Hardening Soil 1re40

Non-Linear Hyperbolic Model & Parameter Selection Short Course on Computational Geotechnics + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder

Contents Introduction Stiffness Modulus Triaxial Data Plasticity HS-Cap-Model Simulation of Oedometer and Triaxial Tests on Loose and Dense Sands Summary Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

Introduction Hardening Soils Most soils behave in a nonlinear behavior soon after application of shear stress. Elastic-plastic hardening is a common technique, also used in PLAXIS.

Usage of the Soft Soil model with creep Creep is usually of greater significance in soft soils.

qf Rf  qa

E ur  3E 50  Hyperbolic stress strain response curve of Hardening Soil model Computational Geotechnics

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Non-Linear Hyperbolic Model & Parameter Selection

Stiffness Modulus Elastic unloading and reloading (Ohde, 1939) We use the two elastic parameters ur and Eur ' m  c cot    3 E urref  ref  c cot  p 

Gur 



pref  100kPa

1 E ur 2(1  )

 Initial (primary) loading m

 '  ref  3  c cot E 50  E 50  ref  p  c cot 

  '3 sin   c cos m  E  ref  p sin  c cos  ref 50

Definition of E50 in a standard drained triaxial experiment

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Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

Stiffness Modulus Oedometer tests

Definition of the normalized oedometric stiffness

Values for m from oedometer test versus initial porosity n 0 ref Normalized oedometer modulusE oed versus initial porosity n 0

Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection 

Stiffness Modulus Normalized oedometric stiffness for various soil classed (von Soos, 1991)

Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

Stiffness Modulus

Values for m obtained from triaxial test versus initial porosity n0 Normalized triaxial modulus E 50ref versus initial porosity n0 

Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

Stiffness Modulus Summary of data for sand: Vermeer & Schanz (1997)

ref E oed  E oed

ref E 50  E 50

 'y p ref

 'x p ref

Comparison of normalized stiffness moduli from oedometer and Triaxial test

Engineering practice: mostly data on Eoed  ref ref E  E Test data: oed 50

Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

Triaxial Data on p  21p 21 



qa q E 50 qa  q

m '    sin   c cos  ref 3 E 50  E 50  ref  p sin   c cos   

Equi-g lines (Tatsuoka, 1972) for dense Toyoura Sand

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

qa 

qf  M( p  c cot  )R1 f Rf

M

6sin  3  sin 

Yield and failure surfaces for the Hardening Soil model

Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

Plasticity Yield and hardening functions

 p  1p  2p  3p  21p  21  21e 

qa q 2q  E 50 qa  q E ur

qa q 2q f   p 0 E 50 qa  q E ur

3D extension In order to extent the model to general 3D states in of stress, we use a modified expression for q in of q˜ and the mobilized angle of internal friction  m

q˜  1'  ( 1) '2   '3 3  sin m  where   3  sin m

Computational Geotechnics

f  q˜  M ˜ ( p  c cot )

˜  6sin m M 3  sin m

Non-Linear Hyperbolic Model & Parameter Selection

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Plasticity Plastic potential and flow rule q  1'  ( 1) '2   '3

with 

3  sin m 3  sin m

g  q  M  ( p  c cotm )

M 

6sin m 3  sin m

    p   1   12  12 sin   12  12 sin       p   g g     p 1 1   2  12  13  12  2  2 sin 13  0  12 13  1 1  p      0      2 2 sin   3    

Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

Plasticity Flow rule  p v  p

 

 p v

 p

 sin m    sin 

with 

sin m 

sin  m  sin  cv 1 sin  m sin  cv

 cv   p   p

Primary soil parameters and standard PLAXIS settings



C [kPa] 0 Eur = 3 E50

Computational Geotechnics

’ [o]

 [o]

30-40 Vur = 0.2

0-10 Rf = 0.9



E50 [Mpa] 40 m = 0.5

Pref = 100 kPa

Non-Linear Hyperbolic Model & Parameter Selection

Plasticity Hardening soil response in drained triaxial experiments

Results of drained loading: stress-strain relation (3 = 100 kPa)

Computational Geotechnics

Results of drained loading: axial-volumetric strain relation (3 = 100 kPa)

Non-Linear Hyperbolic Model & Parameter Selection

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Plasticity Undrained hardening soil analysis Method A: switch to drained Input:

 c ' ; ' ; '  ref E 50    0.2;E  3E ;m  0.5; p ref  100kPa  ur ur 50 Method B: switch to undrained Input:

 c u ; u;  ref E 50    0.2;E  3E ;m  0.5; pref  100kPa  ur ur 50 Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

Plasticity Interesting in case you have data on Cu and not no C’ and ’ m '   ref  3 sin  u  Cu cos u ref E 50  E 50  ref   E 50  const. p sin  u  Cu cos u  m '    sin  u  Cu cos u ref E ur  E urref  ref3   E ur  const. p sin u  Cu cos u 

Assume E50 = 0.7 Eu and use graph by Duncan & Buchignani (1976) to estimate Eu Eu  1.4 E50

2c u

Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

Plasticity Hardening soil response in undrained triaxial tests

Results of undrained triaxial loading: stress-strain relations (3 = 100 kPa)

Computational Geotechnics

Results of undrained triaxial loading: p-q diagram (3 = 100 kPa)

Non-Linear Hyperbolic Model & Parameter Selection

HS-Cap-Model Cap yield surface 2

q˜ 2 2 f c  2  p  pc M

Flow rule

gc  f c

(Associated flow)

Hardening law For isotropic compression we assume  p v





p p 1      p Kc Ks H Computational Geotechnics

with

Kc H Ks Ks  Kc

Non-Linear Hyperbolic Model & Parameter Selection



HS-Cap-Model 



For isotropic compression we have q = 0 and it follows from p  pc  g pc  H   H  c  2H  c p pc 

 p v



For the determination of, we have another consistency condition:

f c f c  fc  pc  0  pc 

 Computational Geotechnics

T 

Non-Linear Hyperbolic Model & Parameter Selection

HS-Cap-Model Additional parameters The extra input parameters are K0 (=1-sin) and Eoed/E50 (=1.0)

The two auxiliary material parameter M and Kc/Ks are determined iteratively from the simulation of an oedometer test. There are no direct input parameters. The should not be too concerned about these parameters.

Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

HS-Cap-Model Graphical presentation of HS-Cap-Model I: Purely elastic response II: Purely f rictiona l hardening with f III: Material f ailu re according to Mohr-Coulomb IV: Mohr- Coulomb and cap fc V: Combin ed friction al hardening f and cap fc VI: Purely cap hardening with fc VII: Isotropic compression

1

2

3

Yield surfaces of the extended HS model in p-q space (left) and in the deviatoric plane (right) Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

HS-Cap-Model 1 = 2 = 3

Yield surfaces of the extended HS model in principal stress space

Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

Simulation of Oedometer and Triaxial Tests on Loose and Dense Sands

Comparison of calculated () and measured triaxial tests on loose Hostun Sand

Comparison of calculated () and measured oedometer tests on loose Hostun Sand Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

Simulation of Oedometer and Triaxial Tests on Loose and Dense Sands

Comparison of calculated () and measured triaxial tests on dense Hostun Sand

Comparison of calculated () and measured oedometer tests on dense Hostun Sand Computational Geotechnics

Non-Linear Hyperbolic Model & Parameter Selection

Summary Main characteristics •Pressure dependent stiffness •Isotropic shear hardening •Ultimate Mohr-Coulomb failure condition •Non-associated plastic flow •Additional cap hardening

HS-model versus MC-model

c,, As in Mohr-Coulomb model ref E 50 Normalized primary loading stiffness

 ur



Unloading / reloading Poisson’s ratio

E urref Normalized unloading / reloading stiffness







m Rf

Computational Geotechnics

Power in stiffness laws Failure ratio Non-Linear Hyperbolic Model & Parameter Selection