Code_Aster ®
Version
5.0
Titrate:
SSLL404 - Flambement of an arch


Date:
23/09/02
Author (S):
J.M. PROIX, F. SOULIE Key
:
V3.01.404-A Page:
1/6

Organization (S): EDF/AMA, SAMTECH
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
V3.01.404 document
SSLL404 - Flambement of an arch

Summary

The applicability of this test is the analysis of stability of the structures. The studied structure is an arch
bent by moments applied at the two ends; it is modelled by elements of beams
straight lines. The goal is to calculate the breaking values of the moments.

The interest of this test lies in the following aspects:

· calculation of a geometrical matrix of rigidity for elements POU_D_E.
· test of modal methods MODE_ITER_SIMULT and MODE_ITER_INV of stability
· presence of close eigenvalues

The calculated clean loads are compared with values obtained analytically for a model of
beam of Euler-Bernoulli.

In this test, one also validates the RAYLEIGH option of command MODE_ITER_INV.

Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLL404 - Flambement of an arch


Date:
23/09/02
Author (S):
J.M. PROIX, F. SOULIE Key
:
V3.01.404-A Page:
2/6

1
Problem of reference

1.1 Geometry
B
y
Y
H X
M
y
B
R
With
X
M


Radius of curvature
R = 0.3 m
Height of the profile
H = 0.015 m
Width of the profile
B = 0.002 m
Section
S = bh
1čre inertia of inflection
IX = bh3/12
the 2nd inertia of inflection
IY = hb3/12
Inertia of torsion
J = hb3/3

1.2
Properties of materials

Young modulus
E = 7. E 10 NR/m ²
Poisson's ratio
= 0.3
Modulus of rigidity
G = E/2 (1+)

1.3
Boundary conditions and loading

The beam Bi-is supported. One prevents the torsion of the section at ends A and B. Pour to respect
the assumptions of the ideal model taken as reference, it is important that the moment is
constant and that the normal effort is null along the beam. This is why free it is left
displacement U according to X at the point B. Les boundary conditions are:

At point a: U = v = W = 0; Y = 0

At point b: v = W = 0; X = 0

The initial state of stress which makes it possible to carry out the analysis of stability is obtained by imposing one
bending moment around axis Z:

At points A and b: M = 1 Nm

1.4 Conditions
initial

Without object in static analysis of stability.
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLL404 - Flambement of an arch


Date:
23/09/02
Author (S):
J.M. PROIX, F. SOULIE Key
:
V3.01.404-A Page:
3/6

2
Reference solution

2.1
Method of calculation used for the reference solution

The reference solution is obtained analytically for a beam of Euler-Bernoulli. Aspects
theoretical are developed in the reference [bib1].

By using the notations of the paragraph [§1], the values criticize are given by the expression:

I.E.(INTERNAL EXCITATION) + GJ
I.E.(INTERNAL EXCITATION) - GJ 2
I.E.(INTERNAL EXCITATION) GJ
M
X
X
±
4n2
X
= -
N
CR
1 2
, 3
,….
R


R

+
=
2
2
R2

The plus sign corresponds to positive moments such as they are indicated on the figure of [§1.1].


2.2
Results of reference

The first 5 critical loads are classified by command of increasing module.

Mode Moment criticizes (Nm)
1 2.86074
2 8.63207
3 ­ 8.78382
4 14.4147
5 ­ 14.5551

With Code_Aster, one finds the opposites of these critical loads (what is logical compared to
formulation of the problem to be solved).

2.3
Uncertainty on the solution

Analytical solution

2.4 References
bibliographical

[1]
TIMOSHENKO Stephen P., GERE James Mr., Theory off Elastic Stability, McGraw-Hill,
International Edition, 1963, pp. 313-318.

Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLL404 - Flambement of an arch


Date:
23/09/02
Author (S):
J.M. PROIX, F. SOULIE Key
:
V3.01.404-A Page:
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3 Modeling
With

3.1
Characteristics of modeling

Y, DY
19
1
X, DX


The arch is with a grid by means of elements of right beam of type POU_D_E.

Boundary conditions:

At point A such as X = R, Y = 0:

DX = DY = DZ = 0 and RY = 0

At the point B such as X = 0, Y = R:

DY = DZ = 0 and X-ray = 0
For the static analysis, unit moments around Z are defined in nodes 1 and 19.


3.2
Characteristics of the grid

A number of nodes: 19
A number of meshs: 18 POU_D_E

3.3 Functionalities
tested

Commands
AFFE_MODELE
AFFE
MODELISATION
“POU_D_E”

AFFE_CHAR_MECA
DDL_IMPO

AFFE_CARA_ELEM
POUTRE

CALC_MATR_ELEM
OPTION
“RIGI_GEOM”

“RIGI_MECA”
MODE_ITER_SIMULT
METHODE
“SORENSEN”

CALC_FREQ
OPTION
PLUS_PETITE'

NMAX_FREQ

MODE_ITER_INV
CALC_FREQ
OPTION
“PROCHE”

CHAR_CRIT

CALC_MODE
OPTION
RAYLEIGH


Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLL404 - Flambement of an arch


Date:
23/09/02
Author (S):
J.M. PROIX, F. SOULIE Key
:
V3.01.404-A Page:
5/6

4
Results of modeling A

Critical load

4.1
MODE_ITER_SIMULT with METHODE = “SORENSEN”

Identification
Reference
Code_Aster %
difference
N° critical load
(multiplied by - 1)
1 ­ 2.86074
­ 2.75137
3.823
2 ­ 8.63207
­ 8.30613
3.776
3
8.78382 8.39554 4.420
4 ­ 14.4147
­ 13.93216
3.348
5
14.5551 14.01104 3.738

4.2
MODE_ITER_INV with OPTION = “PROCHE”

Identification
Reference
Code_Aster %
difference
N° critical load
(multiplied by - 1)
1 ­ 2.86074
­ 2.75137
3.823
2 ­ 8.63207
­ 8.30613
3.776
3 8.78382
8.39554
4.420
4 ­ 14.4147
­ 13.93216
3.348
5 14.5551
14.01104
3.738

Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A

Code_Aster ®
Version
5.0
Titrate:
SSLL404 - Flambement of an arch


Date:
23/09/02
Author (S):
J.M. PROIX, F. SOULIE Key
:
V3.01.404-A Page:
6/6

5
Summary of the results

The methods of Sorensen and the iterations opposite give identical and satisfactory results
since the maximum change with the analytical solution is lower than 4.5%.On recalls than the solution
analytical takes into account the curvature of the structure.

Elements MEPOUCT could not be used in this test because the calculation of the matrix of rigidity
geometrical is not available for this type of element.

Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HT-66/02/001/A

Outline document