3. Modeling A#

3.1. Characteristics of modeling and meshing#

Line element: POU_D_TGM.

Mechanical characteristics of the section (units homogeneous to meters):

\(A\)	\(\mathrm{IY}\)	\(\mathrm{IZ}\)	\(\mathrm{AY}\)		\(\mathrm{AZ}\)	\(\mathrm{JX}\)	\(\mathrm{JG}\)
8.0e-04	2.666667e-08	1.066667e-07	1.191790e+00	1.172840e+00	7.093682e-08	7.093682e-08	1.438125e-12

Loading at point \(B\).

	\(\mathrm{Fx}\)	\(\mathrm{Fy}\)	\(\mathrm{Fz}\)
Instant 1	\(\mathrm{80 000}\text{N}\)	\(–\mathrm{150N}\)	\(–\mathrm{200N}\)
Instant 2	\(\mathrm{80 000}\text{N}\)	\(–\mathrm{280N}\)	\(–\mathrm{400N}\)

At time 1 the section remains elastic, at time 2 the section is partially plasticized.

3.2. Fiber numbering#

The main inertia coordinate system \((\mathrm{Y0},\mathrm{Z0})\) of the beam must be rotated by 90° for the greatest inertia \(\mathrm{IZ}\) to be along the \(Y\) axis of the global coordinate system. The aim is to test the ORIENTATION keyword from the AFFE_CARA_ELEM command.

In the local frame of reference of the beam	In the global study benchmark

Several « types » of fibers are used:

fibers whose numbers range from 1 to 32. They are assigned a non-zero area in the command file,
fibers whose numbers are 33, 34, 34, 35, 36. They are located at the 4 corners of the section. These fibers, in the command file, are assigned a cross section equal to zero. They are only used for post-processing,
fibers whose numbers range from 37 to 60. They are located on the outer edge of the section. These fibers, in the command file, are assigned a cross section equal to zero. They are only used for post-processing.

3.3. Tested sizes and results#

The quantity tested and analyzed is SIEF_ELGA, at the first Gauss point of element \(\text{E00200}\). It is the Gauss point closest to the embedment, \(\mathrm{Lo}=\mathrm{2.95491933m}\).

3.3.1. Elastic behavior#

Gauss point constraints: SIEF_ELGA

The constraints calculated by equation [éq 2.1-1] and by*Code_Aster* are given in the two tables below (values in \(\text{MPa}\)). The layout of the tables is based on the fiber layout diagram in the global coordinate system. The most used fiber is No. 36 with a constraint of \(\mathrm{390 }\text{MPa}\).

Stresses calculated in fibers by equation [éq 2.1-1]

Stresses calculated in fibers by*Code_Aster*.

The relative error between the two calculations is given in the table below. Whatever the fiber, it is still less than \(0.095\text{\%}\).

-0.022%	-0.013%	-0.009%	-0.009%	-0.008%	-0.007%	-0.007%
-0.026%	-0.012%	-0.009%	-0.009%	-0.008%	-0.007%	-0.007%
0.014%	-0.010%	-0.007%	-0.007%	-0.006%	-0.006%	-0.006%
-0.010%	0.010%	-0.004%	-0.004%	-0.005%	-0.005%	-0.005%
-0.013%	-0.029%	0.000%	-0.003%	-0.004%	-0.004%	-0.004%
-0.014%	-0.021%	0.012%	0.012%	0.000%	-0.002%	-0.003%
-0.014%	-0.019%	0.069%	0.069%	0.004%	-0.001%	-0.002%
-0.015%	-0.018%	-0.094%	0.012%	0.002%	0.000%
-0.015%	-0.018%	-0.043%	-0.043%	0.029%	0.006%	0.003%
-0.015%	-0.018%	-0.036%	0.036%	0.049%	0.009%	0.005%

3.3.2. Plastic behavior#

Gauss point constraints: SIEF_ELGA

The table below gives the stress values, after partial plasticization of the section, obtained with*Code_Aster*. The behavior of the material is « almost perfect » elasto-plastic, the work-hardening slope is low. The maximum stress, which is beyond the elastic limit, therefore remains very close to the elastic threshold of \(\mathrm{400MPa}\).

8	79	223	36	366	400	400
-16	55	199	199	342	400	400
-64	7	151	151	294	400	400
-112	-41	103	103	246	390	400
-160	-89	55	55	198	342	400
-208	-137	7	7	150	294	365
-256	-185	-41	41	102	246	317
-305	-233	-89	-89	54	198	269
-353	-281	-137	-137	6	150	221
-377	-305	-161	-161	-18	126	197

This calculation is carried out for the Code_Aster non-regression test.