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1.4 Rock slope design methods
This section summarizes four different procedures for designing rock slopes, and shows the basic data that is required for analyzing slope stability. The design methods and the design data are common to both mining and civil engineering.

Tabel 1.1 Definition of landslides features

1.4.1 Summary of design methods
A basic feature of all slope design methods is that shear takes place along either a discrete sliding surface, or within a zone, behind the face. If the shear force (displacing force) is greater than  the shear strength of the rock (resisting force) on this surface, then the slope will be unstable. Instability could take the form of displacement that may or may not be tolerable, or the slope may collapse either suddenly or progressively. The definition of instability will depend on the application. For example, an open pit slope may undergo several meters of displacement without effecting operations, while a slope supporting a bridge abutment would have little tolerance for movement. Also, a single rock fall from a slope above a highway may be of little consequence if there is an adequate ditch to contain the fall, but failure of a significant portion of the slope that  reaches the traveled surface could have serious consequences.

Tabel 1.2 Definition of landslides dimensions

Based upon these concepts of slope stability, the stability of a slope can be expressed in one or
more of the following terms:
(a) Factor of safety, FS—Stability quantified by limit equilibrium of the slope, which is stable
if FS > 1.
(b) Strain—Failure defined by onset of strains great enough to prevent safe operation of the
slope, or that the rate of movement exceeds the rate of mining in an open pit.
(c) Probability of failure—Stability quantified by probability distribution of difference
between resisting and displacing forces, which are each expressed as probability distributions.
(d) LRFD (load and resistance factor design)—Stability defined by the factored resistance
being greater than or equal to the sum of the factored loads.

Tabel 1.3 Values of Minimum total safety

At this time (2003), the factor of safety is the most common method of slope design, and there is wide experience in its application to all types of geological conditions, for both rock and soil. Furthermore, there are generally accepted factor of safety values for slopes excavated for different purposes, which promotes the preparation of reasonably consistent designs. The ranges of minimum total factors of safety as proposed by Terzaghi and Peck (1967) and the Canadian Geotechnical Society (1992) are given in Table 1.3.

In Table 1.3, the upper values of the total factors of safety apply to usual loads and service conditions, while the lower values apply to maximum loads and the worst expected geological conditions. For open pit mines the factor of safety generally used is in the range of 1.2–1.4, using either limit equilibrium analysis to calculate directly the factor of safety, or numerical analysis
to calculate the onset of excessive strains in the slope.

Although probabilistic design methods for rock slopes were first developed in the 1970s (Harr, 1977; Canada DEMR, 1978), they are not widely used (as of 2003). A possible reason for this lack of acceptance is that terms such as “5% probability of failure” and “consequence of failure Principles of rock slope design 11 expressed as lives lost” are not well understood, and there is limited experience on acceptable probabilities to use in design (see Section 1.4.4).

The calculation of strain in slopes is the most recent advance in slope design. The technique has resulted from the development of numerical analysis methods, and particularly those that can incorporate discontinuities (Starfield and Cundall, 1988). It is most widely used in the mining field where movement is tolerated, and the slope contains a variety of geological conditions (see Chapter 10).

The load and resistance factor design method (LRFD) has been developed for structural design,
and is now being extended to geotechnical systems such as foundations and retaining structures.
Further details of this design method are discussed in Section 1.4.5.

The actual factor of safety, probability of failure or allowable strain that is used in design
should be appropriate for each site. The design process requires a considerable amount of judgment because of the variety of geological and construction factors that must be considered.
Conditions that would require the use of factors of safety at the high end of the ranges quoted in
Table 1.3 include the following:
• A limited drilling program that does not adequately sample conditions at the site, or
drill core in which there is extensive mechanical breakage or core loss.
• Absence of rock outcrops so that mapping of geological structure is not possible, and there
is no history of local stability conditions.
• Inability to obtain undisturbed samples for strength testing, or difficulty in extrapolating
laboratory test results to in situ conditions.
• Absence of information on ground water conditions, and significant seasonal fluctuations
in ground water levels.
• Uncertainty in failure mechanisms of the slope and the reliability of the analysis method.
For example, plane type failures can be analyzed with considerable confidence, while
the detailed mechanism of toppling failures is less well understood.
• Concern regarding the quality of construction, including materials, inspection and weather
conditions.
• The consequence of instability, with higher factors of safety being used for dams and
major transportation routes, and lower values for temporary structures or industrial roads
for logging and mining operations.

This book does not cover the use of rock mass rating systems (Haines and Terbrugge, 1991;
Durn and Douglas, 1999) for slope design. At this time (2003), it is considered that the frequent
influence of discrete discontinuities on stability should, and can be, incorporated directly into
stability analyses. In the rock mass rating, the geological structure is only one component of the rating and may not be given an appropriate weight in the rating.

A vital aspect of all rock slope design is the quality of the blasting used in excavation. Design
assumes that the rock mass comprises intact blocks, the shape and size of which are defined by
naturally occurring discontinuities. Furthermore, the properties of these discontinuities should be predictable from observations of surface outcrops and drill core. However, if excessively heavy blasting is used which results in damage to the rock behind the face, stability could be dependent on the condition of the fractured rock. Since the properties of the fractured rock are unpredictable, stability conditions will also be unpredictable. Blasting and the control of blast damage are discussed in Chapter 11.

1.4.2 Limit equilibrium analysis (deterministic)
The stability of rock slopes for the geological conditions shown in Figure 1.4(a) and (f) depends on the shear strength generated along the sliding surface.

Figure 1.8 Method of calculating factor of safety of sliding block: (a) Mohr diagram showing shear strength defined by cohesion c and friction angle φ; (b) resolution of force W due to weight of block into components parallel and perpendicular to sliding plane (dip ψp).

For all shear type failures, the rock can be assumed to be a Mohr–Coulomb material in which the shear strength is expressed in terms of the cohesion c and friction angle φ. For a sliding surface on which there is an effective normal stress σ
acting, the shear strength τ developed on  this surface is given by

τ = c + σ
tan φ (1.1)

Equation (1.1) is expressed as a straight line on a normal stress—shear stress plot (Figure 1.8(a)),
in which the cohesion is defined by the intercept on the shear stress axis, and the friction angle is
defined by the slope of the line. The effective normal stress is the difference between the stress due to the weight of the rock lying above the sliding plane and the uplift due to any water pressure
acting on this surface. Figure 1.8(b) shows a slope containing a continuous joint dipping out of the face and forming a sliding block. Calculation of the factor of safety for the block shown in Figure 1.8(b) involves the resolution of the force acting on the sliding surface into components acting perpendicular and parallel to this surface. That is, if the dip of the sliding surface is ψp, its area is A, and the weight of the block lying above the sliding surface is W, then the normal and shear stresses on the sliding plane are

Normal stress : σ = W cos ψp/A
and shear stress : τs = W sin ψp/A (1.2)
and equation (1.1) can be expressed as
τ = c + (W cos ψp tan φ)/A (1.3)
or
τsA = W sin ψp and
τA = cA + W cos ψp tan φ (1.4)

In equations (1.4), the term [W sin ψp] defines the resultant force acting down the sliding plane
and is termed the “driving force” (τsA), while the term [cA + W cos ψp tan φ] defines the shear
strength forces acting up the plane that resist sliding and are termed the “resisting forces” (τA).
The stability of the block in Figure 1.8(b) can be quantified by the ratio of the resisting and driving forces, which is termed the factor of safety, FS. Therefore, the expression for the factor of
safety is
FS = resisting forces/driving forces (1.5)
FS = (cA + W cos ψp tan φ)/(W sin ψp) (1.6)

The displacing shear stress τs and the resisting shear stress τ defined by equations (1.4) are plotted on Figure 1.8(a). On Figure 1.8(a) it is shown that the resisting stress exceeds the displacing stress, so the factor of safety is greater than one and the slope is stable.

If the sliding surface is clean and contains no infilling then the cohesion is likely to be zero and  equation (1.6) reduces to
FS = (cos ψp · tan φ)/sin ψp (1.7)
or
FS = 1 when ψp = φ (1.8)

Equations (1.7) and (1.8) show that for a dry, clean surface with no support installed, the block
of rock will slide when the dip angle of the sliding surface equals the friction angle of this surface,
and that stability is independent of the size of the sliding block. That is, the block is at a condition
of “limiting equilibrium” when the driving forces are exactly equal to the resisting forces and the
factor of safety is equal to 1.0. Therefore, the method of slope stability analysis described in this
section is termed limit equilibrium analysis.

Figure 1.9 The effect of ground water and bolt forces on factor of safety of rock slope: (a) ground water and bolting forces acting on sliding surface; (b) Mohr diagram of stresses acting on sliding surface showing stable and unstable stability conditions.

Limit equilibrium analysis can be applied to a wide range of conditions and can incorporate forces such as water forces acting on the sliding surface, as well as external reinforcing forces supplied by tensioned rock anchors. Figure 1.9(a) shows a slope containing a sliding surface with area A and dip ψp, and a vertical tension crack. The slope is partially saturated such that the tension crack is half-filled with water, and the water  table exits where the sliding surface daylights on the slope face. The water pressures that are generated in the tension crack and on the sliding surface can be approximated by triangular force diagrams where the maximum pressure, p at the base of the tension crack and the upper end of the sliding surface is given by
p = γw hw (1.9)
where γw is the unit weight of water and hw is the vertical height of water in the tension crack.
Based on this assumption, the water forces acting in the tension crack, V , and on the sliding plane, U, are as follows:
V =1/2.γw.h2w and U = 1/2.γw.hwA (1.10)
and the factor of safety of the slope is calculated by modifying equation (1.6) as follows:
FS = cA + (W cos ψp − U − V sin ψp) tan φ / (W sin ψp + V cos ψp) (1.11)
Similarly, an equation can be developed for a reinforced slope in which a tensioned rock anchor  has been installed with the anchor below the sliding plane. If the tension in the anchor is T and it is installed at an angle ψT below the horizontal, then the normal and shear forces acting on the sliding plane due to the anchor tension are respectively:
NT = T sin(ψT + ψp) and
ST = T cos(ψT + ψp) (1.12)
and the equation defining the factor of safety of the anchored, partially saturated slope is
FS = cA + (W cos ψp − U − V sin ψp + T sin(ψT + ψp)) tan φ/(W sin ψp + V cos ψp − T cos(ψT + ψp))
(1.13)
Figure 1.9(b) shows on a Mohr diagram the magnitude of the normal and shear stresses on
the sliding surface developed by the water and bolting forces, and their influence on the factor
of safety. That is, destabilizing forces (e.g. water) decrease the normal stress and increase the shear stress, and tend to cause the resultant of the forces to be above the limiting strength line, indicating instability (Point B). In contrast, stabilizing forces (bolting and drainage) increase the normal stress and decrease the shear stress, and cause the resultant to be below the line, indicating stability (Point C).

The force diagram in Figure 1.9(b) can also be used to show that the optimum dip angle for the
bolts, that is, the dip that produces the greatest factor of safety for a given rock anchor force is
ψT(opt) =
φ − ψp or φ = ψp + ψT(opt) (1.14)

Strict application of equation (1.14) may show that the anchor should be installed above the
horizontal, that is, ψT is negative. However, in practice, it is usually preferable to install anchors below the horizontal because this facilitates drilling and grouting, and provides a more reliable installation.

These examples of limit equilibrium analysis to calculate the stability of rock slopes show that this is a versatile method that can be applied to a wide range of conditions. One limitation of the limit equilibrium method is that all the forces are assumed to act through the center of gravity of the block, and that no moments are generated.

This analysis described in this section is applicable to a block sliding on a plane. However, under certain geometric conditions the block may topple rather than slide, in which case a different form of limit equilibrium analysis must be used. Figure 1.10 shows the conditions that differentiate stable, sliding and toppling blocks in relation to the width x and height y of the block, the dip ψp of the plane on which it lies and the friction angle φ of this surface. Sliding blocks are analyzed either as plane or wedge failures (see Chapters 6 and 7 respectively), while the analysis of toppling failures is discussed in Chapter 9. Figure 1.10(b) shows that there are only limited conditions under which toppling occurs, and in fact this is a less common type of failure compared with sliding failures.

References : Wyllie, Duncan C. And Mah, Christopher W  (2004) Rock slope engineering – civil and mining 4th edition, London and New York

1.2 Principles of rock slope engineering
This section describes the primary issues that need to be considered in rock slope design for civil projects and open pit mines. The basic difference between these two types of project are that in
civil engineering a high degree of reliability is required because slope failure, or even rock falls,
can rarely be tolerated. In contrast, some movement of open pit slopes is accepted if production
is not interrupted, and rock falls are of little consequence.

Figure 1.2 Relationship between slope height and slope angle for open pits, and natural and engineered slopes: (a) pit slopes and caving mines (Sjöberg, 1999); and (b) natural and engineered slopes in China (data from Chen (1995a,b)).

As a frame of reference for rock slope design, Figure 1.2 shows the results of surveys of the
slope height and angle and stability conditions for natural, engineered and open pit mine slopes
(Chen, 1995a,b; Sjöberg, 1999). It is of interest to note that there is some correspondence between the steepest and highest stable slopes for both natural and man-made slopes. The graphs also show that there are many unstable slopes at flatter angles and lower heights than the maximum values because weak rock or adverse structure can result in instability of even low slopes.

1.2.1 Civil engineering
The design of rock cuts for civil projects such as highways and railways is usually concerned
with details of the structural geology. That is, Principles of rock slope design 5 Figure 1.3 Cut face coincident with continuous, low friction bedding planes in shale on Trans Canada Highway near Lake Louise, Alberta. (Photograph by A. J. Morris.) the orientation and characteristics (such as length, roughness and infilling materials) of the joints, bedding and faults that occur behind the rock face. For example,

Figure 1.3 Cut face coincident with continuous, low friction bedding planes in shale on Trans Canada Highway near Lake Louise, Alberta. (Photograph by A. J. Morris.)

Figure 1.3 shows a cut slope in shale containing smooth bedding planes that are continuous over the full height of the cut and dip at an angle of about 50◦ towards the highway. Since the friction angle of these discontinuities is about 20–25◦, any attempt to excavate this cut at a steeper angle than the dip of the beds would result in blocks of rock sliding from the face on the beds; the steepest unsupported cut that can be made is equal to the dip of the beds. However, as the alignment of the road changes so that the strike of the beds is at right angles to the cut face (right side of photograph), it is not possible for sliding to occur on the beds, and a steeper face can be excavated.

For many rock cuts on civil projects, the stresses in the rock are much less than the rock strength so there is little concern that fracturing of intact rock will occur. Therefore, slope design is primarily concerned with the stability of blocks of rock formed by the discontinuities. Intact rock strength, which is used indirectly in slope design, relates to the shear strength of discontinuities and rock masses, as well as excavation methods and costs.

Figure 1.4 shows a range of geological conditions and their influence on stability, and illustrates the types of information that are important to design. Slopes (a) and (b) show typical conditions for sedimentary rock, such as sandstone and limestone containing continuous beds, on which sliding can occur if the dip of the beds is steeper than the friction angle of the discontinuity surface. In (a) the beds “daylight” on the steep cut face and blocks may slide on the bedding, while in (b) the face is coincident with the bedding and the face is stable. In (c) the overall face is also stable because the main discontinuity set dips into the face. However, there is some risk of instability of surficial blocks of rock formed by the conjugate joint set that dips out of the face, particularly if there has been blast damage during construction. In (d) the main joint set also dips into the face but at a steep angle to form a series of thin slabs that can fail by toppling where the center of gravity of the block lies outside the base. Slope (e) shows a typical horizontally bedded sandstone–shale sequence in which the shale weathers considerably faster than the sandstone to form a series of overhangs that can fail suddenly along vertical stress relief joints. Slope (f) is cut in weak rock containing closely spaced but low persistence joints that do not form a continuous sliding surface. A steep slope cut in this weak rock mass may fail along a shallow circular surface, partially along joints and partially through intact rock.

Figure 1.4 Influence of geological conditions on stability of rock cuts: (a) potentially unstable—discontinuities “daylight” in face; (b) stable slope—face excavated parallel to discontinuities; (c) stable slope—discontinuities dip into face; (d) toppling failure of thin beds dipping steeply into face; (e) weathering of shale beds undercuts strong sandstone beds to form overhangs; (f) potentially shallow circular failure in closely fractured, weak rock.

 

1.2.2 Open pit mining slope stability
The three main components of an open pit slope
design are as follows (Figure 1.5).

Figure 1.5 Typical open pit slope geometry showing relationship between overall slope angle, inter-ramp angle and bench geometry.

 

First, the overall pit slope angle from crest to toe, incorporates all ramps and benches. This may be a composite slope with a flatter slope in weaker, surficial materials, and a steeper slope in more competent rock at depth. In addition, the slope angle may vary around the pit to accommodate both differing geology and the layout of the ramp. Second, the inter-ramp angle is the slope, or slopes, lying between each ramp that will depend on the number of ramps and their widths. Third,  the face angle of individual benches depends on vertical spacing between benches, or combined multiple benches, and the width of the benches required to contain minor rock falls.

Some of the factors that may influence slope design are the slope height, geology, rock strength, ground water pressures and damage to the face by blasting. For example, with each successive push-back of a slope, the depth of the pit will increase and there may need to be a corresponding decrease in the overall slope angle. Also, for slopes on which the ramp is located, the slope angle may be flatter to limit the risk of failures that take out the ramp, compared to slopes with no ramp where some instability may be tolerated. Where there is significant water pressure in the slope, consideration may be given to installing a drainage system if it can be shown that a reduction in water pressure will allow the slope angle to be increased. For deep pits where an increase in slope angle of one or two degrees will result in a saving of several million cubic meters of rock excavation, an extensive drainage system may be justified. Such drainage systems could comprise fans of holes with lengths of hundreds of meters drilled from the slope face, or a drainage adit with holes drilled into the rock above the tunnel.

With respect to the bench face angle, this may be governed by the orientation of a predominant joint set if there are joints that dip out of the face at a steep angle. If this situation does not exist, then the bench angle will be related to the overall slope geometry, and whether single benches are combined into multiple benches. One factor that may influence the maximum height of individual benches is the vertical reach of excavating equipment, to limit the risk accidents due to collapse of the face.

In order to provide a guideline on stable pit slope angles, a number of studies have been carried out showing the relationship between slope angle, slope height and geology; the records also distinguished whether the slopes were stable or unstable (see Figure 1.2). These studies have been made for both open pit mine slopes (Sjöberg, 1999), and natural and engineered slopes in China (Chen, 1995a,b). As would be expected, if the slopes were not selected according to geology, there is little correlation between slope height and angle for stable slopes. However, sorting of the data according to rock type and rock strength shows a reasonable correlation between slope height and angle for each classification.

1.3 Slope features and dimensions
The International Association of Engineering Geology has prepared definitions of landslide features and dimensions as shown in Figures 1.6 and 1.7 (IAEG, 1990; TRB, 1996). Although the diagrams depicting the landslides show soil-type slides with circular sliding surfaces, many of these landslide features are applicable to both rock slides and slope failures in weak and weathered rock. The value of the definitions shown in  Figures 1.6 and 1.7 is to encourage the use of consistent terminology that can be clearly understood by others in the profession when investigating and reporting on rock slopes and landslides.

Figure 1.6 Definitions of landslide features: upper portion, plan of typical landslide in which dashed line indicates trace of rupture surface on original ground surface; lower portion, section in which hatching indicates undisturbed ground and stippling shows extent of displaced material. Numbers refer to dimensions defined in Table 1.1 (IAEG Commission on Landslides, 1990).

Figure 1.7 Definitions of landslide dimensions: upper portion, plan of typical landslide in which dashed line is trace of rupture surface on original ground surface; lower portion, section in which hatching indicates undisturbed ground, stippling shows extent of displaced material, and broken line is original ground surface. Numbers refer to dimensions defined in Table 1.2 (IAEG Commission on Landslides, 1990).

 

References : Wyllie, Duncan C. And Mah, Christopher W  (2004) Rock slope engineering – civil and mining 4th edition, London and New York