MS TNS Text Oct 18 2008

SHEAR STRENGTH OF ROCK JOINTS AND ROCK MASSES

Mahendra SinghProfessor, Department of Civil Engineering

IIT Roorkee, Roorkee-247 667

1. GENERAL

Assessment of shear strength of rock joints and rock masses is probably the most

important aspect in limiting equilibrium analysis of rock slopes. A small change in shear

strength may substantially affect the safe angle of slope. Whereas the prediction of shear

stress acting along a failure surface is relatively easy, it is very difficult and cumbersome

to assess the shear strength with confidence. Some of the reasons behind this are:

i) The failure of rock mass may take place due to sliding along the existing

joints, shearing of the rock material, rotation of blocks of rock material, or due to

a complex combination of all of these modes.

ii) Jointed rocks and rock masses may behave anisotropically due to the presence

of discontinuities, even though the rock material itself may be isotropic.

iii) The shear strength along a failure surface depends on the level of normal

stress, and varies in a non-linear manner. This aspect is much significant for

slopes, as compared to other structures like foundations and tunnels. Non-linearity

in shear strength behaviour is very high at low normal stress levels, the condition

which generally prevails in case of slopes.

iv) Direct shear test results, even if performed in-situ, may not be directly

applicable due to scale effect, kinematics and in-homogeneity of the slope.

This paper attempts to address some of these issues. It is assumed herein that the

stresses involved are effective stresses and the shear strength parameters applicable are

expressed in terms of effective stresses. For simplicity, the prime sign (‘) generally used

to denote effective stress, and parameters in terms of effective stress will be dropped

1

throughout this chapter. Further, the stresses are considered static and not dynamic in

nature.

1.1. Classes of Rock Shear Strength

The shear strength aspect of rocks may be grouped into the following two major

categories:

i) Shear strength along planar discontinuity (Discontinuity shear strength)

ii) Shear strength of jointed rock mass (Rock mass shear strength)

(a) Isotropic rock mass

(b) Anisotropic rock mass

A careful selection of one of the above classes is essential for a reliable estimate

of the shear strength. Failure may take place along planar discontinuities when a

persistent discontinuity or a set of parallel discontinuities dip towards valley. An example

of a hillside failure along planar discontinuities is shown in Fig. 1. Kinematically, such a

failure will be possible if the joints strike parallel to the slope face and dip into the valley.

A more general and commonly occurring case is the one, when a pair of intersecting joint

planes forms a wedge that slides along the line of intersection of the joint planes.

The second class of rock shear strength pertains to the rock mass as a whole. This

class of shear strength is required in cases where the potential sliding surface lies partially

on discontinuity surfaces, and partially passes through the intact rock. The sliding surface

is generally curved just like in case of soils. Such failures occur when the extent of the

slope is very large compared to the discontinuity size and spacing. The rock mass

generally consists of blocks of intact rock separated by discontinuities. At the time of

slope failure, these blocks may slide, translate or rotate and dilation occurs as the normal

stresses are generally low. Further, the rock mass may behave isotropically or

anisotropically depending upon the number, orientation and spacing of discontinuities.

Though, there is no hard and fast rule, if a rock mass consists of more than four or five

sets of discontinuities, which are equally distributed in all directions, and no particular

discontinuity is substantially different from the others in shear strength response, then the

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mass may be treated isotropic. On the other hand, if there are only about one to three sets

of discontinuities, the rock mass may be considered anisotropic.

The inherent anisotropy in the rock material itself is not a matter of serious

concern in rock slope problems, unless the failure is entirely through the intact rock. In

general, the stresses associated with rock slope stability problems are not that large to

cause failure of intact rock. Therefore, even if the intact rock material is inherently

anisotropic, the rock blocks forming the rock mass may be considered isotropic for all

practical purposes in rock slope problems.

1.2. Some Important Terms

Having gone through the preliminary discussion in the previous section, it is

worth defining some of the terms, which will be used frequently in the subsequent text.

1.2.1. Intact Rock

The rock material separated by discontinuities and free from any defect is termed

as the intact rock. Except in case of weak massive rock, intact rock failure generally does

not occur during failure of rock slopes. The strength of the intact rock is however,

required to assess the rock mass strength.

1.2.2. Discontinuity

Any defect in the continuity of intact rock material is termed discontinuity. The

examples are bedding planes, joints and faults. The most common discontinuities

influencing rock slope stability are the joints. The term has sometimes been used

synonimically to cover all other types of the discontinuities. The discontinuities may be

tight or open, infilled or clear, smooth or rough. If the discontinuities are filled with a

material like clay or fault gouge, and there is no wall-to-wall contact, the shear strength

along the discontinuity will be governed by the characteristics of the infilling material.

The friction angle in such cases is likely to be low, and some cohesion may exist

depending on the type of infilling material. If the discontinuities are clean, the surfaces

will have low cohesion and the friction angle will depend on the roughness of the

material surface and bridges (gaps) between adjacent joints.

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1.2.3. Jointed Rock and Rock Mass

Rock mass is an assemblage of blocks of intact rock. Jointed rock is a general

term, which will be used in this text to indicate that the rock is fractured. It may not

necessarily represent a mass. For example, a laboratory cylindrical specimen having

single fracture will be termed as a jointed rock but not a rock mass. In case of failure of a

rock mass, intact rock blocks forming the mass may slide, translate and rotate. It results

in dilation of the mass if the normal stress level across the failure surface is low.

Consequently, the friction angle of mass at low normal stress is high, whereas the

cohesion is low. As the normal stress is increased, the dilation is suppressed and shearing

of the intact rock material commences. This results in a relatively higher cohesion and

lower friction angle. Due to continuous change in mechanism of failure, the strength

envelope of rock mass is highly curvilinear, especially in low normal stress range.

2. DISCONTINUITY SHEAR STRENGTH

If the probable mechanism of failure indicates a possibility of sliding along the

discontinuity surfaces (planar or wedge failure), it is essential to assess the shear strength

of discontinuity surfaces under the prevailing normal stress. Various models are available

to determine the shear strength of a planar discontinuity at a given normal stress. Some

most commonly referred shear strength models are discussed in the following text.

2.1. Coulomb’s Model

The most commonly used model for assessing the shear strength along the

discontinuities is the conventional Coulomb’s linear model. In this model, shear strength

of the sliding surface is expressed in terms of cohesion, c j and friction angle, j. These

shear strength parameters may be obtained by performing direct shear tests on the

discontinuity surfaces. It is, however, difficult to prepare jointed rock test specimens of

regular geometry. A portable field shear box (Fig. 2a) as suggested by Ross-Brown and

Walton (1975), may be used to perform direct shear tests on rock joints. In this test, an

irregular rock chunk is encased in a cementing material in such a way that the

discontinuity surface coincides with the pre-defined shearing plane of the box. A normal

force is applied on to the discontinuity surface, and the specimen is sheared by subjecting

it to increasing shear force. Shearing force and shear displacements are monitored, and

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the test carried to failure. A plot between shear stress and shear displacement is obtained.

A typical plot for a rough discontinuity tested under a constant normal load is presented

in Fig. 2b. The plot indicates that the shear displacement increases with increase in shear

stress. Initially the plot is linear indicating elastic behaviour of the specimen. Gradually

the force resisting movement is overcome; the curve becomes non-linear and reaches a

maximum that represents the peak shear strength of the discontinuity. With further

increase in shear displacement, the shear stress causing displacement decreases; and at

relatively large displacements, the shear stress reaches a constant value termed the

residual shear strength. The peak shear strength represents the maximum shear stress

required to overcome the resistance of a discontinuity before failure. After failure, the

cohesion of the discontinuity is lost; the minor irregularities on the rock surface are

sheared off, and further displacement requires relatively lesser shear stress.

A number of tests are conducted by adopting different values of normal loads. A

set of peak and residual shear strength values for different normal stresses will be

available from these tests. This data is used to obtain a plot between shear strength and

normal stress (Fig. 2c). The plot is termed as failure envelope. The diagram shows two

plots: one for peak strength and the other for residual strength. The shear strength

parameters cj, j for peak, and r for residual strength may be obtained from these plots.

The shear strength of the discontinuity is defined as:

(peak strength) (1)

(residual strength) (2)

where f is the shear strength along the discontinuity; n is the effective normal

stress over the discontinuity; j is the peak friction angle of the discontinuity surface; cj is

the peak cohesion of the discontinuity surface, and r is residual friction angle for

discontinuity surface.

The above-mentioned expressions are for a rough or infilled discontinuity. If the

discontinuity is planar and clean, its cohesion will be zero and the shear strength will be

governed solely by the friction angle. The surface characteristics of rocks in such cases

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will be governed by the size and shape of the grains exposed on the discontinuity surface

(Wyllie and Mah, 2004). A fine-grained rock with a high mica content aligned parallel to

the surface will tend to have low friction angle; while a coarse-grained rock such as

granite will have a high friction angle. A typical range of basic friction angles for a

variety of rocks is presented in Table 1.

2.2. Patton’s Model

Patton (1966) was probably one of the first research workers who recognized the

importance of surface roughness of natural discontinuities, quantified the roughness and

included its contribution to shear strength of natural discontinuities. Patton observed that

some bedded limestone slopes were stable at steeper angles whereas the other slopes

were unstable even at relatively gentle gradient. The reason for this anomaly was

attributed to the asperities (irregularities) on joint wall surfaces. The asperities were

divided into first order and second order asperities. At very low normal stress level, the

sliding is assumed to occur over second order asperities. With increase in normal stress,

the shearing of higher order asperities occurs, and the first order asperities define the

shearing strength. With further increase in normal stress, first order asperities are also

sheared. Laboratory studies were conducted (Patton, 1966) by simulating the asperities in

the form of saw-tooth specimens (Fig. 3). A bilinear model was suggested to predict the

shear strength as:

(3)

(4)

where i defines the roughness angle, and r is the residual friction angle.

Equation 3 is applicable for low normal stress levels where sliding occurs along

the asperities. At higher normal stress levels, shearing of intact rock occurs, and Equation

4 is suggested for obtaining the shear strength. The model, therefore, considers two

failure modes i.e. either sliding along the discontinuities or shearing of the intact

material.

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The model provides a good basis for explaining the importance of roughness of

surfaces of the natural discontinuities and their influence on the slope stability. However,

from practical standpoint, the limitation of the model is that, it is difficult to assess the

normal stress level at which transition from sliding to shearing takes place. In reality,

there is no such distinct and clear-cut normal stress level, which defines the boundary

between the two failure modes. Due to this limitation, more advanced models have been

proposed which are discussed in the subsequent sections.

2.3. Barton (1973) Model

Barton (1973) model is an extension of the Patton (1966) model. It is probably the

most widely used strength criterion for assessing shear strength along discontinuity

surfaces. It overcomes the limitations of Patton’s model in that, all the parameters used in

the criterion can be easily determined in the field. Major limitation of Patton’s model was

that it considered the shear failure either due to sliding or shearing. In nature, both the

mechanisms occur simultaneously. However, at low normal stress level, the sliding

phenomenon dominates; whereas shearing phenomenon governs the shear strength at

high normal stress level. The shear strength of a joint (Fig. 4) is expressed as:

(5)

where JRC is the joint roughness coefficient, which is a measure of the initial

roughness (in degrees) of the discontinuity surface. JRC is assigned a value in the range

of 0–20, by matching the field joint surface profile with the standard surface profiles on a

laboratory scale of 10 cm (Barton and Choubey, 1977) as shown in Fig. 5. JCS is the joint

wall compressive strength of the discontinuity surface, and n is the effective normal

stress acting across the discontinuity surface.

Equation 5 may also be expressed in the following form:

(6)

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where e is the equivalent friction of the discontinuity at a given normal stress and

is expressed as:

<70 (7)

(8)

It can be seen that, the value of roughness angle, i goes on decreasing with

increasing n. At higher stress levels, when normal stress n equals JCS, the roughness

angle i, becomes zero. At this stress level, the equivalent friction angle, e of the

discontinuity, is equal to the basic friction angle, . At very low stress levels, the ratio

(JCS/n) tends to approach infinity. This may result in an unrealistic value of the

equivalent friction angle e. For practical purposes, this value should not exceed about

50o (Wyllie and Mah, 2004).

Joint roughness coefficient, JRC indicates initial roughness of the joint. This

value is obtained by comparing the profile of discontinuity surface with standard

roughness profiles, which have been prepared on a 10 cm base scale (Fig. 5). It has been

found that, with small displacements, the second order asperities are sheared off, and the

first order asperities come into picture. The roughness of joints is therefore a scale

dependent property. It is also found that JCS is also a scale dependent property. Barton

and Bandis (1982) proposed the following expressions to take into account the scale

effect on the roughness and wall strength of discontinuity surface:

(9)

(10)

where Lo is the dimension of the surface used to measure JRC and Ln is the

dimension of sliding surface.

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2.4. Ladanyi and Archambault (1972) Criterion

Ladanyi and Archambault (1972) criterion is a mechanics based strength criterion

and has been derived from principles of strain energy. It is a criterion that is applicable to

both rock joints and jointed rock masses. To derive this criterion, the authors equated the

external work done in shearing a jointed rock to the internal energy stored in the rock. It

was considered that during shearing, the external force does the work to meet the energy

requirements for the following components:

i) Component of work due to external work done in dilating against the external

force, N.

ii) Additional internal work done in friction due to dilation only.

iii) Work done in internal friction if the specimen did not change in volume in shear.

iv) Work done in shearing asperities.

The criterion is expressed as:

(11)

where f is the shear strength; as is sheared area ratio equal to As/A; A is the total

area of joint surface; As is the sheared area of joint surface; n is the mean applied normal

stress; is the rate of dilation at failure and is equal to ; is the basic friction angle

joint surface, and ci, i are Mohr-Coulomb parameters for intact rock.

The sheared area ratio, as and dilation rate, are difficult to estimate in the field.

The authors have suggested on the basis of limited experimental data, the following

approximate expressions for determining these parameters:

(12)

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(13)

where K1 and K2 are equal to 1.5 and 4 respectively; trn is the brittle –ductile

transition stress which may be taken equal to the UCS of intact rock; and i is the initial

roughness of the joints.

The criterion has been developed based on very sound principles of mechanics,

and is applicable to rock joints and rock masses as well, but lacks in simplicity due to the

fact that the parameters cannot be determined with ease in the field. Due to this reason,

the criterion could not become as popular amongst the practicing engineers, as the

empirical criterion suggested by Barton (1973).

3. SHEAR STRENGTH OF ROCK MASS

Geological conditions in many situations of rock slopes are such that the

discontinuity surfaces along which sliding could take place are not persistent. The

potential failure surface in such cases lies partly on discontinuity surface and partly

passes through the intact rock. The shear strength of rock mass is different from that of a

discontinuity and is very difficult to obtain experimentally, as the sample size would be

prohibitively large (> 1m diameter).

If the number of discontinuity sets in rock mass is large enough (say more than

four or so), and no particular discontinuity is significantly different from the others, and

spacing of discontinuities is small, the rock mass may be treated as isotropic. However, if

the number of joint sets is not large, the behaviour of rock mass will be anisotropic. If the

spacing of the discontinuities is large, it will be more appropriate to analyse the problem

by performing a wedge analysis. It may be realised that mobilised strength parameters of

rock mass are very low in rock slopes due to unrestricted rotation of blocks and freedom

to dilation in wedge failure in rock slopes, unlike that in tunnels (see article 3.3.2).

One of the most reliable methods to estimate rock mass shear strength is the back

analysis. It involves carrying out stability analysis of a failing or failed slope using

available information on different factors affecting the stability. A suitable value of factor

10

of safety is assigned and values of cohesion are calculated for a range of values of friction

angle. A realistic combination of shear-strength parameters c and is then selected for

analyzing slopes under similar geological conditions.

In the absence of back analysis, rock mass strength criteria are used to simulate

triaxial strength tests on the rock mass. Using the results of simulated triaxial strength

tests, a relationship between the normal stress across the failure surface and the

corresponding shear strength is derived. These relationships are generally non-linear.

Most of the non-linear strength criteria for rock masses are generally expressed in terms

of the major and minor principal stresses. Whereas, the stability analysis of the slopes is

generally carried out in terms of normal stress, shear stress and shear strength along the

failure surface. It is, therefore, necessary to convert strength values from (3, 1) space to

(,) space. Moreover, many standard procedures of slope stability problems e.g.

Bishop’s method (Singh and Goel, 2002) represent the solution in terms of Mohr-

Coulomb shear strength parameters, c and . To use these procedures, it becomes

necessary to assess parameters c, at a given normal stress. It should be noted that the

failure envelope for rock masses is highly non-linear especially in low normal stress

range. To incorporate the non-linearity in shear strength behaviour, Hoek (2000) has

introduced the concept of instantaneous values of parameters, c and . The instantaneous

shear strength parameters, cisnt and inst are the values of cohesion intercept and angle of

the tangent drawn to the non-linear failure envelope at a point representing the normal

stress at which the shear strength is sought (Fig. 6). The instantaneous cohesion, cinst and

friction angle, inst are, therefore, normal stress dependent. Once the instantaneous

parameters are available, the shear strength may be obtained by using the linear

Coulomb’s model (Equation 1). The following expressions (Balmer, 1952) may be used

to convert the strength data from (3 ,1) space to (n ,f) space.

The strength criterion is expressed as:

1 = f(3) (14)

then

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(15)

(16)

(17)

(18)

(19)

where n is normal stress on failure plane; f is shear strength along the failure

plane; is the angle between the normal to failure plane and the major principal stress

direction; and inst, cinst are instantaneous Coulomb parameters. It should be noted that the

instantaneous shear strength parameters, cinst and inst vary with the level of normal stress

n acting on the failure plane.

3.1. Linear Strength Criterion

Coulomb’s linear strength criterion is the most widely used criterion for jointed

rock and rock masses as well. The criterion is also referred to as Mohr-Coulomb

criterion. According to this criterion, rock mass is treated as an isotropic material and the

shear strength along the failure surface is expressed as follows:

(20)

where cj and j are Mohr-Coulomb shear strength parameters for jointed rock or

rock mass. For rock mass in field, the values of cj and j may be obtained from

classification approaches as given in the next section.

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3.1.1. Rock Mass Rating

Bieniawski (1973, 1989 and 1993) has suggested values of shear strength

parameters cm, m for five levels of rock mass ratings. Table 2 reproduces

recommendations of Bieniawski (1989) for cohesion and friction angle of rock mass

depending upon the rock mass rating, RMR. It can be seen that the values of cm have been

suggested to be varying from less than 100 kPa to more than 400 kPa and m values

varying from less than 15o to more than 45o respectively. Experience of Indian project

sites (Mehrotra, 1992) has however indicated that the shear strength is under predicted by

expressions suggested by Bieniawski (1989). Figure 7 (Mehrotra, 1992) shows results of

extensive block shear tests to get a relationship between RMR and shear strength

parameters. The figure shows that i) cm and m increase with increase in RMR, ii) values

obtained from field tests are higher than those suggested by Bieniawski (1989), iii) there

is a significant effect of the saturation of rock mass, and iv) friction angle varies non-

linearly with RMR. Further, for very poor rock masses (RMR = 0 – 20), friction angle, m

obtained from field test was always more than 15o whereas the same has been suggested

by Bieniawski (1989) to be less than 15o and approaching zero (or very small value) for

RMR 0. It has been suggested by Singh and Goel (1999) that for very poor rock

masses, the minimum value of m should always be taken greater than 15o. Figure 7 may

therefore be used for assessing the friction angle of rock masses.

A caution is needed here that there should be no double accounting for parameters

in the classification of rock mass and stability of a rock slope. Since ground water table

and joint orientations are considered in the stability analysis of slopes; the RMR should

be assessed on the basis of RQD, UCS, joint spacing, joint conditions and dry rock mass

(rating = +15). The strength parameters so obtained for the dry rock mass should be used

in stability analysis with water pressure being considered separately.

3.1.2. Q index

Rock mass quality index, Q (Barton et al., 1974) can also be used to get the Mohr-

Coulomb shear strength parameters as suggested by Barton (2002) as:

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(21)

(22)

where cm is the cohesion of the undisturbed rock mass; , the friction angle of

the mass; RQD, the rock quality designation (Deere, 1963); Jn, the joint set number; Jr,

the joint roughness number; Ja, the joint alteration number; Jw, the joint water reduction

factor, and σci is the uniaxial compressive strength of intact rock material.

It may be noted that Q system has been used extensively and validated for tunnels

in Indian conditions. In the opinion of this author the shear strength parameters obtained

from Q should be preferred for analyzing underground openings. If used for slopes, an

overestimation in the strength may be expected. For slopes, it is felt that the relationship

between shear strength parameters and RMR as suggested by Mehrotra (1992), and

Singh and Goel (1999) will be more appropriate for Indian conditions and especially the

Himalayan rock masses. The relationships were developed based on extensive in-situ

direct shear tests on saturated rock masses in Himalayas.

3.2. Non-Linear Strength Criteria

Mohr-Coulomb strength criterion considers the rock mass shear strength as a

linear function of normal stress n. It has been well established by now that at very low

normal stress level, which generally prevails in slopes, the shear strength response is

highly non-linear. Consequently, several non-linear strength criteria have been proposed

for jointed rock and rock masses to account for non-linearity in strength behaviour. In

general, these strength criteria have been presented in (1, 3) space, i.e. the major

principal stress at failure is expressed in terms of the minor principal stress. There is a

large number of rock strength criteria which are available in literature which include not

only non-linearity but also the effect of intermediate principal stress (Murrell,1963;

Weibols and Cook,1968; Mogi,1971; Chang and Haimson, 2000; Haimson and Chang,

14

2000; Colmenares and Zoback, 2002; and Al-Ajmi and Zimmerman, 2005). These

criteria are, however, beyond the scope of the present discussion.

From field point of view, only those criteria, whose parameters are easy to obtain

in the field, have been discussed in the subsequent sections. Due to simplicity and ease in

parameter estimation, the empirical failure criteria are widely used in practice. Only the

empirical criteria have, therefore, been presented herein. In the discussion that follows,

non-linear strength criteria have been grouped into two categories, i) strength criteria

applicable only to isotropic rock masses, and ii) the criteria applicable to anisotropic rock

masses as well. Under the first category Hoek-Brown criterion has been discussed and

under the second category Ramamurthy criterion and critical state concept based criterion

(Singh and Singh, 2004; Singh and Rao, 2005a) have been discussed.

3.2.1. From Rock Mass Classification RMR or Q

Based on extensive in-situ direct shear testing of rock masses in the Himalayas,

Mehrotra (1992) has suggested the non-linear variation of shear strength as:

(23)

where A, B and C are empirical constants and depend on RMR or Q. Their values

for different moisture contents, RMR and Q index are presented in Table 3.

3.2.2. Hoek-Brown Strength Criterion

A non-linear strength criterion was initially proposed for intact rocks by Hoek and

Brown (1980). The criterion is expressed as:

(24)

where is the effective major principal stress at failure; is the effective minor

principal stress at failure; mi is a criterion parameter; and ci is the UCS of the intact rock

which is also treated as a criterion parameter.

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A few triaxial tests may be conducted on the intact rock specimens, and the

criterion may be fitted into the triaxial test data to obtain ci and mi. It may be noted that

the computed value of ci should be used in further analysis and not the experimentally

obtained ci. In the absence of triaxial strength test data on intact rock, use of Table 4

(Hoek, 2000) has been suggested for approximate estimation of the parameter, mi.

The strength criterion proposed for intact rocks was also extended to rock masses

(Hoek and Brown, 1980). The criterion was later modified with more data available for

its applicability in the field for different rock types. The latest form of the criterion (Hoek

et al., 2002) is expressed as:

(25)

where is the effective major principal stress at failure; is the effective minor

principal stress at failure; is the uniaxial compressive strength of intact rock within the

mass to be obtained from triaxial strength tests performed on intact rock specimens; is

an empirical constant which depends upon the rock type; and is an empirical constant

which varies between 0 to 1 depending upon the degree of fracturing.

To obtain parameter mj and sj, the use of a classification index, Geological

Strength Index (GSI), has been suggested (Hoek and Brown, 1997; Hoek et al., 2002).

The expressions for mj and sj are given as:

(26)

(27)

(28)

where GSI is the Geological Strength Index; mi is the Hoek-Brown parameter for

intact rock to be obtained from triaxial test data; D is a factor which depends upon the

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degree of disturbance to which the rock mass has been subjected by blast damage and

stress relaxation. It varies from 0 for undisturbed in situ rock masses to 1 for very

disturbed rock masses. For blasted rock slopes, D is taken in the range 0.7 to 1.0.

The Geological strength index (GSI) is assigned from visual inspection of the

rock mass in the field. The index depends upon the structure of the rock mass (massive,

blocky, very blocky, disturbed, seamy, disintegrated, laminated and sheared) and the

surface roughness conditions (very good, good, fair, poor and very poor) of

discontinuities. Like rock mass rating, RMR, the GSI values vary on a scale of zero to

one hundred. A chart for estimating GSI from Marinos et al., (2005) has been reproduced

in Fig. 8.

After assigning GSI, the Hoek-Brown parameters mj and sj are computed. These

parameters can then be used to simulate triaxial strength tests i.e., the 1 values can be

generated for various 3 as is done in the laboratory triaxial tests. From this data, the

failure envelope in vs. space can be obtained and Mohr-Coulomb parameters cinst and

inst may be determined at a given normal stress level. Alternatively Balmer’s expressions

(Balmer, 1952) may be used to generate f at a given n. Experience in Himalayan slopes

shows that cinst obtained from GSI, is too high in the case of rock slopes.

3.2.3. Ramamurthy criterion

Ramamurthy and co-workers (Ramamurthy, 1993; Ramamurthy, 1994;

Ramamurthy and Arora, 1994; Ramamurthy, 2007) have suggested the following non-

linear strength criterion for intact isotropic rocks:

(29)

where 3 and 1 are the minor and major principal stresses at failure; t is the

tensile strength of intact rock; ci is the UCS of the intact rock; and i, i are the criterion

parameters.

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Parameters i and Bi should be obtained by fitting the criterion into the laboratory

triaxial test data for intact rock. In the absence of triaxial test data, the following

approximate correlations may be used (Ramamurthy, 2007):

(30)

For jointed rocks and rock masses, the strength criterion proposed for intact rocks

has been extended to jointed rocks as:

(31)

where j and j are the criterion parameters for jointed rock; and cj the is UCS of

the jointed rock.

Based on extensive laboratory testing, the following correlations were suggested

to obtain criterion parameters j and j:

(32)

(33)

where parameters i and i are obtained from laboratory triaxial tests performed

on intact rock specimens. The UCS of the rock mass, cj which is popularly known as

rock mass strength is an important input parameter to this strength criterion and has been

discussed separately in section 3.3.

3.2.4. Critical state concept based criterion

As discussed in a preceding section, Mohr-Coulomb linear strength criterion is the

most widely used strength criterion in geotechnical engineering practice. Singh and Singh

(2004), and Singh and Rao (2005a) have suggested a strength criterion as an extension of

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Mohr-Coulomb strength criterion. The linear Mohr-Coulomb strength criterion may be

expressed in the following form:

(34)

where c0 and 0 are the instantaneous Mohr-Coulomb shear strength parameters at

30.

Figure 9 shows a plot of linear Mohr-Coulomb criterion. The actual response of a

rock mass is non-linear, which is also shown in the plot. The actual non-linear strength

response can be represented by modifying Mohr-Coulomb strength criterion as:

(35)

where A’ is an empirical constant, which is always positive.

Singh and Singh (2004), and Singh and Rao (2005a) have reduced the above

criterion into a simple form as:

(36)

where A is an empirical parameter, which is always, negative and is given as:

A = -1.23 (ci)-0.77 (37)

By using the values of rock mass strength, cj and intact rock strength, ci in the

criterion, a set of (3,1) values may be generated which then could be used to develop

failure envelope in (n,f) space. Alternatively, Balmer’s equations may be used in

stability analysis to determine instantaneous values of c and or shear strength for a

given normal stress level.

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3.3. Rock Mass Strength, cj

It can be observed from Equations 30, 31, 32 and 35, that rock mass strength, cj

is an important input parameter in the strength criteria. The estimation of rock mass

strength, therefore, becomes an important aspect in a rock slope problem. The accuracy

of shear strength prediction depends on how precisely cj has been estimated.

Ramamurthy criterion and parabolic criterion (Singh and Singh, 2004; Singh and Rao,

2005a) consider the rock mass to be anisotropic in strength response. The anisotropy is

introduced through rock mass strength, cj i.e. if an anisotropic value of rock mass

strength is used in the strength criterion, the resulting shear strength will also be

anisotropic. The following methods can be used to determine the UCS of the rock mass:

i) Joint Factor concept

ii) Rock mass quality, Q

iii) Strength reduction factor

3.3.1. Joint Factor Concept

Ramamurthy and co-workers (Arora, 1987; Ramamurthy, 1993; Ramamurthy and

Arora, 1994; Singh, 1997; Singh et al., 2002) have suggested that the presence of joints

greatly influences the rock mass strength. Joint Factor concept was evolved to define the

effect of joints on the strength of rock mass. The concept was derived from a large

number of uniaxial compressive strength tests conducted on jointed specimens of

artificial and natural rocks. Orientation of joints, their frequency and surface roughness

were varied during the tests. It was argued that the most important properties of joints

that affect the rock mass strength are frequency, orientation and shear strength along the

joints. An empirical parameter called joint inclination parameter, n (Table 5), was

evolved to represent the effect of orientation of joints on engineering response of jointed

rocks. A weakness coefficient, Joint Factor was defined by considering the combined

effect of frequency, orientation and shear strength of joints as:

(38)

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where Jn is the number of joints per metre in the major principal stress direction;

n is an inclination parameter that depends on the orientation of critical joints (Table 5); r

is the joint shear strength parameter, which depends on the conditions of the joint i.e.

roughness, cementation, tightness, aperture, weathering of walls, and nature of in-filled

material, if any (Ramamurthy, 2007).

The joint strength parameter, r is obtained from direct shear tests conducted along

the joint surface at low normal stress levels and is given by:

(39)

where j is the shear strength along the joint; nj is the normal stress across the

joint surface; j is the equivalent value of friction angle incorporating the effect of the

asperities (Ramamurthy, 2007). The tests should be conducted at very low normal stress

levels so that the initial roughness is reflected through this parameter. For cemented

joints, the value of j includes the effect of cohesion intercept also. In case the direct

shear tests are not possible and the joint is tight, a rough estimate of j may be obtained

from Table 6 (Ramamurthy, 1994). If the joints are filled with gouge material and have

reached the residual shear strength, the value of r may be assigned from Table 7

(Ramamurthy, 1994).

The Joint Factor may be determined and the UCS of the rock mass (rock mass

strength) may be obtained as:

cj= ci exp(-0.008 Jf) (40)

where cj is the uniaxial compressive strength of the jointed rock, and ci is the

UCS of intact rock. If there are more than one joint set available in the rock mass, J f

should be computed by considering the individual joint set and the maximum value

should be considered to assess the rock mass strength, cj.

The major part of the experimental work involved in deriving expression for rock

mass strength, cj in the above expression was performed on cylindrical jointed

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specimens of size 38 cm diameter x 76 cm height. Singh (1997) performed an extensive

experimental study on relatively larger sized jointed blocky mass specimens of size 15

cm x 15 cm x 15 cm and having more that about 260 elemental blocks in each of the

specimens. It was observed that rock mass under uniaxial loading could fail due to the

following failure modes (Singh et al., 2002): i) splitting of intact material, ii) shearing of

intact material, iii) rotation of blocks and iv) sliding along the critical joints. An

assessment of the probable failure mode in the field can be made by using guidelines

suggested by Singh (1997) and Singh and Rao (2005a). The guidelines are reproduced in

the following section. It may be noted that in these guidelines, the joints are considered to

be continuous and persistent.

Let be the angle between the normal to the joint plane and the major principal

stress direction:

(i) For = 0 to 10

The failure is likely to occur due to splitting of the intact material of blocks.

(ii) For = 10 to 0.8 j

The mode of failure shifts from splitting (at = 10) to sliding (at 0.8 j).

(iii) For = 0.8j to 65

The mode of failure is expected to be sliding only. Theoretically, the mass should

slide due to its own weight if > j; however, the experimental observations (Singh,

1997) indicate that the mass fails due to its own weight only if it has a single joint set. If

there are more joint sets, the mass deforms to some extent and becomes stable. This is

due to a small amount of rotation of blocks, which generates a small interlocking in the

mass. The net result is that the mass retains a small amount of strength.

(iv) For = 65 to 75

The mode of failure shifts from sliding (at = 65) to rotation of blocks (at =

75).

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(v) For = 75 to 85

The mass fails due to rotation of blocks only. Geometry of the blocks is an

important parameter in governing the strength behaviour of the mass. In the present

study, it is assumed that the mass consists of blocks of square section. In case of slender

columns, the mass can fail due to buckling if the joints are open. Theory of long columns

can be used in this case and in the present analysis, this mode is excluded.

(vi) For = 85 to 90

The failure mode shifts from rotation at = 85 to shearing at = 90. It may be

noted that sharp changes take place in the strength response in this range of orientations.

Table 8 has been suggested to roughly assess the failure mode in the field. The

table can be used for rock mass having one persistent joint set and a cross joint set.

Depending upon the orientation of persistent joints and the interlocking level of the mass,

a failure mode can be assigned from Table 8.

The following equation was suggested to determine the rock mass strength:

cj= ci exp(a Jf) (41)

where ‘a’ is an empirical multiplying constant depending upon the failure mode

(Table 9). If it is not possible to assess the failure mode, an average value of this

empirical constant may be taken -0.017.

Based on the assessed mode of failure, the value of coefficient ‘a’ may be

assigned from Table 9. The value of Jf can be computed based on field observations (i.e.

frequency, orientation and friction angle along the joints). Equation 41 can now be used

to compute the rock mass strength.

The Joint Factor concept discussed above is quite useful in analysing anisotropic

strength behaviour of rock masses. For example, consider a case of a river valley with

joints dipping at some angle, and striking parallel to the river axis. For one bank, the

joints will dip towards valley; whereas on the other bank they will dip away from the

23

valley. Accordingly, the shear strength along the failure surface for a given normal stress

level, will be different for the two banks. That is why, on one side, a very steep slope may

be more stable due to large value of shear strength parameters whereas, the other bank

despite having gentle slope may be unstable due to low shear strength parameters.

3.3.2. Rock Mass Quality, Q

Rock mass classification techniques have not only been used for classifying the

rock masses, but as design methodologies as well. Rock mass quality index, Q (Barton et

al., 1974) has been used extensively to classify rock masses. An estimate of rock mass

strength can also be made by using the rock mass quality index, Q. It should be noted that

the approach treats the rock mass as an isotropic medium.

Singh et al. (1997) have proposed correlations of rock mass strength, cj with Q

by analysing block shear tests in the field. The following expression has been suggested

to predict the rock mass strength for rock slopes in hilly areas:

(42)

where is the unit weight of rock mass in gm/cm3; and Q is the Barton’s rock

mass quality index.

It may be noted that this expression is applicable for rock slopes only. In case of

tunnels, the rock mass is subjected to a constrained dilatancy, which results in strength

enhancement and accordingly, the rock mass strength in tunnels is about 18 times larger

than that given by this expression (Singh and Goel, 2002).

3.3.3. Strength Reduction Factor

Theoretically, the best estimates of rock mass strength, cj in the field can only be

made through large size field-testing in which the mass may be loaded upto failure to

determine rock mass strength. It is, however, extremely difficult, time consuming and

expensive to stress a large volume of jointed mass in the field upto ultimate failure. Singh

and Rao (2005b) have discussed that a better alternative is to get the deformability

properties of rock mass by stressing a limited area of the mass upto a certain stress level,

24

and then relate the ultimate strength of the mass to the laboratory UCS of the rock

material through a strength reduction factor, SRF. It has been shown by Singh and Rao

(2005b) that the modulus reduction factor, MRF and strength reduction factor, SRF are

correlated with each other by the following expression approximately:

SRF = (MRF)0.63 (43)

(44)

where SRF is the strength reduction factor i.e. the ratio of rock mass strength to

the intact rock strength; MRF is the modulus reduction factor i.e. ratio of rock mass

modulus to the intact rock modulus; cj is the rock mass strength; ci is the intact rock

strength; Ej is the elastic modulus of the rock mass; and E i is the intact rock modulus

available from laboratory tests and taken equal to the tangent modulus at stress level

equal to 50% of the intact rock strength.

The elastic modulus of rock mass, Ej may be obtained in the field by conducting

uniaxial jacking tests (IS:7317, 1974). The test consists of stressing two parallel flat rock

faces on the opposite walls of a drift by means of a hydraulic jack (Mehrotra, 1992). The

stress is generally applied in two or more cycles as shown in Fig. 10. The second cycle of

the stress deformation curve is recommended for computing the field modulus as:

(45)

where Ej is the elastic modulus of the rock mass in kg/cm2; is the Poisson’s

ratio of the rock mass (= 0.3); P is the load in kg; e is the elastic settlement in cm; A is

the area of plate in cm2; and m is an empirical constant =0.96 for circular plate of 25mm

thickness.

A number of methods for assessing the rock mass strength, cj have been

discussed above. It is desirable that more than one method be used for assessing the rock

mass strength and then generating the failure envelopes for the rock mass in - space. A

25

range of values will thus be obtained and design values may be taken according to

experience and confidence of the designer.

4. CONCLUSION

The stability of a rock slope is primarily governed by the characteristics of

discontinuities present in the mass. An appropriate understanding of shear strength along

the potential failure plane is essential to carry out limiting equilibrium analysis of rock

slopes. The potential failure plane may pass through existing joints or partly through the

joints and partly through the intact rock material. Based on potential failure mode, the

determination of shear strength has been divided into two broad categories i.e. joints and

jointed rocks / rock masses. Various approaches have been discussed to determine the

shear strength along the joints or jointed rock masses. Non-linear strength criteria, whose

parameters can easily be obtained in the field, have been discussed in detail. Both

isotropic and anisotropic strength responses of rock masses have been considered. An

important input parameter to the shear strength determination is the uniaxial compressive

strength of the rock mass or simply termed rock mass strength. Various recent methods to

determine the rock mass strength, developed from field and laboratory investigations,

have been discussed in detail. Using the strength criteria, one can generate shear strength

vs. normal stress plots for a potential failure surface in any direction, and use these results

in stability analysis. It is the opinion of the author that a range of values, rather than a

single value, of shear strength should be worked out to solve a real life problem in the

field. An appropriate value may then be chosen based on experience of the designer to

carry out the stability analysis. It may be realised that very low shear strength parameters

are mobilised in jointed rock mass slopes due to rotation of blocks.

ACKNOWLEDGEMENT

The author gratefully acknowledges the contributions made by Dr. T.

Ramamurthy, Professor (Retd.) and Dr. K.S. Rao Professor from IIT Delhi; Dr. Bhawani

Singh, Professor (Retd.), Prof. M. N. Viladkar and Prof. N. K. Samadhiya from IIT

Roorkee, Roorkee for their valuable technical inputs during the research presented in this

26

paper. Some part of the research presented in this paper has been conducted under a

research project (Project No. DST-209-CED, IIT Roorkee, 2005-2008) sponsored by

Department of Science and Technology (DST), New Delhi. The author sincerely puts on

record the appreciation for the financial support from DST New Delhi, and the co-

operation and encouragement from Dr. Bhoop Singh, Director NRDMS, DST, New

Delhi, in carrying out research related to rock slope stability problems.

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