Centrifuge Physical Modeling &

Scaling Laws

Tarek Abdoun

RPI/UCD NEES Centrifuge Research and Training

Workshop 2011

Geotechnical Centrifuge

Ng

Ground Centrifuge Modeling

Concept

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

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

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

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.

Application Domain : Loadings

• Static gravity loads

• Earthquake shaking

• Blasting

• Ground deformation

• Water waves

• Contaminant transport

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

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

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

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.)

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

NEES-Pipelines “Evaluation of Ground Rupture Effects on Critical Lifelines”

Numerical

Modeling

Centrifuge

Modeling Full scale

Testing

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

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

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

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

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

-63.5o Strike-Slip (Tension)

HDPE2overhead2.wmv

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 )

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

Modeling of Shear Bands

J. DeJong, U. Mass Amherst web page

The shear band thickness

depends on particle size, not

on L* (N)

Modeling of Shear Bands

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

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.

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

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)

Blast Modeling

• 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

• 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.)

Boundary/Container effects

• Flexible Containers

– Hinged plate, Laminar boxes

• Ideal for gently sloping

or level ground

– Complementary Shear issue

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)

Ground motion selection

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

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.

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.

Thank You