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Centrifuge Physical Modeling &
Scaling Laws
Tarek Abdoun
RPI/UCD NEES Centrifuge Research and Training
Workshop 2011
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Geotechnical Centrifuge
Ng
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Ground Centrifuge Modeling
Concept
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Radial g-field
• At which radius do you calculate g = w2r?
• Pick a point in the model where you are most concerned about accurately modeling the effective stress. Set g accordingly.
– For level ground: s = r (gavg overburden)(d)
• Document the RPM and the radius to a reference point on the model container
• Might need to account for g variation in deep models
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Why Physical Model Tests?
• Complex, nonlinear stress-strain behavior
of soil (made of interacting particles, air,
water)
• Difficulty of numerical simulation of soil
and soil-structure systems at large strains
and failure
• Validate and calibrate numerical methods
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Why Centrifuge Model Tests?
• Small-scale models are cost-effective
• Soil properties are highly stress-dependent
• Centrifuge produces equal confining stresses
in model and prototype, therefore same soil
properties
• Then, reasonable assumption that strains and
deformations are also equal in model and
prototype
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Application Domain: Systems
• Natural or artificial soil deposits, different
soil types, different geometries, earth
dams and dykes
• Soil-foundation and soil-structure systems:
– foundations of buildings, bridges
– buried pipes and tunnels, basements
– earth levees with sheetpiles
– etc.
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Application Domain : Loadings
• Static gravity loads
• Earthquake shaking
• Blasting
• Ground deformation
• Water waves
• Contaminant transport
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Centrifuge Modeling Limitations
• Useful only for systems containing
soil or other pressure-dependent
material
• Models allow limited detail
• Effect of model boundaries
• Time scale and strain-rate issues
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Scaling Laws (N = number of g’s)
• Stress & Pressure σ * = 1
• Density ρ * = 1
• Length 1/N
• Velocity 1
• Acceleration N
• Volume 1/N3
• Mass 1/N3
• Force 1/N2
• Time (dynamic) 1/N
• Time (diffusion) 1/N2
Scaling Laws
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Catalogue of scaling laws and
similitude questions in
centrifuge modelling
• Technical Committee TC2 –Physical
Modelling in Geotechnics 2007
• Covers: dynamics, fluid flow in soils, heat
transfer and ice, particle size effects, rate
effects
• About 60 references
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Concerns regarding scale
effects and scaling laws
• Unsaturated soil, Turbulent flow,
Erosion, Shear bands
• Effect of transducer or model container
on the experiment
• Range of scaling laws applicability (50g,
100g, 150g, etc.)
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Modeling Structural Elements
• Very challenging task:
– D & t (N)
– Area (N2)
– Inertia (N4)
– E (1) for same material
• Usually very difficult to maintain the same scale
for all parameters or to use same material in
both model and prototype (easier if no specific
prototype)
• Need to prioritize (EA, EI, t/D, etc.)
– EI for flexure or bending
– EA for axial loading
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NEES-Pipelines “Evaluation of Ground Rupture Effects on Critical Lifelines”
Numerical
Modeling
Centrifuge
Modeling Full scale
Testing
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EA vs. EI for Structural Elements
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.02 0.04 0.06 0.08 0.1 0.12
tm/D
m
tp/Dp
EA curve
EI curve
Em/Ep= 0.6
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EA vs. EI for Structural Elements
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.02 0.04 0.06 0.08 0.1 0.12
tm/D
m
tp/Dp
EA curve
EI curve
Em/Ep= 0.6
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EA vs. EI for Structural Elements
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.02 0.04 0.06 0.08 0.1 0.12
tm/D
m
tp/Dp
EA curve
EI curve
Em/Ep= 0.6
tm/Dm = 2 tp/Dp
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Other Factors: Strain Rate
0 1 2 3 4
Axial Strain (%)
0
5
10
15
20
25
Axia
l S
tres
s (
MP
a)
HDPE Material Stress-Strain Behavior
0.1%/min
1%/min10%/min
1%/min
0.16%/min
130%/min
300%/min
Hypobolic Fit (Merry & Bray, 1997)
RPI Uniaxial Tension Test
100%/min
300%/min
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Comparison with Full Scale Test
Results (-63.5o Tension Test)
-6 -4 -2 0 2 4 6
Distance from Fault (m)
0
2
4
6
8
10S
pri
ng
lin
e S
train
(%
)Full Scale, f = 1.06 m
Full Scale, f = 0.49 m
Centrifuge, f = 1.06 m
Centrifuge, f = 0.49 m
Springline Strain Comparison
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Time Scaling Conflict
• Dynamic Time L = 0.5 a t2 L* = a* t*2 t* = sqrt(L*/a*)
t*dyn = sqrt(L*/(1/L*)) = L* or 1/N
• Diffusion Time, consider time factor, T For similarity, T* = 1 = cv* t* /L*2
t*dif = L*2 / cv*
If cv* = 1 (same soil in model and prototype) then:
t*dif = L*2 or 1/N2
• Conflict t*dif ≠ t*dyn
• Conflict Resolution – By increasing viscosity of the fluid (m* = 1/L* or N)
– Decreasing the particle size of the soil (k* = C (D10*)2 )
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Time Scaling Conflict
• Sometimes, conflict can be neglected without
changing cv
– both model and prototype are undrained during dynamic
event
– both model and prototype are drained during dynamic event
• we may want to systematically vary viscosity to cover
an interesting range. (Reviewers may have difficulty
with this concept)
• It takes time to saturate a large model with viscous
pore fluid. For practical purposes, we may knowingly
violate time scale factor similarity, and then account
for the different cv by analysis
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Modeling of Shear Bands
J. DeJong, U. Mass Amherst web page
The shear band thickness
depends on particle size, not
on L* (N)
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Modeling of Shear Bands
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Particle Size Reduction
0
10
20
30
40
50
60
70
80
90
100
0.001 0.01 0.1 1
Particle size, mm
% S
oil p
ass
ing
Scaled SandOttawa Sand F#55
Centrifuge
Modeling
Full Scale Testing
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Particle Size effect
• Most basic requirement is that there are a sufficient number of particles across the dimensions of a model so that we can model the soil as a continuum. – Required Dmodel/Dparticle depends on the problem.
– Footings: Dfooting/Dparticle > 30 (minimizes particle size effect)
• To model contact stress and capillary rise most accurately, need to use same particle size (pore size) and fluid. The Ability to model capillary rise is an advantage of centrifuge high g modeling.
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Explosions are Volumetric
• Explosions Scale as N3
• 1 gram of explosive tested at
100g is equivalent to one million
(106) grams of prototype
explosive, or one metric ton
(2200 lb)
• Scale effects also include
particle size effects and
differences in radial acceleration
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Application of High Speed
Camera to Blasting Tests
1.E-02
1.E-01
1.E+00
1.E+01
1.E-04 1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06
Scaled Charge Mass (kg)
Scale
d D
ep
th (
m)
S&H su-ho bu-ve su-ve Pow er (S&H)
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Blast Modeling
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• Time Scales as g2 – E.G., 24 Hour test @ 105g = 30 years prototype time
• Advection (Hydraulic flow) – No theoretical
problems
• Dispersivity (Diffusion, Dispersion) – more
complicated, but can be done
Groundwater/Contaminant
Transport
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• General: Single contaminant, conservative
contaminant – models acceptable
• The robot gives us a unique opportunity to
determine the transport and concentration with
time of multiple contaminants
Groundwater/Contaminant
Transport (cont.)
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Boundary/Container effects
• Flexible Containers
– Hinged plate, Laminar boxes
• Ideal for gently sloping
or level ground
– Complementary Shear issue
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Boundary/Container effects
• Rigid containers
– P-waves from
ends of the container
• Side friction
– Avoid narrow containers (width < height)
– Reduce sides friction
– Move structures e.g., away from boundaries
• Lateral stiffness (maintaining Ko)
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Ground motion selection
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Sine waves, step waves or realistic
ground motions?
• Small step waves – Useful to check that sensors are working
• Sine waves are easier to understand than real ground motions – Because they only reveal information about part of
the problem (one frequency from the possible spectrum)
• Sine sweeps – Useful because they cover all frequencies, but
amplitude is not random.
• Ground motion provides more realistic conditions but could be difficult to analyze
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Final Thoughts • Centrifuge Modeling is a tool that makes model tests more
accurate because it reproduces prototype stress levels in a small scale model but be mindful of it’s limitations
• Centrifuge Modeling is useful to:
– Test the validity of a numerical model
– Perform systematic parameter studies
– Discover mechanisms of behavior
• Model testing is valuable for problems where field data is insufficient – can obtain data that is impossible to obtain in other ways.
• Advanced instruments of NEES (robotics, shakers, instrumentation) enable more accurate and more detailed models than was possible in the past.
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NEES centrifuge research
• Complementary NEES Centrifuges
– UCD: larger container, V&H shaker, more sensors per test, multiple tests per container
– RPI: medium size, H&H shaker, more tests per month, Robot, split box.
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Thank You