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Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

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Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Module 7:

Lecture - 5 on Geotechnical Physical Modelling

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Limitations of centrifuge-based Physical Modelling

Non-homogeneity and anisotropy of soil profiles

Limitations of modelling tools

Variation of g-level with horizontal distance and depth of the model

Boundary effects

Scale effects

Inconsistency of scale factor for time

Coriolis effect

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

The Coriolis force arises from any movement thatoccurs within the centrifuge model.

For example, if we try to construct an embankmentby raining sand in flight or if we drop a ball fromthe center of the centrifuge to study projectilemotion and impact on the soil, or simulation ofrainfall, etc.,

Then, the moving object will have Coriolisacceleration 2vω and Coriiolis force is 2mvω.

Coriolis effect

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Machine configurations Balanced Beam Centrifuge

Beam Centrifuge

Drum Centrifuge

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

• Balanced beam/Beam centrifuge

• Drum Centrifuge

Typical balanced beam centrifuge

R

Types of centrifuges

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Selected centrifuges in the worldMoscow Railway Transport University, Russia

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

RUB Centrifuge (Z1)

R = 4.125 m; Max. g = 250; Max pay load = 2 tonnes; Capacity = 400 g-tonnes

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

(Madabushi, 2014)

The Turner beam centrifuge

R = 4.125 m; Max. g = 150; Capacity = 150 g-tonnes

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

The HKUST Centrifuge

R = 4.2 m; Max. g = 150; Capacity = 400 g-tonnes

Ng(2014)

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

The RPI Centrifuge

R = 3 m; Max. g = 160; Max pay load = 1.5 tonnes;

Capacity = 150 g-tonnes (at 100g)

http://www.nees.rpi.edu/equipment/centrifuge/

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

The IFSTTAR Centrifuge

http://www.series.upatras.gr/LCPCR = 5.5 m; Max. g = 160; Max pay load = 2 tonnes;

Capacity = 200 g-tonnes (at 100g)

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

The KAIST Centrifuge

After Kim et al. (2013)

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

R = 8 m; Max. g = 150; Capacity = 800 g-tonnes ( 8 tonnes at 100 g)

The K-water CentrifugeKim et al. (2014)

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

After Viswanadham and Tripathi (2006)

Small Geotechnical Centrifuge at IITB (1994-2008)

R = 0.32 m; Max. g = 200; Capacity = 0.4 g-tonnes

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Major Attributes

♦Indigenously built

♦Regenerative braking

♦In-flight balancing

♦High payload capacity

R = 4.5 m; Max. g = 200; Capacity = 250 g-tonnes

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Capacities of major balanced/beam centrifugesin the world

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Drum centrifuge at TIT Japan

(Gurung et al., 1998)Payload:0.6t; amax. = 484; Capacity = 290 g-tonnes

Aspect ratio of soilchannel: 1.2 mφ; 0.15m deep and 0.3 mwide.

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Preparation of sand sample in a drum centrifuge

a) By feeding sand through nozzle

b) By feeding sand on to spinning disc

After Wood (2004)

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

(http://www.geotechnics.ethz.ch/centrifuge/)

The ETH Zurich Drum CentrifugeAspect ratio of soil channel: 2.2 mφ; 0.3 m deep and 0.7 m wide.

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Relationship between ramp angle, θ with RPM, ωfor beam centrifuges Rh

θ

NTr

y

mg

R = Rh+lsinθ

T = mω2 (R + l sinθ )

(neglecting vertical acceleration)

If there is no angularacceleration of the modelabout its centre of mass (i.e. P)then there can be no momentsabout P. ⇒ Resultant of T and Npass through P

P

NTanθ = T

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

IITB Large Centrifuge in action

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Variation of ramp angle, θ with RPM, ω for 4.5 m radius beam centrifuge

Rh = 3.3 m; l = 0.815 m

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Variation of ramp angle, θ with RPM, ω for 1.1 m radius beam centrifuge

Beam centrifuges can only be used above a certain ω

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Scaling laws in centrifuge modelling

Assumptions:

Soil can be treated as a continuum(Different parts of model are many times larger than the soil grains)

Soil properties are not affected by a change in acceleration.

(The force of gravity acts at the centre of mass ofeach atom and does not significantly affect theelectron shells which determine all materialproperties other than self-weight)

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Similitude in Geotechnical Engineering

Am/Ap = 1/N2; Vm/Vp = 1/N3

For an identical soil inthe model and prototype,ρm = ρp ⇒NM = NV = 1/N3.

For identical effective stresses in the model and prototype:Nσ′ = Nσ = Nu = σm′ / σp′ = σm/σp = um/up = 1

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

For example,

δm = δp/N

⇒As we have scaled down all the length dimensions ofthe prototype by a factor of N in the centrifugemodel, the scaling law for settlement would be 1/N,that is, settlements in the centrifuge model are 1/Ntimes in the prototype.

⇒Similarly, Am/Ap = 1/N2 and Vm/Vp = 1/N3.

Scaling laws in centrifuge modelling

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Scaling laws in centrifuge modelling

We have already proved that the stresses areidentical in the centrifuge model and the prototype.

As δm = δp/N, ε = (dδ)/δ

With (dδ)m = (dδ)p/N ⇒ εm/εp = 1

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Stress-strain behavior of soilLet us consider the shear stress - shear strain for loose and dense soils

Soil is highly NON-LINEAR and PLASTIC

Shea

r stre

ss τ

Shear strain ϒ

Model behaviour at small stresses and strains

Full-scale structure behaviour atlarge stresses

Full-scale structure behaviour at large stresses when large strains are mobilized

∴ Creation of full-scale stresses and strains in small-scale physical models are important…

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Scaling laws in centrifuge modellingForce, work, and energy Considering the basic definition of Newton’s second

law of motion, a force F acting on a body of mass m will cause an acceleration a, such that: F = ma

⇒ Fm/Fp = 1/N2

For example, a 3000 kN force in a 50 g centrifuge test scales downto be only 1200 N. This is another advantage of a centrifuge test, aswe can easily make actuators to load piles, retaining walls, slopes,etc., and forces that need to be applied by these actuators arerelatively small.

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Scaling laws in centrifuge modellingForce, work, and energyThe basic definition of work done W is the product of a force F moving through a distance d.

⇒ Wm/Wp = 1/N3

⇒This scaling law for work done suggests that the workdone in a centrifuge model is relatively small comparedto that in a prototype. This is also advantageous forcentrifuge modellers.

Prof. B V S Viswanadham, Department of Civil Engineering, IIT Bombay

Scaling laws in centrifuge modellingForce, work, and energyConsider the definition of potential energy PE normallyexpressed as energy lost by a falling mass m througha height h, PE = mgh

⇒ PEm/PEp = 1/N3

Thus, centrifuge modelling can offer a very effective way ofinvestigating the effects of explosions on buildings, earthen dams,dams, retaining structures, etc., without the need to conduct thesestudies at full scale, which can be both expensive and damaging tothe environment.