Hello,
I have faced with a challenge to represent a dense 12 by 12 matrix into a standard beam element stiffness matrix format.The dense stiffness matrix represented by K have 72 elements. Whereas the standard beam element stiffness matrix have only 40 elements.
In other words, Can I transform this dense stiffness matrix(K) into the same sparsity as the standard beam element stiffness matrix without losing much fidelity? More information regarding the dense matrix is given in the attached excel sheet.
Any help will be appreciated.
Thank You,
Paul
Hi Paul
For what it is worth.
It depends. It is doable if both carry the same information but otherwise you must compromise and be compromized.
The way I would address the problem is to set up the 12x12 matrix and use it as a reference solution. I would then adjust inputs of a beam model and iterate until a best match is found.
Weighting of a best match is easy as it should be weighted wrt to impact, i.e. it should be weighted wrt to matrix number size. A caveat is comparing translation with rotation and a practical workaround to this might be to normalize these.
The above is a forward rather than an inverse solution technique. I recommend this as inverse solutions where you assume a pattern easily go awry when conditions are not met. Differently stated, if your 12x12 matrix is seen as, say, a 3rd order equation and your beams only can handle, say, a 2nd order equation. The inverse solution would either stall or come out erroneous. The forward approach allows a best fit where you would be in control of what is considered as 'best'.
In the latter case, the limit would be that beauty lies in the eye of the beholder.
I would start running just numbers. If it truly works out, I would let an analytic maths program like mathematica or similar loose on the problem. If it not works out to 100%, I would delete the items of the lowest importance in the 12x12 matrix and match or approximate those that can be matched.
Just my 2 cents.
Sincerely
Claes
Dear Mr Thomas,
A method could be to minimize the difference between work, defined as xTKx for the first FE and yTKby for the beam, where x are the dofs of the first FE and y those of the beam. Then a matrix to transform (like y=Mx) is required. That calculates e.g. a torsion shear angle from the displacements of the FE, and the bending angles. The axial is the same displacement. But I don't know the dofs of the first, and what is done to it before, according to the excel file. Could you extract those?
Hello Lena J-T Strömberg,
Thank you for taking time out to answer my question.
I could not depend on the initial structure for taking the information like displacement in order to solve for the beam stiffness parameters like EA, GJ and EI. Please look into the attached excel-calculation tab for more details. I have explained the FEM details along with the 12 dofs for each matrices. I understood only little about the idea you have proposed above and i have prepared a round about idea for this. Please let me know whether you have meant the same or different. But this method have more unknowns than equations. Solving will be impossible.
Also i would like to to about what you have meant by minimizing the work difference of work.
Thank You,
Paul
Hi,
I read about the Guyan reduction yesterday. Turning a solid 3D to a solid 2D would be easy, only zeroing dofs. But the beam has other dofs than the 3D. Therefore, I thought they started with a plate element, some way, and got the matrix in your excel sheet. All entries which should be zero are small except those of power 10^4.
You could decompose the I and J in A and inertia radius, and G with and Poisson's ratio.
In my idea, there would be options for the matrix M, on how to distribute the displacement dofs. The beam is more flexible in bending, than a solid, but I don't know how to capture that in a matrix equation. Minimizing the difference in work seams possible, bcz work has the same dimension for both elements.
I 'll take a look at the info in the new. xls later today.
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