Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
www.elsevier.com/locate/soildyn
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. q 2000 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.: 1 1-613-541-6000, ext. 6347; fax: 1 1613-545-8336.
E-mail 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 0.02
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PII: S0267-726 1(00)00010-5
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K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
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, T1, of a reinforced-soil retaining
wall with a level surface:
T1 ˆ CH
1†
where T1 is in seconds, H the height of the wall in meters and
C a coefficient that ranges from 0.020 to 0.033 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 9 m† under strong ground motion (peak ground acceleration, ag ˆ 0:4g† whereas both theoretical approaches
resulted in satisfactory frequency predictions for shorter
H ˆ 3 m† reinforced-soil wall models. The theoretical
predictions were satisfactory for all model heights under
moderately strong …ag 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):
f11vˆ0 ˆ f1 ·GFvˆ0
1
f1 ˆ
4H
GFvˆ0
s
G
@
s
8…1 2 n† H 2
ˆ 11
1 2 2n B
2†
3†
4†
K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
139
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):
svˆ0
ˆ f 1 ·GFsvˆ0
f11
GF svˆ0
s
22n H 2
ˆ 11
12n B
5†
6†
vˆ0
svˆ0
and f11
are the frequencies
In the above equations, f 11
(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 modulus; @ the density and n
is Poisson’s ratio of the soil. The parameters GFvˆ0 and
GFsvˆ0 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, f1, 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 11S ˆ f1 ·GFS
s
2
64 1 2 n
H
GFS ˆ 1 1 2
B
p 1 2 2n
7†
8†
S
is the frequency (in Hz) corresponding to the first
where f11
(two-dimensional) mode shape of the backfill medium
(superscript S denotes Scott’s formula) and GFS is the
geometric factor according to Scott’s solution.
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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:
W
ˆ f1 ·GFW
f11
s
2
2
H
GFW ˆ 1 1
12n
B
9†
10†
W
is the frequency (in Hz) corresponding to the first
where f11
(two-dimensional) mode shape of the soil medium (superscript W denotes Wu’s formula) and GFW 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 f1. 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
K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
141
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, f11 =Hvs ;
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, vs. 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]:
f11R ˆ
38:1
H
11†
R
where f11
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.
2.2. Numerical approach
The two-dimensional, finite difference program Fast
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Table 1
Parametric values used in the evaluation of the fundamental frequency of wall models (notes: reinforcement spacing Sv ˆ 1:0 m; viscous damping ratio j ˆ
5%†
Wall height
(m)
Model width Width to height
B (m)
ratio B=H
Reinforcement Toe Restraint
ratio L=H
condition
3
6
9
30
42
18, 36, 54 a
0.4, 1.0
0.4, 1.0
0.4, 1.0
a
10
7
2, 4, 6
Reinforcement
stiffness J (kN/m)
Fixed
500, 10 000, 69 000
Fixed, Sliding 500, 10 000, 69 000
Fixed
500, 10 000, 69 000
Friction
angle (8)
Peak ground
Input frequency
acceleration a g (g) fg (Hz)
0.2, 0.4
20, 35, 45 0.2, 0.4
0.05, 0.2, 0.4
3, 5, 6, 7
1, 2.5, 3, 3.5, 4
1, 1.5, 2, 2.5, 3, 4
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 10000 kN/m; (c) J ˆ 69000 kN/m.
K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
143
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].
2.3. 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 Sv 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
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K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
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].
2.4. 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 Kw ˆ 11 430 MPa and Gw ˆ
10 430 MPa†; respectively. Poisson’s ratio for the panel
material was taken as nw 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 358; dilatancy angle c ˆ 68;
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
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145
Fig. 5. Variation of response ratio of maximum wall lateral displacement (response ratio between ag ˆ 0:2g and ag ˆ 0:4g loadings) with normalized
frequency of input ground motion.
and shear modulus Gs 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 fi ˆ 208 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 0.002 m 2.
The equivalent linear elastic stiffness values for the reinforcement layers were taken as J ˆ 500; 10 000 and
69 000 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 Ty 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
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Fig. 6. Variation of maximum reinforcement incremental load with normalized frequency of input ground motion: (a) J ˆ 500 kN/m; (b) J ˆ 10000 kN/m;
(c) J ˆ 69000 kN/m.
with a grout material of negligible thickness and with an
interface friction angle dg ˆ 358: The grout’s bond stiffness
and bond strength values were taken as kb ˆ 2 × 10 3 MN/
m/m and sb ˆ 1 × 103 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.
2.5. 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
 ˆ b e2at tz sin…2pft†
12†
u…t†
K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
147
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
ag 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.
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K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
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.
3.1. Effect of reinforcement length and stiffness
3.1.1. 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 v H/vs where v is the input circular frequency and vs
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
K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
149
Fig. 7. Variation of response ratio of maximum reinforcement incremental load (response ratio between ag ˆ 0:2g and ag 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 over-
estimate 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 ag ˆ 0:4g: The normalized
frequency value according to Richardson’s empirical equation (Eq. (11)) is v11 H=vs 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
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Fig. 8. Influence of toe restraint condition on frequency response of wall to input ground motion …H ˆ 6 m; L=H ˆ 1; ag 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 …ag 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 ag 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
K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
151
Fig. 9. Variation of maximum wall response with normalized frequency of input ground motion …H ˆ 6 m; L=H ˆ 1; ag 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 ag 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.
3.1.2. Reinforcement loads
Fig. 6 summarizes the variation of maximum dynamic
reinforcement load, Tmax, 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
152
K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
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, Tmax, 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 ag 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. 5.
3.2. 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 ag ˆ 0:4g: The stronger input
ground motion …ag 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.
3.3. 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 69 000 kN=m representing a metallic reinforcement product. The calculated
K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
153
Fig. 10. Variation of maximum wall response with normalized frequency of input ground motion for different backfill width cases
H ˆ 9 m; L=H ˆ 1; J ˆ 10 000 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.
3.4. 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
10 000 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 ag 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
154
K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
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. ag ˆ 0:2g
and 0:4g compared to ag 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 p vs =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
K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
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 ) 21/2 in wall response compared to the value
(2j ) 21 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 f1 ·GF
13†
where GF ˆ f …n; B=H† represents the modification of f1 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. ag 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
155
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
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K. Hatami, R.J. Bathurst / Soil Dynamics and Earthquake Engineering 19 (2000) 137–157
retaining walls with uniform backfill soils and constant
material properties. Nonlinear effects such as interface slip
or rupture of the reinforcement and stress-dependent properties of soil were not addressed in this study. Stress-dependent (e.g. hyperbolic model proposed by Duncan et al. [42])
or strain-dependent modulus values of the backfill as
proposed by Seed and Idriss [8] may result in an even
greater influence of ground motion intensity on the resonance frequency of retaining wall systems than reported
here. Richardson [25] introduced a frequency correction
factor, FCF, to modify the predicted fundamental frequency
of reinforced-soil walls from linear elastic theory according
to the expected peak dynamic strain amplitude in the retaining wall-backfill system. A strain-dependent stiffness model
for the backfill and further development of frequency
correction factors are reserved for future study.
Finally, the intensity of the input ground motion is characterized in this study by its peak acceleration value. Alternatively, ground motion intensity can be represented by
other intensity parameters (e.g. as defined by Arias [43] or
spectral intensity defined by Housner [44]). These effects
are believed to have a quantitative influence on the response
of retaining wall systems to ground motion. However, the
major conclusion of this study regarding the negligible
influence of structural component values on the predicted
resonance frequency of reinforced-soil retaining wall
systems is not likely to be changed.
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
Acknowledgements
[18]
The funding for the work reported in the paper was
provided by grants from the Natural Sciences and Engineering Research Council and Department of National Defence
(Canada).
[19]
[20]
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