Talk:Second moment of area

Polygon
This section must be checked by an expert.


 * Deutsch page has any reference and show a - sign for Ixy, which means there are wrong formulae on the web.
 * Spanish page refers the same Spanish book ... But I could not trust the formulae against the English one using symbolic computation (Sympy):

E1 = sympify('(yi**2 + yi*yi1 + yi1**2)*(xi * yi1 - xi1 * yi)')

E2 = sympify('(xi1-xi)*(yi1+yi)*(yi1**2+yi**2)') — Preceding unsigned comment added by 90.43.11.93 (talk) 21:18, 2 October 2018 (UTC)

first comments
The calculation of "y" the distance to the centroid is incorrect. It should be:

y=(h+h1)/4

John.g.taylor 04:30, 28 May 2007 (UTC)

I think the second moment of area is different to the second moment of inertia.

John.g.taylor 04:29, 28 May 2007 (UTC)

Specifically:

Second moment of inertia = $$I_x = \int y^2 dm$$ rather than dA. ..

The second moment of area is a more general concept. For example the static roll stability of a ship depends on the second moment of area of the waterline section - short fat ships are stable, long thin ones are not. Gordon Vigurs 08:10, 11 July 2006 (UTC)

As a Swede I find it strange that "area moment of inertia" is the last choice. In Swedish that is what we call it: "yttröghetsmoment". Compare to moment of inertia = sum up all the infinitesimal mass elements multiplied by the lever squared, versus area moment of inertia = sum up all the infinitesimal area elements multiplied by the lever squared. — Preceding unsigned comment added by 83.223.9.100 (talk) 11:51, 23 October 2023 (UTC)

reverting
I had several problems with the changes made by 69.251.146.46: Nephron T|C 02:49, 25 August 2006 (UTC)
 * The lead sentence shouldn't be about what it is confused with.
 * Polar moment of area -- much less frequently used than polar moment of inertia... dead linkage

'I' in the stress bending eqn should be for 2nd moment of area. I referred this from 'Mechanics of solids and structures', Benham, P.P. ISBN: 0273361910

Need diagrams
This article needs some diagrams to go with the formulas, to make the variables clear. Some body needs to translate and lift the diagrams from the De page on this

Comment moved from article to talk (RJFJR 15:14, 2 May 2007 (UTC)) "I-beam" cross section
 * A suitably dimensioned and annotated I-beam diagram would greatly enhance the usefulness and applicability of this forula.


 * I added diagrams for the I-beam section. My lettering, which didn't look that bad in the original stands out now.  I can redraw if there is interest.  I could also add a variable for the the thickness of the top and bottom pieces.  Before I do it, are there any other changes? (you can drop a comment on the diagrams on my talk page.) RJFJR 19:45, 13 May 2007 (UTC)
 * Diagrams look good- however, they should have the coordinate system axis for completeness.Mollynet (talk) 18:43, 14 September 2011 (UTC)

Product Moment of Area
I have found the math here too sloppy to be understandable. I suggest to put (7) from http://www.dynamore.de/documents/papers/euro4/implicit-new-developments/ls-dyna-beam-elements-default-and-user-defined as an example of rectangular cross section beam. This was what made me understand the concept. —Preceding unsigned comment added by 145.236.252.34 (talk) 10:45, 14 October 2009 (UTC)

This material on the "product moment of area" was int eh article but seems to lack context and explanation:


 * $$I_{xy}= \sum yxA$$


 * $$I_{xy}$$ is also known as Product Moment of Area

$$Insert formula here$$

What is this used for? Is it really related to the second moment because an object with both x-axis and y-axis symmetry would have a value of zero, whcih is certainly not true of the second moment (it more resembles a first moment). Can we clearify this before we put it back in teh article, if this is where it belongs. And why does this have a redlink? RJFJR 13:48, 6 May 2007 (UTC)


 * It is used in the bending formula, if for some reason you have a poor choice of coordinate system. The bending formula in the general form is relatively complicated:
 * $$\sigma=-\frac{M_y I_x + M_x I_{xy}}{I_x I_y - I_{xy}^2 } x + \frac{M_x I_y + M_y I_{xy}}{I_x I_y - I_{xy}^2} y$$
 * The trick is to find a coordinate system, which has a product moment of area equal to zero. This is always possible, and as you have noticed, it is trivial with a symmetric cross section. Then the bending formula becomes a lot simpler:
 * $$\sigma=-\frac{M_y}{I_y} x + \frac{M_x}{I_x} y$$
 * The reason it has a redlink, is probably because all common cross sections have symmetry, and then there is no point in calculating it, so in real life it is only rarely used. Hemmingsen 19:22, 13 May 2007 (UTC)

Why is there a discussion of mast calculations?
There is a discussion of mast calculations in here under the section on the second moment of area of a circular cross section. Everything the section mentions is true but are mast calculations really an important enough consideration to warrant a mention here? I think it adds unnecessary length to the article. I am cleaning the section up some but it needs to be considered for deletion anyway.

The definition is incomplete
The definition does not state what areas are to be summed in the integral.


 * The elements of area comprising the surface of a body?
 * The cross sectional areas perpendicular to an axis, such that with a thickness, the volume of the body is accounted for?

At each point with coordinates (x,y,z), there are infinitely many surfaces including that point, each with a different orientation. Which of them should be included in the sum?

Can the sum include area elements that overlap?

The definition lacks every hint of context. It is understandable only for those who already know what it means. Cacadril (talk) 16:25, 24 November 2009 (UTC)


 * I think the bit that needed clarifying was that this a property of a cross section of body, not of a body. The area to include in the integral is simply the area of the cross section. Do you have any suggestions for further clarifications? 77.215.191.91 (talk) 17:00, 24 November 2009 (UTC)

Rename the article
This page is not about second moment of area any more, it is about bending stress and strain.

I think it should be possible to define this concept mathematically, and then proceed to show how it is applied in various contexts, and notably for computation of bending stresses, with limitations and conditions. Perhaps the article should be renamed, and a new article about the specific mathematical concept be written. Cacadril (talk) 05:15, 15 February 2010 (UTC)


 * As far as I know, beam bending analysis is the predominant (only?) use of second moment of area, so I think it's fine that the emphasis is on beam bending. I don't really care much about separating the pure math from the applied math.  But I'm an engineer, not a mathematician.


 * The lead sentences says "This article is about the moment of inertia as related to the bending of a plane. For the moment of inertia dealing with rotation of an object, see Moment of inertia." I think it is great to distinguish the area moment of inertia from the mass moment of inertia.  However, I find it strange to talk about "bending of a plane".  The plane is not bending.  The plane is a section through a beam that is bending.  One of the assumptions in the derivation of beam bending equations is that plane sections remain plane.  That is, they don't bend.  So I'm inclined to edit that sentence.  I just haven't decided what I want it to say instead yet, so if anyone has any suggestions, I'd like to hear them.


 * Kimaaron (talk) 22:21, 14 May 2010 (UTC)


 * I agree, this article is terribly written to the point that it completely lost focus from what was supposed to be its subject. Whoever wrote it clearly had a limited grasp on the subject, and his limited understanding added to the confusion of mixing up the second moment of area with beam theory.  This article needs a complete rewrite to stay meaningful. -- Mecanismo | Talk 12:31, 10 February 2013 (UTC)

Parallel axis theorem
Something should be said about the product moment of area in this section. If I remember correctly, the formula used for that transformation is:

$$I_{xy}'= I_{xy} + Axy$$

-- Fernando Estel ☆ · 星 (Talk: here- commons- es) 10:21, 13 June 2010 (UTC)

Intuition
As a non-expert I would like to know why the 'moment of inertia' of the cross-section of a beam is important in relation to stiffness or laibility to failure (Strength). I found the introduction of elastic stored energy unhelpful. Its relevance is not at all clear from what is written. What is required is an explanation of 2 independent relationships to radius. For example, in the case of rotational moment of inertia, I could explain how moment of inertia relates to torque and angular acceleration and then show that a particle of mass at radius r accelerates more as r is increased, and also how the moment of the accelerating force increases with r, thus giving an overall dependence on r^2. I would appreciate an analogous approach with 2nd moment of area, bending moment, and deflection (or condition of failure), or whatever the concepts are which are most relevant. Why is integral(r^2)da a significant metric; why is the square of the radius of an element of area important ? At present the underlying principles of the analysis remain obscure (to me at least). I think that the effective explanation of concepts in simple terms is a particular skill and needs more careful thought than is being used at present. —Preceding unsigned comment added by G4oep (talk • contribs) 09:04, 24 October 2010 (UTC)
 * I agree with everything above, but furthermore the second paragraph of 'Intuition' is not consistent with the first. Rewrite is needed: watch, as they don't say, that space! 86.176.166.181 (talk) 00:37, 17 April 2012 (UTC)

Questionable link
In the section "Product moment of area", the linked reference to "Pilkey 2002" does not seem to work, and I cannot find a book by that title or an book on a related subject by "Pilkey". Please clarify.

NextThreshold (talk) 12:31, 5 July 2011 (UTC)

Readership?
I'd be interested to know for whom the article in its current state - with 'a symmetric tensor' in the third paragraph and 'Minimum total potential energy principle' in the fourth - is intended. You really don't need to know about the Kronecker delta before you can form a working understanding of second moment of area. 86.185.64.17 (talk) 01:55, 2 April 2012 (UTC)

Major cleanup
I've cleaned up the article, which included a whole lot of nonsense and stuff which was simply wrong. The criteria for the cleanup was:
 * if it mentioned any form of beam theory then it doesn't belong here.
 * if it mentioned any form of stress then it has no place in this article
 * examples must be exclusively about an area's geometric properties, because that's what any moment of area does

This article is still a bit incomplete, but at least now it doesn't include blatant errors and misconceptions about its subject. -- Mecanismo | Talk 13:00, 10 February 2013 (UTC)


 * < if it mentioned any form of beam theory then it doesn't belong here. > Why is that? It says right on this talk page that this is a Civil Engineering topic! 109.151.253.181 (talk) 21:17, 13 November 2013 (UTC)

If your work removed the blatant errors, etc, could you change the cleanup tag to reflect its current state?165.121.80.228 (talk) 10:35, 1 April 2013 (UTC)

Major Rework
I've reworked a lot of what was in the article to get rid of mentions to cross-section (still remnant of the beam-theory days). Also fixed some of the linkages. Added simple sections about parallel and perpendicular axis theorems and composite shapes. And removed the unnecessary example that just pulls from the list. I only kept (and reorganized) examples that exploit some calculation of the second moment of area.

The only thing I have a question about is the product of inertia. Should this article include this? Or should we rewrite the existing page for product of inertia that just links here?

Also, I didn't remove the messages at the top. I would like someone else to review the article and clear them if they feel it's cleared up the issues. — Preceding unsigned comment added by BeaumontTaz (talk • contribs) 04:13, 14 April 2013 (UTC)

Annulus
So I changed some stuff in the Annulus section of the page. One thing to keep in mind (and I clarified this in a later edit) is that with that integral, I was calculating $$J_z$$ directly, not $$I_x$$ or $$I_y$$. I think someone who edited it after me got confused about what I was calculating and changed some stuff to $$I_z$$ if I recall correctly. I just wanted to justify why I switched it back. -- Chiraag.nataraj (talk) 23:00, 27 April 2013 (UTC)

- Hey. I'm the user who edited it after you changed it. Honestly, I think I was looking at an older version of your series of edits, not the final one. What I saw reflected this edit. My apologies for the confusion. I believe the article looks good. Thank you for the edit! — Preceding unsigned comment added by BeaumontTaz (talk • contribs) 02:18, 28 April 2013 (UTC)

Ah I see. Thanks for the clarification! :) -- Chiraag.nataraj (talk) 18:02, 28 April 2013 (UTC)

Current State of Momentum
I have looked at the recent edits and can see major improvements, that result in a clarity of understanding. Keep up the Good Work! DPHutchins (talk) 05:44, 11 July 2013 (UTC)

Updated
I have read through the formulae and confirmed that all are correct (up to the polygons -- I didn't check that one.)

It's worth noticing here that there are two communities playing tug-of-war over this article: the math/physics people who keep saying that it is fundamentally wrong and the engineering-type folks who are trying to make it useful for us. The formulae that are here now match both the basic fundamental mathematics and principally the useage in engineering. I'd propose that the big warning at the top of the page be removed.

I also tried to clean up some of the references. Annahoward (talk) 13:54, 24 March 2015 (UTC)annahoward

Polygon
I found an appropriate reference.

https://apps.dtic.mil/dtic/tr/fulltext/u2/a183444.pdf

I will program these equations and verify them against known results.

Also, there are some explanations on how to deal with disconnected polygons and polygons with holes.

Diego Torquemada (talk) 16:48, 19 May 2019 (UTC)