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by solving the LE equations. A majority of the existing dis-placement-based pseudo-static LE methods (e.g. Cai & Bath-urst, 1996; Ling et al., 1997; Nakajima et al., 2010) weredeveloped by considering only a translational mode of failure(i.e. external sliding stability) for calculating earthquake-induced displacement, and consequently do not provide ameans for assessing the seismic displacement of GRESs due torotational movements (internal stability). This observation is of

particular concern, as Leshchinsky et al. (2009) have demon-strated that internal stability may control the required tensilestrength of the reinforcement for GRESs. Moreover, internalrotational failure can degenerate to a translational one should itbe more critical (e.g.Leshchinsky & Zhu, 2010); however, thereverse is not true, which makes rotational failure a moregeneric mechanism.

To address this issue, this paper presents a new analyticalnumerical framework for the displacement-based design ofGRESs, which assesses the potential for earthquake-induceddisplacements via an internal stability (rotational) failuremechanism. For design purposes, in order to determine thesuperimposed force in the reinforcement due to seismicityand its associated displacement, the proposed approach ex-amines two limiting conditions.

(a) The upper-bound force that can be mobilised in thereinforcement, as determined by pseudo-static LE. For agiven earthquake acceleration, this would be the maxi-mum force that can be induced in the reinforcement bythe inertial response of the sliding body.

(b) The force that can be induced in the reinforcement by agiven earthquake acceleration applied over a finite timeincrement. This second limiting condition examines thetime limitation effect, which accounts for the transientcharacteristics of the applied ground motion.

The smaller of the forces yielded from these two conditionscontrols the seismically induced force and displacement inthe reinforcement for each time increment. In order to relate

the force and displacement in the geosynthetic reinforce-ment, a set of pullout simulations was performed using FEanalysis for various geosynthetic stiffnesses. The summationof the incremental displacements and induced forces in thereinforcement over the entire time series analysis providesthe necessary information to guide the design of a GRES.

METHODOLOGY AND FORMULATIONLeshchinsky et al. (2012) presented an analytical solution

based on LE and using a log-spiral failure surface to extendthe M-O method for unreinforced slopes. Using a similarformulation,Vahedifardet al. (2012) proposed a pseudo-staticLE approach for assessing the internal stability of GRESs that

can be utilised for a given reinforcement strength to determinethe yield acceleration required for calculating seismic displa-cements in Newmark-type methods. However, for designpurposes, it is also desirable to have a displacement-basedmethod to assess the required reinforcement strength underseismic conditions. The method proposed by Vahedifardet al.(2012) is utilised in this paper to determine the static resultantforce in the reinforcement. For brevity, some fundamentalsrelated to this paper are not reiterated here; interested readersare referred toVahedifardet al. (2012).

The proposed model assumes a rotational (internal) modeof failure along a log-spiral failure surface (Fig. 1). Theproposed model treats the reinforced soil mass inside thefailure surface as a rigid body, and represents the reinforce-

ment acting outside the failure surface as a spring. Thisapproach assumes that the reinforcement layers hold thesliding mass coherent during shaking, and that consequentlythe internal deformations are not significant.

Figure 2 shows the flow of calculations that are necessaryto perform the displacement-based design process utilised inthis paper. The associated assumptions that are made foreach of the steps in the calculation are discussed in moredetail in the following sections. As shown in Fig. 2, aftercalculating the mobilised resultant reinforcement force understatic loading conditions (Tstat), in a parallel attempt theseismic-induced displacement and force in the reinforcementare calculated using two approaches: a pseudo-static LEapproach and an acceleration time-series integrationapproach. Since the induced force that is predicted by the

time series approach cannot exceed the required force tosatisfy equilibrium at each time increment, the LE equationsprovide an upper limit to the force and displacement thatcan be mobilised. As an auxiliary tool, a set of pulloutsimulations using FE analysis was performed to establish arational relationship between reinforcement displacement andforce during shaking. The scope of the proposed approach islimited to cases where pullout failure of the reinforcementdoes not control the internal design. This limitation is not aconcern for most GRESs constructed using extensible rein-forcement (e.g. geogrids or geotextiles), as this mode offailure is not an issue for nearly all typical design cases.

Step 1: Static reinforcement force from limit equilibriumLeshchinsky et al. (2010) present stability charts for the

static design of GRESs that fail internally via a rotationalfailure mechanism. These charts were developed for GRESs

( , )X Yc c

2

d tan

R2

d 2

Aexp( )

H

1

R1

d1

Aex

p(

)

1

2

Tstat

D

W

q

Rh

3

(0, 0)

Y

X

Tmax-j

( , )X Yc c

2 d tan

R2

d 2

Aexp( )

1

R1

d

1

Aex

p(

)

r

Rr

dr

Aexp(

)

1

2

H

D

W

q

Rh

3

(0, 0)

Y

X

K Wh

Tdyn

K Wv Tmax- md-j j T

(a)

(b)

Fig. 1. Notation and convention: (a) static condition, state of LE;

(b) dynamic condition, rotation initiated

452 VAHEDIFARD, LESHCHINSKY AND MEEHAN

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constructed using cohesionless soils, and for horizontal back-slope conditions. The general approach that was used todevelop these stability charts can be extended as shown inFig. 1(a) and equation (1) to allow for the static design ofGRESs with an inclined backslope.

For a given log-spiral failure surface, static failure of thesliding mass is prevented by the reactive static force that ismobilised in the reinforcement. In order to determine thereinforcement force that is required to resist sliding, an LE

analysis approach is applied. In this analysis, the trace ofthe log-spiral is defined by the radius vector R A exp(d), where A is the log-spiral constant,d tan(design), and is the angle in polar coordinates.For a given log-spiral failure surface (Fig. 1(a)), the static

reinforcement force that is needed for equilibrium (Tstat) canbe determined by applying moment equilibrium about thepole of the log-spiral (Leshchinsky et al., 2010; Vahedifardet al., 2012)

(eq. 2)

T T Tmd1( ) dyn1( ) stati i

d 0Tmd1( )i

x2( ) 2( ) ( 1) r i i i(d )cos

T Tmd1( ) md( 1)i i

Find for

(from finite-element curves)

T xdyn2( ) 2( )i i

dT T Tmd1( ) md1( ) md( 1)i i i T T Tmd2( ) dyn2( ) stati i

T Tmd2( ) md( 1)i i d 0Tmd2( )i Find d for d

(from finite-element curves)

x T1( ) md1( )i i

d d /cos 1( ) 1( ) r i i x dT T Tmd2( ) md2( ) md( 1)i i i

d min(d ; d )

d min(d ; d )

T T Tmd( ) md1( ) md2( )

( ) 1( ) 2( )

i i i

i i i

T T Tmd( ) md( ) md

( ) ( )

i i

i i

d

d

( 1)

( 1)

i

i

T T

T T

T T

dyn1( ) md1( )

dyn2( ) md2( )

dyn( ) md( )

i i

i i

i i

, , . . . : from LE

, , . . . : from integration

, , . . . : prevailing values

Step 3

Step 2-1 Step 2-2

Step 1

Input , D, , , , , q K KH(t) V(t)

CalculateTstat(eq. (1) and max. process)

t i

Read andK Kh( ) v( )i i

CalculateTdyn1( )i Calculate d from2( )iintegration (eqs. (7) (10))

t i 1

No

No

Yes

Yes

IntegrationLimit equilibrium (LE)

Fig. 2. Flow of calculations for proposed approach

INTERNAL DESIGN OF GEOSYNTHETIC-REINFORCED EARTH STRUCTURES 453

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Tstat

21

Aed cos Aed2 cos2

Aed sin

Aed cos dsin d

H2

2 tan

H

3tan Aed1 sin1

Htan Aed1 cos1 Aed2 cos2 H

Aed1 sin1

H

2tan

2 Aed2 sin2 Ae

d1 sin1 Htan

Aed1 cos1 Aed2 cos2 H

3 Aed1 sin1 Htan 13

Aed2 sin2 Aed1 sin1 Htan

h i

q

2 Aed2 sin2 2

Aed1 sin1 Htan 2h i

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>;

Aed1 cos1 D

Rh

TstatAed1 cos1

(1)

where is the unit weight of reinforced soil; H is the heightof the slope; is the backslope angle;1 and2 are the anglesof the points where the log-spiral enters and exits the slope(Fig. 1(a)); is the batter; q is the surcharge; D is theelevation of line of action of the resultant reinforcement force(Tstat), and Rh is the horizontal resistance of the facing at thebottom of the slope. In this study, the impact of facing is notconsidered, and Rh is ignored; this is a legitimate assumptionin certain structures, such as wrapped-face GRESs or small-block facing walls. Because of publication space restrictions,the effects of toe resistance are ignored in the current study;toe resistance effects and their implications in the context ofdesign of reinforced earth structures are addressed by Lesh-chinsky & Vahedifard (2012). The elevation ofD is a function

of the reinforcement spacing and the assumed distribution ofthe individual reinforcement forces; the effect of the assumedlocation ofD is examined in more detail by Vahedifardet al.(2012).

The maximum required static reinforcement force,max(Tstat), can be calculated by utilising a maximisationprocess in conjunction with equation (1); this process isexplained in more detail by Leshchinsky et al. (2010). Theobjective of this process is to identify the test body thatrequires the maximum reactive force from the reinforcementthat is needed to satisfy the LE state. This process yieldsvalues that are necessary to determine the critical slipsurface (Xc, Yc and A) that corresponds to the value ofmax(Tstat). Note that the equivalent reinforcement force

under the static condition is assumed to act horizontally.

Step 2-1: Seismic reinforcement force and displacement fromlimit equilibrium

If unfactored soil strengths and geosynthetic reinforcementstrengths equal to the reactive resistances are used in theequilibrium calculations, the GRES shown in Fig. 1(a) willbe in a state of incipient static failure (i.e. it will be at itslimit state, with its factor of safety (FS) 1.0). If anyadditional earthquake-induced forces are added to this sys-tem, the sliding mass will move. For GRES design purposes,additional reinforcement strength is needed to resist theearthquake-induced forces that are imposed to the system that is, the reinforcement needs to provide more strength tostabilise the GRES.

During earthquake shaking, the reinforced soil mass is

subjected to additional seismic forces. As shown in Fig.1(b), this shaking at each time increment can be representedby inertial forces that act at the centre of gravity of thesliding mass. In response to these inertial forces, the resul-tant force in the reinforcement (Tdyn1) changes, and needs tobe recalculated in response to the dynamic loading condi-tion. For the purposes of the analyses herein, it is assumedthat this resultant dynamic reinforcement force acts horizon-tally (Fig. 1(b)).

In a similar fashion as the calculations that were per-formed to determine Tstat, we can use the equation formoment equilibrium about the pole of the log-spiral todetermine the associated value of the resultant dynamicforce in the reinforcement from the LE approach (Tdyn1) for

each time increment in the analysis.


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Tdyn1(i)

1 Kv(i)

21

Aed cos Aed2 cos2

Aed sin

Aed cos dsin d

H2

2 1 Kv(i) tan

H

3tan Aed1 sin1

H 1 Kv(i) tan Aed1 cos1 Ae

d2 cos2 H

Aed1 sin1

H

2tan

2 1 Kv(i) Ae

d2 sin2 Aed1 sin1 Htan

Aed1 cos1 Ae

d2 cos2 H

3 Aed1 sin1 Htan 13

Aed2 sin2 Aed1 sin1 Htan

h i

q

2 Aed2 sin2 2

Aed1 sin1 Htan 2h i

2Kh(i)

21

Aed cos 2

Aed2 cos2 2h i

Aed cos dsin d

H2

2 Kh(i)tan Ae

d1 cos1 HH

3

HKh(i)tan Aed1 cos1 Aed2 cos2 H

3 Aed2 cos2 12


h i

2Kh(i) Ae

d1 cos1 Aed2 cos2 H

Aed2 sin2 Ae

d1 sin1 Htan

3 Aed2 cos2 13


h i

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

Aed1 cos1 D

Rh

Tdyn1(i)Aed1 cos1

(2)

where Kh(i) and Kv(i) are the horizontal and vertical seismiccoefficients at time increment (i), and the other variables areas defined previously.

Entering the calculatedXc, Ycand A values from the staticcase (equation (1)) into equation (2), one can calculateTdyn1(i), which corresponds to a known log-spiral along whichthe static strengths, and Tstat, are fully mobilised. Unlikethe procedure followed with equation (1), a maximisationprocess is not utilised for this step in the analyses; insteadthe critical log-spiral from the static case is used. Thisassumption is shown to be reasonable by the experimentalresults provided by Leshchinsky et al. (2009), and is alsoconsistent with the failure surface assumptions that are madeby FHWA (2009) and AASHTO (2007) for assessing the

internal stability of earth-retaining structures under seismicloading conditions.

From the geometry shown in Fig. 1(b), one can calculatethe angle where the radius intersects the equivalent rein-forcement force, r, from the relationship

Aedr cosr Aed1 cos1 D (3)

As shown in Fig. 2, Tdyn1 is determined for each timeincrement (i) of the earthquake time series. For each step, itsvalue should be checked to ensure that it does not exceedthe pullout capacity. If a value of Tdyn1 greater than thepullout capacity is calculated, the displacement results fromthe model should not be used, as the scope of the current

paper is limited to cases where pullout does not occur. Inthis case, the force in the reinforcement will be the pulloutforce.

For each time increment, the superimposed force in the

reinforcement due to seismicity from the LE equations(Tmd1) can be determined as

Tmd1(i) Tdyn1(i) Tstat (4)

Note that Tmd1(i) > 0 and

dTmd1(i) Tmd1(i) Tmd(i1) (5)

where dTmd1 is the incremental superimposed force in thereinforcement from the LE equations. Note that dTmd1 isdetermined by taking the difference between the superim-posed force in the reinforcement in the current time incre-

ment from LE (Tmd1(i)) and the prevailing value from theprevious time step (Tmd(i1)).

Since LE analysis cannot directly relate displacement,stiffness and force in the reinforcement, an intermediatecalculation tool is needed for this purpose. As will bediscussed later in this paper, a set of FE analyses will beused to determine the displacement that corresponds todTmd1:

As shown in Fig. 1, Tstat and Tdyn1 act horizontally.Consequently, Tmd1 and dTmd1 also act horizontally. Fromthese forces, it is possible to determine the horizontalcomponent of incremental displacement, d x1, using the FEcharts that are provided later in this paper. The resultantdisplacement from the LE approach, d1, can be calculated

using the equation (see Fig. 3)

d1(i)d x1(i)

cosr(6)


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Step 2-2: Seismic reinforcement force and displacement fromintegration

The force-based procedure that was explained in theprevious section yields the upper-bound force that can bemobilised in the reinforcement, as determined by pseudo-static LE. For a given earthquake acceleration, this is themaximum force that can be induced in the reinforcementby the inertial response of the sliding body. However, sincethe body is accelerating for a limited time, it may notreach the force and displacement values that are determinedusing the LE equations. Therefore it is also necessary toassess the force that can be induced in the reinforcementby a given earthquake acceleration applied over a finitetime increment. Following this approach, the force anddisplacement in the reinforcement can be determined usingan integration procedure that is applied to the earthquakeacceleration time series.

As shown inFig. 2, for each time increment i, we have

a(i) ah(i)cosr av(i)sinr (7)

where a(i) is the acceleration perpendicular to the radiusleading to rotational acceleration, and ah(i) and av(i) are theinput horizontal and vertical accelerations, respectively(ah Khg, av Kvg, andg gravitational acceleration).

One can calculate angular velocity, v, for each timeincrement as

v(i)

ffiffiffiffiffiffia(i)

R

r

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia(i)

Aedr

r (8)

Since the movement (i.e. rotation) is small and the changein R is negligible, R is constant in equation (8) during theseismic event. This assumption is proved to be reasonableby a set of large-scale shaking table test results provided byLeshchinsky et al. (2009), which show that in a rotationalmode of failure the reinforcement holds the failure masscoherent, and internal deformations are not significant.

The perpendicular velocity of the rotating mass, v, foreach time increment is equal to

v(i) Rv(i)

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia(i)Ae

drq (9)

and the incremental resultant displacement from the integra-tion procedure, d2, is

d2(i)v(i) v(i1)

2 dt (10)

where dt ti ti1.The horizontal component of displacement from the inte-

gration approach, x2, can be determined using the equation

x2(i) d2(i) (i1)

cosr (11)

Note that the horizontal component of displacement fromthe integration approach ( x2) is determined by summationof the incremental displacement in the reinforcement in thecurrent time increment from integration (d2(i)) and theprevailing value from the previous time step ((i1)).

For the next step in the analysis, a series of curves thatrelate the force and displacement in the reinforcement priorto pullout is used to determine the dynamic force (Tdyn2) thatcorresponds to the calculated horizontal component of dis-placement from the integration approach ( x2). These curveswere developed using FE analysis, following the procedurethat is discussed in more detail later in this paper.

The superimposed force in the reinforcement (Tmd2) and

incremental superimposed force in the reinforcement (dTmd2)from the integration procedure are calculated using the equa-tions

Tmd2(i) Tdyn2(i) Tstat (12)

dTmd2(i) Tmd2(i) Tmd(i1) (13)

In these analyses, it is assumed that the reinforcement willnot experience any retraction during the seismic event.Consequently, negative values of Tmd2 will be set to zero(Fig. 2).

In a similar fashion to the LE approach, the incrementalsuperimposed force in the reinforcement (dTmd2) is calcu-lated by taking the difference between the superimposed

force in the reinforcement in the current time incrementfrom the integration approach (Tmd2(i)) and the prevailingvalue from the previous time step (Tmd(i1)).

Step 3: Seismic displacementAt each step in an acceleration time series, the LE ap-

proach (step 2-1, equations (2) (6)) can be used to deter-mine the upper-bound force and associated displacement thatcan be mobilised in the reinforcement. In addition, theintegration approach described herein (step 2-2, equations(7) (13)) provides a means to calculate the displacementand associated force that can be induced in the reinforce-ment over a finite time increment. The smaller of the forcesyielded from these two conditions controls the seismicallyinduced force and displacement in the reinforcement foreach time increment. In the analysis, this is determinedusing the following calculation process (Fig. 2).

dTmd(i) min dTmd1(i); dTmd2(i) (14)

d(i) min d1(i); d2(i)

(15)

where dTmdis the prevailing incremental superimposed forcein the reinforcement, and d is the prevailing incrementaldisplacement. The summation of the incremental displace-ments and induced forces in the reinforcement over theentire time series analysis provides the necessary informationto guide the design of a GRES. This is achieved by perform-ing the following calculations for each time increment (i).

Tmd(i) dTmd(i) Tmd(i1) (16)

(i) d(i) (i1) (17)

( , )X Yc c

Rr r

x

y

r

Fig. 3. Rotational failure of GRES, and seismic rotation anddisplacement components


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In addition to the resultant displacement along the shearplane (), it can also be instructive to look at the totalangular rotation () that occurs as a result of shaking (seeFig. 3). This can be achieved by performing the followingcalculation for each time increment (i).

(i)(i)

R

(i)Aedr

(18)

From the geometry shown in Fig. 3, the horizontal ( x) andvertical ( y) components of the resultant displacement canbe calculated using the equations

x(i) R(i)sinr

Aedr(i)sinr(19)

y(i) R(i)cosr

Aedr(i)cosr

(20)

At the end of a given shaking event, we have a certainseismic-induced superimposed force, Tmd, that remains in thereinforcement, and a corresponding displacement. Over time,this superimposed force will relax to its original value.However, the displacement will remain permanent.

The displacement-based design approach presented hereinis not intended to calculate the displacement of GRESs foranalysis purposes, but will provide the reinforcementstrength and pullout resistance for design purposes. Thedesign approach that is presented ensures that the reinforce-ment will not be ruptured that is, its stretching in terms ofGRES displacement will always be small, or else the rein-forcement will be ruptured or pulled out.

It is widely recognised that the shear strength of soilunder seismic conditions can be degraded by a shaking

event. To account for this degradation, the authors recom-mend the use of residual (or cv) for design; this contrastswith the recommendation made by FHWA (2009) and AASHTO (2007) to use peak for design purposes (althoughAASHTO limits to 408). The authors consequently recom-mend the use ofcv for use with the design approach that isproposed in this paper. The use of cv (or residual) in theseismic design of reinforced earth structures has been advo-cated by a number of researchers (e.g. Bolton, 1986; Jewell,1996; Liu & Ling, 2012) to account for the strengthdegradation and strain-softening of backfill soil that canoccur during a seismic event. Further discussion regardingthe use of peak against residual shear strength for GRESdesign purposes can be found in the papers by Leshchinsky

(2001) andLiu & Ling (2012).

RELATIONSHIP BETWEEN FORCE ANDDISPLACEMENT IN EQUIVALENT GEOSYNTHETICLAYER

As explained in the previous sections, an intermediate-stage tool is needed that can directly relate displacement,stiffness and force in the reinforcement. For a given rein-forcement stiffness, this tool will be utilised in the currentapproach to find the horizontal displacement ( x1) thatcorresponds to the superimposed force calculated from theLE equations (Tmd1). In parallel, the tool will also beemployed to determine the dynamic force (Tdyn2) that corre-

sponds to the horizontal displacement calculated using theintegration process ( x2). This relationship can be found bycalibrating the model based on experimental test results (e.g.Paulsen & Kramer, 2004) or by using numerical tools. In

this study, two-dimensional FE analyses were performedusing the program ABAQUS (2007) to develop a series offorcedisplacement curves.

As shown in Fig. 1(b), displacement of the sliding mass(which is assumed to be a rigid body) becomes feasible onlywhen the anchored reinforcement stretches. The reinforce-ment can stretch from pullout (slippage between the geosyn-thetic and surrounding soil), from true deformation (strain in

the geosynthetic itself), or from both mechanisms actingtogether. In this study, a series of FE analyses was per-formed to simulate the behaviour of a long geosyntheticthat is stretched to simulate force displacement behaviourof a geosynthetic that is not experiencing pullout. So, in theFE analysis, a pullout test with very long reinforcement wassimulated. Fig. 4, which is not drawn to scale, shows aschematic view of the model that was used in the analyses.If the reinforcement is embedded to a sufficiently longlength, its rear end will not move (i.e. it will not be able tofeel the load applied at the intersection with the log-spiral).

As shown in Fig. 4, FE analyses were used to simulatethe behaviour of a very long equivalent geosynthetic layerthat was embedded between two confining soil layers. Thegeosynthetic was modelled using a uniform mesh comprisingtwo-node linear two-dimensional truss elements that obeyedlinear elastic behaviour. The behaviour of the interface be-tween the geosynthetic and the reinforced soil was simulatedusing interface elements that utilised a penalty formulationapproach (ABAQUS, 2007).

The geosynthetic/soil interface angle was assumed to begiven by tan 0.8 tan, where is the interface frictionangle. The soil layers surrounding the geosynthetic weremodelled using four-node bilinear plane-strain quadrilateralelements that utilised the MohrCoulomb constitutive mod-el. Poissons ratio was assumed to be 0.3 for both thereinforced soil and the equivalent geosynthetic layer. TheYoungs modulus of the reinforced soil was assumed to be35 MPa. Parametric studies using this model showed that the

Poissons ratios that were chosen for the geosynthetic andsurrounding soil layers had little or no impact on the modelresults within the displacement range of interest.

In the FE analyses, incremental pullout displacementswere applied to one end of the geosynthetic reinforcementlayer at a very slow rate (1 mm/min), causing the reinforce-ment to stretch along its length. The reaction force at thefront edge of the geosynthetic (where the displacement wasapplied) was monitored, and the resulting data were used toplot the associated curve of pullout force against displace-ment. For each displacement increment that was applied, thereinforcement strains the most at the point of application,with the strain decreasing along the length of the reinforce-ment as load is shed into the surrounding soil layers. For

sufficiently long reinforcement, there is zero reinforcementstrain at the far end of the layer. At a given location, theload carried by the reinforcement can be determined by

Interface, 08 tan

Reinforced soil, ,

L H100

H/2

/10 H

Reinforcement,Jeq

H/2Reinforced soil, ,

Interface, 08 tan

Fig. 4. Schematic view of the FE model utilised for pulloutsimulation


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multiplying the local strain by the modulus of the reinforce-ment (J).

Numerous pullout simulations were performed over a widevariety of input parameters. However, as space is limitedherein, only the results for two friction angles of thereinforced soil ( 308 and 408) and two different heights(H 3 m and 6 m) are provided (Figs 5 and 6). Results arepresented for variations in the geosynthetic modulus (J) over

a range from 300 to 3000 kN/m per m. The results of theFE simulation for H 3 m and H 6 m are illustrated inFigs 5and6 respectively.

In Figs 5 and 6, an equivalent normalised modulus ofgeosynthetic, Jeq, was used, which is equal to

Jeq nJ

H2 (21)

where n is the number of geosynthetic layers in a GRES, Jis the geosynthetic modulus, is the unit weight of thereinforced soil, and H is the height of the earth structure.

The geosynthetic modulus can be calculated by multiplyingYoungs modulus, E, by the geosynthetic thickness.

The authors would like to reiterate that the FE analysis

utilised in the current study is just one of several possibleways to develop forcedisplacement curves (e.g. experimen-tal data, linear spring-slider models or more complex FEmodels). As explained previously, these curves act as anauxiliary tool within the proposed analytical framework.

ILLUSTRATIVE DESIGN EXAMPLEThe following example shows how the methodology that

is presented can be utilised for the seismic design of aGRES. For this example, a strong ground motion recordedat the Kakogawa station during the Kobe (1995) earth-quake (Mw 6.9) is used (Fig. 7). The peak groundacceleration (PGA) for this strong motion is 0.345g. As

shown, just the horizontal component is considered in thecurrent example. In order to design a GRES to resist thisapplied ground shaking, the approach that is presentedherein will be utilised to determine the superimposed force

Jeq 10

010008006004002

Tensile

force,

/T

H

2

010008006004002

Displacement, /

(b)

H

0

Jeq 10

Displacement, /

(a)

H

0

Tensile

force,

/T

H

2

0

05

10

15

20

25

20

100

30

0

04

08

12

16

20

24

28

32

20

100

30

Fig. 5. Tensile force against displacement for H 3 m from theFE pullout simulation: (a) 308; (b) 408

Jeq 5

010008006004002

Tensile

force,

/T

H

2

010008006004002

Displacement, /

(b)

H

0

Jeq 5

Displacement, /

(a)

H

0

Tensile

forc

e,

/T

H

2

0

03

06

09

12

15

18

10

50

15

0

04

08

12

16

20

24

10

50

15

Fig. 6. Tensile force against displacement for H 6 m from theFE pullout simulation: (a) 308 (b) 408


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in the reinforcement due to seismicity (Tmd), the resultantdisplacement along the shear plane (), and the totalrotation (). A reinforcement force distribution functionwill then be used to determine the maximum unfactoredreinforcement force under the static condition (Tmax-j) andthe superimposed force due to seismicity (Tmd-j) in eachof the reinforcement layers.

For purposes of this example, the following structure isconsidered: a wrapped-face GRES with 08, a horizontalcrest, H 6 m, and Sv 0.5 m, where Sv is the uniformvertical spacing between the reinforcement layers. The spa-cing from the base to the first layer is taken to be 0.5Sv,which leads to a total of 12 reinforcement layers (n 12).Wrapped facing does not generate toe resistance, Rh, andconsequently Rh is zero in this example. The contributoryarea of each layer is taken as the vertical height of thewrapped portion, and is consequently equal to 0.5 m. Forcalculating Tmax at each reinforcement layer, the location ofeach geosynthetic layer is approximated to be at the middleof the contributory area. A uniform (D H/2) distributionfunction was chosen to distribute the force among the rein-forcement layers. For the reinforced soil, two types of

cohesionless backfill material were examined ( 308 and408), which were both assumed to have a 20 kN/m3: Forpurposes of design, two types of geosynthetic were examined,with different stiffnesses (J 300 and 3000 kN/m per m).

To solve this problem, the step-by-step procedure shownin Fig. 2 was carried out. In order to select the appropriateFE forcedisplacement curves for use in this design process,the equivalent reinforcement stiffness for each type of rein-forcement is calculated based on the reinforcement type.Using a geosynthetic with J 300 kN/m per m Jeq will be

Jeq nJ

H2

12 3 300 (kN=m per m)

20 (kN=m3) 3 62 (m2 )

5 m1

Similarly, Jeq 50 m1 for a geosynthetic with

J 3000 kN/m per m.The failure surfaces for the different values that were

examined are shown in Fig. 8. Values of Tmd, and forfriction angles of 308 and 408 are shown in Figs 9 and 10respectively. Table 1 provides a summary of the results forthe cases that were investigated. As noted before, a uniformdistribution function (D H/2) was employed in this exam-

ple, and the maximum tensile force under the static condi-tion for the jth reinforcement layer (Tmax-j), and thesuperimposed force due to seismicity for the jth reinforce-ment layer (Tmd-j) were determined accordingly.

As explained previously, the current approach is intendedto determine the value of Tmd that corresponds to a certainseismic displacement. For comparison purposes, the Tmdvalues calculated using the proposed displacement-based ap-

proach are compared with those obtained using a pseudo-static approach. Following a pseudo-static methodology, theseismically induced force in the reinforcement can be calcu-lated by multiplying the weight of the failure mass (W) by

Kh

403530252015105

t: s

004

02

0

02

04

Component: 90, PGA 0345 g

Fig. 7. Strong ground motion recorded at Kakogawa stationduring Kobe (1995) earthquake (Mw 6.9)

30

40

Y

H/

120804

X H/

0

0

04

08

12

0, backslope: horiz.

Fig. 8. Failure surfaces for design example using 308

and 408

Tm

d:kN/m

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

0 5 10 15 20 25 30 35 40

:cm

30, 5 Jeq

30, 50 Jeq

10

:degrees

2

0

5

10

15

20

25

t: sec

0

10

2030

40

50

60

70

0

2

4

6

8

10

Fig. 9. Superimposed reinforcement force due to seismicity, Tmd,displacement,, and rotation, , for design example using 308


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the pseudo-static coefficient (Kh PGA/g). For 308,utilising the geometry of the failure surface shown in Fig. 8and a PGA of 0.345g (Fig. 7), one can calculate theseismically induced force in the reinforcement from a pseu-do-static approach as

WKh AH2Kh

0:2673 3 62 (m2=m)3 20 (kN=m2) 3 0:345

66:4 kN=m

where A is the area of the failure mass determined from Fig.8, and the other variables are as defined previously. Using asimilar calculation for 408, WKh 54.1 kN/m. As shown

inTable 1, for the cases that were examined in this example,the proposed displacement-based approach yields Tmd valuesthat are 36% to 182% smaller than those that result from apseudo-static approach.

SUMMARY AND CONCLUSIONSThe significant importance of post-earthquake serviceabil-

ity of GRESs has led to an increasing trend towards the useof displacement-based design methods within the past dec-ade. This paper presents a new analyticalnumerical frame-work for the displacement-based design of GRESs, whichassesses the potential for earthquakeinduced displacementsvia a rotational failure mechanism. While the majority of

the existing analytical approaches for displacement-basedseismic design of GRESs are developed by considering onlya translational mode of failure (i.e. external sliding stability),the proposed approach provides a rational framework forassessing the seismic displacement of GRESs due to internalstability (rotational movement). This failure mechanism maycontrol the required tensile strength of the reinforcement forGRESs. Moreover, an internal rotational failure mode candegenerate to a translational one should it be more critical;however, the reverse is not true, which makes rotationalfailure a more generic mechanism.

For design purposes, in order to determine the super-imposed force in the reinforcement due to seismicity and itsassociated displacement, the proposed approach examinestwo limiting conditions.

(a) The upper-bound force that can be mobilised in thereinforcement, as determined by pseudo-static LE. For agiven earthquake acceleration, this would be the maxi-mum force that can be induced in the reinforcement bythe inertial response of the sliding body.

(b) The force that can be induced in the reinforcement by agiven earthquake acceleration applied over a finite timeincrement. This second limiting condition examines thetime limitation effect, which accounts for the transientcharacteristics of the applied ground motion.

The smaller of the forces yielded from these two conditionscontrols the seismically induced force and displacement inthe reinforcement for each time increment. In order to relate

the force and displacement in the geosynthetic reinforce-ment, a set of pullout simulations was performed using FEanalysis for various geosynthetic stiffnesses. The summationof the incremental displacements and induced forces in thereinforcement over the entire time series analysis providesthe necessary information to guide the design of a GRES.

Using a Kobe earthquake record, a design example wassolved to illustrate the application of the proposed method.The superimposed force due to seismicity, seismic displace-ment, and seismic rotation were calculated for various soilfriction angles and geosynthetic stiffnesses. For each casethat was examined, a uniform distribution function is used todetermine the required unfactored geosynthetic strength inthe individual reinforcement layers. For the examples that

were provided, the proposed displacement-based approachyielded superimposed forces in the reinforcement due toseismicity that were 36% to 182% smaller than those thatresulted from a pseudo-static approach.

Tm

d:kN/m

403530252015105

:cm

4035302520151050

10

2:degrees

4035302520151050

t: s

40, 5 Jeq

40, 50 Jeq

00

10

20

30

40

50

0

5

10

15

20

25

30

0

1

2

3

4

5

Fig. 10. Superimposed reinforcement force due to seismicity,Tmd,displacement,, and rotation, , for design example using 408

Table 1. Summary of results for design example

Case : degrees J: kN/mper m

Jeq: m1 Tstat: kN/m WKh: kN/m Tmd: kN/m Tdyn: kN/m Tmax-j+ Tmd-j:

kN/m: cm 3 102:

degrees

1 30 300 5 130.1 66.4 23.5 153.6 12.8 8.5 60.5

2 30 3000 50 130.1 66.4 23.5 153.6 12.8 1.7 12.33 40 300 5 83.0 54.1 32.0 115.0 9.6 3.9 26.24 40 3000 50 83.0 54.1 39.6 122.6 10.2 1.1 7.2


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ACKNOWLEDGEMENTSThis material is based upon work supported by the

National Science Foundation under Grant No. CMMI-0844836. This National Science Foundation grant partiallysupported the first and third authors in conjunction with theirwork on this project. The authors would also like to extendtheir thanks to Mr Fan Zhu for his valuable assistance withthe formulation and programming that was conducted during

this project.

NOTATIONA log-spiral constanta(i) acceleration perpendicular to radius of

log-spiral failure surface at timeincrement i

ah(i), av(i) horizontal and vertical accelerations attime increment i

D elevation of line of action of resultantreinforcement force, measured from toeof structure

E Youngs modulusg gravitational acceleration

H height of slope

i increment in earthquake accelerationtime series

J, Jeq geosynthetic modulus, equivalentgeosynthetic modulus

Kh(i), Kv(i) horizontal and vertical seismiccoefficients at time increment i

n number of geosynthetic layersq uniform surcharge acting on crest

R log-spiral radiusRh horizontal resistance of facing at bottom

of slopeRr radius where the trace of log-spiral

intersects the equivalent reinforcementforce

Sv uniform vertical spacing between

reinforcement layersTdyn( i), Tdyn1(i), Tdyn2(i) seismic force in reinforcement at timeincrement i (prevailing value), fromlimit equilibrium equations and fromintegration approach respectively

dTdyn(i), dTdyn1(i), dTdyn2(i) incremental seismic force inreinforcement at time increment i(prevailing value), value from limitequilibrium equations, and value fromintegration approach respectively

Tmax-j maximum tensile force under staticcondition for jth reinforcement layer

Tmd(i), Tmd1(i), Tmd2(i) superimposed force in the reinforcementdue to seismicity at time increment i(prevailing force), from limitequilibrium equations and from

integration approach respectivelydTmd(i), dTmd1(i), dTmd2(i) incremental superimposed force in

reinforcement due to seismicity at timeincrement i (prevailing value), valuefrom limit equilibrium equations, andvalue from integration approachrespectively

Tmd-j superimposed force due to seismicity forjth reinforcement layer

Tstat resultant reinforcement force understatic loading

dt time step in earthquake accelerationtime series; dt t(i) t(i1)

v(i) perpendicular velocity of rotating massat time increment i

v(i) angular velocity of rotating mass at time

increment iW weight of failure mass

Xc, Yc coordinates of pole of log-spiral inCartesian coordinate system

backslope angle of crestr angle of rotation to point where radius

intersects equivalent reinforcement force1, 2 angle of rotation to points where log-

spiral enters and exits slope unit weight of soil

(i), 1(i), 2(i) resultant seismic displacement at timeincrement i (prevailing value), valuefrom limit equilibrium equations, and

value from integration approachrespectively

d(i), d1(i), d2(i) incremental resultant seismicdisplacement at time increment i(prevailing value), value from limitequilibrium equations, and value fromintegration approach respectively

x(i), x1(i), x2(i) horizontal component of seismicdisplacement at time increment i(prevailing value), value from limitequilibrium equations, and value fromintegration approach respectively

d x(i), d x1(i), d x2(i) incremental horizontal component ofseismic displacement at time increment i(prevailing value), value from limitequilibrium equations, and value from

integration approach respectively y(i), y1(i), y2(i) vertical component of seismic

displacement at time increment i(prevailing value), value from limitequilibrium equations, and value fromintegration approach respectively

d y(i), d y1(i), d y2(i) incremental vertical component ofseismic displacement at time increment i(prevailing value), value from limitequilibrium equations, and value fromintegration approach respectively

(i), 1(i), 2(i) seismically induced rotation at timeincrement i (prevailing value), fromlimit equilibrium equations and fromintegration approach respectively

d(i), d1(i), d2(i) incremental seismically induced rotationat time increment i (prevailing value),value from limit equilibrium equations,and value from integration approachrespectively

cv constant-volume angle of frictiondesign design internal angle of frictionpeak peak angle of friction

residual residual angle of frictiond tan slope batter (slope face

inclination 908 )

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