What size crane beam do I need

[Portable Welder]

I am fabricating a bridge crane for my shop, my run way beam is a 24" WF at 55Lbs a ft with a 39' span that I installed the when I built the building.
My run way beams are 24' on center
I have a 10" I-beam on hand that weighs 25.4 lbs a ft that is 26' long, I am doing a underhung bridge beam and for latteral loading of the beam, I will weld a 6" channel toed down to the top flange of the bridge.
I want to put a 2 Ton electric chain hoist on this beam.
So my question is, do we have any engineers that could figure this one out.
I am trying to keep my bridge beam as light as possible because I will be manually pulling it up and down the shop.

[Prototype3a]

It would probably help if you had a sketch for us to look at. If my textbooks from college weren't back home I could do it as they have the tables of beam specs and such in the back.

[Portable Welder]

I am looking at buying a chain hoist for my crane, there is CM Load star, coffing, budgit, Milwakee. These are a few different ones that I have seen.
Can any one here give some input on which one is the best and most reliable.
I want to get the 2 ton single phase hoist.

[MMW]

We have three 3-ton & one 1-ton Coffing electric hoists in the shop. All about 20 years old & still working fine. I have never used the other brands so I can't comment on them.---MMW---