Code_Aster
®
Version
4.0
Titrate:
SSLL403 Buckling of a beam under the effect of its actual weight
Date:
22/01/98
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.403-A
Page:
1/6
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/01/010/A
Organization (S)
: EDF/IMA/MN, IAT St CYR
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
Document:
V
V
3
3
.
.
0
0
1
1
.
.
4
4
0
0
3
3
SSLL403 - Buckling of a beam under the effect of
its actual weight
Summary:
This test makes it possible to validate in linear elasticity the loading due to the forces of gravity for a modeling
of right beam type of Euler (POU_D_E). It also allows the implementation and the validation of the calculation of
stamp geometrical rigidity.
The reference solution is analytical and the results considered to be satisfactory.
Code_Aster
®
Version
4.0
Titrate:
SSLL403 Buckling of a beam under the effect of its actual weight
Date:
22/01/98
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.403-A
Page:
2/6
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/01/010/A
1
Problem of reference
1.1 Geometry
With
0
Z
X
total reference mark
Appear 1.1-A
Rectangular section: H
y
= 0.01 m, Hz = 0.01 m
Length: L = 1 m
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:
Embedded end (0): DX = DY = DZ = DRX = DRY = DRZ = 0.
Loading:
Force gravity: p weight per unit of length with G = (0 0
- 9,81) (given in total reference mark).
Code_Aster
®
Version
4.0
Titrate:
SSLL403 Buckling of a beam under the effect of its actual weight
Date:
22/01/98
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.403-A
Page:
3/6
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/01/010/A
2
Reference solutions
2.1
Method of calculation used for the reference solutions
In local reference mark, X following axis OA of the beam, the bending moment, with X-coordinate X, has for
expression:
[
]
M
X
p
v
v X D
Fy
X
L
()
()
()
=
-
.
The arrow
()
v X
satisfied thus the equation:
[
]
E I D v
dx
p
v
v X D
p
v
D
L X v X
Z
X
L
X
L
2
2
=
-
= -
+
-
()
()
()
(
) ()
By deriving the two members, one obtains the differential equation:
D v
dx
p
E I
L X FD
dx
Z
3
3
0
+
-
=
(
)
.
The function
()
v X
FD
dx
'
=
satisfied the linear and homogeneous differential equation with the second command:
D v
dx
p
E I
L X v
Z
2
2
0
'
(
) '
+
-
=
,
who can be solved using the functions of Bessel.
One finds the value of the linear weight then criticizes equalizes with:
p
E I
L
C
Z
= 7 837
3
,
.
The analytical solution gives numerically:
p
C
=
=
·
-
7.837 2 10
10
12
1 3061667 10
11
8
3
,
,
.
2.2
Results of reference
The value criticizes multiplier
:
C
C
P
Sg
=
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 “
Code_Aster
®
Version
4.0
Titrate:
SSLL403 Buckling of a beam under the effect of its actual weight
Date:
22/01/98
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.403-A
Page:
4/6
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/01/010/A
3 Modeling
With
3.1
Characteristics of modeling
The model is composed of 10 elements right beam of Euler.
3.2
Characteristics of the mesh
It consists of 10 elements POU_D_E.
3.3 Functionalities
tested
Controls
Keys
AFFE_CARA_ELEM
BEAM
SECTION
RECTANGLE
[U4.24.01]
AFFE_CHAR_MECA GRAVITY
[U4.25.01]
CALC_CHAM_ELEM SIEF_ELGA_DEPL
[U4.61.01]
MODE_ITER_SIMULT METHOD
JACOBI
[U4.52.02]
4
Results of modeling A
4.1 Values
tested
Eigenvalue of the system (K +
KG) X = 0:
Reference
Aster Variation
%
170.701 170.0005
0.408
4.2 Notice
Since
p
S G
C
=
, (
S G
represent linear prestressing), we have like
critical loading:
p
NR m
C
=
-
1300 84
1
,
.
4.3 Parameters
of execution
Version: 4.02.13
Machine: CRAY C90
Overall dimension memory:
8 MW
Time CPU to use:
5.8 seconds
Code_Aster
®
Version
4.0
Titrate:
SSLL403 Buckling of a beam under the effect of its actual weight
Date:
22/01/98
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.403-A
Page:
5/6
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/01/010/A
5
Summary of the results
The results are very close to the analytical solution (variation: 0,4% per 10 elements). This variation is
function of the smoothness of discretization being given assumptions used for rigidity
geometrical (cf [R3.08.01]). This thus validates this type of loading for the buckling of Euler.
Code_Aster
®
Version
4.0
Titrate:
SSLL403 Buckling of a beam under the effect of its actual weight
Date:
22/01/98
Author (S)
:
J.M. PROIX, m.t. BOURDEIX, P. HEMON, O. WILK
Key:
V3.01.403-A
Page:
6/6
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/01/010/A
Intentionally white left page.