Research ArticleSolution of Strain-Softening Surrounding Rock in Deep TunnelIncorporating 3D Hoek-Brown Failure Criterion and Flow Rule

Jin-feng Zou Song-qing Zuo and Yuan Xu

School of Civil Engineering Central South University No 22 Shaoshan South Road Central South University Railway CampusChangsha Hunan 410075 China

Correspondence should be addressed to Jin-feng Zou zoujinfeng csu163com

Received 19 March 2016 Accepted 13 June 2016

Academic Editor John D Clayton

Copyright copy 2016 Jin-feng Zou et al This 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

In order to investigate the influence of the intermediate principal stress on the stress and displacement of surrounding rock anovel approach based on 3D Hoek-Brown (H-B) failure criterion was proposed Taking the strain-softening characteristic of rockmass into account the potential plastic zone is subdivided into a finite number of concentric annulus and a numerical procedurefor calculating the stress and displacement of each annulus was presented Strains were obtained based on the nonassociated andassociated flow rule and 3D plastic potential function Stresses were achieved by the stress equilibrium equation and generalizedHoek-Brown failure criterion Using the proposed approach we can get the solutions of the stress and displacement of thesurrounding rock considering the intermediate principal stress Moreover the proposed approach was validated with the publishedresults Compared with the results based on generalized Hoek-Brown failure criterion it is shown that the plastic radius calculatedby 3DHoek-Brown failure criterion is smaller than those solved by generalizedH-B failure criterion and the influences of dilatancyeffect on the results based on the generalized H-B failure criterion are greater than those based on 3D H-B failure criterion Thedisplacements considering the nonassociated flow rule are smaller than those considering associated flow rules

1 Introduction

A reasonable assessment of plastic failure scope and itsdeformation is the key to evaluate the safety and stability oftunnel Analytical and semianalytical solution based on linearand nonlinear failure criteria have been studied by manyresearchers such as Yu et al [1 2] Carranza-Torres [3 4] andPark and Kim [5] Yu et al [1 2] presented a nonlinear unifiedstrength criterion for rock material which took the effect ofintermediate principal stress into account Carranza-Torresand Fairhurst [6] were one of the earliest scholars applyingthe Hoek-Brown failure criterion to the engineering practiceOn the basis of this elastic-brittle-plastic solutions basedon the Mohr-Coulomb and Hoek-Brown (119886 = 05) failurecriteria were proposed by Carranza-Torres [3 4] Accordingto the research of Carranza-Torres [3] Sharan [7 8] presenteda new solution and calculation method of critical plasticzone using Newton-Raphson method Due to its feasibilityof using computer to obtain numerical solution and simpleexpression it has been accepted by most scholars However

the above results are limited to ignore the influence of theintermediate principal stress on the distribution of stress anddisplacement of surrounding rock since they are based ongeneralized Hoek-Brown failure criterion Hence there aresome deviations with exact solutions because the deep buriedtunnel is in the three-dimensional stress state

Although lots of solutions of surrounding rock wereproposed based on the assumption of plane strain problemthe influence of axial stress should not be ignored Theinfluence of axial stress on the distribution of stress anddisplacement of surrounding rock was proposed by Reed[9] he studied the relationship between the axial stress andthe major intermediate and minor principal stresses Onthe basis of Reed [9] Pan and Brown [10] considered theeffects of the axial stress and dilation on the convergenceand stability of the surrounding rock In particular it ispointed out that the case of stress exchange can occur in thecalculation Wang et al [11] improved Reedrsquos approach andpresented an analytical solution of surrounding rock underdifferent axial stresses which is based on Mohr-Coulomb

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 7947036 12 pageshttpdxdoiorg10115520167947036

2 Mathematical Problems in Engineering

failure criterion Particularly the stress and displacement ofsurrounding rock under three different axial stress states werediscussed in his paper The effects of axial stress and in situstress on stress displacement and strain of surrounding rockwere studied by Lu et al [12] and Zhou et al [13] Moreoverthese approaches can be supplemented and verifiedmutuallyZou and Su [14] presented an analytical solution of thesurrounding rock based on the generalized Hoek-Brownfailure criterion and elastic-brittle-plastic model and thissolution is compared with Wang et al [11] to verify itscorrectness using the method of parameter transformationThe theoretical solutions for the elastic-brittle-plastic andelastic-plastic rockmass incorporating the out-of-plane stressand seepage force were proposed by Zou et al [14ndash18]

Comparedwith the elastic-plasticmodel strain-softeningmodel is closer to failure of rock mass model in engineer-ing practices Taking into account that rock yield strengthdoes not drop to residual strength instantaneously strain-softening process can be regard as a gradual decline in theprocess of yielding of rock mass So the strain-softeningmodel can better fit the test curve of rock For instance Leeand Pietruszczak [19] proposed a numerical procedure forcalculating the stresses and radial displacements around a cir-cular tunnel excavated in a strain-softening Mohr-Coulombor generalized Hoek-Brown media In this approach thepotential plastic zone is divided into a finite number ofconcentric rings and it is assumed that all the strengthparameters are linear functions of deviatoric plastic strainWang et al [20] proposed a new closed strain-softeningmethod considering softening process as a series of brittle-plastic and plastic flow process and presented a new methodto describe the strain-softening process of rock soil massAlonso et al [21] standardized the process of modeling andthe problem was transformed into the initial value problemof the Runge-Kutta method Zou and Li [22] proposed animproved numerical approach to analyze the stability of thestrain-softening surrounding rock with the considerationof the hydraulic-mechanical coupling and the variation ofelastic strain in the plastic region Moreover Zou and He [15]proposed a numerical approach that considers the effect ofout-of-plane stress for circular tunnels excavated in strain-softening rock

At present the generalized H-B failure criterion is widelyused [23ndash25] but it is difficult to obtain a relatively accuratesolutionThemajority of scholars have donemany researchesabout three-dimensional failure mechanism including PanandHudson [26] Singh et al [27] Priest [28] Zhang andZhu[29] and Yang and Long [23 30] They proposed different3D failure mechanisms based on different experimental ortheoretical models respectively Among these researchesthe model proposed by Zhang and Zhu [29] which can becompared with the two-dimensional Hoek-Brown model iswidely recognized

Although the 3D Hoek-Brown failure criterion has beenwidely recognized the theoretical analysis for deep tunnelis still little discussed The paper focus on the influences ofthe axial stress on the stress and strain of strain-softeningsurrounding rock in deep tunnel considering 3D Hoek-Brown failure criterion Strains are obtained by the 3D plastic

potential function and stresses are given by plane strainmethod Moreover the results are compared with thosebased on generalized Hoek-Brown failure criterion to finddifferences between the two methods

2 Failure Criterion

Hoek et al [31] modified the previous Hoek-Brown failurecriterion and proposed the generalized H-B failure criterion

1205901minus 1205903= 120590119888(119898119894

1205903

120590119888

+ 1)

119886

(1)

where 120590119888is the unconfined compressive strength of the rock

mass 1205901and 120590

3are the major and minor principal stresses

respectively 119898 119904 and 119886 are the H-B constants for the rockmass before yielding which are expressed as follows

119898 = 119898119894exp [(GSI minus 100)

(28 minus 14119863)

]

119904 = exp [(GSI minus 100)(9 minus 3119863)

]

119886 = 05 +

1

6

[exp (minusGSI15

) minus exp(minus203

)]

(2)

where119863 is a factor that depends on the degree of disturbanceto which the rock has been subjected in terms of blast damageand stress relaxation which varies between 0 and 1 and GSIis the geological strength index of the rockmass which variesbetween 10 and 100

The generalized Hoek-Brown failure criterion has beenwidely used but the influence of the intermediate principalstress on distribution of stress and strength is neglectedHence a 3D Hoek-Brown failure criterion is proposed byZhang and Zhu [29] on the basis of the generalized Hoek-Brown and Mogi failure criteria

1

120590(1119886minus1)

119888

(

3

radic2

120591oct)1119886

+

119898119894

2

(

3

radic2

120591oct) minus 1198981198941205901198982 = 119904120590119888 (3)

where 120591oct is octahedron deviatoric stress and 12059010158401198982

is averageeffective stress

120591oct =1

3

radic(1205901minus 1205902)2

+ (1205902minus 1205903)2

+ (1205901minus 1205903)2

1205901015840

1198982=

1205901+ 1205903

2

(4)

The 3D generalized H-B failure criterion not only con-siders the influence of the intermediate principal stress butalso inherits the merit of the H-B failure criterion Underthe triaxial compression and triaxial tension conditions theparameters of the H-B failure criterion can be directly usedfor 3D generalized H-B failure criterion

3 Computational Model

As shown in Figure 1 a circular opening with an initial radius(1199030) is subjected to a three-dimensional and uniform in situ

Mathematical Problems in Engineering 3

y

x

z

1205900

1205900

1205900

Residual zone

Softening zone

Elastic zone

Elasto-plasticinterface

pin

rp

rs

r0

Figure 1 Stress state of the surrounding rock

pressure (1205900) at infinity and an internal support pressure (119901in)

in the tunnel wall The surrounding rock mass is consideredas continuous homogeneous isotropic and initially elasticBecause the axial stress (120590

119911) along the axis of the deep tunnel

is also considered the analyzed model of surrounding rockcan be regarded as spatial axial symmetry model

During excavation of the deep tunnel 119901in is graduallyreducedWhen the internal support pressure (119901in) is less thana critical support pressure plastic region will appear aroundthe surrounding rock Because the yielding extent of thesurrounding rock is different the plastic zone of surroundingrock is divided into softening zone and residual zone Thepaper considers the surrounding rock to be homogeneousand isotropic material Therefore the radial normal strain120576119903 the circumferential normal strain 120576

120579 and the radial

displacement 119906119903can be obtained by small strain assumption

as follows

120576119903=

120597119906119903

120597119903

120576120579=

119906119903

119903

(5)

Outside the plastic zone the stress and displacement ofsurrounding rock in elastic zone can be obtained by thesolution presented by Reed [9]

120590119903= 1205900minus (1205900minus 120590119877) (

119903119901

119903

)

2

(6a)

120590119903= 1205900+ (1205900minus 120590119877) (

119903119901

119903

)

2

(6b)

When considering the influence of intermediate principalstress the axial stress is equal to the in situ stress in deeptunnel as follows

120590119911= 1205900 (7a)

119906119903=

1

119864

(1 + 120583) (1205900minus 120590119903119901)

1199032

119901

119903

(7b)

4 Strain-Softening Model

For strain-softening model the stress-strain curves can besimplified to 3-slope as shown in Figure 2 [15 22]

The strength and deformation parameters of the strain-softening rock mass are evaluated based on plastic deforma-tion and are controlled by the deviatoric strain [19 21]

120574119901= 120576119901

1minus 120576119901

3 (8)

where 1205761199011and 120576119901

3are the major and minor plastic strains

respectivelyThe physical parameters of the surrounding rockmass are

described according to the bilinear function of plastic shearstrain as follows [19 21]

120596 (120574119901) =

120596119901minus (120596119901minus 120596119903)

120574119901

120574119901

119903

0 lt 120574119901lt 120574119901

119903

120596119903 120574

119901ge 120574119901

119903

(9)

where 120596 represents a strength parameter such as 120593 119888 119898 119904119886 120601 and 119864 120574119901

119903is the critical deviatoric plastic strain from

which the residual behavior is first observed and should beidentified through experimentation the subscripts 119901 and 119903represent the peak and residual values respectively

To avoid the influence that the curve of stress and strain instrain-softening stage is nonlinear on theoretical analysis thestrain-softening process can be simplified to a series of brittle-plastic processes [20] as shown in Figure 3 The whole plasticzone of the deep tunnel is separated into 119899 rings and theadjacent rings have different parameters Hence the actualconstitutive model is replaced by piecewise linear 119894 (119894 + 1)and each segment has different negative slope it seems likethere are 119899 brittle-plastic processesThe greater 119899 is the moreaccurate the strain-softening model is

5 Solutions in Plastic Zone

The total plastic region can be divided into 119899 connectedannuli which are bounded by annuli of the radii 119903

(119894minus1)and 119903(119894)

The 119894th annulus is determined by the outer radius 119903(119894minus1)

andthe inner radius 119903

(119894)which is adjacent to the elastic region

The parameter values of the outmost ring are equal to thevalues of innermost elastic region So the (119899 minus 1) iterationsof brittle-plastic analysis are conducted on the inner plasticannulus until the residual strength is reached

The stress equilibrium equation of an element in tunnelwall can be represented by

120597120590119903

120597119903

+

120590119903minus 120590120579

119903

= 0 (10)

where 120590119903is the radial stress and 120590

120579is the tangential stress

4 Mathematical Problems in Engineering

A

O1205761

1205901 minus 1205903

B C

120590cr

120590c

(a) (1205901 minus 1205903)-1205761 curves

+

minus

O1205761

120576

(b) 1205761-120576V curves

+

minus

O1205761

1205763

(c) 1205761-1205763 curves

A

120590cr

O1205761

1205901 minus 1205903

120590c

120573120576e1120576e1

B C

(d) (1205901 minus 1205903)-1205761 simplified curves

H

+

minus

O

120576p1

minus120576v1

1

1205761

120576v

(e) 1205761-120576V simplified curves

+

minus

O

120576e1

1

1205761

1205763

120573120576e1

120576p1

minus120576p3

h

(f) 1205761-1205763 simplified curves

Figure 2 Strain-softening material behavior model

Calculate constitutive model

Actual constitutive model

O

0

i minus 1

i + 1i

middot middot middot

⋱

120576

120590

Figure 3 Strain-softening model of rock mass

Based on the plane strain assumption the axial strainshould satisfy

120576119911= 120576119890

119911+ 120576119901

119911= 0

120576119890

119911=

1

119864

[120590119911minus V (120590

119903+ 120590120579) minus (1 minus 2V) 120590

0]

(11)

Hence the relationship between radial normal stress 120590119903

circumferential normal stress 120590120579 and axial normal stress 120590

119911

is expressed as

120590119911= V (120590

120579+ 120590119903) minus (2V minus 1) 120590

0minus 119864120576119901

119911 (12)

The radial normal stress and the circumferential normalstress at the elastoplastic interface satisfy the failure criterionand can be given by

119865 (1205901 1205903 120574119901) = 1205901minus 1205903minus 119867 (120590

1 1205903 120574119901) (13)

The radius of the first ring is 119903(0)= 119903119901which is at the

interface between the elastic region and the equivalent plasticzone

120588(0)=

119903(0)

119903119901

119903(0)= 119903119901

(14)

The stress and strain at the elastoplastic interface can begiven by

120590119903(0)

120590120579(0)

120590119911(0)

=

120590119877

21205900minus 120590119877

1205900

Mathematical Problems in Engineering 5

120576119903(0)

120576120579(0)

120576119911(0)

=

1 + V119864

119889119906

119889119903

119906

119903

0

(15)

where

120576119903(0)= minus

1

2119866

1205900minus 120590119877

120576120579(0)=

1

2119866

1205900minus 120590119877

(16)

The normalized inner radius can be expressed as

120588(119894)=

119903(119894)

119903119901

(17)

120590119903on both inner and outer boundaries of the plastic zone

are known a priori and 120590119903decreases from 120590

119877to 119901in The

increment of radial normal stress is given by

Δ120590119903=

(119901in minus 120590119877)

119899

(18)

So the radial normal stress at each ring can be representedby

120590119903(119894)= 120590119903(119894minus1)

+ Δ120590119903 (19)

The stress equilibrium equation can be expressed inanother way as follows

119889120590119903

119889120588

+

120590119903minus 120590120579

120588

= 0 (20)

From above equations the following expressions can beobtained

120588(119894)

Δ120590119903(119894)

Δ120588(119894)

= 120590120579(119894)minus 120590119903(119894) (21)

The stress equilibrium differential equation for the 119894thannulus is derived by using (13) and (20) expressed as (21)and (22)

120590119903(119894)minus 120590119903(119894minus1)

120588(119894)minus 120588(119894minus1)

minus

119867 (120590119903(119894) 120574119901)

120588(119894)

= 0 (22)

120588(119894)=

119867 (120590119903(119894) 120574119901) + Δ120590

119903

119867(120590119903(119894) 120574119901)

120588(119894minus1) (23)

The radius of each ring is a known quantity and the radialstress of the 119894th ring can be obtained from (17) So the axialnormal stress 120590

119911can be given by

120590119911(119894)= V (120590

120579(119894)+ 120590119903(119894)) minus (2V minus 1) 120590

0minus 119864120576119901

119911(119894) (24)

Combining (3) (17) and (22) stress at the outer ring canbe obtained and the radial stress at the 119894th ring is calculatedby using linear interpolation

119876119909=

119876119906(119894)minus 119876119906(119894minus1)

120588(119894)minus 120588(119894minus1)

(120588119909minus 120588(119894minus1)) + 119876119906(119894) (25)

where119876119909is stress or displacement of rockmass119876

119906(119894)is stress

or displacement of rock mass at 119894th ring 119876119906(119894minus1)

is stress ordisplacement of rock mass at (119894 minus 1)th ring and 120588

119909is the

distance between center of the tunnel wall and any point ofthe surrounding rock

The compatibility equation can be written in the generalform as follows

119889120576120579

119889119903

+

120576120579minus 120576119903

119903

= 0 (26)

Equation (26) can be transformed into the followingform

119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

120576119890

120579minus 120576119890

119903

120588

(27)

It can be rewritten as119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

1 + ]119864

119867(120590119903 120574119901)

120588

(28)

51 Associated Flow Rule According to the geotechnicalplastic mechanics there is an equipotential surface of plasticpotential at any point 119872 in stress space Its mathematicalexpression is called the plastic potential function and theplastic potential function can be presented by

119892 (120590119894119895 119867119886) = 0 (29)

where119867119886is hardening parameter and 120590

119894119895is principal stress

The plastic strain increment 119889120576119901119894119895can be obtained by

119889120576119901

119894119895=

120597119865

120597120590119894119895

119889120582 (30)

where 119889120582 is multiplication operator 120576119901119894119895is plastic strain and

120590119894119895is principal stressIt is called the associated flow rule when 119865 and 119876 are

completely the same And the physical meaning of yieldfunction 119865 is a judgment criterion whether a point ofsurrounding rock reaches the yield state And the physicalmeaning of plastic potential function 119876 is the relationshipbetween plastic strain increment and loading surface

While the strains of rock and soil mass satisfy theassociated flow rule its plastic potential function is given by

119876 (120590) = minus

119899

3

1198681+

3

120590119888

1198692+

radic3

2

119899radic1198692 (31)

where 119899 is dilation parameter

1198681= 1205901+ 1205902+ 1205903

1198692=

1

6 [(1205901minus 1205902)2

+ (1205902minus 1205903)2

+ (1205903minus 1205901)2

]

(32)

6 Mathematical Problems in Engineering

The differential of three-direction strain is expressed as

119889120576119901

120579=

120597119891

120597120590120579

119889120582 = [(

radic3 (2120590120579minus 120590119903minus 120590119911)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590120579minus 120590119903minus 120590119911)] 119889120582

119889120576119901

119911=

120597119891

120597120590119911

119889120582 = [(

radic3 (2120590119911minus 120590120579minus 120590119903)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590119911minus 120590120579minus 120590119903)] 119889120582

119889120576119901

119903=

120597119891

120597120590119903

119889120582 = [(

radic3 (2120590119903minus 120590120579minus 120590119911)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590119903minus 120590120579minus 120590119911)] 119889120582

(33)

If 1198711 1198712 and 119871

3are defined as follows

[(

radic3 (2120590120579minus 120590119903minus 120590119911)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590120579minus 120590119903minus 120590119911)]

= 1198711

[(

radic3 (2120590119911minus 120590120579minus 120590119903)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590119911minus 120590120579minus 120590119903)]

= 1198712

[(

radic3 (2120590119903minus 120590120579minus 120590119911)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590119903minus 120590120579minus 120590119911)]

= 1198713

(34)

then (33) can be simplified to

119889120576119901

1=

120597119891

1205971205901

119889120582 = 1198711119889120582

119889120576119901

2=

120597119891

1205971205902

119889120582 = 1198712119889120582

119889120576119901

3=

120597119891

1205971205903

119889120582 = 1198713119889120582

(35)

The relationship between radial plastic normal strain 120576119901119903

circumferential plastic normal strain 120576119901120579 and axial plastic

normal strain 120576119901119911can be represented by

120576119901

120579

1198711

=

120576119901

119911

1198712

=

120576119901

119903

1198713

(36)

119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

1 + ]119864

119867 (120590119903 120574119901)

120588

(37)

Combination of (36) and (37) leads to

Δ120576119901

120579= (minus

119889120576119890

120579(119894)

119889120588(119894)

minus

1 + ]119864

119867 (120590119903(119894) 120574119901)

120588(119894)

minus

120576119901

120579(119894minus1)(1 minus 119871

3(119894)1198711(119894))

120588(119894)

)(

1

Δ120588(119894)

+

(1 minus 1198713(119894)1198711(119894))

120588(119894)

)

(38)

The increment of the radial plastic strain at 119894th annuluscan be obtained by (36)

Δ120576119901

119903(119894)=

1198713(119894)

1198711(119894)

Δ120576119901

120579=

1198713(119894)

1198711(119894)

(minus

119889120576119890

120579(119894)

119889120588(119894)

minus

1 + ]119864

119867(120590119903(119894) 120574119901)

120588(119894)

minus

120576119901

120579(119894minus1)(1 minus 119871

3(119894)1198711(119894))

120588(119894)

)

sdot (

1

Δ120588(119894)

+

(1 minus 1198713(119894)1198711(119894))

120588(119894)

)

(39)

The displacement at 119894th annulus can be obtained

119906119903(119894)= 119903(119894)120576120579(119894)= 119903(119894)(120576119890

120579(119894)+ 120576119901

120579(119894))

= 119903(119894)(120576119890

120579(119894)+ 120576119901

120579(1)+

119894

sum

119894=1

Δ120576119901

119903(119894))

(40)

52 Nonassociated Flow Rule The plastic potential functionof the surrounding rock is redefined by Reed [9] by replacingthe internal friction angle with the angle of internal friction

120573120576119901

1+ 120576119901

3= 0 (41)

where 120573 = (1 + sin120595)(1 minus sin120595) and 120595 is dilation angleAccording to the nonassociated flow rule we can know

120576119901

2= 0 (42)

So (12) can be rewritten as

120590119911= V (120590

120579+ 120590119903) minus (2V minus 1) 120590

0 (43)

Therefore the relationship of strains is no longer deter-mined by the failure criterion considering the nonassociatedflow rule but is determined directly by the dilatation coeffi-cient 120573 which is defined by the dilation angle 120595 Because theintermediate principal strain 120576

119911= 0 the relationship between

the major and minor principal strains is determined by 1198711

and 1198713using the associated flow rule Hence in order to

compare the results calculated by the associated flow rule andnonassociated flow rule the results of displacement using thenonassociated flow rule can be replaced by using

minus

1198713

1198711

= 120573 =

1 + sin1205951 minus sin120595

(44)

Mathematical Problems in Engineering 7

6 Validation

To validate the correctness of the proposed approach andcalculation program the results of the proposed approachare compared with the results of Sharan [8] According todifferent types of the surrounding rock Sharan provided thestrength parameters of indoor test for many groups of rocksUnder different surrounding rock conditions Sharan [8]calculated the radius of plastic zone and radial displacementof the surrounding rock The specific parameters are shownin Table 1 and calculation results are shown in Table 2

If the intermediate stress 120590119911is equal to major principal

stress 120590120579or minor principal stress 120590

119903 3D Hoek-Brown failure

criterion can be simplified to two-dimensional Hoek-Brownfailure criterion which is used by Sharan

As shown in Table 2 the results of the paper are in goodaccordance with Sharan [8] when strain-softening of rockmass is not considered Under the condition that parametersof surrounding rock are the same the results show that thegreater the stress is the greater the plastic radius of thesurrounding rock is the larger the internal support pressureis the smaller the radius of the plastic zone of surroundingrock is And we found that the results calculated by elastic-brittle-plastic model are larger than those calculated byelastoplastic model As its strength parameters immediatelyfall after reaching the peak value the elastic-brittle-plasticmodel can be regarded as a special case of strain-softeningmodel

7 Numerical Calculation and Discussion

71 Computational Examples To analyze and compare theproposed approach which considers the intermediate prin-cipal stress and the 3D H-B failure criterion with thosebased on the generalized H-B failure criterion the followingparameters obtained from Sharanrsquos experimental results [8]are adopted 120590

119888= 30MPa 119904 = 00039 119886 = 055 119864 = 55GPa

119864119903= 55GPa 119898 = 17 119904

119903= 00019 119886

119903= 06 119898

119903= 16

V = 025 119901in = 5Mpa and 1199030= 5m The calculation results

are shown in Table 3Through the comparison results in Table 3 we can

find that the plastic zone radius and radial displacementcalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterionThus itproves that the results may overestimate the plastic zone radiiand radial displacement of the surrounding rock withoutconsidering intermediate principal stress

As shown in Table 3 the radial displacement of sur-rounding rock calculated by associated flow rule is muchgreater than those calculated by nonassociated flow ruleWith different softening parameters the plastic zone radiicalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterion Forexample elastoplastic model strain-softening model (120574119901 =0006) and elastic-brittle-plastic model would be reducedby 1457 1498 and 1600 respectively The results ofradial displacement using nonassociated flow rule are shownas follows elastoplastic model strain-softening model andelastic-brittle-plastic model (120595 = 0

∘ and 120595 = 20∘) would

120590120590

0

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

02

04

06

08

10

12

14

16

12 14 16 18 20 22 24 2610rr0

Figure 4 Stresses of surrounding rock mass (elastoplastic model)

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16120590p

0

Figure 5 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 004)

be reduced by 2085 3467 2178 3629 2190 and3653 respectively From above results it can be obtainedthat the calculated deviations between 3D H-B failure crite-rion and generalized H-B failure criterion increase with theincreasing of dilation angle

At the same time the radial displacement differencesbetween three-dimensional associated flow rule and nonas-sociated flow rule considering dilatancy effect are comparedThe differences between two calculation approaches areanalyzed for predicting the displacement of plastic zone Theresults of stress and displacement are shown in Figures 4ndash11As shown in Figures 4ndash7 120590

119903 120590120579 120590119911 and 119906

119903are calculation

results based on 3DH-B failure criterion120590101584011990312059010158401205791205901015840119911 and 1199061015840

119903are

calculation results based on generalizedH-B failure criterion

8 Mathematical Problems in Engineering

Table 1 Calculation parameters [8]

Quality of rock mass V 1199030(m) 120590

119888(MPa) 119904 119904

119903119886 119886

119903119898119887

119898119887119903

119864 (GPa) 119864119903(GPa)

Average 025 5 80 00039 0 051 053 201 034 9 5Very poor 03 5 25 00039 00019 055 06 17 085 57 57

Table 2 Comparisons between the results of this paper and Sharan without considering the strain-softening and intermediate principalstress

Parameter groups Elastoplastic model 1205900

119901in 1199031199011199030(Sharan) 119903

1199011199030(this paper)

1 e-p 40 0 177 177141 e-p 80 0 259 25881 e-p 80 1 231 23121 e-p 80 5 191 19071 e-b-p 40 3 253 25261 e-b-p 40 5 203 20301 e-b-p 80 18 233 23332 e-b-p 15 0 390 39012 e-b-p 15 3 151 15102 e-b-p 30 0 784 78362 e-b-p 30 5 233 2327

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

00

02

04

06

08

10

12

14

16

120590p

0

Figure 6 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 0006)

Figures 4ndash7 show the stress comparisons between theideal elastoplastic strain-softening and elastic-brittle-plasticmodel based on 3D and generalized Hoek-Brown failurecriteria respectively When considering the influence of theintermediate principal stress on different failure criterion thevalue of plastic radius calculated by the 3D H-B criterionis smaller than those calculated by the generalized H-Bcriterion

Under the condition that the geotechnical parameters arethe same the influence of dilatation coefficient on generalizedH-B failure criterion is larger than 3D H-B failure criterionParticularly as the softening coefficient is the same the

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16

120590p

0

Figure 7 Stresses of surrounding rock mass (elastic-brittle-plasticmodel)

residual radius is smaller than the plastic radius of thesurrounding rock if it is calculated by generalized H-B failurecriterion For example if 120574119901 = 004 the plastic radiuscalculated by generalized failure criterion is 14660 and theresidual radius does not exist Based on 3D H-B failurecriterion the plastic radius is 12841 and the residual radiusis 10372 Hence if the softening coefficient is the same theplastic radius calculated by different failure criteria is quitedifferent

The radial displacement comparisons of surroundingrock under different flow rules softening parameters anddilation angles are shown in Figures 8ndash11 Summary results

Mathematical Problems in Engineering 9

Table 3 Plastic radius critical stresses and displacements of surrounding rock

Softening parameters 1199031199011199030

1199031199041199030

Associated flowrule

1199061199031199030()

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 75

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D H-B

e-p 14591 1 06793 07245 07906 08534120574119901= 004 14660 1 06879 07373 08114 08839

120574119901= 0006 14945 12401 07239 07819 08676 09500e-b-p 15091 15091 07358 07947 08819 09658

3D H-B

e-p 12681 1 07877 05621 05815 06088 06337120574119901= 004 12841 10372 09599 05669 05884 06194 06484

120574119901= 0006 12980 12198 10685 05944 06219 06610 06970e-b-p 13010 13010 10873 06036 06314 06710 07074

Table 4 Calculation parameters

Rock mass 119904 119904119903

a 119886119903

119898119887

119898119887119903

1 00039 00039 055 06 17 172 00039 00019 055 055 17 173 00039 00039 055 055 17 164 00039 00039 055 055 17 085

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

03

04

05

06

07

08

09

urr

0(

)

105 110 115 120 125 130100rr0

Figure 8 Displacements of surrounding rock mass (elastoplasticmodel)

can be seen from Table 3 The plastic zone radii calculated byassociated flow rule and nonassociated flow rule are almostconsistent However the radial displacement of rock massincreases 1064 (e-p model) 1184 (120574119901 = 004) 1391 (120574119901 =0006) and 1413 (e-b-p model) if it adopted associatedflow rule Obviously with the continuous deteriorations ofstrength parameters the result differences between associatedflow rule and nonassociated flow rule will increase gradually

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

urr

0(

)

Figure 9 Displacements of surrounding rock mass (strain-softening model 120574119901 = 004)

72 Parameter Analysis In order to analyze the effect ofdifferent softening parameters on plastic radius and radialdisplacement of surrounding rock when considering 3D orgeneralized H-B failure criterion the following parametersare cited 120590

119888= 30MPa 119864 = 55GPa V = 025 119901in = 5Mpa

and 1199030= 5mThe detailed data are shown in Table 4

Elastic-brittle-plastic model (dilation angle is equal to20∘) was selected to study the stresses and displacementsof surrounding rock as shown in Tables 4 and 5 Com-pared with the softening process of three parameters thesoftening process with only one parameter was conductedThe following conclusions can be drawn as follows thedifferences of plastic radius are 188 (2D) and 137 (3D)when only parameter ldquo119886rdquo is softening the differences of radialdisplacement are 628 (2D) and 577 (3D) when onlyparameter ldquo119886rdquo is softening the differences of plastic radius

10 Mathematical Problems in Engineering

Table 5 Parameter analyses

H-B Softening parameters Groups 1199031199011199030

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D e-p 1 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14771 07008 08276 08996

3D 120574119901= 0006 12803 05773 06331 06629

2D e-b-p 14807 07053 08327 090513D e-b-p 12832 05809 06367 066662D e-p 2 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14602 06796 07919 08553

3D 120574119901= 0006 12688 05629 06102 06354

2D e-b-p 14604 06800 07923 085573D e-b-p 12690 05632 06105 063572D e-p 3 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14727 06953 08195 08901

3D 120574119901= 0006 12775 05738 06282 06574

2D e-b-p 14777 07016 08273 089883D e-b-p 12814 05787 06337 066322D e-p 4 14591 06783 07897 085253D e-p 12531 05877 06446 067512D 120574

119901= 0006 16778 09724 13501 15894

3D 120574119901= 0006 14509 08046 10370 11740

2D e-b-p 17339 10539 14379 167813D e-b-p 14882 08584 10797 12092

03

04

05

06

07

08

09

10

11

urr

0(

)

105 110 115 120 125 130100rr0

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (associated flow rule 120595 = 75∘)ur4 (associated flow rule 120595 = 15∘)ur5 (associated flow rule 120595 = 20∘)

Figure 10 Displacements of surrounding rock mass (strain-softening model 120574119901 = 0006)

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3

ur4

ur5

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

11

urr

0(

)

(associated flow rule 120595 = 75∘)(associated flow rule 120595 = 15∘)(associated flow rule 120595 = 20∘)

Figure 11 Displacements of surrounding rock mass (elastic-brittle-plastic model)

Mathematical Problems in Engineering 11

are 323 (2D) and 246 (3D) when only parameter ldquo119904rdquo issoftening the differences of radial displacement are 1140(2D) and 1014 (3D) when only parameter ldquo119904rdquo is softeningthe differences of plastic radius are 208 (2D) and 151 (3D)when only parameter ldquo119898rdquo was softening the differences ofradial displacement are 693 (2D) and 625 (3D) whenonly parameter ldquo119898rdquo is softening

The above results show that the softening of surroundingrock is commonly decided by many parameters The order ofrock mass parameters affecting the stress and displacementsis shown as follows 119904 gt 119898 gt 119886

8 Conclusions

(1) Incorporating the 3D Hoek-Brown failure criterionassociated flow rule nonassociated flow rule and thestrain-softening model numerical solutions of stressdisplacement and plastic radius were proposed

(2) Comparison results show that the plastic radius andradial displacement calculated by 3D Hoek-Brownfailure criterion are smaller than those based ongeneralized Hoek-Brown

(3) Radial displacement calculated by nonassociated flowrule is smaller than those considering associated flowrule The influences of dilatancy parameter on theresults based on generalized H-B failure criterion arelarger than those based on 3D H-B failure criterion

Notations

119886 Parameter of HB failure criterion for peakstrength [mdash]

119886119903 Parameter of HB failure criterion for

residual strength [mdash]1199030 Radius of the tunnel opening [L]119863 HB constants for the rock mass [mdash]119864 Youngrsquos modulus of the rock mass [FLminus2]119898119887 Parameter of HB failure criterion for peakstrength [mdash]

119898119887119903 Parameter of HB failure criterion forresidual strength [mdash]

119901in Critical internal pressure [FLminus2]119903 Radial distance from the center of opening

[L]119903119901 Plastic radius [L]

119903119904 Residual radius [L]119904 Parameter of the HB failure criterion for

peak strength [mdash]119904119903 Parameter of the HB failure criterion for

residual strength [mdash]119906119903 Radial displacement [L]

1205900 Initial in situ stress [FLminus2]

120590119888 Uniaxial compressive strength of the rock

[FLminus2]120590119903 Radial normal stress [FLminus2]

120590120579 Circumferential normal stress [FLminus2]

120590119911 Axial normal stress along the axis of thetunnel [FLminus2]

1205901 Major principal stresses [FLminus2]

1205902 Middle principal stresses [FLminus2]

1205903 Minor principal stresses [FLminus2]

120590119877 Radial normal stress at the elastoplasticinterface [FLminus2]

120576119903 Radial normal strain [mdash]120576120579 Circumferential normal strain [mdash]120576119911 Axial normal strain [mdash]119901 Plastic parts of normal strain and stress [mdash]119890 Elastic parts of normal strain and stress [mdash]120595 Dilation angle [mdash]120573 Dilation coefficient [mdash]120574119901 Softening coefficient of the surrounding

rock [mdash]V Poissonrsquos ratio of the rock mass [mdash]119865 Yield function [mdash]119876 Plastic potential function [mdash]

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors are grateful to the 973 Program (2013CB036004)National Natural Science Foundation of China (no51208523)

References

[1] M H Yu N L He and L Y Song ldquoTwin shear stress theoryand its generalizationrdquo Scientia Sinica (Sciences inChina Englishedition) A vol 28 no 11 pp 1113ndash1120 1985

[2] M-H Yu Y-W Zan J Zhao and M Yoshimine ldquoA UnifiedStrength criterion for rock materialrdquo International Journal ofRock Mechanics andMining Sciences vol 39 no 8 pp 975ndash9892002

[3] C Carranza-Torres ldquoDimensionless graphical representation ofthe exact elasto-plastic solution of a circular tunnel in a Mohr-Coulomb material subject to uniform far-field stressesrdquo RockMechanics amp Rock Engineering vol 36 no 3 pp 237ndash253 2003

[4] C Carranza-Torres ldquoElasto-plastic solution of tunnel problemsusing the generalized formof theHoek-Brown failure criterionrdquoInternational Journal of Rock Mechanics amp Mining Sciences vol41 supplement 1 pp 629ndash639 2004

[5] K-H Park and Y-J Kim ldquoAnalytical solution for a circularopening in an elastic-brittle-plastic rockrdquo International Journalof Rock Mechanics and Mining Sciences vol 43 no 4 pp 616ndash622 2006

[6] C Carranza-Torres and C Fairhurst ldquoThe elasto-plasticresponse of underground excavations in rockmasses that satisfythe Hoek-Brown failure criterionrdquo International Journal of RockMechanics andMining Sciences vol 36 no 6 pp 777ndash809 1999

[7] S K Sharan ldquoElastic-brittle-plastic analysis of circular open-ings in Hoek-Brown mediardquo International Journal of RockMechanics andMining Sciences vol 40 no 6 pp 817ndash824 2003

[8] S K Sharan ldquoAnalytical solutions for stresses and displace-ments around a circular opening in a generalized Hoek-Brown

12 Mathematical Problems in Engineering

rockrdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 1 pp 78ndash85 2008

[9] M B Reed ldquoThe influence of out-of-plane stress on a planestrain problem in rock mechanicsrdquo International Journal forNumerical amp Analytical Methods in Geomechanics vol 12 no2 pp 173ndash181 1988

[10] X-D Pan and E T Brown ldquoInfluence of axial stress anddilatancy on rock tunnel stabilityrdquo Journal of GeotechnicalEngineering vol 122 no 2 pp 139ndash146 1996

[11] S Wang Z Wu M Guo and X Ge ldquoTheoretical solutions ofa circular tunnel with the influence of axial in situ stress inelastic-brittle-plastic rockrdquo Tunnelling and Underground SpaceTechnology vol 30 pp 155ndash168 2012

[12] A-Z Lu G-S Xu F Sun and W-Q Sun ldquoElasto-plasticanalysis of a circular tunnel including the effect of the axial insitu stressrdquo International Journal of Rock Mechanics and MiningSciences vol 47 no 1 pp 50ndash59 2010

[13] X-P Zhou H-Q Yang Y-X Zhang and M-H Yu ldquoTheeffect of the intermediate principal stress on the ultimatebearing capacity of a foundation on rock massesrdquo Computersamp Geotechnics vol 36 no 5 pp 861ndash870 2009

[14] J F Zou and Y Su ldquoTheoretical solutions of a circular tunnelwith the influence of the out-of-plane stress based on the gen-eralized Hoek-Brown failure criterionrdquo International Journal ofGeomechanics (ASCE) vol 16 no 3 2016

[15] J F Zou and Z He ldquoNumerical approach for strain-softeningrock with axial stressrdquo Proceedings of the Institution of CivilEngineersmdashGeotechnical Engineering vol 169 no 3 pp 276ndash290 2016

[16] J-F Zou S-S Li Y XuH-CDan andL-H Zhao ldquoTheoreticalsolutions for a circular opening in an elasticndashbrittlendashplastic rockmass incorporating the out-of-plane stress and seepage forcerdquoKSCE Journal of Civil Engineering vol 20 no 2 pp 687ndash7012016

[17] J F Zou and S Q Zuo ldquoAn approximate solution for the cylin-drical cavity expansion problem under the non-axisymmetricdisplacement boundary condition on hypotenuserdquo Interna-tional Journal of Geotechnical Engineering In press

[18] J F Zou and Z Q Xia ldquoSolutions for displacement and stressin strain-softening surrounding rock incorporating the effectsof hydraulic-mechanical coupling and rockbolts effectivenessrdquoGeotechnical amp Geological Engineering 2016

[19] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[20] S Wang X Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[21] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[22] J Zou and S Li ldquoTheoretical solution for displacement andstress in strain-softening surrounding rock under hydraulic-mechanical couplingrdquo Science China Technological Sciences vol58 no 8 pp 1401ndash1413 2015

[23] X Yang and Z Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[24] X L Yang J S Xu Y X Li andRM Yan ldquoCollapsemechanismof tunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[25] X L Yang and RM Yan ldquoCollapse mechanism for deep tunnelsubjected to seepage force in layered soilsrdquo Geomechanics andEngineering vol 8 no 5 pp 741ndash756 2015

[26] X D Pan and J A Hudson ldquoA simplified three dimensionalHoek-Brown yield criterionrdquo in Proceedings of the ISRM Inter-national Symposium International Society for RockMechanicsMadrid Spain 1988

[27] B Singh R K Goel V K Mehrotra S K Garg and MR Allu ldquoEffect of intermediate principal stress on strengthof anisotropic rock massrdquo Tunnelling and Underground SpaceTechnology vol 13 no 1 pp 71ndash79 1998

[28] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[29] L Zhang andHZhu ldquoThree-dimensional hoek-brown strengthcriterion for rocksrdquo Journal ofGeotechnical andGeoenvironmen-tal Engineering vol 133 no 9 pp 1128ndash1135 2007

[30] X-L Yang and Z-X Long ldquoRoof collapse of shallow tunnelswith limit analysis methodrdquo Journal of Central South Universityvol 22 no 5 article no 2712 pp 1929ndash1936 2015

[31] E Hoek D Wood and S Shah ldquoA modified Hoek-Brownfailure criterion for jointed rock massesrdquo in Proceedings ofthe International Conference on Eurock vol 92 pp 202ndash214September 1992

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

failure criterion Particularly the stress and displacement ofsurrounding rock under three different axial stress states werediscussed in his paper The effects of axial stress and in situstress on stress displacement and strain of surrounding rockwere studied by Lu et al [12] and Zhou et al [13] Moreoverthese approaches can be supplemented and verifiedmutuallyZou and Su [14] presented an analytical solution of thesurrounding rock based on the generalized Hoek-Brownfailure criterion and elastic-brittle-plastic model and thissolution is compared with Wang et al [11] to verify itscorrectness using the method of parameter transformationThe theoretical solutions for the elastic-brittle-plastic andelastic-plastic rockmass incorporating the out-of-plane stressand seepage force were proposed by Zou et al [14ndash18]

Comparedwith the elastic-plasticmodel strain-softeningmodel is closer to failure of rock mass model in engineer-ing practices Taking into account that rock yield strengthdoes not drop to residual strength instantaneously strain-softening process can be regard as a gradual decline in theprocess of yielding of rock mass So the strain-softeningmodel can better fit the test curve of rock For instance Leeand Pietruszczak [19] proposed a numerical procedure forcalculating the stresses and radial displacements around a cir-cular tunnel excavated in a strain-softening Mohr-Coulombor generalized Hoek-Brown media In this approach thepotential plastic zone is divided into a finite number ofconcentric rings and it is assumed that all the strengthparameters are linear functions of deviatoric plastic strainWang et al [20] proposed a new closed strain-softeningmethod considering softening process as a series of brittle-plastic and plastic flow process and presented a new methodto describe the strain-softening process of rock soil massAlonso et al [21] standardized the process of modeling andthe problem was transformed into the initial value problemof the Runge-Kutta method Zou and Li [22] proposed animproved numerical approach to analyze the stability of thestrain-softening surrounding rock with the considerationof the hydraulic-mechanical coupling and the variation ofelastic strain in the plastic region Moreover Zou and He [15]proposed a numerical approach that considers the effect ofout-of-plane stress for circular tunnels excavated in strain-softening rock

At present the generalized H-B failure criterion is widelyused [23ndash25] but it is difficult to obtain a relatively accuratesolutionThemajority of scholars have donemany researchesabout three-dimensional failure mechanism including PanandHudson [26] Singh et al [27] Priest [28] Zhang andZhu[29] and Yang and Long [23 30] They proposed different3D failure mechanisms based on different experimental ortheoretical models respectively Among these researchesthe model proposed by Zhang and Zhu [29] which can becompared with the two-dimensional Hoek-Brown model iswidely recognized

Although the 3D Hoek-Brown failure criterion has beenwidely recognized the theoretical analysis for deep tunnelis still little discussed The paper focus on the influences ofthe axial stress on the stress and strain of strain-softeningsurrounding rock in deep tunnel considering 3D Hoek-Brown failure criterion Strains are obtained by the 3D plastic

potential function and stresses are given by plane strainmethod Moreover the results are compared with thosebased on generalized Hoek-Brown failure criterion to finddifferences between the two methods

2 Failure Criterion

Hoek et al [31] modified the previous Hoek-Brown failurecriterion and proposed the generalized H-B failure criterion

1205901minus 1205903= 120590119888(119898119894

1205903

120590119888

+ 1)

119886

(1)

where 120590119888is the unconfined compressive strength of the rock

mass 1205901and 120590

3are the major and minor principal stresses

respectively 119898 119904 and 119886 are the H-B constants for the rockmass before yielding which are expressed as follows

119898 = 119898119894exp [(GSI minus 100)

(28 minus 14119863)

]

119904 = exp [(GSI minus 100)(9 minus 3119863)

]

119886 = 05 +

1

6

[exp (minusGSI15

) minus exp(minus203

)]

(2)

where119863 is a factor that depends on the degree of disturbanceto which the rock has been subjected in terms of blast damageand stress relaxation which varies between 0 and 1 and GSIis the geological strength index of the rockmass which variesbetween 10 and 100

The generalized Hoek-Brown failure criterion has beenwidely used but the influence of the intermediate principalstress on distribution of stress and strength is neglectedHence a 3D Hoek-Brown failure criterion is proposed byZhang and Zhu [29] on the basis of the generalized Hoek-Brown and Mogi failure criteria

1

120590(1119886minus1)

119888

(

3

radic2

120591oct)1119886

+

119898119894

2

(

3

radic2

120591oct) minus 1198981198941205901198982 = 119904120590119888 (3)

where 120591oct is octahedron deviatoric stress and 12059010158401198982

is averageeffective stress

120591oct =1

3

radic(1205901minus 1205902)2

+ (1205902minus 1205903)2

+ (1205901minus 1205903)2

1205901015840

1198982=

1205901+ 1205903

2

(4)

The 3D generalized H-B failure criterion not only con-siders the influence of the intermediate principal stress butalso inherits the merit of the H-B failure criterion Underthe triaxial compression and triaxial tension conditions theparameters of the H-B failure criterion can be directly usedfor 3D generalized H-B failure criterion

3 Computational Model

As shown in Figure 1 a circular opening with an initial radius(1199030) is subjected to a three-dimensional and uniform in situ

Mathematical Problems in Engineering 3

y

x

z

1205900

1205900

1205900

Residual zone

Softening zone

Elastic zone

Elasto-plasticinterface

pin

rp

rs

r0

Figure 1 Stress state of the surrounding rock

pressure (1205900) at infinity and an internal support pressure (119901in)

in the tunnel wall The surrounding rock mass is consideredas continuous homogeneous isotropic and initially elasticBecause the axial stress (120590

119911) along the axis of the deep tunnel

is also considered the analyzed model of surrounding rockcan be regarded as spatial axial symmetry model

During excavation of the deep tunnel 119901in is graduallyreducedWhen the internal support pressure (119901in) is less thana critical support pressure plastic region will appear aroundthe surrounding rock Because the yielding extent of thesurrounding rock is different the plastic zone of surroundingrock is divided into softening zone and residual zone Thepaper considers the surrounding rock to be homogeneousand isotropic material Therefore the radial normal strain120576119903 the circumferential normal strain 120576

120579 and the radial

displacement 119906119903can be obtained by small strain assumption

as follows

120576119903=

120597119906119903

120597119903

120576120579=

119906119903

119903

(5)

Outside the plastic zone the stress and displacement ofsurrounding rock in elastic zone can be obtained by thesolution presented by Reed [9]

120590119903= 1205900minus (1205900minus 120590119877) (

119903119901

119903

)

2

(6a)

120590119903= 1205900+ (1205900minus 120590119877) (

119903119901

119903

)

2

(6b)

When considering the influence of intermediate principalstress the axial stress is equal to the in situ stress in deeptunnel as follows

120590119911= 1205900 (7a)

119906119903=

1

119864

(1 + 120583) (1205900minus 120590119903119901)

1199032

119901

119903

(7b)

4 Strain-Softening Model

For strain-softening model the stress-strain curves can besimplified to 3-slope as shown in Figure 2 [15 22]

The strength and deformation parameters of the strain-softening rock mass are evaluated based on plastic deforma-tion and are controlled by the deviatoric strain [19 21]

120574119901= 120576119901

1minus 120576119901

3 (8)

where 1205761199011and 120576119901

3are the major and minor plastic strains

respectivelyThe physical parameters of the surrounding rockmass are

described according to the bilinear function of plastic shearstrain as follows [19 21]

120596 (120574119901) =

120596119901minus (120596119901minus 120596119903)

120574119901

120574119901

119903

0 lt 120574119901lt 120574119901

119903

120596119903 120574

119901ge 120574119901

119903

(9)

where 120596 represents a strength parameter such as 120593 119888 119898 119904119886 120601 and 119864 120574119901

119903is the critical deviatoric plastic strain from

which the residual behavior is first observed and should beidentified through experimentation the subscripts 119901 and 119903represent the peak and residual values respectively

To avoid the influence that the curve of stress and strain instrain-softening stage is nonlinear on theoretical analysis thestrain-softening process can be simplified to a series of brittle-plastic processes [20] as shown in Figure 3 The whole plasticzone of the deep tunnel is separated into 119899 rings and theadjacent rings have different parameters Hence the actualconstitutive model is replaced by piecewise linear 119894 (119894 + 1)and each segment has different negative slope it seems likethere are 119899 brittle-plastic processesThe greater 119899 is the moreaccurate the strain-softening model is

5 Solutions in Plastic Zone

The total plastic region can be divided into 119899 connectedannuli which are bounded by annuli of the radii 119903

(119894minus1)and 119903(119894)

The 119894th annulus is determined by the outer radius 119903(119894minus1)

andthe inner radius 119903

(119894)which is adjacent to the elastic region

The parameter values of the outmost ring are equal to thevalues of innermost elastic region So the (119899 minus 1) iterationsof brittle-plastic analysis are conducted on the inner plasticannulus until the residual strength is reached

The stress equilibrium equation of an element in tunnelwall can be represented by

120597120590119903

120597119903

+

120590119903minus 120590120579

119903

= 0 (10)

where 120590119903is the radial stress and 120590

120579is the tangential stress

4 Mathematical Problems in Engineering

A

O1205761

1205901 minus 1205903

B C

120590cr

120590c

(a) (1205901 minus 1205903)-1205761 curves

+

minus

O1205761

120576

(b) 1205761-120576V curves

+

minus

O1205761

1205763

(c) 1205761-1205763 curves

A

120590cr

O1205761

1205901 minus 1205903

120590c

120573120576e1120576e1

B C

(d) (1205901 minus 1205903)-1205761 simplified curves

H

+

minus

O

120576p1

minus120576v1

1

1205761

120576v

(e) 1205761-120576V simplified curves

+

minus

O

120576e1

1

1205761

1205763

120573120576e1

120576p1

minus120576p3

h

(f) 1205761-1205763 simplified curves

Figure 2 Strain-softening material behavior model

Calculate constitutive model

Actual constitutive model

O

0

i minus 1

i + 1i

middot middot middot

⋱

120576

120590

Figure 3 Strain-softening model of rock mass

Based on the plane strain assumption the axial strainshould satisfy

120576119911= 120576119890

119911+ 120576119901

119911= 0

120576119890

119911=

1

119864

[120590119911minus V (120590

119903+ 120590120579) minus (1 minus 2V) 120590

0]

(11)

Hence the relationship between radial normal stress 120590119903

circumferential normal stress 120590120579 and axial normal stress 120590

119911

is expressed as

120590119911= V (120590

120579+ 120590119903) minus (2V minus 1) 120590

0minus 119864120576119901

119911 (12)

The radial normal stress and the circumferential normalstress at the elastoplastic interface satisfy the failure criterionand can be given by

119865 (1205901 1205903 120574119901) = 1205901minus 1205903minus 119867 (120590

1 1205903 120574119901) (13)

The radius of the first ring is 119903(0)= 119903119901which is at the

interface between the elastic region and the equivalent plasticzone

120588(0)=

119903(0)

119903119901

119903(0)= 119903119901

(14)

The stress and strain at the elastoplastic interface can begiven by

120590119903(0)

120590120579(0)

120590119911(0)

=

120590119877

21205900minus 120590119877

1205900

Mathematical Problems in Engineering 5

120576119903(0)

120576120579(0)

120576119911(0)

=

1 + V119864

119889119906

119889119903

119906

119903

0

(15)

where

120576119903(0)= minus

1

2119866

1205900minus 120590119877

120576120579(0)=

1

2119866

1205900minus 120590119877

(16)

The normalized inner radius can be expressed as

120588(119894)=

119903(119894)

119903119901

(17)

120590119903on both inner and outer boundaries of the plastic zone

are known a priori and 120590119903decreases from 120590

119877to 119901in The

increment of radial normal stress is given by

Δ120590119903=

(119901in minus 120590119877)

119899

(18)

So the radial normal stress at each ring can be representedby

120590119903(119894)= 120590119903(119894minus1)

+ Δ120590119903 (19)

The stress equilibrium equation can be expressed inanother way as follows

119889120590119903

119889120588

+

120590119903minus 120590120579

120588

= 0 (20)

From above equations the following expressions can beobtained

120588(119894)

Δ120590119903(119894)

Δ120588(119894)

= 120590120579(119894)minus 120590119903(119894) (21)

The stress equilibrium differential equation for the 119894thannulus is derived by using (13) and (20) expressed as (21)and (22)

120590119903(119894)minus 120590119903(119894minus1)

120588(119894)minus 120588(119894minus1)

minus

119867 (120590119903(119894) 120574119901)

120588(119894)

= 0 (22)

120588(119894)=

119867 (120590119903(119894) 120574119901) + Δ120590

119903

119867(120590119903(119894) 120574119901)

120588(119894minus1) (23)

The radius of each ring is a known quantity and the radialstress of the 119894th ring can be obtained from (17) So the axialnormal stress 120590

119911can be given by

120590119911(119894)= V (120590

120579(119894)+ 120590119903(119894)) minus (2V minus 1) 120590

0minus 119864120576119901

119911(119894) (24)

Combining (3) (17) and (22) stress at the outer ring canbe obtained and the radial stress at the 119894th ring is calculatedby using linear interpolation

119876119909=

119876119906(119894)minus 119876119906(119894minus1)

120588(119894)minus 120588(119894minus1)

(120588119909minus 120588(119894minus1)) + 119876119906(119894) (25)

where119876119909is stress or displacement of rockmass119876

119906(119894)is stress

or displacement of rock mass at 119894th ring 119876119906(119894minus1)

is stress ordisplacement of rock mass at (119894 minus 1)th ring and 120588

119909is the

distance between center of the tunnel wall and any point ofthe surrounding rock

The compatibility equation can be written in the generalform as follows

119889120576120579

119889119903

+

120576120579minus 120576119903

119903

= 0 (26)

Equation (26) can be transformed into the followingform

119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

120576119890

120579minus 120576119890

119903

120588

(27)

It can be rewritten as119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

1 + ]119864

119867(120590119903 120574119901)

120588

(28)

51 Associated Flow Rule According to the geotechnicalplastic mechanics there is an equipotential surface of plasticpotential at any point 119872 in stress space Its mathematicalexpression is called the plastic potential function and theplastic potential function can be presented by

119892 (120590119894119895 119867119886) = 0 (29)

where119867119886is hardening parameter and 120590

119894119895is principal stress

The plastic strain increment 119889120576119901119894119895can be obtained by

119889120576119901

119894119895=

120597119865

120597120590119894119895

119889120582 (30)

where 119889120582 is multiplication operator 120576119901119894119895is plastic strain and

120590119894119895is principal stressIt is called the associated flow rule when 119865 and 119876 are

completely the same And the physical meaning of yieldfunction 119865 is a judgment criterion whether a point ofsurrounding rock reaches the yield state And the physicalmeaning of plastic potential function 119876 is the relationshipbetween plastic strain increment and loading surface

While the strains of rock and soil mass satisfy theassociated flow rule its plastic potential function is given by

119876 (120590) = minus

119899

3

1198681+

3

120590119888

1198692+

radic3

2

119899radic1198692 (31)

where 119899 is dilation parameter

1198681= 1205901+ 1205902+ 1205903

1198692=

1

6 [(1205901minus 1205902)2

+ (1205902minus 1205903)2

+ (1205903minus 1205901)2

]

(32)

6 Mathematical Problems in Engineering

The differential of three-direction strain is expressed as

119889120576119901

120579=

120597119891

120597120590120579

119889120582 = [(

radic3 (2120590120579minus 120590119903minus 120590119911)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590120579minus 120590119903minus 120590119911)] 119889120582

119889120576119901

119911=

120597119891

120597120590119911

119889120582 = [(

radic3 (2120590119911minus 120590120579minus 120590119903)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590119911minus 120590120579minus 120590119903)] 119889120582

119889120576119901

119903=

120597119891

120597120590119903

119889120582 = [(

radic3 (2120590119903minus 120590120579minus 120590119911)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590119903minus 120590120579minus 120590119911)] 119889120582

(33)

If 1198711 1198712 and 119871

3are defined as follows

[(

radic3 (2120590120579minus 120590119903minus 120590119911)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590120579minus 120590119903minus 120590119911)]

= 1198711

[(

radic3 (2120590119911minus 120590120579minus 120590119903)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590119911minus 120590120579minus 120590119903)]

= 1198712

[(

radic3 (2120590119903minus 120590120579minus 120590119911)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590119903minus 120590120579minus 120590119911)]

= 1198713

(34)

then (33) can be simplified to

119889120576119901

1=

120597119891

1205971205901

119889120582 = 1198711119889120582

119889120576119901

2=

120597119891

1205971205902

119889120582 = 1198712119889120582

119889120576119901

3=

120597119891

1205971205903

119889120582 = 1198713119889120582

(35)

The relationship between radial plastic normal strain 120576119901119903

circumferential plastic normal strain 120576119901120579 and axial plastic

normal strain 120576119901119911can be represented by

120576119901

120579

1198711

=

120576119901

119911

1198712

=

120576119901

119903

1198713

(36)

119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

1 + ]119864

119867 (120590119903 120574119901)

120588

(37)

Combination of (36) and (37) leads to

Δ120576119901

120579= (minus

119889120576119890

120579(119894)

119889120588(119894)

minus

1 + ]119864

119867 (120590119903(119894) 120574119901)

120588(119894)

minus

120576119901

120579(119894minus1)(1 minus 119871

3(119894)1198711(119894))

120588(119894)

)(

1

Δ120588(119894)

+

(1 minus 1198713(119894)1198711(119894))

120588(119894)

)

(38)

The increment of the radial plastic strain at 119894th annuluscan be obtained by (36)

Δ120576119901

119903(119894)=

1198713(119894)

1198711(119894)

Δ120576119901

120579=

1198713(119894)

1198711(119894)

(minus

119889120576119890

120579(119894)

119889120588(119894)

minus

1 + ]119864

119867(120590119903(119894) 120574119901)

120588(119894)

minus

120576119901

120579(119894minus1)(1 minus 119871

3(119894)1198711(119894))

120588(119894)

)

sdot (

1

Δ120588(119894)

+

(1 minus 1198713(119894)1198711(119894))

120588(119894)

)

(39)

The displacement at 119894th annulus can be obtained

119906119903(119894)= 119903(119894)120576120579(119894)= 119903(119894)(120576119890

120579(119894)+ 120576119901

120579(119894))

= 119903(119894)(120576119890

120579(119894)+ 120576119901

120579(1)+

119894

sum

119894=1

Δ120576119901

119903(119894))

(40)

52 Nonassociated Flow Rule The plastic potential functionof the surrounding rock is redefined by Reed [9] by replacingthe internal friction angle with the angle of internal friction

120573120576119901

1+ 120576119901

3= 0 (41)

where 120573 = (1 + sin120595)(1 minus sin120595) and 120595 is dilation angleAccording to the nonassociated flow rule we can know

120576119901

2= 0 (42)

So (12) can be rewritten as

120590119911= V (120590

120579+ 120590119903) minus (2V minus 1) 120590

0 (43)

Therefore the relationship of strains is no longer deter-mined by the failure criterion considering the nonassociatedflow rule but is determined directly by the dilatation coeffi-cient 120573 which is defined by the dilation angle 120595 Because theintermediate principal strain 120576

119911= 0 the relationship between

the major and minor principal strains is determined by 1198711

and 1198713using the associated flow rule Hence in order to

compare the results calculated by the associated flow rule andnonassociated flow rule the results of displacement using thenonassociated flow rule can be replaced by using

minus

1198713

1198711

= 120573 =

1 + sin1205951 minus sin120595

(44)

Mathematical Problems in Engineering 7

6 Validation

To validate the correctness of the proposed approach andcalculation program the results of the proposed approachare compared with the results of Sharan [8] According todifferent types of the surrounding rock Sharan provided thestrength parameters of indoor test for many groups of rocksUnder different surrounding rock conditions Sharan [8]calculated the radius of plastic zone and radial displacementof the surrounding rock The specific parameters are shownin Table 1 and calculation results are shown in Table 2

If the intermediate stress 120590119911is equal to major principal

stress 120590120579or minor principal stress 120590

119903 3D Hoek-Brown failure

criterion can be simplified to two-dimensional Hoek-Brownfailure criterion which is used by Sharan

As shown in Table 2 the results of the paper are in goodaccordance with Sharan [8] when strain-softening of rockmass is not considered Under the condition that parametersof surrounding rock are the same the results show that thegreater the stress is the greater the plastic radius of thesurrounding rock is the larger the internal support pressureis the smaller the radius of the plastic zone of surroundingrock is And we found that the results calculated by elastic-brittle-plastic model are larger than those calculated byelastoplastic model As its strength parameters immediatelyfall after reaching the peak value the elastic-brittle-plasticmodel can be regarded as a special case of strain-softeningmodel

7 Numerical Calculation and Discussion

71 Computational Examples To analyze and compare theproposed approach which considers the intermediate prin-cipal stress and the 3D H-B failure criterion with thosebased on the generalized H-B failure criterion the followingparameters obtained from Sharanrsquos experimental results [8]are adopted 120590

119888= 30MPa 119904 = 00039 119886 = 055 119864 = 55GPa

119864119903= 55GPa 119898 = 17 119904

119903= 00019 119886

119903= 06 119898

119903= 16

V = 025 119901in = 5Mpa and 1199030= 5m The calculation results

are shown in Table 3Through the comparison results in Table 3 we can

find that the plastic zone radius and radial displacementcalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterionThus itproves that the results may overestimate the plastic zone radiiand radial displacement of the surrounding rock withoutconsidering intermediate principal stress

As shown in Table 3 the radial displacement of sur-rounding rock calculated by associated flow rule is muchgreater than those calculated by nonassociated flow ruleWith different softening parameters the plastic zone radiicalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterion Forexample elastoplastic model strain-softening model (120574119901 =0006) and elastic-brittle-plastic model would be reducedby 1457 1498 and 1600 respectively The results ofradial displacement using nonassociated flow rule are shownas follows elastoplastic model strain-softening model andelastic-brittle-plastic model (120595 = 0

∘ and 120595 = 20∘) would

120590120590

0

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

02

04

06

08

10

12

14

16

12 14 16 18 20 22 24 2610rr0

Figure 4 Stresses of surrounding rock mass (elastoplastic model)

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16120590p

0

Figure 5 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 004)

be reduced by 2085 3467 2178 3629 2190 and3653 respectively From above results it can be obtainedthat the calculated deviations between 3D H-B failure crite-rion and generalized H-B failure criterion increase with theincreasing of dilation angle

At the same time the radial displacement differencesbetween three-dimensional associated flow rule and nonas-sociated flow rule considering dilatancy effect are comparedThe differences between two calculation approaches areanalyzed for predicting the displacement of plastic zone Theresults of stress and displacement are shown in Figures 4ndash11As shown in Figures 4ndash7 120590

119903 120590120579 120590119911 and 119906

119903are calculation

results based on 3DH-B failure criterion120590101584011990312059010158401205791205901015840119911 and 1199061015840

119903are

calculation results based on generalizedH-B failure criterion

8 Mathematical Problems in Engineering

Table 1 Calculation parameters [8]

Quality of rock mass V 1199030(m) 120590

119888(MPa) 119904 119904

119903119886 119886

119903119898119887

119898119887119903

119864 (GPa) 119864119903(GPa)

Average 025 5 80 00039 0 051 053 201 034 9 5Very poor 03 5 25 00039 00019 055 06 17 085 57 57

Table 2 Comparisons between the results of this paper and Sharan without considering the strain-softening and intermediate principalstress

Parameter groups Elastoplastic model 1205900

119901in 1199031199011199030(Sharan) 119903

1199011199030(this paper)

1 e-p 40 0 177 177141 e-p 80 0 259 25881 e-p 80 1 231 23121 e-p 80 5 191 19071 e-b-p 40 3 253 25261 e-b-p 40 5 203 20301 e-b-p 80 18 233 23332 e-b-p 15 0 390 39012 e-b-p 15 3 151 15102 e-b-p 30 0 784 78362 e-b-p 30 5 233 2327

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

00

02

04

06

08

10

12

14

16

120590p

0

Figure 6 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 0006)

Figures 4ndash7 show the stress comparisons between theideal elastoplastic strain-softening and elastic-brittle-plasticmodel based on 3D and generalized Hoek-Brown failurecriteria respectively When considering the influence of theintermediate principal stress on different failure criterion thevalue of plastic radius calculated by the 3D H-B criterionis smaller than those calculated by the generalized H-Bcriterion

Under the condition that the geotechnical parameters arethe same the influence of dilatation coefficient on generalizedH-B failure criterion is larger than 3D H-B failure criterionParticularly as the softening coefficient is the same the

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16

120590p

0

Figure 7 Stresses of surrounding rock mass (elastic-brittle-plasticmodel)

residual radius is smaller than the plastic radius of thesurrounding rock if it is calculated by generalized H-B failurecriterion For example if 120574119901 = 004 the plastic radiuscalculated by generalized failure criterion is 14660 and theresidual radius does not exist Based on 3D H-B failurecriterion the plastic radius is 12841 and the residual radiusis 10372 Hence if the softening coefficient is the same theplastic radius calculated by different failure criteria is quitedifferent

The radial displacement comparisons of surroundingrock under different flow rules softening parameters anddilation angles are shown in Figures 8ndash11 Summary results

Mathematical Problems in Engineering 9

Table 3 Plastic radius critical stresses and displacements of surrounding rock

Softening parameters 1199031199011199030

1199031199041199030

Associated flowrule

1199061199031199030()

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 75

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D H-B

e-p 14591 1 06793 07245 07906 08534120574119901= 004 14660 1 06879 07373 08114 08839

120574119901= 0006 14945 12401 07239 07819 08676 09500e-b-p 15091 15091 07358 07947 08819 09658

3D H-B

e-p 12681 1 07877 05621 05815 06088 06337120574119901= 004 12841 10372 09599 05669 05884 06194 06484

120574119901= 0006 12980 12198 10685 05944 06219 06610 06970e-b-p 13010 13010 10873 06036 06314 06710 07074

Table 4 Calculation parameters

Rock mass 119904 119904119903

a 119886119903

119898119887

119898119887119903

1 00039 00039 055 06 17 172 00039 00019 055 055 17 173 00039 00039 055 055 17 164 00039 00039 055 055 17 085

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

03

04

05

06

07

08

09

urr

0(

)

105 110 115 120 125 130100rr0

Figure 8 Displacements of surrounding rock mass (elastoplasticmodel)

can be seen from Table 3 The plastic zone radii calculated byassociated flow rule and nonassociated flow rule are almostconsistent However the radial displacement of rock massincreases 1064 (e-p model) 1184 (120574119901 = 004) 1391 (120574119901 =0006) and 1413 (e-b-p model) if it adopted associatedflow rule Obviously with the continuous deteriorations ofstrength parameters the result differences between associatedflow rule and nonassociated flow rule will increase gradually

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

urr

0(

)

Figure 9 Displacements of surrounding rock mass (strain-softening model 120574119901 = 004)

72 Parameter Analysis In order to analyze the effect ofdifferent softening parameters on plastic radius and radialdisplacement of surrounding rock when considering 3D orgeneralized H-B failure criterion the following parametersare cited 120590

119888= 30MPa 119864 = 55GPa V = 025 119901in = 5Mpa

and 1199030= 5mThe detailed data are shown in Table 4

Elastic-brittle-plastic model (dilation angle is equal to20∘) was selected to study the stresses and displacementsof surrounding rock as shown in Tables 4 and 5 Com-pared with the softening process of three parameters thesoftening process with only one parameter was conductedThe following conclusions can be drawn as follows thedifferences of plastic radius are 188 (2D) and 137 (3D)when only parameter ldquo119886rdquo is softening the differences of radialdisplacement are 628 (2D) and 577 (3D) when onlyparameter ldquo119886rdquo is softening the differences of plastic radius

10 Mathematical Problems in Engineering

Table 5 Parameter analyses

H-B Softening parameters Groups 1199031199011199030

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D e-p 1 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14771 07008 08276 08996

3D 120574119901= 0006 12803 05773 06331 06629

2D e-b-p 14807 07053 08327 090513D e-b-p 12832 05809 06367 066662D e-p 2 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14602 06796 07919 08553

3D 120574119901= 0006 12688 05629 06102 06354

2D e-b-p 14604 06800 07923 085573D e-b-p 12690 05632 06105 063572D e-p 3 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14727 06953 08195 08901

3D 120574119901= 0006 12775 05738 06282 06574

2D e-b-p 14777 07016 08273 089883D e-b-p 12814 05787 06337 066322D e-p 4 14591 06783 07897 085253D e-p 12531 05877 06446 067512D 120574

119901= 0006 16778 09724 13501 15894

3D 120574119901= 0006 14509 08046 10370 11740

2D e-b-p 17339 10539 14379 167813D e-b-p 14882 08584 10797 12092

03

04

05

06

07

08

09

10

11

urr

0(

)

105 110 115 120 125 130100rr0

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (associated flow rule 120595 = 75∘)ur4 (associated flow rule 120595 = 15∘)ur5 (associated flow rule 120595 = 20∘)

Figure 10 Displacements of surrounding rock mass (strain-softening model 120574119901 = 0006)

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3

ur4

ur5

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

11

urr

0(

)

(associated flow rule 120595 = 75∘)(associated flow rule 120595 = 15∘)(associated flow rule 120595 = 20∘)

Figure 11 Displacements of surrounding rock mass (elastic-brittle-plastic model)

Mathematical Problems in Engineering 11

are 323 (2D) and 246 (3D) when only parameter ldquo119904rdquo issoftening the differences of radial displacement are 1140(2D) and 1014 (3D) when only parameter ldquo119904rdquo is softeningthe differences of plastic radius are 208 (2D) and 151 (3D)when only parameter ldquo119898rdquo was softening the differences ofradial displacement are 693 (2D) and 625 (3D) whenonly parameter ldquo119898rdquo is softening

The above results show that the softening of surroundingrock is commonly decided by many parameters The order ofrock mass parameters affecting the stress and displacementsis shown as follows 119904 gt 119898 gt 119886

8 Conclusions

(1) Incorporating the 3D Hoek-Brown failure criterionassociated flow rule nonassociated flow rule and thestrain-softening model numerical solutions of stressdisplacement and plastic radius were proposed

(2) Comparison results show that the plastic radius andradial displacement calculated by 3D Hoek-Brownfailure criterion are smaller than those based ongeneralized Hoek-Brown

(3) Radial displacement calculated by nonassociated flowrule is smaller than those considering associated flowrule The influences of dilatancy parameter on theresults based on generalized H-B failure criterion arelarger than those based on 3D H-B failure criterion

Notations

119886 Parameter of HB failure criterion for peakstrength [mdash]

119886119903 Parameter of HB failure criterion for

residual strength [mdash]1199030 Radius of the tunnel opening [L]119863 HB constants for the rock mass [mdash]119864 Youngrsquos modulus of the rock mass [FLminus2]119898119887 Parameter of HB failure criterion for peakstrength [mdash]

119898119887119903 Parameter of HB failure criterion forresidual strength [mdash]

119901in Critical internal pressure [FLminus2]119903 Radial distance from the center of opening

[L]119903119901 Plastic radius [L]

119903119904 Residual radius [L]119904 Parameter of the HB failure criterion for

peak strength [mdash]119904119903 Parameter of the HB failure criterion for

residual strength [mdash]119906119903 Radial displacement [L]

1205900 Initial in situ stress [FLminus2]

120590119888 Uniaxial compressive strength of the rock

[FLminus2]120590119903 Radial normal stress [FLminus2]

120590120579 Circumferential normal stress [FLminus2]

120590119911 Axial normal stress along the axis of thetunnel [FLminus2]

1205901 Major principal stresses [FLminus2]

1205902 Middle principal stresses [FLminus2]

1205903 Minor principal stresses [FLminus2]

120590119877 Radial normal stress at the elastoplasticinterface [FLminus2]

120576119903 Radial normal strain [mdash]120576120579 Circumferential normal strain [mdash]120576119911 Axial normal strain [mdash]119901 Plastic parts of normal strain and stress [mdash]119890 Elastic parts of normal strain and stress [mdash]120595 Dilation angle [mdash]120573 Dilation coefficient [mdash]120574119901 Softening coefficient of the surrounding

rock [mdash]V Poissonrsquos ratio of the rock mass [mdash]119865 Yield function [mdash]119876 Plastic potential function [mdash]

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors are grateful to the 973 Program (2013CB036004)National Natural Science Foundation of China (no51208523)

References

[1] M H Yu N L He and L Y Song ldquoTwin shear stress theoryand its generalizationrdquo Scientia Sinica (Sciences inChina Englishedition) A vol 28 no 11 pp 1113ndash1120 1985

[2] M-H Yu Y-W Zan J Zhao and M Yoshimine ldquoA UnifiedStrength criterion for rock materialrdquo International Journal ofRock Mechanics andMining Sciences vol 39 no 8 pp 975ndash9892002

[3] C Carranza-Torres ldquoDimensionless graphical representation ofthe exact elasto-plastic solution of a circular tunnel in a Mohr-Coulomb material subject to uniform far-field stressesrdquo RockMechanics amp Rock Engineering vol 36 no 3 pp 237ndash253 2003

[4] C Carranza-Torres ldquoElasto-plastic solution of tunnel problemsusing the generalized formof theHoek-Brown failure criterionrdquoInternational Journal of Rock Mechanics amp Mining Sciences vol41 supplement 1 pp 629ndash639 2004

[5] K-H Park and Y-J Kim ldquoAnalytical solution for a circularopening in an elastic-brittle-plastic rockrdquo International Journalof Rock Mechanics and Mining Sciences vol 43 no 4 pp 616ndash622 2006

[6] C Carranza-Torres and C Fairhurst ldquoThe elasto-plasticresponse of underground excavations in rockmasses that satisfythe Hoek-Brown failure criterionrdquo International Journal of RockMechanics andMining Sciences vol 36 no 6 pp 777ndash809 1999

[7] S K Sharan ldquoElastic-brittle-plastic analysis of circular open-ings in Hoek-Brown mediardquo International Journal of RockMechanics andMining Sciences vol 40 no 6 pp 817ndash824 2003

[8] S K Sharan ldquoAnalytical solutions for stresses and displace-ments around a circular opening in a generalized Hoek-Brown

12 Mathematical Problems in Engineering

rockrdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 1 pp 78ndash85 2008

[9] M B Reed ldquoThe influence of out-of-plane stress on a planestrain problem in rock mechanicsrdquo International Journal forNumerical amp Analytical Methods in Geomechanics vol 12 no2 pp 173ndash181 1988

[10] X-D Pan and E T Brown ldquoInfluence of axial stress anddilatancy on rock tunnel stabilityrdquo Journal of GeotechnicalEngineering vol 122 no 2 pp 139ndash146 1996

[11] S Wang Z Wu M Guo and X Ge ldquoTheoretical solutions ofa circular tunnel with the influence of axial in situ stress inelastic-brittle-plastic rockrdquo Tunnelling and Underground SpaceTechnology vol 30 pp 155ndash168 2012

[12] A-Z Lu G-S Xu F Sun and W-Q Sun ldquoElasto-plasticanalysis of a circular tunnel including the effect of the axial insitu stressrdquo International Journal of Rock Mechanics and MiningSciences vol 47 no 1 pp 50ndash59 2010

[13] X-P Zhou H-Q Yang Y-X Zhang and M-H Yu ldquoTheeffect of the intermediate principal stress on the ultimatebearing capacity of a foundation on rock massesrdquo Computersamp Geotechnics vol 36 no 5 pp 861ndash870 2009

[14] J F Zou and Y Su ldquoTheoretical solutions of a circular tunnelwith the influence of the out-of-plane stress based on the gen-eralized Hoek-Brown failure criterionrdquo International Journal ofGeomechanics (ASCE) vol 16 no 3 2016

[15] J F Zou and Z He ldquoNumerical approach for strain-softeningrock with axial stressrdquo Proceedings of the Institution of CivilEngineersmdashGeotechnical Engineering vol 169 no 3 pp 276ndash290 2016

[16] J-F Zou S-S Li Y XuH-CDan andL-H Zhao ldquoTheoreticalsolutions for a circular opening in an elasticndashbrittlendashplastic rockmass incorporating the out-of-plane stress and seepage forcerdquoKSCE Journal of Civil Engineering vol 20 no 2 pp 687ndash7012016

[17] J F Zou and S Q Zuo ldquoAn approximate solution for the cylin-drical cavity expansion problem under the non-axisymmetricdisplacement boundary condition on hypotenuserdquo Interna-tional Journal of Geotechnical Engineering In press

[18] J F Zou and Z Q Xia ldquoSolutions for displacement and stressin strain-softening surrounding rock incorporating the effectsof hydraulic-mechanical coupling and rockbolts effectivenessrdquoGeotechnical amp Geological Engineering 2016

[19] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[20] S Wang X Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[21] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[22] J Zou and S Li ldquoTheoretical solution for displacement andstress in strain-softening surrounding rock under hydraulic-mechanical couplingrdquo Science China Technological Sciences vol58 no 8 pp 1401ndash1413 2015

[23] X Yang and Z Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[24] X L Yang J S Xu Y X Li andRM Yan ldquoCollapsemechanismof tunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[25] X L Yang and RM Yan ldquoCollapse mechanism for deep tunnelsubjected to seepage force in layered soilsrdquo Geomechanics andEngineering vol 8 no 5 pp 741ndash756 2015

[26] X D Pan and J A Hudson ldquoA simplified three dimensionalHoek-Brown yield criterionrdquo in Proceedings of the ISRM Inter-national Symposium International Society for RockMechanicsMadrid Spain 1988

[27] B Singh R K Goel V K Mehrotra S K Garg and MR Allu ldquoEffect of intermediate principal stress on strengthof anisotropic rock massrdquo Tunnelling and Underground SpaceTechnology vol 13 no 1 pp 71ndash79 1998

[28] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[29] L Zhang andHZhu ldquoThree-dimensional hoek-brown strengthcriterion for rocksrdquo Journal ofGeotechnical andGeoenvironmen-tal Engineering vol 133 no 9 pp 1128ndash1135 2007

[30] X-L Yang and Z-X Long ldquoRoof collapse of shallow tunnelswith limit analysis methodrdquo Journal of Central South Universityvol 22 no 5 article no 2712 pp 1929ndash1936 2015

[31] E Hoek D Wood and S Shah ldquoA modified Hoek-Brownfailure criterion for jointed rock massesrdquo in Proceedings ofthe International Conference on Eurock vol 92 pp 202ndash214September 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

y

x

z

1205900

1205900

1205900

Residual zone

Softening zone

Elastic zone

Elasto-plasticinterface

pin

rp

rs

r0

Figure 1 Stress state of the surrounding rock

pressure (1205900) at infinity and an internal support pressure (119901in)

in the tunnel wall The surrounding rock mass is consideredas continuous homogeneous isotropic and initially elasticBecause the axial stress (120590

119911) along the axis of the deep tunnel

is also considered the analyzed model of surrounding rockcan be regarded as spatial axial symmetry model

During excavation of the deep tunnel 119901in is graduallyreducedWhen the internal support pressure (119901in) is less thana critical support pressure plastic region will appear aroundthe surrounding rock Because the yielding extent of thesurrounding rock is different the plastic zone of surroundingrock is divided into softening zone and residual zone Thepaper considers the surrounding rock to be homogeneousand isotropic material Therefore the radial normal strain120576119903 the circumferential normal strain 120576

120579 and the radial

displacement 119906119903can be obtained by small strain assumption

as follows

120576119903=

120597119906119903

120597119903

120576120579=

119906119903

119903

(5)

Outside the plastic zone the stress and displacement ofsurrounding rock in elastic zone can be obtained by thesolution presented by Reed [9]

120590119903= 1205900minus (1205900minus 120590119877) (

119903119901

119903

)

2

(6a)

120590119903= 1205900+ (1205900minus 120590119877) (

119903119901

119903

)

2

(6b)

When considering the influence of intermediate principalstress the axial stress is equal to the in situ stress in deeptunnel as follows

120590119911= 1205900 (7a)

119906119903=

1

119864

(1 + 120583) (1205900minus 120590119903119901)

1199032

119901

119903

(7b)

4 Strain-Softening Model

For strain-softening model the stress-strain curves can besimplified to 3-slope as shown in Figure 2 [15 22]

The strength and deformation parameters of the strain-softening rock mass are evaluated based on plastic deforma-tion and are controlled by the deviatoric strain [19 21]

120574119901= 120576119901

1minus 120576119901

3 (8)

where 1205761199011and 120576119901

3are the major and minor plastic strains

respectivelyThe physical parameters of the surrounding rockmass are

described according to the bilinear function of plastic shearstrain as follows [19 21]

120596 (120574119901) =

120596119901minus (120596119901minus 120596119903)

120574119901

120574119901

119903

0 lt 120574119901lt 120574119901

119903

120596119903 120574

119901ge 120574119901

119903

(9)

where 120596 represents a strength parameter such as 120593 119888 119898 119904119886 120601 and 119864 120574119901

119903is the critical deviatoric plastic strain from

which the residual behavior is first observed and should beidentified through experimentation the subscripts 119901 and 119903represent the peak and residual values respectively

To avoid the influence that the curve of stress and strain instrain-softening stage is nonlinear on theoretical analysis thestrain-softening process can be simplified to a series of brittle-plastic processes [20] as shown in Figure 3 The whole plasticzone of the deep tunnel is separated into 119899 rings and theadjacent rings have different parameters Hence the actualconstitutive model is replaced by piecewise linear 119894 (119894 + 1)and each segment has different negative slope it seems likethere are 119899 brittle-plastic processesThe greater 119899 is the moreaccurate the strain-softening model is

5 Solutions in Plastic Zone

The total plastic region can be divided into 119899 connectedannuli which are bounded by annuli of the radii 119903

(119894minus1)and 119903(119894)

The 119894th annulus is determined by the outer radius 119903(119894minus1)

andthe inner radius 119903

(119894)which is adjacent to the elastic region

The parameter values of the outmost ring are equal to thevalues of innermost elastic region So the (119899 minus 1) iterationsof brittle-plastic analysis are conducted on the inner plasticannulus until the residual strength is reached

The stress equilibrium equation of an element in tunnelwall can be represented by

120597120590119903

120597119903

+

120590119903minus 120590120579

119903

= 0 (10)

where 120590119903is the radial stress and 120590

120579is the tangential stress

4 Mathematical Problems in Engineering

A

O1205761

1205901 minus 1205903

B C

120590cr

120590c

(a) (1205901 minus 1205903)-1205761 curves

+

minus

O1205761

120576

(b) 1205761-120576V curves

+

minus

O1205761

1205763

(c) 1205761-1205763 curves

A

120590cr

O1205761

1205901 minus 1205903

120590c

120573120576e1120576e1

B C

(d) (1205901 minus 1205903)-1205761 simplified curves

H

+

minus

O

120576p1

minus120576v1

1

1205761

120576v

(e) 1205761-120576V simplified curves

+

minus

O

120576e1

1

1205761

1205763

120573120576e1

120576p1

minus120576p3

h

(f) 1205761-1205763 simplified curves

Figure 2 Strain-softening material behavior model

Calculate constitutive model

Actual constitutive model

O

0

i minus 1

i + 1i

middot middot middot

⋱

120576

120590

Figure 3 Strain-softening model of rock mass

Based on the plane strain assumption the axial strainshould satisfy

120576119911= 120576119890

119911+ 120576119901

119911= 0

120576119890

119911=

1

119864

[120590119911minus V (120590

119903+ 120590120579) minus (1 minus 2V) 120590

0]

(11)

Hence the relationship between radial normal stress 120590119903

circumferential normal stress 120590120579 and axial normal stress 120590

119911

is expressed as

120590119911= V (120590

120579+ 120590119903) minus (2V minus 1) 120590

0minus 119864120576119901

119911 (12)

The radial normal stress and the circumferential normalstress at the elastoplastic interface satisfy the failure criterionand can be given by

119865 (1205901 1205903 120574119901) = 1205901minus 1205903minus 119867 (120590

1 1205903 120574119901) (13)

The radius of the first ring is 119903(0)= 119903119901which is at the

interface between the elastic region and the equivalent plasticzone

120588(0)=

119903(0)

119903119901

119903(0)= 119903119901

(14)

The stress and strain at the elastoplastic interface can begiven by

120590119903(0)

120590120579(0)

120590119911(0)

=

120590119877

21205900minus 120590119877

1205900

Mathematical Problems in Engineering 5

120576119903(0)

120576120579(0)

120576119911(0)

=

1 + V119864

119889119906

119889119903

119906

119903

0

(15)

where

120576119903(0)= minus

1

2119866

1205900minus 120590119877

120576120579(0)=

1

2119866

1205900minus 120590119877

(16)

The normalized inner radius can be expressed as

120588(119894)=

119903(119894)

119903119901

(17)

120590119903on both inner and outer boundaries of the plastic zone

are known a priori and 120590119903decreases from 120590

119877to 119901in The

increment of radial normal stress is given by

Δ120590119903=

(119901in minus 120590119877)

119899

(18)

So the radial normal stress at each ring can be representedby

120590119903(119894)= 120590119903(119894minus1)

+ Δ120590119903 (19)

The stress equilibrium equation can be expressed inanother way as follows

119889120590119903

119889120588

+

120590119903minus 120590120579

120588

= 0 (20)

From above equations the following expressions can beobtained

120588(119894)

Δ120590119903(119894)

Δ120588(119894)

= 120590120579(119894)minus 120590119903(119894) (21)

The stress equilibrium differential equation for the 119894thannulus is derived by using (13) and (20) expressed as (21)and (22)

120590119903(119894)minus 120590119903(119894minus1)

120588(119894)minus 120588(119894minus1)

minus

119867 (120590119903(119894) 120574119901)

120588(119894)

= 0 (22)

120588(119894)=

119867 (120590119903(119894) 120574119901) + Δ120590

119903

119867(120590119903(119894) 120574119901)

120588(119894minus1) (23)

The radius of each ring is a known quantity and the radialstress of the 119894th ring can be obtained from (17) So the axialnormal stress 120590

119911can be given by

120590119911(119894)= V (120590

120579(119894)+ 120590119903(119894)) minus (2V minus 1) 120590

0minus 119864120576119901

119911(119894) (24)

Combining (3) (17) and (22) stress at the outer ring canbe obtained and the radial stress at the 119894th ring is calculatedby using linear interpolation

119876119909=

119876119906(119894)minus 119876119906(119894minus1)

120588(119894)minus 120588(119894minus1)

(120588119909minus 120588(119894minus1)) + 119876119906(119894) (25)

where119876119909is stress or displacement of rockmass119876

119906(119894)is stress

or displacement of rock mass at 119894th ring 119876119906(119894minus1)

is stress ordisplacement of rock mass at (119894 minus 1)th ring and 120588

119909is the

distance between center of the tunnel wall and any point ofthe surrounding rock

The compatibility equation can be written in the generalform as follows

119889120576120579

119889119903

+

120576120579minus 120576119903

119903

= 0 (26)

Equation (26) can be transformed into the followingform

119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

120576119890

120579minus 120576119890

119903

120588

(27)

It can be rewritten as119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

1 + ]119864

119867(120590119903 120574119901)

120588

(28)

51 Associated Flow Rule According to the geotechnicalplastic mechanics there is an equipotential surface of plasticpotential at any point 119872 in stress space Its mathematicalexpression is called the plastic potential function and theplastic potential function can be presented by

119892 (120590119894119895 119867119886) = 0 (29)

where119867119886is hardening parameter and 120590

119894119895is principal stress

The plastic strain increment 119889120576119901119894119895can be obtained by

119889120576119901

119894119895=

120597119865

120597120590119894119895

119889120582 (30)

where 119889120582 is multiplication operator 120576119901119894119895is plastic strain and

120590119894119895is principal stressIt is called the associated flow rule when 119865 and 119876 are

completely the same And the physical meaning of yieldfunction 119865 is a judgment criterion whether a point ofsurrounding rock reaches the yield state And the physicalmeaning of plastic potential function 119876 is the relationshipbetween plastic strain increment and loading surface

While the strains of rock and soil mass satisfy theassociated flow rule its plastic potential function is given by

119876 (120590) = minus

119899

3

1198681+

3

120590119888

1198692+

radic3

2

119899radic1198692 (31)

where 119899 is dilation parameter

1198681= 1205901+ 1205902+ 1205903

1198692=

1

6 [(1205901minus 1205902)2

+ (1205902minus 1205903)2

+ (1205903minus 1205901)2

]

(32)

6 Mathematical Problems in Engineering

The differential of three-direction strain is expressed as

119889120576119901

120579=

120597119891

120597120590120579

119889120582 = [(

radic3 (2120590120579minus 120590119903minus 120590119911)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590120579minus 120590119903minus 120590119911)] 119889120582

119889120576119901

119911=

120597119891

120597120590119911

119889120582 = [(

radic3 (2120590119911minus 120590120579minus 120590119903)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590119911minus 120590120579minus 120590119903)] 119889120582

119889120576119901

119903=

120597119891

120597120590119903

119889120582 = [(

radic3 (2120590119903minus 120590120579minus 120590119911)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590119903minus 120590120579minus 120590119911)] 119889120582

(33)

If 1198711 1198712 and 119871

3are defined as follows

[(

radic3 (2120590120579minus 120590119903minus 120590119911)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590120579minus 120590119903minus 120590119911)]

= 1198711

[(

radic3 (2120590119911minus 120590120579minus 120590119903)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590119911minus 120590120579minus 120590119903)]

= 1198712

[(

radic3 (2120590119903minus 120590120579minus 120590119911)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590119903minus 120590120579minus 120590119911)]

= 1198713

(34)

then (33) can be simplified to

119889120576119901

1=

120597119891

1205971205901

119889120582 = 1198711119889120582

119889120576119901

2=

120597119891

1205971205902

119889120582 = 1198712119889120582

119889120576119901

3=

120597119891

1205971205903

119889120582 = 1198713119889120582

(35)

The relationship between radial plastic normal strain 120576119901119903

circumferential plastic normal strain 120576119901120579 and axial plastic

normal strain 120576119901119911can be represented by

120576119901

120579

1198711

=

120576119901

119911

1198712

=

120576119901

119903

1198713

(36)

119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

1 + ]119864

119867 (120590119903 120574119901)

120588

(37)

Combination of (36) and (37) leads to

Δ120576119901

120579= (minus

119889120576119890

120579(119894)

119889120588(119894)

minus

1 + ]119864

119867 (120590119903(119894) 120574119901)

120588(119894)

minus

120576119901

120579(119894minus1)(1 minus 119871

3(119894)1198711(119894))

120588(119894)

)(

1

Δ120588(119894)

+

(1 minus 1198713(119894)1198711(119894))

120588(119894)

)

(38)

The increment of the radial plastic strain at 119894th annuluscan be obtained by (36)

Δ120576119901

119903(119894)=

1198713(119894)

1198711(119894)

Δ120576119901

120579=

1198713(119894)

1198711(119894)

(minus

119889120576119890

120579(119894)

119889120588(119894)

minus

1 + ]119864

119867(120590119903(119894) 120574119901)

120588(119894)

minus

120576119901

120579(119894minus1)(1 minus 119871

3(119894)1198711(119894))

120588(119894)

)

sdot (

1

Δ120588(119894)

+

(1 minus 1198713(119894)1198711(119894))

120588(119894)

)

(39)

The displacement at 119894th annulus can be obtained

119906119903(119894)= 119903(119894)120576120579(119894)= 119903(119894)(120576119890

120579(119894)+ 120576119901

120579(119894))

= 119903(119894)(120576119890

120579(119894)+ 120576119901

120579(1)+

119894

sum

119894=1

Δ120576119901

119903(119894))

(40)

52 Nonassociated Flow Rule The plastic potential functionof the surrounding rock is redefined by Reed [9] by replacingthe internal friction angle with the angle of internal friction

120573120576119901

1+ 120576119901

3= 0 (41)

where 120573 = (1 + sin120595)(1 minus sin120595) and 120595 is dilation angleAccording to the nonassociated flow rule we can know

120576119901

2= 0 (42)

So (12) can be rewritten as

120590119911= V (120590

120579+ 120590119903) minus (2V minus 1) 120590

0 (43)

Therefore the relationship of strains is no longer deter-mined by the failure criterion considering the nonassociatedflow rule but is determined directly by the dilatation coeffi-cient 120573 which is defined by the dilation angle 120595 Because theintermediate principal strain 120576

119911= 0 the relationship between

the major and minor principal strains is determined by 1198711

and 1198713using the associated flow rule Hence in order to

compare the results calculated by the associated flow rule andnonassociated flow rule the results of displacement using thenonassociated flow rule can be replaced by using

minus

1198713

1198711

= 120573 =

1 + sin1205951 minus sin120595

(44)

Mathematical Problems in Engineering 7

6 Validation

To validate the correctness of the proposed approach andcalculation program the results of the proposed approachare compared with the results of Sharan [8] According todifferent types of the surrounding rock Sharan provided thestrength parameters of indoor test for many groups of rocksUnder different surrounding rock conditions Sharan [8]calculated the radius of plastic zone and radial displacementof the surrounding rock The specific parameters are shownin Table 1 and calculation results are shown in Table 2

If the intermediate stress 120590119911is equal to major principal

stress 120590120579or minor principal stress 120590

119903 3D Hoek-Brown failure

criterion can be simplified to two-dimensional Hoek-Brownfailure criterion which is used by Sharan

As shown in Table 2 the results of the paper are in goodaccordance with Sharan [8] when strain-softening of rockmass is not considered Under the condition that parametersof surrounding rock are the same the results show that thegreater the stress is the greater the plastic radius of thesurrounding rock is the larger the internal support pressureis the smaller the radius of the plastic zone of surroundingrock is And we found that the results calculated by elastic-brittle-plastic model are larger than those calculated byelastoplastic model As its strength parameters immediatelyfall after reaching the peak value the elastic-brittle-plasticmodel can be regarded as a special case of strain-softeningmodel

7 Numerical Calculation and Discussion

71 Computational Examples To analyze and compare theproposed approach which considers the intermediate prin-cipal stress and the 3D H-B failure criterion with thosebased on the generalized H-B failure criterion the followingparameters obtained from Sharanrsquos experimental results [8]are adopted 120590

119888= 30MPa 119904 = 00039 119886 = 055 119864 = 55GPa

119864119903= 55GPa 119898 = 17 119904

119903= 00019 119886

119903= 06 119898

119903= 16

V = 025 119901in = 5Mpa and 1199030= 5m The calculation results

are shown in Table 3Through the comparison results in Table 3 we can

find that the plastic zone radius and radial displacementcalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterionThus itproves that the results may overestimate the plastic zone radiiand radial displacement of the surrounding rock withoutconsidering intermediate principal stress

As shown in Table 3 the radial displacement of sur-rounding rock calculated by associated flow rule is muchgreater than those calculated by nonassociated flow ruleWith different softening parameters the plastic zone radiicalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterion Forexample elastoplastic model strain-softening model (120574119901 =0006) and elastic-brittle-plastic model would be reducedby 1457 1498 and 1600 respectively The results ofradial displacement using nonassociated flow rule are shownas follows elastoplastic model strain-softening model andelastic-brittle-plastic model (120595 = 0

∘ and 120595 = 20∘) would

120590120590

0

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

02

04

06

08

10

12

14

16

12 14 16 18 20 22 24 2610rr0

Figure 4 Stresses of surrounding rock mass (elastoplastic model)

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16120590p

0

Figure 5 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 004)

be reduced by 2085 3467 2178 3629 2190 and3653 respectively From above results it can be obtainedthat the calculated deviations between 3D H-B failure crite-rion and generalized H-B failure criterion increase with theincreasing of dilation angle

At the same time the radial displacement differencesbetween three-dimensional associated flow rule and nonas-sociated flow rule considering dilatancy effect are comparedThe differences between two calculation approaches areanalyzed for predicting the displacement of plastic zone Theresults of stress and displacement are shown in Figures 4ndash11As shown in Figures 4ndash7 120590

119903 120590120579 120590119911 and 119906

119903are calculation

results based on 3DH-B failure criterion120590101584011990312059010158401205791205901015840119911 and 1199061015840

119903are

calculation results based on generalizedH-B failure criterion

8 Mathematical Problems in Engineering

Table 1 Calculation parameters [8]

Quality of rock mass V 1199030(m) 120590

119888(MPa) 119904 119904

119903119886 119886

119903119898119887

119898119887119903

119864 (GPa) 119864119903(GPa)

Average 025 5 80 00039 0 051 053 201 034 9 5Very poor 03 5 25 00039 00019 055 06 17 085 57 57

Table 2 Comparisons between the results of this paper and Sharan without considering the strain-softening and intermediate principalstress

Parameter groups Elastoplastic model 1205900

119901in 1199031199011199030(Sharan) 119903

1199011199030(this paper)

1 e-p 40 0 177 177141 e-p 80 0 259 25881 e-p 80 1 231 23121 e-p 80 5 191 19071 e-b-p 40 3 253 25261 e-b-p 40 5 203 20301 e-b-p 80 18 233 23332 e-b-p 15 0 390 39012 e-b-p 15 3 151 15102 e-b-p 30 0 784 78362 e-b-p 30 5 233 2327

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

00

02

04

06

08

10

12

14

16

120590p

0

Figure 6 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 0006)

Figures 4ndash7 show the stress comparisons between theideal elastoplastic strain-softening and elastic-brittle-plasticmodel based on 3D and generalized Hoek-Brown failurecriteria respectively When considering the influence of theintermediate principal stress on different failure criterion thevalue of plastic radius calculated by the 3D H-B criterionis smaller than those calculated by the generalized H-Bcriterion

Under the condition that the geotechnical parameters arethe same the influence of dilatation coefficient on generalizedH-B failure criterion is larger than 3D H-B failure criterionParticularly as the softening coefficient is the same the

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16

120590p

0

Figure 7 Stresses of surrounding rock mass (elastic-brittle-plasticmodel)

residual radius is smaller than the plastic radius of thesurrounding rock if it is calculated by generalized H-B failurecriterion For example if 120574119901 = 004 the plastic radiuscalculated by generalized failure criterion is 14660 and theresidual radius does not exist Based on 3D H-B failurecriterion the plastic radius is 12841 and the residual radiusis 10372 Hence if the softening coefficient is the same theplastic radius calculated by different failure criteria is quitedifferent

The radial displacement comparisons of surroundingrock under different flow rules softening parameters anddilation angles are shown in Figures 8ndash11 Summary results

Mathematical Problems in Engineering 9

Table 3 Plastic radius critical stresses and displacements of surrounding rock

Softening parameters 1199031199011199030

1199031199041199030

Associated flowrule

1199061199031199030()

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 75

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D H-B

e-p 14591 1 06793 07245 07906 08534120574119901= 004 14660 1 06879 07373 08114 08839

120574119901= 0006 14945 12401 07239 07819 08676 09500e-b-p 15091 15091 07358 07947 08819 09658

3D H-B

e-p 12681 1 07877 05621 05815 06088 06337120574119901= 004 12841 10372 09599 05669 05884 06194 06484

120574119901= 0006 12980 12198 10685 05944 06219 06610 06970e-b-p 13010 13010 10873 06036 06314 06710 07074

Table 4 Calculation parameters

Rock mass 119904 119904119903

a 119886119903

119898119887

119898119887119903

1 00039 00039 055 06 17 172 00039 00019 055 055 17 173 00039 00039 055 055 17 164 00039 00039 055 055 17 085

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

03

04

05

06

07

08

09

urr

0(

)

105 110 115 120 125 130100rr0

Figure 8 Displacements of surrounding rock mass (elastoplasticmodel)

can be seen from Table 3 The plastic zone radii calculated byassociated flow rule and nonassociated flow rule are almostconsistent However the radial displacement of rock massincreases 1064 (e-p model) 1184 (120574119901 = 004) 1391 (120574119901 =0006) and 1413 (e-b-p model) if it adopted associatedflow rule Obviously with the continuous deteriorations ofstrength parameters the result differences between associatedflow rule and nonassociated flow rule will increase gradually

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

urr

0(

)

Figure 9 Displacements of surrounding rock mass (strain-softening model 120574119901 = 004)

72 Parameter Analysis In order to analyze the effect ofdifferent softening parameters on plastic radius and radialdisplacement of surrounding rock when considering 3D orgeneralized H-B failure criterion the following parametersare cited 120590

119888= 30MPa 119864 = 55GPa V = 025 119901in = 5Mpa

and 1199030= 5mThe detailed data are shown in Table 4

Elastic-brittle-plastic model (dilation angle is equal to20∘) was selected to study the stresses and displacementsof surrounding rock as shown in Tables 4 and 5 Com-pared with the softening process of three parameters thesoftening process with only one parameter was conductedThe following conclusions can be drawn as follows thedifferences of plastic radius are 188 (2D) and 137 (3D)when only parameter ldquo119886rdquo is softening the differences of radialdisplacement are 628 (2D) and 577 (3D) when onlyparameter ldquo119886rdquo is softening the differences of plastic radius

10 Mathematical Problems in Engineering

Table 5 Parameter analyses

H-B Softening parameters Groups 1199031199011199030

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D e-p 1 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14771 07008 08276 08996

3D 120574119901= 0006 12803 05773 06331 06629

2D e-b-p 14807 07053 08327 090513D e-b-p 12832 05809 06367 066662D e-p 2 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14602 06796 07919 08553

3D 120574119901= 0006 12688 05629 06102 06354

2D e-b-p 14604 06800 07923 085573D e-b-p 12690 05632 06105 063572D e-p 3 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14727 06953 08195 08901

3D 120574119901= 0006 12775 05738 06282 06574

2D e-b-p 14777 07016 08273 089883D e-b-p 12814 05787 06337 066322D e-p 4 14591 06783 07897 085253D e-p 12531 05877 06446 067512D 120574

119901= 0006 16778 09724 13501 15894

3D 120574119901= 0006 14509 08046 10370 11740

2D e-b-p 17339 10539 14379 167813D e-b-p 14882 08584 10797 12092

03

04

05

06

07

08

09

10

11

urr

0(

)

105 110 115 120 125 130100rr0

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (associated flow rule 120595 = 75∘)ur4 (associated flow rule 120595 = 15∘)ur5 (associated flow rule 120595 = 20∘)

Figure 10 Displacements of surrounding rock mass (strain-softening model 120574119901 = 0006)

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3

ur4

ur5

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

11

urr

0(

)

(associated flow rule 120595 = 75∘)(associated flow rule 120595 = 15∘)(associated flow rule 120595 = 20∘)

Figure 11 Displacements of surrounding rock mass (elastic-brittle-plastic model)

Mathematical Problems in Engineering 11

are 323 (2D) and 246 (3D) when only parameter ldquo119904rdquo issoftening the differences of radial displacement are 1140(2D) and 1014 (3D) when only parameter ldquo119904rdquo is softeningthe differences of plastic radius are 208 (2D) and 151 (3D)when only parameter ldquo119898rdquo was softening the differences ofradial displacement are 693 (2D) and 625 (3D) whenonly parameter ldquo119898rdquo is softening

The above results show that the softening of surroundingrock is commonly decided by many parameters The order ofrock mass parameters affecting the stress and displacementsis shown as follows 119904 gt 119898 gt 119886

8 Conclusions

(1) Incorporating the 3D Hoek-Brown failure criterionassociated flow rule nonassociated flow rule and thestrain-softening model numerical solutions of stressdisplacement and plastic radius were proposed

(2) Comparison results show that the plastic radius andradial displacement calculated by 3D Hoek-Brownfailure criterion are smaller than those based ongeneralized Hoek-Brown

(3) Radial displacement calculated by nonassociated flowrule is smaller than those considering associated flowrule The influences of dilatancy parameter on theresults based on generalized H-B failure criterion arelarger than those based on 3D H-B failure criterion

Notations

119886 Parameter of HB failure criterion for peakstrength [mdash]

119886119903 Parameter of HB failure criterion for

residual strength [mdash]1199030 Radius of the tunnel opening [L]119863 HB constants for the rock mass [mdash]119864 Youngrsquos modulus of the rock mass [FLminus2]119898119887 Parameter of HB failure criterion for peakstrength [mdash]

119898119887119903 Parameter of HB failure criterion forresidual strength [mdash]

119901in Critical internal pressure [FLminus2]119903 Radial distance from the center of opening

[L]119903119901 Plastic radius [L]

119903119904 Residual radius [L]119904 Parameter of the HB failure criterion for

peak strength [mdash]119904119903 Parameter of the HB failure criterion for

residual strength [mdash]119906119903 Radial displacement [L]

1205900 Initial in situ stress [FLminus2]

120590119888 Uniaxial compressive strength of the rock

[FLminus2]120590119903 Radial normal stress [FLminus2]

120590120579 Circumferential normal stress [FLminus2]

120590119911 Axial normal stress along the axis of thetunnel [FLminus2]

1205901 Major principal stresses [FLminus2]

1205902 Middle principal stresses [FLminus2]

1205903 Minor principal stresses [FLminus2]

120590119877 Radial normal stress at the elastoplasticinterface [FLminus2]

120576119903 Radial normal strain [mdash]120576120579 Circumferential normal strain [mdash]120576119911 Axial normal strain [mdash]119901 Plastic parts of normal strain and stress [mdash]119890 Elastic parts of normal strain and stress [mdash]120595 Dilation angle [mdash]120573 Dilation coefficient [mdash]120574119901 Softening coefficient of the surrounding

rock [mdash]V Poissonrsquos ratio of the rock mass [mdash]119865 Yield function [mdash]119876 Plastic potential function [mdash]

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors are grateful to the 973 Program (2013CB036004)National Natural Science Foundation of China (no51208523)

References

[1] M H Yu N L He and L Y Song ldquoTwin shear stress theoryand its generalizationrdquo Scientia Sinica (Sciences inChina Englishedition) A vol 28 no 11 pp 1113ndash1120 1985

[2] M-H Yu Y-W Zan J Zhao and M Yoshimine ldquoA UnifiedStrength criterion for rock materialrdquo International Journal ofRock Mechanics andMining Sciences vol 39 no 8 pp 975ndash9892002

[3] C Carranza-Torres ldquoDimensionless graphical representation ofthe exact elasto-plastic solution of a circular tunnel in a Mohr-Coulomb material subject to uniform far-field stressesrdquo RockMechanics amp Rock Engineering vol 36 no 3 pp 237ndash253 2003

[4] C Carranza-Torres ldquoElasto-plastic solution of tunnel problemsusing the generalized formof theHoek-Brown failure criterionrdquoInternational Journal of Rock Mechanics amp Mining Sciences vol41 supplement 1 pp 629ndash639 2004

[5] K-H Park and Y-J Kim ldquoAnalytical solution for a circularopening in an elastic-brittle-plastic rockrdquo International Journalof Rock Mechanics and Mining Sciences vol 43 no 4 pp 616ndash622 2006

[6] C Carranza-Torres and C Fairhurst ldquoThe elasto-plasticresponse of underground excavations in rockmasses that satisfythe Hoek-Brown failure criterionrdquo International Journal of RockMechanics andMining Sciences vol 36 no 6 pp 777ndash809 1999

[7] S K Sharan ldquoElastic-brittle-plastic analysis of circular open-ings in Hoek-Brown mediardquo International Journal of RockMechanics andMining Sciences vol 40 no 6 pp 817ndash824 2003

[8] S K Sharan ldquoAnalytical solutions for stresses and displace-ments around a circular opening in a generalized Hoek-Brown

12 Mathematical Problems in Engineering

rockrdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 1 pp 78ndash85 2008

[9] M B Reed ldquoThe influence of out-of-plane stress on a planestrain problem in rock mechanicsrdquo International Journal forNumerical amp Analytical Methods in Geomechanics vol 12 no2 pp 173ndash181 1988

[10] X-D Pan and E T Brown ldquoInfluence of axial stress anddilatancy on rock tunnel stabilityrdquo Journal of GeotechnicalEngineering vol 122 no 2 pp 139ndash146 1996

[11] S Wang Z Wu M Guo and X Ge ldquoTheoretical solutions ofa circular tunnel with the influence of axial in situ stress inelastic-brittle-plastic rockrdquo Tunnelling and Underground SpaceTechnology vol 30 pp 155ndash168 2012

[12] A-Z Lu G-S Xu F Sun and W-Q Sun ldquoElasto-plasticanalysis of a circular tunnel including the effect of the axial insitu stressrdquo International Journal of Rock Mechanics and MiningSciences vol 47 no 1 pp 50ndash59 2010

[13] X-P Zhou H-Q Yang Y-X Zhang and M-H Yu ldquoTheeffect of the intermediate principal stress on the ultimatebearing capacity of a foundation on rock massesrdquo Computersamp Geotechnics vol 36 no 5 pp 861ndash870 2009

[14] J F Zou and Y Su ldquoTheoretical solutions of a circular tunnelwith the influence of the out-of-plane stress based on the gen-eralized Hoek-Brown failure criterionrdquo International Journal ofGeomechanics (ASCE) vol 16 no 3 2016

[15] J F Zou and Z He ldquoNumerical approach for strain-softeningrock with axial stressrdquo Proceedings of the Institution of CivilEngineersmdashGeotechnical Engineering vol 169 no 3 pp 276ndash290 2016

[16] J-F Zou S-S Li Y XuH-CDan andL-H Zhao ldquoTheoreticalsolutions for a circular opening in an elasticndashbrittlendashplastic rockmass incorporating the out-of-plane stress and seepage forcerdquoKSCE Journal of Civil Engineering vol 20 no 2 pp 687ndash7012016

[17] J F Zou and S Q Zuo ldquoAn approximate solution for the cylin-drical cavity expansion problem under the non-axisymmetricdisplacement boundary condition on hypotenuserdquo Interna-tional Journal of Geotechnical Engineering In press

[18] J F Zou and Z Q Xia ldquoSolutions for displacement and stressin strain-softening surrounding rock incorporating the effectsof hydraulic-mechanical coupling and rockbolts effectivenessrdquoGeotechnical amp Geological Engineering 2016

[19] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[20] S Wang X Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[21] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[22] J Zou and S Li ldquoTheoretical solution for displacement andstress in strain-softening surrounding rock under hydraulic-mechanical couplingrdquo Science China Technological Sciences vol58 no 8 pp 1401ndash1413 2015

[23] X Yang and Z Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[24] X L Yang J S Xu Y X Li andRM Yan ldquoCollapsemechanismof tunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[25] X L Yang and RM Yan ldquoCollapse mechanism for deep tunnelsubjected to seepage force in layered soilsrdquo Geomechanics andEngineering vol 8 no 5 pp 741ndash756 2015

[26] X D Pan and J A Hudson ldquoA simplified three dimensionalHoek-Brown yield criterionrdquo in Proceedings of the ISRM Inter-national Symposium International Society for RockMechanicsMadrid Spain 1988

[27] B Singh R K Goel V K Mehrotra S K Garg and MR Allu ldquoEffect of intermediate principal stress on strengthof anisotropic rock massrdquo Tunnelling and Underground SpaceTechnology vol 13 no 1 pp 71ndash79 1998

[28] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[29] L Zhang andHZhu ldquoThree-dimensional hoek-brown strengthcriterion for rocksrdquo Journal ofGeotechnical andGeoenvironmen-tal Engineering vol 133 no 9 pp 1128ndash1135 2007

[30] X-L Yang and Z-X Long ldquoRoof collapse of shallow tunnelswith limit analysis methodrdquo Journal of Central South Universityvol 22 no 5 article no 2712 pp 1929ndash1936 2015

[31] E Hoek D Wood and S Shah ldquoA modified Hoek-Brownfailure criterion for jointed rock massesrdquo in Proceedings ofthe International Conference on Eurock vol 92 pp 202ndash214September 1992

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

A

O1205761

1205901 minus 1205903

B C

120590cr

120590c

(a) (1205901 minus 1205903)-1205761 curves

+

minus

O1205761

120576

(b) 1205761-120576V curves

+

minus

O1205761

1205763

(c) 1205761-1205763 curves

A

120590cr

O1205761

1205901 minus 1205903

120590c

120573120576e1120576e1

B C

(d) (1205901 minus 1205903)-1205761 simplified curves

H

+

minus

O

120576p1

minus120576v1

1

1205761

120576v

(e) 1205761-120576V simplified curves

+

minus

O

120576e1

1

1205761

1205763

120573120576e1

120576p1

minus120576p3

h

(f) 1205761-1205763 simplified curves

Figure 2 Strain-softening material behavior model

Calculate constitutive model

Actual constitutive model

O

0

i minus 1

i + 1i

middot middot middot

⋱

120576

120590

Figure 3 Strain-softening model of rock mass

Based on the plane strain assumption the axial strainshould satisfy

120576119911= 120576119890

119911+ 120576119901

119911= 0

120576119890

119911=

1

119864

[120590119911minus V (120590

119903+ 120590120579) minus (1 minus 2V) 120590

0]

(11)

Hence the relationship between radial normal stress 120590119903

circumferential normal stress 120590120579 and axial normal stress 120590

119911

is expressed as

120590119911= V (120590

120579+ 120590119903) minus (2V minus 1) 120590

0minus 119864120576119901

119911 (12)

The radial normal stress and the circumferential normalstress at the elastoplastic interface satisfy the failure criterionand can be given by

119865 (1205901 1205903 120574119901) = 1205901minus 1205903minus 119867 (120590

1 1205903 120574119901) (13)

The radius of the first ring is 119903(0)= 119903119901which is at the

interface between the elastic region and the equivalent plasticzone

120588(0)=

119903(0)

119903119901

119903(0)= 119903119901

(14)

The stress and strain at the elastoplastic interface can begiven by

120590119903(0)

120590120579(0)

120590119911(0)

=

120590119877

21205900minus 120590119877

1205900

Mathematical Problems in Engineering 5

120576119903(0)

120576120579(0)

120576119911(0)

=

1 + V119864

119889119906

119889119903

119906

119903

0

(15)

where

120576119903(0)= minus

1

2119866

1205900minus 120590119877

120576120579(0)=

1

2119866

1205900minus 120590119877

(16)

The normalized inner radius can be expressed as

120588(119894)=

119903(119894)

119903119901

(17)

120590119903on both inner and outer boundaries of the plastic zone

are known a priori and 120590119903decreases from 120590

119877to 119901in The

increment of radial normal stress is given by

Δ120590119903=

(119901in minus 120590119877)

119899

(18)

So the radial normal stress at each ring can be representedby

120590119903(119894)= 120590119903(119894minus1)

+ Δ120590119903 (19)

The stress equilibrium equation can be expressed inanother way as follows

119889120590119903

119889120588

+

120590119903minus 120590120579

120588

= 0 (20)

From above equations the following expressions can beobtained

120588(119894)

Δ120590119903(119894)

Δ120588(119894)

= 120590120579(119894)minus 120590119903(119894) (21)

The stress equilibrium differential equation for the 119894thannulus is derived by using (13) and (20) expressed as (21)and (22)

120590119903(119894)minus 120590119903(119894minus1)

120588(119894)minus 120588(119894minus1)

minus

119867 (120590119903(119894) 120574119901)

120588(119894)

= 0 (22)

120588(119894)=

119867 (120590119903(119894) 120574119901) + Δ120590

119903

119867(120590119903(119894) 120574119901)

120588(119894minus1) (23)

The radius of each ring is a known quantity and the radialstress of the 119894th ring can be obtained from (17) So the axialnormal stress 120590

119911can be given by

120590119911(119894)= V (120590

120579(119894)+ 120590119903(119894)) minus (2V minus 1) 120590

0minus 119864120576119901

119911(119894) (24)

Combining (3) (17) and (22) stress at the outer ring canbe obtained and the radial stress at the 119894th ring is calculatedby using linear interpolation

119876119909=

119876119906(119894)minus 119876119906(119894minus1)

120588(119894)minus 120588(119894minus1)

(120588119909minus 120588(119894minus1)) + 119876119906(119894) (25)

where119876119909is stress or displacement of rockmass119876

119906(119894)is stress

or displacement of rock mass at 119894th ring 119876119906(119894minus1)

is stress ordisplacement of rock mass at (119894 minus 1)th ring and 120588

119909is the

distance between center of the tunnel wall and any point ofthe surrounding rock

The compatibility equation can be written in the generalform as follows

119889120576120579

119889119903

+

120576120579minus 120576119903

119903

= 0 (26)

Equation (26) can be transformed into the followingform

119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

120576119890

120579minus 120576119890

119903

120588

(27)

It can be rewritten as119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

1 + ]119864

119867(120590119903 120574119901)

120588

(28)

51 Associated Flow Rule According to the geotechnicalplastic mechanics there is an equipotential surface of plasticpotential at any point 119872 in stress space Its mathematicalexpression is called the plastic potential function and theplastic potential function can be presented by

119892 (120590119894119895 119867119886) = 0 (29)

where119867119886is hardening parameter and 120590

119894119895is principal stress

The plastic strain increment 119889120576119901119894119895can be obtained by

119889120576119901

119894119895=

120597119865

120597120590119894119895

119889120582 (30)

where 119889120582 is multiplication operator 120576119901119894119895is plastic strain and

120590119894119895is principal stressIt is called the associated flow rule when 119865 and 119876 are

completely the same And the physical meaning of yieldfunction 119865 is a judgment criterion whether a point ofsurrounding rock reaches the yield state And the physicalmeaning of plastic potential function 119876 is the relationshipbetween plastic strain increment and loading surface

While the strains of rock and soil mass satisfy theassociated flow rule its plastic potential function is given by

119876 (120590) = minus

119899

3

1198681+

3

120590119888

1198692+

radic3

2

119899radic1198692 (31)

where 119899 is dilation parameter

1198681= 1205901+ 1205902+ 1205903

1198692=

1

6 [(1205901minus 1205902)2

+ (1205902minus 1205903)2

+ (1205903minus 1205901)2

]

(32)

6 Mathematical Problems in Engineering

The differential of three-direction strain is expressed as

119889120576119901

120579=

120597119891

120597120590120579

119889120582 = [(

radic3 (2120590120579minus 120590119903minus 120590119911)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590120579minus 120590119903minus 120590119911)] 119889120582

119889120576119901

119911=

120597119891

120597120590119911

119889120582 = [(

radic3 (2120590119911minus 120590120579minus 120590119903)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590119911minus 120590120579minus 120590119903)] 119889120582

119889120576119901

119903=

120597119891

120597120590119903

119889120582 = [(

radic3 (2120590119903minus 120590120579minus 120590119911)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590119903minus 120590120579minus 120590119911)] 119889120582

(33)

If 1198711 1198712 and 119871

3are defined as follows

[(

radic3 (2120590120579minus 120590119903minus 120590119911)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590120579minus 120590119903minus 120590119911)]

= 1198711

[(

radic3 (2120590119911minus 120590120579minus 120590119903)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590119911minus 120590120579minus 120590119903)]

= 1198712

[(

radic3 (2120590119903minus 120590120579minus 120590119911)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590119903minus 120590120579minus 120590119911)]

= 1198713

(34)

then (33) can be simplified to

119889120576119901

1=

120597119891

1205971205901

119889120582 = 1198711119889120582

119889120576119901

2=

120597119891

1205971205902

119889120582 = 1198712119889120582

119889120576119901

3=

120597119891

1205971205903

119889120582 = 1198713119889120582

(35)

The relationship between radial plastic normal strain 120576119901119903

circumferential plastic normal strain 120576119901120579 and axial plastic

normal strain 120576119901119911can be represented by

120576119901

120579

1198711

=

120576119901

119911

1198712

=

120576119901

119903

1198713

(36)

119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

1 + ]119864

119867 (120590119903 120574119901)

120588

(37)

Combination of (36) and (37) leads to

Δ120576119901

120579= (minus

119889120576119890

120579(119894)

119889120588(119894)

minus

1 + ]119864

119867 (120590119903(119894) 120574119901)

120588(119894)

minus

120576119901

120579(119894minus1)(1 minus 119871

3(119894)1198711(119894))

120588(119894)

)(

1

Δ120588(119894)

+

(1 minus 1198713(119894)1198711(119894))

120588(119894)

)

(38)

The increment of the radial plastic strain at 119894th annuluscan be obtained by (36)

Δ120576119901

119903(119894)=

1198713(119894)

1198711(119894)

Δ120576119901

120579=

1198713(119894)

1198711(119894)

(minus

119889120576119890

120579(119894)

119889120588(119894)

minus

1 + ]119864

119867(120590119903(119894) 120574119901)

120588(119894)

minus

120576119901

120579(119894minus1)(1 minus 119871

3(119894)1198711(119894))

120588(119894)

)

sdot (

1

Δ120588(119894)

+

(1 minus 1198713(119894)1198711(119894))

120588(119894)

)

(39)

The displacement at 119894th annulus can be obtained

119906119903(119894)= 119903(119894)120576120579(119894)= 119903(119894)(120576119890

120579(119894)+ 120576119901

120579(119894))

= 119903(119894)(120576119890

120579(119894)+ 120576119901

120579(1)+

119894

sum

119894=1

Δ120576119901

119903(119894))

(40)

52 Nonassociated Flow Rule The plastic potential functionof the surrounding rock is redefined by Reed [9] by replacingthe internal friction angle with the angle of internal friction

120573120576119901

1+ 120576119901

3= 0 (41)

where 120573 = (1 + sin120595)(1 minus sin120595) and 120595 is dilation angleAccording to the nonassociated flow rule we can know

120576119901

2= 0 (42)

So (12) can be rewritten as

120590119911= V (120590

120579+ 120590119903) minus (2V minus 1) 120590

0 (43)

Therefore the relationship of strains is no longer deter-mined by the failure criterion considering the nonassociatedflow rule but is determined directly by the dilatation coeffi-cient 120573 which is defined by the dilation angle 120595 Because theintermediate principal strain 120576

119911= 0 the relationship between

the major and minor principal strains is determined by 1198711

and 1198713using the associated flow rule Hence in order to

compare the results calculated by the associated flow rule andnonassociated flow rule the results of displacement using thenonassociated flow rule can be replaced by using

minus

1198713

1198711

= 120573 =

1 + sin1205951 minus sin120595

(44)

Mathematical Problems in Engineering 7

6 Validation

To validate the correctness of the proposed approach andcalculation program the results of the proposed approachare compared with the results of Sharan [8] According todifferent types of the surrounding rock Sharan provided thestrength parameters of indoor test for many groups of rocksUnder different surrounding rock conditions Sharan [8]calculated the radius of plastic zone and radial displacementof the surrounding rock The specific parameters are shownin Table 1 and calculation results are shown in Table 2

If the intermediate stress 120590119911is equal to major principal

stress 120590120579or minor principal stress 120590

119903 3D Hoek-Brown failure

criterion can be simplified to two-dimensional Hoek-Brownfailure criterion which is used by Sharan

As shown in Table 2 the results of the paper are in goodaccordance with Sharan [8] when strain-softening of rockmass is not considered Under the condition that parametersof surrounding rock are the same the results show that thegreater the stress is the greater the plastic radius of thesurrounding rock is the larger the internal support pressureis the smaller the radius of the plastic zone of surroundingrock is And we found that the results calculated by elastic-brittle-plastic model are larger than those calculated byelastoplastic model As its strength parameters immediatelyfall after reaching the peak value the elastic-brittle-plasticmodel can be regarded as a special case of strain-softeningmodel

7 Numerical Calculation and Discussion

71 Computational Examples To analyze and compare theproposed approach which considers the intermediate prin-cipal stress and the 3D H-B failure criterion with thosebased on the generalized H-B failure criterion the followingparameters obtained from Sharanrsquos experimental results [8]are adopted 120590

119888= 30MPa 119904 = 00039 119886 = 055 119864 = 55GPa

119864119903= 55GPa 119898 = 17 119904

119903= 00019 119886

119903= 06 119898

119903= 16

V = 025 119901in = 5Mpa and 1199030= 5m The calculation results

are shown in Table 3Through the comparison results in Table 3 we can

find that the plastic zone radius and radial displacementcalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterionThus itproves that the results may overestimate the plastic zone radiiand radial displacement of the surrounding rock withoutconsidering intermediate principal stress

As shown in Table 3 the radial displacement of sur-rounding rock calculated by associated flow rule is muchgreater than those calculated by nonassociated flow ruleWith different softening parameters the plastic zone radiicalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterion Forexample elastoplastic model strain-softening model (120574119901 =0006) and elastic-brittle-plastic model would be reducedby 1457 1498 and 1600 respectively The results ofradial displacement using nonassociated flow rule are shownas follows elastoplastic model strain-softening model andelastic-brittle-plastic model (120595 = 0

∘ and 120595 = 20∘) would

120590120590

0

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

02

04

06

08

10

12

14

16

12 14 16 18 20 22 24 2610rr0

Figure 4 Stresses of surrounding rock mass (elastoplastic model)

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16120590p

0

Figure 5 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 004)

be reduced by 2085 3467 2178 3629 2190 and3653 respectively From above results it can be obtainedthat the calculated deviations between 3D H-B failure crite-rion and generalized H-B failure criterion increase with theincreasing of dilation angle

At the same time the radial displacement differencesbetween three-dimensional associated flow rule and nonas-sociated flow rule considering dilatancy effect are comparedThe differences between two calculation approaches areanalyzed for predicting the displacement of plastic zone Theresults of stress and displacement are shown in Figures 4ndash11As shown in Figures 4ndash7 120590

119903 120590120579 120590119911 and 119906

119903are calculation

results based on 3DH-B failure criterion120590101584011990312059010158401205791205901015840119911 and 1199061015840

119903are

calculation results based on generalizedH-B failure criterion

8 Mathematical Problems in Engineering

Table 1 Calculation parameters [8]

Quality of rock mass V 1199030(m) 120590

119888(MPa) 119904 119904

119903119886 119886

119903119898119887

119898119887119903

119864 (GPa) 119864119903(GPa)

Average 025 5 80 00039 0 051 053 201 034 9 5Very poor 03 5 25 00039 00019 055 06 17 085 57 57

Table 2 Comparisons between the results of this paper and Sharan without considering the strain-softening and intermediate principalstress

Parameter groups Elastoplastic model 1205900

119901in 1199031199011199030(Sharan) 119903

1199011199030(this paper)

1 e-p 40 0 177 177141 e-p 80 0 259 25881 e-p 80 1 231 23121 e-p 80 5 191 19071 e-b-p 40 3 253 25261 e-b-p 40 5 203 20301 e-b-p 80 18 233 23332 e-b-p 15 0 390 39012 e-b-p 15 3 151 15102 e-b-p 30 0 784 78362 e-b-p 30 5 233 2327

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

00

02

04

06

08

10

12

14

16

120590p

0

Figure 6 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 0006)

Figures 4ndash7 show the stress comparisons between theideal elastoplastic strain-softening and elastic-brittle-plasticmodel based on 3D and generalized Hoek-Brown failurecriteria respectively When considering the influence of theintermediate principal stress on different failure criterion thevalue of plastic radius calculated by the 3D H-B criterionis smaller than those calculated by the generalized H-Bcriterion

Under the condition that the geotechnical parameters arethe same the influence of dilatation coefficient on generalizedH-B failure criterion is larger than 3D H-B failure criterionParticularly as the softening coefficient is the same the

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16

120590p

0

Figure 7 Stresses of surrounding rock mass (elastic-brittle-plasticmodel)

residual radius is smaller than the plastic radius of thesurrounding rock if it is calculated by generalized H-B failurecriterion For example if 120574119901 = 004 the plastic radiuscalculated by generalized failure criterion is 14660 and theresidual radius does not exist Based on 3D H-B failurecriterion the plastic radius is 12841 and the residual radiusis 10372 Hence if the softening coefficient is the same theplastic radius calculated by different failure criteria is quitedifferent

The radial displacement comparisons of surroundingrock under different flow rules softening parameters anddilation angles are shown in Figures 8ndash11 Summary results

Mathematical Problems in Engineering 9

Table 3 Plastic radius critical stresses and displacements of surrounding rock

Softening parameters 1199031199011199030

1199031199041199030

Associated flowrule

1199061199031199030()

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 75

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D H-B

e-p 14591 1 06793 07245 07906 08534120574119901= 004 14660 1 06879 07373 08114 08839

120574119901= 0006 14945 12401 07239 07819 08676 09500e-b-p 15091 15091 07358 07947 08819 09658

3D H-B

e-p 12681 1 07877 05621 05815 06088 06337120574119901= 004 12841 10372 09599 05669 05884 06194 06484

120574119901= 0006 12980 12198 10685 05944 06219 06610 06970e-b-p 13010 13010 10873 06036 06314 06710 07074

Table 4 Calculation parameters

Rock mass 119904 119904119903

a 119886119903

119898119887

119898119887119903

1 00039 00039 055 06 17 172 00039 00019 055 055 17 173 00039 00039 055 055 17 164 00039 00039 055 055 17 085

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

03

04

05

06

07

08

09

urr

0(

)

105 110 115 120 125 130100rr0

Figure 8 Displacements of surrounding rock mass (elastoplasticmodel)

can be seen from Table 3 The plastic zone radii calculated byassociated flow rule and nonassociated flow rule are almostconsistent However the radial displacement of rock massincreases 1064 (e-p model) 1184 (120574119901 = 004) 1391 (120574119901 =0006) and 1413 (e-b-p model) if it adopted associatedflow rule Obviously with the continuous deteriorations ofstrength parameters the result differences between associatedflow rule and nonassociated flow rule will increase gradually

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

urr

0(

)

Figure 9 Displacements of surrounding rock mass (strain-softening model 120574119901 = 004)

72 Parameter Analysis In order to analyze the effect ofdifferent softening parameters on plastic radius and radialdisplacement of surrounding rock when considering 3D orgeneralized H-B failure criterion the following parametersare cited 120590

119888= 30MPa 119864 = 55GPa V = 025 119901in = 5Mpa

and 1199030= 5mThe detailed data are shown in Table 4

Elastic-brittle-plastic model (dilation angle is equal to20∘) was selected to study the stresses and displacementsof surrounding rock as shown in Tables 4 and 5 Com-pared with the softening process of three parameters thesoftening process with only one parameter was conductedThe following conclusions can be drawn as follows thedifferences of plastic radius are 188 (2D) and 137 (3D)when only parameter ldquo119886rdquo is softening the differences of radialdisplacement are 628 (2D) and 577 (3D) when onlyparameter ldquo119886rdquo is softening the differences of plastic radius

10 Mathematical Problems in Engineering

Table 5 Parameter analyses

H-B Softening parameters Groups 1199031199011199030

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D e-p 1 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14771 07008 08276 08996

3D 120574119901= 0006 12803 05773 06331 06629

2D e-b-p 14807 07053 08327 090513D e-b-p 12832 05809 06367 066662D e-p 2 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14602 06796 07919 08553

3D 120574119901= 0006 12688 05629 06102 06354

2D e-b-p 14604 06800 07923 085573D e-b-p 12690 05632 06105 063572D e-p 3 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14727 06953 08195 08901

3D 120574119901= 0006 12775 05738 06282 06574

2D e-b-p 14777 07016 08273 089883D e-b-p 12814 05787 06337 066322D e-p 4 14591 06783 07897 085253D e-p 12531 05877 06446 067512D 120574

119901= 0006 16778 09724 13501 15894

3D 120574119901= 0006 14509 08046 10370 11740

2D e-b-p 17339 10539 14379 167813D e-b-p 14882 08584 10797 12092

03

04

05

06

07

08

09

10

11

urr

0(

)

105 110 115 120 125 130100rr0

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (associated flow rule 120595 = 75∘)ur4 (associated flow rule 120595 = 15∘)ur5 (associated flow rule 120595 = 20∘)

Figure 10 Displacements of surrounding rock mass (strain-softening model 120574119901 = 0006)

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3

ur4

ur5

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

11

urr

0(

)

(associated flow rule 120595 = 75∘)(associated flow rule 120595 = 15∘)(associated flow rule 120595 = 20∘)

Figure 11 Displacements of surrounding rock mass (elastic-brittle-plastic model)

Mathematical Problems in Engineering 11

are 323 (2D) and 246 (3D) when only parameter ldquo119904rdquo issoftening the differences of radial displacement are 1140(2D) and 1014 (3D) when only parameter ldquo119904rdquo is softeningthe differences of plastic radius are 208 (2D) and 151 (3D)when only parameter ldquo119898rdquo was softening the differences ofradial displacement are 693 (2D) and 625 (3D) whenonly parameter ldquo119898rdquo is softening

The above results show that the softening of surroundingrock is commonly decided by many parameters The order ofrock mass parameters affecting the stress and displacementsis shown as follows 119904 gt 119898 gt 119886

8 Conclusions

(1) Incorporating the 3D Hoek-Brown failure criterionassociated flow rule nonassociated flow rule and thestrain-softening model numerical solutions of stressdisplacement and plastic radius were proposed

(2) Comparison results show that the plastic radius andradial displacement calculated by 3D Hoek-Brownfailure criterion are smaller than those based ongeneralized Hoek-Brown

(3) Radial displacement calculated by nonassociated flowrule is smaller than those considering associated flowrule The influences of dilatancy parameter on theresults based on generalized H-B failure criterion arelarger than those based on 3D H-B failure criterion

Notations

119886 Parameter of HB failure criterion for peakstrength [mdash]

119886119903 Parameter of HB failure criterion for

residual strength [mdash]1199030 Radius of the tunnel opening [L]119863 HB constants for the rock mass [mdash]119864 Youngrsquos modulus of the rock mass [FLminus2]119898119887 Parameter of HB failure criterion for peakstrength [mdash]

119898119887119903 Parameter of HB failure criterion forresidual strength [mdash]

119901in Critical internal pressure [FLminus2]119903 Radial distance from the center of opening

[L]119903119901 Plastic radius [L]

119903119904 Residual radius [L]119904 Parameter of the HB failure criterion for

peak strength [mdash]119904119903 Parameter of the HB failure criterion for

residual strength [mdash]119906119903 Radial displacement [L]

1205900 Initial in situ stress [FLminus2]

120590119888 Uniaxial compressive strength of the rock

[FLminus2]120590119903 Radial normal stress [FLminus2]

120590120579 Circumferential normal stress [FLminus2]

120590119911 Axial normal stress along the axis of thetunnel [FLminus2]

1205901 Major principal stresses [FLminus2]

1205902 Middle principal stresses [FLminus2]

1205903 Minor principal stresses [FLminus2]

120590119877 Radial normal stress at the elastoplasticinterface [FLminus2]

120576119903 Radial normal strain [mdash]120576120579 Circumferential normal strain [mdash]120576119911 Axial normal strain [mdash]119901 Plastic parts of normal strain and stress [mdash]119890 Elastic parts of normal strain and stress [mdash]120595 Dilation angle [mdash]120573 Dilation coefficient [mdash]120574119901 Softening coefficient of the surrounding

rock [mdash]V Poissonrsquos ratio of the rock mass [mdash]119865 Yield function [mdash]119876 Plastic potential function [mdash]

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors are grateful to the 973 Program (2013CB036004)National Natural Science Foundation of China (no51208523)

References

[1] M H Yu N L He and L Y Song ldquoTwin shear stress theoryand its generalizationrdquo Scientia Sinica (Sciences inChina Englishedition) A vol 28 no 11 pp 1113ndash1120 1985

[2] M-H Yu Y-W Zan J Zhao and M Yoshimine ldquoA UnifiedStrength criterion for rock materialrdquo International Journal ofRock Mechanics andMining Sciences vol 39 no 8 pp 975ndash9892002

[3] C Carranza-Torres ldquoDimensionless graphical representation ofthe exact elasto-plastic solution of a circular tunnel in a Mohr-Coulomb material subject to uniform far-field stressesrdquo RockMechanics amp Rock Engineering vol 36 no 3 pp 237ndash253 2003

[4] C Carranza-Torres ldquoElasto-plastic solution of tunnel problemsusing the generalized formof theHoek-Brown failure criterionrdquoInternational Journal of Rock Mechanics amp Mining Sciences vol41 supplement 1 pp 629ndash639 2004

[5] K-H Park and Y-J Kim ldquoAnalytical solution for a circularopening in an elastic-brittle-plastic rockrdquo International Journalof Rock Mechanics and Mining Sciences vol 43 no 4 pp 616ndash622 2006

[6] C Carranza-Torres and C Fairhurst ldquoThe elasto-plasticresponse of underground excavations in rockmasses that satisfythe Hoek-Brown failure criterionrdquo International Journal of RockMechanics andMining Sciences vol 36 no 6 pp 777ndash809 1999

[7] S K Sharan ldquoElastic-brittle-plastic analysis of circular open-ings in Hoek-Brown mediardquo International Journal of RockMechanics andMining Sciences vol 40 no 6 pp 817ndash824 2003

[8] S K Sharan ldquoAnalytical solutions for stresses and displace-ments around a circular opening in a generalized Hoek-Brown

12 Mathematical Problems in Engineering

rockrdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 1 pp 78ndash85 2008

[9] M B Reed ldquoThe influence of out-of-plane stress on a planestrain problem in rock mechanicsrdquo International Journal forNumerical amp Analytical Methods in Geomechanics vol 12 no2 pp 173ndash181 1988

[10] X-D Pan and E T Brown ldquoInfluence of axial stress anddilatancy on rock tunnel stabilityrdquo Journal of GeotechnicalEngineering vol 122 no 2 pp 139ndash146 1996

[11] S Wang Z Wu M Guo and X Ge ldquoTheoretical solutions ofa circular tunnel with the influence of axial in situ stress inelastic-brittle-plastic rockrdquo Tunnelling and Underground SpaceTechnology vol 30 pp 155ndash168 2012

[12] A-Z Lu G-S Xu F Sun and W-Q Sun ldquoElasto-plasticanalysis of a circular tunnel including the effect of the axial insitu stressrdquo International Journal of Rock Mechanics and MiningSciences vol 47 no 1 pp 50ndash59 2010

[13] X-P Zhou H-Q Yang Y-X Zhang and M-H Yu ldquoTheeffect of the intermediate principal stress on the ultimatebearing capacity of a foundation on rock massesrdquo Computersamp Geotechnics vol 36 no 5 pp 861ndash870 2009

[14] J F Zou and Y Su ldquoTheoretical solutions of a circular tunnelwith the influence of the out-of-plane stress based on the gen-eralized Hoek-Brown failure criterionrdquo International Journal ofGeomechanics (ASCE) vol 16 no 3 2016

[15] J F Zou and Z He ldquoNumerical approach for strain-softeningrock with axial stressrdquo Proceedings of the Institution of CivilEngineersmdashGeotechnical Engineering vol 169 no 3 pp 276ndash290 2016

[16] J-F Zou S-S Li Y XuH-CDan andL-H Zhao ldquoTheoreticalsolutions for a circular opening in an elasticndashbrittlendashplastic rockmass incorporating the out-of-plane stress and seepage forcerdquoKSCE Journal of Civil Engineering vol 20 no 2 pp 687ndash7012016

[17] J F Zou and S Q Zuo ldquoAn approximate solution for the cylin-drical cavity expansion problem under the non-axisymmetricdisplacement boundary condition on hypotenuserdquo Interna-tional Journal of Geotechnical Engineering In press

[18] J F Zou and Z Q Xia ldquoSolutions for displacement and stressin strain-softening surrounding rock incorporating the effectsof hydraulic-mechanical coupling and rockbolts effectivenessrdquoGeotechnical amp Geological Engineering 2016

[19] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[20] S Wang X Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[21] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[22] J Zou and S Li ldquoTheoretical solution for displacement andstress in strain-softening surrounding rock under hydraulic-mechanical couplingrdquo Science China Technological Sciences vol58 no 8 pp 1401ndash1413 2015

[23] X Yang and Z Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[24] X L Yang J S Xu Y X Li andRM Yan ldquoCollapsemechanismof tunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[25] X L Yang and RM Yan ldquoCollapse mechanism for deep tunnelsubjected to seepage force in layered soilsrdquo Geomechanics andEngineering vol 8 no 5 pp 741ndash756 2015

[26] X D Pan and J A Hudson ldquoA simplified three dimensionalHoek-Brown yield criterionrdquo in Proceedings of the ISRM Inter-national Symposium International Society for RockMechanicsMadrid Spain 1988

[27] B Singh R K Goel V K Mehrotra S K Garg and MR Allu ldquoEffect of intermediate principal stress on strengthof anisotropic rock massrdquo Tunnelling and Underground SpaceTechnology vol 13 no 1 pp 71ndash79 1998

[28] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[29] L Zhang andHZhu ldquoThree-dimensional hoek-brown strengthcriterion for rocksrdquo Journal ofGeotechnical andGeoenvironmen-tal Engineering vol 133 no 9 pp 1128ndash1135 2007

[30] X-L Yang and Z-X Long ldquoRoof collapse of shallow tunnelswith limit analysis methodrdquo Journal of Central South Universityvol 22 no 5 article no 2712 pp 1929ndash1936 2015

[31] E Hoek D Wood and S Shah ldquoA modified Hoek-Brownfailure criterion for jointed rock massesrdquo in Proceedings ofthe International Conference on Eurock vol 92 pp 202ndash214September 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

120576119903(0)

120576120579(0)

120576119911(0)

=

1 + V119864

119889119906

119889119903

119906

119903

0

(15)

where

120576119903(0)= minus

1

2119866

1205900minus 120590119877

120576120579(0)=

1

2119866

1205900minus 120590119877

(16)

The normalized inner radius can be expressed as

120588(119894)=

119903(119894)

119903119901

(17)

120590119903on both inner and outer boundaries of the plastic zone

are known a priori and 120590119903decreases from 120590

119877to 119901in The

increment of radial normal stress is given by

Δ120590119903=

(119901in minus 120590119877)

119899

(18)

So the radial normal stress at each ring can be representedby

120590119903(119894)= 120590119903(119894minus1)

+ Δ120590119903 (19)

The stress equilibrium equation can be expressed inanother way as follows

119889120590119903

119889120588

+

120590119903minus 120590120579

120588

= 0 (20)

From above equations the following expressions can beobtained

120588(119894)

Δ120590119903(119894)

Δ120588(119894)

= 120590120579(119894)minus 120590119903(119894) (21)

The stress equilibrium differential equation for the 119894thannulus is derived by using (13) and (20) expressed as (21)and (22)

120590119903(119894)minus 120590119903(119894minus1)

120588(119894)minus 120588(119894minus1)

minus

119867 (120590119903(119894) 120574119901)

120588(119894)

= 0 (22)

120588(119894)=

119867 (120590119903(119894) 120574119901) + Δ120590

119903

119867(120590119903(119894) 120574119901)

120588(119894minus1) (23)

The radius of each ring is a known quantity and the radialstress of the 119894th ring can be obtained from (17) So the axialnormal stress 120590

119911can be given by

120590119911(119894)= V (120590

120579(119894)+ 120590119903(119894)) minus (2V minus 1) 120590

0minus 119864120576119901

119911(119894) (24)

Combining (3) (17) and (22) stress at the outer ring canbe obtained and the radial stress at the 119894th ring is calculatedby using linear interpolation

119876119909=

119876119906(119894)minus 119876119906(119894minus1)

120588(119894)minus 120588(119894minus1)

(120588119909minus 120588(119894minus1)) + 119876119906(119894) (25)

where119876119909is stress or displacement of rockmass119876

119906(119894)is stress

or displacement of rock mass at 119894th ring 119876119906(119894minus1)

is stress ordisplacement of rock mass at (119894 minus 1)th ring and 120588

119909is the

distance between center of the tunnel wall and any point ofthe surrounding rock

The compatibility equation can be written in the generalform as follows

119889120576120579

119889119903

+

120576120579minus 120576119903

119903

= 0 (26)

Equation (26) can be transformed into the followingform

119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

120576119890

120579minus 120576119890

119903

120588

(27)

It can be rewritten as119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

1 + ]119864

119867(120590119903 120574119901)

120588

(28)

51 Associated Flow Rule According to the geotechnicalplastic mechanics there is an equipotential surface of plasticpotential at any point 119872 in stress space Its mathematicalexpression is called the plastic potential function and theplastic potential function can be presented by

119892 (120590119894119895 119867119886) = 0 (29)

where119867119886is hardening parameter and 120590

119894119895is principal stress

The plastic strain increment 119889120576119901119894119895can be obtained by

119889120576119901

119894119895=

120597119865

120597120590119894119895

119889120582 (30)

where 119889120582 is multiplication operator 120576119901119894119895is plastic strain and

120590119894119895is principal stressIt is called the associated flow rule when 119865 and 119876 are

completely the same And the physical meaning of yieldfunction 119865 is a judgment criterion whether a point ofsurrounding rock reaches the yield state And the physicalmeaning of plastic potential function 119876 is the relationshipbetween plastic strain increment and loading surface

While the strains of rock and soil mass satisfy theassociated flow rule its plastic potential function is given by

119876 (120590) = minus

119899

3

1198681+

3

120590119888

1198692+

radic3

2

119899radic1198692 (31)

where 119899 is dilation parameter

1198681= 1205901+ 1205902+ 1205903

1198692=

1

6 [(1205901minus 1205902)2

+ (1205902minus 1205903)2

+ (1205903minus 1205901)2

]

(32)

6 Mathematical Problems in Engineering

The differential of three-direction strain is expressed as

119889120576119901

120579=

120597119891

120597120590120579

119889120582 = [(

radic3 (2120590120579minus 120590119903minus 120590119911)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590120579minus 120590119903minus 120590119911)] 119889120582

119889120576119901

119911=

120597119891

120597120590119911

119889120582 = [(

radic3 (2120590119911minus 120590120579minus 120590119903)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590119911minus 120590120579minus 120590119903)] 119889120582

119889120576119901

119903=

120597119891

120597120590119903

119889120582 = [(

radic3 (2120590119903minus 120590120579minus 120590119911)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590119903minus 120590120579minus 120590119911)] 119889120582

(33)

If 1198711 1198712 and 119871

3are defined as follows

[(

radic3 (2120590120579minus 120590119903minus 120590119911)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590120579minus 120590119903minus 120590119911)]

= 1198711

[(

radic3 (2120590119911minus 120590120579minus 120590119903)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590119911minus 120590120579minus 120590119903)]

= 1198712

[(

radic3 (2120590119903minus 120590120579minus 120590119911)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590119903minus 120590120579minus 120590119911)]

= 1198713

(34)

then (33) can be simplified to

119889120576119901

1=

120597119891

1205971205901

119889120582 = 1198711119889120582

119889120576119901

2=

120597119891

1205971205902

119889120582 = 1198712119889120582

119889120576119901

3=

120597119891

1205971205903

119889120582 = 1198713119889120582

(35)

The relationship between radial plastic normal strain 120576119901119903

circumferential plastic normal strain 120576119901120579 and axial plastic

normal strain 120576119901119911can be represented by

120576119901

120579

1198711

=

120576119901

119911

1198712

=

120576119901

119903

1198713

(36)

119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

1 + ]119864

119867 (120590119903 120574119901)

120588

(37)

Combination of (36) and (37) leads to

Δ120576119901

120579= (minus

119889120576119890

120579(119894)

119889120588(119894)

minus

1 + ]119864

119867 (120590119903(119894) 120574119901)

120588(119894)

minus

120576119901

120579(119894minus1)(1 minus 119871

3(119894)1198711(119894))

120588(119894)

)(

1

Δ120588(119894)

+

(1 minus 1198713(119894)1198711(119894))

120588(119894)

)

(38)

The increment of the radial plastic strain at 119894th annuluscan be obtained by (36)

Δ120576119901

119903(119894)=

1198713(119894)

1198711(119894)

Δ120576119901

120579=

1198713(119894)

1198711(119894)

(minus

119889120576119890

120579(119894)

119889120588(119894)

minus

1 + ]119864

119867(120590119903(119894) 120574119901)

120588(119894)

minus

120576119901

120579(119894minus1)(1 minus 119871

3(119894)1198711(119894))

120588(119894)

)

sdot (

1

Δ120588(119894)

+

(1 minus 1198713(119894)1198711(119894))

120588(119894)

)

(39)

The displacement at 119894th annulus can be obtained

119906119903(119894)= 119903(119894)120576120579(119894)= 119903(119894)(120576119890

120579(119894)+ 120576119901

120579(119894))

= 119903(119894)(120576119890

120579(119894)+ 120576119901

120579(1)+

119894

sum

119894=1

Δ120576119901

119903(119894))

(40)

52 Nonassociated Flow Rule The plastic potential functionof the surrounding rock is redefined by Reed [9] by replacingthe internal friction angle with the angle of internal friction

120573120576119901

1+ 120576119901

3= 0 (41)

where 120573 = (1 + sin120595)(1 minus sin120595) and 120595 is dilation angleAccording to the nonassociated flow rule we can know

120576119901

2= 0 (42)

So (12) can be rewritten as

120590119911= V (120590

120579+ 120590119903) minus (2V minus 1) 120590

0 (43)

Therefore the relationship of strains is no longer deter-mined by the failure criterion considering the nonassociatedflow rule but is determined directly by the dilatation coeffi-cient 120573 which is defined by the dilation angle 120595 Because theintermediate principal strain 120576

119911= 0 the relationship between

the major and minor principal strains is determined by 1198711

and 1198713using the associated flow rule Hence in order to

compare the results calculated by the associated flow rule andnonassociated flow rule the results of displacement using thenonassociated flow rule can be replaced by using

minus

1198713

1198711

= 120573 =

1 + sin1205951 minus sin120595

(44)

Mathematical Problems in Engineering 7

6 Validation

To validate the correctness of the proposed approach andcalculation program the results of the proposed approachare compared with the results of Sharan [8] According todifferent types of the surrounding rock Sharan provided thestrength parameters of indoor test for many groups of rocksUnder different surrounding rock conditions Sharan [8]calculated the radius of plastic zone and radial displacementof the surrounding rock The specific parameters are shownin Table 1 and calculation results are shown in Table 2

If the intermediate stress 120590119911is equal to major principal

stress 120590120579or minor principal stress 120590

119903 3D Hoek-Brown failure

criterion can be simplified to two-dimensional Hoek-Brownfailure criterion which is used by Sharan

As shown in Table 2 the results of the paper are in goodaccordance with Sharan [8] when strain-softening of rockmass is not considered Under the condition that parametersof surrounding rock are the same the results show that thegreater the stress is the greater the plastic radius of thesurrounding rock is the larger the internal support pressureis the smaller the radius of the plastic zone of surroundingrock is And we found that the results calculated by elastic-brittle-plastic model are larger than those calculated byelastoplastic model As its strength parameters immediatelyfall after reaching the peak value the elastic-brittle-plasticmodel can be regarded as a special case of strain-softeningmodel

7 Numerical Calculation and Discussion

71 Computational Examples To analyze and compare theproposed approach which considers the intermediate prin-cipal stress and the 3D H-B failure criterion with thosebased on the generalized H-B failure criterion the followingparameters obtained from Sharanrsquos experimental results [8]are adopted 120590

119888= 30MPa 119904 = 00039 119886 = 055 119864 = 55GPa

119864119903= 55GPa 119898 = 17 119904

119903= 00019 119886

119903= 06 119898

119903= 16

V = 025 119901in = 5Mpa and 1199030= 5m The calculation results

are shown in Table 3Through the comparison results in Table 3 we can

find that the plastic zone radius and radial displacementcalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterionThus itproves that the results may overestimate the plastic zone radiiand radial displacement of the surrounding rock withoutconsidering intermediate principal stress

As shown in Table 3 the radial displacement of sur-rounding rock calculated by associated flow rule is muchgreater than those calculated by nonassociated flow ruleWith different softening parameters the plastic zone radiicalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterion Forexample elastoplastic model strain-softening model (120574119901 =0006) and elastic-brittle-plastic model would be reducedby 1457 1498 and 1600 respectively The results ofradial displacement using nonassociated flow rule are shownas follows elastoplastic model strain-softening model andelastic-brittle-plastic model (120595 = 0

∘ and 120595 = 20∘) would

120590120590

0

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

02

04

06

08

10

12

14

16

12 14 16 18 20 22 24 2610rr0

Figure 4 Stresses of surrounding rock mass (elastoplastic model)

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16120590p

0

Figure 5 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 004)

be reduced by 2085 3467 2178 3629 2190 and3653 respectively From above results it can be obtainedthat the calculated deviations between 3D H-B failure crite-rion and generalized H-B failure criterion increase with theincreasing of dilation angle

At the same time the radial displacement differencesbetween three-dimensional associated flow rule and nonas-sociated flow rule considering dilatancy effect are comparedThe differences between two calculation approaches areanalyzed for predicting the displacement of plastic zone Theresults of stress and displacement are shown in Figures 4ndash11As shown in Figures 4ndash7 120590

119903 120590120579 120590119911 and 119906

119903are calculation

results based on 3DH-B failure criterion120590101584011990312059010158401205791205901015840119911 and 1199061015840

119903are

calculation results based on generalizedH-B failure criterion

8 Mathematical Problems in Engineering

Table 1 Calculation parameters [8]

Quality of rock mass V 1199030(m) 120590

119888(MPa) 119904 119904

119903119886 119886

119903119898119887

119898119887119903

119864 (GPa) 119864119903(GPa)

Average 025 5 80 00039 0 051 053 201 034 9 5Very poor 03 5 25 00039 00019 055 06 17 085 57 57

Table 2 Comparisons between the results of this paper and Sharan without considering the strain-softening and intermediate principalstress

Parameter groups Elastoplastic model 1205900

119901in 1199031199011199030(Sharan) 119903

1199011199030(this paper)

1 e-p 40 0 177 177141 e-p 80 0 259 25881 e-p 80 1 231 23121 e-p 80 5 191 19071 e-b-p 40 3 253 25261 e-b-p 40 5 203 20301 e-b-p 80 18 233 23332 e-b-p 15 0 390 39012 e-b-p 15 3 151 15102 e-b-p 30 0 784 78362 e-b-p 30 5 233 2327

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

00

02

04

06

08

10

12

14

16

120590p

0

Figure 6 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 0006)

Figures 4ndash7 show the stress comparisons between theideal elastoplastic strain-softening and elastic-brittle-plasticmodel based on 3D and generalized Hoek-Brown failurecriteria respectively When considering the influence of theintermediate principal stress on different failure criterion thevalue of plastic radius calculated by the 3D H-B criterionis smaller than those calculated by the generalized H-Bcriterion

Under the condition that the geotechnical parameters arethe same the influence of dilatation coefficient on generalizedH-B failure criterion is larger than 3D H-B failure criterionParticularly as the softening coefficient is the same the

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16

120590p

0

Figure 7 Stresses of surrounding rock mass (elastic-brittle-plasticmodel)

residual radius is smaller than the plastic radius of thesurrounding rock if it is calculated by generalized H-B failurecriterion For example if 120574119901 = 004 the plastic radiuscalculated by generalized failure criterion is 14660 and theresidual radius does not exist Based on 3D H-B failurecriterion the plastic radius is 12841 and the residual radiusis 10372 Hence if the softening coefficient is the same theplastic radius calculated by different failure criteria is quitedifferent

The radial displacement comparisons of surroundingrock under different flow rules softening parameters anddilation angles are shown in Figures 8ndash11 Summary results

Mathematical Problems in Engineering 9

Table 3 Plastic radius critical stresses and displacements of surrounding rock

Softening parameters 1199031199011199030

1199031199041199030

Associated flowrule

1199061199031199030()

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 75

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D H-B

e-p 14591 1 06793 07245 07906 08534120574119901= 004 14660 1 06879 07373 08114 08839

120574119901= 0006 14945 12401 07239 07819 08676 09500e-b-p 15091 15091 07358 07947 08819 09658

3D H-B

e-p 12681 1 07877 05621 05815 06088 06337120574119901= 004 12841 10372 09599 05669 05884 06194 06484

120574119901= 0006 12980 12198 10685 05944 06219 06610 06970e-b-p 13010 13010 10873 06036 06314 06710 07074

Table 4 Calculation parameters

Rock mass 119904 119904119903

a 119886119903

119898119887

119898119887119903

1 00039 00039 055 06 17 172 00039 00019 055 055 17 173 00039 00039 055 055 17 164 00039 00039 055 055 17 085

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

03

04

05

06

07

08

09

urr

0(

)

105 110 115 120 125 130100rr0

Figure 8 Displacements of surrounding rock mass (elastoplasticmodel)

can be seen from Table 3 The plastic zone radii calculated byassociated flow rule and nonassociated flow rule are almostconsistent However the radial displacement of rock massincreases 1064 (e-p model) 1184 (120574119901 = 004) 1391 (120574119901 =0006) and 1413 (e-b-p model) if it adopted associatedflow rule Obviously with the continuous deteriorations ofstrength parameters the result differences between associatedflow rule and nonassociated flow rule will increase gradually

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

urr

0(

)

Figure 9 Displacements of surrounding rock mass (strain-softening model 120574119901 = 004)

72 Parameter Analysis In order to analyze the effect ofdifferent softening parameters on plastic radius and radialdisplacement of surrounding rock when considering 3D orgeneralized H-B failure criterion the following parametersare cited 120590

119888= 30MPa 119864 = 55GPa V = 025 119901in = 5Mpa

and 1199030= 5mThe detailed data are shown in Table 4

Elastic-brittle-plastic model (dilation angle is equal to20∘) was selected to study the stresses and displacementsof surrounding rock as shown in Tables 4 and 5 Com-pared with the softening process of three parameters thesoftening process with only one parameter was conductedThe following conclusions can be drawn as follows thedifferences of plastic radius are 188 (2D) and 137 (3D)when only parameter ldquo119886rdquo is softening the differences of radialdisplacement are 628 (2D) and 577 (3D) when onlyparameter ldquo119886rdquo is softening the differences of plastic radius

10 Mathematical Problems in Engineering

Table 5 Parameter analyses

H-B Softening parameters Groups 1199031199011199030

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D e-p 1 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14771 07008 08276 08996

3D 120574119901= 0006 12803 05773 06331 06629

2D e-b-p 14807 07053 08327 090513D e-b-p 12832 05809 06367 066662D e-p 2 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14602 06796 07919 08553

3D 120574119901= 0006 12688 05629 06102 06354

2D e-b-p 14604 06800 07923 085573D e-b-p 12690 05632 06105 063572D e-p 3 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14727 06953 08195 08901

3D 120574119901= 0006 12775 05738 06282 06574

2D e-b-p 14777 07016 08273 089883D e-b-p 12814 05787 06337 066322D e-p 4 14591 06783 07897 085253D e-p 12531 05877 06446 067512D 120574

119901= 0006 16778 09724 13501 15894

3D 120574119901= 0006 14509 08046 10370 11740

2D e-b-p 17339 10539 14379 167813D e-b-p 14882 08584 10797 12092

03

04

05

06

07

08

09

10

11

urr

0(

)

105 110 115 120 125 130100rr0

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (associated flow rule 120595 = 75∘)ur4 (associated flow rule 120595 = 15∘)ur5 (associated flow rule 120595 = 20∘)

Figure 10 Displacements of surrounding rock mass (strain-softening model 120574119901 = 0006)

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3

ur4

ur5

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

11

urr

0(

)

(associated flow rule 120595 = 75∘)(associated flow rule 120595 = 15∘)(associated flow rule 120595 = 20∘)

Figure 11 Displacements of surrounding rock mass (elastic-brittle-plastic model)

Mathematical Problems in Engineering 11

are 323 (2D) and 246 (3D) when only parameter ldquo119904rdquo issoftening the differences of radial displacement are 1140(2D) and 1014 (3D) when only parameter ldquo119904rdquo is softeningthe differences of plastic radius are 208 (2D) and 151 (3D)when only parameter ldquo119898rdquo was softening the differences ofradial displacement are 693 (2D) and 625 (3D) whenonly parameter ldquo119898rdquo is softening

The above results show that the softening of surroundingrock is commonly decided by many parameters The order ofrock mass parameters affecting the stress and displacementsis shown as follows 119904 gt 119898 gt 119886

8 Conclusions

(1) Incorporating the 3D Hoek-Brown failure criterionassociated flow rule nonassociated flow rule and thestrain-softening model numerical solutions of stressdisplacement and plastic radius were proposed

(2) Comparison results show that the plastic radius andradial displacement calculated by 3D Hoek-Brownfailure criterion are smaller than those based ongeneralized Hoek-Brown

(3) Radial displacement calculated by nonassociated flowrule is smaller than those considering associated flowrule The influences of dilatancy parameter on theresults based on generalized H-B failure criterion arelarger than those based on 3D H-B failure criterion

Notations

119886 Parameter of HB failure criterion for peakstrength [mdash]

119886119903 Parameter of HB failure criterion for

residual strength [mdash]1199030 Radius of the tunnel opening [L]119863 HB constants for the rock mass [mdash]119864 Youngrsquos modulus of the rock mass [FLminus2]119898119887 Parameter of HB failure criterion for peakstrength [mdash]

119898119887119903 Parameter of HB failure criterion forresidual strength [mdash]

119901in Critical internal pressure [FLminus2]119903 Radial distance from the center of opening

[L]119903119901 Plastic radius [L]

119903119904 Residual radius [L]119904 Parameter of the HB failure criterion for

peak strength [mdash]119904119903 Parameter of the HB failure criterion for

residual strength [mdash]119906119903 Radial displacement [L]

1205900 Initial in situ stress [FLminus2]

120590119888 Uniaxial compressive strength of the rock

[FLminus2]120590119903 Radial normal stress [FLminus2]

120590120579 Circumferential normal stress [FLminus2]

120590119911 Axial normal stress along the axis of thetunnel [FLminus2]

1205901 Major principal stresses [FLminus2]

1205902 Middle principal stresses [FLminus2]

1205903 Minor principal stresses [FLminus2]

120590119877 Radial normal stress at the elastoplasticinterface [FLminus2]

120576119903 Radial normal strain [mdash]120576120579 Circumferential normal strain [mdash]120576119911 Axial normal strain [mdash]119901 Plastic parts of normal strain and stress [mdash]119890 Elastic parts of normal strain and stress [mdash]120595 Dilation angle [mdash]120573 Dilation coefficient [mdash]120574119901 Softening coefficient of the surrounding

rock [mdash]V Poissonrsquos ratio of the rock mass [mdash]119865 Yield function [mdash]119876 Plastic potential function [mdash]

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors are grateful to the 973 Program (2013CB036004)National Natural Science Foundation of China (no51208523)

References

[1] M H Yu N L He and L Y Song ldquoTwin shear stress theoryand its generalizationrdquo Scientia Sinica (Sciences inChina Englishedition) A vol 28 no 11 pp 1113ndash1120 1985

[2] M-H Yu Y-W Zan J Zhao and M Yoshimine ldquoA UnifiedStrength criterion for rock materialrdquo International Journal ofRock Mechanics andMining Sciences vol 39 no 8 pp 975ndash9892002

[3] C Carranza-Torres ldquoDimensionless graphical representation ofthe exact elasto-plastic solution of a circular tunnel in a Mohr-Coulomb material subject to uniform far-field stressesrdquo RockMechanics amp Rock Engineering vol 36 no 3 pp 237ndash253 2003

[4] C Carranza-Torres ldquoElasto-plastic solution of tunnel problemsusing the generalized formof theHoek-Brown failure criterionrdquoInternational Journal of Rock Mechanics amp Mining Sciences vol41 supplement 1 pp 629ndash639 2004

[5] K-H Park and Y-J Kim ldquoAnalytical solution for a circularopening in an elastic-brittle-plastic rockrdquo International Journalof Rock Mechanics and Mining Sciences vol 43 no 4 pp 616ndash622 2006

[6] C Carranza-Torres and C Fairhurst ldquoThe elasto-plasticresponse of underground excavations in rockmasses that satisfythe Hoek-Brown failure criterionrdquo International Journal of RockMechanics andMining Sciences vol 36 no 6 pp 777ndash809 1999

[7] S K Sharan ldquoElastic-brittle-plastic analysis of circular open-ings in Hoek-Brown mediardquo International Journal of RockMechanics andMining Sciences vol 40 no 6 pp 817ndash824 2003

[8] S K Sharan ldquoAnalytical solutions for stresses and displace-ments around a circular opening in a generalized Hoek-Brown

12 Mathematical Problems in Engineering

rockrdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 1 pp 78ndash85 2008

[9] M B Reed ldquoThe influence of out-of-plane stress on a planestrain problem in rock mechanicsrdquo International Journal forNumerical amp Analytical Methods in Geomechanics vol 12 no2 pp 173ndash181 1988

[10] X-D Pan and E T Brown ldquoInfluence of axial stress anddilatancy on rock tunnel stabilityrdquo Journal of GeotechnicalEngineering vol 122 no 2 pp 139ndash146 1996

[11] S Wang Z Wu M Guo and X Ge ldquoTheoretical solutions ofa circular tunnel with the influence of axial in situ stress inelastic-brittle-plastic rockrdquo Tunnelling and Underground SpaceTechnology vol 30 pp 155ndash168 2012

[12] A-Z Lu G-S Xu F Sun and W-Q Sun ldquoElasto-plasticanalysis of a circular tunnel including the effect of the axial insitu stressrdquo International Journal of Rock Mechanics and MiningSciences vol 47 no 1 pp 50ndash59 2010

[13] X-P Zhou H-Q Yang Y-X Zhang and M-H Yu ldquoTheeffect of the intermediate principal stress on the ultimatebearing capacity of a foundation on rock massesrdquo Computersamp Geotechnics vol 36 no 5 pp 861ndash870 2009

[14] J F Zou and Y Su ldquoTheoretical solutions of a circular tunnelwith the influence of the out-of-plane stress based on the gen-eralized Hoek-Brown failure criterionrdquo International Journal ofGeomechanics (ASCE) vol 16 no 3 2016

[15] J F Zou and Z He ldquoNumerical approach for strain-softeningrock with axial stressrdquo Proceedings of the Institution of CivilEngineersmdashGeotechnical Engineering vol 169 no 3 pp 276ndash290 2016

[16] J-F Zou S-S Li Y XuH-CDan andL-H Zhao ldquoTheoreticalsolutions for a circular opening in an elasticndashbrittlendashplastic rockmass incorporating the out-of-plane stress and seepage forcerdquoKSCE Journal of Civil Engineering vol 20 no 2 pp 687ndash7012016

[17] J F Zou and S Q Zuo ldquoAn approximate solution for the cylin-drical cavity expansion problem under the non-axisymmetricdisplacement boundary condition on hypotenuserdquo Interna-tional Journal of Geotechnical Engineering In press

[18] J F Zou and Z Q Xia ldquoSolutions for displacement and stressin strain-softening surrounding rock incorporating the effectsof hydraulic-mechanical coupling and rockbolts effectivenessrdquoGeotechnical amp Geological Engineering 2016

[19] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[20] S Wang X Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[21] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[22] J Zou and S Li ldquoTheoretical solution for displacement andstress in strain-softening surrounding rock under hydraulic-mechanical couplingrdquo Science China Technological Sciences vol58 no 8 pp 1401ndash1413 2015

[23] X Yang and Z Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[24] X L Yang J S Xu Y X Li andRM Yan ldquoCollapsemechanismof tunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[25] X L Yang and RM Yan ldquoCollapse mechanism for deep tunnelsubjected to seepage force in layered soilsrdquo Geomechanics andEngineering vol 8 no 5 pp 741ndash756 2015

[26] X D Pan and J A Hudson ldquoA simplified three dimensionalHoek-Brown yield criterionrdquo in Proceedings of the ISRM Inter-national Symposium International Society for RockMechanicsMadrid Spain 1988

[27] B Singh R K Goel V K Mehrotra S K Garg and MR Allu ldquoEffect of intermediate principal stress on strengthof anisotropic rock massrdquo Tunnelling and Underground SpaceTechnology vol 13 no 1 pp 71ndash79 1998

[28] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[29] L Zhang andHZhu ldquoThree-dimensional hoek-brown strengthcriterion for rocksrdquo Journal ofGeotechnical andGeoenvironmen-tal Engineering vol 133 no 9 pp 1128ndash1135 2007

[30] X-L Yang and Z-X Long ldquoRoof collapse of shallow tunnelswith limit analysis methodrdquo Journal of Central South Universityvol 22 no 5 article no 2712 pp 1929ndash1936 2015

[31] E Hoek D Wood and S Shah ldquoA modified Hoek-Brownfailure criterion for jointed rock massesrdquo in Proceedings ofthe International Conference on Eurock vol 92 pp 202ndash214September 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

The differential of three-direction strain is expressed as

119889120576119901

120579=

120597119891

120597120590120579

119889120582 = [(

radic3 (2120590120579minus 120590119903minus 120590119911)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590120579minus 120590119903minus 120590119911)] 119889120582

119889120576119901

119911=

120597119891

120597120590119911

119889120582 = [(

radic3 (2120590119911minus 120590120579minus 120590119903)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590119911minus 120590120579minus 120590119903)] 119889120582

119889120576119901

119903=

120597119891

120597120590119903

119889120582 = [(

radic3 (2120590119903minus 120590120579minus 120590119911)

12radic1198692

minus

1

3

) 119899

+

1

120590119888

(2120590119903minus 120590120579minus 120590119911)] 119889120582

(33)

If 1198711 1198712 and 119871

3are defined as follows

[(

radic3 (2120590120579minus 120590119903minus 120590119911)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590120579minus 120590119903minus 120590119911)]

= 1198711

[(

radic3 (2120590119911minus 120590120579minus 120590119903)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590119911minus 120590120579minus 120590119903)]

= 1198712

[(

radic3 (2120590119903minus 120590120579minus 120590119911)

12radic1198692

minus

1

3

) 119899 +

1

120590119888

(2120590119903minus 120590120579minus 120590119911)]

= 1198713

(34)

then (33) can be simplified to

119889120576119901

1=

120597119891

1205971205901

119889120582 = 1198711119889120582

119889120576119901

2=

120597119891

1205971205902

119889120582 = 1198712119889120582

119889120576119901

3=

120597119891

1205971205903

119889120582 = 1198713119889120582

(35)

The relationship between radial plastic normal strain 120576119901119903

circumferential plastic normal strain 120576119901120579 and axial plastic

normal strain 120576119901119911can be represented by

120576119901

120579

1198711

=

120576119901

119911

1198712

=

120576119901

119903

1198713

(36)

119889120576119901

120579

119889120588

+

120576119901

120579minus 120576119901

119903

120588

= minus

119889120576119890

120579

119889120588

minus

1 + ]119864

119867 (120590119903 120574119901)

120588

(37)

Combination of (36) and (37) leads to

Δ120576119901

120579= (minus

119889120576119890

120579(119894)

119889120588(119894)

minus

1 + ]119864

119867 (120590119903(119894) 120574119901)

120588(119894)

minus

120576119901

120579(119894minus1)(1 minus 119871

3(119894)1198711(119894))

120588(119894)

)(

1

Δ120588(119894)

+

(1 minus 1198713(119894)1198711(119894))

120588(119894)

)

(38)

The increment of the radial plastic strain at 119894th annuluscan be obtained by (36)

Δ120576119901

119903(119894)=

1198713(119894)

1198711(119894)

Δ120576119901

120579=

1198713(119894)

1198711(119894)

(minus

119889120576119890

120579(119894)

119889120588(119894)

minus

1 + ]119864

119867(120590119903(119894) 120574119901)

120588(119894)

minus

120576119901

120579(119894minus1)(1 minus 119871

3(119894)1198711(119894))

120588(119894)

)

sdot (

1

Δ120588(119894)

+

(1 minus 1198713(119894)1198711(119894))

120588(119894)

)

(39)

The displacement at 119894th annulus can be obtained

119906119903(119894)= 119903(119894)120576120579(119894)= 119903(119894)(120576119890

120579(119894)+ 120576119901

120579(119894))

= 119903(119894)(120576119890

120579(119894)+ 120576119901

120579(1)+

119894

sum

119894=1

Δ120576119901

119903(119894))

(40)

52 Nonassociated Flow Rule The plastic potential functionof the surrounding rock is redefined by Reed [9] by replacingthe internal friction angle with the angle of internal friction

120573120576119901

1+ 120576119901

3= 0 (41)

where 120573 = (1 + sin120595)(1 minus sin120595) and 120595 is dilation angleAccording to the nonassociated flow rule we can know

120576119901

2= 0 (42)

So (12) can be rewritten as

120590119911= V (120590

120579+ 120590119903) minus (2V minus 1) 120590

0 (43)

Therefore the relationship of strains is no longer deter-mined by the failure criterion considering the nonassociatedflow rule but is determined directly by the dilatation coeffi-cient 120573 which is defined by the dilation angle 120595 Because theintermediate principal strain 120576

119911= 0 the relationship between

the major and minor principal strains is determined by 1198711

and 1198713using the associated flow rule Hence in order to

compare the results calculated by the associated flow rule andnonassociated flow rule the results of displacement using thenonassociated flow rule can be replaced by using

minus

1198713

1198711

= 120573 =

1 + sin1205951 minus sin120595

(44)

Mathematical Problems in Engineering 7

6 Validation

To validate the correctness of the proposed approach andcalculation program the results of the proposed approachare compared with the results of Sharan [8] According todifferent types of the surrounding rock Sharan provided thestrength parameters of indoor test for many groups of rocksUnder different surrounding rock conditions Sharan [8]calculated the radius of plastic zone and radial displacementof the surrounding rock The specific parameters are shownin Table 1 and calculation results are shown in Table 2

If the intermediate stress 120590119911is equal to major principal

stress 120590120579or minor principal stress 120590

119903 3D Hoek-Brown failure

criterion can be simplified to two-dimensional Hoek-Brownfailure criterion which is used by Sharan

As shown in Table 2 the results of the paper are in goodaccordance with Sharan [8] when strain-softening of rockmass is not considered Under the condition that parametersof surrounding rock are the same the results show that thegreater the stress is the greater the plastic radius of thesurrounding rock is the larger the internal support pressureis the smaller the radius of the plastic zone of surroundingrock is And we found that the results calculated by elastic-brittle-plastic model are larger than those calculated byelastoplastic model As its strength parameters immediatelyfall after reaching the peak value the elastic-brittle-plasticmodel can be regarded as a special case of strain-softeningmodel

7 Numerical Calculation and Discussion

71 Computational Examples To analyze and compare theproposed approach which considers the intermediate prin-cipal stress and the 3D H-B failure criterion with thosebased on the generalized H-B failure criterion the followingparameters obtained from Sharanrsquos experimental results [8]are adopted 120590

119888= 30MPa 119904 = 00039 119886 = 055 119864 = 55GPa

119864119903= 55GPa 119898 = 17 119904

119903= 00019 119886

119903= 06 119898

119903= 16

V = 025 119901in = 5Mpa and 1199030= 5m The calculation results

are shown in Table 3Through the comparison results in Table 3 we can

find that the plastic zone radius and radial displacementcalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterionThus itproves that the results may overestimate the plastic zone radiiand radial displacement of the surrounding rock withoutconsidering intermediate principal stress

As shown in Table 3 the radial displacement of sur-rounding rock calculated by associated flow rule is muchgreater than those calculated by nonassociated flow ruleWith different softening parameters the plastic zone radiicalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterion Forexample elastoplastic model strain-softening model (120574119901 =0006) and elastic-brittle-plastic model would be reducedby 1457 1498 and 1600 respectively The results ofradial displacement using nonassociated flow rule are shownas follows elastoplastic model strain-softening model andelastic-brittle-plastic model (120595 = 0

∘ and 120595 = 20∘) would

120590120590

0

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

02

04

06

08

10

12

14

16

12 14 16 18 20 22 24 2610rr0

Figure 4 Stresses of surrounding rock mass (elastoplastic model)

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16120590p

0

Figure 5 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 004)

be reduced by 2085 3467 2178 3629 2190 and3653 respectively From above results it can be obtainedthat the calculated deviations between 3D H-B failure crite-rion and generalized H-B failure criterion increase with theincreasing of dilation angle

At the same time the radial displacement differencesbetween three-dimensional associated flow rule and nonas-sociated flow rule considering dilatancy effect are comparedThe differences between two calculation approaches areanalyzed for predicting the displacement of plastic zone Theresults of stress and displacement are shown in Figures 4ndash11As shown in Figures 4ndash7 120590

119903 120590120579 120590119911 and 119906

119903are calculation

results based on 3DH-B failure criterion120590101584011990312059010158401205791205901015840119911 and 1199061015840

119903are

calculation results based on generalizedH-B failure criterion

8 Mathematical Problems in Engineering

Table 1 Calculation parameters [8]

Quality of rock mass V 1199030(m) 120590

119888(MPa) 119904 119904

119903119886 119886

119903119898119887

119898119887119903

119864 (GPa) 119864119903(GPa)

Average 025 5 80 00039 0 051 053 201 034 9 5Very poor 03 5 25 00039 00019 055 06 17 085 57 57

Table 2 Comparisons between the results of this paper and Sharan without considering the strain-softening and intermediate principalstress

Parameter groups Elastoplastic model 1205900

119901in 1199031199011199030(Sharan) 119903

1199011199030(this paper)

1 e-p 40 0 177 177141 e-p 80 0 259 25881 e-p 80 1 231 23121 e-p 80 5 191 19071 e-b-p 40 3 253 25261 e-b-p 40 5 203 20301 e-b-p 80 18 233 23332 e-b-p 15 0 390 39012 e-b-p 15 3 151 15102 e-b-p 30 0 784 78362 e-b-p 30 5 233 2327

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

00

02

04

06

08

10

12

14

16

120590p

0

Figure 6 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 0006)

Figures 4ndash7 show the stress comparisons between theideal elastoplastic strain-softening and elastic-brittle-plasticmodel based on 3D and generalized Hoek-Brown failurecriteria respectively When considering the influence of theintermediate principal stress on different failure criterion thevalue of plastic radius calculated by the 3D H-B criterionis smaller than those calculated by the generalized H-Bcriterion

Under the condition that the geotechnical parameters arethe same the influence of dilatation coefficient on generalizedH-B failure criterion is larger than 3D H-B failure criterionParticularly as the softening coefficient is the same the

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16

120590p

0

Figure 7 Stresses of surrounding rock mass (elastic-brittle-plasticmodel)

residual radius is smaller than the plastic radius of thesurrounding rock if it is calculated by generalized H-B failurecriterion For example if 120574119901 = 004 the plastic radiuscalculated by generalized failure criterion is 14660 and theresidual radius does not exist Based on 3D H-B failurecriterion the plastic radius is 12841 and the residual radiusis 10372 Hence if the softening coefficient is the same theplastic radius calculated by different failure criteria is quitedifferent

The radial displacement comparisons of surroundingrock under different flow rules softening parameters anddilation angles are shown in Figures 8ndash11 Summary results

Mathematical Problems in Engineering 9

Table 3 Plastic radius critical stresses and displacements of surrounding rock

Softening parameters 1199031199011199030

1199031199041199030

Associated flowrule

1199061199031199030()

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 75

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D H-B

e-p 14591 1 06793 07245 07906 08534120574119901= 004 14660 1 06879 07373 08114 08839

120574119901= 0006 14945 12401 07239 07819 08676 09500e-b-p 15091 15091 07358 07947 08819 09658

3D H-B

e-p 12681 1 07877 05621 05815 06088 06337120574119901= 004 12841 10372 09599 05669 05884 06194 06484

120574119901= 0006 12980 12198 10685 05944 06219 06610 06970e-b-p 13010 13010 10873 06036 06314 06710 07074

Table 4 Calculation parameters

Rock mass 119904 119904119903

a 119886119903

119898119887

119898119887119903

1 00039 00039 055 06 17 172 00039 00019 055 055 17 173 00039 00039 055 055 17 164 00039 00039 055 055 17 085

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

03

04

05

06

07

08

09

urr

0(

)

105 110 115 120 125 130100rr0

Figure 8 Displacements of surrounding rock mass (elastoplasticmodel)

can be seen from Table 3 The plastic zone radii calculated byassociated flow rule and nonassociated flow rule are almostconsistent However the radial displacement of rock massincreases 1064 (e-p model) 1184 (120574119901 = 004) 1391 (120574119901 =0006) and 1413 (e-b-p model) if it adopted associatedflow rule Obviously with the continuous deteriorations ofstrength parameters the result differences between associatedflow rule and nonassociated flow rule will increase gradually

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

urr

0(

)

Figure 9 Displacements of surrounding rock mass (strain-softening model 120574119901 = 004)

72 Parameter Analysis In order to analyze the effect ofdifferent softening parameters on plastic radius and radialdisplacement of surrounding rock when considering 3D orgeneralized H-B failure criterion the following parametersare cited 120590

119888= 30MPa 119864 = 55GPa V = 025 119901in = 5Mpa

and 1199030= 5mThe detailed data are shown in Table 4

Elastic-brittle-plastic model (dilation angle is equal to20∘) was selected to study the stresses and displacementsof surrounding rock as shown in Tables 4 and 5 Com-pared with the softening process of three parameters thesoftening process with only one parameter was conductedThe following conclusions can be drawn as follows thedifferences of plastic radius are 188 (2D) and 137 (3D)when only parameter ldquo119886rdquo is softening the differences of radialdisplacement are 628 (2D) and 577 (3D) when onlyparameter ldquo119886rdquo is softening the differences of plastic radius

10 Mathematical Problems in Engineering

Table 5 Parameter analyses

H-B Softening parameters Groups 1199031199011199030

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D e-p 1 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14771 07008 08276 08996

3D 120574119901= 0006 12803 05773 06331 06629

2D e-b-p 14807 07053 08327 090513D e-b-p 12832 05809 06367 066662D e-p 2 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14602 06796 07919 08553

3D 120574119901= 0006 12688 05629 06102 06354

2D e-b-p 14604 06800 07923 085573D e-b-p 12690 05632 06105 063572D e-p 3 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14727 06953 08195 08901

3D 120574119901= 0006 12775 05738 06282 06574

2D e-b-p 14777 07016 08273 089883D e-b-p 12814 05787 06337 066322D e-p 4 14591 06783 07897 085253D e-p 12531 05877 06446 067512D 120574

119901= 0006 16778 09724 13501 15894

3D 120574119901= 0006 14509 08046 10370 11740

2D e-b-p 17339 10539 14379 167813D e-b-p 14882 08584 10797 12092

03

04

05

06

07

08

09

10

11

urr

0(

)

105 110 115 120 125 130100rr0

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (associated flow rule 120595 = 75∘)ur4 (associated flow rule 120595 = 15∘)ur5 (associated flow rule 120595 = 20∘)

Figure 10 Displacements of surrounding rock mass (strain-softening model 120574119901 = 0006)

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3

ur4

ur5

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

11

urr

0(

)

(associated flow rule 120595 = 75∘)(associated flow rule 120595 = 15∘)(associated flow rule 120595 = 20∘)

Figure 11 Displacements of surrounding rock mass (elastic-brittle-plastic model)

Mathematical Problems in Engineering 11

are 323 (2D) and 246 (3D) when only parameter ldquo119904rdquo issoftening the differences of radial displacement are 1140(2D) and 1014 (3D) when only parameter ldquo119904rdquo is softeningthe differences of plastic radius are 208 (2D) and 151 (3D)when only parameter ldquo119898rdquo was softening the differences ofradial displacement are 693 (2D) and 625 (3D) whenonly parameter ldquo119898rdquo is softening

The above results show that the softening of surroundingrock is commonly decided by many parameters The order ofrock mass parameters affecting the stress and displacementsis shown as follows 119904 gt 119898 gt 119886

8 Conclusions

(1) Incorporating the 3D Hoek-Brown failure criterionassociated flow rule nonassociated flow rule and thestrain-softening model numerical solutions of stressdisplacement and plastic radius were proposed

(2) Comparison results show that the plastic radius andradial displacement calculated by 3D Hoek-Brownfailure criterion are smaller than those based ongeneralized Hoek-Brown

(3) Radial displacement calculated by nonassociated flowrule is smaller than those considering associated flowrule The influences of dilatancy parameter on theresults based on generalized H-B failure criterion arelarger than those based on 3D H-B failure criterion

Notations

119886 Parameter of HB failure criterion for peakstrength [mdash]

119886119903 Parameter of HB failure criterion for

residual strength [mdash]1199030 Radius of the tunnel opening [L]119863 HB constants for the rock mass [mdash]119864 Youngrsquos modulus of the rock mass [FLminus2]119898119887 Parameter of HB failure criterion for peakstrength [mdash]

119898119887119903 Parameter of HB failure criterion forresidual strength [mdash]

119901in Critical internal pressure [FLminus2]119903 Radial distance from the center of opening

[L]119903119901 Plastic radius [L]

119903119904 Residual radius [L]119904 Parameter of the HB failure criterion for

peak strength [mdash]119904119903 Parameter of the HB failure criterion for

residual strength [mdash]119906119903 Radial displacement [L]

1205900 Initial in situ stress [FLminus2]

120590119888 Uniaxial compressive strength of the rock

[FLminus2]120590119903 Radial normal stress [FLminus2]

120590120579 Circumferential normal stress [FLminus2]

120590119911 Axial normal stress along the axis of thetunnel [FLminus2]

1205901 Major principal stresses [FLminus2]

1205902 Middle principal stresses [FLminus2]

1205903 Minor principal stresses [FLminus2]

120590119877 Radial normal stress at the elastoplasticinterface [FLminus2]

120576119903 Radial normal strain [mdash]120576120579 Circumferential normal strain [mdash]120576119911 Axial normal strain [mdash]119901 Plastic parts of normal strain and stress [mdash]119890 Elastic parts of normal strain and stress [mdash]120595 Dilation angle [mdash]120573 Dilation coefficient [mdash]120574119901 Softening coefficient of the surrounding

rock [mdash]V Poissonrsquos ratio of the rock mass [mdash]119865 Yield function [mdash]119876 Plastic potential function [mdash]

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors are grateful to the 973 Program (2013CB036004)National Natural Science Foundation of China (no51208523)

References

[1] M H Yu N L He and L Y Song ldquoTwin shear stress theoryand its generalizationrdquo Scientia Sinica (Sciences inChina Englishedition) A vol 28 no 11 pp 1113ndash1120 1985

[2] M-H Yu Y-W Zan J Zhao and M Yoshimine ldquoA UnifiedStrength criterion for rock materialrdquo International Journal ofRock Mechanics andMining Sciences vol 39 no 8 pp 975ndash9892002

[3] C Carranza-Torres ldquoDimensionless graphical representation ofthe exact elasto-plastic solution of a circular tunnel in a Mohr-Coulomb material subject to uniform far-field stressesrdquo RockMechanics amp Rock Engineering vol 36 no 3 pp 237ndash253 2003

[4] C Carranza-Torres ldquoElasto-plastic solution of tunnel problemsusing the generalized formof theHoek-Brown failure criterionrdquoInternational Journal of Rock Mechanics amp Mining Sciences vol41 supplement 1 pp 629ndash639 2004

[5] K-H Park and Y-J Kim ldquoAnalytical solution for a circularopening in an elastic-brittle-plastic rockrdquo International Journalof Rock Mechanics and Mining Sciences vol 43 no 4 pp 616ndash622 2006

[6] C Carranza-Torres and C Fairhurst ldquoThe elasto-plasticresponse of underground excavations in rockmasses that satisfythe Hoek-Brown failure criterionrdquo International Journal of RockMechanics andMining Sciences vol 36 no 6 pp 777ndash809 1999

[7] S K Sharan ldquoElastic-brittle-plastic analysis of circular open-ings in Hoek-Brown mediardquo International Journal of RockMechanics andMining Sciences vol 40 no 6 pp 817ndash824 2003

[8] S K Sharan ldquoAnalytical solutions for stresses and displace-ments around a circular opening in a generalized Hoek-Brown

12 Mathematical Problems in Engineering

rockrdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 1 pp 78ndash85 2008

[9] M B Reed ldquoThe influence of out-of-plane stress on a planestrain problem in rock mechanicsrdquo International Journal forNumerical amp Analytical Methods in Geomechanics vol 12 no2 pp 173ndash181 1988

[10] X-D Pan and E T Brown ldquoInfluence of axial stress anddilatancy on rock tunnel stabilityrdquo Journal of GeotechnicalEngineering vol 122 no 2 pp 139ndash146 1996

[11] S Wang Z Wu M Guo and X Ge ldquoTheoretical solutions ofa circular tunnel with the influence of axial in situ stress inelastic-brittle-plastic rockrdquo Tunnelling and Underground SpaceTechnology vol 30 pp 155ndash168 2012

[12] A-Z Lu G-S Xu F Sun and W-Q Sun ldquoElasto-plasticanalysis of a circular tunnel including the effect of the axial insitu stressrdquo International Journal of Rock Mechanics and MiningSciences vol 47 no 1 pp 50ndash59 2010

[13] X-P Zhou H-Q Yang Y-X Zhang and M-H Yu ldquoTheeffect of the intermediate principal stress on the ultimatebearing capacity of a foundation on rock massesrdquo Computersamp Geotechnics vol 36 no 5 pp 861ndash870 2009

[14] J F Zou and Y Su ldquoTheoretical solutions of a circular tunnelwith the influence of the out-of-plane stress based on the gen-eralized Hoek-Brown failure criterionrdquo International Journal ofGeomechanics (ASCE) vol 16 no 3 2016

[15] J F Zou and Z He ldquoNumerical approach for strain-softeningrock with axial stressrdquo Proceedings of the Institution of CivilEngineersmdashGeotechnical Engineering vol 169 no 3 pp 276ndash290 2016

[16] J-F Zou S-S Li Y XuH-CDan andL-H Zhao ldquoTheoreticalsolutions for a circular opening in an elasticndashbrittlendashplastic rockmass incorporating the out-of-plane stress and seepage forcerdquoKSCE Journal of Civil Engineering vol 20 no 2 pp 687ndash7012016

[17] J F Zou and S Q Zuo ldquoAn approximate solution for the cylin-drical cavity expansion problem under the non-axisymmetricdisplacement boundary condition on hypotenuserdquo Interna-tional Journal of Geotechnical Engineering In press

[18] J F Zou and Z Q Xia ldquoSolutions for displacement and stressin strain-softening surrounding rock incorporating the effectsof hydraulic-mechanical coupling and rockbolts effectivenessrdquoGeotechnical amp Geological Engineering 2016

[19] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[20] S Wang X Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[21] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[22] J Zou and S Li ldquoTheoretical solution for displacement andstress in strain-softening surrounding rock under hydraulic-mechanical couplingrdquo Science China Technological Sciences vol58 no 8 pp 1401ndash1413 2015

[23] X Yang and Z Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[24] X L Yang J S Xu Y X Li andRM Yan ldquoCollapsemechanismof tunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[25] X L Yang and RM Yan ldquoCollapse mechanism for deep tunnelsubjected to seepage force in layered soilsrdquo Geomechanics andEngineering vol 8 no 5 pp 741ndash756 2015

[26] X D Pan and J A Hudson ldquoA simplified three dimensionalHoek-Brown yield criterionrdquo in Proceedings of the ISRM Inter-national Symposium International Society for RockMechanicsMadrid Spain 1988

[27] B Singh R K Goel V K Mehrotra S K Garg and MR Allu ldquoEffect of intermediate principal stress on strengthof anisotropic rock massrdquo Tunnelling and Underground SpaceTechnology vol 13 no 1 pp 71ndash79 1998

[28] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[29] L Zhang andHZhu ldquoThree-dimensional hoek-brown strengthcriterion for rocksrdquo Journal ofGeotechnical andGeoenvironmen-tal Engineering vol 133 no 9 pp 1128ndash1135 2007

[30] X-L Yang and Z-X Long ldquoRoof collapse of shallow tunnelswith limit analysis methodrdquo Journal of Central South Universityvol 22 no 5 article no 2712 pp 1929ndash1936 2015

[31] E Hoek D Wood and S Shah ldquoA modified Hoek-Brownfailure criterion for jointed rock massesrdquo in Proceedings ofthe International Conference on Eurock vol 92 pp 202ndash214September 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

6 Validation

To validate the correctness of the proposed approach andcalculation program the results of the proposed approachare compared with the results of Sharan [8] According todifferent types of the surrounding rock Sharan provided thestrength parameters of indoor test for many groups of rocksUnder different surrounding rock conditions Sharan [8]calculated the radius of plastic zone and radial displacementof the surrounding rock The specific parameters are shownin Table 1 and calculation results are shown in Table 2

If the intermediate stress 120590119911is equal to major principal

stress 120590120579or minor principal stress 120590

119903 3D Hoek-Brown failure

criterion can be simplified to two-dimensional Hoek-Brownfailure criterion which is used by Sharan

As shown in Table 2 the results of the paper are in goodaccordance with Sharan [8] when strain-softening of rockmass is not considered Under the condition that parametersof surrounding rock are the same the results show that thegreater the stress is the greater the plastic radius of thesurrounding rock is the larger the internal support pressureis the smaller the radius of the plastic zone of surroundingrock is And we found that the results calculated by elastic-brittle-plastic model are larger than those calculated byelastoplastic model As its strength parameters immediatelyfall after reaching the peak value the elastic-brittle-plasticmodel can be regarded as a special case of strain-softeningmodel

7 Numerical Calculation and Discussion

71 Computational Examples To analyze and compare theproposed approach which considers the intermediate prin-cipal stress and the 3D H-B failure criterion with thosebased on the generalized H-B failure criterion the followingparameters obtained from Sharanrsquos experimental results [8]are adopted 120590

119888= 30MPa 119904 = 00039 119886 = 055 119864 = 55GPa

119864119903= 55GPa 119898 = 17 119904

119903= 00019 119886

119903= 06 119898

119903= 16

V = 025 119901in = 5Mpa and 1199030= 5m The calculation results

are shown in Table 3Through the comparison results in Table 3 we can

find that the plastic zone radius and radial displacementcalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterionThus itproves that the results may overestimate the plastic zone radiiand radial displacement of the surrounding rock withoutconsidering intermediate principal stress

As shown in Table 3 the radial displacement of sur-rounding rock calculated by associated flow rule is muchgreater than those calculated by nonassociated flow ruleWith different softening parameters the plastic zone radiicalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterion Forexample elastoplastic model strain-softening model (120574119901 =0006) and elastic-brittle-plastic model would be reducedby 1457 1498 and 1600 respectively The results ofradial displacement using nonassociated flow rule are shownas follows elastoplastic model strain-softening model andelastic-brittle-plastic model (120595 = 0

∘ and 120595 = 20∘) would

120590120590

0

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

02

04

06

08

10

12

14

16

12 14 16 18 20 22 24 2610rr0

Figure 4 Stresses of surrounding rock mass (elastoplastic model)

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16120590p

0

Figure 5 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 004)

be reduced by 2085 3467 2178 3629 2190 and3653 respectively From above results it can be obtainedthat the calculated deviations between 3D H-B failure crite-rion and generalized H-B failure criterion increase with theincreasing of dilation angle

At the same time the radial displacement differencesbetween three-dimensional associated flow rule and nonas-sociated flow rule considering dilatancy effect are comparedThe differences between two calculation approaches areanalyzed for predicting the displacement of plastic zone Theresults of stress and displacement are shown in Figures 4ndash11As shown in Figures 4ndash7 120590

119903 120590120579 120590119911 and 119906

119903are calculation

results based on 3DH-B failure criterion120590101584011990312059010158401205791205901015840119911 and 1199061015840

119903are

calculation results based on generalizedH-B failure criterion

8 Mathematical Problems in Engineering

Table 1 Calculation parameters [8]

Quality of rock mass V 1199030(m) 120590

119888(MPa) 119904 119904

119903119886 119886

119903119898119887

119898119887119903

119864 (GPa) 119864119903(GPa)

Average 025 5 80 00039 0 051 053 201 034 9 5Very poor 03 5 25 00039 00019 055 06 17 085 57 57

Table 2 Comparisons between the results of this paper and Sharan without considering the strain-softening and intermediate principalstress

Parameter groups Elastoplastic model 1205900

119901in 1199031199011199030(Sharan) 119903

1199011199030(this paper)

1 e-p 40 0 177 177141 e-p 80 0 259 25881 e-p 80 1 231 23121 e-p 80 5 191 19071 e-b-p 40 3 253 25261 e-b-p 40 5 203 20301 e-b-p 80 18 233 23332 e-b-p 15 0 390 39012 e-b-p 15 3 151 15102 e-b-p 30 0 784 78362 e-b-p 30 5 233 2327

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

00

02

04

06

08

10

12

14

16

120590p

0

Figure 6 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 0006)

Figures 4ndash7 show the stress comparisons between theideal elastoplastic strain-softening and elastic-brittle-plasticmodel based on 3D and generalized Hoek-Brown failurecriteria respectively When considering the influence of theintermediate principal stress on different failure criterion thevalue of plastic radius calculated by the 3D H-B criterionis smaller than those calculated by the generalized H-Bcriterion

Under the condition that the geotechnical parameters arethe same the influence of dilatation coefficient on generalizedH-B failure criterion is larger than 3D H-B failure criterionParticularly as the softening coefficient is the same the

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16

120590p

0

Figure 7 Stresses of surrounding rock mass (elastic-brittle-plasticmodel)

residual radius is smaller than the plastic radius of thesurrounding rock if it is calculated by generalized H-B failurecriterion For example if 120574119901 = 004 the plastic radiuscalculated by generalized failure criterion is 14660 and theresidual radius does not exist Based on 3D H-B failurecriterion the plastic radius is 12841 and the residual radiusis 10372 Hence if the softening coefficient is the same theplastic radius calculated by different failure criteria is quitedifferent

The radial displacement comparisons of surroundingrock under different flow rules softening parameters anddilation angles are shown in Figures 8ndash11 Summary results

Mathematical Problems in Engineering 9

Table 3 Plastic radius critical stresses and displacements of surrounding rock

Softening parameters 1199031199011199030

1199031199041199030

Associated flowrule

1199061199031199030()

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 75

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D H-B

e-p 14591 1 06793 07245 07906 08534120574119901= 004 14660 1 06879 07373 08114 08839

120574119901= 0006 14945 12401 07239 07819 08676 09500e-b-p 15091 15091 07358 07947 08819 09658

3D H-B

e-p 12681 1 07877 05621 05815 06088 06337120574119901= 004 12841 10372 09599 05669 05884 06194 06484

120574119901= 0006 12980 12198 10685 05944 06219 06610 06970e-b-p 13010 13010 10873 06036 06314 06710 07074

Table 4 Calculation parameters

Rock mass 119904 119904119903

a 119886119903

119898119887

119898119887119903

1 00039 00039 055 06 17 172 00039 00019 055 055 17 173 00039 00039 055 055 17 164 00039 00039 055 055 17 085

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

03

04

05

06

07

08

09

urr

0(

)

105 110 115 120 125 130100rr0

Figure 8 Displacements of surrounding rock mass (elastoplasticmodel)

can be seen from Table 3 The plastic zone radii calculated byassociated flow rule and nonassociated flow rule are almostconsistent However the radial displacement of rock massincreases 1064 (e-p model) 1184 (120574119901 = 004) 1391 (120574119901 =0006) and 1413 (e-b-p model) if it adopted associatedflow rule Obviously with the continuous deteriorations ofstrength parameters the result differences between associatedflow rule and nonassociated flow rule will increase gradually

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

urr

0(

)

Figure 9 Displacements of surrounding rock mass (strain-softening model 120574119901 = 004)

72 Parameter Analysis In order to analyze the effect ofdifferent softening parameters on plastic radius and radialdisplacement of surrounding rock when considering 3D orgeneralized H-B failure criterion the following parametersare cited 120590

119888= 30MPa 119864 = 55GPa V = 025 119901in = 5Mpa

and 1199030= 5mThe detailed data are shown in Table 4

Elastic-brittle-plastic model (dilation angle is equal to20∘) was selected to study the stresses and displacementsof surrounding rock as shown in Tables 4 and 5 Com-pared with the softening process of three parameters thesoftening process with only one parameter was conductedThe following conclusions can be drawn as follows thedifferences of plastic radius are 188 (2D) and 137 (3D)when only parameter ldquo119886rdquo is softening the differences of radialdisplacement are 628 (2D) and 577 (3D) when onlyparameter ldquo119886rdquo is softening the differences of plastic radius

10 Mathematical Problems in Engineering

Table 5 Parameter analyses

H-B Softening parameters Groups 1199031199011199030

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D e-p 1 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14771 07008 08276 08996

3D 120574119901= 0006 12803 05773 06331 06629

2D e-b-p 14807 07053 08327 090513D e-b-p 12832 05809 06367 066662D e-p 2 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14602 06796 07919 08553

3D 120574119901= 0006 12688 05629 06102 06354

2D e-b-p 14604 06800 07923 085573D e-b-p 12690 05632 06105 063572D e-p 3 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14727 06953 08195 08901

3D 120574119901= 0006 12775 05738 06282 06574

2D e-b-p 14777 07016 08273 089883D e-b-p 12814 05787 06337 066322D e-p 4 14591 06783 07897 085253D e-p 12531 05877 06446 067512D 120574

119901= 0006 16778 09724 13501 15894

3D 120574119901= 0006 14509 08046 10370 11740

2D e-b-p 17339 10539 14379 167813D e-b-p 14882 08584 10797 12092

03

04

05

06

07

08

09

10

11

urr

0(

)

105 110 115 120 125 130100rr0

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (associated flow rule 120595 = 75∘)ur4 (associated flow rule 120595 = 15∘)ur5 (associated flow rule 120595 = 20∘)

Figure 10 Displacements of surrounding rock mass (strain-softening model 120574119901 = 0006)

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3

ur4

ur5

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

11

urr

0(

)

(associated flow rule 120595 = 75∘)(associated flow rule 120595 = 15∘)(associated flow rule 120595 = 20∘)

Figure 11 Displacements of surrounding rock mass (elastic-brittle-plastic model)

Mathematical Problems in Engineering 11

are 323 (2D) and 246 (3D) when only parameter ldquo119904rdquo issoftening the differences of radial displacement are 1140(2D) and 1014 (3D) when only parameter ldquo119904rdquo is softeningthe differences of plastic radius are 208 (2D) and 151 (3D)when only parameter ldquo119898rdquo was softening the differences ofradial displacement are 693 (2D) and 625 (3D) whenonly parameter ldquo119898rdquo is softening

The above results show that the softening of surroundingrock is commonly decided by many parameters The order ofrock mass parameters affecting the stress and displacementsis shown as follows 119904 gt 119898 gt 119886

8 Conclusions

(1) Incorporating the 3D Hoek-Brown failure criterionassociated flow rule nonassociated flow rule and thestrain-softening model numerical solutions of stressdisplacement and plastic radius were proposed

(2) Comparison results show that the plastic radius andradial displacement calculated by 3D Hoek-Brownfailure criterion are smaller than those based ongeneralized Hoek-Brown

(3) Radial displacement calculated by nonassociated flowrule is smaller than those considering associated flowrule The influences of dilatancy parameter on theresults based on generalized H-B failure criterion arelarger than those based on 3D H-B failure criterion

Notations

119886 Parameter of HB failure criterion for peakstrength [mdash]

119886119903 Parameter of HB failure criterion for

residual strength [mdash]1199030 Radius of the tunnel opening [L]119863 HB constants for the rock mass [mdash]119864 Youngrsquos modulus of the rock mass [FLminus2]119898119887 Parameter of HB failure criterion for peakstrength [mdash]

119898119887119903 Parameter of HB failure criterion forresidual strength [mdash]

119901in Critical internal pressure [FLminus2]119903 Radial distance from the center of opening

[L]119903119901 Plastic radius [L]

119903119904 Residual radius [L]119904 Parameter of the HB failure criterion for

peak strength [mdash]119904119903 Parameter of the HB failure criterion for

residual strength [mdash]119906119903 Radial displacement [L]

1205900 Initial in situ stress [FLminus2]

120590119888 Uniaxial compressive strength of the rock

[FLminus2]120590119903 Radial normal stress [FLminus2]

120590120579 Circumferential normal stress [FLminus2]

120590119911 Axial normal stress along the axis of thetunnel [FLminus2]

1205901 Major principal stresses [FLminus2]

1205902 Middle principal stresses [FLminus2]

1205903 Minor principal stresses [FLminus2]

120590119877 Radial normal stress at the elastoplasticinterface [FLminus2]

120576119903 Radial normal strain [mdash]120576120579 Circumferential normal strain [mdash]120576119911 Axial normal strain [mdash]119901 Plastic parts of normal strain and stress [mdash]119890 Elastic parts of normal strain and stress [mdash]120595 Dilation angle [mdash]120573 Dilation coefficient [mdash]120574119901 Softening coefficient of the surrounding

rock [mdash]V Poissonrsquos ratio of the rock mass [mdash]119865 Yield function [mdash]119876 Plastic potential function [mdash]

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors are grateful to the 973 Program (2013CB036004)National Natural Science Foundation of China (no51208523)

References

[1] M H Yu N L He and L Y Song ldquoTwin shear stress theoryand its generalizationrdquo Scientia Sinica (Sciences inChina Englishedition) A vol 28 no 11 pp 1113ndash1120 1985

[2] M-H Yu Y-W Zan J Zhao and M Yoshimine ldquoA UnifiedStrength criterion for rock materialrdquo International Journal ofRock Mechanics andMining Sciences vol 39 no 8 pp 975ndash9892002

[3] C Carranza-Torres ldquoDimensionless graphical representation ofthe exact elasto-plastic solution of a circular tunnel in a Mohr-Coulomb material subject to uniform far-field stressesrdquo RockMechanics amp Rock Engineering vol 36 no 3 pp 237ndash253 2003

[4] C Carranza-Torres ldquoElasto-plastic solution of tunnel problemsusing the generalized formof theHoek-Brown failure criterionrdquoInternational Journal of Rock Mechanics amp Mining Sciences vol41 supplement 1 pp 629ndash639 2004

[5] K-H Park and Y-J Kim ldquoAnalytical solution for a circularopening in an elastic-brittle-plastic rockrdquo International Journalof Rock Mechanics and Mining Sciences vol 43 no 4 pp 616ndash622 2006

[6] C Carranza-Torres and C Fairhurst ldquoThe elasto-plasticresponse of underground excavations in rockmasses that satisfythe Hoek-Brown failure criterionrdquo International Journal of RockMechanics andMining Sciences vol 36 no 6 pp 777ndash809 1999

[7] S K Sharan ldquoElastic-brittle-plastic analysis of circular open-ings in Hoek-Brown mediardquo International Journal of RockMechanics andMining Sciences vol 40 no 6 pp 817ndash824 2003

[8] S K Sharan ldquoAnalytical solutions for stresses and displace-ments around a circular opening in a generalized Hoek-Brown

12 Mathematical Problems in Engineering

rockrdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 1 pp 78ndash85 2008

[9] M B Reed ldquoThe influence of out-of-plane stress on a planestrain problem in rock mechanicsrdquo International Journal forNumerical amp Analytical Methods in Geomechanics vol 12 no2 pp 173ndash181 1988

[10] X-D Pan and E T Brown ldquoInfluence of axial stress anddilatancy on rock tunnel stabilityrdquo Journal of GeotechnicalEngineering vol 122 no 2 pp 139ndash146 1996

[11] S Wang Z Wu M Guo and X Ge ldquoTheoretical solutions ofa circular tunnel with the influence of axial in situ stress inelastic-brittle-plastic rockrdquo Tunnelling and Underground SpaceTechnology vol 30 pp 155ndash168 2012

[12] A-Z Lu G-S Xu F Sun and W-Q Sun ldquoElasto-plasticanalysis of a circular tunnel including the effect of the axial insitu stressrdquo International Journal of Rock Mechanics and MiningSciences vol 47 no 1 pp 50ndash59 2010

[13] X-P Zhou H-Q Yang Y-X Zhang and M-H Yu ldquoTheeffect of the intermediate principal stress on the ultimatebearing capacity of a foundation on rock massesrdquo Computersamp Geotechnics vol 36 no 5 pp 861ndash870 2009

[14] J F Zou and Y Su ldquoTheoretical solutions of a circular tunnelwith the influence of the out-of-plane stress based on the gen-eralized Hoek-Brown failure criterionrdquo International Journal ofGeomechanics (ASCE) vol 16 no 3 2016

[15] J F Zou and Z He ldquoNumerical approach for strain-softeningrock with axial stressrdquo Proceedings of the Institution of CivilEngineersmdashGeotechnical Engineering vol 169 no 3 pp 276ndash290 2016

[16] J-F Zou S-S Li Y XuH-CDan andL-H Zhao ldquoTheoreticalsolutions for a circular opening in an elasticndashbrittlendashplastic rockmass incorporating the out-of-plane stress and seepage forcerdquoKSCE Journal of Civil Engineering vol 20 no 2 pp 687ndash7012016

[17] J F Zou and S Q Zuo ldquoAn approximate solution for the cylin-drical cavity expansion problem under the non-axisymmetricdisplacement boundary condition on hypotenuserdquo Interna-tional Journal of Geotechnical Engineering In press

[18] J F Zou and Z Q Xia ldquoSolutions for displacement and stressin strain-softening surrounding rock incorporating the effectsof hydraulic-mechanical coupling and rockbolts effectivenessrdquoGeotechnical amp Geological Engineering 2016

[19] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[20] S Wang X Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[21] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[22] J Zou and S Li ldquoTheoretical solution for displacement andstress in strain-softening surrounding rock under hydraulic-mechanical couplingrdquo Science China Technological Sciences vol58 no 8 pp 1401ndash1413 2015

[23] X Yang and Z Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[24] X L Yang J S Xu Y X Li andRM Yan ldquoCollapsemechanismof tunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[25] X L Yang and RM Yan ldquoCollapse mechanism for deep tunnelsubjected to seepage force in layered soilsrdquo Geomechanics andEngineering vol 8 no 5 pp 741ndash756 2015

[26] X D Pan and J A Hudson ldquoA simplified three dimensionalHoek-Brown yield criterionrdquo in Proceedings of the ISRM Inter-national Symposium International Society for RockMechanicsMadrid Spain 1988

[27] B Singh R K Goel V K Mehrotra S K Garg and MR Allu ldquoEffect of intermediate principal stress on strengthof anisotropic rock massrdquo Tunnelling and Underground SpaceTechnology vol 13 no 1 pp 71ndash79 1998

[28] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[29] L Zhang andHZhu ldquoThree-dimensional hoek-brown strengthcriterion for rocksrdquo Journal ofGeotechnical andGeoenvironmen-tal Engineering vol 133 no 9 pp 1128ndash1135 2007

[30] X-L Yang and Z-X Long ldquoRoof collapse of shallow tunnelswith limit analysis methodrdquo Journal of Central South Universityvol 22 no 5 article no 2712 pp 1929ndash1936 2015

[31] E Hoek D Wood and S Shah ldquoA modified Hoek-Brownfailure criterion for jointed rock massesrdquo in Proceedings ofthe International Conference on Eurock vol 92 pp 202ndash214September 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 1 Calculation parameters [8]

Quality of rock mass V 1199030(m) 120590

119888(MPa) 119904 119904

119903119886 119886

119903119898119887

119898119887119903

119864 (GPa) 119864119903(GPa)

Average 025 5 80 00039 0 051 053 201 034 9 5Very poor 03 5 25 00039 00019 055 06 17 085 57 57

Table 2 Comparisons between the results of this paper and Sharan without considering the strain-softening and intermediate principalstress

Parameter groups Elastoplastic model 1205900

119901in 1199031199011199030(Sharan) 119903

1199011199030(this paper)

1 e-p 40 0 177 177141 e-p 80 0 259 25881 e-p 80 1 231 23121 e-p 80 5 191 19071 e-b-p 40 3 253 25261 e-b-p 40 5 203 20301 e-b-p 80 18 233 23332 e-b-p 15 0 390 39012 e-b-p 15 3 151 15102 e-b-p 30 0 784 78362 e-b-p 30 5 233 2327

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

00

02

04

06

08

10

12

14

16

120590p

0

Figure 6 Stresses of surrounding rock mass (strain-softeningmodel 120574119901 = 0006)

Figures 4ndash7 show the stress comparisons between theideal elastoplastic strain-softening and elastic-brittle-plasticmodel based on 3D and generalized Hoek-Brown failurecriteria respectively When considering the influence of theintermediate principal stress on different failure criterion thevalue of plastic radius calculated by the 3D H-B criterionis smaller than those calculated by the generalized H-Bcriterion

Under the condition that the geotechnical parameters arethe same the influence of dilatation coefficient on generalizedH-B failure criterion is larger than 3D H-B failure criterionParticularly as the softening coefficient is the same the

120590r

120590120579

120590z

120590998400r

120590998400z

120590998400120579

12 14 16 18 20 22 24 2610rr0

02

04

06

08

10

12

14

16

120590p

0

Figure 7 Stresses of surrounding rock mass (elastic-brittle-plasticmodel)

residual radius is smaller than the plastic radius of thesurrounding rock if it is calculated by generalized H-B failurecriterion For example if 120574119901 = 004 the plastic radiuscalculated by generalized failure criterion is 14660 and theresidual radius does not exist Based on 3D H-B failurecriterion the plastic radius is 12841 and the residual radiusis 10372 Hence if the softening coefficient is the same theplastic radius calculated by different failure criteria is quitedifferent

The radial displacement comparisons of surroundingrock under different flow rules softening parameters anddilation angles are shown in Figures 8ndash11 Summary results

Mathematical Problems in Engineering 9

Table 3 Plastic radius critical stresses and displacements of surrounding rock

Softening parameters 1199031199011199030

1199031199041199030

Associated flowrule

1199061199031199030()

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 75

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D H-B

e-p 14591 1 06793 07245 07906 08534120574119901= 004 14660 1 06879 07373 08114 08839

120574119901= 0006 14945 12401 07239 07819 08676 09500e-b-p 15091 15091 07358 07947 08819 09658

3D H-B

e-p 12681 1 07877 05621 05815 06088 06337120574119901= 004 12841 10372 09599 05669 05884 06194 06484

120574119901= 0006 12980 12198 10685 05944 06219 06610 06970e-b-p 13010 13010 10873 06036 06314 06710 07074

Table 4 Calculation parameters

Rock mass 119904 119904119903

a 119886119903

119898119887

119898119887119903

1 00039 00039 055 06 17 172 00039 00019 055 055 17 173 00039 00039 055 055 17 164 00039 00039 055 055 17 085

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

03

04

05

06

07

08

09

urr

0(

)

105 110 115 120 125 130100rr0

Figure 8 Displacements of surrounding rock mass (elastoplasticmodel)

can be seen from Table 3 The plastic zone radii calculated byassociated flow rule and nonassociated flow rule are almostconsistent However the radial displacement of rock massincreases 1064 (e-p model) 1184 (120574119901 = 004) 1391 (120574119901 =0006) and 1413 (e-b-p model) if it adopted associatedflow rule Obviously with the continuous deteriorations ofstrength parameters the result differences between associatedflow rule and nonassociated flow rule will increase gradually

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

urr

0(

)

Figure 9 Displacements of surrounding rock mass (strain-softening model 120574119901 = 004)

72 Parameter Analysis In order to analyze the effect ofdifferent softening parameters on plastic radius and radialdisplacement of surrounding rock when considering 3D orgeneralized H-B failure criterion the following parametersare cited 120590

119888= 30MPa 119864 = 55GPa V = 025 119901in = 5Mpa

and 1199030= 5mThe detailed data are shown in Table 4

Elastic-brittle-plastic model (dilation angle is equal to20∘) was selected to study the stresses and displacementsof surrounding rock as shown in Tables 4 and 5 Com-pared with the softening process of three parameters thesoftening process with only one parameter was conductedThe following conclusions can be drawn as follows thedifferences of plastic radius are 188 (2D) and 137 (3D)when only parameter ldquo119886rdquo is softening the differences of radialdisplacement are 628 (2D) and 577 (3D) when onlyparameter ldquo119886rdquo is softening the differences of plastic radius

10 Mathematical Problems in Engineering

Table 5 Parameter analyses

H-B Softening parameters Groups 1199031199011199030

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D e-p 1 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14771 07008 08276 08996

3D 120574119901= 0006 12803 05773 06331 06629

2D e-b-p 14807 07053 08327 090513D e-b-p 12832 05809 06367 066662D e-p 2 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14602 06796 07919 08553

3D 120574119901= 0006 12688 05629 06102 06354

2D e-b-p 14604 06800 07923 085573D e-b-p 12690 05632 06105 063572D e-p 3 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14727 06953 08195 08901

3D 120574119901= 0006 12775 05738 06282 06574

2D e-b-p 14777 07016 08273 089883D e-b-p 12814 05787 06337 066322D e-p 4 14591 06783 07897 085253D e-p 12531 05877 06446 067512D 120574

119901= 0006 16778 09724 13501 15894

3D 120574119901= 0006 14509 08046 10370 11740

2D e-b-p 17339 10539 14379 167813D e-b-p 14882 08584 10797 12092

03

04

05

06

07

08

09

10

11

urr

0(

)

105 110 115 120 125 130100rr0

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (associated flow rule 120595 = 75∘)ur4 (associated flow rule 120595 = 15∘)ur5 (associated flow rule 120595 = 20∘)

Figure 10 Displacements of surrounding rock mass (strain-softening model 120574119901 = 0006)

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3

ur4

ur5

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

11

urr

0(

)

(associated flow rule 120595 = 75∘)(associated flow rule 120595 = 15∘)(associated flow rule 120595 = 20∘)

Figure 11 Displacements of surrounding rock mass (elastic-brittle-plastic model)

Mathematical Problems in Engineering 11

are 323 (2D) and 246 (3D) when only parameter ldquo119904rdquo issoftening the differences of radial displacement are 1140(2D) and 1014 (3D) when only parameter ldquo119904rdquo is softeningthe differences of plastic radius are 208 (2D) and 151 (3D)when only parameter ldquo119898rdquo was softening the differences ofradial displacement are 693 (2D) and 625 (3D) whenonly parameter ldquo119898rdquo is softening

The above results show that the softening of surroundingrock is commonly decided by many parameters The order ofrock mass parameters affecting the stress and displacementsis shown as follows 119904 gt 119898 gt 119886

8 Conclusions

(1) Incorporating the 3D Hoek-Brown failure criterionassociated flow rule nonassociated flow rule and thestrain-softening model numerical solutions of stressdisplacement and plastic radius were proposed

(2) Comparison results show that the plastic radius andradial displacement calculated by 3D Hoek-Brownfailure criterion are smaller than those based ongeneralized Hoek-Brown

(3) Radial displacement calculated by nonassociated flowrule is smaller than those considering associated flowrule The influences of dilatancy parameter on theresults based on generalized H-B failure criterion arelarger than those based on 3D H-B failure criterion

Notations

119886 Parameter of HB failure criterion for peakstrength [mdash]

119886119903 Parameter of HB failure criterion for

residual strength [mdash]1199030 Radius of the tunnel opening [L]119863 HB constants for the rock mass [mdash]119864 Youngrsquos modulus of the rock mass [FLminus2]119898119887 Parameter of HB failure criterion for peakstrength [mdash]

119898119887119903 Parameter of HB failure criterion forresidual strength [mdash]

119901in Critical internal pressure [FLminus2]119903 Radial distance from the center of opening

[L]119903119901 Plastic radius [L]

119903119904 Residual radius [L]119904 Parameter of the HB failure criterion for

peak strength [mdash]119904119903 Parameter of the HB failure criterion for

residual strength [mdash]119906119903 Radial displacement [L]

1205900 Initial in situ stress [FLminus2]

120590119888 Uniaxial compressive strength of the rock

[FLminus2]120590119903 Radial normal stress [FLminus2]

120590120579 Circumferential normal stress [FLminus2]

120590119911 Axial normal stress along the axis of thetunnel [FLminus2]

1205901 Major principal stresses [FLminus2]

1205902 Middle principal stresses [FLminus2]

1205903 Minor principal stresses [FLminus2]

120590119877 Radial normal stress at the elastoplasticinterface [FLminus2]

120576119903 Radial normal strain [mdash]120576120579 Circumferential normal strain [mdash]120576119911 Axial normal strain [mdash]119901 Plastic parts of normal strain and stress [mdash]119890 Elastic parts of normal strain and stress [mdash]120595 Dilation angle [mdash]120573 Dilation coefficient [mdash]120574119901 Softening coefficient of the surrounding

rock [mdash]V Poissonrsquos ratio of the rock mass [mdash]119865 Yield function [mdash]119876 Plastic potential function [mdash]

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors are grateful to the 973 Program (2013CB036004)National Natural Science Foundation of China (no51208523)

References

[1] M H Yu N L He and L Y Song ldquoTwin shear stress theoryand its generalizationrdquo Scientia Sinica (Sciences inChina Englishedition) A vol 28 no 11 pp 1113ndash1120 1985

[2] M-H Yu Y-W Zan J Zhao and M Yoshimine ldquoA UnifiedStrength criterion for rock materialrdquo International Journal ofRock Mechanics andMining Sciences vol 39 no 8 pp 975ndash9892002

[3] C Carranza-Torres ldquoDimensionless graphical representation ofthe exact elasto-plastic solution of a circular tunnel in a Mohr-Coulomb material subject to uniform far-field stressesrdquo RockMechanics amp Rock Engineering vol 36 no 3 pp 237ndash253 2003

[4] C Carranza-Torres ldquoElasto-plastic solution of tunnel problemsusing the generalized formof theHoek-Brown failure criterionrdquoInternational Journal of Rock Mechanics amp Mining Sciences vol41 supplement 1 pp 629ndash639 2004

[5] K-H Park and Y-J Kim ldquoAnalytical solution for a circularopening in an elastic-brittle-plastic rockrdquo International Journalof Rock Mechanics and Mining Sciences vol 43 no 4 pp 616ndash622 2006

[6] C Carranza-Torres and C Fairhurst ldquoThe elasto-plasticresponse of underground excavations in rockmasses that satisfythe Hoek-Brown failure criterionrdquo International Journal of RockMechanics andMining Sciences vol 36 no 6 pp 777ndash809 1999

[7] S K Sharan ldquoElastic-brittle-plastic analysis of circular open-ings in Hoek-Brown mediardquo International Journal of RockMechanics andMining Sciences vol 40 no 6 pp 817ndash824 2003

[8] S K Sharan ldquoAnalytical solutions for stresses and displace-ments around a circular opening in a generalized Hoek-Brown

12 Mathematical Problems in Engineering

rockrdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 1 pp 78ndash85 2008

[9] M B Reed ldquoThe influence of out-of-plane stress on a planestrain problem in rock mechanicsrdquo International Journal forNumerical amp Analytical Methods in Geomechanics vol 12 no2 pp 173ndash181 1988

[10] X-D Pan and E T Brown ldquoInfluence of axial stress anddilatancy on rock tunnel stabilityrdquo Journal of GeotechnicalEngineering vol 122 no 2 pp 139ndash146 1996

[11] S Wang Z Wu M Guo and X Ge ldquoTheoretical solutions ofa circular tunnel with the influence of axial in situ stress inelastic-brittle-plastic rockrdquo Tunnelling and Underground SpaceTechnology vol 30 pp 155ndash168 2012

[12] A-Z Lu G-S Xu F Sun and W-Q Sun ldquoElasto-plasticanalysis of a circular tunnel including the effect of the axial insitu stressrdquo International Journal of Rock Mechanics and MiningSciences vol 47 no 1 pp 50ndash59 2010

[13] X-P Zhou H-Q Yang Y-X Zhang and M-H Yu ldquoTheeffect of the intermediate principal stress on the ultimatebearing capacity of a foundation on rock massesrdquo Computersamp Geotechnics vol 36 no 5 pp 861ndash870 2009

[14] J F Zou and Y Su ldquoTheoretical solutions of a circular tunnelwith the influence of the out-of-plane stress based on the gen-eralized Hoek-Brown failure criterionrdquo International Journal ofGeomechanics (ASCE) vol 16 no 3 2016

[15] J F Zou and Z He ldquoNumerical approach for strain-softeningrock with axial stressrdquo Proceedings of the Institution of CivilEngineersmdashGeotechnical Engineering vol 169 no 3 pp 276ndash290 2016

[16] J-F Zou S-S Li Y XuH-CDan andL-H Zhao ldquoTheoreticalsolutions for a circular opening in an elasticndashbrittlendashplastic rockmass incorporating the out-of-plane stress and seepage forcerdquoKSCE Journal of Civil Engineering vol 20 no 2 pp 687ndash7012016

[17] J F Zou and S Q Zuo ldquoAn approximate solution for the cylin-drical cavity expansion problem under the non-axisymmetricdisplacement boundary condition on hypotenuserdquo Interna-tional Journal of Geotechnical Engineering In press

[18] J F Zou and Z Q Xia ldquoSolutions for displacement and stressin strain-softening surrounding rock incorporating the effectsof hydraulic-mechanical coupling and rockbolts effectivenessrdquoGeotechnical amp Geological Engineering 2016

[19] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[20] S Wang X Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[21] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[22] J Zou and S Li ldquoTheoretical solution for displacement andstress in strain-softening surrounding rock under hydraulic-mechanical couplingrdquo Science China Technological Sciences vol58 no 8 pp 1401ndash1413 2015

[23] X Yang and Z Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[24] X L Yang J S Xu Y X Li andRM Yan ldquoCollapsemechanismof tunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[25] X L Yang and RM Yan ldquoCollapse mechanism for deep tunnelsubjected to seepage force in layered soilsrdquo Geomechanics andEngineering vol 8 no 5 pp 741ndash756 2015

[26] X D Pan and J A Hudson ldquoA simplified three dimensionalHoek-Brown yield criterionrdquo in Proceedings of the ISRM Inter-national Symposium International Society for RockMechanicsMadrid Spain 1988

[27] B Singh R K Goel V K Mehrotra S K Garg and MR Allu ldquoEffect of intermediate principal stress on strengthof anisotropic rock massrdquo Tunnelling and Underground SpaceTechnology vol 13 no 1 pp 71ndash79 1998

[28] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[29] L Zhang andHZhu ldquoThree-dimensional hoek-brown strengthcriterion for rocksrdquo Journal ofGeotechnical andGeoenvironmen-tal Engineering vol 133 no 9 pp 1128ndash1135 2007

[30] X-L Yang and Z-X Long ldquoRoof collapse of shallow tunnelswith limit analysis methodrdquo Journal of Central South Universityvol 22 no 5 article no 2712 pp 1929ndash1936 2015

[31] E Hoek D Wood and S Shah ldquoA modified Hoek-Brownfailure criterion for jointed rock massesrdquo in Proceedings ofthe International Conference on Eurock vol 92 pp 202ndash214September 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 3 Plastic radius critical stresses and displacements of surrounding rock

Softening parameters 1199031199011199030

1199031199041199030

Associated flowrule

1199061199031199030()

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 75

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D H-B

e-p 14591 1 06793 07245 07906 08534120574119901= 004 14660 1 06879 07373 08114 08839

120574119901= 0006 14945 12401 07239 07819 08676 09500e-b-p 15091 15091 07358 07947 08819 09658

3D H-B

e-p 12681 1 07877 05621 05815 06088 06337120574119901= 004 12841 10372 09599 05669 05884 06194 06484

120574119901= 0006 12980 12198 10685 05944 06219 06610 06970e-b-p 13010 13010 10873 06036 06314 06710 07074

Table 4 Calculation parameters

Rock mass 119904 119904119903

a 119886119903

119898119887

119898119887119903

1 00039 00039 055 06 17 172 00039 00019 055 055 17 173 00039 00039 055 055 17 164 00039 00039 055 055 17 085

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

03

04

05

06

07

08

09

urr

0(

)

105 110 115 120 125 130100rr0

Figure 8 Displacements of surrounding rock mass (elastoplasticmodel)

can be seen from Table 3 The plastic zone radii calculated byassociated flow rule and nonassociated flow rule are almostconsistent However the radial displacement of rock massincreases 1064 (e-p model) 1184 (120574119901 = 004) 1391 (120574119901 =0006) and 1413 (e-b-p model) if it adopted associatedflow rule Obviously with the continuous deteriorations ofstrength parameters the result differences between associatedflow rule and nonassociated flow rule will increase gradually

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (non-associated flow rule 120595 = 75∘)ur4 (non-associated flow rule 120595 = 15∘)ur5 (non-associated flow rule 120595 = 20∘)

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

urr

0(

)

Figure 9 Displacements of surrounding rock mass (strain-softening model 120574119901 = 004)

72 Parameter Analysis In order to analyze the effect ofdifferent softening parameters on plastic radius and radialdisplacement of surrounding rock when considering 3D orgeneralized H-B failure criterion the following parametersare cited 120590

119888= 30MPa 119864 = 55GPa V = 025 119901in = 5Mpa

and 1199030= 5mThe detailed data are shown in Table 4

Elastic-brittle-plastic model (dilation angle is equal to20∘) was selected to study the stresses and displacementsof surrounding rock as shown in Tables 4 and 5 Com-pared with the softening process of three parameters thesoftening process with only one parameter was conductedThe following conclusions can be drawn as follows thedifferences of plastic radius are 188 (2D) and 137 (3D)when only parameter ldquo119886rdquo is softening the differences of radialdisplacement are 628 (2D) and 577 (3D) when onlyparameter ldquo119886rdquo is softening the differences of plastic radius

10 Mathematical Problems in Engineering

Table 5 Parameter analyses

H-B Softening parameters Groups 1199031199011199030

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D e-p 1 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14771 07008 08276 08996

3D 120574119901= 0006 12803 05773 06331 06629

2D e-b-p 14807 07053 08327 090513D e-b-p 12832 05809 06367 066662D e-p 2 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14602 06796 07919 08553

3D 120574119901= 0006 12688 05629 06102 06354

2D e-b-p 14604 06800 07923 085573D e-b-p 12690 05632 06105 063572D e-p 3 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14727 06953 08195 08901

3D 120574119901= 0006 12775 05738 06282 06574

2D e-b-p 14777 07016 08273 089883D e-b-p 12814 05787 06337 066322D e-p 4 14591 06783 07897 085253D e-p 12531 05877 06446 067512D 120574

119901= 0006 16778 09724 13501 15894

3D 120574119901= 0006 14509 08046 10370 11740

2D e-b-p 17339 10539 14379 167813D e-b-p 14882 08584 10797 12092

03

04

05

06

07

08

09

10

11

urr

0(

)

105 110 115 120 125 130100rr0

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (associated flow rule 120595 = 75∘)ur4 (associated flow rule 120595 = 15∘)ur5 (associated flow rule 120595 = 20∘)

Figure 10 Displacements of surrounding rock mass (strain-softening model 120574119901 = 0006)

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3

ur4

ur5

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

11

urr

0(

)

(associated flow rule 120595 = 75∘)(associated flow rule 120595 = 15∘)(associated flow rule 120595 = 20∘)

Figure 11 Displacements of surrounding rock mass (elastic-brittle-plastic model)

Mathematical Problems in Engineering 11

are 323 (2D) and 246 (3D) when only parameter ldquo119904rdquo issoftening the differences of radial displacement are 1140(2D) and 1014 (3D) when only parameter ldquo119904rdquo is softeningthe differences of plastic radius are 208 (2D) and 151 (3D)when only parameter ldquo119898rdquo was softening the differences ofradial displacement are 693 (2D) and 625 (3D) whenonly parameter ldquo119898rdquo is softening

The above results show that the softening of surroundingrock is commonly decided by many parameters The order ofrock mass parameters affecting the stress and displacementsis shown as follows 119904 gt 119898 gt 119886

8 Conclusions

(1) Incorporating the 3D Hoek-Brown failure criterionassociated flow rule nonassociated flow rule and thestrain-softening model numerical solutions of stressdisplacement and plastic radius were proposed

(2) Comparison results show that the plastic radius andradial displacement calculated by 3D Hoek-Brownfailure criterion are smaller than those based ongeneralized Hoek-Brown

(3) Radial displacement calculated by nonassociated flowrule is smaller than those considering associated flowrule The influences of dilatancy parameter on theresults based on generalized H-B failure criterion arelarger than those based on 3D H-B failure criterion

Notations

119886 Parameter of HB failure criterion for peakstrength [mdash]

119886119903 Parameter of HB failure criterion for

residual strength [mdash]1199030 Radius of the tunnel opening [L]119863 HB constants for the rock mass [mdash]119864 Youngrsquos modulus of the rock mass [FLminus2]119898119887 Parameter of HB failure criterion for peakstrength [mdash]

119898119887119903 Parameter of HB failure criterion forresidual strength [mdash]

119901in Critical internal pressure [FLminus2]119903 Radial distance from the center of opening

[L]119903119901 Plastic radius [L]

119903119904 Residual radius [L]119904 Parameter of the HB failure criterion for

peak strength [mdash]119904119903 Parameter of the HB failure criterion for

residual strength [mdash]119906119903 Radial displacement [L]

1205900 Initial in situ stress [FLminus2]

120590119888 Uniaxial compressive strength of the rock

[FLminus2]120590119903 Radial normal stress [FLminus2]

120590120579 Circumferential normal stress [FLminus2]

120590119911 Axial normal stress along the axis of thetunnel [FLminus2]

1205901 Major principal stresses [FLminus2]

1205902 Middle principal stresses [FLminus2]

1205903 Minor principal stresses [FLminus2]

120590119877 Radial normal stress at the elastoplasticinterface [FLminus2]

120576119903 Radial normal strain [mdash]120576120579 Circumferential normal strain [mdash]120576119911 Axial normal strain [mdash]119901 Plastic parts of normal strain and stress [mdash]119890 Elastic parts of normal strain and stress [mdash]120595 Dilation angle [mdash]120573 Dilation coefficient [mdash]120574119901 Softening coefficient of the surrounding

rock [mdash]V Poissonrsquos ratio of the rock mass [mdash]119865 Yield function [mdash]119876 Plastic potential function [mdash]

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors are grateful to the 973 Program (2013CB036004)National Natural Science Foundation of China (no51208523)

References

[1] M H Yu N L He and L Y Song ldquoTwin shear stress theoryand its generalizationrdquo Scientia Sinica (Sciences inChina Englishedition) A vol 28 no 11 pp 1113ndash1120 1985

[2] M-H Yu Y-W Zan J Zhao and M Yoshimine ldquoA UnifiedStrength criterion for rock materialrdquo International Journal ofRock Mechanics andMining Sciences vol 39 no 8 pp 975ndash9892002

[3] C Carranza-Torres ldquoDimensionless graphical representation ofthe exact elasto-plastic solution of a circular tunnel in a Mohr-Coulomb material subject to uniform far-field stressesrdquo RockMechanics amp Rock Engineering vol 36 no 3 pp 237ndash253 2003

[4] C Carranza-Torres ldquoElasto-plastic solution of tunnel problemsusing the generalized formof theHoek-Brown failure criterionrdquoInternational Journal of Rock Mechanics amp Mining Sciences vol41 supplement 1 pp 629ndash639 2004

[5] K-H Park and Y-J Kim ldquoAnalytical solution for a circularopening in an elastic-brittle-plastic rockrdquo International Journalof Rock Mechanics and Mining Sciences vol 43 no 4 pp 616ndash622 2006

[6] C Carranza-Torres and C Fairhurst ldquoThe elasto-plasticresponse of underground excavations in rockmasses that satisfythe Hoek-Brown failure criterionrdquo International Journal of RockMechanics andMining Sciences vol 36 no 6 pp 777ndash809 1999

[7] S K Sharan ldquoElastic-brittle-plastic analysis of circular open-ings in Hoek-Brown mediardquo International Journal of RockMechanics andMining Sciences vol 40 no 6 pp 817ndash824 2003

[8] S K Sharan ldquoAnalytical solutions for stresses and displace-ments around a circular opening in a generalized Hoek-Brown

12 Mathematical Problems in Engineering

rockrdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 1 pp 78ndash85 2008

[9] M B Reed ldquoThe influence of out-of-plane stress on a planestrain problem in rock mechanicsrdquo International Journal forNumerical amp Analytical Methods in Geomechanics vol 12 no2 pp 173ndash181 1988

[10] X-D Pan and E T Brown ldquoInfluence of axial stress anddilatancy on rock tunnel stabilityrdquo Journal of GeotechnicalEngineering vol 122 no 2 pp 139ndash146 1996

[11] S Wang Z Wu M Guo and X Ge ldquoTheoretical solutions ofa circular tunnel with the influence of axial in situ stress inelastic-brittle-plastic rockrdquo Tunnelling and Underground SpaceTechnology vol 30 pp 155ndash168 2012

[12] A-Z Lu G-S Xu F Sun and W-Q Sun ldquoElasto-plasticanalysis of a circular tunnel including the effect of the axial insitu stressrdquo International Journal of Rock Mechanics and MiningSciences vol 47 no 1 pp 50ndash59 2010

[13] X-P Zhou H-Q Yang Y-X Zhang and M-H Yu ldquoTheeffect of the intermediate principal stress on the ultimatebearing capacity of a foundation on rock massesrdquo Computersamp Geotechnics vol 36 no 5 pp 861ndash870 2009

[14] J F Zou and Y Su ldquoTheoretical solutions of a circular tunnelwith the influence of the out-of-plane stress based on the gen-eralized Hoek-Brown failure criterionrdquo International Journal ofGeomechanics (ASCE) vol 16 no 3 2016

[15] J F Zou and Z He ldquoNumerical approach for strain-softeningrock with axial stressrdquo Proceedings of the Institution of CivilEngineersmdashGeotechnical Engineering vol 169 no 3 pp 276ndash290 2016

[16] J-F Zou S-S Li Y XuH-CDan andL-H Zhao ldquoTheoreticalsolutions for a circular opening in an elasticndashbrittlendashplastic rockmass incorporating the out-of-plane stress and seepage forcerdquoKSCE Journal of Civil Engineering vol 20 no 2 pp 687ndash7012016

[17] J F Zou and S Q Zuo ldquoAn approximate solution for the cylin-drical cavity expansion problem under the non-axisymmetricdisplacement boundary condition on hypotenuserdquo Interna-tional Journal of Geotechnical Engineering In press

[18] J F Zou and Z Q Xia ldquoSolutions for displacement and stressin strain-softening surrounding rock incorporating the effectsof hydraulic-mechanical coupling and rockbolts effectivenessrdquoGeotechnical amp Geological Engineering 2016

[19] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[20] S Wang X Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[21] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[22] J Zou and S Li ldquoTheoretical solution for displacement andstress in strain-softening surrounding rock under hydraulic-mechanical couplingrdquo Science China Technological Sciences vol58 no 8 pp 1401ndash1413 2015

[23] X Yang and Z Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[24] X L Yang J S Xu Y X Li andRM Yan ldquoCollapsemechanismof tunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[25] X L Yang and RM Yan ldquoCollapse mechanism for deep tunnelsubjected to seepage force in layered soilsrdquo Geomechanics andEngineering vol 8 no 5 pp 741ndash756 2015

[26] X D Pan and J A Hudson ldquoA simplified three dimensionalHoek-Brown yield criterionrdquo in Proceedings of the ISRM Inter-national Symposium International Society for RockMechanicsMadrid Spain 1988

[27] B Singh R K Goel V K Mehrotra S K Garg and MR Allu ldquoEffect of intermediate principal stress on strengthof anisotropic rock massrdquo Tunnelling and Underground SpaceTechnology vol 13 no 1 pp 71ndash79 1998

[28] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[29] L Zhang andHZhu ldquoThree-dimensional hoek-brown strengthcriterion for rocksrdquo Journal ofGeotechnical andGeoenvironmen-tal Engineering vol 133 no 9 pp 1128ndash1135 2007

[30] X-L Yang and Z-X Long ldquoRoof collapse of shallow tunnelswith limit analysis methodrdquo Journal of Central South Universityvol 22 no 5 article no 2712 pp 1929ndash1936 2015

[31] E Hoek D Wood and S Shah ldquoA modified Hoek-Brownfailure criterion for jointed rock massesrdquo in Proceedings ofthe International Conference on Eurock vol 92 pp 202ndash214September 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

Table 5 Parameter analyses

H-B Softening parameters Groups 1199031199011199030

Nonassociatedflow rule120595 = 0

∘

1199061199031199030()

Nonassociatedflow rule120595 = 15

∘

1199061199031199030()

Nonassociatedflow rule120595 = 20

∘

1199061199031199030()

2D e-p 1 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14771 07008 08276 08996

3D 120574119901= 0006 12803 05773 06331 06629

2D e-b-p 14807 07053 08327 090513D e-b-p 12832 05809 06367 066662D e-p 2 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14602 06796 07919 08553

3D 120574119901= 0006 12688 05629 06102 06354

2D e-b-p 14604 06800 07923 085573D e-b-p 12690 05632 06105 063572D e-p 3 14591 06783 07897 085253D e-p 12681 05621 06088 063372D 120574

119901= 0006 14727 06953 08195 08901

3D 120574119901= 0006 12775 05738 06282 06574

2D e-b-p 14777 07016 08273 089883D e-b-p 12814 05787 06337 066322D e-p 4 14591 06783 07897 085253D e-p 12531 05877 06446 067512D 120574

119901= 0006 16778 09724 13501 15894

3D 120574119901= 0006 14509 08046 10370 11740

2D e-b-p 17339 10539 14379 167813D e-b-p 14882 08584 10797 12092

03

04

05

06

07

08

09

10

11

urr

0(

)

105 110 115 120 125 130100rr0

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3 (associated flow rule 120595 = 75∘)ur4 (associated flow rule 120595 = 15∘)ur5 (associated flow rule 120595 = 20∘)

Figure 10 Displacements of surrounding rock mass (strain-softening model 120574119901 = 0006)

ur1 (associated flow rule)ur2 (non-associated flow rule 120595 = 0∘)ur3

ur4

ur5

105 110 115 120 125 130100rr0

03

04

05

06

07

08

09

10

11

urr

0(

)

(associated flow rule 120595 = 75∘)(associated flow rule 120595 = 15∘)(associated flow rule 120595 = 20∘)

Figure 11 Displacements of surrounding rock mass (elastic-brittle-plastic model)

Mathematical Problems in Engineering 11

are 323 (2D) and 246 (3D) when only parameter ldquo119904rdquo issoftening the differences of radial displacement are 1140(2D) and 1014 (3D) when only parameter ldquo119904rdquo is softeningthe differences of plastic radius are 208 (2D) and 151 (3D)when only parameter ldquo119898rdquo was softening the differences ofradial displacement are 693 (2D) and 625 (3D) whenonly parameter ldquo119898rdquo is softening

The above results show that the softening of surroundingrock is commonly decided by many parameters The order ofrock mass parameters affecting the stress and displacementsis shown as follows 119904 gt 119898 gt 119886

8 Conclusions

(1) Incorporating the 3D Hoek-Brown failure criterionassociated flow rule nonassociated flow rule and thestrain-softening model numerical solutions of stressdisplacement and plastic radius were proposed

(2) Comparison results show that the plastic radius andradial displacement calculated by 3D Hoek-Brownfailure criterion are smaller than those based ongeneralized Hoek-Brown

(3) Radial displacement calculated by nonassociated flowrule is smaller than those considering associated flowrule The influences of dilatancy parameter on theresults based on generalized H-B failure criterion arelarger than those based on 3D H-B failure criterion

Notations

119886 Parameter of HB failure criterion for peakstrength [mdash]

119886119903 Parameter of HB failure criterion for

residual strength [mdash]1199030 Radius of the tunnel opening [L]119863 HB constants for the rock mass [mdash]119864 Youngrsquos modulus of the rock mass [FLminus2]119898119887 Parameter of HB failure criterion for peakstrength [mdash]

119898119887119903 Parameter of HB failure criterion forresidual strength [mdash]

119901in Critical internal pressure [FLminus2]119903 Radial distance from the center of opening

[L]119903119901 Plastic radius [L]

119903119904 Residual radius [L]119904 Parameter of the HB failure criterion for

peak strength [mdash]119904119903 Parameter of the HB failure criterion for

residual strength [mdash]119906119903 Radial displacement [L]

1205900 Initial in situ stress [FLminus2]

120590119888 Uniaxial compressive strength of the rock

[FLminus2]120590119903 Radial normal stress [FLminus2]

120590120579 Circumferential normal stress [FLminus2]

120590119911 Axial normal stress along the axis of thetunnel [FLminus2]

1205901 Major principal stresses [FLminus2]

1205902 Middle principal stresses [FLminus2]

1205903 Minor principal stresses [FLminus2]

120590119877 Radial normal stress at the elastoplasticinterface [FLminus2]

120576119903 Radial normal strain [mdash]120576120579 Circumferential normal strain [mdash]120576119911 Axial normal strain [mdash]119901 Plastic parts of normal strain and stress [mdash]119890 Elastic parts of normal strain and stress [mdash]120595 Dilation angle [mdash]120573 Dilation coefficient [mdash]120574119901 Softening coefficient of the surrounding

rock [mdash]V Poissonrsquos ratio of the rock mass [mdash]119865 Yield function [mdash]119876 Plastic potential function [mdash]

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors are grateful to the 973 Program (2013CB036004)National Natural Science Foundation of China (no51208523)

References

[1] M H Yu N L He and L Y Song ldquoTwin shear stress theoryand its generalizationrdquo Scientia Sinica (Sciences inChina Englishedition) A vol 28 no 11 pp 1113ndash1120 1985

[2] M-H Yu Y-W Zan J Zhao and M Yoshimine ldquoA UnifiedStrength criterion for rock materialrdquo International Journal ofRock Mechanics andMining Sciences vol 39 no 8 pp 975ndash9892002

[3] C Carranza-Torres ldquoDimensionless graphical representation ofthe exact elasto-plastic solution of a circular tunnel in a Mohr-Coulomb material subject to uniform far-field stressesrdquo RockMechanics amp Rock Engineering vol 36 no 3 pp 237ndash253 2003

[4] C Carranza-Torres ldquoElasto-plastic solution of tunnel problemsusing the generalized formof theHoek-Brown failure criterionrdquoInternational Journal of Rock Mechanics amp Mining Sciences vol41 supplement 1 pp 629ndash639 2004

[5] K-H Park and Y-J Kim ldquoAnalytical solution for a circularopening in an elastic-brittle-plastic rockrdquo International Journalof Rock Mechanics and Mining Sciences vol 43 no 4 pp 616ndash622 2006

[6] C Carranza-Torres and C Fairhurst ldquoThe elasto-plasticresponse of underground excavations in rockmasses that satisfythe Hoek-Brown failure criterionrdquo International Journal of RockMechanics andMining Sciences vol 36 no 6 pp 777ndash809 1999

[7] S K Sharan ldquoElastic-brittle-plastic analysis of circular open-ings in Hoek-Brown mediardquo International Journal of RockMechanics andMining Sciences vol 40 no 6 pp 817ndash824 2003

[8] S K Sharan ldquoAnalytical solutions for stresses and displace-ments around a circular opening in a generalized Hoek-Brown

12 Mathematical Problems in Engineering

rockrdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 1 pp 78ndash85 2008

[9] M B Reed ldquoThe influence of out-of-plane stress on a planestrain problem in rock mechanicsrdquo International Journal forNumerical amp Analytical Methods in Geomechanics vol 12 no2 pp 173ndash181 1988

[10] X-D Pan and E T Brown ldquoInfluence of axial stress anddilatancy on rock tunnel stabilityrdquo Journal of GeotechnicalEngineering vol 122 no 2 pp 139ndash146 1996

[11] S Wang Z Wu M Guo and X Ge ldquoTheoretical solutions ofa circular tunnel with the influence of axial in situ stress inelastic-brittle-plastic rockrdquo Tunnelling and Underground SpaceTechnology vol 30 pp 155ndash168 2012

[12] A-Z Lu G-S Xu F Sun and W-Q Sun ldquoElasto-plasticanalysis of a circular tunnel including the effect of the axial insitu stressrdquo International Journal of Rock Mechanics and MiningSciences vol 47 no 1 pp 50ndash59 2010

[13] X-P Zhou H-Q Yang Y-X Zhang and M-H Yu ldquoTheeffect of the intermediate principal stress on the ultimatebearing capacity of a foundation on rock massesrdquo Computersamp Geotechnics vol 36 no 5 pp 861ndash870 2009

[14] J F Zou and Y Su ldquoTheoretical solutions of a circular tunnelwith the influence of the out-of-plane stress based on the gen-eralized Hoek-Brown failure criterionrdquo International Journal ofGeomechanics (ASCE) vol 16 no 3 2016

[15] J F Zou and Z He ldquoNumerical approach for strain-softeningrock with axial stressrdquo Proceedings of the Institution of CivilEngineersmdashGeotechnical Engineering vol 169 no 3 pp 276ndash290 2016

[16] J-F Zou S-S Li Y XuH-CDan andL-H Zhao ldquoTheoreticalsolutions for a circular opening in an elasticndashbrittlendashplastic rockmass incorporating the out-of-plane stress and seepage forcerdquoKSCE Journal of Civil Engineering vol 20 no 2 pp 687ndash7012016

[17] J F Zou and S Q Zuo ldquoAn approximate solution for the cylin-drical cavity expansion problem under the non-axisymmetricdisplacement boundary condition on hypotenuserdquo Interna-tional Journal of Geotechnical Engineering In press

[18] J F Zou and Z Q Xia ldquoSolutions for displacement and stressin strain-softening surrounding rock incorporating the effectsof hydraulic-mechanical coupling and rockbolts effectivenessrdquoGeotechnical amp Geological Engineering 2016

[19] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[20] S Wang X Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[21] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[22] J Zou and S Li ldquoTheoretical solution for displacement andstress in strain-softening surrounding rock under hydraulic-mechanical couplingrdquo Science China Technological Sciences vol58 no 8 pp 1401ndash1413 2015

[23] X Yang and Z Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[24] X L Yang J S Xu Y X Li andRM Yan ldquoCollapsemechanismof tunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[25] X L Yang and RM Yan ldquoCollapse mechanism for deep tunnelsubjected to seepage force in layered soilsrdquo Geomechanics andEngineering vol 8 no 5 pp 741ndash756 2015

[26] X D Pan and J A Hudson ldquoA simplified three dimensionalHoek-Brown yield criterionrdquo in Proceedings of the ISRM Inter-national Symposium International Society for RockMechanicsMadrid Spain 1988

[27] B Singh R K Goel V K Mehrotra S K Garg and MR Allu ldquoEffect of intermediate principal stress on strengthof anisotropic rock massrdquo Tunnelling and Underground SpaceTechnology vol 13 no 1 pp 71ndash79 1998

[28] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[29] L Zhang andHZhu ldquoThree-dimensional hoek-brown strengthcriterion for rocksrdquo Journal ofGeotechnical andGeoenvironmen-tal Engineering vol 133 no 9 pp 1128ndash1135 2007

[30] X-L Yang and Z-X Long ldquoRoof collapse of shallow tunnelswith limit analysis methodrdquo Journal of Central South Universityvol 22 no 5 article no 2712 pp 1929ndash1936 2015

[31] E Hoek D Wood and S Shah ldquoA modified Hoek-Brownfailure criterion for jointed rock massesrdquo in Proceedings ofthe International Conference on Eurock vol 92 pp 202ndash214September 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

are 323 (2D) and 246 (3D) when only parameter ldquo119904rdquo issoftening the differences of radial displacement are 1140(2D) and 1014 (3D) when only parameter ldquo119904rdquo is softeningthe differences of plastic radius are 208 (2D) and 151 (3D)when only parameter ldquo119898rdquo was softening the differences ofradial displacement are 693 (2D) and 625 (3D) whenonly parameter ldquo119898rdquo is softening

The above results show that the softening of surroundingrock is commonly decided by many parameters The order ofrock mass parameters affecting the stress and displacementsis shown as follows 119904 gt 119898 gt 119886

8 Conclusions

(1) Incorporating the 3D Hoek-Brown failure criterionassociated flow rule nonassociated flow rule and thestrain-softening model numerical solutions of stressdisplacement and plastic radius were proposed

(2) Comparison results show that the plastic radius andradial displacement calculated by 3D Hoek-Brownfailure criterion are smaller than those based ongeneralized Hoek-Brown

(3) Radial displacement calculated by nonassociated flowrule is smaller than those considering associated flowrule The influences of dilatancy parameter on theresults based on generalized H-B failure criterion arelarger than those based on 3D H-B failure criterion

Notations

119886 Parameter of HB failure criterion for peakstrength [mdash]

119886119903 Parameter of HB failure criterion for

residual strength [mdash]1199030 Radius of the tunnel opening [L]119863 HB constants for the rock mass [mdash]119864 Youngrsquos modulus of the rock mass [FLminus2]119898119887 Parameter of HB failure criterion for peakstrength [mdash]

119898119887119903 Parameter of HB failure criterion forresidual strength [mdash]

119901in Critical internal pressure [FLminus2]119903 Radial distance from the center of opening

[L]119903119901 Plastic radius [L]

119903119904 Residual radius [L]119904 Parameter of the HB failure criterion for

peak strength [mdash]119904119903 Parameter of the HB failure criterion for

residual strength [mdash]119906119903 Radial displacement [L]

1205900 Initial in situ stress [FLminus2]

120590119888 Uniaxial compressive strength of the rock

[FLminus2]120590119903 Radial normal stress [FLminus2]

120590120579 Circumferential normal stress [FLminus2]

120590119911 Axial normal stress along the axis of thetunnel [FLminus2]

1205901 Major principal stresses [FLminus2]

1205902 Middle principal stresses [FLminus2]

1205903 Minor principal stresses [FLminus2]

120590119877 Radial normal stress at the elastoplasticinterface [FLminus2]

120576119903 Radial normal strain [mdash]120576120579 Circumferential normal strain [mdash]120576119911 Axial normal strain [mdash]119901 Plastic parts of normal strain and stress [mdash]119890 Elastic parts of normal strain and stress [mdash]120595 Dilation angle [mdash]120573 Dilation coefficient [mdash]120574119901 Softening coefficient of the surrounding

rock [mdash]V Poissonrsquos ratio of the rock mass [mdash]119865 Yield function [mdash]119876 Plastic potential function [mdash]

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors are grateful to the 973 Program (2013CB036004)National Natural Science Foundation of China (no51208523)

References

[1] M H Yu N L He and L Y Song ldquoTwin shear stress theoryand its generalizationrdquo Scientia Sinica (Sciences inChina Englishedition) A vol 28 no 11 pp 1113ndash1120 1985

[2] M-H Yu Y-W Zan J Zhao and M Yoshimine ldquoA UnifiedStrength criterion for rock materialrdquo International Journal ofRock Mechanics andMining Sciences vol 39 no 8 pp 975ndash9892002

[3] C Carranza-Torres ldquoDimensionless graphical representation ofthe exact elasto-plastic solution of a circular tunnel in a Mohr-Coulomb material subject to uniform far-field stressesrdquo RockMechanics amp Rock Engineering vol 36 no 3 pp 237ndash253 2003

[4] C Carranza-Torres ldquoElasto-plastic solution of tunnel problemsusing the generalized formof theHoek-Brown failure criterionrdquoInternational Journal of Rock Mechanics amp Mining Sciences vol41 supplement 1 pp 629ndash639 2004

[5] K-H Park and Y-J Kim ldquoAnalytical solution for a circularopening in an elastic-brittle-plastic rockrdquo International Journalof Rock Mechanics and Mining Sciences vol 43 no 4 pp 616ndash622 2006

[6] C Carranza-Torres and C Fairhurst ldquoThe elasto-plasticresponse of underground excavations in rockmasses that satisfythe Hoek-Brown failure criterionrdquo International Journal of RockMechanics andMining Sciences vol 36 no 6 pp 777ndash809 1999

[7] S K Sharan ldquoElastic-brittle-plastic analysis of circular open-ings in Hoek-Brown mediardquo International Journal of RockMechanics andMining Sciences vol 40 no 6 pp 817ndash824 2003

[8] S K Sharan ldquoAnalytical solutions for stresses and displace-ments around a circular opening in a generalized Hoek-Brown

12 Mathematical Problems in Engineering

rockrdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 1 pp 78ndash85 2008

[9] M B Reed ldquoThe influence of out-of-plane stress on a planestrain problem in rock mechanicsrdquo International Journal forNumerical amp Analytical Methods in Geomechanics vol 12 no2 pp 173ndash181 1988

[10] X-D Pan and E T Brown ldquoInfluence of axial stress anddilatancy on rock tunnel stabilityrdquo Journal of GeotechnicalEngineering vol 122 no 2 pp 139ndash146 1996

[11] S Wang Z Wu M Guo and X Ge ldquoTheoretical solutions ofa circular tunnel with the influence of axial in situ stress inelastic-brittle-plastic rockrdquo Tunnelling and Underground SpaceTechnology vol 30 pp 155ndash168 2012

[12] A-Z Lu G-S Xu F Sun and W-Q Sun ldquoElasto-plasticanalysis of a circular tunnel including the effect of the axial insitu stressrdquo International Journal of Rock Mechanics and MiningSciences vol 47 no 1 pp 50ndash59 2010

[13] X-P Zhou H-Q Yang Y-X Zhang and M-H Yu ldquoTheeffect of the intermediate principal stress on the ultimatebearing capacity of a foundation on rock massesrdquo Computersamp Geotechnics vol 36 no 5 pp 861ndash870 2009

[14] J F Zou and Y Su ldquoTheoretical solutions of a circular tunnelwith the influence of the out-of-plane stress based on the gen-eralized Hoek-Brown failure criterionrdquo International Journal ofGeomechanics (ASCE) vol 16 no 3 2016

[15] J F Zou and Z He ldquoNumerical approach for strain-softeningrock with axial stressrdquo Proceedings of the Institution of CivilEngineersmdashGeotechnical Engineering vol 169 no 3 pp 276ndash290 2016

[16] J-F Zou S-S Li Y XuH-CDan andL-H Zhao ldquoTheoreticalsolutions for a circular opening in an elasticndashbrittlendashplastic rockmass incorporating the out-of-plane stress and seepage forcerdquoKSCE Journal of Civil Engineering vol 20 no 2 pp 687ndash7012016

[17] J F Zou and S Q Zuo ldquoAn approximate solution for the cylin-drical cavity expansion problem under the non-axisymmetricdisplacement boundary condition on hypotenuserdquo Interna-tional Journal of Geotechnical Engineering In press

[18] J F Zou and Z Q Xia ldquoSolutions for displacement and stressin strain-softening surrounding rock incorporating the effectsof hydraulic-mechanical coupling and rockbolts effectivenessrdquoGeotechnical amp Geological Engineering 2016

[19] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[20] S Wang X Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[21] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[22] J Zou and S Li ldquoTheoretical solution for displacement andstress in strain-softening surrounding rock under hydraulic-mechanical couplingrdquo Science China Technological Sciences vol58 no 8 pp 1401ndash1413 2015

[23] X Yang and Z Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[24] X L Yang J S Xu Y X Li andRM Yan ldquoCollapsemechanismof tunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[25] X L Yang and RM Yan ldquoCollapse mechanism for deep tunnelsubjected to seepage force in layered soilsrdquo Geomechanics andEngineering vol 8 no 5 pp 741ndash756 2015

[26] X D Pan and J A Hudson ldquoA simplified three dimensionalHoek-Brown yield criterionrdquo in Proceedings of the ISRM Inter-national Symposium International Society for RockMechanicsMadrid Spain 1988

[27] B Singh R K Goel V K Mehrotra S K Garg and MR Allu ldquoEffect of intermediate principal stress on strengthof anisotropic rock massrdquo Tunnelling and Underground SpaceTechnology vol 13 no 1 pp 71ndash79 1998

[28] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[29] L Zhang andHZhu ldquoThree-dimensional hoek-brown strengthcriterion for rocksrdquo Journal ofGeotechnical andGeoenvironmen-tal Engineering vol 133 no 9 pp 1128ndash1135 2007

[30] X-L Yang and Z-X Long ldquoRoof collapse of shallow tunnelswith limit analysis methodrdquo Journal of Central South Universityvol 22 no 5 article no 2712 pp 1929ndash1936 2015

[31] E Hoek D Wood and S Shah ldquoA modified Hoek-Brownfailure criterion for jointed rock massesrdquo in Proceedings ofthe International Conference on Eurock vol 92 pp 202ndash214September 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

rockrdquo International Journal of Rock Mechanics and MiningSciences vol 45 no 1 pp 78ndash85 2008

[9] M B Reed ldquoThe influence of out-of-plane stress on a planestrain problem in rock mechanicsrdquo International Journal forNumerical amp Analytical Methods in Geomechanics vol 12 no2 pp 173ndash181 1988

[10] X-D Pan and E T Brown ldquoInfluence of axial stress anddilatancy on rock tunnel stabilityrdquo Journal of GeotechnicalEngineering vol 122 no 2 pp 139ndash146 1996

[11] S Wang Z Wu M Guo and X Ge ldquoTheoretical solutions ofa circular tunnel with the influence of axial in situ stress inelastic-brittle-plastic rockrdquo Tunnelling and Underground SpaceTechnology vol 30 pp 155ndash168 2012

[12] A-Z Lu G-S Xu F Sun and W-Q Sun ldquoElasto-plasticanalysis of a circular tunnel including the effect of the axial insitu stressrdquo International Journal of Rock Mechanics and MiningSciences vol 47 no 1 pp 50ndash59 2010

[13] X-P Zhou H-Q Yang Y-X Zhang and M-H Yu ldquoTheeffect of the intermediate principal stress on the ultimatebearing capacity of a foundation on rock massesrdquo Computersamp Geotechnics vol 36 no 5 pp 861ndash870 2009

[14] J F Zou and Y Su ldquoTheoretical solutions of a circular tunnelwith the influence of the out-of-plane stress based on the gen-eralized Hoek-Brown failure criterionrdquo International Journal ofGeomechanics (ASCE) vol 16 no 3 2016

[15] J F Zou and Z He ldquoNumerical approach for strain-softeningrock with axial stressrdquo Proceedings of the Institution of CivilEngineersmdashGeotechnical Engineering vol 169 no 3 pp 276ndash290 2016

[16] J-F Zou S-S Li Y XuH-CDan andL-H Zhao ldquoTheoreticalsolutions for a circular opening in an elasticndashbrittlendashplastic rockmass incorporating the out-of-plane stress and seepage forcerdquoKSCE Journal of Civil Engineering vol 20 no 2 pp 687ndash7012016

[17] J F Zou and S Q Zuo ldquoAn approximate solution for the cylin-drical cavity expansion problem under the non-axisymmetricdisplacement boundary condition on hypotenuserdquo Interna-tional Journal of Geotechnical Engineering In press

[18] J F Zou and Z Q Xia ldquoSolutions for displacement and stressin strain-softening surrounding rock incorporating the effectsof hydraulic-mechanical coupling and rockbolts effectivenessrdquoGeotechnical amp Geological Engineering 2016

[19] Y-K Lee and S Pietruszczak ldquoA new numerical procedurefor elasto-plastic analysis of a circular opening excavated in astrain-softening rock massrdquo Tunnelling and Underground SpaceTechnology vol 23 no 5 pp 588ndash599 2008

[20] S Wang X Yin H Tang and X Ge ldquoA new approachfor analyzing circular tunnel in strain-softening rock massesrdquoInternational Journal of Rock Mechanics and Mining Sciencesvol 47 no 1 pp 170ndash178 2010

[21] E Alonso L R Alejano F Varas G Fdez-Manin and CCarranza-Torres ldquoGround response curves for rock massesexhibiting strain-softening behaviourrdquo International Journal forNumerical and Analytical Methods in Geomechanics vol 27 no13 pp 1153ndash1185 2003

[22] J Zou and S Li ldquoTheoretical solution for displacement andstress in strain-softening surrounding rock under hydraulic-mechanical couplingrdquo Science China Technological Sciences vol58 no 8 pp 1401ndash1413 2015

[23] X Yang and Z Long ldquoSeismic and static 3D stability oftwo-stage rock slope based on Hoek-Brown failure criterionrdquoCanadian Geotechnical Journal vol 53 no 3 pp 551ndash558 2016

[24] X L Yang J S Xu Y X Li andRM Yan ldquoCollapsemechanismof tunnel roof considering joined influences of nonlinearity andnon-associated flow rulerdquo Geomechanics and Engineering vol10 no 1 pp 21ndash35 2016

[25] X L Yang and RM Yan ldquoCollapse mechanism for deep tunnelsubjected to seepage force in layered soilsrdquo Geomechanics andEngineering vol 8 no 5 pp 741ndash756 2015

[26] X D Pan and J A Hudson ldquoA simplified three dimensionalHoek-Brown yield criterionrdquo in Proceedings of the ISRM Inter-national Symposium International Society for RockMechanicsMadrid Spain 1988

[27] B Singh R K Goel V K Mehrotra S K Garg and MR Allu ldquoEffect of intermediate principal stress on strengthof anisotropic rock massrdquo Tunnelling and Underground SpaceTechnology vol 13 no 1 pp 71ndash79 1998

[28] S D Priest ldquoDetermination of shear strength and three-dimensional yield strength for the Hoek-Brown criterionrdquo RockMechanics and Rock Engineering vol 38 no 4 pp 299ndash3272005

[29] L Zhang andHZhu ldquoThree-dimensional hoek-brown strengthcriterion for rocksrdquo Journal ofGeotechnical andGeoenvironmen-tal Engineering vol 133 no 9 pp 1128ndash1135 2007

[30] X-L Yang and Z-X Long ldquoRoof collapse of shallow tunnelswith limit analysis methodrdquo Journal of Central South Universityvol 22 no 5 article no 2712 pp 1929ndash1936 2015

[31] E Hoek D Wood and S Shah ldquoA modified Hoek-Brownfailure criterion for jointed rock massesrdquo in Proceedings ofthe International Conference on Eurock vol 92 pp 202ndash214September 1992

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of