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3D and 2D simulations of liquefaction-induced settlements of shallow foundations using Ta-Ger model

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A multiaxial constitutive model for sand (Ta-Ger model) based on a smooth-hys-teresis plasticity framework, implemented in the finite difference code FLAC and FLAC3D, is used for the 2D and 3D simulation of liquefaction-induced settlements of shallow foundations in centrifuge experiments. The same set of calibrated parameters were used in both 2D and 3D sim-ulations. It is shown that 3D analyses are in close agreement with the experimental results both in terms of foundation settlements and excess pore pressures, whereas the 2D analyses tend to overpredict the measured foundation settlements.
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1 INTRODUCTION
Recent advances in numerical modeling combined with closer interaction between research and
practice and the increased availability of computational power have led to the increasing popu-
larity of performance-based design approaches. While, considerable progress has been made in
the field of constitutive modeling of highly nonlinear phenomena, such as liquefaction (e.g.:
Prevost 1985, Pastor et al. 1990, Elgamal et al. 2002, Dafalias & Manzari, 2004, Park & Byrne
2004, Andrianopoulos et al. 2010, Boulanger & Ziotopoulou 2013) and the implementation of
these models in finite element and finite difference codes for use in practice, the availability of
such constitutive models in 3D remains limited.
In this study, the Ta-Ger constitutive model (Tasiopoulou & Gerolymos, 2016a,b), a multiaxial
constitutive model for sand based on a smooth-hysteresis plasticity framework is implemented in
the finite difference codes FLAC (Itasca, 2016) and FLAC3D (Itasca, 2012), calibrated generi-
cally, and used to simulate the liquefaction-induced behavior of shallow foundations observed in
centrifuge experiments performed by Dashti et al. (2010a,b). The numerical results from both 2D
and 3D analyses are compared against the measured responses.
2 TA-GER CONSTITUTIVE MODEL FOR SAND
2.1 Formulation
The Ta-Ger constitutive model for sand is described in Tasiopoulou & Gerolymos (2016a,b). The
model is based on an alternative plasticity framework combining perfect plasticity with Bouc-
Wen type smooth hysteresis. While the basic framework of the model, implemented in FLAC and
FLAC3D, is as described in Tasiopoulou & Gerolymos (2016a,b), three types of modifications
have been made in: i) the evolution of both the bounding and the phase transformation (dilatancy)
surface, ii) the evolution of the exponent n of the hardening parameter, ζ and iii) the incorporation
of fabric effects on cyclic mobility in the dilatancy stain ratio. The first two modifications aim to
improve the calibration process by providing more independent and less coupled model parame-
ters that target specific mechanisms, such as the number of cycles of load required to trigger
liquefaction, or the extent of post-liquefaction shear strain accumulation. The purpose of the third
modification is to improve model performance under irregular earthquake shaking where high
shear stress ratios can be followed by very low ones and vice versa, after the onset of liquefaction.
Herein, only the modifications discussed above are presented.
The dilatancy, d, has been adjusted to the following:
3D and 2D simulations of liquefaction-induced settlements of
shallow foundations using Ta-Ger model
P. Tasiopoulou, Y. Chaloulos
GR8 GEO, Athens, Greece
N. Gerolymos
National Technical University of Athens, Greece
A. Giannakou, J. Chacko
GR8 GEO, Athens, Greece
ABSTRACT: A multiaxial constitutive model for sand (Ta-Ger model) based on a smooth-hys-
teresis plasticity framework, implemented in the finite difference code FLAC and FLAC3D, is
used for the 2D and 3D simulation of liquefaction-induced settlements of shallow foundations in
centrifuge experiments. The same set of calibrated parameters were used in both 2D and 3D sim-
ulations. It is shown that 3D analyses are in close agreement with the experimental results both
in terms of foundation settlements and excess pore pressures, whereas the 2D analyses tend to
overpredict the measured foundation settlements.
,
2
3:
d d z pt
dRA A M




=−rn
(1)
where Mpt,θ is the phase transformation stress ratio dependent on Lode angle, θ, r is the stress ratio
tensor and n is the normalized stress ratio tensor. The densification factor Rd is part of the original
formulation by Tasiopoulou & Gerolymos (2016a), while the dilatancy factor, Ad and fabric fac-
tor, Az have been introduced to the definition of the dilatancy strain ratio, d. The fabric factor Az,
is calculated as a function of the deviatoric fabric-dilatancy tensor-valued variable z, (Eqs. 2-3)
as suggested by Dafalias & Manzari (2004):
1
z
A=+z:n
(2)
( )
max
p
zv
d c d z
− − +=z n z
(3)
The evolution of the exponent n, the bounding (Ms) and the phase transformation (Mpt) stress
ratios, as a function of the cumulative deviatoric strain increments, Σdεq, is obtained from Equa-
tions 4-6, respectively, slightly modifying the original equations suggested by Tasiopoulou &
Gerolymos (2016a), while retaining the basic concept. Both the exponent n, and the bounding
ratio, Ms start from an initial value, n0, and Ms0, respectively. As they evolve, they reach a peak
value, after which they gradually decrease, asymptotically reaching residual values, nf and Mcs
(critical stress ratio) respectively. The model parameters npeak and Msp are the maximum values
that can potentially be reached depending primarily on the model parameter c and secondarily the
parameters AR and ac. The phase transformation ratio, Mpt, starts from an initial value Mpt0 and
gradually reaches the critical stress ratio, Mcs.
( ) ( )
4
00.5 R
q
dq
peak peak f Ad
fcc
n n e n nn n e
+ −
=+
(4)
( ) ( )
0c
dq
sp sp cs
s
s cs dq
ac
c
M M e M MM M e
+ −
=+
(5)
(6)
2.2 Generic Calibration of model parameters
An effort was made to provide a generic calibration for a significant number of model parameters
based on (semi-)empirical relationships found in literature. The elastic shear modulus is given by
the following relationship:
2max
1000 m
kpG=
(7)
where p is the current mean effective stress in kPa and k2max = 0.13Dr0 +3.6 is a coefficient adapted
from Seed & Idriss (1970), dependent on initial relative density, Dr0 in %.
The exponent m is a stress-dependent dimensionless parameter defined as:
0
0
pp
mm




=
(8)
where m0 is the initial value of m, usually obtaining values within a range of 0.33-0.5 (Richart et.
al. 1970), and which is used for the calculation of the small strain shear modulus. Herein, a value
of m0 equal to 0.4 is considered. Equation 8 ensures that as the mean effective stress tends to zero
at liquefaction occurrence, the value of exponent m tends to unity, indicating a linear dependency
of the shear modulus on the confining stress (Ishihara 1996). The elastic bulk modulus, K, is equal
to G, assuming a Poisson ratio of ν = 0.15.
The model parameters related to the evolution of the bounding and phase transformation stress
ratios, as well as the dilatancy have been calibrated based on Bolton’s state parameter, the relative
dilatancy index, IR, (Bolton 1986):
( )
( )
0
0ln
Rr
ID Q p R= −
(9)
where Dr0 is the initial relative density of the sand, p0 is the initial mean effective stress, and Q,
R, are constants with values equal to 10 and 1, respectively. The initial value of the bounding
stress ratio, Ms0, is obtained as:
( )
( )
06sin (0.8 5 )
3 sin (0.8 5 )
b
b
cs R
scs R
I
MI


+
=−+
(10)
where ab is a factor accounting for principal stress rotation and intermediate stress effects (Tasi-
opoulou & Gerolymos 2016b). This factor is taken equal to 0.86 which is representative of simple
shear loading.
The model parameter, Msp, in Equation 5, is calculated through an iterative process at the ini-
tialization of the model so that the maximum value of the bounding stress ratio, Ms, is equal to
Mspeak, given by Equation 11. The bounding stress ratio is equal to Mspeak when the derivative of
Ms with respect to Σdεq, is equal to zero (at Σdεqpeak). Mspeak is obtained as:
()
()
6sin
3 sin
cs R
b
speak cs R
b
gI
MgI
+
=−+
(11)
where φcs is the critical friction angle, considered to be equal to 33 degrees, and factor, gab, ac-
counting for intermediate stress effects, has values in the range of 3 to 5 for triaxial and plain
strain conditions (Bolton 1986, Tasiopoulou & Gerolymos 2016b). Herein, this factor is consid-
ered equal to 5.
The estimation of the dilatancy factor, Ad, in Equation 1, is based on the dilatancy strain ratio
suggested by Bolton (1986) given by:
( )
()
,
3(0.3 )
3 0.3 qspeak
pt peak
R
dRd
I
AMMI
=+
(12)
where Mpt,Σdεqpeak is the value of the phase transformation stress ratio at Σdεqpeak, when Ms=Mspeak.
The initial value of the phase transformation stress ratio is obtained as:
00
3(0.3 )
3 (0.3 )
R
pt s R
I
MM I
=+
(13)
The evolution of exponent n (Eq. 4) determines the number of cycles to liquefaction (Tasio-
poulou & Gerolymos 2016a). Greater values of the parameter npeak lead to larger number of cycles
to liquefaction. To allow for calibrating to liquefaction triggering curves with different slopes,
this model parameter, npeak, is made dependent on a cyclic stress ratio variable, CSRR:
a
pR
peak f
n n n CSR=+
(14)
where, np and a are dimensionless parameters. The parameter, a, controls the slope of the lique-
faction triggering curve obtained by the model. The variable CSRR is defined as:
0
0
31
12 ref
p
RCSR p
kq
CSR
+
=
(15)
where CSRref is a reference cyclic stress ratio on the liquefaction resistance curve given by empir-
ical relationships or empirical data, e.g. the cyclic stress ratio at 15 cycles, CSR15, while qp is the
deviatoric stress at the last loading reversal, defined as:
( ) ( )
00
1:
2
p p p
q−−=s s s s
(16)
where sp is the deviatoric stress tensor at the last loading reversal and s0 is the initial deviatoric
stress tensor. The parameter AR, in Equation 4 is calculated as:
max R
CSR
R
A ru=
(17)
where rumax is the current maximum value of the excess pore pressure ratio, ru=1-(p/p0).
Finally, the model parameter c in Equations 4 to 6 is calculated as:
0.7
344 R
I
ce
=
(18)
3 SIMULATIONS OF LIQUEFACTION-INDUCED FOUNDATION SETTLEMENTS
3.1 Description of centrifuge experiments
Dashti et al. (2010a,b) performed a series of centrifuge experiments to study the liquefaction-
induced settlement mechanisms of buildings founded on shallow footings. The centrifuge models
included three types of structures with varying dimensions and static bearing pressures, placed on
a 1-m-thick rigid mat foundation made of aluminum. The centrifuge models were initially spun
at a nominal centrifuge acceleration of 55g and subsequently, they were subjected to a sequence
of three shaking events with increasing intensity (small, moderate and large). The sequence of the
input motions consists of scaled versions of the north-south, fault-normal Kobe Port Island re-
cording from the 1995 Kobe Earthquake. Shaking was applied to the base of the models parallel
to their long side.
This paper focuses on the centrifuge experiment T3-30, which includes a 3m-thick liquefiable
layer of Nevada sand with a nominal relative density, Dr, of 30%, underlain by a 21m-thick layer
of dense Nevada sand with a nominal relative density, Dr, of 90%. The liquefiable layer of loose
Nevada sand is overlain by a 2m-thick Monterey sand layer with Dr ≈ 85%. The water level is
located at a depth of 1.1m below the surface. All units used in this paper are in prototype scale.
3.2 Numerical modeling in FLAC 2D and 3D
Numerical simulations of the centrifuge experiment T3-30 were performed in FLAC and
FLAC3D using Ta-Ger soil constitutive model. The analyses focus on Structure A of the experi-
ment, which represents a 1-storey (5-m high) building designed with a static bearing pressure of
80 kPa and founded on a rigid mat foundation with width, W=6 m and length L=9 m. The footings
were simulated with 1-m thick elastic solid elements both in FLAC and FLAC3D, assigned the
elastic Young’s modulus of aluminum, i.e. 68 GPa. Interface elements were placed both at the
base and at the vertical side of each footing following an elastic-perfectly plastic constitutive law
with a maximum interface friction angle of 35 degrees. The structure was simulated in FLAC
using linear elastic beam elements described in terms of mass density, ρ; Young’s modulus, E;
cross-section area, A, and moment of inertia, I, while in FLAC3D the structure was modeled with
linear elastic shell elements described in terms of mass density, ρ; Young’s modulus, E; and thick-
ness, t. The properties of the beam elements in FLAC and the shell elements in FLAC3D, respec-
tively, were chosen so that the pressure transmitted to the soil is 80 kPa and the fixed-base natural
period of the structure is equal to 0.21 s. The geometric characteristics of the simulated structure
(e.g. height, width and length) were based on the data provided by Dashti et al. (2007) and Karimi
& Dashti (2016).
The soil mesh created in FLAC comprises 0.5m and 1m-thick elements for the top loose and
the bottom dense sand layers, respectively. The width of the zones was 0.6m close to the founda-
tion area and gradually increased towards the model boundaries. The same mesh discretization
was used in FLAC3D, while in the out-of-plane direction the length of the elements increased
with the distance from the center of the model from 1 m to 1.5 m. Half of the model was simulated
in FLAC3D taking advantage of the symmetry with respect to the direction parallel to the long
side of the model where the input motion is applied. The nodes of the lateral boundaries of the
models (perpendicular to the shaking direction) were tied to each other to simulate the kinematics
of the laminar box employed in the test.
As for the permeability, the values reported by Dashti & Bray (2013) were used i.e., 1.88x10-4
m/s for the loose Nevada sand layer, 5.63x10-5 m/s for the dense Nevada sand layer and 1.32x10-
3 m/s for the dense Monterey layer. These values are based on experimental measurements on
Nevada sand specimens of various densities (Arulmoli et al. 1992), properly scaled to account for
the effects of centrifugal acceleration and the viscosity of the fluid employed in the tests. The Ta-
Ger model, was used for the saturated sand layers. The top 1.0 m of dry dense Monterey sand
layer was modeled with the Itasca S3 hysteretic model (Itasca 2012) in combination with a Mohr
Coulomb failure criterion assuming a friction angle of 40 degrees. The input motion correspond-
ing to the large seismic event with a PGA equal to 0.55g was applied at the base of the numerical
models. The 3D finite difference mesh and a detail of the numerical model close to the area of the
structure are presented in Figure 1.
Figure 1. Numerical model in FLAC 3D (left) and details of the modeled structure (right).
3.3 Calibration against laboratory data
Calibration of model parameters was performed against available laboratory cyclic undrained di-
rect simple shear (DSS) tests on Nevada sand. Attention was given to calibration of model pa-
rameters to capture both liquefaction triggering and post-liquefaction shear strain accumulation
(Tasiopoulou et al. 2018). We note that the latter is expected to play a significant role in the
predicted settlements since the loose sand layer liquefies early on during shaking. Figure 2a il-
lustrates the liquefaction triggering curves obtained from single element tests for Dr = 30%, 40%
and 85%. The curves corresponding to Dr = 30% and 85% were used for the simulation of the
loose and dense sand layers, respectively. Figure 2b depicts the comparison between single ele-
ment simulations and undrained DSS tests performed on Nevada sand for Dr = 40%. The values
of the model parameters are presented in Table 1.
Figure 2. a. Comparison between liquefaction triggering curves (lines) obtained from single element simu-
lations and laboratory data obtained from cyclic undrained DSS tests on Nevada sand (symbols). b. Com-
parison between single element simulation results (left) and laboratory data obtained from undrained cyclic
DSS tests on Nevada sand with Dr = 40%.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
110 100
CSR
Number of cycles to liquefaction (3% shear strain)
Nevada Sand Dr = 40% (Arulmoli et al.,
1992)
Dr=40% (Doygun, 2009)
Dr=40% (Kano, 2008)
Dr=90% (Kammerer et al.,
2000)
Ta-Ger Dr=85-90%
Ta-Ger Dr=40%
Ta-Ger Dr=30%
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
110 100
CSR
Number of cycles to liquefaction (3% shear strain)
Nevada Sand Dr = 40% (Arulmoli et al.,
1992)
Dr=40% (Doygun, 2009)
Dr=40% (Kano, 2008)
Dr=90% (Kammerer et al.,
2000)
Ta-Ger Dr=85-90%
Ta-Ger Dr=40%
Ta-Ger Dr=30%
Table 1. Values of model parameters used in the numerical simulations.
Model Parameters
Loose sand Dr = 30%
Loose sand Dr = 40%
Dense sand Dr =85-90%
Dr0
0.3
0.4
0.85
n0
0.1
0.1
0.1
nf
0.07
0.08
0.07
np
0.8
0.8
0.3
CSRref
0.045
0.08
0.3
α
4
4
3
ac
0.006
0.01
0.012
cz
300
200
500
zmax
4
4
4
3.4 Comparison of numerical versus experimental results
The numerical results from 2D and 3D analyses are compared against experimental measurements
in terms of foundation settlements at the center of the footing in Figure 3a. As shown on this
figure the experimental measurements and 3D numerical results are in good agreement, indicating
settlements on the order of 48 cm and 44 cm at the end of shaking, respectively. By contrast, the
2D numerical analysis predicts significantly higher settlement of the footing on the order of 70
cm. The contours of the settlements at the end of shaking, depicted in Figure 3b, indicate that both
the upper loose sand layer and the deeper dense sand layer contribute to the overall foundation
settlement. This trend, which is also observed in the 2D numerical analysis, is attributed to the
liquefaction of both the top loose sand layer and a significant part of the dense bottom sand layer,
as explained subsequently from the comparison of recorded and predicted excess pore pressure
time histories.
Figure 3. a. Time histories of foundation settlements obtained from experimental measurements and both
2D and 3D numerical analyses. b. Contours of settlement at the end of shaking obtained from the 3D
numerical model.
The loose sand layer liquefies early in the earthquake in both 2D and 3D numerical analyses,
especially in the free field (sensor P14 mid height of the layer); a trend consistent with the
experimental observations (see Fig. 4a). Figure 4b shows the comparison between numerical and
experimental results in terms of excess pore pressure time histories within the loose sand layer
below the footing (sensor P16). The measured and computed free-field excess pore pressure time
histories at the middle of the dense sand layer (sensor P22 location) are plotted in Figure 4c indi-
cating liquefaction of even the dense sand layer at that location. This explains the significant
shear-strain-induced settlements (on the order of 20 cm) that develop within the zone of influence
of the foundation within the dense sand layer.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
110 100
CSR
Number of cycles to liquefaction (3% shear strain)
Nevada Sand Dr = 40% (Arulmoli et al.,
1992)
Dr=40% (Doygun, 2009)
Dr=40% (Kano, 2008)
Dr=90% (Kammerer et al.,
2000)
Ta-Ger Dr=85-90%
Ta-Ger Dr=40%
Ta-Ger Dr=30%
Figure 4. Excess pore pressure time histories obtained experimentally and numerically at three locations:
a. at the middle of the bottom dense sand layer in the free field (P14), b. below the footing within the loose
sand layer (P16) and c. at the middle of the dense sand in the free field (P22).
Overall, both numerical and experimental results are generally in good agreement in terms of
excess pore pressure time histories. Additionally, the trends of the foundation deformation are
similar and consistent with the ratcheting mechanism observed for shallow foundations on lique-
fiable soil (Dashti et al. 2010a). The difference in the predicted foundation settlements between
2D and 3D numerical analyses is most likely due to limitation of modeling the geometry of the
problem in 2D. In the 2D analyses, an infinite strip footing is modeled instead of a rectangular
one. Since, in the 3D analyses (but not in 2D), soil can deform in the out-of-plane direction, shear
resistance can develop along the side walls of the foundation. Thus, it appears logical that the
settlements from the 3D analyses are lower for a given set of soil properties.
4 DISCUSSION AND CONCLUSIONS
The Ta-Ger constitutive model was implemented in FLAC2D and FLAC3D, and generic calibra-
tions developed against commonly used parameters. The model and its implementation were
tested by simulating a centrifuge experiment examining liquefaction-induced settlements of a
shallow foundation resting on loose (i.e. Dr =30%) and dense sand (i.e. Dr =85%) layers and
subjected to strong shaking. The constitutive model parameters were calibrated using available
cyclic laboratory tests in order to capture both triggering and post-liquefaction shear strain accu-
mulation, the latter being particularly important for the problem at hand, since the loose sand layer
liquefies early in the earthquake. The same calibrated parameters were used in both 2D and 3D
simulations.
It was shown that the results of the 3D analyses are in close agreement with the experimental
results both in terms of foundation settlements and excess pore pressures. By contrast, the 2D
analyses tend to overpredict the measured foundation settlements even though the excess pore
pressures estimated numerically are in agreement with the centrifuge measurements. The differ-
ence in the predicted footing settlements between 2D and 3D numerical analyses can be attributed
to geometry effects and modeling limitations in 2D. Therefore, 2D and 3D simulations of this
problem are not equivalent.
The Ta-Ger model, as implemented in FLAC3D (a code commonly used in geotechnical prac-
tice), appears to be capable of capturing the important phenomena associated with complex soil-
structure interaction problems and liquefaction. This tool can be particularly useful for under-
standing and evaluating problems which cannot easily be simplified to a 2D geometry.
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0
0.1
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accumulation in sands. Journal of Geotechnical and Geoenviromental Engineering ASCE (under re-
view).
... The basic formulation of the model is described in Tasiopoulou and Gerolymos (2016a,b). Modifications, regarding: (i) the evolution of both the bounding and the phase transformation (dilatancy) surface, (ii) the evolution of the exponent n of the hardening parameter, f and (iii) the incorporation of fabric effects on cyclic mobility in the dilatancy strain ratio, have been recently made and described in Tasiopoulou et al. (2019). A brief summary of these modifications, that also apply in the current version of the model implemented in FLAC and FLAC3D and used in this study, is given below. ...
... where M pt is the phase transformation stress ratio dependent on Lode angle, h, r is the stress ratio tensor and n is the normalized stress ratio tensor. The densification factor R d is part of the original formulation by Tasiopoulou and Gerolymos (2016a), while the dilatancy factor, A d and fabric factor, A z have been recently introduced to the definition of the dilatancy strain ratio, d (Tasiopoulou et al., 2019). The fabric factor A z , is calculated as a function of the deviatoric fabric-dilatancy tensor-valued variable z (Eqs. ...
... Thus, calibration formulation of these model parameters is presented in the following, initially for the two extreme drainage conditions separately and last it is shown how these two types of calibration formulation can be bridged for partially drainage conditions. Details on the context of this calibration formulation are given by Tasiopoulou et al. (2019). ...
Article
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Bucket foundations have been increasingly used to support offshore wind turbines as alternatives to monopiles. Despite the substantial research effort on the bearing capacity and stiffness of such foundations in recent year, there is still a lack of knowledge regarding their cyclic response in saturated sand. In this paper, the multiaxial sand constitutive model Ta-Ger implemented in the finite deference code FLAC3D is employed in the analysis of the lateral response of bucket (skirted) foundations subjected to wind/wave loading. The model is reformulated to reproduce the cyclic response of sand for undrained, fully drained and partially drained conditions, using a unique set of calibration parameters. Having been calibrated against laboratory data available in literature, it is then used to predict the long-term cyclic lateral response of a bucket foundation in dry medium dense sand from a centrifuge experiment. After building confidence in the numerical approach the drainage effects are investigated. To gain qualitative insights into the effect of drainage conditions, the 3D numerical model, used to simulate the centrifuge test, was analyzed under saturated conditions and a range of soil permeabilities. It was shown that when flow is allowed the response up to a number-of-cycles threshold resembles that of fully drained conditions. Above this threshold, significant and abrupt increase of excess pore water pressures occurs causing liquefaction. Increasing the permeability delays the occurrence of liquefaction and the associated development of large deformations
... The basic formulation of the model is described in Tasiopoulou & Gerolymos (2016a,b). Modifications, regarding: i) the evolution of both the bounding and the phase transformation (dilatancy) surface, ii) the evolution of the exponent n of the hardening parameter, ζ and iii) the incorporation of fabric effects on cyclic mobility in the dilatancy strain ratio, have been recently made and described in Tasiopoulou et al. (2019). A brief summary of these modifications, that also apply in the current version of the model implemented in FLAC and FLAC3D and used in this study, is given below. ...
... where Mpt,θ is the phase transformation stress ratio dependent on Lode angle, θ, r is the stress ratio tensor and n is the normalized stress ratio tensor. The dilatancy factor, Ad and fabric factor, Az, incorporated to the dilatancy strain ratio are described in detail by Tasiopoulou et al. (2019). The fabric factor Az, defined by Dafalias & Manzari (2004), results to two model parameters, cz and zmax. ...
... The fabric factor Az, defined by Dafalias & Manzari (2004), results to two model parameters, cz and zmax. The dilatancy factor, Ad, is calculated based on empirical relationships by Bolton (1986) as a function of Bolton's state parameter, the relative dilatancy index, IR (Tasiopoulou et al., 2019). Last, the densification factor Rd, which is essential for loading under drained conditions, is part of the original formulation by Tasiopoulou & Gerolymos (2016a) and it is defined as: ...
Conference Paper
Bucket foundations have been increasingly used to support offshore wind turbines as alternatives to monopiles. Despite the substantial research effort on the bearing capacity and stiffness of such foundations in recent year, there is still a lack of knowledge regarding their cyclic response in saturated sand. In this paper, the multiaxial sand constitutive model Ta-Ger (Tasiopoulou and Gerolymos, 2016a) implemented in the finite deference code FLAC3D is employed in the analysis of the lateral response of bucket (skirted) foundations subjected to wind/wave loading. The model has been adjusted to reproduce the cyclic response of sand for undrained, fully drained and the intermediate conditions, using a unique set of calibration parameters. Numerical validation is conducted against centrifuge test including a bucket foundation in dry sand, by building a 3D numerical model. In order to gain qualitative insights into the effect of drainage conditions, the same problem used for validation was analyzed under saturated conditions and a range of soil permeabilities. It was shown that when flow is allowed the response up to a cycle threshold resembles that of fully drained conditions. Increasing the permeability delays the occurrence of liquefaction and the associated development of large deformations.
... Such models typically contain a large number of parameters, which have to be carefully calibrated against soil elements tests (e.g., Ref. [18]). Before extracting reliable results, validation against centrifuge model tests (or real scale shake table tests) is necessary (e.g., Ref. [19,20]). Based on such numerical analyses, Karamitros et al. ...
Article
This paper studies the effect of structure–soil–structure interaction (SSSI) on the seismic response of neighboring structures with shallow foundations on liquefiable sand. The problem is studied through coupled hydromechanical analyses. Nonlinear soil response is modeled with PM4Sand, calibrated on the basis of soil element tests of Hostun sand. The numerical methodology has been compared against six centrifuge model tests, showcasing its ability to predict the settlements. Three idealized structures of width B are considered, of different aspect ratio and foundation bearing pressure q, founded on two liquefiable layer depths, DL/B = 1 and 2. Initially, the response of a single building is studied, offering insights on the developing failure mechanisms. While the settlement increases with q in the case of a deep (DL/B = 2) layer, this is not the case for the shallow (DL/B = 1) layer, where the increased soil confinement leads to the development of a stiffer soil column, which offers increased support to the structure. Pairs of identical structures are subsequently analysed, revealing the effect of SSSI on settlement (w) and rotation (ϑ). While its effect on w is beneficial, its effect on ϑ is detrimental, leading to a dramatic increase compared to the single structure. The detrimental effect of SSSI on θ is shown to be a function of the gap (s/B) between the buildings and the depth of the liquefiable layer (DL/B). In the case of the shallow layer, the two structures rotate away from each other. This is not the case of the deeper layer, where they may either rotate away or towards each other, depending on s/B.
... Numerical modelling of foundation-structure systems during earthquake-induced liquefaction requires nonlinear deformation analyses (NDAs), employing advanced fully-coupled constitutive models that can capture the nonlinear stress-strain response of liquefiable soil (e.g., Refs. [1,2,4]. Such advanced constitutive models typically contain a large number of model parameters, requiring extensive soil element testing for proper calibration. ...
Article
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Despite recent advancements in predicting the response of shallow strip foundations during earthquake-induced liquefaction, significant modelling–related uncertainties remain, which are the focus of this paper. The problem is analysed through coupled hydromechanical analyses, employing an advanced constitutive model. The model is calibrated based only on the initial void ratio, and then validated against 6 centrifuge model tests, conducted at the University of Cambridge. Through a strict validation procedure, based on pore pressures, settlement and rotation time histories, as well as deformation mechanisms, the strengths and weaknesses of the numerical model are identified. It is shown that final settlement and rotation can be predicted with adequate accuracy, but more work is needed to achieve accurate predictions of settlement rate, maximum rotation, and pore pressures in the vicinity of the foundation. The numerical model is then used to investigate key modelling uncertainties. After revealing the sensitivity to initial soil density and to parasitic vertical acceleration, the effects of the centrifuge model container and of the distance of lateral model boundaries (L) are parametrically investigated. Boundary effects are minimized with a laminar container, where a normalized boundary distance L/DL≥1 is shown to be adequate for all liquefiable layer depths (DL) examined. The use of a rigid container is proven problematic, as it always imposes an unrealistic wave propagation pattern. The use of Duxseal inclusions offers a major advantage, allowing accurate reproduction of foundation settlement even with L/DL≥1, a key conclusion for the design of centrifuge tests.
... Before conducting parametric studies, careful validation against experimental results of boundary value problems is necessary (e.g. Bray & Macedo, 2017;Bullock et al., 2019;Tasiopoulou et al., 2019). ...
Conference Paper
The Ta-Ger constitutive model for sand implemented in the finite difference code FLAC 3D was used to simulate selected field tests of monopiles in Dunkirk sand conducted as part of the PISA program. The test piles simulated were medium diameter piles subjected to both monotonic and cyclic lateral and overturning loading. Comparison between numerical results and test measurements show that the simulations can reproduce the basic mechanisms of the monopile/soil system response, under both monotonic and cyclic loading. Features reproduced in the simulations include the increase of the system stiffness during cyclic loading and the associated decreasing rate of accumulation of lateral displacement and rotation. Lateral soil support in terms of p-y curves as well as distributed moment due to vertical shear stresses along the pile perimeter were obtained from the numerical analysis. It is shown that the numerical methodology can be used to gain insight into soil-OWT foundation interaction mechanisms and derive the soil reactions acting on the foundations as a result of lateral and overturning loads.
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The recently developed constitutive model for sand, designated as Ta-Ger sand (as described by the present authors in another recent paper), is reformulated to account for intrinsic and stress-induced anisotropy. The model combines features of bounding surface plasticity and Bouc–Wen type hysteresis, and incorporates Rowe’s dilatancy theory and the critical state concept. A physically motivated calibration methodology is developed with the aim to reduce the number of internal model parameters by expressing them as functions of four state variables: (a) Bolton’s relative density index, Ir, for measuring the ‘distance’ between the current stress state and the critical state; (b) cumulative absolute deviatoric strain increment for controlling the evolution in stress space of the bounding and phase transformation lines; (c) principal stress rotation angle; and (d) intermediate stress ratio accounting for stress-induced anisotropy. The calibration procedure is step-by-step validated against experimental data for various drained and undrained loading paths (including triaxial compression and extension, plane-strain compression and direct simple shear) and initial states (relative density and pressure). Three different types of sand are examined to account for diverse behaviour, even for the same initial states and applied stress paths, due to intrinsic anisotropy attributed to fabric effects (e.g. grain size,shape and packing).
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Numerical simulations of a centrifuge test, in which dissipation patterns, lateral spreading, and shear strain localization were measured and recorded, are performed for validation of the numerical modeling approach and insight on the deformation mechanisms. The constitutive model calibration process is performed based on available laboratory data and is described in detail. The model container and construction sequence are simulated to provide good approximations of the boundary conditions and initial stress distributions. Two approaches for simulating successive shaking events are presented and compared. The key observations and mechanisms from the centrifuge test, namely (1) the dynamic response and onset of liquefaction, (2) the amount and pattern of surface deformation, (3) the patterns of pore pressure dissipation and void ratio redistribution, and (4) the difference in response between a sand profile treated and not treated with liquefaction drains, are all reasonably captured and bounded by the simulations.
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Seismically induced settlement of buildings with shallow foundations on liquefiable soils has resulted in significant damage in recent earthquakes. Engineers still largely estimate seismic building settlement using procedures developed to calculate postliquefaction reconsolidation settlement in the free-field. A series of centrifuge experiments involving buildings situated atop a layered soil deposit have been performed to identify the mechanisms involved in liquefaction-induced building settlement. Previous studies of this problem have identified important factors including shaking intensity, the liquefiable soil's relative density and thickness, and the building's weight and width. Centrifuge test results indicate that building settlement is not proportional to the thickness of the liquefiable layer and that most of this settlement occurs during earthquake strong shaking. Building-induced shear deformations combined with localized volumetric strains during partially drained cyclic loading are the dominant mechanisms. The development of high excess pore pressures, localized drainage in response to the high transient hydraulic gradients, and earthquake-induced ratcheting of the buildings into the softened soil are important effects that should be captured in design procedures that estimate liquefaction-induced building settlement.
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A framework for the estimation of coseismic deformations in the postliquefaction regime is developed based on an extensive database of available cyclic undrained stress-controlled tests on clean sand samples without static shear bias, covering a wide range of relative densities. Based on fundamental experimental observations, a compliance rate is defined as the postliquefaction shear strain rate per cycle over the shear stress amplitude. Semiempirical relationships of the compliance rate as a function of relative density are developed to provide guidance for estimating postliquefaction shear strains. The proposed framework provides a basis for the calibration of advanced constitutive models capable of capturing postliquefaction strain accumulation. A calibration methodology is proposed using both existing liquefaction resistance curves and the newly developed semiempirical relationships for estimating postliquefaction shear strain accumulation. The validity of the proposed methodology is demonstrated by numerical simulations, using the PM4Sand model, of two well-documented centrifuge tests focusing on liquefaction-induced demands on engineering structures.
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The effective design of earthquake-resistant structures and liquefaction mitigation techniques requires an improved understanding of the development and consequences of liquefaction. In this paper, the results from centrifuge experiments of structures with shallow foundations on liquefiable sand were used to evaluate the predictive capabilities of a state-of-the-practice numerical tool. Fully-coupled numerical simulations with the UBCSAND model implemented in FLAC-2D captured building settlements measured in these experiments reasonably well for one scaled input motion, mostly within factors of 0.7 and 1.8. The soil model captured the overall contribution of deviatoric displacement mechanisms and localized volumetric strains during partially drained cyclic loading. The primary limitation of the model became evident for slower rates of earthquake energy buildup, when the extent of soil softening and building displacement was overestimated by up to a factor of 4. The observations from recent case histories, the results of the experiments, and the insights gained from the numerical analyses are combined to provide guidance on the evaluation of building response on liquefiable sand and the performance of liquefaction remediation strategies.