2.Similitude Laws in Centrifuge Modeling
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Transcript of 2.Similitude Laws in Centrifuge Modeling
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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
33
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
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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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• Scaling law of water flow in centrifuge models
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***
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*
* ==μ
ρ R
e*
*
fl
p
f u F =
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Flow re imen Flow medium Flow domain
• Scaling law of water flow in centrifuge models
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 law of water flow in centrifuge models
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
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v fl
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• Scaling law of water flow in centrifuge models
-
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
dynamic time t* dyn 1/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*
****
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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
dynamic time t* dyn 1/n
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
• Scaling law of water flow in centrifuge models
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
(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, 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
(Garnier & König, 1998)
If B/d50 > 100 scale effect on τp is limited
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