Redoing an attic

[Garald]

1 hour ago, SteamyTea said:

@Garald

I have been pondering your mathematics and how it fits in with structures.

 

 

I found this, but thought I would let you digest it and write up a in less than 500 words for the rest of us to read in the morning.

 

https://cdn.ima.org.uk/wp/wp-content/uploads/2018/02/MT-2007-Nonlinear-Mathematics-in-Structural-Engineering.pdf

 

This one is interesting as well.  It is about teaching.

 

https://peer.asee.org/using-engineering-mathematics-to-learn-structural-analysis.pdf

 

Cool, thanks - let me see!

 

In the meantime: my girlfriend just raised the difficulty level - she asks whether I can request an aerial rig on the top beam.

 

[Garald]

3 hours ago, SteamyTea said:

@Garald

I have been pondering your mathematics and how it fits in with structures.

 

 

I found this, but thought I would let you digest it and write up a in less than 500 words for the rest of us to read in the morning.

 

https://cdn.ima.org.uk/wp/wp-content/uploads/2018/02/MT-2007-Nonlinear-Mathematics-in-Structural-Engineering.pdf
 

 

OK. This is not really a math paper as such - it is written in the hope that the "general professional public" (meaning members of a maths-and-engineering society) understands it. Here is my attempt at a summary/commentary.

 

Most things are linear if you look closely, in the sense that the graph of a differentiable curve looks more and more like a line the more you zoom in. There are cases, however, where a linear approximation will not do:

1. because of the material - for some materials, Hooke's law (which is linear) is a very good approximation; for others, not so much.

2. because of the geometry (so to speak). This is the focus of the text.

 

Here's a classic example of 2 (not given in the text). You've heard of Galileo discovered that the time a pendulum takes to swing is independent of how wide the oscillation is. Well, this is only truish. It would be true if we had sin alpha = alpha (where alpha is an angle measured in radians), but this is of course only approximately true, and only for small alpha; more precisely, \sin \alpha = \alpha - \frac{\alpha^3}{6} + ... There you see how the approximation sin alpha = alpha is very good for alpha tiny, and less so for larger alpha. The same approximation alpha to sin alpha lies beneath some approximate estimates in engineering that becoming less exact as alpha gets larger.

 

Non-linearity is particularly important when we talk about equilibria. Whether an equilibrium is stable or unstable depends on second derivatives  - the first derivatives are all zero, that's why we are at equilibrium. In general, when linear terms vanish, the quadratic and higher terms become much more important.

 

Things buckle after reaching a critical stress? If the resulting situation is a stable equilibrium, breathe a sigh of relief. If the resulting equilibrium is unstable, oopsie.

 

There's an important made at the end of page 105/beginning of page 106: if buckling results in a stable equilibrium, that also means, in general, that impurities and imperfections will not result in much of a change; if it results in an unstable equilibrium, imperfections will lower the critical load considerably. This is not just empirical - there's an actual mathematical reason, which I wish were explained at greater length (it was new to me - I know nothing).

 

The last section is about *how* things buckle - what sort of shape they tend to take. Here I think you need a mathematician other than me (one who knows much more than what the article gives away) to summarize and frame things nicely and answer your questions. You need someone from dynamical systems.

[SteamyTea]

6 hours ago, Garald said:

Here is my attempt at a summary/commentary

I was only teasing about summarising it in 500 words.

Do you ever sleep.

 

Odd you mention pendulums, I have always thought that the simple harmonic motion was not intuitively correct.