Plate four points bending equation for a plate

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Discussion Overview

The discussion centers around the analytical calculation of normal stress in a plate subjected to four-point bending. Participants explore the applicability of beam theory to plates and seek resources or methodologies for solving this problem, including references to literature and alternative approaches.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses a lack of mechanical engineering background and seeks guidance on calculating normal stress in a plate under four-point bending, noting their familiarity with Euler-Bernoulli beam theory.
  • Another participant suggests that the problem may not be well-covered in standard references like Roarke's Stress and Strain, recalling challenges with unsolvable partial differential equations related to plate theory.
  • A different participant proposes using Finite Element Analysis (FEA) as a potential solution, acknowledging that it may not yield an analytical result but could be validated through experimentation.
  • A reference to "Theory of Plates and Shells" by Timoshenko & Krieger is provided, specifically pointing to a section that discusses rectangular plates with four edges supported elastically.
  • Several participants express appreciation for the references shared, indicating a collaborative effort to find suitable resources.
  • Another participant mentions a more modern book, "Formulas for Stress Strain and Structural Matrices" by Pilkey, as a potentially useful resource.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the best approach to the problem, with multiple competing views on the applicability of existing literature and methodologies. The discussion remains unresolved regarding the specific analytical solutions for four-point bending of plates.

Contextual Notes

Participants note limitations in existing references and the complexity of the equations involved, indicating that the problem may depend on specific assumptions or configurations of the plates being analyzed.

pierebean
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Hello,

I am no mechanical engineer and my knowledge in bending is limited to Euler-Bernouilli beam theory.

I wish to analytically calculate the normal stress of a plate bent by four points bending. I have already calculated this stress for a beam.

However I cannot apply the beam theory to a plate.

Does someone know a tutorial or an article giving specifically the four-points bending of a plate and its solution?

Thank you
 
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pierebean said:
Hello,

I am no mechanical engineer and my knowledge in bending is limited to Euler-Bernouilli beam theory.

I wish to analytically calculate the normal stress of a plate bent by four points bending. I have already calculated this stress for a beam.

However I cannot apply the beam theory to a plate.

Does someone know a tutorial or an article giving specifically the four-points bending of a plate and its solution?

Thank you
:eek: That's a tough one that i don't think even Roarke's Stress and Strain covers. This old timer took graduate courses in Theory of Plates and Theory of Elasticity eons ago. All I remember is a bunch of unsolvable partial differential equations.
Stay away from square or rectangular plates supported on 4 corners. Consider a statically determinate circular plate instead, supported at 3 equidistant simple supports, then get a copy of Roark's book. Beyond that, i cannot help, sorry.
 
Well Roark's is definitely the go to book but like PhantomJay I don't remember seeing that in there but you never know.

My solution would be use FEA. I know you won't get an analytical solution but you may be able to validate it via a simple experiment.
 
"Theory of Plates and Shells", Timoshenko & Krieger

Article 49, p 218 in my copy

"Rectangular Plates Having Four Edges Supported Elastically or Resting on Corner Points with All Edges Free"
 
Nice reference thanks.
 
Nice reference thanks.

Roark is oft quoted at PF, (as here) however a much more modern book ( and at least twice as thick) is

Formulas for Stress Strain and Structural Matrices

Pilkey

go well
 

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