Simple cantilever beam 2

[aerohead56]

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.

 

[israelkk]

DRW75:

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.

 

[jhardy1]

DRW75 / israelkk,

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!

 

[hacksaw]

sounds like the "unobtainium" must have solved the problem

 

[DRW75]

JulianHardy,

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

 

[babilu]

hmm
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.

 

[DRW75]

A fair statement babilu, as the beam gets deeper, you start to get an internal strut effect, and the beam analogy starts to be conservative. I assume though in Khan's problem, he won't go with a beam that is 674mm deep with a 750mm projection as he probably won't have that much of a base to connect to. You would more efficiently use a struted frame if you actually have that much base connection capability. It was also more of an excercise to show the fact that there is always deflection present.

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DRW75