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It is not possible to support a horizontal beam from one end, and the same is true for an I-Beam. I believe this can also be proven for a pipe, but that proof is just outside of my reach.

But all of these proofs depend on uniform density of the beam. If we remove that assumption, the very method of proof no longer holds.

Can we design a boxy* truss of reasonable materials that when supported from one end is horizontal within margin of error of construction? That is, we can ignore Agravado's number, imperfect cuts, imperfect welds, uneven density, etc, but we cannot simply make it so strong that the bending is below margin of error. I'm expecting any such truss must consist of struts of different thicknesses.

Any truss would have the bottom row of struts bending down between the pins on both ends. We can ignore this and say it's horizontal when all the pins are at the same height.

This problem as no practical use. For practical purposes we would just mount a tiny bit higher and use turnbuckles on the diagonals to fine-tune the resulting beam height at the end.

* I mean to eliminate the trivial solution of running a guywire from the end back to the support beam high above.

OK truss formation: XXXX>

Cheating: >

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    $\begingroup$ I'm not sure I understand the premise, we can support a horizontal beam from one end. That is called a cantilever. You can't make it so it doesn't bend at all, but you can definitely control it to make horizontal construction. Cantilevers $\endgroup$ – JMac Sep 28 '19 at 15:44
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    $\begingroup$ Even if you support a beam at both ends, it will deform at least slightly due to its load (or even its own weight). Whether supported at one or both ends, if the load is fixed, you could in principle design a pre-stressed beam to become horizontal once loaded. $\endgroup$ – The Photon Sep 28 '19 at 15:49
  • $\begingroup$ You could certainly pre-bend it so that it is perfectly horizontal on installation Edit @The Photon for the win :) $\endgroup$ – Dale Sep 28 '19 at 15:50
  • $\begingroup$ If you are looking for practical solutions rather than physical limits, you might want to ask for the question to be migrated to engineering stackexchange. $\endgroup$ – The Photon Sep 28 '19 at 15:51
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    $\begingroup$ Of course it is possible to make a beam supported from one end that is straight. You just make the beam slightly curved, so that its own weight straightens it out. "How to design a (lightweight, flexible) structure that has a given shape when it is loaded" is a standard type of optimization problem - but probably not one you will learn about in a physics course, because dealing with "real world" problems tends to be done by engineers, not physicists. $\endgroup$ – alephzero Sep 28 '19 at 23:47

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