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Hazelton, Kansas | Pokey,
Most of what the others posted is spot on, except for Paul's first paragraph, which is...well...just wrong (sorry, Paul).
If you want to calculate the I value for a tube, just run the b*h*h*h/12 number for the outside dimensions of the tube, then do the same calculation for the inside dimensions, and subtract the inside from the outside value.
So, to your example:
If we assume a 3/8 inch wall thickness, the answer to your question is
For the 12x12x3/8 tube, I = (12x12x12x12/12) - (11.25x11.25x11.25x11.25/12) = 393in^4
For the 8x12x3/8 tube, I = (8x12x12x12/12) - (7.25x11.25x11.25x11.25/12) = 292in^4
So the 8x12 tube is about 3/4 as strong in bending as the 12x12.
This calculation procedure assumes sharp corners, so the book values for actual tubes with rounded corners will likely be a little less.
If the statement that "all the strength comes from the flanges" were totally true, the 8x12 would only be 2/3 as strong as the 12x12. So the webs DO contribute a little bending strength (but not a lot).
There are a bazillion ways to mess up strength calculations, and one of the big areas of uncertainty is that we seldom really know the actual load that will be applied to the structure. Anyone who has ever cut a corner a little too short and run a folded chisel plow or planter off of a culvert can appreciate a little good, old fashioned over-design. Not that I've ever done that. :-)
FWIW
MDS
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Square vs. rectangle? Tube bridge physics.
