689
Technical Committee 103 /
Comité technique 103
the literature (Bui et al. 2008-2012), and reader can refer to
these references for further details on the validation process.
2.2
Rigid body motion of retaining wall blocks
The segmental retaining wall simulated in this paper is
comprised of individual rectangular blocks; each is assumed as
a rigid body and has complete degrees-of-freedom. The motion
of the block can be determined by specifying the motion of the
central mass and the rotation about its mass central. The
equation of motion of the central mass is given as follows,
F
dt
dVM
(14)
where
M
is the central mass,
V
is the velocity vector of the
central mass,
F
is total force vector acting on the body. The
equation of rotation about the central mass is,
T
dt
dI
(15)
where
I
is the inertial moment,
is the angular velocity
which is perpendicular to the plane of the motion, and
T
is the
total torque about the central mass.
In the computation, the rectangular block is represented by
the set of boundary particles that are equi-spaced around the
boundary. Denoting the force vector acting on each boundary
particle
i
located on the moving block is
f
i
, Equations (14) and
(15) can be rewritten, respectively, as follows,
i
i
f
dt
dVM
(16)
k
i
i
f
R r
dt
dI
)
(
(17)
where
r
i
and
R
are vector coordinates of boundary particle
and central mass, respectively. The rigid body boundary
particles move as a part of the rigid body, thus the change on
position of boundary particle
i
is given by,
)
(
R r
V
dt
dr
i
i
(18)
The force
f
i
acting on a boundary particle on the rigid body is
due to the surrounding soil particles or boundary particles that
belong to different rigid bodies. This force can be calculated
using a suitable contact model which will be described in the
next section.
2.3
Contact force model
In this paper, a soft contact model based on a concept of the
spring and dash-pot system is proposed to model the interaction
between soil and retaining wall blocks and between blocks.
Accordingly, the radial force acting between two particles can
be calculated using the following equation,
ai
i
a
ai
i
a
n
ak n
n ai
n
i a
d h h
d h h vc
K
f
2 )
(
0
2 )
(
2/3
(19)
where
K
is the radial stiffness;
n
is the allowable
overlapping distance between two particles;
c
n
is the radial
damping coefficient;
v
n
is the relative radial velocity vector
between particle
a
and particle
i
;
h
a
and
h
i
are the initial
distance (so-called smoothing length in SPH) between soil
particles and between boundary particles, respectively; and
d
ai
is
the distance between two particles. The stiffness, overlapping
distance and damping coefficient can be calculated using the
following relationships,
3/
eff
eff
ai
h E K
(20)
2/)
(
i
a
ai
n
h h d
(21)
ai
ai
n
Km c
2
(22)
where
E
eq
and
h
eq
are equivalent Young’s modulus and
smoothing length, respectively. The contact force in the shear
direction which is perpendicular to the radial direction can be
calculated in the same manner,
ai
i
a
ai
i
a
s
ai
s
s ai
s
i a
d h h
d h h vc
k
f
2 )
(
0
2 )
(
(23)
where
k
ai
is the shear stiffness;
s
is the relative displacement
between the two particles in the shear direction;
c
s
is the shear
damping coefficient;
v
s
is the relative shear velocity vector
between particle
a
and particle
i.
These unknown variables can
be calculated using the following relationships,
n eq
eq
ai
h G k
4
(24)
dt v
s
ai
s
(25)
ai
ai
n
km c
2
(26)
where
G
eq
is the equivalent shear modulus. The current shear
force must satisfy Coulomb’s friction law which implies that the
maximum shear force must not exceed the maximum resisting
force,
n
i a
s
s
s
i a
f
f
(27)
Finally, these forces are converted to the conventional
coordinate system and added to Equations (2), (16) and (17) to
progress the motion of soil and rigid bodies.
3 OUTLINE OF MODEL TEST
Two-dimensional experiments of retaining wall collapse were
conducted to validate the SPH numerical results. Figure 1
shows a schematic diagram of the two-dimensional
experimental setup and Figure 2 shows the initial setup
condition of the model ground and retaining wall blocks in the
experiment. The size of the model ground is 15cm in height and
50cm in width at the base. Aluminum rods of 5cm in length,
having diameters of 1.5 to 3mm and mixed with the ratio 3:2 in
weights, are used as the model ground. The total unit weight of
the model ground is 23kN/m
3
. The retaining block is 3.2cm in
width, 2.5cm in height, and 5cm in length, which is
manufactured from aluminum (Young’s modulus of 69GPa and
unit weight of 26.5kN/m
3
). In the experiment, the segmental
retaining wall system was constructed by successively placing
one block on the top of the other with an overlapping of 1.9cm,
followed by filling the model ground at each layer. To visualize
the failure pattern of the model ground, square grids
(2.5
2.5cm) were drawn on the specimen. The experiments
were initiated by quickly removing the stopper stand and digital
photos were taken to record the failure process as well as the
final configuration of the retaining wall system after collapse.
