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Name: Gabe
Status: student
Grade: 9-12
Location: CA
Country: USA
Date: Fall 2011


Question:
lets say you had a hollow sphere and a pyramid (rectangular or triangular), equal in height, width and length, and put a single board on top of them. these two shapes are on a flat surface so the sphere will not roll. If you keep adding weight on top of the board which would break first.



Replies:
Hi Gabe,

This is an educated guess, not having done a structural analysis. I believe the spherical shape will be the strongest in spite of the fact that its loading resolves into more complex compression and tensile forces, as well as bending stress at the points of contact with the weight and the floor.

The pyramidal shape would appear at first glance to be stronger, because all forces resolve into pure compression. However, contact with the weight is at a single point, meaning that stress at this point is infinitely high.

It would be interesting to "build" a CAD model of each and "test" it with FEA (Finite Element Analysis) software. Must try this! Regards,

Bob Wilson


Gabe - I think it is difficult to address this question in a precise intellectual manner. The results will depend on many details of the real materials used. All materials have at least three parameters of strength to consider: compressive strength, tensile strength, and elasticity. Some materials have more of one and less of another. Breakage can be gradual or abrupt, i.e., yielding or ductile vs. brittle.

The material parameters of the board must be compared to those of the shapes, too.

In the case of a long-skinny rod between the boards, breakage likely depends more on elastic stiffness than on material-strength. When compressive stress reaches a certain level, the rod will suddenly bow out sideways, in any random direction. Bowing will then accelerate until the yield strain is exceeded and the rod breaks. It is a case often covered in college physics classes and I am sure you could find it in a Schaum's study guide or somewhere online. " http://en.wikipedia.org/wiki/Buckling" looks about right. "elastic collapse of a column" might be a phrase to look for. I am not sure a pyramid can do elastic collapse, and I think maybe a thin-walled sphere can, in the sense that the point with the pressure on it might suddenly invert and bulge in towards the center instead of pressing against the board.

Think about this: you should specify that the sphere and the triangle have the same thickness. Thicker is usually stronger, and with zero thickness there is always zero strength, in any real or realistic-imaginary material. Which collapses first might actually depend on the thickness, on the ratio of the wall-thickness to your outer size parameter.

And what kind of breakage you be allowing for first? If the board is made of hard strong stone, and the sphere and the triangle of hard but not-so-strong plastic, and the thickness was substantial (how to define that?...), then the pointy top of the pyramid would probably get crushed off to some modest extent before the ball broke. Does blunting the tip count as breaking? If the board is made of soft wood, and the shapes of hard glass, then the glass pyramid-tip would poke into the wood and make a soft nesting contact. Then the sphere might break first. But you have to specify a board that is stronger than either shape... does it have to be harder and stiffer too? Maybe it would be fairest to place two pyramids point-to-point, one being upside down, so the pyramid-point vs. the board-face is no longer a question. Now it would be a pyramid-point, blunted to a certain size, vs. its identical twin.

So you see there are almost too many issues to decide on before actually tackling this question. If you actually try it with a real choice of material and thickness, and observe a repeatable breakage mode, then you can probably start modelling the why's of all the behavior you observe. Some engineers would probably find that approach interesting.

Unfortunately, I am not sure there is any workable theme here, about which ultimate shape is the strongest. And if there is, I think the cone and the tetrahedron should be included in the challenge. I also wonder if the icosahedron and dodecahedron would act like better versions of the sphere. They do not tend to roll, and each would have the area of one flat face against the board, so that there is no one point of highly-concentrated pressure to flex, bend, or break first. But those are just more thoughts in a very messy debate.

Jim Swenson



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