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Numerical Modelling of Deep Mixed Columns
Harald KrennUniversity of Strathclyde
Urs VoglerUniversity of Glasgow
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
Columns under embankment fill
– Improve stability– Reduce settlements– Reduce the time
for settlements– Reduce vibrations
c
Embankment
Column
c
Deep mixed columns
2D Numerical Modelling
2D model– PLAXIS 2D v8.2 finite element
code– Axisymmetric unit cell– Radii of the unit cell
dependent on the c/c –spacing
Restriction:– Not a true geometric
representation
πcR =
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
Volume averaging technique
Columns and soil Composite system
Idea: model 3D column behaviour within 2D calculations
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 data
available
S-CLAY1S Model
q
p’ pm’ pmi’
M1
M1
α 1
CSL
CSL
p’σ’y
σ’x
σ’z
α
mim 'p)x1('p +=Intrinsic yield surface (Gens & Nova 1993)
{ } { }[ ] { } { } [ ] 0'p'p'p23M'p'p
23F md
Td
2dd
Tdd =−⎥⎦
⎤⎢⎣⎡ αα−−α−σα−σ=
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
Deep-Stabilised Columns
Drained and undrained triaxial tests– Stiffness is highly non-
linear and dependent on confining pressure
Hardening Soil model
020406080
100120140160180
0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0
Axial strain, ε 1, %
q [k
Pa]
CADC C29HS-model
E50ref Eoed
ref Eurref 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
Predicted Settlements
c/c - spacing [m]
0.8 1.0 1.2 1.4 1.6
Dis
plac
emen
ts [m
]
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
2D MCC2D S-CLAY12D S-CLAY1S3D S-CLAY1 Preliminary
Vertical Stress Distributions
1 m c/c 1.2 m c/c 1.4 m c/c
dσ'v [kN/m²]
-250-200-150-100-500
Dep
th [m
]
-12
-10
-8
-6
-4
-2
0
dσ'v [kN/m²]
-250-200-150-100-500
dσ'v [kN/m²]
-250-200-150-100-500
Soil
ColumnSoil SoilColumnColumn
2D MCC 2D S-CLAY1 2D S-CLAY1S 3D S-CLAY1 Preliminary
Principal Stress Directions
Conclusions (numerical study)
• Anisotropy and destructuration have a – minor effect on the predicted vertical
stresses– 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”
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)
Volume averaging technique
Columns and soil Composite system
- Assumptions for volume averaging technique- Determination of equivalent constitutive material matrix- Solution strategy- Example- Further work
Assumptions- Perfect bonding between in-situ soil and columns- Volume ratio of the columns is not negligible- Columns have a regular pattern
( ) soilsoilsoil εDσ && =′
( ) columncolumncolumn εDσ && =′
J.-S. Lee, 1993:Finite Element Analysis of Structured Media
Assumptions
( ) eqeqeq εDσ && =′
( ) soilsoilsoil εDσ && =′
( ) columncolumncolumn εDσ && =′
- 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
Equilibrium and KinematicsLocal equilibrium conditions:
columnyz
soilyz
eqyz
columnxy
soilxy
eqxy
columnz
soilz
eqz
columnx
soilx
eqx
τ=τ=τ
τ=τ=τ
σ=σ=σ
σ=σ=σ
&&&
&&&
&&&
&&&x
yz
AcolumnAsoil
Kinematic conditions (bonding):
columnzx
soilzx
eqzx
columny
soily
eqy
γ=γ=γ
ε=ε=ε
&&&
&&&
Averaging Rules
columncolumn
soilsoil
eq
columncolumn
soilsoil
eq
εεε
σσσ&&&
&&&
µ+µ=
µ+µ=
Volume fraction of soil / column:
AA;
AA column
columnsoil
soil =µ=µ
Determination of Deq
By combining the constitutive equations with the kinematic and equilibrium conditions:
column1columncolumn
soil1
soilsoil
eq SDSDD µ+µ=
( )columnsoilsoil
column,soil1 ,,f DDS µ=
With the material matrixes S1soil and S1
column :
Solution Strategy-Calculate equivalent material matrix
Deeq or Dep
eq
-Calculate strain increment
δBεPKδ&&
&&
=
= −
eq
1
eqcolumncolumneqsoilsoil εSεεSε &&&& 11 ==
-Calculate stress increments
( ) ( ) columncolumncolumnsoilsoilsoil εDσεDσ &&&& =′
=′
-Trial stresses
( ) ( ) ( ) ( ) ( ) ( )′+′
=′′
+′
=′
−−column
ncolumn
ncolumnsoil
nsoil
nsoil σσσσσσ && 11
Solution Strategy-Check yielding
( ) ( ) 00 ≤≤ columncolumnsoilsoil FF σσ-Return mapping soil/column-Adjust stress components if necessary
columnyz
soilyz
columnyz
columnxy
soilxy
columnxy
columnz
soilz
columnz
columnx
soilx
columnx
dddd
ττττττ
σσσσσσ
−=−=
−=−=
( ) ( ) ( )′+′
=′
−column
ncolumn
ncolumn d σσσ 1
-Recheck column yielding
( ) 0≤columncolumnF σ-Calculate stress in equivalent material
columncolumn
soilsoil
eq σσσ &&& µµ +=
First ExampleSingle integration point program – triaxial loadingSoil: Mohr-Coulomb model, linear elastic – ideal plasticColumns: 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
σ2
ε 2
equivalentsoilcolumnSig_soil(1)Sig_column(1)
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 into2D finite element code (Plaxis)
comparison of volume averaging method with full 3D simulations
Thank you very much for your attention