Code_Aster
®
Version
4.0
Titrate:
Elastic SSLL103 Buckling of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A
Page:
1/8
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A
Organization (S):
EDF/IMA/MN
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
Document: V3.01.103
SSLL103 - Elastic buckling 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 the 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].
Code_Aster
®
Version
4.0
Titrate:
Elastic SSLL103 Buckling of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A
Page:
2/8
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A
1
Problem of reference
1.1 Geometry
Z
y
L = 1200 mm
With
2
P
M
X
P
With
1
M
C
With
G
has
y
Z
E
= 120 mm have
E = 8 mm
CG = 41.012 mm
Section A
1
and A
2
Characteristics of the section:
To = 1856 mm
2
I
y
= 4167339 mm
4
I
Z
= 1045547 mm
4
J = 39595 mm
4
I
= 44398819 mm
6
I
yr2
= 84948392 mm
5
y
C
= 41.012 mm
Z
C
= 0
1.2
Material properties
E = 2.1 10
5
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
bending (according to OZ for cases 2 following OY for case 3) to offset the effort out of C (or in A).
Code_Aster
®
Version
4.0
Titrate:
Elastic SSLL103 Buckling of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A
Page:
3/8
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A
2
Reference solution
2.1
Method of calculation used for the reference solution
With taking into account of the roll, the calculations made by V. Of City De Goyet [bib1] give:
that is to say:
(
)
(
)
(
)
(
)
(
) (
)
(
)
(
)
I
Z dA
I
y dA
I
y y
Z
dA
I
Z y
Z
dA
Pcry
E I
L
Pcrz
E I
L
Pcrx
GJ
E I
L Macaw
Arc
I
I
With
y
Z
y
I
I
y
Z
I
I
Z
Macaw
I
I
With
y
Z
y
I
I
y
Z
I
y
Z
yr
Zr
With
With
With
With
Z
y
y
Z
C
C
C
yrz
Z
C
C
Zr
y
C
y
Z
C
C
has
yrz
Z
C
has
=
=
=
+
=
+
=
=
=
+
=
+
+
+
+
-
+
-
=
+
+
+ +
-
+
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
;
;
;
/
;
/
;
/
/
/
/
/
/
(
)
Zr
y
C
I
Z
2
2
/
-
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:
(
)
(
) (
)
(
)
(
)
(
)
(
)
Macaw Pcry
P Pcrz
P Pcrx
P
P
Pcrz
P Z
Z
P
Pcry
P y
y
C
has
C
has
-
-
-
-
-
-
-
-
-
=
2
2
2
2
0
Case 4:
Critical moment
Mcr
(around the axis y) is worth:
(
)
(
)
Mcr
GJ
E I
L Pcry
= ±
+
2
2
1 2
/
/
By neglecting the roll: 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 =
10
8
).
2.4 References
bibliographical
[1]
V. OF TOWN OF GOYET “Analyzes static nonlinear by the finite element method of
formed space structures of beams with nonsymmetrical section " - Thesis of doctorate
University of Liege, MSM, academic year (1988-1989).
[2]
P. PENSERINI “elastic Instability of the beams with open mean profile: theoretical aspects and
numerical " Notes EDF/DER/HM77/112.
[3]
J. CHERRY TREE “Propagation of two cases tests of modeling of the calculation of the beams in
elastic buckling in
Code_Aster
“HM77/184
Code_Aster
®
Version
4.0
Titrate:
Elastic SSLL103 Buckling of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A
Page:
4/8
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A
3 Modeling
With
3.1
Characteristics of modeling
8 elements
POU_D_TG
With
1
With
2
3.2
Characteristics of the mesh
A number of nodes: 9
A number of meshs and types: 8 SEG2
3.3 Functionalities
tested
Controls
Keys
CALC_MATR_ELEM
“RIGI_GEOM”
[U4.41.01]
MODE_ITER_SIMULT
“PLUS_PETITE”
[U4.52.02]
Code_Aster
®
Version
4.0
Titrate:
Elastic SSLL103 Buckling of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A
Page:
5/8
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A
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
Overall dimension memory:
8 megawords
Time CPU To use:
11 seconds
Code_Aster
®
Version
4.0
Titrate:
Elastic SSLL103 Buckling of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A
Page:
6/8
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A
5 Modeling
B
5.1
Characteristics of modeling
8 elements
POU_D_T
With
1
With
2
5.2
Characteristics of the mesh
A number of nodes: 9
A number of meshs and types: 8 SEG2
5.3 Functionalities
tested
Controls
Keys
CALC_MATR_ELEM
“RIGI_GEOM”
[U4.41.01]
MODE_ITER_SIMULT
“PLUS_PETITE”
[U4.52.02]
Code_Aster
®
Version
4.0
Titrate:
Elastic SSLL103 Buckling of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A
Page:
7/8
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A
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 roll (modeling A).
6.3 Parameters
of execution
Version: 3.6
Machine: CRAY C90
System:
UNICOS 8.0
Overall dimension memory:
50 megawords
Time CPU To use:
15 seconds
Code_Aster
®
Version
4.0
Titrate:
Elastic SSLL103 Buckling of an angle
Date:
19/01/98
Author (S):
J. PELLET
Key:
V3.01.103-A
Page:
8/8
Manual of Validation
V3.01 booklet: Linear statics of the linear structures
HI-75/96/039 - Ind A
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
find the good critical loads, but in the middle of much of others… for example for
case 3, the 3 sought critical loads corresponds to
nume_mode
: 1, 10 and 19!
This is true for two modelings.