Sand cone in the attic My question is based on Halliday books Tom 2, chapter 13 (exercise 42).
There is a cone of sand laying in the attic. Consider stress of the floor under the sand cone.
Why the biggest stress is in radius $r_m$ from the middle of sand cone?
I can't understand why stress function of radius $\sigma (r)$ is like it is shown on the graph.
Can it be proven theoretically (mathematically)?

 A: See Nature 382, 336 (1996), "An explanation for the central stress minimum in sand piles". Their abstract:
"A KEY component of a continuum-mechanical description of a sand pile (viewed as an assembly of hard particles in frictional contact) is the requirement of stress continuity—the forces acting on a small element of material must balance. But an additional physical postulate is required to close the equations. Standard continuum approaches postulate that the material is everywhere just at the point of slip failure1,2. But these approaches have been unable to explain a startling experimental observation3—that the weight exerted by a conical sand pile on a surface has a minimum, not a maximum, below the apex. Here we propose a new closure, which embodies an intuitive model of arching4,5 within a fully consistent continuum theory. Our assumption is that the principal stress axes have a fixed angle of inclination to the vertical. In two dimensions, this is sufficient to close the equations. In three dimensions, a second closure relation is required, but our results are relatively insensitive to the choice made. Our model, which contains no adjustable parameters, can account for the vertical stress distribution in real sand piles. This supports the idea that stresses propagate within a granular medium according to local rules that depend on its construction history."
To get an idea of "arching", see the picture at http://www-personal.umich.edu/~rlmich/research/projects/arching.html .
The abstract of Science  02 Aug 1996, Vol. 273, Issue 5275, pp. 579-580, commenting on the Nature article:
"For the past 15 years, physicists have tried—and failed—to explain a perplexing mystery of granular mechanics: A conical pile of poured sand does not exert its maximum pressure on the ground in the middle, directly under the apex, but in a ring around the center point; there is actually a dip in pressure in the middle. Now a group of British and French researchers has proposed a new model of sand piles, in which the main compressive stresses lie along fixed parallel lines. These stress lines, angled precisely halfway between the slope of the pile surface and the vertical, steer the pile's weight away from the center, giving a central pressure dip. Despite its simplicity, the model does a respectable job of reproducing experimental data."
A: The builders of ancient dome structures, like the Rome Pantheon, managed to make them using a temporarily wooden frame to support the stones.
Once all stones were in place, the wooden frame was removed, and the compressive horizontal components of the stresses between the stones were supported by the friction between them and the columns and walls at the sides of the dome.
If we think of several domes being built that way, each one inside the other like Russian dolls, the remaining height at the center becomes smaller and smaller due to the layers of domes. Finally, when the center part is filled, it consists only of a small vertical pile of stones.
For this extreme example, the loads are minimum at the center because the weight of the central stone of each dome doesn't add to the final central column. All of them are supported off-center. All that design needs enough friction in the ground to hold the horizontal stresses.
In the case of the pile of sand, if all the grains were cubic and organized as bricks, all the loads would be vertical, and the maximum load would be at the center.
But its formation is randomic, and between the model of domes and the model of pile up of bricks.
If there is enough friction in the soil, and between sand grains it is possible to have centripetal horizontal stresses as in the case of the domes.
The result is some transfer of vertical weight in the center to the periphery, following the principle of the layers of domes.
