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My goal is to meet the code requirement for the strength of the floors on a 2nd story for a remodel.

I am remodeling a house where the floor joists supporting the 2nd floor are nominal 4x4 (3.5" x 3.5"). My problem is that by adding lumber of the correct depth (presumably 2x8) the ceiling height will be too low for taste, and potentially for code. The current ceiling height measured to the joist is 7' 10". My goal is to keep the current ceiling height by using aluminum or other metal as low depth joists to strengthen the floor (fitting into the current 3.5" depth). 

I'm not an engineer, but I think my problem is that I'm exceeding L/360 deflection limit. Here's the scenario:

The 2nd floor area is a 23' x 20' rectangle (rounding up to the nearest foot). The floor joists run along the 23' length toward the center of the home, and they sit on top of a wall and beam combo with a 3.5" bearing. The joists run to 10' 6" on one side then sit on the wall/beam, and they run 12' 5" on the other side before they lap over the wall/beam. The 4x4 joists are placed at 13" on center. Very odd. Apparently, the 2nd story addition happened in the 70's and survived the 89 earthquake in Oakland CA.

What are your suggestions for keeping the current floor depth, but strengthening the floors so as to reduce deflection specifically, but more generally to meet code requirements for floor strength?

I was thinking of inserting some 3.5" x3.5" aluminum joists alongside the 4x4 lumber to increase strength and reduce deflection. 

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    It would open a lot of options if you could bump height to nominal 2x6. A 7' 6" ceiling still isn't terrible by any means, and you'll have a much easier time finding solutions if you can spare 5-1/2". I'd consider doubled 2x6 on 12" centers. – isherwood Dec 9 '20 at 20:45
  • Aluminum would be extremely expensive. If anything you'd probably use a steel C-channel or box, but you'd need an engineer's stamp of approval. – isherwood Dec 9 '20 at 20:54
  • Even steel beams get their strength from the depth of the web. On paper, I suspect S3x7.5 I-beams could do that span as floor joists, but on top of costing $200 each, you then have to attach subfloor to the steel flange? On the other hand, 2x6 Structural Select lumber would probably work with normal construction methods. – Jeff Wheeler Dec 9 '20 at 21:44
  • I meant something more like this, not full-on I-beams. – isherwood Dec 9 '20 at 21:46
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    FYI - 4x4s are almost twice as good as carrying a floor load, but almost half as good as a 2x6. There is that big of a difference. If no architect signs off on it I would take the 4x4 out of equation. – DMoore Dec 9 '20 at 21:46

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