on fatigue damage computation in random loadings with threshold level and mean value influence

LUGLIO 2005 ING/ 139

ON FATIGUE DAMAGE COMPUTATION IN RANDOM LOADINGS WITH

THRESHOLD LEVEL AND MEAN VALUE INFLUENCE

DENIS BENASCIUTTI, ROBERTO TOVO

Dipartimento di Ingegneria, Università degli Studi di Ferrara, via Saragat 1, 44100 Ferrra, Italy

REPORT DEL DIPARTIMENTO DI INGEGNERIA N. 139

Luglio 2005

Dipartimento di Ingegneria, Università degli Studi di Ferrara

via Saragat 1, 44100 Ferrara tel. +39 – 532 973800, fax. +39 – 532 974870

Dipartimento di Ingegneria, Università di Ferrara Report n. 139 (luglio 2005)

1

INDEX

1. Introduction ...................................................................................................................................1 2. Preliminary defintions ...................................................................................................................1 3. Formulae for damage computation................................................................................................3

3.1. Theoretical formulae for damage computation ...................................................................3 3.1.1. Narrow-band approximation............................................................................4 3.1.2. TB method .......................................................................................................4

3.2. Damage formulae including a threshold level SL................................................................5 3.2.1. Narrow-band approximation............................................................................6 3.2.2. TB method .......................................................................................................6

3.3. Effect of mc on damage .......................................................................................................7 3.3.1. Narrow-band approximation............................................................................8 3.3.2. TB method .......................................................................................................8

3.4. Effect of mc and mr on damage............................................................................................8 3.4.1. TB method .......................................................................................................9

4. Probability of threshold crossing occurrence ................................................................................9 4.1. Poisson approximation ......................................................................................................10 4.2. Linear combination ...........................................................................................................11 4.3. Overall probability of threshold crossing..........................................................................12

5. Conclusions .................................................................................................................................12 6. References ...................................................................................................................................12

1. INTRODUCTION

The classical time-domain approach for estimating the fatigue damage in random loadings is based on count-ing methods and damage accumulation rules (e.g. rainflow count and linear damage accumulation rule).

On the opposite, the frequency-domain approach models the irregular loading as a random process, de-scribed in the frequency-domain by a power spectral density (PSD), and it characterises the statistical vari-ability of rainflow counted cycles by means of a probability density function, since the fatigue damage under the linear rule is easily calculated from the cycle distribution.

The existing frequency-domain methods compute the fatigue damage by referring to amplitude or ampli-tude-mean probability density functions defined over infinite domains, i.e. cycles having an infinitely large value of its peak or valley are virtually possible. In addition, very often such methods estimate the fatigue damage usually neglecting the influence of the mean stress of each rainflow counted cycle and then only fo-cusing on the statistical variability of the amplitudes.

However, a more physically meaningful approach we should account for the existence of a threshold level for the systems (actually representing a ultimate static strength or simply a limit state condition for the system functionality), and also we should provide a more reliable damage prediction by including the influ-ence of the mean stress of counted cycles, since it is well-known the more damaging effect of cycles with a positive mean stress.

The present report proposes a theoretical framework developed to formulate suitable criteria for fatigue damage assessment, which estimate the fatigue damage by including the existence of a system threshold level, and that further include the effect on fatigue damage of mean stresses of counted cycles. This paper presents two approaches are proposed with an increasing complexity: the first one evaluates the effect of the global mean stress value of the random loading, cm (constant), while the second also includes the effect of the random mean stress component rm of rainflow cycles, evaluated in respect to cm (see Figure 1)

2. PRELIMINARY DEFINTIONS

Let )(tX be a stationary random process with mean value cm (see Figure 1) and power spectral density (PSD) )(ωS , characterised by the set of spectral moments [6]:


2

∫+∞

=0

d)( ωωωλ Sii (1)

and bandwidth parameters:

40

22

20

11 λλ

λαλλ

λα == (2)

If )(tX is Gaussian, the mean upcrossing rate 0ν and the rate of peak occurrence pν are:

2

4p

0

20 2

121

λλ

πν

λλ

πν == (3)

Further, if process )(tX is Gaussian with mean cm , its probability density of peaks is given as:

−

−Φ−+−

=−−

−−−

22

22)(

22)1(2

)(22

p1

)()(21

)(2

2

22

2

2

ασα

σα

σπα σασ

X

cmu

X

cmu

X

muemueup X

c

X

c

(4)

The probability density of valleys is symmetrical to that of peaks in respect to cm , that is )v2(v)( pv −= cmpp . The cumulative distribution function of peaks finally is:

−

−Φ−

−

−Φ=−−

22

22)(

222

p1

)(

1)(

2

2

ασαα

ασσ

X

cmu

X

c muemuuF X

c

(5)

Each fatigue cycle counted in a random loading can be described by its peak u and valley v (being always v≥u ), or equivalently by its amplitude s and mean m :

2v,

2v +=−= umus (6)

Counted cycles are clearly random events and should be characterised in a probabilistic sense. A simple way to describe the statistical variability of counted cycles is to define a joint probability density function (PDF), say )v,(uh , depending on peak u and valley v levels; note that )v,(uh is defined only for v≥u . The cor-responding distribution function (CDF):

∫ ∫∞− ∞−

=u

yxyxhu,Hv

dd),(v)( (7)

then gives the probability to count a cycle with peak lower or equal to level u and valley lower or equal to level v .

In the engineering field we are more familiar with other probability densities, as the amplitude-mean PDF (obtained from )v,(uh through a simple variable change):

),(2),( smsmhmsp −+= (8)

and the amplitude PDF (i.e. the marginal distribution associated to ),( msp ):

∫+∞

∞−

= mmspsp d),()( (9)


3

In our analysis we are mainly interested in estimating the distribution of rainflow cycles; note that the rain-flow count is a "complete counting method" and therefore its distribution must satisfy the following “com-pleteness condition” [9]:

=

=

∫

∫

∞+

∞−

vRFv

RFp

d)v,()v(

dv)v,()u(

uuhp

uhpu

(10)

Another cumulative distribution often used in the fatigue analysis of random loadings is the count intensity )v,(uµ , which gives the number of rainflow cycles with peak equal or higher than u and valley equal or

lower than v [8]. The count intensity related to the joint distribution )v,(RF uh is:

∫ ∫+∞

=

=

∞−

=ux

y

yxyxhu,v

RFpRF dd),(v)( νµ (11)

3. FORMULAE FOR DAMAGE COMPUTATION

3.1. Theoretical formulae for damage computation

Each rainflow cycle counted in )(tX is characterised, besides its amplitude s , also by a mean value m , equal to the sum of the global mean value component cm (constant) and the random mean stress component

rm , evaluated in respect to cm (see Figure 1). If the fatigue behaviour is characterised by the S-N relation CNs k = , defined for 0=m , the fatigue

damage rate under the linear rule (neglecting the mean value m of rainflow cycles) is:

∫+∞

=0

RFpa

RF d)( sspCsD

k

ν (12)

where )(RF sp is the probability density of amplitudes; the damage in Eq. (12) does not consider the statisti-cal variability of the mean stress m of rainflow cycles and it also represents a completely theoretical for-mula, since it integrates the amplitude distribution )(RF sp over an infinite domain.

Figure 1: Amplitude s , global mean value cm and random mean stress component rm of a counted cycle.


4

In order to obtain a fatigue damage estimate, which includes also the statistical variability of mean stress m of rainflow cycles, it is necessary to update Eq. (12) by using the amplitude-mean joint probability den-sity ),(RF msp , as:

∫ ∫+∞ +∞

∞−

=0

RFpma,

RF dd),( msmspCsD

k

ν (13)

The formulae presented above show that the fatigue damage depends on the statistical distribution of counted cycles through )(RF sp or ),(RF msp distributions.

3.1.1. Narrow-band approximation If )(tX is Gaussian, distribution )(RF sp is Rayleigh and the fatigue damage is [1]:

( )

+Γ=2

12 00

NBk

CD

kλν (14)

where )( ⋅Γ is the gamma function:

∫∞

−−=Γ0

1 d)( ueua ua (15)

We know that in wide-band processes )(RF sp is not Rayleigh and taht Eq. (14) gives a conservative damage estimate, therefore we need other methods which give a more accurate estimation of )(RF sp [1].

3.1.2. TB method A linear combination is used to estimate the rainflow cycle distribution:

)v,()1()v,()v,( RMLCRF uhbuhbuh −+= (16)

where b is a suitable weight depending on the process PSD and approximated as [1]:

( ) ( ) ( )[ ]( )2

2

2111.2

212121app 1

)(1112.1 2

−−++−+−=

ααααααααα αeb (17)

and where )v,(LC uh and )v,(RM uh are the joint distributions associated to the level-crossing counting:

[ ]

≤−

>−+−+−=

c

cc

muuupmuuupmuupup

uhif)v()(

if)v()()2v()()()v,(

p

vvpLC δ

δδ (18)

and the range-mean counting:

−

−= −−−−−

−−−

−+−

)1(4

v22

1)v,(220

)1(2)v(

221

)1(4v)(

)1(4v

220

RM220

22

22

22

220

2

22

220

22

αλαλπαλα

αααλααλ ueeeuh

ummuu cc

(19)

A linear combination similar to Eq. (16) can be used also for estimating the joint amplitude-mean distribu-tion ),(RF msp , using the distribution associated to the level-crossing counting:

[ ]

≤+

>++−−=

c

cc

mmssmpmmssmpmmspsp

mspif)()(

if)()()()()(),(

p

vvpLC δ

δδ (20)

and to the range-mean counting:


5

( )0

22

2

220

2

2220

)1(2

220

RM)1(2

1),( λααλ

αλαλπ

smm

esemspc −

−−−

−= (21)

which are both symmetric in respect to cmm = . The above formulae are derived from those presented in [1] by including the symmetry in respect to the

global mean value cm . A linear combination is also used for the damage:

RMLCTB )1( DbDbD −+= (22)

where LCD is the damage from the level-crossing counting (which is equal to NBD [1]) and RMD is the (ap-proximated) damage from the range-mean counting, see [1].

3.2. Damage formulae including a threshold level SL

All the above formulae assume that amplitudes and mean values are defined over an infinite domain. How-ever, a more realistic model should consider the existence for process )(tX of a threshold level LS (being

LS− the corresponding symmetric negative value), which could indicate an ultimate static strength or sim-ply a limit state condition for the system.

This threshold level divides the domain of amplitudes (and mean values) into two distinct regions (see for example Figure 2). Therefore, the damage can be expressed as:

excRF,inRF,thrRF, DDD += (23)

in which excRF,D is the damage of cycles, having a maximum and/or minimum greater than LS . The formula for damage calculation which accounts for LS is obtained from Eq. (13):

∫∫+∞

−

− −+=

c

c

mS

kc

mS k

sspC

mSssp

CsD

L

L

d)()(

d)( RFL

p0

RFpa

thrRF, νν (24)

Note that the maximum allowable amplitude (i.e. the limit of integration for amplitudes) is cmS −L and that all amplitudes greater than this limit its are transformed into cmS −L .

s

mmc-SL

LS

LS Figure 2: Schematic representation of the level curves of the function ),(RF msp .


6

If we refer instead to the joint amplitude-mean distribution ),(RF msp , we note that the domain defined by LSsm ≥+ actually corresponds to rainflow cycles for which the process )(tX crosses the threshold level LS , see Figure 2.

Therefore, as done in the previous formula, we have to distinguish between rainflow cycles inside the no-crossing domain and rainflow cycles associated to a threshold crossing occurrence.

Consequently, the formula for damage computation depending on the amplitude-mean joint distribution ),(RF msp and accounting for the threshold level LS is the sum of two contributions:

ma,excRF,

ma,inRF,

ma,thrRF, DDD += (25)

in which ma,inRF,D is the damage corresponding to rainflow cycles for which LSsm <+ (no threshold cross-

ing):

+= ∫ ∫∫ ∫

−

−

+ L L

L

L

0 0RF

0

0RF

pma,inRF, dd),(dd),(

C

S mSk

S

mSk msmspsmsmspsD

ν (26)

while ma,excRF,D is the damage for rainflow cycles with LSsm ≥+ (threshold crossing), which are trans-

formed into cycles with the same mean value m and amplitude mSs −= L :

( )

−++= ∫ ∫∫ ∫

∞+

−−

∞+

+

L

LL L 0RFL

0

RFLpma,

excRF, dd),()(dd),(C

S

mS

k

S mS

k msmspmSmsmspmSDν

(27)

In both equations, the first integral term refers to cycles with 0<m , while the second to cycles with 0>m .

3.2.1. Narrow-band approximation The damage of the narrow-band approximation including the threshold level LS specialises as:

( ) ( ) ( )( )

−+

−+=

−−

0

2L

2L

0

2L

00

thrNB, 2,

212 λ

λγλν cmS

kc

ckemS

mSkC

D (28)

where ),( ⋅⋅γ is the incomplete gamma function:

∫−−=

xua ueuxa

0

1 d),(γ (29)

3.2.2. TB method

This method can give two damage estimates: athrTB,D , depending only on amplitudes, and ma,

thrTB,D , depending on amplitudes and mean values.

Referring only to amplitudes, we apply Eq. (24), obtaining:

athrRM,

athrLC,

athrTB, )1( DbDbD −+= (30)

where the damage contribution from the level-crossing is given explicitly as:


7

( ) ( ) ( )( )

−+

−+=

+=

−−

0

2L

2L

0

2L

02p

aexcLC,

ainLC,

athrLC,

2,

212 λ

λγλ

αν cmSk

cck

emSmSk

C

DDD

(31)

and it is obviously equal to thrNB,D given for the narrow-band approximation, see Eq. (28), since 02p ναν = , while the damage contribution from the range-mean counting is given as:

( ) ( )( )

−+

−+

=

+=

−−

022

2L

2L

022

2L

022

p

aexcRM,

ainRM,

athrRM,

2,

212 λα

λαγλα

ν cmSk

cc

kemS

mSkC

DDD

(32)

Referring instead to the amplitude-mean joint distribution, we apply Eqs. (25)-(27):

( ) ( )

+−+

+

+=

+−++=

−+=

∫ ∫∫ ∫

∫ ∫∫ ∫

−

−

+

−

−

+

L L

L

L

L L

L

L

0 0RM

0

0RM

0 0LC

0

0LC

p

ma,excRM,

ma,inRM,

ma,excLC,

ma,inLC,

ma,thrRM,

ma,thrLC,

ma,thrTB,

dd),(dd),(d)1(

dd),(dd),(

)1(

)1(

S mSk

S

mSk

S mSk

S

mSk

msmspsmsmspsmb

msmspsmsmspsbC

DDbDDb

DbDbD

ν (33)

The damage contribution, ma,thrLC,D , associated to the level-crossing counting is the same given in Eq. (31),

while the damage contribution associated to the range-mean counting is given by the following integrals:

( ) ( )

( )( ) ( )

⌡

⌠ −−

=

⌡

⌠

−+

−

=

−

−−

−−

−

−

−−

−

L

L

220

2L

220

2

L

L

220

2

d)1(2

d2

,2

1)1(2

2

2)1(2L2

20

pthrexcRC,

220

2L)1(2

220

022p

thrinRC,

S

S

mSmmk

S

S

mmk

meemSC

D

mmSke

CD

c

c

αλαλ

αλ

αλπ

ν

αλγ

αλπ

λαν

(34)

which can be solved by numerical integration.

3.3. Effect of mc on damage

The dependence of damage on cm is obtained by inserting the Haigh correction in Eq. (24):

[ ]

−

−−

+

−= ∫

−

)(1)(1

d)()(1C LRF

L

L

0RF

L

paRF,

L

SPSmm

mSssp

SmmsD

k

cc

cmS k

ccm

c

c IIν

(35)

in which )(RF sP is the cumulative distribution function of amplitudes. The indicator function ( 1)( =xI if 0≥x , 0)( =xI elsewhere) is used to specify that the mean value correction is applied only when 0>cm .


8

The formula for the correction of cm , Eq. (35), even if approximated, is easily applicable to all existing spectral methods (e.g. narrow-band approximation, TB method [1, 2], Dirlik method [3], Zhao-Baker method [10]) which provide an estimate of the amplitude distribution )(RF sp .

The error given by the proposed formula in neglecting in the damage estimate the random mean stress component rm depends on the relative importance of this component in respect to the global mean value component cm . Such error should diminish when the frequency bandwidth of process )(tX decreases, since in narrow-band processes fatigue cycles are virtually symmetric respect to cm , and all have 0≅rm .

3.3.1. Narrow-band approximation Writing explicitly Eq. (35) using a Rayleigh distribution for )(RF sp gives:

( ) ( )

−−

+

−+

−=

−−

0

2L

2

L

L

0

2L

L

00,NB )(12

,2

1)(1

2 λ

λγ

λν c

c

mSk

cc

cc

k

ccm e

SmmmSmSk

SmmCD

II (36)

3.3.2. TB method We refer to the damage of the TB method calculated in terms of amplitudes:

aRC,

aLC,

aTB, )1(

ccc mmm DbDbD −+= (37)

in which aLC, cmD and a

RM, cmD are the damage of the level-crossing and range-mean countings, computed as a

function of amplitudes; in particular, aLC, cmD coincides with

cmDNB, given in Eq. (36), while aRM, cmD , com-

puted according to the distribution )(RM sp derived from Eq. (21), is:

( ) ( )

−−

+

−+

−=

−−

022

2L

2

L

L

022

2L

L

022pa

RC, )(12,

21

)(1

2 λα

λαγ

λαν c

c

mSk

cc

cc

k

ccm e

SmmmSmSk

SmmCD

II (38)

3.4. Effect of mc and mr on damage

The theoretical damage estimation could be further improved by inserting in formulae also the influence of the random mean stress component rm . The formulae for damage computation, given in Eq. (25)-(27), ex-pressed as a function of ),(RF msp , are modified by inserting the Haigh correction for cycles with 0>m .

The formula of the rainflow damage depending on ),(RF msp then is:

ma,,exc

ma,,in

ma,RF, mmm DDD += (39)

in which ma,,in mD is the damage calculated for cycles with LSsm <+ :

−+= ∫ ∫∫ ∫

−

−

+ L L

L

L

0 0RF

L

0

0RF

pma,in dd),(

1dd),(

C

S mS k

S

mSk

,m msmspSm

smsmspsDν

(40)

while ma,,exc mD is the damage calculated for cycles with LSsm ≥+ :


9

( )

++= ∫ ∫∫ ∫

∞+

−−

∞+

+

L

LL L 0RFL

0

RFLpma,

,exc dd),(dd),(C

S

mS

k

S mS

km msmspSmsmspmSD

ν (41)

The Haigh correction is applied to cycles with 0>m . The previous formulae are applicable only to those methods which provide an estimate of the joint distribution ),(RF msp [1, 2, 4, 5, 7, 9].

As an example, in the following section we apply the formula to the TB method.

3.4.1. TB method The rainflow damage can be written as:

−−+−+

+

−+=

−+=

∫ ∫∫ ∫

∫ ∫∫ ∫

−

−

+

−

−

+

L L

L

L

L L

L

L

0 0RC

L

0

0RC

0 0LC

L

0

0LC

p

ma,RC,

ma,LC,

ma,TB,

dd),(1

)1(dd),(d)1(

dd),(1

dd),(

)1(

S mS k

S

mSk

S mS k

S

mSk

mmm

msmspSm

sbmsmspsmb

msmspSm

sbmsmspsbC

DbDbD

ν

(42)

in which the first two terms are the damage of the level-crossing, which is equal to the damage cmDNB, given

in Eq. (36), while the other two are the damage associated to the range-counting and given as:

mmm DDD excRC,inRC,

ma,,RC += (43)

in which mD inRC, is the damage of the range-counting cycles associated to the condition LSsm <+ :

( )

( )( )

⌡

⌠

−+

−−

=

−

−−−

L

L

220

2

d2

,2

1)(1)1(2

2220

2L

L

)1(2

220

022p

RC,in

S

S

k

mmk

m mmSk

Smme

CD

c

αλγ

αλπ

λαν αλ

I (44)

while mD excRC, is the damage of the range-counting cycles associated to the condition LSsm ≥+ :

( ) ( )

⌡

⌠

−−

−=

−

−−

−−

−L

L

220

2L

220

2

d)(1)1(2

2)1(2

L

L

220

pexcRC,

S

S

mSmmkm mee

SmmmS

CD

c

αλαλ

αλπ

νI

(45)

The above integrals can be solved by numerical integration.

4. PROBABILITY OF THRESHOLD CROSSING OCCURRENCE

In the preceding sections, we computed a fatigue damage also for all rainflow cycles associated to a thresh-old occurrence, i.e. cycles in which the maximum and/or minimum exceeds the threshold LS . Another possibility is to discard all such cycles from the fatigue damage computation and to compute the probability

fP that such cycles will produce an immediate fracture of the system.

Let 1fP be the probability of occurrence of a threshold crossing of the level LS for a single rainflow

cycle counted in )(tX . This probability can be computed by integrating the )v,(RF uh distribution outside the domain LSu ≥ and Lv S≤ .


10

The threshold levels LS and LS− are assumed symmetric; if process )(tX is assumed Gaussian, it is symmetric and therefore its distribution )v,(RFC uh is also symmetric. Then the probability of threshold crossing occurrence for a single rainflow cycle can be computed as:

∫ ∫∫ ∫+∞

=

−=

−∞=

+∞

=

=

−∞=

−⋅=L

L

L

v

vRF

v

vRF

1 dvd)v,(dvd)v,(2Su

S

Su

u

f uuhuuhP (46)

By means of the "completeness condition", see Eq. (10), and by symmetry, we can use the probability den-sity of peaks, by writing the first integral in terms of the cumulative distribution function of peaks, )(p uF :

∫ ∫∫+∞

=

−=

−∞=

+∞

=

−⋅=L

L

L

v

vRFp

1 dvd)v,(d)(2Su

S

Suf uuhuupP (47)

Note that the second integral gives the probability of threshold crossing for the joint event LSu ≥ and

Lv S≤ . This probability can be expressed in terms of the rainflow count intensity, see Eq. (11):

( ) ( )p

LLRFLp

1 ,)(12ν

µ SSSFf−−−⋅=P (48)

The expression given in Eq. (48) is exact and it gives the probability that a single rainflow cycle will pro-duce a threshold crossing.

However, even if Eq. (48) has general validity, it can not be solved explicitly, since we do not know the analytical expression of the rainflow count intensity )v,(RF uµ .

In the following sections two possible approximations are proposed; to simplify all proposed equations, we will assume that 0=cm .

4.1. Poisson approximation

The approximation of the rainflow count intensity based on the Poisson convergence of the level upcrossing spectrum is [4]:

)v()()v()()v,(Pois

RF µµµµµ

+≈

uuu (49)

where )(xµ is the upcrossing spectrum. This approximation is valid for 0>>u and 0v << . We define the rainflow cumulative probability associated to the rainflow count intensity )v,(RF uµ as:

( ))v()()v()()v,()v,(

pp

PoisRFPois

RF µµνµµ

νµ

+≈=

uuuuq (50)

In a Gaussian load, the upcrossing spectrum is given by the Rice’s formula [2]:

−= 2

2

0 2exp)(

X

xxσ

νµ (51)

and therefore the approximate rainflow cumulative probability is:

−+

−

≈

2

2

2

22Pois

RF

2vexp

2exp

)v,(

XX

uuq

σσ

α (52)

since p02 ννα = . Calculating the cumulative distribution for LSu = and Lv S−= gives:


11

2

2L

2

2LL

PoisRF

2

),(X

S

e

SSqσ

α−

≈− (53)

Consequently, the explicit expression for the probability of fracture (for 1 rainflow cycle) is:

)2exp(21112 22

L

222

L2222

2

L1 2

2

XX

S

Xf S

SeSX

L

σα

ασαα

ασσ −

−Φ−

−Φ−⋅=

−P (54)

4.2. Linear combination

In the TB method, we used a linear combination to estimate the distribution )v,(RF uh , see Eq. (16). Simi-larly, a linear combination can be used to estimate the cumulative distribution )v,(RF uq :

)v,()1()v,()v,( RMLClinRF uqbuqbuq −+= (55)

where the cumulative distributions for the narrow-band approximation and the range-mean count are, re-spectively [2]:

( ) ( )

+−++=

−−v)(v)v,(

2

2

2

2

2v

22LC ueueuq XX

u

II σσα (56)

and:

−

−−Φ+

−

−−Φ−=

−−

)1(2

)21(v

)1(2

)2(1v1)v,(222

222

222

222

v

2RM2

2

2

2

ασα

α

ασα

αα σσ

X

u

X

ueueuq XX (57)

Calculating the cumulative distribution for LSu = and Lv S−= gives:

2

2L

22LLLC ),( X

S

eSSq σα−

=− (58)

and:

−

−−Φ+

−

−Φ−=−−

)1(

)1(

)1(

)(11),(222

22L

222

22L2

2LLRM2

2L

ασαα

ασααα σ

XX

SSSeSSq X (59)

By using the fact that )(1)( xx Φ−=−Φ , we have:

−

−Φ−=−−

)1(

)(112),(222

22L2

2LLRM2

2L

ασααα σ

X

SSeSSq X (60)

The final formula for the cumulative distribution then is:

−

−Φ−−+=−−

)1(

)(11)1(2),(222

22L2

2LLlinRF

2

2L

ασααα σ

X

SSbbeSSq X (61)

Consequently, the explicit expression for the probability of fracture (for 1 rainflow cycle) is:


12

−

−Φ−−+

−

−Φ−

−Φ−⋅=

−

−

)1(

)(11)1(2

1112

222

22L2

2

22

L2222

2

L1

2

2L

2

2L

ασα

αα

ασ

ααασ

σ

σ

X

S

X

S

X

f

Sbbe

SeS

X

XP

(62)

4.3. Overall probability of threshold crossing

In the hypothesis that rainflow cycles are independent, the probability of survival after N cycles is simply the product of probabilities:

( )Nfs11 PP −= (63)

For example, the number of rainflow cycles for which there is a probability of survival of 50% then is:

( )( ) ( )1ln693.01ln

ln 11 −=

−= f

f

sfN P

PP (64)

5. CONCLUSIONS

This report provides an analytical framework for the frequency-domain damage assessment of a random process )(tX , which includes the effect of a threshold level and the effect of mean stress values on the fa-tigue damage computations. Approximate approaches for evaluating the effect of the mean value on the fa-tigue damage are presented. The rainflow cycles counted in a given random process )(tX have a mean value stress rc mmm += , which is the sum of the global mean value cm (constant) of process )(tX , and the ran-dom mean stress component rm evaluated in respect to cm . The first approach considers only the effect of the global mean value cm and it is applicable to all such spectral methods existing in the literature which provide an estimate only of the amplitude distribution of rainflow cycles (e.g. narrow-band approximation, Dirlik method, Zhao-Baker method). The second approach evaluates the effect of both cm and rm mean values, based on the joint distribution ),(RF msp and it has been explicitly applied to the TB method.

6. REFERENCES

1 Benasciutti D., Tovo R. Spectral methods for lifetime prediction under wide-band stationary random processes. Int. J. Fatigue, 2005, 27(8): 867-877.

2 Benasciutti D., Fatigue analysis of random loadings. PhD Thesis, Department of Engineering, Univer-sity of Ferrara (Italy), March 2005.

3 Dirlik T. (1985) Application of computers in fatigue analysis. PhD Thesis, University of Warwick, UK. 4 Johannesson P., Thomas J. Extrapolation of rainflow matrices. Extremes, 2001, 4(3): 241-262. 5 Lindgren G., Broberg K.B. Cycle distributions for Gaussian processes – exact and approximate results.

Extremes, 2005, 7(1): 69-89. 6 Lutes L.D., Sarkani S. Stochastic analysis of structural and mechanical vibrations, Prentice-Hall, 1997. 7 Nagode M., Klemenc J., Fajdiga M. Parametric modelling and scatter prediction of rainflow matrices.

Int. J. Fatigue, 2001, 23: 525-532. 8 Rychlik I. Note on cycle counts in irregular loads. Fatigue Fract. Engng. Mater. Struct., 1993, 16(4),

377-390. 9 Tovo R. Cycle distribution and fatigue damage under broad-band random loading. Int. J. Fatigue, 2002,

24(11): 1137-1147. 10 Zhao W., Baker M.J. On the probability density function of rainflow stress range for stationary Gaus-

sian processes. Int. J. Fatigue, 1992, 14(2): 121-135.

on fatigue damage computation in random loadings with threshold level and mean value influence

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