Effect of structural design on fundamental frequency of reinforced-soil retaining walls

Transcription

1 Soil Dynamics and Earthquake Engineering 19 (2000) Effect of structural design on fundamental frequency of reinforced-soil retaining walls K. Hatami*, R.J. Bathurst Civil Engineering Department, Royal Military College of Canada, Kingston, Ont., Canada K7K 7B4 Accepted 5 March 2000 Abstract The results of a numerical study on the influence of a number of structural design parameters on the fundamental frequency of reinforcedsoil retaining wall models are presented and discussed. The design parameters in the study include the wall height, backfill width, reinforcement stiffness, reinforcement length, backfill friction angle and toe restraint condition. The intensity of ground motion, characterized by peak ground acceleration, is also included in the study as an additional parameter. The study shows that the fundamental frequency of reinforced-soil wall models with sufficiently wide backfill subjected to moderately strong vibrations can be estimated with reasonable accuracy from a few available formulae based on linear elastic wave theory using the shear wave speed in the backfill and the wall height. Numerical analyses showed no significant influence of the reinforcement stiffness, reinforcement length or toe restraint condition on the fundamental frequency of wall models. The strength of the granular backfill, characterized by its friction angle, also did not show any observable effect on the fundamental frequency of the reinforced-soil retaining wall. However, the resonance frequencies of wall models were dependent on the ground motion intensity and to a lesser extent, on the width to height ratio of the backfill Elsevier Science Ltd. All rights reserved. Keywords: Fundamental frequency; Reinforced-soil; Retaining walls; Seismic response; Dynamic analysis; Geosynthetics; FLAC 1. Introduction Dynamic lateral earth pressure behind a reinforced-soil retaining wall subjected to an intensive ground motion can be significant. This additional (incremental) horizontal pressure may induce excessive wall lateral displacement and reinforcement load which can result in damage to or collapse of the structure. Damage to bridge superstructures, as a result of excessive lateral movement of abutment retaining walls due to seismic loading has been reported [1 4]. An essential step in seismic design of both conventional and reinforced-soil retaining walls is to determine the natural frequencies of the structure. Reinforced-soil retaining walls of typical heights (e.g. H 10 m and backfill material are generally considered as short-period structures (e.g. see Ref. [5]). Soil damping also significantly reduces the contribution of higher modes in total dynamic response of retaining wall systems [6]. Therefore, the response of the wall to ground motion is dominated by the fundamental frequency of the structure (also, see Ref. [3]). The fundamental frequency of * Corresponding author. Tel.: , ext. 6347; fax: address: hatami-k@rmc.ca (K. Hatami). a retaining wall-backfill system is often estimated according to a one-dimensional shear beam analogy based on the height of the wall and the speed of shear wave in the backfill material [1,5 10]. In contrast to an infinitely long uniform soil layer, a reinforced-soil retaining wall system includes structural components such as reinforcement layers and a vertical-facing panel supported on a footing. The vertical wall face suggests that a two-dimensional approach to fundamental frequency response analysis may be more appropriate than the onedimensional shear beam approach. Dynamic response of reinforced-soil retaining walls to ground motion has been the subject of several studies [11 16]. However, little can be found in the available literature that specifically addresses the influence of structural design (e.g. reinforcement stiffness, length and spacing, facing panel type and thickness, and toe restraint condition at the panel footing), geometry and material properties of the backfill, intensity level of shaking and duration of excitation on the fundamental frequency of reinforced-soil retaining wall structures. Richardson and Lee [14] conducted a series of shaking table studies on small-scale (380 mm high) reinforced-soil wall models. They subjected the retaining wall models to harmonic motions with different amplitudes and frequencies. The maximum base acceleration varied between /00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S (00)

2 138 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) and 0.50g and the frequency ranged between about 3 and 40 Hz. The results of frequency sweep of the input base acceleration for each acceleration level provided welldefined frequency response curves. The fundamental frequency and the magnitude of acceleration amplification at the surface of the backfill model increased with decreasing input acceleration level. Richardson and Lee concluded that the backfill model responded as a damped, single mode, nonlinear elastic oscillator within the examined range of frequency and acceleration level of the shaking. They also calculated the fundamental period of various reinforced-soil wall models using the finite element-based program QUAD4B. Based on the results of their finite element simulations they proposed the following empirical equation for the fundamental period, T 1, of a reinforced-soil retaining wall with a level surface: T 1 ˆ CH 1 where T 1 is in seconds, H the height of the wall in meters and C a coefficient that ranges from to depending on the shear modulus of the backfill. Bathurst and Hatami [11,12] carried out numerical simulations of the response of 6 m high reinforced-soil wall models to variable-amplitude, harmonic input ground motions with a range of frequencies. The backfill was modeled as a cohesionless, elastic plastic material with Mohr Coulomb failure criterion. They presented frequency response plots of the reinforced-soil retaining wall models that were obtained using the calculated maximum wall displacement and reinforcement load. The peak ground acceleration, a g, was set to 0.2g where g is the acceleration of gravity. Their results indicated that the fundamental frequencies of reinforced-soil retaining wall models were close to the predicted values based on the conventional one-dimensional shear beam model for cases with wide backfill. Their results also showed that a proposed formula by Wu [17] for a two-dimensional backfill model provided a reasonable estimate of the fundamental frequency of the reinforced-soil wall models with sufficiently wide backfill, B (e.g. B=H 5 : Hatami and Bathurst [18] also examined the accuracy of the one-dimensional shear beam analogy and Wu s two-dimensional solution for retaining wall fundamental frequency for different wall height and ground motion intensity values. They showed that both the onedimensional shear beam equation and Wu s formula overestimated the fundamental frequencies of the tall walls H ˆ 9m under strong ground motion (peak ground acceleration, a g ˆ 0:4g whereas both theoretical approaches resulted in satisfactory frequency predictions for shorter H ˆ 3m reinforced-soil wall models. The theoretical predictions were satisfactory for all model heights under moderately strong a g ˆ 0:2g ground acceleration. The current paper extends the preliminary work of Hatami and Bathurst [18] by examining the influence of a wider range of model backfill width to height ratio, soil strength (friction angle), reinforcement stiffness and peak ground acceleration values on the predicted fundamental frequency of idealized reinforced-soil wall systems. The current study reviews closed-form solutions for the prediction of the fundamental frequency of one-dimensional and two-dimensional linear elastic media. The results of these closed-form solutions to predict the fundamental frequency of the model retaining walls are compared with values from the results of numerical analyses. 2. Frequency response analysis of retaining wall models 2.1. Predicted frequencies from closed-form solutions for linear elastic soil models The fundamental frequencies of the retaining wall models were evaluated based on the backfill soil height and shear wave speed in order to obtain the appropriate frequency range for the parametric analysis. The closed-form solutions provided by Wood [19], Scott [20], Wu [17], and Matsuo and Ohara [21] were examined for this purpose. The two theoretical solutions reported by Matsuo and Ohara [21] for the fundamental frequency of a linear elastic soil, subjected to horizontal ground motion, are based on two different assumptions: (i) the soil is restricted from any vertical displacement v ˆ 0 throughout the backfill domain and; (ii) no vertical normal stress exists to restrict the backfill soil from vertical displacement s v ˆ 0 over the entire domain. Matsuo and Ohara argued that the solution for the real case lies between these two extreme cases. They derived the solutions for soil horizontal displacement and lateral pressure on a rigid wall subjected to harmonic loading for the above two limiting cases in a common expression with different coefficients representing each case. Their solutions apply to a wall retaining an infinitely wide backfill. The solutions for displacement showed a decay of amplitude with distance from the wall towards the far field for loading frequencies below the fundamental frequency of the soil wall system. On the other hand, the solution indicated radiation of waves towards the far field for high frequency loading. Matsuo and Ohara found the calculated pressure at the bottom of the wall to be about 10% higher in case (i) than in case (ii) for Poisson s ratio n ˆ 0:3: The fundamental frequencies of the soil rigid wall system for the two cases described above can be expressed as follows [17,19,21] (see also, Ref. [22]): Case (i): f11 vˆ0 ˆ f 1 GF vˆ0 2 f 1 ˆ 1 4H s s 8 1 n H 2 GF vˆ0 ˆ 1 1 2n B 3 4

3 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 1. Variation of the geometric factors from different theoretical solutions with normalized backfill width for different values of backfill Poisson s ratio. Case (ii): f svˆ0 11 ˆ f 1 GF svˆ0 5 s GF svˆ0 ˆ 1 2 n H n B In the above equations, f11 vˆ0 and f11 svˆ0 are the frequencies (in Hz) of the first (two-dimensional) mode shape of the elastic medium corresponding to the cases (i) and (ii), respectively; G is the shear the density and n is Poisson s ratio of the soil. The parameters GF vˆ0 and GF svˆ0 are geometric factors defined here to represent the two-dimensional effect of a limited-width backfill on the fundamental frequency of the soil-wall system. The frequency of an infinitely long, uniform soil layer, f 1,is given by Eq. (3). Wood [19] numerically calculated the roots of the frequency equations associated with the two-dimensional boundary value problem of a uniform backfill contained between rigid walls. He presented plots of backfill natural frequencies as a function of backfill width for different Poisson s ratio values. The backfill was modeled as a plane-strain, homogeneous elastic soil. Scott [20] derived the equation for natural frequencies of a rigid retaining wall assuming the backfill as a one-dimensional shear beam attached to the wall with elastic springs. He calculated the stiffness of the springs by comparing his equation for fundamental frequency of the retaining wall with the equation given by Wood [19]. This comparison included the effect of backfill width on the calculated frequency of the retaining wall system. Scott s equation for the fundamental frequency of a two-dimensional soil wall system with uniform depth is given by: f S 11 ˆ f 1 GF S 7 s GF S ˆ n H 2 p n B where f11 S is the frequency (in Hz) corresponding to the first (two-dimensional) mode shape of the backfill medium (superscript S denotes Scott s formula) and GF S is the geometric factor according to Scott s solution.

4 140 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 2. Variation of the fundamental frequency of retaining walls with normalized backfill width from closed-form and empirical solutions n ˆ 0:3 : Wu [17] and Wu and Finn [23,24] developed an approximate closed-form solution for dynamic earth pressure on a rigid wall with the assumption of a homogenous, elastic backfill. Their formulation included approximations based on shear beam analogy with zero shear stress at the surface and assumption of no vertical normal stress throughout the backfill. The wall-backfill system was assumed to be a plane-strain model. The wall and the foundation were both assumed to be rigid. Accordingly, the displacement field across the backfill could be approximated as a summation of sinusoidal mode shapes in horizontal and vertical directions. Wu [17] derived a closed-form expression for the undamped natural frequencies of the backfill model subject to the above boundary conditions and under small amplitude vibration. Wu argued that the nonlinear behavior of the backfill under strong ground motion would reduce the fundamental frequency of the soil-wall system and therefore would alter the peak seismic thrust on the wall compared to the case of a linear soil model. However, no quantitative evaluation of the expected difference in the response was provided. Wu suggested that his analysis could be extended to stronger input ground motions by evaluating the reduced shear modulus of the backfill at larger strain levels and calculating the modified fundamental frequency of the soil-retaining wall system. The fundamental frequency of the soil model behind a rigid wall under small-amplitude vibrations according to Wu [17] is given by: f W 11 ˆ f 1 GF W s 2 H 2 GF W ˆ 1 1 n B 9 10 where f W 11 is the frequency (in Hz) corresponding to the first (two-dimensional) mode shape of the soil medium (superscript W denotes Wu s formula) and GF W is the geometric factor according to Wu s solution. The variation of geometric factors with normalized width of the backfill B=H from theoretical solutions (i.e. Eqs. (4), (6), (8), and (10) and the reproduced plots of Wood [19]) is plotted in Fig. 1. It is seen that for the case of an infinitely wide backfill B=H! ; the geometric factors based on all the above solutions approach unity and the corresponding frequencies converge to f 1. It is also seen that the predicted fundamental frequency of the wall for different values of B=H and Poisson s ratio according to all the above solutions fall between the predicted values based on the two limiting cases of v ˆ 0 and s v ˆ 0: The difference between the two cases v ˆ 0 and s v ˆ 0 increases significantly as n approaches the value 0.5. According to Fig. 1, the v ˆ 0 approximation results in a relatively large predicted frequency of the soil-wall system for n close to 0.5. However, for Poisson s ratio values of typical granular soils used in reinforced-soil retaining walls n 0:3 and sufficiently wide backfill (e.g. B=H 5 ; the above limiting approximations show almost no difference in predicted values for the fundamental frequency of soil-retaining wall systems. The value of GF using Scott s solution also increases significantly for large values of Poisson s ratio n! 0:5 and narrow backfill (e.g. B=H 3 (Fig. 1). Large Poisson s ratio values and narrow backfill result in strong two-dimensional effects which violate the v ˆ 0 condition and the one-dimensional approximation used by Scott (shear beam analogy). Fig. 1 also shows that Scott s prediction of the retaining wall fundamental frequency is very close to the prediction based on the v ˆ 0 approximation. The close agreement can be expected since Scott [20] used the same approximation to determine

5 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 3. Example numerical grid for reinforced-soil wall with fixed toe condition. the stiffness of the springs representing the soil wall interaction. Non-dimensionalized fundamental frequencies, f 11 =Hv s ; for retaining wall models with different backfill width are plotted in Fig. 2 and have been calculated using the geometric factors shown in Fig. 1 (case of n ˆ 0:3 : The backfill properties assumed for numerical models of this study (described later in section 2.4.) are used to calculate the shear wave speed, v s. The two limiting cases v ˆ 0 and s v ˆ 0 are not included in this figure for brevity. However, they would show the same relative trends as in Fig. 1 with respect to other solutions. Fig. 2 also includes the nondimensionalized form of the fundamental frequency of retaining wall models using the empirical formula proposed by Richardson [25]: f11 R ˆ 38:1 11 H where f11 R is the estimated fundamental frequency of the reinforced-soil wall in Hz (superscript R denotes Richardson s formula) and H is the wall height in meters. The empirical formula in Eq. (11) was proposed based on the measured response of a full-scale (6.1 m high) test wall excited by a buried explosive charge and forced vibration tests of four existing reinforced-soil walls of different heights ranging from 2.3 to 8.5 m [8,25]. Richardson et al. [8] showed a good agreement between Eqs. (3) and (11) using their estimate of the backfill shear modulus values of their wall models. Hence, Eq. (11) provided a simple and satisfactory estimation of the fundamental frequency of the reinforced-soil walls that were tested. The predicted fundamental frequency of reinforced-soil retaining walls based on Eq. (11) is consistent with the suggested range in Eq. (1) proposed earlier by Richardson and Lee [14]. However, Eq. (11) is valid for a particular sandy soil compacted to medium relative density and may not apply for other soils. In addition, the formula does not explicitly include the effect of backfill aspect ratio (width to height) on the predicted fundamental frequency, which may be significant for a narrower backfill as shown in Figs. 1 and 2. Fairless [26] reviewed the results of physical tests reported in the literature and demonstrated that Eq. (11) typically overestimated the measured fundamental frequencies of model reinforced-soil retaining walls. Overall, Richardson s empirical estimation of the fundamental frequency of reinforced-soil retaining walls is within the range of predicted results using the theoretical solutions for conventional retaining walls (Fig. 2) but may result in values significantly greater than the fundamental frequency values predicted using the theoretical solutions with B=H 3: The above theoretical predictions for the fundamental frequency of soil-retaining wall systems are based on linear theory and do not include the influence of the shaking intensity level which can generate nonlinear and plastic response of the structure. Other attempts to predict the fundamental frequency of gravity and earth dam structures are also cited in the literature that are valid for low-amplitude ground motions where linear elastic theory is applicable [27]. In following text, the results of theoretical solutions described in this section are compared against the results of numerical evaluation of the fundamental frequency of reinforced-soil retaining wall models subjected to different ground motion intensities Numerical approach The two-dimensional, finite difference program Fast

6 142 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Table 1 Parametric values used in the evaluation of the fundamental frequency of wall models (notes: reinforcement spacing S v ˆ 1:0m; viscous damping ratio j ˆ 5% Wall height (m) Model width B (m) Width to height ratio B=H Reinforcement ratio L=H Toe Restraint condition Reinforcement stiffness J (kn/m) Friction angle ( ) Peak ground acceleration a g (g) Input frequency f g (Hz) , 1.0 Fixed 500, , , 0.4 3, 5, 6, , 1.0 Fixed, Sliding 500, , , 35, , 0.4 1, 2.5, 3, 3.5, , 36, 54 a 2, 4, 6 0.4, 1.0 Fixed 500, , , 0.2, 0.4 1, 1.5, 2, 2.5, 3, 4 a Reference wide-width backfill model case. Fig. 4. Variation of normalized maximum lateral displacement of wall crest with normalized frequency of input ground motion: (a) J ˆ 500 kn/m; (b) J ˆ kn/m; (c) J ˆ kn/m.

7 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 4. (continued) Lagrangian Analysis of Continua (FLAC 3.40 [28]) was used to carry out the numerical experiments. The program is widely used in geotechnical engineering applications and is attractive for seismic analysis of reinforced-soil retaining walls because it can model large distortions and nearcollapse conditions [11,12,29] Numerical grid and problem boundaries The numerical grid for a typical wall model used in the study is illustrated in Fig. 3. Numerical simulations represent a backfill of constant depth retained by a continuous panel wall with uniformly spaced reinforcement layers. The width of the backfill, B, in the cases representing an infinitely wide backfill was extended to a large distance beyond the back of the facing panel (Table 1) so as to contain the shear wedge (plastic zone) that develops behind the reinforced zone during base shaking. The vertical spacing between reinforcement layers was kept constant at S v ˆ 1:0 m: The height of the wall models H ˆ 3; 6 and 9 m) and the spacing between reinforcement layers are typical of actual structures in the field. The toe restraint condition (wall footing) 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 of a previous study [11] show that the lateral displacement of the wall and the magnitude and distribution of reinforcement load can be significantly affected by the wall toe restraint condition. For sliding cases, the wall model was seated on a thin (0.05 m thick) layer of soil that was extended across the

8 144 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 4. (continued) entire width of the numerical grid. This layer performed a similar function to a sliding interface and was required to ensure that models representing walls without horizontal toe restraint (i.e. sliding-wall cases) were not artificially restrained along their contact area with the foundation during shaking. For the fixed-toe condition, the wall and soil regions were connected directly to a foundation base comprising of a 1m-thick layer of very stiff material (Fig. 3). Two values for the reinforcement length to wall height ratio were selected to represent narrow L=H ˆ 0:4 and wide L=H ˆ 1 reinforced zones in the parametric analysis. These reinforcement ratio values capture the range of values reported in the literature for actual structures in the field as well as experimental studies on reduced-scale retaining wall models. Reinforcement length to wall height ratios as low as L=H ˆ 0:33 and 0.4 have been used in some shaking table studies [30 33] Material properties 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 K w ˆ MPa and G w ˆ MPa ; respectively. Poisson s ratio for the panel material was taken as n w ˆ 0:15: The granular backfill was modeled as a purely frictional, elastic plastic soil with a Mohr Coulomb failure criterion. The reference friction angle of the soil was f ˆ 35 ; dilatancy angle c ˆ 6 ; and unit weight g ˆ 20 kn=m 3 : The soil material was assigned constant values of bulk modulus K s ˆ 27:5 MPa

9 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 5. Variation of response ratio of maximum wall lateral displacement (response ratio between a g ˆ 0:2g and a g ˆ 0:4g loadings) with normalized frequency of input ground motion. and shear modulus G s ˆ 12:7 MPa: The foundation zone for fixed-toe cases was assigned the same material properties as the concrete facing panel. The panel-soil interface was modeled using a thin (0.05 m thick) soil column directly behind the facing panel. The friction angle and the dilatancy angle of the interface soil column between the reinforced zone and the facing panel were set to f i ˆ 20 and c i ˆ 0; respectively. The remaining soil properties of the interface soil column were the same as the properties of the backfill soil. The reinforcement layers were modeled using linear, elastic plastic cable elements with negligible compressive strength and an equivalent cross-sectional area of m 2. The equivalent linear elastic stiffness values for the reinforcement layers were taken as J ˆ 500; and kn/m (Table 1). The lower, intermediate and higher stiffness values represent an extensible (polymeric) geotextile reinforcement, a very stiff (polymeric) geogrid reinforcement material and a steel strip reinforcement, respectively. A large range of stiffness values was included in the parametric study to identify any possible stiffness effects on the resulting resonance frequency of the retaining walls with the input ground motion. The yield strength of the reinforcement in all cases was kept constant at T y ˆ 200 kn=m; which is well above the magnitude of the maximum reinforcement load recorded in the simulations. Consequently, reinforcement rupture was not a possible failure mechanism in this study. The interface between the reinforcement (cable elements) and the soil was modeled

10 146 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 6. Variation of maximum reinforcement incremental load with normalized frequency of input ground motion: (a) J ˆ 500 kn/m; (b) J ˆ kn/m; (c) J ˆ kn/m. with a grout material of negligible thickness and with an interface friction angle d g ˆ 35 : The grout s bond stiffness and bond strength values were taken as k b ˆ MN/ m/m and s b ˆ kn=m; respectively. These interface and grout properties were selected to simulate a perfect bond between the soil and reinforcement layers. The end of each cable element was connected to a single grid point at the back surface of the facing panel region to simulate a fixed reinforcement connection in the field Seismic loading The staged construction of each wall model was simulated by placing the backfill and the reinforcement in layers while the continuous wall facing panel was braced horizontally using rigid external supports. The panel supports were then released in sequence from the top to the bottom of the structure as is done in the field. After static equilibrium was achieved (end of construction stage), the full width of the foundation was subjected to the variable-amplitude harmonic ground motion record illustrated in Fig. 3 (inset). This acceleration record was applied horizontally to all nodes at the bottom and the right-hand side (truncated) boundary of the backfill region at equal time intervals of Dt ˆ 0:05 s: The mathematical expression for input acceleration is given by: q u t ˆ b e at t z sin 2pft 12

11 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 6. (continued) where a ˆ 5:5; b ˆ 55; and z ˆ 12 are constant coefficients, f is the base acceleration frequency and, t is the time. The resulting peak amplitude, a g, using these parameters is 0.2g, where g is the acceleration of gravity. Coefficient terms were adjusted to give a peak amplitude a g ˆ 0:4g representing a stronger earthquake. The variableamplitude input ground motion from Eq. (12) was chosen over simple harmonic acceleration because it simulates the rise and time decay of an idealized accelerogram. In addition, it does not lead to excessive response of the wall model that can be expected in the case of a sustained, constant amplitude input base acceleration. The input acceleration in Eq. (12) is similar to the Tsang signal function reported elsewhere [34]. The application of uniform acceleration at the vertical truncated boundary was based on the assumption of uniform distribution of horizontal acceleration over the depth of the backfill away from the facing panel. This assumption also represents the case of reduced-scale reinforced-soil retaining wall models on shaking tables where a rigid back boundary moves in phase with the base [14]. A viscous damping ratio of j ˆ 5% was chosen for both the soil and facing panel regions in the parametric analyses. This damping ratio value may appear conservative but it is comparable to the range of values estimated based on measured response of retaining walls to dynamic loading in a number of experimental studies [5,8]. In addition, a major portion of input seismic energy is dissipated through the hysteresis and plastic deformation of the soil.

12 148 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 6. (continued) A relatively low value for the backfill damping ratio j ˆ 5% also ensures a detectable difference in dynamic response of retaining wall models for different parametric cases. 3. Response of wall models to input ground motion The calculated response of wall models to the introduced base motion (Eq. (12)) is represented in terms of maximum normalized lateral displacement of the facing panel and maximum reinforcement load during shaking. These parameters are selected as representative parameters in the present frequency response analysis due to their significance in design and performance of retaining wall structures under both static and seismic loading conditions Effect of reinforcement length and stiffness Wall displacements Fig. 4 summarizes the variation of normalized maximum lateral displacement of the wall crest with normalized frequency of input ground motion. The datum for all displacement plots in this study is taken with respect to the end-ofconstruction condition following prop release. The toe restraint condition is fixed in all the cases presented in Fig. 4. The input frequency is nondimensionalized in the form vh/v s where v is the input circular frequency and v s is the speed of shear wave propagation in the backfill material. Progressive outward displacement of the wall facing was observed during all the parametric simulation runs. The accumulated, dynamic portion of the wall lateral

13 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 7. Variation of response ratio of maximum reinforcement incremental load (response ratio between a g ˆ 0:2g and a g ˆ 0:4g loadings) with normalized frequency of input ground motion. displacement was significant in all analysis cases. A shear failure wedge developed and extended well beyond the reinforced zone. The observed wedge angle, in general, showed good agreement with the predicted value according to the Mononabe Okabe theory considering amplification of acceleration over the height of the wall [11]. Despite significant wall displacement and evidence of a plastic region in the backfill, the frequency response curves in Fig. 4 indicate that the fundamental frequency of wall models with wide backfill (e.g. B=H 5 subjected to moderately strong ground motion a g ˆ 0:2g is predicted satisfactorily using one-dimensional theory (Eq. (3)) and Wu s solution (Eqs. (9) and (10)). The fundamental frequencies of wall models based on Wood and Scott solutions generally overestimate the numerical results. This overestimation is more significant at lower B=H ratio values (e.g. compare theoretical predictions in the plots for cases H ˆ 3 m (where B=H ˆ 10 and H ˆ 9 m (where B=H ˆ 6 in Fig. 4). The predicted fundamental frequencies of wall models using any of the linear elastic theories become less accurate for the stronger ground motion a g ˆ 0:4g: The normalized frequency value according to Richardson s empirical equation (Eq. (11)) is v 11 H=v s ˆ 3:03 which clearly overestimates the fundamental frequency of the retaining wall models plotted in Fig. 4. Comparison of the frequency response curves shown in Fig. 4a c reveals almost no influence of the reinforcement stiffness or length on the predicted low-amplitude

14 150 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 8. Influence of toe restraint condition on frequency response of wall to input ground motion H ˆ 6m; L=H ˆ 1; a g ˆ 0:4g : fundamental frequency of a wall model of given height and backfill material. This conclusion is consistent with observations by Wolfe [16] who found that for a given height of model wall tested on a shaking table, the reinforcement density did not influence the observed low-amplitude fundamental frequency of the structure (also, see Ref. [26]). However, the actual magnitude of wall displacement is sensitive to reinforcement stiffness value and reinforcement length as demonstrated in parametric analyses presented by Bathurst and Hatami [11]. Compared to the cases with L=H ˆ 1:0; greater displacements were observed in wall models with short reinforcement length L=H ˆ 0:4 under strong a g ˆ 0:4g ; lowfrequency input ground motion (Fig. 4). This is mainly due to insufficient reinforcement length for the case L=H ˆ 0:4: Design codes such as FHWA [35] recommend a minimum L=H ˆ 0:7 for the reinforcement length in reinforcedsoil walls under static loading. The wall response to input ground motion with a frequency lower than the wall fundamental frequency is essentially a quasi-static response to lateral earth pressure. Accordingly, a wall with under-designed reinforcement length may undergo significant lateral displacement and possible instability due to the short width of the reinforced zone. A documented case of translational slip at the base of a reinforced-soil wall in Japan during the Kobe earthquake has been attributed to insufficient reinforcement length [36]. In the current parametric analyses, some cases of instability in numerical models with L=H ˆ 0:4 and J ˆ 500 kn=m subjected to a g ˆ 0:4g ground motion were observed (Fig. 4a). These cases are also consistent with observations by Wolfe [16] in shaking table studies of model reinforcedsoil walls where large lateral displacements of models with short reinforcement length under strong shaking were observed. Wolfe noted that the wall response is significantly more sensitive to the loading intensity than the loading frequency when the predominant frequency of the input motion is lower than the fundamental frequency of the retaining wall. Wolfe suggested that a minimum reinforcement length can be determined by treating the reinforced soil zone as a gravity mass and determining the width of the gravity block required to resist dynamic lateral earth forces acting against the sliding mass and inertial force of the mass. This approach has been adopted in recent Newmark-type [37] and seismic pseudo-static design methods for

15 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 9. Variation of maximum wall response with normalized frequency of input ground motion H ˆ 6m; L=H ˆ 1; a g ˆ 0:2g; fixed-toe condition): (a) normalized lateral displacement of wall crest; (b) reinforcement load. reinforced soil walls constructed with modular block facings [38 40]. Fig. 5 shows another representation of the variation of the wall displacement response with input frequency. The ratio of the calculated wall lateral displacement under the two ground motion intensity levels a g ˆ 0:2g and 0.4g generally depends on both the ratio of input intensity levels and the input frequency. The influence of the input loading intensity approaches a minimum at the resonance frequency of the retaining wall with ground motion. This phenomenon is clearly observed in Fig. 5 which is consistent with the fundamental frequency of the wall models inferred from Fig. 4. The minimum response ratio value of about one in Fig. 5 is also another indication of nonlinearity and plastic response of the retaining wall system when subjected to moderately strong to strong ground motion. In contrast to the response of a purely linear elastic system, in which the response magnitude is proportional to the input intensity level, the plastic deformation of the backfill at resonance is great enough to control the system response for both values of input ground motion intensity used in the current study Reinforcement loads Fig. 6 summarizes the variation of maximum dynamic reinforcement load, T max, with normalized loading frequency. The reported reinforcement forces are incremental values taken with respect to values calculated at the end of construction and following prop release. The maximum load in each reinforcement layer was observed at the connection point to the facing panel. However, contrary to values for wall lateral displacement that was the largest at

16 152 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 9. (continued) the wall crest, the layer with the maximum reinforcement incremental load varied between parametric cases. This may explain why the frequency response curves of wall lateral displacement (Fig. 4) are better than the corresponding reinforcement load plots (Fig. 6) with respect to defining the fundamental frequency value of the retaining wall structures. Typically the maximum incremental load, T max, was observed in lower reinforcement layers for stiffer reinforcement, shorter reinforcement length and, stronger input ground motion for a given set of other parameters. Cases with L=H ˆ 0:4 resulted in large incremental load at lower frequencies which can be attributed to the under-designed reinforcement length and quasi-static loading condition. Nonetheless, the frequency response of the incremental load also shows a maximum (with better accuracy for moderately strong ground motion of a g ˆ 0:2g in the vicinity of the predicted fundamental frequency based on linearelastic analysis. The observations noted earlier regarding the accuracy of theoretical solutions to predict wall lateral displacement frequency response (Fig. 4) are also applicable to frequency response of reinforcement load. Fig. 7 shows the variation of reinforcement load response ratio with input frequency. The influence of the input loading intensity approaches a minimum at the resonance frequency of the retaining wall with ground motion in a similar fashion to that observed for the data in Fig Influence of toe restraint condition Frequency responses of relative lateral displacement of the wall crest and maximum reinforcement incremental load for the two cases of fixed-toe and sliding-toe condition are shown in Fig. 8. The results are shown for 6m-high wall models with L=H ˆ 1 and a g ˆ 0:4g: The stronger input ground motion a g ˆ 0:4g was chosen in order to magnify any possible influence of the toe restraint condition on the fundamental frequency of the wall. The frequency responses of lateral displacement and reinforcement load do not show any observable dependence of fundamental frequency on toe restraint condition of the wall models Influence of soil friction angle Fig. 9a shows the calculated frequency response of wall lateral displacement for three different values of friction angle, f, for the backfill soil. Two reinforcement stiffness cases were investigated: J ˆ 500 kn=m representing a polymeric reinforcement material and, J ˆ kn=m representing a metallic reinforcement product. The calculated

17 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 10. Variation of maximum wall response with normalized frequency of input ground motion for different backfill width cases H ˆ 9m; L=H ˆ 1; J ˆ kn=m : (a) normalized lateral displacement of wall crest; (b) reinforcement load. maximum dynamic lateral displacement of the wall generally increases with decreasing value of backfill friction angle and lower reinforcement stiffness. However, neither of the above parameters shows a measurable influence on the predicted fundamental frequency of wall displacement for the reinforced-soil wall models. A similar conclusion can be made with respect to frequency response of maximum reinforcement load data in Fig. 9b. Consistent with the results of Fig. 4, excessive lateral displacement in the lower frequency range was observed for cases with low friction angle as a result of excessive failure of the backfill soil (plasticity). However, all analyses were numerically stable Influence of backfill width and ground motion intensity Frequency responses of normalized lateral displacement of the wall crest and maximum reinforcement incremental load for the two backfill aspect ratio values B=H ˆ 2 and 4 are given in Fig. 10a and b, respectively. The results are shown for 9m-high wall models with L=H ˆ 1:0 and J ˆ kn=m: The predictions of the fundamental frequency of wall models in Fig. 10 using two dimensional solutions show significant differences from the prediction based on onedimensional elastic theory for the case of a narrow backfill (e.g. B=H ˆ 2 : The prediction of fundamental frequency based on Wu s approach shows the closest agreement with the frequency values inferred from the numerical results in Fig. 10 of all the two-dimensional solutions evaluated. The agreement is most satisfactory for small-amplitude vibrations (i.e. case of a g ˆ 0:05g and is better than the onedimensional prediction for the retaining walls with a narrow backfill (Fig. 10a and b with B=H ˆ 2 : The theoretical predictions of the fundamental frequency of wall models (vertical lines in Fig. 10a and b) for the cases B=H ˆ 2 and 4 show a strong dependence on the backfill B=H ratio (consistent with the plots in Figs. 2 and 4). The fundamental frequency values inferred from large response amplitude

18 154 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) Fig. 10. (continued) regions in the plots of numerical results show much less influence of the B/H ratio than the closed-form solution results, particularly for the stronger input ground motions. The fundamental frequencies of the wall models from numerical results consistently shift toward lower frequencies under stronger input ground motions (i.e. a g ˆ 0:2g and 0:4g compared to a g ˆ 0:05g; also, compare cases of 0.4g and 0.2g in Fig. 4) and happen to approach the predicted value based on the one-dimensional solution (Eq. (3)). This observation is attributed to the decrease in the magnitude of the modulus of the backfill material at larger strain levels which has also been observed in experimental studies [14]. The equivalent viscous damping ratio of the backfill increases with increasing strain level. A larger damping ratio value also reduces the fundamental frequency of the retaining wall structure under strong ground motion to values less than the predicted value based on undamped, linear elastic analysis. The dependence of fundamental frequency of the wall on ground motion intensity is a nonlinear characteristic of the structure under severe excitation. This shift of frequency was more pronounced for taller wall models. This may be attributed to larger strain levels that are developed in 9 m high models as compared to the 3 m high models. The frequency response curves of wall lateral displacement and reinforcement load show welldefined peak characteristics for all wall heights and reinforcement stiffness values, which is in accordance with the results of shaking table studies reported by Richardson and Lee [14]. The response of the reinforced-soil retaining wall models in the vicinity of resonance (i.e. proximity to v ˆ pv s =2H was difficult to calculate due to numerical stability problems. In addition, due to plastic behavior and excessive local deformation of the backfill at the surface away from the wall crest, the calculated lateral displacement at the wall crest does not increase in the vicinity of the wall resonance frequency with increasing ground motion intensity. Accordingly, the displacement response values very close to the

19 K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) resonance frequency cannot be reliably used to complete the well-defined frequency response curves by numerical calculations. However, some theoretical studies on conventional retaining walls [6,41] indicate a resonance amplification factor of (2j) 1/2 in wall response compared to the value (2j) 1 which corresponds to a simple linear oscillator with viscous damping ratio, j. For j ˆ 5%; the magnitude of amplification factor at resonance will be about three which is considerably less than the value of 10 associated with a viscously damped linear oscillator. 4. Conclusions Retaining walls of typical heights (e.g. H 10 m are considered as short-period structures and therefore, their seismic response is dominated by their fundamental frequency. The paper first summarizes theoretical solutions for evaluating the fundamental frequency of conventional retaining walls (typically rigid retaining walls). These solutions can be applied to continuum, plane-strain models of retaining wallbackfill systems and are presented in the general form: f 11 ˆ f 1 GF 13 where GF ˆ f n; B=H represents the modification of f 1 to obtain the fundamental frequency of a two-dimensional retaining wall model from the one-dimensional frequency formula for an infinitely long uniform soil layer. The results of theoretical solutions are compared to the results of numerical modeling of a wide range of reinforced-soil retaining wall models subjected to base excitation using a variable-amplitude harmonic input acceleration record with a range of frequencies in the vicinity of the predicted values according to linear elastic analysis. Parametric seismic analyses on reinforced-soil retaining wall models were carried out to investigate the influence of different structural components on their fundamental frequency. The structural components included the reinforcement stiffness and length, the restraining condition at the toe (footing) of the facing panel and the friction angle of the granular backfill soil. Problem geometry parameters included the wall height and the backfill width. The intensity of ground motion, characterized by the peak ground acceleration, was also varied. The results of the analyses showed that the fundamental frequency of reinforced-soil retaining wall systems with a sufficiently wide uniform backfill subjected to moderately strong ground motion (e.g. a g ˆ 0:2g in the present study) can be estimated with reasonable accuracy from a commonly used one-dimensional solution based on linear elastic theory. Among the two-dimensional approaches examined, the frequency formula proposed by Wu [17] and Wu and Finn [23,24] gave the closest agreement to the fundamental frequency value inferred from numerical results. The fundamental frequency values from two-dimensional continuum models were shown to approach values based on one-dimensional theory for significantly wide backfill (e.g. B=H 10 : Earlier numerical simulation work by the writers [11] has demonstrated that reinforcement stiffness, reinforcement length and toe restraint condition can have a significant influence on the magnitude of reinforcement forces and lateral displacements of reinforced-soil wall models during a simulated seismic event. However, the results of the current study using the same numerical models demonstrate that these variables do not significantly affect the fundamental frequency of reinforced-soil wall models with a wide range of structural component values. Large response magnitude was observed at low frequencies for the cases where the reinforcement length, reinforcement stiffness or backfill friction angle was very low. The fundamental frequency of a retaining wall with narrow backfill according to theoretical predictions can be significantly higher than the case of a wall with an infinitely wide backfill. However, the numerical results of this study on fundamental frequency of model reinforced-soil walls were relatively less sensitive to the backfill width of the reinforced-soil wall system. The reason for the reduced effect of the backfill width is attributed to soil plasticity that develops in the near-field behind the facing panel. In contrast, purely elastic soil response is more significantly influenced by the geometry, i.e. backfill width, of the model. Another reason for the difference may be partly due to the type of boundary condition at the truncated far-end boundary of the backfill. The uniform acceleration applied coherently with the base acceleration at the truncated far-end boundary of reinforced-soil wall models in this study may subdue the influence of backfill width compared to the case with a nonaccelerated, radiating boundary. However, since the same far-end boundary condition was used in all parametric cases, the type of boundary condition does not effect the relative response of the different models investigated. It is also noted that the available theoretical approaches do not result in satisfactory estimation of the fundamental frequency of retaining walls with limited-width backfill for the case where the retaining wall structure is expected to experience a severe ground motion. The intensity of the input ground motion, represented by the magnitude of the peak acceleration showed the most dominant influence on the resonance frequency of a given retaining wall model. The resonance frequencies were lower under a stronger input acceleration and were different from the predicted values based on low-amplitude, linear elastic analysis. Also, the difference between numerical and theoretical predictions of fundamental frequency is larger for retaining wall models subjected to stronger input ground motion. 5. Additional remarks The conclusions of this study are limited to model