probabilistic slope analysis for practice

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Probabilistic slope stability analysis for practice

H. El-Ramly, N.R. Morgenstern, and D.M. Cruden

Abstract: The impact of uncertainty on the reliability of slope design and performance assessment is often significant.

Conventional slope practice based on the factor of safety cannot explicitly address uncertainty, thus compromising the

adequacy of projections. Probabilistic techniques are rational means to quantify and incorporate uncertainty into slope

analysis and design. A spreadsheet approach for probabilistic slope stability analysis is developed. The methodology is

based on Monte Carlo simulation using the familiar and readily available software, Microsoft® Excel 97 and @Risk.

The analysis accounts for the spatial variability of the input variables, the statistical uncertainty due to limited data,

and biases in the empirical factors and correlations used. The approach is simple and can be applied in practice with

little effort beyond that needed in a conventional analysis. The methodology is illustrated by a probabilistic slope

analysis of the dykes of the James Bay hydroelectric project. The results are compared with those obtained using the

first-order second-moment method, and the practical insights gained through the analysis are highlighted. The

deficiencies of a simpler probabilistic analysis are illustrated.

Key words: probabilistic analysis, slope stability, Monte Carlo simulation, spatial variability.

Résumé : L'impact de l'incertitude sur la fiabilité de la conception et de l'évaluation de la performance des talus estsouvent significative. La pratique conventionnelle des talus basée sur le coefficient de sécurité ne peut pas traiter de fa-

çon explicite l'incertitude, compromettant ainsi la précision des projections. Les techniques probabilistes constituent des

moyens rationnels pour quantifier et incorporer l'incertitude dans la conception et l'analyse des talus. On développe une

approche de tableur pour l'analyse de la stabilité des talus. La méthodologie est basée sur la simulation de Monte Carlo

au moyen de logiciels familiers et facilement accessibles, Microsoft® Excel 97 et @Risk. L'analyse tient compte de la

variabilité spatiale des variables entrées, de l'incertitude statistique due aux données limitées et aux distorsions des fac-

teurs empiriques et des corrélations utilisés. L'approche est simple et peut être mise ne pratique avec peu d'effort en

plus de celui requis par une analyse conventionnelle. La méthodologie est illustrée par une analyse probabiliste des ta-

lus des digues du projet hydroélectrique de la Baie James. Les résultats sont comparés avec ceux obtenus en utilisant

la méthode du second moment de premier ordre, et les éclaircissements pratiques obtenus par cette analyse sont mis en

lumière. On illustre les déficiences d'une analyse probabiliste plus simple.

Mots clés : analyse probabiliste, stabilité de talus, simulation de Monte Carlo, variabilité spatiale.

[Traduit par la Rédaction] El-Ramly et al. 683

Introduction

Slope engineering is perhaps the geotechnical subject mostdominated by uncertainty. Geological anomalies, inherentspatial variability of soil properties, scarcity of representativedata, changing environmental conditions, unexpected failuremechanisms, simplifications and approximations adopted ingeotechnical models, and human mistakes in design and con-struction are all factors contributing to uncertainty. Conven-tional deterministic slope analysis does not account forquantified uncertainty in an explicit manner and relies on con-

servative parameters and designs to deal with uncertain condi-tions. The impact of such subjective conservatism cannot beevaluated, and past experience shows that apparently conser-vative designs are not always safe against failure.

The evaluation of the role of uncertainty necessarily re-quires the implementation of probability concepts and meth-

ods. Probabilistic analyses allow uncertainty to be quantifiedand incorporated rationally into the design process. Probabil-istic slope stability analysis (PSSA) was first introduced intoslope engineering in the 1970s (Alonso 1976; Tang et al.1976; Harr 1977). Over the last three decades, the conceptsand principles of PSSA have developed and are now well es-tablished in the literature. In addition to accounting for un-certainty, PSSA is also a useful approach for estimatinghazard frequency for site-specific quantitative risk analyses,particularly in the absence of representative empirical data.

The merits of probabilistic analyses have long been noted

(Chowdhury 1984; Whitman 1984; Wolff 1996; Christian1996). Despite the uncertainties involved in slope problemsand notwithstanding the benefits gained from a PSSA, theprofession has been slow in adopting such techniques. Thereluctance of practicing engineers to apply probabilisticmethods is attributed to four factors. First, engineers’ train-

Can. Geotech. J. 39: 665–683 (2002) DOI: 10.1139/T02-034 © 2002 NRC Canada

665

Received 22 May 2001. Accepted 7 December 2001. Published on the NRC Research Press Web site at http://cgj.nrc.ca on 16 May 2002.

H. El-Ramly.1 AMEC Earth and Environmental, 4810-93 Street, Edmonton, AB T6E 5M4, Canada.N.R. Morgenstern and D.M. Cruden. Geotechnical and Geoenvironmental Group, Department of Civil and EnvironmentalEngineering, University of Alberta, Edmonton, AB T6G 2G7, Canada.

1Corresponding author (e-mail: [email protected]).


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ing in statistics and probability theory is often limited to ba-sic information during their early years of education. Hence,they are less comfortable dealing with probabilities thanthey are with deterministic factors of safety. Second, there isa common misconception that probabilistic analyses requiresignificantly more data, time, and effort than deterministicanalyses. Third, few published studies illustrate the imple-

mentation and benefits of probabilistic analyses. Lastly, ac-ceptable probabilities of unsatisfactory performance (orfailure probability) are ill-defined, and the link between aprobabilistic assessment and a conventional deterministic as-sessment is absent. This creates difficulties in comprehend-ing the results of a probabilistic analysis. All of these issuesare addressed in detail in El-Ramly (2001).

This paper addresses the integration of probabilistic meth-ods into geotechnical practice, focusing on the first three ob-stacles of the factors listed in the previous paragraph. First,an overview of some of the concepts involved in quantifyinguncertainty of soil properties is presented. The paper thendescribes the development of a practical methodology forprobabilistic slope analysis based on Monte Carlo simula-

tion. The methodology makes use of familiar and readilyavailable software, namely Microsoft® Excel 97 and @Risk.The implementation of the methodology is illustratedthrough a probabilistic analysis of the stability of the JamesBay dykes. Lastly, the results are compared with those of thefirst-order second-moment (FOSM) method and the practicalinsights gained through the analysis are highlighted.

Statistical analysis of soil data

Components of statistical analysisStatistical analysis of soil data comprises two main stages:

data review and statistical inference. Data review is largely judgmental and encompasses a broad range of issues. First,

the consistency of the data, known as the condition of stationarity, should be ensured. Inconsistency can arise frompooling data belonging to different soil types, stress condi-tions, testing methods, stress history, or patterns of sampledisturbance (Lacasse and Nadim 1996). The histogram is aconvenient tool in this regard. A multimodal histogram is anindication of inconsistent data, or a nonstationary condition.Data review also aims at identifying trends in the data, bi-ases in measurements, errors and discrepancies in the re-sults, and outlier data values. Decisions should be madewhether to reject outliers or to accept them as extreme val-ues. Baecher (1987) warned that care should be exercised inthis process to avoid rejecting true, important information.Another concern is any clustering of data in a zone within

the spatial domain of interest which may render the data un-representative.

Statistical inferences are commonly based on a simplifiedmodel (Vanmarcke 1977a; DeGroot and Baecher 1993) whichdivides the quantity, x i, at any location, i, into a deterministictrend component, t i, and a residual component, ε i, where

[1] xi = t i + ε i

The trend is evaluated deterministically using regressiontechniques and is usually a function of location. The residualcomponent is a random variable whose probability densityfunction (PDF) has a zero mean and a constant variance,

σ2 [ε], independent of location and estimated from the scatterof observations around the trend. The residual component atlocation i is spatially correlated with the residual compo-nents at surrounding locations.

Uncertainty in trendSite investigation programs are usually controlled by bud-

get, which limits the number of tests performed. Since thetrend function is estimated from the available sample of ob-servations, it is uncertain. The uncertainty due to the samplesize (number of observations) is known as statistical uncer-tainty and is typically quantified by considering the regres-sion coefficients of the trend function as variables withmeans and non-zero variances, which are inversely propor-tional to the number of observations, n.

The method of least squares is the most common tech-nique for estimating the means and variances of trend-function coefficients. Neter et al. (1990) discussed estimat-ing the uncertainties of regression coefficients for lineartrends. Li (1991) pointed out some limitations of the methodof least squares. Baecher (1987) recommended that trend

equations be linear, as the higher the order of the trend func-tion, the more uncertain the regression coefficients. Wherethe soil properties do not exhibit a clear trend, a constantequal to the mean of the observations is assumed.

Bias is another source of uncertainty that could affect theestimated trend. The measured soil property can be consis-tently overestimated or underestimated at all locations. Fac-tors such as the testing device used, boundary conditions,soil disturbance, or the models and correlations used to in-terpret the measurements may contribute to biases. As an ex-ample, Bjerrum (1972) noted that the field vane consistentlyoverestimated the undrained shear strength of highly plasticclays. An empirical correction factor is used to adjust the bi-ased measurements. The additional uncertainty introduced

by the use of Bjerrum’s vane correction factor is, however,significant. This could also be the case with other empiricalfactors.

Spatial variabilitySoil composition and properties vary from one location to

another, even within homogeneous layers. The variability isattributed to factors such as variations in mineralogical com-position, conditions during deposition, stress history, andphysical and mechanical decomposition processes (Lacasseand Nadim 1996). The spatial variability of soil properties isa major source of uncertainty.

Spatial variability is not a random process, rather it iscontrolled by location in space. Statistical parameters suchas the mean and variance are one-point statistical parametersand cannot capture the features of the spatial structure of thesoil (Fig. 1). The plot in Fig. 1 compares the spatial variabil-ity of two artificial sets of data generated using the geo-statistics software GSLIB (Deutsch and Journel 1998). Bothsets have similar histograms. The upper plot is highly erraticand the data are almost uncorrelated, whereas the lower plotis characterized by high spatial continuity. Additional statis-tical tools, such as the autocovariance and semivariogram,are essential to describe spatial variability. The pattern of soil variability is characterized by the autocorrelation dis-tance (or scale of fluctuation; Vanmarcke 1977a). A large

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autocorrelation distance reflects the spatial continuity shownin the lower plot of Fig. 1, whereas a short distance reflectserratic variability (Fig. 1, upper plot). DeGroot (1996) andLacasse and Nadim (1996) illustrated the estimation of auto-correlation distance.

Spatial averagingThe performance of a structure is often controlled by the

average soil properties within a zone of influence, rather thansoil properties at discrete locations. Slope failure is morelikely to occur when the average shear strength along the fail-ure surface is insufficient rather than due to the presence of some local weak pockets. The uncertainty of the averageshear strength along the slip surface, not the point strength, istherefore a more accurate measure of uncertainty. Baecher(1987) warned, however, that depending on performancemode, average properties are not always the controlling factor.Internal erosion in dams and progressive failure are examplesof cases where extreme values control performance.

The variance of the strength spatially averaged over somesurface is less than the variance of point measurements. As

the extent of the domain over which the soil property isbeing averaged increases, the variance decreases. For quanti-ties averaging linearly, the amount of variance reduction de-pends on the autocorrelation distance. The mean, however,remains almost unchanged. The numerical data in Fig. 2 (up-per right plot) are discretized based on grids of sizes 1 × 1,5 × 5, and 10 × 10. For each scheme, the averages withingrid squares are calculated and the histogram of the local av-erages is plotted as shown in Fig. 2. As the averaging area isincreased, the coefficient of variation of the local averagesdrops from 1.55 to 0.28 and the minimum and maximumvalues become closer to the mean. If the averaging area isconstant, the amount of variance reduction increases as theautocorrelation distance decreases, i.e., soil variability be-comes more erratic.

Theory of random fieldsThe theory of random fields (Vanmarcke 1977a, 1977b,

1983) is a common approach for modeling spatial variabilityof soil properties. It is also the basis of our probabilisticslope analysis methodology; to illustrate its rationale, some

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El-Ramly et al. 667

Fig. 1. A highly erratic spatial structure (upper right) and a highly continuous structure (lower right), both with similar histograms.


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of the basic concepts of the theory of random fields are pre-sented here.

At any location within a soil layer, a soil parameter is anuncertain quantity, or a random variable, unless it is mea-sured at that particular location. Each random variable ischaracterized by a probability distribution and is correlatedwith the random variables at adjacent locations. The set of random variables at all locations within the layer is referredto as a random field and is characterized by the joint proba-bility distribution of all random variables. A random field iscalled stationary (or homogeneous) if the joint probabilitydistribution that governs the field is invariant when trans-lated over the parameter space. This implies that the cumula-tive probability distribution function, the mean, and thevariance are constant for any location within the randomfield and that the covariance of two random variables de-

pends on their relative locations, regardless of their absolutelocations within the random field.

The point to point variation of a random field is very diffi-cult to obtain in practice and is often of no real interest

(Vanmarcke 1983). Local averages over a spatial or temporallocal domain are of much greater value. For example, thehourly or daily rainfall is of interest to hydrologists ratherthan the instantaneous rate of fall. Similarly, the averageshear strength of a soil over the area of the slip surface is of more interest to geotechnical engineers than the variation of strength from point to point within the layer.

Figure 3 shows a one-dimensional (1D) stationary randomfield of a variable, x, with a mean, E [ x], a variance, σ2 , anda cumulative probability distribution function, F ( x); thegraph could also portray the variation along a line in the pa-rameter space of a homogeneous k -dimensional random

© 2002 NRC Canada


Fig. 2. Variance reduction due to spatial averaging over blocks of sizes 1 × 1, 5 × 5, and 10 × 10.


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field. For this type of random field, Vanmarcke (1977a) de-fined the dimensionless variance function Γ (∆ z) as

[2] Γ ∆

∆

( ) z

z

= σσ

2

2

where σ∆ z2 is the variance of the local average, X (∆ z), of thevariable, x, over an interval of length ∆ z. The variance func-tion is a measure of the reduction in the point variance, σ 2 ,under local averaging. For most commonly used correlationfunctions, Vanmarcke (1983) showed that Γ (∆ z) can be ap-proximated by

[3] Γ ∆∆

∆ ∆( ) z

z

z z

=≤

≥

1 δδ δ

where δ is the scale of fluctuation. In concept, the scale of fluctuation has the same meaning as the autocorrelation dis-

tance but differs in numeric value. For the common expo-nential and Gaussian autocorrelation models, δ is 2.0 and1.77 times the autocorrelation distance, r o, respectively.

The model indicates no variance reduction, Γ (∆ z) = 1, dueto local averaging up to an averaging interval ∆ z equal to δ.The rationale of this approximation is that modeling a ran-dom phenomenon at a level of aggregation more detailedthan the way the information about the phenomenon is ac-quired or processed is impractical and unnecessary(Vanmarcke 1983). In site investigation programs, the spac-ing or the time interval between observations is often large.Characterizing the correlation structure at small intervals be-comes unreliable and justifies the approximation by a perfectautocorrelation.

Local averages of the variable x over intervals ∆ z and ∆ z′(Fig. 3), X (∆ z) and X ′(∆ z′), are spatially correlated. The cor-relation coefficient, ρ( X ∆ z, X z∆ ′′ ), between X (∆ z) and X ′(∆ z′)is given by eq. [4] (Vanmarcke 1983). It is a function of thelengths of the two intervals, the separation, Z o, betweenthem (Fig. 3), and the variance function of the variable beingaveraged:

[4] ρ( , ) X X z z∆ ∆ ′′ =

Z Z Z Z Z Z Z Z

z z z

o o 12

1 22

2 32

32

2

Γ Γ Γ Γ ∆ ∆ Γ ∆ Γ ∆

( ) ( ) ( ) ( )

[ ( ) (

− + −′ z′ )]0.5

where Z 1 is the distance from the beginning of the first inter-val to the beginning of the second interval, Z 2 is the distancefrom the beginning of the first interval to the end of the sec-ond interval, and Z 3 is the distance from the end of the first

interval to the end of the second interval.

Random testing errorRandom testing errors arise from factors related to the

measuring process (other than bias), such as operator erroror a faulty device. They can be directly measured onlythrough repeated testing of the same specimen by differentoperators and devices. This is not practical, as most geo-technical field and laboratory tests disturb the sample, sorandom measurement errors cannot be known. In data analy-sis, measurement error is regarded as a random variable witha zero mean and a constant variance, referred to as randomerror variance. The uncertainty due to testing errors is not atrue variation in the soil property and should be discarded in

evaluating parameter variability. Approximate analytical pro-cedures (Baecher 1987) are used to estimate the random er-ror variance. Jaksa et al. (1997), however, pointed toimportant inaccuracies associated with such procedures.

Reliable estimation of random error variance requires dataat very small separation distances (r ≈ 0), not available inpractice. As a result, the distinction between noise, or ran-dom error, and real, short-scale variability (over distancesless than data spacing) is not possible. Hence, the analyti-cally computed variance is composed of random error vari-ance and short-scale variability. Jaksa et al. (1997) alsoshowed that the computed value of the random error vari-ance is sensitive to data spacing. Kulhawy and Trautmann(1996) commented that quantitative assessment of testing er-

rors of in situ measurements is rare because of the difficul-ties in separating natural spatial variability from testuncertainty. So, the reliability of the analytical estimation of random error variance is in question. Random testing errorsare best addressed through standardization of testing equip-ment and procedures, proper personnel training, and contin-uous inspection.

Probabilistic slope analysis methods

Probabilistic procedures for slope stability analysis varyin assumptions, limitations, capability to handle complex

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Fig. 3. A realization of a 1D random field of a variable x with a mean E [ x], variance σ2, and cumulative probability distributionfunction F ( x), showing local averages over intervals ∆ z and ∆ z′.


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problems, and mathematical complexity. Most of them,however, fall into one of two categories: approximate meth-ods (the FOSM method, the point estimate method), andMonte Carlo simulation.

Approximate methods make simplifying assumptions thatoften limit their application to specific classes of problems.For example, some used very simple slope models such as

the ordinary method of slices (Tang et al. 1976; Vanmarcke1980; Honjo and Kuroda 1991; Bergado et al. 1994). Othersdealt only with frictionless soils (Matsuo and Kuroda 1974;Vanmarcke 1977b; Bergado et al. 1994). A restriction to acircular (or cylindrical) slip surface is common in manystudies (Alonso 1976; Vanmarcke 1977b; Yucemen and Al-Homoud 1990; Bergado et al. 1994). The spatial variabilityof soil properties and pore-water pressure is often ignored,assuming perfect autocorrelation (Nguyen and Chowdhury1984; Wolff and Harr 1987; Duncan 2000).

Approximate methods allow the estimation of the meanand variance of the factor of safety but do not provide anyinformation about the shape of the probability density func-tion, so the failure probability can only be obtained by as-

suming a parametric probability distribution of the factor of safety (typically normal or log-normal). Estimates of lowprobabilities, required for safe structures, are sensitive to theassumed distribution.

Few studies used Monte Carlo simulation in slope analysiswhen compared to approximate methods. The extensive com-putational efforts involved in the simulations required research-ers to develop their own software to solve slope stabilityproblems. Computer times needed for the simulations werealso significant. As a result, a Monte Carlo simulation was con-sidered, until recently, to be uneconomic (Tobutt 1982; Priestand Brown 1983). Studies applying Monte Carlo simulationalso rarely addressed the spatial variability of soil properties(Major et al. 1978; Tobutt 1982; Nguyen and Chowdhury

1985) because of difficulties in generating random values inways that preserved their correlations. Today, rapid develop-ments in software are significantly changing this situation.

The limitations, and sometimes the complexities, of probabilistic methods combined with the poor training of most engineers in statistics and probability theory have sub-stantially inhibited the adoption of probabilistic slope stabil-ity analysis in practice. To facilitate the integration of probabilistic methods into practice, the methodologies andtechniques, while being consistent with principles of logicand mechanics, should be simple, capable of solving realslope problems, and in formats familiar to engineers.

An integrated probabilistic slope analysismethodology

Outline of the methodologyThe probabilistic slope analysis methodology based on

Monte Carlo simulation developed here has the advantagesof being simple and not requiring a comprehensive statisticaland mathematical background. The methodology is spread-sheet based and makes use of the familiar and readily avail-able Microsoft® Excel 97 (Microsoft Corporation 1997) and@Risk (Palisade Corporation 1996) software. Other similarsoftware, e.g., Lotus 1-2-3 and Crystal Ball (DecisioneeringInc. 1996), is likely capable of undertaking the task.

The slope problem (geometry, stratigraphy, soil properties,and slip surface) and the selected method of analysis are firstmodeled in a Microsoft® Excel 97 spreadsheet. Availabledata are examined and uncertainties in input parameters areidentified and described statistically by representative proba-bility distributions. Only those parameters whose uncertain-ties are deemed significant to the analysis need to be treated

as variables. A Monte Carlo simulation is then performed asillustrated schematically in Fig. 4. @Risk draws at random avalue for each input variable from within the defined proba-bility distributions. These values (input set 1, Fig. 4) areused to solve the spreadsheet and calculate the correspond-ing factor of safety. The process is repeated sufficient times,m, to estimate the statistical distribution of the factor of safety. Statistical analysis of this distribution allows estimat-ing the mean and variance of the factor of safety and theprobability of the factor of safety being less than one. Theprocedure is applicable to any method of limit equilibriumanalysis that can be represented in a spreadsheet.

Statistical characterization of input variables

Probability distributionsDifferent approaches can be adopted to estimate the prob-

ability distribution of each input variable. Where there areadequate amounts of data, the cumulative distribution func-tion (CDF) of the measurements can be used directly in thesimulation process. In geotechnical practice, the observedCDF (or histogram) may show spikes that would not appearwere more observations available. The CDF is better ob-tained by resetting the probability associated with each ob-servation to the average of its cumulative probability andthat associated with the next lower observation (Deutsch andJournel 1998). This procedure smooths unwanted spikes andallows assigning finite probabilities for values of the input

parameter less than the minimum value measured and morethan the maximum value.

Where observations are scarce or absent, parametric dis-tributions can be assumed from the literature. Studies haveestimated coefficients of variation and probability densityfunctions of soil properties (Lumb 1966; Chowdhury 1984;Harr 1987; Kulhawy et al. 1991; Lacasse and Nadim 1996).Care should be exercised, however, to ensure that the mini-mum and maximum values of the selected distribution areconsistent with the physical limits of the parameter beingmodeled. For example, shear strength parameters should nottake negative values. If the selected distribution implies nega-tive values, then the distribution is truncated at a practicalminimum threshold. Alternatively, variability can be assumed

to follow a triangular distribution with estimates of minimum,maximum, and most likely values based on expert opinion.

@Risk built-in functions allow great flexibility in model-ing input variables. Parametric and nonparametric distribu-tions (e.g., experimental CDF) can be modeled using @Risk functions. Desired truncations can be easily imposed on anydistribution either through @Risk or using Microsoft® Ex-cel 97 functions.

Modeling parameter uncertaintyThe probabilistic slope analysis methodology accounts for

the uncertainty of the input parameters based on the conceptsdiscussed in previous sections and the following equation:

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[5] V = B(t + ε)

where V is the input variable adjusted for statistical errorsand bias, B is a bias correction factor, t is the trend compo-nent, and ε is the residual component. The statistical uncer-tainty in the trend is estimated by standard statisticalmethods (Ang and Tang 1975; Neter et al. 1990). The vari-ance of the residual component is equal to the variance of the measured data, with no explicit consideration of the un-certainty due to random testing errors for two reasons. Esti-mates of random error variance are unreliable, as discussedin an earlier section, so any reduction of the observed vari-ance could be unsafe. In addition, random errors fluctuate inmagnitude above and below zero and are uncorrelated spa-tially. By taking the spatial average of the input variable over

the whole area of interest, such as the slip surface, positiveand negative random errors at the different locations withinthe averaging domain tend to cancel out. As a result, the ran-dom error variance associated with the averaged quantity islargely reduced. It is emphasized, however, that a criticalreview of the observations should ensure that appropriatetesting standards and procedures are followed. The bias cor-rection factor is also considered a variable with a mean andvariance evaluated by experience and comparison with fieldperformance or by observations from a more accurate testingprocedure. It does not have to be a multiplier as in eq. [5]; itcould be an addition [V = B + (t + ε)].

The spatial variability of an input variable is representedby the correlation structure of the residual component, ε, andmodeled using the theory of random fields. The spatial vari-ability of ε along the slip surface is approximated by a 1Dstationary random field. Vanmarcke’s (1983) approximatemodel (eq. [3]) is adopted for the analysis of the randomfield. The point to point variability of the residual compo-nent along the slip surface is resembled by the variability of its local averages over segments of the surface. The portionof the slip surface within the subject layer is divided intosegments of length l not exceeding the scale of fluctuation δ.The local average of the residual component over the lengthof any of these segments, ε(l), is considered a variable. TheCDF of the variable ε(l) is the same distribution function of the residual component, F (ε), with no variance reduction.

This is because the variance function Γ (l), eq. [3], equalsone for l ≤ δ. The correlation coefficients between the vari-ables representing the local averages over different segmentsof the slip surface are estimated using eq. [4]. Choosing thelength of segments equal to δ eliminates the correlation coef-ficients between most of the variables and greatly simplifiesthe simulation. Figure 5 is a schematic illustration of themodel, where ρ is the correlation coefficient between vari-ables.

Critical slip surface

For any slope, there is an unlimited number of potentialslip surfaces. The slope may fail, or perform unsatisfactorily,along any of these surfaces. The total probability of unsatis-

factory performance of a slope is, thus, the joint probabilityof failure occurring along the admissible slip surfaces. Thesesurfaces, however, are highly correlated, since they are allanalyzed using the same input variables and the same analyt-ical model. Furthermore, the close proximity of many of them add an element of spatial correlation. Evaluating thetotal probability of unsatisfactory performance of the slopeis a mathematically formidable task.

The strong correlation between slip surfaces, however,significantly reduces the difference between the total proba-bility of unsatisfactory performance and that of the mostcritical surface (Mostyn and Li 1993). Hence, the probabilityestimate from the most critical slip surface is considered areasonable estimate of slope reliability (Vanmarcke 1977b;

Alonso 1976; Yucemen and Al-Homoud 1990). In this study,the probability of unsatisfactory performance is obtained byindependently analyzing a number of predetermined slip sur-faces. The highest estimated probability value is consideredrepresentative of the total probability of unsatisfactory per-formance of the slope. The question, however, is which sur-faces to consider?

Chowdhury and Tang (1987) and Hassan and Wolff (1999)indicated that the deterministic critical slip surface is not al-ways the most critical surface in a probabilistic analysis.Hassan and Wolff (1999) proposed a search algorithm for lo-cating the slip circle with the minimum reliability index. Itsearches for the slip surface with the longest length within

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Fig. 4. Monte Carlo simulation procedure using Microsoft® Excel 97 and @Risk software.


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the soil layer whose uncertainty contributes the most to theuncertainty of the factor of safety. Their algorithm, however,does not directly account for the spatial variability of soilproperties and pore-water pressure. In locating the probabil-

istic critical slip surface, variance reduction due to spatialaveraging of soil parameters over the length of the slip sur-face has to be considered. The reduction depends on theautocorrelation distance and the length of slip surface, whichis not known beforehand. As a result, an increase in the vari-ance of the factor of safety, due to a longer portion within ahighly uncertain layer, could be offset by a variance reduc-tion due to spatial averaging. Hence, in slopes dominated byuncertainty due to spatial variability, the slip surface basedon the Hassan and Wolff algorithm may not be the most crit-ical surface in a probabilistic analysis.

An essential part of the probabilistic analysis is, thus, toconsider all slip surfaces that may be hazardous. These mayinclude the deterministic critical slip surface, the minimum

reliability index surface according to Hassan and Wolff (1999), noncircular, structurally controlled surfaces, and sur-faces through weak or presheared layers.

Spreadsheet modelingThe spreadsheet software Microsoft® Excel 97 is used in

modeling the slope geometry, stratigraphy, and slope analysismethod. Using analytical geometry, the equations describingthe ground profile, the boundaries between soil layers, andthe slip surface are first established with reference to a coor-dinate system. The equations are then modeled in aMicrosoft® Excel 97 spreadsheet. The coordinates of thepoints of intersection of the different boundaries, for exam-ple, where the slip surface meets the borders between layers,can be obtained within the spreadsheet and used in calculat-ing the slice information (width, coordinates of mid-basepoint, total height, and thickness in each soil type) for factorof safety computations.

Next, the slip surface within each soil layer is divided intosegments of length δ in addition to a residual segment of smaller length, similar to the illustration in Fig. 5. The localaverages of a soil parameter over the length of each segmentare considered variables. The spatial variability of the soilparameter along the slip surface is modeled by the correla-tions between these variables. The variables are character-ized by the CDF of the residual component, F (ε), with no

variance reduction, since the variance function Γ (∆ z ≤ δ) isequal to one (eq. [3]). @Risk functions are used to assignappropriate probability distributions for the spreadsheet cellscontaining the input variables. The correlations betweenvariables are modeled using @Risk functions “IndepC” and“DepC” or by creating a correlation coefficients matrix us-ing the “Correlate” command. In the simulation process,

correlated variables are sampled so as to preserve input cor-relation coefficients.

Slope analysis methods, such as the Bishop, Janbu, andSpencer methods, can be easily modeled in a spreadsheet.However, more advanced and complex methods such as theMorgenstern–Price method and three-dimensional (3D)methods, are, for now, cumbersome to model in a spread-sheet. In this paper, the Bishop method of slices is used. Thespreadsheet model is built by creating a calculation tablesimilar to that proposed by Bishop (1955) for mechanicalcomputations. The appropriate soil parameters for each sliceare automatically chosen by a series of nested “IF” state-ments that compares the X coordinate of the mid-base pointwith the coordinates of the intersections of the slip surface

with soil layers. The Microsoft® Excel 97 “circular refer-ence” feature is used for the iteration process until the differ-ence between the assumed and the calculated factors of safetyis within the user-specified range. Further discussions and anillustration of the structure of the spreadsheet are presented inthe context of the case study analyzed in a later section.

Issues in simulationMonte Carlo simulation requires the generation of random

numbers, which are used in sampling the CDFs of the inputvariables. @Risk software (Palisade Corporation 1996) al-lows two sampling techniques, Monte Carlo sampling (orrandom sampling) and Latin Hypercube sampling. The latterallows sampling the entire CDF curve, including the tails of

the distribution, with fewer iterations, and consequently lesscomputer time. It is adopted in this study.

The output of a Monte Carlo simulation is sensitive to thenumber of iterations, m. When m is large, the random samplesdrawn for each input variable are also large and the match be-tween the CDF recreated by sampling and the original inputCDF is more accurate. Hence, the level of noise in the simu-lation diminishes and the output becomes more stable at theexpense of increasing computer time. The optimum numberof iterations depends on the sizes of the uncertainties in theinput parameters, the correlations between input variables,and the output parameter being estimated. A practical way tooptimize the simulation process is to repeat the simulation us-ing the same seed value and an increasing number of itera-tions. A plot of the number of iterations, m, against theprobability of unsatisfactory performance indicates the mini-mum number of iterations at which the probability value sta-bilizes.

Interpretation of the output

Probability of unsatisfactory performanceThe output of the simulation is the probability density func-

tion of the factor of safety from which the mean and standarddeviation can be estimated. The probabilistic safety measureused in this study is the failure probability. It is the probabil-

© 2002 NRC Canada


Fig. 5. Modeling the spatial variability of an input parameter

along the failure surface using Vanmarcke’s (1983) approxima-

tion of a 1D random field.


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ity of the factor of safety, FS, being less than one, or simplythe number of iterations with FS ≤ 1.0 relative to the totalnumber of iterations, m. The term failure, however, impliesthat the collapse of the slope is the event of concern to the de-signer, which is not necessarily the case. The serviceability of the slope is as important as slope collapse and requires thor-ough evaluation and assessment. Serviceability issues include,

among others, slope movement and cracking (without theslope collapsing), high water seepage, and surface erosion.The term failure may also imply to clients, particularly non-professional clients, that the slope would fail. We recommendusing a different terminology. The U.S. Army Corps of Engi-neers (1992, 1995) used probability of unsatisfactory perfor-mance, Pu, instead of failure probability. The same term,unsatisfactory performance, is adopted here to address failuremechanisms. Another safety measure, perhaps probability of unsatisfactory serviceability, should be considered to addressserviceability criteria. The evaluation of slope serviceabilityis, however, beyond the scope of this study.

Because the simulation process uses random sampling of the input variables, the calculated probability of unsatisfac-

tory performance is also a variable. The probability valuefrom a single simulation could differ from the true value.Law and McComas (1986) described relying on the resultsof a single simulation run as one of the most common andpotentially dangerous simulation practices. It is, therefore,essential to repeat the simulation using different seed valuesto assess the consistency of the estimates. By running thesimulation many times, the histogram of the probability of unsatisfactory performance, the mean probability, and the95% confidence interval around the mean could also be esti-mated. Using @Risk “macro” functions, the process of run-ning a number of simulations can be fully automated. Asimple macro file can be designed to run a number of simula-tions and save the output of each simulation in a separate file.

In reflecting on the computed probability of unsatisfactoryperformance, three points should be noted. First, the com-puted probability of unsatisfactory performance does notaddress temporal changes in input variables. If pore-waterpressures, as an example, vary with time, the impact of suchvariations on stability can only be assessed by undertakingfurther analyses using the new pore-pressure distributions.

Second, the computed probability of unsatisfactory perfor-mance represents the probability of occurrence of a hazardousevent over the lifetime of the conditions considered in theanalysis, such as loading conditions and environmental condi-tions. The calculated probability is, thus, associated with atime frame that varies from one problem to another. If the an-nual probability of unsatisfactory performance is required, asis the case in quantitative risk analyses, the reference timeneeds to be estimated. For example, if the pore-water pressureused in the analysis is the result of a rainstorm, the annualprobability of unsatisfactory performance is estimated basedon the return period of the storm. In the case that none of theinput variables is time dependent, the annual probability couldbe referenced to the lifetime of the slope.

Third, despite efforts to include all sources of uncertainty,there is the possibility of undetected uncertainties such ashuman mistakes affecting slope performance. The contribu-tion of these unknown uncertainties to the probability of un-satisfactory performance is not considered, so the computed

probability could be a lower bound to the actual probabilityof unsatisfactory performance. That is why comparison withcomputed probabilities of different designs is believed to beof greater value.

Having estimated the probability of unsatisfactory perfor-mance, the next step is to assess whether it is acceptable.Typically, this is achieved through comparing the computed

values with a probabilistic slope design criterion. Commonly,a criterion based on the observed frequency of slope failuresand judgement is used. A major drawback to this approachis that the site- and case-specific features such as slope ge-ometry, site conditions, and sources and levels of uncertaintyof the case histories constituting the database are not ad-dressed. Applying such a global criterion to any slope is asignificant generalization.

The authors are not aware of any well-founded probabilis-tic criteria for the acceptability of a design. A reliable ap-proach to estimate such criteria is to calibrate the computedprobabilities of unsatisfactory performance of slopes withtheir observed field performance. This requires probabilisticslope stability analyses of cases of failed slopes and of

slopes performing satisfactorily. Through the comparison of the probabilities associated with the two classes, a probabil-istic slope design criterion could be established. This ap-proach has been developed by El-Ramly (2001) and will bepublished separately.

Reliability indexThe reliability index, β , another common probabilistic

safety measure, is given by

[6] βσ

= − E [[

FS]

FS]

1

where E [FS] and σ[FS] are the mean and standard deviationof the factor of safety, respectively. The above definition isaccurate if the performance function (factor of safety equa-tion) is linear, which is not the case for slope analysis mod-els. Mostyn and Li (1993) suggested, however, that theperformance functions of slopes are reasonably linear and rec-ommended ignoring the nonlinearity when calculating β. Angand Tang (1984) provided a convenient theoretical backgroundof the meaning and estimation of the reliability index.

Sensitivity analysisThrough the software @Risk, Spearman rank correlation

coefficients between the factor of safety and the input vari-ables can be calculated for a sensitivity analysis. TheSpearman coefficient is a correlation coefficient based on therank of the data values within the minimum–maximumrange, not the actual values themselves. It varies between 1and –1. A value of zero indicates no correlation between theinput variable and the output, a value of 1 indicates a com-plete positive correlation, and a value of –1 indicates a com-plete negative correlation. Spearman correlation coefficientscould be used as measures of the relative contribution of each input variable to the uncertainty in the factor of safety.This contribution comprises two elements, the degree of un-certainty of the input parameter and the sensitivity of thefactor of safety to changes in that parameter.

The results of sensitivity analyses are of significant practi-cal value, as they quantify the contributions of the various

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El-Ramly et al. 673


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sources of uncertainty to the overall design uncertainty. Re-sources, whether intellectual or physical, can thus be ratio-nally allocated towards reducing the uncertainty of thevariables with large impacts on design. Also, the relative im-pacts of systematic uncertainty (statistical uncertainty andbias) and uncertainty due to inherent spatial variability canbe estimated. This information is of interest because system-

atic uncertainty has a consistent effect at all locations withinthe domain of the problem and can have a major impact ondesign. Unlike the uncertainty due to inherent spatial vari-ability, systematic uncertainty can be reduced by increasingthe size of data sample and by avoiding highly uncertain em -pirical correlations and factors.

Probabilistic analyses of James Bay dykes

Spreadsheet-based probabilistic slope analysis

The application of the probabilistic slope analysis methodol-ogy described earlier in the paper is illustrated through theanalysis of the well-documented dykes of the James Bay hy-droelectric project in Quebec, Canada. Although the dykeswere not built, the design was the subject of extensive studiesquantifying the sources of uncertainty, analyzing the spatialvariability of soil properties (Ladd et al. 1983; Soulié et al.1990), and evaluating the stability of the dyke probabilisticallyusing the FOSM method (Christian et al. 1994).

The project called for the construction of 50 km of earthdykes in the James Bay area. Various design options wereinvestigated. Only one design is considered in this study, thesingle stage construction of a 12 m high embankment with aslope angle of 18.4° (3h:1v) and a 56 m wide berm at mid-height. Figure 6 shows the geometry of the embankment andthe underlying stratigraphy.

The embankment is on a clay crust, about 4.0 m thick,overlying a sensitive marine clay which in turn is underlainby a lacustrine clay. The marine clay is about 8.0 m thick and the lacustrine clay is about 6.5 m thick. The undrainedshear strength of both clays, measured by field vane tests at1.0 m depth intervals, exhibited a large scatter. The meanundrained shear strength of the marine clay is about 35 kPaand that of the lacustrine clay is about 31 kPa. The lacus-trine clay is underlain by a stiff till.

The uncertainty in soil parameters was quantified by Laddet al. (1983) and Christian et al. (1994). They identified theinput parameters whose uncertainties are deemed importantto the stability of the dykes and estimated their statistical pa -rameters. The following probabilistic stability analyses are

based on the conclusions of these studies. Other case studiesdemonstrating the complete analysis starting with field andlaboratory data, through the quantification of parameter un-certainty, the probabilistic assessment, and the estimation of the probability of unsatisfactory performance will be pub-lished separately by the present authors.

Eight input parameters are considered variables: the unitweight and friction angle of embankment material, the thick-ness of the clay crust, the undrained shear strength andBjerrum vane correction factors for the marine and lacus-trine clays, and the depth of till layer. Table 1 summarizesthe means and variances of all input variables.

The variances of the unit weight and friction angle of theembankment material are evaluated judgementally (Christianet al. 1994) to reflect potential variability in fill properties.The bias in vane measurements is adjusted by Bjerrum’svane correction factor, which is also considered uncertaindue to the scatter in Bjerrum’s database. The uncertainty inthe depth of the till layer, Dtill, is entirely statistical uncer-

tainty due to the limited number of borings. In the absenceof the actual probability distributions of the data, all vari-ables are assumed normal. The spatial variability of all un-certain soil parameters is characterized by an isotropicautocorrelation distance of about 15 m, which was inferredfrom Christian et al. (1994).

As the strength of the lacustrine clay is relatively low, thedepth to the till layer controls the location of the critical slipcircle. The uncertainty in Dtill, thus, introduces uncertaintyin the location of the slip circle. To examine the impact of this uncertainty, deterministic stability analyses were per-formed varying Dtill incrementally between 15.5 and 21.5 m(±3 standard deviations). The minimum factor of safety var-ied between 1.69 and 1.30, implying that the uncertainty in

Dtill could have an important impact on the reliability of thedesign. All the critical slip circles were tangent to the top of the till, daylighted within a short distance at the top of theembankment, and shared almost the same x coordinate forthe centres.

The equations describing the dyke profile and soil layerswith reference to a coordinate system are estimated andmodeled in a Microsoft® Excel 97 spreadsheet. The generallayout and structure of the spreadsheet are illustrated in Ap-pendix A. Since the depth of the till is considered a variable,a different value is sampled for each simulation iteration andconsequently the critical slip circle varies from one iterationto another. To minimize the computer time, some restrictionswere imposed on the geometry of the slip circles based onthe results of the previous parametric study. The slip circlesare assumed tangent to the till layer, daylight at a fixed pointat the top of the embankment ( X 1 = 4.9, Y 1 = 36.0), and havea common X coordinate for the centres ( X 0 = 85.9). Theequation of the slip circle is then added to the spreadsheet asa function of the Y coordinate of the centre, Y 0, and the ra-dius, R; both depend on the sampled value of Dtill. In eachiteration the sampled value of D till thus defines only one crit-ical slip circle. Using the principles of analytical geometry,the intersections of the slip circle and the boundaries be-tween layers and the breakage points in ground profile arecomputed.

The spatial variabilities of soil parameters are modeled as

1D random fields, assuming exponential autocovariancefunctions. The spatial variability of the unit weight of theembankment is, in fact, a two-dimensional (2D) randomfield. Vanmarcke (1983) showed that the process of averag-ing a 2D random field over a rectangular area with one sideof the area smaller than the scale of fluctuation in the samedirection could be approximated by averaging a 1D randomfield in the perpendicular direction. If the cross section of the embankment is regarded as a rectangle, the variability of unit weight could be approximated by a 1D random field inthe horizontal direction. This approximation is not likely tohave any effect on the analysis, as the impact of the spatial

© 2002 NRC Canada



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variability of unit weight on design reliability is minimal(Alonso 1976; Nguyen and Chowdhury 1984). This is attrib-uted to the small spatial variability of the unit weight (a co-efficient of variation of only 5% in the James Bay case) andthe insensitivity of stability calculations to variations in unitweight.

The slip surface within each soil layer is divided into seg-ments of lengths not exceeding δ = 30 m (δ = 2r o for expo-nential autocovariance functions), as illustrated in Fig. 6.Hence, the average undrained shear strength over the length

of each segment has the same variance as the input data (Ta-ble 1) with no reduction. The undrained shear strength of thelacustrine clay, for example, is modeled by three variablesrepresenting the average strength over three segments of theslip surface, as illustrated in Fig. A1 in Appendix A. Simi-larly, the embankment fill is divided into five zones and theaverage unit weight within each zone is considered a randomvariable. The correlation coefficients between the variablesare estimated using eq. [4], and the correlation coefficientmatrices are shown in Appendix A. Additional variables areadded to the spreadsheet to model statistical uncertaintiesand the uncertainties in Bjerrum vane correction factors. In

© 2002 NRC Canada

El-Ramly et al. 675

Fig. 6. Cross section and stratigraphy of the James Bay dykes, showing the approach adopted to account for spatial variability of soil

properties.

Soil parameters

Variance Bias factors

Input variable Mean Inherent variability Statistical uncertainty Mean Variance

Fill friction angle, φfill (°) 30.0 1.00 3.00 — —Fill unit weight, γ fill (kN/m3) 20.0 1.00 1.00 — —Clay crust thickness, t cr (m) 4.0 0.19

a 0.04 — —

Shear strength of marine clay, S uM (kPa) 34.5 66.26 0.90 — —

Bjerrum vane correction factor for S uM, µM — — — 1.0 0.006Shear strength of lacustrine clay, S uL (kPa) 31.2 74.82 3.00 — —

Bjerrum vane correction factor for S uL, µL — — — 1.0 0.023Depth of till, D

till (m) 18.5 — 1.00 — —

aVariance is reduced by 80% to account for spatial averaging based on the assessment of Christian et al. (1994).

Table 1. Input variables and their statistical parameters (based on Christian et al. 1994)

Fig. 7. Estimating the optimum number of iterations for Monte

Carlo simulation.


12/19

total, 19 variables are defined in the spreadsheet (shadedcells in Fig. A1 in Appendix A) to account for the variouscomponents of parameter uncertainty.

Using @Risk options, truncation limits are imposed onthe probability distributions of the undrained shear strengthto prevent sampling negative values. Truncation limits at themean ±3σ are also imposed on the probability distributionsof the thickness of the clay crust and the depth of the tilllayer. The purpose of these limits is to avoid sampling ex-treme low or high values that may cause discrepancies in thesequence of layers.

The stability calculations are based on the Bishop methodof slices. The tables used for factor of safety computations are

shown in Fig. A2 in Appendix A. For each set of sampledinput parameters, the iterative process for factor of safety cal-culations is repeated until the difference between the factorsof safety in two consecutive calculations is less than 0.01.

Trial simulations indicated that the optimum number of it-erations is 32 000 (Fig. 7). Using a seed number of 31 069(an arbitrary value), the mean factor of safety is estimated tobe 1.46, with a standard deviation of 0.20. The probability of unsatisfactory performance is 4.70 × 10–3. Figure 8 showsthe histogram and the probability distribution function of thefactor of safety. The histogram is slightly right skewed, witha coefficient of skewness of 0.29. After 25 simulations, themean factor of safety is 1.46, with a standard deviation of 0.20 and a coefficient of skewness of 0.30. The mean probabil-ity of unsatisfactory performance is 4.70 × 10–3, with the 95%confidence interval between 4.50 × 10–3 and 4.90 × 10–3. Fig-ure 9 shows the histogram of the probability of unsatisfactoryperformance. The reliability index is calculated to be 2.32.

A sensitivity analysis (Fig. 10) shows Spearman rank cor-relation coefficients for all 19 input variables. It is interest-ing that many of the factors with major contributions to theuncertainty of the factor of safety are not related to soilproperty measurements. For example, Bjerrum’s correctionfactor for the undrained shear strength of the lacustrine clay,the statistical uncertainty in the depth of the till, and the sta-tistical uncertainty in the unit weight of the fill (which was

evaluated judgementally) are among the main factors affect-ing the reliability of the design. The results highlight thesignificance of the additional uncertainty that could be intro-duced by the designer through the use of empirical factorsand subjective estimates of uncertainty. Although subjectiveestimates based on experience are acceptable in practice,quantitative estimates of uncertainty, such as variance, arenot yet reliable. Therefore care should be exercised in mak-ing such judgements.

First-order second-moment (FOSM) analysis

The FOSM method is an approximate approach based onTaylor’s series expansion of the performance function g( x1,

© 2002 NRC Canada


Fig. 8. Histogram and probability distribution function of the factor of safety.

Fig. 9. Histogram of the probability of unsatisfactory perfor-

mance based on the results of 25 simulations.


13/19

© 2002 NRC Canada

El-Ramly et al. 677

x2,…, xn) around its mean value. The performance function gcould be Bishop’s method of slices, which is a function of anumber of input variables xn that represents soil properties,

pore pressure, and slope geometry. For uncorrelated inputvariables, the mean and variance of the factor of safety aregiven by eqs. [7] and [8], respectively:

[7] E [FS] ≅ g( E [ x1], E [ x2],…, E [ xn])

[8] σ σ22

1

2[ [ ]FS] ≅ ∂∂

=∑ g

x x

ii

j

i

where E [–] and σ2 [–] denote the mean and variance, respec-tively. For most geotechnical models, the analytical evalua-tion of the derivatives (∂g / ∂ xi) is cumbersome. A finitedifference approach is commonly used to approximate thepartial derivatives as follows:

[9] ∂

∂ ≅ −

− =g

x x x xi i i i

FS FS FS1 2

1 2 ∆∆

where FS1 and FS2 are the factor of safety calculated for val-ues of the input variable xi equal to xi

1 and xi2 , respectively,

with all other input variables assigned their mean values.Usually, x i

1 and xi2 are taken equal to one standard deviation

above and below the mean E [ xi]. To account for spatial vari-ability and the reduction in the variance of the average pa-rameters, the variance of measured data, σ 2 [ xi], is reducedby a reduction factor f . Vanmarcke (1977a) suggested thatthe variance reduction factor can be approximated by

[10] f r

L≅ 2 o

where L is the length over which the parameter of interest isbeing averaged.

When the stability of James Bay dykes is analyzedprobabilistically using the FOSM method, the mean factor of safety is estimated to be 1.46. Table 2 summarizes the calcu-lations of the variance of the factor of safety. For computingthe variance reduction factor f , the length L is taken equal tothe length of the slip surface within each layer, except forthe fill unit weight, where L is taken equal to the length of the embankment in cross section. The standard deviation of the factor of safety is computed to be 0.19, thus the reliabil-ity index is 2.42. To estimate the probability of unsatisfac-tory performance, a form of the probability density functionof the factor of safety must be assumed. For normal and log-

normal probability distributions, the probability of unsatis-factory performance is estimated to be 8.4 × 10 –3 and 2.5 ×10–3, respectively. Table 3 compares the results of the FOSMmethod and the spreadsheet-based Monte Carlo simulation.The FOSM method appears to be a reasonable approach forestimating the mean and variance of the factor of safety.However, the uncertainty about the shape of the probabilitydensity function of the factor of safety introduces uncertain-ties in estimating the probability of unsatisfactory perfor-mance, as illustrated in Table 3.

Simplified analysisThe spatial variability of soil properties and pore-water

Fig. 10. Sensitivity analysis results. Spearman rank correlation coefficients for all input variables.


14/19

pressure is a major source of parameter uncertainty. How-ever, probabilistic analyses ignoring the issues of spatialvariability and statistical uncertainty and relying only on themeans, variances, and probability distributions of measureddata are not uncommon (e.g., Nguyen and Chowdhury 1984;Wolff and Harr 1987; Duncan 2000). Such an approach, re-ferred to hereafter as a simplified analysis, can be erroneousand misleading.

To illustrate the errors incurred by the simplified approach,the James Bay dykes are reanalyzed probabilistically, ignor-ing the spatial variability of soil properties. The analysis isbased directly on the probability distributions of the datawith no variance reduction and no considerations of statisti-cal uncertainties. Table 4 summarizes the input variables andthe statistical parameters used in the analysis, based onChristian et al. (1994). All variables are assumed to be nor-mally distributed.

The deterministic critical slip surface almost coincidedwith the surface of minimum reliability index according to

the algorithm of Hassan and Wolff (1999). Based on trialsimulations, the latter slip surface yielded a slightly higherprobability of unsatisfactory performance and was subse-quently used in the probabilistic assessment. A Microsoft®Excel 97 model and a Monte Carlo simulation using 32 000iterations and a seed number of 31 069 gave a mean factorof safety of 1.46, with a standard deviation of 0.25 and a co-efficient of skewness of 0.32. The probability of unsatisfac-tory performance is calculated to be 2.39 × 10–2. Based onthe results of 25 simulations, the mean probability of unsat-isfactory performance is 2.37 × 10–2, with the 95% confi-dence interval between 2.34 × 10–2 and 2.41 × 10–2. Thereliability index is 1.84. Table 3 compares the results of thesimplified analysis with those of the previous analyses. Thesimplified analysis overestimates the probability of unsatis-factory performance by a factor of 5. This is attributed to thesignificant reduction in the uncertainty due to soil variabilityas a result of spatial averaging, which is taken into accountin all the analyses but the simplified analysis. In slopes dom-

© 2002 NRC Canada


Method of analysis E [FS] σ[FS] Skewness Pu β

Spreadsheet-based probabilistic slope analysis 1.46 0.20 0.30 4.70×10–3 2.32

FOSMa 1.46 0.19 Not available 8.40×10–3; 2.50×10–3 2.42

Simplified analysis 1.46 0.25 0.32 2.37×10–2 1.84

aThe value 8.40 × 10–3 is based on the assumption that the probability density function of the factor of safety is normal, and the value 2.50 × 10 –3

assumes the probability density function of the factor of safety is log-normal.

Table 3. Comparing the outputs of different analysis approaches.

Soil parameters Bias factors

Input variable Mean Variance Mean Variance

Fill friction angle, φ fill (°) 30 4 — —Fill unit weight, γ fill (kN/m3) 20 2 — —

Clay crust thickness, t cr (m) 4 1 — —Shear strength of marine clay, S uM (kPa) 34.5 66.26 — —

Bjerrum vane correction factor for S uM, µ M — — 1 0.006Shear strength of lacustrine clay, S uL (kPa) 31.2 74.82 — —

Bjerrum vane correction factor for S uL, µ L — — 1 0.023

Table 4. Input variables and statistical parameters for the simplified analysis (based on Christian

et al. 1994).

Variance, σ 2 [ x]

Input variable

Inherent

variability

Systematic

uncertainty ∆FS/ ∆ x f ≈ 2r o / LInherent variability

(∆FS/ ∆ x)2 f σ2 [ x]Systematic uncertainty

(∆FS/ ∆ x)2σ2 [ x]

φfill (°) 1.00 3.00 0.009 1.00 0.0001 0.0003γ fill (kN/m3) 1.00 1.00 0.061 0.24 0.0009 0.0037t cr (m) 0.19a 0.04 0.007 — 0.0000 0.0000S uM-1 (kPa) 66.26 7.60 0.005 1.00 0.0019 0.0002

S uM-2 (kPa) 66.26 7.60 0.005 1.00 0.0019 0.0002

S uL (kPa) 74.82 24.90 0.022 0.37 0.0129 0.0115

Dtill (m) 0.00 1.00 0.055 0.00 0.0000 0.0030

Output

Σ 0.0180 0.0190σ2[FS] 0.0370σ[FS] 0.1920

aVariance is reduced by f = 0.2 based on the assessment of Christian et al. (1994).

Table 2. FOSM calculations of the variance of the factor of safety.


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inated by uncertainties due to the spatial variability of soilproperties, the simplified analysis could significantly overes-timate the probability of unsatisfactory performance.

Conclusions

Conventional slope design practice addresses uncertainty

only implicitly and in a subjective manner, thus compromis-ing the adequacy of projections. Without proper consider-ation of uncertainty, the factor of safety alone can give amisleading sense of safety and is not a sufficient safety indi-cator. Probabilistic slope stability analysis is a rationalmeans to incorporate quantified uncertainties into the designprocess. An important conclusion of this study is that proba-bilistic analyses can be applied in practice without an exten-sive effort beyond that needed in a conventional analysis.The stated obstacles impeding the adoption of such tech-niques into geotechnical practice are more apparent than real.

The developed probabilistic spreadsheet approach, basedon Monte Carlo simulation, makes use of Microsoft® Excel97 and @Risk software, which are familiar and readily avail-

able to most engineers. The underlying procedures and con-cepts are simple and transparent, requiring only fundamentalknowledge of statistics and probability theory. At the sametime, the analysis accounts for the spatial variability of theinput variables, the statistical uncertainty due to limited data,and the bias in the empirical factors and correlations used.The approach is flexible in handling real slope problems, in-cluding various loading conditions, complex stratigraphy, c–φ soils, and circular and noncircular slip surfaces.

Probabilistic slope stability analysis provides practicingengineers with valuable insights that cannot be reached oth-erwise. For example, the level of uncertainty in the factor of safety is quantified through the variance of the factor of safety and the probability of unsatisfactory performance.

This could have an important impact on decisions about adesign factor of safety. If the reliability of the computed fac-tor of safety is deemed high, the profession may be willingto adopt lower design factors of safety than usual, providedthat the serviceability of the slope is not compromised. Withincreasingly sparse funds, many agencies and organizationsprioritize slope repair and maintenance expenditures, as isthe case of hydraulic structures, based on safety levels.Comparing the safety of different structures based on thefactor of safety alone is inadequate because the underlyingsources and levels of uncertainty are not addressed. By com-bining the most likely value of the factor of safety and theuncertainty in that value, the probability of unsatisfactoryperformance and the reliability index provide a sounder ba-sis for the comparison.

The practical value of quantifying the relative contribu-tions of the various sources of uncertainty to the overall un-certainty of the factor of safety through sensitivity analyses,using Spearman rank correlation coefficient, cannot be un-derestimated. Such information allows available resources,whether intellectual or physical, to be rationally allocated to-wards reducing the uncertainties of the variables with thelargest impact on design. The sensitivity analyses undertakenin this study showed that the uncertainty of Bjerrum’s vanecorrection factor is substantial and could have a large impacton the reliability of the design. This warns that the reliability

of a design could be significantly reduced by the use of empirical factors and correlations without the designer evenrealizing this. Understanding the limitations and, more im-portantly, the reliability of such factors and correlationsprior to using them is essential.

It is our view that combining conventional deterministicslope analysis and probabilistic analysis will be beneficial to

slope engineering practice and will enhance the decision-making process. It is important to note, however, that simpli-fied probabilistic analyses can be erroneous and misleading.For example, ignoring the spatial variability of soil propertiesand assuming perfect autocorrelations as in our simplifiedanalysis can significantly overestimate the probability of un-satisfactory performance.

Acknowledgements

The authors would like to thank the Natural Sciences andEngineering Research Council of Canada for providing thefinancial support for this research. We are also grateful foruseful discussions with colleagues at the University of Al-berta and elsewhere.

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List of symbols

b slice width

B bias correction factor

C cohesion

Dtill depth of till

E [–] mean

f variance reduction factor

F ( x) cumulative probability distribution function of variable x

FS factor of safety

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g( x1, x2,

…, xn)

performance function

h total height of slice

hcr height in clay crust

hfill height in embankment fill

hL height in lacustrine clay

hM height in marine clay

l1, l2, l3 segments of slip surface of lengths l1, l2, and l3 L length over which soil parameters are averaged

m number of iterations in Monte Carlo simulation

n number of observations

Pu probability of unsatisfactory performance

r separation distance between two variables in the space

of a random field

r o autocorrelation distance

r u pore-pressure ratio

S u undrained shear strength

S uL undrained shear strength of lacustrine clay

S uM undrained shear strength of marine clay

t trend component for variable x

t cr clay crust thickness

t i trend component at location iu pore-water pressure

V input variable adjusted for statistical uncertainty and bias

W total height of slice

x random variable

xi value of random variable x at location i

X (∆ z) local average of variable x over a length ∆ z z distance

Z 0 separation distance between two intervals ∆ z, ∆ z′ Z 1 distance from the beginning of the first interval to the

beginning of the second interval

Z 2 distance from the beginning of the first interval to the

end of the second interval

Z 3 distance from the end of the first interval to the end of

the second interval

α angle to the horizontal of slice baseβ reliability indexδ scale of fluctuation

ε residual component, equal to difference between x and t ε i residual component at location iφ friction angle

φ′ effective friction angleφfill fill friction angle

Φ variable representing soil stratigraphyγ bulk unit weight

γ fill fill unit weightξ soil parameterµ Bjerrum vane correction factor

µL, µ M Bjerrum vane correction factor for S uL and S uM, respec-tively

ρ correlation coefficient between two variablesσ[–] standard deviation

σ2

[–] varianceσ∆ z2 variance of X (∆ z)∆ z interval of length ∆ z

∆ z′ interval of length ∆ z′Γ (∆ z) variance function

Appendix A

This appendix illustrates the general structure of thespreadsheet model used in the probabilistic analysis of theJames Bay dyke.

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Fig. A2. Spreadsheet model, section 2: tables of factor of safety computations. G.S., ground surface; S.S., slip surface.

probabilistic slope analysis for practice

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