background image
Code_Aster
®
Version
6.4
Titrate:
SSNV138 - Plate Cantilever in great rotations subjected to one moment
Date
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX
Key
:
V6.04.138-C
Page:
1/12
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A
Organization (S):
EDF-R & D/AMA,
SAMTECH















Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.138



SSNV138 - Plate Cantilever in great rotations
subjected to one moment




Summary:

Quasi-static calculation of an elastic plate embedded on a side and subjected to one bending moment to the other
side, leading to great rotations of the plate.

Interest:

To test the geometrical nonlinear finite elements
COQUE_3D
(modelings A and C) and
POU_D_T_GD
(modeling B) using the algorithm of update of great rotations 3D
GREEN_GR
in
STAT_NON_LINE
.
Note:
This test is the version plates case test of beam SSNL103. The mechanical characteristics were
modified in order to support a surface modeling.
background image
Code_Aster
®
Version
6.4
Titrate:
SSNV138 - Plate Cantilever in great rotations subjected to one moment
Date
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX
Key
:
V6.04.138-C
Page:
2/12
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A
1
Problem of reference
1.1 Geometry
Z
X
1
P
3
P
2
P
m
y
4
P
L=10 m
H = 0.1 m
B = 1 m
Rectangular plate embedded in
4
1
P
P
and subjected in
3
2
P
P
with a linear couple:
0
;
>
-
=
m
m
y
E
m

1.2
Properties of materials and characteristic of section
Elastic behavior:
0
;
10
12
6
=
×
=
AP
E
The fact that the Poisson's ratio is null makes the solution of plate identical to that of beam.
y
I
is the inertia of the section with a model of beam:
3
3
10
12
1
12
-
×
=
=
H
B
I
y

1.3
Boundary conditions and loading
Embedding in
4
1
P
P
. One seeks the successive states of balance under the loading made up of
linear couple in
3
2
P
P
:
()
T
T
m
100
=
;
T
pseudo-time.
One is interested particularly in displacements horizontal and vertical and rotation of the line
3
2
P
P
.
background image
Code_Aster
®
Version
6.4
Titrate:
SSNV138 - Plate Cantilever in great rotations subjected to one moment
Date
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX
Key
:
V6.04.138-C
Page:
3/12
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A
2
Reference solution
2.1
Method of calculation used for the reference solution
With a kinematics of beam and a model in resulting efforts, curvature (in great rotations)
cantilever subjected to the bending moment
mb
M
=
is, with the preceding numerical data:
L
T
I.E.(internal excitation)
mb
dx
D
y
=
=
It is the solution of Euler.
2.2
Results of reference
L
R
=
L
)
cos
1
(
-
R
sin
R
According to the solution of Euler, the deformation is an arc of circle. With the section
3
2
P
P
)
(
L
X
=
, rotation
is worth:
(
)
.t
L
X
=
=
In the absence of normal effort, average surface remains inextensible and the radius of curvature is given
by:
T
L
dx
D
R
=


=
-
1
Horizontal displacement is then
)
1
sin
(
sin
-
=
-
=
T
T
L
L
R
U
and vertical displacement is
)
cos
1
(
)
cos
1
(
T
T
L
R
v
-
=
-
=
background image
Code_Aster
®
Version
6.4
Titrate:
SSNV138 - Plate Cantilever in great rotations subjected to one moment
Date
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX
Key
:
V6.04.138-C
Page:
4/12
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A
2.3 References
bibliographical
[1]
Mr. Al MIKDAD: Statics and Dynamics of the Beams in Great Rotations and Resolution of
Problems of Nonlinear Instability. Thesis of Doctorate, University of Technology of
Compiegne (1998).
[2]
J.C. SIMO and L. CONSIDERING QUOC: In Three-dimensional Finite Strain Rod Model. Leaves II:
Computational Aspects. Comput. Meth. Appl. Mech. Engrg. 58, 79-116 (1986).
[3]
J.C. SIMO, D.D. FOX and Mr. S. RIFAI: There are Resulting Stress Geometrically Exact Shell
Model. Leaves III: Computational Aspects off the Nonlinear Theory. Comput. Meth. Appl. Mech.
Engrg. 79, 21-70 (1990).
background image
Code_Aster
®
Version
6.4
Titrate:
SSNV138 - Plate Cantilever in great rotations subjected to one moment
Date
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX
Key
:
V6.04.138-C
Page:
5/12
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A
3 Modeling
With
3.1
Characteristics of modeling
Modeling
COQUE_3D

3.2
Characteristics of the mesh
elements
10
element
1
A number of nodes: 54
A number of meshs and type: 10 QUAD9 and 1 SEG3

3.3 Functionalities
tested
·
Modeling
COQUE_3D
into nonlinear geometrical.
·
The static algorithm of update of great rotations
GREEN_GR
of
STAT_NON_LINE
.
Controls
STAT_NON_LINE
COMP_ELAS
RELATION: “ELAS”
COQUE_NCOU: 1
DEFORMATION: “GREEN_GR”
background image
Code_Aster
®
Version
6.4
Titrate:
SSNV138 - Plate Cantilever in great rotations subjected to one moment
Date
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX
Key
:
V6.04.138-C
Page:
6/12
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A
4
Results of modeling A
4.1 Values
tested
The incremental analysis is carried out in the interval of pseudo-time [0: 2.4] in fourteen pitches of
charge.

4.1.1 History of horizontal rotation DRY to the nodes charged
Moment
M couples
Aster
DRY (radians)
Reference
DRY (radians)
% difference
0.6 60.
­ 0.6000E+00
­ 0.6000E+00
+0.000
1.2 120
­ 1.20003E+00
­ 1.2000E+00
+0.003
1.8 180
­ 1.80011E+00
­ 1.8000E+00
+0.006
2.4 240
­ 2.40021E+00
­ 2.4000E+00
+0.009

4.1.2 History of horizontal displacement DX to the nodes charged
Moment
M couples
Aster
Reference
% difference
0.6 60.
­ 5.89343E-01
­ 5.8929E-01
+0.009
1.2 120
­ 2.23316E+00
­ 2.23300E+00
+0.012
1.8 180
­ 4.59041E+00
­ 4.58973E+00
+0.015
2.4 240
­ 7.1866E+00
­ 7.18557E+00
+0.015

4.1.3 History of vertical displacement DZ to the nodes charged
Moment
M couples
Aster
Reference
% difference
0.6 60.
2.91108E+00
2.91107E+00
+0.000
1.2 120
5.31370E+00
5.31368E+00
+0.000
1.8 180
6.817704E+00
6.81778E+00
­ 0.001
2.4 240
7.2386
E+00
7.23914E+00
­ 0.007
We present hereafter a visualization of the deformation during 14 pitch of load:
4.2 Remarks
One uses
COEF_RIGI_DRZ
= 0.001. The value of the angle swing reached is 135 degrees.
background image
Code_Aster
®
Version
6.4
Titrate:
SSNV138 - Plate Cantilever in great rotations subjected to one moment
Date
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX
Key
:
V6.04.138-C
Page:
7/12
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A
5 Modeling
B
5.1
Characteristics of modeling
POU_D_T_GD
(beam 3D in great rotations).
modeling
POU_D_T_GD
.

5.2
Characteristics of the mesh
elements
10
A number of nodes: 11
A number of meshs and type: 10 SEG2

5.3 Functionalities
tested
·
The geometrical nonlinear element
POU_D_T_GD
.
·
The static algorithm of update of great rotations
ELAS_POUTRE_GD
of
STAT_NON_LINE
.
Controls
STAT_NON_LINE
COMP_ELAS
RELATION: “ELAS_POUTRE_GD”
DEFORMATION: “GREEN_GR”
background image
Code_Aster
®
Version
6.4
Titrate:
SSNV138 - Plate Cantilever in great rotations subjected to one moment
Date
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX
Key
:
V6.04.138-C
Page:
8/12
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A
6
Results of modeling B
6.1 Values
tested
The incremental analysis is carried out in the interval of pseudo-time [0: 6] in 60 pitches of load.

6.1.1 History of horizontal rotation DRY (radians) to the nodes charged
Moment
Moment m
Aster
Reference
% difference
0.3 30.
­ 0.3000E+00
­ 0.3000E+00
+0.000
0.6 60.
­ 0.6000E+00
­ 0.6000E+00
+0.000
3.0.300.
­ 3.0000E+00
­ 3.0000E+00
+0.000
6.600 - 6
- 6 0

6.1.2 History of horizontal displacement DX (m) to the nodes charged
Moment
Moment m
Aster
Reference
% difference
0.3 30.
­ 1.4895E-01
­ 1.4932E-01
­ 0.247
0.6 60.
­ 5.8788E+01
­ 5.8934E+01
­ 0.240
3.0.300. ­ 9.5278 ­ 9.5296
­ 0.02
6 600
­ 10.4729
­ 10.4657
0.07

6.1.3 History of vertical displacement DZ (m) to the nodes charged
Moment
Moment m
Aster
Reference
% difference
0.3 30.
1.4888E+01
1.4887E+01
+0.004
0.6 60.
2.9115E+00
2.9110E+00
+0.015
3.0.300.
6.6582 6.6333 0.38
6 600
6.74377
6.638286
1.6
background image
Code_Aster
®
Version
6.4
Titrate:
SSNV138 - Plate Cantilever in great rotations subjected to one moment
Date
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX
Key
:
V6.04.138-C
Page:
9/12
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A
7 Modeling
C
7.1
Characteristics of modeling
Modeling
COQUE_3D

7.2
Characteristics of the mesh
elements
10
element
1
A number of nodes: 64
A number of meshs and type: 20 TRIA7 and 1 SEG3

7.3 Functionalities
tested
·
Modeling
COQUE_3D
into nonlinear geometrical.
·
The static algorithm of update of great rotations
GREEN_GR
of
STAT_NON_LINE
.
Controls
STAT_NON_LINE
COMP_ELAS
RELATION: “ELAS”
COQUE_NCOU: 1
DEFORMATION: “GREEN_GR”
background image
Code_Aster
®
Version
6.4
Titrate:
SSNV138 - Plate Cantilever in great rotations subjected to one moment
Date
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX
Key
:
V6.04.138-C
Page:
10/12
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A
8
Results of modeling C
8.1 Values
tested
The incremental analysis is carried out in the interval of pseudo-time [0: 2.2] in eight pitch of load.

8.1.1 History of horizontal rotation DRY to the nodes charged
Moment
M couples
Aster
Reference
% difference
0.6 60
­ 0.5993E+00
­ 0.6000E+00
­ 0.107
1.2 120
­ 1.1950E+00
­ 1.2000E+00
­ 0.411
1.8 180
­ 1.7843E+00
­ 1.8000E+00
­ 0.868
2.2 220
­ 2.2000E+00
­ 2.1728E+00
­ 1.235

8.1.2 History of horizontal displacement DX to the nodes charged
Moment
M couples
Aster
Reference
% difference
0.6 60
­ 5.88149E-01
­ 5.8929E-01
­ 0.194
1.2 120
­ 2.21699E+00
­ 2.23300E+00
­ 0.717
1.8 180
­ 4.52666E+00
­ 4.58973E+00
­ 1.374
2.2 220
­ 6.21278E+00
­ 6,3250163463
­ 1.774

8.1.3 History of vertical displacement DZ to the nodes charged
Moment
M couples
Aster
Reference
% difference
0.6 60
2.90827E+00
2.91107E+00
­ 0.096
1.2 120
5.29785
E+00
5.31368E+00
­ 0.298
1.8 180
6.79264E+00
6.81778E+00
­ 0.369
2.2 220
7.207771E+00
7,22046E+00
­ 0.176
We present hereafter a visualization of the deformation during 8 pitches of load:
8.2 Remarks
One uses
COEF_RIGI_DRZ
= 0.001. The value of the angle swing reached is 125 degrees.
background image
Code_Aster
®
Version
6.4
Titrate:
SSNV138 - Plate Cantilever in great rotations subjected to one moment
Date
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX
Key
:
V6.04.138-C
Page:
11/12
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A
9
Summary of the results
One notices difficulties of convergence which disappear by multiplying the thickness by 3 or 4.
It is necessary to increase the value of
COEF_RIGI_DRZ
who allots a rigidity around
normal of the elements of hull which is worth by defect 10
­ 5
(the smallest rigidity of bending around
directions in the plan of the hull) in order to be able to increase the value of the swing angle that
one can reach. Values of this coefficient up to 10
­ 3
remain licit.
During the iterations of Newton, deformations of membrane appear and are cancelled with
convergence.
Speeds of convergence of the algorithms of NEWTON are comparable for modelings
POU_D_T_GD
and
COQUE_3D
.
background image
Code_Aster
®
Version
6.4
Titrate:
SSNV138 - Plate Cantilever in great rotations subjected to one moment
Date
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX
Key
:
V6.04.138-C
Page:
12/12
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A


























Intentionally white left page.