What would be the ideal cross section of an axially loaded cantilever beam?

[Johnstonator]

Pretty much the title. Just some brain teasers I'm trying to figure out.

I can't think of how a cross section would come into play when it comes to axial loading. Buckling? Since the critical force for buckling is proportional to moment of inertia, so theoretically if I have a high moment of inertia about a specfic axis (like an I beam) the greater the critical force I can apply, thus "ideal"?

But then again, I could have a very large rectangular or circular solid beam and handle axial loading but how would I determine if it's ideal or not?

 

[jrmichler]

The answer is an application specific optimization. A beam is loaded with a bending moment. It may have an axial load. You have to analyze for all possible failure modes.

Some (not all) failure modes include tensile yield, compressive yield, compressive buckling, and web crippling. The optimal structural member will be designed so that it almost fails by every possible failure mode at the same time when it is subjected to the maximum load. You take a cross section, such as I-beam or rectangular box, then optimize it. Then take another cross section and optimize that. Repeat until you find the lightest possible beam.

How to optimize: Start with a simple shape, such as a circular tube. A thick wall tube will fail by yielding. A thin wall tube will fail by buckling. Somewhere in between is a wall thickness and diameter where the yield failure load will be the same as the buckling failure load. That tube will be the optimal circular tube. Then repeat with a rectangular box tube, an I-beam, etc.

An example of an optimal structure is the well known poem about the (fictional) One Hoss Shay: https://rpo.library.utoronto.ca/content/deacons-masterpiece-or-wonderful-one-hoss-shay-logical-story.

 

[Baluncore]

But then again, I could have a very large rectangular or circular solid beam and handle axial loading but how would I determine if it's ideal or not?

Your analysis has made an assumption that the beam section will remain constant. A tapered truss able to do the same job, would cost more to fabricate, but would weigh less than a standard section.