a design oriented reliability methodology for fatigue life under stochastic...
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1 Copyright © 2013 by ASME
Proceedings of the ASME 2013 International Design Engineering Technical Conferences &
Computers and Information in Engineering Conference
IDETC/CIE 2013
August 4-7, 2013, Portland, OREGON, USA
DETC2013-12033
A DESIGN ORIENTED RELIABILITY METHODOLOGY FOR FATIGUE LIFE UNDER
STOCHASTIC LOADINGS
Zhen Hu Department of Mechanical and Aerospace
Engineering Missouri University of Science and Technology
Rolla, MO, USA, 65401
Xiaoping Du1
Department of Mechanical and Aerospace Engineering
Missouri University of Science and Technology Rolla, MO, USA, 65401
1400 West 13th Street, Toomey Hall 290D,Rolla, MO 65401, U.S.A., Tel: 1-573-341-7249, e-mail: [email protected]
ABSTRACT Fatigue damage analysis is critical for systems under
stochastic loadings. To estimate the fatigue reliability at the
design level, a hybrid reliability analysis method is proposed in
this work. The First Order Reliability Method (FORM), the
inverse FORM, and the peak distribution analysis are
integrated for the fatigue reliability analysis at the early design
stage. Equations for the mean value, the zero upcrossing rate,
and the extreme stress distributions are derived for problems
where stationary stochastic processes are involved. Then the
fatigue damage is analyzed with the peak counting method. The
developed methodology is demonstrated by a simple
mathematical example and is then applied to the fatigue
reliability analysis of a shaft under stochastic loadings. The
results indicate the effectiveness of the proposed method in
predicting fatigue damage and reliability.
1. INTRODUCTION In engineering applications, stochastic loadings are
commonly encountered, for example, offshore structures under
stochastic wave and wind loadings, hydrokinetic turbine
systems under stochastic river flow loadings, and aircraft under
stochastic aerodynamic loadings [1-6]. As nonlinear functions
of stochastic loadings, the stress responses of a component or a
system are also stochastic processes. The fatigue life of the
component or the system is of great interest to its designers and
users.
In the past decades, many progresses have been made in
fatigue life analysis. The methodologies can be classified into
three categories. The first category is concerned with the fatigue
life under one single stochastic loading. One example is the
extreme and fatigue analysis method proposed by Winterstein
[7] based on the moment matching method and the assumption
of monotonic responses with narrow-band loading.
Winterstein’s method has been further investigated by Azaïs [8],
Braccesi [9], Chen [10], and other researchers [11-15] for
different applications.
The second category includes fatigue life analysis methods
based on field stress or strain data. These methods are widely
used. For instance, Liou, et al. [16] developed a model for the
estimation of fatigue life by using the vibration theory to
analyze the stress history. Sofia, et al. [17] proposed a Laplace
driven moving average method for the fatigue damage
assessment of non-Gaussian random loads. Bengtsson and
Rychlik [18] discussed how to handle uncertainties in fatigue
life estimation. Gladskyi and Shukaev [19] proposed a new
model for low cycle fatigue of metal alloys by investigating the
fatigue life under different strain levels. Yin, et. al [20] studied
the fatigue behavior of case-hardened steels by predicting the
fatigue life based on standard strain-life curve. Many other
approaches for fatigue life prediction based on stress and strain
data can be found in [21-27].
Methods in the third category are for fatigue damage
analysis. For example, Liu [28] developed a stochastic S-N
curve model to overcome the limitation of the constant
amplitude fatigue testing method. Tovo [29] compared different
cycle counting methods in fatigue life analysis.
The above methods have limitations. For example, the
Winterstein’s [7] method assumes narrow-band loadings and
monotonic responses. The dominating fatigue life analysis
methods [16-18, 21-25, 28, 29] rely on the availability of stress
or strain data. These methods may not be directly applied to the
fatigue life estimation at the design level.
The analysis at the design level means that the fatigue life
can be predicted for a given set of design variables. This can be
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achieved with computational analysis models, for example,
CAE simulation models, such as the model of Finite Element
Analysis. With the computational models, design variables are
linked with the response variables, such as stresses or strains. It
is therefore possible to predict fatigue life at the design level.
As there are many uncertainties, such as stochastic loadings and
manufacturing variations, the fatigue life is also uncertain. Since
more uncertainties should be considered at the design level, the
fatigue analysis at the design level is much more complicated
than the fatigue analysis at the stress or strain level.
Consequently, the objective of this research is to develop a
fatigue life analysis method at the design level.
The focus of this work is to predict the fatigue life under
non-linear combination of loadings in the form of stationary
stochastic processes at the design level. The new development
includes three research tasks. They are the estimation of the
mean value of a stress response, the calculation of the number
of cycles per unit time, and the approximation of the damage for
a given design. Relevant studies have been performed for the
three tasks, but they have limitations. For instance, Gupta [30]
developed a method for the peak distribution analysis for the
combination of only Gaussian stochastic loads. The method is
based on importance sampling, and its efficiency can be further
improved. Lutes [31] investigated the method for the joint
distribution analysis of peaks and valleys of stochastic
processes. The method considers only one Gaussian stochastic
process. The present work performs fatigue life analysis under
multiple Gaussian and non-Gaussian loadings. As mentioned
previously, more uncertainties are considered at the design level
in this work.
The contributions of this work include three components.
First, the fatigue life analysis is performed at the design level.
Second, an efficient method for the estimation of mean value of
the peak stress distribution is implemented. And third, an
accurate and efficient method is proposed for the peak stress
distribution analysis in the presence of both Gaussian and non-
Gaussian stochastic processes.
In the next section, we review the fatigue damage analysis
and its challenges at the design level. Following that, in
Section 3, we first summarize the main procedure of the
proposed method and then discuss it in details. In Section 4,
two numerical examples are employed to demonstrate the
proposed method. Conclusions are made in Section 5.
2. PROBLEM STATEMENT
2.1. Fatigue life assessment under stochastic
loadings There are many fatigue damage accumulation rules. The
most commonly used one is the Palmgren-Miner’s rule [32],
which formulates the total fatigue damage as follows
1
( )
( )
j
i
F
i i
n sD
N s
( 1)
where ( )in s is the number of stress cycles at stress level is ,
and ( )iN s is the number of cycles to failure at stress level is .
With the S-N curve approach, Eq. (1) is rewritten as
( )
1/ ( )
i
F
i i
n sD
s ( 2)
where and are two parameters, which are usually
obtained from fatigue test.
Eqs. (1) and (2) express the fatigue damage in a discretized
form, when given in a continuous form, the expected fatigue
damage FD over a time duration T is [29]
0 ( )FD v T s p s ds ( 3)
where 0v is the mean upcrossing rate of the stress process, and
( )p s is the probability density function (PDF) of stress cycle.
To estimate the fatigue damage and fatigue life using Eq.
(3), one needs stress/strain data because the current fatigue
damage analysis is based on the stress/strain responses. The
data are usually from field or experiments. They are, however,
unavailable at the design stage for many applications. In this
case, we need to perform stress/strain analysis by using
computational models to predict stresses/strains for a given set
of design variables.
2.2. Fatigue life estimation in the early design stage With responses obtained from computational models,
fatigue life analysis is feasible in the early design stage. Fig. 1
shows that computational models produce stress response ( )S t
for a given set of input variables, including random variables
x and stochastic processes, ( )ty .
A computational model can be an explicit function. In most
cases, it is a black-box model, such as simulation models of
Finite Element Analysis (FEA), Computational Fluid Dynamics
(CFD) analysis, and other Computer Aided Engineering (CAE)
simulations.
The stress response is given by
( ) ( , ( ))S t g t X Y ( 4)
in which ( )g is the computational model,
1 2[ , , , ]nX X XX is a vector of random variables, which
represent uncertainties in the form of randomness that do not
Computational Models
x
( )ty
( )S t
Fig. 1. Computational models
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vary with time, and 1 2( ) [ ( ), ( ), , ( ) ]mt Y t Y t Y tY is a vector
of stochastic processes, which represent uncertainties in the
form of randomness that vary with time. Stochastic loadings
are part of ( )tY .
The research task is now to estimate the fatigue life given
X , ( )tY , and ( )g in the design stage. The Monte Carlo
Simulation (MCS) can be used for this task, but it is practical
only for problems with cheap computational models. Since
MCS calls the computational models many times, it is not
applicable for problems involving FEA or CFD simulations. For
problems under one stochastic loading, which is monotonic to
the stress and strain responses, one possible way is constructing
cheap metamodels to replace the sophisticated simulation
models. Based on the low fidelity models, MCS is then applied.
However, this method is not satisfactory when there are
nonlinear combinations of stochastic loadings involved in the
problems. First, the construction of metamodels for problems
under non-linear combination of stochastic loadings and other
uncertain parameters is difficult. Second, MCS is still
computationally expensive for the low fidelity models (as
demonstrated in the numerical examples). This work devotes its
efforts to explore an efficient way to approximate the fatigue
life.
To efficiently analyze the fatigue life, we should rely on
approximations. We then face three challenges.
(1) Calculate the mean value upcrossing rate 0v , which is
an important parameter for fatigue reliability analysis. To
calculate 0v , we need at first to determine the mean value of
the stress response for a given set of input variables.
(2) Obtain the stress cycle distribution ( )p s . Since in the
design stage, the stress data are often unavailable, we should
estimate the stress cycle distribution with the computational
model ( )g and its inputs X and ( )tY .
(3) Determine the integration region for the damage
analysis. This is required for the fatigue life analysis that
integrates the stress over the damages region as indicated in Eq.
(3).
Addressing the above challenges is a difficult task. In this
work, we only focus on the problems that involve only
stationary input stochastic processes.
3. FATIGUE LIFE RELIABILITY ANALYSIS Due to uncertainties in the fatigue life analysis, it is
reasonable to express the fatigue life in a probabilistic form
rather than in a deterministic form. Fatigue life reliability
analysis is a tool for doing this. In the past decades, progresses
have been made in fatigue life reliability analysis [33-37]. For
instance, a time-dependent fatigue reliability analysis method
was proposed by Liu and Mahadevan [28]. The uncertainties of
the stochastic nature in the S-N curve have been investigated in
their method. Guo, et. al. [38] developed a fatigue reliability
analysis method for the steel bridge based on the combination
of an advanced traffic load model and Finite Element Analysis.
Their method is applicable for problems of a special group, but
cannot be applied to other general problems. Rajaguru, et.al.
applied the kriging and radial basis function to the fatigue
reliability analysis of a wire bond structure [39]. Their methods
focus on problems under only one stochastic loading. Different
from the prior works, this work aims to develop a new fatigue
reliability analysis method for problem with inputs of
generalized random variables and stochastic processes.
In this section, we first give the main implement procedures
of the proposed method. After that, we discuss details of the
new probabilistic analysis method for fatigue reliability.
3.1. Overview of the proposed method Fig. 2 shows the seven main steps of the proposed method.
The steps are explained as below.
• Step 1: Initialization ─ transform the non-Gaussian
random variables X into standard Gaussian random
variables Xu . Input an initial point 0
Xu for the MPP
search in step 2.
• Step 2: Mean value evaluation ─ for given values of
Xu , approximate the mean value of the stress response
( ( ), ( ))XS T tu Y .
• Step 3: Zero upcrossing rate analysis ─ calculate the
mean-value upcrossing rate of the stress response under
given values of Xu .
• Step 4: Stress cycle analysis ── for given values of
Xu , perform the stress peak distribution analysis.
• Step 5: Fatigue damage analysis ─ compute the fatigue
damage with the results from zero upcrossing rate analysis
and stress cycle analysis.
• Step 6: Convergence study ─ check if the reliability
index F X u converges or not. If converge, go to next
step, otherwise, update Xu and go to Step 2.
• Step 7: Fatigue life reliability analysis ─ once the MPP *
Xu is identified, the fatigue life reliability is
approximated.
3.2. First Order Reliability Method for Fatigue
Reliability Analysis As described previously, a stress response function is given
by ( ) ( , ( ))S t g t X Y , which is dependent on random variables
X , stochastic processes ( )tY , and time t. Consequently, the
fatigue life FT is also a random variable. The probability that
FT is larger than T is the fatigue reliability at T and is given by
Pr{ }F FR T T ( 5)
where Pr{} stands for a probability.
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In the design stage, the statistics of fatigue life is not
available. If the stresses are available, FR can be estimated by
the fatigue damage analysis. According to the Palmgren-Miner’s
rule [32], a fatigue failure is defined as the event that the
accumulative damage is greater than one. The fatigue reliability
is then given by
0Pr{ ( ) 1}F FR D v T s p s ds ( 6)
With the link between stresses and basic input variables X and ( )tY , the fatigue reliability is computed by
( ) 1
P r { ( ) 1} ( )
F
F F
D
R D f d
X
X X X ( 7)
The probability in Eq. (7) can be estimated by the
Importance Sampling (IS), First Order Reliability Method
(FORM), Second Order Reliability Method (SORM), and other
methods. In this work, we use FORM because of its good
accuracy and efficiency. The random variables X are
transformed into standard normal variables, XU . After the
transformation, the fatigue damage function becomes
( ) ( ( ))F F XD D TX U ( 8)
in which ( )T denotes the transformation.
We then search for the Most Probable Point (MPP), *
X Xu u by solving the following optimization problem [40,
41]
Min
s.t.
( ( )) 1
X
F XD T
u
u
( 9)
Once the MPP is identified, the probability that
Pr{ }F FR T T is approximated as follows
Pr{ } ( )F F FR T T ( 10)
where ( ) is the Cumulative Density Function (CDF) of a
standard normal variable, and F is given by
*
F X u ( 11)
in which stands for the determinant of a vector.
The optimization model given in Eq. (9) can be solved
numerically. The critical task is the estimation of the fatigue
damage FD given inputs of
Xu . Next we discuss how to use
the mean value upcrossing rate and the stress cycle distribution
analysis for the task.
3.3. Mean value analysis As mentioned above, the mean value upcrossing rate or the
zero upcrossing rate is essential for the fatigue damage analysis.
To estimate the mean value upcrossing rate, we first need to
obtain the mean value of the stress response s . Given the PDF
of s , the mean value s is computed by
0
( )s sf s ds
( 12)
To use FORM, we rewrite Eq. (12) as
1
1
0( )s s sF P dP ( 13)
in which
( )sP F s ( 14)
where ( )F s is the Cumulative Distribution Function (CDF) of
stress response s, and 1( )F is the inverse function of ( )F s .
We use the Gauss-Legendre quadrature (GLQ) method to
calculate the integral. GLQ estimates the integral by summing
up weighted the integrand evaluated at the optimized Gauss
points as follows:
1
11
( ) ( )r
j j
j
h d w h
( 15)
where jw are the weights of Gaussian points, j are
Gaussian points, and r is the number of Gaussian points.
0
Xu
Stress cycle
distribution
analysis
Mean value evaluation
Zero upcrossing Rate
Analysis
Accumulated Fatigue Damage
Converge?
*
Xu
Y
N
New Xu
0v
Fig. 2. Flowchart for the proposed method
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The weights and Gaussian points for r =1, 2, 3, 4, 5, and 6,
are listed in Table 1. More points for higher order quadratures
can be found in [42].
Table 1. Weights and Gaussian points for Gauss– Legendre quadrature
r j jw
1 0 2
2 ±0.5773502692 1
3 0 0.8888888889
±0.7745966692 0.5555555555
4 ±0.3399810435 0.6521451548
±0.8611363116 0.3478548451
5
0 0.5688888889
±0.5384693101 0.4786286705
±0.9061798459 0.2369268850
6
±0.2386191861 0.4679139346
±0.6612093865 0.3607615730
±0.9324695142 0.1713244924
By applying GLQ to Eq. (13), we have
1
1 1
01
11( )
2 2
rj
s s s j
j
F P dP w F
( 16)
where r is number of Gaussian points used for Eq. (13), and
j are the Gaussian point listed in Table 1.
For every Gaussian point, 1, 1, 2, ,j j r , there is no
straightforward form available for 11
2
jF
. To
approximate 11
2
jF
, we use the inverse FORM. Given a
specified Gaussian point, the associated stress response is
approximated by the following optimization model:
( )
1
( )
1
( )
1Max ( ( ), ( ))
2
s.t.
1
2
Y t
j
X Y t
j
Y t
s F g T T
uu u
u
( 17)
in which ( )Y tu is the standard normal variables associated with
stochastic loadings ( )tY .
3.4. Mean-value upcrossing rate analysis To estimate the mean-value upcrossing rate of the stress
response, we define the following limit-state function:
( ) ( ( ), ( ))X sZ t S T t u Y ( 18)
We then search for the MPP for Eq. (18) by solving the
following optimization model:
( )
( )
Min
s.t.
( ( ), ( )) 0
Y t
X Y t sS T T
u
u u
( 19)
Then the mean-value upcrossing rate 0v can be calculated
using the Rice’s formula [5, 43, 44] as follows:
0 ( ) ( ) / ( )v t t t t ( 20)
in which ( )t is the reliability index of limit-state function
( )Z t , ( )t is the first derivative of ( )t , and ( ) is a
function defined by
( ) ( ) ( )x x x x ( 21)
and
( )
( )t
tt
( 22)
where ( ) is the PDF of a standard normal variable.
2 ( )t is given by [5]:
2
12( ) ( ) ( ) ( ) ( , ) ( )T Tt t t t t t t C ( 23)
in which
*
, ( )
*
, ( )
( )t
t
t
Y
Y
u
u ( 24)
1 2
2
1 2
12
1 2
( , )( , )
t t t
t tt t
t t
CC ( 25)
and
1
1 2
1 2
1 2
( , ) 0 0
0 0( , )
0 0 ( , )m
Y
Y
t t
t t
t t
C ( 26)
where 1 2( , )iYt t is the autocorrelation coefficient function of
stochastic loading ( )iY t .
It should be noted that the principle of the above discussed
upcrossing rate method is the same as that of the PHI2 method
[45]. The implementation of Eqs. (18)-(26) however, is much
easier than the PHI2 method as no extra random variables are
introduced.
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3.5. Stress cycle distribution analysis For the fatigue damage analysis, it is critical to obtain the
stress cycle distribution. In the past decades, various counting
methods have been proposed. Among these methods, the most
commonly used four methods [29] are peak counting (PC),
level crossing counting (LCC), range counting (RC), and
rainflow counting (RFC). RFC can provide results with the best
agreement with experiments. Due to its complicated
implementation, it is hard to apply RFC to the fatigue damage
analysis in the design stage. One possible way is to use Monte
Carlo simulation (MCS) with RFC, but it is inefficient. The
other three methods, PC, LCC, and RC, are applicable. Several
methods have recently proposed for special cases with Gaussian
stochastic loadings [31, 46].
For general problems subjected to nonlinear combinations
of stochastic loadings, in this work, we use PC. Similar to the
mean value analysis in Sec. 3.2, the stress response
( ) ( , ( ))XS t g t u Y is transformed into the standard normal
space. Then ( )S t becomes
( ) ( ( ), ( ( )))X YS t g T T t u U ( 27)
For a given stress level is , we formulate a limit-state
function ( )iZ t as
( ) ( ( ), ( ( )))i X Y iZ t g T T t s u U ( 28)
By solving the optimization model in Eq. (19), the MPP *
, ( )i tYu for ( )iZ t is identified. Then, we have
*
, ( )i i t Y
u ( 29)
( ) ( ) ( )T
i YL t t t α U ( 30)
and
Pr{ ( ) ( , ( ( ))) 0}
Pr{ ( ) ( ) ( ) } ( ), if
Pr{ ( ) ( ) ( ) } ( ), otherwise
i X Y i
T
i Y i i i s
T
i Y i i
Z t g T t s
L t t t s
L t t t
u U
α U
α U
( 31)
where
*
*( ) i
i
i
t u
αu
( 32)
It can be found from above transformation that a higher
stress level is corresponds to a larger value of i in the
standard normal space. Since the transformation from is to
i is monotonic for both stress levels i ss and i ss ,
we can get the following conclusion. The probability that the
peak of stress response, ( )pZ t , is smaller than a stress level is
is equivalent to the probability that the equivalent stress
response peak ( )pL t is less than the associated i (for
i ss ) or i (for
i ss ). Mathematically, it can presented
as
Pr{ ( ) }, if
Pr{ ( ) }Pr{ ( ) }, otherwise
p i i s
p i
p i
L t sZ t s
L t
( 33)
in which ( )pZ t and ( )pL t are the peaks of ( )iZ t and
( )L t , respectively.
To estimate the peak distribution, Pr{ ( ) }p iL t or
Pr{ ( ) }p iL t , we need to know the statistics of the
transformed stochastic process ( ) ( ) ( )T
i YL t t t α U . Its
autocorrelation coefficient between two time instants, 1t and
2t , is given by
1 2 1 1 2 2( , ) ( ) ( , ) ( )T
L i it t t t t t α C α ( 34)
where 1 2( , )t tC is given in Eq. (26).
The occurrence of a peak indicates a downcrossing of zero
level by 0( ) / ( )L t t . The occurrence rate of peaks of ( )L t is
then given by
( )
( )2
p
p
tv t
( 35)
where
1 2
2
2 1 2
0
1 2
( , )( ) L
t t t
t tt
t t
( 36)
and
1 2
222 1 2
1 2 0 1 0 2 1 2
( , )1( )
( ) ( )
L
p
t t t
t tt
t t t t t t
( 37)
Substituting Eq. (34) into Eqs. (36) and (37), we have
1 2
2
0
1 2 1 2 2 1 1 2 2
1 12 1 2 2 1 1 1 2 2
( )
( ) ( , ) ( ) ( ) ( , ) ( )
( ) ( , ) ( ) ( ) ( , ) ( )
T T
T T
t t t
t
t t t t t t t t
t t t t t t t t
C C
C C
( 38)
and
1 2
2
' ' '2 3
0 1 0 2 0 11 2 1 2
2 2 2 2
1 20 1 0 2 0 1 0 2 1 2
' 3 4
0 2 1 2 1 2
2 2 2 2
0 1 0 20 1 0 2 1 2 1 2
( )
( ) ( ) ( )( , ) ( , )
( ) ( ) ( ) ( )
( ) ( , ) ( , )1
( ) ( )( ) ( )
p
L L
L L
t t t
t
t t tt t t t
t tt t t t t t
t t t t t
t tt t t t t t
( 39)
1 2
1 1 2
1
( , )( , )
t tt t
t
CC ( 40)
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1 2
2 1 2
2
( , )( , )
t tt t
t
CC ( 41)
2
1 2
12 1 2
1 2
( , )( , )
t tt t
t t
CC ( 42)
As the stochastic processes ( )tY are assumed to be
stationary and t does appear explicitly in the stress response, we
have
( ) 0t ( 43)
After further derivations and simplifications, we have
1 2
2
0 1 12 1 2 2( ) ( ) ( , ) ( )T
t t tt t t t t
C ( 44)
' 1 112 1 2 2
0 1
0 1
( ) ( , ) ( )( )
2 ( )
Tt t t tt
t
C ( 45)
' 1 122 1 2 2
0 2
0 2
( ) ( , ) ( )( )
2 ( )
Tt t t tt
t
C ( 46)
3
1 2
1 122 1 2 22
1 2
( , )( ) ( , ) ( )TL t tt t t t
t t
C ( 47)
3
1 2
1 112 1 2 22
1 2
( , )( ) ( , ) ( )TL t tt t t t
t t
C ( 48)
and
4
1 2
1 1112 1 2 22 2
1 2
( , )( ) ( , ) ( )TL t tt t t t
t t
C ( 49)
Substituting Eqs.(36) and (45) through (49) into Eq. (39),
we obtain
1 2
1122 1 2
0
( ) ( , ) ( )
( )( )
T
t t t
p
t t t t
tt
C
( 50)
With Eqs. (44) and (50), the regularity factor of stochastic
process ( )L t is computed by
1 2
0 12
1122 1 2
( ) ( ) ( , ) ( )
( ) ( ) ( , ) ( )
T
Tp
t t t
t t t t t
t t t t t
C
C ( 51)
The regularity factor is defined as the ratio between the
rate of zero-upcrossings and the rate of local maxima. If the
regularity factor tends to be one, the associated stochastic
process is “narrow-band”. For the standard Gaussian stochastic
process with regularity factor , the CDF and PDF of the peak
is given by the Rice distribution as follows [47]:
2
2
2 2( )
1 1PF e
( 52)
and
2
2 2
2 2( ) 1
1 1pf e
( 53)
Fig. 3 shows the PDFs of the peaks of standard Gaussian
stochastic processes with different regularity factors.
-4 -2 0 2 4 60
0.1
0.2
0.3
0.4
0.5
0.6
0.7
PD
F=0
=1
Increase
Fig. 3. PDFs of the peaks of standard Gaussian stochastic processes with different regularity factors
Since ( ) ( , ( ( )))i X Y iZ t g T t s u U are transformed into
( ) ( ) ( )T
i YL t t t α U at every time instant, we have
( )
( ) 0Y t
T
L i t U
α μ ( 54)
and
( )
2 ( ) 1Y t
T
L i t U
α σ ( 55)
We can therefore use Eqs.(52) and (53) to estimate the
probabilities given in Eq. (33). Combining Eq. (33) with Eqs.
(52) and (53), we have
2
2
2 2Pr{ ( ) }
1 1
i
i i
p iL t e
( 56)
or
2
2
2 2Pr{ ( ) }
1 1
i
i i
p iL t e
( 57)
where i and are given in Eqs. (29) and (51),
respectively.
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With Eqs. (28) through (57), the CDF of the stress peak is
obtained. The PDF ( )p s in the integral ( )s p s ds is then
also available. We now discuss how to evaluate the integral
numerically. To use the CDF of the peak directly, we perform
the following transformation:
1
1( ) ( ( ))p p
s s
s sP
s p s ds P P dP
( 58)
where ( )ps pP P s is the CDF of the peak
ps , which is given
in Eq. (57), 1( )P is the inverse function of ( )P , and
2
2
2 21 1s
P e
( 59)
in which is the regularity factor by plugging ( )t into
Eq. (51).
Based on the transformation, we also solve the integral by
GLQ, and the integral is computed by
1
1 1
1
1( ( )) ( ( ))
2
s
p ps
r
s s i piP
i
PP P dP w P P
( 60)
in which r is the number of Gaussian points, iw is the
weights, and piP is the i-th probability point corresponding to
the i-th Gaussian point i .
piP is given by
1
(1 )2 s s
i
piP P P
( 61)
For every Gaussian point, 2, 1, 2, ,i i r , there is no
analytical form available for 1( )piP P . To compute Eq. (60),
we propose an inverse stress cycle distribution analysis method
based on the inverse FORM. Given a specified value of i , the
associated stress peak ps is approximated by the following
optimization model:
( )
1 2
1
( )
( )
( )
( )
0 12
1122 1 2
1
Max ( ) ( , , )
s.t.
( )
( ) ( ) ( , ) ( )
( ) ( ) ( , ) ( )
( )
Y t
p pi X Y t
i Y t
Y t
i
Y t
T
i i
iT
mi i
t t t
obj P pi i
i obj
s P P g t
t
t t t t t
t t t t t
F P
uu u
u
uα
u
C
C
( 62)
After getting the integral for the stress cycle distribution
analysis, we substitute it into the fatigue damage constraint
equation in Eq. (9). Given the value of Xu , the fatigue damage
over time duration T is then approximated as
2
1
0 0
1
( ( ), ( ))
( ) ( ( ))
F X
r
i pi
i
D T t
v T s p s ds v T w P P
u Y
( 63)
where 0v is given in Eq. (20).
With Eqs. (12) through (63), the optimization model in Eq.
(9) can be solved. The probabilistic fatigue life in Eq. (5) can
then be estimated.
In the proposed method, the reliability analysis method, the
peak distribution analysis method, and the upcrossing rate
analysis are integrated. In the following section, we summarize
the main numerical procedure.
4. EXAMPLES In this section, we use two examples to demonstrate the
proposed method. The first is a mathematical problem and the
second is a shaft problem subjected stochastic force and torque.
In the first example, the response is a function of Gaussian and
non-Gaussian stochastic processes. It is used to verify the
proposed mean value estimation, zero upcrossing rate analysis,
and stress cycle distribution analysis. The second example
shows the practical application of the new method. The example
involves a mechanical component under a nonlinear
combination of stochastic loadings. The problem has both
random variables and stochastic processes. The fatigue life
reliability of the shaft is also predicted in the second example.
4.1. A mathematical example A limit-state function is given by
1 2 3( ) ( , ( )) ( ) ( ) ( )S t g t Y t Y t Y t X Y ( 64)
The three stochastic processes in the function are given in
Table 2.
Table 2 Stochastic processes of the mathematical example
Variable Mean Standard
deviation Process type
Auto-
correlation
1( )Y t 1 0.3 Gaussian Eq. (65)
2 ( )Y t 2 0.5 Gaussian Eq. (66)
3 ( )Y t 1.5 0.2 Lognormal Eq. (68)
The auto-correlation functions of the two Gaussian
processes 1( )Y t and 2 ( )Y t are
1
2
1 2 2 1( , ) exp[ ( ) ]Y t t t t ( 65)
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2 1 2 2 1( , ) cos[ ( )]Y t t t t ( 66)
The non-Gaussian stochastic process 3 ( )Y t is a function
of a standard Gaussian stochastic process 3 ( )U t as below:
3 3( ) exp[0.2 ( ) 1.5]Y t U t ( 67)
The auto-correlation function for the underlying standard
Gaussian stochastic process 3 ( )U t is given by
3
2 2
1 2 2 1( , ) exp[ ( ) / 0.5 ]U t t t t ( 68)
We calculated the mean value and zero upcrossing rate of
the response g using the Monte Carlo simulation (MCS) and the
proposed method. For MCS, the time interval [0, 20] years is
divided into 500 time instants and 1×105 samples are generated
at each time instants. The total functioncall for MCS is
therefore 5×107. The latter method used six Gaussian points.
Table 3 shows the results.
Table 3 Results of Example One
Variable MCS Proposed Error (%)
Mean Value 7.5720 7.5423 0.39
Zero upcrossing rate 0.4420 0.4475 1.24
Function calls 5×107 488 -
The errors in the table are those from the proposed method
with respect to those from MCS. The small errors indicate that
the proposed method is accurate. We also compared the peak
distributions of S from both methods. The PDFs and CDFs from
the two methods are plotted in Figs. 4 and 5, respectively. The
results show that the proposed method is also accurate in
estimating the peak distribution.
4 6 8 10 12 14 16 180
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Peaks of g
PD
F
MCS
Proposed
Fig. 4. PDFs of the peak values
4 6 8 10 12 14 16 180
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Peaks of g
CD
F
MCS
Proposed
Fig 5. CDFs of Peak values
Figs. 4 and 5 show that the proposed method can accurately
predict the peak distribution. The method is also therefore able
to predict fatigue reliability accurately fir this problem.
4.2. A shaft subjected to stochastic loadings Fig. 6 shows a shaft subjected to a force and a torque, and
both of them are stochastic processes. The stress response at the
root of the shaft is also a stochastic process. The designed
fatigue life of the shaft is 20 years. The task is to estimate the
fatigue reliability by accounting for uncertainties in the
geometry and S-N curve of the material, in addition to the
stochastic loadings.
The stress response of the shaft is given by
2 22
3
16( , ( )) 4 ( ) 3 ( )sS t l F t Q t
d X Y ( 69)
where [ , ]s d lX and ( ) [ ( ), ( )]t F t Q tY .
After considering the uncertainties in the parameters of S-N
curve, the fatigue reliability is
Pr{ ( , ( )) }f Fp T t T X Y ( 70)
in which more uncertain variables are involved. The new
variables are [ , , , ]s d l X and ( ) [ ( ), ( )]t F t Q tY , where
Fig. 6. A shaft under stochastic force and torque
F(t) l
Q(t)
d
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and are parameters of S-N curve. 20 yearsT is the
designed fatigue life. There are totally six independent variables
involved in this example. (If the variables were dependent, they
should be transformed into independent ones before performing
the reliability analysis.)
The random variables and stochastic loadings are given in
Table 4.
Table 4 Random Variables and stochastic processes of the shaft example
Variable Mean Standard
deviation Distribution
Auto-
correlation
d 59 mm 2.95 mm Gaussian N/A
l 200 mm 10 mm Gaussian N/A 2.87×10
-13 5.74×10
-15 Gaussian N/A
1.0693 0.0214 Gaussian N/A
ln( ( ))F t 7.59 0.11 Lognormal Eq. (71)
( )Q t 250 Nm 25 Nm Gaussian Eq. (72)
The auto-correlation functions of the underlying standard
normal process ( )FU t of ( )F t and ( )T t , are given as
below:
2
1 2 2 1( , ) exp[ ( ) /1.44]FU t t t t ( 71)
and
1 2 2 1( , ) cos[ ( )]4
Q t t t t
( 72)
We estimated the probability of failure for the fatigue life
using the proposed method and MCS. The MCS results are used
as a benchmark to evaluate the accuracy of the proposed
method. The percentage error is given by
100%
MCS
f f
MCS
f
p p
p
( 73)
where fp is the probability of failure obtained from the new
method, and MCS
fp is the probability of failure from MCS.
The number of samples of MCS was 1×105, and the time
interval was discretized into 400 instants. A sample path of the
stochastic stress is depicted in Fig. 7. Table 5 shows the
probabilities of failure, the numbers of function call, and the
actual computational time by the proposed method and MCS.
The analysis was performed on a Dell personal computer with
Intel (R) Core (TM) i5-2400 CPU and 8GB system memory.
Table 5 Results of fatigue reliability analysis
Probability
of failure
Error
(%)
Function
call Time
Proposed 0.0518 2.26 1288779 About 1 hour
MCS 0.0530 N/A 4×1012
About 376 hours
0 50 100 150 200 2501.8
2
2.2
2.4
2.6
2.8
3x 10
7
t (hour)
Str
ess
(P
a)
Fig. 7. A sample path of the stress
The results show that the proposed is accurate in predicting
the fatigue reliability. Its computational time was much less than
that of MCS. The mean values, the zero upcrossing rates, and
the fatigue damage calculated from the two methods are also
given in Table 6.
Table 6 Fatigue damage analysis for the shaft example
Variable MCS HRM-FRA Error
(%)
Mean Value 2.205×107 Pa 2.200×10
7 Pa 0.227
Zero upcrossing
rate 0.1644 0.1836 11.68
( )s p s ds 3.157×10-5
3.108×10-5
1.55
5. CONCLUSIONS Fatigue reliability is a critical issue for problems under
stochastic loadings. In this work, a fatigue reliability analysis
method is developed for the fatigue life reliability analysis of
structures under stochastic loadings. The analysis is performed
at the design level instead of the stress or strain level. With the
link between basic design variables and fatigue life of
structures, the method allows for a prediction of fatigue life and
reliability during the design stage. The method is based on
FORM, inverse FORM, peak distribution analysis, and
numerical integration. Numerical examples show its accuracy in
estimating the peak distribution, the mean value, the zero
upcrossing rate, and fatigue life reliability for problems
involving stationary stochastic processes.
Peak counting method is employed to analyze the fatigue
damage. This method may result in conservative fatigue damage
estimation. However, since the peak counting method is the
basis for range counting (RC) and level crossing counting
(LCC) methods, the developed method lays a foundation for the
fatigue life analysis using LCC and RC at the design level.
Using other counting methods will be our future work. With the
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comparison with MCS, it shows that the proposed method is
much more efficient than MCS. However, it does not mean that
the proposed method is the only way to improve the efficiency.
The advanced MCS method may also be applied to reduce the
computational effort. It will also be one of our future works.
The developed method is based on the assumption that the
stochastic loadings are stationary. It may be extended to non-
stationary stochastic loadings. It is our other future research will
be the investigation of the extension.
ACKNOWLEDGMENTS This material is based upon work supported in part by the
Office of Naval Research through contract ONR
N000141010923 (Program Manager – Dr. Michele Anderson),
the National Science Foundation through grant CMMI
1234855, and the Intelligent Systems Center at the Missouri
University of Science and Technology.
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