Code_Aster ®
Version
4.0
Titrate:
SSLL403 Flambement 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

Organization (S): EDF/IMA/MN, IAT St CYR
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
Document: V3.01.403

SSLL403 - Flambement 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.

Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/01/010/A

Code_Aster ®
Version
4.0
Titrate:
SSLL403 Flambement 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


1
Problem of reference

1.1 Geometry

With
Z
0
X
total reference mark

Appear 1.1-A

Rectangular section: Hy = 0.01 m, Hz = 0.01 m
Length: L = 1 m

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:

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).

Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/01/010/A

Code_Aster ®
Version
4.0
Titrate:
SSLL403 Flambement 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


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:

L
M
(X) = p
[v () - v (X)]D
Fy

.
X

Arrow v (X) satisfied thus the equation:

L
L
D 2v


E I
= p
() - () = -
() + (-) ()



2
[v
v X] D
p
v
D
L X v X
Z dx


X
X



By deriving the two members, one obtains the differential equation:

D 3v
p
FD
+
(L - X)
= 0.
dx3
E I
dx
Z

FD
The function v' (X) =
satisfied the linear and homogeneous differential equation with the second command:
dx
D 2v'
p
+
(L - X) v' = 0,
dx2
E Iz

who can be solved using the functions of Bessel.

One finds the value of the linear weight then criticizes equalizes with:

E I
p
Z
C = 7 837
,
.
L3

The analytical solution gives numerically:

-
10 8
p
11
3
C = 7 837
,
2 10 ·
= 1 3061667
,
10.
12

2.2
Results of reference

The value criticizes multiplier:
C
P
C =
Sg

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 Aster beams “
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/01/010/A

Code_Aster ®
Version
4.0
Titrate:
SSLL403 Flambement 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


3 Modeling
With

3.1
Characteristics of modeling

The model is composed of 10 elements right beam of Euler.

3.2
Characteristics of the grid

It consists of 10 elements POU_D_E.

3.3 Functionalities
tested

Commands


Keys
AFFE_CARA_ELEM
POUTRE
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 represents 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



Obstruction memory:
8 MW
Time CPU to use:
5.8 seconds
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/01/010/A

Code_Aster ®
Version
4.0
Titrate:
SSLL403 Flambement 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


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.

Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/01/010/A

Code_Aster ®
Version
4.0
Titrate:
SSLL403 Flambement 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

Intentionally white left page.
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/01/010/A

Outline document