Mechanics of materials centroidal question

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Discussion Overview

The discussion revolves around a homework problem concerning the mechanics of materials, specifically focusing on finding the neutral axis, area centroid, and the distance between them for a rectangular beam section. The scope includes conceptual understanding and technical reasoning related to beam mechanics.

Discussion Character

  • Homework-related
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant calculates the neutral axis as Yc = 5 cm, using the formula Yc = h/2, but expresses uncertainty about finding the area centroid of the rectangle.
  • Another participant questions the clarity of the original question and suggests that the neutral axis depends on the loading direction, implying that the problem may be trivial.
  • A different participant believes that the neutral axis and area centroid should yield the same value (h/2), but mentions a discrepancy with a reference that states the area centroid is at h/4, raising the possibility that the area centroid is half the distance from the neutral axis.
  • Another participant references a proof from a strength of materials book that states the neutral axis coincides with the centroidal axis under pure bending, but notes exceptions when axial load and bending are combined, suggesting the original question may be intended to provoke deeper thought about fundamental concepts.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between the neutral axis and area centroid, with no consensus reached on the correct values or interpretations. The discussion remains unresolved regarding the specifics of the calculations and the underlying principles.

Contextual Notes

There are limitations in the clarity of the original question, potential dependence on definitions of neutral axis and area centroid, and unresolved mathematical interpretations regarding the distances involved.

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Homework Statement


a beam is 12 cm wide and 10 cm long. Find the neutral axis, the area centroid and the distance between them.


Homework Equations


Yc=h/2


The Attempt at a Solution


Yc=10/2=5 cm.
I can't seem to find the area centroid of a rectangle. Any help? Thanks!
 
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Your question isn't clear to me. Do you mean "beam section"? The neutral axis would be a neutral plane, depending on the direction of loading. On the other hand, it seems a very trivial question. What do you think the answer is? If you were answering this in an exam, and had 15 seconds to answer it, what would you say to get a possible mark?
 
Well I thought that the answer would be the same for the neutral axis as for the area centroid, h/2. But the answer in the back says the area centroid is actually h/4. So the area centroid is half of the distance from the neutral axis?
 
There is a proof (in my strength of materials book) that, under pure bending, the neutral axis is coincident with the centroidal axis. The circumstance in which this is not the case is when there is a combination of axial load and bending, The book could be wrong, as many authors in this forum will attest. On the other hand, the question could be a clumsy attempt to get you to look hard at the fundamentals. See if you can find that proof, and then go and teach it to an interested friend.
 

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