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I have a cantilever beam that is resting against a wall (not fixed) and will not slide (strictly looking at bending moments). The beam of length = $0.3 \;\text{m}$ is under a distributed load of $4000 \;\text{N}$ and a preload is applied to the free end of the beam to hold the beam. The cross-section is an annulus with outer diameter $60 \;\text{mm}$ and inner diameter $30 \;\text{mm}$.

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How can I find what minimum preload I need to hold the cantilever beam level? Does this change base on the cross-section of the beam?

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The first step is to identify the failure mode if the preload is absent: Since sliding is precluded, the beam would pivot around the lowest point where it contacts the wall. Therefore, we must apply a sufficient preload such that the net moment around that point is zero or counterclockwise (to press the highest point into the wall) rather than counterclockwise (which would allow the highest point to detach from the wall).

Balancing the moments, we have (4000 N)(0.15 m) = P(0.03 mm). Note that I took the equivalent point load and moment arm to the distributed load; in addition, the 0.03 mm comes from the outer radius, or the distance from the center preload position to the potential pivot point (inasmuch as this moment arm stays constant, the size and shape of the cross section is irrelevant). I find P > 20 kN for the necessary preload. It's good practice to then check whether this new load would have any adverse effect, such as buckling the beam or exceeding its compression strength.

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