Hardening Soil Model (HS)

Constitutive Model in FE Analysis

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Constitutive Model - PlasticityYield SurfaceFlow RuleHardening RuleExpansion 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

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Constitutive Model - PlasticityM-C model has a fixed yield surface, a yield surface fully defined by model parameters and not affected by strainVariation of yield surface

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2k1k

Hardening Soil Model (HS)

strain or displacement)

(stress) Real soil response

Idealised soil model – MC model

Hardening Soil Model

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Hardening Soil Model (HS)

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Yield Surface of HS Model

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p’

qMC Model Failure Line

With increasing hardening parameter

Shear Hardening

Yield Surface

7

q

p

3

2

1

q

p

3

2

1

q

p

3

2

1

Shear hardening

Compression hardening

2 yield surface

Yield Surface Cap in HS Model

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p’

q

cc cot

Elastic Zone

Yield Surface Cap in HS Model

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Dilatancy Cut-off in HS Model

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Dilatancy cut-off on emax

1

v

HS Model

MC Model

2 sin1 - sin

Strain-Hardening Types

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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

zTriaxial Test

y

Oedometer Test

Features of HS ModelAllows for non-linearity of the stress-strain curve (Hyperbolic)Differentiate between first loading and unloadingStiffness depends on stressesYield surface expands (harden) in the space due to plastic strainThe yield surface has a cap to allow for hardening due to volumetric strain

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Input Parameters of HS ModelStress-dependent stiffness according to a power law [input parameter: m]

Plastic straining due to primary deviatoric loading [input parameter: ( )]

Plastic straining due to primary compression loading [input parameter ( )]

Elastic unloading/reloading [input parameter: ( , )]

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

refE50

refoedE

refurE

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From triaxial test

From oedometer test

Unloading/reloading test

Stress Dependent E50

qaqf

qf/2

150E

AsymptoteFailure line

1urE

Axial strain

Deviator stress

m

refref

pccEE

sincos'sin'cos' 3

5050

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y

x= z

z

Triaxial Test

’3 = x = z

f

fa

f

Rq

q

cq

and

sin1

sin2)'cot'( 3

refE50 When ’ = pref = 100 kPa

Rf = 0.9 qf = 0.9qa

Stress Dependent Eur

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qaqf

qf/2

150E

AsymptoteFailure line

1urE

Axial strain

Deviator stress

m

refrefurur pc

cEEsincos'sin'cos' 3

y

x= z

z

Triaxial Test

Stress Dependent Eoed

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v’

Axial strain

pref

1

refoedE

m

refrefoedoed p

EE 1

Oedometer Test

v

1 = v

Application of HS Model

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When shearing is dominant (more than compression)

When the problem involves substantial unloading

When the stiffness varies with stress

Selection of Parameters in HS Model

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: Secant modulus in standard drained triaxial test

: Tangent stiffness for primary oedometer loading

: unloading/reloading modulus ( 3 )

ur : Poisson’s ratio for unloading/reloading (default ur = 0.2 )

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

: K0-value for normally consolidation (default = 1-sin )

m 1 for clays and m 0.5 for sands

refE50refoedErefurE ref

urE refE50

NCK0NCK0

Hardening Soil ModelAdvantages

Better nonlinear formulation of soil behaviour in general (both soft soil and harder soil types)Distinction between primary loading and unloadingMemory of preconsolidation stressesDifferent stiffness for different stress paths based on standard testsWell suited for unloading situations with simultaneous deviatoric loading

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Hardening Soil ModelLimitation

No peak strength and softeningNo secondary compressionNo anisotropyE50/Eoed > 2 difficult to inputStiffness at small strain is underestimated

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Hardening Soil ModelThe 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 modelA total stress analysis maybe performed with both undrained strength (Cu and friction angle=0). However, no stress dependent stiffness and no compression hardening.

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Remarks* on Finite Element Analysis

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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