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Titrate:
SSNV138 - Plaque Cantilever in great rotations subjected to one Date moment
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX Key
:
V6.04.138-C Page:
1/12

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

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

SSNV138 - Plaque 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.

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A

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Titrate:
SSNV138 - Plaque Cantilever in great rotations subjected to one Date moment
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX Key
:
V6.04.138-C Page:
2/12

1
Problem of reference

1.1 Geometry


y
Z
L=10 m
4
P
3
P
B = 1 m
X
m
1
P
H = 0.1 m
2
P


Rectangular plate embedded into 1
P 4
P and subjected into 2
P 3
P with a linear couple:


m = - me
; m
y
> 0

1.2
Properties of materials and characteristic of section

Elastic behavior:

E = 12 ×106 Pa; = 0

The fact that the Poisson's ratio is null makes the solution of plate identical to that of beam.
I is there the inertia of the section with a model of beam:

3
B
1
- 3
I y = H =
×10
12
12

1.3
Boundary conditions and loading

Embedding in 1
P 4
P. One seeks the successive states of balance under the loading made up of
linear couple in 2
P 3
P:

m (T) = 100 T; T pseudo-time.

One is interested particularly in displacements horizontal and vertical and rotation of the line
2
P 3
P.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A

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SSNV138 - Plaque Cantilever in great rotations subjected to one Date moment
:
17/06/03
Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX Key
:
V6.04.138-C Page:
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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 M = mb is, with the preceding numerical data:

D
mb
T
=
=
dx
I.E.(internal excitation)
L
y

It is the solution of Euler.

2.2
Results of reference

L

R =
Rsin
R 1
(- cos)
L

According to the solution of Euler, the deformation is an arc of circle. With section 2
P 3
P (X = L), rotation
is worth:

(X = L) = T.

In the absence of normal effort, average surface remains inextensible and the radius of curvature is given
by:

D - 1

L
R =
=
dx
T
Horizontal displacement is then

sin
U =
T
R sin - L = L (
-)
1
T


and vertical displacement is

L
v = R 1
(- cos) =
1
(- cost)
T
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A

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SSNV138 - Plaque Cantilever in great rotations subjected to one Date moment
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P. MASSIN, Mr. Al MIKDAD, J.M. PROIX Key
:
V6.04.138-C Page:
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2.3 References
bibliographical

[1]
Mr. Al MIKDAD: Statics and Dynamique of Poutres in Grandes Rotations and Résolution of
Problems of Instabilité Non Linéaire. Thesis of Doctorat, Université de Technologie 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 is Stress Resultant Geometrically Exact Shell
Model. Leaves III: Computational Aspects off the Nonlinear Theory. Comput. Meth. Appl. Mech.
Engrg. 79, 21-70 (1990).
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A

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SSNV138 - Plaque Cantilever in great rotations subjected to one Date moment
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:
V6.04.138-C Page:
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3 Modeling
With

3.1
Characteristics of modeling

Modeling COQUE_3D

3.2
Characteristics of the grid

1 element
10 elements


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.

Commands
STAT_NON_LINE
COMP_ELAS
RELATION: “ELAS”
COQUE_NCOU: 1
DEFORMATION: “GREEN_GR”

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A

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SSNV138 - Plaque Cantilever in great rotations subjected to one Date moment
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:
V6.04.138-C Page:
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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 steps of
charge.

4.1.1 History of horizontal rotation DRY to the nodes charged

Moment
M couples
Aster
Reference
% difference
DRY (radians)
DRY (radians)

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 step of load:



4.2 Remarks

One uses COEF_RIGI_DRZ = 0.001. The value of the angle swing reached is 135 degrees.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A

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Titrate:
SSNV138 - Plaque Cantilever in great rotations subjected to one Date moment
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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 grid

10 elements


A number of nodes: 11
A number of meshs and type: 10 SEG2

5.3 Functionalities
tested

· Nonlinear element geometrical POU_D_T_GD.
· The static algorithm of update of great rotations ELAS_POUTRE_GD of
STAT_NON_LINE.

Commands
STAT_NON_LINE
COMP_ELAS
RELATION: “ELAS_POUTRE_GD”
DEFORMATION: “GREEN_GR”

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A

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Titrate:
SSNV138 - Plaque Cantilever in great rotations subjected to one Date moment
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:
V6.04.138-C Page:
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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 steps 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

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7 Modeling
C

7.1
Characteristics of modeling

Modeling COQUE_3D

7.2
Characteristics of the grid

1 element
10 elements


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.

Commands
STAT_NON_LINE
COMP_ELAS
RELATION: “ELAS”
COQUE_NCOU: 1
DEFORMATION: “GREEN_GR”

Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A

Code_Aster ®
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Titrate:
SSNV138 - Plaque Cantilever in great rotations subjected to one Date moment
:
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Author (S):
P. MASSIN, Mr. Al MIKDAD, J.M. PROIX Key
:
V6.04.138-C Page:
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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 step 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 steps of load:



8.2 Remarks

One uses COEF_RIGI_DRZ = 0.001. The value of the angle swing reached is 125 degrees.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/03/008/A

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Titrate:
SSNV138 - Plaque Cantilever in great rotations subjected to one Date moment
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:
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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 the COEF_RIGI_DRZ which allots a rigidity around
normal of the elements of hull which is worth by defect 10­5 (the smallest rigidity of inflection 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 until 10­3 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.
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
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SSNV138 - Plaque Cantilever in great rotations subjected to one Date moment
:
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:
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