2006_43_1074

Stochastic stability analysis of a test excavationinvolving spatially variable subsoil

Yu-Jie Wang and Paul Chiasson

Abstract: A stochastic slope stability analysis method is proposed to investigate the short-term stability of unsupportedexcavation works in a soft clay deposit having spatially variable properties. Spatial variability of undrained shearstrength is modelled by a stochastic model that is the sum of a trend component and a fluctuation component. Theundrained shear strength trend, which is also spatially variable, is modelled by kriging or a random function. Slopestability analyses are performed on the stochastic soft clay model to investigate the contribution of spatial variability ofundrained shear strength to a disagreement among high factors of safety computed from deterministic methods forslopes that have failed. Probabilities of failure as computed from the stochastic analyses give a better assessment offailure potential. Probability of failure values also correlate with time delay before failure. This phenomenon may berelated to progressive failure or creep and to pore pressure dissipation with time.

Key words: slope stability analysis, failure probability, spatial variability, stochastic modelling, geostatistics, vane tests,sensitive clay.

Rsum : Une mthode danalyse stochastique de stabilit de pente est propose afin dtudier la stabilit courtterme de fouilles non blindes dans un dpt dargile molle prsentant des proprits variables dans lespace. La variabilitspatiale de la rsistance non draine est modlise par un modle stochastique form de la somme dune tendance etdune fluctuation alatoire. La tendance, qui est aussi variable dans lespace, est modlise soit par krigeage soit parune fonction alatoire. Des analyses de stabilit de pentes sont excutes sur le modle stochastique dargile molle afindvaluer la contribution de la variabilit spatiale de la rsistance non draine un dsaccord entre des pentes qui ontcd et une situation juge stable par des facteurs levs de scurit, tels que calculs par mthode dterministe. Lesanalyses montrent que les probabilits de rupture calcules par approche stochastique prvoient les instabilits observes.Lanalyse des rsultats montre une corrlation entre lintensit de la probabilit de rupture et le dlai en temps avantlamorce du glissement. Ce dlai avant glissement pourrait sexpliquer par un phnomne de rupture progressive, parfluage et par dissipation de la pression interstitielle dans le temps.

Mots cls : analyse de stabilit des pentes, probabilit de glissement, variabilit spatiale, modlisation stochastique, gosta-tistique, essais au scissomtre, argile sensible.

Wang and Chiasson 1087

Introduction

The short-term stability of unsupported excavation worksis of great interest in geotechnical practice. An accurateevaluation of this type of stability is intimately dependentupon the applied analysis methods and the representation ofsoil shear strength given as `an input. Undrained shearstrength as measured by field vane testing is commonly usedas an input for soil shear strength in total stress analyses.Field undrained shear strengths, however, especially onsites involving soft, sensitive clay deposits, often presentstrong spatial variability (Souli et al. 1990; Chiasson et al.1995). In addition, estimates of undrained shear strengths at

locations where no measurements are available pose manyquestions. These two factors cannot be considered effec-tively in the traditional deterministic total stress analysis, ex-cept, maybe, by the experience of the designer.

Through detailed back-analyses of a number of fieldembankments, Bjerrum (1972) found that direct use ofundrained shear strength in total stress analyses over-estimated the level of stability of embankments. Cases ofcut-slope failures were reported where a prior stability analysisgave factors of safety that were greater than 1.0 (Bjerrum1972; Lafleur et al. 1988). Many explanations for the dis-agreement between site observation and stability analysishave been postulated, such as time effect and anisotropy(Bjerrum 1972) or progressive failure (Dascal and Tournier1975). Spatial variability (Souli et al. 1990) and uncer-tainty involved in estimation of undrained shear strengthcould be other factors that are responsible for this disagree-ment.

As an attempt at estimating uncertainty and variability ofsoil shear strength, reliability analysis considers shear strengthas a random variable. A probability of failure or reliabilityindex is obtained as a supplement to the deterministic factorof safety (Duncan 2000). Spatial variability of soil shear

Can. Geotech. J. 43: 10741087 (2006) doi:10.1139/T06-062 2006 NRC Canada

1074

Received 20 August 2003. Accepted 28 April 2006. Publishedon the NRC Research Press Web site at http://cgj.nrc.ca on8 November 2006.

Y.-J. Wang. Department of Geotechnical Engineering, ChinaInstitute of Water Resources and Hydropower Research (IWHR),Beijing 100044, China.P. Chiasson.1 Gnie Civil, Facult dingnierie, Universit deMoncton, Moncton, NB E1A 3E9, Canada.1Corresponding author (e-mail: [email protected]).

strength is not considered in a reliability analysis, however,except maybe by an ad hoc approach (El-Ramly et al. 2002).Using a stochastic model for spatial variability of undrainedshear strength (Chiasson et al. 1995), Chiasson and Chiasson(2000) suggested a stochastic slope stability method to eval-uate the failure risk rather than the deterministic factor ofsafety. Undrained shear strength values were modelled asoutcomes of a stochastic process.

Based on a well-recorded test excavation work in spatiallyvariable soft clay, this paper presents a model for the spatialvariability of undrained shear strength that combines a trendcomponent with a stochastic fluctuation (or residual) compo-nent. A kriged, local linear trend and a random trend of theundrained shear strength are explored. The stochastic slopestability method is applied to investigate (i) how spatial varia-bility and uncertainty of shear strength of soil are modelled,and (ii) how spatial variability of shear strength can explainthe disagreement between high deterministic factors of safetycomputed from a total stress analysis and the fact that slopefailures were observed.

Description of study site

To investigate short-term stability of unsupported excava-tion works in soft sensitive clay deposits of eastern Canada,a test excavation was constructed at a site near Saint-Hilaire,approximately 50 km east of Montral, Canada (Lafleur etal. 1988). The crest of the excavation formed a 60 m 60 msquare in plan and had four slopes: 18 (south slope), 27(north slope), 34 (east slope), and 45 (west slope). Excava-tion depth was 6 m on the 45 west slope side and 8 m onthe three other slopes (Fig. 1). The site is underlain by threedistinctive soil layers. From the top, a thin 0.6 m layer offine uniform beach sand overlies a layer of 0.92.5 m thickness

of weathered and fissured clay. Below these two layers lies30 m of sensitive and lightly overconsolidated ChamplainSea clay. Extensive vane tests (27 vane profiles as shown inFig. 1) were carried out to determine the undrained shearstrength of the clay deposit (Fig. 2). These profiles extendgenerally to 15.5 m in depth, with readings spaced every0.5 m. The horizontal spacing between profiles is typically10 m.

Following careful data review, a high spatial variability inundrained shear strength of the clay deposit was found. Inaddition, the depth of the top layer of fissured and weatheredclay varies from 1.5 to 3.1 m; such a layer is generally con-sidered a tension-crack zone in stability analyses.

During and after construction of the test excavation, eightslope failures were observed. Major slope failures occurredon the two steepest slopes (45 west slope and 34 eastslope), and a small 3 m wide rotational slide was observedon the 27 south slope (Fig. 1).

Geostatistical modelling of undrained shearstrength

Undrained shear strength measured by vane tests at thestudied site displays considerable scatter and variability inspace, with a minimum value of 10.7 kPa and a maximumvalue of 68.1 kPa within the top 15 m of intact clay (Fig. 2).For each vane profile, measured undrained shear strengthshows a trend of increasing strength with increasing depth,but this trend varies from one vane profile to another. Thisvariability of undrained shear strength can be illustrated bythree adjoining vane profiles following a line perpendicularto the crest of the 45 west slope (Fig. 3a) and another threeprofiles following a line parallel to the crest (Fig. 3b).Measurements show that undrained shear strength is spatiallyvariable and possesses two characteristics: (i) a local, random,erratic variation; and (ii) a general (or average) structuredvariation. The undrained shear strength cannot be simplyrepresented by a single value or a number of values as in a

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Fig. 1. Site plan of the Saint-Hilaire test excavation showinglocations of vane profiles and slope failure scars (AF). Scale inmetres.

Fig. 2. Scatterplot of measured undrained shear strength cu ver-sus depth.

traditional deterministic method. It also becomes cumber-some (if not doubtful) when trying to model the undrainedshear strength through curve fitting. A proper formulationthat may provide a simple representation of the spatial varia-bility must take these two aspects into account, namely random-ness and structure. One such formulation is the probabilisticinterpretation provided by random function theory (Journeland Huijbregts 1978).

Viewed as a regionalized variable with a spatial structure,undrained shear strength Cu at location xi = (xi, yi, zi) can bepictured as a random variable Cu(xi) that follows a certaindistribution function. The undrained strength values, Cu(xi),that could be measured at all locations xi within the domainD of the actual clay deposit are modelled as outcomes of theset of random variables Cu(xi). Thus, the complete set of fieldvalues of undrained strengths spreading throughout the do-main of the subsoil is modelled as the outcome of the set ofrandom variables Cu(xi). This set of random variables definesthe random function Cu(x) = {Cu(xi), xi D}. The spatialstructure of the random function Cu(x) is characterized byits variogram function (x, h) = E{[Cu(x) Cu(x + h)]2}/2for different separation vectors or lags h (modulus and direc-tion). Undrained shear strength at locations where no mea-surements are available is then either estimated by kriging orsimulated (Journel and Huijbregts 1978).

Through calculation of the mean first-order increment ofmeasured undrained shear strength in the vertical direction,Chiasson et al. (1995) found that undrained shear strength atthe studied site is nonstationary. Undrained shear strengthmay be considered as the sum of a stationary random functionand a vertical trend (the latter being the nonstationary expec-tation of Cu(x) = {Cu(xi), xi D}). Undrained shearstrength is hence modelled as a local trend added to a ran-dom fluctuation component, giving

[1] C x y z m x y z x y zi i i i i i i i iu( , , ) [ ( , , ) ( , , )]= + where Cu(xi, yi, zi), m(xi, yi, zi), and (xi, yi, zi) are, respec-tively, the undrained shear strength, the local trend value,

and the fluctuation component at point xi = (xi, yi, zi). In thispaper, two types of trend components are presented: one is alocal kriged trend, and the other is a random trend. Theprocedures for modelling these two types of trend compo-nents are presented later in the paper.

A structural analysis, performed with the experimentalgeneralized covariance method described by Chiasson andSouli (1997), suggested that in the vertical direction thefluctuation component of Cu can be modelled with a sphericalmodel having a range of 2 m and a sill of 24 (kPa)2 with nonugget effect (Chiasson et al. 1995). A theoretical model forthe horizontal direction was not proposed, however, due toexcessive spacing in the horizontal direction between vaneprofiles. In this paper, the spatial variability of the fluctuation(residual) component in the horizontal direction is alsomodelled with a spherical model having a sill of 24 (kPa)2(no nugget effect) and a range of 10 m, which correspondsto the minimum horizontal spacing between two vane pro-files. This value is also comparable with horizontal rangesmeasured on fine-grained soils (Lunne et al. 1977; El-Ramlyet al. 2003).

Slope stability analyses of the test excavation reportedhere are focused on two-dimensional (2D) cases. A numberof cross sections are analysed for stochastic slope stability.Cross sections passing through a given vane profile areselected to increase analysis accuracy. For example, a crosssection passing through vane profile Su13 (Fig. 1) for the45 west slope is identified as cross section Su13. Terms ofeq. [1] depend on the location within the cross section. Itwould be time-consuming to generate them at every point,so a grid spaced by 0.25 m horizontally and 0.15 m verticallyis used to define the 2D cross section. This has been foundsufficiently accurate in past studies (Djebbari 1996). Onlyterms in eq. [1] at grid points defining the cross section needto be computed. The local trend at each grid point is thendetermined by kriging or by joint simulation (random trendmodel), and the fluctuation component is simulated condition-ally to available data using sequential Gaussian simulation

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1076 Can. Geotech. J. Vol. 43, 2006

Fig. 3. Plot of measured undrained shear strength versus depth for two cross sections: (a) vane profiles following a perpendiculartransect to the crest of the 45 slope; (b) vane profiles following a transect parallel to the crest of the 45 slope.

(Deutsch and Journel 1998). Measurements at profile loca-tions are detrended using the trend value at that location ascomputed by one of the two spatial variability models (krigedlocal trend model or random trend model). This yields a setof detrended measured fluctuations. Detrended fluctuationshave a zero mean and a transition variogram with a sillvalue. Fluctuations are therefore simulated by a stationaryrandom function. Detrended fluctuations are used to condition-ally simulate fluctuations. Conditioning ensures that a simu-lated fluctuation at a measurement location is equal to thedetrended measured fluctuation at the same location. Further-more, within the autocorrelation distance, simulated data arecorrelated with the detrended measured fluctuations. The termconditioning means that simulation outcomes are constrainedby measurements at and around their location of measure-ment.

Undrained shear strength is the sum of the trend componentand the simulated fluctuation (residual) component. This formsone outcome map of undrained shear strength. Repeatedconditional simulation of the fluctuation component (and, ifit is the case, repeated conditional simulation of the randomtrend component) permits the generation of a number of out-comes of undrained shear strength maps. It should bementioned here that conditional simulation generates valuesat measurement locations that are identical to field values.Values of the generated cu at locations where no measure-ments are available are different from one outcome to another,but all outcome maps have the same spatial structure.

Generation of the trend component

Kriged local trendAs discussed previously, the mathematical expectation, or

trend, found from measured undrained shear strength dependsnot only on depth but also on position in plan. It is a func-tion of location in space. Generally, the estimation of thetrend at point (xi, yi, zi) can be made from all the availabledata within a set neighbourhood N(xi, yi, zi) around the point(xi, yi, zi). It is usually enough to express the trend in thisneighbourhood N(xi, yi, zi) in the form of polynomials ofdegree k with unknown coefficients. In this paper, the trendat point (xi, yi, zi) is represented by a linear function of loca-tion (xi, yi, zi) at the studied site:[2] m x y z b b x b y b zi i i i i i( , , ) = + + +0 1 2 3where the coefficients b0, b1, b2, and b3 are coefficients thatare a function of a sliding neighbourhood N(xi, yi, zi). Theyare relevant to the neighbourhood N(xi, yi, zi) selected.Focusing on 2D cross sections for slope stability analysis,the value for the linear trend at every grid point is estimatedusing the kriging technique. This type of trend is identifiedin this paper as the kriged local linear trend of undrainedshear strength (consult Deutsch and Journel 1998 for athorough description of the method local kriging the trend).Figure 4 illustrates one outcome map of the undrained shearstrength for cross section Su13 based on kriged local lineartrends.

Slope stability analysis is generally concerned with aneffective area that is between or close to the crest and toe ofa slope (Fig. 4). Those areas which are out of the effectivearea have little influence on the slope stability analysis.

There are few vane profiles available within this effectivearea, however, to determine shear strength for slope stabilityanalyses. The closest vane profile is 9 m behind the crest ofthe 45 west slope and 12 m behind the crest of the 34 eastslope. In addition, the adopted horizontal range of the spher-ical model that represents the spatial variability of undrainedshear strength is 10 m. This means that only values of un-drained shear strength measured from the closest vane pro-file principally contribute to kriging and analysis results.This also results in little variation of the kriged trend in thehorizontal direction for selected cross sections (kriged trendin Fig. 5). This small amount of variability of the krigedtrend in the horizontal direction cannot effectively reflectthe characteristics of the undrained shear strength trend atthe studied site. A technique that better models the spatialvariability of the trend is hence warranted.

Random trendAt a vane profile location in plan (or a point (xi, yi)),

measured undrained shear strength increases with an increasein depth zi, a trend that can be expressed mathematically as[3] m (xi, yi) = b0(xi, yi) + b1(xi, yi) ziwhere b0(xi, yi) is the intercept of the vertical trend, andb1(xi, yi) is the slope. Both coefficients are constant alongdepth z. Chiasson et al. (1995) estimated these coefficientsby the kriging trend method (Deutsch and Journel 1998) fora neighbourhood including all data within a vertical profile.Thus, each profile has its b0 and b1 set of coefficients. Bothcoefficients are only dependent on location (xi, yi) in plan.Scatterplotting of values of b0 and b1 for all vane profiles atthe studied site show that b0 and b1 varies considerably inplan and are negatively correlated (Fig. 6). Intuitively, valuesof b0 and b1 at locations in plan where no vane profiles areavailable may be estimated by kriging or simulation. Correla-tion between b0 and b1 has to be considered, however. Thejoint behaviour of b0 and b1 hence needs to be taken intoaccount. Similar to the procedure for generating undrainedshear strength, coefficients b0 and b1 are viewed as location-dependent random variables. Geostatistical analyses showthat the spatial variability of b0 can be modelled by a Gaussianmodel with a sill C(0) of 21.0 (kPa)2, a range of 55.0 m, and

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Fig. 4. Plot of an undrained shear strength realisation map forcross section Su13.

no nugget effect (Fig. 7). Omnidirectional, northsouth andeastwest variograms for b1 display considerable scatter andhave a high nugget effect (Fig. 8). For modelling purposes,the eastwest variogram is preferred because most lag couplesare within the neighbourhood of the 34 and 45 slopes. Themodel used for b1 is a Gaussian model with a sill C(0) of0.12 (kPa/m)2, a range of 55.0 m, and a nugget effect of 0.09(kPa/m)2.

Random variables b0 and b1 are assumed to be bivariatenormal. In particular, the conditional distribution of b1 givenb0 is normal with mean and variance as follows:

[4]

b b b bb

bb1 0 11

0

00/ ( )= +

[5] b b b1 0 12 2 21/ ( )=

where b0 , b1 , b0 , and b1 are the mean and standard devia-tion of random variables b0 and b1, respectively; b b1 2/ andb b1 2/ are the conditional mean and standard deviation of ran-dom variable b1 given b0; and is the correlation coefficientof random variables b0 and b1.

The generation of a random trend of undrained shear strengthconsidering correlation of b0 and b1 proceeds as follows:(1) using the Gaussian variogram model that represents thespatial variability of random variable b0, conditionally simu-late b0 by applying sequential Gaussian simulation (Deutschand Journel 1998); (2) transform variable b1 at vane profilelocations to the standardized normally distributed variable using the equation = ( )// /b b b b b1 1 0 1 0 , and form a data-base for the next conditional simulation of the standardizedvariable ; (3) using the Gaussian variogram model that rep-resents the spatial variability of random variable b1, condi-tionally simulate the standardized variable by sequentialGaussian simulation (Deutsch and Journel 1998); (4) back-

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Fig. 5. Spatial variation of kriged and random trend models and generated outcomes of undrained shear strength along a horizontaltransect at 6 m depth through profile Su13.

Fig. 6. Scatterplot of intercept b0 and slope b1 of the undrainedshear strength trend at vane profile locations.

Fig. 7. Variogram analyses of linear trend intercept b0 andGaussian model: sill C(0) = 21 (kPa)2; effective range a = 55 m(the number of omnidirectional lag couples is given in parenthe-ses).

transform the simulated standardized variable to randomvariable b1 using b b b b b1 1 0 1 0= +( )/ / , where b b1 0/ andb b1 0/ in this step are the conditional mean and standard de-viation, respectively, of variable b1 given the correspondingsimulated b0 in step 1; and (5) calculate values of trends atgrid points for cross sections using eq. [3]. In this paper, thetrend value generated by following this procedure is referredto as random trend. Figure 5 illustrates variations of thegenerated random trend at 6 m depth in the horizontal direc-tion following cross section Su13. Compared to the krigedlinear trend, the random trend (Fig. 5) presents a better repre-sentation of the true trend with strong spatial variability.

Stochastic slope stability method

Upper bound methodHaving obtained a number of outcome maps of undrained

shear strength for the selected cross section, an upper boundmethod (Donald and Chen 1997) is used to perform slopestability analysis. Through constructing a kinematicallyadmissible velocity field, the upper bound method iterativelydetermines the factor of safety by solving a workenergybalance equation. An optimization method called the randomsearch simplex method searches for the critical slip surfaceassociated with the global minimum factor of safety (Donaldand Chen 1997). Theoretically, the factor of safety deter-mined by the upper bound method is an upper bound valueand thus should always be greater than or equal to the truefactor of safety. Michalowski (1995) and Donald and Chen(1997) proved, however, that the upper bound method isequivalent to limit equilibrium methods using multislice,translational mechanisms of sliding masses, with the advan-tages of avoiding complicated calculation of forces and anumber of simplifying assumptions.

Technique generating different initial slip surfacesIn a stochastic slope stability analysis where spatial vari-

ability of soil strength parameters is considerable, the abilityto find the most critical slip surface corresponding to theglobal minimum of the factor of safety is of the utmost

importance. Even though a simplex method along with arandom search technique are used to search for the minimumvalue of factor of safety, the method often fails to find theglobal minimum factor of safety (Wang 2001). Hence, a newtechnique was developed to increase the success rate in findingthe global minimum. A large number of initial slip surfaceswere generated to act as initial iterations for the simplexmethod. The critical slip surface can be located using thisscheme and the associated global minimum factor of safety(Fos) can be determined.

A cross-sectional diagram of the slope passing throughvane profile Su13 is shown in Fig. 9a. A grid defining thecircle centre is used to generate initial slip surfaces withvariable radii. The circle-centre grid is symmetrical aboutthe line passing through the midpoint of the slope surface tomake sure the generated initial slip surfaces can cover thewhole domain of the studied slope. One set of sides of thegrid is parallel to the slope surface. The other set is perpen-dicular to the slope surface. At each circle centre the largestand the smallest radius covering the analysis domain are

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Fig. 8. Variogram analyses of slope b1 of linear trend andGaussian model: sill C(0) = 0.12 (kPa/m)2; nugget effect c0 =0.09 (kPa/m)2; and effective range a = 55 m (the number of lagcouples is given in parentheses).

Fig. 9. (a) Illustration presenting initial seed slip surfaces thatcover the analysis area and the critical slip surface obtained forcross section Su13 (6 m deep cut). (b) Cumulative frequency versuslocal minimum factor of safety for different schemes generatingdifferent initial seed slip surfaces.

determined and a number of radius increments are used. Thecircle-centre grid is defined by iv increments following theside perpendicular to the slope surface, ih increments on theother side, and ir increments of radius. A total of (iv +1)(2ih + 1)(ir + 1) initial slip surfaces are hence automatic-ally generated as input for the slope stability analysis. Foreach initial slip surface, a critical slip surface and the corres-ponding local minimum factor of safety are found using thesimplex method. The global minimum factor of safety ischosen as the smallest value among all computed local minimavalues obtained from the initial slip surfaces set. Figure 9aillustrates a case where a number of different initial slipsurfaces were generated. It also displays the resulting criti-cal slip surface associated with the computed most likelyglobal minimum factor of safety. The optimization methodyields a local minimum Fos for each initial slip surface. Cu-mulative frequency distributions for three sets of local mini-mum Fos obtained from three different schemes ofgenerating initial slip surfaces show considerable scatter(Fig. 9b). For example, the optimization method yielded localminimum Fos values between 1.17 and 2.22 for the schemeof ih = 3, iv = 4, and ir = 5. This illustrates well how minimavalues optimized through the simplex method are local mini-mum values that are function of the location of the first iter-ation slip surface. The scheme of ih = 2, iv = 2, and ir = 5gave the best global minimum Fos (the minimum of the setof local minimum Fos) and is used in the following stochas-tic analyses.

Definition of failure threshold FfDiscrepancies between factors of safety calculated using

undrained shear strengths and observed performances of large-scale cuts and embankments have long been recognized(Bjerrum 1972, 1973; Azzouz et al. 1983; Aas et al. 1986).Leroueil et al. (1990) revisited data from these and othersources. They noted that the short-term stability of cuts innon-fissured clays, having liquidity indices greater than 0.4and plasticity indices (IP) less than 50, could be estimated onthe basis of the undrained shear strength measured using thefield vane or the undrained shear strength from un-consolidated undrained triaxial (UU) tests when the porepressure redistribution is not significant. A least-squares lineobtained from their data is plotted in Fig. 10 and has thefollowing equation:

[6] FfST = 0.0107IP + 0.7686

This line is close to the trend first proposed by Bjerrum(1972, 1973). At Saint-Hilaire, the average plasticity indexfor a depth range of 2.58.0 m is 29.7 1.8%. Thus, basedon eq. [6], slopes that have a computed, short-term, totalstress factor of safety of 1.087 will on average fail at Saint-Hilaire. In this paper, this value is defined as the factor ofsafety at failure threshold (Ff). If the computed factor ofsafety Fos is equal to or less than Ff, the slope is in a state offailure.

Definition of probability of failureFor N cross-sectional outcome maps of undrained shear

strength, there are accordingly N factors of safety that areobtained. If there are M factors of safety less than the failure

threshold Ff, a calculated probability of failure is definedas Pfc = M/N. If there are no factors of safety less than thefailure threshold Ff or not enough for a reliable statisticalprediction within the N outcomes, a statistical distribution ofthe factor of safety needs to be estimated first. A nominalprobability of failure Pfn is then defined as the probabilityof the event of Fos Ff based on the approximate distribu-tion. For example, if the factor of safety is assumed to benormally distributed, the nominal probability of failure wouldbe determined by

[7] P P x F x xFfn fFos exp df= = =

( )

( ).

12 20 0

2

2

In this paper, all values of probability of failure presentedare of the calculated type.

Convergence study of mean and standard deviation offactor of safety

To determine how many cross-sectional outcome mapsshould be generated to get a stable prediction of the proba-bility of failure (or mean factor of safety), a study of itsconvergence and that of the mean and standard deviation ofthe factor of safety was performed at cross section Su13.Estimated mean and standard deviation of factor of safetyare relatively stable after 200 outcomes or more (Fig. 11a).The same is observed for the probability of failure (Fig. 11b).The probability of failure and estimated mean and standarddeviation of factor of safety are stable at values of 41%, 1.1,and 0.1, respectively, when the number of outcome maps isgreater than 200. In this paper, 300 outcome maps ofundrained shear strength are generated for each cross sectionto be analysed. Unlike the mean and standard deviation, theconfidence interval of the probability of failure will not onlydepend on the number N of generated outcome maps, butalso on the number M of outcome maps that yield a factor ofsafety less that the failure threshold Ff. Since a small proba-bility of failure means a small number M, small probabilityof failure values will need a higher number of outcome maps

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Fig. 10. Factors of safety obtained by total stress analyses ofshort-term failures (FfST) versus the plasticity index (IP) for thecase where liquidity index IL > 0.4 and IP < 50 (adapted fromLeroueil et al. 1990). The broken line represents the trend firstproposed by Bjerrum (1972, 1973).

to be generated. For example, an analysis using N = 300 thatyields 1% for probability of failure means that only threeoutcome maps gave Fos < Ff. Thus if N is not increased, onemust not consider a small computed probability of failure asa precise statistic but rather as an appreciation that the out-come of failure of the slope is relatively low.

The procedures for stochastic slope stability analysisproposed in this paper could be summarized as (i) modellingthe spatial variability of soil strength (the undrained shearstrength) based on measured data; (ii) identifying slope crosssections to be analysed; (iii) generating 300 different out-come maps of the undrained shear strength grid that definesthe selected cross section; (iv) performing slope stabilityanalysis to obtain the statistical distribution of the factor ofsafety; and (v) determining statistics based on this distribu-tion (for example, probability of failure and mean and standarddeviation of factor of safety).

Influence of top layer of fissured clay onslope stability analysis

Site investigation shows that the top layer of clay at thestudied site is weathered and fissured. Failures were precededby the appearance of tension cracks on the crests of the 34and 45 slopes, and rainfall occurred during and afterexcavation. During the 45 days over which the excavation

work spanned, it rained for a total of 15 days, halting thework four times (Lafleur et al. 1988). Based on these obser-vations, stability analyses consider the top layer of clay as atension crack zone filled with water. The measured thicknessof the fissured clay (or depth of tension cracks) variesconsiderably with location in plan, ranging from 1.50 to3.10 m with a mean value of 2.47 m and a standard deviationof 0.38 m.

To evaluate the sensitivity of the factor of safety to tensioncrack depth, three uniform tension crack depths of 2.47 m(), 2.85 m ( + , = 0.38 m), and 2.09 m ( ) areanalysed. Tension cracks are assumed to be full of water asper field conditions. A stochastic slope stability analysis ofthe 45 slope at 6 m depth is performed at cross sectionSu13 by considering these three different depths (the krigedtrend model is used only). Statistics listed in Table 1 showthat the mean factor of safety, standard deviation, and proba-bility of failure do not vary in order of magnitude whenaltering the tension crack depth from 2.09 to 2.85 m. Thisresult proposes that the variation in thickness of the top layerof fissured clay within the investigated range does not have asignificant impact on the results of stochastic slope stabilityanalyses. This is why a uniform tension crack depth of 2.47 mis introduced in the following stochastic slope stability analyses.

Results from stochastic analysis

Figure 1 shows that eight slope failures occurred along45, 34, and 27 slopes during and after excavation. Toexplore the excavation behaviour in terms of cut depth, 2Dstochastic slope stability analyses are performed for a numberof cross sections passing through vane profiles Su9, Su13,Su12, Su14, and Su15 for the 45 slope, Su19, Su20, andSu21 for the 34 slope, and Su25 for the 27 slope. Thesecross sections are relatively well centred with observed failurescars. For comparison, undrained shear strength generatedfrom a kriged local trend and a random trend are analysed.In addition, water contents for the top fissured clay and theunderlying intact clay deposit are assumed to be 40% and50%, respectively. These are average site values. Watercontent for the studied site ranges between 34% and 99%,with a standard deviation of 7%. Average water contents of40% (top fissured clay) and 50% (intact clay) correspond tounit weights of 18.2 and 17.2 kN/m3. Tension cracks areassumed to be full of water. Mean factors of safety, failureprobabilities, and other statistics for these selected crosssections were computed for undrained shear strength mapsbased on a kriged local trend (Table 2) and a random trend(Table 3).

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Fig. 11. Statistics for the function of number of outcome mapsat cross section Su13 (6 m deep excavation): (a) estimated meanand standard deviation of factor of safety; (b) probability offailure. Factor of safety (Fos)

Depth of tensioncrack (m) Mean SD Min. Max.

Probability offailure (%)

2.09 1.11 0.10 0.74 1.44 44.02.47 1.10 0.10 0.73 1.44 41.02.85 1.11 0.11 0.77 1.47 38.7

Table 1. Stochastic results for the 6 m deep 45 slope for differentdepths of tension cracks at cross section Su13.

Excavation of the steepest 45 slope progressed from northto south during a 5 day period (Lafleur et al. 1988). Slopefailures also occurred chronologically in the same orderonce the excavation reached a depth of 6 m. Two majorslope failure zones can be identified: one is centred at crosssection Su13 between cross sections Su9 and Su12, and theother is between cross sections Su12 and Su15. Mean factorsof safety at cross sections Su9 and Su13 are 0.92 and 1.10for the kriged trend model and 1.04 and 1.05 for the randomtrend model (Tables 2, 3). All these values are close to oneanother and below the failure threshold Ff = 1.09 (as definedearlier). Mean factors of safety at cross sections Su12, Su14,and Su15 are greater than Ff, however. They are close to1.30 and 1.26 for the kriged local trend model and therandom trend model, respectively (Tables 2, 3). Probabilitiesof failure determined from the kriged local trend model are95.0% and 41.0% at cross sections Su9 and Su13, respec-tively, whereas they range between 1.3% and 3.3% at crosssections Su12, Su14, and Su15 (Table 2). Probabilities offailure determined with the random trend model, however,are 60.7% and 58.0% at cross sections Su9 and Su13 and inthe order of 17.0%25.7% at cross sections Su12, Su14, andSu15 (Table 3). These analyses suggest that there are twodistinctive slope failure zones, which is in agreement withsite observations (Fig. 1). It is also interesting to note thathigher probabilities of failure are associated with the zoneneighbouring cross sections Su9 and Su13 where slopefailures occurred immediately after the completion ofexcavation. Lower probabilities of failure are associated with

the zone between cross sections Su12 and Su15 where slopefailures occurred 3 days after the completion of excavation.Variations of the mean factor of safety and probability offailure for cross sections along the crest follow the sametrend as that for both the local kriged trend model and therandom trend model (Fig. 12). Mean factors of safety for thekriged local trend model, however, are generally greater thanthose based on the random trend model, whereas the inverseis true for probabilities of failure.

Work on the 34 slope on the east side also progressedfrom north to south. One larger failure (scar A in Fig. 1,14 m wide by 23 m long) centred near vane profile Su19occurred when the slope was cut down to a depth of 5 m.The mean factor of safety and probability of failure at crosssection Su19 are 1.35 and 3.3%, respectively, for the krigedlocal trend model and 1.29 and 26.0%, respectively, for therandom trend model (Tables 2, 3). The other larger slideoccurred at the designed excavation depth of 8 m and iscentred near cross section Su21 (scar B, Fig. 1). At thiscross section, a mean factor of safety of 1.16 and a probabilityof failure of 15% are obtained with the kriged local trendmodel and 1.14 and 44.7% from the random trend model(Tables 2, 3). At cross sections Su19 and Su21, stochasticanalyses based on the random trend model present lowervalues of the mean factor of safety and higher probabilitiesof failure than those computed from the kriged trend model.Mean factors of safety for the random trend model, however,are still higher and probabilities of failure are lower thanexpected for cross sections where slope failures occurred.

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Fos

Slope ()Neighbouringvane profile

Depth ofcut (m) Mean SD

Probability offailure (%) Observation (delay to failure in days)

45 Su9 6 0.92 0.100 95.0 Small failure C1 (2)45 Su13 6 1.10 0.100 41.0 Small failure C2 (0) and large failure C3 (1)45 Su12 6 1.33 0.120 1.3 Limit of scar C3 (1) and C4 (1)45 Su14 6 1.30 0.100 3.3 Within limits of large failure D (3)45 Su15 6 1.30 0.110 3.0 Limit of scar E (3)34 Su19 5 1.35 0.130 3.3 Large failure A (11)34 Su20 8 1.17 0.067 8.0 No failure34 Su21 8 1.16 0.079 15.0 Large failure B (7 and 29)27 Su25 8 1.63 0.056 0.0 Small spoon-shaped failure (26)

Table 2. Results from stochastic slope stability analyses and slope behaviour observations (kriged trend).

Fos


Depth ofcut (m) Mean SD

Probability offailure (%)

45 Su9 6 1.04 0.21 60.745 Su13 6 1.05 0.22 58.045 Su12 6 1.26 0.22 20.745 Su14 6 1.28 0.21 17.045 Su15 6 1.23 0.22 25.734 Su19 5 1.29 0.28 26.034 Su20 8 1.21 0.21 29.034 Su21 8 1.14 0.18 44.727 Su25 8 1.59 0.25 3.0

Table 3. Results from stochastic slope stability analyses (random trend).

An average mean factor of safety closer to the failure thresholdFf = 1.09 would have been expected and the average proba-bility of failure for both these slopes, which is 35%, shouldhave been closer to 50%. At cross section Su20 where noslope failure was reported, the kriged trend model gives 8.0%probability of failure and 1.17 for mean factor of safety. Therandom trend model yields a higher probability of failure of29% and mean factor of safety of 1.21. Variations in themean factor of safety and probability of failure along thecrest of the 34 slope for the kriged trend and random trendmethods are also illustrated in Fig. 13.

Excavation of the 27 south slope advanced from east towest and necessitated 34 days, corresponding to the totalduration of the construction period. A small, spoon-shapedfailure occurred 26 days later (scar F, Fig. 1). Both krigedand random trend models give high values of mean factor ofsafety (1.63 and 1.59). The probability of failure is conse-quently low. The kriged trend model gives no cases wherethe computed factor of safety is below the failure thresholdFf (i.e., Pf < 1/300 = 0.3%) while a probability of failure of3.0% is obtained for the random trend model (Tables 2, 3).

Discussion

Comparison between kriged local trend and randomtrend models

Stochastic slope stability analyses were performed withtwo types of simulated undrained shear strength generatedfrom a kriged local trend model and a random trend model.The random trend model is proposed mainly because the

kriged local trend model shows little variations followingthe horizontal direction (Fig. 5), which does not representthe actual variability in the trend as disclosed by measuredundrained shear strength data (Figs. 3, 6). Since the randomtrend model uses the same technique of conditional simula-tion as that used in the kriged local trend model to generatethe fluctuation component, undrained shear strength fromthe random trend model shows larger scatter than the krigedtrend model. This larger scatter of generated undrained shearstrength is closer to that of the measured undrained shearstrength. This can be illustrated by comparing the undrainedshear strength from neighbouring site measurements withsimulated cross-sectional maps through profile Su13 obtainedfrom the random trend model and another from the krigedtrend model (Fig. 14). Following the preceding stochasticanalyses, one can observe that the random trend model giveshigher values of the standard deviation of factor of safetythan the kriged trend model while the mean values of thefactor of safety are somewhat comparable for the twomodels (Tables 2, 3). This results in generally higher valuesof the probability of failure for the random trend model,which better reflects site observations of slope failures. Anexception to this rule is found at cross section Su20, whereno failure was observed, even though the probability offailure is rather high. This can be explained by two factors.First, debris from failure scars A and B (Fig. 1) created aberm at the foot of cross section Su20, increasing the resistingmoment. Second, high undrained strength values weremeasured at depths of 10, 8, and 7 m at vane profiles Su19,Su20, and Su21, respectively. These measurements indicatethe presence of a zone of higher strength clay that extendsbetween these three profiles. The highest undrained strength

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Fig. 12. Comparison of stochastic analysis results based onkriged trend model with those based on random trend model forthe 45 slope: (a) mean factor of safety; (b) probability of failure.

Fig. 13. Comparison of stochastic analysis results based on krigedtrend model with those based on random trend model for the 34slope: (a) mean factor of safety; (b) probability of failure.

values, reaching up to 70 kPa, are found at profile Su20.Although it is unknown how this zone extends in the eastwest direction, the fact that the slope at cross section Su20was still standing after a number of years indicates that thezone of high undrained strength clay spreads up to this sec-tion of the slope. This would explain why the slope at Su20is still standing, even though the probability of failure ishigh. Thus, this slope location falls with the set of inverseevents where the slope does not fail and where the probabil-ity that it does not fail is 1 Pf.

Comparison of results from stochastic analyses anddeterministic analyses

Deterministic slope stability analyses were performed atthe same cross sections to explore how they are related tostochastic slope stability analyses. Undrained shear strengthis determined by linear interpolation of vane profilemeasurements. For example, the undrained shear strength

map at cross section Su13 corresponds to measurementsobtained from vane profile Su13. Undrained shear strengthon horizontal transects at locations (or grid points) where nomeasurements are available are equal to vane profile Su13measurements at the same depths (no spatial variability inplan). These analyses are identified as deterministic Fosvalues based on measured undrained shear strength. In addi-tion, kriging (Journel and Huijbregts 1978), a local estima-tion technique that provides the best linear unbiased estimatorfrom measured data of the studied unknown characteristics,was also used to compute undrained shear strength at gridpoints. These analyses are identified as deterministic Fosvalues based on kriged undrained shear strength. Stochasticmean factor of safety and deterministic factors of safetyobtained from measured and kriged maps of undrained shearstrength are given in Table 4.

The factor of safety obtained from the kriged undrainedshear strength is comparable with that from the measuredundrained shear strength, but stochastic mean factors of safetyare generally smaller than those from the measured undrainedshear strength. Note that the kriged undrained shear strengthin the zone of interest for slope stability analysis is predomi-nantly dependent on measured values from the closest vaneprofile. This is why there is no significant differencebetween factors of safety based on measured or krigedundrained shear strength.

Relation between probability of failure and mean factorof safety

Mean factor of safety (expectation of the random variableFos) and probability of failure are presented to evaluate therisk of instability of excavated slopes. The mean factor ofsafety from stochastic analysis is designed to be the bestlinear estimation of the true factor of safety. The resultsshow that it is generally smaller but comparable to the deter-ministic factor of safety. Mean factors of safety computedusing both trend models for 45, 34, and 27 slopes (Table 2,3) are generally higher than the failure threshold Ff = 1.09.Thus, it can be concluded that, on average, most slopesshould have been stable. The mean factor of safety, like thedeterministic counterpart, does not assess the probability ofa slope failure at this site. A factor of safety that is safe onaverage does not preclude instances where its value is lowerthan the failure threshold. The probability of failure, on theother hand, takes into account the influence of spatialvariability of undrained shear strength on the statisticaldistribution of the factor of safety. The probability of failureis a function of both mean and statistical dispersion (measuredby standard deviation) of the factor of safety.

To further explore the relation between probability of failureand mean factor of safety, stochastic slope stability analyseswere performed for 51, 40, 34, and 30 slopes followingcross sections Su9, Su13, Su12, Su14, and Su15. A plot ofthese analyses and the preceding results for the 45 and 34slopes (Fig. 15) shows a nonlinear relation between themean factor of safety and the probability of failure. Thisgood relationship between the probability of failure and theaverage factor of safety is a consequence of the spatial varia-bility model, which is the same throughout the site. Therelationship between probability of failure and average (ordeterministic) factor of safety is thus site dependent. A

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Fig. 14. Comparison of measurements within an area centred atprofile Su13 with a 40 m radius with one realisation map ofundrained shear strength for cross section Su13 as obtained from(a) the kriged trend model and (b) the random trend model.

different site will display a different relationship. For thesite investigated, Fig. 15 can be used for a fast estimation ofprobability of failure when the average factor of safety isestimated through a deterministic analysis.

Relation between probability of failure and delay forslope failure

Extensive stochastic slope stability analyses show that thefactor of safety at this specific site follows a normal distribu-tion except for some deviation at the upper and lower tails ofthe statistical distribution (Fig. 16). This signifies that for alarge number of failed slopes, the mean factor of safety fromstochastic analyses should be, on average, equal to the failurethreshold Ff with a 50% of probability of failure. In the cur-rent study, calculated mean factors of safety are on average closeto or above the failure threshold Ff = 1.09 for analysed crosssections at excavated depths where failures were observed.The average probabilities of failure for the 45 and 34 slopeswhere failures were observed are 36.4% and 33.2%, respec-tively. Although there should be some statistical scatter in themean probability of failure, it would have been expected thatthe estimated values will approach 50%. There is, thus, somedisagreement between the computed probability of failure andfailure observations. There are other possible sources that mayhave contributed to this discrepancy between the probability offailure and observed performance, with time between the end ofconstruction and failure being one. For example, it is wellknown that the factor of safety of slopes excavated in clay is atits highest value immediately after completion of construction

(Bishop and Bjerrum 1960). As negative excess pore pressures(generated by total stress release from soil excavation) dissi-pate, the factor of safety decreases. Another contributing factorfor the discrepancy with the average probability of failurecould be attributed to progressive failure due to strain softeningwith time. Plotting the probability of failure as a function ofslope failure delay in days shows that slopes with lowerprobability values failed following a longer time delay after thecompletion of construction (Fig. 17). Although data are scarce,two curves are plotted for time delay to failure as a function ofprobability of failure. The 45 slope falls on curve 1, and the34 slope on curve 2. Both of these curves indicate that thelower the probability of failure, the longer the delay for slopefailure. The influence of progressive failure and pore pressuredissipation or of another time-dependent factor needs further in-vestigation, however.

Spatial variability and statistical scatter of thecomputed Fos

Spatial variability is found to produce significant scatterin Fos (Fig. 16). A histogram of Fos values obtained fromthe random trend model for profiles from the 34 and 45

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Fig. 15. Probability of failure as function of mean factor ofsafety.

Fig. 16. Normal plot of factor of safety for cross section Su13(6 m deep cut).

Deterministic Fos


Depth(m)

Stochasticmean Fos Measured cu Kriged cu

45 Su9 6 1.04 0.94 0.9345 Su13 6 1.05 1.16 1.1545 Su12 6 1.26 1.34 1.3145 Su14 6 1.28 1.32 1.3645 Su15 6 1.23 1.33 1.3634 Su19 5 1.29 1.29 1.3934 Su20 8 1.21 1.34 1.4634 Su21 8 1.14 1.33 1.3427 Su25 8 1.59 1.65 1.53

Table 4. Comparison of factors of safety between stochastic method (random trend model) anddeterministic method.

slopes, plotted in Bjerrums chart, clearly show how the sto-chastic nature of the spatial variability influences statisticalscatter of the Fos (Fig. 18). Figure 18 also indicates that sta-tistical scatter present in Bjerrums chart may be partly trib-utary to stochastic spatial variability of the sites used todevelop the chart.

Conclusions

A stochastic slope stability analysis method is proposed toinvestigate the short-term stability of unsupported excavationworks involving spatially variable subsoil. Random trend andkriged trend were used to model the spatial variability of thetrend that is apparent at the studied site. Stochastic slopestability analyses based on these two types of trends wereperformed to investigate the contribution of the spatial varia-bility of undrained shear strength to the disagreement betweenthe high factor of safety computed from deterministicmethods and the fact that slope failures occurred. Analysisof the results led to the following conclusions: (i) the spatialvariability of the undrained shear strength has a significantinfluence on excavation stability; (ii) stochastic mean factorsof safety are comparable with deterministic factors of safety;(iii) statistical variations in thickness of the weathered toplayer of clay within have little influence on the stochasticresults; (iv) progressive failure and pore pressure dissipationmay be responsible for the correlation between probabilityof failure and delay in time for failure, with lower probabilitiesof failure indicating a longer delay before failure will occur.

Acknowledgements

Funding for this research work was provided by grantOGP013720 from the Natural Sciences and EngineeringResearch Council of Canada (NSERC) and grants 50539100and 50509027 from the China National Natural ScienceFoundation. The authors are also grateful to the Universitde Moncton for its financial and technical support and to theChina Institute of Water Resources and Hydropower Research(IWHR).

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