Structural Safety 32 (2010) 64–76

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Structural Safety

journal homepage: www.elsevier .com/ locate/s t rusafe

A Bayesian analysis of fatigue data

Maurizio Guida a,*, Francesco Penta b

a Department of Information and Electrical Engineering, University of Salerno, 84084 Fisciano (SA), Italyb Department of Mechanics and Energetics, University of Naples ‘‘Federico II”, Naples, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 4 August 2008Received in revised form 5 June 2009Accepted 7 August 2009Available online 13 September 2009

Keywords:Fatigue testingPSN curvesStatistical analysisBayesian inferenceFatigue design curves

0167-4730/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.strusafe.2009.08.001

* Corresponding author. Tel.: +39 089 964282; fax:E-mail address: mguida@unisa.it (M. Guida).

The aim of the present paper is to bring arguments in favour of Bayesian inference in the context of fati-gue testing. In fact, life tests play a central role in the design of mechanical systems, as their structuralreliability depends in part on the fatigue strength of material, which need to be determined by experi-ments. The classical statistical analysis, however, can lead to results of limited practical usefulness whenthe number of specimens on test is small. Instead, despite the little attention paid to it in this context,Bayes approach can potentially give more accurate estimates by combining test data with technologicalknowledge available from theoretical studies and/or previous experimental results, thus contributing tosave time and money. Hence, for the case of steel alloys, a discussion about the usually available techno-logical knowledge is presented and methods to properly formalize it in the form of prior credibility den-sity functions are proposed. Further, the performances of the proposed Bayesian procedures are analysedon the basis of simulation studies, showing that they can largely outperform the conventional ones at theexpense of a moderate increase of the computational effort.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The development of systems with a high structural reliabilitytogether with low manufacturing and maintenance costs requiresthe mechanical strength of materials be accurately known. This le-vel of knowledge, however, is difficult to obtain as far as the fatiguephenomenon is concerned, for several reasons: the inherent com-plexity of the fatigue mechanism, the variety of influential factorsand the laboriousness of fatigue testing.

Even if the laboratory fatigue tests are carried out under con-trolled operating conditions, fatigue data are characterized by ahigh variability and require a statistical analysis. This analysisessentially consists in: assuming a probabilistic model for the life-times, estimating its unknown parameters and constructing a ‘‘fa-tigue design curve” which should take into account both theinherent randomness of fatigue phenomenon and the uncertaintyon the true values of model parameters due to the finite samplesize. The classical (frequentist) statistical analysis, however, canlead to results of limited technical usefulness when the sample sizeis small. This circumstance often happens at the beginning of thedevelopment phase of new systems or components, when limitedexperimental data are usually available for both cost and timereasons.

A lot of theoretical studies and experimental life testing resultshave been produced by engineers and manufacturers in order to

ll rights reserved.

+39 089 964218.

evaluate the properties of different materials. When new constantamplitude life tests are carried out, however, this body of techno-logical knowledge is completely ignored by the conventional esti-mation procedures which are usually adopted. In principle, theavailable technological information, when properly formalized asa prior credibility on model parameters, might be combined viaBayes theorem with test data to make more accurate inferenceon the quantities of interest (for a comprehensive reference onBayes inferential approach see, as an example, [1]).

Although a number of papers have appeared which exploitedBayesian inference in the analysis of the propagation of fatiguecracks, very few attempts have been made in the past to use theBayes approach in the context of SN fatigue tests. In fact, as faras the authors are aware, only two papers have specifically ad-dressed this topic. For the case of offshore structural joints, Madsen[2] considered a Bayesian linear regression analysis where theprior information was derived from the behaviour of similar joints.The posterior distribution of model parameters was then used topredict the fatigue life. Edwards and Pacheco [3] in their paper crit-icised two limitations of the conventional procedures for establish-ing design SN curves from laboratory fatigue tests, namely the useof design curves based on point estimates of regression modelparameters and the impossibility of accounting for fatigue run-outs in a rational manner. Then, they proposed the use of Bayesianmethod (in its noninformative setting) in order to handle boththese limitations.

The little attention paid to Bayesian methods in the context ofSN fatigue testing is probably due to two main reasons: (a) the

M. Guida, F. Penta / Structural Safety 32 (2010) 64–76 65

greater computational burden usually associated to Bayesianmethods when compared to the classical methods available forSN tests, and (b) the suspiciousness still existing in many peopletowards ‘‘subjective” probability which is the basis of Bayesianinference.

The aim of this paper is to bring arguments in favour of Bayes-ian inference in the context of constant amplitude fatigue testing,by taking into account that the nowadays computational capabili-ties make the use of Bayesian viewpoint feasible and by showing,on the basis of simulation studies, that the performances of Bayes-ian procedures can sensibly outperform the conventional ones,when the available technical information is properly formalized.

The paper is organized as follows: the next section reviews thebasic concepts of SN tests and related classical estimation proce-dures. In Section 3 the Bayes point of view is briefly outlined andsuitable prior and posterior probability density functions (pdf)are introduced. Section 4 reviews technical knowledge on the cor-relations among static and fatigue properties of steels, which iscommonly available. Section 5 is, then, devoted to the mathemat-ical formalization of this technical knowledge in the form of priorpdf’s on material parameters. Finally, Sections 6 and 7 present cri-teria to compare Bayesian and classical estimators along withnumerical results obtained by Monte Carlo simulation.

2. SN tests and the related classical inferences

The lifetime of mechanical systems depends in part on the fati-gue strength of the material, whose properties need to be deter-mined by experiments. In constant amplitude fatigue tests, testspecimens are subjected to alternating load until failure. The mag-nitude of the load amplitude is the controlled (independent) vari-able and the number of cycles to failure is the response(dependent) variable. In the finite life region, the number of cyclesto failure Ni is usually assumed to depend on the load amplitude Si

by the Basquin’s relation [4]

Ni ¼ AS�Bi ð1Þ

where Að> 0Þ and Bð> 0Þ are (usually unknown) materialparameters.

Because of inherent microstructural inhomogeneity in thematerials properties, differences in the surface and in test condi-tions of each specimen and other factors, the number of cycles tofailure exhibits a random behaviour. To model this experimentalvariability, a multiplicative random error is introduced in (1) giving

Ni ¼ AS�Bi ei ð2Þ

where ei is usually assumed to be distributed as a log-normal ran-dom variable (rv) with (unknown) variance independent of Si. In re-peated testing, the errors ei are also assumed to be stochasticallyindependent rv’s.

For statistical analysis it is convenient to rewrite model (2) bytaking the logarithm (either the log base-10 or the natural log) ofboth sides of (2) and subtracting the sample mean of log Si timesB, arriving to Yi ¼ a� bðui � �uÞ þ rei, where Yi ¼ log Ni, ui ¼ log Si,�u ¼ ð1=nÞ

Pni¼1 log Si, b ¼ B, a ¼ log A� B�u, and ei � Nð0;1Þ.

By letting xi ¼ ðui � �uÞwithPn

i¼1xi ¼ 0, the simple linear regres-sion model is obtained

Yi ¼ a� bxi þ rei ð3Þ

Under the above assumptions, the Yi are independently distrib-uted Normal rv’s with mean lxi

¼ a� bxi and constant variance r2.After having observed the couples ðxi; yiÞði ¼ 1; :::;nÞ, point esti-

mates of the unknown parameters a and b are obtained by theLeast Squares method or, equivalently, by the maximum likelihoodmethod, which both give

a ¼ 1n

Xn

i¼1

yi b ¼ �Pn

i¼1xiyiPni¼1x2

i

ð4Þ

Thus, a and b are linear estimators of a and b, i.e. they are linearcombinations of the observed data yi. Since Yi are Normal rv’s, inrepeated sampling a and b also are Normal rv’s. They are unbiasedminimum variance estimators of a and b.

The error variance r2 is usually estimated by

s2 ¼Pn

i¼1½yi � lxi�2

n� 2ð5Þ

where lxi¼ a� bxi. This is a minimum variance unbiased estimator

of r2.From the frequentist (classical) point of view, the uncertainty

about an unknown quantity (model parameters or functions there-of, or future values of observable random variables) is measuredthrough a random interval generated by a rule which has in re-peated sampling a known probability ð1� cÞ of generating intervalsthat include the unknown quantity of interest. Depending on theparticular quantity to be estimated, random intervals are given dif-ferent names: confidence intervals, when population parameterssuch as mean or variance are involved; tolerance intervals, whenpercentages of a population are involved; and prediction intervals,when future values of observable rv’s are involved.

For the linear regression model (3), the following exact resultshold (see, for example, [5,6]).

Confidence interval on the median lx ¼ a� bx: The rv

ðlx � lxÞ=slx , where lx ¼ a� bx and slx ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=nþ x2=

Pni¼1x2

i

q, has

a Student distribution with m ¼ n� 2 degrees of freedom (df), sothat the ð1� cÞ two-sided equal tailed confidence interval is

lx � st1�c=2;n�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=nþ x2=

Xn

i¼1

x2i

vuut : ð6Þ

Confidence interval on the 100p-quantile ypx ¼ a� bxþ rzp (toler-

ance interval): The 100p-quantile of the distribution of a rv Y is de-fined as the value yp such that PrðY 6 ypÞ ¼ p. It can be readilyshown that the rv ðlx � yp

xÞ=slx has a non-central Student distribu-tion with m ¼ n� 2 df and non-centrality parameterd ¼ �zp=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=nþ x2=

Pni¼1x2

i

q. Thus, the ð1� cÞ two-sided equal

tailed confidence interval for the 100p-quantile ypx is given by

lx � std;1�c=2;ðn�2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=nþ x2

Xn

i¼1

x2i

,vuut ;

lx � std;c=2;ðn�2Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1=nþ x2

Xn

i¼1

x2i

,vuut : ð7Þ

In particular, a lower confidence limit of the 100p-quantile ypx is

usually of interest (a one-sided c size confidence limit). Relative tosuch a limit, we may say that ‘‘we are ð1� cÞ confident that thetrue 100p-quantile yp

x is greater than this limit”. This is equivalentto say that ‘‘we are ð1� cÞ confident that (at the most) a fraction pof all future observations of the sampled population will not ex-ceed this limit” or equivalently that ‘‘(at least) a fraction ð1� pÞof all future observations of the sampled population will exceedthis limit”. When the latter formulation is used, this limit is oftencalled a lower tolerance limit.

Although (under the model hypothesis) the only correct way todefine a tolerance interval is via Eq. (7), in the practice different orapproximate intervals are often used. As an example, practitionersoften refer to a ‘‘design curve” obtained shifting the estimatedmedian SN curve in the logarithm coordinates to the left by twoor three times the estimated standard deviation. This approach,however, being based on point estimates, suffers of the obvious

66 M. Guida, F. Penta / Structural Safety 32 (2009) 64–76

limitation that it does not take into account the sample variabilitywhich, in case of small or moderate samples, may even be the pre-dominant part of the experimental uncertainty. Also, the approxi-mate Owen one-side tolerance limit [7] has been proposed toaccount for the uncertainty in regression analysis.

Prediction interval on a future value of the rv Y: The rv

ðY � lxÞ=sY�lx , where sY�lx ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1=nþ x2=

Pni¼1x2

i

q, has a Student

distribution with m ¼ n� 2 df, so that the ð1� cÞ two-sided equaltailed prediction interval is

lx � st1�c=2;n�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1=nþ x2

Xn

i¼1

x2i

,vuut ð8Þ

Fig. 1 depicts the aforementioned classical lower limits. Notethat the usual representation of SN tests assigns the horizontal axisto the rv Y. Coherently with this setting, the proper regression anal-ysis is to minimize horizontal distances.

We recall for future use, that it is common practice among prac-titioners to write the deterministic link equation between the loadamplitude and the number of cycles to failure in the followingparametric form

S ¼ S0f ð2NÞ�b or S=Su ¼ ðS0f =SuÞð2NÞ�b ð9Þ

where 2N is the number of reversals, Su is the ultimate tensilestrength and S0f is a material parameter, known as the ‘‘fatiguestrength coefficient”, which represents the hypothetical fatiguestrength value which one would obtain by extrapolating the SN lineat one reversal. Then, material fatigue properties can also be set interms of the parameters a ¼ logðS0f =SuÞ and b which are related tothe parameters a and b in (3) by

a ¼ C2 þ b�1ða� C1Þb ¼ b�1

(ð10Þ

where C1 ¼ � log 2 and C2 ¼ ð1=nÞPn

i¼1 logðSi=SuÞ is the mean of thelogarithm of the relative load levels that will be used in the fatiguetesting.

Finally, we recall that it is often of interest to explore the behav-iour of steel alloys at load levels close to the fatigue limit, that is thecut-off point at which the SN curve changes to a nearly horizontalline. In fact, for many components the fatigue limit is a design cri-terion. In such a case, the straight line may be a poor approxima-tion and S-shaped SN curves might be considered. Also, theassumption of constant standard deviation of the random log-life-time at all stress levels is untrue. Hence, statistical approaches dif-

123

log10

(N)

log 10

(S)

4

1 - Estimated median life;

2 - 90% LCL for median life;

3 - 90% LPL for future life;

4 - 90% LCL for the 5%-quantile

of lifetimes distribution;

Fig. 1. Data points, estimated SN line and classical lower limits

ferent from the one described above should be considered. It is tobe stressed, however, that data samples with a relatively limitednumber of specimens, which are often common at the beginningof the development phase of new systems or components, cannotbe used for the purpose of determining the whole shape of a SNcurve. In fact, in those cases it is hardly possible to estimate eventhe slope of the high cycle fatigue straight line portion of the fati-gue curve, with an acceptable accuracy [8].

3. Bayesian measures of uncertainty

Let Y be a rv with pdf pðyjHÞ, indexed by a vector of parametersH. Given a random sample of observations y ¼ fy1; :::; yng , Bayes-ian inference on H is obtained via Bayes’ rule

pðHjyÞ ¼ pðyjHÞpðHÞRH pðyjHÞpðHÞdH

; ð11Þ

where pðHÞ is the personal degree of belief that a coherent personhas about H before observing the sample y, pðyjHÞ is the likelihoodfunction of the observed data, and pðHjyÞ is the updated personaldegree of belief about H after having observed the sample y.

Since a coherent person’s degree of belief satisfies the Kolmogo-rov’s axioms of probability theory, we may think of pðHÞ andpðHjyÞ as (subjective) probability density functions. In particular,pðHÞ is the prior pdf and pðHjyÞ is the posterior pdf of H. Thedenominator in (11) is simply a normalizing factor which ensuresthat, over the support of H, the posterior pdf integrates to one.

Inference on H can be made on the basis of posterior credibilityregions, that is regions over the support of H, with a given posteriorprobability content. Inference on the scalar components of vectorH can be obtained via marginal posterior pdf’s. For such compo-nents both ð1� cÞ two-sided equal tailed probability intervals orð1� cÞ highest posterior density (HPD) intervals can be derived.

Point estimates of scalar components of H are often taken to bethe expected value of the corresponding marginal posterior pdf.This is because the posterior mean h ¼ EðhjyÞ minimizes the meanquadratic loss function E½ðh� hÞ2�.

3.1. Noninformative Bayesian inference for the simple linear model

Different kind of prior functions can arise depending on the de-gree of initial personal knowledge about model parameters. In par-ticular a noninformative (or reference) prior assumes that ‘‘verylittle is known about H ”. Although there have been various pointsof view about how to express the notion of ‘‘knowing little”, it iscommonly recognised that a noninformative prior must at least en-sure a consistent inference under simple parameter’s transforma-tions. Noninformative priors are often improper, that is they donot integrate to one. This is usually accepted, provided that theposterior is a proper density. The regression model (3) is indexedby the parameters vector H ¼ ½abr�. For this model a and b are loca-tion parameters, while r is a scale parameter. Assuming indepen-dence among parameters information, the noninformativeJeffrey’s prior (which is consistent under parameters’ transforma-tions) is [9]

pða; b;rÞ / 1=r a; b�R; r�Rþ ð12Þ

which is an improper prior.When using prior (12), it can be shown (see, as an example, [9])

that the ð1� cÞ two-sided noninformative credibility intervals forlx, yp

x and for a future observation Y, numerically coincide withthe corresponding classical intervals (6)–(8), respectively. Recall,however, that credibility intervals and confidence intervals arequite different inferential objects, in that a ð1� cÞ credibility levelhas to be thought of as a ‘‘final” measure of precision (the precision

M. Guida, F. Penta / Structural Safety 32 (2010) 64–76 67

after the experiment is run), whereas a ð1� cÞ confidence level hasto be thought of as an ‘‘initial” measure of precision (the precisionof the ‘‘generating rule” before the experiment is even run). Thismight appear as just a ‘‘philosophical” difference with no conse-quence from a practical point of view, thus considering not soappealing the use of noninformative Bayesian inferential frame-work. It is to be noted, however, that, even from a practical pointof view, noninformative inference should be considered as an use-ful tool when censored sampling is involved. Indeed, in presence ofType I censoring, classical inferential procedures fail to be appliedwhereas Bayesian approach does not suffer from the experimentalcontext.

3.2. Informative Bayesian inference for the simple linear model

Informative Bayesian inference assumes that a subject is able toexpress his/her personal knowledge about an unknown quantity Hin a quantitative manner. The available information is formalizedin terms of a prior pdf pðHÞ, which encapsulates all relevantknowledge one has about H. For model (3), pðHÞ ¼ pða; b;rÞ. Sucha pdf could be based upon some or all of the following:

(1) engineering, physics, etc. information,(2) mathematical or physical models,(3) expert’s judgements,(4) corporate memory, commercial databases, historical data.

After having observed a random sample of (transformed) num-ber of cycles to failure y ¼ fy1; :::; yng at (transformed) loadsx ¼ fx1; :::; xng , one can combine the prior pða; b;rÞ with thelikelihood

pðy; xja; b;rÞ ¼ 1

ð2pÞn=2rnexp � 1

2r2

Xn

i¼1

ðyi � aþ bxiÞ2( )

; ð13Þ

thus obtaining the posterior density

pða; b;rjy;xÞ ¼ r�n

Dexp � 1

2r2

Xn

i¼1

ðyi � aþ bxiÞ2( )

pða; b;rÞ

ð14Þ

where D is the normalizing factor given by

D ¼Z Z Z

r�n exp � 12r2

Xn

i�1

ðyi � aþ bxiÞ2( )

pða; b;rÞda db dr:

The marginal pdf’s of a, b and r can be obtained integrating overthe other variables. Moreover, by changing variables in (14), theposterior pdf of lx and yp

x can be derived. Also, by combining(14) with the likelihood of a future observation Y and integratingover the parameters’ space, the predictive pdf of Y can be obtained.In general a triple integration will be required to obtain a 100p-quantile of the above quantities and the use of simulation methods(such as Monte Carlo Markov Chain) could be considered.

In this paper we will assume that estimates of the materialparameters of steel alloys belonging to the same ‘‘steel family” ofthe one to be tested are available and empirical relationships canpossibly be established between these parameters or functionsthereof (see the next section for details). Thus, when formulatinga prior pdf based on this information, model parameters a and bneed usually to be viewed as correlated rv’s, while parameter rcan usually be considered as a rv a priori independent of a and b.Then, the joint prior can be factored as pða; b;rÞ / pða; bÞpðrÞ,pða; bÞ being the joint prior of a and b, and pðrÞ the prior for r.Moreover, the available information about r is often more vaguethan that about a and b. Then, we will assume

pðrÞ / 1=r rL < r < rU ð15Þ

since this pdf appears to be a very flexible one, while keeping sim-ple mathematics. In fact, when rL ! 0 and rU !1 this pdf con-verges to the noninformative Jeffrey’s prior, whereas it canconveniently describe less vague knowledge about r by simplyadjusting the interval ðrL;rUÞ in a suitable manner.Then, the priorpdf results in

pða; b;rÞ / pða; bÞ=r a 2 A; b 2 B;r 2 R ð16Þ

where A, B and R are suitable subsets of R

By combining the prior (16) with the likelihood (13), the poster-ior (14) results in

pða; b;rjy; xÞ ¼ r�ðnþ1Þ

Dexp � 1

2r2

Xn

i¼1

ðyi � aþ bxiÞ2( )

pða; bÞ

ð17Þ

The normalizing constant D is given by

D ¼ 2ðn�2Þ=2Z

A

ZB

gða; bÞ�n=2G½gða; bÞ; R�pða; bÞdadb

where gða; bÞ ¼Pn

i¼1ðyi � aþ bxiÞ2 and

G½gða; bÞ; R� ¼Cðn=2Þ when R ¼ ð0;1ÞIG n

2 ;gða;bÞ2r2

L

� �� IG n

2 ;gða;bÞ2r2

U

� �when R ¼ ðrL;rUÞ

8<:

with IGðm; fÞ ¼R f

0 zm�1e�zdz (the incomplete gamma function).On the basis of this prior pdf, the following posterior pdf’s are

obtained:Joint posterior pdf of a and b: Integrating (17) over r, the joint

posterior pdf of a and b is obtained

pða; bjy; xÞ ¼ 2ðn�2Þ=2D�1gða; bÞ�n=2G½gða; bÞ; R�pða; bÞ: ð18Þ

Posterior pdf of lx: By making the change of variable a ¼ lx þ bxin (17) and integrating over b and r, the posterior pdf of lx isobtained

pðlxjy;xÞ ¼ 2ðn�2Þ=2D�1Z

Bgðlx; bÞ

�n=2G½gðlx; bÞ; R�pðlx; bÞdb ð19Þ

where gðlx; bÞ ¼Pn

i¼1½yi � lx þ bðxi � xÞ�2 and

G½gðlx; bÞ; R� ¼Cðn=2Þ when R ¼ ð0;1ÞIG n

2 ;gðlx ;bÞ

2r2L

� �� IG n

2 ;gðlx ;bÞ

2r2U

� �when R ¼ ðrL;rUÞ

8<:

Posterior pdf of ypx: By making the change of variable

r ¼ ðypx � aþ bxÞ=zp in (17) and integrating over a and b, the pos-

terior pdf of ypx is obtained

pðypx jy; xÞ ¼ D�1jzpj�1

ZA

ZB

gðypx ; a; bÞpða; bÞdadb ð20Þ

where gðypx ;a;bÞ¼ ½zp=ðyp

x �aþbxÞ�nþ1 expf�12 ½zp= ðyp

x �aþbxÞ�2Pn

i¼1

ðyi�aþbxiÞ2g. The integration domain in (20) is

A¼ðyp

x þ bx�rUzp;ypx þ bx�rLzpÞ when zp > 0

ðypx þ bx�rLzp;y

px þ bx�rUzpÞ when zp < 0

(

B¼ ð�1;þ1Þ

Predictive pdf of a future value of Y: By combining (17) with thelikelihood of a future observation and integrating over the param-eters space, the predictive pdf of Y is derived

pðyjy;xÞ¼2ðn�1Þ=2D�1Z

A

ZB½gðy;a;bÞ��ðnþ1Þ=2G½gðy;a;bÞ;R�pða;bÞdadb

ð21Þ

where gðy; a; bÞ ¼ ðy� aþ bxÞ2 þPn

i¼1ðyi � aþ bxiÞ2 and

68 M. Guida, F. Penta / Structural Safety 32 (2009) 64–76

G½gðy;a;bÞ;R� ¼C½ðnþ1Þ=2� when R¼ð0;1ÞIG nþ1

2 ;gðy;a;bÞ2r2

L

� �� IG nþ1

2 ; gðy;a;bÞ2r2

U

� �when R¼ðrL;rUÞ

8<:

4. Empirical knowledge on material parameters

Many studies have been addressed in the past to establishingempirical relations between static and fatigue properties of steel,aluminium and other alloys. Most of them dealt with data gener-ated under completely reversed strain cycling. Indeed, besidesthe traditional method based on nominal stresses and Basquin’sequation, a more recent approach for the prediction of crack initi-ation or nucleation life is now commonly used, which is based ondata obtained by strain controlled fatigue tests. The fundamentalrelation, often referred to as the Manson–Coffin equation, relatesthe reversals to failure, 2Nf , to the total strain amplitude De=2:

De2¼ Dee

2þ Dep

2¼

S0fE� ð2Nf Þ�b þ e0f � ð2Nf Þ�c ð22Þ

where Dee=2 and Dep=2 are the elastic and plastic strain amplitude,respectively, and E is the Young modulus. In Eq. (22), two additionalparameters with respect to Eq. (9) are needed, which define theplastic strain amplitude, namely the fatigue ductility coefficient e0fand the fatigue ductility exponent c.

Recall that, in the region of high cycles fatigue, i.e. Nf P 105,Dep � De ’ Dee. Moreover, the way the test is controlled has littleeffect on the lifetimes [10], so that SN ’ EDe=2 ’ EDee=2, and theestimates obtained by strain controlled tests are also valid for loadcontrolled tests.

The main empirical formulae that have been proposed for esti-mating the parameters S0f and b [11–18] are listed in Table 1, whererf is the true fracture stress, ef is the ductility or fracture elonga-tion, n0 is the hardening exponent and HB is the Brinnel hardness.Even if the various proposed relations are not always in a closeagreement, most of them assume that a correlation exists betweenthe fatigue coefficient S0f and the ultimate tensile strength Su.Experimental results also seem to show that, for several types ofsteel alloys, the quantities b and logðS0f =SuÞ are correlated, too.

In particular, it is worth mentioning the recent paper [18],where a sample of 845 different structural alloys is analysed. Theauthors conclude that, on the average, steels present significantlyhigher b exponent than aluminium and titanium alloys, thereforedifferent estimates of this parameter should be considered for eachalloy family. Furthermore, correlation between b and tensile prop-erties resulted poor, while the fatigue strength coefficient S0f pre-sented a fair correlation with the ultimate tensile strength. Also

Table 1Empirical estimation methods for Basquin’s formula parameters.

Estimation method S0f b

Morrow [11] – n0=ð1þ 5n0ÞManson’s universal

slopes [12]1:9Su 0.12

Manson’s four points[12]

1:25rf � 2b

rf ’ Suð1þ ef Þlogð0:36rf =SuÞ1=5:6

Mitchell et al. [13] Su þ 345 (MPa) log½ðSu þ 345Þ=ð0:5SuÞ�1=6

Muralidharan andManson [14]

0:623EðSu=EÞ0:832 0.09

Baumel and Seeger[15]

1:5Su 0.087

Ong [16] Suð1þ ef Þ log½ð6:25rf =EÞ=ðSu=EÞ0:81�1=6

Roessle and Fatemi[17]

4:25HBþ 225 (MPa) 0.09

Meggiolaro and Castro[18]

1:5Su 0.09

in [15], some years before, it was recognised the importance of sep-arating the estimation method by alloy families. These results,along with the fact that tensile characteristics of many kinds ofsteels show a correlation with several metallurgical factors whichaffect the fatigue behaviour, motivated us to analyse the fatigueproperties by ‘‘steel families”, defined on the basis of common ‘‘pri-mary alloy elements and thermal treatment”.

Some empirical data from the literature regarding steel alloys[19] are reported and analysed in the following for an illustrativepurpose. In particular, data referring to some hot-rolled (HR) andquenched and tempered (QT) carbon steels are presented inFig. 2. From this figure it can be inferred that, for HR and QT steelalloys, log S0f and log Su as well as b and logðS0f =SuÞ appear to behighly correlated quantities. Moreover, the point estimates ofmaterial parameters of HR and QT carbon steels appear to formwell separated clusters on the graphs in Fig. 2. Data referring tosome low alloys steel (AISI 41xx) are also given in Fig. 3 and corre-lations among material parameters are analysed. Even if, for thesealloys, log S0f and log Su appear to be highly correlated, neverthelessb and logðS0f =SuÞ appear to be not. Finally, data referring to somestainless steels SUS 3xx-B are given in Fig. 4: in this case it appearsthat log S0f and log Su are uncorrelated, whereas b and logðS0f =SuÞseem to be correlated.

These empirical observations suggest to consider HR, QT, AISI41xx and SUS 3xx-B steels as different ‘‘steel families”, in orderto increase the estimation accuracy when empirical regressionlines based on a steel population are used to predict the valuesof S0f and b, once the ultimate tensile strength of the material is gi-ven. The observed variability in the material parameters is poten-tially due to (at least) two non deterministic factors: (a) therandomness of their estimates due to finite sampling, and (b) aninherent random variation of material parameters over the popula-tion of steels, which would determine a departure from a meantrend even in absence of estimation error. In order to detect thenature of the underlying variability, specific tests should be carriedout in which the estimation error is made negligible by using largesample sizes. It is to be stressed, however, that the present paper isnot focused on proposing or establishing relationships betweenmaterial parameters. Data in Figs. 2–4 are simply reported withthe aim of suggesting that a lot of information about materialparameters is often available, which is totally ignored by conven-tional estimation procedures. In the framework of the Bayes para-digm instead, this information, when properly formalized, could beused in conjunction with test data in order to render more efficientthe inference on quantities of interest. To this end, some Bayesianprocedures will be developed and discussed in the next section.

5. Formalizing the empirical knowledge on material parametersas a prior pdf

Assume that fatigue tests have to be run in order to estimate thematerial parameters b and S0f of a certain steel alloy with a knownultimate tensile strength Su. Let h denote the data set of the esti-mates of these material parameters observed in a population ofsteel alloys characterized by the same ‘‘primary alloy elementsand thermal treatment” as the steel to be tested, which we callhereinafter the ‘‘reference population” for this steel. In the frame-work of Bayes inference, the quantities b and a ¼ logðS0f =SuÞ are rv’sand their joint pdf, given h and Su, say pða; bjh; SuÞ, can be factoredas pða; bjh; SuÞ ¼ f ðbja;h; SuÞf ðajh; SuÞ.

Without loss of generality, assume that data pertaining to thereference population, which the steel under investigation belongsto, support the hypothesis that a linear relationship exists betweenb and a. Then, it is reasonable to model the prior uncertaintyabout the parameter b of the steel to be tested (given a and Su)

Fig. 2. Carbon steels: (a) plot of log S0f vs log Su; (b) plot of b vs logðS0f =SuÞ.

2.9 3.0 3.1 3.2 3.3 3.4 3.5

2.6

2.8

3.0

3.2

3.4

3.6

regression line ( ):

y = 0.7803 x + 0.7808 (R2

=0.773)

log(

S' f)

log(Su)

-0.05 0.00 0.05 0.10 0.15 0.20

0.00

0.04

0.08

0.12

0.16

0.20β

log(S'

f / S

u)

regression line ( ): y = 0.1423 x + 0.0729 (R2= 0.408)

ba

Fig. 3. AISI 41xx steels: (a) plot of log S0f vs log Su; (b) plot of b vs logðS0f =SuÞ.

ba

0.10

0.15

0.20

0.25

0.30

β

log(S'

f / S

u)

regression line ( ): y = 0.2328 x + 0.0666 (R2=0.617)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.92.65 2.70 2.75 2.80 2.85 2.90

2.6

2.8

3.0

3.2

3.4

3.6

log(

S' f)

log(Su)

regression line ( ):y = - 0.1764 x +3.7489 (R2=0.0036)

Fig. 4. SUS 3xx-B steels: (a) plot of log S0f vs log Su; (b) plot of b vs logðS0f =SuÞ.

M. Guida, F. Penta / Structural Safety 32 (2010) 64–76 69

70 M. Guida, F. Penta / Structural Safety 32 (2009) 64–76

by assuming that f ðbja;h; SuÞ is the Bayes predictive pdf for thesimple (Normal) linear regression model bi ¼ A1 þ A2ai þ rbei

ði ¼ 1; :::;mÞ, under the noninformative prior (12) on the parame-ters A1, A2 and rb. This predictive pdf implies that the rvðb� lbjaÞ=½s

ffiffiffiffiffiffiffiffifficðaÞ

p� is Student with m ¼ m� 2 df, where m is the

number of steel alloys in the reference population,lbja ¼ A1 þ A2a, A1 and A2 being the estimated coefficients of theregression line, and

sb ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPmi¼1ðbi � A1 � A2aiÞ2

m� 2

s;

cðaÞ ¼ 1þ 1mþ ða� �aÞ2Pm

i¼1ðai � �aÞ2with �a ¼

Xm

i¼1

ai=m:

Thus, the proposed conditional prior pdf of b, given a and Su, is

f ðbja;h; SuÞ ¼Cffiffiffiffiffiffiffiffiffiffiffiffiffi

m� 2p

rbja1þ b� lbjaffiffiffiffiffiffiffiffiffiffiffiffiffi

m� 2p

rbja

" #28<:

9=;�ðm�1Þ=2

ð23Þ

where C ¼ C½ðm� 1Þ=2�=fCð1=2ÞC½ðm� 2Þ=2�g and rbja ¼ sb

ffiffiffiffiffiffiffiffifficðaÞ

p.

Now, analyse log S0f and log Su and assume that data in the refer-ence population support the hypothesis that a linear relationshipexists between these variables, too. Then, following the sameabove-mentioned arguments, it can be assumed that, given theultimate tensile strength of the material to be tested, the priorknowledge on the rv c ¼ log S0f can be expressed by the Bayes pre-dictive pdf for the simple (Normal) linear regression modelci ¼ B1 þ B2 log Si

u þ rcei ði ¼ 1; :::;mÞ, under the noninformativeprior on the parameters B1, B2 and rc.

Thus, the conditional prior pdf of c, given Su, is

f ðcjh; SuÞ ¼Cffiffiffiffiffiffiffiffiffiffiffiffiffi

m� 2p

rcjSu

1þ c� lcjSuffiffiffiffiffiffiffiffiffiffiffiffiffim� 2p

rcjSu

" #28<:

9=;�ðm�1Þ=2

where lcjSu ¼ B1 þ B2 log Su, B1 and B2 being the estimated coeffi-cients of the regression line,

rcjSu ¼ scffiffiffiffiffiffiffiffiffiffifficðSuÞ

p; sc ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPmi¼1ðci � B1 � B2 log Si

uÞ2

m� 2

s;

and

0.100.30

0.60

0.90

0.100.30

0.60

0.90

0.04

0.08

0.12

0.16

0.20a

β

α

Su=1000 MPa

Su=350 MPa

0.0 0.1 0.2 0.3 0.4 0.5

Fig. 5. Prior joint degree of belief on a and b (a) and on a and

cðSuÞ ¼ 1þ 1mþ ðlog Su � log SuÞ2Pn

i¼1ðlog Siu � log SuÞ2

; with log Su ¼Xm

i¼1

log Siu=m

Then, given Su, it readily follows that the conditional pdf of therv a ¼ logðS0f =SuÞ ¼ c� logðSuÞ is

f ðajh; SuÞ ¼Cffiffiffiffiffiffiffiffiffiffiffiffiffi

m� 2p

rajSu

1þ a� lajSuffiffiffiffiffiffiffiffiffiffiffiffiffim� 2p

rajSu

" #28<:

9=;�ðm�1Þ=2

ð24Þ

where lajSu ¼ lc � log Su ¼ B1 � ð1� B2Þ log Su, and rajSu � rcjSu ¼sc

ffiffiffiffiffiffiffiffiffiffifficðSuÞ

p.

By combining (23) and (24), the joint prior pdf of a and b, givenh and Su, is obtained

pða;bjh; SuÞ ¼ f ðbja;h; SuÞf ðajh; SuÞ

¼ C2

ðm� 2ÞrajSu rbja1þ b� lbjaffiffiffiffiffiffiffiffiffiffiffiffiffi

m� 2p

rbja

" #28<:

9=;

24

1þ a� lajSuffiffiffiffiffiffiffiffiffiffiffiffiffim� 2p

rajSu

" #28<:

9=;35�ðm�1Þ=2

ð25Þ

whose parameters can be estimated from data pertaining to thereference population of steel alloys the one to be tested belongsto.

As an example, the prior knowledge (25) on the materialparameters a and b of a new HR steel alloy with ultimate tensilestrength Su ¼ 350 MPa or Su ¼ 1000 MPa, is depicted in Fig. 5a.

The joint prior pdf (25) was derived under the assumption thatboth the couples of variables ðb;aÞ and ðlog S0f ; log SuÞ are linearlydependent, as historical data suggest that it happens for HR andQT steel alloys. This prior pdf, however, can be easily adapted tothe situation where one (or even both) of the above-mentionedcouples of variables are uncorrelated.

In particular, when b and a are uncorrelated, as for example itseems to happen for AISI 41xx steel alloys (see Fig. 3), thepredictive pdf of b implies that the rv ðb� lbÞ=rb is Student withm ¼ m� 1 df, where lb ¼

Pmi¼1bi=m ¼ �b, and rb ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1=m

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPm

i¼1ðbi � �bÞ2=ðm� 1Þq

.

b

0.1

0.30.6

0.1

0.3

0.5

3 4 5 6 7 8 9

5.0

7.5

10.0

12.5

15.0

17.5

0.9

0.9b

a

0.7

Su=1000 MPa

Su=350 MPa

b (b) for HR steels with Su ¼ 350 MPa and Su ¼ 1000 MPa.

M. Guida, F. Penta / Structural Safety 32 (2010) 64–76 71

Hence

f ðbjh; SuÞ ¼C½m=2�

Cð1=2ÞC½ðm� 1Þ=2�

1ffiffiffiffiffiffiffiffiffiffiffiffiffim� 1p

rb

1þ b� lbffiffiffiffiffiffiffiffiffiffiffiffiffim� 1p

rb

" #28<:

9=;�m=2

ð26Þ

Instead, when log S0f and log Su are uncorrelated, as for exampleit seems to happen for SUS 3xx-B steel alloys (see Fig. 4), the priorpdf of a is

f ðajh; SuÞ ¼C½m=2�

Cð1=2ÞC½ðm� 1Þ=2�

1ffiffiffiffiffiffiffiffiffiffiffiffiffim� 1p

ra1þ b� lajSuffiffiffiffiffiffiffiffiffiffiffiffiffi

m� 1p

ra

� �2( )�m=2

ð27Þ

where lajSu ¼Pm

i¼1ci=m� log Su ¼ �c� log Su, ci being the log fatiguestrength coefficient of the ith steel in the reference population, and

ra ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1=m

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPmi¼1ðci � �cÞ2=ðm� 1Þ

q.

The parameters a and b are related to the parameters a and b ofmodel (3) by Eq. (10). For fixed values of a and b, the equations sys-tem (10) has a unique solution. Moreover, the Jacobian of thetransformation is b3. Then, by the transformation theorem, thepdf of a and b can be converted into the pdf of a and b. It results

pða; bjh; SuÞ ¼C2b�3

ðm� 2ÞrajSu rbja1þ b�1 � lbjaffiffiffiffiffiffiffiffiffiffiffiffiffi

m� 2p

rbja

" #28<:

9=;

24

1þ b�1ða� C1Þ þ C2 � lajSuffiffiffiffiffiffiffiffiffiffiffiffiffim� 2p

rajSu

" #28<:

9=;35�ðm�1Þ=2

ð28Þ

where lajSu � lajSu , rajSu � rajSu , lbja ¼ A1 þ A2½b�1ða� C1Þ þ C2� and

rbja ¼ sb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 1

mþ ½b

�1ða� C1Þ þ C2 � �a�2Pmi¼1ðai � �aÞ2

vuut :

With reference to HR steels with ultimate tensile strengthsSu ¼ 350 MPa and Su ¼ 1000 MPa, respectively, in Fig. 5b the priorknowledge about a and b, derived from the reference population, ispresented, when the k ¼ 3 relative loads Si=Su ¼ 0:35;0:375;0:40are used in the fatigue testing.

Expressions for the prior pdf analogous to (27) or (28) can bederived by using one of (or even both) Eqs (28) and (27) insteadof (23) and (24), when it is the case.

6. Comparison of classical and Bayesian estimation methods

When choosing among different inferential procedures, opti-mality criteria are usually adopted. This is possible, however, onlywithin the same inferential framework. As already observed, in-stead, classical and Bayesian inference represent quite differentpoints of view about ‘‘knowing from experience”. Thus, in principleit makes no sense trying to compare these two approaches. Never-theless, it is a common practice to carry out simulation studies inorder to analyse the behaviour of Bayesian procedures in repeatedsampling. Although philosophically questionable, this approach ap-pears to be the only available tool to compare classical and Bayes-ian methods.

To this end, for a given set of (transformed) load amplitudes xi

ði ¼ 1; :::; kÞ, constant amplitude fatigue tests are simulated where,for each xi, m (transformed) number of cycles to failure Yi are ob-tained by sampling m pseudo-random values from a Normal distri-bution with mean lxi

¼ a� bxi and variance r2 both known. Thus, asample of pseudo-random observations of size n ¼ km is ob-tained and Bayesian point and interval estimates of the regression

model parameters and functions thereof are calculated, under a gi-ven prior pdf. By repeating this procedure a large number NS oftimes (keeping fixed the simulation context and the prior pdf),the sampling properties of Bayesian estimators (i.e., bias, mean-squared error, covering percentage) can be empirically evaluatedand compared with the homogeneous precision measures of thecorresponding classical estimators.

In particular, we recall that for regression model (3), exact valuesof bias and mean-squared error of classical point estimators of a, band yp

x (for any value of p) are available from statistical theory. Infact, a, b and s2 are unbiased minimum variance estimators withknown sampling distributions. They also are stochastically inde-pendent rv’s. These theoretical results are also useful for calibratingthe size NS of simulated tests to be carried out for the comparativeanalysis. In the present case, it was found that NS ¼ 2000 simulatedtests ensure a good agreement between exact and empirical resultsboth in terms of mean-squared errors and covering percentages.

As to the estimation of the material parameters b and S0f in theclassical framework, we recall that they are known functions of themodel parameters a and b. Hence, maximum likelihood estimators,say b and S0f , of these quantities are easily obtained from a and b,due to the invariance property of the maximum likelihood estima-tion principle. The sampling distributions of b and S0f are unknown,however, and bias and variance of these estimators have to beempirically calculated by simulation.

The following precision measures are used when comparingpoint estimators: the normalized bias (NB), defined as the ratio ofthe bias of the estimator to the true value of the parameter; andthe relative efficiency (RE), defined as the ratio of the root mean-squared error (rmse) of the classical estimator to the rmse of theBayesian estimator.

Comparison between the 90% Bayesian lower credibility limitand the 90% lower confidence limit on the 100p-quantile is givenin terms of: the relative mean distance (RMD) defined as the ratioof the mean distance of the classical lower limit from the truequantile to the corresponding mean distance of the Bayesian lowerlimit; the relative standard deviation (RSDV) defined as the ratio ofthe standard deviation of the classical lower limit to the standarddeviation of the Bayesian limit; and the covering percentage (CP),defined as the fraction of times the Bayesian limit is less than orequal to the true 100p-quantile, in repeated sampling.

7. Analysis of the simulation study

A large Monte Carlo study was carried out to assess Bayes esti-mators performances with respect to classical ones, when chang-ing: number of specimens on test, true values of regressionmodel parameters and prior information.

In particular, fatigue tests at k ¼ 3 different constant amplitudeload levels were simulated. The values 0.35, 0.375 and 0.40 wereselected for the ratio Si=Su ði ¼ 1;2;3Þ. Then, for each load level,m pseudo-random determinations of the number of cycles to fail-ure were generated by assuming known values of the materialproperties ðSu; S

0f ; bÞ –which in turn determine known values of

the regression model parameters a and b– and a known value ofthe regression model paramerer r. Letting m vary from 2 to 5, fa-tigue tests with sample sizes n ¼ 6, 9, 12 and 15 were considered.

The pseudo-random lifetimes were generated from two hypo-thetical steel alloys, namely: a HR steel with Su ¼ 1000 MPa,S0f ¼ 1300 MPa and b ¼ 0:085 (HR1000, for short) and a SUS steelwith Su ¼ 535 MPa, S0f ¼ 1870 MPa and b ¼ 0:195 (SUS535). BothHR1000 and SUS535 have material properties very close to thatof some real steels included in the respective reference population.Also, the material properties are close to the corresponding esti-mated values from the respective reference population.

72 M. Guida, F. Penta / Structural Safety 32 (2009) 64–76

As to the regression model parameter r, a consolidated empir-ical evidence of fatigue testing on steels (see, for example, [20])indicates that the variation coefficient of the lifetimes ranges from0.3 to 0.5, which implies the standard deviation of the log-lifetimesto range from 0.13 to 0.20. Thus, the central value of this intervalwas assumed for r in the simulation study.

The prior pdf defined by Eq. (28) was assumed for modelling theuncertainty about the model parameters a and b for HR1000. ForSUS535, instead, the prior pdf was obtained by combining Eq.(23) with Eq. (27), to take into account that for SUS 3xx-B steellog S0f and log Su appear to be uncorrelated. Given the referencepopulation of steels, both pdf’s are completely defined by the Su va-lue pertaining to the steel alloy on test, only. No indications werefound in the literature of a correlation between r and materialparameters, so that an independent prior was used to model theuncertainty about the parameter r. In particular, two differentprior were used, namely: the prior pðrÞ / 1=r over Rþ (i.e., thenoninformative Jeffrey’s prior) and the same prior pðrÞ / 1=r overthe finite interval 0.13–0.20, which is an informative prior.

7.1. Point estimation of model and material parameters

In Table 2, classical and Bayes (under the noninformative prioron r) point estimation of parameters a, b, S0f and b is analysed,when simulating from HR1000 and SUS535 steel. It is immediatelyevident that Bayes estimators of parameters related to the ‘‘slope”of the regression line are exceedingly better than classical ones,achieving rmse’s which are up to hundreds times smaller. The mostcritical situation for classical estimation particularly occurs in caseof steels with a high b value. In fact, when simulating from SUS535(b ¼ 0:195), near to zero, possibly negative, estimates of b were ob-served in a fraction of over 20% of the simulated tests, and it wasnot even possible to evaluate the empirical mean of the classicalpoint estimator of S0f . Instead, even in case of samples where clas-sical estimation fails, Bayes method provides accurate estimates.Both methods show the same efficiency in the estimation of a,however.

This can be explained as it follows. Recall that under the fullynoninformative Bayes approach, i.e. under the Jeffrey’s priorpða; b;rÞ / 1=r, Bayes point and interval estimates of the regres-sion model parameters numerically coincide with the correspond-ing classical estimates. This is because the quantities

ffiffiffinpða� aÞ=s

andffiffiffiffiffiffiffiffiffiffiffiffiffiffiPn

i¼1xi

qðb� bÞ=s are distributed as Student rv’s with ðn� 2Þ

df under both Bayes and classical framework, although with a quitedifferent meaning of the quantities involved. Thus, the joint nonin-formative posterior is a bivariate Student pdf with mode at (a, b),and (for the cases being treated here) with an uncertainty on amuch smaller than that on b. At the same time, this pdf, whencentered on the true a and b values, also describes the sam-pling fluctuation of a and b estimators. When using the priorpða; b;rÞ / pða; bÞ=r, it can be readily shown that the joint poster-ior (18) can be factored as the product of the joint noninformative

Table 2HR1000 and SUS535 – noninformative prior on r: comparison of point estimates of regre

Steel Sample size Parameter estimates

a b

NBc NBb RE NBc NBb

HR1000 6 0 �0.000 1.02 0 �0.0039 0 �0.000 1.01 0 �0.00212 0 �0.000 1.01 0 �0.00215 0 �0.000 1.01 0 �0.002

SUS535 6 0 �0.000 1.01 0 0.01415 0 0.000 1.00 0 0.011

posterior times the prior pða; bÞ. In particular, the priors forHR1000 and SUS535 steels, derived from their reference popula-tions in Figs. 2 and 4 and used for the simulation study, are approx-imately centred on the true material parameters, with anuncertainty on a (on b) greater (smaller) than that conveyed bythe noninformative posterior. In repeated sampling, the noninfor-mative posterior mode presents small fluctuations around theprior mean of a and dominates the prior in terms of this variableso that the informative posterior tends to have the mode close toa. Instead, the noninformative posterior presents large fluctuationsaround the prior mean of b and it is dominated by the prior interms of this variable so that the informative posterior tends tohave the mode close to the prior mean of b. This explains why, inrepeated sampling, the Bayes point estimate of a practically hasthe same properties of a, whereas the Bayes point estimate of bhas dramatically better properties than b, in terms of mean-squared error.

In Table 3 classical and Bayes point estimation of the 50% (med-ian) and the 5%-quantile of the log-lifetime distribution (usuallyknown as the R50 and the R95 quantile, respectively) is analysed,for HR1000 and SUS535. For both steels, Bayes point estimatesshow very similar properties. In term of bias, Bayes estimators be-have as classical ones. Moreover, under the noninformative prioron r, the Bayes point estimator appears to be more efficient thanthe classical one in correspondence of ‘‘low” and ‘‘high” load levels,showing root-mean-squared errors from about 45% to 75% lowerfor the R50 quantile, and from about 10% to 20% for the R95quantile.

For both steels, under the informative prior pðrÞ / 1=r over theinterval 0.13–0.20, only a slight improvement was observed inBayes point estimation of a; b; S0f ; b and of the median log-life,whereas the point estimator of the 5%-quantile of the log-life dis-tribution showed to be exceedingly more efficient than classicalone in correspondence of all the load levels, with rmse’s fromabout 50% to 75% lower in correspondence of the intermediateload, and from about 75% to 125% in correspondence of the two ex-treme loads.

7.2. Lower tolerance limit estimation

In Table 4, for HR1000 and SUS535 steels, the 90% lower toler-ance limit and the Bayes 90% lower credibility limit on the 50% andon the 5%-quantile of the log-lifetimes distribution (usually knownas R50C90 and R95C90, respectively) are analysed.

It appears that, for both steels, Bayes limits greatly outperformclassical ones. In fact, under the noninformative prior on r, in cor-respondence of the ‘‘low” and the ‘‘high” load level, the BayesR50C90 limit is (in the mean) closer to the population median from38% to 56% than the classical one for HR1000 (47% to 50% forSUS535); and the Bayes R95C90 limit is (in the mean) closer tothe 5%-quantile from about 21% to 31% for HR1000 (25% to 38%for SUS535), depending on the sample size. Further improvements

ssion model and material parameters (subscript c for classical, b for Bayes).

S0f b

RE NBc NBb RE NBc NBb RE

9.30 0.534 0.015 234.53 0.066 0.009 30.136.99 0.126 0.014 16.98 0.044 0.008 8.715.68 0.071 0.014 8.64 0.029 0.008 6.464.95 0.048 0.014 6.18 0.020 0.008 5.3210.91 0.073 0.0095.90 0.073 0.194 0.009 33.46

Table 3HR1000 and SUS535 – comparison of point estimation of 50%- and 5%-quantile (subscript c for classical, b for Bayes).

Steel Quantile Prior on r Sample size Quantile estimates

Low load Intermediate load High load

NBc NBb RE NBc NBb RE NBc NBb RE

HR1000 R50 Noninform 6 0 �0.001 1.55 0 �0.001 1.01 0 �0.000 1.749 0 �0.000 1.53 0 �0.000 1.01 0 �0.000 1.6912 0 �0.000 1.48 0 �0.000 1.01 0 �0.000 1.6315 0 �0.000 1.46 0 �0.000 1.00 0 �0.000 1.59

Informative 6 0 �0.000 1.55 0 �0.000 1.01 0 �0.000 1.749 0 �0.000 1.53 0 �0.000 1.01 0 �0.000 1.6912 0 �0.000 1.49 0 �0.000 1.01 0 �0.000 1.6415 0 �0.000 1.46 0 �0.000 1.00 0 �0.000 1.59

R95 Noninform 6 0.003 �0.006 1.10 0.003 �0.007 0.90 0.003 �0.005 1.139 0.002 �0.003 1.17 0.002 �0.003 0.94 0.002 �0.003 1.1912 0.001 �0.002 1.18 0.001 �0.002 0.97 0.001 �0.002 1.2115 0.001 �0.001 1.20 0.001 �0.002 0.98 0.001 �0.001 1.22

Informative 6 0.003 0.001 2.04 0.003 0.001 1.73 0.003 0.001 2.259 0.002 0.001 1.95 0.002 0.001 1.61 0.002 0.001 2.0612 0.001 0.001 1.76 0.001 0.001 1.48 0.001 0.001 1.8715 0.001 0.001 1.77 0.001 0.001 1.49 0.001 0.001 1.86

SUS535 R50 Noninform 6 0 �0.000 1.59 0 �0.001 1.02 0 �0.001 1.7115 0 0.000 1.50 0 �0.000 1.01 0 �0.000 1.58

Informative 6 0 0.000 1.59 0 �0.001 1.01 0 �0.001 1.7115 0 0.000 1.50 0 �0.000 1.01 0 �0.001 1.58

R95 Noninform 6 0.003 �0.008 1.12 0.003 �0.009 0.90 0.003 �0.010 1.1315 0.001 �0.002 1.22 0.001 �0.002 0.98 0.001 �0.003 1.21

Informative 6 0.003 0.001 2.09 0.003 0.001 1.74 0.003 0.001 2.2215 0.001 0.002 1.80 0.001 0.001 1.49 0.001 0.001 1.86

Table 4HR1000 and SUS535 – comparison of 90% lower tolerance limit estimation on the 50%- and 5%-quantile.

Steel Quantile Prior on r Sample size 90% Lower tolerance limit estimates

Low load Intermediate load High load

RMD RSDV CP RMD RSDV CP RMD RSDV CP

HR1000 R50 Noninform 6 1.42 1.60 92.6 1.00 1.04 91.2 1.56 1.73 92.09 1.41 1.58 92.6 1.00 1.04 90.8 1.53 1.69 92.512 1.38 1.54 92.6 1.01 1.02 90.3 1.49 1.65 92.615 1.38 1.54 92.4 1.02 1.05 90.2 1.50 1.67 92.9

Informative 6 1.61 1.76 91.5 1.15 1.17 90.4 1.77 1.89 91.39 1.52 1.65 92.2 1.09 1.07 90.3 1.66 1.77 92.012 1.44 1.57 92.7 1.06 1.05 90.7 1.58 1.70 92.115 1.42 1.58 92.4 1.08 1.07 90.1 1.55 1.67 92.5

R95 Noninform 6 1.27 1.29 90.8 1.15 1.15 90.1 1.31 1.34 90.19 1.22 1.25 90.7 1.08 1.07 90.0 1.26 1.29 90.412 1.21 1.24 90.4 1.06 1.04 89.3 1.26 1.27 90.215 1.21 1.23 90.4 1.06 1.05 89.7 1.26 1.29 90.6

Informative 6 2.88 3.27 92.4 2.77 3.00 91.4 3.14 3.55 92.29 2.32 2.64 92.9 2.20 2.36 91.5 2.51 2.87 92.812 2.08 2.36 92.8 1.94 2.08 91.2 2.24 2.53 92.815 1.93 2.16 92.5 1.84 1.91 91.0 2.11 2.35 92.5

SUS535 R50 Noninform 6 1.50 1.63 92.2 1.00 1.05 91.3 1.50 1.71 92.215 1.47 1.58 91.5 1.02 1.03 90.6 1.43 1.63 93.4

Informative 6 1.70 1.81 91.3 1.14 1.15 90.6 1.70 1.91 91.915 1.54 1.57 91.5 1.07 1.03 89.9 1.50 1.61 93.2

R95 Noninform 6 1.33 1.37 90.7 1.19 1.24 90.2 1.38 1.48 90.715 1.25 1.26 90.0 1.08 1.08 89.7 1.34 1.54 90.9

Informative 6 3.03 3.33 91.8 2.76 3.03 91.4 3.02 3.78 92.415 2.08 2.24 91.7 1.82 1.92 91.0 2.01 2.35 93.0

M. Guida, F. Penta / Structural Safety 32 (2010) 64–76 73

are obtained under the informative prior on r, especially for theR95C90 Bayes limit which is (in the mean) closer to the 5%-quan-tile than the classical one from about 93% to 111% when n ¼ 15 andfrom about 188% to 214% when n ¼ 6 for HR1000, and from about101% to 108% when n ¼ 15 and from about 202% to 214% whenn ¼ 6 for SUS535. It is worth noting that these results are obtainedwith a covering percentage that is even greater than 90%. Theabove-mentioned results can be explained as it follows. Recall thatunder the fully noninformative approach, Bayes and classical limits

numerically coincide. Hence, when we compare the two inferentialmethods in repeated sampling, the properties of the two estima-tors coincide, too. If one looks at Eq. (7), he/she immediately recog-nises that the sampling fluctuation of the classical lower limitdepends on the sampling fluctuation of lx (which, in turn, dependson the fluctuation of a and b) and on the sampling fluctuation of s.Roughly speaking, we may say that, when using an unbiased infor-mative prior pdf on a and b it is as if, in a sense, we contrasted -through this prior - the sample variability of the estimates of a

74 M. Guida, F. Penta / Structural Safety 32 (2009) 64–76

and b, and when adding information on r it is as if we contrastedthe sample variability of the estimate of r. In particular, it is theinformation on r which mainly contributes to reduce the uncer-tainty on yp, by dramatically reducing the long left tail which char-acterizes the two posterior pdf’s obtained under a noninformativeprior and an informative prior on a and b, only. Obviously, whenthe sample size increases, these effects tend to decrease, thoughto an extent which is difficult to quantify in advance, however.

A different very impressive way to measure the global efficiencyof the Bayes estimation method is in terms of specimens savedwith respect to the classical method. As an example, it results thatBayes estimation of the R95C90 limit based upon samples of sizen ¼ 6 have a precision, both in terms of mean distance from thetrue quantile and in terms of standard deviation, that classical esti-mation attains with samples of size n ¼ 12 or n ¼ 24, dependingupon whether prior information on r is available or not. Thus,Bayes estimation of the R95C90 design curve results in a testingtime and cost which is 2 or 4 times smaller than in case of classicalestimation, respectively. In Fig. 6, for HR1000 steel, Bayes and clas-sical average (90% confidence) design curves for the 5%-quantile ofthe lifetimes distribution are presented, for the sample sizesn ¼ 6;9;15. The average performances of the two estimation meth-ods and the effect of using an informative prior on r against a non-informative one are immediately evident.

7.3. Robustness analysis

It is to be noted that previous results are obtained with a priorknowledge about model parameters which is centered on the truevalues, i.e. the values used to generate the pseudo-random deter-minations of the number of cycles to failure. This is in keeping withthe hypothesis that observed departures of material properties

2.52

2.54

2.56

2.58

2.60

2.62

5.00 5.25 5.50

n=9n=6n=15

n=9

n=6

n=15

R95classical design curves bayesian design curves (non informa

bayesian design curve (informative

log

log 10

(S)

Fig. 6. HR1000: classical and Bayes average R95C90 design curves in case of bo

Table 5QT 1300 – comparison of point estimates of regression model and material parameters (s

Prior on r Sample size Material parameters estimates

a b

NBc NBb RE NBc NBb

Noninform 6 0 0.001 1.01 0 0.09915 0 0.000 1.01 0 0.087

Informative 6 0 0.000 1.01 0 0.09715 0 0.000 1.01 0 0.086

from the regression lines characterizing reference population ofsteels, essentially depend on estimation errors. Since, however, itcannot be excluded that an inherent random effect is possibly pres-ent, it is necessary to test the proposed Bayesian approach in casewhen the prior pdf is not centered on the true values of the mate-rial parameters. To this end, we analysed a situation where theprior pdf is chosen through Eq. (28), but the number of cycles tofailure are generated from a hypothetical steel, whose materialparameters values are far from the values that reference popula-tion regression lines would assign to it, on the basis of its ultimatetensile strength.

In particular, we chose the worst case actually observed in thepopulations of both HR and QT steels. This is represented by theQT steel with (rounded) material parameters Su ¼ 1300 MPa,S0f ¼ 1800 MPa and b ¼ 0:090 (QT 1300, for short), to which –givenSu – the regression lines instead assign parameter valuesS0f ¼ 1482 MPa and b ¼ 0:066. Then, the case where the prior iscentered on S0f ¼ 1482 MPa and b ¼ 0:066 while fatigue tests aresimulated from S0f ¼ 1800 MPa and b ¼ 0:090 is analysed.

For this worst case, in Tables 5 and 6, a comparison of classicaland Bayes point estimates (under both the noninformative and theinformative prior on r) is presented.

Unlike the centered prior situation, Bayes point estimators of S0fand b parameters related to the ‘‘slope” of the SN regression line(Table 5) appear to be moderately biased, with a normalized biasranging from 7.5% to 10%. It is to be noted that, except the classicalpoint estimator of b which is unbiased, the corresponding classicalpoint estimators have similar or even much greater biases. Thermse’s of the Bayes point estimators, however, are much smallerthan that of the classical ones. Moreover, Bayes estimation ofparameter a is not affected by the prior, showing the same effi-ciency as classical estimation.

5.75 6.00 6.25

n=6

n=15

n=9tive prior on σ )prior on σ)

10(N)

th noninformative and informative prior on r, for sample sizes n ¼ 6;9;15.

ubscript c for classical, b for Bayes).

S0f b

RE NBc NBb RE NBc NBb RE

2.61 4.86 �0.100 1485.6 0.093 �0.084 6.041.71 0.059 �0.090 2.95 0.023 �0.075 2.192.66 4.86 �0.100 1502.3 0.093 �0.083 6.131.73 0.059 �0.089 2.98 0.023 �0.074 2.21

Table 6QT 1300 – comparison of 50% and 5%-quantile point estimates (subscript c for classical, b for Bayes).

Quantile Prior on r Sample size Quantile estimates

Low load Intermediate load High load

NBc NBb RE NBc NBb RE NBc NBb RE

R50 Noninform 6 0 0.006 1.35 0 0.001 1.01 0 �0.005 1.6415 0 0.005 1.21 0 0.000 1.01 0 �0.005 1.37

Informative 6 0 0.005 1.38 0 0.000 1.02 0 �0.005 1.6215 0 0.005 1.22 0 0.000 1.01 0 �0.005 1.37

R95 Noninform 6 0.003 �0.001 1.16 0.003 �0.006 0.93 0.003 �0.012 1.0715 0.001 0.003 1.17 0.001 �0.002 0.98 0.001 �0.007 1.08

Informative 6 0.003 0.007 1.76 0.003 0.001 1.74 0.003 �0.004 2.1915 0.001 0.006 1.43 0.001 0.001 1.50 0.001 �0.004 1.75

Table 7QT 1300 – comparison of 90% lower tolerance limit estimation on the 50% and 5%-quantile.

Quantile Prior on r Sample size 90% Lower tolerance limit estimates

Low load Intermediate load High load

RMD RSDV CP RMD RSDV CP RMD RSDV CP

R50 Noninform 6 2.20 1.62 82.5 2.80 1.09 90.3 2.37 1.85 96.615 2.46 1.51 79.1 2.00 1.04 90.6 1.55 1.71 98.2

Informative 6 2.48 1.73 81.2 3.11 1.15 89.7 2.56 2.01 96.515 2.60 1.50 78.2 2.11 1.04 90.1 1.59 1.68 98.0

R95 Noninform 6 1.56 1.35 86.7 1.19 1.20 90.4 1.17 1.39 93.615 1.61 1.25 83.5 1.05 1.05 89.8 0.97 1.29 95.8

Informative 6 4.41 3.19 82.1 2.90 3.04 90.6 2.43 3.71 96.815 3.29 2.14 79.8 1.84 1.93 91.3 1.47 2.45 98.2

M. Guida, F. Penta / Structural Safety 32 (2010) 64–76 75

From Table 6 it also appears that Bayes point estimators of 50%and 5%-quantile are practically unbiased and with quite better rel-ative efficiencies, especially for the extreme loads and when theinformative prior on r is used. Hence, the Bayes point estimationapproach appears to be quite robust with respect to the departureof the material properties from the reference population regressionlines.

In Table 7, Bayes 90% lower credibility limits on the 50% and the5%-quantile of the log-lifetimes distribution are analysed (underboth the noninformative and informative prior for r). Bayes limitsresult much closer (in the mean) to the corresponding true valuesthan classical ones. Also, their standard deviations are quite smal-ler than that of classical estimates. The covering percentages, how-ever, suffer of a distorting effect, in the sense that for the ‘‘high”load the covering level is sensibly greater than 90%, whereas forthe ‘‘low” load the covering level is about 80%. Nevertheless, inthe whole the Bayes method appears to be quite robust even in asuch unfavourable situation.

8. Conclusions

A Bayesian analysis of SN fatigue data has been presented forestimating material properties and for establishing fatigue designcurves from small size samples.

Posterior distributions for the linear regression model parame-ters and function thereof have been derived on the basis of priortechnological knowledge available on steel alloys with primary al-loy elements and thermal treatment analogous to the one to betested. Both the case when only prior information on regressionmodel parameters is considered and the case when informationon the log-lifetimes variance is also available, have been analysed.

Bayes procedures have been compared against conventionalones by using Monte Carlo simulation. It was found that the pro-posed Bayes approach largely outperforms the classical one forboth point and interval estimation. In particular, exceedingly bet-

ter performances were observed for Bayes point estimation ofmaterial parameters related to the slope of the regression lineand for Bayes estimation of R95C90 lower limit. In general, itwas observed that the presence of an informative prior on thelog-lifetime variance significantly increases the performance ofBayes estimators. For example, Bayes estimation of R95C90 curvefrom a sample of size n ¼ 6 is as efficient as classical estimationis in case of n ¼ 12 or n ¼ 24 specimens on test, depending uponwhether prior information on r is available or not. Thus, efficiencybeing equal, Bayes estimation of a R95C90 design curve results in atesting time and cost which is 2 or 4 times smaller, respectively.

For the application of the proposed Bayes estimation approach,an ad hoc mathematical software is needed, however, which re-quires a once and for all effort. Computational methods involvedessentially are multidimensional quadrature (up to the third order)and the search for a zero of a real function. The computing time re-quired for the complete analysis of a fatigue lifetimes data set is inthe order of a few minutes on a personal computer. Thus, whenrepetitive analyses are required, computing costs appear to beinsignificant compared with the costs of obtaining the data.

It is also worth noting that the Bayes estimation procedure hasshown to be fairly robust with respect to the formulation of abiased prior information on material parameters.

Therefore, the proposed Bayesian approach appears to be a veryinteresting alternative to the conventional procedures for estimat-ing material fatigue properties and lower bounds of designquantiles.

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