SSNL502 - Beam in buckling

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SSNL502 - Beam in buckling

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V6.02 booklet: Non-linear statics of the linear structures

HT-66/03/008/A

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EDF-R & D/AMA, DeltaCAD

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

SSNL502 - Beam 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 (Buckling of Euler)

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

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SSNL502 - Beam in buckling

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Problem of reference

1.1 Geometry

With

Length:

L = 0.5 m

Width:

B = 0.075 m

Thickness:

H = 0.0045 m

Moment of inertia I

= 5.7 10

- 10

(bh

/12)

X, U

y, v

Z, W

1.2

Properties of material

The properties of material constituting the plate are:

= 2. 10

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

()

with

T pseudo_temps

critical load of Euler

The load applied corresponds to the critical load of Euler

I.E.(internal excitation)

1124 21

1.4 Conditions

initial

Without object

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SSNL502 - Beam in buckling

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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/P

/L W

With

L/I.E.(INTERNAL EXCITATION)

Charge

(NR)

(m)

With

(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

0,00

0,10

0,20

0,30

0,40

0,50

0,60

0,70

0,80

Displacement (m)

P/Pcr

W
U

250

500

750

1000

1250

1500

1750

2000

Bending moment (N.m)

P/Pcr

Moment

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 University of Technology of Compiegne 1981.

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3 Modeling

With

3.1

Characteristics of modeling

y, v

Z, W

Modeling COQUE_3D (TRIA7)

X, U

With

0.03

Boundary conditions:
- Dimensioned AD:

U = v = W =

3.2

Characteristics of the mesh

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

3.3 Functionalities

tested

Controls Key word

factor

Key word

AFFE_MODELE

AFFE

MODELING = “COQUE_3D”

AFFE_CARA_ELEM HULL

THICK
COEF_RIGI_DRZ = 0.001

AFFE_CHAR_MECA

FORCE_ARETE

FX, FZ

STAT_NON_LINE PILOTING

TYPE=' LONG_ARC'

Modeling

COQUE_3D

(TRIA7)

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Results of modeling A

4.1 Values

tested

Identification Moments Reference

Aster %

difference

 0.0150

1.04532 0.1100

0.1025 0.230

1.04532 0.0150 0.0154 2.498

ETA_PILOTAGE

1.04532 1.015

1.0380 2.271

 0.0595

1.09778 0.2110

0.2081 1.340

1.09778 0.0595 0.0580 2.572

ETA_PILOTAGE

1.09778 1.063

1.098 3.274

 0.22

1.20824 0.3595

0.3579 0.451

1.20824 0.22

 0.22 0.010

ETA_PILOTAGE

1.20824 1.293

1.39

7.483

 0.3255

1.26646 0.396

0.3937 0.581

1.26646 0.3255 0.3298

1.32

ETA_PILOTAGE

1.26646 1.518

1.688 11.182

 0.5535

1.38521 0.375

0.3668 2.186

1.38521 0.5535

 0.554

0.094

ETA_PILOTAGE

1.38521 2.541

3.038 19.562

 0.67

1.46121 0.3125

0.2971 4.924

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

(option

DDL_IMPO

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 the control

TEST_RESU

, since these values cannot be imposed. For

to define the values of reference, we sought the values of

closest possible

those listed in the table of [§2.2] and we deferred the values of the parameter of piloting
and of

that one was to obtain for the values of

in question.

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5 Modeling

5.1

Characteristics of modeling

y, v

Z, W

Modeling COQUE_3D (QUAD9)

X, U

With

0.03

Boundary conditions:
- Dimensioned AD:

U = v = W =

5.2

Characteristics of the mesh

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

5.3 Functionalities

tested

Controls Key word

factor

Key word

AFFE_MODELE

AFFE

MODELING = “COQUE_3D”

AFFE_CARA_ELEM HULL

THICK
COEF_RIGI_DRZ = 0.001

AFFE_CHAR_MECA

FORCE_ARETE

FX, FZ

STAT_NON_LINE PILOTING

TYPE=' LONG_ARC'

Modeling

COQUE_3D

(QUAD9)

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SSNL502 - Beam in buckling

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Results of modeling B

6.1 Values

tested

Identification Moments Reference

Aster %

difference

 0.0150

1.03356 0.1100

0.1086

 1.255

1.03356 0.0150

 0.0149

 0.656

ETA_PILOTAGE

1.03356 1.015

1.028

1.366

 0.0595

1.08921 0.2110

0.2115

0.228

1.08921 0.0595

 0.0599

0.689

ETA_PILOTAGE

1.08921 1.063

1.087

2.256

 0.22

1.20259 0.3595

0.3618

0.653

1.20259 0.22

 0.226

2.932

ETA_PILOTAGE

1.20259 1.293

1.36

5.215

 0.3255

1.25521 0.396

0.3944

 0.384

1.25521 0.3255

 0.3247

 0.231

ETA_PILOTAGE

1.25521 1.518

1.594

5.034

 0.5535

1.37521 0.375

0.374

 0.099

1.37521 0.5535

 0.5501

 0.608

ETA_PILOTAGE

1.37521 2.541

2.6875

5.767

 0.67

1.45321 0.3125

0.3088

 1.194

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

(option

DDL_IMPO

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 the control

TEST_RESU

, since these values cannot be imposed. For

to define the values of reference, we sought the values of

closest possible

those listed in the table of [§2.2] and we deferred the values of the parameter of piloting
and of

that one was to obtain for the values of

in question.

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SSNL502 - Beam in buckling

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8 Modeling

8.1

Characteristics of modeling

y, v

Z, W

Modeling POU_D_T_GD (SEG2)

X, U

With

Boundary conditions:
- Not a:

U = v = W =

8.2

Characteristics of the mesh

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

8.3 Functionalities

tested

Controls 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

PILOTING

TYPE = “LONG_ARC”

Modeling

POU_D_T_GD

(SEG2)

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SSNL502 - Beam in buckling

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Results of modeling C

9.1 Values

tested

Identification Moments Reference

Aster %

difference

 0.22

1.18684 0.3595

0.3587 0.64

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

1.24521 0.396

0.397

0.1

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

1.30521 0.4015

0.4038

0.56

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

(option

DDL_IMPO

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

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10 Summary of the results

Displacement following X according to the critical load

0,8

1,0

1,2

1,4

1,6

1,8

2,0

0,00

0,05

0,10

0,15

0,20

0,25

0,30

Displacement DX (m)

Tank

Ge C

R
itic (

P
/PC

R
)

Displacement following Z according to the critical load

0,8

1,0

1,2

1,4

1,6

1,8

2,0

0,00

0,02

0,04

0,06

0,08

0,10

Displacement DZ (m)

Critical load (P/Pcr)

The critical load is well detected. The first two results corresponding to the loads
P/P

=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 area where displacements in

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=10

xH/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.

TRIA7

QUAD9

Reference

TRIA7

QUAD9

Reference