WAI)(’ TE(IINICAI4 R} PORT59-400PART I

c~4~STATISTICAL EVALUATION OF DATA FROM4.fl~ FATIGUE AND CREEP-RUPTURE TESTS

PART I. FUNDAMENTAL CONCEPTS AND GENERAL METHODS

c~.Wuloddi

BoeA’urn i;lIa n, Bri’.a rp~Station, Sweden

ASTIA ‘

APL~.JGTON HALL STATION

ARLINGTON ~2, VIRGINIA

SEPTEMBER1959

ASTIA

r~’t~nrp

~ .~.•._ ~ —

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WRAGHT AIR DEVELOPMENTCENTER

WAI)C TECHNICAL REPORT59-40O

PART I

STATISTICAL EVALUATION OF DATA FROMFATIGUE AND CREEP-RUPTURE TESTS

PART U. FUNDAMENTAL CONCEPTS AND GENrPAL METHODS

WafoddiWeibull

Boekam~illan,Br~~sarpsStation,Sweden

SEPTEMBER1959

MaterialsLaboratory

Cor~tractNo. AF 6l(5l4)-I2~ProjectNo. 7360

1/1—

WRIGHT AIR DEVELOPMENTCENTERAIR RESEARCHAND DEVELOPMENTCOMMAND

UNITED STATESAIR FORCE

WRIGHT-PATTERSONAIR FORCEBASE, OHIO

500 — Odob.r 1959— $-238

FOT~1oR

This report was preparedby Dr. ~Inloddi deibull, Bockarn~llen,BroaarpsStation, 3weden, un~ierU3AF Contract No. A? 61(514)4208.The contract was initiated under Project No. 7360, ‘Materials Analyaiecu~i EvFlluatton Teehnlpies0, Taak No. 736OL~, ‘Fati~’ue and Creep ofZ~ateria1s.’ The contract was administered by the European Office,Air Research and tieveloiinent Cc*xniand The work was monitored by theMaterials Laboratory, Directorate of Labcratoriea, bIri~ht AixDevelopiient Center, under the directic~n of Mr. .1. 3. Trepp.

This report covers work performed during t)~ period of 1 March1958 to 28 Febru9xy 1959.

~JADCTB 59-t~OO ~t I

AP.3ThACP

Fati~ue tests arc clnssified into three types with considerationto the appropriate method of evaluating the data. Ftindamental eta—tiE~tical concept~, general methods, and useful tools are presented.

The poasibilites of estimating distribution parameters have beenexamined.. Starting from the concept of information available masample,, various methods of estimating +~eparametersof location, scale,and shape are discussed, completed by a comparison of the efficiency ofvarious estimates.

General principles of fitting curves to observations are outlinedand applied tä the methods 9f maxiis.~mlikelihood, linear regression,an~best linear cistimators.

PUBLICATIOJ REVIEW

This report has been reviewed aM is approved:

FOR THE CO~IPJ~DERs

WALTER 3. TRAPPChief, Strength end Dyneznici BrnchMatale e.nd CeramtcaDivi.ionMaterials Laboratory

iiiWADC TB 59—400 ‘h I

TABLE OF CONTENTS

sECTION ‘ PAGE

Introduction . , . . . . , . . . . . . -. . . . . . . . . 1

II Classification of Test Series . . . . . . . . . . . . . 2

2.1 Gc,neral , . . . . . . . . .‘ ~ . . . . . . . . . . . 22.2 Short Life Tests . . . . . . . . . . . . . . . . . . 22.3MediumLifeTesta ................. 32.4 Long Life Tests . . . . . . . . . . . . . . . •. . . 3

statistical Methods and Tools . . . a . . • . . . . .

IV Estimation of Distribution Parameters . . . . . . . . . 20

4.1 Information Providedby a Sampis . . . . . . . .

4.2 Methods for Estimating Location and ScaleParameters4.2.1 General . . . . . . . . . . . . . . . . .

4.2.2 Method of Moments . . . . . . . . . . . .

4.2.3 Method of Maxi~mmLtkelihood. . . . . . . . .

4.2.4 A MechanicalModel . . . . . . . . . . .

4.2.5 Best Linear Estimates . . . . . . . . . . .

4.2.6 Nearly Best Linear Estimates . . . . . . . .

4.3 Methods of Estimating Shape Parameters . . . . .

4.4 Efficiency of.Eetimates . . . . . . . . . . .

4.4.1 General . . . . . . . . . . . . . . . .

4.4.2 Maxinum LIkelihood Method . . . . . . .

4.4.3 I.iast SquaresMethod . . . . . . . . . . . . .

4.4.4 Nonparametrio Method . . . . . . . . . .

V Fitting Q.irvee to Observations . . . . . . . . . .

5.1 General . . . . . . . . . . .

5.2. Maxiimiin Likelihood Conditions5.3 Linear RegressionForn,.ilas . . . . .

5~4Linear Estimators for Indepandent Valuen

Bibliography, . . . . . • • •., .‘. . . . . . . . . 48

III

3.1 Fundamental Concepts and F~ruialas . . . . .

3.2 Transformation of Random Variables . . . . . .

3.3 Probability Papers. . . . . . . . . . . .

3.4 Random Number Tables . . . . . . . . . . . . .

3.5 The Statistical Concept of Fatigue’Strength .

4

a. 4• . 11• . 12.. 15

18

2027272729323436394040404141

42

42444446

• . . S • S S .~ • •

• . . • 5 • • S S •~

• 5 • 5 S S

• . • . S •

VADC TB 59.4~0O Pt I iv

Ti

LI ST OP TABLES

TABLE PAGE

I Traraformation of percentages P to probits T . . • . . 50

II Tranr~formationof percentages P to values

of z,_j.1og5(i_P~71 • • •• . . . . • • • • • . • . 51

Ifl Means of order otatistics from the normal distribution • 54

IV Mee.ue of order statistics from the distribution

~ UIl_OP(_Z1/a) . S • S • • S • • • • S • S S • S S 5 55

V Splitting a sampleinto random sub—samples • . • • • . • 58

VI Coefficients of nearly best 1inea~ estimates ~1 and

for the distribution F_l_exp~((z_ja)/P)h/~~J. • • 59

LIST OF ILlUSTRATIONS

FIQJRE PAGE

1 Graphical representationof a random experiment . . • . 65

2 Cuxa.Llatjye distribution function of the discrete type . . 66

3 Cumulative distribution function and frequency functionof the continuoustype •......... .. • . . . . 67

4 Transformation of random variables , • , , • • • • • , , 63

5 The principle of probaDility papc~xs. . . • . • . . . • . 69

6 A graphical construction for the Blom definition ofthe amount of information, • , . • • . • . . • • •.. • 70

7 Graphical solution of the ma~1mu.mlikelihood equations

for the f(z) — o.5.1”~’. • . S • ~ . • • . , • . • • 71

8 ‘Graphical solution of the maximum likelihood squation~for the normal distribution • . . • • . . . • . . . • • 72

9 Graphical solution of the maximum likelihood equation’2

‘for F(s)—l—e’ • • . . • • • . • . • • . • . • . . • 73

WADCTB59-1~00PtI V

1’ •

SECTION I • II~T~O1JJCTION

Fatigue testing is an c~p~nsive and timeabsorbing task due to thefollowirg fdCts8 the test pi~cs has to be p~rpare4 with rroat care; eachsingle test usually take8 a long time; and the large sc~ttrr in fatiguelife demands a lr~rge numbrr of tests. For this reason efficient methodsare of particular importance in fatigue testing.

No substantial sa’~’inp’ ii time and cost can he attained in connec-tion with the two fir°t mertioned sources of eYpensns, at least not com-pared to the possibility of reducing the number of’ te9ta by appropriateprocedures.

For this purpose 1wo different ways are available. ~irst, an im-provement of the testing technique, in particular the testing machine,which frequently introduces, say, fifty pcr cent of the total scatter orperhaps more. Considering the fact that a doubled atardard deviation re-quires four times as many tests for equal accuracy, it is obvious thatpoor testing machines are extremely erponsi’ve to use. Secondly, the useof efficient statistical methods for ev~1uatio,i of the test data. ThisIs the only alternative that will be examined in the present report,which will also be restricted to methods based on known or assumed dis-tributions, while nonparametric methods will be analysed in a followingpart of the Investigation,

The importaice of using efficient nethods for evaluation purposesis Illuatrat~d by a drastic example presented in Section IV, where it isproved that a ~ in the number of tests of some thirty percent or mo~ecan be attaIned by the simple measure of’ usirg the proper formula for com-puting an average value. In many *~ases no increase in computing time arises by such a measure, but, even if so, it Is to be noted that the costof one single specimen of simple shape is about equal to the cost of fif-teen minutes computing by ‘an electrcnio digital computer (data of theSwedish computer BESK) during which time a fair ;mount of computing workcan be performed. The concept and use of efficient estimates will there-fore be tieated at some length,

The initial scope of the present investigation was to state the pos-sibility and merits of applying llrzear regression theory to fatigue andcreep-rupturedata. I~iringthe progressof this work, the program was e.-tended to involve also an analysis of the scatter In fatigue life and fa-tigue strength and to investigate the applicability of maximum-likelihoodand nonparamotric methods.

Under these circumstances It was deemed better and more convenientto the reader, if fundamental concepts and gene~ul methods be collOctedinto a separate part.

Manuscript released by the author 20 April 1959 for publication as aWADCTechnical Report.

~IADCTB 59-400 Pt I —1.-

It has been the author’s ambition to make the deductionsreadableal~c~to those unfamilI~.r with advanced mathematical statistics and. withoutaC~S5Bto suitable handbooks, by using as far as Tofisible mocharioal inter-pretations and models c~..rrent to engineers u.nd designers of’ averagematbe—mattoal ability and experience.

SECTION II. CLA$SIFICATWJ OF TEST SERIES

2.1 General

Fatigue teats may be classified into different categories from ‘a—rious viewpoints, one of them, which will now he taken, is relating tothe consequences of’ the test design on the evaluation method.

Covering the complete field from the cycle life NsO to Ns ~,

three different types of tests can be distinguisheds.

(1) Tests using stress levels close to the static strength of the testpiece, including one or more failures during the first stress cycle.Tests of this type w.ll be called short-life tests.

(2) ‘lests using stress levels below the lower bound of the staticstrength but well above the fatigue limit of the test piece, re-sulting in fatiguo failures of all specimenstested. Teatsofthis type will be called medium—life tests.

~3) Pestsusing stress levels close above 1;he fatigue limit, includingone or more run—outs at a preassigned cycle life. Teste of thistype will be called, long—life tests.

Some featu~es of the vari~is types and the proper w&y of evaluatingthe data are given in the sequel.

2.2 5hort Life Tests

For the pres’~nt, static tests are used as a substitute, not fullj8atisfactory, for the short-life tests. The objaotion.against this pra~e(highly preferable to having no short—life testo at all) is explained bythe fact that a static tensile test takes a time of’ one or tr~ ~nutes,while the duiation of a fatigue stress cycle may be of the magnitude of,say, twenty milliseconds. It is therefore to be expectedthat data fromthese two ty’pcs of tests will belong to different average S-N curves. Aconvincing example is preserted ifl a report by (~Th~MING,~TUL~1 & SCRULTE~l955). Tests of SAE 4340 steel rotating—beamspecimenswith two speeds,130 and 10,000 rpm resulted in average S-li curves differing from eachother. It is beLieved that the discrepancywould be still more accentu..

~iR59-1~OOIt I —2-.

atod by comparingstatic and fatigue tests of materials such as 1i~ht-alloys and plastics.

It is proposed that real short-life tests should be performedbyi~ubjecting. the test piece to one single stress cycle of the same durationas uaed for the fatigue tests and observing the percentage only of failedspecimens. Survivals sh~uld, of course, be subjected to continued iden-tical stress cycling. This would b, a quanta] response tsst of exactlythe samE type as used in tests for determinuL.iun uf fatigue s’rength atpreassigned cycle life.

Data from static tests should be eva1u~ted by the methods of linearestimators (Section IV.2) or of maximum likolihood. (Section IV.2), where-as proper ahort-life tests should be evaluated by means of the method ofleast squares (Section V.3) or that of maximum likelihood applied to pro-portions (Section V.2).

2.3 Medium Life Tests

Thin type of tests is the most common of all fatigue tests. Stresslevels are chosen in such a way that neither fractures during the firstatiess crcle nor run—outs will occur. An average S-N curve is fitted tothe data points by means of the methods of l~st squares (Section V.3),maximum likelihood (Section V.2), or by nonparametric methods.

Deviations from the avexage curve aze computed and pooled, if pos-sible. Their distribution is finally determLned by means of some methodpresentel in Section IV.

2.4 ~ Life Tests

ThIE type of test is mostly ~ised for determination of the fatiguestrE~ngth and its distribution at a preassigned cycle life. By choosingseveral stress levels producing a sufficiently wide range of tractionfailed, t~iedistribution can be determined by a least—squares (Sectionv.3) or by a maximum-likelihood (probit) method (Section V.2).

I

1WADCTN 59-400 Pt I .3..

3

~i~ci’ici~II.L. ~3TJ~I~TI~~L I~’i9~0DJAID P00L~

The result of this experiment can bealso be graphically representedby’ a stepsteps of the height 1/n at each x~ as

/fi~ur. 1. Clearly the tensile strength X( less the whole step furction is given.

ISuch a quantity as X. ~vLich has

fuLction 1’~ called a random variable.t~rs while specific values of it will

to be described by a distributionIt will be denoted by capital let—

‘te denoted ‘by small letters.

Let us i~oworder the specimensaccording to the values x from leastto greatest and assign to each of them an order number, denotesby (i).The ordered values will be denoted ‘by ~ If several specimenswithequal x—values occur, all of them are gi’~ex~the same order number, equalto the highest order number within this group.

By’ this procedure we have attained t1:~t the number of specimenshaving a value equal to ~ less than x., denoted by 1, is equal tothe order number Ci). Dividing these ~wo quantities by the sample sizen, a relative number i/n and a relative order number (1)/n are ob-tained v:hic!~ satisfy the equation

i/n - (i)/n

The function F(x) is called the curailative uistribution function(abbreviated cdt.) and will be defined by

(1)

F(x1) — (i)/n (2,)

The prohsbility that X,from the sample, takes a vah~erelative number i/n and thus

beiong~ngto a specimen drawn at randomequal to or less than x~ is equal .to theby (1) and (2)

Denoting the left—hand member of (3) by P~, we have

F(x~) (4)

3.1 F\indamer.tal Concepts and ~r~las

Let us su~posethat we wish to know some rroperty X attributedto eao}~individual of a large population, for exar~ple,t!e ultimate ten—sUe strength of a batch of specimensof specified material and dimensions.For thin purpose the strerigth,of n nominally identical specimensis mea—~ .,~ulting in n different values x..

given in a table, bat it canfur.ction ‘ F(x) consisting o~dem~onstratcd b~the diagram in

is’tthcomp1ot~lydescribed un—

— i/n . (i)/ni — F(Zj) (3)

WADCTB 59-400 Pt I —4—

A general property of the cdf. F(x), immediately following from

(2), is

0 ~ F(x)~l (5)

The difforcnce betweAn the two hounds sh4mld h~nntsl. 1+ ~~rnybeexpressedby the statemont that P(x) belongs to a halfopen interval(0,1), closed on the right.

Consider now the case that the sample size n tends to infinity.I~oneof the preceding statemen has to be changed, but the function maybe transformod into a continunu.., po4n’t function. The condition (5) isstill valid and the general properti’~sof ~he cdf. F(x) is then thatit is a non—decreasingpoint fur.ctioi. which is everywhere continuous tothe right and satisfies the conditions F(—o~) — 0 and F(~ u 1, whichdoes not exclude the possibility of finite bounds, a lower bound x de-fined by F(x0) 0, and an upper bound x~ defined by F(x~) —

Even if ~ the cdf. may be a stop functIon. The random van—able X is then i’aid to be of the discrete type and to p~cscssa di~—crete cdi’. A quantity of this type is, for example, the proportion offailures occuring in a sample of size 5 repeatedly tested, where the onlyvalues possible are the six values: 0/5, 1/5, ...,5/5.

Suppose that a large number of samples, each óontaining 5 spccimenshave been tested in fatigue at the same stress level. Let p denote‘the percentage of samples having no failure, p1 the percentRge of thosehaving exactly one failure, etc. If X signifies the observed proportion,then the probability that X takes a value x1 is

— x1) — ~ , (.6)

and the probability that X takes a value equal to or less than

p(x f x1) - P~,- F(x ) - E p~ . . (7)i

Clearly

Epsi Ii_i i

The distribution of the random variable X may be graphicallypresentedby the step function F(x.) as demonstrated in figure 2,another convenient diagram is obta1~iedby taking an ordinate of ~

over each point x~.

These two diagrams have been constructed on the basis that twenty perc.mt of the specisers belonging to the parent population, from which thesamplesare drawn at random, have a fatigue strength less than the appliedstress level, that is, if a very large sample be testcd, twenty per cent

(8)

re-buthoight

WADC~59-4oOPtI —5...

it

of the specimens are expected” to fail. The cdf. F(x1) is identicalwith a cumulative b1ncmi:il probability distribution. This distributionhas been extensively tabulated. An excellent table has, for example, beenprepared and published ‘by the Computation Laboratory of Harvard University(1955).

The values of’ p~ and are for this specific example given inthe table below.

It may be instructive to learn thatthe probability of having no failure isp - 32.8 %, of }ing one failure is

~ 40.9, etc.,and that 94.2 ~of the to-tal number of aam~plea may be expected tocontain two failures or less.

The relatIonship between3. Evidently, the inflection

value of f(x).

I p~ P1

o 32.81 40.92 20.53 5.14 0.75 0.0

32.873.794.299• 3

100.0100 •0

/

(c)

Turning now to the case that the cd.f. is a continuous function and.

that the function

1(x) — F~(x)- dF(x)/d.x

exists arid is continuous for all values except possibly in certain points,then we have the probability

The functionand sometimes thebill ty

and

P(X~ x) - F(x) -fl(s) dx (io)

f(x) is called the f~iency function (abt~r~viated. fr.f.)

density function. From (10) it follows that the proba—

P(a~X4b)a. Fb)—F(a) _j’f(x)dx (11)

(12)

4,

fc(X)dX -1

F(x) ani 1(x) is demonstrated in figurepoint of F(x) corresponds to a maxizmim of

It is Convenient to interpret the probability as a unIt mass distri-buted on the x—axia in such a way that the density everywhereis propor-tional to f(z,),that is, the mass belonging to the interval d.x is d.F —f(x) dx which by (10) is identical with the cor4ition that the q~uantityof mass allotted to the interval X ~x be equal to F(z).

There are three characteristics of” the distribution which may serve

VADC TR 59-400 Pt I —6—

as a measureof location urd w!ich will be frequently encounteredin thesequel, ~ ~he rear, the median, and the mode.

The mean may be interpreted as the centre of gravity of the mass dis-tributed over the x—axi~. fence, tt~e means denoted 1,

1 dF(x) , • (13)

which in the case of a one—dimer.sional distributir~n of the discretetype takes the form

— Ex~p1 (l4~

while for the continuous type __

I _fxf(x)dx (15)

Sometimes the mean of the population is denotedby in, while.‘! denotes the arithmetic mean of the sample values £x1/n.

Another notation, frequently used, is E(X), ir~dicating that themean is regarded as an “expected value” of the rai~dom v~riah1e X.

This symbol may be extended to a function of’ X. Let g(X). be afunction integrable over ( —~ +o) with respect •to F, then

E(g(X) ) aJg(x)dF(x) (16)

may be interpreted as a weighted mean of the values of g(x).

The median, denoted by ~, is the middlemost po~..nt, defined by.

F(~) — 1/2 (il)

that is, fifty per e~nt of’ the mass is situated to the left and fiftyper cent to the right of the median.

The mode, denoted by ~‘, is the point of~maxinR1mdensity, de-fined by’

- df(!)/d.x — 0 (18)

If but only if the distribution is symmetrical

— ~

x —x—xThe preceding quantities rre characteristic values of the distni—.

bution indicating its location. Another characteristio~providing a mea-sure of its dispersion~is the variance, mechanically interpreted as the

WADC TR 59-400 Pt I

woi~ientof inertia of the mass distribution with’respect to a perpen-dicular axis through tze centre of graVity.

From this ana1o~r w have the variance~2 — T)2dl’(x) (19)

which for the distribution of the discrete type takes the form

— z(x — x)2 .p1, (20)

and. in the continuous case

2 2.a a.~(x—x) f(x)dx ~21,

Using the notation E defined by (16) we have

o2.~(X.E(7Q72—E(X2)—E2(X) . . (22)

The variance will also be dencted by n2(x) and var(X).

The non—negative square root of the variance, that is, a . or

D(X) is called the standard. deviation.

By definition it follows that

D(aX+b)./a/D(X) (23)

and on the condition that X and I are ir4e~pendentvariables that

D2(aX+bY)_a2Dh~(X)+b2D2(Y) (24)

which may be extendedto any finite number of terms.

Three continuous distributions, frequently used in the sequel anddenoted by F1, F2, and F~, will now be examined. The first oneis the rectangular distribut2~on which is the basis of some fundamentaldeductions. The two others are the normal distribution and. the func-tion ~_1_01(_5m) which have the general property of reproducingthemselves, the forme’r by addition arid the latter by isiltiplication.Bepro~iuctIonby i1tIp1ica~ionis particularly suitable for the studyof material strength problems.

The rectangular distrIbution is defined for the interval (o,i) bythe most siuzle cdf.

(25)

WADCTP. 59-400 Pt I -8-

and the fr.f.

(26)

This function is discontinu~R,having a positive saltus at u—Oand a negative one at u—i. Observed.values u~ are thus uniformlydistributed over the interval (0,1).

A most important and useful exampleof the rectangular distribu-tion is presented by any cumulative distribution function F(x) itself.

To prove this statementit t.s to be noted that by (3) equation (25)is to beinterpreted as the probability P(Ufu) — F(u)—u. Applyingthis condition to an arbitrary odf. F(x) we have

P(F(x)~p)~p (27)

The general validity of this expressionis obvious ‘vrhen using theinterpretation indicated in (3) according to which (27) can be trans-lated ii~t~ the following statement:The relative nu~nberof individualshaving a relative order number equal to or less than p is equal. to p,which is an obvious truth considering the equality of i and. (i) in(1).

Thi~q the random variables P -F(x1) are independentlyand uni-formly distributed over the inter~a1 (0,1).

The normal distribution F2 is defined for the interval (—co,Ico)by the cdf,

F2(z) — ___~._~/2a~ (28)V27~

where

(29)

The parameters p and. a are called the parametersof locationand of’ scale, r~spectively. flare p is the mean, a the standard de-viation, and a the variance of tI~pcpulatlon. E*ie to symmetry, themedium and the mode are equal. By (28) we hays 2

,~ 1 ,,~ 1 —z/2f~xj— -i-F tZJ - ~ (30)

(31)

f’(x)/f(x). -s/a (32)

I‘dI.Ai)C TR 59-400 Pt I —9—

As montionud above any normal distribution reproduces itself byaddition. Fhis implies that the sum of any finite number n of’ ind.epen—dent normally diatributed variable3 X1, X2 etc., is itself normallydistributed with a moan

and a variance

C — C~ + 02 .+ ... + 03~ . (34)

The function F3 is defined. for the inte’v~il (0,u~)by

i—.”5 (35)

whers

z— (x—p)/p . (36)

Rere the parameter p is the lower bound of the distribution.

The parameter ~ is related to the standard deviation of the po-pulation a by

p2., ~~2z)!_(a!)2]

where the s~iuibo1 / denotes the complete gamma—function and u.1/m.The parttmetor u is a shape parameter which in practical applications be-longs to the Interval (0,1).

By (35) we have

t(x) — in 5m—i• s (38)

f~(1)___%7~_.1)1m2_2m_2]_s (39)

f’(x)/f(x)~-~~m—1)s~’— ms~] (40)

The property of reprothicing itself by nxiltiplication, implies thatthe shape of the function does riot change when multiplying an arbitrarynumber of probabilities (1—P ). For eiample, if the probability of non—failure within eerh of t~ pa~ts of a solid is 1—P .eZ , then theprobability of2simultMneous non—failure within both ~f them isl—P2.(l—P1) •e’2~

, This property, suitable for the study of ma-terial strength, i~ satisfied by a large class of functions of which Fis the most simple one. It was proposed. for this specific purpose asbeing an adequatemathematIcal representation of the weakest—link prin-ciple by WEI~JLL (1939), unaware of earlier investigations by FISHER &TIPPETT (1928) wh~approachedthe problem from a different starting—point,

bIADC TB 59-400 Pt 1

leading to three azymptotic extreme—valuedistributions, including thefunction F3 but not covering the above—mentionedlarge class of func-tions.

A valuable and comprehensiveme.nual ~‘Stz~istics of Extieme&’, in—cluding the function F3, has recei~tly been presented by W~BEL(1958).

3.2 Transformation of RandomVarIables

Consider a random variable X defined by a b~ownodf. H(x) or bythe fr.f. h(x). This random variabl6 Ia to be transformed into anotherrandom variable Z by meansof a given function g(x),which uniquelyrelates x and z. For a:~y value 14 there is a value

z~. g(r.) (41)

The quantities x~ and. will be called corrcsponding values or points.

The transformation may be motivated, for example, by thO fact that itis more convenient to deal with Z than with X becauseof a simplified.cdt. of Z. For example, the logarithm of a strength ~a2- logX may beprefei~~to I itself, or the random variable U — N to the fatiguelife ~.

The problem now arises to det’rmine the cdt. F(s) and the fr.f.f(s) of the transform Z from the known functicns H(x), k.(x), and. g(x).

By (41) there is an one—to—onecorrespondencebetweenthe vzilues ;and. z~e Accord~rig1y, to each individual of one of the dIstributions thereis always a corresponding individual of the other distribution and~the to-tal number of indivIduals within each poDulation are consequently equal.This equality is valid also for numbers of individuals within correspondingarbitrary intervals, that is, intervals bounded by correspondingpoints.Taking in particular the intervals (z , —co) and (z , —cc) and assumingg(x) to be a non—decreasingfunction ~extenaion to a ~Ion—increasing func-tion is immediate), then obviously the relative order numbersof x andZj are identical and, by definition,

r(zj).F~(x~)j_ H(x1) (42)

Applying thi8 argumentation to the finite Intervals end

u].’ ~2~’ it follows that

F(z2)—P(s1)~~‘2~ — H(x1)

and. to th. infinitesimal intervals dx and. d.z, it follows that

f(s). dx — h(z). dx (44)

WADCTB 59-400Pt I —11—

Since dz.g’(x).d.x, we he.v•

f(s) uih(x)/g’(x) (45)

Thus, the fr.f. of the transform is uniquely detormin~dby the known

functj~n~ h(x) and g’(x).

The transformation of h(x) into f(s) may be mecI~anical1yinter-preted, as do~onatratedin figure 4, as the proo&hir~of moving every massparticle on the x~a~1sin vertIcal direction urtil it reaches the curvez.g(x) and. then horizontally until It reachesthe s—axis.

This procedure i~ pr.rticularly simple in tbe caseof a distributionof the discrete type. ~~verymass p is then moved in this way from14 to without changing its val~e. The mean and the variance of thethnsformed distribution are then easily computed.

It is readily seen that the medians ~ and I of any distributionsare corresponding points, just as any other percentagepoint of the twodistributions, that is,

Ysg(X) (46)

Thie condition does not apply in general to the means and the inod~s,

except in the particular case of a linear relationship between x and a.

Putting

z-g(x).(x—,i)/p (~)

it follows immediately that

E~-(! —p)/P (48)

and

var(!)-var(Z)/fi2 (4~)

3.3 Probability Paper.

The odf. of a d.istribution is frequently given in the form

• (so)

where p is a parameter of location, fl a parameterof scale, and. thefunction may or may not involve one or more parameterswhich will becalled paramctersof shape.

~JADC~ 59-400Pt I —12—

Putting

• z - (x_p)/p (51)

the variable a is called the reducedvariable, dependingonly on shapeparameters, it any.

A convenient tool for examilling the cdf. (5) and. for a.preliminarydetermination of the parameters p and f~ is furnished by the probabi-lity paper.

The principle of this graphical tool is illustrated by the diazrami~i fi~ure5. The Aaildrawn curve representsthe distribution functionP - F(z) but. also its inver~ ~r~tion a— c(r) dependingupon the sid.’~from which it is. regarded.

If now the values of a — c(p) are computed. for an appropriate setof P—values, say P-i to 99 per cent, a grid can be constructed and mar-ked with the P—values at corresponding s—points.

Tabies giving tra.nformations of percentages P to values of aare given for the normal distribution in table I and for the functionin table II for P — 0.l(O.l)].o(l)99(0.1)99.9. The normal deviates(~a— 0, a — 1) in table I are increased by 5 to avoid negative values andare then called p~bita. In table II values are presented. for a • 0.1(0.1)0.6(0.2)1.0. .. .

Since the grid, has been constructed, the ordered samplevalues xare plotted against properly chosen values P4 and. a straight line isfitted by eye to the points. On the condition that the function F(s)corresponds to the true distribution, the plotted. points will fall approxi-mately on the line, the deviations being due to sampling errors which tendto zero when the sample size tends to infinity. The parameter p is thendeterminedby the intc.rsection with the x—axia while the parameter ~3 isdeterminedby the slope of the line.

By (51) we have

1(j)11+ P~$(j) (52)

Applying the operation E to (5?) we obtain

EX(~).u. jl+ ~E5(~) (53)

from whIch It follows that the graphical estimate is unbiasedif the

plotting position P1 is defined by Es1 — 0(P1) or:1 — F(Ez(1)) (54)

bIADC TB 59-400 Pt I —13-

Aj

The probability that z,..~ belongs to the interval (F(s) F(z)+4F(s))is composed of the eimultaneô?i~ and independent probabilities: 1l)~ that(i—i) values belong to th~ ~nt’u~val (0,F(x)) which is F(s)(2) that ~n—i) valuus belong to’the interval (F(z),1) which is(1~p(5))f, and (3) that one single value belongs to the interval dF(z)which is f(z).d.z. Considering that this i,~rrange~ent can be obtained. in

different ways, the fr.f, of 5(j) is

(i—i)!(n—iJ? ~4~(~)t.1~ (l—F(z)r1.f(z)j (55)Formulas for the mean of any nv,~ment z~ ~(k—l,2,3.....), have been

derived by LIEBLEIN (1955) re3uiting in the46llowing expression

E(7~)1. ~,Z’( +kU)~ (_1)?l (~1i~l(n+p—i+l)~ (56)

The means are obtained by putting k - 1 and the variances‘by

(57)

Several other plotting positions have been used. One of them

P1.i/(n+1) (58)

which corresponds to the mean of P( .~ was proposed by WEI~LL (1939).It has been such recommended by GUJ~t (1954). As being independen~ ofthe odf. P(z), it will necessarily Introduce some bias~but it may beacceptablefor preliminary purposes.

Another well—known rule is

(59)

A more general fori~ala,including the two preceding, was proposedbyBLOM (1958) who has presenteda thorough snalysie of the plotting problem.

wLd’ire Ez~ ~ is the mean of the or~ier ~tati.r~tic which is depen-dent on ai~’unique1y determii~adby the odf. P(a). ‘ ‘

1sane of order statisticsin table .111 for sample sizesfor sample sizes 1(1)15 andbeen computedby G.Blom underbeen computed.using the fr.f.argurnantations

from the normal distribution are given1(1)26 and from the cdf. P in table IVa. o.1~o.l)o.6(o.2)].o. This2table has

the present contract. Thcse values haveof 5(j) which is obtainedby the following

where I’ia the Gamma—function and. C~~’is the binomial coefficient.

JADC TR 59-400 Ft I -14-

1:is formula is

(60)

By choosing the values of a and ~3 with respect to the cdt.,nearly unbiased estimates are obtainable. For the normal distributionthe following plotting rule is recommended

(61)

which leads to a practIc~ly unbiaeed. estimate of a.

3.3 Rando~a“umber Tables •

If it is required to select at random a number of specimens froma given sample or a hypothetical population or to split up a given sampleinto randomsubsamples,this can be done in many different ways, forexample, by drawing numbered cards from a pack or drawing ta~a from a box.Very often there will, however, be introduced a bias in favour of certainindividuals or groups of individuals.

The most convenient and reliable method consists in the use of ran-dom—number tables. These tables furnish a sequence of numbers represen-ting the result of a simple random sampling. All the natural numbersO,1,2...,9 appear with approximately the same frequency. By combining num-bers in pairs, thu numbers from 00 to 99 are obtained. These two—figure numbers are arrant-ed in rows and columns, ueiially 50 rows and 25columns on eachpage. A table containing 6 pages of this s~zeis pre-pared and. publishedby FI3HEI~& YATE5 (1943). Another table containing5 similar pages is reproduced.in the textbook by DIXON & MASSEY (1957),being part of a very extensive table containing one million random digitswith 100,000 normal deviates, which was preparedby the RANDC0RP0RAT’C~(1955). Among other random—number tables may be mentionedthose by TIP—PETT (1927) and KE!~DALL & BLBBINGPON—SLIITH (1940).

When using a random—number table, it ehould be. entered. in a randommanner, by throning a die, drawing a numbered. card., or, most cimply, byoLosing the eyes and placing a finger at an arbitrary point of the table.After this procedure a c~~nsecutive sequence may be taken, since the num-bers are ~oughly mixed. in the tabla.

Methods of uaing the table will be illustrated. by the following fiveexamples,

Example 1: Required ~o select at random five specimens from a sample ofsixteen

Sixteen specimens have been tested. at a certain stress level. Obser—

~!~C T~59-4u0 Pt 1 -15-

*

I

vod cycle lIves are listed in Col.l of tablo V. To cach of these apeci-rnens i~ assi~nod an order number (Col.2). P~romthe table of FI~flER &YAT~3(1.c.) a sequenceof sixteen two—digit number; are taken at random.The obtained values are recorded in Col.3. Considering them as decimalplaces, they are given order numbers according to magnitude as recordedin Col.4. To increase the randomness, it is possible to start arbitrarilyeither from the least or from tbs greatest value. If a numbor comes uptwice (in this particular case .16), one or more decimal places taken fromthe table are added, sufficient to decide between the values (.167 and162). For small samples gencz~.lly a three—digit number will suffice, forlarge samples even four digits may be required. Another praxis, frequentlyused, cnnsI~tsin skipping thu secondnumber and adding a new number atthe end of the sequence,but th~srule may introduce some tias and is notrecommended.

The sampleis now arrangedby means of the new random order numbers.Two alternatives are possible. Aucording to the first one the randomor—dei numbers in Col.4 are assumedto be the new order number of correspon-ding sample values in Col.1.. According to the second alternative the ran-dom number in Cci.4 is assumedto indicate the position of the spocimenhaving equal assigned.order number (Col.2). The re~lt of the first ruleis given in Col.5 and that of the second rule in Col.6.

Sinoe in this way ~ completely disordered e~mp1ehas been obtained.,the first five values are chosen as the. raquired selection, or the lastfive, but the choice should be fixed before starting the procedure.

Example 2: Required to split up at random a sample of size_sixteen intothree m2.bsamplnsof size five

The procedureof the preceding exampLe is directly applicable. Sincethe sample has been disordered, the firat five values are allotted to thefirst subsample, the next five to the second subsample, and the followingfive to the third subsample, while the sixteenth value is skipped.

The purpose of this procedure may be to allocate groups from a batchof specimens at random to different stress levels of fatigue testing.Another purpose which will be encountered. in the sequel is related to theexperimental determination of variancesof computed. parameters. ~‘hen thesample is sufficiently large, it may be split up into subsamples,the com-puting technique applied to each subsample, and. the scatter in computed.values taken as a measureof the accuracy, fc: example, by means of thevariance.

~xamp~ 3: Required a random sample of size n from a population havinga rectangular distiibution

According to (26) the rectangular distribution is defined by fr.f.f(u)~1 for the interval (0,1). This implies that each individual of this

~IAiC ‘FR 59—lOO• t I —16-

‘I,

population has equal chance of being selected..parently approximately satisfied by any randombe a probability or 1/1000 of ae1ectin~ anyone000 to 999.

This condition is ap—number table. There willof the three—figure numbers

The required sample is thus obtainud. by selecting a sequence Of3

nthroe—digit numbers from the ta~le and ‘to multiply eaoh value by 10This is obviously a grouped. sample. A finer gro’iping can be obtained byselecting a sequenceof four—digit numbers and multiply them by 10 ‘.

This proceduremay be extended to a variable rcctangulz~rlydistri-buted ovor an arbitrary interval (a,b) by intro~iucingthe linear transfor-mation

x.(b —a)u+a (62)

The randcmvariable X is evidently uniformly diatributed over theinterval (a,b) and a sample from this distribution is obtained.by nulti—plying the values taken from the table by (b — a) and adding a.

Example 4: Required a random sample of size n from a normal distribution

tables consisting of numberspopulations. The numbers withnumbers further and further away

It is, of course, possible to preparechosen at random from normally distributedvalues near zero will occur frequently andfrom ze~o Will occur less and less frequently.

Such tables, which are called tables of random normal numbers, havebeen prepared. by the R.A~DCORPOI~ATI0N (l.c.) containing 100,000 ~orma1 de-viates. A selection of these tables is reproduced in the text—book by.DIXON & MASSEY (i.c.), viz., Table A—2 cuiru;ponding to the populationmean 0 and the population standard deviation a — 1; Table ~.—23 cor-responding to ~ 2• a—1, and Table A—24 correspondingto p — 0,a .2. Each of these tables contains 50 columns and 50 rows of deviatesgiven to threa decimals.

Example 5: Required a random sample of size n from an arbitrary distri

—

button

As a consequence of the general validity of (27) it has been provedthat P .F(z~) for any cdt. P is uniformly distributed over the inter-val (o,i). For this reason, any sequence of numbers taken from a random—number table may be considered a samp!~ of P~ and, the corresponding va-lues z ,and any linear transform x~sf3z +p a sample drawn fromthe pop~lation defined by the cdt. F. DeJting, as usual, the inverse ofF by G, we have

xi,— ~0(P1)+p (63)

WADC‘FR 59-400 Pt I —17—

z~d. th~ ~rucedure simply consists in chosi’~g t~a values of P fromthe table aiid computing the corro3ponding va1ue~ of x~ by m~ansof (63).

ifl many cases sufficient accuracy is obtained by doing this graphi-cally. If a table giving the transformation of P to values of z, asfor example table I for the normal distribution and table II for the dts-’tributiwa. is available, this procedure is easily performed.

Taking as a numerical example1/a

F(z)_1_e_Z ; a— 0.5; I1~i; ~ —2.

then

1/2z. — c(ç) — (log0 1/(l_P~)) . ~64

From the r~ndo~table a sequence of five values P4 has been se-lected at random, ordered., and recorded in the table below. The valuesof z are cotputed by (64)or tahn from table II, arid. ~, xfinally the va.uea of x i i 1.are computed by (63 1 0.034 0.186 1.372

2 0.125 0.365 1.73

3 0.167 0.427 1.864 0.559 0.905 2.815 0.977 1.942 4.88

3.4 The Statistical Coi~cept of Fatigue Strength

In the preceding, the static tensile strength, S,~, was taken as a,typical example of a random variable, the magnitude of which can bo pro-perly defined by a statisttcal distribution function only. Another quan-tity of a similar character, is the fatigue life at a given stress level

‘Is.

Since the values of these two quantities can be measured directly byexperiment for each ind~~dua1specimenswithin a sample, the determinationof the distribution ~s a straight—forward prr,c,dure, the accuracy of itsresult depending ent~re1y on the number of specimens available.

Of a somewhat different character is the quantity called the fatiguestrength at a given cycle life, insofar as the vnlre of it cannot be di-rectly meas.ire~.for a single specimen, because the cycle life aimed at ‘willnever be exactly attained.

Nevertheless, this quantity is as gooa a random variable as the twopreviously uu~ntioned and it will presently be demonstrated. that its distri-bution funct~.on can be determined with any des~red accuracy, provided asufficiently large number of specimens is available.

iIA~iC TB 59-400 Pt I

There ha3 been soule di~cuz~aon about thi3 quantity bcing a randomvariable seemin~~.ydue to the ambiguity of the notation S. Th1~3ymbOli~ unfortunately used in two completely different senses,viz., as anapplied load and as a strength. In the first alte~’n:ttivu S is certainlynot a random variable, while in the seconda]tern~tive it is, the confu—zion being introduced by Lhe fa~L that the strength is equal to the loadwhen failure occurs.

For the further diccussicn, the concept of ~zi 1nd~vidua1 ti—N curvewill be introduced. It is thus postulated that there exists for everyspecimen an individua] S-N curve involving parameters taking valuec v~ry—ing from specimen to specimen. It is, in addition, postulated that every-where

dS/dN.<O~ . (65)

This assumption cannot, of course, b~verified. by experiment because thevalues of the parain~ters are changing during the fatigue damage processuntil finally the tensile strength of the specimen attains a value equalto the applied load. At this moment failure occurs. Consequently onlyone single point of the ii~dividua1 S—N curve can be erperimentally deter-mined but the assumpt!on (65) is almost self—evident since it implies thata shorter fatigue life is e;pected when the specimen in subjected to an in-creased load and vice versa.

Suppose now, that it is required to determine the fatigue strength ata given cycle life lI~. For this purpose n specimens are tested at thestress level S~ wits the re~u1t that I specimens failed at a shorterlIfe than N~, and (n—i) survived this number of cycles.

It can be proved that the ratio i/n is an unbiased estimate of theproportio.i p of the population having fatigue lives less than Ni,, andconsequently that

urn i/n. p (66)

Frbm the aes’i’~ption (65) it Immediately follows that all specimenwhichfailed prior to Yl~, t~at is I spec~r~ens,will h~vea fatigue strength

~an v~hr1eall speci~cncwhich sustainea Ni cycles, that ~(n—i) specimens,will have a fatigue strength greater than S • Thepoint (s,,Ni) is consequently a common percentagapoInt of th~fatIgue—life dist~ibution at etrese level S arid of the fatigue—strength distri-bution at cycle life To exemplhy, if 190 specimenshave been sub-jected to a ~epeated tensIle load of 10 kg/mm and 75 of them failed at1s~sthan 10 cycles,then it can be ~onc1uded by (65)2that 75 per cent ofthe spe3imenstested had a fatigue strength ~1O kg/mm at N 10~cycles.In the same way, percentagepoints at arbitrarily chosen stress levels canbe estimated for any preassignedcycle life.

I‘JADC TB 59-400 Pt T 19—

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