3. Modeling A#
3.1. Characteristics of modeling#
10 items TUYAU.
3.2. Characteristics of the mesh#
10 SEG3 stitches. The beam is oriented according to the vector \((\mathrm{4,}\mathrm{3,}0)\).
3.3. Notes on field content#
The fields at the Gauss points for element TUYAU, EPSI_ELGA, and SIEF_ELGA, which provide the deformations and constraints at the integration points in the element’s local coordinate system, are organized as follows:
We store the values:
for each Gauss point in the length, \((n=\mathrm{1,}3)\)
for each integration point in the thickness, \((n=\mathrm{1,}{\mathrm{2N}}_{\mathrm{COU}}+1=7)\)
for each integration point on the circumference, \((n=\mathrm{1,}{\mathrm{2N}}_{\mathrm{SECT}}+1=33)\)
6 deformation or stress components:
EPXX EPYY EPZZ EPXY EPXZ EPYZ or
SIXX SIYY SIZZ SIXY SIXZ SIYZ
where \(X\) refers to the direction given by the element’s two vertex nodes, \(Y\) represents the angle \(\phi\) describing the circumference, and \(Z\) represents the radius. EPZZ and EPYZ correspond to \({\epsilon }_{\mathrm{rr}}\), \({\epsilon }_{r\phi }\) in the case of deformations and SIZZ and SIYZ corresponding to \({\sigma }_{\mathrm{rr}}\), \({\sigma }_{r\phi }\) in the case of constraints are taken equal to zero.
For MECA_STATIQUE or MACRO_ELAS_MULT, the number of layers is fixed, and equal to 3, and the number of sectors is equal to 16.
EFGE_ELNO represents the efforts generalized to the 3 nodes in the classical way: N, VY, VZ, MT, MFY, MFZ.
3.4. Quantities tested and Results of modeling A#
Load case |
Size |
Reference |
% difference |
\({F}_{X}=4.0E+02\) |
DX |
5.53E—06 |
—0.04 |
\({F}_{Y}=3.00E+02\) |
DY |
4.14E—06 |
—0.04 |
\({F}_{X}=–3.0E+02\) |
|
2.63E—02 |
—0.04 |
\({F}_{Y}=4.0E+02\) |
DX |
—5.27E—02 |
—0.056 |
DY |
7.02E—02 |
—0.056 |
|
\({F}_{Z}=5.0E+02\) |
|
1.58E—02 |
—0.04 |
DRY |
—2.11E—02 |
—0.039 |
|
DZ |
8.78E—02 |
—0.056 |
|
\({M}_{X}=4.0E+02\) |
|
1.10E—02 |
0 |
\({M}_{Y}=3.0E+02\) |
|
8.21E—03 |
0 |
\({M}_{X}=–3.0E+02\) |
|
—6.32E—03 |
—0.04 |
\({M}_{Y}=4.0E+02\) |
|
8.42E—03 |
—0.04 |
DZ |
—2.63E—02 |
—0.04 |
|
\({M}_{Z}=5.0E+02\) |
|
1.05E—02 |
—0.039 |
XX |
—1.58E—02 |
—0.04 |
|
DY |
2.11E—02 |
—0.039 |
|
7: pressure |
WO |
7.38E—06 |
—2.946 |
8: gravity |
DZ |
—4.646E-02 |
0.09 |
9: distributed load |
DZ |
—4.646E-02 |
0.09 |
Case of load |
Field |
Mesh |
Point |
Component |
Reference |
% difference |
|
1 |
|
|
1 |
1 |
N |
5.00E+02 |
0.136 |
1 |
|
|
1 |
1 |
|
1.38E—06 |
—0.031 |
1 |
|
|
1 |
1 |
|
2.76E+05 |
—1.159 |
4 |
|
|
1 |
1 |
MT |
5.00E+02 |
0 |
4 |
|
|
1 |
1 |
|
—8.77E—05 |
—0.102 |
4 |
|
|
693 |
693 |
|
—1.09E—04 |
0.049 |
4 |
|
|
1 |
1 |
|
—6.75E+06 |
—0.159 |
4 |
|
|
693 |
693 |
|
—8.42E+06 |
0.049 |
5 |
|
|
1 |
1 |
|
|
0.123 |
5 |
|
|
479 |
479 |
|
6.74E—05 |
—0.046 |
5 |
|
|
479 |
479 |
|
1.35E+07 |
—1.288 |
6 |
|
|
1 |
1 |
|
5.00E+02 |
0.123 |
6 |
|
|
471 |
471 |
|
6.74E—05 |
—0.046 |
6 |
|
|
471 |
471 |
|
1.35E+07 |
—1.288 |
7 |
|
|
1 |
1 |
|
2.28E—04 |
—1.716 |
7 |
|
|
693 |
693 |
|
|
0.741 |
7 |
|
|
1 |
1 |
|
4.56E+07 |
—0.641 |
7 |
|
|
693 |
693 |
|
|
—0.371 |
8 |
|
|
1 |
1 |
|
1764.3 |
2 |
9 |
|
|
1 |
1 |
|
1764.3 |
2 |
Generalized deformations DEGE_ELNO:
Case of load |
Loads |
Size |
Reference |
% difference |
1 |
|
|
1.38155E-06 |
—0.04 |
\({F}_{Y}=3E+02\) |
||||
2 |
|
|
3.5920E-06 |
32.0 |
\({F}_{Y}=4E+02\) |
KZ |
1.0530E—02 |
—1.2 |
|
3 |
|
|
3.5920E-06 |
32 |
KY |
—1.0530E—02 |
—1.2 |
||
4 |
|
|
2.73783E-03 |
0 |
\({M}_{Y}=3E+02\) |
||||
5 |
|
KY |
2.1060E-03 |
—0.04 |
\({M}_{Y}=4E+02\) |
||||
6 |
|
KZ |
2.1060E-03 |
—0.04 |
Natural frequency |
Reference |
% difference |
1 |
2.90229 |
0.05 |
2 |
2.90229 |
0.05 |
3 |
18.18967 |
0.08 |
4 |
18.18967 |
0.08 |
5 |
50.99367 |
0.02 |
6 |
50.99367 |
0.02 |
7 |
99.81783 |
0.2 |
8 |
99.81783 |
0.2 |
9 |
157.0190 |
0.001 |
10 |
164.9922 |
0.3 |
11 |
164.9922 |
0.3 |
12 |
253.185 |
2 |
3.5. notes#
The shear values corresponding to the shear force are not accurate for this modeling. This is due to the second-order interpolation functions of this element, for beam movements and girder rotations. Since transverse beam shears are obtained by: \({\gamma }_{\mathit{xy}}\mathrm{=}{\theta }_{z}\mathrm{-}\frac{{\mathit{du}}_{y}}{\mathit{dx}}\), and because for simple bending, the rotations vary like polynomials of order 2, but the displacements, like polynomials of order 3, which is poorly approximated by the interpolation functions. The derivative of the displacements is therefore not precise.