Metamorphic and sedimentary rocks Rock
type Argillaceous Arenaceous Chemical Igneous rocks
Siltstone Shales Sandstone Quartzite Limestone Marble Andesite Granite Clays Slates Anhydrite Dolomite Diorite Charnockite
where σcm is the rock mass strength in unconfined compression; Bm is a material constant for rock mass; and αm is the slope of the plot between (σ′1−σ′3)/σ′3 and σcm/σ′3, which can be assumed to be 0.8 for rock masses as well. σcm and Bm can be obtained by
(4.88)
(4.89)
in which σc is the unconfined compressive strength of intact rock strength; and Br is a material constant for intact rock, as in Equation (4.83).
(e) Comments
In addition to the four empirical strength criteria for rock masses described above, there are many other criteria. All these criteria are purely empirical and thus it is impossible to say which one is correct or which one is not. However, the Hoek-Brown strength criterion is the most representative one of the empirical strength criteria for rock masses, because it is the mostly widely referred and used. Since its advent in 1980, considerable application experience has been gained by its authors as well as by others. As a result,
this criterion has been modified several times to meet the needs of users who have applied it to conditions which were not visualized when it was originally developed.
It is noted that all the empirical strength criteria for rock masses have the following limitations:
1. The influence of the intermediate principal stress, which in some cases is important, is not considered.
2. The criteria are not applicable to anisotropic rock masses. So they can be used only when the rock masses are approximately isotropic, i.e. when the discontinuity orientation does not have a dominant effect on failure.
4.4.2 Equivalent continuum approach for estimating rock mass strength
(a) Model of Jaeger (1960) and Jaeger and Cook (1979)
Figure 4.42(a) shows a cylindrical rock mass specimen subjected to an axial major principal stress σ′1 and a lateral minor principal stress σ′3. The rock mass is cut by well-defined parallel discontinuities inclined at an angle β to the major principal stress σ′1. The strength of both the intact rock and the discontinuities are described by the Coulomb criterion, i.e.
(4.90) (4.91)
where τr and τj are respectively the shear strength of the intact rock and the discontinuities; cr and are respectively the cohesion and internal friction angle of the intact rock; cj and are respectively the cohesion and internal friction angle of the discontinuities; and σ′n is the effective normal stress on the shear plane.
For the applied stresses on the rock mass cylinder, the effective normal stress σ′n and the shear stress τ on a plane which makes an angle β′ to the σ′1 axis are respectively given by
(4.92)
(4.93)
If shear failure occurs on the discontinuity plane, the effective normal stress σ′n and the shear stress τ on the discontinuity plane can be obtained by replacing β′ in Equations (4.92) and (4.93) by β. Adopting the obtained stresses on the discontinuity plane to substitute for σ′n and τj in Equation (4.91) and then rearranging, we can obtain the effective major principal stress required to cause shear failure along the discontinuity as follows
(4.94)
If shear failure occurs in the intact rock, the minimum effective major principal stress can be obtained by
(4.95)
The model of Jaeger (1960) and Jaeger and Cook (1979) assumes that failure during compressive loading of a rock mass cylinder subject to a lateral stress σ′3 [see Fig.
4.42(a)] will occur when σ′1 exceeds the smaller of the σ′1f values given by Equations (4.94) and (4.95). Figure 4.42(b) shows the variation of σ′1f with β, from which we can clearly see the anisotropy of the rock mass strength caused by the discontinuities.
(b) Model of Amadei (1988) and Amadei and Savage (1989, 1993)
As seen above, the model of Jaeger (1960) and Jaeger and Cook (1979) assumes that the jointed rock mass is under axisymmetric loading, so the effect of the intermediate principal stress is not involved in their formulations. To address the limitation of the model of Jaeger (1960) and Jaeger and Cook (1979), Amadei (1988) and Amadei and Savage (1989, 1993) derived solutions for the strength of a jointed rock mass under a variety of multiaxial states of stress. As in the model of Jaeger (1960) and Jaeger and Cook (1979), the modeled rock mass is cut by a single discontinuity set. In the formulations of Amadei (1988) and Amadei and Savage (1989, 1993), however, the intact rock strength is described by the HoekBrown strength criterion and the discontinuity strength is modeled using a Coulomb criterion with a zero tensile strength cut-off.
The principle used by Amadei (1988) and Amadei and Savage (1989, 1993) to derive the expressions of the jointed rock mass strength is the same as that used by Jaeger (1960) and Jaeger and Cook (1979). However, since the effect of the intermediate principal stress is included and since the nonlinear Hoek-Brown strength criterion is used, the derivation process and the final results are much more complicated. For reasons of space, only some of the typical results of Amadei and Savage (1989, 1993) are shown here.
Consider a jointed rock mass cube under a triaxial state of stress σ′x, σ′y and σ′z. The orientation of the discontinuity plane is defined by two angles β and Ψ with respect to the xyz coordinate system (see Fig. 4.43). Let nst be another coordinate system attached to the discontinuity plane such that the n-axis is along the discontinuity upward normal and the s-and t-axes are in the discontinuity plane. The t-axis is in the xz plane. The upward unit vector n has direction cosines