A Bayesian method for establishing fatigue design curves

Structural Safety, 2 (1984) 27-38 Elsevier Soence Pubhshers B.V., Amsterdam - Prmted m The Netherlands

27

A BAYESIAN METHOD FOR ESTABLISHING FATIGUE DESIGN CURVES G. Edwards and L.A. Pacheco Shell Research B V, Rqswqk (The Netherlands)

(Received May 18, 1983; accepted m revised form November 9, 1983)

Key words: Fatigue testing, censored data, S - N curves, fatigue reliability, statistical analysis, Bayesian inference.

ABSTRACT From a rehablfity vtewpomt, the simple conventlonal procedure for establishing design S - N curves from laboratory fattgue test data suffers from two tmportant hmitattons. Ftrstly, the calculated fattgue rehabdity on the destgn curve only reflects the observed "'physwal'" uncertainty associated with the fatigue process itself "Statistical'" uncertamttes, connected with estimatmg the parameters of the fatigue model

are not considered. Secondly, tt is not possible to account for fatigue run-outs (non-fadures) m a rattonal manner. A new procedure, based on Bayesian statistical mference, is presented whtch is capable of handhng both the above problems. Its use ts tllustrated vta the analysts of some typtcal fatigue data sets and a number of general points arising from the results are discussed.

INTRODUCTION

suring adequate fatigue reliability during the planned service life.

The large scatter observed in the results from laboratory fatigue tests (Fig. 1) has led, over the years, to the practice of analysing fatigue data statistically, in order to produce design S - N curves. An S - N curve for a particular structural component or detail relates a nominal, usually constant, amplitude applied stress range (S) to an allowable number of cycles ( N ) which the component or detail can safely withstand at that stress range. Such a curve is used as part of the overall structural design process and plays a major role in e n 016%4730/84/$03.00

LEGEND S CONSTANT AMPLITUDE CYCLIC STRESS RANGE N NUM~R OF STRESS CYCLES x A FATIGUE FAILURE

1 s

(logS)

XXXXXXXX~ X

--

XXXXXX~ X XXX

A FATIGUE RUN-OUT

XX~

n

(log N)

Fig. 1 Typical laboratory fatigue data.

© 1984 Elsevier Soence Pubhshers B.V.

2~

(log S)

\ CLASSICAL

~20-~ __~

S-NCURVE

MEAN S-N

RELATIONSHIP

.[

\ xL~

(Io; N) Fig 2 T h e L N C V fatigue m o d e l

In order to establish an S - N curve from a set of experimental fatigue data, the following methodology is generally adopted [1,2]: (i) a probabilistic model describing the underlying physical process is assumed, (ii) the parameters of this model are estimated from the available data set, (iii) the model and parameter estimates are then used to produce the design S - N curve. In step (i) above, the well k n o w n L i n e a r / N o r m a l / C o n s t a n t Variance ( L N C V ) model is most often used. Here, as illustrated in Fig. 2, the mean S - N relationship is assumed to be linear on logarithmic scales of S and N and the scatter in log N is described by a normal distribution with constant variance. In step (ii), the conventional approach is to employ the classical statistical procedures of linear regression analysis and variance estimation to make point estimates of the model parameters (#, o and k in this case). In step (iii), the design S - N curve is then usually selected at two standard deviations below the mean (Fig. 2), implying a notional fatigue reliability of 0.977 under constant amphtude cycling. This simple "classical" method of analysis has gained widespread acceptance [3]. However, from a reliability viewpoint, it may be considered to suffer from two important limitations: (a) The notional fatigue reliability on the design curve only reflects the observed physt-

~al uncertainty associated with the fatigue process itself. However. when the fatigue data ~et is relatively small, for example when tesl~ng large-scale structural components, then stattsttcal uncertainties concerning the pc,nt parameter esumates could be equally .~gmficant. These should ,tlso be reflected m the rehabihty esumate on the design curve. (b) Fatigue test results (Fig. 1) often contain 'run-outs', i.e. values of cycles reached without failure. Typical causes of run-outs include "series" testing of sets of components, termination of testing at some pre-determlned limit or termination due to some other part of the specimen failing. Such data are said to be censored. The simple method of analysts described above cannot take proper account of censored data and because of this, the run-outs are often treated as failures In view of these factors, an ~mproved method for using the L N C V model to analyse experimental fatigue data is desirable. Thxs paper presents such a method, which is based on Bayesian statistical inference [4-6]. In the past, a number of attempts have been made to treat the above difficulties within the framework of classical statistics, but these have invariably considered only one of the two aspects [1,2] or involved arbitrary simphfymg assumptions [3]. The Bayesian approach ]s very general, revolves no unnecessary assumptions and is readily programmable on a digital computer. The physical uncertainty, statistical uncertainties and censored data are treated m a rational manner in the same analys~s and their combined effects are reflected in the reliability estimate on the final design curve.

THE BAYESIAN ANALYSIS PROCEDURE In the Bayesian procedure, the unknown parameters in the L N C V fatigue model (/~, o and k) are considered explicitly as random quantities. This is a fundamental conceptual difference from the simple classical approach

29 described earlier. A discrete joint probability mass function (PMF) of/~, X and k (Note: = In[o]) is then developed from the available data set, using Bayes' theorem. This expresses the probability that various sets of parameters are correct. (The reason for working with the transformed variable ~ wdl become clear later). Finally, an S-N curve ~s established on which the Bayesian expected rehabihty is equal to a specified value. The expected reliability is the sum of all possible reliabilities (each associated with a particular set of parameters) multiplied by their respective probabilities of being correct.

Joint P M F of (!~, X, k) Bayes' theorem [5] is used to develop a joint PMF of (/.t, 7~, k) from the available data set as follows:

Pr( l~,, hj, kqlD }

Pr{ Oltl,, Xj, kq) "Pr( [t,, ~ky, kq} E E EPl'{Ol[d't, ~kj, kq)"er(~t,~kj, kq} t j q

(1) In equation (1):

Pr( }

denotes the probability that the event { } is true I denotes "conditional upon" D is the event of observing the data set Pr( Dll~,, ~j, kq}- is the likelihood function of the data set, conditional upon a particular set of parameters (see below) Pr{l~,, )tj, kq} is a "prior" PMF of ~, and k (see below) ~,t~jY.q - d e n o t e s summations over discrete values of g,, Xj, kq for which the likelihood function takes on significant values. Qualitatively, eqn. (1) has a simple mean-

ing. It states that "belief" in a particular set of parameter values following observance of the data set is a combination of previously held belief (represented by the prior) and the new evidence provided by the data (represented by the likelihood function). The summation term in the denominator is simply a normalising constant.

Prior P M F of (F,)~, k) In general, the "prior" in a Bayesian analysis can be used to express any subjective belief which exists before new information (data) is obtained. However, in the present application, it is necessary for the parameter estimates (and hence the S-N curve) to be based only on the Information contained in the data set. Therefore, a locally non-informattve prior PMF of (#, X, k) is used which contains negligible information compared with that in the data set. As outlined elsewhere [5], a suitable non-informative prior is:

PF{ I~t, )kj, kq } ~ constant

(2)

Note, the reason for working with the transformed variable 2~= ln[o] is connected with this requirement for a non-informative prior. A uniform prior in (/z, ~, k) can be shown [5] to convey less prior information than one in (t~, o, k). Substitution of (2) into (1) leads to the following simplified form of Bayes' theorem for use in the subsequent analysis.

Pr{ l~,, ~j, kqlD } Pr{ D]l~, , Xj, kq) --- E E EPr( Dll~ ,)t ,kq } t

j

(3)

q

Likelihood function In order to evaluate the required joint PMF of (/~, ~, k) from (3), the likelihood function

30 Pr{ D[/~, X~, kq} must be constructed. Yhl~ expresses the probability of observing the data set. given a particular combinauon of parameter values (kt,, X/, kq). As outlined in the Appendix, for a censored data set consisting of a total of t results w~th / failures and t - - / run-outs, the likelihood function IS given by

exp( X, )

x

I-I

a =f+ 1

Q

exp-~ )

(4)

In eqn. (4): suffix a

- denotes fatigue test a (1 ~< a ~
.~lmple classical approach described m the Introduction. In this case. as discussed 111 the Appendix, tile rehabfllt\ associated with any point (n, s)IS given b\

R=QIn+/'s-#

(5 )

If R is set equal to 0.977, eqn. (5) becomes equivalent to the classical procedure for constructing an S- N curve, Illustrated in Fig 2 Consider, now, the present s~tuanon m which ~, X and k are random and many different combinations of the parameters are poss~ble. each with a probability of occurrence given by (3). In this case, the rehablhty given by (5) is also a random quannty and can take on many different values, each corresponding to a particular (kt, k. k) combination. The probablhty of occurrence of each reliability value is also, therefore, given by (3). In th~s situation, the expected value of the random rellablhty can be used to pro,ade a rational definition of a design curve. The expected rehabihty is obtained by summing all possible rehabflltles multiphed by their respective probabilities of occurrence. Equating the expected rehablht) to a required ~.alue, say R*, then leads to

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I

q

×Pr{#,,k ,kq[D}=R*

(6)

Substitution of (3) and (5) into (6) then gives

[

Bayesian S - N curve Equations (3) and (4) allow a discrete joint of (#, X, k) to be developed from the available data set. This, then, provides the basis for establishing points (n e, s~) on the Bayesian S-N curve. (Note: n~ = log~0N~, s~, --- logl0S~). To explain the method, it is useful first to consider the case where determtmstw point estimates of #, ?~ and k are available, as in the

]

exp( X )

n~ + kqs, - i.t~ ]

PMF

×

EPr{Dlt~,,X,,kq}

=R*

(7)

q

which defines points (n,,, s e) on the reqmred curve. In (7), the likelihood funcnon, Pr( D[l~,, X,, kq} is given by (4).

31

COMPUTING DETAILS AND SOLUTION STRATEGY In order to use (7) to establish an S - N curve from any set of fatigue data, a computer program is necessary. Such a program should be designed to perform the summations in (7) for any specified input pair (n e, Se) and thus evaluate the expected reliability corresponding to this pair. Because of the implicit nature of (7), an iterative approach is then required to establish a point on the required S - N curve. For a selected value of s~, "trial" values of n~ should be varied until the computed reliability is equal to R*. Repeating this procedure at other selected s~ values then leads to other points on the curve. The most critical aspect of the above analysis is the specification of the three-dimensional grid of points for performing the summations in (7). In general, this grid should cover parameter ranges for which the joint P M F (3) takes on significant values and should be sufficiently refined to give a convergent solution to (7). In order to establish these ranges and grid refinement, preliminary investigations are necessary on any data set before the analysis proper is commenced. To facilitate these investigations, the computer program should be designed to enable the grid refinement to be varied easily and should also be capable [7] of producing numerical and visual representations of the separate marginal P M F ' s of the three parameters from equation (3). The marginal P M F of a parameter is obtained from the joint P M F by summing the probability values in the other two senses, thus providing a "summary" of all available information on the parameter.

Firstly, some notional, simulated fatigue data sets are treated. These are uncensored. Secondly, the procedure is applied to a real, censored data set resulting from tests on threaded tubing connectors used in the oil industry.

Simulated, uncensored data Four uncensored data sets, consisting respectively of 15, 30, 45 and 60 results, were simulated using a normal random number generator and the parameters /t = 15.0, X = -0.693, k = 4 . 0 . The advantage of using simulated data is that the parameters defining the imaginary fatigue process are known and the performance of the analysis procedure can be judged in an objective manner. The four simulated data sets are given in Table 1. In each, an equal number of "results" are present at three stress ranges: 199.5 MPa,

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ILLUSTRATIVE ANALYSIS -2 0

Two sets of analysis, using a program of the type described above, are presented here.

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32 IAB[F 1

f-~BLE 1 (~.ontmucd)

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Iest % n, ~5 V number (]ogauS.) logmN .) (MPa) (C}cles) 1 2 ~ 4 ~, 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 3(~ 37 28 29 41) 41 42 43 44 45 46 47 38 4~

23 23 23 23 23 24 24 24 24 24 25 25 25 25 25 23 23 2.3 23 23 24 24 24 24 24 25 25 25 25 25 23 23 23 23 23 24 24 24 24 2.4 25 25 25 2.5 25 23 23 23 23

64208 5 9087 5 4220 61113 65454 5 9162 4 9661 5 7666 5 3638 61754 47901 48976 4.9477 5 6699 50367 6 3485 56183 5.6633 49158 5 3968 5 7954 4 9974 5.0632 5 9375 48187 5 1237 5 0168 55164 5 7980 5 1288 52452 5.4718 60684 5 8149 5 0957 5 0293 4.6974 5 0604 54291 47269 4 3689 5 1475 4 9012 4 9673 4 9018 6.3832 5.9271 5 8235 5 8122

1995 199 5 199 5 1995 1995 251 2 251 2 251 2 251 2 2512 3162 3162 316 2 316 2 3162 199 5 199.5 199 5 1995 199 5 251 2 251 2 251 2 251 2 251 2 316 2 316 2 3162 316.2 316 2 1995 t99 5 1995 199 5 199 5 251 2 251 2 251 2 251 2 251 2 316 2 316 2 316 2 316 2 316 2 199.5 199 5 199.5 199 5

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33

LEGEND:

x

SIMULATED FATIGUE TEST RESULT CLASSICAL R= 0 977

0

S

0

(log s)

S-N

CURVE

BAYESIAN S-N CURVE E (R)= 0 9 7 7

50 28 26 24 22 20 18 16 30

I

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5 0

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60

70

n

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( l o g N)

Fig. 5 Classical and Bayesian S - N curves from simulated data set 1.

LEGEND

x

SIMULATED FATIGUE TEST RESULT CLASSICAL R= 0 9 7 7

s

0

0

( l o g S)

S-N

BAYESIAN S-N E {R) = 0 9 7 7

CURVE CURVE

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Fig. 6. Classical and Bayesian S - N curves from sxmulated data set 2.

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parameter ranges for the function evaluauon grid Le.

',; N cur~ea o b t a i n e d using the classw,d anal\s~s p r o c e d u r e (see I n t r o d u c t i o n ) on ~ h w h the notional rehabfllt~ ~s al,~o () 977

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were established in the prehmlnary mvesttgattons by "trtal and error", using the crttenon that outside the ranges, the marginals should be three orders of magnitude lower than their peak values. The marginals in Fig. 3 were calculated using an 80 × 80 × 80 grid of points to perform the summatmns. Further grid refinement or extensmn of the ranges produced no significant change in the expected rehabihtv according to (7). The marginal P M F ' s for the remaining three data sets were produced using a s~milar approach to that above. Figure 4 shows, for each data set, the ranges of the marginals containing 99% of the probabIhty mass Finally, Figs. 5 and 6 show Bayesian S - N curves obtained from data sets 1 and 2 using the criterion that the expected rehabihty R* (eqn. (7)) should be 0.977. Also shown are the

Censored data from threaded tubing connectors Th~s data set was obtained from faugue tests on a total of 13 threaded tubing connectors. Such connectors are used m the off industry to join flowlmes containing hydrocarbons. Frequently, they are subjected to fatigue loading, both due to the laying process and subsequent wave actton.

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35

LEGEND: x

FATIGUE

FAILURE

FATIGUE RUN-OUT ~

CLASSICAL S-N CURVE R= 0 9 7 7 (RUNOUTS ~ FAILURES) BAYESIAN S - N CURVE E (R) = 0 9 7 7

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Laboratory testing was carried out at stress ranges of 244 MPa, 368 MPa and 480 MPa (s a = 2.39, 2.57 and 2.68), the stress range being measured at the outer surface of the bare pipe adjacent to the connector. At the termination of the various tests there were 7 failures and 6 run-outs, the run-outs being caused by the fact that sets of specimens were tested in "series". The data set, which is interesting both because of its small size and the high level of censoring, is summarised in Table 2. The marginal P M F ' s of /~, ~ and k are shown in Fig. 7. These were obtained using a similar approach to that described earlier. Figure 8 shows the Bayesian S - N curve, again for an expected reliability of 0.977. Also shown is the corresponding classical curve, calculated by treating the run-outs as failures. This assumption is sometimes made in engineering practice in the absence of a capability for treating the runouts in a more rational manner.

DISCUSSION OF RESULTS The above results provide a good illustration of a number of points relating to the performance of the Bayesian analysis procedure. The marginal P M F ' s in Figs. 3 and 7 demonstrate how the procedure provides a complete summary of all information contained in a data set on the parameters of a fatigue process. On the basis of any data set, there will always be a "best" (most likely) estimate of each parameter and also some degree of uncertainty surrounding this estimate. The best estimates (Figs. 3 and 7) are indicated by the peak values of the marginals. The uncertainties are represented by the respective "spreads". As more data become available, the marginals would therefore be expected to centre on the true parameter values and become less dispersed. These trends are illustrated, for the simulated data sets, in Fig. 4. Here, the range containing 99% of the

36 probability mass for each parameter i,~ noticeably smaller for the larger data sets and tends to centre on the specified input parameter value It is to be noted, however, exen for data set 4 (60 results) that there is still considerable uncertainty surrounding the parameter estimates. Considering, now, the Bayesmn S - N curves shown in Figs. 5, 6 and 8, they are seen to lie below those derived using the simple classical procedure, although the notional rehabfllty level associated with Bayesian and classical curves ~s the same. This ~s due to the fact that the Bayesmn curves reflect two sources of uncertainty, i.e. physical and statistical, whereas the classical curves only consider the former. Another ~mportant feature of the Bayesian curves is that they are indeed "curves" and not straight lines. This is explamed by the fact that the parameter uncertainties play a varying role, depending on the distance from the average stress level m the data set. In regions close to the average stress level, the curvature is relatively small and the Bayesmn and classical curves are closest. However, in other regions, the differences become more significant. The above factors suggest that, as more test results become available m any s~tuatlon and statistical uncertainties reduce, then Bayesian and classical curves should tend to comcide. This trend can be observed for simulated data sets 1 and 2 (Figs. 5 and 6) and was confirmed by spot-checks using data sets 3 and 4, whmh are not presented here. In general, the trend is strongest within the range of stress levels present in the data set, because this is where stat~stmal uncertainties are least ~mportant. Outside this region, it is less marked and more sxgmficant differences are present. This last point has important implications because it is often necessary, m engineering practice, to use an S - N curve outside the range of stress levels present m the data set. If the same underlying L N C V fatigue model is assumed to be valid in such a regmn, the

Bayesian curve 1~ clearly the most rational 1~, employ. However. because it hes beh3~ ~tN classical counterpart, ltN u',e may well be unattractive - for obxums reasons. 111 ~uch a situation, the &fference between Bayesian and classical curves can be used to g~ve an red,cation of the need for more experimental work at the relevant stress range(s}. Alternatlx.elx, the designer may dec~de to accept the more stringent Bayesian design cnterm if the costs revolved are less than those of the addmonal testing Considering, now, the effects of censoring in fatigue testing, some mteresting insights are prowded from a comparison of the S N curves in Fig. 5 (s~mulated, uncensored data) and Fig. 8 (censored tubing connector data). In the two cases, the data sets are rather s~milar m size and stress ranges--the most ~mportant difference being that the data set m F~g. 8 has a very high censoring level. Clearly, therefore, the censored data contains much less " i n f o r m a t i o n " on the parameters of the fatigue process than the uncensored one. This is reflected by the fact that the difference between Bayesmn and classical curves ~s greatest m Fig. 8. Also, the "'spread" of the marginals m Fig. 7 (particularly In[o]) is greater than m F~g. 3. Therefore, m this case, it may be concluded that the "information loss" is mainly assocmted with o. More work would be needed to generatise this conclusmn, however. Finally, apphcatlon of the Bayesmn analysis procedure, solar, has shown it to be relatively costly in computer ttme. As an m&cat~on, about 21 hours CPU t~me were reqmred on a Univac 1100 series machine to establish the curve in Fig. 8 for the tubing connectors. However, when such costs are compared w~th those involved in obtaining the data, they are likely, m most cases, to be relatively inslgmflcant.

37

CONCLUDING REMARKS (1) A Bayesian analysis procedure has been presented for establishing fatigue design curves from small censored data sets. The procedure only uses information available m the data set and treats the underlying physical uncertainties, statistical uncertainties and censored data in a consistent manner in the same analysis. Their combined effects are reflected m the reliability estimate on the final design curve. (2) The procedure allows S-N curves to be established from small censored data sets such that average reliability levels are consistent with those in existing (larger data base) curves, derived using classical methods. (3) As more data become available in any situation and statistical uncertainties reduce, Bayesian and classical S-N curves will tend to coincide--particularly within the range of stress levels present in the data set. In general, differences between Bayesian and classical curves can be used to make decisions on the relative benefits of accepting the Bayesian design criterion or carrying out more experimental work. (4) The Bayesian analysis procedure provides important insight into the effects of censoring in fatigue testing. In particular, the information loss associated with the censoring can be examined in connection with the individual parameters of the fatigue model and its overall influence on the resulting design curve studied. (5) A large number of repetitive calculations are required to apply the procedure and computing costs are relatively high. However, in most practical applications, these are likely to be insignificant compared with the costs of obtaining the data.

ACKNOWLEDGEMENTS The authors would like to thank the management of Shell Research B.V. for per-

mission to publish this paper. They are also indebted to Professor A.P. Dawid (Dept. of Statistics, University College, London) and Professor A.F.M. Smith (Dept. of Mathematics, University of Nottingham) for a number of stimulating discussions.

REFERENCES 1 P.H. Wlrsching, Probablhty based fatigue design criteria for offshore structures. API-PRAC ProJect 80-15 Final Report 2nd Year, 1981. 2 J.E. Splndel and E. Halbach, The method of maximum likelihood applied to the statistical analysis of fatigue data. Int. J. Fatigue (April 1979). 3 N W. Snedden, Background to Proposed Design Rules for Steel Welded Joints in Offshore Structures. Dept. of Energy, U.K., May 1981. 4 T Bayes, An essay towards solving a problem in the doctrine of chances. Phil. Trans. Royal Soc. (1763) 53.370 (Reprinted in Blometrica, 45, Dec. 1958). 5 G.E.P. Box and G.C. Tlao, Bayesian Inference in Statistical Analysis. Addison-Wesley, 1973. 6 P.M. Redly, The numerical computation of posterior distributions in Bayesian statistical inference Appl. Stats. 25 (3) (1976). 7 J.C. Naylor and A.F.M. Srmth, Application of a Method for the Efficient Computation of Posterior Distributions University of Nottingham, Dept. of Mathematics (Stats. Group), 1981.

APPENDIX The likelihood function Pr( DI~,, Xj, kq} expresses the probability of observing the data set conditional upon a particular set of the parameters (/~, Xs, kq). It is derived under the assumption that the run-outs and the failures are drawn from a single population described by the LNCV fatigue model. The probability of observing the data set is the probability of the joint occurrence of the failures ( F ) and the run-outs (RO). Assuming independence of all observations, the likelihood function is therefore

Pr(Dllx ,•s, kq)=Pr{F}.Pr{RO }

(A1)

3~ i s

(log S)

/-

A

.....

P o,

. AREA A

(log S)

ORDINATE h

PDF OF rl o

o._ _ .

So ~----

NORMAL {k.t., kcl,o, 0"~}

So

no

/'J*l

n

no

Pr

n

}ii

(loqN)

( ~o9N)

[FA'LURE AT (nO, 5o)] ~ ORDINATE h

~'t [RUNOUT AT (n0,S0) ] o~ AREA A

F~g A1. The probablhty of a fatigue failure at (n,,, ~,)

Fg

A2 T h e p r o b a b l l l t ) o f a fatigue r u n - o u t at ( f t , . ~, ).

The probability of the failures, Pr{ F}, IS considered to be the product of a number of terms, each representing the probability of an individual failure. These individual failure probabilities are calculated from the ordinate of the normal distribution of endurance assumed in the L N C V fatigue m o d e l - - s e e F g . A1. Thus, if (n a, s,) represents an individual fatigue failure (1 ~ a ~
which the run-out is observed. For a run-out at (n., s.), this area is given by

: 1 Pr { r } = .=IH v/~w exp( x: )

A = Q[v.]

×exp

-~

ex-~ X, )

*' (A2)

The probability of the run-outs, Pr{ RO } is also considered to be a product of a number of terms, each representing the probability of an individual run-out. As illustrated m Fig. A2, these individual probabilities are calculated from the area under the normal distribution of endurance to the right of the point at

A=

f[.

1

~/~¢rexp(X,) 1 { n + kqs. - / x -2, e~p(X~

×exp

)

dn

(A3)

It can be shown that (A3) is equivalent to (14)

where r. = (n. + k q s . - ~,)/exp(X:) and Q[ ] = the standard normal integral ( 1 - - t h e standard cure. normal function). Thus. l

Pr{RO}=

I-I Q[v.] a=/+ 1

_He/,,o++o-.,l

/A5)

Finally, substitution of (A5) and (A2) into (A1) leads to eqn. (4) for the likelihood function of the data set.