The seismic buffer is modelled as a linear elastic material and the soil wedge shear surface by a stress-dependent linear spring.. Keywords: Geofoam; Expanded polystyrene; Earthquake; Se
Trang 1Soil Dynamics and Earthquake Engineering 27 (2007) 344–353
A simple displacement model for response analysis of
EPS geofoam seismic buffers Richard J Bathursta, , Amin Keshavarzb,1, Saman Zarnanic, W Andy Takeb
a GeoEngineering Centre at Queen’s-RMC, Department of Civil Engineering, 13 General Crerar, Sawyer Building, Room 2085,
Royal Military College of Canada, Kingston, Ont., Canada K7K 7B4
b
GeoEngineering Centre at Queen’s-RMC, Queen’s University, Kingston, Ont., Canada K7L 3N6
c
GeoEngineering Centre at Queen’s-RMC, Department of Civil Engineering, Queen’s University, Kingston, Ont., Canada K7L 3N6
Received 28 July 2006; accepted 31 July 2006
Abstract
A simple displacement-type block model is proposed to compute the compression–load–time response of an idealized seismic buffer placed against a rigid wall and used to attenuate earthquake-induced dynamic loads The seismic buffer is modelled as a linear elastic material and the soil wedge shear surface by a stress-dependent linear spring The model is shown to capture the trends observed in four physical reduced-scale model shaking table tests carried out with similar boundary conditions up to a base excitation level of about 0.7g
In most cases, quantitative predictions are in reasonable agreement with physical test results The model is simple and provides a possible framework for the development of advanced models that can accommodate more complex constitutive laws for the component materials and a wider range of problem geometry
r2006 Elsevier Ltd All rights reserved
Keywords: Geofoam; Expanded polystyrene; Earthquake; Seismic buffer; Displacement model; Shaking table; Numerical modelling; Physical testing
1 Introduction
The concept of reducing the magnitude of static earth
pressures against rigid wall structures by placing a
compres-sible vertical inclusion between the wall and the retained soil
has been demonstrated in the laboratory[1,2]and in the field
[3] A suitably selected compressible vertical inclusion will
allow sufficient lateral expansion of soil (controlled yielding)
such that the retained soil is at or close to active failure and
hence the earth pressures against the rigid structure are
(according to classical earth pressure theory) at a minimum
value Today, the compressible vertical inclusion material of
choice is block-moulded low-density expanded polystyrene
(EPS), which is classified as a ‘‘geofoam’’ material in modern
geosynthetics terminology[4]
Karpurapu and Bathurst [5] used a non-linear finite element code to numerically simulate the controlled yielding concept for static load conditions The accuracy
of the code was verified using the results of the small-scale model test results reported by McGown and co-workers
[1,2] The code was then used to carry out a numerical parametric study to develop a series of design charts for the proper selection of the modulus and thickness of the compressible layer for a given wall height and soil type The first reported field installation of a compressible inclusion to attenuate seismic-induced lateral earth forces against a rigid wall structure was described by Inglis et al
[6] Panels of EPS from 450 to 610 mm thick were placed against rigid basement walls up to 9 m in height at a site in Vancouver, BC Analyses using the program FLAC [7]
showed that a 50% reduction in lateral loads could be expected during a seismic event compared to a rigid wall solution
Hazarika et al [8], Zarnani et al.[9]and Bathurst et al
[10] performed physical shaking table tests on reduced-scale models to examine the hypothesis that the concept of
www.elsevier.com/locate/soildyn
0267-7261/$ - see front matter r 2006 Elsevier Ltd All rights reserved.
doi: 10.1016/j.soildyn.2006.07.004
Corresponding author Tel.: +1 613 541 6000x6479/6347/6391;
fax: +1 613 545 8336.
E-mail address: bathurst-r@rmc.ca (R.J Bathurst).
1
Permanent address: Civil Engineering Department, School of Engineering,
Shiraz University, Iran.
Trang 2earth force reduction using a compressible inclusion placed
against a rigid wall can be extended to the case of dynamic
earth loading The test data from Hazarika et al [8]
showed that the peak lateral loads acting on the
compressible model walls were reduced from 30% to
60% of the value measured for the nominally identical
structure but with no compressible inclusion The test
results by Zarnani et al [9] show that the magnitude of
dynamic lateral earth force attenuation increased with
decreasing geofoam modulus For the best case, the total
earth force acting against the rigid wall during seismic
shaking was reduced to 60% of the value for the nominally
identical structure without a geofoam buffer inclusion The
results of numerical modelling reported in the current
paper are compared to physical test results reported by
Zarnani et al.[9]and Bathurst et al.[10]
2 Numerical model
2.1 Block wedge
A simple one-block model is proposed for calculating the
dynamic response analysis of seismic buffer retaining walls
The soil wedge is modelled as a rigid block under plane
strain conditions Fig 1 shows the problem geometry
representing an idealized seismic buffer placed between a
rigid retaining wall and soil Parameters H and b are the
height and width of the geofoam buffer, respectively A
linear failure plane is assumed to propagate through the
backfill soil from the heel of the buffer at a to the horizontal
The system of forces acting on the block is shown in
Fig 2 Here, m is the mass of the soil block and u¨gand g are the corresponding accelerations in the horizontal and vertical directions, respectively Here we assume that only seismic-induced horizontal ground motions occur Hence, the model is excited using a forcing function u¨g(t) representing a prescribed horizontal ground acceleration record The normal and shear forces acting at the block boundaries are denoted as Niand Si, respectively
Nomenclature
A,B,C,D,E constant coefficients
b width of the geofoam buffer
d width of direct shear box
Eb modulus of elasticity of buffer
Es modulus of elasticity of soil
Fi total force acting on soil wedge in the i direction
g acceleration due to gravity
H height of wall
k spring stiffness
kmax maximum spring stiffness
kNi normal spring stiffness at boundary i
kSi shear spring stiffness at boundary i
ksoil soil–soil interface stiffness
L length of soil shear surface
m mass of soil wedge
Ni normal force at boundary i
Si shear force at boundary i
t time
W width of soil wedge and buffer in plane strain
direction
€ug horizontal acceleration
xi horizontal displacement of soil wedge in the i
direction _
xi horizontal velocity of soil wedge in the i direction
€
xi horizontal acceleration of soil wedge in the i
direction
a inclination angle of soil shear surface
Dd horizontal shear displacement in direct shear
box test
Dni incremental normal displacement in the
direc-tion of Ni
Dsi incremental sliding displacement in the direction
of Si
Dt time step
Dtc critical time step
b mass damping factor
d interface friction angle between soil wedge and
buffer
f soil friction angle
rb density of EPS geofoam buffer
sn normal stress
t shear stress
m interface friction coefficient
w soil stress factor
Fig 1 General arrangement for geofoam buffer wall.
Trang 3The forces at the boundaries are computed using linear
spring models (Fig 3) described later in the paper The
compression-only force developed at the boundary between
the soil wedge and geofoam buffer is computed using a
single linear compression-only spring (kN1) The linear
normal spring acting at the soil–soil wedge boundary (kN2)
permits tension and compression but was observed to
develop only compressive forces during computation
cycles
The shear springs at block boundaries are modelled as
stress-dependent linear-slip elements to permit plastic
sliding
The displacements and forces for the model are shown in
Fig 4 Forces Fiare the force components assumed to act
at the centre of gravity of the soil wedge The displacement
of the soil wedge can be defined by horizontal and vertical
displacements computed at the centre of gravity of the
mass In reality the soil wedge is constrained to displace
laterally The small volume of soil at toe of the wedge is
neglected to simplify the model
2.2 Numerical approach The solution scheme used in this investigation is based
on an explicit time-marching finite difference approach, which is commonly used for the solution of discrete element problems (e.g [11–13]) The approach has been modified in this study to consider the compressible geofoam-soil boundary condition and changes in geometry
of the soil wedge (block) At each time step, the numerical scheme involves the solution of the equations of motion for the block followed by calculation of the forces
2.2.1 Equations of motion
If xi is the displacement of the soil wedge in the i direction and Fi is the force acting on the block, then Newton’s second law can be written as
€
xi¼d _xi
dt ¼
Fi
where €xi, _xi and m are the acceleration, velocity and mass
of the wedge and t is time Using a central difference approximation with time step Dt
d _xi
dt ¼
_
xi
ð ÞtþDt=2ð Þx_i tDt=2
Substituting Eq (2) into (1) _
xi
ð ÞtþDt=2¼ð Þx_i tDt=2þFi
The updated displacements at the end of each time step can be calculated from
xi
ð ÞtþDt¼ð Þxi tþDt _ð Þxi tþDt (4) 2.2.2 Force equations
The normal and shear forces at the block boundaries are calculated from the following force–displacement laws:
Ni
ð ÞtþDt¼ðNiÞtDnikN, (5)
Fig 3 Springs used in the block model.
Fig 4 Block forces and deformations.
Fig 2 Forces acting on soil wedge.
Trang 4ð ÞtþDt¼ðSiÞtDsikSi, (6a)
Si
ð Þ ¼min mNð i; Sj ijÞsign Sð iÞ, (6b)
where Dni and Dsi are the incremental normal and shear
displacements in the direction of Niand Si, respectively; kN i
and kS i are the normal and shear spring stiffness values
with units of force/length For the soil–soil boundary,
kS 2¼ksoil Eq (6b) describes the implementation of a
limiting Coulomb strength for the shear surface at block
boundaries The interface friction coefficient is computed
as m ¼ tan f for the soil–soil interface where f is the soil
friction angle, and m ¼ tan d for the geofoam–soil interface
where d is the geofoam–soil interface friction angle The
incremental normal and shear displacements Dni and Dsi
are computed from horizontal and vertical components
of incremental displacements from the previous time step
(Eq (4))
For the compression-only forces described earlier, values
of Nithat become positive are set to zero at each time step
Updated horizontal and vertical total forces (Fig 4)
acting on the soil wedge are calculated as follows:
F1 ¼N1m €ugN2 sin a þ S2 cos a b m _u1
tþDt, (7)
F2 ¼ Sð 1mg þ N2 cos a þ S2 sin a b m _u2ÞtþDt (8)
Included in the formulation above is the provision for mass
damping where _ui is velocity and b is a mass damping
factor
2.3 Calculation of material parameters
2.3.1 Interface stiffness values
The values of the normal spring constants at the soil
wedge boundaries are calculated based on assumed values
for the modulus of elasticity of the materials and the depth
of the zone in the direction of normal force The stiffness of
spring kN 1 (soil–buffer interface) is calculated as
kN 1¼Eb
H W
where W ¼ 1 m is the unit width and Ebis the modulus of
elasticity of the buffer The corresponding normal spring
stiffness for the soil–soil interface is
kN2¼Es 2W
cos a sin a, (10)
where Esis the modulus of elasticity of the soil assumed as
a constant The normal stiffness value varies non-linearly
with orientation of the soil shear surface The minimum
value of the normal spring constant occurs at a ¼ 451
The stress-dependent shear stiffness of the soil is
computed here as
kS2¼wsnW ¼ wN2sin a
where w is a dimensionless stress factor, snis normal stress
and N is the force acting normal to the soil shear surface
This stress factor parameter is used to vary shear stiffness
of the soil in proportion to normal load (stress level)
2.4 Evolution of soil wedge geometry during excitation Pseudo-static analyses using closed-form solutions pre-dict that the critical orientation of the sliding surface (a) will become shallower as the magnitude of horizontal accelera-tion ( €ug) increases Angle a for a soil wedge with a single linear failure surface can be computed as follows[14]:
a ¼ f tan1 €ug
g
þtan1 D A
E
where
A ¼ tan f tan1 €ug
g
,
D ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A A þ Bð ÞðB C þ 1Þ
p
,
E ¼ 1 þ C A þ Bð Þ,
B ¼ 1=A,
C ¼ tan d þ tan1 €ug
g
Positive values of €ug correspond to outward acceleration
of the soil wedge and compression of the seismic buffer In this numerical approach, if the applied horizontal accelera-tion is less than the maximum previously computed value, the value of a remains unchanged (Fig 5)
2.5 Numerical implementation
A computer code written in Visual Basic was used to implement the calculations described above Numerical instability was prevented by selecting a time step Dt ¼ 0.1
Dtcwhere the critical time step Dtcis computed as
Dtc¼2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m=kmax
p
Fig 5 Evolution of soil failure angle with input horizontal accelerogram.
Trang 5Here, kmax is the maximum computed spring stiffness
during a computation cycle Since, the normal and shear
spring stiffness for the soil wedge will change during a
simulation run as a decreases, the value of Dt will also
change over the course of a simulation run
3 Numerical examples
3.1 Physical tests
A series of shaking table tests were carried out at the
Royal Military College of Canada [9,10] The physical
models were 1 m high (H) and 1.4 m wide (W) as shown in
Fig 6 The models comprised of a very stiff aluminium
bulkhead (wall) rigidly attached to the shaking table
platform (2.7 m 2.7 m in plan area) A compressible
EPS layer of thickness 0.15 m was attached to the wall
The wall and retained soil mass were contained in a rigid
strong box affixed to the table platform The soil extended
2 m from the back of the seismic buffer The length of the
soil volume behind the buffer was sufficient to prevent
intersection of any soil failure mechanism with the back
boundary of the rigid soil container The strong box was
excited in the horizontal direction only
An artificial sintered silica-free synthetic olivine sand was
used as the retained soil The soil properties are
summar-ized in Table 1 The same material has been used in
experimental work that investigated the response of
reduced-scale models of geosynthetic reinforced soil walls
under simulated earthquake loading[15,16] All tests in the
current investigation were performed with the same soil
volume and placement technique
Four of the physical tests reported by Zarnani et al.[9]
are considered here The properties of the geofoam buffer
materials were varied between tests and are summarized in
Table 2 The EPS panels for walls 2–4 were commercially
available products Wall 6 was constructed with a reduced
gross density by removing 50% of the EPS material by
coring The elasticized product is produced from solid EPS
block that is subjected to a cycle of compression load–unloading in order to increase the linear elastic range
of the material behaviour Non-elasticized EPS products are linear elastic up to about 1% strain Elasticized EPS materials have a linear elastic range up to 40% strain but have a lower elastic modulus[4]
The soil was placed in thin lifts and compacted by lightly shaking each lift using the shaking table Thereafter, the same target stepped-amplitude sinusoidal record with a frequency of 5 Hz was used as the horizontal base excitation history in all tests The model base excitation was increased at 5-second intervals to peak acceleration amplitude of 0.8 g, at which point the test was terminated
[10] A 5 Hz frequency (i.e 0.2 s period) at 1
6 model scale corresponds to 2 Hz (i.e 0.5 s period) at prototype scale according to the scaling laws proposed by Iai [17] Frequencies of 2–3 Hz are representative of typical predominant frequencies of medium to high frequency earthquakes Nevertheless, this simple base excitation record is more aggressive than an equivalent true earth-quake record with the same predominant frequency and amplitude
The total horizontal force transmitted to the rigid wall was measured by load cells attached to supports used to
Fig 6 General arrangement of shaking table test configuration and instrumentation (after Bathurst et al [10] ).
Table 1 Backfill soil properties
Soil–buffer interface friction angle 151
Soil elastic modulus, E s 15.2 MPa Soil shear stiffness, k soil (varies) (Eq (11)) Soil stress factor (dimensionless) 500–2000 Note: Soil shear strength parameters and shear stiffness determined from 0.1 0.1 m direct shear box tests [15,20]
Trang 6rigidly restrain the wall in the horizontal direction The
wall footing support was comprised of frictionless linear
bearings to decouple horizontal and vertical wall forces
Four potentiometer-type displacement transducers located
at different elevations above the base of the wall were
connected to metal plates embedded in the surface of the
geofoam layer to record lateral displacements
(compres-sion) of the geofoam Details of the experimental design,
test configurations and interpretation of results can be
found in the paper by Bathurst et al.[10]
3.2 Selection of model input parameters
The general model explained earlier was implemented
with d ¼ 151 to compute a in Eq (12) This value was
assumed based on recommendations by Xenaki and
Athanasopoulos[18]and Kramer[19] However, interface
friction was set to zero at the soil–wall boundary (i.e shear
force S1¼0 inFig 2) This assumption is reasonable since
interface friction could not be mobilized due to cyclic
normal force load–unloading at this interface during base
shaking
The friction angle of the soil (f) in the numerical model
was taken as the peak friction angle value The stiffness
factor w used to adjust the shear stiffness of the soil
(Eq (11)) was back-calculated from the results of
conven-tional direct shear tests on specimens 0.1 0.1 m in plan
area [15] under a range of normal stress sn (Fig 7) The
calculation was carried out as follows:
w ¼ t=sn
where stress ratio t=snwas computed over the initial linear
portion of the plots inFig 7(0.05–0.2% strain) Values of
w from 500 to 2000 were used in the numerical simulations
The lower value was used in tests 4 and 6 that were
constructed with the most compressible buffer materials It
is believed that this more compressible boundary condition
during model construction may have led to a less stiff soil placement The elastic modulus of the sand (Es) was taken from a value reported by El-Emam et al [20] to reproduce the direct shear box test data using a numerical FLAC code
Values for the dynamic elastic modulus of the geofoam buffer materials were calculated by plotting the measured dynamic compressive stress–strain response at each of the four displacement transducers arranged along the height of the wall models The stress was calculated as the dynamic horizontal force recorded by the horizontal load cells divided by the surface area of the geofoam inclusion The dynamic strain was calculated by dividing the dynamic displacement readings by the geofoam thickness (0.15 m)
An example of dynamic stress–strain response is shown in
Fig 8 The datum for stress and strain values is the end of construction The ranges of values using maximum and minimum measured dynamic deflections are summarized in
Table 2
EPS geofoam buffer properties
Geofoam
Walla Typeb Density, r b (kg/m3) Dynamic elastic modulus, E b (MN/m2)
a Numbering scheme from [9] ; Wall 1 was a control wall with no seismic buffer (rigid wall case).
b ASTM classification system [22]
c
50% of material removed by coring.
d
From correlation with ESP density reported by Negussey [21]
e
Manufacturer’s literature.
Fig 7 Approximation to stress-dependent shear stiffness and peak strength behaviour of direct shear box tests on shaking table sand (physical tests from El-Emam and Bathurst [15] ).
Trang 7Table 2 Comparison of these values with values from other
sources is problematic since there are a large number of
correlations between elastic modulus and EPS density in
the literature Furthermore, the stiffness of these materials
is sensitive to specimen size and rate of loading For
example, Bathurst et al [10] showed that published
correlations for 16 kg/m3 (Type I) and 12 kg/m3 (Type
XI) density materials gave a range (mean71 standard
deviation) of 5.171.9 and 3.371.5 MPa, respectively The
ranges quoted here capture the range of values used later in
the numerical simulations Nevertheless, Negussey [21]
showed that a 10-fold increase in specimen cube size
resulted in a doubling of the specimen elastic modulus The
values used in numerical simulations are also larger than
the values for the two unmodified EPS materials using a
linear correlation between elastic modulus and EPS density
reported by Negussey [21] Consistent with the trend for
non-elasticized EPS, is the observation that the elastic
modulus for the elasticized material reported by the
manufacturer is also less than the range of values
back-calculated from the physical tests The compressive strains
computed for the tests with unmodified EPS buffer
materials were less than 1%, which is within the elastic
limit of the materials Only at the end of the excitation
record for test 6, did the maximum compressive strain
approach 2%
Each of the numerical models was excited by the
horizontal accelerogram recorded by the accelerometer
mounted on the shaking table platform (Fig 6) The
accelerograms were adjusted using a conventional linear
baseline correction
During model initialization, mass damping (parameter b
in Eqs (7) and (8)) was set to a large number to bring the
model to static equilibrium within a reasonable number of
computation cycles During dynamic loading b was set to
0.05 However, the solutions were not very sensitive to
values of b in the range of 0–0.05 compared to the influence
of the magnitude of other input values Additional system
damping also occurs when the shear strength of the soil is reached (Eq (6b))
3.3 Comparison of physical and numerical results For clarity only the peak values from the input acceleration–time records for each numerical simulation are plotted inFigs 9a, 10a, 11a and 12a These data are taken from accelerations recorded at the shaking table platform in each corresponding physical test Computed compression–time and load–time responses of the four test cases are presented in the other plots in these figures The datum for the plots is the end of construction Hence, these values are the result of dynamic loading only The measured compressions are plotted as the average of peak values recorded from the four displacement transducers arranged along the height of the wall and the maximum
Fig 8 Measured average post-construction dynamic compressive stress–
strain response of EPS geofoam buffer (test 4).
Fig 9 Comparison of physical and numerical results (test 2): (a) input accelerogram; (b) peak buffer compression; (c) peak buffer compressive force.
Trang 8and minimum values The peak dynamic horizontal forces
from the physical tests were computed from the sum of
readings from the horizontal load cells mounted against the
back of the walls
As may be expected, the compression–time response
from the physical tests is generally greater with deceasing
buffer elastic modulus Comparison of theFigs 9c and 12c
shows that the measured peak horizontal force acting on
the wall was less for the most compressible buffer
(Eb¼0.4 MN/m2) compared to the stiffest material
(Eb¼4.1 MN/m2) This is consistent with the hypothesis
that peak dynamic loads can be attenuated by the use of a
vertical compressible inclusion (seismic buffer)
There is generally good agreement between the physical
and numerical models for all configurations up to peak
base input acceleration of about 0.7g At higher
accelera-tions there are likely more complex system responses that
cannot be captured by the simple displacement model employed For example, there are likely higher wall deformation modes at higher levels of base excitation The poor predictions at peak base excitation levels likely led to the overestimation of buffer compression and loads
at the end of the tests when the walls were returned to the static condition Nevertheless, the trends in the measured data for the four walls with respect to buffer force and compression are generally captured by the numerical model
up to about 0.7g, and in many instances there is good quantitative agreement
4 Conclusions This paper describes a simple displacement-type block model that can be used to predict the compression and dynamic force response of an idealized system comprised of
Fig 10 Comparison of physical and numerical results (test 3): (a) input
accelerogram; (b) peak buffer compression; (c) peak buffer compressive
force.
Fig 11 Comparison of physical and numerical results (test 4): (a) input accelerogram; (b) peak buffer compression; (c) peak buffer compressive force.
Trang 9a linear-elastic seismic buffer placed against a rigid wall.
Input parameters used to simulate the physical tests were
estimated from independent laboratory direct shear box
testing of the sand backfill and reasonable estimates of the
dynamic modulus of the EPS materials used to construct
the seismic buffers The model is shown to capture the
trends observed in reduced-scale model shaking table tests
carried out with similar boundary conditions up to peak
acceleration levels of 0.7g The model is simple and
provides a possible framework for the development of
advanced models that can accommodate more complex
constitutive laws for the component materials, other modes
of deformation, and a wider range of problem geometry
For example, possible non-uniform contact stress
distribu-tions and rotation of the soil–geofoam interface surface
may have to be introduced into the model if the general
approach is to be applied to field-scale structures
Acknowledgements The second author would like to acknowledge the financial support provided by the Iranian Ministry of Science, Research and Technology, for a Ph.D Visiting Fellowship held at the GeoEngineering Centre at Queen’s-RMC The work reported in this paper was also supported
by grants from the Natural Sciences and Engineering Research Council of Canada (NSERC) awarded to the first and fourth authors
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