Centrifuge Modeling, Scaling Laws and Scale Effects Bruce L. Kutter Director, Center for...

Centrifuge Modeling, Scaling Laws and Scale Effects

Bruce L. Kutter

Director, Center for Geotechnical Modeling

University of California

Davis

UCD/RPI NEES Centrifuge Research and Training Workshop

September 2008

Centrifuge can be used to study many problems affected by gravity

• Geotechnical/foundation engineering• Frost heave• Dynamic problems, for example:

– Earthquake engineering– Explosive cratering

• Water wave loading• Contaminant transport processes• Sea Ice formation and ice floe

Reasons for Centrifuge Modeling• Need model test data

– Verification of analyses– Test an hypothesis– Explore problems, not

sure of mechanisms, see what happens

• Specific prototypes– Design/retrofit options

• Investigate scale effects– G-knob

Research Trends• More attention to details

– layering, heterogeneity

• Increasing instrumentation.– 100’s of sensors and several cameras to measure– earthquake response– evolution of properties in the experiment

• Soil-foundation-structure systems– Buildings and bridges on piles and shallow

foundations– Groups of buildings

How does the centrifuge

work?

How does

the centrifuge

work?

10 - on-board computers

5 - servo controllers

fiber optic switch

And much, much, more

High speed video cameras

Model Container

About 1 g, 10 RPM

About 2 g, 14 RPM

About 5 g, 22 RPM

About 75 g, 90 RPM

The centrifugal force increases the “weight” of the model to simulate weight of full scale Civil Structures

To simulate earthquakes, we to shake the models while they spin

S

A

N

D

Horizontal actuator

75 g Centrifugal Force


Horizontal actuator


A building or other structure may be placed on the sand

2 ft of soil spinning at 75 g represents 150 ft of soil at 1g

S

A

N

D


Slowing down

Vertical actuators

S A N D

Basic Scaling Laws (Kutter, 1995, Recent Advances in Centrifuge Modeling of Seismic Shaking )

Let * = m/ p = 1 (soil properties depend on ’)

Let L* = Lm/Lp = 1/N (definition of scale factor, N)

Let * = m/ p = 1 (same materials)

And because [ = [g][L] [x] = units of x* = * g* L*

1 = (1)(g*)(L*) g* = 1/L* = N

* = 1 is important because strength, stiffness, dilatancy, and void ratio of soil have nonlinear dependence on effective stress. Modeling similarity is enhanced by stress similarity.

Stress Distribution in 1/2 Scale Model Under 2g

d/2 g)(d/2) = gd

Prototype Stress Distribution

d

g d

PRINCIPLE OF CENTRIFUGE MODELLING

Idea is to produce a realistic stress and realistic stress distribution in controlled experiments with well defined boundary conditions and well defined material properties

Lp

Lm

L* = Lm/Lp = 0.5

g* = gm/gp = 2

* = m/ p = 1

Common scale factors for various quantities

Stress, Moduli, Pressure: σ * = 1Density: ρ * = 1Length, Displacement: L* = L*Gravity: g* = 1/L*Force F* = σ* (L*)2 = (L*)2

Mass M* = ρ * (L*)3 = (L*)3

Dynamic Time: t*dyn = (L*/g*)2 = L*

Dynamic Frequency f*dyn = t*dyn = 1/L*

Dynamic Velocity v*dyn = L*/t*dyn= 1

Dynamic Acceleration a*dyn = g* = 1/L*

Diffusion Time t*dif = (L*)2

Catalogue of scaling laws and similitude questions in centrifuge modelling

• J. Garnier et al. TC2 Catalogue IJPMG September 2007– dynamics, fluid flow in soils, heat transfer and

ice, particle size effects, rate effects, container effects

• About 100 references

TC2- Catalog

Scale effects and other issues I plan to address today

• Dynamic vs diffusion time scaling conflict• Boundary effects: friction, added mass, lateral flexibility

– e.g., lubricants, light weight, shear beam containers• Radial g-field and Coriolis forces

– At which radius do you calculate g = 2r?• Particle size effects

– if Lparticle/Lmodel is large– Shear band thickness and strain softening– applies to centrifuge, lab testing, field tests, prototype

structures• Rate effects (especially for clay and partly drained sand)• Ground motion selection and scaling

Time Scaling Conflict• Dynamic Time

[L] = [a] [t]2 L* = a* t*2 t* = sqrt(L*/a*)t*dyn = sqrt(L*/(1/L*)) = L*

• Diffusion Time, consider time factor, TFor 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

• Conflict t*dif ≠ t*dyn if cv* = 1

• Conflict ResolutionIf t*dif ≠ t*dyn if cv* = 1 is unacceptable, then for precise modeling, we

must requireL*2 / cv* = L* cv* = L* by increasing viscosity of the fluid (* = 1/L*) or decreasing the particle size of the soil (k* = C(D10*)2 )

Pore pressure contours in sloping ground with an impermeable (silt) layer

Shaking and dissipation movies

Sometimes, conflict can be neglected without changing cv

• Case 1. tdis >> tgen both model and prototype; both are undrained during dynamic event. Example:

– At prototype scale, tgen = 10 s, tdiss=10000 s– At model scale (N = 10), tgen = 1 s, tdiss=100 s the conflict can be neglected.

• Case 2. tdis << tgen in model and prototype; both are drained during dynamic event the conflict can be neglected

• Case 3. tdis ≈ tgen in either the model or prototype, or tdis – tgen has opposite sign in model and prototype. Example:

– At prototype scale, tgen = 10 s, tdiss=100 s– At model scale (N = 100), tgen = 0.1 s, tdiss=0.01 sCentrifuge model is “drained”, prototype is “undrained” the conflict cannot be neglected.

If we are not precisely modeling a specific prototype anyway, why should we worry about the time

scaling conflict?• we may want to systematically vary tgen/tdis by

scaling viscosity systematically to cover an interesting range.

• 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.

Boundary/Container effects

• Flexible Containers– Hinged plate, Laminar boxes

• Ideal for gently sloping

or level ground

– Complementary Shear issue

Flexible Shear Beam (FSB2)

Soil-Pile-Embankment Wharf System – an unsymmetrical model in an FSB

Dickenson, McCullough, Schlecter (OSU)

Kutter, Boland (UCD)

Hinged plate container to study liquefaction and lateral spreading

Boundary/Container effects

• Rigid containers– P-waves from ends of the container

• Side friction (silo effect)– Avoid narrow containers (width < height)– Move structures and make important

measurements away from boundaries

• Lateral stiffness (maintaining Ko)

stress history and construction processes

• Clay OCR and consolidation • Installation effects possibly modeled by robotics

– Piles– Ground improvement– Excavation in flight

• Site investigation in flight to monitor changes in stresses and density due to construction processes and seismic history.

Consolidation press

Radial g-field • At which radius do you calculate

g = 2r?• Pick a depth, zref in the model

where you are most concerned about accurately modeling the effective stress. Set g accordingly. – For level ground:

• Document the RPM and the radius to a reference point on the model container

• Note that g vector is radial, not vertical – one advantage or having a large centrifuge

r

dMrdF 2refz

refzz zr

zg

ref)( 2

2

Coriolis forces

relcor va 2

e.g.

vrel = velocity relative to the centrifuge

= angular velocity of the centrifuge

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/Dparticledepends on the problem.

– Footings: Dfooting/Dparticle >~ 35 minimizes particle size effect.

– Trap door problem: size effect has been observed for Dtrap door/Dparticle as high as 1000.

– (see TC2 catalog)

Particle size effects

• To model contact stress and capillary rise most accurately, 0.1 mm diameter sand in a model represents 0.1 mm diameter sand in a prototype:– F*contact ≈ ’ * Dpl 2 ; contact force depends on effective

stress and particle size; if we desire the same contact force, we should use the same particle size

– Height of capillary rise: hc*= 2Ts*/*w r*effective

same pore size and fluid r*effective= 1; Ts* = 1 g* = 1/L* *w = ρ*w g* = 1/L*

hc*= (1)/((1/L*)(1)) = L* Ability to model capillary rise is an advantage of centrifuge

modeling over 1 g modeling

– Caveat: there is an effect of g* on the relationship between water content and pore water suction for soils with low degree of saturation

Water table

Top of Capillary fringe

Impermeable bottom

Vadose zone

Ground surface

Unconfined aquifer

z

zaw

z1

zcp

zan

z2

1

Srw

Srn

0Saturation,Sr

Water

LNAPL

Air

Residual saturation is known to depend on g*

Particle size effects

Shear band thickness

From this perspective, sand with 0.1 mm diameter in the model represents sand with 4 mm diameter in a prototype tested at 40 g

Jason DeJong

stress

Macroscopic strain

Rate of strain-softening in a shear band in dilatant sand depends on strains in the shear band, but the shear band thickness depends on particle size, not on L*

Particle size effects are important for other aspects of

geotechnical engineering

• Laboratory test results also suffer from particle size effects.

• Designers ignore particle size effect in design.• Numerical models have a difficult time with strain

softening on a shear band and require “characteristic length” parameter.

• Centrifuge testing allows us to study particle size effects We can vary D50/Lmodel by increasing Lmodel and decreasing g at the same time so that identical stress is obtained, identical soil properties, – the only difference is D50/Lmodel

Strain-rate effects on clay strength

• Can be an issue for any lab test that is conducted at a rate different than that in the prototype.

• Undrained shear strength of clay increases about 5 or 15% for every log cycle of strain rate

• Strains are the same in model and prototype, time is scaled in the model tests. If test is done at 50 g, the time is scaled by a factor of 50 and strain rate is increased by a factor of 50.

• If the soils in the model and prototype have the same void ratio, the shear strength may increase by

• Which has a value of 1.2 for t* = 1/100

)log()1.0(1 *

,

, pu

mu

c

c

Consolidation time effects

• In field, soil is consolidated for years and undergoes strengthening due to volumetric creep. If soil is freshly consolidated in the centrifuge press, it only undergoes volumetric creep for a matter of hours or days.

• The increase in strength due to strain-rate effect is counterbalanced by reduced strength due to small consolidation time in the model.

• In-flight testing using robotics is valuable for monitoring strength and should be interpreted with time effects in mind.

Ground motions in

1st centrifuge

test

Ground motion selection

(b)

Period (sec)0.01 0.1 1 100

1

2

3

BAM01

BAM02

BAM03

BAM04

BAM05

BAM06

Spec

tral

acc

eler

atio

n (g

)Repeatability and versatility of frequency content of input motions

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.

• Strain softening of model structures can result in peculiar results if a model is shaken at a single frequency.

Ground motions – Data from Boulanger et al. (1999)

Sensitivity of simulations of model

tests to assumed BC’s (Ilankatharan and

Kutter 2008) Structural model

Vertical bearings

FSB container

Soil model

Centrifugal force

Actuator

Shaking table

Reaction mass

(a)

erro

r in

pea

k su

rfac

e ac

c (%

)

0

50

100absorbing-baserigid-base

(c)

error in Gassumed (%)

-40 -20 0 20 40

erro

r in

pea

k ba

se a

cc (

%)

0

50

100

(b)

erro

r in

pea

k su

rfac

e A

RS

(%)

0

50

100

(d)

error in Gassumed (%)

-40 -20 0 20 40er

ror

in p

eak

base

AR

S (%

)

0

50

100absorbing-baserigid-base

absorbing-baserigid-base

absorbing-baserigid-base

FIG. 10. Sensitivity of peak & peak spectral accelerations of surface and base motions to error in Gassumed of elastic soil material in the absorbing-base, and the rigid-base boundary simulations (Ilankatharan and Kutter 2008)

Conclusions (1)

• Centrifuge Models or Centrifuge Tests – The centrifuge is a tool that makes model tests more

accurate because it accurately reproduces prototype stress levels in a small scale model. The interesting geotechnical problems include effects of dilation, contraction, stiffness, and failure. A good way to get realistic combinations of these things is to accurately model the stresses

– But - the centrifuge does not make the model tests perfect – be mindful of potential scale effects.

• If modeling laws fail, the centrifuge may still be a useful “g-knob” to adjust the stress level to– Test the validity of a numerical model– Perform systematic parameter studies– Discover mechanisms of behavior

Conclusions (2)• Model testing is valuable for problems where field

data is insufficient – can obtain data that is impossible to obtain in other ways.

• The centrifuge enhances accuracy of model testing.

• Advanced instruments of NEES (robotics, shakers, instrumentation) enable more accurate and more detailed models than was possible in the past.

Do you have another concern about scale effects and scaling

laws?• Unsaturated soil, Turbulent flow, Erosion,

Shear bands?

• Effect of transducer or model container on the experiment?

• Are any of the above an issue that you would like me to address in this presentation?

• Any other topics?

Centrifuge Modeling, Scaling Laws and Scale Effects Bruce L. Kutter Director, Center for...

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