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Assume an ideal board, supported by two joists. Where should those two joists be optimally placed?

Instinctively, I'd say at somewhat less than 25% and somewhat more than 75% of the extend of the board.

Presumably this also depends on whether the board can move freely with respect to the joists, or whether the board is attached to the joists.

I'm unclear even where to start analyzing this problem ...

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  • $\begingroup$ Optimal is a though word. Optimal so that it wont fall over in the joist direction? Optimal for bearing a heavy load without getting imbalanced? There are many criteria on which you can base you question. If you want to do this in real life, it will be much better to ask it on the home improvement portal, the people there have experience with the practical aspects of it. Otherwise improve your question to include the criteria for being optimal. $\endgroup$ – WalyKu May 31 '14 at 21:58
  • $\begingroup$ I guess if we had a way of analyzing the problem, it should be straightforward to optimize for different criteria? E.g. @Jack Wakefield below wants to equal two bending moments. Another would be identical vertical displacement of middle and end of the board. $\endgroup$ – Johannes Ernst Jun 4 '14 at 20:09
  • $\begingroup$ Yes, you could optimize it for different criteria. You can't optimize it for an arbitary number of criteria(if they are in correlation/conflict). I suggest that you look at what the application requires, and what the worst case scenarios are. Based on that form the requirements. I had a course where we used linear programming for this kind of problems. If the system you get is not linear look for related methods. Why are you interested in the problem anyway? Do you want to build something or is it just theoretical? $\endgroup$ – WalyKu Jun 4 '14 at 21:26
  • $\begingroup$ I have shelving parts ready to be assembled, and I hate that I don't know where to put the supports. (I realize that 25% and 75% is going to be just fine for my purposes.) $\endgroup$ – Johannes Ernst Jun 4 '14 at 22:02
  • $\begingroup$ Yes that will be fine then. If you wanted to get some real feedback from the community you should have asked it in a more scientific/engineering fashion. For the practical side the home improvement guys (as I suggested) would be more helpful, because well they do such stuff :-D $\endgroup$ – WalyKu Jun 4 '14 at 22:05
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This is an structural engineering problem. To optimize the placement of the two "joists" (supports), the absolute value bending moment at the center of the board (beam) must be equal to the absolute value bending moment over the support. Solution Method: Solve for M1, solve for M2, Equate M1 to M2, solve for X (the unknown distance of the support from the beam centerline.

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  • $\begingroup$ I'm not certain why you say M1 must be be M2? $\endgroup$ – Johannes Ernst Jun 4 '14 at 22:03

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