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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.