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