TITLE: Sand cone in the attic
QUESTION [2 upvotes]: 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)?

REPLY [1 votes]: 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.
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