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Consider a thin surface and hold it tightly and vertically at a point on its boundary. What it is the shape it assumes?

As a particular example let's take a circular plate clamped on a plane, as shown below. The clamping condition can be imposed by demanding that the normal vector to the surface at that point lies completely in the plane, however,

How to approach the problem ? Which theories/equations are suitable for this kind of problem (the goal is to determine the height z as a function of some coordinates on the plane) ? The material of course has some rigidity and know elastic properties (paper would be prototypical), so apart from gravity which forces are present?

enter image description here

(This question was initially posted in Physics SE, where it was deleted for being "a numerical calculation", according to a comment of one of the users; however, the contributor did not mention which equations should be numerically solved, under what conditions and why are they applicable. Also, in the closure message the moderator(s) indicated that the question was about "application of scientific knowledge to construct a solution to solve a specific problem", again without pointing out which piece of knowledge is suitable to the problem. Such sub-questions I'm trying to answer.)

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  • $\begingroup$ If the surface is flexible, holding it just at one point does not make sense, because such "connection" has zero stiffness. Of course if you use FEA and fix one node in place, it works, but that is because the FEA node doesn't represent just a single point but also area around it. $\endgroup$
    – Tomáš Létal
    Commented Nov 4, 2022 at 8:43
  • $\begingroup$ @TomášLétal Of course, in reality one holds it not at a single point, but with that I mean an area small compared to the overall area of the plate. $\endgroup$
    – Daniel Castro
    Commented Nov 6, 2022 at 8:30
  • $\begingroup$ The size and shape of the small area that is fixed will matter a lot, I think you should specify it in the question. $\endgroup$
    – Tomáš Létal
    Commented Nov 6, 2022 at 12:52
  • $\begingroup$ @TomášLétal I understand. But I would like to know which equations and, or theories (preferably with name and surname) are suitable for this kind of problems. I am not really interested in the circular case but something more intricate (tree leafs, actually); this is just a toy model that I'm trying to analize to get a grasp of the general methods. Thank you. $\endgroup$
    – Daniel Castro
    Commented Nov 6, 2022 at 13:22
  • $\begingroup$ From such general sense, I think your only option is nonlinear finite element analysis with large deflections. You could also check out "large deflection plate theory". $\endgroup$
    – Tomáš Létal
    Commented Nov 6, 2022 at 14:28

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