Help with differential equation

[StevenKatzeff]

Hi,

The following is the differential equation for a beam on a varying-stiffness elastic foundation. Does anybody know how to solve such an equation as a closed form solution (if possible)?

E*I*y(x)''''+k*y(x)/x^3=0

where k is a constant.

 

[MiketheEngineer]

Not sure what this is - but I have often written a very simple computer program or Excel to iterate to a solution. Works well.
Probably takes less time then trying to figure it out - at least for me.

 

[PicoStruc]

Why solving that by hand ???

Every good foundation book should have solution to beam/slab on elastic foundation.

 

[Denial]

I don't think you'll get a closed form solution to that one.

If your actual problem really IS a beam on an elastic foundation, and you have numerical values for E,I,k and the location of the beam relative to the origin (and some loads of course), then you could solve that specific problem using any of many FE programs that have a beam-on-elastic-foundation element. You would have to break the overall beam up into an adequate number of sub-beams, each with a tailored (constant) foundation stiffness. You'd need to take care to keep each beam's length sufficiently small, and you'd have to take even more care near the origin.

Be aware that many (most? all?) such programs do not treat BEF rigorously, but simply lump half of the distributed transverse stiffness at each of the element's two end nodes. This means that elements above a certain length cannot be modelled accurately.

However I suspect that your problem is not really a beam on an elastic foundation, but some other phenomenon with analogous underlying mathematics. An axisymmetric cylindrical pipe with tapering wall thickness perhaps? I would be interested to know.

PicoStruc. Those "foundation books" are unlikely to cover the case where the foundation stiffness varies with the inverse cube of the distance from the origin.

 

[Denial]

If you are going to cross-post (foundation engineering), then at least use the same description in each of them.

Your problem description in the other post cancels my second last paragraph above. However I do not think your differential equation comes close to applying to the problem you describe there. A Winkler approach assumes zero continuity in the "foundation", which most certainly does NOT apply to your "beam on beam" problem.

 

[StevenKatzeff]

Thanks for the advice Denial, and sorry for cross-posting.

The problem I am investigating is a beam, fastened on one side to a back-up cantielever beam. The overhang footing beds into the backup beam and the moment arm to the centroid of the "bedding-in" distribution determines the prying force on the fastener.

If the backup beam is idealised as springs, then they vary to the third power. Hetenyi in his seminal work on beams on elastic foundations outlines an approach for a linearly varying elastic foundation, but I require one that varies to the third power.

 

[Denial]

It has been several decades since I read Hetenyi's tour de force on beams on elastic foundation, and I no longer have access to a copy. My recollection is that for the linearly varying foundation stiffness he gives a solution composed of four independent power series, each multiplied by an arbitrary constant, with the four arb constants being chosen to match the boundary conditions. Not very analyst-friendly.

However the DE you have posted does not have a foundation stiffness that varies linearly along the beam. You do not even have a foundation stiffness that varies "to the third power" along the beam (despite what your words say). You have a foundation stiffness that varies as the inverse of the third power. And, as I said above, I do not think you will find any solution in analytical form.

However my bigger worry is whether the DE you have posted can represent to problem you are trying to solve. I might be misunderstanding things here, in the absence of a diagram, but I do not think there is any way you can Winkler-ise the behaviour of the cantilever upon which your beam is sitting. The key aspect of Winkler behaviour is that the deflection at one point (in the foundation) is completely independent of the deflection at any other point. In other words, Winkler assumes there is NO continuity in the foundation. This assumption is dubious enough in soils, where there is a degree of continuity. It is even more dubious for a cantilever beam, where continuity is a dominant behaviour.

Can you look at your problem as a contact / lift-off problem?

 

[StevenKatzeff]

Interesting insight into the problem... You're right about the continuity in the backup beam, but I thought it would be adequate to assume that the springs act independently.

You would know from textbooks such as Niu and Flabel that they assume a triangular pressure distribution on the overhang. I thought that by modelling the backup using the Winkler model, I could investigate the nature of this distribution.

You're right again that it is a contact problem, but I thought that the Winkler foundation model could predict the contact ditribution accurately enough to make some headway in the investigation.

I would rather leave FE as a final check, as I have had some trouble modelling 1-D contact in Nastran, especially in modelling contact with the cantilever backup.

 

[rb1957]

i'd use FEA to solve ... it'd allow me to explore different contact geometries, different issues that come up once you see a solution.

 

[prex]

If the solution to that equation is all you need, it's easy to find a series expansion for it. Should be in the form
[ζ]=[Σ]₃^[∞](-1)^n-1[ξ]ⁿ
where [ζ] and [ξ] are adimensional, and respectively proportional to y(x) and x.
Don't see at the moment what type of function this series will give. It is obviously zero for [ξ]=0 and [ξ]=1 and rapidly converges for values in between.

prex

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[StevenKatzeff]

Hi again.

I am now using a different approach. Denial, I think that you are right-the varying Winkler foundation is not the way to go. I now add the stiffness of the beam and backup cantilever together and obtain the elastic curve. Then I use Excel's Goalseek to predict the bolt force and spring forces required to match that elastic curve for the beam alone.

I may need to extend the number of spring that I use as I am getting lift-off-which I then reiterate assuming no spring at that position. It seems that the stiffer the backup cantilever relative to the beam, the larger the prying load.

What do you gentlemen think?

 

[BAretired]

I recommend Newmark's Numerical Procedures for a problem such as this. It is easy to understand what you are doing and straightforward to apply.

BA

 

[BAretired]

Oops, that wasn't right.
Try thread507-267603

BA

 

[RFreund]

Steven -
Unfortunately I do not have any insightful information to help however can you post a sketch of your problem. I can't picture what you analyzing but I'm very interested.
Thanks

EIT

http://www.HowToEngineer.com