Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 1/16

Organization (S): EDF-R & D/AMA
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
V6.02.106 document

SSNL106 - Elastoplastic Poutre in traction
and pure inflection

Summary:

This test validates elastoplasticity in a right beam in traction and inflection, for a behavior
elastoplastic perfect or with linear work hardening.

·
Analyze static
·
Elastoplastic behavior of beam: VMIS_POUTRE or of TUYAU
·
3 sections: rectangular, circular full, hollow circular
·
3 types of work hardening: no one, linear (ECRO_LINE), nonlinear (ECRO_FLEJOU)

8 modelings make it possible to test elements POU_D_TG, POU_D_E, POU_D_T, TUYAU (3 and 4 nodes)
COQUE_3D, POU_D_EM and POU_D_T_GM.

The test makes it possible to validate the operation of the integration of these laws of behavior and of the algorithm of
resolution until complete plasticization of the beam.

One also tests plasticization in alternating bending. Moreover, the two methods of resolution are tested
for modelings of beams with total plasticity (POU_D_TG, POU_D_E, POU_D_T): implicit integration or
by Runge-Kutta of command 4.

The ninth modeling makes it possible to test the operation of DYNA_NON_LINE on a calculation
quasi-static of traction of a beam modelled in POU_D_TG.
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 2/16

1
Problem of reference

1.1 Geometry

Right beam length L = 1, direction X.

y
L
O
X
O
B
L = 1
Z


One calculates simultaneously 2 types of section:

y
1 rectangular section
2 v = 0.2
Z
B = 0.1
1 circular section
R
R = 0.1


R = 0.1
E
For modeling D, one calculates 1 section of thin tube:
E = 0.001

1.2
Material properties

E = 2. 1011 Pa
= 0.3

ECRO_LINE:
ECRO_FLEJOU:
SY = y = 150.106 Pa
EP = 2.E10
H = D_SIGM_EPSI = 2.109 Pa or 0
SY = 150.E6

KNOWN = 160.E6

PUISS = 1

VMIS_POUTRE: rectangular section:

normal effort limits NP
= 3. 106 NR
elastic moment MEZ
= 105 Nm
limiting moment MPZ
= 1.5.105 Nm
coefficient AZ
= 0.84
coefficient BZ =
0.0013
MEY, MPY, MT, AY, BY
= without object

VMIS_POUTRE: circular section:

normal effort limits NP
= 4.712389 106 NR
MEZ = MEY
= 117809.72 Nm
MPZ = MPY
= 200.000 Nm
MT =
without
object
AY = AZ
= 0.84
BY = BZ
= 0.0012
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 3/16

1.3
Boundary conditions and loadings

Embedding out of O

Displacement imposed out of B

L.
DX E
y
=
= 0.75 10 ­ 3 m
E

DX vari
DX E
E of
with 3DX E

Rotation imposed out of B

DRZ E = 0.75 10 ­ 2 m

DRZ varies
D
RZ E to 20 × DRZ E then decrease until - 2 × DRZ E

Note:

In pure inflection, MZ and DRZ do not depend on X. Curvature
D (DRZ)


=
= DRZ
dx
L

like L = 1
= DRZ ()
B

Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 4/16

2
Reference solution

2.1
Method of calculation used for the reference solution

2.1.1 Pure inflection - linear Ecrouissage

Analytical solution:

y
v
v
xx
U
xx
0
X
0
X
­ U
­ v
­ v
xx=. y:curvature


Calculation of the moment by:

M (U) = xx (y). y ds
S
= E for 0 y U
xx
xx


= + H
y
xx
y
xx -


E
for U < y v

One obtains:

for the rectangular section:

M

H 3 1 2
H
E
=

1 -



M


E
-
2 2 + E
E
E



M
I
y
E
.
front

EC.
=
M
Z
E
=
I.E.(internal excitation)
E
v

for the circular section:

R3 y
H 4
3/2
M (µ) =

+ (E - H) (1 - Μ2)
E
4 µ
3
E - H

+
sin µ - µ 1 - 2µ 2 1 - µ 2

2µ (Arc
(
)
)
U
y
E
front
µ
EC. =
=
=
R
ER


In discharge, after having reached the limiting load charges some, one obtains a limiting load of sign
opposite.
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 5/16

for the tubular section:
(assumption of beam of Navier-Bernoulli)

The limiting load (H = 0) is worth:

M
4
=
Me


The complete solution for a thin tube is [bib1]:

M (µ)

(
2 1 -)
=
+
arcsin µ + µ 1 - µ 2
µ
µ (
)
Me

E
H
T
with =
=
E
E + H

2.1.2 Traction - linear Ecrouissage

H
HS
Analytical solution: there is immediately NR = S
1 -
. DX
y

.
E + L

2.1.3 Traction - nonlinear Ecrouissage

The analytical solution is obtained here for a particular case: the model of work hardening
(ECRO_FLEJOU) is written into uniaxial:

-
with = U
y
U
p
p
E. p
E
= +
y
1



p


E.H

p
1+

and
E =


E H
U
-




If = 1, one finds according to while solving:

(

p
-
p
p
y)

1 +
= E.

U

What leads to the equation of the 2nd degree in:

2 - (E (+ U) + U) + E (+
y U
U) = 0
Then:
= (E (+
p
U) - U) 2 + 4th. E. 2u

and
1
= (E (+ U) + ±
U
)
2

The solution corresponding to the beginning of plasticization (growing starting from y) is that
corresponding to the sign -.

Thus the normal effort in the beam is:


2

S (
U B)

(
U B)


2
NR =
E
+

E

+ 4th E p





L
U
U

+
-
+
L
U
U
U

-


2


2.2 References
bibliographical

[1]
J.H. LAU and T.T. LAU: Newspaper off Pressure Vessel Technology Vol. 106 p188-195 - Mai 1984.
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 6/16

3 Modeling
With

3.1
Characteristics of modeling

2 elements POU_D_TG by type of section. There are thus 2 groups of elements comprising each one 2
elements.

Group GR1:
rectangular section

GC1:
circular section
Simple traction:
on GR1 (ECRO_FLEJOU) and GC1 (ECRO_LINE)
Pure inflection:
on GR1 and GC1 without work hardening

3.2
Characteristics of the grid

2 X 2 elements POU_D_TG

3.3 Functionalities
tested

Commands



STAT_NON_LINE COMP_INCR RELATION
:
“VMIS_POUTRE”
DEFI_MATERIAU VMIS_POUTRE



ECRO_LINE


ECRO_FLEJOU


STAT_NON_LINE CONVERGENCE
RESO_INTE
:
“RUNGE_KUTTA_4”


4
Results of modeling A

4.1 Values
tested

Traction

DX (B)
N°ordre
GROUP_MA
Identification Reference Aster Difference
DX E
(%)
2 11
GR1 NR
3.123 106 3.123
106
0
3 21
GR1 NR
3.1529 106 3.1532
106
0.01
2 21
GC1 NR
4.75951 106 4.75951
106
0
3 31
GC1 NR
4.80664 106 4.80664
106
0

Inflection

DRZ (B)
N°ordre
GROUP_MA
Identification Reference Aster Difference
DRZ E
(%)
1 1
GR1
MFZ
(Nm)
105 105
0
5 21
MFZ 1.48 105 1.477
105
­ 0.23
10 31
MFZ 1.495 105 1.489
105
­ 0.39
20 41
MFZ 1.499 105 1.495
105
­ 0.26
1 1
GC1 MFZ 1.1781 105
1.1781 105
0
5 21
MFZ 1.9602 105
1.955 105
­ 0.27
10 31
MFZ 1.99 105 1.98
105
­ 0.53
20 41
MFZ 1.998 105 1.99
105
­ 0.36
­ 2 71 GR1 MFZ ­ 1.5 105 ­ 1.499
105
­ 0.08
­ 2 71 GC1 MFZ ­ 2. 105 ­ 1.994
105
­ 0.32
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 7/16

5 Modeling
B

5.1
Characteristics of modeling

2 elements POU_D_T by type of section. There are thus 2 groups of elements comprising each one 2
elements.

Group GR1:
rectangular section

GC1:
circular section
Simple traction:
on GR1 (ECRO_FLEJOU) and GC1 (ECRO_LINE)
Pure inflection:
on GR1 and GC1 without work hardening

5.2
Characteristics of the grid

2 X 2 elements POU_D_T

5.3 Functionalities
tested

Commands
STAT_NON_LINE COMP_INCR
RELATION
:
“VMIS_POUTRE”

DEFI_MATERIAU VMIS_POUTRE



ECRO_LINE



ECRO_FLEJOU



STAT_NON_LINE CONVERGENCE
RESO_INTE
:
“IMPLICITE”


6
Results of modeling B

6.1 Values
tested

Traction

DX (B)
N°ordre
GROUP_MA
Identification Reference Aster Difference
DX E
(%)
2 11
GR1 NR
3.123 106 3.125
106
0.08
3 21
GR1 NR
3.1529 106 3.15
106
0.09
2 21
GC1 NR
4.75951 106 4.7596
106
0.004
3 31
GC1 NR
4.80664 106 4.8068
106
0.004

Inflection

DRZ (B)
N°ordre
GROUP_MA
Identification Reference Aster Difference
DRZ E
(%)
1 1
GR1
MFZ
(Nm)
105 105
0
5 21
MFZ
1.48 105 1.477
105
­ 0.20
10 31
MFZ
1.495 105 1.489
105
­ 0.37
20 41
MFZ
1.499 105 1.495
105
­ 0.24
­ 2 71 GR1 MFZ
­ 1.5 105 ­ 1.496
105
­ 0.23
­ 2 71 GC1 MFZ
­ 2 105 ­ 1.993
105
­ 0.33
1 1
GC1 MFZ
1.1781 105 1.1781
105
0
5 21
MFZ
1.9602 105 1.955
105
­ 0.24
10 31
MFZ
1.99 105 1.98
105
­ 0.50
20 41
MFZ
1.998 105 1.99
105
­ 0.33
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 8/16

7 Modeling
C

7.1
Characteristics of modeling

2 elements POU_D_E by type of section. There are thus 2 groups of elements comprising each one 2
elements.

Group GR1:
rectangular section

GC1:
circular section
Simple traction:
on GR1 (ECRO_FLEJOU) and GC1 (ECRO_LINE)
Pure inflection:
on GR1 and GC1 without work hardening

7.2
Characteristics of the grid

2 X 2 elements POU_D_E

7.3 Functionalities
tested

Commands



STAT_NON_LINE COMP_INCR RELATION
:
“VMIS_POUTRE”

DEFI_MATERIAU VMIS_POUTRE



ECRO_LINE



ECRO_FLEJOU



STAT_NON_LINE CONVERGENCE
RESO_INTE
“IMPLICITE”


“RUNGE_KUTTA_4”


8
Results of modeling C

8.1 Values
tested

Traction

DX (B)
N°ordre
GROUP_MA
Identification Reference Aster Difference
DX E
(%)
2 11
GR1 NR
3.123 106 3.123
106
0
3 21
GR1 NR
3.1529 106 3.1532
106
0.01
2 21
GC1 NR
4.75951 106 4.75951
106
0
3 31
GC1 NR
4.80664 106 4.80664
106
0

Inflection

DRZ (B)
N°ordre
GROUP_MA
Identification Reference Aster Difference
DRZ E
(%)
1 1
GR1
MFZ
(Nm)
105 105
0
5 21
MFZ
1.48 105 1.477
105
­ 0.23
10 31
MFZ
1.495 105 1.489
105
­ 0.39
20 41
MFZ
1.499 105 1.495
105
­ 0.26
­ 2 71 GR1 MFZ
­ 1.5 105 ­ 1.499
105
­ 0.08
­ 2 71 GC1 MFZ
­ 2 105 ­ 1.994
105
­ 0.32
1 1
GC1 MFZ
1.1781 105 1.1781
105
0
5 21
MFZ
1.9602 105 1.955
105
­ 0.27
10 31
MFZ
1.99 105 1.98
105
­ 0.53
20 41
MFZ
1.998 105 1.99
105
­ 0.36
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 9/16

9 Modeling
D

9.1
Characteristics of modeling

2 elements TUYAU for the tubular section.

Simple traction:
(ECRO_LINE)
Pure inflection:
without work hardening

Moreover, one blocks the DDL which correspond to the mode 3d' ovalization: U03 V03 W03

9.2
Characteristics of the grid

2 elements TUYAU (METUSEG3)

9.3 Functionalities
tested

Commands



STAT_NON_LINE COMP_INCR
RELATION
: “VMIS_ISOT_LINE”


TUYAU_NCOU
:
3


TUYAU_NSEC
:
16

AFFE_MODELE
MODELISATION
:
TUYAU

AFFE_CHAR_MECA

DDL_IMPO
U03: 0.

V03: 0.
W03: 0.

10 Results of modeling D

10.1 Values
tested

Traction

DX (B)
N°ordre Identification Reference Aster Difference
DX E
(%)
2 11
NR
9.47 104 9.47
104
0
3 21
NR
9.565 104
9.565 0

Inflection

DRZ (B)
N°ordre Identification Reference Aster Difference
DRZ E
(%)
1 1
MFZ
4.642 103 4.642
103
0
5 21
MFZ
5.9106 103 5.9365
103
0.4

Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 10/16

11 Modeling
E

11.1 Characteristics of modeling

2 elements TUYAU with 4 nodes for the tubular section.

Simple traction:
(ECRO_LINE)
Pure inflection:
without work hardening

Moreover, one blocks the DDL which correspond to the mode 3d' ovalization: U03 V03 W03

11.2 Characteristics of the grid

2 elements TUYAU (meshs SEG4)

11.3 Functionalities
tested

Commands



STAT_NON_LINE COMP_INCR
RELATION
: “VMIS_ISOT_LINE”


TUYAU_NCOU
:
3


TUYAU_NSEC
:
16

AFFE_MODELE
MODELISATION
:
TUYAU

AFFE_CHAR_MECA

DDL_IMPO
U03: 0.

V03: 0.
W03: 0.
MODI_MAILLAGE
OPTION
SEG3_4


12 Results of modeling E

12.1 Values
tested

Traction

DX (B)
N°ordre Identification Reference
Aster Difference
DX E
(%)
2 11
NR
9.47 104 9.47
104
0
3 21
NR
9.565 104
9.565 0

Inflection

DRZ (B)
N°ordre Identification Reference
Aster Difference
DRZ E
(%)
1 1 MFZ 4.642 103 4.642
103
0
5 21 MFZ 5.9106 103 5.9365
103
0.4
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 11/16

13 Modeling
F

13.1 Characteristics of modeling

112 elements COQUE_3D for the tubular section, and 2 elements pipe to apply the conditions
with the limits. The length of the grid hulls is of 0.98m. the length of each element pipe is
of 0.01m.

A connection COQUE_TUYAU is applied at each end of the grid hulls, with an element
pipe. Moreover, one blocks the DDL of the pipes which correspond to the mode 3d' ovalization: U03 V03
W03

Simple traction:
(ECRO_LINE)
Pure inflection:
without work hardening

13.2 Characteristics of the grid

112 meshs QUAD9 and 2 meshs SEG3.

13.3 Functionalities
tested

Commands



STAT_NON_LINE COMP_INCR RELATION
: “VMIS_ISOT_LINE”

COQUE_NCOU
:
1

AFFE_MODELE

MODELISATION:
COQUE_3D, TUYAU_3M

AFFE_CHAR_MECA
DDL_IMPO
Ui3: 0.


Vi3: 0.
Wi3: 0.
Uo3: 0.
Vo3: 0.
Wo3: 0.
AFFE_CHAR_MECA LIAISON_ELEM
OPTION
“COQ_TUYAU”


14 Results of modeling F

14.1 Values
tested

Traction

DX (B)
N°ordre Identification Reference
Aster Difference
DX E
(%)
2 11
NR
9.47 104 9.473
104
0.02
3 21
NR
9.565 104
9.569 E+04
0.04

Inflection

DRZ (B)
N°ordre Identification Reference
Aster Difference
DRZ E
(%)
1 1 MFZ 4.642 103 4.6415
103
0.01
2.8 19
MFZ 5.7824 103 5.7836
103
0.02
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 12/16

15 Modeling
G

15.1 Characteristics of modeling

2 elements POU_D_EM for the tubular section.
The section is with a grid in QUAD4: it is discretized by a mesh in the thickness, and 90 meshs
on the circumference.

Simple traction:
(ECRO_LINE)
Pure inflection:
without work hardening

15.2 Characteristics of the grid

2 meshs SEG2 for the beam. 90 meshs QUAD4 for the section.

15.3 Functionalities
tested

Commands



STAT_NON_LINE COMP_INCR
RELATION
: “VMIS_ISOT_LINE”

AFFE_CARA_ELEM BEAM SECTION
GENERALE

AFFE_SECT
MAILLAGE

AFFE_MODELE
MODELISATION
:
POU_D_EM


16 Results of modeling G

16.1 Values
tested

Traction

DX (B)
N°ordre Identification Reference
Aster Difference
DX E
(%)
2 11
NR
9.47 104 9.471
104
0.01
3 21
NR
9.565 104
9.5647 0.01

Inflection

DRZ (B)
N°ordre Identification Reference
Aster Difference
DRZ E
(%)
1 1 MFZ 4.642 103 4.641
103
0.01
5 21 MFZ 5.9106 103 5.90
103
0.02
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 13/16

17 Modeling
H

17.1 Characteristics of modeling

2 elements POU_D_TGM for the tubular section.
The section is with a grid in QUAD4: it is discretized by a mesh in the thickness, and 90 meshs
on the circumference.

Simple traction:
(ECRO_LINE)
Pure inflection:
without work hardening

17.2 Characteristics of the grid

2 meshs SEG2 for the beam. 90 meshs QUAD4 for the section.

17.3 Functionalities
tested

Commands



STAT_NON_LINE COMP_INCR
RELATION
: “VMIS_ISOT_LINE”

AFFE_CARA_ELEM BEAM SECTION
GENERALE

AFFE_SECT
MAILLAGE

AFFE_MODELE
MODELISATION
:
POU_D_TGM


18 Results of modeling H

18.1 Values
tested

Traction

DX (B)
N°ordre Identification Reference
Aster Difference
DX E
(%)
2 11
NR
9.47 104 9.471
104
0.01
3 21
NR
9.565 104
9.5647 0.01

Inflection

DRZ (B)
N°ordre Identification Reference
Aster Difference
DRZ E
(%)
1 1 MFZ 4.642 103 4.641
103
0.01
5 21 MFZ 5.9106 103 5.90
103
0.02
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 14/16

19 Modeling
I

19.1 Characteristics of modeling

2 elements POU_D_TG by type of section. There are thus 2 groups of elements comprising each one 2
elements.

Group GR1:
rectangular section

GC1:
circular section
Simple traction:
on GR1 (ECRO_FLEJOU) and GC1 (ECRO_LINE)
Pure inflection:
on GR1 and GC1 without work hardening

19.2 Characteristics of the grid

2 X 2 elements POU_D_TG

19.3 Functionalities
tested

The characteristic of modeling I is to test the operation of DYNA_NON_LINE in calculation
of quasi-static traction of a beam modelled in POU_D_TG. This type of modeling has for
characteristic to reveal null pivots on the lines of the matrix of mass corresponding
with the degrees of freedom of warping. In this case, the initialization of the diagram of NEWMARK
fact more by inversion of the matrix of mass, which is singular, but by resetting of acceleration
initial.

Commands



DYNA_NON_LINE COMP_INCR
RELATION
:
“VMIS_POUTRE”
DEFI_MATERIAU VMIS_POUTRE


ECRO_LINE


ECRO_FLEJOU


STAT_NON_LINE CONVERGENCE RESO_INTE
:
“RUNGE_KUTTA_4”


20 Results of modeling I

20.1 Values
tested

Traction

DX (B)
N°ordre
GROUP_MA
Identification
Reference
Aster Difference
DX E
(%)
2 11 GR1 NR
3.123 106 3.123
106
0.009
3 21 GR1 NR
3.1529 106 3.153
106
0.012
2 21 GC1 NR
4.75951 106 4.760
106
0.0003
3 31 GC1 NR
4.80664 106 4.807
106
0.0003

20.2 Observations

It is noticed that the results in traction resulting from DYNA_NON_LINE are identical to those given
by STAT_NON_LINE.

An alarm informs the user who the matrix of mass is singular, which should attract sound
caution.
Indeed for the majority of the finite elements the matrix of mass is definite positive. A message invites it
thus to check the assignments of its model. In the framework of this modeling, the presence of
degrees of freedom of warping naturally involves the singularity of the matrix of mass.
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 15/16

21 Summary of the results

The results obtained for a traction correspond exactly to the analytical solution. On the other hand
in inflection, the difference reaches 0.5%. This is due to the criterion of plasticity chosen, whose form does not allow
not to find the analytical solution exactly. It is necessary to adjust as well as possible parameters AY, BY, AZ,
BZ.

With regard to modelings pipe and hull, the conclusions are the same ones, but this time
the difference with the analytical solution comes from this solution which is valid for a beam of
very mean tubular section, without effect of ovalization. This ovalization is blocked at the ends, it
who allows to obtain a solution with less than 0.4% of the analytical solution.

Two modelings in multifibre beams provide a solution to less than 0.02% of
analytical solution, for a very weak time calculation, compared to modeling pipe and hull,
but without it being necessary to adjust parameters as for modelings of beam with
total plasticity. The only approximation comes from the grid of the section.

Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Code_Aster ®
Version
7.3
Titrate:
SSNL106 - Elastoplastic Poutre in traction and pure inflection
Date:
25/10/04
Author (S):
J.M. PROIX, J.L. FLEJOU
Key: V6.02.106-C Page: 16/16

Intentionally white left page.
Handbook of Validation
V6.02 booklet: Nonlinear statics of the linear structures
HT-66/04/005/A

Outline document