The response of a layered liquefiable soil profile, with granular columns as a mitigation strategy, was evaluated via numerical and centrifuge modeling. Comparisons were made for a level site containing a single granular column and for a pair of gentle slopes, one of which was mitigated with a network of dense granular columns. The results reveal the abilities and limitations of two state-of-the-art soil constitutive models. All simulations were performed in 2-dimensions using: 1) the pressure-dependent, multi-yield-surface, plasticity-based soil constitutive model (PDMY02); and 2) the bounding surface, plasticity-based, Manzari-Dafalias (M-D) soil constitutive model, both implemented in OpenSees. Numerical model parameters were previously calibrated via element testing. Both constitutive models under-predicted PGA near the surface at different distances from the granular column, but they better predicted spectral accelerations at periods exceeding 0.5 s (particularly M-D). The M-D model generally predicted seismic settlements well, while PDMY02 notably underestimated soil's volumetric compressibility and strains. Both models accurately predicted the peak value and generation of excess pore pressures during shaking for the unmitigated slope, leading to a successful prediction of lateral deformations. However, lateral movement of the treated slope was poorly predicted by both models due to inaccuracies in predicting the dissipation rate in the presence of drains. Both models came close to predicting the performance of gently sloping, liquefiable sites when untreated. But further advances are required to better predict the rate of excess pore pressure dissipation and seismic performance when the slope is treated with granular columns.

1. Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering
journal homepage: www.elsevier.com/locate/soildyn
Evaluating 2D numerical simulations of granular columns in level and gently
sloping liquefiable sites using centrifuge experiments
Ping Lia
, Shideh Dashtib,⁎
, Mahir Badanagkib
, Peter Kirkwoodb
a
Institute of Disaster Prevention, Sanhe, Hebei province 065201, China
b
University of Colorado Boulder, Dept. of Civil, Env. and Arch. Engineering, Boulder, CO 80309, United States
A R T I C L E I N F O
Keywords:
Soil liquefaction
Granular columns
Drains
Liquefaction mitigation
Centrifuge modeling
Numerical modeling
Lateral spreading
Slope performance
A B S T R A C T
The response of a layered liquefiable soil profile, with granular columns as a mitigation strategy, was evaluated
via numerical and centrifuge modeling. Comparisons were made for a level site containing a single granular
column and for a pair of gentle slopes, one of which was mitigated with a network of dense granular columns.
The results reveal the abilities and limitations of two state-of-the-art soil constitutive models. All simulations
were performed in 2-dimensions using: 1) the pressure-dependent, multi-yield-surface, plasticity-based soil
constitutive model (PDMY02); and 2) the bounding surface, plasticity-based, Manzari-Dafalias (M-D) soil con-
stitutive model, both implemented in OpenSees. Numerical model parameters were previously calibrated via
element testing. Both constitutive models under-predicted PGA near the surface at different distances from the
granular column, but they better predicted spectral accelerations at periods exceeding 0.5 s (particularly M-D).
The M-D model generally predicted seismic settlements well, while PDMY02 notably underestimated soil's vo-
lumetric compressibility and strains. Both models accurately predicted the peak value and generation of excess
pore pressures during shaking for the unmitigated slope, leading to a successful prediction of lateral deforma-
tions. However, lateral movement of the treated slope was poorly predicted by both models due to inaccuracies
in predicting the dissipation rate in the presence of drains. Both models came close to predicting the performance
of gently sloping, liquefiable sites when untreated. But further advances are required to better predict the rate of
excess pore pressure dissipation and seismic performance when the slope is treated with granular columns.
1. Introduction
Earthquake induced soil liquefacction can cause extensive damage
to buildings, structures, slopes, and retaining walls. Examples include
the 1964 Niigata (Japan), 1990 Dagupan City (Philippines), 1999 Chi-
Chi (Taiwan), 1999 Kocaeli (Turkey), and 2008 Wenchuan (China)
earthquakes among others. Remediation methods are often required to
limit liquefaction-induced soil strains to acceptable levels. Dense
granular columns reduce soil strains by enhancing drainage and in-
creasing (to different degrees) shear stiffness. In addition, some in-
stallation methods lead to significant densification of the surrounding
soils, which can help further reduce the potential for generating large
excess pore pressures and excessive deformations. Hausler and Sitar
[11] compiled over 90 case histories on the performance of improved
sites from 14 earthquakes in Japan, Taiwan, Turkey, and the United
States. The collected data indicated that drains made of stones, gravel,
or sand generally improved site performance in terms of observed de-
formations.
Beyond case history observations, full scale tests conducted by
Ashford et al. [2] have shown that installation of stone columns can
notably increase the relative density of the surrounding ground and
limit generation of excess pore pressures, while simultaneously pro-
viding shear reinforcement. Adalier et al. [1] conducted centrifuge tests
to assess the response of granular columns as a liquefaction counter-
measure in non-plastic silty soils. They showed that granular columns
can reduce net seismic settlements in silty deposits, particularly under
shallow foundations.
Numerical simulations may be used to evaluate the generation and
dissipation of excess pore pressures, accelerations, and deformations of
liquefiable, level or sloping sites when treated with dense granular
columns. Previous numerical studies of the response of treated sites
have used both two- and three-dimensional (2D and 3D) approaches.
The program FEQ-Drain [22], for example, models a unit cell under 3D
axisymmetric conditions. But it does not compute lateral soil de-
formations in slopes. Effective stress, coupled, 2D, dynamic simulations
were used by Seid-Karbasi and Byrne [29] to investigate the
https://doi.org/10.1016/j.soildyn.2018.03.023
Received 31 December 2017; Received in revised form 16 March 2018; Accepted 20 March 2018
⁎
Corresponding author.
E-mail addresses: chinaliping1981@126.com (P. Li), shideh.dashti@colorado.edu (S. Dashti), Mahir.Badanagki@Colorado.edu (M. Badanagki),
peter.kirkwood@colorado.edu (P. Kirkwood).
Soil Dynamics and Earthquake Engineering 110 (2018) 232–243
0267-7261/ © 2018 Elsevier Ltd. All rights reserved.
T

2. development of excess pore pressures and deformations in sloping sites
with a low-permeability barrier, and the effectiveness of drains under
such conditions. Elgamal et al. [8], on the other hand, performed a 3D
numerical parametric study to evaluate the effectiveness of liquefaction
mitigation through granular columns and pile-pinning approaches.
Results showed that such methods could be effective in reducing lateral
displacements by enhancing drainage and shear reinforcement. Raya-
majhi et al. [26,27] subsequently evaluated the influence of granular
columns through 3D, nonlinear, finite element simulations. The results
showed that dense granular columns may be effective in reducing lat-
eral spreading of gentle slopes, even if liquefaction triggering is not
prevented. Later, Howell et al. [14] analyzed lateral spreading in gently
sloping sites treated with prefabricated vertical drains (PVDs) using 2D,
fully-coupled finite element models, which could capture the 2D rota-
tional modes of deformation experienced by the slopes considered.
However, the capabilities and limitations of different soil constitutive
models in capturing the influence of granular columns on slopes have
not been sufficiently and systematically evaluated based on physical
model studies. This is a necessary step before these models can be used
in engineering design.
In summary, field case histories are insightful, yet limited in quality
(due to lack of instrumental recordings) and quantity for mitigated
conditions. Full-scale tests can demonstrate the complexities of soil
response under realistic conditions (e.g., pressure, heterogeneity, fines
content, mitigation construction techniques, etc.), but their cost and
logistics are often limiting. Centrifuge experiments can simulate rea-
listic stresses in a scaled model and subject it to realistic (albeit typi-
cally 1D horizontal) earthquake loads in a cost-effective manner.
Layering in soil can be simulated, and the results can provide critical
insights into the underlying mechanics and help validate numerical
simulations. However, realistic soil heterogeneities and complexities as
well as installation and construction processes may be difficult to re-
create in centrifuge. Therefore, for example, densification caused by the
installation of granular columns under increased gravity is often not
captured in centrifuge. Fully-coupled, effective stress numerical simu-
lations with nonlinear elasto-plastic soil constitutive models (if well
calibrated and validated) can provide insight into the effects of various
liquefaction remediation techniques on slopes and structures in terms of
the key engineering demand parameters of interest. However, valida-
tion through comparison with physically obtained measurements is
necessary, if results are to be relied upon.
In this paper, we evaluate the predictive capabilities and limitations
of two different state-of-the-art, nonlinear, elasto-plastic soil con-
stitutive models appropriate for modeling liquefaction and employed in
2D using the OpenSees finite element platform, based on their com-
parison with centrifuge experimental results. The simulations are per-
formed in 2D to provide guidance on the limitations and capabilities of
a practical numerical tool. The numerical results presented here are
Class-C predictions (i.e., [18]), in that they were performed after the
experiment, but the modeler did not have access to the centrifuge re-
sults other than the recorded base motions. The simulations were per-
formed using: 1) the pressure-dependent, multi-yield-surface, plasticity-
based soil constitutive model (PDMY02) developed and implemented in
OpenSees by Elgamal et al. [9] and Yang et al. [33]; and 2) the
bounding surface, plasticity-based soil constitutive model developed by
Dafalias and Manzari [6], here referred to as M-D, implemented in
OpenSees by Ghofrani and Arduino [10]. The soil model parameters
were previously calibrated using a series of monotonic and cyclic,
drained and undrained trixial tests as well as a free-field centrifuge test
involving the same soil types and conditions used in this study (detailed
by Ramirez et al. [24,25]).
The results of the numerical study are compared to results from two
centrifuge experiments performed by Badanagki et al. [4]. Each test
measured the response of a layered soil profile including a liquefiable
layer of clean sand overlain by a thin silt cap. The first test comprised a
single granular column at the center of a level site, to evaluate its
influence on acceleration, pore pressure, and settlement patterns at
different radial distances during 1D, horizontal earthquake loading. The
second test contained a network of granular columns on a gently
sloping site.
Appropriate adjustments were made to convert soil's hydraulic
conductivity from 3D (axisymmetric) flow conditions to 2D plane strain
in the simulations involving granular columns. The numerically com-
puted response was compared to experimental measurements in terms
of lateral and vertical displacements, net excess pore pressures during
and after shaking, and accelerations developed in gentle slopes with
and without granular columns. Overall, this study reveals the strengths
and weaknesses of two state of the art soil constitutive models and one
numerical platform in modeling the effectiveness of granular columns
as a liquefaction countermeasure in a level and gently sloping, layered
site. This understanding is essential for future planning of these models
in parametric studies, and the design of liquefaction mitigation using
granular columns that improve the site's overall performance.
2. Centrifuge experiments
Two centrifuge experiments were conducted at the University of
Colorado Boulder's (CU) 400 g-ton (5.5 m-radius) centrifuge facility to
investigate the influence of dense granular columns on site performance
when installed in level and gently sloping, layered, liquefiable ground
[4]. The first experiment (Test 1) simulated the response of a unit
granular column in a level site. The second test (Test 2) simulated
gently sloping liquefiable soils with and without granular columns.
Figs. 1 and 2 show the elevation and plan view geometry and in-
strumentation layout of the two tests. The models were spun to 70 g of
centrifugal acceleration and subject to a series of 1D horizontal earth-
quake motions in flight, in the same order. All the results presented in
this paper are in prototype scale, unless stated otherwise.
Ottawa and Monterey sand layers were prepared by air pluviation
using the automated pluviator at CU. From the bottom, the soil profile
Fig. 1. Schematic drawing (elevation and plan view) and instrumentation
layout of Test 1 with one drain.
P. Li et al. Soil Dynamics and Earthquake Engineering 110 (2018) 232–243
233

3. consisted of a dense layer of Ottawa sand F-65 (Gs = 2.65, Cu = 1.56,
emax = 0.81, emin = 0.53, k = 1.19e-04 m/s, Dr = 90%) with a
thickness of 8 m inside a flexible-shear-beam (FSB) container con-
structed of aluminum and rubber. Subsequently, the same sand with a
Dr = 40% and thickness of 8 m was dry pluviated as the liquefiable
material. A thin layer of Silica silt (Gs = 2.65, Cu = 7.3, emax = 1.35,
emin = 0.5, k = 3e-07 m/s) was subsequently pluviated and compacted
with a static pressure of 5 kPa to achieve a thickness of 0.5 m (detailed
by [4]). Monterey sand 0/30 (D50 = 0.04 mm, Cu = 1.3, emax=0.84,
emin = 0.54, k = 5.29e-04 m/s) was then pluviated at a Dr = 90% and
thickness of 1.5 m as the surface, non-liquefiable crust. After model
preparation and flushing with CO2, the specimen was saturated under
vacuum with a solution of hydroxypropyl methylcellulose prepared to a
viscosity 70 times that of water, thus satisfying the dynamic scaling
laws [32]. Arrays of accelerometers (Acc's), pore pressure transducers
(PPT's), and linear variable differential transformers (LVDT's) were
placed at three different radial distances from the single drain in Test 1
and at different depths to track wave propagation, net pore water
pressure generation, and volumetric deformations throughout the soil
profiles, as shown in Fig. 1.
In Test 2, two symmetric gentle slopes (3°) were constructed with
the head of the slopes at opposite ends of the model container, and the
slope toes separated by an open channel (shown in Fig. 2). One slope
was treated with a grid of 1.75 m-diameter dense granular columns,
separated by 3.5 m (center-to-center) with an area replacement ratio
(Ar) of 20%. Ar is defined as the cumulative area of columns normalized
by the total treatment area [3]. To avoid clogging between subsequent
motions applied experimentally, these columns were encased with geo-
textile filters. The granular columns in this study were made of rela-
tively uniform, clean, medium gravel (Cu = 1.54, emax= 0.92, emin =
0.62). The achieved dry unit weight of the granular columns was
17 kN/m3
and their hydraulic conductivity, measured during constant
head tests, was k = 2.9 cm/s. The facing slope in Test 2 was left un-
treated, in order to evaluate the effectiveness of granular columns as a
mitigation technique in terms of overall slope performance. The in-
strumentation layout, shown in Fig. 2, was designed to monitor accel-
erations, pore pressures, settlements, and lateral displacements on the
two sides of the slope with and without mitigation.
A series of motions was applied to the base of the container in flight
during the two tests. However, in this paper, we focus only on the first
428mm 428mm
[30m] [30m]
Treated side Untreated side
Test 2
[26.3m]
376mm
Shaking table
228mm
328mm
[16m]
[23m]
Shaking
150 mm Model
10.5 m [Prototype]
0 75
5.30
Acc. (A)
PPT (P)
LVDT (D)
21mm7mm
[1.5m][0.5m]
Ottawa sand
Dr=90%
Ottawa sand
Dr=40%
[8m]
115mm
[8m]
115mm
3°
60°
Fig. 2. Schematic drawing (elevation and plan view) and instrumentation
layout of Test 2 with gently sloping ground on the two sides of a channel with
and without granular columns.
Fig. 3. The acceleration and Arias Intensity (Ia) time histories as well as the acceleration response spectra (5%-damped) of the Kobe earthquake motion recorded at
the base of the container in centrifuge during Tests 1 and 2 and used as input to numerical simulations.
Fig. 4. Strength-corrected, normalized shear modulus reduction curves of
Ottawa sand (Dr of 40% and 90%) manually implemented in OpenSees as part
of their calibration when using the PDMY02 model.
P. Li et al. Soil Dynamics and Earthquake Engineering 110 (2018) 232–243
234

4. Table 1
Summary of PDMY02 model parameters for different soil layers.
Parameter Ottawa sanda
Silica siltb
Monterey sanda
Granular
columnsc
Unit Description
Dr 40 90 85 90 80 % Relative density
e 0.70 0.55 0.88 0.56 0.68 – Void ratio
ρ 1.94 2.03 1.86 2.01 1.97 ton/m3
Saturated unit weight
Gmax 108.5 130 87.6 87.6 137.4 MPa Octahedral low-strain shear modulus
γmax 0.1 0.1 0.1 0.1 0.1 – Maximum octahedral shear strain
Br (MPa) 283.0 339.0 233.8 264.0 201.5 MPa Bulk modulus
ψtxc 31.3 41.5 41.0 42.0 43.3 deg. Triaxial friction angle used by model
ψPT 27.5 28.0 36.0 32.0 36.5 deg. Phase transformation angle
c1 0.045 0.070 0.30 0.014 0.005 – Control the shear-induced volumetric change, contraction tendency based on the
dilation history, and overburden stress effect, respectivelyc2 1.5 4.0 5.0 2.0 0.50 –
c3 0.50 0.95 1.5 0.15 0.0 –
d1 0.03 0.010 0.02 0.36 0.40 – Reflect dilation tendency, stress history, and over burden stress, respectively
d2 3.0 3.0 3.0 3.0 3.0 –
d3 0.0 0.0 0.0 0.0050 0.0 –
NYS 44 99 20 20 20 – Number of yield surfaces generated by model
liq1 1.0 1.0 1.0 1.0 1.0 – Account for permanent shear strain (slip strain orcyclic mobility) in Sloping ground
liq2 0 0 0 0 0 –
Pr 101 101 101 101 101 kPa Reference effective confining pressure
k 1.41e-04 1.19e-04 3.00 e-07 5.29e-04 2.90e-02 m/s Hydraulic conductivity
a
Ramirez et al. [24,25].
b
Karimi and Dashti [16].
c
Rayamajhi et al. [26,27].
Table 2
Summary of Manzari-Dafalias model parameters for different soil layers.
Parameter Ottawa sanda
Silica silta
Monterey Sanda
Granular columnb
Description
Elasticity G0 100 100 130 135 Bulk modulus constant
ν 0.05 0.05 0.05 0.05 Poisson ratio
Critical state Mc 1.26 1.26 1.27 1.62 Critical state stress ratio
c 0.73 0.73 0.712 0.7 Ratio of critical state stress ratio in extension and compression
λc 0.0287 0.0287 0.02 0.018 Critical state line constant
e0 0.78 0.78 0.858 0.59 Critical void ratio at p = 0
ksi 0.70 0.70 0.69 0.86 Critical state line constant
Yield surface m 0.02 0.02 0.02 0.05 Yield surface constant (radius of yield surface in stress ratio space)
Plastic modulus h0 5 5 8.5 10 Constant parameter
ch 0.968 0.968 0.968 0.768 Constant parameter
nb
0.64 0.64 1.05 2.14 Bounding surface parameter
Dilatancy A0 0.45 0.45 0.6 0.8047 Dilatancy parameter
nd
0.5 0.5 2.5 2.98 Dilatancy surface parameter
Fabric-dilatancy tensor zmax 11 11 4 10 Fabric-dilatancy tensor parameter
cz 500 500 50 60 Fabric-dilatancy tensor parameter
a
Ramirez et al. [31,32].
b
Choi [7].
Fig. 5. a) The small-strain Vs profile of soil layers in Tests 1 and 2 based on an empirical procedure (Seed and Idriss [28]); b) maximum allowed and selected element
size to enable shear waves to propagate vertically through the soil column with frequencies as high as 10 Hz.
P. Li et al. Soil Dynamics and Earthquake Engineering 110 (2018) 232–243
235

5. major motion, prior to which soil geometry and properties were known
with greater accuracy. This first motion was a modified version of the
horizontal component of acceleration recorded during the 1995 Kobe
earthquake at the Takatori station, here referred to as the Kobe motion.
The acceleration and Arias Intensity time histories as well as the ac-
celeration response spectra (5%-damped) of this motion as recorded at
the container base during the two tests are shown and compared in
Fig. 3. The motion recorded on the container base during each test was
used as input to the numerical simulation of the corresponding test.
More details on soil and ground motion properties were provided by
Badanagki et al. [4], which are not repeated here for brevity.
3. Numerical simulations
Numerical simulations of the two centrifuge experiments (presented
above) were performed using the Open System for Earthquake
Engineering Simulation (OpenSees) finite element program [20] in
prototype scale. The nonlinear, elasto-plastic response of soil and
granular columns was modeled with two constitutive models: 1)
PDMY02 [9,33]; and 2) Manzari-Dafalias (M-D) [6]. The PDMY02
model, developed by Yang et al. [33,34], is based on the multi-yield
surface plasticity model initially introduced by Iwan [15] and Mroz
[21] and later implemented for soils by Prevost [23]. The M-D model
was initially proposed by Manzari and Dafalias [19], later modified to
account for fabric change effects [6], and was implemented in Open-
Sees by Ghofrani and Arduino [10]. The model uses a state parameter to
link the stress-strain-strength properties of soil to its evolving void ratio
and stress conditions as it approaches the critical state.
The model parameters were calibrated previously for Ottawa and
Monterey sand by Ramirez et al. [24,25], and for Silica silt by Karimi
and Dashti [16] based on the available monotonic and cyclic, drained
and undrained triaxial tests and cyclic, undrained simple shear tests.
Ramirez et al. [24,25] also provided recommendations on the calibra-
tion of both PDMY02 and M-D model parameters for Ottawa sand based
on Class-C and C1 simulations of a free-field centrifuge experiment,
which are implemented in this study. For example, the calibration of
PDMY02 parameters for Ottawa sand included manual implementation
of strength-corrected, normalized shear modulus reduction curves (G/
Gmax versus shear strain) shown in Fig. 4, which helped improve cali-
bration results with respect to element tests in small to medium ranges
of strain as well as predictions of site response in centrifuge [25]. The
properties of granular columns were obtained based on recommenda-
tions of Rayamajhi et al. [26,27] and Choi [5] as well as the strength
and permeability tests conducted by the authors. The parameters
adopted in this study for each of the soil layers and constitutive models
are summarized in Tables 1, 2, and details of calibration are not re-
peated here for brevity.
In all simulations, 2D QuadUP quadrilateral elements were used.
Three degrees of freedom at each node, two for displacement in dif-
ferent directions and one for fluid pressure, were expressed by these
elements. The element size was selected to allow for shear wave pro-
pagation in the frequency range of interest. Fig. 5a shows the small-
strain shear wave velocity (Vs) of the soil profile estimated empirically
based on Seed and Idriss [28]. A maximum frequency (fmax) of 10 Hz (in
prototype scale) was conservatively assumed for the vertically propa-
gating shear waves during dynamic loading, which was beyond the
capacity of the shaking table under increased gravity. The maximum
allowable element size at each depth was then estimated as: hallowable
= (minimum wavelength, λmin)/(4xN) = (Vs/fmax)/(4xN). The factor N
was obtained as 6 in a numerical sensitivity study, to account for soil
nonlinearity and strength loss due to excess pore pressure generation at
larger strains, reducing soil's effective Vs. The selected mesh size
(hselected) should always be smaller than the maximum allowable size at
different depths (hallowable), which was the case in this study as shown
in Fig. 5b.
An equal-degree-of-freedom boundary condition (through the
master-slave command in OpenSees) was employed to tie the left ele-
ments to the right. This condition was expected to roughly simulate the
boundaries in a flexible-shear-beam container in centrifuge. A small-
strain damping ratio of 3% was assigned at the first and third modal
frequencies of the far-field, level site (as recommended by [12,17]).
Material damping at larger strains was automatically provided by the
constitutive model, which could affect the results and predictions as
discussed in the next sections.
3.1. Modification of soil hydraulic conductivity for plane strain conditions
For simulating undrained loading conditions in saturated soils, the
hydraulic conductivity (k) may not be of great concern, provided it is
small enough to ensure that the rate of drainage is significantly slower
than the rate of generation or loading. The proper representation of k
becomes particularly important when evaluating the region mitigated
with granular columns, because they affect the flow of pore water to-
wards the columns during dynamic loading and hence, the slope's
seismic performance. Moreover, k values appropriate for a 3D flow
problem cannot be directly used in a 2D plane strain simulation. The 2D
representation of slopes with granular columns effectively models drain
walls that extend infinitely in the y-direction (horizontal direction or-
thogonal to shaking), increasing the drainage capacity significantly. In
order to equate the average dissipation rate and degree of consolidation
in an axisymmetric unit cell to a plane strain unit cell, Hird et al. [13]
introduced modifications to soil hydraulic conductivity based on drain's
radius of influence. The original and modified values of k for each of the
Table 3
Original and modified hydraulic conductivity values of different soil layers for
3D (axisymmetric) and 2D (plane strain) conditions when simulating Test 1.
Soil (Dr%) Original k, axisymmetric
(m/s)
Modified ka
, plane strain
(m/s)
Monterey sand (90%) 5.29e-04 6.00e-05
Silica sand 3.00 e-07 1.35e-07
Loose Ottawa sand (40%) 1.41e-04 3.03e-05
Dense Ottawa sand (90%) 1.19e-04 3.32e-05
a
Hird et al. [13].
Fig. 6. Numerically computed and experimentally measured excess pore pres-
sure time histories during the Kobe motion at three radial distances from the
single granular column in Test 1.
P. Li et al. Soil Dynamics and Earthquake Engineering 110 (2018) 232–243
236

6. soil layers used in centrifuge are summarized in Table 3 for the con-
ditions in Test 1, using the following equations:
=k k
μ
μ
pl ax
pl
ax (1)
⎜ ⎟= ⎛
⎝
+ − + − ⎞
⎠
μ
n
s
k
k
z l z
k
k r
In In(s)
3
4
(2 )ax
s w w
2
(2)
= + −μ z l z
k
BQ
2
3
2 (2 )pl
w (3)
• kpl is the soil hydraulic conductivity in a plane strain model.
• kax is the soil hydraulic conductivity in an axisymmetric model.
• z is the depth.
• l is the drain length.
• s = rs/rw, where rs is the radius of the smear zone, and rw is the
radius of the well.
• k is the horizontal hydraulic conductivity of soil, which was as-
sumed to be isotropic and equal to the original k measured for the
uniform, clean, and homogeneous soil layers used in centrifuge.
• ks is the horizontal hydraulic conductivity of the smear zone. In this
paper, we neglected the effects of the smear zone, making s = rs/rw
= 1, and therefore =In(s) 0
k
ks
.
• kw is the vertical or longitudinal hydraulic conductivity of the drain,
assumed to be equal to k of the granular column measured by the
authors.
• B is half the width of the plane strain unit cell (equal to R).
• Qw is the discharge capacity of drain.
• n = R/rw, where R is the radius of an axisymmetric unit cell, and rw
is the radius of the well.
3.2. Modeling of a level site with a unit granular column
To evaluate the modified hydraulic conductivity values, the cen-
trifuge experiment with a unit granular column was first simulated in
2D using the PDMY02 and M-D constitutive models and the modified k
values in Table 3. QuadUP quadrilateral elements were used to re-
present soil and granular columns, adding to 4995 nodes and 4824
elements. The acceleration time history recorded at the base of the
container in the centrifuge was applied directly to the base nodes in the
simulations, assuming a rigid base.
Fig. 6 compares the numerically simulated and experimentally
measured excess pore pressures at different depths and radial distances
from the unit cell. Similarly, Figs. 7 and 8 compare the numerical and
experimental results in terms of 5%-damped acceleration response
spectra and vertical displacements (or settlements), respectively. The
centrifuge recordings showed that a single drain could not reduce peak
excess pore pressures or prevent triggering of liquefaction (e.g., defined
as ru=Δu/σzo’=1.0), even at a short radius of 2.5 m. However, it in-
creased the rate of dissipation especially at greater depths. The ex-
perimental results showed a reduction in acceleration amplitudes at
periods between 0.3 and 3 s and an increase at greater periods as waves
traveled from the base of the container toward the surface, due to soil
softening and lengthening of site's fundamental period. Soil's dilation
tendencies at large excursions of shear strain also amplified the PGA in
some cases, particularly near the surface closer to the granular column.
Settlements recorded on the soil surface were greatest at locations away
from the drain (e.g., radius = 17.4 m) and reduced substantially as the
radius decreased to 8.5 and 2.5 m. The longer duration in which excess
pore pressures were kept at their peak at greater distances from the
drain (particularly at lower elevations) helped amplify volumetric
strains due primarily to sedimentation, despite the reduction in volu-
metric strains due to partial drainage. A slight increase in the rate of
dissipation at shorter distances to drains appeared to have a notable
influence on reducing net surface settlements.
In general, 2D elasto-plastic, fully-coupled OpenSees simulations
with either of the two constitutive models could successfully capture
the peak magnitude and rate of excess pore pressure generation at
different locations. The M-D model could better capture the rate of pore
pressure generation and dissipation compared to PDMY02, so it could
predict vertical displacements more reliably (as shown in Fig. 8). It,
however, underestimated the drainage rate closer to the soil surface
compared to the experiment. To improve the prediction of dissipation
rate by the M-D model, a variable hydraulic conductivity (k) may be
required over time that increases with ru, as suggested by Shahir et al.
Fig. 7. Numerically computed and experimentally measured acceleration re-
sponse spectra (5%-damped) during the Kobe motion at three radial distances
from the single granular column in Test 1.
Fig. 8. Numerically computed and experimentally measured vertical displace-
ments during the Kobe motion at three radial distances from the single granular
column in Test 1.
P. Li et al. Soil Dynamics and Earthquake Engineering 110 (2018) 232–243
237

7. [31]. In the presented simulations, the permeability was not varied
temporally for consistent comparisons between the two constitutive
models. The PDMY02 model, on the other hand, is known to over-es-
timate the rate of dissipation by greatly underestimating the coefficient
of volumetric compressibility and over-estimating rate of consolidation
[14]. The comparisons in pore pressures after strong shaking improved
for both models in the far-field away from the drain, where the dis-
sipation rate was expected to be slower and primarily vertical.
Sudden drops in excess pore pressures associated with soil dilation
(and the corresponding increase in PGA) were captured slightly better
by the PDMY02 model, because it was able to predict the recovery of
shear strength and stiffness in each cycle at larger shear strains [9]. The
acceleration response spectra were generally overestimated by the
PDMY02 model in periods ranging from 0.3 to 3 s and underestimated
in periods less than about 0.3 s. The M-D model predicted the accel-
erations at periods greater than about 0.3 s with reasonable accuracy,
while it largely underestimated accelerations at lower periods.
The increase in settlements away from the granular column was
captured by both constitutive models, although notably better by the M-
D model. The PDMY02 model underestimated volumetric strains at all
locations, due to its tendency to underestimate the coefficient of volu-
metric compressibility. Overall, the relatively successful simulation of
the generation and dissipation of excess pore pressures and the corre-
sponding effects on accelerations and settlements for a layered, lique-
fiable soil profile incorporating a single drain helped validate the
modified permeability of different layers in 2D prior to simulating the
slopes with a grid of granular columns.
3.3. Modeling of a gentle slope with granular columns
Centrifuge Test 2 included two gentle slopes, one with no mitigation
and one with a grid of granular columns with Ar = 20%. Numerically,
QuadUP quadrilateral elements were used to model soil, with 4845
total nodes and 3434 elements (see Fig. 9). The hydraulic conductivity
(k) values of all soil layers were modified on the treated side according
to Eqs. (1) through (3) for these 2D simulations, as discussed in the
previous section. Meanwhile, the untreated side was separately simu-
lated with the original k values of each soil layer, since no drain was
present. Table 4 presents the modified k values for the 2D simulation of
Test 2. Note that a different effective drain radius (e.g., 1.97 m in Test 2
compared to approximately 8.17 m in Test 1) led to a different mod-
ification of k values compared to what was shown in Table 3. The nu-
merical results were compared to those measured during the Kobe
motion in terms of accelerations and excess pore pressure ratios (ru) as
well as horizontal and vertical displacements on the slopes with and
without drains. The comparisons were made at locations where in-
strumental recordings were available.
Fig. 10 compares the experimentally measured and numerically
Fig. 9. Schematic of the finite-element model simulating the response of gentle slopes with and without granular columns (Test 2). Note: The treated side was
simulated with the modified k (Table 4), while the untreated side was simulated with the original k values of each soil layer.
Table 4
Original and modified hydraulic conductivity values of different soil layers for
3D (axisymmetric) and 2D (plane strain) conditions when simulating Test 2.
Soil (Dr%) Original k, axisymmetric
(m/s)
Modified ka
, plane strain
(m/s)
Monterey sand (90%) 5.29e-04 5.30e-05
Silica silt 3.00e-07 9.06e-08
Loose Ottawa sand (40%) 1.41e-04 8.79e-07
Dense Ottawa sand (90%) 1.19e-04 8.11e-07
a
Hird et al. [13].
Fig. 10. Numerically computed and experimentally measured excess pore
pressure ratio time histories at different depths on the two sides of slope with
and without drains in Test 2.
P. Li et al. Soil Dynamics and Earthquake Engineering 110 (2018) 232–243
238

8. computed excess pore pressure ratio (ru) time histories from the Kobe
motion on both treated and untreated slopes. The experimental results
showed that liquefaction (defined as ru = 1.0) was observed on both
sides of slope within the looser layer of Ottawa sand. They also showed
a notable increase in drainage rates throughout the slope treated with
granular columns (Ar = 20%), successfully limiting the extent and
duration of large pore pressures compared to the untreated side.
Both numerical models accurately captured peak ru values within
the liquefiable layer on both slopes. However, as also noted for Test 1,
and for the same reasons, the PDMY02 model overestimated the rate of
pore pressure dissipation compared to the experiment and the M-D
model at most locations. Overall, the PDMY02 model performed best
within the dense sand layers, and better for the untreated slope com-
pared to the mitigated slope.
The M-D model showed a reasonably good agreement with recorded
excess pore pressures both during and after shaking at most depths
particularly on the treated side. In the untreated region and at greater
depths, the M-D model underestimated the rate of drainage after strong
shaking, leading to notably larger excess pore pressures sustained for a
longer time after strong shaking compared with experimental record-
ings. This may have been due to the constant soil permeability assumed
numerically. In reality, soil permeability (particularly after
Fig. 11. Numerically computed and experimentally measured Arias Intensity (Ia) time histories and acceleration response spectra (5%-damped) at different depths on
the two sides of slope with and without drains in Test 2.
P. Li et al. Soil Dynamics and Earthquake Engineering 110 (2018) 232–243
239

9. liquefaction) does not remain constant [30]. Instead, it is expected to
increase as excess pore pressures increase. Similarly, due to excessive
disturbance, and possibly development of a water film beneath the thin
silt cap, the hydraulic conductivity (k) of silt also tends to increase
during strong shaking, as noted by Dashti et al. [7] and Karimi and
Dashti [16]. In these simulations, a constant k was assumed for both
sand and silt layers, which may explain why the rate of dissipation was
underestimated greatly by the M-D model on the untreated side at
lower elevations.
Fig. 11 compares the Arias Intensity (Ia) time histories and accel-
eration response spectra (5%-damped) obtained numerically using the
two soil constitutive models at different depths with data from Test 2.
The experimental results showed an increase in low period spectral
accelerations towards the surface on both slopes, but the increase was
more significant on the treated side. This was due to the faster rates of
pore pressure dissipation, which increased the shear stiffness of the soil
near the granular columns even during shaking. The numerical results
(with both constitutive models and both sides of slope) compared well
with experimental recordings in terms of accelerations within the dense
layer of Ottawa sand. The results diverged within the liquefiable layer.
The spectral accelerations predicted by PDMY02 were generally un-
derestimated at lower periods (between 0.1 and 0.3 s) and over-
estimated at greater periods (between about 0.3–1.5 s), particularly on
the treated side, due to the model's excessive dilative tendencies (also
evident in pore pressure comparisons). The M-D model provided
slightly better predictions of the high period spectral accelerations, but
like PDMY02 underestimated Sa at lower periods (and hence, under-
estimated PGA). The quality of predictions, particularly (PGA) im-
proved on the side without drains (similar to the patterns seen pre-
viously in Test 1). The notable increase in PGA experienced within the
soil next to granular columns (due to enhanced drainage, increasing soil
shear modulus and decreasing soil's damping over time) was not ac-
curately captured by either model, even though they had generally
captured the excess pore pressure response near drains during shaking.
The surface Arias Intensities on the treated side were also typically
over-estimated by the PDMY02 and underestimated by the M-D model.
Fig. 12 compares the vertical and horizontal displacement time
histories obtained numerically and experimentally. Vertical
displacements (or settlements) are reported and compared at the top of
the two slopes, while lateral displacements are reported at the toe,
where more reliable measurements were obtained during the experi-
ment [4]. Settlement of gentle slopes are dominated by volumetric
strains (arising from sedimentation, partial drainage, and consolida-
tion), with a minor contribution from shear strains driven by static and
dynamic shear stresses in the gentle slopes [4]. As noted previously, the
PDMY02 is known to largely underestimate the coefficient of volu-
metric compressibility [14], which led to a notable underestimation of
volumetric strains and net settlements (Dv) in both treated and un-
treated slopes. The M-D model, which more accurately simulated vo-
lumetric compressibiilty, better captured settlements on both treated
and untreated slopes. It, however, overestimated settlements on the
untreated side, due to the overprediction of excess pore pressures at
lower elevations over an extended time after strong shaking.
Horizontal deformations were better captured on the untreated side
by both models compared to the mitigated side, which was consistent
with the pore pressure and acceleration predictions. On the treated side,
the horizontal LVDT core disconnected from its holder at about 15 s.
Hence, the actual lateral displacement of the treated slope was likely
slightly larger than what is shown in Fig. 12, continuing through the
end of shaking (about 25 s). Nevertheless, both PDMY02 and M-D
models initially predicted lateral displacements well, because they
captured the rate of pore pressure generation and softening in soil re-
latively accurately. However, the post peak excess pore pressures were
largely under-predicted by PDMY02 due to an excessive rate of dis-
sipation. Therefore, the subsequent lateral displacements resulting from
softening were similarly underestimated. The M-D model better cap-
tured but slightly overestimated lateral deformations compared to the
experiment on the treated side (noting the measurement error beyond
15 s).
4. Summary of numerical and experimental comparisons
The accuracy of numerical predictions was assessed in terms of re-
siduals for different response parameters of interest:
⎜ ⎟= ⎛
⎝
⎞
⎠
X
Residual(X) log
X
numerical
experimental (5)
Where X refers to a given quantity obtained numerically or experi-
mentally. The range of residuals in Arias Intensity (Ia) time histories,
acceleration response spectra (Sa), excess pore water pressure ratios,
and horizontal and vertical displacements at different depths are shown
in Fig. 13. A positive residual indicates an over-prediction of the ex-
perimental response, and vice versa.
The acceleration response spectra (5% damped), in general, were
predicted relatively well on the untreated slope by the two soil models
for periods ranging from 0.5 to 4 s, with mean residuals within ap-
proximately +/−0.2. The quality of predictions worsened slightly on
the treated side (mean residuals within about +/−0.3). Negative re-
siduals (meaning under-prediction of acceleration response spectra)
were evident for both models and both sides at shorter periods (0–0.5 s)
and higher elevations, leading to poor predictions of PGA by both
models. The M-D model's tendency to slightly overestimate excess pore
pressures during shaking on the untreated side led to an overestimation
of softening and damping in soil and hence, underestimation of spectral
accelerations and Arias Intensities. The PDMY02 model's tendency to
overestimate the dissipation rate, on the other hand, led to a slight
overestimation of soil stiffness and underestimation of damping. Hence,
it overestimated accelerations in periods between 0.5 and 2 s and also
Arias Intensities.
Both models tended to overestimate the rate of dissipation after
strong shaking when granular columns were present. Therefore, ru re-
siduals were particularly large (average residuals of up to −2, in-
dicating a notable underestimation) on the treated side after about 20 s
Fig. 12. Numerically computed and experimentally measured vertical and
horizontal displacement time histories along the two sides of slope with and
without drains in Test 2.
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240

10. Fig. 13. Range of residuals in the prediction of Arias Intensity (Ia) time histories, acceleration response spectra (5%-damped), excess pore pressure ratios, and
horizontal and vertical slope displacements using two soil constitutive models in gently sloping sites with granular columns of Ar = 0% and 20%.
P. Li et al. Soil Dynamics and Earthquake Engineering 110 (2018) 232–243
241

11. for PDMY02 and 40 s for M-D models.
The M-D model predicted vertical displacements (settlements) well,
with residuals tending to 0 and 0.1 on the treated and untreated sides,
respectively. Residuals of vertical displacement for the PDMY02 model
tended to about −0.9 and −0.4 on the treated and untreated sides,
respectively, due to this model's under-prediction of volumetric com-
pressibility (discussed previously). Lateral slope displacements were
predicted well by both M-D and PDMY02 models for the untreated side
(with residuals tending to 0 after shaking). On the treated side, lateral
displacements were over-estimated and under-estimated by the M-D
and PDMY02 models, respectively (residuals of less than +0.4 for M-D
and exceeding −0.4 for PDMY02). Note, however, that measurements
of lateral slope displacement were not reliable in the centrifuge ex-
periment after about 15 s. In general, both models had difficulty cap-
turing the slope's performance on the treated side across most response
parameters of interest.
5. Concluding remarks
Two centrifuge experiments were performed to evaluate the influ-
ence of granular columns on the performance of level and gently
sloping liquefiable sites. The predictive capabilities and limitations of
two state-of-the-art soil constitutive models (one multi-yield surface
and one critical-state based) and a 2D finite element computational
platform (OpenSees) were subsequently evaluated by comparing with
results from centrifuge experiments.
Soil constitutive model parameters were determined based on ele-
ment scale laboratory testing. The numerical results presented in this
study were Class-C predictions, in that they were performed after the
experiment without having seen the experimental measurements. The
simulations were performed in 2D, requiring conversion of soil hy-
draulic conductivities from 3D axisymmetric conditions around drains
to 2D plane strain. Hydraulic conductivity (k) of different soil layers
was not varied temporally, to enable direct comparison of the two
constitutive models.
Both constitutive models predicted spectral accelerations with rea-
sonable accuracy for periods between 0.5 and 4 s, particularly on the
untreated slope. But importantly, neither model could capture the no-
table increase in PGA due to mitigation with granular columns. This
may result in under-prediction of seismic demand for slopes mitigated
with granular columns (e.g., in terms of PGA or Cyclic Stress Ratio, CSR,
used in liquefaction triggering analyses of a treated site or in design of
overlying structures).
The M-D model generally predicted settlements well, with residuals
tending to about 0–0.1 on both treated and untreated sides. The
PDMY02 model, on the other hand, was observed to notably under-
estimate volumetric strains. This was due to its inherent under-
estimation of coefficient of volumetric compressibility (which also
amplified the rate of consolidation and excess pore pressure dissipation
during and after shaking). For both models, the ability to accurately
predict soil strains depended upon good prediction of excess pore
pressures and the extent of softening at different times. Pre-peak excess
pore pressures were generally well predicted by both models. However,
PDMY02 especially poorly predicted the rate of dissipation after strong
shaking. This affected the overall quality of displacement predictions in
a gentle slope.
For the untreated slope, lateral deformations were predicted rea-
sonably well (residuals tending to about 0 after strong shaking for both
models). However, lateral slope deformations on the treated side were
over-estimated by the M-D model (residuals less than about +0.4) and
underestimated by the PDMY02 model (residuals exceeding −0.4).
This was attributed to errors in the prediction of excess pore pressures
within the liquefiable layer in the presence of granular columns.
This study shows that both M-D and PDMY02 constitutive models can
predict many of the key response parameters in liquefiable gentle slopes,
particularly when no mitigation measures are employed. However,
improvements are required to correctly predict the dissipation rate of excess
pore pressures. This, in turn, will improve predictions of surface settlement,
lateral displacement, and near surface seismic demand parameters such as
PGA and CSR. Improvements in predicting the excess pore pressure re-
sponse will also result in a more accurate prediction of the seismic perfor-
mance of slopes when reinforced with granular columns.
Acknowledgements
This work was partly funded by the National Science Foundation for
Young Scientists of China (No. 51508096). The authors would also like
to thank Dr. Zana Karimi and Ms. Jenny Ramirez Calderon for their
contributions to the numerical simulations presented in this paper. The
centrifuge experiments presented were conducted at the University of
Colorado Boulder's Center for Infrastructure, Energy, and Spacing
Testing (CIEST).
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