Fatigue analysis of structures during random loadings
We have developed and described a special approach for long life fatigue analysis of welded structures under random multi-axial loadings, which has a real physical meaning and which is very easy to perform. The principal idea is to find an equivalent simple loading suitable to represent the real ran...
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Інститут проблем міцності ім. Г.С. Писаренка НАН України
2009
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| Назва видання: | Надійність і довговічність машин і споруд |
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| Цитувати: | Fatigue analysis of structures during random loadings / J. Zarka, H. Karaouni // Надійність і довговічність машин і споруд. — 2009. — Вип. 32. — С. 18-34. — Бібліогр.: 6 назв. — англ. |
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irk-123456789-369082012-08-05T12:09:05Z Fatigue analysis of structures during random loadings Zarka, J. Karaouni, H. We have developed and described a special approach for long life fatigue analysis of welded structures under random multi-axial loadings, which has a real physical meaning and which is very easy to perform. The principal idea is to find an equivalent simple loading suitable to represent the real random loading Разработан и описан специальный подход для анализа длительной усталости сварных конструкций при случайных многолетних аксиальных нагрузках, который имеет реальный физический смысл, и который очень легко выполнить. Основная идея заключается в том, чтобы найти подходящий эквивалент простой нагрузки, представить реальную случайную нагрузку с точки зрения повреждения на уровне материала и затем на уровне структуры. Розроблено і описано спеціальний підхід для аналізу тривалої втоми зварних конструкцій при випадкових багаторічних аксіальних навантаженнях, що має реальний фізичний зміст, і який дуже легко виконати. Основна ідея полягає в тому, щоб знайти підходящий еквівалент простого навантаження, представити реальне випадкове навантаження з точки зору пошкодження на рівні материалу і потім на рівні структури. 2009 Article Fatigue analysis of structures during random loadings / J. Zarka, H. Karaouni // Надійність і довговічність машин і споруд. — 2009. — Вип. 32. — С. 18-34. — Бібліогр.: 6 назв. — англ. 0206-3131 http://dspace.nbuv.gov.ua/handle/123456789/36908 539.4 en Надійність і довговічність машин і споруд Інститут проблем міцності ім. Г.С. Писаренка НАН України |
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We have developed and described a special approach for long life fatigue analysis of welded structures under random multi-axial loadings, which has a real physical meaning and which is very easy to perform. The principal idea is to find an equivalent simple loading suitable to represent the real random loading |
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Article |
| author |
Zarka, J. Karaouni, H. |
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Zarka, J. Karaouni, H. Fatigue analysis of structures during random loadings Надійність і довговічність машин і споруд |
| author_facet |
Zarka, J. Karaouni, H. |
| author_sort |
Zarka, J. |
| title |
Fatigue analysis of structures during random loadings |
| title_short |
Fatigue analysis of structures during random loadings |
| title_full |
Fatigue analysis of structures during random loadings |
| title_fullStr |
Fatigue analysis of structures during random loadings |
| title_full_unstemmed |
Fatigue analysis of structures during random loadings |
| title_sort |
fatigue analysis of structures during random loadings |
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Інститут проблем міцності ім. Г.С. Писаренка НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/36908 |
| citation_txt |
Fatigue analysis of structures during random loadings / J. Zarka, H. Karaouni // Надійність і довговічність машин і споруд. — 2009. — Вип. 32. — С. 18-34. — Бібліогр.: 6 назв. — англ. |
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Надійність і довговічність машин і споруд |
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AT zarkaj fatigueanalysisofstructuresduringrandomloadings AT karaounih fatigueanalysisofstructuresduringrandomloadings |
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2025-07-03T18:38:48Z |
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| fulltext |
ISSN 0206-3131. Надійність і довговічність машин і споруд, 2009. Вип. 32 18
УДК 539.4
2009 J. Zarka and H. Karaouni
CADLM, Gif Sur Yvette, France
FATIGUE ANALYSIS OF STRUCTURES DURING RANDOM LOADINGS
We have developed and described a special approach for long life fatigue analysis of
welded structures under random multi-axial loadings, which has a real physical meaning
and which is very easy to perform. The principal idea is to find an equivalent simple
loading suitable to represent the real random loading in terms of damage produced at the
level of the material then at the level of the structure.
Keywords: fatigue analysis, structure, random loading, software, welded structures.
Abstract. We have developed and a special approach for long life fatigue
analysis of welded structures under random multi-axial loadings. During classical
analysis, a linear or non-linear finite element simulation is usually performed, and
then eventually with some special methods to extract and to count the cycles from
the random stress path (very often, the Rainflow method), it is needed to cumulate
"damage" based on some S-N diagrams and the linear Miner-Palmgreen rule or
some more elaborated non linear rules. The "damage" factor (scalar or tensor) is
still the object of many researches; it is impossible to measure it during the whole
loading path, even if, before failure; some slight changes on the elastic properties
may be experimentally detected. Moreover, the representations of the loading and
the counting methods are purely based on mathematical aspects and ignore the
particular mechanical behavior of the present materials in the structure.
The objective of our paper is at first to describe an approach which has a real
physical meaning and which is very easy to perform. The principal idea is to find
an equivalent simple loading suitable to represent the real random loading in terms
of damage produced at the level of the material then at the level of the structure.
We introduce a global representation of any loading at the scale of the
material to give the equivalence rule between any two loadings at the scale of the
material and to allow to represent any random loading by an equivalent periodic
loading. We chose as a "measure" of any one-parameter random loading, the
cumulated plastic strain (or equivalently the dissipated energy) at a new very local
level (similarly to what it is done during a seismic analysis). Two loadings are
equivalent if they induce the same cumulated plastic strain (or the same dissipated
energy).
The other objectives refer to that during fatigue analysis:
1) we have to take into account the in certainties on geometry, material
properties, loadings;
2) we have no knowledge of the real initial state;
3) and also, we can not control the errors during numerical simulations or
measurements.
To cross over these difficulties a new framework for fatigue analysis and
reliability of structures has been built which allows the analysis to be reduced to
ISSN 0206-3131. Надійність і довговічність машин і споруд, 2009. Вип. 32 19
the analysis on a 2-D window with a Characterized radial cyclic loading. It is given
one description for the smooth specimen and any general structure with any
loading conditions; simplified analysis of inelastic structures and simple rules for
damage and accumulation are taken but the results are coupled with results from
returns on real structures.
All these procedures and framework were introduced in a Software,
FATPRO, that is developed and distributed by CADLM.
http://www.cadlm.fr/page.php?page=coit_log_fatpro&lang=anglais
Review on the inelastic analysis of structures. We are concerned by a
general structure which is subjected to a general complex cyclic static or dynamical
loading. It is well known that even if there is no collapse or ratcheting or excessive
deformation, after a while, there will be shakedown and then failure of the structure
by fatigue. We have proposed a very simple and practical framework for this
analysis. All details may be found in the book [1].
Loadings on the structure. The structure has the volume with the surface
. It is subjected to the following loading:
– on a part Uj de : given displacements Uj
d(t);
– on the complementary part Fi de : given surface loads Fi
d(t);
– in the volume : specific load Xd(t);
– in the volume : initial strains EI(t).
The full data Ud( ) , Fd( ) , Xd( ) , EI( ) for any [ 0 , t] characterizes the
loading path on the structure.
Purely elastic response. At first the structure is assumed made of linear
elastic materials. At any time t, the elastic response is given by :
– the elastic displacement Uel(t);
– the associated strains Eel(t);
– the elastic stresses el(t) by using any tool ELAS which has the
arguments: ELAS {, Fi, Uj, Xd, Fi
d, Uj
d, EI,M}, where M is the elastic
coefficient matrix (M –1 = L)
el(t) = L(Eel(t) - EI(t)).
Real response. There will be in the structure the fields: U(t),E(t)
Kinematically Admissible with Uj
d(t), on Uj
E(t) =1/2(U(t) + TU(t)).
(t), Statically Admissible with Xd(t) in and Fj
d(t), on Fi
(t) = L(E(t) - EI(t) - Ep(t)).
These fields may be split into one elastic part (previously analyzed) and one
inelastic part:
U(t) =Uel(t) + Uine(t);
E(t) = Eel(t) + Eine(t);
(t) = el(t) + R(t),
where Uine(t), Eine(t) = M R(t) + Ep(t) K.A. with 0 sur Uj; R(t) is S.A. with 0 in
and 0 on Fi.
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These inelastic fields may be obtained, when the plastic strain filed is
known, with: ELAS{ , Fi , Uj ,0,0,0,Ep,M}, which means that:
R Z0Ep,
where Z is an integral operator on the plastic strain field.
Particular case of the cinematic hardening material. This material
associated to the Mises criterion is classically defined by:
f( , y) = f(S-y) =1/2 (S-y)T(S-y) - 0
2,
where S is the deviatory part of the stress tensor and which defines in the stresses
space the moving convex elastic surface C(y) = C0 + y.
The internal parameters are here the plastic strain tensor; the transformed
internal parameters y are linked to Ep by: y =C Ep Ep =C-1 y; C, is the work-
hardening matrix.
The normality flow rule is classically written symbolically: E.p(t) C(y) ( (t))
i.e. external normal cone to C(y) at .
By using el + R the yield criterion may be rewritten:
f( el + R – C Ep)
and then as
f( el - Y),
where Y = C Ep - R is the structural transformed parameters field.
It is obtained with the procedure ELAS (, Fi, Uj, 0, 0, 0, Ep, M),
which means Uine(t), Eine(t), R and then Y.
The main point is that: since at any time el is known independently of the
previous load history, the Y is a locally known position
C(el) = C0 + el !!
Very often, it is possible to guess or to obtain exactly, the actual position of
the structural transformed parameters field Y, but it is, then, necessary to come back
to the classical fields in order to characterize the eventual damage of the structure.
So, if the field Y is assumed known, the same operator with the new
arguments ELAS ( , Fi , Uj , 0, 0, 0, Y/C, M' ) gives R and then Ep. M' is the
modified elastic matrix (L' =M' -1) given by:
M' = (M + C-1dev).
The one to one relation in the structure between the 2 fields Ep, Y is then
elementary to expressed.
During monotonic radial loading, an ultimate state is reached and is
constructed exactly.
During cyclic radial loading, bounds are constructed with elementary rules.
Even during contacts loadings such as successive rolling or impacts (which
are not classical loadings) or dynamical loadings such as seismic loadings, it is
possible to propose special simple rules.
Review on the intelligent optimal design of materials and structures [2].
Automatic learning expert systems. The engineers have to face very important
problems in the design, the test, the survey and the maintenance of their structures.
ISSN 0206-3131. Надійність і довговічність машин і споруд, 2009. Вип. 32 21
These problems did not yet get full answer even from the best people in the world.
Usually in these problems (such as no satisfactory constitutive modeling of materials,
no real control of the accuracy of the numerical simulations, no real definition of the
initial state and/or the effective loading of the structure), there is no solution and the
experts do not understand the problem in its whole. Moreover, the available data may
be not statistically representative (i.e. are in limited number), and may be fuzzy,
qualitative and/or missing in part. Other methods need to be used!
An automatic learning expert systems generator is able to automatically
extract the rules from the raw examples base given by the expert. The experts know
they do not know the full solution but they are able to build an examples base, for
which the solution is known experimentally or numerically. Their main problem is
to provide a good description of such an examples base. (By analogy, we can say
that the data base is the program, the automatic learning tool is the compiler and
the execution gives the knowledge).
Basically such a automatic learning system includes five main functions:
PREPARE: to transform the example files from user format (ASCII, Excel,
...) into the own format of the system and to handle the discretization of the
descriptors and the splitting of the initial data base into a training set and a test set;
LEARN: to automatically extract a rules base from the training set according
to the quality of available information (noise, sparseness of the training set,..) using
Statistical, Symbolic, Numerical, Fuzzy logic, Neural Network or Genetic
Algorithm learning;
TEST: to experimentally evaluate the quality of the extracted rule on the test set;
INCLEAR: to allow the expert to visualize IN CLEAR the extracted rules
with the initial user format and to say what descriptors are the pertinent descriptors
which are kept;
CONCLUDE: from the description of a new case, to deliver a conclusion
based on the extracted rules.
In all problems, it is necessary to consider one conclusion which may be a
class or any continuous real number. Moreover, often, several conclusions may be
considered together. The rules have to be automatically generated for each one of them.
Then an optional but fundamental sixth function, OPTIMIZE, based on
genetic algorithms and other special optimization techniques, may be used to solve
the inverse problem i.e. when some conclusions and some descriptors have to
belong to some given sets (or constraints), what are the possible solutions and in
some particular cases what is the best solution if an objective function is given
(cost, weight)?
Several automatic learning systems are now available. For example, at the
Ecole Polytechnique in France, the LES program has been developed based on
symbolic learning and numerical learning (by M. Sebag and M. Terrien) but also
the Wards systems Group at Frederick, MD in USA, proposes the Neuroshell
program witch is based on neural networks. These two programs are coupled with
optimization add ones.
General principle of the approach. We have defined a new approach where
it is needed:
1) To build a DATABASE of examples i.e. to obtain some experimental,
real or simulated results where the EXPERTS indicate all variables or descriptors
this may take a part. This is, at first, done with some PRIMITIVE descriptors x,
ISSN 0206-3131. Надійність і довговічність машин і споруд, 2009. Вип. 32 22
which are usually in a limited number and which are often in a different number
and type for each example. Then, the data are transformed with the introduction of
some INTELLIGENT descriptors XX, with the actual whole knowledge thanks to
(but often insufficient) beautiful theories and models. These descriptors may be
number, Boolean, strings, names of files which give access to data bases, or
treatments of curves, signals and images. But for all examples, their number and
their type are always the same, which is the only one way to allow the fusion of
data. The results or conclusions may be classes (good, not good, ...) or numbers.
Usually, it is hoped to be possible to get ~50 to 500 examples in the data
base with 10 to 1000 descriptors, 1 to 20 conclusions for each case. This is the
most important (and difficult) task;
2) To generate the RULES with any Automatic Learning Tool. Each
conclusion is explained as function or set of rules of some among the input intelligent
descriptors with a known reliability or accuracy. If this reliability is too low, that
means that there is not enough data or we have bad or missing intelligent descriptors;
3) To optimize at two levels (Inverse Problems).
Considering the intelligent descriptors as independent; it is possible to get
the OPTIMAL SOLUTION satisfying the special required properties and allowing
the DISCOVERY OF NEW MECHANISMS.
Considering the intelligent descriptors linked to primitive descriptors for a
special family; it is possible to obtain the optimal solution that is technologically
possible.
So, not only a Practical Optimal Solution is obtained but also the Experts may
learn the missing parts, may build models or theories based only on the retained
intelligent descriptors and guided by the shapes of the rules or relationships.
General description of mechanical problems. Several tools were built to
describe most of the mechanical problems. It is necessary to describe them in an
"intelligent" way.
Material descriptors. We use classical terms according the problem elastic
coefficients, plastic properties, rupture and fatigue properties. All these descriptors
are obtained on smooth specimen.
Field descriptors. At any given time, any scalar field M varying between
Mmin and Mmax may be characterized with:
1) the interval [Mmin, Mmax] is divided in n equal segments;
2) the relative surfaces in the window for which the scalar M is included in
the interval n, (Sn /S0), where S0 is the surface of the domain.
Very often, we assume that the stress field at a given time is represented by
I1, its first invariant, J2, the second invariant of its deviatory part and the norm of
the gradient of this J2 (the problem being the fatigue life).
It is also necessary to describe the evolution of the stress field with the time.
Hablot [3] introduced:
– 1 = Supt [M / Sup (M)] envelop during the time of the relative M
(included between 0 and 1);
– 2 = Sup t [M] envelop along the time of the absolute value;
– 3 = rr Supt [|| grad {(M) / Sup (M)} ||] envelop of the norm of the gradient
of the relative equivalent stress. This field shows the location of the maxima of the
gradient during the evolution;
ISSN 0206-3131. Надійність і довговічність машин і споруд, 2009. Вип. 32 23
– 4 = rr Supt [|| grad (M) ||]envelop of the norms of the absolute gradient,
which expresses the most important gradients reached during the evolution;
– rr is a characteristic dimension.
Many other quantities were also defined to represent curves, signals and
pictures.
In order to follow our "intelligent" approach for the fatigue analysis, we need
o create a data base of examples/problems. Each case/problem is at first defined by
its primitive description. There is no way, with such a description, to use the results
obtained from one case into another case/problem. The intelligent description has
then to be performed.
Since, we can not rely to the numerical (or experimental) results for the
classical LOCAL dimensioning criteria, we proposed to perform only the cheapest
elastic analysis or our simplified analysis of inelastic structures and to use only the
simplest rules for damage criteria and cumulating of damage for materials but we
shall couple them with the results from returns on real structures.
Fundamental data on materials. On a smooth tensile specimen, it is easy
to perform the following characterizations:
At first, the cyclic curves at ~± 1% of deformation are obtained (this is the
best choice since the first tensile curve is a function of the initial state of the
material, but it implies that the material is cyclically stable) eventually for different
temperatures. Usually, these curves are represented by:
= e+ p= /E+[ /K' ] 1/n' ,
where K' and n' are constant.
Then, the Woehler curves are constructed during cyclic uniaxial loading,
associated to various mean radial stresses and probabilities of failure:
max = a + 0 ,
is the maximum or peak stress, where a is the stress amplitude and 0 is the mean
stress.
As proposed by the experts, they are represented by the stress-life equation:
a= f (2Nf)b-Sd,
or, more practically, as in the Morrow's representation, by the strain-life equation:
/2 = ( f
' - 0)/E(2Nf)b + f
' (2Nf)c.
In this equation, Sd is the endurance limit during alternating cycles; /2 is the
strain amplitude; 2Nf is the number of cycles to failure with the constitutive
constants for the material; f
' is the fatigue strength coefficient; E is the Young
modulus; f
' is the fatigue ductility coefficient; c is the fatigue ductility exponent;
b is the fatigue strength exponent.
Then, the endurance diagram are constructed. A modified Goodman's
diagram is usually employed for a given number of cycles, N, in the plane (m, a),
a bounded domain is defined:
ISSN 0206-3131. Надійність і довговічність машин і споруд, 2009. Вип. 32 24
Rupture after N cycles if : for
max-2 m y - y m ( N- y)
max- m N ( N- y) m 0
max-( 1-r) m N 0 m ( y - N)( 1-r)
max y ( y - N)( 1-r) m y
with r N/u , u is the tensile rupture limit.
Next, the creep time to rupture is extracted.
During complex uniaxial loading, different deterministic mathematical
representations are based on peak-valley or range-mean matrix type to define
groups of constant amplitude cycles, each with a fixed load amplitude and mean
value. Such as cumulative exceeding curves (with the same reduction or counting
of cycles) or sequential variable amplitude histories (usually the rainflow method
of counting the cycles is preferred).
During each of these groups of constant amplitude cycles some "damage" is
induced. Various rules are proposed to cumulate the "damage" Linear ones by
Palgreen-Miner:
Di=Ni/NRi and failure when Di = 1,
(where Ni is the number of cycles realized during the cyclic loading i for which the
number of cycles to rupture is NRi) or Non-linear ones such as those by Lemaitre-
Chaboche.
The probabilistic representation (with the power spectra density to measure
the load amplitude intensity in the frequency range) is often employed during
random (stationary Gaussian) uniaxial loadings; it may be only combined with the
linear cumulating rule for damage. Several experts are thinking that non Gaussian
unixial loadings are more realistic and need special treatments.
During cyclic multi-axial loading (several loading parameters inducing out-
of-phase stresses), the critical plane approach seems to be reasonable:
max (1 + nn
max /y) = (1 + ) f
' /E(2Nf)b + n/2(1 + ) (f
' )2/E y(2Nf)2b +
+(1 + p) f
' (2Nf)c + n/2(1 + p) (f
' f
' )/ y(2Nf)b+c,
where max is the maximum shear strain on the maximum shear strain amplitude
plane; n
max is the maximum stress normal to the maximum shear strain amplitude
plane with the supplementary constitutive constants for the material; is the elastic
Poisson ratio; y is the elastic limit or yield stress; p is the plastic Poisson ratio
(0.5 for metals); n is a material constant to correlate the tensile data to the torsion
data.
For non-cyclic multi-axial loadings, we shall only keep the simplest
endurance criteria such that proposed by Dang Van or Kakuno-Kawada. Generally,
it has the form:
Ca + Pa + Pmean < b,
where Ca is the deviatory stress amplitude Pa, the amplitude and Pmean, the mean
value of the pressure (first invariant of the stress tensor); , and b are some
phenomenological constants of the material.
The Dang Van's criterion is for example, written locally:
ISSN 0206-3131. Надійність і довговічність машин і споруд, 2009. Вип. 32 25
eq = (t)+aDV.p(t) bDV,
where (t) is the local shear stress on a critical slip plane, p(t) is the local pressure
and aDV , bDV are constants identified from endurance limits during alternate flexion
and torsion tests.
These local quantities may be expressed from the global ones by making
some simplifications:
IE = nmax (tmax{ || Calt|| +aDV.P(t)/bDV} ) < 1
where Calt is the alternate part of the tangential stress and P(t) is the global
pressure.
But there are still many open problems:
– what is the validity of these Fatigue/Endurance criteria?
– what is the real physical meaning of the Damage parameter(s)?
– how to extract or represent of the random loading?
– the cumulation rules for Damage?
Indeed, it seems for us, that there is no difference between «GOD» with
several religions and churches and DAMAGE with several models/theories and
schools! That is why, we shall take all of them.
Intelligent representation of any loading. The most difficult part is to
describe the loading. The main idea of the approach is to find an Equivalence rule
between TWO loadings relative to «Damage» which may be used as a
«Quantification» or norm of any loading relative to one particular structure made
of particular materials. This rule needs to give a physical meaning of fatigue
analysis during any loading and must allow for construction of simple and practical
tools for fatigue analysis or accelerated fatigue tests. This representation has to be
considered at the level of the materials and then at the level of the structure.
The materials have a Macroscopic behavior with its macroscopic elastic
limit and workhardening modulus with which Direct or incremental analysis of the
structure are performed. As only High cycles Fatigue is considered in this work,
elastic shakedown of the structure is reached after some time.
But the materials have a Microscopic local behavior with its microscopic
elastic or Endurance limit and local workhardening modulus. Although the global
elastic shakedown, there is a local plastic shakedown.
Level of the Material.
One parameter random loadings. Any one dimensional loading has its
Center C0 = (max + min)/2, and fluctuation F = (max - min)/2.
The local Cumulated plastic strain or local Dissipated energy during the real
random loading is computed on the model representing the material such as, for
example, the Kinematic Hardening behavior.
TWO LOADINGS ARE EQUIVALENT if they induce the SAME
cumulative plastic strain (Epc) or the SAME dissipated energy, they have the same
center and almost the same fluctuation.
A particular family of cyclic loadings with a constant amplitude and a
number of cycles may be deduced in Fig. 1.
ISSN 0206-3131. Надійність і довговічність машин і споруд, 2009. Вип. 32 26
similar to the "Wohler" curves (F2 = k, N).
Fig. 1. A particular family of cyclic loadings with a constant amplitude and a number of cycles.
Multi-parameter random stress path. The same concept is used to define one
equivalent radial cyclic loading to the general one: it will produce the same
cumulated plastic strain, it will have the center and its fluctuation will be higher or
equal the real fluctuation.
Fig. 2. Scheme “circular” stress path.
As we keep the fluctuation always the same, the range of microscopic plastic
strains will be the same (to keep the same unknown damage mechanisms), as it
may be seen in the figures: the response to any stress path is bounded by the
"circular" stress path.
=
-1.00
-0.75
-0.50
-0.25
-0.00
0.25
0.50
0.75
1.00
0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5
Evolution of : cyc1
TO ORDER CALL 817-431-8470
1- pc1 = pc2
2- C1 = C2
3- F2 F1
pc1 ; C1 ; F1
pc2 ; C2 ; F2
Fi
Ci
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In the stress space In the transformed parameters space In the plastic strain space
Fig. 3. Species stress paths is bounded by the "circular" stress path.
Intelligent description of one general fatigue case. It is needed to transfer
the primitive, passive descriptions of the geometry of structure, of its various
material and of one block of its random loading parameters to an intelligent, active
description such that the experimental results on one structure may be used to any
other new one!
Here, in any point of the structure, the Output Primitive descriptors or
conclusions are:
– failure or no failure;
– when there is failure, the number of blocks of the loading.
A multi-scale elastic analysis is considered, for example for a naval structure
(Fig. 4):
Fig. 4. Example for a naval structure.
1) at the scale of the hull with rather crude mesh (Fig. 5,a); elastic analysis is
performed due to the various loadings;
2) then at the level of a substructure 3-D with a refined mesh (with the initial
loading and the real loading, elastic, elastoplastic or simplified inelastic analysis
are performed); the equivalent radial cyclic loading is characterized;
3) then at the scale of a 2-D detail (one slice by a plane of the substructure)
with a much refined mesh (Fig. 5,b); a mobile window allows to focus at the
different hot zones. The materials: base material, weld, and HAZ, are described
from the material property database (smooth specimen data). The moving window
is similar to a filter. Its size is such that the quantities which are needed become
rather insensitive to the the errors associated to the mesh size and distribution.
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a b
Fig. 5. Design scheme: a – with rather crude mesh; b – with a much refined mesh and
mobile windows.
These quantities are related to the description of the stress and the gradient
of this stress field. Among them, there are:
– the Maximum, Minimum and Average value of any field, such as I1 and
J2, the Dang Van criterion ...;
– inside the window, volumes where any fatigue criterion is violated by a
factor of 50%, 80% or 100%.
Generation of the design rules to fatigue
Available experimental data. All the experimental data used in the study are
coming from the tests which were performed at the Illinois University. These tests
were done in order to compare various design methods and to analyze the influence of
some loading parameters and of the geometry on marine welded structures [4] (Fig. 6).
Fig. 6. Classical weld details [4].
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Several structures, indeed several characteristical details were tested. They
were selected among the classical ones in function of their relative importance.
One the main details is the detail No. 20 (Fig. 7) which is also selected for its
simplicity.
Fig. 7. Detail No. 20 with the critical points [4].
Within all details, three materials were mechanically characterized: the base
metal, the weld and the HAZ heat affected zone.
Detail No. 20. It is made of one central plate on which two lateral plates are
welded. Moreover the real geometry of the weld were measured. Twenty four
specimens with three different plate thicknesses were tested.
Three thickness sizes of specimens were used in the experiments: 6.35, 12.7
and 25.4 mm for loading plates and 7.9, 15.9 and 31.8 mm for corresponding
centre plates.
Three different materials were always considered: base material (for detail
No. 20, ASTM A-36 steel), weld metal (with Shielded Metal Arc Welding
method), heat affected zone.
The experimental set up in the tension-compression machine, implies that
the grips induce important initial stresses which were measured thanks to four
strain gages located in the vicinity of the weld on the lateral plates.
During each test, a fluctuating tensile load was applied: a mean stress level
varying 0 and 144.5 MPa with a random amplitude block. (these stresses result
from a long analysis on real naval structure). In principle, such a block was applied
several times until one crack is initiated or the maximum number of 1500 times
(which represents 125 years on the real structure). During the tests, special cares on
the critical zones were taken, as these zones are the most sensitive to the crack
initiation damage. For each sample, we have thus the geometry, the real loading,
the place where cracks appear (but unhappily, the initial stresses due to the
weld were not measured).
Modeling and Meshing with ALGOR. Script files for the design system SD3
from Algor were written to allow the systematical input of the geometry and the
mesh of the specimen according the given experimental data. It was taken into
account that 3 materials had to be differentiated.
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The Heat Affected Zone was not perfectly described in the report. It was
guessed according the other geometrical information. The system Algor proposes
several mesh generators. For a 2D structure, it uses Supergen.
In order to take into account the different materials and to have a thinner
mesh around the sensitive points, it is necessary to give some more information.
Fig. 8. Mesh and regions for 2D analysis Detail No. 20.
Elastic Analysis with NISA from EMRC. It is necessary to edit the nisa file
within the preprocessor DISPLAY, to define the materials, the boundary
conditions.
It is also necessary to introduce at first the initial stresses due to the grips and
the misalignment. As underlined in the report, these stresses may be important
although the care brought by the technicians.
It is, at last, necessary to define the loading on the specimen. For elastic only
some unit loadings are introduced and then combined. For the elastoplastic
analysis, the full time history of the loading is introduced.
Without giving details, we obtained the results for the 8 sensitive zones (IJP
and TOE) and also for some 4 other randomly chosen zones within the specimen.
The size of the zones or windows has been taken as a fundamental parameter of the
study, 4 sizes were retained (1, 2, 4 and 8 mm).
The result of these analysis consists into the deduction of the initial stress
field due to the grips and the applied stress field due to the random traction for each
of the 12 regions and for each of the 3 differentiated materials.
Fatigue analysis with NISA. From these last results, it is possible to use the
post processor ENDURE from EMRC. Only the crack initiation option of the
program was used: the number of applied blocks necessary to induce fatigue in any
point of the structure.
This last file was used to produce one of the intelligent descriptors, the mean
value of the life taken on all the nodes within a zone and for one material.
Definition of the intelligent descriptors. As underlined all along this text,
our objective being to be able to describe this detail 20 as any general structure. We
have to make its analysis at 3 successive multi-levels:
1) the first one is the global analysis of the structure;
2) the second one is the local analysis of the critical details which have been
detected as highly solicited;
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3) the third one is at the level of the region i.e. a moving window with a
given size.
These analysis have to be made with corresponding mesh sizes and some
submodelling option where the results on the boundary of the coarse mesh are
applied to the thinner mesh.
For us, one case, will consist at the particular behavior of one material within
one actual window.
At our actual stage of analysis of the detail No. 20, we have performed
already levels 2 and 3; it is now necessary to describe in an intelligent way the
results obtained in each window for each material:
– the material characteristics during cyclic loadings, cyclic curves at
different strain rate, Woehler curve and eventually also the rupture properties;
– the stress field with its evolution;
– some other factors such as given by the experts in fatigue/rupture or
deduced from general tools such as ENDURE.
With such a description, the conclusions will be
– Failure or Not failure;
– if Failure, what is the fatigue life.
There are some natural methods to represent the material properties with
some discrete parameters or descriptors.
The most difficult part is the representation of the evolution of the stress
field which are here the initial stresses due to the welding (unknown in this study),
the initial stresses due to the grips, and the applied random tensile load.
Material descriptors. We used only these 7 classical terms: (su ) ultimate
stress; (s’y) elastic cyclic limit; (b) fatigue strength exponent; (c) fatigue ductility
exponent; (n’) cyclic hardening exponent; (k’) coefficient cyclic strength
coefficient; (s’f) fatigue strength coefficient.
Stress field descriptors. At any given time, any scalar field M varying
between Mmin and Mmax may be characterized with:
– the interval [Mmin, Mmax] is divided in n equal segments;
– the relative surfaces in the window for which the scalar M is included in
the interval n, (Sn /S0), where S0 is the surface of the window.
In this study, we have taken n equal to 5 and we have assumed that the stress
field at a given time is represented by I1, its first invariant, J2, the second invariant
of its deviatoric part and the norm of the gradient of this J2 (the problem being the
fatigue life).
It is also necessary to describe the evolution of the stress field with the time.
Hablot [3] introduced:
– 1 = Supt [M / Sup (M)] envelop during the time of the relative M
(included between 0 and 1);
– 2 = Supt [M] envelop along the time of the absolute value;
– 3 = rr Supt [|| grad {(M) / Sup (M)} ||] envelop of the norm of the gradient
of the relative equivalent stress. This field shows the location of the maxima of the
gradient during the evolution;
– 4 = rr Supt [|| grad (M) ||]envelop of the norms of the absolute gradient.
which expresses the most important gradients reached during the evolution;
– rr is a characteristic dimension and many other quantities.
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Although it was indicated in the Illinois report that the tensile load was a
random load, during their tests, the real applied load was almost with a constant
amplitude. We have then considered that it was sufficient for our study to
characterize the stress field at 3 positions:
1) after the mean applied load;
2) at the maximum of the applied load;
3) at the minimum of the applied load.
57 descriptors are so introduced to represent the evolution of the stress field
within the window.
Elasto-plastic analysis. The same samples with the same meshes but with a
special loading path were analyzed. Indeed we added to the initial stress field due
to the grip, the particular real load history as described in the report and we
assumed that the stress field due to the welding (which is not known) was equal to
zero.
As with all the samples, the elastic shakedown behavior was reached, we
kept from the elastoplastic analysis, only the residual stress field which can then be
treated as the initial stress field due to the grips during the elastic analysis.
The same intelligent descriptors were computed within the same windows
around the same points in the specimens.
We also performed the incremental inelastic analysis during the random
loading as decribed in the report and we considered the stress field at the end of the
loading.
The execution times were very important with NISA.
The simplified inelastic analysis, ZAC was at last performed.
Data base and generation of the rules.
Compilation of data and automatic learning. Several details were tested. For
each size of the window, several text files are induced. All these text files are
merged. The observed conclusions are added. The data base has been so created. In
the Excel files name1.xls , name2.xls, name4.xls and name8.xls, name10.xls the
results of all the analysis corresponding to a window size of 1, 2, 4, 8 and 10 mm
are given.
We recall that in our approach, each subregion i.e. one material in one
region, is considered as an independent case/example for which we have its own
intelligent descriptors and its two conclusions Failure or No failure and when there
is failure, the number of cycles or number of blocks.
We used L.E.S. from the Ecole Polytechnique and Neuroshell from Wards
systems to extract automatically the rules. Only the learning of Failure or No
Failure has been done for the moment.
We also used the Probabilistic Neural Networks (PNN) which are able to
train on sparse data sets and to separate data into a specified number of output
categories.
Conclusions. From these results, the best window size has to be taken equal
to 8 mm during the elastic analysis. Moreover, the simplified and elastoplastic
analysis give almost the same reliability than the elastic one. Then only elastic
analysis for any new structure with a window size of 8 mm have to be taken.
The rules which were generated on this detail, can now be used to analyze
any other structure.
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However, more experimental tests are still necessary to qualify fully our
approach.
Comparaison between the reliabilities of the various learnings
File Des_Ela1Pro.out C1 C2 File Desc_zac1pro.out C1 C2
True positive proportion: 0.8381 0.9348 True positive proportion: 0.8730 0.9130
False positive proportion: 0.0652 0.1619 False positive proportion: 0.0870 0.1270
File Des_Ela2Pro.out C1 C2 File Desc_zac2pro.out C1 C2
True positive proportion: 0.8404 0.9796 True positive proportion: 0.8333 0.9388
False positive proportion: 0.0204 0.1596 False positive proportion: 0.0612 0.1667
File Des_Ela4Pro.out C1 C2 File Desc_zac4pro.out C1 C2
True positive proportion: 0.8003 0.9800 True positive proportion: 0.8312 0.9800
False positive proportion: 0.0200 0.1997 False positive proportion: 0.0200 0.1688
File Des_Ela8Pro.out C1 C2 File Desc_zac8pro.out C1 C2
True positive proportion: 0.8408 0.9800 True positive proportion: 0.8177 0.9200
False positive proportion: 0.0200 0.1592 False positive proportion: 0.0800 0.1823
File Des_Ela10Pro.out C1 C2 File Desc_zac10pro.out C1 C2
True positive proportion: 0.7392 0.9400 True positive proportion: 0.8333 0.9388
False positive proportion: 0.0600 0.2608 False positive proportion: 0.0612 0.1667
File Desc_cyc1pro.out C1 C2 File nDes_Ran1pro.out C1 C2
True positive proportion: 0.8770 0.9111 True positive proportion: 0.9280 0.9333
False positive proportion: 0.0889 0.1230 False positive proportion: 0.0667 0.0720
File Desc_cyc2pro.out C1 C2 File nDes_Ran2pro.out C1 C2
True positive proportion: 0.8710 0.9388 True positive proportion: 0.8737 0.8776
False positive proportion: 0.0612 0.1290 False positive proportion: 0.1224 0.1263
File Desc_cyc4pro.out C1 C2 File nDes_Ran4pro.out C1 C2
True positive proportion: 0.8771 0.9400 True positive proportion: 0.8886 0.8400
False positive proportion: 0.0600 0.1229 False positive proportion: 0.1600 0.1114
File Desc_cyc8pro.out C1 C2 File nDes_Ran8pro.out C1 C2
True positive proportion: 0.8035 0.9800 True positive proportion: 0.8359 0.9600
False positive proportion: 0.0200 0.1965 False positive proportion: 0.0400 0.1641
File Desc_cyc10pro.out C1 C2 File nDes_Ran10pro.out C1 C2
True positive proportion: 0.8402 0.9200 True positive proportion: 0.7380 0.9167
False positive proportion: 0.0800 0.1598 False positive proportion: 0.0833 0.2620
Acknowledgments. This paper was written in part during the regular visits
of the first author to UCSD's Center of Excellence for Advanced Materials, under
ONR contract N00014-96-1-0631 (R. Barsoum, Coordinator) to the University of
California San Diego and for the other part by the second author during his Phd
ISSN 0206-3131. Надійність і довговічність машин і споруд, 2009. Вип. 32 34
thesis under the sponsorship of Ligeron S.A (A. Azarian, Coordinator). We want to
thank Prof S. Nemat-Nasser for his helpful discussions.
Резюме
Разработан и описан специальный подход для анализа длительной усталости
сварных конструкций при случайных многолетних аксиальных нагрузках,
который имеет реальный физический смысл, и который очень легко выполнить.
Основная идея заключается в том, чтобы найти подходящий эквивалент простой
нагрузки, представить реальную случайную нагрузку с точки зрения
повреждения на уровне материала и затем на уровне структуры.
Ключевые слова: усталостный анализ, структура, случайная нагрузка,
программное обеспечение, сварные конструкции.
Резюме
Розроблено і описано спеціальний підхід для аналізу тривалої втоми зварних
конструкцій при випадкових багаторічних аксіальних навантаженнях, що має
реальний фізичний зміст, і який дуже легко виконати. Основна ідея полягає в
тому, щоб знайти підходящий еквівалент простого навантаження,
представити реальне випадкове навантаження з точки зору пошкодження на
рівні материалу і потім на рівні структури.
Ключові слова: втомний аналіз, структура, випадкове навантаження,
програмне забезпечення, зварні конструкції.
1. Zarka J. and al. A New Approach in Inelastic Analysis of Structures. –
CADLM Editor (1990).
2. Zarka J. and Navidi P. Clever Optimal Design of Materials and Structures //
Second French-Korean Conference on Numerical Analysis of Structures,
Seoul, September 1993.
3. Hablot J. M. Construction de solutions exactes en elastoplasticite, application
a l’estimation d’erreurs par apprentissage // These de l’Ecole Nationale des
Ponts et Chaussees, 1990.
4. Park S. K. and Lawrence F. V. Fatigue Characterisation of fabricated Ship
Details for Design – Phase 2 – University of Illinois at Urbana-Champaign,
1988.
5. Zarka J. and Navidi P. Optimal design of reliable structures // SMIRT: Post
Conference Seminar N° 13 Paris August, 1997, Intelligent software Systems
in Inspection and Life Management of Power and Process.
6. Zarka J. and Navidi P. Intelligent Optimal Design of Materials and Structures
– CADLM Editor (2000), to be freely downloaded from
http://www.cadlm.fr/page.php?page=coit_ress_ine
Поступила 13.04.2009
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