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Organization (S): EDF-R & D/AMA

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
Document: V3.04.130

SSLV130 - Hollow Cylindre into incompressible

Summary:

This test makes it possible to validate the quasi-incompressible elements in statics for a three-dimensional problem,
axisymmetric or two-dimensional (plane deformations). One considers a hollow roll subjected to a pressure
intern. The material has a Poisson's ratio equal to 0.4999 and one uses the elements quasi
incompressible (modeling INCO). In all the cases of modeling, one carries out the test or not while imposing
the condition of perfect incompressibility (DDL_IMPO and GONF=0)

Four modelings are carried out for this problem. Modelings A and B make it possible to test
quasi-incompressible modeling 3D (3d_INCO), on the one hand with HEXA20 (A) and on the other hand with
TETRA10 (B). Modelings C and D are studies 2D being based on mixed grids QUAD8 and
TRIA6. Modeling C is the study in plane deformations (D_PLAN_INCO), modeling D is a study
axisymmetric (AXIS_INCO).

This test is similar to test SSLV100.

The numerical results are satisfactory for all modelings. The fact of imposing explicitly
condition of incompressibility influences only very little the results.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/03/008/A

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1
Problem of reference

1.1 Geometry
y
R
E
F

C
45°
D
P
With
B
Z
X
Internal radius
= 0.1 m has
External radius
B = 0.2 m



Co-ordinates of the points

WITH B
E
F
C
D
X 0.1.0.2.0.1 * cos (45) 0.2 * cos (45) 0.1 * cos (22.5) 0.2 * cos (22.5)
y
0 0.0.1 * sin (45)
0.1 * sin (45)
0.1 * sin (22.5) 0.1 * sin (22.5)
Z
0 0
0
0
0
0

1.2
Properties of material

E = 2.105 MPa
= 0.4999

1.3
Boundary conditions and loadings

Pressure interns P = 60 MPa.
Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/03/008/A

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2
Reference solution

2.1
Method of calculation

The general solution in displacement is as follows:


2
Pa

2
B
U =
1
(+)
R
1
(- 2) R +


E (2
B - 2
has)


R

U = U = 0

Z
In deformations:

2
Pa

2
B
=
1
(+)
rr
1
(- 2) -


E (2
B - 2
has)


2
R


2
Pa

2
B

=
1
(+

) 1
(- 2) +


E (2
B - 2
has)


2
R


=

= 0
R
zz


In constraints:

2
has

2
B
= P
rr
2
2 1 -
2

B - has
R

2
2

has

B
= P
1+

2
2
2
B - has
R


2
has
= 2 P
zz
2
B - 2

has
= 0

R

One obtains for a perfectly incompressible cylinder (= 0.5):

in R = 0 1
. : U = 6. 105
in R = 0.2:
= 3. 105
R
ur
= 6
-. 10-4
= 1
- 5
. 10-4
rr
rr

= 6. 10-4


= 1 5
. 10-4


= 60
-.
= 0.
rr
rr

= 100.


= 40.

= 20.
= 20.
zz
zz

The passage in the Cartesian system is done using the following relations:

= cos2 +
2
xx
rr
sin - 2 sin cos
R

= sin2 +
2
yy
rr
cos + 2 sin cos
R

= sin cos -
2
2
xy
rr
sin cos - 2
R (cos - sin)
Handbook of Validation
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HT-66/03/008/A

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2.2
Sizes and results of reference

One compares with the values of reference:

·
displacements (U, v) at points A and F,
·
the strains (xx, yy, xy) and the stresses (xx, yy, zz, xy) at points A and F,
·
equivalent strains and stresses equivalent to point A.

Lastly, to test the passage of the sizes of the points of Gauss to the nodes for the nodes mediums,
one also tests the nonnull strains and stresses in a node medium of the structure.

2.3 References
bibliographical

[1]
Y.C. FUNG: Foundations off solid mechanics. Prenctice-hall, Inc. Englewood Cliffs.
NJ. 1965, p. 243-245
[2]
[V3.04.100] hollow Cylindre in plane deformations

3 Modeling
With

3.1
Characteristics of modeling

Grid with incompressible elements 3D of type HEXA20 only


y
B
Face blocked in dx
F
With
Normally blocked face
E
Face with imposed pressure
45°
X


Along axis Z:

·
total thickness E = 0.01
·
2 layers of elements

For the needs for examination in a node medium, one defines the node NOEUMI = A + (0. 0. E/4) where
the strains and the stresses are the same ones as in A.
Handbook of Validation
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HT-66/03/008/A

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Limiting conditions:

DDL_IMPO =
GROUP_NO = ' FACSUP'
DZ =
0.
GROUP_NO
= ' FACINF'




DZ =
0.
faces AEFD (z=0 and Z = 0.01)




GROUP_NO = ' FACEAB' DX =
0.
face AB
FACE_IMPO = GROUP_MA = ' FACEEF'
DNOR =
0.
face EFF
PRES_REP = GROUP_MA = ' FACEAE'
PRES =
60.
face AE

3.2
Characteristics of the grid

A number of nodes: 1501 nodes
A number of meshs: 240 HEXA20

3.3 Functionalities
tested

Commands



DEFI_GROUP CREA_GROUP_NO
CRIT_NOEUD
“SOMMET”
AFFE_MODELE MODELING “3D_INCO”
GROUP_MA
DEFI_MATERIAU ELAS


AFFE_CHAR_MECA DDL_IMPO
GROUP_NO
FACE_IMPO
GROUP_MA

PRES_REP
GROUP_MA

STAT_NON_LINE COMP_INCR
RELATION ELAS
NEWTON
REAC_ITER
1

3.4
Sizes tested and results

For A and F, one notes the result obtained

·
first column without imposing GONF = 0
·
second column while imposing GONF = 0

Identification
Reference
Aster
% difference
With U
0. 3.29
10-22 - 1.39
10-21
- -
v
6. 10­5
5.9998 10-5
5.9995 10-5 ­ 0.003 - 0.008
xx
100. 99.92
99.92
­ 0.078
- 0.078
yy
­ 60. ­ 59.91
- 59.91
- 0.155
- 0.155
zz
20. 19.98
19.99
- 0.076
- 0.056
xy
0. - 8.48
10-3 - 8.48
10-3
- -
xx
6. 10­4
5.9948 10-4
5.9945 10-4 ­ 0.086 - 0.091
yy
­ 6. 10­4
- 5.9915 10-5
- 5.9918 10-4 - 0.141 - 0.136
xy
0. - 6.36
10-8 - 6.36
10-8
- -
INVA_2
6.9282 10-4
6.9211 10-4
6.9211 10-4 - 0.102
- 0.102
PRIN_1
­ 6. 10­4
- 5.9936 10-4
- 5.9939 10-4 - 0.107
- 0.102
PRIN_2
0. 9.38
10-16
- 1.3394 10-15
- -
PRIN_3
6. 10­4 5.9942
10-4
5.9939 10-4 - 0.097
- 0.102
VMIS
138.5641 138.1433 138.1433 - 0.304
- 0.304
TRESCA
160. 159.5142
159.5142
- 0.304
- 0.304
PRIN_1
- 60. - 59.7571
- 59.7571
- 0.405
- 0.405
PRIN_2
20. 19.9961
20.0001
- 0.019
0.0007
PRIN_3
100. 99.7571
99.7571
- 0.243
- 0.243
VMIS_SG
138.5641 138.1433 138.1433 - 0.304
- 0.304

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/03/008/A

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Identification
Reference
Aster
% difference
F U
­ 2.1213 10­5 - 2.1216
10-5 - 2.1212 10-5
0.014 - 0.006
v
+2.1213 10­5 2.1216
10-5 2.1212
10-5
0.014 - 0.006
xx
20. 20.01
20.01
- 0.104 0.04
yy
20. 19.99
19.99
- 0.035 - 0.035
zz
20. 19.999
19.999
- 0.027 - 0.007
xy
20. 20.02
20.02
0.087 0.087
xx
0. 1.01
10-7 7.11
10-8
- -
yy
0. - 1.08
10-8 - 4.07
10-8
- -
xy
1.5 10­4 1.5012
10-4 1.5012
10-4
0.08 0.08

Checking of the passage to the node for the nodes mediums (only for the result obtained without
to impose GONF = 0) - value on node NOEUMI:

Identification
Reference
Aster
% difference
xx
100. 99.92
­ 0.076
yy
­ 60. ­ 59.92 - 0.127
zz
20. 19.996
- 0.076
xx
6. 10­4 5.9942
10-4 ­ 0.019
yy
­ 6. 10­4 - 5.9936
10-4 - 0.107

3.5 Remarks

One obtains very good results since for all the examined sizes, the difference between
solution obtained with the code and the analytical solution is lower than 0.2%. It is seen that the variation enters
the solutions obtained by imposing or not the condition tr = 0 is unimportant.

Handbook of Validation
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HT-66/03/008/A

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4 Modeling
B

4.1
Characteristics of modeling

Grid with incompressible elements 3D of type TETRA10 only

F
Normally blocked face
y
D
E
C

45°

With
B
X
Face with imposed pressure
Face blocked out of Dy


AB is on axis OX (contrary to modeling A).

Cutting:

6 equidistant nodes on segments AB, CD and EFF,
5 equidistant nodes on arcs ACE and BDF.
Along axis Z:
·
total thickness E = 0.01
·
2 layers of elements

For the needs for examination, one defines the node NOEUMI = A + (0. 0. E/4) where deformations
and the constraints are the same ones as in A.

Limiting conditions:

DDL_IMPO =
GROUP_NO = ' FACSUP'
DZ =
0.
GROUP_NO
= ' FACINF'




DZ =
0.
faces AEFD (z=0 and Z = 0.01)




GROUP_NO = ' FACEAB' DY =
0.
face AB
FACE_IMPO = GROUP_MA = ' FACEEF'
DNOR =
0.
face EFF
PRES_REP = GROUP_MA = ' FACEAE'
PRES =
60.
face AE

4.2
Characteristics of the grid

A number of nodes: 703
A number of meshs: 356 TETRA10
Handbook of Validation
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4.3 Functionalities
tested

Commands



DEFI_GROUP CREA_GROUP_NO
CRIT_NOEUD
“SOMMET”
AFFE_MODELE MODELING “3D_INCO”
GROUP_MA
DEFI_MATERIAU ELAS


AFFE_CHAR_MECA DDL_IMPO
GROUP_NO

FACE_IMPO
GROUP_MA

PRES_REP
GROUP_MA

STAT_NON_LINE COMP_INCR
RELATION
ELAS
NEWTON
REAC_ITER
1

4.4
Sizes tested and results

For the points A and F, one notes the result obtained without imposing GONF = 0 (column 1) and while imposing
GONF = 0 (column 2)

Identification
Reference
Aster
% difference
With U
6. 10­5 6.011
10-5 6.011
10­5 0.19
0.18
v
0. 2.65
10­23 - 3.06
10­22 - -
xx
­ 60. - 59.51
­ 61.1452
- 0.82
1.90
yy
100. 100.95
101.345
0.95
1.34
zz
20. 20.42
19.813
2.08
­ 0.93
xy
0. - 2.104
­ 2.60
- -
xx
- 6. 10­4 ­ 6.02
10­4 ­ 6.02
10­4 0.36
0.36
yy
+6. 10­4 6.01
10­4 6.01
10­4 0.20
0.19
xy
0. - 1.58
10
­ 5 ­ 1.58
10­5 - -
INVA_2
6.9282 10-4
6.9509 10-4
6.9509 10-4
0.328 0.328
PRIN_1
­ 6. 10­4
- 6.0241 10-4
- 6.0244 10-4
0.401 0.406
PRIN_2
0. - 2.85
10-6
- 2.86 10-6
- -
PRIN_3
6. 10­4 6.0153
10-4
6.0150 10-4
0.255 0.250
VMIS
138.5641 137.4852
137.4852
- 0.779 - 0.779
TRESCA
160. 158.7541 158.7543 - 0.779 - 0.779
PRIN_1
- 60. - 58.6561 - 58.6560 - 2.240 - 2.240
PRIN_2
20. 20.4167 20.4209 2.083 2.104
PRIN_3
100. 100.0980 100.0980 0.098 0.098
VMIS_SG
138.5641 137.4852
137.4852
- 0.779 - 0.779

Identification
Reference
Aster
% difference
F U
2.1213 10­5 2.1210
10-5
2.1206 10-5 ­ 0.02 ­ 0.04
v
2.1213 10­5 2.1210
10-5
2.1206 10-5 ­ 0.02 ­ 0.04
xx
20. 20.13 20.12 0.627 0.60
yy
20. 19.96 19.98
- 0.209 ­ 0.11
zz
20. 20.025 20.035
0.127 0.17
xy
­ 20. - 19.98 ­ 19.97 0.105 ­ 0.13
xx
0. 6.85
10-7 6.55
10­7
- -
yy
0. - 5.68
10-5 ­ 5.98
10­7
- -
xy
2.1213 10­5 2.1210
10-5
2.1206 10-5 ­ 0.02 ­ 0.04

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Checking of the passage to the node for the nodes mediums (only for the result obtained without
to impose GONF = 0) - value on node NOEUMI:

Identification
Reference
Aster
% difference
xx
- 60. 58.988
­ 1.687
yy
100. 100.501
0.501
zz
20. 20.685
3.424
xx
- 6. 10­4 - 5.97503
10-4 ­ 0.416
yy
6. 10­4 5.98584
10-4 - 0.236

4.5 Remarks

The results obtained here are a little worse than in the case of modeling A, but
discretization is coarser since there are approximately 2 times less nodes in this case-test.
results are satisfactory all the same since the variations are lower than 0.2% for
displacements, lower than 0.5% for the deformations and lower than 1% for the constraints. One
note again that there is no significant improvement of the result when one imposes
explicitly tr = 0.
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5 Modeling
C

5.1
Characteristics of modeling

Grid with incompressible elements 2D of type QUAD8 and TRIA6

y
B
Face blocked in dx
F
With
Normally blocked face
E
Face with imposed pressure
45°
X


Limiting conditions:

DDL_IMPO =
GROUP_NO = ' GRNM11'
DX =
0.
side AB
FACE_IMPO = GROUP_MA = ' GRMA12'
DNOR =
0.
dimensioned EFF
PRES_REP = GROUP_MA = ' GRMA13'
PRES =
60.
face AE

Name of the nodes:

WITH = N2, B = N361, C = N121, D = N584, E = N155, F = N503

5.2
Characteristics of the grid

A number of nodes: 591
A number of meshs: 200 TRIA6, 50 QUAD8.

5.3 Functionalities
tested

Commands



DEFI_GROUP CREA_GROUP_NO CRIT_NOEUD
“SOMMET”
AFFE_MODELE MODELING
“D_PLAN_INCO”
GROUP_MA
DEFI_MATERIAU ELAS


AFFE_CHAR_MECA DDL_IMPO
GROUP_NO
FACE_IMPO
GROUP_MA

PRES_REP
GROUP_MA

STAT_NON_LINE COMP_INCR
RELATION ELAS
NEWTON
REAC_ITER
1

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5.4
Sizes tested and results

For the points A and F, one notes the result obtained without imposing GONF = 0 (column 1) and while imposing
GONF = 0 (column 2).

Identification
Reference
Aster
% difference
With U
0. 1.59
10-21
9.53 10-22
- -
v
6. 10­5
5.9935 10-5
5.9932 10-5 - 0.108
- 0.113
xx
100. 99.93 99.93
- 0.071
- 0.071
yy
­ 60. - 59.72 - 59.72
- 0.464
- 0.464
zz
20. 20.09 20.10
0.470
0.490
xy
0. 0.224
0.224
- -
xx
6. 10­4
5.987 10-4
5.987 10-4 - 0.212
- 0.217
yy
­ 6. 10­4
- 5.986 10-4
- 5.986 10-4 - 0.237
- 0.232
xy
0. 1.68
10-6
1.68 10-6
- -
INVA_2
6.9282 10-4
6.9127 10-4
6.9127 10-4 - 0.224
- 0.224
PRIN_1
­ 6. 10­4
- 5.9858 10-4 - 5.9861 10-4 - 0.237
- 0.232
PRIN_2
0. 0.0
0.0
- -
PRIN_3
6. 10­4 5.9873
10-4
5.9870 10-4 - 0.212
- 0.217
VMIS
138.5641 138.1093 138.1093 - 0.328
- 0.328
TRESCA
160. 159.4749
159.4749
- 0.328
- 0.328
PRIN_1
- 60. - 59.6335
- 59.6335
- 0.611
- 0.611
PRIN_2
20. 20.0940
20.0980
0.470
0.490
PRIN_3
100. 99.8414
99.8414
- 0.159
- 0.159
VMIS_SG
138.5641 138.1093 138.1093 - 0.328
- 0.328

Identification
Reference
Aster
% difference
F U
­ 2.1213 10­5
- 2.1195 10-5 - 2.1191 10-5
- 0.086
- 0.106
v
+2.1213 10­5
2.1195 10-5
2.1191 10-5
- 0.086
- 0.106
xx
20. 20.03
20.03
0.153
0.153
yy
20. 19.97
19.97
- 0.134
- 0.134
zz
20. 19.99
19.997
- 0.036
- 0.016
xy
20. 20.01
20.01
0.051
0.051
xx
0. 2.84
10-7
2.54 10-7
-
-
yy
0. - 1.46
10-7
- 1.76 10-7
-
-
xy
1.5 10­4 1.5007
1.5007 0.044
0.044

5.5 Remarks

As for modeling 3D, the results obtained are completely satisfactory.

Handbook of Validation
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6 Modeling
D

6.1
Characteristics of modeling

Incompressible elements axi (TRIA6 + QUAD8)

Center cylinder
Node blocked out of Dy
y
E
F
0.01m
C
D
With
B
X
Face with imposed pressure


For the needs for examination, one defines the nodes:

·
NOEUMIA = A + (0. 0.01/4) where the strains and stresses are the same one as in A
·
NOEUMIB = B + (0. 0.01/4) where the strains and stresses are the same one as out of B

Limiting conditions:

DDL_IMPO =
GROUP_NO = ' FACSUP'
DY =
0.
y=0.1
GROUP_NO
= ' FACINF'




DY =
0.
y=0
PRES_REP = GROUP_MA = ' FACEAE'
PRES =
60.
face AE

6.2
Characteristics of the grid

A number of nodes: 175.
A number of meshs and types: 20 QUAD8, 40 TRIA6.

6.3 Functionalities
tested

Commands



DEFI_GROUP CREA_GROUP_NO
CRIT_NOEUD
“SOMMET”
AFFE_MODELE MODELING
“AXIS_INCO”
GROUP_MA
DEFI_MATERIAU ELAS


AFFE_CHAR_MECA DDL_IMPO
GROUP_NO

FACE_IMPO
GROUP_MA

PRES_REP
GROUP_MA

STAT_NON_LINE COMP_INCR
RELATION
ELAS
NEWTON
REAC_ITER
1

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Key S. MICHEL-PONNELLE
:
V3.04.130-C Page:
13/14

6.4
Sizes tested and results

For A and F, one notes the results obtained without imposing GONF = 0 (column 1) and while imposing
GONF = 0 (column 2).

Identification
Reference
Aster
% difference
With U
6. 10­5
6.0000 10-5 5.9996
10-5
- 0.002 - 0.007
v
0. 9.93
10­24 - 8.93
10-23
- -
xx
­ 60. - 59.91
- 59.91
- 0.150 - 0.150
yy
20. 19.99
19.997 - 0.036 - 0.016
zz
100. 99.91
99.91
- 0.086 ­ 0.086
xy
0. 3.73
10-9
3.73 10-9
- -
xx
­ 6. 10­4
­ 5.99 10­4 - 5.99
10-4
- 0.180 - 0.123
yy
0. - 1.69
10­15 - 1.69
10-15
- -
xy
0.
2.80 10­14 2.80
10-14
- -
INVA_2
6.9282 10-4
6.9201 10-4
6.9201 10-4
- 0.117 - 0.117
PRIN_1
­ 6. 10­4
- 5.9923 10-4
- 5.9926 10-4
- 0.128 - 0.123
PRIN_2
0. 6.45
10-16
6.40 10-16
- -
PRIN_3
6. 10­4 5.9937
10-4
5.9934 10-4
- 0.105 - 0.110
VMIS
138.5641 138.4116
138.4116
- 0.110 - 0.110
TRESCA
160. 159.8239 159.8239 - 0.110 - 0.110
PRIN_1
- 60. - 59.9101 - 59.9101 - 0.150 - 0.150
PRIN_2
20. 19.9928 19.9968 - 0.036 - 0.016
PRIN_3
100. 99.9138 99.9138 - 0.086 - 0.086
VMIS_SG
138.5641
138. 4116
138. 4116
- 0.110 - 0.110

Identification
Reference
Aster
% difference
F U
3 10­5 3.0004
10-5
2.9998 10-5
0.014 ­ 0.006
v
0. - 4.96
10-22
- 5.69 10-22
­ ­
xx
0. 2.59
10-2
2.58 10-2
- -
yy
20. 19.997 20.001 - 0.014 0.006
zz
40. 39.99 39.99
- 0.025 - 0.025
xy
0. - 4.87
10-3
- 4.87 10-3
- -
xx
­ 1.5 10­4 - 1.498
10-4 - 1.498
10-4
- 0.129 - 0.109
yy
0. - 3.20
10-8 - 3.20
10-8
- -
xy
0.
- 3.65 10-8
- 3.66 10-8
­ ­

Checking of the passage to the node for the nodes mediums (only for the result obtained without
to impose GONF = 0)

NOEUMIA

Identification
Reference
Aster
% difference
xx
- 60. - 59.91 ­ 0.150
yy
20. 19.99
- 0.036
zz
100. 99.91
- 0.086
xx
- 6. 10­4
- 5.9923 10-4 ­ 0.128
zz
6. 10­4
5.9937 10-4 - 0.105

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/03/008/A

Code_Aster ®
Version
7.2
Titrate:
SSLV130 - Hollow Cylindre into incompressible


Date:
23/09/03
Author (S):
Key S. MICHEL-PONNELLE
:
V3.04.130-C Page:
14/14

NOEUMIB

Identification
Reference
Aster
% difference
xx
0. - 1.39
10-2
­
yy
20. 19.9999
- 0.002
zz
40. 39.9993
- 0.002
xx
- 1.5 10­4 - 1.4988
10-4 ­ 0.078
zz
1.5 10­4 1.4999
10-4 - 0.008

6.5 Remarks

The precision obtained is very good.

7
Summary of the results

With a Poisson's ratio very close to 0.5, one finds the results of the solution
analytical incompressible with a weak difference. It is noticed that it is not necessary
to impose explicitly the condition of incompressibility tr = 0 to obtain good results
since the results are quasi-identical that one activates or not, condition GONF=0 with
DDL_IMPO.

Handbook of Validation
V3.04 booklet: Linear statics of the voluminal structures
HT-66/03/008/A

Outline document