Non-Linear Hyperbolic Model & Parameter Selection

Short Course on Computational Geotechnics + DynamicsBoulder, ColoradoJanuary 5-8, 2004

Stein StureProfessor of Civil EngineeringUniversity 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

IntroductionHardening 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 creepCreep is usually of greater significance in soft soils.

R f q f

qa

Eur 3E50

Hyperbolic stress strain response curve of Hardening Soil modelComputational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

Stiffness Modulus

Definition of E50 in a standard drained triaxial experiment

E50 E50ref 3

' c cotpref c cot

m

E50ref 3

' sin c cospref sin c cos

m

Eurref c cot 3

'

c cot pref

m

Gur 1

2(1)Eur

pref 100kPa

Elastic unloading and reloading (Ohde, 1939)We use the two elastic parameters ur and Eur

Initial (primary) loading

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

Stiffness Modulus

Definition of the normalized oedometric stiffness

Values for m from oedometer test versus initial porosity n0

Normalized oedometer modulus versus initial porosity n0

Oedometer tests

Eoedref

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

Stiffness ModulusNormalized 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 versus initial porosity n0

E50ref

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

Stiffness Modulus

Comparison of normalized stiffness moduli from oedometer andTriaxial test

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

Eoed Eoedref y

'

pref

E50 E50ref x

'

pref

Engineering practice: mostly data on Eoed

Test data:

Eoedref E50

ref

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

Triaxial Data on p 21p

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

Yield and failure surfaces for the Hardening Soil model

21 qa

E50

qqa q

E50 E50ref 3

' sin c cospref sin c cos

m

qa q f

R f

M( p c cot)R f 1

M 6sin

3 sin

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

PlasticityYield and hardening functions

p 1p 2

p 3p 21

p 21 21e

qa

E50

qqa q

2qEur

f qa

E50

qqa q

2qEur

p 0

3D extensionIn order to extent the model to general 3D states in terms of stress, we use a modified expression for in terms of and the mobilized angle of internal friction

q

˜ q

m

˜ q 1' ( 1) 2

' 3'

3 sinm

3 sinm

f ˜ q ˜ M ( p c cot)

˜ M 6sinm

3 sinmwhere

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

Plasticity

q 1' ( 1) 2

' 3'

3 sinm

3 sinm

g q M( p c cotm )

M 6sinm

3 sinm

Plastic potential and flow rule

with

p

1p

2p

3p

12g12

13g13

12

12

12 sin

12

12 sin0

13

12

12 sin0

12

12 sin

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

PlasticityFlow rule

with

C [kPa] ’ [o] [o] E50 [Mpa]  0 30-40 0-10 40  Eur = 3 E50 Vur = 0.2 Rf = 0.9 m = 0.5 Pref = 100 kPa

vp

p sinm v

p

sinp

sinm sinm sincv

1 sinm sincv

cv p p

Primary soil parameters and standard PLAXIS settings

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

Plasticity

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

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

Hardening soil response in drained triaxial experiments

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

Plasticity

c '; '; '

E50ref

ur 0.2;Eur 3E50;m 0.5; pref 100kPa

cu;u;E50

ref

ur 0.2;Eur 3E50;m 0.5; pref 100kPa

Undrained hardening soil analysis

Method A: switch to drainedInput:

Method B: switch to undrainedInput:

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

2cu

Eu 1.4 E50

PlasticityInteresting in case you have data on Cu and not no C’ and ’

E50 E50ref 3

' sinu Cu cosu

pref sinu Cu cosu

m

E50ref const.

Eur Eurref 3

' sinu Cu cosu

pref sinu Cu cosu

m

Eurref const.

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

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

Plasticity

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

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

Hardening soil response in undrained triaxial tests

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

HS-Cap-Model

fc ˜ q 2

M 2 p2 pc2

gc fc

vp

p

Kc

p

Ks

1H

p

H Kc

Ks Kc

Ks

Cap yield surface

Flow rule

Hardening lawFor isotropic compression we assume

(Associated flow)

with

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

HS-Cap-ModelFor isotropic compression we have q = 0 and it follows from

For the determination of, we have another consistency condition:

pp

c

p

c Hvp

H

cgpc

2H

c p

f

c f c

T

f c

pc

p

c 0

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

HS-Cap-ModelAdditional 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 user should not be too concerned about these parameters.

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

HS-Cap-Model

1

2 3

Graphical presentation of HS-Cap-ModelI: Purely elastic response

II: Purely frictional hardening with f

III: Material failure according to Mohr-Coulomb

IV: Mohr-Coulomb and cap fc

V: Combined frictional hardening f and cap fc

VI: Purely cap hardening with fc

VII: Isotropic compression

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-Model1 = 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

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

HS-model versus MC-modelAs in Mohr-Coulomb modelNormalized primary loading stiffnessUnloading / reloading Poisson’s ratio

Normalized unloading / reloading stiffnessPower in stiffness lawsFailure ratio

c,,

E50ref

ur

Eurref

m

R f

Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection