Cyclic behaviour of cohesionless soils under seismic loading

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Context From reality to laboratory From laboratory to numerical modelling Conclusion

Cyclic behaviour of cohesionless soils under seismic

loading

Cerfontaine Benjamin - Boursier FRIA

University of Li`ege - GEO3

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Outline

1 Context

Don’t forget the soil !

2 From reality to laboratory

Equivalence in-situ/triaxial test Monotonic behaviour charaterization Cyclic behaviour characterization

3 From laboratory to numerical modelling

Summary Prevost’s model

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Context From reality to laboratory From laboratory to numerical modelling Conclusion Don’t forget the soil !

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Context From reality to laboratory From laboratory to numerical modelling Conclusion Don’t forget the soil !

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Context From reality to laboratory From laboratory to numerical modelling Conclusion Equivalence in-situ/triaxial test

Outline

1 Context

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Stress state in the soil

Ground

Ground Water Table σ'=σv K0σv before earthquake σ3 she ar _stress normal stress K0=1

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Stress state in the soil

Ground

Ground Water Table σ'=σv K0σv σ'=σv K0σv τh σ'=σv K0σv τh during earthquake before earthquake σ3 σv R τh -τh K0=1 she ar _stress she ar _stress normal

stress normal stress

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Stress state in laboratory

Essai triaxial :

• compresssion/extension

• monotonic/cyclic

• drained/undrained

And also :

• torsional shear test

σ1 σ3 σ2=σ3 σ1-σ3 σ3 u u

u

N

T

u

torque

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Stress state in laboratory

Essai triaxial : • compresssion/extension • monotonic/cyclic • drained/undrained And also : • simple shear

σ1 σ3 σ2=σ3 σ1-σ3 σ3 u u

u

N

T

u

torque

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Stress state in laboratory

Essai triaxial : • compresssion/extension • monotonic/cyclic • drained/undrained And also : • simple shear

σ1 σ3 σ2=σ3 σ3 u u

u

N

T

u

torque

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Stress state in a triaxial test

σ'=σv K0σv τh σ'=σv K0σv τh during earthquake σv R τh -τh K0=1 she ar _stress normal stress X Y 45° 45° σ3+1/2σd σ3 σ3 σ3-1/2σd 1/2σd 1/2σd σ3+1/2σd σ3-1/2σd R 45° X X Y Y XX +1/2σd YY -1/2σd σ3 she ar stress normal stress 45° 45° σ3-1/2σd σ3 σ3 σ3+1/2σd 1/2σd 1/2σd σ3+1/2σd σ3-1/2σd 45° R X X Y Y XX +1/2σd YY -1/2σd σ3 she ar stress normal stress

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Outline

1 Context

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Context From reality to laboratory From laboratory to numerical modelling Conclusion Monotonic behaviour charaterization

Monotonic undrained test

Steady State : • continuous deformation • constant p’ • constant q • constant velocity Contractancy : ∆V < 0 under shearing (a) (b) De via to ric st re ss q

Effective mean stress p'

De via to ric st re ss q Axial strain εa PT line Steady State SSS

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Monotonic undrained test

Quasi Steady State :

• transient state • p’ minimum • q minimum Contractancy : ∆V < 0 under shearing (a) (b)

Effective mean stress p' Axial strain εa

De via to ric st re ss q

De via to ric st re ss q Axial strain εa PT line Steady State

Quasi Steady State

SQSS

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Monotonic undrained test

Dilative state • strain hardening • no instability Dilatancy : ∆V > 0 under shearing (a) (b) De via to ric st re ss q

De via to ric st re ss q Axial strain εa PT line Steady State

Quasi Steady State

SQSS

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Triaxial undrained test examples (1)

Deformation controlled triaxial undrained tests : Toyoura sand [Verdugo and Ishihara, 1996]

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Triaxial undrained test examples (1)

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Triaxial undrained test examples (1)

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Triaxial undrained test examples (2)

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Outline

1 Context

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Context From reality to laboratory From laboratory to numerical modelling Conclusion Cyclic behaviour characterization

The cyclic triaxial test

load/deformation controlled

drained/undrained until failure : liquefaction or not

q

t

qs qcycl qs qcycl

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The cyclic triaxial test

drained/undrained

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The cyclic triaxial test

drained/undrained until failure : liquefaction or not

Liquefaction failure (general term)

Liquefaction and liquefaction failures encompass all phenomena involving excessive deformations of saturated cohesionless soils.

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Cyclic behaviour

after [Hyodo and al. 1994] q p' q εa q εa q p' Dr=50% Dr=70% Dr=50% Dr=50% Initial liquefaction

Transient state when the soil sample is submitted to zero mean effective stress (u = pconf) and zero deviatoric stress for the first

time. This phenomenon involves very large deformations. Liquefaction (specific term)

True liquefaction occurs when the soil reaches the steady state and deforms continuously.

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Cyclic behaviour

after [Hyodo and al. 1994] q εa q q p' p' Dr=50% Dr=70% Dr=50% Dr=50% Cyclic mobility

The cyclic mobility denotes the undrained cyclic soil response where the soil undergoes strain softening which is mainly a consequence of the build up of pore water pressure.

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Cyclic behaviour

after [Hyodo and al. 1994] flow q p' FL FL flow Δu time flow εa time loose sand εa q p' qs 0 flow flow flow Δu time time loose sand Flow deformation

Flow deformation is an instability characterized by a quick

development of strain and pore pressure. If after this phenomenon, the strain increases slowly again, this behaviour is called limited

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Flow deformation

Structural collapse

In a loose sand in undrained conditions, the structural collapse is the specific response of the loose structure which is exhibited as vigorous pore pressure generation.

Arc grosse (a) cavité Metastable structure 1 2 3 X E

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Flow deformation

Structural collapse

Arc

grosse

(a) (b)

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Flow deformation

Structural collapse

Arc

grosse

(a) (b) (c)

cavité

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Flow deformation

Structural collapse

FL PT FL PT qcycl qs SPT SPT q=σ a -σr p'=(σ'a+2σ'r)/3 FL PT FL PT qcycl qs SPT SPT q=σ a -σr p'=(σ'a+2σ'r)/3 (a) (b)

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Flow deformation examples

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Outline

1 Context

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Context From reality to laboratory From laboratory to numerical modelling Conclusion Summary

The model has to take into account :

the contractive/dilative transition and softening ;

the pore pressure build up ;

the different modes of failure ;

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the contractive/dilative transition and softening ;

the pore pressure build up ;

the failure anisotropy ;

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Context From reality to laboratory From laboratory to numerical modelling Conclusion Summary

the contractive/dilative transition and softening ; the pore pressure build up ;

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the contractive/dilative transition and softening ; the pore pressure build up ;

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Context From reality to laboratory From laboratory to numerical modelling Conclusion Prevost’s model

Outline

1 Context

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Characterization

N nested conical yield surfaces associated with H’, M, α ;

Kinematic hardening in the stress-space ; Calibration using monotonic triaxial tests ; Non-associative volumic plastic potential

Sophistication : p’ dependency, Lode angle dependency, ...

σ'

₁

σ'

₁

σ'

3

p'

₀₀ 0

p'

(p'+p')α

σ'

₁ 3 2

σ'

₂ 3 2

σ'

₃ 3 2

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Characterization

N nested conical yield surfaces associated with H’, M, α ;

Kinematic hardening in the stress-space ;

Calibration using monotonic triaxial tests ; Non-associative volumic plastic potential

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Characterization

N nested conical yield surfaces associated with H’, M, α ; Kinematic hardening in the stress-space ;

Calibration using monotonic triaxial tests ;

Non-associative volumic plastic potential

M1 M2 M M α p' q εy p' q q

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Characterization

Calibration using monotonic triaxial tests ;

Non-associative volumic plastic potential

P” = 1 − η ¯ η 2 1 + η ¯ η 2 η = q p0 p'

q

contractive Dilative

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Characterization

Calibration using monotonic triaxial tests ; Non-associative volumic plastic potential

H0 = H₀0 · p0 pref n , K = K0· p0 pref n , G = G0· p0 pref n 1 σ 2 σ 3 σ 3 2 1σ σ σ= =

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Numerical examples

0 20 40 60 80 100 120 140 160 180 200 -100 -50 0 50 100 150 200 250 p [kPa] q [ kP a]

Influence de p'_init(kPa)

90 70 60 40 PT

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Numerical examples

0 10 20 30 40 50 60 70 80 90 100 -40 -20 0 20 40 60 80 p [kPa] q [ kP a]

Influence de p'_init(kPa)

90 70 60 40 PT

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Numerical examples

100 150 200 250 300 350 400 450 -200 -100 0 100 200 300 400 500 600 700 p [kPa] q [ kP a]

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Numerical examples

-25 -20 -15 -10 -5 0 5 10 15 20 25 -200 -150 -100 -50 0 50 100 150 200 250 Essais Drainés p'=cst ε_y[%] q [ kP a]

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Numerical examples

-25 -20 -15 -10 -5 0 5 10 15 20 25 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 Essais Drainés p'=cst ε_y[%] εv [% ]

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Numerical examples

0 50 100 150 200 250 300 350 400 -400 -300 -200 -100 0 100 200 300 400

500 Essais Non Drainés

p [kPa]

q [

kP

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Numerical examples

-5 -4 -3 -2 -1 0 1 2 3 4 5 -400 -300 -200 -100 0 100 200 300 400

500 Essais Non Drainés

ε_y[%]

q [

kP

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Numerical examples

-10 0 10 20 30 40 50 60 70 80 90 -20 -10 0 10 20 30 40 50 60 70 p' [kPa] q [ kP a]

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Context From reality to laboratory From laboratory to numerical modelling Conclusion

To conclude

1 _{Very complex actual behaviour}

• pore pressure buildup

• different modes of failure

• contractive/dilative transition

• instability

2 Prevost model : qualitatively OK...

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