A Hoek–Brown criterion with intrinsic material strength factorization

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International Journal of Rock Mechanics & Mining Sciences 45 (2008) 210222

A HoekBrown criterion with intrinsic material strength factorization

Thomas Benza,, Radu Schwabb, Regina A. Kautherb, Pieter A. Vermeera

aInstitute of Geotechnical Engineering, Universitat Stuttgart, Pfaffenwaldring 35, 70569 Stuttgart, GermanybFederal Waterways Engineering and Research Institute, Kussmaulstr. 17, 76187 Karlsruhe, Germany

Received 24 October 2006; received in revised form 24 March 2007; accepted 6 May 2007

Available online 21 June 2007

Abstract

Probably the most common failure criterion for rock masses is the HoekBrown (HB) failure criterion. The HB criterion is anempirical relation that extrapolates the strength of intact rock to that of rock masses. For design purposes, the HB criterion is often fitted

using equivalent Coulomb failure lines. However, equivalent MohrCoulomb (MC) shear strength parameters cannot yield the same

failure characteristics as the HB criterion. The curvilinear HB criterion automatically accommodates changing stress fields; the MC

criterion does not. The extended HB criterion proposed in this paper provides a solution to this problem by incorporating an intrinsic

material strength factorization scheme. The original HB criterion is additionally enhanced by adopting the spatial mobilized plane (SMP)

concept, first introduced by Matsuoka and Nakai (MN). The SMP concept accounts for the experimentally proven, influence of

intermediate principal stresses on failure, which is disregarded in the original HB criterion. A small set of examples provided at the end of

the article gives a good indication of the merits of using the extended HB criterion in practical applications.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: HoekBrown; MatsuokaNakai; Material strength factorization; Slope failure

1. Introduction

This paper is concerned with the numerical simulation of

rock masses using the empirical HoekBrown (HB) failure

criterion, which has been found very useful in engineering

practice. The HB criterion takes into account the proper-

ties of intact rock and introduces factors to reduce these

properties on the basis of joint characteristics within the

rock mass. Originally derived from studies on the behavior

of jointed rock masses [1], the original criterion was

subsequently changed in order to extend its use to the

behavior of weak rock masses. In the following, thegeneralized HB criterion (2002-Edition) presented in Hoek

et al. [2] is adopted. Hoek et al. derived equivalent

parameters for the MohrCoulomb (MC) failure criterion

using a best-fit procedure within a given stress domain.

Although often used in practice, these parameters cannot

reflect the non-linear features of the failure criterion they

have been derived for. Examples in the literature prove that

the ultimate load in a boundary value problem employing

the HB criterion might significantly differ from that found

in an equivalent MC analysis e.g. [3]. The incorporation of

a material strength reduction scheme in the HB criterion

directly is therefore of high value, particulary for its design

oriented use.

The brief description of the HB criterion and its basic

numerical implementation in the first part of this paper are

followed by two sections which detail the new features

added to the constitutive model: First the original HB

criterion is extended by the Spatial Mobilized Plane (SMP)

concept proposed by Matsuoka and Nakai (MN) [4,5]. It isshown here that the resulting criterion accounts reasonably

well for the intermediate principal stresss influence on

failure. Second, an internal material strength reduction

scheme is introduced. Without the need to extract

equivalent MC parameters, the HB criterion with internal

strength reduction can be used in ultimate load design

directly.

Although rock masses typically show both inherent

anisotropy and stress induced anisotropy caused by the

evolution of crack systems in an inhomogeneous stress

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Corresponding author.

E-mail address: [email protected] (T. Benz).
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field; the HB criterion was initially introduced to predict

failure in quasi-isotropic rock masses. The constitutive

model proposed here is likewise isotropic. In the following,

the sign convention of soil mechanics is used: Compressive

stress and strain is taken as positive. Tensile stress and

strain is taken as negative. All stresses are taken to be

effective values.

2. The constitutive model and implementationbasic form

2.1. Governing equations of the Generalized HB criterion

At failure, the Generalized HB criterion relates the

maximum effective stress s1 to the minimum effective stress

s3 through the equation:

s1 s3 sci mbs3

sci s

a

, (1)

where mb extrapolates the intact rock constant mi to the

rock mass:

mb mi exp GSI 100

28 14D

, (2)

sci is the uniaxial compressive strength of the intact rock,

and s and a are constants which depend upon the rock

masss characteristics:

s exp GSI 1009 3D

, 3

a 12 1

6exp

GSI15 exp

203 . 4

The Geological Strength Index (GSI), introduced by Hoek

[6] provides a system for estimating the reduction in rock

mass strength under different geological conditions. The

GSI takes into account the geometrical shape of intact rock

fragments as well as the condition of joint faces. Finally, D

is a factor that quantifies the disturbance of rock masses. It

varies from 0 (undisturbed) to 1 (disturbed) depending on

the amount of stress relief, weathering and blast damage as

a result of nearby excavations. For the significance of the

parameters and their values see [7].

The HB criterion can be more conveniently written as

fHB s1 s3 ~fs3 with ~f sci mb s3sci s

a. (5)

2.2. Basic implementation

The HB failure surfaces in 3D principal stress space can

be written piecewise as

fHB;13 s1 s3 ~fs3. (6)

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Fig. 1. HoekBrown failure criterion in principal stress space (left) and in the deviatoric plane (right). (a) Basic HoekBrown criterion (HB). (b) Extended

HoekBrown criterion (HBMN).

T. Benz et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 210222 211


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The resulting geometric representation is showm in Fig. 1.

Trial stresses outside the yield surface are returned to it

with a non-associated flow rule. The plastic potential

function is defined after [8] as

gHB;13 S1 1 sin cmob1

sin cmob S3, (7)

where Si are the so-called transformed stresses

Si si

mbsci s

m2b(8)

and cmob is the mobilized angle of dilatancy. With

increasing minor principal stress, the initial angle of

dilatancy c is reduced to 0 in a linear manner:

cmob sc s3 st

sc; cX0, (9)

where sc is the minor principal stress at which zero

volumetric plastic flow is reached and stscis=mb is the

maximum allowable tensile stress. The initial angle ofdilatancy c and the threshold stress sc are model

parameters. In the basic implementation, the model

behavior is assumed to be linear elasticperfectly plastic.

The elastic stiffness of the model is defined by two elastic

constants: the shear modulus G and Poisons ratio n.

However, a linear elastic law cannot accurately describe the

deformation characteristics of rock masses under low stress

levels. At low stress levels, the influence of microcracks on

the stressstrain law should also be considered. The pre-

failure deformation characteristics of rock masses due to

reversible closure of microcracks are discussed briefly in

Appendix A.

3. The influence of the intermediate principal stress on

failure

The influence of the intermediate principal stress on rock

failure has been experimentally investigated by numerous

researchers, e.g. [914]. From that, it is commonly

acknowledged, that such an influence exists for most rocks.

As a consequence of the experimental findings, a number of

failure criteria that account for the influence of all three

principal stresses on failure have been proposed, e.g.

[12,13,15,16]. Yet, the somewhat simpler MC and HB

criteria are still the most often used failure criteria in

engineering practice.

Recently, Al-Ajmi and Zimmermann [17,18] pointed

out the relation between the linear form of the Mogi

criterion, which accounts for the intermediate principalstress, and the MC criterion. With their MogiCoulomb

relation it is possible to overcome the cumbersome

process of parameter selection in the original Mogi

criterion, and hence, to obtain a criterion which is as

simple to use as the MC criterion and which at the same

time accounts for the intermediate principal stress

influence on failure. In the following, the HB criterion is

similarly enhanced. However, rather then using Mogis

theory, the SMP concept by MN [4,5] is applied to the HB

criterion. The model simulations presented at the end of

this section reveal that the SMP concept is not only a

reasonable assumption for soils (for which it was originally

proposed) but also for rocks. In contrast to Mogis theory,

the SMP concept guarantees the convexity of the resulting

failure surface and therefore fulfills Druckers stability

postulate [19].

3.1. The concept of SMPs by MN

MN [4,5] proposed the concept of a SMP, which defines

the plane of maximum spatial mobilization in principal

stress space. The SMP is geometrically constructed by

deriving the mobilized friction angles for each principal

stress pair separately (Fig. 2, left) and sketching therespective mobilized planes in principal stress space

(Fig. 2, right). MN derived their failure criterion by

limiting the averaged ratio of spatial normal stress to

averaged spatial shear stress on this plane. The resulting

failure stress ratio can be expressed in stress invariants:

fMN I1I2

I3 a 0 with a 9 sin

2 j

1 sin2 j , (10)

where I1, I2, and I3 are the first second and third invariant

of the stress tensor respectively and a is defined such that

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mob23mob13

mob12

3 2 1

2

Mobilized

Planes

45+mob12

2

II

III

3

1I

45

45

mob232

mob132

+

+

SMP

Fig. 2. The SMP concept after Matsuoka and Nakai. Left: Three mobilized planes where the maximum shear stress to normal stress ratio is reached for

the respective principal stresses. Right: SMP in principal stress space.

T. Benz et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 210222212


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the MN failure criterion is identical to the MC criterion in

triaxial compression and extension.

3.2. A mixed formulation of HB and MN

For a mixed formulation of the HB and the MN criteria,

the HB criterion is first expressed in triaxial p2

q space. TheRoscoe invariants p and q are functions of the first and

second invariant of the stress tensor respectively. In triaxial

conditions, they simplify to:

p 13saxial 2slateral,

q saxial slateral. 11In triaxial compression and extension, the mean stress p is

given as

pc s1 2s3

3,

pe 2s

1 s

33 , 12

respectively. When substituting Eq. (12) in Eq. (6), the

following slopes are obtained:

Msc;HB q

pc 3s1 s3

s1 2s3 3

~f

~f 3s3, 13

Mse;HB q

pe 3s1 s3

2s1 s3 3

~f

2 ~f 3s3, 14

where the superscript s indicates secant values, and the

subscripts c and e indicate triaxial compression and

extension, respectively. Tangents to the HB yield criterioncan be obtained by differentiation

dq

dp dq

ds3

dp

ds3

1, (15)

which leads to:

Mtc;HB 3 ~f

0

~f0 3

, 16

Mte;HB

3 ~f0

2 ~f0 3.

17

The slopes of the MC failure criterion on the other hand

are constant:

Mc;MC 6sin j3 sin j , 18

Me;MC 6sinj

3 sin j , 19

where again, the subscripts c and e indicate triaxial

compression and extension, respectively. A graphical

illustration of the different slopes calculated above is givenin Fig. 3. By equating the slopes derived for the HB

criterion with the MC slopes, the following locally fitted

equivalent friction angles can be found:

sin js ~f

~f 2s3 st, 20

sin jt ~f0

~f0 2

, 21

where the subscripts s and t again distinguish between the

different local fitting procedures introduced above. Forboth, the secant and the tangent fitting approach, the

resulting equivalent friction angle is a function of the minor

principal stress s3.

For a given equivalent friction angle sin j, the SMP

concept of MN, can be expressed as function of the Lode

ARTICLE IN PRESS

1[kPa]

3[kPa]

t

300

200

100

50 100 150 200

Triaxial compression

q

=

13

[kPa] 200

100

1

1

50 100 150 200

p [kPa]

Mtc,HB Ms

c,HB

ptt pt

s

Evaluation point

p = f(3)+3/3~

Fig. 3. The HB criterion in a s12s3 plane (a) and its representation in a triaxial p

2q plane (b).



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angle y as follows [20]:

Ly ffiffiffi

3p

d

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2 d 1

p 1cosW

with

W 16arc cos 1 27d

21 d22d2 d 1sin

23y

for yp0;

W p3 1

6arc cos 1 27d21 d2

2d2 d 1sin23y

for y40;

8>>>>>>>:

and d 3 sinj3 sinj , 22

where the Lode angle y is defined as

y 13arc sin 3

ffiffiffi3

pJ3

2J3=22

!(23)

and J2 and J3 are the second and third invariant of the

deviatoric stress tensor, respectively.

By way of Eq. (22), the deviatoric shape of the MN

criterion can now be assigned to the HB criterion bywriting

fHB;MN q LMc;HBp p, (24)where

p pst

scis

mbfor Mc;HB:Msc;HB

ptt ~f

Mc;HB

~f

3 s3 for Mc;HB:Mtc;HB

8>>>>>: (25)

and L is varying between 1 andMe;MCMc;MC

for triaxial

compression and extension respectively. For a graphical

interpretation of the mean stresses pst and ptt see Fig. 3.In Eq. (24) the influence of the intermediate stress in the

new failure criterion can either be defined as equal to that

in a MN criterion with identical apex (Mc;HB Msc;HB), orequal to that in a MN criterion that fits tangentially to the

HB criterion (Mc;HB Mtc;HB) in triaxial compression andextension. The tangential approach guarantees identical

instantaneous friction angles of both criteria being

combined. The secant approach does not. Therefore, the

tangential approach is employed in the remainder of this

paper. The tangential approach is from here on referred to

as the HBMN criterion.

In the numerical scheme, the previously applied potentialfunction (Eq. (7)) is not used in combination with the

HBMN model as it would introduce corner problems.

A DruckerPrager potential is employed instead:

gHB;MN q p6sincmob

3 sincmob

, (26)

where again, p and q represent the Roscoe invariants and

cmob is the mobilized angle of dilatancy as defined

previously in Eq. (9). The plastic potentials sole function

is to give the plastic strain increments after brittle failure. It

should be noted that the above choice of a radial deviatoric

flow direction is primarily a model assumption for

simplicity. The radial deviatoric flow may be replaced by

another flow direction if experimental evidence suggests to

do so.

3.3. Evaluation of the HBMN criterion

A comparison of true triaxial test data with failure

predictions of the HB and the HBMN criterion are given inFigs. 4 and 5, respectively. All experimental data for brittle

rock failure used in this comparison are taken from [21].

In their own study, Colmenares and Zoback applied a

grid search for material parameters that minimize the

mean standard deviation misfit of the criteria investigated.

Their findings for the best fit material parameters of the

HB criterion are summarized in Table 1. Fig. 4 illustrates

the resulting fit of the HB criterion to the available

test data.

As a consequence of the applied best fit procedure, the

unconfined compressive strength is overestimated by

the HB criterion. This can for example be clearly obser-

ved in the KTB Amphibolite test. In this test, the

unconfined compressive strength is explicitly tested to

158pscip176 MPa. The best fit for the original HB

criterion yields an unconfined compressive strength of

sci 250 MPa. The HBMN criterion however, givesvery reasonable results when using the actual tested

unconfined compressive strength of sci 175 MPa asinput to the model. Similar observations can be made

for the four remaining rock types considered (Fig. 5).

Therefore, only the unconfined compressive strength

input to the HBMN model is addressed in a first

model evaluation step. All other material parameters

are taken to be equal to the best fit parameters given inTable 1.

Table 2 summarizes mean standard deviation misfits of

the two failure criteria to the available test data. In

conclusion, the HBMN criterion performs better than the

HB criterion in all tests. The HBMN criterions minimum

misfit can be further reduced when the input parameter miis addressed, too: the minimum misfits shown in Table 2

were obtained in a model parameter optimization process,

similar to the one described in [21].

4. Ultimate limit state design with the HB criterion

For slopes, the factor of safety is traditionally defined as

the ratio of the actual shear strength to the minimum shear

strength required to prevent failure [22]. As the HB

criterion makes no use of Coulomb shear strength

parameters, its application in ultimate limit state design is

not straight forward. Almost all approaches to apply the

HB criterion in ultimate limit state design found in

literature rely on fitting procedures. Sometimes the HB

criterion is locally fitted using a MC criterion (e.g. [23]). In

reducing the locally fitted MC criteria, a reduced HB

criterion can be computed. Hammah et al. [24] propose a

procedure to fit such a point-wise reduced HB criterion

using the generalized HB equation in combination

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with a new set of HB parameters. Most often however, a

best MC fit of the HB criterion within a specified stress

domain is strived for (e.g. [2,25]). Then the problem is

reduced to the determination of two equivalent MC

parameters.

An ingenious solution to the problem of finding

equivalent strength parameters is proposed by Hoek et al.

[2]. They calculate equivalent shear strength parameters

within the stress domain stpsigma3ps3max by balancing

the areas above and below the straight MC failure line

enclosed by the curved HB criterion:

sinj 6ambs mbs3na1

21 a2 a 6ambs mbs3na1, 27

c sci1 2as 1 ambs3ns mbs3na1

1 a2 affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 6ambs mbs3na1=1 a2 aq ,

28where s

3n s3max

sci.

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Dunham Dolomite1200 750

625

500

375

250

125

00 125 250 375 500 625 750

1000

800

12001000800

600

600

400

400

200

2000

0

300

250

200

150

100

50

0

300

250

200

150

100

50

00 100

1500

1500

1250

1250

1000

1000

750

750

500

500

250

2500

0

50 150 200 250 300

1[M

Pa]

1[M

Pa]

2 [MPa] 2 [MPa]

2 [MPa]

0

2 [MPa]

50 100 150 200 250 300

2 [MPa]

1[MPa]

1[MPa]

1[MPa]

3 = 145 3 = 80

3 = 60

3 = 40

3 = 20

3 = 50

3 = 25

3 = 40

3 = 30

3 = 20

3 = 15

3 = 18

3 = 5

3 in [MPa]

3 in [MPa]

3 in [MPa]

3 = 150

3 = 100

3 = 60

3 = 30

3 = 0

3 = 0

3 = 30

3 = 60

3 = 100

3 = 150

3 in [MPa]

3 in [MPa]

3 = 25

3 = 45

3 = 65

3 = 85

3 = 105

3 = 125

3 = 145

3 = 20

3 = 40

3 = 60

3 = 80

3 = 25

3 = 50

3 = 8

3 = 15

3 = 20

3 = 30

3 = 40

3 = 5

3 = 1053 = 125

3 = 85

3 = 65

3 = 453 = 25

Shirahama Sandstone Yuubari Shale

Solenhofen Limestone

KTB Amphibolite

Fig. 4. Best-fitting solution after [21] when employing the HB criterion. Continuous lines give calculated results for the specified minor principal stress s3.



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Equivalent shear strength parameters, as for example

those by Hoek et al. can subsequently be used in

conventional limit-equilibrium analysis and in numerical

analysis employing the MC failure criterion. However,

it is clear that they generally will only approximate the

ultimate strength of the HB criterion, which they have been

derived for. Even for tangentially fitted parameters it has to

be considered that the minor principal stress s3 in

gravitational stress fields is not a constant. A real

equivalence of HB and MC parameters can only be given

when defining them as a function of the minor principal

stress. In conventional limit-equilibrium analysis (Bishop,

Janbu, Spencer,...) this poses a problem, but not in

automated numerical material strength reduction schemes

(e.g. j2c reduction). Dawson et al. [23] for example

explicitly calculate tangentially fitted MC parameters for

each single finite element in their numerical calculation.

However, the load steps applied in calculations with such

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Dunham Dolomite

1200

1000

800

600

400

200

00

1[MPa]

1[MPa]

2 [MPa]

1[MPa]

2 [MPa]

2 [MPa]

1[MPa]

1[MPa]

2 [MPa]

3 = 85

3 = 145

3 = 65 3 = 45

3 = 25

3 = 25

3 = 80

3 = 603 = 40

3 = 20

3 = 20

3 = 40

3 = 60

3 = 80

3in [MPa]

3 in [MPa]

3in [MPa]

3in [MPa]

3in [MPa]

2 [MPa]

3 = 45

3 = 85

3 = 40

3 = 5

3 = 50

3 = 25

3 = 25

3 = 50

3 = 8

3 = 15

3 = 20

3 = 30

3 = 40

3 = 150

3 = 100

3 = 60

3 = 303 = 0

3 = 0

3 = 30

3 = 60

3 = 100

3 = 150

3 = 30

3 = 20

3 = 15

3 = 83 = 5

3 = 125

3 = 145

3 = 105

3 = 65

200 400 600 800 1000 1200

750

625

500

375

250

125

00 125 250 375 500 625 750

Solenhonfen Limestone

Shirahama Sandstone300

250

200

150

100

50

00 50 100 150 200 250 300

300

250

200

150

100

50

0

Yuubari shale

0 50 100 150 200 250 300

KTB Amphibolite

1500

1250

1000

750

500

250

00 250 500 750 1000 1250 1500

Fig. 5. Results from the HBMN criterion. Continuous lines give calculated results for the specified minor principal stress s3.



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piecewise linear yield functions have to be small in order to

be accurate.

A more robust and reliable method to incorporate shear

strength reduction into the HB criterion is to include a

material strength reduction factor Z in its yield function:

fHB

s1

s3

~fd

s3

with ~fd

~f

Z

sci

Z

mbs3

sci s

a

.

(29)

As yet, the magnitude ofZ is hard to appraise as it is not

related to those factors commonly used in shear strength

reduction schemes. Therefore, in the following Z is related

to the shear strength reduction factors proposed in

Eurocode 7. These are:

tan jd tan jc=gj, 30cd cc=gc, 31

where the subscript c and d indicate characteristic and

design values respectively. The strength reduction factors

for the friction angle and for cohesion are generallyassumed to be equal: gj gc g. Hence, for a quantitativeinterpretation of Z, its relation to g is to be derived. It

should be noted that with the condition gj gc g, theapproach of Eurocode 7 is compatible with the idea by

Bishop, and hence g can be directly considered to be a

factor of safety in the traditional sense.

4.1. Material strength reduction in the HB criterion

In the previous section, the instantaneous friction angle

sin jt was introduced (Eq. (21)) by locally fitting a

tangential MC criterion. This friction angle is now applied

to relate the strength reduction factor Z to g: In a p2q

representation, only the slope of a MC failure line

decreases with increasing g; its apex remains constant at

p c cot j. Reducing the q over p ratio of a HB criterionequally to that of a tangentially fitted MC criterion yields

in triaxial compression:

Mc;HB

Mc;MC Mc;HBdMc;MCd

(32)

and hence,

3 ~f03 sin j

3Z ~f03g sin sin j

, (33)

where

gsin gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 sin2 j 1g2

1 s

(34)

expresses the material strength reduction factor g when

applied to sin j instead of tan j.When substituting Eqs. (21) and (34) in (33), finally an

expression for the strength reduction factor Z can be

derived. Note that Z could likewise be evaluated in triaxial

extension with the same result:

Z 12

g2 ~f0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

1g2

1

~f02

2 ~f02

vuuut ~f00BB@

1CCA. (35)

4.2. Evaluation of the material strength reduction scheme

Fig. 6 illustrates HB yield curves in triaxial p2q space

which vary due to different shear strength reduction

factors. The material parameters used in the example are

taken from a slope problem discussed in Hammah et al.

[25], which will be looked at in more detail in the next

section. As a reference, Fig. 6 also shows MC yield curves,

that are derived according to Eq. (27) for s3max 189kPa.The equivalent strength parameters are: j 20:89 andc 20kPa. The factorized shear strength parametersapplied to the MC criterion were calculated according to

Eq. (30).

From Fig. 6 it can be concluded that both, the unreduced

MC criterion, as well as the reduced MC criterion fit the

respective HB criteria reasonably well within the desired

stress domain. However, in the apex region of the HB

criterion they do not. Here, cohesion is overestimated and

the angle of friction is underestimated. The crucial issue in

using a global MC fitting procedure is clearly to specify a

suitable stress domain s3 max in which it can be applied. If

the s3max value is chosen too big, the criteria will

increasingly deviate in the apex region. If it chosen value

is too small, the criteria will considerably deviate for higher

stresses. For example ifs3max 47:5 kPa, is chosen insteadofs3 max

189kPa as shown in Fig. 6, the result will be as

shown in Fig. 7. Displayed as dashed lines in Fig. 6 are

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Table 1

Material parameters

Rock type sci(MPa) mi () GSI () D ()

Dunham Dolomite 290 (400) 8.0 100 0

Solenhofen Limestone 310 (370) 4.6 100 0

Shirahama Sandstone 45 (65) 18.2 100 0

Yuubari Shale 78 (100) 6.5 100 0

KTB Amphibolite 175 (250) 30.0 100 0

Values in brackets are used in the basic HB criterion only.

Table 2

Mean standard deviation misfit to test data in MPa

Rock type HB HBMNa HBMNb

Dunham Dolomite 56 24 21

Solenhofen Limestone 38 22 21

Shirahama Sandstone 9 8 7

Yuubari Shale 13 9 9

KTB Amphibolite 89 72 64asci, mi acc. Table 1.bsci, mi optimized.



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factorized HB criteria that result from neglecting Eq. (35),

that is setting Z g. Obviously, the latter method should beavoided ifg is defined as a shear strength reduction factor.

Then, only the transformation between Z and g defined in

Eq. (35) yields consistent results.

Appreciating the fact that global MC fitting procedures

will introduce errors, it is best to avoid them completely.

The intrinsic material strength factorization proposed here

gives the possibility to do so. A quantitative discussion of

the errors introduced by the global MC fitting procedure is

given in the following section.

5. Slope failure examples

The merits of the proposed extensions to the HB failure

criterion can best be illustrated in slope failure examples.

Results from the HB and the HBMN criteria with intrinsic

material strength reduction are compared to results from

analyses that employ equivalent MC criteria. The examples

chosen are simple, excluding soil layering, ground water flow,

etc. First, a relatively flat slope (35:5

) in a homogeneous

weathered rock layer is investigated. Second, a steeper slope

of 75:0 in likewise homogeneous rock is discussed.The first example including rock data is taken from

Hammah et al. [25]. The geometry of the slope and the

meshing used in the FE calculation is shown in Fig. 8.

Material data are given in Table 3. The equivalent MC

parameters are calculated from Eq. (27). The stress domain

for fitting is set to s3max 237 kPa following the recom-mendation by Hoek et al. [2].

All plane strain analyses were performed with the FE

code Plaxis V8 using six noded triangular elements. The

applied load stepping scheme relies on an arc-length

method. Specifically, slope stability was determined for the

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Triaxial compression Triaxial compression Triaxial compression

200 200 200

100

100

50 100 150 200

100

100

50 100 150 200

100

100

50 100 150 20013[kPa]

= 1 = 1 = 1

= 5= 2

= 1.4

P [kPa]P [kPa] P [kPa]

Triaxial extension Triaxial extension Triaxial extension

Fig. 6. Shear strength reduction of the HB criterion for three different factors of safetys3max

189kPa

.

Triaxial compression Triaxial compression Triaxial compression

200

100

100 150 200

200

100

50 100 150 200

200

100

50 100 150 20050

100 100 100

13[kpa]

Triaxial extension Triaxial extension Triaxial extension

P [kPa] P [kPa] P [kPa]

=1

=1.4

=1

=2

= 1

= 5

Fig. 7. Shear strength reduction of the HB criterion for three different factors of safety s3max 47:5kPa.



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HB, the mixed HBMN, and the MC failure criteria. For the

latter the commercially available phi-c reduction procedure

within Plaxis V8 was applied. The factors of safety for

the HB and the mixed HBMN were derived by varying the

material strength reduction factor g in steps of 0.01 and

subsequently applying gravity load to the slope. The highest

factor that leads to a convergent solution is considered the

ultimate strength reduction factor, or the factor of safety.

The equivalent MC parameters were additionally applied in

a Bishop slope stability calculation. The Bishop analysis

reproduced the results of the j-c reduction scheme reason-

ably well. Differences to results from the j-c reductionscheme were found to be well below 2%.

Results of the FE slope analysis are shown in Fig. 9. The

geometrical shapes of localized shear strains are almost

identical in all analyses. At the same time they are also in

reasonably good agreement with the circular failure surface

assumed in the simplified Bishop analysis. The material

strength reduction factors at which failure occurs are

however not in close agreement (Table 4). A detailed

discussion of the results follows after the next example,

which is a steeper slope of 75:0. Calculation proceduresare the same as outlined above. The geometry is given in

Fig. 10, material parameters are shown in Table 3. Both,

geometry and material parameters are selected in close

agreement to an example presented in Wyllie and Mah [26].

The results from the steeper slope calculation are illustrated

in Fig. 11 and quantified in Table 4.

5.1. Discussion of results

In both slope examples, the factorized HB calculation

gives least slope stability. When including the influence of

the intermediate principal stress on failure (HBMN), the

factor of safety increases. In one example, the analysis with

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0.00

10.00

20.00

30.00

40.00

50.00

0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00

Fig. 8. 35:5 slope geometry and mesh.

Table 3

Material parameters used in the slope failure analyses

Parameter Weight MN=m3 sci (MPa) mi() GSI () D () j (1) c (kPa)

35:5 slope 0.025 30 2.0 5 0.0 21 2075:0 slope 0.026 40 10.0 45 0.9 38 180

Fig. 9. 35:5 slope at failure. (a) Bishop slip circle. Incremental displacements: (b) MC, (c) HB, (d) HBMN. Shear strain: (e) HB, (f) HBMN.



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equivalent MC parameters that were determined by the

procedure outlined in [7] gives higher slope stabilities than

the HB analysis. Especially, in the steep slope example, the

differences are significant.

The geometrical shapes of localized shear strains are

almost identical in the different analyses of the flat slope. In

the analysis of the steep slope, the localized shear strains

in the HB and HBMN differ notably from those observed in

the equivalent MC calculation. As the friction angle in the

latter is generally underestimated in the apex region, it is

reasonable to obtain somewhat steeper bands of localizedshear strains in the factorized HB and HBMN calculations.

Finally, the following conclusions can be drawn from the

examples presented: (a) An equivalent MC calculation can

suggest higher slope stabilities than the HB criterion it is

derived for. Ambiguities in deriving equivalent MC

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Table 4

Slope failure analyses results

Calculation Bishop MC HB HBMN

35:5 slope 1.37 1.37 1.51 1.7275:0 slope 1.55 1.54 1.00 1.02

0.00 10.00 20.00 30.00 40.00 50.00 60.00

0.00

10.00

20.00

30.00

40.00

50.00

60.00

30.0

7.0

2.0

2.0

60slo

pe

75slop

e

Excavation

Fig. 10. 75:0 slope geometry and mesh.

Fig. 11. 75:0 slope at failure. Incremental displacements: (a) MC, (b) HB, (c) HBMN. Shear strain: (d) MC, (e) HB, (f) HBMN.



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parameters can be avoided when the HB criterion is

factorized directly. (b) The intermediate principal stresss

influence on material strength indicates higher slope

stabilities. The use of the mixed HBMN criterion may

therefore result in a more economic but yet save design.

6. Summary

For its use in numerical limit state design, the Generalized

HB criterion (2002-Edition) presented in Hoek et al. [2] has been

extended twofold. First, the influence of the intermediate

principal stress on failure was considered in a combined

formulation of the HB and the MN failure criteria. Second, an

intrinsic material strength reduction scheme for the HB criterion

was developed. In a small set of slope stability examples, the

merits of the extended HB criterion were illustrated.

The material strength reduction scheme developed is

compatible with design approaches that employ factorized

shear strength (e.g. Eurocode 7). Error prone fitting proceduresthat relate HB to equivalent MC parameters are no longer

needed. Instead, the HB criterions non-linear failure char-

acteristics can be fully considered in ultimate limit state design.

Particularly for steep slopes, this may lead to notably steeper

failure mechanisms and notably smaller factors of safety.

Appendix A. Pre-failure deformation characteristics of rock

masses

Rock mass properties, e.g. rock mass stiffness and

permeability are highly influenced by cracks. In the past,

the deformation characteristics of rock masses has been

studied by numerous researchers [27]. Based on the

stressstrain behavior shown in Fig. 12, Bienawski [28]

defines five stages in the stressstrain behavior of rock

masses: (a) Crack closure gradually occurs until the normal

stress reaches a threshold value scc. During crack closure,

stiffness increases as pre-existing cracks successively close.

Crack closure is particularly important in near-surface

structures; (b) Almost linear elastic behavior is encountered

once the majority of existing cracks are closed; (c) Stable

micro-fracturing is found after the crack initiation stress sciis reached; (d) Crack growth then becomes unstable for

stresses exceeding the threshold value scd; (e) Rock masses

may show either ductile or brittle failure depending on

geology, confining pressure and temperature.

A.1. A simple hypo-elastic law for crack closure

Within the crack closure domain the stiffness of rock

masses is a function of the normal stress acting on partially

opened cracks. With increasing stress and decreasing crack

separation, the stiffness of the entire rock mass increases.

The effect of crack closure on rock mass properties is

extensively studied in Geophysical literature, e.g. [29].

Anisotropy of mechanical rock mass properties induced by

the crack closure process is for example discussed in [30].

The linear elastic law introduced above cannot capture

these effects. A simple non-linear elastic law could be usedinstead. However, for the sake of simplicity and compat-

ibility to the isotropic HB model we propose here to use a

non-linear isotropic elastic law. Assuming that either the

mean stress p or the major principal stress s1 drive the

crack closure process, a hypo-elastic law which incorpo-

rates the effect of crack closure could be formulated as

E Ei Em Ei scc pscc

pEm or A:1

E Ei Em Eiscc s1sccpEm

, A:2

respectively. Here Ei denotes the initial Youngs modulus

for p s1 0, and Em is the maximum Youngs modulusthat is reached upon closure of all cracks in the rock mass,

i.e. pXscc or s1Xscc.

A.2. Evaluation of the crack closure law

Fig. 13 gives the result of nine unconfined compression

tests on weak siltstone samples. All test results could be

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1

ci

cd

peak

cc

peak strength

crack damage threshold

crack initiation

threshold

crack closure

threshold

1

Fig. 12. Behavior of fractured rock in uniaxial compression (after [27]).

Axialstress11[kPa]

3000

2000

1000

0.00 0.50 1.00 1.50 2.00 2.500

Axial strain 11 [%]

Experimental results

Simulation cc = 1000 kPa

Simulation cc = 400 kPa

Fig. 13. Simulation of nine unconfined compression tests with two

different crack closure stresses.



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reasonably well reproduced in the pre-failure domain by

specifying the lower and upper crack closure stress to scc 400 kPa and to scc 1000 kPa, respectively. All remainingmaterial parameters were assumed to be constant. To what

extent the two different crack closure stresses found in the

back-analysis are related to the anisotropic rock-mass

features observed in field tests on the same site, is to bedetermined in future work.

References

[1] Hoek E, Brown ET. Empirical strength criterion for rock masses.

J Geotech Eng Div, ASCE 1980;106(GT9):101335.

[2] Hoek E, Carranza-Torres C, Corkum B. Hoek-Brown failure

criterion2002 edition. In: Proceedings of the 5th North American

symposiumNARMS-TAC, Toronto, 2002.

[3] Merifield RS, Lyamin AV, Sloan SW. Limit analysis solutions for the

bearing capacity of rock masses using the generalised HoekBrown

yield criterion. Int J Rock Mech Min Sci 2006;43:92037.

[4] Matsuoka H. Stressstrain relationships of sands based on the

mobilized plane. Soils and Foundations 1974;14(2):4761.[5] Matsuoka H, Nakai T. A new failure criterion for soils in three

dimensional stresses. In: IUTAM conference on deformation and

failure of granular materials, Delft, 1982. p. 253263.

[6] Hoek E. Strength of rock and rock masses. ISRM News 1994;2(2):

416.

[7] Hoek E. Practical rock engineeringan ongoing set of notes. Available

on the Rocscience website; hhttp://www.rocscience.comi; 2004.[8] Carranza-Torres C, Fairhurst C. The elasto-plastic response of

underground excavations in rock masses that satisfy the Hoek

Brown failure criterion. Int J Rock Mech Min Sci 1999;36:777809.

[9] Chang C, Haimson BC. True triaxial strength and deformability of

the German Continental deep drilling program (KTB) deep hole

amphibolite. J Geophys Res 2000;105:189999013.

[10] Dehler W, Labuz JF. Stress path testing of an anisotropic sandstone.

J Geotech Geoenviron Eng 2007;133(1):1169.[11] Haimson BC, Chang C. A new true triaxial cell for testing mechanical

properties of rocks, and its use to determine strength and deform-

ability of Westerly granite. Int J Rock Mech Min Sci 2000;37:28597.

[12] Mogi K. Effect of the intermediate principal stress on rock failure.

J Geophys Res 1967;72(20):511731.

[13] Mogi K. Fracture and flow of rocks under high triaxial compression.

J Geophys Res 1971;76(5):125569.

[14] Smart BGD, Somerville JM, Crawford BR. A rock test cell with true

triaxial capability. Geotech Geol Eng 1999;17:15776.

[15] Ewy R. Wellbore-stability predictions by use of a modified Lade

criterion. SPE Drill Completion 1999;14(2):8591.

[16] Wiebols GA, Cook NGW. An energy criterion for the strength of

rock in polyaxial compression. Int J Rock Mech Min Sci 1968;5:

52949.

[17] Al-Ajmi AM, Zimmerman RW. Relation between the Mogi andthe Coulomb failure criteria. Int J Rock Mech Min Sci 2005;42:

4319.

[18] Al-Ajmi AM, Zimmerman RW. Stability analysis of vertical bore-

holes using the MogiCoulomb failure criterion. Int J Rock Mech

Min Sci 2006;43:120011.

[19] Drucker DC. On uniqueness in the theory of plasticity. Q Appl Math

1956;14:3542.

[20] Bardet JP. Lode dependences for isotropic pressure-sensitive elasto-

plastic materials. Trans ASME 1990;57(9):498506.

[21] Colmenares LB, Zoback MD. A statistical evaluation of intact rock

failure criteria constrained by polyaxial test data for five different

rocks. Int J Rock Mech Min Sci 2002;39:695729.

[22] Bishop AW. The use of the slip circle in the stability analysis of

slopes. Ge otechnique 1955;5(1):717.

[23] Dawson E, You K, Park Y. Strength-reduction stability analysis ofrock slopes using the HoekBrown failure criterion. In: Labuz JF,

Glaser SD, Dawson E. editors, Trends in rock mechanics. New York;

ASCE; 2000. p. 695729.

[24] Hammah RE, Yacoub TE, Corkum B. The shear strength reduction

method for the generalized HoekBrown criterion. In: Alaska rocks

2005, Anchorage; 2005.

[25] Hammah RE, Curran JH, Yacoub TE, Corkum B. Stability

analysis of rock slopes using the finite element method. In: Schubert,

editors. Eurock 2004 and Geomechanics Colloquium, Salzburg;

2004.

[26] Wyllie DC, Mah CC. Rock slope engineering. London: Spon Press;

2004.

[27] Eberhardt E, Stead D, Stimpson B, Read RS. Identifying crack

initiation and propagation thresholds in brittle rock. Can Geotech J

1998;35:22233.[28] Bienawski ZT. Mechanics of brittle rock fracture. Int J Rock Mech

Min Sci 1967;4:395423.

[29] Gibson RL, Toksoz MN. Permeability estimation from velocity

anisotropy in fractured rocks. J Geophys Res 1990;95:1564365.

[30] Schwartz LM, Murphy III WF, Berryman JG. Stress-induced

transverse isotropy in rocks. In: Stanford exploration project,

SEP80, 1994.

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A Hoek–Brown criterion with intrinsic material strength factorization

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