Variables in probabilistic design 167

that the mean and standard deviatior of the final stress acting (Haugen, 1980):

",._, = fui.,/*, + pr*, . ,4. + i". i*,)o'

Byreplacing r? with K' in the above equations, the strcssfornotch sensitive materialscan be modelled ifinformation is known about the variables involved.

4.3.3 Service loads

(4.23)

\4.24)

Lpled

One of the topical problems in the field of rcliability and fatigue analysis is theprediction ofload ranges applied to the structural component during actuaioperatingconditions (Nagode and Fajdiga, 1998). Service loads exhibit staiisricat vaitatitityand uncertainty that is hard to predict and this inffuences the adequacv of th;design {Bury. l97J; (;11161. 1997i M6rup, t99J: Rice, i997r. Mechflnic;l lo;dr mavnot hc well characterized out of ignorance or sheer dilficulry tCruse. l99rbj.Empirical methods in determining load distribulion are curently superior to statistical-based methods (Carrer, 1997) and this is a key problem in lhe development ofreliability prediction methods. probabjlistic design then, rather than a dete;ministicapproach, becomes more suitable when there are large variations in the anticipatedloads (Welling and Lynch. 1985) and rhe loads should be considered as Leingrandom variables in the same way as the material strength (Bury, 1975).

Loads can be borh internal and extemal. They can bi due to weighr. mechanicalIbrces(axial tension orcompression, shear, bnding or torsional). inertial forces, elec_trical forces, metallurgical forces, chemioal or biological effects; due to temperature,environrnental effecrs, dimensional changes or a combination of these lctarter. 1986;lreson et a1.,1996; Shigley and Mischke, 1989; Smirh, 1976). ln fact some erviron_mcnts may impose greater stresses than those in normal operation. for exampleshock or vibration (Srnith, 19?6). These factors may well be as importanl as;nyload.in corvnlional operation and can only be formulated with fuli knowle

168 Designing reliable products

Some of the important consideratioN surrounding static loading conditions an'l

slatic design are discussed next lnitially focusing on static design will aid the

development of the more complex dynamic analysis of componnts in service. for

example fatigue design. FatiSue design. although of great praclical application, will

not be considered here.

fhe concePts of static designThe most sig;ificant factor in mcctanical failure anaiysis is lhe character oflotdinS'

whether staiic or dynamic static loads are applicd slowly and remain essentially

constant with time, whereas dynamic loads are eithcr suddenly applied (impacl

lordsl or repe:rtedly var ied ui lh t ime rfat iSue loadsi or hoth Thc degree of imptrcl

is reiated to the rapidity of loading and fte natural frequenc, of lhe structure lf

the timc for loading is three times th fundamental natural frequoncy' static loading

may be assumcd (Juvinall, 1967) tmpact loading requires the structure to absorb-a

giuen a.ount of energy: slalic loading requlres Ihal il resist given loads .rJuvlndll'196?, rnd thrs is a fundamentaldlfferenc when selecllng the theory ol lallure ro ne

used. An analysis guidewith respct to the load classification ispresentcd in Table 4 8

A stalic load, i;terms ofa deterministic approsch, is a stationary force or moment

aclin'r on a member' To be glatronary il must be unchangmg in magnitude point or

poini ofapplication and direclion (shigley ind Mischke l9S9r' It rJ a uriqu lalue

,aora.ant,ng what the designer regards as the maximum load in praclice that the

product wiij be subjecled lo in '.rvic Kirkpdlrick {19?01 defi.ne' a limit of strain

rate for static loading as lesr than l0-L hut greater than l0 ' strain rate tecond

This fits in with lhe slrain rate at which lensile properties of matetiah are tested

for static conditions (lo-r) With regard to probabilistic design, a load can also be

considrd static when som variation is expected (Shigley and Mischke' 1989)'

Sutic loading also requires thlrt there ar less than 1000 repetitions of th load

d uring its des-igned service (Edwards 4nd McKec' I 99 | ), which introduces ihi?oncepl

nitt"-auty"y"i" upptoptiate to reliability engineering static loads induce reactionsin

compone;h and equilibrium usually devebps. Where the total static loading on a

mechanical system- arises from more than one independent sourcc' a sutistical

model comb;ning ttre loads may be written for the resultant loading statistics using

the algebta ofra;dom variables The correlation ofthe loading variablcs is important

in thii respect. that is the load on the component may be the function of anothet

applicd load ot associated somehow (Haugen, 1980)'For static design 10 b valid inpractic. wemust assume situations where ther-e is no

delerioration ofihe material strength within the time priod being considered for the

loading history ofthe product. With a large number of cyclic loadslhe malerial will

"""tttt"olly futigu" Witi rn assumed static analysis, stress rupture is the mechanism of

failure tnbe considere

Variabls in probabilistic design 169

be rgnored if the number is snlall, to creare a steady stress or qrarLrldll. condition.bxr one whcre a variarion in loadins slill exists (welling and Ly;ch, r9S5). Atthous;,ifthe loading is irealod as a random v.riabl. then thn coutd inrply a dynanric analy-sis(Freudenihal c/,/1.. 1966). While staric or quasi static looding is oiicn the basi; ofengrncering dcsign prrctice, ir is often imporrant io address the implicfttions olrepeatcd or nuctuaring to ds. Such badins condirions can resrtt in iaiigue liiiLrre,l|.s menlioned above (Cbllins. Igi ).

ln a lypicallord hisbry of I machine elcmenl. most ofthc appticd loads are retalively.nr l l l JnJ lheir (LrmLhI\ . . marer iJt JrmiE( etrect, urc ncgl i r ib lc. when rhe rpptredh ir( l rnp \ l re_sse\.rr( rbo\c the mrlLr rJI \ eq r \alenr cnJJrrnc. l inr iL. rhe rc,ulr ing,;(cu_mlrlation ofdamage implics that the component fails by fatigue at some linite n mberol load rcpetidons. Retarivcly largc lords occur onty occisionalty suggcsting thcircumulativc damagc effecls are negtigible (Bury. 1975) Dvidcnrly. failurt*itt o".u, o"smn as a single load oxceeds the valuc ofthe applicablc strengih critcrion.

Some typicdl load historics lbr mcch.lnical componcnts and systcms arc shownin Figure 4.21. Thc load (dc d load. pressurc. bending moment. ctc.) is subjecrlo vrrialion in all crscs. rathcr than a uniquc vtlue. the likcly shaDe of the fi-nal

B6nding

,,,**",1 llr n r il l l ,lFigur 4.21 Typ ca load hnores fornginern9 componentvs\atems

Slalic lvol2.3 kN

170 Designing reliable Products

67a9Tim

- mlnuls

Figure 4.22 Traneenl loads may bp man)] nmes the 5talr oad n operalion (Nkholson eta/ 1993)

distdbLrl ion shown schc r l ic l l ly lo lhc r ighl ot ctrch loud hislorv Liven pcrmrrncnl

(lcnd lon(ls thal shoukl rrlrrintrin const nt mugnilLrdc sho\r' rchlivclv sDl.lll rrd

slow f l l0( lom !ur i l l ion i l l prrcl icc Al thc othcr cnd of lhe k)r tdrng typc splcl f t rnr t i rc

l f r l l r icnl lo l( ls which occlrr in l icqucnl lv l ln( l lus( lbf ! short pcr i(x l ol t in)0 r l r

inrpulsc. l i ) r cxnmplc c\ l fcrrrc wiDd trn( l c l l r lhqurrkc lo ds (ShinrrT!kl l : tnd tLrn l984)

ihc aborc ivoukl lssonrc lhrr thc lo ct dis l f ibut ions {ot s l . r l ic dcsigrr\ : r lc ol tcD

highly unsyDnrelr icnl . i r ld icrr t iDg thr l there is i r nnir l l p() for l ion ol l ( ) ' rds thl t t t rc

rci . rr ively largc ( tsury. 1975). During thccondi l ions ofusc. cnvi f t ' rncnl ' t l t rb{ isct! 'ct

vrr inr ions give r isc to lemporary ovc o0ds of t rdnsienls c. iusing i r i lurcc (Kl i l ?/ i / / 'l99l) . Drt col lcctcci f roln nrechlnicdl equifmcni in scNice hxs sh(^!n thrr l thcsc

lr.rnsicnl lo.rds dclcloped cluring opcration nliv bc sclcrltl timcs rhe nominrtl loud

s shown in Figure 4.22.' rhcrc h vcry l i t t le inl i ) rrml ioD on thc vrr i . r l ional n.r turc of louds conlmonlv

cncounlered in nechrnical engiieering. several rcfcrences frovide guidance in

lcrms ol the cocl l ic ienl ol vtr i r r l ion. a, . l i , r s ne common lo 'rding ivpcs as shown

in frblc 49 (BLrry. 1975r El l ing$ood nd G.rhnbos, 1984; F' t i rs. 19651 Lrncoln

T,hle :1.9 Tyri.rl co.llici.nt oJ vrltion. a,. l.' !rfiou: liidn'e

a,

115III

-5

002

o:s

. / d/ . . I98.1: S

ably. whcn r(( ; :0.05. Trel i rbi l i ly ol

Example -of variatio

estirnrlcd by

Aer(lymmic lords in xitcriiSnnng IbreBoh prc l.id usng l.scf.d sdc*drversPo$cred $rcnch lrtrquc

Mcchniil dcvicc\ h s.rvre. rigure 4.23 Te

Variabtes in probabit ist ic design t7T

./ rr l . . 1984r Shiglcy and Mischke. 1996r Smirh. t995: WonlsoD./ d/ . . 1992). Thcle. ,1, . \ . , re,rrr

172 Designing Ieliable Ploduds

r : lord

u and , = me.ln sectidral dimcnsions o{ thc bar'

Assuning thc vari tion encounlefed in dinrcnsions 4 and /) is negligiblc' lhen the

mern stress. lir.. on th b r rs:

j , , tooooo 6b {)7 MIh' ' , , , , . , , , . o.o, . o.o.

The load coelllcien! ol variution (, = 0 l Rcarranging lhe equnlrcn fof C! to givc the

st ndard dcviation ol lhe bading slress viclds:

d/ : ( r x / t r :01 r 6667:667MPa

Thc slress, a, ! . tn therel i )re bc .Lpproximated bv a Normal distr ibul ion wiLh p f

l , - N(66 67,6.67) MPa

P rcbabil istic cons i de rati on sl i ) r m:rny ycrrs in nccb nical dcsign, lond vcd t ions hrve bcen m sked bv using

Iuctors oi st l t ty in a dctcrninisl ic npprorrch. t ts shown bek)w ( Ul lnrr tn l9t)2) l

. l (ol . l Load wcl ldc{ jncd s str l rc or f lucludt ing i l lhere urc no rnlrcrpr lcd ovcr '

lo l |ds or shock loLrds rnd i f rn rrccufLrtc nrclhod ol anr lvsing lhe slrcss hrrs ?'cn

.1.21()1.3 Nl l lurcofbrrdisdcl incdin n. tverngcnrannerwithovcr lo ' tdsol 2{) io

50"/" tlnd lhe slrcss rnrlysis mcthod mnv resull in orrors lcss thun 50'/0'

. 1.4 io 1.7 Lond nol wel l known .rndror slrcss rnal) \ is nelhod of doubitul

'Ihcsc frctors coukl be in addirion to thc factols ol s'rfelv lvpiMllv cmplovcd rs

discussed at the hcginning of this section tscruse of thc dimcultv in finding the

exact d istributio n ai nat urc of lhc load. lhis lpproach lo design w'ts considered

ndcqurle nd conomical. lr has bcen afgued iht! il is far k)o t nc consuNDg rnd

..'.tiyto n,""tur" rhe load dislribLrtions compfehcnsivelv l(larler. 1997), bul this

should not prevcnt a reprcsenrrlion beiDg deviscd. as lhe alternirtrvc

rcprehensibie (Carler, 1986). Pucticallv speaking. decidiDg on ihc servicc brds is

liequenrly challcnging. but some sor! of estimates arc essential and i! is oficn

lound lhrt thc rcsulis are closc to thc irue loads (laircs. 1965)

Sircc staric liljlures are caused bv infrequcnt lrrgc loads, as discusscd rbole 'r rs

imporlanr th l lhe distributions thal nodel these extrcme evcnls at thc righr'hand

side of thc tail are rn nccuratc represcntation of aciual load licquencics Conlrol ol

thc lord tril would appcar to be the nrosl effcclj\ mcthod ofconrrolling reliabilil:/'

bccause the hil of thc lo0d distribution diclates thc shrpe ol rhe dislibuilon 'indhcnce tuil re Inte (Cnrler. 198(r' Ihc NorDral disiribution is usuall) inrppropriate'

although somc rpplicd lo.ds do follow r Normd dislribution. for cxamplc rocket

nloiorihrust (Ha;gcn. 1965) orrhc grs prcssure in the cvlinderheads ofreciprocaiins

cngines lLipson er ai.. 196?). Other loads mav be skcwed or possess verv iitllc scatter

Figure 4.24

rnd in litc

Nornxlan

high coel l iis Nonn"l .

Experim

although

in Figl iol

relirbilitl

probibiLl

Variables in probabilistic design t7l

Figure 4.24 A oad ng nr5s d nr buron w{h extrenre cvsns

rnd iD l i rc l lo ld spcclrums lcnd to bc highty skcwcd lo rhc t r t i wi lh hjgh lot ldsoccurf ins only occr is ionnl ly ls s lx) ' rn in I ' igurc 4 t4 tRury. t974). ( . iv i t cnsinccr ing

usLrt l l ty sLrhiccrcd to i l t .ognormrl ty l )c ot k)r td. bur in gcncrat rhcNornrr l l lnd l -pur l rmcrcr wcibul td isrr ibur ions urc conrnronly cnrpk)ycd t in. ; rcchl , r ic l lc(n poncnls. lndthcl ixrorrcnl i r i ) rctccl f i ! | tcor)rponcnrs(N4urly l ln( iNl ik n. t ( . )97).

l l rs Lrscl i r l in lhc l iNt instnncc lo ( lcscr ihc rhc tor td iD lcrnrs ol N () |nrnl d is l f ibut ionhccrLrsc ol lhc Dccessi ly to t ransta n i l int(J thc to ding srrcss pnf lmclc| ] \ lhrough lhcv!r innce cctunr io l l nd dirr)cnsionl l l v i , f i lb lcs. Atrhorsh rhe rccuf cy ot . rhc v i l ; i locccqur l 'on is dcpcndcnl on using lo nel f Nornr l r t ( i jsrr ibur ions nnd vur i b lcs ol towcoel l ic icnt o l v f i r ion (( \ .0.2). i t rs sr i usel i r t rvheD rhe rnl ic ipr lcd torc ls h \ ,chigh cocl l ic icnrs ol va nr i (nr . Ft !cn i l rhc dis l f ibLrLion lccounrrDg tbr thc st r ic kJ drs Nonn l . thc st fcss nrodct is usLra ) Logno, lnr t 0t lugcn. tg l t0tdue to rhc conttexnr lurc ol the v l r i r th les rhrt n kc up thc sLrcss tuDclron.

Experimental load analysisIn l i ) rrn t ion on lord distr ibut io|s w|s vidua y nor cxisrcnr uul i l quire fccenl lyI l rhough lr l lch of i t i r \ery fudimenlafy. protrabty bcotuse coltect ing Arra is veryexpcnsl\ ,e, thc Drer{r ng tr . rnsdLrcers bcing di | ] icul t ro insl .r l t on thc rcst pr.oduct ofprototype (( n er. 1997). I r has been circd thr l t le.rsl one prolol-rpc is required lomrkc I rel i . rbi l i ty e! . l lunl ion (Fddign i , / r / . , t996). r ind rhis Inun surcly be to understand the loads thar could be cxperienccd in serlicc as crose rs posstbic.

ln experimenr1l l lo.rd sludies. the mersurable !af i rbtcs rre ol ien suft . . rce s1r in.ncceleft t ion. weight. pressurc or lcmperalufc (Haugen. t9t0). A discusstun ol .rheLccnnrqucs on lrow to mc sure thc differenL types ofload prLftrmcrcrs caD be ibLrndin I,igliolr nd Bctlsley (19q5). The nersufenrenl of slrcls dircctly wolrld bcndv.rnrageous. )ou would rssumc. Ibr usc in subscquent catcularions ro predjclrolirbiliry. Ho\,!e!cr. no rr.rnstarion of the dimenstunal v.rridbiljty ol the pari coutdthcn be accoLrnred lbf 1n thc probrbitisric nrodel to gile rbc srress distribulion. AbcLter te-sr rvould bc Lo ourpLrl the load direcrty r! showD rurd thcn use the appfopriareprobabilislic model ro derernrine the slrcss distribuilon.

. A kc]' problcn in expcrimental toad analysis is lhc lr.rrslalion oflhe data yielded liom

the measurcmenr syslcm (as rcpresenred bt the tond hislories in f.igurc,i.:ty into an

174 Designing feliable products

tigure 4.25 Determinat on oi the PDF for a random process (adapted from Newland, 1975)

rppropriale distributional form for use in the probabilistic calculations. Mcasuring thedistribution ofthe load pcak ampliludes is a useful model for both strtic and dynamicloads. as peaks are meaningful values in the lo4d history (Haugon, 1980). A much sim-pler method, described below, analyses th continuous dala from the lo8d history.

Assuming thal the statistical characteristics of a load function. ir(l). are notchanging with time. tben we can use the load-time plot, as shown in Figure 4.2t,ro derermine thc PDF for n(r. The tigure shows i sample hislory for a rando/nprocss with the times for whioh x < x(t < ('I + r/.r). identified by the shadedstrips. During the time interval, I, x(0 lies in the band of values .r 10 (,y + d.t) fora total t imc of (r / r l +dt2+dt j+dt4). \Necansaythat i fTislongenough(inf ini te).the PDF or l(x) is given by:

/("r)dr = fraction of total elapsed time for wbich ir(4 lies in the 'I to (n + r/.r) band

k | + dtz + dt j +. . . ) Li-t,tt,T

tigure 4.25

Giren tl.tpplicI|troshges ol I

AnthroFSome faill

lhcreby brcliability

thc differrHunan

given in '

This typedevialionstudies ha

'" :01!For Engl

age rangtthis probtype ol d

(4.2s)

The fraction oftime clapsed for each increment ofr can be expressed as a percentagerounded to th narest whole number for use in the plotting proceduro to 6nd thecharacterizing distribulional model. Eslimation of the dislribution parameters andthe corrlation coemcient, /, for sevral distribulion types is tben performed byusing linear reclilication and the least squarcs rcchnique. The distnbutional modlwith a correlation coemcient closest to unity would then be chosn as the moslapprop ate PDF represnting the load history. The above can be asily traNlatedinto a computer code providing an ellcctive link between the prototype and loadmodel. See the case study la&r employing this technique for staiistically modellinga load history.

For discrete data resulting from many individual load tests. for example springforce for a given deflection as shown in Figure 4.26, a histogram is besl constructed.Tbe optimum number of classes can b determined from the rules in Appendix LAgain, the best distribution characterizing the sample data can be selected usingthe aDDroach in Section 4.2.

Variabls in probabilistic design j75

Forco pll unft daflccflon

Figure 4.25 D nribuiion of sprng iorce for a givn def ection

Civcn the lack ofovailuble standards und lhc degree ofdiversity and types ofloadapplicalion, thc pproacb adopted hrs lbcused on supporting thc enginee; in rhe earlystages ol product developrncni wirh limited cxperiment l load dria.

Anthropometric dataSome failures arc caused by human rchted evcnts such as installation. oDeration orl l ln inreninrc errors rurher rhrn by i rn) int f ln j( t ropef l) of the componcnr\ (Kl i ret al.. 1993'). A numbcr of mechanical systems rcquire opcration by a hunran, wholhcrcby becoDres an integral part of the system and has a signific;nr effect on thcreliabiliry (Snrith. 197(t. Consumcr producls are nolorious ftr the ways in whichthe.consumer can and will misusc rhe product. tt therfbre becomcs important tonukc a reasonable dercrminalion of the likely loading condirions a""ociatc.l *iththe diffrent users ofthe product in servicc (Cruse. 199?b).

Human forcc dara is rypicauy given as anthropometrjc slrengrh. An example isgiven in Table 4.10 for arm strenglh for matcs'right arms in G.

"tttinq oo"it;oo.

Thi, r \ ne oldira r . reldrt ] r \ r i t rble JnJ pre,enr(J in rerm. otrhr: mean rni , randarddeviation for the property of irteresr (pheasant. 1987; Woodson

"/ l7/.. 1992). Manv

\ luJie, hcvc been unJerr: lken lo (ol tJre lh i , di ra. parr icutrr ly rn rhc US a rneLJ torctritnd spacc programmes.

Anthropometric data is also provided for popularion weights. For exanrple,C":0.18 for the weiShr ofmales aged between l8 and ZO (Wooason er al . . tCgZt.F, ' f I nglNh mrle, $ i rh in rhi , . rge r ingc. rhe mcrn. / , .o5tg. ind rhe,rrndJrddevialion, d = 1 1.7kg. Therefore. al +jd, you wo l{:l cxpect I ; 1350 males in thisage range to be over l00kg in weight. as derermined tiom SND rheorv. ln rhe US_thie nrobabiLr l rppl ie. to mate, ot Skg in $eiphr Ihe de,rgner,houtd u,c rhi ,tl,pe ofdata whenever human inreracrion wilh the producr is anriciDalcd.

'

ll

i

116 Designing reliable products

Trble 4,10 IIusan amr stfcnslh d.h (w..dson.r 41. 199:)

. Givcs.

. lnteffer

4.4.1 Dlst.ndard dcvrtron d(Nl

t20

I l0 't50

l8l

54ll6t5llt l

l0 l:18L0l

B8

loadins sbeing rcln

the hadir

This rs d

r992). Th

4,4 Str$s-Strength lnterference (SSl) analyslsHavinS prcvi rsly introduccd thc key melhods to dctermine the imporl nl vrriiblo(with rcspcct t() stress rnd strcnglh distribuiions. the most acceptrblc way to predict

mchrnic.r l componcnl rc l iabi l i ty is by rpplying SSI theory (Dhi l lon. l9U0) SSl.tn.rlysis is onc of lhe oldcst mcthods tu asscss slructurrtl reliability. ,tnd is the moslcommonly used method bccause of its simPlicity, ease and cconomy (Murty ndNrikrn. lg97t Sundaf lrruJan .ncl wi i l . 1995). I1is.r proct ical enginccr ing tool uscdfor qu nlit.rively prcdicting th reliabilily of mechnnical conponenls subjeclcd t(,mcchanical loading (Sadlon. 1993) and his been dcsoribcd as a sinul.rtive modcl off i lurc (Dlsgupta nd Pechr. I9ql) .

The rheory is concerned w(h thc problem ofdclcrmining the prob.rbilily ol fiilurof a nrn which is subjected to a loading stress, 1.. and which has a strenglh. S. lt isnssumed ihat bolh L rnd S nrc r.tndom variablcs wilh known PDFS. rcprcsentedby/(.s) and l(L) (Disney./ d/ . . l96l t) . The prcbnbi l i ty of fai lur, and hcnce thercliability. can lhcn be estimatcd as the area ol inlcrference between thcsc slressrnd slrength iunct ions (Murly and Naikan. 1997)

SSI is rccommended lbr situations whcrc a considerabl polcnljal for varialionexisis in any ofthe design t,trameters involved. for example whcrc large dimensionalvariutions exist or wheD the anticip tcd design loads have a large range. The mainreasons li)r performing il are (lreson ?r a/. 1996):

. To ensure sulJiciem strength to operare in ils cnvironment and under spccified loads

. To ensurc Do excess of naterirl or overdcsign occun (high costs. weight, mass.

Some advantagcs and disadv.inlages of SSI an.tlysis have : so becn summarized(Sadlon. 1993):

The probr

Figure 4.21

stress-Strength Interferen(e (SSt) analysit 177

. Addresses variability ofloading stress and material strength

. Gi\e\ d quanrr lar i !e e, l rmate of rel iabi l i t ' .

. Interference is often at the exlremes ofthe dislribution tails

. The stress variable is not always available

4.4.1 Derivation of reliability equations

A mechaniatl componnt is considered safe and reliable when the strncth of thecomponenr..S. exceed' the value of loading stress. Z. on it {Rao. l9q2r.

-When the

loading.stress exceeds the strength, failure oc{urs, the reliability of ihe part. _R.being relaled ro this failure probabiliry, p, by equation 4.26:

P(S > l , ) =R

h is also evident that rhe reliability can also be derived by finding the probability ofthe loading stress being less than the strngth of the component:-

P(a

178 Designing reliable products

Thc probability oI thcse two evenls occurring simultaneously is the product ofequaljons 4.28 and 4.29 which gjvcs lhe elementi reliabi)ilt. i1R, as:

letuhandof the aplprinciple.

Il we lr

i/R : /(s,)d,tr . .l' 1,,rtrt(1 l t )

for all possible values ol strength. rhe rcliability, R. is then given by thc integral

(4.I )

This equation can be simplillcd by omitting ncgllivc limits to giYe thc gcncralcquation for rcliability using lhc SSI approach. For stalic design (i.e. no strenglhdeterioration) wilh a single npplication ofthe lo.Ld, cquation 4.31 becomcsl

Equ lion 4.34 reprcsents probably one of the mosl inportanl rheorics in relirbilit)(Carrcr. I 986.). The n umber of lo,td applications dclines the useful life o{ lhe compo-ncD t and is of appropriate conccrn to the designcr ( Bury, 1974) Thc n unber of timesa load is applied has an ellccl on the failurc rate ofthe quipmcnt due to the fdct thalthe probability of experioncing higbef loids from ihe distribution population hrsincreascd. Each load application in sequence is indepcndenl and belongs to thesame load distribution and it is assuned that the malcrial suffrs no strcngth

-: J* = ['-rr"r(J -.r

r,.t a,) u'

^= | r'r(J"rr.r rr)r' This can(4:r2)

Whcn the lo,ld ing strcss ccn bedctcnniDed by a CDI- in cbscd ibrn. this simplilics to:

n = .[- ,.1r1.r1s;,ts(4. ] ] )

ln gencrul both lhe brd &nd strength rrc functions of time morc oflcn than not(Vinogradov. l99l): howcver. in gencrul mcchanical engineering lhc control of theopcrAtion is more rclaxcd and so il bccomes more dillioull to dsfinc t duly cyclc(cu rlcr, I 997). H owcvcr. in most dcsign applicol ions the n unr hcr o I load a pplicutio n son r conponent/systcm during i ls contemplaled l i le is,r l r rgc number (Bury, I975)and this has a mljor c0cci on the prcdictcd relixbiljly. Thc timc dcpendence ol thebad becomes a litctor th!t trnnsforms ll problcnl ofprobabilisric design inlo one ofrcliabiliiy. Thc rcliability nllcr muliipl indcpcndenl load applicnlions in sequenceR,,. is given by:

n f ' , ' ( [ ' , , ,a)a '

a : numbcr of independcnl load applicaiicrrs in sequence

or l rom equat ion 4.13:

n, : lo r(r)'./(.s) rs

(4.34)

(4.35)

4.4.2 R

thcy courppry roNormal r

Figur

lind the I

I

th

l

hc) ld

i )

Stress*Strength Interference (SSt) analysis j79

deterld.rrion. ID fact, the srrengrh dislriburion lvould change due 10 the climin,rdonofdre wc.lk irens fron lhe population. and wouta in cfecrlecomc trurc,rea at rtrel . ' i h.rnJ.r( le. H.,{e!er. nL' r . r ! ,Lnl ot \ l ,Hrerh J l rer . , r rurr u i r t r r r rnc r , inctuJ

180 Designing reliable products

Losdin9

t(L)

(L>s)

, '/a;4,

t ts- l1L=o ! ts- lL

ct =! !

The fehabi l i t ) . R. i ( therefofe:

p'116

lnterl.snca msasudby rh dilf nc6 dblrlbuilon

s|t3ngrh

Figure 4.2

Normal,Weibul lobtained

is the ./,storts by

F _ 1 osNo(z)

Figure4.28 Drivatonol lhecouplngequalonforthecaselrhenbothloadingstressandrnatera nrenqtharc a Norma distrbuton

and (1" : qI (lhc mean and strndard dviation determincd

from Appendix IX).

Using the SND theory from Appendix l. the probability of failutc. P. can be deter-nined from lhe Stnndard Normrl variate, :, by:

P: osND(:)

R: l P:1 OSND(:)

ind

By deter

Using tbe Normal distribuiion 1() nodel both stress and strenglh is thc mosl commonand the mathematics is easily managed. giving a quick ard economical $lution. Theconvenience and flexibility ol using a single disrribution, such as the Normal. is veryattractive and has found favour wilh Dany praclitioners. However, it has beenobservd thai loading stresses and material slrcnglhs do not necessarily follow the

(4.40)

(4.41) Equ.rliol

tbr a disEquatiortions for

For tllunclion

Stress-Strength tnteference (SSt) analysis 181

0

Figure 4.29 Pol of G velsLr! H torlrre rnlegr,rl tr.rnslorm mthod

i,'lirli;il ili,l,liltiliU'ti'ilitltil il)l:; ifi,t,Tt,:"):lililli"itiiiili,l it,il,ltif;iiiilllJ',i:;:t1ltil:\i'lti:1t,,:ll,t,liiiii,llt,iilffillr,*::iltn,,:]1:il . r l r (

, rr l , r ' ,d/ t , i , t , ! ,n, t U, t t t , , , t t . | t f r . r . I ,n, ,{r . t t l ! . , , , , . r - . , f , ,

" , , . , , , , " ,

, , i , . , i " l . i\ r : I r r n) r ( t , t . . f r ' r ! t ' r ( r f r , . , I (o.r . r . r , r r 4. l r , $r t I r t r ( t . . , ,$ rC

l4. ,1_1)

n I ' t t . ' ' ,u ' 'l l

l ly . rcr . r l ' l l l l lnp , t .H- ! r . , t r . r t r r . i t r . , rsc, r t , ( Ln, \ nr Inr( ! r , , . , , ,n . . , f l

\ r ro$I In l - r !ur( , r .20. ru gr\c lh( fetr . . t , i I r ) . R..r , .

(4.4:r)

1() I . l ls

\4.411a l"r;da

' l,ou" (145)

,, l,

Equrrion ,14,1.is n)tved using Siml,nrr t! R,h ro iDlegrire rhc arca of o\rlap. The

":",'If ,:t:i:li ;"Jl*ill it:::,i::';T:i; J:l,l:l*::ul" i:: t r:;mi;,.quation:144 pcf irs the calcutatton ofreri biri,y r,.

"ry *",'r,i,,",;",," ,,i.ii,iiU,"r i ' , r r ' 1, ' r , rL.* , rnd-, t r rsrh t r , , \ rued rhe nrr r . . . r re,r . . , t , , hnJ H cJn a. to rnJ.

, ,l.ll ,ll'i, l'iili,l#:i;i';,:, ;l'$iI'::: ;l.:i;t:: i,: i;;;:,'i l:$il;

182 Designing rel iable product!

melhd h{s rlso becn applied to c.tscs whefe no b.rsis exists for trlsuming rnv specili'l

d is l r ibut ions fof c i thcr s i fess or strcrrgth. but where exper imcnt!1ion has becn pcL

fonncd yielding sul i ic ient enrpir ical dal f i rpur al ld L mbcAon- 1977r VernlL rnd

Murt l . 1989).

4.4.3 Reliability determination with multiple load application

l -be rpproacb trkl :n by ( ' l r ter (1986. 199?) !o. lctcrnr ioe the rcl i .Lbi l i tv when lnulr iplclord rppl icr l rcns nrc cxperienccd (cqurt ion 4 : l ' l ) is 6rst to prcscnl . r Salt tv Mrfgrn.SM. t l Doni i rDcnsionrl quanl i ry ro indicir te thc scpxrat ion o' lhc nrcss and slrcnsthdisl f ibut iorrs ns gi \cn by:

5y : 4j-11-

V,"i I ";' t h is is cssc ( i I l ly {hc coupl ing cqul l io lr l i ) r thc ursc whcn bolh nrcss rnd nr!nsrh rrrc

r Nonrul dis lr ibul(r) . A ptrr t l r rc let lo dcl inc lhe fr l r t t ivc shupcs ol lhc slrcs\ undslrcnglh disl f iburors is r lso prcscrr lcd. c l l lcd lhc t .otrdins Roughrrcss. l . l t . s i \cn bv:

LR , 11-

/"i r "i

(.r.'16 )

(4.171

Figufe 4.30

Bury ( 191.1

i . i l in l lo. td

lhc rypior l loudiDg roughncss l i ) r l cchrrnic l nf t )dtrc ls is high. lvpictr l lv 0 9 (( t ' cr .

l98a)) . A high lo.rdir)g foughDess r)rc ln\ lh t the vtrr i t rb i l i ly o l the lor t ( l is h iSh col t r -

purcd $i lh th l | t of lhe nfcDgrh. bul docs Dol necesstrr i ly nrcun lhr l lhc lor tc l i tsc l l is

high ( t -e i lch. l t )95) (scc I ' igurc,1. l0) . Thc genefr t l run ol r rechlr icrr l cquipDrcrr t is

sLrbicct 10 nruch roughcr loxding lhnD. s! l r - . e lccl l1nr ic cquipmenl. r l lhough r1 htrs

bcen rrgucd thr l c lcclr ln ic equipnrcDt cun r ' ln) bc s bjectcd 1r) rough lor id ing

undcr soDe cordi l ioDs (Lol l . 1t)87). l r t t r iscs [ i rgcl ] due ro lhc lcss prccise knowlcdgc

rnd conLr) lof the cDvirotr enl . r rndt lsofr 'om thc wide r .ugc ol d i l lcrcnt rpplrcr t lx)ns

ol mosl mecbrnic.Ll conpdlcnls. Arr xnf l ic . r r ion of th is is th. t1 rhe rel iabi l i tv of lhe

systcm is fe lat ivcly insensi l ivc 10 ihe numbcr of componcnts rnd the rc l i rb i l i lv ot

mechanicul syslcms is detcml incd bJ- i ls wcrkesi l i rk (Broadbent. 1993: Carter.

1986: Furm n. l98l i l {oysid. 1992). l l n lso meLus lhr l lhc probrbi l i l } o l l t t i l re

pcr rppl icr l io of load r i l h igh SM rrnd LR vrhres $hcrc ) i is much gfcatcr thar I

is lcry c losc to that fof s ingle lond nppl icr t io|s | ror th is relsor. Curtcr (1986)

noles thrl i1is mpossihlc 1() design lin r specified rcli.tbilit)' wilh lowcr LR vahres

For a gi \cn SM. LR aDd duty cycl . . "

(as del lncd br the number ol lnnes the load is

applied). the faiLlnc pcr rpplic tion of the lo.td. /,. can be delemined from Figut

4.1I. Thc llgure hrs bccn construclcd tbr near Nomal siress tnd strcngth urterlcrcDcc

conditions.rnd whefc ti is nuch grerler thtn l. The final reli.rbilitv is given by

(O'ConDor. 1995):

R,: ( l i ) l '

l' : probability of tuilure per applicalion of lord 1,'

(.r.48)

Stress-Strength hterference (SSt) analysis t8:

rh

g

I

IJ

:fl,'"""';:,:,li*;".** t road nsnre55and nr.nqth dnnbLrron\torva.ols oadns rouqhness!iid

BLrry.(1974i1975i t978;1999) in l roduccsrtrcconccprofdLr lycycteslost . t r icdcsignjnn (liR-crcnr^ pproadr. The dury clctc of mission tc"g,r,

"r, ,r."ig,, ir;

"q";,"i._ ii ir,"nurnber of l,ad applic,ri.'rs.,j howe\,cf , n i''

""r! rr,"

'.,,i"ii,", ;,;; ;r,r:; j,':,;dcsrgn sisnilicrncc. rf rhe lord. /.. is a ftnltonr vlriablc rher s" i. iill ,,,;,;;,,",;,\ , r lLe. / . . : ( l rhf PDt- , , , r r lch i , rL.e. . r ted t r r , r hr .h^$ .h:rr r , , . . ( Dt . / , r

' . r rhi | | r . { . lu , I . r ! i , rn. tenL Je, ot . ( . f r t rr i , , , , , l l I i .

' , . , i . r t . , 44. , .( i (Z) isoftcndi l l icLrh Lo dercrnr inc for a gi \ ,cn load djslr ibur ion. bLrr when .r i is targc.an approxrmit ion is gi \ ,cD by lhe Maximun Extremc Value Type I drsr l ibur ion oiar l-n

iilii i"tfiiltii :*rri,il,""lii ; Ti:,1;,il;;iil l:;:,r:.tit,;u.lxlitil:.rnrcler Wcib ll or I prraDrerer Weibult distibution. thc cxrrcmat modcl prrancrersc.rn be drrernined by rbc cquarions iD Tabie 4.I L rrr*" "q."r-".

;r.r*f. t"r,r. i.".the numbcr ofbrd ,tpplica rions. ,. Thc exl,emat m.ad tlr ir" r,,,,,ar"s.ri;"r-"",, ;h;be rscd in the SSt an. ysrs to dcternrine rhc reli.rbility.

. tor exaDpte. Lo derc ninc the reliability, R,. li)r .,. independcnr load applications.rc c:rn use cquarion 4.3-l $,hcn thc loading stress ts moaerrea .,,i,c 1r,. ii;il.Extremc Value Type I disribution. as foirhe above approrch. rtrc COI f,,r rf,l

184 Detigning feliable products

loadmg

of load {p)

1

01

001

0.001

0 0001

0 00001

0.000001

0.0000001

0 00000001

0.00000000r

0.0000000001

10'

lo '

10'

10

10"

'I hc l

4.4.4

' l he rss

ir) l i ! rLsalsty marsln (sM)

Flgure 4.31 Farure probabilty (per applcilion ol oad)veEU, !afely margin lorvaro!s loadinq roughn!sva ue5 (adapld lrom Carter 1997)

T{blc 4. l l l - r r fcmnl uluc nr.rDrclcB lRrn nr i t r l lo.dins nresr distr iburr .ns

l lh(1, 0.5lnhr( , i ) I 2655,-"1 ;:r,,.,,)-

/ l \: ' r . i r l r ;J, l ln( , i )1" '

f ( ; ) \' , t , .1\=' ' ' ( , Ji

(1)r" , ' , ' '(?) '",,r " t igure 4

Srress Strength Interference (SSl)analysis 185

q

lo ld i rg stress is gi lcn br:

r / , dp( . . ' ( t " ' ) )

rnd rhc s l rcnglh. / ( . t ] . is g i !cn b,"" x Nornrr l ( l i \ t f ibLrt ion. lbr . \ lnrple t r \ :

ro ., .-""( ! r l ' )

1: l 50l

1.1.5l l

th. f r l ixbr l l l f is g i r (n b\ . r luulron. l l . l 1)r ' sLrhst i lu l ing in thrsc tcnns:

| / . . i/ r | \ r , { . . ( ' l ) . . . , r , { \ . t .' v . l\hrc i r r r r hc !o l \c( l nrnr i r i ( r1 l r usirrg Sirrrpson s l tLr ic ls shor!n Ln At) | .n( l i \ XI I

lh( nrnjcr i .x l ( )hrr jon ol ( t ur t i , ,n '1.15 rs sUl i ierc. l i r r nron crscs to t fo! i ( lc r lr .u!)rrrbl ( r rs! . r l i ) r rc l i i rb i l i l \ $ i lh r l r r l t r I lc lord r !p l i . I t iorrs l i r r r r t eonrbinrr i , r r rt ) j l (J i r l rn! s l r (ss l r ( l s t r . I ! lh { l inr lh t ion ( l . fc!( l .n lh i r l .7, / / . t i )66).

4.4.4 Reliability determination when the stress is a maximumvalue and strength is variable

' lhc rs{rrr t ) l io. th i r l x rL i r i ( t ( nrrr i ru l r lox, l ing: l r . \ \ ( i ( . \ r r i rh i l i t } ) is , ts\ i t i r ( ( t rsbcirg rcIr .s.r l i r l i \c i r thc l ) inrrbi l i \ t r . I r ( !1cl \ !hcr ! i | | i i r l r l r t ) cxis ls I r strel ! l t rn)rrrc l nrc\ r t ) |hc\ . r rd thLs is 1fc l1r( l xs r : t )ecix l . . rsc hcf . lhc I roblcnr i \ \h(Nr ltn I rgrLr 'c.1 r l . We .xn rc lcr t ( ) th is nlL\ i ru i I \ l rc\ \ . / . , , , , . l ronr lhc b.glrr inf r ' lr f t ) l r . r l i , ,n ur l i l i l is rcnro\cd S\. \ ( . f rL l t i r ) r . ( lcpc.( lcDl lo ld lng pir IcrDs .r i r \ hclr . , r1c( l i rs I r r \ I l ru i r l { ) r (hn! . rscs. hr . \ l I r l ) l ( thc torqrLc l I t ) l i ! ( l lo x b() l t o lprc\ \ur . r ! f t ) l rc( l lo r r i \c l . l l lhc r tp l i . ( l 1,) i r ( t is s l ror l c lough in du troD nor 1ocrus. \ !crkcning i ) l lhc nfr lg l I due lo lxt igue. thfn i l I r . r ! bc rcf fcsc| t rd b! rLr)rr \ rnt | l l r lord Ihc rc\r l r ing rc l i lh i l l { } ( locs ol ( tcpcn( l on t i rDcxn( l rssinrf l } thc

rigure 4.32 fvlar r!n slat.lo,rdrnq n.es5 and rarabe nrenqrh

186 Designing reliable products

probability thai thc sr'stem survi!cs rhc applicaiic)ll ol lhc load However. tn renlitl

lo{ds are subject to variabilit} rDd il the conpoDenl docs no1 have the strength tL'

sustain rnv onc ol rhese. i l lv i l l l { i l (Lewis, 1996).Therefore, thc rcliabilil,\ .RL ,. is given by:

r:Il;l':L35(

5

o

fr,,.".R, =t f : t | / (s) t / ,s. , ,

l0tJ sr l

/ ( 's) : s l reDgrh dinr ibut ion

| ]quNl lon 4.53 crn bc solvcd b]- intcgral ing the / (J) usiDg Simpson s Rule of by using

thc C t)F l in thc strcrrgth dircr l ly wherr in c lo lcd l i r rm. i .c. R : I / " ( / - , , , , . ) lnthc

c.rse ol thc Nornr i r l .urd l -ognonnr l d istr ihul ions. (hc use of SNI) lhcory f lukcs

thc ct l lcuhLion slr l r ighl ibNrrd. The l lbovc l i ) rn lu lat iorr suggesls lhNl r l l nrcngth

vr lLrcs lcss lhan the nr ix i lnunr lording strcss wi l l fa i l i r respcct i !c ol i rny rc lLrrr l

! r r i r l ion on thc londing condi l rons which nlr tJ,- occur in prtrc l icc

4.4.5 Example - calculation of reliability using different loadingca5es

( onsi( lcr rhc s iLuul ion shcfc thc lo ld ing s l rcss on rr colnpor\ i jn l ls grvcn rrs

/ . - . N(1150, '10)MPn rchr i r rg lo n Norrrr i r l d isLr ibut ion wi lh r t nrurn o1'

/ / / l50MI ' r t r rnd s l r r rdrfd dcl ia l ioo o/ = '10MPrr ' Thc sl f toglh ( i isLf ibLr l ioD ol

thc conrponcnl is s - , \ (500.50)Ml '1. 11 is fccluircd lo l jnd the fc l i lb i l i ty i i ) r thesc

cordiLions usiDg crclr rppfonch r tbovc. givcn lhat lhc lord wi l l bc rrppl icd l0(X)

t i r res duf ing n dcl i r red duly cyclc.

Maximum static loading strcss, Lnax, with va able stengthI l wc ns{une thLt l lhc nrnxinrum slrcss ppl icd is ' - l ' l iom lhc nrcrn st fess. wherc thisloldiDg slress vr lLrc colcrs 99.87"i thosc rppl ied in servicc:

/ . , , , , , 150 + l ( '10) 470 MPt'

Bccruse the st fenglh disrr ibut ion is N{)rnr l . we crrn dclcrminc the St.rndrrrd Nornral

/4-o :no\\o / \ )U /

Ffom Tablc I in Appendix I . the probrbi l i ty ol i l i lure P: 0.17'{251. The rel iabi l i t }R. is gi lcn blr

R:r P: l 0.271253

Rr,." , :0.7257,17

The relhbi l i ly. R/, , , . as r funcLion oi the mlximunr srrcss \ 'a lue used i ! shosn 1nFigure ,1.33. Thc rellability fnpidly firlls oII at highcr rrlues of stress choscD. such

Figure 4.1

Single ,strcngt

I . rom lr

'l'hrs vrlr

Variabapplic.(1974),Li i ing (

Stress-Strength Inte{erence (SSt) analysis 187

:n

I

09

08

Q7

06

05

a2

01

0

2a 36

350 360 370 380 390n00 410 4?A 43A 44a 45t 16A 47a 4A0 490 500 510 520 530 s40 550slrcss (1, , , , ,1(MPa)

Figura 4.33 Nr, l ,b lyr , r r rnLrtr i r l | !x fLrr rnr i ,

i r :1. A| , r i1r(ulxr ( l r l l ieLr l l r L j r rh isr l ) t ) , . r ( | s lh. I . th.(h() iceoi n j , r \ i r l rU r l , ' i r ( l I !\1rc\s lh, i l r . l lcr1\ 1 l l ( 1rUc: l rcs\ (J l th. l ) r ( ib lc i l

Single application of a vaiable static loading strcss with variablestrengthSub\ l i l rL l ing l I f ! r lc i r t ) iLrr I rL lcrs l i ! sr1.s\ rLrr( i s l rcI f lh LI thr c()rLI l in! ( ( t l r r l i ( ) I( ( ( tLr i r l i , )n I l f ) g i lcs:

_ io l ' t io

\ /5or t '10' I11

l i ( !n l rb lcLirAt) t )c l r ( l r \ l lhepn)birbi l i l ! ( ) l t r i l ! rct 0 0t l ' ) l i l t thc f ( .Lrrhi l i t \ rs

, f i t / , t0.()()r i ) t l

R 0.990r5N

I l r is \ r lu. is nrLrch n)ofe oDlInr\ t ic . ls !oLr rvoLrkl lssunrc. hcclLrsr rhc fct i i rb i t i r ! is l t rcf fobxbi l i l ) o j hoth s l fess l l ( l , i t r .ngLh bcing in lef l i ' Ig nol . iLrst lh. \ l rc.g1h b( ingccluir l lo r r r r r rn lLnr \ i t luc.

va able static loading stress with a defined duty cycle of 'n' loadapplications with variable strength using apprcaches by Bury(r974), Carter (1997) and Freud;nthal e{at. (1966)l ls ing ( r l tef s .rptrorch Ji f \1. 11oln eqIrr lon .1. ,17 $c ci l | c.r tcu|Lrc LR to bc:

IR: ' . .

061/50 +.10-

We r1rc.rdy kDo\\ ' SM:2.3,1 becxusc i t i \ thc posit ive vi lue ol rhe Str1ld.rrd Nofmlt!arate. - . cr lculr tcd rbove Thc probabi l i t ! of i r i lurc pcf uppt icul ioD ot lo ld

188 Designing reliable products

/, ! 0.0009 irom figure 4.11. Using cqualion 21.48. and given rhal , : 1000. gives thcreliabilitlr as:

R,, : ( l l , ) , ' : ( t 0.0009)rofrr

Rrooo =' 0 '106405

Ncxl using aur) 's rpproach. from Trbte, l . I I thc cxtrenrr l pnramcrcr! . 1j and e). f tonran jni l i r l Normri lot ldirg strcss distr ibLrLion arc dctelnincd froml

[:n(,) o.s rn rn(,t r.:osslL v ' 'n ' ' ' I

. . . . "^ f : tn ' I , ,no1 [ . In l r ) r l rnxrr | :b:51r, , L 4u 4-4 , r , Vt , r

/ l ln l0((r lrnd

er. -.L -_J.:= - to76Mr,./ : l , r a / - ' l r r \ l inr l

Subst i lu l ing in thc pl f i rmclcfs l i l r borh s l ress t i r rd s l fcngrh in lo cqur lR)n 4.51 lndsolvrng Usrng Si lnpn,ns l tu lc ( integf l l jng bcrwcc,r thc t inr i ls ot . t n( i t0t)0. t ; lcxrnrplc) g i !cs th:r t the fc l i r lb i l i ty is :

/l|)00 0.645

This v lue is nlorc opr i rn is l ic rh o lh:r l dcref ln incd by ( r r rcr .s . rppro ch.Ncxl . solv ing cqurtron 4.15 direcr ly using Sinpson.s I tutc l i r r le, , s descr ihcd by

Frcudcntbr l ( / d/ (1966) l

l r , [ ' l . i . r , r . ts,0

A l-prrrnrclcr Weibull .tpproximrres lo a Normal distributi(nr when iJ . :1.44. .rnd sowe crn corvert the Nonnrl s lress 1( ' WcibUtlpnr ' rnrclers by using:

\ol -

t t t 3. t394111dt

,z! l1r+0.:153018,1o.

i t . : t .41

Therelbc. the loading stres\ CDIr qrn be represcDrcd by a :t paramerer Weibult dis-

r r7 r . . rpf ( : "") )' \ \/), ('J /and lhe strcngth is represented by a Nomal distriburion. lhc PDF being ,ts lbllows:

;

.9E

g

1

09

0.8

0.7

06

0.5

0.4

0.3

0.2

0.1

0

Figure4.34

R,, iI

Figure 4.-1

s - N(50(

natcly l0(

rcli.tbilil\

thal the c

(Bury.197i (s) : ; ; . " ' ( -- . t f )

0.9

^ 08

.i-

5 06!

: ' "

3 oa

!0.P

o2

Stress-Strength Intedrence (SSt) anatysis j89

1

0I

cane( 1997

- Bn t,1974

10 100 1000 r0 000Nlmber of toad apptcal ons.,

f igure4-34 ldrblry,rr l ,n( jonolrrmbroth.rdrf t ) raror!Lrr t r l { j t f { rentauro. . t t rs lorLR 06,)lixirLrn ord nq krnrhnr$tiid tM 2 l4

rulrrg lhcsc rnt() cqul lkD 4 lJ 8i lcs the rc l i rb i l i t ! . / t , , xs l

, , ' ' ( ( , ; , , l l r l ] . . " r , , J. , , ,ol thrscqu.r l ioI nUnrc .u|) t i ) r , l l )00. thcrct i h i l i t l is j i )urr t l ( ,

r ' IR, t l

.Js ,a I

{r0r = 0'69( l

I r r r . r . l . , I , r r ,q, .he r . t r . , 1 t \ . , . . . r t | | . | | , . | | . , t r t .L r , r r . ,cr , . t t , , . r . t . ,nnt , ( : , | | , , , . t, , ' , r , l ( r r ) ,Jr : , r r ' , , . r , r . - L i r . , 1e, , . , ,J(r , r , . , ( R . . i , . r , . r i . . , j : , . . , " .

, , . ,brr$ecn rhe rct i rbi l i ry v! lucs cr lcuhrc( l t in. , . . roi ,o. nq., ." , i "* , r i : "_. , . . j1" ' i l , i .: l ] " '1,11' , : . . \ , id i , lC i rrcss. . - . \ ( : r50,,10)Mp!. Dur wrl l r l strcnglh drslr ib tron ol ..J- , \ (50(1,10)Mt,r incre.rscs rbc l_t{ ut l re 10 0.1i9 n( i SM:]15 Frqurr . t35

shows t l rrL r higher LR vr lucs, rhc resul ls. l rc in bcl ler.Lg.." ln.nr. up uiupp., ,r i_m,rclr 100(l lord nppl icrnio s, \ \ ,hich is rhc l i r ) iL t i r sLl t jc design.

Thc rbole ercrci le \uggesLs th. i f t lc hro used cqurlron ,1. ]2 n) detcrmine lherel i ib i l i ty ol I componcnt r ,"hcn i t is known thl ] t rhc tor( l nld). Ue appt i"r t rn, ,nr ll rmes dufrng i t ! l i1c.r l r overopl imisl ic vatuc \ \oLrkl ha!c bec. " t"" i*A.

i . f r ;" , , , " , ," ' rthrt rhc comporenr cortd expcrie cc l ] ] (rc fr i lures ,r ," , ,U,

"" , i . - ,n,u.a,, i - i r , .drsrgn sL ge. Thrs rs commor prlcticc rnd fundaricnta y inc(rLrecr approactr( B ufy. l 975). A hjgh confidcnce rn r he rctil b irl ert;. "t..

tr ".."pra

r;r rr,. riiuntin,,rhere.r srnglc applictiion ofthe toad is cxperienccd. R . Howci,er. rlle confidorce r-s

190 Designing reliable produds

1000Numbr ol load applicaio.s n'

Figure4.35 Relabl tyasafun. lonolnlmbroi oad appl ca1 ons ui nq dr l ler-"nl approa.hes for tR 089(roLrqh o.d ng) and Sl\rl : 35

lowcf whcn dctcfmining thc rc l i rb i l i ly t ls l l !ncl() l l o l lhc nrmbcr ( ,1 lord t lpnl ier-r ions. f i , , . whcn, > l . usirg thc rur ious xfpfonchcs oul l iDcd r l kN LR \nf t rcs. Athighcf l -R vlr lucs. thc thrce rppftxchcs lo dctcmriDc thc rc l i t |b i l i ly lor r r , l ( lo

givc s inr ihr rcsulrs rp lo, : 1000. lh is bciDg lhc l ln i t o l lhc nunrbcr ol lor td uppl icr lrnrs val ic l lbf str t ic desigrr . 11 ct ln l lso bc sccn lhnl a vofy high in i l in l fo l inbi l i l ) isrcqurred iom the design 11 R 10 bc nblc L{) survi lc nrnDy load nppl icalnnrs rndsl i l l mt l i r t r in.r h igh rc l i . rb i l i l r - n l n, , .

4.4.5 Extensions to 5Sl theory

a

.9!E os

3 04

5 03

u 0.2

(Bcn-H:r lnrdlElculi tot-ails to Dro

Colrlidcf

r i lc l t in th

Lhc l i l lc /

4.s ElThe c.rlculi

rd thc liL

str l is l icr l I

/ (L) . rh ls

Use ol t

fling the ll

(Shigle) ru

10 1o ooo

' l hc qucsl i inr . t r ises iom the boYe.rs 1o thc nmourl ol crror in rcl i rhi l i ty c lcul l ionsdtrc tr) the rssumpi ion of ronrr l ly dis lr ibulcd slrcnglh iDd paft icul r l l thc l () ! id iDgsress. when rr f rct onc or bolh could bc Lognormal or wcibul l . Dislr ibrt ions withsmNll coemcients ol \ . | f i L ions (4" < 0.I for t l re Logrormal distr ibur lon) rend lo bes,\nluretricrl llnd huvc r gcneral shrpe slmihr to thal of rh Normal type wirh.rppfoxiftrlcly lhc snmc nrcrn .rnd standafd deviaiion. Differences do occuf at rherril probrbililies (utpcr rail lbr stress. lower tail for stfengih) rnd signilicant efrorscould occur liom substilLlting lhe symmeirical form of the Normal distribuiioD folr skewed dislr ibut ion.

I l lhc lbrm of lhe dislr ibut ional nodel is only appror imalcly cofrcct. thcn thc t . r i lsm.ry drlTer substantially from the tails ol llrc actu.rl di\tribulion. This is bcc.rusc thcmodel p.lramerers. related to low ofdcr mon]ents. afc dctcrnrined l;onl tynicalnther than r e events. In lhis casc. design decisions will be srtisfacbrr- tbr bulk

Elements of siress analysis and failure theory t9l

crtuslrophcs. lbr cxample. $hich rfc ot'rcn ol.greaLcsl concern to tr,e a"rign. ifrcsubopriDral i ly ma) be ml]ni icsled as ei thcr I( Bcn-H.i.'. r eeai v.",,,,,,"r,.", r,",*'*::T::'li:"'j:il: T ,iii jil1i""'ff ;di l i icul l 1o ver i l ,v or just i t ) $ i1 l r concrcte dr l r r r lhe r i ls of thc djsrr ib" , i , , " . l f . , " l

fa i ls to model rhe r . r t behal iour of rhc bajc \anublc\ invotved -*""r1, .

i r , . " i i , .resul t 'ng rc l iabi t i ry tc let is qucsr jon.rbte rs norcd (Maes t l l ,a n*t , , , rg. t , i , i l . ' r iIn. \F. : rJ\ .nr , r r \1 iu\ lh( f t . ,1, f . . . r . , r . r , t . r t . . r r ! . \ r r i r . t . . , , , " . , r , f . , ._, i , , i r , , . " , , . , , , ,r ' , i l . l , : , f . , l . r . . rh\ , ,Lt ,< r . , , rd, . , , , \ : ,1, ,ht( . . t : , , t ,pp, , r . , , , . . r : . , , : , " . ; , . t . ,1, . . : ' i ; . , :( le lc loped sfeci l icul ly lo rchievc th is (Kiefcngrrccn and (bmer. t99O.(, . r h, l . r r . < k\ . r . , n t IJ r ' ' . . r : ,h. , r t \ ( i tLr . , tL. \ I . , .n, . l r . .SSt In, \ l ( t , . , r . t . ( r ( ,11r.n. \ l: ra rr . . , \ ' t , r t s hrr l t r . . t , t ) t , t t . r , t r ( , , . , I . t . t r ( ! tn . . i ( h. , \ rd . , , . l t \ . n r . r , | . , rh,UnL'r l . . l . l r , , r { l ( j ( ( r ! r t r

192 Designing rel iable products

in lailurc thcory is thc chrracter of loading. whether st0tic or dynamic. In the discussion thrLl lbllows. $c conslrar lhe :rrgument io frilure by siatic Ioadirr only

Olicn in slrcss inal)sis we may be requifed !o mak! sim pliJicd .lss uDrpt()n s. nd ns arcsult. nccnirinties or loss of accurncy tuc iDlroduccd (Bury. 1975). Thc accur'{c} ol'calc latior decreases rs the complcxi!) iDcrcascs lionr rhc sinplc c.rsc. but ultilrr.rtel)thc component prrrt will still brc.rk rt its wcakesl section. Theoretical failure foflnulaeare devised under rssumptions ol idcal nrxtcriai ho ogcneity and isotropic behaliour.Ho ogeneou! neons tht|t thc rnrlcrirls properlies are uriform throughouti isotfoprenrerns thrt the mrtcrirl propcrtics arc independenl of orientrlion or directior. Onlym tbe simples! d crLses c.Ln rhcy lunrish us wilh the cornplele solution of the shessdistdbut ion problcm. In thc mi jor i ly of crses. engnreers hrve to use lpproxinlr t rsolut ions rLnd l lny ol lhc rcal s i tuat i rs ibal ar ise urc so compl io l l led lhnt (hcy

clnnor bc lu l ly rcnrcscntcd by r s ingle mrthenrnt icr l nulel (Go on. l99l ) .' Ihc l : r i lnrc dclcrrr in ing s l resse$ nfc r lsr) o l lcn k)cr tcd in locr l rcgl(ms ol th(

lonr i roncnt rnd r i rc not ers i ly represcnlc( l by stxndnrd strcss rnr l r -s is mcthod:(Schr l . / . /d| . . 197,1). l -ordsin(rvoofnrorcrxcsgcnc l lyprovi( lc1hcsrc Lcst nrcssc' .rnd s l r ( ,u ld bc resolvcd into t r incipr l strcsscs ( l rcsolr .? d/ . . l ( . ) {Xt . In s lLr t ic l r r i lufclhcory. thc crror c n be fctrcscntcd by r cocl l ic icnl o l ! l r iu l ( ) r ) . .urd lus bccnploposc( l |s ( , = {){}2. lh is nnrgin ol cr(n incrclscs $i th dyrramic odcls rDdj i ) r s l i l t ic l in i lc c lcDrcnl rnalysis. lhc cocl l ic icnl o l vr l i r l ion is c i lcd l ls ( , . .0.05

(Snr i lh, 1995i t l lh l l r r ) . 199?).t lndcrshndins lhc polcr) t i t | l iu i lufc mcchrnisrDs ol n pfoducl is . lso ncccssrry I , ,

dcvck)p fc l i rb lc producrs l rLr i lufc nrcchnnis 'ns cnn be bfol |c l ly gfouped i r i to o!cfst f tss(tur cxrmplc. br i t t lc l i t |cLulc, ducl i lc l incrure. y ie ld, buckl inS) rnd werf l t (werf .

corfosor. crcep) nrcchlnisms ( l )nsguptr rn. l Pechl . l99l i . cordon (191) l ) t l rgucsthr l rhe nunrbcf o l l r i lL lc mocics obsefvcd probubly increrses with complcxi t ) o lthe systcN. thcrcl i ) rc c l l tc l ivc l i r i luf t rnt l l r -s is is t ln csscnl iu l pur l o l rc l i l lb i l i l ! - work(Buf l rs. 1994). I hc l i r i lurc governir)g st fess musl be dctcrnincd l i r lhc l l i lLr fc lnodcin c lucstknr rnd iht use of t MLA in dctcmining possiblc l i l i lurc m)dcs is cfuci . t lin lh is rcspccr

Thc l in lnuht i {in Young( l9 i i

r probl lb i l is t r .

4.5.2 ComF

4.5.1 Simple stress systems

ln posruht ing r s lat ist ical model tbr a sht ic stress vnr iable. i l is imporlrnl 1odist ingr sh bctwccr br i l l lc and ducl i le mnler inls (8ury. I975). I ror s implc strcsssysrcnrs. i.c. uDi.txial or pure torsiolr, whcrc only oDc lypc ol strcss rLcts on theconrponenr. the followimg equrlion! dctcnninc thc litilDrc crirenon tbr ductile andhri t t lc l r -pes to predict lhe rel iabi l i ly (Haugcn. 1980):

For ductjle materials in unirxirl lcnsion. rhc relirhilily is the probabilistic require-mcnl to avoid yield:

l lcdict ing l l i l

( though lhef t

d i l lcrenr l . l i lu

Ductile fra