# Technical Report 129, c4e-Preprint Series, Cambridge

## Lifting a buried object: reverse hopper theory

ref: Technical Report 129, c4e-Preprint Series, Cambridge

Associated Theme: Particle Processes

A theory is given to predict the upward force, *F*, to lift an object buried at depth *H* in granular material. The 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, σ_{2}, is *K* x (horizontal stress σ_{1}); here *K* = (1 + sin φ)/(1 - sin φ), φ being the angle of friction for the granular material. This gives the vertical stress, σ_{20}, on the disc. An additional lift force is needed to overcome the frictional stress, τ, at the conical interface between stationary and upward moving particles: it is assumed that τ = μ σ_{1}, μ being the internal friction coefficient. For consolidated granules, μ = tan φ, but for the sheared material, μ < tan φ. The total lift force *F* is the sum of (i) the effect of σ_{20} plus (ii) the effect of τ; this sum gives an equation to predict the *breakout factor* *N*_{qf} = F/(γ^{′}*AH*), where γ^{′} = bulk weight density and *A* = π *D*^{2}/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 +α and -α to the vertical. Similar assumptions as to the stress distribution, i.e. two-dimensional *active* failure, give an equation for *N*_{qf}. The two predictive equations for cases (1) and (2) agree well with relevant published measurements of *N*_{qf}.

Material from this preprint has been published in: Chemical Engineering Science 105, 198-207, (2014)

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