1 s2.0-s014211230600137 x-main

Probabilistic high cycle fatigue behaviour prediction based
on global approach criteria q
R. Ben Sghaier a
, Ch. Bouraoui a
, R. Fathallah b,*, T. Hassine a
, A. Dogui a
a
Laboratoire de Génie Mécanique, Ecole Nationale d’Ingénieurs de Monastir, Avenue Ibn El Jazzar, 5019 Monastir, Tunisia
b
Laboratoire de Génie Mécanique, Ecole Nationale d’Ingénieurs de Sousse, Cité Taffala, 4003 Sousse, Tunisia
Received 15 July 2005; received in revised form 27 March 2006; accepted 28 March 2006
Available online 6 June 2006
Abstract
This paper presents an approach to predict the reliability of high cycle fatigue (HCF) behaviour of metallic parts using the multi-axial
HCF criterion of Crossland, for the case of normally distributed in-phase fully reversed torsion and bending loading and HCF material
characteristic parameters. The dispersions of: (i) the HCF criterion material characteristic parameters and (ii) the applied loading have
been taken into account. The reliability of the HCF resistance was determined by using the ‘‘Strength Load’’ with first order reliability
method (FORM). This approach gives iso-probabilistic Crossland diagrams (PCD) corresponding to different coefficient of variation
(COV) of loading and material HCF characteristic parameters. An application has been carried out on a hard steel metal submitted
to a fully reversed torsion and bending loading. Two types of various dispersed loadings, having different COV, are studied: (i) only
random torsion amplitude loading and (ii) both random torsion and bending amplitude loading. The proposed method allows evaluation
of the influences of different dispersions on the reliability of the HCF behaviour. It has been observed that, the proposed method is qual-
itatively consistent with the physical observations and leads to a more reliable HCF prediction compared to the deterministic approach,
which takes into account separately two fatigue limits corresponding to a given reliability value, in the HCF criterion of Crossland.
Ó 2006 Elsevier Ltd. All rights reserved.
Keywords: Probabilistic fatigue; High cycle fatigue; Fatigue reliability; FORM; Global approach fatigue criteria; Fatigue scattering parameters
1. Introduction
Practical computation of HCF reliability of mechanical
components, used in automotive, aerospace, naval struc-
tures, nuclear plants etc. is much needed [1–4] for a better
secure design of mechanical parts. The HCF reliability is
usually affected by many uncertainties and characterised
by several random variables. Various causes of uncertainty
having influence on the HCF behaviour are summarised by
Sevensson [5], as follows: (i) material properties, (ii) struc-
tural properties of components, (iii) load variation, (iv)
parameter estimation; and (v) model error.
The prediction of the HCF behaviour of a representative
volume element of a mechanical component is carried out,
in the majority of cases, by deterministic multi-axial HCF
criteria, such as Sines [6], Crossland [7], Dang Van [8]
and Papadopoulos [9,10]. Their applications are carried
out in a deterministic way, and generally use experimental
fatigue limits corresponding to failure probability of 50%.
They do not include the stochastic effects, particularly the
loading and the material dispersions effects, on the HCF
prediction. As a consequence, the calculations may predict,
but not correctly, the total reliability of the HCF behaviour
of mechanical parts.
Several works were carried out to take into account the
probabilistic effects on the fatigue behaviour. The majority
of them have been used in the one-dimensional loading
0142-1123/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.ijfatigue.2006.03.015
q
This work is a part of the Phd thesis of Mr. R. Ben Sghaier supervised
by C. Bouraoui and R. Fathallah.
*
Corresponding author. Tel.: +216 73 332 657; fax: +216 73 332 65.
E-mail addresses: raouf.fathallah@gmail.com, raouf.fathallah@
issatso.rnu.tn (R. Fathallah).
www.elsevier.com/locate/ijfatigue
International Journal of Fatigue 29 (2007) 209–221
International
Journalof
Fatigue

case, giving therefore a probabilistic stress–number of
cycles curves [11,12]. The weakest-link theory is the first
approach treating the statistical effect in fatigue. Such an
approach uses the Weibull model [13]. This model has been
used recently, by Bomas et al. [14] on bearing steel and by
Hild and Roux [15] and Chantier et al. [16] on a cast iron to
explain the statistical distribution on fatigue strength.
More recently, Morel [4] has used a combination of the
concept of the weakest-link together with a critical plane
damage model based on a microplasticity analysis to
describe the distributions of the fatigue limit and the fati-
gue life under different loadings.
The aim of this work is to develop a practical probabilis-
tic method to evaluate the HCF reliability of metallic parts.
The HCF Crossland criterion has been used by taking into
account the following scatterings: (i) the dispersion of the
HCF Crossland material characteristic parameters and (ii)
the dispersion of the applied loading. The reliability has
been computed by using the approximation method of first
order reliability method (FORM) based on the strength load
method [17–21]. Our proposed method was applied to deter-
mine PCD, corresponding to two different types of loadings.
The first one is characterised by a constant value of maximal
hydrostatic pressure and the second one represents the case
of an in-phase fully reversed torsion-bending loadings.
2. Theoretical background
2.1. HCF Crossland criteria
The Crossland criterion [7], based on the plasticity crite-
rion of Von Mises, has the advantage of being multi-axial
and corresponds with experimental results [9]. It is defined
by the limitation of an equivalent stress (req) expressed by a
linear relationship between the amplitude of octahedral
shear stress (DCoct,a) and the maximum hydrostatic pres-
sure (Pmax) for a representative volume element. The
HCF resistance defined for a specified number of cycles
(generally 107
cycles) is given by the following inequality:
req ¼ DCoct;a þ aCPmax 6 bC ð1Þ
where (DCoct,a) is obtained by a double maximisation
DCoct;a ¼
1
2
ffiffiffi
2
p max
ti2T
max
tj2T
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
SðtiÞ SðtjÞ : ðSðtiÞ SðtjÞÞ
q

ð2Þ
With; SðtÞ ¼ rðtÞ
1
3
trðrðtÞÞ I ð3Þ
S(t) is the cyclic stress deviator tensor at two instantaneous
times ti and tj varying within [0,T], and I is the unit tensor
Pmax ¼
1
3
max
t2T
½trðrðtÞÞ ð4Þ
where, aC and bC are two material parameters which can be
identified by a fully reversed torsion test (i.e., DCoct,a = t1;
Pmax = 0) and a fully reversed bending test (i.e., DCoct;a ¼
f1=
ffiffiffi
3
p
; Pmax = f1/3) on smooth specimens, where t1
and f1 are, respectively, the fatigue limits under fully re-
versed torsion and fully reversed bending. Then, the mate-
rial parameters are expressed as follow:
aC ¼ 3
t1
f1

ffiffiffi
3
p
ð5Þ
and; bC ¼ t1 ð6Þ
Two deterministic principle cases of prediction can be iden-
tified: (i) a safe prediction, when the representative point of
the loading is strictly under the HCF limit line of the Cross-
land criterion and, (ii) a failure prediction, when the repre-
sentative point is above the Crossland line.
2.2. First order reliability method review
To compute the reliability, various approximation meth-
ods are used. Among these, The FORM is considered to be
one of the most reliable computational and basic method
for structural reliability [22,23,12].
For a vector of random variables {xi}, representing
uncertain structural quantities, a function of performance
G, separating the security and the failure fields, is given by:
GðxiÞ ¼ SðxiÞ LðxiÞ: ð7Þ
Where S and L are the strength and the load functions,
respectively, and xi represents an element of the random
vector {xi}.
In that case, if the inequality G(xi) 0 is verified, this
indicates a structural safety condition. In the opposite case,
however, if G(xi) 0, this means a failure of such a struc-
ture. G(xi) = 0 represents the limit state.
Using the performance function G, the probability of
failure can be determined by the following expression:
Pf ¼ PrðLðxiÞ P SðxiÞÞ ¼ PrðGðxiÞ 6 0Þ
¼
Z
GðxiÞ60
ffxigdx1 . . . :dxn ð8Þ
where ffxig denotes the joint probability density function of
{xi}.
The probability of failure can be also expressed as
follows:
Pf ¼ Pr
GðxiÞ mGðxiÞ
rGðxiÞ
6
mGðxiÞ
rGðxiÞ

ð9Þ
where, mGðxiÞ and rGðxiÞ are, respectively, the mean value
and the standard deviation of G function.
By introducing the standard normalised function UGðxiÞ,
and the reliability index Ic, which are expressed as follow:
UGðxiÞ ¼
GðxiÞ mGðxiÞ
rGðxiÞ
ð10Þ
Ic ¼
mGðxiÞ
rGðxiÞ
ð11Þ
The expression of Pf is then:
Pf ¼ PrðUGðxiÞ 6 IcÞ ð12Þ
Assuming that S and L obey a normal distribution, it is
established that, when the load and the strength function
210 R. Ben Sghaier et al. / International Journal of Fatigue 29 (2007) 209–221

are independent, the reliability index can be expressed as
follows:
Ic ¼
mS mL
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
S þ r2
L
p ð13Þ
Where, (mS,mL) and (rS,rL) are the mean values and the
standard deviations of the functions S and L, respectively.
Finally, the probability of failure Pf can be determined
by the following expression:
Pf /ðIcÞ ð14Þ
Where, / is the cumulative distribution function of normal
random variable.
The reliability R, which represents the probability of
safety, is given by the following relationship:
R ¼ 1 Pf ð15Þ
3. Probabilistic HCF Crossland criterion
In this work, we assume that, when taking into account
the dispersions effects of the loading and the materials
parameters, the representative point of the cyclic loading
and the strength Crossland limit line will be transformed
into random distributions.
3.1. Modelling of the material properties random distribution
The dispersions due to the HCF material properties can
be characterised by the dispersions of the two Crossland
material parameters aC and bC, which are obtained from
t1 and f1 (Eqs. 5 and 6). It is well established, that the
different HCF stochastic experimental methods, allow hav-
ing fatigue limits, with normal distributions from the S–N
curves [24–26]. To simulate this dispersion, we assume that
t1 and f1 to be random variables normally distributed
characterised by their mean and standard deviation values.
We proceed by a normal random sampling of t1 and
f1, leading to a sample of the material parameters aC
and bC. The use of all combinations gives a beam of lines
(Fig. 1) representing two-dimensional distributions in the
Crossland diagram.
It is worth noticing that generally, for each category of
metals (mild, hard or brittle) a physical condition results
in a relationship between t1 and f1. In particular, for
hard metals, where the ultimate strength is higher than
680 MPa, the following condition is proposed by Papado-
poulos et al. [9]:
0:577 6
t1
f1
6 0:8 ð16Þ
This physical condition will be taken into account in the sim-
ulation of the material dispersion. The influence of this con-
dition on the adopted statistical hypotheses must be verified.
The Crossland diagram is, then, transformed into three
zones: a quasi-security zone, a quasi-failure zone, and an
intermediate probable failure zone (corresponding to the
confidence level of six standard deviations).
3.2. Modelling of the loading random distribution
In this work, the amplitudes of the in phase fully
reversed torsion-bending loading stresses, are supposed to
be normally distributed. It is evident that the dispersion
of the maximum hydrostatic pressure Pmax is normally dis-
tributed. However, it is not the case for DCoct,a. In our
approach, we assume that Pmax and DCoct,a are supposed
to be a Gaussian correlated set. In this case, we suppose
that, Pmax and DCoct,a are defined by their mean values
l1 and l2 and their standard deviations r1 and r2, respec-
tively. The probabilistic density function (PDF) is repre-
sented by a three-dimensional bell-shaped surface. It is
expressed as follows:
uðPmax; DCoct;aÞ ¼
1
2pr1r2
ffiffiffiffiffiffiffiffiffiffiffiffi
1 r2
p
e
1
2ð1r2Þ
ð
ðPmaxl1Þ
r1
Þ2
þ
ðDCoct;al2Þ
r2
2
2r
ðPmaxl1Þ
r1

ðDCoct;al2Þ
r2

ð17Þ
where, r is the correlation coefficient between Pmax and
DCoct,a, defined by:
r ¼
r1;2
r1 r2
ð18Þ
r1, 2 is the co-variance between Pmax and DCoct,a.
The projection of the PDF surface on the Crossland
plan is represented by concentric ellipses (Fig. 2), having
the following equation:
ðPmax l1Þ
r1
2
þ
ðDCoct;a l2Þ
r2
2
2r
ðPmax l1Þ
r1

ðDCoct;a l2Þ
r2
¼ k2
; k is a constant ð19Þ
The obtained elliptical surfaces shape corresponding to the
loading dispersion surface will be verified.
Fig. 1. The material dispersion surface in the probabilistic Crossland
criterion.
R. Ben Sghaier et al. / International Journal of Fatigue 29 (2007) 209–221 211

4. Computations of the HCF reliability
In this work, the strength and the load functions, S and
L, are defined by the intersection between a specific direc-
tion, depending on the cyclic loading, and the both disper-
sion surfaces. The two functions are assumed to be
independent and obeying normal distributions. The nor-
mality assumption of the L and S functions will be quali-
fied and justified. The general procedure, in the case of
random in-phase fully reversed torsion-bending loading,
is developed by the following steps:
Step 1: Input data
The input data are normally distributed. They are as
follow: (i) the material characteristics: the fully reversed
bending f1 and the fully reversed torsion t1, fatigue
limits, with their corresponding COV values, (ii) the
applied loading characteristics (the amplitude of the tor-
sion loading sa, the amplitude of the bending loading ra
and their corresponding COV values).
Step 2: Modelling of the HCF material parameters dis-
persion
A N normal random sampling of values, of t1 and
f1, defined by their mean and COV values, respectively,
has been carried out. A set of ðaCi
; bCi
Þ constitutes, then,
an event obtained from (f1(i),t1(i)) by using Eqs. (5)
and (6). It leads to a Crossland fatigue limit line.The N
events of ðaCi
; bCi
Þ, generated by the normal random
sampling, represent a beam of lines (Fig. 1). The restric-
tive condition introduced by Papadopoulos et al. [9]
given by the inequality (16), reduces the number of events
from N to N0
(N0
is lower than N). A zone of probabilistic
failure, between the quasi-security and the quasi-failure
zones is then obtained by a beam of N0
lines. The used
number N, for the normal random sampling, is optimised
to have a normal distribution law of the number of lines,
crossed by any axis in the Crossland diagram. For each
case, different calculations have been made by changing
the number N. The selected N is chosen when the 99%
confidence limit value of t1 obtained by the proposed
approach is saturated (Table 1).
Step 3: Modelling of the loading function dispersion (L)
A N normal random sampling of values, of sa and ra
amplitudes defined by their mean and COV values,
respectively, has been carried out. Each event ðrai
; sai
Þ
corresponds to a point ðmxi
; myi
Þ within the loading dis-
persed surface. The coordinates of this point are
expressed as follow:
mxi ¼
rai
3
ð20Þ
and; myi
¼
ffi
r2
ai
3
þ s2
ai
s
ð21Þ
The N events of ðmxi
; myi
Þ, generated by the normal ran-
dom sampling, lead to obtain an elliptical surface cha-
racterising the loading dispersion in the Crossland
diagram (Fig. 2). This loading dispersion surface is de-
fined by: (i) the coordinates of its center (mx,my) where,
mx and my are, respectively, the mean values of
mxi and myi
, and (ii) its principals axes (nPmax,fDCoct,a)
defined by an angle a between the major axis nPmax
and the Pmax axis of the Crossland diagram (Fig. 2),
which is expressed as follows:
tanð2aÞ ¼
2rr1r2
ðr1Þ2
ðr2Þ2
ð22Þ
where, r1 and r2 are the standard deviation values of mxi
and of myi
, respectively.And, r is the correlation coeffi-
cient defined in (18).
In this work, the major axis of the elliptical surface is
called the correlation direction. It corresponds to the
most unfavourable case (the most probable failure
point). It is used to compute the reliability. The PDF
given in Eq. (17), when transformed in the principal axis
(nPmax,fDCoct,a) is expressed as follows:
uðnPmax; fDCoct;aÞ ¼
1
2p:rn:rf
e
1
2
nPmax
rn
2
þ
fDCoct;a
rf
2

ð23Þ
where, rn and rf are the standard deviations of the load-
ing, along the major nPmax and the minor axis fDCoct,a
of the dispersion elliptical surface, respectively. They
are given by the following expressions:
rn ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
1 cos2 a þ rr1r2 sin 2a þ r2 sin2
a
q
ð24Þ
and; rf ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r2
1 sin2
a rr1r2 sin 2a þ r2 cos2 a
q
ð25Þ
In this work, the standard deviation rn is assumed to be
equal to the loading standard deviation rL. Finally, the
load function L used in the FORM to evaluate the reli-
ability is then characterised by the center point (mx,my)
and its standard deviation rL.
mx
my
Pmax
ΔCoct,a
Correlation direction
kσξ
kσζ
ξ Pmax
ζΔCoct,a
α
Fig. 2. The elliptical dispersion surface of the both random torsion and
bending amplitudes loading.

Step 4: Modelling of the strength function S
In order to characterise the dispersion due to the
HCF material properties, we use the intersection points
(xi,yi) between the dispersed Crossland lines and the
correlation direction. These points are supposed nor-
mally distributed and characterised by their mean and
standard deviation values.The coordinates (xi,yi) are
expressed as follow:
xi ¼
bCi
my þ mx tanðaÞ
aCi
þ tanðaÞ
ð26Þ
and; yi ¼ bCi
aCi
xi ð27Þ
where, aCi
and bCi
are defined in the Step 2.The coordi-
nates of the intersection point (xm,ym), between the
50%-deterministic Crossland line obtained by the mean
values of t1 and f1, and the correlation direction.
These coordinates are expressed as follow:
xm ¼
bC my þ mx tanðaÞ
aC þ tanðaÞ
ð28Þ
and; ym ¼ bC aC xm ð29Þ
where, aC and bC correspond to the 50%-deterministic
Crossland line as expressed in Eqs. (5) and (6).
Let di be the distance between a point (xi, yi) and the
deterministic point (xm,ym). It is expressed as follows:
di ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðxi xmÞ2
þ ðyi ymÞ2
q
ð30Þ
di corresponds to an element of a random variable d
having a standard deviation rd expressed as follows:
rd ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
P
ðdiÞ2
N0
1

P
di
N0
1
2
s
ð31Þ
In this work, the standard deviation rd is supposed to be
equal to the strength standard deviation rS.
Finally, the strength S used in the FORM to evaluate
the reliability is then characterised by the deterministic
point (xm,ym) and its standard deviation rS.
Step 5: Calculation of the reliability index Ic
In this work, the functions S and L are supposed to
be normal and independent. The reliability index IC is
then expressed as follows:
Ic ¼
d0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
r2
L þ r2
S
p ð32Þ
where, d0 is the algebraic distance between the determin-
istic point (xm,ym) and the ellipse’s center point (mx,my),
given by the following equations:
d0 ¼ þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðmx xmÞ
2
þ ðmy ymÞ
2
q
if mx xm ð33Þ
and;
d0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðmx xmÞ2
þ ðmy ymÞ2
q
if mx xm ð34Þ
Finally, according to the FORM we deduce the reliabil-
ityR by using Eq. (15)
R ¼ 1 /
d0
r2
L þ r2
S
p
!
ð35Þ
where, / is the cumulative distribution function of nor-
mal random variable.
Table 1
The selected numberN for the normal random sampling to characterise the material dispersion surfaces
COVt1 (%) COVf1 (%) N rS rS calculated on N N0
rS (at 99%) calculated
on N0
t1 experimental 99%
limit confidence
t1 calculated
on N0
(99%)
1 1 5.103
1.962 1.990 5.103
1.990 191.63 191.61
104
1.971 104
1.971 191.67
15.103
1.960 15.103
1.960 191.67
2.104 a
1.952 2.104
1.952 191.67
2 2 5.103
3.924 3.939 4980 2.578 187.06 190.25
104
3.936 9973 3.080 189.08
15.103
3.962 14961 3.537 187.98
2.104 a
3.946 19938 3.986 187.12
3 3 5.103
5.886 5.857 4846 4.398 182.49 186.02
104
5.912 9666 4.842 183.16
15.103
5.892 14482 5.233 183.15
2.104 a
5.930 19320 5.634 183.10
5 5 5.103
9.81 9.765 4296 8.506 173.4 176.45
104
9.746 8664 8.424 176.63
15.103
9.835 12915 8.455 176.57
2.104 a
9.804 17340 8.507 176.45
7 7 5.103
13.734 13.656 3953 9.227 164.2 174.79
104
13.689 7724 9.877 173.25
15.103
13.750 11723 10.503 171.79
2.104
13.781 15608 11.038 170.56
5.104
13.739 39107 11.860 168.64
105 a
13.763 77670 11.879 168.58
a
Corresponds to the selected N.

5. Iso-probabilistic Crossland diagram
The PCD, corresponding to constant values of reliabil-
ity, can be obtained as a direct application of this proposed
probabilistic HCF approach. These diagrams are a func-
tion of various dispersed material characteristics and differ-
ent types of random loading.
For a given type of random loading and for a dispersed
material, the PCDs are obtained by using repetitive compu-
tations corresponding to different loadings. They are car-
ried out according to the following steps: (i) determining
the Crossland uncertainty zone characterising the material
as indicated in Section 4 Step 2, (ii) determining the char-
acteristics of the dispersion loadings zones (centers of ellip-
tical surface, directions of correlation and the standard
deviations) according to Eqs. (20)–(25), (iii) recording the
different centers of the dispersed loadings zones corre-
sponding to the target reliability, and finally (iv) fitting
the iso-reliability recorded centers.
Two cases of an in-phase fully reversed torsion-bending
multi-axial loading are considered: (i) only random torsion
amplitude loading and (ii) both random torsion and bend-
ing amplitudes loading.
5.1. Case of both random amplitudes of torsion and bending
loading
This loading is defined by the two amplitudes of the fully
reversed bending ra and of the fully reversed torsion sa. Let
h be the angle characterising the radial loading direction in
the Crossland diagram (Fig. 3). It is expressed as follows:
tanðhÞ ¼
DCoct;a
Pmax
ð36Þ
For this type of loading the angle h varies between 0° (case
of purely torsion loading) and 60° (case of purely bending
loading). For different values of the angle h, we increase
incrementally the level of the dispersed loading until having
the target reliability.
5.2. Case of only random amplitudes of torsion loading
(constant Pmax)
The loading dispersion surface is reduced, in this case, to
a vertical segment, corresponding to the dispersion of
DCoct,a (Fig. 4). In this case, for different values of Pmax,
we increase incrementally the level of the dispersed loading
until having the target reliability.
6. Applications
The studied material, here, is a hard steel, having mean
value of t1 equal to 196.2 MPa, and mean value of f1
equal to 313.9 MPa [9]. Two applications were carried
out: (i) a first one, to evaluate the HCF reliability, in a case
of random in-phase fully reversed torsion-bending loading.
And (ii) a second one, to determine the PCD for different
COV values of the HCF material characteristics and of
the applied loading parameters.
6.1. Application 1: HCF reliability evaluation
In this application, the COV of the material parameters:
t1 and f1 were both taken equal to 1%. The in-phase fully
reversed torsion-bending loading are: the mean amplitudes
of ra and of sa are both chosen equal to 160 MPa. Three cases
have been studied, by changing the COV values of ra and sa
(2% and 5%) to show their influences on the reliability. The
results are reported in Table 2 and illustrated in Fig. 5.
6.2. Application 2: PCD
6.2.1. Case of deterministic loading
The PCD have been determined for different reliabilities
in the case of deterministic in-phase fully reversed torsion-
Fig. 3. The load and the strength distributions (case of both random
torsion and bending amplitudes loading).
Fig. 4. The load and the strength distributions (case of only random
torsion amplitude loading).

bending loading. Different cases using various COV values
of HCF material characteristics t1 and f1 were carried
out (Table 3). In this application, it is worth noticing, that
for this particular case, the obtained PCD were obtained as
proposed in Section 5.1 (calculation 1), for the case of both
random loading amplitudes, and as proposed in Section 5.2
(calculation 2), for the case of only random torsion ampli-
tude. In the two cases (calculation 1 and calculation 2), the
obtained results are identical, as shown in Figs. 6.a and 6.b.
6.2.2. Case of random loading
The 99% PCD’s reliability has been determined for dif-
ferent COV values of the material and the loading param-
eters. Two types of random loading were considered: (i)
only random torsion amplitude loading and (ii) both tor-
sion and bending amplitudes loading, as detailed in Table
3. The different obtained PCD are presented in Fig. 7, for
the first loading and in Fig. 8, for the second one.
7. Discussion
For all the cases of the application, the principle hypoth-
esis leading to have elliptical surfaces of the loading disper-
sion zone has been clearly observed and verified. The
normality distributions of the load and the strength func-
tions, along the correlation direction, have been verified
according to ‘‘Henry’s line’’ within a linear correlation
coefficient higher than 99.8% for all the cases of the
application.
The first application shows that, for the three studied
cases, the loading deterministic representative points (cen-
ters of the elliptical surfaces), are located in the security
zones of the deterministic Crossland diagrams (Fig. 5),
leading to a secure HCF behaviour. However, the pro-
posed probabilistic approach permits to compute the three
HCF reliability values, corresponding to each case. Also, it
is observed that, for the same mean values of the applied
loading amplitudes, the HCF reliability decreases, when
the applied loading COV values increase as shown in
(Table 2). These results are physically consistent.
The deterministic method commonly used to introduce
the probability aspect in the Crossland criterion, consists
to consider fixed values of t1 and f1, at a given confident
level (a fixed probability). This method is called, here, usual
engineering methods (UEM). The second application, car-
ried out in the case of deterministic loading (Figs. 6.a and
6.b), presents a comparison between the PCD and the
UEM Crossland limit lines, corresponding to 75% and
99% reliabilities. It is observed that the PCD curves have
a parabolic form, whereas the UEM lines are linear. The
PCD curvature seems to be function of the HCF material
parameters dispersions. It increases when the COV values
Table 2
The studied cases for the application 1
Studied cases COV loading values Results
COVra (%) COVsa (%) d0 rS rL Ic R (%)
Case 1 2 2 3.8003 1.4088 2.9272 1.1698 87.8
Case 2 3 3 3.7459 1.3871 4.3425 0.8217 79.4
Case 3 5 5 3.6582 1.3737 7.2891 0.4932 68.9
Fig. 5. Probabilistic HCF Crossland criterion (case 1-application 1).
Table 3
The studied cases for the application 2
Studied cases COV material parameters
values
COV loading parameters
values
Figures
COVt1 (%) COVf1 (%) COVra (%) COVsa (%)
Cases of deterministic loading Case 1 1 1 0 0 6.a
Case 2 7 7 0 0 6.b
Cases of only random torsion amplitude loading Case 3 1 1 0 1 7
0 2
0 3
Cases of both random torsion and bending amplitudes loading Case 4 1 1 1 1 8
2 2
3 3

of t1 and f1 increase. This result seems to be more phys-
ically consistent, since the probabilistic effect in the Cross-
land criterion depends on the dispersions of aC and bC
parameters and not on the dispersions of t1 and f1. Eq.
(6) shows that, for a given confidence level, the confidence
limits of t1 and bC are the same, whereas, the confidence
limit of aC, obtained by Eq. (5), is not corresponding to
those of t1 and f1.
Also, it is observed that the values of bC, for 99% confi-
dence limit (Pmax = 0 in the Crossland diagram), corre-
sponding to the UEM (bC (99%-UEM)) and the PCD
curves (bC (99%-PCD)), are slightly different (less than
4% in the case 1; application 2) (Fig. 6.b; Table 1). This
is contrary to Eq. (6), leading to have the same values. This
difference is more important for the higher COV values of
t1 and f1 (case 2; application 2). This can be explained by
the elimination of some events due to the physical HCF
material parameters condition, expressed by the inequality
(16), in the probabilistic approach.
The results of only random torsion amplitude loading
application (case 3; application 2), are presented in
Fig. 7. They show a difference between the pattern of the
PCD and the UEM curves. The security zone decreases,
when the COV of the material and the COV of the torsion
amplitude increase. For the case 3 of the application 2
(Fig. 7), the PCD curves, obtained with different COV val-
ues of the torsion amplitude, converge to a particular
point. This point seems to correspond to the loadings
Fig. 6.b. PCD for the case of deterministic in-phase fully reversed torsion-bending loading (case 2-application 2).
Fig. 6.a. PCD for the case of deterministic in-phase fully reversed torsion-bending loading (case 1-application 2).

having Pmax approximately about 100 MPa. This may be
explained by the fact that, when the bending amplitude
increases, the loading standard deviation rL decreases
and will be less and less affected by the COV of the random
torsion amplitude. And the reliability index, as defined in
Eq. (35), will depend only on the material dispersion char-
acteristics (rS).
The results of the application 2, for the case of both ran-
dom torsion and bending amplitudes loading random tor-
sion amplitude loading, are reported in Fig. 8. The
comparison between the 99%-PCD curves and the 99%-
UEM Crossland limit lines shows that the security zone
decreases when the dispersion of the applied loading ampli-
tudes and the dispersion of the material parameters
increase. These results are very interesting. They have the
advantage to take into account the loading dispersion,
which are generally taken intro account by using empirical
security factors in the design of the mechanical compo-
nents. This proposed approach can be used to calibrate
more qualitatively, such as, coefficient of security.
8. Conclusion
In the present work, a probabilistic approach has been
developed to evaluate the HCF reliability, by using the
fatigue criterion of Crossland, and taking into account
Fig. 7. PCD for the case of only random torsion amplitude loading (case 3-application 2).
Fig. 8. PCD for the case of both random torsion and bending amplitudes loading (case 4-application 2).

the dispersions due to the material and the applied loading.
The usual Crossland criterion has been transformed. The
HCF limit line has been transformed into a dispersion zone
characterised by a beam of lines, corresponding to the dif-
ferent cases pulled out from N values, while respecting a
physical condition between the t1 and f1 for the studied
material. The loading representative point is transformed
into elliptical surface in the Crossland diagram, for the case
of both random torsion and bending amplitudes loading,
or into a vertical segment, for the case of only random tor-
sion amplitude loading with constant Pmax. The HCF reli-
ability has been calculated by using the FORM. A general
procedure to determine PCD taking into account, the HCF
characteristic material parameters and the applied loading
dispersions, is detailed, in Section 4.
The probabilistic proposed method has been applied for
a hard metal. A first example was carried out, by using
COV values of the HCF material characteristic parameters,
in the case of both random torsion and bending amplitudes
loading, with several loading COV values. It has been
observed that the reliability decreases when the COV val-
ues of the loading increases. This is seems to be physically
consistent. A second application has been carried out to
determine different PCD for, different studied loadings
and HCF material characteristic parameters, COV values.
It has been observed that the security zone is reduced, when
the dispersion parameters of the loading and/or of the
material increases.
The proposed approach allows having a better qualita-
tive evaluation of the HCF reliability for designing
mechanical components, leading to have a more secure pre-
diction of HCF behaviour, when the effects of the material
and the loading dispersions were significant, and economi-
cally design solutions for the cases corresponding to
reduced and controlled dispersions. Calibrating coefficient
of security, depending on the applied loading dispersion,
can be achieved by the proposed method.
Appendix. Verification of the approximation of the normal
distribution hypothesis of L and S functions
To check the approximation of the normal distribution
assumptions of the L and S functions, we determine two
fields of points obtained by intersection between the corre-
lation direction and the corresponding dispersions surfaces.
Then, empirical distributions histograms are plotted. And
finally, we proceed to check the approximation of normal-
ity by using the ‘‘Henry Line’’. Its worth noticing, that we
use the approximate First Order Reliability Method to
evaluate the reliability, where only the first and the second
moment’s distributions (mS; mL; r2
S; r2
L) are needed.
Determination of the empirical distribution histograms
In the case of the S function, the generated points corre-
spond to the intersection between the ‘‘beam of lines’’ and
the correlation direction. However, for the case of L func-
tion, they are obtained by the projection of the loading ran-
dom sampling events (the elliptical surface), on the
correlation direction (Fig. A.1). For the two cases the scat-
tering’s range is obtained along the correlation direction.
Each scattering range is then limited by a lower (Xmin)
and higher (Xmax) values. This range is, then, divided into
k equal intervals (k is chosen equal to the square root of
the chosen sampling size N). The number of points ni
belonging to each divided interval is computed. Finally,
the histograms of L and S, along the correlation direction,
are plotted (Figs. A2b,d, A4b,d).
Normality verification of the empirical distributions
Let X denote the random variable characterising the
position of each point along the correlation direction.
Let F(x) denote the cumulative density function defined
by:
Fig. A.1. Determination of empirical distribution histograms within the correlation direction.

F ðxjÞ ¼ PrðX 6 xjÞ
According to the empirical distribution histogram, we ob-
tain:
F ðxjÞ ¼
X
j
i¼1
ni
N
where, ni is the number of points belonging to each divided
interval and N is the total number of the events.
Then, the standard value uj is determined as follows:
uj ¼ erfðF ðxjÞÞ
where, erf is the inverse cumulative Normal distribution
function.
The approximation of the normality distribution is
checked by the ‘‘Henry line’’ method leading to verify the
linearity’s degree between U and X variables.
Empirical distribution is called normal, when a linear
regression between U and X gives a correlation coefficient
equal to 1.
Verifications results
Figs. A2–A4 show verifications results carried out for
three different cases:
Fig. A2.a. Load and strength dispersion surfaces (case 1).
Fig. A2.b. L distribution histogram (case 1).
Fig. A2.c. L Henry line (case 1).
Fig. A2.d. S distribution histogram (case 1).
Fig. A2.e. S Henry line (case 1).

t1 = 196.2 MPa; f1 = 313.9 MPa; COVt1 = COVf1 =
1%.
Case 1: ra = 130 MPa; sa = 120 MPa; COVra = 2%;
COVsa = 2%.
COVsa = 2%.
COVsa = 2%.
For all the cases we obtain, for L and S distribution
functions, the correlation coefficients higher than 99.8%
(Fig. A2–A4). Hence, the adopted hypotheses in our
approach seem to be not influent on the results.
References
[1] Wirsching PH. Fatigue reliability in welded joints of offshore
structures. Int J Fatigue 1980;2(2):77–83.
[2] Shen M. Reliability assessment of high cycle fatigue design of gas
turbine blades using the probabilistic Goodman Diagram. Int J
Fatigue 1999;21(7):699–708.

[3] White P, Molent L, Barter S. Interpreting fatigue test results using a
probabilistic fracture approach. Int J Fatigue 2005;27(7):752–67.
[4] Morel F, Flacelière L. Data scatter in multi-axial fatigue: from the
infinite to the finite fatigue life regime. Int J Fatigue
2005;27(9):1089–101.
[5] Sevensson T. Prediction uncertainties at variable amplitude fatigue.
Int J Fatigue 1997;17(Suppl. 1):S295–302.
[6] Sines G. Behavior of metals under complex static and alternating
stresses. In: Sines G, Waisman JL, editors. Met fatigue. New York:
McGraw-Hill; 1959. p. 69–145.
[7] Crossland B. Effect of large hydrostatic pressures on the torsional
fatigue strength of an alloy steel. In: Proceeding of the international
conference on fatigue of metals. London: Institution of Mechanical
Engineers; 1956.
[8] Dang Van K et al. Criterion for high Cycle fatigue Failure under
multi-axial Loading. In: Biaxial and Multi-axial Fatigue. London:
Mechnical Enginneering publications; 1989. p. 459–78.
[9] Papadopoulos IV, Davol P, Gorla C, Filippini M, Bernasconi A. A
comparative study of multi-axial high cycle fatigue criteria of metals.
Int J Fatigue 1997;19(3):219–35.
[10] Papadopoulos IV. Long life fatigue under multi-axial loading. Int J
Fatigue 2001;23(10):839–49.
[11] Nisitani H, Chen DH. Stress intensity factor for a semi-elliptical
surface crack in a shaft under tension. Trans Jpn Soc Mech Eng
1984;50:1077–82.
[12] Zhao YG, Ono T. Moment for structural reliability. Structural Safety
2001;23:47–75.
[13] Weibull W. A statistical theory of the strength of materials. R Swed
Inst Eng Res 1939;151.
[14] Bomas H, Linkewitz T, Mayr P. Application of a weakest-link
concept to the fatigue limit of the bearing steel SAE 52100 in a
bainitic condition. Fatigue Fract Eng Mater Struct
1999;22(9):733–41. ISSN 8756-758X.
[15] Hild F, Roux S. Fatigue initiation in heterogeneous brittle materials.
Fatigue Fract Eng Mater Struct 1991:409–14.
[16] Chantier I, Bobet V, Billardon R, Hild F. A probabilistic approach to
predict the very high cycle fatigue behaviour of spheroidal graphite
cast iron structures. Fatigue Fract Eng Mater Struct 2000;23:173.
[17] Dettinger MD, Wilson JL. First order analysis of uncertainty in
numerical models of groundwater flow: Part 1. Mathematical
development. Water Resour Res 1981;17(1):149–61.
[18] Madsen HO, Krenk S, Lind NC. Methods of Structural Safety.
Englewood Cliffs (NJ): Prentice-Hall, Inc.; 1986.
[19] Tang J, Zhao J. A practical approach for predicting fatigue reliability
under random cyclic loading. Reliab Eng Syst Safe 1995;50:7–15.
[20] Ditlevsen O, Madsen HO. Structural reliability methods. London:
Wiley; 1996.
[21] Karadeniz H. Uncertainty modeling in the fatigue reliability calcu-
lation of offshore structures. Reliab Eng Syst Safe 2001;74:S323–35.
[22] Bjerager P. Methods for structural reliability computation. I. In:
Casciati F, editor. Reliability problems: general principles and
applications in mechanics of solid and structures. New York: Springer
Verlag; 1991. pp. 89–136.
[23] Lemaire M, Chateauneuf A, Mitteau JC. Fiabilité des structures:
couplage mécano-fiabiliste statique, Edit. Hermes Paris. Cote: Rez-
de-chaussée; 2005, 620.004 52 LEM [In French].
[24] Schijve J. Statistical distribution functions and fatigue of structures.
Int J Fatigue 2005;27(9):1031–9.
[25] Lieurade HP. La pratique des essais de fatigue. Pyc Livres (January 1,
1982). ISBN: 2853300536 [In French].
[26] Bathias C, Bailon J-P. La fatigue des matériaux et des structures.
Hermes 1997 [In French].

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