Help with Bending of a Plate with unique boundary conditions

  1. Hi,

    Can anybody help me withg the following problem:

    A rectangular plate with points starting from top left corner and going clockwise:: A B C D. Sides CD and DA are simply supported, and a point force F is applied anywhere on the surface. I am looking for the bending stress distribution in the plate.

    I have looked in all the relevant textobooks and online, but have yet to come across an example of such a situation.

    Thanks,

    hushish
     
  2. jcsd
  3. Baluncore

    Baluncore 2,965
    Science Advisor

    What a beautiful problem. I would have expected a very elegant general analytic solution.

    Unfortunately, if a force is applied to a point then the infinite pressure will punch through the plate. For that reason the force must be applied to a small finite area of the plate. This upsets the analytic solution of your idealised model somewhat, it may explain why it is not found in the literature.

    The obvious numerical solution would be to use finite element analysis.

    I suspect you may find an analogue solution in the electrical field. Consider a rectangular resistive sheet, grounded along two adjacent edges. A current is injected into one small patch of the resistive sheet. The potential and current distributions give the solution you are seeking.
     
  4. Thanks Baluncore,

    I am not familiar with electrical field applications, can you point me in the right direction?

    If it would help, the force can be distributed over a finite circle of radius R. What would the displacement and load distribution look like then? I assume it needs to be of the following form to solve the differential equation:

    Capture2.PNG
     
  5. SteamKing

    SteamKing 9,624
    Staff Emeritus
    Science Advisor
    Homework Helper

    In general, while plate problems can be described rather elegantly, their solutions can be somewhat difficult and messy, even numerical solutions.

    Take a gander here:

    http://www.efunda.com/formulae/solid_mechanics/plates/theory.cfm

    For small deflections, the plate problem is an example of a bi-harmonic problem. Except for a few trivial examples, these problems have to be solved numerically, using some type of finite element approach.
     
  6. Thanks SteamKing,

    I was hoping that my case fell inside the "trivial" solution side, but it seems like FE is the only answer.
     
  7. SteamKing

    SteamKing 9,624
    Staff Emeritus
    Science Advisor
    Homework Helper

    I'm sorry I couldn't be more helpful, but I think your research on this problem has barely scratched the surface.

    The link I enclosed mentioned 'solid mechanics', which is one discipline which studies plate problems, among others.

    While most of the information on the solution to such problems once came from actual test results on real plates, FE and Boundary Element techniques have closed the gap in recent years, with some codes accurately reproducing test results done on real plates. There's a large body of work out there, because plate structures are very important in the aerospace and shipbuilding industries.
     
  8. What you have is a mechanism (i.e. the number of degrees of freedom is less than the number of equations . As the only two supported sides are simply supported only (i.e. have no fixity). If the supports are truly supported and have no fixity the beam will simply be allowed to rotate at its free end.

    If you have a FE program try it out and the analysis will fail. Even simpler, try a simple beam with only a pin restraining translation only in x and y directions. The analysis will fail.
     
  9. Hi Henneh,

    The following boundary conditions definitely do work in FE.

    Regards,
     

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