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.

 

[PJC01]

I would approach this problem by first considering the forces acting on the bending plate. In this case, we have the weight of the plate acting downwards and the reaction force from the support acting upwards. These two forces must be balanced in order for the plate to remain in static equilibrium.

Next, I would consider the moments acting on the plate. Since the plate is only supported at the middle, there will be a moment acting on the plate causing it to bend. This moment is caused by the weight of the plate acting at its center of gravity.

Using the principles of statics, we can set up equations to determine the shape of the plate. By considering the forces and moments acting on the plate, we can use the equations of equilibrium to solve for the unknown variables such as the shape of the plate.

In order to prove that the shape of the bending plate is an inverted parabola or catenary, we can use mathematical techniques such as differential equations or calculus. By modeling the plate as a continuous system, we can use these mathematical tools to derive the equation of the curve and solve for its shape.

Additionally, we can also conduct experiments or simulations to validate our mathematical proof. By measuring the deflection of the plate at different points and comparing it to the predicted shape, we can confirm that our mathematical proof is accurate.

In conclusion, as a scientist, I would approach the problem of discovering the shape of a bending plate by considering the forces and moments acting on the plate and using mathematical techniques to derive the equation of the curve. This mathematical proof can then be validated through experiments or simulations.