Video lecture for mca ! Edhole

Video Lecture for MCA
By:
video.edhole.com

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
video.edhole.com

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 video.edhole.com & 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.

Rf =
qf
qa

Eur=3E50
Hyperbolic stress strain response curve of Hardening Soil model

Stiffness Modulus
Elastic unloading and reloading (Ohde, 1939)
We use the two elastic parameters nur and Eur
Definition of E50 in a standard drained triaxial experiment

æ
è
ref ccotj -s3
æ
è
ref s3
ç
E50 =E50
m
' +ccotj
pref +ccotj
ç
m
ö
÷
ø
æ
è
ref s3
=E50
' sinj +ccosj
pref sinj +ccosj
ç
ö
ø
m
÷

Eur
'
ccotj + pref
ö
÷
ø
Gur = 1
2(1+n)
Eur

pref=100kPa
Initial (primary) loading

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

ref
Eoed
Computational Geotechnics Non-Linear Hyperbolic Model videos.edhole.com & Parameter Selection

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

Stiffness Modulus
Values for m obtained from triaxial test versus initial porosity n0
Normalized triaxial modulus versus initial porosity n0

ref
E50

Stiffness Modulus
Summary of data for sand: Vermeer & Schanz (1997)
Comparison of normalized stiffness moduli from oedometer and
Triaxial test

ref s y
Eoed = Eoed
'
pref
ref s x
E50 = E50
'
pref
Engineering practice: mostly data on Eoed
Test data:

ref »E50
Eoed
ref

Triaxial Data on gp » 2e1
p
Equi-g lines (Tatsuoka, 1972) for dense Toyoura Sand
Yield and failure surfaces for the Hardening Soil model

= qa
E50
2e1
q
qa-q

æ
è
ref s3
E50=E50
m
' sinj+ccosj
pref sinj+ccosj
ç
ö
÷
ø

qf
qa=
Rf
-1
=M(p+ccotj)Rf

M= 6sinj
3-sinj

Plasticity
Yield and hardening functions

gp =e1
p-e2
p-e3
p »2e1
p =2e1
e = qa
-2e1
E50
q
qa-q
-2q
Eur
f = qa
E50
q
qa-q
-2q
Eur
-gp =0
3D extension
In 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˜

jm
˜ q =s1'+(a-1)s2
3
'-a'
sa=3+sinjm
3-sinjm
˜ = 6sinjm
f=q˜ -M˜ (p+ccotj)
M3-sinjm
where
Compvutiadtioenaol Gseo.teechdnichs ole.com Non-Linear Hyperbolic Model & Parameter Selection

Plasticity

q*
=s1
'+(b-1)s2
3
'-b'
s
b=3+sinym
3-sinym

g=-m
y(p+ccot)
* M*
q
M*= 6sinym
3-sinym
Plastic potential and flow rule
with

ê
ê
ê
ê
ê
·
=
ep
·
e1
p
·
ú
ú
ú
ú
ú
e2
p
·
e3
p
é
ë
ù
=L ·
û
¶g
¶s12
12
+L ·
¶g
¶s13
13
=L ·
ê
ê
ê
12
1
2-1
2siny
-1
ú
ú
ú
+L ·
2-1
2siny
0
é
ë
ù
û
ê
ê
ê
13
1
2-1
2siny
0
-1
ú
ú
ú
2-1
2siny
é
ë
ù
û

Plasticity
Flow rule
with
·
· =sinym
·
=sinygp
=sinjm
C [kPa] j’ [o] y [o] E50 [Mpa]
0 30-40 0-10 40
Eur = 3 E50 Vur = 0.2 Rf = 0.9 m = 0.5 Pref = 100 kPa

ev
p
gp
Þp
v
e·

sinym
-sinjcv
1-sinjm
sinjcv

jcv=jp
-yp
Primary soil parameters and standard PLAXIS settings

Plasticity
Hardening soil response in drained triaxial experiments
Results of drained loading:
stress-strain relation (s3 = 100 kPa)
Results of drained loading:
axial-volumetric strain relation (s3 = 100 kPa)

Plasticity

Undrained hardening soil analysis
Method A: switch to drained
Input:
c';j';y'
Eref
50
ì
í
ï
ï
î
nur=0.2;Eur=3E50;m=0.5;pref =100kPa

Method B: switch to undrained
Input:
cu;ju
;y
ref
E50
ì
í
ï
îï
nur=0.2;Eur=3E50;m=0.5;pref =100kPa

Plasticity
Interesting in case you have data on Cu and not no C’ and f’
2cu
m
=E50
m
=Eur
Eu » 1.4 E50
50 =E50
Eref s3
' sinju
+Cucosju
pref sinju
+Cucosju
æ
ç
è
ö
÷
ø
ref =const.
+Cucosju æ
è
ref s3
Eur =Eur
' sinju
+Cucosju
ç
pref sinju
ö
÷
ø
ref =const.
Assume E50 = 0.7 Eu and use graph by Duncan & Buchignani (1976) to estimate Eu

Plasticity
Hardening soil response in undrained triaxial tests
Results of undrained triaxial loading:
stress-strain relations (s3 = 100 kPa)
Results of undrained triaxial loading: p-q
diagram (s3 = 100 kPa)

HS-Cap-Model

fc=q˜ 2
M2+p2-pc
2
gc=fc
·
= p ·
ep
v
Kc
- p ·
Ks
(Associated flow)
=1
H
p ·

H= Kc
Ks-Kc
Ks
Cap yield surface
Flow rule
Hardening law
For isotropic compression we assume
with

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

p ·
=p ·
¶g
¶pc
·
=HL ·
For the determination of, we have another consistency condition:
c

p ·
v
=Hep
cc
=2HL ·
cp

T
·
c=¶fc
f
¶s
+¶fc
¶pc
s ·
p ·
c=0

HS-Cap-Model
Additional parameters
The extra input parameters are K0 (=1-sinf) 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.

HS-Cap-Model
1
Graphical presentation of HS-Cap-Model
I: 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
2 3
Yield surfaces of the extended HS model in p-q space (left) and in the deviatoric plane (right)

HS-Cap-Model
s1 = s2 = s3
Yield surfaces of the extended HS model in principal stress space

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

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

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
As in Mohr-Coulomb model
Normalized primary loading stiffness
Unloading / reloading Poisson’s ratio
Normalized unloading / reloading stiffness
Power in stiffness laws
Failure ratio

c,j,y

ref
E50

nur

ref
Eur

m
Rf

Video lecture for mca ! Edhole

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