Bsi bs en 61649 2008

BS EN 61649:2008 BSI British Standards Weibull analysis NO COPYING WITHOUT BSI PERMISSION EXCEPT AS PERMITTED BY COPYRIGHT LAW raising standards worldwide Copyright European Committee for Electrotechnical Standardization ELEC ™ BS EN 61 649:2008 BRITISH STANDARD National foreword This British Standard is the UK implementation of EN 61 649:2008 It is identical to IEC 61 649:2008 It supersedes BS IEC 61 649:1 997 which is withdrawn The UK participation in its preparation was entrusted by Technical Committee DS/1 , Dependability and terotechnology, to Subcommittee DS/1 /1 , Dependability A list of organizations represented on this committee can be obtained on request to its secretary This publication does not purport to include all the necessary provisions of a contract Users are responsible for its correct application © BSI 2009 ISBN 978 580 54368 ICS 03.1 20.01 ; 03.1 20.30 Compliance with a British Standard cannot confer immunity from legal obligations This British Standard was published under the authority of the Standards Policy and Strategy Committee on February 2009 Amendments issued since publication Amd No Copyright European Committee for Electrotechnical Standardization ELEC Date Text affected EUROPEAN STANDARD NORME EUROPÉENNE EUROPÄISCHE NORM EN 61 649 BS EN 61 649:2008 November 2008 ICS 03.1 20.01 ; 03.1 20.30 English version Weibull analysis (IEC 61 649:2008) Analyse de Weibull (CEI 61 649:2008) Weibull-Analyse (IEC 61 649:2008) This European Standard was approved by CENELEC on 2008-1 0-01 CENELEC members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European Standard the status of a national standard without any alteration Up-to-date lists and bibliographical references concerning such national standards may be obtained on application to the Central Secretariat or to any CENELEC member This European Standard exists in three official versions (English, French, German) A version in any other language made by translation under the responsibility of a CENELEC member into its own language and notified to the Central Secretariat has the same status as the official versions CENELEC members are the national electrotechnical committees of Austria, Belgium, Bulgaria, Cyprus, the Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, the Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and the United Kingdom CENELEC European Committee for Electrotechnical Standardization Comité Européen de Normalisation Electrotechnique Europäisches Komitee für Elektrotechnische Normung Central Secretariat: rue de Stassart 35, B - 050 Brussels © 2008 CENELEC - All rights of exploitation in any form and by any means reserved worldwide for CENELEC members Ref No EN 61 649:2008 E Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 EN 61 649:2008 –2– Foreword The text of document 56/1 269/FDIS, future edition of IEC 61 649, prepared by IEC TC 56, Dependability, was submitted to the IEC-CENELEC parallel vote and was approved by CENELEC as EN 61 649 on 2008-1 0-01 The following dates were fixed: – latest date by which the EN has to be implemented at national level by publication of an identical national standard or by endorsement (dop) – latest date by which the national standards conflicting with the EN have to be withdrawn (dow) 201 -1 0-01 2009-07-01 Annex ZA has been added by CENELEC Endorsement notice The text of the International Standard IEC 61 649:2008 was approved by CENELEC as a European Standard without any modification In the official version, for Bibliography, the following notes have to be added for the standards indicated: IEC 60300-1 NOTE Harmonized as EN 60300-1 :2003 (not modified) IEC 60300-2 NOTE Harmonized as EN 60300-2:2004 (not modified) IEC 60300-3-1 NOTE Harmonized as EN 60300-3-1 :2004 (not modified) IEC 60300-3-2 NOTE Harmonized as EN 60300-3-2:2005 (not modified) IEC 60300-3-4 NOTE Harmonized as EN 60300-3-4:2008 (not modified) IEC 61 703 NOTE Harmonized as EN 61 703:2002 (not modified) Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 EN 61 649:2008 –3– Annex ZA (normative) Normative references to international publications with their corresponding European publications The following referenced documents are indispensable for the application of this document For dated references, only the edition cited applies For undated references, the latest edition of the referenced document (including any amendments) applies NOTE When an international publication has been modified by common modifications, indicated by (mod), the relevant EN/HD applies Publication IEC 60050-1 91 Year 990 IEC 60300-3-5 2001 IEC 61 81 0-2 - 1) ISO 2854 976 ISO 3534-1 2006 1) Un dated reference 2) Valid edition at date of issue Copyright European Committee for Electrotechnical Standardization ELEC Title International Electrotechnical Vocabulary (IEV) Chapter 91 : Dependability and quality of service Dependability management Part 3-5: Application guide - Reliability test conditions and statistical test principles Electromechanical elementary relays Part 2: Reliability Statistical interpretation of data - Techniques of estimation and tests relating to means and variances Statistics - Vocabulary and symbols Part : General statistical terms and terms used in probability EN/HD - Year - - - EN 61 81 0-2 2005 2) - - - - BS EN 61 649:2008 –2– 61 649 © I EC: 2008 CON TENTS I N TRODU CTI ON Scope N orm ati ve references Term s, defi n iti ons, abbrevi ations an d sym bols Term s and d efi n i ti ons Abbrevi ati ons 3 Sym bols Appl icati on of the techn i q ues 1 The Weibu l l d istribu tion 1 The two-param eter Weibu ll distri bu tion 1 The three-param eter Weibu ll d istributi on Data consi derations Data types Tim e to first fail ure Material ch aracteristics and th e Weibu ll d istri bution Sam ple size Censored an d suspend ed d ata Graph ical m ethods an d g ood ness-of-fi t Overview H ow to m ake the probabil ity pl ot Rankin g 2 The Weibu l l probabil i ty pl ot Dealin g wi th suspensi ons or censored d ata Probabil i ty plotti n g 7 Ch eckin g the fit 7 H azard plotti n g I n terpreti n g th e Weibu l l probabi li ty pl ot The bathtub curve General < – I m pli es earl y fai lu res β β β 8 8 = – I m pl i es constan t i n stantaneous fai l ure rate 20 > – I m pl i es wear-out 20 U nkn own Weibu ll m od es m ay be "m asked" 20 Sm al l sam ples 21 Outli ers 22 I n terpretation of non-l in ear plots 22 Distribu ti ons oth er than the Weibu ll 25 Data inconsistenci es an d m ultim od e fai l ures 25 Com pu tati on al m ethod s an d g ood n ess-of-fit 25 I n trod ucti on 25 Assum ptions and d iti ons 26 Lim itations an d accuracy 26 I n put an d ou tpu t d ata 26 Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 61 649 © I EC: 2008 9 –3– Good n ess-of-fi t test 27 MLE – poi nt estim ates of the d istribu ti on param eters β an d η 27 Poi n t estim ate of th e m ean tim e to fai lure 28 Poi n t estim ate of th e fractil e (1 %) of th e tim e to fail ure 28 9 Poi n t estim ate of th e rel i abi l ity at tim e t ( t ≤ T) 28 Software program s 28 Confid ence intervals 28 1 I n terval estim ation of β 28 I n terval estim ation of η 29 M RR Beta-bin om ial boun ds 30 Fisher's M atrix boun d s 30 Lower confi dence l im it for B 31 Lower confidence l im it for R 31 1 Com parison of m ed ian rank regression (MRR) an d m axim um likel ih ood (MLE) estim ation m eth od s 31 1 Graph ical d ispl ay 31 1 B l ife estim ates som etim es kn own as B or L percenti l es 31 1 Sm all sam ples 32 1 Shape param eter β 32 1 Confi d ence i ntervals 32 1 Sin g l e fail ure 32 1 Mathem atical ri gor 32 1 Presen tation of results 32 WeiBayes approach 33 Descri ption 33 2 Method 33 WeiBayes wi th ou t fai l ures 33 WeiBayes wi th fai lures 33 WeiBayes case stu d y 34 Sud d en d eath m eth od 35 Oth er d istributi ons 37 Ann ex A (inform ati ve) Exam pl es and case stud i es 38 Ann ex B (inform ative) Exam pl e of com putati ons 40 Ann ex C (inform ative) M ed i an rank tables 42 Ann ex D (n orm ati ve) Statistical Tables 47 Ann ex E (inform ati ve) Spreadsheet exam pl e 48 Ann ex F (i nform ative) Exam ple of Weibu ll probabi l ity paper 55 Ann ex G (i nform ati ve) M ixtures of several fai l ure m odes 56 Ann ex H (i nform ative) Three-param eter Weibu l l exam pl e 59 Ann ex I (i nform ati ve) Constructi n g Weibu l l paper 61 Ann ex J (i nform ative) Techn ical backgroun d an d references 64 Bibli ograph y 67 Fi gu re – Th e PDF sh apes of th e Weibu l l fam il y for ? = , Fi gu re – Total test tim e (in m in utes) Fi gu re – Typi cal bath tu b curve for an item Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 –4– 61 649 © I EC: 2008 Fi gu re – Weibu l l fai l ure m od es m ay be “m asked ” 21 Fi gu re – Sam ple size: 21 Fi gu re – Sam ple size: 00 22 Fi gu re – An exam ple showi n g lack of fit with a two-param eter Weibu l l d istri buti on 23 Fi gu re – Th e sam e d ata plotted wi th a three-param eter Weibu ll distri bu tion sh ows a good fit with m onths offset (locati on – 2, 99 m on ths) 24 Fi gu re – Exam pl e of estim ati ng t0 by eye 25 Fi gu re – N ew com pressor d esi g n WeiBayes versus old d esi g n 35 Fi gu re A – M ain oil pu m p low tim es 38 Fi gu re A – Au gm enter pum p bearin g fail ure 39 Fi gu re A – Steep β valu es h id e problem s 39 Fi gu re B – Pl ot of com pu tations 41 Fi gu re E – Weibu ll plot for g raphical anal ysis 49 Fi gu re E – Weibu l l pl ot of censored d ata 51 Fi gu re E – Cum ul ati ve hazard plot for d ata of Table E 52 Fi gu re E – Cum ul ative hazard pl ots for Table E 54 Fi gu re H – Steel-fracture tou gh n ess – Curved d ata 59 Figu re H – t0 im proves th e fit of Fi g ure H d ata 60 Tabl e – G u id ance for u si ng th is I n tern ati onal Stan d ard 1 Tabl e – Ranked fl are failure rivet d ata Tabl e – Adj usted ranks for suspen ded or censored data Table – Subgrou p si ze to estim ate tim e to X % fail ures usi ng th e sud d en death m ethod 36 Tabl e – Ch ain d ata: cycl es to fai lure 36 Tabl e B – Tim es to fai l ure 40 Tabl e B – Summ ary of resu lts 41 Tabl e D – Val u es of th e g amm a function 47 Tabl e D – Fracti les of the norm al d istri bu tion 47 Tabl e E – Practical anal ysis exam pl e 48 Tabl e E – Spreadsheet set-u p for anal ysis of censored d ata 50 Tabl e E – Exam pl e of Weibull an al ysis for suspen d ed data 50 Table E – Exam pl e of Spreadsh eet appl ication for censored d ata 51 Tabl e Table Tabl e Tabl e E – Exam pl e spread sheet 52 E – A relay d ata provi d ed by I SO/TC94 an d H azard anal ysis for fail ure m od e 53 I – Construction of ord i nate ( Y) 62 I – Constructi on of abscissa ( t) 62 Tabl e I – Content of d ata en tered into a spreadsheet 62 Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 61 649 © I EC: 2008 –7– I N TRODUCTI ON The Weibu l l distribu tion is used to m odel data reg ard less of wh eth er th e fai lure rate is i ncreasing , d ecreasi ng or constant The Weibu l l d i stri bution is fl exibl e an d ad aptable to a wi d e ran g e of d ata The tim e to fai l ure, cycles to fail u re, m i l eag e to fai l ure, m ech an ical stress or sim il ar tin u ous param eters n eed to be record ed for all i tem s A l ife d istri bu tion can be m odell ed even if n ot all the item s have fai led Gu id ance is gi ven on h ow to perform an an al ysis usin g a spreadsh eet program G u i d ance is also g i ven on h ow to an al yse d ifferen t fai l ure m odes separatel y an d i den tify a possible weak popu lation U sin g th e three-param eter Weibu l l d istri buti on can g i ve inform ati on on tim e to first fai l ure or m in im um en d urance i n the sam pl e Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 –8– 61 649 © I EC: 2008 WEIBULL ANALYSIS Scope This I n tern ati onal Stan d ard provi d es m ethod s for an al ysin g data from a Weibul l d istri bu tion usin g conti nu ous param eters such as tim e to fai lure, cycles to fai l ure, m ech an ical stress, etc This stand ard is appl icable wh en ever d ata on stren gth param eters, e g tim es to fai l ure, cycl es, stress, etc are avail able for a rand om sam ple of item s operati ng un d er test di tions or i n-service, for th e purpose of estim ati n g m easures of reli abi lity perform ance of the popu lation from wh ich th ese item s were drawn This stan dard is appl icabl e wh en th e d ata bei ng an al ysed are i n depend entl y, id entical l y d istribu ted Th is sh ou l d eith er be tested or assum ed to be true (see I EC 60300-3-5) I n th is stan d ard , num eri cal m ethods and graphi cal m ethods are d escribed to pl ot d ata, to m ake a goodn ess-of-fit test, to estim ate the param eters of th e two- or three-param eter Weibull d istri buti on an d to plot confi dence l im its Guid ance is g i ven on h ow to in terpret th e plot i n term s of risk as a function of tim e, fai l ure m odes and possi bl e weak popu l ation an d tim e to first fai lure or m i n im um en durance Normative references The fol l owi ng referenced d ocum ents are i n d ispen sabl e for th e appl icati on of th is d ocum ent For d ated references, on l y the ed i ti on ci ted appli es For u n dated references, th e l atest ed i tion of th e referenced d ocum en t (i nclu d ing an y am end m ents) appl ies I EC 60050-1 91 : 990, In tern a tion a l Electrotech n ica l Voca b ula ry – Pa rt 91 : De pe n da b ility a n d qua lity of service I EC 60300-3-5: 2001 , De pe n da b ility ma n a ge me n t – Pa rt 3-5: A p p lica tion guide – Relia b ility test conditions a nd sta tistica l test princip les I EC 61 81 0-2, Electrom ech a n ica l ele m en ta ry re la ys – Pa rt 2: Re lia b ility I SO 2854: 976, Sta tistica l in te rpreta tion of da ta – Te ch n ique s of estima tio ns and tests re la tin g to m ea ns a n d va ria nces I SO 3534-1 : 2006, Sta tistics – Voca b ula ry a n d sym b ols – Pa rt : Ge n era l sta tistica l terms a n d te rms in pro b a b ility Terms, definitions, abbreviations and symbols For the purposes of th i s d ocum ent, th e d efin iti ons, abbrevi ati ons an d sym bols g i ven in I EC 60050-1 91 and I SO 3534-1 appl y, together wi th th e fol l owi ng 3.1 Terms and definitions 3.1 censoring term i natin g a test after eith er a g i ven durati on or a g i ven num ber of fail ures N OTE A test term in ated wh en there are stil l u nfai l ed item s m ay be cal l ed a “cen sored test", an d test tim e d ata from such tests m ay be referred to as “censored d ata” Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 – 56 – 61 649 © I EC: 2008 Annex G (informative) Mixtures of several failure modes G.1 Description A Weibu l l pl ot contai ni n g a d og l eg ben d is a cl ue to the potenti al of mu lti pl e com petiti ve fai l ure m od es An example of this was a probl em in a com pressor start bleed system U pon exam i nation of the data, ou t of fail ures had occurred at one i nstal lation base I t was concl u ded th at th e l ocati on of th is base was tribu tin g to the probl em The base was l ocated on th e ocean an d th e salt air was th e factor Th e data were categori zed i nto separate Weibul l plots wi th th is eng i n eeri n g knowl ed ge The first Weibu l l had a sl ope of 0, 75 This cou ld be consi d ered an i nfant m ortal ity problem , wh il e th e ocean base Weibu l l had a stress corrosi on wear-out fail ure m echanism wi th β = 1 , M ore atten tion to m n ten ance resol ved th e probl em Dog leg Weibu l ls are cau sed by m ixtures of more th an on e fail ure m ode These are usu al l y com peti ti ve fail ure m od es, com peting to prod uce fai l ure H owever th ere are several types of m ixtures d escribed i n th is an nex For i nstance, fuel pum p fail ures can be d ue to bearin gs, housi ng cracks, leaks, etc I f these d ifferen t fail ure m odes are pl otted on one Weibu ll pl ot, on e or m ore d og leg ben ds wi ll resu l t When th is occurs, a cl ose exam i nation of th e fai l ed parts is th e best way to separate the d ata in to d ifferent fail ure m od es I f th is is d one correctl y, separate good Weibul ls wi l l resu lt Th ere can be m ixtures of m odes an d popu l ati ons, perh aps batches and com peti ng fai l ure m od es A steep sl ope fol lowed by a sh al low slope usu al l y i nd icates a batch probl em , as there are som e "perpetu al survi vors" that are n ot subj ect to th e fai l ure m od e For exam pl e, th ere m ay be d efects i n som e, but n ot al l , parts; a batch probl em I t is al ways preferable to separate the fai lure mod es based on anal ysis of th e parts (an d en vironm ent) and to an al yse th em separatel y, rath er th an rel y on statistical m ethods Supposin g a d ata set of 50 parts, and 20 of th em have on e fai l ure m od e an d th e other 30 have a different failure m ode The first set shou ld be an al ysed as 20 fai l ures (of F1 ) an d 30 suspensions (for F2 ) Th e secon d set wou ld be 30 fai l ures (of F2 ) and 20 suspensions (for F1 ) These two sets of param eters can th en be used to pred ict th e cum ul ati ve fai l ure d istribu tion When parts are not avai l abl e for ph ysical an al ysis, th e d ata m ay be spl it in to grou ps based on plotting posi tion Th is can cause errors, becau se a sm all percentage of wear-out fai lures wi l l occur at an "earl y" l ife, and a percentage of i nfan t m ortali ty fai l ures wi l l occur at later l ife A m in im um of 20 fail ures is need ed for cred i ble resu l ts from a m ixture of two fail ure m od es, and 50 or m ore fai l ures for th e oth er m ixtu res The fol lowi ng are bri ef descripti ons of the m ore comm on m eth ods for h and l i ng m ixtures: – p i nd icates th e porti on or batch of th e total popu l ati on th at has a parti cu lar fai l ure d istribu tion ( F1 in th e sim pl e m ixture); – F , F , an d F – R , R a-n d R are th e correspondi n g rel iabi l ity d istri bu tions; i nd icate fai l ure d i stri buti ons; The popu lation cum u lati ve fai l ure d istri bu tions are Copyright European Committee for Electrotechnical Standardization ELEC F and R BS EN 61 649:2008 61 649 © I EC: 2008 – 57 – The descri ptions are g iven with out d escribi n g th e particu l ar d istri buti on shape (e g Weibull , log-n orm al , n orm al, or expon en tial) An appropriate distribu tion sh ape n eed s to be su bstitu ted for each Fn G.2 Competing risk F = – (1 – F1 )(1 – F2 ) (G ) Com petin g risk occurs wh en a populati on h as two or m ore fail ure m odes and th e en tire popu lation is at risk from either fail ure m od e Even thou gh a Weibu l l pl ot of th ese d ata wi l l appear curved, th is is not a m ixture of subpopu l ati ons; it is a hom ogen eou s popu lation I f on e defin es a m ixture to be a m ixture of fai lure m od es, th en th is m od el wou l d be a m ixture as wel l , si nce th ere are two differen t fai l ure m od es N OTE ? Thi s i s sim pl y a seri es rel i abi l i ty probl em : R = R1 * R2 An exam pl e of com petin g risk is an ASI C com pon ent in a plastic encapsu lation wi th m icro BG A sold er connections The ASI C m ay fail throug h crack propagation i n th e sold er bal ls or throug h m oisture pen etrati on th rou gh th e plastic The two fail ure m od es are in depen d en t of each oth er, bu t wi l l com pete i n causin g the ASI C to fai l G.3 Simple mixture F = p F1 + (1 – p ) F2 (G 2) This is a m ixture of two ind epend ent subpopu l ati ons wi th no com m on fai lure m od es Each subpopu l ati on h as its own u ni q ue fai lu re m od es Alth ou g h l isted as a sim ple m ixture, th ere are very few applicati ons th at tru l y fit th is m od el Most m ixtu res have at least on e com m on fail ure m ode Th e sim ple m ixture m ay be used as an approxim ati on for m ore com pl ex d istri butions, su ch as th e com peti ng risk m ixture, d escri bed i n Cl ause G An exam ple cou ld be the sol d er balls of the m icro BGA of th e ASI C Som e of th e sold er balls have one or m ore voi ds A crack wi ll propagate to th e voi d red uci ng the l ife tim e si g n ifican tl y, com pared to th e sol d er bal ls where the crack has to propagate throu gh th e wh ole len gth of th e sol der G.4 Competing risk mixture F = p [1 – (1 –F1 )(1 – F2) ] + (1 – p) F2 (G 3) Most m ixtures of subpopu l ations are com petin g risk m ixtures There is at l east one fai l ure m ode ( F1 ) th at is u n iq u e to one subpopu l ati on , and th ere is a fai lu re mod e ( F2 ) th at is comm on to both su bpopu l ati ons I n th is case, on e subpopu lation is su bj ect to fai l ure m odes and 2, as ind icated by the porti on of th e eq u ati on i n brackets [ ] This su bpopulation by itself has competin g risk As an exam ple, a tyre of a car m ay wobble d u e to bei n g out of rou n d ( F1 ), but th e tyre m ay also g et a pu ncture I n both si tu ati ons, th e tyres are wi th in specification of rou n dness and th e tyre that wobbl ed m ay g et a pu ncture So i t is possi ble to get a pu ncture ( F2 ) on th e way to th e d ealer to h ave the wobbl in g tyre replaced Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 – 58 – 61 649 © I EC: 2008 M ixtures of m ore th an three fai lure m od es wi l l have a better fit, the dog l egs will d isappear an d ? wi l l ten d toward one Thus Weibu l ls for a system or com ponen t wi th m an y m od es m ixed tog ether wi l l ten d toward a ? of on e These Weibu lls sh ou l d n ot be em ployed if there is an y way to categ ori ze th e d ata in to separate, m ore accurate fai l ure m odes U si ng a Weibu ll plot with m ixtures of m an y fai l ure m odes is th e equ ival en t of assum in g the exponen ti al d istri buti on appl ies Expon enti al resu lts are often m islead in g and yet th is is com m on practice Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 61 649 © I EC: 2008 – 59 – Annex H (informative) Three-parameter Weibull example H.1 Example Fi gu re H is a typi cal exam pl e of a three-param eter Weibu ll d istri bu ti on usi ng fracture tou gh n ess of steel plate as th e d ata of in terest The m odel i nd icates it is ph ysicall y im possi bl e to fail th e plate at a l ow l evel of stress (see Fi g ure H for th e effect of th e t0 sh ift) Th ere are m an y possible reasons for an ori g in sh ift The m an ufacturer m ay have pu t time or m ileage on th e system as part of production acceptance, bu t reported th at th e item s are "zero tim e " Th e purpose of prod ucti on acceptance is to elim in ate th e infant m ortal ity failures Electronic com pon ents m ay be su bj ected to burn-in or en vironm en tal stress screening for the sam e purpose I n these cases, th e i tem s have aged before bein g d eli vered as "zero tim e" system s Spare parts such as ru bber, chem icals an d ball bearings m ay ag e in storag e an d use part of th eir l ife on th e sh elf, req u iring a n egati ve t0 For m ateri al properti es, where the Weibul l abscissa is stress or strain , i t m ay be im possibl e for fracture or creep or other properti es to prod uce fai l ure n ear the orig i n on th e scal e 99 90 70 Weibull CDF 50 30 Specimen failures 10 10 00 Fracture toughness 000 IEC 340/08 Figure H.1 – Steel-fracture toughness – Curved data Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 – 60 – 99 61 649 © I EC: 2008 W/rr t0 (1 ) = 77,75 90 70 Weibull CDF 50 30 Specimen failures Specimen failures 10 η β 7,05 ,479 r 0,978 n/s 25/0 0,1 Fracture toughness 00 10 IEC 341 /08 F i g u re H – t0 i m p rove s th e fi t of F i g u re H d ata For th ese reasons an d others, th e Weibu l l plot m ay be curved an d needs an ori g in sh ift, from zero to t0 β and Three param eters, t0 , shown i n Equ ati on (H ): η , are incl u ded i n th e Weibu l l cum ul ati ve d istri bu ti on fu nction as F(t) = − e −(( t − t0 ) / η) β (H ) wh ere t t0 is th e fail ure tim e; is th e starti ng poi nt or ori g in of the d istribu ti on When the t0 correcti on is applied to the data, th e resu ltin g plot wi l l fol low a strai g ht l in e if th e correcti on is appropriate Fig ure H sh ows th e fracture d ata i n Fig u re H wi th th e t0 correction N ote th at the Weibu l l ord inate scal e an d the ch aracteristic l ife are now in th e t0 dom ain To vert back to real tim e, add t0 back Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 61 649 © I EC: 2008 – 61 – Annex I (informative) Constructing Weibull paper I.1 Weibull probability plotting paper Al l probabi l i ty papers have scales that transform the cum u lati ve d istributi on fu ncti on in to a strai ght li n e I f d ata are pl otted on th e transform ed scale an d if they conform to a strai gh t li n e, th en th is su pports th e contenti on th at the distri bu ti on is appropri ate Weibul l g raph paper can be constructed usin g th e transform ation as d escribed i n th e foll owi n g paragraphs The Weibu l l d istribu tion m ay be d efi n ed m athem atical l y as sh own in Eq uati on (I ): = - e - ( /? ) F (t) t ? (I ) wh ere F( t) d efi nes th e cum ul ati ve fraction of item s that wi l l fai l by a tim e t Th e fracti on of item s th at has not fai l ed up to tim e t is – F (t) This can be wri tten as (I 2): 1 − F( t ) = e ( t/? ) β (I 2) Taking n atural l ogarithm s of both sid es twice (d ecim al l ogarithm s can also be used) g i ves an eq u ati on of a strai gh t l i ne, as sh own i n (I 3) below: ? ln ln ? ? ? ? − F( t ) ?? ?? ?? ?? ? ? ? = ? ln( t) − ? ln( n ) (I 3) The eq u ati on above is a strai g ht l i ne of th e form y = m x + c Weibu l l paper is constructed by plotti ng the cum ulati ve probabili ty of fai l ure usi ng a log-log reciprocal scal e against t on a l og scale The slope of th e strai gh t l i ne pl otted i n this m ann er wi l l be the shape param eter, ? , as shown i n (I 4) y = ln ? ? ? ln ) ? ? ? ? ? m x c 1 − F( t ) ?? ?? ?? ?? =β (I 4) =ln( t) =–β · ln( η ) The scal e param eter is then calcu l ated from the intercept (val u e of t = ) as shown i n Eq u ati on (I 5): ?=e Copyright European Committee for Electrotechnical Standardization ELEC − intercept ? y for x = ln( t) = 0, i e for (I 5) BS EN 61 649:2008 – 62 – 61 649 © I EC: 2008 Table I.1 – Construction of ordinate ( ) Y F( ) t lnln(1 /(1 - F( ))) Col Value + 6,91 t 0, 001 0, 01 0, 0, 0, 0, 99 0, 999 2, 31 4, 66 6, 54 7, 74 8, 44 8, 84 –6, 91 –4, 60 –2, 25 –0, 37 0, 83 , 53 , 93 Table I.2 – Construction of abscissa ( ) t ln( ) h t t 0, 69 ,1 , 39 , 61 2, 30 2, 71 3, 00 4, 61 6, 91 10 15 20 00 000 The Weibu ll param eter paper or pl ot β is estim ated by m easuri ng th e sl ope of the lin e on the Weibu ll MRR is th e techn iq ue wh ich com bines th e m ed ian rank as a plotti n g position and th e least squ are regressi on on th e Weibu ll paper as a fitti n g criterion I.2 Using a spreadsheet to construct Weibull plots Weibull an al ysis can be carried out usin g an y com m ercial spreadsheet compu ter software in a m anner sim ilar to th e struction of th e probabi l ity paper H ere, th e paper is n ot prepared for m anual plottin g , bu t th e spreadsh eet graph presen ts th e d ata i n a m an ner appropri ate for determ in ation of Weibu l l param eters using lin ear regression Table I.3 – Content of data entered into a spreadsheet Failure No i Failure time ti t1 t2 t3 t4 i ti n tn Copyright European Committee for Electrotechnical Standardization ELEC Median rank F ( ) ( - 0,3)/( +0,4) i t i yi xi n (1 -0,3)/( n +0,4) (2-0,3)/( n +0,4) (3-0,3)/( n +0,4) (4-0,3)/( n +0,4) ( i-0,3)/( n +0,4) ( n -0,3)/( n +0,4) ln( t1 ) ln( t2) ln( t3) ln( t4) ln( t ) ln( t ) ln{ln[1 /(1 -F ( t)]} ln{ln[1 /(1 -F ( t)]} ln{ln[1 /(1 -F 3( t)]} ln{ln[1 /(1 -F 4( t)]} i ln{ln[1 /(1 -F ( t)]} n ln{ln[1 /(1 -F ( t)]} i n BS EN 61 649:2008 61 649 © I EC: 2008 – 63 – A practical exam ple is sh own i n An nex E I.3 C o m m e rc i a l s o ftw a re Com m ercial software is also avai l abl e for anal ysis an d Weibu ll pl ottin g Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 – 64 – 61 649 © I EC: 2008 Annex J (informative) Technical background and references Ann ex J g ives inform ation on the orig i n of th e proced ures of Clause of th is stan d ard Th e references q u oted are all l isted in Cl ause J J.1 Goodness-of-fit test This is the M an n-Scheu er-Ferti g (1 973) test i n the form presented by Lawl ess (1 982) Th e expected val u es of th e stan dard extrem e val ue order statistics, necessary for th e calcu lation of th e ? i , in 5, h ave been approxim ated as sug g ested by Blom (1 958) Th is test has been shown to h ave power com parabl e to th e Sh apiro an d Brai n test (1 987) and to Tiku's test as described by Lawless (1 982) Th e latter was slig h tl y better th an an y avai lable em pirical d istribu tion fu ncti on tests I n ad d iti on , th e M ann-Scheu er-Ferti g test can deal wi th censored sam ples J.2 Maximum likelihood estimates of ? and ? The equ ations are those comm onl y used for si n g l y censored sam ples At presen t, they are th e m ost wi despread n um erical tech niq u es to obtai n Weibul l param eters Th e form presented i n th is stan d ard is th at of Man n, Sch afer and Sin g pu rwall a (1 974) Si nce th e n um erical proced ure of th is stand ard onl y appl i es to sam ple si zes greater th an 0, th e statistical bias is sm all J.3 Confidence intervals and lower confidence limits The approach ad opted is that of Bain an d En g elh ard t (1 981 ) for com plete sam ples an d Bain an d En gelh ard t (1 986) for censored sam ples These references have coefficients g enerated by Mon te Carlo m eth od s, an d use asym ptotic approxim ati ons to adj ust th e resu lts Som e sim ple l in ear an d non-lin ear functions have been fitted to these tables el im inatin g th e need for auxi l i ary tabl es The d ifferences are, i n all cases, very m inu te (~1 %) An alternati ve wou ld h ave been to use Lawl ess's (1 978) d iti onal m eth ods, but th is approach , althoug h th eoreticall y m ore appeali n g, wou ld h ave led to a m uch m ore com plicated proced ure, requ irin g extensive n um erical i ntegrati on The purel y asym ptotic approach was rej ected because th e proced ure n eeds to be robust for relati vel y sm al l sam pl es J.4 Accuracy of the standardized procedures The proced ures of th is stan dard have been com pared to resu lts publ ish ed usin g sim il ar an d d ifferent techn i q ues Al l th e exam ples anal ysed obtain th e sam e m axim um likeli h ood estim ates as th e procedure of th is stan d ard The on l y d ifferences are in th e confidence i ntervals an d l ower l im its The foll owi n g is a summary of th ese com parisons J.4.1 Bain and Engelhardt (1 986) Since this is th e orig i n of th e stan d ard i zed proced ure, th e n eed to com pare th e results cou ld be q u estion ed The i n terest of th e com parison lies in the accuracy of th e approxim atin g functi ons used i n th is stand ard The com parison is as fol lows: Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 61 649 © I EC: 2008 Bain & Engelhardt Standardized procedure 90 % confi d ence in terval for ? [1 , 34 ; 2, 73] [1 , 34 ; 2, 74] 90 % confi d ence in terval for ? [70, ; 05, 9] [70 ; 08] 0, 801 0, 800 ( = 32, 46) R 0, t J – 65 – Lawless (1 978) The sam ple an al ysed h as 28 failu res for a sam ple size n = 40 Lawl ess on l y g i ves 90 % confid ence intervals for k an d 95 % lower confi d ence lim i ts for B an d for R ( t = e – ) The resu l ts are as foll ows: Lawless Standardized procedure 90 % confi d ence in terval for ? [0, 783 ; , 381 ] [0, 785 ; , 370] 95 % lower confid ence li m it for B 0, 066 0, 074 0, 647 0, 644 ( = 0, 368) R 0, 95 t J 4.3 M eeker and N el son (1 976) This is an asym ptotic techn iq ue Th e exam ple treated is a sam pl e of 96 l ocom oti ves, 37 of th em havin g fai led The censori ng tim e T is sl i gh tl y greater than th e tim e of th e last failure Since th e sam pl e si ze is fairl y larg e, the asym ptoti c approach sh ou l d be accurate in th is case The auth ors on l y g i ve a 95 % confi dence in terval for k and , since th ere is a 95 % confi d ence i nterval for B , we can d eri ve th e 97, % l ower confi dence l im it for th is q u anti ty J 4 Meeker & Nelson Standardized procedure 90 % confi d ence in terval for ? [1 , 72 ; 3, 6] [1 , 61 ; 3, 04] 97,5 % lower confidence limit for B1 55, 54, Guid a (1 985) This paper contains M on te Carlo gen erated tabl es to obtai n exact l ower l im its for th e m axim um likel ih ood estim ates of th e reliabil ity i n sm all censored sam pl es ( n ? 20) Som e ran d om l y g en erated Weibu l l distri bu ted sam ples were u sed to com pare th e lower lim its of th e reliabi li ty calcu lated accord ing to th is stan d ard an d those obtai ned by G u i d a I n all cases, the d ifferences were of th e order of % or l ess J.5 Reference documents BAI N , L J an d EN G ELH ARDT, M (1 981 ), Simple Approximate Distributional Results for Confidence and Tolerance Limits for the Weibull Distribution Based on Maximum Likelihood Estimators , Techn om etri cs, Vol 23, N o , pp 5-20 BAI N , L J an d EN G ELH ARDT, M (1 986), Approximate Distributional Results Based on the Maximum Likelihood Estimators for the Weibull Distribution , J ourn al of Qu al i ty Tech n olog y, Vol 8, N o 3, pp 74-1 81 BLOM, G (1 958), Statistical Estimates and Transformed Beta-Variables, New York, J Wiley & Sons Copyright European Committee for Electrotechnical Standardization ELEC BS EN 61 649:2008 – 66 – GU I DA, M (1 985), of Ma ximum 61 649 © I EC: 2008 On th e Co nfidence L imits for Weib ull Re lia b ility a n d Qua n tiles: Th e Ca se L ike lih ood Estim a tion Eng i neeri ng , Vol 2, pp 21 7-240 from Sma ll Size LAWLESS, J F (1 978), Confide nce Interva l Estima tio n , Tech n om etrics, Vol 20, N o 4, pp 355-368 Ce nsore d Sa mp les , Reliability for th e Weib ull a n d Extre me Va lue Distrib utions LAWLESS, J F (1 982) Sons Statistical Models and Methods for Lifetime Data , New York, J Wiley & M AN N , N R , SCH EU ER, E M and FERTI G, K W (1 973), A Ne w Goodn ess-of-fit test for th e , Com m un Stat , Vol 2, pp 383-400 Two-p aram e ter Weib ull or Extre me Va lue Distrib ution M AN N , N R , SCH AFER, E an d SI N G PU RWALLA, N (1 974), of Re lia b ility a nd L ifetime Da ta , N ew York, J Wiley & Sons M EEKER, W Q an d N ELSON , W (1 976), Meth ods for Sta tistica l A n a lysis Weib ull Perce ntile Estima te s a nd Confide nce , I EEE Trans on Reliability, Vol R-25, L im its from Singly Censored Data by Maximum Likelihood No , pp 20-24 SH API RO, S S and BRAI N , C W (1 987), Statist -Sim ul a , Vol 6, N o , pp 209-21 Copyright European Committee for Electrotechnical Standardization ELEC W-Test for th e Weib ull Distrib ution , Comm un BS EN 61 649:2008 61 649 © I EC: 2008 – 67 – Bibliography [1 ] ABERN ETH Y, R B , “Th e N ew Weibu l l H an d book”, 2003, th ed iti on [2] Defence Stan dard 00-40, “Reli abi li ty and Mai ntai n abi l ity”, 2003 [3] Defence Stand ard 00-971 , “G eneral Specification for Aircraft Gas Turbi ne En g in es”, 987 [4] I EC 60300-1 , Depend abi l ity m an ag em en t – Part : Depend abil i ty m an agem en t system s [5] I EC 60300-2, Depen d abil ity m an ag em en t – Part 2: G ui deli nes for depen d abi l ity m anag em en t [6] I EC 60300-3-1 : 2003, Depen dabi l ity m an ag em en t – Part 3-1 : Appl ication g uid e – Anal ysis tech n i qu es for d epend abili ty – Gu i d e on m ethod olog y [7] I EC 60300-3-2, Depen dabil i ty m an ag em en t – Part 3-2: Applicati on g u id e – Col l ecti on of depen d abi l ity d ata from the fi eld [8] I EC 60300-3-4: 2007, Depen dabi l ity m an ag em en t – Part 3-4: Appl icati on gu id e – Gu i de to th e specification of depen dabi l ity req u irem en ts [9] I EC 60605-4: 2001 , Eq ui pm ent rel iabi l ity testin g – Part 4: Statistical proced ures for exponen ti al d istri bu tion – Poin t estim ates, confi dence i n tervals, predicti on i ntervals an d tol erance intervals [1 0] I EC 60605-6: 2007, Equ i pm ent reli abi l ity testi ng – Part 6: Tests for th e vali d ity an d estim ation of th e constan t fai l ure rate and constan t fai l ure in tensity [1 ] I SO 1 453: 996, Statistical in terpretation of d ata – Tests an d confi d ence in tervals relatin g to proporti ons [1 2] I SO 2854: 976, Statistical i nterpretati on of d ata – Tech n iq ues of estim ati on and tests relatin g to m eans and variances [1 3] J EN SEN , F an d PETERSEN , N E , “Burn-I n”, J ohn Wiley, 982 [1 4] J OH N SON , L G , “The Statistical Treatm ent of Fatigu e Experim ents”, Elsevier, 974 [1 5] J OH N SON , N L , Kotz, S an d BALAKRI SH N AN , Distribu ti ons Volum e ”, n d ed i tion, J oh n Wiley, 994 [1 6] LAWLESS, J F , “Statisti cal Mod els an d M eth ods for Lifetim e Data”, J oh n Wiley & Sons 982 [1 7] M EEKER, W Q and ESCOBAR, L A , “Statistical Methods for Reliability Data”, J oh n Wiley, 998 [1 8] MI SCH KE, C R , “A Distribution-i nd epen dent Pl otti ng Ru l e for Ordered Fail ures”, ASM E Desi gn Eng i neeri ng Tech n ical Conference, 979 [1 9] MU RTH Y, Xi e & J ian g, “ Weibu ll M odels”, J ohn Wiley 2004 [20] N ELSON , W , “Appl i ed Li fe Data An al ysis”, J oh n Wiley, 982 Copyright European Committee for Electrotechnical Standardization ELEC N., “Con ti n u ou s U ni variate BS EN 61 649:2008 – 68 – 61 649 © I EC: 2008 [21 ] N ELSON , W , “Accelerated Testi ng ”, J oh n Wiley, 990 [22] O’ CON N OR, P D T , “Practical Rel i abi l ity En g i neerin g”, J oh n Wiley, 2002 [23] I EC 61 703, Mathem atical expressi ons for rel iabi l ity, avai l abil i ty, m ntai nabi lity an d m ainten ance su pport term s [24] I SO/TR 3425: 2006, Gu id el in es stan dard izati on an d specificati on for the _ Copyright European Committee for Electrotechnical Standardization ELEC selection of statistical m ethods in Copyright European Committee for Electrotechnical Standardization ELEC This page deliberately left blank British Standards Institution (BSI ) BSI is the independent national body responsible for preparing British Standards It presents the UK view on standards in Europe and at the international level It is incorporated by Royal Charter Revisions British Standards are updated by amendment or revision Users of 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