Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
1/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
Organization (S):
EDF/IMA/MN
Manual of Reference
R7.07 booklet: Analyze limit
R7.07.01 document
Calculation of load limits by the method
of Norton-Hoff-Friaâ
Summary:
The limiting analysis makes it possible to determine the acceptable loadings D `a structure, of geometry fixes given,
composed of a material having a criterion of resistance. One considers the case of loadings made up of
summon of a continued load and of another parameterized by the load factor.
After a recall of the theoretical formulation, one presents the regularized kinematic approach applied to the criterion
of resistance of Von Mises (method of Norton-Hoff-Friaâ) and implemented in Code_Aster. One will be able
to refer to [bib4] for the various possible methods of regularization suggested in the literature. One
expose then the calculation of the solutions of this nonlinear problem and postprocessing providing one
estimate by higher value of the limiting load.
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
2/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
Contents
1 theoretical Formulation of the analysis limits ............................................................................................... 3
1.1 Definition of the load limits ........................................................................................................... 3
1.2 Calculation of the load limits by a kinematic approach ............................................................... 3
1.3 Regularization of the kinematic approach by the method of Norton-Hoff-Friaâ ........................... 5
2 numerical Aspects of the calculation of the load limits ................................................................................. 7
2.1 Establishment in Code_Aster .................................................................................................... 7
2.2 Relation of behavior of Norton-Hoff in
STAT_NON_LINE
............................................... 8
2.3 Postprocessing of the calculation of the load limits ................................................................................. 10
3 an example of validation ..................................................................................................................... 12
3.1 Problem of reference .................................................................................................................. 12
3.2 Plane case ........................................................................................................................................ 12
3.2.1 Solution analyzes limit of it ..................................................................................................... 12
3.2.2 Solution analyzes regularized limit of it .................................................................................. 13
3.3 Axisymmetric case ......................................................................................................................... 13
3.3.1 Solution analyzes limit of it ..................................................................................................... 14
3.3.2 Solution analyzes regularized limit of it .................................................................................. 14
3.4 Three-dimensional case ....................................................................................................................... 15
3.4.1 Solution analyzes limit of it ..................................................................................................... 15
3.4.2 Solution analyzes regularized limit of it .................................................................................. 15
4 Bibliography ........................................................................................................................................ 16
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
3/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
1
Theoretical formulation of the limiting analysis
1.1
Definition of the limiting load
One considers a solid occupying a field
limited subjected to surface loadings
F F
+
0
on the edge
F
and of the loadings of volume
F F
+
0
on
. The loading is distinguished
(,)
F F
, parameterized by positive reality
, and the permanent loading
(,)
F F
0
0
. Conditions of
Dirichlet homogeneous are applied to the complementary edge
U
of
(an imposed displacement
or an initial anelastic deformation - thermics, plastic… - do not have an effect on the field of
safe loads). One can find in [bib5] several other useful properties.
The material constitutive of the solid has a criterion of resistance expressed by a scalar function of
stresses, negative for working stresses. The criterion used for a material of the type
elastoplastic perfect with threshold of von Mises and selected here is:
()
()
(
) (
) (
)
G
J
.
.
.
=
-
=
-
=
-
+
-
+
-
-
y
D
D
y
y
32
2
2
1
2
2
2
3
2
1
3
2
D
is the diverter of the tensor of the stresses,
y
is the threshold of resistance in simple traction (like an elastic limit), possibly
variable according to areas' of the solid considered.
I
being main stresses of
.
Being given this criterion of resistance one seeks to calculate the value limits
, called limiting load
lim
, for which the structure can support the loadings
lim
F F
+
0
and
lim
F F
+
0
.
Strictly speaking, the value
lim
indicate the limit of the bearable loadings, but for
materials obeying the Principle of Maximum Plastic Work, this value is the limit of
supported loadings.
1.2
Calculation of the load limits by a kinematic approach
In design the collapse two approaches are possible: static approach (in variables of stresses)
and kinematic approach (in variables speeds). These approaches provide terminals of
charge limit: undervaluing for the approach static and raising for the kinematic approach. When them
two provide the same result, the limiting load obtained is exact.
The kinematic approach is that used in Code_Aster using finite elements in displacements.
For the loading given
(F, F)
, one defines the space speeds kinematically acceptable and
standardized by:
V
L
has
U
D
ds
F
1
1
=
=
=
+
=
,
,
acceptable
on
v
v
0
v
F. v
F. v
()
Power of the permanent loading
(,)
F F
0
0
is noted:
L
0
()
v
.
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
4/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
From the criterion of resistance in stresses
G
()
, one defines:
·
the whole of working stresses by:
()
()
()
(
)
{
}
G
X
X
X
=
,
G
0
(
()
G
X
is convex for the criterion
G
)
·
the indicating function:
()
(
)
()
()
()
()
G
if
G
if
G
X
X
X
X
X
=
+
0,
,
·
the function of support:
()
[
]
=
-
Sup
()
R
6
.
G
Sup
in
()
can be reached only if
is selected in
()
G
X
, such as:
µ
=
+
D
Id
(what ensures
//
D
). The optimum corresponds to
()
(
)
G
.
.
/
=
=
-
0
2 3
1 2
y
D
D
G
0
·
·
D
D
D
D
1
2
0
()
Appear 1.2-Error! Argument of unknown switch. : Optimum
and graph of the function
()
in 1D
From where the function of support:
()
()
() ()
(
)
µ
µ
v
v. v
v
=
+
.
Sup
.div
y
2
3
R
. It is observed that
function
()
is not differentiable in
0
.
One to date does not treat in Code_Aster possible internal surfaces of discontinuity with the center
solid
[bib 4].
The kinematic approach is defined using the convex functional calculus
()
S
E
v
, positively homogeneous
of degree one, for
v
V
has
1
defined on the whole field:
()
()
S
L
E
D
v
v
v
=
-
()
()
0
This functional calculus is the integral on the field of the function of support
the convex one
()
G
X
, calculated
in
()
v
and is interpreted like the maximum resistant power in the field speeds
v
(
contribution of resistance of interface on surfaces of discontinuity is supposed to be null). The function
of support
is positively homogeneous degree 1, and thus the functional calculus
()
S
E
v
also by
consequence.
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
5/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
With the criterion of Von Mises the functional calculus of power
()
S
E
v
is:
()
(
)
S
L
E
y
Q
Q
D
v
v. v
v
v
=
+
-
.
() () Sup .div
()
2
3
0
R
éq 1.2-1
where it is noted that only the fields
v
belonging to
{
}
C
has
=
=
v
v
V
1
0
, div
in
provide finished values. Fields
v
must thus check the condition known as of incompressibility
()
div
tr
v
v
=
=
0
. This is why it is necessary to use the incompressible elements for a calculation of
charge limit with the criterion of Von Mises. The processing of the incompressibility is detailed in
[R3.06.05].
The limiting load
lim
given by the kinematic approach is:
()
()
()
()
()
()
(
)
(
)
lim
Inf
Inf
Sup Inf
=
=
=
-
-
>
>
v
v
v
v
v
v
v
v
v
V
V
V
S
S
L
S
L
L
has
E
has
E
has
E
1
0
0
1
1
With the optimum one obtains a solution
U
and the limiting load
lim
(not unicity of
U
but unicity of
lim
).
Thus, any loading
L
L
0
()
()
v
v
+
with
0
lim
is bearable. Beyond
lim
, it
problem of balance does not have a solution.
Note:
There are situations where, even if
L
0
()
v
is not bearable only,
combination
L
L
0
()
()
v
v
+
, for
1
2
, becomes it on a certain interval, and not
only for two parallel loadings.
Note:
The limiting load calculated for a two-dimensional problem, in plane deformations, is
necessarily higher than that obtained for this problem modelized in plane stresses. It
result thus provides one raising. If one wishes to deal with the problem in plane stresses, it is
necessary then to make the kinematic approach on a three-dimensional modeling.
1.3 Regularization of the kinematic approach by the method of
Norton-Hoff-Friaâ
Implementation the numerical of the kinematic approach requires the minimization of the functional calculus
not-differentiable
()
S
E
v
. Many techniques of regularization exist [bib4]. Method of
Norton-Hoff-Friâa is used here [bib2], [bib7]. It rests on precursory work of Casciaro in
1971. It consists in replacing the function of support
()
by the function of support regularized and
differentiable
()
NH
. It is adjustable by a parameter of regularization
m
(
1
2
m
), of which
limiting value
m
+
1
conduit with convergence towards the function of support
()
:
()
()
(
)
NH
m
m
K
m
=
-
1
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
6/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
In Code_Aster one uses the parameter of regularization
N
with
N
m
=
-
1
1
(
)
and one chooses
K
y
= µ
2
3
to find the incompressible elastic problem when
m
=
2
that is to say
N
=
1
(
2
µ
being the coefficient of
rigidity of shearing).
One notes the space acceptable speeds adapted to the problem of viscous flow for the law of
Norton-Hoff of command
m
:
()
()
()
()
{
}
V
L
amndt
m
m
U
L
and
L
on
1
0
1
=
=
=
v
v
v
v
,
,
,
One defines on this space the regularized functional calculus
()
S
EM
v
:
()
()
()
S
L
EM
m
m
K
m
D
v
v
v
=
-
-
1
0
()
The problem of minimization
()
[
]
Inf
v
v
V
S
amndt
EM
1
is well posed thanks to the properties of
spaces
()
L
m
and a solution has
U
m
, for which the value of
Inf
reached is:
m
. One shows
that this problem can be also written in the form of the search of the point-saddle
(
)
m
m
m
p
,
,
U
Lagrangian following:
()
()
(
)
(
)
Max Inf
Sup
L
.
() ()
.div
()
()
+
-
-
-
R v
v. v
v
v
v
V
L
L
has
m
Q
With m
m
D
Q
D
2
0
1
éq 1.3-1
with:
()
()
()
()
With m
K
m
m
y m
y
m
m
m
=
=
-
-
-
1
2
3
2
2
1 2
3
2
3
/
/
µ
. In practice the continuation is taken:
m
I
= +
-
1 10
2
1,1
1,01
1,001
…
1
N
m
= -
1
1
1
10
100
1000
…
()
With m
2
µ
…
y
2
3
It is noticed that
()
With m
is increasing with
m
(if
E
y
) and homogeneous with a stress, and remains
limited when
N
+
. If one chooses
E
y
=
then
()
()
With m
y
m
=
2
3
2
/
. This Lagrangian allows
to impose directly in the operator the condition of incompressibility and standardization on
1
power of the loadings. One builds a decreasing continuation then of
m
and the limiting load
lim
is the limit of this continuation when
m
+
1
(either
N
+
):
()
[
]
()
(
)
lim
lim
Inf
lim
=
=
m
amndt
EM
m
EM
m
1
1
1
v
v
U
V
S
S
éq 1.3-2
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
7/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
For the demonstration one will refer to [bib4] and [bib7]. One shows also the following property of
solutions of [éq 1.3-1]. Thus if one ampifie the loading
L
L
, when
L
0
0
=
, the solutions have
following dependence according to
:
()
()
()
()
()
(
)
()
(
)
()
(
)
()
(
)
U
U
U
U
U
U
m
m
m
m m
D
m
m D
m
EM
m
m EM m
p
p
=
=
=
=
-
-
-
-
1
1
1
1
1
1
1
;
;
S
S
One of the essential assets of this method of regularization resides in the properties of nesting
spaces
()
L
m
, which makes it possible to have properties interesting for the continuation of
m
, to see it
paragraph 2.3. Thus one shows the following properties [bib7], for a limited field
, in
calling
=
D
and
()
m
With m D
=
:
For all
1
m
and
1
R S
and any function
U
of
V
amndt
, one a:
()
()
(
)
(
)
()
(
)
(
)
With m
D
With m
D
With m
D
m R
R
R
m S
S
S
() ()
() ()
() ()
/
/
U. U
U. U
U. U
-
-
1 1
2
1
1 1
2
1
éq 1.3-3
y
m
y m
m
m
D
D
.
() ()
.
() ()
2
3
1 1
2
3
1
U. U
U. U
U
-
éq 1.3-4
These properties are interesting because applicable if the material is heterogeneous, and one can
to consider the limit of resistance either as measures (density) or like belonging to
the deformation energy.
2
Numerical aspects of the calculation of the limiting load
2.1
Establishment in Code_Aster
To carry out a calculation in Code_Aster in analysis limits with the method of regularization of
Norton-Hoff-Friaâ with the criterion of resistance of Von Mises, it is necessary:
·
to define the model 2D (plane or axis) or 3D with the incompressible finite elements,
·
to define the characteristics of the materials (Young modulus
E
y
, Poisson's ratio
near to
0 5
,
to ensure the operation of the incompressible finite elements [R3.06.03],
limit of resistance
y
and coefficient of Norton-Hoff
N
), the coefficient of shearing is
deduced:
2
2
3
µ =
E/
. It should be noted that the limiting load is independent of
E
and
,
·
to define the permanent loading and that which is parameterized by
,
·
to define the standardization of the power of the loading parameterized (key word
LIAISON_CHAMNO
control
AFFE_CHAR_MECA
),
·
to carry out a non-linear calculation with the relation of behavior of Norton-Hoff with
order
STAT_NON_LINE
[U4.32.01],
·
post-to treat calculation to obtain the load limits with the control
POST_ELEM
[U4.61.04].
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
8/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
2.2
Relation of behavior of Norton-Hoff in
STAT_NON_LINE
The problem is written in variational form in the following way:
For one
(
)
N
m
=
-
1
1
/
given, to find
(
)
()
m
m
m
has
p
,
,
L
U
×
×
R
V
2
such as:
()
() ()
(
)
()
()
()
()
With m
D
p
D
Q
D
Q
m
m
m
m
m
m
has
m
m
.
.div
()
.div
L
(
)
U. U
U. v
v
v
v
v
U
U
-
+
-
=
=
=
2
0
2
0
1
L
L
V
L
éq 2.2-1
This problem admits a single solution for all
N
1
(see [bib4]). For
m
=
2
or
N
=
1
the problem
is of incompressible linear elasticity type.
One thus obtains an estimate of the limiting load by higher value, the field
U
m
giving one
idea of a mode of ruin.
For the processing of the incompressibility, one will refer to the document [R3.06.05]. The treated equation is
in fact:
()
Q
D
Q p D
Q
.div
.
,
L
U
+
=
0
2
, the term
Q p D
.
allowing to avoid putting
at fault the solvor employed and agent with a choice of the Poisson's ratio
=
0 4999
.
…
solutions are thus only quasi-incompressible.
The principle of operation of the general algorithm of
STAT_NON_LINE
is described precisely in
[R5.03.01]. One leads to the following incremental problem:
To find
(
)
,
,
U p
×
×
R
V
Q
0
such as:
(
)
()
(
)
()
(
) ()
(,
)
()
(
,)
U
U. v
B v
v
v
v
B U
U
D
D
v
+
+
+
-
+
=
+
+
=
+
=
D
p
p
Q
Q
L
L
L
V
Q
L
1
0
0
0
1
1
·
B
who is a linear operator contains the boundary conditions homogeneous of Dirichlet,
incompressibility,
·
D
described the data imposed on the solution (boundary conditions of Dirichlet, incompressibility),
·
L
0
is the permanent loading and
L
1
the loading controlled by the parameter
,
·
V
0
and
Q
are spaces of functions discretized on the basis of finite element, and are thus
defined by a vector
(,)
U P
degrees of freedom.
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
9/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
The tensor of the stresses
()
U
check the relation of behavior of Norton-Hoff. The diverter of
stresses associated at the speed of deformation is:
()
()
(
)
()
()
(
)
D
D
D
m
D
D
N
D
D
N
D
With m
With m
U
U.
U
U
U
.
U
.
()
()
.
()
.
.
()
=
=
-
-
2
1
éq 2.2-2
The problem is solved by the method of Newton, after direct implicit discretization of the relations
of behavior [R5.03.02].
The phase of prediction consists in solving the following system, starting from the current state
(U, p)
for
to obtain the reiterated first:
(
,
)
U
0
0
p
.
()
(
)
()
(
) ()
(,
)
()
(
,)
U
.
U. v
B v
v
v
v
B U
D
U
U
D
D
D
p
Q
Q
O
0
0
1
0
0
1
0
1
+
-
=
=
+
=
L
L
V
Q
L
The option
RIGI_MECA_TANG
provides the tangent operator
()
U
D
D
.
One uses the tangent rigidity, applied to a tensor deviatoric
E
:
()
(
)
(
)
D
D
With R
R
D
D
D
D
R
D
D
D
D
.
.
()
()
.
()
()
()
().
E
U.
U
E
U
U
U.
U
E
=
+
-
-
2
2
Then one treats the phase of correction, for the iteration
I
:
(
) (
)
()
(
)
()
[
]
(
)
()
(
)
D
D
D
p
D
p
Q
Q
Q
I
I
I
I
O
I
I
I
(
) ()
,
()
(
) ()
,
,
,
(
)
.
U
U. v
B v
v
v
U U. v
B v
v
B U
D
D B U
U U
U
U
+
+
+
+
+
-
+
- +
=
+
-
+
+
=
+
-
+
=
1
1
1
0
0
1
1
1
1
L
L
L
V
Q
L
The option
RAPH_MECA
provides the second member
(
) ()
U
U. v
+
I
D
. The aforementioned is built by call
with the subroutine of Code_Aster of the type
NICOMP
, which calculates the stress field suitable for
law of behavior used. The option
FULL_MECA
provides in more the tangent operator
(
)
U
U
+
I
D
D
. One
can decide updating of this operator with certain iterations
I
only, to avoid one
too frequent expensive assembly.
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
10/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
In our case, the resolution can be made in a nonincremental direct way, but there is interest with
to reactualize the tangent rigidity of time to other to accelerate convergence. The solvor to be used
is: `
LDLT
', while having taken care to have specified that standardization is placed at the end of the system
(key word:
NUME_LAGR
: `
AFTER
'in
LIAISON_CHAMNO
).
One can also decide to make recoveries, starting from a solution
(,)
U p
obtained previously,
even coming from a resolution for another parameter
N
(calculations less expensive and better
convergence).
In all the cases, it is advised to start to make a calculation on a coarse mesh for
to evaluate the effect of the parameter
N
on
m
.
2.3
Postprocessing of the calculation of the limiting load
Having obtained the solution
(
)
m
m
m
p
,
,
U
, for one
N
given, it remains to use the continuation of
m
for
to build the approximation of the limiting load. For that one exploits the properties [éq 1.3-2], [éq 1.3-3],
the fact that
()
With m
is increasing and the property resulting from minimization [éq 1.3-1] (see [bib7]).
Of these two last, with
1
R S
, one deduces that for and
U
R
and
U
S
respective solutions (checking
also the condition of incompressibility and standardization) of [éq 1.3-1] for
m R
=
and
m S
=
:
()
(
)
(
)
()
(
)
(
)
With R
D
With S
D
R
R
R
S
S
S
(
) (
)
(
) (
)
/
/
U. U
U. U
2
2
Associated the property [éq 1.3-2], one fires for
1
R S
:
()
()
(
)
(
)
()
(
)
(
)
With R
D
With R
D
With S
D
R
R
R R
R
R
R
R
S S
S
S
S
S
(
) (
)
(
) (
)
(
) (
)
/
/
U. U
U. U
U. U
-
-
1 1
2
1
1 1
2
1
éq 2.3-1
The terms are noted
~
m
continuation, which one calculates in practice by postprocessing using
U
m
(
external power being unit:
()
(
)
(
)
~
(
) (
)
(
)
/
m
m m
m
m
m
m
m
With m
D
=
-
-
1 1
2
1
0
U. U
U
L
éq 2.3-2
This continuation
~
m
is thus decreasing for
N
+
and it is shown [bib7] that it converges towards
lim
,
what allows a good control. As one can undervalue (knowing that
()
With
y
1
2
3
=
) the first term
of [éq 2.3-1]:
lim
(
) (
)
(
)
~
-
y
m
m
m
m
D
2
3
0
U. U
U
L
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
11/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
the continuation is thus calculated
!
m
decreasing for
N
+
(parameter '
CHAR_LIMI_SUP
'of the table
exit of
POST_ELEM
[U4.61.04]) and also converging towards
lim
,
lim
!
(
) (
)
(
)
=
-
m
y
m
m
m
D
2
3
0
U. U
U
L
éq 2.3-3
One judges quality of the approximation of the limiting load
lim
by comparison of different
values of
!
m
who converge towards
lim
by excess (in
N
+
). These terms are calculated by
numerical integration at the points of Gauss of the finite elements.
Another interpretation of the interest which this continuation brings lies in the fact that it exploits
directly the expression of the function of support of convex of resistance, i.e. power
dissipable in the modes of potential ruin, applied to the incompressible and standardized solutions
calculated
U
m
.
If the permanent loading (parameter '
PUIS_PERMANENTE
') is null:
L
0
0
=
, one can
easily to exploit the stress field (almost statically acceptable) calculated with
solution
U
m
and to obtain a value by estimate of the limiting load, which would be him would necessarily be
lower if balance were checked exactly (see [bib4]). The continuation is thus calculated
m
(parameter
'
CHAR_LIMI_ESTIMEE
'of the table resulting from
POST_ELEM
[U4.61.04]), which does not have on the other hand
properties of monotony:
()
() ()
(
)
() ()
m
m
m
m
D
m
D
m
y
m
With m
m
D
=
-
.
. Sup
!
U. U
U.
U
X
3
2
1
éq 2.3-4
This maximization (of the function called gauge of convex of resistance) is calculated only with
points of Gauss of the finite elements. Also the value obtained, for each
m
, lower than
!
m
[bib4],
can be regarded only as one indication.
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
12/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
3
An example of validation
3.1
Problem of reference
A rectangular plate or a hexahedron or an axisymmetric cylinder are considered. The criterion of
resistance of homogeneous constitutive material checks the criterion of von Mises (with for threshold
y
).
structure is subjected to pressures on the horizontal edge
-
F
and on the vertical edge
(
)
- -
1
F
with
0.5
. On this very simple problem, an analytical calculation makes it possible to obtain
charge exact limit in the direction of the loading, as well as the estimates produced by
method of regularization. For more details one will refer to [bib4] and [bib5]. This example of
validation corresponds to test SSNV124 [V6.04.124].
The geometry is defined by:
·
Interior radius:
has
mm
=
1
,
·
External radius:
B
mm
=
2
, thickness
B has
mm
- =
1
,
·
Height
:
H
mm
=
4
.
Z or y
R or X
With
B
D
C
B
has
H
0
3.2 Case
plan
The structure is subjected to pressures on the edges horizontal:
-
F
and vertical:
-
1
-
(
)
F
, with:
1/2
, and one exerts a blocking in
Z
. One considers two ways of controlling the loading:
·
cas1: the two pressures horizontal and vertical are parameterized by
,
·
cas2: the horizontal pressure is parameterized by
, while the vertical pressure is
constant
(
)
- -
1
0
F
, with
F
F
0
0
=
.
3.2.1 Solution analyzes limit of it
The solution is homogeneous (biaxées stresses
:
xx
=
F,
yy
=
1
-
(
)
F
,
xy
=
0
,
plane deformations
). One obtains [bib4] the limiting load in these directions of loading, for
criterion of von Mises, in plane deformations, with the threshold
y
:
cas1:
lim
.
.
F
y
=
-
2 3
3 2
1
éq 3.2.1-1
cas2:
lim
.
.
.
F
F
y
=
+ -
2 3
3
1
0
éq 3.2.1-2
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
13/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
It is checked that if one takes
0
=
lim
in the cas2, one finds the cas1 then.
3.2.2 Solution analyzes regularized limit of it
The solution is homogeneous. The plane deformations are necessarily of the form:
()
() ()
U
U
U
=
=
-
1
0
0
0
1 0
0
0
0
2
;
.
éq 3.2.2-1
By the law of Norton-Hoff, one obtains the deviatoric stresses:
()
()
D
m
m
D
VM
m
m
With m
With m
=
=
-
-
-
-
-
2
2
3
2
2
1
0
0
0
1 0
0
0
0
2
1
.
;
éq 3.2.2-2
The standardization of the loading leads to:
cas1:
(
) (
)
F
H B has
=
-
-
1
2
1
éq 3.2.2-3
cas2:
(
)
F
H B has
=
-
1
éq 3.2.2-4
Terms of the continuation
!
m
of limiting load in these two parameter settings of the loading are then:
cas1:
! .
.
m
y
F
m
=
-
2 3
3 2
1
éq 3.2.2-5
cas2:
! .
.
.
m
y
F
F
m
=
+ -
2 3
3
1
0
éq 3.2.2-6
Invariance according to
m
observed here (what is a particular case) results owing to the fact that one is
in an isostatic situation. In the cas1, one can also exploit the continuation of
m
:
cas1:
m
y
F
m
.
.
=
-
2 3
3 2
1
éq 3.2.2-7
One thus obtains the limiting load
lim
exact when
m
+
1
.
3.3 Case
axisymmetric
In axisymmetric 2D one considers the same geometry, but the solid, on which one imposes a bloquage
axial complete, is only subjected to a pressure on the internal wall:
F
parameterized by
.
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
14/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
3.3.1 Solution analyzes limit of it
One obtains [bib5] the limiting load in this direction of loading, for the criterion of von Mises, in
axisymmetric and null axial deformations, with the threshold
y
:
lim
.
ln
F
B
has
y
=
2 3
3
éq 3.3.1-1
3.3.2 Solution analyzes regularized limit of it
The solution is homogeneous. Displacement being only radial, the isochoric deformations are
necessarily of the form:
()
()
() ()
U R
R
R
R
R
=
=
=
-
;
;
.
U
U
U
2
1 0 0
0
0 0
0
0 1
2
2
éq 3.3.2-1
By the law of Norton-Hoff, one obtains the deviatoric stresses:
()
()
D
m
m
m
D
VM
m
m
m
With m
R
With m
R
=
=
-
-
-
+
-
-
-
-
+
2
2
3
2
2
2
2
1 0 0
0
0 0
0
0 1
2
1
2
2
.
.
;
.
.
éq 3.3.2-2
The equilibrium equations axial and radial result in determining the mean stress:
()
()
tr
.
.
.
R
With m
R
m
m
m
m
m
=
-
- +
-
-
-
+
3
2
2
1
3
2
2
2
2
éq 3.3.2-3
where
is a constant, which is calculated starting from the boundary condition of null pressure in wall
external. The components of the stresses then are obtained:
()
(
)
()
(
)
(
)
()
(
)
(
)
()
(
)
()
rr
m
m
zz
m
m
m
m
m
m
R
B
R
R
B
m R
R
B
m R
with
With m
m
fH
=
-
=
- -
=
- -
=
-
-
+
-
+
-
+
-
+
-
+
-
+
-
-
2
2
2
2
2
2
2
2
2
2
2
2
2
1
2
3 2
2
1
:
éq 3.3.2-4
The standardization of the loading leads to:
F
H
=
1
.
Terms of the continuation
!
m
of limiting load for this loading are then:
! .
ln
m
y
has
B
y
F
H R rdr
B
has
m
=
=
2 3
3
2 3
3
2
éq 3.3.2-5
Terms of the continuation
m
of limiting load for this loading are:
] [
(
)
(
)
m
y
m
has
B
B has
m
y
m
m
m
F
m
R
rdr max R
m
m
B
has
has
.
.
,
=
=
-
-
-
-
+
-
-
+
-
+
-
+
2 3
3
3
3 1
2
2
2
1
2
2
2
2
2
2
éq 3.3.2-6
In
m
+
1
, one finds:
1
2 3
3
+
=
.
ln
F
B
has
y
, i.e. the same value as
!
m
and
lim
.
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
15/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
3.4 Case
three-dimensional
In 3D one considers the same geometry, but the solid, unit thickness, is free in the direction
antiplane
Z
. The solid is subjected to pressures on the walls horizontal:
-
F
and vertical:
-
1
-
(
)
F
, with:
1/2
. The two pressures horizontal and vertical are parameterized by
.
3.4.1 Solution analyzes limit of it
The solution is homogeneous (biaxées stresses
:
xx
=
F,
yy
=
1
-
(
)
F
,
xy
=
0
,
zz
=
0
,
deformations
). One obtains the limiting load in this direction of loading [bib5], for the criterion
of von Mises, with the threshold
y
:
lim
. F
y
=
-
+
3
3
1
2
éq 3.4.1-1
3.4.2 Solution analyzes regularized limit of it
The solution is homogeneous. The isochoric deformations are necessarily of the form:
()
() ()
(
)
U
U
U
=
=
+ +
- -
1 0
0
0
0
0 0
1
2
2 1
;
.
éq 3.4.2-1
By the law of Norton-Hoff, one obtains the deviatoric stresses:
(
)
()
(
)
D
D
VM
m
m
with
With m
=
=
+ +
=
+ +
- -
-
-
.
;
:
1 0
0
0
0
0 0
1
2
2
2
2
3 1
2 1
éq 3.4.2-2
One deduces from
zz
=
0
:
(
)
tr
= -
+
3 1
. From where stresses:
=
+
+
.
2
0
0
0
1 2
0
0
0
0
.
The balance of the solid imposes that
(
)
xx
yy
.
.
1
-
=
. One deduces the parameter from it
=
-
-
3
2
1 3
.
The standardization of the loading leads to:
(
)
(
)
(
)
F
H B has
=
-
+
-
1
1
éq 3.4.2-3
Terms of the continuation
!
m
of limiting load in this case of loading are thus identical to:
(
)
(
)
! .
.
m
y
y
F
=
+ +
+
-
=
-
+
2 3
3
2 1
1
3
3
1
2
2
éq 3.4.2-4
Code_Aster
®
Version
4.0
Titrate:
Calculation of load limits by the method of Norton-Hoff-Friaâ
Date:
12/11/97
Author (S):
F. VOLDOIRE, E. SCREWS
Key:
R7.07.01-A
Page:
16/16
Manual of Reference
R7.07 booklet: Analyze limit
HI-74/97/023/A
4 Bibliography
[1]
ANGLES J., VOLDOIRE F., Modeling and calculation of the load limits of a fissured component,
CR-MN 1522-07, Sept. 96.
[2]
FRIAA A., Law of Norton-Hoff generalized in plasticity and viscoplasticity, Thesis of doctorate,
1979.
[3]
FRIAA A., FREMOND Mr., the methods statics and kinematics in design the collapse and in
analyze limit, Journal of Mechanics theoretical and applied, vol. 11, NO5, 881-905, 1982.
[4]
VOLDOIRE F., Design the collapse and analyze limit of the structures, notes EDF HI-74/93/082.
[5]
VOLDOIRE F., Analyzes limit of the fissured structures and criteria of resistance, notes
EDF/DER HI-74/95/026.
[6]
SCREW E., VOLDOIRE F., finished Eléments incompressible, note EDF/DER HI-75/95/019.
[7]
VOLDOIRE F., Implemented of the method of regularization of Norton-Hoff-Friaâ for
the analysis limits structures, notes EDF/DER HI-74/97/026.