Rak-50_3149_l._l12-_hardening_soil_model.pdf

Hardening Soil Model

Prof. Minna Karstunen University of Strathclyde With thanks to Prof. Pieter Vermeer, Prof. Thomas Benz, Prof. Steinar Nordal & Dr Ronald Brinkgreve

Introduction The aim is to discuss the formulations for the double-hardening models in PLAXIS: • Hardening Soil Model • HSsmall (small strain stiffness model)

Introduction The topics to be covered include: • Hardening Soil model in the context of constitutive models • Hardening Soil model formulation • Limitations of HS model • Small strain stiffness • HSsmall model formulation • Limitations of HSsmall model • Parameter selection for HS models

Mohr Coulomb Model τ

linear elastic

σ'0

γ

τ

τ G 1

τmax

elastic

γ σ‘ • Associated flow (φ’=ψ’) – simple, numerically efficient, but too much dilation • Non-associated flow (φ’ ≠ ψ’) – non-symmetric stiffness matrix not numerically efficient, still far from real soil behaviour

MC vs Real Soil Behaviour

Stress path in oedometer loading-unloading

Critical State Models q, τ

τ

yield surface elastic

γ Stress-Strain Curve

Modified Cam Clay & Soft Soil Model: • Based on volumetric hardening: works in the “wet” side when volumetric strains are dominating • Not representative in the “dry” side (e.g. excavations)

p′′, σ‘

Hardening Soil Model • Volumetric hardening is complemented by deviatoric hardening • Deviatoric hardening surface evolves with deviatoric strains until failure is reached τ

γ

Model characteristics: • Hyperbolic stress-strain relationship in axial compression • Plastic strain in mobilizing friction (shear hardening) • Plastic strain in primary compression (volumetric hardening) • Stress-dependent stiffness according to a power law • Elastic unloading/ reloading compared to virgin loading • Memory of pre-consolidation stress • Mohr-Coulomb (MC) failure criterion • Dilatancy below MC line • Small strain stiffness (HS-small only)

Hyperbolic stress-strain relationship Duncan-Chang or hyperbolic model in (tri)axial loading:

For q < qf:

where

and

ε1 = ε 50 qf = qa =

q qa − q

2sin ϕ (σ 3' + c cot ϕ ) 1 − sin ϕ qf Rf

≥ qf

Rf = ‘failure ratio’ (standard value: 0.9)

q q q

asymptote a

f

1 E

50

½ qf

E

ur

1 ε1

The hyperbolic model with Mohr-Coulomb failure criterion constitutes the basis for the HS and HS-small models. In contrast to the Duncan-Chang model, the HS models are elasto-plastic models.

Shear hardening s Yield function (cone): f =

qa q 2q − − γ ps E50 qa − q Eur

where γps is a state parameter that is tracking the opening of the cone:

q

Elastic

MC failure line

q plastic

Increasing plastic shear strains -a = c cot ϕ

p’

γ

Evolution law for γps: d γ ps = d λ s where dλs is the plastic multiplier for the cone type yield surface of the model.

Volumetric (density) hardening c

Yield function (cap): f =

q% 2

α2

− p '2 − p p 2

where pp is a state parameter that remembers the position of the cap: q αpp’

K0’

cap

pp’ σc’ Volumetric or vertical strain, ε1

p’

σ1’

Volumetric (density) hardening

αpp

K0NC is controlled by α. By inputting a realistic value of K0NC as a result αvalue in large and the yield cap is relatively steep.

Cap is closing the MC cone in principal stress space

Therefore: q% = f (σ 1 , σ 2 , σ 3 , ϕ ) m

K s Kc  σ 3 + a  p Evolution law: dp p =   d ε v K s − K c  p ref + a  Eurref where K s = and the bulk stiffness of the cap Kc is determined 3(1 − 2ν ) from Eoed and K0nc

Stress dependent moduli q

asymptote: qa E50

qf=0,9 qa

1

m

εy σc’

σy’ Eoed

εy

 σ '+ a  E50 = E50ref  3   pa '+ a  (Secant modulus)

+a ref  σ '3 / K Eoed = Eoed   ' + p a a   nc 0

1

m

(Tangent modulus)

• All stiffness moduli are updated according to current stress level. • Input stiffness are values at reference stress, e.g. pa’ = 100 kPa.

Effect of modulus exponent m

Elastic unloading/ reloading q

Unloading, reloading by:

E50 1

Eur >E50 1

Eur = Eurref (

εy σc’

σ 3 '+a pa '+ a

)m

σy’

ν ur = low value

Eoed 1

εy

Inside cone and cap: Bulk modulus

Kur =

Eur 3(1−2νur)

Shear modulus

Gur =

Eur 2(1+νur)

Oedometer

Eoed = ur

Eur(1−νur) (1−2νur)(1+νur)

Initial conditions for the HS model d e p t h

d e p t h

σc’ σy0’

σy0’ σc’

Preconsolidation is entered by OCR or POP relative to initial vertical stress and is then converted to pp.

σ 'c = σ y 0 '+ POP

σ 'c = OCR ⋅ σ y 0 ' σ’yy Prestress

σ’c

Initial CAP

1

POP σ’y0 1

ν ur 1−ν ur

Initial horizontal stress:

σ x 0 ' = K 0 '⋅σ c '−(σ c '−σ y 0 ' ) ⋅ 1ν−ν

ur

Default:

Current stress

K '0 = 1 − sin ϕ

Modified if the MC yield criterion is violated

K’0

σ’x0

σ’xx

ur

Initial conditions for the HS model

Initial stresses: q

MC failure line

Output:

' OCR ' = OCRiso

K0nc line α pc

p eq =

Cap p’0, q0

peq0

pc,0

p

( p' )

2

pc = eq p

+ q2 /α 2

Dilatancy ~ e max 2sin ψ

ψ

1-sin ψ

εy εv sin ϕ cv =

sin ϕ '− sin ψ 1 − sin ϕ ' sin ψ

sin ϕ m =

σ '1 −σ '3 σ '1 +σ '3 −2c' cot ϕ '

sin ψ m =

sin ϕ m − sin ϕ cv 1 − sin ϕ m sin ϕ cv

Dilatancy formulation: Rowe (1962) modified

Dilatancy from cone (most important part):

q

ψm

Nonassociated cone flow: Increasing dilatancy ψm from zero at ϕcv to input value ψinput on MC line (Rowe).

p’

Dilatancy set to zero for sinϕm < ¾·sinϕ, see Material Model Manual.

ψinput

Stress state

ϕcv

Contractancy from cap during shearing (less dominating): q

-a

Associated cap flow: Increasing contractancy from zero to a maximum value at MC line, but only when cap moves! pp’

p’

HS input parameters

Parameter

Description

E50ref

Stiffness modulus for primary loading in drained triaxial test

Eoedref

Stiffness modulus for primary loading in oedometer test

Eurref

Stiffness modulus for unloading/reloading in drained triaxial test

m

Modulus exponent for stress dependency

νur

Poisson’s ratio for loading/unloading

c’ φ’ ψ

Effective cohesion at failure Effective friction angle at failure Dilatancy angle at failure

POP: (σp – σvo’) OCR: σp/σvo’ Konc = 1 – sinφ’

Initial preconsolidation stress, σp Earth pressure coefficients at rest

Example: Settlement of strip footing

Monotonic loading

Example: Vertical displacement of a retaining wall Wall pulled up

odulus G/G0 [-] Shear mo

Limitation: No small strain stiffness 1

Retaining walls Foundations Tunnels

Very small strains

Conventional soil testing

Small strains

Larger strains 0 -6

10

-5

10

-4

10

-3

10

Dynamic methods Local gauges

-2

10

-1

10

Shear strain γ[-]

Limitation: Unloading/reloading stiffness is the same q=σ1-σ3 [kPa]

ε1 [%] • Truly elastic behaviour on for very small loops • At small strains stiffness increases • Hysteresis increases with increasing strains

Limitation: Non-monotonic loading in heavily OC clays Need to use artificially low POP/OCR value to trigger plasticity within ‘yield surface’ in order to represent different stiffness for loading/unloading for nonmonotonic loading. However the stress path may still be wrong when approaching to failure.

q

OC clay

HSmodel p’

Recommended procedure for application MC model: for simple estimates and for safety factors (stability) Advanced soil models: for more accurate deformation predictions Hardening Soil model: •

Use previous experience from lab, field and case records for strength and stiffness (E50 etc)

•

Simulate an oedometer or/and a triaxial test to calibrate your soil parameter set

•

Run your design problem

•

Check the results and compare to hand calculations or other estimates / experience

Comparison HS models and MC model Isotropic compression test: H-S

p' [kN/m2] 1000

M-C

800

600

400

200 Illustration by Brinkgreve, R.B.J. 0 0

0.01

0.02

0.03 eps-v

0.04

0.05

Comparison HS models and MC model Drained triaxial test: q' [kN/m2] 300 M-C 250 H-S 200

150

100

50 Illustration by Brinkgreve, R.B.J. 0 0

0.02

0.04 eps-1

0.06

0.08

Comparison HS models and MC model

Drained triaxial test:

eps-v 9.00E-03 6.00E-03 M-C

3.00E-03 0.000

H-S

-3.00E-03 -6.00E-03 0

0.02

0.04 eps-1

0.06

0.08

Illustration by Brinkgreve, R.B.J.

Comparison HS models and MC model

One-dimensional compression test (oedometer): sig'-yy [kN/m2] 1000

800

600 M-C 400 H-S 200 Illustration by Brinkgreve, R.B.J. 0 0

0.005

0.010

0.015 eps-1

0.020

0.025

0.030

Comparison HS models and MC model

One-dimensional compression test (oedometer): Stress state after unloading HS

MC

Illustration by Brinkgreve, R.B.J.

Small Strain Stiffness and HSsmall

Introduction The aim is to discuss the extension of the HS model to account for small strain stiffness. The topics to be covered include: • Small strain stiffness • HSsmall model formulation • Limitations of HSsmall model

Soil Stiffness at Small Strains • The strain range in which soils can be considered truly elastic is very small indeed • On exceeding this domain irrecoverable strains start to develop

Soil Stiffness at Small Strains

Soil Stiffness at Small Strains There are several different approaches to model soil stiffness at small strains: – Non-linear elastic models (e.g. Jardine et al. 1986, Benz et al. 2005) – Elasto-plastic models include multi-surface (e.g. Mroz 1967) and bounding surface models (e.g. Dafalias and Herrmann 1982, Al Tabbaa & Wood 1989, Stallebrass and Taylor 1997)

Soil Stiffness at Small Strains q=σ1-σ3 [kPa]

ε1 [%] • Truly elastic behaviour on for very small loops • At small strains stiffness increases • Hysteresis increases with increasing strains

Soil Stiffness at Small Strains q=σ1-σ3 [kPa]

ε1 [%]

Small-strain stiffness or Gur vs. G0 Soil stiffness derived from laboratory testing is commonly plotted as (secant-) shear modulus G over shear strain γ. G = G(γ) is a function of the applied shear strain after the last load reversal.

E0 = 2 (1 + νur) G0 q = σ1-σ3

Gγ = ∆q/∆γ

G0

Gur

∆q γ = ε1-ε3 normal γ-axis

∆γ scaled γ-axis

γ = ε1-ε3

Why is small-strain stiffness important? Not accounting for small strain stiffness in geotechnical analyses may potentially result in overestimating foundation settlements and retaining wall deflections consequently underestimating stresses. The gradient of settlement troughs behind retaining walls or above tunnels may be underestimated. Piles or anchors within the working load range may show a too soft response. Analysis results are also less sensitive to the choice of proper boundary conditions. Large meshes no longer cause extensive accumulation of displacements, because marginally strained mesh parts are very stiff.

Experimental evidence and data for small-strain stiffness True elastic stiffness was first observed in soil dynamics. Back then, the apparent higher soil stiffness in dynamic loading applications was attributed to the nature of loading, e.g. inertia forces and strain rate effects. Nowadays, static small-strain measurements are available as well. These show only little differences to dynamic measurements. Still, the term dynamic soil stiffness is sometimes used when true elastic or small-strain stiffness is meant.

odulus G/G0 [-] Shear mo

Soil Stiffness at Small Strains 1

Retaining walls Foundations Tunnels

Very small strains

Conventional soil testing

Small strains

Larger strains 0 -6

10

-5

10

-4

10

-3

10

Dynamic methods Local gauges

-2

10

-1

10

Shear strain γ[-]

Experimental evidence and data for small-strain stiffness Empirical relationships between G0 or E0 =2(1+νur)G0 and void ratio e: p' = 100 kPa

Most relationships are of the form:

250

Hardin & Black

m

A simple relationship for soils with wl < 50 % is proposed by Biarez & Hicher*:

E0 = Eref 0 with E *

ref 0

p' pref 140 MPa = e

Iwasaki & Tatsuoka

Shear moduluss G0 [MPa]

 p'   G0 = G   pref  k with Gref 0 = function (e ) ⋅ OCR ref 0

Biarez & Hicher

200

Kim et al.

150

Kokusho et al. Sand & Gravel

100

Kenya sand Clay

50

0 0.5

1

1.5

Void ratio e [-]

J. Biarez, P.-Y. Hicher. Elementary Mechanics of Soil Behaviour. Balkema, 1994

2

2.5

Experimental data & empirical relationships (E0) The relationship between E0 and Eur can be estimated from the chart by Alpan* assuming Edynamic/Estatic ≅ E0/Eur (10 kg/m²=1 MPa):

E dynamic E ≈ 0 E static Eur

E static [kg/cm²] ≈ Eur *

I. Alpan, The geotechnical properties of soils, Earth-Science Reviews 6 (1970), 5-49.

Experimental data & empirical relationships Stiffness reduction curves according to Seed & Idris* (left) and Vucetic & Dobry** (right)

* H.B. Seed, I.M. Idriss, Soil moduli and damping factors for dynamic response analysis. Report 70-10, EERC (Berkeley, Cali-fornia), 1970. ** M. Vucetic, R. Dobry, Effect of soil plasticity on cyclic response, Journal of Geotechnical Engineering, ASCE 117 (1991), no. 1, 89-107.

Empirical relationship for γ0.7 Based on statistical evaluation of test data, Darandeli* proposed correlations for a hyperbolic stiffness reduction model, similar to the one used inside the HSS model. Correlations are given for different plasticity indices. Based on Darendeli‘s work, γ0.7 can be estimated to: IP = 0:

γ 0.7 = 0.00015

p' pref

IP = 30:

γ 0.7 = 0.00026

p' pref

IP = 100: γ 0.7 = 0.00055

p' pref

Note: The indicated stress dependency of γ0.7 is not implemented in the commercial HSS model. If needed, the stress dependency of γ0.7 can be incorporated into boundary value problems through definition of sub-layers. *Darendeli,

Mehmet Baris, Development of a New Family of Normalized Modulus Reduction and Material Damping Curves. PhD Dissertation (supervisor: Prof. Kenneth H. Stokoe, II), Department of Civil Engineering. The University of Texas at Austin. August, 2001.

HS-Small model 1.0

G/G0 [-]

0.8 0.6 0.4 0.2 0.0 -3 1e

-2

-1

0

1

1e 1e 1e 1e 1e Normalisierte Scherdehnung γ/γ0.7 [-]

2

1e

3

Small-strain stiffness in the HS model (HSsmall) Strain(path)-dependent elastic overlay model: Gs =

G0 1 + 0.385 γ / γ 0.7

Gt =

Gur

G0

(1 + 0.385 γ / γ 0.7 )2

≥ Gur

G starts again at G0 after full strain reversal

Small-strain stiffness in the HS model (HSsmall) τ

Cyclic loading leads to Hysteresis

Gt G0

Gs

-γc +γc G0

CiTG, Geo-engineering, http://geo.citg.tudelft.nl

G0

γ


Energy dissipation
Damping

Small-strain stiffness in the HS model (HSsmall) γ0.7

G0

Gt

Gs

Gur

HS-small extension 1-dimensional The 1-dimensional model by Hardin & Drnevich*:

τ Hardin & Drnevich:

G0 τf

G 1 = G0 1+ γ / γ r

1

Modified HS-Small:

γ

G 1 = G0 1 + (3γ ) /(7γ 0.7 ) Note: γr in the original approach by Hardin & Drnevich relates to the failure shear stress τf.

*

B.O. Hardin, V.P. Drnevich, Shear modulus and damping in soils: Design equations and curves, ASCE Journal of the Soil Mechanics and Foundations Division 98 (1972), no. SM7, 667-692.

HS-small model – stiffness reduction Left: Secant modulus reduction → Parameter input Right: Tangent modulus reduction → Stiffness reduction cut-off If the small-strain stiffness relationship that is implemented in the HS-Small model predicts a tangent stiffness lower than Gurref, the model‘s elastic stiffness is set constant as then hardening plasticity accounts for further stiffness reduction.

G0ref

40000 Tangent modulus G [kN/m²]

Secant modulus G [kN/m²]

40000

γ0.7

30000

20000

10000 HS-Small

30000

20000

Gurref 10000

Hardin & Drnevich

0

1E-5

0.0001 Shear strain γ [-]

0.001

0.01

0

1E-5

0.0001 Shear strain γ [-]

0.001

0.01

Difference HS and HS-small Model response in a standard triaxial test. Here: Dense Hostun sand σ3 = 300 kPa CD

σ1/σ3

GSecant [kN/m2]

εVol[-] -0.20

4

-0.16

160000

HS (original) 120000

HS-Small

-0.12 2

-0.08 -0.04

0 0.00

0.00 0.02

0.04

0.06

0.08

ε1[-]

Experiment

80000 40000 0 0.0001

0.001

0.01

Parameter: Eurref = 90 MPa, E0ref = 270 MPa, m = 0.55, γ0.7=2x10-4 The stress-strain curves of the Hardening Soil model and the HS-Small model are almost identical (Figure left-hand side). However, in zooming into the first part of the curve, the difference in the two models can be observed (Figure right-hand side).

ε1-ε3[-]

Excavation example Distance to wall [m]

Limburg excavation: Settlement trough

0

20

40

60

80

0 -0.004 -0.008 -0.012

MC (E50) MC (Eur)

-0.016

Settlement [m]

A comparison: MC (E50): MC calculation with E = E50 MC (Eur): MC calculation with E = Eur HS: HS calculation with Eoed = E50 HSS: Same as HS but with small strain stiffness

Distance to wall [m] 0

20

40

60

80

0 -0.004 -0.008 -0.012

HS HSS

-0.016

Settlement [m]

Excavation example Limburg excavation: Horizontal wall displacement

MC-Model (E50) -0.04 -0.03 -0.02 -0.01

MC Model (Eur) 0

-0.04 -0.03 -0.02 -0.01

Hardening Soil Model 0

-0.04 -0.03 -0.02 -0.01

Hardening Soil Small

0

-0.04 -0.03 -0.02 -0.01

0

0

0

0

0

-5

-5

-5

-5

-10

-10

-10

-10

-15

-15

-15

-15

-20

-20

-20

-20

-25

-25

-25

-25

Depth below surface [m]

Depth below surface [m]

Depth below surface [m]

Depth below surface [m]

Excavation example Limburg excavation: Bending moments in [kNm/m]

MC-Model (E50) -600 -400 -200

0

200 400

MC Model (Eur) -600 -400 -200

0

200 400

Hardening Soil Model -600 -400 -200

0

200 400

Hardening Soil Small -600 -400 -200

0

200 400

0

0

0

0

-5

-5

-5

-5

-10

-10

-10

-10

-15

-15

-15

-15

-20

-20

-20

-20

-25

Depth below surface [m]

-25

Depth below surface [m]

-25

Depth below surface [m]

-25

Depth below surface [m]

Tunnel example Steinhaldenfeld - NATM

Distance to tunnel axis [m] 0

10

20

30

0

-0.01

Measurement -0.02

HS (original) HS-Small

Settlement [m]

40

HS-Small model 1.0

G/G0 [-]

0.8 0.6 0.4 0.2 0.0 -3 1e

-2

-1

0

1

1e 1e 1e 1e 1e Normalisierte Scherdehnung γ/γ0.7 [-]

2

1e

3

HS model (HSsmall) Relevance of small-strain stiffness: • Very stiff behaviour at very small strains (vibrations) • Reduction of stiffness with increasing strain; restart after load reversal • Hysteresis in cyclic loading: – Energy dissipation – Damping

Also relevant for applications like: – Excavations (settlement trough behind retaining wall) – Tunnels (settlement trough above tunnel)

Parameters of the HS(small) model Parameters: E50ref Secant stiffness from triaxial test at reference pressure Eoedref Tangent stiffness from oedometer test at pref Eurref Reference stiffness in unloading / reloading G0ref Reference shear stiffness at small strains (HSsmall only) γ0.7 Shear strain at which G has reduced to 70% (HSsmall only) m Rate of stress dependency in stiffness behaviour pref Reference pressure (100 kPa) νur Poisson’s ratio in unloading / reloading c’ Cohesion ϕ’ Friction angle ψ Dilatancy angle Rf Failure ratio qf /qa like in Duncan-Chang model (0.9) K0nc Stress ratio σ’xx/σ’yy in 1D primary compression

Selected references • Brinkgreve, R.B.J. et al (20xx): Users manual for PLAXIS 2D. • Schanz,T. Vermeer, P.A., Bonnier P.G. (1999): ”The Hardening Soil Model: Formulation and verification”, Beyond 2000 in Computational Geotechnics – 10 years of PLAXIS, Balkema. • Benz, T: Small-Strain Stiffness of Soils and its Numerical Consequences, PhD Thesis. IGS, Universität Stuttgart, Mitteilung 55.

HS-small model application Elastic stiffness properties of the HS-Small model can be visualized in state variable 10.

0

3.0 20

G m [-]

40 60

2.0

Gm=Gref /Gurref

80

1.0 100 -20

0

20

40

60

80

100

120

The dark (blue) area is the strain area where G = Gur. The light gray (yellow) area is the very-small-strain area with G ≈ G0. In between Gur and G0 is the area where shear strains are small but not very small according to the definition by Atkinson.

Limitation: Heavily OC clays Need to use artificially low POP/OCR value to trigger plasticity within ‘yield surface’ in order to represent different stiffness for loading/unloading for nonmonotonic loading. However the stress path may still be wrong when approaching to failure. 200 180 160

q (kPa)

140 120 100 100 kPa Stress Path

80 60

HSsmall Model Prediction

40 20 0 0

50

100 150 P` (kPa)

200

250

Limitation: Heavily OC clays By default, the initial stiffness is set to G0.

Care needs to be taken when the geologic loading history of a soil is modeled. If, for example, a vertical surcharge was applied and removed in order to model OCR, the model remembers the vertical heave upon unloading including its decreased small-strain stiffness. The initial stiffness at the onset of loading the footing might then look as the one shown at the left-hand side of the above figure. Here, the material should be exchanged or a reverse load step applied.

Selection of parameters for the Hardening Soil model

Based on materials by Dennis Waterman Plaxis bv Prof. Steinar Nordal - NTNU Norwegian University of Science and Technology Nubia González – UPC Universidad Politécnica de Cataluña

Input parameters - Hardening Soil model Parameter

Description

E50ref

Reference modulus for primary loading in drained triaxial test

Eoedref

Reference modulus for primary loading in oedometer test

Eurref

Reference modulus for unloading/reloading in drained triaxial test

m νur = 0.2 c’ φ’ ψ

Modulus exponent for stress dependency Poisson’s ratio for loading/unloading Effective cohesion at failure Effective friction angle at failure Dilatancy angle at failure

POP: (σp – σvo’) Initial preconsolidation stress, σp OCR: σp/σvo’ Ko Konc = 1 – sinφ’

Earth pressure coefficients at rest

G0ref

Reference shear stiffness at very small strain levels, (HSsmall only)

γ0.7

Shear strain at which G has reduced to 70% of G0ref , (HSsmall only)

Input parameters - HS model • Important: – The Hardening Soil is completely defined in effective stresses and therefore needs both effective stiffness parameters and effective strength parameters – A total stress analysis may be performed with the Hardening Soil model using both undrained strength (φ=0 and c=cu) and undrained stiffnesses but with the following limitations: – No stress dependent stiffness, only constant stiffness – No compression hardening, only elastic compression

Stiffness parameters - Hardening Soil model Stress dependent stiffness for primary shear, primary compression and unloading/reloading behaviour Plastic Cone hardening secant modulus: Plastic Cap hardening, tangent modulus: Elastic, unloading, reloading tangent modulus:

E50 = E50ref

Eoed

 σ 3 '+ a    p + a  ref 

m

 σ '1 + a  ref = Eoed   p + a  ref 

Eur = Eurref

 σ 3 '+ a    p + a  ref 

a = ccot(ϕ) m

m

Parameters of the HS model Parameters:

q qult (ϕ, c)

σ3=pref E50ref Eurref

σc σ1=pref qf=Rf qult 0.5 qf

Triaxial test

1 Eoedref

ε1

εv

Oedometer test

σ1

Stiffness of Sand

Parameters of the HS model For sands (m≈0.5):

Schanz (1998)

Stiffness of sand, drained triaxial testing: σy - σx

E50

E50

εy

∆σy’ ∆σx’

σy - σx E 50

σy - σx

Test 1: σ’x= 50kPa

E50

Test 2: σ’x= 100kPa

E50 = E50ref

ref 50

E

εy

εy

Test 3: σ’x= 200kPa

σ 'x pref

Loose sands: E50ref ≈ 15 MPa pref = 100kPa

σx’

Dense sands: E50ref ≈ 50 MPa

Stiffness of sand, K0 test: σy’ Eoed

εy Laboratory experience: σy’ dependency of Eoed ref E oed = E oed

σ 'y p ref ref Loose sands: Eoed ≈ 15 MPa

Oedometer results confirm:

ref Dense sands: Eoed ≈ 50 MPa

Stiffness of sand ref 50

E ≈E

ref oed

ref Eoed ≈ RD • 60MPa

emax − e RD = emax − emin

Correlation by Lengkeek for pref=100 kPa

Stiffness of sand ref E50ref ≈ Eoed

How can this be true?

p ref -σ’1= p ref -σ’3 = p ref

-σ’3

-σ’1

Eoed Cone tip Soil type of sand resistance at σ’v = σ’v0 Stiffness Eoed = 4qc qc < 10 MPa Loose sands: Eoed = 2qc+20 10 MPa < qc < 50 Unaged and MPa MPa uncemented, predominantly Eoed = 120 MPa qc > 50 MPa silica Eoed = 5qc qc < 50 MPa Dense sands Eoed = 250 MPa qc > 50 MPa

Lunne et al. (1997)

Stiffness for unloading-reloading loops

Triaxial tests: Unloading is purely elastic in HS model

Stiffness for unloading-reloading loops From oedometer tests for elastic behaviour with low Poisson’s Ratio:

(1−νur ) Eur,oed = Eur ≈1.1⋅ Eur (1−2νur )(1+νur ) Triaxial test Alternatively:

Eur ,oed = α Eoed and Eur ≈ Eur ,oed : m

E

ref ur

m

 σ 3 '+ a   σ 1 '+ a  ref   = α ⋅ Eoed   ⇒  pref '+ a   pref '+ a  m

Eurref

m ref  σ 1 ' + a  ref α = α ⋅ Eoed  = ⋅ E / K ( )  0 oed σ ' + a  3 

Sand (m=0.5): Eur ,oed ≈ 3Eoed ref ref Eurref ≈ 3Eoed / K0 ≈ 4Eoed

EXAMPLE: Triaxial test results by Shaoli (2004) Dense Hokksund sand at 40 kPa, n = 35,9% (initial) – 39,6% (end of test)

D e v ia t o r ic s t r e s s , q [ k P a ]

dense 40

200 150 dense 40

100 50

E50ref = E50 0 0

1

2

3

Axial strain [%]

4

pref + a

σ 'x + a

5

= 20000kPa

100kPa = 32MPa 40kPa

EXAMPLE Triaxial test results by Shaoli (2004) Dense Hokksund sand at 40 kPa, n = 35,9% (initial) – 39,6% (end of test) Dense 40

Axial strain [%] Volumetric strain, [%]

-4

1-sin ψ

-3

2sin ψ

-2 -1

0

1

2

Dense 40

3

4

5

1 − sinψ 1− 5 = = 1,2 2 sinψ 4,2 sinψ = 0,29

0 1

ψ = 17°

EXAMPLE Oedometer test dense Hokksund sand, n = 39% , (Moen, 1975) 0

Loading:

Vertical strain [%]

-0,2

Test data

-0,4

ref Eoed = Eoed

-0,6 -0,8 ref Eoed = Eoed

-1 -1,2 -1,4 0

200

400

600

800

1000

1200

Vertical effective stress [kPa]

Unloading:

Eur oed ≈ 1.1⋅ Eur , Eur = Eurref

Low Poisson Ratio

Eurref ≈ 0.9 ⋅ Eur oed

σ 1 '+ a pa '+ a

pa ''+ +a σ 1 '+ a

850kPa 100kPa 0,008 400kPa = 53MPa =

σ 3 '+ a pa '+ a

pa '+ a 850kPa 100 = 0.9 ⋅ = 195MPa σ 3 '+ a 0,0028 200

HS Material parameters for dense Hokksund sand from fitting PLAXIS results to experimental data (after trial and error, starting with estimated parameters):

γ=0

Axial symmetry

pw = 0

E50ref = 35 MPa (estimated 32 MPa) Eoedref = 45 MPa (estimated 53 MPa) Eurref = 180 MPa (estimated 195 MPa) m = 0,6 c = 1 kPa ϕ = 440 ψ = 180 (estimated 170) K0NC = 0,4 ν ur = 0,2 Triaxial tests by Shaoli (2004)

Triaxial test results and PLAXIS simulation Dense Hokksund sand at 40 kPa,

Deviatoric stress, q [kPa]

n = 35,9% (initial) – 39,6% (end of test) 200 180 160 140 120 100 80

Plaxis 40 dense 40

60 40 20 0 0,00

1,00

2,00

3,00

Axial strain [%]

4,00

5,00

Triaxial test results and PLAXIS simulation Dense Hokksund sand at 40 kPa, n = 35,9% (initial) – 39,6% (end of test Axial strain [%]

Volumetric strain, [%]

-4 -3,5 -3 -2,5 -2

from PLAXIS 40 Dense 40

-1,5 -1 -0,50,00 0 0,5 1

1,00

2,00

3,00

4,00

5,00

Oedometer test and PLAXIS simulation dense Hokksund sand, n = 39% , (Tore Ingar Moen, 1975)

Vertical strain [%]

0

Test data

-0,2

Plaxis

-0,4 -0,6 -0,8 -1 -1,2 -1,4 0

200

400

600

800

Vertical effective stress [kPa]

1000

1200

Stiffness of Clay

Stiffness of clay Drained stiffness from oedometer tests σy’

σc’

Basically

σ 'y  ref  Eoed = Eoed   p  ref  εy

1

E

Eoed

σy’

Eoed

E

ref oed

1 = mv

Eoed = E

ref oed

for

ref oed

 σ 'y    p  ref 

Soft NC clays: pref = 100kPa

σy’

pref = σ y '

≈ 1 MPa

Hard NC clays: ≈ 3 MPa

Stiffness of clay Ideal: drained triaxial test or consolidated undrained triaxial test Practice: undrained triaxial test σy - σx

E50 = f ⋅ Eu50

Eu50 1

1+ ν 1+ νu

cu: undrained shear strength

f =

cu

f = 1/ 3 0.25 < f < 0.35

εy

CUR 195 (2000) COB (1997) Lambe & Whitman (1969)

Stiffness for unloading-reloading loops From oedometer tests for elastic behaviour with low Poisson’s Ratio:

Eur,oed = αEoed and Eur ≈ Eur,oed : m

 σ1 '+ a  m ref E =α ⋅ E   = α ⋅ Eoed / ( K0 )  σ3 '+ a  ref ur

ref oed

Clay (m=1): ref ref / K0 ≈ 20Eoed NC : Eur ,oed ≈ 10Eoed ⇒ Eurref ≈ 10Eoed ref ref OC : Eur ,oed ≈ 3Eoed ⇒ Eurref ≈ 3Eoed / K0 ≈ 3Eoed

Stiffness of clay Other estimations for stiffness of normally consolidated clays (m=1):

ref oed

1 2

E ≈ E

ref 50

Order of magnitude (very rough)

ref oed

50000 kPa ≈ Ip

Correlation with Ip for pref =100 kPa

ref oed

500 kPa ≈ wL − 0.1

Correlation by Vermeer

E

E

Drained stiffness from oedometer tests on intact lean Norwegian NC clays for σy’ > σc’

E

ref oed

Oedometer modulus

Stiffness of clay

5 MP a 4 ref Eoed = 230 ⋅ (

3

2

1

0 Based on Janbu (1963)

1 + e0 ) Cc

Stiffness of sand and clay After Janbu (1963)

Janbu :

Eoed

 σ′  ref  = Eoed ⋅    pref 

m

more general:

 σ′ + a    Eoed = Eref ⋅ oed   pref + a  with a = c´ cotϕ´

m

Eoed [MPa] for NC-soils soils and σ´ = 100 kPa

105

rock 104

103

sandy gravel 102

sand 10

Norwegian clays

1

Mexico City Clay 0

50

porosity porosityn n[%][%]

100

Stiffness of clay

Tomlinson (1995): Foundation Design and Construction, Pitman Publishing Inc.

Parameters of the HS model For normally consolidated clays (m=1): ref Eoed ≈

1 2

ref E50

Order of magnitude (very rough)

ref Eoed ≈

50000 kPa Ip

Correlation with Ip for pref=100 kPa

ref Eoed ≈

500 kPa wL − 0.1

Correlation by Vermeer

ref Eoed = p ref λ*

Relationship with Soft Soil model

Oedometer test simulation, soft clay. Initial stresses, preconsolidation and parameters GW σyy

10 m

σxx

γ = 20kN/m3

High “oedometer” cut out as a vertical column from the site in question. Start the “test” from in situ stresses and specified preconsolidation for the sample studied.

Sample pw σ’yy

pw σ’xx

5m Soft clay by Hardening soil model:

ref E 50 = 2 MPa

m=1

ϕ = 25o

ref E oed = 2 MPa

νur = 0,2

Ψ = 0o

E urref = 10 MPa

c = 5 kPa

K0NC = 0.577

Oedometer test results, soft clay Strain 0,00

Note the preconsolidaton levels at:

OCR = 3

-0,02 -0,04

εyy

OCR = 1,5

-0,06

σ’yy = 150 kPa σ’yy = 300 kPa

-0,08 -0,10 0

-100 -200 -300 -400 -500 -600

σ’yy

Stress [kN/m2]

For OCR = 1,5: 0

-200

-400

-600

0 -74

σ x 0 ' = K 0 '⋅σ c '−(σ c '−σ y 0 ' ) ⋅ 1ν−ν

ur

OCR = 1,5

-123

σ’xx-200 -300

ur

σ x 0 ' = 0.577 ⋅150 − (150 − 100) ⋅ 1−00, 2, 2 = 74 kPa

OCR = 3

Note that for OCR = 3: σ’xx > σ’yy

Konc 11

-400

σ’yy

Drained triaxial test simulation soft clay Soft clay by Hardening soil model: γ=0

ref E 50 = 2 MPa

pw = 0

ref E oed = 2 MPa

Axial symmetry

ref E ur = 10 MPa

m=1 νur = 0,2 c = 5 kPa

ϕ = 25o Ψ = 0o K0NC = 0.577

Zero initial stresses, preconsolidation generated by preloading

Drained triaxial test on soft clay results q [kN/m2]

Stress paths:

q [kN/m2]

200

200

160

160

120

120

80

80

40

40

Initial cap OCR = 1

OCR = 2 0

0 0

-40

-80

-120

-160

0

-50

160

q

Top curve OCR = 3 OCR = 2 OCR = 1

140

Effect of OCR:

-150

p’ [kN/m2]

p' [kN/m2] 180

-100

120 100 80 60 40

εy

20 0 -20 0

0,05

0,1

0,15

0,2

0,25

0,3

εyy

-200

-250

Parameter limitations HS model has internal parameters that are computed from our ”engineering” input parameters.

Not all combinations of input parameters can be used q

α pc

E50

Eur

E oed

β double hardening

p´ E50 / Eoed > 2 difficult to input

Small-strain stiffness in the HS model (HSsmall) Strain (path)-dependent elastic overlay model:

G starts again at G0 after full strain reversal

Input parameters of HSSmall: • G0ref • γ0.7

Gur

ref 0

G0 = G

 c cos ϕ − σ 3 sinϕ   c cos ϕ + p ref sinϕ   

m

Small-strain stiffness in the HS model (HSsmall) τ

Cyclic loading leads to Hysteresis

Gt G0


Energy dissipation
Damping

Gs

-γc +γc G0

γ

Gs =

G0 1 + 0.385

G0

Gt =

γ γ 0.7

G0  γ   1 + 0.385  γ 0.7  

2

Small-strain stiffness parameters in the HSsmall model Drained triaxial test, HS vs. HSsmall model 160 140

HSsmall

q [kN/m²]

120

E0

ref E 0 G0ref = 2(1+νur )

HS

100 80 60 40 20 0 0

Et ≥ Eur -0.002

-0.004

-0.006

-0.008

εyy

-0.01

-0.012

-0.014

Small-strain stiffness parameters in the HSsmall model ref 0

G

G0ref

(2.97 − e)2 = 33 [MPa] 1+ e ≈ RD • 70MPa + 60MPa

γ 0.7 =

Hardin & Black (1969) Lengkeek

0.385 [2c(1 + cos(2ϕ )) − σ 1(1 + K0 )sin(2ϕ )] 4G0

Benz (2007)

Order of magnitude:

G0ref = (2.5 to10)Gurref

γ 0.7 = (1 to 2) ⋅10 −4

where

ref ur

G

Eurref = 2(1 + ν ur )

Recommended procedure for application MC model: for simple estimates and for safety factors (stability) Advanced soil models: for more accurate deformation predictions Hardening Soil model: •

Use previous experience from lab, field and case records for strength and stiffness (E50 etc)

•

Simulate an oedometer or/and a triaxial test to calibrate your soil parameter set

•

Run your design problem

•

Check the results and compare to hand calculations or other estimates / experience

Which model in which situation? Soft soil (NC-clay, Hard soils (OCpeat) clay, sand, gravel) Primary load. (surcharge)

Soft Soil (Crp), HS / HSsmall

HS / HSsmall

Unloading + deviatoric load (excavation) Deviatoric loading

HS / HSsmall

HS / HSsmall

Soft Soil (Crp), HS / HSsmall

HS / HSsmall

Secondary compression

Soft Soil Creep

n/a