!

Ftlh-b

D.r,tII

l-I

,I

| "se stldies 20i

I rbiliry to meer the rarger requiremenrs. Evatuating and comparing alternative desiqnsI and choosing rhe one wirh the grearerr predicred retiabrl iry ui;t p., i ; ; ; ; ' ; ;"I

effecri\e desrgn solurion. rnd rns is rtre approucl ua,ocu;; ;";; i;;;;:r:;;;:"

| , , , i f i i l l l l ; :nvorhers\rorkinsinrhisarel)eieaaanaH' lu," . i is; ' r , i i ; ' ""

lJlg/..rAn^alternativeapproachlorhedesignerselecringrhedesignwiththehighestrcl ia-[, " !ll']t

f,:l: ""rberofdesrsn schemes is ro make smalrredeiign impr""".1",ln ii._ ongrnaldesign, especially ifproducr devetopmennmEi. c urOb ro maximize theimprovement in reliability. this t.i"g""fri.*a UV .",iV ,Vrrc.r ti.changes to the design configuration (Cla using, I 99qf. attf,"ret lried J ir-difliy

";;"",be measured effecrively, the design parameters th"t a"t "^ii"

*ii"Oiiitv l"ibe.-""0the control and verification of these parameters (along wirtr an efieqive proJucrdevetopmenr slralegy) wilt lead ro rhe aflainment "r "

.!rla . J..;g"V?r"r"l'"i,t996). The-designer shourd keep rhis in-Eind wli" d"riil;;;;i;C,;rij'g;;:;alggchjdaqlaria&abour rhe criricat parametelytrroulloui itre pioau"i J.ffi:-ment process before proceeding w-ii6'iilliii$Gfr,. *lr-i.*rn."i ir r,ifi ..ri-"iiiii,at the design stage is mainly the apptr(luon or aoglneeflnt common sense couDled$fh alnericulous attention ro rrivial details (Ca e;. l9g6r'l\lhe range of iroblems rhar;E6abilisric techniques c/n te apptied to is vasr,ba.sicalty anywhere where variabiiry dominarss tr,", i,rotr., aorniii,. iiri" iornio-nenl is critical and if the nriii66Gi-arinot wetttd", ,h"; lh;;;;;;;j;;;;il,be included in the analysis. Under these sorrs of requirements, it i,

"rr""li"i tquandfy rhe retiabitiry and safety of engineering .;p"";"i;:';;"0;ffi;;

analysis must b performed (Weber and rinny. test;. tn rerrni of SSI ;;;iil;;"main application modes ar:. Stress rupture - ductile and brittle fracture for simple and comDlex sressee

_ . Assembty.fearures _ torqued connections. .r,.inr.'nt., ,o"p nii. .i,;;r-;i;.

"".. ) other weak link mechanisms.

i (9iutrcri "" """ "rGI'"r I",.resr. stress distributions injoints due to the matinc

- ::.*:::::-i.::T^bly-T to b investisared. Stresses are induced by d"-;;;;i;operation and have efects similarto residual stre"* e"i"., ljljl. iii.l, * i.oiill* $::,: iT ;:il J":Jffi Ht: tT:":ili r.jl;n*r;.1^: *i:;li*;

[:ll;:5.J**"* buckring or vibration i6 made p*,iur.

"ii"e ,i.'#i#]

ffxill1;".:ff.".1'1, ill:;t rate the application of the merhodolosv to a number of

4.8.1 Solengid torque setting

ff; ffii"?"'';.T:ixl"J;ll'J:X",:1.:i:Tiifi"i,:Tffi JITJj: *T;ii:

204 Darigning rellabl produds

0??

Ta

Flglr 4.40 Solenoid arEngement on assembiv

could fail at the weakest section by ,trcss nsecondry rhat thc pre.ro;d, i;;; ;i" ;;;iii"i:s:.1:ffi:Tlj,ll ::i:";i::i:Tw;ffi :lr#:,l.ll','ii.Jil,J +i:x",,fl'f"in:i:.,:*.;: n:'*jlil":l;";#;"1".n:"t"tt

these two competins failure modes "'-i' "-i'.iiliiiiri

The makrial used for the solenoid body is 220MOZ frce cutting stecl. It has a mini-mum yietd srrcnsrh srmn = J4OMpa ani a rninir". p-"r'rii;"r, lp., j j,io iiilifo^r rhe si2e of bar srock (BS 970. l99 t ). The outsia. aii..,.., o. u, iriii.l;.f ...,ioiof th,M14 x 1.5 thread is tumed ro th; tolerance .p""in.a "na

tfr" n.la" !i"rn"i"r,;is drilled ro toteranco, Both thc solenoid body ana-frouring "." ""arrriurn

plliialfi,lsolenoid tu assembled using an aL toot with ; clut"f, **friri.. gi"irg u-ioii *"it*il Jl:ff .':f iJfJ:f isi:*'"i:$T'fiT?; ;Tl; r'" tr'i"iil"ieii J'eJe?'":'iiProbabi I irtic design apqoech

Sarcrs or flst sssenbly

il!^T.",1.il shows rh Suess-strengrh lnrcrference (SSI) diag]rams for rhc lwoassembty operadon fajlure modes, The instanraneous stress on tie relief sccrion on

I::,n[T''J'.":*ffi;,1 :ij1j;"H*:* "ppr.a rc".;r.'i'.* l. a"" i" tii,u, "na tr,i. i,,io*n li irel;;," ;i"iiiir;filhl'"llJ:,*:,'ffi fi ii,:'ji?j,';ar a maxrmum during rhe assembl' operdlion. If th. "ornpon.ni

."_f"i, ,ii"str$s.it will not fail by stress rupture lat;r in life

(b

P

Flgur.4.4l SS mo(

'=7=;@=4

An apprcxrmateM, for a given prfor bolts and falfound in the con

wherel

Theftfore, combMr

(4.10)

(4.11)

Cate studies 205

(a) Situa on on fl't asmbty

POF

st&/Btrngtil

(b) Sliuafgn atter r6taradon ot shsar 6trssPDF

Slr!3dstr.nglh

FaIurs

Flgore 4.41 SSI mode s for the soleno d asrmbly lailure modes

,{ = cross_sectional area

"/ = polar second moment ofarea.An approximale relar ionshipiscommonly used todelermine lhelorque forassemblv.

"v. for.r grven pre-toad. r(shistey and Mischke. t9s6r ,, i,

" ",;;;;,;;;;',";ror bohs and fasrener, derermined from experimcnr ""d

i. ,.biJi; ;;';;i"i;;;found in the contacting suriaces of th parts on assembly.

V KFD ta1)\uhere:

1{ : rorque coefficient (or nut factor).Thereforc. combining the abov equations in rerms ofthe shear stress gives:

32KFDr1(D4 d!) \4.73)

206 Dsigning roliable products

The principai stresscs at the reliefsccrion. .! and 12, are tOund from:

\ , - 05. . ! , / t l5\ , r_. \4.74)

lT"1'T,j,:*il:i;:"i[T1'::;];:i':i::.iii;*:li,ilxiillii'iii,ii,LllitTilt;,";'1i14.75)

o=r(sr ' \ , f ; f ; ,Stress in seftice

lffi ?i6s'I$ilffffi'fri$#fiti;'{i-.l:*"**q*;:rt*a"Jrilff t',,.,H:''i*i1f [il#,#]::'j]:'"r':'"",::Determining the design variablesBltore a prohihilistic model r.rn be dcveluped. thc variablcs invol\cd mun bedcrrmined. lr is assumed lhal the variables ril foff"* ,f,. l"r."l aiurii"ii"" ,,iithar thcy urc st ir i* icr y indepcndenr. i .c. noL correhled i" ,"1*"1.- i i ,"."", i .r '" ithc prc-toad. F. usrng an.rir lool with a clurch,,

"ppro*,r"t. ly;. i i ; ;_;; ' , t l , imcan, which givcs thc coefficient of variarion, C,, = 1y. i, 35sg.;n, !t ;;;";; ,;t.range. thereforei

n=p(a>0.5.st ." , )14.76)

(4.77)

Flgur 4.42 P

The shi l lc

For thc torque coemcient. f. reported values range from 0.153 (o 0.128 for cadmium

i;:tl"iJ:::,;1kfiilor 0 24 (shisrev and rii*t't"' r's;) iiffi'"' ;F"i;i;;px :0.24 and oK = 0.0292

i:J.,:::;*t1li,tff :;ilill;?;ll#i; ]if i;i"ffi ::,liTH?:,;iff 'J;:li;*[:lnrgesl stimare (or wo'.l case) anricipaled durins "

p,.ar"iri,, |".l-C;i"" ,r,iiii'l:lT:lsl:n, 1,

= U l2-+ 0.03 mm is lurned, rhe material to p,o"""" ,t"r "un

'r,"

:ilJl ffiA"=*]. "nd the geometry to process .i.r, so : l. en oajr,;rei ;r;:

Adjusted rolerance: Desien tolerance (r) =**=0,,

Thcrclolc:

li follows rh

Similffly, Iiprovided in

isk. to gjveThe stand

*1,*,,,'',.;i;i,1ffi :itdf #i1il1i+',r:ili:*tilr$:*i

1,75)

a.74)

Fr-lhelie.lnlicek-

lor

f:F

IFI

;

)I

P, toDrnc

Case studi.s 207

runmloaonrro -PROCESa cAPAgtLtw ritAP

0,

o

o0ot

DTUGEA/DUEitsroN lm)Flgure 4.42 process capabil ty m6p for tuming/borirg

* The shifred standard deviadon. /, for th

,. can rnen be predicred f.or .qu"tion 4.2idittnsional tolerance on lhe diameter.

"; = ';t'" - o or :-r 'oY = o.oo28 mm-12t2

Therefore:

pD = l2ftm and oD = 0.002gmmIt follows that for the radius, f, the variability .. halfofthat ofthe diameter, Di

l r , = 6mm and a,=0.0014mmSimilarly. for the inrernal diameter. ,,L thepro, idea in Figure.+.+i. n;i;;;'il,;i:'."p#;' j:XXtJ':l ff :,,:T.*l'lj";ls glven as a .+' only,

Adjusted tolence = Design tolerance (7) - +0.2hp.Ep i l i iJx l l .5=+0162m6A risk value.,4 = 1.02. is inlerpolated for arorerance- This vdtue again defauhs ," ,n.o]T:t]:l-:f

onmm and rhe adiusredrisk, to give 4- = 1.02.

'' "'- component manufacturing variabilityThe standard deviation for one hslf of the rcleranc can be estimated by:

.,, _ | r / ?!. qr. = \+g = o.oo87 mmTherefore,

pd t 9.1mm and o/ = 0.0087mm

208 Designing reliable product5

DRILIING PROCESS CAPAATLIY [,tAPindcpcnden

The variablNormal dis

oro,n.l" r*r I

Figure 4.43 Process capabiiry map for diUing

Determining the strcss distribution using the va ance equationAssuming thar al l rhc var iabtcs fol low a Normaldistr ibut ion. I proba; i l is l ic modcl! . rn f 'c crerrcLl to dcrermrnc !he slrc, , dishbut i , ,n for Lhe dr, t ia i lure n), \ lc usin, lrhe \JnJncc c-qurl i ( ,n Jnd n, l ! ing u, ing lhc Frnrr" Oiner*ce V.rfr .A r*e nppenJi iXl)._Thefunction for the yl, Mlrcr stress. 1_. on tirrt asc.tty ur *,e sj.noijiJi,,nis takcn from cquation 4.75 .tnd is givcn by:

. : l(0.5r + y6-r,, + ;;. + 10.s, v6.zt, +;1.- 105,, /025..?, - ) , ,0s._ /u25\: . - : ,4?bl

4F 32K1:Dr-- ' .ndt \ tD. - t , t - lD" , f )

As there are five variablcs involvcd. f.. ](. ,, I and 4 the variance equarion becomes:

Consider lh

and

For the firsl

Letting rhe

",:lG*| ,,.(f)' rr.(,#l *"t(,a)' e +(#l 4",',r,,n, Repeating IThe Finite Ditrerencc Method can bc used 10 approximatc each lrm in this equarionby srng the diffrence equrtion for rhe first partinl derivarive. The values of thelunction a1.rwo poinrs cirher side ofthc poinl ofinterest, k, are derermined, fr, rind)k r. Theseare equxlty spaced by an jncrement Ar. The finite dl ference equitionapproximares rhe value of rhe parrial dcrivarivc by taking rhe a;tf"."n"" of tn""values and dividing by the incrcment range. The rerms subscriprea by i ind;caie

bnc modellc usingppendrx

Case studies

j : : :_o' l t ] . . r . i , Inci :menred by a\, ror catcular ins.rL,r , and f . . hordin8 rhe orhcrInoenudent var irbles consrrnr at lherr I \ajue pornl \ .

/9)" ' " , , . "\ox, / \ 2AyTle van.rbles /'. K. D, . and ,1are ,r assurNormarris,rihu,,on \air; iil;;;";:;;;:li,:l;,'"..'j;lT T.Tl,[:i.lj:;1",,*

r.^ N(t,r,0. l trF) NK _ N(0.24,0.0292)

, _ ,v(t2.0.0028)mm

-R - /v(6,0.0014) mmD _ lr'(9.1,0.001i7) mm

::l-.j:: rl: poinr r, ro he rhe mean varue of I variabre llnd x, _. dnd x, , rheer(rremes otthe variable. Theexlremescan berhe] exist +4' a$r) rrom th.;;;:;il:'.::::-rned

ror each varrable bv arsuminsFor cxampre. for the prc.il;;;;;.;':;:'.:::::'pproximarer' ee eeoo orsiruat,ons:(Iremes become (for pr. = l0off)N)l

Frrr=14000N

4-r = 6000 Nand

AF = 4(1000) - 4000 NFor the firsr variable, F. in Equation 4.ti4.

f0L)*r . r , r_2, I\'J? / 2Al'

Letting the variable i. be ils maximum valurmcan varues. sives L, ;.;;;;;;;; i";;:..,Ti lffl[: ;1.','" kepr anhe,r

L*, t=423.6x 106 and Z* r= t8t .Sx 106Thereforc,

14 78)

(41e)

rf the

lnese

Repeating the

210 Designing reliable products

The mean valuc ol the ro, Mh6 srrcss can be approximored by subsritlrring in lhcmean values ol-cnch variable in eq alion 4.78 ro give:

Pz:3026MPa

Therefore, lhe loading stress can be approximated by:

Z - N(302.6.35.6) MP.r

Thc !,, Mircr stress, a. is thcn dctermined forvarious values ol pretoad. F. using theabovc ncthod. Equally, wc could hirve used Monlc Carlo Simularior ro derermine uranswcr Ior the stress standard dcviation. The answer using this approach js in h!.td/

- 16 MPa over a numbcr ol tr i t runs.

Stress-Strcngth lnterference (SSt) modelsA statislical rcprcsentrtion ofthe yicld strength for BS 220M07 is nol .rvaihbtei how-cver. the coeflicicnl ol varintion, C,. for thc yiekj strenglh ofsteels is conrmonly givnas0.08(Furman, I98l) . I .or conveDicncc. the paramelcrs ofthe Nornral distr ibutrcnwi l l be calculalcd by iNsuming that rhc minimum valuc h ,3 standard devial ionsfiom the xpcctcd mcun value (Cable and Virene. 1967):

It.,n = irs, 3o.r,

340 : / , r j - 3(0.01i/ , t , )

/rr, : :-- = 447 MPr

lrnd

d\. , .=0.08x447=t6Mpr

The yicld strength for 220M07 san b approximarcd by:

s) - N(417.36)Mpr

ln the stress rupture case. the intcrfcrcnce of rhe strcss, Z, rnd slrcnglh, S'r. bothfollowing .r Nonnal distribuiion can be dtermined from the coupling equrtioni

t;V-sr +di

and the reliabilily. R. can be determincd as:

n: I osND(,

Fof rhe loosening casc. the probabiiity rhal rhe loading slrcss is lss tban a uniquevalue of stress (0.5S1nin) is used, given by the lbllowing cquaiion:

/o s V''" /r\l - l l - lUr ./

the reliability. R. again being determined by:

n : I O5p(z)

0000000

o

0000000

ti90re 4.4

The reli.'

isscmblyof 0.999

DetermTh maxlypically

The mca

Case studies 2l lI the '|

0.990.980.970,960.950.940.930,920,91

l*-----------.--- i

25 30

U.an to|quo (it ) (Nm)

pF = 12976 3(0.1x 129?6) = 9081N

! rhe

facrF

itn!on

0,00.890.880,870.860.850.840.830.820.E1

olh

Iue

4A

Figure 4.44 opt mun value ol meaf app ted tooue

The reliabilities dueto the competing failure modes of stress ruptur and looseningcan be superjmposed on a graph againsr mean rorque. M. as shown in Fiaure 4.1+i^n

opumum varue can then be selecred bssed.onJh;:_"r:$;Ifi;J,i:;:,lHl

:I1X!?';:l'T,ii^= i'"i iT ?::'# ilil;scheme. an obviou; im;;;;";; ;;,i;";i"1f;,J',t1""',ff:1:::'fi [: ::,',,,::internaldiameter by redming lo f in irhed srTeDeteministic design apprcach

;xi.:lift?[,T:;,j:i;i;::";ilI liiil,l,li pr6opo(ion or ,he proo, ,oar. Fe

Fm* = o.eFe = 0.es?nin ito, - ar\

=o.n "too x 106 x;(0.012,

_ 0.00elr)= 12976N

The mean preload, pF, applied car be found from:

212 Derlgning reliable products

\%)

FKDtd

O3tgn vadabt

Flgure 4,45 Sensitivity analysis oi the strcr5 on I rst assembiy (for F = 1 0 000 N)

The mesn or tsrget assembly torque is calculated from equation 4.72:pM = lLK. pF. pD

pM = 0.24 x 9083 x 0.012 = 26.2NmThe assembly torque, M, is l07o lower using the deterministic approach than thatusing probabilistic anslysis. This would result in more of the solenoids becominsIoose and failing m service. approximatc\ 2qo from Figure 4.44.

Sensitivity analysisThe contribution ofeach variable to the frnal stress dhtributio[ in the case of stressrupture can be examined using sensitivity analysh, From the variance equation:

4. = t.2674 x lorr=(9.ls5xto'a) + (3.5417 x 1ot4\ + (t17 265,2)+ (13?8.3) + (439235.7)

Therefore, !h contribution of the pre-load, F, to the overall stress is given by:9.155 x l0 ' "Lz6:,4;7ir';xtoo=721%

The same process can be prformed with the other variables to provide Figure 4.45.Itshows th6t tle largest varianc contdbution is provided by the torque on assemblywhich afrects the Dre"load. a

4,8.2 Foot pedal optimum design

The objective of this case study is to determine the optimum depth, I, of the foot pedalsection as shown in Figure 4.40 using sraric probabilistic design merbod,. w; are

f l '1i'l

Flgure 4.46

glven ut4 describ(the sectiorsection d!material (ithe load (statubfy r(s)=6,rsafety baclthat l0 00will be fcsv - w(2

Figure 4ally durint

Lo.d (N)+

e00J---:

,ao=+31ef=-

450F-

!00F

I5O T=TF

o,_IFt

Figur 4.47

Case studies 21i

ft'

l*- - ^o-.- - ,,]

I l< 6=' '?1o5mm

l-atlW

ii;;lh1;.ili*.-.i#i#filli,l.-1t.;l:i;"rr",.r{{l,irj{Hrr+

iii*$*g$g*H+'"-.t;l'll il{fqtitr**ri$r+H#':;;4,:ff '#;::#f ['.","^i.iffi il,j,:lffi,,; liffi ; l,'.'ffil ;#:i:'lTj"l ;":?,:i,i:Jf.T i:[.; l.j;l::ff ::1H1X:

Figwe 4,45 toot pedalarangement

Lo.d (N)

Figure 4.47 Typica oad h story durng operarion ol the roor peoal

214 Designing reliable productt

pedal will be subJectcd to no more th.n i000 indcpendent toad appticarjons during itsclesigned scrvice life.

Determining the load distributionThere is sulilcient informalion to gneralc a pDF for the load assumins th.r the loadhi , rof) o!(r rh( f ime inrervat. l i \ represenrr l i \eofrhracrual t , ,aJhi , ioryrnsenice.Thc approach uscd was discusscd in Section 4.3.3. Firsl we can divide the load on rhr-ai(is into classcs with a class widrh, 1,, of 30N for conveniencc. Bv summins theJmounr of r imr. , . rh.rr lhe r ,) : ld , j rnat fa s u rrhrn erth.ta,s, uc crn oirain rr rel-ur i r .mcasure of thc load frequncy wirh respecr to the limc interval. Z For examDle. th.shaded strip in Figure 4.47 represents thc Inrd class tiom :7lr rd .100 N \ irh d mid_class value of 285 N. The approximxte amounl of timc that th sisnnl is wilhin thrsload class is approximarely ?.9% ol rhe lotr l Lrme. / Rcpeat in; lh i ! pfoce,\ torexch load class builds up a frequency distribution tbr the bad in perccnuge of rhctotal . as shown in F_igurc 4.48.

From a visual inspccrion of the hislogram, it is evident that th load frequenc,approachcs zero al zero lo,ld and is slightly skcwed to rhc lefi. Thc 2-paramcrcrWeibull dislribution can be effeclivcly used 1() modcl this shape offrequency dislribu-tion with a zcfo rhreshold. For conciscncss, it is thc only disrribution typc considercd,a lthough comparison ofthc load data with the Lognormal d istribution may also be peFlbrmed. The frcquency wrlucs in ligure 4.48 are in pcrcentages to one decimrl pl;c..but we ctn simplily for the process of filling the dhtributional model to the data b,multjplying thc liequency by l0 to convert ro whole numbrs. thcrefore N : 1000.

The Lmalysisol the frequency dala isshown inTable 4.12. Notc thc us ofthc MedianR:rnk equalion, commonly used for bolh Weibull distribulions. Linear rcctilicationcquarions providd in Appendix X for thc 2-paramctcr Weibull model arc used t(r

75105|]5

2252552lJ5: l15

175

525

5r5

10

2-param

E

d_

sK pF p p* F H; g g$ g $ HHLoad mid-ctass (neMoirs)

Figure 4.48 Load trequency d stribuiion for the foot peda

6A63CR

Figure 4.r

case studies 215Trbl .a, lz AnalvnsoftoJdftequemydardandpto rhstoyrrnnstnr lhc:-par.merer\ le ihdldiqrnbulon

(%) (x l0)(4 -

, 0 l ln imrd_clAsr In ln l l r l - 4,l l a\ i r l

15t05r35165195225

285315345175405435

49552s5555ti5

615?05735165

0.J0,92.62.4

7,9

6.8

2.92.72.2t ,8

t .8

I

5

26

5979

8376

5235292?22l l il4

l8l lt0

5

t08t72241100379

557631701750E02ti3?8668939t5933947961919990r000

0,00470.01370.0:t970,06370.1071

0.24060.29960.37850.47350 55650.63240.7004

0.E0t40.It3640.86540,89230.914309323

0,96010,9?E30.y)030.9993

1.80614,31754,65404.$535.r0595 21345.4t615.54t35,65255.?5265l i4t55,92706.00196,01536.14206,20466.26346 31906,17t6

6.46936,51416.55826.59996,6J99

-5.35784.283J

,3.2062-2 7208-2. t720

1.6700-1,2902-1.0325-07431

-0 20700,00080.t8610.32490.48020 59350.69590l i0t30,89890.9905L073tt . t7t4r.34201,5338r,91130

:,"ffi' ;TllTi{r#ffi ,*",-l# *{t{:i'.*.:n,r*ll'l u:Xl*::: *"t"*l'***1i"ilff 1'i",,J1.**Lt ti :il',f"Tl,T:n:ff l,:

-1

! -3

3.5 45 5.5|n(mrd{ra$) (newlons)

Figure 4.49 Linear regression fo. the 2_paramter Weibul trarsrormed toad frequency dara

y=2.4415t-14.A13

216 Designing reliable produds

The charactcristic value. P, and shape paramter, /r, lbr the 2,parametr WibuUdistribution can bc dctcrmined from the equation for thc lire in the formI = ,.11r +,10 and liom the equations given in Appendix X, lvherel

. / 40\ / / t4qt \ \ ,, r . -n( 4r)=*P(- | 24 . J) r . / r r \

i=At=2.48

Therefore, thc io.d, .li can be characterized by a 2-paramclcr Weibull distribulion

F- " / (39r.3,2.48)NThe equivaient mean l1I = 347.1N and o/ = 149.6N.

Wc can compnre the calculated 2-parameler Weibull di$tribution with the originalircqucncy distribution by multiplying lhe /(.r) or PDF by the scdling faclor. Nr1.. .rsdclcrmined Iiom equation 4.86. The variarc..y, is lhc mid-cldss value ovr the loadrange. Also note thal the population, N. is dividcd by l0 to change the frcqucncyback to a percentage value. The resul!s of this cxcrci$e are shown in Figurc 4.50.

, r=N"f : ) ( l ) ' * r ( f { ) )\r"/ \''l \ \''l '/

l00O . i 2.48 \ / . ' " t ' 2d(=l; , ' , ( f r ) ( , - l * ' ( - (+) . )

. r io

's p p p p p N $ H: $ I c $ E q B s H g: E F p FLoad mid-class (newtons)

Figure 4.50 Comparrson oi the ca cu ated 2 paramelr Weibu d nribulion with the or gin.l load Irqunry

Deteml

t ion 4.81:

10

rs grvcn o

Thcrclbr(

Equationmined tht50mm Iofth foc

The widl

foreins rihe weigl

Bccau.geomelr)

-Ee&-

:

E3

0

2-paramt6r

dislribution

!

4t".

Case studies 2i7

Determining the stress vaiableThe slress, a. due to pure bending at the section A A on the pedal is given by equa-tron 4.811

(4.81),MY

i-1 - bending moment

/ = distance from x_x axis to extreme fibre/** = second moment ofarea about the axis x_xF = load

I = couple length

d = depth of section.

F'or the elliptical cross-sction specified, the scond moment ofarea, ,, about x_x axisls given byl

I*. * o.o49o9di bwher:

, = breadth of the section.Therfore, substituting equation 4.82 into equation 4.81 givesl

. t0. t8517F{t l -

d.h

(4.82)

(4.83)

Equation 4.8J srates rhat there ere four variables involved. We hdve alreadv deter-mined the load variable. F. earlier. The load is applied at { mean distance, //r,;fI50mm repr$enling the couple lenglh. snd is no-itty dkn;but.a ubou, th.'*iattof the foot pad. The standard deviarion ofthe couple te;gth, d.r, can be appr;;im;i;;by assuming that 6a cove$ the pad, therefore:

' r=?=8333nlnI - x(150,8.133)mrn

The width of the elliptical cross-section, ,, has a mean 11, : l2mm. The standarddevidrion can be delermjned from equarion 4.t8 an,t reierence l. the .t..;t;i;rorgng proceq( capabi l i ly map for lo\r to medium carbon and tou ul lo) steels forlhe weight range given. shown in Figure 4.51.

Because rhe width of the section is over a parting line, ihe only process dsk is agometry to process risk gp : 1.7 which gives rhe adjusted tolerance as:

Adjusred tolerance = Elq-iglgllce (r) 1 2

-,.so- =

i x lJ = o.7o6mm

218 Designing reliable products

fi. closeo or: roncuqc pnocEsslcaPAgtLtTY MAP FOR LOW TO MEDTUMI cAFBoN alo Low aLloy sreELs lL (wErclr tpro 1k!) |

t lgure 4.51 ooces, (aoaoJr/ -*,oL

or.o o"-or"n o,rL" * t

From f isure 4 51. rhc r isk , l und hcnce 9.. = 15 heLruie rhcre i rc no,,Lher r iskr i rctors to l r tc rnto tc{ounl. The strnJl fd Jevisr ion crn be i tnpro\ imuted t iom:

, \ r 2. i^ u6. ts2d + 12.. :=- 12-=r) l l1mm

Thcreforc.

, - N(12.0. I l t ) ' rmThe original objectivc of thc cxercisc was lo frnd lhe optimum depth of the sectionsrth rcgrrd l ( , i r , f . ture.( \cnry. Vatues rrngrng from d = t4 ro:5mm in sren. oflmnr $i l l be ured rn rhe c.r lculnrron ufrhc rel iahi l i ty f f , .

" . , rr . .pnn,f ing r"nirr , i

Je\ iaf iun, lof ca,h d(prh .rrc lgain (atcul ed fn,m rhc procss u.r lubi tr i t m:,n rbr( lo.ed Jie fofging. rh! ror i r t roteranee for erch r .r iue rr ien r io. i t . e ' t . i 1""grvmg 4m : 1.7 as rhcre are no other process risks 10 trke into account. ts-or example.rl d : l4mm rhe rotal rolerrncc at ,4 : t.? is Z = 0.66 nrm. Thcrefore. ttre srandiradeviation is given by:

, . / 2r . , / i , o. l l 1.7:1/

- l : i -U 0.o80mnr

t ' - N(14' 0 080) nm

This can be repcdtd lbr rhe rcmaining valucs ofrhe secrion depth chosen.We can use a Monie Carlo simularion ofrhc ranrtom variables in quation 4.83 to

delermine rhe likely mean ard standard dcviarion ofrhe loading srrcss, assumin;tha;l tu ' \ \ i l l be J \ormrl ( l i . rnhu on roo. LrLcpr i , , r rhe to.rd, f , u nicr, r , mo.fafejr , , , ,l -pdramerer weibul l Jrsrr ihu on. rhe fenraining rrr iabte. rrr chara.renrc. l bv iheNornral distribution. The 3-paranrerer Weibull disrribution

"r, t"

"".a t",i,,jJ

goa

EiF

the Noincludir

Tflbl

val ic l .

Calcul

In thc p

closed Idistribx

Case studies 219

Trble 4,13 Loadins stEs Nomal distribulionparamebrs lo. ! ranse ots4tior dcprh lalues

I4l5

11t8

20l2l222324

225.0

112.2152,6l16. I122.1I t0,2100.09I. Ir i l416,670.5

911.2ll5 775.1

51.348141,6

31.4308

lliii':" t#['";'Tlil'il"1lj:l]i *! iiiT:;]ii Ili"tii "'l"l;''rT:rI'*n,,nf ;ll;*iliffil i';[t*ili r)ltl1!,"'.il]iill;tl;.1'ili lT:;l::'#tril'*ir ffi ''ri*""i"ii"ry,11"il#i ji{ *::r'i::& T.::':il ilT

*p{li*i*fr i'ill+t*'+,**rri"'rfiffierrlier in Seclion 4.2..r. bur

"n appro\imaLion ,ri"g fvf."" i*f" ,i."i",i"" ir'rr;ii

Calculating the reliabitity

'L".illrlli',lliiliilLi.,irs ro determine the rerrabir'|t) ror a given number or

../1s)./s (4.Js)

R, = rliabiliry ar l,lh appljcation of loadn : number of independenr load applications in sequenc

I(a) = loadinc stress CDF

l(s): srrengrh pDF.

Ir the problem.hre. tte loading srress is a Normal dislriburion and the strenqrh is a 3-parameter Weibult distriburion. Becaus the Normal ai",.tU",i..;, ib-ei,,.i;'"ciosed form, .he j-parametr Weibull disrribld s ribur ion whin ; : r ; il ;; ;:;:;;:l',JJ i?: ::;,f ;::1 fr .1 ;,lii:lil,l *:

o=f. t"y

220 Designing reliable products

io and d, can be estimated given thc mean. rl, and sl.ndrrd deviation. o,1br a Normaldistribution (assuniins, : 3.44) by:

la x ! 3.13944',73o

//=p+0.353011144

for cxample. the loading stress at d : I 4 mm in terms of Wcibull paramerers bccomcs:

xo1 = y1 1.13944'73o1: 225 - 1.1394473(98.2) : Sl lMPa

01 : p1+ 0.3530184o1 = 225 + 0 3530184(98 2) : 259 7 MPa

i: \ =34q

Allhough -!oL is ncgative. it willnol aflect the detcrmination ofthe reliability bccausewe are only inlcrcsted in thc righl-hand side ol lhe dislribution lbr slrcss strengthinlerference analysis,

The linal rcliability formulalron for the inlcrfrnce of two 3-prrameler Weibulldistributbns subjcted to mulliple load applications is given in cqu,rtion 4.84i

I I / / ! - \ , r \ "1 ' l ' . f f ( \ / ' - l ! r \ "R"= | lL-cxn{- l -' - J._. . . 1 - ' \ \pr- . , . , / i l L\ , r - r , , . i ' \ r . . i . r . /

, Lr .c, . \ t \ l.*n l_(,, _ ,*,/ /1., 'Thc limits of inlcgrution are liom the expclcd minimum value ol yicld slrength,.ros : 272.4 MPa to 1000 MP.r, rcpresenting .o. Thc solution ofthis equalion numeri-cnlly using Simpson s Rule is dcscribed in Appendix Xll For th casc whcn / = 2{l mmand the numbcr of load applicutions n = 1000. the reli.tbility. R,, is lbund 1o be;

Rro00 :0'q97856

Finding an optimum designThe boidcst line on Figurc 4.52 shows R1o0o ctlcuhted for cach lalue of the scthndepth fron 14 io 25mm. The approth dcscribed above is based on lreudenthal

et al. (t9(.( 1997) we

The relevidnt tt

rcliabilitYtarget vrl

bc juslilie

Determ

from the

J.i,, in IdcviLrtronol lhe pe(

Rearmnl

Repatithe opli

10.90.8o.7

E::d;;

030.20.1

0

f igure 4

14 t5 16 17 18 19 20 21 22 23 24 25 From Tgreaterl

50%.

Section dprh, d(mn)

figure 4.52 Relablty, nrcm, asa functon of sction depth, d, caculated usnq three different approaches

caie siudis 221

et al (1966). Two further approaches ro determine R, by Bury (1974) and Carter(1997) were also identified in Section 4.4.3, the results f.om th"i" app.oach", arealso shown on Figure 4-{2. Note that the approsch by Carter requires tiatan equiva_lentman and standard deviation is calculated for thematerial's lelal strength, whichare found to be ls = 342 MPa and os = 26Mpa

The results from the three different methods are in partial agreement and it isevident that a suitable design exish with a section depth,4 griater than 20mm,but probably less than 24mm to avoid overdesigning the p;rt. Consulting th;reliability targer map in Figure 4.53, for an fVbA Severity Rating (S) =-0, atarget value of R = 0.99993 is required for an acceptable design. fhi reliatitiiiescnllulated.for .1 - 22mm-show quile a large spread. reflecting rlie differences of rhemernoos. However. tne finat selection of a pedal ,eclion depth d = 22mm wouldbejustified.

Debrministic approachwhen dcligning urng a determinisric approach. ir is a fair assumption rhar a lenerousticror of safery rFS) would be allocared lo dercrmine th allowahle worki;g strcssfrom the minimum material strength. For the variables in theproblem such asiimen_sions, mean values would be chosen. except for the load, which would be themaximum lodd e\pected rhroughout the service life. The minimum vi.fa

"..n",n.Lm in this case can be approximared from the mean value minus *r..,r.na-"iJdc!iarions. us discussed in Section 4.3.1. The working stress and the section deothofrhe pedalcan be calculakd for a range off{cror ofsafety values, where the workincstress is substituted for Z in equation 4.83. Therefore:

,S.;"=i ,r .e_3og

Snn=342-3(26)=264MPa

Re:uranging equation 4.83 for the section depth, 4 we obtain:

(4.84)

Fil:JXliit for FS=3 and anticipating thar the maximum load is ?65N from

= 0.03327m = 33.3 mm0 012

Repearrng thi . exercFe for a ranSe of FS !atues grves Figure 4.54. To ha!e derermrnedrne op||mum sectron deprh ol22mm. I S t . l would need ro hale been spri f reJ.f l"Tl"!1.1l ,Inrc-at tacror or safery apptred mighr h",. h..",;; l i ;;;gred(er:) ror lhr\ lypeot prohlem dnd sub.equenl ly rhe pdrr $outd ha\c been over-oesrgneo. Increasrng the rotume dnd lhercfore the marerial co,r by approximatel)500k.

10.18537x765x0.15

264 x 106\

' - )

222 Designing Ieliable Products

Loglolailure RellabilityPpm

Figure 4.5

4.8.3 r

123456910

FMEA Severity Ratlng (S)

Fiqure 4.53 opiob.,ro tF op lon-o lordlopo ' " .U' " . p P 0 o Ltcd- o ' r ' " "

;::;;;, ;;; "" i"

er al (1966))

by shfursl ight ly I

iiom thrl l is r(

t ) :o1rcngrn csteel SA

GoverlThe iorihe holchub .nd

Acceptabledesign

Conservatlvode!l9n

Case studis 223

403938

363534

E32;31r30"* 29n2a

'i 26,9 25-' 24

2322

2019.18

Facror ot e.tty (FS)Figufe a.54 Section depth, d, based on facror of safety

4.8.3 Torque transmifted by a shrink fit

An ellective way ofassembling a machinc elemenr suc[ as a gear or pulley to a shafl isly:hdnk dlr ing This invotves selr ing up a rrdiat presuri b"r*e"n u,f,ofr _i i t n\trgn ) trrger Lrlamcter rhan thc in

224 Designing reliable

Interference flt

Determi

bore H7+0.053 rnr

Figure 4.55

The radial1966) ' :

The hub ir

With refersion of Z

The shifte

Thereforel

Similffly Iand the sh

Arrangement ol the hub and shait

MH = holding torque (hub)

/ = coefficicnt of friction betwccn hub and shafr

rr = radial pressure

ds = shaft diametr

/H : lcngth of hub.

pressure, p! sel up on assembly is given by equation 4.86 (Tjmoshcnko.

(4.86)

, : Modulus of Elasticity

4r = hub outsid diameter

1 : inlerfernce between the shafr and hub.

Suhsl i rul ing equarion 4.kt ' rnlo cqurlron 4.85 giveq:

MH o 78s4t. r . r . , ,+(r (*) ' ) (4.1r7)

' The rudial presure is nor.on{rnr over Lhe lengdr of thc hub. but in fact pats at thc prolecliru portjonsof thc shalt which rsist conprcssion rcsulling in dn incrcasd prcsure rr lhe ends ofthc hub, or slressconenralron. I-.r thrs re!$n. frennrs fatiguc failurc na! be a.ticipatcd Nhen ibe apptied torqnc is

Case studies 225

Determ i n i ng the va ria b lesFof an arlequatc inaerlcrence fir selccted on th basis of the hole. the tolerances arctaken fron BS 45004 (1970) as H7-s6. This transtates ro

" ,.f".,,"."., ,i,.-f,rt,

bore H7 : +0.030. +0.000mm and for the shafi diameter, s6: +0.072.,o.05lnrm Ci\(n rhc norJtr i ,n fnr huh bore di m(r(.r i , / /H, JnJ conrertrnu r( ,Ine: ln \rhrc, . rnd hr l rerr t loter|rnce,. rhe,e drmen,i , ,n, h. .(omc;

. /H = 250.0t5 + 0.01S lnm

dsrU50.063+0.010mm

and l iom Figure 4.55.

D| = @,t0 +o.O'mm

/H = 100+0.05mm

Thc hub inside diltmetcr. outsid dirnlerer and lcngth are machined using a lathe andr, \ te emplo). th rurning hof lng jn f r \ shn\ n prc\roust) rn frguie a.42. Thcnrr lef i rr \pecrtrcd l , , r lhe hub i , mi ld.reel gi \ ing J m.r lerul lo procc\) nsk,nr = 1..r . l lc fcornLtry ro pru(ess n,k (n = Lt j2 duc ,o " : .

f f "ng,f , ' , , a;" ,n. i . : r .r tur, . An rdju\rc\ i tolcrJnc( ' l i , r thc hUh h(,re..4, . r \ rhcn givcn hy.

n dJusted toleftince = Design tolerance (r) =i#L=00,,With rclerencc to thc proccss cap.r bility map for turning/boring. ,4 = l.? for a dimcn_sion of U50rnm. This valuc delaults t,, th.

"n-pon"nr mmulacturina variabilitv

rjsk. qD. when lhre is no considcrurion of surt.ace tjnish capabitity i" ; ^;"i;;;.,'.Thc shiftcd srandard dcvialrtlIr, d,. tbr lhc dimcnsionat rot,irancc on rh; il;;;;can then bc predicred tioln equa on 4.28:

, , .a: , o.ul5 . t t '

Thereforc:

.41 - N(50.015,0.004) mm

Sirnilarly lbr an analysis on the hub outsidc diametr aDd lengtb, which are rurned,rnd the shaft di.imeter, which is nnishd using cyl;ndricat gririiing. we gct:

, rr _ r(70,0.005)mn

lH - N(100.0.006)tnm

ds - N(50.063.0.0012) mm

\1er1. ' r ' te.r f l r

226 Designing reliable producls

Thc mcan ol-thc intcrl-crcncc bclwccn thc shalt and hub bore is giver by:

h: t t^ Pdtt : 50 063 50 015 : 0 0413 mm

Thc sl.indard dcvi,ilion ol thc intcrlcrencc is givcn by:

. 1. . . " '^) (0.0012' 0 004-) (r . ( i04)mnr

1- N(0.048,0.0042) mm

Thc maxnnum cocnicicnt of variation for thc Modulus ol Elnstioity. ,. for crrbonslcel was given in Table 4.5 as Cv : 0 03. Typically, t = 208 CPa and therefore wecan infer rha! t is represented by a Normal distribution wilh par.rnrclcrs:

r - N(208.6.24)GPa

V,rlucs typically rangc lionr 0.077 to 0.33 li)r thc s1a1ic coellicient offriclion fof sleelon slccl undcr an inlcrltrcncc lit wilh no lubrication (Kutz, 1986). The inlerferenceund coellicicnl ol lriclion rrc corrclatcd in practicc but for the example here. weassume st.tislical independence. Also. assuming that 6 stxndard deviations covcrthe rangc givcn. wc c n dcrivc thal:

/ _ N(0.2,0.04)

Determining the probability of interferenceThe mean v,rlue lbr the holding lorquc capaci!y. trM,,, is found by inserting lhe meanvalues ol all thc voriables into equation 4.87. To lind the standurd deviation of theholding torque. d,r,,. wc cun apply thc Finitc Dillbrence Method. !inolly, we arrive

MH - N(3.84.0.85) kNm

At this stuge ir is worrh highlighting the relative contribuiion ofeach variablc 1o thcholding iorqu v.ri.mce. The results liom ihe Finite Difference Method rre used toconsrruct Figure 4.56 which shows rhe sensitivity analysis of all the variablcs !o thcvarirnce of the holding torque. 11 is clear thar ac(urat and representative data forth coefrcient offriction. l; for a particular situation and the control ofthe interfer-ence litdinnsions are crucialin lhc dclcnninalion ofth holding torque distribulion.

Thc lorquc thit can bc lransmilted by thc shali withour yielding. ,'Us, is given by:

M. : :)--:

I : shear Yield strength

./ : polar second momenr ol-arca

/ : radius ol shaft.

In terms of the shaft diameter, this simplifies ro:

l'/")

Figure 4.55

The shcar$trenglh.

Applyinggivcs:

Using Molbund 1o t

Bodr ol- lh

From Tab

the hub fr(hub slips.

Rathcrthc yicld srin thc shal

yield usinlvs:{r ly6l4e5rr . r t r (1.88)

Case studier 227

lY.)

D..tg. v.rtabt.Flgure 4.56 Senstvtty anayss fof the holdrng iorque valaDres

Tbe shefir yierd strenglh for ductire metnrs is a linear function of the uniaxiar yierdstrength. Thercfbrc, for purc lorsioh from equation 4.56:

7Y = 0 5775''Applying this conversion to the Nornnl distribudon paramcters for SAE 1035 sleelglvcs:

ry _ N(t97.3, r5.2)MPaUsing MonLe Curlo , in lul i r f i , ,n anpl ied ro cqu.r l ron 4.8t. thc . ,h f t r(yquc crtacir) r \

Ms _ X(4.86.0.38) tNm

.B!th of the torquc capacilies calculated. the holding torque ol.the hub and thc shaft

llfl1"-i .litd.t l.".,"l..sented by rhe Normal di",;tl*t;rr. ,rr"*r_. *" "1" ","

ti"coupxng equatron ro derermjnc rhe probability ofinterferencc. wherc:4.86 - 3.84

::_l lo/0..18: + o. l l5r

il'$l;[fp*$#i[t*i'*fi*f *ififfi mll#"ii"',r':ffiiljTflff:fr1T#i:j:i'=isn

as a wh're a smarr incrcase in

ir.:i'"Tir:L'kji',ft *s{tt";l:#il1i#iift': ji:!ffi iiif

228 De5igning rliable prodocts

Shsit ma| att SAE 1035 steol

IntortolEr

Figure 4.57 D !1r butions and re ative inrederence ofthe ho ding torque and shalr torque capactes tor iro

a reliabihy n = 0.999896. Although nor safcty crirical, rhe dcgree ofprorecrion 1iomthe shaft yielding is now dequate after rcl-erence to the rcliability targel map. Therlativ shape oi thc torque distributions and degree of intrfrence irc shown inFigure 4.57 for thc two situatioDs wherc diffrnr shafi slcels re used.

4.8.4 Weak link design

The major assumption in wcak link design is that rhc cost offailure of rhe machinethat is to be prolccted from an overload situ.ttion in service is nuch greater rhanthe cost offailure of a weak link placed in thc system which is dcsigDed to fail firsr.The situalion is p marily driven by various costs which musr be bal.nced to avoidat one exlreme thc cost of lailure of the system, and the other overdesign of theelements in the syslcn. The cost factors involved rre lypically:

. Cost ofintroducing the weak link

. Cost of replacement ofthe wenk link

Flgure 4.sg

. Cost ol'l ink

. Cost du(

. Cbst ol i

Figurc 4.58

Thc rpprotsrrength rslink fails p,Ratings (.slFMEA Se\

critical t0 tmode beingcrprbi l i ty. l llevel offaiiu

'Ihe varial

the weak linlthan thal bc

snaft mar6ial: sAE 10183te61

,n'-"liXlNnn,n o,

Case studies 229

Slr3s/st16ngih

.n*l?* "

r(s)strngth dl

t\L)

. Cost of failure with the weak link relative to the cost offailure without the weakI ink

. Cost due ro machine downtirne if the weak link fails prematurely

. Cost ofincreasing strength ofmachine elemenrs ro accommodate the weak link.F'igure 4.58 shows the concept ofweak link design. The loading stress distribution isJerermined from rhe normal operaring condirionis l"*a t" ,r,.iy.i..

"J'iiiir'rir.r,ls useo lo dekrm,nc the dimensions of rhe wcrk link. The fallure mode for th weaklin k is srress rupture rnd so rhe ulrimatc rensile or utr,rnare ,t ea, ,trencth ls ,D;cln.l.The appropriare le\el of inrerfcrence berween lhe toadlng srresi ;; u'Ja'k- lrnk.trength is dckrmined from the consequences of machine downtime rf rhe ueakln(. tart \ premdturcty. The use of lhe Larger rel iabi t i ty map \^ irh FMEA Severi lvKaung< tJl tor production processes is useful ln thls respect lsee the proce_FMEA Sevedry Rrt ingr provided b) Chr ' . ter Corporar ion cr at . , fSqSi iore\ampte. rn Seclton 2,6.4. thc charactenst i( dimen\ion on lhe cover suppo lea wascnri(at ro rhe succerr of rhe auromared ascembty n-*". , r , . pot l i , i r i i i i r r r ."modc

_being a maJor disrupt ion ro the producLion l ine on fai lure ro meer lhe reoui; ; ;(npabrtr ly. theretbre.S=8\ra.al locared. l f lheweakl inkwardesiCnedto \e; \ lowle\el ol la i lure nrobrbi l r ty, meaning rhe \eparar ion l . roe.n r fr . r*o-ai .r ; i r r ;oni *a,greater. rhe mnchrne etemenr( (ould be overdesisned.Jhe.variabi |ry oI rhe weak t ink rrrenglh sboutJ be we kno\rn becau\e r t is dcnlrcal component In lhe slstem and cxpef imental rest ing Lo derermine the ul t imalercnsre r l renglh. l r . rs recommended where dala is lacking. The inrer lerence belween

lhe weal l ink \rrength and machine srrengrh i . usua y at;a)s ,ma er comFd;;r iv; l ;rnan rhr l belueen lhe toading \ l ress and weaI l ink slrength. This is due to r tr"i-act that we musr se the failure ofthe weak link b"fo." tf,..i*f,in"in uff,itu"ti"n..

Figure 4.58 The wet hnl concepr

230 Designing reliable products

Coupllng

Figure 4.59 Wak Ink afiangement

Howcver. too great a scparation, and overdesign may occor. Thc overload conditj(rlrs rcpresnted by a uoiquc stress. which is vcry much grearcr thltn lhc workingstress. rpplied suddenly which c{uscs onty thc we k liDk to failuro drlc to stresi

In thc lbl lowing c se srudy wc wi l lexnmine the corrcepts ot werk l ink design. f igurc4.59 shows lhe {rrangemenl o{ l coupl ing wirh lL shear pin, lct ing as a wcak-l inkbetwccn ir l rAnsnrissiou syslcm 0nd n pump. lhc.ssomption being th t lhc cost ot .fui lurc ol the punrp is much grcaler lh n the cosr of fai lure of ihc weak l ink. Inthe evcnt thdt thc punrp suddcnly stops due to a blockagc, thc shc r pin Dlustpfotect thc systcm l iom drm.rgc The ppl ied t(xque rhrouSh lhc rrunsmissionshaf l undcr normit l running condi l ions is M = 3.2 kNnr wi!h a caeff ic ieDt ofvirr i t ioo(" =0.1, rnd lhc diaDreter v.rr i b les ot lhc transmission nd pulnp sh,r f ts rreD-N(60.0.004)mnr. The fMEA Severiry Rrt ing (S) =5, relal iDg lo.r minordisruption il lhe wcnk link fxils prcmoturcly ctusing the pump to expcrience down_time. Slccl is to be uscd s lhe Dr rcr.irl for rll the ff chine clcDrents.

Experimental determination ol ultimate tensile stength of theweak link materialBecausc lhe design ofthe shcnr pin is cr i l icr l , ihe ut! imrle bnsi le slrcngrh of lhc steelselectcd lbr the wc k link nabrial was in rhis casc Dleasrrcd statisticaliy by pcrtbrnr,ing .i simple expcrinenral hardness tesr. The grade of srecl setecred is 226M07 colddra'"vn liee curling steel. Thc size testcd is U t6nDr, estimared as the approximrtedianrclcr ofthe pin. lrnd 30 snmples arc sclected lioln rhe srock lnalcrial. Thc BrinnelHardness (Hts) valuc oferch s. rple is rnclsured. thc resulls of which nfe showD inTrble 4.14. Rarher rhan develop a histogram for rhc drra. wc can dcrcrmine rhcNornal dislribution pblting posilions using the mern rank equation for rhe individual values. The resulrs ,rre plotted iD Figurc .{.60, the cquation oI rhe slraighr ljnc.rnd rhe corrclation cocllicient. /. dcren ned.

The mean .tnd srand.rrd deviation ofrhe h.trdnss for rhe srecl can be determincdliom the regrcssion conllanrs,40 and ll as:

/ ,40\ / - l4.r i0r\\ qt) \ n. i ro J

*no'

/ t 4n\ / ln\ / t , r4.80t\ r l4.d0t\' " " I r r - J ' t ; rJ I o j r r . , / - l . , ,zr in. / o. t ,

2

1.5

I 0.5

P 05

r5

2

Shsar pln

Figur 4.60 Lrn

Case studies 231

'":\"'# :":ll;i i,l'"*- dd'd Jnd pro"ns r'n.o1' rr,

HB(rr' : l0) t

-* '

I i r i ( rr

igurclrnk

k. ln

rt ior

t4t . lJ

145.4

l4s 9

t47.8t4t .8

150 :lt50.1150.1t50It50.:lr50 3t50.1150.1l5l lt52.8l52. l t152 l1t52.8

155.4t56,5157 0t5lr . l

I2

5

I(l

l2

I5

t7li

2|2l222124252621:8

l0

0.0121

0 09680. l ]90

0. t9350.225r10 258t0.29010.1226

0.38710.41920.45t6

06t290.6452

n7142

0.811170 87t00.9012

r . r i49t .5 t8r.300I BI

0.75:l

-0 552

-0.312-021!?

0.244-0. t21-0.041

0.0410,1210.2040.2ta7o 172

0,552

0.751

L l3 l| 3001.5lr

2

1.5IJ

I 05

En6-

2

25

i \ i '

Bnnlt hardn6ss (HB)tigure 4.50 Llnear regression forthe No.mal distrtbLt on transforrned hardnss daia

r = 0.94G

____l] -_

151 152 153 154 155 156 157 153 1

232 Designing reliable produds

From equtrtions 4.12 and 4.13, the mean and standard deviation for th ultimalctensil strength, Sr, for stl can bc derived:

/ ,( , : l45tHB : 3.45 x I50.07 : 517.7MPa

'", = (3.45' : . oiB + 0. ls2r. piu + 0.152,. oi ts)o5

= (3.45'? x 4.31: + 0.152, x 150.0?r + 0.1522 x 4.31?)r5

ds' = 27 2MPa

.tr - x(517.7,27.2) MPa

Typicauy for duclilc slccls. the ultimate shcrr strcngth. 'u

is 0.75 of t!1 (Crccn. t99:.,.

n, - N(313u.3,20.4) Mpa

Determining the diameter of the shear pinAssuming that an adcqualc transilion 6t is specified for both the pin hole andcoupling bore, the pin is in double pure shcar with the borc of the coupling, and sothc shc4r stress. ,, is givcn byi

F = lLngential forcc

,1 = rrca ol-pin.

The tangential lirce acting on the pin it a r:rdius, /. duc 10 the appticd torque. M. isgrven by:

.M

Threfore, combining the above equ.ttions and substituting rhe radius for rh dia-meter variablc gives:

. t .27324M' D., !z

Rearranging cquation 4.89 in ternts ofrhe pin diamcter, 1, gives:

(4.e0)

FoI FMEAR : 0.999 w10 o foilure 1Table I in Icouplrng cqr

couplrng eqr

0.1. .rnd tha

zero gives:

which yields

Thc pin is mdislr ibut ion+0.002mm

Solvins ,

.rv(281,28.31given slressthat specilie

SeledingThc torque (shear pin in

link materia

Sdving equ

The coem

(4.8e)

M = lpplied torque

-D : shaft diamctcr

Z = loading stress.

| .27124M

Case studies 233

For FMEA (S) = 5. rhc retiabiiity of the wak tink in servicc is rcquircd to beR rr .aq' l wrrh relcrence to lhe r( l iahi tr l ' lar lel map gr!en in Frgrrre 4. 16..1hr, reldlerro a lJrrure pfohJhj l r t ) P o 001 or $hen $orlrng qrrh the SND, I 0q tromrable I in Appendjx L For a siven v. ue of rhc itandard N."*i *.i"i.-,. ii.couplrng equalion for the interfcrence of two Normat Aistriburions can Ue usei rodctcrminc th loading stress. and hencc the diamcrer of thc shear pin. from thccoupling equalion we get:

1,.\ I r kn,N thrt rhe^ci

234 Designing reliable products

lorqu capacity ofrhe pump shaft will approximarely be the same, becaus rhe dinen_sronat v.triablc of thc shai diameler is very smnll in comparison to th. ,h;"; ;i;,"strength. We also nced ro specutate ah,ur lh( trket) faiturc proUabititl ae.eriabrehcr$een rhe $cJk t ink and fL,mn chal ro.que,.rpac, i ie, . e-umi"g i i , r i - i r ," "*rr"r"srtuatjon is only likety to occur once in 1000 operarjng cycles, th"'.oupting equorioncan be wrirten as:

3.09: ]:1t ]1!!L

| "i" + r"",

, ,Rarr.rngingequarion 4.91 to se1 rhe right-hand side to zero and substjruting in the(nown paramcters $vesi

(4.e3)

t4.94)

+ 4421.7 - r4:0which yiclds t/rr : 6092.5 Nl1l by ilerarion.

Therelbre. rhc Normal disrributim parametcrs for the pump shaft rorque capacily

Mp - N(6092.5.487.4) NrnTh!. lorquc thf l r crn-hc rrunrmrrted hy r shrf l ni lhouL yleldrng u&, gncn in c, . turt ionc.n^. KeitrrJngrng lor the rhcrr y ic lLl strcngth rnd the !rr jah, l rs rn rhrs ei .rmplcSrvcsl

5.09296't,.,=_ _r__Solving equntion 4.94 using Monte Carlo simulation for the variablcs involved. thesherr ) ie ld slrenglh required f , , r lhe punrt shj f t marcr ir l rs found t . , hr,rc.rr !orm3t dtsl f l hutton wrth nirr . rmelcr\ .

r , ' _ N( 143.7, I 1.6)MPa

Tier(t ; re-- thL \ormrtdr.rr ibulx,n p.rrrrnelers for rhe m ren.r l . , ten.r leyr( ld,rrengthrre t /u.) / / l tme\ grenter. giv ing:

s] , _ N(24e,20. l) MPa

A suitable material would be hot rollcd mild steet 070M20, which has a mininumyreld slrengdr. Si ' r rr :2t5Mpa (BS 970, l99t) By considering rh"t ,h"; i ; i rn; ;yjeld strengrh is 3 slandard deviitions from rhc mean and that the typicrl cocmcienrof variation_ C,:0.08 for the yicld slrengrh of srcel, the N--^r Air"ltr,i""parumetcrs for 070M20 can be approximateo Dy:

s_r _ N(282.9, 22.6) MPaFrom equrtion 4.94 and using Monte Carlo sirnuhlion again, the actual torquecapacity ol-th pump shlft using rhe 0?0M20 steel gradc is found ro be:

MP _ N(6922.9,560.1) Nmfinal l j . lhe inrcf leren(e bel$een lhe r t f l ied rorqu( rnd lhe tumn \hatt rorou(crprcrr) (an he anJt\ \cJ to derermrne i f \ (prralron herueen rhem rs roo grea r ieading

1.09

Figur 4.61

design. Th

4.8.5 De

glven srluaFigure 4

self-driven

hanger rs r

lable misal

(o.otpy")r+234.l,

Case rtudies 235

roqL,e (Nm)6922.9

Figure4.6l Weak ink rorquecapacty shown relatve ro th applied lorqueand lhe torque capac ty oi thep!mp snan

to ovcrdcsign. The Saltly M,rrgin (SM) wrs given in cquation 4.46 (Carrer, l9it6):

6922.9 , 1200=:=5.17V560.lr + l20r

As a guidc. SM should be lcss than l0 for all cascs offoilure scverity Io avoid over-JL.. iSn. I he dislr lbut ions of the dtpl id rorque. $c k l ink lorque l j rprc y and pumporquc capacity arc plottd 10 scale in Figure 4.61 for comparison.

4.8,5 Design of a structural member

The usc of probabilislic concprs in structural sieelwork design could porentiallyreduce malcrial costs by dclivcring oprimized dcsigns wirh srandard seclion sizes.such seclions bcing rypically uscd in large volumes and repetirive applications.Hcre we will dernonstrare this poinr by slecring thc oprimum section size tbr agiven situalion where a standard strucrural member must be uiilized.

Figure 4.62 shows the arrangcnent of one of i pair of hangers. which suspend aself-driven belt conveyor unit lm in length above a factory lioor. The conveyorunit is part ol-50 rhar comprisc rhe matrials handling system in the facrory. Thhanger is essentially a cantilever bcam made from an unequat angte section and issecurely attached to a column. lt is nominally l250mm in lengrh from the columnlace 1o the hole for the vcrlical tie rod, and has a fabricarion rolerance of +5mm_At its liee end. the hanger carries a load. F. which hangs verrically. bui has an accep-table misalignment tolerance of+1.5", based on installarion xprience.

SM

4421,7

o1,, + },

236 Designing reliable products

H6nger

!:-J?5!r5!lL>l

tigure 4.62 l.langer arrangement

The load carried comprises approximately balf $e mass of the conveyor unil(50 + 9kg). and half thc mass of thc items being conveyed at any one time. Themass of thc items being conveyed on half of ench conveyor unit ffuctuates from 0to 72k9, approximately following a Normal distribution. The material specined lbrrhe hanger is hoi rolled Grade 43C structural steel, which has a minimum vieldJrrength .S,,min = 275 M Pa for a thrcknes I S 16 mm (BS 4:160. t990).

From 3n FMEA of rhc rlsrem de$gn. ! SeveriD Rating (SJ = 7 was ullocffrcd.rlating to a salbty critical fnilure in servic. lt is required to find the oplirnum unequalangle section size from the slandard sizes available. It is assumed that the load iscarried :rt the section's centre ofgravity, G. and only stresses duc to bending of thesection ar coosidcred, that is, the torsionnl effcts are minimal. The combindweight of the beam and tie rod arc not to be laKelr lnLo account,

The gcneral dimensional propcrties ofthe unequalangle section used for thehanserare \hown in FiguR 4.61. Nole lhar .7 < 6 and r - d and Lhal the lcg radir rrr : ,quirelor mathcmatical simplification ol the problem.

Determining the stress va ablesIn lerms of the dimensions, d, , and I for rh section, several area Dror,erties can befound .rboul thc r \ dnd./ r a\e,. ,uch r\ rhe sccond momenr of i rer. /" , . cndthe product moment of area, 1*y. Howevr, because the section has no axes ofsymmetry, unsymmtdcal bending theory must be applied and it is required to findthe principal axes, ,

'/ and u r', about which rhc second momnrs oj ara are a

maximum and minimum respectively (Urry and Turner, 1986). The principal axesare again perpendiculur and na* lhrough lhe cenrre of gravrry. bur are a dl .ptaceddngle.

^ from \ v rs sbo$n rn Frgure 4.h1. The objecr i le rs ro frnd rhe ol ;ne in

$hich rhc pr incrpal d\( . l ie and cdlculaLe the \e(ond moment\ . f afea about lheseaxes. The lbllowing fornulae will be usd in rhe devetopmenr of the Droblem.

Figure 4.53

The pol

The secor

The prod

The prin,

Case studies 237

Figure 4,53 Gefecldmensions loran Lrnequa angl section

Thc position ol the cenlre ol gravity, C. is located by:

u '1+lh-t) l2la+(.b_l)

b1 + (a t) t2(a+lb-t))

The second momenls of area about .r r and l, _r respcsnvely arc:

1*=+lr(b-r)r+l t ) r ("- r0 D. l1yy = + fr(a r)r+b/r (1, 4(j r),1

The product moment ofarca is:

, l ' . , , I f , t , t . A / \2l l r 2 l ,1t11 ) lx ) l to '

- : . / '

The principal second moments of area abour the principal a)tes, , Irespeclively are given by:

6 r)t1l

(4.ee)

(4.95)

(4.e6)

(4.e7)

(4.e8)

(4.100)

(4.r 01)

/ "" : j (1--+/ ,y) +j

/ "" : j (1--+/yy) l

(/,, 1:,))'+4/:y

(1-" - 1,y)r+4/i

238 Designing reliable products

where thc principal ptan is al an angle:

Lln r l ' " ' I

2 (1. r02)

(4.r03)

Final ly. i t i !

Thc tcnsile

A simi lar lsection aB I

Stress-.'tl

nndanglcsc

of tsritish sl

assumlng It

The tbove cquatir)ns can a b written in terms ofthe nominal dimcnsions, d,, and l.for rhr,ecr jr ,n S.tLl ion, tof rhc Inean dnd sr.rndard devr.r l ion ofeiuh pronerry. for.rn) sc(t ion ( .rnhcr iundunnsMonreCarto\ imut,r i "" l ' l t r t ln""r" lg. , , r ' i r , . i i i . i i! . r l la lron lor hol rol t ing ot . l ruclural sreel ,ecLion\ . lht cocrhcienL oi\ur j r t inn for lh i ' proiess m rer idl comhrnrl ion i . C, 0.0r,hl (Hauren. lgai tr

T. ' Jerennrne rhc i t re, , at rny point on rhe,cct ion requires t f iL r te tr . . f f , "rcrol !cJ inr( ' compr'nenl\ n rdl lct ro rhc pr incjpuf ," . , . to. i .orpnn.nr ,r i t t . "u. .ncnorng rn thc pldnc ot a pr inci fr l dxi , . rnd Lhe rolr l srrc\s ar a givcn r 'n inl i . lhesum of the stress due (o the load componcnts considerecl scparatel-y. ffow"ucr. tirriwe must considcr the naturc of thc toading distribution and how ir is rcsolvedabout thc principal axcs.

The mean of the mass of half the conveyor unit was 50kg. Assuming thal 6srtrnJard deviaLionr rdequalely chrractcr izes rhe rulerance range ol r9-kq. rhcrrndrrd Jevralion anproximutes ro jkg. Thc mar, of Lhe i,";, ;.,";y..; ;;:srirc\r to range liom 0 ro -2lg and it wds 3ssumed rhrs mass rrfles r.rndomly accord_ing_to.r Normal dtstr ihut ion, Again we assume rhut this mass range anproximatesto 6 nrndard devirrionr. which gives a mein of 16 kg and

" ,,r"aii,t lii,i"ii""

"ir lKg._rne tolal f t rJs can be represnted b) a Normal dlstr ibunon with a f ie irnp.=50+36=86kg, and rhe standsrd deviat ion of t t" rotut *urr, o, . . . i . i f , "statistical sum ofthe independent variabls given by:

",, = t/l' + tz' = tz.lt te

Convert ing to nc$lons force h) muhiplying by 9.807 giveq lhe load in lerm< uf rhe1\Ormul dtstnbulron r(

F- i r ' (843,I2l)newtons

The allLtwable misalignmenr toterance for fte verticat tie rod, {r = +t.5., is atso(onsidered Io be normal ly distr ibured in pract ice. Wirh tne as,umprion rhat , rpproximutely 6 \ landard devial ion. are covering lhi , range. rhe srand:rrd der iu r ion beiomesd, : 0.5'.. The man oflhe angte on which rhe princ;pal ptane ties is 7r", and ttre loaJsmust be resolved for rhis angle. but i1s standard d.ui"ti,in i" tr," .t"iiiii""r ,u_.i""and or, as given by equation 4.103:

l t . , I '

The Normal distribution parameters of the lengrh, l, can be developed in the samemanner as above to grv:

,t" + 4,

? - f f(r250, 1.667) mm

2)

bf

aI

aI

Finally, it is evident from Figure 4.6j that the maximum tDsile stress on rhe sectiondue 1o the load components about the principalaxes will te atpointe. themaximumcompressive stress will be at point B. From trigonometry, if," alrtun"", Irorn if.,"centre ofgravity to point A on the sectjon in the directioni of ttre principat axes arc:

r./A = tcos rr + tsin o,rA =tcoso-ts in. t

Tle tensile sttess at point A on the section Z^ can then be determincd by applyingsimplc bending theory:

t _ Mw,u^ M", ,u^*--1.-- 1*

Therefore, th bending moments resolved about the principal axes arei

-41"' = F' cos Q' ' l

Mv":r .s ina, . l

,B = cosaltana(, _t) _ (r _ 4]/b- t \

' "= \"" . . -J - s jn al lana( ' - t ) - ( ' t - r ) l

r _ U"! . uB M,, , uB'" - -- /,, - --i,

rL, (MPa) 'L!

(MPa)

Stress-ifength lnterterence analysissevcra I standd rd secrion rizes for uneq url anglcs are listed in Tablc 4. I 5 ( BS 4160. I 9D0 ).for each standard section. first lhe \tdtistical variation ofthe area properties, distancisandanglescan be estimatd using MonteCarlo simulaton, *ti"t

"r"tt.n ur"aio J"t.r,rnme stresses at pojnts A and B on the section from solutionofequations 4.lOg and 4.1I l.The stresses found at points A and B for the sections tisteA are aso snown in iatf" L f f.A specifc sralistical representalion of the yreld nrengrn for C.aae +:C trot ,oitea(leet rs nol a!atlahlc: howe\er. the coemcienl of vdriat ion. C*. for rhe ] ield srrenflhof Brir i .h nructural sleels is given as 0.05 for a Lhickness r S Ir.?#ir i ; ; . id; j ;For convenienc. the parameters of rhe Normal distribudo; will b" cal;;Ed ;;assuming that the minimum value is _3 standard aeviatio"" frorn ttre lxpecti

Case studies 239

(4.1ft)

(4.105)

(4.r06)(4.lu1)

(4.r09)

(4. I ]0)

(4. I l r )

(4. r08)

A similar approach can be used to determine the compressive stress at point B on thesection ZE wherc:

til

I

II

iln

{I

,lli

t05.2264.22 | I .1133,0148.2r42.6114.0

45139.2:L227.022.02l .0t6.ll

291.u264.8225.9

-176.2112.2114.5

436

32.926.025.1

r,.6487240.9189510.9992990.9s99951.000000r.000000I 000000

0.3303910.7151t90.9r86480.996 2

1.0000001.000000

240 Designing reliable products

nrcrn v l lLrr rs de$r lhcd by equrr ion I l : l :

) f r , , t ,s, l r \ r

t75 - / r \ r io 05i i r , )r75

1," ..ai i ,s l : .r.5 MI,r, arrd ", t).05 r r: t .5 = l6.t N4t)r' Ihc \ ic l l s l rcr ! lh ol ( ; r rd. .1. l ( \ t l re l l ] [ r ] \ t .c l c ln b! lpt l r ( ) \ inrr lc( l b\ :

.Sr - , \ ( l l r 5. l ( ) l ) l \ ' l l , r l

I l r ( e()Ut l in! : (quxtr( f lar lhc s l l ( | \ : \ l rcn! t l in l r r t ( , fcIc. , | l r t ]srs t i ) f l t rc pr$tcIr i \

: 11\ 1t ! '

\ "

$h.r( l l i ( Sl l rx | l | ( l Norn l !x i r lc . : . ( r I hc rctr ler t t i ) t t rc rct iuhr l i l ! . t . r \ !h() \ \t ) rc\ i , )Lr \ l \ lh. f ( { r l rs rr . t ) r ( t r i (1.( t i r I . rhtr , l . t5 l1 lc tMtrA Sc!$i t \ t . t , r r rnr( . \ l I J i ) r lh\ . s l \ lc I r 1n( l ,cr lcs r t i , r ! ( l rct i .b i t r ly / i i , 0 919995 trrrr I igrrc .1 rr ,l l i rsc( l , ) , r l I i \ s I fc i l i .x lhn. 1hc 5l) / 65 \ | scel io l h:rs I r ( t rxtr i t i t \ / i r r 0 i ) , ) ( )99!\ ! / r . r lh. t l rcr lcsl \1,( \ :

's 'Lk.( l r lhc. , ) rLILint f ( tur lhn r is rssU rcs ot ( ( )ursc.

lhr l l l 'c . ( ! l l t ) ressi \c \ ic l ( l r l r .net t r ( ) t lhc j r r r ler i i r t is cr t i \ r l ( r r l lo lhc tcDsi t ( , \ ic t ( lr , r . r l , \ l i .hr ' l r ' . . . . . t , , r r ' . " r , t , . | | t , , r , , . . . , t .

4.8.6 Bimetallic strip deflection

Figure 4.64

,rnr lysi \ . I

Determinl inr . l r l l ic c lcrr .nt \ i [ . $ i r lc l ! U\e( l I r instr 'L l lcnts sUetr r \ rher Inost i r ts l ( ) {n\c ( ! .onlrol lcnrf . | r turcs lhcrcrr . \ . \ . r i r ] brnrclul t j ( . tcnrcnlr \ 'pcsr i ! l i t rbtc.suctr i r \ \ t l lL ighls lnps. corh rrx l drses. brLl ! l l rc l ) , , r ) t t re {nrc workrn! f r inciplc. lD r t \ nx, \ l b l \ ( . tor nrlhe bi l rcrr l l ie s lnp eo|ntr isc\ ot lNo ( i is \ i rn i t r i r r ) rc l i r t st f r Is t ronde(t l ( jsothrf . u\ l l t , ! , ) llhc \ r rnc sUri ic. . Icr l . hLr l l lo l n1jcc\s l r i t ] , t thr srnrc rhicknrss th.rn)str t . I hc e(r IIosncnrcl ' t s l r rp rs eLlrDpc( j r l r )nc ( | rd t ( j i ret i r i r c lnt i lc !cf bcun). !n( l is hor i /ontr t r lr i f r r l ' .u | ' r lcr ] r l1err lUfe. Whcn thc r0l l fc | l r tL|re is incrersc(I . lhe \ l r ip ( lef l .c ls i r ] th.di fccl ion ol rhc urt l l l wi lh thc lcrsr cocf l lc icnl o incxrcxpl sur t ts $ort ing pr inet t tc

'c l rcs ou lhc luct t l r l t lhc nrct l ts wi l t c\ t l |d r t d i lL l1:nl r l tes !s the nr i t is hcr lcct t :hc

purtosc ol lhrs dcl lccl ion i \ to l l t i .x l l \ c lusc lhc st f ip lo nlrkc conl iLct r r rh l *r , i lch () fconrplctr . r c lccl f ic c i rcui l u l l p!rrrcUl l f scl t r ) in l teml)cr tuf . Ibole lh. r inrhicDL

Frgurc '164 shor\ I thcrnrosr l r conrIr is ing r h i lneru ic sr f ip t i , rmed br- bonding uslr ' r o l cold rol lcd 60' .10 br i rss \ \ i th ! s i ]n i l r s i /c sr f jp otcold ro cd nr i lc tsrcet BoLhIhc bfrss lnd sLccl s l r ips ! fc exrct t \ lhr l r l lncNi( l rhot t5ml l ] .Thcbi cr | l t icstr ip jspr.c l \ch locrrccl rnd f ig ic i l i c lunrpecl s i r ingr tcn.{ thol 70 + 0 {)S mm. Wh.n herredlroor rn rL 'nbie| t or d i r tum tcnrruf l turc. /0 : l5 ( ' . rher) rhc str ip is hof izonLat. ro j lssdpolnl t rnrperr tufc. 7: rhc srr ip deUccts dorn\ ! r i fus . tDd complcrc\ a1r ctcclr iceircLr i r 1 l i \ rcquired lo deter lnrnc rhe scL}) int rer ipcr ture lor ihc himerr l l ic s l f iplbr r g i \en dei lect ion ol 2 _L 0.05 nnn. which is thc disr lncc t l l . l t scp r teq rhc brsc

Case srudies 241

''

lh ickness ol br ls! str lp

Nlodulus of l - last ic i l ] l i ) r

Nlodulus of Elast ic i lv for

ContactEtectricalinsubrion

Figure 4.64 Tl f t rmnrl in i r i , f t r t f r

( )J rhc nf i t ) r r ( l lhc r l l j ( l r icr l . ( jnt i rct . t l rs . r ln, fc( tu c( l l ( ) dclcr n) jnc I hc nr()s l ( r r l ie i r l\ r j | l rb l fs i I \o l \ ( l i r r r r rccrrn,r r I is rcqUircr ] r ( I t th i )Ugh pcrtaf l l l rg. sc s i t ru l !i I r . r l r \ is lhclhiekn(ssol thc br i rs\ r l Ip rr05 t00]nl l l . i rn{ l lhc nl t ( t s lc( t s l r i t )rhr .kDcssi \0. ,1 00tnIL Ihr NI(x iuturot l i t i rst icr l ) / t l )5( i t ' l t i r 60r:10 l , r rss{r lh . ' eocl l ic ic l l o l u rx l ( )D (1. , { )01. i rn( l / r t0t( i t ' rL t i ) r r l r i td stc. t $ i lh( , : : i l )01 thc (oci l ierc i r t ot t rcxr f \ t ) r I \ ron . , l t . j ( t0 , / , ( ) t i ) r .o l ( l fo c( lhfrss. i r r l J , ) r e( ,k l ro l lc( l nr ik l nc. l . , , 1.1.7(t0 , ' , , ( , ) . h,n]r $ i lh l coc ierc0t ( ) t .! i r f r r l r ( ) I ( \ : ,001, c\ t1nl I . ( i t ro l r cngrncc. I ! t j lc f t turc t r is rss nrc(t th i r l lhcslrcsscs \c l rLp i , r crch r l l | ( , ( lc l leetro,r rLrc {c l1 bct , r t r lhLj i r v ickt s l rc elhs i Ix llhr l r r ) resi( lu i r l \ ( r$s$,! . l ( l i Ig I r thf i ( t i ! id l t st f i t I r i tc f ix l .

Detemining the temperature variation' lhcrrr \ inrulr( l . l lcel j , ' r r l1hct iccc ( lot lhcbinrctr I i rst f i t ) . r , , , , . . ( lUetoxlrnr l ) r r rl r fc rn.rcrsc l io l r / , t { ) / is gr \cr hy (Younr. t98,)) :

(1 | | r )

/ : lcrgrh ol the hirnclal l ic s l r i l

r ' ( l iDu coef l lcrcnt o l erprnsion of lnr td steel

, {B: l incxr coelf ic ieor

242 Derigning teliable Ptoduds

Replacing the change in lemperature (I - 70) with AI' which is a random variabl

i l 'e l f , dnd reurrdnging for thr\ rerm gi \L '

(4.113)and the rn

SensitivFor the prsetpoint Ithermosta

The variance equation to determine the standard devialiofl of the change in tempera_

lure can be wrtlrcn as:

Itai.:) '",. ,(H)'"r. (ff) ' a *(H)'d.l 'l-tgl' r"-(*i)' "r (ff)' ",.-(H)' ";,J,,.

The mean value, !A?, can be approximaled by substituting the mean valuts foreach variable inlo equation 4 l 14 All eigbt variables are assumed to be r4ndom innature following the Normal distribution. A summary of the parameters for eachvariable is givei belo\4. The standard deviation of each has been drivcd, in thecase of 1,,n,", and l. from assuming thal 6 standard deviations cover the tolrancerange givJn, in the casc of the individual sttip thicknesss, from an appraisal usingC,{ *na io tt'" case of th elastic constants and coefrcients of linear cxpansion.direcrly from the collicints of variation provided.

]'* - /v(0 002' 0'000017) m

' ds - N(0.0003,0000004)m

dB - N(0 0005, 0.000002) rn

ts - N(208,6.24)GPa

,'ts - N(los,2 l )GP!

es - N(12.7,0127) l0 6/"c

.rB - N(1s.s.0185) l0 6/"c

I - N(0.07,0.000017) m

With relerence to Appendix Xl, we can solve each parlial derivaiive term in cquation4.1 14 using rhe Frnrle Drlference melhod lo give:

From th

variables

.L

For the fivariance c

f (z.o,rs ' too)'?,(r.r ' to ')?+(t.oer, ro')"(+ '

lo 6)'

I +(a.,rr x toa)':x (z x to 6)'?+(t.trz

' ro ")" (e z+ "

ton)'?

| +(-2.+rs ' to 'r)2,.(z.t x lo')'?+(t.+rz x to')"1o tzr ' lo

u1'

l+1 r .+ts*to ' ; " (o. t rs ' l0 u)?+( 2421 xLoa) '?x(t .zxto 5) 'z

Repeatin!lng oroer

for the nsettilg thproperties

minimal cThis ca

than fordevelopmlthe basic

and it isproblem

Dle

, l ] )

Itar-

Case studies 243

and

oAr = 3 41"C

and the mean pAr = 84.74'C. Therefore, thc setpoint temperature is given byl

T - AT + To = 84.74 + t5 =99.14"c

Sensitivity analysisFoa the purposes ofmeeting a customr spccification, a tolerance for the thernostatsetpoint temperature can be estimated at +3aAr, from which the approximatethermostat spcifi cation becomesi

r-100+l l "C

From tle Finitc Difrerence method results above, the contribution of the vari4nc ofeach variable to the temperaturc variance can be estimated to focus in on the keyvariables bounding the problem, This is an attempt to rcduce the thermostat spcifi-calion tolerance to around half its cuffent design valuc,

,1r = (0.203) + (0.t82) + (0.02E) + ( j .345 x lo-3) + (2.579 x l0r)

+ (3.49?) + (7.5s0)+ (0,169)

= t1.637

For the first term, th variance contribution of the dflection variable, ,mrx, to thevariance of th temperaturc as a percentage btomes:

0.203 _^^l t tn/1$=t. . t4%

Rcpating the above for each variable and ranking the percentage values in descend-ing order gives the Pareto chart in Figure 4.65. It is evident tlat over 90% of thevariarce in the tchprature is due to the varianc of the expansion comciertsfor the metals usd. These are the most critical variables in the problem ofsetling the tolerance specifrcatiod, A ttter ulderstandilg and control of theseproperties of the metals! or even the us of mor exotic materials than those firstchorn, will lead to a reduction in th achievable variation. This is because theother variables in the problem, such as the elastic constants of the metals. have aminimal contribulion.

This case study has highlightd the use of probabilistic design principles otherthan for the pu.pose of SSI analysis. For lrearly every engineeriDg problem, thedevelopment of an answer using probabilistic tEchriques relies on manipulatingthe basic governing function for tle parameter of interst, as would b dode ina deterministic approach. A development from first principles is not necssary,and it is commonly tle determidation of the random variables involved in theproblem which requires further and a more thorough investigation. For thesituation above, a standa.d formulation is available, which is the case for manvengineering problems. Albough rbe malhematics needed for a probabil ist ic

[4)

i fort intchthe

hceItlgbn.

rion

244 Designing rel iabl ptoducls

Dcsq. var abe

Figure 4.65 ! f i , I ! r r r [ 7r ! l ( r r ] r l rJI r i r l r ro ! . t r . r

rn i r l \s i \ Ls Inorc (() | r r t )1( \ th:rn x { l . l . r r r r r r r is l i ( , )nc. i rny cnslrrcerLrrg l r ( ) l ) l . r r r cxr i h(

f ) ro,rehe( l i r r r sr I r l l l f nr . .h i r ls l i ( lx \ l r ion Lrsr. ! lhc lc.hDit lLr .s rrr( l r r r r l l i , \ lo l , ) ! \

4.8.7 Design of a con-rod and pin

lh is.rse s lu( l l ( l is !Lrsscs lh. ( lc tgr) , , l t r r . . i f f tJert iDg rrrcehxrr lcr l | rcss l i , r lh( '

rnn.ulr . lureol crn l r ( l ' ( | i r \ ! r r 1r( l r t shccl \ lcc l nrrr lcr i r r l lhc!ul lors$(r(rrr \ ( ,1!e( lin rhr exr l \ s l rgcs oi lhc I ro( luel Jc\ f l ( ) f r r r tn l Pn)ccss lo x( l ! isc th. e( !n l r r | ! dfs ign

rng rhc t res\ r r choosing bcl$eerr r r ru rbu r) ! ( ic \ ign rr l lcrDr l i lcs wr ih l Ie gort l o l

cDsur l l lg r ls r . ' l i rhr l i l t Ihc rulL(,rs usc( l l fn)bubi l is l re r t t f r l ) rc l r 1(r lhc f roblcm Lo

ffor ide lhr ncccs\rr , - dee| l : . o l c l i r r i r r ' bcl$ccn lhc.orrr l . l ing \ ( i l t r l r , i . \

the f f .s\ hnl l b.er c l .s ig|e( l $ i rh r crpr. i r ) ro c l t l i \ef l80kN f fcss l ; fcc rnd t( l

wofk r1 rL t roJuelur rr ic ol , l0 l i , l \ l )err)rrnulc (x lcu1lr l ions l t ) ( lc lcrnirn. t l rccl is_

t f ihLr l ion ol l i , rnrug lo lds fcqrr i r .d r .d icr led thtr l the rucss ! i rprrc i l ! $!s rc lcqtr t t l t to

i i fnr thc l lnr i l } - o l steel l1( l \ l , i b. proclucecl orr the mrrchinc. a) .c ol rh. nrNlor r fers

ol inrcrcsl rn thc d.s ign \ \ ' rs lhc con n u d t i r (sec I i rurc I66) Thc l i rs l opt io l r

cons( lcrcd Nls brtd on a prc\ ious c lcsig| whefe the con nr i $! \ nrxDul ict t r fed

l iom crsr r fon $irh phosphor bronze berf ings r t l rh. b ig aDCt snrxl l crds. Ho\e\cr .\ . ! l tDc\ses io rhis upprorch r lecessi(r lcd thc !onidcrrLiolr of o lhef ot l ions. Ihc

cr\c studt presenrs lhc rnr l \s is of lhc pin nnd con rocl using sunf le pfohrtbi l isrrc

Lcehnictue\ in i rn r l lcnrtL to pnn c in 'cr l icc

r . l r tb l t prerr opefr t ior . Ihc we) t

$crk l ink wrs intrx luccd 1(] ensufc crsc t i l nr i inrcnxDcc rod repr i f i r thc c lcnL

t igure 4.6

b.r( l l ! t j j

l l ( rx l l \

Design

shdr. I

thc s\s lc

Ihe1I

sl)erf . Tl

Case studies 2.15

Fiqure4.66 ,r Indt { , rp, , i l , i , ,1

" l i ro l f r jo. ' ( t ! tu l l ionr\ i r lso(, , \cr . ( l ( ) r r r t ( )u( ts l \ , t r icr !oefrLr r t . l t ressf( l | i I t islxdl l orurtr lcr l r i t f i i r thc f rc* di .s ()r l t l l ; reLeD tro, l1 i , c:rughr , r r rhc i tLc scr.

l '1df l l ' , ( lc \cf ibcd:rs r \ t , I l | r , j r in rhc Iorks. Anir t !s is t )J.{ t rc botts rs nol .o\cf t ( t

Design strategyI r ' f rx lLrc l r )n thcprcssNi l tb!o|) | r r l ingxrrni io lsotcyelcs/)r ! r , / r r Ihcrel ; rc. i lr l r { ,u ld h. ( l ! \ i ! I .d ugr insl j r l rguc lur tur . . xncl thc eon,11)d ln( i p in l l rust bfengrncefcr l In thc lSht ol lhc din.hur io l r ) j th. l \ ! r . i r tcd cn( lLr f r r )ec st fcDtth inshcrr l r ( r thc purto\rs oj thc x.uly\ is i l wl \ !s\ul ] rc( t thr l rhc rppl ic( l sLrc\sroulc l h l !c.D cxlrcnrc \ i r tu. coresPof( t i l ! L{) the r f f t rcxr ion of l Ic t i0kN l ( , r (1.A so. iD c l r r r - ing oLrt thc l rut !s is, f f r (x iorrr . s l l l l rsLrcl t rodet\ wl j rc r .c( lcd t i ] fthc nlr ter i ls enduf i l rce \ r fcrgrh bused (Jn the r ! l i [ rb l . ( t r t tu. A\ n]cnLioned.uf l ier .thc s\sterD \ ! ls to bc deng.cd r i rh l u, . r t t ink. T,) s l l rst j th is rcquircnrent rhc pirwls desig.cd such lb l t i r \oLr lc l tx i t in x. o!ef tond. i i t l [ t ior .

Ihe rrr lcf iL i l sc lccrecl l i ) r rhe pin qls 070M10 rorml l ized nikt s leel . The frr $us tobc uru'rul lctLrrcd hr o !h i t r l l rg t iorn brf rncl ! \ ls . rssLrmcct to br!r non cr i r ic ! ld 'Drcrs()nr l | i l r ixr ion in lcrns ot lhc stress distr ibut io l r . lnd lhcfc ibrc rhc o! . r toadnrc\s could bc rcprcscnrccl b l - I unique \ ! luc Thc pur s iTe routd be ( tcternr i .cdbrsed on th. i s landafd dcl j l r lon t jmi t ot the r ia lcr j r l .s e|duf nce \rrcngrh in\herr This i r t . rsthrt rhc prob!bi t i t ) 01 J l ihrc of lhc con_rod rystcm t lue Lo I .ur iguc$'ould bc ver! k)$ ' . l round l : l50ppnr rs\umi|g r t Nor.rrr l c j isrr ibur ion t .or Lhc end-Lnancc dfcrgrh in she.tr . This retr les to r rc l i rb i I ry R ! 0 999 $ hich is r( jequr lc for rhc

246 Deiigning rliable prodocts

3lnglh

Flgure 4.57 Stess-strength inteffercnce lor the pin

non-safety critical nature of the failure mode. The situation is represented in thestress-strength interfernce diagram given in Figure 4.6?.

Detemination of the pin diameter and con-rcd seclion sizeThere is no data available on the endurance strength ill shear for the material chosenfor the pin. An approximate method for determining the paramercrs ofthis malerialproperty for low carbon steels is given next. The pin sleel for the approximate sectionsize has the following Normal distribution parameters for the ultimate tensilestrengtt, Sr:

sI,] - N(50s.9,25.3) MPa

The endurance strength in bending, Se, is commonly found by multiplying sx by anempirical facior, typically 0.5 for steels. For mild steel, tle relationship is (Watermanand Ashby, 1991):

S? = 0.47Sr] (4.r l5)The relationship between the enduralce strength in shear to that in bending is givenby (Haugen, 1980):

,c ^r

0.577Se

Therefore, substituting quation 4.115 into equation 4.116 gives:

(4.116)

p.. = o.27 ps, : 0.2',t(s05.9) = l36.6Mpa

Typjcally for the variation of the endurance strcngth in bending, at 106 cycles ofoperation (Fuman, 1981):

Substitutir

Finally, su

Summarizmild sleel

The pin is

The mininstandard d

Thereforei

The final sln desigr

individualmellt. ln Oof th pin.ultimste sltensile and

Thereforel

standard d

tF

iD the

iosenllcnalDCtronlasile

Dy annDan

l , r r tglven

Lr l6)

ls of

Case studis 247

Substituting equation 4.115 into the above gives:

ds. = 0.038pr].

Finally, substituting equation 4.116 into the above gives the stanalard deviation of theendurance strength in shear to be:

o,, = 0.022ps, = 0.022(505.9) = I l.l Mpa

Summarizing, the parameters for the endumnce strength for 070M20 normalizedmild steel in shear are:

' " -x(136.6, IL l )MPaThe pin is in double shear in ser1r'ice. The diameter, 4 is determined by:

(4.|7)

F = shear force

icmin = minimum endurance strength in shear.

The minimum endurance strength for the problem stated earlier is set at the _3standafd deviations limit. thereforel

i . .nin = pr, - 3dr,= 136.6 - 3( l l r 1) = t03.6Mpa

Therforel

= 0.0415m = 41.5 mm

Th final selection ofpin diameter based on preferred numbers gives d = Z42mm.h designirg rhe con-rod, we wish to ensure rhat the pin wil f;il, in tle;;se ofan

overload, in preference to the con-rod. To realize this, the mean values of thejrindividual strength distributions are to be set apart bya margin toensure this rcquire-m-ent. In this way, the probability of con-rod failure will become insignifrcanl to thatof the pin. The force to shear the pin in an overload situation is a-function of theultimate shear strength, 7u, of the material. The relationship between tle ultimatetensile and shear properties for steel js (Green. 1992):

t" = 0 75S!

Thelefore:

z" - N(3?9.4, 19) MPa

We assume that the maximum r timate shar strergth, rurox, of the pin is +3standard deviations from the mean value, therefore

,rn * = p," +3o,, = 379.4 + 3(19) :435.4Mpa

F = 1.5708',-"*d' : 1.5108 '<

436.4 x 106 x 0.0422 = l.2l MN

0.63662x280x10r10l.6 r t06

2,18 Desigfling reliable pfoduds

fa52

3

Figure 4.68 8r!r r l r t rJtr or \ o l l l r I o,r n lnfr Icf l l ( (Ni ! t ror ,

' lhc n.r l s l r ! :c i : l ( ) ( lcrLhlc tht eon ro( l s i / . hxsc{ l orr 1 l l i \ l ( , rcc l | rn\rnL11.( l l ronrth. | | r rn lhc, \ . r lor( l \ r r rLrr io. l I t h i rsre ( l i r r rerrs iorr \ o l lnrLl l (n{ l . r (J\ \ scelronrr .or tor i r l ing r Ih, ,s l )h0r br()r /c l )crrrg rr( sh()rvrr r r i I rgrr t r i : l68

( iL! . r lhc ((^ l (onslrr i r ls i r r f l \ ( ( l ( r r lhf f (ss hrrnrc r lctgtr . rD( l lhc ( le:Lrc lonr in i r r i / . drrr(rr \k)nl l ! , r r i r r1 i ( )n. l I ( {xrrre r i r t l f f i r t l r l ls \c lc. lct l t i ) r lhc c( , r r r ( { le iosss(clror. l , ) ( ldcrnr l rc l l re nirrerrs i ( ) r r . / ) . rs in! lhc s i r r r . s lcel lor l l rc(orrr( \ l :

Solv ing J i ) r / ) sr lcs:

iy !11 lli 1 1111520.051410: l0" l

5(,5 9 l l ls l ) : l lo Nl l ' rL

.l ( ,05(/) r ) 0 i l l

I i rcx ol lh. scct ion

Figure 4 t

Obsen

4.9

\ idc rhr

rppIcul

{) .108: l nr 108.1n r

Agrin. l l prc lcrred uloe i i f thc l lcLur l sccl io l r $ idth would be /) : l l t )nnn In r

nrorc s iDrr l i t iecl \ \ay th i ln lhuL trescnled in Sect ion,1.8.1. wc hu!c scplrxrcd rhcf i i lurc ol the prn i l1nn th. con rod b, \ rpproxi l r l r le ly l+ l strcnglh st .n. l r rdde\r . ] l lons. lhe rc lunl scf l rx l ior c. l l l be modcl lcd l i ) r thc dinr ibur ion ol lhe sherrlirrce in the pin nDd rcllsile force in thc con r1)d. .Ls illusrrrted ir Figlrfe ,1.69.

The s l -ety mufgin. S\4. is cr lcLr l r ted to b. .1.61. or dclnred another $ly. thcrelirbillt)' { - 0.999991J which is rd.qu

tc li)r rhc .rpplicxtion to rvoid oveftlcsign

Summary 249

0 105

Figure 4.69 5epf l ron ofr t r . rn ind (on ({ t f ( t r (c\

Observations

^n I lpor l r r t ls | .c l (J l l t rc s inr f tc I r ( )brhj t is l r . rPf f ( ) rch usc( l rbo\c srs lhr t i l

t l i ) r r i r i l r ' t r ' . t , . r r .1rr . rn. , . , . , t , t . r I : , r , . r t r r . , r , . t r . ) r th. . , . : r_I . . . . . t r r t l t r r

: l i l : i i i : '";trtri:t: :1t;., , l l ;:r::;,1.1,1.'; l ; l l):r.]],,:: l i , ' . ' i ' ' , '- 'l r r l , r . r l r t r . r r l . . r ! \ r r . i t .$, , r r \ \ r , , r | ! t r1t , r r . t r . , t . . ,hr . r . r r . r t , f r , ,ht . | .ur , r , . , r l

. . ' , r . , , . , i , . , , . . . , , , r . r , , , , . r . , , r . . . , , , . , . \ . ,h r , . . . , , . . , . r , , . , , ,1r . , , , . . . , . i , . , , . , ' , r ' ,1book A t l l iguc rrrr tysis t i ) , . lhc coI nr l wout(t r)cc( l t () l lkc into lcco rr l r i l t relor\r l lc ' r i r )g rhc rrr i !ue r i i i . such rs nress con.cIrr !r i1!rs i | | )cr srr rrcc r in]srr . l lowcvcr. i l: ,1, . ' : , , , ' . ' . . ,

r t , . . r . . . r ! , . , , . ,a. . r . r r . , { ( i , , . , . r r , r .n. . , . . r , . , . : , | | . . r r r . , t , . . , r 1, : , . . . l l r !rc. , rcr . In. i . L i t r . t r r . , , . , , j , r \ t t . . ,h. ,hr t . . rh. r . . l r ! rJ Jr \ , r lr rc ( l l | cetcd rd ( r r lcr ( t916). l l ruScn ( t r ) t ( ) ) uncl Mischkc ( I99l)

4.9 Summary

l " l : l l : l l i . . l l l :1, . , , , : , , , , , , . , , . , r , r r r r . , . . z , , r ,1 \c,- , , r1, . , , . ,nr , r , . . , .' /n) ! on l r ' 5,rn . . . , . t Jn: r ( r ! n, , r , . r t r . r . , D.r . . , I . r r J.- i r : r r r r t , r . ,or , ,: l l l i l :. ' ,1i;;; iri l; ' i i i :: l i :r; ",i.: ;,, i.: 'J,,l,:.: l i, lr,:,tr., i; l l l ,t;rppl icr l ion. bLrt s incr t i rctoA of\ l terr

"r . nu,p.nn,n,un." rc lrred, easure\ rberc, nn $ , \ h! sh, .h. Inu$ $te,hl rF. l Jr . i ! , , r . , , . , , , , " . , , , , , ; ; , , , .. ' \J f , , . , .e, \ , r i \ . , . , . r i r , . . , .1, t i ( : , , , , ,n. t ,1, t" ,h, t , r . , ,pn, , , , . r , , : , . , i r i . , . ' , " , . i ,

ior. but hale i . r ro be t . rken up wiJely br " , , , ; ; ; " . , ; ; ; ;

Virtually rlldesjgD paranercrs sLr.h rs rolerances. lnalcfial propcfries lrrcl servicelo.Lds e)ihibit s(nne sratisticll variabiliry and Lrnccr,";tl, ,rr"t ;"nr.,,..ir," "i.qr,r.yot 1le dcsign. A key reqrirement in Lhe pfobrbilistic approach is dcrailcd knowiedg,e

250 Designing rel iable produds

.rbout the di l l r ibu( ions in\ol lcd. 1o enable pl . rusible resul ts to be p|(Xluced. Ihcr l rour l of i r l i rnrrr ion r l r ihblc.r l these e.rr ly stages is l i ru i ied. iud ihe del ignelnlrkes expef ience( l judgcm.nrs $hcrc infofmrl ion is lacking Ihis is$hylhedelernr i r r is t ic rpprouch is s l r l l Dopul . r r . bccruse nr.ury of the vr f i rb le i r fe t rken undclthc unrbfc l l r 'o l onc l rctor. I f krowledge ol thecr i l ic l l uf i rb lcs in thc dcsign crnbc csum!rcd $i th in r cert l i r !onf idcncc lc!c l . thcn lhc probrbi l is t ic rp l l forchhcconrcs rnorc sui t rh lc I ' fobrbi l is t ic ( lcs ign thcn tro\ idcs r m()rc rcr l isr lc rrry ollh iDking rbout thc dcsign pr)blcnr.

This chxt tcf hxs out l inc( l lhc nrr in concctt \ t l | rd lcchniqucs associ : r lcd wi th c icsign'ins lor rc l i rh i l i l ! th i r t hr !c bcerr ( lc !c l ( )pc(t \v i lh rcgrrd 1o r t robrhi l is l ic dcsign. I ) rs lwor l i 1,n ( ( , r rJonnt lb i l i l l - Arr l ls is (( A ) hrs rrrdc i l fossiblc ro csLi r r tc din)cnsiorr lvr l i r l ( )n. r k(y conr|oncn1 iD lhc probrbi l is l ic cr lculr i t io l ls usc( l ls pr o l lhcnrclho( lo l t )g! A lc\ problcn) i f i pr i )bt l l l i l is l ic d$ign is rhc Scn.fr l ion ol rh.ch. f rcrcnTiDg r l is l r rbul ion\ l i ) f cxt) ! r inrcnl i r l dr l r ' lhc c l lcc l i !c s lx l is l ic . l rn(x lc l l rng,r l nulcrr l propcrr \ (h lx ur i l scr ! icc lor( ls is l i rn(hnrcnlr l lo lhc f rohxhr l is l r .xfproxeh Whct( l l r i \ ( | r1r .cc( ls lo bc nrodcl lc( I . lh. nhl c l i icrcnl lcchrrquc\n kc r l t ( ) \s ib lc l ( ) (s l i r r r lc thc p.r : rmclcrs l i ) r sc!cr . l inr for l r r l ( l is l lhul iors

Wc rcc( l i r stccir l r lgcbrt lo ot)cfr lc oD lhc cDgrrrcctrnA slrcss cqur l ions r : prr l o l

t r ( )brhi l isrr .dcsigr I hrouAh lhr Isc,)1 1hc\!r i !n.ccqu! l ion.rnr{r !o icsl inrr l i rgrhe slrcs: \ r r i r l ) lc is fn) \ i ( lcr l by rc l . l r r ! : gcoDrclr ic rDd lo i rd drslr lhulror is $r lh lhclrr lure sovr ' ) ing s l rcss cqur l ior . Mcthods ro !J lv( th( r r r i r r )cc r . I r l iooshi j rs l i r l

cur Nofmrl con(| l iors hr!c bccr disctLss(( i : rn( l cxlnrplcs givcr | ; r rn,) i , j !onrPlcrcrs( \ using lhc 1 Inr lc I ) i l l . r . rce Mclh({ l rnd Monlc ( l f lo s inrul i r l i { ,n lhc ut f i t l rcccqL|. |1 i ( )n x lnr p i ) ! i ( les r \ r lur l ) lc (Jol wi lh $hieh 1o ( l f rw scnsi l iv i l ] i r t lc tcnccs t r tg i r f thre,)nrrrbul iorol crc l rvrr i rb lc lo lhtovrrr l l ! l f i rb i l i ly inrhctr(nr lcDr Scrsrt i ! i1! unt l lys is is p!r I o l lhe srr ( i r fd fc l i lh i l i l r - u iu l ts i \ rn( l lhrnrgh i1s Is. probxhi lis t ic mclhods provr lc l Inorc c l l ic l i !c wrry l i ) ( lc tcf ln inc k.y dcsign trr rnrc lcrs i r r r( l$rgn. l i1)rn th is rnd olhcf in l i ,nrx lk)D rrr I ) r rc lo chrf t 1 i )nu. the ( icsrgrr . f crrrquickl l l i ,cLrs on lhc ( lonrnrnr r ! rxhlcs lof r .dc\ igu purposes.

Srcss \ r fcDgrh inrcr lcrencc ror lysir o l lc fs : r pr i rc l icr l cnginccr inS ntpn)rch l i ) fc lesignirg rn( l qulnl i l r l i \c ly predir t i r )g thc rel iubi l i ly o l comtoncnls sLrbicclcd 1o,rcchrnicr l lording i 'n( l hrs becn descf ibed t ls t l s iuruLrr i \ 'e mi)dcl o l j l i i l 'L tc. lhcprobrbi l i l ) o l l i r i lLrrc. anr l hcuct the rc l i rb i l i l ) . c n bc csl inrr lcd rs thc r f tn ol 'i r tcf le|cncc hcl$t tn lhc strcs\ xnd sl fength distr ibut idrs. Howcvcr. lhc unulr ' \ i \ o lrc l inhi l i l r using this. tpprorch is motc ol icn thrn nol incorccLl) pcf l i ) fnrcd xnJ rlhon)ugh undefqtrndiog of thc londiDg tr-pc is rcqui fcd b! Lhc f r rcr i l io ler Tbischapler hrs reviewed lhc rppl icnr ion ol stress qtrelrgl l r in ler lereoce rnr lysis 1osorne irDp(rtrnt crsc:i in stntic dcsign xnd hrs pforided melllods to sohe tlre ffoblemli)r .rny combi[rtirD d strcss nn(l srrcngth using cbsed-lbflD equrtions of rulreficrl

when designing r pr ix lucl iL is uscl i r l to hale r re l iabi l i ty targel . in order to x l t r i incustomer sat isfudion und fcducc lc le ls of ron-conlbrmrnce rnd r t tendanl t r i lurccosts. A c lcrrr nDd concisc . tppr oach to i - r l l l r fe mode descr ip l ioni t rnd r1r l i l lh i l i t r tnfgctsdetef |n inaLkD is r lso crucial in the development ol rc l i tb le dcsigDs.. ind Lhc in lcgr. cdusc ol I -MEA in set l lng le l i rb i l i ty t rgc! lc lc ls is a kcy bcDcl i l o l -Lhc rpprorch in th isrclpcc!. A ker obiective of the methftiology is lo prctidc thc dcsigncr wirh .t deeperundcfsLnDdins ol lhese crirical design prfnmctcri rnd how Lhcr" inllucncc thc

thrLl thtr

usiug l Ipfovidin

Summary

adequacy oi the design in its operating environment. The design intent musl bc toproduce del.tilcd designs that roflect ahigh rcliabilitywhcn hservice_ It is apprecialcdthai there is not always published data on cngineering variables. However, uch cnnbe done by rpproaching engincering problems with a probabilistjc lnindset..

Probabilislic design provides a traNparent means ofexplaining to a business mo.cabout the saltly aspects of enginccring design dccisions with a dgree ot slariry norprovidcd by the'faclor of safety' anprodoh. The measures of pcrtorma,ce dcrcrminedusing a pfobabilistic upproach give rhc designrs moce conndcnce in their dcsigns byprovjding better undcrstlnding of rhc variables involved and quantitative csrimaresfor rcliabilitv.

)51

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