A Statistical Distribution Function of Wide …

1 AStatisticalDistributionFunctionofWideAp plicabilityByWALODDIWEIBULL,lSTOCKHOLM, ,lst'VlD2thisconditionI .FavariableXisattributedtotheindividuals ofapopula~ion, thedistributionfunction(df)ofdenotedF(x) ,maybedefinedasthenumberofallindividuals havinganX~x, ,andthuswehaveP(X~x)=F(x)[1]F(x)=1-e(x-x u)mXI.[51 Anydistributioniunctionmaybewritteninthe form(1p)n=e-nrp(x)..[3]Theonlymeritofthi sdfistobefoundinthefactthatitsimplestmat hematicalexpressionoftheappropriateform, tion[2J, ,inmanycases, ~ ' ,appliedtorealpopulationsfromnaturalbiol ogicalfields, ,itutterlyhopelesstoexpectatheoreticalba sisfordistributionfunctionsofrandomvaria blessuchasstrengthofma-terialsorofmachin epartsorparticlethe"Dl~rtlCJ~~S"flyash,C yrtoideae,orevenadultmales, , [5],hasbeennotonlytopopulations,forwhich itwasoriginallyalsotopopulationsfromwide lydifferentfields,and, ,"~:l,.'trverymuchdoubtsthesenseofoftheb utionfunction,justasisnomeaningincorrect strengthvaluesofanbutalsouponite-</I(X). ]]

2 [2]F(x)=1'Themeritsofthisformulawillbede monstratedonasimpleprob~ ,bytesting,theprobabilityoffailurePatany loadxappliedtoa"single"link,andifwewantt ofindtheprobabilityoffailurePnofa chainconsistingofnlinks,wehavetobaseourd eductionsuponthepropositionthatthechaina sa wholehasfailed, ,theproba-bilityofnonfailureofthechain,( 1Pn), (1-P,J=(1-p) singlelinktakestheformEquation[21,weobta inPn=1 -e-rnp(x)..[4][4]givestheappropriatemath ematicaleX]:) ,or,moregen-erally, , ~~.-+---+ ~1-+---+---f--+---'-----'----t0' ( )ObservedvaluesnYIELDSTRENGTHOFABOFORSST EELT heobservedvaluesareobtainedasroutinetest sofa Boforssteel, a formaloneorrealone,butfactitselfmaybea valuablestimulustoa givesthecurve, ,Xo= ,m= offare13-3= 'QUE~nCYandTheundividedmarkedN1 Itistothatone, , :Xu= ,Xo= ,m= ,9-10,and11-13givesX2= offare73=4,andP= :mp, ,Xo= , ,which,ontheotherhand, (x-xu). +2oftheSizeofCyrtoideaeinAlbatrossOcean, " ,Nature, "AStatisticalCoresFromtheEast164, "Derdauerndundunterbrochenwirkender, ::hlLngaufdesDauerbruchs," !

3 H~:enJfOr8Ch'L~na.(March,1938), + +---~12345678910111213141516"01----+---- +-1----+--+--+----TABLELENGTHOFCYRTOIDEA E(xlengthinmicrons),.. +132135374998218538355718764892861041611 1531158011801119111196811992119981200028 nl+ ,..-------r----.,.. "O' "----"---I--L+-----11---~ +---+-~-i3"-----+--;I--~--I---~+ ~--++---+--4---f---i--+--t---+ 'Component' '0800m ,Nu 30S00m 'log(x-xu) .. ) tmaybeofinteresttohaveexamplesofthisandf orthisreason, ,thesamplingerrorsoftheob-servedvaluesin Table5 havebeeneliminatedalmostentirely(without affectingthefunction), , easytoseethatthematerial,probablynotbein gkilled,. yieldstrengthoflessthan25 ,weobtain14speci-mensoutof20, ,as165/235= ,ofcourse, ,thevalueofX' ,andthedoff9 - 31/,-51/, ,atthefirstlook,theagreementwiththenorma ldistributionseemsverysatisfactory,butth atacloserexaminationshowsasmallnegatives kewnessanda ' :NormaldistributionX'" < ' off12P< ' , Weibullpublished"AStatisticalDistributio nFunctionofWideApplicability"intheASMEJ ournalofAppliedMechanics,Transactionsoft heAmericanSocietyOfMechanicalEngineers,S eptember1951, ,TransactionsoftheAmericanSocietyOfMecha nicalEngineers,June1952, '.

4 1:Theauthorshouldbecongratulated(orhavin gde-visedadistributionfunctionoftrulywid eapplicability, )'concerned~itbtheproblemsofparticle-siz edistributioninaerosols, ' \" 'sdistributionfunctionitisnecessarytodet erminetheparametersx.,Xo, , ' :appliedtoanUOknOVl' 'sfunctionifitaapplicationoouldresultina savingofex ' ,2,and3inthepaper,appeartorepresenttheeq uation(r-ro).P-l-enratberthanCouldthatbe Ii,misprint?Thevaluesforlog( ) "inthesecondex-wnpJe(l! ), ,wouldtheauthorkindl:,'flhowflcorrectedf igure? 'strealmentisdefinitelya(:on- " \ , , ,.ember,19b1,iseueoftheJO'tl'RNALOFApJ' , , 'AeeociatePro(e8llOrorEnpneeriD&Research ,ThePennlJ)' , , ,thereasonforintroducingtheminimumvalueI ",andignoringthemaximum:r", (llatestotheoriginalapplications,", ", ,onewouldexpecttbemaximumparticletobemor etangible,andalsomoresignifi-cantpractic ally, :ximumandaminimum"alueofxwillbringEqUAti on[5 Jintotheform-'(~)'F(x)=l- ~whichwillngllinredu('(> ,andtotheRosin-Rammlertypeof('Quation4as x" $ ' ;yVanUven'andmorerecentlybyMugeleandEvan s.)))))]

5 ' :~ ,whenthesearecorrected, " ,theea!le~thwhichitcanbeappliedtostudyin gthesizeeffect, 'l-tributionimpartially,however, ,x". , (lJDifI'erentiationgivesthefrequencydill iributionf_dF_"!.(x_-x.). [_(x_-x.) ~~~Thenthmomentahout. ""'-/.'(x-x.)'fdx-/.'(x-x.),"!.(x-x.) ":..X" [ -(X:. [ "FeiohcituodBtrukturdesKohlenstAuba," .HAmmier, :urY," ,1927, "Ske"FrequencyCurvcs," , , "Drople:.SiMDistributionioSpraYlI," :\{ ,IndlUtriolandKnqinurinqCh~ ,1951, 'eMOror:'dechanicalEngineerina:,Muaachul lettaInstituteofTechcology, ,1952' ~,(x. -r(1+2(mJ-P(I+I(m)181 ACTHOR'SCLOSURET heauthorappreciat-esthecommentsmadebythe dh, :J; ' ,buttheprocc< ,Le.,byputtingz=(x-x)lu, :Uldelimi-natingtwooftheparameters,forin stance,.r" !-I,v!.-(2a)-"-'(a)+,,-(a)I';a!wherea= {ordifferentvaluesofa, ardizcdGaussiandistribution, (P,z)onthispaper,it iseasytodecidewbetberthedis tributionissimpleorcomplexandtoestimate, ,thevalueof0'.Astothethirdquestion, (x-x,,) ~mentionedthatthex-valuesaremid-pointval uesandshouldcorrectlyhavebeenincreasedby 1 "=30~ ) geleisa , ~possibilityofcomputingtheparametersandh asmentionedit(withsomedifferentnotationf orthcgammafunction) ,however, complex, ,ofcourse,notvery83tisfactory,buta simpleelectroniccomputingmachine,recentl yeompleted,facilitatestheotherwi,<:;eted iouscornpuw.)))))]]]}}

6 '00.'02-LO0'.020T7&~"':l<o"'-['F:::f--~ ,Ii, 'lUf(>oftheskewness('anbpobtained0'3:= '/z.' , ~,Pot/x.,!,and1l1' :Ji<.5 Since~isafunctionofmonly, ,thatis,thesquareofthestandarddeviation,oftheexperimentaldatais known,therelationEcanbesolvedforXo'Finally,therelation~,' ( (x,-r(1+I(m)[9121J.:=-x.,2[f(J+21m)-rt(I+11m)] [f(I+31m)-3r(I+21m)r(l+11m)+2r~(1+l/m)]16 JThisintegralcanbeexpressedinterms,oftheGammafunction1J.,,'=x/'r(1+nlm)15 JThesecondandthirdmomentsaboutthempanare1'0,.'=xo"fo'" (-71)dJ1:I4] ,preferablysystem-atic,shouldbefollowedinthecaseofa"complex" , ' ,itwouldbeinterestingtoseewhethertheotherdataontheST-37steelreportedby~ "ThePhenomenonofRuptureinSolids," , , \'LShaffer,publishedintheDecember, 'ApPLn:nM! , !AS.'>istantProfe&5orofMechanicalEngineering,U nh'ersityofIIIinoi6,Urbana.)))))]