For a gym set, a couple of solid tack welds would hold as much as you can put on the box tubing. Using a 70xx electrode or even MIG would mean that with just a 1/4" long tack would hold several tons of force.
I think you are overthinking the welding issue. Just weld it with a fillet around the joint OR put a hole in one side and tack weld everything together. Then turn up on edge so it is a flat weld and weld it the best you can. Just a bit of solid weld is all that you need to keep it together for a gorilla swinging on it.
The WEAK link is going to be the span of the box tubing. If it holds without bending, your rods aren't going to be a problem even with just small tack weld on 2 sides.
I was definitely over-thinking it. I did the math on a 3.14159" fillet and it's pretty much gonna stand any test so long as I don't use chewing gum. But even then, some brands of gum will probably hold. Wrigley's pretty much turns to solid gunk after a few minutes, and when it gets wet it might as well have been filler metal. I think I'm just going to use the rain this week to build a jig, and then when it clears I'll put it together. The wife wants it in the basement this winter for the kids.
My primary welding is furniture (racking, etc.), tractor junk (heavy plate), logging tools and some structural. Otherwise the thin stuff is all just tacked together - nothing I ever had to put weight on (but none of it ever broke, either). All rod and some flux core. Getting into more delicate stuff with MIG and TIG, so not sure I can 'Kentucky Windage' things like monkey bars, yet. Hence the idea of cutting it and finding out if my guesses were close enough. Thanks for your advice; I will heed it.
XMS. The 1" RotaCut requires a larger arbor than the smaller ones in the set. You may have to purchase it separately. I also purchased extra arbors so that each cutter had its own. Just thought I'd help you spend your money

Terry
That's good to know. Thank you.
This is going to end up much stronger than those wooden ladder things that rot over time. I never really cared for the monkey bars, but I liked the thing everybody jumped on and pushed in a circle like crazy.
Don't forget the splinters. Hate the splinters.
Those rotating carrousels are monstrously awesome, which is why they are now banned from pretty much every new playground today. There is still one at the park we enjoy, but our new school asked for help in putting up a new playground and I guess the list of "cannot insure" is so long they just said 'eff it and built a 'natural playground'. Key feature is a massive sandbox built using my tractor, some logs and about 24,000 pounds of play sand. The kids were digging in it before we were done hand-leveling the thing.
They pulled a bunch of swings and monkey bars from a Florida beach public playground where we go in the winter and where the wife & kids spent most of this summer. The rebuild is a big pirate boat that is pretty cool, but it has nothing to hang from or to swing from. It's basically nothing more than stairs and a totally enclosed slide, which gets about the temperature of the sun in the summer. At least that part is the same from my childhood.
Part of this project is building all the cool stuff they are yanking from the schools and parks. Our kids are monkeys, and I'd rather them climb the monkey bars over a sand pit, then climb those oaks I got out back in the woods. Sad to say, but if I catch them climbing the stuff I did as a kid...
It's in the their DNA. I climbed and my wife spent half her life looking at cliffs in places like Yosemite and finding her way to the top one hand over the other. My three year old saw a rock climbing wall last week and gravitated to it like a moth to a flame. So we call this a controlled compromise. For now.