Deflection at any point formula

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Discussion Overview

The discussion revolves around determining the precise stress at any point along a beam based on known deflections. Participants explore the relationship between deflection and stress, particularly focusing on bending stress distribution while considering the complexities involved in the calculations.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • RMX seeks assistance in relating deflection to stress at any point along a beam.
  • Some participants reference existing threads that discuss the relationship between deflection and internal moment, noting that assumptions about small angle deflection are necessary.
  • There is a suggestion that RMX may be looking for the bending stress distribution while ignoring shear stresses.
  • One participant mentions the complexity of determining stress at every point, indicating that it may be almost impossible.
  • Another participant describes a method involving integration and symbolic manipulation using MathCAD, detailing the steps required to derive the beam bending formula.
  • Concerns are raised about the assumption that maximum deflection and bending moment occur at the center of the beam, suggesting that this may not always be the case.
  • Participants discuss the challenges of copying formulas from MathCAD to Physics Forums, with suggestions on how to share documents effectively.

Areas of Agreement / Disagreement

There is no consensus on the best approach to relate deflection to stress, and multiple competing views and methods are presented. The discussion remains unresolved regarding the most effective way to handle the complexities of the problem.

Contextual Notes

Participants express uncertainty about the assumptions involved in their calculations, particularly regarding the distribution of stress and the conditions under which certain simplifications can be made.

RMXByker
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Good Afternoon Everyone,

I am need of a little assistance. I am working on determing the precise stress at any point along a beam and would like a little help. I know the deflection at every point and would like an easy way to relate this back to stress. Any help?

[PLAIN]http://sphotos.ak.fbcdn.net/hphotos-ak-snc3/hs358.snc3/29503_523051459059_53800516_30961326_1066764_n.jpg

Let me know your thoughts,
RMX
 
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Hey RMXByker,

It is a good question and Studiot provided you a good link talking about the relationship between deflection and internal moment. With this equation, however, you are assuming small angle deflection, so be sure you remember that everything I am about to say only applies for small angle deflection.

Also, to determine all stress at every point in the beam would be almost impossible. I am going to interrupt that you are looking for the bending stress distribution (ignoring all shear stresses). Please correct me on what you are looking for if this is wrong.

In the attached document (cause I can't seem to make the build in TeX to work), I have made up a little sheet for you. I hope it helps.
 

Attachments

I solve a very similar problem in this thread:

Did your formula in that thread come from MathCad?
I can't get it to work with PF
 
Studiot said:
Did your formula in that thread come from MathCad?
I can't get it to work with PF

The last attachment I posted is basically a fully symbolic derviation of the beam bending formula using the integration procedure. MathCAD did all the heavy lifting for me in terms of symbolic manipulation, but it can at least give you an idea of what you're in for (a lot of work).

You'll end up having to split the beam into three sections, integrate three times for each section, and then solve a system of 9 equations with 9 unknowns at the end. It's not pretty. A pdf of the MathCAD sheet you'll be most interested in is attached to the following post:

https://www.physicsforums.com/showpost.php?p=1600375&postcount=19

The integration tree you'll need is as follows:

[tex]\nu''''=\frac{q(x)}{EI}[/tex]

[tex]\nu'''=\frac{V(x)}{EI}[/tex]

[tex]\nu''=\frac{M(x)}{EI}[/tex]

[tex]\nu'=\theta(x)[/tex]

[tex]\nu=\delta(x)[/tex]
 
Thank you, I know how to solve a beam.

I was asking how to copy from MathCad to Physics Forums.
At present I use MathType, but this has no word processor or picture (graph) capabilities.
 
One thing that will make your problem more difficult is you won't be able to make the assumption that your max deflection & bending moment will always be in the middle. You'll also have to think about what you want to do about cases where a is near one side of the beam or the other.
 
Studiot said:
Thank you, I know how to solve a beam.

I was asking how to copy from MathCad to Physics Forums.
At present I use MathType, but this has no word processor or picture (graph) capabilities.

You can't directly copy MathCAD to PF, you have to print the MathCAD sheet to pdf and attach the pdf file.
 

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