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848
DISCUSSION / DISCUSSION
Discussion of  Probabilistic slope stability analysis
for practice 1
J. Michael Duncan, Michael Navin, and Thomas F. Wolff
Duncan et al. 850
The authors have made an important contribution to the that are available in practice. The amount of data available
literature on the reliability of slopes. This discussion focuses for the James Bay project is unusually large. Methods that
on four aspects of the paper, in this order: (1) the importance require such large amounts of data will not find widespread
of spatial correlation in reducing variance associated with use in practice.
the physical properties of soil strata, (2) the relationship be-
tween reliability index (²) and probability of unsatisfactory
performance (Pu), (3) the importance of identifying the criti- Relationship between and Pu
cal failure mechanism, and (4) the advantages and disadvan-
Although the differences are of little practical signifi-
tages of using Microsoft® Excel and @Risk for evaluating
cance, the discussers point out that they have not been able
the reliability of slopes.
to confirm the values of Pu shown in Table 3. Those values
are shown in Table D1, together with the values the
Variance reduction due to spatial averaging
discussers believe are correct.
In the opinion of the discussers, the most significant con- Also, for the FOSM analysis with assumed lognormal dis-
tribution of the paper is that it shows very clearly the impor- tribution of the factor of safety, the discussers find Pu =
2.13 × 10 3, rather than the value Pu = 2.5 × 10 3 shown in
tance of the reduction in the uncertainty due to soil
the note below Table 3.
variability as a result of spatial correlation of soil properties.
This is illustrated by the results in Table 3, where the com- Although differences in values if Pu of these magnitudes
puted value of Pu is reduced 70 80% when variance reduc- (27 63%) are of marginal significance with respect to practi-
tion is incorporated in the calculations to account for spatial cal applications, they could lead to confusion for readers
correlation. interested in using the information in Table 3.
It is not necessary to use Microsoft® Excel and @Risk to
take this reduction in variance into account. It can be done
as well using existing slope stability programs combined Importance of identifying the critical failure
with the first-order second-moment (FOSM) method. As
mechanism
noted below, using the FOSM method has significant advan-
Because the authors used the Bishop method of slices,
tages.
they were only able to analyze circular slip surfaces. The
In the opinion of the discussers, the greatest need in this
minimum factor of safety from their analyses was 1.46. The
area is for simple methods for evaluating autocorrelation dis-
discussers used Spencer s method (Spencer 1967), with
tance. If this important concept is to be widely incorporated
UTEXAS4 (Shinoak Software, Austin, Tex.), to compute
in probabilistic analyses of slope stability, it is imperative
factors of safety for wedge-shaped and curved noncircular
that simple methods be available for estimating auto-
surfaces, and found a more critical failure mechanism, with
correlation distance, using the types and amounts of data
a factor of safety equal to 1.17. The critical circular,
wedge-shaped and curved noncircular surfaces are shown in
Received 29 November 2002. Accepted 13 March 2003.
Fig. D1.
Published on the NRC Research Press Web site at
The lower factor of safety for the curved noncircular sur-
http://cgj.nrc.ca on 11 August 2003.
face results in a significantly higher probability of unsatis-
J.M. Duncan2 and M. Navin. Department of Civil and factory performance, as shown in Table D2 and Fig. D2.
Environmental Engineering, Virginia Tech, Blacksburg, VA
Values of Pu for the curved noncircular surface range from
24061, U.S.A.
0.19 to 0.26, i.e., 8 90 times the values for the circular slip
T.F. Wolff. Michigan State University, East Lansing, MI
surface. This large difference shows the critical importance
48824, U.S.A.
of correctly identifying the critical failure mechanism. No
1
matter how sophisticated the methodology used for comput-
Appears in Canadian Geotechnical Journal, 39: 665 683.
2
Corresponding author (e-mail: [e-mail]). 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
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Duncan et al. 849
Table D1. Values of reliability index, ², and corresponding values of unsatisfactory perfor-
mance, Pu.
Discussers values of Pu, from Ang and Tang
Value of ² shown Value of Pu shown (1975), and the Microsoft® Excel function
in Table 3 in Table 3 NORMSDIST; 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 D2. Values of Pu for circular and noncircular failure surfaces computed using vari-
ous assumptions regarding variance reduction and distribution of factor of safety.
Variance reduction Ratio of Pu:
due to spatial Distribution of Failure noncircular/
Pu
averaging? factor of safety mechanism circular
No Normal Noncircular 2.5×10 1 8
Circular 3.3×10 2
Yes Normal Noncircular 1.9×10 1 24
Circular 7.8×10 3
No Lognormal Noncircular 2.6×10 1 16
Circular 1.6×10 2
Yes Lognormal Noncircular 1.9×10 1 90
Circular 2.1×10 3
Note: Variance reduction using the reduced value of standard deviation shown in Table 3.
Fig. D1. Critical circular, wedge, and noncircular slip surfaces for the James Bay dykes.
on a noncritical failure mechanism can greatly underes-
be meaningful unless they are based on the critical failure
timate the value of Pu, as shown in Table D2 and
mechanism.
Fig. D2.
(2) The very important effect of variance reduction due to
Advantages and disadvantages of using spatial averaging, shown clearly in the paper, does not
require the use of Microsoft® Excel and @Risk. As
Microsoft® Excel and @Risk for evaluating
shown in Table 3, essentially the same value of Pu is
reliability of slopes
calculated when variance reduction due to spatial aver-
In the opinion of the discussers, the advantages of using aging is included in the FOSM method. The important
Microsoft® Excel and @Risk for evaluating reliability of effects illustrated in the paper are due to variance reduc-
slopes are outweighed by the disadvantages, for the follow- tion caused by spatial averaging, not by the use of
ing reasons: Microsoft® Excel and @Risk.
(1) Microsoft® Excel with @Risk can only be used to ana- (3) The simplified method that has been described by the
lyze circular slip surfaces in the form that is described discussers (Wolff 1994; U.S. Army Corps of Engineers
in the paper. In many conditions, like those at the James 1999; Duncan 2000), combining a slope stability pro-
Bay dykes, noncircular surfaces are significantly more gram with the FOSM method, is applicable to a wider
critical than circular surfaces. Basing reliability analyses variety of problems and conditions than are Microsoft®
© 2003 NRC Canada
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850 Can. Geotech. J. Vol. 40, 2003
Fig. D2. Variation of Pu with factor of safety (FS) for circular and noncircular slip surfaces.
Spencer, E. 1967. A method of analysis of the stability of embank-
Excel and @Risk. The FOSM method can be used as an
ments assuming parallel inter-slice forces. Géotechnique, 17(1):
adjunct to any method of deterministic analysis, includ-
11 26.
ing analyses performed using any slope stability com-
U.S. Army Corps of Engineers.1999. Risk based analysis in geo-
puter program.
technical engineering for support of planning studies, ETL
1110 2 556, Department of the Army, U.S. Army Corps of
References
Engineers, Washington, DC. [Available online at www.usace.
Ang, A.H-S., and Tang, W.H. 1975. Probability concepts in engi- army.mil/usace-docs.]
neering planning and design. Vol. 1. Basic principles. John Wolff, T.F.1994. Evaluating the reliability of existing levees. Re-
Wiley, New York. port of a research project entitled: Reliability of existing levees,
Duncan, J.M. 2000. Factors of safety and reliability in geotechnical prepared for U.S. Army Engineer Waterways Experiment Sta-
engineering. Journal of Geotechnical and Geoenvironmental tion Geotechnical Laboratory, September 1994.
Engineering, ASCE, 126: 307 316.
© 2003 NRC Canada