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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.
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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.
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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?
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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:
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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.
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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
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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.
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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?
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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?
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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.
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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!
1648:- 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
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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
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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:
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465:'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
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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.
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Something should be said about the product moment of area in this section. If I remember correctly, the formula used for that transformation is:
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Spanish page refers the same
Spanish book ... But I could not trust the formulae against the English one using symbolic computation (Sympy):
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1000:{\displaystyle \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}
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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:
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I have looked at the recent edits and can see major improvements, that result in a clarity of understanding. Keep up the Good Work!
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This article is still a bit incomplete, but at least now it doesn't include blatant errors and misconceptions about its subject. --
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mathematics and principally the useage in engineering. I'd propose that the big warning at the top of the page be removed.
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A suitably dimensioned and annotated I-beam diagram would greatly enhance the usefulness and applicability of this forula.
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pulls from the list. I only kept (and reorganized) examples that exploit some calculation of the second moment of area.
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The cross sectional areas perpendicular to an axis, such that with a thickness, the volume of the body is accounted for?
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I have read through the formulae and confirmed that all are correct (up to the polygons -- I didn't check that one.)
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http://www.dynamore.de/documents/papers/euro4/implicit-new-developments/ls-dyna-beam-elements-default-and-user-defined
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The definition lacks every hint of context. It is understandable only for those who already know what it means.
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1606:. I think someone who edited it after me got confused about what I was calculating and changed some stuff to
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examples must be exclusively about an area's geometric properties, because that's what any moment of area does
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need to know about the
Kronecker delta before you can form a working understanding of second moment of area.
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If your work removed the blatant errors, etc, could you change the cleanup tag to reflect its current state?
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This material on the "product moment of area" was int eh article but seems to lack context and explanation:
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Deutsch page has any reference and show a - sign for Ixy, which means there are wrong formulae on the web.
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Also, there are some explanations on how to deal with disconnected polygons and polygons with holes.
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the article. I am cleaning the section up some but it needs to be considered for deletion anyway.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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edit. My apologies for the confusion. I believe the article looks good. Thank you for the edit!
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as an example of rectangular cross section beam. This was what made me understand the concept.
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This page is not about second moment of area any more, it is about bending stress and strain.
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Diagrams look good- however, they should have the coordinate system axis for completeness.
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This article needs some diagrams to go with the formulas, to make the variables clear.
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I have found the math here too sloppy to be understandable. I suggest to put (7) from
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Why is that? It says right on this talk page that this is a Civil
Engineering topic!
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if I recall correctly. I just wanted to justify why I switched it back. --
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The definition does not state what areas are to be summed in the integral.
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Some body needs to translate and lift the diagrams from the De page on this
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1090:{\displaystyle \sigma =-{\frac {M_{y}}{I_{y}}}x+{\frac {M_{x}}{I_{x}}}y}
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if it mentioned any form of stress then it has no place in this article
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I will program these equations and verify them against known results.
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if it mentioned any form of beam theory then it doesn't belong here.
501:. Specific illustrations, plots or diagrams can be requested at the
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E1 = sympify('(yi**2 + yi*yi1 + yi1**2)*(xi * yi1 - xi1 * yi)')
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I had several problems with the changes made by 69.251.146.46:
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The lead sentence shouldn't be about what it is confused with.
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infinitesimal area elements multiplied by the lever squared.
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This article needs a complete rewrite to stay meaningful. --
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this is where it belongs. And why does this have a redlink?
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https://apps.dtic.mil/dtic/tr/fulltext/u2/a183444.pdf
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1679:Ah I see. Thanks for the clarification! :) --
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1655:— Preceding
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1496:— Preceding
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1482:Major Rework
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283:— Preceding
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40:WikiProjects
1736:annahoward
1661:BeaumontTaz
1502:BeaumontTaz
1395:Readership?
1332:—Preceding
593:—Preceding
503:Graphic Lab
310:y=(h+h1)/4
289:90.43.11.93
121:CE articles
30:Start-class
1778:Categories
1726:Annahoward
1700:DPHutchins
1133:of a body?
1103:Hemmingsen
1439:Mecanismo
1324:Intuition
1231:Mecanismo
426:reverting
1669:contribs
1657:unsigned
1510:contribs
1498:unsigned
1346:contribs
1334:unsigned
1214:Kimaaron
1188:Cacadril
1148:Cacadril
595:unsigned
565:Mollynet
495:included
491:diagrams
408:unsigned
285:unsigned
1743:Polygon
1713:Updated
1521:Annulus
1317:☆ · 星
1131:surface
449:Nephron
263:Polygon
244:on the
217:Physics
208:Physics
164:Physics
139:on the
36:scale.
1402:don't
1338:G4oep
772:RJFJR
556:RJFJR
531:RJFJR
1766:talk
1730:talk
1704:talk
1685:talk
1665:talk
1650:this
1639:talk
1506:talk
1473:talk
1458:talk
1443:Talk
1410:talk
1386:talk
1363:talk
1342:talk
1235:Talk
1218:talk
1192:talk
1168:talk
1152:talk
603:talk
569:talk
416:talk
293:talk
131:High
1579:or
1313:--
493:be
489:or
236:Mid
1780::
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1667:•
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1344:•
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1028:−
1022:σ
971:−
870:−
796:−
790:σ
639:∑
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1058:+
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1038:y
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1025:=
995:y
987:2
982:y
979:x
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840:x
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830:x
826:M
822:+
817:x
813:I
807:y
803:M
793:=
755:e
752:r
749:e
746:h
743:a
740:l
737:u
734:m
731:r
728:o
725:f
722:t
719:r
716:e
713:s
710:n
707:I
679:y
676:x
672:I
648:A
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636:=
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628:x
624:I
601:(
567:(
457:C
455:|
453:T
414:(
378:m
375:d
370:2
366:y
359:=
354:x
350:I
291:(
248:.
143:.
42::
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