A Bayesian analysis of fatigue data

Structural Safety 32 (2010) 64–76

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Structural Safety journal homepage: www.elsevier.com/locate/strusafe

A Bayesian analysis of fatigue data Maurizio Guida a,*, Francesco Penta b a b

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

a r t i c l e

i n f o

Article history: Received 4 August 2008 Received in revised form 5 June 2009 Accepted 7 August 2009 Available online 13 September 2009 Keywords: Fatigue testing PSN curves Statistical analysis Bayesian inference Fatigue design curves

a b s t r a c t The aim of the present paper is to bring arguments in favour of Bayesian inference in the context of fatigue testing. In fact, life tests play a central role in the design of mechanical systems, as their structural reliability depends in part on the fatigue strength of material, which need to be determined by experiments. The classical statistical analysis, however, can lead to results of limited practical usefulness when the 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 technological knowledge available from theoretical studies and/or previous experimental results, thus contributing to save time and money. Hence, for the case of steel alloys, a discussion about the usually available technological knowledge is presented and methods to properly formalize it in the form of prior credibility density functions are proposed. Further, the performances of the proposed Bayesian procedures are analysed on the basis of simulation studies, showing that they can largely outperform the conventional ones at the expense of a moderate increase of the computational effort. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction The development of systems with a high structural reliability together with low manufacturing and maintenance costs requires the mechanical strength of materials be accurately known. This level of knowledge, however, is difficult to obtain as far as the fatigue phenomenon is concerned, for several reasons: the inherent complexity of the fatigue mechanism, the variety of influential factors and the laboriousness of fatigue testing. Even if the laboratory fatigue tests are carried out under controlled operating conditions, fatigue data are characterized by a high variability and require a statistical analysis. This analysis essentially consists in: assuming a probabilistic model for the lifetimes, estimating its unknown parameters and constructing a ‘‘fatigue design curve” which should take into account both the inherent randomness of fatigue phenomenon and the uncertainty on the true values of model parameters due to the finite sample size. The classical (frequentist) statistical analysis, however, can lead to results of limited technical usefulness when the sample size is small. This circumstance often happens at the beginning of the development phase of new systems or components, when limited experimental data are usually available for both cost and time reasons. A lot of theoretical studies and experimental life testing results have been produced by engineers and manufacturers in order to * Corresponding author. Tel.: +39 089 964282; fax: +39 089 964218. E-mail address: [email protected] (M. Guida). 0167-4730/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.strusafe.2009.08.001

evaluate the properties of different materials. When new constant amplitude life tests are carried out, however, this body of technological knowledge is completely ignored by the conventional estimation procedures which are usually adopted. In principle, the available technological information, when properly formalized as a prior credibility on model parameters, might be combined via Bayes theorem with test data to make more accurate inference on the quantities of interest (for a comprehensive reference on Bayes inferential approach see, as an example, [1]). Although a number of papers have appeared which exploited Bayesian inference in the analysis of the propagation of fatigue cracks, very few attempts have been made in the past to use the Bayes approach in the context of SN fatigue tests. In fact, as far as the authors are aware, only two papers have specifically addressed this topic. For the case of offshore structural joints, Madsen [2] considered a Bayesian linear regression analysis where the prior information was derived from the behaviour of similar joints. The posterior distribution of model parameters was then used to predict the fatigue life. Edwards and Pacheco [3] in their paper criticised two limitations of the conventional procedures for establishing design SN curves from laboratory fatigue tests, namely the use of design curves based on point estimates of regression model parameters and the impossibility of accounting for fatigue runouts in a rational manner. Then, they proposed the use of Bayesian method (in its noninformative setting) in order to handle both these limitations. The little attention paid to Bayesian methods in the context of SN fatigue testing is probably due to two main reasons: (a) the

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greater computational burden usually associated to Bayesian methods when compared to the classical methods available for SN tests, and (b) the suspiciousness still existing in many people towards ‘‘subjective” probability which is the basis of Bayesian inference. The aim of this paper is to bring arguments in favour of Bayesian inference in the context of constant amplitude fatigue testing, by taking into account that the nowadays computational capabilities make the use of Bayesian viewpoint feasible and by showing, on the basis of simulation studies, that the performances of Bayesian 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 the basic concepts of SN tests and related classical estimation procedures. In Section 3 the Bayes point of view is briefly outlined and suitable prior and posterior probability density functions (pdf) are introduced. Section 4 reviews technical knowledge on the correlations among static and fatigue properties of steels, which is commonly available. Section 5 is, then, devoted to the mathematical formalization of this technical knowledge in the form of prior pdf’s on material parameters. Finally, Sections 6 and 7 present criteria to compare Bayesian and classical estimators along with numerical results obtained by Monte Carlo simulation. 2. SN tests and the related classical inferences The lifetime of mechanical systems depends in part on the fatigue strength of the material, whose properties need to be determined by experiments. In constant amplitude fatigue tests, test specimens are subjected to alternating load until failure. The magnitude of the load amplitude is the controlled (independent) variable and the number of cycles to failure is the response (dependent) variable. In the finite life region, the number of cycles to failure N i is usually assumed to depend on the load amplitude Si by the Basquin’s relation [4]

Ni ¼

ASB i

ð1Þ

where Að> 0Þ and Bð> 0Þ are (usually unknown) material parameters. Because of inherent microstructural inhomogeneity in the materials properties, differences in the surface and in test conditions of each specimen and other factors, the number of cycles to failure exhibits a random behaviour. To model this experimental variability, a multiplicative random error is introduced in (1) giving

^¼ a

n 1X y n i¼1 i

Pn ^ ¼  Pi¼1 xi yi b n 2 i¼1 xi

ð4Þ

^ are linear estimators of a and b, i.e. they are linear ^ and b Thus, a combinations of the observed data yi . Since Y i are Normal rv’s, in ^ also are Normal rv’s. They are unbiased ^ and b repeated sampling a minimum variance estimators of a and b. The error variance r2 is usually estimated by

s2 ¼

Pn

^ xi 2 l n2

i¼1 ½yi

ð5Þ

^ . This is a minimum variance unbiased estimator ^  bx ^ xi ¼ a where l i 2 of r . From the frequentist (classical) point of view, the uncertainty about an unknown quantity (model parameters or functions thereof, or future values of observable random variables) is measured through a random interval generated by a rule which has in repeated sampling a known probability ð1  cÞ of generating intervals that include the unknown quantity of interest. Depending on the particular quantity to be estimated, random intervals are given different names: confidence intervals, when population parameters such as mean or variance are involved; tolerance intervals, when percentages 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 results hold (see, for example, [5,6]). Confidence interval on the median lx ¼ a  bx: The rv qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ^ and s ^ ¼ s 1=n þ x2 =Pn x2 , has ^  bx ^x ¼ a ^ x  l Þ=s ^ , where l ðl x

lx

lx

i¼1 i

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

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n X u l^ x  st1c=2;n2 t1=n þ x2 = x2i :

ð6Þ

i¼1

Confidence interval on the 100p-quantile ypx ¼ a  bx þ rzp (tolerance interval): The 100p-quantile of the distribution of a rv Y is defined as the value yp such that PrðY 6 yp Þ ¼ p. It can be readily ^ x  ypx Þ=sl^ x has a non-central Student distribushown that the rv ðl m ¼ nP  2 df and non-centrality parameter tion with qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ¼ zp = 1=n þ x2 = ni¼1 x2i . Thus, the ð1  cÞ two-sided equal tailed confidence interval for the 100p-quantile ypx is given by

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , u n X u l^ x  std;1c=2;ðn2Þ t1=n þ x2 x2i ; i¼1

Ni ¼ ASB i ei

ð2Þ

where ei is usually assumed to be distributed as a log-normal random variable (rv) with (unknown) variance independent of Si . In repeated testing, the errors ei are also assumed to be stochastically independent rv’s. For statistical analysis it is convenient to rewrite model (2) by taking the logarithm (either the log base-10 or the natural log) of both sides of (2) and subtracting the sample mean of log Si times  Þ þ rei , where Y i ¼ log N i , ui ¼ log Si , B, arriving to Y i ¼ a  bðui  u P  , and ei  Nð0; 1Þ.  ¼ ð1=nÞ ni¼1 log Si , b ¼ B, a ¼ log A  Bu u P  Þ with ni¼1 xi ¼ 0, the simple linear regresBy letting xi ¼ ðui  u sion model is obtained

Y i ¼ a  bxi þ rei

ð3Þ

Under the above assumptions, the Y i are independently distributed Normal rv’s with mean lxi ¼ a  bxi and constant variance r2 . After having observed the couples ðxi ; yi Þði ¼ 1; :::; nÞ, point estimates of the unknown parameters a and b are obtained by the Least Squares method or, equivalently, by the maximum likelihood method, which both give

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , u n u X t 2 ^ lx  std;c=2;ðn2Þ 1=n þ x x2i :

ð7Þ

i¼1

In particular, a lower confidence limit of the 100p-quantile ypx is usually of interest (a one-sided c size confidence limit). Relative to such a limit, we may say that ‘‘we are ð1  cÞ confident that the true 100p-quantile ypx is greater than this limit”. This is equivalent to say that ‘‘we are ð1  cÞ confident that (at the most) a fraction p of all future observations of the sampled population will not exceed this limit” or equivalently that ‘‘(at least) a fraction ð1  pÞ of all future observations of the sampled population will exceed this limit”. When the latter formulation is used, this limit is often called a lower tolerance limit. Although (under the model hypothesis) the only correct way to define a tolerance interval is via Eq. (7), in the practice different or approximate intervals are often used. As an example, practitioners often refer to a ‘‘design curve” obtained shifting the estimated median SN curve in the logarithm coordinates to the left by two or three times the estimated standard deviation. This approach, however, being based on point estimates, suffers of the obvious

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limitation that it does not take into account the sample variability which, in case of small or moderate samples, may even be the predominant part of the experimental uncertainty. Also, the approximate Owen one-side tolerance limit [7] has been proposed to account for the uncertainty in regression analysis. Prediction interval on a future value of the ffirv Y: The rv qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ^ x Þ=sYl^ x , where sYl^ x ¼ s 1 þ 1=n þ x2 = ni¼1 x2i , has a Student ðY  l distribution with m ¼ n  2 df, so that the ð1  cÞ two-sided equal tailed prediction interval is

ferent from the one described above should be considered. It is to be stressed, however, that data samples with a relatively limited number of specimens, which are often common at the beginning of the development phase of new systems or components, cannot be used for the purpose of determining the whole shape of a SN curve. In fact, in those cases it is hardly possible to estimate even the slope of the high cycle fatigue straight line portion of the fatigue curve, with an acceptable accuracy [8].

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi , u n X u l^ x  st1c=2;n2 t1 þ 1=n þ x2 x2i

3. Bayesian measures of uncertainty

ð8Þ

i¼1

Fig. 1 depicts the aforementioned classical lower limits. Note that the usual representation of SN tests assigns the horizontal axis to the rv Y. Coherently with this setting, the proper regression analysis is to minimize horizontal distances. We recall for future use, that it is common practice among practitioners to write the deterministic link equation between the load amplitude and the number of cycles to failure in the following parametric form

S ¼ S0f ð2NÞb or S=Su ¼ ðS0f =Su Þð2NÞb

ð9Þ

where 2N is the number of reversals, Su is the ultimate tensile strength and S0f is a material parameter, known as the ‘‘fatigue strength coefficient”, which represents the hypothetical fatigue strength value which one would obtain by extrapolating the SN line at one reversal. Then, material fatigue properties can also be set in terms of the parameters a ¼ logðS0f =Su Þ and b which are related to the parameters a and b in (3) by

(

a ¼ C 2 þ b1 ða  C 1 Þ b¼b

ð10Þ

1

log10(S)

P where C 1 ¼  log 2 and C 2 ¼ ð1=nÞ ni¼1 logðSi =Su Þ is the mean of the logarithm of the relative load levels that will be used in the fatigue testing. Finally, we recall that it is often of interest to explore the behaviour of steel alloys at load levels close to the fatigue limit, that is the cut-off point at which the SN curve changes to a nearly horizontal line. In fact, for many components the fatigue limit is a design criterion. In such a case, the straight line may be a poor approximation and S-shaped SN curves might be considered. Also, the assumption of constant standard deviation of the random log-lifetime at all stress levels is untrue. Hence, statistical approaches dif-

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

pðHjyÞ ¼ R H

pðyjHÞpðHÞ ; pðyjHÞpðHÞdH

where pðHÞ is the personal degree of belief that a coherent person has about H before observing the sample y, pðyjHÞ is the likelihood function of the observed data, and pðHjyÞ is the updated personal degree of belief about H after having observed the sample y. Since a coherent person’s degree of belief satisfies the Kolmogorov’s axioms of probability theory, we may think of pðHÞ and pðHjyÞ as (subjective) probability density functions. In particular, pðHÞ is the prior pdf and pðHjyÞ is the posterior pdf of H. The denominator in (11) is simply a normalizing factor which ensures that, over the support of H, the posterior pdf integrates to one. Inference on H can be made on the basis of posterior credibility regions, that is regions over the support of H, with a given posterior probability content. Inference on the scalar components of vector H can be obtained via marginal posterior pdf’s. For such components both ð1  cÞ two-sided equal tailed probability intervals or ð1  cÞ highest posterior density (HPD) intervals can be derived. Point estimates of scalar components of H are often taken to be the expected value of the corresponding marginal posterior pdf. This is because the posterior mean ^ h ¼ EðhjyÞ minimizes the mean quadratic loss function E½ðh  ^ hÞ2 . 3.1. Noninformative Bayesian inference for the simple linear model Different kind of prior functions can arise depending on the degree of initial personal knowledge about model parameters. In particular a noninformative (or reference) prior assumes that ‘‘very little is known about H ”. Although there have been various points of view about how to express the notion of ‘‘knowing little”, it is commonly recognised that a noninformative prior must at least ensure a consistent inference under simple parameter’s transformations. Noninformative priors are often improper, that is they do not integrate to one. This is usually accepted, provided that the posterior is a proper density. The regression model (3) is indexed by the parameters vector H ¼ ½abr. For this model a and b are location parameters, while r is a scale parameter. Assuming independence among parameters information, the noninformative Jeffrey’s prior (which is consistent under parameters’ transformations) is [9]

pða; b; rÞ / 1=r a; bR; rRþ

1 - Estimated median life; 2 - 90% LCL for median life; 3 - 90% LPL for future life; 4 - 90% LCL for the 5%-quantile

4

3

2

1

of lifetimes distribution;

log10(N) Fig. 1. Data points, estimated SN line and classical lower limits

ð11Þ

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 for lx , ypx and for a future observation Y, numerically coincide with the corresponding classical intervals (6)–(8), respectively. Recall, however, that credibility intervals and confidence intervals are quite different inferential objects, in that a ð1  cÞ credibility level has to be thought of as a ‘‘final” measure of precision (the precision

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

after the experiment is run), whereas a ð1  cÞ confidence level has to be thought of as an ‘‘initial” measure of precision (the precision of the ‘‘generating rule” before the experiment is even run). This might appear as just a ‘‘philosophical” difference with no consequence from a practical point of view, thus considering not so appealing the use of noninformative Bayesian inferential framework. It is to be noted, however, that, even from a practical point of view, noninformative inference should be considered as an useful tool when censored sampling is involved. Indeed, in presence of Type I censoring, classical inferential procedures fail to be applied whereas Bayesian approach does not suffer from the experimental context. 3.2. Informative Bayesian inference for the simple linear model

pðrÞ / 1=r rL < r < rU

since this pdf appears to be a very flexible one, while keeping simple mathematics. In fact, when rL ! 0 and rU ! 1 this pdf converges to the noninformative Jeffrey’s prior, whereas it can conveniently describe less vague knowledge about r by simply adjusting the interval ðrL ; rU Þ in a suitable manner.Then, the prior pdf results in

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

engineering, physics, etc. information, mathematical or physical models, expert’s judgements, corporate memory, commercial databases, historical data.

After having observed a random sample of (transformed) number of cycles to failure y ¼ fy1 ; :::; yn g at (transformed) loads x ¼ fx1 ; :::; xn g , one can combine the prior pða; b; rÞ with the likelihood

pðy; xja; b; rÞ ¼

(

1 ð2pÞn=2 rn

) n 1 X 2 exp  2 ðy  a þ bxi Þ ; 2r i¼1 i

ð13Þ

rn D

( exp 

n 1 X ðy  a þ bxi Þ2 2 2r i¼1 i

)

pða; b; rÞ

where D is the normalizing factor given by

Z Z Z

(

rn exp 

n 1 X ðy  a þ bxi Þ2 2 2r i1 i

D

)

pða; bÞ

The normalizing constant D is given by

D ¼ 2ðn2Þ=2

Z Z A

gða; bÞn=2 G½gða; bÞ; Rpða; bÞdadb

B

Pn

i¼1 ðyi

 a þ bxi Þ2 and

8 < Cðn=2Þ when R ¼ ð0; 1Þ     G½gða; bÞ; R ¼ n gða;bÞ : IG n2 ; gða;bÞ  IG when ; R ¼ ðrL ; rU Þ 2 2r2 2r2 L

Rf

U

zm1 ez dz

with IGðm; fÞ ¼ 0 (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 2ðn2Þ=2 D1 gða; bÞn=2 G½gða; bÞ; Rpða; bÞ:

pðlx jy; xÞ ¼ 2ðn2Þ=2 D1 ð14Þ

D¼

n 1 X ðy  a þ bxi Þ2 exp  2 2r i¼1 i

ð18Þ

Posterior pdf of lx : By making the change of variable a ¼ lx þ bx in (17) and integrating over b and r, the posterior pdf of lx is obtained

thus obtaining the posterior density

pða; b; rjy; xÞ ¼

(

rðnþ1Þ

ð17Þ

where gða; bÞ ¼ (1) (2) (3) (4)

ð16Þ

where A, B and R are suitable subsets of R By combining the prior (16) with the likelihood (13), the posterior (14) results in

pða; b; rjy; xÞ ¼

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

ð15Þ

)

pða; b; rÞda db dr:

The marginal pdf’s of a, b and r can be obtained integrating over the other variables. Moreover, by changing variables in (14), the posterior pdf of lx and ypx can be derived. Also, by combining (14) with the likelihood of a future observation Y and integrating over the parameters’ space, the predictive pdf of Y can be obtained. In general a triple integration will be required to obtain a 100pquantile 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 material parameters of steel alloys belonging to the same ‘‘steel family” of the one to be tested are available and empirical relationships can possibly be established between these parameters or functions thereof (see the next section for details). Thus, when formulating a prior pdf based on this information, model parameters a and b need usually to be viewed as correlated rv’s, while parameter r can 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 vague than that about a and b. Then, we will assume

where gðlx ; bÞ ¼

Z

Pn

i¼1 ½yi

gðlx ; bÞn=2 G½gðlx ; bÞ; Rpðlx ; bÞdb

B

ð19Þ

 lx þ bðxi  xÞ2 and

8 < Cðn=2Þ when R ¼ ð0; 1Þ     G½gðlx ; bÞ; R ¼ gðlx ;bÞ gðlx ;bÞ n n : IG 2 ; 2r2  IG 2 ; 2r2 when R ¼ ðrL ; rU Þ L

U

Posterior pdf of ypx : By making the change of variable

r ¼ ðypx  a þ bxÞ=zp in (17) and integrating over a and b, the posterior pdf of ypx is obtained

pðypx jy; xÞ ¼ D1 jzp j1

Z Z A

B

gðypx ; a; bÞpða; bÞdadb

ð20Þ

where gðypx ; a; bÞ ¼ ½zp =ðypx  a þ bxÞnþ1 expf 12 ½zp = ðypx  a þ bxÞ2 ðyi  a þ bxi Þ2 g. The integration domain in (20) is

(

A¼

Pn

i¼1

ðypx þ bx  rU zp ; ypx þ bx  rL zp Þ when zp > 0 ðypx þ bx  rL zp ; ypx þ bx  rU zp Þ when zp < 0

B ¼ ð1; þ1Þ Predictive pdf of a future value of Y: By combining (17) with the likelihood of a future observation and integrating over the parameters space, the predictive pdf of Y is derived

pðyjy;xÞ ¼ 2ðn1Þ=2 D1

Z Z A

½gðy;a;bÞðnþ1Þ=2 G½gðy;a;bÞ; Rpða;bÞdadb

B

ð21Þ 2

where gðy; a; bÞ ¼ ðy  a þ bxÞ þ

Pn

i¼1 ðyi

2

 a þ bxi Þ and

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

8 < C½ðn þ 1Þ=2 when R ¼ ð0; 1Þ     G½gðy;a;bÞ; R ¼ nþ1 gðy;a;bÞ nþ1 gðy;a;bÞ : IG 2 ; 2r2  IG 2 ; 2r2 when R ¼ ðrL ; rU Þ L

U

4. Empirical knowledge on material parameters Many studies have been addressed in the past to establishing empirical relations between static and fatigue properties of steel, aluminium and other alloys. Most of them dealt with data generated under completely reversed strain cycling. Indeed, besides the traditional method based on nominal stresses and Basquin’s equation, a more recent approach for the prediction of crack initiation or nucleation life is now commonly used, which is based on data obtained by strain controlled fatigue tests. The fundamental relation, often referred to as the Manson–Coffin equation, relates the reversals to failure, 2N f , to the total strain amplitude De=2: 0 De Dee Dep Sf ¼ þ ¼  ð2N f Þb þ e0f  ð2N f Þc 2 2 2 E

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 additional parameters with respect to Eq. (9) are needed, which define the plastic strain amplitude, namely the fatigue ductility coefficient e0f and the fatigue ductility exponent c. Recall that, in the region of high cycles fatigue, i.e. N f P 105 , Dep  De ’ Dee . Moreover, the way the test is controlled has little effect on the lifetimes [10], so that SN ’ EDe=2 ’ EDee =2, and the estimates obtained by strain controlled tests are also valid for load controlled tests. The main empirical formulae that have been proposed for estimating the parameters S0f and b [11–18] are listed in Table 1, where rf is the true fracture stress, ef is the ductility or fracture elongation, n0 is the hardening exponent and HB is the Brinnel hardness. Even if the various proposed relations are not always in a close agreement, most of them assume that a correlation exists between the fatigue coefficient S0f and the ultimate tensile strength Su . Experimental results also seem to show that, for several types of steel alloys, the quantities b and logðS0f =Su Þ are correlated, too. In particular, it is worth mentioning the recent paper [18], where a sample of 845 different structural alloys is analysed. The authors conclude that, on the average, steels present significantly higher b exponent than aluminium and titanium alloys, therefore different estimates of this parameter should be considered for each alloy family. Furthermore, correlation between b and tensile properties resulted poor, while the fatigue strength coefficient S0f presented a fair correlation with the ultimate tensile strength. Also

Table 1 Empirical estimation methods for Basquin’s formula parameters. Estimation method Morrow [11] Manson’s universal slopes [12] Manson’s four points [12] Mitchell et al. [13] Muralidharan and Manson [14] Baumel and Seeger [15] Ong [16] Roessle and Fatemi [17] Meggiolaro and Castro [18]

S0f

b 0

0

– 1:9Su

n =ð1 þ 5n Þ 0.12

1:25rf  2b rf ’ Su ð1 þ ef Þ Su þ 345 (MPa)

logð0:36rf =Su Þ1=5:6

0:623EðSu =EÞ0:832

log½ðSu þ 345Þ=ð0:5Su Þ1=6 0.09

1:5Su

0.087

Su ð1 þ ef Þ 4:25HB þ 225 (MPa)

log½ð6:25rf =EÞ=ðSu =EÞ0:81 1=6 0.09

1:5Su

0.09

in [15], some years before, it was recognised the importance of separating the estimation method by alloy families. These results, along with the fact that tensile characteristics of many kinds of steels show a correlation with several metallurgical factors which affect the fatigue behaviour, motivated us to analyse the fatigue properties by ‘‘steel families”, defined on the basis of common ‘‘primary alloy elements and thermal treatment”. Some empirical data from the literature regarding steel alloys [19] are reported and analysed in the following for an illustrative purpose. In particular, data referring to some hot-rolled (HR) and quenched and tempered (QT) carbon steels are presented in Fig. 2. From this figure it can be inferred that, for HR and QT steel alloys, log S0f and log Su as well as b and logðS0f =Su Þ appear to be highly correlated quantities. Moreover, the point estimates of material parameters of HR and QT carbon steels appear to form well separated clusters on the graphs in Fig. 2. Data referring to some low alloys steel (AISI 41xx) are also given in Fig. 3 and correlations among material parameters are analysed. Even if, for these alloys, log S0f and log Su appear to be highly correlated, nevertheless b and logðS0f =Su Þ appear to be not. Finally, data referring to some stainless steels SUS 3xx-B are given in Fig. 4: in this case it appears that log S0f and log Su are uncorrelated, whereas b and logðS0f =Su Þ seem to be correlated. These empirical observations suggest to consider HR, QT, AISI 41xx and SUS 3xx-B steels as different ‘‘steel families”, in order to increase the estimation accuracy when empirical regression lines based on a steel population are used to predict the values of S0f and b, once the ultimate tensile strength of the material is given. The observed variability in the material parameters is potentially due to (at least) two non deterministic factors: (a) the randomness of their estimates due to finite sampling, and (b) an inherent random variation of material parameters over the population of steels, which would determine a departure from a mean trend even in absence of estimation error. In order to detect the nature of the underlying variability, specific tests should be carried out in which the estimation error is made negligible by using large sample sizes. It is to be stressed, however, that the present paper is not focused on proposing or establishing relationships between material parameters. Data in Figs. 2–4 are simply reported with the aim of suggesting that a lot of information about material parameters is often available, which is totally ignored by conventional estimation procedures. In the framework of the Bayes paradigm instead, this information, when properly formalized, could be used in conjunction with test data in order to render more efficient the inference on quantities of interest. To this end, some Bayesian procedures will be developed and discussed in the next section.

5. Formalizing the empirical knowledge on material parameters as a prior pdf Assume that fatigue tests have to be run in order to estimate the material parameters b and S0f of a certain steel alloy with a known ultimate tensile strength Su . Let h denote the data set of the estimates of these material parameters observed in a population of steel alloys characterized by the same ‘‘primary alloy elements and thermal treatment” as the steel to be tested, which we call hereinafter the ‘‘reference population” for this steel. In the framework of Bayes inference, the quantities b and a ¼ logðS0f =Su Þ are rv’s and their joint pdf, given h and Su , say pða; bjh; Su Þ, can be factored as pða; bjh; Su Þ ¼ f ðbja; h; Su Þf ðajh; Su Þ. Without loss of generality, assume that data pertaining to the reference population, which the steel under investigation belongs to, support the hypothesis that a linear relationship exists between b and a. Then, it is reasonable to model the prior uncertainty about the parameter b of the steel to be tested (given a and Su )

69

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

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

3.6

b 0.20

3.4

0.16

3.2

0.12

regression line ( ): 2 y = 0.1423 x + 0.0729 (R = 0.408)

β

'

log(Sf )

a

3.0

0.08

regression line (

): 2 y = 0.7803 x + 0.7808 (R =0.773)

2.8

0.04

2.6

0.00 2.9

3.0

3.1

3.2

3.3

3.4

3.5

-0.05

0.00

0.05

0.10

0.15

0.20

'

log(Sf / Su)

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

a

b

3.6

0.30

regression line ( ): 2 y = 0.2328 x + 0.0666 (R =0.617)

3.4

0.25

β

'

log(Sf)

3.2

0.20

3.0

2.8

0.15

regression line ( ): 2 y = - 0.1764 x +3.7489 (R =0.0036)

2.6 2.65

0.10 2.70

2.75

2.80

2.85

2.90

0.1

0.2

0.3

0.4

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

0.5 '

log(Sf / Su)

0.6

0.7

0.8

0.9

70

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

by assuming that f ðbja; h; Su Þ is the Bayes predictive pdf for the simple (Normal) linear regression model bi ¼ A1 þ A2 ai þ rb ei ði ¼ 1; :::; mÞ, under the noninformative prior (12) on the parameters A1 , A2pand ffiffiffiffiffiffiffiffiffi rb . This predictive pdf implies that the rv ^ bja Þ=½s cðaÞ is Student with m ¼ m  2 df, where m is the ðb  l number of steel alloys in the reference population, l^ bja ¼ A^ 1 þ A^ 2 a, A^ 1 and A^ 2 being the estimated coefficients of the regression line, and

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm 2 ^ ^ i¼1 ðbi  A1  A2 ai Þ sb ¼ ; m2 2

cðaÞ ¼ 1 þ

Þ 1 ða  a þ Pm m  Þ2 ð a a i i¼1

¼ with a

m X

ai =m:

i¼1

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

8 " #2 9ðm1Þ=2 < = ^ bja bl C f ðbja; h; Su Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ^ bja ; ^ bja : m  2r m  2r

^1 þ B ^ 2 log Su , B ^ 1 and B ^ 2 being the estimated coeffi^ cjSu ¼ B where l cients of the regression line,

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm i 2 ^ ^ i¼1 ðci  B1  B2 log Su Þ sc ¼ ; m2

and

a

Then, given Su , it readily follows that the conditional pdf of the rv a ¼ logðS0f =Su Þ ¼ c  logðSu Þ is

8 " #2 9ðm1Þ=2 < = a  l^ ajSu C f ðajh; Su Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ^ ajSu ; ^ ajSu : m  2r m  2r

ð24Þ

^ 1  ð1  B ^ 2 Þ log Su , and r ^ ajS ¼ l ^ c  log Su ¼ B ^ ajSu  r ^ cjSu ¼ where l pffiffiffiffiffiffiffiffiffiffiffi u sc cðSu Þ. By combining (23) and (24), the joint prior pdf of a and b, given h and Su , is obtained

28 " #2 9 < = ^ bja b  l C2 4 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ^ ^ : ðm  2ÞrajSu rbja ^ bja ; m  2r 8 " #2 93ðm1Þ=2 < = a  l^ ajSu 5 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi : ^ ajSu ; m  2r

ð23Þ

8 " #2 9ðm1Þ=2 < = c  l^ cjSu C f ðcjh; Su Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi ^ cjSu ; ^ cjSu : m  2r m  2r

r^ cjSu

m X 1 ðlog Su  log Su Þ2 ; with log Su ¼ log Siu =m þ Pn 2 i m i¼1 i¼1 ðlog Su  log Su Þ

pða; bjh; Su Þ ¼ f ðbja; h; Su Þf ðajh; Su Þ

pffiffiffiffiffiffiffiffiffi ^ bja ¼ sb cðaÞ. where C ¼ C½ðm  1Þ=2=fCð1=2ÞC½ðm  2Þ=2g and r Now, analyse log S0f and log Su and assume that data in the reference population support the hypothesis that a linear relationship exists between these variables, too. Then, following the same above-mentioned arguments, it can be assumed that, given the ultimate tensile strength of the material to be tested, the prior knowledge on the rv c ¼ log S0f can be expressed by the Bayes predictive pdf for the simple (Normal) linear regression model ci ¼ B1 þ B2 log Siu þ rc ei ði ¼ 1; :::; mÞ, under the noninformative prior on the parameters B1 , B2 and rc . Thus, the conditional prior pdf of c, given Su , is

pffiffiffiffiffiffiffiffiffiffiffi ¼ sc cðSu Þ;

cðSu Þ ¼ 1 þ

ð25Þ

whose parameters can be estimated from data pertaining to the reference population of steel alloys the one to be tested belongs to. As an example, the prior knowledge (25) on the material parameters a and b of a new HR steel alloy with ultimate tensile strength Su ¼ 350 MPa or Su ¼ 1000 MPa, is depicted in Fig. 5a. The joint prior pdf (25) was derived under the assumption that both the couples of variables ðb; aÞ and ðlog S0f ; log Su Þ are linearly dependent, as historical data suggest that it happens for HR and QT steel alloys. This prior pdf, however, can be easily adapted to the situation where one (or even both) of the above-mentioned couples of variables are uncorrelated. In particular, when b and a are uncorrelated, as for example it seems to happen for AISI 41xx steel alloys (see Fig. 3), the ^ b Þ=r ^ b is Student with predictive pdf of b implies that the rv ðb  l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P  and r ^b ¼ m ^ b ¼ 1 þ 1=m m ¼ m  1 df, where l b =m ¼ b, i¼1 i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Pm  2 i¼1 ðbi  bÞ =ðm  1Þ.

b 17.5

0.20

15.0

Su=350 MPa

0.16

0.1

Su=1000 MPa

Su=1000 MPa

Su=350 MPa

0.10 0.30 0.60

0.3 0.5

12.5

0.7

0.90 0.12

β

b

0.9

10.0

0.1 0.3

0.90

0.6

0.08

7.5

0.60 0.30 0.10 0.04

0.9

5.0 0.0

0.1

0.2

0.3

α

0.4

0.5

3

4

5

6

7

a

Fig. 5. Prior joint degree of belief on a and b (a) and on a and b (b) for HR steels with Su ¼ 350 MPa and Su ¼ 1000 MPa.

8

9

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

Hence

C½m=2 f ðbjh; Su Þ ¼ Cð1=2ÞC½ðm  1Þ=2

8 " #2 9m=2 < = ^b 1 bl 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi ^b ; ^b : m  1r m  1r

ð26Þ

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

f ðajh; Su Þ ¼

C½m=2 Cð1=2ÞC½ðm  1Þ=2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi ^a m  1r

(



^ ajS bl 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi u ^a m  1r

2 )m=2 ð27Þ

P  ^ ajSu ¼ m where l i¼1 ci =m  log Su ¼ c  log Su , ci being the log fatigue strength coefficient of the ith steel in the q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi reference population, and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pm  2 r^ a ¼ 1 þ 1=m i¼1 ðci  cÞ =ðm  1Þ. The parameters a and b are related to the parameters a and b of model (3) by Eq. (10). For fixed values of a and b, the equations system (10) has a unique solution. Moreover, the Jacobian of the 3 transformation is b . Then, by the transformation theorem, the pdf of a and b can be converted into the pdf of a and b. It results

28 " 1 #2 9 3 < ^ bja = b  l C2b 4 1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi pða; bjh; Su Þ ¼ ^ bja : ^ ajSu r ðm  2Þr ^ bja ; m  2r 8 " 1 #2 93ðm1Þ=2 < ^ ajSu = b ða  C 1 Þ þ C 2  l 5 pffiffiffiffiffiffiffiffiffiffiffiffiffi 1þ : ; ^ ajSu m  2r

ð28Þ

^1 þ A ^ 2 ½b1 ða  C 1 Þ þ C 2  and ^ ajSu  l ^ ajSu , r ^ ajSu  r ^ ajSu , l ^ bja ¼ A where l

r^ bja

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 u  2 1 ½b ða  C 1 Þ þ C 2  a : ¼ s b t1 þ þ Pm m  2 i¼1 ðai  aÞ

With reference to HR steels with ultimate tensile strengths Su ¼ 350 MPa and Su ¼ 1000 MPa, respectively, in Fig. 5b the prior knowledge about a and b, derived from the reference population, is presented, when the k ¼ 3 relative loads Si =Su ¼ 0:35; 0:375; 0:40 are used in the fatigue testing. Expressions for the prior pdf analogous to (27) or (28) can be derived by using one of (or even both) Eqs (28) and (27) instead of (23) and (24), when it is the case. 6. Comparison of classical and Bayesian estimation methods When choosing among different inferential procedures, optimality criteria are usually adopted. This is possible, however, only within the same inferential framework. As already observed, instead, classical and Bayesian inference represent quite different points of view about ‘‘knowing from experience”. Thus, in principle it makes no sense trying to compare these two approaches. Nevertheless, it is a common practice to carry out simulation studies in order to analyse the behaviour of Bayesian procedures in repeated sampling. Although philosophically questionable, this approach appears to be the only available tool to compare classical and Bayesian 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 Y i are obtained by sampling m pseudo-random values from a Normal distribution with mean lxi ¼ a  bxi and variance r2 both known. Thus, a sample of pseudo-random observations of size n ¼ k m is obtained and Bayesian point and interval estimates of the regression

71

model parameters and functions thereof are calculated, under a given prior pdf. By repeating this procedure a large number N S of times (keeping fixed the simulation context and the prior pdf), the sampling properties of Bayesian estimators (i.e., bias, meansquared error, covering percentage) can be empirically evaluated and compared with the homogeneous precision measures of the corresponding classical estimators. In particular, we recall that for regression model (3), exact values of bias and mean-squared error of classical point estimators of a, b and ypx (for any value of p) are available from statistical theory. In ^ and s2 are unbiased minimum variance estimators with ^, b fact, a known sampling distributions. They also are stochastically independent rv’s. These theoretical results are also useful for calibrating the size N S of simulated tests to be carried out for the comparative analysis. In the present case, it was found that N S ¼ 2000 simulated tests ensure a good agreement between exact and empirical results both in terms of mean-squared errors and covering percentages. As to the estimation of the material parameters b and S0f in the classical framework, we recall that they are known functions of the model parameters a and b. Hence, maximum likelihood estimators, ^ ^ and ^ ^ and b, say b S0 , of these quantities are easily obtained from a f

due to the invariance property of the maximum likelihood estima^ and ^ tion principle. The sampling distributions of b S0f are unknown, however, and bias and variance of these estimators have to be empirically calculated by simulation. The following precision measures are used when comparing point estimators: the normalized bias (NB), defined as the ratio of the bias of the estimator to the true value of the parameter; and the relative efficiency (RE), defined as the ratio of the root meansquared error (rmse) of the classical estimator to the rmse of the Bayesian estimator. Comparison between the 90% Bayesian lower credibility limit and the 90% lower confidence limit on the 100p-quantile is given in terms of: the relative mean distance (RMD) defined as the ratio of the mean distance of the classical lower limit from the true quantile to the corresponding mean distance of the Bayesian lower limit; the relative standard deviation (RSDV) defined as the ratio of the standard deviation of the classical lower limit to the standard deviation of the Bayesian limit; and the covering percentage (CP), defined as the fraction of times the Bayesian limit is less than or equal 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 estimators performances with respect to classical ones, when changing: number of specimens on test, true values of regression model parameters and prior information. In particular, fatigue tests at k ¼ 3 different constant amplitude load levels were simulated. The values 0.35, 0.375 and 0.40 were selected 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 failure were generated by assuming known values of the material properties ðSu ; S0f ; bÞ –which in turn determine known values of the regression model parameters a and b– and a known value of the regression model paramerer r. Letting m vary from 2 to 5, fatigue tests with sample sizes n ¼ 6, 9, 12 and 15 were considered. The pseudo-random lifetimes were generated from two hypothetical 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 steel with Su ¼ 535 MPa, S0f ¼ 1870 MPa and b ¼ 0:195 (SUS535). Both HR1000 and SUS535 have material properties very close to that of some real steels included in the respective reference population. Also, the material properties are close to the corresponding estimated 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 empirical evidence of fatigue testing on steels (see, for example, [20]) indicates that the variation coefficient of the lifetimes ranges from 0.3 to 0.5, which implies the standard deviation of the log-lifetimes to range from 0.13 to 0.20. Thus, the central value of this interval was assumed for r in the simulation study. The prior pdf defined by Eq. (28) was assumed for modelling the uncertainty about the model parameters a and b for HR1000. For SUS535, instead, the prior pdf was obtained by combining Eq. (23) with Eq. (27), to take into account that for SUS 3xx-B steel log S0f and log Su appear to be uncorrelated. Given the reference population of steels, both pdf’s are completely defined by the Su value pertaining to the steel alloy on test, only. No indications were found in the literature of a correlation between r and material parameters, so that an independent prior was used to model the uncertainty about the parameter r. In particular, two different prior were used, namely: the prior pðrÞ / 1=r over Rþ (i.e., the noninformative Jeffrey’s prior) and the same prior pðrÞ / 1=r over the 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 prior on r) point estimation of parameters a, b, S0f and b is analysed, when simulating from HR1000 and SUS535 steel. It is immediately evident 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 most critical situation for classical estimation particularly occurs in case of 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 observed in a fraction of over 20% of the simulated tests, and it was not even possible to evaluate the empirical mean of the classical point estimator of S0f . Instead, even in case of samples where classical 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 fully noninformative Bayes approach, i.e. under the Jeffrey’s prior pða; b; rÞ / 1=r, Bayes point and interval estimates of the regression model parameters numerically coincide with the correspondpffiffiffi ^Þ=s ing classical estimates. This is because the quantities nða  a qffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn ^ and i¼1 xi ðb  bÞ=s are distributed as Student rv’s with ðn  2Þ df under both Bayes and classical framework, although with a quite different meaning of the quantities involved. Thus, the joint nonin^ ^, b), formative posterior is a bivariate Student pdf with mode at (a and (for the cases being treated here) with an uncertainty on a much smaller than that on b. At the same time, this pdf, when centered on the true a and b values, also describes the sam^ estimators. When using the prior ^ and b pling fluctuation of a pða; b; rÞ / pða; bÞ=r, it can be readily shown that the joint posterior (18) can be factored as the product of the joint noninformative

Table 2 HR1000 and SUS535 – noninformative prior on Steel

Sample size

SUS535

6 9 12 15 6 15

7.2. Lower tolerance limit estimation In Table 4, for HR1000 and SUS535 steels, the 90% lower tolerance limit and the Bayes 90% lower credibility limit on the 50% and on the 5%-quantile of the log-lifetimes distribution (usually known as R50C90 and R95C90, respectively) are analysed. It appears that, for both steels, Bayes limits greatly outperform classical ones. In fact, under the noninformative prior on r, in correspondence of the ‘‘low” and the ‘‘high” load level, the Bayes R50C90 limit is (in the mean) closer to the population median from 38% to 56% than the classical one for HR1000 (47% to 50% for SUS535); and the Bayes R95C90 limit is (in the mean) closer to the 5%-quantile from about 21% to 31% for HR1000 (25% to 38% for SUS535), depending on the sample size. Further improvements

r: comparison of point estimates of regression model and material parameters (subscript c for classical, b for Bayes).

Parameter estimates a

HR1000

posterior times the prior pða; bÞ. In particular, the priors for HR1000 and SUS535 steels, derived from their reference populations in Figs. 2 and 4 and used for the simulation study, are approximately centred on the true material parameters, with an uncertainty on a (on b) greater (smaller) than that conveyed by the noninformative posterior. In repeated sampling, the noninformative posterior mode presents small fluctuations around the prior mean of a and dominates the prior in terms of this variable so that the informative posterior tends to have the mode close to ^. Instead, the noninformative posterior presents large fluctuations a around the prior mean of b and it is dominated by the prior in terms of this variable so that the informative posterior tends to have the mode close to the prior mean of b. This explains why, in repeated sampling, the Bayes point estimate of a practically has ^, whereas the Bayes point estimate of b the same properties of a ^ in terms of meanhas dramatically better properties than b, squared error. In Table 3 classical and Bayes point estimation of the 50% (median) and the 5%-quantile of the log-lifetime distribution (usually known as the R50 and the R95 quantile, respectively) is analysed, for HR1000 and SUS535. For both steels, Bayes point estimates show very similar properties. In term of bias, Bayes estimators behave as classical ones. Moreover, under the noninformative prior on r, the Bayes point estimator appears to be more efficient than the classical one in correspondence of ‘‘low” and ‘‘high” load levels, showing root-mean-squared errors from about 45% to 75% lower for the R50 quantile, and from about 10% to 20% for the R95 quantile. For both steels, under the informative prior pðrÞ / 1=r over the interval 0.13–0.20, only a slight improvement was observed in Bayes 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 distribution showed to be exceedingly more efficient than classical one in correspondence of all the load levels, with rmse’s from about 50% to 75% lower in correspondence of the intermediate load, and from about 75% to 125% in correspondence of the two extreme loads.

S0f

b

b

NBc

NBb

RE

NBc

NBb

RE

NBc

NBb

RE

NBc

NBb

RE

0 0 0 0 0 0

0.000 0.000 0.000 0.000 0.000 0.000

1.02 1.01 1.01 1.01 1.01 1.00

0 0 0 0 0 0

0.003 0.002 0.002 0.002 0.014 0.011

9.30 6.99 5.68 4.95 10.91 5.90

0.534 0.126 0.071 0.048

0.015 0.014 0.014 0.014 0.073 0.073

234.53 16.98 8.64 6.18

0.066 0.044 0.029 0.020

0.009 0.008 0.008 0.008 0.009 0.009

30.13 8.71 6.46 5.32

0.194

33.46

73

M. Guida, F. Penta / Structural Safety 32 (2010) 64–76 Table 3 HR1000 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

HR1000

R50

Noninform

Informative

R95

Noninform

Informative

SUS535

R50

Noninform Informative

R95

Noninform Informative

6 9 12 15 6 9 12 15 6 9 12 15 6 9 12 15 6 15 6 15 6 15 6 15

Intermediate load

High load

NBc

NBb

RE

NBc

NBb

RE

NBc

NBb

RE

0 0 0 0 0 0 0 0 0.003 0.002 0.001 0.001 0.003 0.002 0.001 0.001 0 0 0 0 0.003 0.001 0.003 0.001

0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.006 0.003 0.002 0.001 0.001 0.001 0.001 0.001 0.000 0.000 0.000 0.000 0.008 0.002 0.001 0.002

1.55 1.53 1.48 1.46 1.55 1.53 1.49 1.46 1.10 1.17 1.18 1.20 2.04 1.95 1.76 1.77 1.59 1.50 1.59 1.50 1.12 1.22 2.09 1.80

0 0 0 0 0 0 0 0 0.003 0.002 0.001 0.001 0.003 0.002 0.001 0.001 0 0 0 0 0.003 0.001 0.003 0.001

0.001 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.007 0.003 0.002 0.002 0.001 0.001 0.001 0.001 0.001 0.000 0.001 0.000 0.009 0.002 0.001 0.001

1.01 1.01 1.01 1.00 1.01 1.01 1.01 1.00 0.90 0.94 0.97 0.98 1.73 1.61 1.48 1.49 1.02 1.01 1.01 1.01 0.90 0.98 1.74 1.49

0 0 0 0 0 0 0 0 0.003 0.002 0.001 0.001 0.003 0.002 0.001 0.001 0 0 0 0 0.003 0.001 0.003 0.001

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.005 0.003 0.002 0.001 0.001 0.001 0.001 0.001 0.001 0.000 0.001 0.001 0.010 0.003 0.001 0.001

1.74 1.69 1.63 1.59 1.74 1.69 1.64 1.59 1.13 1.19 1.21 1.22 2.25 2.06 1.87 1.86 1.71 1.58 1.71 1.58 1.13 1.21 2.22 1.86

Table 4 HR1000 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

HR1000

R50

Noninform

Informative

R95

Noninform

Informative

SUS535

R50

Noninform Informative

R95

Noninform Informative

6 9 12 15 6 9 12 15 6 9 12 15 6 9 12 15 6 15 6 15 6 15 6 15

Intermediate load

High load

RMD

RSDV

CP

RMD

RSDV

CP

RMD

RSDV

CP

1.42 1.41 1.38 1.38 1.61 1.52 1.44 1.42 1.27 1.22 1.21 1.21 2.88 2.32 2.08 1.93 1.50 1.47 1.70 1.54 1.33 1.25 3.03 2.08

1.60 1.58 1.54 1.54 1.76 1.65 1.57 1.58 1.29 1.25 1.24 1.23 3.27 2.64 2.36 2.16 1.63 1.58 1.81 1.57 1.37 1.26 3.33 2.24

92.6 92.6 92.6 92.4 91.5 92.2 92.7 92.4 90.8 90.7 90.4 90.4 92.4 92.9 92.8 92.5 92.2 91.5 91.3 91.5 90.7 90.0 91.8 91.7

1.00 1.00 1.01 1.02 1.15 1.09 1.06 1.08 1.15 1.08 1.06 1.06 2.77 2.20 1.94 1.84 1.00 1.02 1.14 1.07 1.19 1.08 2.76 1.82

1.04 1.04 1.02 1.05 1.17 1.07 1.05 1.07 1.15 1.07 1.04 1.05 3.00 2.36 2.08 1.91 1.05 1.03 1.15 1.03 1.24 1.08 3.03 1.92

91.2 90.8 90.3 90.2 90.4 90.3 90.7 90.1 90.1 90.0 89.3 89.7 91.4 91.5 91.2 91.0 91.3 90.6 90.6 89.9 90.2 89.7 91.4 91.0

1.56 1.53 1.49 1.50 1.77 1.66 1.58 1.55 1.31 1.26 1.26 1.26 3.14 2.51 2.24 2.11 1.50 1.43 1.70 1.50 1.38 1.34 3.02 2.01

1.73 1.69 1.65 1.67 1.89 1.77 1.70 1.67 1.34 1.29 1.27 1.29 3.55 2.87 2.53 2.35 1.71 1.63 1.91 1.61 1.48 1.54 3.78 2.35

92.0 92.5 92.6 92.9 91.3 92.0 92.1 92.5 90.1 90.4 90.2 90.6 92.2 92.8 92.8 92.5 92.2 93.4 91.9 93.2 90.7 90.9 92.4 93.0

are obtained under the informative prior on r, especially for the R95C90 Bayes limit which is (in the mean) closer to the 5%-quantile than the classical one from about 93% to 111% when n ¼ 15 and from about 188% to 214% when n ¼ 6 for HR1000, and from about 101% to 108% when n ¼ 15 and from about 202% to 214% when n ¼ 6 for SUS535. It is worth noting that these results are obtained with a covering percentage that is even greater than 90%. The above-mentioned results can be explained as it follows. Recall that under the fully noninformative approach, Bayes and classical limits

numerically coincide. Hence, when we compare the two inferential methods in repeated sampling, the properties of the two estimators coincide, too. If one looks at Eq. (7), he/she immediately recognises that the sampling fluctuation of the classical lower limit ^ x (which, in turn, depends depends on the sampling fluctuation of l ^ and on the sampling fluctuation of s. ^ and b) on the fluctuation of a Roughly speaking, we may say that, when using an unbiased informative 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 contrasted the sample variability of the estimate of r. In particular, it is the information on r which mainly contributes to reduce the uncertainty on yp , by dramatically reducing the long left tail which characterizes the two posterior pdf’s obtained under a noninformative prior and an informative prior on a and b, only. Obviously, when the sample size increases, these effects tend to decrease, though to an extent which is difficult to quantify in advance, however. A different very impressive way to measure the global efficiency of the Bayes estimation method is in terms of specimens saved with respect to the classical method. As an example, it results that Bayes estimation of the R95C90 limit based upon samples of size n ¼ 6 have a precision, both in terms of mean distance from the true quantile and in terms of standard deviation, that classical estimation attains with samples of size n ¼ 12 or n ¼ 24, depending upon whether prior information on r is available or not. Thus, Bayes estimation of the R95C90 design curve results in a testing time and cost which is 2 or 4 times smaller than in case of classical estimation, respectively. In Fig. 6, for HR1000 steel, Bayes and classical average (90% confidence) design curves for the 5%-quantile of the lifetimes distribution are presented, for the sample sizes n ¼ 6; 9; 15. The average performances of the two estimation methods and the effect of using an informative prior on r against a noninformative one are immediately evident. 7.3. Robustness analysis It is to be noted that previous results are obtained with a prior knowledge about model parameters which is centered on the true values, i.e. the values used to generate the pseudo-random determinations of the number of cycles to failure. This is in keeping with the hypothesis that observed departures of material properties

from the regression lines characterizing reference population of steels, essentially depend on estimation errors. Since, however, it cannot be excluded that an inherent random effect is possibly present, it is necessary to test the proposed Bayesian approach in case when the prior pdf is not centered on the true values of the material parameters. To this end, we analysed a situation where the prior pdf is chosen through Eq. (28), but the number of cycles to failure are generated from a hypothetical steel, whose material parameters values are far from the values that reference population regression lines would assign to it, on the basis of its ultimate tensile strength. In particular, we chose the worst case actually observed in the populations of both HR and QT steels. This is represented by the QT steel with (rounded) material parameters Su ¼ 1300 MPa, S0f ¼ 1800 MPa and b ¼ 0:090 (QT 1300, for short), to which –given Su – the regression lines instead assign parameter values S0f ¼ 1482 MPa and b ¼ 0:066. Then, the case where the prior is centered on S0f ¼ 1482 MPa and b ¼ 0:066 while fatigue tests are simulated from S0f ¼ 1800 MPa and b ¼ 0:090 is analysed. For this worst case, in Tables 5 and 6, a comparison of classical and Bayes point estimates (under both the noninformative and the informative prior on r) is presented. Unlike the centered prior situation, Bayes point estimators of S0f and b parameters related to the ‘‘slope” of the SN regression line (Table 5) appear to be moderately biased, with a normalized bias ranging from 7.5% to 10%. It is to be noted that, except the classical point estimator of b which is unbiased, the corresponding classical point estimators have similar or even much greater biases. The rmse’s of the Bayes point estimators, however, are much smaller than that of the classical ones. Moreover, Bayes estimation of parameter a is not affected by the prior, showing the same efficiency as classical estimation.

2.62 n=6 n=

n=9

15

2.60

n=

n= 15 9 n= 6

log10(S)

2.58

2.56

R95 classical design curves bayesian design curves (non informative prior on σ ) bayesian design curve (informative prior on σ)

2.54

2.52 5.00

5.25

5.50

n= 6

n= 9 n= 15

5.75

6.00

6.25

log10(N) Fig. 6. HR1000: classical and Bayes average R95C90 design curves in case of both noninformative and informative prior on

r, for sample sizes n ¼ 6; 9; 15.

Table 5 QT 1300 – comparison of point estimates of regression model and material parameters (subscript c for classical, b for Bayes). Prior on r

Sample size

Material parameters estimates a

Noninform Informative

6 15 6 15

S0f

b

b

NBc

NBb

RE

NBc

NBb

RE

NBc

NBb

RE

NBc

NBb

RE

0 0 0 0

0.001 0.000 0.000 0.000

1.01 1.01 1.01 1.01

0 0 0 0

0.099 0.087 0.097 0.086

2.61 1.71 2.66 1.73

4.86 0.059 4.86 0.059

0.100 0.090 0.100 0.089

1485.6 2.95 1502.3 2.98

0.093 0.023 0.093 0.023

0.084 0.075 0.083 0.074

6.04 2.19 6.13 2.21

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M. Guida, F. Penta / Structural Safety 32 (2010) 64–76 Table 6 QT 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

R50

Noninform Informative

R95

Noninform Informative

6 15 6 15 6 15 6 15

Intermediate load

High load

NBc

NBb

RE

NBc

NBb

RE

NBc

NBb

RE

0 0 0 0 0.003 0.001 0.003 0.001

0.006 0.005 0.005 0.005 0.001 0.003 0.007 0.006

1.35 1.21 1.38 1.22 1.16 1.17 1.76 1.43

0 0 0 0 0.003 0.001 0.003 0.001

0.001 0.000 0.000 0.000 0.006 0.002 0.001 0.001

1.01 1.01 1.02 1.01 0.93 0.98 1.74 1.50

0 0 0 0 0.003 0.001 0.003 0.001

0.005 0.005 0.005 0.005 0.012 0.007 0.004 0.004

1.64 1.37 1.62 1.37 1.07 1.08 2.19 1.75

Table 7 QT 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

R50

Noninform Informative

R95

Noninform Informative

6 15 6 15 6 15 6 15

Intermediate load

High load

RMD

RSDV

CP

RMD

RSDV

CP

RMD

RSDV

CP

2.20 2.46 2.48 2.60 1.56 1.61 4.41 3.29

1.62 1.51 1.73 1.50 1.35 1.25 3.19 2.14

82.5 79.1 81.2 78.2 86.7 83.5 82.1 79.8

2.80 2.00 3.11 2.11 1.19 1.05 2.90 1.84

1.09 1.04 1.15 1.04 1.20 1.05 3.04 1.93

90.3 90.6 89.7 90.1 90.4 89.8 90.6 91.3

2.37 1.55 2.56 1.59 1.17 0.97 2.43 1.47

1.85 1.71 2.01 1.68 1.39 1.29 3.71 2.45

96.6 98.2 96.5 98.0 93.6 95.8 96.8 98.2

From Table 6 it also appears that Bayes point estimators of 50% and 5%-quantile are practically unbiased and with quite better relative efficiencies, especially for the extreme loads and when the informative prior on r is used. Hence, the Bayes point estimation approach appears to be quite robust with respect to the departure of the material properties from the reference population regression lines. In Table 7, Bayes 90% lower credibility limits on the 50% and the 5%-quantile of the log-lifetimes distribution are analysed (under both the noninformative and informative prior for r). Bayes limits result much closer (in the mean) to the corresponding true values than classical ones. Also, their standard deviations are quite smaller than that of classical estimates. The covering percentages, however, suffer of a distorting effect, in the sense that for the ‘‘high” load the covering level is sensibly greater than 90%, whereas for the ‘‘low” load the covering level is about 80%. Nevertheless, in the whole the Bayes method appears to be quite robust even in a such unfavourable situation. 8. Conclusions A Bayesian analysis of SN fatigue data has been presented for estimating material properties and for establishing fatigue design curves from small size samples. Posterior distributions for the linear regression model parameters and function thereof have been derived on the basis of prior technological knowledge available on steel alloys with primary alloy elements and thermal treatment analogous to the one to be tested. Both the case when only prior information on regression model parameters is considered and the case when information on the log-lifetimes variance is also available, have been analysed. Bayes procedures have been compared against conventional ones by using Monte Carlo simulation. It was found that the proposed Bayes approach largely outperforms the classical one for both point and interval estimation. In particular, exceedingly bet-

ter performances were observed for Bayes point estimation of material parameters related to the slope of the regression line and for Bayes estimation of R95C90 lower limit. In general, it was observed that the presence of an informative prior on the log-lifetime variance significantly increases the performance of Bayes estimators. For example, Bayes estimation of R95C90 curve from a sample of size n ¼ 6 is as efficient as classical estimation is in case of n ¼ 12 or n ¼ 24 specimens on test, depending upon whether prior information on r is available or not. Thus, efficiency being equal, Bayes estimation of a R95C90 design curve results in a testing 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 requires a once and for all effort. Computational methods involved essentially are multidimensional quadrature (up to the third order) and the search for a zero of a real function. The computing time required for the complete analysis of a fatigue lifetimes data set is in the order of a few minutes on a personal computer. Thus, when repetitive analyses are required, computing costs appear to be insignificant compared with the costs of obtaining the data. It is also worth noting that the Bayes estimation procedure has shown to be fairly robust with respect to the formulation of a biased prior information on material parameters. Therefore, the proposed Bayesian approach appears to be a very interesting alternative to the conventional procedures for estimating material fatigue properties and lower bounds of design quantiles.

References [1] O’Hagan A, Forster J. Kendall’s advanced theory of statistics: Bayesian inference. London: Arnold; 2004. [2] Madsen HO. Bayesian fatigue life prediction. In: Eddertz S, Lind NC, editors. Probabilistic methods in the mechanics of solids and structures. Stockholm: Proc IUTAM Symp; 1984. p. 395–406.

76

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

[3] Edwards G, Pacheco LA. A Bayesian method for establishing fatigue design curves. Struct Saf 1984;2:27–38. [4] Basquin OH. The exponential law of endurance tests. ASTM 1910;10:625–30. [5] Draper NR, Smith H. Applied regression analysis. New York: Wiley; 1966. [6] Owen DB. A survey of properties and applications of the noncentral tdistribution. Technometrics 1968;10:445–72. [7] Shen CL, Wirshing PH, Cashman GT. Design curve to characterize fatigue strength. ASME J Eng Mater Technol 1996;118:535–41. [8] Spindel JE, Haibach E. Some considerations in the statistical determination of the shape of S–N curves. In: Little RE, Ekvall JC, editors. Statistical analysis of fatigue data, ASTM STP 744; 1981. p. 89–113. [9] Box GEP, Tiao GC. Bayesian inference in statistical analysis. Reading: AddisonWesley; 1973. [10] Ruiz C, Koenigsberger F. Design for strength and production. Macmillan; 1970. [11] Morrow JD. Cyclic plastic strain energy and fatigue of metals. Internal friction, damping and cyclic plasticity – ASTM STP 738. Philadelphia (PA): American Society for Testing and Materials; 1964. p. 45–87. [12] Manson SS. Fatigue: a complex subject – some simple approximations. Exp Mech J Soc Exp Stress Anal 1965;5(7):193–226.

[13] Mitchell MR, Socie DF, Caulfield EM. Fundamentals of modern fatigue analysis. Fracture Control Program Report No. 26. University of Illinois; 1977, p. 385– 410. [14] Muralidharan U, Manson SS. Modified universal slopes equation for estimation of fatigue characteristics. ASME J Eng Mater Technol 1988;110:55–8. [15] Baumel Jr A, Seeger T. Materials data for cyclic loading – supplement I. Amsterdam: Elsevier; 1990. [16] Ong JH. An improved technique for the prediction of axial fatigue life from tensile data. Int J Fatigue 1993;15(3):213–9. [17] Roessle ML, Fatemi A. Strain-controlled fatigue properties of steels and some simple approximations. Int J Fatigue 2000;22:495–511. [18] Meggiolaro MA, Castro JTP. Statistical evaluation of strain-life fatigue crack initiation predictions. Int J Fatigue 2004;26:463–76. [19] Mitchell MR. Parameters for estimating fatigue life. In: Lampman S, editor. Fatigue and fracture. ASM handbook, vol. 19. ASM International; 1996. p. 963– 79. [20] Wirsching PH. Probabilistic fatigue analysis. In: Sundararajan C, editor. Probabilistic structural mechanics handbook (Theory and industrial applications). Chapman & Hall – ITP; 1995. p. 146–65.