v3.04.004 SSLV04 - Hollow cylinder with plane stresses#
Summary:
This test is taken from Guide VPCS (test SSLV04 /89) and its object is a hollow cylinder loaded with internal pressure.
This three-dimensional problem is treated with various models:
in \(3D\): 9 models (pentahedra, hexahedra, tetrahedra and pyramids, degrees 1 and 2),
in \(2D\) plane constraints: 4 models (triangles and quadrangles degrees 1 and 2, quadrangles with 9 knots),
in \(2D\) axisymmetric: 3 models (triangles and quadrangles degrees 1 and 2, quadrangles with 9 knots).
The tested functionalities are:
distributed pressure,
shape effect (with fixed or variable pressure),
mandatory trips,
stiffness matrices,
deformations and stresses at the nodes,
nodal reactions (K modeling),
use of MACR_LIGN_COUP on a mult_elas concept and a comb_fourier concept
(modeling I)
There are 16 models.
Reference Issue
Geometry
Coordinates of the points:
\(>\) |
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|
|
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||
\(>\) |
0.100 |
0.200 |
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|
|
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|
\(>\) |
|
|
|
|
|||
\(>\) |
0 |
Material Properties
The Young’s modulus of the material is equal to \(>\).
The Poisson’s ratio is equal to \(>\).
Boundary conditions and loads
Internal pressure:
\(>\)
Variable internal pressure (P modeling only):
\(>\) varies linearly from \(>\) to \(>\) to \(>\) to \(>\)
Reference solution
Calculation method used for the reference solution
In plane stress (cylinder with free edges at the ends)
\(>\)
We get:
for \(>\): \(>\); for \(>\): \(>\)
Transition to the Cartesian axis system:
\(>\)
with:
\(>\) in points \(>\) and \(>\),
\(>\) in points \(>\) and \(>\),
\(>\) at points \(>\) and \(>\).
Benchmark Results
Displacements \(>\) and constraints \(>\) at points \(>\).
Bibliographical references
Guide VPCS. SSLV04 /89
Y.C. FUNG. Foundations of solid mechanics. Prentice-hall, Inc. Englewood Cliffs. NJ. 1965 p.243-245.
COURBON. Material strength p 649
Modeling A
Modelling characteristics
3D elements (PENTA6 and HEXA8).
Mesh obtained by extrusion from a \(>\) mesh similar to the mesh below (30 elements in the radial direction with gradual de-refinement and \(>\) elements in the circumferential direction).
+———————————+——————————-+——————-+———+———+———++++ |Following axis :math:`>`: |1 layer of elements | | +---------------------------------+-------------------------------+-------------------+---------+---------+---------++++ |Total thickness: |:math:`>` | | +---------------------------------+-------------------------------+-------------------+---------+---------+---------++++ | | | | +---------------------------------+-------------------------------+-------------------+---------+---------+---------++++ |Boundary conditions: | | | +---------------------------------+-------------------------------+-------------------+---------+---------+---------++++ | | | | +---------------------------------+-------------------------------+-------------------+---------+---------+---------++++ |Bow :math:`>`: :math:`>` | | | +---------------------------------+-------------------------------+-------------------+---------+---------+---------++++ |Face :math:`>` stuck in :math:`>` | | | +---------------------------------+-------------------------------+-------------------+---------+---------+---------++++ | |Face :math:`>` blocked normally | +---------------------------------+-------------------------------+-------------------+---------+---------+---------++++ |Pressure on the face :math:`>` |:math:`>` | | +---------------------------------+-------------------------------+-------------------+---------+---------+---------++++ | | | | | || +———————————+——————————-+——————-+———+———+———++++ |Node names: |:math:`>`|\(>`|:math:\)>`| || +———————————+——————————-+——————-+———+———+———++++ | |:math:`>`|\(>`|:math:\)>`| || +———————————+——————————-+——————-+———+———+———++++
Mesh characteristics
Number of knots: 1922
Number of meshes and types: 900 PENTA6, 450 HEXA8 and 90 QUAD4 (inner skin sides).
Tested quantities and results
Location |
Size |
Reference |
Reference Type |
Tolerance |
|
Field nor_ DNOR Point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
0.1% |
|
Field nor_ DNOR point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
0.1% |
|
Field nor_ DNOR point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
0.1% |
|
Field nor_ DNOR point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
0.1% |
|
Field nor_ PRES point \(>\) |
Normal vector, component \(>\) |
1.0 |
“ANALYTIQUE” |
1.5% |
|
Field nor_ PRES point \(>\) |
Normal vector, component \(>\) |
0.0 |
“ANALYTIQUE” |
1.5% |
|
Field nor_ PRES point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
1.5% |
|
Field nor_ PRES point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
1.5% |
Location |
Size |
Reference Value |
Reference Type |
Tolerance |
point \(>\) |
Field DEPL, comp. \(>\) |
5.9 10—5 |
“ANALYTIQUE” |
0.1% |
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-10 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—60. |
“ANALYTIQUE” |
2% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
0.5 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
2 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—4.5 10—4 |
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
5.9 10—4 |
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
\(>\) |
Field DEPL, comp. \(>\) |
4 10 —5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-10 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
2 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
2% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
2 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—6 10—5 |
“ANALYTIQUE” |
5% |
|
Field EPSI_NOEU, comp. \(>\) |
|
“ANALYTIQUE” |
5% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
\(>\) |
Field DEPL, comp. \(>\) |
3.69552 10—5 |
“ANALYTIQUE” |
0.1% |
Field DEPL, comp. \(>\) |
1.53073 10—5 |
“ANALYTIQUE” |
1% |
|
\(>\) |
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
0.1% |
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
0.1% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
6% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
6% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
0.1 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—80. |
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
10% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
10% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—5.2 10 —4 |
“ANALYTIQUE” |
1% |
|
\(>\) |
Field DEPL, comp. \(>\) |
2.82843 10 —5 |
“ANALYTIQUE” |
0.1% |
Field DEPL, comp. \(>\) |
2.82843 10—5 |
“ANALYTIQUE” |
0.1% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
6% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
0.2 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—20. |
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
8% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
2% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—1.3 10—4 |
“ANALYTIQUE” |
1% |
B Modeling
Modelling characteristics
3D elements (PENTA15 and HEXA20).
Mesh obtained by extrusion from mesh \(>\) below (F modeling)
+———————————+——————————-+——————–++ |Following axis :math:`>`: |2 layers of elements|| +———————————+——————————-+——————–++ |Total thickness: |:math:`>` || +———————————+——————————-+——————–++ | | || +———————————+——————————-+——————–++ |Boundary conditions: | || +———————————+——————————-+——————–++ | | || +———————————+——————————-+——————–++ |Bow :math:`>` | || +———————————+——————————-+——————–++ |Face :math:`>` stuck in :math:`>` | || +———————————+——————————-+——————–++ | |Face :math:`>` blocked normally | +---------------------------------+-------------------------------+--------------------++ |Pressure on the face :math:`>` |:math:`>` || +———————————+——————————-+——————–++
Node names: |
\(>\) |
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\(>\) |
|
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Mesh characteristics
Number of knots: 2115
Number of meshes and types: 400 PENTA15, 100 HEXA20, 100, 40 QUAD8 (inner skin sides)
Tested quantities and results
Location |
Size |
Reference |
Reference Type |
Tolerance |
|
Mesh MA751, point \(>\) |
|||||
SIRO_ELEM |
|
0.0 |
“ANALYTIQUE” |
||
Normal stress at element face |
\(>\) |
60.0 |
“ANALYTIQUE” |
0.5% |
|
\(>\) |
0.0 |
“ANALYTIQUE” |
0.01 |
||
SIRO_ELEM |
|
0.0 |
“ANALYTIQUE” |
0.1 |
|
Tangential stress in the element plane |
\(>\) |
0.0 |
“ANALYTIQUE” |
0.1 |
|
\(>\) |
0.0 |
“ANALYTIQUE” |
0.1 |
||
SIRO_ELEM |
|
0.0 |
“ANALYTIQUE” |
0.01 |
|
First constraint value |
\(>\) |
0.0 |
“ANALYTIQUE” |
“” |
0.01 |
tangential in the element plane |
\(>\) |
0.0 |
“ANALYTIQUE” |
“” |
0.05 |
SIRO_ELEM |
|
100.0 |
“ANALYTIQUE” |
0.15% |
|
Second constraint value |
\(>\) |
0.0 |
“ANALYTIQUE” |
“” |
4.5 |
tangential in the element plane |
\(>\) |
0.0 |
“ANALYTIQUE” |
“” |
0.20 |
“ANALYTIQUE” |
|||||
Mesh MA769, point \(>\) |
“ANALYTIQUE” |
||||
SIRO_ELEM |
|
-42.426 |
“ANALYTIQUE” |
6.0% |
|
Normal stress at element face |
\(>\) |
42.426 |
“ANALYTIQUE” |
4.0% |
|
\(>\) |
0.0 |
“ANALYTIQUE” |
0.01 |
||
SIRO_ELEM |
|
0.0 |
“ANALYTIQUE” |
0.01 |
|
Tangential stress in the element plane |
\(>\) |
0.0 |
“ANALYTIQUE” |
0.01 |
|
\(>\) |
0.0 |
“ANALYTIQUE” |
0.03 |
||
SIRO_ELEM |
|
0.0 |
“ANALYTIQUE” |
0.01 |
|
First constraint value |
\(>\) |
0.0 |
“ANALYTIQUE” |
“” |
0.01 |
tangential in the element plane |
\(>\) |
0.0 |
“ANALYTIQUE” |
0.01 |
|
SIRO_ELEM |
|
70.711 |
“ANALYTIQUE” |
3.5% |
|
Second constraint value |
\(>\) |
70.711 |
“ANALYTIQUE” |
4.5% |
|
tangential in the element plane |
\(>\) |
0.0 |
“ANALYTIQUE” |
“” |
0.1 |
Location |
Size |
Reference Value |
Reference Type |
Tolerance |
point \(>\) |
||||
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field DEPL, comp. \(>\) |
5.9 10—5 |
“ANALYTIQUE” |
1% |
|
Mesh MA251 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1% |
|
Mesh MA251 |
Champ SIGM_ELNO, comp. \(>\) |
— 60. |
“ANALYTIQUE” |
1% |
Mesh MA251 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.02 (absolute) |
|
Mesh MA251 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.5 (absolute) |
|
Mesh MA251 |
Champ EPSI_ELNO, comp. \(>\) |
5.9 10—4 |
“ANALYTIQUE” |
1% |
Mesh MA251 |
Champ EPSI_ELNO, comp. \(>\) |
— 4.5 10—4 |
“ANALYTIQUE” |
1% |
Mesh MA251 |
Champ EPSI_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1E-2 (absolute) |
|
Mesh MA251 |
Champ EPSI_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1E-2 (absolute) |
|
Point \(>\) |
||||
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field DEPL, comp. \(>\) |
4 10—5 |
“ANALYTIQUE” |
1% |
|
Mesh MA551 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.1% |
|
Mesh MA551 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.1 (absolute) |
|
Mesh MA551 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.015 (absolute) |
|
Mesh MA551 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.15 (absolute) |
|
Mesh MA551 |
Champ EPSI_ELNO, comp. \(>\) |
|
“ANALYTIQUE” |
1% |
Mesh MA551 |
Champ EPSI_ELNO, comp. \(>\) |
— 0.6. 10—4 |
“ANALYTIQUE” |
1% |
Mesh MA551 |
Champ EPSI_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1E-2 (absolute) |
|
Mesh MA551 |
Champ EPSI_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1E-2 (absolute) |
|
\(>\) |
||||
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
1% |
|
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
1% |
|
Mesh MA399 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
10% |
|
Mesh MA399 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
10% |
|
Mesh MA399 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.5 (absolute) |
|
Mesh MA399 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1% |
|
Mesh MA399 |
Champ EPSI_ELNO, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
1% |
Mesh MA399 |
Champ EPSI_ELNO, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
1% |
Mesh MA399 |
Champ EPSI_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1E-2 (absolute) |
|
Mesh MA399 |
Champ EPSI_ELNO, comp. \(>\) |
5.2 10 —4 |
“ANALYTIQUE” |
1% |
\(>\) |
||||
Field DEPL, comp. \(>\) |
—2.82843 10—5 |
“ANALYTIQUE” |
1% |
|
Field DEPL, comp. \(>\) |
2.82843 10—5 |
“ANALYTIQUE” |
1% |
|
Mesh MA695 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1% |
|
Mesh MA695 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1% |
|
Mesh MA695 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.05 (absolute) |
|
Mesh MA695 |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.1% |
|
Mesh MA695 |
Champ EPSI_ELNO, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
1% |
Mesh MA695 |
Champ EPSI_ELNO, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
1% |
Mesh MA695 |
Champ EPSI_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1E-2 (absolute) |
|
Mesh MA695 |
Champ EPSI_ELNO, comp. \(>\) |
1.3 10 —4 |
“ANALYTIQUE” |
1% |
C Modeling
Modelling characteristics
3D elements (TETRA4).
\(>\) is on the \(>\) axis
+———————————+——————————-+——————————————————————–++ |Die-cutting: |21 equidistant nodes on segments :math:`>`, :math:`>`, and :math:`>`|| +———————————+——————————-+——————————————————————–++ | |21 knots equidistant on arcs :math:`>` and :math:`>` || +———————————+——————————-+——————————————————————–++ | | || +———————————+——————————-+——————————————————————–++ |Following axis :math:`>`: |1 layer of elements || +———————————+——————————-+——————————————————————–++ |Total thickness: |:math:`>` || +———————————+——————————-+——————————————————————–++ | | || +———————————+——————————-+——————————————————————–++ |Boundary conditions: | || +———————————+——————————-+——————————————————————–++ | | || +———————————+——————————-+——————————————————————–++ |Bow :math:`>`: :math:`>` | || +———————————+——————————-+——————————————————————–++ |Face :math:`>` stuck in :math:`>` | || +———————————+——————————-+——————————————————————–++ | |Face :math:`>` blocked normally | +---------------------------------+-------------------------------+--------------------------------------------------------------------++ |Pressure on the face :math:`>` |:math:`>` || +———————————+——————————-+——————————————————————–++
Node names: |
\(>\) |
|
|
||
\(>\) |
|
|
Mesh characteristics
Number of knots: 1115
Number of meshes and types: 3724 TETRA4 and 1760 TRIA3 (inner skin sides)
Tested quantities and results
Location |
Size |
Reference Value |
Reference Type |
Tolerance |
|
Field nor_ DNOR Point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
0.1% |
|
Field nor_ DNOR point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
0.1% |
|
Field nor_ DNOR point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
0.1% |
|
Field nor_ DNOR point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
0.1% |
|
Field nor_ PRES point \(>\) |
Normal vector, component \(>\) |
1.0 |
“ANALYTIQUE” |
1.5% |
|
Field nor_ PRES point \(>\) |
Normal vector, component \(>\) |
0.0 |
“ANALYTIQUE” |
1.5% |
|
Field nor_ PRES point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
1.5% |
|
Field nor_ PRES point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
1.5% |
Location |
Size |
Reference Value |
Reference Type |
Tolerance |
point \(>\) |
Field DEPL, comp. \(>\) |
5.9 10—5 |
“ANALYTIQUE” |
0.1% |
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-10 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—60. |
“ANALYTIQUE” |
5% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
5% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
0.1 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
3 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—4.5 10—4 |
“ANALYTIQUE” |
5% |
|
Field EPSI_NOEU, comp. \(>\) |
5.9 10—4 |
“ANALYTIQUE” |
5% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
\(>\) |
Field DEPL, comp. \(>\) |
4 10 —5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-10 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
2 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
5% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—6 10—5 |
“ANALYTIQUE” |
15% |
|
Field EPSI_NOEU, comp. \(>\) |
|
“ANALYTIQUE” |
5% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
\(>\) |
Field DEPL, comp. \(>\) |
3.69552 10—5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
1.53073 10—5 |
“ANALYTIQUE” |
1% |
|
\(>\) |
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
11% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
10% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
0.5 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—80. |
“ANALYTIQUE” |
5% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
20% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
20% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—5.2 10 —4 |
“ANALYTIQUE” |
3% |
|
\(>\) |
Field DEPL, comp. \(>\) |
2.82843 10 —5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
2.82843 10—5 |
“ANALYTIQUE” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
10% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
0.4 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—20. |
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
11% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
5% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—1.3 10—4 |
“ANALYTIQUE” |
1% |
D Modeling
Modelling characteristics
3D element (TETRA10).
\(>\) is on the \(>\) axis
+———————————+——————————-+——————————————————————-++ |Die-cutting: |11 equidistant nodes on segments :math:`>`, :math:`>` and :math:`>`|| +———————————+——————————-+——————————————————————-++ | |11 knots equidistant on arcs :math:`>` and :math:`>` || +———————————+——————————-+——————————————————————-++ | | || +———————————+——————————-+——————————————————————-++ |Following axis :math:`>`: |1 layer of elements || +———————————+——————————-+——————————————————————-++ |Total thickness: |:math:`>` || +———————————+——————————-+——————————————————————-++ | | || +———————————+——————————-+——————————————————————-++ |Boundary conditions: | || +———————————+——————————-+——————————————————————-++ | | || +———————————+——————————-+——————————————————————-++ |Bow :math:`>`: :math:`>` | || +———————————+——————————-+——————————————————————-++ |Face :math:`>` stuck in :math:`>` | || +———————————+——————————-+——————————————————————-++ | |Face :math:`>` blocked normally | +---------------------------------+-------------------------------+-------------------------------------------------------------------++ |Pressure on the face :math:`>` |:math:`>` || +———————————+——————————-+——————————————————————-++
Node names: |
\(>\) |
|
|
||
\(>\) |
|
|
Mesh characteristics
Number of knots: 1395
Number of meshes and types: 652 TETRA10 and 480 TRIA6 (inner skin sides)
Tested quantities and results
Location |
Size |
Reference Value |
Reference Type |
Tolerance |
point \(>\) |
Field DEPL, comp. \(>\) |
5.9 10—5 |
“ANALYTIQUE” |
0.1% |
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—60. |
“ANALYTIQUE” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—4.5 10—4 |
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
5.9 10—4 |
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
\(>\) |
Field DEPL, comp. \(>\) |
4 10 —5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—6 10—5 |
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
|
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
\(>\) |
Field DEPL, comp. \(>\) |
3.69552 10—5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
1.53073 10—5 |
“ANALYTIQUE” |
1% |
|
\(>\) |
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
5% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
5% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
2 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—80. |
“ANALYTIQUE” |
5% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
5% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
2% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—5.2 10 —4 |
“ANALYTIQUE” |
1% |
|
\(>\) |
Field DEPL, comp. \(>\) |
2.82843 10 —5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
2.82843 10—5 |
“ANALYTIQUE” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
2E-3 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—20. |
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
2% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
2% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—1.3 10—4 |
“ANALYTIQUE” |
1% |
E Modeling
Modelling characteristics
Elements C_PLAN (TRIA3 + QUAD4)
Mesh \(>\) similar to the mesh below (30 elements in the radial direction with gradual deraffination and \(>\) elements in the circumferential direction).
Boundary conditions:
+———————————+——————————+++ |Side :math:`>` stuck in :math:`>` ||| +———————————+——————————+++ | |side :math:`>` locked normally | +---------------------------------+------------------------------+++ |Pressure on :math:`>` :math:`>` ||| +———————————+——————————+++
Node names: |
\(>\) |
|
|
\(>\) |
|
|
Mesh characteristics
Number of knots: 961
Number of meshes and types: 900 TRIA3, 450 QUAD4
Tested quantities and results
Location |
Size |
Reference Values |
Reference Type |
Tolerance |
|
Field nor_ DNOR Point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
0.1% |
|
Field nor_ DNOR point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
0.1% |
|
Field nor_ DNOR point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
0.1% |
|
Field nor_ DNOR point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
0.1% |
|
Field nor_ PRES point \(>\) |
Normal vector, component \(>\) |
1.0 |
“ANALYTIQUE” |
1.5% |
|
Field nor_ PRES point \(>\) |
Normal vector, component \(>\) |
0.0 |
“ANALYTIQUE” |
1.5% |
|
Field nor_ PRES point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
1.5% |
|
Field nor_ PRES point \(>\) |
Normal vector, component \(>\) |
-0.707 |
“ANALYTIQUE” |
1.5% |
Location |
Size |
Reference Value |
Reference Type |
Tolerance |
point \(>\) |
Field DEPL, comp. \(>\) |
5.9 10—5 |
“ANALYTIQUE” |
0.1% |
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-10 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—60. |
“ANALYTIQUE” |
2% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-5 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
2 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—4.5 10—4 |
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
5.9 10—4 |
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
\(>\) |
Field DEPL, comp. \(>\) |
4 10 —5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-10 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1.5 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
2% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-5 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—6 10—5 |
“ANALYTIQUE” |
12% |
|
Field EPSI_NOEU, comp. \(>\) |
|
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
\(>\) |
Field DEPL, comp. \(>\) |
3.69552 10—5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
1.53073 10—5 |
“ANALYTIQUE” |
1% |
|
\(>\) |
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
7% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
7% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-5 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—80. |
“ANALYTIQUE” |
1% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
15% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
15% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—5.2 10 —4 |
“ANALYTIQUE” |
1% |
|
\(>\) |
Field DEPL, comp. \(>\) |
2.82843 10 —5 |
“ANALYTIQUE” |
1% |
Field DEPL, comp. \(>\) |
2.82843 10—5 |
“ANALYTIQUE” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
3% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
2% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-5 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—20. |
“ANALYTIQUE” |
7% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
5% |
|
Field EPSI_NOEU, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
5% |
|
Field EPSI_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field EPSI_NOEU, comp. \(>\) |
—1.3 10—4 |
“ANALYTIQUE” |
7% |
F Modeling
Modelling characteristics
Elements C_Plan **** (**** QUAD8 **** **+** TRIA6 ****) **
Boundary conditions:
side \(>\) stuck in \(>\) |
|
side \(>\) locked normally |
|
press on \(>\) \(>\) |
Node names: |
\(>\) |
|
|
\(>\) |
|
|
Mesh characteristics
Number of knots: 591
Number of meshes and types: 200 TRIA6, 50 QUAD8
Tested quantities and results
G Modeling
Modelling characteristics
Modeling C_PLAN **** (**** QUAD9 **** )
Boundary conditions:
side \(>\) stuck in \(>\) |
side \(>\) locked normally |
pressure on \(>\) \(>\) |
Node names: |
\(>\) |
|
|
\(>\) |
|
|
Mesh characteristics
Number of knots: 441
Number of meshes and types: 100 QUAD9
Tested quantities and results
H Modeling
Modelling characteristics
Elements AXIS **** (**** TRIA3 + QUAD4 **** )
Boundary conditions:
node \(>\) stuck in \(>\) press on \(>\) \(>\) |
Node names: |
\(>\) |
|
|
\(>\) |
|
|
Mesh characteristics
Number of knots: 113
Number of meshes and types: 40 QUAD4, 80 TRIA3
Tested quantities and results
Location |
Size |
Reference Value |
Reference Type |
ORIN (0.1, 0) |
Membrane Stress*Pm* |
82.9180024359435 |
“ANALYTIQUE” |
ORIN (0.1, 0) |
Bending Stress*Pb* |
62.3994636932228 |
“ANALYTIQUE” |
ORIN (0.1, 0) |
Pm + Pb |
145.3170033 |
“ANALYTIQUE” |
EXTR (0.2, 0) |
Membrane Stress*Pm* |
82.9180024359435 |
“ANALYTIQUE” |
EXTR (0.2, 0) |
Flexural Stress*Pb* |
47.1217312842318 |
“ANALYTIQUE” |
EXTR (0.2, 0) |
Pm + Pb |
35.80513084 |
“ANALYTIQUE” |
Modeling I
Modelling characteristics
Elements AXIS **** (**** TRIA6 **** + QUAD8 )
Boundary conditions:
Node \(>\) stuck in \(>\) press on \(>\) \(>\) |
Node names: |
\(>\) |
|
|
\(>\) |
|
|
Mesh characteristics
Number of knots: 175
Number of meshes and types: 20 QUAD8, 40 TRIA6
Tested quantities and results
At the end of this test, a loading in rotation in pure axisymmetry was compared with the same loading in Fourier mode 0. There are many identical results.
Results from MACR_LIGN_COUP on the AE segment:
The NON_REGRESSION tests are done on the tables from MACR_LIGN_COUP applied to a CHAMP_GD. They are the “AUTRE_ASTER” reference values for tables from MACR_LIGN_COUP applied to a mult_elas result.
Location |
Size |
Reference value |
Reference type |
Precision |
||
Point |
ABSC_CUR |
Field |
Component |
|||
\(>\) |
0000 |
DEPL |
DX |
“NON_REGRESSION” |
||
\(>\) |
0.005 |
DEPL |
DX |
“NON_REGRESSION” |
||
\(>\) |
0.1 |
DEPL |
DX |
NON_REGRESSION “ |
||
\(>\) |
0000 |
DEPL |
DX |
1.35511346318 |
“AUTRE_ASTER” |
|
\(>\) |
0.005 |
DEPL |
DX |
1.35531703991 |
“AUTRE_ASTER” |
|
\(>\) |
0.1 |
DEPL |
DX |
1.35511346318 |
“AUTRE_ASTER” |
|
The same procedure is used to test the result of MACR_LIGNE_COUPE applied to a comb_fourier concept.
Location |
Size |
Reference value |
Reference type |
Precision |
||
Point |
ABSC_CUR |
Field |
Component |
|||
\(>\) |
0000 |
DEPL |
DX |
“NON_REGRESSION” |
||
\(>\) |
0.005 |
DEPL |
DX |
“NON_REGRESSION” |
||
\(>\) |
0.1 |
DEPL |
DX |
NON_REGRESSION “ |
||
\(>\) |
0000 |
DEPL |
DX |
5.8999716105E-05 |
“AUTRE_ASTER” |
|
\(>\) |
0.005 |
DEPL |
DX |
5.90000189066E-05 |
“AUTRE_ASTER” |
|
\(>\) |
0.1 |
DEPL |
DX |
5.89997161037E-05 |
“AUTRE_ASTER” |
|
J Modeling
Modelling characteristics
Elements AXIS **** (**** QUAD9 **** )
Boundary conditions:
node \(>\) stuck in \(>\) press on \(>\) \(>\) |
Node names: |
\(>\) |
|
|
\(>\) |
|
|
Mesh characteristics
Number of knots: 205
Number of meshes and types: 40 QUAD9
Tested quantities and results
K-Modeling
Modelling characteristics
Elements 3D **** (**** PENTA6 **** **** and HEXA8 )
Mesh obtained by extrusion from mesh \(>\) below (modeling E)
+———————————————————–+——————————-+—————————————-++ |Following axis :math:`>`: |2 layers of elements || +———————————————————–+——————————-+—————————————-++ |Total thickness: |:math:`>` || +———————————————————–+——————————-+—————————————-++ | | || +———————————————————–+——————————-+—————————————-++ |Boundary conditions: | || +———————————————————–+——————————-+—————————————-++ | | || +———————————————————–+——————————-+—————————————-++ |Node :math:`>`: :math:`>` face :math:`>` stuck in :math:`>` | || +———————————————————–+——————————-+—————————————-++ | |Face :math:`>` blocked normally | +-----------------------------------------------------------+-------------------------------+----------------------------------------++ |Face :math:`>` |radial displacement imposed on :math:`>`|| +———————————————————–+——————————-+—————————————-++
Node names: |
\(>\) |
|
|
plan \(>\) |
|
|
|
plan \(>\) |
|
|
|
Node names: |
\(>\) |
|
|
plan \(>\) |
|
|
|
plan \(>\) |
|
|
|
Mesh characteristics
Number of knots: 513
Number of meshes and types: 400 PENTA6, 100 HEXA8, 100, 40 QUAD4 (inner skin sides)
Remarks
In this case, the load is in forced displacement, unlike other models. We are testing the reactions.
Tested quantities and results
Location |
Size |
Reference Value |
Reference Type |
Tolerance |
\(>\) |
Field REAC_NODA, comp FX |
1.0884 E—3 |
“AUTRE_ASTER” |
0.7% |
Field REAC_NODA, comp FY |
4.5084 E—4 |
“AUTRE_ASTER” |
1.7% |
|
\(>\) |
Field REAC_NODA, comp FX |
2.1768 E—3 |
“AUTRE_ASTER” |
1% |
Field REAC_NODA, comp FY |
9.0170 E—4 |
“AUTRE_ASTER” |
1.3% |
|
\(>\) |
Field REAC_NODA, comp FX |
1.0884 E—3 |
“AUTRE_ASTER” |
0.7% |
Field REAC_NODA, comp FY |
4.5084 E—4 |
“AUTRE_ASTER” |
1.7% |
|
\(>\) |
Field REAC_NODA, comp FX |
1.1636 E—3 |
“AUTRE_ASTER” |
0.7% |
Field REAC_NODA, comp FY |
1.8429 E—4 |
“AUTRE_ASTER” |
0.6% |
|
\(>\) |
Field REAC_NODA, comp FX |
1.0045 E—3 |
“AUTRE_ASTER” |
1% |
Field REAC_NODA, comp FY |
6.1550 E—4 |
“AUTRE_ASTER” |
1% |
|
\(>\) |
Field REAC_NODA, comp FX |
2.3272 E—3 |
“AUTRE_ASTER” |
0.5% |
Field REAC_NODA, comp FY |
3.6858 E—4 |
“AUTRE_ASTER” |
0.6% |
|
\(>\) |
Field REAC_NODA, comp FX |
2.0090 E—3 |
“AUTRE_ASTER” |
0.7% |
Field REAC_NODA, comp FY |
1.2310 E—3 |
“AUTRE_ASTER” |
0.8% |
Remarks
We check that the nodal forces of reactions are zero at all nodes, except on the nodes of surface \(>\) and surfaces \(>\) and \(>\).
L Modeling
Modelling characteristics
Elements 3D **** (**** PYRAM5 **** )
+———————————————————–+——————————-+———————————————-++ |Following axis :math:`>`: |each parallelepiped is divided into 6 pyramids|| +———————————————————–+——————————-+———————————————-++ |Total thickness: |:math:`>` || +———————————————————–+——————————-+———————————————-++ | | || +———————————————————–+——————————-+———————————————-++ |Boundary conditions: | || +———————————————————–+——————————-+———————————————-++ | | || +———————————————————–+——————————-+———————————————-++ |Node :math:`>`: :math:`>` face :math:`>` stuck in :math:`>` | || +———————————————————–+——————————-+———————————————-++ | |Face :math:`>` blocked normally | +-----------------------------------------------------------+-------------------------------+----------------------------------------------++ |Pressure on the face :math:`>` |:math:`>` || +———————————————————–+——————————-+———————————————-++
Node names: |
\(>\) |
|
|
|
Mesh characteristics
Number of knots: 342
Number of meshes and types: 600 PYRAM5 620 QUAD4 (inner skin sides)
Tested quantities and results
M Modeling
Modelling characteristics
Elements 3D **** (**** PYRAM13 **** )
+———————————————————–+——————————-+———————————————-++ |Following axis :math:`>`: |each parallelepiped is divided into 6 pyramids|| +———————————————————–+——————————-+———————————————-++ |Total thickness: |:math:`>` || +———————————————————–+——————————-+———————————————-++ | | || +———————————————————–+——————————-+———————————————-++ |Boundary conditions: | || +———————————————————–+——————————-+———————————————-++ | | || +———————————————————–+——————————-+———————————————-++ |Node :math:`>`: :math:`>` face :math:`>` stuck in :math:`>` | || +———————————————————–+——————————-+———————————————-++ | |Face :math:`>` blocked normally | +-----------------------------------------------------------+-------------------------------+----------------------------------------------++ |Pressure on the face :math:`>` |:math:`>` || +———————————————————–+——————————-+———————————————-++
Node names: |
\(>\) |
|
|
|
Mesh characteristics
Number of knots: 1703
Number of meshes and types: 600 PYRAM13 620 QUAD8 (inner skin sides)
Tested quantities and results
Location |
Size |
Reference Value |
Reference Type |
Tolerance |
point \(>\) |
||||
Field DEPL, comp. \(>\) |
5.90 10-5 |
“ANALYTIQUE” |
1% |
|
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
M6 mesh |
Champ SIGM_ELNO, comp. \(>\) |
—60. |
“ANALYTIQUE” |
1% |
M6 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1% |
|
M6 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.1 (absolute) |
|
M6 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.1 (absolute) |
|
M6 mesh |
Champ EPSI_ELNO, comp. \(>\) |
—4.5 10—4 |
“ANALYTIQUE” |
1% |
M6 mesh |
Champ EPSI_ELNO, comp. \(>\) |
5.9 10—4 |
“ANALYTIQUE” |
1% |
M6 mesh |
Champ EPSI_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
|
M6 mesh |
Champ EPSI_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
|
\(>\) |
||||
Field DEPL, comp. \(>\) |
4 10-5 |
“ANALYTIQUE” |
1% |
|
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
M58 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.1 (absolute) |
|
M58 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.12% |
|
M58 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.1 (absolute) |
|
M58 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.1. (absolute) |
|
M58 mesh |
Champ EPSI_ELNO, comp. \(>\) |
—6 10—5 |
“ANALYTIQUE” |
1% |
M58 mesh |
Champ EPSI_ELNO, comp. \(>\) |
|
“ANALYTIQUE” |
1% |
M58 mesh |
Champ EPSI_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
|
M58 mesh |
Champ EPSI_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
|
\(>\) |
||||
Field DEPL, comp. \(>\) |
2.82843 10—5 |
“ANALYTIQUE” |
1% |
|
Field DEPL, comp. \(>\) |
2.82843 10—5 |
“ANALYTIQUE” |
1% |
|
M546 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1% |
|
M546 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1% |
|
M546 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.1 (absolute) |
|
M546 mesh |
Champ SIGM_ELNO, comp. \(>\) |
—80. |
“ANALYTIQUE” |
0.12% |
M546 mesh |
Champ EPSI_ELNO, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
1% |
M546 mesh |
Champ EPSI_ELNO, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
2% |
M546 mesh |
Champ EPSI_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
|
M546 mesh |
Champ EPSI_ELNO, comp. \(>\) |
—5.2 10 —4 |
“ANALYTIQUE” |
1% |
\(>\) |
||||
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
1% |
|
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
1% |
|
M599 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1% |
|
M599 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1% |
|
M599 mesh |
Champ SIGM_ELNO, comp. \(>\) |
“ANALYTIQUE” |
0.01 (absolute) |
|
M599 mesh |
Champ SIGM_ELNO, comp. \(>\) |
-20 |
“ANALYTIQUE” |
0.12% |
M599 mesh |
Champ EPSI_ELNO, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
1% |
M599 mesh |
Champ EPSI_ELNO, comp. \(>\) |
0.7 10—4 |
“ANALYTIQUE” |
1% |
M599 mesh |
Champ EPSI_ELNO, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
|
M599 mesh |
Champ EPSI_ELNO, comp. \(>\) |
—1.3 10—4 |
“ANALYTIQUE” |
1% |
N Modeling
Modelling characteristics
Elements 3D **** (**** PENTA15 **** **** and HEXA20 )
Mesh obtained by extrusion from a \(>\) mesh similar to the mesh below (8 elements in the radial direction, \(>\) elements in the circumferential direction) and duplicated to have a complete section of the cylinder (over 360°).
+—————————————————————+———+——————-+———+———+++ |Following axis :math:`>`: |1 layer of elements | +---------------------------------------------------------------+---------+-------------------+---------+---------+++ |Total thickness: |:math:`>` | +---------------------------------------------------------------+---------+-------------------+---------+---------+++ | | | +---------------------------------------------------------------+---------+-------------------+---------+---------+++ |Boundary conditions: | | +---------------------------------------------------------------+---------+-------------------+---------+---------+++ | | | +---------------------------------------------------------------+---------+-------------------+---------+---------+++ |Face :math:`>` stuck in :math:`>` face :math:`>` stuck normally | +---------------------------------------------------------------+---------+-------------------+---------+---------+++ | | | +---------------------------------------------------------------+---------+-------------------+---------+---------+++ |Pressure on the face :math:`>` |:math:`>` | +---------------------------------------------------------------+---------+-------------------+---------+---------+++ |shape effect on sections |:math:`>` | +---------------------------------------------------------------+---------+-------------------+---------+---------+++ | | | +---------------------------------------------------------------+---------+-------------------+---------+---------+++ |Node names: |:math:`>` |:math:`>`|\(>`||| +---------------------------------------------------------------+---------+-------------------+---------+---------+++ | |:math:\)>` |:math:`>`|:math:`>`||| +—————————————————————+———+——————-+———+———+++
Mesh characteristics
Number of knots: 8832
Number of meshes and types: 1024 PENTA15, 512 HEXA20, 512, 1176 QUAD8 and 2048 TRIA6.
Remarks
Contrary to previous models, the background effect applying to the sections at the ends of the cylinder is taken into account here.
Tested quantities and results
Location |
Size |
Reference Value |
Reference Type |
Tolerance |
point \(>\) |
||||
Field DEPL, comp. \(>\) |
5.60 10-5 |
“NON_DEFINI” |
0.1% |
|
Field DEPL, comp. \(>\) |
“NON_DEFINI” |
1E-4 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—60. |
“NON_DEFINI” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1E-4 (absolute) |
||
\(>\) |
||||
Field DEPL, comp. \(>\) |
3.4 10-5 |
“NON_DEFINI” |
0.1% |
|
Field DEPL, comp. \(>\) |
“NON_DEFINI” |
1E-4 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
0.1 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1E-4 (absolute) |
||
\(>\) |
||||
Field DEPL, comp. \(>\) |
2.82843 10—5 |
“NON_DEFINI” |
1% |
|
Field DEPL, comp. \(>\) |
2.82843 10—5 |
“NON_DEFINI” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
2% |
||
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
—80. |
“NON_DEFINI” |
1% |
|
\(>\) |
||||
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“NON_DEFINI” |
1% |
|
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“NON_DEFINI” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
-20. |
“NON_DEFINI” |
1% |
O Modeling
Modelling characteristics
Elements C_PLAN **** (**** QUAD8 **** + TRIA6 )
Mesh \(>\) similar to the mesh below (8 elements in the radial direction, \(>\) elements in the circumferential direction) and duplicated to have a complete section of the cylinder (over 360°).
Boundary conditions:
side \(>\) stuck in \(>\) |
side \(>\) locked normally |
pressure on \(>\) \(>\) |
Node names: |
\(>\) |
|
|
\(>\) |
|
|
Mesh characteristics
Number of knots: 3840
Number of meshes and types: 1026 TRIA6, 512 QUAD8
Tested quantities and results
Location |
Size |
Reference Value |
Reference Type |
Tolerance |
point \(>\) |
||||
Field DEPL, comp. \(>\) |
5.90 10-5 |
“ANALYTIQUE” |
1% |
|
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—60. |
“ANALYTIQUE” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
0.3 (absolute) |
||
\(>\) |
||||
Field DEPL, comp. \(>\) |
4 10-5 |
“ANALYTIQUE” |
1% |
|
Field DEPL, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
0.1 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
0.5% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
0.015 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
0.1. (absolute) |
||
\(>\) |
||||
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
1% |
|
Field DEPL, comp. \(>\) |
4.17193 10—5 |
“ANALYTIQUE” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
5% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
5% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
-80 |
“ANALYTIQUE” |
1% |
|
\(>\) |
||||
Field DEPL, comp. \(>\) |
2.82843 10—5 |
“ANALYTIQUE” |
1% |
|
Field DEPL, comp. \(>\) |
2.82843 10—5 |
“ANALYTIQUE” |
1% |
|
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1% |
||
Field SIGM_NOEU, comp. \(>\) |
“ANALYTIQUE” |
1E-4 (absolute) |
||
Field SIGM_NOEU, comp. \(>\) |
—20. |
“ANALYTIQUE” |
1% |
P modelling
Modelling characteristics
Elements 3D (PENTA15 and **** HEXA20 **** ) — same mesh as N modelling
Mesh obtained by extrusion from a \(>\) mesh similar to the mesh below (8 elements in the radial direction, \(>\) elements in the circumferential direction) and duplicated to have a complete section of the cylinder (over 360°).
+—————————————————————————————————————-+——————————-+——————-++ |Following axis :math:`>`: |1 layer of elements|| +—————————————————————————————————————-+——————————-+——————-++ |Total thickness: |:math:`>` || +—————————————————————————————————————-+——————————-+——————-++ | | || +—————————————————————————————————————-+——————————-+——————-++ |Boundary conditions: | || +—————————————————————————————————————-+——————————-+——————-++ | | || +—————————————————————————————————————-+——————————-+——————-++ | |Face :math:`>` blocked normally | +----------------------------------------------------------------------------------------------------------------+-------------------------------+-------------------++ |Face :math:`>` stuck in :math:`>` | || +—————————————————————————————————————-+——————————-+——————-++ |Pressure on the face :math:`>` |:math:`>` || +—————————————————————————————————————-+——————————-+——————-++ |shape effect on sections |:math:`>` || +—————————————————————————————————————-+——————————-+——————-++ | | || +—————————————————————————————————————-+——————————-+——————-++ |With :math:`>`: pressure as a linear function of time equal to :math:`>` to :math:`>` and :math:`>` to :math:`>` || +—————————————————————————————————————-+——————————-+——————-++
Node names: |
\(>\) |
|
|
||
\(>\) |
|
|
Mesh characteristics
Number of knots: 8832
Number of meshes and types: 1024 PENTA15, 512 HEXA20, 512, 1176 QUAD8 and 2048 TRIA6.
Remarks
Contrary to N modeling, here we test a pressure and a shape effect that vary as a function of time. A linear variation in pressure leads to a linear variation in stresses.
Tested quantities and results
Location |
Size |
Reference Value |
Reference Type |
Tolerance |
point \(>\) |
||||
Instant 1.0 |
Field DEPL, comp. \(>\) |
5.60 10-5 |
“NON_DEFINI” |
0.1% |
Instant 1.0 |
Field DEPL, comp. \(>\) |
“NON_DEFINI” |
1E-10 (absolute) |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
—60. |
“NON_DEFINI” |
1% |
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1E-4 (absolute) |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
—120. |
“NON_DEFINI” |
1% |
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1E-4 (absolute) |
|
\(>\) |
||||
Instant 1.0 |
Field DEPL, comp. \(>\) |
3.4 10-5 |
“NON_DEFINI” |
0.1% |
Instant 1.0 |
Field DEPL, comp. \(>\) |
“NON_DEFINI” |
1E-10 (absolute) |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
0.1 (absolute) |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1E-4 (absolute) |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
0.1 (absolute) |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1E-4 (absolute) |
|
\(>\) |
||||
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
2% |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
2% |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
—80. |
“NON_DEFINI” |
1% |
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
2% |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
2% |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
—160. |
“NON_DEFINI” |
1% |
\(>\) |
||||
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 1.0 |
Field SIGM_NOEU, comp. \(>\) |
-20. |
“NON_DEFINI” |
1% |
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
“NON_DEFINI” |
1% |
|
Instant 2.0 |
Field SIGM_NOEU, comp. \(>\) |
-40. |
“NON_DEFINI” |
1% |
Summary of results
Summary of max errors in %
3D |
Location |
mod A |
mod A |
mod B |
mod D |
mod L |
mod M |
mod M |
mod N |
mod P |
|
elem |
pe6, h8 |
pe15, h8 |
pe15, h20 |
te4 |
te10 |
py5 |
py13 |
pe15, h20 |
pe15, h20 |
pe15, h20 |
|
geom |
45° |
45° |
45° |
45° |
45° |
45° |
360° |
360° |
360° |
||
nb no |
1922 |
2115 |
2115 |
1115 |
1395 |
342 |
1703 |
8832 |
8832 |
||
Depl. |
A, E B, F |
0.08 0.10 |
0.09 0.07 |
0.17 0.30 |
0.04 0.04 |
0.21 0.16 |
0.00 0.00 |
0.00 0.00 |
0.00 0.00 |
||
\(>\) |
A, E B, F |
4.59 5.24 |
0.39 0.07 |
10.45 7.78 |
4.41 0.95 |
11.74 5.24 |
0.11 0.11 |
1.64 0.09 |
1.64 0.09 |
||
\(>\) |
A, E B, F |
5.70 1.89 |
0.92 0.01 |
9.46 2.46 |
1.80 0.49 |
57.57 4.75 |
0.81 0.08 |
1.64 0.09 |
1.64 0.09 |
||
\(>\) |
A, E B, F |
Good Good |
Good Good |
Good Good |
Good Good |
Good Good |
Good Good |
0.87 0.04 |
0.87 0.04 |
||
\(>\) |
A, E B, F |
0.15 0.90 |
0.26 0.06 |
2.89 0.81 |
0.27 0.28 |
3.46 4.95 |
0.07 0.09 |
0.03 0.01 |
0.03 0.01 |
C_PLAN |
Location |
mod E |
mod F |
mod G |
mod O |
|
Type of elements |
tria3, quad4 |
tria6, quad8 |
quad9 |
tria6, quad8 |
||
Modeled geometry |
45° |
45° |
45° |
45° |
360° |
|
Number of nodes |
961 |
961 |
591 |
441 |
384 |
|
Travel |
A, E B, F |
0.07 0.09 |
0.09 0.07 |
0.00 0.00 |
0.01 0.00 |
|
Constraints \(>\) |
A, E B, F |
4.65 5.19 |
0.39 0.06 |
0.27 0.02 |
1.44 0.08 |
|
Constraints \(>\) |
A, E B, F |
5.68 1.40 |
0.90 0.04 |
0.16 0.05 |
1.44 0.08 |
|
Constraints \(>\) |
A, E B, F |
Good Good |
Good Good |
Good Good |
Good Good |
|
Constraints \(>\) |
A, E B, F |
0.16 1.23 |
0.23 0.07 |
0.20 0.09 |
0.10 0.00 |
AXIS |
Location |
mod H |
mod I |
mod J |
|
Type of elements |
tria3, quad4 |
tria6, quad8 |
quad9 |
||
Number of nodes |
113 |
113 |
175 |
205 |
|
Travel |
A, E B, F |
0.01 0.01 |
0.00 0.00 |
0.00 0.00 |
|
Constraints \(>\) |
A, E B, F |
5.66 Good |
0.17 Good |
0.17 Good |
|
Constraints \(>\) |
A, E B, F |
Good Good |
Good Good |
Good Good |
|
Constraints \(>\) |
A, E B, F |
1.29 1.00 |
0.09 0.07 |
0.09 0.02 |
|
Constraints \(>\) |
A, E B, F |
Good Good |
Good Good |
Good Good |
The results are more accurate with items of order 2.
The problem is more suited to axisymmetric modeling. The results are better.
The meshes remain insufficient for 3D elements of order 1: constraints and deformations of the A, C, E and L models (especially for the L modeling in PYRAM5).
Pyramids give results similar to other 3D elements, with equivalent meshes.
N and P models with constant or variable shape effect and pressure give good results.