beam shear deflection
beam shear deflection
(OP)
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...., 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(y) 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!
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...., 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(y) 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!





RE: beam shear deflection
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
RE: beam shear deflection
RE: beam shear deflection
PDF of the relevant section:
http://files.engineering.com/download.aspx?folder=...
RE: beam shear deflection
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
www.espcomposites.com
RE: beam shear deflection
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.
RE: beam shear deflection
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
www.espcomposites.com