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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


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


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.


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


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.


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)


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.


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)


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


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


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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