�Slope Stability Santiago Chile, November 2009

Bench - inter-ramp - overall: A guide to statistically designing a rock slope

The proper design and evaluation of the catch bench angle, inter-ramp slope angle, and overall slope angle, individually as well as in combination, are required for successful excavation and economic optimization of a rock slope. In many slopes at least one, if not more, of the above controlling angles are essentially ignored, resulting in a slope properly designed for one facet of the excavation but ignoring the other components. Bench face angles can be accurately described statistically utilizing engineering predictions from the rock mass discontinuity network and discontinuity shear strengths. Together with the required bench width, the bench controlled inter-ramp angle is determined. Inter-ramp angles can be accurately determined by careful construction of a structural geologic model, noting location and orientations of discrete intermediate and large planes of weakness for the excavation in question. The location and orientation of the overall slope is dependent upon the slope as determined by the bench controlled inter-ramp angle and the stability controlled inter-ramp angle. Given advances in data collection and analytical techniques and continuing moves to increase mining safety while simultaneously attempting to minimize excavation costs, the only possible way to truly optimize slopes is through rigorous analytical methods combined with probabilistic techniques.

Abstract

James I. Mathis

Zostrich Geotechnical

INTRODUCTION

A rock slope consists of up to three stability controlled slope components. These are bench face angle, inter-ramp angle, and overall slope angle (Figure 1). Depending on the situation, these can be excavation specified or maximum attainable angles. It is, however, critical that the interaction of each of these components is incorporated in the resulting slope design.

Let’s clarify this interaction. Assume an open pit mine with a substantial overall slope height. Now, assume a bench height of 15m, a required catch bench width of 8m with a stability determined face angle of 70°. Assume the inter-ramp slope has been determined to be acceptably stable at an angle of 55° with a ramp width of 33m and a mean height of 150m. Structural considerations determine that the overall slope shall not exceed an angle of 44°.

What to all these numbers tell us? Well, in order to maintain the required catch bench angle, the inter-ramp angle cannot exceed the geometrical constraints imposed by the bench geometry. In this case, the bench determined inter-ramp angle is:

(1)

Now, it is obvious that the bench controlled inter-ramp angle (48°) is less than the angle at which the inter-ramp slope has been determined to be stable (55°), thus the slope must be designed to accommodate the required bench geometry.

How about the inter-ramp versus overall slope? The inter-ramp angle cannot exceed 48°, as noted above. If the ramp width is added, the overall slope, as dictated by inter-ramp constraints is:

(2)

The overall slope was determined to be sufficiently stable at 44°. Yet, the bench geometry dictates the inter-ramp angle. This angle, together with the inter-ramp height and the required ramp width impose an overall slope angle of 42°. Therefore, the bench geometry, for this specific case, dictates the overall slope angle as well.

Of course, some variables can be modified. Bench height can at times be adjusted, as can excavation methods (especially blasting), ramp widths can be adjusted based on equipment selection, artificial support may be contemplated, etc. In fact, a multitude of possibilities exist to adjust the individual components of the slope geometry. Still, in order to reach that point one must first understand how one engineers each of the critical slope components: bench, inter-ramp, and overall slope angles.

tan-1 (15m/(8m+(15m/tan(70°))) = 48°

tan-1 (150m/(33m+(150m/tan(48°))) = 42°

� Santiago Chile, November 2009 Slope Stability

CATCH BENCH DESIGN

The catch bench consists of a bench face and a catch bench. A bench face is the vertical to intermediate dipping wall created in rock by excavation actions (Figure 2). This wall, or face, will also have the added component of a “bench”. This bench, located at the base of the rock face, will be an area reserved for catching (restraining) rocks that detach from the excavated face, thus the term bench face or the face above the bench.

What is bench face design? A basic definition would be the engineering design of a rock face above a bench such that the general standing, stable angle of the face is quantified. This should incorporate at a minimum the input parameters listed in Figure 2.

Figure 1 - Bench controlled inter-ramp angle, inter-ramp angle, and overall slope angle.

Figure 2 - Anatomy of a bench design

For a “standard” bench design, this information is then compiled and the following conducted:

Identify potential failure modes;Determine the population of potential structural orientations that may occasion bench failure;

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�Slope Stability Santiago Chile, November 2009

Calculate the stability of the structurally defined failure blocks by standard kinematic analysis;Adjust the bench face angle until an appropriate safety factor is realized against sliding and/or;Calculate potential reinforcement for the sliding blocks, if required;Determine the requisite bench width to retain failed rock from the bench face.

Yet, a variety of questions arise with this “standard” design. Amongst these are:

Most benches are drilled and blasted vertically. To what minimum angle will the bench fail to and how much material will fall from the crest?What is the distribution of face angles and how will this distribution of face angles affect the bench catch width design?How was the variability of the shear strengths incorporated in the analysis?What is the impact of the discontinuity length and spacing on the face angle and what is the sensitivity of the design to these parameters?What would the impact be of utilizing the entire structural orientation distribution instead of point values? How much backbreak can be eliminated by drilling angle holes and is it warranted?

Catch bench face angle design, as conducted by this author, utilizes a rock fabric simulation to determine bench face angle reliability (2). Discontinuity spatial characteristics obtained from a rigorous sampling method and obtained either from physical or photogrammetric mapping are simulated in a three dimensional, Monte Carlo generated, discontinuity model (Figure 3). The three dimensional model is then cut by a simulated bench face and statistical failure analyses of wedge and plane shear failures are conducted on the daylighting features that transect the bench crest. This provides not only the bench face angle distribution as a function of bench height, but also provides a large number of simulated face profiles for analysis and allows for the effect of an excavated face angle of something less than 90°.

Rock fall is analyzed using simulated face profiles to determine/verify the required bench width to accommodate rock fall. Note that rock fall described herein is material physically falling to the bench, not volumetric failure accumulation as considered by some engineers.

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Figure 3 - Persistence modeling of structures for bench scale analysis

Of course, the analysis incorporates known structural domains (areas of similar geologic structure including lithologic variations), variations in face orientation (design sectors), and the complete discontinuity shear strength distribution (peak and residual) in the design. Blasting effects are accommodated as adjustments in the discontinuity spatial characteristics. At times, rock reinforcement may be considered to modify the bench face angle distribution.

Once the bench face angles distributions are defined, the reliability of the bench face angle is utilized for selection of the design bench geometry. The face angle reliability can vary from 70% for areas not often frequented by man or machine to values >90% for areas where bench failure may substantially impact operations or potentially endanger personnel. For open pit mines a reliability of 80% appears to be somewhat standard as this appears to contain most rock that escapes a single bench.

Face angles at the chosen reliability, and segregated by external effects (excellent vs. poor excavation techniques, etc.) are compiled into a table. This table includes the structural domain and design sector (face orientation). Note that this table can also include varying bench height and catch bench width. A geometrically constrained bench controlled inter-ramp angle, as noted in the introduction to this article, is then calculated.

� Santiago Chile, November 2009 Slope Stability

Note that the described methodology answers all of the questions posed above, including the effect of discontinuity length (persistence) and center density (spacing). Note further these questions can only be answered using probabilistic techniques.

One of the aforementioned points, discontinuity persistence, is absolutely critical to proper face angle design. As can be seen in Figure 4, discontinuities that are assumed to be continuous through the bench will not honor the rock mass. Continuous structures would result in a bench face angle that would not change as a function of bench height. Of course, this is not the case, as double and triple benches are nearly always steeper than single benches due to the interaction of discontinuity persistence occasioning failures as they transect the bench crest. Thus, only a method honoring the mapped discontinuity persistence will provide an accurate estimate for bench face design if the discontinuity persistence is substantially less than the bench height. This variation in bench face angles is demonstrated for a 10m and 20m bench height utilizing the program Z-Fabric (Zostrich Geotechnical) (Figure 5).

Experience has shown that it is always prudent, as well as good engineering, to validate any slope design. Verification of bench

scale performance utilizing the above described methodology has been conducted using a multitude of individual face profiles obtained from surveying as well as photogrammetric techniques. One of the advantages of utilizing photogrammetric techniques is that the entire imaged slope is available for face profiling to compare with the analytical bench face and slope angles. While some blind areas may exist due to camera location, the accuracy of the slope topography is far superior to that obtained by any other easily applied methodology. The verification process allows one to detect errors in discontinuity data collection, failure mode analysis, and blasting practices such that the bench design may be refined.

Figure 4 - effect of discontinuity persistence as a function of bench height

Figure 5 - comparison of bench face distributions as a function of bench height

INTER-RAMP DESIGN

As was discussed previously, bench scale design is, for most part, predicated on relatively simple failure modes with low applied stresses on the sliding surfaces. The controlling geologic structures (rock fabric) can be dealt with statistically as has been done above.

However, inter-ramp slope design is more much complex, incorporating intermediate faults, rock fabric, and at times, rock mass strength characteristics. Due to this varying height of the inter-ramp slope, the required slope analyzes can fall anywhere between fabric stability analyzes for benches and the individual failure analyzes required for overall slope stability.

�Slope Stability Santiago Chile, November 2009

The inter-ramp portion of a pit slope is that slope between: The crest of the excavation and any intermediate ramp;Between two sections of ramp, or;The ramp and the base of the excavation

A term that has come into common usage in the last few years is “stack height” or the continuous vertical “stack” of benches that will maintain stability given specific design parameters and engineering analyses. This term is essentially synonymous with the inter-ramp height as it is specifically addressed in inter-ramp design as slope height.

Stability of the inter-ramp slope is generally still, as for benches, controlled by relatively simple structural failure modes. Yet, when attempting to proceed utilizing similar logic as for bench design, two major problems immediately arise:Stability constraints. The frequency, continuity, and location of the geologic structures that may control any potential inter-ramp scale failures must be defined for analysis. The output of potential number of failures together with their associated size, defined not only by the expected number but a probable range, is available only through statistical analysis.

Economic constraints. Unlike bench design, the operator must decide when an inter-ramp scale failure occasions economic impact on the operation. Other than the obvious safety concerns, which must be addressed, this latter actually dictates the inter-ramp slope angle chosen for design. Again, as the input to any economic analysis of this sort requires a range of failure occurrences and volumes this will only lend meaningful results through probabilistic stability analyses.

Stability constraintsInter-ramp stability analyses examine the stability of potential failure geometries greater than one bench up to, and including, the

entire inter-ramp height. As noted previously, the inter-ramp height can at times be the entire rock slope, at which point the inter-ramp is equivalent to the overall slope.

Similar input values are required as for bench design (Figure 2) with the substitution of the inter-ramp height and bearing for the equivalent bench values. Adjustments may be required in the discontinuity shear strengths, as fault values may be required. Assessments of rock mass strength values may be required as well.

One of the greatest differences between inter-ramp and bench design is that equivalent structure defining the failure geometry must generally be greater than the height of the bench in persistence. However, such structures are difficult if not impossible to map as a single feature as they traverse multiple benches in an existing pit or transect outcrop in natural terrain prior to mining. Extrapolation of fabric mapping data for such analysis is possible but somewhat unproven. Further, some assessment of the feature density (spacing) must be provided as well. At present, geologic structural interpretation is required to fulfill this requirement.

The requisite structural interpretation is extensive requiring pit wall/outcrop mapping of continuous structures and intermediate faults, drillhole interpretation of faults/broken zones, and a comprehensive assessment of the entire structural picture (3). Note this is not a standard geologic interpretation, but has been developed as a function of advances in photogrammetric and modeling software. This provides the structural underlay for geotechnical assessment (Figure 6).

For a pushback, or additional excavation immediately behind an existing wall or outcrop, the mapped structures together with their physical characteristics can be projected directly onto the wall. Multiple failure geometries can be analyzed together with exterior influences such as water and blasting. This allows slope geometries to be adjusted, rock mass strength influence assessed, and potential rock reinforcement considered. Such an analysis indirectly addresses persistence as the projection of the geologic structures is not large given their persistence. This is the preferred inter-ramp slope design environment. Note that the related stability analyses, even for discrete structures, utilize the shear strength distributions on the defining structural features. In addition, as noted above, variations in water level, statistical representations of release structures, etc. can and should be included in the design if they exhibit a demonstrable effect.

Alternatively, where the design slope is substantially behind the mapping slope or in areas where no structural projections may be made, the designer is forced to utilize statistical representations of structural populations. The orientation distribution for the design structure population is relatively simple to obtain, either by projection of fabric mapping data, analysis of major features, or both. However, two critical parameters for the stability analyses are persistence and density (spacing).

Density (spacing): The spacing of the major geologic structures impacting inter-ramp stability can be obtained from analysis of the local structural interpretation, as can be the variation of this spacing (Figure 6). Both can be utilized for predictive models. It has been observed by this author that the periodicity of interpreted “major” structures within a geologic mass appears to be essentially normally distributed. This contradicts with the multitude of distributions proposed for rock fabric spacing (periodicity). More research is needed in this area as such assumptions have a substantial impact on the slope design, especially in terms of failure probability. Continuity (persistence): It is difficult, if not impossible, to accurately assess continuity for major structures that do not exhibit direct exposure on a rock face. In this case, while assessments may be made of the continuity, it is by no means certain these assessments are

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accurate. As was noted above, the direct impact of utilizing infinitely persistent bench scale (fabric) features was expressly discouraged as it had a demonstrably detrimental effect on the accuracy of the bench face angles. In that case the persistence of the fabric features could be obtained relatively simply. For the inter-ramp case, this may not be possible (although it has been done for some larger scale exposures). Thus, for this specific case where there is a lack of information, an estimate may be made of the mean and variation of the persistence of the feature or an infinite persistence determined that honors the density (spacing) of the discontinuity set.

One of the interesting facets of analyzing inter-ramp stability in the fashion described above is that the release planes for failures can be analyzed in both a discrete and probabilistic fashion. The greater the persistence and the number of the required release planes to define a kinematically viable failure block, the smaller the probability that block will actually exist (Figure 7).

For analytical purposes, the ramp, crest, or toe of an inter-ramp slope section provide obvious, pre-defined release planes. This is similar to the previously discussed bench geometry where the structures defining the failure block were required to transect the crest in order for a viable failure block to be defined.

Figure 7 - Effect of multiple structural intersections on probability of failure geometry occurrence

Figure 6 - Interpreted major structure paralleling a design pit wall

The rapid change in wall curvature at a pit bottom (Figure 8), or a “nose” developed in a pit wall can also be analyzed statistically. Where these two special slope cases have often been described as being more, or less, stable because of degrees of freedom of motion that is only partially true. As the plan curvature of the slope wall increases, it is more difficult to create a viable geometry as the release structures must be found in a specific locale and have a substantial, and increasing, persistence for a viable failure block to be defined. Alternatively, for a nose, the requirements for persistence and location of the release structures are substantially reduced. These can both be addressed analytically, using probabilistic methods, with the aforementioned interpretative structural work thus improving slope stability assessments.

Economic constraintsAn inter-ramp slope analysis, if conducted using statistical methods, will be either expressed as a probability of failure, the expected

number of failures, or both. The location and extent of the failure will have some economic impact on the operation.

�Slope Stability Santiago Chile, November 2009

For example, if the inter-ramp scale failure is on a slope where it will have little impact either on mining operations, traffic, or facilities, then the economic impact of inter-ramp failure is minimal. The probability of failure can likely be quite high without substantial economic consequences.

However, if the design inter-ramp slope is below a haul road to the bottom of a pit with no alternative methods for access if the ramp is obliterated by failure, the failure is an end of mine life event with large economic consequences. In this situation, the probability of failure, including the accuracy of the failure estimate, must be carefully considered. Slope angles may be flattened to account for this risk in specific portions of the pit specifically to address this factor.

Figure 8 - Effect of structure persistence for inter-ramp failures as a function of slope geometry

All inter-ramp slope failures carry an impact on operation. Thus, inter-ramp slope designs should always consider not only the stability of the slope, but the impact on operations if the slope is to fail. This can only be done with techniques specifically addressing the volume and probability of slope failure. Many operations fail to consider this in their design rendering them exposed to risks which should have been assessed during slope design.

Inter-ramp designs are conducted as are bench designs, with accommodation of structural domains, lithologic bounds, design sectors, etc. Existing inter-ramp slopes can be back analyzed and compared to theoretical values. Of course this is more difficult than for benches as the number of inter-ramp failures will provide a less reliable sample than multiple benches provide. However, such an analysis is certainly worthwhile, especially as any failures provide information not only regarding potential failure geometries in future walls but on the shear strength of the features/zones involved.

COMPARISON OF BENCH AND INTER-RAMP SLOPE ANGLES

As was noted in the introduction, the geometry of the bench width and the bench face can determine the inter-ramp slope angle. The inter-ramp angle may not only be dictated by simple stability analyses, as the probability of failure coupled with any economic consequences of failure can have a profound impact on the design angle for the inter-ramp.

The comparison of the bench determined inter-ramp angle with the inter-ramp determined economically stable angle is still required as both criteria must be met. The necessary criterion for acceptance is the shallower of the two slope angles.

Once these values have been determined for all structural domains and slope angles, the slope should be re-optimized. This may result in substantial slope translation from the original design if those estimates were in error.

However, this design stage provides for an optimization and accommodation of individual geologic structures and structural zones that may actually improve the slope. For example, as seen in Figure 6, the structural interpretation indicates the design slope falls along a major structure. However, stepping the slope back slightly into the wall behind the structure allows a portion of the slope to be excavated steeper than the original design allowed as the rock fabric has been accommodated in the bench design. Whether this is economic is unknown, but it certainly should be compared against reducing the slope angle in front of the existing structure. Similar analyses should be made regarding changes in wall orientation, bench height, ramp location, etc. Economic benefits of such result analyses/combinations/adjustments can be substantial. In order to realize such benefits, active participation of a knowledgeable geotechnical engineer who recognizes potential opportunities and pitfalls is required in the planning process.

� Santiago Chile, November 2009 Slope Stability

OVERALL SLOPE DESIGN

Of the three stability controlled slope components, the overall slope is generally the simplest in terms of conceptual framing and analysis.

This comment is not made without considered examination of its implications. Let us examine what we have already conducted to arrive at the overall slope design:

The overall slope design should only be conducted after the bench and inter-ramp designs have been conducted, with final interactions of both considered. Thus, slopes have been adjusted to reflect inter-ramp and bench stability concerns, including major structures and zones of low rock mass strength.Inter-ramp design requires a detailed structural geologic interpretation for the inter-ramp areas. As these encompass the entire design pit, a detailed structural model is available for the overall slope design.

Thus, a piecewise stable slope has been designed, with the location of these individual slope components being generally correct spatially. In other words, there is no real guesswork as to final location of the slope compared to overall wall failure geometries.

The complete structural and lithologic model for the design slope is then examined for potential failure geometries. It makes no difference as to the analytical methods utilized, as these are dependent upon the failure modes and stresses involved. As long as the designer incorporates the uncertainty in the model within the analytical process, the probability of failure for any portion of the design pit may be calculated. This variability is, of course, more easily accommodated in simpler models as compared to three-dimensional numeric models. Even in the latter case, the structural and strength variability can, and must, be incorporated in the final analysis.

Again, economics come into play. In the case of mining, if an overall slope fails resulting in pit closure three years into a five year mine life, the economic results may be devastating. Yet, if a structural zone that substantially increases the probability of overall slope failure is only exposed in the last 6 months of the 5 year mine life, and steeper slopes can be maintained for the preceding 4.5 years by not exposing the critical structure, it may be that the cost savings are worth the increased risk near the end of mine life.

In other words, as for the inter-ramp slopes, the simple stability of the rock slopes at a fixed point in time may not be the driving design factor. Economics must also be included.

CONCLUSION

One may argue that this is not the way most rock slopes are currently designed. In fact, for an open pit mine the geotechnical engineer is often asked for the overall stable slope angle. Bench face and inter-ramp angles are then back calculated from that angle. While this can be done at times, and approximate overall slope angles may be utilized to generally locate the pit extents and depth this is the reverse of this author’s approach.

The methodology for determining the influence of slope components on each other is not the only concept that may appear upside down. As was demonstrated throughout the article, many analyses simply cannot be conducted without the use of probabilistic techniques. This includes not only bench design, where all the components of the discontinuity spatial characteristics, including persistence, are analyzed in conjunction with the discontinuity shear strength distribution, but also inter-ramp and overall design, where the probability of discontinuity density (spacing) and persistence comes into play not only for the main sliding plane but associated failure release planes.

Economics are a critical, and generally understated, portion of slope design. Again, these can generally only be addressed with probabilistic failure analyses and the associated distribution of the probability of failure occurrence and volume. In order to truly optimize the slope, economics must be included.

REFERENCES

1. Mathis, J.I. (1988) “Development and verification of a three-dimensional rock joint model”. Doctoral Thesis 1988:63 D, University of Luleå, Sweden, May.

2. Mathis, J.I. (2002) “Bench face design in rock”. http://www.edumine.com

3. Mathis, J.I. (2007): “Pit slope design and structural analysis at the Jericho diamond mine utilizing digital photogrammetry”, Slope Stability 2007, Perth, W. Australia, Sept, 2007, pp93-104

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