Code_Aster ®
Version
4.0
Titrate:
Transitory SDLL100 dynamic Réponse of a beam
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.02.100-D Page:
1/6
Organization (S): EDF/EP/AMV
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
Document: V2.02.100
SDLL100 - Transitory dynamic Réponse
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.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
Transitory SDLL100 dynamic Réponse of a beam
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.02.100-D Page:
2/6
1
Problem of reference
1.1 Geometry
y, v
With
B
F (T) = (T). Fx
X, U

1
R
X
T
R = 0.05 m I = 1. m
y
N01
N02
X
1.2
Material properties
E = 98 696.044 MPa
= 0.
= 3. 106 kg/m3
Without damping
: C = 0. or with damping proportional of Rayleigh
:
C = K + µ M
- 4
, = 5.10, µ = 5.
1.3
Boundary conditions and loadings
Force applied to the N02 node in b: Fx = 1. 106 NR
Function (T) evolution of the loading: (T) = 1., T 0.
1.4 Conditions
initial
Initial displacement no one.
Null initial speed.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
Transitory SDLL100 dynamic Réponse of a beam
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.02.100-D Page:
3/6
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:
F
X
X
=
1 - cos
B (T)
2 (
(T
0
)
m 0

1
3rd
2

2
m
=
S
I, 0 =
, T
=
O
3
I 2
0
where S is the surface of the section (R2).
· With damping: the analytical solution of the problem with 1 element is:
F

µ + 2
2
0
µ +

X (T)
X
=
1 - exp-
T
0 sin
cos
2


(T1) + (T
B
1
)
m
2
2
0


1




, µ coefficient damping proportional C = µ M + K

(- µ) 2 2 2 4
4
2
-
-
0
µ


0
=
1
2
2.2
Results of reference
xB displacement with T = I T0 I = 1,…, 10.

10
with:
T
= 2
0


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.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
Transitory SDLL100 dynamic Réponse of a beam
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.02.100-D Page:
4/6
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:
10­5 S.
Integration NEWMARK
= 0.25, = 0.5
WILSON integration
= 1.4
3.2
Characteristics of the grid
A number of nodes: 2
A number of meshs and types: 1 mesh SEG2
3.3 Functionalities
tested
Commands
Keys
COMB_MATR_ASSE
[U4.53.01]
DYNA_LINE_TRAN
NEWMARK
[U4.54.01]
WILSON
MATR_AMOR
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
Transitory SDLL100 dynamic Réponse of a beam
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.02.100-D Page:
5/6
4
Results of modeling A
4.1 Values
tested
Without damping:
Moment in dryness.
Reference
Aster
% diff.
WILSON aster
% diff.
NEWMARK
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
% diff.
WILSON aster
% diff.
NEWMARK
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 steps 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
Obstruction memory:
8 megawords
Time CPU To use:
150 seconds
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
Transitory SDLL100 dynamic Réponse of a beam
Date:
01/12/98
Author (S):
A.C. LIGHT
Key:
V2.02.100-D Page:
6/6
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 steps of time.
This problem requires a step of time of integration of 10­5 S.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A