# Bending of a parabolic plate with some thickness

• oldk
In summary: I don't think there's a simpler way to do this.In summary, you need to find the equation for the deflection of a parabolic plate clamped at one end, with a concentrated load along the center line at the "apex" end. You can find the equation by integrating the deflection equation for a beam of the same cross-section.
oldk
Hi,

I am trying to obtain a relation for calculating the deflection of a parabolic plate of thickness (say 't') clamped at one end, with a concentrated load along the center line at the "apex" end.

Any direct links for the formula?

Thanks!

Sounds complicated, especially because of the non-axisymmetric clamp on just one side. I'd use a Finite Element model together with some limiting simplified cases, such as treating it as a straight beam with the same thickness, and as a channel-section beam with the same overall thickness. That's if it's reasonably flat shaped. Would be different if it's a deep bowl.

That's if it's a revolved parabola. If it's just a 2D shape with arbitrary "thickness" then you can do hand calculations all the way.

If I'm getting it right you have a parabolic cross-section. In that case, I believe that you can use the deflection equation from beam theory:

$$\frac{dw^2}{dx^2}=\frac{M}{EI}$$

http://en.wikipedia.org/wiki/Euler–Bernoulli_beam_equation

I'm not sure if I understand the geometry of your problem correctly, so I can't really tell you more :) Be careful with the moment of inertia though, it should be calculated in respect to the center of mass of your cross-section.

Hi,

I am attaching the geometry of the plate. It is not a parabolic cross section but a parabolic shaped plate. The moment of inertia would be a function of x (width of the plate is a function of x -- see attached) and I would like to obtain a deflection function as a function of x (i.e, if I know the position along the center line where I apply the force, I should know the deflection from that positional force).

I hope the attached helps.

Thanks again!

#### Attachments

• fig 1.pdf
53 KB · Views: 299

Then this should be pretty straightforward to calculate, you just replace the equation for the moment of inertia into the equation of the deflection and you'll get the equation you need by integrating twice.

$$w(x=0)=0 , \frac{\partial w(x=0)}{\partial x}=0 , \frac{\partial^2 w(x=0)}{\partial x^2}=0$$

## What is the concept of bending in a parabolic plate with some thickness?

The bending of a parabolic plate with some thickness refers to the deformation of a parabolic-shaped plate when a force is applied to it. This deformation can be seen as a change in the curvature of the plate's surface.

## What factors affect the bending of a parabolic plate?

The bending of a parabolic plate can be affected by various factors such as the material properties of the plate, the magnitude and direction of the applied force, the thickness of the plate, and the boundary conditions.

## How is the bending behavior of a parabolic plate predicted?

The bending behavior of a parabolic plate can be predicted using mathematical models such as the Euler-Bernoulli beam theory or the Kirchhoff plate theory. These models take into account the plate's material properties, dimensions, and applied loads to calculate the resulting deflection and stress.

## What are the applications of studying the bending of a parabolic plate?

The study of the bending of a parabolic plate has various applications in engineering and science. It is commonly used in the design and analysis of structures such as bridges, aircraft wings, and shells. It also has applications in fields such as optics, where curved mirrors and lenses are used.

## How can the bending of a parabolic plate be controlled?

The bending of a parabolic plate can be controlled by adjusting the material properties, thickness, and boundary conditions of the plate. Additionally, reinforcing the plate with stiffeners or using different support structures can also help control its bending behavior.

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