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I'm not an engineer, but I'm trying to understand the correct analysis for what seems like a pretty simple case.

I'd like to know what the effective load on the middle joist is.

                   |             |
                   |     LOAD    |
          _________|_____________|_________
subfloor |_________________________________|
         | |             | |             | |
         Joist          Joist          Joist

The joists are evenly spaced and the load is centered on the middle joist. The load is uniform and runs the length of the joist. For concreteness, let's say the joists are 16' long and 24" apart, the load width is 12" and the load is 150 psf (pounds per square foot) so also 150 plf (pounds per linear foot). Let's say a joist can support 100 plf, so the load would be supported even without the middle joist, and each of the outer joists would support half the load (assuming the subfloor supports the 150 psf load with the doubled span between joists).

Obviously having the middle joist helps, but it can't support the full load. So what part of the load would it effectively support? I'm pretty sure it wouldn't be its max (100 plf) and I'd like to know how to calculate or approximate it. In reality, I'd just like to be able to figure out the maximum possible load (in plf) for this configuration, with the maximum allowable load for each joist as the input. In particular, I'm interested in the case where the middle joist is "weaker" than the outer joists (eg. 40 vs 100 plf max allowable load).

Questor
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    It depends on the stiffness of your flooring. The more flexible it is the more load on the center joist. for typical 5/8" or 3/4" thk. plywood flooring almost all load is carried by the center joist. – kamran Aug 03 '23 at 17:38
  • Okay, I understand that. So what "happens" when the center joist can't support the load by itself (but the outer joist without the central joist can)? Let's say I have 2x14 joists. My question is what would placing a 2x4, 2x6, 2x8, etc joist in between them add in terms of max allowable load (for the load as described above - not for a uniform load over the entire surface [I understand how tributary areas work for that case]). – Questor Aug 03 '23 at 19:42
  • You will have large (maybe crack-inducing) deflections under the central joint. If the load is going to be applied slowly and the ambient humidity is not overly high to let the wood creep, it will be tolerable! If it is easy to do, double the joist, or, block the joists with 2x solid @ 24" OC. and use Simpson MST 60 at the bottom of the cross blockings to tie them up for the negative moment. If you want I put this as an answer so I can add diagram. – kamran Aug 03 '23 at 23:02
  • Thank you, but it's more about getting an understanding than finding a practical solution. Maybe I'm wrong but wouldn't the central joist deflection be at most the deflection of the subfloor (produced without the central joist present). Once it deflects that far wouldn't all the weigh be distributed to the outer joists? It almost seems as if adding a weaker middle joist really does nothing for a heavy load. Consequently a heavy load should be put between joists rather than centered above a single one, which seems rather counterintuitive. – Questor Aug 04 '23 at 01:54
  • To answer your last question: the weak center joist would act as a spring. You can calculate how much deflection happens for a given unit of load for this spring. If the spring is weak enough, yes, it will be pretty much equivalent to having nothing there at all. But it is not going to be quite that weak- it's going to carry some amount of load, though it might deflect quite a bit to carry it before transferring to the outer joists. But it will never deflect more than the unsupported subfloor would without it, of course. The more it deflects, the more it is carrying. – Rick Aug 17 '23 at 19:01

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