Lifting a buried object: Reverse hopper theory
A theory is given to predict the upward force, F, to lift an object buried at depth H in granular material. Explicit formulae give F in terms of the material friction coefficient of the granular material and the geometric parameters. The lifted object is either (1) a horizontal disc of diameter D or (2) a horizontal plate of width B and length L, where L >> B. In case (1), the lifted disc is assumed to cause axi-symmetric upward particle motion, reverse hopper flow, within an inverted cone. Active failure is assumed: the vertical stress, Ïƒ2, is K x (horizontal stress Ïƒ1); here K = (1 + sin Ï†)/(1 - sin Ï†), Ï† being the angle of friction for the granular material. This gives the vertical stress, Ïƒ20, on the disc. An additional lift force is needed to overcome the frictional stress, Ï„, at the conical interface between stationary and upward moving particles: it is assumed that Ï„ = Î¼ Ïƒ1, Î¼ being the internal friction coefficient. For consolidated granules, Î¼ = tan Ï†, but for the sheared material, Î¼ < tan Ï†. The total lift force F is the sum of (i) the effect of Ïƒ20 plus (ii) the effect of Ï„; this sum gives an equation to predict the breakout factor Nqf = F/(Î³â€²AH), where Î³â€² = bulk weight density and A = Ï€ D2/4. For case (2), relevant to the uplift of a long buried pipe, the theory is similar: the two failure surfaces are flat, inclined at angles +Î± and -Î± to the vertical. Similar assumptions as to the stress distribution, i.e. two-dimensional active failure, give an equation for Nqf. The two predictive equations for cases (1) and (2) agree well with relevant published measurements of Nqf.