Weight Distribution of granular piles

Question

I was recently at a farm watching piles of grain being formed in the shape of enormous cones. Each of these grain stacks weighs in excess of 50 tonnes. I am not a physicist but I am very curious as to why the grain at the bottom of the stack does not get crushed by the weight on top of it.

If i place 1 tonne on a single piece of grain it will crush into powder, however when over 50 tonnes of grain is tipped into a pile there is no damage. What is the physics principle that explains this phenomenon?

Answer 1

The physical processes that control the structure of conical piles are fascinating and imperfectly understood even today. However we can approach your question in approximate way.

The angle that the surface of the pile makes with the ground is called the angle of repose. Predicting this theoretically is hard because it is is sensitive to the exact nature of the material in the pile. However the angle of repose is generally in that range 30º to 45º. You don't say what grain you saw, but wheat is actually at the low end of this range at 27º, and I'd guess corn is fairly similar. This means the piles are quite broad so the weight is spread out over a large area.

The volume of a conical pile of height $h$ and base radius $r$ is given by:

$$ V = \tfrac{1}{3}\pi r^2 h $$

and the area of the base is just $\pi r^2$, so the average pressure at the base turns out to be just dependent on the height:

$$ P_\text{av} = \frac{Mg}{A} = \frac{V\rho g}{A} = \tfrac{1}{3}h\rho g $$

For wheat the average density of a pile is around 750 kg/m$^3$, and putting the numbers into the equation above we get:

$$ P_\text{av} \approx 2500 h \,\text{Pa} $$

For comparison, atmospheric pressure is 101325 Pa, so the pile would have to be 40 metres high for the average pressure at the base to be even equal to one atmosphere. And there's your answer. Even ina large grain pile the pressure is simply too small to crush the grains.

Answer 2

The force straining a grain (which may soon be squashed) is equal and opposite to the weight (gravity force) supplied by the grain pile above. That's Newton's third law. If ALL the weight of the pile were held up by one single kernel,

F_grain ~= Mass_of_pile * g

and it would be easy for the ton of grain to smash the bottom kernel. It doesn't work that way, because ALL the grains on the bottom of the pile share the support of the weight above. If that support were equal,

N_bottom_layer * F_grain ~= Mass_of_pile * g

But the grains at the edge of the bottom of the pile undoubtedly are under lower stress than those at the center; the peak force at the center is likely to follow the familiar formula from hydraulics

F_grain_max/Area_granule = Density_of_pile * g * height_of_pile

Which indicates that a force on a grain is proportional to the density, height of the pile, and the area OF A SINGLE GRAIN. The one-lonely-grain's load doesn't at all scale with the weight of the whole pile.