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Titrate:
Criteria of structural stability

Date
:

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:
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Organization (S): EDF-R & D/AMA
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Document: R7.05.01
Criteria of structural stability

Summary:

This document presents the various criteria of stability, with the direction buckling of structure, available in
Code_Aster. One can classify them according to two categories:

· criterion of Euler on linearized problem,
· nonlinear criteria.

These criteria make it possible to detect the loss of unicity in solution of the quasistatic problem.
They are directly transposable with the framework of dynamics, but as they do not take account nor of
stamp of mass nor of that of damping, one cannot speak about dynamic criterion of stability to the direction
traditional (for example, of negative or null damping becoming).

The nonlinear choice of criteria fulfills the requirements of:

· versatility (general method for any relation of behavior and being able to accept all
tensor of deformation available in the code),
· minimization of cost CPU and the additional obstruction memory.

The criterion presented is a generalization of the criterion of Euler, based on the analysis of the matrix of total stiffness
reactualized. It is called within operator STAT_NON_LINE, to be able to be evaluated with each step of
nonlinear quasi-static incremental resolution.
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Criteria of structural stability

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Count

matters

1 Introduction ............................................................................................................................................ 3
2 Study of the stability of a structure ....................................................................................................... 3
2.1 General concept of buckling ........................................................................................................... 3
2.2 Writing of the mechanical problem ..................................................................................................... 5
2.3 Study of stability of the system .......................................................................................................... 6
2.3.1 Writing of the elastic geometrical nonlinear problem ...................................................... 7
2.3.2 Study of stability into nonlinear geometrical .................................................................... 10
2.3.2.1 Stability condition of a nonlinear elastic balance ...................................... 11
2.3.2.2 Case of small displacements: charge of Euler ......................................................... 11
2.3.2.3 Case of the imposed forces depend on the geometry ............................................. 12
2.3.2.4 Vibrations under prestressed ................................................................................... 13
2.3.2.5 Processing of the elastoplastic behavior (plastic buckling) .................... 13
3 Establishment in the code .................................................................................................................... 15
3.1 Criterion of Euler ................................................................................................................................ 15
3.2 Nonlinear criterion ........................................................................................................................ 16
3.2.1 Impact on operator STAT_NON_LINE ............................................................................... 16
3.2.1.1 Algorithm of STAT_NON_LINE ............................................................................... 16
3.2.1.2 Impact on the structure of data result of STAT_NON_LINE .............................. 17
3.2.2 Characteristics related to the tensor of deformation ..................................................................... 18
3.2.2.1 In linearized deformations: PETIT and PETIT_REAC .............................................. 19
3.2.2.2 In great displacements: GREEN, GREEN_GR and SIMO_MIEHE ................................ 19
3.2.2.3 Case of mixed modelings .................................................................................. 20
3.2.3 Improvement of the performances of the criterion ............................................................................ 20
3.3 Generalization with dynamics ...................................................................................................... 21
3.4 Syntax of call of the criterion ............................................................................................................. 21
3.5 Validation of the developments ..................................................................................................... 22
4 Conclusions ......................................................................................................................................... 22
5 Bibliography ........................................................................................................................................ 23

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1 Introduction

Code_Aster allows the search of the modes of linear buckling, qualified method of Euler.
It is enough to solve a problem generalized with the eigenvalues (thanks to operator MODE_ITER_INV
or MODE_ITER_SIMULT and key word TYPE_RESU=' MODE_FLAMB'). The two matrices arguments of
generalized problem are the matrix of rigidity and the matrix of geometrical rigidity, resulting from a calculation
linear elastic precondition (operator MECA_STATIQUE).

In all the cases where one cannot neglect nonthe linearities, which they are geometrical or
behavioral, the Euler approach is not valid any more.
We thus propose an ad hoc criterion, that one can regard as a generalization of the criterion
of Euler on reactualized configuration. This criterion is built on the matrix of tangent stiffness
assembly, which is calculated in the algorithm of the Newton type to solve the problems quasi
nonlinear statics (operator STAT_NON_LINE).
This criterion, into nonlinear, makes it possible to treat the elastic relations of behavior rigorously
nonlinear. On the other hand, the laws which present a dissipative aspect are treated rigorously
that if the loading, in any point of the structure, follows a monotonous evolution (that corresponds to
the assumption of Hill [bib4]).

2
Study of the stability of a structure

2.1
General concept of buckling

Buckling is a phenomenon of instability [bib6]. Its appearance can be observed in particular on
slim elements of low stiffness of inflection. Beyond of a certain level of loading,
structure undergoes an important change of configuration (which can appear by the appearance
sudden of undulations, for example).
One distinguishes two types of buckling: buckling by junction and buckling by limiting point
([bib1], [bib7], [bib8]). To describe the behavior of these two types of buckling, one is considered
structure of which the parameter µ is characteristic of the loading and whose parameter is
characteristic of displacement.

A'
stable
µ
unstable
B
µcr
With
B'
unstable
stable
O

Appear 2.1-a: Flambage by junction
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Between the point O and point A, the structure admits only one family of curve (µ,), it can, for example
to act of traditional linear elasticity or elastoplasticity, where if the problem is well posed (cf
2.2]), there is the traditional result of existence and unicity of the solution.
On the other hand, beyond point A, several families of curves are solution of the problem of balance.
This loss of unicity is accompanied by an instability of the initial branch (known as fundamental).
connect secondary can be stable (curved AB) or unstable (curve AB'). The load beyond
which there is junction is called the critical load µcr.
Buckling by junction is characterized by the fact that the mode (or direction of buckling), which
initiate the secondary branch, does not generate additional work in the loading applied:
mode of buckling being orthogonal to him.
An example of buckling per junction with instability of the secondary branch is in
case of a circular cylindrical hull under axial compression [bib10]. Examples of buckling
by junction with stability of the secondary branch are in elastic beams in
axial compression, of the circular rings in radial compression and the rectangular plates in
longitudinal compression.
µ
With
stable
unstable
O

Appear 2.1-b: Flambage by limiting point
µ
With
µcr
A'
stable
unstable
stable
O

Appear 2.1-c: Buckling by point limits with breakdown
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On figures [Figure 2.1-b] and [Figure 2.1-c], which illustrates buckling by limiting point, the structure
does not admit that only one family (µ,) of solution of the equilibrium equations. At point A, there is loss of
stability of the solution with total loss of rigidity in the case of the figure [Figure 2.1-b] and with one
phenomenon of breakdown in the case of the figure [Figure 2.1-c] (the solution becomes again stable after one
discontinuity of displacement; case of a segment of a sphere under external pressure). Point A is then
called not limits.

The problem thus amounts in all the cases seeking the load from which the branch
fundamental of balance becomes unstable or of dubious stability. That generally mobilizes them
great displacements.
One can finally have the case of the ruin by plastic flow which is connected at the limiting point
[Figure 2.1-b].

2.2
Writing of the mechanical problem

This chapter aims to introduce the general formalism of structural analysis adapted to
nonlinear mechanical problem which we wish to tackle.
To start, we thus briefly will point out the setting in equation of a standard problem of
structural analysis. To simplify, we place ourselves, all at least at the beginning, within the framework of
small disturbances.



fd
S1
Fd
Ud
S
S2
Appear 2.2-a: Représentation of a problem of structural analysis

The structure S is subjected to imposed voluminal efforts fd, surface efforts D
F on
edge
S2 and of the displacements imposed Ud on the remainder of the edge of S, noted
S1.
The unknown factors of the problem of reference on the solid are the field of displacement U and the field
of constraint of Cauchy. The solution (U,) of the problem structure where the heating effects are
neglected is defined as:

To find (U) 1
,
H (
2
S) × L (S) which checks:

· Equations of connections:
U
= Ud











éq 2.2-1
S1
· Relation of behavior:
= F () with which is the tensor of deformation


éq 2.2-2
1
= (
T
U + U) in assumption of small disturbances éq

2.2-3
2
If a linear elastic behavior is supposed:
=
C









éq 2.2-4
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· Equilibrium equations:


2
D U
= + fd with =

D T 2



éq 2.2-5

N
= F

D
S2

2.3
Study of stability of the system

The object of this chapter is to present the methods making it possible to determine the stability of balance
quasi-static nonlinear of a structure. To start, we are interested only in
detection of instability, or more exactly with the loss of unicity of the solution [bib6]. Among work of
synthesis recent, one can quote [bib9] or [bib7] and [bib8] which presents very complete papers on
nonlinear analysis of stability of the structures.
The calculation of the post-critical solution will not be approached.

To make the analysis of stability, we introduce an initial configuration of reference S0, one
current configuration S and a disturbed configuration S1:

S1
u1


S
S0
u0

U
Appear 2.3-a: Définition of the various configurations
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That is to say U the field of displacement of the points of the structure. The behavior is supposed, for
moment, linear elastic isotropic. The structure subjected to imposed displacements and efforts goes
to deform and become the structure located by the current configuration S. Nous let us seek with
to determine a state of balance characterized by the field of displacement between the initial configuration
S0 and current configuration S, as well as a stress field of Cauchy, noted, or of
Piola-Kirchhoff II, noted:
F = U + I:
tensor gradient

transformed



of

tion

- 1


= det F F
0
with
det F =



éq

2.3-1
:
tensor
Piola

of

- Kirchhoff I

0 - 1
T
-
=
F F


In this expression, one sees appearing the relationship between the initial density 0 and masses it
voluminal current.
The following stage is the prediction of the stability of this balance.

To this end, we will seek a criterion allowing to determine if there is only one field of
displacement balancing the efforts applied. We will suppose that the efforts increase
gradually and we will seek to find as from which moment there are two configurations
S and S1 which respect the equations of the problem: we seek a point of junction,
i.e. a loss of unicity of the solution. This moment will be described as moment of buckling.

2.3.1 Writing of the elastic geometrical nonlinear problem

The solution U, of the problem structure without heating effects checks ([bib1], [bib7], [bib2]):

· Equations of connections:
U
= U








éq 2.3.1-1
D
S10
· Elastic relation of behavior:

=, () with
tensor



is

who
die

of

formation éq

2.3.1-2
comporteme

one

suppose

one

If
liné

rubber band

NT
surface:
=
C







éq 2.3.1-3
· Equilibrium equations:

2
D U
= + F

with


D
=

dt2




éq 2.3.1-4
F n0
=

Fd
S 20
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The associated tensor of deformation is that of Green-Lagrange (referred with the initial configuration):

(
1
U) = (FT F - I) with

F = U + I
2
(
1
U)
L
= (U) +
Q
(U, U)



éq 2.3.1-5
2

1
L
(U) = (U + You)
linear

part
:
With
:
2

Q
(U, U) = T U U
quadratiqu

part
:
E

We can now write Principe of Puissances Virtuelles in nonlinear elasticity
geometrical and into quasi-static:

Pint - Pext =,
0 *
U CA

0
int
P
= Tr (
)
L
1 Q


D =
L
*
Q
*
Tr U
U, u.a. U
U, U
D



() +
() (() + (
)





S 0
S0
:
With

2


ext.
*
*
P
=
D
F U dS +
fd U

D




S 0
S0
éq 2.3.1-6

In order to obtain a discretized formulation, one can rewrite the tensor of deformation:


L 1 NL

(U) = B + B (U) U



2

éq

2.3.1-7

=
C (U) with



tensor



is

who
Piola

of

- Kirchhoff II


The power of the internal efforts becomes:

int
P
= Tr
T
L
NL
*
B
B
U U D




éq 2.3.1-8

[[+ ()] ]
S 0

By taking account of the relation of behavior [éq 2.3.1-3]:


int
L 1 NL
T
P
= Tr
L
NL
*
B
B
U
C B
B
U U U D

éq 2.3.1-9


+
() [+

()]


S 0


2




After discretization by the finite elements, one can put this equation in matric form:

*
U [
L
K





éq 2.3.1-10
0 + K (U)
Q
+ K (U)]
ext.
U = P
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The matrix
L
K is symmetrical and there are the following expressions:


T
K
B CB
0 =
L
L

D

S0

1
L

NL
T
T
L
L
NL

K (U) =
B
U CB
B CB
U D


éq 2.3.1-11


()
+
()


S 0 2


1
Q
K (U) =
NL
B (U) T
NL
CB (U)
D


2 S0

One obtains directly what precedes the writing in matric form by balance:

[
L
K






éq 2.3.1-12
0 + K (U)
Q
+ K (U)]
ext.
U = F

That is to say still, in an equivalent way:

Fint =
T
ext.
F with Fint = [L
B + NL
B
(U)]
D

éq 2.3.1-13
S0

We can just as easily formulate Principe of Puissances Virtuelles starting from the state of
constraint of Cauchy and the tensor of deformation of Almansi (thus on the current configuration). One
obtains then:


((*
Tr
U) D =
*
*
D
F U dS +
fd U
D


éq 2.3.1-14


S



S
S

That one can also put in the form, after discretization:

T
int
ext.
B D F
=
= F







éq 2.3.1-15
S

That is to say still, by supposing the elastic relation of behavior:

Ku = ext.
F with K
= T
B CB
D



éq 2.3.1-16
S

The integrals of these equations are calculated on volume running S which depends, of course,
field of displacement solution U. In the same way, the operator B must be calculated on the configuration
current S and not on the initial configuration S0, as it was the case previously.
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2.3.2 Study of stability into nonlinear geometrical

One will seek if there is a second field of displacement kinematically acceptable which checks
equilibrium equations: one thus seeks to know if there will be junction.
This second field will be written as the sum of a disturbance added to the first solution,
that is to say: U + u1, with which is a very small reality and that one will make tend towards 0. The field u1 is
chosen kinematically acceptable to 0.

Principe of Puissances Virtuelles will be then written for this new field.
The field of deformation is put in the form:
(


L
1
U + U = U + U
Q
+ U, U
Q
+ U, U
Q
+
U, U éq

2.3.2-1
1)
() (1) ((1) ()
2
1

(1 1)

2

2
The virtual deformations are given by:


L
= (*
U
Q
+ U, U
Q
+
U, U = U
Q
+
U, U

éq 2.3.2-2
1
) (*)
(*
1
) (*)
(*
1
)

In the same way, if we choose S0 like configuration of reference, the constraints become:



L
1
= + C U
Q
+ U, U
Q
+ U, U +
C Q
U, U éq 2.3.2-3
1
(1)
((1) ()
2
1

(1 1)

2

2

We can now express Principe of Puissances Virtuelles for the field of
disturbed displacement. Let us take as assumptions that the imposed forces do not depend on
displacement and that the initial configuration is selected like reference.

int
int
P
P
1
=



Q
*

*
L
1 Q
Q


+ Tr ((U, U D
Tr u.a. U
U, U
U, U
D
O
1
) +
1


() (1) + ((
1) +
(
)

+ ()

S0
S0


2



ext.
ext.
P
P
1
=

int
ext.
P
P
1
- 1 = 0
éq 2.3.2-4

For sufficiently small, it will be enough that the term proportional to in the expression [éq 2.3.2-4] is
no one so that Principe of Puissances Virtuelles is checked for the field U + u1. In this case, it
more unicity of the solution will not thus have there, which will translate the loss of stability of the system.
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When the imposed efforts do not depend on the geometrical configuration, the study of stability
thus state yourself like:

Knowing the current state, i.e the field of displacement U kinematically acceptable and the field
of constraint, if there is a field of u1 displacement kinematically acceptable to 0 and such as,
for any displacement *
U kinematically acceptable with 0, one has:

Tr

(Q
(U, *
1 U)
D
S0
+
Tr

[L (* U) L
C
(U) Q
+ (U, *
L
L
Q
Q
Q
1
U)
C
(U) + (*
1
U)
C
(U, U) + (U, *
1
U)
C
(U, u1)]
D
S0
= 0
éq 2.3.2-5
Then the problem considered is unstable.

One can express this condition of junction in matric form while introducing, moreover,
stamp geometrical stiffness K () which discretizes the first term of it:

T
U
* CA 0, U * K U
T1 = 0

éq 2.3.2-6
With K
= K0 + K L
T
(U) + KQ (U) + K ()
stiffness



is

who
tangent


If one writes the condition of junction on the current configuration S, then one a:

*
U

CA0,

*
U T [K + K ()]U = 0





éq 2.3.2-7
1

The constraint to be considered is then the constraint of Cauchy and all the integrals are evaluated on
the field running S.

2.3.2.1 Stability condition of a nonlinear elastic balance

It comes immediately, that if there is a state such that the tangent matrix K T defined above is
singular, we will have exhibé a field of u1 displacement well not no one which shows the loss
of unicity of the solution of the mechanical problem. This field of displacement is the mode of buckling.
One can notice that the condition of junction is well checked, whatever the standard and it
sign u1: in this direction, one thus speaks about mode of buckling, like direction, because one is
limited in [éq 2.3.2-4] to the first command in.

2.3.2.2 Case of small displacements: charge of Euler

When displacements can be qualified the small ones before buckling, one can confuse
initial configuration with the current geometry. Matrices
L
K and Q
K can then be
neglected. Moreover, the constraint can be confused with the usual constraint; equations
of buckling are written then:
[K + K U =
éq
2.3.2.2-1
0
()]
0
1
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It is advisable to notice that the matrix K () is proportional to and thus to the loading
applied to the structure. If one multiplies the constraint by, one obtains:
[K + K
U =
éq
2.3.2.2-2
0
()]
0
1
This equation immediately makes think of a problem generalized with the eigenvalues, the same one
type that in the case of the search of the modes of vibration, which is written:
[
2
K - M v =
éq
2.3.2.2-3
0
] 0
1
The matrix K () is replaced by the matrix of mass M, and one sees appearing the own pulsation
, whereas v1 is the associated mode of vibration.

If one wishes to study buckling under loading of which only a part is controlled (part
variable of the loading), by a principle of superposition, the contribution, constant, loading
not controlled must be added at the end K 0 and only the constraint generated by the loading
controlled will be in the term in. Formally, the following problem is thus posed:

[K0 + K (cte) + K (VAr)]u1 = 0

gene

constraint
:
rée by

loading



controlled

not

éq
2.3.2.2-4
cte
:
With
gén

constraint
:
érée by

loading



controlled

VAr

The two stress fields are obtained by resolution of two linear problems, one for
loading not controlled, the other for the controlled part of the total loading. Documentation [U2.08.04]
[bib17].

2.3.2.3 Case of the imposed forces depend on the geometry
Example of the following pressures:
When the external forces depend on the configuration, that involves that the work of the forces
external intervenes under the stability condition. Let us take the example of a pressure applied to
structure. This pressure will be supposed to be constant during buckling: in other words, the value of
pressure does not change during displacement.
This assumption corresponds to two types of real problems. The first type is that where volume
fluid imposing the pressure on the structure is very large in front of the variations of volume
generated by the displacement of the solid. The problems of pressure tanks intern where them
displacements of walls are considerable compared to dimensions of the structure itself
thus do not re-enter within this framework.
The second case corresponds to the existence of a source of fluid which makes it possible to maintain the pressure with
a constant value. It is not then necessary any more to worry about the amplitude of the displacement of
solid.
The value of pressure being taken fixes, the variation of the normal in the course of time are to be taken in
count. This variation is due to the field of displacement which modifies the surface of the structure. Of
even, if one reasons in terms of resultant and thus of integral, the element of surface can too
to change surface. Consequently, the resultant of the compressive forces will vary and it is advisable to hold some
count.
We see quickly that the power of the efforts, expressed on the current configuration, associated
to a pressure is given by the following equation (see for example [bib11]):
ext.
P
N
1
*
N
U
éq
2.3.2.3-1
pressure =

dS

p
+
1




dS
SP

dS

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In this equation, we notice that the power of the external efforts is modified in
u1 displacement. We will have then:
ext.
P
P
N U
éq
2.3.2.3-2
1
= ext. + p 1 * dS

1
SP
Finally, the matrix K T is enriched by an additional term, function of the pressure:
L
K = K + K U + K U + K + K

éq
2.3.2.3-3
T
0
()
Q ()
() (p)
If one writes the operators on the current geometry, one leads to:
K = K + K () + K (p
éq
2.3.2.3-4
T
)
When we are in the presence of following compressive forces, same methods as those
presented previously will be able to apply to calculate the buckling loads: it will be enough to
to supplement the matrix K T with the new term K (p). One can show that the matrix K (p) is
symmetrical if the compressive forces do not work on the “edge” of the model.

2.3.2.4 Vibrations under prestressing

Same methodology can also apply to the study of the vibrations of the structure in
configurationcourante S. Cette structure is prestressed and deformed. It is enough to write Principe
of geometrical Puissances Virtuelles nonlinear [éq 2.3.1-6] by taking account of the effects of inertia and
by injecting the assumption there that displacements are of the periodic functions of the type:
U
= v

éq
2.3.2.4-1
1 (T)
sin
1
(T)
It results from this:
[K +KL (U) +KQ (U) +K () +K (p) 2
- M v = éq 2.3.2.4-2
0
] 0
1

Let us interest in this equation.
First of all, we notice that when we have a state the Eigen frequency criticizes then of
vibration of the structure corresponding to the mode of buckling is null.
Moreover, we observe that the Eigen frequencies of the structure charged are different from those
initial structure for two reasons:
The own pulsation is modified by prestressing p: it is the principal effect which is used, by
example, to grant a violin. The tension of the cord exploits the height of the note
corresponding, therefore on its Eigen frequency.
A second effect is the variation of the frequency by modification of the geometry: the matrix of
geometrical starting stiffness K 0 is replaced by the matrix of stiffness on the current geometry:
L
Q
K 0 + K + K. What causes to modify the vibratory equations.

2.3.2.5 Processing of the elastoplastic behavior (plastic buckling)

Far from any exhaustiveness, we will present only the simplest approaches here, for their
easy establishment in the code.
When the structure functions in an elastoplastic mode, buckling is affected by the loss
of resistance due to plasticity [bib2]. The modification comes from the relation of behavior during
additional displacement u1.
The constraint becomes, in incremental form:
=

+ C [L

+
+



2.3.2.5-1
T
(U)
Q (U, U)]
2
C
Q U, U
1
T
(1 1)
2
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In this expression, the matrix of behavior is tangent matrix CT. The choice of this
matrix is not immediate: indeed, the matrix depends on u1 and is thus not known as long as it
mode is unknown. One can, for example, to discharge during buckling if the mode develops
in a direction and to charge if it develops in the opposite direction. It is thus necessary to do one
assumption for the behavior during plastic buckling. , We will start
to apply the assumption of Hill [bib4] who leaves the principle that the structure continues to charge
plastically during buckling.
Let us consider an elastoplastic law of Von Mises type. We define the three modules: E which is
the Young modulus, AND the tangent module, and ES the secant module. These modules are recalled on
following figure:



AND
ES
E

Be reproduced 2.3.2.5-a: Représentation of the various modules on a traction diagram 1D

Then we propose three possible methodologies.
The assumption of the tangent module simply consists in replacing the Young modulus by the module
tangent in the relation of behavior. One obtains then:
E
C =
C
2.3.2.5-2
T
AND
This method is very rudimentary, but it is always pessimistic, which can constitute one
favor, if one places oneself from the point of view of dimensioning.

The method used usually consists in using the tangent matrix of incremental calculation
(operator STAT_NON_LINE [bib14]). We thus have the following equation in the case of plasticity
of Von Mises [bib15]:






T
D
D
With
AC






CT = CI -

T
D
D




AA

H +

D




VM



2.3.2.5-3
D
vector
:
diverter

constraint



S



T

With
D
D
D


With
intervenan

stamp
:
Settings

Von
of

normalizes



in
T



=

With




VM



E E

T
H
by

defined

plastic

slope
:
: h=



E AND
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This method is perfectly rigorous only in nonlinear elasticity or if one respects
the assumption of Hill: it does not make it possible to predict the junctions in the ways of loading. As of
that the relation of behavior is dissipative, the critical loads calculated will not be exact that
if one can check that the loading is monotonous, in any point of the structure (Hill [bib4]).

The most realistic method consists in using the finished theory of the deformation only for
to calculate the plastic load of buckling. The tangent matrix of behavior is given by
the equation below:

1
-
1
1
T
D
D


C
2.3.2.5-4
T =
With [
]

With


1
-
1
1
-
+ C +
-
With
E
E
D



E
E
T
S
S


VM


Compared with the method based on the matrix of tangent stiffness [éq 2.3.2.5-3], this criterion requires
construction and assembly of a specific total matrix. This expensive operation comes to weigh down
the incremental resolution.
For considerations of general information and minimization of the development cost and cost of
calculation (CPU and memory), we thus choose the criterion based on the tangent module [éq 2.3.2.5-3].

3
Establishment in the code

In any rigor, in order to secure the analysis of stability of a nonlinear quasi-static calculation, it is necessary
to use the criterion of ad hoc stability to each step of incremental calculation. Any criterion of stability not
linear must thus be intrinsically the least expensive possible in time CPU and place memory.
Speaking Algorithmiquement, it appears judicious to establish the call to the criterion inside even of
routine corresponding to operator STAT_NON_LINE [bib14]. Indeed, the principle of call to each step
put up badly with a completely externalized operation of the incremental method of
resolution of the nonlinear mechanical problem.

3.1 Criterion
of Euler

This criterion (cf [§2.3.2.2]) requires only the resolution of a linear static problem, then
construction and assembly of the geometrical matrix of stiffness. This one and stamps it stiffness
assembly are then to pass like argument of a solvor [bib12] for the problem to the values
clean [éq 2.3.2.2-2].
At output one thus recovers the modes of buckling and the critical loads corresponding. For
more details, the user will be able usefully to consult the document [U2.08.04] [bib17].
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3.2
Nonlinear criterion

3.2.1 Impact on operator STAT_NON_LINE

Let us start by briefly pointing out the operation of the incremental method of resolution
nonlinear problems of structure [bib14].

3.2.1.1 Algorithm
of
STAT_NON_LINE

One will use index I (like “moment”) to note the number of an increment of load and exhibitor N
(like “Newton”) to note the number of the iteration of Newton in progress. The algorithm used in
operator STAT_NON_LINE can then be written schematically in the following way:

(U, and known
0
0)
0

Loop over moments T (or increments of load): loading L = L T
I
(I)
I

·
(U,
known
I

1
-
I 1
-)
· Prediction: calculation of
0
U and
0

I
I
· Loop on iterations of Newton: calculation of a continuation (N
N
U,
I
I)
·
(N N
U, and (N
N
U, known
I
I)
I
I)
· Calculation of the matrices and vectors associated with the following loads
· Expression of the relation of behavior
·
calculation of constraints N
and of internal variables N
starting from the values
I
I
1
-
and with preceding balance (T) and of the increment of displacement
1
-
I 1
-
N
U
U
U since this balance
I =
N
I -
I 1
-
·
calculation of the “nodal forces”:

N
T
N
Q + B
I
I
·
possible calculation of the matrix of tangent stiffness:
N
K = K (N
U)
I
I
· Calculation of the direction of search
N 1
+
(U,
N 1
+
) by resolution of a system
I
I
linear
· Iterations of linear search:
· Updating of the variables and their increments:

n+1
N
n+1
U
=u + U
U =u + U
I
I
I


n+1
N
n+1

and
I
I
I

n+1
N
n+1
= +


= +
I
I
I

n+1
N
n+1
I
I
I
· Test of convergence
· Filing of the results at the moment T
I

U = U
U
1 +
I
I
I
= 1 +
I
I
I
I

I

It is noticed that there are three overlapping levels of loops: a loop external on the steps of time,
a loop of iterations (qualified the total ones) of Newton and subloops possible for
linear search (if it is asked by the user) and certain relations of behavior
requiring iterations (known as interns), for example for elastoplasticity in plane constraints.

If one chooses the criterion based on the assembled tangent matrix [éq 2.3.2.5-3], it is necessary to have this
stamp reactualized with each step where one wants to make the analysis of stability.
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It is the case when one uses a method of the Newton type, and not a method of the Newton type
modified.

One leads then to the following algorithm:

(U, and known
0
0)
0

Loop over moments T (or increments of load): loading L = L T
I
(I)
I

·
(U,
known
I

1
-
I 1
-)
· Prediction: calculation of
0
U and
0

I
I
· Loop on iterations of Newton: calculation of a continuation (N
N
U,
I
I)
·
(N N
U, and (N
N
U, known
I
I)
I
I)
· Calculation of the matrices and vectors associated with the following loads
· Expression of the relation of behavior
·
calculation of constraints N
and of internal variables N
starting from the values
I
I
1
-
and with preceding balance (T) and of the increment of displacement
1
-
I 1
-
N
U
U
U since this balance
I =
N
I -
I 1
-
·
calculation of the “nodal forces”:

N
T
N
Q + B
I
I
·
possible calculation of the matrix of tangent stiffness:
N
K = K (N
U)
I
I
· Calculation of the direction of search
N 1
+
(U,
N 1
+
) by resolution of a system
I
I
linear
· Iterations of linear search:
· Updating of the variables and their increments:

n+1
N
n+1
U
=u + U
U =u + U
I
I
I


n+1
N
n+1

and
I
I
I

n+1
N
n+1
= +


= +
I
I
I

n+1
N
n+1
I
I
I
· Test of convergence
· Filing of the results at the moment T
I

U = U
U
1 +
I
I
I
= 1 +
I
I
I
I

I

·
Criterion of stability, function of the reactualized tangent stiffness:
N
K = K (N
U)
I
I

The criterion is calculated at the end of the step, just after filing. It thus has like arguments the quantities
converged with the current step. Moreover, this choice of position of call makes it possible to hold account correctly
following loadings, since their calculation is done at the time of the iterations of Newton. The criterion
could thus be called before the end of these iterations.

3.2.1.2 Impact on the structure of data result of STAT_NON_LINE

The call of the nonlinear criterion of stability will induce the resolution of a problem to the eigenvalues.
The result of this calculation will be thus a whole of couples critical load/mode of buckling.
The critical loads are scalars and the associated modes are fields of displacement, which
will come to enrich the structure of data result by STAT_NON_LINE.
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3.2.2 Characteristics related to the tensor of deformation

In the code, it is advisable to distinguish two great families from description of the deformations.

On the one hand the linearized tensor corresponds to the case of the small disturbances (argument PETIT of the word
key DEFORMATION), but also with the case of the small disturbances reactualized (Lagrangian reactualized with
each step of incremental calculation: argument PETIT_REAC of key word DEFORMATION).

The tensor of deformation is written then (like [éq 2.2-3]):
1
= (U
T
+ U) éq
3.2.2-1
2
The use of PETIT_REAC implies a resolution of the balance of the structure on its geometry
current with a tensor of deformations linearized. One thus calculates the increment of deformation by
report/ratio with position X, displacement U and the increment of displacement U in the following way:

1 U
U

=
I

éq
3.2.2-2
ij
+
J


2 (X + U)
X U
J
(+)
I

In addition, the code offers tensors of deformation of the type Green-Lagrange (GREEN or
GREEN_GR) for the processing of great displacements (and the rotations finished for certain elements
of structure) but under assumption of small deformations. The tensor used is the traditional tensor
according to [éq 2.3.1-5]:
1
(U) =
U
+ U + U U
ij
(I, J J, I K, I K, J) éq
3.2.2-3
2
Key word GREEN corresponds to modelings in 3D whereas key word GREEN_GR applies to
modelings beam or hull.

Lastly, the framework of modeling in great transformations most complete accessible in
Code_Aster results from the theory of Simo and corresponds to key word SIMO_MIEHE. It takes into account
great rotations and great deformations since the law of behavior is written in
great deformations. For more precise details on the fundamental differences between the different ones
types of deformations, the documentation [bib16] of Code_Aster presents in detail modeling
SIMO_MIEHE.

Code_Aster does not allow calculations in configuration eulerienne: as with the tensor
of Almansi, for example. All the tensors of deformation available are of Lagrangian type.

The fundamental difference, as for the writing of the criterion, is between the linearized deformations
(PETIT and PETIT_REAC) and deformations GREEN, GREEN_GR and SIMO_MIEHE.
Indeed, Code_Aster has need to make its search for balance of the tangent matrix. This one
is written according to the equation ([§2.2.2.1] of documentation on STAT_NON_LINE [bib14]):

T


Q
T
K
éq
3.2.2-4
T = Q
:
+
:
U

U

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T
T

Q
However, Q:
corresponds at the end traditional of material rigidity and
: corresponds at the end
U

U

of geometrical rigidity which is present only in great displacements. Thus when one wants to apply one
criterion of buckling of the type (formally assimilable to [éq 2.3.2.2-2]): (K + K ()) = 0.
This criterion is valid only in small deformations since the geometrical term of rigidity is
regarded as negligible in the tangent matrix.
One thus can, with reason, to make a traditional search of the eigenvalues and clean vectors of
buckling type of Euler.

On the other hand in great transformations, the evaluation of this criterion consequently method is
problems for two reasons. On the one hand, in the tangent matrix, the term of rigidity
geometrical is already calculated and, in addition, the matrix K () which would possibly have to be added is
obtained under Code_Aster in small deformations. For these reasons, it is necessary to evaluate
manner different the criterion according to the type of tensor of deformation asked by the user.

If one made the choice of a description eulerienne, the development of a criterion of the Euler type
reactualized would be facilitated on the level of the calculation of the term K (), whatever the tensor of deformation.

3.2.2.1 In linearized deformations: PETIT and PETIT_REAC

As we said previously, this case of figure does not pose a major problem. It is enough to
to calculate the geometrical matrix of rigidity and to make a traditional search for modes and values
clean, of type Euler [éq 2.3.2.2-2]:

(K + K ()) = 0
éq
3.2.2.1-1

K is the tangent matrix reactualized at the end of the step of time.
In this case, one can thus speak indeed about criterion of reactualized the Euler type.
As one is in small deformations, the matrix of geometrical rigidities is proportional to
loading. Therefore, when the critical coefficient is obtained, it is enough to multiply it by the load
real with the step of current time to obtain the critical load buckling. The case =1 corresponds
thus with the loss of stability.

3.2.2.2 In great displacements: GREEN, GREEN_GR and SIMO_MIEHE

The traditional method does not apply any more in this case. Indeed, Code_Aster calculates like
stamp tangent the matrix of material rigidity plus the geometrical matrix of rigidity (and
possibly, the contribution due to the following pressures).
One in the manners of checking buckling then is to make a search of the eigenvalues of
only stamp tangent. If one of the eigenvalues is negative, it is that the matrix became
singular and that an instability occurred between the moment when all its eigenvalues were
positive and moment when one of it became negative.
The problem to be treated is thus slightly different since in the case of the small deformations
(PETIT and PETIT_REAC), there is the following system to solve [éq 3.2.2.1-1]: , (K + K ()) = 0 then
that in case GREEN, GREEN_GR and SIMO_MIEHE it is necessary to solve:

(K + I) = 0 éq
3.2.2.2-1

With I which is the matrix identity and is, this time, of physical size equivalent to K, then
that in the case of the small deformations, the eigenvalue is adimensional (from where sound
direct interpretation as a multiplying coefficient of the loading).
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One of the defects inherent in this method compared to more explained more traditional search
high [§ 2.3.2] is that one can have forecasts of buckling only when one approaches “close”
critical load, even when it is exceeded. Far from this load, the first eigenvalue
found does not have really physical significance since nonlinearities can appear
between the step running and the calculated critical load. The coefficient report/ratio criticizes on load at moment I
is thus different from that at the moment i+1 whereas in small deformations this report/ratio remains constant.
Moreover, for all the steps of time, all the eigenvalues and clean vectors except more
low do not have any physical significance since, for a clean couple vector eigenvalue
(V, one a:
I
I)
(K U
() + K ())V = V éq
3.2.2.2-2
I
I
I
This has clear direction only as from the moment when
, in which case one finds the critical load and
I 0
the clean vector criticizes associated.
Always compared with the criterion of Euler (reactualized [éq 3.2.2.1-1] or not [éq 2.3.2.2-2]), it is noticed that
the eigenvalue of the problem [éq 3.2.2.2-1]: (K + I) = 0 are not adimensionnée. It results from this
a greater difficulty of interpretation as for knowing if the value is “small” or not. Otherwise
known as, when can one say that one is close to a junction?
To define a relevant interval and of general use, in order to limit the vicinity of an instability, it
would be interesting of adimensionner the eigenvalues.

3.2.2.3 Case of mixed modelings

As Code_Aster makes it possible to assign several types of deformations to the same structure, it is necessary
to consider the case where one uses several types of tensors of deformation in same calculation.
The differentiation of the various elementary matrices being of no utility, it is appropriate of
to solve to slice at the total level enters a method or the other. One chose to extract the values and
clean vectors of the tangent matrix without adding geometrical matrices of stiffnesses. All
pass as if the structure were in deformation of the Green-Lagrange type from the point of view of the criterion.
Indeed, let us consider an unspecified solid made up of two parts I and II. On part I, the tensor
of deformation which was adopted is the tensor linearized PETIT and on part II that of
Green-Lagrange. The tangent matrix resulting from the assembly of the two submatrices becomes:

K
*
0
I



*
*
*
éq
3.2.2.3-1
0
* K
K
II +
(
II)



The spangled terms represent the nodes common to both parts and are thus a combination
linear of the values of the two matrices. In this configuration, it appears that none the solutions
is not satisfactory but that less penalizing is to make a search for “
type
Green-Lagrange “[§ 3.2.2.2] i.e. to use (K + I) = 0 [éq 3.2.2.2-1].

This solution not being exact but nevertheless the only able one to be carried out simply, it is
envisaged to add a message of alarm informing the user who the results obtained are not
guaranteed due to the juxtaposition of several types of tensors of deformations.

3.2.3 Improvement of the performances of the criterion

During the incremental resolution of a nonlinear quasi-static problem, in the ideal and if one
admits that the discretization in time is sufficiently fine, it would be necessary to make an analysis of stability with
each step of calculation. With each step, that induces the resolution of a problem to the eigenvalues,
admittedly limited in the search of some modes. The analysis of stability thus brings a overcost CPU
important, with a nonlinear calculation already being able to be long.
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The idea is to call upon the resolution of a problem to the eigenvalues only when it is
really necessary, therefore when the current configuration is “close” to an instability. If one
can define this vicinity by a preset interval, then one can call upon a test of Sturm [bib12].
This test makes it possible to know if there is at least an eigenvalue on the interval of search. In
the affirmative, one will be able to then carry out modal search. In the contrary case, one continues
quasi-static incremental resolution, without solving problem with the eigenvalues.
The cost of a test of Sturm is notably lower than the cost of search of the critical loads.
The interval of search for the test of Sturm can, either to be given by the user, or to have one
default value in the code.
In the case of a criterion of reactualized Euler (case of the small deformations [§ 3.2.2.1]), where the problem with
to solve is written: (K + K ()) = 0 [éq 3.2.2.1-1], the interval of search must be centered on
eigenvalue = 1 (which corresponds to the value - 1 for the algorithm of MODE_ITER_SIMULT, because it
solves in fact a problem of the type: K = µK ())).
The terminals of the interval are the terminals of the multiplying coefficient of the loading, therefore
adimensional quantities, which are a function of the safety coefficients and the evaluation of
uncertainties for the problem given. The test of Sturm is implemented within this framework.
In the specific case adapted to the tensor of Green-Lagrange [§ 3.2.2.2], where one solves:
(K + I) = 0 [éq 3.2.2.2-1], the interval are centered on 0. Moreover, terminals of the interval of test
, contrary to the preceding case, are not adimensionnées [§ 3.2.2.2]. It is thus more difficult
to identify relevant and general values (for the case of the default values). The test of Sturm
is not currently established for this case.

3.3
Generalization with dynamics

We will not approach here the framework of the criteria of dynamic instability (negative damping…). It
just acts to announce that the nonlinear criterion presented here can completely apply directly in
nonlinear dynamics. It will then detect any potential buckling of the structure, within the meaning of
singularity of the total matrix of reactualized tangent stiffness.
In order to be exhaustive in terms of analysis of stability on a nonlinear dynamic study, the user
should use two criteria:

· a criterion of buckling (criterion on the stiffness),
· a dynamic criterion (criterion on damping or the quadratic linearized problem
total [bib13], for example).

For the moment, the criterion is not available in DYNA_NON_LINE.

3.4
Syntax of call of the criterion

In operator STAT_NON_LINE, the call to the criterion is done in the following way:

CRIT_FLAMB = _F (
CHAR_CRIT = (- 1.1, 0. ),
NB_FREQ
=
3
.
)
,

Key word CHAR_CRIT defines the field on which one will make the test of Sturm, into small
linearized deformations. If one finds at least an eigenvalue on the interval, then, one carries out
resolution of the problem to the eigenvalues corresponding, if not, one does not do anything and calculation
incremental can continue.
If one uses modelings GREEN, GREEN_GR or SIMO_MIEHE, the modal resolution has inevitably
place, and one seeks the eigenvalues smallest.
Key word NB_FREQ makes it possible to specify the number of modes which one seeks (default value:
3). It can be useful to seek more than one mode, mainly to be able to detect the cases
“pathological” such as multiple modes or very close relations.

The mode of buckling corresponding to the smallest eigenvalue (in absolute value) is stored in
the structure of data RESULTAT (eigenvalues named CHAR_CRIT, fields of displacement
named MODE_FLAMB, which one can visualize via IMPR_RESU).
Handbook of Référence
R7.05 booklet: Instabilities
HT-66/03/005/A

Code_Aster ®
Version
7.2
Titrate:
Criteria of structural stability

Date
:

15/12/03
Author (S):
NR. GREFFET, J.M. PROIX, L. SALMONA Key
:
R7.05.01-A Page
: 22/24

3.5
Validation of the developments

The cases tests of validation are: SSNL126 and SSLL105D.
More precisely, the cases tests SSNL126 treat the case of a beam fixed at an end and
subjected to a compression at the other end. Modeling is three-dimensional, with relation of
elastoplastic behavior with linear isotropic work hardening. Two representations kinematics are
presented:

· modeling a: linearized deformations,
· modeling b: deformations of Green-lagrange.

The case test SSLL105D is based on a problem of beam in L, which one studies buckling
rubber band. The finite elements are of beam type.

4 Conclusions

Code_Aster offers two criteria of stability, within the meaning of buckling, for the structural analyzes.

On the one hand, whenever a linearized approach is enough, one can apply a criterion of the Euler type
([bib12] and [bib17]), by call to an operator of resolution of the problem to the eigenvalues generalized
(for example MODE_ITER_SIMULT with key word TYPE_RESU=' MODE_FLAMB').

In addition, for all the cases where it is essential to take account of nonthe linearities, which they are
had with the relation of behavior or the great transformations, the user can employ one
adapted criterion, of generalized Euler type. The call of this criterion is done during the resolution
incremental of the quasi-static problem (operator STAT_NON_LINE [bib14]).
With each step of time, the criterion is based on the resolution of a problem to the eigenvalues [bib12]
on the matrices of brought up to date total stiffnesses.
This criterion, which is declined in two different forms, according to the tensor of deformation chosen,
base on a linearization around the step of current calculation. It accepts any type of tensor of
deformation, as any type of relation of behavior for which one is able of
to build the matrix of total stiffness, at every moment. Moreover, the selected criterion is perfectly
rigorous in the case of relations of nonlinear elastic behavior, and in the case of
elastoplasticity associated with the assumption with Hill [§ 2.3.2.5].

Handbook of Référence
R7.05 booklet: Instabilities
HT-66/03/005/A

Code_Aster ®
Version
7.2
Titrate:
Criteria of structural stability

Date
:

15/12/03
Author (S):
NR. GREFFET, J.M. PROIX, L. SALMONA Key
:
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: 23/24

5 Bibliography

[1]
C. CHAVANT, A. COMBESCURE, J. DEVOS, A. HOFFMANN, Y. MEZIERE: “Buckling
rubber band and plastic of the thin hulls ", Cours IPSI, 1982.
[2]
S. DURING, A. COMBESCURE: “Analysis of junction in great deformations
elastoplastic: elementary formulation and validation ", Rapport intern n°190,
LMT-Cachan, 1997.
[3]
NR. GREFFET: “Coupled Simulation fluid-structure applied to the problems of instability not
linear under flow ", Thèse of doctorate, LMT, ENS-Cachan, 2001.
[4]
R. HILL: “A general theory for uniqueness and stability in elastic-plastic solids”, J. Mech.
Phys. Solids, vol. 6, 236-249, 1958.
[5]
T.J.R. HUGHES, W.K. LIU, I. LEVIT: “Nonlinear dynamic finite element analysis off shells,
Nonlinear Finite Element Analysis in Structural Mechanics ", W. Wunderlich, E. Stein &
K.J. Bathe editors, Berlin, Springer, 151-168, 1981.
[6]
G. LOOSS: “Elementary stability and junction theory”, Springer-Verlag, 1990.
[7]
A. LIGHT, A. COMBESCURE, Mr. POTIER-FERRY: “Junction, buckling, stability in
Mechanics of the structures ", Cours IPSI, 1998.
[8]
A. LIGHT: “Junction, buckling, stability in mechanics of the structures”, Note EDF-DER
HI-74/98/024/0.
[9]
J. SHI: “Structural Computing critical points and secondary paths in nonlinear stability
analysis by finite element method ", Computer & Structures, Vol. 58, n°1, 203-220, 1996.
[10]
J.C. WOHLEVER, T.J. HEALEY: “A group theoretic approach to the total junction
analysis off year axially compressed cylindrical Shell ", Comp. Meth. In Applied Mech. And
Engrg., vol. 122, 315-349, 1995.
[11]
E. LORENTZ: “Efforts external of pressure in great displacements”, Manuel of reference
of Code_Aster [R3.03.04].
[12]
O. BOITEAU: “Algorithm of resolution for the generalized problem”, Manuel of reference
of Code_Aster [R5.01.01].
[13]
D. SELIGMANN, R. MICHEL: “Calculation algorithm of the problem quadratic of values
clean ", Manuel of reference of Code_Aster [R5.01.02].
[14]
NR. TARDIEU, I. VAUTIER, E. LORENTZ: “Quasi-static nonlinear Algorithm (operator
STAT_NON_LINE) “, Manuel of reference of Code_Aster [R5.03.01].
[15]
J.M. PROIX, E. LORENTZ, P. MIALON: “Integration of the elastoplastic relations”, Manuel
of reference of Code_Aster [R5.03.02].
[16]
V. CANO, E. LORENTZ: “Model of Rousselier in great deformations”, Manuel of
reference of Code_Aster [R5.03.06].
[17]
NR. GREFFET: “Note of calculation to buckling”, Manuel of use of Code_Aster [U2.08.04].

Handbook of Référence
R7.05 booklet: Instabilities
HT-66/03/005/A

Code_Aster ®
Version
7.2
Titrate:
Criteria of structural stability

Date
:

15/12/03
Author (S):
NR. GREFFET, J.M. PROIX, L. SALMONA Key
:
R7.05.01-A Page
: 24/24

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Handbook of Référence
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