FACTOR OF SAFETY AND RELIABILITY

IN GEOTECHNICAL ENGINEERING"

Discussion by Claudio Cherubini'

This paper is very interesting. highlighting the imponanceof the variability of soil properties in geotechnical calculations.The statistical and probabilistic techniques proposed are sim­ple and very useful in current practice.

Some observations and considerations can. however. bemade:

Regarding the variability of soil properties. some furtherpapers can be mcmioned where coefficients of variationare reported and commented upon (Becker 1996; Cheru­bini 1997: Phoon and Kulhawy 1999).

From Cherubini (1997) for example. the range of co­efficients of variation (COV) of effective cohesion (be­twecn 13 and 70%) can be deduced wherever. from thepaper of Becker (1996). among others. are reported co­efficients of variation of design model uncenainty (5­15%). design decision uncertainty (15-45%). and con­struction variability (5-15%).

Also. in their paper, Phoon and Kulhawy evidencesome COY values regarding "inherent" or "intrinsic" soilvariability. without any external influence (incorrect sam­pling, errors of calculation, and so on) tllat could causeadjunctive variability. From this paper and that by Che­rubini (1997). it is evidenced thaL the COY of frictionangle for clay is higher lhan the COY for sand. Dealingwilh standard deviation, all the variabilities for frictionangle are in t.he range between 1.5 and 5°.Concerning the possibility to obtain coefficients of vari­ation and means from estimated values of a propcny, wecan lIi'e the following expressions (Cherubinj and Orr1999).

For the mean:

0+ 4b + cX'" i:

6

which. togelher with (5) from the original paper. gives lhecoefficient of variation

c-aCOV. '" -=----=--­

a+4b+c

where a = estimated minimum vaJue: b = most likelyvalue: and c =estimated maximum value.

Alternatively, knowing the range (minimum minusmaximum value) for a limited set of data. it is possiblelO use the lable given by Snedecor and Cochran (1964)to obtain the standard deviation in a function of the num­ber of values considered.

Knowing the mean and standard deviation. it is possibleto determine the so called "characteristic value" accord­ing to Eurocode 7 (Cherubini and Orr 1999: Orr and Far­rell 1999).A comparison bel ween calculated and measured values of-for eX:lmple. the settlement of :l shallow foundation,can be rationally perfomled (Cherubini and Orr 2(00)evaluating

-April 2000. Vol. 126. No.4. by J. M. Duncan (Paper 20950).~FuIJ Prof. of Engrg. Gcol.; formerly. Assoc. Prof. of Soil Mcch .. Tech.

Univ. of Bad. 70125 Bolri. Italy.

ith calculated valuek =====--==, ith measured value

with" v::rrying from I to /I and calculating the mean of k"values as a measure of "accuracy" and its standard de­viation as a measure of "precision:' Some indexes havebeen proposed, considering accuracy and precisionlumped logether. as the ranking index (Briaud and Tucker1998) and the ranking dislance (Cherubini and Orr 2000).The author's st3temenl "that the probability of failureshould not be viewed a~ a replacement for factor of safetybUI as a supplement" is acceptable at present. but the dis­cusser thinks that. in the fUlure. complelely detenninisticevaluations of safety will disappear. According to Lacasse(1994). "the fact thal one finds difficult the quantifyingof the uncertainties is not a valid reason 10 evade definingthe uncertainties or establishing their significance on theresults obtained. On the contrary. the greater the uncer­tainties. the more urgent is the need for reliability analy­sis:'

For all these reasons. this p:lper represents a significant con­tribution towards reliability evaluation in geotechnics.

REFERENCES

Becker. D. E. (1996). "Eiglueenlh Canadian Geotechnical Colloqium:Limit Slates design for foundalions. II: Development for the NationalBuilding Code of Canada." Call. Gp01n.:h. J., Onawa. 33. 984-1007.

Bri'lUd. J. L.. and Tucker. L. M. (1998). "Measured and predicled axialresponse of 98 piles:' 1. Geo/l'ch. Engrg.. ASCE. 114(9),984-1001.

Cherubini. C. (1997). "Data and con:.idcrations on the variabilily of geo~technical properties of soils:' Proc.. ESR£L '97. Lisbon. Portugal. 2.t583-1591.

Cherubini. c.. and Orr. T. L. L. (1997). "Considerations on the applica­bility of semi-probabilistic Bayesian ll1elhods to geotcchnical design."Pmc.. 20,h COIl\'egl1o Na:imwle di Geofecllica. Panna. Italy, 421--426.

Cherubini. C.. and Orr. T. L. L. (2000)...A ntlional procedure 10 comparemeasured and calculated values in geotcchnics.·· Pmc., IS· )'okohamll,Yokohama. Jllpan. in press.

Lacasse. S. (199-1). "ReliabililY and probabilistic lllclhods:' Proc.• /3thI",. Cmif. 011 Soil. Mech. lind Found. £"g~.• New Delhi. India. 225­227.

Orr. T. L. L.. and F:lITell. E. R. (1999). Geotechnical desig" to £lIrQcode7. Springer. London.

Snedecor. G. W.. and Cochmn. W. G. (1964). SflIfisricaJ methods. Uni·versity of Iowa Press. Iowa City. Iowa.

Discussion by John T. Christian,'Fellow, ASCE, and Gregory B. Baecher:

Member ASCE

The discussers strongly agree with the author's position thatthe reliability of geotechnical systems can be evaluated bycombining modest computat.ional effort with data that are read­ily available on the engineering project. The discussers aJsosuppon the conclusion thai reliability analysis provides aframework for establishing approprime factors of safety andother design targets and leads to a better apprecimion of therelative importance of uncertainties in different parameters.Geotechnical engineers have long recognized thai they dealwith an uncertain world: reliability amllysis is a rational wayto grapple with it.

Daunting tcmlinology and nOlaLion encumber the fields ofreliability. probabilily, and statistics. and this has needlessly

'Consulting Engr.. Waban. MA.·Prof. and Chair, Depr. of Civ. and Envir. Engrg .. Univ. of Maryland.

Cullcgu Park. MD 20742.

700 I JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING I AUGUST 2001

prevented many practicing geotechnical engineers from takingadvamage of these powerful tools. The discussers commendthe author for his courage in presenting a straightforward ex­position of reliability methods without mystification and hopethe paper will encourage wider use of the techniques.

This discussion addresses issues raised by the paper in thehope that they might be subjects of further considerationwithin Ihe geotechnical community. They are the implicationsof the three-sigma rule, the assumption that factors of safetyare lognormally distributed, and the computational method ex­pressed by (2a).

THREE-SIGMA RULE

The paper lists three methods for establishing standard de­viations of uncertain variables in the order: (I) statistical com­putation from data; (2) reliance on published values; and (3)the three-sigma rule. The first two methods are clearly sound.However, it should be noted that many of the published valuesof variances in geotechnical properties are too large, becausethey include the measuremenl noise in addition to lhe actualvariability in the properties. As regards the three-sigma rule,the concept of bracketing an uncertain quantity and then as­sulning that the range corresponds to some number of standarddeviations is also sound. Certainly, when there are not enoughdala to compute the parametric values needed in an analysis,one must rely on approximations, and the three-sigma rule isone way to approximate slandard deviations. However, weknow from psychological studies that people have difficultyassessing the extent of their own uncertainty. When asked tostate the greatest and least values an uncertain quantity canhave, even statisticians report highly overconfident intervals-overconfident by manyfold. Morgan and Henrion (1990)summarize much of the work on this problem. The discusserssuspect that, whereas highly experienced engineer who arealso sophisticated in statistical issues might be able to inter­rogate themsel ves to obtain reasonable upper and lowerbounds on a soil property, it is too much to hope that lessexperienced engineers could do so without systematic and un­conservative bias. Thus, we are wary of the three-sigma rulein practice.

The literature on order statistics provides a useful first ap­prox.imation for estimating standard deviations when somedata are available. Consider that a sample of 11 observations ismade from a nomlal distribution (olher distributions could beused). One can calcu.late the expected values of the maximumand minimum of the n sample values and their difference,which is the range. This expected sample range is a functionof the standard deviation. Table 9 shows the expected rangein standard deviation units as a function of sample size n (Bur~

lington and May 1970). So, if we make, say, ten tests andobserve some range of values, we can estimate the standarddeviation as thai range divided by 3.078. Note that the three­sigma rule divides the range by 6.0.

DISTRIBUTION OF FACTOR OF SAFETY

The factor of safety is often assumed to be lognormallydistributed. Ahhough arguments on lhis topic often seem to belike the warfare among the Lilliputians, for whom the issuewas whether one should eat a boiled egg from the big end orthe small end, consequences do follow from the lognormalassumption.

There seem to be three reasons for assuming a lognormaldistribution. First, a lognormal distribution avoids negativevalues of factor of safety. In practice, the probability that anegative factor of safety will arise is insignificanl. For exam­ple, in the LASH case. the mean and the standard deviationof the factor of safety are 1.17 and 0.18, respectively. If thefactor of safety is normally distributed, the probability of anegative factor of safety is 4 X IQ-IJ. Second, there are manymultiplications and divisions in geotechnical computations,and by the Central Limit Theorem. the lognonllal distributionis a good representation for lhis situation. However, there aremany additions and subtractions as well. Furthermore. for theargument to be valid for small numbers of variables, the in­dividual variables should also be lognonnally distributed. Thethree-sigma rule assumes that variables are normally distrib­uted.

Third. it might be true that factors or margins of safetycalculated in practice really do have a lognormal distribution.The consolidation problem in the paper has a complicated in­leraction of parameters. If the maximum past pressure is heldconstant, C" and C,_ contribute linearly; if they are normallydistributed, so is the settlement. On the other hand, the max­imum past pressure contributes in a nonlinear manner. It is notimmediately clear what the distribution of the settlementshould be. but Monte Carlo simulation indicates it is closer toa nomlal than a lognormal distribut.ion. In the time-dependentanalysis. c.. appears in an exponential; when it is normal anddorninates, the settlemenl will closely approximate a lognor­mal distribution. Since the result of a realistically complicatedcalculation is not likely to follow a simple distribution, thediscussers prefer to assess the probability of failure by: (I)assuming a nomlal distribulion in the absence of other infor­mation: (2) using a margin of safety instead of a factor ofsafety as illustrated below: or (3) using the Hasofer-Lind(1974) approach to computing the reliability index without re~

tying on the distribulion of the margin or factor of safety.

COMPUTATIONAL METHOD

The derivation of (20) Slarts with the first-order approxi­mation for the variance of the factor of safety:

When the partial derivatives cannot be evaluated analytically,a central difference approximation is used:

TABLE 9. Number of Standard Deviations in Expected SampleRa.nge after Burlington and May (1970) (cr = Range/Nil)

n

23456789

1011

1.1281.6932.0592.3262.5342.7042.8472.9703.07H3.173

"12131415161718192030

3.2583.3363.4073.4723.5323.5883.6403.6893.7354.09

of F(x + ax,) - F(x - at;)-~

ox, 2'LU,

When lhe increments in each of the variables are made equalto the standard deviations. (2a) results. The arguments for us­ing this fonn of the first-order second-momenL (FOSM)method include that it is simple, that il eliminates multiplica­tion by the variances, and that it ensures that the evaluation iscarried out over a significant range of values of the variables.The discussers prefer to use. small values of the variables 10

compute the central differences in order to obtain the best es­limate of the derivalives. The additional effort is small, ~md

the technique avoids errors that arise from taking a secantapproximation to a tangent. In most cases lhe differences be-

JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / AUGUST 2001 /701

tween the two approaches are significant. but the followingexample shows that this is not always so.

EXAMPLE

The failure load P on a mine pillar is described by Karzu­lovic (1999) as

P = k. \V~I~HU~

where k ;;: design rock mechanics strength (DRMS) in MPa:H = height of the pillar in m; and Wc:ff;;: effect'ive width of thepillar in 111, defined as four times the cross-sectional area di­vided by the perimeter. The fonnula is used when Weff:S; 4.5H.Since the fonnula is empirical. it is likely to have bias anduncertainty. and the exponents also have uncertainty. Theseuncertainties are ignored in the following calculaLions.

Le[ lhe load per unit area on [he pillar be L. The safety ofthe pillar could be described by the factor of safely F (=PIL)or by the margin of safely M (=P - L). Obviously. the failurecondition can be expressed equally well as F = I or M = O.Data for pillars in several mines provided to the discussers byDr. Antonio Karzulovic (personal communicarion. 1999) yieldthe statistics presented in Table 10. All four variables are rea­sonably presented by normal distributions. For present pur­poses. the correlations among the variables are ignored.

The partiaJ derivatives of F or M can be evaluated directly,so the FOSM resuJts can be computed exactly. With the failurecriterion expressed in tenns of M. the probability of failurewas computed five ways: (I) FOSM with [he exact panialderivatives; (2) FOSM using (20); (3) [he Rosellblueth (1975,1981) poin[-estimatc method; (4) the Hasofer-Lind first-orderreliability method: and (5) Monte Carlo simulation with im­portance sampling. The results are shown in Table II. In thefirst three cases. the probability of failure was computed fromi3 with [he assumption [hat M was nomlUlly distributed. Allmethods give substantially the same results. and (2a) is asaccurate as <lny of them.

With the failure criterion expressed in terms of F, six com­pu[ations of probabili[y of failure were made using: (I) FOSMwith the exacl panial derivatives: (2) FOSM with (2a): (3)

TABLE 10. Mean!> and Standard Deviations of Parameters for MinePillar Analy!>is

Variable Mean Standard deviation Units

k 49.13 12.21 MPaWttl 13.85 2.9t 111H 4.00 0.20 111

L 33.66 [6.44 MPa

Method IJ..., a .. ~ PI

FOSM -direct 35.623 25.012 1.424 0.077FOSM-(2n) 35.623 25.012 1.424 0.077P-E 35.236 24.906 10415 0.079H-L 1.454 0.073M-C 0.076

FOSM with small increments in [he variables; (4) [he Rosen­blueth point-estimale method; (5) lhe Hasofer-Lind method;and (6) Monle Carlo simulation. In the third case. the incre­mems were 1.0 for k, 0.5 for W, 0.1 for H, and 1.0 for L. Inthe first four cases, the probability of failure was computedfrom rl with the assumption that F was normally distributed.The results are in Table 12. Tt is clear that. except for theHasofer-Lind and Monte Carlo results. the methods based onthe factor of safety disagree with each other and from thosebased on the margin of safety. Since the latter are consistentfor all methods, including Mont.e Carlo simulation. the errorsmust lie in the methods based on the factor of safety. Fur­thermore, the most inaccurate method is that using the factorof safely and (2ll).

It might also be assumed that the factor of safety is log­normally distributed. The mean and standard deviation of thelogarithm of F can be calculated from the mean and standarddeviation of F itself. The failure criterion is then In F;;: 0, and[he value of i3 for In F leads [0 a reviscd probability of failure.This procedure is codified in Table 2 of the original paper. Forthe present problem, i3 based on In F is 1.123, and the prob­ability of failure is 0.131.

A strong argument for a lognonnal distribution of F is thatit is computed by multiplying and dividing four variables. Fora very large number of variables, the Central Limit Theoremimplies that this is true for almost any distributions of theindividual variables. but, when the number of variables issmall, each variable should also be lognormally distributed. Ifeach variable is lognonnally distributed with the same meansand standard deviations used before, the expression for In Fbecomes a linear combination of nonnalty distributed varia­bles. The mean and standard deviation of F can be calculateddirectly. leading [0 i3 ; 1.429 and PI ; 0.077.

CONCLUSIONS

The author has presented a cogent argument for more wide­spread use of reliability methods in geotechnical engineeringand has demonstrated that it can be applied easily to a varietyo.f.geotechnical problems. The discussers agree with this po­SUlon and hope that the paper encourages wider use of relia­bility methods.

The discussers caution against uncritical use of the three­sigma approach. Experts of all sons have great difficulty es­timating the ranges of uncertain parameters. An alternativebased on a limited number of data has been discussed.

Analysis of the safely of mine pillars suggest that, whenone is dealing with a failure criterion such as F that is cal­culated primarily by multiplications and divisions. [he FOSMmethod with F nonnally distributed may be in error, and thesimplified version expressed by (2a) is least accurate. Assum­ing that F is lognonnally distributed improves the accuracybut still gives results that differ significantly from the correctvalues. Accurate results based on the factor of safety requireeither: (I) using the Hasofer-Lind procedure: (2) using MOilleCarlo simulation: or (3) describing each of the variables aswell as F by a lognonnal distribution. Failure estimates basedon the margin of safety are mutually consistent and accurate.

REFERENCES

Method fL. a .. ~ PI

FOSM--direct 2.058 1.151 0.920 0.179FOSM-(2a) 2.058 1.434 0.738 0.230FOSM-small increments 2.058 !.I5 [ 0.9t9 0.t79poE 2.688 1.544 1.093 0.t37H-L 1.454 0.073M-C 0.076

Burington. R. S.. and May. D. C.. Jr. (1970). Handbook of probabilityand statistics. McGrllw-Hili. ew York.

Hasofer. A. M .. and Lind. N. (1974). "Exact and inv:uiant M:Cond-mo·mentcode forma!.'·J. Engrg. Mech.• ASCE. 1000t). 111-121.

Morgan. M. G.. and Hennon. M. (1990). Uncemliflt)': II guide to dealingwith wlcertaintv in qIlOlltiwlil'l' risk mId policv cmafysis. CambridgeUniversity Press. Cambridge. U.K.

Roscnblucth. E. (1975). "Point estim:lIes for probabililY mOl1lents,"Prot:.. Nat. Academy of Sci .. 72(10). 3812-3814.

7021 JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGtNEERtNG 1AUGUST 2001

Rosenblueth. E. (1981). '"Two-point estimates III probabilities. ,. Al'pl. 6Mmht'maricol Model/ing. 512), 329-335.

,

FIG. 7. Probability An"lysis for Setllement Using SPT Dutre Average= 0.3 in.

0

IPROBABILITY ANALYSIS FOR SETTLEMENT,"- USING SPT DATA-AVERAGE=0.3IN

0 I... .~,I .. logN~ -

••••••• NonnM

0

,0

•• , ,, .0

~\f.4 I

.-,

J "i'" "0

.. '

, N.., ,1~

,

01.., .--- ", .

where O""~",,ll ;:: overall standard deviation: crmooel ;:: standarddeviation from model uncertainty: (J'not.<e ;:: standard deviationfrom measurement noise: and O"..,...al ;:: 'itandard deviation fromspatial variability.

The uncertainty from measurement noise for SP'T can be asbigh as 45-100% (Schmertmann 1978; Kuhawy 1996). Wick­remesinghe (1989) showed that measurement noise for pi­ezocone (CPTU) equaled 5% and dilatometer tests (DMT)equaled 6% at the McDonald Farm test site in Vancouver.From case study data (Schmertmann 1986), the coefficient ofvariation for the DMT model for predicting settlement was21 % when the dilmometer is pushed and excluding quickclayey silts (Failmezger et al. 1999).

As shown in Table 13 and Figs. 7 and S. the discusser an­alyzed the different probability distributions and the test and

thor uses seu..lement predictions based on SPT NbO values. Heevaluates the coefficient of vari:ltion of bow well the modelpredicts what has been measured based on case study data as67%. This high coefficient of variation is panly due to usinga dynamic penetration test to predict the static deformationproperties of sand. In addition to the uncertainty of the m.odel.there is uncenainty from measurement noise (test repeatabil­ity) and the spatial (subsurfaee) variabiliry of the site. Thediscusser believes that these sources of uncertainty are inde­pendent and should be summed using the following equation:

" ....., = Vl(".......)' + (,,_.j' + (",..,,,)'1 (12)

o

o.10 -08 -06 -04 -02000204 06 08 10 1.2 14 16 18 20

SETTLEMENT (inch)

•Discussion by Roger A, Failmezger,'Member, ASCE

'Pres. In-Situ Soil Testing. L.c.. 2762 White Chapel Rd.. Lancaster.VA 22503. E-mail: in.!>itusoil@prodigy.nel

The author has presented a practical approach for usingprobability in geotechnical engineering design. The numericexamples are very beneficial for undersumding probabilityconcepts.

The discusser believes there is an errOr in some of tile valuescontained in Table 2. The probability of failure should nOl

exceed 50% for an F/tI1V that exceeds 1.0. For a uniform prob­ability distribution (the distribution with the highest coefficientof variation and all possibilities equally likely) with an F~1Il

= 1.05 and limits from 0 to 2.1. the probability of failure =1.012.1, or 48%. In Table 7, the discusser feels that il is notpossible for the coefficient of variation to increase and theprobability of failure lO decrease as is the case for SR :::; 1.10.

The discusser also questions the use of the lognonnal prob­ability distribution for geotechnical design applications. Thelognormal distribution has limits of zero and positive infinity,and thus the distribution is always skewed to the left. Withprobability design. the engineer evaluates the area beneath theprobability distribution function at the tail ends. The failurezone will be the area beneath the left tail below 1.00 for faetorof safety based designs (Ihe factor of safety is the abscissa).For settlement based design, the failure zone wilJ be the areabeneath the right tail above a maximum settlement thresholdvalue (the settlement is the abscissa). Because of the left skew­ness of the lognonnal distribution, designs where the failurezone is along the left tail will tend to be conservative and thosewith the failure zone along the right tail will tend to be un­conservative.

The normal probability distribution function is symmetricalabout its mean and has limits from negative infinity to po itiveinfinity. These limits are not realistic and probably cause someerror when evaluating the failure zone. The discusser sugge tsthat a beta probability distribution be used because its limitscan be realiSiically chosen by the engineer (HaIT 1977). (Thenormal distribution is a subset of the beta distribution.) Wherethe average value occurs with respect to those limits will de­termine the skewness of the beta distribution.

In the example for determining the probability of unsatis­factory performance for settlement of footing on sand. the au-

TABLE 13. Analysis of Probabilily Distributions and Tesl and Analysis Melhods

Prob<tbility Test and Average Standard Devkltion From Overall Threshold Probabililyofdistribution analysis selllcmcl1t Spulial Measurement Model Sland<lrd settlemenl unsatisfaclory Probabilityfunction method (in.) variability noise error deviation" (in.) performance of success

Betti SPT 0.30 0.059 0.148 0.198 0.254 0.50 0.21 0.79Lognormal SPT 0.30 0.059 0.148 0.198 0.254 0.50 0.14 0.86Nonnal SPT 0.3U 0.059 0.148 0.198 0.254 0.50 0.21 0.79Bem SPT 0.30 0.059 0.t48 0.198 0.254 0.90 0.02 0.98Lognonnal SPT 0.30 0.059 0.148 0.198 0.254 0.90 0.03 0.97Normal SPT 0.30 0.059 0.148 0.198 0.254 0.90 0.01 0.99Beta DMT 0.30 0.059 0.Ot8 0.062 0.087 0.50 0.00 1.00Lognormal DMT 0.30 0.059 0.018 0.062 0.087 0.50 0.02 0.98Nonnal DMT 0.30 0.059 0.Ot8 0.062 0.087 0.50 0.01 0.99Beta DMT 0.30 0.059 0.Ot8 0.062 0.087 0.90 0.00 1.00Lognom1al DMT 0.30 0.059 0.018 0.062 0.087 0.90 0.00 1.00Normal DMT 0.30 0.059 0.018 0.062 0.087 0.90 0.00 1.00

·Sec Eq. (12).

JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / AUGUST 2001 /703

,

6Oc---,----------=-====-rl I PROBABILITY ANALYSIS FOR SETTLEMENT..I--I USING DILATOMETER DATA-AVERAGE:O.3IN

I I; .I---l--+-<.-:...~.-.. -+-~--+It.:.::,. :::........f-

;i~~\+--1---;

:/ \\ II---+-----;.Jf-t_"e.- -+--1'-

i------j!---E-I- ~~--+---:'--i-i-j----1+--+---,\.1!1-J--i---

1

1--t---t-

'\ I

0203040.50607080.910SETTLEMENT (inch)

FIG. 8. Probability Analysis for Sculcment U~ing Dilatomcler DUla:Average = 0.3 in.

analysis methods to determine their effects on the probabilityof unsatisfactory performance of exceeding a threshold seuJe­ment. The probability of success equaJs 1.0 minus the proba­biliry of unsatisfactory performance. Because the overall stan­dard deviation was so high in comparison to the average valuefor the SPT case, me left side of all three distributions wasdistorted (Fig. 7). The probability analysis For settlement. how­ever. focuses on (he riglll side of the distribution curve. Be­cause of its left skewness. the lognormal distribution for SPTwith a tlueshold settlement of 0.5 in. gave a higher probabilityof success (86%) (unconserv:.ltive) than bela or normal dislri­bULions (79%). A higher threshold selliemenl lessens Ihe effectof the probabilily distribution.

However, the choice of test and analysis method in Table13 had a much more signifieanL eFfect than the probabilitydistribution. The standard devialion from spatial variabilit)was assumed to be equal to 20% of the average settlementvalue for all the SPT and OMT cases. The standard deviationsfrom measurement noise and model uncenainty from SPTwere much larger than those from DMT. In fact. they werehuge! The overall standard deviation for the SPT was 86% ofthe average value. as compared with only 29% for the DMT.The discusser questions the value of using the SPT as amethod to compute settlement altogether.

The high SPT variability shown above emphasizes that. forgeotechnical design. the engineer should select the best avail­able test <lnd analysis method and attempt to minimize modeluncertainty and measurement noise. The engineer should thenfocus on and quantify the spatial variability of the site, whichis often beyond hi!o> or her control. Probabilistic design meth­ods provide a good means to address variability. The proba­bility distribution chosen For analyses should provide an ap­proprime result. The more heterogeneous the site is. the moreuncertainty there is. the flatter thc probability distribution willbe. and the more conservative the design should be. The re­verse is al,>o true.

REFERENCES

Failmc7gcr. R. A.. Rom. D., and Ziegler. . B. (19Y9). "SPT! A benerapproach 10 sile charaClcriz:lIIOn of residual soils u'iing olher in-:-ilulesL~." Behlll'ioral characteristics of residual Wi/.f. 8. Edelen. ed..A$CE. Reston. Vrl.. 158-175.

Harr. M, E. (1977). MN'lumic.1: of particillari' metli(l: {/ rntJnobili\lic IIp­proal'h. McGraw-Hili. New York.

Klllhawy. F. H.. and Tralllm.ulO. C. H. r1996). "E:-timalion of in-~ilU ICSIutlccnaimy.·· Ullcerraimy i/l 'he geologic t!lII';m",lIe1ll: from ,hl'Ory w

pl'm:llCl'. Vol. J. C. D. Shackelford. P. P. Nehon. am.! M. J. S. Roth.eds .. ASCE. New York. 269-2M6.

chmcnman. J. H. f 19781. "Use the SPT 10 mea<;urc dynamic :.011 prop­enies'!-Ye~. bUI . _,.. Dwwmic geou!ch"'l'al 't'stm~l:. American So­ciclY for T~ling and Mlllerial ... Wesl Con~hohocken. Pa.. 341-355.

Schmenmann. J. H. (1986). "Dilalomeler 10 compute foundation ~ettle­

men I .. Pnx·.. In Siru '86: ASC£ Spet.-·ialt)' Con! 011 Use of In Silll Testsllml GeOlech. Engrg.. ASCE. Reston. Va._ 303-311,

Wickrcmesin£he. D. S (1989). "Sl3ustical chara~teri7ation of soil pro­file .. U'illlg -in-situ I~IS:' PhD Ihesi ... Dept. of Civ. Engrg .. The Uni­versity of British Columbia. Vancouver.

Discussion by John A. Focht Jr."Fellow, ASCE, and John A. Focht III,'

Member, ASCE

The author is to be commended for developing a rationaltechnique for incorporating reliability into rouline factor ofsafety analyses that can be understood and effectively utilizedby a geotechnical engineering practitioner. Most practitioners,including the discussers. do not have enough confidence in"reliability based design" (RBO) to substitute it for their moreconventionaJ detenninislic approaches. Most RHD papers sug­gest the blind application of statistical 3nalyses of data withoutmuch engineering judgment regarding individual data points.trends in data, the type of design problem. or spatial variationswithin the data. The discussers believe that the applicatjon ofRBD-based design approaches does nOf eliminate the need forsound engineering judgment. The author's proposed approachwill certainly enhance thc valuc of problem solutions for theengineering practitioner. The author also assumcd that soundengineering judgment would be applied to both the data andthe engineering problem. but still seemed to use the numericalaverage as the "most likely value." The discussers concur withthe author's belief that the use of sound engineering judgmentis always a criteria for properly evaluating engineering prob­lems. This view is neither new nor unique: Karl Terzaghi verypointedly addressed the importance of sound engineeringjudg­ment in his May 1936 Presidential Address to the First Inter­national Conference on Soil Mechanics and Foundation En­gineering (ICSMFE):

The miljor pan of the college tmining of civil engineersconsi~lS in the absorption of the laws and rules which applyto relatively simple and well-defined materials. such as steelor concrete. This type of education breeds the illusion thateverything connected with engineering should and can becomputed on the basi:,> of a priori assumptions. As a COIl­

sequence. cngineer~ imagincd that the future science offoundations would consist in carrying Ollt the followingprogram: Drill a hole into the ground. Send the soil samplesobtained from the hole through a hlbor:.llOry with standard­ized apparatus served by conscientious human automatons.Colleci Ihe figures, introduce them illlo the equations. andampUle the result. Since Ihe Ihinking was already done by

the man who derived the equation. the brain~ are merelyrequired to secure the conl.r,Jct and to invest the money. Thelasl remnants of thi~ period of unwarranted optimism areMill found in allempts to prescribe simple fonnula.s for com­puting the settlement of buildings or of the safel) factor ofdams againsl piping. No such fonnulas can possibly be ob­tained except by ignoring a consider<lble number of vitalfactors.

bSr. Consult.. Focht Cunsu1t:mls. Inc.. 12226 Penhshire. Houston. TX77024.

'Chi. Engr.. Focht COI1\ultanls. Inc .. 12961 Park Celltrul, Stc. 1390.San Antonio. TX 78216.

704 I JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING I AUGUST 2001

1 fl. 0.305 m. 1 psI· 47 9 pascals. 1 loch ~ 25" mm

Shear Strength - S upounds per square foot (psI)

FIG, 9. Comparison of Various Shear Strength Profiles for Sun Fran­cisco Bay Mud at LASH Termin:"!1 Site in San Francisco ["fter Duncanand Buchignani (1973) and Duncan (1999))

200- ~ '00 , 000 ,,... 1 • incIl (lnmmed) UU tab;

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lines, may cross a number of geologic units. Frequently, theremight be only a few borings in each unit, so that "averaging"involves limited data at widely dispersed locations within adistinct unit. The junior discusser has been involved in thedesign and construction of a number of electric transmissionlines in south and central Texas. The most recent long trans­mission line was a 138 kY line about 90 mi long. and thealignment traversed at least four distinct geologic units. Someof the borings were d.ry; others were near rivers and wereprobably wet for at least a portion of the year. The soil boringswere typically a mile apart with towers about 700 fl apan.Even with the use of aerial photography, geologic maps, andground reconnaissance, the choice of design parameters wasdifficult. The majority of structures were steel monopolesplaced in drilled holes with concrete backfill. The client wasa regional electric utility company with in-house design ca­pabilities, and the client's representative indicated the utilityrecognized that the risk of failure of a given structure increasesas the design factor of safety decreases. The client was willingto accept "some" risk of failure; the acceptable risk of failurewas (in retrospect) not really defined. The design procedurefor the monopoles, chosen jointly by the client's representativeand the junior discusser, used a combination of expected av­erage soil parameters at the monopole site, performance limitsat full design loading (by limiting calculated ground-line de­flection, pier-tip deflection, near-surface bearing pressures andpier rotation), and rotational stability with the design loadsdoubled (for an effective minimum design factor of safetyagainst catastrophic failure of 2). The procedure utilized amodified soft clay p-y criteria [proposed by Evans and Duncan(1982), which permits the use of both cohesion and <I> in cal­culating pl, which resulted in some conservatism with respect

.,

.,

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PROJECT TYPE

The author included three problem examples in his paper.The first problem concerned the actual failure of underwaterslopes in San Francisco Bay, while the other problems con­cerned hypothetical problems involving predictions of consol­idation settlement in soft clays and footing settlements insands. The slope stability example is different from the othertwo examples. because the problem involves a linear project.Other types of linear projects include tunnels, many wharf andbulkhead projects. open-cul trenches for pipelines. and electrictransmission lines. The discussers have considerable experi­ence with linear projects. which have (in the discussers' ex­perience) always encountered spatially diverse conditions. Thediscussers believe that the author's approach is particularlyuseful for these types of projects.

The discussers believe that the missed prediction of theslope failure illustrates the difficully in selecting the properstrength for analysis despite a very comprehensive investiga­tion. Part of the error may have been with the tacit assumptionthat the "most likely" strength is the average of all the strengthdata, 2.8 in. specimens, 1.4 in. specimens, and field vane, Thediscussers would nOl have included the field vane results inthe averaging process, because past experience suggests theywould be too large; however, we would have been inclined torely more on the 1.4 in. data than that from 2.8 in. specimens.A more significant departure from simple averaging wouldhave been to anticipate spatial variations in the 2,000 ft lengthand pick a "most likely" strength line for design less than theaverage for all the borings. The borings may have not beendrilled at locations where lesser strengths were more predom­inant. Selection of a "l11ost likely" profile should also takeinto account that the actual failure surface can deviate for its"theoretical" position 10 pass through weaker material so thatthe average mobilized strength is less than the "apparent av­erage strength It at that location. The senior discusser, based on50 plus years of experience, would today have picked ProfileI in Fig. 9 as the "most likely" strength profile based on the1.4 in. data. Fig. 9 shows the reported strength data from 1.4in. specimens and five potential interpretations of the availabledata: (I) the senior discusser's "most likely" line for the 1.4in. data; (2) the 1973 Duncan average profile for all of thestrength data: (3) the 2000 Duncan average profile for all ofthe data; (4) the linear regression of the 1.4 in. data; and (5)the linear regression of the 2.8 in. data. Profile 2 was takenfrom Fig. 5 of Duncan and Buchignani (1973), and an equiv­alent approximation of it was used for the original stabilityanalyses. Profile 3 was taken from Fig. 2(b) of Duncan (2000).It would have been interesting to compare Profile I with onesdrawn for individual borings for "average" and "most likely"interpretations. It is also significant to note that Profile I (our"most likely") actually differs only slightly from the "average<7" profile shown in Fig. 2(b). Profile I would have yielded alesser computed factor of safety than those computed for Pro­files 2 and 3, but perhaps still more than 1.0. This comparisonagain supports the discussers' conclusion that, for linear pro­jects, use of the average of all data from all borings can be anoverly optimistic prediction of the "most likely" strength pre­vailing somewhere along the project length.

The discussers are strong proponents of using a "mostlikely" strength profile based on judgment rather than a simplenumerical average or the computed regression line, and inmost instances the most likely strength profile will fall underthe average profile. At the same time, they recognize that thereis not a simple, unifonnly applicable way to select the "mostlikely" profile other than to consider the scatter of the data,the character of the site geology. the project, and the type ofdesign problem.

Some other linear problems, such as electric transmission

JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / AUGUST 2001 /705

to deflection and rotation predictions. However. the true riskof failure was (and still is) unclear. The usc of the author'sprocedure would have allowed the junior discusser to providethe client with clearer choices regarding the economic tradeoffbetween initial construction cost and risk of failure of a portionof the structures. The junior discusser is unaware of any per·f rmance problems with any of the structures built as a partof this project: however, it is also doubtful that the structureshave experienced the design loads.

HISTORICAL PERSPECTIVE

A historical recounting of Arthur Casagrande's procedure torecognize probability or analysis reliability in earth dam sta­bility is appropriate for this discussion. Livingston Dam, de­signed in the I960s on the Trinity River not far from Houston,has a 14.000-ft-long, 90-ft-high earth embankment and a 584­ft-long gated spillway in the embankment. The geotechnicalconsultant was Associated Soil Engineers. a joint venture ofMcClelland Engineers and National Soil Services. both ofHouston. Ralph Reuss and the senior discusser were the prin­cipal . The critical foundation stratum. which is overlain bythe usual alluvial deposits of clay over sand, is a stiff to verystiff. slickensided. highly plastic, heavily overconsolidated Mj­

ocene clay. As a result of the Waco Dam failure. we wereconcerned about the implications of residual strength of theMiocene clay in reference to embankment stability. Dr. ArthurCasagrande was rct:lined as a special consultant. and he fo­cused on the strength of the slickensided surfaces in the Mi­ocene clay. We had already nm a large number of consoli­dated-drained direct shear tests to determine both peak andresidual strengths. At Arthur's suggestion, we ran three testson specimens carefully trimmed to have a slickensided surfacein the center of each. Because the stratum was deep. the testswere run under a vertical load of either 3.0 or 6.0 ksf and thestrength envelopes were drawn Ihrough the origin (c; 0). Wejointly agreed. based on those test results. that a strength of 'P; 17°, c;; 0 could be assigned to the Miocene clay. This valuelies near the lower boundary of the peak strength envelopes(cp ; 14°. c ;; 0) from the routine tests. The average peakstrength parameters were cp ; 24°. c ; 0.4 ksf. Residualstrengths had been found 10 be in Ihe range of 10-13°. Basedon triaxial tests. the pore pressure response in the Mioceneclay to embankment loading was predicted to be 50% (whichwas later confimled by piezometers during construction).

Because he believed the combination of assumptions wasconservative, Anhur recommended a factor of safety of 1.4 fordesign. However, he added a second criterion for acceptability.While the results would not be reported, the proposed em­bankment mllst have a factor of safety of 1.0 computed for aresidual strength of 'P ; 10°, C ;; O. Anhur indicated that it wasa routine procedure for him to seek a factor of safety of atleast 1.0 for a least likely strength. His approach to selectionof design strengths and acceptable factors of safety convincedthe senior discusser that he understood at that time the conceptthat the author is proposing. The senior discusser funher be­lieves thal Anhur Casagrande would have remained a propo­nent of relying heavily on judgment rather than just sophisti­Caled computer analyses and would have endorsed theproposed concept.

SUMMARY

The discussers believe that the author's approach. when ap­plied with good judgment, will improve the value of geotech­nical analyses for many practitioners. The proposed approachprovides the practitioner with the simple, direct tools neededto evaluate the risk of failure for a wide variety of geotechnicalproblems. The method is rational, integrates easily with exist-

ing design approaches, and provides the geotechnical engi­neering practitioner with the tools he/she needs to effectivelycommunicate relative risk-and benefit-to the client. Thediscussers hope that the authors modified RBD approach willbecome widely used in geotechnical engineering practices fora wide variety of de ign problems.

REFERENCES

Evans. L. T.. and Duncan. J. M. (1982). "Simplified analysis of lalerallyloaded piles:' Rep. No. UCB/GTI82·04. Dept. of Civ. Engrg.. Univer­sity of California at Berkeley. Berkeley. Calif.

Ter;.aghi. K. (1936). "Presidential address: Relation belween soil me­chanics and foundation engineering:' Pmc.. ISf 1m. CO/if. on SoilMet'''. lind Found. Engrg.. Cambridge, Mass.

Discussion by Demetrious C. Koutsoftas,"Member, ASCE

The author presents interesting applications of statistics andreliability theory to problems involving stability and settlementanalyses in geotechnical engineering. The basic concepts pre­sented in the paper provide a useful tool for geotechnical en­gineers in evaluating the risk associated with their design rec­ommendations. However. the discusser is concerned that someengineers may be attempted to use this tool as a substitute forthorough investigations with high quality data that are essen­tial ill assessing risk and reliability for important projects. Thefollowing two examples illustrate this concern.

STRESS HISTORY OF BAY MUD

One of the examples used by the aUlhor to illustrate the useof the proposed simplified reliability atlalysis methodology in­volves the stress history of a deposil of soft Bay Mud al IheHamilton Air Force Base in Marin County. California. In Fig.2(a), the author uses the results of a limited number of con­solidation tests to assess the variability of the preconsolidationstress and estimated settlements.

Several years ago, the discusser's firm performed a seriesof Oeonor vane shear tests at the Hamilton Air Force Base,which are summarized in Fig. 10. The results of the tests aredivided into two groups to illustrate likely variations ill theundrained strength and stress history over the site. On the leftside are the results of four strength profiles that seem to befairly consistent in lenns of strength variations with depth aswell as thickness of the layer. On the right side are the resultsof two other profiles where the Bay Mud is deeper, and wherelhe strength is fairly constant within the depth interval of 10­40 fl (3.05-12.19 m).

The site has been recl:limed by placing approximately 4 ft(1.22 m) of fill afler Ihe author performed his investigation(Duncan 1965). The Slress profiles in Fig. 10 have been mod­ified to reflect the increase in in situ overburden stresses due10 Ihe fill.

The discusser has found from numerous cases in the BayArea. where vane shear tests and consolidation tests are avail­able, that the preconsolidation stress IT;. can be approximatelyestimated as four times the undrained vane shear strength afterit has been corrected to make the strengths correspond to thedirect simple shear. This is reasonably consistent with Mesri's(1989) empirical equation of

(Su)rf\<!b ; 0.22(1;

~Prin .. URS/Domcs & Moore. 221 Main St.. $tc. 600. Sun Francisco.CA 94105.

7061 JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING 1 AUGUST 2001

VERTICAL EFFECTIVE STRESSES, a'vo, a~hIAX'KSFo 1.0 2.0 J~ 4~ ~ ~o

I I I I i I I

VERTICAL EFFECTIVE STRESSES, a'vo' a'VM' KSFo 1.0 2.0 3.0 4.0 $.0 6.0I I I I I I I

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FIG. 10. Vune Shear Tests and Interpreted Stress Hislory of Bay Mud: Hal1lihon Air Force Base

Fig. IO(a) shows the profiles of initial venical effectivestress, the venical effective stress after full consolidation under4 ft (1.22 m) of fill, and the profiles of the highest and lowestconceivable preconsolidation stress estimated by the author. Itis clear that within the depth interval of 8-25 ft (2.44-7.6201), the highest conceivable preconsolidation stress profile isconsi tent with the upper limit indicated by the vane sheardata. However, at depths greater than 25 ft (7.62 m). the vaneshear data indicate that the preconsolidation stresses are sig­nificantly higher than the highesl conceivable values assumedby the author. The same general behavior is evident in Fig.100b), although the differences are now much greater thanshown in Fig. 10(a).

The vane shear data also reveal some delails that cannOI bedetected from conventional investigations. At test locationnumber 4. there appear to be lenses of softer material that arebelieved to be highly organic clays. Exploralions conductedby the discusser in a nearby site show that random layers oforganic clay are sandwiched between the Bay Mud. Also. thedata show clearly a transition zone near the base of the BayMud layer where strengths increase rapidly with depth and thesoil is heavily preconsolidated. The author's data extend downto a depth of only 20 ft (6.1 m). even though Figs. 2(a) and5 indicate that the mud extends at least to a depth of 30 ft(9.14 m). The transition zone at the base of the layer was notdetected by the author's investigation. The presence of thistransition zone is important in evaluating both the magnitudeand the rate of settlements. The stiff clay at the base of themud fonns an impermeable drainage boundary. which is crit­ical in assessing time rates.

It is evident from the data presented in Fig. 10 that

I. Vane shear data provide very valuable information fromwhich the stress history of the Bay Mud can be evalu­ated. Even if one does not wanL to accept the numericalvalues of the a;, calculated from the vane strengths. the

data cenainly provide a much more reliable guide forestimating upper and lower limits for the preconsolida­tion stress than those obtained from limited laboratorytesting.

2. The vane shear data reveal variations in the strength andstress history of the Bay Mud that conventional investi­gations are likely to miss. such as the potential presenceof lenses of soft organic clay that may be more com­pressible than Bay Mud, as well as the presence of tran­sition zones near the base of the Bay Mud. which affectboth the magnitude and the rate of settlements.

3. Vane shear tests provide an economical and practicalmethod for evaluating substantial variations in stress his­tory and undrained shear strength across a site. Hence.they constitute a very valuable supplement to laboratorytesting.

The above example illust.rates the usefulness of the vaneshear test and points out that even the 1110s1 experienced geo­technical engineer may not be able to assess reliably the char­acteristics of sof[ clay deposits from "meager data." In thiscase, a much more thorough site investigation would be re­quired to improve the reliability of the geotechnical analysesand recommendations.

UNDERWATER SLOPE FAILURE

Another case used to illustrate the application of reliabilityanalysis involved a deep Cut in soft San Francisco Bay Mudat the LASH terminal in San Francisco. which failed duringexcavation. as detailed by Duncan and Buchignani (1973).Their explanation was thai the slope failed primarily becauseof loss of strength due to undrained creep. The author reex­amines this case and concludes that the estimated variationsin Ihe undrained shear strength and buoyant unit weight of theBay Mud resulted in a probability of failure of 18% associatedwith the calculated factor of safely of 1.17. The author con-

JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING I AUGUST 2001 1707

cludes that a probabilily of faHure of 18% was "100 high tobe acceplable," and, perhaps, if il was realized that the prob­ability of failure was that high. the design might have beenchanged to reduce the probability of failure.

This is a very interesting case. because. in spite of the facithat the 1973 paper presented a comprehensive assessment ofa number of faclors affecting the undrained strength of themud. there are several other factors not discussed in the paperthat might have affected the observed failure. and which arerelevant to the reliability analysis presented by the author.These factors are discussed below.

Stress History and Undrained Strength Parameters

The slope stability analyses were based un undrained shearstrengths measured from unconsolidated. undrained (UU) tri­axial compression tests and from field vane shear tests. Theinterpreted strengths from UU and field vane shear tests werewithin 8% of e..1ch other. which, given the scalier in the data,is interpreted by the discusser to suggest that the UU strengthsand vane shear strengths are essentially the same. The resultsof lest comparisons on many projects performed by the dis­cusser in the last 16 years. involving San Francisco Bay Mud,are consistent with this finding. However. the agreement instrengths is entirely fonuitous. for the following reasons: (I)Undrained strengths of Bay Mud measured from UU tests.even on high quality samples. where strains to failure are onthe order of 3% or less, underestimate the in situ strength incompression. The primary reason for this is that the effectivestresses, even in high quality samples, are significantly lowerthan the in situ effective stresses. as explained by Ladd andLambe (1963). The discusser's experience has been that UUstrengths underestimate the in situ strength of Bay Mud incompression by about 30%. (2) Vane shear strengths adjustedusing a factor 0.9 are on the average equal to the undrainedshear strengths determined from direct simple shear tests. andLhey also underestimate the slIength of the mud in compres­sion. because of anisolIopic effects. by about 30%. This rela­tionship between vane shear strengths and strengths in com­pression is consistent. although somewhat lower than thecorrection factor of 1.35 recommended for Bay Mud by Bjer­rum et aJ. (1972). Fig. I I shows the results of three vane shearsoundings performed along the [slais Creek, not far from theLASH terminal, and they are compared with Ihe undrainedstrength profiles in compression. extension. and simple shearestimated based on SHANSEP (Ladd and Foott 1974) princi­ples. 1l is evident that the vane shear strengths adjusted by afactor of 0.9 agree reasonably well with the strengths for directsimple shear. As seen in Fig. II, and as discussed in moredetail by Koutsoftas CI aJ. (2000), the Bay Mud is anisotropicwith respect to its undrained shear strength. It is therefore nec­essary to carefully consider how the resuhs of the UU strengthlests and the field vane shear tests apply to the stability of thecut slope.

Another imponant fact revealed from the data shown in Fig.II is that the mud appears to be lightly preconsolidated downto a depth of 35 ft (10.67 m), even though the mud is under20 fl (6. I m) of fill placcd in the I920s. This suggests 'hat themud must have been preconsolidated prior to the plilcementof the fill. probilbly due to aging. The three vane soundingsshown in Fig. II were perfomled right along the nonh bankof Islais Creek. where the mud extends down to 100- t20 ft(30.48-36.58 m) deep. The SLTesS history of the mud. prior tothe placement of the fill al this location. could not have beenmuch different from that of the mud at the LASH terminal.Therefore. based on the results shown in Fig. II. it could beconcluded that the mud at the LASH terminal must have beenpreconsolidated also. Close examination of the consolidationand vane shear data included in the 1973 paper by Duncan

UNDRAINED SHEAR STRENGTH, Su, PSF

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and Buchignanj also suggest that the mud is preconsolidated.The uncorrected vane shear strenglhs (figure 6 of the 1973paper) indicate approximate undrained strength ratios (S"Iu:.,,)of 0.66.0.46,0.41, and 0.38, respectively. at elevations of - 30fl (-9.14 m), -40 ft (-12.19 m), -50 ft (-15.24 m). and-60 ft (- 18.29 m). These values are significantly higher thanthe corresponding undrained strength ratio of 0.25 (uncor­rected) for normally consolidated Bay Mud and lead to theconclusion that the mud at the LASH terminal was overcon­solidated. as suggested by the data in Fig. I I. Given the ageof the Holocene mud. and that the mud at the LASH tenninalsite is not likely to have been exposed to desiccation or otherexterna.lly applied stresses from human activity. it would bereasonable to conclude that the mud in the Bay would beoverconsolidated due to aging (Bjemlm 1973), with probableoverconsolidation ratios in the range of 1.6-2.0. Fig. 12 showsthe index propenies and stress history of Bay Mud at a site inthe Bay located just outboard of the end of the runways at SanFrancisco Airpon. The mud is below water and the site is farenough from the old shoreline that it is unlikely it had beenexposed to desiccation or had been subjected to externalstresses from human activities. The results of five consolida­tion tests performed on undisturbed samples obtained withinthe upper 30 ft (9.1401) of the mud indjcate preconsotidationstresses that range between 1.6 and 2.0 times the in situ ver­tical effective slIesses. The results shown in Fig. 12. panicu­larly the trend for increasing preconsolidation stress withdepth, lead to the conclusion that the preconsolidation is theresult of aging. Given the geologic history of the Bay Area, itis the discusser's assessment that the stress history of the mudat the airpon site would be quite similar to the stress hiswryof the mud at the LASH temtinal. The results shown in Fig.12 reinforce the conclusion that the mud in the Bay, includingthe LASH terminal site. is overconsolidated, most likely duelO aging.

7081 JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING I AUGUST 2001

2.5

EFFECTIVE STRESSES, a'vo' a., KSF

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(b) STRESS HISTORY

FIG. 12. Index Propcnies and Stress History of Bay Mud from Site Offshore S:lTl Francisco Airport Runway No. 28

If the mud was indeed overconsoJidated, !.he dissipat"ion ofnegative excess pore pressures, generated by the excavationprocess, could have been faster than anticipated by the authorand could have contributed to loss of strength due to swelling.

Stability Analysis

Fig. J3 shows the critical failure as it was depicted in figure9 of the paper by Duncan and Buchignani (1973). Much ofthe critical failure surface shown in Fig. 12 represents shearin compression (designated the "compression unloading"zone), while shear along the reSt of critical failure surface canbe approximated by simple shear. As indicated by the stresspaths in Fig. 13(b), negmive excess pore pressures are gener­aled during shear in the compression unloading zone (Fig. 13).It is not intuitively obvious that the loss of strength due tocreep. under conditions that involve generation of positivepore pressures (in triaxial compression tests like those per­formed by the author), is applicable under conditions that in­volve generation of negative excess pore pressures. Hence. theassumption of loss of strength due to creep is questionable.

If the vane shear strengths were corrected for anisotropy (asper Fig. II) before being used to represent the in situ strengthsalong the portion of the critical failure surface that passesthrough the "compression-unloading" zone, the calculated fac­tor of safety would be significantly higher than the vaJue of1.17 calculated by the author. If that was the case, then themud must have experienced a significant loss of strength dur­ing excavation of the trench to cause the failure. The faiJurecould not be explained either by the statistical variations instrength and buoyant unit weight. or by the strength reductiondue to creep. The required reduction in undrained strength to

cause the failure couJd have occurred only if there was sig­nificant dissipation of negative excess pore pressures. As de­scribed below, the geologic conditions at the LASH site mighthave been favorable, at least over some portions of the site,for rapid dissipation of negative excess pore pressures andmight have caused the failure.

The discusser has conducted numerous explorations withinthe lslais Creek basin and has reviewed borehole records frommany previous explorations conducted by Dames and Mooreand others in the area, including Piers 80. 90, and 92 in thevicinity of the LASH tenllinal. In a large number of cases, amarine sand layer was encountered directly below the mud.The marine sand varies from fairly clean fine sand to clayeysand with stringers of Bay Mud. In some instances the marinesand is less than 5 ft (1.52 m) thick and is underlain either bypermeable dense (Colma) sand or Old Bay Clay. Often, thepresence of the thjn marine sand layer may not be detected,especially when sampling at depth intervals of 5 ft (1.52 m)or more, which is fairly standard in the Bay Area at depths of100ft (30.48 m) or more. It is less common to find Old BayClay directly below the Bay Mud. However. in some of theboreholes drilled at Pier 80, which is located jusl north of theIslais Creek Channel near the LASH terminal. Old Bay Claywas found d.irectly below the mud. acting as an impermeabledrainage boundary at the base.

If a layer of reasonably permeable sand (i.e., much morepermeable than Bay Mud) was present below the Bay Mud,as many of the boreholes show, the negative excess pore pres­sures near the base of the mud could dissipate very rapidly.Approximately 30% of lhe critical failure surface (and perhapsalso of the actual failure surface) passes directly over or within

JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING 1AUGUST 2001 1709

ESP = Effective Stress Path

CONCLUSION

Discussion by Charles C. Ladd;Honorary Member, ASCE, and

Gregory Da Re,l. Associate Member, ASCE

Undrained strengths determined from UU tests rarely providerealistic assessments of in situ strengths because of sampledisturbance. strength anisotropy. and other factor, as dis­cussed by Germaine and Ladd (19 8) and Ladd et al. (1998).Likewise. vane shear strengths need to be corrected for theeffects of rate of shear and anisotropy before they are used forstability analysis, as discussed by Bjerrum (1972. 1973) andLadd et al. (1977).

Finally. the discusser would modify the author's claim that··an essential component of the an of geotechnical engineeringis the ability to estimate reasonable values of p3ramelers basedon meager data." It appears from the two cases reviewed inthis discussion thm using meager clat..l often involves signifi·canl risks lhat onc might reach the wrong conclusions. If mea­ger data arc used for design, one should also recognize theinherent risks and utilize higher factors of safely to accountfor the uncenainties. The discusser believes that the true anof geotechnical engineering should also include knowing whena situation demands comprehensive exploration and testing 10

correctly characterize a given field situation. When this is ac­complished systematically. we can then expect a real improve­ment in the practice of geotechnical engineering and improvedreliability of our designs.

\

BJcrrum. L.. Frimman Claus..en. C L and Duncan. J. M. (1972). "Earthpre..sure.!l on fJexible .!tIructure~-a state-of-the-art report.'· Pmc.• 5111Ellmpl'flll COli! Oil Soil Mech. alfd FOLllfd. E"grg.. 2. 169-196.

Bjcrrum. L. (1973), ··SI:ttc-of-Lhc-an report: problems of soil mechanic~

and con<;truction on soft clays and structlJrnlly unswbte soils:' Pmc..8th 1111. Con! Oll Soil Mech. fllld FOlIl/d. £ngl'g.. l\·!oscow. 3. 111- 159.

Germ:linc. J. T.. and Ladd. C. C. (1988). "St:llc-of-tbc·art: lriaxialtcsLingof saturated cohesive soih..·· Adval/ced rr;ruiflll(>\'I;"g olsoil lind rock.ASTM. We!'1 Conshohocken. Pa.• 421........t59

KOUl.!>oftns. D. C. Frobeniu... I>.. WU. C. L.. Mcycr,ohn. D.• and Kulesza.R. (2000). "Deforl1lation~ during cUI·:lOd-(;o\,er conSLnlclion of MUNIMetro Tumback Project:' J. Ceo/echo and Geoef1\·;r:. £n~'!rg .. ASCE.12(,(4). 3-1+-359.

Ladd. C. C.. and FOOL!. R. (1974). "New design procedure for stabilityof son days:' J. Gemecll. £ngrg. Di,'.. ASCE. 100(7).763-786.

Ladd. C. C.. Foot!. R.. Ishihura. K.• Schlosser. F.. and Poulos, H. G.(1977). ··Stress. defonmuJon. and ~trength characteristics:' Proc.. 9th1m ConI. on Soil Met:1I. and FO/lnd. Engrg.• Tokyo. 2. 421-494.

Ladd. C C. and Lambe. T. W. (1963). "The strength of undi'olurbed cia)determined from undmined test .... ·' Laboratory .~hear testing of soils.ASTM. West Con"hohocken. Pa.. 342-371.

Ladd. C C. Young. G. A.. Kraemer. S. R.. and Burke. D. 1\1. (1998)."Engineering properties of Boston blue clay from special testing pro­grolnl.·· Proc.. Geo,Collgr:. 98. ASCE. ReMon. Va.. 1-24.

Lambe. T. W. (1967). "Slress palh method:' Soil Mech. and Found.£"8'8" ASeE. 93«(,). 309-331.

Mesri. G. (1989). "A reevaluliLion of S.''''''''' = O.22a; u~ing labor.:uQryshe:lr tests:' Call. Ge01ec:h. 1.. OUawa. 29( 1). 162- 164.

REFERENCES

The author presents convincing arguments for using relia­bility analyses to estimate the probability of failure (Pt ) asso­cimed with factors of safety (F) obtained from conventionalstability analyses. The discussers especially endorse his viewthat election of design values of F should reflect the uncer­tainties in the stability analyses and the consequences of fail­ure, rather than the "one size fits air· approach commonly

Qprof.. Dept. of Civ. and Envir. Engrg .. MIT, Cambridge. MA 02139.'\1POSldocl. Assoc .. Dept. of Ci". and En"ir. Engrg .. MIT. CambridgL:.

MA 02139.

STRESS CHANGES

ELEMENT No.1 (cr,), =(cr",,),(ah), = (altO), - 6.ahl INOTE: 1 Foot· 0.328 M

+2'

~+,

= -...J -2ll COMPRESSION· ·2'

!. UNLOADING... ZOOEW ...W...Z0 ·'00;:: ·120

~ .~.' .. FIRMW

.,,,,SOILS

...J APPROXIMATELYW DIRECT SIMPLE SHEAR ZONE

.180 (OSS ZONE)

AVERAGE NORMAL STRESS

P=Ya(O'+(}3)' P=1Ji,(J,+(J3)

zoiii

)(~ r--;::===t===r=====:±:;----L---jw (T-UO)SP =Total Stress Path

Minus the InitialHydrostatic Pressure

The two cases reviewed in this discussion le:ld to the con­clusion that reliability analyses can be of lillie benefit to thegeotechnical engineer, unless such analyses :.lre based on are:llistic assessment of field conditions and employ deSign pa­rameters that truly represent the in situ behavior of the soil.

(0) APPROXIMATE STRESS STATESALONG CRITICAL FAILURE SURFACE

(b) EFFECTIVE STRESS PATHELEMENT No.1

a few feel of the interface with the underlying "firm'- soils.Dissipation of negative excess pore pressures along the baseof the mud could have caused loss of strength over a signifi­cant portion of the failufC surface to cause the observed failure.

Based on the above observations. it is the discusser's con­clusion that the stabililY of Ihe underwater slopes at the LASHterminal was critically dependent on the drainage conditionsat the base of the mud. Where the marine sand layer waspresenl. rapid drainage must have occurred, leading to the ob­served failures. Where Old Bay Clay. or clayey sand, werepresent below the mud. drainage must have been inhibited.thus avoidjng loss of strength due to dissipation of excess porepressures. and the slopes remained stable. Given the generalstratigraphic conditions that prevail in the Islais Creek area, itis perhaps fonunate that slope failures were not more preva­lent.

FIG. 13. Approximate Stress Siaies and Stre.'is Path" alont; FailureSurface

710/ JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / AUGUST 2001

specified in regulations or adopted by practitioners. Unfortu­nately, many engineers do not appreciate the fact that a com­puted factor of safely per se has liltle physical meaning, unlessone also has some measure of its accuracy. As clearly statedin the paper. a reliability analysis provides a systematicmethod for evaluating the combined influence of uncertaintiesin the parameters affecting F. so that one can assess the actualdegree of safety (at least in relative terms) via the correspond­ing probability of failure.

The author recommends using published values (Table 3)and/or the three-sigma rule for estimating the standard devia­tion (a) of parameters for ,·the common situation in whichlimited amounts of data are available and many properties areestimated using correlations." i.e., in lieu of using (3), whichdefines a. These values of a are used to compute the resultingstandard deviation in the factor of safety (ar). The values ofar and F4W (the most likely value of F based on best estimatesfor all parameters) are then used with Table 2 to obtain theprobability of failure (PI)' The discussers feel that this simpleapproach may lead to misleading resuhs unless the user alsounderstands the different sources of uncertainty, which can af­fect election of both the best estimate of parameters and rea­sonable values for their standard deviation. These differentsources also affect the physical meaning of PI'

The discussion focuses on undrained slope stability analysesand starts with a description of the different sources of uncer­tainty in undmined shear strength (S,,). The discussers thenapply this approach to the soils data for the LASH case history.leading to estimates of PI based on statistical analyses of thedata rather than the three-sigma rule. The effect of "modelerror" is also introduced.

SOURCES OF UNCERTAINTY IN S. AND MEANINGOF P,

A reliability analysis should distinguish between the twobasic types of uncenainty in a soil property such as 5" (e.g..Christian et al. 1994). One type is the uncertainty in the bestestimate value or mean trend with depth of S", which is calledthe systematic uncenainty (denoted by a~ or the coefficient ofvariation, COV,,.). The second type of uncertainty is the spatialvariation in S" about the mean trend and, in theory. shouldreflect both the magnitude of the fluctuation about the meantrend (a.. or COV,,) and the scale (distance) over which thisfluctuation occurs. Note thal identifiable zones of stronger orweaker oil should be assigned different mean trends.

Suppose that a reliability analysis produces a best estimatefactor of safety of Ful.\ :;:: 1.25 and aF:; 0.25. which leads toPI = 15% from Table 2. If a, = a". = 0.25 (i.e., all due touncenainty ill the best estimate of SIt). this implies that thereis a 15% chance that the entire length of slope will fail. Butif ar:;:: a,p::: 0.25 (i.e., all due to spatial variation in SIt abouta mean trend having no error), this implies that, depending onthe scale of the fluctuations. up to 15% of the length of theslope will fail.

Selection of the best estimate of SIt and analyses to obtainthe systematic and spatial uncertainty must address four prob­lems in evaluating strength data. Using SIt from field vane tests,S.(FV). as the example, they are

I. Scalier in S,,(FV) due to real spatial variability2. Scatter in S,,(FV) due to random testing errors (noise)3. Error in the mean S,,(FV) due to the limited number of

tests. called the statistical uncertainty4. Error in the mean S,,(FV) due to measurement bias

Items 3 and 4 produce the systematic uncertainty (i.e., thepOlential error in the selected best estimate of S,,): a;v :;:: u;, +a~;u, where U J ' and UbI., denOle the statistical and bias com-

ponents. all is easily obtained from conventional statistics.even though the data may be meager. For a layer with a con­stant S". a;, :; a;..tn, where as" :;:: standard deviation of 11 valuesof the measured 5... Estimates of U hl,"" are generally far moredifficult. However. when using field vane data. one would usu­ally base the best estimate on values of ~S"(FV), where ~ isBjerrum's (1972) empirical correction factor. For this case. theuncertainty (bias error) comes from scatter about the recom­mended IJ. versus PI correlation (as discussed laler).

For items 1 and 2, it is difficult to separate real spalial var­iability from random scaner, and even more difficult to esti­mate the scale of spatial fluctuations about the mean. For sim­plicity. the first discusser often assumes that a;, equalsone-half of the total scatter. a;". The total uncertainty in S"then becomes a~ :; a~. + 0.5a;" and values of PI using aTrepresent the likelihood of having "small" failures within theunknown scale of spatial fluctuations. In any case, the dis­cussers believe that situations in which the systematic uncer­tainty dominates will generally be of greater concern thanthose involving mainly spatial variations in SIt. In other words,serious errors in the best estimate of SIt are generally moreimportant than fluctuations about the selected mean trend.

The above discussion raises questions about the coefficientsof variation cited in Table 3. Are the ranges meant to reflectthe total uncertainty (aT +- best estimate) or only the system­atic error in the mean? Also. should the quoted values for therange in S" depend on the type of test and heterogeneity ofthe soil and that for S..ta~ also depend on the overconsolidationratio (OCR)?

LASH UNDERWATER SLOPE FAILURE

Available Soils Data

Fig. 14 plots measured strengths from 18 field vane testsand 21 UUC tests run on 1.4 in. (36 mm) diameter specimens.These data were scaled from Duncan and Buchignani (1973)and exclude results from 2.8 in. (71 mm) diameter UUC teststhat were considered "somewhat disturbed." Line I was se­lected by these authors to represent the UUC trengths and isvery similar to a linear regression (LR) line that excludes thetwo deepest tests. Fig. 15 plots stress history data and shows

The effective overburden stress, a:.,. based on buoyantunit weights having an average equal to that selected bythe author.Values of minimum-maximum and average preconsoLi­dation pressure (cr;) obtained via Casagrande from con­ventional "one-day" incremental oedometer tests. A linearregression line is also ploued (it excluded two of the IItests), which indicates that the deposit is actually slightlyoverconsolidated.Values of a; predicted from the field vane tests usingChandler's (1988) equation:

OCR = (S.(FV)la~,) "" (13)

S"with S,.", :;:: 0.20 for PI = 20%. Linear regression on thesedata produce a line parallel to, but significantly higher than.the lab a; line.

Reliability Analysis

Table 14 summarizes the results of three reliability analysesfor the LASH underwater slope having a height of 100 ft (30.5m). The author's is listed first. 1t predicts that most of the PI= 18% comes from the estimated uncertainty in SIt as comparedwith the estimated uncertainty in buoyant unit weight. Becauseone does not know the systematic versus spatial uncertainty,

JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING 1AUGUST 2001 1711

oo

a'"

-50

-50

-40

-80

·90

·70

·30

Stress History, d". and d, (ks!)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0·20 IT'--"'-V---.;=r::=r=:::c:=c::::::r:=::;l

-'00 L..>c..L_L---'-_-'----'-_.L.!::ll:lI'--...L.--"_...J

FIG. 15. Stress History from Lnbormory Consolidation and Field VaneTests. Note: Small Symbols Excluded for LR Annlysis (I ft = 0.305 m:t ksf=47.9kPa: I pcf=0.157 kN/m·')

• 1.4 in. UUCo Field Vane

o•

oo

·50

·40

-90

·70

·30

·60co

~'"w

,'e

'0,\ .~

., \\ \':il~ \ \1\ \ \ ,\ \ ~ ~

\ \ \

\ \\

\~ ,~ ~

\ "~\ \\ \ \ 0 ~\ 0,\\ \ '

·80 ~-,--,--,,-----,\\ \Un. Profil. \\ ~ ,

1-'='T'''''-lics.=,0<l:;ed'=''''by,",,0&;;B'"'(''',9''7'''3):-1 \ 4 ,L!.J IOf UUC .. \

[ZJ LA on .S.IFY).•• 1.0 ~. \ ',1

~ Uno'.a. \\ • ,m SHANSEP S.IO) \ \ \ ~lID Une4.a" \\ \ \

·1 00 CI::=c:::i==r::=L-Lili--.i.L:~~

FIG. 14. Undrained Strengths from 1.4 in. UUC and Field VnneTestsLInd SHANSEP. Note: Small ymbols Excluded for LR Analysis (I in. =25.4 mm; I ft = 0.305 m: 1 psf = 0.0479 kPa)

Shear Strength, Su (pst)

o 100 200 300 400 500 600 700 800 900 1000·20 "'--.'<""T,,--,--,--.-,-----,-....-,--,

(14)

the fact that 22% of the slope length failed is indeed fonuilOus.Also, Line I in Fig. 14 produces F= 1.25 using the UTEXAS3computer program (Wright 1991). as compared with F = 1.17reponed by Duncan and Buehignani (1973). who replaced theline by 20 ft (6.t m) thick layers of constant S" material.

Analysis Using Field Vane Strengths

The second analysi is based on statistical treatment of thefield vane (FV) data using a correction factor of IJ. = 1.00based on the reported PI = 20% for Bay Mud. Hence, Ihclinear regression Line 2 in Fig. 14 represents the best estimate.which produces PM!.v = 1.19. The sLatistical uncertainty in the

location of Line 2 reOecls the vertical distribution and scatlerin S.(FV) and equals

( [(EI. - 48)'])"

u.(psf) = 646 1 + 3t8

whieh varies with depth between abou' 25 and 105 psf (1.2and 5.0 kPa). Based on the data in Figure 51 of Ladd et al.(1977). the uncertainty in ,... at PI = 20% has a coefficient ofvariation of COVlj..L] = 20%. Thus. the total systematic uncer­tainty at any depth equals

un(psf) = [17:, + (0.20·S. on Line 2)']"' (15)

The mean minus a A,. profile produces P = 0.93, a,. = 0.26, and

TABLE 14. Results from Reliability Analysis

Basis for..maly!ois

AUlhor's T;Jble 5

Corrected fieldvane for ..... =­1.00

SHA SEP DSSS = 0.22.m = 0.8

Mean S~ and factor ofsafety

Fig. 2. FNII = 1.17

Fig. 14. Line 2. FI.I1X =1.192

Fig. 14. Line 4. F,IIU =­(0.997)( 1.10) =1.0')7

Source of uncertainty

COVIS.I = 13%Plus COVI-y.1 = 8.7%Statistical and COVI,...) = 20%Statistical and' COV[ JL) = 25%

Systcmatic uncenaintl in S..Plus modcl error. COV[3D] =

5%Plus spalial uncertainlY in S~

PIfrom

Partial err Total u,. Table 2(COY. %) (COY. ON) % Remarks

0.155 (13.2) t3 S. contribUles most of PI0.10 (8.5) 0.185 (15.8) 18

0.262 (22.0) 24 Probability of "large" failure0.317 (26.6) 29

0.t33 (12.1) 24 Probability of "large" Failure0.055 (5.0) 0.14-1 (13.1) 26

0.116' (10.6) 0.185 (16.9) 32 Probabilily of "small" Failure

·Su = Fig. 14. Line 3.~S~ =- Fig. 14. Line 5. ,.. = (0.876)(1.10) = 0.964.'0.116 = (0.1852 - 0.1442)"$.

7121 JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING 1 AUGUST 2001

Parameters for the best estimate of the direct simple shear(DSS) strength [S.(D) = Line 4 in Fig. 14] are:

PI = 24%. Line 3 in Fig. 14 equals the mean minus "".S.computed with COV[".] = 25% and gives F = 0.875 and PI =29%. The actual". could be significantly less than 0.75, basedon the h.igh CT; predicted from field vane tests relative to thelaboratory (Fig. 15). A lower than usual". would be expectedif the field vane tests were run with a torque wrench ratherthan a gear system (failure in seconds rather than minutes), Inany case, this analysis predicts an unacceptable level of risk,even wilhout consideration of the spatial uncertainty in Su. Thediscussers also believe that this approach is morc realistic andless dependent on judgment than the one based on the lhree­sigma rule lines in Fig. 2(b). Does the author agree that therewere sufficient field vane data for statistical analyses?

Allalysis Using SHANSEP

The third analysis used the SHANSEP equation

S; = S(OCR)";",.,,'

OCR =.2-0':"

(16)

creases the total standard deviation in F to CT,,' = (0.1332 +0.055')°' = 0.144, and PI increases (slighUy) to 26%. (Note:For more complex stability analyses, such as embankmentswith staged construction, the total model error may increaseto COy '" 10-15%.)

The last SHANSEP analysis in Table 14 includes a rathersubjective estimate of the spatial varintions about the best es­timate S,,(D) line. This uncertainty was obtained by increasingCOV[Sl from 10 to 14% and adding a spatial fluctuation incr; of cr,pl";] = 0.125 ksf (6.0 kPa), which equals one-half ofthe total scatter in the data about the u;' LR line. The resultingPI implies a 32% probability of having a "small" slope failure,as compared with 25% for a "Iarge" failure.

The SHANSEP reliability analysis appears rather compli­cated because the discussers purposely included a rather com­prehensive set of factors to illustrate different sources of errorsand how they contribute to the overall uncertainty in the sta­bility analyses. Nevertheless, after entering the soils data in aspreadsheet. the calculations of best estimates and uncertain­ties in S". plus the stabi.lity analyses for each profile, shouldnot require more than a day or so.

CONCLUSIONSS = 0.220 from three CKoUDSS tests run by MIT onnormally consolidated Bay Mud having a PI = 25%(DeGroot et al. 1992)m = 0.8, recommended by Ladd (1991), p. 584,,:., = line in Fig. 15,,; = LR line in Fig. 15. which is then increased by 10%to convert the one-day compression curve to an estimatedend-of-primary curve

The UTEXAS3 program with S.(D) gives F =0.997. Thisvalue is then increased by 10% to model three~dill1ensional

end-effects based on the mean value from case histories offailures in Azzouz et al. (1983). Hence, the best estimate be­comes FMI.V = 1.097. ( ate: Four significant figures are shownfor numerical consistency. not for implied accuracy. Also,since Bjerrum's jJ.. factor already includes end-effects. this 10%correction was not applied to the FV factor of safety.)

Baecher and Ladd (1997) show that the uncenainty in theSHANSEP S. profile, assuming no error in ":., equals

COV'[S.l = COV'ISj + m'COV'r,,;J + In'(OCR)'''~ (17)

Values used to compute the systematic uncertainty in S,,(D)were:

I. COV[S] = 10% estimated for bias (the three DSS testsproduced a negligible statistical error of less than 2%)

2. ,,[m] = "" = 0.05 estimated for bias3. COV[,,;] decreased from 29 to 7% with depth, mainly

due to the increasing mean CT; and includes one statisticaland two bi3s components:

u.. from LR analysis, which varies with depth in afashion similar to (14) and ranges between 0.05 and0.15 ksf (2.4 and 7.2 kPa)" = 0.15 ksf (7.2 kPa) for error in the Casagrandeestimates of u; (reflects min-max range in Fig. 15)COV = 5% for error in converting from the one-dayto end-of-primary curve.

The resulting S.(D) minus "n profile (Line 5 in Fig. 14) givesF = (0.876)(1.10) = 0.964, ",' = 0.133, and PI = 24%, due touncenainty in the best estimate of the SHANSEP strength pro­file.

Azzouz et al. (1983) indicate a COV[3Dl of about 5% forthe increase in factor of safety due to three-dimensional end­effects. This model error of "3D = (0.05)(1.097) = 0.055 in-

The reliability analy es using the field vane data with Bjer­rum's (1972) correction and using SHANSEP, both of whichrelied heavily on statistics. predict a 25-30% probability of alarge failure, such as actually occurred. due to uncertainty inthe undrained shear strength of the Bay Mud. In contrast, theauthor's approach predicts a PI of less than 15% due to theuncertaimy in S... which appears to be too low. This apparentunderestimate based on the three-sigma rule may have oc­curred because the :!:3" lines in Fig. 2(b) ignored the S.(FV)data above EI. - 50 (values "believed to be erroneous") andfocused on the UUC strengths. It is preferable to stan withobjective statistical analysis techniques to compute mean vaJ­ues and amount of scatter, then use judgment in selecting bestestimates of parameters and their uncertainties. Sole relianceon the three-sigma rule should be discouraged.

If one has only UUC strength data, the discussers believethat estimates of uncenainty are extremely difficult, if not im­possible. unless one has extensive prior experience with thesame soil. This problem occurs because estimates of the S"from UUC strengths appropriate for stability analyses dependon a totally fortuitous cancellation of three major errors: theincreased S.. due to rapid shearing (strain rate effects) and dueto failure in the vertical direction (ignores anisotropy) must beoffset by an S" reduction due to significant sample disturbance.The first discusser's experience shows that vue strengths canrange from being several times too high [e.g., Table 7 of Ger­maine and Ladd (1988)] to being well less than one-half of anappropriate design S".

REFERENCES

Azzouz, A. S.. BaUgh. M. M.. and Ladd. C. C. (1983). "Correcled fieldvane slrenglh for embankment design." J. Georet:h. £ngrg., ASCE.t09(5). 730-734.

Baecher. G. B.. and Ladd. C. C. (1997). "Fonnal observaliona1approachto staged loading." Transp. Res. Nee. No. /582. Transportation Re~

search Board. Washington. D.C.. 49-52.Bjerrum. L. (1972). "Emb.mkmenlS on soft ground: SOA Report." Proc..

Specialty Con! 0" Performa"ce of Eanh alld Eanh·Stlpported StrllCl..ASCE. New York. 2, I-54.

Chandler. R. J. (1988). "The in-situ measurement of Ihe undr.lined shearstrenglh of clays using Ihe field vane: SOA paper." Va"e shear strengthtesting in ~·Qils. field and laboratory studie~·. STP /014. ASTM. WestConshohocken. Pa.. 13-44.

DeGroot. D. J.• Ladd. C. C.. and Gcnnaine, J. T. (1992). "Direci simpleshear testing of cohesive soils." Res. Rep. R92·18. Dept. of Civ. Engrg..MIT. Cambridge, Mass.

JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / AUGUST 2001 /713

Gennaine. J. T.. and Ladd. C. C. (1988). "Triaxial testing of saturatedcohesive soils: SOA paper." Adwmeed triaxial It!stit.g of soil and rock.STP 977. ASTM. We~t Conshohocken. Pa., 421-459.

Ladd. C. C. (1991). "Stability cvuluatjon during staged construction:' J.CeO/ceh. EIIgrg.. ASCE. 117(4).540-615.

Wright. S. G. (1991). UT£XAS3: (I eomplller program for slope ""abilitycalcllimiolls. Shinoak Software. Austin. Te~.

Discussion by K. S. Li," Member, ASCE,and Joley Lam 12

Use of probabilistic methods in geotechnical engineeringhas been advocated by many researchers since the 19605. mostnotably by Lumb (1966. 1967, 1968). It is gratifying to seethe author. a long-time prominent figure in the determinist'scamp, publish a paper to promote the wider use of probabilisticmethods in geotechnical engineering.

Probabilistic methods are often regarded as being quitemathematical and difficult to learn by determinisLs who areused to the simple concept of factor of safety. Despite this.probabilists seem to enjoy developing increasingly complexmathematical techniques of probabi listie analyses. Conse­quently, geotechnical engineers would find it difficult to come10 grips with difficult probabilistic concepts and jargons suchas nommJ-tail approximation. Rosenblatt's transfonnation, andzero-upcrossing. to name a few. The end-result is obvious­geotechnical practitioners are disinterested in using probabilis­tic methods.

The author ~hould be commended for presenting a meth­od logy for qUIck assessment of Ihe probability failure of geo­technical structures, which is simple to undersland :lnd be usedby practicing geotechnical engineers. although it may be re­garded by some pure probabilists as being not very rigorousmathematically. As discussed by Li and Lumb (1987) and a­dim and Lacasse (1999), the factor of safety is nOl a goodmeasure of risk of failure. A geOlechnical structure with ahigher factor of safety can have a higher risk of failure thana similar structure with a lower factor of safety. depending onthe accuracy of the model used for analysis and the uncertain­ties of the input parameters. The discussers consider that acrude probabilistic analysis would be far more fruitful in giv­ing the designer an idea of the uncertainties and risks associ­ated with a geotechnical design than a very precise calculationof the factor of safety. The three-sigma rule proposed by theauthor is UllIS regarded as being good enough for applicationin geotechnical engineering for most practicaJ cases.

The failure probability given by the three-sigma rule de­pends significantly on the values of sigma to be used in theanalysis. A soil property is regarded as a spatially random var­iability, as the actual values (i.e.. realiz<ltions) of the soil prop­erties can vary from one poilll LO another within an apparentlyhomogeneous soil. The statistical propenies of a soil propertyat any panicular point [which is called the poinl property byLi and Lumb (1987)] can be characterized by constant meanvalue J..L and standard deviation u.

The sigma of <1 soil property is oflen taken to be standarddeviation of the point property. as illustrated in Fig. 2 of theauthor's paper. As discussed by Li and Lumb (1987). Li (1991,1992), and Li and Lee (1991), the appropriale value of sigmato be used for probabilistic analyses may not necessarily bethe standard deviation of the point propeny. The discusserswould like to take this opportunity 10 discuss some conceptsof soil variability using a simple language. It is hoped that the

"0ir.. Victor Li lInd Assoc .. LId .. and Assoc. Prof.. Dept. of Civ.Engrg.. Univ. of Hong Kong. Hong Kong. China.

11Engr.. Victor Li and A:>isoc.. LId.. Hong Kong. China.

discussion could give some guidance to practicing engineersi~ choosing suitable values of sigma when using the three­sigma rule for probabilistic analyses. To enable a better un­derstanding of the concepts. we will consider the simple caseof a homogeneous slope.

CONCEPT 1: POINT PROPERTY VERSUSSPATIAL AVERAGE

~or a nonbriule soil, yielding or failure of soil at a singlePOlOt along a potential failure slip surface will not lead to anuJtimate faiJure. Failure will occur when the sum of the resis­tance.s of all individual soil elements along the entire slip sur­fa~e tS less than disturbing force. Therefore. the probability offaiJ.ure tends to be controlled by the variability of the averagereSistance along the slip surface (i.e.. spatially averaged prop­erty) rather than the variability of the soil resistance at anyparticular point (i.e., the point property). The variability of thespalially averaged property is usually significantly less thanthe point property as a result of variance reduction due tospatial averaging (Vannarcke 1977). We Can illustrate this us­ing a simple example. For sake of discussion. let us assumethat there are only two soil elements along the slip surface. Ifwe denote the soil resistance as X, using a detenninistic ap­proach, the total resistance Xr can be expressed as 2X in adeterministic approach. A mistake is commonly made in as­suming that the statistical properties of X are the same as thoseof the point propeny. if such a mistake is made. the standarddeviation of the totaJ resistance Xr will become 2u.

In reality. the soil resistances along the slip surface will varyfrom point to point. The two soil elements may have differentvalues of soil resistance, and they should be treated as twoseparate random variables, to be denoted as Xl and X2• Assum­ing ;] unifonn soil in a statistical sense, the standard deviationof Xl and Xl can each be taken to be equal to u. The totalresistance Xr along the slip surface should be given by XT :;;;:

Xl + Xz :;;;: 2X. where X is the spatially averaged soil resistancealong the slip surface. If the soil resistances of the two soilelements are independent of each other. the sigma value for Xis equal to a/0. as is discussed in many elementary textbooksin probability. Therefore. the sigma value of XT becomes0u instead of 2u. In generaJ, the standard deviation of aspatially averaged soil property can be wOllen as ur(L). wherer(L) is a reduction factor that depends on the dimension ofthe domain of the spatial average. For most geotechnical struc­tures, the variance reduction is very significant and the totaluncenainty will be dominated by other factors, as will be dis­cussed below.

If the sigma value of Xr is incorrectly taken to be equal tothe larger standard deviation of the point property u, the fail­ure probability of the slope will be overestimated. The errorcan be very significant, particularly when the dimension of theslip surface is large relative to the scale of fluctuation of thesoil property (Li and Lumb J987).

CONCEPT 2: SAMPLE UNCERTAINTY VERSUSINNATE VARIABILITY

In the preceding section. it is assumed that the true meanvalue of the soil property is known. In practice. the true meanvalue of a soil property is never known. and it can only beestimated by, for example, the sample mean value, 111, of thesoil property determined from soil measurements. The vari­ance of a spatiaJly averaged soil property estimated by Ulesample mean value is given (Li and Lumb 1987: Li 1989) by

varIX I = varlm J + u'r'(L) (18)

The first tenn on the right-hand side of (18) is associated withthe sampling uncertainty. while the second term is associated

714/ JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / AUGUST 2001

CONCEPT 3: JUDGMENTAL VALUES

REFERENCES

CONCEPT4:MODELERROR

The expression in (19) covers both the sampling uncertaintyand the uncertainty associated with the determination of thecorrection factor. The sigma value to be used in the three­sigma rule is then the square root of the expression on theright-hand side of (19).

The author has presented a simplified approach to faciHtatethe application of reliability analysis for a variety of geotech­nical problems. The discussers believe that the geoteChnicalengineering community needs to start embracing such a reli­ability-based approach, which explicitly recognizes the uncer­tainties so inherent in geotechnical engineering. Because theauthor's paper is easy to follow. the discussers sincerely hopethat it would facilitate this process.

The author's approach was recently applied to a slope sta­bility problem for which the discussers were commissioned toperform a third-party review. Through this case history. thediscussers would like to provide some discussions on issuesinvolved in applying the author's simplified approach in prac­tice, restricting the comments on slope stability problems in­volving back-calculation of bedding plane materials.

A slope in Southern California had been cut and buttressedadjacent to a roadway with the resulting critical factor of safetyof about 1.3. The geotechnical engineer of record had arguedthat this factor of safety was adequate. rather than the gener­ally required factor of safety of 1.5. The reasons given werethat the slope had been thoroughly investigated and tested andthat a small landslide had occurred during construction fromwhich the effeclive friction angle of the bedding plane mate­rials, considered to be the most critical material in the slope,was back-calculated.

In situations like this. the discussers believe that applicationof the reliability-based approach is very useful. The slope sta­biUty analysis can be performed using any method that satis­fies all the equilibrium conditions; the Spencer's method wasused here. Beyond that, the discussers found it vital to thinkabout each parameter affecting the slope stability as appliedto the particular case before starting the analysis. One con­venient way is 10 think in terms of epistemic and aleatoryuncertainties (as used in the probabilistic seismic hazard anal­ysis). The former uncenainty could, in theory. be reduced bymore studies; shear strength may belong to this category. Thelatter is random uncertainty. not amenable to reduction throughadditional studies; future pore water pressure conditions maybelong more to this category. It is noted that one of the benefitsof reliability-based slope stability analysis is the ability toquantify the contribution of various parameters to the com­puted factor of safety. allowing for rational choice on the mostproductive aspects of investigation to focus. The discussersfound discussions on the uncertainty considerations associated

Discussion by Yoshi Moriwaki l3 andJohn A. Barneich,t4 Members, ASCE

13Prin.. GeoPentech. Santa Ana. CA 92705.lofPrin .. GcoPentech. Santa Ana. CA 92705.

Li. K. S.. and Lee. I. K. (1991). "The assessment of geotechnical safety'"Selt'cted toph'~' ill geolechnical t:ng;neel'ing-Lumb \loillme. K. S. Li.ed., University of New South Wales, Kensington, Australia. 195-229.

Li. K. S.. and Lumb. P. (1987). "Probabilistic design of slopes'" Call.CeO/echo J.. Onawa. 24(4). 520-535.

Lumb. P. (1966). ''The variability of natural soils." ClUl. Ceo/ech. J ..Ottawa. 3. 74-97.

Lumb. P. (1967). "Statistical methods in soil invesligations:' Proc.. 51hAustralian-New Zealand COli/. Oil Soil Mech. alld Foulld. Engrg.. 26­33.

Lumb. P. (1968). "Statistical aspects of soil measurements'" Proc.. 41hAustralian Road Res. COllf. 4.1761-1770.

Nadim, F.. and Lacasse. S. (1999). "Probabilistic slope stability evalua­tion." Ceou:ch. Risk Mgml .• Proc., Alln. Seminar of Ceofech. IJh,.•Hong Kong Institution of Engineers. Hong Kong. 179-186.

Vanmarckc, E. H. (1977). "Probabilistic modeling of soil profile." J.Geof/!c!l. ElIgrg.• A$CE. 103(11). 1227-1246.

( 19)variX"} =N'varlm} + m'var{N}

Measurements of soil properties are often subjected to bias.The corrected soil property X' is often expressed in tenns ofthe measured soil property X using the simple relationship ofX' ;;;; NX, where N is the correction factor. The same correctionfactor can be applied to obtain the "corrected" spatially av­eraged soil property. If the contribution from spatial variabilitycan be ignored. the variance of corrected spatial average X" isgiven as follows (Li 1989):

Sometimes, very limited or virtually no data exist's for aparticular random soil parameter. Certain judgmental valuesmay then be assigned by the designers for the mean value andstandard deviation of the random soil parameter to enable aprobabili tic analysi to be pert rmed. The uncertainty asSo­ciated with the judgmental value is similar in concept to thesampling uncertainty associated with the estimation of themean value of the soil property. There is no variance reductiondue to spatial averaging for judgmental design values. One cantherefore treat the uncertainties for judgmental values in a sim­ilar way as the sampling uncertainty discussed in (18). Thesigma value to be used for the three-sigma rule is then simplythe judgmental standard deviation of the soil property.

Li. K. S. (1989). "Discussion of 'Stability of gravity platfoml: reli;lbilityanalysis .. ·· Geolechniqlle. London, 39(4). 561-562.

Li. K. S. (1991). "Discussion of 'Embankment reliability versus factorof safety: before and after slide repak' .. I1u. 1. for Nl/lller. and Ana­Iylical Methods ill Ceomech.. 15( 12). 893-895.

Li. K. S. (1992). "Some common mistakes in probabilistic analysis ofslopes." Proc.. 61h 1m. Symp. all ul.1ldslides. Christchurch. New Zea­land. I. 475-~8().

with the spmial variability of the soil properlY. Taking a simpleexample, if the average soil strength is estimated from thesample mean value calculated from a total of n independentmeasurements, the sampling variance var{m} becomes (J2In.The standard deviation of the sample mean is then gi ven byalvn.

The first tenn of (18) would not be affected by the numberof soil measurements, but not the dimension of the domain ofspatial average. The concept of variance reduction due to spa­tial averaging. therefore. does not apply for sampling uncer­tainty. For a large spatial domain. the variance reduction dueto spatial averaging is significant and the total uncertainty ofthe spatially averaged soil property will then be dominated bythe sampling uncenainty [i.e., the first term of (J 8)].

The sample uncertainty can be reduced by more soil mea­surements. However, when the sampl.ing uncertainty [i.e., thefirst term of (18)) becomes comparable WitJl the innate varia­bility of the soil property [i.e., the second leml of (19)], it willnot be cost effective to further reduce the sampling uncertaintyby taking more measurements. In most geotechn.ical designs.soil data is usually limited and the effect of spatial averagingis significant, with the result that the total uncertainty is nor­mally dominated by the samp.ling uncertainty, i.e., the (irstterm of (18). The sigma value to be used for the three-sigmarule is therefore simply the standard deviation of the samplemean.

JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING I AUGUST 2001 /715

with paramelers presented by Christian et al. (1994) to be veryuseful in thinking about these issues.

As part of this third-party review. the discussers first re­peated the back-caJculation using the Spencer's method 10 con­firm the general reasonableness of the back-calculated effcc·live friction angle of 17°. In Southern California, the shearstrength associated with many landslides involving beddingplane and preshcared malcrials can often be represented by aneffective friction angle wiLholiL an artificial effective cohesionintercept. Though 1101 used in this case. the discussers believethat back-calculation should in general result in a distributionof back-calculated friction angle. Because adequate details ofactual landslide are rarely known. one should apply the au­thor's approach in back-calculation as well: forcing the back­calculated factor of safety of one Wilh the best estimate andplus and minus one standard deviation of parameters that arenot well known at the time of land~lide to develop this distri­bution. The resulting range of back-c~llculatecl shear strengthcall be used with presumably greater. future ranges of otherparameters to perform forward reliability-based slope ~tability

analysis u~ing the author's approHch.One caution regarding the use of the author's approaCh 10

slope stability problems is the lack of spatial variability con­siderations in the author's approach. This may resulL in over­estimation of failure probability. Although discussions pre­sented in Christian et al. (1994) should be helpful in thisregard. one can bound the potential problem by assuming thetwo extremes of perfect correlation and no correlation of prop­erties over space without gClIing into analysis involving au­tocorrelation distance.

Using the author's approach together with the ranges in ma­terial properties and geologic conditions. the discussers per­formed parametric slope stability analyses of the slopes at sev­eral sections along the 1.000 ft long slope to compute a bestestimate overall static factor of safety of about 1.3 and a 90­92% reliability that the actual factor of safely would be greaterthan 1. The results were not very dependcnt on the assumeddistributions (lognormal or normal relations), as expected forthe range of the reliability valucs involved. Funher, it wasfound that. if the computed factor of safety had been 1.5 (forthe same ranges of material parametcrs and geologic condi·tions), this reliability would have increased to 96-98%. If.however, less were known about the geologic conditions andmaterial properties, the reliability likely could have been inthe 90-92% range, even with a computed best-estimate factorof safety of 1.5. Thus, in essence, the author's approach wasused to quantify the benefit of detailed investigation and back­calculation as applied to this particulur program. The com­puted factor of safety of 1.3 in this case was equivalent to thecomputed factor of safety of 1.5 without such additional in­formation.

The above evaluation. were computed assuming thm thewater table stayed below the base of the critical slide surface.which was considered reasonable duc to restriction to devel·apmcnt upslope of the critical slide surfClcc (no significam wa­tering) ancl drainage provisions incorporated into the slope.However, because so many landslides in SOllthern Californiaare caused by unexpectedly adverse water conditions and be­cause the future waler condilions tend to be "random'" anadditional analysis was performed using extremely highgroundwater conditions as the best estimale: it was found thatthe computed factor of safety would be slightly greater than1.0. reducing the reliability of the actua] factor of safely to begreater than 1.0 to 50-60%. Such conditions. though ratherunlikely. would not be acceptable in general. However, if thelikelihood of such conditions from ever developing were re­duced further by a water monitoring/observation system, theacceptance would be easier. Such a monitoring/observationsystem was suggested.

Based on the resulls of this third-party review and consid­eration that lhe consequence of the rai lure of slope would notbe life threatening. the owner decided to accept the slope withthe provision that the design bui Id contractor provide a 10·year insurance policy covering stability of the slope.

tn closing, the discussers believe that the amhor has donethe profession a great service. This easy-lo-use procedure al­lows the practicing geotechnical engineers and engineering ge­ologists to quickly put factor of safety estimates in perspeclivewith respect to uncertainty in analysis parameters. At the sametime, the results would provide a means of communicating thisperspective to client and possibly regulatory agencies, so thatinformed discussion can be started. Though, as applied toslope stability problems, a more complete reliability approachsuch as the one by Christian et al. (1994) may be preferablefor some problems. it should be easier to apply the author'sapproach, With the emphasis in practice being more on therelative reliability (as was the case in the example use of theauthor's approach presented herein). the lower-bound estimateof the probability of failure provided by these simplified pro­cedures should not be critical. For significant projects, theselimitations can be overcome by an even more formal approach(e.g .• Whitman 1984: Barneich et al. 1996) using fault trees.etc. In whatever ways possible. the discussers would like tosee the geotechnical engineering profession move more in thedirection of reliability-based approach, at least for certaintypes of problems.

REFERENCES

Barneich. J.• Majors. D.. Moriwaki, Y.. Kulkarni. R.. and Davidson, R.(1996). "The reliability analysis of a major dam projccl." Pmc.. VII­caw;",.\' ·9r'5. ASCE. Reston. Va.. 1367-1382.

Chrisl.i:m. J. T.. Uldd. C. C. and Baecher. G. B. (1994). "Reliabilityapplied to slope stability analysis." J. GeQtl:cll. £lIgrg.. ASCE. 120( 12).2t80-2207.

Whitman. R. Y. (1984). "Evaluating calculated risk in geolcchnical en­gineering." J. Geotech. Ellgrg.. ASCE, 110(2). 145-188.

Discussion byJohn H. Schmertmann,ts Fellow, ASCE

The author has made an excellent and welcome contributiontowards simplifying and thus encouraging the use of the in­terrelationship between factor of safety (F) and reliability(R). which depends on the variability of F. Tables 2 and 75hould prove very useful for evaluating low·end-tail and high­end-tail reliability, respectively. for the lognormaJ distributionof F considered reasonable and used by the author.

At the expense of added complexity, which hopefully willnot discourage use of the methods described in this paper,some warning seems appropriate to help users of the proposedmethods avoid, or at least recognize, some items thai canproduce unconservative results (R lower than expecled for agiven F).

The potentially unconserv3live items briefly discussedherein include Lhe distribution assumed for the variability inF. ullconserv3tive bias. and three items 1113t can increase VI'-non-independent paramelcrs, soil variability, and model un­certainty. The following uses the author's (h = 0.25. F = 1.50.R = 99% example [(I), Fig. I. and Table IJ for a convenientdemonstration of the effect of these items. Although the au~

thor's example uses the low-end tail of a lognormal distribu­tion. the items discussed remain appropriate in principle for

I~St.:hlllerlmann & Cmpps. Inc.. 4509 W 23rd Ave .. SIC. 19. Gaines­ville. FL 32606.

716/ JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING J AUGUST 2001

both tails of any distribution. The following considers the ef­fect of each item separately.

DISTRIBUTION OF F

The F required for a given R depends on the shape of thetail of the distribution used. For example, using the nonnalversus the lognormal increases the required F to 1.58 for R =99%. The increase becomes more dramatic as V,.· increases.Doubtless, nature provides many distributions that our ideal­ized ones can only approximate. Perhaps we can include theapproximation as part of the model uncertainty noted subse­quently?

BIAS

Users should also consider the possibility of unconservativebias in one or more of the parameters used to calculate F. Forexample, if the test for tan 8 in (I) produced values consis­tently 10% too high because of undetected or uncorrected ap­paratus friction, then the required F increases to 1.65.

NON-INDEPENDENCE OF PARAMETERS

The use of the Taylor series to obtain (2a) assumes that eachof the parameters in Table 1 varies independently of the others(often questionable; for example, "Y~fand "Ylifmay correlate witheach other). Eq. (2a) provides one approx.imate value for (TF'

The upper bound for (TF results form adding the absolute val­ues of (t>.F/2) in Table I-in this case, giving Up ~ 0.405.This would require F ~ 1.75 for R ~ 99%. How best to com­bine the parameter standard deviations depends on the judg­ment of the engineer. Perhaps the author can comment andprovide some guidance.

SOIL VARIABILITY

The author presents Table 3 as a guide for choosing V forvarious geotechnical parameters, as obtained from the litera­ture and his experience. If the author knows, it would help toknow which Vs in Table 3 include soil (material) as well astest variabiLity. WouJd the author suggest that the low V valuesin each range apply primarily to sites judged to have relativelylow soil variability and the high V values LO those with highsoil variability? When previously considering such questionsin Schmertmann (1989, see Tables 2 and 3), this discusserapplied an approximalc 1.5 [acLOr to VI"~U; to estimate V(I",,~+ ... 'l)

for all relative site soil variabilities (low, average, high). As­suming Table I includes only test variability, applying a 1.5factor LO the standard deviation in Table 1 gives a new rI,.· ;;;;

0.375, which gives a new required F ~ 1.70 for R ~ 99%.

MODEL UNCERTAINTY

Every mathematical model, includ.ing (1), presents a sim­plified picture of reality. [FQfCIUaJ - F(!)] varies from site to siteand, thus, the model of reality itself includes uncertainty. De­noLe the standard devialion of this uncertainty as (TM' One de­termines {TM from comparisons of Fu and Flo with Fa generallyonly known when equal to I at some defined failure. Or, theengineer can estimate CfM from the literature and personal ex­perience. Perhaps Table 1 already includes model uncertainty?Perhaps the author can suggest typical values for (J"lot or VM?For the purpose of this discussion, assume no bias and CfM ;;;;

0.20 when F;;;; I, or VAl ;;;; 20%. Further assuming the modeluncertai.nty independent of the Table I parameter uncertainties,and lIsing the Taylor series approx.irnate value for a combinedVp , gives V, ~ 26% and F ~ 1.88 for R ~ 99%.

COMBINATIONS

Each of the above items produces a relatively small, butpossibly importanl, change in the F required to maintain R =

99%. But they can occur in combination. If they all occurredtogether, namely [normal distribution and 10% bias and Canon-independent upper bound (TF and a 1.5 soil variability fac·tor) and (an independent V. ~ 20%)], Lhen the required Fincreases to 3.28 for R = 99%. Using the lognonnal distribu·tion reduces the required F to 2.48. Also, using the assumptionof parameter independence further reduces the required F to2.15. As shown, the items presented can increase F substan­tially from the author's 1.50 and may require consideration inhis and other applications of the proposed meLhods.

REFERENCE

Schmenmann. J. H. (1989). "Density tests above zero air voids line:' J.Geotech. £ng'g.. ASCE, 115(7). 1003-1018.

Closure by J. M. Duncan'·

INTRODUCTION

The writer is very pleased that the paper has drawn suchextensive discussion, reflecting a high degree of interest in thetopic, and is grateful to all the discussers for their valuableCOmments. Their views of the benefits of reliability analysisin geotechnical engineering practice are enlightening and wel­come. Their discussions of the necessary prerequisites for de­terministic and probabilistic analyses will be of lasting value.Finally, their discussions of various methods for calculatingprobability of failure and unsatisfactory perfonnance providea more comprehensive view of these techniques than was con­tained in the paper. The writer appreciates each of these con­tributions and extend his sincere thanks to all of the discussers.

BENEFITS OF RELIABILITY ANALYSES

The discussions describe a number of benefits of reliabilityanalyses in geotechnical engi.neering applications. These aresummarized briefly in Table 15.

APPROPRIATE USE OF RELIABILITY METHODS

The discussions offer usefu.l insights infO appropriate (andinappropriate) use of reliability analyses with regard to theneed for good data, choices of su.itable test methods and in­terpretation techniques, and the need for sound judgmenl inmany aspects of geotechnical analyses.

Need for Good Data

Koutsoftas expressed the concern that some engineers maybe tempted to use reliability analyses as a substitute for thor­ough investigations with high quality data. The writer hopesthat this will not be the case. It would be inappropriate to usereliability analyses to justify inferior investigations or substan­dard data. Quite the contrary, the type of reliability analysesdescribed in the paper are most appropriately used to evaluatethe combined effects of uncertainties in anaJyses, to identifythose aspects of analyses where uncertainties are most signif­icant, where further investigation would be mosl valuable, andwhere further investigation would or would not significantlyreduce the degree of uncertainty involved in the final result.

Test Methods and Interpretatiort

Although there is undoubted.ly uniform agreement lhat gooddata is needed for either detenninistic or probabilistic analyses,preferences for different test methods to obtain "good dala"

'(lUniv. Distinguished Prof.. Dept. of Civ. and Envir. Engrg.. VirginiaTech., Blacksburg. VA 24061.

JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / AUGUST 2001/717

Need for Judgment

As noted by Focht and Focht, the lise of reliability tech­niques does not eliminate the need for sound engineering judg­ment. rn faci. reliability melhods cannol- be applied withoulmaking judgmenls regarding both Ihe techniques and the dataon which the analyses are based. The necessity t use soundengineering judgmcnt in applying rcliability analysis pervadesall of the discussions of the paper and is clearly of paramountimportance.

The discussions include many specific examples of the useof judgments to interpret and supplement available test data.It is worthwhile to summarize a few of Ihem here, to illustratetheir importance and their pervasive nature:

vary widely, based on an individual's experience. As noted inthe next section, some engineers eschew SPT data and preferthe DMT, some rejeci UU triaxia.l lesls and favor field vaneshear, and some rejeci vane shear and prefer UU triaxial tests,based on their individual experiences.

It is well to remember that all of the methods of collectingand interpreting data thaI are used in geotechnjcaJ engineeringanalyses are semiempirica.l, and that their use is justified byfavorable experience. It is doublful that any type of geotech­nical investigation and analysis can be found where approxi­mations and assumptions are not required to link the resultsof field investigations to conclusions regarding likely perfor­mance. Experience with the use of particular tests and methodsof interpretation is an important component of a geotechnicalengineer's personal art and forms Ihe necessary core for mak­ing the important judgmenls that are inevitably required.

Noted benefits of reliability analysis forgeotechnical engincering

Provides systenHuic method for evaluating combinedinfluence of uncertainties in parameters affeclingfactor of safety.Provides systemic method of assessing degr~e ofsafety, at least in relative tem1S.

Provides useful 1001 for evalu:ujng risk associatedwith design recommendations.

Provides framework for establishing appropriate fac­tors of safety and leads to better understanding ofrelative importance of uncertainties.

Provides lools for communicating relative risk andbenefit to client, and provides client with clearerchoices regarding economic tradeoff between initiollconstruction COSls and risk of failure.Method described in paper integrates easily withconventional approaches.When used with judgment. melhod described in pa­per will improve value of geotechnical analyses.

Method described in paper is easy to understand andapply, "allhough it may be regarded by some ... 3S

being not very rigorous m31hcmatic:llly,"Even crude probabilistk analysis is more useful fordeveloping idea of uncertainties and risks thun veryprecise deterministic calculation of factor of ..afcty.

Quantifies contributions 10 overall uncertainty ofeach parameter.Provide means of detcnnining consequences of un~

certainly. benefit of detailed investigation. and bene·fits of back analysis.Put.. computed values of factor of sM'cty into per·spcctive with regard to unccI1ainties in analysis pa·ramclers.Provide means of communicating reliability of fac­tor of safety to c1ienl.'i and perhaps regulators, sothat infonned discussion can begin.

TABLE 15. Benefits of Reliability Analyses

Ladd andDa Re

Koutsoftas

Focht andFocht

Discussers

Moriwuki antiBarneich

Christian andBuecher

Ladd and Da Re employed engineering judgmenl in sev­eral aspects of their reanalyses of the LASH TerminaJslope failure. They lIsed SHANSEP parameter correla­tions. developed for other clays. which they judged to beapplicable to San Francisco Bay Mud. They adjusted val­ues of OCR based on previous experience. They increasedtheir computed two-dimensional factor of safety as an al­lowance for end effects. And they used a subjective esti­mate of spatiaJ variation in undrained strength to supple­ment the available data. It is always necessary to usejudgments and adjustments such as these to fill gaps indata and to improve the reasonableness of parameter val­ues for geotechnical analyses.Christian and Baecher and Li and Lam noted that onemust rely on approximations to estimate values of neededparameters when insufficient data are available. Examplesin this category would include parameters such as frictionangles for sands and gravels (no undisturbed samples. andtherefore no laboratory test data), and equivalent fluid unitweights for use in wall design (no data because valuesare based entirely on experience and judgment). Use ofjudgments, approximations. and correlations is essentialin many circumstances and should not be construed as asign of poor engineering practice.Focht and Focht noted that. for linear projects that crossa number of geologic units or spatially diverse conditions,judgment is necessary 10 choose conditions that aredeemed to be representative for each segment of the proj­ect, based on an understanding of the geologic conditionsand information from widely spaded borings.Koutsoftas used correlations between vane shear strengthand preconsolidation pressu.re developed from other sitesin the Bay Area to estimate preconsolidation pressures atHamihon Air Force Base. Allhough it has been deler­mined that the site at Hamilton described by Koutsoftasis not the one where the data shown in Fig. 2(a) of thepaper were obtained (Koulsoftas, personal communica­tioll. 200 I), the use of vane shear correlations as a toolfor evaJuating stress history is nevertheless of interest andillustrates the use of judgment and correlations to supple­ment site-specific data.Failmezger uses judgment and experience to discount Iheuse of SPT data for estimating settlement of foundationson sand.Ladd and Da Re and Koutsoftas use judgment and ex­perience to rule out the use of UU triaxial compressiontests to evaluate undrained shear strengths of clays.Fochl and Focht use judgment and experience to rejectthe use of field vane shear lests to evaluate undrainedshear strengths of clays.Ladd and Da Re used judgment and experience to con­clude ,hal three CKoUDSS lests run by MIT on normallyconsolidated Bay Mud fr0111 another site in the Bay Areaprovided a sufficient basis for reanaJysis of stability of theLASH Temlinal slope.Moriwaki and Bameich used judgment and experience asthe basis for including assumed extremely high ground­water as a condition in their analysis of the stability of aslope in southern California.Koutsoftas used the results of his reanalysis of the LASHTerminal slope to conclude thai the undrained factor ofsafety would have been significantly higher than the valuecalculated in the paper. and therefore the strength of themud must have been reduced during excavation of thetrench, most likely by drainage at the base of the deposit.Schmenmnnn used his experience with reliabililY analysesto identify several items that could lead to higher esti­males of the probability of failure for the retaining wallexample in the paper.

718/ JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING / AUGUST 2001

Li and Lam

Thus, although reliability analysis can provide a valuablesupplement to conventional detenninistic analyses of stabilityand settlement, neither deterministic nor probabilistic analysescan be applied without experience and judgment.

METHODS AND ASSUMPTIONS FORCALCULATING P,

Many of the discussions focused on reliability analysis tech­niques and suggested enhancements and altemati ves to themetbod described in the paper. These comments concernsources of uncenaimy, the three-sigma rule, model uncertainty,and the lognormal distribution.

Sources of Uncertainty, Types of Uncertainty, Effectson Computed P,

Evaluation of geotechnical parameters is fraught with manytypes of uncertainties. Given a collection of test data, geo­technical engineers are faced with a number of imponant ques­tions. Among them are

Does the scatter in the data represent variations in Lheproperties of the ground, variations due to random testerrors, or both?Should the data be corrected for systematic effects beforethey can be used in analyses to estimate performance?Would the uncertainties in the value of a needed param­et.er be significantly reduced if more tests were per­formed?Is the data possibly deficient because the sampling or testlocations missed some important feature of the site­some important geologic detail?[s it appropriate to select the "most likely value" of aneeded parameter as t.he average measured value, orshould the "most likely value" be higher or lower thanthe average?Do the results of the analysis to be perfonned depend onquantities not reflected in the data, such as possible futurechanges in groundwater levels?

Ladd and Oa Re noted that it is important to consider thesources of uncertainty involved, or else the results of reliabilityanalyses will be misleading. They suggest that uncertaintiesbe considered in the following categories, which provide auseful framework for evaluation and judgment:

I. Systematic variations of properties from one location toanother

2. Scatter due to spatial variations about the mean trend,and the scale over which this fluctuation occurs

3. Scatler due to random test errors4. Statistical uncertainty due to limited numbers of tests5. Error in the mean due to measurement bias

While it is useful to realize that these types of uncertainty allaffect test data, it is important to understand that these affectscan only be identified, separated, and evaluated through theexercise of judgment.

One issue that emerges clearly from the discussions by Laddand Oa Re and by Li and Lam is that the variability of theaverage value of a property for a large mass of soil is signif­icantly less than the variability of the "point values" measuredin a series of tests. Li and Lam pointed out that, aU other thingsbeing equal. using the standard deviation of (he point value asthe standard deviation of the spatial average will result in avalue of probability of failure that is too large.

Focht and Focht pointed out that an additional uncertaintyshould be considered when selecting an appropriate "most

likely" spatial average value of a property such as shearslrength, because the failure mechanism will find and tend topass through the weakest material.

Moriwaki and Bameich prefer to group uncertainties in twocategories, following the convention in probabilistic seismichazard analysis:

I. Epistemic uncertainties are those that are reduciblethrough beuer or more extensive testing. Hypothetically.at least, uncertainties about properties such as the resid­ual friction angle of a clay deposit would fall in thiscategory. There are, however, practical limits on the ex­lent of investigation. with tbe result that epistemic un­certainties cannot be completely eliminated and must bedealt with through use of theory and judgmental evalu­ation.

2. Aleatory uncertainties are those that cannot be evaluatedthrough investigation, no matter how thorough. An ex­ample in this category would be the highest possible fu­ture water level wit.hin a slope.

Moriwaki and Barneich suggested Lhat the emphasis in prac­tice is on values of relative uncertainty, rat.her than absoluteuncertainty. Useful results for such comparative analyses canbe achieved using approximate vaJues of properties and theirstandard deviations and using simplified analyses, providedthey are used consistently.

Coefficients of Variation and Three-Sigma Rule

Schmertrnann raised a number of questions about the co­efficients of variation listed in Table 3: Which of the valuesinclude both soil and test variability? fs it correct that thesmaller values pertain to sites with less variability and thelarger values to sites with more variability? The values of co­efficient of consolidation from the writer's own files representstatistics for C" evaluated using Casagrande's method and Tay­lor's method for several diJferent clays and therefore representboth soil and test variability. Likewise, where values are at­l'ributed to more than a single source. they would reflect morethan one soil, and test variability would also inevitably beinvolved.

Ladd and Da Re asked if the coefficient of variation ofSJp should depend on OCR. and if the coefficient of variationof 5" should depend on the type of teSt and the heterogeneityof the deposit? The writer believes tlle answer to both ques­tions is yes, but has no data to support that judgment

As noted in the paper, the values of coefficient of variationlisted i.n Table 3 cover extremely wide ranges of values forthe same parameter. and the conditions of sampling and testingarc not specified. The values in Table 3, therefore, provideollly a rough guide for any given case. It is important to usejudgment in applying coefficients of variation from publishedsources.

Christ.ian and Baecher provided very useful information re·lating lO the use of the three-sigma rule. showing that people(experienced engineers included) tend to be overconfidentabout their ability to estimate values, and estimate possibleranges of values that are narrower than the acnlal range. If therange between the highest conceivable value (HeY) and thelowest conceivable vaJue (LeY) is too small, values of coef­ficient of variation estimated using the three-sigma rule willalso be too small, introducing an unconservative bias in reli­ability analysis.

Their Table 9 is very interesting in this regard. Based all

statistics. it shows that the expected range of values in a sam­ple of 20 values is 3.7 times the standard deviation, and theexpected range of values in a sample of 30 to 4.1 times thestandard deviation. This information could be used to improve

JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGtNEERING I AUGUST 2001/719

estimated values of standard deviaLion by modifying the three­sigma rule. If the experience of the person making the estimateencompasses examination of sample sizes in the range of 20­30 values, a beuer estimate of standard deviation would bemade by dividing the range between HCY and LCY by 4rather than 6:

HCY - LCY(J" = ---,-----

4(20)

sure, the base friction, and the unit weights of the concreteand the backfill. The effect of the assumed lognormal distri­bution, while deserving of consideration, is not part of thedeterministic calculation, which the writer considers consti­tutes the model. The writer believes it would be preferable toexamine the effect of the assumed distribution by computingprobabilities of failure using more than one assumed distri­bution. The following paragraphs provide an easy means fordoing this.

Based on Christian and Baecher's discussion, and the ideathat experience will often be limited to examination of samplesincluding no more than 30 values. this "two-sigma rule"would be 3n improvement on the three-sigma rule discussedin the paper. It could be extended easily to a "graphical IWO­sigma rule." placing the average minus sigma and average plussigma lines halfway between the most likely variation and theextremes.

Model Uncertainty

Model uncertainty was included in only one of the fourexamples in the paper-the one involving estimalion of thesettlement of a footing on sand based on SPT blow count. Thedata shown in Fig. 6 were used to estimate the coefficient ofvariation = 67% associated with the use of (8), which in thiscase is the "model," The uncertainty is due to scatter in SPTresuhs and approximations made in deriving (8).

Model uncertainty was not tncluded in the other examplesin the paper, but it probably should have been. Li and Lam,Failmezgcr, and Schmertmann all mentioned model uncer­tainty (or model error) in their discussions and suggested waysof including it in analyses. Schmel1mann asked if the writercould suggest coefficients of variation for model uncertaintyfor the retaining wall example. and if the assumed lognormaldistribution of [<lCWr of safety would be considered to be anaspect of model uncertainty. In the writer's opinion, if move­ment of the retaining wall is not resisted by passive pressureon the front of the footing. the equation of horizontal equilib­rium on which (I) is based provides a highly reliable model,which would need no additional allowance for uncertainty be­yond the coefficients of variation assigned to the earth pres-

Assumed Distributions of F

Christian and Baecher discuss the characteristics of the log­normal distribution and indicate their preference for the normaldistribution. There i no way of detemlining which of these isbest in a given case, as is clear from their discussion. It mayoften be useful, therefore, to be able to compute P, using boththe normal and the lognormal distributions.

Table 16 can be used to determine PI based on an assumednormal distribution in the same way that Table 2 is used todetermine PI based on an assumed lognomml distribution. Bycomparing the tables, it can be seen that values of PI based ona lognormal distribution, and values of PI based on a normaldistribution, are considerably different in some regions of thetables. In Table 16, Ihe boldface values of PJ based on anassumed normal distribution are greater than those based onan assumed lognormal distribution. In the lightface values thereverse is true.

As examples of the effect on PI involved in assuming thatfactor of safety is normally or lognonnally distributed. con­sider the values from the paper and the discussions given inTable 17.

Failmezger suggesled thal. because Ihe lognormal distribu­tion is skewed to the left, estimates of the probability that thefactor of safety could be less than 1.0 will tend to be conser­vative, and estimates of the probability that settlements couldbe larger than computed will tend to be unconsetvative, ascompared with an assumed normal distribution. Table 16shows that this is not the case. Assuming a lognonnal distri­bution of factor of safety can be either less conservative ormore conservative than assuming a normal distribution. de-

TABLE 16. Probabilities that Factor of Safely Is Smaller than 1.0, Based on Nomlal Distribution of Factor of Safety

Coefficient of Varialion of Factor of Safety

F",\ 21l 41< 61l 8% 10% 12% 14% 16% 20% 25% 30% 40% 50% 60% 80%

1.05 0.9% 11.7% 21.4% 27.61l 31.7% 34.6% 36.71l 38.3'11- 40.6% 42.4% 43.7% 45.3% 46.2% 46.8% 47.61l1.10 0.0% 1.2% 6.5% 12.8% 18.2% 22.4% 25.81l 28.5% 32.5~ 35.8% 38.1% 41.0% 42.8% 44.0% 45.5%1.15 0.0% 0.1% 1.5% 5.2% 9.6% t3.9% 17.6% 20.7% 25.7% 30.1% 33.2% 37.2% 39.7% 41.4tK- 43.51<1.16 0.0% 0.0% 1..1% 4.2% 8.4% 12.5% 16.2% 19.4% 24.5% 29.1% 32.3% 36.5% 39.1% 40.9% 43.2\l-1.18 0.0% 0.0% 0.6% 2.8% 6.4% 10.2% 13.8% 17.0% 22.3% 27.1% 30.6% 35.1% 38.0% 40.0% 42.4%1.20 0.0% 0.0% 0.3% 1.9% 4.8% 8.2% 11.7% 14.9% 20.2% 25.2~ 28.9% 33.8% 36.9% 39.1% 41.7%1.25 0.0% 0.0% 0.09'1l 0.6% Z.3% 4.8% 7.7% 10.6% 15.9% 21.2% 25.2% 30.9% 34.5% 36.9% 40.1%1.30 0.0% 0.0% 0.0% 0.2% 1.1% 2.7% 5.0% 7.5% 12.4% 17.8% 22.1% 28.2% 32.2% 35.0'l!- 38.6%1.35 0.0% 0.0% 0.0% 0.1% 0.5% 1.5% 3.2% 5.3% 9.7% 15.0% 19.4% 25.8% 30.2% 33.3% 37.3>;<1.40 0.0% 0.0% 0.0% 0.0% 0.2% 0.9% 2.1% 3.7% 7.7% 12.7% 17.0% 23.8% 28.4% 31.7% 36.0%1.50 0.0% 0.0% 0.0% 0.0% 0.0% 0.3% 0.9% 1.9% 4.8% 9.1% 13.3% 20.Z% 25.2% 28.9% 33.8%1.60 0.0% 0.0% 0.0% 0.0% 0.0% 0.1% 0.4% 1.0% 3.0% 6.7% 10.6% 17.4% 22.7% 26.6% 32.011-1.70 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.2% 0.5% 2.0% 5.0% 8.5% 15.2% 20.5% 24.6% 30.3%1.80 0.0% 0.0% 0.0% O.Otrll 0.0% 0.0% 0.1% 0.3% 1.3% 3.8% 6.9% 13.3% 18.7% Z2.9% 28.9\l-1.90 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.2% 0.9% 2.9% 5.7% 11.8% 17.2% 21.5% 27.7%2.00 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.1% 0.6% 2.3% 4.8% to.6% 15.9% 20.2% Z6.6%2.20 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% O.Oo/ll 0.0% 0.3% 1.5% 3.5% 8.6% 13.8% 18.2% Z4.8%2.40 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.2% 1.0% 2.6% 7.2% 12.2% 16.5% 23.3%2.60 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.1% 0.7% 2.0% 6.2% 10.9% 15.3% 22.1%2.80 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.1% 0.5% 1.6% 5.4% 9.9% 14.2% 21.1 %3.00 0.0% 11.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.0% 0.4% 1.3% 4.8% 9.1% 13.3% 20.2%

Note: F6 /f.\ = fuctor of ~afelY computed using mos1. likely vlIlues of parameters. Where table values lire boldface. vnlue of PJ computed assumingnomull distribution is greater than value of PI computed assuming lognonnal dislribution. Where table v<llues are lightface. nonnul PI i~ smaller thanthe lognormal PJ'

720 I JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING I AUGUST 2001