Note of calculation to buckling

Code_Aster

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6.4

Titrate:

Note of calculation to buckling

Date

20/12/02

Author (S):

NR. GREFFET

Key

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Organization (S):

EDF-R & D/AMA

Instruction manual
U2.08 booklet: Advanced functions and control of calculations
Document: U2.08.04

Note of calculation to buckling

Summary:

The objective of this documentation is to not present a methodological guide for an analysis of buckling
linear of a structure. One approaches mainly two functionalities of Code_Aster there:

analysis of buckling linear, known as of Euler, through

MODE_ITER_SIMULT

, (option

TYPE_RESU: “MODE_FLAMB”

the calculation of the quasi-static evolution (operator

STAT_NON_LINE

) of the structure which presents

not geometrical and behavioral linearities, which one seeks a limiting point, even the answer
post-critical.

The first stage is, generally, a calculation of buckling of Euler, who will allow to know the modes of
buckling and corresponding critical loads. From the point of view of the designer, the knowledge of
first mode and of its critical load is often sufficient, in order to be defined a margin of operation
compared to the imposed loading: the multiplying coefficient enters the imposed loading and the critical load
weakest the safety margin gives.

Remarks

The knowledge of the first mode of buckling can also be used as indication to optimize
management of nonlinear incremental calculation carried out thereafter. Indeed, with the approach of the load
critical, one can then decide to modify piloting or to reduce the pitch of time, even
to increase the iteration count of checking of balance in the method of residue, with
each pitch of load.

The pace of the mode of buckling of Euler can also be used for to impose a geometrical defect
initial on the structure, in order to make sure, amongst other things, that incremental nonlinear calculation
this mode will fork of course.

The analysis of Euler being per linear definition, it does not make it possible to take into account relations of
behavior inelastic or of the contact. It is then necessary to make a nonlinear calculation, which in
quasi-static will be based on the control

STAT_NON_LINE

of Code_Aster. It is the conventional method

incremental by residue in balance. The particular points of its use will be approached thereafter.

Code_Aster

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Note of calculation to buckling

Date

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Author (S):

NR. GREFFET

Key

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Analyze buckling of Euler

The calculation of the modes of buckling within the meaning of Euler [bib5] can be done by the operator of resolution
problems with the eigenvalues

MODE_ITER_SIMULT

(or

MODE_ITER_INV

). Within the framework of

buckling, one with following typical syntax:

MODP1 = MODE_ITER_SIMULT (MATR_A = RAMEP1,
MATR_B = RAGEP1,
TYPE_RESU = “MODE_FLAMB”,
CALC_FREQ = _F (OPTION = “TAPE”,
CHAR_CRIT = (- 2.4, - 2.2,),
DIM_SOUS_ESPACE = 80,
NMAX_ITER_SOREN = 80,),)

The argument of the key word

MATR_A

must be the matrix of rigidity known as material, whereas the key word

MATR_B

waits the geometrical matrix of rigidity. If the operator had been employed

MODE_ITER_INV

arguments of the key words

MATR_A

and

MATR_B

would be the same ones.

For recall, the modes of buckling are the clean modes of the problem to the eigenvalues
according to:

(

)

(

tor

multiplica

coeffician

with

clean

value

géométriqu

rigidity

stamp

material

rigidity

stamp

With

Material rigidity (or rubber band) is calculated with the option `

RIGI_MECA'

CALC_MATR_ELEM

Geometrical rigidity is calculated starting from the stress field solution of the linear problem (option
`

RIGI_GEOM'

CALC_MATR_ELEM

). Thus should have been carried out a static linear calculation

before the use of

MODE_ITER_SIMULT

for buckling.

If the loading is composed of a fixed part (not controlled) and of a variable part, the coefficient
multiplier of the loading should not, of course, relate that to the variable part. The contribution of
the other part of the loading is found in the first member. Let us note

the fixed loading and

the controlled loading (proportional to

). The problem with the eigenvalues becomes:

(

)

(

)

()

(

)

()

)

(

clean

value

variable

for

géométriqu

rigidity

stamp

controlled

not

for

géométriqu

rigidity

stamp

material

rigidity

stamp

With

In this case, it is thus necessary to solve two preliminary linear elastic problems, to be able
to calculate the two different geometrical matrices of rigidity.
In order to be exhaustive, the presentation will relate to a structure subjected to imposed displacements
as well as efforts, which will be the combination of a fixed loading and a variable loading that
one will control with a coefficient growing being able to lead to buckling.

Code_Aster

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Note of calculation to buckling

Date

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NR. GREFFET

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1.1

Stage 1: Calculation (S) linear (S) preliminary (S)

One will be useful oneself of

MECA_STATIQUE

. The structure, with a grid in elements of the type hull (elements of

voluminal hulls type [bib3]), is subjected to boundary conditions of Dirichlet (

CONDLIM

) and of

Neumann. These last break up into:

WEIGHED

: field of gravity,

PRESPH

: field of pressure imposed not controlled,

PRESPS1

: field of variable pressure imposed.

For the analysis of buckling, it is necessary to separate the constant efforts from those which variable (are controlled by
a coefficient). One thus will make two linear static calculations. The first will be the case of the structure
subjected to imposed displacements and the constant efforts, the second will see the structure subjected to
displacements imposed and on the variable efforts.

Controlled loading:

RESC11P1 = MECA_STATIQUE (MODEL = MODEL,
CHAM_MATER = CHMAT,
CARA_ELEM = CARAELEM,
EXCIT = (_F (LOAD = CONDLIM,),
_F (LOAD = PRESPS1,),),
OPTION = “SIEF_ELGA_DEPL”,
PLAN = “MOY”,)

Loading not controlled:

RESC12P1 = MECA_STATIQUE (MODEL = MODEL,
CHAM_MATER = CHMAT,
CARA_ELEM = CARAELEM,
EXCIT = (_F (CHARGE=CONDLIM,),
_F (LOAD = WEIGHED,),
_F (LOAD = PRESPH,),),
OPTION = “SIEF_ELGA_DEPL”,
PLAN = “MOY”,)

One will use the stress field to calculate the associated matrices of geometrical rigidity, for
two loadings:

SIGC11P1 = CREA_CHAMP (TYPE_CHAM = “ELGA_SIEF_R”,
OPERATION = “EXTR”,
RESULT = RESC11P1,
NOM_CHAM = “SIEF_ELGA_DEPL”,
TYPE_MAXI = “MINI”,
TYPE_RESU = “VALE”,)
#
REGC11P1 = CALC_MATR_ELEM (OPTION = “RIGI_GEOM”,
MODEL = MODEL,
CARA_ELEM = CARAELEM,
SIEF_ELGA = SIGC11P1,)

REGC11P1

is thus the geometrical matrix of stiffness associated the variable case of loading

(

PRESPS1

Code_Aster

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One calculates, in the same way, the geometrical matrix of stiffness for the constant loading (

WEIGHED

and

PRESPH

), from

RESC12P1

SIGC12P1 = CREA_CHAMP (TYPE_CHAM = “ELGA_SIEF_R”,
OPERATION = “EXTR”,
RESULT = RESC12P1,
NOM_CHAM = “SIEF_ELGA_DEPL”,
TYPE_MAXI = “MINI”,
TYPE_RESU = “VALE”,)
#
REGC12P1 = CALC_MATR_ELEM (OPTION = “RIGI_GEOM”,
MODEL = MODEL,
CARA_ELEM = CARAELEM,
SIEF_ELGA = SIGC12P1,)

It remains to calculate the matrix of material rigidity for the total loading:

REMEP1 = CALC_MATR_ELEM (OPTION = “RIGI_MECA”,
MODEL = MODEL,
CHAM_MATER = CHMAT,
CARA_ELEM = CARAELEM,
CHARGE = (CONDLIM, WEIGHED,
PRESPH, PRESPS1,),)

All the elementary matrices are calculated, the following stage is thus their assembly:

NUP1 = NUME_DDL (MATR_RIGI = REMEP1,)
#
RAMC1P1 = ASSE_MATRICE (MATR_ELEM = REMEP1,
NUME_DDL = NUP1,)
#
RAGEP1 = ASSE_MATRICE (MATR_ELEM = REGC11P1,
NUME_DDL = NUP1,)
#
RAGC12P1 = ASSE_MATRICE (MATR_ELEM = REGC12P1,
NUME_DDL = NUP1,)

One summons then the matrices of material rigidity (

RAMC1P1)

and geometrical (

RAGC12P1)

corresponding to the case of constant loading:

RAMEP1 = COMB_MATR_ASSE (COMB_R = (_F (MATR_ASSE = RAMC1P1,
COEF_R = 1.0,),
_F (MATR_ASSE = RAGC12P1,
COEF_R = 1.0,),),)

The two matrices necessary to the calculation of the modes of buckling are thus built.

Code_Aster

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Note of calculation to buckling

Date

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NR. GREFFET

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1.2

Stage 2: Calculation of the modes of Euler

He can be useful to make tests of STURM (operator

IMPR_STURM

) on the interval of search on

which one wants to find the cases of buckling. Thus, that will make it possible to optimize the size of the interval and
to control the good course of later modal calculation since one will know the number in advance of
existing modes. Syntax is:

IMPR_STURM (MATR_A = RAMEP1,
MATR_B = RAGEP1,
TYPE_RESU = “MODE_FLAMB”,
CHAR_CRIT_MIN = - 2.4,
CHAR_CRIT_MAX = - 2.2,)

Once the interval of search for critical load of buckling chosen, one can then implement

MODE_ITER_SIMULT

as follows:

Notice

If the algorithm does not converge or if the number of modes is not that predicted by

IMPR_STURM

, it can be useful to increase the values of

DIM_SOUS_ESPACE

and

NMAX_ITER_SOREN

One normalizes the modes [bib6], only while being useful oneself of the degrees of freedom of translation:

MODP1 = NORM_MODE (reuse = MODP1
MODE = MODP1,
= “TRAN NORMALIZES”,)

The modes can then be post-treaties.

Remarks

It is essential to check that the geometrical stiffness of the selected model is well one
option available in Code_Aster (for example, it is not the case of the DKT).

A finer discretization leads normally to a fall of the critical loads.

The discretization must be ready to collect the modes of buckling, knowing that these
modes can generate localized deformations (folds). The preliminary calculation of
dynamic modes can constitute a first indication on the quality of the mesh,
although these modes can be very different from the modes of buckling.

The critical loads of the various modes are proportional to the Young modulus E.

Code_Aster

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Note of calculation to buckling

Date

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Quasistatic nonlinear study of the structure

This stage is justified if the structure has strong not linearities, whose analysis of Euler cannot
to hold account. The operator of resolution of the nonlinear problems into quasi-static names himself

STAT_NON_LINE

[bib7].

These nonlinearities can be related to the material which can have an elastoplastic behavior
[bib8], as in the example which will follow. The taking into account of the contact, even of friction, is
another source of nonlinearities. One can also quote the case of the following loadings, like
pressure ([bib1] and [bib2] for the elements of voluminal the hulls type), which requires an approach
nonlinear.

For the study of a structure potentially unstable or likely to know a limiting point, which
be thus likely to meet a junction in solution during the evolution of the loading, it is
often useful to be able to choose a branch of particular solution (often the physical solution
when it is a priori defined without ambiguities). For that, the user can have to introduce a defect
initial which “will force” the structure to fork on the branch of particular solution.
Several methods exist to define this defect.

One the most adapted of is of prédéformer slightly the structure according to the pace of the mode
of Euler of buckling corresponding to the branch which one wants to follow. The amplitude of this
predeformation must be weak, for example less than 1/10

ème

thickness for one

mean structure. The ideal being to find the defect minimal which is compatible with one
satisfactory performance of the algorithm of residue in balance. Indeed, a too weak defect
can involve a difficulty of convergence of the residue, mainly in the case of one
piloting in effort.

The geometrical defect can also be defined by experimental measurements of the real part
whose geometry could not be perfect.

The defect can also take the form of a disturbance of the loading (misalignment,
addition of a loading located,…) or of the mechanical characteristics of material
(local attenuation of the Young modulus, for example). He can nevertheless be then more
difficult to adapt the defect to the mode of wished buckling, especially if the structure presents
relatively close modes.

Notice

In certain cases, even on the nondisturbed problem, the loading is such as it causes
desired junction.

One of the other particular points, related to instability, is the choice of the technique of piloting of

the algorithm

STAT_NON_LINE

. Indeed, conventional piloting in effort is not adapted any more because it cannot

to collect an unstable branch of solution. In the same way, with the approach of a limiting point, convergence with

piloting in effort will become increasingly difficult, the matrix of tangent rigidity becoming

singular. It is then necessary to reduce the increment of load and to increase the maximum number

of iteration to continue calculation.

There are techniques of piloting [bib9] making it possible to circumvent these numerical difficulties. Among

methods suggested by Code_Aster, that called by length of arc [bib12] (option

TYPE=' LONG_ARC'

key word

PILOTING

STAT_NON_LINE

), which is adapted for

instabilities of the buckling type, in the case of “soft” snap-backs possible [bib13]. In the case of

snap-backs more brutal, Crisfield proposes an alternative [bib13], nonavailable in version 6 of

Code_Aster.

Other methods exist, like that of Riks [bib14] (nonavailable either), which treats also it

dynamic case.

If one wants only to obtain the point limits, including with a good precision, a piloting in

loading can be enough, with the proviso of managing well the parameters of pitch of increment of load

(

SUBD_PAS

and

SUBD_PAS_MINI

key word

INCREMENT

) and of maximum iteration count authorized

(

ITER_GLOB_MAXI

CONVERGENCE

). It can also be useful, with the approach of the limiting point, of more

to use the tangent matrix reactualized for the solvor, since it is quasi-singular. One can

then to be satisfied not to reactualize this matrix with each calculation (parameters

REAC_INCR

and

REAC_ITER

) or, in the worst of the cases, to adopt the basic elastic matrix

(

PREDICTION=' ELASTIQUE'

and

MATRICE=' ELASTIQUE'

key word

NEWTON

Code_Aster

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Note of calculation to buckling

Date

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Here an example of use of

STAT_NON_LINE

for an elastoplastic calculation into large

displacements ([bib4] for the elements employed, which are of voluminal hulls type), with piloting
in efforts:

RESU = STAT_NON_LINE (MODEL = MODEL,
CHAM_MATER = CHMAT,
CARA_ELEM = CARAELEM,
EXCIT = (_F (LOAD = CONDLIM,
TYPE_CHARGE = “FIXE_CSTE”,),
_F (LOAD = WEIGHED,
TYPE_CHARGE = “FIXE_CSTE”,),
_F (LOAD = PRESPH,
FONC_MULT = FONCMUL2,
TYPE_CHARGE = “SUIV”,),
_F (LOAD = PRESPS1,
FONC_MULT = FONCMUL,
TYPE_CHARGE = “SUIV”,),),
COMP_INCR = (_F (RELATION = “VMIS_ISOT_TRAC”,
COQUE_NCOU = 1,
DEFORMATION = “GREEN_GR”,
GROUP_MA = (“RING”, “ROOF”,
“RINGS”, “SGOU”),
),),
COMP_ELAS = _F (RELATION = “ELAS”,
COQUE_NCOU = 1,
DEFORMATION = “GREEN_GR”,
GROUP_MA = “LTIGE”,),
INCREMENT = _F (LIST_INST = L_INST1,
NUME_INST_FIN = 14,
SUBD_PAS = 4,
SUBD_PAS_MINI = 1.E-9,),
NEWTON = _F (REAC_INCR = 1,
PREDICTION = “TANGENT”,
STAMP = “TANGENT”,
REAC_ITER = 1,),
CONVERGENCE = _F (RESI_GLOB_RELA = 1.E-06,
ITER_GLOB_MAXI = 40,
STOP = “YES”,),
SOLVEUR = _F (METHOD = “MULT_FRONT”,
RENUM = “MONGREL”,),)

Remarks

One uses the tangent matrix reactualized with each calculation, while authorizing under
pitch division of load.

The imposed pressures are following efforts (

TYPE_CHARGE=' SUIV'

In the case of a modeling in solid elements, the tensor of deformation
recommended in great displacements is `

SIMO_MIEHE'

Code_Aster

Version

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Titrate:

Note of calculation to buckling

Date

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Author (S):

NR. GREFFET

Key

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If one wants to replace piloting in effort by a method by length of arc, it is enough to write:

RESU = STAT_NON_LINE (MODEL = MODEL,
CHAM_MATER = CHMAT,
CARA_ELEM = CARAELEM,
EXCIT = (_F (LOAD = CONDLIM,
TYPE_CHARGE = “FIXE_CSTE”,),
_F (LOAD = WEIGHED,
TYPE_CHARGE = “FIXE_CSTE”,),
_F (LOAD = PRESPH,
FONC_MULT = FONCMUL2,
TYPE_CHARGE = “SUIV”,),
_F (LOAD = PRESPS1,
TYPE_CHARGE = “FIXE_PILO”,),),
COMP_INCR = (_F (RELATION = “VMIS_ISOT_TRAC”,
COQUE_NCOU = 1,
DEFORMATION = “GREEN_GR”,
GROUP_MA = (“RING”, “ROOF”,
“RINGS”, “SGOU”),
),),
COMP_ELAS = _F (RELATION = “ELAS”,
COQUE_NCOU = 1,
DEFORMATION = “GREEN_GR”,
GROUP_MA = “LTIGE”,),
INCREMENT = _F (LIST_INST = L_INST1,
NUME_INST_FIN = 14,
SUBD_PAS = 4,
SUBD_PAS_MINI = 1.E-9,),
NEWTON = _F (REAC_INCR = 1,
PREDICTION = “TANGENT”,
STAMP = “TANGENT”,
REAC_ITER = 1,),
CONVERGENCE = _F (RESI_GLOB_RELA = 1.E-06,
ITER_GLOB_MAXI = 40,
STOP = “YES”,),

PILOTING = _F (GROUP_NO = “G”,
TYPE = “LONG_ARC”,
NOM_CMP = (“DY”,),
COEF_MULT = 7. ),)

Remarks

In version 6 of Code_Aster, one cannot control following forces.

For piloting by length of arc, it, in general, is recommended that

GROUP_NO

all the structure contains.

Code_Aster

Version

6.4

Titrate:

Note of calculation to buckling

Date

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NR. GREFFET

Key

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3 Bibliography

[1]

E. LORENTZ: Code_Aster, reference material, [R3.03.04], 1996

[2]

P. MASSIN, Mr. Al MIKDAD: Code_Aster, reference material, [R3.03.07], 2000

[3]

P. MASSIN, A. LAULUSA: Code_Aster, reference material, [R3.07.04], 2000

[4]

P. MASSIN, Mr. Al MIKDAD: Code_Aster, reference material, [R3.07.05], 2000

[5]

O. BOITEAU: Code_Aster, reference material, [R5.01.01], 2001

[6]

B. QUINNEZ, J.R. LEVESQUE: Code_Aster, reference material, [R5.01.03], 1997

[7]

NR. TARDIEU, I. VAUTIER: Code_Aster, reference material, [R5.03.01], 2001

[8]

J.M. PROIX, E. LORENTZ: Code_Aster, reference material, [R5.03.02], 2001

[9]

E. LORENTZ: Code_Aster, reference material, [R5.03.80], 2001

[10]

C. ROSE: Code_Aster, reference material, [R6.02.02], 2001

[11]

Mr. A. CRISFIELD, G. JELENIC, SEMI Y., H.-G. ZHONG & Z. FAN: Summon aspects off the
non-linear finite element method, Finite Elements in Analysis and Design, vol. 27, 19-40,
1997

[12]

Mr. A. CRISFIELD: With fast incremental iterative solution procedure that handles snap through,
Computers & Structures, vol. 13, 55-62, 1981

[13]

H.-B. HELLWEG & Mr. A. CRISFIELD: With new arc-length method for handling sharp snap-
backs, Computers & Structures, vol. 66, 705-709, 1998

[14]

E. RIKS, D.C. RANKIN & F.A. BROGAN: One the solution off mode jumping phenomena in
thin-walled Shell structures, comp. Meth. In Applied Mech. And Engrg., vol. 1367, 59-92,
1996

[15]

J. SHI & Mr. A. CRISFIELD: Combining arc-length and line searches in path-following, Com.
Numer. Meth. Engrg, vol. 11, 793-803, 1995