Code_Aster
®
Version
3.0
Titrate:
Estimate of fatigue under random stress
Date:
24/05/96
Author (S):
A. Mr. DONORE, P. MORILHAT
Key:
R7.04.02-A
Page:
1/8
Manual of Reference
R7.04 booklet: Evaluation of the damage
HI-75/96/034/A
Organization (S):
EDF/IMA/MN, EP/REME
Manual of Reference
R7.04 booklet: Evaluation of the damage
Document: R7.04.02
Estimate of fatigue under random stress
Summary:
This note presents two methods of counting of cycles of stresses which lead to an expression
analytical of the mechanical damage generated by a random loading:
·
method of counting of the peaks of stresses,
·
method of counting of the goings beyond of a given level.
The first method calls upon the signal, its derivative first and its derivative second. The second requires
only the knowledge of the signal and its derivative first.
The cycles of stresses being known by these methods, one determines the average damage over the duration of
signal, using the method of Wöhler.
Code_Aster
®
Version
3.0
Titrate:
Estimate of fatigue under random stress
Date:
24/05/96
Author (S):
A. Mr. DONORE, P. MORILHAT
Key:
R7.04.02-A
Page:
2/8
Manual of Reference
R7.04 booklet: Evaluation of the damage
HI-75/96/034/A
Contents
1 Introduction ............................................................................................................................................ 3
2 Evaluation of the damage ........................................................................................................................ 4
2.1 Diagram of endurance ................................................................................................................. 4
2.2 Elastoplastic coefficient of concentration .................................................................................. 4
3 a Number of cycles of stresses ........................................................................................................... 5
3.1 Recalls: spectral moments and factor of irregularity .................................................................... 5
3.2 Method of counting of the peaks of stresses ............................................................................... 5
3.3 Method of counting of the goings beyond of level ..................................................................... 7
4 statistical Estimate of the damage ....................................................................................................... 7
4.1 Average damage ............................................................................................................................ 7
5 Conclusion ............................................................................................................................................. 8
6 Bibliography .......................................................................................................................................... 8
Code_Aster
®
Version
3.0
Titrate:
Estimate of fatigue under random stress
Date:
24/05/96
Author (S):
A. Mr. DONORE, P. MORILHAT
Key:
R7.04.02-A
Page:
3/8
Manual of Reference
R7.04 booklet: Evaluation of the damage
HI-75/96/034/A
1 Introduction
The evaluation of the damage is based on the use of curves of endurance of material, associating one
variation of stresses of amplitude given to a number of acceptable cycles defined by a curve
of fatigue.
The curves of fatigue of material are established by subjecting test-tubes to stresses
sinusoidal of constant amplitude since the beginning of the test until the rupture.
To use these curves starting from a real loading, it is necessary to identify cycles in
the history of the stresses, which is done by methods of counting of cycles. Many
methods exist: the document [R7.04.01] presents two methods of counting of cycles in
case of a deterministic real loading.
However, of many real mechanical loadings affecting the nuclear components
present a randomness which results in privileging the use of statistical methods for
to evaluate the damage undergone by these structures.
Certain methods of counting of cycles of stresses were the interpretation object
statistics:
·
method of counting of the peaks of stresses
·
method of counting of goings beyond of a given level.
The field of application of these two methods [bib1] [bib2] is limited to random loadings
ergodic (the analysis of only one sample is enough to characterize the parameters of the process) and
Gaussian (the values of the measured signal are distributed according to a normal law).
In addition, the evolution of the signal is comparable with a random process characterized by its
statistical parameters (spectral moments of command 0, 2 and 4) [R7.10.01].
In both cases, the statistical event to take into account is simple to define:
·
a peak of stresses is defined by a null slope and a negative acceleration for a peak
positive, a positive acceleration for a negative peak,
·
a going beyond of level of stresses S
0
is characterized by a value of the signal equalizes with
S
0
and by a positive slope.
The cycles of stresses being known by these methods, one passes then to the calculation of the number of
cycles with the rupture starting from a curve of fatigue. The curves of Wöhler which are established
in experiments were approached by various mathematical expressions characterizing
more or less correctly various areas of these curves.
Three mathematical expressions are available in Code_Aster: a discretized form and two
analytical forms.
Knowing the number of cycles of stresses (given by one of these two methods of counting
cycles) and elementary damage associated (determined by interpolation on the curve with Wöhler with
material), one can calculate an average damage over the duration of the signal.
Code_Aster
®
Version
3.0
Titrate:
Estimate of fatigue under random stress
Date:
24/05/96
Author (S):
A. Mr. DONORE, P. MORILHAT
Key:
R7.04.02-A
Page:
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R7.04 booklet: Evaluation of the damage
HI-75/96/034/A
2
Evaluation of the damage
For a structure without defect geometrical and subjected to a pure alternate stress, the number of
cycles with the rupture is given starting from a diagram of endurance, still called curve of
Wöhler or curve SN.
The number of cycles to the rupture is determined by interpolation of the curve of Wöhler of material
for a level of alternate stress (unidimensional) given (to each elementary cycle
a level of amplitude of stress corresponds
=
-
max
min
and an alternate stress
S
alt
=
1 2
/
).
The damage of an elementary cycle is equal contrary to the number of cycles to the rupture:
D
S
NR
R
alt
=
1
(
)
2.1 Diagram
of endurance
The diagram of endurance, also called curve of Wöhler or curve SN (curve forced
a number of cycles to the rupture) is obtained in experiments by subjecting test-tubes to
periodic cycles of efforts (generally sinusoidal) of normal amplitude
and of frequencies
constants, and by noting the number of cycles
R
NR
with the end of which the rupture occurs [R7.04.01].
The various mathematical shapes of the curve of Wöhler are described in the document
“Estimate of fatigue to great numbers of cycles”, [R7.04.01] as well as the way of introducing them
in Code_Aster.
2.2
Elastoplastic coefficient of concentration
It can also be necessary to balance the value of the stress determined by the method of
counting by the elastoplastic coefficient of concentration
K
E
.
The elastoplastic coefficient of concentration Ke (aimed to the B3234.3 articles and B3234.5 of the RCC_M
[bib4]) is defined as being the relationship between the amplitude of real deformation and the amplitude of
deformation determined by the elastic analysis.
The value of the coefficient
K
E
is given in the document [R7.04.01].
Code_Aster
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Version
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Titrate:
Estimate of fatigue under random stress
Date:
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Author (S):
A. Mr. DONORE, P. MORILHAT
Key:
R7.04.02-A
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R7.04 booklet: Evaluation of the damage
HI-75/96/034/A
3
A number of cycles of stresses
3.1
Recalls: spectral moments and factor of irregularity
One calls spectral moment of command
I
following quantity [R7.10.01]:
()
I
I
S
G
D
=
-
+
where
is the pulsation and
G
S
spectral concentration of power or DSP.
One has in particular:
0
2
2
2
4
2
=
=
=
S
S
S
who are the standard deviations of
S
and of its
first derived.
The factor of irregularity translates the frequential pace of the signal. Ranging between 0 and 1, it tends towards 1
when the process is with narrow tape, on the other hand it tightens towards 0 for a broad band process.
Its expression is:
I
S
S S
=
=
'
''
2
2
2
0
4
We point out these definitions because the evolution of the signal is comparable with a random process
characterized by its statistical parameters (spectral moments of command 0, 2 and 4).
For the method of counting of the peaks of stresses, the random signal is entirely characterized
by the three spectral moments of command 0, 2 and 4.
In the case of the method of counting of the goings beyond of level, the spectral moments of command
0 and 2 are enough to characterize the random signal.
In a practical way, these values are determined by the control
POST_DYNA_ALEA
[U4.76.02] which
operate statistical processing on a random loading. The definition of the various parameters
is given in the document [R7.10.01].
The operator of fatigue analysis in the random field
POST_FATI_ALEA
[U4.67.05] uses them
three spectral moments values calculated by
POST_DYNA_ALEA
and calculates the average damage and
the standard deviation of the damage by the methods described in this document.
3.2
Method of counting of the peaks of stresses
The principle of this method consists in counting the local maxima (in absolute value) located
leaves and other of the average value of the stresses.
The stationary Gaussian signal, being centered compared to its average value, the distribution of the peaks
is symmetrical compared to this average. One is thus interested in the distribution of the positive peaks.
In the general case, the distribution of the peaks of amplitude
S
positive is written in the form:
Code_Aster
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()
(
)
()
P
S
I
I
E
IS
E
I
S
I
I
peak
S
S
I
S
S
E
S
S S
S
S
S
T
dt
+
-
-
-
=
+
-
+
=
=
-
-
-
2
2
1
1
1
2
2
1
2
2
2
2
2
2
2
2
2
2
with
'
“
This distribution of the positive peaks is in the case of simplified the signals for which the factor
of irregularity is worth
I
I
=
=
0
1
or
. :
·
Signal with broad band: law of Gauss or normal law
(
)
I
=
0
()
P
S
E
peak
S
S
S
+
-
=
2
2
2
2
2
2
·
Signal with narrow tape: law of Rayleigh
(
)
I
=
1
()
P
S
S.E.
peak
S
S
S
+
-
=
2
2
2
2
The method of counting of peaks of stresses associates each peak of positive amplitude
S
a cycle
of amplitude
S
= 2
S
(one thus has directly
S
Salt
=
).
The number of peaks of positive amplitude is written:
()
()
N
S
P
S
NR
peak
peak
peak
+
+
+
=
×
where
NR
I
peak
S
S
+
=
+ ×
1
4 1
1
(
)
“
'
= an average number of positive peaks per unit of time.
From where the number
NR
cycles to be taken into account is:
NR S
I
P
S
S
S
peak
()
()
“
'
= +
+
1
4
Note:
It is noticed well that the expression of the number
NR
cycles to be taken into account does not depend
that of
S
(for the calculation of the factor of irregularity
I
),
S
'
and
S
“
.
Code_Aster
®
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Titrate:
Estimate of fatigue under random stress
Date:
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Author (S):
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Key:
R7.04.02-A
Page:
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R7.04 booklet: Evaluation of the damage
HI-75/96/034/A
3.3
Method of counting of the goings beyond of level
This method requires the variation division of stress in classes of amplitude.
The number of cycles
()
NR S
is obtained starting from the difference of the numbers of goings beyond of
level with a positive slope between two successive classes, on the basis of the class of amplitude
maximum.
For a Gaussian process centered the expression of
NR S
()
is:
NR S
S.E.
S
S
S
S
()
'
=
-
1
2
3
2
2
2
Note:
The expression of the number
NR
cycles to be taken into account requires only knowledge
of
S
and
S
'
(independence with respect to
S
“
).
In this method the coefficient of irregularity does not intervene
I
.
4
Statistical estimate of the damage
The mechanical damage is calculated by using the linear rule of office plurality To mine.
Damage
D
generated by
NR
cycles of half amplitude
S
express yourself by
D
NR
NR S
=
()
where
NR S
()
is the acceptable number of cycles determined by the curve of endurance of material.
The mechanical damage is a random variable which one determines the average.
4.1 Too bad
means
The average damage is written in the form of the expectation:
E D
T
NR S
NR S
S
S
dS
R
()
()
()
min
max
=
where
T
is the duration of the signal
The two methods of counting suggested calculate the number of cycles
()
NR S
from
stresses of positive amplitude from where
min
S
= 0 (except when the curve of Wöhler is given under
form “current area”, in which case
min
S
=
S
L
with
S
L
limit of endurance of material).
In addition, laws of distributions used being continuous,
max
S
=
. However, the experiment
show that the expression to be integrated attenuates quickly and one thus takes
max
S
= 10.
S
where
S
is
the standard deviation of the signal.
The calculation of
E D
()
is realized by numerical integration, by the method of the trapezoids while taking
for pitch of integration
S
S
max
min
.
-
300
.
Note:
In the case of the method of counting of goings beyond of level and for a curve of
Wöhler expressed in the mathematical form suggested by Basquin, the average damage by
unit of time to an analytical expression (this expression is not used in
order
POST_FATI_ALEA
).
Code_Aster
®
Version
3.0
Titrate:
Estimate of fatigue under random stress
Date:
24/05/96
Author (S):
A. Mr. DONORE, P. MORILHAT
Key:
R7.04.02-A
Page:
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R7.04 booklet: Evaluation of the damage
HI-75/96/034/A
5 Conclusion
In the case of a Gaussian loading, ergodic and stationary, two methods of counting of
cycles find an interpretation statistical and provide an analytical expression of the damage
means, utilizing only the formulation of the curve of endurance of material and standard deviations
signal and of its derivative first and seconds.
In Code_Aster, the calculation of the damage under random stress is carried out by the control
POST_FATI_ALEA
[U4.67.05].
The user can determine the damage by the method of counting of the peaks of stresses
(
COUNTING: “PEAK”
) or by the method of counting of goings beyond of a given level
(
COUNTING: “LEVEL”
).
According to the adopted method, the random signal will have to be introduced by the data of the moments
spectral of command 0 and 2 or by the data of the spectral moments of command 0, 2 and 4 (key words
MOMENT_SPEC_0
,
MOMENT_SPEC_2
and
MOMENT_SPEC_4
). Values of the spectral moments
can also be recovered in a table created by
POST_DYNA_ALEA
[U4.76.02] (key word
COUNT
).
The curve of Wöhler of material can be introduced in three distinct forms (in accordance with
order
POST_FATIGUE
[U4.67.01] (calculation of the damage to great numbers of cycles) and software
POSTDAM).
The given size is the average damage over the duration of the signal which is stored in a table
of type
POST_FATI_ALEA
.
6 Bibliography
[1]
P. MORILHAT: Thermal faience manufacturing: Estimate of the mechanical damage. Note
HP-14/90.07
[2]
P. MORILHAT: Random mechanics: Statistical estimate of the mechanical damage
generated by nonstationary loadings. Note HP-14/91.19A
[3]
A. DUMOND: Operator
POST_DYNA_ALEA
[R7.10.01]
[4]
RCC_M Edition 1983