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INVESTIGATION OF SEISMIC RESPONSE OF REINFORCED SOIL RETAININGWALLS

Kianoosh HatamiRoyal Military College of CanadaKingston, Ontario K7K 7B4

Richard J. BathurstRoyal Military College of CanadaKingston, Ontario K7K 7B4

ABSTRACT

Dynamic response of a segmental (modular block) retaining wall system to recorded ground mo tions is inves tigated. The magnitudeand characteristics of wall response are compared to those obtained under harmonic input base acceleration. The calculated maximumlateral displacem ent and reinforcement load of the segmental retaining wall mode1 subjected to a single frequency, harmonic inputacceleration were considerably larger than the corresponding values obtained using a number of earthquake accelerograms with

comparable predominant frequencies. It is concluded that the random characteristic of actual ground acceleration may partly explainthe relatively good performance of reinforced-soil retaining wall systems that were designed without seismic considerations or at bestusing simple pseudo-static limit equilibrium methods. Nevertheless, it was found that low-frequency ground motions with highintensity values can result in significant structural response magnitude of short-period retaining wall systems.

KEYWORDS

Retaining walls, Reinforced soil, Seism ic response, Earthquake characteristics, FLAC

INTRODUCTION

Reinforced soil retaining walls are composite structurescomprised of horizontal layers of geosynthetic or metallic

reinforcement extending into a soil backfill and typicallyattached to a hard facing. The hard facing may comprise full-height reinforced concrete panels, articulated incrementalconcrete panels, gabions or modular block systems. Wheregeosynthetic polymeric reinforcement m aterials such asgeogrids have been used, these systems have proven to be verycost effective particularly with respect to traditional gravity-type structures. Within the family of reinforced so il walltechnologies, reinforced segmental (modular block ) retainingwalls constructed with a dry-stacked column of masonryconcrete units to form the facing are the most economical. Inaddition, these walls offer ease of construction for thecontractor and a wide range of aesthetic appearances or the

architect.Design and analysis methods, albeit conservative, are wellestablished for reinforced soil walls under static loadingconditions (FHWA 1996, NCMA 1996, PWRI 1992,AASHTO 1998). However, increasing numbers of reinforcedsoil retaining wall systems are being constructed inseismically active areas. Simultaneous ly, m ore significantearthquake ground motion events are being recorded aroundthe world and national seismic hazard maps (e.g., NBCC

1995, Leyendecker 2000, Leyendecker et al. 2000) are beingcontinuously modified to accommodate the most recent majorseismic records and to reflect updated seismic hazard levels.Reinforced soil retaining walls have generally shown a good

performance record under seismic loading when comparedconventional gravity retaining wall systems (Tatsuoka et1995, Bathurst and Alfaro 1997). Nevertheless, seismic designmethods are not well advanced for reinforced soil walstructures. The most common approach for seismic designreinforced soil retaining walls is to use Mononabe-Okabtheory and a selected peak ground acceleration to calculatemodified lateral earth pressure coefficient (Bathurst andAlfaro 1997). Typically, the backfill dynamic incremental lo(i.e., the load in addition to the static part) is calculated andempirically partitioned into soil reinforcemen t layer loads (e.AASHTO 1998, Bathurst 1998, PWRI 1992). The shorcomings of pseudo-static methods are recognized inUnited States where these methods are limited to sites wherethe peak ground acceleration is not expected to exceed 0.29g.For greater accelerations, sliding block displacement methodshave been proposed for reinforced soil walls (Cai and Bathurst1996, Ling et al. 1997). However, displacement methodslimited to internal and external sliding mechanism s and doaddress potential internal failure mechanisms suchreinforcement over-stressing or pullout.

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Proceedings: Fourth International Conference on Recent Advances in Geotechnical Earthquake Engineeringand Soil Dynamics and Symposium in Honor of Professor W.D. Liam FinnSan Diego, California, March 26-31, 2001

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ModularBlock Facing

0.30 7.00L

\ 1 1

-Iiiiiiiiiiiiiiiii i i i i i i i i i I,

3.60

Interfaces1 ,

7.50 11

GO Fixed Bounda; (acceleratedunder dynamic loading)

Fig. 1 Discretised model of segmental modular block) retaining wall system at end of conslruction dimensions are inmetres).

Due to the shortcomings of the limit-equilibrium methodsdescribed above, numerical methods that allow the entireresponse of a structure to be simulated hold promise as a toolto develop new seismic design guidelines for reinforced soilretaining wall structures and to quantify levels of safetyagainst collapse and serviceability under site-specific seismicloading.As a first step in this direction, the writers investigated theinfluence of wall height, backfill width, reinforcementproperties and toe restraint condition among other parameterson characteristic frequency and seismic respons e of idealizedfull-height propped-panel walls using a numerical approach(Bathurst and Hatami 1998, Hatami and Bathurst 2000). Tosimplify the dynamic loading, the wall models wer e subjectedto a variable-amplitude, harmonic input ground motion.Numerical simulation results showe d that retaining wall

models developed significant lateral displacement andreinforcement load when subjected to a single-frequency baseexcitation in the vicinity of retaining wall fundamentalfrequency. The wall fundamental frequency was predictedusing available c losed-form solutions and interpretation of theresponse of the numerical wall model to a range of inputfrequencies.This paper extends the earlier work by the writers byexamining the response of a reinforced soil retaining walls to anumber of recorded earthquake ground motions. Themagnitude and characteristics of the retaining wall response toreal accelerogra ms viz. single-frequency, harmonic inputaccelerations are also presented and discussed.

NUMERICAL MODEL

General Description and Geometry

The segmental (modular block) wall model d epicted in Fig. 1was used as the example structure in the current paper. The

wall is 3.6m high and includes 24 concrete block couwhich are stacked to produce an 8 batter angle fromvertical. The wall geometry represents a full-scale segmeretaining wall that was recently constructed at the RMilitary C ollege and tested under surcharge loading (Bathet al. 2000). The wa ll consists of six geosynthereinforcement layers with a length of 2.52m from the frothe facing into the backfill. This reinforcement length provthe minimum reinforcement length to wall height ratio L/H=0.7) recomme nded by current design codes for sstability (e.g., FHWA 1996, AASH TO 1998). The vertspacing, S ,, between the reinforcement layers is constant equal to 0.6m which is the maximum spacing permitted foexample wall structure ac cording to AASHT O (1guidelines. The backfill width is extended to 7.25m (avevalue over the depth) behind the facing giving a width

height ratio, B /H=2 for the backfill model. A fixed boundcondition representing a rigid foundation is assumed abottom of the backfill.

Material Properties

The backfill soil is modelled as a purely frictional, elaplastic material with Mohr-Coulomb failure criterion. Thebackfill unit weight is assumed as y =17 kN/m. The modulus and shear modulus values o f the backfill m atassumed during wall construction are calculated using stress dependent, hyperbolic model proposed by Duncan

1980). Thereafter, constant linear elastic-plastic properwere assumed in order to reduce computation time anensure numerical stability for all seismic simulation runs. material properties assumed for the backfill soil are presein Table I. The reinforcement is modelled using linear elaperfectly plastic FLAC cable elements (seeNumericaApproach below) with negligible compressive strength.

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Table 1. Backfill soil properties.

Stiffness Properties(Hyperbolic model)

K, (Elastic modulus number)Kb (Bulk modulus number)n (Elastic modulus exponent)m (Bulk modulus exponent)Rr (Failure ratio)v (Poissons ratio)

Strength PropertiesI (Peak friction angle)c (Cohesion)v (Dilation angle)Notes: So il stiffness parameters are all dimensionless.

Value

200020000.250.250.730.15

460

6

reinforcement stiffness is assume d to be J=lOO O kN/m whichrepresents a typical stiff geogrid reinforcem ent.

Numerical Approach

The numerical simulations were carried out using the programFLAC (Ita sca 1998). The retaining wall model wa s assumed in

plane-strain condition and was constructed in layers. Thebackfill and wall facing were elevated in lifts of 0.15m and thereinforcement layers we re placed in the model at designatedelevations. The numerical model a t each stage was solved toequilibrium with a prescribed tolerance before placing the nextfacing block, soil lift and reinforcement layer. After the wallmodel was constructed (Fig. l), it was subjected to differenthorizontal input accelerations across the foundation. Theexcitation inputs were applied in terms of velocity historieswith base line correction to ensure zero displacement at thebase at the end of shaking. The base input velocities weresimultaneously applied to the far-end boundary of the backfillmodel ba sed on the assumption that the acceleration in the

Table 2. Fundamental frequency of model wall fromnumerical (FLAC) modelling and from closed-form solutions.Method f, (Hz)Matsuo and Ohara (1960) (Neglecting vertical 4.65dynamic pressure in the backfill - o, =0)Wu and Finn (1996) 4.71Free vibration response FLAC) 5.2

Scott 1973) 6.44

Matsu o and Ohara (1960) (Neglecting vertical 6.92vibration amplitude in the backfill - ~0)Wood (1973) 7.5

Richardson (1978) 10.58

Note: The methods are described by Hatami and Bathurst 2000).

Table 3. Range and mean values of classified ground motionrecords (N aumoski et al. 1993).Ensemble VH NH NI NL NVLAN 2.63 1.60 0.82 0.62 0.36Range to to to to

3.52 :q43 1.21 0.79 0.59(AN),,, 3.03 1.98 1.02 0.70 0.48

0

0 20 40 60 80 100

f I Hz)

Fig. 2 Fundamental frequency of modular block retainingwall model from displacement response to sinusoidalimpulse.

backfill depth was uniform at a sufficient distance fromwall facing.

SEISMIC LOADING

Wall Fundamental Frequency

The fundamental frequency of the model segmental wallevaluated before it was subjected to the actual recoground motions. One full sinusoidal impulse with the pT=O.ls was applied at the base and far-end boundary omodel. The fundamental frequency of the wall from vibration response of facing lateral displacement determined to be fi=5.2 Hz (Fig. 2). From the closed-fsolutions for fundamental frequency of rigid retaining shown in Table 2, the formula proposed by Wu and (1996) ga ve a value closest to the numerical predicted valcan also be noted that the observed wall fundamefrequency value falls between limiting values from soluby Matsu o and Ohara in Table 2 (see also Hatami and Bat2000).

Earthouake Records

A set of 6 recorde d ground motions were selec ted as accelerog rams to the retaining wall model. These records chosen from a database that is classified according toaccelerog ram AN ratio values (Naumoski et al. 1988, where A is the peak ground acceleration in g (i.e., acceleraof gravity) and V is the peak ground velocity of the recoground motion in m/s. Results of a number of studies indthat the AN ratio of a recorde d ground motion correlates its frequency content (Seed et al. 1976, McGuire Sawada et al. 1992). The database of records by Naumoskal. (1993) contains a total of 75 recorde d ground motions around the world that are categorise d into five different gwith 1 5 records in each group according to their AN

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s0.0 .

2 1:3

-0.5 1.

0 10 20 30 40 50

_ ._ __ ___~ ~_~__

0 5 10 15

0 10

_ _20 30 40 50 b

- 0 . 2 I -

O 10 20 30 40 50 60

432 l- ~~ I

0 5 10 15

-0.2 l~___--.-.~--..---.-.. __ ~~ - ._~ ._._.J

5 10 15

t s)

Fig. 3 Input acceleration histories applied to the modu larblock wall model number in each jgure refers to groundmotion record in Table 4).

values (Table 3). The notations VH , NH, NI, NL and NV L in

the table stand for very high, new high, new intermediate, low and new very low categories in terms of the AiV values of the records , respec tively. The new groups a rrefined versions of a previous classification of a smdatabase comprising of only 3 different ensembles (Naumet al. 1988). The entire e nsemble of records in the newincludes seismic events with magnitudes between M= 5M=S and epicentral distances between 3 and 500 km. Rewith larger A/V ratio values are statistically associate d moderately strong to strong earthquak es at short epice

distances. Reco rds with lower A& ratio values typicorrespond to large earthquak es at large epicentral dist(Naum oski et al. 1993). Accordingly, record s with higherratio values typically have higher predominant frequenThe perfect positive correlation between the AiV ratio and the predominant frequency of an acceleration recoevident for the case of a simple harmonic motion (e.g. re7 and 8 in Table 4). Since an accelero gram can be geneconsidered as a summation of harmonic componentscorrelation between A/V ratio and predominant frequencygeneral acc eleration record can be expec ted.The ground motions chosen for this study are shown in 4 and plotted in Fig. 3. Ground motion records from theNH and NI categories were chosen for seismic input tmodel retaining wall. These categories were selected bethey represent ground motion record s with higher predomifrequencies that appear to be more aggressive to the period retaining wall model under study. Each of the rein Table 4 has a relatively narrow and well-defined peak spectral acceleration curves as given by Naum oski (1993). The data in Table 4 includes three main charac teriof the ground motion records namely, duration, intensity frequency content. In theduration section, the param eters tr and td represent the time interval between recorde d points, the total record ed time of the accelero gram ancalculated duration of strong ground motion of the rerespectively. The duration of strong ground motion calculated according to the method propose d by TrifimacBrady (19 75). One of the advantages of this method (e.g.,the brackete d definition propose d by Bolt 1969 ) is thacalculated duration is not affecte d by scaling of the reaccording to their peak acceleration amplitude. In theintenssection of the table, V, is the peak ground velocity orecord a fter scaling to a reference peak ground acc elerA=O. 15g. All the input ground motion record s were s caledc~rnm~n peak ground acceleration magnitude to isolateinfluence of other ground motion characteris tics on response. Param eter IA in Table 4 denotes the intensity ofground motion re cord according to the following equpropose d by Arias (1969):

I,4 5) =cos- 5) IT.

s_d i,t)dt (g l-5The parameter IA* is a modified intensity measu re equa(IA)*. This intensity measure is introduced as an additparam eter because, in contrast to the Arias intensity, linearly propo rtional to the ground accelera tion ampli

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0.06

0 10 20 30 40 50 60

Fig. 4 History offacing lateral displacement at the top ofwall subjected to input ground motions in Table 4 (all scaled to A=O.l5g).

The intensity parameters IA and IA 2 are calculated using aviscous damping ratio value of 5 =lO . The parameter fs inthe table is the predominant frequency of each record which isdetermined from its Fourier transform. Two sinusoidal

functions (records 7 and 8 in Table 4) were also used as inputmotions to compare the wall response to harmonic viz.recorded ground accelerations (Fig. 3). The sinusoidal inputmotions are defined by:

ii,(t) = Jpe- t c sin(27t-f, t) (2)

Where a = 5.5, l3 = 31 and < = 12 are constants, t is time andfs is the frequency of base acceleration. The resulting peakacceleration amplitude, A, using the selected values forconstant parameters is 0.15g.

RESPONSE OF RETAINING WALL TO GROUNDMOTIONS

The response of the model modular block (segmental)retaining wall system to input ground motion is examined interms of facing lateral displacement and reinforcement load.

Table 4. Ground motion records used in the current study.Record No. of Duration Intensity ( ) Frequency Foundation

No. Naumoski Naumoski data At tT f A(*) v(*) v (1)n I*( ) IAl * ( ) fs fl Conditionet al. et al. points set set set g m/s m/s m/s (m/s) Hz g.s/m

(1988) (1993)1 H9 - 1851 0.02 37.0 12.60 0.146 0.085 0.087 0.190 0.436 5.0 1.72 Rock2 H14 VH15 578 0.02 11.6 5.44 0.042 0.016 0.057 0.111 0.333 6.4 2.63 Rock3 H2 NH2 2202 0.02 44.0 6.74 0.434 0.255 0.088 0.096 0.309 2.8 1.70 Rock4 H7 NH7 612 0.02 12.2 3.52 0.084 0.044 0.079 0.138 0.372 5.2 1.91 Rock5 13 N13 2719 0.02 54.4 30.54 0.156 0.157 0.151 0.474 0.689 1.5 0.99 Rock6 114 N113 6001 0.01 60.0 14.80 0.105 0.116 0.166 0.281 0.530 1.4 0.91 Rock7 Sl 400 0.025 10.0 2.15 0.150 0.080 0.080 0.265 0.515 3.0 1.88 -8 s2 480 0.0125 6.0 2.15 0.150 0.040 0.040 0.265 0.515 6.0 3.75 -

Notes: 1) Peak ground velocity V, and Arias intensity IA including the modified form IA ) are reported for scaled records. 2) The reported correspond to the records before being scaled to A=O. 1 Sg

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Figure 4 shows the histories of lateral displacement at the topof the facing column when the wall was subjected to thescaled input base accelerations in Table 4. The parameter Xdin Fig. 4 denotes the wall lateral displacement due to dynamicloading of ground motion in excess of the amount of lateraldisplacement at the end of construction. The effects of groundmotion characteristics on wall response are discussedseparately in the following sections.

Wall Lateral Displacement

Effect of Predominant Freouencv. The effect of groundmotion fundamental frequency on wall response can be readilyexamined by comparing the displacement histories of records7 and 8 in Fig. 4. The records have iden tical Arias intensityand strong motion duration (Table 4). The lower frequencyrecord 7 has a larger peak ground velocity. However, theresults of Fig. 4 clearly show that record 8 with a predominantfrequency closer to the fundamental frequency of the wall(fs=6 Hz vs. f,=5.2 Hz) induces a significantly larger lateraldisplacement in the wall (0.035m vs. 0.019m).Earthquake records 2 and 3 have comparative characteristicsof predominant frequency, peak ground velocity (V,), scaled

intensity and strong ground motion duration that are closest tothe corresponding values for harmonic records 8 and 7,respectively. However, maximum wall displacement due torecord 2 (0.015m) is not much greater than the maximumdisplacement value resulting from earthquake record 3(0.012m). Quantitatively, the difference in wall response torecords 3 and 2 is proportionally smaller than the difference inwall response to records 7 and 8. It follows that in the case ofactual recorded ground motions -as opposed to harmonic inputaccelerations- the predominant frequency is not the soledominant parameter that determines the magnitude of wallstructural response.The effect of ground motion predominant frequency on wall

(cl (44 (b).I_~1

T4 1

----7

- - - - .

----?

:,~

- - - - -,

-----

------7irn-----

-----

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response can be examined by comparing displacement resultsfor records 3 and 4. The records have comparable scaled peakvelocity values. Since the records are scaled to the same peakground acceleration, the A/V ratios of the records are alsoclose in magnitude. The magnitude of maximum wall responseto record 4 (0.021m) is considerably greater than the responseto record 3 (0.012m) even though the intensity (i.e., IA ) record 4 is only about 20 larger than the intensity of record3. The strong motion duration of record 4 is also smaller thanthat of record 3. However, this does not appear to be a

dominating factor for the magnitude of maximum wadisplacement because neither of the two displacement historieshows a gradual growth of displacement esponse with time.

Harmonic Motion vs. Recorded Earthquake Accelerogram.Comparison of wall displacement esponse o harmonic inputs7 and 8 against records 1 and 4 shows that the singlefrequency harmonic records induced relatively large wallresponse lthough their frequencies are not as close to the wallfundamental frequency as the predominant frequencies ofrecords I and 4. This observation is in agreement with theresults of Bathurst and Hatami (1998) who compared thedynamic response of a 6m-high propped-panel model wall toharmonic and 1940 El Centro earthquake acceleration ecords.It may be concluded that single-frequency input accelerationsare typically more aggressive to the structure than actualearthquake records with identical predominant frequency andpeak ground acceleration. This conclusion s also supported bcomparing the displacement esponse of the wall to records 8and 2 which have practically equal predominant frequencies.The magnitude of I,+ or record 8 is about 55 greater thanthe I* value of record 2. However, the magnitude ofmaximum wall displacement response to harmonic inputacceleration 8 (0.035m) is substantially larger than themaximum wall response o accelerogram 2 (0.015m).

(4.-.-__5

.--- ---

(Cl (8) (h)6

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1 I I I a,

----

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0 2 4 6 0 2 4 6 0 2 4 6 0 2 4 6 0246024602460246T,,, Wm) T,,, &N/m) T,,, WW T,,, WW T,,, &N/m) T,,, W/m) T,,, VW T,, WJW

Fig. 5 Distribution of maximum reinforcement incremental load over the height of wall subjected to different input ground motions(number n eachjigure refers to ground motion record in Table 4).

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Effect of Inter&v and Strong Ground Motion Duration. Theeffect of ground motion intensity and strong motion durationcan be examined by comparing the wall response to records 5and 6. The two records have comparable scaled peak velocityand A/V ratio values. The predominant frequencies of the tworecords are almost equal and considerably lower than thefundamental frequency of the model wall. The IA * value ofrecord 5 is about 30 larger than that of record 6. However,the strong ground motion duration of record 5 is significantlyhigher than the duration of record 6. Figure 4 shows that the

wall displacement response to record 5 is substantially largerthan its response to record 6. The wall displacement responseto record 5 is the only response history in Fig. 4 that shows aclear influence of the ground motion duration on themagnitude of wall response. Comparison of wall response toother ground motion records in Fig. 4 does not show a cleardependence of the wall response magnitude on either theground motion duration (e.g., cf. records 1 and 4) or intensity(e.g., cf. records 2 and 6). The effects of ground motionintensity and duration on the magnitude of wall response arediff icult to isolate for records with a predominant frequencyclose to the fundamenta l frequency of the wall. For example,comparison of wall displacement response to records 1 and 4shows that the magnitudes of maximum wall displacement forwalls subjected to acceleration records with predominantfrequencies in the vicinity of the wall natural frequency arenot significantly different (0.02 lm vs. 0.026m) in spite of theirdifferent intensity and strong motion dura tion values. It maybe concluded that a short-period retaining wall structuresubjected to a low-frequency ground acceleration can stilldevelop a large response as a result of the combined effect oflarge intensity and duration of the ground motion.Records classified as having intermediate A/V ratio values(i.e., records 5 and 6) typically have larger strong motionduration values than records in the high A/V ratio category(Table 4). Therefore, reinforced-soil retain ing walls that maybe subjected to ground motions from large distant earthquakes(e.g., in the range of a few hundred kilometres from a majorfault) can be susceptible to failure or excessive deformationunder seismic loading.

Reinforcement Load

Figure 5 shows plots of reinforcement incremental load (i.e.,the axial force in the reinforcement due to dynamic loadingonly) in the retain ing wall subjected to the scaled groundmotions listed in Table 4. The relative performance of the wallmodel based on displacement response for the 8 base

excitation cases applies equally to the relative performance ofthe wall based on maximum reinforcement load. In addition ,inspection of maximum reinforcement loads in Fig. 5 showsthat for ground motion records with fs I fr (i.e., all recordsexcept 8 and 2), the distribution of reinforcement incrementalload over the wall height can be considered to be essentiallyuniform. The lowermost reinforcement layer may develop asignificant load under a severe dynamic loading (Fig. 5e). Asimilar observation was made in a previous study on propped-panel type retaining wall models (Bathurst and Hatami 1998).For the cases with fs > fi in Fig. 5 (i.e., input records 8 and 2),

the distribution of maximum reinforcement incremental loashows comparatively low values at the toe and at an elevationof about 0.6H above the toe. This distr ibution pattern iconsistent with the second vibration mode shape of acantilever shear beam model. The reinforcement incrementalload otherwise shows an overall parabolic distribution shapewith the largest magnitude of reinforcement load at about mid-height of the wall.

CONCLUSIONS

Seismic response of a segmental (modular block) retain ingmodel to recorded and harmonic ground motions is studiedusing a numerical modelling approach. The wall response ispresented in terms of lateral displacement histories of the wallfacing and maximum values of reinforcement incrementalload . The displacement responses of the segmental retain ingwall model subjected to single frequency, harmonic inputaccelerations were considerably larger than the responsemagnitude to earthquake records with comparablpredominant frequencies. This result confirms a statementmade by the writers in an earlier parametric study (Bathurstand Hatami 1998) that the use of simple harmonic functions tosimulate the seismic behavior of reinforced soil walls may buseful to establish the relative performance of differentretaining wall systems, although the magnitude of response islikely to be excessive. The predominant frequency of scaledearthquake ground motion records, in general, showed adominant effect on the magnitude of wall response to seismicloading . In addition, it was found that low-frequency groundmotions with high intensity and strong motion duration valuecan result in significant structural response magnitude ofshort-period retaining wall systems. For the combination olow intensity earthquake ground motion with predominantfrequency below the fundamental frequency of the structure,the maximum incremental reinforcement loads were uniformlydistributed over the wall he ight. This result is considered useful observation for the future refinement of empirical-basedseismic design methods for reinforced soil retain ing walls.The numerical model in the current study was selected tosimulate a segmental retaining wall structure. Nonetheless, theconclusions from this preliminary study are believed to bapplicable to other types of hard-faced concrete reinforced soilwall structures where the facing can add considerable stiffnessand toe restraint to the composite gravity mass. Accordingly, iis concluded that the random characteristic of actual groundacceleration may explain the documented good performanceof different reinforced-soil retain ing wall systems durinrecent earthquakes (Tatsuoka et al. 1995, Bathurst and Alfaro1997).

ACKNOWLEDGEMENTS

Funding for the research program described here has beenprovided by the Natural Sciences and Engineering ResearchCouncil of Canada and grants from the Department ofNational Defence (Canada).

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