2518
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
of sands to silty sands, with a high degree of heterogeneity,
characterized by USCS as SP, SW, SM, SM-SP, SM-SW. In
some cases they appear as clayey sand (SC) to sandy clay (CL),
whereas in other cases, they turn out to be gravel layers, such
as: GP, GW, GM, GP-GM. According to the almost 200 SPTs
performed, the mean value of blows was calculated about 23,
with a standard deviation of +11. The whole area, where the
bridge is founded, has been initially divided into three sub-
regions represented each by a different geotechnical design
section (ITSAK & Gazetas 2003), and finally a design
geotechnical section has been attributed to each bridge pier
(Edafomichaniki 2007) used for dynamic analyses purposes.
From various simplified design geotechnical sections, each
per bridge pier, it has been chosen one, for the needs of the
present project, corresponding to a precise pier of the bridge, as
being the most representative of the area, but not the most
conservative one. The soil profile used in the present work, can
be described as follows:
Layer S
1A
(0 το 2m): loose to medium dense gravels with sand
and sand or silty sand with local presence of gravels (GP,
SW-SM, SP): N
SPT
22, γ=20.5kN/m
3
, φ’=36
0
, c’=3kPa,
E
s
=10MPa, ν=0.33
Layer S
1B
(2 to 5m): medium dense gravels with sand and sand
to silty sand with local presence of gravels (GP, SW-SM, SP):
N
SPT
23, γ=20.5kN/m
3
, φ’=37
0
, c’=5kPa, E
s
=12MPa, ν=0.32
Layer S
2A
(5 to 12m): medium dense gravels with silt and sand
to silty sand with presence of gravels (GM-GP, SP-SM, SM):
N
SPT
25, γ=21.0kN/m
3
, φ’=39
0
, c’=6kPa, E
s
=16MPa, ν=0.31
Layer S
2B
(12 to 19m): medium dense silty gravels, silty sand
with presence of gravels to silty sand (GM-GP, SP-SM, SM):
N
SPT
28, γ=21.0kN/m
3
, φ’=40
0
, c’=8kPa, E
s
=20MPa, ν=0.30
Layer S
3A
(19 to 23m) and layer S
3B
(23 to 35m): medium dense
clayey sand-gravels mixture to sandy clay with gravels, or silty
sand-gravels mixture (GC-GM, SM, CL): N
SPT
26,
γ=21.2kN/m
3
, φ’=37
0
, c’=12kPa, E
s
=15MPa, ν=0.31.
From 35 to almost 48m the weathering zone of the gneissic
bedrock or highly weathered gneiss is met.
4. METHODOLOGICAL APPROACH
The analysis was carried out with FLAC 3D numerical code of
finite differences.
4.1 Modeling Procedure
By considering the construction of a stone column in the above
soil profile, simulation of two distinct stages of the construction
of a stone column is attempted using a three-dimensional (3D)
model which represents the stone column and the surrounding
soil. Simulation of soil materials is realized by a 3-diamensional
polyhedral grid with use of the finite difference method. The
mathematical model adopted, incorporates geometry and
boundary conditions of the problem, the profile of soil layers,
physical, deformational and mechanical properties, constitutive
laws for the geomaterials, as well as, initial conditions of
stresses and deformations of the subsoil stratums of the area
under study.
Geometry of the problem is simplified to axial symmetry. A
vertical plane through stone column axis is a plane of symmetry
for the analysis. Model grid is shown in figure (1). Coordinate
axes are located with origin at the base of the grid, whereas y-
axis is oriented along vertical column axis and upward. The
initial grid is assigned by 5.0m and 50 units in x-direction, by
5.0m and 50 units in z-direction and by 28.0m and 56 units of in
y-direction. A Mohr-Coulomb constitutive model elastoplastic
behavior is assigned to all zones of soil surrounding stone
column, whilst linear elastic one is assigned to stone column
backfilling crushed material. Boundary conditions consist of
roller boundaries along the external grid sides of column axis
and a fixed base. Equilibrium conditions for initial stresses are
based on earth pressure coefficient at rest K
o
=v/(1-ν), where ν:
Poisson’s ratio.
The modeling sequence consists of the following stages:
Stage I
: Initial stresses
Establish equilibrium conditions to initialize stresses
Stage II
: Excavation
Stone column excavation at full penetration depth was decided
to be numerically simulated in one and only stage, since in
reality, excavation was accomplished in about 30 min for a
typical stone column of the project, and also, because no steps
of excavation during its construction, could be discretized.
Stage III : Stone Column Construction
In reality, construction of cylindrical stone columns of the
project with a theoretical diameter D=0.8m and a length
L=23.0m, is realized by ascending steps of 0.5m; at each step,
the crushed geomaterials are driven through the top of the stone
column downwards (top feed method), and then, the vibrational
torpedo is sinked into the excavated cyclic area, reaches the top
of the crushed material and starts vibrating harmonically at a
frequence of 30Hz, in order to achieve an harmonically applied
normal stress of 30 to 35MPa. However, our choice of
computational ascending steps to simulate stone column
construction was of 1.0m, since an initial comparative study
between 0.5m and 1.0m ascending steps, revealed no significant
differences, whereas computational time difference was
important. Therefore, Stage III is sub-divided in two distinct
calculation steps, ever after named as “Sub-stage IIIa and IIIb”
Sub-stage IIIa : Simulation of Vibration and Compaction
Based on the construction procedure concerning the one stage
of excavation of the stone column to be realized, which affects
significantly the mechanical properties of the surrounding zone,
a weak zone boundary has been created, by reducing φ’ & c’, in
a distance of 0.60m surrounding column lateral sides, in order to
simulate relaxation due to excavation. The width of the weak
zone, the reduced values of the mechanical parameters and the
elastic deformation modulus, resulted from a “trial and error”
back calculating procedure, based on the quantity of the crushed
material measured in situ, during the construction of a stone
column of the project. Namely, we tried to match the increase of
the “as built” diameter of the examined stone column, in
agreement with the quantity of the crushed material used for the
construction of the stone column, by adjusting the values of
mechanical and deformational parameters of the disturbed zone.
Vertical normal stress, harmonically applied on top of filling
crushed material in order to compact the crushed fill material,
per numerical ascending step of the stone column construction,
is transferred as a lateral pressure “p” to simulate subjected
compressive lateral loads of material due to gravel compaction,
in terms of an “equivalent static” lateral (radial) pressure, as
explained in the following paragraph.
Sub-stage IIIb : Simulation of Crushed Stone Material filling
This sub-stage simulates filling of the stone column crushed
material taking under consideration the preceding compaction
procedure. In order to maintain the shape of the “deformed
diameter” per constructed step of the stone column, crushed fill
material, considered as a linear elastic one, it has been attributed
a very high modulus of elasticity, avoiding thus a rebound of
the plastic lateral displacements obtained from sub-stage IIIa.
4.2 Assessment of equivalent lateral static loading
It is widely known in Mechanics, that a dynamic system
responds to an harmonic external loading, according to the
following equation:
2 2
2
1
1
1 /
4
st
u f
u
f f
(1)
where, u(f): dynamic displacement, u
st
: equivalent static
displacement (=P/K), ω: frequency of the input motion, ω
1
:
