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A couple of comments from Nk[edit]

A couple of comments:

  • The stress maxima are opposite but not allways equal (non-symmetrical sections)
  • I don't understand the last sentence: "Because of this area with no stress and the adjacent areas with low stress, bending is not a particularly efficient..." From my point of view that is simply the reason to use I-profiles, truss girders, etc.

--Nk 15:26, 9 Dec 2004 (UTC)


  • The phrasing in question did indeed assume a symmetric section. I changed it to make it more universally relevant.
  • I'm not entirely fond of the current phrasing, but I haven't yet come up with a better way to address the inefficiency of bending (as compared to tension, which uniformly and almost fully--some would say optimally--uses the material at its disposal). As a stop-gap, I mentioned the rationale behind the use of wide-flange beams and truss girders.

--Spindustrious 20:58, 9 Dec 2004 (UTC)

It may be worth mentioning the beams produced by cutting the web of an RSJ in a toothed pattern, offseting and re welding along the neutral axis, leaving (usually) hexagonal gaps down the centre of the web. These show rather graphically how little the web contributes to the strength of the beam, and emphasising the lack of stress along the welded join where the "teeth" are rejoined. I'll look for a good picture.


Categorizing Bending[edit]

I maintain that Continuum Mechanics is the best category for this article in light of the recommendations in Wikipedia:Categorization:

  • Since the article only peripherally addresses specific subjects within engineering, any such categorization fails this test: "If you go to the article from the category, will it be obvious why it's there? Is the category subject prominently discussed in the article?"
  • "An article should not be in both a category and its subcategory."
  • "An article will often be in several categories. Restraint should be used, however — categories become less effective the more there are on a given article."

Many fields in engineering (including mechanical and structural engineering) draw from the fundamentals of continuum and fluid mechanics. I think it's a stretch to say that mechanical or structural engineering isn't, in this case, a subcategory of continuum mechanics. Given this doubt, it's appropriate to use restraint as called for in the style manual.

--Spindustrious 15:26, 13 Mar 2005 (UTC)

I do agree that this article isn't appropriate for the Mechanical engineering category, however the rationale is tenuous. The Mechanical and Structural engineering categories are currently not subcategories of Continuum Mechanics, nor am I sure they should be. If so, should they be added as subcategories? Or, perhaps a category for Engineering mechanics should be created instead, that seems more appropriate to me; also Engineering mechanics is currently a stub which could use some expansion anyway. Comments? Commander 16:09, Mar 13, 2005 (UTC)
I too am more comfortable with the category of Engineering Mechanics, but the more I think about it, the less confident I am of the distinction between Continuum and Engineering Mechanics. The preferred term here seems to be the former as evidenced by the strong article and amply populated category... More opinions please! --Spindustrious 16:54, 13 Mar 2005 (UTC)
To me, Continuum Mechanics seems for the most part a pure science field, whereas Engineering Mechanics is more closely related to engineering disciplines. Maybe it's because I had to take Engineering Mechanics as a part of an EE curriculum (the curriculum of most engineering fields require at least a class in statics, whereas mechanical engineers and some others have to take more classes on dynamics). I therefore think that both categories (Continuum Mechanics and Engineering Mechanics) would be appropriate. Commander 17:27, Mar 13, 2005 (UTC)
I agree and that's what I did for the Bulgarian translation - a new Engineering mechanics category. In de: they also have a nice de:Kategorie:Technische Mechanik. --Nk 10:08, 26 Mar 2005 (UTC)

Figure 2 indicates shear stress in the top figure and no shear stress in the bottom figure. It has to be in both or neither, correct? I believe it should be in neither. Rtdrury 03:53, 5 December 2005 (UTC)

I think the shear stress has been left out of the bottom figure because it is constant throughout the cross-section. It's also hard to show shear stress and normal stress in the same diagram without confusing people. Bending moment is the integral of the shear force, so it is important although it isn't discussed in the article yet. Toiyabe 19:24, 5 December 2005 (UTC)
While Toiyabe's interpretation is spot on, I've amended Figure 2's caption to make it clear that the shear stress has intentionally been left out of the diagram. --Spindustrious 01:57, 7 December 2005 (UTC)

Can you guys tell me the Mcaulays method of solving the curvature and deflection equation ? and does the method fall under the scope of this article ? Poticecream (talk) —Preceding undated comment added 03:36, 10 May 2011 (UTC).


Assuming it only undergoing elastic bending, what is the general shape of a bending beam? Arc, Bezier curve, catenary, parabola, or none of the above? I don't know, but those seem four likely candidates... I don't think it's an arc, and catenary seems the most likely. Timeroot (talk) 14:46, 2 July 2009 (UTC)

It depends on how the ends are constrained. That's all I would's a good question. Wizard191 (talk) 16:27, 2 July 2009 (UTC)
Catenary works if the load is evenly distributed across a beam that is fixed at both ends (I think), though you would only see it in items with very low flexural rigidity (i.e., rope). I've never spent much time thinking about what the shape was called for different solutions. One commonly thought-of shape, though, is of a beam (like a simplified diving board) that is pinned at one end and loaded by some mass at the other. There's a fourth-order differential equation commonly known as the plate equation, which actually isn't on this page (I should add it) that can be used to describe any flexural situation; it might be useful to solve it for a variety of situations that you think may give the characteristic shapes that you're looking for. Awickert (talk) 16:55, 2 July 2009 (UTC)
I was considering a beam, slightly bent, being compressed from either side, where the ends can rotate, but not move. I guess you could say this is a beam, with both ends fixed, with a force being applied from each side. Specifically, I had just been wondering what shape beams take when they buckle, if that's of any help. And thanks for the fast replies... Timeroot (talk) 16:14, 6 July 2009 (UTC)

What you are describing is a column rather than a beam (or at least from an analysis point of view) and the shape is sinusoidal. For a beam in bending due to a lateral load rather than the axial load, which you described, can bend in several shapes depending on the load type, for example a point load produces a quadratic deflection curve and a uniformly distributed load produces a cubic deflection curve. (talk) 22:16, 29 July 2009 (UTC)

Compression and tension relative to direction of bending axis[edit]

In the section on Euler-Bernoulli, it says:

Compressive and tensile forces develop in the direction of the beam axis under bending loads. These forces induce stresses on the beam. The maximum compressive stress is found at the uppermost edge of the beam while the maximum tensile stress is located at the lower edge of the beam.

Unfortunately, the direction of bending is not mentioned. Which part of the beam is considered "up"? This is confusing for me, and perhaps I have misunderstood it, but intuitively, assuming x the beam axis and y the bending axis, I would expect tensile stress to be on the side above the center of the beam (y+) and compressive below the center of the beam (y-). This image ( and this other article (Area_moment_of_inertia) seem to suggest the same. Maybe we should change the wording to specify the direction of bending? Unfortunately I am still learning, so I wouldn't dare -- I need to understand it first :) --Rhombus (talk) 09:53, 24 October 2010 (UTC)


In the section "Timoshenko bending theory", the rotation of the normal is calculated: [[1]] Is this correct? shoudn`t it be dw / dx insted of d2w / dx2 ? — Preceding unsigned comment added by Etilev (talkcontribs) 12:46, 2 November 2011 (UTC)

Links to Beam Bending Stress/Strain Calculators[edit]

In the external links section there is currently a link to a page "" with a few tools for calculating beam stresses with limited functionality. Further to this I would like to link a shareware tool which can do this for any arbitrary beam cross-section assembly with any combination of materials and plot the stress/strain contours across the entire section for any applied bending. I think this would be useful for anyone coming to this wikipage looking to increase their understanding of basic engineering beam theory, especially college engineering students whom I had in mind when I wrote this tool. I have had this link removed already and I am trying to understand exactly what the issue with this type of external link is, because I see similar links at many pages related to engineering topics like this. — Preceding unsigned comment added by (talk) 19:42, 9 October 2013 (UTC)

Here is a link to the tool in question:;1 (talk) 19:49, 9 October 2013 (UTC)

Hi, WP:OTHERCRAPEXISTS is not a valid argument to make. There are many messed up links and articles in WP, but that doesn't give a license to add more. As mentioned by other editors to you earlier, as per WP:ELNO links that are not really recognized by notable people are discouraged. A m i t  웃   20:18, 9 October 2013 (UTC)
See also WP:SPS. Apuldram (talk) 22:52, 9 October 2013 (UTC)

Ok thanks, as an author of peer-reviewed publications in international journals on applied mechanics I understand the notability argument.

So after some research I found that, per WP:ACADEMIC, the first criteria for academic notability as it pertains to Wikipedia is “1. The person's research has made significant impact in their scholarly discipline, broadly construed, as demonstrated by independent reliable sources.” In 2001 I published one of the first papers of a finite-element based implementation of the meso-scale damage mechanics model for the prediction of delamination propagation in composites. This paper has since been cited by composite mechanics researchers across the world and I was awarded the Louis T Rader Civil Engineering award for it. Feel free to search “Damage development in composites with large stress gradients” at Google scholar.

So do I think that this achievement qualifies me for a paragraph dedicated to me on the damage mechanics or composite materials wikipages, no I don’t, but do I think this is enough to satisfy the academic notability criteria and justify linking a novel tool for helping interested students and engineers learn more about beam bending theory (the topic of this wikipage), yes85.171.171.189 (talk) 18:32, 10 October 2013 (UTC)

No. Wikipedia is not the place to publish your own work. A shareware distributor doesn't count as an independent publisher. Congratulations on your award. However it doesn't alter the situation here. Apuldram (talk) 09:20, 11 October 2013 (UTC)

Continuum mechanics[edit]

I have removed the image continuum mechanics from the article as it isn't appropriate. It's not an infobox, but an illustration created to illustrate Bernoulli's Law for the article on Bernoulli's principle. It isn't relevant here. The illustration has nothing to do with bending. The sections of the image are misleading: Are there only four laws in continuum mechanics? Have there only been eleven scientists involved?
Instead, I have added a link to Continuum mechanics in the See also section. Apuldram (talk) 10:41, 28 February 2014 (UTC)

Um, no it's a navbox that is collapsed. If you click "show" under "Solid Mechanics" Bending is one of the topics in Continuum mechanics. I'm going to re-add it. I'll see if there's an option to have the relevant section automatically expanded so people don't think it's an image. These sorts of navboxes are very common on wikipedia, I'm surprised you haven't seen them before. Every other entry in the Continuum Mechanics navbox has one on there. 0x0077BE [talk/contrib] 16:42, 2 March 2014 (UTC)
I'm not actually sure how to get it auto-expanded, I'll look into it in more detail later. If you want more examples of how these navboxes (not infoboxes) are extremely widespread, see the "what links here" link on {{sidebar with collapsible lists}}. Generally they have a small illustrative image of the overall concept, which is why the Bernoulli's principle image is there. 0x0077BE [talk/contrib] 16:50, 2 March 2014 (UTC)

My criticism was not of the inclusion of a navbox here. It was that Continuum mechanics is not an appropriate navbox for this article, for the reasons I've already given. The problem is that continuum mechanics is too all-embracing and, by covering everything, it succeeds in telling us nothing. A more specific navbox is needed. I am replacing it with the Classical mechanics box. That shows Continuum mechanics as a branch, so the link is not lost. A future development might be the creation of a box for Structural engineering or for Statics. Bending would fall naturally into either. Apuldram (talk) 12:00, 9 March 2014 (UTC)

It definitely is, because Bending is one of the topics in the navbox. Every single other article that's in the navbox has the navbox on there. That's how navboxes work. I'm not trying to get a link to Continuum Mechanics prominently displayed, it's not trying to "tell us something" (not to mention Classical Mechanics is a broader term than Continuum Mechanics), I'm trying to put the convenient navbox there which makes for easy navigation between continuum mechanics topics. If you have an issue with the continuum mechanics template in general, go take it up on the talk page there. 0x0077BE [talk/contrib] 17:55, 9 March 2014 (UTC)
Bending is definitely a topic in solid mechanics and solid mechanics is definitely a topic in continuum mechanics, so the navbox has a reasonable hierarchy. Continuum mechanics as an organizing principle also seems reasonable. For instance, both fluids and solids have stress-strain tensors defined under the continuum assumption. All the equations of this article are based on calculus and ultimately the continuity and piecwise differentiability of the properties of the solid under investigation. I agree that if one wants to change how the continuum mechanics topics are arranged into navboxes (and the associated categories, too) it is best to try this on the navbox talk page. Given the navbox as it stands, it is perfectly reasonable to include it in this article. --Mark viking (talk) 02:51, 10 March 2014 (UTC)
OK Apuldram (talk) 09:45, 10 March 2014 (UTC)
Alternatively you could add the list at the bottom of the page (as I have done for this page). Bbanerje (talk) 21:09, 10 March 2014 (UTC)
Either one works for me. 0x0077BE [talk/contrib] 22:27, 10 March 2014 (UTC)
I prefer the alternative suggested by Bbanerje, because I feel the image showing the derivation of Bernoulli's Law is not in itself relevant to bending and so is confusing. I'll wait a week to see if there are any other comments. Apuldram (talk) 10:30, 12 March 2014 (UTC)
I think the image thing really shouldn't be a problem - I've never heard of people confused by this sort of thing before, and these inline navboxes usually have some sort of image that is illustrative of the general concept, even if that's not a relevant image to the article in question. I think the reason it is very unlikely to be confusing is that if you see some image and you don't understand why it's there, you'll go to check the caption only to find that it's a navbox and you'll realize that it's an icon, not an image designed to illustrate something.
That said, on Wikiproject Physics, RockMagnetist suggested that the concept is too broad for sidebar nav, which also seems to be follow your intuition Apuldram. I'm not intimately familiar with the details of continuum mechanics and I haven't been involved in sidebar nav creation, so I think it's probably a good idea to make a decision about whether the sidebar nav is relevant writ large, and if not, switch it over to bottom-of-the-page nav for the other continuum mechanics topics as well (or at least all the ones that are formatted like Bending. RockMagnetist suggests that the template page over there doesn't get enough attention for that kind of discussion, which is a solid point, so I think we should formulate an RfC and post notices at WikiProject Physics and WikiProject Engineering, as well as on the talk pages of most if not all of the affected pages. I'll formulate one tonight and post it there.
Certainly we can make a decision about the navbox on this page independent of the result of that discussion (unless it results in the deletion of the sidebar, I suppose), but I think it's preferable if the navboxes are used consistently. Often times I'm interested in navigating the space of a navbox like this, and so I'll click one article, then when I'm done with it I'll locate the navbox and click the next one. If half the articles have sidebar boxes and half have bottom-of-the-page boxes, I find it breaks the flow of the navigation (because I have to find the navbox in the new location and recognize that it's the same navbox that I've been exploring and that I'm not moving on to a new topic). So that's my argument for being consistent with the other pages in whatever we choose. 0x0077BE [talk/contrib] 15:09, 12 March 2014 (UTC)
I have just noticed that there is another navbox, {{Topics in continuum mechanics}}, at the bottom of this page already. The two navboxes largely duplicate each other, which is grounds for a TFD (see Reasons to delete a template). If someone were to create a sidebar for this page, an appropriate topic might be deformation. RockMagnetist (talk) 15:45, 12 March 2014 (UTC)
I have proposed a merger (see Wikipedia:Templates for discussion/Log/2014 March 12). RockMagnetist (talk) 17:47, 12 March 2014 (UTC)


I removed the word "isotropic" from the discussion of the Euler equation because there is no need for the beam to be fully isotropic. The Euler equation works just fine for orthotropic beams as it does for isotropic beams. I added the word "straight" because the Euler equation does in fact rely on a beam that is originally straight (or only slightly curved). As a fix, I'll change "iso" to "ortho", and add "straight." Hermanoere (talk) 17:37, 1 July 2015 (UTC)

Addition of 'and cantilever'[edit]

I reverted the edits by Mandloiabhishek554, who inserted multiple additions of and cantilever, showing lack of understanding that a cantilever is a type of beam. Among other errors Mandloiabhishek554 wrote "In a horizontal cantilevered beam supported at the ends", showing ignorance that a cantilever is not supported at the ends.

My reversion was undone without explanation by, who has already received three recent warnings for vandalism.

I have restored the position before the edit by Mandloiabhishek554. Apuldram (talk) 17:59, 22 October 2015 (UTC)