"I would first check for track flatness"
This thread is a great example of how engineering is often NOT a solution to problems, classic "hammer and nail" territory here. And how engineers often ofterthink things unnecessarily ;)
Then, what’s the issue? “Too much tension” is the question. A reasonable definition of “too much” is possible damage or that it affects performance.
Having experience with these, if it’s sitting on the ground flat, and it’s not being help there, then it’s about an order of magnitude away from “too much”, for damage.
“No tension” is a different question.
Again, if it's flat on the ground, it's far from the point where something breaking is a concern.
Far from a concern, as long as it’s stationary.
I'm afraid it is not accurate at all because it is not answering the question as asked. It verifies that the track is under tension, but it doesn't attempt to answer if that tension is "too much". Which is what the question asks.
Spoiler alert: it didn't. Nowhere does the mathematical answer address the question of "too much".
And what do you mean by "progressed from stress to strain?" Stress doesn't turn into strain, they exist simultaneously. You're probably trying to say progressed from elastic deformation to plastic deformation.
I think you (and many others in this thread) are confused because you read the title but not the body of the OP. Quoted:
> 1. Is there any way to quickly see if there is any tension, and why? (I know I could just take one piece out, and put it back in to feel it myself, but I am looking for a more logical way, so I am able to reason it.)
> 2. Suppose I want to update the track in the picture to have less tension. If you have to take away exactly 1 rail piece (straight or curved), which one is the best, and why? If you have to add exactly 1 rail piece (straight or curved), what is the optimal place to insert one?
The accepted answer attempts to address these questions.
Edit: Re-reading the rest of the "look for track flatness" comment; the second and third sentences about tolerances and bowed joints are spot on. For example, looking at the final track layout for the "mathematical" approach, I can tell you that I'd have no problem shifting that track down an inch and snapping it in place.
As an aside, these articles are the gems that keep me coming back to HN.
Now because I work in a textile mill, the tolerance I usually get is 0.125 inches which is huge. I usually go all the way down to 0.0001 inches because I think it's funny, and also I do have aspirations beyond just working with textiles.
Several days and derivations later, the theorist reports the volume, after which the experimentalist tosses the shape into a volumetric flask and determines the volume by looking at the difference in flask volume levels. (I am unable to track this story down to its original)