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Constitutive Model in FE Analysis Hardening Soil Model (HS)

1

Constitutive Model - Plasticity Yield Surface Flow Rule Hardening Rule Expansion or shrinkage of the loading or yield surface. Predicts change in the yield surface due to plastic strains. Link changes in stresses and strains to the size of the Loading Surface

2

Constitutive Model - Plasticity M-C model has a fixed yield surface, a yield surface fully defined by model parameters and not affected by strain Variation of yield surface

k2 k1

3

Hardening Soil Model (HS) Hardening Soil Model

(stress)

Real soil response

Idealised soil model – MC model

strain or displacement) 4

Hardening Soil Model (HS)

5

Yield Surface of HS Model Shear Hardening

q

MC Model Failure Line

With increasing hardening parameter

p’ 6

Yield Surface q

q

q

p

p

1

1

3

2

7

1

3 2

Shear hardening

p

3 2

Compression hardening

2 yield surface

Yield Surface Cap in HS Model q

Elastic Zone

c c cot

8

p’

Yield Surface Cap in HS Model

9

Dilatancy Cut-off in HS Model v

MC Model Dilatancy cut-off on emax 1 - sin 2 sin

HS Model 1

10

Strain-Hardening Types Strain-hardening has two types: Shear hardening: plastic strain is primarily due to deviatoric loading Compression hardening: plastic strain is primarily due to compression (oedometer) and isotropic loading

y x= z

z

Triaxial Test y

Oedometer Test 11

Features of HS Model Allows for non-linearity of the stress-strain curve (Hyperbolic) Differentiate between first loading and unloading Stiffness depends on stresses Yield surface expands (harden) in the space due to plastic strain The yield surface has a cap to allow for hardening due to volumetric strain

12

Input Parameters of HS Model Stress-dependent stiffness according to a power law [input parameter: m] Plastic straining due ref to primary deviatoric loading [input parameter: ( E50 )] From triaxial test Plastic straining due to primary compression loading ref [input parameter ( Eoed )] From oedometer test Elastic unloading/reloading [input parameter: ( E ref , )]Unloading/reloading test ur

Failure according to the Mohr-Coulomb model [input parameter: (c, and )] 13

y

Stress Dependent E50 E50

ref E50

Deviator stress

c' cos c ' cos ref E50

qa qf

When

E50 1

Triaxial Test

qf

’3 =

(c ' cot

and qa

Eur

x

=

z

2 sin '3 ) 1 sin

qf Rf

1 Axial strain

z

z

’ = pref = 100 kPa

Asymptote Failure line

qf/2

14

'3 sin p ref sin

m

x=

Rf = 0.9

qf = 0.9qa

Stress Dependent Eur

y

Deviator stress

x=

qa qf

Asymptote Failure line

z

z

Triaxial Test

qf/2

Eur

E50 1

1

Eur Axial strain

15

ref ur

E

c' cos c' cos

'3 sin p ref sin

m

Stress Dependent Eoed m

Eoed

ref Eoed

p

1

1 ref

=

v v

v’

pref

Oedometer Test

ref Eoed

1 Axial strain 16

Application of HS Model When shearing is dominant (more than compression) When the problem involves substantial unloading When the stiffness varies with stress

17

Selection of Parameters in HS Model ref : Secant modulus in standard drained triaxial test E50 ref Eoed

: Tangent stiffness for primary oedometer loading

ref Eurref : unloading/reloading modulus ( Eur ur

ref 3 E50 )

: Poisson’s ratio for unloading/reloading (default

ur

= 0.2 )

pref : Reference stress for stiffness (default pref = 100 kPa)

K 0NC: K0-value for normally consolidation (default K 0NC= 1-sin ) m

18

1 for clays and m

0.5 for sands

Hardening Soil Model Advantages Better nonlinear formulation of soil behaviour in general (both soft soil and harder soil types) Distinction between primary loading and unloading Memory of preconsolidation stresses Different stiffness for different stress paths based on standard tests Well suited for unloading situations with simultaneous deviatoric loading 19

Hardening Soil Model Limitation No peak strength and softening No secondary compression No anisotropy E50/Eoed > 2 difficult to input Stiffness at small strain is underestimated

20

Hardening Soil Model The hardening soil model is completely defined in effective stresses and therefore need both effective strength parameters and effective stiffness parameters in order to take advantages of the model A total stress analysis maybe performed with both undrained strength (Cu and friction angle=0). However, no stress dependent stiffness and no compression hardening.

21

Remarks* on Finite Element Analysis The ability of the Finite Element Method to accurately reflect field conditions essentially depends on the ability of the constitutive models to represent real soil behaviour and the ability of the geotechnical engineer to assign appropriate boundary conditions to the various stages of construction. Advantages over the conventional methods are the effects of time on the development of pore water pressures can be simulated by including coupled consolidation/swelling, dynamic behaviour can be accounted for, and – perhaps most importantly no postulated failure mechanism or mode of behaviour of the problem is required, as these are predicted by the analysis itself. *Potts, D. M. (2003). Geotechnique 53, No.6, 535-573

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1

Constitutive Model - Plasticity Yield Surface Flow Rule Hardening Rule Expansion or shrinkage of the loading or yield surface. Predicts change in the yield surface due to plastic strains. Link changes in stresses and strains to the size of the Loading Surface

2

Constitutive Model - Plasticity M-C model has a fixed yield surface, a yield surface fully defined by model parameters and not affected by strain Variation of yield surface

k2 k1

3

Hardening Soil Model (HS) Hardening Soil Model

(stress)

Real soil response

Idealised soil model – MC model

strain or displacement) 4

Hardening Soil Model (HS)

5

Yield Surface of HS Model Shear Hardening

q

MC Model Failure Line

With increasing hardening parameter

p’ 6

Yield Surface q

q

q

p

p

1

1

3

2

7

1

3 2

Shear hardening

p

3 2

Compression hardening

2 yield surface

Yield Surface Cap in HS Model q

Elastic Zone

c c cot

8

p’

Yield Surface Cap in HS Model

9

Dilatancy Cut-off in HS Model v

MC Model Dilatancy cut-off on emax 1 - sin 2 sin

HS Model 1

10

Strain-Hardening Types Strain-hardening has two types: Shear hardening: plastic strain is primarily due to deviatoric loading Compression hardening: plastic strain is primarily due to compression (oedometer) and isotropic loading

y x= z

z

Triaxial Test y

Oedometer Test 11

Features of HS Model Allows for non-linearity of the stress-strain curve (Hyperbolic) Differentiate between first loading and unloading Stiffness depends on stresses Yield surface expands (harden) in the space due to plastic strain The yield surface has a cap to allow for hardening due to volumetric strain

12

Input Parameters of HS Model Stress-dependent stiffness according to a power law [input parameter: m] Plastic straining due ref to primary deviatoric loading [input parameter: ( E50 )] From triaxial test Plastic straining due to primary compression loading ref [input parameter ( Eoed )] From oedometer test Elastic unloading/reloading [input parameter: ( E ref , )]Unloading/reloading test ur

Failure according to the Mohr-Coulomb model [input parameter: (c, and )] 13

y

Stress Dependent E50 E50

ref E50

Deviator stress

c' cos c ' cos ref E50

qa qf

When

E50 1

Triaxial Test

qf

’3 =

(c ' cot

and qa

Eur

x

=

z

2 sin '3 ) 1 sin

qf Rf

1 Axial strain

z

z

’ = pref = 100 kPa

Asymptote Failure line

qf/2

14

'3 sin p ref sin

m

x=

Rf = 0.9

qf = 0.9qa

Stress Dependent Eur

y

Deviator stress

x=

qa qf

Asymptote Failure line

z

z

Triaxial Test

qf/2

Eur

E50 1

1

Eur Axial strain

15

ref ur

E

c' cos c' cos

'3 sin p ref sin

m

Stress Dependent Eoed m

Eoed

ref Eoed

p

1

1 ref

=

v v

v’

pref

Oedometer Test

ref Eoed

1 Axial strain 16

Application of HS Model When shearing is dominant (more than compression) When the problem involves substantial unloading When the stiffness varies with stress

17

Selection of Parameters in HS Model ref : Secant modulus in standard drained triaxial test E50 ref Eoed

: Tangent stiffness for primary oedometer loading

ref Eurref : unloading/reloading modulus ( Eur ur

ref 3 E50 )

: Poisson’s ratio for unloading/reloading (default

ur

= 0.2 )

pref : Reference stress for stiffness (default pref = 100 kPa)

K 0NC: K0-value for normally consolidation (default K 0NC= 1-sin ) m

18

1 for clays and m

0.5 for sands

Hardening Soil Model Advantages Better nonlinear formulation of soil behaviour in general (both soft soil and harder soil types) Distinction between primary loading and unloading Memory of preconsolidation stresses Different stiffness for different stress paths based on standard tests Well suited for unloading situations with simultaneous deviatoric loading 19

Hardening Soil Model Limitation No peak strength and softening No secondary compression No anisotropy E50/Eoed > 2 difficult to input Stiffness at small strain is underestimated

20

Hardening Soil Model The hardening soil model is completely defined in effective stresses and therefore need both effective strength parameters and effective stiffness parameters in order to take advantages of the model A total stress analysis maybe performed with both undrained strength (Cu and friction angle=0). However, no stress dependent stiffness and no compression hardening.

21

Remarks* on Finite Element Analysis The ability of the Finite Element Method to accurately reflect field conditions essentially depends on the ability of the constitutive models to represent real soil behaviour and the ability of the geotechnical engineer to assign appropriate boundary conditions to the various stages of construction. Advantages over the conventional methods are the effects of time on the development of pore water pressures can be simulated by including coupled consolidation/swelling, dynamic behaviour can be accounted for, and – perhaps most importantly no postulated failure mechanism or mode of behaviour of the problem is required, as these are predicted by the analysis itself. *Potts, D. M. (2003). Geotechnique 53, No.6, 535-573

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