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Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
1
Possibilities and limitations of the Prevost model for the modelling of cohesionless
soil cyclic behaviour.
Possibilités et limitations du modèle de Prevost pour la modélisation du comportement cyclique des
sols sans cohésion.
B. Cerfontaine
a,b
, R. Charlier
a
& F. Collin
a
a
University of Liège, Department of Architecture, Geology and Constructions, Geotechnical Engineering Division, Chemin
des chevreuils, 1, B52/3, 4000 Liège, Belgium
b
FRIA, F.R.S.-FNRS, National Fund for Scientific Research, 1000, Bruxelles, Belgium
ABSTRACT: The Prevost’s model is currently used to model cyclic behaviour of soils especially in earthquake engineering. The
original model is able to capture the main features of cyclic behaviour: pore pressure build up and plastic deformation accumulation.
But accurate modelling of laboratory tests requires improvements. Enhanced models exist but require a lot of parameters that make
them cumbersome for practical purpose. A suction caisson, part of a tripod offshore foundation for wind turbines is modelled.
Possibilities of the Prevost’s model are highlighted compared with a classical Drucker-prager model.
RÉSUMÉ : Le modèle de Prévost est couramment utilisé pour modéliser le comportement cyclique des sols, notamment dans
l’ingénierie sismique. Le modèle original permet la représentation des caractéristiques principales de ce genre de comportement : une
accumulation des pressions d’eau et déformations plastiques. Cependant, la représentation précise d’essais de laboratoire nécessite des
modifications du modèle. Ces améliorations existent mais au prix d’un grand nombre de paramètres additionnels, ce qui rend malaisée
son utilisation en pratique. Un caisson à succion, partie d’une fondation tripode d’éolienne offshore a été modélisé. Le sol est
représenté alternativement par un modèle classique de Drucker-Prager puis par le modèle de Prevost afin de souligner les apports de
celui-ci.
KEYWORDS: soil mechanics ; cyclic behaviour ; foundations ; constitutive behaviour
1 INTRODUCTION
Modelling the cyclic behaviour of soils is a crucial issue for
earthquake engineering as well as for designing offshore wind
turbines. This topic of interest is still an ongoing domain
(Houlsby, et al., 2005). Despite its drawbacks, the Prevost’s
model, based on nested surfaces and non-associated plasticity is
able to capture hysteretic behaviour of soils under cyclic
loading.
1.1 Definitions
The sign convention of soil mechanics is applied, compressive
stresses and strains are positive. The Macauley brackets
are
defined according to
(1)
‘:’ indicates a dot product between two tensors (in bold
characters) of the same order: for example
in
index notation. If is the effective (Cauchy) stress tensor and
is the mean effective stress, then the deviatoric
stress tensor () is defined through
(2)
where is the identity tensor.
1.2 Constitutive equations
The Prevost’s model lies within the framework of elasto-
plasticity. Constitutive equations are written in incremental
form. The equation below links the effective stress rate to the
elastic deformation rate
(3)
where is the isotropic fourth-order tensor of elastic
coefficients. The plastic rate of deformation
is defined
through
(4)
where is a symmetric second order tensor, defining a plastic
potential. The plastic loading function is a scalar that depicts the
amount of plastic deformation and is defined in the following
(5)
where is a second-order tensor defining the unit outer normal
to the yield surface and the plastic modulus associated to this
surface. This normal tensor can be decomposed into its
deviatoric and dilationnal parts as
(6)
1.3 Yield functions
The model is made of conical nested surfaces in principal stress
space (Prevost, 1985). Their apex are fixed at the origin of axes
but could be translated on the hydrostatic axis to take a small
cohesion into account (for numerical purpose). The i-th surface
is defined through
(7)
where
is a kinematic deviatoric stress tensor defining the
coordinates of the yield surface centre in deviatoric space and
is a material parameter denoting the aperture of the cone.
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
2
1.4 Plastic flow rule
The plastic potential
is decomposed into its
deviatoric part which is associative,
(8)
and its dilationnal part which is non-associative
and
(9)
where is a material parameter that takes into account the
phase transformation line (Ishihara, et al., 1975). This parameter
rules the dilationnal behaviour and separates the p’-q plane into
two zones. Stress ratios () lower than imply a plastic
contractive behaviour under shear loading while the other zone
depicts a dilative plastic behaviour.
1.5 Hardening rule
The hardening rule is purely kinematic. During loading, active
surface moves up to contact the next one. All surfaces inside the
active one stay tangential at the current stress state. The
relationship between plastic function and kinematic hardening is
determined through the consistency condition (Prevost, 1985)
and leads to
(10)
where μ is a tensor defining the direction of translation of the
active surface in the deviatoric space. At this step, any direction
of translation could be used. The only requirement is that the
outermost activated surface has to be at most tangential to the
next one, at the end of a given step. The overlapping of surfaces
is then avoided. In the original paper, the Mroz rule was chosen.
This choice enforces an explicit integration of the law.
(11)
1.6 Refinements
Dependency on stiffness (bulk, shear and plastic moduli) for the
mean effective stress is taken into account through (Prevost,
1985)
(12)
where
is a reference pressure and
is the corresponding
stiffness at
. Typically n equals 0.5 for sands (Prevost, 1985) .
Other shapes of surfaces can be considered. The Lode-angle
dependency is taken into account through the use of Van
Eekelen surfaces (Yang, et al., 2008; Zerfa, et al., 2003).
2 COMPARISON WITH LABORATORY TESTS
2.1 Calibration
A series of tests on Nevada sand, available in the scope of
VELACS project (Arulmoli, et al., 1992), was used as
experimental background. It provides a large number of triaxial
monotonic and cyclic tests at several densities.
Elastic parameters for modelling are available in (Popescu,
et al., 1993). Monotonic triaxial tests in both compression and
extension are used to calibrate the basic plastic parameters of
the model (Zerfa, et al., 2003). For a triaxial test, Equation (7)
describing the i-th surface depends only on 2 scalars parameters
(
et
) and is transformed into
(13)
Fig. 1. Comparison between experimental p’-constant drained triaxial
tests (Arulmoli, et al., 1992) and numerical modelling for Nevada Sand
(Dr=40%) : deviatoric stress vs. vertical deformation. Tests are
available for 3 different initial mean effective pressures (40-80-160 kPa)
in compression and extension.
Parameters related to the surfaces are obtained from drained
tests:
1. An experimental q-ϵ
y
curve (deviatoric stress vs. vertical
deformation) in compression is delimited into linear
segments along which plastic moduli are constant. The
number of segments corresponds to the number of
surfaces. Transitions from a surface to another give initial
upper bounds of surfaces.
2. The procedure is repeated for an experimental extension
curve at the same initial mean effective pressure but
plastic modulus are already known. The lower bound of
each surface is then found. Aperture (M
i
) and initial
position (α
i
) of the centre of each surface are then
computed.
Figure 2. Comparison between experimental drained triaxial tests
(Arulmoli, et al., 1992) and numerical modelling for Nevada Sand
(Dr=40%) volumetric vs. vertical deformation. Tests are available for 3
different initial mean effective pressures (40-80-160 kPa).
The p’=80kPa curve was adopted as a reference curve for
calibration. The model is then applied to other initial mean
pressures. Results in the q-ϵ
y
plane are given in Fig. 1.
Numerical and experimental curves fit relatively well. However,
regarding the volumetric deformation, the numerical curves
don’t match the tendency of experimental ones. The only
parameter that rules the plastic potential is the cause of those
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
3
discrepancies. The phase transformation line is accurately
defined in the p’-q plane but the amount of contractancy or
dilatancy is not adjustable.
Fitting of the monotonic curves can be considerably
improved considering a modified plastic potential (in Figure 3).
The potential is split into a near field (small ) in which the
potential is the original one and a far-field (large ) in which the
potential is modified. Only two new parameters are necessary.
(14)
Figure 3. Comparison between experimental drained triaxial tests
(Arulmoli, et al., 1992) and numerical modelling for Nevada Sand
(Dr=40%) volumetric vs. vertical deformation. Tests are available for 3
different initial mean effective pressures (40-80-160 kPa).
Insufficiency of this adaptation is obvious in Figure 4. A
cyclic behaviour is modelled, based on parameters obtained
above. Stress path in p’-q plane is quite different but moreover,
the q-ϵ
y
curve doesn’t match and isn’t represented here. An
accurate modelling of cycles in both q-p’ and q-ϵ
y
planes
definitely requires a more complex expression of the plastic
potential such as in (Elgamal, et al., 2003). The immediate
consequence is an increase in the number of state parameters
necessary to describe the model, though the number of
laboratory tests increases.
Figure 4. Comparison between experimental drained triaxial tests
(Arulmoli, et al., 1992) and numerical modelling (with modified plastic
potential) for Nevada Sand (Dr=40%) deviatoric vs. mean effective
stress.
Despite its drawbacks, the Prevost’s model can qualitatively
capture the main features of the cyclic behaviour of a soil: pore
pressure build up (Zerfa, et al., 2003) and plastic deformation
accumulation. Hence it’s better than classical isotropic
hardening models to represent cyclic behaviour of soils.
3 PRACTICAL EXAMPLE: SUCTION CAISSON
The advantages of the model are shown through an application,
based on (Vertseele, 2012). The case study is a suction caisson
part of a tripod foundation for wind turbine firstly loaded by the
dead weight and then submitted to a cyclic loading. The latter
consists of two phases of loading with different amplitudes
(period=10s). The second amplitude is twice the first one.
Results obtained from the Prevost’s model (PR) are compared
to the classical Drucker-Prager model (DP). The PR model is
used in its basic form since the proposed modification of the
plastic potential doesn’t improve the fitting of the cyclic
behaviour. Furthermore, the p’-dependency of the stiffness is
not taken into account ( because the DP model
implemented in the code is not able to represent it.
3.1 Geometry
In order to simplify comparison, the mesh is supposed
axisymmetric and the horizontal load is neglected. The loading
is only a compression/decompression vertical force. The caisson
has a diameter of 8m and a skirt depth of 4m. The total mesh
size is 24mx22m to avoid problems with boundaries. There are
1892 finite coupled elements (from FE code LAGAMINE).
Parameters are those obtained for a Nevada Sand isotropically
consolidated at a relative density of 60% (Table 1).
Table 1. Parameters characterizing the models. Elastic parameters (E,ν)
and permeability are common to both models.
and
are
respectively the initial and final friction angle of the DP model whilst
is the dilatancy angle.
Com.
Elastic
E (kPa)
2,7.10
5
ν (/)
0.25
k (m/s)
1.10
-5
PR model
Surface
1
2
3
4
5
6
7
8
9
H’ (kPa)
5.10
4
4.10
4
3.10
4
2.10
4
1.10
4
7.10
3
2400
1000
300
(/)
0.065
0.135
0.220
0.250
0.325
0.400
0.400
0.400
0.385
(/)
0.165
0.265
0.380
0.450
0.575
0.700
0.900
1.050
1.165
0.7
DP
(°)
10
(°)
42
(°)
4.5
3.2 Results
Figure 5. Comparison between vertical displacement at 0.5m depth
under the top of the suction caisson for PR and DP models.
The first comparison in Figure 5 depicts the accumulation of
vertical displacement of the soil under the top of the caisson.
The weight of the wind turbine causes the first initial
displacement. Afterwards, the behaviour under cyclic loading is
quite different. For the DP model, the displacement oscillates
between nearly fixed boundaries and the soil lies within the
elastic zone most of the time. Actually the median displacement
is not exactly constant but changes very slightly at the
beginning of both loading phases.
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
4
On the other hand the PR’s model clearly shows an
accumulation of vertical displacement at each cycle. Moreover
during the first phase, transient and stationary behaviours can be
distinctly observed. The change in amplitude of loading implies
a change in the tendency of accumulation.
Figure 6. Comparison between mean effective and deviatoric stresses at
0.5m depth under the top of the suction caisson for PR and DP models.
Corresponding evolutions of mean effective and deviatoric
stresses are shown in Figure 6 while pore pressure accumulation
is given in Figure 8. During the first part of the loading, the
difference between both models is limited. For the Prevost’s
model, the transient phase is short and a drop in mean effective
stress is coupled with an increase of pore water pressure. Then a
stationary phase takes place, corresponding to an
accommodation phase (see in Figure 7) and the mean effective
pressure gets back to its first value when pore water pressure
dissipates. The difference is quantitatively greater for the
deviatoric stress but qualitatively the behaviour is identical.
Figure 7. Stress paths in the p’-q plane at 0.5m depth (PR model). The
first step of loading ends after 1000s when the second begins.
During the second part of loading, the soil characterized by
PR model shows a continuous decrease of mean effective stress
without reaching a stationary state. Greater amplitude of loading
entails greater plasticity effects and then contractancy. After
about 1600s, the stress path reaches the phase transformation
line and the mean effective stress increases for a while before
going down again. This continuous contractancy appears
because the model involves plasticity in loading and unloading
as well as a transition between contractive and dilative zones.
On the other hand, the soil described by DP model behaves
elastically most of the time because once the greatest deviatoric
stress is reached, the stress path lies within the plasticity surface
when unloaded and reloaded.
4 CONCLUSION
The Prevost’s model is simple, elegant and able to qualitatively
take into account the main features of cyclic loading. Basic
parameters are easy to obtain from classical laboratory tests.
Nevertheless an accurate modelling of cyclic tests requires
additional parameters and a new form of plastic potential.
A suction caisson was modelled as a practical case study.
Capabilities of the Prevost’s model compared with a classical
Drucker-Prager model appear clearly. The transient modelling
depicts pore pressure and plastic deformation accumulation
which the Drucker-Prager model is unable to represent.
Figure 8. Comparison between pore pressure evolutions at 0.5m depth
under the top of the suction caisson for PR and DP models.
5 ACKNOWLEDGEMENTS
I would warmly acknowledge all people that help me daily to
achieve this PhD and my colleagues that suffer my little
idiosyncrasies. I would also thank the FNRS for its financial
support.
6 REFERENCES
Arulmoli, K., et al. 1992. Verification of Liquefaction Analyses by
Centrigue Studies, Laboratory Testing Program, Soil Data Report. .
Elgamal, Ahmed, et al. 2003. Modeling of cyclic mobility in saturated
cohesionless soils. Internation Journal of Plasticity. 2003, Vol. 19,
pp. 883-905.
Houlsby, G. T., Ibsen, L. B. et Byrne, B. W. 2005. Suction caissons for
windturbines. Perth , Australia : International Symposium on
Frontiers in Offshore Geotechnics (ISFOG).
Ishihara, K., Tatsuoka, F. et Yasuda, S. 1975. Undrained deformation
and liquefaction of sand under cyclic stresses. Soils and
foundations. 1975, Vol. 15, 1, pp. 29-44.
Popescu, R. et Prevost, J.-H. 1993. Centrifuge validation of a numerical
model for dynamic soil liquefaction. Soil Dynamics and
Earthquake Engineering. 1993, Vol. 12, pp. 73-90.
Prevost, J.H. 1985. A simple plasticity theory for frictional cohesionless
soils. Soil Dynamics and Earthquake Engineering. 1985, Vol. 4, 1,
pp. 9-17.
Vertseele, H. 2012. Cyclic loading of suction caisson foundations for
offshore wind turbines. University of Liège. 2012. Master Thesis.
Yang, Zhaohui et Elgamal, Ahmed. 2008. Multi-surface Cyclic
Plasticity Sand Model with Lode Angle Effect. Geotechnical and
Geological Engineering. June 2008, Vol. 26, 3, pp. 335-348.
Zerfa, F. Z. et Loret, B. 2003. Coupled dynamic elastic-plastic analysis
of earth structures. Soil dynamics and Earthquake Engineering.
2003, Vol. 23, pp. 435-454.