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Video lecture for mbaTRANSCRIPT
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
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Contents
Introduction
Stiffness Modulus
Triaxial Data
Plasticity
HS-Cap-Model
Simulation of Oedometer and Triaxial Tests on Loose and Dense Sands
Summary
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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 Selectionvideo.edhole.com
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
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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
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Stiffness ModulusNormalized oedometric stiffness for various soil classed (von Soos, 1991)
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Stiffness Modulus
Values for m obtained from triaxial test versus initial porosity n0
Normalized triaxial modulus versus initial porosity n0
E50ref
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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
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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
q
qa 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
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PlasticityYield and hardening functions
p 1p 2
p 3p 21
p 21 21e
qa
E50
q
qa q
2q
Eur
f qa
E50
q
qa q 2q
Eur
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 sinm
where
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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
12g
12
13g
13
12
12
12 sin
12
12 sin0
13
12
12 sin0
12
12 sin
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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
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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
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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:
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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
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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
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HS-Cap-Model
fc ˜ q 2
M 2 p2 pc
2
gc fc
vp
p
Kc
p
Ks
1
Hp
H Kc
Ks Kc
Ks
Cap yield surface
Flow rule
Hardening lawFor isotropic compression we assume
(Associated flow)
with
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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
cg
pc
2H
c p
f
c fc
T
fc
pc
p
c 0
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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.
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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 Selectionvideos.edhole.com
HS-Cap-Model
1 = 2 = 3
Yield surfaces of the extended HS model in principal stress space
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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
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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
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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 model
Normalized primary loading stiffness
Unloading / reloading Poisson’s ratio
Normalized unloading / reloading stiffness
Power in stiffness laws
Failure ratio
c,,
E50ref
ur
Eurref
m
R f
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