course notes on nonlinear hyperbolic model and parameter selection.doc

8/13/2019 Course Notes on Nonlinear Hyperbolic Model and Parameter Selection.doc

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Non-Linear Hyperbolic Model & Parameter Selection

NON-LINEAR HYPERBOLIC MOEL & 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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Contents

Introduction

Stiffness Modulus

Triaxial Data

Plasticity

HS-Cap-Model

Simulation of Oedometer and Triaxial Tests on Loose and

Dense SandsSummary

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Stiffness Modulus

#lastic unloading and reloading 0Ode% +4/43

5e use te t)o elastic parameters urandEur:

cot

cot

m

3refur ref

c - 'E

c + p

+

60+ 3ur ur

ur

G E

=+ +..

refp kPa=

Initial 0primary3 loading

7 7

/ /-. -. -.

cot sin cos

cot sin cos

m m

ref ref

ref ref

c cE E E

p c p c

+ += = + +

*ig" 6, Definition ofE50in a standard drained triaxial experiment

&ourse '&omputational %eotechnics %


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Stiffness Modulus

Oedometer tests

*ig" /, Definition of te normali8ed oedometric stiffnessref

oedE

*ig" 9, 1alues for mfrom oedometer test !ersus initial porosity n.

*ig" , :ormali8ed oedometer modulusref

oedE !ersus initial porosity n.

&ourse '&omputational %eotechnics


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Stiffness Modulus

Triaxial tests

*ig" ;, :ormali8ed oedometric stiffness for !arious soil classes 0!on Soos% +44+3

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Stiffness Modulus

*ig" 2, 1alues for mobtained from triaxial test !ersus initial porosity n.

*ig"


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Stiffness Modulus

Summary of data for sand, 1ermeer = Scan8 0+4423

*ig" 4, Comparison of normali8ed stiffness moduli from oedometer and triaxial

tests

7

yref

oed oed refE E

p

=

7

-. -.

ref x

refE E

p

=

#ngineering practice, mostly data onEoed

Test data, -.ref ref

oedE E

0standard setting PL&'IS !ersion 23

&ourse '&omputational %eotechnics )


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Triaxial Data on p6+p

*ig" +., #$ui-g lines 0Tatsuo>a% +4263 for dense Toyoura Sand

+

-.

6 a

a

q q

E q q =

7

/-. -.

sin cos

sin cos

m

ref

ref

cE E

p c

+= +

+0 cot 3f

a f

f

qq M p c R

R = = +

;sin

/ sinM

=

*ig" ++, ?ield and failure surfaces for te Hardening Soil model

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Plasticity

?ield and ardening functions

+ 6 / + + +

-.

66 6 6p p p p p e a

a ur

q q q

E q q E

= = =

{+ +

-.

6 6

6.

e

pa

a ur

q q qf

E q q E

= =

142 43

/D extension

In order to extent te model to general /D states in terms of stress% )e use a

modified expression for q in terms of %q and te mobili8ed angle of internal

friction m

7 7 7

+ 6 /0 +3q = + %

)it

/ sin

/ sin

m

m

+=

0 cot 3mf q M p c = +%%

)ere

;sin

/ sin

m

m

M

=

%

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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

mM

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 sinp

p pvm v m

p

= =

)it

sin sinsin

+ sin sin

m cvm

m cv

=

cv p p =

Table +, Primary soil parameters and standard PL&'IS settings

C A>PaB AoB AoB E50 AMpaB

. /.-9. .-+. 9.

Eur /E. Vur ."6 Rf ."4 m ." Pref +.. >Pa

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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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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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HS-Cap-Model

Cap yield surface6

6 6

6c c

qf p p

M

= + %

*lo) rule

c cf=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 fromcp p

=

6p ccv ccp # # # pp

= = = g

*or te determination of% )e use anoter consistency condition,

.

$

c cc c

c

f ff p

p

= + =

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HS-Cap-Model

&dditional parameters

Te extra input parameters are . 0 + sin 3" = and -.H 0 +".3oedE E =

Te t)o auxiliary material parameter M and cs are determined iterati!ely fromte simulation of an oedometer test" Tere are no direct input parameters" Te

user sould not be too concerned about tese parameters"

Jrapical presentation of HS-Cap-Model

I, Purely elastic response

II, Purely frictional ardening )itf

III, Material failure according to Mor-Coulomb

I1, Mor-Coulomb and capfc

1, Combined frictional ardeningfand capfc

1I, Purely cap ardening )itfc

1II, Isotropic compression

*ig" +2, ?ield surfaces of te extended HS model in p-$-space 0left3 and in te

de!iatoric plane 0rigt3

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+

6 /


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HS-Cap-Model

*ig" +


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Dense Sands

*ig" +4, Comparison of calculated 0K3 and measured triaxial tests on loose Hostun

Sand

*ig" 6., Comparison of calculated 0K3 and measured oedometer tests on loose

Hostun Sand

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Dense Sands

*ig" 6+, Comparison of calculated 0K3 and measured triaxial tests on dense Hostun

Sand

*ig" 66, Comparison of calculated 0K3 and measured oedometer tests on dense

Hostun Sand

&ourse '&omputational %eotechnics "*


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Summary

Main caracteristics

Pressure dependent stiffness

Isotropic sear ardening

(ltimate Mor-Coulomb failure condition

:on-associated plastic flo)

&dditional cap ardening

HS-model !ersus MC-model

% %c &s in Mor-Coulomb model

-.

refE :ormali8ed primary loading stiffness

ur (nloading reloading PoissonGs ratioref

urE :ormali8ed unloading reloading stiffness

m Po)er in stiffness la)sfR *ailure ratio

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#xercise +, Calibration of te HS-Cap-Model for Loose

and 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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#xercise +, Calibration of te HS-Cap-Model for Loose

and Dense Sand

esults for dense sand

*ig" 6, Triaxial tests on dense Hostun Sand

*ig" 6;, Oedometer tests on dense Hostun Sand

course notes on nonlinear hyperbolic model and parameter selection.doc

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