I have a one piece X thickness of acrylic and I want it to be a dome. There is a formula that has gravity, density and all, but what becomes to acrylic, what is the limit of radius? Let’s say 30mm, 100mm and 300mm acrylic and an overall formula.
Not sure I understand your question, but if the formula you have doesn’t contain the thickness of the material my guess is that’s because the maximum size is theoretically independent of the thickness. Think of it this way: if you have a dome of thickness x of maximal size, you can presumably make a slightly smaller dome of equal thickness, which will also be able to stand on its own, and which fits exactly inside the former, and fuse the two together into a single piece of thickness 2x. On the other hand, if you try to make a slightly larger dome designed to fit exactly outside the first, it would already be in a state of collapsing, thus putting more strain on the structure beneath, causing the whole thing to crumble…
The logic of this reasoning is probably not irrefutable, as multiple layers isn’t really the same thing as a solid piece of the same material, but I guess the original formula might be a bit simplistic anyhow.
You would want to relate the weight/mass of the dome of a given thickness to the forces imposed by this weight to the strength of acrylic sheets of that thickness
I think he’s got a formula for size of the dome given a thickness .
But what sets the thickness ? Is there a limit ?
thickness = t
radius=r
So you have r=f(t)
You don’t recognise the shape of this curve really well, eg its got multiple parts involving different curve types, you can’t just determine it by inspection.
Now see if there was going to be a maximum radius, then the radius vs thickness graph is going to have to turn and drop. Reach a peak … a ^ …
How to find this peak ? The easy to find the peak, and even find if it exists, is to differentiate the right hand side of r (radius) … and where that is 0… where dr/dt = 0 !!! Thats the very top of the peak. Thats how to know.
If there is no zero of the slope, it never reaches a peak… your thickness can keep increasing for ever … but you have to check that for sanity, is it still capable of making a dome ? has the thickness become large compares to the radius ? If you plug in a billion thickness, and the radius then is a billion, the formula is based on an assumption radius being much larger than thickness and it has becomes inaccurate for very large thickness.
Also check any result for sanity… if thickness is close to radius, its really not a dome any more… its a big lump with a bit of artwork on the edges.
It may be that your definition of dome vs a big lump is subjective, and your limit on that is up to you. ( How does the formula deal with the shape of the ends ? because as thickness increases to be within a tenth of radius,
then the shape of the end becomes relevant. )
Ok - if I wanted to build an acrylic semisphere of height 100 meters. Now, what would be ghe thickness?