Code_Aster
®
Version
7.2
Titrate:
SSNL126 - Elastoplastic buckling of a right beam
Date:
28/10/03
Author (S):
J.M. PROIX, NR. GREFFET, L. SALMONA
Key
:
V6.02.126-A
Page:
1/6
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/03/008/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
Document: V6.02.126
SSNL126 - Elastoplastic buckling of one
right beam
Summary:
A slim right beam of circular section is subjected to a compressive force at an end, and is
embedded at the other end. The behavior of material is elastoplastic, with an isotropic work hardening
linear. During the climb in load, one calculates the critical loads of elastic buckling, then
plastic.
Two modelings make it possible to test the criterion of buckling in elastoplasticity:
Voluminal modeling a: mesh, small deformations and small displacements.
Modeling b: voluminal mesh, small deformations and great displacements (GREEN).
Code_Aster
®
Version
7.2
Titrate:
SSNL126 - Elastoplastic buckling of a right beam
Date:
28/10/03
Author (S):
J.M. PROIX, NR. GREFFET, L. SALMONA
Key
:
V6.02.126-A
Page:
2/6
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/03/008/A
1
Problem of reference
1.1 Geometry
Right beam, length L = 1m
Circular section of radius R = 0.01m.
1.2
Material properties
Elastoplastic material with isotropic linear work hardening:
Young modulus: E = 210000 MPa
Poisson's ratio:
= 0. (assumption of beam of Euler-Bernoulli)
Elastic limit:
Y
= 4 MPa
Tangent module: E
T
= 70000 MPa
1.3
Boundary conditions and loadings
·
C.L. : embedding on all basic surface
·
Surface force on the higher face: with T = 1s, F = 6.5 MPa
This load is applied in 10 pitches of time équirépartis.
Embedding
Surface force
Code_Aster
®
Version
7.2
Titrate:
SSNL126 - Elastoplastic buckling of a right beam
Date:
28/10/03
Author (S):
J.M. PROIX, NR. GREFFET, L. SALMONA
Key
:
V6.02.126-A
Page:
3/6
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/03/008/A
2
Reference solution
2.1
Method of calculation used for the reference solution
Analytical solution:
In small displacements:
· in elastic mode (for F<
Y
) the theoretical breaking value corresponds to the load of Euler.
Within the framework of a kinematics of beam, the critical load is worth:
2
2
4 L
I.E.(internal excitation)
=
F
Cr
, therefore critical pressure:
2
Cr
SSL
I.E.(internal excitation)
=
P
4
2
with
4
4
R
=
I
and
2
R
=
S
that is to say
2
2
16 L
ER
=
P
2
Cr
· in elastoplastic mode, as one considers a uniform compression without discharge
rubber band and because of law of behavior, the critical load of buckling is worth:
2
2
4
.
L
I
And
=
F
Cr
that is to say a pressure criticizes:
2
2
16L
EtR
=
P
2
Cr
2.2
Results of reference
Values of the critical load for the two loading cases.
In elastic mode, for F < 4 MPa, is T < 0.61538462, one must obtain: P
Cr
= 12.95 MPa.
In plastic mode the critical value of pressure of buckling is: 4,32 MPa.
The critical coefficients according to the loading are:
No time
Surface force
(in MPa)
Critical coefficient
Critical load
(in MPa)
1
0.65
19.9290 12.9539
2
1.3
9.9645 12.9539
3
1.95
6.6430 12.9539
4
2.6
4.9823 12.9539
5
3.25
3.9858 12.9539
6
3.9
3.3215 12.9539
7
4.55
0.9490
4.3180
8
5.2
0.8304
4.3180
9
5.85
0.7381
4.3180
10
6.5
0.6643
4.3180
Code_Aster
®
Version
7.2
Titrate:
SSNL126 - Elastoplastic buckling of a right beam
Date:
28/10/03
Author (S):
J.M. PROIX, NR. GREFFET, L. SALMONA
Key
:
V6.02.126-A
Page:
4/6
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/03/008/A
3 Modeling
With
3.1
Characteristics of modeling
Voluminal mesh 3D.
3.2
Characteristics of the mesh
A number of nodes: 600
A number of meshs and types: 90 HEXA20
3.3 Functionalities
tested
Controls
STAT_NON_LINE CRIT_FLAMB
STAT_NON_LINE COMP_INCR DEFORMATION
GREEN
STAT_NON_LINE COMP_INCR RELATION
VMIS_ISOT_LINE
4
Results of modeling A
4.1 Values
tested
Moment Reference Aster %
difference
0.2 9.9645
9.9763
0.16
1 0.6643
6.752
2.3
Code_Aster
®
Version
7.2
Titrate:
SSNL126 - Elastoplastic buckling of a right beam
Date:
28/10/03
Author (S):
J.M. PROIX, NR. GREFFET, L. SALMONA
Key
:
V6.02.126-A
Page:
5/6
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/03/008/A
5 Modeling
B
5.1
Characteristics of modeling
Voluminal mesh 3D. Great displacements and deformations (but small rotations)
The surface force applied is worth 20 here MPa with T = 1s, in order to pass, during the evolution of
loading, by the critical point.
This load is applied in 10 pitches of time équirépartis.
Two complete calculations are carried out: one with a purely elastic behavior, in order to be able
to compare the result with the elastic solution of reference, and the other with a behavior
elastoplastic.
5.2
Characteristics of the mesh
A number of nodes: 600
A number of meshs and types: 90 HEXA20
5.3 Functionalities
tested
Controls
STAT_NON_LINE CRIT_FLAMB
STAT_NON_LINE COMP_INCR DEFORMATION
GREEN
STAT_NON_LINE COMP_INCR RELATION
VMIS_ISOT_LINE
STAT_NON_LINE COMP_INCR RELATION
ELAS
6
Results of modeling B
6.1 Values
tested
In elastic behavior
One tests the end value of the critical coefficient: (test of nonregression)
Moment Reference Aster %
difference
1 19.0657
19.0657
0
In the case of great displacements or great deformations, the value of the critical coefficient must
to be interpreted differently of the case small displacements: the structure becomes unstable when
“critical load” is cancelled.
Code_Aster
®
Version
7.2
Titrate:
SSNL126 - Elastoplastic buckling of a right beam
Date:
28/10/03
Author (S):
J.M. PROIX, NR. GREFFET, L. SALMONA
Key
:
V6.02.126-A
Page:
6/6
Manual of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/03/008/A
The evolution of this coefficient in the course of time is as follows:
No time
Surface force
(in MPa)
Critical coefficient
Aster
Critical load Euler
1
2
27.5797 12.9539
2
4
22.8250 12.9539
3
6
17.9808 12.9539
4
8
13.0407 12.9539
5
10
7.9975 12.9539
6
12
2.8434 12.9539
7
14
2.4301
12.9539
8
16
7.8324
12.9539
9
18
13.3738
12.9539
10
20
19.0657
12.9539
The critical coefficient thus passes well by 0 between moments 6 and 7, and more precisely (curved cf
following) in the neighborhoods of moment 6.5, which corresponds well to the critical load in elasticity.
Coefficient criticizes Aster
- 30.0000
- 20.0000
- 10.0000
0.0000
10.0000
20.0000
30.0000
0
1
2
3
4
5
6
7
8
9
10
In elastoplasticity, one tests the moments when the critical coefficient changes sign. The tests are of
not regression since one does not have analytical solution in this case.
Moment Reference Aster %
difference
0.4 4.9917
4.9917
0
0.5 1.3186
1.3186
0
7
Summary of the results
The results as of modeling in small displacements is in conformity with the analytical reference
(less than 2% of variation in plasticity). The results in great displacements cannot be
compared with a reference solution, but the change of sign of the critical coefficient is
conform to the awaited solution.