Overhang beam pressure distribution

[StevenKatzeff]

Hi,

I am investigating the overhang beam pressure distribution when part of the beam is resting on an elastic foundation. We are all aware of the assumption that when a moment is reacted at the foundation, the reacting end beds-into the foundation and creates a triangular pressure distribution; the centroid of which is at 2/3 of the edge distance L.

If I model this beam, assuming the foundation to be a Winkler elastic foundation, as the elastic foundation constant increases, the centroid of the distributed load approaches the simple support. Essentially, the stiffer the foundation, the closer it approaches to a second simple support very close to the simple support reacting the tension load. However, if this beam is modelled using linear contact in Patran/Nastran, and the foundation is modelled as infinitely stiff, the centroid does not decrease to less than 27% of L, the edge distance of the beam on the foundation. I have even considered lift-off of the beam in the Winkler model.

Is there an explanation for this? Obviosuly, the Winkler foundation model breaks down at some point, but what other mathematical model would yield the FE result?

Regards,

 

[prex]

You should be more specific about your assumptions: moment of inertia, beam length, modulus of foundation, ...
Also it is unclear to me what an infinitely stiff foundation means in your calculation. If the foundation was really infinitely stiff, then you would have a perfect fixity at the support.
In the first site below, here, you can experiment with the mathematical model of the Winkler foundation (just replace the load by arm length with the corresponding moment).

prex
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[StevenKatzeff]

Hi prex,

By imposing my boundary conditions on the differential equation for the deflection of a beam supported on an elastic foundation, I get the value of the four constants of the characteristic equation:

y = e^lambda*x(C1*cos(lambda*x)+C2*sin(lambda*x))+e^-lambda*x(C3*cos(lambda*x)+C4*sin(lambda*x))

where lambda = (k/(4EI))^0.25, and k is the elastic subgrade reaction constant.

Now, as lambda gets large (larger than 0.001), my beam starts to lift off of the foundation. Eventually, as lambda gets very large, the reaction will apprach a simple support very close to the actual simple support. This, as mentioned above, is at odds with FE results, and also intuitively does not make sense. Clearly the Winkler model breaks down at some point and another model is required. My question is which model...

Does that make sense?

 

[prex]

First point is that the linear equation you wrote does consider Winkler springs as acting both ways, push and pull, otherwise the equation wouldn't be linear anymore.
Second point is that, with lambda becoming large, the reaction of the subgrade approaches the behavior of a clamped support because of the point above.
Otherwise your description is correct (and intuitively does make sense to me), but I don't know how your FEM model is built. Did you consider non linearity due to the subgrade reacting only in compression?
Besides that, Winkler model is correct and is the only one I know that is widely used (because of its simplicity of course). To judge whether a more sophisticated model would be useful we should know what's your purpose and what kind of phenomenon you wish to represent: if this is a theoretical exercise I guess you can stick with Winkler, possibly adding to it the unidirectional behavior.

prex
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[URL unfurl="true"]http://www.megamag.it[/url] : Magnetic brakes and launchers for fun rides
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[StevenKatzeff]

I am investigating prying action in a riveted joint subjected to a tension load. The centroid of the overhang pressure distribution determines the prying load on the rivet. To model the contact effect in Patran, I modelled the overhang beam and simply varied the E of the foundation from soft to very stiff relative to the top beam. I used HEX8 elements and SOL101 running linear contact boundary conditions.

The model seems correct to me.

Here is a little experiment I did and the reason why I said that "this does not make intuitive sense to me":
When I fix a plastic ruler on a rigid surface and apply a force at the overhang, the ruler does not immediately lift off the base, as it should according to the Winkler model. Not very scientific, but it does illustrate the point IMHO.

 

[prex]

"does not immediately lift off the base": if you refer to a transient effect, this has nothing to do with a statics calculation. Otherwise if you mean that for a small force the ruler does not lift off, this is simply because you must first counterbalance the weight (assuming the ruler is initially more that 50% on the support, otherwise it would fall down by itself).
It is quite evident that, for a rigid foundation, if the foundation reacts only in compression, your beam cannot be stable, and if the foundation is bidirectional, then it will react as a fixed support.
There must be some problem in your model, as this is the only intuitive and physical possible behavior!
Also this does not seem a correct model for your riveted joint: in Winkler model the foundation reacts to external forces, whilst the compression in the joint is due to the rivet action.


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[rb1957]

"When I fix a plastic ruler on a rigid surface and apply a force at the overhang, the ruler does not immediately lift off the base," ...

the ruler should pivot about the restraint, no? if you clamp the ruler (with a C-clmap), that point will be fixed (for all intents and purposes) and if you pull up on the ruler it should deflect about this clamped point.

i'd've thought that if the foundation got stiffer, then the support got "better", that the 2/3L couple arm (based on the "elastic" triangular distribution) got smaller and the couple forces increased. maybe this is what you're seeing too ? ultimately you'd've an infinite couple acting over a zero length ...

 

[ishvaaag]

I don't know if this can be the problem in the Winkler model of Steven but certainly if separation is expected, the constraints must show that in the model; typically for foundations this is obtained through compression-only springs, that only show stiffness when compressed against the ground, and nil when outwards from it. RISA 3D can do that easily.

 

[StevenKatzeff]

rb1957 - Your conclusion of an infinite couple is correct if we are using a standard Winkler foundation model. However, this is at odds with my contact FE model of the same system, and that is where the problem lies. I suspect that the FE is correct in this case.

 

[rb1957]

"this beam is modelled using linear contact in Patran/Nastran, and the foundation is modelled as infinitely stiff"

this implies a gap, small though it may be, between the beam and the foundation ... yes ?

the rigid foundation would be modelled by constraining the beam directly, no?

 

[StevenKatzeff]

No gap, beams are directly on top of each other. Foundation is modelled as a solid block of material and constrained at its base. I'll post some pictures of the model and results tomorrow.

 

[rb1957]

if i was modelling a rigid connection i'd have some (large-ish) volume of ground modelled, share nodes between the beam and the foundation, and constrain the foundation away from the beam ... but i'm willing to bet that'd produce some tensions ... so i guess you've put zero length springs/gap elements between the beam and the ground.

the typical triangular distribution assumes (i think) that the foundation is less stiff than the beam, or maybe that the ground stiff is constant, and the beam is elastic.

if the foundation becomes in-elastic, that the reaction in the beam will peak towards the far end ... yes?

if the beam goes in-elastic, then the ground reaction moves closer to the surface, reducing the 2/3*L arm; or the beam starts to rotate at the surface (reducing the end moment) ... yes?