Shear Shape Factor

[MagicFarmer]

Good morning,

I am currently working through the analysis of a series of laced/battened columns. I am using:
Guide to Stability Design Criteria for Metal Structures - Galambos
Behavior of Laced and Battened Structural Members - Lin et. al. 1970

Both the book and the paper utilize a Shear Shape Factor, n, in their calculations.

In Stephen Timoshenko's Theory of Elastic Stability, there is a brief paragraph in Section 2.17 that states the following:

"For a rectangular cross section the factor n = 1.2, and for a circular cross section n = 1.11. For an I beam bent about the minor axis of the cross section (that is, bent in the plane of the flanges) the factor n ~= 1.2A/Af, where Af is the area of the two flanges. This value lies within the range 1.4 to 2.8 for the usual I beam and plate girder sections. If an I beam bends in the plane of the web(about he major axis) the factor n~= A/Aw, where Aw is the area of the web. For this case, values of n from 2 to 6 are typical for rolled steel section."

I can not find any other description of the shear shape factor in any other text. Is this concept covered somewhere in a more extensive manner? What about channels? WHY is the shape factor 1.2 for a rectangular cross section? How is this parameter derived?

I have a lot of questions and not a lot of answers.

As always, any help would be greatly appreciated.

Thanks in advance,
MF

 

[JoshPlumSE]

I don't believe the numbers quoted there are quite correct. My reference on this subject was Stress, Strain, and Structural Matrices by Walter D. Pilkey. That reference gives formulas for deriving the shear shape factor. So, you might want to check out that reference.

Note that this is really saying not all the area of a member is effective in resisting shear deflection. Take an I beam for example. In strong axis bending, only the web resists shear deflection. That's why the n factor can be estimated as A/Aw.

 

[MagicFarmer]

@JoshPlum

Thank you for the reference, I am still seeing what I can dig up for Pilkey.

Do you have the section or page at hand?

I am seeing that
Buckling Strength of Metal Structures - Bleich
Steel Structures - McGuire
Both list n = 1.2 for rectangular sections and approximately 2 for w sections...

Thanks
MF

 

[JoshPlumSE]

I don't have Pilkey anymore. That reference was the property of my former employer.

Tonight I can check my notes on the subject (which are at home). Not sure I have a page number. But, I have various formulas that I got from Pilkey for different shear shape factors for stress and deflection. Some I had to derive myself based on the theory presented in Pilkey, others were copied directly from the book.

 

[prex]

You should find the derivation of the shear factor in a book on the theory of elasticity (but not Timoshenko's Theory of Elasticity, I don't think it is there).
The derivation of the shear factor is by calculating the work of deformation due to the shear strain in a bent beam and is not so simple to be reproduced here. For the circular and the elliptical sections the value normally used is 10/9.
For a channel the same approximate value A/Aw as for an I beam should be used.

prex
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[MagicFarmer]

Using the values given for w-sections, does it stand to reason that back-to-back angles would be approximately 1.2A/Af and toe-to-toe angles A/Aw?

I see that my local university library has a copy of Pilkey that I am going to try and snag tonight.

 

[BAretired]

I am not clear on the definition of Shear Shape Factor. Is it as defined below?

The "Shape Factor" of a cross-section of a beam is defined as the fully plastic strength divided by the force or moment causing first yield. Thus the Shape Factor for bending is Mp/My and similarly for shear, is Vp/Vy.

BA

 

[Hokie93]

Roark's Formulas for Stress & Strain has a decent discussion on the factor to be used for shear deflection. Roark's calls it an "F" factor. The book provides an equation for I-shaped beams and box beams with flanges and webs of uniform thickness. The topic is covered in the section "Beams of Relatively Great Depth" in the 6th edition.

 

[canwesteng]

I'm interpreting shape factor here as the ratio of peak shear stress to average shear stress, although with another factor to account for only the web to be effective in shear in open section. i.e. peak shear in a rectangle is 1.2 times average shear.

 

[BAretired]

I'm interpreting shape factor here as the ratio of peak shear stress to average shear stress, although with another factor to account for only the web to be effective in shear in open section. i.e. peak shear in a rectangle is 1.2 times average shear.

That is not as I remember it. Peak shear stress in a rectangular beam is 1.5 times average shear.

BA

 

[BAretired]

"Roark's Formulas for Stress and Strain" talks about shape factors but I found nothing about shear shape factors. The attached article by Tom Irvine discusses shape and form factors. It seems that the term "shear shape factor" is not defined consistently by everyone. Timoshenko and Gere in the attached Table 3 refer to "Shear Coefficient" and "Form Factor" for a rectangular section to be 1.5 and 1.2 respectively.

At the moment, I am not sure how the above form factor is to be used in calculating shear deformations.

BA

 

[JoshPlumSE]

I'm interpreting shape factor here as the ratio of peak shear stress to average shear stress, although with another factor to account for only the web to be effective in shear in open section. i.e. peak shear in a rectangle is 1.2 times average shear.

Well, there are two different shear shape factors. One for stress, one for deflection. Years ago (when I first started at RISA), the program used a single factor for both and it was user entered (SAy and SAz). Users used to get this confused with the section modulus and enter in values that were really high. Therefore, their shear stresses (and shear deflections) would be greatly over estimated.

Now, if they had done it properly, users they would have used 1.5 (as BARetired pointed out) for a solid rectangle. But, when we re-wrote the program to automatically account for these shear factors we realized that there were different values that applied for shear stress and shear deflection. That's when we bought the Pilkey book... which was very clear about the different values for stress and deflection.

 

[Hokie93]

The concept is covered in both the 5th and 6th editions of Roark's Formulas for Stress and Strain in the "Beams of Relatively Great Depth" section. The factors given in Roark's match up with the "Form Factors" in your attachment (e.g., 6/5 for a rectangle, 10/9 for a solid circular section, and 2 for a thin-walled hollow circular member).

I agree with BA that the maximum shear stress in a rectangular section is 1.5 times the average shear stress.

 

[MagicFarmer]

Thank you all so much for the information.

I have been comparing the values derived from the formula given in Roark's to the ones given in the excerpt I posted originally.
The formula in Roark's is given in the attachment for anyone reading this in the future, or whom may be interested now.

The approximate values are great for weak axis I-section bending, so-so for strong axis.

I am using these values in the pursuit of calculating effective K values for laced/battened channels and angles. The process is outlined in Guide to Stability Design Criteria for Metal Structures - Galambos. The shape factor appears in an equation used to derive the shear flexibility parameter for columns with battens. Again, for anyone else whom may encounter this in the future, in Galambos, Example 11.1, there is a combined section that is 2-W840x299 with W460x97 battens, oriented on their sides (y-y becomes x-x). In that example, when calculating F for the column (Galambos uses nu), the width of the flange (D2 in Roark's) is taken as the total flange width (2*B + spacing), and rx is taken as 2*rx for the W840x299.

Thank you all again.
MF

 

[MagicFarmer]

Wait a minute... I'm now looking at Galambos' assumed use of rx = 2*rx...

rx comp. for two side-by-side members be the same as rx for the single member.

A comp = 2*A
Ix comp = 2*Ix
rx=SQRT(2I/2A) = SQRT(I/A)...

I am beginning to believe that I have used this formula beyond it's function.

The 2-W840x299 are spaced 210 mm apart, flange tip-to-flange tip.
Timoshenko Predicts F = 2.45
Galambos Lists F = 1.6

I am going to try and acquire that copy of Pilkey, and/or try and work my way through the integral given in @BAretired's attachment.

- MF