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Numerical Modelling ofDeep Mixed Columns
Harald Krenn
University of Strathclyde
Urs Vogler
University of Glasgow
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Outline of Presentation
- Deep mixed columns under embankment fill
- Numerical modelling deep mixed columns
- Results of numerical study
- Enhanced numerical 2D model – volume averaging technique
- Future work
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Columns under embankment fil l
– Improve stability – Reduce settlements
– Reduce the time
for settlements
– Reduce vibrations
c
Embankment
Column
c
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Deep mixed columns
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2D Numerical Modelling
2D model – PLAXIS 2D v8.2 finite element
code
– Axisymmetric unit cell
– Radii of the unit celldependent on the c/c –
spacing
Restriction: – Not a true geometric
representation
π
c R =
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3D Numerical Modelling
3D model
– PLAXIS 3D beta version – True unit cell
– All calculation phases
fully drained
Restrictions: – Idealisation of columns in
square/triangular grid
under the centreline of an
embankment
Column
Soil
Soil
Column
Embankment
fill
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Volume averaging technique
Columns and soil Composite system
Idea: model 3D column behaviour within 2D calculations
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Idealised Soil Profile
Vanttila clay (Finland)
– Dry crust (0-1m depth)• over-consolidated (POP
30kPa)
• Limited lab data available
– WT at 1 m depth – Soft Vanttila clay (1-12 m
depth)
• Lightly over-consolidated(POP 10 kPa)
• Plenty of lab dataavailable
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S-CLAY1S Model
q
p’ pm’ pmi’
M
1
M1
α1
CSL
CSL
p’σ’y
σ’x
σ’z
α
mim ' p)x1(' p +=Intrinsic yield surface (Gens & Nova 1993)
{ } { }[ ] { } { } [ ] 0' p' p' p2
3
M' p' p2
3
F md
T
d
2
d d
T
d d =−⎥⎦
⎤
⎢⎣
⎡
αα−−α−σα−σ=
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Soil Parameters
Soil Depth e0 POP [kPa] α x
Dry crust 0 - 1 1.7 30 0.63 90
Vanttila clay 1 - 11 3.2 10 0.46 20
Soil γ[kN/m3]
κ ν’ λ M kx= ky
[m/day]
Dry crust 13.8 0.029 0.2 0.25 1.6 -
Vanttila clay 13.8 0.032 0.2 0.88 1.2 6.9E-5
Soil β µ λi a b
Dry crust 1.07 15 0.07 11 0.2
Vanttila clay 0.76 40 0.27 11 0.2
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Deep-Stabilised Columns
Drained and undrainedtriaxial tests – Stiffness is highly non-
linear and dependent onconfining pressure
Hardening Soil model
020
40
60
80
100
120
140
160
180
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
Axial strain, 1, %
q [
k P a ]
CADC C29
HS-model
E50ref Eoed
ref Eur ref ν’ur M c’ ϕ’ γ’
[kPa] [kPa] [kPa] - - kPa [ ° ] [kN/m3]
12000 12000 27000 0.35 0.8 27 36 15
Reference stress for stiffness, pref =100kPa
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Predicted Settlements
c/c - spacing [m]
0.8 1.0 1.2 1.4 1.6
D i s p l a c e m e
n t s [ m ]
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
2D MCC
2D S-CLAY1
2D S-CLAY1S
3D S-CLAY1 Preliminary
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Vertical Stress Distributions
1 m c/c 1.2 m c/c 1.4 m c/c
d 'v [kN/m²]
-250-200-150-100-500
D e p t h [ m ]
-12
-10
-8
-6
-4
-2
0
d 'v [kN/m²]
-250-200-150-100-500
d 'v [kN/m²]
-250-200-150-100-500
Soil
ColumnSoil SoilColumn
Column
2D MCC
2D S-CLAY12D S-CLAY1S
3D S-CLAY1 Preliminary
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Principal Stress Directions
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Conclusions (numerical study)
• Anisotropy and destructuration have a
– minor effect on the predicted verticalstresses
– greater effect on the predicted settlements
• Hardening soil model gives a realistic stress-strain relationship for deep-stabilized columns
2D - unit cell
3D model versus 2D - unit cell• Preliminary simulation “less settlements”
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Volume averaging technique
Columns and soil Composite system
Aim: model 3D column behaviour within 2D calculations- Obtain overall response of system
- Save computational costs
- Feed model with known behaviour of constituents
(soil and column)
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Volume averaging technique
Columns and soil Composite system
- Assumptions for volume averaging technique- Determination of equivalent constitutive material matrix
- Solution strategy
- Example
- Further work
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Assumptions- Perfect bonding between in-situ soil and columns
- Volume ratio of the columns is not negligible
- Columns have a regular pattern
( ) soil soil soil εDσ && =
′
( ) columncolumncolumnεDσ && =′
J.-S. Lee, 1993:
Finite Element Analysis of Structured Media
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Assumptions
( ) eqeqeqεDσ && =
′
( ) soil soil soil εDσ && =
′
( ) columncolumncolumnεDσ && =′
- Equilibrium and kinematics satisfied between constituents
- Analysis with equivalent stress/strain relationship
- Separate yield function for soil and column
J.-S. Lee, 1993:
Finite Element Analysis of Structured Media
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Equilibrium and KinematicsLocal equilibrium conditions:
column
yz
soil
yz
eq
yz
column
xy
soil
xy
eq
xy
column
z
soil
z
eq
z
columnxsoilxeq x
τ=τ=τ
τ=τ=τ
σ=σ=σσ=σ=σ
&&&
&&&
&&&
&&& xy
z
Acolumn A soil
Kinematic conditions (bonding):
column
zx
soil
zx
eq
zx
column
y
soil
y
eq
y
γ=γ=γ
ε=ε=ε
&&&
&&&
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Averaging Rules
column
column
soil
soil
eq
column
column
soil
soil
eq
εεε
σσσ
&&&
&&&
µ+µ=
µ+µ=
Volume fraction of soil / column:
A
A
;A
A columncolumn
soilsoil =µ=µ
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Determination of D
eq
By combining the constitutive equations
with the kinematic and equilibrium conditions:
column
1columncolumn
soil
1
soil
soil
eq SDSDD µ+µ=
columnsoil
soil
column,soil
1 ,,f DDS µ=
With the material matrixes S1soil and S1
column :
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Solution Strategy-Calculate equivalent material matrix
Deeq or Dep
eq
-Calculate strain increment
δBε
PK δ
&&
&&
=
= −
eq
1
eqcolumncolumneq soil soil
εSεεSε
&&&&11 ==
-Calculate stress increments
( ) ( ) columncolumncolumn soil soil soil
εD
σεD
σ &&&&
=
′
=
′
-Trial stresses
( ) ( ) ( ) ( ) ( ) ( )
′+
′=
′′+
′=
′−−
columnn
columnn
column soil n
soil n
soil σσσσσσ && 11
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Solution Strategy-Check yielding
00 ≤≤ columncolumn soil soil F F σσ
-Return mapping soil/column-Adjust stress components if necessary
column
yz
soil
yz
column
yz
column
xy
soil
xy
column
xy
column
z
soil
z
column
z
column
x
soil
x
column
x
d d
d d
τ τ τ τ τ τ
σ σ σ σ σ σ
−=−=
−=−=
( ) ( ) ( )′+′
=′
− column
ncolumn
ncolumn d σσσ 1
-Recheck column yielding
0≤columncolumn F σ
-Calculate stress in equivalent material
column
column
soil
soil
eq
σσσ &&&
µ µ +=
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First ExampleSingle integration point program – triaxial loading
Soil: Mohr-Coulomb model, linear elastic – ideal plastic
Columns: Linear elastic columns with 50% area ratioEcolumn = 2 Esoil
-0.018
-0.016
-0.014
-0.012
-0.01
-0.008
-0.006
-0.004
-0.002
0
-400-300-200-1000 2
2
equivalent
soil
column
Sig_soil(1)
Sig_column(1)
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Future Work
full 3D simulations of embankments on deep mixed columns
use of advanced constitutive models for soil and columns
for homogenisation technique (S-CLAY1S, …)
implementation of averaging technique as constitutive model into
2D finite element code (Plaxis)
comparison of volume averaging method with full 3D simulations
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Thank you very much for your attention