P ATHS OF INCORPORATION

Knowledge of the stress state in a slope is essential in order to understand the mechanics of slope behavior. The stresses acting upon a structure in comparison to the strength of the structure govern the stability of that structure. The virgin stresses (before excavation) in the rock mass are, in almost all cases, compressive, and are primarily a combination of:

• Gravitational stresses arising from the weight of the overlying rock

• Tectonic stresses which stem from external tectonic forces

• Stresses caused by previous glaciation

• Residual stresses

The gravitational and tectonic components are in most cases the major contributors to the overall virgin stress state. The virgin vertical stress can usually be assumed to be solely due to the weight of the overlying rock mass. The virgin horizontal stress, on the other hand, is more difficult to quantify due to the tectonic component normally present. The tectonic stresses vary significantly in different regions of the world, and this only further emphasizes the importance of conducting stress measurements to determine the virgin stress state before mining. Stress measurement techniques are not discussed here, but can be found in, for example, Brady and Brown (1985).

In general, an excess of horizontal stresses compared to vertical stresses are found at shallower depths, in particular in old shield regions such as Scandinavia. At deeper depths (>1000-2000 meters) the horizontal stresses decrease and at very deep depths the horizontal stresses are smaller than the vertical stresses (see e.g., Hoek et al., 1995). For an open pit mining operation in Scandinavia and other old shield regions, one will in most cases have a virgin stress state in which the horizontal stress is higher than the vertical stress, whereas in younger, sedimentary rock strata, the opposite can be true.

The virgin stress state is altered as the open pit is being excavated. The void created forces the stress to redistribute around the open pit. To understand this, consider a two-dimensional section of a very long open pit, in which the virgin stress state is characterized by a vertical, gravitational stress, and a horizontal stress which is higher than the vertical stress.

Theoretically, the horizontal stress is forced under the bottom of the pit, resulting in stress

concentrations at the toe of the slope, and destressing of the pit walls, as is illustrated in Figure 3.1. The vertical stress after excavation will still mainly be due to the gravitational load. The resulting principal stresses around the open pit will thus have reoriented themselves compared to the virgin stress state. For a very long open pit, the principal stresses will be oriented parallel and perpendicular to the slope outline (Figure 3.1).

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Figure 3.1 Two-dimensional representation of the redistribution of horizontal stress around a long open pit (large dimension perpendicular to this plane).

Compressive stress concentrations at the toe of the slope promote stress-induced failure in this region. A zone of increased shear stresses also develops under the toe of the slope. Close to the pit wall, the principal stresses will be lower than the virgin stresses (destressing) which promotes joint opening and shear failure along pre-existing discontinuities in the rock mass, due to low normal stress.

In the literature on stability of rock slopes, it is often stated that stresses in slopes are low.

Such a general statement is very dangerous since it only applies to the region close to the pit walls, and at the toe of small scale slopes. For large scale slopes, the stress state is much more complex with zones of both low and high stresses. There are, unfortunately, very few studies on the stress state in open pit slopes. Most of what has been learned is through photoelastic analysis, see e.g., Hoek and Pentz (1968), and from numerical analysis, see e.g., Stacey (1970, 1973). In the two-dimensional analyses carried out by Stacey, the slope geometry and the

virgin stress state were varied. It was found that destressing from horizontal stresses became more pronounced for steeper rock slopes. The width of the pit bottom influenced the toe stresses but not the stress state higher up in the slope. Varying the lateral (horizontal) virgin stress had a very large effect on both magnitudes and orientations of the resulting principal stresses. This suggests that the horizontal virgin stress is important to determine accurately.

Later studies, for example, by Hustrulid and Kuchta (1995) and by the author, have shown that varying the horizontal virgin stress only affect the stress state in the toe region, whereas the region close to the slope face still will only be subjected to gravitational loads (the

destressed region in Figure 3.1). Stacey's study also showed that tensile stresses developed at the crest of the slope. The tensile zone increased in extent with increasing virgin horizontal stress and steeper slope angles. This phenomenon was also found in analyses by Nilsen (1979) and Coulthard et al. (1992), but is yet to be confirmed by measurements.

Comprehensive evaluations and verifications of the overall stress state in and around open pits are relatively scarce. A comparison between measured and calculated displacements was presented by Blake (1968), in which good agreement was found, thus indicating indirectly that the calculated stresses were reasonable. Stacey (1970, 1973) compared calculated stresses with measured stresses at the Kimbley pit (Blake, 1968) and found relatively good agreement between the two. Nilsen (1979) and Broch and Nilsen (1982) compared calculated stresses around the Ørtfjell open pit, with measured stresses in a drift close to the pit. A qualitative agreement was found, indicating the same orientation and the same order of magnitude of the principal stresses.

In the above analyses of the stress state around open pits, homogeneous, isotropic and elastic material behavior was assumed. In reality, one can also expect some local stress

redistributions around pre-existing discontinuities, as well as stress redistributions due to local failure and yielding of the rock material. A more serious assumption is that the pit geometry was simplified to enable a two-dimensional representation. The actual curvature of an open pit will change the way in which the virgin stresses are redistributed around the pit. In the two-dimensional case considered in Figure 3.1, the horizontal stress acting perpendicular to the cross-section is almost unaffected by the pit. For an open pit with finite dimensions, the intermediate principal stress will no longer be oriented parallel to the slope face, and its magnitude will also change significantly.

Some of the dimensional stress effects were quantified by Stacey (1973) using a three-dimensional numerical model. In this model two different slope curvatures (different radii) were analyzed with different virgin horizontal stresses. The slope angle was kept constant at 45°. The results showed that the stress redistribution in a vertical cross-section was of the

same pattern as for the two-dimensional case. However, due to the added confinement from the three-dimensional structure, the stress magnitudes at the toe of the slope were markedly lower. Furthermore, when the magnitude of the two virgin horizontal stresses were equal, no tensile zones were detected in the models. This is also an effect of the added confinement in a three-dimensional geometry. However, when the virgin horizontal stresses were unequal, tensile zones developed at the slope crest and behind the face of the slope, with more

extensive tensile zones with larger differences between the two horizontal stress components.

The extent of low-stress zones in the pit walls also depends on the orientation of the pit in relation to the virgin stress orientations. In general, for an elliptical or elongated pit, the three-dimensional effects will be most pronounced on the short-axis of the pit. In these concave regions, added confinement by the intermediate principal stress will result in less destressing of the pit walls. The principal stresses will all be compressive, except perhaps for the minor principal stress (Long, 1964; Hoek and Pentz, 1968), (Figure 3.2).

These effects have been confirmed by field observations (Piteau, 1970; Hoek and Bray, 1981).

These observations indicated an increase in stable slope angles of a little more than 10° when the radius of slope curvature decreased from 1000 feet (300 meters) to 200 feet (60 meters), for a slope height of 320 feet (100 meters). This was for a concave slope, but there could also be smaller portions of an open pit with a convex slope curvature. For this case, the stress state will be much more unfavorable with the minor principal stress, σ3, being tensile and oriented tangential to the slope face (Figure 3.2). The intermediate principal stress can also be very low or tensile in this case. For a convex slope with a radius of curvature less than the slope height, Hoek and Bray (1981) suggested a flattening of the slope angle by 10° compared to a stability analysis in which only a two-dimensional geometry is considered. Bear in mind though, that such recommendations are only rough guidelines based on very few observations.

The non-uniformity of the stress state in a slope also affects the strength of the rock mass.

The shape and location of the failure surface will be described in detail later, but for now assume that failure in a high rock slope occurs either along a curved failure surface which is deep seated in the slope, or along a planar, but more shallow, failure surface. The stress state at different points in the slope is illustrated in Figure 3.3, along with an assumed curved failure envelope. Low compressive, or even tensile, normal stresses are found at the crest of the slope. At the toe, the normal stress is moderate whereas the shear stress is relatively high. In the interior of the slope, the normal stress is high which implies that the resulting strength of the curved failure surface is high in this portion. Similar conclusions were drawn by Jennings and Steffen (1967).

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Figure 3.2 Principal stress state in different portions of an open pit mine with varying slope curvature (partly after Hoek and Pentz, 1968 and Long, 1964).

The different stress states and the different strengths along various portions of the failure surface indicate that the mechanism of failure can be significantly different along the failure surface. In Figure 3.3, only a two-dimensional stress state is considered with the major and minor principal stresses oriented parallel to the cross-section of the slope. As was discussed earlier for a long open pit, the horizontal stress parallel to the long axis of the pit is less affected by the excavation of the pit slopes. For some cases it can even exceed the other two stress components in magnitude. This leads to the possibility of compressive failure in the direction perpendicular to the cross-section shown in Figure 3.3.

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Figure 3.3 Schematic illustration of the stress state at different points along a potential planar and a potential curved failure surface in a rock slope.