-- or a soffit to contain the new ductwork.

(That's the way we had to do it.)

I actually just looked up the joist guide from the joist manufacturer (TJI 230 14”) and it looks like a 6” hole is perfectly acceptable. I’m pretty much in the center of the joist length wise. so now I’m thinking of going through 4 joists to get to the exterior wall in the other direction. Has anyone else tried this?

Hole size and location depends on a few things that are hard to see in this image. Is the photo right-side-up? Are we looking straight on to the side of the beam? If so, is that beam in two parts or just cracked? Is the beam a 2-by-something or thicker? What is the beam's height?

Is the beam hung at the left with some kind of inadequate and up-side-down hanger, or is that there just to keep the two beam halves aligned? How far to the left of the left joist does the beam terminate? Is it supported by a wall somewhere to the right? How far? Structural analysis is tougher than normal with no data.

The type of joist I see is strong in tension in the vertical plane, but depends on both top and bottom being constrained so there is no twisting. Whatever ceiling is being removed to access this joist path -- when replaced -- must provide adequate retention of the bottom of the joists.

Beyond the beam issues, I wonder whether a suitably sized island hood volumetric flow rate can be supported by a mere six-inch duct. Some hood parameters would be helpful.

The picture is right side up. that is the side of the beam and the right side of the picture is the sprinkler. The range hood is 400 CFM zline ke2i-36, and indeed is designed for a 6” duct. My last comment I found out that The joists appear to support 6” holes no problem (I’m basically near the center length-wise) and i see no other major holes along the ones I’ve looked at so far. So I am thinking of going through them. It’s a short distance anyways - around 7 ft compared to 15. Any thoughts on that? Here’s a picture of the situation.

I think that if all of the conditions for joist loading and installation are met and the data allow six-inch holes (have to be on the centerline where the tension/compression neutral point is), then you can safely bore them. Note that a six-inch duct may require a slightly larger hole to fit, and that alignment of holes may be critical for minimal excess hole size.

What I don't know is whether the code includes some rule about gap between a hood duct and combustible materials (in case of a grease fire, I suppose). I would expect in a residential setting that duct grease fires are low probability, but I haven't seen any data on the topic. Maybe someone here informed on this detail can pipe up.

I have a strong preference for overhead hoods, but where structure precludes venting overhead, a down-draft vent system is an option to be considered.