# LSSVM-Based Rock Failure Criterion and Its Application in Numerical Simulation

Authors: Changxing Zhu; Hongbo Zhao; Zhongliang Ru
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A rock failure criterion is very important for the prediction of the failure of rocks or rock masses in rock mechanics and engineering. Least squares support vector machines (LSSVM) are a powerful tool for addressing complex nonlinear problems. This paper describes a LSSVM-based rock failure criterion for analyzing the deformation of a circular tunnel under differentin situ stresses without assuming a function form. First, LSSVM was used to represent the nonlinear relationship between the mechanical properties of rock and the failure behavior of the rock in order to construct a rock failure criterion based on experimental data. Then, this was used in a hypothetical numerical analysis of a circular tunnel to analyze the mechanical behavior of the rock mass surrounding the tunnel. The Mohr-Coulomb and Hoek-Brown failure criteria were also used to analyze the same case, and the results were compared; these clearly indicate that LSSVM can be used to establish a rock failure criterion and to predict the failure of a rock mass during excavation of a circular tunnel.


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## Body

## 1. Introduction

The failure of rock has been one of the most active research areas in the engineering of rock structures since the establishment of the International Society for Rock Mechanics (ISRM) in 1962. Ulusay and Hudson gave a detailed description of the importance of the ability to predict rock failure in practical engineering [1]. Rock failure criteria include the theoretical basis of methods of predicting when and how rock materials fail under the action of external loads and are therefore essential for optimization of engineering design and construction in rock masses. Over the past several decades, a number of failure criteria have been proposed (e.g., Mohr-Coulomb, Hoek-Brown, and Griffith’s and Drucker-Prager criteria) and developed to predict the failure of rock [2–11]. Recently, ISRM suggested methods have been published as criteria for the prediction of rock failure [12, 13]. Both the Mohr-Coulomb and Hoek-Brown criteria are generally considered to be reliable predictors of rock failure. In these two models, the major principal stress σ1 is seen as function of the uniaxial compressive strength (σc), the minor principal stress (σ3), and some constants, that is, f(σ1,σ3)=0. These constants can often be determined by laboratory experiment,in situ tests, and back-analysis.Empirical failure criteria established under specific conditions such as rock type, stress state, or functional form are not generic and cannot be adopted uncritically for any set of conditions [10] but must be established experimentally for any given set of rock formation and stress conditions.For this reason, artificial neural network (ANN) computational models are widely used to establish rock failure criteria for geotechnical engineering problems. This is due to the fact that ANN perform function approximation without requiring additional assumptions of the function form. More detailed descriptions of the ANN-based failure criterion method have been reported by, for example, Meulenkamp and Grima, Singh et al., Çanakci and Pala, Tiryaki, Zorlu et al., and Rafiai and Jafari [14–20]. However, a disadvantage of ANN for establishing rock failure criteria is its slow convergence and overfitting. To overcome these limitations, in the present study a LSSVM-based rock failure criterion was developed based on experimental data. Support vector machines (SVM) have been widely used in geotechnical engineering for nonlinear mapping [21–23].This paper is organized as follows. Section2 focuses on the formulation of LSSVM. A LSSVM-based rock failure criterion is described in detail in Section 3. In Section 4, a hypothetical numerical experiment is used to verify the LSSVM-based rock failure criterion by comparing with earlier failure criteria. Finally, conclusions are given in Section 5.


## 2. Least-Squares Support Vector Machine (LSSVM)

LSSVM models are an alternative formulation of SVM regression proposed by Suykens and Vandewalle [24]. Suppose there is a given training set of N data points {xk,yk}(k=1,2,…,N) with input data xk∈RN and output yk∈r, where RN is an N-dimensional vector space and r is a one-dimensional vector space; LSSVM models in feature space are given by
(1)yx=wTϕx+b,
where the nonlinear mapping φ(·) represents the input data in a higher-dimensional feature space; w∈Rn; b∈r; w is an adjustable weight vector; and b is the scalar threshold. For function estimation in LSSVM, the following optimization problem is formulated:
(2)Minimizehh12wTw+γ12∑k=1Nek2Subjectedtoyx=wTφx+b+ek,k=1,…,N,
where γ is the regularization parameter for determining the trade-off between the fitting error minimization and smoothness, and ek is an error variable.The Lagrangian(L(w,b,e,α)) for the above optimization problem (2) is
(3)Lw,b,e;α=12wTw+γ12∑k=1Nek2-∑k=1NαkykwTφ(xk)+b-1+ek,
where αk is the Lagrange multiplier. The conditions for optimality are given by
(4)∂L∂w=0⟹w=∑k=1Nαkφxk,∂L∂b=0⟹∑k=1Nαk=0,∂L∂ek=0⟹αk=γek,k=1,…,N,∂L∂αk=0⟹wTφxk+b+ek-yk=0,k=1,…,N
without regard to ek and w; the solution is given by the following set of linear equations:
(5)Φbα=0y,
where y=[y1,…,yN], 1=[1,…,1], α=[α1,…,αN], and Mercer’s theorem is applied within the Ω matrix, Ω=φxkTφ(xl)=K(xk,xl), k,l=1,…,N, where K(xk,xl) is the kernel function, and the matrix Φ is invertible by choosing γ>w and can be written as
(6)Φ=01T1Ω+γ-1I.
Then the analytical solution of α and b is given by
(7)bα=Φ-10y.
The LSSVM model can then be expressed as
(8)yx=∑k=1NαkKx,xk+b.
In (8), K(x,xk) is the kernel function, composed of the following three forms.(1)
Polynomial kernel:(9)KX,Y=X·Y+1d,d=1,2,…,n.(2)
Radial kernel:(10)KX,Y=exp⁡-X-Y22σ2.(3)
Sigmoidal kernel:(11)KX,Y=tanhφX·Y+θ.The brief procedure LSSVM is the following.Step 1.
Build the training samples(xk,yk) based on rock mechanical experiment.Step 2.
Get (6) according to the LSSVM algorithm.Step 3.
Get the analytical solution ofα and b in (7).Step 4.
Get the LSSVM model (8).


## 3. LSSVM-Based Rock Failure Criteria

In order to better predict the behavior failure of rock masses, LSSVM was used in this work to represent the nonlinear relationship between rock properties and rock failure and thus to establish a LSSVM-based rock failure criterion. Similar to traditional failure criteria in form, LSSVM-based rock failure criteria can be defined as(12)SVM(σ1,σ3)=0,
where σ1 and σ3 are the major and minor principal stresses, respectively. These can be obtained by laboratory experiment and field testing. The LSSVM-based rock failure criterion can be rewritten in the form:
(13)σ1=∑k=1NαkKσ3,σ3k+b,
where σ3k is the series of experimental data and αk and b are calculated using the algorithm introduced in Section 2. Once a LSSVM-based rock failure criterion is established, it can be used in numerical analysis to predict rock failure. In this work, Excel VBA software was used to execute the code of the LSSVM-based rock failure criterion.To implement the LSSVM-based failure criteria in a numerical model, it is useful to express it as an equivalent Mohr-Coulomb criterion with variable parameters. The parameters of the equivalent criterion are obtained as(14)σceq=σ1-qeqσ3=∑k=1NαkKσ3,σ3k+b-qeqσ3,(15)qeq=∂σ1∂σ3=∂∑k=1NαkK(σ3,σ3k)∂σ3=∑k=1Nαk∂K(σ3,σ3k)∂σ3.
The RBF kernel function (Equation (10)) was adopted:
(16)∂Kσ3,σ3k∂σ3=-1σ2Kσ3,σ3k.
Substituting (16) into (15) gives the following form:
(17)qeq=-1σ2∑k=1NαkKσ3,σ3k,
and the equivalent Mohr-Coulomb criterion may be calculated from the following equations:
(18)ceq=σceq2qeq,(19)φeq=2tan-1qeq-π2,
where ceq and φeq are the instantaneous values of cohesion and friction angle, respectively.LSSVM-based rock failure criterion presents the mechanical character of rock through combining L-SVM and rock failure criterion. The brief procedure is showed in Figure1.Figure 1
The flowchart of LSSVM-based failure criterion.


## 4. Numerical Experiment

### 4.1. Matching the Experimental Data

In this work, the RBF kernel function was adopted for pattern analysis or recognition, with the following parameters:σ=2. The value of b is 14.5061. The values of αk are listed in Table 1. As in any other use of LSSVM, the SVM must be trained and tested; 98 data sets from triaxial compression tests, consisting of 79 training samples and 19 testing samples, were used as the experimental data in this study (seen in Table 1). Once the performance of the SVM model was satisfactory, it was used to represent the nonlinear relationship between rock properties and the mechanical behavior of the rock to establish the LSSVM-based rock failure criterion.Table 1
Experimental data and the values of Lagrange multiplierαk.
Number
σ
3 (Mpa)
σ
1 (Mpa)
α
k
Number
σ
3 (Mpa)
σ
1 (Mpa)
α
k
1
0.0000
0.0000
−138.4852
41
3.8699
17.8247
0.1287
2
0.0853
2.7979
97.6169
42
3.9471
18.0397
0.0845
3
0.2398
4.0913
5.5845
43
4.0244
18.2534
0.0837
4
0.3170
4.6287
2.2133
44
4.1016
18.4657
0.0829
5
0.3943
5.1224
1.6698
45
4.1788
18.6769
0.0822
6
0.4715
5.5829
1.3160
46
4.2561
18.8868
0.1220
7
0.5487
6.0173
1.5167
47
4.4105
19.3033
0.1210
8
0.7032
6.8251
1.2040
48
4.4878
19.5098
0.0800
9
0.7804
7.2046
0.6538
49
4.5650
19.7153
0.1193
10
0.8577
7.5708
0.5713
50
4.7195
20.1231
0.1186
11
0.9349
7.9254
0.7297
51
4.7967
20.3255
0.0787
12
1.0894
8.6045
0.6295
52
4.8740
20.5269
0.0785
13
1.1666
8.9311
0.3662
53
4.9512
20.7273
0.1175
14
1.2439
9.2501
0.4865
54
5.1057
21.1255
0.1173
15
1.3983
9.8678
0.4344
55
5.1829
21.3232
0.0781
16
1.4756
10.1677
0.2606
56
5.2601
21.5200
0.0781
17
1.5528
10.4620
0.3552
57
5.3374
21.7160
0.1171
18
1.7073
11.0360
0.3247
58
5.4919
22.1055
0.1172
19
1.7845
11.3162
0.1989
59
5.5691
22.2990
0.0783
20
1.8617
11.5922
0.1876
60
5.6463
22.4918
0.0784
21
1.9390
11.8644
0.2615
61
5.7236
22.6838
0.1178
22
2.0935
12.3979
0.2442
62
5.8780
23.0655
0.1182
23
2.1707
12.6597
0.1526
63
5.9553
23.2553
0.0790
24
2.2479
12.9183
0.1459
64
6.0325
23.4443
0.1190
25
2.3252
13.1740
0.1398
65
6.1870
23.8203
0.1195
26
2.4024
13.4268
0.1342
66
6.2642
24.0073
0.0800
27
2.4796
13.6769
0.1914
67
6.3415
24.1937
0.1206
28
2.6341
14.1696
0.1826
68
6.4959
24.5644
0.1212
29
2.7113
14.4124
0.1165
69
6.5732
24.7488
0.0813
30
2.7886
14.6529
0.1678
70
6.6504
24.9327
0.0816
31
2.9431
15.1275
0.1616
71
6.7276
25.1159
0.0820
32
3.0203
15.3617
0.1040
72
6.8049
25.2985
0.1237
33
3.0975
15.5940
0.1015
73
6.9593
25.6620
0.1244
34
3.1748
15.8245
0.0992
74
7.0366
25.8429
0.0835
35
3.2520
16.0532
0.1447
75
7.1138
26.0232
0.0839
36
3.4065
16.5054
0.1410
76
7.1911
26.2030
0.1267
37
3.4837
16.7290
0.0918
77
7.3455
26.5610
0.1276
38
3.5609
16.9511
0.0903
78
7.4228
26.7392
0.0856
39
3.6382
17.1717
0.0889
79
7.5000
26.9169
15.4377
40
3.7154
17.3908
0.1309As shown in Figures2 and 3, the predicted LSSVM values and experimental values of σ1 were almost identical, indicating that the LSSVM-based failure criterion model effectively simulated the failure behavior. Figure 4 shows a comparison between the failure envelopes determined by the LSSVM, Mohr-Coulomb, and Hoek-Brown models and those from the experimental data. It can be seen from Figure 4 that σ1 predicted by the LSSVM model agreed well both with the experimental data and the Hoek-Brown model at large stresses but differed a little from the Mohr-Coulomb model; however, σ1 differed a little from the Hoek-Brown model at small stresses. These results indicate that the LSSVM-based rock failure criterion trained on experimental data predicts the failure behavior of rock with reasonable accuracy.Figure 2
Comparison of major principal stressσ1 between the experimental data and predicted value for training samples.Figure 3
Comparison of major principal stressσ1 between the experimental data and predicted value for testing samples.Figure 4
Comparison of failure envelopes for different rock failure criteria.


### 4.2. Numerical Analysis Using the LSSVM-Based Rock Failure Criterion

To verify the feasibility of the LSSVM-based rock failure criterion in numerical analysis, it was combined with FLAC3D modeling code to simulate the failure behavior of a 5 m radius circular rock tunnel as a hypothetical numerical case. An initialin situ stress and gravity was presupposed in the numerical model, and the experimental data from Section 4.1 was used. The properties of the rock mass listed in Figure 5 are based on the Mohr-Coulomb model. The numerical model is built using FLAC3D together with the algorithm in Section 3. The horizontal displacements are almost the same as the value of Mohr-Coulomb by FLAC3D (seen in Figure 6). The stress of surrounding rock is in well agreement with the law of Mohr-Coulomb by FLAC3D (seen in Figure 7). The horizontal displacements and the horizontal, vertical, and shear stresses induced in the surrounding rock by the excavation show that the LSSVM model results were almost identical to those obtained by the Mohr-Coulomb model. The induced horizontal and vertical stresses calculated by the two models are also shown as contour plots in Figures 8 and 9. It shows the mechanical character of rock was presented by LSSVM-based rock failure criteria. Overall, the results show that the LSSVM-based rock failure criteria present well the mechanical behavior and character and can be used for numerical analysis.Figure 5
A circular tunnel model, stress condition, and rock mass properties.Figure 6
Comparison of surrounding rock displacement of tunnel obtained by LSSVM and Mohr-Coulomb model.Comparison of horizontal stress of tunnel obtained by LSSVM and Mohr-Coulomb model (a)σx, (b) σy, (c) σxy.
(a)
(b)
(c)Comparison of horizontal stress contour in surrounding rock mass obtained by LSSVM and Mohr-Coulomb model.
(a)
Mohr-Coulomb model
(b)
LSSVM modelComparison of vertical stress contour in surrounding rock mass obtained by LSSVM and Mohr-Coulomb model.
(a)
Mohr-Coulomb model
(b)
LSSVM modelTo verify the LSSVM-based rock failure criterion for different initialin situ stress states, the circular tunnel was investigated at different vertical stress σyy=5, 10, and 15 MPa and corresponding horizontal stress σxx=1.2σyy in each case. The maximum error and maximum relative error of horizontal displacements are about 0.08 and 2.8%, respectively. The maximum relative error of vertical stress and shear stress is about 18% at the beginning of excavation (seen Figure 11(b)) and then it will be less than 5% with the excavation. Using the LSSVM and Mohr-Coulomb models, the calculated horizontal displacements in the rock surrounding the tunnel are shown in Figure 10 and the major principal stresses are shown in Figure 11. Those show it is feasible to combine LSSVM rock failure criterion with numerical analysis.Figure 10
Comparison of surrounding rock mass displacement obtained by LSSVM and Mohr-Coulomb model in differentin situ stress level (syis the vertical in situ stress).Comparison of surrounding rock stress of tunnel between LSSVM and Mohr-Coulomb model in differentin situ stress level (a) σx, (b) σy, (c) τxy (syis the vertical in situ stress).
(a)
(b)
(c)The displacements for both models agree well for the three initialin situ stress states (Figure 10), and the principal stresses are also similar for both models (Figure 11), with σy and τxy showing small differences near the wall of tunnel. These results generally reflect the above results (in which σyy=1 MPa).The LSSVM and Hoek-Brown models were also compared. Figures12–14 show the results for the same initial stress states as above, and it is seen that displacements in the surrounding rock agreed well with both the Hoek-Brown and Mohr-Coulomb results. Some small differences are evident in σxx and τxy compared to the Mohr-Coulomb model, but the results are consistent with the Hoek-Brown model because, as shown in Figure 4, the LSSVM and Hoek-Brown failure envelopes are similar. Thus, the proposed failure criterion can be used in numerical analysis to effectively reflect the mechanical behavior of rock.Displacement and stresses induced by excavation atσy=5MPa.
(a)
Displacement
(b)
σ
x
(c)
σ
y
(d)
τ
x
yDisplacement and stresses induced by excavation atσy=10 MPa.
(a)
Displacement
(b)
σ
x
(c)
σ
y
(d)
τ
x
yDisplacement and stresses induced by excavation atσy=15 MPa.
(a)
Displacement
(b)
σ
x
(c)
σ
y
(d)
τ
x
y


## 4.1. Matching the Experimental Data

In this work, the RBF kernel function was adopted for pattern analysis or recognition, with the following parameters:σ=2. The value of b is 14.5061. The values of αk are listed in Table 1. As in any other use of LSSVM, the SVM must be trained and tested; 98 data sets from triaxial compression tests, consisting of 79 training samples and 19 testing samples, were used as the experimental data in this study (seen in Table 1). Once the performance of the SVM model was satisfactory, it was used to represent the nonlinear relationship between rock properties and the mechanical behavior of the rock to establish the LSSVM-based rock failure criterion.Table 1
Experimental data and the values of Lagrange multiplierαk.
Number
σ
3 (Mpa)
σ
1 (Mpa)
α
k
Number
σ
3 (Mpa)
σ
1 (Mpa)
α
k
1
0.0000
0.0000
−138.4852
41
3.8699
17.8247
0.1287
2
0.0853
2.7979
97.6169
42
3.9471
18.0397
0.0845
3
0.2398
4.0913
5.5845
43
4.0244
18.2534
0.0837
4
0.3170
4.6287
2.2133
44
4.1016
18.4657
0.0829
5
0.3943
5.1224
1.6698
45
4.1788
18.6769
0.0822
6
0.4715
5.5829
1.3160
46
4.2561
18.8868
0.1220
7
0.5487
6.0173
1.5167
47
4.4105
19.3033
0.1210
8
0.7032
6.8251
1.2040
48
4.4878
19.5098
0.0800
9
0.7804
7.2046
0.6538
49
4.5650
19.7153
0.1193
10
0.8577
7.5708
0.5713
50
4.7195
20.1231
0.1186
11
0.9349
7.9254
0.7297
51
4.7967
20.3255
0.0787
12
1.0894
8.6045
0.6295
52
4.8740
20.5269
0.0785
13
1.1666
8.9311
0.3662
53
4.9512
20.7273
0.1175
14
1.2439
9.2501
0.4865
54
5.1057
21.1255
0.1173
15
1.3983
9.8678
0.4344
55
5.1829
21.3232
0.0781
16
1.4756
10.1677
0.2606
56
5.2601
21.5200
0.0781
17
1.5528
10.4620
0.3552
57
5.3374
21.7160
0.1171
18
1.7073
11.0360
0.3247
58
5.4919
22.1055
0.1172
19
1.7845
11.3162
0.1989
59
5.5691
22.2990
0.0783
20
1.8617
11.5922
0.1876
60
5.6463
22.4918
0.0784
21
1.9390
11.8644
0.2615
61
5.7236
22.6838
0.1178
22
2.0935
12.3979
0.2442
62
5.8780
23.0655
0.1182
23
2.1707
12.6597
0.1526
63
5.9553
23.2553
0.0790
24
2.2479
12.9183
0.1459
64
6.0325
23.4443
0.1190
25
2.3252
13.1740
0.1398
65
6.1870
23.8203
0.1195
26
2.4024
13.4268
0.1342
66
6.2642
24.0073
0.0800
27
2.4796
13.6769
0.1914
67
6.3415
24.1937
0.1206
28
2.6341
14.1696
0.1826
68
6.4959
24.5644
0.1212
29
2.7113
14.4124
0.1165
69
6.5732
24.7488
0.0813
30
2.7886
14.6529
0.1678
70
6.6504
24.9327
0.0816
31
2.9431
15.1275
0.1616
71
6.7276
25.1159
0.0820
32
3.0203
15.3617
0.1040
72
6.8049
25.2985
0.1237
33
3.0975
15.5940
0.1015
73
6.9593
25.6620
0.1244
34
3.1748
15.8245
0.0992
74
7.0366
25.8429
0.0835
35
3.2520
16.0532
0.1447
75
7.1138
26.0232
0.0839
36
3.4065
16.5054
0.1410
76
7.1911
26.2030
0.1267
37
3.4837
16.7290
0.0918
77
7.3455
26.5610
0.1276
38
3.5609
16.9511
0.0903
78
7.4228
26.7392
0.0856
39
3.6382
17.1717
0.0889
79
7.5000
26.9169
15.4377
40
3.7154
17.3908
0.1309As shown in Figures2 and 3, the predicted LSSVM values and experimental values of σ1 were almost identical, indicating that the LSSVM-based failure criterion model effectively simulated the failure behavior. Figure 4 shows a comparison between the failure envelopes determined by the LSSVM, Mohr-Coulomb, and Hoek-Brown models and those from the experimental data. It can be seen from Figure 4 that σ1 predicted by the LSSVM model agreed well both with the experimental data and the Hoek-Brown model at large stresses but differed a little from the Mohr-Coulomb model; however, σ1 differed a little from the Hoek-Brown model at small stresses. These results indicate that the LSSVM-based rock failure criterion trained on experimental data predicts the failure behavior of rock with reasonable accuracy.Figure 2
Comparison of major principal stressσ1 between the experimental data and predicted value for training samples.Figure 3
Comparison of major principal stressσ1 between the experimental data and predicted value for testing samples.Figure 4
Comparison of failure envelopes for different rock failure criteria.


## 4.2. Numerical Analysis Using the LSSVM-Based Rock Failure Criterion

To verify the feasibility of the LSSVM-based rock failure criterion in numerical analysis, it was combined with FLAC3D modeling code to simulate the failure behavior of a 5 m radius circular rock tunnel as a hypothetical numerical case. An initialin situ stress and gravity was presupposed in the numerical model, and the experimental data from Section 4.1 was used. The properties of the rock mass listed in Figure 5 are based on the Mohr-Coulomb model. The numerical model is built using FLAC3D together with the algorithm in Section 3. The horizontal displacements are almost the same as the value of Mohr-Coulomb by FLAC3D (seen in Figure 6). The stress of surrounding rock is in well agreement with the law of Mohr-Coulomb by FLAC3D (seen in Figure 7). The horizontal displacements and the horizontal, vertical, and shear stresses induced in the surrounding rock by the excavation show that the LSSVM model results were almost identical to those obtained by the Mohr-Coulomb model. The induced horizontal and vertical stresses calculated by the two models are also shown as contour plots in Figures 8 and 9. It shows the mechanical character of rock was presented by LSSVM-based rock failure criteria. Overall, the results show that the LSSVM-based rock failure criteria present well the mechanical behavior and character and can be used for numerical analysis.Figure 5
A circular tunnel model, stress condition, and rock mass properties.Figure 6
Comparison of surrounding rock displacement of tunnel obtained by LSSVM and Mohr-Coulomb model.Comparison of horizontal stress of tunnel obtained by LSSVM and Mohr-Coulomb model (a)σx, (b) σy, (c) σxy.
(a)
(b)
(c)Comparison of horizontal stress contour in surrounding rock mass obtained by LSSVM and Mohr-Coulomb model.
(a)
Mohr-Coulomb model
(b)
LSSVM modelComparison of vertical stress contour in surrounding rock mass obtained by LSSVM and Mohr-Coulomb model.
(a)
Mohr-Coulomb model
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LSSVM modelTo verify the LSSVM-based rock failure criterion for different initialin situ stress states, the circular tunnel was investigated at different vertical stress σyy=5, 10, and 15 MPa and corresponding horizontal stress σxx=1.2σyy in each case. The maximum error and maximum relative error of horizontal displacements are about 0.08 and 2.8%, respectively. The maximum relative error of vertical stress and shear stress is about 18% at the beginning of excavation (seen Figure 11(b)) and then it will be less than 5% with the excavation. Using the LSSVM and Mohr-Coulomb models, the calculated horizontal displacements in the rock surrounding the tunnel are shown in Figure 10 and the major principal stresses are shown in Figure 11. Those show it is feasible to combine LSSVM rock failure criterion with numerical analysis.Figure 10
Comparison of surrounding rock mass displacement obtained by LSSVM and Mohr-Coulomb model in differentin situ stress level (syis the vertical in situ stress).Comparison of surrounding rock stress of tunnel between LSSVM and Mohr-Coulomb model in differentin situ stress level (a) σx, (b) σy, (c) τxy (syis the vertical in situ stress).
(a)
(b)
(c)The displacements for both models agree well for the three initialin situ stress states (Figure 10), and the principal stresses are also similar for both models (Figure 11), with σy and τxy showing small differences near the wall of tunnel. These results generally reflect the above results (in which σyy=1 MPa).The LSSVM and Hoek-Brown models were also compared. Figures12–14 show the results for the same initial stress states as above, and it is seen that displacements in the surrounding rock agreed well with both the Hoek-Brown and Mohr-Coulomb results. Some small differences are evident in σxx and τxy compared to the Mohr-Coulomb model, but the results are consistent with the Hoek-Brown model because, as shown in Figure 4, the LSSVM and Hoek-Brown failure envelopes are similar. Thus, the proposed failure criterion can be used in numerical analysis to effectively reflect the mechanical behavior of rock.Displacement and stresses induced by excavation atσy=5MPa.
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Displacement
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yDisplacement and stresses induced by excavation atσy=10 MPa.
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Displacement
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σ
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τ
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yDisplacement and stresses induced by excavation atσy=15 MPa.
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Displacement
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## 5. Conclusions

The results of comparisons between the proposed LSSVM-based rock failure criterion and the Mohr-Coulomb and Hoek-Brown criteria and experimental data demonstrated clearly that LSSVM provided an effective rock failure criterion for the purpose of numerical analysis. Comparisons of the displacements in the rock surrounding a circular tunnel from the LSSVM, Mohr-Coulomb and Hoek-Brown models, and experimental data showed that the LSSVM model mapped the nonlinear relationship between the mechanical properties of the rock and its failure behavior.



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*Source: 246068-2015-02-23.xml*