Simple cantilever beam
Simple cantilever beam
(OP)
I recently came across a problem that has all of the FEM engineers at our company stumped. I generated a cantilever beam from CBARs and CQUAD4s in PATRAN (using NASTRAN 2001 for the analysis). The beam is 100mm tall, 1000mm long, has a web of starting thickness 2mm and spar caps with properties of Area=256mm^4 and I=1306666.6mm^4 (in the primary bending direction) for the caps. Bar elements run only along the top and bottom surfaces to represent the caps. The beam has 10 elements through its thickness and 100 elements down its length, making each quad perfectly square. I applied a pure up load on one end and constrained each of the 11 nodes on the other end with simple supports. This model produced results that are within 5% of beam theory. I then lowered the thickness of the web to 1mm and recalculated the model and the beam theory answers. This resulted in an error of about 9% with beam theory. A further reduction of the web thickness to .5mm was calculated and produced an error of approximately 21%. This error is on displacements. I haven't even bothered checking stress/strain yet. We have tried adding K6ROT. Epsilon for this model is less than 10^-10. The free-body balances in the model. If anyone has any advice I would appreciate it.





RE: Simple cantilever beam
corus
RE: Simple cantilever beam
RE: Simple cantilever beam
please clarify the beam theory model being referenced?
RE: Simple cantilever beam
RE: Simple cantilever beam
You are most in the field of shell and plate theory, and a theoretical solution is not easy to calculate, maybe impossible.
A way to validate your analysis can be mesh refinement / solution convergence.
gelu
RE: Simple cantilever beam
RE: Simple cantilever beam
RE: Simple cantilever beam
There are at least a half-dozen theoretical beam equations(pde) to choose from.
We found significant departures with simple beams for L/D<100.
RE: Simple cantilever beam
If you don't have a copy of Roark I would highly recommend it. It is a standard in the Aircraft Industry, but would be highly useful for anyone doing strength calculations.
RE: Simple cantilever beam
RE: Simple cantilever beam
RE: Simple cantilever beam
RE: Simple cantilever beam
RE: Simple cantilever beam
We generally dont use shell models for complicated geometry and if the geometry is simple it does not take more time to make a solid model. Besides this in using solid models we make less assumptions. So why not use a solid model and avoid all the trouble. It takes less time to run a 2D model but since we only do simple geometries in 2D, creating the same in solid should not take a lot of time and the number of DOF is not that large. Could you gurus please enlighten me on this. Thanks
YM
RE: Simple cantilever beam
RE: Simple cantilever beam
corus
RE: Simple cantilever beam
khan
RE: Simple cantilever beam
You can not prevent deflection! Every material in the world will deflect under a load, this is physics!. You can only limit the deflection it to a specific deflection that you desire.
RE: Simple cantilever beam
you can always prevent deflection, all you need is a beam with an infinite moment of inertia. Although they can get a bit pricey ;)
as a note, shehzadkhan, usually i treat a 'zero' deflection as less than 0.5mm... but this is quite a relative thing. I am often asked by the pipers to provide a support with no deflection, and i usually use 0.5mm as my cutoff, as you will ALWAYS have deflection.
DRW75
RE: Simple cantilever beam
I assume you are kidding when you offer an infinite moment of intertia. Continueing this spirit it will not just be pricy but infinitely heavy. I do not think that you will find an engine to power such an automobile.
RE: Simple cantilever beam
An alternative to the impractical solution of using a beam with an infinite moment of inertia is to make a finite sized beam out of Unobtainium. In your finite element analysis, simply define a new material called Unobtainium, with zero density and an EXTREMELY high modulus of elasticity (say 1E100). (Some FEA systems come with Unobtainium already defined in their standard library of materials.) Now when you analyse your beam, you will find that you have eliminated the deflections totally - and your beam has zero mass as well!
Getting hold of Unobtainium can be a bit tricky, however - and if you DO get a piece, you will find that machining and forming is EXTREMELY difficult!
A Google search for "Unobtainium" shows that most suppliers are in far-flung corners of the universe, and you usually need to battle Klingons and various other aliens to get your hands on a sample. (A Google search for "Unobtainium" but excluding "Star Trek" turns up far fewer hits!)
Hope this helps!
RE: Simple cantilever beam
RE: Simple cantilever beam
Ha, I like that, unobtanium...
but I would still think you would need to have an E of infinity to get our super-stiff beam to work in theory.
Seriously though Khan, here is a table of values that i have calculated based on Mg -> E=45 GPa, density=1.74 t/m3
for: (no FEA, just standard formulae)
defl(mm) d_required(mm)
0.5 84.955
0.25 107.038
0.1 145.275
0.05 183.036
0.01 312.998
0.005 394.3614
0.001 674.3977
And so on...
But be cautioned, if you are considering deflections much less than 0.1mm, you need to ensure that the cantilever has a high degree of fixity at the base. Once again, in reality, there is no such thing as a perfect cantilever, as some degree of rotation will occur at the base (as everything in this world behaves as a spring). A small rotation could have an effect if you are looking for tolerances in the thou. range.
As an aside,
Self weight has little effect on the calcs... i didn't know Mg was so light. I am usually dealing with steel or the light weight aluminum.
DRW75
RE: Simple cantilever beam
If do you use Euler Benoulli beam, you must know that Euler Bernoulli beam is neglecting the shear transverse effect. The Timoshenko hencky beam is calculating the shear transverse effect but is haveing the shear locking for tall beam. I recommended Timoshenko variant "Discrete Shear Beam" for calculating beam.
RE: Simple cantilever beam
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DRW75