research article lssvm-based rock failure criterion and

Research ArticleLSSVM-Based Rock Failure Criterion and Its Application inNumerical Simulation

Changxing Zhu Hongbo Zhao and Zhongliang Ru

School of Civil Engineering Henan Polytechnic University Jiaozuo 454003 China

Correspondence should be addressed to Hongbo Zhao bxhbzhaohotmailcom

Received 19 November 2014 Revised 3 February 2015 Accepted 11 February 2015

Academic Editor Erik Cuevas

Copyright copy 2015 Changxing Zhu et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A rock failure criterion is very important for the prediction of the failure of rocks or rock masses in rock mechanics andengineering Least squares support vector machines (LSSVM) are a powerful tool for addressing complex nonlinear problemsThis paper describes a LSSVM-based rock failure criterion for analyzing the deformation of a circular tunnel under different in situstresses without assuming a function form First LSSVMwas used to represent the nonlinear relationship between the mechanicalproperties of rock and the failure behavior of the rock in order to construct a rock failure criterion based on experimental dataThen this was used in a hypothetical numerical analysis of a circular tunnel to analyze the mechanical behavior of the rock masssurrounding the tunnel The Mohr-Coulomb and Hoek-Brown failure criteria were also used to analyze the same case and theresults were compared these clearly indicate that LSSVM can be used to establish a rock failure criterion and to predict the failureof a rock mass during excavation of a circular tunnel

1 Introduction

The failure of rock has been one of the most active researchareas in the engineering of rock structures since the establish-ment of the International Society for RockMechanics (ISRM)in 1962 Ulusay and Hudson gave a detailed descriptionof the importance of the ability to predict rock failure inpractical engineering [1] Rock failure criteria include thetheoretical basis of methods of predicting when and howrock materials fail under the action of external loads and aretherefore essential for optimization of engineering design andconstruction in rock masses Over the past several decades anumber of failure criteria have been proposed (eg Mohr-Coulomb Hoek-Brown and Griffithrsquos and Drucker-Pragercriteria) and developed to predict the failure of rock [2ndash11] Recently ISRM suggested methods have been publishedas criteria for the prediction of rock failure [12 13] Boththe Mohr-Coulomb and Hoek-Brown criteria are generallyconsidered to be reliable predictors of rock failure In thesetwo models the major principal stress 120590

1is seen as function

of the uniaxial compressive strength (120590119888) theminor principal

stress (1205903) and some constants that is 119891(120590

1 1205903) = 0 These

constants can often be determined by laboratory experimentin situ tests and back-analysis

Empirical failure criteria established under specific con-ditions such as rock type stress state or functional form arenot generic and cannot be adopted uncritically for any setof conditions [10] but must be established experimentally forany given set of rock formation and stress conditions

For this reason artificial neural network (ANN) com-putational models are widely used to establish rock failurecriteria for geotechnical engineering problems This is due tothe fact that ANN perform function approximation withoutrequiring additional assumptions of the function form Moredetailed descriptions of the ANN-based failure criterionmethod have been reported by for example Meulenkampand Grima Singh et al Canakci and Pala Tiryaki Zorluet al and Rafiai and Jafari [14ndash20] However a disadvantageof ANN for establishing rock failure criteria is its slowconvergence and overfitting To overcome these limitations

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 246068 13 pageshttpdxdoiorg1011552015246068

2 Mathematical Problems in Engineering

in the present study a LSSVM-based rock failure criterionwas developed based on experimental data Support vectormachines (SVM) have been widely used in geotechnicalengineering for nonlinear mapping [21ndash23]

This paper is organized as follows Section 2 focuses onthe formulation of LSSVM A LSSVM-based rock failurecriterion is described in detail in Section 3 In Section 4a hypothetical numerical experiment is used to verify theLSSVM-based rock failure criterion by comparing with ear-lier failure criteria Finally conclusions are given in Section 5

2 Least-Squares Support VectorMachine (LSSVM)

LSSVMmodels are an alternative formulation of SVMregres-sion proposed by Suykens and Vandewalle [24] Supposethere is a given training set of 119873 data points 119909

119896 119910119896 (119896 =

1 2 119873) with input data 119909119896isin 119877119873 and output 119910

119896isin 119903

where 119877119873 is an 119873-dimensional vector space and 119903 is a one-dimensional vector space LSSVMmodels in feature space aregiven by

119910 (119909) = 119908119879120601 (119909) + 119887 (1)

where the nonlinear mapping 120593(sdot) represents the input datain a higher-dimensional feature space 119908 isin 119877

119899 119887 isin 119903 119908 isan adjustable weight vector and 119887 is the scalar threshold Forfunction estimation in LSSVM the following optimizationproblem is formulated

Minimize 1

2

119908119879119908 + 120574

1

2

119873

sum

119896=1

1198902

119896

Subjected to 119910 (119909) = 119908119879120593 (119909) + 119887 + 119890

119896 119896 = 1 119873

(2)

where 120574 is the regularization parameter for determiningthe trade-off between the fitting error minimization andsmoothness and 119890

119896is an error variable

The Lagrangian (119871(119908 119887 119890 120572)) for the above optimizationproblem (2) is

119871 (119908 119887 119890 120572) =

1

2

119908119879119908 + 120574

1

2

119873

sum

119896=1

1198902

119896

minus

119873

sum

119896=1

120572119896119910119896[119908119879120593 (119909119896) + 119887] minus 1 + 119890

119896

(3)

where 120572119896is the Lagrange multiplier The conditions for

optimality are given by

120597119871

120597119908

= 0 997904rArr 119908 =

119873

sum

119896=1

120572119896120593 (119909119896)

120597119871

120597119887

= 0 997904rArr

119873

sum

119896=1

120572119896= 0

120597119871

120597119890119896

= 0 997904rArr 120572119896= 120574119890119896 119896 = 1 119873

120597119871

120597120572119896

= 0 997904rArr 119908119879120593 (119909119896) + 119887 + 119890

119896minus 119910119896= 0 119896 = 1 119873

(4)

without regard to 119890119896and 119908 the solution is given by the

following set of linear equations

Φ[

119887

120572

] = [

0

119910

] (5)

where 119910 = [1199101 119910

119873] 1 = [1 1] 120572 = [120572

1 120572

119873]

and Mercerrsquos theorem is applied within the Ω matrix Ω =

120593(119909119896)119879120593(119909119897) = 119870(119909

119896 119909119897) 119896 119897 = 1 119873 where 119870(119909

119896 119909119897)

is the kernel function and the matrix Φ is invertible bychoosing 120574 gt 119908 and can be written as

Φ = [

0 1119879

1 Ω + 120574minus1119868

] (6)

Then the analytical solution of 120572 and 119887 is given by

[

119887

120572

] = Φminus1[

0

119910

] (7)

The LSSVMmodel can then be expressed as

119910 (119909) =

119873

sum

119896=1

120572119896119870(119909 119909

119896) + 119887 (8)

In (8) 119870(119909 119909119896) is the kernel function composed of the

following three forms

(1) Polynomial kernel

119870 (119883 119884) = ((119883 sdot 119884) + 1)119889 119889 = 1 2 119899 (9)

(2) Radial kernel

119870 (119883 119884) = expminus|119883 minus 119884|2

21205902

(10)

(3) Sigmoidal kernel

119870 (119883 119884) = tanh (120593 (119883 sdot 119884) + 120579) (11)

The brief procedure LSSVM is the following

Step 1 Build the training samples (119909119896 119910119896) based on rock

mechanical experiment

Step 2 Get (6) according to the LSSVM algorithm

Step 3 Get the analytical solution of 120572 and 119887 in (7)

Step 4 Get the LSSVMmodel (8)

Mathematical Problems in Engineering 3

Start

LSSVM algorithm

LSSVM model eq (8) Rock failure criterion

Equivalent rock failure criterion eq (18) and (19)

End

Experiment data of rock (xk yk)

Figure 1 The flowchart of LSSVM-based failure criterion

3 LSSVM-Based Rock Failure Criteria

In order to better predict the behavior failure of rock massesLSSVM was used in this work to represent the nonlinearrelationship between rock properties and rock failure andthus to establish a LSSVM-based rock failure criterionSimilar to traditional failure criteria in form LSSVM-basedrock failure criteria can be defined as

SVM (1205901 1205903) = 0 (12)

where 1205901and 120590

3are the major and minor principal stresses

respectivelyThese can be obtained by laboratory experimentand field testingThe LSSVM-based rock failure criterion canbe rewritten in the form

1205901=

119873

sum

119896=1

120572119896119870(1205903 1205903119896) + 119887 (13)

where 1205903119896is the series of experimental data and 120572

119896and 119887 are

calculated using the algorithm introduced in Section 2 Oncea LSSVM-based rock failure criterion is established it canbe used in numerical analysis to predict rock failure In thiswork Excel VBA software was used to execute the code of theLSSVM-based rock failure criterion

To implement the LSSVM-based failure criteria in anumerical model it is useful to express it as an equiva-lent Mohr-Coulomb criterion with variable parameters Theparameters of the equivalent criterion are obtained as

120590eq119888= 1205901minus 119902

eq1205903=

119873

sum

119896=1

120572119896119870(1205903 1205903119896) + 119887 minus 119902

eq1205903 (14)

119902eq=

1205971205901

1205971205903

=

120597 (sum119873

119896=1120572119896119870(1205903 1205903119896))

1205971205903

=

119873

sum

119896=1

120572119896

120597 (119870 (1205903 1205903119896))

1205971205903

(15)

The RBF kernel function (Equation (10)) was adopted

120597 (119870 (1205903 1205903119896))

1205971205903

= minus

1

1205902119870(1205903 1205903119896) (16)

Substituting (16) into (15) gives the following form

119902eq= minus

1

1205902

119873

sum

119896=1

120572119896119870(1205903 1205903119896) (17)

and the equivalent Mohr-Coulomb criterion may be calcu-lated from the following equations

119888eq=

120590eq119888

2radic119902eq (18)

120593eq= 2tanminus1 (radic119902eq) minus 120587

2

(19)

where 119888eq and 120593eq are the instantaneous values of cohesion

and friction angle respectivelyLSSVM-based rock failure criterion presents themechan-

ical character of rock through combining L-SVM and rockfailure criterion The brief procedure is showed in Figure 1


Table 1 Experimental data and the values of Lagrange multiplier 120572119896

Number 1205903(Mpa) 120590

1(Mpa) 120572

119896Number 120590

3(Mpa) 120590

1(Mpa) 120572

119896

1 00000 00000 minus1384852 41 38699 178247 01287

2 00853 27979 976169 42 39471 180397 00845

3 02398 40913 55845 43 40244 182534 00837

4 03170 46287 22133 44 41016 184657 00829

5 03943 51224 16698 45 41788 186769 00822

6 04715 55829 13160 46 42561 188868 01220

7 05487 60173 15167 47 44105 193033 01210

8 07032 68251 12040 48 44878 195098 00800

9 07804 72046 06538 49 45650 197153 01193

10 08577 75708 05713 50 47195 201231 01186

11 09349 79254 07297 51 47967 203255 00787

12 10894 86045 06295 52 48740 205269 00785

13 11666 89311 03662 53 49512 207273 01175

14 12439 92501 04865 54 51057 211255 01173

15 13983 98678 04344 55 51829 213232 00781

16 14756 101677 02606 56 52601 215200 00781

17 15528 104620 03552 57 53374 217160 01171

18 17073 110360 03247 58 54919 221055 01172

19 17845 113162 01989 59 55691 222990 00783

20 18617 115922 01876 60 56463 224918 00784

21 19390 118644 02615 61 57236 226838 01178

22 20935 123979 02442 62 58780 230655 01182

23 21707 126597 01526 63 59553 232553 00790

24 22479 129183 01459 64 60325 234443 01190

25 23252 131740 01398 65 61870 238203 01195

26 24024 134268 01342 66 62642 240073 00800

27 24796 136769 01914 67 63415 241937 01206

28 26341 141696 01826 68 64959 245644 01212

29 27113 144124 01165 69 65732 247488 00813

30 27886 146529 01678 70 66504 249327 00816

31 29431 151275 01616 71 67276 251159 00820

32 30203 153617 01040 72 68049 252985 01237

33 30975 155940 01015 73 69593 256620 01244

34 31748 158245 00992 74 70366 258429 00835

35 32520 160532 01447 75 71138 260232 00839

36 34065 165054 01410 76 71911 262030 01267

37 34837 167290 00918 77 73455 265610 01276

38 35609 169511 00903 78 74228 267392 00856

39 36382 171717 00889 79 75000 269169 154377

40 37154 173908 01309


0

5

10

15

20

25

30

0 5 10 15 20 25 30

Pred

icte

d va

lue o

f1205901

(MPa

)

Experiment value of 1205901 (MPa)

Figure 2 Comparison of major principal stress 1205901between the experimental data and predicted value for training samples

0

5

10

15

20

25

30

0 5 10 15 20 25 30

Pred

icte

d va

lue o

f1205901

(MPa

)


Figure 3 Comparison of major principal stress 1205901between the experimental data and predicted value for testing samples

4 Numerical Experiment

41 Matching the Experimental Data In this work the RBFkernel function was adopted for pattern analysis or recogni-tion with the following parameters 120590 = 2 The value of 119887 is145061 The values of 120572

119896are listed in Table 1 As in any other

use of LSSVM the SVM must be trained and tested 98 datasets from triaxial compression tests consisting of 79 trainingsamples and 19 testing samples were used as the experimentaldata in this study (seen in Table 1) Once the performance

of the SVM model was satisfactory it was used to representthe nonlinear relationship between rock properties and themechanical behavior of the rock to establish the LSSVM-based rock failure criterion

As shown in Figures 2 and 3 the predicted LSSVMvalues and experimental values of 120590

1were almost identical

indicating that the LSSVM-based failure criterion modeleffectively simulated the failure behavior Figure 4 shows acomparison between the failure envelopes determined by theLSSVMMohr-Coulomb andHoek-Brownmodels and those


0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8

Experiment dataHoek-Brown model

Mohr-Coulomb modelSVM-based model

1205901

(MPa

)

1205903 (MPa)

Figure 4 Comparison of failure envelopes for different rock failure criteria

120590xxR = 5 cm

120590yy

O

In situ stress120590xx = minus12MPa 120590yy = minus1MPa 120590zz = minus15MPa

Rock mass propertiesE = 30GPa 120583 = 03 c = 15272MPa 120593 = 304∘

120574 = 2500kNm3

Figure 5 A circular tunnel model stress condition and rock mass properties

from the experimental data It can be seen from Figure 4that 120590

1predicted by the LSSVMmodel agreed well both with

the experimental data and the Hoek-Brown model at largestresses but differed a little from the Mohr-Coulomb modelhowever 120590

1differed a little from the Hoek-Brown model at

small stresses These results indicate that the LSSVM-basedrock failure criterion trained on experimental data predictsthe failure behavior of rock with reasonable accuracy

42 Numerical Analysis Using the LSSVM-Based Rock FailureCriterion To verify the feasibility of the LSSVM-based rockfailure criterion in numerical analysis it was combined withFLAC3D modeling code to simulate the failure behavior of a5m radius circular rock tunnel as a hypothetical numericalcase An initial in situ stress and gravity was presupposedin the numerical model and the experimental data from

Section 41 was usedThe properties of the rockmass listed inFigure 5 are based on theMohr-CoulombmodelThe numer-ical model is built using FLAC3D together with the algorithmin Section 3 The horizontal displacements are almost thesame as the value of Mohr-Coulomb by FLAC3D (seen inFigure 6) The stress of surrounding rock is in well agree-ment with the law of Mohr-Coulomb by FLAC3D (seen inFigure 7) The horizontal displacements and the horizontalvertical and shear stresses induced in the surrounding rockby the excavation show that the LSSVM model results werealmost identical to those obtained by the Mohr-Coulombmodel The induced horizontal and vertical stresses calcu-lated by the two models are also shown as contour plots inFigures 8 and 9 It shows the mechanical character of rockwas presented by LSSVM-based rock failure criteria Overallthe results show that the LSSVM-based rock failure criteria


0

05

1

15

2

25

3

5 10 15 20 25 30 35Distance to center of circular tunnel (m)

Mohr-CoulombSVM

Disp

lace

men

t (10

minus4

m)

Figure 6 Comparison of surrounding rock displacement of tunnel obtained by LSSVM and Mohr-Coulomb model

0

02

04

06

08

1

12


120590x

(MPa

)

(a)

0

02

04

06

08

1

12

14

16

18


120590y

(MPa

)

(b)

0

002

004

006

008

01

012

014

016

018


Mohr-CoulombSVM

120591xy

(MPa

)

(c)

Figure 7 Comparison of horizontal stress of tunnel obtained by LSSVM and Mohr-Coulomb model (a) 120590119909 (b) 120590

119910 (c) 120590

119909119910


(a) Mohr-Coulomb model (b) LSSVMmodel

Figure 8 Comparison of horizontal stress contour in surrounding rock mass obtained by LSSVM and Mohr-Coulomb model


Figure 9 Comparison of vertical stress contour in surrounding rock mass obtained by LSSVM and Mohr-Coulomb model

0

1

2

3

4

5

6

7

8


M-Csy = 5MPa SVMsy = 5MPaM-Csy = 10MPa SVMsy = 10MPaM-Csy = 15MPa SVMsy = 15MPa

Disp

lace

men

t (10

minus3

m)

Figure 10 Comparison of surrounding rock mass displacement obtained by LSSVM and Mohr-Coulomb model in different in situ stresslevel (119904

119910is the vertical in situ stress)


0

2

4

6

8

10

12

14

16


120590x

(MPa

)


(a)

0

5

10

15

20

25


120590y

(MPa

)


(b)

0

02

04

06

08

1

12


120591xy

(MPa

)


(c)

Figure 11 Comparison of surrounding rock stress of tunnel between LSSVM and Mohr-Coulomb model in different in situ stress level (a)120590119909 (b) 120590

119910 (c) 120591

119909119910(119904119910is the vertical in situ stress)

present well the mechanical behavior and character and canbe used for numerical analysis

To verify the LSSVM-based rock failure criterion for dif-ferent initial in situ stress states the circular tunnel was inves-tigated at different vertical stress 120590

119910119910= 5 10 and 15MPa and

corresponding horizontal stress 120590119909119909

= 12120590119910119910

in each caseThe maximum error and maximum relative error of hori-zontal displacements are about 008 and 28 respectivelyThe maximum relative error of vertical stress and shearstress is about 18 at the beginning of excavation (seenFigure 11(b)) and then it will be less than 5 with the exca-vation Using the LSSVM and Mohr-Coulomb models thecalculated horizontal displacements in the rock surroundingthe tunnel are shown in Figure 10 and the major principalstresses are shown in Figure 11 Those show it is feasibleto combine LSSVM rock failure criterion with numericalanalysis

The displacements for both models agree well for thethree initial in situ stress states (Figure 10) and the principalstresses are also similar for both models (Figure 11) with 120590

119910

and 120591119909119910

showing small differences near the wall of tunnelThese results generally reflect the above results (in which120590119910119910= 1MPa)The LSSVM and Hoek-Brown models were also com-

pared Figures 12ndash14 show the results for the same initialstress states as above and it is seen that displacements in thesurrounding rock agreed well with both theHoek-Brown andMohr-Coulomb results Some small differences are evident in120590119909119909

and 120591119909119910

compared to the Mohr-Coulomb model but theresults are consistent with the Hoek-Brown model becauseas shown in Figure 4 the LSSVM and Hoek-Brown failureenvelopes are similar Thus the proposed failure criterioncan be used in numerical analysis to effectively reflect themechanical behavior of rock


0

02

04

06

08

1

12

14

16

18


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

1

2

3

4

5

6

5 15 25 35Distance to center of circular tunnel (m)

120590x

(MPa

)

(b) 120590119909

0

1

2

3

4

5

6

7

8

9


Hoek-BrownSVMMohr-Coulomb

120590y

(MPa

)

(c) 120590119910


0

01

02

03

04

05

06

07


120591xy

(MPa

)

(d) 120591119909119910

Figure 12 Displacement and stresses induced by excavation at 120590119910= 5MPa

5 Conclusions

The results of comparisons between the proposed LSSVM-based rock failure criterion and the Mohr-Coulomb andHoek-Brown criteria and experimental data demonstratedclearly that LSSVM provided an effective rock failure cri-terion for the purpose of numerical analysis Comparisonsof the displacements in the rock surrounding a circulartunnel from the LSSVM Mohr-Coulomb and Hoek-Brownmodels and experimental data showed that the LSSVMmodel mapped the nonlinear relationship between themechanical properties of the rock and its failure behavior

List of Symbols

119873 Number of samples119877119873 119873-dimensional vector space

119903 One-dimensional vector space119909119896 119910119896 Input and output of training samples

120593(sdot) Nonlinear mapping in ahigher-dimensional feature space

119908 Adjustable weight vector119887 Scalar threshold120574 Regularization parameter119890119896 Error variable


0

05

1

15

2

25

3

35

4

45


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12


120590x

(MPa

)

(b) 120590119909

0

2

4

6

8

10

12

14

16



120590y

(MPa

)

(c) 120590119910

0

01

02

03

04

05

06

07

08



120591xy

(MPa

)

(d) 120591119909119910


120572119896 Lagrange multiplier

119896( ) Kernel functionΦ Matrix of kernel function119897 1 times 119873matrix1205901 Major principal stress

1205903 Minor principal stress

120590119910119910 Vertical in situ stress

120590119909119909 Horizontal in situ stress

120590119909 Horizontal stress of surrounding rock

mass120590119910 Vertical stress of surrounding rock mass

120591119909119910 Shear stress

1205903119896 Experimental data

119888eq Instantaneous values of cohesion

120593eq Instantaneous values of friction angleSVM( ) LSSVM-based rock failure criteria120590 Parameter of RBF kernel function

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Support by the Program for Innovative Research Team (inScience and Technology) in University of Henan Province


0

1

2

3

4

5

6

7

8


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12

14

16


120590x

(MPa

)

(b) 120590119909

0

5

10

15

20

25



120590y

(MPa

)

(c) 120590119910


0

02

04

06

08

1

12


120591xy

(MPa

)

(d) 120591119909119910


(no 15IRTSTHN029) and National Fund of Science in China(nos 51104057 41172244) are gratefully acknowledged

References

[1] R Ulusay and J A Hudson ldquoSuggested methods for rockfailure criteria general introductionrdquo Rock Mechanics and RockEngineering vol 45 no 6 p 971 2012

[2] C Fairhurst ldquoOn the validity of the lsquoBrazilianrsquo test for brittlematerialsrdquo International Journal of Rock Mechanics and MiningSciences amp Geomechanics Abstracts vol 1 no 4 pp 535ndash5461964

[3] D W Hobbs ldquoThe strength and the stress-strain characteristicsof coal in triaxial compressionrdquoThe Journal of Geology vol 72no 2 pp 214ndash231 1964

[4] S A F Murrell ldquoThe effect of triaxial stress systems on thestrength of rock at atmospheric temperaturerdquo International

Journal of Rock Mechanics andMining Sciences vol 3 pp 11ndash431965

[5] J A Franklin ldquoTriaxial strength of rock materialsrdquo RockMechanics Felsmechanik Mecanique des Roches vol 3 no 2 pp86ndash98 1971

[6] Z T Bieniawski ldquoEstimating the strength of rock materialsrdquoJournal of The South African Institute of Mining and Metallurgyvol 74 no 8 pp 312ndash320 1974

[7] E Hoek and E T Brown Underground Excavations in RockInstitution of Mining amp Metallurgy London UK 1980

[8] T Ramamurthy G V Rao and K Rao ldquoA strength criterion forrocksrdquo in Proceedings of the Indian Geotechnical Conference pp59ndash64 Roorkee India 1985

[9] I W Johnston ldquoStrength of intact geomechanical materialsrdquoJournal of Geotechnical Engineering vol 111 no 6 pp 730ndash7491985


[10] P R Sheorey A K Biswas and V D Choubey ldquoAn empiricalfailure criterion for rocks and jointed rock massesrdquo EngineeringGeology vol 26 no 2 pp 141ndash159 1989

[11] N Yoshida N R Morgenstern and D H Chan ldquoA failurecriterion for stiff soils and rocks exhibiting softeningrdquoCanadianGeotechnical Journal vol 27 no 2 pp 195ndash202 1990

[12] J F Labuz andA Zang ldquoMohr-Coulomb failure criterionrdquoRockMechanics and Rock Engineering vol 45 no 6 pp 975ndash9792012

[13] E Eberhardt ldquoThe Hoek-Brown failure criterionrdquo RockMechanics and Rock Engineering vol 45 no 6 pp 981ndash9882012

[14] F Meulenkamp and M A Grima ldquoApplication of neuralnetworks for the prediction of the unconfined compressivestrength from Equotip hardnessrdquo International Journal of RockMechanics and Mining Sciences vol 36 no 1 pp 29ndash39 1999

[15] V K Singh D Singh and T N Singh ldquoPrediction of strengthproperties of some schistose rocks frompetrographic propertiesusing artificial neural networksrdquo International Journal of RockMechanics andMining Sciences vol 38 no 2 pp 269ndash284 2001

[16] H Canakci and M Pala ldquoTensile strength of basalt from aneural networkrdquo Engineering Geology vol 94 no 1-2 pp 10ndash18 2007

[17] B Tiryaki ldquoPredicting intact rock strength formechanical exca-vation using multivariate statistics artificial neural networksand regression treesrdquo Engineering Geology vol 99 no 1-2 pp51ndash60 2008

[18] K Zorlu C Gokceoglu F Ocakoglu H A Nefeslioglu and SAcikalin ldquoPrediction of uniaxial compressive strength of sand-stones using petrography-based modelsrdquo Engineering Geologyvol 96 no 3-4 pp 141ndash158 2008

[19] H Rafiai and A Jafari ldquoArtificial neural networks as a basis fornew generation of rock failure criteriardquo International Journal ofRockMechanics andMining Sciences vol 48 no 7 pp 1153ndash11592011

[20] H Rafiai and A Jafari ldquoImplementation of ANN-based rockfailure criteria in numerical simulationsrdquo in Proceedings of the12th International Congress on Rock Mechanics of the Inter-national Society for Rock Mechanics (ISRM rsquo11) pp 501ndash506Beijing China October 2011

[21] H-B Zhao ldquoSlope reliability analysis using a support vectormachinerdquo Computers and Geotechnics vol 35 no 3 pp 459ndash467 2008

[22] H-B Zhao and S Yin ldquoGeomechanical parameters identi-fication by particle swarm optimization and support vectormachinerdquo Applied Mathematical Modelling vol 33 no 10 pp3997ndash4012 2009

[23] H-B Zhao and S Yin ldquoA CPSO-SVM model for ultimatebearing capacity determinationrdquo Marine Georesources andGeotechnology vol 28 no 1 pp 64ndash75 2010

[24] J A K Suykens and J Vandewalle ldquoLeast squares supportvector machine classifiersrdquo Neural Processing Letters vol 9 no3 pp 293ndash300 1999

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


in the present study a LSSVM-based rock failure criterionwas developed based on experimental data Support vectormachines (SVM) have been widely used in geotechnicalengineering for nonlinear mapping [21ndash23]

This paper is organized as follows Section 2 focuses onthe formulation of LSSVM A LSSVM-based rock failurecriterion is described in detail in Section 3 In Section 4a hypothetical numerical experiment is used to verify theLSSVM-based rock failure criterion by comparing with ear-lier failure criteria Finally conclusions are given in Section 5

2 Least-Squares Support VectorMachine (LSSVM)

LSSVMmodels are an alternative formulation of SVMregres-sion proposed by Suykens and Vandewalle [24] Supposethere is a given training set of 119873 data points 119909

119896 119910119896 (119896 =

1 2 119873) with input data 119909119896isin 119877119873 and output 119910

119896isin 119903

where 119877119873 is an 119873-dimensional vector space and 119903 is a one-dimensional vector space LSSVMmodels in feature space aregiven by

119910 (119909) = 119908119879120601 (119909) + 119887 (1)

where the nonlinear mapping 120593(sdot) represents the input datain a higher-dimensional feature space 119908 isin 119877

119899 119887 isin 119903 119908 isan adjustable weight vector and 119887 is the scalar threshold Forfunction estimation in LSSVM the following optimizationproblem is formulated

Minimize 1

2

119908119879119908 + 120574

1

2

119873

sum

119896=1

1198902

119896

Subjected to 119910 (119909) = 119908119879120593 (119909) + 119887 + 119890

119896 119896 = 1 119873

(2)

where 120574 is the regularization parameter for determiningthe trade-off between the fitting error minimization andsmoothness and 119890

119896is an error variable

The Lagrangian (119871(119908 119887 119890 120572)) for the above optimizationproblem (2) is

119871 (119908 119887 119890 120572) =

1

2

119908119879119908 + 120574

1

2

119873

sum

119896=1

1198902

119896

minus

119873

sum

119896=1

120572119896119910119896[119908119879120593 (119909119896) + 119887] minus 1 + 119890

119896

(3)

where 120572119896is the Lagrange multiplier The conditions for

optimality are given by

120597119871

120597119908

= 0 997904rArr 119908 =

119873

sum

119896=1

120572119896120593 (119909119896)

120597119871

120597119887

= 0 997904rArr

119873

sum

119896=1

120572119896= 0

120597119871

120597119890119896

= 0 997904rArr 120572119896= 120574119890119896 119896 = 1 119873

120597119871

120597120572119896

= 0 997904rArr 119908119879120593 (119909119896) + 119887 + 119890

119896minus 119910119896= 0 119896 = 1 119873

(4)

without regard to 119890119896and 119908 the solution is given by the

following set of linear equations

Φ[

119887

120572

] = [

0

119910

] (5)

where 119910 = [1199101 119910

119873] 1 = [1 1] 120572 = [120572

1 120572

119873]

and Mercerrsquos theorem is applied within the Ω matrix Ω =

120593(119909119896)119879120593(119909119897) = 119870(119909

119896 119909119897) 119896 119897 = 1 119873 where 119870(119909

119896 119909119897)

is the kernel function and the matrix Φ is invertible bychoosing 120574 gt 119908 and can be written as

Φ = [

0 1119879

1 Ω + 120574minus1119868

] (6)

Then the analytical solution of 120572 and 119887 is given by

[

119887

120572

] = Φminus1[

0

119910

] (7)

The LSSVMmodel can then be expressed as

119910 (119909) =

119873

sum

119896=1

120572119896119870(119909 119909

119896) + 119887 (8)

In (8) 119870(119909 119909119896) is the kernel function composed of the

following three forms

(1) Polynomial kernel

119870 (119883 119884) = ((119883 sdot 119884) + 1)119889 119889 = 1 2 119899 (9)

(2) Radial kernel

119870 (119883 119884) = expminus|119883 minus 119884|2

21205902

(10)

(3) Sigmoidal kernel

119870 (119883 119884) = tanh (120593 (119883 sdot 119884) + 120579) (11)

The brief procedure LSSVM is the following

Step 1 Build the training samples (119909119896 119910119896) based on rock

mechanical experiment

Step 2 Get (6) according to the LSSVM algorithm

Step 3 Get the analytical solution of 120572 and 119887 in (7)

Step 4 Get the LSSVMmodel (8)


Start

LSSVM algorithm



End





SVM (1205901 1205903) = 0 (12)

where 1205901and 120590



1205901=

119873

sum

119896=1

120572119896119870(1205903 1205903119896) + 119887 (13)


119896and 119887 are



120590eq119888= 1205901minus 119902

eq1205903=

119873

sum

119896=1

120572119896119870(1205903 1205903119896) + 119887 minus 119902

eq1205903 (14)

119902eq=

1205971205901

1205971205903

=

120597 (sum119873

119896=1120572119896119870(1205903 1205903119896))

1205971205903

=

119873

sum

119896=1

120572119896

120597 (119870 (1205903 1205903119896))

1205971205903

(15)


120597 (119870 (1205903 1205903119896))

1205971205903

= minus

1

1205902119870(1205903 1205903119896) (16)


119902eq= minus

1

1205902

119873

sum

119896=1

120572119896119870(1205903 1205903119896) (17)


119888eq=

120590eq119888

2radic119902eq (18)


2

(19)






Number 1205903(Mpa) 120590

1(Mpa) 120572

119896Number 120590

3(Mpa) 120590

1(Mpa) 120572

119896

1 00000 00000 minus1384852 41 38699 178247 01287

2 00853 27979 976169 42 39471 180397 00845

3 02398 40913 55845 43 40244 182534 00837

4 03170 46287 22133 44 41016 184657 00829

5 03943 51224 16698 45 41788 186769 00822

6 04715 55829 13160 46 42561 188868 01220

7 05487 60173 15167 47 44105 193033 01210

8 07032 68251 12040 48 44878 195098 00800

9 07804 72046 06538 49 45650 197153 01193

10 08577 75708 05713 50 47195 201231 01186

11 09349 79254 07297 51 47967 203255 00787

12 10894 86045 06295 52 48740 205269 00785

13 11666 89311 03662 53 49512 207273 01175

14 12439 92501 04865 54 51057 211255 01173

15 13983 98678 04344 55 51829 213232 00781

16 14756 101677 02606 56 52601 215200 00781

17 15528 104620 03552 57 53374 217160 01171

18 17073 110360 03247 58 54919 221055 01172

19 17845 113162 01989 59 55691 222990 00783

20 18617 115922 01876 60 56463 224918 00784

21 19390 118644 02615 61 57236 226838 01178

22 20935 123979 02442 62 58780 230655 01182

23 21707 126597 01526 63 59553 232553 00790

24 22479 129183 01459 64 60325 234443 01190

25 23252 131740 01398 65 61870 238203 01195

26 24024 134268 01342 66 62642 240073 00800

27 24796 136769 01914 67 63415 241937 01206

28 26341 141696 01826 68 64959 245644 01212

29 27113 144124 01165 69 65732 247488 00813

30 27886 146529 01678 70 66504 249327 00816

31 29431 151275 01616 71 67276 251159 00820

32 30203 153617 01040 72 68049 252985 01237

33 30975 155940 01015 73 69593 256620 01244

34 31748 158245 00992 74 70366 258429 00835

35 32520 160532 01447 75 71138 260232 00839

36 34065 165054 01410 76 71911 262030 01267

37 34837 167290 00918 77 73455 265610 01276

38 35609 169511 00903 78 74228 267392 00856

39 36382 171717 00889 79 75000 269169 154377

40 37154 173908 01309


0

5

10

15

20

25

30

0 5 10 15 20 25 30

Pred

icte

d va

lue o

f1205901

(MPa

)



0

5

10

15

20

25

30

0 5 10 15 20 25 30

Pred

icte

d va

lue o

f1205901

(MPa

)












0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8



1205901

(MPa

)

1205903 (MPa)


120590xxR = 5 cm

120590yy

O



120574 = 2500kNm3










0

05

1

15

2

25

3


Mohr-CoulombSVM

Disp

lace

men

t (10

minus4

m)


0

02

04

06

08

1

12


120590x

(MPa

)

(a)

0

02

04

06

08

1

12

14

16

18


120590y

(MPa

)

(b)

0

002

004

006

008

01

012

014

016

018


Mohr-CoulombSVM

120591xy

(MPa

)

(c)


119910 (c) 120590

119909119910






0

1

2

3

4

5

6

7

8



Disp

lace

men

t (10

minus3

m)




0

2

4

6

8

10

12

14

16


120590x

(MPa

)


(a)

0

5

10

15

20

25


120590y

(MPa

)


(b)

0

02

04

06

08

1

12


120591xy

(MPa

)


(c)


119910 (c) 120591




119910119910= 5 10 and 15MPa and


= 12120590119910119910



119910

and 120591119909119910



and 120591119909119910



0

02

04

06

08

1

12

14

16

18


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

1

2

3

4

5

6


120590x

(MPa

)

(b) 120590119909

0

1

2

3

4

5

6

7

8

9



120590y

(MPa

)

(c) 120590119910


0

01

02

03

04

05

06

07


120591xy

(MPa

)

(d) 120591119909119910


5 Conclusions


List of Symbols






0

05

1

15

2

25

3

35

4

45


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12


120590x

(MPa

)

(b) 120590119909

0

2

4

6

8

10

12

14

16



120590y

(MPa

)

(c) 120590119910

0

01

02

03

04

05

06

07

08



120591xy

(MPa

)

(d) 120591119909119910









120591119909119910 Shear stress






Acknowledgments



0

1

2

3

4

5

6

7

8


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12

14

16


120590x

(MPa

)

(b) 120590119909

0

5

10

15

20

25



120590y

(MPa

)

(c) 120590119910


0

02

04

06

08

1

12


120591xy

(MPa

)

(d) 120591119909119910



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

LSSVM algorithm



End





SVM (1205901 1205903) = 0 (12)

where 1205901and 120590



1205901=

119873

sum

119896=1

120572119896119870(1205903 1205903119896) + 119887 (13)


119896and 119887 are



120590eq119888= 1205901minus 119902

eq1205903=

119873

sum

119896=1

120572119896119870(1205903 1205903119896) + 119887 minus 119902

eq1205903 (14)

119902eq=

1205971205901

1205971205903

=

120597 (sum119873

119896=1120572119896119870(1205903 1205903119896))

1205971205903

=

119873

sum

119896=1

120572119896

120597 (119870 (1205903 1205903119896))

1205971205903

(15)


120597 (119870 (1205903 1205903119896))

1205971205903

= minus

1

1205902119870(1205903 1205903119896) (16)


119902eq= minus

1

1205902

119873

sum

119896=1

120572119896119870(1205903 1205903119896) (17)


119888eq=

120590eq119888

2radic119902eq (18)


2

(19)






Number 1205903(Mpa) 120590

1(Mpa) 120572

119896Number 120590

3(Mpa) 120590

1(Mpa) 120572

119896

1 00000 00000 minus1384852 41 38699 178247 01287

2 00853 27979 976169 42 39471 180397 00845

3 02398 40913 55845 43 40244 182534 00837

4 03170 46287 22133 44 41016 184657 00829

5 03943 51224 16698 45 41788 186769 00822

6 04715 55829 13160 46 42561 188868 01220

7 05487 60173 15167 47 44105 193033 01210

8 07032 68251 12040 48 44878 195098 00800

9 07804 72046 06538 49 45650 197153 01193

10 08577 75708 05713 50 47195 201231 01186

11 09349 79254 07297 51 47967 203255 00787

12 10894 86045 06295 52 48740 205269 00785

13 11666 89311 03662 53 49512 207273 01175

14 12439 92501 04865 54 51057 211255 01173

15 13983 98678 04344 55 51829 213232 00781

16 14756 101677 02606 56 52601 215200 00781

17 15528 104620 03552 57 53374 217160 01171

18 17073 110360 03247 58 54919 221055 01172

19 17845 113162 01989 59 55691 222990 00783

20 18617 115922 01876 60 56463 224918 00784

21 19390 118644 02615 61 57236 226838 01178

22 20935 123979 02442 62 58780 230655 01182

23 21707 126597 01526 63 59553 232553 00790

24 22479 129183 01459 64 60325 234443 01190

25 23252 131740 01398 65 61870 238203 01195

26 24024 134268 01342 66 62642 240073 00800

27 24796 136769 01914 67 63415 241937 01206

28 26341 141696 01826 68 64959 245644 01212

29 27113 144124 01165 69 65732 247488 00813

30 27886 146529 01678 70 66504 249327 00816

31 29431 151275 01616 71 67276 251159 00820

32 30203 153617 01040 72 68049 252985 01237

33 30975 155940 01015 73 69593 256620 01244

34 31748 158245 00992 74 70366 258429 00835

35 32520 160532 01447 75 71138 260232 00839

36 34065 165054 01410 76 71911 262030 01267

37 34837 167290 00918 77 73455 265610 01276

38 35609 169511 00903 78 74228 267392 00856

39 36382 171717 00889 79 75000 269169 154377

40 37154 173908 01309


0

5

10

15

20

25

30

0 5 10 15 20 25 30

Pred

icte

d va

lue o

f1205901

(MPa

)



0

5

10

15

20

25

30

0 5 10 15 20 25 30

Pred

icte

d va

lue o

f1205901

(MPa

)












0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8



1205901

(MPa

)

1205903 (MPa)


120590xxR = 5 cm

120590yy

O



120574 = 2500kNm3










0

05

1

15

2

25

3


Mohr-CoulombSVM

Disp

lace

men

t (10

minus4

m)


0

02

04

06

08

1

12


120590x

(MPa

)

(a)

0

02

04

06

08

1

12

14

16

18


120590y

(MPa

)

(b)

0

002

004

006

008

01

012

014

016

018


Mohr-CoulombSVM

120591xy

(MPa

)

(c)


119910 (c) 120590

119909119910






0

1

2

3

4

5

6

7

8



Disp

lace

men

t (10

minus3

m)




0

2

4

6

8

10

12

14

16


120590x

(MPa

)


(a)

0

5

10

15

20

25


120590y

(MPa

)


(b)

0

02

04

06

08

1

12


120591xy

(MPa

)


(c)


119910 (c) 120591




119910119910= 5 10 and 15MPa and


= 12120590119910119910



119910

and 120591119909119910



and 120591119909119910



0

02

04

06

08

1

12

14

16

18


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

1

2

3

4

5

6


120590x

(MPa

)

(b) 120590119909

0

1

2

3

4

5

6

7

8

9



120590y

(MPa

)

(c) 120590119910


0

01

02

03

04

05

06

07


120591xy

(MPa

)

(d) 120591119909119910


5 Conclusions


List of Symbols






0

05

1

15

2

25

3

35

4

45


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12


120590x

(MPa

)

(b) 120590119909

0

2

4

6

8

10

12

14

16



120590y

(MPa

)

(c) 120590119910

0

01

02

03

04

05

06

07

08



120591xy

(MPa

)

(d) 120591119909119910









120591119909119910 Shear stress






Acknowledgments



0

1

2

3

4

5

6

7

8


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12

14

16


120590x

(MPa

)

(b) 120590119909

0

5

10

15

20

25



120590y

(MPa

)

(c) 120590119910


0

02

04

06

08

1

12


120591xy

(MPa

)

(d) 120591119909119910



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Number 1205903(Mpa) 120590

1(Mpa) 120572

119896Number 120590

3(Mpa) 120590

1(Mpa) 120572

119896

1 00000 00000 minus1384852 41 38699 178247 01287

2 00853 27979 976169 42 39471 180397 00845

3 02398 40913 55845 43 40244 182534 00837

4 03170 46287 22133 44 41016 184657 00829

5 03943 51224 16698 45 41788 186769 00822

6 04715 55829 13160 46 42561 188868 01220

7 05487 60173 15167 47 44105 193033 01210

8 07032 68251 12040 48 44878 195098 00800

9 07804 72046 06538 49 45650 197153 01193

10 08577 75708 05713 50 47195 201231 01186

11 09349 79254 07297 51 47967 203255 00787

12 10894 86045 06295 52 48740 205269 00785

13 11666 89311 03662 53 49512 207273 01175

14 12439 92501 04865 54 51057 211255 01173

15 13983 98678 04344 55 51829 213232 00781

16 14756 101677 02606 56 52601 215200 00781

17 15528 104620 03552 57 53374 217160 01171

18 17073 110360 03247 58 54919 221055 01172

19 17845 113162 01989 59 55691 222990 00783

20 18617 115922 01876 60 56463 224918 00784

21 19390 118644 02615 61 57236 226838 01178

22 20935 123979 02442 62 58780 230655 01182

23 21707 126597 01526 63 59553 232553 00790

24 22479 129183 01459 64 60325 234443 01190

25 23252 131740 01398 65 61870 238203 01195

26 24024 134268 01342 66 62642 240073 00800

27 24796 136769 01914 67 63415 241937 01206

28 26341 141696 01826 68 64959 245644 01212

29 27113 144124 01165 69 65732 247488 00813

30 27886 146529 01678 70 66504 249327 00816

31 29431 151275 01616 71 67276 251159 00820

32 30203 153617 01040 72 68049 252985 01237

33 30975 155940 01015 73 69593 256620 01244

34 31748 158245 00992 74 70366 258429 00835

35 32520 160532 01447 75 71138 260232 00839

36 34065 165054 01410 76 71911 262030 01267

37 34837 167290 00918 77 73455 265610 01276

38 35609 169511 00903 78 74228 267392 00856

39 36382 171717 00889 79 75000 269169 154377

40 37154 173908 01309


0

5

10

15

20

25

30

0 5 10 15 20 25 30

Pred

icte

d va

lue o

f1205901

(MPa

)



0

5

10

15

20

25

30

0 5 10 15 20 25 30

Pred

icte

d va

lue o

f1205901

(MPa

)












0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8



1205901

(MPa

)

1205903 (MPa)


120590xxR = 5 cm

120590yy

O



120574 = 2500kNm3










0

05

1

15

2

25

3


Mohr-CoulombSVM

Disp

lace

men

t (10

minus4

m)


0

02

04

06

08

1

12


120590x

(MPa

)

(a)

0

02

04

06

08

1

12

14

16

18


120590y

(MPa

)

(b)

0

002

004

006

008

01

012

014

016

018


Mohr-CoulombSVM

120591xy

(MPa

)

(c)


119910 (c) 120590

119909119910






0

1

2

3

4

5

6

7

8



Disp

lace

men

t (10

minus3

m)




0

2

4

6

8

10

12

14

16


120590x

(MPa

)


(a)

0

5

10

15

20

25


120590y

(MPa

)


(b)

0

02

04

06

08

1

12


120591xy

(MPa

)


(c)


119910 (c) 120591




119910119910= 5 10 and 15MPa and


= 12120590119910119910



119910

and 120591119909119910



and 120591119909119910



0

02

04

06

08

1

12

14

16

18


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

1

2

3

4

5

6


120590x

(MPa

)

(b) 120590119909

0

1

2

3

4

5

6

7

8

9



120590y

(MPa

)

(c) 120590119910


0

01

02

03

04

05

06

07


120591xy

(MPa

)

(d) 120591119909119910


5 Conclusions


List of Symbols






0

05

1

15

2

25

3

35

4

45


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12


120590x

(MPa

)

(b) 120590119909

0

2

4

6

8

10

12

14

16



120590y

(MPa

)

(c) 120590119910

0

01

02

03

04

05

06

07

08



120591xy

(MPa

)

(d) 120591119909119910









120591119909119910 Shear stress






Acknowledgments



0

1

2

3

4

5

6

7

8


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12

14

16


120590x

(MPa

)

(b) 120590119909

0

5

10

15

20

25



120590y

(MPa

)

(c) 120590119910


0

02

04

06

08

1

12


120591xy

(MPa

)

(d) 120591119909119910



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5

10

15

20

25

30

0 5 10 15 20 25 30

Pred

icte

d va

lue o

f1205901

(MPa

)



0

5

10

15

20

25

30

0 5 10 15 20 25 30

Pred

icte

d va

lue o

f1205901

(MPa

)












0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8



1205901

(MPa

)

1205903 (MPa)


120590xxR = 5 cm

120590yy

O



120574 = 2500kNm3










0

05

1

15

2

25

3


Mohr-CoulombSVM

Disp

lace

men

t (10

minus4

m)


0

02

04

06

08

1

12


120590x

(MPa

)

(a)

0

02

04

06

08

1

12

14

16

18


120590y

(MPa

)

(b)

0

002

004

006

008

01

012

014

016

018


Mohr-CoulombSVM

120591xy

(MPa

)

(c)


119910 (c) 120590

119909119910






0

1

2

3

4

5

6

7

8



Disp

lace

men

t (10

minus3

m)




0

2

4

6

8

10

12

14

16


120590x

(MPa

)


(a)

0

5

10

15

20

25


120590y

(MPa

)


(b)

0

02

04

06

08

1

12


120591xy

(MPa

)


(c)


119910 (c) 120591




119910119910= 5 10 and 15MPa and


= 12120590119910119910



119910

and 120591119909119910



and 120591119909119910



0

02

04

06

08

1

12

14

16

18


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

1

2

3

4

5

6


120590x

(MPa

)

(b) 120590119909

0

1

2

3

4

5

6

7

8

9



120590y

(MPa

)

(c) 120590119910


0

01

02

03

04

05

06

07


120591xy

(MPa

)

(d) 120591119909119910


5 Conclusions


List of Symbols






0

05

1

15

2

25

3

35

4

45


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12


120590x

(MPa

)

(b) 120590119909

0

2

4

6

8

10

12

14

16



120590y

(MPa

)

(c) 120590119910

0

01

02

03

04

05

06

07

08



120591xy

(MPa

)

(d) 120591119909119910









120591119909119910 Shear stress






Acknowledgments



0

1

2

3

4

5

6

7

8


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12

14

16


120590x

(MPa

)

(b) 120590119909

0

5

10

15

20

25



120590y

(MPa

)

(c) 120590119910


0

02

04

06

08

1

12


120591xy

(MPa

)

(d) 120591119909119910



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5

10

15

20

25

30

0 1 2 3 4 5 6 7 8



1205901

(MPa

)

1205903 (MPa)


120590xxR = 5 cm

120590yy

O



120574 = 2500kNm3










0

05

1

15

2

25

3


Mohr-CoulombSVM

Disp

lace

men

t (10

minus4

m)


0

02

04

06

08

1

12


120590x

(MPa

)

(a)

0

02

04

06

08

1

12

14

16

18


120590y

(MPa

)

(b)

0

002

004

006

008

01

012

014

016

018


Mohr-CoulombSVM

120591xy

(MPa

)

(c)


119910 (c) 120590

119909119910






0

1

2

3

4

5

6

7

8



Disp

lace

men

t (10

minus3

m)




0

2

4

6

8

10

12

14

16


120590x

(MPa

)


(a)

0

5

10

15

20

25


120590y

(MPa

)


(b)

0

02

04

06

08

1

12


120591xy

(MPa

)


(c)


119910 (c) 120591




119910119910= 5 10 and 15MPa and


= 12120590119910119910



119910

and 120591119909119910



and 120591119909119910



0

02

04

06

08

1

12

14

16

18


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

1

2

3

4

5

6


120590x

(MPa

)

(b) 120590119909

0

1

2

3

4

5

6

7

8

9



120590y

(MPa

)

(c) 120590119910


0

01

02

03

04

05

06

07


120591xy

(MPa

)

(d) 120591119909119910


5 Conclusions


List of Symbols






0

05

1

15

2

25

3

35

4

45


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12


120590x

(MPa

)

(b) 120590119909

0

2

4

6

8

10

12

14

16



120590y

(MPa

)

(c) 120590119910

0

01

02

03

04

05

06

07

08



120591xy

(MPa

)

(d) 120591119909119910









120591119909119910 Shear stress






Acknowledgments



0

1

2

3

4

5

6

7

8


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12

14

16


120590x

(MPa

)

(b) 120590119909

0

5

10

15

20

25



120590y

(MPa

)

(c) 120590119910


0

02

04

06

08

1

12


120591xy

(MPa

)

(d) 120591119909119910



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

05

1

15

2

25

3


Mohr-CoulombSVM

Disp

lace

men

t (10

minus4

m)


0

02

04

06

08

1

12


120590x

(MPa

)

(a)

0

02

04

06

08

1

12

14

16

18


120590y

(MPa

)

(b)

0

002

004

006

008

01

012

014

016

018


Mohr-CoulombSVM

120591xy

(MPa

)

(c)


119910 (c) 120590

119909119910






0

1

2

3

4

5

6

7

8



Disp

lace

men

t (10

minus3

m)




0

2

4

6

8

10

12

14

16


120590x

(MPa

)


(a)

0

5

10

15

20

25


120590y

(MPa

)


(b)

0

02

04

06

08

1

12


120591xy

(MPa

)


(c)


119910 (c) 120591




119910119910= 5 10 and 15MPa and


= 12120590119910119910



119910

and 120591119909119910



and 120591119909119910



0

02

04

06

08

1

12

14

16

18


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

1

2

3

4

5

6


120590x

(MPa

)

(b) 120590119909

0

1

2

3

4

5

6

7

8

9



120590y

(MPa

)

(c) 120590119910


0

01

02

03

04

05

06

07


120591xy

(MPa

)

(d) 120591119909119910


5 Conclusions


List of Symbols






0

05

1

15

2

25

3

35

4

45


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12


120590x

(MPa

)

(b) 120590119909

0

2

4

6

8

10

12

14

16



120590y

(MPa

)

(c) 120590119910

0

01

02

03

04

05

06

07

08



120591xy

(MPa

)

(d) 120591119909119910









120591119909119910 Shear stress






Acknowledgments



0

1

2

3

4

5

6

7

8


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12

14

16


120590x

(MPa

)

(b) 120590119909

0

5

10

15

20

25



120590y

(MPa

)

(c) 120590119910


0

02

04

06

08

1

12


120591xy

(MPa

)

(d) 120591119909119910



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














0

1

2

3

4

5

6

7

8



Disp

lace

men

t (10

minus3

m)




0

2

4

6

8

10

12

14

16


120590x

(MPa

)


(a)

0

5

10

15

20

25


120590y

(MPa

)


(b)

0

02

04

06

08

1

12


120591xy

(MPa

)


(c)


119910 (c) 120591




119910119910= 5 10 and 15MPa and


= 12120590119910119910



119910

and 120591119909119910



and 120591119909119910



0

02

04

06

08

1

12

14

16

18


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

1

2

3

4

5

6


120590x

(MPa

)

(b) 120590119909

0

1

2

3

4

5

6

7

8

9



120590y

(MPa

)

(c) 120590119910


0

01

02

03

04

05

06

07


120591xy

(MPa

)

(d) 120591119909119910


5 Conclusions


List of Symbols






0

05

1

15

2

25

3

35

4

45


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12


120590x

(MPa

)

(b) 120590119909

0

2

4

6

8

10

12

14

16



120590y

(MPa

)

(c) 120590119910

0

01

02

03

04

05

06

07

08



120591xy

(MPa

)

(d) 120591119909119910









120591119909119910 Shear stress






Acknowledgments



0

1

2

3

4

5

6

7

8


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12

14

16


120590x

(MPa

)

(b) 120590119909

0

5

10

15

20

25



120590y

(MPa

)

(c) 120590119910


0

02

04

06

08

1

12


120591xy

(MPa

)

(d) 120591119909119910



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

2

4

6

8

10

12

14

16


120590x

(MPa

)


(a)

0

5

10

15

20

25


120590y

(MPa

)


(b)

0

02

04

06

08

1

12


120591xy

(MPa

)


(c)


119910 (c) 120591




119910119910= 5 10 and 15MPa and


= 12120590119910119910



119910

and 120591119909119910



and 120591119909119910



0

02

04

06

08

1

12

14

16

18


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

1

2

3

4

5

6


120590x

(MPa

)

(b) 120590119909

0

1

2

3

4

5

6

7

8

9



120590y

(MPa

)

(c) 120590119910


0

01

02

03

04

05

06

07


120591xy

(MPa

)

(d) 120591119909119910


5 Conclusions


List of Symbols






0

05

1

15

2

25

3

35

4

45


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12


120590x

(MPa

)

(b) 120590119909

0

2

4

6

8

10

12

14

16



120590y

(MPa

)

(c) 120590119910

0

01

02

03

04

05

06

07

08



120591xy

(MPa

)

(d) 120591119909119910









120591119909119910 Shear stress






Acknowledgments



0

1

2

3

4

5

6

7

8


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12

14

16


120590x

(MPa

)

(b) 120590119909

0

5

10

15

20

25



120590y

(MPa

)

(c) 120590119910


0

02

04

06

08

1

12


120591xy

(MPa

)

(d) 120591119909119910



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

02

04

06

08

1

12

14

16

18


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

1

2

3

4

5

6


120590x

(MPa

)

(b) 120590119909

0

1

2

3

4

5

6

7

8

9



120590y

(MPa

)

(c) 120590119910


0

01

02

03

04

05

06

07


120591xy

(MPa

)

(d) 120591119909119910


5 Conclusions


List of Symbols






0

05

1

15

2

25

3

35

4

45


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12


120590x

(MPa

)

(b) 120590119909

0

2

4

6

8

10

12

14

16



120590y

(MPa

)

(c) 120590119910

0

01

02

03

04

05

06

07

08



120591xy

(MPa

)

(d) 120591119909119910









120591119909119910 Shear stress






Acknowledgments



0

1

2

3

4

5

6

7

8


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12

14

16


120590x

(MPa

)

(b) 120590119909

0

5

10

15

20

25



120590y

(MPa

)

(c) 120590119910


0

02

04

06

08

1

12


120591xy

(MPa

)

(d) 120591119909119910



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

05

1

15

2

25

3

35

4

45


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12


120590x

(MPa

)

(b) 120590119909

0

2

4

6

8

10

12

14

16



120590y

(MPa

)

(c) 120590119910

0

01

02

03

04

05

06

07

08



120591xy

(MPa

)

(d) 120591119909119910









120591119909119910 Shear stress






Acknowledgments



0

1

2

3

4

5

6

7

8


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12

14

16


120590x

(MPa

)

(b) 120590119909

0

5

10

15

20

25



120590y

(MPa

)

(c) 120590119910


0

02

04

06

08

1

12


120591xy

(MPa

)

(d) 120591119909119910



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

1

2

3

4

5

6

7

8


Disp

lace

men

t (10

minus3

m)

(a) Displacement

0

2

4

6

8

10

12

14

16


120590x

(MPa

)

(b) 120590119909

0

5

10

15

20

25



120590y

(MPa

)

(c) 120590119910


0

02

04

06

08

1

12


120591xy

(MPa

)

(d) 120591119909119910



References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article lssvm-based rock failure criterion and

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