First, your question
how do I calculate what that new beam has to be given the existing joists and the desired post options?
This is a bit long winded, but it is how we figure it out accurately.
Figure out a design load per square foot (PSF), we typically use 40 PSF live load and 10 PSF dead load for floors and decks. In a simple span, the beam carries the load halfway to the next support or tributary width; in this case 5'. Multiply the PSF x the tributary width to get the pounds per lineal foot (PLF) on the beam, in this case 50 PSF x 5 Ft = 250 PLF. Then, using software or load tables find a beam that can carry that load based on the span of the beam; allowable loads reduce as the span increases. In your case this requires (3) 2x10 DF #2.
I typically use Forte for beam sizing instead of load tables.
For the post sizes, calculate the tributary load which, in simple spans, is the load halfway between supports. In your case, this would be half the beam length between post multiplied by PLF. For your center post that is 250 PLF x 11.5 FT = 2875 LB. Then, using load tables or calculations find a post that can support that load based on height and bracing of the column. Take into account bearing area of the 2x10 at about 625 PSI perpendicular to the grain for a required bearing area of 4.6 Sq. In. In your case a 4x4 is plenty.
Try a column load calculator here
For multiple span beams, cantilevers, and anything outside of "simple span" things are a little different. For this reason I typically use the beam sizing program.
If this is too intense, maybe just be guided by the following.
If I read correctly, you have deck joist spanning about 10' and on one end a single 2x10 "beam" or rim joist that the joist hang into that spans approximately 11'-6" between (3) 4x4 post.
Based on this, your deck joist are structurally OK although maxed out at a 40/10 load, but are deflecting (bending) a good amount giving part of the bouncy feel. You can add 2x6 or larger joist down the middle of the existing joist to firm things up, or just live with a safe bouncy deck.
The 2x10 is definitely undersized and should be addressed. It also contributes to the bounciness. The simplest solution to this is to add (2) intermediate post mid span reducing the span of the 2x10 to below 6'. OR beef up the 2x10 by adding (2) more 2x10's to it nailing 3 16D nails per foot per beam lamination would easily transfer the load between beams.
Finally, the 4x4 post are more than adequate to support a load of 2,875lb. An 8'-0" length of 4x4 can support over 6,000 lb in wet service.
We applied over 10,000lb to a 7' 4x4 in jacking up a floor with no indication of possible buckling; we were compression limited on the member being jacked.
Best Answer
You don’t give all the required information, but I’ll make some assumptions and you tell me where I’m wrong.
I’m going to assume the water is 3’ deep. Water weighs about 62 lbs. per cubic foot so the water in the tub will weigh a total of:
7’-10” x 9’-2” x (62 x 3’ deep) = 13,400#
I did not add weight for equipment because I assume there are cutouts for seats, which would reduce the amount of water (weight).
In addition, I did not add weight for occupants, because that is considered an impact load (load that is not on the structure for 7 days). This weight is calculated differently.
Assuming the hot tub sits across the 2x10’s in the 7’-10”direction, the 2x10’s at 12” on center spanning 7’ can support about 450 lbs. per foot depending on the species and grade of wood. The water and tub weighs about 3’ x 62 = 186 lbs. per foot. Adding 4 people each weighing about 200 lbs. each adds another 12 lbs. per square foot. Therefore, the total weight is:
Water = 186 psf (water 3’ deep) plus Lumber = 8 psf (deck and joists) plus People = 12 psf (4 people in tub) for a Total = 206 psf (well within a safety factor)
However, in addition to supporting the tub, the double 2x12’s need to support a live load adjacent to the tub too. Therefore, the total load on the double 2x12’s is:
Tub/deck/4 people: 206 x 2.5 = 515 plf plus Live load adjacent to tub, which is 40 psf x 7’/2 = 140 plf plus Lumber (deck, etc.) 8 psf x 7’/2 = 28 plf for a Total on double 2x12’s = 683 plf
If the 2-2x12’ are spanning 5’, then they’ll support about 1,800 lbs. per foot depending on the species and grade of wood. This is also well within a safety factor.
However, the total weight on the 4-4x6’s is about 13,400 lbs. / 4 posts = 3,350 lbs. plus about 1,700 lbs. live load for a total load of 5,050 lbs.
The 2x12’s are bolted to the 4x6’s in a double shear loading pattern. Assuming those are 5/8” bolts, they can only support about 510 lbs. each or about 1,020 lbs. at each 4x6 post.
Summary: The framing lumber will easily support the required loads, but the connection at the 4x6 posts is grossly inadequate. You’ll need to install a 2x6 vertical under each side of each 4x6 that is under the hot tub. Make sure it bears on the footing and fits tight under each 2x12.