# Talk:Section modulus

## Hollow Section

Shouldn't hollow solid circle be $S = \cfrac{\pi (d_o^3 - d_i^3)}{32}$ and hollow rectangle be $S = \cfrac{bd^2 - b_1d_1^2}{6}$? - 77.44.16.71 (talk) 14:28, 8 September 2010 (UTC)

Because S = I/c you first find I for the entire section and then divide by the maximum c. Bbanerje (talk) 23:25, 8 September 2010 (UTC)

## Definition

The very definition of the section modulus as given in this article is wrong. It is not necessarily at the extreme compression fibre, it could be at the level of any fibre. Please make the change. Adityarn (talk) 13:47, 31 January 2010 (UTC)

I agree (though generally it is calculated on the extreme tensile or compressive fibres in a bending beam), but generally compression is the most critical case due to onset of flexural torsional buckling. Tensile extreme fibres usually have a higher allowable stress or capacity. In the case of T-sections if you have tensile fibres at the bottom of the T they may still be more critical than the compressive fibres at the top due to a generally much larger distance from the neutral axis so despite having a higher allowable the elastic section modulus is also lower (though F/T buckling is also effected by beam length and restraint). There may also be a number of different cases that might be worth mentioning in the article, such as there being different values for orthogonal and principal axes and in the case of unequal angles in the principal axes there is a section modulus for each corner. Engineers are generally concerned with the lowest elastic section modulus (to be conservative) although if the loading on a beam is well understood you can take advantage of different section modulus for tension and compression to get a bit more out of the design. Perhaps a bit of detail regarding how section modulus is used in structural engineering might help clear things up. 203.129.23.146 (talk) 01:02, 6 July 2013 (UTC)

## S or Z

Machinery's Handbook uses Z for elastic section modulus. That seems to go against the article's assertion that S is used. Both letters are used in different US texts and I haven't been able to discern a standard usage between the various books at my disposal. --207.70.169.36 (talk) 18:18, 18 June 2009 (UTC)

Z is used for elastic modulus in UK design codes and S for plastic modulus, although I recently read and article that used Zp for plastic modulus.

Eurocode 3 (steel) uses Wpl for plastic modulus and Wel for elastic modulus. —Preceding unsigned comment added by 77.89.161.122 (talk) 09:41, 31 March 2011 (UTC)

The Australian Standard for Steel Structures (AS4100) uses S for plastic and Z for elastic — Preceding unsigned comment added by 130.95.131.64 (talk) 05:36, 8 June 2011 (UTC)

## The table could use a few more figures

Anybody want to take on creating figures for the remaining entries in the table? I don't have software that will generate a vector format image. Khakiandmauve (talk) 19:35, 10 May 2011 (UTC)

Will these work: File:Area moment of inertia of a I-beam.svg, File:Area moment of inertia of a square2.svg, File:Area moment of inertia of a rectangle2.svg, File:Area moment of inertia of a channel.svg? Wizard191 (talk) 20:17, 10 May 2011 (UTC)
They're mostly usable. Since we're using c as the distance to the farthest fiber, I'd like to remove c from the center on the figures. The diamond could be more general, as a rhombus instead of a square. And we'll have to recast the equations in the table to match the labeled dimensions. Care to check my math on the section moduli?
For the I-beam, strong axis, $I = \cfrac{BH^3}{12} - \cfrac{bh^3}{12}$ and $S = \cfrac{BH^2}{6} - \cfrac{bh^3}{6H}$. Same equations for the hollow rectangle. For the channel, the existing expression doesn't even have the right units. From your figure, I get the same expressions. Given that they're all the same expressions, and that the reasons for that are somewhat instructive, I'd suggest reordering the table to group these rows, and merge the cell where the expression for section modulus is given.
I'm not real sure where the other expression for the I-beam comes from. Maybe weak axis? If so, we'd need a different figure showing the different neutral axis, but I don't get anything like that equation for the weak axis modulus. With the new figure, $I = \cfrac{B^3(H-h)}{12} + \cfrac{(B-b)^3h}{12}$ and $S = \cfrac{B^2(H-h)}{6} + \cfrac{(B-b)^3h}{6B}$.
For the diamond, again, I don't see where the current equation came from. With a rhombus, with b = the point to point width and h = the point to point height, about the horizontal neutral axis I get $I = \cfrac{bh^3}{48}$ and $S = \cfrac{bh^2}{24}$. For a square of side H as shown in your figure, I get $I = \cfrac{H^4}{12}$ and $S = \cfrac{\sqrt{2}H^3}{12}$.Khakiandmauve (talk) 16:43, 12 May 2011 (UTC)
I'm not sure how you got your equations for the I beams. It doesn't seem that simple to me. Here's a website with two different equations that work for an I beam in the strong direction: http://www.eng-tips.com/viewthread.cfm?qid=238050&page=2. I can rework the images any way you like to make them more applicable, just let me know what you need. Wizard191 (talk) 17:46, 12 May 2011 (UTC)
Superpostion. For the strong axis, a solid rectangle B x H centered on the neutral axis is BH^3 / 12, then subtract off the "negative rectangles" on either side of the web, with dimensions b/2 x h and centered on the neutral axis; the moment contribution is -2 x bh^3/24. So, I = BH^3/12 - bh^3/12. With c = H/2, the section modulus is S = I/c = BH^2/6 - bh^3/(6H). For the weak axis, it's just three rectangles superposed. I = 2 * [(H-h)/2]B^3/12 + h(B-b)^3/12 = (H-h)B^3/12 + h(B-b)^3/12. c = B/2, and S = I/c = (H-h)B^2/6 + h(B-b)^3/(6B). Note that these aren't the most natural ways to define an I section, but they do result in relatively simple expressions for I and S. If you convert to the dimensions labeled on the eng-tips thread, they should would out the same.
As for mods to the images, can you stretch the diamond vertically, and instead of labeling the side length H, label the corner to corner width B and the height H? Then, on all of them, remove the C marking the center. Also, for the weak axis I section, rotate the neutral axis line 90° and change the label to a y. (It would be nice to redo the other figures in the same style, but I don't want to take up too much of your time!) Khakiandmauve (talk) 18:33, 12 May 2011 (UTC)
I know it's taken a little while, but here's what I've got done so far: File:Section modulus-rectangular tube.svg, File:Secion modulus-diamond.svg, File:Section modulus-I-beam-weak axis.svg, File:Section modulus-C-channel.svg. Wizard191 (talk) 18:46, 19 May 2011 (UTC)
No problem, thanks for putting in the work! Two requests - on the diamond section, can you rotate the H so it's in the vertical orientation, for consistency with the others? And we need a way to make a clearer distinction between the neutral axis arrow and the dimension arrows. Maybe use dashed lines for the dimensions, similar to the existing figures in the article?Khakiandmauve (talk) 14:03, 20 May 2011 (UTC)
OK, can do. Wizard191 (talk) 19:10, 20 May 2011 (UTC)
I updated these: File:Section modulus-I-beam-weak axis.svg, File:Section modulus-C-channel.svg, and created this one: File:Section modulus-I-beam-strong axis.svg. Hopefully tomorrow I can get to the other two. Wizard191 (talk) 18:08, 25 May 2011 (UTC)
Thanks! I'll go ahead and add them to the table, but I'm going to try to rename them. The strong axis of the I section should be horizontal, and the weak axis vertical - those are reversed. Khakiandmauve (talk) 20:40, 25 May 2011 (UTC)
Actually, that appears to be beyond my Wikipedia abilities. I'll leave it the way it is, and put them in the article. Next time you get a chance, can you switch the pictures, and change the table? Khakiandmauve (talk) 20:55, 25 May 2011 (UTC)
The arrow that define strong/weak axis are actually confusing to me because I think of them as a force, but you are thinking of them as the axis about which the cross-section is being loaded. Is there any way we could more clearly distinguish these? Wizard191 (talk) 16:06, 26 May 2011 (UTC)
Good call - the original figures used the solid line with arrowhead to define the bending axis rather than the direction of force (although it only makes a difference for the rectangular section). Maybe drop the arrowhead and label the line NA for neutral axis, and we'll note that in the text above the table. Oh, and for some reason, the current version of the File:Section_modulus-I-beam-weak_axis.svg isn't showing up - in the article, I'm seeing the old one with sideways H. Same on the image page, although the File History table shows the correct current version thumbnail, and when I click on that, I get the corrected figure. Any idea why? I've cleared my cache.... Khakiandmauve (talk) 20:13, 26 May 2011 (UTC)

────────────────────────────────────────────────────────────────────────────────────────────────────I like the "NA" idea. I'll try and get those images updated on Tuesday. As for the lagging image, that's probably because of issues they've been having at the commons recently. I'm sure by the time you are reading this, the image will look right to you. Wizard191 (talk) 22:10, 29 May 2011 (UTC)

Fixed: File:Section modulus-I-beam-weak axis.svg, File:Section modulus-I-beam-strong axis.svg, File:Secion modulus-diamond.svg, File:Section modulus-rectangular tube.svg, File:Section modulus-C-channel.svg. Wizard191 (talk) 17:43, 1 June 2011 (UTC)
Great work! I've updated the table. Khakiandmauve (talk) 13:46, 2 June 2011 (UTC)

## Limit states

Both elastic and plastic modulii are used in the limit states design method per AS 4100-1998. Even if just a link is added to Limit state design to start with might be useful. — Preceding unsigned comment added by 203.129.23.146 (talk) 02:07, 27 August 2011 (UTC)

In the article, the following is said: "For general design, the elastic section modulus is used" and "The Plastic section modulus is used for materials where (irreversible) plastic behaviour is dominant. The majority of designs do not intentionally encounter this behaviour." Are these sentences correct? In limit states design, Mp = ZFy is often used for the design of steel beams per CSA S16-09, and My = SFy is used when the section class is non-compact/Class 3/etc. LM 04:55, 30 November 2011 (UTC) — Preceding unsigned comment added by LanaMohinder (talkcontribs)

Limit states design can be based on elastic or plastic analysis (both are covered in AS 4100-1998, with elastic analysis using moment amplification factors), but it differs from older allowable stress design by factoring loads up rather than material strengths down (although there are general capacity factors in AS 4100 that reduce capacities arbitrarily to account for material strength uncertainties). Factoring loads generally allows designing more close to the bone by applying different safety factors for different degrees of certainty for various load cases (permanent loads such as structural self-weight are more certain and predictable and aren't factored up as much as uncertain and unpredictable imposed loads such as traffic). The capacities in limit states design have much less safety than allowable stresses, but margin of safety is maintained by the use load factors. Usually a structure designed using limit states would not be designed to fail plastically, even though the plastic modulus is used in the calculation of capacities. Section capacity (Ms in AS 4100) is equal to Ze x fy, where Ze is effective section modulus, which depending on section slenderness can be a function of elastic or plastic section modulus or both. For compact sections, Ze = MIN(S,1.5Z), non-compact Ze is a function of Z and compact Ze, and slender Ze is a function of Z. It's a pity that AS 4100 is not freely available as a source for Wikipedia. Another good source for limit states design to AS 4100 is Steel designers' handbook http://steel.org.au/bookshop/steel-designers-handbook/ 203.129.23.146 (talk) 12:24, 5 July 2013 (UTC)

## javascript error

Starting with IE 11, AttachEvent is not longer supported and has to be converted to switched over to AddListenerEvent. — Preceding unsigned comment added by 67.165.114.13 (talk) 13:30, 2 December 2013 (UTC)