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that's not really how engineering works
It's certainly not how practical bicycle wheels are designed, mostly because of the problems in computational analysis which we're talking about. What actually happens is that people take some known good design and progressively shave bits off it until it fails endurance testing, then send the last version which passed endurance testing to market 🙂
the tension in spokes being different is not what second-order effects refer to, that's just what happens due to the preload you mentioned earlier, and the way any wheel is built
Spoke tension dropping to zero is a second order effect, it can only happen if the rim moves quite a long way from its starting point relative to the hub, because the spoke is elastically strained by up to 0.3% under "no load" conditions.
you don't want to be that close to failure anyway
Mostly you don't. In a competitive environment, you sometimes do; I'd rather win all the races I finish than finish all the races I start 🙂
gbj_tester-
Spoke tension dropping to zero is a second order effect, it can only happen if the rim moves quite a long way from its starting point relative to the hub, because the spoke is elastically strained by up to 0.3% under "no load" conditions.
Fair enough, that is a good point.
But otherwise, no, you definitely don't want to be that close to failure, and no one is, even in pro cycling. Otherwise, you'd get a LOT more incidents of parts failing in a catastrophic way, and a LOT more riders getting seriously injured, or killed. Especially things like wheels are certainly always overengineered quite a bit. And you mentioned another good point - it doesn't just have to withstand one load cycle, but many many many of them. However, how many exactly is a bit of an arbitrary decision of course, in the grand scheme of things. But especially big brands are usually extremely keen on their name not becoming indelibly connected to randomly failing hardware, so they tend to be on the generous side as far as I know. I will definitely follow them in that in stuff I make...
SwissChap
Ok that sounds good in theory, but that's not really how engineering works though. In practice I would not trust anyone claiming to have designed a wheel like that, because you don't really know the 'maximal load condition', and you always want some kind of safety factor anyway. In actual fact, almost everything in life is overengineered to a varying degree. The stuff that isn't is the stuff that occasionally breaks way too early and people get annoyed about. So you want to make sure you're not going to the limit on anything. I'd also rather pay for stuff to have a safety factor, than brake my bike with my face.
In any case, the tension in spokes being different is not what second-order effects refer to, that's just what happens due to the preload you mentioned earlier, and the way any wheel is built. That's too complicated to work out quickly on a sheet of paper, but there are computer programs that would allow you to figure it out relatively 'easily'. The really complicated stuff happens when you start to consider how the non-rigidity of the material subtly changes the way the forces apply, and how that in turns changes the deformations themselves again. You can then move on to third-order effects, but honestly it gets a bit pointless at that point.
But the thing is, because you always put on a safety factor and because those second-order effects are comparatively small unless your rim was to flex significantly, it's not really worth putting too much time into working out their exact impact, you don't want to be that close to failure anyway.