beam shear deflection

[tomsing]

I'm looking at a significantly short beam, cantilevered, and wanted to run a hand calc to validate a FEM, so I figure I ought to include shear deformation in a deflection calc.

Blodgett, section 2.6, gives a cantilevered beam, concentrated load at the tip, deflection at the tip of aPL/GA, where a is a shape factor.  Blodgett says a is the ratio of max shear stress to average stress in the section.  For simplicity, call it a rectangular section, then tau_max / tau_avg = (VQ/It) / (V/A) = 3/2.  And, if you take Blodgett's eqn 5 and take it to the limit of a rectangular section, you get the same, 1.5.  So, Blodgett tells me that the shear deflection at the tip of the beam should be 3/2 PL/GA.

But that doesn't match my FEM.  I even got down to just a stick model, make it super simple.  And in doing that (I rarely work with bar elements), I notice that the transverse shear stiffness for unit length is KAG, so K here is like the inverse of Blodgett's a.  But K for a rectangular section is showing up as = 5/6 (as written out by the preprocessor, FEMAP), not 2/3.

So, not having done this in a while, I dug up

https://www.fpl.fs.fed.us/documnts/fplrn/fplrn210.pdf,

which gives a decent refresher on how to do Castigliano's theorem, and lo and behold, I can work my way to, shear deflection at the tip of a cantilevered beam is kPL/GA.  This looks the same as Blodgett, so I looked into how they get k.  They give k not as a ratio of max to average shear stress, but as an area integral, Q^2 A/(I^2 b^2) dA.  For a rectangular section, Q works out to b/2*(h^2/4 - y^2), and when you crank through the algebra and the integral and some more algebra, you come up with k = 6/5, which (when you invert it) matches what's in the FEM.

So, does Blodgett have an error? Am I missing something in how I'm applying his formula? Or is there an error in FEMAP?

Thanks for your help!

 

[Sparweb]

I've always used shape factors based on cross-section properties - which would agree with the FPL procedure.
If your book from Blodgett makes any mention of the formula "VQ/It" then consideration of flexural shear is being included, and you are specifically trying not to use that.

Check out the following books from Roark and Bruhn which you might find valuable for detailed stress analysis hand-calcs.
If you're keen on classic engineering texts, you can also look for ANC-18 which covers beam shape factors in more detail than any other source I've seen.
I haven't checked, but it's possible that all of these procedures trace their roots to either Timoshenko or Castigliano. Or both.

STF

 

[SWComposites]

You are mixing up shear stress and shear deflection K (shape) factors. Two different things.

 

[tomsing]

SWComposites, if I'm mixing them up, then so is Blodgett. He gives a cantilever beam deflection = PLα/AE_s, and α = τ_max/τ_avg.

PDF of the relevant section:

http://files.engineering.com/downlo...le=Blodgett_2.6_Shear_Deflection_of_Beams.pdf

 

[ESPcomposites]

For a section like an I-beam, and assuming the web is in a constant state of shear (or nearly so), the deflection due to shear is 1.0*PL/(AG). For a rectangular section it would not be reasonable to take the maximum shear stress of 1.5*avg, to account for the deflection, since this stress only occurs a single plane in the cross section (it is less than this everywhere else). The factor of 1.5 for shear deflection would only be possible if the vast majority of the cross section has a stress of 1.5*avg (but this is not the case).

So what gives? Have a look at the note in FIG. 5 (Beam sections for which Eq. 5 applies - an I-beam and another section with a web). There is also a comment "the following formulas are valid for several types of beams..." and FIG. 4 states for "build-up beams". A rectangular section does not meet these conditions. The equations are probably only appropriate (though I am not familiar with the formulas) for I-beams, Channels, box beams, etc. where the web is a state of shear that is relatively uniform (much like the figure on page 2.6-2).

Brian

http://www.espcomposites.com

 

[tomsing]

Brian, you're saying you expect that Blodgett's average stress in alpha = τ_max / τ_avg is basically the average of the stress in the web? I don't get that from Fig 4. He's got alpha = (Vay/It) / (V/A), where A is the total area of the section. So the denominator is an average shear stress over the whole section. And if you do alpha / A = ay/It = Q/It and work it out for the I beam, you do get precisely Q = 1/8 * (bd² - bd₁² + td₁²). That is, Blodgett's eqn 5 is precisely what you'd get if you took alpha = τ_max / τ_avg, with τ_avg calculated for the entire section.

In any case, if we run through the calc for a 4 in wide x 4 in tall x 0.1 web and flange beam, Blodgett's formula gives an alpha that is about 9% higher than the exact calculation. Is that a reasonable amount of error? It feels big to me, although my background is aerospace.

 

[ESPcomposites]

No, I am not saying anything in regard to the specific formulas (I am not familiar with those). I am saying that by simple deduction we can not expect the shear deflection coefficient to be 1.5 for a rectangular section (it is 1.0 for a section with a theoretically constant shear stress). In addition, Blodgett states that you should only use the formulas for built-up sections, presumably with a flange on each side (I-beam, channel, box, etc.). I can't say if the formulas are correct, but either way, the formulas do not appear to be applicable to a rectangular section. That would address the relatively large discrepancy you stated in the original post. But I don't know if the they are accurate when the appropriate beam section is used.

I don't know the actual theoretical solution, but I would be surprised if it was too much different than 1.0 for typical sections with a flange on each side (where the area in PL/(AG) is based on just the web's area). If a rectangular section is 1.2 (parabolic shear stress distribution along the entire section) and a section with constant shear stress is 1.0, then a deep beam with flanges (much closer to constant through the section) should be not be too far from 1.0.

Brian

http://www.espcomposites.com