Transcript Non-Linear Hyperbolic Model & Parameter Selection

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

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 
1
Gur 
E ur
2(1  )
pref 100kPa

Initial (primary) loading

m
'




c
cot

ref
3
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
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




qf
qa 
 M( p  c cot )R1
f
Rf
Equi-g lines (Tatsuoka, 1972) for dense Toyoura Sand

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 
f 
qa
q
2q

E 50 qa  q E ur
qa
q
2q

p 0
E 50 qa  q E ur
3D extension
In order to extent the model to general 3D states in terms of stress, we use
a modified expression for q in terms of q˜ and the mobilized angle of
internal friction  m
q˜  1'  ( 1) '2   '3
3  sin  m 

˜ ( p  c cot )
f  q˜  M
where  

Computational Geotechnics
3  sin  m
˜  6sin m
M
3  sin m
Non-Linear Hyperbolic Model & Parameter Selection


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

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; p ref  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
ref
3 sin  u  Cu cos u
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˜
f c  2  p 2  pc2
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  p c

g
pc  H   H  c
 2H  c p
pc


p
v

For the determination of, we have another consistency condition:
f c T  f c 
fc

pc  0

pc

 Computational Geotechnics
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 user 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 frictiona l hardening with f
III: Material f ailure according to Mohr-Coulomb
IV: Mohr- Coulomb and cap fc
V: Combin ed frictional hardening f and c ap 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 Power in stiffness laws
R f Failure ratio



Computational
Geotechnics
Non-Linear Hyperbolic Model & Parameter Selection