In this work, the software program ProFatigue is presented as a practical tool for derivation of probabilistic S-N and ε-N fields from experimental fatigue data. The program provides an estimation of the parameters involved in the regression probabilistic Weibull fatigue model developed by the authors, allowing an advantageous application of both fields to the stress or strain based approaches in the fatigue design of structures and mechanical components. An extension to the analysis of more complex and varied lifetime problems as thermomechanical, multiaxial and fretting fatigue is possible by adopting suitable damage parameters proposed in the literature as driving force. Application to probabilistic assessment of cumulative damage and further program enhancement are now envisaged

Castillo Ron, Enrique

Fernández Canteli, Alfonso

López Aenlle, Manuel

Nogal Macho, María

Przybilla, Constanze

Procedia Engineering

English

AbstractIn this work, the software program ProFatigue is presented as a practical tool for derivation of probabilistic S-N and ɛ-N fields from experimental fatigue data. The program provides an estimation of the parameters involved in the regression probabilistic Weibull fatigue model developed by the authors, allowing an advantageous application of both fields to the stress or strain based approaches in the fatigue design of structures and mechanical components. An extension to the analysis of more complex and varied lifetime problems as thermomechanical, multiaxial and fretting fatigue is possible by adopting suitable damage parameters proposed in the literature as driving force. Application to probabilistic assessment of cumulative damage and further program enhancement are now envisaged

Fernández-Canteli, A.

Przybilla, C.

Nogal, M.

Aenlle, M. López

Castillo, E.

Elsevier - Publisher Connector 

ProFatigue: A Software Program for Probabilistic Assessment of Experimental Fatigue Data Sets 

A. Fernández-Canteli

C. Przybilla

M. Nogal

M. López Aenlle

E. Castillo

Crossref

ProFatigue: A Software Program for Probabilistic Assessment of Experimental Fatigue Data Sets

 Procedia Engineering  74 ( 2014 )  236 – 241 
1877-7058 © 2014 Elsevier Ltd. 
Selection and peer-review under responsibility of the Politecnico di Milano, Dipartimento di Meccanica
doi: 10.1016/j.proeng.2014.06.255 
ScienceDirect
Available online at www.sciencedirect.com
XV11 International Colloquim on Mechanical Fatigue of Metals (ICMFM17)
ProFatigue: A software program for probabilistic assessment of
experimental fatigue data sets
A. Ferna´ndez-Cantelia,∗, C. Przybillaa, M. Nogalb, M. Lo´pez Aenllea, E. Castillob
aDepartment of Construction and Manufacturing Engineering, University of Oviedo, Spain
bDepartment of Applied Mathematics and Computational Sciences, University of Cantabria, Spain
Abstract
In this work, the software program ProFatigue is presented as a practical tool for derivation of probabilistic S-N and ε-N ﬁelds from
experimental fatigue data. The program provides an estimation of the parameters involved in the regression probabilistic Weibull
fatigue model developed by the authors, allowing an advantageous application of both ﬁelds to the stress or strain based approaches
in the fatigue design of structures and mechanical components. An extension to the analysis of more complex and varied lifetime
problems as thermomechanical, multiaxial and fretting fatigue is possible by adopting suitable damage parameters proposed in the
literature as driving force. Application to probabilistic assessment of cumulative damage and further program enhancement are
now envisaged.
c© 2014 The Author y
Selection and peer-review under responsibility of the Politecnico di Milano, Dipartimento di Meccanica.
Keywords: Fatigue software; Probabilistic assessment; S-N ﬁeld; ε-N ﬁeld
1. Introduction
In most of the models used for the derivation of S-N and ε-N ﬁelds from assessment of experimental fatigue data
the probability of failure is not explicitly included though the high scatter of the lifetime at the diﬀerent stress ranges
requires its consideration. This has consequences in the consideration of both fatigue ﬁelds by the stress and strain
approaches for fatigue design of structures and mechanical components. As a suitable alternative, the probabilistic
fatigue model developed by Castillo and Ferna´ndez-Canteli [1] fulﬁls both physical (weakest link principle and lim-
ited range) and statistical conditions related to extreme value analysis (stability, limit behaviour). In particular, the
necessary compatibility between the cumulative distribution functions (cdfs) for fatigue lifetime at given stress range
and stress range at given lifetime is fulﬁlled as an indispensable prerequisite allowing a functional equation to be
established. Its solution provides only two possible Weibull models for minima:
Model I: F(N,Δσ) = 1 − exp
⎡⎢⎢⎢⎢⎣−
(
(log N − B)(logΔσ −C) − λ
δ
)β⎤⎥⎥⎥⎥⎦ ; (log N − B)(logΔσ −C) ≥ λ, (1)
∗ Corresponding author. Tel.: + 34 985 18 20 54; fax: + 34 985 18 20 55
E-mail address: afc@uniovi.es
 2014 Elsevier Ltd. 
Selection and peer-review under responsibility of the Politecnico di Milano, Dipartimento di Meccanica
Open access under CC BY-NC-ND license. 
Open access under CC BY-NC-ND license. 
237 A. Ferna´ndez-Canteli et al. /  Procedia Engineering  74 ( 2014 )  236 – 241 
Model II: F(N,Δσ) = 1 − exp
⎡⎢⎢⎢⎢⎣−
(
(log N − B)(logΔσ −C)γ
δ
)β⎤⎥⎥⎥⎥⎦ ; log N ≥ B, logΔσ ≥ C, (2)
where B is a threshold value of the lifetime, C is the endurance limit, or fatigue limit for N → ∞, λ, δ, β are location,
scale and shape Weibull parameters, respectively, and γ is a parameter that scales the normalization of the S-N ﬁeld.
The percentile curves are hyperbolas sharing the asymptotes N = exp(B) and Δσ= exp(C), whereas the zero percentile
curve represents the minimum possible required number of cycles to achieve failure for diﬀerent values of logΔσ. If
the lower bound of the results, represented by the location parameter in the Weibull model, can be relaxed, a Gumbel
model may be assumed as a limiting case of the Weibull model, taking the forms:
Model I: F(N,Δσ) = 1 − exp
[
− exp
(
(log N − B)(logΔσ −C) − λ
δ
)]
; (log N − B)(logΔσ −C) ≥ λ, (3)
Model II: F(N,Δσ) = 1 − exp
[
− exp
(
(log N − B)(logΔσ −C)γ
δ
)]
; log N ≥ B, logΔσ ≥ C. (4)
These models oﬀer the advantage of requiring one parameter less than the Weibull one, avoiding the always com-
promising task of assuming a lower bound associated with zero probability of failure and cover the whole lifetime
region from the ﬁrst cycle. The assumption of a Gumbel model is fully justiﬁed when β is greater than, say, 6, based on
the practical coincidence of the data evaluation for both probability distributions, at least up to fairly low probabilities
of failure, see [1].
A similar solution as model II, though based on micromechanical considerations, is proposed by Bolotin [2] in
both Weibull and Gumbel alternatives:
Weibull: F(N|s) = 1 − exp
⎡⎢⎢⎢⎢⎣− MM0
(
N − N0
Nc
)β ( s − sth
sc
)α⎤⎥⎥⎥⎥⎦ ,
Gumbel: F(N|s) = 1 − exp
⎡⎢⎢⎢⎢⎢⎣− MM0 exp
(
N − N0
Nc
)1/m ( s − sth
sc
)⎤⎥⎥⎥⎥⎥⎦ ,
(5)
where s can be understood as logΔσ, N as log N, and the quotient M/M0 represents the scale eﬀect.
The size eﬀect is handled by assuming the weakest-link principle and statistical independence, so that simultaneous
evaluation of experimental results obtained from testing specimens of diﬀerent size is feasible. The consideration of
the normalizing variable V = (log N −B)(logΔσ−C) allows the whole S-N ﬁeld to be reduced to a unique cumulative
distribution function (cdf).
The potentiality of the model, initially restricted to the assessment of fatigue results obtained at constant stress level,
becomes evident by successive interpretations, improvements, new applications and extensions to more advanced
models since its initial proposal.
2. The ProFatigue program
In order to facilitate the applicability of the model to the practical assessment of the S-N ﬁeld from fatigue data,
the free use software program ProFatigue is developed [3]. The model parameters are estimated for any appropriate
fatigue data thus allowing a probabilistic prediction of the lifetime as a function of the stress range for ﬁxed stress level
(R, σmean, etc.). The ProFatigue program can ﬁt the parameters of model I of the Weibull (Eq. (1)) and the Gumbel
(Eq. (3)) type to the fatigue data. Implementing model II as well as an extension to the case of varying stress level,
this having been already developed by the authors [4], is envisaged as a future program enhancement. In the following
only the Weibull model I is considered.
The product V = (log N − B)(logΔσ − C) becomes a normalizing variable allowing the whole S-N ﬁeld to be
reduced to a unique cdf what suggests the model parameters being estimated in a two-step method: ﬁrstly, B and
C, then the other three parameters of the normalized Weibull distribution using a standard procedure. It permits all
238   A. Ferna´ndez-Canteli et al. /  Procedia Engineering  74 ( 2014 )  236 – 241 
S-N data to be pooled as pertaining to a single distribution so that the model parameters are estimated with higher
reliability. As soon as the ﬁve parameters are estimated, the analytical expression of the whole S-N ﬁeld is known.
The percentile curves can be interpreted as representing diﬀerent initial crack sizes, for the moment unknown. Run-
outs are considered in the parameter evaluation by means of the E-M algorithm [1]. Conﬁdence intervals of the S-N
ﬁeld can be estimated by simulation using the boot-strap method in addition to the ProFatigue program. Since the
model extends the Wo¨hler ﬁeld up to an inﬁnite number of cycles, an extrapolation of the lifetime outside the scope
of number of cycles tested is possible allowing an evaluation of data results in the HCF or even VHCF region [1].
The weakest-link principle and statistical independence were assumed in order to take into account the size eﬀect;
in particular, simultaneous evaluation of experimental results obtained from testing specimens of diﬀerent size can be
considered by including the relation between the tested length Li and a reference length L0 leading to
F(N,Δσ) = 1 − exp
⎡⎢⎢⎢⎢⎣− LiL0
(
(log N − B)(logΔσ −C) − λ
δ
)β⎤⎥⎥⎥⎥⎦ = 1 − exp
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝
(log N − B)(logΔσ −C) − λ
δ
(
L0
Li
)1/β
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
β⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ . (6)
As an example of application, fatigue data obtained for prestressing wires of lengths 1960 mm and 8540 mm (see
Fig. 1) are evaluated by the proposed probabilistic model following Eq. (6) and shown in Figs. 2a and 3a together with
the S-N ﬁeld represented by curves of constant failure probability. Run-outs are represented once as circles at the
number of cycles where the test was suspended and once as stars at the expected lifetimes estimated by the program.
Figures 2b and 3b represent the cumulative distribution functions of the normalizing variable V corresponding to each
length.
Since the fundamentals of the model rest on general internal functional properties of the Wo¨hler ﬁeld, ProFatigue
can be applied to a broad spectrum of lifetime problems of diﬀerent nature, as it is conﬁrmed by satisfactory appli-
cation to a number of diﬀerent materials as reinforcing and prestressing steels, various steel and aluminium alloys,
concrete, polyethylene and composites, including size eﬀect and extrapolation to the VHCF region, as reported in pre-
vious works of the authors furnishing reliable results. The results conﬁrm the potentiality of the model for subsequent
practical design, irrespective of the problem handled.
Compared to conventional models such as that of Basquin or the up-and-down method, the ProFatigue program
proves to be advantageous providing higher reliability and capability to reproduce the whole Wo¨hler ﬁeld.
Fig. 1: Input data for S-N fatigue analysis for prestressing wires with two diﬀerent lengths
239 A. Ferna´ndez-Canteli et al. /  Procedia Engineering  74 ( 2014 )  236 – 241 
104 105 106 107
200
250
300
350
400
450
500
550
600
650
Lifetime [cycles]
St
re
ss
 [M
Pa
]
Threshold lifetime
Endurance limit
Pf = 0%
Pf = 5%
Pf = 50%
Pf = 95%
Data L=1960mm
Run−outs
Transformed run−outs
(a) S-N ﬁeld
2 2.5 3 3.5 4 4.5 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
V = (ln N − B) ⋅ (ln Δσ − C)
P f
 
[ ]
        Stress (MPa)
260
280
300
320
340
360
380
420
460
630
Data without run−outs
Run−outs
(b) Cdf of the normalizing variable V
Fig. 2: Model application on fatigue data for prestressing wires - length= 1960 mm
104 105 106 107
200
250
300
350
400
450
500
550
600
650
Lifetime [cycles]
St
re
ss
 [M
Pa
]
Threshold lifetime
Endurance limit
Pf = 0%
Pf = 5%
Pf = 50%
Pf = 95%
Data L=8540mm
Run−outs
Transformed run−outs
(a) S-N ﬁeld
2 2.5 3 3.5 4 4.5 5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
V = (ln N − B) ⋅ (ln Δσ − C)
P f
 
[ ]
        Stress (MPa)
260
280
300
320
340
360
380
420
460
630
Data without run−outs
Run−outs
(b) Cdf of the normalizing variable V
Fig. 3: Model application on fatigue data for prestressing wires - length= 8540 mm
3. Applications to the ε-N ﬁeld
Since the basic physical and statistical considerations used to derive the Weibull S-N model also apply in the case
of the ε-N ﬁeld, due to the parallelism in the functional problem being dealt with, the same Weibull regression model
(a) (b)
Fig. 4: (a) ε-N ﬁeld for SAE1137 steel using the proposed model [5]; (b) Resulting S-N ﬁeld from the ε-N ﬁeld [6]
240   A. Ferna´ndez-Canteli et al. /  Procedia Engineering  74 ( 2014 )  236 – 241 
has been implemented in the ProFatigue program for the strain-life case simply by considering strain range as driving
force. Nevertheless, the analytical probabilistic deﬁnition of the whole strain-life ﬁeld as percentile curves applies
both for the low cycle and high cycle fatigue regions, because the upper limit in the ε-N, opposite to the S-N ﬁeld, is
practically unrestricted, see [1].
Similarly to the former case handled for the S-N ﬁeld, Fig. 4 (a) represents the ε-N ﬁeld using ProFatigue as an
advantageous alternative to the Coﬃn-Manson approach [5]. Moreover, using the σ-ε cyclic curves considering a
Ramberg-Osgood model, it is possible to derive the corresponding stress life curves, i.e. the probabilistic S-N ﬁeld,
which in this case presents the typical change in curvature in the low-cycle fatigue region, see Fig. 4 (b), thus
reproducing the LCF region.
A possible alternative at disposal to overcome the apparent inability of the model to reproduce the LCF region
consists in setting energy parameters as the driving force [7].
4. Further achievements of the ProFatigue program
The following achievements are also provided by the ProFatigue program:
4.1. Support for fatigue program planning
The ProFatigue program facilitates optimizing planning of testing programs by means of an adequate testing strat-
egy (cluster, cascade or mixed methodology have been already investigated). A preliminary estimation of the median
S-N curve, i.e. that related to a failure probability Pf = 50%, can be also provided by the program when the number
of experimental results is yet scarce (say between 5 and 10). Without requesting the Weibull parameters this gives
advice for the choice of a suitable stress range for subsequent fatigue tests.
4.2. Applicability of the ProFatigue program to assessment of complex fatigue problems
The compatibility condition is a prerequisite for guaranteeing validity in the present model the same as for the
derivation of any models proposed for the assessment of the diﬀerent lifetime problems. As a consequence, an
application of the ProFatigue program to complex fatigue problems is possible if the adequate damage parameter,
for instance Fatemi-Socie, SWT, ΔJ, etc., is adopted as generalized driving force according to the problem handled,
for assessment of data in multiaxial fatigue, thermomechanical fatigue, fretting fatigue, UniGrow model, etc. and
subsequent use in stress or strain based approaches. This allows the results of all these phenomena to be handled in a
probabilistic way. In particular, an application to the deﬁnition of the LCF region can be contemplated [7].
The necessity of using energetic parameters seems to conﬁrm the unsuitability of the stress range, as conven-
tional primary parameter, when local plasticity eﬀects are present in the fatigue process and the presumable energetic
character of the damage induced by fatigue [8].
4.3. Support for probabilistic cumulative damage calculation
The deﬁnition of the S-N ﬁeld, identifying percentile curves with curves representing diﬀerent initial crack sizes,
i.e. of damage, and the normalization of the S-N ﬁeld as a unique cumulative distribution function by means of the
normalizing variable V = (log N − B)(logΔσ − C) given by the ProFatigue program, enables us to consider damage
as probability of failure, in accordance to the conventional concept of ultimate limit state, and predict lifetime under
varying loading using a probabilistic concept in the analysis of the Miner number. Simultaneous calculation of the
Miner number and the normalized variable, which is related to probability of failure, allows to relate the cumulative
damage provided by the Miner number to probability of failure.
5. Conclusions
• The probabilistic fatigue model of Castillo et al., based primarily on fulﬁlling the compatibility condition be-
tween both lifetime and stress range probability distributions, proves to be adequate to avoid primary errors in
241 A. Ferna´ndez-Canteli et al. /  Procedia Engineering  74 ( 2014 )  236 – 241 
establishing fatigue models according to the stress and strain based conventional approaches promoting suc-
cessive advances in their complexity and facilitating its application for the development of the fatigue software
program ProFatigue.
• The ProFatigue program allows estimation of the model parameters and, accordingly, to deﬁne analytically an
expression of the S-N and ε-N ﬁeld to be applied subsequently in the application of the stress based and strain
based approaches in fatigue design of structures and mechanical components.
• Further extensions or improvements of the model can be envisaged allowing successive advances in its com-
plexity, as for instance, for more complex fatigue cases as in UniGrow model, multiaxial, thermomechanical,
fretting, etc. for any appropriate experimental fatigue data set to be estimated, as well as for determination of
conﬁdence levels and optimization of fatigue program planning or probabilistic damage assessment in practical
fatigue design.
Acknowledgements
The authors gratefully acknowledge ﬁnancial support of this work by the MICINN (Ministry of Science and Inno-
vation) under project BIA2010-19920.
References
[1] E. Castillo, A. Ferna´ndez-Canteli, A Uniﬁed Statistical Methodology for Modeling Fatigue Damage, Springer, 2009.
[2] V. V. Bolotin, Mechanics of Fatigue, volume 11, CRC Press, 1999.
[3] ProFatigue: Software program for assessment of fatigue results, 2013. (Free download at dcif.uniovi.es/profatigue/ProFatigue pkg.exe. Users
Guide: dcif.uniovi.es/profatigue/ProFatigue-UserGuide-2013.pdf).
[4] R. Koller, M. L. Ruiz-Ripoll, A. Garcı´a, A. Ferna´ndez-Canteli, E. Castillo, Experimental validation of a statistical model for the Wo¨hler ﬁeld
corresponding to any stress level and amplitude, International Journal of Fatigue 31 (2009) 231–241.
[5] E. Castillo, A. Ferna´ndez-Canteli, H. Pinto, M. Lo´pez-Aenlle, A general regression model for statistical analysis of strain-life fatigue data,
Materials Letters 62 (2008) 3639 – 3642.
[6] A. Ferna´ndez-Canteli, E. Castillo, H. Pinto, M. Lo´pez-Aenlle, Estimating the S-N ﬁeld from strain-lifetime curves, Strain 47 (2011) e93–e97.
[7] A. Ferna´ndez-Canteli, C. Przybilla, J. Correia, A. de Jesus, E. Castillo, Extending the applicability of probabilistic S-Nmodels to the LCF region
using energetic damage parameters, in: XVI International Colloquium ”Mechanical Fatigue of Metals” Brno, Czech Republic, September 24-
26, 2012, 2012.
[8] J. A. Correia, A. M. de Jesus, A. Ferna´ndez-Canteli, Local uniﬁed probabilistic model for fatigue crack initiation and propagation: Application
to a notched geometry, Engineering Structures 52 (2013) 394 – 407.


In this work, the software program ProFatigue is presented as a practical tool for derivation of probabilistic S-N and E-N fields from experimental fatigue data. The program provides an estimation of the parameters involved in the regression probabilistic Weibull fatigue model developed by the authors, allowing an advantageous application of both fields to the stress or strain based approaches in the fatigue design of structures and mechanical components. An extension to the analysis of more complex and varied lifetime problems as thermomechanical, multiaxial and fretting fatigue is possible by adopting suitable damage parameters proposed in the literature as driving force. Application to probabilistic assessment of cumulative damage and further program enhancement are now envisaged.The authors gratefully acknowledge financial support of this work by the MICINN (Ministry of Science and Innovation) under project BIA2010-19920

Aenlle-López, Manuel

UCrea

 Procedia Engineering  74 ( 2014 )  236 – 241 1877-7058 © 2014 Elsevier Ltd. Selection and peer-review under responsibility of the Politecnico di Milano, Dipartimento di Meccanicadoi: 10.1016/j.proeng.2014.06.255 ScienceDirectAvailable online at www.sciencedirect.comXV11 International Colloquim on Mechanical Fatigue of Metals (ICMFM17)ProFatigue: A software program for probabilistic assessment ofexperimental fatigue data setsA. Fernández-Cantelia,∗, C. Przybillaa, M. Nogalb, M. López Aenllea, E. CastillobaDepartment of Construction and Manufacturing Engineering, University of Oviedo, SpainbDepartment of Applied Mathematics and Computational Sciences, University of Cantabria, SpainAbstractIn this work, the software program ProFatigue is presented as a practical tool for derivation of probabilistic S-N and ε-N fields fromexperimental fatigue data. The program provides an estimation of the parameters involved in the regression probabilistic Weibullfatigue model developed by the authors, allowing an advantageous application of both fields to the stress or strain based approachesin the fatigue design of structures and mechanical components. An extension to the analysis of more complex and varied lifetimeproblems as thermomechanical, multiaxial and fretting fatigue is possible by adopting suitable damage parameters proposed in theliterature as driving force. Application to probabilistic assessment of cumulative damage and further program enhancement arenow envisaged.c© 2014 The Author ySelection and peer-review under responsibility of the Politecnico di Milano, Dipartimento di Meccanica.Keywords: Fatigue software; Probabilistic assessment; S-N field; ε-N field1. IntroductionIn most of the models used for the derivation of S-N and ε-N fields from assessment of experimental fatigue datathe probability of failure is not explicitly included though the high scatter of the lifetime at the different stress rangesrequires its consideration. This has consequences in the consideration of both fatigue fields by the stress and strainapproaches for fatigue design of structures and mechanical components. As a suitable alternative, the probabilisticfatigue model developed by Castillo and Fernández-Canteli [1] fulfils both physical (weakest link principle and lim-ited range) and statistical conditions related to extreme value analysis (stability, limit behaviour). In particular, thenecessary compatibility between the cumulative distribution functions (cdfs) for fatigue lifetime at given stress rangeand stress range at given lifetime is fulfilled as an indispensable prerequisite allowing a functional equation to beestablished. Its solution provides only two possible Weibull models for minima:Model I: F(N,Δσ) = 1 − exp⎡⎢⎢⎢⎢⎣−((log N − B)(logΔσ −C) − λδ)β⎤⎥⎥⎥⎥⎦ ; (log N − B)(logΔσ −C) ≥ λ, (1)∗ Corresponding author. Tel.: + 34 985 18 20 54; fax: + 34 985 18 20 55E-mail address: afc@uniovi.es 2014 Elsevier Ltd. Selection and peer-review under responsibility of the Politecnico di Milano, Dipartimento di MeccanicaOpen access under CC BY-NC-ND license. Open access under CC BY-NC-ND license. 237 A. Fernández-Canteli et al. /  Procedia Engineering  74 ( 2014 )  236 – 241 Model II: F(N,Δσ) = 1 − exp⎡⎢⎢⎢⎢⎣−((log N − B)(logΔσ −C)γδ)β⎤⎥⎥⎥⎥⎦ ; log N ≥ B, logΔσ ≥ C, (2)where B is a threshold value of the lifetime, C is the endurance limit, or fatigue limit for N → ∞, λ, δ, β are location,scale and shape Weibull parameters, respectively, and γ is a parameter that scales the normalization of the S-N field.The percentile curves are hyperbolas sharing the asymptotes N = exp(B) and Δσ= exp(C), whereas the zero percentilecurve represents the minimum possible required number of cycles to achieve failure for different values of logΔσ. Ifthe lower bound of the results, represented by the location parameter in the Weibull model, can be relaxed, a Gumbelmodel may be assumed as a limiting case of the Weibull model, taking the forms:Model I: F(N,Δσ) = 1 − exp[− exp((log N − B)(logΔσ −C) − λδ)]; (log N − B)(logΔσ −C) ≥ λ, (3)Model II: F(N,Δσ) = 1 − exp[− exp((log N − B)(logΔσ −C)γδ)]; log N ≥ B, logΔσ ≥ C. (4)These models offer the advantage of requiring one parameter less than the Weibull one, avoiding the always com-promising task of assuming a lower bound associated with zero probability of failure and cover the whole lifetimeregion from the first cycle. The assumption of a Gumbel model is fully justified when β is greater than, say, 6, based onthe practical coincidence of the data evaluation for both probability distributions, at least up to fairly low probabilitiesof failure, see [1].A similar solution as model II, though based on micromechanical considerations, is proposed by Bolotin [2] inboth Weibull and Gumbel alternatives:Weibull: F(N|s) = 1 − exp⎡⎢⎢⎢⎢⎣− MM0(N − N0Nc)β ( s − sthsc)α⎤⎥⎥⎥⎥⎦ ,Gumbel: F(N|s) = 1 − exp⎡⎢⎢⎢⎢⎢⎣− MM0 exp(N − N0Nc)1/m ( s − sthsc)⎤⎥⎥⎥⎥⎥⎦ ,(5)where s can be understood as logΔσ, N as log N, and the quotient M/M0 represents the scale effect.The size effect is handled by assuming the weakest-link principle and statistical independence, so that simultaneousevaluation of experimental results obtained from testing specimens of different size is feasible. The consideration ofthe normalizing variable V = (log N −B)(logΔσ−C) allows the whole S-N field to be reduced to a unique cumulativedistribution function (cdf).The potentiality of the model, initially restricted to the assessment of fatigue results obtained at constant stress level,becomes evident by successive interpretations, improvements, new applications and extensions to more advancedmodels since its initial proposal.2. The ProFatigue programIn order to facilitate the applicability of the model to the practical assessment of the S-N field from fatigue data,the free use software program ProFatigue is developed [3]. The model parameters are estimated for any appropriatefatigue data thus allowing a probabilistic prediction of the lifetime as a function of the stress range for fixed stress level(R, σmean, etc.). The ProFatigue program can fit the parameters of model I of the Weibull (Eq. (1)) and the Gumbel(Eq. (3)) type to the fatigue data. Implementing model II as well as an extension to the case of varying stress level,this having been already developed by the authors [4], is envisaged as a future program enhancement. In the followingonly the Weibull model I is considered.The product V = (log N − B)(logΔσ − C) becomes a normalizing variable allowing the whole S-N field to bereduced to a unique cdf what suggests the model parameters being estimated in a two-step method: firstly, B andC, then the other three parameters of the normalized Weibull distribution using a standard procedure. It permits all238   A. Fernández-Canteli et al. /  Procedia Engineering  74 ( 2014 )  236 – 241 S-N data to be pooled as pertaining to a single distribution so that the model parameters are estimated with higherreliability. As soon as the five parameters are estimated, the analytical expression of the whole S-N field is known.The percentile curves can be interpreted as representing different initial crack sizes, for the moment unknown. Run-outs are considered in the parameter evaluation by means of the E-M algorithm [1]. Confidence intervals of the S-Nfield can be estimated by simulation using the boot-strap method in addition to the ProFatigue program. Since themodel extends the Wöhler field up to an infinite number of cycles, an extrapolation of the lifetime outside the scopeof number of cycles tested is possible allowing an evaluation of data results in the HCF or even VHCF region [1].The weakest-link principle and statistical independence were assumed in order to take into account the size effect;in particular, simultaneous evaluation of experimental results obtained from testing specimens of different size can beconsidered by including the relation between the tested length Li and a reference length L0 leading toF(N,Δσ) = 1 − exp⎡⎢⎢⎢⎢⎣− LiL0((log N − B)(logΔσ −C) − λδ)β⎤⎥⎥⎥⎥⎦ = 1 − exp⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣−⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝(log N − B)(logΔσ −C) − λδ(L0Li)1/β⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠β⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ . (6)As an example of application, fatigue data obtained for prestressing wires of lengths 1960 mm and 8540 mm (seeFig. 1) are evaluated by the proposed probabilistic model following Eq. (6) and shown in Figs. 2a and 3a together withthe S-N field represented by curves of constant failure probability. Run-outs are represented once as circles at thenumber of cycles where the test was suspended and once as stars at the expected lifetimes estimated by the program.Figures 2b and 3b represent the cumulative distribution functions of the normalizing variable V corresponding to eachlength.Since the fundamentals of the model rest on general internal functional properties of the Wöhler field, ProFatiguecan be applied to a broad spectrum of lifetime problems of different nature, as it is confirmed by satisfactory appli-cation to a number of different materials as reinforcing and prestressing steels, various steel and aluminium alloys,concrete, polyethylene and composites, including size effect and extrapolation to the VHCF region, as reported in pre-vious works of the authors furnishing reliable results. The results confirm the potentiality of the model for subsequentpractical design, irrespective of the problem handled.Compared to conventional models such as that of Basquin or the up-and-down method, the ProFatigue programproves to be advantageous providing higher reliability and capability to reproduce the whole Wöhler field.Fig. 1: Input data for S-N fatigue analysis for prestressing wires with two different lengths239 A. Fernández-Canteli et al. /  Procedia Engineering  74 ( 2014 )  236 – 241 104105106107200250300350400450500550600650Lifetime [cycles]Stress [MPa]Threshold lifetimeEndurance limitPf = 0%Pf = 5%Pf = 50%Pf = 95%Data L=1960mmRun−outsTransformed run−outs(a) S-N field2 2.5 3 3.5 4 4.5 500.10.20.30.40.50.60.70.80.91V = (ln N − B) ⋅ (ln Δσ − C)Pf [ ]        Stress (MPa)260280300320340360380420460630Data without run−outsRun−outs(b) Cdf of the normalizing variable VFig. 2: Model application on fatigue data for prestressing wires - length= 1960 mm104105106107200250300350400450500550600650Lifetime [cycles]Stress [MPa]Threshold lifetimeEndurance limitPf = 0%Pf = 5%Pf = 50%Pf = 95%Data L=8540mmRun−outsTransformed run−outs(a) S-N field2 2.5 3 3.5 4 4.5 500.10.20.30.40.50.60.70.80.91V = (ln N − B) ⋅ (ln Δσ − C)Pf [ ]        Stress (MPa)260280300320340360380420460630Data without run−outsRun−outs(b) Cdf of the normalizing variable VFig. 3: Model application on fatigue data for prestressing wires - length= 8540 mm3. Applications to the ε-N fieldSince the basic physical and statistical considerations used to derive the Weibull S-N model also apply in the caseof the ε-N field, due to the parallelism in the functional problem being dealt with, the same Weibull regression model(a) (b)Fig. 4: (a) ε-N field for SAE1137 steel using the proposed model [5]; (b) Resulting S-N field from the ε-N field [6]240   A. Fernández-Canteli et al. /  Procedia Engineering  74 ( 2014 )  236 – 241 has been implemented in the ProFatigue program for the strain-life case simply by considering strain range as drivingforce. Nevertheless, the analytical probabilistic definition of the whole strain-life field as percentile curves appliesboth for the low cycle and high cycle fatigue regions, because the upper limit in the ε-N, opposite to the S-N field, ispractically unrestricted, see [1].Similarly to the former case handled for the S-N field, Fig. 4 (a) represents the ε-N field using ProFatigue as anadvantageous alternative to the Coffin-Manson approach [5]. Moreover, using the σ-ε cyclic curves considering aRamberg-Osgood model, it is possible to derive the corresponding stress life curves, i.e. the probabilistic S-N field,which in this case presents the typical change in curvature in the low-cycle fatigue region, see Fig. 4 (b), thusreproducing the LCF region.A possible alternative at disposal to overcome the apparent inability of the model to reproduce the LCF regionconsists in setting energy parameters as the driving force [7].4. Further achievements of the ProFatigue programThe following achievements are also provided by the ProFatigue program:4.1. Support for fatigue program planningThe ProFatigue program facilitates optimizing planning of testing programs by means of an adequate testing strat-egy (cluster, cascade or mixed methodology have been already investigated). A preliminary estimation of the medianS-N curve, i.e. that related to a failure probability Pf = 50 %, can be also provided by the program when the numberof experimental results is yet scarce (say between 5 and 10). Without requesting the Weibull parameters this givesadvice for the choice of a suitable stress range for subsequent fatigue tests.4.2. Applicability of the ProFatigue program to assessment of complex fatigue problemsThe compatibility condition is a prerequisite for guaranteeing validity in the present model the same as for thederivation of any models proposed for the assessment of the different lifetime problems. As a consequence, anapplication of the ProFatigue program to complex fatigue problems is possible if the adequate damage parameter,for instance Fatemi-Socie, SWT, ΔJ, etc., is adopted as generalized driving force according to the problem handled,for assessment of data in multiaxial fatigue, thermomechanical fatigue, fretting fatigue, UniGrow model, etc. andsubsequent use in stress or strain based approaches. This allows the results of all these phenomena to be handled in aprobabilistic way. In particular, an application to the definition of the LCF region can be contemplated [7].The necessity of using energetic parameters seems to confirm the unsuitability of the stress range, as conven-tional primary parameter, when local plasticity effects are present in the fatigue process and the presumable energeticcharacter of the damage induced by fatigue [8].4.3. Support for probabilistic cumulative damage calculationThe definition of the S-N field, identifying percentile curves with curves representing different initial crack sizes,i.e. of damage, and the normalization of the S-N field as a unique cumulative distribution function by means of thenormalizing variable V = (log N − B)(logΔσ − C) given by the ProFatigue program, enables us to consider damageas probability of failure, in accordance to the conventional concept of ultimate limit state, and predict lifetime undervarying loading using a probabilistic concept in the analysis of the Miner number. Simultaneous calculation of theMiner number and the normalized variable, which is related to probability of failure, allows to relate the cumulativedamage provided by the Miner number to probability of failure.5. Conclusions• The probabilistic fatigue model of Castillo et al., based primarily on fulfilling the compatibility condition be-tween both lifetime and stress range probability distributions, proves to be adequate to avoid primary errors in241 A. Fernández-Canteli et al. /  Procedia Engineering  74 ( 2014 )  236 – 241 establishing fatigue models according to the stress and strain based conventional approaches promoting suc-cessive advances in their complexity and facilitating its application for the development of the fatigue softwareprogram ProFatigue.• The ProFatigue program allows estimation of the model parameters and, accordingly, to define analytically anexpression of the S-N and ε-N field to be applied subsequently in the application of the stress based and strainbased approaches in fatigue design of structures and mechanical components.• Further extensions or improvements of the model can be envisaged allowing successive advances in its com-plexity, as for instance, for more complex fatigue cases as in UniGrow model, multiaxial, thermomechanical,fretting, etc. for any appropriate experimental fatigue data set to be estimated, as well as for determination ofconfidence levels and optimization of fatigue program planning or probabilistic damage assessment in practicalfatigue design.AcknowledgementsThe authors gratefully acknowledge financial support of this work by the MICINN (Ministry of Science and Inno-vation) under project BIA2010-19920.References[1] E. Castillo, A. Fernández-Canteli, A Unified Statistical Methodology for Modeling Fatigue Damage, Springer, 2009.[2] V. V. Bolotin, Mechanics of Fatigue, volume 11, CRC Press, 1999.[3] ProFatigue: Software program for assessment of fatigue results, 2013. (Free download at dcif.uniovi.es/profatigue/ProFatigue pkg.exe. UsersGuide: dcif.uniovi.es/profatigue/ProFatigue-UserGuide-2013.pdf).[4] R. Koller, M. L. Ruiz-Ripoll, A. Garcı́a, A. Fernández-Canteli, E. Castillo, Experimental validation of a statistical model for the Wöhler fieldcorresponding to any stress level and amplitude, International Journal of Fatigue 31 (2009) 231–241.[5] E. Castillo, A. Fernández-Canteli, H. Pinto, M. López-Aenlle, A general regression model for statistical analysis of strain-life fatigue data,Materials Letters 62 (2008) 3639 – 3642.[6] A. Fernández-Canteli, E. Castillo, H. Pinto, M. López-Aenlle, Estimating the S-N field from strain-lifetime curves, Strain 47 (2011) e93–e97.[7] A. Fernández-Canteli, C. Przybilla, J. Correia, A. de Jesus, E. Castillo, Extending the applicability of probabilistic S-N models to the LCF regionusing energetic damage parameters, in: XVI International Colloquium ”Mechanical Fatigue of Metals” Brno, Czech Republic, September 24-26, 2012, 2012.[8] J. A. Correia, A. M. de Jesus, A. Fernández-Canteli, Local unified probabilistic model for fatigue crack initiation and propagation: Applicationto a notched geometry, Engineering Structures 52 (2013) 394 – 407.

Profatigue: a software program for probabilistic assessment of experimental fatigue data sets

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ProFatigue: a software program for probabilistic assessment of experimental fatigue data sets

ProFatigue: a software program for probabilistic assessment of experimental fatigue data sets

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