My nephew has a Gravity Liberty flat bar with disc brakes Save up to 60% off new Flat Bar Road Bikes - Gravity Liberty FB | Save up to 60% off new road bikes. The outside width of the rims is 19mm so I am assuming the inside width is 14mm or 15mm. (I have not measured). He wants wider tires- e.g 40mm. According to Sheldon Brown's chart 40mm tires would be too wide. In searching these forums, I get all kind of conflicting information. If run at pressure not exceeding 50psi, e.g., could I safely mount 40mm tires for him or are they too wide. Thank you in advance.

Sounds like you've got pretty narrow road rims. Weinmann LP18's are another rim with a 19 mm external width and their internal width is 13.6 mm -- maybe 14 and change if you discount the hook. I'd expect yours to be in the same ballpark. According to the ETRTO, whose chart has been reused by Schwalbe and Sheldon Brown among others, the widest tire that should be used on a rim with a 15 mm inside width is 32 mm. The chart's pretty conservative and Sheldon notes that many people have exceeded those recommendations, but 40 mm seems like quite the stretch.

Schwalbe's web site has a blurb specifically about using their Big Apple wide tires on narrow rims, along with pressure recommendations. For their 54 mm tire, they recommend going no higher than 55 psi on narrow (17c) rims. I'd guess (and I'm not calling it an "educated" guess!) that similar pressures would probably be safe with a 40 mm tire on your rims. By safe, I mean safe for the rim, as in the pressure from the tire probably won't kill it. Not necessarily ideal, though.

Personally, I'd try to err on the side of caution and see if he might be willing to go slightly smaller on tire size. 35, maybe?

On the bright side, the frame shouldn't be a limiting factor. The BikesDirect site says the bike has clearance for 29" x 1.7" tires. Discounting possible inaccuracy in tire sizes, that should work out to 43 mm width.

i looked at a Cannondale 29er outside the local Sports Basement the other day. the rims outside width are only a millimeter or two wider than my oldschool road rims. the tires were at least 2" wide. i've decided i'm not going to be concerned much about tire width vs. rim width anymore.

A 40mm tire will be fine on those rims. Basically, if the tire will fit in the frame, almost anything up to 3 times the rim width will be OK. My Crosstrail, came OEM with 45mm tires on 18mm(internal) rims, speced with 85PSI. Run the pressure to low, and you might have issues with the tire trying to roll off the rim in turns. But not very likely!