International Journal of Rock Mechanics & Mining Sciences 51 (2012) 43–52

Contents lists available at SciVerse ScienceDirect

International Journal ofRock Mechanics & Mining Sciences

1365-16

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ijrmms

Modified Mohr–Coulomb criterion for non-linear triaxial and polyaxialstrength of jointed rocks

Mahendra Singh n, Bhawani Singh

Department of Civil Engineering, IIT Roorkee, Roorkee 247667, India

a r t i c l e i n f o

Article history:

Received 18 March 2011

Received in revised form

4 December 2011

Accepted 17 December 2011Available online 5 February 2012

Keywords:

Jointed rock

Triaxial

Polyaxial

Strength criterion

Rock burst

09/$ - see front matter & 2012 Elsevier Ltd. A

016/j.ijrmms.2011.12.007

esponding author. Tel.: þ91 1332 285651; fa

ail address: singhfce@iitr.ernet.in (M. Singh).

a b s t r a c t

Rocks encountered in civil and mining engineering fields are invariably jointed and act under triaxial or

polyaxial stress conditions. The Mohr–Coulomb shear strength criterion is the most widely used

criterion for jointed rocks. In its present form there are two major limitations of this criterion; firstly it

considers the strength response to be linear, and, secondly the effect of the intermediate principal

stress on the strength behaviour is ignored. A modified non-linear form of Mohr–Coulomb strength

criterion has been suggested in this study to overcome these limitations. Barton’s concept of critical

state for rocks has been imbibed in the linear Mohr–Coulomb criterion to deduce a semi-empirical

expression for non-linear criterion. However, the shear strength parameters of the conventional Mohr–

Coulomb criterion are used in the proposed criterion. The proposed criterion is a simple and rational

nonlinear polyaxial strength criterion for anisotropic jointed rocks. In an earlier publication [1] the

applicability of the criterion was evaluated for intact rocks. In present paper the criterion is extended to

jointed rocks, which are anisotropic in nature. The applicability of the proposed criterion has been

verified by applying it to extensive experimental data on triaxial and polyaxial test results on jointed

rocks available from literature. Applicability of the criterion, to explain rock burst conditions for some

Indian rocks, is also demonstrated.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Rocks encountered in civil and mining engineering applica-tions are, in general, jointed and anisotropic in nature. Theirstrength, under prevailing confining stress conditions, is neededwhile analysing problems related to deep tunnels, undergroundexcavations and foundations. The strength of jointed rock, as awhole, depends on strength of the intact rock, joint geometry andsurface characteristics of the joints. Depending on joint geometryand joint strength characteristics the rock blocks may undergosliding, shearing, splitting or rotation at the time of failure.Substantial research has been carried out in past to understandthe mechanical behaviour of rock joints [2–6]. The outcome ofthese studies may be used to analyse the rock mass behaviour ifthe joints are modelled explicitly.

Classification approaches [7–8] consider the rock mass as anequivalent continuum, and the effect of the joints is consideredimplicitly. These approaches have found wide acceptability in thefield. Laboratory studies on rocks and model materials have alsobeen used to represent rock mass as an isotropic or anisotropicequivalent continuum [9–13]. These equivalent continuum

ll rights reserved.

x: þ91 1332 275568.

approaches can be used to characterize the rock mass, fromwhich the rock mass strength under unconfined state may beobtained. The effect of confinement (triaxial or polyaxial) maythen be included using an appropriate strength criterion. Themain objective of the present study is to suggest an approach inwhich the effect of minor and intermediate principal stress on thestrength of jointed rock mass can be obtained with adequateaccuracy at any given confining pressure.

The strength behaviour of the rocks is generally expressed by astrength criterion. Mohr–Coulomb strength criterion is the mostwidely used criterion for intact and jointed rocks as well.As discussed in earlier publication [1], the criterion in its presentform suffers from two major limitations: (i) it ignores the non-linearity in strength behaviour, and (ii) the effect of intermediateprincipal stress is not considered in its conventional form. A non-linear strength criterion for intact rocks was suggested by Singhet al. [1], which is an extended form of the conventionalMohr–Coulomb criterion. The intermediate principal stress wasalso incorporated in the criterion. Using available extensive datafrom triaxial and polyaxial tests it was shown that the proposedsimple criterion works better than the other popular criteria invogue. An important advantage of the proposed criterion is thatthe conventional Mohr–Coulomb shear strength parameters areretained as such. In present paper, the criterion proposed forintact rock [1] is extended to jointed rocks. The applicability of

30.0

M. Singh, B. Singh / International Journal of Rock Mechanics & Mining Sciences 51 (2012) 43–5244

the proposed criterion has been verified by applying it to database available from literature.

0.0

5.0

10.0

15.0

20.0

25.0

0.0 10.0 20.0 30.0 40.0Normal stress (MPa)

She

ar s

tress

(MP

a)

Unjointed (constant weight)

Zone 1

T60 (weakest model)

Single joint

H30 (strongest model)

Zone 2

34.5°

Zone 3

Fig. 1. Mohr envelopes for intact and jointed specimens (redrawn from [16]).

σ3

σ 1-σ

3

σcrtj

Aj'(σ3)2

σcj

σci

Intact

Jointed

Mohr-Coulomb

Fig. 2. Non-linear variation of strength of intact and jointed rock.

2. Triaxial conditions

2.1. Modified Mohr–Coulomb criterion

The complete derivation of the criterion for intact rocks hasalready been presented in [1]. The criterion was deduced fromBarton’s concept of critical state in rocks [14]. Barton [14] statesthat ‘‘critical state for any intact rock is defined as stress condition

under which Mohr-envelope of peak shear strength of the rocks

reaches a point of zero gradient. This condition represents the

maximum possible shear strength of the rock. For each rock, there

will be a critical effective confining pressure above which the shear

strength cannot be made to increase’’. Modified Mohr–Coulombcriterion for intact rock [1], in its general form, was expressed as

ðs1�s3Þ ¼ sciþ2sinfi0

1�sinfi0

s3�A0s23 for 0rs3rscrti ð1Þ

where s3 and s1 are the effective minor and major principalstresses at failure; sci is the UCS of the intact rock, defined as

sci ¼2ci0cosfi0

1�sinfi0

ð2Þ

where ci0 and fi0 are the Mohr–Coulomb shear strength para-meters obtained by conducting triaxial strength tests on rockspecimens at low confining pressures (s3-0); A0 is an empiricalconstant for the rock type under consideration, and scrti is thecritical confining pressure for the rock. By employing critical stateconcept [14], Eq. (1) was reduced to the following form:

ðs1�s3Þ ¼ sciþ2sinfi0

1�sinfi0

s3�1

scrti

sinfi0

ð1�sinfi0Þs2

3 for 0rs3r scrti

ð3Þ

or

ðs1�s3Þ ¼ sciþ2mi0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þm2i0

q�mi0

s3�1

scrti

mi0ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þm2

i0

q�mi0

s23

for 0rs3rscrti ð4Þ

where mi0 is the coefficient of internal friction (tan fi0) of the rockunder low stresses.

The critical confining pressure will depend on the particularrock type and its lithology. However statistical analysis of morethan 1100 triaxial test data in [1] has shown that the criticalconfining pressure for an intact rock can be taken nearly equal toits UCS without introducing error of engineering significance inthe prediction of confined strength. The modified Mohr–Coulombcriterion for intact isotropic rocks under triaxial stress conditionmay therefore be expressed as

ðs1�s3Þ ¼ sciþ2sinfi0

1�sinfi0

s3�1

sci

sinfi0

ð1�sinfi0Þs2

3 for 0r s3 r sci

ð5Þ

Brown [15] conducted a number of triaxial tests on intact andjointed rock mass specimens under triaxial stress condition. Fig. 1(redrawn from Brown [15]) shows a plot indicating variation ofshear stress at failure with normal stress for intact and jointedrock. It is observed from this plot, that the failure envelopes forjointed and intact rocks tend to merge with each other atsufficiently high confining pressure (zone 3, Fig. 1). The failureenvelope for the intact rock represents the upper limit of all thefailure envelopes. Non-linearity in strength behaviour at low s3,is very high in case of jointed rock as compared to the intact rock.Consequently, the difference between the strength values of

intact and jointed rock is high at low confining pressures.As confining pressure increases, the difference between thestrength of the intact and the jointed rock decreases. At suffi-ciently high confining pressure this difference tends to reduce toalmost zero. In other words beyond this confining pressure, theeffect of joints ceases to exist, and the jointed rock behaves like anintact rock; as the intact rock follows critical state concept, thejointed rock will also follow the same.

Taking the idea from the above discussion, a model forstrength behaviour of jointed rocks is shown in Fig. 2. Thestrength of jointed rock is assumed to increase with confiningpressure following a parabolic variation and the strength envel-ope merges with that of intact rock at a sufficiently high confining

Table 1Triaxial test data base for jointed rocks.

S. no. Citation Rock type sci (MPa) No. of triaxial

tests on jointed

rock

1. Brown [15] Gypsum plaster 20.75 20

2. Brown and

Trollope [16]

Gypsum plaster 20.75 16

3. Einstein and

Hirschfeld [17]

Concrete bricks 29.83 42

4. Hoek [18] Andesite 265.5 4

5. Yaji [19] Plaster of Paris 9.5 120

6. Yaji [19] Granite 123.0 12

7. Yaji [19] Kota sandstone 70.0 20

8. Arora [20] Plaster of Paris 11.3 256

9. Arora [20] Jamrani

sandstone

55.10 64

10. Arora [20] Agra sandstone 110.0 92

11. Roy [21] Plaster of Paris 9.46 80

12. Soni [22] Sandstone 63.40 6

Total 732

M. Singh, B. Singh / International Journal of Rock Mechanics & Mining Sciences 51 (2012) 43–52 45

pressure. Further, according to the critical state concept [14], thefailure envelope will become nearly horizontal at critical confin-ing pressure of the jointed rock. In critical state, the deviatorstress at failure (s1�s3) will be nearly same for both intact andjointed rocks. So, the semi-empirical expression for triaxialstrength criterion for the jointed rock may be written as

ðs1�s3Þ ¼ scjþ2sinfj0

1�sinfj0

s3�A0js23 for 0rs3rscrtj ð6Þ

where s1, s3 are the effective major and minor principal stressesat failure for jointed rock, scrtj is the critical confining pressure ofthe jointed rock, scj is the anisotropic UCS of the jointed rock inthe direction of major principal stress, hereafter termed rock massstrength, and defined as

scj ¼2cj0cosfj0

1�sinfj0

ð7Þ

where cj0 and fj0 are the Mohr–Coulomb shear strength para-meters of the anisotropic jointed rock at low confining pressure(s3-0); A0j is an empirical constant which accounts for thedeviation in strength of jointed rock from conventional linearMohr–Coulomb criterion.

It has already been shown in [1] that for all practical purposes,the average value of critical confining pressure for an intactisotropic rock may be taken equal to its UCS, sci. The failureenvelope of the jointed rock may merge with the failure envelopeof the intact rock either at a confining pressure less than sci ormore than sci. However, when both intact and jointed rock reachcritical state, the strength will be equal as it is evident thatmerger of the two failure envelopes has to take place at a highconfining pressure. Now considering the critical confining pres-sure of the jointed rock equal to scrtj, the gradient of the failureenvelope of jointed rock should be asymptotic to a horizontal lineat s3¼scrtj. Differentiating Eq. (6) with respect to s3 yields

@ðs1�s3Þ

@s3¼

2sinfj0

1�sinfj0

�2A0js3

for s3-scrtj;@ðs1�s3Þ

@s3-0

) Aj

0

¼1

scrtj

sinfj0

1�sinfj0

ð8Þ

The modified Mohr–Coulomb criterion for jointed rock maynow be expressed as

ðs1�s3Þ ¼ scjþ2sinfj0

1�sinfj0

s3�1

scrtj

sinfj0

ð1�sinfj0Þs2

3 for 0rs3rscrtj

ð9Þ

The maximum deviatoric stress at failure for jointed rock willbe reached when s3¼scrtj

) ðs1�s3Þmax ¼ scjþsinfj0

1�sinfj0

scrtj ¼ scjþBj

2scrtj ð10Þ

where

Bj ¼2sinfj0

1�sinfj0

ð11Þ

The maximum deviatoric stress (s1�s3) at failure for theintact rock may be obtained by putting s3 equal to sci in the RHSof the Eq. (5) as

ðs1�s3Þmax ¼ sciþsinfi0

1�sinfi0

sci ¼ sciþBi

2sci ð12Þ

where

Bi ¼2sinfi0

1�sinfi0

ð13Þ

As discussed earlier, the strength of intact and jointed rockswill be nearly equal in critical state. Equating the maximumdeviatoric stresses at failure for intact and jointed rock, thefollowing expression is obtained:

Bj

2¼ 1þ

Bi

2�SRF

� �sci

scrtjð14Þ

where

SRF ¼ Strength reduction f actor¼scj

scið15Þ

Using expression (14) the friction angle of the jointed rock fj0

may now be predicted. It needs intact rock properties sci and fi0.The parameter fi0 may be obtained by conducting few triaxialtests at low s3 values. Once fj0 is available, Eq. (9) may be used tosimulate the triaxial strength of the jointed rock for givenconfining pressure. For this purpose two more parametersi.e. scrtj and scj will be required.

2.2. Critical confining pressure of jointed rocks

Since no method is available to obtain the critical confiningpressure of jointed rocks, it was decided to fit the proposedcriterion into a data-base collected from literature. The data wascollected from the sources, where triaxial strength tests wereconducted on intact as well as the jointed rock. The details of thedata base i.e. rock type, UCS of intact rock, confining pressure forwhich triaxial tests were conducted on the jointed specimens aregiven in Table 1. The data base for jointed condition comprised of153 sets of triaxial data tests. The total number of triaxial testdata was more than 730. The complete data base is available inAppendix-I as supplementary electronic material (SEM) on thewww.ScienceDirect.com20website. The UCS, scj was available formajority cases, however, if not available; the same was obtainedby extrapolating the triaxial test results. A computer program waswritten to back-analyse this data base.

The following steps are used by the computer program:

i.

Let there be a number of rock-types for which triaxial testdata is available. For each rock type, first data set comprisesof triaxial test results for intact rock in the form of minor and

10

100

A

M. Singh, B. Singh / International Journal of Rock Mechanics & Mining Sciences 51 (2012) 43–5246

major principal stresses at failure (s3,s1). This data set isfollowed by a number of data sets where results on (s3,s1) atfailure are available for different joint configurations. So, foreach rock type there is one set of triaxial test results forintact rock, and more than one sets of data for jointedcondition.

O

ii. Consider first rock-type of the data base.

e C

iii.

rag

Assume first trial vale of scrtj, say scrtj¼0.25sci where sci isthe UCS of the intact rock.

ve

iv.

0.1

1

0.1 1 10

A

σcrtj/σci

100

Use triaxial test results of intact rock to obtain the optimisedvalue of fi0 as explained in [1] using the following expressions:

Ai ¼

Pðs1�s3�sciÞPðs2

3�2scis3Þ; 0os3rsci ð16Þ

Bi ¼�2Aisci ð17Þ

sinfi0 ¼Bi

2þBið18Þ

v.

Consider first set of jointed rock data. vi.

CO

A

Use scj to obtain fj0 from expressions (11) and (14). Now putfj0 and scj in Eq. (9) and predict triaxial strength s1 for givenvalues of s3 for this data set.

10n

in

vii.

0.1

1

0.1 1 10

Sta

ndar

d de

viat

io

For the data set under consideration, compute the indextermed coefficient of accordance (COA) c2 defined as fol-lows:

c2¼

Pðs1exp�s1calÞ

2

Pðs1exp�s1avÞ

2ð19Þ

where s1exp is the experimental value of major principal stressat failure; s1cal is the predicted value of s1 for a given s3; ands1av is the average of the experimental s1 values for the triaxialdata set under consideration. A lower value of COA indicates abetter prediction.

σcrtj/σci

viii. Repeat steps (vi) and (vii) for all other triaxial data sets forthis rock type.

Fig. 3. (a) Variation of average coefficient of accordance with assumed critical

confining ratio. and (b) Variation of standard deviation in coefficient of accordance

ix.

with assumed critical confining ratio.

Consider next rock type and repeat steps (iii) to (viii) for allrock types to get COA for all 153 sets of all rock types.

x.

Compute the average of all the 153 values of COA. This valueindicates how good the predictions are for the entire data baseif the critical confining pressure is taken equal to 0.25sci.The standard deviation of all COA values is also computed.

xi.

Now consider the next trial value of scrtj (¼0.5sci). Againrepeat steps (iv) to (x) to get the average and the standarddeviation of COA values. The computations are repeated till amaximum value of scrtj (¼5sci) is reached.

Fig. 3a and b show the variations of average COA and standarddeviation in COA with assumed scrtj. Similar to what was observedfor intact rocks [1], it is seen here that the average value of COAdecreases with increasing scrtj and reaches a minimum at scrtjEsci

and then increases for scrtj4sci. Standard deviation in COA alsohas been found to attain a minimum near scrtjEsci. It was alsoseen in case of intact rocks [1] that for all practical purposes thecritical confining pressure for intact rocks could be assumed equalto sci. It is, therefore, suggested that in case of jointed rocks also,the critical confining pressure may be assumed nearly equal to sci

for strength prediction using the proposed criterion. Furtherresearch is however needed on this aspect.

Now taking scrtj equal to sci, the simplified form of theproposed criterion may be written as

ðs1�s3Þ ¼ scjþ2sinfj0

1�sinfj0

s3�1

sci

sinfj0

ð1�sinfj0Þs2

3 for 0rs3rsci

ð20Þ

where scj and fj0 are the criterion parameters. The value of theanisotropic friction angle fj0 may be obtained from Eq. (14) as

sinfj0 ¼ð1�SRFÞþ sinfi0

1�sinfi0

ð2�SRFÞþ sinfi0

1�sinfi0

ð21Þ

Eqs. (20) and (21) were used to predict s1 values for all thetriaxial tests with inputting only sci, scj and fi0. A comparison ofthe predicted and experimental strength values for the entiretriaxial test data for jointed rocks is shown in Fig. 4. The figureshows a plot between the experimental and computed values ofs1 at failure for all the triaxial tests. Majority of data points in thisplot are found to be located near the line having 1:1 slope. Also avery good correlation coefficient of 0.982 has been obtained. Thusit can be inferred that reasonably good estimates for s1 at failureare made by the proposed criterion with only input of scj forjointed rock and sci and fi0 for the intact rock.

The above analysis indicates that the proposed critical-state-concept-based-Mohr–Coulomb criterion does have good potentialof assessing triaxial strength of jointed rocks also. Eq. (21)indicates that the friction angle of jointed rock mass (fj0) isslightly higher than that of intact rock material (fi0). Thisequation considers indirectly the effect of interlocking betweenthe blocks of rocks during shearing. Hence, the initial angle ofdilatation (D0) of jointed rock mass may be of the orderof (fj0�fi0)/2, which may be used in numerical modelling.

Fig. 4. Comparison of predicted and laboratory values of triaxial strength (for

all data).

0

0.2

0.4

0.6

0.8

1

-60 -40 -20 0 20 40 60Permissible error, %

Pro

babi

lity

Fig. 5. Probability of error in predicting triaxial strength of jointed rocks.

M. Singh, B. Singh / International Journal of Rock Mechanics & Mining Sciences 51 (2012) 43–52 47

This school of thought on the critical state mechanics needs to besimulated in numerical modelling of deep underground openings,drill holes, foundations and fault-slip analysis, etc.

2.3. Probability of error in prediction of triaxial strength

A designer will have more confidence in his analysis if he hasan idea of the probable error in predicted strength of the rockmass. A probabilistic analysis has been done in this section tohave insight into the probable error in prediction, if the presentcriterion is used to predict triaxial strength of the jointed rockmass. The available triaxial test data base for jointed rock withmore than 730 data points has been used for this purpose. Thetriaxial strength of jointed rocks predicted through inputting scj,

sci and fi0 were used to compute the error in prediction for all thedata points as given below:

Error in prediction ð%Þ ¼s1cal�s1exp

s1exp� 100 ð22Þ

The probabilities of occurrence of these errors were computedusing the data base. The results of this analysis are shown inthe form of probability distribution function (Fig. 5). The y

co-ordinate of any point on this plot indicates the probabilityand the x co-ordinate indicates the corresponding error. The valueof y co-ordinate gives probability for which the error will be equalto or less than the corresponding x value. It is seen that theprobability of error to be equal to or less than �20% is 0.059. Alsothe probability of error to be equal to or less than þ20% is 0.936.This indicates that the probability of error to lie within �20 toþ20% is 0.936�0.059 i.e. 0.877, which indicates that reasonablygood estimates are possible using the present criterion.

2.4. Comments on parameter scj

For a realistic prediction of strength, it is essential that thecriterion parameters are assessed with good accuracy. The para-meter fj0 can be obtained as discussed in the previous section. Theother parameter scj, in the present study, was available fromlaboratory model tests. An excellent discussion on obtaining rockmass strength scj in field is presented by Zhang [23]. The followingapproaches may be used to get the rock mass strength in the field:RQD [23], Rock Mass Rating [7], Rock Mass Quality Q [8],Joint Factor [9–11], Fracture Tensor [12,13] or the Rock MassIndex, RMi [24].

In the opinion of the authors, the most reliable results can beobtained through field testing only. It is, however, extremelydifficult to load a rock mass in the field up to its failure stress.Alternatively, the rock mass in the field may be loaded only upto apre-defined stress level and its modulus may be obtained. It isshown by Singh and Rao [25] that under uniaxial loading condi-tions, there is a strong correlation between the strength reductionfactor and the modulus reduction factor of a rock mass. Thiscorrelation may be used to get the strength of the rock mass asfollows:

SRF ¼ ðMRFÞ0:63ð23Þ

where

SRF ¼ Strength reduction f actor ¼scj

sci

MRF ¼Modulus reduction f actor¼Ej

Ei

and Ej and Ei are the elastic modulii of the rock mass and theintact rock, respectively, in the principal stress direction. Intactrock modulus Ei may be obtained from laboratory tests. Rockmass modulus Ej may be obtained from uniaxial jacking tests asexplained in [25].

Amongst classification approaches, Q system is the mostwidely used approach for tunnelling projects in India. Based onthe back analysis of case studies of 39 tunnelling projects fromIndia and abroad, Singh et al. [26] proposed the followingcorrelation, which can be used with confidence, for UCS of therock mass for tunnels:

scj ¼ 7gQ1=3ðfor Q o10, 2oscio100MPa, SRF ¼ 2:5Þ ð24Þ

where scj is rock mass strength in MPa, g is the unit weight of therock mass in gm/cc and Q is the post construction rock massquality.

Barton [27] has suggested the following expression to computethe rock mass strength:

scj ¼ 5g Qsci

100

� �1=3

MPa ð25Þ

where g is the unit weight of the rock mass in gm/cc, and scj andsci are in MPa.

3. Extension to polyaxial stress conditions

The importance of considering the strength of rocks in poly-axial condition has been highlighted in the first part of thisstudy [1]. It has been supported from literature that there isgrowing concern amongst the geotechnical fraternity about thepolyaxial strength of rocks and rock masses. In the followingsection the modified Mohr–Coulomb criterion has been extendedto polyaxial stress conditions and it has been shown that reason-ably good estimates of the polyaxial strength of jointed rocks can

Table 2Polyaxial strength test data with model material sand lime brick from Tiwari and

Rao [29] (sci¼13.5 MPa, fi0¼44.701).

s3 (MPa) s2 (MPa) s1 at failure (MPa)

Joint inclination, y1

0 20 40 60 80 90

0.31 0.31 6.81 4.23 2.92 1.84 4.91 7.28

0.31 0.59 7.78 5.01 5.32 2.85 6.76 8.68

0.31 0.95 7.65 6.05 7.34 3.67 7.69 8.34

0.31 1.22 9.32 7.41 7.96 4.24 8.27 7.33

0.31 1.62 9.32 8.68 8.54 7.53 8.54 9.18

0.78 0.78 9.76 7.23 6.26 3.54 7.73 9.54

0.78 1.22 10.60 9.01 9.38 4.99 8.59 10.65

0.78 2.24 11.45 10.60 11.91 7.66 10.86 11.97

1.22 1.22 12.51 9.32 9.13 4.93 9.36 13.13

M. Singh, B. Singh / International Journal of Rock Mechanics & Mining Sciences 51 (2012) 43–5248

be made with input from the UCS and triaxial tests on intact rockspecimens and UCS of the jointed rock.

3.1. Polyaxial strength criterion

On lines similar to intact rock criterion [1] the strengthcriterion for jointed rocks (Eq. 20) is extended to polyaxial stresscondition purely on trial basis. The criterion for polyaxial strengthis expressed as

ðs1�s3Þ ¼ scjþ2sinfj0

1�sinfj0

s2þs3

2

� ��

1

sci

sinfj0

ð1�sinfj0Þ

s22þs2

3

2

� �

for 0rs3rs2rsci ð26Þ

where scj is the anisotropic strength of the rock mass underuniaxial loading condition (s3¼s2¼0) in the direction of s1,which will depend on the characteristics of the joints (frequency,orientation and surface roughness) and the properties of theintact rock; fj0 is the anisotropic friction angle of the rock massat low confining stress level and may be obtained as a function ofSRF and fj0 using Eq. (21).

3.2. Application to polyaxial strength test data

It is very difficult and time consuming to prepare models of rockmass and test them in laboratory. Though several studies havebeen reported on jointed rocks under uniaxial, triaxial and directshear condition; the studies on rock masses (involving largenumber of intact rock blocks) are rather limited. The first authorof this paper conducted an extensive study [28] on a jointed blockymass under uniaxial loading conditions. The test specimens wereprepared by arranging cubical blocks of 25 mm�25 mm�25 mmsize (Fig. 6) and forming a mass of 150 mm�150 mm�150 mmsize. Tiwari and Rao [29] extended this study to polyaxial stressconditions. The stresses were applied as shown in Fig. 6. The detailsof the tests are available in the works of Tiwari and Rao [29,30]. Inall, fifty four tests were conducted in polyaxial and triaxial stressconditions. The test results are reproduced in Table 2.

The s3 values in these tests were varied between 0.31 MPa to1.22 MPa and s2 values were varied from 0.31 to 2.24 MPa(Table 2). It is reported by Tiwari and Rao [30] that differentfailure patterns were observed depending upon joint configura-tion, stress ratio and stress orientations. In case of specimens withy¼0, 20, 80 and 901, shearing of intact material was observed,where y is the dip of the continuous joint set. The shear planeswere observed to be dipping in s3-direction. In specimens with

Fig. 6. Joint configurations tested in [28,29].

y¼401, joint dilation and shear of some blocks occurred, whereassliding of the blocks along joint planes mixed with shearing wasobserved for y¼601. The shifting of failure mode from sliding anddilation to shearing was also observed with increasing s2/s3 ratio.It is also reported that brittleness of rock mass increased withincrease in s2 and the mass became very brittle at higher s2. Itspost peak curve also became steeper.

To validate the applicability of the proposed polyaxial strengthcriterion to the tests results from Tiwari and Rao [29], thefollowing input parameters, available from laboratory tests, wereused: sci¼13.5 MPa; fi0¼44.71; scrtj¼sci¼13.5 MPa.

The anisotropic UCS of the rock mass scj, was obtained byextrapolating the test results pertaining to triaxial conditions. Thepolyaxial strength for all the test specimens was computed usingthe proposed polyaxial criterion. A comparison of the predicted s1

values with laboratory s1 values is presented in Fig. 7. Thepercent error for different specimen types is also given inTable 3. It can be seen that with only input parameters sci, fi0

and scj, the predictions are reasonably good for specimens withy¼0, 20, 40, 80 and 901 and have average percent error withinabout 15%.

The predictions for specimens with y¼601 are not good(average per cent error¼49.6%). It may be noted that, for thesespecimens, the failure pattern observed by Tiwari and Rao [29]was sliding of blocks along joint planes mixed with shearing. Atthis orientation the joints are critically inclined and the wedgeformed by the intersecting joints is likely to slide under its ownweight at low confining pressure. The intact rock properties areless significant in this case. It is the orientation at which geometryof the blocks and joint characteristics play dominant role infailure mechanism. The rock mass strength scj was of the orderof 4% of intact rock strength, sci. The present criterion is, there-fore, not applicable for this case and a wedge analysis byincorporating an appropriate strength criterion for joints (sayBarton and Chaubey [31]) will be more justified. For y¼401, thescj was about 6.3% of sci and the average percent error was 15.3%.A limiting value of rock mass strength, scj for applicability of theproposed criterion may, therefore, be taken to be about 7% ofintact rock strength sci.

It is also worth mentioning that in the study by Tiwari and Rao[29] the stress s2 was acting in the dip direction of joint planes.Despite this, the failure planes for specimens with y¼0, 20, 80and 901 dipped in the direction of s3. The predictions of theproposed polyaxial criterion were very good for these specimens.The criterion is, therefore, most suitable for block jointed rockmasses where the failure planes dip in s3 direction, and failure isthrough joints and intact rock as well.

0

5

10

15

0 5 10 15σ1exp, MPa

σ 1ca

l, M

Pa

1:1

θ = 00°

0

5

10

15

0 5 10 15σ1exp, MPa

σ 1ca

l, M

Pa

1:1

θ = 20°

0

5

10

15

0 5 10 15σ1exp, MPa

σ 1ca

l, M

Pa

1:1

θ = 40°

0

5

10

15

0 5 10 15σ1exp, MPa

σ 1ca

l, M

Pa

1:1

θ = 60°

0

5

10

15

0 5 10 15σ1exp, MPa

σ 1ca

l, M

Pa

1:1

θ = 80°

0

5

10

15

0 5 10 15σ1exp, MPa

σ 1ca

l, M

Pa

1:1

θ = 90°

y = 1.0706xR2 = 0.8268

0

5

10

15

0 5 10 15σ1exp, MPa

σ 1ca

l, M

Pa

1:1

All data (except θ = 60°)

Fig. 7. Comparison of predicted and experimental polyaxial strength values.

M. Singh, B. Singh / International Journal of Rock Mechanics & Mining Sciences 51 (2012) 43–52 49

M. Singh, B. Singh / International Journal of Rock Mechanics & Mining Sciences 51 (2012) 43–5250

3.3. Application to rock burst conditions

One of the most important applications of the polyaxialcriterion is in analysing rock burst conditions in deep under-ground openings. It has been observed in Indian conditions thatsqueezing occurs in tunnels under high overburden if the jointfriction angle obtained from Q(¼ tan�1ðjr=jaÞ) is less than 301;whereas rock burst occurs if the joint friction angle is more than301 and the overburden is more than about 900 m. At the time offailure of the rock, the strain energy released depends upon stressdifference (s1�s3). For same s3, the strain energy released will,therefore, be high if s2 is high. In case of underground openings,the minor principal stress s3 at the periphery of the opening willbe nearly zero. If the out of plane stress along axis of the opening(s2) is high, the strain energy released will be high, and the rockwill be more brittle and this may result in rock burst conditionrather than squeezing condition. Therefore, the following criter-ion for rock burst in tunnels is proposed for tunnel periphery byputting s3 equal to 0 in Eq. (26):

syZscjþ2sinfj0

1�sinfj0

s2

2

� ��

1

sci

sinfj0

ð1�sinfj0Þ

s22

2

� �for 0rs2rsci

ð27Þ

where sy is the circumferential stress at the periphery of theopening. If the intermediate principal stress at the site is not

Table 3Average percent error in predicting polyaxial strength of jointed rock mass.

s3 (MPa) s2 (MPa) Percent error in predicting s1

Joint inclination, y1

0 20 40 60 80 90

0.31 0.31 3.2 15.2 9.1 60.2 17.6 1.2

0.31 0.59 0.8 14.4 23.3 35.1 2.1 7.7

0.31 0.95 15.9 12.6 29.1 35.7 0.2 8.2

0.31 1.22 3.1 2.6 24.3 36.9 2.1 33.3

0.31 1.62 14.7 0.7 15.5 7.1 11.9 18.1

0.78 0.78 4.7 13.6 6.1 81.4 17.1 8.7

0.78 1.22 7.9 5.5 14.8 55.9 20.1 8.9

0.78 2.24 23.3 16.4 8.2 40.2 20.5 19.2

1.22 1.22 4.8 20.6 7.1 94.2 28.5 1.0

Average 8.7 11.3 15.3 49.6 13.3 11.8

Table 4Computations for rock burst potential in NJPC tunnel (g¼2.7 g/cc).

Section H

(m)

Q sci,

(MPa)fi0

(deg.)

scj,

(MPa)

SRF fj0

(deg.)

s2

(MPa)

1 1430 4.7 50 45 31.7 0.633 47.3 38.61

2 1420 4 32 37 30.0 0.938 37.7 38.34

3 1420 4.5 50 45 31.2 0.624 47.4 38.34

4 1320 1.8 32 37 23.0 0.718 39.9 35.64

5 1300 3.5 50 45 28.7 0.574 47.7 35.1

6 1300 2 60 45 23.8 0.397 48.7 35.1

7 1300 1.8 55 45 23.0 0.418 48.6 35.1

8 1300 3.3 50 45 28.1 0.563 47.8 35.1

9 1230 2.2 50 45 24.6 0.492 48.2 33.21

10 1180 4.7 42 55 31.7 0.754 55.8 31.86

11 1180 2 34 30 23.8 0.7 34.4 31.86

12 1180 3.4 42 45 28.4 0.677 47.1 31.86

13 1100 7.5 42 45 37.0 0.881 45.8 29.7

14 1090 7 50 45 36.2 0.723 46.8 29.43

15 1060 3.8 50 45 29.5 0.59 47.6 28.62

a Critical state.

known, then an approximate value equal to the overburdenpressure may be used for s2.

A case study of tunnel for a power project in Indian Hima-layas has been discussed here to demonstrate the influence ofconsidering effect of intermediate principal stress on thestrength of the rock mass. The NJPC tunnel passes through augengneisses having Q values in the range of 1.8 to 7.5 and the ratioof parameters Jr/Ja is more than 0.75 [32]. According toMohr–Coulomb theory rock burst was expected for more than900 m overburden. However, when the tunnel was constructed,there was slabbing of rocks and mild bursts only. Minor rocksupports were sufficient though heavy rock burst conditionswere feared. An approximate analysis of the tunnel sections at15 locations is presented in Table 4. The intermediate principalstress at the periphery has been taken equal to the overburdenpressure gH, and the maximum circumferential stress at theperiphery is taken equal to 2gH. The rock mass strength wascalculated using Q (Eq. 24). Friction angle fi0 for the rockmaterial was available. Strength reduction factor (SRF) wascomputed using scj and sci, and friction angle of rock masswas computed using Eq. (21). The RHS of Eq. (27) was computed,which gives the biaxial strength s1 of the rock mass at theperiphery of the tunnel. The computations (Table 4) show thatfor two sections s2 was more than sci, which indicates criticalstate reached at these sections. To compute strength in case ofcritical state, sci was used in place of s2 in the RHS of Eq. (27). IfMohr–Coulomb criterion is used, the strength of the rock at theperiphery will be equal to scj. Finally the biaxial strength tostress ratio was computed for both i.e. Mohr–Coulomb andpresent criterion respectively. It is seen that in case of thepresent criterion, the biaxial strength-to-stress ratio varies from0.72 to 1.98, and is less than one in three cases out of total fifteensections. This implies that mild rock bursting is likely to occuronly at few places. If the Mohr–Coulomb criterion is used, thestrength-to-stress ratio varies between 0.32 and 0.62, whichsuggests severe rock bursting at all the sections. As stated above,only mild rock burst was observed, thus the results using theproposed criterion are more realistic.

Another important application of critical state mechanics andpolyaxial strength criterion may be in the field of seismology.The depth of brittle earth crust may be derived approximately assci/(Kg), where K is the ratio of minor in-situ horizontal andvertical stress. The most of the epicentres of the earthquakes arewithin this brittle earth crust.

Ratio

s2/sci

s1

(MPa)

symax

(MPa)

Strength to stress ratio

Present criterion Mohr–Coulomb criterion

0.77 97.6 77.22 1.26 0.41

1.20a 55.2 76.68 0.72 0.39

0.77 97.2 76.68 1.27 0.41

1.11a 51.7 71.28 0.73 0.32

0.70 93.5 70.2 1.33 0.41

0.59 98.8 70.2 1.41 0.34

0.64 94.7 70.2 1.35 0.33

0.70 93.2 70.2 1.33 0.40

0.66 89.5 66.42 1.35 0.37

0.76 126.2 63.72 1.98 0.50

0.94 45.8 63.72 0.72 0.37

0.76 82.6 63.72 1.30 0.45

0.71 85.7 59.4 1.44 0.62

0.59 92.1 58.86 1.56 0.61

0.57 87.2 57.24 1.52 0.52

M. Singh, B. Singh / International Journal of Rock Mechanics & Mining Sciences 51 (2012) 43–52 51

4. Limitations and assumptions

The following limitations and assumptions apply to the pro-posed strength criterion for jointed rocks:

i.

The suggested criterion is more suitable for those failurepatterns where assumption of equivalent continuum is valid,and the equivalent properties are function of intact rockproperties and joint characteristics. The joints are assumedto be tight with no infilling. If the joints are filled with gougematerial, there will not be any wall-to-wall contact andthe gouge material will govern the strength behaviour. Thus,in-situ direct shear tests may be conducted to obtain thestrength parameters.

ii.

There is no rotation of blocks during failure. In case of rotationof blocks, the geometry of the blocks becomes more importantthan the strength properties of the intact rock.

iii.

If the joints are critically oriented and sliding occurs alongthese joint planes, the mass will behave more as a disconti-nuum than a continuum and this criterion will be of limiteduse. The strength reduction factor should be more than about0.07 for the proposed polyaxial criterion to be applicable. If scj

is less than about 0.07sci, either the joints are criticallyoriented or block rotation is occurring. The geometry of theblocks/wedges will be more important and wedge analysisincorporating a suitable strength criterion for joints will bemore appropriate.

iv.

The proposed criterion is more suitable for deep tunnels,caverns and rock foundations within block jointed rockmasses. In case of slopes with critically oriented joints, wherescj is very low, and failure occurs along dominant joint planes,wedge analysis or discontinuum approaches should be usedwith appropriate criterion for joints.

5. Concluding remarks

Rock masses encountered in civil and mining engineeringapplications are invariably jointed. Mohr–Coulomb linearstrength is the most widely used strength criterion to assess thestrength behaviour of geological materials. However, in its con-ventional form, the criterion considers the strength behaviour tobe linear and also ignores the effect of the intermediate principalstress. In present study the Mohr–Coulomb criterion has beenmodified to consider the non-linearity in assessing the strengthbehaviour. Barton’s critical state concept, which was originallysuggested for intact rocks, has been used to deduce the non-linearstrength criterion. Back analysis of a triaxial test data basecomprising of more than 730 triaxial test results for variety ofrocks (sci¼9.5 to 123 MPa) suggests that the critical state conceptis applicable to jointed rocks as well. The back analysis alsoindicates that the critical confining pressure equal to the UCS ofthe intact rock may be used in the criterion without introducingerror of engineering significance. The parameters of the proposedtriaxial strength criterion (scj and fj0) can be easily obtainedusing classification approaches and the laboratory test results ofintact rocks. Statistical analysis of the triaxial test data baseshows that there is 87.7% probability of the predicted results tolie within 720% error, with only input of scj, sci and fi0.

The criterion has also been extended to polyaxial stressconditions. A set of 54 polyaxial test results has been analysedto check the applicability of the criterion to polyaxial stressconditions in anisotropic jointed rocks. The criterion has beenfound to work well for those failure patterns where assumption ofequivalent continuum is valid and the equivalent properties are

function of intact rock properties and joint characteristics. Forsituations where wedge formed by intersecting joints is likely toslide along dominating joint planes, an appropriate criterion forrock joints should be used. As a rough guide-line it is recom-mended that the proposed polyaxial criterion for jointed rockscan be used with confidence for such situations having scj morethan about 7% of sci. The polyaxial strength criterion has beenused to derive expression for analysing rock burst condition.A case study from Indian tunnel has been analysed using theproposed criterion and the results of the proposed criterion arefound to be more realistic. It is suggested that the simplepolyaxial strength criterion (Eq. 26) may be used in the non-linear stress analysis of underground openings in natural rockmasses.

Appendix A. Supporting information

Supplementary data associated with this article can be foundin the online version at doi:10.1016/j.ijrmms.2011.12.007.

References

[1] Singh M, Raj A, Singh B. Modified Mohr–Coulomb criterion for non-lineartriaxial and polyaxial strength of intact rocks. Int J Rock Mech Min Sci2011;48:546–55.

[2] Patton FD. Multiple modes of shear failure in rock. In: proceedings of the 1stinternational congress on rock mechanics, Lisbon; 1966. p. 509–513.

[3] Barton NR. Review of new shear strength criterion for rock joints. Eng Geol1973;7(4):287–332.

[4] Ladanyi B, Archambault G. Evaluation of shear strength of a jointed rockmass. In: Proceedings 24th international geological congress, Montreal; 1972.p. 249–270.

[5] Kulatilake PHSW, Shou G, Huang TH, Morgan RM. New peak shear strengthcriteria for anisotropic rock joints. Int J Rock Mech Min Sci Geomech Abstr1995;32(7):673–97.

[6] Grasselli G, Egger P. Constitutive law for the shear strength of rock jointsbased on three-dimensional surface parameters. Int J Rock Mech Min Sci2003;40(1):25–40.

[7] Bieniawski ZT. Engineering classification of jointed rock masses. Trans SouthAfr Inst Civ Eng 1973;15(12):335–44.

[8] Barton N, Lien R, Lunde J. Engineering classification of rock masses for thedesign of tunnel support. Rock Mech 1974;6(4):183–236.

[9] Ramamurthy T. Strength and modulus response of anisotropic rocks. In:Husdon JA, editor. Comprehensive Rock Engineering, vol. 1. Oxford: Perga-mon; 1993. p. 313–29.

[10] Ramamurthy T, Arora VK. Strength prediction for jointed rocks in confinedand unconfined states. Int J Rock Mech Min Sci Geomech Abstr 1994;31(1):9–22.

[11] Singh M, Rao KS, Ramamurthy T. Strength and deformational behaviour ofjointed rock mass. Rock Mech Rock Eng 2002;35(1):45–64.

[12] Kulatilake PHSW, He W, Um J, Wang H. A physical model study of jointedrock mass strength under uniaxial compressive loading. Int J Rock Mech MinSci 1997;34(3–4): paper No. 165.

[13] Kulatilake PHSW Liang J, Gao H. Experimental and numerical simulations ofjointed rock block strength under uniaxial loading. J Eng Mech 2001;127(12):1240–7.

[14] Barton N. The shear strength of rock and rock joints. Int J Rock Mech Min SciGeomech Abstr 1976;13(9):255–79.

[15] Brown ET. Strength of models of rock with intermittent joints. J Soil MechFound Div ASCE 1970;96(SM6):1935–49.

[16] Brown ET, Trollope DH. Strength of a model of jointed rock. J Soil Mech FoundDiv ASCE 1970;96(SM2):685–704.

[17] Einstein HH, Hirschfeld RC. Model Studies on mechanics of jointed rock. J SoilMech Found Div ASCE 1973;90:229–48.

[18] Hoek E. An empirical strength criterion and its use in designing slopesand tunnels in heavily jointed weathered rock. In: Proceedings of the 6th

southeast Asian conference on soil engineering, Taipei; 19–23 May 1980.p. 111–158.

[19] Yaji RK. Shear strength and deformation response of jointed rocks. PhDthesis. Indian Inst Tech, Delhi; 1984.

[20] Arora VK. Strength and deformational behaviour of jointed rocks. PhD thesis.Indian Inst Tech, Delhi; 1987.

[21] Roy N. Engineering behaviour of rock masses through study of jointedmodels. PhD thesis. Indian Inst Tech, Delhi; 1993.

[22] Soni DS. Grout jointed rock strength behaviour with variation in jointorientation. ME thesis. MNR Engineering College, Allahabad, India; 1997.

M. Singh, B. Singh / International Journal of Rock Mechanics & Mining Sciences 51 (2012) 43–5252

[23] Zhang L. Estimating the strength of jointed rock masses. Rock Mech Rock Eng2010;44:391–402.

[24] Palmstrom A. Collection and use of geological data in rock engineering. ISRMNews J 1997;4(2):21–5.

[25] Singh M, Rao KS. Empirical methods to estimate the strength of jointed rockmasses. Eng Geol 2005;77:127–37.

[26] Singh B, Viladkar MN, Samadhiya NK, Mehrotra VK. Rock mass strengthparameters mobilised in tunnels. Tunnelling Underground Space Technol1997;12(1):47–54.

[27] Barton N. Some new Q-Value correlations to assist in site characteristics andtunnel design. Int J Rock Mech Min Sci 2002;39:185–216.

[28] Singh M. Engineering behaviour of jointed model materials. PhD thesis.Ind Inst Tech, New Delhi; 1997.

[29] Tiwari RP, Rao KS. Response of an anisotropic rock mass under polyaxial

stress state. ASCE J Mater Civ Eng 2007;19(5):393–403.[30] Tiwari RP, Rao KS. Post failure behaviour of a rock mass under the influence

of triaxial and true triaxial confinement. Eng Geol 2006;84:112–29.[31] Barton N, Chaubey V. The shear strength of rock joints in theory and practice.

Rock Mech 1977;10:1–54.[32] Singh B, Goel RK. Tunnelling in weak rocks. Elsevier; 2006.