The Use of Rock Mass Classification Systems to Estimate the Modulus and Strength of Jointed Rock

Abstract

Three-dimensional, elastic and elasto-plastic finite element (FE) programs have permitted calculation of the displacements and the factor of safety (FOS) for the excavation for a tower, 132.70 m high (above foundation) on the island of Tenerife. The tower is supported by a 2 m thick reinforced concrete slab on jointed, vesicular and weathered basalt and scoria. The installation of rod extensometers at different depths below the slab has permitted comparison between measured and calculated displacements and the estimation of in situ deformation modulus. The moduli deduced from the simple empirical equations proposed by Hoek et al. (In: NARMS-TAC, 2002) and Gokceoglu et al. (Int J Rock Mech Min Sci 40:701–710, 2003) as a function of GSI, and Nicholson and Bieniawski (Int J Min Geol Eng 8:181–202, 1990) as a function of RMR, provide an acceptable fit with the measured settlements in this type of rock. Good correlation is also obtained with the empirical equation presented by Verman et al. (Rock Mech Rock Eng 30(3):121–127, 1997) that incorporates the influence of confining stress in the deformation modulus. The FOS obtained from different correlations with geomechanical classifications is within a relatively narrow range. These results increase our confidence in the use of classification schemes to estimate the deformation and stability in jointed rock.

Appendix: Estimate of Rock Mass Strength

The generalized Hoek–Brown failure criterion for jointed rock masses is defined by the equation:

$$ \sigma^{\prime}_{ 1} = \sigma^{\prime}_{3} + \sigma_{\text{ci}} (m_{\text{b}} \sigma^{\prime}_{ 3} /\sigma^{\prime}_{\text{ci}} + s)^{a} $$

(10)

where m _b is a reduced value of the intact rock constant m _i and is given by

$$ m_{\text{b}} = m_{\text{i}} \,\exp [({\text{GSI}} - 100)/(28 - 14D)] $$

(11)

s and a are constants for the rock mass given by the following relationships:

$$ s = { \exp }[({\text{GSI}} - 100)/(9 - 3D)] $$

(12)

$$ a = 0. 5+ [{\text{exp(}} - {\text{GSI}}/15 )- {\text{exp(}} - 20/ 3 )]/ 6 $$

(13)

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 0 for undisturbed rock masses to 1 for very disturbed rock masses. The excavation in Tenerife was carried out by mechanical means and a value of 0.7 for the strata in the front of the excavation and 0 for the layers below the foundation have been used.

Hoek (1983) showed that, for brittle materials, the uniaxial tensile strength is equal to the biaxial strength. The tensile strength is obtained by setting $\sigma^{\prime}_{1} = \sigma^{\prime}_{3} = \sigma_{\text{t}}$ in Eq. 10 which represents a condition of biaxial tension.

$$ \sigma_{\text{t}} = - s\sigma_{\text{ci}} /m_{\text{b}} $$

(14)

Normal and shear stresses at failure are related to principal stresses by the equations given by Balmer (1952):

$$ \sigma^{\prime}_{n} = \sigma^{\prime}_{3} + (\sigma^{\prime}_{1} - \sigma^{\prime}_{3} )/(1 + \partial \sigma^{\prime}_{1} /\partial \sigma^{\prime}_{3} ) $$

(15)

$$ \tau = (\sigma^{\prime}_{n} - \sigma^{\prime}_{3} )(\partial \sigma^{\prime}_{1} /\partial \sigma^{\prime}_{3} )^{0.5} $$

(16)

Deriving Eq. 10 and operating:

$$ \partial \sigma^{\prime}_{1} /\partial \sigma^{\prime}_{3} = 1 + m_{\text{b}} a(m_{\text{b}} \sigma^{\prime}_{3} /\sigma_{\text{ci}} + s)^{a - 1} $$

(17)

Since most geotechnical software is still written in terms of the Mohr–Coulomb failure criterion, it is necessary to determine equivalent angles of friction and cohesive strengths for each rock mass and stress range. This is done by fitting an average linear relationship to the curve generated by solving Eq. 10 for a range of minor principal stress values defined by $\sigma_{\text{t}} < \sigma^{\prime}_{3} < \sigma^{\prime}_{{3{ \max }}}$ as illustrated in Fig. 15a.

Fig. 15

Relationships between Hoek–Brown and linear Mohr–Coulomb criteria for rock type d1: σ _t < σ _min < 1 MPa. a Major and minor principal stresses. b Normal and shear stresses

The well-known linear relationship for the Mohr–Coulomb criterion is

$$ \sigma^{\prime}_{1} = 2c^{\prime}\,{ \cos }\,\Upphi^{\prime}/(1 - { \sin }\,\Upphi^{\prime}) + [(1 + { \sin }\,\Upphi^{\prime})/(1 - { \sin }\,\Upphi^{\prime})]\sigma^{\prime}_{3} $$

(18)

which may be simplified as

$$ \sigma^{\prime}_{1} = \sigma_{\text{cm}} + k\sigma^{\prime}_{3} $$

(19)

where σ _cm is the uniaxial compressive strength of the rock mass corresponding to the linear Mohr–Coulomb relationship, and k is the slope of the straight line relating $\sigma^{\prime}_{1}$ and $\sigma^{\prime}_{3}$.

The values of Φ′ and c′ can be calculated equalizing the first and the second terms respectively of the right hand side of Eqs. 18 and 19, and operating:

$$ { \sin }\,\Upphi^{\prime} = (k - 1)/(k + 1) $$

(20)

$$ c^{\prime} = \sigma_{\text{cm}} /(2\sqrt k ) $$

(21)

The problem of fitting the linear relationship (19) to Eq. 10 has been solved analytically by Hoek et al. (2002) who have found the following equations:

$$ { \sin }\,\Upphi^{\prime} = \frac{{6am_{\text{b}} (s + m_{\text{b}} \sigma^{\prime}_{3n} )^{a - 1} }}{{2(1 + a)(2 + a) + 6am_{\text{b}} (s + m_{\text{b}} \sigma^{\prime}_{3n} )^{a - 1} }} $$

(22)

$$ c^{\prime} = \frac{{\sigma_{{{\text{c}}i}} [(1 + 2a)s + (1 - a)m_{\text{b}} \sigma^{\prime}_{3n} ](s + m_{\text{b}} \sigma^{\prime}_{3n} )^{a - 1} }}{{(1 + a)(2 + a)(1 + (6am_{\text{b}} (s + m_{\text{b}} \sigma^{\prime}_{3n} )^{a - 1} )/((1 + a)(2 + a)))^{1/2} }} $$

(23)

where $\sigma^{\prime}_{3n} = \sigma^{\prime}_{{3{ \max }}} /\sigma_{\text{ci}}$.

The normal stress-shear stress relationships are represented in Fig. 15b.

The selected level of $\sigma^{\prime}_{{3{ \max }}}$ has a great influence on the values of c′ and Φ′ obtained in this analysis. In some rock stability problems, the effective normal stress on some parts of the failure surface can be quite low. In these cases, the value of $\sigma^{\prime}_{{3{ \max }}}$, the upper limit of confining stress over which the relationship between the Hoek–Brown and the Mohr–Coulomb criteria is considered has to be determined from an appropriate stress analysis technique. Using Bishop’s circular failure analysis for a wide range of slope geometries and rock mass properties, Hoek et al. (2002) present the following equation for slopes:

$$ \frac{{\sigma^{\prime}_{3\max } }}{{\sigma^{\prime}_{\text{cm}} }} = 0.72\left( {\frac{{\sigma^{\prime}_{\text{cm}} }}{\gamma H}} \right)^{ - 0.91} $$

(24)

where H is the height of the slope.

This approach has been used to find the Mohr–Coulomb parameters for the different basalt types included in Tables 13 and 10. As the upper basalt is in the front of the excavation, Eq. 24 has been employed to find $\sigma^{\prime}_{{3{ \max }}}$. For basalt types d1 to d3, which are interspersed below the foundation, the maximum $\sigma^{\prime}_{3}$ value obtained in the elastic FE calculations (Sect. 5.1) has been chosen (1 MPa). The calculations have been carried out using the RocLab 1.0 program (Hoek 2006). The results are included in Table 13.

Table 13 Mohr–Coulomb parameters of jointed basalt from Hoek–Brown failure criterion