1999 CGJ Phoon Kulhawy Evaluation of Variability

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Evaluation of geotechnical property variability

Kok-Kwang Phoon and Fred H. Kulhawy

Abstract: To evaluate geotechnical variability on a general basis that will facilitate the use of reliability-based design

procedures, it is necessary to assess inherent soil variability, measurement error, and transformation uncertainty

separately. The inherent variability and measurement error are addressed in a companion paper, and transformation

uncertainty is addressed herein. A second-moment probabilistic approach is applied to combine these uncertainties

consistently based on the manner in which the design soil property is derived. The design properties considered in this

paper are undrained shear strength, effective stress friction angle, in situ horizontal stress coefficient, and Young’s

modulus. This paper concludes with specific guidelines on the typical coefficients of variation for these common design

soil properties as a function of the test type and the type of correlation used.

Key words: transformation uncertainty, undrained shear strength, friction angle, in situ horizontal stress coefficient,

Young’s modulus, geotechnical variability.

Résumé : Pour évaluer la variabilité géotechnique d’une manière générale permettant d’utiliser des procédures de

calcul basée sur la fiabilité, il est nécessaire de prendre en compte séparément la variabilité intrinsèque du sol,

l’erreur de mesure et l’incertitude de transformation. On a traité de la variabilité intrinsèque et de l’erreur de mesuredans un article conjoint et l’incertitude de transformation fait l’objet de la présente communication. Puis on applique

une approche probabiliste d’ordre deux pour combiner ces incertitudes, en accord avec la façon dont on acquiert la

propriété du sol en question. Les propriétés étudiées dans cet article sont : (i) la résistance au cisaillement non drainé,

(ii) l’angle de frottement effectif, (iii) le coefficient horizontal de poussée des terres en place et (iv) le module

d’Young. On conclut par des recommandations sur les coefficients de variation typiques à employer pour des

paramètres courants de calcul, en fonction du type d’essai et de corrélation utilisés.

Mots clés : incertitude de transformation, résistance au cisaillement non drainé, angle de frottement, coefficient horizon-

tal de poussée des terres en place, module d’Young, variabilité géotechnique.

[Traduit par la Rédaction] P hoon and K ulhaw y 63 9

Introduction

Since the mid-1970s, numerous reliability-based design(RBD) codes have been put into practice for routine struc-tural design (e.g., ACI 1983; BSI 1972; CSA 1974; NKB1978). However, the geotechnical design community hasbeen slow in assimilating this new design methodology.Part of the reason lies in the difficulty of assessing the vari-ability of design soil properties that are needed for thesenew RBD procedures. Unlike the variability of manufac-tured materials used in structures, geotechnical variabilityis a complex attribute that results from many disparatesources of uncertainties. The three primary sources are in-herent soil variability, measurement error, and transforma-tion uncertainty. The degree of uncertainty arising from

these sources generally depends on factors such as the vari-ability of the soil profile at the site, the degree of equip-ment and procedural control maintained during testing, andthe precision of the correlation model used to transform thetest result measurement into the desired soil property.

A number of the soil property statistics reported in the

geotechnical literature have been determined from total vari-ability analyses that implicitly assume a uniform source of uncertainty. Clearly, these lumped statistics are only applica-ble to the specific set of circumstances (site conditions, mea-surement techniques, correlation models) for which thedesign soil properties were derived. To evaluate geotechnicalvariability on a more general basis that will facilitate the useof RBD procedures, it is necessary to assess the three pri-mary component uncertainties separately and to develop asystematic approach that can combine these uncertaintiesconsistently, based on the manner in which the design soilproperty is derived. Statistics on inherent soil variability andmeasurement error were reviewed and summarized in thecompanion paper of this two-part series (Phoon and

Kulhawy 1999). This paper first describes the evaluation of transformation uncertainty. Then the uncertainties in the de-sign soil properties are evaluated rationally by combiningthe appropriate component uncertainties using a second-moment probabilistic approach. The paper concludes with

Can. Geotech. J. 36 : 625–639 (1999) © 1999 NRC Canada

62 5

Received March 11, 1998. Accepted February 16, 1999.

K.-K. Phoon. Department of Civil Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260.F.H. Kulhawy.1 School of Civil and Environmental Engineering, Hollister Hall, Cornell University, Ithaca,NY 14853-3501, U.S.A.

1 Author to whom all correspondence should be addressed.



specific guidelines on the typical coefficients of variation(COVs) for some common design soil properties as a func-tion of the type of test and the type of correlation used.

Transformation uncertainty

The direct measurement from a geotechnical test typicallyis not directly applicable to design. Instead, a transformationmodel is needed to relate the test measurement to an appro-priate design property. Some degree of uncertainty will be

introduced, because most transformation models are ob-tained by empirical data fitting. Transformation uncertaintystill would be present even for theoretical relationships be-cause of idealizations and simplifications in the theory. Thedata scatter about the transformation model can be quanti-fied using probabilistic methods, as illustrated in Fig. 1. Inthis approach, the transformation model typically is evalu-ated using regression analyses. The spread of data about theregression curve is modeled as a zero-mean random variable(ε). The standard deviation of ε (SDε) is an indicator of themagnitude of transformation uncertainty, as shown in Fig. 1.

An example of a probabilistic transformation model isgiven below:

[1] s D q m qu

vo

KT vo

vo

DKT vo

voσσ

σ ε σ

σ= − = + −( ) ( ) ( )

in which su is the undrained shear strength (design property);qT is the corrected cone tip resistance (test measurement);σvo and σvo are the effective and total overburden stresses,respectively; DK is the uncertain model slope; mDK is themean of DK; and ε is the zero-mean random variable repre-senting transformation uncertainty. The corrected cone tipresistance, qT, is defined as

[2] qT = qc + (1 – a)ubt

in which qc is the cone tip resistance, a is the net area ratio,and ubt is the pore-water stress behind the cone tip. Ideally,the transformation uncertainty should be evaluated so that itdoes not include extraneous variabilities, such as inherentsoil variability and measurement error. This result can beachieved approximately in practice by using high-qualitydata that satisfy some basic requirements, such as ( i) keep-ing the distances between corresponding pairs of in situproperty (qT) and soil property (su) measurements at a mini-mum to reduce inherent soil variability; and (ii) using thesame cone type, method of obtaining qT, and test to measuresu, where possible, to reduce systematic measurement errors.In a comprehensive study of this type (Kulhawy et al. 1992),it was found that considerable uncertainty still can be attrib-uted to the model slope ( DK), despite efforts to minimize theinfluence of extraneous variabilities, as shown in Table 1.This study demonstrated that transformation uncertainty is asignificant and independent source of geotechnical variabil-ity that cannot simply be explained away as being the resultof inherent soil variability and measurement error.

Transformation models for other design properties andtest measurements can be found in the geotechnical litera-ture, and many have been summarized by Kulhawy andMayne (1990). Tables 2 and 3 summarize the availability of these models in the literature for a range of soil propertiesdeveloped from theory, laboratory test measurements, or

© 1999 NRC Canada

626 C an. G eotech. J. Vol. 36, 1999

su test typea Mean DKb COV of DK (%)

CIUC 0.0789 35

UU 0.0512 29

VST 0.0906 40

aCIUC, consolidated isotropic undrained triaxial compression test; UU,unconsolidated–undrained triaxial compression test; VST, vane shear test.

bsu / σvo = DK(qT – σvo)/ σvo.

Table 1. Second-moment statistics for DK (source: Kulhawy et

al. 1992, p. 5-24).

Fig. 1. Probabilistic characterization of transformation model.

http://nrc.literatumonline.com.nrc.literatumonline.com/action/showImage?doi=10.1139/t99-039&iName=master.img-000.png&w=355&h=239



field test measurements. However, the transformation uncer-tainties associated with these models are seldom analyzedwith the same degree of rigor as that shown in eq. [1]. Themajority are empirical and do not contain sufficient informa-tion for probabilistic evaluation. Although the magnitudes of uncertainty in these empirical models are unknown, they arelikely to be as large as those indicated in Table 1, particu-larly for the case of empirical models where two (or more)items are being linked together that are not directly related.A good example is the standard penetration test (SPT) N value noted in Tables 2 and 3. The N value is the dynamicdriving resistance for a particular type of sampler, yet it hasbeen correlated with the soil consistency, relative density,vertical and horizontal soil stress state, drained and un-

drained strength, modulus, and liquefaction resistance. Al-though these characteristics undoubtedly influence N indi-rectly, it is too much to expect that they all (singly orcollectively) can be predicted reliably without incurring sig-nificant uncertainties.

Variability of design soil properties

From the above observations, it is clear that the uncer-tainty in a design soil property is a function of inherent soilvariability, measurement error, and transformation uncer-tainty. These components can be combined consistently us-ing the simple second-moment probabilistic approachdescribed below (Phoon et al. 1995).

© 1999 NRC Canada

P hoon and K ulhaw y 627

Laboratory or

theory correlation

Field test correlation

Property category Soil property SPT CPT CPTU PMT DMT

Basic characterization Classification × — × × — ×

Unit weight × — — — — —

Relative density — × × × — ×

In situ stress Coefficient of horizontal soil stress × — × — × ×

Strength Effective stress friction angle × × × — × ×

Deformability Poisson’s ratio × — — — — —

Young’s modulus × × — — × ×

Compression index × — — — — —

Constrained modulus × — × — — —

Subgrade modulus × — — — — —

Permeability Hydraulic conductivity × — — — — —

Liquefaction resistance Cyclic stress ratio — × × — — ×

Table 3. Available correlations for cohesionless soils (source: Kulhawy and Mayne 1990, pp. J4–J5).

Property category Soil property

Laboratory or

theory correlation

Field test correlation

SPT CPT CPTU PMT DMT VST

Basic characterization Classification × — × × — × —

Unit weight × — — — — — —

Consistency — × × — — — —

In situ stress Preconsolidation stress × × × × × × ×

Overconsolidation ratio × × — × — × ×

Coefficient of horizontal soil stress × × × × × × —

Strength Effective stress friction angle × — — — — — —

Undrained shear strength × × × × × × ×

Deformability Poisson’s ratio × — — — — — —

Young’s modulus × — — — × — —

Compression indices × — — — — — —

Constrained modulus × × × — — × —

Coefficient of consolidation × — — × — × —

Coefficient of secondary compression × — — — — — —

Permeability Hydraulic conductivity × — — — — — —

Note: CPT, cone penetration test; CPTU, piezocone test; DMT, dilatometer test; PMT, pressuremeter test; SPT, standard penetration test; VST, vaneshear test.

Table 2. Available correlations for cohesive soils (source: Kulhawy and Mayne 1990, pp. J2–J3).



• Assume that the design property (ξd) is predicted from atest measurement (ξm) using the following generic probabil-istic transformation model:

[3] ξd = T (ξm, ε)

in which T () is the transformation function, which might benonlinear; and ε is the transformation uncertainty.

• Introduce inherent soil variability (w) and measurementerror (e) into eq. [3] by substituting ξ m = t + w + e (refer tocompanion paper for details):

[4] ξd = T (t + w + e, ε)

in which t is the deterministic trend function. Note that themean of w, e, and ε is zero.

• Linearize eq. [4] about the mean of (w, e, ε) using a first-order Taylor-series expansion (Benjamin and Cornell 1970):

[5] ξ ∂∂

∂∂

ε ∂∂εd ≈ + + +T t w

T

we

T

e

T

t t t

( , )( , ) ( , ) ( , )

00 0 0

• Estimate the mean and variance of

ξ d by applying second-

moment probabilistic techniques to eq. [5] (Benjamin andCornell 1970):

[6a] m ξ d ≈ T (t ,0)

[6b] SD SD SD Sdξ∂∂

∂∂

∂∂ε

2

2

2

2

2

2

≈

+

+

T

w

T

e

T w e Dε

2

in which mξd and SD dξ2 are the mean and variance of ξ d, re-

spectively; SDw2 is the variance of inherent soil variability;

SDe2 is the variance of measurement error; and SD ε

2 is thevariance of transformation uncertainty.

The approximate mean and variance given in eq. [6] con-stitute the second-moment statistics of the design property at

a point in the soil mass. For foundation design, it is not un-common to evaluate the spatial average of the design prop-erty over some depth interval, rather than use the value of the design property at a point. The spatial average of ξd isdefined as

[7] ξ ξa d d= ∫ 1

L z z

L

( )

in which ξ a is the spatial average, L is the averaging length,and z is the depth.

In principle, the mean and variance of ξ a can be computedby substituting eq. [5] into eq. [7] and applying second-moment probabilistic techniques to the resulting equation. If t and ∂T / ∂w (or ∂ξd / ∂w) are constants, it can be shown thatthe mean of the spatial average is the same as that given ineq. [6a], and the variance of the spatial average (SD aξ

2 ) isgiven by (Vanmarcke 1983)

[8] SD SD SDaξ∂∂

∂∂

∂∂ε

2

2

2 2

2

2≈

+

+

T

w L

T

e

T w eΓ ( )

2

2SDε

in which Γ 2() is the variance reduction function, which de-pends on the length of the averaging interval, L. The follow-ing approximate variance reduction function has beenproposed for practical applications (Vanmarcke 1983):

[9a] Γ 2( L) = 1 for L = δv

[9b] Γ 2 ( ) L L

= δv for L > δv

in which δv is the vertical scale of fluctuation. Equation [9]states that the variance reduction function decreases as the

length of the averaging interval increases. Therefore, the ef-fect of averaging is to reduce the uncertainty associated withinherent soil variability (SDw

2 ), as shown in eq. [8]. For thecase in which t and ∂T / ∂w are general functions of depth, nosimple results are available. However, it can be noted thatthe requirements of constant t and ∂T / ∂w can be satisfied inmost cases because (i) t is approximately constant if the av-eraging interval is not too large, and ( ii) ∂T / ∂w is a constantfor linear transformation models, which are commonly as-sumed in foundation design.

The above general probabilistic approach can be appliedto determine the typical range of variability for some com-mon design soil properties. The design properties consideredherein are undrained shear strength, effective stress friction

angle, in situ horizontal stress coefficient, and Young’smodulus. The compilation of statistical data on inherent soilvariability and measurement error presented in the compan-ion paper serves as realistic inputs for the evaluation of eqs. [6] and [8]. However, eq. [8] only can be approximatelyevaluated for many cases, because the vertical scale of fluc-tuation of most test measurements is not available. In addi-tion, rigorous statistics on transformation uncertaintygenerally are not available, as noted previously. However, afirst-order estimate of SDε can be obtained by noting thatabout two thirds of the data typically fall within ± one stan-dard deviation of the transformation model, because ε is ap-proximately normally distributed (Fig. 1). Unless the datapopulation has been screened carefully to remove extraneous

variabilities as noted above, significant improvement on thisfirst-order estimate is unlikely, even if more sophisticatedstatistical techniques were employed. Even with this simpletechnique of evaluating transformation uncertainty, only alimited number of correlations could be examined becausemost are presented without the supporting data, therebyeliminating the possibility of assessing the transformationuncertainty.

Undrained shear strength

For direct laboratory measurement of the undrained shearstrength (su), eq. [4] simplifies to

[10] ξd = su = t + w + e

The mean and variance for this design property ( su) are de-termined from eq. [6] as follows:

[11a] m ξd = t

[11b] SD dξ2 = SDw

2 + SDe2

Therefore, the COV of ξ d, which is defined by SDξd / mξd, isgiven by

[12] COV dξ2 = COVw

2 + COVe2

© 1999 NRC Canada

628 C an. G eotech. J. Vol. 36, 1999



in which COVw is the COV of inherent variability = SD w / t ,and COVe is the COV of measurement error = SDe / t . TheCOV of the spatial average (ξa) can be obtained from eq. [8]in a similar manner:

[13] COV aξ2 = Γ 2( L) COVw

2 + COVe2

The COVs of inherent variability (COVw) for the uncon-

fined compression test (UC), unconsolidated–undrainedtriaxial compression test (UU), and consolidated isotropicundrained triaxial compression test (CIUC) were estimatedin the companion paper to be between 20 and 55%, 10 and30%, and 20 and 40%, respectively. The COV of measure-ment error (COVe) for undrained strength tests was esti-mated to be between 5 and 15%. The typical vertical scaleof fluctuation for su was judged to be on the order of 1–2 m.Using these numerical data, the total COVs (COVξd) for di-rect measurement of su using UC, UU, and CIUC tests are21–57%, 11–34%, and 21–43%, respectively (eq. [12]). Foran averaging length of 5 m, the amount of variance reduc-tion [Γ 2( L)] would vary from 0.2 to 0.4 (eq. [9]). By substi-tuting this variance reduction into eq. [13], the

corresponding COVs for the spatial averages (COVξa) of thethree tests would be 10–38%, 7–24%, and 10–29%.

Correction from vane shear testAn undrained shear strength also can be estimated using

the in situ vane shear test (VST). However, a correction fac-tor is needed to account for strain-rate effects and soil aniso-tropy (e.g., Kulhawy and Mayne 1990). For this case, eq. [4]can be expressed as follows (e.g., Baecher and Ladd 1985):

[14] ξd = (mµ + ε) (t + w + e)

in which ξd is the design property, which is the corrected su

from the vane shear test; mµ is the mean correction factor; ε

is the uncertainty in the correction factor; and (t + w + e) =su(VST). The mean and variance for this design property aredetermined from eq. [6] as follows:

[15a] m ξd ≈ mµ t

[15b] SD dξ2 ≈ mµ

2 (SDw2 + SDe

2 ) + t 2SD2ε

The COV of ξd is given by

[16] COV dξ2 = COVw

2 + COVe2 + COVε

2

in which COVε is the COV of transformation uncertainty =SDε / mµ . The COV of the spatial average can be obtainedfrom eq. [8] in a similar manner:

[17] COV aξ2

= Γ 2

( L) COVw

2

+ COVe

2

+ COVε2

The COV of transformation uncertainty (COVε) for softclays was estimated to be between 7.5 and 15% (Baecherand Ladd 1985). The COV of inherent variability (COVw)and measurement error (COVe) for the vane shear test wereestimated in the companion paper to be between 10 and 40%and 10 and 20%, respectively. The vertical scale of fluctua-tion for su(VST) was judged to vary between 2 and 6 m.Using these numerical data, the total COV (COVξd) for thecorrected su from the vane shear test is between 16 and 47%(eq. [16]). For an averaging length of 5 m, the amount of variance reduction [Γ 2( L)] would vary from 0.4 to 1.0

(eq. [9]). By substituting this variance reduction intoeq. [17], the corresponding COV for the spatial average(COVξa) would be between 14 and 47%.

Correlation with corrected cone tip resistanceThe correlation between the undrained shear strength and

the corrected cone tip resistance was given in eq. [1]. This

correlation model can be expressed in the form of eq. [4] asfollows:

[18] ξd = su = (mDK + ε)(t + w + e – σvo)

in which (t + w + e) = qT. The mean and variance for this de-sign property are determined from eq. [6] as follows:

[19a] mξd ≈ m DK (t – σvo)

[19b] SD dξ2 ≈ mDK

2 (SDw2 + SDe

2 ) + (t – σvo)2 SDε2

The COV of ξ d is given by

[20] COV COV COV

COVd

vo

ξ εσ

22 2

22

1

≈ +

−

+( )w e

t

in which COVε is the COV of transformation uncertainty =SDε / mDK. The COV of the spatial average can be obtainedfrom eq. [8] in a similar manner:

[21] COV COV COV

COVa

vm

ξ εσ

22 2 2

22

1

≈ +

−

+[ ( ) ]Γ L

t

w e

in which σvm is the average total overburden stress over L.The COVs of transformation uncertainty (COVε) for

CIUC and UU reference tests were found to be 35 and 29%,respectively (Table 1). The COV of inherent variability

(COVw) for the corrected cone tip resistance (qT) in clay wasestimated in the companion paper to be less than 20%. Themeasurement error (COVe) for qT likely would be compara-ble to that produced by the electric cone penetration test,which was estimated to be between 5 and 15%. The verticalscale of fluctuation for qT in clay was found to be less than0.5 m. To evaluate eqs. [20] and [21], it also is necessary todetermine the ratios σvo / t and σvm / t . Note that t is equal tothe mean value of qT, because the mean of w and e is zero.From Fig. A1a in the companion paper, the mean value of qT (= t ) in clay is on the order of 2 MN/m2. For a typical to-tal soil unit weight of 20 kN/m3 and a depth of 10 m, the to-tal overburden stress (σvo or σvm) is 200 kN/m2. Therefore,the ratios σ

vo

/ t and σvm

/ t are on the order of 0.1. Using thesenumerical data, the total COVs (COVξd) for su(CIUC) andsu(UU) predicted from qT would be between 35 and 47%and 30 and 40%, respectively (eq. [20]). For an averaginglength of 5 m, the amount of variance reduction [Γ 2( L)]would be less than 0.1 (eq. [9]). By substituting this variancereduction into eq. [21], the corresponding COVs for the spa-tial averages (COVξa) of the two reference tests would bebetween 35 and 39% and 30 and 34%.

Correlation with SPT N valueReasonable correlation between the undrained shear

strength and the SPT N value can be achieved, provided the

© 1999 NRC Canada




correlation is restricted to one type of geology (Kulhawyand Mayne 1990). An example of such a local correlation isillustrated in Fig. 2, which is applicable to alluvial clays inJapan. The probabilistic equation for this correlation is

[22] log10(su / pa) = log10(0.29) + 0.72 log10( N ) + ε

in which su is the undrained shear strength from UU tests,and pa is the atmospheric pressure. Equation [22] can be re-written in an equivalent exponential form:

[23] su / pa = 0.29 N 0.7210 ε

Equation [23] then is expressed in the form of eq. [4] as fol-lows:

[24] ξd = su = 0.29 pa(t + w + e)0.7210 ε

in which (t + w + e) = N . The mean and variance for this de-

sign property are determined from eq. [6] as follows:[25a] mξd ≈ 0.29t 0.72

[25b] SD dξ2 ≈ mξd

2 [0.722 (COVw2 + COVe

2 )

+ (loge10)2 SDε2 ]


[26] COV dξ2 ≈ 0.722 (COVw

2 + COVe2 ) + (loge10)2 SDε

2

The COV of the spatial average (ξa) can be obtained fromeq. [8] in a similar manner:

[27] COV aξ2 ≈ 0.722 [Γ 2( L) COVw

2 + COVe2 ]

+ (loge10)2 SDε2

The standard deviation of the transformation uncertainty(SDε) was not reported, but about two thirds of the data fallwithin ±0.15 of the transformation equation shown in Fig. 2.Therefore, a first-order estimate of SDε is 0.15. The COV of inherent variability (COVw) and measurement error (COVe)for N were estimated in the companion paper to be between25 and 50% and 15 and 45%, respectively. The vertical scaleof fluctuation for N in clays is not available. However, acomparison between the inherent variability of N in sandsand clays showed that the effect of soil type was not signifi-cant (Fig. A1c in the companion paper). Therefore, it is pos-sible that the vertical scale of fluctuation for N is similar forboth soil types. The vertical scale of fluctuation for N insandy soils was reported as 2.4 m in one study (Vanmarcke

1977). Using these numerical data, the total COV (COVξa)for su(UU) predicted from N is between 40 and 60%(eq. [26]). For an averaging length of 5 m, the amount of variance reduction [Γ 2( L)] would be about 0.5 (eq. [9]). Bysubstituting this variance reduction into eq. [27], the corre-sponding COV for the spatial average (COVξa) would be be-tween 38 and 54%.

Correlation with DMT horizontal stress index

The undrained shear strength also can be correlated to thedilatometer test (DMT) horizontal stress index, K D. One ex-ample of such a correlation for Italian clays is illustrated in

© 1999 NRC Canada

630 C an. G eotech. J. Vol. 36, 1999

Fig. 2. Relationship between su and SPT N value (source: Hara et al. 1974, p. 9). OCR, overconsolidation ratio; r , correlation coefficient.




Fig. 3. The recommended correlation equation is (Marchetti1980)

[28] s

K u

vo DMT

D1.250.22(0.5 )

σ

=

Note that the strength data have not been corrected to thesame reference test. Also, the mean line and the standard de-viation about the mean were not reported by the author. Forsubsequent probabilistic analysis, the following model wasdetermined using linear regression analysis:

[29] log log (10 10s

K u

vo DMT

10 D0.12) +1.24 logσ

ε

= +

The standard deviation of the transformation uncertainty(SDε ) was found to be 0.11. Equations [29] and [22] aresimilar, and therefore the COV for this case can be obtainedby direct comparison with eq. [26]:

[30] COV dξ2 ≈ 1.242 (COVw

2 + COVe2 ) + (loge10)2 SDε

2

Similarly, the COV of the spatial average (ξa) can be ob-tained by direct comparison with eq. [27]:

[31] COV aξ2 ≈ 1.242 [Γ 2( L) COVw

2 + COVe2 ]

+ (loge10)2 SDε2

The COV of inherent variability (COVw) for K D in sandswas found to lie between 20 and 60%. However, a compari-son between the COVs of dilatometer A and B readings re-vealed that the effect of soil type is significant (Fig. A1d inthe companion paper). In the absence of relevant statistics,the COV of inherent variability for K D in clays could be as-sumed to be comparable to those for the A and B readings in

clays, which were estimated to lie between 10 and 35%. Themeasurement error (COVe) for K D could be assumed to bethe same as that produced by the dilatometer test, which wasestimated to be between 5 and 15%. Using these numericaldata, the total COV (COVξd) for su determined from thedilatometer test would be between 29 and 53% (eq. [30]).The vertical scale of fluctuation for K D is not available, but

it is likely to lie between 2 and 6 m, as discussed in thecompanion paper. For an averaging length of 5 m, theamount of variance reduction [Γ 2( L)] would vary from 0.4 to1.0 (eq. [9]). By substituting this variance reduction intoeq. [31], the corresponding COV for the spatial average(COVξa) would be between 27 and 53%.

Correlation with plasticity indexThe undrained shear strength determined from the vane

shear test can be correlated to the plasticity index (PI) as fol-lows (e.g., Chandler 1988):

[32] su

p

(VST)0.11 0.0037 PI

σ = +

in which σp is the preconsolidation stress. The transforma-tion uncertainty was not given, but the author noted that theaccuracy of eq. [32] would be on the order of ±25%. For de-sign, a correction factor should be applied to su(VST), asnoted previously. The COV of the correction factor also wasnoted as lying between 7.5 and 15%. In addition, thepreconsolidation stress is an uncertain parameter. The totalvariability of σp was found to lie between 12 and 39% inone study (Vanmarcke and Fuleihan 1975). A simple proba-bilistic model for the above correlation is

[33] ξd = mσp (1 + ε) [0.11 + 0.0037 (t + w + e)]

in which ξ d is the design property, which is the corrected su

from the vane shear test; mσp is the mean preconsolidationstress; (t + w + e) = PI; and ε is a random variable that ac-counts for the inaccuracy in eq. [32], uncertainty in the VSTcorrection factor, and uncertainty in the preconsolidationstress. The mean and variance for this design property aredetermined from eq. [6] as follows:

[34a] mξd ≈ mσp (0.11 + 0.0037t )

[34b]( )

SDSD SD

SDd dξ ξ ε2 2

2 2

2

2

2973≈

+

+ +

mt

w e

( . )


[35]( )

COVSD SD

SDdξ ε2

2 2

2

2

2973≈

+

+ +

w e

t ( . )

From eq. [33], it can be seen that ∂ξd / ∂w = 0.0037mσp,which is not a constant, because the mean preconsolidationstress (mσp) generally varies with depth. As mentioned pre-viously, there are no simple solutions for the COV of thespatial average (COVξa) in such cases.

The standard deviation of inherent variability (SDw) andmeasurement error (SDe) of PI were estimated in the com-panion paper to be between 3 and 12% and 2 and 6%, re-

© 1999 NRC Canada


Fig. 3. Relationship between su and K D from DMT (source:

Marchetti 1980, p. 317). I D, dilatometer material index.




spectively. The lower and upper bounds on SDε can beestimated from the uncertainty in eq. [32] (≈ 0.25), the un-certainty in the VST correction factor (≈ 0.08–0.15), and theuncertainty in σp (≈ 0.10–0.40), as follows:

[36a] SD 0.25 0.08 0.10 0.282 2 2ε ≥ + + =

[36b] SD 0.25 0.15 0.40 0.502 2 2ε ≤ + + =

Using these numerical data and assuming a typical mean PI(= t ) of 30%, the total COV (COVξd) for the corrected su

from the vane shear test is between 29 and 55% (eq. [35]).

Effective stress friction angle

For direct laboratory measurement of the effective stressfriction angle (φ), eq. [4] simplifies to

[37] ξd = φ = t + w + e

Equation [37] is the same as eq. [10]. Therefore, the mean,variance, and COV of the design property (φ) can be deter-mined from eqs. [11a], [11b], and [12], respectively. Simi-larly, the COV of the spatial average can be determinedusing eq. [13].

The COV of inherent variability (COVw) and measure-ment error (COVe) for typical mean values of φ were bothestimated in the companion paper to be between 5 and 15%.Using these numerical data, the total COV (COVξd) for di-rect determination of φ is between 7 and 21% (eq. [12]). Thevertical scale of fluctuation for φ is not available, but it islikely to lie between 2 and 6 m, as discussed in the compan-ion paper. For an averaging length of 5 m, the amount of variance reduction [Γ 2( L)] would vary from 0.4 to 1.0

(eq. [9]). By substituting this variance reduction intoeq. [13], the corresponding COV for the spatial average(COVξa) would be between 6 and 21%.

Correlation with cone tip resistanceA number of correlations between the triaxial compres-

sion effective stress friction angle [φ (TC)] and the cone tipresistance (qc) have been summarized (Kulhawy and Mayne1990). The following correlation for sands is selected forstudy, because the transformation uncertainty is available forsubsequent probabilistic analysis:

[38] φσ

ε( /

TC) 17.6 11.0 log /

10c a

vo a

= +

+q p

p

The standard deviation of ε (SDε) for this correlation is 2.8°(Kulhawy and Mayne 1990). Equation [38] can be expressedin the form of eq. [4] as follows:

[39] ξ = φσ

εd 10a

vo a

TC) 17.6 11.0 log /

( ( )/ = + + +

+t w e p

p

in which (t + w + e) = qc. The mean and variance for thisdesign property [φ (TC)] can be determined from eq. [6] asfollows:

[40a] m t p

pξ σd 10

a

vo a

17.6 11.0 log /

≈ +

/

[40b] SD dξ2 ≈ 22.8 (COVw

2 + COVe2 ) + SDε

2


[41] COV dξ2 ≈ 22.8 (COV COV ) SD2 2 2

d2

w e

m+ + ε

ξ


[42] COV 22.8 ( )COV COV ] SD

a

2 2 2

d2ξ

ε

ξ

2 ≈ + +[Γ 2 L

m

w e

The COV of inherent variability (COVw) for qc in sandwas estimated in the companion paper to be between 20 and60%. The measurement error (COVe) for q c can be assumedto be comparable to that produced by the electric cone pene-

tration test, which was estimated to be between 5 and 15%.The vertical scale of fluctuation for qc was found to be lessthan 1 m. The mean friction angle (mξd) for most soils is be-tween 20° and 40°. A typical value for mξd is 30°. Usingthese numerical data, the total COV (COVξd) of φ (TC) de-termined from qc is between 10 and 14% (eq. [41]). For anaveraging length of 5 m, the amount of variance reduction[Γ 2( L)] would be less than 0.2 (eq. [9]). By substituting thisvariance reduction into eq. [42], the corresponding COV forthe spatial average (COVξa) would be between 9 and 11%.

Correlation with plasticity indexThe constant-volume effective stress friction angle ( φcv) of

normally consolidated clays can be correlated to the plastic-

ity index (PI) as follows (e.g., Mitchell 1976):

[43] sin φcv = 0.8 – 0.094 log ePI

This correlation is illustrated in Fig. 4. A simple probabilis-tic equation that accounts for the spread of data about thiscorrelation is as follows:

[44] sin φcv = 0.8 – 0.094 log ePI + ε

Equation [44] can be expressed in the form of eq. [4] as fol-lows:

[45] ξd = φcv = sin–1[0.8 – 0.094 loge(t + w + e) + ε]

(degrees)

in which (t + w + e) = PI. The mean and variance for this de-sign property (φcv) can be determined from eq. [6] as fol-lows:

[46a] mξd ≈ s in–1(0.8 – 0.094 loget )

(degrees)

[46b] SD 0.094 (COV COV SD

1 (0.8 0.d

2 2 2 2

ξ ε

π2

2180≈

+ +− −

w e )

094 loget )2

(degrees)2

© 1999 NRC Canada

632 C an. G eotech. J. Vol. 36, 1999



The COV of ξ d simply is given by SDξ d / m ξ d. The COV of

the spatial average (ξa) can be obtained from eq. [8] in asimilar manner:

[47] COV 0.094 [ ( )COV COV SD

a

2 2 2 2 2

d

ξ ε

ξπ2

2180≈

+ +Γ L

m

w e ]2 [1 (0.8 0.094 log− −

e t ) ]2

The standard deviation of the transformation uncertainty(SDε) was not reported, but it can be seen that about twothirds of the data fall within ±0.07 of the transformationequation shown in Fig. 4. Therefore, a first-order estimate of SDε is 0.07. The standard deviation of inherent variability(SDw) and measurement error (SDe) for PI were previouslynoted to be between 3 and 12% and 2 and 6%, respectively.Using these numerical data and assuming a typical mean PI(= t ) of 30%, the total COV (COVξd) for φcv determinedfrom PI is between 16 and 19% (eq. [46]). The vertical scaleof fluctuation for PI is not available, but it is likely to lie be-tween 2 and 6 m, as discussed in the companion paper. Foran averaging length of 5 m, the amount of variance reduc-tion [Γ 2( L)] would vary from 0.4 to 1.0 (eq. [9]). By substi-tuting this variance reduction into eq. [47], thecorresponding COV for the spatial average (COVξa) wouldbe between 16 and 19%.

In situ horizontal stress coefficient

The self-boring pressuremeter test (SBPMT) provides a

direct measurement of the in situ horizontal stress. No corre-lation is needed because the stress is measured directly, tak-ing into account equipment calibrations. The in situhorizontal stress is expressed commonly in terms of the co -efficient K o, which is defined as the ratio of the effective insitu horizontal stress to the corresponding vertical effectivestress. For this direct field measurement, eq. [4] simplifies to

[48] ξd = K o = t + w + e

Equation [48] is the same as eq. [10]. Therefore, the mean,variance, and COV of the design property (K o) can be deter-mined from eqs. [11a], [11b], and [12], respectively. Simi-

larly, the COV of the spatial average can be determined us-

ing eq. [13].The COV of inherent variability (COVw) for the in situ

horizontal stress is not available, but it can be assumed to becomparable to the corresponding COV for the pressuremeterlimit stress, which has been estimated in the companion pa-per to be between 10 and 35% for clay and between 20 and50% for sand. The measurement error (COVe) for K o can beassumed to be comparable to that produced by the self-boring pressuremeter test, which was estimated to be be-tween 15 and 25%. Using these numerical data, the totalCOVs (COVξd) for direct determination of K o using the self-boring pressuremeter in clay and sand are between 18 and43% and 25 and 56%, respectively (eq. [12]). The verticalscale of fluctuation for K o is not available, but it is likely to

lie between 2 and 6 m, as discussed in the companion paper.For an averaging length of 5 m, the amount of variance re-duction [Γ 2( L)] would vary from 0.4 to 1.0 (eq. [9]). By sub-stituting this variance reduction into eq. [13], thecorresponding COVs for the spatial average (COVξa) in clayand sand would be between 16 and 43% and 20 and 56%,respectively.

Correlation with dilatometer horizontal stress index

The dilatometer test provides an indirect measurement of K o because of disturbance caused by the insertion of theblade into the ground. Correlations between K o from theSBPMT and the dilatometer horizontal stress index (K D) are

available for sand and clay. However, only the following cor-relation for clay is selected for study, because the transfor-mation uncertainty is available for subsequent probabilisticanalysis (Kulhawy and Mayne 1990):

[49] K o = 0.27K D + ε

The standard deviation of ε (SDε) for this correlation is 0.48(Kulhawy and Mayne 1990). Equation [49] can be expressedin the form of eq. [4] as follows:

[50] ξd = K o = 0.27 (t + w + e) + ε

© 1999 NRC Canada


Fig. 4. Relationship between φcv and PI for normally consolidated clays (source: Mitchell 1976, p. 284).




in which (t + w + e) = K D. The mean and variance for thisdesign property (K o) can be determined from eq. [6] as fol-lows:

[51a] mξd ≈ 0.27t

[51b] SD dξ2 ≈ 0.272(SDw

2 + SDe2 ) + SDε

2


[52] COV COV + COV SD

d

d

ξ ε

ξ

2 2 2

2

≈ +

w e

m


[53] COV ( )COV + COV SD

a2

d

ξ ε

ξ

2 2 2

2

≈ +

Γ Lm

w e

The COV of inherent variability (COVw) for K D in clays isnot available. As discussed previously, this COV might be

comparable to those for the A and B readings in clay, whichwere estimated to lie between 10 and 35%. The measure-ment error (COVe) for K D could be assumed to be the sameas that produced by the dilatometer test, which was esti-mated to be between 5 and 15%. Using these numerical dataand assuming a typical mean K o (mξd) of 1.5, the total COV(COVξd) for K o determined from a dilatometer test would bebetween 34 and 50% (eq. [52]). As noted previously, thevertical scale of fluctuation for K D is not available, but it islikely to lie between 2 and 6 m. For an averaging length of 5 m, the amount of variance reduction [Γ 2( L)] would varyfrom 0.4 to 1.0 (eq. [9]). By substituting this variance reduc-tion into eq. [53], the corresponding COV for the spatial av-erage (COVξa) would be between 33 and 50%.

Correlation with SPT N valueThe standard penetration test N value also is related indi-

rectly to K o, because it is a measurement of vertical penetra-tion. An example of a correlation between K o and N for clayis given below (Kulhawy and Mayne 1990):

[54] K Np

oa

vo

0.073= +σ

ε

The standard deviation of ε (SDε) for this correlation is 0.43(Kulhawy and Mayne 1990). Equation [54] can be expressedin the form of eq. [4] as follows:

[55] ξ σ εd o a

vo= = + + +K t w e p0073. ( )

in which (t + w + e) = N . The mean and variance for this de-sign property (K o) can be determined from eq. [6] as fol-lows:

[56a] m t p

ξ σda

vo

0.073=

[56b] SD 0.073

(SD + SD SDda

vo

ξ εσ2

2

2 2 2≈

+ p

w e )


[57] COV COV COV SD

d2 2

d

ξ ε

ξ

2

2

≈ + +

w e

m

From eq. [55], it can be seen that ∂ξd / ∂w = 0.073 pa / σvo,which is not a constant, because the effective overburden

stress ( σvo) varies linearly with depth. As mentioned previ-ously, there are no simple solutions for the COV of the spa-tial average (COVξa) in such cases.

The COVs of inherent variability (COVw) and measure-ment error (COVe) for N were estimated previously to be be-tween 25 and 50% and 15 and 45%, respectively. Usingthese numerical data and assuming a typical mean K o (mξd)of 1.5, the total COV (COVξd) predicted from N is between41 and 73% (eq. [57]).

Young’s modulus

The pressuremeter test (PMT) provides a direct measure-ment of the horizontal modulus of soils. This modulus( E PMT) often is presumed to be roughly equivalent to theYoung’s modulus ( E ). For this direct field measurement of the soil modulus, eq. [4] simplifies to

[58] ξd = E PMT = t + w + e

Equation [58] is the same as eq. [10]. Therefore, the mean,variance, and COV of the design property ( E PMT) can be de-termined from eqs. [11a], [11b], and [12], respectively. Sim-ilarly, the COV of the spatial average can be determinedusing eq. [13].

The COV of inherent variability (COVw) for E PMT in sandwas estimated in the companion paper to be between 15 and65%. The measurement error (COVe) for E PMT can be as-

sumed to be comparable to that produced by thepressuremeter test, which was estimated to be between 10and 20%. Using these numerical data, the total COV(COVξd) for direct determination of E PMT in sand using thepressuremeter is between 18 and 68% (eq. [12]). The verti-cal scale of fluctuation for E PMT is not available, but it islikely to lie between 2 and 6 m, as discussed in the compan-ion paper. For an averaging length of 5 m, the amount of variance reduction [Γ 2( L)] would vary from 0.4 to 1.0(eq. [9]). By substituting this variance reduction intoeq. [13], the corresponding COV for the spatial average(COVξa) would be between 14 and 68%.

Correlation with dilatometer modulus

The dilatometer test (DMT) also provides a direct mea-surement of soil modulus. The dilatometer modulus ( E D) isrelated to the Young’s modulus as follows:

[59] E E

D 2(1 v )=

−

in which ν is Poisson’s ratio. For this direct field measure-ment of the soil modulus, eq. [4] simplifies to

[60] ξd = E D = t + w + e

Equation [60] is the same as eq. [10]. Therefore, the mean,variance, and COV of the design property ( E D) can be deter-

© 1999 NRC Canada

634 C an. G eotech. J. Vol. 36, 1999



mined from eqs. [11a], [11b], and [12], respectively. Simi-larly, the COV of the spatial average can be determinedusing eq. [13].

The COV of inherent variability (COVw) for E D in sandwas estimated in the companion paper to be between 15 and65%. The measurement error (COVe) for E D can be assumedto be comparable to that produced by the dilatometer test,

which was estimated to be between 5 and 15%. Using thesenumerical data, the total COV (COVξd) for direct determina-tion of E D using the dilatometer test in sand is between 16and 67% (eq. [12]). The vertical scale of fluctuation for E Dis not available, but it is likely to lie between 2 and 6 m, asdiscussed in the companion paper. For an averaging lengthof 5 m, the amount of variance reduction [Γ 2( L)] would varyfrom 0.4 to 1.0 (eq. [9]). By substituting this variance reduc-tion into eq. [13], the corresponding COV for the spatial av-erage (COVξa) would be between 11 and 67%.

Correlation with SPT N valueCorrelations between soil modulus and N are fairly weak,

as shown in the example between E PMT and N in clays given

in Fig. 5. The appropriate probabilistic equation for this cor-relation is

[61] log log (19.3) 0.63 log10PMT

a

10 10 E

P N

= + + ε

From the data scatter in Fig. 5, a first-order estimate of thetransformation uncertainty (SD ε) is 0.37. Equations [61] and[22] are similar. Therefore, the COV for this case can be ob-tained by direct comparison with eq. [26]:

[62] COV dξ2 ≈ 0.632 (COV COV2 2

w e+ ) + (loge10)2 SDε2


[63] COV aξ2 ≈ 0.632 [Γ 2( L) COV COV2 2

w e+ ]

+ (loge10)2 SDε2

The COV of inherent variability (COVw) and measure-ment error (COVe) for N were previously noted to be be-tween 25 and 50% and 15 and 45%, respectively. Thevertical scale of fluctuation for N in sandy soils also wasnoted above as equal to 2.4 m. Using these numerical data,the total COV (COVξd) for E PMT predicted from N is be-tween 87 and 95% (eq. [62]). For an averaging length of 5 m, and assuming that the vertical scales of fluctuation inclay and sand are comparable as before, the amount of vari-ance reduction [Γ 2( L)] would be about 0.5 (eq. [9]). By sub-stituting this variance reduction into eq. [63], thecorresponding COV for the spatial average (COVξa) wouldbe between 86 and 93%. The large COV provides a quantita-tive measure of the significant inaccuracy in the correlation

model given above.

Importance of local correlations for Young’s modulusLocal correlations that are developed within a specific

geologic setting generally are preferable to generalizedglobal correlations because they are significantly more accu-rate. An example of such a correlation between thedilatometer modulus ( E D) and SPT N value is given inFig. 6. The data in Fig. 6 were obtained in sandy silts of thePiedmont geologic province, in and around the Washington,D.C., area. A simple probabilistic equation for this correla-tion is

© 1999 NRC Canada


Fig. 5. Relationship between E PMT of clays and SPT N value (source: Ohya et al. 1982, p. 129).




[64] log log (22) 0.82 log10D

a

10 10 E

P N

= + + ε

From the data scatter in Fig. 6, a first-order estimate of thetransformation uncertainty (SDε) is 0.12. Equations [64] and[22] are similar. Therefore, the COV for this case can be ob-tained by direct comparison with eq. [26]:

[65] COV dξ2 ≈ 0.822 (COV COV2 2

w e+ ) + (loge10)2 SDε2


[66] COV aξ2 ≈ 0.822 [Γ 2( L) COV COV2 2

w e+ ]

+ (loge10)2 SDε2

The numerical data on inherent variability (COVw), measure-ment error (COVe), and vertical scale of fluctuation for N were noted above. The total COV (COVξd) for E D predictedfrom N is between 37 and 62% (eq. [65]). For a variance re-duction [Γ 2( L)] of 0.5, the COV for the spatial average(COVξa) would be between 34 and 54% (eq. [66]). The sig-nificant reduction in the COVs, in contrast to those obtainedin the preceding section, clearly demonstrates that reason-able correlations with N can be achieved, provided the corre-lation is restricted to one type of geology.

Geotechnical versus structural variability

The results presented in these two companion papers am-ply illustrate the variable nature of the uncertainties ingeotechnical design properties. For example, the COV of theundrained shear strength of clays can vary from 10 to 60%,depending on whether it is measured directly or correlatedempirically with certain field measurements. It is overly sim-plistic to assign “typical” COVs for geotechnical designproperties without proper reference to some of the importantfactors discussed in this paper, such as the soil type, themeasurement process, and the transformation model used. In

contrast, the uncertainties in structural resistances typicallyfall within a narrow range of 10–20% for a wide range of materials (e.g., concrete, steel, aluminum) and resistancemodels (e.g., tension, flexure, shear), as shown in Table 4.Note that the uncertainties in structural material propertiesare even lower because the uncertainties in structuralresistances shown in Table 4 also include uncertainties aris-ing from fabrication and modeling errors. The uncertaintiesin structural loads generally depend on the source of the

© 1999 NRC Canada

636 C an. G eotech. J. Vol. 36, 1999

Fig. 6. Relationship between E D and SPT N value of Piedmont sandy silts (source: Mayne and Frost 1989, p. 22).

COV (%)

Load a

Dead 10

Live 25

Wind 37

Snow 26

Earthquake 138

Resistances

Concrete

Flexure beams 8–14

Short columns 12–16

Slender columns 12–17

Shear beams 17–21

Steel

Tension members 11

Compact beams, uniform moment 13

Axially loaded columns 14

Beam-columns 15

Aluminum

Tension members 8

Beams 8–13

Columns 8–14

Glue-laminated timber beams 18aFifty-year maximum load effects.

Table 4. Typical load and resistance statistics for structures (data

from Ellingwood et al. 1980).




loadings (e.g., dead, live, wind). Typical COVs can be as-signed to each loading type as shown in Table 4. The COVsare approximately 30%, with the exception of the nearly de-terministic dead loads (COV = 10%) and the highly variable

earthquake loads (COV = 138%).The differences between geotechnical and structural

variabilities have a significant impact on the development of reliability-based design procedures for geotechnical engi-neering. As noted above, most of the COVs in structural de-sign are fairly small. Therefore, structural loads andresistances usually can be modeled adequately as lumpedparameters in the reliability calibration process. In addition,the COVs in structural design either fall within narrowranges (e.g., structural resistances) or can be categorizedreadily into a few cases (e.g., loading type). It is thereforerelatively easy to obtain structural load and resistance factorsthat are not dependent on COVs from the reliability calibra-tion process. The situation in geotechnical engineering ismore complex. As discussed herein, there are many existingmeans of evaluating the same design soil property. A practi-cal reliability-based geotechnical design procedure will haveto account for the wide range of COVs resulting from thedifferent evaluation methods and possibly high COVs fromhighly variable site conditions, poor equipment and proce-dural control, and (or) low-quality correlations. It may nolonger be realistic to model geotechnical capacities aslumped parameters or to calibrate geotechnical resistancefactors over the entire range of COVs. A more complete dis-cussion of these important considerations is given elsewhere(Phoon et al. 1995).

Summary and conclusions

Rigorous statistics on transformation uncertainty generallyare not available. A first-order estimate of the transformation

uncertainty can be obtained by noting that about two thirdsof the data typically fall within one standard deviation of thetransformation model. Even with this simple technique, onlya limited number of models could be examined, becausemost models have been presented without their supportingdata.

Second-moment probabilistic techniques were applied tocombine inherent soil variability, measurement error, andtransformation uncertainty in a consistent manner. A sum-mary of the variability of some design properties as a func-tion of the test measurement, correlation equation, and soiltype is presented in Table 5. The variability of the designproperties is given in terms of the COV at a point and theCOV for a spatial average. These ranges of COV are based

on representative statistics of inherent variability and mea-surement error, as described herein. More accurate COVscan be calculated by substituting site-specific data on inher-ent variability and measurement error into the closed-formCOV equations given in this paper. For example, the COVfor the undrained shear strength determined from thedilatometer test is given by eqs. [30] and [31]. However, itmust be emphasized that these COV equations only are ap-plicable to the specific correlations considered herein.

The COVs of the undrained shear strength determined byseveral different methods were found to be in the range of 10–60%. For the undrained shear strength predicted from N ,

© 1999 NRC Canada


Design

propertya Testb Soil type Point COV (%)

Spatial average

COVc (%)

Correlation

equation

su(UC) Direct (lab) Clay 20–55 10–40 —

su(UU) Direct (lab) Clay 10–35 7–25 —

su(CIUC) Direct (lab) Clay 20–45 10–30 —

su(field) VST Clay 15–50 15–50 14su(UU) qT Clay 30–40d 30–35d 18

su(CIUC) qT Clay 35–50d 35–40d 18

su(UU) N Clay 40–60 40–55 23

sue K D Clay 30–55 30–55 29

su(field) PI Clay 30–55d — 32

φ Direct (lab) Clay, sand 7–20 6–20 —

φ (TC) qT Sand 10–15d 10d 38

φcv PI Clay 15–20d 15–20d 43

K o Direct (SBPMT) Clay 20–45 15–45 —

K o Direct (SBPMT) Sand 25–55 20–55 —

K o K D Clay 35–50d 35–50d 49

K o N Clay 40–75d — 54

E PMT Direct (PMT) Sand 20–70 15–70 —

E D Direct (DMT) Sand 15–70 10–70 — E PMT N Clay 85–95 85–95 61

E D N Silt 40–60 35–55 64

a E D, dilatometer modulus; E PMT, pressuremeter modulus; K o, in situ horizontal stress coefficient; s u, undrained shear strength; s u(field), corrected s u fromvane shear test; φ , effective stress friction angle; φcv, constant-volume φ ; TC, triaxial compression; UC, unconfined compression test.

bK D, dilatometer horizontal stress index; N , standard penetration test blow count; PI, plasticity index; q T, corrected cone tip resistance.cAveraging over 5 m.d COV is a function of the mean; refer to COV equations in the text for details.eMixture of s u from UU, UC, and VST.

Table 5. Approximate guidelines for design soil property variability (source: Phoon et al. 1995, p. 4-51).



higher COVs develop when “universal” relationships areused that are not calibrated to a specific geology. The proba-ble range of COV for the undrained shear strength is esti-mated to be between 10 and 70%. The COV of the frictionangle for sand and clay was found to be between 5 and 20%.For the in situ horizontal stress coefficient (K o), the COVwas found to be in the range of 20–80% for clay, depending

on the method of evaluation. The corresponding range of COV for sand, which was found to be in the range of 25–55%, only would be applicable to the direct determination of K o. The COV for indirect methods of evaluation could notevaluated because probabilistic transformation models werenot available. By comparing the range of COV for sand andclay in the case of direct determination, the COV for sandgenerally is higher. It is possible that the COV for indirectdetermination of K o in sand also is higher than that for clay.Based on this consideration and the range of 20–80% forclay as a reference, an overall COV of 30–90% is estimatedto be appropriate for both soil types. The COV of soil modu-lus was found to be highest. Even for direct methods of evaluation, the COV was found to be in the range of 20–

70%. Higher COVs were obtained for correlations with N ,particularly if the correlation is not restricted to a specificgeology. The probable range of COV for soil modulus alsois estimated to be on the order of 30–90%.

Acknowledgments

This research was supported, in part, by the ElectricPower Research Institute (EPRI) under RP1493. The EPRIproject manager was A. Hirany.

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A: dilatometer A reading B: dilatometer B readingCOV: coefficient of variationCOVe: coefficient of variation of measurement errorCOVw: coefficient of variation of inherent variabilityCOVε: coefficient of variation of transformation uncertaintyCOVξa: coefficient of variation of spatial averageCOVξd: coefficient of variation of design soil property

DK: model slope for su and qT correlation E : Young’s modulus E D: dilatometer modulus E PMT: pressuremeter modulus I D: dilatometer material index

K D: dilatometer horizontal stress index

K o: in situ coefficient of horizontal soil stress L: averaging length N : standard penetration test valueOCR: overconsolidation ratioPI: plasticity indexSDe: standard deviation of measurement errorSDw: standard deviation of inherent variabilitySDε: standard deviation of transformation uncertaintySDξa: standard deviation of spatial averageSDξd: standard deviation of design soil property

T (): transformation functiona: net area ratio

© 1999 NRC Canada

638 C an. G eotech. J. Vol. 36, 1999



© 1999 NRC Canada


e: measurement error

mDK: mean of DK

mµ : mean correction factor for vane shear test

mξd: mean design soil property

mσp: mean preconsolidation stress

pa: atmospheric pressure

qc: cone tip resistance

qT: corrected cone tip resistance

r : correlation coefficient

su: undrained shear strength

t (): trend function

ubt: pore-water stress behind the cone tip

w: inherent soil variability

z: depthΓ 2(): variance reduction functionδv: vertical scale of fluctuationε: transformation uncertainty

ν: Poisson’s ratioξa: spatial averageξd: design soil property

ξm: measured soil propertyφ: effective stress friction angleφcv: constant-volume effective stress friction angleσp: preconsolidation stressσvm: average total overburden stress over Lσvo: total overburden stressσvo: effective overburden stress



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1999 CGJ Phoon Kulhawy Evaluation of Variability