course notes on nonlinear hyperbolic model and parameter selection
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
NON-LINEAR HYPERBOLIC MO EL & PARAME!ERSELEC!ION
(Introduction to the Hardening Soil Model)
(following initial development byTom Schanz at Bauhaus !niversit"t #eimar$ %ermany)
&omputational %eotechnics
&ourse '&omputational %eotechnics "
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Contents
IntroductionStiffness ModulusTriaxial DataPlasticityHS-Cap-ModelSimulation of Oedometer and Triaxial Tests on Loose and
Dense SandsSummary
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Introduction
Hardening SoilsMost soils be a!e in a nonlinear be a!ior soon after application of s earstress" #lastic-plastic ardening is a common tec ni$ue% also used in PL&'IS"
(sage of t e Soft Soil model )it creepCreep is usually of greater significance in soft soils"
Hyperbolic stress-strain la) for triaxial response cur!es
*ig" +, Hyperbolic stress strain response cur!e of Hardening Soil model
f f
a
q R
q=
-./ur E E =
0standard PL&'IS setting 1ersion 23
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Stiffness Modulus
#lastic unloading and reloading 0O de% +4/43
5e use t e t)o elastic parameters ur and E ur :
cot
cot
m
3ref ur ref
c - ' E
c + p
+60+ 3ur ur ur
G E
= + +..ref p kPa=
Initial 0primary3 loading
7 7/ /
-. -. -.
cot sin coscot sin cos
m m
ref ref ref ref
c c E E E
p c p c
+ += = + +
*ig" 6, Definition of E 50 in a standard drained triaxial experiment
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Stiffness Modulus
Oedometer tests
*ig" /, Definition of t e normali8ed oedometric stiffness ref oed E
*ig" 9, 1alues for m from oedometer test !ersus initial porosity n .
*ig" , :ormali8ed oedometer modulus ref oed E !ersus initial porosity n .
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Stiffness Modulus
Triaxial tests
*ig" ;, :ormali8ed oedometric stiffness for !arious soil classes 0!on Soos% +44+3
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Stiffness Modulus
*ig" 2, 1alues for m obtained from triaxial test !ersus initial porosity n .
*ig"
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Stiffness Modulus
Summary of data for sand, 1ermeer = Sc an8 0+4423
*ig" 4, Comparison of normali8ed stiffness moduli from oedometer and triaxialtests
7 yref
oed oed ref E E p
=7
-. -.ref x
ref E E p =
#ngineering practice, mostly data on E oed
Test data, -.ref ref oed E E
0standard setting PL&'IS !ersion 23
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Triaxial Data on p 6+ p
*ig" +., #$ui-g lines 0Tatsuo>a% +4263 for dense Toyoura Sand
+-.
6 aa
q q E q q
= 7/
-. -.
sin cossin cos
m
ref ref
c E E
p c
+= +
+0 cot 3 f a f f
qq M p c R
R = = + ;sin
/ sin M
=
*ig" ++, ?ield and failure surfaces for t e Hardening Soil model
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Plasticity
?ield and ardening functions
+ 6 / + + +
-.
66 6 6 p p p p p e a
a ur
q q q
E q q E
= = =
{+ +
-.
6 6
6.
e
pa
a ur
q q q f
E q q E
= =142 43
/D extension
In order to extent t e model to general /D states in terms of stress% )e use a
modified expression for q in terms of %q and t e mobili8ed angle of internal
friction m
7 7 7+ 6 /0 +3q = + %
)it
/ sin/ sin
m
m
+= 0 cot 3m f q M p c = +
%%
) ere
;sin
/ sin
m
m
M
=
%
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Plasticity
Plastic potential and flo) rule
@ 7 7 7+ 6 /0 +3q = +
)it/ sin/ sin
m
m
+=
@ @0 cot 3mq M p c = +g) ere
m
m M
sin/
sin;@
=
+ + + + +6 6 6 6
+/+6 + +6 6 6
+6 +/ +6 +/ + +6 6
/
sin sin
sin .
. sin
p
p p
p
= = + = +
gg
*lo) rule
sin sin p
p pvm v m
p
= =
)itsin sin
sin+ sin sin
m cvm
m cv
= cv p p =
Table +, Primary soil parameters and standard PL&'IS settingsC A>PaB AoB AoB E50 AMpaB. /.-9. .-+. 9.
Eur / E . Vur ."6 Rf ."4 m ." Pref +.. >Pa
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Plasticity
Hardening soil response in drained triaxial experiments
*ig" +6, esults of drained triaxial loading, stress-strain relations 0s/ +.. >Pa3
*ig" +/, esults of drained triaxial loading, axial-!olumetric strain relations 0s/ +.. >Pa3
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Plasticity
(ndrained ardening soil analysis
Met od &, s)itc to drained
Input, 7 7 7
-.
-.
E E
."6E / E ."-E +..
ref
ref ur ur
c
E
E E m p kPa
= = = =
Met od F, s)itc to undrainedInput,
-.
-.
E E .
."6E / E ."-E +..
u u
ref
ref ur ur
c
E E E m p kPa
=
= = = =
Interesting in case you a!e data on C u and not no CG and G7/
-. -. -.
sin cos"
sin cos
m
ref ref u u uref
u u u
c E E E con !
p c
+= = = +
7/ sin cos
"sin cos
m
ref ref u u uur ur ur ref
u u u
c E E E con ! p c
+= = = +
&ssume -. ."2 u E E = and use grap by Duncan = Fuc ignani 0+42;3 to estimate E u
*ig" +9, (ndrained Hardening Soil analysis
&ourse '&omputational %eotechnics "$
6c u
E u +"9 E
.
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Plasticity
Hardening soil response in undrained triaxial tests
*ig" + , esults of undrained triaxial loading, stress-strain relations 0s/ +.. >Pa3
*ig" +;, esults of undrained triaxial loading, p-$ diagram 0s/ +.. >Pa3
&ourse '&omputational %eotechnics "%
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
HS-Cap-Model
Cap yield surface6
6 66c c
q f p p
M
= + %
*lo) rule
c c f =g 0&ssociated flo)3
Hardening la)
*or isotropic compression )e assume
+ pvc
p p p " " #
= =
5it
c
c
" # "
" " =
*or isotropic compression )e a!e $ . and it follo)s fromc p p
=
6 p ccv cc p # # # p p
= = = g
*or t e determination of% )e use anot er consistency condition,
.$
c cc c
c
f f f p
p
= + =
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
HS-Cap-Model
&dditional parameters
T e extra input parameters are . 0 + sin 3 " = and -.H 0 +".3oed E E =
T e t)o auxiliary material parameter M and c s are determined iterati!ely fromt e simulation of an oedometer test" T ere are no direct input parameters" T euser s ould not be too concerned about t ese parameters"
Jrap ical presentation of HS-Cap-Model
I, Purely elastic response
II, Purely frictional ardening )it f
III, Material failure according to Mo r-Coulomb
I1, Mo r-Coulomb and cap f c
1, Combined frictional ardening f and cap f c
1I, Purely cap ardening )it f c
1II, Isotropic compression
*ig" +2, ?ield surfaces of t e extended HS model in p-$-space 0left3 and in t ede!iatoric plane 0rig t3
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+
6 /
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
HS-Cap-Model
*ig" +
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Simulation of Oedometer and Triaxial Tests on Loose andDense Sands
*ig" +4, Comparison of calculated 0K3 and measured triaxial tests on loose HostunSand
*ig" 6., Comparison of calculated 0K3 and measured oedometer tests on looseHostun Sand
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Simulation of Oedometer and Triaxial Tests on Loose andDense Sands
*ig" 6+, Comparison of calculated 0K3 and measured triaxial tests on dense HostunSand
*ig" 66, Comparison of calculated 0K3 and measured oedometer tests on denseHostun Sand
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8/10/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection
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Non-Linear Hyperbolic Model & Parameter Selection
Summary
Main c aracteristics
Pressure dependent stiffness
Isotropic s ear ardening
(ltimate Mo r-Coulomb failure condition
:on-associated plastic flo)
&dditional cap ardening
HS-model !ersus MC-model
% %c &s in Mo r-Coulomb model-.ref E :ormali8ed primary loading stiffness
ur (nloading reloading PoissonGs ratioref ur E :ormali8ed unloading reloading stiffness
m Po)er in stiffness la)s f R *ailure ratio
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Non-Linear Hyperbolic Model & Parameter Selection
#xercise +, Calibration of t e HS-Cap-Model for Looseand Dense Sand
Oedometer and triaxial s ear experimental data for bot loose and dense sands aregi!en in *igs" 6/ 6;"
Table 6, Parameters for loose and dense sand vur m -.Href ref oed E E -.Href ref ur E E -.ref E
loose ."6 ."; /9 o . o +". /". +;dense ."6 ."; 9+ o +9o ."4 /". / MPa
Proceed according to t e follo)ing steps,
(se "o + sin and # oed # . according to Table 6 in t e ad!anced material
parameter input in PL&'IS"
*or bot simulations use an axis-symmetric mes 0+x + AmB3 )it a coarseelement density" C ange loading and boundary conditions according to t etest conditions"
Simulation of oedmoter tests )it unloading for unloading for maximum axialstress"
Loose sand, max+ 6.. kPa =
Dense sand,max
+ 9.. kPa =
If necessary impro!e gi!en material parameters to obtain a more realisticresponse"
C ec> triaxial tests )it t e parameters obtained from t e oedometersimulation"
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Non-Linear Hyperbolic Model & Parameter Selection
#xercise +, Calibration of t e HS-Cap-Model for Looseand Dense Sand
esults for loose sand
*ig" 6/, Triaxial tests on loose Hostun Sand
*ig" 69, Oedometer tests on loose Hostun Sand
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Non-Linear Hyperbolic Model & Parameter Selection
#xercise +, Calibration of t e HS-Cap-Model for Looseand Dense Sand
esults for dense sand
*ig" 6 , Triaxial tests on dense Hostun Sand
*ig" 6;, Oedometer tests on dense Hostun Sand
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