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)
• Scaling law of water flow in centrifuge models
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 law of water flow in centrifuge models
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
• Scaling law of water flow in centrifuge models
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
Diffusion time T*diff 1/n²
Fluid velocity V*fl n
gravity g* n
dynamic time t*dyn 1/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)
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(Goodings, 1994)
• Scaling law of water flow in centrifuge models
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 )
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Scaling law of water flow in centrifuge models
Dynamic domain 0≠∂∂tv Non steady state flow
∂t
Same fluid
Same soilScaling factor
dynamic time t*dyn 1/n
velocity v* 1
Same soil
2nd
approach
Scaling factor
Diffusion time T*diff 1/n²
Fluid velocity V*fl n
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
• Scaling law of water flow in centrifuge models
,
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
(Garnier & König, 1998)
(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
(Garnier & König, 1998)
If B/d50 > 100 scale effect on τp is limited
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