2.similitude laws in centrifuge modeling

8/9/2019 2.Similitude Laws in Centrifuge Modeling

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Similitude laws incentrifuge modelling


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Content

• Principle of scaling laws – Vashy-Bukingham theorem• Scaling laws for centrifuge tests• Scaling laws of water flow in centrifuge•Grain size effects on interface and shear band pattern

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• Vaschy-Buckingham theorem

« If we have a physically meaningful equation involving a certain number, n , of physicalvariables, and these variables are expressible in terms of k independent fundamentalphysical quantities, then the original expression is equivalent to an equation involving aset of = n − k dimensionless arameters constructed from the ori inal variables »

0,...,, 21 = n X X X f

kiii ki r r r X

×××→ ...21 21i=1,n

niii ni X X X

π ×××→ ...21 21i=1,n-k

( ) 0,...,, 21 =− k n f π

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similar (they differ only in scale); they are equivalent for the purposes of the equation,


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• Application on the equation of dynamic equilibrium

0σdiv2

= ∂++

σp: stress tensoru p: distance

L: meter

n=6 k=3

t p ∂p

g p: volumic forcest p: time

ξp: displacement

M: massT: time

- Direct method

p

m

p

m == **ρ

ρρ

σ

σσ 1***

*

= L

σ

m m

p

m

p

m

L L

L

t t

g g

==

==

**

**

ξ

ξξ 12**

*

=

ρ

ξ

44


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- Second method

( ) 0,,,,, = t gu f

MM TT LL

11 --22 --11 uu

11 00 --33

gg 11 --22 00 i ti giiuii t gui

ξ ξ×××××=

dimensional parameters

tt 00 11 00

ξ 00 00 11

Rank 3 → p=3

***1 1 h g

σπ ==

**2 1 t

ξπ == *3 1 u

ξπ ==

55

ρ


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• Scalling law for reduce scale tests

Experimental work on geotechnical structures

• Models at reduce scale ξ* = 1/nsca ing actor L =1 n

• Test at 1g : g*=1

• Material with the same*=

ε*= 1σ* = 1/n* =

Hooke law

−+

+= ij kkijij E δ

ν

νε

νσ

211

''

E n

E 1' = Mechanical characteristics of thematerial must be modified

−+

+= ij kkijij

E

nδ

ν

νε

νσ '' 211

1 ν='

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• Some examples of 1g tests problem on reduced scale model

- Bearing capacity of shallow foundation (De Beer cited by Corté, 1989; Garnier, 2003)

Terza hi 1943

( )γ BN qu 1=

effect of B on Ny for cohesionless soils Kn=(q u(B)/B)/(q u(B 0)/B0)(De Beer cited by Corté, 1989)

nComabrieu (1997)

R a t i o

( ) ( )( )

φ

ψ

γ λγ

γsin1

sin1sin

23

34

21 +

+

+==

B A

B N B BN qu

77Ratio B/B 0Sol : E=E

0(1+ λz)

(Garnier, 2003)


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• Some example of 1g tests problem on reduced scale model

• Suction anchors (Puech A)model Full scale

(Garnier, 2001)

• Shallow foundation – reinforcement with a geotextil (Garnier 1995 – 1997) mo e

(Garnier, 2001)

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Technical committee 2 TC2 of the ISSMGE

• Scaling law for centrifuge tests

Catalogue of scaling laws and similitude questions in geotechnical centrifuge modelling.J. Garnier, C. Gaudin, S.M. Springman, P.J. Cullingan, D. Goodings, D. Konig, B. Kutter, R.Phillips, M.F. Randolph and L. Thorel. IJPMG, Vol. 3 (2007)

Available one line : http://www.tc2.civil.uwa.edu.au

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• Scalling law derived from the equation of equilibrium + σ*=1Scaling factorScaling factor

*

distancedistance u*u* 1/n1/n

stressstress σσ** 11*ξ

***1 1 h gσπ ==

soil densitysoil density (same soil as prototype)(same soil as prototype) ρρ** soilsoil 11

gravitygravity g*g* nn

*3 1 u

ξπ ==

*ξπ

displacementdisplacement ξξ** 1/n1/n

dynamic timedynamic time t* t* dyndyn 1/n1/n

**2 t gπ ==

strainstrain εε** 11

velocityvelocity v*v* 11

*** / dyn tv ξ=

accelerationacceleration a* a* nn

frequencyfrequency f*f* nn

dyn tv a =**

/1 dyn t f =forceforce F*F* 1/n1/n 22

Unit weightUnit weight γγ** nn

2*** u F ×=σ3*** u m ×= ρ

1111

massmass m*m* 1/n1/n 33*** g×= ρ


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• Vertical stress with depth in centrifuge tests

Center of rotation

Surface of the soil layerR0 R1 RN R

z

Schofield A. N. (1980)Bottom of the container

dR

G=2ω For a constant ω, centrifuge n

dR RGdr d v2ω==

Minimum difference for

Rn= H/3 below the soil surface

1212( )2022 20 R R RG dR R n R

Rv −== ∫ ρω


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• Scaling law of water flow in centrifuge models

Navier Stokes equation (incompressible and Newtonien fluid)

p p fl p p p p p p p

p

pu P grad gu grad u

t fl

pΔ+−=+

∂ ν

ρ1

Dimensional analysis (direct method)

ρ **2**2*

p p

f

p p p m m m m m

m

u P grad P g x gu grad u t fl p Δ+−=+∂ ννρ

ρ *1

***

1**

2**

== xu

F fl

r

Froud number Reynolds number

1*

**

* == xu

R fl 1*

2**

= P

u fl fl

ν11 ** == fl and P ρ

1313

1= fl u


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Dimensiona ana ysis secon met o

Hypothesis : not deformable porous media, incompressible and Newtonian fluid

( ) 0,,,,, = L d u P f

p fl

P: fluid pressure

fl: m croscop c u ve oc y n ers a ve oc y

d p: pore diameter

L: length

(Babendrier, 1991; Stephensen, 1979;Menand, 1995)

μ:dynamic viscosityρ:density of the fluid

Reynolds number***ρ

Friction factor***

14141

*

* ==μ

ρ R

e*

*

fl

p

f u F =


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Flow re imen Flow medium Flow domain


Reynolds number***ρ

Friction factor***

* g d iForchheimer equation

i K u fl fl =*

*

μ

ρ Re =*

fl f u

=

t

c bv avi fl fl fl

∂++= 2

Permanent flowFlow in porous media Creeping flow

Transient flow

Flow ex:waves

Laminar flow without orwith non-linear convective

inertia forces

Fully turbulent flow

**

2** u F r =

Froud number Forchheimer equationi K u fl fl =

1515

2

fl fl bu aui +=


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Same fluid


Scaling factordistance L* 1/n

*

Same soil Scaling factorDiffusion time T* diff 1/n²

gravity g* n

dynamic time t* dyn 1/n

u ve oc ty fl n

Dynammicviscosity

μ 1

velocity v* 1

Darcy’s formulation Hydraulic gradient express as a pressure

i g

i K v fl fl μ

== gradP gradP K v flp fl μ==

P*

*1* == i

Li

ni

K ==*

L*

1*

** == flp v

i K

1616

v fl


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-

Same fluidScalingfactor

Scaling factor

distance L* 1/n

soil density ρ* soil 1

Same soil

2 ndapproach

Diffusion time T* diff 1/n²

Fluid velocity V* fl n

gravity g* n


Dynammic viscosity μ 1

Intrinsic permability K* 1

ve oc y v

Grain diameter d* i 1

Pore size * 1

Friction factor eyno s num er

n n d v

R i fl e =××== 11*

****

μ

ρ n n

n nv

g d i F

fl

i f =

××== 1*

****

1717

μ


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Scaling law of water flow in centrifuge models

Static domain – permanent regimen: limit of validity of the Darcy law

(Khalifa et al., 2000 ; Goodings, 1994 ; Bezuijen,2010 ; Wahuydy, 1998)

- Several definition of the reynolds number and friction factor in soils3

( ) nv

R eq fl e −=

13

2

μ

τ

( ))13 3 nv F

fl

eq f −

=τ

(Khalifa et al., 2000 )

μ

ρ

n

d v R fl e

50=22

50

fl f v

gnid F =

(Goodings, 1994)

Re=3.9 -5.5 (5%) Re=3.3

(Khalifa et al., 2000 )Goodin s 1994

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• Scaling law of water flow in centrifuge modelsStatic domain – permanent regimen: limit of validity of the Darcy law

Pokrovski & Fyodorov, 1975 : maximum hydraulic gradient for staticgeotechnical problems : i max =1

Fontainebleau Labenne Hostun Le Rheu Loire

d 50 0.21 0.3 0.35 0.55 0.65

5% error 78 36 21 12 2

10% error 155 72 43 23 5

Maximum gravity level for similar behaviour between model and prototype (Khalifa et al., 2000 )

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Scaling law of water flow in centrifuge models

Dynamic domain 0≠∂v Non steady state flow

Same fluid

Scaling factor


velocity v* 1

2 ndapproach

Scaling factor

Diffusion time T* diff 1/n²

Fluid velocity V* fl n

Forchheimer equation

t c bv avi fl

fl fl ∂++= 2

During the shaking : consolidation+dynamic strains After the shaking : consolidation

2 stages

Dynamic loading of the soil mass :t* dyn =1/n

Pore pressure dissipation : t* diff =1/n²Pore pressure dissipation : t*diff=1/n²

2020


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Stewart et al. 1998


Nondimensional time factor (consolidation)

t km t c diff v diff vHypothesis : Darcy law valuable

²2 h h μ== I μ*=n, t* diff =t* dyn

water HPMC μ×=10

30g centrifuge

gradP k

v fl μ

=1** ==vv fl

**

dyn diff t t =

11s

p m n μ×=

n n

d v R i fl e

1111*

**** =××==

μ

ρ

2121

+4.5s

+45s(Stewart et al., 1998)


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• grain size effects on interfaces and shear band patterns

- Scale effects on shear mobilisation (Garnier & König, 1998)

Pull out testsScaling effects ?

- roughness effect : R/D50

D (1/n) R (selected)

- pile diameter e ect : B/D50

D50 (1)

B (1/n)

(Garnier & König, 1998)

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Roughness effect : R/D50

Normalized roughness: R n=R max /D 50Shear box tests

interface)

Centrifuge pull out

,

., es on ver ca

piles (b=12mm,D50=0,2mm 50g)

Three different zones of roughness


- smooth interface : interfacial shear along the soil-grain contact, small strength, no dilatency

-Intermediate roughness : frictional resistance increases with the increase in roughness, lowdilantency

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- angle independant from the roughness, large dilantency


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Pile diameter effect : B/D50

•Balachoski tests (1995)Hostun sand d50=0.32mm

Pile diameter BB: 16 to 55mm

Centrifuge acceleration : 100g

•LCPC testsFontainebleau sand D50=0.2 mm

Pile diameter B from 2 to 36 mm

Centrifuge acceleration 50g


If B/d50 > 100 scale effect on τp is limited

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2.similitude laws in centrifuge modeling

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