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

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Oswald Ray
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2 Module 7: Lecture - 6 on Geotechnical Physical Modelling

3 Scaling laws in centrifuge modelling Force, work, and energy Consider the definition of potential energy PE normally expressed as energy lost by a falling mass m through a height h, PE = mgh PE m /PE p = 1/N 3 Thus, centrifuge modelling can offer a very effective way of investigating the effects of explosions on buildings, earthen dams, dams, retaining structures, etc., without the need to conduct these studies at full scale, which can be both expensive and damaging to the environment.

4 Scaling laws in centrifuge modelling Force, work, and energy PE m /PE p = 1/N 3 Example 1: Consider a requirement to model an explosion event (in the field) that has an energy release of 1GJ (1 Gega Joule = 10 9 Joules) at 100g E m = E p /N 3 = 10 9 /100 3 = 1000 Joules = 1 Kilo Joule This is approximately equivalent to having an explosion from grams (= (1 x 10 3 )/(4.184 x 10 3 ) of TNT (Using 1 g of TNT = x 10 3 Joules of Energy) TNT = Trinitrotoluene

5 Scaling laws in centrifuge modelling Force, work, and energy Example 2: Consider 3.5 grams of TNT (including detonator) used for modelling an explosion event at 50 gravities. Energy released in the centrifuge model (E m ) = 3.5 x (4.184 x 10 3 joules) = x 10 3 Joules Equivalent energy in the field = x 10 3 x 50 3 = 1.83 x 10 9 Joules = 1.83 Gega Joules This implies that effects of explosions can be modelled on geotechnical structures using quite small charges in a centrifuge model to simulate blasts with extensive energy release in reality.

6 Scaling laws in centrifuge modelling With g m /g p = N = 1/N l and using (l m /t m2 ) / (l p /t p2 ) = N, we can get t m /t p =1/N and v m /v p = (l m /t m )/(l p /t p ) = 1 Scale factor for strain rate [(change in length /original length) in time t ] Nε t = (δ l m / l m t m ) / (δ l p / l p t p ) = (δ l m / δ l p ) (l p /l m ) (t p /t m ) = (1/N) (N) (N) = N (ε t ) m = N(ε t ) p

7 Centrifuge modelling technique for simulation of dynamic compaction After Grundbau Taschenbuck (1996) The pounding creates a depression at each drop location and also produces an areal settlement. PE m /PE p = 1/N 3

8 Example 3: Dynamic compaction PE in prototype = 20,000 x 9.81 x 20 = x 10 6 Joules PE in model = 3.92 x 10 6 /50 3 = Joules 20 tonnes 20 m m m = m p /N 3 = 20/50 3 = 160 gm = (0.16 x 50x9.81x 0.4) = Joules h m = h p /N = 20/50 = 0.40 m =19.81 m/s 50g =19.81 m/s

9 Scaling laws in centrifuge modelling Seepage through a slope (Field situation) H σ p = ρgh h u p = ρ w gh

10 Scaling laws in centrifuge modelling Seepage through a slope (Centrifuge model) σ m = ρng(h/n) Water source H/N h/n Ng u m = ρ w Ng(h/N) = u p

11 Scaling law for time of seepage Scale factor for Seepage force, Seepage pressure and Seepage Velocity Force applied to sand particles h 1 = γ w h 1 ( lx1)-γ w h 2 ( lx1) = γ w ( h/ l) ( l 2 x1) J = γ w i V Seepage pressure = Seepage force/unit volume l l h h 2 p s = iγ w

12 Scale factor for Seepage force, Seepage pressure and Seepage Velocity Using Darcy s law: i = v/k F s = (v/k) γ w V For σ m /σ p = σ m / σ p = u m/ u p = 1 and with W F = γ w V With K m = K p ; (ρ w ) m = (ρ w ) p ; (µ w ) m = (µ w ) p k g for g = 0, k 0?

13 Scale factor for Seepage force, Seepage pressure and Seepage Velocity L If we are using the same soil in the centrifuge model and the prototype, then we should expect the same permeability k for the soil in both. The hydraulic gradient, however, is defined as the change in pressure head over a given distance, and this is different in the model and the prototype.

14 Scale factor for Seepage force, Seepage pressure and Seepage Velocity This scaling law for hydraulic gradients suggests that the centrifuge models will have much higher hydraulic gradients than in the prototypes.

15 Scale factor for Seepage force, Seepage pressure and Seepage Velocity As v s = v/n So the scaling law for seepage velocity is N, which indicates that the seepage velocity in the centrifuge model will be relatively high. This result is quite important and we shall ensure that laminar regime is prevalent in centrifuge models. (i.e. by maintaining R e < 1)

16 Scale factor for Seepage force, Seepage pressure and Seepage Velocity Using k m = k p & For k m k p v m /v p = N(k m /k p ) The above equation is valid, if for some reason the soils in the model and prototype have different permeability's.

17 Scaling law for time of consolidation Consolidation of soil is a diffusion process that occurs when excess pore pressures are generated in the soil due to application of rapid loading. With elapse of time these excess pore pressures decrease and the effective stress in the soil increases. The void ratio of the soil changes allowing for the settlement to take place.

18 Scaling law for time of consolidation The governing equation for consolidation in three dimensions can be written as: (Using rate of change of volume = rate of change of void ratio) Let x = Nx m ; y = Ny m ; z = Nz m u m = N u u; (γ w ) m = N γw γ w ; t m = N t t and k m =N k k

19 Scaling law for time of consolidation After simplification: For similarity in model and prototype: Then, N t = t m /t p = 1/N 2 With the above N k /N γ w =1 This scaling law for time of consolidation suggests the consolidation of soil in a centrifuge model occurs N 2 times faster compared to the prototype.

Civil Engineering Department College of Engineering

Civil Engineering Department College of Engineering Course: Soil Mechanics (CE 359) Lecturer: Dr. Frederick Owusu-Nimo FREQUENCY CE 260 Results (2013) 30 25 23 25 26 27 21 20 18 15 14 15 Civil Geological