Code_Aster ®
Version
4.0
Titrate:
SHLL101 Poutre straight line. Analyze harmonic
Date:
01/09/99
Author (S):
B. QUINNEZ, G. DEVESA
Key:
V2.06.101-B Page:
1/8
Organization (S): EDF/IMA/MN, EP/AMV
Handbook of Validation
V2.06 booklet: Harmonic response of the linear structures
Document: V2.06.101
SHLL101 - Right Poutre. Analyze harmonic
Summary:
This two-dimensional problem consists in calculating the efforts present in a beam subjected to a traction or
with an inflection during a harmonic analysis. The reference solution is obtained starting from the equations
discretized.
This test comprises two modelings.
For the first modeling, four stresses are tested:
· force traction,
· force traction and material presenting a damping,
· flexural strength,
· flexural strength and material presenting a damping.
For the second modeling, two stresses are tested:
· force traction,
· force traction and material presenting a damping.
The second modeling makes it possible to test the complex loadings imposed by the command
AFFE_CHAR_MECA_C.
Handbook of Validation
V2.06 booklet: Harmonic response of the linear structures
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SHLL101 Poutre straight line. Analyze harmonic
Date:
01/09/99
Author (S):
B. QUINNEZ, G. DEVESA
Key:
V2.06.101-B Page:
2/8
1
Problem of reference
1.1 Geometry
y
y
y, v

X, U
With
B
X
Z
The geometrical characteristics of the beam constituting the mechanical model are as follows:
Length: L = 10 m
Cross section
Surface
IZ = IY
JX
3.439 10­3 m2
1.377 10­5 m4
2.754 10­5 m4
The co-ordinates (in meters) of the points characteristic of the beam are:
With
B
X
0.
10.
y
0.
0.
1.2
Material properties
The properties of material constituting the beam are:
E = 1.658 1010 Pa
= 0.3
= 1.3404106 104 kg/m3
= Amor_alpha = 0.001
= Amor_beta = 0.
1.3
Boundary conditions and loadings
The boundary condition which characterizes this problem is the embedding of point A and is written:
U = v = 0.
= 0.
For the loading one a:
Fx = 3000. NR
Fy = Fz = 0.
(tractive effort)
Fx = 0.
Fy = 3000. NR
Fz = 0.
(bending stress)
Handbook of Validation
V2.06 booklet: Harmonic response of the linear structures
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SHLL101 Poutre straight line. Analyze harmonic
Date:
01/09/99
Author (S):
B. QUINNEZ, G. DEVESA
Key:
V2.06.101-B Page:
3/8
2
Reference solution
2.1
Method of calculation used for the reference solution
If the beam is modelled by a beam of Euler-Bernoulli and only one finite element, the problem
harmonic can be written in the following way:
problem in traction:
(
ES
SSL
1 + I)
(
U
)
2
B -
(
U
)
B = F ()
B
L
6
X
from where
(
F B
U
)
()
B = ES
SSL
ES
2
-
+ I
L
6
L
problem in inflection:

13L
-
2
11 L

- L




12th I 1

y
2
v ()
B
F ()

B
y

- 2 35
210

+ (1+ I)


2

=


-
2
3
11 L
L

3
L
- L
L ()


B
0


210
105
2
3



Note:
If the material does not present damping, one has then: Amor_alpha = = 0.
The efforts at the point B are calculated in the following way:
problem in traction:
ES
2 SSL
NR ()
B =
-
(
U
)
B



L
6
problem in inflection:

13L
- 11L2

- L



12
1

VY ()
B
E I

2


y
v ()
B

= - 2 35
210

+


2


MFZ ()
B

- 11L2
L3

L3
- L
L ()
B





210
105
2
3



One analytically solves the systems 2 X 2 to obtain the solution.
2.2
Results of reference
The results of reference are displacements, speeds, accelerations and the efforts
generalized obtained at the point B during the harmonic analysis.
2.3
Notice for modeling B
For modeling B, one wants to test the problem in traction in the case of key word FORCE_POUTRE
who allows to apply efforts distributed. To obtain the same solution as the beam subjected to
nodal force in its end, the relation between the effort distributed constant and the nodal force is:
F L
F ()
B =
X
2
With the values given to the 1.3, one a: F = 600 NR/m
Handbook of Validation
V2.06 booklet: Harmonic response of the linear structures
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SHLL101 Poutre straight line. Analyze harmonic
Date:
01/09/99
Author (S):
B. QUINNEZ, G. DEVESA
Key:
V2.06.101-B Page:
4/8
2.4
Uncertainty on the solution
If the assumptions are checked (beam of Euler-Bernoulli), the solution is analytical.
2.5 References
bibliographical
[1]
Reference material of Code_Aster: Elements of beams “exact” (right and
curves) - [R3.08.01].
Handbook of Validation
V2.06 booklet: Harmonic response of the linear structures
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SHLL101 Poutre straight line. Analyze harmonic
Date:
01/09/99
Author (S):
B. QUINNEZ, G. DEVESA
Key:
V2.06.101-B Page:
5/8
3 Modeling
With
3.1
Characteristics of modeling
y, v
With
B
X, U
The beam consists of only one mesh.
The modeling used for the beam is that of Euler-Bernoulli (POU_D_E).
End A is embedded:
DX = DY = DZ = 0.
DRX = DRY = DRZ = 0.
3.2
Characteristics of the grid
A number of nodes: 2
A number of meshs and types: 1 mesh of the type SEG 2
The points characteristic of the grid are as follows:
Not A = A
Not B = B
3.3 Functionalities
tested
Commands
Keys
AFFE_CARA_ELEM
POUTRE
“GENERALE”
TOUT
[U4.24.01]
AFFE_CHAR_MECA
DDL_IMPO
NOEUD
[U4.25.01]
FORCE_NODALE
NOEUD
FX
FY
DEFI_MATERIAU
ELAS
E, RHO, NAKED
[U4.23.01]
AMOR_ALPHA
AMOR_BETA
CALC_MATR_ELEM
OPTION
“MASS_MECA”
[U4.41.01]
'AMOR_MECA
“RIGI_MECA”
CALC_VECT_ELEM
OPTION
“CHAR_MECA”
[U4.41.02]
DYNA_LINE_HARM
MATR_MASS
[U4.54.02]
MATR_RIGI
MATR_AMOR
EXCIT
VECT_ASSE
CALC_ELEM
OPTION
“EFGE_ELNO_DEPL”
[U4.61.02]
Handbook of Validation
V2.06 booklet: Harmonic response of the linear structures
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SHLL101 Poutre straight line. Analyze harmonic
Date:
01/09/99
Author (S):
B. QUINNEZ, G. DEVESA
Key:
V2.06.101-B Page:
6/8
4
Results of modeling A
4.1
Values tested (reality-imaginary form)
Problem 1: traction
Not/Size
Reference
Aster
% difference
displacement
B
DX
(5.318 10­5, 0.)
(5.318 10­5, 0.)
0.
speed
B
DX
(0., 3.341 10­3)
(0., 3.341 10­3)
0.
acceleration
B
DX
(­ 2.099 10­1, 0.)
(­ 2.099 10­1, 0.)
0.
generalized effort
B
NR
(3000., 0.)
(3000., 0.)
0.
Problem 2: inflection
Not/Size
Reference
Aster
% difference
displacement
B
DY
(1.828 10­2, 0.)
(1.828 10­2, 0.)
0.
DRZ (1.82 10­2, 0.)
(1.82 10­2, 0.)
0.
speed
B
DY
(0., 1.1489)
(0., 1.1489)
0.
DRZ (0., 1.1438)
(0., 1.1438)
0.
acceleration
B
DY
(­ 7.219 10­1)
(­ 7.219 10­1, 0.)
0.
DRZ (­ 7.186 10­1, 0.)
(­ 7.186 10­1, 0.)
0.
generalized effort
B
VY
(3000., 0.)
(3000., 0.)
0.
MFZ (0., 0.)
(­ 1.164 10­10, 0.)
0.
Problem 3: traction + damping
Not/Size
Reference
Aster
% diff
displacement
B D
(5.296 10­5, ­ 3.363 10­3)
(5.296 10­5, ­ 3.363 10­3)
0.
X
speed
B D
(2.113 10­4, 3.327 10­3)
(2.113 10­4, 3.327 10­3)
0.
X
acceleration
B D
(­ 2.091 10­1, 1.327 10­2)
(­ 2.091 10­1, 1.327 10­2)
0.
X
generalized effort
B NR
(2.987 103, ­ 1.8975 102)
(2.987 103, ­ 1.8975 102)
0.
Problem 4: inflection + damping
Not/Size
Reference
Aster
%
diff
displacement
B DY
(1.746 10­2, ­ 4.469 10­3)
(1.746 10­2, ­ 4.469 10­3)
0.
DRZ (1.757 10­2, ­ 3.402 10­3)
(1.757 10­2, ­ 3.402 10­3)
0.
speed
B DY
(2.808 10­1, 1.097)
(2.808 10­1, 1.097)
0.
DRZ (2.138 10­1, 1.104)
(2.138 10­1, 1.104)
0.
acceleration
B DY
(­ 6.895 10­1, 1.764 10­1)
(­ 6.895 10­1, 1.764 10­1)
0.
DRZ (­ 6.94 10­1, 1.343 10­1)
(­ 6.94 10­1, 1.343 10­1)
0.
generalized effort
B VY
(3.021 103, 1.212 102)
(3.021 103, 1.212 102)
0.
MFZ (­ 1.567 102, ­ 8.583 102)
(­ 1.567 102, ­ 8.583 102)
0.
4.2 Parameters
of execution
Version: NEW 3.06
Machine: CRAY C90
Obstruction memory:
8 MW
Time CPU To use:
5.9 seconds
Handbook of Validation
V2.06 booklet: Harmonic response of the linear structures
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SHLL101 Poutre straight line. Analyze harmonic
Date:
01/09/99
Author (S):
B. QUINNEZ, G. DEVESA
Key:
V2.06.101-B Page:
7/8
5 Modeling
B
5.1
Characteristics of modeling
y, v
With
B
X, U
The beam consists of only one mesh.
The modeling used for the beam is that of Euler-Bernoulli (POU_D_E).
End A is embedded:
DX = DY = DZ = 0.
DRX = DRY = DRZ = 0.
5.2
Characteristics of the grid
A number of nodes: 2
A number of meshs and types: 1 mesh of the type SEG 2
The points characteristic of the grid are as follows:
Not A = A
Not B = B
5.3 Functionalities
tested
Commands
Keys
AFFE_CARA_ELEM
POUTRE
“GENERALE”
TOUT
[U4.24.01]
AFFE_CHAR_MECA_C
DDL_IMPO
NOEUD
[U4.25.01]
FORCE_POUTRE
NOEUD
FX
DEFI_MATERIAU
ELAS
E, Rho, Naked
[U4.23.01]
Amor_alpha
Amor_Beta
CALC_MATR_ELEM
OPTION
“MASS_MECA”
[U4.41.01]
'AMOR_MECA
“RIGI_MECA”
DYNA_LINE_HARM
MATR_MASS
[U4.54.02]
MATR_RIGI
MATR_AMOR
EXCIT
CHARGE
FONC_MULT_C
CALC_ELEM
OPTION
“EFGE_ELNO_DEPL”
[U4.61.02]
EXCIT
CHARGE
FONC_MULT_C
Handbook of Validation
V2.06 booklet: Harmonic response of the linear structures
HI-75/98/040 - Ind A

Code_Aster ®
Version
4.0
Titrate:
SHLL101 Poutre straight line. Analyze harmonic
Date:
01/09/99
Author (S):
B. QUINNEZ, G. DEVESA
Key:
V2.06.101-B Page:
8/8
6
Results of modeling B
6.1
Values tested (reality-imaginary form)
Problem 1: traction (effort distributed real: null imaginary part)
Not/Size
Reference
Aster
% difference
displacement
B
DX
(5.318 10­5, 0.)
(5.318 10­5, 0.)
0.
speed
B
DX
(0., 3.341 10­3)
(0., 3.3414 10­3)
0.
acceleration
B
DX
(­ 2.099 10­1, 0.)
(­ 2.0994 10­1, 0.)
0.
generalized effort
B
NR
(3000., 0.)
(3000., 0.)
0.
Problem 2: traction (effort distributed complex: null rélle part)
Not/Size
Reference
Aster
% difference
displacement
B
DX
(0., 5.318 10­5)
(0., 5.318 10­5)
0.
speed
B
DX
(- 3.341 10­3, 0.)
(- 3.3414 10­3, 0.)
0.
acceleration
B
DX
(0., ­ 2.099 10­1)
(0., ­ 2.0994 10­1)
0.
generalized effort
B
NR
(0., 3000.)
(0., 3000.)
0.
Problem 3: traction + damping (effort distributed real: null imaginary part)
Not/Size
Reference
Aster
% diff
displacement
B
DX
(5.296 10­5, ­ 3.363 10­3)
(5.2966 10­5, ­ 3.3637 10­3)
0.
speed
B
DX
(2.113 10­4, 3.327 10­3)
(2.1135 10­4, 3.3279 10­3)
0.
acceleration
B
DX
(­ 2.091 10­1, 1.327 10­2)
(­ 2.091 10­1, 1.3279 10­2)
0.
generalized effort
B
NR
(2.9879 103, ­ 1.897 102)
(2.987 103, ­ 1.8975 102)
0.
Problem 4: inflection + damping (effort distributed complex: null real part)
Not/Size
Reference
Aster
% diff
displacement
B
DX
(3.363 10­3, 5.296 10­5)
(5.296 10­5, ­ 3.363 10­3)
0.
speed
B
DX
(- 3.327 10­3, 2.113 10­4)
(- 3.3279 10­3, 2.1135 10­4)
0.
acceleration
B
DX
(- 1.327 10­2, - 2.091 10­1)
(- 1.3279 10­2, - 2.091 10­1)
0.
generalized effort
B
NR
(1.897 102, 2.9879 103)
(1.8975 102, 2.98794 103)
0.
When the effort distributed is applied as an imaginary part of the loading, the reference solution is
obtained from that of real modeling A while exchanging left and imaginary part and in
changing the sign of the new real parts.
6.2 Parameters
of execution
Version: NEW 4.03
Machine: CRAY C90
Obstruction memory:
16 MW
Time CPU To use:
7.9 seconds
7
Summary of the results
The analytical results well are found.
Handbook of Validation
V2.06 booklet: Harmonic response of the linear structures
HI-75/98/040 - Ind A