545

i) Associate Professor, School of Civil and Environmental Engineering, Yonsei University, Seoul, South Korea (junlee＠yonsei.ac.kr).ii) Graduate Research Assistant, ditto.iii) Assistant Professor, Dept. of Civil Engineering, Dongguk University, Seoul, Korea.iv) Associate Professor, School of Civil Engineering, Purdue University, USA.

The manuscript for this paper was received for review on March 24, 2008; approved on June 8, 2009.Written discussions on this paper should be submitted before March 1, 2010 to the Japanese Geotechnical Society, 4-38-2, Sengoku, Bunkyo-ku, Tokyo 112-0011, Japan. Upon request the closing date may be extended one month.

545

SOILS AND FOUNDATIONS Vol. 49, No. 4, 545–556, Aug. 2009Japanese Geotechnical Society

ESTIMATION OF THE SMALL-STRAIN STIFFNESS OFCLEAN AND SILTY SANDS USING STRESS-STRAIN

CURVES AND CPT CONE RESISTANCE

JUNHWAN LEEi), DOOHYUN KYUNGii), BUMJOO KIMiii) and MONICA PREZZIiv)

ABSTRACT

The initial, linear elastic range of a soil stress-strain curve is often deˆned by the small-strain elastic modulus E0 orshear modulus G0. In the present study, simpler and eŠective methods are proposed for the estimation of the small-strain stiŠness of clean and silty sands; these are based on triaxial compression test results and the CPT cone resistanceqc. In the method based on stress-strain curves obtained from triaxial compression tests, an extrapolation technique isadopted within the small-strain range of a transformed stress-strain curve to obtain estimates of the small-strain elasticmodulus. Calculated small-strain elastic modulus values were compared with the values measured using bender ele-ment tests performed on clean sands and sands containing nonplastic ˆnes. The results showed that the methodproposed produces satisfactory estimates of the small-strain elastic modulus for practical purposes. In the CPT-basedmethod, two G0-qc correlations available in the literature were evaluated. For isotropic conditions, both correlationsproduced reasonably good estimates of G0 for clean sands but overestimated it for silty sands. A G0-qc correlationwhich is proposed takes into account the eŠect of silt content of the sand and stress anisotropy.

Key words: cone resistance, horizontal eŠective stress, hyperbolic stress-strain curve, modulus degradation, silt con-tent, small-strain elastic modulus, triaxial tests (IGC: D6/E2)

INTRODUCTION

It is well known that the stress-strain response of soil ishighly nonlinear. The nonlinear stress-strain curve fromthe origin to the peak in the stress-strain plot may bedivided into the following two representative stages: thesmall-strain linear elastic range and the nonlinear elasticrange ending at the peak state. Post-peak softening maybe observed as well. Within the initial linear elastic range,the soil behaves as a linear elastic material, and inducedstrains are fully recoverable. The initial linear elasticrange can be represented by the small-strain modulus (E0

or G0). After the initial linear elastic range, the stress-strain curve becomes highly nonlinear with the modulusdegrading as a function of stress or strain level (Shibuyaet al., 1992; Fahey and Carter, 1993; Lee and Salgado,2000).

A number of experimental and empirical methods havebeen proposed for the estimation of the small-strain shearmodulus G0 (Hardin and Black, 1966; Yu and Richart,1984; Robertson and Campanella, 1983; Baldi et al.,1989; Viggiani and Atkinson, 1995). In the case of the ex-perimental methods, shear wave velocity measurements

have been used most of the time. Laboratory determina-tion of shear wave velocity requires somewhat sophisti-cated testing devices and high-accuracy data acquisitionsystems. In situ evaluation of shear wave velocity mea-surements is well established, and various techniques, in-cluding down-hole, cross-hole and Spectral Analysis ofShear Wave (SASW) tests, have been developed. While insitu evaluation of shear wave velocity measurements ap-pears to be more appropriate for G0 estimation, it re-quires values of the soil density, which are highly variablein natural soil deposits. Although other geophysical tech-niques, such as well logging tests, may be used for the es-timation of in situ soil density, they require calibrationand a methodology for interpretation of the test results.

Various empirical correlations have been proposed forthe estimation of G0. The correlations can be groupedinto property-based methods (Hardin and Black, 1966;Yu and Richart, 1984), which utilize intrinsic and statesoil variables, and in situ correlations based on ˆeld testresults, such as the SPT blow count number N and theCPT cone resistance qc (Imai and Tonouchi, 1982;Robertson and Campanella, 1983; Rix and Stokoe,1991). The property-based methods are simpler to use

546

Fig. 1. Hyperbolic stress-strain relationship: (a) stress-strain curveand (b) stress-strain curve in transformed space

546 LEE ET AL.

and produce reasonable estimates of G0 when the soilproperties and correlation parameters are known a priori.If either intrinsic or state soil variables are not known,which happens frequently in the ˆeld, the property-basedmethods are no longer applicable. In such cases, use of insitu test results would be more appropriate. This ap-proach, however, requires the development of reliablecorrelations between G0 and in situ test measurements forvarious soil types.

In the present study, two methods are proposed for theestimation of the small-strain modulus: one is based onthe stress-strain relationship obtained from conventionaltriaxial compression tests without measurement of thesmall-strain response using wave-related testing devices,and, the other, on the CPT cone resistance qc. In the ˆrstmethod, transformed stress-strain curves are investigat-ed, and an extrapolation technique within the small-strain range is adopted to estimate E0. In the developmentof the second method, a careful evaluation, for varioussoils and stress states, of two existing correlations be-tween G0 and qc was performed. Estimates of G0 obtainedusing these correlations were compared with estimates ofG0 obtained from bender element test results and an em-pirical, property-based equation that has been widelyveriˆed. Based on this careful evaluation, a correlationbetween G0 and qc is proposed that takes into account thesilt content of the sand and stress anisotropy, two factorsthat are not accounted for in the afore-mentioned G0-qc

correlations.

NONLINEAR STRESS-STRAIN CURVE ANDMODULUS DEGRADATION

It is well known that the stress-strain behaviour of soilis highly nonlinear from the very early stages of loading.Nonlinear soil models have been widely used to representsuch nonlinear soil behaviour over a wide range ofstrains. Since Kondner (1963) ˆrst proposed the originalhyperbolic equation for the stress-strain relationship ofsoil, several modiˆcations have been suggested (Duncanand Chang, 1970; Fahey and Carter, 1993; Tatsuoka etal., 1993; Lee and Salgado, 2000). The hyperbolic equa-tion proposed by Kondner (1963) is written as:

t＝g

a＋bg(1)

where t and g are the shear stress and strain; and a and bare the material constants that deˆne the stress-straincurve. Figure 1 shows stress-strain curves in the originaland transformed spaces. For the hyperbolic stress-strainrelationship given by Eq. (1), the parameters a and b inFig. 1 represent the reciprocal of the initial tangent shearmodulus Gi and of the asymptotic value of the limit shearstress tlim.

Figure 1 and Eq. (1) show that considerable strain isrequired for the stress-strain curve to reach tlim. In orderto ˆt a nonlinear relationship to a real soil stress-straincurve, Duncan and Chang (1970) modiˆed Eq. (1) by in-troducing a material parameter Rf into it. Rf is referred to

as the failure ratio, relating the limit stress tlim of theoriginal nonlinear stress-strain curve to the actual shearstress of soil at failure tf:

tf＝Rftlim (2)

The modiˆed hyperbolic equation is then written as:

t＝g

1Gi＋

g・Rf

tf

(3)

where Gi is the initial tangent shear modulus, t and g arethe current shear stress and strain, respectively. Using t＝G・g, Eq. (3) can be rewritten as:

GGi＝1－

ttf

Rf (4)

where G＝secant shear modulus. According to Duncanand Chang (1970), Rf is typically in the range of 0.75–1.0.If Eq. (4) is applied to triaxial compression stress-straincurves, then Eq. (4) can be rewritten as:

EEi＝1－

(s1?－s3?)(s1?－s3?)f

Rf (5)

where E＝secant Young's modulus; Ei＝initial tangentYoung's modulus; (s1?－s3?)＝deviatoric stress; (s1?－s3?)f＝deviatoric stress at failure.

Equation (4) implies linear degradation of the soil stiŠ-

547547SMALL-STRAIN ELASTIC MODULUS

ness from its initial maximum value G0. However, thedegradation of the elastic modulus obtained experimen-tally for real soils under static or quasi-static loading isnot linear. In order to describe more realistically themodulus degradation relationship, Fahey and Carter(1993) and Lee and Salgado (2000) proposed modiˆedhyperbolic models for 2D and 3D conditions, respec-tively, as follows:

GG0

＝1－f Ø ttmax

»g

(6)

GG0

＝«1－f Ø J2－ J20

J2max－ J20»

g

$Ø I1

I10»

ng

(7)

where J2, J20, and J2max are the second invariants of thedeviatoric stress tensor at the current, initial, and failurestates, respectively; I1 and I10 are the ˆrst invariants of thestress tensor at the current and initial states, respectively;and ng is the material constant. The parameter f in Eqs.(6) and (7) has the same role as Rf in Eq. (4). Theparameter g determines the shape of the degradationcurve as a function of stress level. G0 is used in Eqs. (6)and (7), whereas Gi is used in Eq. (4). As mentionedpreviously, Gi is obtained from the transformed hyper-bolic curve shown in Fig. 1. If the stress-strain curve ofEq. (3) and the modulus degradation relationship givenby Eq. (4) in fact represented the actual stress-strainresponse of soils, then Gi would be the same as G0. As willbe discussed later, however, Gi is signiˆcantly diŠerentfrom G0.

SMALL-STRAIN MODULUS

The small-strain shear modulus G0 is a state soil varia-ble that is observed for strains in the range of 10－6 to 10－5

for sands. Within this small-strain range, soil behaves asa linear-elastic material and strains are recoverable. Thecorresponding Young's modulus is denoted by E0. As G0

is a state soil variable, its value is constant for a given soilcondition, regardless of the nature of the loading type(Shibuya et al., 1992).

There are a number of ways to estimate G0 for a givensoil and stress state. These may be grouped into in situtests, laboratory tests, and empirical equations (Yu andRichart, 1984; Baldi et al., 1989; Viggiani and Atkinson,1995). While the laboratory tests allow accurate determi-nation of the shear wave velocity Vs, these are still consi-dered costly for use in routine projects as they requirecomplex testing systems speciˆcally designed to measurethe wave propagation characteristics or other small-strainsoil properties. In experimental approaches using shearwave velocity measurements, G0 is determined from thefollowing relationship:

G0＝r(Vs)2 (8)

where r＝total mass density of the medium throughwhich the shear wave propagates. Values of r are knownwith certainty in laboratory tests. In the ˆeld, while rela-tive evaluation of soil density using geophysical tech-

niques such as well logging tests would be possible, iden-tiˆcation of absolute values of soil density requires acalibration procedure.

Empirical equations can be used to estimate G0 if therelevant soil parameters are known. Most of the empiri-cal equations proposed for the estimation of G0 are basedon either soil properties or in situ test results such as theSPT blow count number N or the CPT cone resistance qc

(Hardin and Black, 1966; Robertson and Campanella,1983; Yu and Richart, 1984; Rix and Stokoe, 1991). Theproperty-based empirical equations are generally ex-pressed as:

G0

pA＝CF(e)Øs?m

pA»

n

(9)

where pA＝reference stress＝100 kPa; C and n are non-dimensional material constants; F(e) is the function ofthe void ratio; s?m is the mean eŠective stress. An exampleof a property-based empirical equation is that suggestedby Hardin and Black (1966) and given by:

ØG0

pA»＝Cg

(eg－e)2

1＋e Øs?mpA

»ng

(10)

where Cg, eg and ng are the intrinsic soil variables that de-pend only on the nature of the soil and e is the initial voidratio.

Application of Eq. (10) is limited as values of the in-trinsic soil variables are likely to be unknown for in situsoils. In such cases, the use of in situ test results would bemore appropriate since estimation of these variables isnot necessary. This approach, however, requires the de-velopment and validation of correlations between G0 andin situ test measurements for various soil conditions.

ESTIMATION OF ELASTIC MODULUS BASED ONTRANSFORMED STRESS-STRAIN CURVES

Transformed Stress-Strain CurvesIn this section, a methodology to estimate the small-

strain elastic modulus based on conventional triaxialcompression stress-strain curves is proposed. For thispurpose, triaxial compression test results by Salgado etal. (2000) and Lee et al. (2004) are adopted. The drainedtriaxial compression tests were performed under staticconditions using a CKC automatic triaxial testing system(Chan, 1981). The soil samples were prepared with theslurry deposition method of Kuerbis and Vaid (1988). Alltriaxial tests were performed for isotropically-consolidat-ed soil samples at strain rates that were slow enough to al-low full dissipation of pore pressures during loading. Thesoil samples used in the triaxial compression tests weremixtures of Ottawa sand and nonplastic silt (silt contentssco equal to 0, 2, 5, 10, 15, and 20z by weight) preparedin a wide range of relative densities (DR＝10 to 100z).Conˆning stresses (sc?) ranging from 100 to 500 kPa wereused in these tests.

G0 was measured with bender element (BE) tests. TheBE tests were performed using a wave generator andreceiver (the bender elements) attached at the base

548

Table 1. Basic properties of Ottawa sand with diŠerent silt contents (after Salgado et al., 2000)

Silt content (z) gmin (kN/m3) gmax (kN/m3) emin emax Cg eg ng qc

0 14.59 17.55 0.48 0.78 611 2.17 0.44 29.592 15.18 17.91 0.45 0.71 514 2.17 0.58 29.695 15.28 18.29 0.42 0.70 453 2.17 0.46 31.09

10 15.74 19.09 0.36 0.65 354 2.17 0.58 32.0915 (DRÀ38z) 15.93 19.67 0.32 0.63 238 2.17 0.75 32.5915 (DRº38z) 15.93 19.67 0.32 0.63 238 2.17 0.75 32.5920 (DRÀ59z) 16.03 20.13 0.29 0.62 270 2.17 0.69 33.0920 (DRº59z) 16.03 20.13 0.29 0.62 207 2.17 0.81 33.09

548 LEE ET AL.

pedestal and top platen of the triaxial apparatus (Salgadoet al., 2000; Lee et al., 2004). The BE tests were per-formed after consolidation for a number of triaxial sam-ples. G0 values for these samples were obtained using Eq.(8). The shear wave velocity from the test was measuredfrom the eŠective length of the test sample (i.e., the dis-tance between tips of the bender elements) and the traveltime of the shear wave identiˆed from the ˆrst arrival ofthe signal generated by the source bender element. Ac-cording to Viggiani and Atkinson (1995), possible errorsin G0 values from the BE tests could be up to 15z, in ex-treme cases. The errors are mainly due to deviation from1-D wave propagation, wave interference at the caps, anddiŠerent time delays between the generation of the electri-cal signal and its transformation into a mechanical pulse(Salgado et al., 2000). As indicated by Arulnathan et al.(1998), these factors sometimes compensate each other,while sometimes they do not. Therefore, it can be saidthat values of G0 from the BE tests considered in thispaper may include errors of up to 15z, while the actualerrors are likely to be smaller due to the self-compensat-ing eŠects (Salgado et al., 2000). Other basic properties ofthe test sands used in the triaxial and bender element testsare given in Table 1.

Figure 2 shows the stress-strain, modulus degradation,and transformed stress-strain curves obtained fromdrained triaxial compression tests performed on twoclean sand (i.e., sco＝0z) samples with DR＝38 and 63zat s3?＝400 kPa (tests were also performed for other sandswith diŠerent conˆning stress). In Fig. 2(b), the secantelastic modulus E was normalized with respect to E0 (thesmall-strain elastic modulus obtained from the benderelement tests). As expected, the denser sand sampleshows higher strength and lower modulus degradationrate at a given stress level. The curves shown in Fig. 2(c)were plotted until the peak strength was reached (axialstrain levels of around 15.7 and 4.7z were observed forthe sand samples with DR＝38 and 63z, respectively).

Figure 3 shows measured peak deviatoric stresses s?d, p

versus those calculated using Eqs. (1) and (3) by Kondner(1963) and Duncan and Chang (1970), respectively. Asshown in Fig. 3, both approaches are in reasonably goodagreement with the measured values of s?d, p.

Determination of Small-Strain Elastic Modulus fromTransformed Stress-Strain Curves

Figure 4 shows measured small-strain elastic modulus

E0 versus Ei calculated following Duncan and Chang'sprocedure. As shown in Fig. 4, the initial tangent elasticmodulus obtained from Duncan and Chang's procedureis signiˆcantly smaller than the measured small-strainelastic modulus. The degree of underestimation dependson the relative density of the sand. Figure 5(a) shows Ei

/E0 ratios as a function of DR. From Fig. 5(a), it is seenthat the denser the soil, the higher the values of Ei/E0.This indicates that, as the soil becomes more dilative, theunderprediction of the small-strain modulus by Duncanand Chang's (1970) procedure becomes less pronounced.Such tendency can be more clearly observed in Fig. 5(b).

According to Bolton (1986), the dilatancy of sands,which is essentially controlled by relative density and con-ˆning stress can be quantiˆed using the dilatancy index IR

as follows:

IR＝ID«Q－ln Ø100s?mp

pA»$－R (11)

where ID＝relative density as a number between 0 and 1;pA＝reference stress＝100 kPa; s?mp＝mean eŠective stressat peak strength (in the same units as pA); and Q and R＝intrinsic soil variables. Values of Q and R for the cleanand silty sands tested are given by Salgado et al. (2000)and Lee et al. (2004). As shown in Fig. 5(b), the degree ofunderprediction of the small-strain modulus obtainedwith Duncan and Chang's procedure becomes lesspronounced with increasing IR.

The diŠerence between Ei and E0 can be attributed tothe inability of the hyperbolic relationship to representthe rate of modulus degradation in the small-strain range.The transformed stress-strain relationship shown in Fig.1 is given by:

es＝a＋be (12)

The initial tangent elastic modulus is then deˆned as(Tatsuoka et al., 1993):

limeª/

d(e/s)de

＝b＝1

slim＝

Rf

sf(13)

limeª0

es＝a＝

1E0

(14)

where s and e are the axial stress and strain; slim is thelimit axial stress as deˆned in Fig. 1; and sf is the axialstress at failure. If the stress-strain curves of the soils fol-

549

Fig. 2. Stress-strain responses of clean sand samples prepared atdiŠerent relative densities: (a) stress-strain curves, (b) modulusdegradation curves and (c) transformed stress-strain curves

Fig. 3. Measured and calculated peak deviatoric stresses using (a)Kondner's procedure and (b) Duncan and Chang's procedure

Fig. 4. Values of Ei calculated with the hyperbolic model versus E0

from bender element tests

549SMALL-STRAIN ELASTIC MODULUS

lowed well the hyperbolic relationship of Eq. (12), thenthe transformed stress-strain curve (as shown in Fig.1(b)) would be linear in e/s (or g/t ) vs. e (or g) space,and Eqs. (12) and (14) would be deˆned by constantvalues of a and b. This is, however, not observed in thestress-strain curves of typical soils, as in the small-strainrange, the transformed stress-strain relationship is notlinear (Shibuya et al., 1992).

550

Fig. 5. Ei/E0 ratios vs. (a) relative density and (b) dilatancy index

Fig. 6. DiŠerent types of stress-strain responses: (a) original stress-strain curves and (b) transformed stress-strain curves

550 LEE ET AL.

Figure 6 shows typical stress-strain curves obtainedfrom triaxial compression tests and transformed stress-strain curves. As shown in Fig. 6, a stress-strain curvereferred to as type A represents a stress-strain curvewhich is well reproduced by the hyperbolic relationship,except within an initial portion of the small-strain range(a curved shape is observed only initially, as shown in Fig.6(b)); this is due to a higher degree of modulus degrada-tion than that deˆned by the conventional hyperbolicfunction given by Eqs. (4) and (5). For a stress-straincurve as the one referred to as type B in the ˆgure, a rela-tively linear stress-strain response exists before the peak,and the transformed stress-strain curve shows a concaveupward portion, as shown in Fig. 6(b). In both cases,values obtained using Duncan and Chang's (1970) proce-dure do not correspond to E0 or G0 values due to the cur-vature in the initial portion of the transformed hyperbolicstress-strain curve. It can be concluded that, if a proce-dure is capable of predicting values of a compatible withthe initial linear-elastic strain range, then realistic valuesof the small-strain elastic modulus can also be foundfrom these stress-strain curves.

In order to evaluate values of the parameter corre-sponding to the small-strain modulus, the nonlinear por-tion of the transformed stress-strain curves (i.e., see theregion I in Fig. 6(b)) needs to be examined in detail. Asconventional triaxial testing systems cannot accuratelymeasure soil response within the initial, linear-elasticstrain range (typically up to 10－6–10－5), the initial, non-linear portion of the transformed hyperbolic stress-straincurves were reverse-extended down to a strain level equalto zero using an extrapolation technique. From the ex-trapolation procedure, modiˆed values of the parametera (referred to as a*) were determined.

Figure 7 shows an example of the extrapolation proce-dure used in the determination of the parameter a*. Notethat adoption of a fairly well deˆned initial stress-straincurve and an appropriate extrapolation function is keyfor successful implementation of this approach. In thispaper, a 2nd-order polynomial function was adopted forthe extrapolation of the stress-strain response within thesmall-strain range. As shown in Fig. 7, the parameter a*obtained with the extrapolation method (Fig. 7(b)) isequal to 0.0005978, while the parameter a obtained withthe Duncan and Chang's approach is equal to 0.0039

551

Fig. 7. Determination of the parameters a and a* for calculation ofthe initial small-strain elastic modulus with (a) Duncan andChang's method and (b) extrapolation method proposed

551SMALL-STRAIN ELASTIC MODULUS

(Fig. 7(a)). The 2nd order polynomial regression curveshown in Fig. 7(b) is given by:

e(s1?－s3?)

＝－0.0279e2＋0.0157e＋0.0005978 (15)

The values of a*＝0.0005978 and a＝0.0039 correspondto values of the initial tangent elastic modulus Ei equal to167.3 and 25.6 MPa, respectively. The measured E0 valuefor this case is equal to 187.3 MPa.

A suitable range of well deˆned initial strain datashould be used in the extrapolation procedure proposedin this paper since values of a* are sensitive to the strainrange considered and the quality of the strain data. If theupper bound of the strain range considered in the ex-trapolation procedure is too large, a* will approach thevalue of a obtained with Duncan and Chang's procedure,and, hence, result in underestimation of the small-strainelastic modulus. Based on the results of this study, theupper bound of the strain range considered in the ex-trapolation procedure should be about 0.2–0.25z toproduce reasonably satisfactory estimates of E0. Theproposed extrapolation procedure is further evaluated in

the next section.

Comparison of Measured and Calculated Small-StrainElastic Modulus

In order to evaluate the extrapolation methodproposed in this paper, values of the small-strain elasticmodulus obtained from triaxial compression tests com-bined with bender element tests were compared withvalues calculated with the procedure outlined in the previ-ous section, the method of Duncan and Chang (1970),and the tangent method. In the case of the tangentmethod, the initial elastic modulus was estimated fromsimple calculation of the initial tangent modulus usingthe ˆrst numerical data point of the stress-strain responseobtained from triaxial compression tests (note that thetangent method is not considered reliable as it is aŠectedby experimental noise). As discussed earlier, if measure-ment of stresses and strains within the linear-elastic rangewere available, then the tangent method would producevirtually the same value for the small-strain elastic modu-lus that would be obtained using other methods such asthe resonant column and bender element tests. However,when a conventional triaxial test system is used with atypical strain increment equal to more or less 0.1z, theˆrst stress-strain data point is beyond the linear elasticrange. The bedding error observed between the soil sam-ple and the top and bottom platens also contributes sig-niˆcantly to the diŠerence between the small-strainmodulus values measured with diŠerent techniques(Shibuya et al., 1992).

Figure 8 compares the measured values of the small-strain elastic modulus E0 with those calculated usingdiŠerent methods. Results from the proposed method areshown in Fig. 8(a). While some scatter is observed, ap-proximately 80z of the data points (82 data points froma total of 101 data points) fall within the ±30z bounds.The tangent method results are shown in Fig. 8(b). Itunderestimates the small-strain elastic modulus because,as indicated earlier, it is based on the ˆrst data point,which is beyond the initial, linear elastic range. Duncanand Chang's procedure also underestimates the actualvalues of the small-strain elastic modulus. As seen in Fig.8(c), the modulus values calculated with Duncan andChang's procedure are signiˆcantly smaller than themeasured values (no data points were higher than ap-proximately 15z of E0). This is because the hyperbolicmodel neglects the existence of the initial, linear elasticrange and does not adequately represent the rate ofmodulus degradation in the small-strain range. Based onthese results, it can be concluded that the methodproposed in this paper, which is based on a simple ex-trapolation procedure applied to conventional triaxialcompression test data, can be used eŠectively to estimatethe small-strain modulus for practical purposes, particu-larly when equipment with small-strain measurementcapability is unavailable. However, use of the bender ele-ment technique or high-resolution internal strain measur-ing devices is always preferable to obtain more accuratemodulus estimates.

552

Fig. 8. Measured and calculated small-strain elastic modulus obtainedfrom (a) extrapolation method proposed, (b) tangent method and(c) Duncan and Chang's method

552 LEE ET AL.

The performance of the proposed method was quiteconsistent for the sands with diŠerent silt contents (seeFig. 8(a)) considered in this paper, indicating that the

proposed method is applicable to sands containing diŠer-ent silt contents without additional correction proce-dures. The stress-strain curves from the triaxial compres-sion tests depend on the silt content of the sample, andthus the initial portion of the stress-strain curves (used inthe extrapolation procedure) also includes the eŠect ofsilt content.

DETERMINATION OF SMALL-STRAIN SHEARMODULUS FROM CPT CONE RESISTANCE

Correlations between G0 and qc

Determination of the small-strain shear modulus in thelaboratory requires sample preparation and test proce-dures that reproduce in situ conditions. When collectionof soil samples is not a possibility, use of in situ tests is anoption that can be considered. Several correlations havebeen proposed between the small-strain shear modulusand cone resistance qc (Robertson and Campanella, 1983;Rix and Stokoe, 1991). However, there are di‹culties inestablishing reliable correlations between G0 and qc, as qc

is related essentially to the shear strength that is mobi-lized at large strains, whereas G0 is a small-strain proper-ty. Nonetheless, both these quantities depend on similarsoil variables, that is, relative density and stress state.

Robertson and Campanella (1983) proposed the fol-lowing relationship between qc and G0:

G0

qc＝G1ØpA

qc»

0.389

(16)

where G1＝correlation parameter＝50; and pA＝referencestress＝100 kPa＝1 bar. Originally, Eq. (16) wasproposed in terms of the SPT blow count N by Imai andTonouchi (1982). It was later modiˆed by Robertson andCampanella (1983) by converting N to qc. The relation-ship between qc and G0 of Rix and Stokoe (1991), on theother hand, can be given in a normalized form as follows:

G0

qc＝G2 Ø qc

pA

pA

s?v0»－0.75

(17)

where s?v0＝initial vertical eŠective stress; and G2＝corre-lation parameter＝290; and pA＝reference stress＝100kPa. The units in Eqs. (16) and (17) are the same as theunits of pA.

Measured and Calculated Small-Strain Shear ModulusFigure 9 shows measured G0/qc ratios for the triaxial

test dataset used in this study as a function of qc (Fig.9(a)) and qc/(s?v0)0.5 (Fig. 9(b)). Curves corresponding toEqs. (16) and (17) by Robertson and Campanella (1983)and Rix and Stokoe (1991) were also plotted in Figs. 9(a)and (b), respectively. The cone resistance qc for the soilstates in the triaxial compression tests was calculated us-ing the cone resistance analysis program CONPOINT,which has been extensively veriˆed (Salgado and Ran-dolph, 2001). As shown in Fig. 9, for clean sands, bothcorrelations produce a reasonably good prediction of G0.For silty sands, however, these correlations do not seemto work well, as G0/qc ratios plot much below the predict-

553

Fig. 9. G0/qc ratios plotted with results from the correlations by (a)Robertson and Campanella (1983) and (b) Rix and Stokoe (1991)

Fig. 10. Values of the correlation parameters G1 and G2 for sand con-taining diŠerent silt contents

553SMALL-STRAIN ELASTIC MODULUS

ed curves shown in Fig. 9.The curves and data shown in Fig. 9 indicate that an

additional factor that needs to be accounted for in corre-lations between G0 and qc is the ˆnes content of the sand.As can be seen in Fig. 9, both correlations overestimatethe small-strain shear modulus of silty sands. This is be-cause G0 and qc are aŠected by the silt content (sco) of thesand in a diŠerent way. According to Salgado et al.(2000), the small-strain shear modulus decreases with in-creasing silt content while the shear strength (i.e., peakfriction angle qp?) increases; this is due primarily to ahigher degree of particle interlocking. The increase ofshear strength with increasing silt content would producehigher cone resistance qc and thus, as a result, lowervalues of G0/qc.

Based on the triaxial compression test results of Salga-do et al. (2000) and Lee et al. (2004), new values for thecorrelation parameters G1 and G2 were proposed in orderto take into account in Eqs. (16) and (17) the eŠect of thesilt content of the sand. Figure 10 shows values of G1 andG2 obtained in this study as a function of sco. As shown inFig. 10, G1 and G2 decrease with increasing silt content

sco. From Fig. 10, equations for G1 and G2 were found interms of sco as follows:

G1＝25・e－0.24・sco＋25 (18)G2＝150・e－0.23・sco＋140 (19)

Figure 11 compares measured versus calculated G0

values for the same triaxial test datasets shown in Fig. 9.G0 values were calculated with Eqs. (16) and (17), and G1

and G2 were calculated with Eqs. (18) and (19). Figure 11shows that satisfactory estimates of the measured small-strain shear modulus can be obtained by using the corre-lation parameters presented in Fig. 10 in Eqs. (16) and(17). Note that the G1 and G2 values given in Fig. 10 wereobtained from triaxial compression samples that wereconsolidated to an isotropic stress condition (i.e., s?v0＝s?h0). Further research should be conducted to obtain cor-relation parameters that take into account anisotropicstress states, which are most often found in practice. Inaddition, these correlation parameters are valid only forsilty sands with properties similar to the ones consideredin this study.

EŠect of Anisotropic Stress StateIt is known that values of the cone resistance qc are

predominantly in‰uenced by the horizontal eŠectivestress sh? rather than by the vertical eŠective stress sv?(Houlsby and Hitchman, 1988). The small-strain shearmodulus G0 on the other hand is in‰uenced by the meaneŠective stress s?m, re‰ecting eŠects of both sv? and sh?(Hardin and Black, 1966; Iwasaki and Tatsuoka, 1977;Robertson and Campanella, 1983; Ghionna et al., 1994).Stress anisotropy therefore is another factor to be ad-dressed in correlations between G0 and qc.

In order to investigate the eŠect of anisotropic stressstates, a total of 60 soil state conditions were consideredfor clean and silty sands. DiŠerent values were consideredfor the relative density (30z and 70z) of the sand andthe lateral earth pressure coe‹cient K0 (0.4 and 0.8).Table 2 shows the stress and soil conditions used in thecalculations. Basic properties of the clean sand and siltysand mixtures were assumed to be those given in Table 1.For each case, values of G0 and qc were obtained usingEq. (10) and CONPOINT (Salgado and Randolph,

554

Fig. 11. Measured and calculated small-strain shear modulus using thecorrelations by (a) Robertson and Campanella (1983) and (b) Rixand Stokoe (1991) (calculations were done using the correlationparameters given in Fig. 10)

Table 2. Stress and soil conditions used to obtain the G0-qc correlationproposed

sco

(z)DR

(z)

K0＝0.4 K0＝0.8

s?v0 (kPa) s?h0 (kPa) s?v0 (kPa) s?h0 (kPa)

0 30, 70250 100 125 100500 200 250 200750 300 375 300

5 30, 70250 100 125 100500 200 250 200750 300 375 300

10 30, 70250 100 125 100500 200 250 200750 300 375 300

15 30, 70250 100 125 100500 200 250 200750 300 375 300

20 30, 70250 100 125 100500 200 250 200750 300 375 300

Fig. 12. Values of G0/qc for the conditions given in Table 2, as well asthose from Eq. (17)

554 LEE ET AL.

2001), respectively. The parameters required as input inEq. (10) and CONPOINT are given in Table 1 (Salgadoet al., 2000; Lee et al., 2004). As Eq. (10) and CON-POINT have been extensively veriˆed, values of G0 and qc

obtained as outlined in this paper represent closely thosethat would be measured in situ for the corresponding soiltype and soil state conditions considered.

Figure 12 shows values of G0/qc for the stress and soilconditions given in Table 2 as a function of qc/(s?v0)0.5, aswell as the curves given by Eqs. (17) and (19). SomediŠerences are still observed in Fig. 12 between the datapoints of G0/qc and the correlation curves of Eqs. (17)and (19). Contrasting these results with those presented inFig. 9, one can conclude that these diŠerences can be at-tributed to the diŠerent initial stress states considered:isotropic (i.e., K0＝1.0 with constant conˆning stress s3?)in the triaxial compression tests and anisotropic (i.e., K0

º1.0) for the stress states considered in the calculations( see Table 2).

As G0 and qc are governed by diŠerent stress compo-nents, the correlation of Eq. (17) was further investigatedwith respect to diŠerent stress components of s?v0, s?h0,and s??m0. From the investigation of all the cases in Table2, it was found that the correlation based on s?m0 producesimproved predictions with less data scatter. This is be-cause G0 and qc represent dependencies on diŠerent stresscomponents and, therefore, eŠects of both s?v0 and s?h0

need to be included in the correlation. Based on theresults of this study, a modiˆed version of the G0-qc cor-relation of Eq. (17) is proposed:

G0

qc＝G3 Ø qc

pA

pA

s?m0»－0.75

(20)

G3＝110・e－0.23・sco＋160 (21)

where sco＝silt content in z; s?m0＝in situ mean eŠectivestress and pA＝reference stress＝100 kPa＝1 bar.

Note that Eqs. (20) and (21) apply only to recentlydeposited, uncemented clean and silty sands with proper-ties similar to the ones considered in this study. It is welldocumented in the literature that in situ G0-qc correla-tions depend on many factors such as sand compressibili-

555

Table 3. Values of qc from calibration chamber tests and G0 fromresonant column tests for the test conditions considered

DR (z) s?v0 (kPa) s?h0 (kPa) K0 qc (MPa) G0 (MPa)

55 100 27 0.27 3.39 64.655 100 40 0.40 5.39 68.755 100 70 0.70 6.92 77.055 100 100 1.00 8.42 84.255 57 40 0.70 5.36 61.755 150 40 0.27 5.32 75.786 100 40 0.40 18.69 84.986 100 70 0.70 19.65 95.386 100 100 1.00 22.64 104.186 57 40 0.70 15.40 76.386 150 40 0.27 18.68 93.7

Fig. 13. Values of G0/qc for calibration chamber tests in terms of (a)the vertical eŠective stress s?v0, (b) the horizontal eŠective stress s?h0

and (c) the mean eŠective stress s?m0

555SMALL-STRAIN ELASTIC MODULUS

ty, aging and cementation (Baldi et al., 1989; Schnaid etal., 2004), which are not accounted for in the correlationsproposed in this paper.

Comparison with Calibration Chamber Test ResultsIn order to compare the measured and predicted G0/qc

ratios, results from calibration chamber cone penetrationtests by Lee et al. (2008) were used. The sand used to pre-pare the calibration chamber samples was Jumunjinsand, a standard Korean sand. A series of resonantcolumn tests were performed on Jumunjin sand samplesprepared at various soil states to obtain G0 values; theseresults were then used to obtain the intrinsic soil variablesneeded in Eq. (10). A total of 11 calibration chamberCPTs were performed at diŠerent relative densities andstress states. Other experimental details of the calibrationchamber tests can be found in Lee et al. (2008). Table 3shows values of qc from calibration chamber tests and G0

from resonant column tests.Figure 13 shows the values of G0/qc as a function of qc

/(s?v0)0.5, qc/(s?h0)0.5, and qc/(s?m0)0.5 (qc obtained fromcalibration chamber CPTs and G0 from resonant columntests). Curves corresponding to the correlations given byEqs. (17) and (20) were also plotted in Fig. 13. As shownin Fig. 13, results from the approaches based on s?v0, s?h0,and s?m0 are in fairly close agreement with the measuredvalues. Average correlation errors between predicted (us-ing Eqs. (17) and (20); note that the silt content ofJumunjin sand is equal to zero and thus the correlationparameters G2 and G3 are equal to 290 and 270, respec-tively) and measured data were 13.5z, 12.8z, and 7.6zfor the stress components of s?v0, s?h0, and s?m0, respec-tively.

SUMMARY AND CONCLUSIONS

In the present study, simple and eŠective methods areproposed for the determination of the small-strain modu-lus. One method is based on an extrapolation procedure:the initial nonlinear portion of the transformed stress-strain curve is reverse-extended down to a strain levelequal to zero using an extrapolation technique to obtain aparameter (a*), which is then used to calculate the small-strain elastic modulus. The second method is based on

the CPT cone resistance qc.A series of stress-strain curves from triaxial compres-

sion tests performed on clean and silty sands were ana-lyzed to evaluate the method of estimation of the small-strain elastic modulus based on the extrapolation proce-dure proposed in this paper. The following conclusions

556556 LEE ET AL.

were reached:1) Values of s?d, p calculated using the hyperbolic

procedure of Duncan and Chang were in good agreementwith measured values of s?d, p.

2) Values of Ei obtained from the Duncan andChang's procedure, on the other hand, were signiˆcantlysmaller than the measured E0 values. The denser the soil,the higher the values of Ei/E0; this indicates that, as thesoil becomes more dilative (the larger the relative density,the greater the tendency for dilation, as indicated by Eq.(11)), underestimation of the small-strain modulus by theDuncan and Chang's procedure becomes lesspronounced.

3) The extrapolation procedure proposed in thispaper produced satisfactory estimates of the measuredstress-strain responses of clean and silty sands within thesmall-strain range. Measured and calculated small-strainelastic modulus values were found to be in reasonablygood agreement.

Two G0-qc correlations available in the literature(Robertson and Campanella, 1983; Rix and Stokoe,1991) and the one proposed in this paper were evaluated.The following conclusions were reached:

1) For isotropic conditions, the correlations byRobertson and Campanella (1983) and Rix and Stokoe(1991) produced good estimates of G0 for clean sands, butoverestimated G0 for silty sands.

2) For isotropic conditions, very good estimates ofthe measured small-strain shear modulus of silty sandswere obtained with the correlations by Robertson andCampanella (1983) and Rix and Stokoe (1991) by usingthe new values proposed in this paper for the correlationparameters G1 and G2, which account for the eŠect of thesilt content of the sand.

3) Values of G0/qc obtained for clean and silty sandstested under various anisotropic stress states (G0 calculat-ed with the Hardin and Black (1966) correlation and, qc

with CONPOINT) were in good agreement with G0/qc

values calculated using the G0-qc correlation presented inthis paper, which takes into account not only the silt con-tent of the sand but also stress anisotropy.

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