Code_Aster
®
Version
6.4
Titrate:
SDLL311 - Transitory dynamic response of a beam in traction
Date:
13/06/03
Author (S):
E. BOYERE, T. QUESNEL
Key
:
V2.02.311-A
Page:
1/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/03/008/A
Organization (S):
EDF-R & D/AMA, IRCN
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
V2.02.311 document
SDLL311 - Transitory dynamic response of one
beam in traction under imposed displacement
Summary:
This problem-test corresponds to a linear transitory analysis of a bar requested in traction by application
of a displacement imposed at an end, the other end being embedded. Function displacement of time
is of type “Heaviside” imposed as from the initial moment.
The results obtained in the middle of the beam for a modeling with four elements are compared with
analytical solution of the problem discretized by four elements by not taking into account the peaks
instantaneous speed and of acceleration at the initial moment on the level of the end where displacement is imposed.
Code_Aster
®
Version
6.4
Titrate:
SDLL311 - Transitory dynamic response of a beam in traction
Date:
13/06/03
Author (S):
E. BOYERE, T. QUESNEL
Key
:
V2.02.311-A
Page:
2/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/03/008/A
1
Problem of reference
1.1 Geometry
y, v
X, U
R
R
=
0,05 m
=
1 m
With
B
C
/2
/2
U
(C)
= F (T) .u
F
1
T
1.2
Material properties
E = 98 696,044 MPa
= 0
= 3.10
6
kg/m
3
Damping proportional of Rayleigh:
M
K
C
µ
+
=
,
µ
=
=
-
510
5
4
.
,
1.3
Boundary conditions and loadings
Displacement imposed at the end C:
()
u.a.
U F T
=
()
with
U
=
-
10
3
m and
F T
()
evolution in function
time of the Heaviside type:
F T
T
()
,
=
1
0
.
Embedded end A.
1.4 Conditions
initial
Initial displacement no one in any point.
Null initial speed in any point.
Code_Aster
®
Version
6.4
Titrate:
SDLL311 - Transitory dynamic response of a beam in traction
Date:
13/06/03
Author (S):
E. BOYERE, T. QUESNEL
Key
:
V2.02.311-A
Page:
3/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/03/008/A
2
Reference solution
2.1
Method of calculation used for the reference solution
The discretized problem checks:
M
M
M
M
U
U
C
C
C
C
U
U
K
K
K
K
U
U
F
L
ld
ld
T
dd
L
D
L
ld
ld
T
dd
L
D
L
ld
ld
T
dd
L
D
D
+
+
=
&&
&&
&
&
0
,
with index
L
: ddl free
index
D
: ddl imposed
()
F T
D
external loadings applied to the nodes ends and leading to displacements
imposed
U
D
are unknown, one thus eliminates these equations and one obtains:
[]
{}
[]
{}
[]
{}
[]
{}
[]
{}
[]
{}
M
U
C
U
K
U
M
U
C
U
K
U
L
L
L
L
L
L
ld
D
ld
D
ld
D
&&
&
&&
&
+
+
= -
-
-
.
The only nonnull terms of the second member of this system are related to the variables kinematics
relating to the node end where displacement is imposed. However, with t=0,
cd.
cd.
U
U
&
&&
and
are not
defined but in t=0- and t=0+,
cd.
cd.
U
U
&
&&
and
are null. All the complexity of the problem comes from that.
To obtain a reference solution, we considered
cd.
cd.
U
U
&
&&
and
uniformly null it
who amounts not considering that the forces intern elastic at the end C. This is debatable of one
physical point of view but, by adopting the same assumptions at the time of the modeling of the problem,
the validation of Code_Aster can be concluded.
One calculates the reference solution by dealing with the following problem:
[]
{}
[]
{}
[]
{}
[]
{
}
()
{
}
()
{
}
0
0
0
0
)
(
=
=
-
=
+
+
L
L
D
ld
L
L
L
L
L
L
U
U
T
U
K
U
K
U
C
U
M
&
&
&&
and
with
.
With this intention, one transports the problem in the modal base of the system which checks:
[]
{}
[]
{}
M
U
K
U
L
L
L
L
&&
+
=
0
.
Damping being diagonal, the diagonal system is obtained:
[]
{}
[]
{}
[]
{}
()
{}
()
{} {}
0
=
=
+
+
T
G
T
G
T
G
X
K
X
C
X
m
G
G
G
for
where
&
&&
,
with
()
{
}
()
{}
0
0
0
0
=
=
X
X
&
and
.
In modal space, one thus solves three equations (3 ddls free) differential of the second command then
one returns in physical space. One obtains then the displacement of the point medium:
()
()
()
(
)
U T
E
has
T
B
T
B
it
I
I
I
I
I
=
+
-
=
1
3
cos ~
sin ~
,
with
~
I
: I
ème
own pseudo-pulsation of the deadened system.
Code_Aster
®
Version
6.4
Titrate:
SDLL311 - Transitory dynamic response of a beam in traction
Date:
13/06/03
Author (S):
E. BOYERE, T. QUESNEL
Key
:
V2.02.311-A
Page:
4/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/03/008/A
2.2
Results of reference
Displacement, speed and acceleration of the point medium B of the beam.
Displacement of the point medium B
- 2,00E-04
0,00E+00
2,00E-04
4,00E-04
6,00E-04
8,00E-04
1,00E-03
0
0,005
0,01
0,015
0,02
0,025
0,03
time (S)
U
B
(m)
2.3
Uncertainty on the solution
Analytical solution of the problem discretized in four elements length equalizes while considering
speed and acceleration uniformly null at the point C where displacement is imposed.
Code_Aster
®
Version
6.4
Titrate:
SDLL311 - Transitory dynamic response of a beam in traction
Date:
13/06/03
Author (S):
E. BOYERE, T. QUESNEL
Key
:
V2.02.311-A
Page:
5/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/03/008/A
3 Modeling
With
3.1
Characteristics of modeling
Modeling in element of beam 3D: POU_D_T
.
.
.
.
.
With
X
NR
1
NR
3
NR
2
NR
4
NR
5
B
y
C
Cutting:
AC = 4 meshs SEG2 equal length
Limiting conditions:
·
Embedded node N1(A)
DDL_IMPO DX=DY=DZ=DRX=DRY=DRZ=0
·
N5 node (C) in imposed displacement following X
DDL_IMPO DY=DZ=DRX=DRY=DRZ=0 DX (T) =
U
Resolution:
Algorithm of direct integration of Newmark
No time:
T = 10
5
S
Duration of observation: 0,03 S
3.2
Characteristics of the mesh
Node S numbers: 5
A number of meshs and type: 4 meshs SEG2
3.3
Functionalities tested
Controls
DYNA_LINE_TRAN NEWMARK
C.L. DIRICHLET BY VECTASS
Code_Aster
®
Version
6.4
Titrate:
SDLL311 - Transitory dynamic response of a beam in traction
Date:
13/06/03
Author (S):
E. BOYERE, T. QUESNEL
Key
:
V2.02.311-A
Page:
6/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/03/008/A
4
Results of modeling A
4.1 Values
tested
·
Displacement at the point medium B
Time
(S)
Displacement
Reference (m)
Displacement
Aster (m)
Difference
(%)
0,0054
87,376 e3
87,3763 e3
28,6 E 3%
0,0055
87,360 e3
87,3598 e3
21,9 E 3%
0,0108
26,818 e3
26,8178 e3
57,0 E 3%
0,0109
26,800 e3
26,8000 e3
10,3 E 3%
0,0163
64,386 e3
64,3865 e3
84,9 E 3%
0,0164
64,366 e3
64,3663 e3
42,7 E 3%
0,0217
41,083 e3
41,0828 e3
49,6 E 3%
0,0218
41,084 e3
41,0844 e3
94,0 E 3%
0,0271
55,525 e3
55,5247 e3
62,6 E 3%
0,0272
55,530 e3
55,5305 e3
93,3 E 3%
Code_Aster
®
Version
6.4
Titrate:
SDLL311 - Transitory dynamic response of a beam in traction
Date:
13/06/03
Author (S):
E. BOYERE, T. QUESNEL
Key
:
V2.02.311-A
Page:
7/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/03/008/A
5 Modeling
B
5.1
Characteristics of modeling
idem that modeling A
5.2
Characteristics of the mesh
idem that modeling A
5.3
Functionalities tested
Controls
DYNA_LINE_TRAN NEWMARK
C.L. DIRICHLET BY LOAD
6
Results of modeling B
6.1 Values
tested
·
Displacement at the point medium B
Time
(S)
Displacement
Reference (m)
Displacement
Aster (m)
Difference
(%)
0,0054
87,376 e3
87,3763 e3
28,7 E 3%
0,0055
87,360 e3
87,3598 e3
21,9 E 3%
0,0108
26,818 e3
26,8178 e3
56,9 E 3%
0,0109
26,800 e3
26,8000 e3
10,4 E 3%
0,0163
64,386 e3
64,3865 e3
85,0 E 3%
0,0164
64,366 e3
64,3663 e3
42,9 E 3%
0,0217
41,083 e3
41,0827 e3
49,6 E 3%
0,0218
41,084 e3
41,0844 e3
94,0 E 3%
0,0271
55,525 e3
55,5247 e3
62,6 E 3%
0,0272
55,530 e3
55,5305 e3
93,2 E 3%
Code_Aster
®
Version
6.4
Titrate:
SDLL311 - Transitory dynamic response of a beam in traction
Date:
13/06/03
Author (S):
E. BOYERE, T. QUESNEL
Key
:
V2.02.311-A
Page:
8/8
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HT-66/03/008/A
7
Summary of the results
The results given by Code_Aster are in perfect agreement with the results of the analytical model,
that displacement boils about it beam is imposed by a VECTOR ASSEMBLES or by a LOAD.
Caution: questions of Dirichlet in
DYNA_LINE_TRAN
are compatible only with
method of integration of NEWMARK.