Help with differential equation
Help with differential equation
(OP)
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.
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.






RE: Help with differential equation
Probably takes less time then trying to figure it out - at least for me.
RE: Help with differential equation
Every good foundation book should have solution to beam/slab on elastic foundation.
RE: Help with differential equation
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.
RE: Help with differential equation
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.
RE: Help with differential equation
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.
RE: Help with differential equation
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?
RE: Help with differential equation
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.
RE: Help with differential equation
RE: Help with differential equation
ζ=Σ3∞(-1)n-1ξn
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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RE: Help with differential equation
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?
RE: Help with differential equation
BA
RE: Help with differential equation
Try thread507-267603: Newmark's Numerical Procedures
BA
RE: Help with differential equation
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
www.HowToEngineer.com