5. C modeling

5.1. Characteristics of modeling

10 items TUYAU_6M.

5.2. Characteristics of the mesh

10 SEG3 stitches. The beam is oriented according to the vector (4, 3, 0).

5.3. Note on the content of the fields

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 corresponding 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.

5.4. Quantities tested and results of the C modeling

	Load case	Size	Reference	% difference
1	\({F}_{X}=4E+02\)	DX	5.53E—06	—0.04
1	\({F}_{Y}=3E+02\)	DY	4.14E—06	—0.04
2	\({F}_{X}=–3E+02\)	DRZ	2.63E—02	—0.04
2	\({F}_{Y}=4E+02\)	DX	—5.27E—02	—0.056
2		DY	7.02E—02	—0.056
3	\({F}_{Z}=5E+02\)	DRX	1.58E—02	—0.04
3		DRY	—2.11E—02	—0.039
3		DZ	8.78E—02	—0.056
4	\({M}_{X}=4E+02\)	DRX	1.10E—02	0
4	\({M}_{Y}=3E+02\)	DRY	8.21E—03	0
5	\({M}_{X}=–3E+02\)	DRX	—6.32E—03	—0.04
5	\({M}_{Y}=4E+02\)	DRY	8.42E—03	—0.04
5		DZ	—2.63E—02	—0.04
6	\({M}_{Z}=5E+02\)	DRZ	1.05E—02	—0.039
6		XX	—1.58E—02	—0.04
6		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	EFGE_ELNO	M18	1	1	N	5.00E+02	0.136
1	EPSI_ELGA	M18	1	1	EPXX	1.38E—06	—0.031
1	SIEF_ELGA	M18	1	1	SIXX	2.76E+05	—1.159
4	EFGE_ELNO	M18	1	1	MT	5.00E+02	0
4	EPSI_ELGA	M18	1	1	EPXY	—8.77E—05	—0.102
4	EPSI_ELGA	M18	693	693	EPXY	—1.09E—04	0.049
4	SIEF_ELGA	M18	1	1	SIXY	—6.75E+06	—0.159
4	SIEF_ELGA	M18	693	693	SIXY	—8.42E+06	0.049
5	EFGE_ELNO	M18	1	1	MFY	5.00E+02	0.123
5	EPSI_ELGA	M18	479	479	EPXX	6.74E—05	—0.046
5	SIEF_ELGA	M18	479	479	SIXX	1.35E+07	—1.288
6	EFGE_ELNO	M18	1	1	MFZ	5.00E+02	0.123
6	EPSI_ELGA	M18	471	471	EPXX	6.74E—05	—0.046
6	SIEF_ELGA	M18	471	471	SIXX	1.35E+07	—1.288
7	EPSI_ELGA	M18	1	1	EPYY	2.28E—04	—1.716
7	EPSI_ELGA	M18	693	693	EPYY	1.78E—04	0.741
7	SIEF_ELGA	M18	1	1	SIYY	4.56E+07	—0.641
7	SIEF_ELGA	M18	693	693	SIYY	3.56E+07	—0.371
8	EFGE_ELNO	M1	1	1	MFY	1764.3	2
9	EFGE_ELNO	M1	1	1	MFY	1764.3	2

Generalized deformations DEGE_ELNO:

Load case	Loads	Size	Reference	% difference
1	FX= 4.102	EPXX		1.38155E-06	—0.04
	FY= 3.102
2	FX= —3.102	GAXY	3.5920E-06	32
	FY= 4.102	KZ	1.0530E—02	—1.2
3	FZ= 5.102	GAXZ	3.5920E-06	32
		KY	—1.0530E—02	—1.2
4	MX= 4.102	GAT	2.73783E-03	0
	MY= 3.102
5	MX= —3.102	KY	2.1060E-03	—0.04
	MY= 4.102
6	MZ= 5 102	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

5.5. 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: \({\gamma }_{\mathrm{xy}}={\theta }_{z}-\frac{{\mathrm{du}}_{y}}{\mathrm{dx}}\), 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.