Cute little problem about beams

2010 January 1
by Daniel Lakeland

Suppose you have a simply supported beam between two points x=0 and x=1. The beam has a given bending stiffness and is loaded with a uniform load such that the deflection at x=1/2 is y (this all assumes nondimensional variables).

Now suppose that your goal is to minimize the deflection of this beam by adding 1, 2, or 3 pointlike support forces at locations between x=0 and x=1. Where should these supports be placed and what should the forces be? Give a solution for minimization of both maximum deviation from the average as well as minimum integrated squared deviation.

Suppose instead that the goal is to minimize the maximum tensile stress in the cross section? Assume a rectangular cross section. Give a solution for this situation for 1, 2, or 3 pointlike support forces.

4 Responses leave one →
  1. January 1, 2010

    I should be able to approach this with Karush-Kuhn-Tucker, if only I would know the meaning of all the terms 🙂

  2. January 5, 2010

    I am no CivE but the intuitive answer to minimize deflection is to shorten the spans. Therefore the minimum deflection would come with 5 supports total, with the 3 additional supports placed at the 1/4 span points, i.e., x=0.25, 0.50, and 0.75, with each support bearing an equal load of 0.2.

    I would give the same answer for tensile stress, but I have no calculations to back it up, it's just a gut feeling. The max tensile stress in a simply supported beam would be at the mid-point, so it just seems to me like 3 additional supports would achieve the minimization of deflection and of tensile stress.

    Did you ever figure out the answer?

    • Daniel Lakeland
      January 5, 2010

      It is intuitive, and I think you can make an easy case via symmetry that for 1 support it should be in the middle. But how much force to apply (or what deflection to induce at the midpoint?) For 2 and 3 supports it is not obvious that you should in fact use the 1/3 or 1/4 points with equal forces on each, that's definitely going to be a possible case, but for example the minimum integrated squared deviation condition should probably give a different solution than the maximum deviation condition. Also, by causing the beam to wiggle around more you might fit it into a smaller range of deflections, but that might increase the stresses due to bending.

      I'm working on some solution code out of curiosity now. I'll post the results if I can get it to work.

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