ROCKTOPPLE: A spreadsheet-based program for probabilistic block-toppling analysis

ARTICLE IN PRESS Computers & Geosciences 36 (2010) 98–114

Contents lists available at ScienceDirect

Computers & Geosciences journal homepage: www.elsevier.com/locate/cageo

ROCKTOPPLE: A spreadsheet-based program for probabilistic block-toppling analysis$ Bryan S.A. Tatone a,b,, Giovanni Grasselli a a b

Geomechanics Research Group, Lassonde Institute, Department of Civil Engineering, University of Toronto, 35 Saint George Street, Toronto, Ontario, Canada M5S 1A4 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 in f o

a b s t r a c t

Article history: Received 26 September 2008 Received in revised form 1 April 2009 Accepted 28 April 2009

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.

Keywords: Probabilistic slope stability analysis Uncertainty Monte Carlo simulation Limit equilibrium Rock slope engineering

1. Introduction Like all rock engineering problems, slope stability analysis is a data-limited problem that always involves some degree of uncertainty. This uncertainty arises due to the natural spatial and temporal variabilities of rock mass properties, prohibitive cost of obtaining large amounts of data during site investigations, lab testing results not representing in situ properties, modelling assumptions, and human errors. (Baecher and Christian, 2003). Traditionally, slope stability analysis has followed the deterministic approach of calculating resisting and driving forces to arrive at a factor of safety. To address the issue of uncertainty, conservative values of rock mass properties are adopted and a minimum acceptable factor of safety is specified to provide a margin of safety against unexpected performance. Although this approach is widely utilized and accepted, the impact of conservatism cannot be assessed and effects of varying degrees of uncertainty cannot be quantified. As a result, apparently conservative designs are not always safe against failure (El-Ramly et al., 2002). Increasingly, probabilistic methods are being applied

$

Program code and user manual available at: http://www.geogroup.utoronto.ca/

 Corresponding author at: Geomechanics Research Group, Lassonde Institute,

Department of Civil Engineering , University of Toronto, 35 Saint George Street, Toronto, Ontario, Canada M5S 1A4. E-mail addresses: [email protected] (B.S.A. Tatone), [email protected] (G. Grasselli). 0098-3004/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2009.04.014

to rock slope stability analysis as an alternative approach of dealing with uncertainty. Probabilistic slope stability analysis tools not only offer a systematic way of quantifying and evaluating the role of uncertainty, but also provide a useful approach to estimate hazard frequency for quantitative risk analyses, which are 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 of rock slopes were first introduced in the 1970s (McMahon, 1971; Major et al., 1977; Piteau and Martin, 1977; and others). Since then, the concepts and methods have undergone continual development such that methods of probabilistic analysis for translational 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 many others). At present, there are several commercially available software packages capable of performing probabilistic limit equilibrium slope stability analysis. Some of the most popular packages include: SLOPE/W (GEO-SLOPE, 2007), SLIDE, ROCPLANE, SWEDGE (Rocscience, 2008a–c), and RockPack III (RockWare, 2008). These software packages employ Monte Carlo simulations to repeatedly calculate the factor of safety with input parameters that are randomly generated according to user-defined probability distributions. Therefore, instead of obtaining a singular value for the factor of safety from singular input values (deterministic approach), a distribution of values is

ARTICLE IN PRESS B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

obtained, which represents the uncertainty of the input parameters. The probability of failure is defined as the number of Monte Carlo trials producing a factor of safety less than one divided by the total number of trials. Although probabilistic methods for analyzing soil slopes and rock slopes susceptible to planar sliding and wedge sliding are now well established, the toppling failure mode of rock slopes has received relatively little attention. Very few publications can be found that focus on incorporating the uncertainty of input parameters in the analysis of toppling (Muralha, 2003; Scavia et al., 1990) and none have considered the role of kinematic stability on the probability of failure. The objectives of this paper are to: (1) review the conventional deterministic stability analysis of slopes susceptible to block-toppling; (2) introduce a new probabilistic block-toppling analysis procedure that accounts for kinematic stability; (3) describe the implementation of this new procedure in a computer program created in Microsoft Excel using Visual Basic for Applications (VBA); and (4) demonstrate how this program can be used as a tool to analyze slopes with blocktoppling hazard.

2. Conventional deterministic analysis of block-toppling failure Before introducing the probabilistic analysis procedure, it is valuable to review the conventional deterministic approach for analyzing slopes susceptible to block toppling. The evaluation of rock slope stability is typically a two-step process. First a kinematic analysis of structural discontinuities via stereographic techniques is undertaken to identify potentially unstable conditions. Subsequently, if a kinematically unstable condition is found to exist, a kinetic analysis using a limit equilibrium method is used to evaluate the factor of safety (Norrish and Wyllie, 1996; Wyllie and Mah, 2004). Depending on the orientations of discontinuities in relation to the geometry of the slope under consideration, potential slope failures can typically be classified into four modes: circular, planar

99

sliding, wedge sliding, or toppling. This paper focuses on the toppling failure mode, which involves the overturning of rock columns delineated by a well-defined discontinuity set striking sub-parallel to the slope face and dipping steeply into the face. Goodman and Bray (1976) classified toppling failures into three types (Fig. 1). The analysis procedure presented in this paper is intended for the analysis of slopes susceptible to the blocktoppling type of failure only. 2.1. Kinematic conditions for block-toppling Considering a single rock block on an inclined surface subject to no external forces (Fig. 2a), toppling occurs if the block’s centre of gravity acts outside of its base and sliding does not occur along its base. Mathematically, toppling occurs when

Dx=yn o tan c

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 the base plane, respectively. When a series of blocks is considered (Fig. 2b), two additional requirements exist. The first requirement is that the strike of discontinuities defining the base and width of the toppling blocks must be sub-parallel to the slope face ( 7201) such that the blocks are free to topple without restraint from the adjacent rock mass. This requirement is defined mathematically as (Norrish and Wyllie, 1996) jaa  as j o 203 and jab  as jo 203

ð3Þ

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

Fig. 1. Common types of toppling: (a) block toppling of rock columns divided into blocks of finite height by a second, widely spaced, roughly orthogonal joint set;(b) flexural toppling of continuous rock columns; and (c) block-flexural toppling characterized by pseudo-continuous flexure of rock columns with numerous cross-joints that accommodate significant lateral displacements (from Wyllie and Mah, 2004 after Goodman and Bray, 1976).

ARTICLE IN PRESS 100

B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

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).

(Goodman and Bray, 1976, Norrish and Wyllie, 1996) ð903  cb Þ r ðcs  fb Þ

7 ð4Þ

where fb and cb are the friction angle and dip angle of the sub-vertical discontinuities, respectively, and cs is the dip of the slope face. Figs. 2c and d illustrate the lower hemispherical stereographic projection of the slope geometry depicted in Fig. 2b along with envelopes defined by conditions (2)–(4). It is noted that although Cruden (1989) has shown the kinematic limits of block-toppling to extend to cataclinal, underdip slopes, the analysis procedure and computer program presented in this paper is restricted to the anaclinal geometry originally outlined by Goodman and Bray (1976). 2.2. Kinetic (limit equilibrium) analysis of block-toppling Given that the kinematic conditions for block toppling exist in the slope under consideration, the kinetic stability can be evaluated using the limit equilibrium method developed by

6 5 4

3 Toe block

2 Stable 1

Topple Slide

Fig. 3. Example of a system of toppling blocks on a stepped base (Goodman and Bray, 1976).

ARTICLE IN PRESS B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

 a set of short blocks at the toe of the slope (e.g. Block 1) that

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

are pushed by the toppling blocks above. These blocks stable depending on slope geometry. The stability analysis is a step-wise process that begins with establishing the dimensions and calculating the forces acting on each block in the slope. Subsequently, the stability of each block is evaluated starting at the topmost block. Considering the balance of forces and moments acting on the blocks, each block may remain 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 in magnitude to the force needed to maintain the current block in

 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 (ca o fa); 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

Start

Input mean values, standard deviations, and number of trials (Nummtrials) as defined by user (except spacing)

Set ini, Unstable Count, Stable Count, and Kinematic Count = 0

10 8 6 4 2 0

Randomly sample input values from user defined PDF’s Check kinematic stabiliity

Kinematically feasible ?

No

Yes Kinematic Count = Kinematic Count +1 Generate random block geometry Perform limit equilibrium analysis Yes

If FS > 1

ini = ini + 1

N No

No Unstable Count = Unstable Count +1

Stable Count = Stable Count +1

101

ini = Num Trials? Yes

Calculate probabilities of failure: Pkinematic, Pf kinetic|kimematic, and Pf

Stop Fig. 4. Overview of probabilistic block-toppling stability analysis procedure.

ARTICLE IN PRESS 102

B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

limiting equilibrium. However, if a block is stable, no forces are transmitted to the next block in the slope. The overall stability of the slope is controlled by the stability of the lowermost block, or toe block. If the toe block is stable, the entire slope is considered stable and, conversely, if the toe block is unstable, the entire slope is considered unstable. As with all limit equilibrium methods, this method can easily incorporate external forces acting on the slope, including support loads, water pressures, and pseudo-static earthquake loads (Wyllie, 1999).

2.3. Factor of safety

tanfavailable tanfrequired

This section describes the probabilistic block-toppling analysis procedure and its implementation in a computer program called ROCKTOPPLE created in Microsoft Excel using VBA (available at www.geogroup.utoronto.ca). The program logic is described by first providing an overview of the entire analysis procedure followed by a detailed description of each major step in the procedure.

3.1. Overview of probabilistic analysis procedure

Since the toe block is assumed to control the overall stability of a system of toppling blocks, the factor of safety of the toe block is assumed to define the factor of safety of the entire slope. In the absence of cohesion, Goodman and Bray (1976) proposed that the following equation can be used to define a factor of safety against block toppling: FS ¼

3. Development of a probabilistic block-toppling analysis procedure and its implementation in a spreadsheet-based program

ð5Þ

where tan favailable defines the coefficient of friction on the base plane of the toe block and tan frequired defines the coefficient of friction needed for limiting equilibrium. One must be aware that this approach assumes that the critical failure mode of the toe block is sliding (which is the case in the three examples presented in Goodman and Bray, 1976) when, in fact, the critical failure mode of the toe block may also be toppling (Wyllie and Mah, 2004). Moreover, it assumes that the blocks above the toe block push on the toe block when, in fact, there may be multiple blocks of the ‘‘stable’’ mode at the toe that collectively resist the movement of upslope blocks. In these two cases outlined above, a different means of calculating the factor of safety must be adopted. The approach adopted in the current study is presented in a later section of this paper (Section 3.5.2).

The probabilistic approach developed herein (Fig. 4) utilizes Monte Carlo simulation to repeatedly perform the deterministic analysis procedure (kinematic and kinetic analysis) described in the preceding section. Considering Fig. 4, the first step in the analysis procedure requires the user to specify the number of Monte Carlo trials and define the appropriate probability distributions for the parameters characterizing the slope (see Section 3.2). Subsequently, the Monte Carlo simulation procedure is initiated, which involves repeatedly sampling random input parameters from the user-defined probability distributions; checking the kinematic stability conditions (see Section 3.3); generating the random slope geometry (see Section 3.4); and evaluating the kinetic stability (factor of safety) via limit equilibrium analysis (see Section 3.5). Afterwards, the kinematic, kinetic, and total probabilities of block-toppling failure are calculated (see Section 3.7). The methodology outlined in Fig. 4 was coded into an Excelbased program as it allowed the use of Excel’s built-in functions and graphing capabilities, greatly reducing the overall coding effort required. The Excel workbook that houses the ROCKTOPPLE program consists of 6 worksheets or ‘‘tabs’’ named as follows: ‘‘Analysis Input’’, ‘‘Add Support’’, ‘‘Results’’, ‘‘Analysis Details 1’’, ‘‘Analysis Details 2’’, and ‘‘Analysis Details 3’’. The first three of these tabs are

Fig. 5. Screenshot of ‘‘Analysis Input’’ tab of ROCKTOPPLE.

ARTICLE IN PRESS B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

discussed in the subsequent sections of this paper while, descriptions 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 be entered by the user, while the graphic on the right side provides a preview of the slope geometry according to the mean input values. The input values are divided into ‘‘fixed’’ parameters, defined by singular input values, and ‘‘uncertain’’ parameters, defined by probabilistic distributions. The height, H, and orientation of the slope, as, cs, and cts are always considered ‘‘fixed’’ while, all remaining 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 be performed and what support measures, if any, should be considered in the analysis. When the orientations of joint sets A and B are considered ‘‘uncertain’’, they are assumed to be defined by a Fisher distribution (Fisher, 1953), which is a symmetric three-dimensional (3D) distribution often used to describe the angular dispersion 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 of clustering around the mean value. In terms of analyzing blocktoppling, the use of a 3D distribution for joint orientation data allows kinematic analysis of the randomly generated discontinuities according to Section 2.1. However, since the adopted kinetic analysis procedure is two-dimensional (2D), a 2D representation of the 3D orientation data is needed before analysis can be performed. Considering a cross-section perpendicular to the slope face, the difference between the true dip, ctrue, and apparent dip, capparent, of discontinuities that satisfy the kinematic conditions for block toppling (i.e. dip directions within 7201 of the dip direction of the slope face) is very small (tan capparent = 0.94tan ctrue). Therefore, to perform kinetic analysis the true dip angles sampled from the Fisher distributions

103

are assumed to define the apparent dip angles for the 2D section. Since true dip is always greater than apparent dip, this assumption introduces some conservatism into the kinetic analysis. 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, lognormal, or exponential distributions. Estimates of the mean and standard deviation are the only user inputs required to define these distributions. 3.3. Kinematic analysis Based on the randomly sampled values defining the orientation and friction angle of joint sets A and B, ROCKTOPPLE checks if the kinematic conditions for block toppling, as defined in Section 2.1, are satisfied. The condition set out by (1), however, is not enforced since external forces such as water pressures or seismic loads can cause blocks to topple despite having a centre of gravity that lies within their base. If the remaining kinematic conditions are satisfied, the program proceeds with kinetic analysis, as described in the following sections of this paper; otherwise, it advances to the next Monte Carlo trial. In trials where the conditions for block-toppling are not satisfied, the randomly sampled orientations of discontinuity sets A and B may result in one of the following alternative kinematic conditions: 1. Failure not kinematically possible: sliding cannot occur on set A and toppling cannot occur on set B. Therefore, the total probability of failure is 0. 2. Only sliding on joint set A is kinematically possible: friction angle of set A is less than the dip angle; the dip direction of set A is within 7201 of slope dip direction but toppling on set B is not possible. 3. Only toppling on set B is kinematically possible: orientation of set B satisfies requirements for interlayer slip and alignment but the dip direction of set A prevents sliding. Therefore, the toe blocks cannot slide.

Fig. 6. Idealized geometry of a rock slope subject to toppling (after Scavia et al., 1990).

ARTICLE IN PRESS 104

B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

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

Start

Pass values of H,s, ts, a, b, and ground water level Therefore, although block toppling may not be kinematically feasible, the slope geometry may still result in a kinematically unstable condition. To give the user an indication of the likelihood of these other kinematic conditions existing, the program calculates the kinematic probability of each. However, the program does not perform the corresponding kinetic analysis. Hence, it must be emphasized that the total probability of failure calculated by ROCKTOPPLE only represents the total probability of block-toppling failure and does not account for kinetic stability of the other potential failure modes (2–4 above). To evaluate the total probability of slope failure, including these other potential failure modes, kinetic analysis of each kinematic condition would need to be undertaken with appropriate techniques. Subsequently, system reliability methods could be used to calculate the total failure probability. It should be noted that in cases where the geometry of the slope and discontinuities are known with increased certainty (i.e. an existing slope in which several joint measurements have been obtained), it may be desirable to treat the discontinuity orientations as ‘‘fixed’’ input values (accomplished by entering K= 0 on the ‘‘Analysis Input’’ tab). In this situation, given that the fixed discontinuity orientations satisfy kinematic conditions for blocktoppling, the kinematic probability will be 1.0, meaning the total probability of block toppling of failure will be given by kinetic probability of failure. In other words, if desired, the user can effectively skip the kinematic analysis by considering discontinuity orientations as ‘‘fixed’’ inputs. 3.4. Generation of block geometry If the kinematic conditions for block toppling are satisfied, the next step in the analysis procedure involves generating the geometry of the n blocks that form the slope. The procedure adopted in ROCKTOPPLE follows that developed by Scavia et al. (1990). Unlike the original limit equilibrium procedure, which assumes the rock blocks are delineated by evenly spaced, perpendicular discontinuity sets, this approach is capable of generating blocks delineated by non-orthogonal, irregularly spaced joint sets (Fig. 6). In generating the random block geometry, the following assumptions are made (Scavia et al., 1990):

 the joint sets A and B are considered 100% persistent;  the system of blocks sits on a stepped base that represents a

into sub-procedure

i= 1 Define (x, y) of Toe as (0, 0) BlockCount = 1 BlockCount = BlockCount + 1 Randomly sample values of Sa and Sb from probabilistic distributions Calculate coordinates defining i th block

and store in separate arrays

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

No i=i+1

Check if block extends beyond the slope limits

Yes

n = BlockCount

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

the corners of each block are computed, and several forces and dimensions specific to each block, as defined in Fig. 8, are evaluated for use in subsequent limit equilibrium calculations. This process is continued until the stepped base reaches the upper boundary of the slope, forming n blocks. 3.5. Kinetic (limit equilibrium) analysis

‘‘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 such that the problem can be analyzed two dimensionally.

The limit equilibrium analysis procedure requires the calculation of the forces transferred from the uppermost block through to the toe block. These forces are referred to as interblock forces. Once these forces are determined, the factor of safety of the toe block or group of stable toe blocks can be evaluated. The following two sub-sections describe the calculation procedure for inter-block forces and toe block stability, respectively.

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

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

  

ARTICLE IN PRESS B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

joint sets A and B are orthogonal (ct = 0). As this assumption is rarely valid when values of ca and cb are randomly sampled from independent probabilistic distributions, these equations were reformulated to account for non-orthogonal joint sets before being implemented in ROCKTOPPLE. Considering moment and force equilibrium for a typical block (Fig. 8), the revised equations for the force, Pn  1, that is just sufficient to prevent the block from toppling and sliding, are given, respectively, by Pn1:t ¼

Wn Xw þKYw  Pn tanfb Sb  V3 y3 þV1 y1 þ V2 y2 þ Pn Mn Ln ð6Þ

Pn1:s ¼ Pn 

equal to Pn  1:s. Once the appropriate value of Pn  1 is determined, all forces acting on the block are resolved in directions perpendicular and parallel to the base of the block. The normal force, Rn, and shear force, Sn, are calculated, respectively, as Rn ¼ Wn cosca  Ksinca V2 þ ðV3  V1 þPn1  Pn Þsinct þðPn  Pn1 Þtanfb cosct

ð8Þ

Sn ¼ Wn sinca þ Kcosca þ ðV1  V3  Pn1 þ Pn Þcosct þðPn  Pn1 Þtanfb sinct

ð9Þ

Wn ðcosca tanfa  sinca Þ  Kðsinca tanfa þ cosca Þ þ ðV3  V1 Þðcosct þ sinct tanfa Þ  V2 tanfa ðtanfa þtanfb Þsinct þð1  tanfa tanfb Þcosct

Fig. 9 summarizes the methodology used to calculate the interblock forces acting on each block on the slope. Starting at the uppermost block, the forces Pn  1:t required to prevent toppling and 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 current block is considered stable and the force, Pn, transmitted to the next block is set to zero. However, if Pn  1:t 4Pn  1:s, the block is on the point of toppling and Pn  1 is set equal to Pn  1:t. Conversely, if Pn  1:s 4Pn  1:t, the block is on the point of sliding and Pn  1 is set

105

ð7Þ

Subsequently, a check is made to ensure if there is a positive normal force on the base plane and that sliding does not occur: Rn 40

and

jSn j o Rn tan fa

ð10Þ

If the conditions set out by (10) are not satisfied, toppling cannot occur even if Pn  1:t 4 Pn  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 is then repeated for the next block and all subsequent blocks in succession until the force, Pn, acting on each block has been determined. It is noted that due to kinematic constraints, once the transition from toppling to sliding occurs, the critical state for all subsequent blocks is sliding (Wyllie and Mah, 2004). 3.5.2. Analysis of toe block(s) Following the calculation of inter-block forces, the stability mode of each block above the toe block is defined as ‘‘sliding’’, ‘‘toppling’’, or ‘‘stable’’. In the case where the block immediately above the toe block is of the ‘‘sliding’’ mode, the potential failure mode of the toe block is limited to sliding and the factor of safety against toe block sliding is considered to be the factor of safety of the entire system of blocks. In the case where the block immediately above the toe block is of the ‘‘toppling’’ mode, the potential failure mode of the toe block can be sliding or toppling and the critical factor of safety against toe block sliding or toppling is taken as the factor of the safety system of blocks. Considering the forces acting on a typical toe block (Fig. 10), the factor of safety against toe block sliding and toe block toppling are given, respectively, by P Forcesresisting FS ¼ FStoe block ¼ P sliding Forcesdriving ½Wn cosca  Ksinca  V2 þ ðV1  Pn Þsinct þ Pn tanfb costanfa ¼ Wn sinca þ Kcosca þ ðV1 þ Pn Þcosct þ Pn tanfb sinct ¼

Rn tanfa : Sn

P Momentsresisting Pn tanfb Sb þWn Xw FS ¼ FStoe block ¼ P ¼ toppling Pn Mn þV1 y1 þ V2 y2 þ KYk Momentsdriving

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

ð11Þ

ð12Þ

In the case where the block immediately above the toe block is of the ‘‘stable’’ mode, the factor of safety of the toe block is no longer representative of the stability of the entire slope system. Instead, the factor of safety is dictated by the collective ability of the group of stable blocks at the toe to resist the driving forces produced by unstable blocks upslope. The factor of safety, in this case, can be defined as the sum of the resisting forces of each ‘‘stable’’ block divided by the sum of the driving forces for each

ARTICLE IN PRESS 106

B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

‘‘stable’’ block: FS ¼

ðRn1 þ Rn2 þRn3 þ    Rni Þ tanfa Sn1 þ Sn2 þ Sn3 þ    Sni

Start ð13Þ

i=n Pn = 0

where i is the number of ‘‘stable’’ blocks at the toe of the slope. 3.6. Addition of support

ith Block

As previously mentioned, external forces can be easily incorporated into limit equilibrium methods. Hence, the effect of rock support elements on slope stability can be easily assessed in the block-toppling analysis. The support of slopes susceptible to block-toppling through the use of rock support elements can be accomplished in two ways (Wyllie and Wood, 1983): (1) a support force, T, can be added to the toe block, as shown in Fig. 11a or (2) the potential toppling blocks can be bolted together to increase their effective width, as shown in Fig. 11b. Both of these support methods were incorporated into ROCKTOPPLE under the ‘‘Add Support’’ tab shown in Fig. 12. In this tab, the user can specify the magnitude and orientation of toe block support and the effective width of the toppling blocks when the blocks are bolted together. When toe block support is added to the analysis, it is assumed to be installed at the mid-point of the toe block face inclined at a user-defined angle, i, from the horizontal (Fig. 11a). Based on the mean slope geometry, the optimum orientations, iopt, of the toe block support to prevent sliding and toppling of the toe block are given, respectively, as (Goodman and Bray, 1976; Wyllie and Mah, 2004) ioptjsliding ¼ fa  ca

ð14Þ

ioptjtoppling ¼  ca

ð15Þ

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

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

Calculate Sn and Rn

If Rn ≤ 0 or

Pn-1 = Pn-1

Yes Pn-1 = Pn-1:s

Set Pn = Pn-1

No

If i = 1

i = i-1

In addition to the magnitude and orientation of the support, a drop-down menu on the tab allows the user to specify whether it should be considered an active or passive force. If the support force is applied actively, the revised factors of safety against toe block toppling and sliding are given respectively, by Pn tanfb Sb þ Wn Xw FStoe block ¼ toppling Pn Mn þ V1 y1 þV2 y2 þ KYk  TLt

Sn > Rn tan b

No

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

ð16Þ 3.7. Calculation of failure probabilities

FStoe block ¼ sliding

fWn cos ca  Ksin ca  V2 þ ðV1  Pn Þsin ct þ Pn tan fb cosct þ Tsinðca þ iÞgtan fa Wn sin ca þKcos ca þ ðV1 þ Pn Þcos ct þ Pn tan fb sin ct  Tcosðca þiÞ

ð17Þ If it is applied passively, the factors of safety are given by

Following the completion of the specified number of Monte Carlo trials, the failure probabilities are calculated and displayed in the ‘‘Results’’ tab of ROCKTOPPLE (Fig. 13). The ‘‘Results’’ tab provides a detailed summary of the kinematic and kinetic probabilities of failure, the mean and median factor of safety, a histogram of the factors of safety, and a summary of the applied rock support.

FStoe block ¼

Pn tanfb Sb þ Wn Xw þTLt Pn Mn þ V1 y1 þV2 y2 þ KYk

FStoeblock ¼

fWn cosca  Ksinca  V2 þ ðV1  Pn Þsinct þ Pn tanfb cos þ Tsinðca þiÞgtanfa þTcosðca þ iÞ Wn sinca þ Kcosca þðV1 þ Pn Þcosct þ Pn tanfb sinct

toppling

sliding

ð18Þ

In the case where the blocks are bolted together, the effective width of toppling blocks below the crest is increased. ROCKTOPPLE 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 effective width of two times the actual block width is specified, the analysis proceeds by assuming the blocks below the crest are 4 m wide. It should be noted that this simplistic approach does not consider potential failure of the bolts holding the blocks together.

ð19Þ

The probability of kinematic failure is given by Pf

kinematic

¼

Nkinematically feasible Nt

ð20Þ

where Nkinematically feasible is the number of trials in which blocktoppling failure is kinematically feasible and Nt is the total number of Monte Carlo trials. Similarly, probabilities of the other kinematic conditions (as defined in Section 3.3) are calculated by dividing the number of trials in which the conditions occur by Nt.

ARTICLE IN PRESS B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

Considering that toppling failure involves a system of blocks that can topple, slide, or remain stable, the kinetic probability of failure must be defined in terms of the reliability of the system as a whole. For this system of blocks, however, the conventional approach of using event tree or fault tree analysis to examine all possible failure paths of the system becomes quite complex and cumbersome. Recalling that the stability of the toe block ultimately controls the stability of the entire system of blocks independent of the behaviour of other blocks in the system, repeatedly calculating the factor of safety of the toe block while varying the input parameters effectively constitutes the simulation approach for analyzing system reliability (Baecher and Christian, 2003). Thus, the kinetic probability of failure for the system can be defined in terms of the factor of safety of the toe block or group of stable toe blocks. Since kinetic failure can occur via sliding or toppling, the kinetic probability of block-toppling failure is given by

s

b

Pn Pntanb

Sb

V1 K

Mn

(xcm,ycm)

xw

y1

t Wn

Origin

Yk V2

y2

a

Pf

kineticjkinematic

¼

NFS o 1:toe sliding þNFS o 1:toe toppling Nkinematically feasible

s

i T Lt a

ð21Þ

where NFS o I:toe sliding and NFS o I:toe toppling are the number of trials resulting in a factor of safety less than 1 when the critical failure mode of the toe block is sliding and toppling, respectively.

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

b

107

t

Origin Fig. 11. Methods of applying support in ROCKTOPPLE: (a) toe block support and (b) bolting blocks together.

Fig. 12. Screenshot of ‘‘Add Support’’ tab in ROCKTOPPLE.

ARTICLE IN PRESS 108

B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

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

Table 1 Summary of slope characteristics for deterministic example given in Wyllie and Mah (2004). Parameter

Deterministic value

Overall geometry Slope height (m) Slope angle (deg)a Top angle (deg)

92.5 56.6 4

Discontinuity orientations Dip of set A (deg)a Dip of set B (deg)a

30 60

Rock mass characteristics Spacing of joint set A (m) Spacing of joint set B (m) Friction angle of set A (deg) Friction angle of set B (deg) Unit weight of rock (kN/m3)

1 10 38.15 38.15 25

External loads Seismic coefficient (g) Water pressure (%) Rock support

0 0 n/a

4. Deterministic verification of ROCKTOPPLE

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 conditional probability since kinetic analysis is undertaken only for kinematically feasible geometries. Based on the properties of conditional probabilities, the total probability of block-toppling failure is given by the product of (20) and (21) (Glynn, 1979; Park and West, 2001): Pf ¼ Pf

kinematic Pf kineticjkinematic

Section 3.3, are not included in the calculation of the total probability of block toppling. They are provided merely to inform the user that given the slope geometry and discontinuity orientations, instability via other failure modes may be possible. As mentioned previously, calculation of the total probability of slope failure would require a separate kinetic analysis of each unstable kinematic condition via alternate analysis methods and the evaluation of the total failure probability using system reliability methods.

ð22Þ

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

Since examples of probabilistic block-toppling analysis could not be found in the literature, the probability of failure calculated with ROCKTOPPLE was not compared with previous results. The output of the program was, however, compared to a deterministic block-toppling example published in Wyllie and Mah (2004) by considering the input parameters (Table 1) as fixed values. Results as shown in Wyllie and Mah (2004) and those obtained with ROCKTOPPLE are tabulated in Appendix B. When comparing results, it is important to note that the methodology used to define the geometry of the blocks varies between the published example and ROCKTOPPLE. While the published example assumes the blocks are rectangular to simplify calculations (Fig. 14a), ROCKTOPPLE assumes they are trapezoidal (Fig. 14b). As a result, the blocks generated by the program are taller above the slope crest and shorter below the slope crest when compared with rectangular blocks. The largest percent difference in block weight occurs for the uppermost and lowermost blocks. The 4 uppermost blocks vary from 12% to 59% and the 3 lowermost blocks vary from 13% to 46%; all other blocks vary by less than 10%. When the parameters listed in Table 1 were considered as fixed input values in ROCKTOPPLE, it was revealed that the discontinuity orientations did not satisfy the kinematic conditions for block-toppling. Therefore, to obtain kinetic stability results with

ARTICLE IN PRESS B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

12

14 15 16

9 7

8

6 5 4 3 1 2

140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 -10

10

11

12

13

14

15 16

9 8 7 6 5 4 3 1

2

-10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140

11 10

13

Differences in the block geometry also resulted in differing inter-block forces and, consequently, differing values of Rn and Sn (Fig. 15). According to the results obtained from ROCKTOPPLE, the factor of safety according to Eq. (12) is 0.94 compared to 1.00 for the published results. Although the results obtained from ROCKTOPPLE varied from the published results due to differing block geometries, similar behaviour in terms of stability mode and the relative shear and normal forces was predicted for all blocks. The resulting difference in the factor of safety, given the same input parameters, underscores the impact of varying the slope geometry on stability and further illustrates the importance of incorporating geometric uncertainties into the analysis of block toppling.

Vertical Distance (m)

140 130 120 110 100 90 80 70 60 50 40 30 20 10 0 -10

-10 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140

Vertical Distance (m)

ROCKTOPPLE that could be compared with published values, analysis of kinematic stability was temporarily disabled. The results obtained with ROCKTOPPLE, in terms of the stability mode of the blocks, are in close agreement with the published results (i.e. a set of stable blocks at the crest, a set of intermediate toppling blocks, and a set of sliding blocks at the toe). There was, however, a notable discrepancy as ROCKTOPPLE predicted only 2 stable blocks at the slope crest compared to 3 in the published example. This discrepancy is attributed to the differences in block geometry noted earlier. Since trapezoidal blocks above the slope crest are taller relative to their rectangular counterparts, their centroid locations are shifted in the downslope direction relative to their base and are, therefore, more likely to topple.

109

Horizontal Distance (m)

Horizontal Distance (m)

Fig. 14. Geometry of deterministic example problem: (a) rectangular blocks as considered in Wyllie and Mah (2004) and (b) trapezoidal blocks considered by ROCKTOPPLE.

1

2

3

4

5

6

7

Block 8 9

10

11

12

13

14

15

16

0 1000 2000

Force (kN)

3000 4000 5000 6000 7000 8000

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

9000 Fig. 15. Comparison of shear (Sn) and normal (Rn) forces along base of each block as given in Wyllie and Mah (2004) and as calculated by ROCKTOPPLE.

ARTICLE IN PRESS 110

B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

5. Probabilistic application of ROCKTOPPLE To demonstrate the use of the probabilistic stability analysis procedure and computer program ROCKTOPPLE described in the preceding sections, the stability of a 15 m high granite rock cut was analyzed, for which sufficient geotechnical data are available and suitable.

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

Fig. 16. Equal-area stereographic representation of slope and discontinuity geometry considered probabilistically along with envelopes defining kinematic conditions required for block toppling.

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

Mean orientation (dip/dip direction)

Fisher constant K

D1 D2 D3

751/1341 291/0501 721/2251

49 46 24

Table 3 Summary of input parameters for probabilistic analysis. Parameter

Probabilistic distribution

Mean value

Standard deviation

Overall geometry Slope height (m) Slope angle (deg) Top angle (deg) Dip direction of slope face (deg)

Fixed Fixed Fixed Fixed

15 70 2.5 55

– – –

Discontinuity orientations Dip/dip direction of set A (deg) Dip/dip direction of set B (deg)

Fisher Fisher

29/050 72/225

46 (Fisher K) 24 (Fisher K)

Rock mass characteristics Spacing of joint set A (m) Spacing of joint set B (m) Friction angle of set A (deg) Friction angle of set B (deg) Unit weight of rock (kN/m3)

Log normal Log normal Normala Normala Fixed value

0.75 2.5 35 35 26

0.15 0.30 2.5 2.5 –

External loads Seismic coefficient (g) Unit weight of water (kN/m3) Water pressure (%) Rock support

Fixed value Fixed value Log normal Not considered

0 9.81 10

– – 5

value value value value

a Physically, these parameters cannot have values outside the range of 0–901. Therefore, the normal distributions are truncated at these extremes to prevent sampling of non-physical values.

ARTICLE IN PRESS B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

111

Table 4 Comparison of deterministic and probabilistic results obtained with ROCKTOPPLE Analysis description

Factor of safety

Deterministic

1.23 Probabilities of failure Mean factor of safety 1.17

Probability of kinematic failure 0.602

Probability of kinetic failure 0.234

600

1.0 Critical mode of toe block(s) = toppling

500

Critcal mode of toe block(s) = sliding 400 Frequency

Total probability of block toppling failure 0.141

0.8

Cumulative Probability 0.6

300 0.4 200

Cumulative Probability

Probabilistic

0.2

100

0

0.0

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-vertical toppling planes (as previously defined in Fig. 2), it is evident that block-toppling is possible with discontinuity sets D2 and D3 forming the base plane (set A) and sub-vertical planes (set B), respectively. The input parameters required for performing probabilistic analysis of the slope are given in Table 3, including the selected probabilistic distributions, mean values, and corresponding standard deviations (or Fisher constants). Analysis results, both deterministic and probabilistic, are presented in Table 4 and the distribution of the factor of safety obtained via probabilistic analysis is shown in Fig. 17. The deterministic factor of safety of 1.23 would likely be deemed unacceptable for many civil engineering projects due to the high consequence of failure. However, it may be considered sufficient for slopes in some mining operations. The probabilistic results obtained by performing 10000 Monte Carlo trials indicate the mean factor of safety is lower than the deterministic value (1.14) and the probability of failure is 0.141 or 14%. A review of acceptable failure probabilities for rock slopes by Wang et al. (2000) indicated that although there is no universally accepted value, there is agreement that values exceeding 10% are generally not acceptable. Therefore, it has been shown that by including uncertainty in the analysis of block toppling, conclusions regarding the stability of a slope may differ. In this case, the addition of rock support elements or flattening of the slope may be employed to reduce the probability of failure.

6. Conclusion and summary A new probabilistic method for analyzing the stability of rock slopes according to the limit equilibrium method developed by Goodman and Bray (1976) has been coded in an Excel spreadsheet using Visual Basic for Applications. A review of the methodology and logic used in the spreadsheet-based program has been presented in this paper. The program that was created has been shown to calculate the probability of block-toppling failure by considering both kinematic and kinetic failure criteria, while accounting for:

 irregular and uncertain geometry and shear strength parameters;

 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 a published deterministic example and utilized to assess a granite rock cut to demonstrate its ability to perform probabilistic analyses. It is shown that by considering the uncertainty of input values, conclusions regarding the stability of a slope may differ from those drawn from conventional deterministic analyses.

ARTICLE IN PRESS 112

B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

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.

ARTICLE IN PRESS B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

113

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

Wn (kN)

yn (m)

Dx/y

o tanca?

Mn (m)

Ln (m)

Pn (kN)

Pn  1:T (kN)

Pn  1:S (kN)

Pn  1 (kN)

Rn (kN)

Sn (kN)

Mode

16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

1000 2500 4000 5500 7000 8500 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000

4.0 10.0 16.0 22.0 28.0 34.0 40.0 36.0 32.0 28.0 24.0 20.0 16.0 12.0 8.0 4.0

2.5 1.0 0.6 0.5 0.4 0.3 0.3 0.3 0.3 0.4 0.4 0.5 0.6 0.8 1.3 2.5

No No No Yes Yes Yes Yes Yes Yes Yes Yes Yes No No No No

4.0 5.0 11.0 17.0 23.0 29.0 35.0 36.0 32.0 28.0 24.0 20.0 16.0 12.0 8.0 4.0

4.0 10.0 16.0 22.0 28.0 34.0 35.0 31.0 27.0 23.0 19.0 15.0 11.0 7.0 3.0 4.0

0.0 0.0 0.0 0.0 292.5 825.7 1556.0 2826.7 3922.1 4594.8 4837.0 4637.4 3978.0 2825.5 1413.3 471.9

 832.5  457.5  82.5 292.5 825.7 1556.0 2826.7 3922.1 4594.8 4837.0 4637.4 3978.0 2825.5 1103.0  1485.2  1287.3

 470.7  1176.8  1882.9  2588.9  3002.5  3175.4  3151.2  1409.7 156.4 1299.8 2012.7 2283.9 2095.2 1413.3 471.9 1.2

0.0 0.0 0.0 292.5 825.7 1556.0 2826.7 3922.1 4594.8 4837.0 4637.4 3978.0 2825.5 1413.3 471.9 1.2

866.0 2165.1 3464.1 4533.4 5643.3 6787.6 7662.1 6933.8 6399.8 5871.9 5352.9 4848.1 4369.5 3707.3 2471.6 1235.8

500.0 1250.0 2000.0 2457.5 2966.8 3519.7 3729.2 3404.6 3327.4 3257.8 3199.5 3159.4 3152.6 2912.1 1941.4 970.7

Stable Stable Stable Toppling Toppling Toppling Toppling Toppling Toppling Toppling Toppling Toppling Toppling Sliding Sliding Sliding

Table 6 Results of deterministic example obtained with ROCKTOPPLE. n

Wn (kN)

Xw (m)

Mn (m)

Ln (m)

Pn (kN)

Pn  1:T (kN)

Pn  1:S (kN)

Pn  1 (kN)

Rn (kN)

Sn (kN)

Mode

16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

1833 3303 4772 6241 7711 9180 9632 8641 7639 6637 5635 4634 3632 2630 1628 626

1.9 0.7  0.7  2.1  3.5  5.0  5.2  4.2  3.2  2.2  1.2  0.1 0.9 2.0 3.2 4.9

4.9 10.8 16.6 22.5 28.4 34.3 40.2 37.1 33.1 29.1 25.0 21.0 17.0 13.0 9.0 5.0

9.8 15.6 21.5 27.4 33.3 39.2 36.1 32.1 28.1 24.0 20.0 16.0 12.0 8.0 4.0 0.0

0.0 0.0 0.0 144.6 549.3 1152.5 1940.3 3133.2 3992.8 4461.1 4537.5 4221.3 3511.0 2404.2 1166.3 400.0

 365.6  152.7 144.6 549.3 1152.5 1940.3 3133.2 3992.8 4461.1 4537.5 4221.3 3511.0 2404.2 895.8  952.9 –

 862.9  1554.5  2246.2  2793.3  3080.2  3168.6  2593.9  934.3 396.8 1336.7 1884.8 2040.2 1801.5 1166.3 400.0 –

0.0 0.0 144.6 549.3 1152.5 1940.3 3133.2 3992.8 4461.1 4537.5 4221.3 3511.0 2404.2 1166.3 400.0 –

1587.6 2860.1 4019.0 5087.1 6203.7 7331.1 7404.9 6808.3 6248.0 5688.1 5128.9 4570.8 4014.5 3249.8 2011.7 856.3

916.6 1651.3 2241.4 2715.9 3252.1 3802.1 3623.3 3461.0 3351.3 3242.2 3134.0 3027.1 2922.6 2552.8 1580.2 713.0

Stable Stable Toppling Toppling Toppling Toppling Toppling Toppling Toppling Toppling Toppling Toppling Toppling Sliding Sliding Sliding

Acknowledgments The authors wish to acknowledge the individuals who helped with various aspects of the work presented in this paper: Prof. Stephen Evans of the Department of Earth and Environmental Sciences at the University of Waterloo for providing the initial motivation to develop this tool; Dr. Mikko Jyrkama at the Institute for Risk Research at the University of Waterloo for advice during the early stages of coding the analysis procedure, and Dr. Reginald Hammah at Rocscience Inc. for providing assistance with the code needed to sample a Fisher distribution. Furthermore, the authors would like to thank Prof. Robert Pine and an anonymous reviewer for their constructive comments, which improved this paper. Funding for this work was provided in part by the Natural Sciences and Engineering Research Council of Canada in the form of an Alexander Graham Bell Canada Graduate Scholarship held by B.S.A. Tatone.

Appendix A. ‘‘Analysis details’’ tabs of ROCKTOPPLE Figs. 18–20 illustrate the ‘‘Analysis Details’’ tabs of the ROCKTOPPPLE program. ‘‘Analysis Details 1’’ (Fig. 18)

summarizes the (x,y) coordinates defining the mean geometry of the slope, including the overall slope geometry, the geometry of each block, and the coordinates defining the water table. ‘‘Analysis Details 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 kinetic analysis. ‘‘Analysis details 3’’ provides details the kinetic analysis of the first kinematically feasible trial, including all randomly sampled input parameters, the resulting inter-block forces, the stability mode of each block comprising the slope, and a graphic illustrating the randomly generated geometry.

Appendix B. Detailed results of deterministic example Tables 5 and 6 provide detailed results of the deterministic example presented in Section 4. Table 5 illustrates the results as published in Wyllie and Mah (2004), while Table 6 shows the results obtained with ROCKTOPPLE. It is noted that instead of utilizing Eq. (1) to determine if a block’s centre of gravity lies outside of its base (as in Table 5), Table 6 presents the value of Xw, which describes the horizontal distance between a block’s centre

ARTICLE IN PRESS 114

B.S.A. Tatone, G. Grasselli / Computers & Geosciences 36 (2010) 98–114

of mass and its lowermost corner. Negative values of Xw indicate the centre of gravity lies outside the block’s base. References Baecher, G.B., Christian, J.T., 2003. Reliability and Statistics in Geotechnical Engineering. J. Wiley, Hoboken, NJ (605pp.). Carter, B.J., Lajtai, E.Z., 1992. Rock slope stability and distributed joint systems. Canadian Geotechnical Journal/Revue Canadienne de Geotechnique 29 (1), 53–60. Cruden, D.M., 1989. Limits to common toppling. Canadian Geotechnical Journal/ Revue Canadienne de Geotechnique 26 (4), 737–742. Duzgun, H.S.B., 2008. A quantitative risk assessment framework for rock slides. In: Yale, D., Holtz, S., Breeds, C., Ozbay, U. (Eds.), Proceedings of the 42nd US Rock Mechanics Symposium, San Francisco, CA, American Rock Mechanics Association, Alexandria, VA, Paper 08-279 (CD-ROM). Duzgun, H.S.B., Yucemen, M.S., Karpuz, C., 2003. A methodology for reliabilitybased design of rock slopes. Rock Mechanics and Rock Engineering 36 (2), 95–120, doi:10.1007/s00603-002-0034-0. El-Ramly, H., Morgenstern, N.R., Cruden, D.M., 2002. Probabilistic slope stability analysis for practice. Canadian Geotechnical Journal/Revue Canadienne de Geotechnique 39 (3), 665–683, doi:10.1139/t02-034. Fell, R., Ho, K.K.S., Lacasse, S., Leroi, E., 2005. A framework for landslide risk assessment and management. In: Unger, O., Fell, R., Couture, R., Eberhardt, E., (Eds.), Proceedings of the International Conference on Landslide Risk Management, Landslide Risk Management, Vancouver, Canada, Taylor and Francis, London, pp. 3–25. Feng, P., Lajtai, E.Z., 1998. Probabilistic treatment of the sliding wedge with EzSlide. Engineering Geology 50 (1–2), 153–163, doi:10.1016/S0013-7952(98)00007-6. Fisher, R., 1953. Dispersion on a sphere. Proceedings of the Royal Society of London, Series A: Mathematical and Physical Sciences 217 (1130), 295–305. GEO-SLOPE, 2007. SLOPE/W, Calgary, /http://www.geo-slope.com/products/slo pew2007.aspxS (Accessed September 22, 2008). Glynn, E.F., 1979. A probabilistic approach to the stability of rock slopes. Ph.D. Dissertation, Massachusetts Institute of Technology, Cambridge, MA, 463pp. Goodman, R.E., Bray, J.W., 1976. Toppling of rock slopes. In: Proceedings of the Specialty Conference on Rock Engineering for Foundations and Slopes, Boulder, CO, American Society of Civil Engineers, New York, NY, pp. 201–223. Ho, K.K.S., Leroi, E., Roberds, W.J., 2000. Quantitative risk assessment: application, myths and future direction. In: Proceeding of GeoEng 2000—An International Conference on Geotechnical and Geological Engineering, Melbourne, Technomic Publishing, Lancaster, PA, pp. 269–312. Major, G., Kim, H., Ross-Brown, D., 1977. Pit slope manual supplement 5-1-plane shear analysis. CANMET (Canada Centre for Mineral and Engineering Technology), Mines and Resources Canada, Ottawa, Canada, 307pp. McMahon, B.K., 1971. A statistical method for the design of rock slopes. In: Proceedings of the First Australia–New Zealand Conference on Geomechanics, Melbourne, Australia Institution of Engineers, Sydney, pp. 314–321. Morgenstern, N.R., 1997. Toward landslide risk assessment in practice. In: Cruden, D.M., Fell, R. (Eds.), Proceedings of the International Workshop on Landslide Risk Assessment, Honolulu, HI, A. A. Balkema, Rotterdam, pp. 15–24.

Muralha, J., 2003. Parameter variability in the toppling stability of rock blocks. In: Proceedings of the 10th Conference of the International Society for Rock Mechanic, Technology Road Map for Rock Mechanics, Johannesburg, South Africa, The South African Institute of Mining and Metallurgy, Marshalltown, pp. 849–854. Norrish, N.I., Wyllie, D.C., 1996. Rock slope stability analysis. In: Turner, A.K., Schuster, R.L. (Eds.), Landslides Investigation and Mitigation, Transportation Research Board Special Report 247, National Academy Press, Washington, DC, pp. 391–425. Park, H., West, T.R., 2001. Development of a probabilistic approach for rock wedge failure. Engineering Geology 59 (3–4), 233–251, doi:10.1016/S00137952(00)00076-4. Pine, R.J., Roberds, W.J., 2005. A risk-based approach for the design of rock slopes subject to multiple failure modes—illustrated by a case study in Hong Kong. International Journal of Rock Mechanics and Mining Sciences 42 (2), 261–275, doi:10.1016/j.ijrmms.2004.09.014. Piteau, D.R., Martin, D.C., 1977. Slope stability analysis and design based on probability techniques at Cassiar mine. CIM Bulletin 70 (779), 139–150. Priest, S.D., 1993. Discontinuity Analysis for Rock Engineering, first ed Chapman & Hall, New York, NY (473pp.). Quek, S.T., Leung, C.F., 1995. Reliability-based stability analysis of rock excavations. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts 32 (6), 617–620. RockWare, 2008. RockPack III, RockWare-Earth Science and GIS Software, Golden, CO, /http://www.rockware.com/product/overview.php?id=119S (accessed September 22, 2008). Rocscience, 2008a. ROCPLANE 2.0—planar sliding stability analysis for rock slopes, Rocscience Inc., Toronto, Canada, /http://www.rocscience.com/products/Roc Plane.aspS (accessed September 22, 2008). Rocscience, 2008b. SLIDE 5.0–2D limit equilibrium slope stability analysis, Rocscience Inc., Toronto, /http://www.rocscience.com/products/Slide.aspS (accessed September 22, 2008). Rocscience, 2008c. SWEDGE 5.0–3D surface wedge analysis for slopes, Rocscience Inc., Toronto, /http://www.rocscience.com/products/Swedge.aspS (accessed September 22, 2008). Scavia, C., Barla, G., Bernaudo, V., 1990. Probabilistic stability analysis of block toppling failure in rock slopes. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts 27 (6), 465–478. Wang, J., Tan, W., Feng, S., Zhou, R., 2000. Reliability analysis of an open pit coal mine slope. International Journal of Rock Mechanics and Mining Sciences 37 (4), 715–721. Wyllie, D.C., Wood, D.F., 1983. Stabilization of toppling rock slope failures. In: Hynes, J.L. (Ed.), Proceedings of the 33rd Highway Geology Symposium, Engineering Geology and Environmental Constraints, Vail, CO, Colorado Geological Survey, Denver, CO, pp. 103–115. Wyllie, D.C., 1999. Foundations on Rock, second ed E & FN Spon, New York, NY (401pp.). Wyllie, D.C., Mah, C.W., 2004. Rock Slope Engineering: Civil and Mining, fourth ed Spon Press, New York, NY (431pp.).