Display of the element stiffness matrix 2

[huaijuliu]

hey guys! i use the *element matrix output KEYWORDS to output the element stiffness matrix. the result displays:
----
**

** ELEMENT NUMBER 1 STEP NUMBER 1 INCREMENT NUMBER 1

** ELEMENT TYPE C3D8R

*USER ELEMENT, NODES= 8, LINEAR

** ELEMENT NODES

** 46, 47, 56, 55, 1, 2, 11, 10

1, 2, 3

*MATRIX,TYPE=STIFFNESS

139773.98076923 ,

-63100.961538462 , 139773.98076923

63100.961538462 , -63100.961538462 , 139773.98076923

87865.413461538 , -12620.192307692 , 63100.961538462 , 139773.98076923

12620.192307692 , -38336.509615385 , 12620.192307692 , 63100.961538462

139773.98076923 ,

63100.961538462 , -12620.192307692 , 87865.413461538 , 63100.961538462

63100.961538462 , 139773.98076923

37860.576923077 , -12620.192307692 , 12620.192307692 , 87865.413461538

63100.961538462 , 12620.192307692 , 139773.98076923

12620.192307692 , -88341.346153846 , 63100.961538462 , 63100.961538462

87865.413461538 , 12620.192307692 , 63100.961538462 , 139773.98076923

-12620.192307692 , 63100.961538462 , -88341.346153846 , -12620.192307692

-12620.192307692 , -38336.509615385 , -63100.961538462 , -63100.961538462

139773.98076923 ,

87865.413461538 , -63100.961538462 , 12620.192307692 , 37860.576923077

12620.192307692 , 12620.192307692 , 87865.413461538 , 12620.192307692

-63100.961538462 , 139773.98076923

-63100.961538462 , 87865.413461538 , -12620.192307692 , -12620.192307692

-88341.346153846 , -63100.961538462 , -12620.192307692 , -38336.509615385

12620.192307692 , -63100.961538462 , 139773.98076923

-12620.192307692 , 12620.192307692 , -38336.509615385 , -12620.192307692

-63100.961538462 , -88341.346153846 , -63100.961538462 , -12620.192307692

87865.413461538 , -63100.961538462 , 63100.961538462 , 139773.98076923

-38336.509615385 , 12620.192307692 , -12620.192307692 , -88341.346153846

-63100.961538462 , -12620.192307692 , -138346.18269231 , -63100.961538462

63100.961538462 , -88341.346153846 , 12620.192307692 , 63100.961538462

139773.98076923 ,

-12620.192307692 , 87865.413461538 , -63100.961538462 , -63100.961538462

-88341.346153846 , -12620.192307692 , -63100.961538462 , -138346.18269231

63100.961538462 , -12620.192307692 , 37860.576923077 , 12620.192307692

63100.961538462 , 139773.98076923

12620.192307692 , -63100.961538462 , 87865.413461538 , 12620.192307692

12620.192307692 , 37860.576923077 , 63100.961538462 , 63100.961538462

-138346.18269231 , 63100.961538462 , -12620.192307692 , -88341.346153846

-63100.961538462 , -63100.961538462 , 139773.98076923

-88341.346153846 , 63100.961538462 , -12620.192307692 , -38336.509615385

-12620.192307692 , -12620.192307692 , -88341.346153846 , -12620.192307692

63100.961538462 , -138346.18269231 , 63100.961538462 , 63100.961538462

87865.413461538 , 12620.192307692 , -63100.961538462 , 139773.98076923

63100.961538462 , -88341.346153846 , 12620.192307692 , 12620.192307692

87865.413461538 , 63100.961538462 , 12620.192307692 , 37860.576923077

-12620.192307692 , 63100.961538462 , -138346.18269231 , -63100.961538462

-12620.192307692 , -38336.509615385 , 12620.192307692 , -63100.961538462

139773.98076923 ,

12620.192307692 , -12620.192307692 , 37860.576923077 , 12620.192307692

63100.961538462 , 87865.413461538 , 63100.961538462 , 12620.192307692

-88341.346153846 , 63100.961538462 , -63100.961538462 , -138346.18269231

-63100.961538462 , -12620.192307692 , 87865.413461538 , -63100.961538462

63100.961538462 , 139773.98076923

-138346.18269231 , 63100.961538462 , -63100.961538462 , -88341.346153846

-12620.192307692 , -63100.961538462 , -38336.509615385 , -12620.192307692

12620.192307692 , -88341.346153846 , 63100.961538462 , 12620.192307692

37860.576923077 , 12620.192307692 , -12620.192307692 , 87865.413461538

-63100.961538462 , -12620.192307692 , 139773.98076923

63100.961538462 , -138346.18269231 , 63100.961538462 , 12620.192307692

37860.576923077 , 12620.192307692 , 12620.192307692 , 87865.413461538

-63100.961538462 , 63100.961538462 , -88341.346153846 , -12620.192307692

-12620.192307692 , -88341.346153846 , 63100.961538462 , -63100.961538462

87865.413461538 , 12620.192307692 , -63100.961538462 , 139773.98076923

-63100.961538462 , 63100.961538462 , -138346.18269231 , -63100.961538462

-12620.192307692 , -88341.346153846 , -12620.192307692 , -63100.961538462

87865.413461538 , -12620.192307692 , 12620.192307692 , 37860.576923077

12620.192307692 , 63100.961538462 , -88341.346153846 , 12620.192307692

-12620.192307692 , -38336.509615385 , 63100.961538462 , -63100.961538462

139773.98076923 ,

-88341.346153846 , 12620.192307692 , -63100.961538462 , -138346.18269231

-63100.961538462 , -63100.961538462 , -88341.346153846 , -63100.961538462

12620.192307692 , -38336.509615385 , 12620.192307692 , 12620.192307692

87865.413461538 , 63100.961538462 , -12620.192307692 , 37860.576923077

-12620.192307692 , -12620.192307692 , 87865.413461538 , -12620.192307692

63100.961538462 , 139773.98076923

-12620.192307692 , 37860.576923077 , -12620.192307692 , -63100.961538462

-138346.18269231 , -63100.961538462 , -63100.961538462 , -88341.346153846

12620.192307692 , -12620.192307692 , 87865.413461538 , 63100.961538462

63100.961538462 , 87865.413461538 , -12620.192307692 , 12620.192307692

-88341.346153846 , -63100.961538462 , 12620.192307692 , -38336.509615385

12620.192307692 , 63100.961538462 , 139773.98076923

-63100.961538462 , 12620.192307692 , -88341.346153846 , -63100.961538462

-63100.961538462 , -138346.18269231 , -12620.192307692 , -12620.192307692

37860.576923077 , -12620.192307692 , 63100.961538462 , 87865.413461538

12620.192307692 , 12620.192307692 , -38336.509615385 , 12620.192307692

-63100.961538462 , -88341.346153846 , 63100.961538462 , -12620.192307692

87865.413461538 , 63100.961538462 , 63100.961538462 , 139773.98076923

-------
The element is C3D8R, what's the size of the matrix? why it displays like that?
in the first row there's 1 number
in the 2nd row there's 2 number
...
in the 24th row there's 24 number

Can you please tell me what the exact matrix is?

 

[akabarten]

Hi,

The size of stiffness matrix is nxn where n is number of degrees of freedom (DOF) in the model.
If your model is build up with one C3D8R element then you have 24 DOFs.
In each node (8) you have 3 DOF (translation in global z, y & z direction). 3D elements have no rotational DOF.
So you got matrix 24x24.
Stiffness matrix is symmetric matrix so you got only lower part since upper is the same.

Regards,
akaBarten

 

[huaijuliu]

Thanks akaBarten
I think you are right.
However, it's probably for a stiffness matrix to be sparse. why there's no zero element in the matrix?
I tried another cases and no zero appears either.

 

[johnhors]

You are confusing element stiffness matrix with model stiffness matrix.

An element stiffness matrix does not have any zero terms (see a text book).

Zero terms will only appear in the whole model system stiffness matrix, which may be sparse.

http://www.Roshaz.com

 

[huaijuliu]

Thanks johnhors.
An element stiffness matrix has many general characteristics that can be used to check the formulation of a particular stiffness matrix. An element stiffness matrix must have the following properties:

Symmetric - This means that kij = kji. This is always the case when the displacements are directly proportional to the applied loads.
Square - This implies that the number of rows are equal to the number of columns in the matrix.
Singular - The element stiffness matrix is singular (the determinate of the matrix is equal to zero?
Positive Diagonal Terms
How to explain the singular term?

 

[Elabbasi]

The singularity is because of rigid body modes. If the element translates in any of the 3 directions, or rotates about any of the 3 axes the internal forces should be zero and that can only be achieved if the stiffness matrix has a singularity corresponding to this displacement mode.

So, if you evaluate the determinant it will be 0, and if you evaluate all the eigenvalues, 6 out of the 24 will be 0.

Nagi Elabbasi
Veryst Engineering

 

[johnhors]

huaijuliu,

consider the simplest element stiffness matrix possible, that of a spring in 1D with stiffness K.

The element has two degrees of freedom, one at each end of the spring, the element stiffness matrix is a 2 by 2 square.

The stiffess matrix is:-

K -K
-K K

with positive K on the leading diagonal and negative K for the off diagonal coupling terms.

The determinant is k * k - ( -k * -k ) = 0

"ground" the matrix by holding one end of the spring, remove the corresponding row and column and you are left with a single term K. The value of the determinant for the reduced one by one matrix is simply K.

All element stiffness matrices are singular by definition. The whole model system stiffness matrix is also singular, until sufficient boundary conditions have been applied.

http://www.Roshaz.com

 

[huaijuliu]

I got it
thanks very much