Discovering the Shape of a Bending Plate: Mathematical Proof in 2D Statics

[ucsbphysics]

This is sort of statics, but this came up in my physics class before.

You have a plate, supported only in the middle by a simple support. So, this beam is balanced on the support, and is bending under it's own weight. What shape does the beam take? I was thinking either a inverted parabola or catenary shape? But I don't know how to prove this.

How can you show mathematically that this is the shape it has?

 

[Simon Bridge]

Welcome to PF;

You have a plate, supported only in the middle by a simple support. So, this beam is balanced on the support, and is bending under it's own weight.

Is it a beam or a plate? What is it's geometry? What are it's material properties?

Continuing for a beam (metal? rectangular?):

What shape does the beam take? I was thinking either a inverted parabola or catenary shape?

Just on intuition?

A catenary would follow for a flexible beam supported at each end.
A parabola would imply that the vertical deflection is proportional to the square of the distance from the support.

But I don't know how to prove this.

Have you tried an application of the principle of least action?

For a beam where width W and height H are: W,H<<L, then try treating it as a half-length beam bolted to a wall at one end. What shape does that make?

Bottom line - the problem is under-specified.
Have you tried looking it up?

 

[ucsbphysics]

A plate with length L, and yes it is assumed that L>>H. It could be metal. Just has thickness, H and density, rho. That is just on intuition. Could you say something about the half-length beam attached to a wall because I see how you can treat that the same. The difference being the full beam doesn't have moment at the support.

Thank you!

 

[Simon Bridge]

In the full beam, the extra moment at the support is provided by the other half of the beam.
There is no net moment at the support in either case.
The two cases are physically identical provided the horizontal width is too small to have significant distortion.