Conjugation

The strength of a large scale rock mass will ultimately determine whether slope failure will occur or not. It is thus of utmost importance to be able to quantify the rock mass strength for design purposes. To illustrate the problem consider a slope in a homogeneous and isotropic rock mass, and assume that failure occurs as rotational shear failure. Let the strength of the rock mass be characterized by a friction angle, φ, and a cohesion, c, representative for the composite rock mass as a whole, i.e., both intact rock and discontinuities. For this case, how sensitive is the calculated slope stability to changes in the strength properties? Assuming a

friction angle of 30° and a density for the rock mass of 2700 kg/m³, the required cohesion for maintaining the stability of the slope was calculated using the charts for circular failure given by Hoek and Bray (1981). (These and other design methods for rotational shear failure will be described in more detail in Section 4.3.5). The calculated cohesions are shown in Figure 3.24, for slope heights of 100 to 500 meters. The slope was also assumed to be fully drained (no groundwater pressure).

0.00 0.50 1.00 1.50

30 35 40 45 50 55 60 65 70

Slope Angle

Cohesion [MPa]

100 m 200 m 300 m 400 m 500 m

100 m 500 m

400 m

300 m

200 m

Figure 3.24 Required cohesion for the stability of slopes of various heights in a fully drained rock mass with a friction angle of 30°, and subject to rotational shear failure.

One finds that the required cohesion increases strongly with increasing slope height (or pit depth). Furthermore, the cohesions required to maintain slope stability vary markedly with slope angles. For a 500 meter high slope, an increase in cohesion of only 0.3 MPa

corresponds to an increase in stable overall slope angle from 40° to 50°. Small changes in the strength parameters thus correspond to relatively large geometrical changes of the slope geometry which has a large economic impact. Consequently, the accuracy by which the strength properties need to be determined for the design of large scale slopes is thus very high.

This effect is even larger for a slope with high groundwater pressure (fully saturated), as shown in Figure 3.25.

0.00

Figure 3.25 Required cohesion for a 500 meter high slope for a fully drained and a fully saturated rock mass, respectively, and with a friction angle of 30°, and subject to rotational shear failure.

The strength of the rock mass depends upon the strength of (1) the intact rock, and (2) the discontinuities present in the rock mass. Since the discontinuities in general have a much lower strength than the intact rock, a relatively low percentage of intact rock along the failure surface increases the composite rock mass strength dramatically. Nilsen (1979), showed that an increase from 0% to 10% of intact rock along the failure surface resulted in an increase of the safety factor from around 0.3 to 1.7, i.e., more than five times. It follows that a difference in the percentage of intact rock by only one or two percents may be decisive for the stability of the slope. However, in practice it is almost impossible to determine the ratio of intact rock along a failure surface with this accuracy.

To illustrate the effect of scale, take the 500 meter high slope as an example. The required cohesions for slope angles of 40° and 50° are 0.3 and 0.6 MPa, respectively. From the friction angle and the cohesion, the corresponding uniaxial compressive strength can be calculated as:

σ φ

This yields a rock mass compressive strength of 1.0 to 2.1 MPa. For a hard rock, the intact uniaxial compressive strength determined from laboratory tests can be around 200 MPa. The required strengths for a 500 meter high rock slope thus represent a reduction of the intact rock strength by a factor of 100 to 200. The effect of increasing scale on the rock mass strength is illustrated by the curve in Figure 3.26. The decrease in strength with increasing volume is primarily due the increased number of pre-existing discontinuities that are included into the rock mass, from small scale joints to larger faults, see e.g., Krauland, Söder and Agmalm (1986) and Pinto da Cunha (1990, 1993a, 1993b).

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Figure 3.26 Schematic illustration of the relation between strength and volume of a rock mass.

Measured and back-calculated strengths from several Swedish hard rock mines are

summarized in Table 3.2. Strength values for hangingwalls are exclusively from hangingwall failures in sublevel caving mines (see also Section 3.5.4). The relative decrease in strength becomes less prominent for large volumes. However, even on a very large scale it is not certain that the rock mass strength tends toward a constant value. This is obvious from Table 3.2, where even for volumes larger than 15·10⁶ m³, the scale factors vary from 37 to 220.

The scale effect according to Figure 3.26 is prominent in many materials (Weibull, 1939a, 1939b). There is, however, evidence that the strength may become constant for very large volumes. From studies on coal pillars, Bieniawski (1968) concluded that for samples larger than a certain volume the strength remain constant. Samples smaller than the average joint spacing also exhibited constant strength. Moreover, the scatter in strength decreased with

increasing pillar size in the tests by Bieniawski (1968). The same behavior could potentially be expected also for large scale slopes, but this remains to be verified.

The volume above which scale-free property values can be obtained, is commonly referred to as the Representative Elementary Volume (REV), (see e.g. Pinto da Cunha, 1993a). The REV is the smallest volume for which there is equivalence between the real rock mass and an ideal continuum material. However, this is under the assumption that the rock mass on a large scale actually behaves as a continuum. The REV could be different for different rock masses

though, and different for different properties. Although much has been written on REV, the focus has been toward theoretical and laboratory studies of relatively small scale samples.

Practical applications to large scale slopes are yet to be found.

Table 3.2 Large scale rock mass strength and scale factor for hard rocks. Values obtained through stress measurements (pillars) and back-analysis of failures (stope roofs, sill pillars and hangingwalls) from Swedish mines (from

Krauland, Söder and Agmalm, 1986; Herdocia, 1991; Sjöberg et al., 1996).

Uniaxial compressive strength

Pillar Sulfide ore 210 20 500 10

Pillar Keratophyre 130 26 330 5

Stope roof Sulfide ore 250 65-70 840 4

Sill pillar Sulfide ore 220 90 - 2.4

Sill pillar Sulfide ore 260 102 - 2.5

Sill pillar Sulfide ore 225 96 - 2.3

Hangingwall Quartzite 180 1.9 0.3·10⁶ 95

Hangingwall Leptite 100 0.45 15·10⁶ 220

Hangingwall Leptite 108 2.94 29·10⁶ 37

Hangingwall Quartz porphyry 240 2.05 190·10⁶ 117

Hangingwall Granite 186 2.27 280·10⁶ 82

Taken together, these findings have some very practical significance for open pit slopes. One cannot, for instance, safely extrapolate derived strengths from bench or interramp slope failures to the design of overall pit slope angles. Furthermore, it can be questioned as to whether the required high accuracy in rock mass strength determination for the design of high slopes really can be achieved? If, however, the scatter in strength actually decreases with increasing scale, the possibility arises of extrapolating strength values from one slope to another, provided that the failure mechanisms and the geomechanical conditions are similar.

It is necessary to first consider the different factors contributing to the rock mass strength.

The strength of intact rock and pre-existing discontinuities is briefly reviewed in the following sections. Attempts to describe the rock mass strength are then presented, and finally,

approaches for determining the rock mass strength for large scale rock slopes are discussed.