trying to design a straight stair run using steel tubes as stringers. total run = 10'-0". total rise is 8'-8", width 3'-0". Support is only at each end. yes these are steeper than normal, and do not access a living space. would two steel tube 5"x2" x 1/4" thick be stiff enough? if not what size rectangular tube would be required?
Yes, two 5” x 2” x 1/4” steel tube stringers (with A36 steel) spanning 10’ will adequately support your stairway.
Code requires a Live Load of between 40 lbs. per square foot (psf) and 100 psf depending where it’s located. Assuming it’s 100psf x 3’ wide x 10 long = 3,000 lbs. plus a Dead Load of 15 psf x 3’ x 10’ = 450 lbs. for a Total load of 3,450 lbs.
Your tubes will support about 8,000 lbs. each, depending on connections, etc.
If you don’t have the tubes already, you may want to consider a 5” x 1 3/4” channel which will support 5,100 lbs. each.
There is a simple reason you shouldn't use 5" x 2" tube as the stringer for stairs - it won't fit the stair thread that meeting code, or provide good connections for safety in service. See sketch and table blow.
I suggest consulting with a structural engineer, who may suggest a deeper channel section that will provide the required strength and ease of connecting the stair threads and the stair connections at the ends.
Edit: BTW, while the selected tube section meets the strength requirement for both types of stairs - "residential" and "all others" (which require 40 psf and 100 psf live load respectively), it fails in meeting serviceability criteria, which (per IBC) requires the live load deflection of the stairs be limited to L/360. (note that IRC did not explicitly address the deflection limit)