course notes on nonlinear hyperbolic model and parameter selection

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

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Contents

IntroductionStiffness ModulusTriaxial DataPlasticityHS-Cap-ModelSimulation of Oedometer and Triaxial Tests on Loose and

Dense SandsSummary

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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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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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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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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 m obtained from triaxial test !ersus initial porosity n .

*ig"


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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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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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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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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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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

(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

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6c u

E u +"9 E

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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

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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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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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HS-Cap-Model

*ig" +


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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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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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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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#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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esults for loose sand

*ig" 6/, Triaxial tests on loose Hostun Sand

*ig" 69, Oedometer tests on loose Hostun Sand

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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

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