discussion of "probabilistic slope stability analysis for practice"

DISCUSSION / DISCUSSION

Discussion of “Probabilistic slope stability analysisfor practice”1

J. Michael Duncan, Michael Navin, and Thomas F. Wolff

Duncan et al. 850

The authors have made an important contribution to theliterature on the reliability of slopes. This discussion focuseson four aspects of the paper, in this order: (1) the importanceof spatial correlation in reducing variance associated withthe physical properties of soil strata, (2) the relationship be-tween reliability index (β) and probability of unsatisfactoryperformance (Pu), (3) the importance of identifying the criti-cal failure mechanism, and (4) the advantages and disadvan-tages of using Microsoft® Excel and @Risk for evaluatingthe reliability of slopes.

Variance reduction due to spatial averaging

In the opinion of the discussers, the most significant con-tribution of the paper is that it shows very clearly the impor-tance of the reduction in the uncertainty due to soilvariability as a result of spatial correlation of soil properties.This is illustrated by the results in Table 3, where the com-puted value of Pu is reduced 70–80% when variance reduc-tion is incorporated in the calculations to account for spatialcorrelation.

It is not necessary to use Microsoft® Excel and @Risk totake this reduction in variance into account. It can be doneas well using existing slope stability programs combinedwith the first-order second-moment (FOSM) method. Asnoted below, using the FOSM method has significant advan-tages.

In the opinion of the discussers, the greatest need in thisarea is for simple methods for evaluating autocorrelation dis-tance. If this important concept is to be widely incorporatedin probabilistic analyses of slope stability, it is imperativethat simple methods be available for estimating auto-correlation distance, using the types and amounts of data

that are available in practice. The amount of data availablefor the James Bay project is unusually large. Methods thatrequire such large amounts of data will not find widespreaduse in practice.

Relationship between � and Pu

Although the differences are of little practical signifi-cance, the discussers point out that they have not been ableto confirm the values of Pu shown in Table 3. Those valuesare shown in Table D1, together with the values thediscussers believe are correct.

Also, for the FOSM analysis with assumed lognormal dis-tribution of the factor of safety, the discussers find Pu =2.13 × 10–3, rather than the value Pu = 2.5 × 10–3 shown inthe note below Table 3.

Although differences in values if Pu of these magnitudes(27–63%) are of marginal significance with respect to practi-cal applications, they could lead to confusion for readersinterested in using the information in Table 3.

Importance of identifying the critical failuremechanism

Because the authors used the Bishop method of slices,they were only able to analyze circular slip surfaces. Theminimum factor of safety from their analyses was 1.46. Thediscussers used Spencer’s method (Spencer 1967), withUTEXAS4 (Shinoak Software, Austin, Tex.), to computefactors of safety for wedge-shaped and curved noncircularsurfaces, and found a more critical failure mechanism, witha factor of safety equal to 1.17. The critical circular,wedge-shaped and curved noncircular surfaces are shown inFig. D1.

The lower factor of safety for the curved noncircular sur-face results in a significantly higher probability of unsatis-factory performance, as shown in Table D2 and Fig. D2.Values of Pu for the curved noncircular surface range from0.19 to 0.26, i.e., 8–90 times the values for the circular slipsurface. This large difference shows the critical importanceof correctly identifying the critical failure mechanism. Nomatter how sophisticated the methodology used for comput-ing values of Pu, the results of a reliability analysis will not

Can. Geotech. J. 40: 848–850 (2003) doi: 10.1139/T03-030 © 2003 NRC Canada

848

Received 29 November 2002. Accepted 13 March 2003.Published on the NRC Research Press Web site athttp://cgj.nrc.ca on 11 August 2003.

J.M. Duncan2 and M. Navin. Department of Civil andEnvironmental Engineering, Virginia Tech, Blacksburg, VA24061, U.S.A.T.F. Wolff. Michigan State University, East Lansing, MI48824, U.S.A.

1Appears in Canadian Geotechnical Journal, 39: 665–683.2Corresponding author (e-mail: [email protected]).

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be meaningful unless they are based on the critical failuremechanism.

Advantages and disadvantages of usingMicrosoft® Excel and @Risk for evaluatingreliability of slopes

In the opinion of the discussers, the advantages of usingMicrosoft® Excel and @Risk for evaluating reliability ofslopes are outweighed by the disadvantages, for the follow-ing reasons:(1) Microsoft® Excel with @Risk can only be used to ana-

lyze circular slip surfaces in the form that is describedin the paper. In many conditions, like those at the JamesBay dykes, noncircular surfaces are significantly morecritical than circular surfaces. Basing reliability analyses

on a noncritical failure mechanism can greatly underes-timate the value of Pu, as shown in Table D2 andFig. D2.

(2) The very important effect of variance reduction due tospatial averaging, shown clearly in the paper, does notrequire the use of Microsoft® Excel and @Risk. Asshown in Table 3, essentially the same value of Pu iscalculated when variance reduction due to spatial aver-aging is included in the FOSM method. The importanteffects illustrated in the paper are due to variance reduc-tion caused by spatial averaging, not by the use ofMicrosoft® Excel and @Risk.

(3) The simplified method that has been described by thediscussers (Wolff 1994; U.S. Army Corps of Engineers1999; Duncan 2000), combining a slope stability pro-gram with the FOSM method, is applicable to a widervariety of problems and conditions than are Microsoft®

© 2003 NRC Canada

Duncan et al. 849

Value of β shownin Table 3

Value of Pu shownin Table 3

Discussers’ values of Pu, from Ang and Tang(1975), and the Microsoft® Excel functionNORMSDIST; same values from both sources

2.42 8.40×10–3 7.76×10–3

1.84 2.37×10–2 3.29×10–2

Table D1. Values of reliability index, β, and corresponding values of unsatisfactory perfor-mance, Pu.

Variance reductiondue to spatialaveraging?

Distribution offactor of safety

Failuremechanism Pu

Ratio of Pu:noncircular/circular

No Normal Noncircular 2.5×10–1 8Circular 3.3×10–2

Yes Normal Noncircular 1.9×10–1 24Circular 7.8×10–3

No Lognormal Noncircular 2.6×10–1 16Circular 1.6×10–2

Yes Lognormal Noncircular 1.9×10–1 90Circular 2.1×10–3

Note: Variance reduction using the reduced value of standard deviation shown in Table 3.

Table D2. Values of Pu for circular and noncircular failure surfaces computed using vari-ous assumptions regarding variance reduction and distribution of factor of safety.

Fig. D1. Critical circular, wedge, and noncircular slip surfaces for the James Bay dykes.



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850 Can. Geotech. J. Vol. 40, 2003

Excel and @Risk. The FOSM method can be used as anadjunct to any method of deterministic analysis, includ-ing analyses performed using any slope stability com-puter program.

References

Ang, A.H-S., and Tang, W.H. 1975. Probability concepts in engi-neering planning and design. Vol. 1. Basic principles. JohnWiley, New York.

Duncan, J.M. 2000. Factors of safety and reliability in geotechnicalengineering. Journal of Geotechnical and GeoenvironmentalEngineering, ASCE, 126: 307–316.

Spencer, E. 1967. A method of analysis of the stability of embank-ments assuming parallel inter-slice forces. Géotechnique, 17(1):11–26.

U.S. Army Corps of Engineers.1999. Risk based analysis in geo-technical engineering for support of planning studies, ETL1110–2–556, Department of the Army, U.S. Army Corps ofEngineers, Washington, DC. [Available online at www.usace.army.mil/usace-docs.]

Wolff, T.F.1994. Evaluating the reliability of existing levees. Re-port of a research project entitled: Reliability of existing levees,prepared for U.S. Army Engineer Waterways Experiment Sta-tion Geotechnical Laboratory, September 1994.

Fig. D2. Variation of Pu with factor of safety (FS) for circular and noncircular slip surfaces.



discussion of "probabilistic slope stability analysis for practice"

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