I am trying to make a king-size bed frame. I have 2"x6"x80" pine boards that were free (always best). I planed them down to 1" thick to use for the head/foot/side rails. If I rip these down to 1"x4" with the 4" vertical, if only supported at the ends on 3" legs will they be strong enough to support the box springs, 14" mattress, and us (2 adults & a dog - 390lbs). The link provided is the bed I am using as a base design (per the wife). Thanks in advance.
Board strength of true 1"x4"
Further to what I say above in my Comment, because wood varies so much it's impossible to give real-world figures for this type of query. You can find tables for wood strength in various places online1 but they aren't really of much help when it comes down to the material at hand and the strength of individual boards is important, as it is here.
It's particularly hard to rely on published data with typical softwood material these days, one reason being the frequent presence of knots. Knots can be relatively inconsequential if within a board2 but locate one near the middle of a span along the bottom edge and just one can fatally undermine the strength of a board.
Or to put it another way, forget the data, what's this board like?
If I rip these down to 1"x4" with the 4" vertical... will they be strong enough to support the box springs, 14" mattress, and us (2 adults & a dog - 390lbs).
Without seeing photos of your material to better judge its qualities I think it's still possible to make a judgement call here, and mine is: a definite NO.
My back-of-envelope calculations put the total load here, conservatively, at 600lb (!) or >270kg for my metric brethren. For that kind of weight I'd very much want to err on the side of strength if working with an unknown softwood, no matter how straight-grained and clear it was.
Also you're converting from a hardwood prototype to softwood, and the rule of thumb is you beef up material when you do this (often by going both thicker and wider).
And I hadn't spotted this myself, thanks to @VolframK and @FreeMan, but the commercial item itself appears not to be built strongly enough3..... so yeah, definitely beef up the design if you're going ahead:
1 Much of it apparently sourced from the FPL since you usually see their numbers reproduced verbatim.
2 And live, not dead (loose).
3 Despite costing more than a grand O_O