Beam bending - calculating effects of shear deformation 4

[electricpete]

If we are given a set of loadings w(x) on a beam, we can integrate 4 times (with a few constants) to get V(x), M(x), Slope(x), Yb(x).

This is the displacement due to bending... dependent upon E, and assumes G infinite (no shear deflection). The error is small for anything other than very short fat beam.

If we now want to calculate the deflection due to shear alone, couldn't we analyse a beam with infinite E and finite G. It seems that deflection would be:
Ys(x) = Integral [Gamma dx]
where Gamma = shear strain = Tau/G = V*A/G
Ys(x) = Integral[V(x) * A dx] / G
I think this requires an assumption that shear stress is uniformly distributed... seems reasonable.

Then we could find Y(x) = Yb(x) + Ys(x).
Yb = displacement from typical bending cal
Ys = displacement from shear calc

Would this be exactly correct? Or close to correct? Or totally wrong?
Thanks in advance.


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[unclesyd]

Here is a paper from several that I looked up for friend sometime ago. He was working on a vibration problem at the time.

Shear correction factor equation on page 12.

http://soliton.ae.gatech.edu/people/dhodges/papers/sheartheory.pdf

 

[TERIO]

I took another look at the link I originally posted. The same fixed end BC is pointed out there on page 148.

 

[prex]

electricepete, sure, you can always superpose a pure shear deformation to the pure bending deformation and get a total deformation.
The point is that you can't calculate the shear deformation independently of the bending one: you need the bending solution first or you can calculate both in a single step.
This is formally proved by those two 'uncoupled ' equations in the wiki article, where the equation containing shear contains also bending, but the first equation contains bending only, and can be solved alone. Can't provide at the moment a non formal, logically based, explanation of this fact.
Another way of looking at it: when we use the method of strain energy minimization, we can minimize the energy associated with bending alone and get the pure bending solution, but we can't minimize the energy of shear alone, only the sum of bending and shear energies, otherwise we get a wrong solution or no solution at all.
The only explanation that I can offer of this (but I'm not fully satisfied of it) is that shear is intrinsically associated to bending: the shear stress parallel to the section has its physically required counterpart in the longitudinal shear stress, that's an indispensable element of bending behavior, as it is originated by the varying longitudinal direct stress on section height.

prex

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[desertfox]

hi electricpete

here is another article dealing with shear deformation:-

http://docs.google.com/viewer?a=v&q=cache:qLJnOVRZceEJ:

http://www.bth.se/tek/amt/mtd009.ns...8eRtn7&sig=AHIEtbS3o1Sc0gT5KfWj9WDdvYlZrKew3Q

 

[unclesyd]

http://www.bth.se/tek/amt/mtd009.ns...0CE00529A64/$FILE/przemieniecki_p_201_208.pdf