Principles of centrifuge p gmodeling

Dr Gopal Madabhushi

TNA Workshop on Centrifuge Modelling – 3/4 March 2011

Dr Gopal Madabhushi, Reader in Geotechnical Engineering,

University of Cambridge

IntroductionIntroduction

Modern day geotechnical engineer uses Modern day geotechnical engineer uses the following methods in design:the following methods in design:g gg g

For simple problemsFor simple problemsl Sl S l h dl h dEmpirical or SemiEmpirical or Semi--empirical methodsempirical methods

Closed form solutionsClosed form solutions

For complex problemsFor complex problemsFi it El t A l i Fi it El t A l i Finite Element Analysis Finite Element Analysis

When using the FE method we need to When using the FE method we need to When using the FE method, we need to When using the FE method, we need to know the constitutive behaviour of soil know the constitutive behaviour of soil i.e. the stress strain behaviour of the i.e. the stress strain behaviour of the i.e. the stress strain behaviour of the i.e. the stress strain behaviour of the soil under the type of loading it soil under the type of loading it experiences needs to be known.experiences needs to be known.pp

If Yes, we can program this into the FE If Yes, we can program this into the FE If Yes, we can program this into the FE If Yes, we can program this into the FE codes and proceed with the designcodes and proceed with the design

If No, ???If No, ???

Let us look at the stress strain behaviour Let us look at the stress strain behaviour of soilof soilof soilof soil

StressStress Strain behaviour of soilStrain behaviour of soilStress Stress –– Strain behaviour of soilStrain behaviour of soilLet us consider the shear stress Vs shear Let us consider the shear stress Vs shear

strain for loose and dense soilsstrain for loose and dense soils

ShearDense Sand

Stressτ

Loose Sand

Shear box test

Shear Strain γ

Soil is highly NON-LINEAR and PLASTIC

Physical Modelling in GeotechnicsPhysical Modelling in GeotechnicsPhysical Modelling in GeotechnicsPhysical Modelling in Geotechnics

As soil is nonAs soil is non--linear we cannot simply model linear we cannot simply model our problem (say a caisson subjected to lateral our problem (say a caisson subjected to lateral load) at small scaleload) at small scale

Shear

Dense Sand

Stressτ Loose Sand Prototype behaviour

at large stresses

Model behaviour at

Prototype behaviour atlarge stresses

and when large strains are mobilised

Shear Strain γ

Model behaviour at small stresses and strains

Shear Strain γWe need to create prototype stresses and strains in our models !!!

Centrifuge Modelling TechniqueCentrifuge Modelling TechniqueConsider, for simplicity, we have a block structure of dimensionsL × B × H sitting on a horizontal soil bed.

Principle of Centrifuge Modellingif d d l d l i bj d if l l i h• In a centrifuge a reduced scale model is subjected to centrifugal acceleration so that

correct prototype stresses and strains are created in the model.Stress under the block is given by

Idealised FieldStructure

CentrifugeModel

1g

M/N3H/N

Ng

MB

L

HM/N

L/N

H/N

B/N

σ =MgLB ε δ

=LL

ε δ δ= =

L NL N

LL

//

σ =×

×=

MN Ng

LN

BN

MgLB

3

Centrifugal Acceleration:The easiest way to create high gravity is by spinning our soil models in a centrifuge.

When the centrifuge is rotating with an angular velocity of ‘ω‘, the centrifugal acceleration at any radius ‘r’ is given by

centrifugal acceleration = r × ω2

We wish to match this centrifugal acceleration to be the same factor as the one we used to scale down our prototype by ie ‘N’ (geometrical scaling factor).

⇒ N × g = r × ω2

For example, the Sudarshan Centrifuge at IIT, Bombay willbe doing;be doing;

In a 100g Centrifuge Test:In a 100g Centrifuge Test:

2254819100 225.481.9100 ω×=×

145.. ≅⇒ RPM

Various centrifuges around the worldVarious centrifuges around the worldVarious centrifuges around the worldVarious centrifuges around the world

T B C t if 10 di t d 150 t itTurner Beam Centrifuge – 10 m diameter and 150 g-ton capacity

University of Cambridge

University of California, Davis

IFSTTAR / LCPC Centrifuge NantesIFSTTAR / LCPC Centrifuge - Nantes

The beam Centrifuge at LCPCRadius 5.50 m

1111

2T in the basket at 100 g

US Army Waterways Experiment Station

Takenaka Construction Corporation

Sudarshan – National Geotechnical Centrifuge Facility, Bombay

Scaling LawsScaling Laws

We need a ‘set of rules’ which link the behaviour We need a ‘set of rules’ which link the behaviour we observe in the model to that of an we observe in the model to that of an we observe in the model to that of an we observe in the model to that of an equivalent prototype in the fieldequivalent prototype in the field

These rules are called ‘Scaling Laws’These rules are called ‘Scaling Laws’These rules are called Scaling LawsThese rules are called Scaling Laws

We already know that the Scaling Laws for We already know that the Scaling Laws for a) stress and straina) stress and strainb) dimensionsb) dimensionsc) massc) massd) acceleration due to gravityd) acceleration due to gravity

Scaling Laws

Parameter Model/Prototype

Stress 1

St i 1Strain 1

Length 1/Ng /

Area 1/N2

Volume 1/N3

Mass 1/N3

Acceleration NAcceleration N

Scaling law for forceScaling law for force

By definition Force = Mass X By definition Force = Mass X AccelerationAccelerationAccelerationAcceleration

Using dimensional analysis:Using dimensional analysis:Us g d e s o a a a ys sUs g d e s o a a a ys s

aMFprototype ××= 1prototype

( )M ( )aNNMF el ××= 3mod

2model 1F

= 2NFprototype

Suppose we have a situation in the field where we are expectinga 200 ton load on a pile capa 200 ton load on a pile cap.

Let us say we want to model this load in a 100g centrifuge test

Then:

MNtFprototype 2200 ==

102 6× NF el 200100

1022mod =

×=

Thus we only need to apply 200 N or 20 kg(f) in the model

Scaling law for energyScaling law for energy

By definition energy = work done = Force By definition energy = work done = Force X distanceX distanceX distanceX distance

Using dimensional analysis:Using dimensional analysis:Us g d e s o a a a ys sUs g d e s o a a a ys s

dFE prototype ×=prototype

dFdFE ×=⎟

⎞⎜⎛×= 32mod NNN

E el =⎟⎠

⎜⎝

×=

3model 1E

= 3NEprototype

Let us suppose an explosion from a given device ispp p ganticipated to yield 1 TJ of energy

If we model this explosion in a 100g centrifuge test, then

JE 12101

p g g ,

JEprototype12101×=

kJE el .1000100

1013

12

mod =×

=100

Thus, we would need a much smaller explosion in the, pcentrifuge test, that can be produced by few grams ofTNT !!!

Effects of blast on structures can be studied in this way

Scaling law for consolidationScaling law for consolidationScaling law for consolidationScaling law for consolidation

Consolidation of clayey soils plays an important Consolidation of clayey soils plays an important part in geotechnical designpart in geotechnical design

Long term settlement of many structures is a Long term settlement of many structures is a function of the consolidation settlementsfunction of the consolidation settlements

We can write the 3We can write the 3--D consolidation equation as D consolidation equation as f llf llfollows:follows:

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

+∂∂

=∂∂

zu

yu

xuC

tu

v 2

2

2

2

2

2

⎠⎝ ∂∂∂∂ zyxt

We can identify the coefficient of consolidation Cv as ani t t timportant parameter

We know from Soil Mechanics that degree of consolidation is linked to co-efficient of consolidation Cv, drainage path ‘d’ and time of consolidation ‘t’.

Time factor Tv indicates the degree of consolidation and is linked to the above parameters as follows:

tCT 2.dtCT vv =

We wish to achieve the same degree of consolidation in the centrifuge model as that has occurred in the field

i.e. the Tv should be the same in the model and prototype

In the prototype:

( ) ( ) ( ) t t

prototypeprototypevprototypev d

tCT 2.=

In the model:

( )prototyped

( ) ( ) ( )eltCT mod=( ) ( ) ( ) el

elvelv dCT

mod2modmod .=

Dividing the above equations with one another and notingthat on the LHS we will get ‘1’ as we are aiming for samedegree of consolidation, we get;degree of consolidation, we get;

lt d

( )( )

( )( ) ( )

prototype

el

elvelv

dtt

CC

TT

2

mod

modmod .1 ==( ) ( ) ( )( )prototype

elprototypevprototypev

ddCT

2mod

2

If we use the same soil in the centrifuge model as that is present in the field, then we will have the same co-efficient Cv both in the model and the prototype, therefore:

( )( ) 1mod =elvC( ) 1

prototypevC

Therefore, the time of consolidation in the model and the prototype are related by:

2⎞⎛

2modmod 1

Ndd

tt elel =⎟

⎟⎠

⎞⎜⎜⎝

⎛=

Ndt prototypeprototype⎟⎠

⎜⎝

L h 10 l f M i l ff M b i hLet us suppose that a 10m layer of Marine clay off Mumbai shoretakes about 10 years to reach 95% consolidation

In a 100g test the same cla ill each 95% consolidation inIn a 100g test the same clay will reach 95% consolidation in

st el 2mod60602436510 ××××

=el 2mod 100

hoursst el .8.831536mod ==

This is ‘very attractive’ to us, as we can now start to model l l d b h f flong term consolidation events by running the centrifuge forrelatively short periods of time.

Thus Centrifuge Acts as a Time Machine !!!

Advantages of Centrifuge ModellingAdvantages of Centrifuge ModellingAdvantages of Centrifuge ModellingAdvantages of Centrifuge ModellingWe create CORRECT prototype stresses and strains in We create CORRECT prototype stresses and strains in the centrifuge model thereby the soil will mobilise the the centrifuge model thereby the soil will mobilise the the centrifuge model, thereby the soil will mobilise the the centrifuge model, thereby the soil will mobilise the RIGHT stiffness and TRUE behaviour is capturedRIGHT stiffness and TRUE behaviour is captured

Using centrifuge modelling new materials with unknown Using centrifuge modelling new materials with unknown Using centrifuge modelling new materials with unknown Using centrifuge modelling new materials with unknown stressstress--strain relationships or known materials with new strain relationships or known materials with new stress paths can be tested to reproduce prototype stress paths can be tested to reproduce prototype behaviourbehaviourbehaviourbehaviour

The big advantage of Centrifuge Modelling is that the The big advantage of Centrifuge Modelling is that the f l h ll b df l h ll b dTRUE failure mechanism will be capturedTRUE failure mechanism will be captured

The Centrifuge Modelling can model TRUE ThreeThe Centrifuge Modelling can model TRUE Three--The Centrifuge Modelling can model TRUE ThreeThe Centrifuge Modelling can model TRUE ThreeDimensional Behaviour (say in Pile Groups), which can Dimensional Behaviour (say in Pile Groups), which can be very expensive and time consuming using FE be very expensive and time consuming using FE analysis even todayanalysis even todayy yy y

SoilSoil Structure Interaction is automatically accounted for Structure Interaction is automatically accounted for SoilSoil--Structure Interaction is automatically accounted for Structure Interaction is automatically accounted for in centrifuge modelsin centrifuge models

Long term events such as consolidation settlements can Long term events such as consolidation settlements can be modelled in relatively short durations in a centrifuge be modelled in relatively short durations in a centrifuge test This is also true for other diffusion processestest This is also true for other diffusion processestest. This is also true for other diffusion processestest. This is also true for other diffusion processes

Forces required in the centrifuge model are small Forces required in the centrifuge model are small compared to the prototype, so centrifuge model tests compared to the prototype, so centrifuge model tests are ‘cheap’ compared to the field testingare ‘cheap’ compared to the field testing

Energy scaling law dictates that we can model events Energy scaling law dictates that we can model events such as blast loading using very small charges in such as blast loading using very small charges in

t if d lt if d lcentrifuge models.centrifuge models.