Measurement of Small-Strain Shear Moduli of Partially …unh.edu/geotech/papers/Ghayoomi and...

Majid Ghayoomi,1 and John S. McCartney2

Measurement of Small-Strain Shear Moduli ofPartially Saturated Sand During Infiltration in aGeotechnical Centrifuge

ABSTRACT: This paper describes the use of bender elements to measure changes in small strain shear modulus, Gmax, of sand layers due tothe change in degree of saturation during centrifuge tests. The goal of the measurements is to verify that steady-state infiltration is an appropriatetechnique to control the effective stress in centrifuge physical modeling of partially saturated sands. Specifically, the suitability of infiltration isassessed by checking if the measured values of Gmax of partially saturated sand layers follow a similar trend to dry and saturated sand layers whenthe effective stress is defined from the suction and degree of saturation profiles during steady-state infiltration. Three pairs of bender elements wereinstalled at different depths in a container of Ottawa sand, and the shear wave velocities of the sand were measured during steady-state infiltrationinto the sand layer. The applied infiltration rate was varied to obtain different uniform distributions of degrees of saturation with depth. Consistentwith results from suction-controlled resonant column tests performed on the same sand, the values of Gmax measured from the bender element testsvaried nonlinearly with degree of saturation with a peak value at a degree of saturation between 0.3 and 0.4. When interpreted in terms of meaneffective stress, the values of Gmax from the bender element tests on partially saturated sands followed a unique trend consistent with measurementsfor dry and saturated sands.

KEYWORDS: partially saturated sand, small-strain shear modulus, bender elements, geotechnical centrifuge testing

Introduction

Hardin and Richart (1963) and Hardin and Drnevich (1972)observed that the shear modulus of soil is generally constant forshear strain amplitudes less than 10�4 and is sensitive primarily tothe void ratio and effective stress state. Knowledge of the small-strain shear modulus Gmax is particularly useful in the predictionof the seismic compression of soil layers during earthquake shak-ing (Tokimatsu and Seed 1987; Ghayoomi 2011). Ghayoomi et al.(2011) developed a new centrifuge modeling approach to evaluatethe seismic compression of partially saturated sand layers in whichinfiltration was used to control the degree of saturation and matricsuction, and thus the effective stress state. Although Ghayoomiet al. (2011) found that trends in seismic compression of sand dur-ing steady-state infiltration correlate well with expected changesin Gmax estimated from empirical equations, independent meas-urements of Gmax during variations in infiltration rate would helpincrease confidence in the use of steady-state infiltration for suc-tion control in centrifuge testing.

Bender elements have been used extensively for laboratorymeasurement of Gmax of saturated and dry soils (Shirley andHampton 1978; Dyvik and Madshus 1985; Bates et al. 1989;Argawal and Ishibashi 1991; Brignoli et al. 1996; Arulnathan

et al. 1998; Pennington et al. 2001; Leong et al. 2005) as well aspartially saturated soils (Cabarkapa et al. 1999; Inci et al. 2003;Marinho et al. 1995; Alramahi et al. 2007; Ng and Yung 2008;Sawangsuriya et al. 2009, Ng et al. 2009). They are typicallyincorporated into laboratory tests because they induce and mea-sure shear strains with amplitudes less than 10�4 and are relativelycompact in size.

The objective of this paper is to describe how bender elementscan be used to verify that infiltration is an appropriate tool to con-trol the effective stress state in centrifuge physical modeling ofpartially saturated soils. Although bender elements have beenused in centrifuge tests in the past (Ismail and Hourani 2003; Leiet al. 2004; Brandenberg et al. 2006; Rammah et al. 2006; Fuet al. 2009; Kim and Kim 2010), their use in partially saturatedsoils in the centrifuge deserves further investigation. Accordingly;in addition to describing the details of the bender element setupfor the centrifuge, the measured values of Gmax from the benderelements used in this study are compared with those from suction-controlled resonant column tests on the same sand reported byKhosravi et al. (2010).

Background

Bender elements apply shear waves having small strain magnitudesto soil layers through the use of piezoelectric ceramics. Piezoelec-tric ceramics deform during application of a voltage difference, orgenerate a voltage difference when deformed. When a voltage sig-nal is applied to a transmitting element, a shear wave will beinduced in the surrounding soil, which can be detected using areceiving element located at a known distance from the transmitting

Manuscript received November 19, 2010; accepted for publication June 13,2011; published online July 2011.

1Ph.D., Research Associate, Dept. of Civil, Environmental, and ArchitecturalEngineering, Univ. of Colorado Boulder, e-mail: [email protected]

2Ph.D., P.E., Assistant Professor, Barry Faculty Fellow, Dept. of Civil,Environmental, and Architectural Engineering, Univ. of Colorado Boulder,e-mail: [email protected]

Copyright VC 2011 by ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. 1

Geotechnical Testing Journal, Vol. 34, No. 5Paper ID GTJ103608

Available online at: www.astm.org

Copyright by ASTM Int'l (all rights reserved); Mon Oct 28 18:51:20 EDT 2013Downloaded/printed byUniversity of New Hampshire pursuant to License Agreement. No further reproductions authorized.

element. Shear wave velocity can be measured using a pair ofbender elements by measuring the time difference between trans-mitting and receiving shear waves from one bender element toanother. The shear wave velocity, Vs, is related to Gmax as follows:

Gmax ¼ qV 2s (1)

where q is the total soil densitySince Shirley and Hampton (1978) introduced bender elements

to soil testing, they have been incorporated into many conventionalgeotechnical tests, including triaxial tests (Bates 1989; Brignoli etal. 1996; and Pennington et al. 2001), resonant column tests (Dyvikand Madshush 1985), oedometer tests (Dyvik and Madshush 1985;Kawaguchi et al. 2001), direct simple shear tests (Dyvik and Mad-shush 1985), true-triaxial tests (Agarwal and Ishibashi 1991), andlarge container tests (Blewett et al. 2000). Concerns such as electro-magnetic cross-talk, in-plane and out-of-plane directivity, benderresonant frequency, and effects of reflections from boundaries havealso been addressed in laboratory and theoretical studies (Arulna-than et al. 1998; Greening and Nash, 2004; Lee and Santamarina2005; Leong et al. 2005). Several studies have used bender ele-ments to study the impacts of matric suction, degree of saturation,and hydraulic hysteresis on the magnitude of Gmax for partially sat-urated soils (Cabarkapa et al. 1999; Inci et al. 2003; Marinho et al.1995; Alramahi et al. 2007; Ng and Yung 2008; Hoyos et al. 2008;Sawangsuriya et al. 2009, Ng et al. 2009).

Bender elements have been used in centrifuge modeling testsas well, primarily for shear wave velocity tomography to inferzones of densified soil (Ismail and Hourani 2003; Lei et al. 2004;Brandenberg et al. 2006; Rammah et al. 2006; Kim and Kim2010). Fu et al. (2009) used bender elements in centrifuge tests toevaluate the liquefaction resistance of soils based on their shearwave velocity. Due to difficulties in maintaining a steady partiallysaturated soil condition in centrifuge testing, the application ofbender elements in geotechnical centrifuge tests involving par-tially saturated soils is not extensive. This barrier has been over-come after Ghayoomi et al. (2011) showed that steady-stateinfiltration leads to a uniform matric suction distribution suitablefor mechanical or seismic testing of partially saturated specimensunder controlled conditions.

One of the reasons that careful control of the effective stressstate in partially saturated soils is necessary is that Gmax is particu-larly sensitive to the mean effective stress. Specifically, experi-mental studies on Gmax of saturated and dry soils found that Gmax

and mean effective stress are related in terms of a power function(Hardin and Richart, 1963; Seed and Idriss, 1970; Hardin andDrnevich, 1972). The relationships typically have the followingform for sands

Gmax ¼ AF eð Þ r0mPa

� �n

(2)

where r0m is the mean effective stress, A is a fitting parameter hav-ing the same units as Gmax, n is a dimensionless fitting parameter,and F(e) is a void ratio function. Khosravi and McCartney (2009)observed that Eq. (2) can be extended to partially saturated soils ifa single-value effective stress parameter is used to combine theeffects of matric suction and total stress on the inter-particlestress.

Experimental Setup

Centrifuge and Container

The 400 g-ton geotechnical centrifuge facility at the Univ. of Col-orado Boulder (Ko, 1988) was used for the physical modelingexperiments in this investigation. A laminar container developedby Law (1991) having an inside length of 58.42 cm, width of24.13 cm, and depth of 15.87 cm was used in this study. Althoughthis study did not involve shaking, the laminar container was usedto be consistent with seismic compression results from Ghayoomiet al. (2010). This container is suitable for modeling a prototypesoil layer having a thickness of 0.1587N meters at a centrifugeacceleration N-times that of earth gravity. This container wasmodified by Ghayoomi et al. (2011) to permit control of steady-state infiltration and drainage of pore fluid through the soil layerby adding a drainage plate to the bottom of the container andinstalling a series of spray nozzles suspended above the top of thesoil layer. Elevation views of the container are shown in Figs.1(a), 1(b), and 1(c). A detailed explanation of infiltration setup isavailable in Ghayoomi et al. (2011). The container was modifiedby incorporating pedestals to support the transmitting and receiv-ing bender elements.

Bender Element Setup and Calibration

Three pairs of bender elements were manufactured for this studyto measure the shear wave velocity at different depths. The benderelements were manufactured from a single layer of T226-A4-303X type piezo-ceramic from PiezoSystems, Inc. having a lengthof 31.75 mm, a width of 12.7 mm, and a thickness of 0.67 mm.After connection of coaxial cables to each side of the piezo-ceramics, several coats of nonconductive polyurethane (M-CoatA) were applied to prevent corrosion or short-circuiting in par-tially saturated soils. A layer of Silver Print paint obtained fromMG Chemicals was then applied to mitigate electrical noise frombeing transmitted to the piezo-ceramics. After connecting ground-ing cables to the Silver Print paint, several additional coats of pol-yurethane were applied to the piezo-ceramic. The completedbender elements were potted with nonconductive epoxy within cy-lindrical steel pipes (having lengths of 38.1 mm and inside diame-ters of 15.24 mm) so that the free vibrating length of the exposedpiezo-ceramic was 17.78 mm.

A picture of the bender elements and support pedestals duringplacement of a sand layer is shown in Fig. 2. Each pair of benderelements were installed on the vertical support pedestals so thatthey would be at depths of 3.51, 7.31, and 11.11 cm from the soilsurface. The bender elements were oriented in a vertical directionso that they would generate SH waves. This is important in centri-fuge testing so that the self-weight of the soil does not provide adownward reaction on the flat side of the bender elements. A 100-mm square flange at the base of the pedestals was affixed to thebase of the container using rubber cement, which helped to absorbany vibrational noise from the centrifuge platform. After aligningthe two support pedestals, the horizontal distances (tip-to-tip)between the bender elements were 7.25, 7.55, and 7.75 cm for thetop, middle, and bottom pairs of benders, respectively.

2 GEOTECHNICAL TESTING JOURNAL


A data acquisition system capable of exciting the transmittingbender elements and receiving signals from the receiver benderelements was developed so that the bender elements could beused during flight in the centrifuge. A schematic of the connec-tions in the data acquisition system is shown in Fig. 3. A NationalInstruments PXI chassis with a NI-8176 DAQ controller was usedin this study, similar to that used by Kim and Kim (2010). An 8-channel PXI-6251 module with sampling rate of 1MS=s was usedfor output signal generation and input data measurement. Themaximum amplitude of the generated signal from the PXI-6251module is 610 V, which is not sufficient to generate a signalwhich can be measured by the receiving bender element above thenoise of the centrifuge. Accordingly, an EPA-104 linear amplifiermanufactured by Piezo Systems, Inc., was used to increase theamplitude of the generated wave to 6200 V. Since the amplifierhas only one input and one output channel, a PXI-2527 switchwas used to guide the generated wave to the desired transmittingbender element. Possible delays due to presence of switch or

cross-talk effects were found to be negligible and less than accu-racy of the data acquisition system (i.e., 10�6 s).

The bender element pairs were calibrated to account for thepossible delay due to measurement system. Specifically, a signalwas sent from the transmitting bender element to the receiving ele-ment when the tips of the two bender elements were in contact.The time delay between the transmitted and received signal wasthen measured to be 5.4� 10�6, 5.7� 10�6, and 4.7� 10�6 s forthe top, middle, and bottom bender element pairs, respectively.This calibration time was subtracted from the measurements madeduring bender element tests in the soil.

Infiltration Setup and Calibration

Similar to Ghayoomi et al. (2011), steady-state infiltration wasused to control the degree of saturation and matric suction in thepartially saturated soil layer during centrifugation. Specifically,during steady-state infiltration toward a water table at the bottom

FIG. 1—Elevation views of the container highlighting locations of bender elements and dielectric sensors: (a) side view; (a) section 1-1 highlighting the location ofthe bender elements; (b) section 2-2 highlighting location of three of the dielectric sensors.

GHAYOOMI AND MCCARTNEY ON SHEAR MODULUS IN A GEOTECHNICAL CENTRIFUGE 3


of a soil layer, the matric suction and degree of saturation areapproximately uniform with height in the soil layer (Zornberg andMcCartney 2010). In order to apply infiltration, water from a pres-surized storage tank on the centrifuge was sprayed uniformly overthe upper surface of the soil layer through a series of six fine-mistspray nozzles. A freely-draining water table was imposed at thebottom of the soil layer using a series of eight drainage ports atthe base of the container. Water overflows from the top of a drain-age layer of gravel having a uniform particle size of 6.35 mm. Athin geotextile filter was placed on top of the gravel layer to pre-vent any loss of overlying soil through the drainage ports. Waterpassing through the drainage ports at the bottom of the specimenwas collected in a reservoir to measure the outflow rate.

A schematic of the infiltration and drainage systems is shownin Fig. 4. The changes in volumetric water content in the soil layerwere measured using five EC-TM dielectric sensors from DecagonDevices of Pullman, WA. The sensors were arrayed as shown inFig. 1(a). These sensors were placed in a horizontal orientation, asfar as possible from the bender elements, at depths of 1.27, 5.08,7.30, 8.89, and 12.7 cm from the surface of the soil layer. Sincethe dielectric sensors are far from the benders and their placementdepths are different the benders’ depths, they do not affect theshear wave travel time. A more detailed explanation of the infiltra-tion setup is available in Ghayoomi et al. (2011).

The dielectric sensors were calibrated for use in Ottawa F75sand using a compaction mold. The volumetric water content was

measured using the dielectric sensors and also calculated using themeasured density and gravimetric water content. By correlatingthe measured and actual values, a unique linear equation wasfound to represent the relationship between the actual volumetricwater content and the value measured using the sensors (i.e.,hactual¼ 0.854�hmeasuredþ 0.036).

Materials

F-75 Ottawa sand was used as the sand in this study, to be consistentwith the seismic compression tests performed by Ghayoomi et al.(2011). A target void ratio of 0.66 was used in all of the tests in thisstudy, which corresponds to a relative density of 45 %. When satu-rated, the sand at this void ratio has a permeability 6� 10�4 cm=s.The geotechnical properties of the F-75 fine silica sand are summar-ized in Table 1 and the grain size distribution obtained from thesieve analysis test is shown in Fig. 5(a). The Soil Water RetentionCurve (SWRC) of F-75 Ottawa sand was measured using a hangingcolumn test with controlled outflow described by McCartney et al.(2008). The SWRC is shown in Fig. 5(b) along with the van Gen-uchten (1980) SWRC model fitted to the experimental data usingleast-squares regression. The fitting parameters for the van Gen-uchten (1980) model are also listed in Fig. 5(a). Even through F-75Ottawa sand has a relatively uniform particle size distribution, it hassufficient fine soil particles to retain water to suctions up to 10 kPa.The combination of a high saturated permeability and the shape ofthe SWRC of this sand facilitate the use of infiltration to control thesuction and degree of saturation during testing. An infiltrationapproach would not be appropriate for soils with lower permeabilitythan the smallest infiltration rate that can be applied with an infiltra-tion system, which is why this study is focused on sand.

The SWRC for the sand measured at 1g was used to infer thesuction values during centrifugation from the volumetric water con-tent measurements from the dielectric sensors. This is possiblebecause Dell’Avanzi et al. (2004) observed that matric suction val-ues in a model and prototype scale 1:1 for the case that the ratio ofthe centrifuge arm to the soil layer height is greater than 10. This ra-tio is equal to 33 for the centrifuge and soil layer evaluated in thisstudy. Further, because capillarity is independent of gravity, it canalso be assumed that the SWRC in a model and prototype scale 1:1.

Procedures

The sand layer was placed at the target void ratio using dry pluvia-tion (Whitman and Lambe 1988) in lifts of 1 cm atop the geotex-tile filter and around the bender element pedestals. A flexibleplastic membrane glued to the bottom of the container was used toseparate the sides of the container from the sand and water so thatthey do not penetrate into the gaps between the laminar containerplates. The final thickness of the sand layer was 15.87 cm. Thedielectric sensors were placed in a horizontal orientation in thesand during pluviation.

After placement into the centrifuge, bender element measure-ments were performed on the dry sand at both 1 and 40 g. Specifi-cally, a 10 V-amplitude sinusoidal pulse wave with an excitationfrequency of 10 kHz was generated and was then passed through

FIG. 2—Picture of the bender element system during placement of the firstsand lift.

FIG. 3—Schematic of the on-board centrifuge data acquisition system configu-ration for the bender elements.



the amplifier to increase the amplitude of the wave to 200 V. Thiswave was applied to the transmitting bender element, and theshear wave passed through the soil to the receiving bender ele-ment. The transmitted signal and a typical received signal areshown in Figs. 6(a) and 6(b), respectively. The travel times weremeasured several times and the values were averaged. Repeatmeasurements were consistently within 1.5 % of each other. Afast-Fourier transform of the received signal for the windowshown in Fig. 6(a) is presented in Fig. 6(c). This figure indicatesthat the main frequency of the received signal is approximately 5kHz corresponding to the first deflection while there is also a sig-nificant frequency at 3 kHz corresponding to the first peak. Theother peaks can be attributed to reflections and centrifuge noise.

In this study, the shear wave travel time was calculated by sub-tracting the first arrival time of the shear wave recorded by re-ceiver benders from the departure time of the transmitted signal.This approach was selected as Leong et al. (2005) observed thatthe travel time defined using the first deflection point provides thebest match with independent shear wave velocity measurementsfrom ultrasonic pulse tests. Ghayoomi (2011) showed that the cal-culated Gmax based on the shear wave velocity measured using thefirst deflection point method arrival time selection is more consist-ent with the estimated Gmax from the available empirical relationsand resonant column test data. Consequently, the first deflectionpoint of the received signal was selected as the shear wave arrivaltime in this study as shown in Fig. 6(b).

Near-field effects affect the shape of the received signal andalso the arrival time measurement, so the tip-to-tip length (Ltt) andsignal wavelength (k) were selected to minimize these effects. Theratio Ltt=k was used as the criterion to evaluate the likelihood ofnear-field effects. Minimum values of Ltt=k reported in the litera-ture to minimize near field effects range from 1.00 (Arulnathanet al. 1998) to 3.33 (Leong et al., 2005). The length Ltt in thisstudy was in the range of 7.25 to 7.75 cm and the measured tip totip travel times ranged from 0.0004 to 0.0008 s. Since Ltt=k¼ f� t(where f is the wave frequency and t is the tip to tip wave traveltime), an excitation frequency of 10 kHz, corresponding to areceived wave with a main frequency of 5 kHz, resulted in wave-lengths corresponding to Ltt=k ratios between 2 to 4. This fre-quency is consistent with the range of 5 to 25 kHz used by Leonget al. (2005). An excitation frequency of 20 kHz was also eval-uated, but because the measured travel times were consistent, thefrequency of 10 kHz was used throughout the testing program.

After the bender element measurements were made on the drysand layer, the centrifuge was stopped. The sand layer was thensaturated from the bottom by applying an upward gradientthrough the soil layer. A de-aired water reservoir outside of thecentrifuge was connected to the end of the drainage line while theoutflow proportional control valve was open. The dielectric sen-sors were used to infer changes in volumetric water content duringthe saturation process. After saturation was completed (i.e., whenwater reached the sand surface and the dielectric sensors indicateda uniform volumetric water content equal to the porosity), the out-flow proportional control valve was closed and the de-aired waterreservoir was disconnected from the container.

After a bender element measurement on the saturated sandlayer at 1 g, the centrifuge was spun up to 40 g. Another benderelement measurement was performed on the saturated sand beforethe outflow proportional control valve was opened fully whilecommencing infiltration. The rate of infiltration was controlledusing an inflow proportional control valve. The volume of waterexiting the pressurized tank was metered using a differential pres-sure transducer (DPT). In this manner, water was sprayed on sur-face of the soil until the measured outflow was equal to theapplied inflow (i.e., steady-state conditions). Further, steady-state

FIG. 4—Schematic of the infiltration system and the soil container.

TABLE 1—Geotechnical properties of F-75 Ottawa sand.

Property Description

Mineralogy Quartz, 99.8 % SiO2

Grain shape Rounded

Specific gravity, Gs 2.65

Cu 1.71

Cc 1.01

emin, emax 0.49, 0.80

qmin, qmax 1469, 1781 kg=m3

Ksat at e¼ 0.66 6�10�4 cm=s



conditions were also defined when the measured volumetric watercontent profile was uniform with depth. Bender element measure-ments were made after reaching steady-state conditions for dis-charge velocities ranging from 10�12 to 10�5 cm=s, whichcorrespond to degrees of saturation ranging from 0.11 to 0.71. Af-ter performing a series of tests at 40 g, the centrifuge accelerationwas increased to 50 g and several additional infiltration tests andbender element measurements were performed.

Results

Infiltration Results and Estimated Effective StressProfiles

Profiles of degree of saturation with height in the sand layer atsteady-state conditions under different infiltration rates are shownin Fig. 7(a). These profiles were calculated from the volumetric

water content values measured from the dielectric sensors. Thedegree of saturation values were also converted to matric suctionprofiles using the SWRC in Fig. 5(b), and are shown in Fig. 7(b).In all of the situations, a nearly uniform degree of saturation wasobtained for the different infiltration rates. The location of thebender elements are noted in Figs. 7(a) and 7(b) for reference.

The profiles of degree of saturation and matric suction obtainedfrom solutions to Richards’ equation in the centrifuge presentedby Dell’Avanzi et al. (2004) are also included in Figs. 7(a) and7(b) as solid lines. In general, the dielectric sensor measurementsmatch well with the theoretical predictions, confirming thatsteady-state infiltration leads to nearly uniform distributions indegree of saturation with height in soil layers in the centrifuge.Although suction is constant with depth during steady-state infil-tration, the effective stress is not constant with depth because thetotal stress increases with depth in proportion to the total unitweight of the sand.

FIG. 5—Properties of Ottawa F-75 sand: (a) Grain size distribution; (b) SWRC.

FIG. 6—Typical signals from the bender elements: (a) generated shear wave; (b) received shear wave; (c) received signal frequency domain response spectrum.



Profiles of effective stress in the sand layer were estimatedfrom the known total stress and matric suction profiles using theapproach proposed by Lu et al. (2010). They found that the effec-tive saturation (the degree of saturation weighted between full sat-uration and residual saturation) was an appropriate estimate forthe value of v in Bishop’s equation for effective stress of partiallysaturated soils (Bishop 1959). They incorporated the equation forthe van Genuchten (1980) SWRC model into Bishop’s equationto predict the relationship between the vertical effective stress r0

and matric suction w, as follows:

r0 ¼ r� uað Þ þ w

1þ aw½ �N� � N�1ð Þ=N

(3)

where a and N are parameters for the van Genuchten (1980) SWRCmodel, r is the total stress, and ua is the pore air pressure (assumedto be zero). Profiles of vertical effective stress estimated using Eq.(3), incorporating the total stress induced by centrifugation and theinferred matric suction profile from Fig. 7(b), are shown in Fig.7(c). The values of effective stress for the different locations of thebender elements are noted in this figure for reference. This figureindicates that the total stress induced by the self-weight of the sandlayer is the primary variable in the definition of the effective stress,but changes in suction due to varying infiltration rates will lead toshifts in the effective stress profile with depth.

Measurements of Gmax for Partially Saturated Sands

Pulse wave tests were performed for each pair of bender elementsindependently after reaching steady-state conditions under a given

infiltration rate. The measured shear wave velocities (Vs) fromeach pair of bender elements are shown as a function of degree ofsaturation in Fig. 8(a). In addition, the relationship between totaldensity and degree of saturation is shown in this figure. The meas-urements of Vs show a nonlinear trend with degree of saturationwhile the total density increases linearly. The total density wasassumed only to depend on changes in the degree of saturation asGhayoomi et al. (2011) observed negligible changes in height ofsimilar sand layers during spin-up and infiltration as part of a com-prehensive series of seismic compression tests. This figure alsoshows data for the saturated soils (Sr¼ 1), although it should benoted that these data points are representative of hydrostatic con-ditions while the others are for steady-state infiltration. The posi-tive pore water pressure at the depths of the bender elements leadto a lower effective stress and lower Gmax value than for the testsunder steady-state infiltration. The Vs measurements and total den-sities are also shown as a function of inferred matric suction Fig.8(b). A nonlinear trend between total density and suction wasobtained because the suction was inferred from the SWRC. Thedata points from the saturated tests are not included in this figure.

The values of Gmax were calculated from the measured valuesof Vs and total density using Eq. (1). The Gmax values calculatedfor different degrees of saturation at 40 g are shown in Fig. 8(c)for the bender elements at depths of 3.51 and 11.11 cm. Signalswere detected from the receiver bender element a depth of 7.31cm, but the signals were not sufficiently greater than the noiselevel in the centrifuge. This was attributed to a possible short-cir-cuit in the bender element electrical connections after it was sub-merged in water. Accordingly, the results from a depth of 7.31 cm

FIG. 7—Typical infiltration results from tests performed at 40 g along with theoretical predictions: (a) degree of saturation; (b) matric suction; (c) effective stressprofiles. (Note: Lm is the thickness of the specimen, zm is the height from the bottom of the container, Nr,mid is g-level at mid-height of the container, Ks is the hy-draulic conductivity of the saturated sand, and vm is the discharge velocity.)



are not reported. When comparing the results from the bender ele-ments at the two depths, the bender element deeper in the soil pro-file not only showed a greater magnitude of Gmax due to thegreater total stress, but also showed both a greater variation inGmax with the degree of saturation. Although the peak value ofGmax for the two depths occurred at different values of degree ofsaturation, the peak values occurred between degrees of saturationof 0.3 and 0.4 (suction values of 4.5 to 5.5 kPa).

The calculated values of Gmax as a function of matric suction areshown in Fig. 8(d). The predicted trend between Gmax and matricsuction obtained by combining Eqs. (2) and (3) are also shown inthis figure for comparison. The parameters A and n were definedusing values from Hardin and Richart (1963) reinterpreted in termsof metric units [A¼ 69.77, n¼ 0.5, and F(e)¼ (2.17� e)2=(1þ e),where e¼ 0.66]. Although there is some scatter in the results espe-cially at lower suction values, the trends and magnitudes of the dataand the predicted relationship match reasonably.

Analysis

Comparison of Bender Element and ResonantColumn Results

Khosravi et al. (2010) used a fixed-free Stokoe-type resonant col-umn apparatus with suction control using the hanging columntechnique (McCartney et al. 2008) to measure the value of Gmax

for the same sand used in this study, albeit at a relative density of50 %. Although the strain level in the resonant column test wasnot measured directly, a parametric evaluation indicated that the

value of Gmax was constant for the lowest strain values. The strainlevels in resonant column test are likely larger than those in thebender element test, but both are within the small strain region(less than 10�4). A comparison between the measured Gmax val-ues from the bender element (BE) and resonant column (RC) fortests for different degrees of saturation is shown in Fig. 9(a),while the same Gmax data is plotted as a function of the matricsuction (inferred from the SWRC of the sand) in Fig. 9(b).Because of the different densities in the BE and RC tests, theGmax values were normalized using the void ratio function definedby Hardin (1978) [F(e)¼ 1=(0.3þ 0.7e2)]. Although there is aslight difference in the trends between the two sets of data, themagnitudes are consistent for the measurements from the upperbender element pair and those from the resonant column tests athigher total stress values. The difference in trends between Gmax

and degree of saturation obtained from the two tests can be attrib-uted to the slight differences in relative density and net normalstresses, as well as experimental errors in the outflow measure-ment system for the hanging column test incorporated into the res-onant column test used by Khosravi et al. (2010). Despite thesedifferences, this comparison indicates that steady-state infiltrationleads to similar stress state conditions as those induced in partiallysaturated sands in element scale tests such as the resonantcolumn.

Relationships between Gmax and Mean EffectiveStress

The mean effective stress with depth in the sand layer was calcu-lated by multiplying the vertical effective stress by (1þ 2K0)=3,

FIG. 8—Bender element results from tests at 40 g: (a) shear wave velocity and total density as a function of degree of saturation; (b) shear wave velocity and totaldensity as a function of matric suction; (c) relationship between Gmax and degree of saturation at 40 g; (d) relationship between Gmax and matric suction 40 g alongwith predicted trends.



where K0 is the coefficient of earth pressure at rest. The value ofK0 was estimated using a friction angle / of 35� for the sand cor-responding to a relative density of 45 %. The variation in the esti-mated values of Gmax for dry sand with effective stress measuredusing the bender elements is shown in Fig. 10(a). Results areshown for bender element tests performed during centrifugation at40 g as well as under 1 g. To check the consistency in the trend ofthe data for dry sands with mean effective stress, the data wascompared with the predicted Gmax values from the empirical equa-tions of Hardin and Richart (1963), Seed and Idriss (1970), andHardin and Drnevich (1972). The values of A and n for thesesands were taken directly from their papers, so the predictionswere not expected to provide an exact fit. Nonetheless, the valuesof Gmax for dry sand measured by the bender elements are consist-ent with the trends expected from these relationships.

The results for dry, saturated and partially saturated Ottawasand are plotted together as a function mean effective stress inFig. 10(b). This figure contains tests performed at 1 (saturated anddry tests), 40, and 50 g. The mean effective stress for the partiallysaturated sands was calculated in a similar manner as above, butusing Eq. (3) to define the vertical effective stress. Although there issome scatter in the trend of Gmax with effective stress for the par-tially saturated sands, the data tends to follow a unique relationshipfor dry, saturated, and partially saturated sands. Specifically, a singlepower function may be used to predict the value of Gmax with mean

effective stress for Ottawa sand. The data points fall within a toler-ance of 610 % around the power function. Because the Gmax valuesfor partially saturated sands follow the same relationship with meaneffective stress as dry and saturated sands, this observation confirmsthat steady-state infiltration is an appropriate technique to controlthe stress state in partially saturated sand layers.

Conclusions

This paper describes the details behind a testing program involv-ing the use of bender elements to measure changes in small strainshear modulus, Gmax, of sand layers during variations in degree ofsaturation induced by steady-state infiltration in a geotechnicalcentrifuge. Consistent with results from resonant column tests per-formed on the same sand with suction control using the hangingcolumn approach, Gmax varied nonlinearly with degree of satura-tion and showed a peak value at a degree of saturation between0.3 and 0.4. The measured values of Gmax of partially saturatedsand layers follow the same trend with mean effective stress whenthe vertical effective stress is defined from the suction and degreeof saturation profiles during steady-state infiltration. Accordingly,these observations indicate that steady-state infiltration is an effec-tive tool to control the effective stress in partially saturated sandsin centrifuge physical modeling.

FIG. 9—Values of Gmax normalized with respect to the void ratio function obtained from bender element (BE) and resonant column (RC) tests at similar net normalstress values, plotted as a function of: (a) degree of saturation; (b) matric suction.

FIG. 10—Relationships between Gmax and mean effective stress: (a) comparison of Gmax values for dry sand with empirical relationships from the literature; (b)general relationship and bounds for dry, saturated, and partially saturated Ottawa sand.



Acknowledgments

The writers would like to thank Min Jae Jung and Dr. Kenneth H.Stokoe, II for their assistance in manufacturing the bender ele-ments used in this study, and Kent Polkinghorne for his assistancewith the bender element data acquisition system.

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