Code_Aster
®
Version
8.3
Titrate:
SDNV106 Analyze with the eigenvalues in DYNA_NON_LINE
Date:
09/05/06
Author (S):
NR. GREFFET
Key
:
V5.03.106-A
Page:
1/4
Manual of Validation
V5.03 booklet: Nonlinear dynamics of the voluminal structures
HT-62/06/005/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V5.03 booklet: Nonlinear dynamics of the voluminal structures
Document: V5.03.106
SDNV106 Analyze with the eigenvalues in
DYNA_NON_LINE (stability and oscillatory modes)
Summary:
This case test makes it possible to validate the analysis of buckling, as well as the vibratory modal analysis in
DYNA_NON_LINE.
Only one modeling is used:
Massive modeling A 3D made up of meshs HEXA8.
Code_Aster
®
Version
8.3
Titrate:
SDNV106 Analyze with the eigenvalues in DYNA_NON_LINE
Date:
09/05/06
Author (S):
NR. GREFFET
Key
:
V5.03.106-A
Page:
2/4
Manual of Validation
V5.03 booklet: Nonlinear dynamics of the voluminal structures
HT-62/06/005/A
1
Problem of reference
1.1 Geometry
One considers a cube on side length 2 m subjected to a uniform traction according to the direction
vertical Z:
For reasons of symmetry, one will consider only one eighth of the structure, which will be with a grid by one
only cubic linear voluminal element.
1.2
Properties of material
The structure is supposed to be homogeneous, composed of an isotropic elastoplastic material, with work hardening
isotropic linear:
· E = 2x10
4
MPa
· = 0.49999
· = 7900 kg/m
3
·
y
= 0,1 Mpa (elastic threshold SY)
· E
T
= 200 Mpa (tangent module plastic D_SIGM_EPSI)
One thus chooses a material which remains always almost incompressible, that one is in mode
rubber band or plastic. Moreover, one imposes a relationship 100 between the elastic stiffness and the stiffness
plastic tangent.
1.3
Boundary conditions
One imposes a uniform loading of traction type imposed according to Z on the higher face of the cube.
This force imposed, initially null, grows linearly with time.
The other boundary conditions are of Dirichlet type and translate the conditions of symmetries of
problem (according to the 3 orthogonal plans (xOy), (xOz) and (yOz)).
These boundary conditions are sufficient to lock all the movements of rigid body of
system.
1.4 Conditions
initial
The first calculation being quasistatic, one imposes just an initial displacement no one.
Z
X
Y
Code_Aster
®
Version
8.3
Titrate:
SDNV106 Analyze with the eigenvalues in DYNA_NON_LINE
Date:
09/05/06
Author (S):
NR. GREFFET
Key
:
V5.03.106-A
Page:
3/4
Manual of Validation
V5.03 booklet: Nonlinear dynamics of the voluminal structures
HT-62/06/005/A
2
Reference solution
2.1
Method of calculation
One wants to check two types of quantities:
· the first critical load buckling,
· the first Eigen frequency of the system in vibration.
The value of reference of the critical load required is obtained by a quasistatic calculation (word
key CRIT_FLAMB of STAT_NON_LINE). One takes this value obtained with the last pitch of calculation
quasistatic, which corresponds at the moment T = 1 S.
The number stored under CHAR_CRIT in the structure of data result (it is the coefficient
minimal multiplier of the loading forced to obtain the buckling load) being proportional
with the imposed loading which is monotonous growing linearly with time, one corrects it to have
the true value at the first moment of transitory dynamic calculation, is 1,001 S.
One has, by definition of multiplying coefficient CHAR_CRIT:
()
()
I
ext.
I
critical
T
F
T
CRIT
TANK
F
=
_
The external force is proportional to time:
()
I
ext.
I
ext.
T
F
T
F
=
, therefore
()
I
ext.
I
critical
T
F
T
CRIT
TANK
F
=
_
.
The assumption is made that on a pitch, the loading evolves/moves very slowly and thus that one can
to compare the result of dynamic calculation to a quasistatic evolution during this pitch. One can then
to write, for the first dynamic pitch, which follows quasistatic calculation:
()
()
()
()
()
001
,
1
1
_
_
_
_
_
_
_
1
_
_
1
_
_
1
1
_
_
_
_
=
=
+
+
+
+
I
LINE
NOT
STAT
I
I
I
LINE
NOT
STAT
I
LINE
NOT
DYNA
I
ext.
I
LINE
NOT
DYNA
I
ext.
I
LINE
NOT
STAT
critical
T
CRIT
TANK
T
T
T
CRIT
TANK
T
CRIT
TANK
T
F
T
CRIT
TANK
T
F
T
CRIT
TANK
F
For the vibratory analysis, one will make two tests:
· by using the elastic matrix of stiffness,
· by using the matrix of tangent stiffness plastic.
The two values of reference are obtained by two linear modal calculations carried out with
operator MODE_ITER_SIMULT.
To obtain the first Eigen frequency corresponding to the elastic case, one makes an elastic design
linear with definite initial Young MODE_ITER_SIMULT and material above (of being worth modulus
2.10
4
Mpa).
To obtain the first Eigen frequency corresponding to the tangent plastic case, a calculation is made
linear rubber band with MODE_ITER_SIMULT and a fictitious elastic material of which the Young modulus
is worth the definite plastic tangent module above: 200 Mpa, is 100 times less than the module
real rubber band. One will thus have an Eigen frequency 10 times weaker than the preceding one.
One knows also the analytical solution of our problem (cubic length 1 made up of only one
linear finite element) which is brought back to a case 1D of traction compression:
=
plastic.
material
:
rubber band,
material
:
S
rad
S
rad
E
/
0358128
,
0
/
358128
,
0
2
2.2
Sizes and results of reference
Sizes Values
Unit
Multiplying coefficient of the first
critical load buckling
2.85714E+01/1.001
First elastic Eigen frequency
3.58128E-01
Hz
First plastic Eigen frequency
3.58128E-02
Hz
Code_Aster
®
Version
8.3
Titrate:
SDNV106 Analyze with the eigenvalues in DYNA_NON_LINE
Date:
09/05/06
Author (S):
NR. GREFFET
Key
:
V5.03.106-A
Page:
4/4
Manual of Validation
V5.03 booklet: Nonlinear dynamics of the voluminal structures
HT-62/06/005/A
3 Modeling
With
3.1
Characteristics of the mesh
A number of meshs: 1 HEXA8
A number of nodes: 8
3.2 Functionalities
tested
One tests the key word factor CRIT_FLAMB and MODE_VIBR of DYNA_NON_LINE.
One also tests in postprocessing the recovery of scalars CHAR_CRIT and FREQ at moments
given, in an object of the evol_noli type which is the result of DYNA_NON_LINE:
Controls
Key word factor
Single-ended spanner word
Argument
DYNA_NON_LINE CRIT_FLAMB
NB_FREQ
1
CHAR_CRIT
(- 100.0, 100.)
DYNA_NON_LINE MODE_VIBR
NB_FREQ
3
MATR_RIGI
“TANGENT”
“ELASTIC”
TEST_RESU RESU
PARA
“CHAR_CRIT”
“FREQ”
3.3
Sizes tested and results
Identification Reference
Aster
% difference
Eigen frequency
vibratory plastic
Tps = 1.01
3.58128E-02 3.5812661359567D-02 - 3.87E-04
Tps
=
1.06
3.58128E-02 3.5812661359997D-02 - 3.87E-04
Tps
=
1.25
3.58128E-02 3.5812661358541D-02 - 3.87E-04
Tps
=
1.49
3.58128E-02 3.5812661355801D-02 - 3.87E-04
Eigen frequency
vibratory rubber band
Tps = 1.51
3.58128E-01 3.5812779545194D-01 - 5.71E-05
Tps
=
1.52
3.58128E-01 3.5812779545194D-01 - 5.71E-05
Tps
=
1.56
3.58128E-01 3.5812779545194D-01 - 5.71E-05
Tps
=
1.75
3.58128E-01 3.5812779545194D-01 - 5.71E-05
Tps
=
1.99
3.58128E-01 3.5812779545194D-01 - 5.71E-05
Coefficient of
first critical load
Tps = 1.001 2.854285714E+01 2.8570189972986E+01
0.096
4
Summary of the results
This test makes it possible to validate calculations of critical loads of buckling and Eigen frequencies
vibratory in DYNA_NON_LINE.