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Meta-Analysis of 301 Slope Failure Calculations. II: Database Analysis

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DOI: 10.1061/(ASCE)GT.1943-5606.0000463

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Meta-Analysis of 301 Slope Failure Calculations. II:Database Analysis

Quentin B. Travis, M.ASCE1; Mark W. Schmeeckle2; and David M. Sebert3

Abstract: In response to the growing need for statistical information regarding slope stability risk analysis, this work applies inferentialanalysis to a compiled database of 157 failed slopes and corresponding 301 safety factor (SF) calculations. As presented in the companionpaper, this database also includes a number of slope stability factors, including analytical method used, stress approach (effective versus total),assumed slip surface geometry, slope type, applied correction factors, and soil Atterburg limits. Although the SF data were found to be fairlywell fit by a lognormal distribution, pronounced curvature of the residuals was observed, likely related to various unaccounted slope factors.In response, inferential statistics are used in this paper to analyze the effects of analytical method, slope type, soil plasticity, and effectiveversus total stress analysis. ANOVA hypothesis testing indicated significant differences between analytical methods and significant inter-actions between slope types and pore-water stress approaches. Direct SF calculation methods, such as infinite slope, wedge, and the ordinarymethod of slices were found to produce SF near 1 as expected, but higher order methods in general, and force methods in particular, predictedsafety factors significantly greater than 1. Clay content alone was not a discernible influence on SF calculations. A reduced factor ANOVAmodel was developed to predict SF, given analytical method (a main effect) and the interactions between analytical method with both slopetype and pore-water pressure approach. DOI: 10.1061/(ASCE)GT.1943-5606.0000463. © 2011 American Society of Civil Engineers.

CE Database subject headings: Dam failures; Data analysis; Data collection; Databases; Embankments; Landslides; Limit equilibrium;Risk management; Slope stability.

Author keywords: Dam failures; Data analysis; Data collection; Stability; Embankment stability; Landslides; Limit equilibrium;Risk management; Slope stability; Slopes.

Introduction

This paper’s companion paper introduced a database of 157 slopefailures and corresponding 301 safety factor (SF) calculationsculled from available literature. This database included not onlyminimum safety factor calculations for each failure but also tabu-lated the analysts’model assumptions, specifically pore-water pres-sure approach (effective versus total stress), slip surface geometry(planar, circular, and irregular), two-dimensional limit equilibrium(2DLE) method, and applied correction factors, if any. The SF data-base was approximated by a lognormal distribution, but consider-able curvature of the residuals indicated significant unmodeledslope factors. The general conclusion was that SF calculationfor failed slopes is inherently uncertain, and responsible modelingof the data must include the contributions of the slope factors.

Unfortunately, some of these factors are controllable by the an-alyst, and some are not. Analysis method and slip surface geometryassumptions are elements that the analyst can control. Givenenough information, the analyst may be able to adequately modelpore-water effects. The analyst cannot control soil heterogeneity,

which is likely the most significant complicating factor (Christianet al. 1994). Directly affecting that latter are differences betweenthe actual and sampled soils (sample disturbance, strengthanisotropy, and vane strength bias in clay soils), temporal changes(strain rate, consolidation, and creep), hydraulic conductivity varia-tion, slope integrity (cracking), and strain softening (peak versusmean versus reduced versus residual strength).

The uncertainty and biases of some of the specific factors in-volved in SF calculation have been investigated in the literature.Therefore, the first section of this paper uses this work to predicthow and to what extent these factors are expected to influence theSF database. These predictions are then tested against the data byhypothesis testing. In particular, ANOVA testing is used to identifythose factors that have significant main effects on SF calculationand evaluate possible interaction effects. Finally, a reducedANOVA model of SF calculations is established, and the resultingpractical implications for slope stability analysis are explored.

Expected Uncertainties/Biases

General

It is reasonable to assume that uncertainty and bias is introduced to2DLE analysis from the uncertainty and bias of its inputs. Yet, evenif the effects of these inputs are disregarded, there may be uncer-tainty and bias that are fundamental to the 2DLE approach. Kimet al. (1999) note that 2DLE does not satisfy flow, compatibility,and prefailure constitutive requirements and (at best) satisfiesmoment and force equilibrium in only a global sense, whereas arigorous solution must meet equilibrium requirements at everypoint along a potential slip surface (see Krahn 2003).

1Senior Hydraulic Engineer, WEST Consulting, Inc., Tempe, AZ,85284 (corresponding author). E-mail: btravis@westconsultants.com

2Professor, School of Geographical Sciences, Arizona State Univ.,Tempe, AZ 85287-5306. E-mail: mark.schmeeckle@asu.edu

3Adjunct Professor, Dept. of Mathematics, Columbus State CommunityCollege, Delaware, OH 43015. E-mail: sebertdavid@yahoo.com

Note. This manuscript was submitted on March 8, 2010; approved onOctober 10, 2010; published online on October 20, 2010. Discussion periodopen until October 1, 2011; separate discussions must be submitted for in-dividual papers. This paper is part of the Journal of Geotechnical andGeoenvironmental Engineering, Vol. 137, No. 5, May 1, 2011. ©ASCE,ISSN 1090-0241/2011/5-471–482/$25.00.

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Another fundamental limitation of 2DLE is that actual slopesare three-dimensional, not two-dimensional. In response to thislimitation, limit equilibrium has been generalized to three dimen-sions. Three-dimensional limit equilibrium (3DLE) is reportedby Azzouz et al. (1981) to increase predicted SF compared with2DLE by as much as 30%, a minimum of 7%, and an averageof 14%. Ugai (1988) reports that an analysis of the Ontake landslideyielded a 3DLE SF of 5–30% greater than that predicted by 2DLE.Arellano and Stark (2000) reported a 10% increase between 3DLEand 2DLE SF analysis for the Oceanside Manor landslide.However, Byrne et al. (1992) found only a modest increase ofapproximately 5% for 3DLE over 2DLE when analyzing theKettleman Hills landfill failure, and Sainak (2004) showed thatthree-dimensional finite-element (3DFEM) analysis can actuallyproduce safety factors 30% lower than predicted by 2DLE,3DLE, and two-dimensional finite-element (2DFEM) analysis.This finding was supported by Seed et al. (1990), which showeda reduction of approximately 9% for a 3DFEM reanalysis of theKettleman Hills landfill failure. Despite the possibility thatthree-dimensional (3D) slope stability calculations can producelower SF than 2DLE, the 3D analyses in the literature usually arguethat 2DLE is inherently conservative (e.g., Arellano and Stark2000; Christian et al. 1994).

Failure Surface Model

Christian et al. (1994) point out that there is an inherent noncon-servative bias to 2DLE methods because the analyst must find thefailure surface that will produce the lowest SF; they estimate thisbias to be 5% of the computed SF. For steep slopes, this bias isfurther compounded by an observed tendency for the critical geom-etry to lie near nonconvergent geometries for some analytical meth-ods (Krahn 2003). Therefore, it is expected that, for a givenanalytical method, modeling of the slip surface with simple geom-etry assumptions (circle, planar, and logarithmic spiral) will pro-duce higher SF values than using the more versatile irregularslip surface. However, a number of studies indicate that most fail-ures are circular (e.g., Flaate and Preber 1974), and there is someevidence that even a noncircular failure surface may actually resultfrom rapid progressive circular failures (Burridge 1987).

Analytical Method

Beyond general criticisms of 2DLE, the particular analytical meth-ods of 2DLE have both advocates and detractors. A brief discussionof some of the issues specific to particular methods follows.

Infinite SlopeThe infinite slope procedure allows a direct solution for SF by as-suming an infinite failure plane parallel to an assumed perfectlylinear slope surface. The solution is typically achieved by rotatingthe Cartesian axis to align with the slope, with the x0-axis parallel tothe slope and the y0-axis perpendicular. The requirement that allapplied forces are independent of x0 (Iverson 1990, 1997) is impliedby the coordinate transformation but sometimes neglected in prac-tice and the literature. When applied correctly, the infinite slopeprocedure satisfies both force and moment equilibrium.

WedgeThe wedge method assumes one, two, or three planar failure sur-faces and thus collapses to the infinite slope procedure if orientedparallel with the slope. Otherwise, forces and moments are summedabout the failure surface to satisfy force and moment equilibrium.Safety factors are usually obtained directly by using simple algebra.

Logarithmic SpiralThe only direct-solution 2DLE method that allows a curved failuresurface and satisfies both force and moment equilibrium is thelogarithmic spiral (Duncan and Wright 2005). The restriction offailure surfaces to a logarithmic spiral appears to be its only reallimitation. Although no more difficult to apply by computer thansome of the higher-order methods, it has not been used much foranalysis of actual slope failures; the present database contains onlythree logarithmic spiral calculations.

ϕ � 0 (Swedish Circle)The ϕ ¼ 0 method assumes that the Coulomb friction angle ϕ iszero. Also called the Swedish circle method, the ϕ ¼ 0 methodhas historically been a popular total stress analysis procedure, par-ticularly for clay slopes where cohesion is typically much greaterthan the friction angle effect. The method is equivalent to the log-arithmic spiral analysis, where a zero friction angle is assumed(Duncan and Wright 2005). Thus, the resulting failure surface iscircular, which simplifies the calculation, and complete equilibriumis satisfied. Unfortunately, the appeal of this simple procedure hasled to analysts occasionally ignoring significant friction angles,resulting in inaccurate (but conservative) safety factors. For exam-ple, the procedure has been applied to fill slopes on clay, which notonly neglects the friction angle of the clay (which can be signifi-cant) but also the friction angle of the fill, which is typically muchgreater than its cohesion. Thus, the ϕ ¼ 0 method is expected togenerate SF values less than the actual value for soils with signifi-cant friction angles.

Ordinary Method of SlicesThe ordinary method of slices (OMS) satisfies moment equilibriumfor a circular failure surface and directly calculates SF by assumingthat the interslice forces are zero. For an infinite radius, OMS thuscollapses to the undivided wedge method. Taylor (1948) andWhitman and Moore (1963) both argue that OMS is inaccurate.Duncan and Wright (1980) note that OMS typically underestimatesSF compared with other methods of slices, particularly for slopeswith high pore-water pressures modeled by effective stress analy-sis. The assessment that OMS is conservatively biased appearsto be so widely held that some slope failure analysts automaticallyconclude so even when they have found evidence to the contrary(see Moore 1970 and Sundaram and Bell 1972). One complicatingfactor is that some analysts report SF calculations as based onOMS when in fact they have assumed a zero friction angle, andthus their method of solution is actually the ϕ ¼ 0 method (e.g.Hanzawa et al. 2000).

Simplified BishopThe simplified Bishop method (henceforth described as just theBishop method) is a method of slices that assumes that the intersliceforces are horizontal. The Bishop method satisfies moment equi-librium but satisfies force equilibrium in the vertical direction only.Like OMS, a circular slip surface is assumed. The safety factor iscomputed by iteration. The Bishop method has been historicallyvery popular, particularly because it is relatively simple to applyyet usually reported to closely agree with the higher-order2DLE methods.

The Bishop method, like all the higher-order methods, cansometimes indicate inappropriate stresses along the slice bounda-ries and at the slip surface toe (Duncan and Wright 2005).

JanbuJanbu introduced two methods of 2DLE analysis that satisfy forceequilibrium but not moment equilibrium: the simplified Janbuand Janbu’s generalized procedure of slices (GPS) (the latter is

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sometimes called Janbu’s rigorous method). The GPS methodintroduces a line of thrust description to model the interslice force.The simplified Janbu model is so called because it is the GPS modelsimplified by assuming that the interslice forces are horizontal.Like all of the force equilibrium methods, an irregular failure sur-face is allowed for both methods, and solution must be obtained byiteration. The assumption of horizontal interslice forces used by thesimplified Janbu method was justified by Carter (1971), who foundthat the SF actually varies according to the vertical coordinate usedto sum moments about a slice and therefore concluded that the onlyconsistent vertical coordinate for any force analysis is infinity, cor-responding to horizontal force equilibrium. However, Carter’s con-clusion was contradicted by Boutrup et al. (1979), who argued thatthe SF calculated in this manner is actually maximum when thecircular slip surface intersects the top of the slope at a steep angle.They concluded that Janbu is nonconservative for deep failuresurfaces that intersect the ground surface at the top of the slopeat angles greater than 60°.

Another persistent issue with Janbu’s simplified method is that,in practice, it is not always reported if the original or correctedJanbu procedure is being applied (Duncan and Wright 2005).The Janbu correction, as approximated by Abramson et al. (2002),raises the estimated SF to up to 13%. Therefore, uncorrected sim-plified Janbu SF calculations are expected to be conservativelybiased.

Another challenge of interpreting the database is that analystsdid not always state the Janbu method used. Furthermore, it is dif-ficult to identify the Janbu method used by a cited reference be-cause there appears to be no standard reference for either ofJanbu’s methods. As a result, analysts were found to sometimescite obscure conference papers, titles in other languages, or particu-lar applications of an unspecified Janbu method reported by otherauthors.

Fortunately, identifying the particular Janbu method used is notcritical for the present effort. The limitations of the meta-analysisas applied here requires all force methods to be grouped as a par-ticular subset, and because both of Janbu’s methods are forcemethods, the specific Janbu method used for the SF analysis isnot directly relevant. Therefore, both of Janbu’s methods weredesignated by Janbu’s name only and reported so in the database,although specific information regarding the Janbu method usedwas included in the table footnotes when available (see Table 1in the companion paper).

Lowe and KarafiathThe Lowe and Karafiath procedure satisfies force equilibrium byassuming that the interslice forces are included at the average angleformed between the ground and slip surfaces. Duncan and Wright(2005) recommend Lowe and Karafiath as the force method thatproduces the best results. It is presumed, but not typically discussedin the literature, that the issues expressed by Carter (1971) andBoutrop et al. (1979) would also be applicable to the Lowe andKarafiath procedure.

Corp of Engineers Modified SwedishThe primary limitation of the U.S. Army Corps of Engineer’s modi-fied Swedish (COE) force equilibrium method is an uncertainty asto the procedure’s assumption of interslice force inclination, whichcan be interpreted in several different ways. Unfortunately, forceequilibrium safety factors are quite sensitive to the assumed forceinclination, with differences of up to 25% (Duncan and Wright2005). Thus, significant differences would be expected betweenCOE safety factors produced by different analysts. Coupled withthe observations of Carter (1971) and Boutrop et al. (1979) on forceequilibrium procedures, some inaccuracy of COE would be

expected in practice. It is not clear, however, whether this inaccur-acy would be conservative, nonconservative, or unbiased.

Spencer’s MethodSpencer’s method assumes that all interslice forces are inclinedat the same angle, the value of which is solved by completeequilibrium. Probably the most popular method of slices that sat-isfies complete equilibrium and allows irregular failure surfaces,Spencer’s method can be difficult to apply because of convergenceproblems. Indeed, Boutrup et al. (1979) rejected Spencer’s methodfor their STABL program for this reason.

Spencer’s method, like all the complete equilibrium methods, iscomputationally intensive, requiring simultaneous solution of thethree static equilibrium equations for each slice.

Morgenstern and PriceThe Morgenstern and Price complete equilibrium method extendsSpencer’s method to allow the interslice resultant force angles tochange direction between slices, with the inclination functionspecified by the analyst. The normal force location is also specifiedby the analyst. Despite the seemingly arbitrary nature of theseinputs, the resulting SF calculation seems to be fairly robust todifferent assumptions (Duncan and Wright 2005).

Chen and MorgensternThe Chen and Morgenstern procedure, which also satisfies com-plete equilibrium, adds another analyst-specified function to theMorgenstern and Price algorithm to align the interslice forces withthe slope. The result is reported to restrict the number of solutionsfor the SF.

Sarma’s MethodSarma’s complete equilibrium method was developed to directlycalculate the seismic coefficient given a known SF. When appliedoutside of earthquake modeling (e.g., the seismic coefficient iszero), Sarma’s method becomes a modification of the Morgensternand Price procedure, where the interslice forces are assumed pro-portional to the shear strength. For frictional materials, this requiresthe analyst to specify the distribution of normal forces across eachslice (Duncan and Wright 2005).

Pore-Water Pressure

Pore-water pressure introduces forces that can be destabilizing orrestorative, depending on the failure and slope geometry. The twobasic approaches for including pore-water pressure in the analysisis effective or total stress analysis. For total stress analysis, the pore-water pressures are included implicitly in the strength parameters.For effective stress analysis, the pore-water pressures are intro-duced explicitly as contributing forces in the 2DLE formula. Effec-tive stress analysis is generally perceived as more accurate (e.g.,Pilot et al. 1982). With the advent of the computer, effective stressanalysis has become less computationally difficult and is now oftenused in practice.

Slope Type

Failed slopes may be divided into five categories, depending onavailable information and the control over the slope characteristics.The failed slope with the least-known information is clearly a land-slide, where natural soil conditions are often anisotropic and theslope geometry is typically nonlinear and three-dimensional. Onthe other hand, engineered slopes are better controlled, betterunderstood, and typically more accurately described with simpleparameters. In order of increasing information and control, the en-gineered slopes are cut, test cut, fill, test fill, and experiment slopes.Test cut and test fill slopes are full-scale experiments done in the

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field, whereas experiment slopes here refer to typically smaller-scale slopes brought to failure under strict laboratory conditions.

Greater information and control should reduce uncertainty andraise model accuracy, suggesting that the failure of engineeredslopes should be more predictable than landslides. But engineeredslopes often fail over smaller time scales because of consolidationand transient pore-water effects not typically as active in mostlandslides. Given these complications, 2DLE predictions are notexpected to be equally accurate over all slope types. In particular,Krahn (2003) argued that those methods satisfying completeequilibrium are the most accurate for natural slopes, and Ferkhand Fell (1994) observed that cut slopes and riverbanks fail athigher SF compared with other slopes.

Correction Factors

To account for the perceived limitations of 2DLE, correction factorsare usually introduced when safety factors have been found to de-viate significantly from 1 for a slope at failure. These correctionfactors may be specific to slope location, soil tests, soil types,anisotropy, etc. The most popular correction factors are as follows:• SHANSEP. Developed by Ladd and Foott (1974) and often

referred to by its acronym, the stress history and normalized soilengineering properties (SHANSEP) correction attempts toaccount for sample disturbance and strength anisotropy. Theimportance of this correction is supported byWong et al. (1982),who estimated that soil sampling and laboratory testing uncer-tainty can result in errors of up to 20% in safety factor calcula-tion at slope failure.

• Reduced strength. These correction factors correspond to thefully softened, residual, recompressed, or otherwise reduced soilstrength as opposed to the mean strength indicated by soiltesting. Sometimes, the peak strength is used, correspondingto a correction factor greater than 1 (nonconservative).

• Integrity. The possibility of cracks often involves a judgmentcall by the analyst and may significantly change the com-puted SF.

• Strain rate/consolidation/creep. The mismatch between thefield loading rates (usually occurring over weeks) and thelaboratory loading rates (usually imposed over minutes) canresult in inaccurate test estimates of soil strength. Althoughnot typically accounted for (Duncan and Wright 2005), field-loading correction factors do exist and are sometimes applied(e.g., Wolski et al. 1989; Hanzawa et al. 2000).

• Anisotropy. Duncan and Wright (2005) note that anisotropyarises from inherent soil properties and stress configurationwithin the slope. Corrections for anisotropy have historicallybeen lumped into other corrections, such as vane strength.

• Vane strength.Much work has been done on correction factorsfor vane strength measurements of plastic soils, established toaccount for anisotropy and creep strength loss. The originalvane strength correction factor is usually attributed to Bjerrum(1972), later revised by Bjerrum (1973) and Ladd et al. (1977).The correction factor is empirical and typically based on alinear regression of minimum safety factor versus plasticityindex (PI).

• Pore-water pressure. Some analysts choose to apply correctionfactors to account for rapid drawdown and other effects ofpore-water pressure because either a total stress approach isused or it is argued that the effective stress used underpredictsthe pore-water effect.

Analyst Differences

Different analysts bring their own experience and abilities to slopestability analysis and can reach much different conclusions from

the same data (Christian 2004). These differences can arise fromsimple modeling decisions, such as the number and locations ofinternal slices, interslice force assumptions, slip geometry typeand location, and judgment as to when the minimum SF has beenreached.

Mostyn and Small (1987) reported on a survey of 20 engineerswho solved the same slope problems. They found an average SFstandard deviation (sd) of 0.18 between analysis calculations of thesame circular failure problem. For noncircular failure surfaces, thedifference between analysts was greater, with sd ¼ 0:26. Althoughit is unknown if these results would be consistent for a largerdatabase of analysts, it is likely that analyst differences contributesignificantly to the inherent statistical nature of SF analysis.

Summary

Despite the criticisms of specific 2DLE analytical methods, manyargue that higher-order 2DLE methods (force methods, completemethods, and arguably the Bishop method) produce SF valueswith no significant difference between them (e.g., Bjerrum 1972;Christian et al. 1994). Consistent with this assumption, publicationssometimes report the result of a limit equilibrium analysis but leavethe corresponding 2DLE method unspecified (e.g., Chiasson andWang 2007). Indeed, Yu et al. (1998) argue that Janbu, Spencer,Morgenstern and Price, and any other method satisfying completeequilibrium will produce safety factors within 5% of each other,with the caveat that this consistency does not necessarily implyoverall accuracy of the limit equilibrium approach.

In terms of the 2DLE approach in general, the present positionof the engineering community appears to be that 2DLE safetyfactors are often conservatively biased compared with its true value(e.g., Babu and Bjoy 1999; Singh et al. 2008). One geotechnicalargument for this position is that slope stability modeling doesnot always include matric suction in the analysis, the binding effectof which would tend to increase the calculated SF (although notalways; see Travis et al. 2010).

Database Testing

The database reports 301 minimum SF calculations for 157 failedslopes. Detailed database information may be found in thecompanion paper.

SF Method, Slope Type, and Stress ApproachDifferences

Limit equilibrium analysis uses a number of different factors.Hypothesis testing by ANOVA, as implemented by DesignExpert 7.1.6, was used to investigate potential differences betweenand within these factors. Differences between SF values wereinvestigated as a function of analytical method (grouped into direct,Bishop, force, and complete methods), stress approach (effectiveversus total), and slope type (fill, test fill, experimental, and cutslope, where the cut slope factors included the four “test cut”SF calculations to ensure an adequate number of cell values).

An unbalanced ANOVA analysis was required because the num-ber of replicates was not the same in all cells—a typical character-istic of meta-analysis data. Unbalanced data can reduce statisticalpower and imply false correlations between independent variables.However, unbalanced data effects are negligible in practice (Ryan2007) and, indeed, many well cited publications simply report theresults of an unbalanced ANOVAwithout further explication (e.g.,Boggs 1997; Starck et al. 2000).

Of the four principal types of analysis that can be used forANOVA testing, a Type II analysis tends to minimize the adverse

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effects of unbalanced data (Langsrud 2003; Langsrud et al. 2007),and so is the method adopted here. It is also the default setting inDesign Expert 7.1.6 for categorical variables.

An additional complication of the ANOVA analysis was that thenumber of SF calculations varied per slope. For example, as seen inTable 1 of the companion paper, the Narbonne fill slope failure hadfour published SF calculations, whereas the Sieve River landslidehad only one. Moreover, many studies analyzed more than oneslope, introducing yet another complication.

The appropriate interpretation of multiple responses from thesame source, as seen here, is a typical challenge for meta-analysesin general. Bijmolt and Rik (2001) investigated a number of rea-sonable approaches to the problem, including simply ignoring theissue (assumes all response are independent), introducing weightedfactors, selecting single representative values, and establishingnested relationships between responses. They concluded that thebest method was using a nested model. Unfortunately, a nestedmodel would be difficult to implement in the present work becausethere are not only multiple analyses of some slopes by differentanalysts but also multiple slopes analyzed by the same analysts,and even an overlap between the two (e.g., some slopes analyzedby multiple analysts were also included in studies of multipleslopes by a single analyst).

As an alternative to a fully nested approach, Bijmolt and Rik(2001) recommend simply considering all responses as indepen-dent, even though this approach may introduce unjustified noise.Therefore, although the present analysis proceeds accordingly, itis possible that some of the main effects and interactions will besignificant even if the analysis indicates otherwise. For this reason,although only probability levels (“significance levels”) below 5%(p < 0:05) will be considered significant, probability levels below10% (0:05 ≤ p < 0:10) will also be identified and reported asborderline significant. The assumption of independent data is con-sistent with the apparent homoskedasticity of the database, asshown in the companion paper.

The database contained enough data points to generate all maineffects and first-order interactions. Unfortunately, the overall threefactor interaction could not be computed with the number of avail-able data points (that is, the available degrees of freedom were notsufficiently large enough). In practice, however, higher-order inter-action effects are often negligible and/or redundant (Box et al.1978). Therefore, per industry standards (Montgomery 2009),the overall interaction term was aliased and used as part of the over-all model error estimate. Future work will expand the database andeither confirm or contradict this assumption.

A Box-Cox analysis recommended a log10 transform of the re-sponse (SF) to meet the ANOVA assumptions of residual normality,independence, and constant variance, which is consistent with the

overall database analysis reported in the companion paper. Theresiduals of the transformed model were inspected; no obviousviolation of the ANOVA assumptions was observed, and no evi-dence of a lack of fit was indicated (p ¼ 0:14).

The ANOVA results (Table 1) indicated that the overall modelaccurately described the data (p < 0:0005). The only clearly sig-nificant main effect was analytical method (p < 0:01), althoughthe pore-water approach was also borderline significant (p ¼ 0:09).Slope type was not a significant contributor to the model in termsof main effects (p ¼ 0:70) but was part of a significant interactionwith pore-water approach (p < 0:005) and of a borderline signifi-cant interaction with analytical method (p ¼ 0:06). Analyticalmethod and porewater approach did not significantly interact(p ¼ 0:84).

These results are consistent with the prevalent opinion that thereare differences between safety factor calculation methods and dif-ferences between total and effective pore-water stress analysis.However, the lack of evidence for a significant slope type effectis somewhat surprising, although it does appear to play a significantrole as an interaction term.

The foregoing ANOVA analysis is descriptive but complicatedby insignificant terms. A better understanding of the underlyingfactors as well as the ability to make confident predictions aboutfuture events requires a reduced ANOVA model. ANOVA reduc-tion is accomplished by removing insignificant factors and interac-tions, thus creating a well-defined essential model that minimizesthe risk of overfitting. The reduction process is also characterizedby careful consideration of the residual analysis, ensuring that thefinal model conforms to normality assumptions and is thereforeappropriate for risk analysis.

Model reduction was implemented here by a backward algo-rithm based on significance. This process first eliminated the maineffect of slope type, then the interaction of analytical method versuspore-water approach, and finally the main effect of pore-waterapproach. The reduced ANOVA model is shown in Table 2.

The reduced model is nonhierarchical in the sense that theinteraction terms are included but their corresponding main effectterms are not. There is some controversy in the literature regardinghierarchy in ANOVA and regression models. Peixoto (1987, 1990)argued that an appropriate model requires that the results of stat-istical testing be the same under linear transformations of the data,which can only be achieved if the model includes the main effectsof all factors included in higher-order interactions even if thesemain effects are not significant. However, recent publications, suchas Montgomery et al. (2005), argue that this requirement is overlyrestrictive and may reduce prediction accuracy as a result of over-fitting the data.

Table 1. ANOVA Analysis Results

Source Sum of squares Degrees of freedom Mean square F p-value

Overall model 0.45 27 0.017 2.44 0.0002

Slope type 0.015 4 0.00375 0.55 0.6961

Analytical method 0.086 3 0.029 4.25 0.0059

Porewater approach 0.020 1 0.020 2.96 0.0864

Type × method 0.14 12 0.012 1.72 0.0621

Type × approach 0.10 4 0.026 3.86 0.0045

Method × approach 0.005727 3 0.001909 0.28 0.8385

Residual 1.85 273 0.006772

Lack of fit 0.073 7 0.010 1.57 0.1448

Pure error 1.78 266 0.006675

Corrected total 2.29 300

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The effect of expanding the reduced model to achieve hierarchywas investigated for the reduced model presented here. However,when the main effects of slope type and pore-water approach wereincluded, no difference in the overall analysis conclusions wasfound, and only a minimal benefit to the model was achieved(adj-R2 changed from 0.11 to 0.12). Unfortunately, adding thesemain effects compromises the normality assumptions, resultingin less well-behaved residuals and model sensitivity to a particulardata point ( leverage > 1). For these reasons and to maximize thepredictive accuracy and model parsimony, slope type and pore-water approach were not included as main effects.

The reduced ANOVA model’s overall fit to the data is quite sig-nificant (p < 0:0001). Furthermore, unlike the overall analysis,each factor in the reduced model is of similar significance(0:01 < p < 0:05), indicating that no one factor dominates. Theadequate precision (a measure of signal-to-noise ratio) is 6.85,indicating that the model is appropriate for prediction. Themodel’s R2 values, however, are quite small, with R2 ¼ 0:16,adj-R2 ¼ 0:11, and pred-R2 ¼ 0:04. These low regression values,even by slope stability standards, suggest that the usual determin-istic design methods are wholly inadequate; risk analysis must beused for responsible design.

Risk analysis based on the ANOVA model requires that the re-siduals of the model be normally distributed. The model is consis-tent with this requirement, as shown by the normal probability plotin Fig. 1. The residuals align adequately with the normal probabil-ity line, with their curvature much reduced from the initial distri-bution fit, as shown in the companion paper. The residual analysisalso included residuals versus run order and residuals versus pre-dicted values; like the normal probability plot, these analyses did

not show significant violations. Influence testing did not indicateany significant outliers.

The analytical approach main effects of the reduced ANOVAmodel are shown in Fig. 2. The direct, Bishop, and complete ana-lytical methods all have least-significant difference (LSD) bars(a measure of confidence) that overlap SF ¼ 1. The force methodcalculations, however, are well above SF ¼ 1, suggesting that forcemethods are, on average, significantly nonconservative.

As shown in Fig. 3, slope type and pore-water approach arestrongly interactive. Although effective stress analysis is fairly con-sistent near SF ¼ 1 across all slope types, total stress analysisvaries considerably for different slope types. It has been theorizedthat cut slopes and riverbanks fail at a higher SF than engineeredslopes (e.g., Ferkh and Fell 1994), but Fig. 3 suggests that thisrelationship is more complex, with effective stress analysis predict-ing a high SF for landslide failures but a low SF for cut failures andthe opposite trend for total stress analysis.

Perplexing as well is the strongly correlated nature of the totaland effective stress interaction. For each slope type, the predictedSF calculations are more or less reflected about the line defined bySF ¼ 1:03, with effective stress greater than total stress SF calcu-lations for landslide, fill, and experiment slopes, and total stressgreater than effective stress SF calculations for cut and test fillslopes. The explanation for this interaction is not obvious butmay be related to slip geometry differences. The cut and test fillslope failures tended to be deep and circular, passing throughthe piezometric surface and several different soil strata, whereasthe landslide, fill, and experimental failures tended to be relativelyshallow by comparison. A future paper will further consider thisrelationship.

Table 2. Reduced ANOVA Model

Source Sum of squares Degrees of freedom Mean square F p-value

Overall model 0.38 19 0.020 2.90 < 0:0001

Analytical method 0.073 3 0.024 3.58 0.0143

Type × method 0.15 12 0.013 1.86 0.0387

Type × approach 0.074 4 0.019 2.72 0.0300

Residual 1.92 281 0.00683

Lack of fit 0.14 15 0.00956 1.43 0.1316

Pure error 1.78 266 0.00668

Corrected total 2.29 300

Fig. 1. Normal probability plot for reduced ANOVA model Fig. 2. Main effects plot: Analytical method

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Fig. 4 shows the interaction effects between slope type andanalytical method. This relationship appears quite complicated,but much of the interaction is within error bar range and thereforesomewhat inconclusive. The exception is specific to the landslideslope type, where the force method is seen to have LSD bars wellabove 1 and also above the LSD bars of all the other analyticalmethods.

SF Deviations from 1

Overall, the ANOVA model predicted a least squares SF mean of1.03 or slightly higher than SF ¼ 1, as would be expected. This isinconsistent with the prevailing engineering opinion that 2DLEsafety factors are conservative, and thus one would suspect that,on average, the minimum safety factors would be significantly lessthan 1. The SF database was tested against the expected value ofSF ¼ 1, with the database partitioned by analytical method (theonly main effect contributor found in the ANOVAmodel). Indepen-dent one-sample t-tests were used for hypothesis testing of thelog-transformed SF values. The results are shown in Table 3.

Based on the t-tests of the log10 SF values, all methods exceptdirect method deviated significantly from SF ¼ 1. The Bishopmethod and the complete method both had mean SF values near1.04, an effect that was significant at p ¼ 0:05. The force methodSF values were a bit higher, with a mean of 1.10, which was sig-nificant at p ¼ 0:005. The direct method SF mean was 0.98(p ¼ 0:357). Assuming a normal distribution for the SF (e.g.,not transforming the values) increased the significance of the differ-ence with SF ¼ 1 for all methods except direct method (where itactually decreased), and thus did not change any of the hypothesistest conclusions.

Plasticity Index

Although the vane strength correction factor has been in existencefor more than three decades, only 24% (10 of 43) of the SFcalculations that were based on vane strength were corrected ac-cording to Bjerrum (1972, 1973) or any other vane strength correc-tion factor formulas. Also, the SF calculated with uncorrected vanestrength averaged 1.03, whereas the corresponding corrected SFaveraged 1.12, a difference that is opposite of what would beexpected.

Despite the seeming lack of success regarding the vane strengthcorrections, the general principle of correction by regression is sup-ported by Fig. 5, which shows both the corrected and uncorrectedsafety factors versus PI for the vane strength based data [only stud-ies published after Bjerrum’s (1972) paper were considered]. Thecorrected values are seen to be closely oriented about its regressedline (R2 ¼ 0:61), whereas the uncorrected values exhibit wide scat-ter (R2 ¼ 0:16). Surprisingly, the regressed lines onto the correctedand uncorrected SF are nearly coincident. Although the implica-tions of this relationship are not clear, the results inarguably indi-cate that the vane strength correction alone does not account for SFdeviation.

Soils without Clay

The bulk of the literature on failed slopes is specific to clay soils;indeed, almost 90% of the slopes in the current database had soilswith significant amounts of clay. The calculated SF (averagedacross methods and analysts) for the 17 slope failures in the data-base that did not have appreciable clay are shown in Fig. 6. Awiderange of SF is seen with an average of 1.03 (based on log-trans-formed SF) and sd ¼ 0:076, consistent with the overall results.

Discussion

Analytical Method

The ANOVA testing and one-factor t-tests indicate that the directmethods of solution averaged closer to the expected value of SF ¼1 than alternative methods for calculating SF for unstable slopes. Inparticular, SF calculation based on the force method appeared tohave a nonconservative bias, whereas the average deviation from

Fig. 3. Interaction effects plot: Slope type versus pore-water approach

Fig. 4. Interaction effects plot: Slope type versus analytical method

Table 3. T-Test Results of SF Database (Based on log10 SF)

Statistic Method

Direct Bishop Force Complete

Median 1.00 1.02 1.08 1.03

Mean 0.98 1.04 1.10 1.05

sd (of log10 SF) 0.093 0.087 0.089 0.064

n 83 134 43 41

T 0.93 2.11 3.19 2.08

p 0.357 0.036 0.003 0.044

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1 for the SF calculations from the Bishop and complete methodsof analysis, although significant, tended to be more moderate.Considerable scatter was evident across all methods.

A number of factors likely contribute to these observed differ-ences between methods, including the following:1. All higher-order analytical methods were developed to account

for interslice forces. The direct methods, however, are all in-dependent of the interslice forces except OMS, which assumesthe interslice forces equal zero. This assumption of zero inter-slice forces is obviously inaccurate for slopes with an SF that issignificantly greater than one but may approximate reality as aslope nears failure and the pore-water pressure within the slopesignificantly changes (Moore 1970). Pore-water pressure canrapidly increase at failure of a saturated slope, possibly leadingto liquefaction of the soil and thus complete loss of the inter-slice forces, as noted by Iverson (1997). Of course, in the samestudy, Iverson also found that the pore-water pressure droppedand soil dilation occurred at failure in unsaturated soils; thus,one might expect the corresponding interslice forces to in-crease. Therefore, the only sure conclusion is that regardless ofthe saturated condition of the soil, the interslice forces change

rapidly near failure and are not likely to be modeled correctlyby any limit equilibrium method that uses interslice forcemodeling but fails to account for force changes near failure.

2. The direct methods have few applicability limits and produce asingle solution, whereas higher-order methods have associatedapplicability limitations and multiple solutions, the interpreta-tion of which may require the analyst to apply engineeringjudgment and/or physical reasoning.

3. One of the advantages of higher-order methods is the ability tomodel irregular and thus arbitrary failure surfaces, whichwould seem to allow better modeling of failure and a lowerpredicted SF. However, unless the analyst already has an ideaof the potential critical failure surface geometry, it may takemany trials to find the critical failure surface. Moreover, thehigher-order methods are computationally intensive, requiringnumerous iterations to achieve a solution to even a single fail-ure surface. Therefore, time and patience act as constraintsagainst achieving the critical failure geometry in a higher-ordermodel. Thus, it may be that determining the critical slip surfaceis more important for accurate SF calculation than typicallyassumed in practice.

Fig. 5. Slope minimum safety factors versus plasticity index

Fig. 6. Failed slopes in soils without significant clay

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4. The apparent accuracy of the direct methods may actuallyresult from inherent compensating errors, perhaps from highlysimplified failure geometries, which would tend to increase thecalculated SF. This is balanced by the inadequate representa-tion of the interslice forces, which may decrease the calcu-lated SF.Whatever the reason, the direct methods do appear to predict SF

near 1 for failed slopes more successfully than other methods.However, it should not be presumed that this result would alsoapply to stable slopes. Indeed, as previously discussed, at highSF values, soil response to pore-water pressures would be less pro-nounced, making the interslice forces more likely to be consistentwith the assumptions of the higher-order methods.

Correction Factors

Historically, high SF values for failed slopes have been attributed tothe complicating presence of clay; however, it appears that calcu-lated SF values above 1 also occur on average for slopes with soilswithout significant clay. Moreover, although the vane strength cor-rection factors applied in the database did reduce the calculatedminimum SF, these corrected SF values were still significantlyabove 1. Therefore, although the database supports the ongoinguse of SF-reducing correction factors, it appears that the correctionfactors as currently applied are not sufficient by themselves toreduce SF calculations to the expected value of SF ¼ 1. The data-base also does not support the use of correction factors that raiseSF, such as those sometimes introduced to account for three-dimensional effects or vane strength at low PI values.

Risk Analysis

Risk analysis as applied to stability analysis requires two criticalcomponents. The first is an accurate deterministic model thatproduces a mean SF prediction of 1.0 for real slope failures.The second is an accurate statistical model that can be used toproduce probabilities of failure.

With regard to the first requirement, the database analysis indi-cates that direct methods of solution are more successful than othermethods for predicting an average SF close to 1 for slope failures.However, although the Bishop and the complete methods of solu-tion were found to significantly deviate from 1 from a statisticalstandpoint, the actual extent of this deviation is small enough thatit is probably not relevant from a practical standpoint. Force meth-ods of solution, however, must be applied with caution becausethere is evidence of significant nonconservative bias to the calcu-lated SF.

As for the second requirement, both a Box-Cox analysis and asubsequent residual analysis indicate that a lognormal distributionprovides an adequate fit.

In terms of the general implications for risk analysis, a log10transform of all of the data results in a mean of log10 1.03 anda standard deviation of 0.087. Ignoring the effects of the analyticalapproach, method, or slope type, the SF corresponding to a 99%chance of safety would therefore be the antilog of the total of log101.03 plus a standard deviation of 2.3, or approximately 1.65. This isclose to several values calculated by Christian et al. (1994), whoused a direct error analysis approach (propagation of error) and as-sumed a normal distribution for SF (as opposed to a lognormal dis-tribution). The consistency of the solutions from these two quitedifferent approaches suggests that if comparable values for meanand standard deviation are used, SF risk analysis may be robustwith regard to assumed distribution.

Beyond allowing statistical observations and factor reduction,the reduced ANOVA model may be used to predict SF failure val-ues as a function of slope type, analytical method, and pore-water

approach. To discern trends and identify extreme predictions, sev-eral charts were generated from the reduced ANOVA model. Theseare shown in Figs. 7 and 8, corresponding to the total stressand effective stress pore-water approaches, respectively. Table 4tabulates the predicted mean SF.

The prediction charts show that the total stress analysis tendsto predict failure at higher SF values than the effective stressapproach. However, the maximum values of both pore-water ap-proaches were similar and specific to the force analytical method,with the total stress approach predicting failure at SF ¼ 1:23 for cutslopes and the effective stress method predicting failure at SF ¼1:27 for landslides. The minimum values were less consistent, withthe total stress approach showing a minimum SF of 0.91 for fillslopes analyzed by the direct analytical method, whereas the effec-tive stress approach showed a minimum SF of also 0.91 for cutslopes analyzed by the Bishop method. The direct method pre-dicted failure closer to SF ¼ 1 (across all slope types) better thanthe other methods, although the prediction accuracy of the Bishopmethod was also close to SF ¼ 1. The variability was comparablebetween effective and total stress methods in general. Averagedover all slope types, the direct method was the least variable

Fig. 7. ANOVA predictive model for average failure SF (total stresspore-water approach)

Fig. 8. ANOVA model for average failure SF (effective stress pore-water approach)

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method for the effective stress approach but the most variable forthe total stress approach. The force method was the most variableprediction method for the effective stress approach. The completemethod was the least variable prediction method for the total stressapproach.

For design, it is possible to extend the reduced ANOVA modelto predict the SF values corresponding to a given risk of failure.Although any desired risk of safety can be considered with themodel as given, Table 5 lists the minimum predicted SF neededfor a 1% failure risk. The wide range of predicted values justifiesthe need to account for the contributing SF factors. The total andeffective stress analyses show little deviation, both averaging to apredicted SF ¼ 1:75. In general, the force methods predict thehighest SF values, averaging across all slope types SF values of1.89 and 1.88 for the effective and total stress approaches, respec-tively. The largest single predicted value is SF ¼ 2:16 for land-slides analyzed by the force method and the effective stressapproach. At the low end, fill slopes analyzed by direct methodsand the total stress approach predict SF ¼ 1:50. Direct methodsusing the effective or total stress approach have the lowest averagevalues of any method, with predicted SF values of 1.65 and 1.66,respectively. Direct method prediction using the effective stressapproach also shows the least variability across all slope types,whereas complete methods show the least variability across allslope types for the total stress approach.

Conclusions

The database compiled here, while broad in terms of time and geog-raphy, tells only half of the story. Many slopes with low safetyfactors do not fail, and a full statistical consideration of SF shouldconsider a random selection of these stable slopes as well. None-theless, the following conclusions may be made from the failuredata considered in this paper:1. Different limit equilibrium algorithms produce different safety

factors. For failed slopes, the direct methods of SF calculationappear to be the most successful at predicting SF ¼ 1. TheBishop method and the complete methods of solution appearto have a slight (but significant) nonconservative bias, but themagnitude of this bias is so small that it is likely undetectablein field applications. Force methods, however, demonstrated alevel of bias that may have a significant, nonconservative effecton SF calculation. It is not known, however, if the bias anduncertainty shown by the failure database can be generalizedto stable slopes.

2. Clay content complicates SF analysis. The database indicatedthat correction factors for vane strength are not adequate toreduce predicted SF values to average at SF ¼ 1, as expected.The relationship between plasticity index and safety factor ap-pears to be more complicated than historically assumed. Thatbeing said, there was no evidence that soils without clay weredifferent from soils with clay with regard to SF uncertaintyor bias.

3. Overall, the database was best described by a log10 lineardistribution with a mean value of 1.03 and a standard deviation(of the log-transformed values) of 0.087. A 1% failure risk forSF of approximately 1.65 was calculated from the overall da-tabase. Moreover, the reduced ANOVA model can be appliedin a general way to risk analysis, providing predictions for agiven failure risk as a function of analytical method, slope type,and pore-water stress approach.Regardless of interpretation, the results of the meta-analysis

of these 301 slope failure calculations show that although slopestability analysis is unavoidably uncertain, it is also well describedby statistical modeling. The present effort considered a globalapproach to understand and model this underlying stochastic for-mation, and although this is useful, it cannot take the place of site-specific risk analysis by error propagation, which directly accountsfor project-specific observations, such as soil heterogeneity andpore-water pressure uncertainty. The primary benefits of the presenteffort is that it is applicable both at the site level, where it can beused as a check on the slope-specific risk analysis, and at the regu-latory level to guide responsible and informed policy decisions onslope stability issues in general.

Notation

The following symbols are used in this paper:adj-R2 = adjusted R2;

F = F-test statistic;p = probablity;

pred-R2 = predicted R2;R2 = regression coefficient;sd = standard deviation;T = t-test statistic;x0 = Cartesian horizontal axis rotated to align with

slope surface;

Table 4. Mean Failure SF ANOVA Model Predictions

Analysis Slope type

Landslide Cut Fill Test fill Experiment

Complete

Effective 1.07 0.91 1.06 1.00 1.14

Total 0.96 1.09 1.05 1.07 1.01

Force

Effective 1.27 1.03 1.06 1.04 1.15

Total 1.15 1.23 1.05 1.11 1.02

Bishop

Effective 1.03 0.91 1.13 0.95 1.07

Total 0.93 1.09 1.12 1.01 0.95

Direct

Effective 1.00 0.93 0.92 1.02 1.04

Total 0.91 1.12 0.91 1.09 0.92

Table 5. 1% Failure Risk SF ANOVA Model Predictions

Analysis Slope type

Landslide Cut Fill Test fill Experiment

Complete

Effective 1.80 1.59 1.75 1.68 1.99

Total 1.62 1.89 1.73 1.79 1.73

Force

Effective 2.16 1.82 1.77 1.73 1.97

Total 1.94 2.12 1.75 1.86 1.74

Bishop

Effective 1.70 1.53 1.86 1.56 1.82

Total 1.56 1.85 1.85 1.69 1.61

Direct

Effective 1.71 1.57 1.52 1.71 1.74

Total 1.54 1.89 1.50 1.81 1.56

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y0 = Cartesian vertical axis rotated perpendicularly toslope surface; and

ϕ = Coulomb friction angle.

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