Y
I have columns that are 7.8cm deep (or wide, if you prefer).

I can choose between two different metal beams.
One is about 6cm deep, the other is 7cm.

The 7cm beam is 3 times as strong as I mathematically need.
- It extends 14cm down from the ceiling towards the "door" opening.
The 6cm beam is 2 times as strong as I need.
- It extends 12cm down from the ceiling towards the "door" opening.
If I choose the 7cm beam (in metal),
then I only have the option to clad the sides with 4mm thick boards (wood).

6cm beam = I can clad with 9mm thick material per side.
What would you choose? I plan to install I-beams in a load-bearing wall and then clad them with masonite or something to make them look a bit nicer. :)

However, I don't want to end up in a situation where the cladding of the beam extends further out than the walls around it, hence the mathematics.

And to fasten these boards, I'm planning to glue a board into the I-beam's deeper sections.
 
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I would have chosen a KKR instead, so you avoid messing around with the cladding so much.
KKR is a square tube and you can get it in, for example, 80x120mm.
 
Y
I don't remember what the engineer said about it, I think I had to take much larger measurements then. The I-beam had the smallest dimensions in relation to its strength... I might be wrong, but he did recommend the I-beam anyway and I don't dare to start changing his calculations, hehe
 
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