Astm stp 91 a 1963

"Standard" tests constant ampl itu e or classical Wohl er m ethod.. Relati ng to Stati sti cal An ly si s: 2.. Relati ng to Stati sti cal An ly si s o Fati gue Data: 4... In g en eral, t

© BY A M ERICA N SO C IETY F O R T EST I NG A D M ATERIAL S 193

Library ofCongress C atal o g C ard N u m ber: 63 -163 31

Pr i nt ed in Bal t imor , Md

T he Fi rst E di on o this Gui de was the c omp os i te work o many

peo l e who contri buted a great deal o tme to the dis c us s io n and writ

-i ng o the tex t under the g ui dance o T as k G roup L eader, F B Stul en

A m ajor p rton o the stati sti cal s e c tio n was w rit t en by Mis s Ma ry N

Torey Ge orge R G ohn not onl y contributed to the dis c uss io n an pl

an-ning , but also ed ed and arranged fr the printing o the ad anc e co p i e s

o the tex t T he co rdi nati on o contributons an dis c us s io ns was done by

H N Cummi ng s Appre i abl e contributions to the stati stic l parts o the

Guide w ere als o made by D H Sh af r In ad i ti on to the a ove, R E

Peterson, H F Dodg e, D P Gaver, R H ooke, W T La n kford, R B

Mu rph y , W C Sch l te, P R Toolin, an M B Wilk contributed to the

di scus si ons at vari ous conference

T he o ri gi nal T as k Grou p was org ani zed under the l eadershi p o J T

Ransom, an a fi rst roug h dr aft was prep red i n 19 4 an re vis ed i n 1955

Oth er contributors to the e d r aft s were E W El l i s , W T Lankford, F A

Mc Cl i nto ck, R E Pe te rs o n, E H Sch e tte, F B Stulen, an E J Ward

I n 1956, F B Stul en be ame L eader o the Tas k G roup an the Gui de was

c ompl eted under hi s di re ti on

Up n the f rm a tion o S bcommi tte VI on the Stati sti cal Aspe ts o

Fati g ue, this subcommi tte was asked to re vi e w the Fi rst E di ti on and to

make any re vis i o ns ne ce s sary to brin g the Gui de up to date As a re ul

o f this study , extensi ve revi si o ns have be n made i n vari ous s e c tio ns as

prin t ed i n this Second E di ti on They include: ( 1) revi si ons i n the defini tions

( Se ti on I) an thei r s parate publ i cati on as AST M Tentative Definitons

E 206,

1

(2) an e xp ans i o n o fSe c ti o n I V on the n u m ber o test s pe c i me ns ,

( 3) changes i n Se ti on V on te ts o sig nificance, a d ( 4) the preparation

of a new s ecti on, Ap en i x IV, on the use o the Wei bu distribution fu

c-ton for fa t igue Me

T hi s w ork was c ari e d out by f ur Task Groups headed by S M Marco,

H E Franke M i ss M N Tor ey, and C A M oyer, re pe ti vel y Oth ers

who ass i s te d i n the preparation o the Se c o nd Edi on w ere W N Fi ndl ey,

R A.H el l er, J H K Kao, H N Cummi ng s, W S H yler, B Ruley, an

G R Goh , Chairman o S bcommi tte VI

1

Defi ni tions o T erms Relatin g to Fatig ue g and the Stati sti c A nalysi s o Fatig ue Data

(E 2 6), 19 S p l ement o Boo k o fAST M Stadards, Par t 3

NO TE.—The Society i s not resp onsible, as a b o dy, fr the statements

an opinions ad anced in this p ub licato n

P AGE

IV Min im um Nu m be o f Test Sp ec imens an Their Selection 16

Appendi ce

Additional T e c hni que fo r Distribution S ap e Not As sumed 6

1.—Alo c atio n ofTest Specimens for "Pro i t" Method of Test 1

2.—M inimum Numbe o Sp ecimens Nee ded fr Detemi ni ng 9 Per C ent C o

n-fi denc Interv a ls o Stated W idth fr a Po ul ati on Mean, p 19

3.—M in im um Num be o Sp ec imens Needed fr Detemini ng 9 Per Cent C o

n-fi denc In tev als o Stated W idth fo r a Po ul ati on Stan ard Devi ati on, a 19

4.—M inim um Nu m be o Sp ec ime ns Needed to Dete t i f the Standard "Dev i ation

o f a Po ul ati on I s a Stated Pec ntag e o f a F ixe d Val ue 2

5.—M inim um Nu mbe o Sp ec ime ns Nee de d i n Each Samp l e to Dete t i f a Stan

-ard Devi ation o One Po ul ation I s a Stated M ul ti pl e o the Standard D e

6.—M i ni mum Nu mbe o Sp ec ime ns Ne e de d to Dete t a Stated Dif ren ce Between

7.—M inimum Num be o Sp e c ime ns Ne e ded to Dete t a Stated Dif ren ce Betwe n

14 —Pe rc enta es Surv i ving 10

8

16.—Computati ons fr Fitting a Re s p o ns e Curv e by Method o Le as t S uares 35

18.—Method o Computin g 9 Per Cen t Co fidence L i mi ts fr Per Cent S rvi val

19 —Metho d o Computing 9 Per Cen t Co fi denc L i mi ts fr Fati g ue Strng th

2 1.—R R M oor Rota t in g Bea m ; Ste p Tests o f4 2 Sp e c imens 5

T ABL E PA G E

2 —M inimum Per Cent o fPo ul ati on E xc e di ng M edian o Low Ra nking P int s 5

2 —Pe rc enti l e s o fthe x

3 —k Fac to rs fr S-N C urve ( Normal Di stri buti on A ssum ed) 6

3 —Ordi nate Locati ons Cor sp n ing to Pe Cent F aied Val u 7

3 —Me an-Ran Esti mate o f the Pe r Cent Po ul ati on F ailed C or s po ndi ng to

37 —Typical F atg e Test D ata

LIST OF FIGURES

F I G U RE

1.—P ro b ab i lity-Stre s s -C yc l e (P-S-N) Curv e fo r P ospho r-Bronze Stri p 10

4.—Reprs ntati on o "Step "Te stin o Single Sp ec imen 14

8.—Per Cent o f Sp ec imens H aving at Least the I ndi cated Fatig ue Streng th at 10

7

13.—C o nstruc ti on o Weibu Pro a i l i ty P ape r fr om Lo g- Log Pap e r 73

14 —Es timatio n o f Wei bu Di stri buti on F ncti on Parametes fo r Data i n Table 3 7

16.—Estimation o f Weib ul D is trib uto n F ncti o n P arame te rs fr D ata i n Table 39 7

Abstracts o fArti cl es o n Fatig ue (STP 9)

F atigue M an al (STP 91) (199)

Stati sti cal Asp ects of Fati gue (STP 121) )

Fa tigue, w ith Emphasi s o n Stati s ti cal h (STP 13) (1952)

Papers on Metals (S TP 19) (196)

Fatig ue o fA ircra ft Structure (STP 2 03) (195 6)

L arge Fati gue T e s ting Machi ne s an T he i r Re u s (STP 216) (19 57 )

B asic Mec hanisms o F atigue (STP 2 7) (19 8)

Fati gue o fA ircra ft Structurs (S TP 2 4 ) (19 9)

Aco usti cal Fati g ue (STP 2 4) (19 0)

Fati g ue ofA ircraft Structure (S TP 3 8) (19 3)

7

GUI D E F O R F AT IGUE TESTING AND

A bou t 15 years ago , AST M Commi tte E -9 on Fati g ue prepared a Ma n u a l

o Fati g ue T e s ti ng

1

That Ma nual a t t em pt ed to standardiz e the s ymbo l s

an nomenclature used i n f t i u e te ti ng, des crib ed the pri nci pal type o

te ti ng machi ne s t h en in us , pre ented de tai l ed instructons f r the prepa ra

-tio n o fte s t specimens, o utli ne d test pro cedures an te c hni que s , an gave

some suggesti ons f r the pre entation and interpretation o fa t igu e da ta

Si nc e the Ma n ua l was firt prep red, a n um ber o new te hniq e have b e e n

de ve l o p e d fr evaluating the ft igu e pro ertie o materi al s Furth ermore,

the a pl i cati on o stati sti cal methods to the anal ysi s o the te t re u s o

s amp le s ofrs a means f r esti mati ng the characteri sti cs o the p pulation

from whi ch the s ampl es were taken To t a ke c o gni zanc e o the e de ve l o p

-me nts , thi s g i de has b e e n prep red

I PURPOSE S OF FATIGUE T E ST I NG

The purp s s o fa t igu e te ti ng are ( 1) to e ti mate the relationship

be-tween s tre s s - ( l oad-, strain -, defle ton-) am pltude and c yc l e lf -t o-fa ilu re

fr a give n mate ri al or c o mp o ne nt, an ( 2) to c o mp are the fa t igu e pro erti es

o two or more materi al s or comp nents I n order to spe i fy the rel i abil i ty

o f the s e e s timate s , they must b e based o n the re s ults o f te s ting a sample o f

fa t igu e spe i mens whi ch have be n draw n at ra n dom from a p pul ati on o

p o s s ib le ft igu e spe i mens and te ted i n ac ordanc wit h acc ptabl e te tng

proc d res The principal ac epta l e proc d re di s c us s e d i n this g uide a re:

A "Standard" tests (constant ampl itu e or classical Wohl er m ethod)

2

1 Sin le te t s p e c i me n at each stre s l evel

2 A group o test s p e c ime ns at each s tre s s l e ve l

B Re s p o ns e tests ( constant ampl u e)

1 "Pro it" m et h od

2 Staircase method

3 Modified stai rcas m et hod

C I ncreasi ng ampl ude te ts

1 Ste p m et hod

2 Prot m et h od

The prim a ry purp s s o the stati sti cal anal ysi s o f t igu e da t a are: ( 1) to

e ti mate c rtai n f t igu e pro ertie o material or a comp nent (togeth er

wi th measure o the re l i abi l i ty) fr om a gi ven set o fa t igu e da ta , o tai ned

by te ti ng a sampl e o fa t igu e s p e c ime ns i n accordanc w it h one o the

previ-ous te s t pro c ed re s , an ( 2) to p ro vi de o b je c ti ve proc d re f r c o mp ari ng

two or m ore s ets o fa t igu e data to determ in e whet her or not the da t a c ome

fr om s imil ar p pul ati ons Statisti cal t h eory also provi de inform a tion on ( a)

the mo s t efi ci ent us o a limite d number o f te s t spe imens an (b) the

num-ber o test s p e c i me ns required to gi ve a spe ci fi ed de gre o confi denc i n the

te st re s ul t

Eve n w it h s o me b as ic training, i t i s d ificu lt to l ocate the te hniq es pa

r-ticularly useul i n f t igu e testi ng i n the statistic l lt era t ure The purp se o

this guide is to descri be some stati sti cal t rea t m en t s t h at are suiable f r the

anal ys i s o ft igu e data o tained i n an one o the foreg oing test m eth ods

and to present the s e statisti cal treatments in a for m u seu l to the test

en-gi ne r Defini tions o c rtain stati stic l term s are i ncl uded, but onl y enough

o f the basic c onc ep ts o f s tatis ti c s are inc lude d to make the methods

under-stan dable; theory is let to the references

Test p ro c e dure s are discused hi S ecton III whi l e te c hni que s fo r anal

yz-i ng the data o tai ned i n the se tests are g i ven i n Se c ti on V and the

Appen-dic s

I DEFIN ITION S, SYM BOL S, A N D A BBRE V IA TION S

Relati ng to Fati gue Te sts and Te st Meth ds:

T o e nc o ura e unifrmity o f te rmino lo gy, the terms dealng p ri mari l y wi th

fa tigue testi ng and test methods are also publ i shed i n ASTM Defi ni ti ons

E 6

3

T he s ymb o ls us e d are, hi g enera thos e re ommen ed i n the A m eri

c an Standard Lett er S mb o l s fr Mechani cs o S old Bo die s

4

1 Fati gue (No te 1) —The proc s of progressive localzed permanent struc

-tural c han e oc uri ng in a material su je ted to co di o s w hich produce

fluctu-atn s tre ss e s an strai ns at so me p in t or p i nts an which may culminate in

c rac ks or c ompl e te fr act u r e at er a sufficien t n mber o fluctuations ( Note 2)

N T E 1.—The term fa t igue i n the materi al s testing fi el d, h s —i n at least one c ase

glass te h ol og y—be n used fr s tati c te s ts o considera le du ra tion ,a t yp o test g

ener-aly des i gnate d as stres-rupture

NO T E 2.—Fl uc tuati o ns may occur b th i n stres an wi t ti me (freq ency), as in the

case o f"an om v i brati on."

2 Fati gue L i fe, N.—Th e n mber o fcycles o fstress o r s train o fa sp ecified

char-a ter t h a t a g i ven s p e c ime n sustai ns befre fa ilu r e ofa s p e c ifi e d nat u r e oc ur

De fi ni ti ons 3 to 19, inc lusive, ap l y to thos e c as e s where the

con-d i o ns i mpo s e d u o a s p e c ime n resul t or are as s ume d to resul t i n

u i axi al p ri nc i p al s tre s s e s o r strai ns whic h fluctuate i n magni t de

Mul ti axi al s tre s s , s e que nti al lo ading,an random lo adin req i re more

rigo ro us de i ni ti ons whic h are , at present, bey n the scope of this

s e c tio n

3 No minal Stres s , S.—The stres s at a t cal cul ated o the net c ro s s - s e c tio n

b simple e las t theory, w ithout taki ng into ac ou t the efct o the s tre s s

pro-d c e d b y geome tri c di sc onti nu i e s suc h as holes, g ro v es,'fi ets, e tc

4 Stre s s C ycle.- ^- The smalest s e gme nt o fthe stres -i me fun ct ion whi ch i s re

-p e ate d periodicaly

3

Dei ni ti ons o Term s Rel ati ng to Methods o M echanical T esti ng , 19 62 Su ple me nt

to 19 61 Bo k ofA TM Sta dards ( E 6), Part 3

5/ Ma xim um Stre s s , S

max

- — The s tre s s havin the hi g hest al g ebraic value in the

s tre s s cycle, te ns i l e s tres s bei ng c o ns i de re d p s i ti ve an co mpre ss i ve s tre ss n ega tiv e

I n thi s deinition, as we l l as i n oth er t h a t flow, the n minal stre ss i s use d m ost

commo l y

6./ M inimum Stres s, S

m,

n — The s tre s s hav ing the l o we s t alg ebraic value in the

cycle, te ns il e s tre s s bei ng c onsi de red p si ve an c o mpre s s i ve stres negatve

? Mea n Stre s s ( or Stead Componen t of Stres ), S

m — The alg ebraic av erage

o the m ax im um an minimu m s tre s s e s i n o e cycle, t hat is,

8 Rang e ofStress,S

r — The algebraic di ffrenc b e twe e n the m a x im um an

min im um s tre s s e s i n o e c yc l e, that i s

9 Stre s s A mpltude ( or V ariable Component o Stre s ), S

a — One half the

rang e of s tre ss, t hat s

10 Stre s s Rato, A or R — The al g ebraic ratio of two s p e cifie d stre ss val ues i n a

stres cycle Two commo l y us e d s tre s s ratios a re:

The rati o o fthe stre s ampl i tu e to the me an stres, that is ,

an the ratio of the minimum s tre s s to the maximum s tre s s , tha t is,

1 S-NDiag ram — A pl ot of s tre s s ag ai ns t the n u m ber of c yc le s to fa ilure T he

s tre s s c n be S

ma, S

m

i, or S

aThe di ag ram indic tes the S-N rel ato shi p for a

s pec i fi e d val ue ofS

m, A, or R an a s p e cified pro abi ty of survi val F r N a lo

scle is almost always us ed Fo r S a li e ar scle is used mo s t o te b ut a lo g sc le

i s s o me ti me s used

12 Stre s s Cycl es Endured, N — T he n um ber of cycles o fa s p e c ifi e d chara ter

(that pro uc fluctuatn s tre s s an strain) whi ch a s p ec ime n has en ured at an

tme i n i ts s tre s s history

1 Fati g ue Streng th at TV Cyc l e s , SN — A h p the ti cal val ue of s tre s s for

fil ure at e xactly N cycles as de te rmi ne d from an S- N diagram T he val ue o f

SN thus determined i s subje t o the s ame c o ndi ti o ns as tho s e whic h ap l y to the

S-Ndiagram

NOTE — T he v alue o SN which is commonly fo n in the lt era t u re is the h y pothetical

value o Smai ,-Sm t ^ror S

a,at which 5 per ce nt o the sp ecimens o a gi ven s amp l e c o ul d

survi ve N stress cycles i n whi c h Sm = 0 This i s also kn wn as the median fatig ue stren g th

at N cycles (se de fi ni ti o n 4 7)

14 Fatig ue Li mi t, S / — T he lmi n val ue o fthe median fa t igu e streng th as N

b e c o me s very l arge

Val ue tabulated as ft igu e lmitsu the ltera ture are frequ en t ly (but not al wa s) v l ue s

6 SN fr 50 ~p e r c nt survival at N cycles o s tre s s i n which S

m

= 0

15 Cycle Rato, C — T he ra tio o the n mber o stres s cycles, n, o a s pe c i fi e d

character to th e hypothetic l fa t igu e lfe, N, o ta in ed fr om th e S-Ndia gra m , for

s tres s cycles of the s ame ch a ra ct er, t ha t i s,

Theoretcal Stre ss Conc ntration Factor (o r Stre ss Concentration Factor,

Kt — The rati o o f the g reatest stress i n the region o fa n tch o r other stress

concen-tra t or, as determined b the theory of el asti ciy ( or b ex perimental proc d res

t hat gi ve eq ivalent v alues), to the coresp n in nominal s tre s s

N TE — T he theory o plasticity s ho uld not b usd to d et er m in e K

t

17 Fati gue N otch Fa tor, K/ — T he ratio o the fa t igu e stren gth o a s pe c ime n

wit h n stre s s concentratio to the fatig ue streng th at the s ame n u mber of cycles

wih s tre s s concen tratio for the s ame condi ons

N T E — In spe i fyi ng Kj i t i s nec essary to s p e c i fy the geom etry ad the v alues o

S

m

ax , S

m

, and N fr whi c h it is com puted

18 Fati g ue Notch Sensi ti vi ty, g — A me sure o the de g re o ag re ment b

e-twe n Kfa d Kt fr a p rti cul ar spe i men o a g i ven si z e a d materi al co tai ni ng

a stres s co c ntrator o f a give n siz an s hap e

NO T E — A common deinitio of fa t igu e notch s nsitiv ity is q = (Kf— i )/(K

t

— 1\ in

whi ch q may v a ry b etwe e n zero (whe re Kf = 1) an one (w h ere Kf = t).K

19 Co stant Lif Fati g ue Di ag ram — A pl o t ( us ual l y o n re tang ular co rdi

-nate s ) o f a family o f curves , e ch o f whic h i s fo r a sin le ftg ue l i fe, N, re lati ng

n

to the me n s tre s s S

m The co stant l if ftigu e diag ram is

ge ne rall y derived fr om a fa m ily o S-N curves, e c h o w hich represents a d ifer ent

s tre s s ratio, A or R,for a 50 per c nt pro abi ty of surv iv al

Relati ng to Stati sti cal An ly si s:

2 Po pul ati o n (o r Uni ver e) — The hypothetical c o llec tio n o fall p os s ib le te t

sp ec imens that could be prepare d i n the sp ecifie d way fr om the ma terial u n der

co siderati on

21 Sample — T he speimens sele ted from the po pul ati o n fo r test p urp oses

NO T E — T he method o selec ting the samp l e determine the p pulati on a bout w hich

s tati s ti c al in feren ce or geneal i zati on can be made

2 Gro p — T he s p e c ime ns tested at o e tme, or c ons ecuti vel y, at o e s tre s s

le ve l A g roup may co mpri s e o e or m ore sp ec ime ns

2 Frequency Distrib ti on — T he way n whi c h the frequen cies o oc urenc o

m em ber of a p p l ati on or s amp le are distributed a cordi ng to the val ues of the

vari ab l e under c o ns i de rati o n

24 P arame te r — A co ns tant (usualy u k own) de nning some pro erty o f the

frequen cy distrib to of a p p lato , s uc h as a p p l ati on median or a p p latio

standard devi ati on

2 Stati sti c — A sum m ary val ue c l cul ated fr om the ob s erved val ues in a

s amp le

numerical value s o f one or m ore u nk nown p p lato pa r a m et ers fr om the o

-s e rve d val ue s in a sample

2 E stmate.—The pa rticula r val ue, or val ue s , of a pa ra m et er com pu t ed b an

esti mati on proc dure for a g iven s ampl e

2 P o i nt E stmate.—The estimate of a pa ra m eter gi ve n b a s i ngl e statistic

2 Sampl e M edi an.—-T he mi ddl e value w hen al l o served values in a sample

are arang ed in order o m a gnit u de ifan od nu mber o s pe c i me ns are tested If

the s amp le siz i s even , i t is the averag e of the two mid lemost val ue s I t i s a p in t

estim a te of the p pula tio m edia n , or 50 per c nt p in t

30 Sampl e A v erage (A rithmetic Mea n).—Th e sum ofal l the o b s e rve d values i n

a s ampl e di vi ded b the s ampl e size It is a p int estimate of the p p lato mean

31 Sampl e V a rian ce, s

2

.—T he sum o fthe sq ares ofthe dif rences betwe n

each o b s e rve d val ue an the s amp le averag e divided b the s amp le siz m inu s o e

It is a p i nt estimate ofthe p p lato v arianc

NOT E —T hi s v alue of s* prov ides b th an u bi ase d p in t estim a te of the p pulation

v aria nce an a statisti c that i s use d i n c m putin g interv al e s timate s an several test

sta-tis ti c s (see defin ition s 34 a d 4 ).So me t ext s defin e s

2

as "the s um o the sq ares o the

dif ren ces b tw een e ac h o s erved v alue and the sampl e averag e divided by the sample

s i ze ," but this stati sti c is not as u seul

— 3 Sampl e Stan dard Dev iatio , s.—The sq are ro t o the sampl e vari anc

I is a p in t estimate of the p p lato standard deviatio , a measure ofthe

"s p re ad" of the frequ en cy dist ribu tio ofa p pulation

NO TE.—This val ue o f 5 p ro vide s a statstic that i s used in c o mputi ng i nterval e s timate s

an s e ve ral test s tati s ti c s (see defin it ion s 3 an 4 ).For s mal l s amp le sizs, j

underesti-mates the p pulation sta n da rd devi ati on (Se the ASTM Manual on Qualty Con t rol o

Mate rials

5

or tex ts on stati sti cs fr an u bi ased esti mate o the stan da rd devi ati on o a

N ormal p pul ation.)

3 Sampl e Perc ntag e.—The perc ntag e o fo se rve d val ue s betw een two sta ted

val ue s o f the vari ab l e u n der c o ns i de ratio n I t i s a p int estmate o f the perc ntage

o the p p lato between the s ame two stated val ue s (O ne stated val ue may be

— ° o r + ° )

34 In terv a l E stmate.—The estmate of a parameter gi ven b two statsti cs,

den i ng the e nd p ints o fan interval

3 Con fiden ce Interv a l.—A n in terv al estmate of a p p lato para m eter

comp ted so that the statement, "the p o p ulatio n pa ra m eter les i n thi s interval,"

will be true, o the averag e, i n a stated pro orto of the tmes s uc h statements

are m ade

36 Con fiden ce Limi t —T he two s tati s tic s t h a t define a co nfi de nc e interval

3 Con fiden ce L ev el ( or Co ficient).—The stated pro orto ofthe tmes the

c o nfi de nc e interv al i s e xpe c te d to i nc l ude the p p l ati on pa ra m eter

38 T o le rance Interval.—An i nterval comp ute d s o th at it wi inc lude at east a

stated perc ntag e of the p p lati on w ith a stated pro abi ty

39 Tol eranc Limi t —T he two s t h a t define a tol e ranc interval (One

val ue may be — « or + °.)

4 Toleranc L e ve l —T he stated pro abi ty th at the to l e ranc interval i

n-c l ude s at least the stated perc ntag e of the p p lato I t i s n t he same as a

con-fi de nc e level b ut the term c onfidence level i s freq ently asociated with to le rance

intervals

41 Sig ni f ant.—Stati sti cal l y si gni f ant An e e t or diferen ce between po

pu-lato s i s said to b e prese nt i f the v alue o f a teststatstc i s sig nificant, t h at i s, lies

outside o predetermined li mi ts

N OT —An e et which is statistic ly si g ni f a t may or may not hav e eng i ne ri ng

s i gni f a c

42 T estStati sti c.—A fu n ct ion of the o b s e rve d val ue s in a s ampl e t h at is used

i n a test o fs ignific anc e

43 T est o fSi g ni f anc —A test w hich, b y us e o f a teststati sti c, pu

r-p rts to provide a test o the h p thesis that the efe t is absent

NOT E.—T he re je cti on o the h p thesi s i ndi cates t ha t the e et i s pre se nt

4 Sig nificanc Leve —T he stated pro abi ty ( risk) th a t a gi ven te s t of

sig-ni ficanc wi l l reje t the h po the s i s that a s p e c ifi e d e e t is absent w h en the

hy-p thesi s i s true

Relati ng to Stati sti cal An ly si s o Fati gue Data:

4 Media n Fatg ue L i fe —T he mi ddlemost of the ob s erve d fa t igu e l if val ue s ,

arang ed in order of m agnitude, of the in div idua l s pe c i me ns in a g ro p tested u d er

i denti cal co di ti ons I n the case where an e ve n n u m ber of spe ci mens are tested i t

is the average of the two mid le mos t values

N T E 1.—The us o the s amp l e m edia n, i nste d o the a rith m et ic m ean (th a t i s, the

a erage), i s usually preferred

NOT E 2.—I n th e lt era tu re, the a brevi ated t erm "fa tigu e lfe" usually h as m ea n t th e

medi an fa tigue lf df the g roup However, when ap p lied to a col l ec t n o da t a w it h out

fu r t her q al i fi cati on the term "ftig ue l i fe" i s amb i guo us

4 Fatig ue Life fr p Per Cen t S rvival.—An estimate o the fa t igue lf

that p

per c nt of the p p lato w ould a t ta in or exc e ed at a gi ve n s tre s s le ve l T he o

-s e rve d val ue o the medi an fa t igue lfe estmates the ft i u e lfe f r 5 p r c nt

survi val Fati g ue lf fr p p r c nt survival val ues , w here p is any n um ber, such as

95, 9 , e tc , may also be e s ti mate d fr om the in i vi dual fa t igu e l ife val ue s

47 Median Fati gue Streng th at NCycl es —An estmate of the s tre s s le ve l at

which 50 pe r c nt ofthe p p l ati on would survive N cycles

NOTE 1.—The esti mate o the medi an fa tigue stren g th i s deri ve d fr om a pa rticula r

p int o the fa tig ue l if distri bu on, s inc e there i s no te s t proc d re by whi ch a frequ en cy

distribution o fa tigue streng ths at N cycles can be dirctl y o s rved

NOT E 2.—T hi s i s a s p e cial case o fthe more geneal defin itio 4

48 Fatg ue Streng th for p Per Cent Surv iv al at N Cycl es.—An estmate ofthe

stres s le vel at which p per c ent of the p p lati on wo l d surv iv e Ncycl es ;p may

b e an n u mber , such as 95, 9 , etc

NOT E.—T he e sti mate s o the fa t igu e streng ths fr p per c nt surv i v al val ue are deri ved

from particular p i nts o the fa tigu e lf distribu on, since th ere is no tes t proc dure b

whi ch a frequency distribution o fa tigue s at N cycles can be di re tl y ob se rved

49 Fati g ue L i mi t for^ Per Cen t S rvi va —T he lmitin g value of fa tigu e

streng th for p per c nt survi val as Nb e c o me s very larg e; p may be any nu m ber,

such as 95, 90 , etc (S ee N ote, deini o 14

50 S-N Curv e for 50 Per Cen t S rvi va —A curv e fi ed to the median val ues of

ft i u e lf at e ach of s e ve ral s tre s s levels I t i s an estimate of the relati onshi p

be-twe en ap plied stress an the n mber o f c yc les - to- failure that 50 p e r c e nt o f the

NOT E 1.—This i s a sp ec ial c as e o fthe mor e g eneal defin it io 51.

NOT 2.—In t h e ltera t ure, th e a bbrv ia ted t erm "S-N Curv e" usua lly h a s m ea n t

either the S-N curve d raw n t h roug h the means ( aveage s) or he medians (5 per c nt

val ue ) fr th e fa t igu e lf val ue Sin ce th e t erm "S-N Cu r ve is a m biguous, it sh ould be

usd i n te h i cal pa pers onl y w h en adequately de sc ri b e d

51 S-N Curv e fr p Per Cent S rvival.—A cu rve fitted to th e fa t igu e l if fr p

per c nt survival v l ues at e ac h of se ve ral stres le ve ls I t i s an esti mate of the rel

a-tio ns hip b e twe e n applied stress an the number o f c yc le s - to - failure that p pe r c ent

o the p pul ati on wo l d surv iv e; p may be any n mber , s uc h as 9 , 90, etc

N OT —Cautio s houl d be us d in dra win g concl usi ons fr om ex tra ol ated p rtions of

the S-N curves In g en eral, th e S-N curv es shoul d not be extrapolated bey ond o bs rved

lf val ue

5 Resp nse Curv e f r NCycl es.—A curv e fi ed to ob served v l ue s o

per-c e ntage surv iv al at Ncycles for s e ve ral stres leve ls , where N is a p re as s i gne d nu

m-b e r s uc h as 10

6

, 10

7

, etc It i s an esti mate o fthe rel ati onshi p betwe n a pl i e d stres

an the perc ntag e of the p pul ati on th a t w ould survive N cycles

N T E 1.—Values o the median ftg e stren gth at Ncycles and the fa t igu e strength

fr p per c nt surv iv al at N c yc le s m ay be de ived from the re sp o ns e curv e fr N c yc les,

i f p fls w ithin the rang e o the per c nt survi val val ue s a ctually o b se rved

N T E 2.—Cautio sho ul d be us e d in dra wing c oncl usi o ns fr om ex trapolated p rtion s

o the re s p o ns curve s I n genea the curve shoul d not b ex trapola ted to othe val ue

o p

LI ST OF SYM BO LS AND ABB E VI AT IONS

The fl l owi ng term s are fr eq u en y us e d i n l i eu o or alo ng w it h the term s

c ove re d by the pre edi ng defini ons In g eneral the symb ol s are those

re ommen ed in the A meri can Standard Let t er Symb ols fr M echanics o

S olid Bodi es ( s ee fot n ot e 4) For s tre s s , the us o S w it h a propriate

low er cas e subscri pts is preferred for g eneral purp ses; for m a th em a tica l

anal ysi s the use o G re k s ymb o l s i s g eneral l y preferred

or ki ps per sq are i nch

Pou ds per sq are i nch

Pro abi l i ty of f iu r e; Per c nt fa ilu re;

Pro abi ty o survi val ; Per c nt survi val

Fatigue notch s nsi ti vi ty

Stre s s ratio

Samp le standard devi ati on

Sample vari anc

Nomi nal s tre s s

Standard deviati on; Stre ss

E stmate o st a n da rd dev iation

U nti re ently, there was onl y one ac epted method o conductng l ab

ra-t ory fa t igue tests on a material or comp nent T hi s " tan ard" test, usi ng

Ma n u a l on Fatig ue T esti ng (STP9 1) E xpe ri e nc showed, however, that

thi s test m eth od did not give adeq ate in form at ion f r m a n y o the

pur-p ses f r whi ch fatigue data are ne e de d Therefre, wi t i n the last ten year ,

a num ber of new m ethods for perf rm in g m ore m eaningful fa t igu e tests

have be n introd c d, each m ethod havi ng c rtain advantag es

T he choice o fte st me tho d de p e nds up on the o bje c ti ve o f the te st an the

n u m ber o avai l abl e test s p e c ime ns W h en the o je tiv e i s to determine an

S-N curve, t he " tandard" tests (Se tons I Al an d A 2) are g enerall y t he

most su a l e To det erm ine the l ong -li fe fa t igu e strength or the f tgue

lm it, re spo ns e tests ( Se cti ons I Bl, B2, and B3) or i ncreasi ng a m plt ude

tests ( Se ti ons I Cl and C2) are re ommended The la t ter m eth ods als o are

us e d for comparing the long -l if ft igu e pro erties of d ifer ent materials or

d ifer ent methods of pro c e s s i ng Al l s e ve n of these ex perim ental fa t igu e

testing techniques are descri bed i n the fo owi ng paragraph s For anal ysi s o

the da ta , s e Se c ti on V

A "ST ANDARD" TESTS (C N STA N T AMP L I T UD E )

1 Si ngle Test Spe i me n at Ea h Stres Lev el

In the " tandard" test method descri bed in ST 9 1, e ac h fa t igu e s pe c

i-men is c yc le d at a d ifer en t constant s tres (or strai n) ampli t de until fr

ac-t u r e oc ur T he s tres le vels are usuall y s e le c te d to cover a series o s tre s se s

rang ing fr om hi g h val ues, at which failure wi oc ur wi thin a l i mite d num ber

o c yc le s , to l ow val ues at which no fa ilu re wi l l oc cur (run outs) or at which

f iu r e wi l l oc ur onl y a t er an ex t rem ely l arg e n u m ber of c yc l e s I f the

pri-mary i nterest i s i n the l o ngl i fe e nd o the S- Nre l atio ns hi p ( often caled the

ftg e lm it ), the i nvesti g ator usual ly has some pre onc pti on o thi s v a lue

fr the m aterial or comp nent o be tested In this c as e, the fi rst s tre s s le ve l

i s s e le c te d som ew h a t ab ove the esti mated fa t igu e l i mi Depen i ng up n the

results o the fi rst test, suc e di ng s p e c ime ns are then tested at stres le ve ls

either a ove or bel ow thi s val ue, unt il a stre ss level is reached at w h ich the

sp e ci me n do e s not fa il w it h in the prescri bed n u m ber o c ycles Nea r the

f t igu e l i mi t, some sp e c ime ns must be ru at s tres l e ve ls hi g h enoug h to

produce fa ilu res in order to hav e data fr om w h ich th e fa t igu e lm it m ay be

esti mated

This method o fte s t is used whe n the inve s tigato r has avaiab le only a

rel ati vel y smal n u m ber o s p e c ime ns fr test S ch i s g enerall y the situation

when ( 1) the fa t igu e s p e c ime ns are expensi ve, ( 2) the sup l y o material i s

l imi te d, o r (3) mac hine p rts, full size sections, o r assemblies are b e ing te s te d

2 Grou o Spe i mens Te sted at Ea h Level

Si nc e the " ta dard" test, usi ng onl y one s p e c ime n at e ach s tre s s l e ve l,

gives v ery ltt le in form a t ion concerning the v a ria bi ty o the m a teria l or

component an test proc dure, i t i s m ore sa t isfa ctory to test s e ve ral

speci-mens at e ac h o a n u m ber o d ifer ent s tre s s levels I n t his proc d re, e ac h

group s ho uld c o ns is t o at l east fo r s p e c i me ns i n order to esti mate the vari

-abi ty o the data Ten or more s p e c ime ns are preferable to o tai n some

i ndi cati on as to the sha e o the distribution o f t igu e lf val ues Thre or

S-N curves fr p per c nt survi val (see F ig I

6

Ge ne ral l y, at le ast fou r or

five stress levels are used i n a test o fthis nature T o de te rmine the fatig ue

l i mi t o the m aterial, a number o g roups also s ho uld be te ted at s tre s s

levels i n the vi ci ni t o the fa t igue l i mi t Thi s woul d i ncreas the total n u m

-b e r re qui re d to at least s ven gro up s F rthermore, to o b tai n ap p ro ximate ly

an eq al deg re o p re c i s io n t h rough out the rang e o the S-N curve, more

s p e c ime ns s ho uld be te s te d i n the lo g -l if than i n the short-l i fe rang e

FIG 1.—Probabilty- Stress- C ycle (P-S-N) Curv e fr P ospho r-Bronz e Stri p

B RESP O NSE T S TS (C N STA N T AMP LI T UD E )

1 "Pro i t" Meth d:

In the "Pro i t" method, one or more g roups o s pe cime ns are te ted fr a

fix ed nu m ber o c yc l e s at fo r or fiv e d ifer ent stre s le ve ls dist ributed a bout

the stres s o i ntere t T his te t has be n us e d primariy fr e ti mati ng t

he-fti g ue limit o fa mate ri al , that i s , the s tre s s at whi ch 5 p e r ce nt o fthe te s t

specimens wi l l fail prior to, an 5 per c nt wi ll s urvi ve , the preas i g ned

cycle lf, N The te st i s not l i mi ted to this a pl i cati on; i t i s just as valuabl e

for e s timating the ftg ue stre ngth or the fatig ue l i mi t at an othe r p e

r-c e ntage s o s urvi va provi ded t h at the s p e c i me ns are pro erly al l o c ate d to

the vari ous s tre s s levels When use d t e tmate the fa t igu e l i mi t at 5 p r

c e nt s urviva at le st two stress levels should b e s le ted s o that the per

c ntage o s p e cimens survi vi ng Nc yc le s wil l be less than 5 an two more

stre s le vels se lec ted at whi ch the perc ntag e o survi vors wi l l be more th an

50 A fi fth s tre s s level prod ci ng a proxi matel y 5 p e r c nt survivors i s

de i ra l e but not e s s e nti al

TABL 1.—AL L OCAT I ON OF TEST S E CIM E NS FOR "P RO BIT "

E xpeted er C e nt Surv iv al

Rel ati ve Group S iz"

T he gro up siz is the n mb r of specimens inc lu ed in a test at o ne stres level

T hus , whatever g roup size i s chos en fo r te s ti ng at stress levels fr whi c h the expeted

p e r c ent survi val i s be twe en 2 an 7 , the sizes o f other g roups mus t b e in eas ed b y the

fa ctor i n the s eco nd c lumn to o tain the number o te st s p ec ime ns req i re d fr te s ti ng

at stress levels fo r whic h the pe r c e nt survi val i s expe c te d to b e l arg e r, o r s malle r, i f

s imilar p re cisio n i s to b e o b tai ne d i n the te st re sult I f the stress levels are chosen suc

-csively, s tarti ng with levels re qui ring the smalest g roup siz, the gro up siz re quire d

fo r the other levels wi be de te rmi ne d more easily Pre vi o us data fr the s ame mate ri al

o r simiar materi als s ho ul d b e us e d as a g ui de fr choo sin the s tress levels, whene ve r th ey

are avai lab le ; o the rwi s e a prel i mi nary te s t s uc h as that des crib ed under Sections III A1 o r

B2 may b e req i red A pro erl y desiged "P ro b i t" te s t wi give more useul fatig ue

data than an o fthe othe r resp onse or in e asing ampl i tu e tests

FIG 2 —Resp onse or S rvi val Tests

I n "Prob i t" tests a g roup s ho uld co nsis t o not les th a n fi ve s p e c i me ns

an the to tal te s te d at all stress levels should b e at le st 50 The di s tri b

u-ti o n o the total n um ber o avai l abl e t sp ec imens wi l l de pe nd up n the

purp se o the test T he rel ati ve g roup sizes fr d iferen t s tre s s levels are

s hown in T ab l e 1 This allo c ati o n is s ug e s te d s o t h a t the ob s e rve d perc

nt-age s urvi val val ue s wi have a proxi matel y eq al weig ht, a c o nd i on

neces-sary fr fi tti ng the re s p o ns e curve by the usual m eth od o le as t s quare s This

al l oc ati on also facil itates the computati on o f confi denc limits o n the re

-s p o n-s e c urve s As an al ternati ve to the us o the rel ati ve grou p sizes

(Ta-weig hting fa ct ors are e mp lo ye d an the anal ysi s c n ucted as i ndi cate d i n

re erences (1-3)

7

Fi g ure 2 presents data t h a t m ight b o tai ne d i n a "Pro i t" test o the

ty e describ d ifthe pressig ed n mb r of cycles we re 10

7

Alt o ug not

req i red f r the "Pro i t" analys is , the a tual number o cycl es-o-fi l ure

should b e re c o rde d fo r each specimen that fi l s beore 10

7

cycles, s o th at the

data may be avai l ab le fr other type s o analys i s , s uc h as the pl oti ng o

P-S-NCurv es

2 The Stai rcase Method:

T he stai rc s e ( or "up-and-down") method o testi ng i s a variation o the

"Pro i t" method I may req ire fwer s pe c ime ns than the latter b t is

l i kel y to be useul onl y when the prim a ry interest is in the mean fa t igue

strength c r esp n i ng to a p re as s igne d cyc le lf, N* The advantag e gai ne d

FIG 3.—Illustratio n of S taircse Me tho d

NO TE—Sp ec ime ns n mb e re d i n chronological order N umb r o f cycles fo r ech te s t i s

c o ns tant uless fi l ure o cur b e fo re han

in re duc ing the n mb r o f spe imens te sted may b e ofset b y an increase in

the ti me req i red to c n uct the test

I n the stairc as e method the s pe cime ns are te s ted seq enti al l y, o e at a

ti me T he fi rst s pe cime n i s te s te d at a s tre s s level eq al to the esti mated

val ue o mean fa t igu e streng th fr the prescri bed number o c yc le s or until

i t fi l s, i fi t fai ls be ore t h a t num ber o cycles I f the specimen fails, the n ext

specimen i s te ste d at a stres s le ve l t h a t i s one increment be l ow the fi rst

stress level Ifthe firs t specimen does no t fail, the sec ond s p ec imen is tested

at a stres s level tha t i s one i ncrement ab o ve the fi rst s tre s s le ve l, an so for th

The data are reco rded as sho wn in 3 The sp cimens that did no t fail

are de s i gnate d by the o's an tho se that fai l ed as #'s The chart s ho ws at a

glance the stres level that should b e used fo r the next te s t

T he s e l e c tio n o fthe pro er i ncrement o f stres leve l i s v ery im portant

I deal l y, mo s t o the te s ts shoul d be made at thre s tre s le ve ls , so c ho s e n

t h a t about 5 p r c nt o the test sp ecimens s urvi ve at the mi ddle s tre s s

le ve l, about 7 p r c nt survi ve at the l ower s tre s s le ve l, an about 3 p r

c nt s urvi ve at the hi g her s tre s s level P re vi ous data fr the s ame or s imi lar

m aterials are ne ded i n order to choose the s tre s s le vels eficien tly I fnone

are avai l abl e, some prelminary te ti ng may be required S c h data are di s

-carde d up to the fi rst p ai r o data giving o p p o s ite re uls; f r e xamp l e , i n

F ig 3, data fr ests 1, 2, an 3shoul d be di scarded

Sinc e the te tng is concentrated at s tres s levels near the mean fa t igu e

streng th val ue, the n um ber o s p e c ime ns te ted may be less t h a n f r the

"Pro i t" m eth od, w hich give s re u s f r a w ider range o stre s val ue In

g enera at le as t 3 s p e c i me ns s ho uld b te ted be aus , at most, only half

o the test e ults are actually us e d i n the computation o the mean f tgu e

st ren gt h Ifdata o tai ned by the stai rcas m eth od are anal yz ed by res po ns e

curv e methods, the re uls may be stati sti cal l y bi ase d be aus o the

se-quential nat u r e o the s tai rc as e m et h od Fu rt h er, ifthe main in tere t lie s

in e tmating the re p ns curv e—rath er t h a n the mean stren gt h —a t N

c yc l es , the s tai rcas m et hod i s n t an eficien t experi mental proc dure

3 Modi fi e d Stai rcase Method:

The ti me required to compl ete a test by the stai rcas method can be

re-d c d by di vi di ng the one l o ng s tai rcas pro ram into s e ve ral shorter, i

nde-pendent s tai rcase s an conductng the e s e ve ral tests si mu aneousl y This

t rea t m en t is know n as the modified stai rcas m et h od In the modified pr

o-c dure, the total n u m ber o s p e c i me ns , T, i s di vi ded into r g roups o n each,

so t ha t rn = T E ach group is te ted as a s p rate stai rcas pro ram , wit h a

s parate cha rt fr e ach g roup Thus s e ve ral machi ne may be us d si mul

-taneousl y In the modified s tai rc as e m ethod, as in an other test in which

s p e c i me ns i n a grou p are te ted on m ore t h a n one machi ne, a che c k s houl d

be made to determine w hether the mac hi ne s give si g ni fi cantl y d ifer ent

re-su s Ifthe re u s are not si g nifi cantl y d ifer ent, the data may be combi ned

fr statistic l analys i s

C INCRE ASI N G AM P L I T UDE T STS

1 Ste p Method:

I n m any cases the "Pro i t" or s tai rc as methods o test require more

s p e c ime ns than are avai l ab l e When onl y a few part s are avai l abl e f r

de-termi ni ng the fa t igu e limit, a na tura l de s i re is to test each pa rt unti it

ac-tual l y fis i ns te ad o just cou ti ng the number o runouts W h en te ti ng a

l i mi ted number o s p e c ime ns s o me time the practi ce i s to ru each specimen

at s e ve ral s tres s levels fr a l arg e num ber o f cyc les , s ay 10

7

I ftyp i cal re

-s p o n-s e curves f r the material are avai l abl e, the test may b started at a

s tre s s le ve l core p n i ng to a perc ntag e survi val o f a proxi matel y 9 pe r

c ent For e ach s uc c e s s ful ly co mpl ete d run , the ap p lie d s tre s s le ve l i s i

n-c re as ed by an amount core p n i ng to a dec re as e i n the pro abi ty o sur

vi val o about 5 per c nt an the test i s repeated un til failure o the

speci-at a stre ss level e qual to about 7 per c nt o the esti mated fa tigue lmi, an

the s tre s s increments s hould be a proximately 5 per c e nt o the e sti mated

fa t igue lmi t

I n the past, thi s method has not b en c nsi de red an a c ep ta l e technique

be aus the fa t igu e streng ths o some materials wi l l be inc re ase d or "c o axe d"

b y stres in them at stres levels below the i r ftg ue lmits, where as the

fa t igu e streng ths o ot her material s may be de c re as e d by damag e d e to

"u de r-s tre s si ng." H owever, in re ent years i has be en o b s e rve d th a t un der

s tre s s ing do es not greatly a ffct the t rue fa tigue l i mi t o some aloys, such

as m a n y o the allo y s te e ls a d a few o the n onferous materials (4 ) For

thos materials w ith w hich neither a p rec i abl e c o axi ng nor damag e o curs,

it i s p s ible to e s timate the ftg ue s tre ngth o feach sp cimen o r part b y

s tre s s i ng i t at c o ns e cuti ve l y hig her le ve ls u t the s p e c ime n fis

F IG 4.—Re resentati on of"Step" Testin ofS inle Specimen

This method i s i ll ustrated gra hi cal l y i n F ig 4 I n thi s man er, the fa t igu e

stren gth c resp n i ng to a p re as s igne d value o N fr e ac h s p e c ime n or

pa rt may be es ti mated The mai n di s ad antage o the proc d re i s that

the-s pe ci me nthe-s are ru ini ally at a sufficiently low stres level so t ha t f iu r e

wil l not o ccur As a resul, a n u m ber o s tres levels o ru n ou t s are usually

nec esary beore fa ilu re of the s p e c i me n o curs

Beore the ste te h iq e o fa tigue testi ng can be sael y used, the efe t

o c o axing or u de r-stres si ng the material must be kn own Certain s te e l s ,

s e ns i ti ve to s trai n-agi ng, will have r fa t igu e l i mi ts artficially rai sed by

c o axing or u derstres i ng at l ow stres le ve ls I n other case s i t i s th ough t

t h a t co axin or u derstres i ng may damag e the material artficially an

c aus e prem a tu re fiures

A l thoug h ste te sts have b e n made w it h a s ingle s p e c ime n, f ur or m ore

are needed to estimate the me dian fatig ue strength A l arge r sample gives

g reater p recision i n the estmates o the median an the variabi ty o the

2 The Prot Meth d (5-8 ):

I n 19 5 M arc l Pro i n Fran ce, de vis e d a ra i d m ethod f r e tmati ng

the f t ig e lmi t o a m aterial By us i ng the Prot m et h od, a go o d e tmate

o the fa t igu e l i mi t ma be o tai ned i n a fra ct io o the ti me req i red b

other me tho ds b ut at the expense o fmore u c rtai nty than i s p re s e nt i n

most o the other te t methods The use o thi s techniq e i s re tricted not

onl y to thos m ateria ls whi ch are not s e ns i ti ve to co xi ng efct s, as di

s-c us s e d in Se c tio n II Cl, but also to material s th a t apparently have a ft igu e

l imit In contrast to the ste p m et hod, i t i s s ug e s te d t h a t at l east 2

te t s p e c ime ns be us e d to o tai n the data nee ded f r the Prot anal ys i s

be-c ause o the wide s atter in fr actur e stre ss usual l y fo n i n Prot f t ig e

da t a To date, i has be n fou nd t h a t , by the use o the Prot proc dure, the

ft ig e l i mi ts o man al l o y s tee l s m ay be o tained w it h in a few per c nt

FI G 5.—Grap hic al Ilustration o Prot Data

« !, « 2 , «s, indicate d if r ent lo ading rates in psi p r c ycle

o the e tmate foun d from constant ampl ude methods It is not c rta in,

howev er, th a t the l ong -l i fe ft ig e stren gths o n on ferrou s aloys can always

be eval uated by thi s m eth od

In the Prot m et h od, the te st on a s p e c ime n i s fi rst started at an alternating

stre s o a bou t 6 to 7 per c nt o i ts e ti mated fa t igu e l imit an the stre s

i s rais e d at a constant rate A num ber o sp ec imens i s te s te d at the s ame ra t e

o l o adi ng unti each s pe ci men fi l s At l east t h re rate o l oadi ng are us e d

to e s tab li s h an c he c k the l i near rel ati onshi p betwe n s tre s s an the p wer

o the l o adi ng rate, whi ch i s req i red i n the Prot analys i s The l owe t ra t e

s ho uld be as smal as practcable an e highest ra t e sho ul d be l ow enoug h

so t h at the s p e c ime n do e s not fa il by yi el di ng be ore fract u re The type o

data ob se rve d is shown i n Fi g 5

One o the simp le st m eth ods fr o tai nin a constant ra t e o lo ading i n a

fa t igu e te t i s to us a stream o w a t er fl owi ng at a constant ra t e into the

l oadi ng container A noth er way i s to arrange fr s mall wei g hts, s uc h as shot,

to b p ured in to a contai ner at a constant rate Fai rl y good res ul ts hav e

equal c yc l e in crem en t s A n y devi ce that i ncreases th e stres at practica lly a

conti nuous, c o ns tant rate c an b us e d

NOT E —F r c o ns tant rates o lo adi ng, al l the p ints o b tai ne d at a g i ven ra te s hould

fa ll o the same straig ht l i ne Small va ria t ion s in the r ate o lo din or v a ria tion s in the

testin sp eed may c ause scter such as that sh wn i n F ig 5

The practicabi ty o f t igu e tests i s b sed up n the assumpti on t hat est

ab-l i shi ng the f tg e ch ara cterist ics o a g i ven m a terial by study in g the

per-form an ce o a random sample selected fr om a l arg er b d o possible speci

mens (t he p pulat ion

or univer e) i s p s i bl e Impl ici t i n these tests i s the

assumpton that the s amp le tested is "representa tiv e" o the p pulation

By ran om se le ctio n an al lo c ati o n o the test specimen us ing a tabl e o

ra n dom n u m bers (9, p 366-370) the in flu en ce o al v aria bi ty inh eren t i n

the m a t eria l and testing proc dures is gi ven a fa ir chanc o bei ng refle ted

i n the te s t data

There are inn um erable stag es in the testing progr a m in which any one

spe ci me n or any one group of spe i mens may be afcted diferen t ly from

ot her s from the s ame p pulation For ex ampl e, if one bar of a ba t ch of bar

stock is tested, i is o ten t acit l assumed that t h e rem a in in g b r s a re th e

s ame as the one tes ted Usual l y they are not bec aus e, f r e xamp le , s uc h

blanks a re h ea t t rea t ed in batches For each ba t ch t h e furna e settings a re

sl i g htl y diferent Wit h in each heat tr eatment ba t ch , those spe i mens near

the wall s o the fu rna ce are under sl i g htl y d ifer ent c o nd i o ns fr om tho s e i n

th e c n t er Spe i mens prepa red at th e st a rt o th e day a re ma chined wih

sharper to l s t han those t hat suc e d t h em Spe i mens tested at the begin

-ni ng o a prog ram may have the advantag e o b e i ng te s te d on newer, more

per fe t testing machines t ha n those that a re tested la t er wh en w ea r o t h e

machi nes has modi fi ed their characteri stcs T he s e are but a fw ex a m ples of

th e m an y fctor that m ay produce sig nificant bi ases in th e resu lt s unl es

controled by a ppropria t e ra n dom iza t ion

The fl l owi ng are s o me o the fa ct ors f r which randomi z ation m ight be

considered:

Posii on of spe i men w it h in the whole ba tch of m a t eria l

H eat-reatment batch

Position ofspe i men in heat r eat i g furna e

Order of quen ch in g

Order of p o lis hing

A ssig nm ent to testing s ( tres l evel an d so forth)

Order of testi ng

As s ignment to testi ng machi ne

Ma ch in e o era t or

T hi s li st wi l l sug g est ot her v ariables o im port a nc in pa rt icu la r progra m s

On p jo of th pnginp r and stati sti ci an is to de c ide how the ran omi z ati on

s b _n^1H h e rqm 'eH r^it A common misconception is t hat ra n dom izat ion ca n_

in t o t h e b x of spe imensjQr^ the^iex tjto^ b^ te ted, b t t is summing h w

The sampl e can be "b i as e d" by unconscious an unre o niz d t ren d s o

hu man behavi or as we l l as by u nk nown p ter ns o a r r angement The best

pro c e dure Js _tQ_ae ±-i i D the progra m o the b si s of ra n dom nu mber s as prev

i-ousl y sug gested ( 9"

To o t a in f tg e da t a that ca n be use d m ost eficien t ly, atr ained^tatisti^

ci an s ho i i ld_b £_£ O Ji au]l e j

J_whe ne ve r p s ibl e, in j la n n in g the ex perim en t s

and spe i men sele tion In most cases the st a t ist icia n wil l be a l e to plan the

e xpe ri me nts to measure not o nly the efct s o the mai n vari abl es under

st u dy but al so t h e efe ts of t h e m ore im port a nt se onda ry v a ria bles a s wel,

a nd do this wi hout requ irin g man , if any, a ddition a l spe imens Tests

con-duct ed in a ccorda nce wih such a plan can be a na ly zed to g i ve an est im at e

of the im port a n ce of each of the kn own variables that cont ribu t e to the

sca tt er in the test resu s The t ech niq u es o ex perim en t a l desi g n are to

in-vo l ve oL however, to be i ncl uded in th is G ui de '

"So me indication of the m in im um n u m ber of spe i mens needed for a g i ven

degre of con fiden ce in the results o t a in ed w h en usi ng the difer ent test

proc dures has be n g i ven in Se ti on I I I For supplementary refren ces on

this su bje t , se r efr en ces (10-12)

9

The folow in g se ti ons di scus the minim um nu mber of spe ci mens needed

fr each ty pe of analysis gi ve n in Se ti on V when the sampl e s i ze is fix ed

b f r e testing A ll sampl es are assumed to be ra n dom ly se l e cte d sampl es

from t he p pulation u nd er con sideration

A L I T E DISTRIBU TION SHAP E NOT AS UMED

1 S-N Curves:

The m in im um n um ber o f tg e test spe i mens ne ded at e ach stres

l evel depends on: (1) which per c n t survi val cu rve is desired an d (2) what

con fiden ce l e ve l i s desired

For a 5 per c nt con fiden ce l e ve l and one grou p tested at each stres l evel ,

T abl e 8 i n Se cti on V A1 shows the n u m ber of spe i me ns needed for several

v alues o per c n t survi val For ex ampl e, a 9 per c n t survival curv e

re-q i re s at le ast 1 specimens at e ac h s tres s level

Table 2

10

provi des s i mi l ar inf rm at ion for on e or m ore grou ps tested at

each stres l eve l a d se ve n val ues of con fidence l e ve l , in cludin g 50 per c nt

For example, from T abl e 2 , one grou p o fi ve spe ci me ns at each stres l e ve l

is needed fr an 8 per c nt survival curv e coresp ndi ng to a 5 per c nt

con fiden ce l eve l For a 9 per c n t confiden ce lev el, at least fi ve groups of 10

sp e c i me ns at each stres l e ve l are d fr an a pproxim a t ely equ iv a len t

S-Ncurv e

When se ve ral S-N curves are to be d ra wn from the same da t a , T abl e 2

shoul d be st udied ca refu lly to find the best com bination of n um ber ofgrou ps

and grou p si ze

9

Se al s o Re om m en ded Pra ctic for Ch ic of S mple Si ze to Estim ate the A v era ge

Quality of a L ot or Proce s s (E 12 ), 19 61 Bo k o A TM Sta dards, Part 3

2 Esti mate s o Paramete s,Si ngle Stres Le v el

T he m in im um n um ber o sp ec i mens ne ded depends up n the desi red

wid t h o the con fiden ce in t erv a l fr each pa ra m et er In g eneral, as the

sam-ple size i ncreases, the co fi dence in terv al for any g i ven confidence le ve l

be-come s n a rrower an d the dif ren ce between the o served v a lue an d the u ni

ver e v alue be omes smal l er

For the m edian at a confidence l evel o 0.9 5, confidence lm its are equa l

to the o served m in im u m and m a xim u m values up to a sampl e size o n ine,

wh en the wid t h o the confidence interv al be comes le ss t h a n the o b s e rve d

ran ge ( Se e Ta l e 9 on p g e 2 ) Ifranges fr prior s amp l e s fr om the same

p pulation are k nown , a sampl e s i ze can be chosen so t ha t the in terv al wil

hav e approx imately the desi red widt h

If the n u m ber o fs p e c ime ns are onl y 3, 4 , or 5, Tabl e 10 (s ee Se c ti on V A2)

gi ves proc dures for computng confidence interv als for the mean

Fo r p e r c nt survi val val ue s , T ab le 1 (Sec ti on V A2) give s val ues o f9

per c nt confidence l imis for fou r s amp le sizes Comparing the w idth s of

the confi denc intervals give s so me ide a o fthe size o f s amp le ne ded I f a

good estm ate o p = per c nt s urvi val /10 i s avai l abl e, the minimum

s amp l e s ize i s approx imately:

w h ere E = one half the desi red widt h o a 9 per c nt confi dence interval

(s e e ASTM Re ommended Practc E 12 )

I t is m ore dificu lt to det erm in e the m in im um n u m ber o sp ec i me ns needed

fr a confidence inter val of a g i ven wid t h for ftg e lfe coresp nding to a

stated v alue o per c n t surv iv al ot h er t han 5 per c nt Se f ot n ot e 15,

pag e 2 , f r equations f r set t in g up tabl es si mi l ar to T abl e 9 (in Se ti on

V A2) for ot her perc ntage p ints

In g eneral , the sampl e s ize s would be larger t h a n fr medi ans

3 Te sts o Si gni f a ce :

T he minim um n u m ber o spe i mens ne ded depends upon the de si red

magnitude o the difer en ce that s houl d be dete ted and the size o the ri sks

t h at can be tolerated

When the ra n k test is us e d to test the dif ren ces o group medians, i t i s

dificu lt to relate the desi red v alues and the crieria f r the si gni fi canc test

g i ven in T ab l e 2 for two groups and in T ab le 2 for m ore th an two g roups

At l east fi ve s p e c i me ns s hould be i ncl uded i n each grou p

For dif ren ces of two or m ore s (oth er t h a n 50 per c nt) no

pre i se esti mate of the m in im u m n m ber of spe i mens needed i s p o s s i b l e

u l es prior estmates o the perc ntag es are avai l abl e At l east 15 s p e c i

-mens shoul d be inclu ed i n each group

4 Re sp nse Curves:

A di s c us s i on of the m in im u m n u m ber of spe i mens and their al l ocati on

B L I T E DI STRI BUT I ON SH AP E ASSUME D

1 Normal Di stri buti on:

It i s as sume d here, as wel l as i n Se ti on V B t hat the ft igu e data can be

transf rmed so that t h ey wi l l be a pprox im ately N ormally dist ribu t ed A

Normal di stri buti on is assumed in all c ses Eac h sample is assumed to b e

dra w n a t random fr om it s p pulation

2 S-N Curv e s:

Tab le 3 gi ves k fa ct ors fr compu ng p ints on 7 , 9 , 9 , 9 , a d 99.9

per c nt survi val curves for f ur v alues of confidence l evel , i ncl udi ng 50 per

c ent, an f r n = 3 to 2 T he minimum number o specimens should i

n-c re ase as the per c nt survi val i ncreases, but t here is no defin it e criterion

fr c ho o s i ng a particular grou p size except for the relativ e m a gn it u des of

the k val ues (Not e

that the ra t e o de rease is les s as,« i ncreases ) The nu m -

be r o spec i me ns tested at eac h stres level can be s malle r th an the group

s ize s ne ede d w h en the lf dist ribu t ion is n t as ume d ( Se ti on V A)

OF S E CIMENS

0

OF S E IMEN S" NE DED FOR

WIDTH FO A POPULA TION

STAND-AR D DEVIATION, a

Some E sti mate o <r Avai l ab l e

W idth of Int erv al

Fo th Me an.—If a goo d esti mate o the p pul ati on standard devi ati on,

0 , is avaiable, Table 2 gives the mi ni mum n mber o f specimens nee ded

fr confi denc i nterval s o stated widt h f r the m ean, p, o the p pulation

larg er, since ^.9 5 val ue fr om T ab l e 2 s ho uld be us e d i nstead of the 1.96

in the eq ati on fr n (Table 2)

Fo the Sta dard Demoti on.—In order to find the minim um nu m ber of

sp ec i mens ne e de d for determ ining confi denc i nterval s of stated w idt h for

the standard devi ati on, <r, of a p pulati on, some e t im a te of a m ust be avai

l-a l e, since the w idt h o the interv al i s measured i n un it s o a Howev er,

Table 3 c an b e used as a gui de even if no good e s ti mate o f a is avaiab le

For e xampl e , i fn = 8, the sampl e-standard devi ati on, us e d to e ti mate the

p pul ati on stan da rd devi ation, may be a ove or bel ow a b 0.5< whereas an

e tmate based on n = 3 will not b expe ted to devi ate fr om the t r u e val ue

by more than 0.2 <r

OF S E CIM E NS" NE DED TO

DE-TE CT IF THE STANDARD DEV IA

ULA-TION I S A ST AT ED MUL IPL OF

AN THER POPU LA TION

D fe re nc Be tw een Two Sta dard Dev i ati ons.—Th e s amp le s ize s for testng

the diferen ce betw een two means are gi ven i n T ab le s 6 an 7 In some c as e s ,

the princip l in tere t is in the dif ren ce betw een stan dard deviatons

1 One Sta dard Dev i ati on a Fi x ed Value.—If o e standard devi ati on i s a

fi xed v a lu e—for exampl e, the l ong -ti me standard devi ati on o data b s d

up n an ol d proc dure—and i fthe other standard devi ati on i s to be c o

m-puted from data based up n a new p ro c e dure that may re duc e the

varia-bil i ty, T ab l e 4 gi ve the mi nimum n um ber o s p e c ime ns ne ded to dete t a

reducton o a stated amount T he s e e size a pl y when the ob s e rve d

stan ard deviati on, s, fr the new proc d re i s i nde d s mall e r than the fix ed

value, and the ratio s /(fix ed v alue)

2

is comp red w it h 1/ F

0.9 , core p n i ng

to °o an n — 1 degrees o fre dom f r numerator an de no minato r re

-s p e c ti ve l y (Se e Se c tio n V B4(a) an T ab le 3 )

2 Two Sample Sta dard Dev i ati ons.—If the pro l em is to test w h et h er

the variab ility of p ro c e dure 1, say, is greater than the variab i li ty of p ro

-b le 5 gives the mi ni mum num ber o s p e cimens ne e de d i n e ach s amp le to

dete t th a t si i s a stated mu i pl e of s% I f the o bs e rve d val ue of si i s i nde d

l arg er than the ob s e rve d val ue o s

z,comp re s?/s<? with ^ 0.9 core sp n i ng

to (HI — 1) de g ree s of freedom for num erator an denom inator (s inc e

n\ = HZ) (S ee Section V B4( a) an Tab le 32 ) I n thi s case i t i s not core t

to m ake the

test

i fs

22

i s grea t er than S i

2

D i ffe re nc Betw ee n Tw o Me ans:

1 On Me n a Fi x ed Vali se —If o e m ean i s a fi xed v alue—for exampl e,

the l ong -tme mean o data b se d on an ol d proc dure or a commonl y use d

m aterial—a nd the other mean is to be com puted fr om da t a b se d upon a new

a = U n kn ow n Standa rd Devi ati on o the

P opul ati on B n E sti mated

THE MEA S OF TWO POPU LA TIONS

< = U n kn ow n Standard Devi ati on of

E ach Po ula tion ; <n = az

proc d re tha t may shift the mean, T ab l e 6 gi ve s the m in im um n um ber o

s p e c ime ns needed to detect a sh ift in either dire ton, measured in term s

o the p pul ati on standard de vi ati on o the ne w proc d re T he s e sampl e

s i ze s a pl y w hen the computed v alue o

i s co mp re d with £ 9 5 i n T ab le 2 e Se c tio n V B4( 6).) No F-ratio test

i s nee de d

2 Two Sample Mea s.—Th e mi ni mum n um ber of s p ec ime ns ne e de d i n

e ac h s ampl e to dete t a dif ren ce i n two p pul ati on means, stated as a

multple o thei r eq al u i ve rs e standard de vi ati ons, i s give n i n T abl e 7

T he s e s amp le sizes a pl y w h en ( 1) the two s amp l e standard devi ati ons are

n t si g ni fi cantl y d ifer en t and (2) the computed v alue o / (s e e Se c tio n

V A N A L Y SIS OF FA TIG U E DA TA

A b ask conc pt o stati sti cs i s th a t a grou p o one or m ore sp ec i me ns i s a

s amp le taken fr om a larger b d or p pul ati on S c h a s amp le i s co ns i de re d

to be ju st one o a "number," oft en v ery l arge , o s amp le s that coul d hav e

be en taken The sampl i ng proc dure us e d del i mi ts t h e-p pu la t ion b e ing

e tmated The re uls o tai ned fr om te sts on a random s amp le from the

p pulation can be used to e tmate the characteri sti cs o the whol e p pul

a-ti on an to measure the pre i si on o the e ti mate

In the case o fa t igu e te ts the da t a o se rved are usual l y the li e s o

s p e c i me ns te ste d at a constant a pl i ed stre s (strain or defl ecti on) ampl i tu e

Sinc e the c yc le li fe vari es from spe i men to spe i men, thi s measurable ch a

r-FI G 6.—"Normal " or Gaus s i an Distri buti on Curve

acteristc i s not a fix ed val ue and is be s t de cri bed by a frequen cy

distribu-ti on The g raphi cal pre entation of the distribution of c yc le li e s for the

p pulation o s p e c i me ns t hat h av e l i ve s bet w een c rtain l imi ts i s kn own as a

frequ en cy distribution curv e S ch a dist ribu t ion curv e m ay be e tim a ted

from t h e ra w te t data or from t ra n sf rm ed test da t a , that is, either from

val ue of Nor fr om val ue ofl og N, l og l og N, N

12

, and so forth

W hen the freq u ency distribution cu rve has a partcular kin d o b e ll sha e,

as shown i n Fi g 6, the data are s aid o hav e a "Normal" or Gaus s ian dis

-tribution T hi s Norm a l pro abii t dist ribu t ion curve, fx ), i s repre ented

by the equa t ion :

ti o n standard deviation (a m easure o the di spers i on).

1

It shoul d be

empha-sized t h a t val ue s o the param eter o the p pulation can onl y be esti mate d

from tests on the spe i mens in the sampl e; to o t a in ex act val ues w ould

re-quire t ha t the total p pulation be tested

Whil e some fa t igu e te s ts , partcularly those made i n the fi ni te l if ra n ge

o f a S-N curv e, ma yi el d a proxi matel y N orm a l distributions of cycl e

lf, generally a t ra n sf rm a t ion to lo c yc le lf is required Ot h er do not

yi el d N ormal distributions, even a t er various t ra n sf rm a t ion s are perform ed

o the da t a Thi s i s particularly t ru e in the cas e o tests made at a pl i ed

stre s s es near th e ftg e lm it wh ere ru n ou t s are o served H enc , ot her

distribut ion s, such as the Wei bul di stri buti on,

12

the "ex treme val ue" di s

-t ribu-t ion wit h and wit hou t lower lmits, as us e d by Freu d en t ha l and Gumbel

( IS), and other distributions, t ha t are just as n orm a l in the usua l sense, as the

N orm al or Gaus s ian distribut ion , hav e be n ap pli e d to the anal ysi s o ftg e

data Whi l e referen ces to s o me ,of these distributions are included i n this

G uide, anal ysi s o the f t igu e da ta has be n confined mostl y to m et h ods t h at

requ ire no assumptons o distribut ion sha e or to the m et h ods b sed u pon

th e assumpti on t hat th e ra w data or t h e t ran sf r m ed data hav e a Norm al

dist ribut ion

As st a t ed previ ousl y, h ow ev er, any set o o serv ations to w h ich these

stati sti cal m ethods are a pl i ed is as umed to c o me from a ra n dom sample

from the p pula t ion of in t erest If a seri es of sampl es is drawn, proc dures

fr te sti ng fr statistical control are g i ven in the ASTM Man al on Quality

Control ofMa terials ( s e e f ot not e 5) L ack of statistical con t rol i n d a t a

in-dicates that the s e ri e s o sampl es doe s not c ome from the same p pulation

A L I FE DISTRIBU TION SHA PE N OT ASSUME D

1 S-N Curv e s:

T hes e techniques shoul d be used w h en the a ctua l shape o the dist ribu t ion

o fa t igu e l if val ues for a gi ven m a t erial is u n k nown or sketch y and the

n u m ber o s pe ci me ns tested at each a pl i ed stres l evel i s to smal l , s ay les

t ha n 5 , to estimate the sha e o the dist ribut ion In such cases , these te

h-niques g i ve conservati ve resuls

(a ) O ne Grou at Ea h Stre ss Lev e l.—U sua lly th e first step in the a n a ly sis

o ft ig e d a t a i s to draw the S-N curv e fr 5 per c nt surv iv al; i is the

curv e fi ed to the medi ans o the groups at the several a pl i ed stres leve l s

The m edian, an "o rde r statistic," is the mid lem ost v a lue when the o served

val ues are a rran ged i n order of m a gn it u de, or the averag e of the two mi ddl

e-most val ues i fthe grou p size i s even

Other S-N curves, thos e for p per t survi val (w h ere p is not 5 ), may

be fi tte d to oth er order stati sti cs i f the group s ize i s great er than 1 I f the

group val ues are arranged i n order o m a gn it u de, NI is the m inim um cycl e

lf val ue, or the fi rst order stati stic, A

7

2 is the s ec ond o served value, or the

s ec ond order stati sti c, an so forth

The estim ated perc n ta ge o survi vor fr the p pulation at cycl e lfe

1

In th e Norm al disribu tion , th e media an d th e mean ar eq u a l

val ues o Ni , or 7V

2, depends upon the group s ize Ta l e 8 gi ves th e median

perc ntages at Ni and Nz fr s e ve ral grou p sizes

2 I fth ree s pe cimens are tested at e ac h a pl i ed stres l evel , the 7 , 5 , an

the 21per c nt survi val curves may be estmated from the entries in T abl e

8 and their compl ements The v alue 7 per c nt i s fou n d o p si te s amp l e

s iz 3 i n the s eco nd col umn, the v alue 5 per c nt i s taken fr om the median

SU VIV RS FOR THE POPU LA TION

S-Ncurv e, and the v alue 21per c nt is o ta in ed by subtracting the v alue

i n the second col umn fr om 10 0 p e r c ent

3 If7 s p e c ime ns are tested at each ap p l i e d stre ss l e ve l , the 9 , 7 , 23, an

10 per c nt survi val curves may be esti mated from the en tries i n Ta l e 8 an

thei r compl ements The 5 per c nt survi val curv e may be estim ated from

the median

At l e as t 13 s p e c ime ns m ust be tested at each ap pl i ed stres l e ve l to esti

-m a t e the 9 per c nt survi val curv e

I n practi ce, val ues o per c nt l less than? 5 usual l y are not w a n ted

H enc , i fal l o the sp ec i mens i n a s amp l e are te s te d si mu aneousl y, the tests

may be s top p e d as soon as the s pe i men havi ng the median value o f tg e

lf for th e s amp l e h a s fa iled, u l es s th e da t a a re required for oth er purp ses

1

T hes e ar c alled "median p ercenta es " because, h alf o f th e t i me, th e tr ue per c enta g e

wi be larg e r, an d fo r th e othe h alf o f th e t i me, sm aller T hey ar clo s e to , but usualy

n ot eq al to, th e "expected" p ercenta e of s u rvivo rs, wh ic h is eqal to 1 — i/( + 1), whe r e

i is th e n mber o f th e orde s tats tic an d n is th e s ample si z e Th e c on fid e n c e l evel as so

ci-ate d with ex pec te d percentaes v ries with th e s am e si z e, wheeas it is co ns tant fo r

As menti oned previ ousl y, the perc ntag e survi val val ues gi ve n i n T abl e 8

are medi an val ues; th ey are b s ed on a "confidence l e ve l " o 5 per c e nt

14

Perc ntage s urvi val values coresp ndi ng to h igher con fidence l e vel s, such

as 9 or 9 per c nt, are gi ve n i n T abl e 2 fr a si ngl e sampl e w h en m = 1

For e xamp l e , i ft hree s p e c ime ns are tested at an ap p li e d stres l e ve l, 7

per c nt o the p pulation are expe ted to s urvi ve N\ c yc le s ( 50 per c nt

co fidenc l evel ), but the statem ent t h a t at l e as t 37 per c e nt o the p

pula-ti on wil survi ve N\ c yc les may be made wih grea t er confidence (confi dence

le ve l = 9 per c n t) Ifestimates o the p pulati on perc ntag e are m ade

fr om a s e rie s o sampl es tested at one a pl i ed stres le ve l and the st a t em en t

i s m ade t ha t at l east 7 per c nt o the p pul ation wi l l survi ve N\ c yc le s ,

50 per c nt o such sta tem ents are ex pe ted to be in cor e t Ifthe st a t em en t

i s made e ach tme t h a t at l east 37 per c nt wil l survi ve N\ c yc le s , only 5

per c nt o suc h statements are expe ted to be incor e t H owever, S-N

curv es coresp n i ng to a 5 per c nt co fidence level are usual l y shown

The efe t o fi ttng a curve to the s ame order statistcs at s e ve ral s tre s s

l eve l s pro ably i ncreases the confidenc l evel ; how m uch i s not kn ow n If

S-Ncurves are b s e d on other con fidence l e ve ls , the f ct s houl d be pl ai nl y

in-di cated on the ch a rt

( Z >) Sev eral Samples, or Gro ps, at Ea h Str s Level.—If i is not p s i bl e

to test al the spe i mens in a s amp l e simultaneously and i f sto pi ng the tests

be ore al l the sp ec i me ns have fa iled i s desi ra l e to save tme, the req ired

sampl e may be divi ded, at random, int o two or m ore g roups (see references

17 a d 18) Then the medi an o the particular order statistcs (t h e fi rst,

se on , a nd so forth) for the s eve ral g roups may be used for con st ruct in g

the S-N curve T abl e 25 gi ves values of perc ntage survival for several nu

m-ber of groups a d s eve ral co fidenc le ve ls

E XAM PL E —Wi th fi ve testng machi nes av aiable, 15 s p e c ime ns were tested at a

co stant applied stress level in thre e gro up s o f5e ac h F o r each group, all mac hi nes

were asumed to be s to p p e d ater the s ec ond fiu r e ( Actualy, al l machines were

allo we d to ru u t fr act ur e oc ured or u n t il 10 mi ll i on cyc le s o ft igu e stresi ng

had b e en ap plie d, s o that the ti me saved could b e e s ti mate d fo r thi s p rti cul ar s e t

of tests.)

The test da t a are:

En t erin g T ab le 25, u nder "Lo we s t Ra nking Poi nts," in the c l umn for m = 3

g ro ps, o p p o s i te n = 5 i n eac h group, an at a confi dence level of 50 per c n t,

1 4

Te hnical ly speaki ng, the S- N curves b as e d on o rde r s tati s ti c s are "nonparametric

tolerance li mits," w hich ar de s c ri b e d by Mu rph y (16 T he pro abil i ty t h a t at leas t p

per c nt o the p pul ati on lies ab o ve Ni cycl es, where Ni is the i th order stati sti c o the

sample, i s properly c alle d a "tol eranc lev el"; but the t er m confidence l eve l appears to

re ad 87.05 p er c e nt This val ue is an e sti mate of the p e rc entage of the p o pulaton

fr om whi c h the ori g i nal 15 sp e c ime ns were s electe d th at will surv iv e 16 kiocycles

Si mi l arl y, at a c nfidenc l e ve l o f9 pe r c nt, 6 7 0 3 pe r c nt o r more o f the po p

ula-ti on are estmated to surv iv e the 16 kiloc yc le s Ag ai n, fo r the "S eco d Rankin

Points," at a c nfidenc le vel of 50 pe r c n 68 61 per c nt of the p p l ati on are

es timate d to s urvive 2 9 kiocycles an , at a 95pe r c e nt c o nfide nc e level, 45.4 p e r

c nt or more ofthe p p l ati on are estmated to survi ve 2 9 kiloc yc le s

Addi o al in form a t io can be o tained from the pre eding test da ta b c o

n-s i de ri ng all 15 sp ec imens as o ne "group" an determining the perc ntag e o fthe

p p l ati on e xpec te d to survive 105 kilo c yc le s , whi ch i s the l owest rankin p int

fo r m = 1a d n = 15 in Table 2 Fo r a 9 p e r ce nt co nfide nc e level, straight-line

interp lato between 7 4.1 per c nt for n = 10 an 86 09 per c nt for n — 20

" Based on a table i n N air (19 )

gi ve s a bout 80 per c nt From this, i t i s estmated t hat at a 95 per c nt c nfi denc

level ab ut 80 p er c e nt ofthe p o p ulatio n will survi ve 10 5 kiocycles

2 Esti mates o Parame te s—Si ngle Str s Lev el

(a) Medi an Fati gue Li fe:

1 Poi nt Esti mate.—A p int estmate o the popul ati on me di an i s the

sample median, described a ove in n V Al(a)

2 Co fi de ce Inte v al Esti mate.—A c onfi de nc i nterval for the medi an

th a t doe s not ass ume a part icular frequen cy distribution f r th e p pulation

may b e co mp ute d ifthe sample size is l arge r than five

The n o served value o fat igu e lf, N, are arran ged in order o m a gn it u de

as fl l ows:

N4

N6N2

The c o nfide nc e limits correspondin to a c o nfi de nc e level o fat le st 0 95 are

give n by the order statsti cs de i g nated i n T ab le 9, p 2

E XAMP LE — Ass ume t h a t ten specimens are te s te d at a particular s tre s s level an

the o b s e rve d val ues o ffa t igue l i fe i n k ocycles are 2 01, 2 4 , 2 6, 2 0, 2 2, 2 38, 2

2 4 4 , 2 4 5, an 2 48 T he p int estmate of median fa tigue lf i s the ave rage o f the

two mi ddl e most val ue s , namely 23 kiloc yc le s T he interval esti mate i s defi ned b

A/2 an Ng (se T ab l e 9), whi c h are 2 4 an 24 5 kilocycles , re s p e c tive ly

T he p p l ati on median may be ab ve or b e lo w the sample median — 2 35

kiocycles — b t the chanc es are at l e as t 95 i n 10 t ha t the statement, "the

median les betw een 2 4 an 2 5 ki l o cyc l e s ," i s core t i fthe s ampl e c me fr om

3 ad the rang e o the o b s erve d values to the largest 3 X rang e

value an su tra t i t fr om the s mal le s t val ue:

that is, Ni - (N

3

- J V i) and N

3+ (N

3

- Ni )

4 ad (range) / 4 to the largest value an s ubtrac t i t lj£ X range

fr om the smalest val ue:

0

Se e Yo ude n (2 ) fo r n = 3

6

Pri v ate c o rre s p one nc e fr om W J You en, fr val ue s o n gre ter t han 3

2 Ap rox i mate Co fi de nce Inte v al Esti mate — An approx im ate confidence

i nterval e ti mate f r the mean t h a t doe s not as s ume a p rti cul ar frequ en cy

distribution fr the p pul ati on may be computed as shown i n T abl e 10, i f

the sample size is 3, 4, or 5

() Per Ce t Surv i v al fr a Stated Value o Fati gue Li fe:

1 Poi nt Esti mate — A p i nt e ti mate o the perc ntag e o the p pul ati on

t ha t has fa t igu e lf val ue eq al to or a ove a stated val ue i s the sampl e

perc ntag e o o se rved val ue eq al to or a o ve the same stated val ue

2 Co fi de nc Inte v al Esi mate — Co fidenc l i mi ts co rre spo ndi ng to

po ss ib le val ue o sample perc ntag e, p , fr fo r s amp le s izes are give n i n

T ab l e 11 Val ue s fr other s amp le s ize s may be read fr om a chart from Di xon

an Mas s e y (9), p 415, from w h ich m a n y val ue i n T abl e 1 were taken

E XAM PL E — Us ing the data gi ve n i n the ab o ve e xampl e o f thi s Section an 2 0

l ati o n val ue of per c e nt surviv al are o tained: ( 1) p int estimate: 7 per c nt an

(2) interv al estmate: 34 to 9 per c nt

A la rger s amp le siz wi l l gi ve a shorter interval esti mate (s e e T abl e 1 )

(d) Fati gue Li fe fr a Staled Value o Pe r Ce nt Surv i v al

1 Poi nt Esti mate —A p int estmate o the p pul ati on valu e o ft ig e

lf fr a stated val ue o pe r c nt survi val i s b sed on orde r stati sti cs as

Wher: p = sampl e pec ntag e (for ex am ple, perc ntag e surv iv ing) Con fiden ce li

m-i ts coresp n in to (10 — p) per c nt a re: low e r: 10 — (a bular value for u pper

lmit c o rres p o nding to p, per c nt); u pe: 10 — (ta ul ar value fr l o we r l imit cor'

rsp ni ng to p,p r c n t)

0

Based on chart fr om Di xon an Massey (9), p 41 , a n d, fr n = 4 , on chart fr om

Pe arson an H artley ( 2), p 2 04

o utl i ne d i n the Section o S- N curves: "One Group at Eac h Stre s s Level"

(Sec ti o n V A ) A partcular v alue is e m edian, c r esp ndin to 5 per

c ent surviva

A nother p int es ti mate m ay be deri ved fr om th e cumulative fr eq u ency

dis trib utio n o fthe observed values In ge ne ra the two p o int estimates woul d

not b exactl y eq a

2 Co fi de nc Inte v al Esti mate.—In terv al e s timate s for me dians '(50 per

c nt survi val ) are des cri b ed i n Se ctio n V A2(a) Interv al esti mates fr fa t igu e

lf val ues coresp ndi ng to other perc ntag e p i nts may be computed by

using reerence (21) ,

1

3 Tests o Si gni f a ce :

(a) D fe re nce s o Grou Medi ans—Si ngle Str s Le v e l.—-If tw o or more

groups of s p e c i me ns are tested, the q esti on of whether the o served difer

-e nc -e s in the val ues are d e to chanc or to s o me dif ren ces i n the p

pula-ti ons fr om whi ch the g roups w ere dra w n oft en ari s es The o b s e rve d d ifer

-e nc -e s , fr exampl e, c oul d ari s e be ause o diferen ces i n m a terial l ots or

dif ren ces i n the characteri sti cs o the testi ng machi nes

The rank tests gi ve n i n thi s se c tio n as sume th a t the s e ve ral groups are i

n-dependently and randomly d ra wn fr om p pulati ons t h a t are o the s ame

s ha e but may d ifer w it h respe t to their medi ans All the o s erved val ues

i n o ne g roup are assumed to come from o ne p op l ati on S ince the p o p ulati o ns

are as umed to be o the same (t h ough u n kn ow n ) sha e, onl y those g roups

t h at are tested at the same stres s le ve l shoul d be compared, s i nc e the for m

o the distribution tends to chang e wit h chang e in stres l evel

1 Ra k Test fr Two Gro ps.—In the ra n k test for two groups the ra n k

o e ac h o servati on i n the two groups combi ned i s determined T he l owest

v alue i s g i ven the r an o 1, the nex t h igh er o s erve d value i s gi ven the r an

o 2, a d so forth I f o e v alue a pear s eve ral ti mes, t hat i s, there i s a te,

the averag e o the ra n ks fr those nu m bers is as s i gne d to each one For ex

-ampl e, i fthe l th, 12th, 1 th, an 14th val ues are al l eq al, they are each

gi ve n the r an of( 11+ 12 + 13+ 14 ) /4 = 12 5 T he ra n ks for the two

g roups are total ed sep ratel y and the total f r one o the groups (th e one

w it h the s mal l e r n um ber of o servations i f the grou p s ize s are unequal) i s

comp red wit h the cri ti cal values give n i n T abl e 2 f r ampl e sizes eq al to

the group sizes

I fthe o s erve d val ue falls w it h in the rang e o val ues gi ve n i n T ab l e 2 fr

the chose n si gni fi canc level ( 5 or 1per c n t ), the groups may be c o ns i de re d

to have c o me fr om one p pulation Ifthe o bs erved val ue falls outsi de the

rang e o val ues give n i n the table, the two groups are s aid to be significantly

d ifer en t h a t is, to have c o me from tw o p pul ati ons wit h difer ent medi ans

T he use o the 1per c nt si g ni fi canc le ve l gi ve s a s mal le r ri sk o c al l i ng the

(1) k is c ho s e n s o th at

(2) m i s cho se n so th a t

g roups si g ni fi cantl y diferen t w hen they are a tual l dra w n fr om one p

pu-l ati on an the o bs erved diferen ce i s d e to chanc

EXAMP LE.—To c o mpare two ma hi nes, the rank tes t was ap p lied to the data

from 2 s pe c i me ns ran omly assig ed to two testi ng mac hi nes (S ee Table 12.)

Ac c o rdi ng to Table 2 , the r an total fo r Mac hi ne A i n Table 12, whic h has "the

s mal l e r n u m ber o fmeasurements," s ho ul d b e between 101 an 17 (Ni = 10 ,

NZ = 17) fr the 5 pe r c nt l e ve l o s i gni fi c anc e , an betw een 8 an 191fo r the

1p e r c e nt level o fsignificance This me ans that the ac tual tota 8 , wo uld no t b e

e xp e c te d to o cc ur as ot en as o nc e i n a h u n dred s amp le s d e to c hanc e al one , i f the

two machines were c ompl e tel y i nterchangeabl e Thus, o the averag e, the ma hi nes

give s i gni fi c antl y dif ren t ftig ue l ife v lues

TAB LE 12.—FATI GU E T E ST D TA

2 Ra k Test fr Mor th n Two Gro ps — The method of as s igni ng ra n ks

i s the s ame as f r the two-g roup test, ranking the o servati ons fr al l the

g roups combi ned The ranks are totaled se aratel y f r e ac h group and the

folow ing test-statistic, H, is computed from th e ra n k totals (2 ):

wher e:

k = n um ber o g roups,

Hi = n mber o o bse rvatio ns i n the ih g roup,

N = y ^,ni , the n um ber o o servati ons i n al l g roups combi ned, an

Ri = sum o the ran ks in the i th g roup

The test-statistc H i s distributed a proximately as x

-co mp re d w it h the val ue o x

2

gi ven i n T ab l e 2 to determine w hether

there may be a si g ni f ant dif ren ce among the p pulati ons from whi ch

the g roups w ere drawn or not IfH is g reater than the x

2

val ue fr k — 1

deg re s o fre dom an the c ho s e n si g ni fi c anc le ve l, the p pul ations are

s ai d to be d iferent; t ha t i s, the g roups may be said to have b een dra w n

from two or more p pul ati ons I nspe ti on o the ra n k total s wil l usual ly

sh w w hich g roups are d if r ent from the ot hers if the dif ren ce is si gni fi

-cant

E XAM P L E.—T o compare fi ve ma hines, the ran k tes t was ap p lied to the da ta

from 2 s p e c ime ns , ran dom ly assig ed to the fi ve machines (s e Tab le 13)

TABL E 13.—FAT I GUE T EST DAT A

= ° 49, coresp n in to a 5 per c nt si g ni f anc level or a perc nt e of

95 S inc the computed value o fH, 2.56 , i s v ery mu ch s malle r than 9.49, the o

b-s e rve d val ues o fa tigue lf may b c onsi de red to b fr om one p pula tion; the

ma-c hi ne s may be c o ns ide re d to be i nterchang eabl e

(t) D fe re nce s o Tw o or Mo e Pec enta es (fr ex ample, pe r c nt surv i v al

values).—Th e teststatistic us d to te t the si g ni f a c e o the diferen ces

among perc ntag e val ue computed from o s rved da t a is x

wh ere n = sampl e size a d x = ^ Xi /k (2 3, p 175 -178) The ot her term s

w ere defin ed previ ousl y

The computed val ue o x

2

may be comp red wit h the t a bula r val ue g i ven

in T ab l e 2 for k — 1 degre es of fre dom (d.f.) If the computed value of

X

2

is l arg er t h a n the tabular v alue core p n i ng to: p rcenti le =

10 — ( chos n s i gni fi canc e l e ve l ), the perc ntag es are s aid to b e si g ni fi

-c antly diferen t;that i s, the samples were drawn fr om diferen t p o p ulatio ns

I fthe computed val ue o x

2

i s s mal le r t h a n the tabular val ue, the s ampl e s

may be consi dered to have come fr om one p pul ati on

An ot h er use o the x

2

test i s to test w h et h er or not the o se rved

per-c ntag e val ue a re si g ni fi cantly d if r ent from a n ar bit r ar y v alue, such a s

5 pe r c nt T he method o f com putation i s the s ame as t h a t g i ven previ ousl y,

ex cept t hat: (1) the fir t w ay of w ritin g the form ula for x

pi = o b s e rve d fr act io f r t he i th sampl e: pi = #*/«; an d

P = Z_,Xi /£ ,ni = averag e fra ct ion fr al samples c o mb i ne d

2 Wh en the sampl e size are equal the formu la reduce to

T ABLE 14.— PERCEN TA G ES SU RVIVING 10