Code_Aster ®
Version
4.0
Titrate:
SDLL401 Poutre straight line inclined 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
Organization (S): EDF/IMA/MN, IAT, CNAM
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
Document: V2.02.401
SDLL401 - Tilted right Poutre 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).
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLL401 Poutre straight line inclined 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
1
Problem of reference
1.1 Geometry
B
With
20°
X
Appear 1.1-a
B
With
20°
X
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. 1011 Pa
Poisson's ratio
= 0,3
Density
= 7800 kg/m3
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 blocked
· For the specific loading [1.1-2]
Embedded node A: DX, DY, DZ, DRX, DRY, DRZ blocked
Loadings:
·
F (T) = 1000 * cos (T) according to direction AB
either distributed or applied at the end B
·
M
T = 1000
(T)
T ()
* cos
applied at the end B
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLL401 Poutre straight line inclined 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
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.

2u
2

U
S
- E S
= F (T

)
t2
x2
U (0


) = 0, U (L) = 0.
To solve, one applies to the equation the transform of Fourier in time:
2 U

=
2
1
4
2
-
U +
F
()
2
X
E
E S
U: transform of Fourier of U,
F: transform of Fourier of F.
Thus, we have for F (T) = F cos (2 T
O):

2 O
sin
X
2

has
F

2



has


U (X, T
O
) =

cos
L
1
2
2


ES 4

has

-


2 O
O
sin
L


has




2


O
-

cos
X
1
cos


(2 T
O).

has

-


E
2
with: has
=.
The use of the law of behavior gives us the tractive effort compression:


2 O
cos
X

F has

2


has


NR (X, T
O
) =
1 -

cos
L
2



has


2 O
O

sin
L


has



2

O
+ sin
X
cos 2 T.

has


O


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

Code_Aster ®
Version
4.0
Titrate:
SDLL401 Poutre straight line inclined 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
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

2u
2

U
S
- ES
= 0

t2
x2

U
1
U (0) = 0


, ()
L =
F (T).
X
ES
The technique of resolution is equivalent to that of the paragraph [§2.1.1.1].
For F (T) = F cos (2 T
O), we have:
2

sin
O X
F has

has


U (,
X T) =

cos (2

O T)
ES 2
2


O
cos
O L

has


E
2
with A
=
2

cos
O X

has


and NR (,
X T) = F
cos (2
O T)
2

cos
O L

has


2.1.2.2 Torsion

2 X


2
G
I
- I
X = F (T
p
)
2


X
X

t2
U (0) = 0, U (L) = 0


E
G = (21+)

2
0
4
sin
X
0 0
, 1
B F
4
B
I
=
m,

,
=
p
X (X T)
(
cos 2
T
0)
2
G I

2


2
p
0
0
cos
L
B

2

0
cos
X

B
I = I
M
,
=
cos
p
T (X T)
F
2
X
(
T
0)

2

0
cos
L

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

Code_Aster ®
Version
4.0
Titrate:
SDLL401 Poutre straight line inclined 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
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 Institut Aérotechnique “Proposition and realization of 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 grid
2 elements POU_D_T
3.3 Functionalities
tested
Commands
Keys
DYNA_LINE_TRAN
NEWMARK
[U4.54.01]
EXCIT
CHARGE
FONC_MULT
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SDLL401 Poutre straight line inclined 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
4
Results of modeling A
4.1 Results
4.1.1 Charge divided into traction
Analytical results
Aster results
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
Aster results
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
Aster results
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
Obstruction 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 grid
only of two elements POU_D_T.
Handbook of Validation
V2.02 booklet: Linear dynamics of the beams
HI-75/98/040 - Ind A