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Belleville type washer buckling

Belleville type washer buckling

Belleville type washer buckling

If you took a Belleville style washer (simply supported on the ID) and applied a load to it till it was flat and then increased the load a small amount would the washer "buckle" in the other direction?  I think it would but I'm not sure of the internal stresses that would cause it.  I'm guessing hoop stresses develop as the washer is flattened and possibly those act to "reverse" the washer.  Or maybe radial compressive stresses are developed and they buckle the washer?  I'm asking because I might have a use for that in  a snap action mechanism.  Assume everything stays elastic.

RE: Belleville type washer buckling

You mean sort of like a modified metal dome switch?  


Just one manufacturer.  I'd say whether or not you get the buckling you want is going to strongly depend on the material thickness.  The thinner the material, the easier to buckle.

-handleman, CSWP (The new, easy test)

RE: Belleville type washer buckling

It would depend on how far past 'flat' you push the washer.  You will have to surpass the yield strength of the metal to have it permanently stay in the reversed direction.

RE: Belleville type washer buckling


If you take the standard catalog belleville washers and go beyond flat as much as you like it still will not snap and stay stably on the other side. For a belleville spring to snap the h/t needs to be over 2.82 where:

t = spring thickness
h = height of spring

The smaller the h/t the more linear the spring behavior. When h/t is 1.41 the spring has a constant force range on both sides of the flat position. For h/t > 1.41 the spring force will start to decrease beyond the flat position but will not stay in position when the load is removes. Only when h/t > 2.82 the spring will snap and will stably stay in the new position. You will need to apply a force to snap it backward. The real trick is to design it to a specified backward snap force.

The standard catalog springs usually have the h/t < 1 to  have a close to linear spring behavior and to allow stacking. Belleville springs with h/t > 1.41 can not be stacked because some may start to decrease the force before others resulting in erratic behavior of the stack.


RE: Belleville type washer buckling


Good info, thanks.  Do you know what internal stresses cause it to snap over?  Now I'm thinking it may simply be the bending stresses acting radially.  When it gets "over center" the compressive stresses pull it further over and the tensile stresses push it further over.  When it has flipped over a certain amount the geometry changes such that the tensile and compressive moments cancel and it stabilizes.  Am I close?

RE: Belleville type washer buckling


I do not have information how the snap is connected with the stresses. But the snap phenomena is derived from the deflection vs force equation of the Belleville spring.

Plotting the stresses too, show that high tensile stress in the the outside diameter and high compressive stresses at the inner diameter are still exist. Note that stresses at three locations are usually checked in the belleville spring. Two are in the inside diameter and the third in the outside diameter. One of the stresses in the inside diameter switch sign when the spring is loaded beyond the flat position, but the other two keep their sign. A snapped spring is under internal high stresses in the stable state.

For example: a belleville spring with inside diameter of 10 mm, outside diameter of 40 mm, thickness t = 0.35 mm and h/t=4 in the stable snapped state, have a compressive stress of ~200 kgf/mm2 at the upper part of the inside diameter, a tensile stress of ~70 kgf/mm2 on the bottom side of the outer diameter and a compressive stress of ~30 kgf/mm2 at the lower side of the inner diameter.

RE: Belleville type washer buckling


   A Belleville washer is a spring, and is very highly stressed.  When you compress one flat, you are pretty close to the yield stress of the material.  If you push it past flat, I would expect the material to fail.

   Reasonably current versions of Roark's Equations for Stress and Strain provide instructions for analyzing Belleville washers.  So does the SAE Spring Design Manual.   


RE: Belleville type washer buckling


BobM3 is interested in the "snap/buckling" behavior or a belleville spring, which is a special case where the height to thickness (h/t) is over 2.82. Those springs are loaded beyond flat to switch to second stable position beyond the flat position. He is asking about the stress condition in the snapped spring position.

RE: Belleville type washer buckling

Yes, israelkk, that is what I was after.  Thanks much for the great input!

RE: Belleville type washer buckling

I have played with these washers in the past and my recollection is that if they are fairly thick compared to diameter, they don't exhibit a bi-stable snap-through state, they always want to spring back to the original cone.

By contrast, I'm sure I have also played with large diameter thin rings which do seem quite happy to snap through. There may be a paper somewhere on this behaviour and the critical dimensions required.

Sorry this is just anecdotal.


RE: Belleville type washer buckling

It smells like the relationship between hoop and bending stiffness. Pre-stress state relating to how it was formed and whether it was stress-relieved post-forming will also be a factor.

RE: Belleville type washer buckling

I wish I had the money for a 3d FEA package.  It would be interesting to see what stresses are what and where.  My gut feeling is that hoop stress plays a role but my analytical skills are letting me down right now.  Simple rectangular beams are about my limit these days.

RE: Belleville type washer buckling

Yes, know the feeling. I wouldn't even contemplate working it out from first principles.

Why not just buy a few of different sizes and thicknesses and play with them to get a feel for the problem?

The unknown state of pre-stress after forming will mess with any calculations which you do anyway, FE would not help you that much there unless you get really detailed in your simulation including the forming process. Doable but a bit OTT.


RE: Belleville type washer buckling


I think you have no idea what is a snap belleville spring. I designed and built few of such devices. If you want to understand the matter you need to dig into Wahl's book and even go further to the Almen's article that was the first to solve the math behind this spring.

I do not see the point for this discussion of hoop stresses because you are trying to get something from pipes and vessels and try to apply it in this unique type of spring.

The formulas for the stresses and deflections are well known and can be found in almost every good spring design books. The trick is to understand the formulations and how to apply them. This takes time, patience, research and experience.

You can not buy such springs from the shelf, all the shelf springs has the h/t<1.41 and therefore will never snap to a stable position.

Linear FEA will not help because this spring is non linear. For a given force the spring can be in two positions (deflection) so how an FEA can help?

I am not even sure that FEA with bucking elements can solve this spring and I have never seen an FEA solution to such a problem. Why use a canon to kill a fly when the closed form formulas gives all you need.

Prestress has nothing to do with the force deflection (snap) behavior of the spring. The final prestressed spring dimensions dictates the force deflection behavior.

The prestress operation can help where tensile stress is the problem but most bellevile springs are governed by compressive stresses. More than that, for a snap spring prestress is useless and even harmful, because the spring is loaded to both directions. Prestress is an excellent process for one direction loading only.  

RE: Belleville type washer buckling


If you think you know so much about this then why don't you post some useful links to the equations which BobM3 actually needs instead of just ranting?


RE: Belleville type washer buckling

I just purchased Wahl's book.  I hope to have it in a week.  The preview on Amazon looked pretty good.  I am having trouble getting the Almen-Laszlo paper.  There is a DIN spec that incorporates it somehow (DIN 2092) but that costs $165.00 to download.

I did find an interstesting FEA study (P.BAGAVATHIPERUMAL-K. CHANDRASEKARAN-S. MANIVASAGAM) that was done to predict the load/deflection characteristics of conical springs.  Not quite what I was after but interesting.  They got around the the "large deflection" nonlinearity by iterating - changing the load each iteration.

I was thinking FEA could be used to see what stresses were getting higher as you approached flatness and then using that to guess what the mechanism was that causes the flipping over of the bi-stable spring/switch (I shouldn't call what I'd be using a Belleville washer).  In my Google searches I did see that several studies have shown that radial stresses are not very high.

I appreciate all the responses.  Each got me thinking a bit more about what I needed.

RE: Belleville type washer buckling


You can definitely use FEA to simulate this, including the bi-stable snap through, forming stress/pre-stress etc. It's just a bit involved and you would need a good nonlinear code like ABAQUS or MARC to do it nicely. I think the hand calcs route is the way forward unless you are an FE expert.

Good luck.


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