Dense Granular Columns in Liquefiable Ground. I: ShearReinforcement and Cyclic Stress Ratio Reduction

Deepak Rayamajhi, A.M.ASCE1; Scott A. Ashford, M.ASCE2;Ross W. Boulanger, F.ASCE3; and Ahmed Elgamal, M.ASCE4

Abstract: The installation of dense granular columns by various construction techniques can be used to mitigate liquefaction through acombination of densification, increase of lateral stresses, reinforcement, and drainage. The contributing mechanism of shear reinforcement isisolated and explored using nonlinear three-dimensional (3D) finite-element (FE) analysis. FE models representing both dry and saturatedconditions were developed to evaluate cases with and without generation and dissipation of excess pore-water pressures. The shear stress andstrain distributions between the granular columns and surrounding soil, and the level of shear stress reduction, were investigated for a practicalrange of treatment geometries, relative stiffness ratios, vertical stresses, and relative densities of the surrounding soil. A set of 10 accelerationtime histories were used as input motions. The FE results show that granular columns undergo a shear strain deformation pattern that isnoncompatible with the surrounding soil. As such, the achieved reduction in cyclic stress ratios imposed on the treated soil is far less than thatpredicted by the conventional shear strain compatibility design approach. Reductions in cyclic stress ratios are insensitive to the appliedsurface pressure, granular column length/diameter ratio (L/D), and relative density of the surrounding soil for the range of area replacementratio and column–soil shear modulus ratio examined. A modified design equation to estimate the reduction in cyclic stress ratio provided bydense granular columns is shown to provide good agreement with the FE simulation results. DOI: 10.1061/(ASCE)GT.1943-5606.0001474.© 2016 American Society of Civil Engineers.

Author keywords: Ground improvement; Liquefaction; Liquefaction mitigation; Cyclic stress ratio.

Introduction

Ground improvement using dense granular columns, such as vibro-replacement stone columns or rammed aggregate piers, can be aneffective method for mitigating liquefaction hazards. Physicalmodel tests and earthquake case histories have demonstrated theeffectiveness of different types of dense granular columns in loosesaturated cohesionless soils (e.g., Priebe 1991; Mitchell et al. 1995;Saxena and Hussin 1997; Ashford et al. 2000; Adalier et al. 2003).The installation of dense granular columns by various constructiontechniques are believed to mitigate liquefaction through a combi-nation of densification, increase in lateral stresses, reinforcement,and drainage effects (e.g., Baez 1995; Ashford et al. 2000; Adalieret al. 2003; Shenthan et al. 2004). A primary objective of construc-tion is often to increase standard penetration test (SPT) or conepenetration test (CPT) penetration resistances, both of whichwill reflect the cumulative effects of densification, increases inthe lateral stress (i.e., coefficient of earth pressure at rest, Ko),

and alterations to the soil’s fabric (including any disruption ofcementation, prior strain history, or ageing effects). There are proj-ects, however, where the additional benefits of shear stress reduc-tions have been relied on because the soils had relatively high finescontents and the desired increase in penetration resistances wasnot achievable. In these cases, the shear reinforcement mechanismhas often been analyzed using design procedures based on theassumption of shear strain compatibility between the granular col-umn and surrounding soil.

The concept of shear strain compatibility, as originally proposedby Baez (1995), assumes that the granular columns only deform inpure shear and, being stiffer than surrounding soil, attract a greatershare of the earthquake-induced shear stresses than the surroundingsoil. However, recent studies suggest that granular columns maydeform in both flexure and shear such that they are far less effectivein reducing shear stresses in surrounding soils than predicted byshear strain compatibility (Goughnour and Pestana 1998; Greenet al. 2008; Olgun and Martin 2008; Rayamajhi et al. 2014).Rayamahji et al. (2014) used linear-elastic three-dimensional(3D) finite-element analyses to develop a simplified relationshipfor estimating the shear stress reduction in soil treated with granularcolumns. They found that 10–30% reductions in shear stressescould still be achieved for a wide range of practical design param-eters, despite the lack of shear strain compatibility. The questionarises, however, as to whether the use of linear-elastic modelswould overestimate or underestimate the potential shear stress re-ductions provided by dense granular soils.

This study is an extension of work by Rayamajhi et al. (2014) toisolate and examine the shear reinforcement mechanisms of densegranular columns for reducing seismically induced shear stressesand strains in the improved surrounding soil. In this paper, the con-tributing mechanism of shear reinforcement is examined using 3Dnonlinear dynamic finite-element (FE) models with an effective

1Postdoctoral Scholar, School of Civil and Construction Engineering,Oregon State Univ., 220 Owen Hall, Corvallis, OR 97331 (correspondingauthor). E-mail: rayamajd@onid.orst.edu

2Professor and Dean, College of Engineering, Oregon State Univ., 101Covell Hall, Corvallis, OR 97331. E-mail: scott.ashford@oregonstate.edu

3Professor, Dept. of Civil and Environmental Engineering, Univ. ofCalifornia, One Shield Ave., Davis, CA 95616. E-mail: rwboulanger@ucdavis.edu

4Professor, Dept. of Structural Engineering, Univ. of California, SanDiego, 9500 Gilman Dr., La Jolla, CA 92093. E-mail: aelgamal@ucsd.edu

Note. This manuscript was submitted on August 25, 2014; approved onDecember 1, 2015; published online on March 17, 2016. Discussion periodopen until August 17, 2016; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Geotechnicaland Geoenvironmental Engineering, © ASCE, ISSN 1090-0241.

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stress-based constitutive model. Simulations are performed for arepresentative unit cell from a repeating grid of granular columns.The numerical modeling procedure is described along with adescription of the investigated geometries, material properties,constitutive model calibration procedures, loading conditions,and model cases. Parameters to interpret the shear stress and straindistribution between granular columns and surrounding soil aredescribed. Typical responses for unimproved (without granular col-umns) and improved soil (with granular columns) under earthquakemotions are presented, followed by shear stress and strain distribu-tions from a broader range of parametric studies. A modified designequation for estimating the reduction in cyclic stress ratios providedby the shear reinforcing effect of dense granular columns is shownto provide reasonable agreement with the results of the 3D FEsimulations.

Numerical Modeling

The purpose of the numerical models used herein is to obtain in-sights on the shear reinforcement mechanism of dense granular col-umns, while recognizing that numerical models cannot presentlysimulate construction processes nor many of their effects on thenative soils. For example, the construction of granular columns willproduce spatially variable degrees of densification, changes in thein situ lateral stresses, intermixing of the column and native soils attheir interface, and a disruption of the native soil’s fabric (includingany cementation or aging effects). In practice, the cumulative ef-fects of construction on the native soils are commonly evaluatedusing posttreatment SPT and CPT tests, whereas additional shearreinforcement or drainage effects during seismic loading can onlybe evaluated through analyses or design relationships. The FE mod-els developed subsequently are intended to isolate and examine the

shear reinforcement effects in sufficient detail to reasonably expectthat the observed trends would not change significantly with furthervariations in the modeling parameters.

FE Model

A unit cell-modeling approach was used similar to Asgari et al.(2013), Elgamal et al. (2009), Olgun and Martin (2008), andRayamajhi et al. (2014). The area replacement ratio (Ar), whichis the area of the granular column divided by the total unit cell area,is varied by adjusting either the diameter or center-to-center spac-ing of the granular columns. For the baseline case herein, a 3Dmodel of one-half of the unit cell for Ar ¼ 20% is shown in Fig. 1.Only half of the unit cell is modeled based on the symmetry alongthe x-axis and considering the periodic boundary conditions(e.g., Elgamal et al. 2009; Law and Lam 2001).

All 3D FE models were created in the OpenSees FE framework(Mazzoni et al. 2009; McKenna et al. 2010) with OpenSeesPL(Lu et al. 2006) used as a preprocessor. All soil materials were mod-eled using the 8-node brick brickUP elements. This 8-node brickelement is based on the solid-fluid (u-p) formulation of Chan(1988) and Zienkiewicz et al. (1990) for saturated soil. More infor-mation about this element can be found in Yang et al. (2003, 2008)and Elgamal et al. (2009). The granular column and surroundingsoil elements were connected at their nodes and no interface ele-ments were employed.

For the baseline case, the soil profile consists of a 9-m-thickloose silty sand layer with an average SPT energy-corrected,overburden pressure-corrected, and fines-corrected blow countðN1Þ60cs ¼ 7, corresponding to a relative density ðDrÞ ≈ 39%[based on the correlation used in Idriss and Boulanger (2008)].The liquefiable layer is underlain by a 2-m-thick dense sandlayer with an average ðN1Þ60cs ¼ 35 (Dr ≈ 87%), and overlain

periodic boundary conditions

line of symmetry

dense sand

X

loose silty sand

dense sand

Z

Y

granular column

(b)

1.98 m

0.99 m

shaking direction

1 m

1

2

Y

X

(a)

Note: points 1 and 2 in circle indicatelocations for comparing shear stressesand strains

0 m

dept

h (m

)

6 m

12 m

1.98 m 0.99 m

A

A

Fixed basecondition

Fig. 1. Soil profile configuration and FE model layout: (a) FE model 3D view; (b) FE model plan, section at A-A

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by a 1-m-thick dense sand layer (Dr ≈ 87%). The water table islocated at the ground surface. The granular columns are 1 m indiameter, extend to bottom of the liquefiable layer, and rest onthe top of the underlying dense sand layer as shown in Fig. 1.The column’s length to diameter ratio, L=D, is 9. The unit cellis discretized into 1.0-m-thick elements. Ground motion frequen-cies up to 15 Hz are considered of greatest interest for the presentstudy, which the 1.0-m-thick elements were found reasonably ableto transmit through the liquefiable soil. In order to investigate themesh sensitivity, analyses were carried out with 0.5-m-thick ele-ments (suitable for transmitting frequency up to ∼30 Hz) andthe overall responses (i.e., acceleration, excess pore-water pressure,stress-strain behavior, lateral displacements) were the same forpractical purposes. Thus, all further analyses were carried out using1.0-m-thick elements to reduce the incurred computational effort.

The following boundary conditions were used: (1) the displace-ments of nodes on the left and right boundaries at the same eleva-tion are forced to be equal in the longitudinal and vertical directions(i.e., x-direction and z-direction as shown in Fig. 1), (2) all nodes onthe inner symmetry plane as well as on the boundary of the oppositeside of the mesh were fixed against out-of-plane displacement(i.e., y-direction) but free to move longitudinally and vertically,(3) nodes at the base below the dense sand layer were fixedand the input ground motions were imposed on the base in thex-direction, (4) the soil surface was modeled with a prescribedzero-pore-pressure condition, and (5) the soil surface was modeledas stress free for the free-field condition and with prescribedsurface normal pressures simulating superstructure loads inother cases.

Soil Constitutive Model and Calibration

All soil materials were modeled using the PressureDependMulti-Yield02 (PDMY02) constitutive model. The PDMY02 model(Yang et al. 2003, 2008) was developed based on the frameworkof multisurface-plasticity theory (Prevost 1995) for cohesionlesssoils. The PDMY02 is an elastic-plastic model for simulatingthe cyclic response characteristics of pressure sensitive soil materi-als (e.g., sand, silt). The PDMY02 uses an associative flow rule forthe deviatoric plastic strain, and the volumetric plastic strain fol-lows a nonassociative rule, which allows for simulating shear-induced contraction and dilation as explained in Elgamal et al.(2003). The model is capable of simulating characteristics suchas nonflow liquefaction (cyclic mobility) and cycle by cycle per-manent shear strain accumulation, which are typically exhibitedin sands or silts during monotonic or cyclic loading (Parra1996; Yang et al. 2003, 2008; Elgamal et al. 2003, 2009).

Three sets of properties for the liquefiable soil layer were con-sidered in this parametric study: loose silty sand and two medium-dense silty sands corresponding to the ðN1Þ60cs of 7 (Dr ≈ 39%),15 (Dr ≈ 57%), and 22 (Dr ≈ 70%), respectively. The propertiesfor the granular column, the underlying dense sand layer, and thesurface dense sand layer were not varied and corresponded to anðN1Þ60cs ¼ 35 (Dr ≈ 87%). The parameters used to model the dif-ferent materials are provided in supplement Table S1. These modelparameters were chosen based on calibration of direct simple shear(DSS) element responses (drained and undrained) to publishedempirical design correlations as described in Khosravifar (2012).The friction angles were based on SPT-based empirical relation-ships in Kulhawy and Mayne (1990). The friction angle for thegranular column material was assigned 48°, similar to that reportedby Duncan et al. (2007) for an effective minor principal stress (σ 0

3)equal to 100 kPa (or 1 atm) in consolidated drained triaxial tests.The small-strain shear modulus (Gmax) for the liquefiable soil and

dense sand layers were based on the SPT-based empirical relation-ship by Andrus and Stokoe (2000). The Gmax for the granularcolumns was specified in terms of shear modulus ratio (Gr), whichis the Gmax for the column materials divided by Gmax for the lique-fiable soil.

The liquefiable soils were calibrated to produce a cyclic resis-tance ratio (CRR) corresponding to the SPT-based liquefactiontriggering correlation of Idriss and Boulanger (2008). The model-specific parameters (e.g., phase transformation angle, contractionand dilation parameter, steady-state parameters) for the liquefiablesoils were selected to produce 3% single amplitude shear strain (ap-proximately corresponds to excess pore pressure ratio of 100%)under undrained cyclic DSS simulation at approximately 15 uni-form loading cycles. For the dense sand and granular column ma-terials, the model-specific parameters were selected similar to thoseof a dense sand as recommended in Yang et al. (2003, 2008). Allthe soil models were calibrated for the reference confining pressureof 100 kPa (or 1 atm).

The results from single element undrained and drained cyclicDSS simulations are provided in the online supplement. Undrainedcyclic DSS simulations for the loose sand [ðN1Þ60cs ¼ 7] for differ-ent static shear stress biases (i.e., sloping ground conditions) arepresented in Fig. S1. The simulated relationship for cyclic stressratio (CSR) versus number of uniform cycles to develop 3% single-amplitude shear strain is shown in Fig. S2; these curves are steeperthan commonly observed in experimental data (e.g., as reviewed inBoulanger and Idriss 2014), but this is not expected to affect find-ings regarding the relative effects of granular columns in the presentanalyses. The stress-strain response for the loose sand underdrained strain-controlled cyclic DSS loading is shown in Fig. S3and the normalized shear modulus (G=Gmax) versus shear strainamplitude and damping ratio versus shear strain amplitude for dif-ferent soil materials are shown in Figs. S4 and S5. The normalizedshear modulus and damping curves for sand and granular columnsare reasonably comparable to empirically based curves (i.e., EPRI1993; Rollins et al. 1998) at smaller shear strains, but the hystereticdamping becomes larger than empirically expected at larger strains.Overall, the constitutive model is believed to sufficiently capturethe behaviors of importance to the present study.

OpenSees Analyses

The initial static stress conditions were established in the modelusing uniform linear-elastic material properties throughout themesh, so as to avoid any stress concentrations or arching effectsin the initial conditions. The coefficient of earth pressure at rest(Ko) was set equal to 0.8 to reflect the likely increase in lateralstresses caused by column construction. Sensitivity analysesshowed that the evaluation of shear stress reduction effects wasnot significantly affected by the choice of Ko, provided that theconstitutive model calibration for different ðN1Þ60cs values utilizedthe same Ko value. After initial stresses were established, allelements were assigned their nonlinear material models andparameters. A numerical Raleigh damping of 0.5% was specifiedat frequencies of 0.3 and 5 Hz to control numerical noise and pro-vide a minimum amount of small-strain damping in the elas-tic range.

Ground Motions

Ten acceleration time series were used as input motions for theanalyses. Baker et al. (2011) compiled a set of 40 ground motionswhose median response spectra matches with the median responsespectra predicted by Boore and Atkinson’s (2008) ground motion

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prediction equation. These motions were obtained for an earth-quake magnitude Mw ¼ 7, a source-to-site-distance R ¼ 10 km,a strike-slip earthquake mechanism, and a rock site condition(additional details in Baker et al. 2011). Ten of the 40 motions wereselected such that the median normalized spectra (normalizedby PGA) of the 10 motions matched with median normalizedspectra of the 40 motions as shown in Fig. 2. The 10 selected mo-tions are listed in Table S2. These 10 motions were then linearlyscaled by PGA from 0.05 to 0.6g for the parametric study de-scribed later.

Model Cases and Parametric Studies

Analyses were performed for a reference free-field condition and acolumn-reinforced condition so that the relative effect of the col-umn could be evaluated. The free-field condition is referred to asthe unimproved case herein because it does not include the granularcolumn, but in fact the native soil is assumed to have the sameðN1Þ60cs as in the column-reinforced (or improved case) so thatthe differences in behavior are due to the shear reinforcementeffect alone.

Cases were also performed with and without the effects of ex-cess pore-water pressure generation. A dry model (DM) was usedto understand the shear stress and strain distribution mechanism ofgranular columns solely based on the nonlinearity of soil (i.e., with-out the generation of any pore-water pressure in the soil profile). Asaturated model (SM) was then used to include the effects of excesspore-water pressure (EPWP) generation. The distribution of shearstress and strain were investigated for different states of excesspore-water pressure ratio (ru ¼ EPWP=σ 0

vo, σ 0vo = initial vertical

effective stress). The hydraulic conductivity for the granular col-umn was set equal to that of surrounding soil (1 × 10−5 m=s) sothat it does not provide enhanced drainage for EPWP dissipation.This choice of hydraulic conductivities perhaps represents the caseof complete intermixing of the granular and surrounding soil duringcolumn construction (e.g., Boulanger et al. 1998), but is mainlyintended as a means to artificially isolate the shear reinforcementeffects from drainage effects.

Parameters for Shear Stress and Shear StrainDistributions

The shear stress distribution in the improved and unimproved casesare compared in terms of the CSRs that develop in the native soil,where the CSR is computed using the form of the simplified pro-cedure proposed by Seed and Idriss (1971). The ratio of CSR for

improved soil (CSRI) over the CSR for unimproved soil (CSRU),RCSR, is expressed as (Rayamajhi et al. 2014)

RCSR ¼ CSRI

CSRU¼

�amax;I

amax;U

��rd;Ird;U

�¼ RamaxRrd ð1Þ

where amax;U and amax;I = peak horizontal acceleration at theground surface for unimproved and improved cases, respectively;rd;U and rd;I = shear stress reduction coefficient for unimprovedand improved cases, respectively; Ramax = ratio of peak surface ac-celerations for improved and unimproved cases, and Rrd = ratio ofshear stress reduction coefficient for improved and unimprovedcases. If Ramax ¼ 1, then dynamic responses on unimproved andimproved conditions are same and Rrd becomes RCSR. These termsare used to describe the relative shear stress distributions for caseswith and without excess pore water pressure generation, includingcases where liquefaction is triggered in the native soil. The valuesobtained for cases involving liquefaction in the native soils arenot appropriate for use in forward analyses of liquefaction poten-tial, but rather serve to further examine how shear stresses are dis-tributed as the native soil progressively approaches a liquefiedcondition.

Rayamajhi et al. (2014) used linear-elastic 3D dynamic FEanalyses to develop a design relationship to estimate the Rrd,which depends upon Ar, Gr, and the shear strain ratio γr = ratioof shear strain in granular column divided by shear strain in sur-rounding soil. The expression for the Rrd for discrete columnreinforcement is

Rrd ¼1

Gr½Arγr þ 1Grð1 − ArÞ�

≤ 1.0 ð2Þ

The γr can be estimated as

γr ¼ 1.04ðGrÞ−0.65 − 0.04 ≤ 1.0 ð3Þ

For shear strain compatibility, γr is assumed equal to 1.0, forwhich Eq. (2) becomes identical to the design equation proposedby Baez (1995). For the improved case, the relative stress ratio be-tween granular columns and surrounding soil can be obtained bymultiplying Gr with γr.

Simulation Example

Typical Responses

Typical ground surface acceleration responses for DM (dry modelwithout pore pressure generation) and SM (saturated modelwith pore pressure generation) cases are shown in Fig. 3 forAr ¼ 20%, Gr ¼ 5, the thickness of liquefiable soil Hliq ¼ 9 m,and the native soil’s ðN1Þ60cs ¼ 7. The input motion was a record-ing during the 1989 Loma Prieta earthquake at Gilroy-GavilanCollege station scaled to a PGA (or amax) of 0.20g. Responsesare shown for the element at Location 1 in Fig. 1. The ru and shearstress-strain responses are shown for elements at a depth of 4 m.

For the DM case, the unimproved and improved soil profiles hadvery similar ground surface accelerations with peak values of∼0.60g (i.e., three times the base PGA). This strong amplificationof base motions is reasonable based on this profile’s natural fre-quency, the input motion’s frequency content, and the use of arigid base.

For the SM case, both the unimproved and improved soilprofiles developed liquefaction in the native soils. The peak accel-erations at the ground surface for the unimproved case (0.20g)

Fig. 2. Normalized response spectra (with respect to PGA) of10 selected input motions with median spectra compared to mediannormalized spectra from Baker et al. (2011)

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and improved case (0.35g) were both smaller than the 0.6g thatdeveloped in the DM cases. The peak surface acceleration forthe improved case was, however, almost twice the peak surfaceacceleration for the unimproved case, illustrating the strong effectthat the granular columns often had on dynamic site response afterliquefaction was triggered in the surrounding soils. EPWPs in theunimproved and improved soil profiles show similar behavior asthe native soil approached ru ≈ 95% early in shaking (at about4.1–4.2 s in Fig. 3), after which the improved case showedstronger transient dips of EPWP, which were attributed to strongdilation of the granular columns and the higher peak surfaceaccelerations.

Stress strain responses at Locations 1 (in front of the column inthe direction of shaking; Fig. 1) and 2 (to the side of the column;Fig. 1) at a depth of 4 m for the above DM and SM cases, with andwithout dense granular columns are shown in Figs. 4(a–d). For theDM case, the improved profile had greater shear strains than theunimproved profile at Location 1 [Fig. 4(a)] but smaller strainsat Location 2 [Fig. 4(c)]. For the SM case, the soil at Locations

1 and 2 showed cyclic mobility responses regardless of whethera dense granular column was present or not. In addition, theimproved profile had greater shear strains than the unimprovedprofile at Location 1 [Fig. 4(b)] but smaller strains at Location 2[Fig. 4(d)]. The development of greater shear strains at Location 1with the granular column is not intuitive, but it is consistent with thefindings of Goughnour and Pestana (1998) and is partly attributableto the actual deformation mechanism not being pure simple shearwhen the granular column is present.

The dynamic response of the improved soil profile was found tooften be significantly different from that of the unimproved soilprofile, with the differences attributable to variations in the fre-quency content of the input motions and effective natural frequen-cies of the soil profiles. The results of the present parametric studieswere insufficient to provide any simple relationships relating thepeak surface accelerations in the unimproved and improved cases.In practice, the effects of dense granular columns on site responsemay need to be estimated using site-specific response studieswith an approximate accounting for the effect of the columns onthe natural frequencies of the soil profile (e.g., Bouckovalas et al.2006).

Rrd and γr Responses for DM Case

Values of Rrd and γr averaged over the plan area are plotted versusdepth in Fig. 5 for the DM case with Ar ¼ 20%,Gr ¼ 5, and differ-ent scaled shaking intensities for the same input motion; γr is com-puted at the instant of maximum average shear stress in theimproved soil profile. For the low shaking intensity (i.e., 0.05g),the models behaves similar to a linear elastic condition and thusthe Rrd and γr responses are similar to the linear-elastic solutionsobtained by Rayamajhi et al. (2014). Generally, Rrd slightly de-creases with depth for different shaking intensities, while the γris relatively constant up to 7 m depth and then increases slightlywith further increases in depth. As the ground shaking intensityincreases, both Rrd and γr tend to slightly decrease. These trendsare expected because at higher shaking, the shear modulus of thegranular column does not decrease as much as the shear modulus ofthe native soil does. The differences in shear modulus reductionbehavior are illustrated in Fig. 6 where the Points A and B corre-spond to the strains induced in the granular column and native soil,respectively, during the same strong shaking event; the normalizedshear modulus ratio for the granular column is G=Gmax ¼ 0.47 at0.05% shear strain (Point A) whereas for the native soil it isG=Gmax ¼ 0.42 at 0.20% shear strain (Point B). Thus, the secantshear modulus ratio increases by a factor of 1.12 (i.e., 0.47=0.42)due to the effects of nonlinearity in the two materials. This increasein the secant Gr with increasing shaking level is considered theprimary reason for the slight decrease in Rrd with increasing shak-ing level (Fig. 5). The trends of Rrd and γr with increasing shakinglevel were also observed to vary with the characteristics of the inputground motion, although the differences were not large and theoverall patterns were similar to those in Fig. 5.

Rrd and γr Responses for SM Case

Values of Rrd and γr averaged over the plan area are plotted versusdepth in Fig. 7 for the SM case with Ar ¼ 20%, Gr ¼ 5, and theGavilan college input motion scaled to a base PGA ¼ 0.20g. TheRrd and γr values are plotted for different time windows based onmaximum ru levels in the native soil to evaluate any potentialdependencies on ru. Rrd and γr were computed from the time his-tories up through the time at which ru first exceeded some specifiedlimit; these limits were taken as ru ≤ 0.50, ru ≤ 0.80, and ru ≤ 0.95

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 3. Example responses for a soil profile with and without densegranular columns (DGC) for a ground motion scaled to 0.20g: (a) inputmotion at the base of the soil profile; (b) response for DM case withoutDGC; (c) response for DM case with DGC; (d) response for SM casewithout DGC; (e) response for SM case with DGC; (f) response for SMcase at 4-m depth with and without DGC

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for both unimproved and improved soils. For the early part ofshaking with ru ≤ 0.5, the average Rrd and γr values are closeto the linear-elastic results by Rayamajhi et al. (2014). As shakingprogressed and ru increased to 0.80, the Rrd values began to reducetoward the bottom of the soil profile. Once ru reached 95%, the Rrd

had reduced significantly and was essentially equal to the Rrd

determined from the full time history of shaking. However, the re-ductions in Rrd as ru values neared 100% are not appropriate for usein procedures that evaluate liquefaction triggering, since they areinstead a consequence of liquefaction having already been essen-tially triggered. The values γr were essentially the same for all rutime windows, which is because the peak shear stress (at whichtime γr is computed) occurred early in shaking.

Incremental Dynamic Analyses

The effect of shaking intensity on the stress and strain distributionswas evaluated using incremental dynamic analyses (IDA) with all10 input motions scaled to PGAvalues ranging from 0.05 to 0.60g.The resulting variations in Rrd, Ramax, and γr are plotted versus theinput motion PGA in Fig. 8 for a DM case with Ar ¼ 20% andGr ¼ 2, 5, and 7. The values of Rrd, Ramax and γr all show slighttrends with increasing shaking intensity (e.g., Rrd increases slightlyand Ramax decreases slightly), but these effects are relatively smallcompared to the effects of Gr or the variability in results for differ-ent input motions. Some input motions produced Rrd greater thanunity, which means the improved soil can experience greater shearstresses than the unimproved soil in some cases.

The variations in Rrd, Ramax, and γr are plotted versus PGA inFig. 9 for a SM case with Ar ¼ 20%, Gr ¼ 2, 5, and 7, and each

(a) (b)

(c) (d)

Fig. 4. Shear stress versus shear strain responses under 1989 Loma Prieta earthquake ground motion at Locations 1 and 2 (locations shown inFig. 1) in liquefiable soil at 4-m depth: (a and c) with and without dense granular columns for DM case; (b and d) with and without dense granularcolumns for SM case

(a) (b)

Fig. 5. Variations of shear stress and strain distribution for DM caseunder 1989 Loma Prieta earthquake motion scaled to different PGA:(a) Rrd versus depth trends; (b) γr versus depth trends

Fig. 6. Shear modulus reduction ratios for dense granular columns andloose sand at a depth of 4 m with Points A and B representing the con-current strains in the column and soil, respectively

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response only computed up through the time when ru ≤ 0.8. Theaverage Rrd tended to slightly decrease, the average Ramax tended toslightly increase, and the average γr tended to remain relativelyconstant as the shaking intensity increased for all values of Gr.Similar to the DM case in Fig. 8, these trends with increasingshaking intensity are relatively small compared to the variationin responses for different input motions.

Parametric Study on Rrd and γr

Effects of Ar and Gr

The variation of the average Rrd values with Ar and Gr is shown inFig. 10(a) for a DM case with Ar ¼ 10, 20, and 30%, Gr ¼ 2, 5,and 7, a native soil (ðN1Þ60cs ¼ 7, Hliq ¼ 9 m, D ¼ 1 m, and inputmotion PGA ¼ 0.05 and 0.20g. The corresponding values of γr areplotted Gr in Fig. 10(b). The Rrd values based on the design rela-tionships by Rayamajhi et al. (2014) and Baez (1995) are alsoshown in Fig. 10(a), and the γr values based on Rayamajhi et al.(2014) are shown in Fig. 10(b). The simulation results show Rrddecreases with increasing Ar or increasing Gr for both 0.05 and0.20g input motions. The design expressions by Rayamajhi et al.(2014) are in reasonable agreement with the simulated values of Rrd[Fig. 10(a)] and γr [Fig. 10(b)] for all cases.

The variation of the average Rrd values with Ar and Gr is shownin Fig. 11(a) for a SM case with the same range of parameters as forthe above DM case and the responses only computed up throughthe time when ru ≤ 0.8. The Rrd values are slightly lower than thosefor the DM case [Fig. 10(a)], but are still reasonably similar to therelationship by Rayamajhi et al. (2014) and significantly higherthan the relationship by Baez (1995). The γr values from the sim-ulations remained in good agreement with the relationship byRayamajhi et al. (2014) [Fig. 11(b)].

Effects of Vertical Surface Pressure

The effect of an additional vertical surface pressure (Δσavs), such asmay come from an overlying structure, was examined for unim-proved or improved cases. These analyses did not account for

(a) (b)

Fig. 7. Variations of shear stress and strain distribution for SM case atdifferent levels of ru under a ground motion scaled to 0.20g: (a) Rrd

versus depth trends; (b) γr versus depth trends

(a) (b) (c)

Fig. 8. Variations of Rrd, Ramax, and γr for DM case with different Gr and Ar ¼ 20%: (a) Rrd versus PGA; (b) Ramax versus PGA; (c) γr versus PGA

(a) (b) (c)

Fig. 9. Variations of Rrd, Ramax, and γr for SM case with different Gr, Ar ¼ 20% and ru ≤ 0.8: (a) Rrd versus PGA; (b) Ramax versus PGA; (c) γrversus PGA

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any additional shear loads that might be associated with the dy-namic response of any overlying structure and did not accountfor any arching of the surface loading onto the stiffer granular col-umns. A concentration of vertical stresses in the granular columnswould increase their initial Gr value; this effect is indirectlyaccounted for in the present study by the specification of the Grvalue itself.

The variation of Rrd with Ar is shown in Fig. 12(a) for a SM casewith Ar ¼ 10, 20, and 30%, Gr ¼ 5, a native soil ðN1Þ60cs ¼ 7,Hliq ¼ 9 m, D ¼ 1 m, PGA ¼ 0.05 and 0.20g, ru ≤ 0.80, andΔσavs ¼ 0, 50, and 100 kPa. On average, the increase in Δσavsslightly increases the Rrd values for the range of parameter studiedherein [Fig. 12(a)]. The γr values are relatively unaffected byΔσavsand remain close to the relationship by Rayamajhi et al. (2014)

(a) (b)

Fig. 10. Comparison of average Rrd and γr for different Ar and Gr in DM case at shaking PGA ¼ 0.05 and 0.20g: (a) Rrd versus Ar; (b) γr versus Gr

Fig. 11. Comparison of average Rrd and γr for different Ar and Gr in SM (ru ≤ 0.8) at shaking PGA ¼ 0.05 and 0.20g: (a) Rrd versus Ar;(b) γr versus Gr

0 10 20 30 40 50Ar, %

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 kPa0 kPa

50 kPa50 kPa

100 kPa (ru 0.8, PGA=0.05 g)100 kPa (ru 0.8, PGA=0.20 g)

1 2 3 4 5 6 7 8 9Gr

0.0

0.2

0.4

0.6

0.8

1.0

(a) (b)

Gr=5Baez (1995) Rayamajhi et al. (2014)

Gr=5Rayamajhi et al. (2014)

Fig. 12. Comparison of average Rrd and γr for different vertical surface pressure for SM case (ru ≤ 0.8) at shaking PGA ¼ 0.05 and 0.20g: (a) Rrd

versus Ar; (b) γr versus Gr

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[Fig. 12(b)]. Similar trends of Rrd versus Ar and γr versus Gr werefound for a DM case with the same range of parameters.

Effects of L/D Ratio for Granular Column

The effect of the granular column’s slenderness ratio (L=D) is illus-trated in Fig. 13 showing Rrd versus Ar [Fig. 13(a)] and γr versusGr [Fig. 13(b)] for a SM case with Ar ¼ 10, 20 and 30%, Gr ¼ 5, anative soil ðN1Þ60cs ¼ 7,Hliq ¼ 3, 6, and 9 m,D ¼ 1 m, ru ≤ 0.80,and PGA ¼ 0.05 and 0.20g. The L=D ratios are 3, 6, and 9 for thesecombinations of Hliq and D. On average, the decrease in L/Dslightly reduces Rrd and slightly increases γr for the range ofparameter studied herein. Similar trends of Rrd versus Ar and γrversus Gr were found for a DM case with the same range ofparameters.

Effects of �N1�60cs in the Native Soil

The effect of different ðN1Þ60cs values for the native soil is illus-trated in Fig. 14 showing Rrd versus Ar [Fig. 14(a)] and γr versusGr [Fig. 14(b)] for a SM case with Ar ¼ 10, 20, and 30%, Gr ¼ 5,Hliq ¼ 9 m, D ¼ 1 m, ru ≤ 0.80, PGA ¼ 0.05 and 0.20g, andðN1Þ60cs ¼ 7, 15, and 22. In application, ðN1Þ60cs would reflectchanges in density, lateral stresses, and changes to the soil fabric.In the present analyses, the value of Ko remains constant and only

the relative density and constitutive model calibrations to providethe appropriate CRR are changed. On average, an increase inðN1Þ60cs resulted in a slight increase in Rrd and almost no effecton γr. Similar results were obtained for a DM case with the samerange of parameters.

Conclusions

Nonlinear dynamic 3D FE simulations were used to isolate andevaluate the shear reinforcement mechanisms of dense granular col-umns in terms of reducing the seismic shear stresses on the treatedsoils. Analyses were performed for unit cells with Ar ¼ 10 to 30%,Gr ¼ 2 to 7, native soil ðN1Þ60cs ¼ 7 to 22, Hliq ¼ 3 to 9 m, D ¼0.6 to 1.5 m, Δσavs ¼ 0 to 100 kPa, dry and saturated conditions,and PGA ¼ 0.05 to 0.60g for 10 different input motions.

The results of these nonlinear FE simulations were consistentwith the results previously obtained by Rayamajhi et al. (2014)using linear-elastic FE simulations for a similarly broad range ofconditions. Both sets of simulations indicate that the reductionin seismic shear stresses provided by dense granular columns is(1) significantly smaller than would be estimated based on theassumption of shear strain compatibility; and (2) reasonably esti-mated using the design relationships proposed by Rayamajhi et al.(2014). These FE simulations involve a number of significant

0 10 20 30 40 50Ar, %

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(N1)60=7(N1)60=7

(N1)60=15(N1)60=15

(N1)60=22 (ru 0.8, PGA=0.05 g)(N1)60=22 (ru 0.8, PGA=0.20 g)

1 2 3 4 5 6 7 8 9Gr

0.0

0.2

0.4

0.6

0.8

1.0

(a) (b)

Gr=5Baez (1995) Rayamajhi et al. (2014)

Gr=5Rayamajhi et al. (2014)

Fig. 14. Comparison of average Rrd and γr for different ðN1Þ60 of liquefiable soils for SM case (ru ≤ 0.8) at shaking PGA ¼ 0.05 and 0.20g: (a) Rrd

versus Ar; (b) γr versus Gr

Fig. 13. Comparison of average Rrd and γr for different liquefiable soils depth for SM case (ru ≤ 0.8) at shaking PGA ¼ 0.05 and 0.20g: (a) Rrd

versus Ar; (b) γr versus Gr

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approximations and cannot recreate the actual construction process,but the results are believed to provide a reasonable basis for evalu-ating the relative magnitude of stress reductions provided by suchcolumns. Accordingly, the design relationships by Rayamajhi et al.(2014) are recommended for use on projects where sufficientincreases in posttreatment SPT or CPT penetration resistancesare not achieved during installation of the granular columns, suchthat the additional benefits of shear stress reduction need to beconsidered.

The shear reinforcement provided by dense granular columnsmay also help reduce liquefaction-induced deformations (e.g., lat-eral spreading) in the event that densification, increases in lateralstresses, shear stress reductions, and any drainage effects are insuf-ficient for preventing liquefaction triggering. Dense granularcolumns may also continue to provide support for overlying struc-tures, reducing their settlement relative to the unimproved case.These aspects of granular columns are investigated in a companionmanuscript (Rayamajhi et al. 2016).

Acknowledgments

This research is funded by Oregon Transportation Research andEducation Consortium (OTREC), Oregon Department of Transpor-tation (ODOT), and Pacific Earthquake Engineering Research(PEER) Center. Any opinions or conclusions expressed hereinare those of the authors and do not necessarily reflect the viewsof any of the above organizations. The authors appreciate theassistance with OpenSeesPL provided by Dr. Jinchi Lu and thesuggestions and comments received from Dr. Lisheng Shao andDr. Armin W. Stuedlein during different phases of this research.

Supplemental Data

Tables S1 and S2 and Figs. S1–S5, along with other supplementalmaterial, are available in the ASCE Library (http://www.ascelibrary.org).

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