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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 Campus,Changsha, Hunan 410075, China

Correspondence should be addressed to Jin-feng Zou; zoujinfeng [email protected]

Received 19 March 2016; Accepted 13 June 2016

Academic Editor: John D. Clayton

Copyright © 2016 Jin-feng Zou et al. This is an open access article distributed under the Creative Commons Attribution License,which 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 practice.On the basis of this, elastic-brittle-plastic solutions basedon the Mohr-Coulomb and Hoek-Brown (𝑎 = 0.5) 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 problem,the 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 Reed’s 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 pageshttp://dx.doi.org/10.1155/2016/7947036

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]. Moreover,these approaches can be supplemented and verifiedmutually.Zou 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 transformation.The theoretical solutions for the elastic-brittle-plastic andelastic-plastic rockmass incorporating the out-of-plane stressand seepage force were proposed by Zou et al. [14–18].

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 strain.Wang 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 mass.Alonso 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 [23–25], but it is difficult to obtain a relatively accuratesolution.Themajority 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 researches,the 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:

𝜎1− 𝜎3= 𝜎𝑐(𝑚𝑖

𝜎3

𝜎𝑐

+ 1)

𝑎

, (1)

where 𝜎𝑐is the unconfined compressive strength of the rock

mass; 𝜎1and 𝜎

3are the major and minor principal stresses,

respectively; 𝑚, 𝑠, and 𝑎 are the H-B constants for the rockmass before yielding, which are expressed as follows:

𝑚 = 𝑚𝑖exp [(GSI − 100)

(28 − 14𝐷)

] ,

𝑠 = exp [(GSI − 100)(9 − 3𝐷)

] ,

𝑎 = 0.5 +

1

6

[exp (−GSI15

) − exp(−203

)] ,

(2)

where𝐷 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 neglected.Hence, 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

𝜎(1/𝑎−1)

𝑐

(

3

√2

𝜏oct)1/𝑎

+

𝑚𝑖

2

(

3

√2

𝜏oct) − 𝑚𝑖𝜎𝑚,2 = 𝑠𝜎𝑐, (3)

where 𝜏oct is octahedron deviatoric stress and 𝜎

𝑚,2is average

effective stress:

𝜏oct =1

3

√(𝜎1− 𝜎2)2

+ (𝜎2− 𝜎3)2

+ (𝜎1− 𝜎3)2

,

𝜎

𝑚,2=

𝜎1+ 𝜎3

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(𝑟0) is subjected to a three-dimensional and uniform in situ

Mathematical Problems in Engineering 3

y

x

z

𝜎0

𝜎0

𝜎0

Residual zone

Softening zone

Elastic zone

Elasto-plasticinterface

pin

rp

rs

r0

Figure 1: Stress state of the surrounding rock.

pressure (𝜎0) at infinity and an internal support pressure (𝑝in)

in the tunnel wall. The surrounding rock mass is consideredas continuous, homogeneous, isotropic, and initially elastic.Because the axial stress (𝜎

𝑧) 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, 𝑝in is graduallyreduced.When the internal support pressure (𝑝in) 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 strain𝜀𝑟, the circumferential normal strain 𝜀

𝜃, and the radial

displacement 𝑢𝑟can be obtained by small strain assumption

as follows:

𝜀𝑟=

𝜕𝑢𝑟

𝜕𝑟

,

𝜀𝜃=

𝑢𝑟

𝑟

.

(5)

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

𝜎𝑟= 𝜎0− (𝜎0− 𝜎𝑅) (

𝑟𝑝

𝑟

)

2

, (6a)

𝜎𝑟= 𝜎0+ (𝜎0− 𝜎𝑅) (

𝑟𝑝

𝑟

)

2

. (6b)

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

𝜎𝑧= 𝜎0, (7a)

𝑢𝑟=

1

𝐸

(1 + 𝜇) (𝜎0− 𝜎𝑟𝑝)

𝑟2

𝑝

𝑟

.(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]:

𝛾𝑝= 𝜀𝑝

1− 𝜀𝑝

3, (8)

where 𝜀𝑝1and 𝜀𝑝

3are the major and minor plastic strains,

respectively.The physical parameters of the surrounding rockmass are

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

𝜔 (𝛾𝑝) =

{{

{{

{

𝜔𝑝− (𝜔𝑝− 𝜔𝑟)

𝛾𝑝

𝛾𝑝

𝑟

, 0 < 𝛾𝑝< 𝛾𝑝

𝑟,

𝜔𝑟, 𝛾

𝑝≥ 𝛾𝑝

𝑟,

(9)

where 𝜔 represents a strength parameter, such as 𝜑, 𝑐, 𝑚, 𝑠,𝑎, 𝜙, and 𝐸; 𝛾𝑝

𝑟is the critical deviatoric plastic strain from

which the residual behavior is first observed and should beidentified through experimentation; the subscripts 𝑝 and 𝑟represent 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 𝑛 rings and theadjacent rings have different parameters. Hence, the actualconstitutive model is replaced by piecewise linear 𝑖, (𝑖 + 1)and each segment has different negative slope; it seems likethere are 𝑛 brittle-plastic processes.The greater 𝑛 is, the moreaccurate the strain-softening model is.

5. Solutions in Plastic Zone

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

(𝑖−1)and 𝑟(𝑖).

The 𝑖th annulus is determined by the outer radius 𝑟(𝑖−1)

andthe inner radius 𝑟

(𝑖)which is adjacent to the elastic region.

The parameter values of the outmost ring are equal to thevalues of innermost elastic region. So the (𝑛 − 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

𝜕𝜎𝑟

𝜕𝑟

+

𝜎𝑟− 𝜎𝜃

𝑟

= 0, (10)

where 𝜎𝑟is the radial stress and 𝜎

𝜃is the tangential stress.


A

O𝜀1

𝜎1 − 𝜎3

B C

𝜎cr

𝜎c

(a) (𝜎1 − 𝜎3)-𝜀1 curves

+

−

O𝜀1

𝜀�

(b) 𝜀1-𝜀V curves

+

−

O𝜀1

𝜀3

(c) 𝜀1-𝜀3 curves

A

𝜎cr

O𝜀1

𝜎1 − 𝜎3

𝜎c

𝛽𝜀e1𝜀e1

B C

(d) (𝜎1 − 𝜎3)-𝜀1 simplified curves

H

+

−

O

𝜀p1

−𝜀v1

1

𝜀1

𝜀v

(e) 𝜀1-𝜀V simplified curves

+

−

O

𝜀e1

1

𝜀1

𝜀3

𝛽𝜀e1

𝜀p1

−𝜀p3

h

(f) 𝜀1-𝜀3 simplified curves

Figure 2: Strain-softening material behavior model.

Calculate constitutive model

Actual constitutive model

O

0

i − 1

i + 1i

· · ·

⋱

𝜀

𝜎

Figure 3: Strain-softening model of rock mass.

Based on the plane strain assumption, the axial strainshould satisfy

𝜀𝑧= 𝜀𝑒

𝑧+ 𝜀𝑝

𝑧= 0,

𝜀𝑒

𝑧=

1

𝐸

[𝜎𝑧− V (𝜎

𝑟+ 𝜎𝜃) − (1 − 2V) 𝜎

0] .

(11)

Hence, the relationship between radial normal stress 𝜎𝑟,

circumferential normal stress 𝜎𝜃, and axial normal stress 𝜎

𝑧

is expressed as

𝜎𝑧= V (𝜎

𝜃+ 𝜎𝑟) − (2V − 1) 𝜎

0− 𝐸𝜀𝑝

𝑧. (12)

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

𝐹 (𝜎1, 𝜎3, 𝛾𝑝) = 𝜎1− 𝜎3− 𝐻 (𝜎

1, 𝜎3, 𝛾𝑝) . (13)

The radius of the first ring is 𝑟(0)= 𝑟𝑝which is at the

interface between the elastic region and the equivalent plasticzone:

𝜌(0)=

𝑟(0)

𝑟𝑝

,

𝑟(0)= 𝑟𝑝.

(14)

The stress and strain at the elastoplastic interface can begiven by

{{

{{

{

𝜎𝑟(0)

𝜎𝜃(0)

𝜎𝑧(0)

}}

}}

}

=

{{

{{

{

𝜎𝑅

2𝜎0− 𝜎𝑅

𝜎0

}}

}}

}

,


{{

{{

{

𝜀𝑟(0)

𝜀𝜃(0)

𝜀𝑧(0)

}}

}}

}

=

1 + V𝐸

{{{{{{{

{{{{{{{

{

𝑑𝑢

𝑑𝑟

𝑢

𝑟

0

}}}}}}}

}}}}}}}

}

,

(15)

where

𝜀𝑟(0)= −

1

2𝐺

{𝜎0− 𝜎𝑅} ,

𝜀𝜃(0)=

1

2𝐺

{𝜎0− 𝜎𝑅} .

(16)

The normalized inner radius can be expressed as

𝜌(𝑖)=

𝑟(𝑖)

𝑟𝑝

. (17)

𝜎𝑟on both inner and outer boundaries of the plastic zone

are known a priori and 𝜎𝑟decreases from 𝜎

𝑅to 𝑝in. The

increment of radial normal stress is given by

Δ𝜎𝑟=

(𝑝in − 𝜎𝑅)

𝑛

. (18)

So the radial normal stress at each ring can be representedby

𝜎𝑟(𝑖)= 𝜎𝑟(𝑖−1)

+ Δ𝜎𝑟. (19)

The stress equilibrium equation can be expressed inanother way as follows:

𝑑𝜎𝑟

𝑑𝜌

+

𝜎𝑟− 𝜎𝜃

𝜌

= 0. (20)

From above equations, the following expressions can beobtained:

𝜌(𝑖)

Δ𝜎𝑟(𝑖)

Δ𝜌(𝑖)

= 𝜎𝜃(𝑖)− 𝜎𝑟(𝑖). (21)

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

𝜎𝑟(𝑖)− 𝜎𝑟(𝑖−1)

𝜌(𝑖)− 𝜌(𝑖−1)

−

𝐻 (𝜎𝑟(𝑖), 𝛾𝑝)

𝜌(𝑖)

= 0, (22)

𝜌(𝑖)=

𝐻 (𝜎𝑟(𝑖), 𝛾𝑝) + Δ𝜎

𝑟

𝐻(𝜎𝑟(𝑖), 𝛾𝑝)

𝜌(𝑖−1). (23)

The radius of each ring is a known quantity and the radialstress of the 𝑖th ring can be obtained from (17). So the axialnormal stress 𝜎

𝑧can be given by

𝜎𝑧(𝑖)= V (𝜎

𝜃(𝑖)+ 𝜎𝑟(𝑖)) − (2V − 1) 𝜎

0− 𝐸𝜀𝑝

𝑧(𝑖). (24)

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

𝑄𝑥=

𝑄𝑢(𝑖)− 𝑄𝑢(𝑖−1)

𝜌(𝑖)− 𝜌(𝑖−1)

(𝜌𝑥− 𝜌(𝑖−1)) + 𝑄𝑢(𝑖), (25)

where𝑄𝑥is stress or displacement of rockmass,𝑄

𝑢(𝑖)is stress

or displacement of rock mass at 𝑖th ring, 𝑄𝑢(𝑖−1)

is stress ordisplacement of rock mass at (𝑖 − 1)th ring, and 𝜌

𝑥is 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:

𝑑𝜀𝜃

𝑑𝑟

+

𝜀𝜃− 𝜀𝑟

𝑟

= 0. (26)

Equation (26) can be transformed into the followingform:

𝑑𝜀𝑝

𝜃

𝑑𝜌

+

𝜀𝑝

𝜃− 𝜀𝑝

𝑟

𝜌

= −

𝑑𝜀𝑒

𝜃

𝑑𝜌

−

𝜀𝑒

𝜃− 𝜀𝑒

𝑟

𝜌

. (27)

It can be rewritten as𝑑𝜀𝑝

𝜃

𝑑𝜌

+

𝜀𝑝

𝜃− 𝜀𝑝

𝑟

𝜌

= −

𝑑𝜀𝑒

𝜃

𝑑𝜌

−

1 + ]𝐸

𝐻(𝜎𝑟, 𝛾𝑝)

𝜌

. (28)

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

𝑔 (𝜎𝑖𝑗, 𝐻𝑎) = 0, (29)

where𝐻𝑎is hardening parameter and 𝜎

𝑖𝑗is principal stress.

The plastic strain increment 𝑑𝜀𝑝𝑖𝑗can be obtained by

𝑑𝜀𝑝

𝑖𝑗=

𝜕𝐹

𝜕𝜎𝑖𝑗

𝑑𝜆, (30)

where 𝑑𝜆 is multiplication operator, 𝜀𝑝𝑖𝑗is plastic strain, and

𝜎𝑖𝑗is principal stress.It is called the associated flow rule when 𝐹 and 𝑄 are

completely the same. And the physical meaning of yieldfunction 𝐹 is a judgment criterion whether a point ofsurrounding rock reaches the yield state. And the physicalmeaning of plastic potential function 𝑄 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

𝑄 (𝜎) = −

𝑛

3

𝐼1+

3

𝜎𝑐

𝐽2+

√3

2

𝑛√𝐽2, (31)

where 𝑛 is dilation parameter:

𝐼1= 𝜎1+ 𝜎2+ 𝜎3,

𝐽2=

1

6 [(𝜎1− 𝜎2)2

+ (𝜎2− 𝜎3)2

+ (𝜎3− 𝜎1)2

]

.

(32)


The differential of three-direction strain is expressed as

𝑑𝜀𝑝

𝜃=

𝜕𝑓

𝜕𝜎𝜃

𝑑𝜆 = [(

√3 (2𝜎𝜃− 𝜎𝑟− 𝜎𝑧)

12√𝐽2

−

1

3

) 𝑛

+

1

𝜎𝑐

(2𝜎𝜃− 𝜎𝑟− 𝜎𝑧)] 𝑑𝜆,

𝑑𝜀𝑝

𝑧=

𝜕𝑓

𝜕𝜎𝑧

𝑑𝜆 = [(

√3 (2𝜎𝑧− 𝜎𝜃− 𝜎𝑟)

12√𝐽2

−

1

3

) 𝑛

+

1

𝜎𝑐

(2𝜎𝑧− 𝜎𝜃− 𝜎𝑟)] 𝑑𝜆,

𝑑𝜀𝑝

𝑟=

𝜕𝑓

𝜕𝜎𝑟

𝑑𝜆 = [(

√3 (2𝜎𝑟− 𝜎𝜃− 𝜎𝑧)

12√𝐽2

−

1

3

) 𝑛

+

1

𝜎𝑐

(2𝜎𝑟− 𝜎𝜃− 𝜎𝑧)] 𝑑𝜆.

(33)

If 𝐿1, 𝐿2, and 𝐿

3are defined as follows

[(

√3 (2𝜎𝜃− 𝜎𝑟− 𝜎𝑧)

12√𝐽2

−

1

3

) 𝑛 +

1

𝜎𝑐

(2𝜎𝜃− 𝜎𝑟− 𝜎𝑧)]

= 𝐿1,

[(

√3 (2𝜎𝑧− 𝜎𝜃− 𝜎𝑟)

12√𝐽2

−

1

3

) 𝑛 +

1

𝜎𝑐

(2𝜎𝑧− 𝜎𝜃− 𝜎𝑟)]

= 𝐿2,

[(

√3 (2𝜎𝑟− 𝜎𝜃− 𝜎𝑧)

12√𝐽2

−

1

3

) 𝑛 +

1

𝜎𝑐

(2𝜎𝑟− 𝜎𝜃− 𝜎𝑧)]

= 𝐿3,

(34)

then (33) can be simplified to

𝑑𝜀𝑝

1=

𝜕𝑓

𝜕𝜎1

𝑑𝜆 = 𝐿1𝑑𝜆,

𝑑𝜀𝑝

2=

𝜕𝑓

𝜕𝜎2

𝑑𝜆 = 𝐿2𝑑𝜆,

𝑑𝜀𝑝

3=

𝜕𝑓

𝜕𝜎3

𝑑𝜆 = 𝐿3𝑑𝜆.

(35)

The relationship between radial plastic normal strain 𝜀𝑝𝑟,

circumferential plastic normal strain 𝜀𝑝𝜃, and axial plastic

normal strain 𝜀𝑝𝑧can be represented by

𝜀𝑝

𝜃

𝐿1

=

𝜀𝑝

𝑧

𝐿2

=

𝜀𝑝

𝑟

𝐿3

, (36)

𝑑𝜀𝑝

𝜃

𝑑𝜌

+

𝜀𝑝

𝜃− 𝜀𝑝

𝑟

𝜌

= −

𝑑𝜀𝑒

𝜃

𝑑𝜌

−

1 + ]𝐸

𝐻 (𝜎𝑟, 𝛾𝑝)

𝜌

. (37)

Combination of (36) and (37) leads to

Δ𝜀𝑝

𝜃= (−

𝑑𝜀𝑒

𝜃(𝑖)

𝑑𝜌(𝑖)

−

1 + ]𝐸

𝐻 (𝜎𝑟(𝑖), 𝛾𝑝)

𝜌(𝑖)

−

𝜀𝑝

𝜃(𝑖−1)(1 − 𝐿

3(𝑖)/𝐿1(𝑖))

𝜌(𝑖)

)(

1

Δ𝜌(𝑖)

+

(1 − 𝐿3(𝑖)/𝐿1(𝑖))

𝜌(𝑖)

) .

(38)

The increment of the radial plastic strain at 𝑖th annuluscan be obtained by (36):

Δ𝜀𝑝

𝑟(𝑖)=

𝐿3(𝑖)

𝐿1(𝑖)

Δ𝜀𝑝

𝜃=

𝐿3(𝑖)

𝐿1(𝑖)

(−

𝑑𝜀𝑒

𝜃(𝑖)

𝑑𝜌(𝑖)

−

1 + ]𝐸

𝐻(𝜎𝑟(𝑖), 𝛾𝑝)

𝜌(𝑖)

−

𝜀𝑝

𝜃(𝑖−1)(1 − 𝐿

3(𝑖)/𝐿1(𝑖))

𝜌(𝑖)

)

⋅ (

1

Δ𝜌(𝑖)

+

(1 − 𝐿3(𝑖)/𝐿1(𝑖))

𝜌(𝑖)

) .

(39)

The displacement at 𝑖th annulus can be obtained:

𝑢𝑟(𝑖)= 𝑟(𝑖)𝜀𝜃(𝑖)= 𝑟(𝑖)(𝜀𝑒

𝜃(𝑖)+ 𝜀𝑝

𝜃(𝑖))

= 𝑟(𝑖)(𝜀𝑒

𝜃(𝑖)+ 𝜀𝑝

𝜃(1)+

𝑖

∑

𝑖=1

Δ𝜀𝑝

𝑟(𝑖)) .

(40)

5.2. 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:

𝛽𝜀𝑝

1+ 𝜀𝑝

3= 0, (41)

where 𝛽 = (1 + sin𝜓)/(1 − sin𝜓) and 𝜓 is dilation angle.According to the nonassociated flow rule, we can know

𝜀𝑝

2= 0. (42)

So (12) can be rewritten as

𝜎𝑧= V (𝜎

𝜃+ 𝜎𝑟) − (2V − 1) 𝜎

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 𝛽 which is defined by the dilation angle 𝜓. Because theintermediate principal strain 𝜀

𝑧= 0, the relationship between

the major and minor principal strains is determined by 𝐿1

and 𝐿3using 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

−

𝐿3

𝐿1

= 𝛽 =

1 + sin𝜓1 − sin𝜓

. (44)


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 rocks.Under 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 𝜎𝑧is equal to major principal

stress 𝜎𝜃or minor principal stress 𝜎

𝑟, 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

7.1. 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 Sharan’s experimental results [8]are adopted: 𝜎

𝑐= 30MPa, 𝑠 = 0.0039, 𝑎 = 0.55, 𝐸 = 5.5GPa,

𝐸𝑟= 5.5GPa, 𝑚 = 1.7, 𝑠

𝑟= 0.0019, 𝑎

𝑟= 0.6, 𝑚

𝑟= 1.6,

V = 0.25, 𝑝in = 5Mpa, and 𝑟0 = 5m. The calculation resultsare shown in Table 3.

Through the comparison results in Table 3, we canfind that the plastic zone radius and radial displacementcalculated by 3D H-B failure criterion are smaller thanthose calculated by generalized H-B failure criterion.Thus, 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 rule.With 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 (𝛾𝑝 =0.006), and elastic-brittle-plastic model would be reducedby 14.57%, 14.98%, and 16.00%, respectively. The results ofradial displacement using nonassociated flow rule are shownas follows: elastoplastic model, strain-softening model, andelastic-brittle-plastic model (𝜓 = 0∘ and 𝜓 = 20∘) would

𝜎/𝜎

0

𝜎r𝜎𝜃𝜎z

𝜎r

𝜎z

𝜎𝜃

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.61.0r/r0

Figure 4: Stresses of surrounding rock mass (elastoplastic model).

𝜎r𝜎𝜃𝜎z

𝜎r

𝜎z

𝜎𝜃

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.61.0r/r0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6𝜎/p

0

Figure 5: Stresses of surrounding rock mass (strain-softeningmodel, 𝛾𝑝 = 0.04).

be reduced by 20.85%, 34.67%, 21.78%, 36.29%, 21.90%, and36.53%, 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 compared.The differences between two calculation approaches areanalyzed for predicting the displacement of plastic zone. Theresults of stress and displacement are shown in Figures 4–11.As shown in Figures 4–7, 𝜎

𝑟, 𝜎𝜃, 𝜎𝑧, and 𝑢

𝑟are calculation

results based on 3DH-B failure criterion;𝜎𝑟,𝜎𝜃,𝜎𝑧, and 𝑢

𝑟are

calculation results based on generalizedH-B failure criterion.


Table 1: Calculation parameters [8].

Quality of rock mass V 𝑟0(m) 𝜎

𝑐(MPa) 𝑠 𝑠

𝑟𝑎 𝑎

𝑟𝑚𝑏

𝑚𝑏𝑟

𝐸 (GPa) 𝐸𝑟(GPa)

Average 0.25 5 80 0.0039 0 0.51 0.53 2.01 0.34 9 5Very poor 0.3 5 25 0.0039 0.0019 0.55 0.6 1.7 0.85 5.7 5.7

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

Parameter groups Elastoplastic model 𝜎0

𝑝in 𝑟𝑝/𝑟0 (Sharan) 𝑟𝑝/𝑟0 (this paper)1 e-p 40 0 1.77 1.77141 e-p 80 0 2.59 2.5881 e-p 80 1 2.31 2.3121 e-p 80 5 1.91 1.9071 e-b-p 40 3 2.53 2.5261 e-b-p 40 5 2.03 2.0301 e-b-p 80 18 2.33 2.3332 e-b-p 15 0 3.90 3.9012 e-b-p 15 3 1.51 1.5102 e-b-p 30 0 7.84 7.8362 e-b-p 30 5 2.33 2.327

𝜎r𝜎𝜃𝜎z

𝜎r

𝜎z

𝜎𝜃

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.61.0r/r0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

𝜎/p

0

Figure 6: Stresses of surrounding rock mass (strain-softeningmodel, 𝛾𝑝 = 0.006).

Figures 4–7 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 criterion.Particularly as the softening coefficient is the same, the

𝜎r𝜎𝜃𝜎z

𝜎r

𝜎z

𝜎𝜃

1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.61.0r/r0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

𝜎/p

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 𝛾𝑝 = 0.04, the plastic radiuscalculated by generalized failure criterion is 1.4660 and theresidual radius does not exist. Based on 3D H-B failurecriterion, the plastic radius is 1.2841 and the residual radiusis 1.0372. 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 8–11. Summary results


Table 3: Plastic radius, critical stresses, and displacements of surrounding rock.

Softening parameters 𝑟𝑝/𝑟0

𝑟𝑠/𝑟0

Associated flowrule

𝑢𝑟/𝑟0(%)

Nonassociatedflow rule𝜓 = 0

∘

𝑢𝑟/𝑟0(%)

Nonassociatedflow rule𝜓 = 7.5

∘

𝑢𝑟/𝑟0(%)


∘

𝑢𝑟/𝑟0(%)


∘

𝑢𝑟/𝑟0(%)

2D H-B

e-p 1.4591 1 0.6793 0.7245 0.7906 0.8534𝛾𝑝= 0.04 1.4660 1 0.6879 0.7373 0.8114 0.8839

𝛾𝑝= 0.006 1.4945 1.2401 0.7239 0.7819 0.8676 0.9500e-b-p 1.5091 1.5091 0.7358 0.7947 0.8819 0.9658

3D H-B

e-p 1.2681 1 0.7877 0.5621 0.5815 0.6088 0.6337𝛾𝑝= 0.04 1.2841 1.0372 0.9599 0.5669 0.5884 0.6194 0.6484

𝛾𝑝= 0.006 1.2980 1.2198 1.0685 0.5944 0.6219 0.6610 0.6970e-b-p 1.3010 1.3010 1.0873 0.6036 0.6314 0.6710 0.7074

Table 4: Calculation parameters.

Rock mass 𝑠 𝑠𝑟

a 𝑎𝑟

𝑚𝑏

𝑚𝑏𝑟

1 0.0039 0.0039 0.55 0.6 1.7 1.72 0.0039 0.0019 0.55 0.55 1.7 1.73 0.0039 0.0039 0.55 0.55 1.7 1.64 0.0039 0.0039 0.55 0.55 1.7 0.85

ur1 (associated flow rule)ur2 (non-associated flow rule 𝜓 = 0

∘)ur3 (non-associated flow rule 𝜓 = 7.5

∘)ur4 (non-associated flow rule 𝜓 = 15


∘)

0.3

0.4

0.5

0.6

0.7

0.8

0.9

ur/r

0(%

)

1.05 1.10 1.15 1.20 1.25 1.301.00r/r0

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 10.64% (e-p model), 11.84% (𝛾𝑝 = 0.04), 13.91 (𝛾𝑝 =0.006), and 14.13% (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.


∘)ur3 (non-associated flow rule 𝜓 = 7.5



∘)

1.05 1.10 1.15 1.20 1.25 1.301.00r/r0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ur/r

0(%

)

Figure 9: Displacements of surrounding rock mass (strain-softening model, 𝛾𝑝 = 0.04).

7.2. 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: 𝜎

𝑐= 30MPa, 𝐸 = 5.5GPa, V = 0.25, 𝑝in = 5Mpa,

and 𝑟0= 5m.The 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 conducted.The following conclusions can be drawn as follows: thedifferences of plastic radius are 1.88% (2D) and 1.37% (3D)when only parameter “𝑎” is softening; the differences of radialdisplacement are 6.28% (2D) and 5.77% (3D) when onlyparameter “𝑎” is softening; the differences of plastic radius


Table 5: Parameter analyses.

H-B Softening parameters Groups 𝑟𝑝/𝑟0


∘

𝑢𝑟/𝑟0(%)


∘

𝑢𝑟/𝑟0(%)


∘

𝑢𝑟/𝑟0(%)

2D e-p 1 1.4591 0.6783 0.7897 0.85253D e-p 1.2681 0.5621 0.6088 0.63372D 𝛾𝑝 = 0.006 1.4771 0.7008 0.8276 0.89963D 𝛾𝑝 = 0.006 1.2803 0.5773 0.6331 0.66292D e-b-p 1.4807 0.7053 0.8327 0.90513D e-b-p 1.2832 0.5809 0.6367 0.66662D e-p 2 1.4591 0.6783 0.7897 0.85253D e-p 1.2681 0.5621 0.6088 0.63372D 𝛾𝑝 = 0.006 1.4602 0.6796 0.7919 0.85533D 𝛾𝑝 = 0.006 1.2688 0.5629 0.6102 0.63542D e-b-p 1.4604 0.6800 0.7923 0.85573D e-b-p 1.2690 0.5632 0.6105 0.63572D e-p 3 1.4591 0.6783 0.7897 0.85253D e-p 1.2681 0.5621 0.6088 0.63372D 𝛾𝑝 = 0.006 1.4727 0.6953 0.8195 0.89013D 𝛾𝑝 = 0.006 1.2775 0.5738 0.6282 0.65742D e-b-p 1.4777 0.7016 0.8273 0.89883D e-b-p 1.2814 0.5787 0.6337 0.66322D e-p 4 1.4591 0.6783 0.7897 0.85253D e-p 1.2531 0.5877 0.6446 0.67512D 𝛾𝑝 = 0.006 1.6778 0.9724 1.3501 1.58943D 𝛾𝑝 = 0.006 1.4509 0.8046 1.0370 1.17402D e-b-p 1.7339 1.0539 1.4379 1.67813D e-b-p 1.4882 0.8584 1.0797 1.2092

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

ur/r

0(%

)

1.05 1.10 1.15 1.20 1.25 1.301.00r/r0


∘)ur3 (associated flow rule 𝜓 = 7.5

∘)ur4 (associated flow rule 𝜓 = 15

∘)ur5 (associated flow rule 𝜓 = 20

∘)

Figure 10: Displacements of surrounding rock mass (strain-softening model, 𝛾𝑝 = 0.006).


∘)ur3ur4ur5

1.05 1.10 1.15 1.20 1.25 1.301.00r/r0

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

ur/r

0(%

)

(associated flow rule 𝜓 = 7.5∘)(associated flow rule 𝜓 = 15∘)(associated flow rule 𝜓 = 20∘)

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


are 3.23% (2D) and 2.46% (3D) when only parameter “𝑠” issoftening; the differences of radial displacement are 11.40%(2D) and 10.14% (3D) when only parameter “𝑠” is softening;the differences of plastic radius are 2.08% (2D) and 1.51% (3D)when only parameter “𝑚” was softening; the differences ofradial displacement are 6.93% (2D) and 6.25% (3D) whenonly parameter “𝑚” 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: 𝑠 > 𝑚 > 𝑎.

8. Conclusions

(1) Incorporating the 3D Hoek-Brown failure criterion,associated flow rule, nonassociated flow rule, and thestrain-softening model, numerical solutions of stress,displacement, 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

𝑎: Parameter of HB failure criterion for peakstrength [—]

𝑎𝑟: Parameter of HB failure criterion for

residual strength [—]𝑟0: Radius of the tunnel opening [L]𝐷: HB constants for the rock mass [—]𝐸: Young’s modulus of the rock mass [FL−2]𝑚𝑏: Parameter of HB failure criterion for peakstrength [—]

𝑚𝑏𝑟: Parameter of HB failure criterion forresidual strength [—]

𝑝in: Critical internal pressure [FL−2]

𝑟: Radial distance from the center of opening[L]

𝑟𝑝: Plastic radius [L]

𝑟𝑠: Residual radius [L]𝑠: Parameter of the HB failure criterion for

peak strength [—]𝑠𝑟: Parameter of the HB failure criterion for

residual strength [—]𝑢𝑟: Radial displacement [L]

𝜎0: Initial in situ stress [FL−2]

𝜎𝑐: Uniaxial compressive strength of the rock

[FL−2]𝜎𝑟: Radial normal stress [FL−2]

𝜎𝜃: Circumferential normal stress [FL−2]

𝜎𝑧: Axial normal stress along the axis of thetunnel [FL−2]

𝜎1: Major principal stresses [FL−2]

𝜎2: Middle principal stresses [FL−2]

𝜎3: Minor principal stresses [FL−2]

𝜎𝑅: Radial normal stress at the elastoplasticinterface [FL−2]

𝜀𝑟: Radial normal strain [—]𝜀𝜃: Circumferential normal strain [—]𝜀𝑧: Axial normal strain [—]𝑝: Plastic parts of normal strain and stress [—]𝑒: Elastic parts of normal strain and stress [—]𝜓: Dilation angle [—]𝛽: Dilation coefficient [—]𝛾𝑝: Softening coefficient of the surrounding

rock [—]V: Poisson’s ratio of the rock mass [—]𝐹: Yield function [—]𝑄: Plastic potential function [—].

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 (no.51208523).

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