Bending of a parabolic plate with some thickness

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oldk
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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!
 
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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:

[tex]\frac{dw^2}{dx^2}=\frac{M}{EI}[/tex]

Here's the link from wiki:

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,

Thanks for the reply.

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!
 
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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.

Your boundary conditions would be:

[tex]w(x=0)=0 , \frac{\partial w(x=0)}{\partial x}=0 , \frac{\partial^2 w(x=0)}{\partial x^2}=0[/tex]