Rock slope stability and distributed joint systems

Rock slope stability and distributed joint systems 1

BRUCEJ. CARTER AND EMERY Z. LAJTAI

Can. Geotech. J. Downloaded from www.nrcresearchpress.com by Guangzhou Jinan University on 06/03/13 For personal use only.

Departments of Civil and Geological Engineering, The University of Manitoba, Winnipeg, Man., Canada R3 T 2N2 Received February 5, 1991 Accepted September 30, 1991 A deterministic (GEOSLIDE) and a probabilistic (PROSLIDE) microcomputer code are introduced to aid in performing rock wedge analyses based on the limit equilibrium method. The deterministic code evaluates the stability of a single rock wedge formed by discontinuities in rock through three-dimensional vector algebra. GEOSLIDE undertakes a full kinematic analysis (daylighting and obstruction), analyzes both wedge and plane sliding, and provides for anchor designs and sensitivity analyses (cohesion, friction, and water forces). Through multiple stability analyses, PROSLIDE evaluates the probability of failure for a rock slope by examining the distribution of the factors of safety from all the potential sliding wedges formed by the discontinuities of the rock mass. The probability of failure is expressed as the ratio of kinematically free wedges that have a factor of safety less than unity to the total number of wedges. PROSLIDE can form and analyze as many as 2000 different pairs of discontinuities in less than 30 min using a 25 MHz 486 IBMcompatible computer. In a worked example, the probability of failure for a fixed slope strike and loading condition is shown to vary with the slope angle, following the characteristic 'S' shape of a cumulative distribution function. The effect of an anchor force is to spread the distribution over a wider range of the factor of safety (SF), pushing many wedges into a potential upslide situation and splitting the distribution about the failure zone of the stability diagram (-1 < SF < 1). Key words: rock slope, rock wedge, stability analysis, factor of safety, probability of failure, Monte Carlo simulation. pour les micro-ordinateurs sont introduits Un code dkterministique (GEOSLIDE) et un code probabilistique (PROSLIDE) pour aider rCaliser des analyses de coins de roc basCes sur 1'Cquilibre limite. Le code dkterministique Cvalue la stabilitk d'un simple coin de roc, form6 par des discontinuitCs dans la roche, au moyen d'algkbre vectorielle tridimensionnelle. GEOSLIDE prockde 2 une analyse cinkmatique complkte (espace libre et obstruction), analyse le glissement du coin et du plan et fournit la possibilitC de calculer des ancrages et de faire des analyses de sensibiliti (cohision, friction et forces de pression d'eau). Au moyen d'analyses multiples de stabilitC, PROSLIDE Cvalue la probabilitk de rupture d'un talus rocheux en examinant la distribution des coefficients de sCcuritC de tous les coins formCs par des discontinuitCs du massif rocheux et pouvant potentiellement glisser. La probabilitk de rupture est exprimCe par le rapport des coins cinCmatiquement libres qui ont un coefficient de sCcuritC infkrieur 2 I'unitC, sur le nombre total de coins. PROSLIDE peut former et analyser jusqu'a 2000 paires differentes de discontinuitCs en moins de 30 min en utilisant un micro-ordinateur compatible a I'IBM 486 de 25 MHz. Dans un exemple de calcul, l'on montre que la probabilitC de rupture pour une direction de pente et une condition de chargement donnCes varie avec l'angle de la pente, se conformant a la forme en S caractkristique d'une fonction de distribution cumulative. L'effet de la force d'un ancrage est de rCpartir la distribution sur une plage plus large du coefficient de sCcuritC (SF), repoussant plusieurs coins dans une situation potentielle vers le haut du talus et coupant la distribution du diagramme de stabilitC autour de la zone de rupture (- 1 < SF > 1). Mots clPs : talus rocheux, coin de roc, analyse de stabilitC, coefficient de stabilitC, probabilitk de rupture, simulation de Monte Carlo. [Traduit par la redaction] Can. Geotech. J. 29, 53-60 (1992)

Introduction One of the most difficult jobs in rock slope engineering is selecting design data for a deterministic stability analysis from a site investigation report prepared by an engineering or structural geologist. The availability of a contoured stereonet plot of joints displaying hundreds of careful field measurements is not necessarily comforting, since the conventional design routine for wedge failure (Hoek and Bray 1977) requires only two planes of weakness. The median orientations of planes form the representative wedge, but perhaps the design should be for the worst condition. Alternatively, why not analyze all the possible wedges that could be formed by all possible combinations of the discontinuities in the rock mass? The latter technique falls into the general category of probabilistic design. Since the geological domain from which the design data must be selected is probabilistic and the geological data consequently widely distributed, a probabilistic Printcd in Canada / Imprime au Canada

approach to slope design is appropriate ( ~ o a t e s1977; Einstein and Baecher 1983). Probabilistic techniques require statistical assessments of multiple analyses. Such analyses can now be conducted relatively quickly using microcomputers. The purpose of this paper is (i) to introduce a microcomputer tool that performs multiple analyses and (ii) to show how this tool can solve problems associated with the design of rock slopes. Deterministic analyses of slope stability Rock slope failure may occur in a variety of ways, involving rigid body movement by translational sliding or rotation of single or multiple rock slices or wedges (Varnes 1978; Martin and Kaiser 1984) or through toppling (Goodman and Bray 1976; Aydan et al. 1989; Scavia et al. 1990). This paper is concerned with only one type of movement, translational wedge or planar sliding (Hoek and Bray 1977).


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The general deterministic method of limit equilibrium analysis for wedge or planar stability dates back to the 1970s (Jaeger 1971; Kutter 1974), and recent techniques are based predominantly on the work of Hoek and Bray (1977) and Goodman (1976). A typical slope-stability analysis consists of collecting structural data, selecting the representative wedge, and a kinematic analysis of this wedge for daylighting and obstruction. The loads are then introduced, i.e., gravity, water forces, earthquake loading, surcharge, and other external forces, and the factor of safety for the most likely or critical sliding mode, either planar or wedge slide along dip vectors or along the line of intersection for wedge slide, is computed by limit equilibrium methods. Sensitivity analyses evaluating the influence of such parameters as cohesion, friction, and water condition may lead to redesign by anchoring or by changing the geometry or loading condition. The final result of the deterministic analysis is a single factor of safety, the ratio of the available resistance and the driving force (the component of the total force that points in the sliding direction). Statistical analyses of slope stability In rock mechanics, practically all the parameters are statistical; the joint set characteristics (Call et al. 1976; Glynn and Einstein 1979), the rock strength properties (Chowdhury 1986), and the loading conditions (Glass et al. 1978; Tao and Hong 1987). Since the introduction of probabilistic techniques for rock slope stability (McMahon 1971), site-specific applications have been published (Piteau and Martin 1977; Savely 1985). In any analysis of rock stability, the first step is always the collection of field data on joint orientation. The rock mass is usually divided into structural domains and within these the discontinuities are separated into sets, with several sets forming a system. The statistical distributions of the orientations and possibly the strength-related structural data can then be established for each set (Coates 1977; Herget 1978; Savely 1985). The interpretation of the field data is not necessarily a simple process. For instance, joint orientation is subject to error and bias (Barton 1978; Herget 1978; Baecher and Einstein 1981). However, the bias in orientation and in other parameters, such as trace length, persistence, and spacing (Baecher and Lanney 1978), can be treated quantitatively to correct for errors in measuring (Terzaghi 1965; Wathugala et al. 1990). Once the correct parameter distributions are known, stability analyses may proceed in several ways: (i) conventional limit equilibrium analysis using a subjectively selected critical wedge (Hoek and Bray 1977); (ii) entirely probabilistic techniques, either Monte Carlo simulations (Einstein et al. 1983; Scavia et al. 1990) or reliability index methods (Nguyen 1985; Ramachandran and Hosking 1985; Barbosa et al. 1989); or (iii) a combination of deterministic and probabilistic techniques (Piteau et al. 1985; Martin et al. 1986). Neither the deterministic nor the probabilistic methods are entirely complete by themselves, as the concepts of cost and risk assessment (Coates 1977; Call 1985; Kirsten and Moss 1985; Einstein 1988) also merit consideration. Probabilistic analyses can take many forms. A truly probabilistic method should accept statistical distributions (corrected for bias) for every parameter that is subject to spatial variability. Monte Carlo simulations can be performed using random values of the parameter distributions (Coates 1977).

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If any of the parameters are correlated, however, the Monte Carlo solution may give incorrect'resultS, in which case the proper solution lies in the reliability index or first-order second moment (level 11) method (Glynn and Ghosh 1982; Nguyen 1985). An additional advantage of this method over Monte Carlo simulations is a reduction in computer time (Barbosa et al. 1989). The Monte Carlo simulation method, however, is very easily adapted to existing limit equilibrium computer codes. All Monte Carlo simulations produce not a single factor of safety, but a distribution of values. The probability of failure may then be defined by dividing the number of values less than one by the total number (Savely 1985; Dershowitz and Einstein 1984). Alternatively, the probability of failure can be found from the fitted normal probability density distribution diagram by calculating the area under the curve for factors of safety less than unity (Savely 1985). Software description The stability analysis codes described in this paper have been developed to enable deterministic wedge or plane sliding using GEOSLIDE and a probabilistic analysis with PROSLIDE. GEOSLIDE is an interactive, computer implementation of the conventional, vector-based, wedge analysis method, and it includes numerous graphical displays of the stereonet, wedge, and force vectors. It has been designed to analyze a single wedge or plane geometry. PROSLIDE is a multiple-analysis version of GEOSLIDE and will run unattended until either a prescribed number or all the possible combinations of discontinuities have been examined. Both codes were written in True BASIC and will run on any type of IBM PC or compatible equipped with a graphics card and monitor. The codes were tested with monochrome and color monitors, on PC, XT, AT-286, AT-386, AT-486, and PS/2 computers. For a complete slope-stability analysis using the codes developed by the authors, three different programs may be required: GEOSLIDE, the solution for the sliding wedge or plane; PROSLIDE, the discontinuity analysis code (Monte Carlo simulation); and some type of a statistics-graphics package. Another routine, DISIT,also written by the authors, can be used to turn the single-variable True BASIC record files of PROSLIDE into histograms and cumulative distributions or to fit Gaussian and Weibull distributions to the factors of safety. DISIT can only do graphics screen dumps. For professional-quality figures, a commercial software that does x-y plots and can recover the ASCII (text) files of PROSLIDE would be a better choice. There have been a number of other computer codes developed to analyze rock slope stability (ISRM 1988; Gibbs 1989), some of which include probabilistic analyses. Other programs that have been referenced in papers include FRACWEDG (Kendorski and Bindokas 1987) and an unnamed Fortran code written for the PC (Ghosh and Haupt 1989). Most of the codes have quite similar descriptions, although actual similarities and differences can only be found by using all of the codes. A comparison of these codes with each other and GEOSLIDE-PROSLIDE is beyond the scope of this paper, however. GEOSLIDE and PROSLIDE were originally developed as learning tools for the undergraduate rock mechanics students. This is the main reason for the two-part program, as the actual mechanics of wedge and planar sliding are the

CARTER AND LAJTAI


main focus, whereas statistics are not taught in detail. Because the programs are learning tools, there are a great number of explanatory screens showing intermediate results as the analyses proceed. Both GEOSLIDE and PROSLIDE have been run successfully for more than a year by both graduate and undergraduate rock mechanics students at the authors' institution. The codes have been verified using the examples of Hoek and Bray (1977). PROSLIDE, GEOSLIDE, and DISIT may be obtained from the authors at the cost of reproduction and postage. A manual may not be necessary but it can also be supplied at extra cost.

Deterministic analysis using GEOSLIDE The sliding wedge problem of the Hoek and Bray (1977) slope-stability analysis is implemented by this code. The wedge is assumed to be continuous and rigid. The driving and resisting forces are calculated as vectors, and the factor of safety is found using three-dimensional vector algebra. The code works with a single set of geometry and loading, although a sensitivity-analysis option allows the loading and strength parameters to be varied once the initial factor of safety is calculated. The numerator in the factor of safety term is the sum of the total cohesive and frictional resisting forces. The denominator is the sliding-direction-parallel component of the resultant force which includes weight, surcharge, water, earthquake, external, and anchor forces. The sliding-direction-parallel component is positive when it points downslope, negative when it points upslope. A negative factor of safety signals a potential upslide (a safety factor between 0 and - 1 is unstable and will slide upwards). The program flow is shown in Fig. 1. The third step, kinematic analysis, consists of two parts: a check for daylighting of the line of intersection and the two dip vectors (Hoek and Bray 1977) and an obstruction analysis (Goodman 1976) if either of the two dip vectors daylights. This process eliminates those dip slides (wedge sliding along the dip vector of plane A or B) where the other inactive plane would block the sliding motion. The daylighting and obstruction analyses are accompanied by explanatory graphic displays. If both the daylighting and the obstruction conditions are passed, i.e., sliding is kinematically possible, the stability analysis begins. The wedge is displayed in vector format and in a threedimensional view which can be rotated. The shape of the wedge can be modified by truncating in the lateral direction and by inserting a tension cutoff at the back of the wedge. The code then accepts a series of loading conditions: external forces (anchors, earthquake loading, etc.) and several options for the water condition. For every wedge, five different sliding modes can be examined: (i) wedge sliding along the line of intersection between plane A and plane B; (ii) wedge sliding along the dip vector of plane A; (iii) wedge sliding along the dip vector for plane B; (iv) plane sliding on plane A; and (v) plane sliding on plane B. The corresponding factor of safety is calculated for each active sliding direction. The last two (plane slide) cases represent unobstructed sliding of a block of rock along the dip vector of either joint plane. The influence of the loading condition and the strength parameters can be evaluated through the sensitivity subroutines within GEOSLIDE. All the forces, including the water

NO SLIDING POSSIBLE

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DISPLAY WEDGE

k TRUNCATE EDGES

I CALCULATE FACTOR O F SAFETY

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