zsoilday2011 hsmodels-course v3.1

The Hardening Soil model with small strian stiffness Rafal Obrzud, GeoMod SA 30.08.2011, Lausanne

The Hardening Soil model with small strain stiffness

in Zsoil v2011

Rafal OBRZUD

GeoMod Ing. SA, Lausanne


Content

Introduction

Framework of the Hardening Soil model

Hardening Soil SmallStrain

Hardening Soil Standard

Model parameters

Parameter Identification

Assistance in Parameter Selection

Practical applications


0

50

100

150

200

250

300

350

400

450

0,00% 2,00% 4,00% 6,00% 8,00% 10,00% 12,00% 14,00%

Dev

iato

ric

stre

ss [

kPa]

q

= σ

1-σ

3

Axial Strain ε1

Triaxial drained compression test

Experiment: Texas sand

Simulation: Hardening Soil

Why do we need advanced constitutive models? Approximation of soil behaviour for reliable simulations of practical cases

- Introduction

Simulation: Mohr-Coulomb

Plasticity (yielding) before reaching the ultimate state


Why do we need advanced constitutive models?

ENGINEERING CALCULATIONS

LIMIT STATE ANALYSIS

Bearing capacity

Slope stability

DEFORMATION ANALYSIS

Pile, retaining wall deflection

Supported deep excavations

Tunnel excavations

Consolidation problems

Basic models e.g. Mohr-Coulomb Advanced soil models

- Introduction


Basic differences between implemented soil models

Mohr-Coulomb

p’

q

K0-line

yield surface

Linear elastic domain

ε1

q

E Eur

E = Eur

p’

q

K0-unloading

- Introduction


Tunneling

Standard Mohr-Coulomb Hardening-Soil

Practical applications of the Hardening Soil model

- Introduction



Mohr-Coulomb Volumetric cap models

p’

q

K0-line

yield surface

p’

q

ε1

q

E Eur

ε1

q

Eur

E=Eur

Stiffness degradation

isotropic hardening

mechanism

E

E < Eur

p’

q

σ1’ constant



- Introduction



Practical applications of the Hardening Soil model

Berlin sand excavation

- Introduction



Mohr-Coulomb Volumetric cap models

Hardening Soil models

p’

q

K0-line

yield surface


p’

q

p’

q

ε1

q

E Eur

ε1

q

Eur

ε1

q

Eur

E0

E=Eur



isotropic hardening

mechanism + shear

hardening mechanism

E

E<Eur Eur


E0

E

NON-LINEAR elastic domain

- Introduction


Small strain stiffness in geotechnical practice

e.g. Atkinson, Jardine and many others

- Introduction


Stiffness in different models

0

20

40

60

80

100

120

140

160

-3,61E-16 0,05 0,1 0,15 0,2

Dev

iato

ric

Stre

ss q

[kPa

]

Axial Strain ε1 [-]

Mohr-Coulomb

Modified Cam Clay

Cap model

HS-Standard

HS-Small A

B

0

0,2

0,4

0,6

0,8

1

1E-06 1E-05 0,0001 0,001 0,01 0,1 N

orm

aliz

ed s

oil s

tiff

ness

G/G

0 [-]

Axial Strain ε1 [-]

Linear elasticity at very small strains

Linear elasticity at small strains

Non-linear elasticity at very small strains

- Introduction


Perceived application of ZSoil models

ULS - Ulitmate Limit State analysis SLS - Serviceability Limit State analysis

- Introduction


Content

Introduction

Hardening Soil model framework



Model parameters





Hardening Soil model to describe macroscopic phenomena exhibited by soils

Densification (decrease of voids volume in soil due to plastic deformations)

Soil stress history (accounting for preconsolidation effects)

Plastic yielding (irreversible strains with reaching a yield criterion)

HS-Standard

- Framework of HS model






Stress dependent stiffness (increasing stiffness moduli with increasing depth or stress level)

HS-Standard


0

200

400

600

800

1000

1200

0.00% 5.00% 10.00% 15.00%

Dev

iato

ric

stre

ss [

kPa]

q=

σ1-

σ3

Axial Strain ε1

Triaxial drained compression test - Texas sand

SIG3 = 34.5kPa

SIG3 = 138kPa

SIG3 = 345kPa

E(1)

E(2)

E(3)






Stress dependent stiffness (increasing stiffness moduli with increasing depth or stress level)

Dilatation (occurrence of negative volumetric strains during shearing)

Stiffness variation (degradation of G0 modulus with increasing shear strain amplitudes between γ ≈ 0.0001 – 0.1%)

Hysteretic, non-linear stress-strain relationship applicable in the range of small strains *

HS-Standard

+ HS-SmallStrain for handling stiffness in a small strain range

* HS-SmallStrain can be used to some extent to model hysteretic behavior under cycling loadings with the exception of gradual softening for increasing number of cycles.



Hardening Soil model in ZSoil




Activation of G0 degradation in the small strain range




Secant stiffness in small strains

Triaxial test simulation

zoom



0

100

200

300

400

500

600

700

800

900

1000

0,019 0,02 0,021 0,022 0,023 0,024 D

evia

tori

c st

ress

q

[kPa

]

Axial strain ε1 [-]

HS-SmallStrain

HS-Standard

0

100

200

300

400

500

600

700

800

900

1000

0 0,01 0,02 0,03 0,04 0,05

Dev

iato

ric

stre

ss

q [k

Pa]

Axial strain ε1 [-]

HS-SmallStrain

HS-Standard


Triaxial test simulation with the unloading/reloading cycle

zoom


Eur Eur E0


Main mechanical features of HS model

Smooth hyperbolic approximation of the stress-strain curve of soil (Duncan-Chang concept)

Two plastic mechanisms:

isotropic (to handle important plastic volumetric strains observed in normally-consolidated soils)

deviatoric (to handle domination of plastic shear strains which are observed in coarse materials and heavily consolidated cohesive soils)

Ultimate state described by Mohr-Coulomb criterion

Rowe’s dilatancy

Nonlinear elasticity which includes hysteretic effects (Hardin-Drnevich concept)

HS-Standard

+ HS-SmallStrain



Content

Introduction




Model parameters





Hyperbolic Hardin-Drnevich relation to describe S-shaped stiffness

Non-linear elasticity for very small strains

a = 0.385

- HS SmallStrain


Content

Introduction




Model parameters





Hyperbolic approximation of the stress-strain curve

E0 – tangent initial stiffness modulus E50 – secant stiffness modulus corresponding to 50% of the ultimate deviatoric stress qf described by Mohr-Coulomb criterion Eur – unloading/reloading modulus (parameter corresponding to in Modified Cam Clay)

For most soils Rf falls between 0.75 and 1

- HS Standard


Hyperbolic approximation of the stress-strain curve

- HS Standard


Stress stiffness dependency

0

200

400

600

800

1000

1200

0,00% 5,00% 10,00% 15,00%

Dev

iato

ric

stre

ss [

kPa]

q

= σ

1-σ

3

Axial Strain ε1

Triaxial drained compression test - Texas sand

SIG3 = 34.5kPa

SIG3 = 138kPa

SIG3 = 345kPa

E(1)

E(2)

E(3)

- HS Standard


Stress stiffness dependency

E0ref , E50

ref, , Eurref correspond to reference minor stress σref

where i.e. stiffness degrades with decreasing σ3 up to the limit minor stress σL which can be assumed by default σL =10kPa

In natural soils: m = 0.3 - 1.0 e.g. Norwegian Sands and silts ≈0.5 (Janbu, 1963) Soft clays m=0.38 - 0.84 (Kempfert, 2006)

- HS Standard


Stiffness stress dependency parameter m

1. Find three values of E50(i) corresponding σ3

(i) to respectively

2. Find a trend line y = ax + b by assigning variables

y =

x =

3. Then the determined slope of the trendline a is the parameter m

- HS Standard


Stress stiffness dependency at initial state

0,0

2,0

4,0

6,0

8,0

10,0

12,0

14,0

16,0

18,0

20,0

0 50000 100000 150000 200000

z [m

]

E [kPa]

m = 0.4

m = 0.6

m = 0.8

0,0

2,0

4,0

6,0

8,0

10,0

12,0

14,0

16,0

18,0

20,0

0 50000 100000 150000 200000

z [m

]

E [kPa]

phi = 20

phi = 30

phi = 40

0,0

2,0

4,0

6,0

8,0

10,0

12,0

14,0

16,0

18,0

20,0

0 50000 100000 150000 200000

z [m

]

E [kPa]

c = 0

c = 15

c = 30

Level of reference stress

- HS Standard


Stiffness exponent m

- HS Standard


Unloading/reloading Poisson’s ratio νur

Experimental measurements from local strain gauges show that the initial values of Poisson’s ratio in terms of small mobilized stress levels q/qmax varies between 0.1 and 0.2 for clays, sands.

0,0

0,1

0,2

0,3

0,4

0,5

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

Pois

son'

s ra

tio ν

[-]

Mobilized stress level q / qmax

Toyoura Sand (Dr=56%) Ticino Sand (Dr=77%) Pisa Clay (Drained Triaxial) Sagamihara soft rock

Typically elastic domain

after Mayne et al. (2009)

- HS Standard


Unloading/reloading Poisson’s ratio νur

Therefore, the characteristic value for the elastic unloading/reloading Poisson’s ratio of νur = 0.2 can be adopted for most soils.

- HS Standard


Failure ratio Rf

0,0

20,0

40,0

60,0

80,0

100,0

120,0

140,0

160,0

0 0,05 0,1 0,15 0,2

q =

σ 1-σ

3

Asymptote qa=1/b

E50 = 2a

1

0,0E+00

2,0E-04

4,0E-04

6,0E-04

8,0E-04

1,0E-03

1,2E-03

1,4E-03

1,6E-03

1,8E-03

0 0,05 0,1 0,15 0,2

ε 1/q

ε1 [-]

a =1/2E50

1

b=1/qa

Identification of Rf

Rf = qf / qa = b·q f

- HS Standard


Double hardening model

r(θ) – follows van Ekelen’s formula in order to assure a smooth and convex yield surface (cf. Modified Cam clay model)

- HS Standard


Double hardening model

p’-q section through shear and cap yield surfaces

σ1

σ2

σ3

p’

cap yield surfaces described with van Ekelen’s formula

Mohr-Coulomb failure surface

- HS Standard


Dilatancy

-0,4

-0,3

-0,2

-0,1

0,0

0,1

0,2

0,3

0,4

0,5

0 10 20 30 40

sinϕ

m [

-]

φm [deg]

Dilatancy domain

Contractancy domain

Contractancy cut-off for

φm < φcs

Rowe's dilatancy for

φm≥ φcs

Rowe's dilatancy

- HS Standard


Dilatancy

- HS Standard


Dilatancy in HS-SmallStrain

Since the cut-off for the contractancy domain could yield too small volumetric strain, the scaling parameter D is introduced.

Contractancy domain

Dilatancy domain

- HS Standard


Parameters M and H

H controls the rate of volumetric plastic strains

Evolution of the hardening parameter pc :

- HS Standard


Typical stress path in the oedometric test

M and H must fulfil two conditions: 1. K0

NC produced by the model in oedometric conditions is the same as K0

NC specified by the user

2. Eoed generated by the model is the same Eoed

ref specified by the user

Parameters M and H

- HS Standard


ASSUMPTIONS: 1. At given σoed

ref which is located at post-yield plastic curve, both shear an volumetric mechanism are active

2. Initial stress point

3. A strain driven oedometer test is run with vertical strain amplitude ∆ε=10-5 and the tangent oedometric modulus is computed as Eoed

= δσ/δε ≅ ∆σ/∆ε

4. Optimisation of M and H must fulfil two conditions: K0

NC produced by the model in oedometric conditions = K0NC specified by the user

Eoed generated by the model = Eoed ref specified by the user

Parameters M and H

- HS Standard


Initial state variables - OCR

0

0,02

0,04

0,06

0,08

0,1

0,12 1 10 100 1000

Vol

umet

ric

stra

in

ε v [-

]

Vertical stress σv' [ kPa]

Simulation

Experiment

Vertical preconsolidation pressure σ‘vc

Notion of overconsolidation ratio:

Typical oedometer test σ‘vc OCR =

σ‘v0

σ‘v0 – current in situ stress

σ‘vc – past vertical preconsolidation pressure

NB. In natural soils, overconsolidation may stem from mechanical unloading such as erosion, excavation, changes in ground water level, or due to other phenomena such as desiccation, melting of ice cover, compression and cementation.

- HS Standard


Initial state variables – OCR and K0

Normally-conslidated soil (OCR=1) Overconslidated soil (OCR>1)

Why does it appear and what should be inserted?

- HS Standard


Initial state variables – K0 and K0NC

A

B

Horizontal effective stress

Verti

cal e

ffect

ive

stre

ss

σ’v

σ’h

A B

σ’v

σ’h

A

B

p’

q

Excavation

- HS Standard


Preconsolidation stress configuration corresponding to K0NC

Current in situ stress configuration corresponding to K0 (at the beginning of numerical simulation)


σ’ SR

σ’0

- HS Standard


Estimation of earth pressure at rest

Normally-consolidated soils

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0 20 40 60

K 0NC

φ' [deg]

1-sin(phi')

Brooker&Ireland(1965)

Simpson(1992)

Overconsolidated soils

0,0

0,5

1,0

1,5

2,0

2,5

0 5 10 15 20

K 0

OCR [-]

phi=20deg phi=30deg phi=40deg

Initial state variables – K0 vs. OCR

- HS Standard



K0NC K0

For most cases when soil was subject to “natural” consolidation before unloading

K0SR = K0

NC

- HS Standard



A B

q

In the case of simulating the triaxial compression test after isotropic loading/unloading: K0

SR ≠ K0NC

but

K0SR =1

p’

C

isotropic loading/unloading

- HS Standard


Initial state variables

1. User specifies initial stress conditions σ’0 2. Zsoil at the beginning of a FE analysis calculates

σ’SR with

and the horizontal stress components 3. Then the hardening parameter pc0 is computed

- HS Standard


Setting initial state variables

1. Through K0

2. Through Initial Stress

σy = γ “depth” σx = σxK0(x)

- HS Standard


Setting initial state variables

If the model does not converge at Initial State for the specified K0

1. try to start Initial State analysis from a small Initial loading Fact. and Increment

2. otherwise, define initial conditions through Initial Stress

- HS Standard


Initial state variables - preconsolidation

1. through OCR (gives constant OCR profile)

At the beginning of FE analysis, Zsoil sets the stress reversal point (SR) with:

2. through qPOP (gives variable OCR profile)

- HS Standard


0

2

4

6

8

10

12

14

16

18

1 3 5 7 9 11 13

Dep

th [m

]

OCR [-]

0

2

4

6

8

10

12

14

16

18

0 500 1000 1500

Dep

th [m

]

Effective stress [kPa]

SigEff vertical SigEff preconsolidation

0

2

4

6

8

10

12

14

16

18

0,00 0,50 1,00 1,50

Dep

th [m

]

Ko [-]

Stress history through OCR Deposits with constant OCR over the depth (typically deeply bedded soil layers)

- HS Standard


Deposits with varying OCR over the depth (typically superficial soil layers)

Profiles for Bothkennar clay

- HS Standard


0

2

4

6

8

10

12

14

16

18

0 0,5 1 1,5

Dep

th [m

]

Ko [-]

0

2

4

6

8

10

12

14

16

18

1 3 5 7 9 11 13

Dep

th [m

]

OCR [-]

0

2

4

6

8

10

12

14

16

18

0 200 400 600

Dep

th [m

]

Effective stress [kPa]

SigEff vertical SigEff preconsolidation

qPOP

Stress history through qPOP

K0OC = K0

NC √OCR

K0OC = K0

NC OCRsin(φ) (Mayne&Kulhawy 1982)

Variable K0 can be introduced merely through Initial Stress option

Deposits with varying OCR over the depth (typically superficial soil layers)

- HS Standard


Content

Introduction

Hardening Soil - SmallStrain

Hardening Soil – Standard

Model Parameters





Hardening Soil model List of parameters to be provided by the user (page 1)

Small stiffness (HS-SmallStrain only)

1. E0ref kPa SCPT, SDMT, geophysical methods

2. γ0.7 - Triaxial test with local gauges, geotechnical evidence

Elastic constants

3. E50ref kPa Triaxial test

4. Eurref kPa Triaxial test

5. νur - Triaxial test

6. m - Triaxial test

Auxiliary variable

7. σ3ref kPa Triaxial test

- Model parameters


Hardening Soil model List of parameters to be provided by the user (page 2)

Shear mechanism

8. c kPa Triaxial test, in situ test

9. φ ˚ Triaxial test, in situ test

10. (Rf) - Triaxial test or the default value 0.9

11. ψ ˚ Triaxial test

12. emax ˚ Triaxial test on dense granular or overconsildated cohesive soil

Volumetric (cap) mechanism

13. Eoedref kPa Oedometric test

14. σoedref kPa Oedometric test

Initial state variable

15. OCR / qPOP

- / kPa

Oedometric test or in situ tests

16. K0SR - K0

SR = 1-sin φ (or Ko-consolidation triaxial test)

17. σ0 kPa Init. stress conditions accounting for K0 = K0

SR (for normally-consolidated soil), K0 = K0

OC (for overconsolidated soil)

- Model parameters


Hardening Soil model Comparison of corresponding soil parameters for selected ZSoil models

Corresponding soil parameters

Hardening-Soil

Cap Mohr-Coulomb

Small strain stiffness E0 and γ0.7

none none

Elastic constants Eur and νur E50 and m

E and ν E and ν

Failure criterion φ and c

φ and c φ and c

Dilatancy ψ and emax ψ ψ and emax

Cap surface parameters

Eoed(can be determined from λ) OCR , K0 and K0

SR

λ

OCR , K0

None

K0

- Model parameters


Content

Introduction



Model Parameters





Parameter identification Report

- Parameter identification


The HS model – a practical guidebook

Contents:

Theory Parameter identification Parameter estimation



Parameter identification Stiffness parameters Determination of G0 from geophysical tests SCPT, SDMT and others with ρ denoting density of sol and Vs shear wave velocity. (ν=0.15..0.25 for small strains)

In situ tests with seismic sensors: • seismic piezocone testing (SCPTU) (Campanella et al. 1986) • seismic flat dilatometer test (SDMT) (Mlynarek et al. 2006, Marchetti et al. 2008)

Geophysical tests: (see review by Long 2008)

• continuous surface waves (CSW) • spectral analysis of surface waves (SASW) • multi channel analysis of surface waves (MASW) • frequency wave number (f-k) spectrum method



Parameter identification Stiffness parameters Determination of G0 for sands from CPT

after Robertson and Campanella (1983)



Parameter identification Stiffness parameters Estimation of E0 from E50



Parameter identification Stiffness parameters Estimation of E0 from Eur



HS-SmallStrain Non-linear elasticity for small strains

Cohesionless soils Influence of voids ratio Influence of confining stress p’

after Wichtmann & Triantafyllidis



HS-SmallStrain Non-linear elasticity for small strains

Cohesive soils

Influence of soil plasticity

after Vucetic and Dobry (from Benz, 2007)

approximation proposed by Stokoe et al.



Parameter identification Stiffness parameters Determination of E50 for sands from CPT

after Robertson and Campanella (1983)



Parameter selection Stiffness parameters

Undrained vs drained moduli – theoretical relationship Since the shear modulus is not affected by the drainage conditions



Parameter identification Oedometric modulus Eoed

where Cc is the compression index

Since we look for tangent Eoed, ∆σ’0, and σ*=2.303σoed

ref

If λ is known



Parameter selection - oedometric modulus Eoed

Eoedref

≈ E50ref

but

Eoedref

E50

ref

σref σoedref = σref / K0

NC



Parameter identification Preconsolidation pressure and OCR

Laboratory: - Oedometer test (Casagrande’s method, Pacheco Silva method (1970)) Field tests: - Static piezocone penetration (CPTU) (see reports by Mayne) - Marchetti flat dilatometer (DMT) (correlations by Marchetti (1980), Lacasse and Lunne, 1988) )




(from Lunne et al. 1997)



Parameter identification

Other information from DMT



Parameter identification Strength parameters



Parameter identification Strength parameters Friction angle φ for granular soils from in situ tests

SPT

CPTU

(Robertson & Campanella, 1983)

DMT Upper bound

Lower bound (Totani et al., 1999)



Content

Introduction



Model Parameters





- Assistance in parameter selection


Initialization menu

Empty field reserved for numeric data (basic soil properties,

specific soil properties, in situ test data) in Zsoil v2012



Initialization menu

Non-cohesive soil setup



Initialization menu

Cohesive soil setup



Initialization menu



HS Menu

Scr

oll



Parameter selection – secant modulus Eur

Typical values of the stiffness modulus which are provided in most textbooks are referred to as the "static" stiffness modulus Es It can be assumed that they represent the values of the unloading/reloading modulus Eur In the basic parameter selection, typical Eur is taken as Es and the reference pressure is assumed σref=100kPa



Parameter selection – secant modulus E50

In most practical cases:

In toolbox - ranges: = 2 to 4



Parameter selection - oedometric modulus Eoed

Eoedref

≈ E50ref

but

Eoedref

E50

ref

σref σoedref = σref / K0

NC



Parameter selection – friction angle φ

Multi-criterion selection depending on quantity of provided input data (“soil keywords”)

Keyword: Density

Keyword: Soil type, Density, Gradation



Content

Introduction



Model Parameters





Practical applications – Excavation in Berlin sand

Engineering draft and the sequence of excavation

- Practical applications




-35

-30

-25

-20

-15

-10

-5

0

-0.04 -0.03 -0.02 -0.01 0

Dis

tanc

e fr

om s

urfa

ce

Y [m

]

Horizontal displacement Ux [m]

HS-smallHS-StdMCMeasurements



Texas sand benchmark

Practical applications – Shallow footing



Texas sand benchmark Practical applications – Shallow footing

0

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20D

epth

[m]

KD [-]

DMT-1

DMT-2

1. DMT data 2. OCR interpreted form DMT data

0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50

Dep

th [m

]

OCR [-]

DMT-1

DMT-2

Variable profile



0

1

2

3

4

5

6

7

8

9

10

0 10 20 30 40 50

Dep

th [m

]

OCR [-]

DMT-1

DMT-2

FE Model

0

1

2

3

4

5

6

7

8

9

10

0 200 400 600 800D

epth

[m]

Vertical stress[kPa]

vertical effectivve stress

preconsolidation pressure

Texas sand benchmark Practical applications – Shallow footing 3. Selecting qPOP 4. OCR based on qPOP

Variable profile

qPOP



Texas sand benchmark Practical applications – Shallow footing 5. Interpreting and Setting K0

0

1

2

3

4

5

6

7

8

9

10

0 0.5 1 1.5 2 2.5

Dep

th [m

]

K0 [-]

DMT-1

DMT-2

FE Model

Variable K0 can be introduced merely through Initial Stress option



Texas sand benchmark

Practical applications – Shallow footing

-0.14

-0.13

-0.12

-0.11

-0.1

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

0 2000 4000 6000 8000 10000

Sett

lem

ent [

m]

Load [kN]

Experiement


Standard Mohr-Coulomb



Summary

Hardening Soil-SmallStrain model:

correctly reproduces a strong reduction of soil stiffness with increasing shear strain amplitudes

is recommended for Serviceability Limit State analyses as it generally predicts soil behavior and field measurements more precisely than basic linear-elasticity models

is essential for the engineering problems with unloading/reloading modes such as excavations, tunneling

accounts for stress stress history which is crucial for problems such as footing, consolidation, water fluctuation

is applicable to most soils as it accounts for pre-failure nonlinearities for both sand and clay type materials regardless of the overconsolidation state

The Hardening Soil model is not as difficult in practical use as you may think. When you run a simulation with the M-C model, as an exercise, try to run the Hardening Soil model as well.


zsoilday2011 hsmodels-course v3.1

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