See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/238790017 Seismic Response Analysis of a GeosyntheticReinforced Soil Retaining Wall Article in Geosynthetics International · August 1998 DOI: 10.1680/gein.5.0117 CITATIONS 102 READS 169 2 authors: Some of the authors of this publication are also working on these related projects: Peformance based design of reinforced soil structure View project Sensor-Enabled Geosynthetics (SEG) View project Richard J. Bathurst Royal Military College of Canada 232 PUBLICATIONS 5,281 CITATIONS SEE PROFILE Kianoosh Hatami University of Oklahoma 78 PUBLICATIONS 980 CITATIONS SEE PROFILE All content following this page was uploaded by Richard J. Bathurst on 06 June 2017. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately.GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 127 Technical Paper by R.J. Bathurst and K. Hatami SEISMIC RESPONSE ANALYSIS OF A GEOSYNTHETIC-REINFORCED SOIL RETAINING WALL ABSTRACT: The paper reports results from numerical experiments that were carried out to investigate the influence of reinforcement stiffness, reinforcement length, and base boundaryconditiononthe seismic response ofanidealized6mhighgeosynthetic-reinforced soil retaining wallconstructed witha verystiff continuous facing panel.The numericalmodels were excited at the foundation elevation by a variable-amplitude harmonic motion with a frequency close to the fundamental frequency of the reference structure. The two-dimensional, explicit dynamic finite difference program Fast Lagrangian Analysis of Continua (FLAC) was used to carry out the numerical experiments. Numerical results illustrate that the seismic response of the wall is very different when constructed with a base that allows the wall and soil to slide freely and when the wall is constrained to rotate only about the toe. Parametric analyses were also carried out to investigate the quantitative influence of the damping ratio magnitude used in numerical simulations and the effects of distance and type of far-end truncated boundary. The response of the same wall excited by a scaled earthquake recordwas demonstratedtopreserve qualitative features of walldisplacement andreinforcement loaddistribution as that generatedusing the reference harmonic ground motionapplied at3Hz.The lessons learnedinthis studyare ofvalue toresearchers usingdynamic numerical modeling techniques to gain insight into the seismic response of reinforced wall structures. KEYWORDS: Seismic analysis, Numerical modeling, Parametric analysis, Finite difference, FLAC, Retaining walls, Geosynthetic reinforcement, Metallic reinforcement. AUTHORS: R.J. Bathurst, Professor, and K. Hatami, Research Associate, Department of Civil Engineering, Royal Military College of Canada, P.O. Box 17000, STN Forces, Kingston, Ontario, K7K 7B4, Canada, Telephone: 1/613-541-6000, Ext. 6479, Telefax: 1/613-545-8336, E-mail: [email protected]. PUBLICATION: Geosynthetics International is published by the Industrial Fabrics Association International, 1801 County Road B West, Roseville, Minnesota 55113-4061, USA, Telephone: 1/612-222-2508,Telefax: 1/612-631-9334. Geosynthetics Internationalis registered under ISSN 1072-6349. DATES: Originalmanuscriptreceived10February1998,revisedversionreceived 9March 1998 and accepted 17 March 1998. Discussion open until 1 September 1998. REFERENCE: Bathurst, R.J. and Hatami, K., 1998, “Seismic Response Analysis of a Geosynthetic-Reinforced Soil Retaining Wall”, Geosynthetics International, Vol. 5, Nos. 1-2, pp. 127-166.BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 128 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 1 INTRODUCTION In North America, geosynthetic- and metal strip-reinforced soil walls are routinely designed using limit-equilibrium pseudostatic methods for sites with peak horizontal ground accelerations ≤ 0.29g (AASHTO 1996; FHWA 1996). A limitation of pseudostatic methods is that they cannot consider the effects of duration of seismic loading, frequency content, acceleration amplification through the backfill soil, and foundation condition on the development of reinforcement loads and structure deformations (Bathurst and Alfaro 1997; Bathurst and Cai 1995). Displacement methods developed from classical Newmark sliding-block models have been proposed to predict seismic load-induced deformation of reinforced structures (Cai and Bathurst 1996; Ling et al. 1997a,b). However, pseudostatic limit-equilibrium methods for design against collapse and pseudostatic displacement methods for design against excessive deformations are not satisfactory if the objective is to investigate the coupled effects of reinforcement properties, structure geometry, and foundation condition on reinforced soil wall performance under a prescribed seismic event. Unfortunately, only limited physical data from reduced-scale shaking table tests are available to guide the development of rational models. Carefully conceived and executed numerical experiments offer the possibility to improve the understanding of the effects of dynamic loading on reinforced soil structures and to demonstrate the influence of the primary component properties (e.g. reinforcement stiffness, number of reinforcement layers, base condition, wall geometry, and facing type) on the system response to an earthquake. Numerical experimentswere carriedout toinvestigate theinfluence ofreinforcement stiffness, reinforcement length, and toe restraint condition on the seismic response of an idealized 6m highgeogrid-reinforced soilretaining wallconstructed witha verystiff continuous facing panel. The wall height, number of reinforcement layers, and reinforced soil volumes are typical of actual structures in the field. The wall was subjected to base excitation using a variable-amplitude harmonic motion with a frequency close to the fundamental frequency of the reference structure. The frequency of the applied input base acceleration is representative of a typical predominant frequency of medium- to high-frequency content earthquakes. The excitation frequency was chosen to generate relatively large displacements and reinforcement loads during base shaking and thus help identify performance differences between models with different properties, but at the same time ensure that the models were numerically stable. The two-dimensional, explicit, dynamic finite difference program Fast Lagrangian Analysis of Continua (FLAC 1995) was used to perform the numerical experiments. Analyses were carried out after first confirming that the FLAC program gives similar predictionsto those reported in the literature using a finite element method (FEM)techniqueappliedtothesame idealizedreinforced wallstructure understatic loading(Rowe and Ho 1997). Asecond set of parametric analyseswas carried out with a range of material damping ratios, variable widths of numerical grid, and different far-end truncated boundary conditions to investigate the effect of these model parameters on the predicted wall response. The seismic response of the reinforced wall model subjected to a range of harmonic ground motion frequencies was also examined. Finally, the response of the wallBATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 129 to a scaled earthquake record with the same peak acceleration asthe reference harmonic ground motion was investigated. 2 PREVIOUS RELATED WORK 2.1 Shaking Table Tests Chida et al. (1985) carried out a seriesof experimentson a half-scale shaking table test model of a metal strip-reinforced soil wall. The physical model was constructed with four equal height incremental concrete panelsfor a total wall height of 3 m. A1.4 m high unreinforced sloped fill was placed over the reinforced section that incorporated eight, 4 m long steel strip reinforcement layers. The physical model was 5.2 m from the toe to the back of the shaking table container. Hence, the width of model to height of facing ratio was approximately 1.7 with only a 1.2 m column of unreinforced soil between the back of the reinforced soil zone and the back of the shaking table container. The back vertical boundary of the experiment comprised a rigid wall attached directly to the shaking table. The toe of the wall was constrained horizontally. The model base was excited in a series of experiments using a sinusoidal input acceleration with different frequencies ranging from f = 2 to 7 Hz and peak base accelerations ranging from approximately amax = 0.1g to 0.4g. Resulting maximum reinforcement loads were approximately the same magnitude inall layersand wereobserved toincrease linearlywith increasingpeak base acceleration for frequencies less than 7 Hz and peak base accelerations less than approximately 0.4g. A maximum frequency of 7 Hz for a half-scale model corresponds to approximately 5 Hz at prototype scale. For tests carried out with an estimated base acceleration of amax = 0.4g at f = 2 Hz, and amax = 0.17g at f = 7 Hz, there was a nonuniform distribution of the dynamic load increment in the reinforcement layerswith large values developed in the layers at the top of the structure. Here, the dynamic load increment in a reinforcement layer isthe difference between the maximum tensile load under seismic loading and the maximum tensile load under static loading. Peak acceleration amplification between the base of the wall and the top of the wall was observed to increase by a factor of two at a frequency f = 5 Hz (f = 3.5 Hz at prototype scale). Murata et al. (1994) reported the results of shaking table tests carried out on a reinforced embankment model with a crest width of 3.45 m and contained by two 2.5 m high walls constructed with gabion baskets and an outer continuous concrete panel. The model was subjected to harmonic and actual earthquake records. The harmonic record was observed to generate larger deformations than the earthquake record. Sakaguchi (1996) carried out shaking table tests on a 1.5 m high model test of a reinforced wall. The tests were constructed with lightweight blocks and five layers of geogrid reinforcement. The tests showed that wall displacements and permanent strains in the reinforcement accumulated with time during a 4 Hz sinusoidal base acceleration record applied for 7 seconds (peak accelerations up to approximately 0.5g were applied to the model). The reinforced zone was observed to act as a monolithic body with no evidence of a yield surface propagating across the reinforcement layers even after large wall displacements developed.BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 130 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 2.2 Dynamic Numerical Modeling Segrestin and Bastick (1988) used the FEM program SUPERFLUSH to generate numerical results that showed excellent agreement with the results of half-scale shaking table tests reported by Chida et al. (1985). However, the material properties used to model these physical tests were assumed values since actual material properties were not reported by Chida et al. Segrestin and Bastick (1988) used the same program to predict the seismic response of two hypothetical full-scale reinforced walls (6 and 10.5 m high) that used large articulated concrete facing panels and steel strips as the soil reinforcement. The walls were constrained at the toe but seated on regions that simulated three different foundation materials (hard rock, stiff soil, and loose soil). The width of the numerical model is not reported by Segrestin and Bastick (1988). The location of maximum loads in the reinforcement layers during simulated seismic shaking was generally not at the connections. The results showed that dynamic load increments carried by reinforcement layers increased with depth below the crest of the wall. Yogendrakumar et al. (1991) used a modified version of the dynamic FEM program TARA-3 to carry out a similar study of the influence of seismic loading on a 6 m high steel strip-reinforced soil wall. The program modifications included the introduction of a hysteretic load-strain model for the reinforcement elements. The width of the finite element mesh was 3.3 times the height of the wall. The numerical results showed that maximum reinforcement loads during shaking occurred at the connections with the wall, and significant dynamic load incrementswere generated when the first 10 seconds of the 1940 El Centro earthquake record (scaled to a peak acceleration of 0.2g) was applied to the base of the model. The magnitude of the dynamic load increment in each reinforcement layer close to the back of the wall facing was observed to increase, almost linearly, with depth below the crest of the wall. The maximum amplification factor (i.e. the ratio of the maximum acceleration in the structure to the peak input base acceleration) was approximately 1.4 and occurred at the top of the wall. Cai and Bathurst (1995) used the same modified version of TARA-3 to investigate the response of a 3.2 m high segmental retaining wall reinforced with a polymeric geogrid subjected to a scaled 1940 El Centro earthquake record. The width to height ratio of the FEM of the structure was approximately 4.2, and the far-end truncated boundary was treated as a free-field energy transmitting boundary (i.e. a boundary condition that simulates an infinitely wide domain with respect to elastic wave transmission). The numerical results showed that wall displacements and reinforcement loads accumulated with time during seismic shaking. Displacement and reinforcement amplitudes were insignificant in magnitude compared to the permanent values predicted at the end of base shaking. In contrast to the earlier work by Yogendrakumar et al. (1991) that investigated wallswithastifferfacingandstifferreinforcement,thenumericalresultsfor thediscrete, deformable segmental retaining wall models showed that reinforcement loads under both static and dynamic loading conditions were attenuated at the connections. Also, the dynamic load increments in the reinforcement layers did not increase linearly with depth below the crest of the wall. Base acceleration amplifications were very small (≤ 1.2) which is likely due to the low height of the walls that were investigated.BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 131 2.3 Summary The very limited number of experimental and numerical studies identified above, the limited scope of each study, thedifferent constitutivemodels andnumerical codesused, and the wide range of results illustrate that the current understanding of the seismic response of reinforced soil walls is incomplete. The shaking table tests carried out by Chida et al. (1985) represent an important set of experimental results. Nevertheless, the extrapolation of their results to the field case must be done with caution. In particular, the very narrow volume of material in the retainedsoilzoneandtherigid far-endboundary willinfluence modelresponse asdemonstrated in the current paper. Current practice in North America (AASHTO 1996; FHWA 1996) with regard to empirical rules to calculate dynamic load increments in reinforcement layers is based on the interpretation of numerical results reported by Segrestin and Bastick (1988). The simulations performed by Segrestin and Bastick (1988) were restricted to one type of reinforcement (metal strips), one reinforcement length, and a common footing condition. Segrestin and Bastick (1988) clearly state that the numerical simulation results do not apply to other reinforcement materials (e.g. geosynthetics). Based on a review of the limited data available in the literature, Bathurst and Alfaro (1997) have also noted that the applicability of empirical rules developed using numerical simulation results of the seismic response of metal strip-reinforced soil walls may not be applicable to nominal identical walls constructed with a less stiff (polymeric) reinforcement. The current study is a first steptoward identifyinga numericalmethod andsystematic approach that canultimately beused tovalidate orimprove currentseismic designpractice for continuous panel walls constructed with a range of soil reinforcement products. 3 FLAC PROGRAM FLACisanexplicit, dynamic, finitedifference codebased onthe Lagrangiancalculation scheme that is well suited for modeling large distortions, material collapse, and the dynamic response of earth structures. Complete descriptions of the numerical formulation are reported by Cundall and Board (1988). Several built-in constitutive models are available in the FLAC package. Users can also implement their own models. Other advantages of using FLAC for seismic analysis is the simplicity of applying seismic loading anywhere within the problem domain and the excellent post-processing capabilities. Prior to the time of this study, FLAC had not been used to investigate the seismic response of reinforced soil walls even though the program is widely used by geotechnical and mining engineers for a range of problems. The results of some initial FLAC modeling for seismic response analysis of reinforced slopes and geofoam seismic buffers are summarized in the paper by Bathurst and Alfaro (1997) and represent the only related work to date. 4 COMPARISON OF FEM RESULTS WITH FLAC RESULTS Prior to carrying out the parametric studies that are the focus of the current paper, selected FEM results for a geosynthetic-reinforced continuous panel wall at the end ofBATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 132 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 construction were compared to results using the FLAC program. This comparison was undertaken to develop confidence that the FLAC program can give reasonably similar results to carefully conceived and executed FEM models simulating a static load condition. The example, plane strain reinforced wall selected for this purpose was the base case example reported by Rowe and Ho (1993, 1997) and Ho (1993). While it would be desirable to compare FLAC results to the physical test results reported by Chida et al. (1985), this is not possible because no material properties were given in the report by Chida et al. 4.1 Continuous Panel Wall Models The reference continuous panel wall is 6.0 m high with six uniformly spaced reinforcement layers (Figure 1). The wall facing was modeled as a continuous concrete panel with a thickness of 0.14 m. The bulk and shear modulus values of the wall were Kw = 11,430 MPa and Gw = 10,430 MPa, respectively. Poisson’s ratio for the panel material was taken as νw = 0.15. The panel was hinged at its base, as illustrated in Figure 1. The soil was modeled as a purely frictional, elastic-plastic material with a Mohr-Coulomb failure criterion and nonassociated flow rule. The friction angle of the soil was Ô = 35_, dilatancy angle ψ = 6_, and unit weight γ = 20 kN/m3. These properties and dimensions were the same as those reported by Rowe and Ho (1993) and Ho (1993). Similarly, the following model was used in the current study and by Rowe and Ho (1993) to calculate the elastic modulus of the soil, Es : Es (1) P a = K P σ3 am where: K = 460 and m = 0.5 are constant coefficients; Pa = atmospheric pressure; and σ3 = minor principal effective stress in the soil. A constant Poisson’s ratio value for the Hinge Figure 1. Numerical grid for the reference static load case. L = 4.25 m H=6 m Reinforcement Interface column Facing panel Fixed boundary Fixed boundary Interface layer B=15 mBATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 133 soil material was assumed (νs = 0.3). However, for simplicity, in the current study, the elasticmoduluswasheldconstantforthe durationofthenumerical experiment,includingconstruction, usingvaluescalculatedimmediatelypriortopropremoval andtaking σ3 asthe horizontal earthpressure ineachsoil layer. Hence, soil modulus andPoisson’s ratio values were not updated as was done by Rowe and Ho (1993, 1997). The reinforcement layers were modeled using linear, elastic-plastic cable elements with negligible compressive strength and an equivalent cross-sectional area of 0.002 m2. The stiffness of the reinforcement was taken as J = 2,000 kN/m. The tensile yield strengthofthereinforcementwassettoTy=200kN/mtoensurethat reinforcementrupture was not a failure mechanism and to be consistent with the Rowe and Ho (1993, 1997) model. The interface between the reinforcement (cable elements) and the soil wasmodeledbya groutmaterial ofnegligible thicknesswithaninterface frictionangle δ g = 35_. The bond stiffness and bond strength of the grout were taken as kb = 2×106 MN/m/m and sb = 103 kN/m, respectively. The interface and grout properties were selected to simulate a perfect bond between the soil and reinforcement layers. The results of the FEM simulation of reinforced continuous panel walls have been demonstratedtobe sensitive tomeshconstructiondetailsandmaterial properties at the reinforcement-wall connections (Rowe and Ho 1997; Andrawes and Yogarajah 1994). Inthecurrentstudy,asimpleconnectionmodelwasadoptedthatinvolvedattachingthe end of the cable elements (reinforcement) to a single grid point at the back surface of the continuous panel region. The wall-soil interface was modeled using a thin soil column, 0.05 m thick, directly behindthe facingpanel. Ano-slipboundarywasusedbetweenthe thinsoil columnand the facing panel. The soil-wall interface column material was assigneda frictionangle Ôi =20_andadilatancyangle ψi =0. Asimilarthinsoil layerwasintroducedat thebase of the soil region but was assigned the same properties as the reinforced and retained soil materials. These interface zones in the numerical grid were introduced to match, as closely as possible, the Rowe and Ho (1993, 1997) model that used interface elements to model the same boundaries. Thesoilandreinforcementelementswereconstructedinlayers,whilethecontinuous panel was braced horizontally using rigid external supports. The panel supports were then released in sequence from the top of the structure. Similar to the work by Rowe and Ho (1993, 1997), no attempt was made to model compaction-induced stresses in this static load simulation (or in the dynamic simulations presented in Section 5). The differences between the current simulations and the reference FEM model are: (i)thefacingpanel-soilinterfaceandsoil-foundationinterfaceweremodeledusingthin layers of soil rather thanzero thicknessinterface elements; and (ii)the elastic modulus of the soil was kept at a constant value (corresponding to the end of soil placement) at all stages during the numerical experiment and was not updated. The choice of thin soil layers to model soil interfaces was made because zero-thickness interfaces are not permissible in FLAC (Version 3.30) in combination with intersecting free-field boundaries used in dynamic modeling. Hence, the use of thin layers of soil to model selected interfaces for static loading was preferred. Finally, the use of a constant elastic modulusfor the soil reducescomputation time for bothstatic anddynamic loading simulations using FLAC.BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 134 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 4.2 Comparison of Results The values of lateral displacement of the wall face and normalized (axial) connection loads in reinforcement layers after release of the external props are plotted in Figure 2. The corresponding results reported by Ho (1993) are also presented in the Figure 2. The displacement profiles from both studies are in close agreement as illustrated in Figure 2a. The computed values of connection load, Tc , in Figure 2b have been normalized with respect to the theoretical value of the Rankine active soil pressure at the bottom of the wall. Connection loads are in close agreement over the top half of the wall with lower loads calculated using the FLAC model for the bottom half of the wall height when compared to the corresponding FEM results. The differences may be due to the calculation of soil elastic modulus and the treatment of the wall-soil interface. While, not attempted here, agreement between results could be improved by adopting an updated stress dependent modulus of elasticity in FLAC models for the soil similar to that Elevation (m) 6 5 4 3 2 1 0 0 5 10 15 20 25 30 35 FLAC simulation (a) 6 5 4 3 2 1 0 0.0 0.1 0.2 0.3 0.4 0.5 FLAC simulation Figure 2. Comparison of FLAC results with finite element model results reported by Ho (1993) for the end-of-construction condition: (a) wall displacements; (b) normalized axial loads in the reinforcement at the wall connections. (b) Normalized connection load, Tc / Ka γ H Ho (1993) Ho (1993) Displacement (mm) Elevation (m)BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 135 adopted by Rowe and Ho (1993, 1997). For the purposes of the comparative parametric analyses in Section 6, this was deemed unnecessary when compared to the benefit of keeping the details of dynamic modeling as simple as possible and to minimize computational time. 5 DYNAMIC MODELING USING FLAC Dynamic analyses of a reinforced soil wall subjected to simulated horizontal foundation shaking due to an earthquake were carried out using numerical models with the same height and number of reinforcement layers as those described for the static load FLAC model in Section 4. The numerical grid for the reference geometry in the current study is illustrated in Figure 3. The test series in the current study can be divided into two sets. One set of data is focused on the influence of material properties and dimensions on seismic response of the numerical models. The corresponding variables include: reinforcement stiffness, reinforcement length, and toe restraint condition (Table 1). A second set of data is focused on the influence of the type and location of the far-end truncated boundary and magnitude of material damping ratio on numerical results (Table 2). Several additional numerical analyses were carried out to investigate the influence of frequency of the reference harmonic base input acceleration function on wall response. Finally, a numerical analysis using the initial six seconds of an actual scaled earthquake accelerogram was carried out to investigate quantitative and qualitative differences in the seismic response of the idealized reinforced wall due to harmonic excitation and a typical earthquake record. L = 4.2 m Thin horizontal soil layer (sliding case only) Hinge Thin soil interface column Very stiff facing panel Figure 3. Numerical grid for the reference reinforced soil wall with a fixed-base condition. H = 6 m B = 40 m Right edge of numerical grid and free-field transmitting boundary Free-field transmitting boundary Very stiff foundation (fixed case only) Base acceleration Reinforcement 5 4 3 2 1 Layer 10 m Non-yielding region 1 m Fixed boundary 6BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 136 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 Table 1. Parametric values for the influence of the reinforcement length, reinforcement stiffness, and the base condition. Run Base condition L/H J (kN/m) Run Base condition L/H J (kN/m) 1 Fixed 0.7 500 11 Fixed 1 500 2 Fixed 0.7 1,000 12 Fixed 1 1,000 3 Fixed 0.7 2,000 13 Fixed 1 2,000 4 Fixed 0.7 9,000 14 Fixed 1 9,000 5 Fixed 0.7 69,000 15 Fixed 1 69,000 6 Sliding 0.7 500 16 Sliding 1 500 7 Sliding 0.7 1,000 17 Sliding 1 1,000 8 Sliding 0.7 2,000 18 Sliding 1 2,000 9 Sliding 0.7 9,000 19 Sliding 1 9,000 10 Sliding 0.7 69,000 20 Sliding 1 69,000 Notes: B = 40 m; c = 5%; free-field, far-end boundary. Table 2. Parametric values for the influence of the base condition, far-end boundary condition, boundary distance, and soil damping ratio. Run Base condition Far-end boundary condition B (m) c (%) 21 Fixed Rigid stationary 7.5 5 22 Sliding Rigid stationary 7.5 5 23 Fixed Rigid stationary 15 5 24 Sliding Rigid stationary 15 5 25 Fixed Rigid stationary 25 5 26 Sliding Rigid stationary 25 5 27 Fixed Rigid stationary 40 5 28 Sliding Rigid stationary 40 5 29 Fixed Free-field 40 10 30 Fixed Free-field 40 20 31 Fixed Rigid forced 40 5 Notes: L/H = 0.7; J = 2,000 kN/m. 5.1 Numerical Grid and Problem Boundaries The numerical grid for the reference geometry was selected to represent an infinitely wide region. The width of the backfill, B, for the reference geometry was extended to 40 m beyond the back of the facing panel and a free-field boundary condition was applied at the vertical truncated boundaries at the left and right edges of the grid to allow for the radiation of elastic waves to the far field. The right edge of the grid contains a 10 m wide non-yielding zone. The width of the reinforced zone, L, for the reference geometry was selected to give L/H = 0.7 where H is the height of the wall. This is a typical minimum reinforcement ratio for static design of reinforced soil walls (e.g. FHWA 1996).BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 137 The base condition of the wall was either fixed (i.e. the toe of the wall was slaved to the foundation but was free to rotate) or free to slide horizontally and rotate about the toe. The results ofexperimentally measuredtoe loadsfrom afull-scale, continuouspanel wall has demonstrated that these loads can be very large for walls with a hinged toe and hence toe restraint can significantly add to the capacity of continuous panel walls to carry earth loads (Bathurst 1993). For sliding cases, the wall model was seated on a thin, 0.05 m thick, region of soil that was extended across the full width of the numerical grid. The layer performed a similar function to sliding interface elements in FEM work and was required to ensure that models representing walls without horizontal toe restraint (i.e. sliding-wall cases) were not artificially restrained during shaking. For the fixed-base condition, the wall and soil regions were connected directly to a foundation base comprising a 1 m thick layer of very stiff material. The length of the reinforcement, L, was varied to give L/H = 0.7 or 1.0 and, thus, the influence of the reinforced soil volume on the system response could be examined (Table 1). Ho and Rowe (1996) have shown that for uniformly spaced reinforcement there is little effect of reinforcement length on the maximum tensile loads in reinforcement layers for L/H ≥ 0.7 and static loading conditions. In order to examine the influence of grid width on numerical results, the location of the far-end boundary behind the wall was varied using values of B = 7.5, 15, and 25 m applied to the reference geometry (Table 2). Finally, several additional runs were carried out to examine the influence of the type of far-end truncated boundary on the system response to simulated seismic loading. 5.2 Material Properties The backfill soil properties used in the dynamic analyses were identical to those reported for the static load FLAC model described in Section 4.1 with the exception that this material was assigned constant values of bulk modulus (Ks = 27.5 MPa or, equivalently, Es = 33 MPa) and shear modulus (Gs = 12.7 MPa). Constant values were selected in order to minimize the number of problem parameters. The non-yielding material was assigned the same properties as the backfill soil with the exception that a very large cohesion was used. A non-yielding region was necessary since FLAC does not allow a free-field boundary to be in contact with yielded material. The location of the left hand boundary of the non-yielding region for most analyses was selected to ensure that it did not intersect the active soil zone that develops immediately behind the reinforced soil zone during seismic shaking. The foundation zone in the fixed-base models was assigned the same material properties as the concrete facing panel. The friction angle of the interface soil column between the reinforced soil zone and the panel wall was set to Ôi = 20_ with the remaining soil properties matching the properties of the backfill soil. For sliding cases, the continuous horizontal thin layer at the toe elevation of the wall was assigned the same soil properties as the backfill soil. The reinforcement was modeled using cable elements and grout material as described in Section 4.1. In the parametric analyses, the linear elastic stiffness value for the cables varied over a range of values from J = 500 to 69,000 kN/m (Table 1). The lowest values in Table 1 are typical of the index stiffness of some low strength polymeric geogrids and the highest value was selected to correspond to steel strip reinforcement. The range ofBATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 138 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 stiffness values investigated also captures the magnitude of the initial stiffness values of a typical, woven polyester geogrid and a typical high density polyethylene (HDPE) geogridundercyclic loadingat a frequencyof3Hz (Bathurst andCai 1994). However, the yield strength of the reinforcement in all cases was kept constant at Ty = 200 kN/m, whichiswell above the magnitude ofthe maximum reinforcement loadrecordedinthe simulations. 5.3 Seismic Loading Soilconstructionandpropreleasewereidenticaltothosedescribedforthestaticloading case in Section 4.1. After static equilibrium was achieved, the full width of the foundationwassubjectedtothe variable-amplitude harmonic groundmotion recordillustrated in Figure 4. This acceleration record was applied horizontally to all nodes at the bottom ofthe soil zone at equal time intervals of ∆t = 0.05 s. The accelerogram has both increasing and decaying peak acceleration portions and is expressed as: u ..(t) = βe−αt tζ sin (2πft) (2) where: α= 5.5, β = 55, and ζ= 12 are constant coefficients; f = frequency; and t = time. The peakamplitude ofthe input accelerationis0.2g, andthe frequency, f =3Hz,was selected to represent a typical predominant frequency of medium- to high-frequency content earthquakes (Figure 4). A frequency of 3 Hz was also chosen because it gave stable numerical results in all simulations while generating large displacements and large reinforcement loads within a relatively short simulation time (6 seconds). Large displacements and reinforcement loads were judged to be useful by the authors of the current paper inorder toidentify the relative influence of primaryvariables onseismic response ofa reinforcedsoil wall oftypical height and reinforcedsoil volume. However, it is important to note that the ground acceleration function selected in the current study is not representative of actual earthquake records that have a range of frequency content and typically longer durations. However, the selection of a single earthquake record, scaled to a target peak acceleration, was judged to add more complexity to the proposedworkandwouldrequiregreaterruntimeswithouttheassurancethatpotentiallylargeseismic-inducedeffectswouldresult.Theauthorsofthecurrentpaperareinves- ---2 ---1 2 1 0 0 1 2 3 4 5 6 Time (s) Figure 4. Base acceleration history. Acceleration (m/s2)BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 139 tigating the seismic response of geosynthetic-reinforced soil retaining walls to actual earthquake records using a faster computer platform. The fundamental frequency of vibration for a two-dimensional, linear elastic medium of width B and height H contained by two, rigid vertical boundaries and a rigid base and subject to horizontal base excitation is given by Wu (1994) as: f11 = 1 (3) 4H Gρ  1 +1 − 2 vH B2 where: f11 = frequency in Hz corresponding to the first mode shape of the medium in both the horizontal and vertical directions;G = shear modulus; ρ = density; andv = Poisson’s ratio of the elastic medium. In the limit of an infinitely wide medium (B→∞), Equation 3 becomes the well known expression for the fundamental frequency of a onedimensional elastic medium with height H. For the one-dimensional case with soil layer height H = 6 m and the elastic soil properties described in Section 5.2, Equation 3 gives f11 = 3.32 Hz. This value is close to the frequency of the harmonic input acceleration record (f= 3 Hz) used in most numerical simulationsin the current investigation. Hence, the large deformations reported in Section 6 may be expected because the aspect ratio (H/B) of the numerical grids used in the simulations is small (i.e. Equation 3 with H/B = 0.15 gives f11 = 3.43 Hz.). Richardson and Lee (1975) proposed that the fundamental period, T1 , of reinforced soil walls constructed with steel strip reinforcement can be estimated empirically using the following nondimensional equation: T1 = 0.020H to 0.033H (4) where H is the height of the wall in metres and T1 is in seconds. According to Equation 4, the expected fundamental frequency of most wall models in the current study ranges from 5.1 to 8.3 Hz, which is significantly greater than the applied harmonic base frequency of f = 3 Hz. A possible explanation for the difference in predicted fundamental frequenciesusing Equations3 and 4 isthat the empirical relationship by Richardson and Lee (1975) is applicable to walls retaining a relatively narrow soil volume beyond the reinforced zone. The influence of the choice of base excitation frequency on the seismic response of the 6 m high wall in the current study and implications of the frequency content on numerical modeling results in general are discussed further in Section 6.3. The total duration of the input excitation was limited to 6 seconds (Figure 4) in order to minimize computation time. Only a horizontal acceleration record was applied whereas, in an actual seismic event, vertical acceleration componentsmay be expected. Vertical accelerations are typically ignored in pseudostatic design of wall structures in North America and Japan (Bathurst and Alfaro 1997). It is important to note that the harmonicgroundmotionusedinthecurrentstudyismuchmoreaggressiveon thesystem response than a true earthquake record with the same peak acceleration and comparable peak velocity, duration, and predominant frequency. Hence, the numerical analyses resultsreportedinthecurrentstudyareinterpretedlargelyin relativeand qualitativeterms. Elgamal et al. (1996) have proposed that for conventional reinforced concrete cantilever wall-backfill systems less than 10 m in height, and subject to typical situations of seismic excitation, a viscous damping ratio of 5% isconservative. Hence, a damping ra-BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 140 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 tio of c = 5% was chosen for the reference case in the parametric analysesreported in the current study. However, a number of analyses were carried out to examine the influence of damping ratio on the system response by using values of c = 10 and 20% (Table 2). In all of the simulations, damping was applied to both the soil and facing panel regions; however, the influence of the relatively small panel mass was considered negligible. The program was executed on a 200 MHz personal computer. Computer runsrequired approximately 3 to 6 hours for construction and prop release, and approximately 12 hours for the 6 second base acceleration loading record. 6 RESULTS OF SEISMIC ANALYSES 6.1 Influence of Reinforcement Stiffness, Reinforcement Length, and Toe Restraint Condition 6.1.1 Wall Displacements Example horizontal displacement histories of the wall facing for typical simulations are presented in Figure 5. The datum for horizontal wall displacements, ∆x, was taken at the end of construction following prop release. Results are shown for the two cases of fixed- and sliding-base conditions with L/H = 1 and reinforcement stiffness J = 1,000 kN/m. The displacement histories show that the permanent outward displacement of the wall increases monotonically with time during application of the input acceleration. The amplitudes of motion are small compared to the magnitude of the permanent outward displacement at the end of seismic shaking. The qualitative displacement-time features described for Figure 5 are typical results for all of the simulation runs. Wall displacement profiles predicted at the end of the excitation period are shown in Figure 6 for the range of conditions summarized in Table 1. The maximum displacement at the top of the wall is greater for the fixed-base condition than for the slidingbase condition due to the larger magnitude of tilting that occurs for the fixed-toe condition. For a given base condition (fixed or sliding), the total wall displacements diminish with increasing reinforcement stiffness. Similarly, for a given base condition and reinforcement stiffness, there is less total wall displacement for L/H = 1 compared to configurations with L/H = 0.7. The variation of maximum lateral displacement at the top of the facing panel with reinforcement stiffness, reinforcement length, and base condition issummarized in Figure 7. At the end of construction, there is essentially no influence of reinforcement length on the maximum wall displacements for the fixed-base condition. For the comparable cases with a sliding base, there is a small effect of reinforcement length on the displacements with the maximum wall displacement being slightly greater for L/H = 0.7 compared to the case with L/H = 1. However, the major influence on wall displacements at the end of construction is the reinforcement stiffness, particularly for stiffness values of 2,000 kN/m or less. Figure 7 shows that permanent maximum wall displacements are significantly larger after base shaking. For the fixed-base condition, the displacements at the top of the wall are influenced by both the reinforcement stiffness and the volume of the reinforced soil zone for the range of parameters investigated. The amount of displacement was lessBATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 141 ---0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 ---0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0 1 2 3 4 5 6 Reinforcement layer Figure 5. History of normalized horizontal wall displacements, ∆x/H, at selected elevations: (a) fixed-base condition; (b) sliding-base condition. Time (s) 6 5 4 3 2 1 Soil surface Reinforcement layer (a) (b) 21 5 3 6 4 Soil surface Foundation surface J = 1,000 kN/m L/H = 1 Sliding base ∆x H H ∆x Bottom Bottom Top Top Note: Datum taken at the end of construction following external prop release. J = 1,000 kN/m L/H = 1 Fixed base Foundation surface Normalized horizontal displacement, Dx / H Normalized horizontal displacement, Dx / H with L/H = 1 compared to shorter reinforcement length models (i.e. L/H = 0.7). For sliding-wall cases, the influence of reinforcement length and reinforcement stiffness on wall displacements was significantly less than for the fixed-base condition. In summary, the plots in Figures 6 and 7 show that the toe restraint condition has a greater influence on the magnitude of maximum wall displacements for the given input acceleration record than reinforcement length and reinforcement stiffness. Wall displacements are greater for the fixed-base condition compared to the sliding-base condi-BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 142 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 6 5 4 3 2 1 06 5 4 3 2 1 0 0.00 0.02 0.04 0.06 0.08 0.10 6 5 4 3 2 1 0 500 1,000 2,000 9,000 69,000 Normalized horizontal displacement, ∆x / H Elevation (m) Elevation (m) Figure 6. Total and relative normalized wall displacement at the end of seismic shaking for walls with different reinforcement stiffness: (a) L/H = 0.7, sliding-base condition; (b) L/H = 1, sliding-base condition; (c) L/H = 0.7, fixed-base condition; (d) L/H = 1, fixed-base condition. (a) (b) 6 5 4 3 2 1 0 Relative Total J (kN/m) 500 1,000 2,000 9,000 69,000 J (kN/m) 500 1,000 2,000 J (kN/m) 500 1,000 2,000 9,000 69,000 9,000 69,000 Note: Datum taken at the end of construction following external prop release. J (kN/m) J (kN/m) Elevation (m) Elevation (m) (c) (d) J (kN/m) 500 1,000 2,000 9,000 69,000 J (kN/m) 500 1,000 2,000 9,000 69,000BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 143 Figure 7. Influence of the reinforcement stiffness, J, reinforcement length, L, and base condition on the maximum wall displacements at the end of construction and after seismic shaking. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 100 1000 10000 100000 Reinforcement stiffness, J (kN/m) End of seismic shaking Fixed base L/H = 0.7, sliding base L/H = 1.0, sliding base L/H = 1.0, fixed base End of construction L/H = 0.7, fixed base Maximum displacement (m) Sliding base tionwhenallotherparametervaluesareequal.However,theinfluenceofthetoerestraint condition on wall displacements reduces as the stiffness of the reinforcement increases. 6.1.2 Reinforcement Loads Across section showing the true-scale deformation of the reinforced zone in an example simulated wall after base shaking is shown in Figure 8. Superimposed on Figure 8 are bar graphs of reinforcement loads at the same point in time. In all of the tests, reinforcement loads were greatest at the connections after base shaking. This trend can be attributed to the progressive downward movement of the reinforced soil zone relative to the continuous wall panel during base excitation and the pinned reinforcement-wall connection detail adopted in the model. An example of normalized axial load histories in the reinforcement layers at the connectionsisshowninFigure9.Theresultsareshownforthefixed-and sliding-baseconditions for walls with J = 1,000 kN/m and L/H = 1. Connection loads, Tc , can be seen to accumulate with time during shaking of the base and this qualitative feature was observed in all simulation runs. However, the distribution and magnitude of reinforcement loads over the entire time record was very different between fixed- and sliding-base cases in the current study. The distributions and magnitudes of the maximum recorded load, Tmax , in each reinforcement layer at the end of construction (static loading) and during base shaking (dynamic loading) are plotted in Figure 10. Each maximum load value for a reinforcement layer corresponds to the maximum tensile load recorded along the entire length of thatBATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 144 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 Figure 8. True-scale deformed numerical grid and the reinforcement loads at the end of seismic shaking (t = 6 s) for the fixed-base condition with L/H = 0.7 and J = 2,000 kN/m. Grid boundary H = 6 m B = 40 m L = 4.2 m Note: Maximum reinforcement load of 65 kN/m in bar graphs occurs in the second layer from the bottom. layer. The maximum dynamic load values generally occurred at or close to the connections at a time corresponding to the end of the excitation record (Figure 10). The static load distributions for all of the reinforcement stiffness cases show a trend toward increasing reinforcement loads with increasing reinforcement stiffness for the fixed-base condition (Figures 10a and 10b). Also shown in Figure 10 are the theoretical values for the normalized static load in each reinforcement layer using limit-equilibrium methods based on Rankine (Ka = f(Ô)) and Coulomb earth pressure theory (Ka = f(Ô, δ)), and a contributory area approach to distribute reinforcement loads. For the fixedbase condition, the linear distribution values from theory contain the range of predicted reinforcement loads but do not capture the general trend which is reasonably uniform. The limit-equilibrium solutions under-predict the reinforcement loads close to the top of the wall and over-predict the magnitude of loads toward the base of the wall. These qualitative observations are consistent with the results reported by Rowe and Ho (1997) who investigated the influence ofreinforcement stiffnesson the distribution andmagnitude of reinforcement loads under static conditions. The static load distributions from numerical analyses with a sliding-base condition (Figures 10c and 10d) are more dispersed than for the fixed case. The trends of the calculated static load values with elevation from numerical analyses for the sliding-base cases are better captured using the linear distributionsfrom Rankine and Coulombearth pressure theories than for the fixed-base cases. However, the magnitudes of the reinforcement load using earth pressure theories are consistently lower than predicted values from numerical simulations. There is no significant influence of reinforcement length on the magnitude and distribution of reinforcement loads under static loading apparent in the data in Figure 10.BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 145 0.0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4 5 6 0.0 0.1 0.2 0.3 0.4 0.5 Figure 9. History of the normalized connection loads in the reinforcement layers: (a) fixed-base condition; (b) sliding-base condition. 342 5 6 (a) (b) J = 1,000 kN/m Reinforcement layer L/H = 1 J = 1,000 kN/m L/H = 1 Time (s) 1 234 5 6 Reinforcement layer Top Bottom Note: Tc is the connection load and Ty = 200 kN/m is the yield strength of the reinforcement. Tc / Ty Tc / Ty Top Bottom 1 The maximum reinforcement load distributions recorded during dynamic loading are highlighted by the hatched regions in Figure 10. In all cases, the reinforcement loads were larger underdynamic loadingthan the calculated loadsfor the end-of-construction (i.e. static) condition. The increase in reinforcement load in any layer increases with reinforcement stiffness. For the fixed-base condition (Figures 10a and 10b), the trend of the reinforcement load can be seen to increase with depth below the wall crest for J = 9,000 and 69,000 kN/m while the trend ofthe reinforcement loads forwalls withlower stiffnessreinforcement is more uniform with depth. Moreover, there is essentially no influence of reinforcement length on the magnitude of the maximum loads recorded for the fixed-foundation cases except for the stiffest reinforcement case with L/H = 1 which gives consistently higher reinforcement loads than the otherwise identical configuration with L/H = 0.7. The trends of the reinforcement load with elevation under dynamic loading for sliding-base cases (Figures 10c and 10d) are qualitatively different from the correspondingBATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 146 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 6 5 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 6 5 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 6 5 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 6 5 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 Figure 10. Influence of the reinforcement stiffness, J, reinforcement length, L, and base condition on the maximum load in each reinforcement layer: (a) L/H = 0.7, fixed-base condition; (b) L/H = 1, fixed-base condition; (c) L/H = 0.7, sliding-base condition; (d) L/H = 1, sliding-base condition. Note: Tmax = maximum tensile load recorded along the entire length of the reinforcement layer and Ty = 200 kN/m is the yield strength of the reinforcement. 1 Elevation (m) 2 1 2 3 4 3 2,000 1,000 500 9,000 69,000 (b) (2) (1) J (kN/m) Dynamic 4 5 6, top reinforcement layer 6, top reinforcement layer 5 2 3 4 1 5 4 3 2 J (kN/m) (c) (d) Normalized maximum reinforcement load, Tmax / Ty 6, top reinforcement layer 5 6, top reinforcement layer Elevation (m) (a) 2,000 1,000 500 9,000 69,000 End of construction (static) J (kN/m) (2) (1) Dynamic End of construction (static) (1) (2) Dynamic End of construction (static) 2,000 1,000 500 9,000 69,000 1 (1) (2) Dynamic End of construction (static) 2,000 1,000 500 9,000 69,000 J (kN/m) (1) Ka = f(f), (2) Ka = f(f, d) (1) Ka = f(f), (2) Ka = f(f, d) (1) Ka = f(f), (2) Ka = f(f, d) (1) Ka = f(f), (2) Ka = f(f, d) results for the fixed cases because dynamic loads are observed to increase in magnitude with depth below the crest of the wall for all reinforcement stiffness values. Figure 11 shows the normalized dynamic load increment , ∆T, recorded in all simulations. Dynamic load increment values were calculated by subtracting from the maximum load, Tmax , used to generate the maximum dynamic load curves in Figure 10, the corresponding maximum initial static load values and then normalizing the dynamic load increment value with the yield strength of the reinforcement, Ty . The predicted dynamic load increments using the current AASHTO (1996) method with a peak horizontal ground acceleration of 0.2g are also shown in Figure 11. For the fixed-base condition (Figures 11a and 11b), the empirical AASHTO (1996) method underestimates the magnitude of dynamic load increments, ∆T. The magnitudeBATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 147 6 5 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 6 5 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 6 5 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 1 Elevation (m) Normalized dynamic load increment, ∆T / Ty (c) 2 3 4 5 6, top reinforcement layer 6 5 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 1 1 Elevation (m) J (kN/m) 2,000 1,000 500 9,000 69,000 (a) (b) J (kN/m) (1) 2 3 4 5 6, top reinforcement layer 2 3 4 5 6, top reinforcement layer Figure 11. Influence of the reinforcement stiffness, J, reinforcement length, L, and base condition on the reinforcement dynamic load increment, ∆T : (a) L/H = 0.7, fixed-base condition; (b) L/H=1, fixed-base condition; (c) L/H = 0.7, sliding-base condition; (d) L/H = 1, sliding-base condition. 2,000 1,000 500 9,000 69,000 (1) (d) (1) (1) J (kN/m) 2,000 1,000 500 9,000 69,000 1 2 3 4 5 6, top reinforcement layer J (kN/m) 2,000 1,000 500 9,000 69,000 Note: (1) = AASHTO (1996) method. of the underestimation increases with stiffness of the reinforcement. In addition, the trend of the linear increase in ∆T with increasing depth below the crest of the wall using the empirical AASHTO method does not reflect the trend of the data except for the stiffest reinforcement cases (i.e. J = 9,000 and 69,000 kN/m). For the sliding-base condition (Figures 11c and 11d), the empirical AASHTO (1996) method predicts valuesof ∆Tthat are in the range ofvalues forthe lowest reinforcement stiffness cases but increasingly underestimates the magnitude of ∆T as reinforcement stiffness becomeslarger. However, the trendof the data forthe magnitude of the dynamic load increment, ∆T, with depth below the crest of the wall canbe arguedto be qualitatively similar to the empirical AASHTO results. The influence ofreinforcement elevation, base condition, and reinforcement stiffness is summarized in Figure 12. The following observations can be made on the data corre-BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 148 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 0.0 0.2 0.4 0.6 0.8 1.0 Figure 12. Variation of the normalized maximum tensile load, Tmax / Ty , in the reinforcement layers during seismic shaking versus reinforcement stiffness, J, reinforcement length, L, and base condition: (a) L/H = 0.7, fixed-base condition; (b) L/H = 1, fixed-base condition; (c) L/H = 0.7, sliding-base condition; (d) L/H = 1, sliding-base condition. 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 100 1000 10000 100000 Reinforcement stiffness, J (kN/m) Bottom Top 123456 (a) (b) (c) (d) Tmax / Ty Bottom Top 1 23456 Tmax / Ty Bottom Top 123456 Tmax / Ty Bottom Top 12 3456 Tmax / Ty sponding to the end of base shaking using the applied harmonic input acceleration record: (i) reinforcement loads increase in magnitude from top tobottom ofthe wall; (ii) reinforcement loads are more sensitive to the magnitude of the reinforcement stiffness at lower reinforcement elevations than at the top of the wall; and (iii) reinforcement loads at the bottom of the wall are more sensitive to reinforcement stiffness for the fixed-base condition than for the sliding-base condition. However, as demonstrated in Sections 6.2 and 6.3, quantitative values from numericalsimulations(suchasreinforcementloads)willvarywidelyasaresult ofcharacteris-BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 149 tics of the base input acceleration record, fundamental frequency of the model, and height to width ratio of the excited structure. In addition, the frequency of the harmonic ground motion, which is just below the fundamental frequency of the reference structures, can be expected to generate large reinforcement loads. The observation that the empirical AASHTO(1996) method underestimates the magnitude of the dynamic load increments, ∆T, suggests that this empirical method may be unsafe for the design of walls that are excited close to the fundamental frequency of the structure during an earthquake. 6.1.3 Distribution of Backfill Accelerations The distribution and magnitude of peak accelerations in the backfill soil is of interest in pseudostatic seismic design methods because a coherent distribution of the ground acceleration is considered to be responsible for the additional destabilizing force that must be resisted by reinforced structures during a seismic event. The acceleration responses in the soil at different elevations were filtered using an elliptic, low-passfilter with a cut-off frequency, flim = 10 Hz (Bellanger 1989). Thisfilter function was applied to all acceleration response records to exclude the spurious, highfrequency acceleration peaks associated with the reflection of waves in the simulation runs. The significant portion of seismic energy during actual earthquakes is also imparted at frequencies below 10 Hz (Elgamal et al. 1996). Hence, the upper cut-off frequency of flim = 10 Hz was used to capture a significant portion of the response frequency content in the numerical simulations. The distribution and magnitude of filtered accelerations recorded in the soil immediately behind the wall-soil interface column (i.e. close to the back of the facing panel) are summarized in Figure 13. Acceleration amplification over the depth of the backfill is noticeably larger at the soil surface for the fixed-base condition as compared to the sliding-base condition. The large amount of acceleration amplification at the surface of the backfill soil may be partly attributed to the assumption of a cohesionless backfill and the relatively low damping value used in most analyses. The generally larger acceleration values for the fixed-base condition are also apparent in Figure 14 where the magnitude of the average amplification value is plotted against reinforcement stiffness. The average acceleration amplification value is in the range of 2.0 to 2.8 and is relatively insensitive to the range of reinforcement stiffness values used in the current study. Analyses of the data confirmed that the back-calculated values of the amplification factor were insensitive to the cut-off frequency above 10 Hz. 6.1.4 Failure Zones Figure 15 shows typical plots of shear zones within the reinforced soil zone and in the retained soil for typical fixed- and sliding-base conditions. In this numerical study, there was no evidence of a well-defined failure surface intersecting all reinforcement layersasmay be expected fromconventional tied-backwedge andnonlinear slipsurface methods of analysis (Bathurst and Alfaro 1997). This was true even for models with the lowest reinforcement stiffness (i.e. J = 500 kN/m). Rather, the reinforced soil zone acted as a parallel-sided monolithic mass. Further investigation isrequired to determine if the pattern of internal failure will change with greater reinforcement spacings.BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 150 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 0.0 1.0 2.0 3.0 4.0 5.0 6.0 Figure 13. Distribution of the peak horizontal accelerations recorded in the reinforced soil zone for different reinforcement stiffness values, J, and base conditions (L/H = 0.7): (a) fixed-base condition; (b) sliding-base condition. 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Acceleration (g) Elevation (m) Elevation (m) J (kN/m) 69,000 9,000 2,000 1,000 500 Peak foundation acceleration Peak foundation acceleration 0.0 0.5 1.0 1.5 2.0 2.5 3.0 100 1000 10000 100000 Figure 14. Variation of the mean base acceleration amplification in the reinforced soil zone versus the reinforcement stiffness, J, and the base condition. Reinforcement stiffness, J (kN/m) Fixed base Sliding base Mean base acceleration amplification J (kN/m) 69,000 9,000 2,000 1,000 500 Note: Acceleration response filter cut-off frequency, flim = 10 Hz. Notes: Acceleration response filter cut-off frequency, flim = 10 Hz; damping ratio c = 5%. (a) (b)BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 151 Figure 15. Shear zones at t = 6 s, with L/H = 0.7 and J = 2,000 kN/m: (a) fixed-base condition; (b) sliding-base condition. 12 m (a) 4.2 m 6.0 m 20_ 40 m Grid boundary 23_ 6.0 m 40 m Grid boundary 4.2 m (b) 13 m 31_ 7_ Note: Dark shading indicates relatively large shear strains. 23_ 31_ Large shear strains were recorded at the wall-soil interface and at the reinforced retained soil interface. The failure volume in each simulation can be approximated by a bilinearwedge with a break point at the back ofthe reinforcedsoil zone. The breakpoint was observed to be at a higher elevation for the fixed-base condition when compared to the sliding-base condition. Also shown in Figure 15 are the linear failure surfaces in the retained soil that are predicted from solutions forslip surface orientation usingMononobe-Okabe earthpressure theory (Okabe 1924; Zarrabi 1979; Bathurst and Alfaro 1997). Orientations of 23 and 31_ from the horizontal correspond to computed mean wedge accelerations of approximately 0.5g and 0.4g, respectively, and are in reasonably good agreement with shear zone boundaries. Hence, pseudostatic equilibrium methods may be useful to estimate minimum widths for numerical grids if the influence of yielded soil zones on the wall response is to be captured in numerical simulations.BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 152 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 6.2 Influence of Boundaries and Damping on Numerical Results 6.2.1 Far-End Truncated Boundary Condition The reference, far-end truncated boundary condition in the current study is afree-field boundary condition applied at the vertical left and right edges of the grid (Figure 3). This boundary condition simulates an infinitely wide domain with respect to elastic wave transmission. Two less complicated boundary conditions were also investigated and applied at the right hand boundary: (i) rigid, forced boundary condition - a rigid vertical boundary with the base acceleration function applied to all grid points from the foundation elevation to the soil surface; (ii) rigid, stationary boundary - a rigid vertical boundary that was fixed horizontally during the entire simulation. The influence of boundary condition is illustrated in Figure 16 for walls with a fixedbase condition. The data show that for the three conditions examined, the free-field Normalized horizontal displacement, ∆x / H 6 5 4 3 2 1 0 0.00 0.02 0.04 0.06 0.08 0.10 Figure 16. Influence of the model far-end condition on the wall response with the fixed-base condition: (a) normalized wall displacements; (b) normalized maximum reinforcement connection loads. Free-field Rigid, forced Rigid, stationary 6 5 4 3 2 1 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Rigid, forced Free-field Rigid, stationary Elevation (m) Elevation (m) Normalized connection load, Tc / Ty J = 2000 kN/m L/H = 0.7 B = 40 m Fixed base J = 2000 kN/m L/H = 0.7 B = 40 m Fixed base (a) (b) Note: Displacement datum taken at the end of construction following external prop release.BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 153 condition results in the largest wall displacements and the largest connection loads at the end of seismic shaking. The rigid stationary condition results in the least wall displacement and reinforcement loads. The latter result is considered to be due to the horizontal constraint imposed on the backfill soil volume at the right hand boundary. The reduced maximum reinforcement load at the base of the wall with a fixed-forced boundary compared to the free-field case is consistent with the trend of the results from dynamic finite element modeling work reported by Richardson and Lee (1975) who investigated boundary effects for steel strip-reinforced walls with thin-wall metallic facings and a fixed base. 6.2.2 Model Width The influence of width B of numerical grids on dynamic response was examined using models with J = 2000 kN/m, L/H = 0.7 and a stationary rigid truncated far-end boundary condition. Figure 17 shows that the magnitude of lateral wall displacements Figure 17. Influence of the model width, B, on the normalized wall displacements: (a) fixed-base condition; (b) sliding-base condition. (a) (b) 6 5 4 3 2 1 06 5 4 3 2 1 0 0.00 0.02 0.04 0.06 0.08 Normalized horizontal displacement, ∆x / H J = 2000 kN/m L/H = 0.7 J = 2000 kN/m L/H = 0.7 Elevation (m) Elevation (m) B = 7.5 m 15 m 25 m 40 m B/H = 1.3 2.5 4.2 6.7 Notes: Stationary rigid far-end boundary condition; displacement datum taken at the end of construction following external prop release.BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 154 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 is greatly influenced by the volume of soil behind the reinforced soil zone for the reference base input acceleration record used in the current study. The larger the mass of backfill soil the greater the wall displacements at the end of seismic shaking. The corresponding maximum axial loads in the reinforcement at different elevations are plotted in Figure 18. Not surprisingly, the maximum axial loads calculated for all reinforcement layers (except the top layer) also increase in value with increasing width B. The loads in the bottom reinforcement layer for each base condition vary by approximately a factor of two over the range of B values examined. Nevertheless, the effect of parameter B on calculated displacements and reinforcement loads can be seen to diminish as the width of the soil model increases (e.g. compare cases with B = 25 and 40 m in Figures 17 and 18). The magnitude of reinforced wall displacements and reinforcement loads increases with the width of the backfill model (for the same input motion) as a direct result of the increase in the massof the model and the width of the excitation boundary at the foundation level. The significant dependence of dynamic response of the wall on the width of the backfill model would not be observed if the excitation boundary of the retaining wall system under horizontal ground motion was limited to the facing panel and the truncated boundary of the model was placed at sufficiently large distance from the facing panel. It follows that the size of the backfill model may be a dominating factor in the response analysis of the reinforced retaining walls subjected to a prescribed ground motion. For the range of parameters investigated in this study, the model width, B, was more important than the type of far-end truncated boundary condition employed in numerical simulations. The shear plots in Figure 15 suggest that a width B = 15 m (or ratio B/H = 2.5 ) is sufficient to capture the influence of the yielded zone in the retained soil during seismic shaking. If the mass of soil beyond the yielded zone is ignored, then the quantitative results for reinforcement loads and wall displacements are very much less than those reported earlier for the reference case B = 40 m (B/H = 6.7). However, the width of the 6 5 4 3 2 1 0 0.0 0.1 0.2 0.3 0.4 0.5 Sliding base Figure 18. Influence of model width, B, and base condition on normalized maximum reinforcement connection loads (J = 2,000 kN/m, L/H = 0.7). B 15 m 25 m 40 m Elevation (m) 7.5 m B/H 6.7 4.2 2.5 1.3 Tc / Ty Fixed base H/B 0.15 0.24 0.40 0.80 Note: Stationary rigid far-end boundary condition.BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 155 yielded zone can be expected to increase with base acceleration according to Mononobe-Okabe theory. Hence, more work remains to be done to develop rules to select a representative volume of retained soil in combination with an appropriate far-end truncated boundary condition. 6.2.3 Damping A dynamic shear damping ratio of c = 5% was selected as the reference value for the soil in the current study (i.e. Table 1). To simplify numerical modeling this value was assigned to all material zones except the cable elements. In reality, dynamic damping of a soil mass significantly increases with the amplitude of vibration and a reduction in effective confining pressure. For example, Saxena et al. (1988) proposed the following equation for small strain dynamic shear damping of uncemented sands expressed in percent: ξ = 9.22 P σ0 a−0.38η0.33 (5) where: σo = effective confining pressure; Pa = atmospheric pressure; and η = dynamic shear strain in percent. Equation 5 illustrates that dynamic shear damping of the soil can theoretically reach very high values toward the surface of the backfill where soil confinement stress approaches zero. Results of other studies (Kramer 1996; Ishihara 1996) also indicate a strong dependence of the damping ratio of sands on both cyclic shear strain amplitude and magnitude of effective confining pressure. Hence, it can be argued that a constant value of c = 5% applied to the entire soil zone may lead to an overestimation of the magnitude of model deformation and acceleration response. The influence of damping ratio was investigated by repeating selected simulation runs with c = 10 and 20%. The effects of damping ratio on wall displacements and reinforcement loads at the end of seismic shaking are shown in Figures 19a and 19b. The lateral displacement of the wall at the end of base shaking is up to 40% less for the case with c = 20% as compared to the reference case of c = 5%. The corresponding reduction in maximum reinforcement loads (Figure 19b) is approximately 20%. Nevertheless, the data show that qualitative features of model deformation and reinforcement load distributions are similar in all three cases. The effect of damping ratio on base acceleration amplification is shown in Figure 20. The data show that increasing the damping ratio from c = 5 to 20% can reduce the peak acceleration at selected locations in the soil zone by up to 40%. The corresponding reduction in mean base acceleration amplification factor is approximately 25%. Nevertheless, qualitative trends are preserved in each case (i.e. generally increasing peak acceleration with height above the foundation base). It is worth noting that the influence of the magnitude of damping ratio on the dynamic response of the models in this study is not unexpected since the frequency of the applied base input acceleration record is close to the fundamental frequency of the reference models. Theoretically, the magnitude of viscous damping ratio is frequency dependent. However, an additional simulation run was carried out that showed that introducing theBATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 156 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 Figure 19. Influence of the soil damping value, c , on the wall displacements and the magnitude and distribution of thenormalized peak reinforcement connection loads: (a) wall displacements; (b) normalized maximum connection loads. 6 5 4 3 2 1 0 0.00 0.02 0.04 0.06 0.08 0.10 6 5 4 3 2 1 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Tc / Ty Elevation (m) Elevation (m) (a) (b) Normalized horizontal displacement, ∆x / H c = 20% J = 2000 kN/m L/H = 1.0 B = 40 m Fixed base J = 2000 kN/m L/H = 1.0 B = 40 m Fixed base Notes: Acceleration response filter cut-off frequency, flim = 10 Hz; free-field, far-end boundary condition; displacement datum taken at the end of construction following external prop release. 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Acceleration (g) J = 2000 kN/m L/H = 1 Fixed base Elevation (m) Peak foundation acceleration Figure 20. Influence of the soil damping value, c, on the magnitude and distribution of peak horizontal accelerations recorded in the reinforced soil zone. c = 20% c = 5% c = 10% c = 20% c = 5% c = 10% Note: Acceleration response filter cut-off frequency, flim = 10 Hz. c = 5% c = 10%BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 157 damping ratio at a slightly different frequency had little effect on the calculated model displacements and reinforcement loads. Finally, it can be noted that the greatest accelerations in Figure 20 correspond to the near-surface soil locations. This observation is consistent with Equation 5 which can be rearranged to show that for a given damping value, the strain amplitude will increase with lower confining pressures. 6.3 Influence of Base Acceleration Record on Numerical Results 6.3.1 Frequency of Harmonic Input Acceleration Record Wave propagation theory for one and two-dimensional linear elastic media discussed in Section 5.3 suggests that the frequency of base excitation adopted in the present study (f = 3 Hz) may be close to the fundamental frequency of the 6 m high wall. The influence of frequency on numerical simulation results was investigated by carrying out a series of runs with an input frequencyf = 2.5, 3, 3.2, 3.4, 3.5, 3.7, 4, and 5 Hz. Figure 21a shows the variation of maximum horizontal displacement at the crest of a wall with input frequency. Numerical simulations were stable for all frequencies except 3.4 and 3.5 Hz. Hence, the selected harmonic input acceleration record adopted as the base case in the present study (f = 3 Hz) is close to but below the fundamental (critical) frequency of the reference structures with B = 40 m (H/B = 0.15). The vertical lines in the hatched zone in the figures correspond to fundamental frequencies predicted by Equation 3 for model dimensions with height to width ratio of H/B = 0 (one dimensional), 0.15 (reference case geometry) and 0.8 (minimum width model). Equation 3 for geometries approaching the one dimensional case (H/B → 0) proved to be a good predictor of resonance in numerical simulations (i.e. the difference between predicted fundamental frequency for the reference case geometry with H/B = 0.15 and the one dimensional case is very small). According to Equation 3, the fundamental frequency of two dimensional elastic media increases with increasingly narrower regions (larger H/B ratios). This effect may explain why progressively lower magnitudes of reinforcement load were recorded with decreasing model width B in Figure 18. Figure 21b illustrates the effect of input frequency on the magnitude of maximum reinforcement loads during harmonic shaking. The results of numerical analyses show that the magnitude of reinforcement loads is influenced by input ground motion frequency with a general reduction in reinforcement loads as the input frequency diverges from the fundamental frequency of the structure. 6.3.2 Example Earthquake Input Acceleration Record A numerical simulation run was carried out on the reference wall model with a fixed-base condition using the first 6 seconds of the horizontal component of the 1940 El Centro earthquake acceleration record scaled to a peak acceleration of 0.2g. The first 6 seconds of the El Centro accelerogram contains the peak ground acceleration and the significant portion of the record. While both the harmonic record and scaled El Centro (truncated) record have the same peak acceleration (0.2g) and the same applied duration (6 seconds) in this study they are different with respect to ground motion characteristics (i.e. frequency content, duration of strong ground motion, and peak groundBATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 158 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 6 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 H = 6 m, B = 40 m H Reference case in current investigation (f = 3 Hz) H/B = 0 Base input frequency (Hz) Figure 21. Influence of the frequency of the harmonic base input record on the wall response: (a) maximum horizontal displacement recorded at the top of the wall, ∆x / H; (b) normalized maximum reinforcement load, Tmax / Ty . (a) J = 69,000 kN/m L/H = 0.7 Fixed base B = 40 m Maximum ∆x H/B = 0.15 H/B = 0.8 B Predicted fundamental frequency from Equation 3 Reinforcement layer 1 2 34 5 6 Bottom Top (b) H/B = 0 H/B = 0.15 H/B = 0.8 3.32 3.43 4.5 Note: Displacement datum taken at the end of construction following external prop release. Normalized horizontal displacement Dx / H Normalized maximum reinforcement load T max / Ty velocity). The intent here is to illustrate that qualitative features of wall displacement and reinforcement loads under an actual input earthquake are similar to those reported in this paper using a variable-amplitude harmonic ground motion with a single frequency. Figure 22a illustrates that wall displacements are cumulative with time during shaking and that the amplitudes of motion are small with respect to the permanent displacements at the end of the simulation (compare Figure 22a to Figure 5a). Figure 22bBATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 159 6 5 4 3 2 1 0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 El Centro record (scaled to 0.2g) (a) 0.00 0.01 0.02 0.03 0.04 0 1 2 3 4 5 6 J = 2000 kN/m L/H = 0.7 Fixed base Tc / Ty Elevation (m) (b) J = 2000 kN/m L/H = 0.7 Fixed base Figure 22. Response of the wall with a fixed-base condition to the initial 6 s of the El Centro base acceleration record scaled to 0.2g: (a) history of wall displacements; (b) reinforcement connection loads at the end of the applied shaking record. Harmonic record Time (s) Top Mid-height H ∆x Normalized horizontal displacement Dx / H Notes: Scaled El Centro record from the S00E component of the Imperial Valley accelerogram (epicentral distance = 8 km); displacement datum taken at the end of construction following external prop release. shows that the trend of the connection loads is the same for the El Centro and harmonic input base acceleration cases. In this study, no attempt was made to compare the effects of synthetic earthquake records and actual earthquake records on the seismic response of idealized reinforced soil walls. However, the magnitude of displacements and reinforcement loads predicted using the reference harmonic record in this study are not unexpected since the frequency of applied ground motion was close to the fundamental frequency of the models. Actual earthquake records contain a range of significant frequencies that will typically be less aggressive on wall response than a harmonic record with a single frequency selected to be close to the fundamental frequency of the model structure. Hence, while harmonicBATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 160 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 earthquake records offer simplicity in parametric analyses of the type reported in this paper, quantitative results may not be typical of actual earthquake response. Finally, the difference in magnitude of reinforcement loads shown in Figure 22b points to a limitation in the current AASHTO (1996) pseudostatic design method used to estimate dynamic reinforcement loads under earthquake. Both excitation records have the same peak ground acceleration and based on the AASHTO method would result in the same magnitude of dynamic load at a reinforcement layer which is clearly not the case in Figure 22b. 7 CONCLUSIONS The results of parametric analyses of a reinforced soil wall using program FLAC have been reported. The program was first demonstrated to give similar results to a well-documented FEM model that was used to simulate the static load response of a 6 m high reinforced soil wall constructed with a continuous facing panel. Parametric seismic analyses were carried out on a similar 6 m high reinforced soil wall constructed with two different foundation conditions, a range of geosynthetic reinforcement stiffness values, and two different reinforcement lengths. Qualitative features of the dynamic response of this idealized wall to a variable-amplitude harmonic base input acceleration record are summarized below: 1. Wall displacements and reinforcement loads accumulated during base shaking. The amplitudes of wall deformation and reinforcement load during base shaking were small compared to permanent values calculated at the end of the input record. Similar qualitative responses were calculated when the first 6 seconds of the El Centro earthquake accelerogram scaled to 0.2g were applied to the structure. 2. The magnitude of total wall displacement at the wall crest and relative wall displacement with respect to the wall toe at the end of base excitation were less for a reinforced wall that was free to slide at the base than for a wall that could only rotate about the toe. 3. The magnitude of permanent wall displacement diminished with increasing reinforcement stiffness and increasing reinforcement length. However, for models subjected to the reference harmonic base input motion at 3 Hz, the greatest influence on the magnitude of wall displacement was the foundation condition (i.e. whether the reinforced soil system was free to slide or was constrained to only rotate at the toe). 4. The introduction of a soil column at the back of the facing panel simulating a wallsoil interface with reduced frictional resistance resulted in the maximum reinforcement loads being generated at the connections. This observation was true at the end of construction of the wall (static loading) and during base shaking. 5. The reference harmonic base input record resulted in additional tensile loads being generated in reinforcement layers that were significantly larger than values resulting from static loading alone. The magnitude of additional dynamic-induced loads in reinforcement layers was observed to increase with increasing stiffness of the reinforcement.BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 161 6. The magnitude and distribution of dynamic-induced reinforcement loads was influenced by the base condition. For walls with a sliding base, dynamic load increments increased in a generally linear fashion with depth below the crest of the wall regardless of reinforcement stiffness. For walls with a fixed base, the dynamic load increments did not increase with depth and were attenuated in the lower layers for walls with a reinforcement stiffness J ≤ 2000 kN/m. For stiffer reinforcements, a linear trend of the dynamic reinforcement loads similar to that noted for sliding-base cases was observed. Further study is required to investigate if these trends are applicable to the same models subjected to other input ground motions. 7. Horizontal ground acceleration was amplified with height above the base foundation. For the reference geometry, boundary conditions and base input motion the mean amplification factor ranged from 2 to 2.8 for models with a damping ratio of 5%. However, the magnitude of amplification was shown to be influenced by the magnitude of damping ratio used in the numerical models. 8. The soil in the retained soil zone was observed to yield during shaking and the inclination of the failure surface in this region was reasonably well predicted by Mononobe-Okabe theory. 9. There was no evidence of an interior shear surface propagating from the heel of the facing panel and intersecting all reinforcement layers as is assumed in conventional pseudostatic methods of analyses. The quantitative results reported here illustrate that the predicted seismic response of reinforced soil walls using program FLAC and the reference variable-amplitude harmonic base input acceleration record are strongly influenced by the magnitude of damping ratio for the soil and the type of far-end boundary condition adopted. However, the greatest influence on wall response is the choice of base ground motion record applied to the structure. The difference between the frequency of the base excitation record and the fundamental frequency of the model is the most important factor determining wall response to seismic excitation. The large displacements and reinforcement loads reported in this paper are due to the observation that the applied frequency of base excitation is close to the fundamental frequency of the model with the reference geometry. Wave propagation theory for two-dimensional elastic media shows that a reduction in model width leads to an increase in the magnitude of the fundamental frequency of the model. This theoretical result explains the large reduction in wall displacements and reinforcement loads calculated for walls with a reduced soil mass width but excited at the reference 3 Hz used in this study. The combination of width of model and frequency of the applied base acceleration record has important implications to dynamic numerical modelling of reinforced soil walls. The use of excessively narrow retained soil volumes to reduce numerical grid size and computation time may result in an under-estimation of wall displacements and reinforcement loads. Conversely, very wide numerical grids may generate excessively large deformations and reinforcement loads. As a preliminary recommendation, it may be prudent to select a numerical grid width that will capture the volume of the yielded soil in the retained soil zone predicted by Mononobe-Okabe theory. The results of the current investigation identify deficiencies in the current empirical AASHTO (1996) method that is commonly used to estimate the magnitude of dynamic load increment, ∆T, in reinforcement layers. The linear trend of the dynamic load incre-BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 162 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 ment with depth below the crest predicted by the AASHTO method was not observed inallsimulationsfor wallsconstructed withstiffness valuescomparable togeosynthetic reinforcementmaterials.Inaddition,theAASHTOmethodpredictsthesamemagnitude of dynamic load in a reinforcement layer for the same peak horizontal ground accelerationregardlessofothercharacteristicvaluesoftheaccelerogram.Incontrast,themagnitude of dynamic load increment, ∆T, was demonstrated to be sensitive to frequency of base excitation and model width. It is expected that other characteristics of ground motion (e.g. frequency content, duration of strong ground motion, and peak ground velocity) will also influence the magnitude of seismic-induced loads in reinforcement layers. In fact, the empirical AASHTO (1996) method may be unsafe for design of walls that are excited close to the fundamental frequency of the structure during an earthquake. Ultimately, numerical simulation results of the type demonstrated here may be used to verify or modify current pseudostatic methods of analysis and design. However, numerical results must be checked against physical measurements from large-scale shaking table tests and further numerical work must be carried out to investigate response features of idealized reinforced soil walls subjected to ground motions representing a range of actual earthquake records. ACKNOWLEDGMENTS The writers wish toacknowledge theefforts ofthe reviewersof thispaper whosecomments greatly improved the original submission. The funding for the work reported in the paper was provided by the Department of National Defence (Canada). 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NOTATIONS Basic SI units are given in parentheses. amax = peak base acceleration (m/s2)BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 165 B = width of numerical grid (m) Es = elastic modulus for soil (Pa) f = frequency (Hz) flim = upper cut-off frequency (Hz) f11 = frequency of first mode shape of a two-dimensional elastic medium (Hz) Gs = shear modulus for soil (Pa) Gw = shear modulus for concrete panel material and foundation base (Pa) H = height of wall (m) J = reinforcement stiffness (N/m) K = constant (dimensionless) Ka = coefficient of active earth pressure (dimensionless) Ks = bulk modulus for soil (Pa) Kw = bulk modulus for concrete panel material and foundation base (Pa) kb = grout-soil interface stiffness (N/m/m) L = base width of reinforced soil zone (m) m = constant (dimensionless) Pa = atmospheric pressure (Pa) sb = grout-soil bond strength (N/m) T = reinforcement load (N/m) Tc = reinforcement-wall connection load (N/m) Tmax = maximum tensile load recorded along the length of a reinforcement layer (N/m) Ty = yield strength of reinforcement (N/m) T1 = fundamental period (s) t = time (s) ü(t) = horizontal acceleration (m/s2) α, β, ζ = constants (dimensionless) c = damping (dimensionless) ∆T = reinforcement dynamic load increment (∆T= Tmax (dynamic) - Tmax (static)) (N/m) ∆t = time step (s) ∆x = horizontal wall displacement following prop release at end of construction (m) δ = interface friction angle (Coulomb earth pressure theory) (_) δ g = grout-soil interface friction angle (_) Ô = friction angle for soil (_) Ôi = wall-soil interface column friction angle (used in numerical modeling) (_)BATHURST AND HATAMI D Seismic Response Analysis of a Geosynthetic-Reinforced Wall 166 GEOSYNTHETICS INTERNATIONAL S 1998, VOL. 5, NOS. 1-2 γ = bulk unit weight of soil (N/m3) η = dynamic shear strain (%) ν = Poisson’s ratio (dimensionless) νs = Poisson’s ratio for soil (dimensionless) νw = Poisson’s ratio for panel wall (dimensionless) ρ = density (kg/m3) σ3 = minor principal effective stress (Pa) σo = effective confining pressure (Pa) ψ = dilatancy angle for soil (_) ψi = wall-soil interface column dilatancy angle (_) ABBREVIATIONS AASHTO: American Association of State Highway and Transportation Officials FEM: Finite Element Method FHWA: Federal Highway Administration FLAC: Fast Lagrangian Analysis of Continua View publication stats