Similitude laws in centrifuge modelling

Content

• Principle of scaling laws – Vashy-Bukingham theorem

• Scaling laws for centrifuge testsg g

• Scaling laws of water flow in centrifuge

•Grain size effects on interface and shear band pattern

22

• Vaschy-Buckingham theorem

« If we have a physically meaningful equation involving a certain number, n, of physical variables, and these variables are expressible in terms of k independent fundamental physical quantities, then the original expression is equivalent to an equation involving a set of p = n − k dimensionless parameters constructed from the original variables »p p g

( ) 0,...,, 21 =nXXXf ( )21 n

kiiiki rrrX ααα ×××→ ...21

21i=1,n

niiini XXX βββπ ×××→ ...21

21i=1,n-k

( ) 0,...,, 21 =−knf πππ

Significance : Two systems for which these dimensionless parameters coincide are called

33

Significance : Two systems for which these dimensionless parameters coincide are called similar (they differ only in scale); they are equivalent for the purposes of the equation,

• Application on the equation of dynamic equilibrium

0ξgρσdiv p2

2

ppp =⎟⎟⎞

⎜⎜⎛

⎟⎠⎞⎜

⎝⎛∂

++⎟⎠⎞⎜

⎝⎛

σp: stress tensorup: distanceρ : density L: meter

n=6 k=3

ξt

gρ p2p

ppp ⎟⎠

⎜⎝

⎟⎠

⎜⎝∂

⎟⎠

⎜⎝ ρp: density

gp: volumic forcestp: timeξp: displacement

M: massT: time

- Direct method

p

m

p

m

tg

== **

ρρρ

σσσ 1***

*

=Lgρ

σ

p

m

p

m

p

m

p

m

LLL

ttt

ggg

==

==

**

**

ξξξ 12**

*

=tg

Lg

ξ

ρ

ppξ tg

44

- Second method

( ) 0,,,,, =ξρσ tguf

MM TT LL

σσ 11 --22 --11

00 00 11 General expression of the none uu 00 00 11

ρρ 11 00 --33

gg 11 --22 00itigiiuii tgui

ξρσ αααααα ξρσπ ×××××=

General expression of the none dimensional parameters

tt 00 11 00

ξξ 00 00 11

Rank 3 → p=3

* * *

***

*

1 1hgρ

σπ ==

**

*

2 1tg

ξπ == *

*

3 1uξ

π ==

55

gρ g

• Scalling law for reduce scale tests

Experimental work on geotechnical structures

• Models at reduce scale l f * /

ξ* = 1/nscaling factor L*=1/n

• Test at 1g : g*=1• Material with the same

density: ρ*=1

ε*= 1

σ* = 1/n

t* = 1/ndensity: ρ =1 t 1/n

Hooke law

⎟⎠⎞

⎜⎝⎛

−+

+= ijkkijij

E δεν

νεν

σ211

⎞⎛E ''

En

E 1' = Mechanical characteristics of the material must be modified

⎟⎟⎠

⎞⎜⎜⎝

⎛

−+

+= ijkkijij

En

δεν

νε

νσ '' 2111 νν ='

66

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

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

Terzaghi (1943)

( )φγ γBNqu 21

=

g ( )

2

effect of B on Ny for cohesionless soils ⇒ Kn=(qu(B)/B)/(qu(B0)/B0)

(De Beer cited by Corté, 1989)

Kn

Comabrieu (1997)

Rat

io K

( ) ( )( )

φψφ

γγ λγ

γsin1sin1sin

23

34

21 +

+

⎟⎠⎞

⎜⎝⎛ +==

BABNBBNqu

77Ratio B/B0

Sol : E=E0(1+λz)

(Garnier, 2003)

• 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)

d l Full scalemodel Full scale

(Garnier, 2001)

88

• Centrifuge tests

- First idea : Phillips (1869) – France

- First tests : Bucky (1931) – USA Pokrovskii (1933) - URSS

1 scaling factor is fixed – same stress in the model and in the prototype

1*σ 1=σ+

Same soil ng =*

991* =ρ

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

1010

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

*

distancedistance u*u* 1/n1/n

stressstress σσ** 11*ξ

***

*

1 1hgρ

σπ ==

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

gravitygravity g*g* nn

*3 1uξ

π ==

*

1ξ

π ==displacementdisplacement ξξ** 1/n1/n

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

**2 1tg

π ==

strainstrain εε** 11

velocityvelocity v*v* 11

*** / dyntv ξ=

*** /accelerationacceleration a*a* nn

frequencyfrequency f*f* nn

*** / dyntva =** /1 dyntf =

forceforce F*F* 1/n1/n22

Unit weightUnit weight γγ** nn

2*** uF ×= σ3*** um ×= ρ

1111massmass m*m* 1/n1/n33

*** g×= ργ

• Vertical stress with depth in centrifuge tests

Center of rotation

Surface of the soil layerR0 R1 RN R

y

z

Schofield A. N. (1980)

Bottom of the containerdR

RG

=2ω For a constant ω, centrifuge acceleration varies with depth

nR acceleration varies with depth

dRRGdrd v2ωρρσ ==

Minimum difference for

Rn= H/3 below the soil surface

1212( )2022

20

RRRGdRR

n

R

Rv −== ∫ρωρσ

• Scaling law of water flow in centrifuge models

( )u∂

Navier Stokes equation (incompressible and Newtonien fluid)

( ) ppflppppppp

p

p uPgradgugradutu

flp

Δ+−=⋅⋅+∂∂ ν

ρ1

Dimensional analysis (direct method)

( ) flflflfl xuuuu∂ ρ **2**2*( ) ppflp

flpp

flm

flmmm

m

m uxu

PgradPu

gxg

uugradu

tu

flp

Δ+−=⋅⋅+∂∂ ν

νρ

ρ *1

***

1**

2** ==

xgu

F flr

Froud number Reynolds number

1*

*** ==

xuR fl

e

1*

2**

=Pu flflρ

g *νe

11 ** == flandP ρ

1*

1313

1* =flu

l l ( d h d)


Dimensional analysis (second method)

Hypothesis : not deformable porous media, incompressible and Newtonian fluid

( ) 0,,,,, =ρμLduPf pfl

P: fluid pressure

U : microscopic fluid velocity (interstitial velocity)Ufl: microscopic fluid velocity (interstitial velocity)

dp: pore diameter

L: length

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

μ:dynamic viscosity

ρ:density of the fluid

Reynolds number***ρ xu

Friction factor*** gdi

14141*

* ==μ

ρ xuRe*

*

fl

pf u

gdiF =

Flow regimen Flow medium Flow domain• Scaling law of water flow in centrifuge models

o g o d u o do

Reynolds number***ρ xu

Friction factor***

* p gdiF

Forchheimer equation

v∂iKu flfl =

**

μρ xu

Re =*fl

pf u

gF =

tv

cbvavi flflfl ∂

∂++= 2

ff

Permanent flow Flow in porous media Creeping flow

Transient flow

Flow (ex:waves)

Laminar flow without or with non-linear convective

inertia forces

( )

Fully turbulent flow

**

2**

xgu

Fr =

Froud number Forchheimer equationiKu flfl =

1515

xg 2flfl buaui +=

Same fluid


Scaling factor

distance L* 1/n

soil density ρ*soil 1

Same soil Scaling factor

Diffusion time T*diff 1/n²

Fl id l i V*soil density ρ soil 1

gravity g* n

dynamic time t*dyn 1/n

Fluid velocity V*fl n

Dynammic viscosity

μ 1

velocity v* 1

First approach Second approachFirst approach Second approach

Darcy’s formulation

k

Hydraulic gradient express as a pressure

k

Δh

ikgiKv flfl μ== gradPkgradPKv flpfl μ

==

nP

i ==*

*1* =⇒

Δ= i

Lhi

niKfl ==*

nL

i == *

1*

** ==

flflp v

iK

1616

vflfl fl


Fluid flow in porous media - permanent flow regimenFluid flow in porous media - permanent flow regimen

Same fluidScaling factor

Scaling factor

distance L* 1/n

soil density ρ*soil 1

Same soil

2nd

approach



gravity g* n


l it * 1

Dynammic viscosity μ 1

Intrinsic permability K* 1

Pressure gradient i nvelocity v* 1 Pressure gradient i n

Grain diameter d*i 1

Pore size p* 1Pore size p 1

R ld bFriction factor

Reynolds number

nndvR ifl

e =××

==111

*

****

μρn

nnn

vgdi

Ffl

if =

××==

1*

****

1717

1μ

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 soils

d23dign

( )ndv

R eqfle −

=13

2μ

τρ( ))13 3 nvdign

Ffl

eqf −

=τ

(Khalifa et al., 2000 )

μρ

ndv

R fle

50=2

250

flf v

gnidF =

(Goodings, 1994)

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

(Khalifa et al., 2000 )(Goodings, 1994)

1818

(Goodings, 1994)


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

Pokrovski & Fyodorov, 1975 : maximum hydraulic gradient for static geotechnical problems : imax=1

Fontainebleau Labenne Hostun Le Rheu Loire

d50 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 )

1919

Scaling law of water flow in centrifuge models

Dynamic domain 0≠∂∂tv Non steady state flow

∂t

Same fluid

Same soilScaling factor


velocity v* 1

Same soil

2nd

approach

Scaling factor



Forchheimer equationv∂

tv

cbvavi flflfl ∂

∂++= 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

Stewart et al., 1998


,

Nondimensional time factor (consolidation)

tkmtcT diffvdiffv

f

Hypothesis : Darcy law valuable

²2 hhT ffff

μ== If μ*=n, t*diff=t*dyn

waterHPMC μμ ×=10

30g centrifuge

gradPkv fl μ=

1** == vv fl

**dyndiff tt =

11s

pm n μμ ×=

nndv

R ifle

1111*

**** =

××==

μρ

2121

+4.5s+45s

(Stewart et al., 1998)

• 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

ff

D (1/n) R (selected)

- pile diameter effect : B/D50

D50 (1)

B (1/n)

(Garnier & König, 1998)

2222

Roughness effect : R/D50

Normalized roughness: Rn=Rmax/D50

Shear box tests (steel/sand (steel/sand interface)

(Garnier & König 1998)

(Paikowski et al 1995)Centrifuge pull out

t t ti l


(Paikowski et al., 1995) test on vertical piles (b=12mm,

D50=0,2mm 50g)

Three different zones of roughness(Garnier & König, 1998)

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

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

Large roughness (rough interface): internal shear localised in shear band into the sand friction

2323

-Large roughness (rough interface): internal shear localised in shear band into the sand friction angle independant from the roughness, large dilantency

Pile diameter effect : B/D50

•Balachoski tests (1995)

Hostun sand d50=0.32mm

Pile diameter BB: 16 to 55mm

Centrifuge acceleration : 100g

•LCPC tests

Fontainebleau 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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