Designing Reliable Products I

: : :* : :;g

Designing reliable products

For many ycars, designers havc applied so called la./o|J o/ ratn in r dctcrministicdsign approach. Thes factors are uscd 1o accountibr unccrtiLintics in thcdesign par'ameters wilh ths aim of generating designs that will ide4lly avoid ldilurc in service.Load and slress concentrations were the unknown conlribuling flictors and this ledto rhc lcfin /aLtor ol ig or.rxx, (Gordon. 199I).Thcltchrol safely. or delenninislicappronch. stili predominrtcs in engineering dcsign cullurc. although the stalislicalnarurc ol engineering problc s has been $ludicd ibr mnny years. Many designs rrcstill based on past expcricnce and intuition, rathcr than on thorough on!lysis andexpcrimcnlat ion (K0lpakj ian, 1995).

In gcneral. the detrministic design approach can bc shown by equalion 4.1. Nolclhrr lhe ' rress rnJ strcngth are in lhe srmc unitr

,s- ,_- L (4.1)

a = loading stress

S: maler i i l s l rength

l.S = facror of safcty.

The factors ol salily were initially determind from the snsilivily experienced mpractice ol n part made from a patlicular marerial to n pirlicular loading situation.and in gcncral the greater thc uncerlainties expericnccd. the greater the faclor ofsafety used (Faires. 1965). Tablc 4.1 shows recommendcd faclor of safety valuespublished 60 ycars apart, frrst by Unwin (..1905) and by Faires (1965). They arevery sinil.tr in nature, and in lact the earlier publishcd values are lower in sonrccases. As cngineers learned more aboul the nalurc of variabiliry in enginecringp amclcrs, but were unable io quaniify thcm srlisfactorily, it seems thal thefaclor of safty was increased 1o accommodalc these uncedainties. It is also notice-ablc that the failure criterion of tensilc fracture sems to have bccn replaced by

ducrile yieldjr

ol y ic ldins

bciDs ppl icdr.rngc lrom Iand McKee.

Figure 4.1differelltly lbprob.rbi l is i ic (

engineering l

1980). There

1986). Thc dr

output. Thc

Probabilistl

(Diter, 1986rs those in tl

Determinhtic versus probabilistic design Ej'l.hh.l,l I rdob ofsxfd! lordtrctilennd brnlte Drai.fii\ rDd rarn)us lordtnscondttrons (ntues stD\n xrbrx.k1|aroni t l )05.s i thoulbrackcrsl ionlg6J)( ' r=utrrm"r"""* ; r . " ,* i r r . .s.r : r" ia. t . ""g,r , r ' - '

Cast ton (brnrL m.tr()

Br\cd on Sf Bns.d o. ,t

RcD.rlc(l onc.lir.crnurmild \hock

Rcl)cf lcl rc!.r:.dnnrld sl!)ck

( t : )

( t0)

( L5)

duct i le ) ' ic ldins (duc ro thc onrissbr of such \ , t ues ( .1905).thc prclerrcd cr i ter ion of ln i turc. This is bccluse in gcneralwrll nol fLrrrction stltishctorily rf-ter pcrmancnl cjefornrationol y ic ldinr.

the yickl slrcss bcingnrost Dachlnc purlscausd by lhc onscl

lhc l i rc lor ol s l fety h d l i11lc scicnr i t ic background. bur h d an undcftyingcnrprr icr l and subjecr ivc nr l lurc. No onc c:rn dispulc lhr l a1 lhe r ime thrr s lc$rrrr lysjs wlrs in i ls inf lncy. this wrs thc bcsr knowtedgc vt l i t r lbte. bul lhcy rc sr ibc'ng nppl icd todlyl l rLrcl()rs ot s l tct) rhl t l l rc rcconlmendcd in rcccnl I i lcf t l tLrrcrrngc l rorrr 1.25 to l0 tar vur ious rrrr lcr i l t lypci l lnd tonding condir ions ( t j ( iwl( tsrn( i McKcc, l9t) l i Hlugcn. I9r0).

. l r igufc4. l .g iv$lnindic l l t ionlhalcoginccrsinrhct95{)swercbcginningtorhir)k( l i l lcrcnt ly lboul dcsign wi lh thc inlrodusl ion ot n .rruc. , r . / r . { , , r , , t " , l l , r .

nrd ^pn)bl lbi l istrc design ppfonch wrs bcing ldvocl l ted. t l shows thl l rhc design probtcm

wns nltlllil:rctofccl Llnd v.rrinbilily brsed. Wirh thc incrc sing use ot iratlsr;cs incDgrneerlDg l l found this l ime, rbc rheodcs ol probrbi l jsr ic design and fcl i rbi l i lywcrc 1o become estnbl ished methods in somc scctors by t l r . t960s.

Thc deterDinistic rptro.rch is nol vcry prccisc and the tendcnc! i\ lo Lrsc it vcr)con-senativel)r fcsuhiDg in ovcfdesigncd cornponerts. high costs anrt sonrcrrrnrxircf fect ivcness (Modl ' :cs. l99l) . Crrrer (19s6) Dolcs rhnr siress fupturc w,rs rcspon\ ible l i ' r a su ic icnr nulnber o|r i tures lbr us ro conclude thrr deieruinist ic designdoes not lw.y\ ensuro iDtrinsic relirhilily, and rh.rt rtxxn for imprtvemcnr sittcxisrs. Incrersing denands for pcdormance, rcsulting otien in operaLion ncar linircoDdit i ( 'ns. has phced increasing emphal is on precision and rert isDr (Haugerr,1980). Thcre bas been i grear disenchrnrment with f.lclors ol safery for nrirnvye rs. rn,r'nly becrusc they disregard rhe t'rct thrr malcrirlproprrios. thedilnensl(n;of thc componenrs and the cxternrlly .rpplicd tolds are srarisiical in naturc (Diclcr.l98at lhe dercrnini\ric approach is. rheretore, nol suirabtc lbr today.s producriwhcrc supcrior f ncliomlity and high customer s,irisfrction should l" u a"rigno lput. Thc need li)r more clicienl, higher performance producrs is encouragiigmorc appl icat ions of probabjt ist ic mcrhods (SDri th, 1995).

Prob bilislic design methods have bcen shown to bc imporlant whcn the designcannoL be-tesrcd ro fiilure 1d when i1 is imporrant ro nininizc weight and,or cosl(Dierer. 1986). In conp.nies whcfe mirriDizing rvcighr is crLrcial. foi cxarnplc suchas thosc in the aerospacc induslry, probrbitisric design techniques can Ue touna.

134 Designing reliable products

Figur 4.1 Th lrue'rnargn ol saletyGdapted lrom Furnrai , 198tandNxon,j958)

although NASA has found rhe dclerministic approach to b adequale for somestnrctural analysis (NASA, t995).

Non-complex and/or ron-critical applications in mechanical dsiAn can also makcuse ol pfobabi l i i ic dcsrgn Lechniques and ius' i ty r more In-deprt i infrorch i f .hcbeneJils are related 1() pracritioners and cusromers alike. Surveys have indicatedthat many products in the indDsrrial sector havc in the last been overdesisned(Kalpaki ian. lsq5r. l har h. lhey$ereerrher loobutky,wer;maleofma,er ia lsroohigh in quality, or were made with unwarranted preision for the inlended use.Overdsign may result frorn uncrtainries in design calculations or the concern ofthe designer and manufacturer ovr product safery in ordr to avoid user iniurv or

death, and c

probabilisliclrade-off bel\everj only li'which facrofrsense (lreud(

that will adeqrelated paran

/(s), and lot '

being rellted

Figurc 4.2 shPProach. N(

designcr usiDwhich would(for argurrcr

ofthe producdegree of int1991\:

. The relatir

. The disper

. The shrpe

the probabiU(Haugen, l9ieng'neenng rutjljzed lo q!

The devel(

methods disc

mcthods is d,aid ol compu

dI

F

3

'-.1

Deterministic versus probabilistic design E5

death, nd can add signific.ntly to the product\ cos1. Achjeving feliabjtity byoverdsign. rhcn, is nor an econorric proposilion (Carter. 1997). The use,,iprobabilistic teduiques could save money in this wry as they provi;c,, b"ri,,];.

"lrade-ofi bet*ecn design ond cost facroLs (Cruse, 1997a). ri ir,"y "*

..p.t.,r. r,,r*evcr. onty th convenhoral dcLerministic dcsjgn approach renrains acsording towhich ficlors ofs. eiy afe setected bascti on cngincJring cxpericnce arld con;onsense (freudenthal ./ d1.. t966).

Tlrc rrndom natrre oflhc propcrties ot cnginccring rn.lierials and ofapptied loadsrs well known to engineers. tnginccrs wilt be familitr wilh thc lypic.rt appcaLlncc ofsels ol strenglh dah fionr lcnsile rcsrs in which nost ot.the daia value.s congrcgaterrcund the nrid-range wirh decrc,rsiDsly lcwer vr ues jn thc uppcr rd low;r iaihon cilher sidc of thc mean. I.or nrathenr.|licat rractability. the erperimcntal rtara\!n hc lnodel l , \ l$ i rh J l , r . ,h:rhrt i r ) DJn, y I rDcr. , , rr r t ,Dt ror, , rrr , rr , . . , , , , t , , t ,11,, , , , , , , ,rhr l$. l l IUequir t ! l )dc\ i f ihelh.pr l rLrnot lhc . r l r , ing iU,t r inf le JqL r i , ,n. ro.r .rcraled pdramcrcr.s. In 10l]l1s ol probabitistic design rhen. the rctiribitity of .r conno-ncnt par l crn bascd on the intcrterencc of i ts inhercnr mrtcr ial srrcngrh


(a) Facio. of salov too hlqh le.din! !o ov.rd4lCn

Strnglh

(s) 4.2.1 Mo

(b) Frcior ol.d.iy t6lry l.ldlng io I hlgh hllur.

POF

(r)from availa

variables f(

distributio!Three sl

Norm$l (s(and 3-paraEach 6gur

and widlhsTypical r

' i Fsyield slr

strcngth

M6.n M6an311633 srfngrh

(c) Faclor or clV .d.qu.i.

Olher distrgiven belo'

ties of all i

. Maximl

. Minimr

. Expone

Lacking n

dislriburio

empidcal (iseisi lytr :

able'l andpoint in llresun mrg

It has b,only poslr

Suon9lh

Figure 4.2 Comparron ol ihe probabiislc and deterrninstic dsign apprcaches

Stathtical methods for probabilistic design l3i

4.2.1 Modelling data using statistical distributions

A kcr problem in probabi l is l ic dcsign is thc genef t ion ol thc st t i l t ical dis(dbul ionsl iom avir i l b le informarion about thc raDdon v.rr iables (Sidd.r l . 1983). Thc randonrv.ri.tblc m{y bc ll sel of real numbers corrcspondinr ro the oulcome of a serics o1expcdmcnrs. 1cs1s. dltn mersurements. ctc. tJsually. irform.rtion rcltlting |o lhcscvrriablcs lbr parlicular design is no1 known beibfehand. Evcn if similar dcsignc.rscs.rrc wcll docuDlented. there rre aiwr]-s p.rfticulor circumslrnccs lflecting rhcdi\ t r ihulr , ' r ) l ln! tr ' r r \ iVinogf.rdo\. l , / t ) l r . /

lhrJ( . r i , r r . r r ( : ,1 Jr ' t r rhr t r , r rs lh. . ' r< ' . , ,nrnr, 'n l \ J.ed i (nf in( . r , r . l |e rhJNo rral (scc Figurc 1.1(a)). I -osnorm l (see l- isurc 4.1(b)) rud WcibLrl l , borh 2,rnd I 'prrrnrclcr (scc Fig rc 4.3(c) f in u represcntr( ion of the 2-par nlcrcr typc).I iach f igure shows thc chur ctcr ist ic shl lpes of lhc distr ibut ions wi(h vulyingprI1rnrctc|s lbf rn r fbi t rury !ur iublc. Thc urea ur i lc l cach dist f ibul()n c sc is:r lways cqual lo uni(y. rcprcscnring lhc l {) t l probrbi l i ry, honle rhc vurying hcighls

Iypicr l Ippl icrt ioDs l i ) r lhc thrcc nrr in dist f ibul ionr huvc beeo ci le( l :

. N,,rrr l Tolcrunccs. ul l imt l te lensi le slrcnglh. uni l lx i i r l y ield slrcnglh nd she fyic ld slrcDglh ol somc mctr l l ic r lby\

. Louon l t l t .onds in cngincering. s irenglh ol srrucrulr l r l loy m ror i ls, [ r l i8 cstreoSlh ol- f ic lals

. t / . l r r l / l . . t l iguc cndunrncc slrength ofrneuls ard r trcngih ofcer lr ln ic mnrcf i l ls.

Olhcr dis lr ibut ions highl ighred as benrg inlport nl in rel i rbi l i ly crginecr ing ure alsogiven bclow. A summury of l lof thesedis l r ibul ion\ intermsof lhcirPI) l : , rot t ionrnd vrrirle bound rics is given in Appendix IX. Thc fc.ider intefesrcd in lhc prope.l ies ofr l l the distr ibul()Ds mcnt ioned is referred to Bury (1999).

. Mrixinum Bxtreme Valuc Typc I

. Mininum Extreme ValLlc Typc I

. Ixponertirl.

Lacking nore detiiled inlirrnralioD rcgnrding the naturc ol an engineering fandomlllrirble. it is ofren rssLrncd th.rt its distfibution can bc rcpfesented by r Normaldislr ibut ion (Rice. 1997). (The Nonnal distr ibut ion was ini t i . r l l r - discussed in dclr i lin Appendix t.) The Normal is the n]osl widely used of all dislribuiions and throughcmpirical evidence nrovidcs a good rcpfesentntion oimany cngineering variablcs nndi! easily rflctable mathemitically ( H.t ugen. I 980). lf the N orm al disiri bx tion docs n o rprove to be a good fit 1o thc da1a. thc quesrion should be rskcd, ls the dara dcpcndrblcl' and if ii is. Are there good rea$ns lbr using a differenl model!'There is nopoint in litting a more sophisticrted model ro uDtruslworthtr drra because the endrecu r Inrehl f r"rL ru be,pecrrcJLr non,en,e

ll has been argucd thal marerial properries such as tho ulrimate rensile slrcngth ha\,eonly positive valucs and so the Normal distribution cannoi be ihe rmc distfiburion.


f igure 4.3 Shdo, or the Drobdol t r 'oer5 t ) ' | , .01,pDF, 'o. t r d,1o.% b toqlo.m3 d1o , ,WFbLoFrouors\4Lr vd1r1q oard^ p e1 ddaotedIonaa1er. 'q801

coefficicnt ,negativc vrLogno|n]alnegati!c. I

propcrtis rdislribuliolwhcn the L

Data th!such as thc

(Rice. 199Normal dr

3-paramcl(in modcl l idiscussed i

Thc pricdistribulio

t ions. Hov1973). l f 1dsta is the

Appendixln order

I (r). ^n

lhe Norn

(Murty.rn

approximUllman

NormaltlanYlhing

Statistical methods for probabilistic design

beouse its range within the statistical model is from -.! to +oo. However. when thecoemcient of variation, a'",


limils are -r

to -,

which produces a conservative estinate for the failure prob_ability when used to model stress .lnd strcngth conpared ro orhcr djstrib;tions(Haugen, 1982a). Real-lifc disiribulions havc iinire upper and lower timits, thoughpreciscly wherc these arc localed may bc diflicull to dercrmine. Iror practicalpurposcs. the Normol distribution is lrsefut within rhe range ot.rhree slandard dcvia_tions above rnd bclow ihe lnean. Predictions b.sed on cxtrapotltion ot.lhe Nornalbeyond thrce slandard devialions musl be regarded as suspecl, ancl lhis wilt ati.lctthe rliability prediction rnrdcr some circumslitnces. The Nornial strcss Norn]alslrenglh distributions give thc largesl prcdicied failure probabitity tbr the shtic toading condilions. The Norma I dislribu tion. rhereforc, has an elemenl ofconscrvatisnr inbcing unbounded. and distriburions such s thc 3-parameter Wcibu .iisrfibutionmdy be bctLcr for prcdicling higher reliabilities (Haugen, 1980). A key prcblem isth t if thc usc ofincorrcct dislributions w:rs made tbf stress or strcnqth. ih; nrcdiclertr( l i i rbr l i r ) InJ) make n(,n\en.e nt rc l i rhi t i r ) t rgers rhrough \ \hich r retr :rht ! Jenpncould be idenl i l ied.

Sonre imporunt considcrations iD rhe use ol statisrical dislributions have becn high_lightcd. both iD terfts ol lhe initi{l data and. more imporlanlly. whcn nrocjellin,r the.tr ! , , : rnd srrcnelh for dLlcrminrng the rehrhrLl) . Slr f , r Srrcnglh Inrcr l . , : rcncc iSSlrnnalysis. which is the nl.rin lechniquc used in this conneclion. will be discusscd lIlter.

4.2.2 Fitting distributions to data

Thc est intal ion of thc mcan and standard deviat ion using rhe nomcnl eqult ions rsdcscribed in Appendix I gives Iirllc indicalion of thc degrec of tir ot. rhe disrributionto the set ol cxperi cnlal dala. We will ncxt develop the concepts l.rom whrch rnv((,nr inuous dr.rr ibul ion (r tn be rnoJel led l ( , r ser , , r ' (ht i r . Thi, ul , imrleiv nru\ ide",rhc nr, ,sr surr iblc q.ry. t dct(rmrnrng lhe d' . l r ihur ior) .r l prr rnclcrr .

Thc merhod ccnrres arcun(l rhe &nul. ive lrcltulr(r,of the cxDerincnhl data. Aryprcr l (urnu ,rrr \c l iequrni) r lor f r .m rn rrhirrury .er , , t r t i , i r t ic Jarr r . shown rnFigure 4.4. The horizont1l

^xir;s rhe ind?rndent rrlnrrle. being rhc discrerc varirblc

vnlue or the mid-class for lhe grouped data. The cuDrulativc frequcncy on thc verticalaxis is gcDcrared by adding subsqunr liequencics and is regardcd as rhe ./qri,rl.trlt'uiuhl". The original hislogram is shown superimposecl undcf the cumularivc

By plottirg rhe culnulative frequency as .t rlative percentage ofthc rotal frcquencyol the dal we gel Figure 4.5. (Altrnalivly, ihc cumnlitive trcquency can bedisplnyed from 0 ro l.) Overlaid on the rops ofthe liequency b.rrs is a curvc thatreprcsents thc curnuhtivc function. This curve is dr.iwn by hand in thc fisurc. butt l . ( us< of n, , l lnomi,r i cune hlr ints io \ , rrc $i l l yr( lJ morc uccur: Ic , . ,utr , fn,typc oi graph and its varianis are used to dclcrmine the paramerers ot.anv dislribu_r"n I-or e\Jmple. I rom rhe poinr. on rhe y-a\rs c, ,r fe\ponJrne r^ rhe perc(nlr te

points -84.1% and -15.9% on thc cumultr l ive f tequency (rhc percenlage piob_abilities at +l of the Standard Nomlal variate. ,. from raUte r in ,qpnenalix it thc\r lnd,r 'd ( le\ i ] | ron ean be e,rrmared J, .hoqn, 'n I igur( 4.r . th. , 5U nrr fenr i te , , .the variale determincs the median value oflhe dala. which for a strnmetrical disrribul ion r .u(h as rhe \ofml l) c,rnddes \ i rh rhe medn.

Figure 4.4

l inr i ts 0 |

thc NornCDF equ

dclinite l1

In real i

rigure 4.5

Statistical methods for probabilistic design t4j

malFectm.rl

n in

rcn

gh,lhcSIr

!=20

10

I rot80

E50!210

220ito

0

Varlebtetigure 4.4 Crmuatve freqrencyand hnoqram lofqroupd daa set

Values lbr thc Cumuhrive Dislr ibur ion funct idr (CDF). givcn the notal l (D F(.y).:trc generaled by integrating thc pDF tbr rhc distribution in,lucstion betwccn rtrcl i ln i ls 0 and the vrr i tc of inrcrcsr .r (sce Appendix tX). The vr lue ot. l - ( . r) rhenrctr . :senls thc fr i lurc Frohirbr l i l ) . P. rr rhut n, , inr. Fi lurc 4.r , sh,,wr rhc \hrne. l,hc Norn)r l CDF $irh,t i t i ( renr srJnd.trL. l Ltevi .rr . ,ns foi . rn rr t , i r r , , ry,- , , i , , r . i . . i i1r"CDF cquat ions for al l but the Normrl "rnd Lognonnal dirr l ibut i ;ns nre s; iJ i ;he fi (lo:tul lom, mcaning the equrtion c0n bc m,lthcmatic.rlly ,r,aniputrteJ ,,"'adefinitc.function rarher than nn inlegml. Ahhough n umericrl' rcctrnii ues can leused to inlegrJte rhc Nonnrl Jnd L,,Bn,)rm:r l pDI { l ( , ubl f l in rhe (unr; l l rv. vr luet,r IntcrNt. th{ Lun)utr t i \ r S\D \ lu. \ . r . pro\ idcJ i . l . ,h l , r I in AppcnJr\ I rresolnmonly used for convenience.

_ In real i ty. i r is impossibl lo know the exacl cuDrulal ive fai lure dislr ibut ion ol .thc random vari ble, bccausR wc are raking only rchrivcty small samples ot.tbe

Vartabts

Figure 4.5 Cumulariv freqLrenq ptorand detrmnalof ofmaf and n ndard deviarion grdphicaiy

inlerl

]

I

I


t=A0+At(x) (4.3)

200

tigure 4.7 Di

A reliabli

the line on

st ol drtato thc Stan

' ln ' as lypi(

X togetherlrnear rgrc

The pra(

rectificrtiolMcKe (19(1997). Mi l

Srraighifoolproof r1973). Ad(

Mischke. 1

grouping Inonlineari

Figur 4.5 Sh.pe oi th C Lrmllative Dist but on Fun( on (C OF) lor an ab tlary ior.nal d stribution withvaryng slandad devation (adaoted from Caner. j 986)

population. We observe that 1001, of the sample taken has failcd. but this failuredistribution does not hecessariiy match that of the entire population. Wc can makereasonuble judgemenrs as to $h.rt rhe nopularion cumulative ldilure di,rtrjburionplorling posilions dre rhrough lhe use ofempiricall) b.rsed cumuliri!e f,rrk,x! (drJ-r/rr.{. By ranking the cumularive frequency on the verrical axis culculrred fromone ofth mfiny different lypes ofranking cquatjon, an improved model for the curnulativefunction can be generated. See Appendix X for a list of the most commonly usedranking equations and thc types ofdistribution they are typically appljed to.

An elficientway ofusingallthe information alldjudgement avail&bic in the estima-lion ofthe distribution parametrs is the use of the .linearized' oumulative frequercyrSidd.r l . I983 r. Essenl ial ly. lh i , involves convef l ing the non-l inear equ. ion describinerhc cumllari!cjrequency rnto a linear onc by suirably changing one or more of rhiaxrs vaflahrer. I ne malhematrcat process is called linctt r(.!inntltrn A

Statifiical mthodt tor probabilistia design 143

crudr.nt . -i . ^i

\

rnrr.cept on {

0

tlgor 4.7 Determination otthe iinear reges5ron eqlatioo mdnla y

ytiit'JFl[iil I'J,: il:.:llli,iti;#,,"""TJnit:i,iiij'i :lfixffiH:t:LTlljt_"b"_:" and below .rhe. iine). The derermination "f

th" li;;-;;;;;

illiHTliii::i: ff ',"'#Io"';;liX:r*"raaient nt rt* tine ana inieicepi-oiior example. to dercrmine whether lhe Normal distriburion is an adequute fit to a

i:';: itil:H''i:l*:i1ili :::i:if::"#?:::n *i::*l*,lmiil*!cumulative frequency is converted as for the Norru aruil"iil"'anJ ,i" ""riiiilon.the x-axis is converted to the Natural Logarithmic value (give;;;;;;;;;;;

i"$:'-.T:li1#::r,i*:?:t"h:Tl;i#t*:ffi:*".'.:;t;*l*'lT"x;x together vdrh rhe equations for determining tr," ai""itrii",ip"i".",".. i:"i"'iriilinear regression constants A0 and Al,The.practical utiljzation of linear rectification is demonstrated later through aworked ex4mple. Fitting statistical distribu

;""'*l;r*m*f i:1r"i#q:t:"C:li;;'i:'i:JH:'ri,rll?). yi,.r,r,. rrcs:,. Rao rree2r. and srrigrelf,l:i'Jliii;.til;;Jit*t t"*"". strargnl lrne plot6 ol the cumularive function are commonly used. but rhere is noii?I:T:ff '.1;1 [xfill.il];; l:::Tfi .:l;*'ulniliryxl*:*;ff ,: Kf::,:;:; "#: i?1.11.,t'f:r:f,il#i:'"il"." u:l:;iix';1.i. i:iirerres on grouping rhe data. Thi\ is because of rhe neea ro comnor. rt . eri ;rr larJi:l"',:,1:il"'"x,ff*ffi :jjf,i.'ii."i",il*1T1'",.,1j1;tr.f:*.x*:frf : j;:ilrfi:::l*jrjrs*.","#g*#jlii;"#:t1.r#,.::*iHTtr:


.= +0.5 of ihe 3 pdistibuti(modelledtion ofth(

initially. Ingalnsr lnreasonablto speeo r

Detennhand usir

examples

Furlhcr

uncertainlthe data. lcontrol thsome inexare closroata anal:(re96), N,

ExamplrWc wi l lnefitling a Nin ihe forstrength I(1997a), a

analysis b

the metho

distributicquency vain Table 4Nole thatJ-axis, F,,in MPa. 1

The estimodelled

Flgure 4,8 Co[e at of coeffkint, r,lor severalreationships between x and y variab es

An alternative method is to fit the .best'straight line rhrough the linearized set ofdata associated with distributional models, for exomple th Normsl and 3-parameterWeibull distributions, and then calculate the (o/f? lation tnellicicht, r, tor e^ch lLipsonand Sheth, 1973). The correlation coemcient is a measure of the degree of (li;ear)association between two variables, Jr and ),, as given by equation 4.4.

D'v-14.4)

1{r = number ofdala pairs.

A correlation cofrcient of I indicates that theIe is a very strong association btweenthe two variables as shown in Figure 4.8. Lower values of ,r' indicate that thevaiables have less ofan association; until at | = 0, no correlation between the variablesis evident. A negativ value indicales an inverse relalionshiD. Therefore. the maximumvalue of correlation coefrcient for each linear rectilicrtion model will give mostappropriate distribution that fits the sample data.

The value of the correlati on coemcient usingtheleast squares techniqueand the useof goodness-of-fit tests (in the nonlinear domain) together prcbably provid themenns to determine which distribution is tle most appropriare (Kececioglu, t99l).However. a more intuitive assessment about rhe nature of the data must also bemade when selecting the correct t)?e oI distribution. for example when there islikely to he a zero threshold.

Having introduced t]le concepts of tbe correlarion coelicient. it becomesstraightlbrwnrd to explain the more involved process ofdetermining the parameters

("-?) (rn-ry)

Statisti(al methods for probabilistic design 145

ofthe 3-parameter Weibull distriburion. The procedure for the 3_paramerer Weibulldistribxtion is more complex, as you would expect, due to the distribui;n ;;i;;modeued b) rhree rarher rhan rwu paramerers. Esent ial ty i r requires t i ; ; ; i . ; ; , r" :nonorrhexnecledmrnjmum\dluc. . (d.dlocattonpar. lmeteronther-axis.Arsho\anIn AppenJix X. t ie t inear recl i f icrr ion equ r ion\ rr . . , fun"r,on of fnr . ,o, * f , . r ..Y,, / r- , . . lhe mrnrmum variable value on lhe d.r la. U e don\ know the \ atue of r ,in iLialb. hul b) rearchinS for d !Jtue of ro,uch lhdr \ahcn ln(r . \n, i . Dlorre. lsain* Inln, t ' t - F.

. . rhe correlar ion coemcicnr is rrs highesi i r r" . . . * i r r"J. .re:rsonirbiy lccurate irn\wer. The process can be cu:il1 rranriared ,o.nrpu,.i.oa.to sf,eed up the Drocss,

. Determining thc purrmeter\ for Lhe common distr ihur ions can t lso be done b\hand using,ui tahty scatcd probabit i ry ptoLLing nuper. , , l r" i ; ; l i l ; , ; ; " ; ; ; , ; :odra pornts. herng determined .by eyc. ar descr ihed earher. See Le$is (199;, forexamlles ofprobability ptotting graph paper for sorn. oi rt

" .tuti.t;"1 airtriOuiion,

Funhcr improvemcnt in the selcction ofthe best linear rectification modcl can beperforftcd by comparing the uncerLarnry in modcl fi, ",,1".rnri".;

i;;lh; ,;;;;c//,r ot the dara {Netson. 1982). Al50. rhe use of.oniden.? ii,zjr.( in derermininc rheuncertainty in rh estimares from linear regression is u.eful tor assessins the nJt;eoiIne odla..part,cutdrly \4 hen small samples are taken and or uhen outlyi-ng data poinLsconrrol the gradrenr ot-rhe regression llne. Confidence limits are generalll widei thanjome inexperienced dar{ analysrs expect. so rhey hap *oia tf,iiling if,'"r.rt.oil,rre ctoser lo lne lrue-value than rh) reall) are. A discussion of theii aDDllcalion ino"Ia analysr\ can be iound in Ayyuh rnd Mccuen (i997). Comer and Kjerengrroen(1996), Nelson (1982), and Rice d997).

-E-la??le - fifting a Normal distribution to a set of existina datawe witl nexr demondrate rhe use ofthe Iine3r recrin"rrion ..tr,nJo.i"rilia il"i" Lfiuing a Normat disrribution to a set ofexperiment"r a"tu. rr,f a.iiio l.-"-nr'iiJi'"

rn lne lorm ot I hjs logram given in Figure 4.9. l t show< the di .rr ihuLion oi v iel , lrrren-gtn tor_a cold drawn cirrbon srcel iSAL. l0lg). The data rs raken lrom ASMI tyy /4,. s reterence lhat provides data in lhe form ofhistograms for several lmDortJnLme(hanical propenies of steels. Dsra collaied in this minner hu. b.."

";;'r;;-i;;analysis because a designer may have to resort to the use of data from existinseources, Also. the anlr l ls is of lht l crse rai .er some inlerest inA qu..r ,on, wtr,ct _uil : ,1.::.t i t. irI

h..1n.1 lhen anaryc,ng d,,ra co dr.a ,n p,,cri i.. ana a;.pr,y.J,.in!rne methods descflbed in ADncndix I_ Afrer a visual inspection. i; is evident thar the SAE l0l8 yield srrength data has adi, t r ihur ion cpproaching lhe Normdt type. akhougtr Ltrere;, ,n atnormatt l t r ;etr i re_quenc). \a luerroDnJLhemid-rdngeofthedala. fu i , f , . r .ono5. i , ,of , t .au,J", i r ,o,n

'^n l l i , ]e 4:: usrnC the cumutalrre trequenct mode rns approach yr(tds rrsure 4.t0.

Nole lhal lhe n. , / rr l equat ion i , u\ed ro derermine the ptol l ing posirro-n, on rbel-a_xis, I'iJhe jr-axh plouing positions beins rhe mia-"rr* unir* r.r1i," yi"tJ ,t*.gtr,in MPa. The class vridth w = lj.s Mpa. The values determined g*phl"Iv t". ir,.mean and , landard deviat ion are al ,o sho\an

The.enimared (umular i te hequency f i rs lhe dala wel l . \ hcfe in facr rhe curve r.mooe ed wrth a htrh order polynomial using commcrciat (urve f iLr inc software.

46 Derigning reliable pfoducts

Yl6ld str.ngth, Sy (MPa)

Figure4.9YedltrngthhisloqramlorSAEl0lScoddrawncarbonneelbar(A5N4, 1997i)

(Qrnmercial sofiware such as MS Exccl is useful in this connection bein-q widelyavailable.) Omissions in thc mnkcd values of 4 in Trble 4.2 reflect the omissionsol thc data in the originAl histogram for scvcral clnsses. As can be judged fromFig rc 4.10, inclusion of the cunrulat ivc prob bi l i t ies f t )r thes chsses woukl nollbl low lhe narur lr l prr tcrn ol rhc dislr ibut ion iDd irrc therefore omirted. Howe!er.whcn r very k)w oumbef ol c lniscs cxist thcir inclusi()n cnn be. iust i l ied.

Line f rect i l i f l l ion ol lhc cumulat ive l iequency. 1,. is perfonned by convering iothc Slundrd Nomul vnr ialc. : . The l inedr plot bgether with the sl l1r i8hl l incequrlion th )ugh thc dnta nnd lhe cofrehlion collicient, r. is slx)wn in I'igurc4.11. l : rom fr igurc 4. I l . i t is cvident lhr l ihe mean is 530 MPr bccausc lhe rcgressionline crosses the Stlndard Norntl variate.:. ul 0 representing the 50 percentilc ormedirn in the non-lincarizcd domrio. The melln.rnd siandnrd devialior cI|n alsobe found liun thc rclniionships given in Appcndix X. Iror thc Norm.rl dislibLrtioD

Tlble 4.2 AMltsis ot [islosrrm dnln litr SAF ll)Ll] t. obtnin drc Nornul dhlribulion plortnrs posilnxrs

F =. i -

1-

-" ; ]

085t108l

075+i 07 |5 065 t

E ol i l " 045 |g 04 +E 035 I,3 or l

0.25 I02 | -015 i

,il 1o1*

Figure 4.10 Cj

-_(

( , ! :52)

Thc conclusi

4.10 and 4.1

the Normalthe original

ing the disll

using the rgivcn in this

distributiondisiributiona

4lt 0

458 6

500.0

527 6541.4555 2

5U2 ll

6212618.0

ll2

t l5

5

I

I It520t l38

5l5l5ti:

0.1I l2 l0.1509,10.20755

0.62264

0.8l l t20.99566

103

l.2t

0. l l0. l l0 5ll0.88t l2

2 olt

E

I

10950.9

0.850.8

o.T50.f

0.650.6

0.50450.t

0.3503

0,250.2

0.1501

0.05o

Statistical methods for probabilisti( design 117

a@ 410 420 alo 440 450 !60 470 a30 a90 50 510 5!O 5!O iO 350 560 SIO 530 590 600 610 .20 6rc 6{0yttd skocth (Mpa)

tlgure 4,10 CumulativeireqoencydistrtbutioniorsAEtOlsyieldstrengthdata

we can calculate the mean and standard deviation froml

, = - ( +\= - f lS) = 53o 14MPa' \At / \ 0.022 ,/ -- ' '

"= r I+)_r,{0) _ r '+ '196r)_r l+s) =4s.45Mpa\ At ) \At / \ 0.022 / \ v.uz_ /Th conclusion is that the Normal distribution is an adequate fit to the SAE l0lgdata, A summary ol th Normal distribution parameters calculated from Figures4.10 and 4.ll and other values for the mean and statrdard deviation from vaiioussources (including commercial software and a package developed at Hull Unive$itycalled FastFiuer^) arc Eiven in Table 4.3. The frequency distributions derived fromthe Normal distribution pammeteN from source are shown graphicauy overlayingthe or iginal histogram in Figure 4.12 for comparison.

Itcan be seen fromTable4.3 that there is no positive orfoolproofway ofdetermin-ing the distribulional parameters useful in probabilistic design, although the linearrectifrcation method is an emcienr approach (Siddal, t983). The choii of rankineequation csn also affect the accuracy of the calculated distribution parameter:usirg the methods described. Reference should be made to the guid;nce notesgiven in this respecl.

The abole procs\ above could also be prformed for rbe l_parameler Weibulldrstnbulton to compare Lhe correlarion coemcienls and delermine rhe be er firtrngdistributional model. Computer-basd rechniques have been dvised as part of thiapproach to support businesses attempting to determine the characterizins distributions

'Thc aar,Itlpf softw,rc is alaiLble arom the authors on roucsr

472 - 492da_*t toMPa


Trble 43 Nomal dhllibrrion Daranele4 aor SAE l0l8 tiom various sources

Rcfcrcne (Mischke. 1992)

Norbal linedr rectilicationcumulative. fieq. Sral'n

541512517510534545

531

t8

a

Ifrom sample data. As shown in Figure 4.13, the users screen fiom the software, calledaarlF tlef, is the selection ofthe best fitting PDF and its parameters representing thesample data, here for the yield slrength data for SAE l0l8. The software selects thebest distribution from the seven common types: Normal, Lognormal, 2-parameterWeibull. 3-parameter weibull, Maximum Extreme value Typ l, Minimum ExtremeValue Tlpe I and the Exponential distribution. Using the FarrFrrl software, it isfound that the 3-parameter Weibull distribution gives the highest correlationcoemcient of all the models, at / = 0.995, compared to | = 0.991 for the Normaldistribution. The mean and standard deviation in Table 4.3 fot FaslFittenrecalculated from the Weibull parametrs, the relevant information is provided inAppendix tX.

4.2.3 The algebra of random variables

Typically. if the stress ordistribution, it is likely to

strenglh has not been laken direclly from the measuredbe a combination of random variables, For example. a

t lgur 4,12 I

failure goveor three-dirmathematicparameters

probabilisti,

applicationvnriables arengineering

Engrneeri

(Haugen. lrore vnriabl

-9_.!

i

qq

Yi'd strength (l,lPa)

Figure 4. l l Normadistr ibunonl inearrect i f ical ionlorSAEl0l8yiedstrenqthdat

y=0.022r-11-663r = 0.991

5l,/tb 30 q0 550

E

E

Statistical methods for probabilistic design 149

Yl.rd 6lrn0th (MPa)

Figu.e 4.12 Normal distribLrtions Jrcm var ous soulces for SAE 1 01 8 y eld nngth data

failure governing stress is a function of the applied load variation and maybe two-or three-dimensional variables bounding th geometry of the ptoblem. Themathematical manipulation of the failure governing equations and distributionalpanmeters ofthe raDdom variables used to detcrmine the loading stress in particularare complex, and require that we introduce a new allebft c lled lhe algebn of rundom

We need this special algebra to operate or the engineering equations as part ofprobabilisiic design, for example the bending stress cquation, because the parametersare random variables ofa distributional naturc ralherthan uniquevalues Whn theserandom variables are mathematically manipulated, the result of th operalion isanother random variabl. The algebra has been almost enlirely developed with theapplication of the Normal distribution, because numerous functions of randomvariables xre normally distributed or are approximately normally distributed inengineering (Haugen, 1980).

Enginering variables are found 10 be either statistically irdependenl or correlatedin some way.ln engineering problems,lhe variables ar usually found to beunrelated,for instanc a dimensional variable is not statistically related to a material strength(Haugen, 1980). Table 4.4 shows some common algebraic functions, typically withone variable, jr, or two statistically independent random variables, r and ). Themean and standard deviation of the functions are given in terms of the algbra of

150 Detigning rellable Productt

Flgur 4. | 3 FarFl[er anal],sis ol SAE 1 018 yreld svength data

random variablcs. Where the varieblcs n and ) are corelated in some way, withcorrclation coefrcient, r, several commofl functions have also tleen included

\ry'hen a function, d, is a combination of two or more statistically independentvariables, x/, thcn equation 4.5 can be effctivcly used to delgrmine tleir combinedvarianc, Zd (Mischke, 1980).

'01 (4.t

To detemine the mean value, p, of the function 0;, rCl

. , . i ,",, . . . .r,".) + iI# ." i , (4.6)

Equation 4.5 is exact for linea! functions, but should only be applied to non-linearfunctions if the random variables have a coeffcicnt of variation' C'

stati*ical mthods for probabilirti( design

Trble 4,4 MeM and srandard deviation of statistically indepmdcnr and co.Elaled randon vanabl$ J andr for sme comon fuctions

151

Standald d*iatioi (dc)

I

") .*

d+roi p.

ptr +a,4 t

n- - pl,-

l t^ n, t l, t6\ d, \pr r4 \/! p!/

" , / ,9r t , r / "" f ]-lK t'- '6\t/

l

I f, *o/g.\'l

l;, + "I,'ld i+oi +zr.r , .d!) t '

(p,".s, + i4.4, +4,."if1

llpl,.4 + p',.5,+ 4.4) + lt + if!

r /&.4 + pi .J" \ 'n\-Fa-)h t" l d i - " , .o, \" '. ' \ -+--4.ni , )

2",.r,.l, * 0.:s(rr) ]

,"".d,[,-("J]

,"..,1.1'-;("J]

, .* 4 ' . l r*o:r f , , f ( - ) ]i,l.[r+o.s,r"- rrtr]']

,e'l' i(^')]

^.1.["i]1f,*r / l ) ' l

*[ . , (^ ' ) ]

(4.8)

Equation 4.? is referred 6 as the 'rariance equation and is commonly used in erroranalysis (Fraser and Miln, 1990), variational dsigl (Morrison, 1998). reliability

p0 ". 6(p,,,

p,., , p,.)


analysis (Haugen, 1980.) and sensitiyity an.lysis (P.rry-Jones, 1999). Most imporlantly in probabilistic design, through thc usc of the variancc cquation wc havc -means ofrelating geometric decisions to rcliability goals by including thc dimcnsionaland load random variables in failure governing stress equations lo determine thestress random variable for nny givcn ptoblem.

The variance equnlion can be solved direclly by using the {dLutus of PartD./lydtlvcr. or lbr more complex cases. using the ftxit? Dilfr&k Metho.l. Anorhervaluable method for'solving' lh variance equation is Moxk,(:dth Simuldtion.Howovcr, ralhcr than solve lhe variance equation directly, it allows us to simulatetho oulpul of rhe variance tor a given function of m.ny random variables. AppendxXl explains in dctail each oflh methods to solve the variance equalion and provides$,orkcd cx^mples.

Thc vrrirnce for any sel of data can be cnlculated without reference to the priordistribution as discussed in Appendix L 11 iblbwli thal thc variance cquntion isalso independenl of a prior distribulion. Hcrc il is assumed that in {ll thc cascs thcourpur function is adequately rcprcscntcd by lhc Normrl distribulion whcn thcrandom variables involved arc rll rcprcscntcd by thc Normsl dislribution. Thcassumption that the output function is robustly Normal in all c8scs does not stricllyrpply. particularly when variablcs arc in ccrhin combinrtbn or whcn thc Lognornraldistr ibut ion is used. See Haugcn ( l9l i0). Shigley and Mischke (1996) nnd Siddal(l9li3) for guidAnce on using thc vrrisncc equation.

The variance cquarion provides n valuable tool with which ro draw sensitivilyinferences to givc rhc conlribution of eoch variable to the overall variability of theproblem. Through its use. probabilistic methods provide a more elTective way lodetermine ky design pnramelers for an optimal solution (Comer and Kjerngtroen.1996). From this nnd other information in Pareto Chart form, the designer canquickly lbcus on thc dominant variables. See Appendix Xl for a worked exompleol scnsilivily analysis in delermining the v.rrirnce contribulion of each ol thcdcsign lariables in a stress analysis problem.

Figul 4. l r

Imporl

collcclion

temperatcitd that

. Matcr

. Dimcr

. Geom

. Servic

4.3.1 1\

Design nodels must account for v,trinbilily in the mosl imForlanl dcsign varjables(Cruse, 1997b). lf an adequalc characterization of these important variables isperforned. this will givc a cosl-effective and a fajrly accDrare solution for mostengineering problcns. Thc main engineering random variables that must be ade-quarely dcsc bcd using the probabilistic approrch are shown in Figure 4.14. Thevariables convcnienlly divide into two rypes: d.rrg}r ./.p.ndrrl. which th designerhas rhc grcalcst control ovet, dnd setrirc &p?nd?rr, which the design has'limited'coDrrol ovcr. Typically. the most important desigr dperdert variables are materialstrength and dimensionrl variabiLity. Material strength can be statislically modelledfrom sample dria for the properly required, as previously demonslratcd; howcvcr.dimculties xist in rh collation ol informalion aboul the properties of interest.Dimensional variabiliry and its cflccls on the stress acting on a component can begrear, but information is typically lacking about its statistical nature and its impacton geometric stress concertration values is rarely assessed.

The larg

strongth (s)

Variables in probabilistic design 153

N Enur,oiF"n,r

/ - - . ' . ^- -- . --- \nn

Opelallng Envlonmental Olmenslonat Strossvrriation concsntration

!_1- -,-w--*.. --l \ _-_Savtc. d.pend.nt D..iCn oepenoenr

Str... (a)

Figure 4.14 Key va abes n a probabiistic design appoacn

lmportanr service dependenr varidbles are related lo rhe loading of rhe componentand stresses resulring from environmenral effecls. These

".. 6.;^it;jm:;il;;determine at the design stage because of the cost of performing cxperimental Jaiacollection. the narure of overloading and abu.e in seruice. an'd ,fr. fr.[ "i

J"i"aDout servlce toads jn gener{|, Also. the effect thal servjce conditions have on thematerial properties is impo ant, the most important considerations arisinc fro;extremes in tcmperalure, as there is a tendency towsrds trittle fracture -at fowremperatures. and creep ruplure at high tcmperatures, To this end, il has beenc,ted lhat lhe qual i r) controlofthe environment is much more important ihan qual i lycontrol of the manufacudng processes in achieving rrigtr relauiiity (cartei, rlsij.'

Among the most dramatic modifiers encountered it! design oi tirose mentionealbove are due to thermal effecrs on strength and stress conccntrarion effects onrocar \tres-r magnirude{ in general tHaugen. 1980). A5 seen from Figure 4.14, thererre se\erdt rmporlanl design dependenl !ar iablcs thal lerms ofan engineering anulyr is

. Material strcnglh (with temperature and residual processilg effects included)

. DiInensionalvariabiliry

. Geometric stress concentrations

. Service ionds.

4.3.1 Material strength

The largest design dependenttensile strcngth (S!r), uniaxial

slrength variable is material strength, eilher urumareyield strength (S)), shearjleTdsrr-rigrh (r") or some

Designing reliable products

other lailure resisting prope(y. For dflection and inslability problems. the Modulusof Elasticity (r') is usually of interesl. Shear yicld strength, typically usd h torsioncalculations. is a linear function of th uniaxial yield strength and is likelv to have

thc same distdbulion type (Haugn. 1980)With mass produced products, exlensivc testing car be carrid out to characlerize

the proprty ofintrest. Wbn prodllction is small. material lesting mav be limited tosimple tension tests or pcrhaps none at all (Ayyub and Mccuen. 1997) Matrialproperties are often nol available wilh a su{ficient Dumbet of test rcpctitions toprovide slalislical relevance, and remain one ofthc challenges ofgreatcr applicationof statisticrl methods. lbr example in aircraft design (Smilh, 1995). Anothr problen

is how closc laboralory tcsl results arc 10 that of the malerial provided to the customer(Welling and Lynch, i9lJ5), becausc material propcrties tend to vary fronr lot to lotand manufacturer to manufacturcr (lreson {/ d/. 1996). However' this can all beregardcd &s making the case for a probabilistic spproach ldeally. information onmaterinl Dropeflies should come from test spcimens that closely resemble lhedesign configuration and size. and tested undcr condilions thdt dupucate theexpectdservice conditions as closely as possible (Bury. 1975). Thc more information we

havc about a situation before thc trial takes place and the data collecled, thc more

conlidence there will be in the linal result (Lcitcb, 1995)Onc of the majff reasons why design should be based on stalistics is that material

proterties vary so widely, and any general theory of reliability must takc lhis into

lccount (Haugcn and Wirsching, 1975) Material proprties exhibit v.triability

because of anisolropy and inhomogencily, imperfection, impurities and defects(Bury. 1975). All matrials are. of coursc. processed in some way so lhat they arin some uselul fabrication condition. The level of variabilily in matcrial propenies

associatd wilh the level ol procssiDg c^n also be 4 major contribution. There Arethree main kinds of randomness in malctial properties that are observed (Bolotin'

t994):

. Within specimen inh{:rcnt wlrhin thc microstructure and causd by imperfec-tions. flaws. ctc,

. Specimn 1() specimen caused by the instabililies and imperfeclions of th

manufacturing processcs with the batch.. Batch lo batch nrlural variations due to processing. such as material quality.

equipmenl. operator, melhod. selup and the environment

Olhcr uncertainlies associalcd with marcrial properties are due to humidity and

ambint chemicals and thc elTects of time and corrosion (Farag. 1997; Haugen.1982b). Briltle marerials are aflected additionally by the presence of imperfections.cracks and inrernal flaws. which create stress raisers For exarnple, cast materialssuch as grey cast iron arc brittle due to the graphite llakes in the matenal causmg

internal stress raisers. Their low tnsile slrength is due to these flaws which act asnuclei for srack formation when in tcnsile loading (Norton, 1996). Subsequentlv.brittle m.lerials terd 10 have a large variation in strength, somelrmes many trmcsthat oi ductil matcrials.

Strain rate also allbcts tensile properties al test An increasing strain rate lcnds to

increase teNile properties such as S and S). However, a high loadirg rrlc tends to

Dromote briltl fraclure (Juvimll, 1967). The average strain rale used in obtaining a

It h,ts

Thc rla(Timoshr

I

is a func

A highWcibull

detaild

nrs

Variables in probabilistic design 155

stress slrain diagram is approximatly l0 I ms/m. and this should be kepr in mindwhen.performing experimental tesling of materials (Shigley, 1986).

It has been showr rhat tie ultimate rensile strengih, Sr, tor brittle materialsdpends upon the size ofthe specimen and will decrease wittr increasing aimensions,srnce the probabilily ofhaving wak spoh is increased. This is tcrmed ihe size cffect.Thh 'size elTect' was invesrigated by Weibul ( l95l.t who suggested a staristicat iunc_tron, the Weibull distribution. describing the number and distribution ofthese flaws.The relationship below models the size clfect for delerministic values of Sa(Timoshenko. 1966).

vlI

brb.

bed

hllortylsre

!t.

he

v.

rdn.

Istglsv.

Su1 \rrl(4.e)

Srr = ultimate tensile strength of rest specimen

,9r2 = ultimate tcnsile strength ofcomponnr

rt = effective volume of test specimen

t,2 = eflbctive volume ofcomponent design

ii = shape parameter from Wejbull analysis of lest specimen data.

As can be sen from the above equation. for brittle materials like glass and ccramics,we can sc-ale_the strength for a proposed design from a test specimen analysis, ln arnore useful form for the 2-parameter Weibull disrriburion. rh; probability ;ffail;r;is a function of the applied stress, a.

(-(*))* ' (4.r0)

p = probability of failure

l- = stress applied to component

d = chancteristic value,

and for the 3-paramter Wibull distriburion:

"='*'c(#)')"'"(4. l r ).r(, : expected minimum value.

A high shape factor in thc 2-parameter model suggests less srrengrh variability. TheWeibutl modl can also be used to model ductilc marerials oi to* t".p"*ru.""\ hrch erhrbi l br i le fai lurf {Faires. t965,. rsee Waterman .rnd A,hbt f lc 'qt t fof ;delrr led dr\cussron on modei l ing br i l l le mater ial str fn! th.)

156 Designing reliable Products

Several researchers and organizations over th last 50 years have accumulaled

slatistical material property data Howver, property data is still not available for

many materiah or is not made generally avnilable by th companies manul-acturingthe stock product. This is a problem if you want to design with a spelific material

in a specific environment. For example, it is not adequate just to say thc slatisticalproperty of one parlicular steel is going to be close to that of another similar steel.

Approximate values for the meafl and standard devialion of the ultimate tensile

strength ofsteel. t!, can be found from a hardness test in Brinnel Hardness (HB)

From mpirical investigation (Shigley and Mischke. 1989):

or, = (3.45':.o'1Hs + 0.152']. p'?Hs + 0.lsz'?. ofrs)or

Unfortunally, the statistical datn in refrences such (ASM, 1997a; ASM' 1997bi

Haugen. 1980i Mischke, 1992) is the best available to the designer who requlres

rabid solutions. An example of such data was shown in Figure 4.9. Although theproprty data strictly applies to US grade fcrrous and non-ferrous matcrials, conver-

aion tablcs are avail8ble which show equivalent material grades for UK, F-rench.

German. Swedish and Japanese grades. However, their casual use could make thc

answers obtained misreprscnhtive of the problem They should be treatd with

csution ns a dircct comparison is questionable beolLuse of small deviations in com-positions and prccessing parametrs. Malerial properties for UK grade materials in

statisrical form would be ldvantageous when using probabilistic design techniques

Howcver, there 8re no immcdiate plrns by the British Slandards Insliture (BSI) toproduce materials property dala in a statistioal formai' and all data currentlypublished is based on valucs (tes ronrrz , 1998)

Table 4.5 shows the coefficient ofvnriation, C". for various mnterial properties at

room temperature compiled from a number of sources (Bury. 1975; Haugen. 19801

Haugen and Wirsohing. 1975; Rao. 1992; shigley and Mischke, 1996; Yokobori'

1965).Furftcr insight into the statistical strcngth propertics of some commonly used

mtals is provided by a data shet in Table 4.6. Agrin caution should be exercised

in thir use. but rfcrenc will be made 1() som of thsc values in lhc probabilistic

design casc studies al the nd of this sction.

Trhle 4.5 Typical cochcinl ofvarirtion, a,, for vanous nalcrials dnd hechani'al prope ics

{Ji, = ultinltc tc.sile stnngrh)

ri, of wroughi iro.=0.04S, ol briltle roteills:0.1

Fractur touehnBs ((") ol nctauic uterills = 0 07Mod. of Elasticity {t) .l noduh sl iron=0 04Mod. of Eusncity (r) or titdiun:0 09Mod. oaELsliciry (r) ofaluniniun =0 03

Yield slrenglb (5}) :005 io 0.08F.ndua.e limil (.t ) :008Bineu Hlrdnc$ (HB):0.05Mod. ol Elasticity (E):0.01 ro 003Mod. of tugdi1y (d) =002 to 004Fraclut touetnes ('(c):0.05 to 0.1Poisson s ratio (,, - 0.02 to 0.26

\4.t2)

(4.r 3)

BS 220M07

tls 0?0Mt0

s E l0 l8(BS 080A ti

sAE t0t5(BS 080A1:

sAE 1045(BS 080M4

(lls lll ?M.t

BS Grrdc 4

BS: 65t6

70?5-T6

Finall)calcul{tethe minil

for thc I

Tablc 4.

The valu

the prob

Variabler in probabilirtic design 157

Trble 4.6 Material sirenslh data sheel

(MPa)

Frcc cutting carbon steelas 220M0?

BS 070M20

sAE l0l8(BS 0{r0Al7)

sAE lot5(8S080432)

sAE 1045(BS 080M46)

sAE 4341)(BS 817M40)

BS Cdde 4lC

ns 3 t6sr6

70?5.T6

QS tqnn

8^r 900

21

40

5t1

506

594

8t2

555

36

27

20

27.5

26

540

342

658

803

324

22

50

Finally, ir is worlh investigarilg how deterministic values ofmaterial strength arec{lculdted as commonl} found in engincering dalu bookr. Equation 4.l4 slir;s rhatlhe mrnlmum maler ial strenglh. s ' n. as used in derermlnist ic calculat lons. equalsthe mean value deiermined from test, minus three standard deviations, cslculatedfor the Normal disrriburion (Cable and Viren. 1967)l

Smin:p-30 (4.14)

For example. the deterministic value for the yield strength, Sl, for SAE l0lg colddrawn steel for rhe r ize range rested is approxim{tely l95Mpa (Green. 19921.raDre 4.6 gves the mean und nandard de\ ist ion as Sj ! N{540.4l lVpa . l -helower bound value as used in deierrninistic design becomesi

S/Dri : 540 - 3(41) = 417MPa

The vnlues are within 5% ofeach other. lfdeterministic values are actually calculatdat thc negative 3d linir fron the mean, 1350 failures in every million could beexpected for an applied strss of the same magnitude as determined from SNDthory. It is evjdent from this thar reliabiliry predicrion and determinisric desilnare nol compatible. becau\e as tbe lacror ofsafe(y is inrroJuced ro reduce tai lur is,the probability aspect of the calculation is lost. (Note tha( rhe ASTM srandard onmaterials testing suggests setting the minimum material property at _2.33d fromthe mean value (Shigley and Mischke, 1989).)

158 Designing Ieliabl Produds

Material proPerties and temperatureA number;fba;ic material properties useful in static design depend most notably on

lemperature(Haugen. 1980r ' Fore\anple Fi tsure4 15'howshowhigh rempetururer

al tcr Lhe rmpon.rnimechdnic lnroperlre 'ofalosc rhon steel dndthe\Jr ial i \ )nlhat

"un t"

"*p"ti"n""a. Tcmperature dependent materiak properties are somclimes

avaltable ln statistical form, as shown in Figure 4 16 where the 3-parameter Weibull

distribution is used to model lhe tcnsile strngth of an alloy steel over a range ol

r"rnoerrrures tL,nson . , d1, lsnt) Thir Iypc of intbrmal ion is r l r ict l ) for hlgh tem'

neruture work *herc the.rppl ical 'on of lhc load lasts appror imalely l5Io 20 minulcs

iTimoshenko. 1966).ExDeriments al high temperstures slso show that tensile tcsts depend ontheduration

ofthe test. because is tim; increases the load neccssary to produce fracture decreases

(Timoshenko, 1 966). This is the onset of thc phenomenon known as cr"4? All materials

bgin to lose strength at some tmprature, and asthe tmperature incrcases' the delor-

ma'tinns cear. to 6c ehstic and become more and more plastic in nature Given suffi'

cient time. lhe material may fail by creep usually occurring at a lemperature belween

:O ana +OZ of lhe mlting temperature in dogrees Kelvin (Ashby and Jones 1989)

In carbon sleels. forexample, design stresses can be solely basd on short-term proper'

lies up to an operaling temprature ofabout 400"C. while at temPeratures greater lhan

this. ;reep behaviour is likely to overrule any other design considerations

creD slresses used for dsign purposes are usually determined based on two

"rit".iui ttt" .t.".. fot a given acceprable creep deformation after a certain number

ofhours. which ranges from O.0l to I % deformation in 1000 bours; and the nominal

e6

8:

6S

Elasticily(GPa)

3loo3Igro

Figlr4.15 :et.1,1967)

stress rquilife. Th creis evident tlture, and irmucn grearcited for rlstatistical (

(1997a), AlMany m

S'r, and ler

150

100

o 1oo 2oo 300 400 500

remP6raure ('c)

Figure 4.1 5 Mechan .41 propedies ol a low carbon neel as a iunclion of temperature (adapied from Water_

man and Ashby, 1991)

Variables in p.obabilistic design 159

t '? lku

Tads t ircngthhiqh-alroY n*r s, . 65 k.i

eile! ot lenperaru4

40

J00

tlgure4.l6 shofiern iens le strength We bu t parametrs lor af at oy steet at va oustmperatures(Lipsoneta|. .1961\

stressrequircd to produce rupture ajlcr a specified rime or at the end ol the requircdlilb. The creep rupture stress lbr several stcels at 1000 hours is shown in Fisur 4.17. fti ' c\ ident Ihar a hrge var ial ion er isrs in lh{ rupture srrc. , r atuer tbr r grvin rempera-ture. and in general crep malerial properties lerd to hnve a coelicient ofvariarionmuch greater than staric propcrties (Bury, 1974). For cxample, C' :0.7 has beencited for the creep time to fraclure for copper (yokobori, 1965). Alrhoush littlesldlrst i (al dala has been found on rhe nrL'penies h'ghl ighred. creep.rr .ngih drruand proprties at high temperarures for various matcrials can be found in ASM(1997a), ASM (1997b). Furnan (1980) and Warerman and Ashby (1991).

Many mechanical componenrs also operate at temperarurcs far lower rhan roomtemperalure. As the temperature is reduced, bolh the uhimat tensilc strenslh_sr. und rensrlc ) ield srrenglh. Sr. generi t t ) increa,c for mo,r rnurer iat . . Horever.temperaturcs below freezing have the efect ofaltering rhe structure of some ductile

60

E5

6t

t0 2.0 g

3,5

r0000 900 l2@ t5oo

l..r ir..rmnt:Cr. 0.5* uo, l0,l-0.5lea Ti .nn..tsd, r55t|F

\

\ \

\ \

-p 4

160 Designing teliable Products

150 -100 -50 0 50 100 150Tempsrslure "C

Figure4.18 Duct liobr i t teiranst iondagramforastructuralstee (Magerand N4auchal l '

TsmPoraturo 'C

Figur 4.17 1000 hour creep rupture stress as a functon ol tempe.aturc for various steels Materman andA5hby,1991)

metals so thy fracture in a briltle m4nncr. The tcmperature at which this occurs rs

cnlled thc t1;clilc-to-htittle l\nsition lcmlle1lutc. Tests to determine this usually

involve impact cnergy tcsls, lbr example. Charyy or lzod' which measurcs the

enerc\ to break the spccimen in joules The plor of the retults for it strucruril rleel

nre , i r 'own in Figure,r . lx. sho\aing thc re$ons ofhntt le and ductr lc beha! i(rur

'l/ ou"r,ra

Ir-*tr/

E

The cdependi

1965). '

perntur(Benbar

nominauselul r

absorbrequire,

Resid.

ln lernal

exbibi tecasting.

fits. allr{reldingonly bethe ftrti

(Faircs.The i

and TnTor

(usuall:by shor

l99l) .

t40

120

100

880

e60

o40

20

1984)

Variables in probabilhtic design

. Theductile-to-b rde transition remperature for some steels can be as.high,as0,C,

depending_on the composirion ofrhe sleel (Ashby andJones, 1989). Howev;r, rhere isno way of using the data directly from impact tests quantitatively in th; designprocess. Design specifications do usually stare a minimum impaci strength, b-utcxperience suggests thaa this does nor necssarily eliminate britile failure-(Faires,1965). The Rohettson l?sa can yield more information than either rhe Charov orIzod tesrs because the transition temperalure is statistically correlated with the iern_perature at which the actual structure has been known to fajl in a brittle manner(Benham and Warnock, 1983; Ruiz and Koenigsbergcr. 1970;. The test uses a sevlrelynotched specimen tested under staric tension, and a plot showing the variation ofthenominal stress-at fracture with the tcst specimen temperature drawn. The test givesuseful results frorn which design calculations can be based; however, the te-st ismore expensive ard complex compared to othcr methods, ln geDeral, it is dangerouslo u$ u material belo\a trs transition Iemperalure h{cduse most of irs caoacitr rorbsorh energ) \rithour rupturc har hcen IosL .rnd cureful design una analysis is

ReJidua, stresses and processingIn a component of uniform temperature not aated upon hy exkrndl loads, an\intemnl stresses thatexist 4recalld residurLl stresses. Tbc malerial strengtf,, tf,"."foriis dependcnt not only on th basic material property. but on the res]dual stresse;exhibited by the manufacturing process itsolf, foi exampte by forging. extrusion oicasting..This could b further affccted by secondary processing in production suchas welding, machining, grinding and surface coating pro""rr.J. f.orn dolibrate orunavoidable hcat treatment, and assmbly operations such as fastening and shrinkfits, all otwhicfi promotc residual stresses. Many taiiures result lrom uisathfactorywlding undjoining of purrs of engjneering comlonents {Heyes. I9g91. and this canonly f,c attribu(able lo residual slrcs\es allccLing either lhe stutic or more commonlvthe fatigue prcperties. Additionally manufacturing processcs result in variations i;surface roughness. sharp corners and other stress raisrs (Farag, 1997). I1 is alsocvident thaL the partern of re\ idual

162 Designing leliable Products

Drobiemahc rf tnsile bccausc the) havc low toughness and this could alcelrate

i"i""-"ir. i.riii" "*,ure

The piesence ofresidual stresses are generallv dctrimen-

i"f to tf'. otoau.t integntv rn servic and should b eliminatcd if cxpectcd to be

harmful (Chrndra, 1997)'--''irt"o.ji* v, the ellects of thc manufacturing nrocess on

th-e muteriul propcrtv

distribution can be dtermined, shown hert for the c \c when Normal orstrrourron

anoiies. F-or an adaitive case of a residual stress' it follows that from the algebra of

r;ndon variables (Cartr. 1997)l

(4.r 5)

(4.16)

4.3.2 |

ps = mean of thc final strenglh

os = standard deviation of the final strength

Ps,, = mean of thc original strength

l'so = stand.lrd deviation of thc original strenglh

pv = additive quantitiss ol strengih from the process

ov = standard deviation of the additile strength from thc pcoccss'

For a proporlional improvemcnt in the strength, the product ofa funclion ol random

variables aPPlics:

"s;("3, ,+" 'u)n '

"" = 0,3"."i' + r,i' ,4" + 4" "i,)u'

(4. l7)

(4.18)

The m.

the prol

Large t,bility btvariabil(Hauge

Comable inlhe effc,

inertra

high rcThe

describVariabof lheircapabimrterii

calculi

Smithconfid

Histo'as r /3

As can bc seen from th above equations. lhc standatd devialion of the strength

ir.t.u... ti""in.r",fv with the numher of processes ured rn manufuclure thal 3re

ad,lins Lheie,idual itre',e' Thrs mry be the r!3son for the 'pparenl reluctrn(e r)I

,Jo-or i?r. io qi ' . pt"" i . . r larr \rrc i ld irrd rhout their produ(r (caf ler ' 1997)

n oraericai ,tifficutrv usng the rbote apFrorch rs thal lhere arc loo mdny proce\

Variables in probabilistic dsign j63

4.3.2 Dimensional variability

The manufacturing process inlroduces variations in that absolute djmensionalaccuracy cannot be altained and variations within specified tolerances area necessarvfeature ofal l mdf lufaclured producs r( aner. fOeO,. erop.r,of"ron.." , ; ; ; ; i ; ;rne pfoper lun-cl lonrng. ret i rbi l i r ) cnd lonB l i fe of r producl and lhe acl of assisninsIorcrance\ rn tact nnalrze5 rct i rbi l i ry (Di\on. l9q?: Ruo. I992 Vinogrador, tucl iLargc torerlncer and or large varirnce can result in significanl dcgradrlion of rellu.biliLy, becruse

.rhc_fJilure probubitily ir a funcrion of ihe m.rgniruie ;i d;;r;;;;vannDrrrty aDd tolerance al locrted. i tnd af lect( lo d Induced strels in.r componcnl(Haugen, 1980; Kluger, 1964).

. Componcnt reliabiliry will \ ary fl, .r firnclron of rhe powcr of a dimcnsional vari_snle In J stress tunct jon, Power\ ofdlmensional tar iables greater thJn unltv malni fvtheffcct. For example. lhe equatlon f,,r the polarrn"r*-f r*" 1".

" .ir"'"i"r. ,"fr"f:,v lnes a5 the t( ,urth power of lhc drdmcrer. Orher simi lar ctses lhhlc lo dimn,ionalvarittion effects include thu radius of glrdllon. crosr-secLional ara and moment ofrnerua propertles, Such tilriitiong illc!t strbllil). deffection. srrains and anIularlwists as well as srrcsres lcvels (Haugen. l9t0l. Ir can be scen rhdir;;i;;;;;;";;_

an(e may be ot rmportdncc for crillcal components which need to he designed to .rhigh reliability (Bury, 1974).

_ The mcasures of dimensional vsriability from Conformabjlity Analysis (CA) (asdesoribed in Chaptcrs 2 and 3), spccifica y ttr" Componeni M;;;f*;;variubitir) tusk. 4h. is useful in rhc alocarion oi,"h.;a;; .;;..;;;;;;;tr:t:of lheir.distriburions in probabrlisric desrgn. fne

"atue a, ,s aeie..ineii;;;;;;cap{0rlrl} mrps lor thc manufacruring process and knowlcdge of thc complnenL.smarcrHt irnd geomerry compalibilit) wirh lhe Frocess. In the specific c{se lo rhe ilhcomponenl bilateral tolcrance, ,i, it was shown in Ctrapt".: tfrat tte stanaara aevia_oon csumates werc:

_, t i .q; i" ,_ nr, .s l l

' t2

(4.1e)

(4.20)The ' in cquation 4.l9 relates to the fact rhat this is nor thc true standard deviation,our an estrmate to measure the process shift (or drift) in the disrribution over theexN.Led du_rdrron of l rroducrirn. Equarron 4.20 r. r t e t . rr esr l .ar. for. r t . . i rniarJoevriruon ot lhe dr\ lnhurion as Jetermined by CA wi lh no nroce* shrn.A popular way of dererminirg thc standard dcviation foi use in the probabilisticcalculat ionr i , to (sl imale i t b, equalron 4.21 $ hrch i , based on rhe bi lareral ralcrancer. dnd vanous empir icf l l factor,d,sholrnin. Iahle4.TrDiercf . tg8b;HauAen.tqbOrsmilh. lqq5r. I h( facrors retale lo lhe facl rbar rhe more pJr l . prudr. .J l , f , . . . . .conlidenc there will be in producing capable tolerancesi

Historically. h probabilisricas l/l (Dieter, 1986; Haugen,

(4.21)

calculations, the srandard devialion, o, is expressed1980; Smith, 1995; Weliing and L],nch, 1985), which

164 Dsigning teliable Products

'hblc l.t Endrical |lcloN lbr dcte.oini.gstandard deliation bascd ot lolcrance

No, of parts nrdulicturcd

0.15

0.14

0.13

o.12

0.11

0'r0.09

0.08

0.07

0.06

0.05

0.04

0.03

0.02

0.01

01 1.5 2 25 3 35

Compon Manufacluing Variabilily Risk' q.

Figure 4.19 Standard deviat on estimates b.sed on qm and t/3

l025

relares ro a ma\imum Proces. Capahit i l ) Index. Cr - I This el trmate dor nol take

inro accounl process 'hi f l . t lp ical l ) t l 5d from the trrget dunng a product lon run

due to tooling and production crrors (Evans, 1975)' and rclies hcavily on th

iolerances beiig within a3,7 during inspection and process control Unless there rs

100% inspecti;. howevr. there wilt be som dimensions that will always be oul

of tolersnce (BurY, 1974).ln

"quations 4.19 and 420' improved eslimates for the standard deviation are

nti..",Ja U"..4 on emPiricdl obiervaoonr' Thrs i' sho\Ln in figure 4lq for i

-0.i.. ,ol"on.. nn

"n arbrlrary dmensional characteristrc bul with 3n increa!ing

^. u. *oria t. a"t"t ined for less capable design schmes ll shows thal increasing

iiik of alloc,rting toterances that are not capable' increases the estimats for the i7'

I1.522.53

of the shil

from CA r

rhroughorFromtl

ing rcliabilarge dimSenliitivit'

Stress cCeometri(Sbigley rin the coof manultions varjare typicexamPle.

coniigurlUnder

dimnsionomrnal

norched

using tl I

The rr

amongdesign (

When tlthe slatiapproxildistortr(

materialbrittle fi

I

EE

Variables in probabilisti( dsign

more so when procss shitt is rakefl intoaccounr. as shown bv ihe term o/. AnticiDationof lhe shi fr or dr i f t ot lhe tolerdn.cs \el on a design j , lbereforc an imponant fJclorwhen predicting reliability. lt can be argued thar thc dimensional vdriability estimaresfrom CA apply to the early part of rhc pfoduct's life,cycle. and may be overconservativewhen applid to the useful life ofthe product. However. vnriability driven failure occursthroughout all life-cycle phases as discussed in Chaprer l.

From the abovearguments, it can b secn that anticipation ofthe process capabilirylevcls set on the imporlant design chamcrcristics is an important facror when predicGing rcliability (Murty trnd Naikan, 1997). lfdcsign tolerances are assigned which bavelarge dimcnsional varialions, the effct on rhc reliabiliry predicted must bc assessed.Sensiljvity analysis is uscful in this respect.

Stress concentration facto6 and dimensional vaiabilityGeometric discontinuilies incrcase the stress lcvel beyond the nominfll stresses(Shigley and Mischke. 1996). Thc ratio of this increased stress to the nominal strssin the componcnt is termed thc stress concenlration factor. K/. Due to the natureof manufacturing processes, geomclric dimensions and thereforc strss concntra-lions vary random ly (Haugen, I 9li0). The stress concentration factor values, howevcr.are typically bascd on nominal dimensional values in tables and handbooks. Forcxample, a cornprchcnsive discussion and source of rcfcrence for the various strssconcentration fac$rs (both theorelical and empirical) for various comDonentconJigurations and loading condilions can be found in Pilkey (1997).

Underestimating the cllects of the component tolerances in conditions of highdimensional variation could be catastrophic. Stress concentration factors based onnorninal dimensions are not sumcient. and should. in addition, include estimatesbascd on the dimensional variation. This is demonstrated by Haugen (1980) for anotched round bar in tension in l igure 4.20. The plot shows +3r confidence limitson the strcss concentration factor. ](1, as a function of the notch radius. gcnratedusing a Montc Carlo simulation. At low notch radii, ihe stress concentration factorprdicted co(ld b :rs much as l0r% in error from the results.

The main cause of mechanical f.ilurc is by fatigue wilh up to 90% failures beingaltributable. Strcss concentrations arc primarily responsible for this, as they nreamong the niost dramatic modificrs of local stress magnitudes encountered indesign (Haugen. 1980). Stress conccntration factors are valid only in dynaniccases, such as fntigue. or when the material is brittle. In duc(ile materials subiect k)stalrc loading. Ihe el lccr. ol stress concenrrur ion dre of l i l t le or no impoirrnce.Whcn the rgion of strcss concenrration is small compared ro the sectjon resistingthe static load, localized yiclding in ductile malerials limirs rhe pak stress ro theapproximitc level of the yield strengrh- The lond is carried wilhour qross Dlasticdi ,rort ion The \ tre\ \ concenrrr l ron does no damage lrnd rn fact srrain hcr jeninsoccur\ , and \o i r c.rn be ignorcd. 3nJ no K/ rs.rnnl ied lo lhe srre.. Iunct ion. Howe\ er.stress raisen wher combined wirh facrors such as low temperatures, impact andmaterjals with marginal duciilily could be very signilicanr wirh the possibilily oIbrittle fracture (Juvinall. 1961.

For vry low duclility or briltle materials. the full rr is applied unless informationto the contrary is available, as governed by rhe sensirivity jndex ofthe material, 4".Fuf exrmple. ca.r i ron. have inlernal dr,cunLinujt ies lhe l l rcss raiser

165


K'=l+s",(1- l )

tr:/ = actual stJess conccntration flrclor for staiic loading

(4.22)

4, = iDdex of sensitivity of the material (for static loading - 0.15 for hardenedsteels, 0.25 for qunchcd but untmpercd steel, 0 2 for cast iron:for irnnact loading - 0.4 to 0.6 for ductile matrials. 0.5 for cast iron,I for britile maleials).

ln the probabitistic desig! calculations. the valu of(l would be dtrmined liom thempirical modls related to the rominal part dimensions. including the dimensionalvariation estimales from quations 4.19 or 42n. Norton (1996) models fl usingpowcr laws for many standard cases. Young (1989) uses fourth order polynomials

In either case, it is a relatively sttaightforward task to include ,Kr in the probabilistic

model by determining the slandard dviation through the variance equationTte distributional parameters for ]K/ in the form of the Normal distribution can

then b uscd as a random variabl product with the loading stress to determine the6nal stress acting due to the stress concenlration. Equations 4.23 and 4 24 show

that the m

By replacir

4.3.3 Se

One of thpredictjoncondition!

design (tsr

Empiricalc4l-basedreliabilityapproaclrIoads (W

trical forcenvironm

shock orlo8d in c(thc intenduly cycl1974), anlations (I

Failurcindustry

rhat fiilufailing wl

1985; La

being delength ol

Fadl$ (f) {lnche3)

Floure 4,20 Vaues of stress concentraton factor, (t, as a funclon of radus, r, wth +3d imits lor aciiumferentally notched round bar n ten5lon Id -fi(0.5, 000266) inches, d,

= 0 00333 inchesl(adaptedfrom NaLrgen, 1980)

itself. 4. approaches zcro and the lull v^lue of lKl is rarely applied under static loadingconditions. The following equalion is used (Edwards snd McKee, l99l; Grecn. I992'Juvinall, 1967; Shiglcy and Miscbke, 1996):

Variabls in probabilistic design 167

thal the mean and stand.ird devi.lion of tbe linal stress acting (Haugn, 1980):

r'.^, : (pt,. r'" + p'*, 4. + "j "i,fs (4.24)Byreplacing Krwilh K'in the above equalions, thc stress for notch snsitiv matrialscan be modlled ifinformation is known about thc variables involved.

4.3.3 Service loads

One of the topical problems in the field of reliability and faligue analysis is theprediction ofload ranges applied to the struclural component during actual operatingcondilions (N4godc und Fajdiga, 1998). Service loads exhibit stsristical variabilityand uncertainty that is hard to predict and this influences the adequacy of thedesign (Bury, 1975; Carter, 1997; Morup. 1993: Rice. 1997). Mechaflicnl loads maynol be $ell charecrerized out of ignorancc or sher dilicuhy (Cruse, 1997b).Empirical methods in determining load distribu!ion are currently superior to stntisti-cal-based ffethods (Cartcr, 1997) nnd rhis is a key problem in the developmnt ol'relifibility prediction mcthods. Probabilistio design then, rather than a detemrinisricapproach. becomes more suitable whcn there arc largc variations in the anlicipatcdloads (Welling and Lynch, 1985) dnd the lords should be considered as beingrandom variables in the same way as the marcrialsrrcngth (Bury, 1975).

Loads can be bolh internal and external. They can bc due 10 wight. mechanicallbrc$ (axial tension orcompression, shear, bending or torsional), inertial forces. elec-irical forces, metallurgical forces, chemical or biological eflccB; due to temperature.environmental efrects, dimensionl changes or t combination oflhese (Carter, 1986;Ireson.r a/.. 1996i Shigley and Mischk, 1989i Smith, 197(t. In fsct some environ-mcnts rnay impose greater slresses thon those in normal operation. lor exampleshock or vibration (Smith. 1976). Thse faotors may well be as importffnl as anyload in convcntional opcration and can only be formulated with full knowledgc ofrhe inlended use (Carler. 198(t. Addilionally, many mechanical systems have aduty cycle which requires ellccti!'cly many applicadons of the load (Schatz er r]/..1974). and this aspccl ol lhc lo.rding in scrvicc is seldom reflected in the design calcu-lal ions (Bury, 1975).

Failures resulting from desigr deficiencies arc relalively commoD occurrences inindustry and sometimes components fail on the 6rsr npplication of the load becauseofpoor design (Nicholson et d1., 1993). Th underlying assunplion ofstatic dsign isrhat failure is governed by the occurrence ofthese occasjonal larg loads, the dsignfailirg wher a sirgle loading stress exceeds th strength (Bury. 1975). The overloadmechanisn oI mechanical failur (distortion. instability, fracture, clc.) is a commonocurrence, accounting for berween l l and 18% of all failures. Design errors ladingto overstressing are a major problem and acount for over 30% ofthe causc (Davies.1985; Larsson cr dl., l97l). The designer has great responsibility to ensurc that theyadequalely accounl lbr tbc loads anlicipated in service. lhe service life of a produc!being dependent on the number of times thc product is usd or operated. thelength ofopcrating lime and how it is used (Cruse, 1997a).

Designing Reliable Products I

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