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Assume there is a rectangular beam of length L resting on a rigid horizontal surface. The beam is of a real homogeneous material, say aluminum. Assume the surface is infinitely rigid. I apply a vertical force F on each end of the beam, clamping the beam to the surface. 1) How do the clamp forces on the surface distribute along the beam? 2) What would be the shape of the curve for the bottom of the beam that would result in equal distribution of forces? This has real applications in gluing of large flat objects together, where standard clamps cannot reach the interior areas. Thanks Kevin

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Let's say you want to apply a uniform load of q over the length of the beam pressuring equally along the length. clamps are holding the ends with the force of F= qL/2 at each end.

Imagine there was no floor supporting the beam and these were the reactions of a distributed loading, $q= 2F/l.$ This q is the pressure you want to apply uniformly upward from the ground.

That load would deflect the beam into a parabola with maximum deflection of $5qL^4/384EI, \\ \text{and the equation of the beam's centerline deflection is} \ y= -\frac{qx}{24EI}(L^3-2Lx^2+x^3)$

This curve is the deflection of a simply supported beam under the uniformly distributed load of q, which looks like a rope hanging at 2 ends.

You have to camber your beam so that the centerline of it is deflected down by the above eq. and form it by the above deflection curve to have the floor distribute a uniform load, q, to it when it is clamped and forced down to perfect straight line touching the surface.

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  • $\begingroup$ I assume you mean camber the top also (constant cross section along the length)? A more complicated analysis if only the bottom is curved? Very insightful answer, BTW. $\endgroup$
    – KevinM
    Dec 20, 2019 at 17:11
  • $\begingroup$ @KevinM, right, basically you need to curve the beam. Also, you need to be careful with your clamps so as to avoid local buckling. and please mark as accepted if you accept my answer. $\endgroup$
    – kamran
    Dec 20, 2019 at 18:23

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