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Titrate:
SSNL502 - Poutre in buckling


Date:
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Author (S):
J.M. PROIX, P. MASSIN, F. LEBOUVIER Key
:
V6.02.502-B Page:
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Organization (S): EDF-R & D/AMA, DeltaCAD

Handbook of Validation
V6.02 booklet: Non-linear statics of the linear structures
Document: V6.02.502

SSNL502 - Poutre in buckling

Summary:

This test represents a calculation of stability of a beam comforts subjected to a compressive force to one
end. It makes it possible to validate modelings finite elements COQUE_3D with meshs TRIA7 and QUAD9 and
modeling POU_D_T_GD with meshs SEG2 in the non-linear quasi-static field into large
displacements and in great rotations in the presence of instability (Flambement d' Euler)

Displacements and the moments obtained are compared with an analytical reference solution.

Handbook of Validation
V6.02 booklet: Non-linear statics of the linear structures
HT-66/03/008/A

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SSNL502 - Poutre in buckling


Date:
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:
V6.02.502-B Page:
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1
Problem of reference

1.1 Geometry

Z, W
P
Z
C
B
H
Length:
L = 0.5 m
Width:
B = 0.075 m
B
L
Thickness:
H = 0.0045 m
Moment of inertia I = 5.7 10-10 m4 (bh3/12)
y
y, v
D
y
With
X, U
X


1.2
Properties of material

The properties of material constituting the plate are:

E = 2. 1011 Pa
Young modulus


= 0.3
Poisson's ratio

1.3
Boundary conditions and loadings

- C.L. : Embedded side AD

- One seeks the successive states of balance under the loading imposed on side BC:

p (T) = p T
Cr


with
T pseudo_temps

p

Cr critical load of Euler
2EI
The load applied corresponds to the critical load of Euler P =
1124 21
.

2
=
NR
Cr
4L

1.4 Conditions
initial

Without object
Handbook of Validation
V6.02 booklet: Non-linear statics of the linear structures
HT-66/03/008/A

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SSNL502 - Poutre in buckling


Date:
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:
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2
Reference solution

2.1
Method of calculation used for the reference solution

The solution of the problem known as of the elastic '' '' is presented in [bib1] by making the assumption of not
extension of the average axis. The analytical solution is obtained by considering integrals
elliptic.

2.2
Results of reference

The results of reference retained for the checks are indicated in boldface characters in
table below. Displacements are defined in the reference mark of definition of the geometry [1.1].

P/Pcr
uB/L wB/L

BADLY/I.E.(INTERNAL EXCITATION)
Charge
P uB
wB
MA
(NR)
(m)
(m)
(N.M)
1,015 0,220 0,030 20
0,56 1141.07
0.1100
0.0150
127.58
1,063 0,422 0,119 40
1,09 1195.03
0.2110
0.0595
248.32
1,152 0,593 0,259 60
1,67 1295.09
0.2965
0.1295
380.45
1,293 0,719 0,440 80
2,28 1453.60
0.3595
0.2200
519.41
1,518 0,792 0,651 100
2,96 1706.55
0.3960
0.3255
674.33
1,884 0,803 0,877 120
3,73 2118.01
0.4015
0.4385
849.74
2,541 0,750 1,107 140
4,70 2856.62
0.3750
0.5535
1070.72
4,029 0,625 1,340 160
6,20 4529.44
0.3125
0.6700
1412.44
9,116 0,421 1,577 176
9,44 10248.29
0.2105
0.7885
2150.55

10
P/Pcr
9
10
W
P/Pcr
8
U
9
Moment
7
8
6
7
5
6
4
5
3
4
2
3
1
2
Displacement (m)
0
1
Bending moment (N.m)
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
0
0
250
500
750
1000
1250
1500
1750
2000

2.3
Uncertainties on the solution

Analytical solution

2.4 References
bibliographical

[1]
S.P. TIMOSHENKO, J.M. MANAGES: Theory of elastic stability, second edition, DUNOD
1966.
[2]
J.L. BATOZ: Great displacements and great rotations of elastic thin beams,
Mechanical department of engineering Universit de Technologie of Compiegne 1981.
Handbook of Validation
V6.02 booklet: Non-linear statics of the linear structures
HT-66/03/008/A

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SSNL502 - Poutre in buckling


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:
V6.02.502-B Page:
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3 Modeling
With

3.1
Characteristics of modeling

Z, W
C
2
Modeling COQUE_3D (TRIA7)
Modeling COQUE_3D (TRIA7)
Z
B
Boundary conditions:
- Dimensioned AD:
U = v = W = = = = 0
y, v
X
y Z
10
y
D
With
0.03

X, U
X


3.2
Characteristics of the grid

A number of nodes: 145
A number of meshs and type: 40 TRIA7

3.3 Functionalities
tested

Commands Key word
factor
Key word
AFFE_MODELE
AFFE
MODELISATION = “COQUE_3D”

AFFE_CARA_ELEM HULL
EPAIS

COEF_RIGI_DRZ = 0.001
AFFE_CHAR_MECA
FORCE_ARETE
FX, FZ

STAT_NON_LINE CONTROL
TYPE=' LONG_ARC'


Handbook of Validation
V6.02 booklet: Non-linear statics of the linear structures
HT-66/03/008/A

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Titrate:
SSNL502 - Poutre in buckling


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:
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4
Results of modeling A

4.1 Values
tested

DZ
Identification Moments Reference
Aster %
difference
0.0150
DX
1.04532 0.1100
0.1025 0.230

DZ
1.04532 0.0150 0.0154 2.498

ETA_PILOTAGE
1.04532 1.015
1.0380 2.271






0.0595
DX
1.09778 0.2110
0.2081 1.340
DZ
1.09778 0.0595 0.0580 2.572

ETA_PILOTAGE
1.09778 1.063
1.098 3.274












0.22
DX
1.20824 0.3595
0.3579 0.451
DZ
1.20824 0.22
0.22 0.010

ETA_PILOTAGE
1.20824 1.293
1.39
7.483












0.3255
DX
1.26646 0.396
0.3937 0.581
DZ
1.26646 0.3255 0.3298
1.32

ETA_PILOTAGE
1.26646 1.518
1.688 11.182












0.5535
DX
1.38521 0.375
0.3668 2.186
DZ
1.38521 0.5535
0.554
0.094

ETA_PILOTAGE
1.38521 2.541
3.038 19.562












0.67
DX
1.46121 0.3125
0.2971 4.924
DZ
1.46121 0.67 0.6711 0.166

ETA_PILOTAGE
1.46121 4.029
5.28 31.053

4.2 Remarks

The strategy of calculation used breaks up into two stages:

Imposed loading: one imposes a disturbing load of 1/1000 of the critical load
according to X to reveal the mode of buckling. This load is applied for
P/Pcr=0.98 and to P/Pcr=1.015.
Imposed displacement: beyond 1.01, the structure became very flexible, one imposes one
increase in displacement DZ (option DDL_IMPO in STAT_NON_LINE) for
to determine the behavior postbuckling.

The use of the technique length of arc makes difficult the definition of the value of reference to
to introduce into command TEST_RESU, since these values cannot be imposed. For
to define the values of reference, we sought the values of the closest possible DZ
those listed in the table of [2.2] and we deferred the values of the parameter of control
and of DX which one was to obtain for the values of DZ in question.
Handbook of Validation
V6.02 booklet: Non-linear statics of the linear structures
HT-66/03/008/A

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SSNL502 - Poutre in buckling


Date:
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:
V6.02.502-B Page:
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5 Modeling
B

5.1
Characteristics of modeling

Z, W
C
2
Modeling CO
Modeling
QUE_3D (
COQUE_3D
QUAD9)
(QUAD9)
Z
B
Boundary conditions:
- Dimensioned AD:
U = v = W = = = = 0
y, v
X
y Z
10
y
D
With
0.03

X, U
X


5.2
Characteristics of the grid

A number of nodes: 105
A number of meshs and type: 20 QUAD9

5.3 Functionalities
tested

Commands Key word
factor
Key word
AFFE_MODELE
AFFE
MODELISATION = “COQUE_3D”

AFFE_CARA_ELEM HULL
EPAIS

COEF_RIGI_DRZ = 0.001
AFFE_CHAR_MECA
FORCE_ARETE
FX, FZ

STAT_NON_LINE CONTROL
TYPE=' LONG_ARC'


Handbook of Validation
V6.02 booklet: Non-linear statics of the linear structures
HT-66/03/008/A

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Titrate:
SSNL502 - Poutre in buckling


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:
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6
Results of modeling B

6.1 Values
tested

DZ
Identification Moments Reference
Aster %
difference
0.0150
DX
1.03356 0.1100
0.1086
1.255

DZ
1.03356 0.0150
0.0149
0.656

ETA_PILOTAGE
1.03356 1.015
1.028
1.366












0.0595
DX
1.08921 0.2110
0.2115
0.228
DZ
1.08921 0.0595
0.0599
0.689

ETA_PILOTAGE
1.08921 1.063
1.087
2.256












0.22
DX
1.20259 0.3595
0.3618
0.653
DZ
1.20259 0.22
0.226
2.932

ETA_PILOTAGE
1.20259 1.293
1.36
5.215












0.3255
DX
1.25521 0.396
0.3944
0.384
DZ
1.25521 0.3255
0.3247
0.231

ETA_PILOTAGE
1.25521 1.518
1.594
5.034












0.5535
DX
1.37521 0.375
0.374
0.099
DZ
1.37521 0.5535
0.5501
0.608

ETA_PILOTAGE
1.37521 2.541
2.6875
5.767












0.67
DX
1.45321 0.3125
0.3088
1.194
DZ
1.45321 0.67
0.672
0.3

ETA_PILOTAGE
1.45321 4.029
4.388
8.903

6.2 Remarks

The strategy of calculation used breaks up into two stages:

Imposed loading: one imposes a disturbing load of 1/1000 of the critical load
according to X to reveal the mode of buckling. This load is applied for
P/Pcr=0.98 and to P/Pcr=1.015.
Imposed displacement: beyond 1.01, the structure became very flexible, one imposes one
increase in displacement DZ (option DDL_IMPO in STAT_NON_LINE) for
to determine the behavior postbuckling.

The use of the technique length of arc makes difficult the definition of the value of reference to
to introduce into command TEST_RESU, since these values cannot be imposed. For
to define the values of reference, we sought the values of the closest possible DZ
those listed in the table of [2.2] and we deferred the values of the parameter of control
and of DX which one was to obtain for the values of DZ in question.
Handbook of Validation
V6.02 booklet: Non-linear statics of the linear structures
HT-66/03/008/A

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Titrate:
SSNL502 - Poutre in buckling


Date:
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:
V6.02.502-B Page:
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7
Graphic results of modelings A and B



Handbook of Validation
V6.02 booklet: Non-linear statics of the linear structures
HT-66/03/008/A

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Titrate:
SSNL502 - Poutre in buckling


Date:
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:
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8 Modeling
C

8.1
Characteristics of modeling

Z, W
Modeling P
Modeling
OU_D_T_G
POU_D_T_GD D (SEG
(SEG2) 2)
Z
B
Boundary conditions:

y, v
- Not a:
U = v = W = x= y = z= 0
y
10

X, U
With
X


8.2
Characteristics of the grid

A number of nodes: 11
A number of meshs and type: 10 SEG2

8.3 Functionalities
tested

Commands Key word
factor
Key word
AFFE_MODELE
AFFE
MODELING = “POU_D_T_GD”
AFFE_CARA_ELEM BEAM
SECTION
RECTANGLE
AFFE_CHAR_MECA
FORCE_NODALE
FX, FZ

STAT_NON_LINE
PILOTAGE
TYPE = “LONG_ARC”


Handbook of Validation
V6.02 booklet: Non-linear statics of the linear structures
HT-66/03/008/A

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SSNL502 - Poutre in buckling


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:
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9
Results of modeling C

9.1 Values
tested

DZ
Identification Moments Reference
Aster %
difference
0.22
DX
1.18684 0.3595
0.3587 0.64

DZ
1.18684 0.22
0.2129 3.2

ETA_PILOTAGE
1.18684 1.293
1.292 0.09

MYY
1.18684 519.41
519.1
0.06






0.3255
DX
1.24521 0.396
0.397
0.1
DZ
1.24521 0.3255
0.321
1.4

ETA_PILOTAGE
1.24521 1.518
1.5171 0.06

MYY
1.24521 674.3
676.5
0.3






0.4385
DX
1.30521 0.4015
0.4038
0.56
DZ
1.30521 0.4385
0.4362 0.54

ETA_PILOTAGE
1.30521 1.884
1.885 0.053

MYY
1.30521 849.74
854.73
0.6

9.2 Remarks


The strategy of calculation used breaks up into two stages:

Imposed loading: one imposes a disturbing load of 1/1000 of the critical load
according to X to reveal the mode of buckling. This load is applied for
P/Pcr=0.98 and to P/Pcr=1.015.
Imposed displacement: beyond 1.01, the structure became very flexible, one imposes one
increase in displacement DZ (option DDL_IMPO in STAT_NON_LINE) for
to determine the behavior postbuckling.
The results are in good adequacy with the reference solution from
ETA_PILOTAGE = 1.293. Before this value, the disturbing load (necessary to obtain
buckling) degrades the solution, and the variations with the analytical solution are important
(up to 80%). The corresponding values are the subject of tests of nonregression. But this
variation is only related to the disturbing load, since by increasing the loading
vertical, the good solution is found.

9.3
Graphic results of modeling C


Handbook of Validation
V6.02 booklet: Non-linear statics of the linear structures
HT-66/03/008/A

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Titrate:
SSNL502 - Poutre in buckling


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10 Summary of the results

Displacement following Z according to the critical load
Displacement following X according to the critical load
2,0
2,0
1,8
1,8
1,6
R
)
1,6
/PC
QUAD9
P
1,4
Reference
TRIA7
QUAD9
1,4
TRIA7
R
itic (
Ge C
1,2
1,2
Critical load (P/Pcr)
Tank
1,0
1,0
Reference
0,8
0,8
0,00
0,05
0,10
0,15
0,20
0,25
0,30
0,00
0,02
0,04
0,06
0,08
0,10
Displacement DX (m)

Displacement DZ (m)


The critical load is well detected. The first two results corresponding to the loads
P/Pcr =1.015 and 1.063 are correct, the maximum error is 3.5% for mesh TRIA7 and 2.2%
for mesh QUAD9. Mesh QUAD9 gives better results.

If one continues calculations with the elements of hulls mesh QUAD9 continues to give
better results. In the zone where displacements in DZ are most important, the error
made on the load 9% reach on the quadrangles and go up to 30% on the triangles. Errors
increase in this area because of the slopes of the curves. The solution beam of the code provides
good results compared to the solution beam of reference.

The coefficient of correction of transverse shearing A_CIS was put at 0.833, corresponding to
thick hulls. The value (9000=106xH/L) which would have being taken into account does not allow
to carry out calculations. It introduces a bad conditioning of the matrices of rigidity in
increasing their disparities.
Handbook of Validation
V6.02 booklet: Non-linear statics of the linear structures
HT-66/03/008/A

Code_Aster
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Titrate:
SSNL502 - Poutre in buckling


Date:
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Author (S):
J.M. PROIX, P. MASSIN, F. LEBOUVIER Key
:
V6.02.502-B Page:
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