What are the best locations for placement of $y$ supports for a beam of length $x$?

This seems like a very basic physics question, but I have been unable to find the answer. Perhaps I am not sure what terms to use to search.

For example, let’s take a $4\times 4$ piece of lumber of length $x$.

For $y=1$, the answer is obviously $x/2$, such that the support should be placed in the center.

For $y=2$, I am not sure what the answer is.

That is, for an 8’ piece of 4×4 lumber, if you have two supports, I don’t think the best support would be at 0 feet and at 8 feet, as it would be weaker in the middle. And obviously, if you put both supports at 4’, then it would be weak at the ends.

So, my gut says placing the beams at about 1.5 feet and 6.5 feet would be best, but I am looking for the physics formulae to prove the true answer.

  • 1
    $\begingroup$ en.wikipedia.org/wiki/Airy_points $\endgroup$
    – G. Smith
    May 16, 2020 at 5:16
  • $\begingroup$ That was precisely what I was looking for! Thanks G. Smith! :) $\endgroup$
    – Chad
    May 16, 2020 at 18:18
  • $\begingroup$ And for the record, it appears that the "best" for my example of an 8 foot piece of lumber with two supports is to place the supports about 4.5' apart (not 5' apart, as per my gut), with 1.75' of overhang on both ends. [Precisely: Airy points for two supports 4.616' apart and Bessel points for two supports 4.472' apart] $\endgroup$
    – Chad
    May 16, 2020 at 18:39


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