eigenvalue/eigenvector of tangent stiffness in abaqus

[Michael_ZY]

Hi,

I was doing snap-through problem with abaqus in static nonlinear setting. I got the load-deflection curve corresponding to snap behavior. But I want to ask for the eigenvector and eigenvalue of tangent stiffness for each configuration(each point on the curve), do you know how to do it?

 

[Mustaine3]

I see two options:

1. One large analysis. So push a little bit in a nonlinear static analysis step, then add the frequency step. The adding another static step with a little bit additional load, then another frequency step. And so on. The eigenfrequencies and -vectors are always based of on the result of the previous nonlinear step.

2. Run the full static analysis as now, but save restart data during the analysis. Then do restart or import analysis from each point and do a frequency analysis each time.

 

[Michael_ZY]

Dear Mustaine,

Thanks for your reply. One thing I was confused about frequency analysis.

Because what I did originally was static analysis, it has nothing to do with mass.

So if I did frequency analysis as you suggested, how to deal with mass, and how much mass should I assign? The value of mass will affect the eigenvalue and -vector,right?

Does it mean that I need to do specific settings to make sure that the mass matrix is identity matrix?

 

[Mustaine3]

Usually you just have to define density in your materials. Abaqus does the rest.
Optionally you can add artificial masses, like point mass or mass smeared across a face.

 

[Michael_ZY]

Mustaine,thanks!

what I want is the eigenvalue and vector of tangent stiffness matrix, so it should be solved like (K – λ I) ψ=0, where K is the tangent stiffness matrix, λ- eigenvalue, I- identity matrix, ψ- eigenvector...

But if I do frequency analysis for each configuration, it will calculate eigenvalue and vector in this way:

If you just look at two equations, it seems that they are different problems(different eigenvalue), because if you want to make them equivalent, it is necessary to make M (mass matrix) equal to I (identity matrix). So is my understanding wrong? what i mean that the eigenvalue/vector from frequency analysis is not the exact eigenvalue of tangentstiffness unless the M matrix can be identical, is that correct?

Sorry for this long question

 

[IceBreakerSours]

I do not know the answer to your question but perhaps stiffness-based computation of modes might be worth looking into.

Also, since you are running a static analysis, you are modeling the snap-through behavior of something without taking inertia in to account. Why?



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[Michael_ZY]

Hi,IceBreakerSours

Thanks for your reply.

I do not exactly understand your point because I think inertia and mass is not needed for static analysis.
For the snap-through behaviors, in experiment, structure will snap at critical point, which means dynamic problems kick in. But here I just want to find and follow equilibrium, which is completely related to static solution, and no dynamics involved in.

 

[IceBreakerSours]

I have never run these analyses before which is why I was curious. I would have thought that at the moment of buckling inertia would make the problem determinate and make the solution feasible with a regular implicit dynamic scheme.

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[Mustaine3]

You are right - the two equations are identical, except that you've replaced M with I.

Structures have stiffness and mass, so that's used by Abaqus. What's the purpose of using the real stiffness and an artificial mass?

 

[Michael_ZY]

Hi,IceBreakerSours. You are right, you can also switch it to dynamic analysis at the critical point (snapping point).

 

[Michael_ZY]

Hi Mustaine, thanks for your good question.

Originally, I want to find the bifurcation point or limit point of snapping process where the eigenvalue of tangent stiffness is equal to zero and the corresponding shape is related to eigenvector.

And the eigenvalue and eigenvector of tangent stiffness have nothing to do with mass, as you know, typical nonlinear static analysis of structures does not need mass assignment, so there is no mass in the original static analysis.

As you mentioned, the extraction of eigenvalue from frequency analysis needs mass definition, so in order to get the exact value of tangent stiffness, I think it's essential to control M(mass matrix) to be identity matrix. Otherwise, the eigenvalue/vector you get from frequency analysis is not the exact eigenvalue of tangent stiffness.

 

[Mustaine3]

Isn't that what the *Buckle procedure is doing?