How could Euler have gone about creating his buckling formula?

[musicgold]

TL;DR Summary

How do some scientist go about finding seemingly simple and elegant formulas to describe complicated situations?

Admittedly Euler was a genius and I am a noob, but I sometimes feel that there must have been a method or process that he followed to go about such problems and came up with these elegant solutions. It couldn't have been just a sudden flash of genius. For example, I wonder how he could have come up with the pi factor or why a square and not a cube for the denominator. If I were to grapple with a `100 times simpler problem, how could I go about creating a formula?

where

, Euler's critical load (longitudinal compression load on column),

, modulus of elasticity of column material,

, minimum area moment of inertia of the cross section of the column,

, unsupported length of column,

, column effective length factor

 

[Delta2]

I don't know this specific formula, i assume its something from beam theory which i am not good at. But anyway I believe the method or process that Euler followed was simply his commitment and dedication to hard work. Einstein said that genius is 99% hard work and only 1% talent. Certainly Euler had that 1% talent factor but his hard work is indisputable. Even when Euler got totally blind from both eyes, he didn't stop working for science, discovering new things. So i just think that was his secret process/method, serious and hard work.

 

[sysprog]

Euler was and is a great light among luminaries $-$

here's just one example $-$

an account of his solution to the 'Bridges of Königsberg' problem: https://www.maa.org/press/periodica...ers-solution-to-the-konigsberg-bridge-problem

 

[etotheipi]

I haven't read it yet, but perhaps this helps
https://arxiv.org/pdf/1512.01171.pdf

 

[jrmichler]

There is a mathematical derivation for that formula. Search Euler column buckling. The Wikipedia link has a derivation that looks like something I saw in a strength of materials class a long time ago, then promptly forgot. Note also the assumptions that go into it.

BTW, I once used an Euler column as an overload spring in a machine. We needed a light weight pushrod that would buckle with a force of about 8 lbs. The Euler column formula really does work. The pushrod was designed for 8 lbs, and it buckled at 8 lbs.