SIMULAZIONE DEL COLLASSO DI UN PENDIO CON IL MATERIAL
POINT METHOD
SIMULATION OF SLOPE FAILURE EXPERIMENT WITH THE
MATERIAL POINT METHOD
Francesca Ceccato
DICEA – Universita’ degli studi di Padova
francesca.ceccato@dicea.unipd.it
Alexander Rohe
Deltares – Delft, The Netherlands
alex.rohe@deltares.nl
Paolo Simonini
DICEA – Universita’ degli studi di Padova
paolo.simonini@unipd.it
Sommario
Metodi numerici agli elementi finiti (FEM) sono stati largamente utilizzati negli ultimi decenni per studiare
un gran numero di problemi geotecnici; tuttavia numerose difficoltà appaiono quando applicati nel campo delle
grandi deformazioni. Il Material Point Method (MPM) è stato sviluppato per superare gli svantaggi dei classici
FEM Lagrangiani in problemi a grandi deformazioni. Nel MPM il continuo viene discretizzato con un insieme
di punti detti Material Points (MP) che si muovono all’interno di una mesh. I MP trasportano tutte le
informazioni del continuo, mentre la mesh è usata per risolvere le equazioni del moto ad ogni incremento
temporale, ma non tiene in memoria alcuna informazione permanente. In questa nota si mostra che il MPM è in
grado di riprodurre le gradi deformazioni coinvolte nei fenomeni di instabilità dei pendii. I risultati numerici
sono inoltre confrontati con quelli sperimentali.
Summary
Finite Element methods (FEM) have been widely used in the last decades for a great number of geotechnical
problems, However, difficulties arises with large displacement problems such as landslides, pile driving,
underground excavations ect. The Material Point Method (MPM) has been specifically developed to overcome
FEM drawbacks in large displacement problems. In MPM the continuum is discretized by material points (MP)
which move through a background mesh, thereby following the large deformations of the solid. The MP carry all
the properties of the continuum, while the background mesh is used to solve the governing equations at each
time step, but does not store any permanent information. In this paper it is shown that with the MPM it is
possible to reproduce the large deformations involved in slope failure. Numerical results are compared to a
reference laboratory test.
1. Introduction
The Finite Element Method (FEM) has been successfully applied in many fields of engineering and
science in the last decades, but shortcomings appear in the field of large displacement problems,
mainly because of mesh distortions.
The Material Point Method (MPM) can be considered as an evolution of the updated Lagrangian
FEM, specifically developed for large displacements problems. The governing equations are solved at
the nodes of the grid, similarly as FEM, but the deformations of the body are simulated by material
points moving through a fixed mesh, therefore preventing issues of mesh distortions. A brief overview
of the method is provided in Section 2.
Thanks to its ability to easily model large displacements, the method can be applied for the
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Chieti e Pescara, 14-15-16 luglio
simulation of problems such as landslides, pile driving, underground excavations ect. In this paper the
MPM is used to simulate the failure of a submerged slope.
The stability of submerged slopes is an important issue in many countries. In the Netherlands the
problem has a great impact in the province of Zeeland, in the south-east of the country, characterized
by numerous islands. The shoreline has been severely damaged by sea attack, and submarine
landslides compromise the safety of the area. The phenomenon needs to be deeply investigated in
order to enforce the design of mitigation techniques.
A set of small scale laboratory tests has been performed at Deltares, moreover, numerical
modelling adds interesting insight in the understanding of the problem. Indeed, the scale effect can be
easily investigated and parametric studies can be done.
Section 3 describes the laboratory test which is considered as a reference for the numerical
simulation. The MPM model is presented in Section 4, followed by the results in Section 5. The paper
ends with conclusions and an outlook on future developments.
1
The Material Point Method
The Material Point Method (MPM) is an advanced numerical method particularly suited to model
large deformations. It was firstly developed at the end of the previous century (Sulsky et al., 1994) for
solid mechanics and later applied to granular materials by Więckowski (1999, 2004) and Coetzee
(2005). It recently found entrance into the field of geotechnical engineering (Beuth, 2012; Alonso &
Zabala, 2011; Al-Kafaji, 2013; Bandara, 2013). In the following, the method is briefly described. For
more detailed information the reader is referred to Al-Kafaji (2013) and Bandara (2013).
In the MPM the continuum soil body is represented by a cloud of Lagrangian points, called
material points (MP). Large deformations are modelled by MP moving through a fixed background
mesh, which covers the entire region where the body is expected to move into. The MP carry all the
physical properties of the continuum such as density, momentum, material parameters, strains,
stresses, as well as external loads, whereas the background mesh is used to solve the governing
equations within each time step, but stores no permanent information.
Very often the soil behaviour highly depends on the solid-water interaction. The implementation of
the fully coupled two-phase approach follows the v-w-formulation (Zienkiewicz, 1999), i.e. the
primary unknowns are the velocity of the water and the velocity of the soil. This formulation, which
takes into account all acceleration terms, has proved to be able to capture the physical response of
saturated soil under dynamic loading (Van Esch et al., 2011).
The current implementation uses an explicit time integration scheme (Al-Kafaji, 2013). For each
time step, the background finite element mesh is used to solve the system of equilibrium equations and
to determine the velocity of the soil and the water. Once the strains are determined at the locations of
the material points, the mesh is usually reset into its original state. Figure 1 illustrates a single
calculation step in MPM.
Figure 1
a) Configuration of material points (red) and background mesh at the beginning of a calculation
step; b) Deformed mesh after solving the equilibrium equations; c) Background mesh in initial position at the end of a
calculation step and new location of material points which transport stresses, strains and material parameters.
F. Ceccato, A. Rohe, P. Simonini
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2
The physical model
The slope is completely submerged in a test flume which is 5.4m long, 2.5m high and 0.5m wide.
The sand is placed in the flume using a fluidisation system: water is injected through the tubes placed
on the bottom of the flume and, due to the upward water flow, the whole sand pack goes into
suspension. After stopping the water supply the sand settles in a very loose state.
The slope is built by slowly sucking the sand through a nozzle, it then flows under a natural slope
of the embankment.
After the preparation phase, the slope has an inclination of 31° and a height of 0.60 m. Afterwards,
in the experiment, the failure is triggered by injection of water under the toe of the slope.
The first macro-scale movement is observed in a superficial layer of about one decimetre sliding
downward. This lasted for a few seconds (5 to 10 seconds). The movement continued slowly in a
thinner layer. The whole process took about a minute (Stoutjesdijk, 2014).
3
The numerical model
The geometry just before triggering the failure is considered as the initial situation of the numerical
analyses, see Figure 2. The discretized domain is shown in Figure 3. The mesh consists of 2 202
elements, 1567 of which are active, i.e. filled with 4 MP each at the beginning of the calculation. The
part of the mesh in which most of the deformation is expected is refined to increase the accuracy of the
results. The final discretization has been chosen after some sensitivity analysis as result of a good
compromise between accuracy and computational effciency.
At the left and right boundary the displacments are constrained in horizontal direction, while at the
bottom no displacments are allowed at all. All boundaries are impermeable for water, except during
triggering of the failure at the location where the water-pressure is applied.
An elaso-plastic model with Mohr-Coulomb failure criterion is used for the sand, whose input
parameters are shown in Table 1. The input parameters are according to specifications derived from
the experiment. A local damping factor of 5% is used for the calculation; this value simulates natural
energy dissipation inside the material, which is not taken into account by the constitutive model.
The failure is triggered by applying an excess pore pressure at the bottom of the domain, below the
toe of the slope. The pore pressure is increased linearly from 0 to 10 kPa in 5.0 s. After that, the pore
pressure is reduced to zero again.
Table 1Material parameters for the sand
Material parameter
saturated unit weight of sand
Young modulus of sand
Poisson ratio of sand
bulk modulus of water
initial porosity
permeability
cohesion
friction angle
dilatancy angle
F. Ceccato, A. Rohe, P. Simonini
Symbol
sat
E
’
Kw
n
k
c’


Value
18.7
5 000
0.2
45 310
0.45
1.0*10-4
0
32
0
Unit
kN/m3
kPa
–
kPa
–
m/s
kPa
deg
deg
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Figure 2
Configuration of the slope immediately before triggering the failure. The location of applied water
pressure is indicated.
Figure 3
Numerical model of background mesh (active elements are shaded, inactive elements are transparent).
The left and right boundaries are horizontally fixed, the bottom boundary is fully fixed.
2. Results
The initial stresses are generated using a gravity loading phase. A quasi-static convergence
criterion is applied, which implies that the slope is in static equilibrium and the kinetic energy and the
unbalance force are below a limit value. The pore pressure distribution is initially hydrostatic.
After the initialisation phase, the failure is triggered as described in the previous section. The
excess pore pressure is applied at the bottom of the mesh below the toe of the slope, and the excess
pore pressures propagate upwards. While the excess pore pressures increase, the effective stresses
decrease causing instability of the slope. The first clearly visible displacements appear at the crest of
the slope after 3.75 s since pore pressure application, and the new equilibrium configuration is reached
after approximately 8.5 s. The failure surface localizes at a depth of about 0.15 m.
F. Ceccato, A. Rohe, P. Simonini
Incontro Annuale dei Ricercatori di Geotecnica 2014 - IARG 2014
Chieti e Pescara, 14-15-16 luglio
The final equilibrium state is compared with the experimental results in Figures 4 and 5. It can be
concluded that the numerical simulation is in very good agreement with the experiment.
Figure 4 Final state of the slope after failure in the experiment (blue line). The initial state is indicated by the red line.
Figure 5 Comparison of numerical results (MPM) with experimental results. The red line indicates the initial situation and
the blue line the final state after failure. Displacements are shown in [mm].
3. Conclusions and future developments
It is demonstrated that with MPM it is possible to model large displacement problems. The failure
of a submerged slope triggered by a suddenly increased water pressure at the bottom can be
successfully simulated using the fully coupled MPM implementation. The numerical results of the
deformed slope in the final equilibrium state are in good agreement with the experimental data (see
F. Ceccato, A. Rohe, P. Simonini
Incontro Annuale dei Ricercatori di Geotecnica 2014 - IARG 2014
Chieti e Pescara, 14-15-16 luglio
Figure 5).
The behaviour of sand is complex; the elasto-plastic model with Mohr-Coulomb failure criteria is a
very simplified way of describing soil behaviour. Deeper understanding of the failure process could be
achieved with more sophisticated material models such as the Hypoplastic model (Niemunis, 1997),
the NorSand model (Jefferies, 1993) and the Mohr-Coulomb with strain softening model(Abbo, 1995).
The latter proved to be able to capture the progressive failure of the slope (Alonso & Zabala, 2011;
Yerro et al., 2014).
It is of great interest to investigate the behaviour of true scale slopes, as found in the region of
Zeeland, whose high ranges between 10 and 50 m. Special attention needs to be paid to the stiffness
and the density of the sand which is not assumable to be constant with depth, especially for slopes of
larger high. Therefore it is expected that slopes with larger scale would behave differently than slopes
in model scale. Numerical simulations can indicate safe inclination angle in the natural conditions.
Revetments of various types have been used to prevent erosion and improve the stability of the
slope. The effect of a stone revetment will be considered in the future also in the numerical
simulations.
4
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F. Ceccato, A. Rohe, P. Simonini