Is it reasonable to average von Mises stress plots at discontinuities? 1

[cbk14]

Suppose I have a FEM with two bodies. One body represents the washer underneath a bolt head; it is circular and small relative to the other body, which is a large, thick, flat plate. The washer is bonded to the plate in a circular region with compatible meshes. The edge of the bonded contact is a discontinuity, and of course the stresses peak at this discontinuity.

In one example, the peak stress observed at the discontinuity exceeds the material allowable. Looking at the stress plot, it appears to me that the peak stress is artificial, due to the discontinuity caused by the boundary condition. I would also argue that if this stress were real, localized yielding would occur and would redistribute the stress until the system was in equilibrium. I don't think "localized yielding will occur" is going to fly with my customer, regardless of whether that is accurate.

One method proposed to resolve this is to use a more realistic contact, such as a spring element with friction between the bodies and no penetration. This of course results in a non-linear model, which takes far longer to solve. Suppose for this discussion that non-linear analyses are prohibitively expensive to run.

A simpler method to resolve the peak stress that has been proposed in our group is to average the stress values in a small area around the peak stress. For example, if the observed peak is at a node, we take the stress values from the surrounding nodes at a distance of 3 element edges away. I would argue this is fairly well supported by the Saint-Venant principle, where stresses are equivalent at sufficient distances from the contact, as long as the load is statically equivalent. This raises the question how far away is sufficient? How far away is too far? Or am I completely misunderstanding the Saint-Venant principle and applying it inappropriately?

Now my core questions for you all are:

1) is it reasonable to average von Mises stresses at nodes to reduce the artificially high stresses observed at boundary condition discontinuities?
2) if the above is reasonable, is an arithmetic mean the correct measure? or would RMS, geometric mean, or something else be more appropriate?
3) are there alternative methods for overcoming artificially high stresses at boundary condition discontinuities (aside from running non-linear studies)?

Thank you.

 

[FEA way]

Similar issues occur when modeling welds and we often take the stresses at a distance from the weld toe. Codes tell us how far is enough. In this case, it could be more of an engineering judgment/experience thing.

I know that you exclude nonlinear analyses from your considerations but I’ll just say that nowadays this type of simulation became a standard and the computational cost is usually low when compared to how valuable results you can get.

 

[Kwan]

The simple solution is to find the fastener load, then bearing stress, and compare to the material bearing allowable.

 

[SWComposites]

Why would you model a washer in a FEM?

 

[rb1957]

So you're running a linear FEA and it is predicting localised nonlinear behaviour ?

You're looking at element results ? or nodal results ?
And the nodal results from different elements have different stresses ?
This is physically "wrong" ... stress is a continuous function.
Averaging stresses at nodes is reasonable, IMHO.

But "surely" you're plotting the results in a post processor (rather than looking at reams of numbers) ?
doesn't your post-processor do this "automatically" ?
Normally you plot either element centroid results or averaged nodal results.

another day in paradise, or is paradise one day closer ?

 

[Ng2020]

Could you make the 'local yielding' argument, and back it up by applying Neuber's method to the local peak stress? Would that be acceptable to your customer?

 

[Coreform_Greg]

You're looking at element results ? or nodal results ?
And the nodal results from different elements have different stresses ?
This is physically "wrong" ... stress is a continuous function.
Averaging stresses at nodes is reasonable, IMHO.

Just to be clear, in most traditional finite element methods the solution is approximated with piecewise continuous basis functions (i.e. C[sup]0[/sup] basis functions), the exception being that some shell and beam elements use smooth (C[sup]1[/sup]) basis functions (e.g. Cubic Hermites). This means that while the solution variable (e.g. displacement) is continuous across element boundaries, derived values such as strain/stress are discontinuous (C[sup]-1[/sup]) across element boundaries and thus it is completely expected that the stress evaluated at a node will be different depending on which element you're evaluating. If you have a node shared by 8 elements, there will be 8 equally-valid stress values at the node. Physically "wrong" but variationally "correct" and satisfies the best-approximation property for the given solution space. Codes like Abaqus automatically use volume-weighted averaging for reporting stress (and other derived values) at nodes and for contour plots.

 

[rb1957]

I can't see why there are 8 equally valid stresses at a node. It is, as you say, an artifact of the FEA assumptions for displacement based elements. There are some stress based elements out there where stress is the continuous function. I would describe the stress results as 8 equally approximate values and it is reasonable, depending on the geometry, to average the element results, and most post processors do this internally.

another day in paradise, or is paradise one day closer ?

 

[Coreform_Greg]

What I mean is that since the predominant weak-formulation of elastostatics assumes that u ∈ H[sup]1[/sup](Ω), that then σ ∈ H[sup]0[/sup](Ω). The N unique stress values that are computed at a node (N being the number of elements that share the node) satisfy this assumption, satisfy the best approximation property, and are thus "valid". The fact that the stress field is discontinuously represented by the solution space does make them more difficult to post-process, but this doesn't make them any less valid than the continuous displacement field.

In the physical world, elastic displacements (on the interior) are not merely continuous, though this is the assumption in the weak-form / FEM, and aren't even "just" smooth (which would allow stress to be continuous). Rather they are infinitely smooth (on the interior). Yet we don't dismiss the displacements computed by FEM as being "physically wrong" -- even though they technically are physically wrong.

 

[rb1957]

FEMs are approximations.

Little from FEMs (other than ridiculously simple FEMs) is truth (or physically right).

It is just a good enough approximation.
And nodal averaged stress is similarly a good enough approximation, although there are times when stress is discontinuous (or, in the real world, very nearly so).

another day in paradise, or is paradise one day closer ?

 

[rickfischer51]

"The washer is bonded to the plate with compatible meshes." Actually, it is no longer a washer on a plate, its functionally a plate with a boss. But analytically its two pieces stuck together with an idealization, and you are looking at stress at the idealization and trying to hand wave goofy stress values away. Great care is needed when evaluating stress at a BC, which is often why analysts invoke St Venant in the first place. Seems like you are crawling farther and farther out on a limb. Also, I dont think St Venant would find a few elements distance to be sufficiently large. Backing off a few elements to find a reasonable stress is a discretization consideration, not a boundary condition thing. If you feel bonding the washer to the plate is a reasonable approximation, I would model the plate and washer as one piece and hand wave the stress away with Glinka's method. Otherwise, Id bite the bullet and model it as two pieces with frictional contact and a nonlinear material. You can greatly reduce your run time if you don't make every element in the model a nonlinear material. Just make elements near your yield zone plastic. Make the washer and the plate that's within a plate thicknesses of the washer edge a plastic material, and leave everything else elastic. And use a reasonably fine mesh in the plastic zone. Mesh distortion and the resulting bisections will probably hurt run time more than a few extra elements.

Rick Fischer
Principal Engineer
Argonne National Laboratory

 

[matty90]

Hi,
I think that whats said above by rickfischer51 is correct. Definetly use Neuber/Glinka's method to sort out the real stress (you can resort to that in order to keep the simulation linear if you cannot run a non linear one).

 

[Nereth1]

I think your proposed use of St Venants principle is miss-applying it quite badly...

Sounds like you're wanting to use it to say "Stress at a distance is a good substitute for stress at this point that's badly modelled" when what it really says is "Stress at a distance is correct, regardless of the stress at this point that's badly modelled".

The stress at the distance will be right, according to St Venant's principle, but if you care about the stress at the badly modelled point, it doesn't matter any less. And it sounds like your client cares about that point (since apparently they won't accept local yielding).

You have to either;

1) Run a more realistic analysis and redesign/submit as appropriate per it's results.
2) Have enough experience to say "That's a stress singularity/geometric discontinuity and is not real" (and have your client believe you).
3) Have enough experience to say "That will yield and you don't care" (and have your client believe you).

If you don't have the experience/clout for items (2) or (3) to work for you, then your only option is (1), as it should be, because clearly, the reality is - you might be wrong. But not all is lost - treat it as a learning experience. Running the extra analysis and seeing how things change, is how you eventually get the experience/clout/number of gray hairs required to make the judgement call one way or the other, and convince people of it.