linear buckling analysis versus preload frequency analysis

[gwena]

Hello,

There is something that I don't understand. A linear analysis of buckling (BUCKLE card) is solved by the research of the load for which the stiffness matrix become singular.
Thus, an eigenvalue analysis with various preloads should be another way to verify the critical load. The value of the frequency should decrease to zero as the load is reaching the critical load.
The problem is that results of the eigenvalue analysis aren’t the same than for the buckling analysis !

I’m performing a linear analysis of buckling on a thin-walled and curved strip (modeled with S4R5 elements).
For example :
for a free – clamped end strip, there is flexural buckling at 199N (with Buckle option, these results seem correct).

With an eigenvalue analysis, the frequency for the flexural mode is closed to zero for a load of –4.5N!

Where is the mistake ? Enclosed, you will find commands that I used for the eigenvalue analysis.
Thanks a lot for any explaination !

*BOUNDARY
bottom,1,6
*STEP,NLGEOM
*STATIC
*CLOAD
5006,3,-2.5
*ENDSTEP
*STEP,NLGEOM
*FREQUENCY,EIGENSOLVER=EIGENSOLVER=LANCZOS
3,0
*EL PRINT,FREQUENCY=0
*MODAL FILE
*END STEP

 

[johnhors]

I am missing something here ?

The eigen solver for frequency analysis solves for eigen values (and thus natural frequencies) and eigen vectors (the mode shape) using the model stiffness and mass matrices, whilst when solving for buckling analysis the eigen solver again uses the stiffness matrix but uses a so called "geometric stiffness" matrix in place of the mass matrix to solve for the eigen values (critical load) and eigen vector (failure mode shape), so how can you possibly expect natural frequencies to have any correlation with buckling critical loads ? Or do you know something I don't ?

 

[gwena]

The problem of finding natural frequencies of a structure can be solved using the mass and elastic stiffness matrices of the structure.
On the other hand, the problem of structural instability of a structure and the critical buckling load is a problem, which can be solved using the elastic and geometric stiffness matrices of the structure.
By including the geometric stiffness (NLGEOM) in the first one, this influences the oscillation frequencies of the structure. This is why in general structural cases there is a relationship between the vibration and buckling behavior. It is possible to make natural frequency measurements under different loads on a structure. As the load increase, the frequency decreases. Thus, it is possible to predict the critical buckling load.
In common with the vibration problem, the linear stability problem is all eigen value problem and the eigen values are the critical values of loading magnitude at which buckling occurs.
Do you see what I mean ?

 

[johnhors]

Okay, yes I do, thanks.

I should have realised that when the stiffness matrix is singular as in a "free-free" natural frequency analysis, the mass matrix does not influence the zero frequency rigid body modes (one for each degree of singularity - thus 6 in a 3D problem).

 

[gwena]

So, now, theory is ok.
But in practice, what's wrong ?
I tried to include *INITIAL CONDITIONS,TYPE=PRESSURE STRESS
but it doesn't change the result.
...