Code_Aster
®
Version
4.0
Titrate:
SDLL401 tilted right Beam with 20°, subjected to sinusoidal efforts
Date:
01/12/98
Author (S):
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V2.02.401-A
Page:
1/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A
Organization (S):
EDF/IMA/MN, IAT, CNAM
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
Document: V2.02.401
SDLL401 - Tilted right beam with 20°, subjected
with sinusoidal efforts
Summary:
This test results from the validation independent of version 4 of the models of beams.
It makes it possible to check the internal efforts on an inclined beam, for sinusoidal loadings in function
time (a modeling with elements
POU_D_T
, right beam of Timoshenko).
Code_Aster
®
Version
4.0
Titrate:
SDLL401 tilted right Beam with 20°, subjected to sinusoidal efforts
Date:
01/12/98
Author (S):
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V2.02.401-A
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
X
20°
With
B
Appear 1.1-a
X
20°
With
B
Appear 1.1-b
Right beam length 1 Mr.
slope 20° compared to X (trigonometrical direction).
Characteristics of the section:
S =
* 0.01 ² m ²
1.2
Properties of materials
Young modulus
E = 2. 10
11
AP
Poisson's ratio
= 0,3
Density
= 7800 kg/m
3
1.3
Boundary conditions and loading
Boundary condition:
·
For the loading distributed [1.1-1]
Embedded nodes A and B: DX, DY, DZ, DRX, DRY, DRZ locked
·
For the specific loading [1.1-2]
Embedded node A: DX, DY, DZ, DRX, DRY, DRZ locked
Loadings:
·
()
F T
T
()
* cos
=
1000
according to direction AB
either distributed or applied at the end B
·
()
M
T
T
T
()
* cos
=
1000
applied at the end B
Code_Aster
®
Version
4.0
Titrate:
SDLL401 tilted right Beam with 20°, subjected to sinusoidal efforts
Date:
01/12/98
Author (S):
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V2.02.401-A
Page:
3/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A
2
Reference solutions
2.1
Method of calculation used for the reference solutions
2.1.1 Loading distributed of traction and compression
A right beam length L working only in traction and compression is subjected to one
loading distributed constant according to X but varying in a sinusoidal way according to time. It is
embedded at its two ends.
S
U
T
E S
U
X
F T
U
U L
2
2
2
2
0
0
0
-
=
=
=
()
()
,
()
.
To solve, one applies to the equation the transform of Fourier in time:
2
2
2
4
1
E
U
E S ()
U
X
F
=
+
2
-
U
:
transform of Fourier of U,
F
:
transform of Fourier of F.
Thus, we have for
(
)
F T
F
T
O
()
cos
=
2
:
(
)
U X T
has
F
ES
has
X
L
has
T
O
O
O
(,)
cos
L
sin
has
sin
has
cos
cos
.
=
-
-
-
2
2
2
4
2
1
2
1
2
2
X
2
O
O
O
with:
has
E
2
=
.
The use of the law of behavior gives us the tensile load compression:
NR X T
F has
has
X
L
has
T
O
O
O
(,)
cos
L
cos
has
sin
has
sin
X
cos
.
=
-
2
1
2
2
2
2
+
2
O
O
O
Code_Aster
®
Version
4.0
Titrate:
SDLL401 tilted right Beam with 20°, subjected to sinusoidal efforts
Date:
01/12/98
Author (S):
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V2.02.401-A
Page:
4/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A
2.1.2 Loadings
specific
A beam comforts length L working only in traction compression (or torsion) is
subjected to a sinusoidal force in time, (or a moment) applied at its loose lead.
2.1.2.1 Traction
S
U
T
U
X
=
U
U
X
=
2
2
2
0
0
0
()
1 ()
2
-
=
ES
()
,
L
ES F T.
The technique of resolution is equivalent to that of the paragraph [§2.1.1.1].
For
(
)
F T
F
T
O
()
cos
=
2
, we have:
(
)
U
has
E
O
(,) =
2
2
O
O
O
with
X T
F has
ES
sin
has
X
cos
has
L
cos
T
2
2
2
=
and
(
)
NR
F
O
(,) =
2
2
O
O
X T
cos
has
X
cos
has
L
cos
T
2
2.1.2.2 Torsion
(
)
()
(
)
()
(
)
G I
X
X
I
T
F T
U
U L
G
E
I
m
X T
B F
G I
B
X
B
L
T
I
I
M
X T
F
B
X
B
L
T
B
G
p
X
p
X
p
p
T
X
X
2
2
2
2
4
4
0
0
0
0
0
0
0
0
0
0
2 1
0 01
2
2
2
2
2
2
2
2
-
=
=
=
=
+
=
=
=
=
=
()
()
,
()
,
,
,
sin
cos
cos
,
cos
cos
cos
with
Code_Aster
®
Version
4.0
Titrate:
SDLL401 tilted right Beam with 20°, subjected to sinusoidal efforts
Date:
01/12/98
Author (S):
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V2.02.401-A
Page:
5/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A
2.2
Results of reference
Interior efforts (NR and MT)
2.3
Uncertainty on the solution
Analytical solution.
2.4 References
bibliographical
[1]
Report/ratio n° 2314/A of the Institute Aerotechnics “Proposal and realization for new cases
tests missing with the validation beams ASTER “
3 Modeling
With
3.1
Characteristics of modeling
The model is composed of 2 elements right beam of Timoshenko.
3.2
Characteristics of the mesh
2 elements
POU_D_T
3.3 Functionalities
tested
Controls
Keys
DYNA_LINE_TRAN
NEWMARK
[U4.54.01]
EXCIT
CHARGE
FONC_MULT
Code_Aster
®
Version
4.0
Titrate:
SDLL401 tilted right Beam with 20°, subjected to sinusoidal efforts
Date:
01/12/98
Author (S):
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V2.02.401-A
Page:
6/6
Manual of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A
4
Results of modeling A
4.1 Results
4.1.1 Charge divided into traction
Analytical results
Results
Aster
Variation (%)
Normal effort for X = 0
T = 1/3 S
4.7247E+02
4.7247E+02
9.12E07
T = 2/3 S
3.92944E+02
3.9294E+02
6.08E07
Normal effort for X = L/2
T = 1/3 S
0.0000E+00
2.1985E12
2.20E12 *
T = 2/3 S
0.0000E+00
2.5087E12
2.51E12 *
* Absolute deviation
4.1.2 Charge
specific
4.1.2.1 Loading in traction
Normal effort for X = 0
Analytical results
Results
Aster
Variation (%)
T = 1/3 S
9.44957E+02
9.44956E+02
7.59E07
T = 2/3 S
7.8588E+02
7.8588E+02
3.01E06
4.1.2.2 Loading in torsion
Torque for X = 0
Analytical results
Results
Aster
Variation (%)
T = 1/3 S
9.4495E+02
9.4495E+02
1.88E06
T = 2/3 S
7.8588E+02
7.8589E+02
7.29E06
4.2 Parameters
of execution
Version: 4.02
Machine: CRAY C90
Overall dimension memory:
8 MW
Time CPU to use:
10 seconds
5
Summary of the results
This test makes it possible to check that the efforts intern elements of beam in dynamics are correct.
The results show a very good agreement with the analytical solution, for a made up mesh
only of two elements
POU_D_T
.