Computers and Geotechnics 75 (2016) 126–134
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Computers and Geotechnics
journal homepage: www.elsevier.com/locate/compgeo
Research Paper
Numerical modelling of seismic slope failure using MPM
Tushar Bhandari a,1, Fursan Hamad b, Christian Moormann b,⇑, K.G. Sharma a, Bernhard Westrich b
a
b
Department of Civil Engineering, Indian Institute of Technology-Delhi, Hauz Khas, New Delhi, India
Institute of Geotechnical Engineering, University of Stuttgart, Stuttgart, Germany
a r t i c l e
i n f o
Article history:
Received 26 September 2015
Received in revised form 12 December 2015
Accepted 17 January 2016
Available online 17 February 2016
Keywords:
Material Point Method
Non-zero kinematic condition
Large deformation
Landslides
Slope failure
a b s t r a c t
The Finite Element Method (FEM) is widely used in the simulation of geotechnical applications. Owing to
the limitations of FEM to model problems involving large deformations, many efforts have been made to
develop methods free of mesh entanglement. One of these methods is the Material Point Method (MPM)
which models the material as Lagrangian particles capable of moving through a background computational mesh in Eulerian manner. Although MPM represents the continuum by material points, solution
is performed on the computational mesh. Thus, imposing boundary conditions is not aligned with the
material representation. In this paper, a non-zero kinematic condition is introduced where an additional
set of particles is incorporated to track the moving boundary. This approach is then applied to simulate
the seismic motion resulting in failure of slopes. To validate this simulation procedure, two geotechnical
applications are modelled using MPM. The first is to reproduce a shaking table experiment where the
results of another numerical method are available. After validating the present numerical scheme for relatively large deformation problem, it is applied to simulate progression of a large-scale landslide during
the Chi-Chi earthquake of Taiwan in which excessive material deformation and transportation is taking
place.
Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction
Slopes stability has been considered as one of the most significant topic to study for the geotechnical society over many years. In
most cases, the stability of slope is what one aims to analyse. For
this reason, the evolution of analysis techniques has taken place
in the regime of mesh based methods that can take into account
the limiting deformations as well as the effect of supporting structures. In some cases, however, it is important to study the behaviour of the slopes beyond failure. In these cases, slope failure is
inevitable and the impact of the sliding mass on low-lying areas
becomes important to investigate. The physical significance of
slope failures can be gauged in the form of landslides leading to
a massive flow of debris. Landslide-debris flow is a very rapid
and massive flow-like motion of soil and fragmented rock. The
material mobility and the impact of the avalanche can cause
⇑ Corresponding author at: Pfaffenwaldring 35, 70569 Stuttgart, Germany.
E-mail
addresses:
bhandaritushar1390@gmail.com
(T.
Bhandari),
fursan.hamad@igs.uni-stuttgart.de
(F.
Hamad),
christian.moormann@igs.
uni-stuttgart.de (C. Moormann), kgsharmaiitd@gmail.com (K.G. Sharma),
bmwest@gmx.de (B. Westrich).
1
Present address: SMEC India Pvt. Ltd., 5th Floor, Bldg. 8, Tower C, DLF Cyber City
Phase II, Gurgaon, Haryana, India.
http://dx.doi.org/10.1016/j.compgeo.2016.01.017
0266-352X/Ó 2016 Elsevier Ltd. All rights reserved.
significant damage to human life and engineering structures that
come in the way of the flow.
Slope failures and landslides, in particular, can be triggered by
many factors including heavy rainfall, imposed loads, strength
degradation due to weathering and seismic excitation. Among
these factors, seismic excitation or earthquake has been recognised to be the major causes of slope failures [1]. Therefore it
becomes more important to analyse the response of a slope to a
seismic event. Many methods have been developed to address
this issue. Jibson [2] classifies the methods for assessing the performance of a slope during earthquakes fall into three phases: (1)
pseudo static analysis, (2) permanent displacement analysis, and
(3) stress–deformation analysis. Pseudo static analysis, used as a
preliminary analysis method can only indicate the safety against
slope failure using a limit equilibrium method in which the seismic shaking is represented by a constant inertial force applied on
a sliding mass. As a significant improvement to pseudo static
analysis, permanent displacement analysis provides more quantitative measure to evaluate the performance of slopes during
earthquakes. A common example of the permanent displacement
analysis is the Newmark rigid-block analysis [3] where the permanent slope deformation induced by earthquakes is estimated
by the permanent displacement of the rigid block sliding along
the inclined plane under a base acceleration. Both pseudo static
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T. Bhandari et al. / Computers and Geotechnics 75 (2016) 126–134
Notations
q
r
g
u
a
e
X
n
mass density
the Cauchy stress tensor
gravitational acceleration vector
displacement
nodal displacement vector
strain tensor
volume
unit normal vector
analysis and permanent displacement analysis are based on
highly simplified geometric and material models. However, these
methods cannot be relied upon to evaluate earthquake-induced
slope deformations under complex geological conditions. For this
purpose, stress–deformation analysis is used which can account
for complex soil behaviours (e.g., non-linear response to dynamic
loading, strain softening and strain rate dependence of material
strength) and geometric conditions. It follows the approach of
computing stresses in a material and its response in form of
deformations based on a defined constitutive relationship
between stress and strain. A stress–deformation analysis is frequently performed using numerical methods like the Finite Element Method (FEM) or Finite Difference Method (FDM).
However, the mesh based methods (e.g. FEM) have difficulties
in modelling large deformations due to problems of mesh distortion and entanglement. As a result, stress–deformation analysis is
currently limited to estimating relatively small seismicallyinduced slope deformations [2]. The drawback of these methods
to deal with large deformations therefore considerably impedes
their application in the analysis of earthquake-induced slope
deformations.
In order to overcome these drawbacks in stress–deformation
analysis, various mesh-free methods have been proposed. The
Material Point Method (MPM) is one of these methods, which
has shown its applicability to model granular materials like soil
in different geotechnical applications [4,5]. Previous MPM research
has focused on modelling of collapsing slopes and landslides using
strength degradation [6]. Andersen and Andersen [7] also studied
collapse of slopes in which the slide is initiated by increasing the
density of material, corresponding to the behaviour during heavy
rainfall. MPM has also been applied to model failure of
geotextile-reinforced slope [8].
Although MPM represents the continuum by material points,
solution is performed on the computational mesh. Thus, imposing
boundary conditions is not aligned with the material representation. A non-zero kinematic condition is introduced in this paper
where an additional set of particles is incorporated to track the
moving boundary. The MPM procedure is applied to simulate
the seismic excitation and dynamic response of a slope. The seismic history is introduced to the MPM model via the rigid boundary condition introduced by Hamad et al. [9,10]. Also, the
proposed simulation approach is tested to model a shaking table
experiment and to compare the results with corresponding
numerical simulation from Hiraoka et al. [11] using another
numerical method called the Smooth Particle Hydrodynamics –
SPH method. Finally, MPM is applied to simulate progression of
a large-scale landslide during the 1999 Chi-Chi earthquake of Taiwan. In this paper, the effect of water is not considered. In many
cases, water can be a triggering factor for landslides and may
impact the simulation results significantly. However, the simulation of dry landslide can be very relevant in cases of dry debris
flow and rock avalanches where the controlling factor is the
seismic motion.
M
B
F
N
C1
Vp
mass matrix
gradient of the shape function
nodal force vector
shape function matrix
relaxation coefficient
p-wave velocity
horizontal component of velocity
time
vx
t
2. Brief description of MPM
MPM can be viewed as an extension of the classical finite element procedure, in which the continuum body is discretised by
Lagrangian material points that can move through a fixed computational mesh as shown in Fig. 1. The momentum equation is
solved on the computational mesh which provides a convenient
means of calculating discrete derivatives.
2.1. Spatial discretisation
We start with the conservation of linear momentum, which
reads
qu€ ¼ r r þ qg
ð1Þ
where r(x, t) is the Cauchy stress tensor at position x and time t,
q(x, t) is the mass density, g is the gravitational acceleration vector,
u(x, t) is the displacement with the superposed dot denoting
differentiation with time.
By taking the virtual displacement du as test function for a
domain of volume X surrounded by boundary S, the weak form
of the momentum equation can be written as
Z
X
€ dX ¼
duT q u
Z
X
deT r dX þ
Z
X
Z
duT q g dX þ
duT t dC
ð2Þ
Ct
where t ¼ r n is the prescribed traction on boundary Ct, n is the
outward unit normal and e is the strain tensor represented in vector
form. The superscript T denotes the transpose. Similar to the standard finite element method, the value of a variable inside the element can be based on the nodal values and the nodal shape
functions. Using these definitions and discretizing the momentum
equation, it takes the form (e.g., [4])
€¼F
Ma
ð3Þ
€ the nodal acceleration
where M is the consistent mass matrix, a
vector, and (F = Fext–Fint) with Fext and Fint being the external and
internal nodal force vectors, respectively. In practice, the lumped
mass matrix is preferred over the consistent mass matrix. This simplifies the computations at the expense of introducing a slight
amount of numerical dissipation [12]. Referring to Eq. (3), the internal force vector is given by
Fint ¼
np
X
xp BT ðxp Þrp
ð4Þ
p¼1
where the quotient of the material point mass and density is the
volume of the material point, xp = mp/qp and B is the gradient of
the shape function, as also used in standard finite element method
[4], rp is a vector containing the stress components at the material
point p. The external nodal force vector is given by
Fext ¼
Z
np
X
mp NT ðxp Þg þ
NT t dC
p¼1
Ct
ð5Þ
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corrected using the velocity of the coupled bodies following
Coulomb friction. This solution scheme uses the concept of comparing the single and combined body velocities for a contact node
and defining its behaviour accordingly. The algorithm is able to
detect whether two bodies in contact are approaching or
separating from each other. If the two bodies are separating,
the algorithm allows free separation where each body moves
according to its own equation of motion. For more details about
the algorithm, the reader is referred to Ref. [15].
2.4. Prescribed velocity in MPM
Fig. 1. MPM representation of a continuum.
where N is the global shape function matrix and np is the total number of material points.
2.2. Time integration
The discretised momentum equation (Eq. (3)) needs to be
solved for discrete time intervals. With the mass matrix being a
diagonal matrix, the system of equations can be solved using the
Euler-forward time integration scheme, i.e.
€t ;
a_ tþDt ¼ a_ t þ Dt a
€t ¼ ½Mtl a
1
Ft
ð6Þ
where Dt is the current time increment, a_ t and a_ tþDt are the nodal
velocities at time t and (t + Dt), respectively and Ml is the lumped
mass matrix. The incremental nodal displacement is obtained by
integrating the nodal velocity by the Euler-backward rule (see for
e.g., Jassim et al. [13])
DatþDt ¼ Dt a_ tþDt
ð7Þ
and the positions of the particles are subsequently updated from
xptþDt ¼ xtp þ Np D atþDt
In traditional dynamic FEM, the prescribed velocity is defined
over nodes. These nodes always define part of the Lagrangian body
boundary. On the other hand, in MPM the continuum is defined by
Lagrangian particles which might change position from one element to another. Hence, there is no defined interface surface where
prescribed velocity are applied.
Within the framework of MPM, prescribed velocity can be
applied directly on the material points of a rigid body representing
the moving boundary for simple one-dimensional problem. For
applications with axial movement, part of the mesh can be displaced as a moving mesh having a prescribed value while the rest
is stretched uniformly [16]. This becomes very complicated and
inconvenient when applied to applications like imposing seismic
motion for a slope problem. As an alternative, an additional set
of particles (Fig. 2) is introduced which tracks the moving boundary by carrying the time-dependent boundary evolution [9]. Following the same methodology, prescribed velocities are applied
as a boundary condition for the rigid wall and a contact is defined
between the rigid wall and soil.
ð8Þ
where xtp and xtp+Dt are the particle positions at time t and (t + Dt)
respectively.
For the present MPM solution procedure, a slightly different
algorithm has been adopted for updating the particles velocity following Sulsky et al. [14]. By solving the equation of motion for the
nodes, the elements deform and the material points in the interior
of the element move in proportion to the motion of the nodes,
based on the nodal shape functions. The position of the material
points is updated using a single-valued continuous velocity field
and hence the interpenetration of material is precluded. This
automatic feature of the algorithm allows simulations of no-slip
contact between different bodies without the need for special
interface tracking and contact algorithms.
After getting the nodal velocities, the strain increment of a
material point p is calculated. The constitutive model is applied
at the material points to get the incremental strain, which allows
direct evaluation and tracking of history-dependent variables.
At the end of time step the material point variables are updated
and a new cycle begins using the information carried by the material points to initialise nodal values on the computational mesh.
Note that at this stage, a new computational mesh can be defined
since all the state variables are carried by the material points. In
practice, however, it is more efficient to use the original mesh.
2.3. Contact algorithm
The frictional contact algorithm proposed by Bardenhagen et al.
[15] is used in this paper. It can be seen as a predictor–corrector
scheme formulated in explicit manner, in which the velocity is
predicted from the solution of each body separately and then
2.4.1. Non-zero kinematic conditions
In this paper, the non-zero kinematic condition is developed as
shown in Fig. 2 where an additional set of particles is introduced,
which tracks the moving boundary by carrying the timedependent boundary evolution. At the beginning of a time step,
the velocity a_ p ðxp ; tÞ of the prescribed particle p is assigned. Next,
the prescribed velocity must be mapped (via the shape functions)
from the prescribed particles to the computational nodes, where
the discrete equations are solved. Nodes belonging to the elements
where the prescribed particles are located are then tagged to be
boundary nodes. It should be appreciated that the thickness of
the boundary corresponds to one computational element. The prescribed values are assigned directly at the boundary nodes. As an
alternative, a weighted mapping procedure can be used, which is
more consistent with the principles of MPM, with the nodal velocity a_ i of boundary node i being obtained from [9,10]
P
_
p Ni ðnp Þwp ap
a_ i ¼ P
p Ni ðnp Þwp
ð9Þ
where a_ p is the prescribed velocity of material point p, Ni(np) is the
shape function of i being evaluated at p, and wp is a weighting function (e.g. mass or volume of p). The summations in this equation run
over the number of prescribed particles. Depending on the location
of the boundary particles, the number of the boundary nodes is
updated constantly as well as their values from Eq. (9).
2.4.2. Validation case: prescribed velocity with contact
To validate the procedure of introducing prescribed velocity
particles involving contact, a (1 1 m) square with a unit weight
of 10 kN/m3 supported by 45 prescribed particles with zero
velocity is considered. After calculating the initial gravitational
stresses, the layer of prescribed particles underneath is moved
suddenly with a horizontal velocity of 2 m/s. Fig. 3 shows three
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T. Bhandari et al. / Computers and Geotechnics 75 (2016) 126–134
Soil
1.0 m
Frictional
Contact
0.35
0.53
Wall
Rigid Wall
0.2
11
(a) Two Bodies in Contact
Prescribed Velocity
Contact
Interface
0.5
45o
2
2
(b) MPM Representation
0.9
Fig. 2. Prescribed velocity in MPM.
Shaking Table
Fig. 4. Experimental model (after Hiraoka et al. [11]).
scenarios: if the standard MPM (top) is applied, the body travels
together with the bottom; for the case where a rough contact is
introduced (middle), the glue condition is broken if the bottom
moves fast enough; finally, the absence of the frictional resistance
in the smooth contact case (bottom) leads to the early separation
of the two bodies as shown.
3. Numerical application
In the geotechnical field, dynamic process of slope failures subjected to seismic loads is often investigated by means of physical
modelling [17–19]. Slope failure under seismic excitation is implemented by a box filled with soil and mounted on a shaking table.
These experiments play a vital role in the calibration of numerical
models for similar applications. For assessing the performance of
MPM to simulate seismic excitation in geo-mechanical problems,
a shaking table experiment is considered here and the simulated
response is compared with published results [11] based on Smooth
Particle Hydrodynamics (SPH) method. The objective here is to test
the proposed numerical scheme in MPM (including the non-zero
kinematic condition) with a simple Mohr–Coulomb failure criterion for simulating a dynamic test in comparison with more established numerical methods like SPH.
The shaking table experiment under consideration consists of a
small-scale cut slope as shown in Fig. 4. A steel box is mounted on
top of the shaking table. The soil slope model (0.9 0.6 0.5 m)
was set in the shaking box, and the slope angle was made as 45°.
The soil used in the experiment was Masa soil which is weathered
granite commonly found in Kansai area in Japan. Laser displacement sensors were used to measure displacement within the slope.
The slope model was subjected to the seismic wave loading shown
in Fig. 5. The test runs for 14 s until the slope completely collapsed.
More details about the test setup and the experiment can be found
in Ref. [11].
3.1. Reference solutions
In order to test the proposed treatment of boundary conditions
in MPM, a comparison with other numerical methods namely the
Finite Element Method (FEM) and the Smooth Particle Hydrodynamics (SPH) method is provided in this paper. The FEM model
used in this research is suitable for the failure initiation where
small deformation theory is applicable. On the other hand, the
SPH model is more appropriate for the large deformation analysis.
Fig. 3. Prescribed velocity boundary with contact: (top) standard MPM, (middle) rough contact, and (bottom) smooth contact.
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T. Bhandari et al. / Computers and Geotechnics 75 (2016) 126–134
Velocity, vx (m/s)
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
0
2
4
6
8
10
12
14
Time (s)
Fig. 5. Velocity–time history of the shaking table (after Hiraoka et al. [11]).
[%]
1.5
3.1.1. FEM model
To set a benchmark for the proposed MPM simulation approach,
dynamic analysis for the slope is carried out using the FEM software, PLAXIS 2D. Keeping in mind that the FE software does not
count for large deformations, the FE simulation is performed up
to 10 s of seismic excitation. The soil is modelled using the
Mohr–Coulomb failure criterion. Soil properties are considered as
given by Hiraoka et al. [11] for SPH model. The density of the soil
is 1680 kg/m3 with Young’s modulus of 2570 kPa and Poisson’s
ratio of 0.33. The stiffness parameters are chosen upon the construction process of the slope, which is built in phases of 5 cm thick
compacted layers. The shear strength of the soil is defined by cohesion of 0.78 kPa and friction angle of 23°. A non-associated constitutive model is adopted with zero dilatancy angle.
The problem is solved in two phases, first for gravity and then
the dynamic phase. In PLAXIS, different boundary conditions are
used for solving the static and dynamic phase. During the static
phase, roller boundary at side and fixed boundary at bottom are
considered for the calculation of gravity stresses. These boundary
conditions are applied by introducing prescribed boundary displacements. The boundary fixities are removed during the transition from static to dynamic phase which means that the
displacement boundary conditions are replaced with another set
of boundary conditions defined for the dynamic phase. By removing these boundary fixities, the boundary starts moving as a result
of initial stresses. To prevent this, the original boundary stress is
converted to an initial (virtual) boundary velocity. When calculating the stress in the dynamic phase, the initial boundary velocity is
subtracted from the real velocity:
rn ¼ C 1 qV p ðu_ n u_ on Þ
ð10Þ
where rn is the normal stress on boundary, C1 is relaxation coefficient, q is the density of material, Vp is the p-wave velocity, u_ n is real
velocity and u_ on is the initial velocity. Prescribed velocities at side as
well as bottom are used to induce the seismic motion as per the
velocity–time history (Fig. 5). The deformed mesh to true scale
and shear strain plot are shown in Fig. 6. The scale for the shear
strain plots is set to be same as in Ref. [11] to facilitate comparison
of results. Large deformation is clearly present at t = 10 s as seen in
Fig. 6 and therefore finite element analysis beyond this deformation
is not considered.
3.1.2. SPH model
A 2D-SPH (Smooth Particle Hydrodynamics) model was used by
Hiraoka et al. [11] to simulate the dynamic behaviour of the slope
model. Soil parameters used for the elastic–plastic Drucker–Prager
constitutive model are the same as given in Section 3.1.1. Boundary
conditions were free-roller at the vertical and full-fixity at the base.
A total of 3245 particles were used to create the slope model.
Details of the modelling framework and simulation approach for
SPH can be found in Ref. [11].
0
Fig. 6. Finite element analysis at t = 10 s: (top) deformed mesh and (bottom) shear
strain.
3.2. Simulation approach for seismic slope modelling in MPM
MPM is a relatively new method for geotechnical applications
and hence the suitability of the method to simulate specific
geotechnical problems (like the considered ones) has not been verified. Since, MPM differs from FEM and SPH in its numerical framework, similar simulation approach cannot be applied for such
applications. For instance, unlike FEM, prescribed velocities cannot
be applied directly as a boundary condition to the soil particles.
Therefore, in this paper, an approach is proposed to deal with such
applications in MPM as discussed in the subsequent sections.
3.2.1. Boundary conditions
In order to model the shaking table problem, rigid particles are
defined to play the role of velocity carriers as shown in Fig. 7. The
rigid wall particles can be related to the steel box as generally used
in the experimental setup. The soil is modelled using the Mohr–
Coulomb constitutive law. Furthermore, two different types of contacts are defined between the soil and the walls. A smooth contact
is provided for side wall to allow free settlement of soil along the
glass wall in vertical direction and a rough contact is provided
for the bottom wall.
3.2.2. Calculation phases
As in most of the numerical methods, the analysis is done in different steps. The model is first solved for gravitational stresses. The
rigid walls are assigned zero velocity and the stresses are built-up
under the self-weight of soil. A quasi static solution is obtained
with the use of local damping. Local damping is applied by assuming that the damping force F damp is proportional to unbalanced
forces [16].
F damp ¼ ajF ext F int jsignðv Þ
ð11Þ
where a being the damping coefficient used as 0.7 for the gravity
phase, and the sign of velocity at the degree of freedom (i) is defined
as signðv Þ ¼ v i =jv i j. Eq. (11) is added to the right hand side of Eq. (3).
To check that equilibrium is achieved, the kinetic energy and the
out of balance forces are evaluated at all computational nodes
and compared to a tolerance, which is 0.005 for the present cases.
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T. Bhandari et al. / Computers and Geotechnics 75 (2016) 126–134
Smooth contact
µ=0
Rigid Particle
Rigid Wall
M-C Soil
M-C Soil
Rough contact
µ=1
Fig. 8. Initial configuration of the MPM model.
[%]
1.5
vx(t)= prescribed velocity (Fig. 5)
vy(t)= 0
Fig. 7. Dynamic boundary conditions in MPM.
After obtaining an equilibrium state of gravity stresses, next
step for calculation is defined for dynamic analysis. The local
damping is switched off and prescribed velocities (Fig. 5) are
assigned to the rigid walls which simulates dynamic motion of
the steel box and consequent behaviour of the soil it contains taking into account the external forces, body forces and the inertial
forces.
3.2.3. Simulated results
To model the seismic excitation of the soil slope in MPM, the
simulation approach as discussed in Sections 3.2.1 and 3.2.2 is
used. Mesh and particle discretisation are illustrated in Fig. 8.
Shear strain at t = 10 s and total displacement at t = 14 s are shown
in Fig. 9.
The dynamic response of the slope can be described in the form
of displacement history of specific points on the slope. The displacement histories of specific control points 1 and 2 as outlined
in Fig. 4 are used to compare the experimental results with the
dynamic response predicted by different numerical models. The
comparison for the vertical displacement history for point 1 and
horizontal displacement history for point 2 are shown in Fig. 10.
It is evident that even with limited time i.e. t = 10 s, FEM in unable
to predict the trend of deformations towards failure. The deformation behaviour of the slope as predicted by MPM is in fair comparison with that predicted by SPH and also close to the experimental
results. Sensitivity of the analysis results to mesh discretization
was also assessed by solving three cases for different mesh sizes.
The details of these cases are presented in Appendix A. It is noted
that like other mesh based methods, MPM results are also dependent on the size of mesh. With a variation of ±60% in the mesh size,
the maximum total displacement varies up to around ±7%.
In the present analyses, the shape of the sliding surface is predicted as almost circular, which is in agreement with the other
continuum-based model using SPH. However, the sliding surface
as observed from the experiment was a curved line with higher
curvature angle. Hiraoka et al. [11] suggest a lack of clarity about
this failure mechanism observed in the experiment and attribute
it to possible technical errors while removing the collapsing soil
to specify the failure surface in the experiments. The SPH analysis
[11] also suggests to assign a non-zero dilatancy angle (w = //2 and
/) in order to come closer to the experiment. Although this
assumption improves the prediction of the failure surface, it
over-predicts the plastic volumetric expansion like the soil is heavily compacted which contradicts the initial state of the considered
0
[m]
0.21
0
Fig. 9. MPM analysis: (top) shear strain at t = 10 s and (bottom) norm of total
displacement at t = 14 s.
soil. Consequently, large run-out distance is observed in the
numerical model. Therefore, these analyses are excluded from this
paper.
Advanced constitutive models that consider the effect of soil
degradation with the stress and density evolutions are supposed
to perform better than the simple elasto-plastic Mohr–Coulomb
model being used. In principle, the implementation of these models in MPM is straightforward, while the related numerical stability
problems are under study and the formulation under development.
Considering that the soil in the experiment is reported to have 10%
water content, the advanced constitutive model should be combined with partially saturated soil model for better simulation of
the progressive failure of the experiment.
4. Earthquake induced landslide debris flow
The dynamic process of landslides induced by earthquakes is
very complex in its nature. Various numerical methods are used
to simulate different activities involved in the whole process of
evolution of a landslide starting from the development of a critical
sliding surface to initiation and triggering of failure to disintegration of the sliding mass to debris flow and finally deposition. In this
section, MPM is used to simulate the last part of the process i.e.
progression of the sliding mass and deposition. For this, the
Chiu-fen-erh-shan landslide of 1999 is used as a case study
example.
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T. Bhandari et al. / Computers and Geotechnics 75 (2016) 126–134
Time (s)
2
4
6
Pre-slide Topography
8
10
12
Post-slide Topigraphy
14
0
-25
500 (m)
-50
-75
Fig. 11. Topography of landslide (after Wu et al. [23]).
-100
-125
-150
Experimental
SPH
PLAXIS
MPM
1.0
Velocity, vx (m/s)
Horizontal Displacement (mm)
Vertical Displacement (mm)
0
200
150
0.8
0.6
0.4
0.2
0.0
-0.2
-0.4
100
0
20
40
60
80
100
Time (s)
50
0
Fig. 12. Time history of the horizontal velocity (after Wu et al. [23]).
0
2
4
6
8
10
12
14
Time (s)
Experimental
PLAXIS
SPH
MPM
Fig. 10. Displacement history of the control points: (top) Point 1 and (bottom)
Point 2.
4.1. The Chiu-fen-erh-shan landslide
The Chiu-fen-erh-shan landslide is one of the major landslides
caused by the disastrous Mw 7.6 Chi-Chi earthquake of 1999 in Taiwan. The landslide has been characterised as a translational rockblock slide on dip slope and known to be disintegrated into fragmented rock avalanche travelling long distance at high velocity
[20]. The landslide debris travelled a distance of more than 1 km
and covered an area of 1.95 km2. Many studies have been conducted since then to understand the mechanism and movement
of the landslide [20–23]. These studies concluded that the slope
had undergone gravitational creep and was unstable even prior
to the earthquake. A slip surface had been developed as a result
of flexural folding. Observations by Wang et al. [22] suggested
the presence of clay seams between the alternating beds of shale
and sandstone in the slope. The existence of the clay seam provides
a very smooth surface for the slide to occur. The slope was retained
in its position because of a sandstone bed that formed resistant
ridges at the foot of the slope. The sandstone bed was most probably damaged seriously during the earthquake giving way to the
massive landslide. The topography of the slope before and after
the slope is depicted in Fig. 11. Detailed geological description of
the landslide is available in Ref. [22].
4.2. Continuum modelling of the rock-block slide
According to Wu et al. [23], the mechanical properties of the
rock in sliding mass, composed of highly jointed rock mass, are:
Fig. 13. Progression of landslide.
density of 2550 kg/m3, Young’s modulus of 7.57 GPa, Poisson’s
ratio of 0.19, whereas the strength parameters of the weak planes
of rock are adopted from the direct shear experimental study conducted by Chen [24]. These strength parameters of the rock joints
are considered to mainly control the strength behaviour of the
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T. Bhandari et al. / Computers and Geotechnics 75 (2016) 126–134
sliding mass as movement is likely to occur along the existing
weak planes in the sliding mass and thus these parameters are
used directly to define the shear strength of the continuum. The
values adopted for the cohesion and friction are 20 kPa and 24°
respectively.
In the present numerical scheme, the aim is to model the rock
joint masses as a continuum material. Therefore, the elastic modulus of the equivalent medium was estimated to be 500 MPa with a
geological strength index range of 25–30 [25].
[%]
1.5
0
4.3. Material point analysis of the landslide
[%]
For the seismic motion, horizontal component of the velocity–
time history of the Chi-Chi earthquake as shown in Fig. 12 was
used. This history was used by Wu et al. [23] in their discrete analysis using velocities projected to the slope direction and corrected
for baseline correction. The sliding surface in MPM is modelled
using rigid particles and the velocities are imposed on these
particles.
1.5
4.4. Simulated results
0
The progression of the flow resulting from the landslide is
depicted in Fig. 13. The progressive displacement of the front of
[%]
1.5
Table A.1
Discretization details for cases of mesh sensitivity analysis.
S. no. Item
Case-1
Case-2
Case-3
1.
2.
3.
4.
101,100
16,174
32,827
1.05 103 m
47,640
7503
15,336
1.89 103 m
24,540
4029
8315
2.97 103 m
Number of particles
Number of elements
Number of nodes
Average element size
0
Fig. A.2. Shear strain at t = 10 s for cases 1, 2 and 3 (top to bottom).
the sliding mass is marked for different time steps. It can be seen
from the stabilizing displacement profile that the sliding mass
reaches an equilibrium state in 120 s. The final configuration as
simulated by MPM is compared with the actual scenario in
Fig. 13 which suggests a fair match between the shape of the debris
deposit except for a slightly different shape at the rare end of the
debris flow.
5. Conclusion and outlook
Fig. A.1. Mesh and particle discretization for cases 1, 2 and 3 (top to bottom).
An important geotechnical application has been modelled using
the MPM. Simulation approach to induce seismic motion in MPM is
tested by modelling a shaking table experiment and comparing the
results with other numerical methods. The deformation behaviour
of the slope as predicted by MPM is in fair comparison with that
predicted by SPH. A variation of the order of 5% in results is
observed and attributed to the fact that both MPM and SPH differ
in their basic formulation. Also, the failure criteria and implementation of boundary conditions used in both the methods are different. Compared to SPH, MPM is seen to predict the dynamic
response of the slope closer to the experimental results. However,
better simulation of progressive failure can be achieved by incorporating advanced constitutive models in the current formulation
of MPM. The limitation of conventional FEM to model large deformation problems is also emphasised by comparing the results with
FEM simulation using PLAXIS. While this limitation can largely be
overcome by Lagrangian–Eulerian methods, remeshing may still be
required and state variables associated with material points need
to be remapped. This is of particular concern when the history of
the material must be taken into account.
134
T. Bhandari et al. / Computers and Geotechnics 75 (2016) 126–134
[m]
Appendix A
0.23
To assess the sensitivity of the model to mesh discretization,
three cases for different mesh sizes were solved. The discretization
details of these cases are presented in Table A.1.
The mesh for the three cases is shown in Fig. A.1. Analysis
results in form of contour plots for shear strain at t = 10 s and total
displacement at 14 s are presented in Figs. A.2 and A.3 respectively
for the three cases.
0
References
[m]
0.21
0
[m]
0.20
0
Fig. A.3. Total displacement at t = 14 s for cases 1, 2 and 3 (top to bottom).
Finally, the potential of the MPM to model extensive
deformation in form of landslide debris-flow is demonstrated.
MPM is seen to simulate the progression of the landslide and
generate a reasonable post-failure configuration. The proposed
simulation approach holds well for the post failure scenario
but detailed study is required to understand the modelling of
complete evolution process of landslides which may include a
better contact algorithm to model the brittle behaviour of rock
joints and precise equivalent continuum modelling of rock mass.
Moreover, the present approach can be extended with further
studies to include the effect of water and improve constitutive
modelling to take into account the behaviour of rock in dynamic
conditions.
Acknowledgements
The authors acknowledge the help of ‘‘German Academic
Exchange Service (DAAD)” and ‘‘Institute of Geotechnical Engineering (IGS), Stuttgart” for providing the financial and physical
resources required to carry out this research. We would also like
to acknowledge ‘‘Deltares, The Netherlands” for providing access
to their MPM source code, which was further developed in this
paper.
[1] Keefer DK. Landslides caused by earthquakes. Geol Soc Am Bull 1984;95
(4):406–21.
[2] Jibson RW. Methods for assessing the stability of slopes during earthquakes – a
retrospective. Eng Geol 2011;122(1):43–50.
[3] Newmark NM. Effects of earthquakes on dams and embankments.
Geotechnique 1965;15(2):129–60.
[4] Wie˛ckowski Z, Youn S-K, Yeon J-H. A particle-in-cell solution to the silo
discharging problem. Int J Numer Meth Eng 1999;45(9):1203–25.
[5] Coetzee C, Vermeer P, Basson A. The modelling of anchors using the material
point method. Int J Numer Anal Meth Geomech 2005;29(9):879–95.
[6] Andersen S, Andersen L. Modelling of landslides with the material-point
method. Comput Geosci 2010;14(1):137–47.
[7] Andersen S, Andersen L. Material-point-method analysis of collapsing slopes.
In: Proceedings of the 1st international symposium on computational
geomechanics (ComGeo I), Juan-les-Pins, France; 2009. p. 817–28.
[8] Hamad F, Vermeer P, Moormann C. Failure of a geotextile-reinforced
embankment using the material point method. In: Proceedings of the 3rd
international conference on particle-based methods-fundamentals and
applications, Stuttgart, Germany; 2013. p. 498–509.
[9] Hamad F, Vermeer P, Moormann C. Development of a coupled FEM-MPM
approach to model a 3D membrane with an application of releasing
geocontainer from barge. In: Proceedings of the 3rd international conference
on installation effects in geotechnical engineering, Rotterdam, The
Netherlands; 2013. p. 176–83.
[10] Hamad F, Stolle D, Moormann C. Material point modelling of releasing
geocontainers from a barge. J Geotext Geomembranes 2016;44(3):308–18.
[11] Hiraoka N, Oya A, Bui HH, Rajeev P, Fukagawa R. Seismic slope failure
modelling using the Mesh-free SPH method. Int J GEOMATE 2013;5:660–5.
[12] Burgess D, Sulsky D, Brackbill J. Mass matrix formulation of the FLIP particlein-cell method. J Comput Phys 1992;103(1):1–15.
[13] Jassim I, Stolle D, Vermeer P. Two-phase dynamic analysis by material point
method. Int J Numer Anal Meth Geomech 2013;37(15):2502–22.
[14] Sulsky D, Zhou S-J, Schreyer HL. Application of a particle-in-cell method to
solid mechanics. Comput Phys Commun 1995;87(1):236–52.
[15] Bardenhagen S, Brackbill J, Sulsky D. The material-point method for granular
materials. Comput Methods Appl Mech Eng 2000;187(3):529–41.
[16] Jassim I, Hamad F, Vermeer P. Dynamic material point method with
applications in geomechanics. In: Proceedings of the 2nd international
symposium on computational geomechanics (COMGEO II), CavtatDubrovnik, Croatia; 2011. p. 445–6.
[17] Kutter BL. Earthquake deformation of centrifuge model banks. J Geotech Eng
1984;110(12):1697–714.
[18] Arulanandan K, Yogachandran C, Muraleetharan K, Kutter B, Chang G.
Seismically induced flow slide on centrifuge. J Geotech Eng 1988;114
(12):1442–9.
[19] Wartman J, Seed RB, Bray JD. Shaking table modeling of seismically induced
deformations in slopes. J Geotech Geoenviron Eng 2005;131(5):610–22.
[20] Huang C-C, Lee Y-H, Liu H-P, Keefer DK, Jibson RW. Influence of surface-normal
ground acceleration on the initiation of the Jih-Feng-Erh-Shan landslide during
the 1999 Chi-Chi, Taiwan, earthquake. Bull Seismol Soc Am 2001;91(5):953–8.
[21] Hung J-J. Chi-Chi earthquake induced landslides in Taiwan. Earthq Eng Eng
Seismol 2000;2(2):25–33.
[22] Wang W-N, Chigira M, Furuya T. Geological and geomorphological precursors
of the Chiu-fen-erh-shan landslide triggered by the Chi-chi earthquake in
central Taiwan. Eng Geol 2003;69(1):1–13.
[23] Wu J-H, Lin J-S, Chen C-S. Dynamic discrete analysis of an earthquake-induced
large-scale landslide. Int J Rock Mech Min Sci 2009;46(2):397–407.
[24] Chen H. Engineering geological characteristics of Taiwan landslides. SinoGeotechnics 2000;79:59–70.
[25] Hoek E, Diederichs MS. Empirical estimation of rock mass modulus. Int J Rock
Mech Min Sci 2006;43(2):203–15.