Closure to “performance of a transparent Flexible Shear Beamcontainer for geotechnical centrifuge modelling of dynamic problems”by Ghayoomi M., Dashti S., and McCartney J.S.

M. Ghayoomi a,n, S. Dashti b, J.S. McCartney b

a University of New Hampshire, USAb University of Colorado Boulder, USA

a r t i c l e i n f o

Article history:Received 24 July 2012Received in revised form1 July 2013Accepted 12 July 2013Available online 13 March 2014

Keywords:Flexible Shear Beam containerCentrifuge modelingPhysical modelingBoundary effectsSeismic soil–structure-interaction

a b s t r a c t

In response to the discussion, this closure presents a new set of analyses to confirm the satisfactoryperformance of the recently-developed transparent Flexible Shear Beam (FSB) container and itslimitations. The lateral deformations of the box estimated using Finite Element analyses and measuredduring centrifuge experiments were compared. The maximum deformation to height ratio wassufficiently small to retain at-rest lateral earth pressures for loose to medium-dense cohesionless soilsand all cohesive materials. In addition, higher frequency modes of vibration were estimated for thecontainer, and were found to occur where the earthquake energy is less significant. Further, the box isexpected to approximately replicate free field conditions under 1-D horizontal shaking for the range ofsoil properties under investigation. Overall, the deformation and vibration analyses indicate that theselection of appropriate rubber material properties and boundary conditions are critical when analyzingthe performance of the container.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

The authors thank Professors Haigh and Madabhushi [1] fortheir interest and constructive feedback on the paper related tothe performance of a recently developed transparent FlexibleShear Beam (FSB)-type container used for dynamic centrifugemodelling [2]. This container permits evaluation of the seismicresponse of soil under softened conditions, easier saturation of soilprofiles than laminar containers, and implementation of systemsfor control of infiltration and drainage processes in partiallysaturated soil layers. Further, this new container permits the useof Particle Image Velocimetry [3] and other types of digital imageanalysis to study the dynamic response of soil layers and buriedstructures while reducing boundary effects. However, there arecertain trade-offs that must be considered in its use. Our closureaddresses the three concerns raised by the discussers, namely,(1) the lateral stiffness of the container and the associated effectsof lateral static deflections on the behavior of the soil specimen;(2) the effects of higher modes of vibration on the simulation of1-D horizontal shaking in soil; and (3) the free-field soil responseunder 1-D shaking as predicted by equivalent-linear site responseanalyses and as measured by the container.

2. Lateral stiffness of container walls

The outside dimensions of the container were limited by thesize of the shaker platform and its payload capacity. The centrifugeshake table performance in terms of the amplitude and frequencycontents of the achieved motions is directly related to the payloadmass [4]. The thickness of acrylic frames of 38.1 mm was selectedto balance the need to maximize the inside dimensions of thecontainer (in order to minimize boundary effects) and minimizetotal container mass, while maintaining an acceptable lateralstiffness. The inside dimensions of the container were selectedto be consistent with previous containers used by Dewoolkar et al.[5] and Ghayoomi et al. [6] in dynamic centrifuge experiments atthe same facility. Although the selected frame thickness balancesthe geometric restraints and payload mass limitations, there ispotential for larger lateral deflections compared to those inprevious containers built with thicker acrylic side walls. A lowerlateral stiffness leads to greater lateral deflections during spin-upof a soil specimen, potentially leading to changes in the soil's staticstress state. Although the discussers predict that the lateraldeflections of the new FSB container may be high enough to causea reduction in earth pressures, measurements of lateral frameddeflections under static conditions do not agree with their pre-dictions. Possible reasons are discussed in this section.

The static lateral deflections of the container were measured inthe middle of the long-span of all container frames when it was

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/soildyn

Soil Dynamics and Earthquake Engineering

http://dx.doi.org/10.1016/j.soildyn.2014.02.0020267-7261 & 2014 Elsevier Ltd. All rights reserved.

n Corresponding author at. Tel.: þ1 6038623997.E-mail address: majid.ghayoomi@unh.edu (M. Ghayoomi).

Soil Dynamics and Earthquake Engineering 67 (2014) 363–369

filled with medium-dense, dry sand under different centrifugeaccelerations. The lateral deflection of the top container frameduring spin up and the maximum lateral deflection profile of allcontainer frames at different g-levels is shown in Fig. 1a and b. Anincrease in the maximum lateral deflection of the container from0.36 to 0.58 mm as it was spun from 30 to 80g is observed in thedata in Fig. 1a. The container showed an increase in lateral stiffnessas the g-level increased (i.e., a smaller increase in lateral deforma-tions at higher g-levels) even though lateral earth pressuresincreased under higher centrifuge accelerations. This is proposedto be due to the stress-dependent stiffness of the rubber layers inbetween frames.

The maximum lateral deflection of 0.58 mm (a prototypelateral deflection of 46.4 mm) observed in Fig. 1a at 80g corre-sponds to a maximum deflection to height ratio (δ/H) of 0.0017 fora model-scale soil layer having a height of 338 mm. From the chartprovided by Clayton and Milititsky [7] this deflection to heightratio is sufficient to lead to active earth pressure conditions, as it ishigher than 0.001. However, the δ/H limit of 0.001 for activeconditions is recommended for dense, cohesionless soils, accord-ing to the Canadian Foundation Engineering Manual [8]. Therequired δ/H ratios to mobilize active pressure for different soilconditions are summarized in Table 1. With the goal of betterunderstanding the limitations of the new FSB container in mind,this container will be most suitable for loose to medium cohesion-less soils and all types of cohesive soils when spinning to g-levelsas high as 80–100g.

A simple beam or plate assumption is not appropriate topredict the lateral deflection of the container. For example, eachframe is supported on both sides (top and bottom) by otherframes, which differentiate its behavior from a simple beamresponse with fixed ends. Further, the plate theory approach isnot appropriate because the lateral earth pressures vary withdepth along the container wall, while the same average stress isimposed in the plate theory throughout the height of the plate.

Although a more complex Finite Element (FE) Abaqus analysiscan provide a more accurate response, the input material proper-ties play a critical role in the predicted lateral deflections, whichappeared to differ between the authors and the discussers. Forexample, the predicted lateral deflection of the system is particu-larly sensitive to the stiffness of the rubber, which is stress-

dependent, as well as Poisson's ratio of the rubber [9]. In addition,the impact of the aluminum frame on top of the container was notconsidered in the analysis performed by the discussers. Thesensitivity of the numerical predictions of lateral static deflectionsby Abaqus to these input parameters is presented below. Relativelysmall lateral deflections on the side of the container and negligiblesoil settlements that were measured during spin-up validate thenumerical simulations performed by the authors.

As a starting point, a container model with the aluminum andacrylic material properties presented in Table 1 of the originalmanuscript was created. Young's modulus (Er) of 4 MPa andPoisson's ratio (υr) of 0.49 were used for the rubber in the analysis(Table 1 in the paper). This choice of elastic parameters assumesthat the stiffness of the rubber is not stress-dependent. The upperbound of the rubber modulus was picked because the rubberstiffness was expected to significantly change the response. Ahydrostatically-distributed lateral earth pressure was applied tothe inside of the container throughout the depth in the analysis.The pressure was calculated based on the sand layer withρ¼1565.8 kg/m3, N¼80g and K0¼0.46 (ϕ¼ 331). The exaggerateddeformed shape of the container is shown in Fig. 2. The predictedmaximum lateral deflection is approximately 4.4 mm, which isclose to that calculated by the discussers.

The soft neoprene used in this container has a stress-dependent Young's modulus. The response of this rubber undercompression was tested during loading and unloading, reported byGhayoomi et al. [9] and repeated in Fig. 3. The stress–strain curveclearly indicates that rubber has a highly nonlinear response. Thecurve shown in Fig. 3 is in accordance with other reported stress–strain curves for rubber [10]. The reported modulus of 2–4 MPa inTable 1 of the original manuscript is associated with a normalstress range of 50–200 kPa, as mentioned in the footnote of thattable. However, considering the weight of the frames at 80g, theapplied normal stress on rubber exceeds this range by a largeextent. The normal stress is estimated to vary from about 560 kPaon the top rubber layer to about 2300 kPa on the bottom layer at80g. The equivalent linear modulus can be approximately esti-mated by taking the slope of the stress–strain curve in Fig. 3, inwhich case the modulus increases from 13 to 52 MPa from the topto bottom rubber layers. The modulus associated with pressuresoutside the range in Fig. 3 was approximated using interpolationof the observed trend.

A new analysis was performed in Abaqus with updated stress-dependent Young's modulus of the rubber. The deformed shape ofthe container is shown in Fig. 4. The maximum lateral deflection ofthe container was approximately 2.3 mm, which is nearly half thevalue of 4.4 mm observed in Fig. 2. This indicates the importancein considering the pressure dependent properties of the rubber.

The assumed value of Poisson's ratio will have an importanteffect on rubber's behavior in shear, as the shear modulus isestimated from G¼ E=2ð1þϑÞ . There is some uncertainty in the

0

20

40

60

80

100

0 0.2 0.4 0.6Cen

trifu

ge A

ccel

erat

ion

(g)

Maximum Lateral Deflection(mm)

0

100

200

300

0 0.2 0.4 0.6

Con

tain

er H

eigh

t (m

m)

Lateral Deflection of Frames(mm)

30g60g80g

Fig. 1. Model-scale frame deflections during spin-up of a dry, medium-dense sand layer to 80g in the centrifuge: (a) top frame and (b) all frames.

Table 1Magnitude of wall deflection to height ratios required toreach active conditions for different soil types [2].

Soil type and condition δ/H for active condition

Dense cohesionless 0.001Loose cohesionless 0.004Stiff cohesive 0.01Soft cohesive 0.02

M. Ghayoomi et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 363–369364

actual value of rubber's Poisson's ratio, although it is typicallyassumed to be close to 0.5 [11] (i.e., rubber is assumed to be anincompressible material). To better understand the effect ofPoisson's ratio, a similar analysis with stress-dependent rubbermodulus was performed for different υr assumptions. The max-imum lateral deflection of the top container frame was calculatedto range from 2.0 to 1.9 mm for Poisson's ratios ranging from 0.495to 0.499 (Fig. 5). Therefore, a small change in the assumed value ofPoisson's ratio has a visible influence on lateral displacements.

In addition to the sensitivity of the computed lateral deflectionsto the elastic properties of the rubber layers, an importantcomponent not considered by the discussers is the aluminumframe rigidly bonded to the top acrylic frame. The aluminumframe provided support for the instrumentation rack and simulta-neously added to the lateral stiffness of the system (shown inFig. 1b). In the next analysis, the container was simulated with thetop frame included to demonstrate its influence. Young's modulusof the rubber was also updated because of the slight increase inthe vertical stress on each rubber layer due to the additionalweight of the top aluminum frame previously not considered. Thedeformed shape of the container shown in Fig. 6 indicates amaximum lateral deflection of approximately 0.85 mm. This valueis closer to the measured lateral deflections in Fig. 1.

As mentioned above, an instrumentation rack is typically usedto support the sensors mounted on the soil deposit and structures(e.g. vertical and horizontal LVDTs). Although these racks were not

designed to provide lateral stiffness, they restrain the top alumi-num frame from expanding laterally. In order to estimate thepotential effects of the instrumentation rack, the top aluminumframe was restrained from lateral deformations in the widthdirection (y-direction). The result of the Abaqus analysis is shownin Fig. 7, and a maximum lateral deflection of 0.47 mm is predictedin the second frame from top of the container.

The analyses presented above show the importance of carefullyconsidering the material parameters of the rubber layers as well asthe effect of the top aluminum plate on the estimated lateraldeflections of the container. The maximum calculated deflection(�0.85 mm) is now closer to the maximum measured deflection(�0.58), and is an order of magnitude lower than the valuecalculated by the discussers. It should be mentioned that themeasured 0.58 mm lateral defection was observed without thepresence of an instrumentation rack, which will serve to furtherdecrease the actual lateral deflections.

3. Higher modes of vibration

The higher modes of vibration noted by the discussers in theirAbaqus analysis may have an effect on the assumption that thecontainer is only evaluating the effects of 1D horizontal shaking onthe soil and structure. However, the authors believe that theseeffects will be small in comparison with the major modes in theshaking direction. The container was analyzed: 1) with constantrubber modulus value Er¼4 MPa, υr¼0.49, and without the topaluminum frame (similar to the assumptions of the discussers,except for the constant Young's modulus of rubber being 4 MPainstead of 2 MPa); and 2) with stress-dependent rubber modulus,υr¼0.49, and the top aluminum frame included. The estimatedvibration modes changed considerably in the two sets of analyses,as shown in Table 2. These higher frequency modes are expectedto exceed 2 Hz in the prototype scale at 80g, beyond which theearthquake energy becomes less significant. The estimated modeshapes for the 4th and 5th vibration modes are shown in Fig. 8.Their high modal frequencies presented in Table 2 imply that thevibration modes governed by the acrylic ring stiffness will nothave a significant adverse effect on 1-D motion simulation. Inaddition, considering a stress-dependent modulus for the rubberalso affects the 1st mode horizontal natural frequency of the

Fig. 2. Deformations of FSB container under geostatic conditions at 80g using the parameters reported in the original manuscript.

0

200

400

600

800

1000

0

Nor

mal

Str

ess

(kPa

)

Axial Strain

0.05 0.1 0.15 0.2 0.25

Fig. 3. Response of soft Neoprene rubber (30A Durometer) under axial compres-sion [9].

M. Ghayoomi et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 363–369 365

Fig. 5. Deformation of the FSB container under geostatic conditions with stress-dependent Er and (a) υr¼0.495 and (b) υr¼0.499.

Fig. 6. Deformation of the FSB container including the aluminum frame under geostatic conditions with stress-dependent modulus.

Fig. 4. Deformation of FSB container under geostatic conditions at 80g with stress-dependent elastic properties for the rubber layers with υr¼0.49.

M. Ghayoomi et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 363–369366

container. It is critical to evaluate the performance of the containerwith regards to lateral stiffness on a project-specific basis, basedon the chosen input motion characteristics, the spin accelerationlevel of interest, and soil properties.

4. Free-field soil response

Although Eq. 6 in the original manuscript is not explicitlyprovided by Hardin and Richart [12], the expression for Gmax

was converted from an expression for shear wave velocity in thatpaper. Specifically, Hardin and Richart [12] developed an empiricalrelationship for the shear wave velocity of round-grained sand atstresses greater than 2000 psf (96 kPa) based on the results ofresonant column tests, as follows:

cs ¼ ð170�78:2 eÞffiffiffiffiffiffiffis0m

4p

ð1Þ

where cs is the shear wave velocity in [ft/s], e is the void ratio, ands0

m is the mean effective stress (or confining pressure). The small-strain shear modulus, Gmax, is related to the shear wave velocity asfollows:

Gmax ¼ ρc2s ð2Þ

where ρ is the total density of the soil. By combining Eqs. (1) and(2), and by considering ρ¼ ðGs=ð1þeÞÞρw for dry sand with arepresentative specific gravity (i.e., Gs¼2.65) and water densityof 1.94 slugs/ft3, an expression may be defined for the value ofGmax for round-grained sand at stresses greater than 2000 psf as

follows [13,14]:

Gmax ¼ ð32:17�14:8eÞ21þe

ffiffiffiffiffiffiffis0m

pð3Þ

where s0m is in psf and Gmax in psi. This expression is the same as

Eq. (1) presented in the discussion. The modulus and meaneffective stress values from this equation may be converted toother units. For example, Das [15] and Pak et al. [16] presented thisequation in other units, as follows:

Gmax ¼ 2630ð2:17�eÞ2

1þe

ffiffiffiffiffiffiffis0m

pð4Þ

Gmax ¼ 700ð2:17�eÞ2

1þe

ffiffiffiffiffiffiffis0m

pð5Þ

where both s0m and Gmax are in psi or kg/cm3. Eq. (5) is identical to

Eq. (6) in the manuscript. Using the Gmax values predicted inEq. (5), the depth-dependent, small-strain, shear modulus of a drysand layer with a void ratio of 0.69 may be calculated as follows:

Gmax ¼ 2:8 � 107 ffiffiffiz

p ð6Þwhere z is the depth in meters and Gmax is in Pa. This expression isthe same as that provided in the discussion and in the originalmanuscript, confirming that our equation for Gmax in Eq. (5) isaccurate.

Using Eq. (6) to estimate the small-strain shear modulus ofNevada sand, a 1-D equivalent-linear site response analysis wasperformed using DEEPSOIL [17], considering all the model pre-paration details explained in the original manuscript. The com-puted and measured lateral deformation profiles in the shaking

Table 2Higher mode natural frequencies of the FSB container in the diagonal and lateral extension of the rings.

Container natural frequencies with Er¼4 MPa, υr¼0.49, andneglecting the top aluminum frame (Hz)

Container natural frequencies with stress-dependent Er,υr¼0.499, and including the top aluminum frame (Hz)

Mode

Model scale Prototype scale (at 80g) Model scale Prototype scale (at 80g)

61.1 0.76 132.6 1.66 1: Shear mode in shaking direction70.1 0.88 143.3 1.79 2: Shear mode in transverse direction74.5 0.93 159.5 1.99 3: Torsional mode92.3 1.15 188.1 2.35 4: Diagonal extension97.5 1.21 186.7 2.33 5: Lateral extension

Fig. 7. Deformation of the FSB container including the restrained aluminum frame under geostatic conditions with stress-dependent modulus.

M. Ghayoomi et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 363–369 367

direction are shown in Fig. 9, which are the same as thosepresented in the original manuscript. In addition, the sameDEEPSOIL analysis was repeated with a Gmax profile obtained fromSeed and Idriss [18] (Eq. (5) in the original paper). The depth-dependent modulus was in the form of Gmax ¼ 3:6 � 107 ffiffiffi

zp

, asindicated in the paper. A lower lateral deformation was obtained inthe latter case, because of the higher shear modulus valuesestimated using the Seed and Idriss [18] equation. Both shapesand ranges of deflection are in accordance with the measuredcontainer deflections.

The input soil properties are expected to play a significant rolein a site response analysis. For example, the selection of anappropriate small-strain shear modulus (Gmax) profile with depth,the choice of shear modulus reduction curves (G/Gmax vs. shear

strain) with depth due to the change in vertical effective stresses,and updating the shear modulus reduction curves to accommo-date a constant friction angle and shear strength in the same soilprofile [19] may result in the different predicted deformationprofiles by the authors and discussers. The details of shearmodulus reduction curves used by the discussers are not knownto the authors for proper comparison.

To better evaluate the performance of the container undergoinga range of input motions, the maximum lateral deformations ofthe container frames during the application of the Izmit motion(Test FF-IZM60) at a spin acceleration of 60g are shown in Fig. 10.Eq. (6) was used to estimate the small-strain shear modulusprofile, and other soil model parameters were estimated in asimilar manner as in previous analyses. The DEEPSOIL analysispredicted an approximately similar deformation profile to themeasured one. In this case, the estimated profile of maximumlateral deformation shows a decrease in change near the top, asindicated by the discussers. This shows the sensitivity of themagnitude and shape of lateral deformations on the container(estimated or measured) to the properties of the base motion andspin-acceleration, as expected.

Fig. 11 compares the measured lateral deformations of the newFSB container at the University Colorado with those reported forprevious ESB or laminar containers. The comparison shows a con-sistent trend between the measured lateral deformations in TestFF-LND77 [2] of the original manuscript and those reported byprevious researchers [20–22]. The lateral deformation profile shownby Thevanayagam et al. [22] was obtained at 1g on a large laminar

0 50 100 150 2000

5

10

15

20

25

Peak Lateral Deformation (mm)

Hei

ght f

rom

the

Bas

e (m

)

ExperimentDeepsoilGmax H&B

DeepsoilGmax S&I

Fig. 9. Maximum lateral deformation profiles in Test FF-LND77: experimentalmeasurements vs. DEEPSOIL estimations.

0 5 10 15 200

5

10

15

20

Peak Lateral Deformation (mm)

Hei

ght f

rom

the

Bas

e (m

)

ExperimentDeepsoil

Fig. 10. Maximum lateral deformation profiles in Test FF-IZM60: experimentalmeasurements vs. DEEPSOIL estimations.

0.00.10.20.30.40.50.60.70.80.91.0

0 100 200 300 400

Nor

mal

ized

hei

ght f

rom

bas

e

Peak lateral deformation (mm)

Ghayoomi et al. (2013)Zeng and Schofield (1996)Hushmand et al. (1988)Thevanayagam et al. (2009)

a = 0.3 g

a =0.28 gdry soil

a ~ 0.3 gdry soil

a ~ 0.05 gsaturated soil

Fig. 11. Maximum lateral deflection profiles in various containers used for simula-tion of earthquake engineering problems.

Fig. 8. The 4th and 5th vibration modes of the FSB container (Abaqus simulation).

M. Ghayoomi et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 363–369368

container. Since the test was performed on a saturated soil specimen,a much higher lateral deformation was observed due to lateralspreading. Zeng and Schofield [21] used an Equivalent Shear Beam(ESB) container to model medium dense sand. Because a 10-m thickprototype soil specimen was simulated in their test, a smaller degreeof lateral deformation was observed compared to those presented bythe authors. Further, an ESB container represents the initial lateralstiffness of a soil column prior to softening, and is therefore expectedto undergo smaller lateral deformations under dynamic loadingcompared to an FSB container. Hushmand et al. [20] tested a 16.5-m thick, loose sand specimen (26% relative density) in a laminarcontainer. Their measured maximum lateral deformation profilefollows the trend presented by the authors closely. For propercomparison, the maximum acceleration (PGA) measured at the baseof the container in each case is noted in Fig. 11.

5. Concluding remarks

The discussers have initiated a valuable discussion on impor-tant aspects of container characteristics when used in dynamiccentrifuge modelling. Despite their concern regarding the magni-tude of possible lateral geostatic deflections of the containerframes, the experimental measurements performed during testingof a free-field medium-dense sand layer at different levels ofcentrifugal acceleration up to 80g indicates an acceptable perfor-mance when modelling loose to medium-dense, cohesionlessmaterials and all types of cohesive soils. Consideration of the roleof the upper aluminum frame and the instrumentation rack, aswell as the stress-dependent modulus of the rubber layers haslikely led to lower predicted lateral geostatic deflections innumerical simulations of the container by the authors comparedto the discussers. In addition, the vibration modes associated withframe stiffness are not expected to fall into the high energyfrequency range of interest for most earthquake records. Theauthors believe that the new container mostly achieves itsobjectives in permitting visualization of buried structures withminimized static and dynamic lateral deflections that are consis-tent with those from other containers reported in the literature.

References

[1] Haigh S, Madabhushi G. Discussion of “performance of a transparent flexibleshear beam container for geotechnical centrifuge modelling of dynamicproblems by Ghayoomi, Dashti & McCartney. Soil Dyn Earthquake Eng 2013.

[2] Ghayoomi M, Dashti S, McCartney JS. Performance of a transparent flexibleshear beam container for geotechnical centrifuge modeling of dynamicproblems. Soil Dyn Earthquake Eng 2013;53:230–9.

[3] White DJ, Take WA, Bolton MD. Soil deformation measurement using particleimage velocimetry (PIV) and photogrammetry. Géotechnique 2003;53(7):619–31.

[4] Ketcham SA, Ko HY, Sture S. An electrohydraulic simulator for centrifugetesting. Centrifuge 1988;88:97–102 (Rotterdam).

[5] Dewoolkar MM, Ko H-Y, Pak RYS. Seismic behavior of cantilever retainingwalls with liquefiable backfills. J Geotech Geoenviron Eng 2001;127(5):424–35.

[6] Ghayoomi M, McCartney JS, Ko H-Y. Centrifuge test for seismic compression ofpartially saturated sands. ASTM Geotech Test J 2001;34(4):1–11.

[7] Clayton CRI, Milititsky J. Earth pressure and earth retaining structures.London: Surrey University Press; 1986.

[8] Canadian foundation engineering manual. 3rd ed. Canadian GeotechnicalSociety; 1992.

[9] Ghayoomi, M, Dashti, S, and McCartney, JS. Design and construction of atransparent flexible-shear-beam container for dynamic geotechnical centri-fuge testing. In: Proceedings of the second international conference onperformance-based design in earthquake geotechnical engineering. Taormina,Italy; 2012.

[10] Chaikumpollert O, Yamamoto Y, Suchiva K, Kawahara S. Mechanial propertiesand cross-linking structure of cross-linked rubber. Polym J 2012;44:772–7.

[11] Swallowe GM. Mechanical properties and testing of polymers. Netherlands:Kluwer Academic Publishers; 1999.

[12] Hardin BO, Richart FE. Elastic wave velocities in granular soils. ASCE J SoilMech Found Div 1963;89(SM1):33–66.

[13] Hardin BO, Black WL. Sand stiffness under various triaxial stresses. ASCE J SoilMech Found Div 1966;92(SM2):27–42.

[14] Hardin BO, Black WL. Vibration modulus of normally consolidated clay. ASCE JSoil Mech Found Div 1968;94(SM2):353–69.

[15] Das BM. Principles of soil dynamics. Boston, MA, USA: PWS-KENT PublishingCompany; 1992.

[16] Pak RYS, Ashlock JC, Kurahashi S, Abedzadeh F. Parametric Gmax sounding ofgranular soils by vibration methods. Géotechnique 2008;58(7):571–80.

[17] Hashash, Y. Deepsoil V5.0, UIUC, Users' Manual; 2012.[18] Seed HB, Idriss IM. Soil moduli and damping factors for dynamic response

analyses. Earthquake Engineering Research Center. Berkeley, California: Uni-versity of California; 1970 (Report No. EERC 70-10).

[19] Hashash, YMA, Phillips, C, Groholski, DR. Recent advances in nonlinear siteresponse analysis. In: Proceedings of the 5th international conference onrecent advances in geotechnical earthquake engineering and soil dynamics.San Diego, California; May 24–29, 2010. p. 1–22.

[20] Hushmand B, Scott RF, Crouse CB. Centrifuge liquefaction tests in a laminarbox. Géotechnique 1988;38(2):253–62.

[21] Zeng X, Schofield AN. Design and performance of an equivalent-shear-beamcontainer for earthquake centrifuge modelling. Géotechnique 1996;46(1):83–102.

[22] Thevanayagam S, Kanagalingam T, Reinhorn A, Tharmendhira R, Dobry R,Pitman M, et al. Laminar box system for 1-g physical modeling of liquefactionand lateral spreading. Geotech Test J 2009;32(5):1–12.

M. Ghayoomi et al. / Soil Dynamics and Earthquake Engineering 67 (2014) 363–369 369