# How to get stiffness matrics for shell element(S4)

Greetings,

I’m seeking assistance in obtaining the stiffness matrix for the S4 shell element in Calculix. I’ve configured a single shell element using the S4 element type and assigned material properties without B.C and loading since I need non reduced stiffness matrix and used *Frequency card, I got the stiffness matrix .sti file and degree of freedom (DOF) mapping .dof files (see below).

Assuming that Calculix features 3 degrees of freedom per node, the anticipated stiffness matrix dimension for this element would be 4 nodes x 3 DOF = 12x12. However, upon examining the generated .sti and .dof files, I observed dimensions of 33x33. Could someone tell how to interpret this stiffness file and why the matrix is 33x33 rather than 12x12?

I also know that in Calculix, shell element S4 is expanded into 3D 8 node brick elements, resulting in 4 nodes expanding to 8 nodes. Consequently, the stiffness matrix would be 8 nodes x 3 DOF = 24x24 but why it is 33x33?.

Thanks,
Anoop

Input file.
**
** Heading +++++++++++++++++++++++++++++++++++++++++++++++++
**
*Heading
Hash: qWBNEwvd, Date: 05/02/2024, Unit system: MM_TON_S_C
**
** Nodes +++++++++++++++++++++++++++++++++++++++++++++++++++
**
*Node
1, 1.00000000E+001, 0.00000000E+000, 0.00000000E+000
2, 1.00000000E+001, 5.00000000E+000, 0.00000000E+000
3, 0.00000000E+000, 5.00000000E+000, 0.00000000E+000
4, 0.00000000E+000, 0.00000000E+000, 0.00000000E+000
**
** Elements ++++++++++++++++++++++++++++++++++++++++++++++++
**
*Element, Type=S4, Elset=Shell_part-1
3, 1, 2, 3, 4
**
** Element sets ++++++++++++++++++++++++++++++++++++++++++++
**
**
** Surfaces ++++++++++++++++++++++++++++++++++++++++++++++++
**
**
** Physical constants ++++++++++++++++++++++++++++++++++++++
**
**
** Materials +++++++++++++++++++++++++++++++++++++++++++++++
**
*Material, Name=Material-1
*Elastic
210000.0, 0.3

*Shell section, Elset=Shell_part-1, Material=Material-1, Offset=0
0.1
*Density
1.0
*Step
*Static
*El Print, Elset=Single_element
EVOL
*End step

*STEP
*FREQUENCY,SOLVER=MATRIXSTORAGE,STORAGE=YES
*END STEP

stiffness matrix file .sti

1 1 4.4905448717949e+06
1 2 -1.6826923076923e+03
2 2 4.4939102564103e+06
1 3 -8.4134615384615e+04
2 3 1.6826923076923e+05
3 3 1.5707371794872e+07
1 4 -4.4854967948718e+06
2 4 -8.4134615384615e+02
3 4 -1.6826923076923e+04
4 4 4.4905448717949e+06
1 5 -8.4134615384615e+02
2 5 -4.4838141025641e+06
3 5 3.3653846153846e+04
4 5 -1.6826923076923e+03
5 5 4.4939102564103e+06
1 6 1.6826923076923e+04
2 6 -3.3653846153846e+04
3 6 -1.5704006410256e+07
4 6 8.4134615384615e+04
5 6 -1.6826923076923e+05
6 6 1.5707371794872e+07
1 7 2.2425801282051e+06
2 7 -3.3653846153846e+02
3 7 -4.2067307692308e+04
4 7 -2.2440945512821e+06
5 7 -1.6826923076923e+02
6 7 8.4134615384615e+03
7 7 4.4905448717949e+06
1 8 3.3653846153846e+02
2 8 2.2375320512821e+06
3 8 -3.3653846153846e+04
4 8 1.6826923076923e+02
5 8 -2.2466185897436e+06
6 8 1.6826923076923e+05
7 8 1.6826923076923e+03
8 8 4.4939102564103e+06
1 9 -4.2067307692308e+04
2 9 3.3653846153846e+04
3 9 7.8509935897436e+06
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5 9 1.6826923076923e+05
6 9 -7.8533493589744e+06
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9 32 2.6923076923077e+05
10 32 0.0000000000000e+00
11 32 4.6566128730774e-10
12 32 -2.6923076923077e+05
13 32 0.0000000000000e+00
14 32 9.3132257461548e-10
15 32 2.6923076923077e+05
16 32 0.0000000000000e+00
17 32 -9.3132257461548e-10
18 32 -2.6923076923077e+05
19 32 0.0000000000000e+00
20 32 -3.7252902984619e-09
21 32 -2.6923076923077e+05
22 32 0.0000000000000e+00
23 32 3.7252902984619e-09
24 32 2.6923076923077e+05
25 32 0.0000000000000e+00
26 32 0.0000000000000e+00
27 32 2.3283064365387e-10
28 32 0.0000000000000e+00
29 32 0.0000000000000e+00
30 32 0.0000000000000e+00
31 32 0.0000000000000e+00
32 32 2.1538461538462e+08
1 33 2.0192307692308e+05
2 33 -4.0384615384615e+05
3 33 -3.7252902984619e-09
4 33 -2.0192307692308e+05
5 33 4.0384615384615e+05
6 33 4.1909515857697e-09
7 33 2.0192307692308e+05
8 33 4.0384615384615e+05
9 33 -7.4505805969238e-09
10 33 -2.0192307692308e+05
11 33 -4.0384615384615e+05
12 33 0.0000000000000e+00
13 33 -2.0192307692308e+05
14 33 4.0384615384615e+05
15 33 0.0000000000000e+00
16 33 2.0192307692308e+05
17 33 -4.0384615384615e+05
18 33 0.0000000000000e+00
19 33 -2.0192307692308e+05
20 33 -4.0384615384615e+05
21 33 0.0000000000000e+00
22 33 2.0192307692308e+05
23 33 4.0384615384615e+05
24 33 0.0000000000000e+00
25 33 0.0000000000000e+00
26 33 4.6566128730774e-10
27 33 0.0000000000000e+00
28 33 -1.7462298274040e-10
29 33 0.0000000000000e+00
30 33 0.0000000000000e+00
31 33 0.0000000000000e+00
32 33 0.0000000000000e+00
33 33 7.5384615384615e+08

DOF mapping file .dof

1.1
1.2
1.3
1.1
1.2
1.3
2.1
2.2
2.3
2.1
2.2
2.3
3.1
3.2
3.3
3.1
3.2
3.3
4.1
4.2
4.3
4.1
4.2
4.3
17.1
17.2
17.3
18.1
18.2
18.3
19.1
19.2
19.3

S4 is expanded to C3D8I. Also, what about the rotational DOFs handled internally by CalculiX ? All in all, CalculiX may not be the best software to generate stiffness matrices of 1D and 2D elements.

Isn’t temperature a DOF. 8 * (3+1)=32
Anyway I would be missing one. ??? Maybe the normal direction is stored for the shell elements?. Not sure sorry.

Temperature is a DOF but I don’t think it’s included in the stiffness matrix in a frequency analysis. Or at least shouldn’t be.

I would impose a temperature to my shell to see if all those zeros become numbers.
I would invert my shell to check if some degree of freedom change sign.

It must have something to do with the internal expansion that CalculiX does from shell to solid element.

Note that in the `.dof` file, the real nodes occur twice for every degree of freedom. That could be the expansion.
It also lists extra nodes 17, 18 and 19 (with 3 DOF’s each) that are not present in the input file.
Maybe those are needed for internal housekeeping.

If I create a single `he20`/`C3D20` element and run the same `*STEP`, I get a `.dof` file with 60 degrees of freedom, which matches the expected 3 degrees of freedom per node.

Comments in the source say S4 is expanded to 11 nodes (8 actual plus 3 extra) which means 33 DOFs. I guess the extra nodes are for some internal use. Those final 3 nodes (17,18,19, matrix rows 25-33) are probably the internal ones. Many of them are zero in the stiffness matrix - the whole bottom 9x9 corner is diagonal.

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