Circular Plate w/ Hole: Boundary Conditions

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Discussion Overview

The discussion revolves around determining the boundary conditions for a circular plate with a central hole that is clamped at the circumference and subjected to a uniform load. Participants explore various scenarios, including the implications of different support conditions at the inner and outer edges of the plate.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant suggests that there will be no displacements at the outer edge and that the slope should also be zero at that edge.
  • Another participant draws parallels between the conditions at the center of the plate and those of a cantilevered beam's free end.
  • A participant expresses difficulty in determining the boundary conditions at the inner edge of the plate.
  • It is proposed that the moment at the free end of a cantilevered beam is zero.
  • One participant notes the need for an additional condition to solve for four constants in their equations.
  • There is a question about whether the shear force can be defined at the fixed edge of the plate by calculating the resultant reaction at that edge.
  • Another participant states that the shear force will be maximum at the fixed end of a cantilever beam and zero at the free end.
  • A reference to Roark's Table 11.2 is made, outlining specific boundary conditions for an annular plate with the outer edge fixed and the inner edge free.
  • One participant requests a link to the source of the boundary conditions mentioned.
  • There is a discussion about the conditions if the inner edge is simply supported and the outer edge is clamped, with suggestions to sketch shear force and bending moment diagrams.
  • A participant expresses difficulty in integrating a specific mathematical expression related to the problem.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the boundary conditions for the inner edge of the plate, and multiple competing views regarding the conditions and their implications remain throughout the discussion.

Contextual Notes

Some participants reference specific tables and equations, but there is no resolution on the assumptions or definitions that may affect the boundary conditions discussed.

harpreet singh
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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..
 
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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?
 
These two were known to me.. I was encountring problem with the inner edge only..
 
Well, if you have a cantilevered beam, what can you say about the moment at the free end?
 
I guess moment will b zero there
 
But then also i will be needing 1 more condition coz i have 4 constants to be determined in my equation..
 
Is the shear force defined at the fixed edge of the plate, perhaps by calculating the resultant reaction at that edge?
 
Will there be any shear force at the fixed edge??
 
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.
 
  • #10
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.

[tex]Q_{a}=\frac{-q}{2a}(a^2-r_{o}^2)[/tex]
 
  • #11
Thanx a lot.. can u please send me that link??
 
  • #13
Ok.. Do u have ebook or hard copy??
 
  • #14
Can u please check wat r d conditions if inner edge is simply supported and outer edge clamped..
 
  • #15
Mech_Engineer said:
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.

[tex]Q_{a}=\frac{-q}{2a}(a^2-r_{o}^2)[/tex]
I was trying to not directly quote that to get the OP to think about the conditions on his own...Make him work!
 
  • #16
Thanx a lot sir for helping me that way..
 
  • #17
FredGarvin said:
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!

harpreet singh said:
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.
 
  • #18
sorry but i got stuck once more.. I am not being able to integrate dw/rdr w.r.t dr
 
  • #19
Mech_Engineer said:
Sorry about that Fred, I just get so excited!
I just can't touch that one...
 

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