What is the joist length on each side of the beam? What is the city and state so I can look at snow loading?
Do you need to clear span the 32'?
Thank you. I will size the beam assuming you detail the joists to brace the be against lateral torsional buckling
The beam will be sized for strength and deflection but I would recommend a full dynamic analysis to check for potential vibration problems for such a long clear span with a light structure
You can plan for a W12x30 ASTM A992 Gr 50. It's 12" deep x 6.5" wide and weighs 30 pounds per foot
That's the right size. 31' is not a short clear span. And with the cantilever, it's the same as if the deck were 12'-6" wide in terms of loading on the beam.
Do you have any other questions for me on this topic? If you could rate my answer, I would appreciate it.