Keywords

1 General

While analysing structures like tunnels, underground caverns, landslides, road cuts and foundations of heavily loaded structures like dams, bridges situated in or on rocks, the engineers and geologists are often required to assess the shear strength of jointed rock masses. The discontinuities e.g. joints, foliations and bedding planes are invariably present in rock masses and induce planes of weakness in the mass. While shearing, failure may occur due to a complex combination of sliding on pre-existing discontinuities, shearing of rock substance, translation and/or rotation of intact rock blocks. Consequently, the jointed rock is quite incompetent and anisotropic in strength and deformational behaviour. In addition, the strength behaviour of jointed rock is non-linear with increase in confining pressure. The assessment of shear strength response is therefore, an extremely difficult task. The present article discusses in brief, how the shear strength of the rocks can be assessed in the field.

From application point of view, two broad categories may occur: In the first category, a single or a set of planar persistent discontinuities exists in the rock. Sliding may occur along the discontinuities depending on the kinematics of the problem. These types of the conditions are commonly encountered in case of slopes. The second category pertains to the failure of the rock mass as a whole. The potential failure surface lies partly along discontinuity surfaces, and partly through the intact rock. Sliding, rotation, translation, splitting or shearing of intact rock blocks occur at the time of failure. The rock mass may behave isotropically or anisotropically depending upon the number, orientation and spacing of discontinuities and level of confining stress. These types of failure are common in tunnels and other underground structures. These two broad categories are discussed in the following sections.

2 Discontinuity Shear Strength

In case of shear strength along the surface of discontinuity, the shear strength is represented as a function of normal stress across the failure surface and the shear strength parameters. The most widely used shear strength model is linear Coulomb’s model. To account for non-linearity in shear strength response, Patton, Ladanyi- Archambault and Barton’s models are commonly used.

2.1 Coulomb’s Model

Coulomb’s shear strength parameters cohesion, cj and friction angle, ϕj are used to estimate the shear strength for given normal stress. The shear strength parameters may be obtained by performing direct shear tests on the discontinuity surfaces. Direct shear tests are conducted for various normal loads and shear stress vs. shear displacement plots are obtained. From these plots, the peak and residual shear strength of the joints are obtained. The failure envelopes of peak and residual shear strength are plotted (Fig. 3.1). The shear strength of the discontinuity is defined as:

$$ {\uptau}_{\mathrm{f}}={\mathrm{c}}_{\mathrm{j}}+{\upsigma}_{\mathrm{n}}\tan {\upphi}_d\kern1em \left(\mathrm{Peak}\kern0.4em \mathrm{strength}\right) $$
(3.1)
$$ {\uptau}_{\mathrm{f}}={\upsigma}_{\mathrm{n}}\tan {\upphi}_{\mathrm{r}}\kern1em \left(\mathrm{Residual}\ \mathrm{strength}\right) $$
(3.2)
Fig. 3.1
figure 1

Failure envelopes of shear strength for rough rock joints

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Where τf is the shear strength along the discontinuity; σn is the effective normal stress over the discontinuity; ϕj is the peak friction angle of the discontinuity surface; cj is the peak cohesion of the discontinuity surface, and ϕr is residual friction angle for discontinuity surface.

2.2 Patton’s Bi-Linear Shear Strength Model

Patton [1] recognised the importance of failure modes and suggested a bi-linear shear strength model. It was postulated that at low normal stress level, sliding occurs along the asperities of the joint surfaces, and at high normal stress, the shearing of the asperities takes place. The model is expressed as:

$$ {\tau}_f\kern0.5em =\kern0.5em {\sigma}_n\tan\ \left({\phi}_{\mu }+\mathrm{i}\right)\kern1em \mathrm{for}\ \mathrm{low}\ {\sigma}_{\mathrm{n}}\kern0.75em (Sliding) $$
(3.3)
$$ {\tau}_f\kern0.5em =\kern0.5em {\mathrm{c}}_{\mathrm{j}}+{\sigma}_{\mathrm{n}}\tan \left({\phi}_r\right)\kern0.75em \mathrm{for}\ \mathrm{high}\ {\sigma}_{\mathrm{n}}\kern0.75em (Shearing) $$
(3.4)

Where i defines the roughness angle, ɸμ is basic friction angle and ϕr is the residual friction angle.

In the field, it is very difficult to assess the normal stress level at which transition from sliding to shearing takes place. In reality, there is gradual transition from sliding to shearing and as such, there is no distinct and clear-cut normal stress level, which defines the boundary between the two failure modes.

2.3 Ladanyi and Archambault Criterion

Based on the principle of strain energy, Ladanyi and Archambault [2] equated the external work done in shearing a jointed rock to the internal energy stored in the rock and expressed the shear strength of a joint or rock mass as:

$$ {\uptau}_{\mathrm{f}}=\frac{\upsigma_{\mathrm{n}}\left(1-{\mathrm{a}}_{\mathrm{s}}\right)\ \left(\dot{\mathrm{v}}+\tan {\upphi}_{\upmu}\right)+{\mathrm{a}}_{\mathrm{s}}\left({\mathrm{c}}_{\mathrm{i}}+{\upsigma}_{\mathrm{n}}\tan {\upphi}_{\mathrm{i}}\right)}{1-\left(1-{\mathrm{a}}_{\mathrm{s}}\right)\ \dot{\mathrm{v}}\ \tan {\upphi}_{\upmu}} $$
(3.5)

where τf 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; σn is the mean applied normal stress; v ̇ is the rate of dilation at failure; ϕμ is the basic friction angle of joint surface, and ci, ϕi are Mohr-Coulomb parameters for intact rock.

The sheared area ratio, as and dilation rate, v ̇ are estimated as:

$$ {\mathrm{a}}_{\mathrm{s}}=1-{\left(1-\frac{\upsigma_{\mathrm{n}}}{\upsigma_{\mathrm{trn}}}\right)}^{{\mathrm{K}}_1} $$
(3.6)
$$ \dot{\mathrm{v}}\approx {\left(1-\frac{\upsigma_{\mathrm{n}}}{\upsigma_{\mathrm{trn}}}\right)}^{{\mathrm{K}}_2}\tan \mathrm{i} $$
(3.7)

Where K1 and K2 are equal to 1.5 and 4 respectively; σtrn 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.

2.4 Barton’s JRC-JCS Model

Barton’s model, also known as JRC-JCS model is very simple and is the most widely used strength criterion for assessing shear strength along discontinuity surfaces. The shear strength of a joint is expressed as:

$$ {\uptau}_{\mathrm{f}}={\upsigma}_{\mathrm{n}}\;\tan \left[{\upphi}_{\mathrm{r}}+\mathrm{JRC}\;{\log}_{10}\frac{\mathrm{JCS}}{\upsigma_{\mathrm{n}}}\right] $$
(3.8)

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 [3] as shown in Fig. 3.2. JCS is the joint wall compressive strength of the discontinuity surface, and σn is the effective normal stress acting across the discontinuity surface.

Fig. 3.2
figure 2

Roughness profiles to estimate JRC [3]

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3 Shear Strength of Rock Mass

Depending on the scale of structure, geometry of discontinuities and interlocking conditions, the failure may occur due to complex interaction of sliding, rotation, splitting, shearing and translation of rock blocks. In such cases, the jointed rock mass may be replaced with an equivalent continuum for mechanical analysis. Several strength criteria are used to express the strength behaviour of such rock masses. Some of the strength criteria are discussed in the followings in two sub-headings i.e. linear and non-linear strength criteria.

3.1 Linear Mohr-Coulomb Criterion

The rock mass is treated as an isotropic continuum and the shear strength along the failure surface is expressed as follows:

$$ {\uptau}_{\mathrm{f}}={\mathrm{c}}_{\mathrm{m}}+{\upsigma}_{\mathrm{n}}\tan {\upphi}_{\mathrm{m}} $$
(3.9)

Where cm and ϕm are Mohr-Coulomb shear strength parameters for jointed rock or rock mass. The values of cm and ϕm may be obtained from field shear tests on rock mass. Alternatively, classification approaches provide a rough estimate of the shear strength and some of these approaches are given below.

3.1.1 Rock Mass Rating (RMR)

The RMR classification system for rock masses was suggested by Bieniawski [4,5,6] to characterise the quality of the rock mass. The following parameters of the rock mass are used to classify the mass:

  1. (a)

    UCS of intact rock material,

  2. (b)

    Rock Quality Designation (RQD),

  3. (c)

    Spacing of discontinuities,

  4. (d)

    Condition of discontinuities, and

  5. (e)

    Groundwater condition.

Basic RMR is obtained based on above five parameters and then rating is adjusted based and orientation of discontinuities with respect to the structure. The values of shear strength parameters cm, ϕm are presented in Table 3.1 [5].

Table 3.1 Mohr-Coulomb parameters from RMR [5]
Full size table

Mehrotra [7], based on experience from Indian project sites, observed that the shear strength is under-predicted by expressions suggested by Bieniawski [5]. Figure 3.3 may be used for assessing the shear strength parameters of rock masses especially for slopes.

Fig. 3.3
figure 3

Estimation of friction angle of rock mass from RMR [7]

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3.1.2 Q System

The classification system Q [8, 9] is very popular for characterisation of rock mass. The rock mass quality index Q, is defined as:

$$ \mathrm{Q}=\left(\frac{\mathrm{RQD}}{{\mathrm{J}}_{\mathrm{n}}}\right)\left(\frac{{\mathrm{J}}_{\mathrm{r}}}{{\mathrm{J}}_{\mathrm{a}}}\right)\left(\frac{{\mathrm{J}}_{\mathrm{w}}}{\mathrm{SRF}}\right) $$
(3.10)

Where, RQD is the rock quality designation [10]; Jn is the joint set number; Jr is the joint roughness number; Ja is the joint alteration number; Jw is the joint water reduction factor, and SRF is stress reduction factor.

The shear strength parameters are obtained as:

$$ {\mathrm{c}}_{\mathrm{m}}=\left(\frac{\mathrm{RQD}}{{\mathrm{J}}_{\mathrm{n}}}\right)\left(\frac{1}{\mathrm{SRF}}\right)\left(\frac{\upsigma_{\mathrm{c}\mathrm{i}}}{100}\right)\kern0.84em \mathrm{MPa} $$
(3.11)
$$ {\upphi}_{\mathrm{m}}={\tan}^{-1}\left(\frac{{\mathrm{J}}_{\mathrm{r}}}{{\mathrm{J}}_{\mathrm{a}}}{\mathrm{J}}_{\mathrm{w}}\right) $$
(3.12)

where cm is the cohesion of the undisturbed rock mass; ϕm, the friction angle of the mass; and σci is the uniaxial compressive strength of intact rock material.

If used for slopes, an overestimation in the strength may be expected. For slopes, this author personally feels that the relationship between shear strength parameters and RMR as suggested by Mehrotra [7] will be more appropriate for 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

The Mohr-Coulomb strength criterion considers the rock mass shear strength as a linear function of normal stress σn. In reality, the shear strength response is highly non-linear and the parameters cm, ϕm change with the range of confining pressure used in estimating these parameters. To resolve this issue, several non-linear strength criteria have been proposed for jointed rocks and rock masses. Some of them are presented in the following section.

3.2.1 Empirical Criteria Based on RMR and Q

Mehrotra [7] utilised results of large number of in-situ direct shear tests on rock masses in the Himalayas, and suggested the non-linear variation of shear strength as:

$$ \frac{\uptau_{\mathrm{f}}}{\upsigma_{\mathrm{ci}}}=\mathrm{A}{\left(\frac{\upsigma_{\mathrm{n}}}{\upsigma_{\mathrm{ci}}}+\mathrm{B}\right)}^{\mathrm{C}} $$
(3.13)

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 3.2.

Table 3.2 Shear strength parameters for jointed rock masses [7]
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3.2.2 Hoek-Brown Strength Criterion

Hoek-Brown [11] proposed a non-linear strength criterion for intact rocks as follows:

$$ {\upsigma}_1={\upsigma}_3+\sqrt{{\mathrm{m}}_{\mathrm{i}}{\upsigma}_{\mathrm{ci}}{\upsigma}_3+{\upsigma}_{\mathrm{ci}}^2} $$
(3.14)

where σ 1 is the effective major principal stress at failure; σ 3 is the effective minor principal stress at failure; mi is a criterion parameter; and σci is the UCS of the intact rock, which is also treated as a criterion parameter. The parameters m and σci are obtained by fitting the criterion into the laboratory triaxial test data. If triaxial test data is not available the approximate values of parameters mi can also be obtained from Table 3.3 [12].

Table 3.3 Approximate estimation of parameter mi [12]
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The criterion was extended to heavily jointed isotropic rock masses [11]. The latest form of the criterion [13] is expressed as:

$$ {\upsigma}_1={\upsigma}_3+{\upsigma}_{\mathrm{ci}}{\left({\mathrm{m}}_{\mathrm{j}}\frac{\upsigma_3}{\upsigma_{\mathrm{ci}}}+{\mathrm{s}}_{\mathrm{j}}\right)}^{\mathrm{a}} $$
(3.15)

Where m j is an empirical constant, which depends upon the rock type; and s j is an empirical constant, which varies between zero (for crushed rock) to one (for intact rock) depending upon the degree of fracturing. The expressions for mj and sj are given as:

$$ {\mathrm{m}}_{\mathrm{j}}={\mathrm{m}}_{\mathrm{i}}\exp \left(\frac{\mathrm{GSI}-100}{28-14\mathrm{D}}\right) $$
(3.16)
$$ {\mathrm{s}}_{\mathrm{j}}=\exp \left(\frac{\mathrm{GSI}-100}{9-3\mathrm{D}}\right) $$
(3.17)

where D is a factor which depends upon the degree of disturbance to which the rock mass has been subjected by blast damage and stress relaxation. It varies from zero for undisturbed in situ rock masses to one for very disturbed rock masses. For blasted rock slopes, D is taken in the range 0.7–1.0.

GSI is the Geological Strength Index [13, 14] which depends on the structure of mass and surface characteristics of the discontinuities (Fig. 3.4).

Fig. 3.4
figure 4

Estimation of geological strength index [15]

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The index a is obtained as:

$$ \mathrm{a}=\frac{1}{2}+\frac{1}{6}\left({\mathrm{e}}^{-\mathrm{GSI}/15}-{\mathrm{e}}^{-20/3}\right) $$
(3.18)

The limitation of the GSI approach is that the GSI is estimated only from geological features and disturbance to the mass, and no measurements e.g. joint mapping are done in the field.

3.2.3 Ramamurthy Criterion

Based on extensive triaxial tests conducted on rocks and model materials, Ramamurthy and co-workers [16,17,18,19] suggested the following non-linear strength criterion for intact isotropic rocks:

$$ \left(\frac{\upsigma_1-{\upsigma}_3}{\upsigma_3+{\upsigma}_{\mathrm{t}}}\right)={\mathrm{B}}_{\mathrm{i}}{\left(\frac{\upsigma_{\mathrm{ci}}}{\upsigma_3+{\upsigma}_{\mathrm{t}}}\right)}^{\alpha_i} $$
(3.19)

where σ3 and σ1 are the minor and major principal stresses at failure; σt is the tensile strength of intact rock; σci is the UCS of the intact rock; and αi, Bi are the criterion parameters.

Parameters αi and Bi are criterion parameters and are suggested to be obtained by fitting the criterion into the laboratory triaxial test data for intact rock. Alternatively, the following approximate correlations may be used:

$$ {\upalpha}_{\mathrm{i}}=2/3;\kern0.5em \mathrm{and}\kern0.5em {\mathrm{B}}_{\mathrm{i}}=1.1{\left(\frac{\upsigma_{\mathrm{ci}}}{\upsigma_{\mathrm{t}}}\right)}^{1/3}\mathrm{to}\kern0.5em 1.3{\left(\frac{\upsigma_{\mathrm{ci}}}{\upsigma_{\mathrm{t}}}\right)}^{1/3} $$
(3.20)

The criterion was extended to jointed rocks and rock masses as:

$$ \left(\frac{\upsigma_1-{\upsigma}_3}{\upsigma_3}\right)={\mathrm{B}}_{\mathrm{j}}{\left(\frac{\upsigma_{\mathrm{cj}}}{\upsigma_3}\right)}^{\upalpha_{\mathrm{j}}} $$
(3.21)

where αj and Bj are the criterion parameters for jointed rock; and σcj is the UCS of the jointed rock.

The criterion parameters αj and Bj are suggested to be obtained from the following correlations:

$$ \frac{\alpha_j}{\alpha_i}={\left(\frac{\sigma_{cj}}{\sigma_{ci}}\right)}^{0.5} $$
(3.22)
$$ \frac{B_i}{B_j}=0.13\exp \left[2.037{\left(\frac{\sigma_{cj}}{\sigma_{ci}}\right)}^{0.5}\right] $$
(3.23)

Where σcj is the UCS of the rock mass.

3.2.4 Modified Mohr Coulomb Criterion

The Mohr-Coulomb criterion, though most widely used criterion, has the limitation in that the non-linear strength response of rocks is not captured by this criterion. Singh and Singh [20] used critical state concept for rocks [21] and suggested the Modified Mohr Coulomb (MMC) criterion to incorporate non-linearity in strength behaviour. The advantage of MMC is that the parameters cm and ϕm are retained as such. The criterion is expressed as:

$$ \left({\upsigma}_1-{\upsigma}_3\right)={\upsigma}_{\mathrm{cj}}+\frac{2\ \sin\ {\upphi}_{\mathrm{m}0}}{1\hbox{-} \sin\ {\upphi}_{\mathrm{m}0}}{\upsigma}_3-\frac{1}{\upsigma_{\mathrm{ci}}}\frac{\sin\ {\upphi}_{\mathrm{m}0}}{\left(1\hbox{-} \sin {\upphi}_{\mathrm{m}0}\right)}{\upsigma}_3^2\kern0.5em \mathrm{for}\ 0\le {\upsigma}_3\le {\upsigma}_{\mathrm{ci}} $$
(3.24)

Where σci and σcj are the UCS of the intact rock and rock mass respectively; ϕm0 is the friction angle of the rock mass corresponding to very low confining pressure range (σ3 →0) and can be obtained as:

$$ \sin {\upphi}_{\mathrm{m}0}=\frac{\left(1-\mathrm{SRF}\right)+\frac{\sin {\upphi}_{\mathrm{i}0}}{1\hbox{-} \sin {\upphi}_{\mathrm{i}0}}}{\left(2\hbox{-} \mathrm{SRF}\right)+\frac{\sin {\upphi}_{\mathrm{i}0}}{1\hbox{-} \sin {\upphi}_{\mathrm{i}0}}} $$
(3.25)

Where SRF = Strength Reduction Factor = σcj/σci; ϕi0 is friction angle for the intact rock and is obtained by conducting triaxial strength tests on intact rock specimens at low confining pressures (σ3→0).

If triaxial test data on intact rock is not available, the following non-linear form of the criterion may be used [22, 23]:

$$ {\sigma}_1=\mathrm{A}{\left({\sigma}_3\right)}^2+\left(1\hbox{--} 2\mathrm{A}{\sigma}_{\mathrm{ci}}\right){\sigma}_3+{\sigma}_{\mathrm{cj}};\kern2em {\sigma}_3\le {\sigma}_{\mathrm{ci}} $$
(3.26)

Where A is criterion parameter and may be estimated from the experimental value of σci, using the following expressions:

$$ \mathrm{For}\ \mathrm{average}\kern0.5em {\sigma}_1\kern2.75em \mathrm{A}=-1.23{\left({\sigma}_{\mathrm{ci}}\right)}^{-0.77} $$
(3.27)
$$ \mathrm{For}\ \mathrm{lower}\ \mathrm{bound}\kern0.5em {\sigma}_1\kern0.75em \mathrm{A}=-0.43{\left({\sigma}_{\mathrm{ci}}\right)}^{-0.72} $$
(3.28)

For design purposes, the lower bound values of σ1 are recommended to be used.

3.3 Rock Mass Strength (σcj)

An important input to the strength criteria for rock masses is the UCS of rock mass σcj. The following approaches may be used to determine the UCS of the rock mass:

  1. (i)

    Joint Factor concept, Jf

  2. (ii)

    Rock quality designations, RQD

  3. (iii)

    Rock mass quality, Q

  4. (iv)

    Rock mass rating, RMR

  5. (v)

    Modulus ratio concept (Strength reduction factor)

3.3.1 Joint Factor Concept

Ramamurthy and co-workers have defined a weakness coefficient that characterises the effect of fracturing on rocks and termed it Joint Factor [16, 18, 24,25,26]. The Joint Factor considers the combined effect of frequency, orientation, shear strength of joints, and is defined as:

$$ {\mathrm{J}}_{\mathrm{f}}=\frac{{\mathrm{J}}_{\mathrm{n}}}{\mathrm{n}\kern0.24em \mathrm{r}} $$
(3.29)

where, Jn = joint frequency, i.e., number of joints/metre; n is inclination parameter, depends on the inclination of sliding plane with respect to the major principal stress direction (Table 3.4); r is a parameter for joint strength; it is obtained from direct shear tests conducted along the joint surface at low normal stress levels and is given by:

$$ \mathrm{r}=\frac{\uptau_{\mathrm{j}}}{\upsigma_{\mathrm{nj}}}=\tan {\upphi}_{\mathrm{j}} $$
(3.30)
Table 3.4 Joint inclination parameter n [16]
Full size table

Where τj is the shear strength along the joint; σnj is the normal stress across the joint surface; and ϕj is the equivalent value of friction angle incorporating the effect of asperities [27]. 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 ϕj 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 ϕj may be obtained from Table 3.5 [27]. If the joints are filled with gouge material and have reached the residual shear strength, the value of r may be assigned from Table 3.6 [27].

Table 3.5 Values of joint strength parameter, r for different values of σci (After Ramamurthy [16, 27])
Full size table
Table 3.6 Joint strength parameter, r for filled-up joints at residual stage (After Ramamurthy [16, 27])
Full size table

The UCS of the rock mass is obtained as:

$$ {\upsigma}_{\mathrm{cj}}={\upsigma}_{\mathrm{ci}}\ \exp \left(\mathrm{a}\ {\mathrm{J}}_{\mathrm{f}}\right) $$
(3.31)

where a is an empirical coefficient equal to −0.008.

Singh [25] and Singh et al. [26] suggested that the failure of the rock mass under uniaxial stress condition may occur due to various failure modes i.e. splitting, shearing, sliding and rotation. The values for different modes of failure are presented in Table 3.7. The failure mode may be decided as per guideline given below [25, 26]. If it is not possible to assess the failure mode, an average value of the empirical constant, ‘a’ may be taken equal to −0.017.

Table 3.7 Empirical constant ‘a’ for estimating σcj
Full size table

Let θ be the angle between the normal to the joint plane and the major principal stress direction:

  1. (i)

    For θ = 0–10°, the failure is likely to occur due to splitting of the intact material of blocks.

  2. (ii)

    For θ = 10° to ≈ 0.8 ϕj, the mode of failure shifts from splitting (at θ = 10°) to sliding (at θ ≈ 0.8 ϕj).

  3. (iii)

    For θ = 0.8ϕj to 65°, the mode of failure is expected to be sliding only.

  4. (iv)

    For θ = 65–75°, the mode of failure shifts from sliding (at θ = 65°) to rotation of blocks (at θ = 75°).

  5. (v)

    For θ = 75–85°, the mass fails due to rotation of blocks only.

  6. (vi)

    For θ = 85–90°, the failure mode shifts from rotation at θ = 85° to shearing at θ = 90°.

3.3.2 Rock Quality Designation, RQD

Zhang [28] has proposed the following correlation for obtaining UCS of rock mass as a function of RQD. It may be noted that joint attributes like frequency and surface roughness have not been given any consideration in this approach.

$$ \frac{\sigma_{cj}}{\sigma_{ci}}={10}^{\left(0.013\; RQD-1.34\right)} $$
(3.32)

3.3.3 Rock Mass Quality, Q

Based on extensive database, Singh et al. [29] have proposed correlations of rock mass strength, σcj with Q by analysing block shear tests in the field.

$$ {\upsigma}_{\mathrm{cj}}=0.38{\upgamma \mathrm{Q}}^{1/3}\mathrm{MPa}\kern1em \mathrm{for}\ \mathrm{slopes} $$
(3.33)
$$ {\upsigma}_{\mathrm{cj}}=7{\upgamma \mathrm{Q}}^{1/3}\;\mathrm{MPa}\kern1em \mathrm{for}\ \mathrm{tunnels} $$
(3.34)

Barton [9] modified the above equation for tunnels and suggested the expression:

$$ {\upsigma}_{\mathrm{cj}}=5\upgamma {\left(\frac{\mathrm{Q}\;{\upsigma}_{\mathrm{ci}}}{100}\right)}^{1/3}\mathrm{MPa}\kern1em \mathrm{for}\ \mathrm{tunnels} $$
(3.35)

Where γ is the unit weight of rock mass in gm/cm3; and Q is the Barton’s rock mass quality index.

3.3.4 Rock Mass Rating, RMR

Rock Mass Rating (RMR) may be used to get the shear strength parameters cm and ϕm from RMR [4,5,6] and the rock mass strength σcj may be obtained as:

$$ {\upsigma}_{\mathrm{c}\mathrm{j}}=\frac{2{\mathrm{c}}_{\mathrm{m}}\ \cos {\upphi}_{\mathrm{m}}}{1\hbox{-} \sin {\upphi}_{\mathrm{m}}} $$
(3.36)

Ramamurthy [19] has suggested that the shear strength parameters recommended by Bieniawski [4,5,6] appear to be on lower side resulting in very low values of σcj.

The other commonly used correlations with RMR are as follows:

  1. (i)

    Kalamaras and Bieniawski [30]

$$ \frac{\upsigma_{\mathrm{cj}}}{\upsigma_{\mathrm{ci}}}=\exp \left(\frac{\mathrm{RMR}-100}{24}\right) $$
(3.37)
  1. (ii)

    Sheorey [31]

$$ \frac{\upsigma_{\mathrm{cj}}}{\upsigma_{\mathrm{ci}}}=\exp \left(\frac{\mathrm{RMR}-100}{20}\right) $$
(3.38)

3.3.5 Strength Reduction Factor

In the opinion of this author the best estimates of rock mass strength, σcj can only be made in the field through large size field-testing in which the mass may be loaded upto failure to determine rock mass strength. It is, however, rarely feasible. An alternative will be to get the deformability properties of rock mass by stressing a limited area of the mass upto a certain stress level, and then relate the ultimate strength of the mass to the laboratory UCS of the rock material through a strength reduction factor, SRF. Singh and Rao [32] have shown that the Modulus Reduction Factor, MRF and Strength Reduction Factor, SRF are correlated with each other by the following expression approximately:

$$ \mathrm{SRF}={\left(\mathrm{MRF}\right)}^{0.63} $$
(3.39)
$$ \Rightarrow \frac{\sigma_{\mathrm{cj}}}{\sigma_{\mathrm{ci}}}={\left(\frac{{\mathrm{E}}_{\mathrm{j}}}{{\mathrm{E}}_{\mathrm{i}}}\right)}^{0.63} $$
(3.40)

Where SRF is the ratio of rock mass strength to the intact rock strength; MRF is the ratio of rock mass modulus to the intact rock modulus; σcj is the rock mass strength; σci is the intact rock strength; Ej is the elastic modulus of rock mass; and Ei 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.

It is recommended that field deformability tests should invariably be conducted on project sites. The elastic modulus of rock mass, Ej may be obtained in the field by conducting uniaxial jacking tests [33] in drift excavated for the purpose. The test consists of stressing two parallel flat rock faces (usually the roof and invert) of a drift by means of a hydraulic jack [7]. The stress is generally applied in two or more cycles as shown in Fig. 3.5. The second cycle of the stress deformation curve is used for computing the field modulus as:

$$ {\mathrm{E}}_{\mathrm{j}}=\frac{\mathrm{m}\left(1-{\upupsilon}^2\right)\mathrm{P}}{\sqrt{\mathrm{A}}\kern0.4em {\updelta}_{\mathrm{e}}} $$
(3.41)

where Ej is the elastic modulus of the rock mass in kg/cm2; υ is the Poisson’s ratio of the rock mass (= 0.3); P is the load in kg; δe 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 25 mm thickness).

Fig. 3.5
figure 5

Field modulus of elasticity [33]

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The size of the drift should be sufficiently large as compared to the plate size so that there is little effect of confinement. The confinement may result in over prediction of the modulus values.

A number of methods for assessing the rock mass strength, σcj have been discussed above. It is desirable that more than one method be used for assessing the rock mass strength and generating the failure envelopes. A range of values will thus be obtained and design values may be taken according to experience and confidence of the designer.

4 Concluding Remarks

Assessment of shear strength behaviour of jointed rocks and rock masses is a difficult task. At the time of failure, the strength may be mobilised along a dominating persistent discontinuity or through blocks of the rock mass. Accordingly, discontinuity shear strength or rock mass strength will govern the design. Some of the approaches available for obtaining the shear strength of an individual discontinuity or of a mass as a whole have been discussed in the present article. The strength behaviour is known to vary non-linearly with confinement and hence special emphasis has been given to non-linear strength criteria. In real life problems, heterogeneity and uncertainty are very common. It may be expected that shear strength obtained from different approaches will vary over a range. It is advisable parametric analysis be done to examine the behaviour of the structure for the range of values to gain more confidence in the design.