Garald

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.

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.

Wait, SE is about the last field in which I would think of using AI. The main advance in AI in the last few years is that it has got really, really good at bullshitting. It can write an Op-Ed for a major newspaper, yes, but designing a roof that will not fall? Finite elements was a legitimate revolution in its own bounded field back in the day.

PS. I just have a regular wooden staircase going up to the first floor and then a *really* narrow staircase going up to the attic. I would have thought there would have to be a way to extend the first staircase instead of cramming a narrow staircase in what must have once been a closet, but hey, I don't want to change something that works!

CPD = Continuing Professional Development?

Right, I'm afraid I'll have to hire someone local (and an architect to sign off on things! brrr!) when I actually go through with this. Still, I'm very grateful for all advice I can get here. One of many reasons to find out which possibilities are possible and sensible is that I will need to ask for permission from townhall. I think they care mainly about the external look of things. (Or possibly structural work is the business of another department within townhall?) Wouldn't be fun if I hired an SE first, paid them to do plenty of work, and then found out townhall does not like how it would look from the street.

By all means do!

Since the system apparently deleted the photos in my original post, let me add a couple here, to make clear what things look like now - the attic has already been renovated, and it's a nice space; it's just that there's not a lot of head room and the insulation is very middling. Attic with cat: (There's also a bathroom and a bedroom/second office.) Here's what the area in the second picture looked like when the two skylights where about to be opened; the insulation material in that bit of the ceiling had been removed and was about to be replaced (by rockwool I think). There's still an issue with cold air circulating within the ceiling.

I'll be glad to pay an SE - and the joiners, and other people doing the physical work. As for some of the people that get in the middle... Of course part of the idea of reusing the existing structure is to reduce waste (it seems a bit to scrap something that has lasted almost 100 years and that looks rather nice in my view) and also to reduce labour (and material) costs by avoiding *unnecessary* manual labour.

Completely agreed. I won't be looking for the cheapest - I'll be looking for an SE who besides doing a good job will not be annoyed by @Garald always asking why. (Oh, and will actually be interested in not doing things in the most standard, boring way. Plenty of people in the Paris area just scrap everything in the attic and build something very standard from scrap, I take. My feeling is that if the task actually forces the engineer to put serious thought into the issue, the end product will be much better.) > An Architect with 40 years experience behind them will save you piles of cash too! Well, it seems that due to legal requirements I'll need to go through an architecture studio no matter what. I suppose the algorithm is to ask around for an SE that works in an architecture studio, and choose the studio in which that SE works?

Otherwise put: OK, maybe there is no such thing as engineering books written for mathematicians, since perhaps not enough mathematicians are curious about statistics. But what is the statistics equivalent of Feynman's lectures or the Berkeley Physics Course? (On these two examples: neither is ideal for me *now*, since they assume less mathematical maturity than I have, but they can be very good fun.)

OK. So, open challenge: recommend books that are reasonably concise and heavy on the reasons behind things; mathematical content a big plus (goes together with being concise).

You mean for us to adapt to it, or for someone to work on it and make it suitable?

From Dr.T.H.G. Megson, in Structural and Stress Analysis (Third Edition), 2014 - looks like fun. The more maths/physics there is, the more readable it will be for me.

Very good - recommendations welcome! Assume the reader is a mathematician who remembers some physics and would love to learn some more (the last course I took back in the day was quantum mechanics, which I didn't really learn, but did teach me how important and natural Hilbert spaces, normal operators and spectra really are - lesson to be learned here). I imagine that, for this little project, it will all be timber, with perhaps a few steel components involved at a crucial place or two. > It is very accessible but uses the published tables. Behind it is much more complex maths which you will find if you want, but it is the stuff of academics and not used as everyday. Well, that's likely to be the interesting part, isn't it.