (%)
Fitness
_
R
2
Parameter number
Linear
14.98
0.856
0
Polynomial
12.41
0.957
3
Gaussian
12.24
0.956
1
Sigmoid
13.44
0.952
2


## 5.2. Comparison with Existing Criteria

To certify the rationality of the proposed model, the two models developed by Grasselli and Egger [5] and Yang et al. [8] have been chosen for comparison.(16)Grasselli’s model:τp=1+exp−θmax∗σn9A0Cσtσntanφb+θmax∗C1.18cosβ,(17)Yang’s model:τp=σntanφb+θmax∗C0.45e−σn/JCSC0.75.The 3D morphology parameters of the 50 rock joints are represented in Table4. The estimated results using equations (16) and (17), the RVM model results, and the measured peak shear strength are listed in Table 4. The prediction results of the peak shear strength of rock joints are compared with the measured data, as shown in Figures 4 and 5.Figure 4
Measured peak shear strength of rock joints by Grasselli and Egger [5], which was calculated using Grasselli’s model, Yang’s model, and the proposed model.Figure 5
Measured peak shear strength of rock joints by Yang et al. [8], which was calculated using Grasselli’s model, Yang’s model, and the proposed model.According to equation (9), the mean errors of the estimated value results calculated using Grasselli’s model, Yang’s model, and RVM model are 16.3%, 8.9%, and 15.4%, respectively. Based on equation (11), the correlation values of the estimated value results calculated using Grasselli’s model, Yang’s model, and RVM model are 0.889, 0.940, and 0.938, respectively. From the mean error and correlation coefficient analysis, the average estimation error values of the three models are less than 20% and the correlation coefficients are greater than 0.85, which indicate that the three models were quite efficient to be used to predict the shear strength of rock joints. Moreover, the proposed model is more accurate than that of Grasselli’s model and has almost equivalent precision to that of Yang’s model. After analysis of the above average errors, the existent limitations of the three models should be pointed out. In Grasselli’s and Yang’s models, the function forms are complex. According to Grasselli’s model (i.e., equation (14)), the ratio (θmax∗/C) was taken as a measure of joint surface roughness, but C cannot be zero (C=0 for saw-tooth joints). Yang’s model is not fully explained on how to use the two parameters θmax∗ and C in combination, and A0 is not even considered. Furthermore, Grasselli and Egger [5] and Yang et al. [8] deemed that the actual contact area was closely related with σn/σt and σn/σc, respectively. Compared with Grasselli’s and Yang’s models, the proposed model was less complex for large samples and comprehensively considered many factors; in particular, the effect of compressive strength and tensile strength on the peak shear strength of rock joints was first investigated. However, the determination of kernel function parameter needs to be further improved and optimized.In this study, the investigation is limited to the shear behavior of unfilled joints. Influencing factors, such as size of joint (scale effect), sampling interval, joint matching coefficient, and joint filling, should be further investigated. Although the proposed model has limitations, it is believed to be a novel research idea for assessing peak shear strength of rock joints.


## 6. Conclusions

The hybrid probabilistic regression assessment algorithm RVM was accustomed to identify the peak shear strength of the rock joint, with Gaussian kernel function being introduced to improve the prediction accuracy. The direct shear tests on joint specimens were conducted to train the new model, and its rationality on assessing the peak shear strength was also validated. Some concluding remarks are presented as follows:(1)
A novel intelligent peak shear strength model combined with RVM was supposed to assess the peak shear strength of rock joints, which can comprehensively consider the elements influencing the peak shear strength, such as normal stress, joint surface roughness, rock strength, and basic friction angle. The 36 natural granite joint specimens were accustomed to examine the suggested model, and the result reflects that the RVM model exhibits good training accuracy and high generalization capacity.(2)
Four types of kernel functions were investigated for their performance in regression accuracy in the RVM model. According to the regression performance and the kernel function parameter number, the Gaussian kernel function was determined to be the better one combined with the RVM model. The optimal value of the kernel function parameterd is 52.43.(3)
The peak shear model of Grasselli’s and Yang’s was chosen for comparison with the proposed model. Based on the 3D morphology parameters of 50 rock joints, the prediction results show that the proposed model can meet the accuracy requirements. In addition, the new model has less complexity and can take into account the effects of compressive strength and tensile strength on the peak shear strength of rock joints.



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*Source: 3182736-2019-12-19.xml*