Texture
*These values are for intact rock specimen tests normal to bedding or foliation. The value of mi will be significantly different if failure occurs along a weakness plane.
where mb is the material constant for the rock mass; and s and a are constants that depend on the characteristics of the rock mass.
The original criterion has been found to work well for most rocks of good to reasonable quality in which the rock mass strength is controlled by tightly interlocking angular rock pieces. The failure of such rock masses can be defined by setting a=0.5 in Equation (4.64), giving
(4.65)
For poor quality rock masses in which the tight interlocking has been partially destroyed by shearing or weathering, the rock mass has no tensile strength or ‘cohesion’ and specimens will fall apart without confinement. For such rock masses the following modified criterion is more appropriate and it is obtained by putting s=0 in Equation (4.64) which gives
(4.66)
Equations (4.64) to (4.66) are of no practical value unless the values of the material constants mb, s and a can be estimated in some way. Hoek and Brown (1988) proposed a set of relations between the parameters mb, s and a and the 1976 version of Bieniawski’s Rock Mass Rating (RMR), assuming completely dry conditions and a very favorable (according to RMR rating system) discontinuity orientation:
(i) disturbed rock masses
(4.67a)
(4.67b)
a=0.5 (4.67c)
(ii) undisturbed or interlocking rock masses
(4.68a)
(4.68b)
a=0.5 (4.68c)
Equations (4.67) and (4.68) are acceptable for rock masses with RMR values of more than about 25, but they do not work for very poor rock masses since the minimum value which RMR can assume is 18 for the 1976 RMR system and 23 for the 1989 RMR system (see Chapter 2 for details). In order to overcome this limitation, Hoek (1994) and Hoek et al. (1995) introduced the Geological Strength Index (GSI). The relationships between mb, s and a and the Geological Strength Index (GSI) are as follows:
(i) For GSI>25, i.e. rock masses of good to reasonable quality
(4.69a)
(4.69b)
a=0.5 (4.69c)
(ii) For GSI<25, i.e. rock masses of very poor quality
(4.70a)
s=0 (4.70b)
(4.70c)
It is noted that the distinction between disturbed and undisturbed rock masses is dropped in evaluating the parameters mb, s and a from GSI. This is based on the fact that disturbance is generally induced by engineering activities and should be allowed by downgrading the values of GSI. The methods for determining RMR and GSI have been discussed in Chapter 2.
Since many of the numerical models and limit equilibrium analyses used in rock mechanics are expressed in terms of the Coulomb failure criterion, it is necessary to estimate an equivalent set of cohesion and friction parameters for given Hoek-Brown values. This can be done using a solution published by Balmer (1952) in which the normal and shear stresses are expressed in terms of the corresponding principal stresses as follows:
(4.71)
(4.72)
For the GSI>25, when a=0.5:
(4.73)
For the GSI<25, when s=0:
(4.74)
Once a set of (σ′n, τ) values have been calculated from Equations (4.71) and (4.72), average cohesion c and friction angle values can be calculated by linear regression analysis, in which the best fitting straight line is calculated for the range of (σ′n, τ) pairs.
The uniaxial compressive strength of a rock mass defined by a cohesive strength c and a friction angle is given by
(4.75)
Water has a great effect on the strength of rock masses. Many rocks show a significant strength decrease with increasing moisture content. Typically, strength losses of 30–
100% occur in many rocks as a result of chemical deterioration of the cement or clay binder. Therefore, it is important to conduct laboratory tests at moisture contents which are as close as possible to those which occur in the field. A more important effect of water is the strength reduction which occurs as a result of water pressures in the pore spaces in the rock. This is why the effective not the total stresses are used in the Hoek-Brown strength criterion.
The Hoek-Brown strength criterion was originally developed for intact rock and then extended to rock masses. The process used by Hoek and Brown in deriving their strength criterion for intact rock (Equation 4.63) was one of pure trial and error (Hoek et al., 1995). Apart from the conceptual starting point provided by the Griffith theory, there is no fundamental relationship between the empirical constants included in the criterion and any physical characteristics of the rock. The justification for choosing this particular criterion (Equation 4.63) over the numerous alternatives lies in the adequacy of its predictions of the observed rock fracture behavior, and the convenience of its application
to a range of typical engineering problems (Hoek, 1983). The material constant mi is derived based upon analyses of published triaxial test results on intact rock (Hoek, 1983;
Doruk, 1991; Hoek et al., 1992). The strength criterion for rock masses is just an empirical extension of the criterion for intact rock. Since it is practically impossible to determine the material constants mb and s using triaxial tests on rock masses, empirical relations are suggested to estimate these constants from RMR or GSI. The RMR and the GSI rating systems are also empirical. For these reasons the Hoek-Brown empirical rock mass strength criterion must be used with extreme care. In discussing the limitations in the use of their strength criterion, Hoek and Brown (1988) emphasize that it is not applicable to anisotropic rocks nor to elements of rock masses that behave anisotropically by virtue of containing only a few discontinuities. Alternative empirical approaches and further developments of the Hoek-Brown criterion which seek to account for some of its limitations are given by Amadei (1988), Pan and Hudson (1988), Ramamurthy and Arora (1991), Amadei and Savage (1993), and Ramamurthy (1993).
(b) Bieniawski-Yudhbir criterion
Bieniawski (1974) proposed a strength criterion for intact rock as follows (4.76)
This was changed by Yudhbir et al. (1983), based on tests on jointed gypsum-celite specimens, to the form
(4.77)
to fit rock masses. Yudhbir et al. (1983) recommended that the parameters α and a be determined from
α=0.65
(4.78a) (4.78b)
where Q is the classification index of Barton et al. (1974) and RMR is Bieniawski’s 1976 Rock Mass Rating (Bieniawski, 1976). Parameter b is determined from Table 4.6.
Kalamaras and Bieniawski (1993) suggested that both a and b should be varied with RMR for better results. They proposed the criterion of Table 4.7 specifically for coal seams.
(c) Johnston criterion
Based on experimental data of a wide range of geotechnical material, from lightly overconsolidated clays through hard rocks, Johnston (1985) proposes the following strength criterion
(4.79)
where σ′1n and σ′3n are the normalized effective principal stresses at failure, obtained by dividing the effective principal stresses, σ′1 and σ′3, by the relevant uniaxial compressive strength, σc; B and M are intact material constants; and s is a constant to account for the strength of discontinuous soil and rock masses in a manner similar to that proposed by Hoek and Brown (1980). However, in the development of the criterion, Johnston (1985) considers only intact materials.