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

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2 Module 7: Lecture -3 on Geotechnical Physical Modelling

3 Hydraulic Gradient Similitude Method Example: Uplift behavior of pile in saturated sand profile

4 Applications of Hydraulic Gradient Similitude Method To study load-settlement behaviour of circular footing resting on sand (After Yan and Byrne, 1989) To study response of piles to static and cyclic lateral loads embedded in sand at various stress levels (Yan, 1990) To study collapse of a strip footing on sand dense sand under vertical eccentric loads (After Musso and Ferlisi, 2009) Modelling Uplift behaviour of piles in sand (After Leshchinsky et al. 2012)

5 Limitations of Hydraulic Gradient Similitude Method It can be used to model only the behaviour saturated sand. Uniform fine sands or uniform silty sands tend to yield more uniform gradients across the soil layers, and more consistent test results. The hydraulic gradient technique was found to be particularly sensitive to the distribution of the hydraulic gradient in the soil profile. Limited to model with horizontal ground surface only. It can not account for self-weight of the foundation materials (like footing/pile/etc.,) In conclusion, the hydraulic gradient technique that is presented herein offers a viable, inexpensive scale modelling approach that may be useful for model simulation of a variety of geotechnical engineering problems.

6 Modelling techniques Mathematical Models Learn mechanism of transport processes Verification/ Improvement of theory [full, simplified] Direct modeling of prototype Scale Physical Model [centrifuge model] Prototype [actual field problem]

7 Mechanics of centrifuge modelling Object moving in steady circular orbit

8 Mechanics of centrifuge modelling If a body of mass m is rotating at constant radius r about an axis with steady speed v then in order to keep it in that circular orbit it must be subjected to a constant radial acceleration v 2 /r or rω 2. In order to produce this acceleration the body must experience a radial force mrω 2 directed towards the axis. By normalizing the centripetal acceleration with earth s gravity g and state that the body is being subjected to an acceleration of Ng where N = rω 2 /g

9 Principle of Centrifuge modelling For example, the string attached to the sphere experiences a tension that can be felt by the hand holding the other end. The tension in the string applies the centripetal force on the sphere pulling it toward the centre of rotation. This force must be balanced by another force, otherwise the sphere would move toward the centre. This can be explained by means of a fictional or inertial force that acts on the sphere radially outward (Madabhushi, 2014)

10 Principle of Centrifuge modelling By Newton s third law the centrifugal force and the centripetal force must be equal and opposite. The centrifugal force causes a centrifugal acceleration that acts radially outward. In the example of the car going around a corner, we can say that the tires of the car will exert centrifugal forces on the road acting radially outward. Centrifugal forces are exploited in a wide variety of engineering devices such as centrifugal pumps and centrifugal clutches (Madabhushi, 2014)

11 Rotation about a fixed axis z A B O φ θ r P v x A y

12 Rotation about a fixed axis Relevance to centrifuge based physical modelling: a t = 0 for tests conducted at constant angular velocity and for uniformly increasing g-level, a t is present.

13 Principle of centrifuge modelling It is to generate similitude of stresses and strains between models and their real or hypothetical prototype by simply increasing acceleration levels proportion to the reduction in linear dimensions. - Can be explained through a slope stability problem At failure ( FS = 1), N s = c u /(γ s H) =c u /(ρ s gh) The stability of the slope depends on c u, H, and γ s

14 Principle of centrifuge modelling In order to model the slope failure for 1/50 physical model at 1g, it becomes necessary to produce a material having [c u /γ] = 1/50 of the prototype material. [i.e. for (N s ) m = (N s ) p ] It is almost impossible to produce a material with c u /γ = 1/50 For a 1/50 physical model at 50 g, N s = c u /[ρ s (50g)H/50] = c u /ρ s gh = c u /γ s H

15 Principle of centrifuge modelling 1g β H ω Ng = R e ω 2 H/N 1g β H/N R e The model slope in a centrifuge will behave similar to that in the field.

16 h/n h σ v = γ(h/n) 1:N (1g model) 1:1 (Prototype) σ v = γh h/n σ v = (γn)(h/n) = γh 1:N (Ng model) Principle of centrifuge modeling technique

17 Stress-strain behavior of soil Let us consider the shear stress - shear strain for loose and dense soils Shear stress τ Model behaviour at small stresses and strains Soil is highly NON-LINEAR and PLASTIC Full-scale structure behaviour at large stresses when large strains are mobilized Full-scale structure behaviour at large stresses Shear strain ϒ Creation of full-scale stresses and strains in small-scale physical models are important

18 Centrifuge modelling ω F1 F2 r F2 - F1 = df = (dm)a r The stress increase must provide the force necessary to generate centripetal acceleration.

19 Centrifuge modelling dm = ρ (dv) = ρ (dz. A) dσ v = df/a = ρ ((dz.a)/a).a r Using a r = Ng = ρgz Thus stresses are identical at geometrically equivalent points in the prototype and centrifuge model, provided the linear scale in the model is the inverse of acceleration scale N g = 1/N l = N

20 Element of soil (a) at surface of the earth and (b) on centrifuge z σ v = ρgdz = 0 ρgz z N σ v = ρngdz = 0 ρgz

21 Element of soil (a) at surface of the earth and (b) on centrifuge Consequently, if the mechanical behavior of the soil is strongly dependent on stress level then such behavior should be correctly reproduced in centrifuge modelling.

22 Types of centrifuges Balanced beam/beam centrifuge Drum Centrifuge R Typical balanced beam centrifuge

23 Drum Centrifuge R R ω Ng Ng

24 Vertical stress in the radial acceleration field dr = r b -r t dr r t r ω r b R t dm = ρ (dv) = ρ (dz. A) dσ v = df/a = ρ ((dz.a)/a).a r Using a r = rω 2 dz z

25 Vertical stress in the radial acceleration field dm = ρ (dv) = ρ (dz. A) dσ v = df/a = ρ ((dz.a)/a).a r Using a r = Ng = (R t + z)ω 2 = ρω 2 (R t + z/2)z This indicated that non-liner variation of vertical stress resulting due to model rotating in a radial acceleration field.

26 Variation of centrifugal acceleration in models of a 10 m soil layer [N =50] 1.0 m Merit of large centrifuge 4.0 m g 47.4g 50g m 48.9g 48.8g 50g 57g g 0.6 m g 51.3g

27 A soil layer in prototype and its corresponding centrifuge model Soil σ v σ h Gravitational acceleration g (σ v ) m