Help with Bending of a Plate with unique boundary conditions

In summary: Thanks Henneh,In summary, the conversation discusses a problem with a rectangular plate with simply supported sides and a point force applied on the surface. The bending stress distribution in the plate is sought after, but no solution has been found in textbooks or online. The use of finite element analysis is suggested as a possible numerical solution, and the analogy with electrical field applications is also mentioned. The conversation ends with a discussion on the difficulty of plate problems and the importance of these structures in industries such as aerospace and shipbuilding. Additionally, the issue of mechanism and the use of boundary conditions in FE programs is raised.
  • #1
hushish
29
0
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
 
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  • #2
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.
 
  • #3
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
 
  • #4
hushish said:
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

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.
 
  • #5
Thanks SteamKing,

I was hoping that my case fell inside the "trivial" solution side, but it seems like FE is the only answer.
 
  • #6
hushish said:
Thanks SteamKing,

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

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.
 
  • #7
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.
 
  • #8
Hi Henneh,

The following boundary conditions definitely do work in FE.

Regards,
 

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What is bending of a plate with unique boundary conditions?

Bending of a plate with unique boundary conditions refers to the physical phenomenon of a flat plate undergoing deformation or flexure when subjected to external forces or loads. This can occur when the plate is supported by different boundary conditions, such as clamped, simply supported, or free edges.

What factors affect the bending of a plate with unique boundary conditions?

The bending of a plate with unique boundary conditions is affected by various factors, including the material properties of the plate, the magnitude and distribution of external loads, and the type of boundary conditions applied. Additionally, the plate's dimensions, thickness, and shape can also influence its bending behavior.

How is the bending of a plate with unique boundary conditions calculated?

The bending of a plate with unique boundary conditions can be calculated using mathematical methods such as the theory of elasticity or finite element analysis. These methods take into account the plate's geometry, material properties, and boundary conditions to determine its bending behavior and stress distribution.

What are some common applications of bending of a plate with unique boundary conditions?

The bending of a plate with unique boundary conditions has many practical applications, including in engineering and construction, where it is used to design and analyze structures such as bridges, buildings, and aircraft wings. It is also relevant in manufacturing processes, such as sheet metal forming and rolling.

How can the bending of a plate with unique boundary conditions be controlled or minimized?

The bending of a plate with unique boundary conditions can be controlled or minimized through various methods, including adjusting the boundary conditions, changing the material properties of the plate, or adding reinforcements. Additionally, using mathematical models and simulations can help predict and optimize the bending behavior of a plate.

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