> I really took the wrong approach to this ... first principles and force vectors
W.W.J.D? No, not Him; none of us are enlightened enough to know what He would do. His human father, Joseph The Carpenter. What Would Joe Do? Joe and his fellow carpenters built wood structures all over the world, before calculators, before beam-tables, before easy arithmetic. (What is the area of a II by VIII beam?) Their structures stood the strain; a few are still standing.
Joe knew wood. A log will span 10 or 15 diameters with little flex or breakage if you space logs 2 to 5 diameters depending on the load above. Lightly loaded planks (incense boxes and shelves) can span maybe 30 times their thickness, but money-changer coin storage needs a smaller span/thickness ratio. If the kid had not been Called to another occupation, he woulda spent 5 or 10 years working under a Master Carpenter, Joe or one of his cohort, learning such proportions, before starting his own carpentry business.
Joe did not have a big cheap saw, we do, so we have Lumber. A good safe size is: depth in inches equals span in feet. This is actually very safe and you rarely find a beam this deep; however strength under uniform load goes as square of span (and stiffness falls as cube of span) so you can't go a whole lot further than 12:1. As for width: if 10% of total width is wood (1.5" lumber on 16" centers) you can carry ordinary residential loads, including a room completely full of people, but not high stacks of solid paper (or water).
You can build a house with these rules. My house was thrown-up with very loose understanding of such rules, and it is still standing (though saggy) 170 years later. You may end up using more wood than a span-table user, and you may get nit-picked by building inspector, but it will work.
> found a great deflection versus load/span calculator here:
Looks fine. In this area, the pessimistic (and most common) framing wood is #2 Spruce-Pine-Fir, the optimistic is #2 or #1 Douglas Fir. Makes only about 10% difference in allowed Span.
It, correctly, calculates Uniform Distributed Load. The assumption is that the original design engineer can not know what kinda stuff future occupants will put in there, or where. Building codes give guidelines: 30psf for bedrooms, 40psf living/party room, 60psf for balconies which "could" be fully-stuffed when a parade goes by, 100psf for heavy office, 200psf for filing rooms and factories, etc.
I don't like your plan. 13'7" feet is, as you say, long for 2x8. The table/calc says #2SPF at 12"OC is just-enough for 40psf+10psf distributed load. In fact I had a 24-foot house beCAUSE #2SPF 2x8 16"OC will span 12'3", and they based the design on that fact.
Hmmmm... the problem is deflection, not strength. But loosening to D=S/180 does not allow much higher load. The beam is very nearly "optimum", for residential strength/sag expectations.
The tub is not uniform all over the floor, apparently exceeds 40psf, and moreover could rest entirely on just two joists (assuming little load-sharing through floorboards and cross-bracing). One end is over (dubious) wall-support, the other is nearly at mid-span. WWJD? He might assume half the tub was concentrated AT mid-span, the other half could be ignored being over support. A foot-wide strip of 13.5' 40psf floor is 540 pounds, so it could support a single 270 lb load, or a single 540 lb load spread over 2 feet. Since your load is maybe 400 lb, that leaves 140 lb which is nominally just 10psf live load over the rest of these two joists.
So I'll let you do more math and let it be on your head. Off the record: IF there are no big flaws in the two beams, IF the header is adequate for a similarly concentrated load in excess of distributed-load assumptions, it may exceed Allowed Stress but it won't fall down in a lifetime. (Longer if the tub is usually empty.) But if you didn't build the framing yerself, that's a lotta "ifs".
Thought experiment: you and three fat friends do a group-hug. That could be 800 pounds in the floor-space of a tub. You don't fear fall-through. However you are somewhat relying on the fact that wood will take short-term overload: my sub-Code porch might stand the 800 pound hug for a moment, but if it went on for hours I'd start to worry.
FWIW: baby-grand pianos are sold into homes every week, and you don't hear of them being found in the cellar.
OTOH: in New Brunswick a few years back, there was a party on the second floor which suddenly fell to the first floor. It was a balloon-frame house. Studs ran up 16 feet. Ledgers were spiked to studs to support the upper floor. The entire second floor pancaked to the first floor. This used to be a known hazard of cheap balloon frames, but kids today only know platform frames, and were perhaps too high to care.
> what the headers are like over the openings, so unless I rip open the walls
My 24' house had a single 2x4 over any door or window. (Even with no tributary load, this was mighty cheap; that was the 1947 housing shortage and they got away with some cheap work.)
You can dis-prove such cheapness by throwing a 1/8" drillbit into the header. In my old house, 1/2" of wallboard then nothing. Common practice would be 1.5" lumber, random 1/2" plywood spacers, and another 1.5" lumber. Do this at 2 or 3 heights so you know if it is a 2x12 or a 2x6 header. But you still don't know if the side studs were properly doubled-up (might need to be triple). And stuff like this is where many builders really get sloppy.
> I am not worried about sag in the 1st floor ceiling so much, since there are compressed pulp ceiling tiles
You usually do not want to be shy of D=S/360 for floors you walk on. They feel cheap. Roofs are allowed D=S/180 because roofers are used to a lively surface. In this case it is a minor issue. The floor could sag but won't feel springy with 800 pounds of concentrated load. It is still unwise to allow large sag because the non-90-degree sagged joints may pull nails or dent the edge of the headers.
