Code_Aster ®
Version
4.0
Titrate:
Elastic SSLL103 Flambement of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A Page:
1/8
Organization (S): EDF/IMA/MN
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
Document: V3.01.103
SSLL103 - Elastic Flambement of an angle
Summary:
A right beam (corner with equal wings) biarticulée is subjected to a normal effort (excentré or not) or to one
bending moment.
One seeks the critical loads of elastic buckling.
· linear elastic mechanics,
· buckling of a beam,
· eccentricity of the center of torsion,
· interest of the test: calculation of the geometrical matrix of rigidity of elements POU_D_TG and POU_D_T,
· 2 modelings.
An uncertainty persists on the number of modes of buckling of the reference solution [§5].
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A

Code_Aster ®
Version
4.0
Titrate:
Elastic SSLL103 Flambement of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A Page:
2/8
1
Problem of reference
1.1 Geometry
Z
M
y
M
P
With
With
P
1
2
X
L = 1200 mm
Characteristics of the section:

To = 1856 mm2
Z
Iy = 4167339 mm4
Section A
I
1 and A2
Z = 1045547 mm4
With
J = 39595 mm4
I = 44398819 mm6
G
I
C
yr2 = 84948392 mm5
y
teststemyç = ­ 41.012 mm
has
zc = 0
= 120 mm have
E
E = 8 mm
CG = 41.012 mm
1.2
Material properties
E = 2.1 10­5 MPa
= 0.3
1.3
Boundary conditions and loadings
C.L. :
A1: DX = DY = DZ = DRX = 0
A2: DY = DZ = DRX = 0
Loading
· case 1: axial load P in G
· case 2: axial load P out of C
· case 3: axial load P in A
· case 4: bending moment M
1.4 Remarks
For cases 2 and 3, one applies in A2 an effort in G, then one superimposes in A1 and A2 one moment of
inflection (according to OZ for cases 2 following OY for case 3) to offset the effort out of C (or in A).
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A

Code_Aster ®
Version
4.0
Titrate:
Elastic SSLL103 Flambement of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A Page:
3/8
2
Reference solution
2.1
Method of calculation used for the reference solution
With taking into account of warping, the calculations made by V. Of City De Goyet [bib1] give:
that is to say:
I =
2
;
=
2
2
2
2
2
;
2 =
+
;
2 =
+
y
Z dA Iz
With
y dA Iyr
With
(yy Z) dA
Izr
With
Z (y Z) dA
With
Pcry = 2
E I/L2; Pcrz = 2
E I/L2; Pcrx =
+ 2
2

Z
y
(GJ I.E.(INTERNAL EXCITATION)/L

) Macaw
Arc = (I + I/+ 2 + 2 +
/
- 2
+
2/
-
y
Z) With y
Z
y
C
C
C (I
I
y
yrz
Z
c)
zc (I
I
2z
Zr
y
c)
Macaw = (I + I/+ 2 + 2 +
/
- 2
+
(
-
Zr
/Iy
Z
2
2 c)
y
Z) With y
Z
y
C
C
(I has
I
y
yrz
Z
c)
Z
I
has
with:
(y, Z
has
has): co-ordinates of the point of load application
(y, Z
C
c): co-ordinates of the center of torsion
Case 1, 2, 3:
One obtains 3 critical loads by solving the equation of the 3° degree out of P:
2
2
Macaw (Pcry - P) (Pcrz - P) (Pcrx - P) - P2 (Pcrz - P) (Z - Z) - P2 (Pcry - P) (y - y
=
C
has
C
has)
0
Case 4:
The moment criticizes Mcr (around the axis y) is worth:
1/2
Mcr = ± (GJ + the 2nd I/L2

) Pcry)
By neglecting warping: the analytical solution of reference is given in [bib2] [bib3].
2.2
Results of reference
Values of the critical loads corresponding to the first modes of buckling for the various cases
of load.
2.3
Uncertainty on the solution
Analytical solution. The values of reference are obtained using NAG (routine C0SAGF, EPS =
10­8).
2.4 References
bibliographical
[1]
V. OF TOWN OF GOYET “nonlinear static Analyze by the finite element method of
formed space structures of beams with nonsymmetrical section " - Thèse of doctorate
University of Liege, MSM, academic year (1988-1989).
[2]
P. PENSERINI “elastic Instabilité of the beams with open mean profile: theoretical aspects and
numerical " Note EDF/DER/HM77/112.
[3]
J. CERISIER “Propagation of two cases tests of modeling of the calculation of the beams in
elastic buckling in Code_Aster " HM77/184
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A

Code_Aster ®
Version
4.0
Titrate:
Elastic SSLL103 Flambement of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A Page:
4/8
3 Modeling
With
3.1
Characteristics of modeling
8 elements POU_D_TG
With
With
1
2
3.2
Characteristics of the grid
A number of nodes: 9
A number of meshs and types: 8 SEG2
3.3 Functionalities
tested
Commands
Keys
CALC_MATR_ELEM
“RIGI_GEOM”
[U4.41.01]
MODE_ITER_SIMULT
“PLUS_PETITE”
[U4.52.02]
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A

Code_Aster ®
Version
4.0
Titrate:
Elastic SSLL103 Flambement of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A Page:
5/8
4
Results of modeling A
4.1 Values
tested
Identification
Reference
Aster
% difference
Case 1
mode 1
­ 6.92531E+05
­ 6.92533E+05
0.000
mode 2
­ 1.50487E+06
­ 1.50492E+06
0.003
mode 3
­ 1.00589E+07
­ 1.00593E+07
0.003
Case 2
mode 1
­ 1.50487E+06
­ 1.50492E+06
0.003
mode 2
­ 5.99812E+06
­ 5.99831E+06
0.003
mode 3
1.47904E+06
1.47904E+06
0.000
Case 3
mode 1
­ 5.72260E+05
­ 5.72265E+05
0.001
mode 2
­ 2.45950E+06
­ 2.45957E+06
0.003
mode 3
­ 1.85673E+07
­ 1.85679E+07
0.003
Case 4
mode 1
7.00631E+07
7.00642E+07
0.002
4.2 Remarks
The precision is excellent with 8 elements in the length.
4.3 Parameters
of execution
Version: 3.02
Machine: CRAY C90
System:
UNICOS 8.0
Obstruction memory:
8 megawords
Time CPU To use:
11 seconds
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A

Code_Aster ®
Version
4.0
Titrate:
Elastic SSLL103 Flambement of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A Page:
6/8
5 Modeling
B
5.1
Characteristics of modeling
8 elements POU_D_T
With
With
1
2
5.2
Characteristics of the grid
A number of nodes: 9
A number of meshs and types: 8 SEG2
5.3 Functionalities
tested
Commands
Keys
CALC_MATR_ELEM
“RIGI_GEOM”
[U4.41.01]
MODE_ITER_SIMULT
“PLUS_PETITE”
[U4.52.02]
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A

Code_Aster ®
Version
4.0
Titrate:
Elastic SSLL103 Flambement of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A Page:
7/8
6
Results of modeling B
6.1 Values
tested
Identification
Reference
Aster
% difference
Case 1
mode 1
­ 6.796E+05
­ 6.8E+05
0.06
mode 2
­ 1.505E+06
­ 1.50492E+06
­ 0.005
mode 3
­ 1.0055E+07
­ 9.968E+07
­ 0.816
Case 2
mode 1
­ 1.505E+06
­ 1.50492E+06
­ 0.005
mode 2
­ 5.998E+06
­ 5.99831E+06
+0.005
Case 3
mode 1
­ 5.638E+05
­ 5.649E+05
0.2
mode 2
­ 2.453E+06
­ 2.443E+06
­ 0.4
mode 3
­ 1.8525E+07
­ 1.7883E+07
­ 3.5
Case 4
mode 1
6.9376E+07
6.982E+07
0.064
6.2 Remarks
The precision is rather good with 8 elements in the length. The solution differs a little that
obtained with warping (modeling A).
6.3 Parameters
of execution
Version: 3.6
Machine: CRAY C90
System:
UNICOS 8.0
Obstruction memory:
50 megawords
Time CPU To use:
15 seconds
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A

Code_Aster ®
Version
4.0
Titrate:
Elastic SSLL103 Flambement of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A Page:
8/8
7
Summary of the results
The analytical solution gives us 3 modes of buckling of which the critical loads are roots
of an equation of the 3° degree.
Teststemyà he of other critical loads inserted between the 3 found values?
Aster finds the good critical loads, but in the middle of much of others… for example for
case 3, the 3 sought critical loads corresponds to the nume_mode: 1, 10 and 19!
This is true for two modelings.
Handbook of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A