Boundary conditions for a clamped circular plate with a central hole

[harpreet singh]

Hey people can u please tell me what will be the boundary conditions for a circular plate with a central hole clamped at the circumference... Plate is axis symmetric and is under uniform load..

 

[FredGarvin]

You'll obviously have no displacements at the outer edge. There should also be a zero slope at the edge. At the center there are similarities for the conditions of a beam's free end. Can you think of what those would be?

 

[harpreet singh]

These two were known to me.. I was encountring problem with the inner edge only..

 

[FredGarvin]

Well, if you have a cantilevered beam, what can you say about the moment at the free end?

 

[harpreet singh]

I guess moment will b zero there

 

[harpreet singh]

But then also i will be needing 1 more condition coz i have 4 constants to be determined in my equation..

 

[Mech_Engineer]

Is the shear force defined at the fixed edge of the plate, perhaps by calculating the resultant reaction at that edge?

 

[harpreet singh]

Will there be any shear force at the fixed edge??

 

[Mech_Engineer]

On a uniformly loaded a cantilever beam, the shear force will be maximum at the fixed end of the beam, and equal to the reaction force at that end. Additionally, since the beam is only supported by one end, the shear force is zero at the free end of the beam.

 

[Mech_Engineer]

According to Roark's Table 11.2 Case 2e (Annular plate, outer edge fixed, inned edge free), the following boundary conditions apply:

• Bending moment at the free end is zero.
• Shear force at the free end is zero.
• Displacement at the fixed end is zero.
• Slope at the fixed end is zero.

They also have a definition of the shear force at the fixed edge (enge "a") that is basically a calculation of the reaction force along that edge.

$$Q_{a}=\frac{-q}{2a}(a^2-r_{o}^2)$$

 

[harpreet singh]

Thanx a lot.. can u please send me that link??

 

[Mech_Engineer]

It's not on the web, the equations are located in this book:

https://www.amazon.com/dp/007072542X/?tag=pfamazon01-20

 

[harpreet singh]

Ok.. Do u have ebook or hard copy??

 

[harpreet singh]

Can u please check wat r d conditions if inner edge is simply supported and outer edge clamped..

 

[FredGarvin]

According to Roark's Table 11.2 Case 2e (Annular plate, outer edge fixed, inned edge free), the following boundary conditions apply:

• Bending moment at the free end is zero.
• Shear force at the free end is zero.
• Displacement at the fixed end is zero.
• Slope at the fixed end is zero.

They also have a definition of the shear force at the fixed edge (enge "a") that is basically a calculation of the reaction force along that edge.

$$Q_{a}=\frac{-q}{2a}(a^2-r_{o}^2)$$

I was trying to not directly quote that to get the OP to think about the conditions on his own...Make him work!

 

[harpreet singh]

Thanx a lot sir for helping me that way..

 

[Mech_Engineer]

I was trying to not directly quote that to get the OP to think about the conditions on his own...Make him work!

Sorry about that Fred, I just get so excited!

Can u please check wat r d conditions if inner edge is simply supported and outer edge clamped..

Its quite easy to interpolate what you've "learned" on the case above to the simply supported case. Perhaps trying to sketch a rough shear force and bending moment diagram could help you... specifically looking at what a simple support's effect looks like on those diagrams.

 

[harpreet singh]

sorry but i got stuck once more.. I am not being able to integrate dw/rdr w.r.t dr

 

[FredGarvin]

Sorry about that Fred, I just get so excited!

I just can't touch that one...