Tatone 2009 Computers & Geosciences

ARTICLE IN PRESS

Computers & Geosciences ] (]]]]) ]]]–]]]

Contents lists available at ScienceDirect

Computers & Geosciences

0098-30

doi:10.1

$Pro� Corr

Departm

Toronto

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giovann

Pleasanaly

journal homepage: www.elsevier.com/locate/cageo

ROCKTOPPLE: A spreadsheet-based program for probabilisticblock-toppling analysis$

Bryan S.A. Tatone a,b,�, Giovanni Grasselli a

a Geomechanics Research Group, Lassonde Institute, Department of Civil Engineering, University of Toronto, 35 Saint George Street, Toronto, Ontario, Canada M5S 1A4b Geological Engineering Program, Department of Civil and Environmental Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1

a r t i c l e i n f o

Article history:

Received 26 September 2008

Received in revised form

1 April 2009

Accepted 28 April 2009

Keywords:

Probabilistic slope stability analysis

Uncertainty

Monte Carlo simulation

Limit equilibrium

Rock slope engineering

04/$ - see front matter & 2009 Elsevier Ltd. A

016/j.cageo.2009.04.014

gram code and user manual available at: http:

esponding author at: Geomechanics Researc

ent of Civil Engineering , University of Toro

, Ontario, Canada M5S 1A4.

ail addresses: [email protected] (B.S.A

[email protected] (G. Grasselli).

e cite this article as: Tatone, B.S.A.,sis. Computers and Geosciences (20

a b s t r a c t

Uncertainty and variability are inherent in the input parameters required for rock slope stability

analyses. Since in the 1970s, probabilistic methods have been applied to slope stability analyses as a

means of incorporating and evaluating the impact of uncertainty. Since then, methods of probabilistic

analysis for planar and wedge sliding failures have become well established in the literature and are

now widely used in practice. Analysis of toppling failure, however, has received relatively little

attention. This paper introduces a Monte Carlo simulation procedure for the probabilistic analysis of

block-toppling and describes its implementation into a spreadsheet-based program (ROCKTOPPLE). The

analysis procedure considers both kinematic and kinetic probabilities of failure. These probabilities are

evaluated separately and multiplied to give the total probability of block toppling. To demonstrate the

use of ROCKTOPPLE, it is first verified against a published deterministic result, and then applied to a

practical example with uncertain input parameters. Results obtained with the probabilistic approach

are compared to those of an equivalent deterministic analysis in which mean values of input parameters

are considered.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Like all rock engineering problems, slope stability analysis is adata-limited problem that always involves some degree ofuncertainty. This uncertainty arises due to the natural spatialand temporal variabilities of rock mass properties, prohibitive costof obtaining large amounts of data during site investigations, labtesting results not representing in situ properties, modellingassumptions, and human errors. (Baecher and Christian, 2003).

Traditionally, slope stability analysis has followed the deter-ministic approach of calculating resisting and driving forces toarrive at a factor of safety. To address the issue of uncertainty,conservative values of rock mass properties are adopted and aminimum acceptable factor of safety is specified to provide amargin of safety against unexpected performance. Although thisapproach is widely utilized and accepted, the impact of con-servatism cannot be assessed and effects of varying degrees ofuncertainty cannot be quantified. As a result, apparently con-servative designs are not always safe against failure (El-Ramlyet al., 2002). Increasingly, probabilistic methods are being applied

ll rights reserved.

//www.geogroup.utoronto.ca/

h Group, Lassonde Institute,

nto, 35 Saint George Street,

. Tatone),

Grasselli, G., ROCKTOPPLE09), doi:10.1016/j.cageo.200

to rock slope stability analysis as an alternative approach ofdealing with uncertainty. Probabilistic slope stability analysistools not only offer a systematic way of quantifying and evaluatingthe role of uncertainty, but also provide a useful approach toestimate hazard frequency for quantitative risk analyses, whichare finding increased popularity in engineering practice (e.g.Duzgun, 2008; Fell et al., 2005; Ho et al., 2000; Morgenstern,1997; Pine and Roberds, 2005).

Simplified probabilistic methods for the stability analysis ofrock slopes were first introduced in the 1970s (McMahon, 1971;Major et al., 1977; Piteau and Martin, 1977; and others). Sincethen, the concepts and methods have undergone continualdevelopment such that methods of probabilistic analysis fortranslational failures are now well established in the literature(Carter and Lajtai, 1992; Duzgun et al., 2003; Feng and Lajtai,1998; Park and West, 2001; Quek and Leung, 1995; and manyothers). At present, there are several commercially availablesoftware packages capable of performing probabilistic limitequilibrium slope stability analysis. Some of the most popularpackages include: SLOPE/W (GEO-SLOPE, 2007), SLIDE,ROCPLANE, SWEDGE (Rocscience, 2008a–c), and RockPack III(RockWare, 2008). These software packages employ Monte Carlosimulations to repeatedly calculate the factor of safety withinput parameters that are randomly generated according touser-defined probability distributions. Therefore, instead ofobtaining a singular value for the factor of safety from singularinput values (deterministic approach), a distribution of values is

: A spreadsheet-based program for probabilistic block-toppling9.04.014

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B.S.A. Tatone, G. Grasselli / Computers & Geosciences ] (]]]]) ]]]–]]]2

obtained, which represents the uncertainty of the input para-meters. The probability of failure is defined as the number ofMonte Carlo trials producing a factor of safety less than onedivided by the total number of trials.

Although probabilistic methods for analyzing soil slopes androck slopes susceptible to planar sliding and wedge sliding arenow well established, the toppling failure mode of rock slopes hasreceived relatively little attention. Very few publications can befound that focus on incorporating the uncertainty of inputparameters in the analysis of toppling (Muralha, 2003; Scaviaet al., 1990) and none have considered the role of kinematicstability on the probability of failure. The objectives of this paperare to: (1) review the conventional deterministic stability analysisof slopes susceptible to block-toppling; (2) introduce a newprobabilistic block-toppling analysis procedure that accounts forkinematic stability; (3) describe the implementation of this newprocedure in a computer program created in Microsoft Excel usingVisual Basic for Applications (VBA); and (4) demonstrate how thisprogram can be used as a tool to analyze slopes with block-toppling hazard.

2. Conventional deterministic analysis ofblock-toppling failure

Before introducing the probabilistic analysis procedure, it isvaluable to review the conventional deterministic approach foranalyzing slopes susceptible to block toppling. The evaluation ofrock slope stability is typically a two-step process. First akinematic analysis of structural discontinuities via stereographictechniques is undertaken to identify potentially unstable condi-tions. Subsequently, if a kinematically unstable condition is foundto exist, a kinetic analysis using a limit equilibrium method isused to evaluate the factor of safety (Norrish and Wyllie, 1996;Wyllie and Mah, 2004).

Depending on the orientations of discontinuities in relation tothe geometry of the slope under consideration, potential slopefailures can typically be classified into four modes: circular, planar

Fig. 1. Common types of toppling: (a) block toppling of rock columns divided into blocks

toppling of continuous rock columns; and (c) block-flexural toppling characterized

accommodate significant lateral displacements (from Wyllie and Mah, 2004 after Good

Please cite this article as: Tatone, B.S.A., Grasselli, G., ROCKTOPPLEanalysis. Computers and Geosciences (2009), doi:10.1016/j.cageo.200

sliding, wedge sliding, or toppling. This paper focuses on thetoppling failure mode, which involves the overturning of rockcolumns delineated by a well-defined discontinuity set strikingsub-parallel to the slope face and dipping steeply into the face.Goodman and Bray (1976) classified toppling failures into threetypes (Fig. 1). The analysis procedure presented in this paper isintended for the analysis of slopes susceptible to the block-toppling type of failure only.

2.1. Kinematic conditions for block-toppling

Considering a single rock block on an inclined surface subjectto no external forces (Fig. 2a), toppling occurs if the block’s centreof gravity acts outside of its base and sliding does not occur alongits base. Mathematically, toppling occurs when

Dx=ynotanc ð1Þ

and

cof ð2Þ

where yn and Dx are the height and width of the block,respectively, and c and f are the dip and friction angle of thebase plane, respectively.

When a series of blocks is considered (Fig. 2b), two additionalrequirements exist. The first requirement is that the strike ofdiscontinuities defining the base and width of the toppling blocksmust be sub-parallel to the slope face (7201) such that the blocksare free to topple without restraint from the adjacent rock mass.This requirement is defined mathematically as (Norrish andWyllie, 1996)

jaa � asjo203 and jab � asjo203ð3Þ

where aa and ab are the dip directions of the discontinuitiesdefining the base and width of the blocks, respectively, and as isthe dip direction of the slope face. The second requirement is thatinterlayer slip can occur along sub-vertical discontinuities defin-ing the width of the blocks. Assuming the in situ stresses close tothe slope face are uniaxial and aligned in a direction parallel to theslope face, the condition for interlayer slip can be expressed as

of finite height by a second, widely spaced, roughly orthogonal joint set;(b) flexural

by pseudo-continuous flexure of rock columns with numerous cross-joints that

man and Bray, 1976).


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Fig. 2. Summary of kinematic conditions required for block-toppling failure: (a) example of a single block on an inclined base plane, (b) example of a series of blocks on a

stepped base plane, and (c) and (d) stereographic representation of a slope face and discontinuities along with envelopes (shaded areas) in which discontinuity poles must

lie to satisfy kinematic conditions for block toppling (adapted from Norrish and Wyllie, 1996).

7

Stable

Topple

Slide

Toe block 2

1

3

45

6

Fig. 3. Example of a system of toppling blocks on a stepped base (Goodman and

Bray, 1976).

B.S.A. Tatone, G. Grasselli / Computers & Geosciences ] (]]]]) ]]]–]]] 3

(Goodman and Bray, 1976, Norrish and Wyllie, 1996)

ð903� cbÞr ðcs � fbÞ ð4Þ

where fb and cb are the friction angle and dip angle of thesub-vertical discontinuities, respectively, and cs is the dip of theslope face. Figs. 2c and d illustrate the lower hemisphericalstereographic projection of the slope geometry depicted inFig. 2b along with envelopes defined by conditions (2)–(4).

It is noted that although Cruden (1989) has shown thekinematic limits of block-toppling to extend to cataclinal, under-dip slopes, the analysis procedure and computer programpresented in this paper is restricted to the anaclinal geometryoriginally outlined by Goodman and Bray (1976).

2.2. Kinetic (limit equilibrium) analysis of block-toppling

Given that the kinematic conditions for block toppling exist inthe slope under consideration, the kinetic stability can beevaluated using the limit equilibrium method developed by

Please cite this article as: Tatone, B.S.A., Grasselli, G., ROCKTOPPLE: A spreadsheet-based program for probabilistic block-topplinganalysis. Computers and Geosciences (2009), doi:10.1016/j.cageo.2009.04.014

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Goodman and Bray (1976). This method considers the interactionof a number of tall rock columns resting on a stepped base (Fig. 3).The blocks forming the slope are classified into three groups basedon their stability mode:

�

Pa

a set of short stable blocks in the upper part of the slope(e.g. Blocks 5, 6) not meeting the toppling criteria defined by(1) and not sliding on their base (caofa);
� a set of taller blocks midway down the slope (e.g. Block 2–4),
which meet the toppling criteria defined by (1) and, as a result,exert a force on subsequent downslope blocks, producing a‘‘domino effect’’ (Wyllie and Wood, 1983); and

Stable Count +1

Nini = ini + 1

Input mean values, standard denumber of trials (Nummtrials) a

Kinematic Count = Kinematic

Yes

No

by user (except spacin

Set ini, Unstable CouStable Count, and

Kinematic Count = 0

Check kinematic stabiliity

Perform limit equilibrium analysis

Generate randomblock geometry

Randomly sample inpvalues from user defined P

Kinematically feasible ?

Yes

If FS > 1

Stable Count =

ini = Num Trials?

Yes

Calculate probabilities of Pkinematic, Pf kinetic|kimematic,

Stop

Start

Fig. 4. Overview of probabilistic block-to

lease cite this article as: Tatone, B.S.A., Grasselli, G., ROCKTOPPLEnalysis. Computers and Geosciences (2009), doi:10.1016/j.cageo.200

�
a set of short blocks at the toe of the slope (e.g. Block 1) thatare pushed by the toppling blocks above. These blocks stabledepending on slope geometry.
The stability analysis is a step-wise process that begins withestablishing the dimensions and calculating the forces acting oneach block in the slope. Subsequently, the stability of each block isevaluated starting at the topmost block. Considering the balanceof forces and moments acting on the blocks, each block mayremain stable, topple, or slide. If a block is found to topple or slide,a force is transmitted to the next block in the slope equal inmagnitude to the force needed to maintain the current block in

viations, ands defined

Count +1

g)

nt,

ut DF’s

No

No

1086420

Unstable Count = Unstable Count +1

failure: and Pf

ppling stability analysis procedure.


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limiting equilibrium. However, if a block is stable, no forces aretransmitted to the next block in the slope. The overall stability ofthe slope is controlled by the stability of the lowermost block, ortoe block. If the toe block is stable, the entire slope is consideredstable and, conversely, if the toe block is unstable, the entire slopeis considered unstable.

As with all limit equilibrium methods, this method can easilyincorporate external forces acting on the slope, including supportloads, water pressures, and pseudo-static earthquake loads(Wyllie, 1999).

2.3. Factor of safety

Since the toe block is assumed to control the overall stability ofa system of toppling blocks, the factor of safety of the toe block isassumed to define the factor of safety of the entire slope. In theabsence of cohesion, Goodman and Bray (1976) proposed that thefollowing equation can be used to define a factor of safety againstblock toppling:

FS¼tanfavailable

tanfrequired

ð5Þ

where tanfavailable defines the coefficient of friction on the baseplane of the toe block and tanfrequired defines the coefficient offriction needed for limiting equilibrium. One must be aware thatthis approach assumes that the critical failure mode of the toeblock is sliding (which is the case in the three examples presentedin Goodman and Bray, 1976) when, in fact, the critical failure modeof the toe block may also be toppling (Wyllie and Mah, 2004).Moreover, it assumes that the blocks above the toe block push onthe toe block when, in fact, there may be multiple blocks of the‘‘stable’’ mode at the toe that collectively resist the movement ofupslope blocks. In these two cases outlined above, a differentmeans of calculating the factor of safety must be adopted. Theapproach adopted in the current study is presented in a latersection of this paper (Section 3.5.2).

Fig. 5. Screenshot of ‘‘Analysis


3. Development of a probabilistic block-toppling analysisprocedure and its implementation in a spreadsheet-basedprogram

This section describes the probabilistic block-toppling analysisprocedure and its implementation in a computer program calledROCKTOPPLE created in Microsoft Excel using VBA (available atwww.geogroup.utoronto.ca). The program logic is describedby first providing an overview of the entire analysis procedurefollowed by a detailed description of each major step in theprocedure.

3.1. Overview of probabilistic analysis procedure

The probabilistic approach developed herein (Fig. 4) utilizesMonte Carlo simulation to repeatedly perform the deterministicanalysis procedure (kinematic and kinetic analysis) described inthe preceding section. Considering Fig. 4, the first step in theanalysis procedure requires the user to specify the number ofMonte Carlo trials and define the appropriate probabilitydistributions for the parameters characterizing the slope (seeSection 3.2). Subsequently, the Monte Carlo simulation procedureis initiated, which involves repeatedly sampling random inputparameters from the user-defined probability distributions;checking the kinematic stability conditions (see Section 3.3);generating the random slope geometry (see Section 3.4); andevaluating the kinetic stability (factor of safety) via limitequilibrium analysis (see Section 3.5). Afterwards, the kinematic,kinetic, and total probabilities of block-toppling failure arecalculated (see Section 3.7).

The methodology outlined in Fig. 4 was coded into an Excel-based program as it allowed the use of Excel’s built-in functions andgraphing capabilities, greatly reducing the overall coding effortrequired. The Excel workbook that houses the ROCKTOPPLE programconsists of 6 worksheets or ‘‘tabs’’ named as follows: ‘‘AnalysisInput’’, ‘‘Add Support’’, ‘‘Results’’, ‘‘Analysis Details 1’’, ‘‘AnalysisDetails 2’’, and ‘‘Analysis Details 3’’. The first three of these tabs are

Input’’ tab of ROCKTOPPLE.


www.geogroup.utoronto.ca




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discussed in the subsequent sections of this paper while, descrip-tions of the ‘‘Analysis Details’’ tabs are reserved for Appendix A.

3.2. Definition of input parameters

Fig. 5 illustrates the ‘‘Analysis Input’’ tab of ROCKTOPPLE.Shaded cells on the left side of the tab indicate values that must beentered by the user, while the graphic on the right side provides apreview of the slope geometry according to the mean input values.

The input values are divided into ‘‘fixed’’ parameters, definedby singular input values, and ‘‘uncertain’’ parameters, defined byprobabilistic distributions. The height, H, and orientation of theslope, as, cs, and cts are always considered ‘‘fixed’’ while, allremaining parameters have the option of being treated as ‘‘fixed’’or ‘‘uncertain’’. In addition to the parameters describing the slope,the user must also specify the number of Monte Carlo trials to beperformed and what support measures, if any, should beconsidered in the analysis.

When the orientations of joint sets A and B are considered‘‘uncertain’’, they are assumed to be defined by a Fisherdistribution (Fisher, 1953), which is a symmetric three-dimen-sional (3D) distribution often used to describe the angulardispersion of joint orientations about a mean value (Priest,1993). It is defined by a mean orientation (dip/dip direction)and the Fisher constant, K, which describes the degree ofclustering around the mean value. In terms of analyzing block-toppling, the use of a 3D distribution for joint orientation dataallows kinematic analysis of the randomly generated disconti-nuities according to Section 2.1. However, since the adoptedkinetic analysis procedure is two-dimensional (2D), a 2Drepresentation of the 3D orientation data is needed beforeanalysis can be performed. Considering a cross-section perpendi-cular to the slope face, the difference between the true dip, ctrue,and apparent dip, capparent, of discontinuities that satisfy thekinematic conditions for block toppling (i.e. dip directions within7201 of the dip direction of the slope face) is very small(tancapparent=0.94tanctrue). Therefore, to perform kinetic analysisthe true dip angles sampled from the Fisher distributions

Fig. 6. Idealized geometry of a rock slope sub


are assumed to define the apparent dip angles for the2D section. Since true dip is always greater than apparent dip,this assumption introduces some conservatism into the kineticanalysis.

When the remaining input parameters, including joint spacing,friction angles, unit weights, and external loads, are considered‘‘uncertain’’, their values can be charcterized by normal, lognor-mal, or exponential distributions. Estimates of the mean andstandard deviation are the only user inputs required to definethese distributions.

3.3. Kinematic analysis

Based on the randomly sampled values defining the orienta-tion and friction angle of joint sets A and B, ROCKTOPPLE checks ifthe kinematic conditions for block toppling, as defined in Section2.1, are satisfied. The condition set out by (1), however, is notenforced since external forces such as water pressures or seismicloads can cause blocks to topple despite having a centre of gravitythat lies within their base. If the remaining kinematic conditionsare satisfied, the program proceeds with kinetic analysis, asdescribed in the following sections of this paper; otherwise, itadvances to the next Monte Carlo trial. In trials where theconditions for block-toppling are not satisfied, the randomlysampled orientations of discontinuity sets A and B may result inone of the following alternative kinematic conditions:

1.

ject

: A9.0

Failure not kinematically possible: sliding cannot occur on set A

and toppling cannot occur on set B. Therefore, the totalprobability of failure is 0.

2.
Only sliding on joint set A is kinematically possible: friction angleof set A is less than the dip angle; the dip direction of set A iswithin 7201 of slope dip direction but toppling on set B is notpossible.
3.
Only toppling on set B is kinematically possible: orientation of setB satisfies requirements for interlayer slip and alignment butthe dip direction of set A prevents sliding. Therefore, the toeblocks cannot slide.
to toppling (after Scavia et al., 1990).

spreadsheet-based program for probabilistic block-toppling4.014

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4.

Pass values of H,�s,

into sub-procedure

Start

i = 1Define (x, y) of Toe as (0, 0)

BlockCount = 1

BlockCount = BlockCount + 1

Calculate coordinates defining i th block and store in separate arrays

Randomly sample values of Sa andSb from probabilistic distributions

Calculate: Wn, K, v1, v2, v3, Xw, y1, y2, y3, Mn, Ln, K, Yk, and, (xcm,ycm), for the ith block and store in an array

Check if block extendsbeyond the slope limits

No

Yes

i = i + 1

n = BlockCount

Stop

�ts, �a, �b, and ground water level

Fig. 7. Flow chart outlining procedure to generate random block geometry.

Sliding on set A and toppling on set B occur simultaneously:friction angle of set A is less than the dip angle; the dipdirection of set A is within 7201 of slope dip direction; and allconditions for toppling on set B are satisfied.

Therefore, although block toppling may not be kinematicallyfeasible, the slope geometry may still result in a kinematicallyunstable condition. To give the user an indication of the likelihoodof these other kinematic conditions existing, the programcalculates the kinematic probability of each. However, theprogram does not perform the corresponding kinetic analysis.Hence, it must be emphasized that the total probability of failurecalculated by ROCKTOPPLE only represents the total probability ofblock-toppling failure and does not account for kinetic stability ofthe other potential failure modes (2–4 above). To evaluate thetotal probability of slope failure, including these other potentialfailure modes, kinetic analysis of each kinematic condition wouldneed to be undertaken with appropriate techniques. Subsequently,system reliability methods could be used to calculate the totalfailure probability.

It should be noted that in cases where the geometry of theslope and discontinuities are known with increased certainty (i.e.an existing slope in which several joint measurements have beenobtained), it may be desirable to treat the discontinuity orienta-tions as ‘‘fixed’’ input values (accomplished by entering K=0 onthe ‘‘Analysis Input’’ tab). In this situation, given that the fixeddiscontinuity orientations satisfy kinematic conditions for block-toppling, the kinematic probability will be 1.0, meaning the totalprobability of block toppling of failure will be given by kineticprobability of failure. In other words, if desired, the user caneffectively skip the kinematic analysis by considering disconti-nuity orientations as ‘‘fixed’’ inputs.

3.4. Generation of block geometry

If the kinematic conditions for block toppling are satisfied, thenext step in the analysis procedure involves generating the geometryof the n blocks that form the slope. The procedure adopted inROCKTOPPLE follows that developed by Scavia et al. (1990). Unlikethe original limit equilibrium procedure, which assumes the rockblocks are delineated by evenly spaced, perpendicular discontinuitysets, this approach is capable of generating blocks delineated bynon-orthogonal, irregularly spaced joint sets (Fig. 6). In generatingthe random block geometry, the following assumptions are made(Scavia et al., 1990):

�

Pa

the joint sets A and B are considered 100% persistent;
� the system of blocks sits on a stepped base that represents a
‘‘failure surface’’;
� the steps in the failure plane are defined by alternating values
Sa and Sb;
� the generated failure surfaces extend from the toe of the slope
to the upper surface; and
� the blocks are long in a direction normal to the cross-section,
but are bounded by zero-strength lateral release surfaces suchthat the problem can be analyzed two dimensionally.

The block generation procedure is summarized in Fig. 7. Tobegin, the ‘‘fixed’’ values of H, cs, and cts, together with therandomly sampled values of ca and cb of the current Monte Carlotrial, are passed to the block generation procedure. Then, startingfrom the toe of the slope (0, 0), alternating values of Sa and Sb aresampled from their respective distributions, coordinates defining

lease cite this article as: Tatone, B.S.A., Grasselli, G., ROCKTOPPLEnalysis. Computers and Geosciences (2009), doi:10.1016/j.cageo.200

the corners of each block are computed, and several forces anddimensions specific to each block, as defined in Fig. 8, areevaluated for use in subsequent limit equilibrium calculations.This process is continued until the stepped base reaches the upperboundary of the slope, forming n blocks.

3.5. Kinetic (limit equilibrium) analysis

The limit equilibrium analysis procedure requires thecalculation of the forces transferred from the uppermost blockthrough to the toe block. These forces are referred to as inter-block forces. Once these forces are determined, the factor ofsafety of the toe block or group of stable toe blocks can beevaluated. The following two sub-sections describe the calcula-tion procedure for inter-block forces and toe block stability,respectively.

3.5.1. Calculation of inter-block forces

Fig. 8 illustrates the position and direction of all forces actingon a typical rock block in a system of toppling blocks. The forcesPn�1 and Pn are what are referred to as inter-block forces.Although equations for calculating the inter-block forces areavailable in several rock mechanics and rock engineering texts(e.g. Wyllie and Mah, 2004; Wyllie, 1999), it is often assumed that


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joint sets A and B are orthogonal (ct=0). As this assumption israrely valid when values of ca and cb are randomly sampled fromindependent probabilistic distributions, these equations werereformulated to account for non-orthogonal joint sets beforebeing implemented in ROCKTOPPLE. Considering moment andforce equilibrium for a typical block (Fig. 8), the revised equationsfor the force, Pn�1, that is just sufficient to prevent the block fromtoppling and sliding, are given, respectively, by

Pn�1:t ¼�WnXwþKYw � PntanfbSb � V3y3þV1y1þV2y2þPnMn

Lnð6Þ

Pn�1:s ¼ Pn�Wnðcoscatanfa � sincaÞ � KðsincatanfaþcoscaÞþðV3 � V1ÞðcosctþsincttanfaÞ � V2tanfa

ðtanfaþtanfbÞsinctþð1� tanfatanfbÞcosct

ð7Þ

Fig. 9 summarizes the methodology used to calculate the inter-block forces acting on each block on the slope. Starting at theuppermost block, the forces Pn�1:t required to prevent topplingand Pn�1:s required to prevent sliding are calculated using Eqs. (6)and (7). If the values of Pn�1:t and Pn�1:s are negative, the currentblock is considered stable and the force, Pn, transmitted to thenext block is set to zero. However, if Pn�1:t4Pn�1:s, the block is onthe point of toppling and Pn�1 is set equal to Pn�1:t. Conversely, ifPn�1:s4Pn�1:t, the block is on the point of sliding and Pn�1 is set

Fig. 8. Summary of forces acting on a typical rock block.


equal to Pn�1:s. Once the appropriate value of Pn�1 is determined,all forces acting on the block are resolved in directionsperpendicular and parallel to the base of the block. The normalforce, Rn, and shear force, Sn, are calculated, respectively, as

Rn ¼Wncosca � Ksinca

�V2þðV3 � V1þPn�1 � PnÞsinct

þðPn � Pn�1Þtanfbcosct ð8Þ

Sn ¼WnsincaþKcoscaþðV1 � V3 � Pn�1þPnÞcosct

þðPn � Pn�1Þtanfbsinct ð9Þ

Subsequently, a check is made to ensure if there is a positivenormal force on the base plane and that sliding does not occur:

Rn40 and jSnjoRntanfa ð10Þ

If the conditions set out by (10) are not satisfied, toppling cannotoccur even if Pn�1:t4Pn�1:s; thus, Pn�1 is set equal to Pn�1:s.Once the value of Pn�1 is finalized, it is assumed to be the force, Pn,acting on the next block of the slope. The calculation of Pn�1 isthen repeated for the next block and all subsequent blocks insuccession until the force, Pn, acting on each block has beendetermined. It is noted that due to kinematic constraints, once thetransition from toppling to sliding occurs, the critical state for allsubsequent blocks is sliding (Wyllie and Mah, 2004).

3.5.2. Analysis of toe block(s)

Following the calculation of inter-block forces, the stabilitymode of each block above the toe block is defined as ‘‘sliding’’,‘‘toppling’’, or ‘‘stable’’. In the case where the block immediatelyabove the toe block is of the ‘‘sliding’’ mode, the potential failuremode of the toe block is limited to sliding and the factor of safetyagainst toe block sliding is considered to be the factor of safety ofthe entire system of blocks. In the case where the blockimmediately above the toe block is of the ‘‘toppling’’ mode, thepotential failure mode of the toe block can be sliding or topplingand the critical factor of safety against toe block sliding ortoppling is taken as the factor of the safety system of blocks.Considering the forces acting on a typical toe block (Fig. 10), thefactor of safety against toe block sliding and toe block toppling aregiven, respectively, by

FS¼ FStoe blocksliding¼

PForcesresistingPForcesdriving

¼½Wncosca � Ksinca � V2þð�V1 � PnÞsinctþPntanfbcos�tanfa

WnsincaþKcoscaþðV1þPnÞcosctþPntanfbsinct

¼Rntanfa

Sn: ð11Þ

FS¼ FStoe blocktoppling

¼

PMomentsresistingPMomentsdriving

¼PntanfbSbþWnXw

PnMnþV1y1þV2y2þKYkð12Þ

In the case where the block immediately above the toe block isof the ‘‘stable’’ mode, the factor of safety of the toe block is nolonger representative of the stability of the entire slope system.Instead, the factor of safety is dictated by the collective ability ofthe group of stable blocks at the toe to resist the driving forcesproduced by unstable blocks upslope. The factor of safety, in thiscase, can be defined as the sum of the resisting forces of each‘‘stable’’ block divided by the sum of the driving forces for each


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i = nPn = 0

Start

ith Block

Pn-1 = MAX (Pn-1:t, Pn-1:s)

Calculate Sn and Rn

If Rn ≤ 0 or

Sn > Rn tan �bNo Yes

Pn-1 = Pn-1:sPn-1 = Pn-1

No

Yes

Stop

Calculate Pn-1:t and Pn-1:s

Set Pn = Pn-1

If i = 1i = i-1

Fig. 9. Flow chart illustrating procedure to calculate inter-block forces.


‘‘stable’’ block:

FS¼ðRn1þRn2þRn3þ � � �RniÞ tanfa

Sn1þSn2þSn3þ � � � Snið13Þ

where i is the number of ‘‘stable’’ blocks at the toe of the slope.

3.6. Addition of support

As previously mentioned, external forces can be easilyincorporated into limit equilibrium methods. Hence, the effectof rock support elements on slope stability can be easily assessedin the block-toppling analysis. The support of slopes susceptible toblock-toppling through the use of rock support elements can beaccomplished in two ways (Wyllie and Wood, 1983): (1) a supportforce, T, can be added to the toe block, as shown in Fig. 11a or (2)the potential toppling blocks can be bolted together to increasetheir effective width, as shown in Fig. 11b. Both of thesesupport methods were incorporated into ROCKTOPPLE under the‘‘Add Support’’ tab shown in Fig. 12. In this tab, the user canspecify the magnitude and orientation of toe block support andthe effective width of the toppling blocks when the blocks arebolted together.

When toe block support is added to the analysis, it is assumedto be installed at the mid-point of the toe block face inclined at auser-defined angle, i, from the horizontal (Fig. 11a). Based on themean slope geometry, the optimum orientations, iopt, of the toeblock support to prevent sliding and toppling of the toe block aregiven, respectively, as (Goodman and Bray, 1976; Wyllie andMah, 2004)

ioptjsliding ¼fa �ca ð14Þ

ioptjtoppling ¼ � ca ð15Þ

In addition to the magnitude and orientation of the support, adrop-down menu on the tab allows the user to specify whether itshould be considered an active or passive force. If the supportforce is applied actively, the revised factors of safety against toeblock toppling and sliding are given respectively, by

FStoe blocktoppling

¼Pntanfb SbþWnXw

PnMnþV1y1þV2y2þKYk � TLtð16Þ

FStoe blocksliding¼

fWncosca � Ksinca � V2þð�V1 � PnÞsinctþPntanfbcosctþTsinðcaþ iÞgtanfa

WnsincaþKcoscaþðV1þPnÞcosctþPntanfbsinct � Tcosðcaþ iÞ

ð17Þ

If it is applied passively, the factors of safety are given by

FStoe blocktoppling

¼Pntanfb SbþWnXwþTLt

PnMnþV1y1þV2y2þKYkð18Þ

FStoeblocksliding¼fWncosca � Ksinca � V2þð�V1 � PnÞsinctþPntanfbcosþTsinðcaþ iÞgtanfaþTcosðcaþ iÞ

WnsincaþKcoscaþðV1þPnÞcosctþPntanfbsinct

ð19Þ

In the case where the blocks are bolted together, the effectivewidth of toppling blocks below the crest is increased. ROCK-TOPPLE models this condition by increasing the value of Sb

for all blocks below the slope crest by the user-specified factor.For example, if the spacing of Set B is 2 m and an effectivewidth of two times the actual block width is specified, theanalysis proceeds by assuming the blocks below the crestare 4 m wide. It should be noted that this simplistic approachdoes not consider potential failure of the bolts holding the blockstogether.


3.7. Calculation of failure probabilities

Following the completion of the specified number of MonteCarlo trials, the failure probabilities are calculated and displayedin the ‘‘Results’’ tab of ROCKTOPPLE (Fig. 13). The ‘‘Results’’ tabprovides a detailed summary of the kinematic and kineticprobabilities of failure, the mean and median factor of safety, ahistogram of the factors of safety, and a summary of the appliedrock support.

The probability of kinematic failure is given by

Pf kinematic ¼Nkinematically feasible

Ntð20Þ

where Nkinematically feasible is the number of trials in which block-toppling failure is kinematically feasible and Nt is the totalnumber of Monte Carlo trials. Similarly, probabilities of the otherkinematic conditions (as defined in Section 3.3) are calculated bydividing the number of trials in which the conditions occur by Nt.


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Pn

V1Sb

Mn

y1

(xcm,ycm)K

xw

Wn

Origin

y2

YkV2

�t

�a

Pntan�b

�s�b

Fig. 10. Example of forces acting on a typical toe block.

T

i

Lt

Origin�a

�t

�b �s

Fig. 11. Methods of applying support in ROCKTOPPLE: (a

Fig. 12. Screenshot of ‘‘Add Sup



Considering that toppling failure involves a system of blocksthat can topple, slide, or remain stable, the kinetic probability offailure must be defined in terms of the reliability of the system asa whole. For this system of blocks, however, the conventionalapproach of using event tree or fault tree analysis to examine allpossible failure paths of the system becomes quite complex andcumbersome. Recalling that the stability of the toe blockultimately controls the stability of the entire system of blocksindependent of the behaviour of other blocks in the system,repeatedly calculating the factor of safety of the toe block whilevarying the input parameters effectively constitutes the simula-tion approach for analyzing system reliability (Baecher andChristian, 2003). Thus, the kinetic probability of failure for thesystem can be defined in terms of the factor of safety of the toeblock or group of stable toe blocks. Since kinetic failure can occurvia slidingor toppling, the kinetic probability of block-toppling failure isgiven by

Pf kineticjkinematic ¼NFSo1:toe slidingþNFSo1:toe toppling

Nkinematically feasibleð21Þ

where NFSo I:toe sliding and NFSo I:toe toppling are the number of trialsresulting in a factor of safety less than 1 when the critical failuremode of the toe block is sliding and toppling, respectively.

) toe block support and (b) bolting blocks together.

port’’ tab in ROCKTOPPLE.


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Fig. 13. Screenshot of ‘‘Results’’ tab of ROCKTOPPLE illustrating typical program output.

Table 1Summary of slope characteristics for deterministic example given in Wyllie and

Mah (2004).

Parameter Deterministic value

Overall geometry

Slope height (m) 92.5

Slope angle (deg)a 56.6

Top angle (deg) 4

Discontinuity orientations

Dip of set A (deg)a 30

Dip of set B (deg)a 60

Rock mass characteristics

Spacing of joint set A (m) 1

Spacing of joint set B (m) 10

Friction angle of set A (deg) 38.15

Friction angle of set B (deg) 38.15

Unit weight of rock (kN/m3) 25

External loads

Seismic coefficient (g) 0

Water pressure (%) 0

Rock support n/a

a The deterministic example assumes the discontinuities and slope face have

the same dip direction. Therefore dip directions are not required as input

parameters.


The probability of kinetic failure is considered a conditionalprobability since kinetic analysis is undertaken only for kinema-tically feasible geometries. Based on the properties of conditionalprobabilities, the total probability of block-toppling failureis given by the product of (20) and (21) (Glynn, 1979; Park andWest, 2001):

Pf ¼ Pf kinematicPf kineticjkinematic ð22Þ

It should be noted that the term Pf kinematic in Eq. (22) refers tothe probability of block toppling being kinematically feasible. Theprobability of the other kinematic conditions, as defined in


Section 3.3, are not included in the calculation of the totalprobability of block toppling. They are provided merely to informthe user that given the slope geometry and discontinuityorientations, instability via other failure modes may be possible.As mentioned previously, calculation of the total probability ofslope failure would require a separate kinetic analysis of eachunstable kinematic condition via alternate analysis methods andthe evaluation of the total failure probability using systemreliability methods.

4. Deterministic verification of ROCKTOPPLE

Since examples of probabilistic block-toppling analysiscould not be found in the literature, the probability of failurecalculated with ROCKTOPPLE was not compared with previousresults. The output of the program was, however, compared to adeterministic block-toppling example published in Wyllie andMah (2004) by considering the input parameters (Table 1) as fixedvalues. Results as shown in Wyllie and Mah (2004) and thoseobtained with ROCKTOPPLE are tabulated in Appendix B.

When comparing results, it is important to note that themethodology used to define the geometry of the blocks variesbetween the published example and ROCKTOPPLE. While thepublished example assumes the blocks are rectangular to simplifycalculations (Fig. 14a), ROCKTOPPLE assumes they are trapezoidal(Fig. 14b). As a result, the blocks generated by the program aretaller above the slope crest and shorter below the slope crestwhen compared with rectangular blocks. The largest percentdifference in block weight occurs for the uppermost andlowermost blocks. The 4 uppermost blocks vary from 12% to59% and the 3 lowermost blocks vary from 13% to 46%; all otherblocks vary by less than 10%.

When the parameters listed in Table 1 were considered as fixedinput values in ROCKTOPPLE, it was revealed that the discontinuityorientations did not satisfy the kinematic conditions forblock-toppling. Therefore, to obtain kinetic stability results with


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ROCKTOPPLE that could be compared with published values,analysis of kinematic stability was temporarily disabled. Theresults obtained with ROCKTOPPLE, in terms of the stability modeof the blocks, are in close agreement with the published results(i.e. a set of stable blocks at the crest, a set of intermediate topplingblocks, and a set of sliding blocks at the toe). There was, however, anotable discrepancy as ROCKTOPPLE predicted only 2 stable blocksat the slope crest compared to 3 in the published example. Thisdiscrepancy is attributed to the differences in block geometrynoted earlier. Since trapezoidal blocks above the slope crest aretaller relative to their rectangular counterparts, their centroidlocations are shifted in the downslope direction relative to theirbase and are, therefore, more likely to topple.

0

1000

2000

1

3000

4000

5000

Forc

e (k

N)

6000

7000

8000

9000

Rn: Wyllie & Mah (2004)Sn: Wyllie & Mah (2004)Rn: ROCKTOPPLESn: ROCKTOPPLE

2 3 4 5 6 7 8

Fig. 15. Comparison of shear (Sn) and normal (Rn) forces along base of each blo

100110120130140

30405060708090

45

67

89

1011

1213 14 15 16

-100

1020

-10

Horizontal Distance (m)

1 23

4

Vert

ical

Dis

tanc

e (m

)

0 10 20 30 40 50 60 70 80 90 100

110

120

130

140

Fig. 14. Geometry of deterministic example problem: (a) rectangular blocks as considere


Differences in the block geometry also resulted in differinginter-block forces and, consequently, differing values of Rn and Sn

(Fig. 15). According to the results obtained from ROCKTOPPLE, thefactor of safety according to Eq. (12) is 0.94 compared to 1.00 forthe published results.

Although the results obtained from ROCKTOPPLE varied fromthe published results due to differing block geometries, similarbehaviour in terms of stability mode and the relative shear andnormal forces was predicted for all blocks. The resulting differencein the factor of safety, given the same input parameters, under-scores the impact of varying the slope geometry on stability andfurther illustrates the importance of incorporating geometricuncertainties into the analysis of block toppling.

Block9 10 11 12 13 14 15 16

ck as given in Wyllie and Mah (2004) and as calculated by ROCKTOPPLE.

90100110120130140

15 16

304050607080

Vert

ical

Dis

tanc

e (m

)

45

67

89

1011

1213 14 15

-100

1020

1 23

4

-10

Horizontal Distance (m)

0 10 20 30 40 50 60 70 80 90 100

110

120

130

140

d in Wyllie and Mah (2004) and (b) trapezoidal blocks considered by ROCKTOPPLE.


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5. Probabilistic application of ROCKTOPPLE

To demonstrate the use of the probabilistic stability analysisprocedure and computer program ROCKTOPPLE described in thepreceding sections, the stability of a 15 m high granite rock cutwas analyzed, for which sufficient geotechnical data are availableand suitable.

Fig. 16. Equal-area stereographic representation of slope and discontinuity geometry

required for block toppling.

Table 2Summary of mean discontinuity orientations and corresponding Fisher constants K.

Set Mean orientation (dip/dip d

D1 751/1341

D2 291/0501

D3 721/2251

Table 3Summary of input parameters for probabilistic analysis.

Parameter Probabilistic distribution

Overall geometry

Slope height (m) Fixed value

Slope angle (deg) Fixed value

Top angle (deg) Fixed value

Dip direction of slope face (deg) Fixed value

Discontinuity orientations

Dip/dip direction of set A (deg) Fisher

Dip/dip direction of set B (deg) Fisher

Rock mass characteristics

Spacing of joint set A (m) Log normal

Spacing of joint set B (m) Log normal

Friction angle of set A (deg) Normala

Friction angle of set B (deg) Normala

Unit weight of rock (kN/m3) Fixed value

External loads

Seismic coefficient (g) Fixed value

Unit weight of water (kN/m3) Fixed value

Water pressure (%) Log normal

Rock support Not considered

a Physically, these parameters cannot have values outside the range of 0–901. Therefo

non-physical values.


The results of detailed discontinuity mapping of the rock cutunder consideration are summarized in the stereographic plot inFig. 16. The discontinuity poles form three distinctive clustersrepresenting three main discontinuity sets denoted as D1– D3.Table 2 summarizes the mean orientations of the threediscontinuity sets and the corresponding Fisher constants.Plotting the average plane describing the slope face along with

considered probabilistically along with envelopes defining kinematic conditions

irection) Fisher constant K

49

46

24

Mean value Standard deviation

15 –

70 –

2.5 –

55

29/050 46 (Fisher K)

72/225 24 (Fisher K)

0.75 0.15

2.5 0.30

35 2.5

35 2.5

26 –

0 –

9.81 –

10 5

re, the normal distributions are truncated at these extremes to prevent sampling of


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Table 4Comparison of deterministic and probabilistic results obtained with ROCKTOPPLE

Analysis description Factor of safety

Deterministic 1.23

Probabilities of failure

Mean factor of safety Probability of kinematic failure Probability of kinetic failure Total probability of block toppling failure

Probabilistic 1.17 0.602 0.234 0.141

0.8

1.0

500

600

Critical mode of toe block(s) = topplingCritcal mode of toe block(s) = sliding

0.4

0.6

200

300

400

Cum

ulat

ive

Pro

babi

lity

Freq

uenc

y

Cumulative Probability

0.0

0.2

0

100

Factor of Safety

Fig. 17. Distribution of factor of safety obtained from ROCKTOPPLE using input values given in Table 3.


the envelopes defining potential base planes and sub-verticaltoppling planes (as previously defined in Fig. 2), it is evident thatblock-toppling is possible with discontinuity sets D2 and D3forming the base plane (set A) and sub-vertical planes (set B),respectively.

The input parameters required for performing probabilisticanalysis of the slope are given in Table 3, including the selectedprobabilistic distributions, mean values, and correspondingstandard deviations (or Fisher constants). Analysis results, bothdeterministic and probabilistic, are presented in Table 4 and thedistribution of the factor of safety obtained via probabilisticanalysis is shown in Fig. 17.

The deterministic factor of safety of 1.23 would likely bedeemed unacceptable for many civil engineering projects due tothe high consequence of failure. However, it may be consideredsufficient for slopes in some mining operations. The probabilisticresults obtained by performing 10000 Monte Carlo trials indicatethe mean factor of safety is lower than the deterministic value(1.14) and the probability of failure is 0.141 or 14%. A review ofacceptable failure probabilities for rock slopes by Wang et al.(2000) indicated that although there is no universally acceptedvalue, there is agreement that values exceeding 10% are generallynot acceptable. Therefore, it has been shown that by includinguncertainty in the analysis of block toppling, conclusions regard-ing the stability of a slope may differ. In this case, the addition ofrock support elements or flattening of the slope may be employedto reduce the probability of failure.


6. Conclusion and summary

A new probabilistic method for analyzing the stability of rockslopes according to the limit equilibrium method developed byGoodman and Bray (1976) has been coded in an Excel spreadsheetusing Visual Basic for Applications. A review of the methodologyand logic used in the spreadsheet-based program has beenpresented in this paper. The program that was created has beenshown to calculate the probability of block-toppling failure byconsidering both kinematic and kinetic failure criteria, whileaccounting for:

�

: A9.0

irregular and uncertain geometry and shear strength para-meters;
� external forces, including horizontal ground accelerations and
water pressures; and
� rock support in the form of securing the toe block or bolting
the toppling blocks together.

The ROCKTOPPLE program has been verified against apublished deterministic example and utilized to assess a graniterock cut to demonstrate its ability to perform probabilisticanalyses. It is shown that by considering the uncertainty of inputvalues, conclusions regarding the stability of a slope may differfrom those drawn from conventional deterministic analyses.

spreadsheet-based program for probabilistic block-toppling4.014

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Fig. 18. Screenshot of ‘‘Analysis Details 1’’ tab of ROCKTOPPLE summarizing coordinates defining mean slope geometry and water levels.

Fig. 19. Screenshot of ‘‘Analysis Details 2’’ tab of ROCKTOPPLE summarizing outcome of each Monte Carlo trial.

Fig. 20. Screenshot of ‘‘Analysis Details 3’’ tab of ROCKTOPPLE illustrating details of kinetic analysis of first kinematically feasible Monte Carlo trial.


Please cite this article as: Tatone, B.S.A., Grasselli, G., ROCKTOPPLE: A spreadsheet-based program for probabilistic block-topplinganalysis. Computers and Geosciences (2009), doi:10.1016/j.cageo.2009.04.014

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Table 5Results of deterministic example as presented in Wyllie and Mah (2004).

n Wn yn Dx/y o tanca? Mn Ln Pn Pn�1:T Pn�1:S Pn�1 Rn Sn Mode

(kN) (m) (m) (m) (kN) (kN) (kN) (kN) (kN) (kN)

16 1000 4.0 2.5 No 4.0 4.0 0.0 �832.5 �470.7 0.0 866.0 500.0 Stable

15 2500 10.0 1.0 No 5.0 10.0 0.0 �457.5 �1176.8 0.0 2165.1 1250.0 Stable

14 4000 16.0 0.6 No 11.0 16.0 0.0 �82.5 �1882.9 0.0 3464.1 2000.0 Stable

13 5500 22.0 0.5 Yes 17.0 22.0 0.0 292.5 �2588.9 292.5 4533.4 2457.5 Toppling

12 7000 28.0 0.4 Yes 23.0 28.0 292.5 825.7 �3002.5 825.7 5643.3 2966.8 Toppling

11 8500 34.0 0.3 Yes 29.0 34.0 825.7 1556.0 �3175.4 1556.0 6787.6 3519.7 Toppling

10 10000 40.0 0.3 Yes 35.0 35.0 1556.0 2826.7 �3151.2 2826.7 7662.1 3729.2 Toppling

9 9000 36.0 0.3 Yes 36.0 31.0 2826.7 3922.1 �1409.7 3922.1 6933.8 3404.6 Toppling

8 8000 32.0 0.3 Yes 32.0 27.0 3922.1 4594.8 156.4 4594.8 6399.8 3327.4 Toppling

7 7000 28.0 0.4 Yes 28.0 23.0 4594.8 4837.0 1299.8 4837.0 5871.9 3257.8 Toppling

6 6000 24.0 0.4 Yes 24.0 19.0 4837.0 4637.4 2012.7 4637.4 5352.9 3199.5 Toppling

5 5000 20.0 0.5 Yes 20.0 15.0 4637.4 3978.0 2283.9 3978.0 4848.1 3159.4 Toppling

4 4000 16.0 0.6 No 16.0 11.0 3978.0 2825.5 2095.2 2825.5 4369.5 3152.6 Toppling

3 3000 12.0 0.8 No 12.0 7.0 2825.5 1103.0 1413.3 1413.3 3707.3 2912.1 Sliding

2 2000 8.0 1.3 No 8.0 3.0 1413.3 �1485.2 471.9 471.9 2471.6 1941.4 Sliding

1 1000 4.0 2.5 No 4.0 4.0 471.9 �1287.3 1.2 1.2 1235.8 970.7 Sliding

Table 6Results of deterministic example obtained with ROCKTOPPLE.

n Wn Xw Mn Ln Pn Pn�1:T Pn�1:S Pn�1 Rn Sn Mode

(kN) (m) (m) (m) (kN) (kN) (kN) (kN) (kN) (kN)

16 1833 1.9 4.9 9.8 0.0 �365.6 �862.9 0.0 1587.6 916.6 Stable

15 3303 0.7 10.8 15.6 0.0 �152.7 �1554.5 0.0 2860.1 1651.3 Stable

14 4772 �0.7 16.6 21.5 0.0 144.6 �2246.2 144.6 4019.0 2241.4 Toppling

13 6241 �2.1 22.5 27.4 144.6 549.3 �2793.3 549.3 5087.1 2715.9 Toppling

12 7711 �3.5 28.4 33.3 549.3 1152.5 �3080.2 1152.5 6203.7 3252.1 Toppling

11 9180 �5.0 34.3 39.2 1152.5 1940.3 �3168.6 1940.3 7331.1 3802.1 Toppling

10 9632 �5.2 40.2 36.1 1940.3 3133.2 �2593.9 3133.2 7404.9 3623.3 Toppling

9 8641 �4.2 37.1 32.1 3133.2 3992.8 �934.3 3992.8 6808.3 3461.0 Toppling

8 7639 �3.2 33.1 28.1 3992.8 4461.1 396.8 4461.1 6248.0 3351.3 Toppling

7 6637 �2.2 29.1 24.0 4461.1 4537.5 1336.7 4537.5 5688.1 3242.2 Toppling

6 5635 �1.2 25.0 20.0 4537.5 4221.3 1884.8 4221.3 5128.9 3134.0 Toppling

5 4634 �0.1 21.0 16.0 4221.3 3511.0 2040.2 3511.0 4570.8 3027.1 Toppling

4 3632 0.9 17.0 12.0 3511.0 2404.2 1801.5 2404.2 4014.5 2922.6 Toppling

3 2630 2.0 13.0 8.0 2404.2 895.8 1166.3 1166.3 3249.8 2552.8 Sliding

2 1628 3.2 9.0 4.0 1166.3 �952.9 400.0 400.0 2011.7 1580.2 Sliding

1 626 4.9 5.0 0.0 400.0 – – – 856.3 713.0 Sliding


Acknowledgments

The authors wish to acknowledge the individuals who helpedwith various aspects of the work presented in this paper: Prof.Stephen Evans of the Department of Earth and EnvironmentalSciences at the University of Waterloo for providing the initialmotivation to develop this tool; Dr. Mikko Jyrkama at the Institutefor Risk Research at the University of Waterloo for advice duringthe early stages of coding the analysis procedure, and Dr. ReginaldHammah at Rocscience Inc. for providing assistance with the codeneeded to sample a Fisher distribution. Furthermore, the authorswould like to thank Prof. Robert Pine and an anonymous reviewerfor their constructive comments, which improved this paper.

Funding for this work was provided in part by the NaturalSciences and Engineering Research Council of Canada in the formof an Alexander Graham Bell Canada Graduate Scholarship held byB.S.A. Tatone.

Appendix A. ‘‘Analysis details’’ tabs of ROCKTOPPLE

Figs. 18–20 illustrate the ‘‘Analysis Details’’ tabs of theROCKTOPPPLE program. ‘‘Analysis Details 1’’ (Fig. 18)


summarizes the (x,y) coordinates defining the mean geometry ofthe slope, including the overall slope geometry, the geometry ofeach block, and the coordinates defining the water table. ‘‘AnalysisDetails 2’’ summarizes the outcome of each Monte Carlo trial,including the orientations of randomly generated discontinuities,the results of kinematic analysis, and the results of kineticanalysis. ‘‘Analysis details 3’’ provides details the kinetic analysisof the first kinematically feasible trial, including all randomlysampled input parameters, the resulting inter-block forces, thestability mode of each block comprising the slope, and a graphicillustrating the randomly generated geometry.

Appendix B. Detailed results of deterministic example

Tables 5 and 6 provide detailed results of the deterministicexample presented in Section 4. Table 5 illustrates the results aspublished in Wyllie and Mah (2004), while Table 6 shows theresults obtained with ROCKTOPPLE. It is noted that instead ofutilizing Eq. (1) to determine if a block’s centre of gravity liesoutside of its base (as in Table 5), Table 6 presents the value of Xw,which describes the horizontal distance between a block’s centre


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of mass and its lowermost corner. Negative values of Xw indicatethe centre of gravity lies outside the block’s base.

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dx.doi.org/10.1007/s00603-002-0034-0.3d

dx.doi.org/10.1139/t02-034.3d

dx.doi.org/10.1016/S0013-7952(98)00007-6.3d

http://www.geo-slope.com/products/slopew2007.aspx

http://www.geo-slope.com/products/slopew2007.aspx

dx.doi.org/10.1016/S0013-7952(00)00076-4.3d

dx.doi.org/10.1016/S0013-7952(00)00076-4.3d

dx.doi.org/10.1016/j.ijrmms.2004.09.014

http://www.rockware.com/product/overview.php?id=119

http://www.rocscience.com/products/RocPlane.asp

http://www.rocscience.com/products/RocPlane.asp

http://www.rocscience.com/products/Slide.asp

http://www.rocscience.com/products/Swedge.asp

dx.doi.org/10.1016/j.cageo.2009.04.014

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