Code_Aster
®
Version
4.0
Titrate:
Transitory SDLL100 Dynamic response of a beam
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.02.100-D
Page:
1/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A
Organization (S):
EDF/EP/AMV
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
Document: V2.02.100
SDLL100 - Transitory dynamic response
of a beam in simple traction
Summary:
This problem-test corresponds to a direct transitory analysis of a deadened linear system or not, made up
of a beam in simple traction, subjected to a loading of the Heaviside type applied as from the initial moment.
The problem discretized with 1 single element of beam has an analytical reference solution.
Code_Aster
®
Version
4.0
Titrate:
Transitory SDLL100 Dynamic response of a beam
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.02.100-D
Page:
2/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A
1
Problem of reference
1.1 Geometry
y, v
X, U
With
B
F (T) =
(T). F
X
R
X
1
T
R = 0.05 m I = 1. m
y
N01
N02
X
1.2
Material properties
E = 98 696.044 MPa
= 0.
= 3. 10
6
kg/m
3
Without damping
: C = 0. or with damping proportional of Rayleigh
:
C
=
K
+
µ
M,
=
5.10
-
4
,
µ =
5
.
1.3
Boundary conditions and loadings
Force applied to the N02 node in b: Fx = 1. 10
6
NR
Function
(T) evolution of the loading:
(T) = 1., T
0.
1.4 Conditions
initial
Initial displacement no one.
Null initial speed.
Code_Aster
®
Version
4.0
Titrate:
Transitory SDLL100 Dynamic response of a beam
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.02.100-D
Page:
3/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A
2
Reference solution
2.1
Method of calculation used for the reference solution
·
Without damping: the analytical solution of the problem with 1 element is:
()
()
(
)
X
T
F
m
T
m
IF
E
I
T
B
X
O
=
-
=
=
=
0
2
0
0
2
2
0
1
1
3
3
2
cos
,
,
where
S
is the surface of the section
()
R
2
.
·
With damping: the analytical solution of the problem with 1 element is:
()
()
()
X
T
F
m
T
T
T
B
X
=
-
- +
+
+
µ
µ
0
2
0
2
0
2
1
1
1
1
2
2
exp
sin
cos
,
µ
damping coefficient proportional
C
M
K
=
+
µ
(
)
µ
µ
1
0
2
2
2
0
4
4
2
2
=
-
-
-
2.2
Results of reference
Displacement
X
B
with
T
=
I T
0
10 I
=
1,…, 10.
with:
T
0
=
2
0
2.3
Uncertainty on the solution
Analytical solution.
Note:
The reference solution corresponds to the solution obtained with the discretization to an element
and by keeping a matrix masses full. That makes it possible to validate the algorithm but it is not
the solution of the physical problem.
Code_Aster
®
Version
4.0
Titrate:
Transitory SDLL100 Dynamic response of a beam
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.02.100-D
Page:
4/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A
3 Modeling
With
3.1
Characteristics of modeling
POU_D_T
y
N01
N02
X
Cutting:
N01
N02
1 mesh SEG2
Limiting conditions:
DDL_IMPO
with the N01 node:
DX:0., DY:0., DZ:0., DRX:0, DRY:0, DRZ:0
No time:
10
5
S.
Integration NEWMARK
= 0.25,
= 0.5
WILSON integration
= 1.4
3.2
Characteristics of the mesh
A number of nodes: 2
A number of meshs and types: 1 mesh SEG2
3.3 Functionalities
tested
Controls
Keys
COMB_MATR_ASSE
[U4.53.01]
DYNA_LINE_TRAN
NEWMARK
[U4.54.01]
WILSON
MATR_AMOR
Code_Aster
®
Version
4.0
Titrate:
Transitory SDLL100 Dynamic response of a beam
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.02.100-D
Page:
5/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A
4
Results of modeling A
4.1 Values
tested
Without damping:
Moment in dryness.
Reference
Aster
NEWMARK
% diff.
Aster
WILSON
% diff.
2.E3
2.4638E04
2.4519E04
0.5
2.4424E04
0.86
4.E3
8.9141E04
8.8948E04
1.93
8.8794E04
0.38
6.E3
1.6887E03
1.6868E03
0.11
1.6852E03
0.20
8.E3
2.3337E03
2.3325E03
0.05
2.3316E03
0.09
1.E2
2.5801E03
2.5801E03
0.03
2.5801E03
0
1.2E2
2.3337E03
2.3349E03
0.05
2.3359E03
0.09
1.4E2
1.6887E3
1.6906E03
0.43
1.6922E03
0.21
1.6E2
8.9141E04
8.9334E04
0.21
8.9489E04
0.4
1.8E2
2.4638E04
2.4758E04
0.48
2.4854E04
0.87
2.E2
0.0000
3.1989E09
-
9.3188E09
-
With damping:
Moment in dryness.
Reference
Aster
NEWMARK
% diff.
Aster
WILSON
% diff.
2.E3
2.3775E04
2.3662E04
0.47
2.3572E04
0.85
4.E3
8.3189E04
8.3015E04
0.21
8.2877E04
0.37
6.E3
1.5307E03
1.5290E03
0.11
1.5277E03
0.2
8.E3
2.0704E03
2.0694E03
0.04
2.0686E03
0.09
1.E2
2.2721E03
2.2721E03
0.
2.2720E03
0.004
1.2E2
2.0976E03
2.0984E03
0.04
2.0991E03
0.07
1.4E2
1.6488E03
1.6501E03
0.08
1.6511E03
0.14
1.6E2
1.1164E03
1.1176E03
0.11
1.1186E03
0.2
1.8E2
7.0165E04
7.0241E04
0.11
7.0302E04
0.19
2.E2
5.4263E04
5.4266E04
0.005
5.4269E04
0.01
4.2 Remarks
After the first two pitches of time, the solution with damping is obtained with an error
lower than 0.2%.
4.3 Parameters
of execution
Version: 3.02.19
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
8 megawords
Time CPU To use:
150 seconds
Code_Aster
®
Version
4.0
Titrate:
Transitory SDLL100 Dynamic response of a beam
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.02.100-D
Page:
6/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A
5
Summary of the results
The two algorithms give a solution with an error lower than 0.2% of the solution of
reference after the first two pitches of time.
This problem requires a pitch of time of integration of 10
5
S.