Avalabl onln at www.scncdrct.com cncdrct Procda Engnrng 0 (05 ) 9 6 3rd Intrnatonal Confrnc on Matral and Componnt Prformanc undr Varabl Ampltud oadng, VA05 Dsgn stratgs of tst cods for durablty rqurmnt of dsk braks n truck applcaton Bngt Johannsson a, *, Tatana mrnova b, taffan Johansson a 0F* a Volvo Group Truck Tchnology, E-40508 Gothnburg, wdn b Altn wdn AB, E-430 Gothnburg, wdn Abstract Durablty rqurmnts for dsk braks n truck applcaton ar mprovd basd on th actual customr usag. A scond momnt rlablty ndx s usd to dsgn tst cods for th assssmnt of varabl ampltud dsk brak fatgu lf. Th approach s basd on th man stmats of logarthms of quvalnt strngth of th brak and customr load varabls. Th ndx gvs possblts to tak all uncrtants n th fatgu lf assssmnt nto account, ncludng scattr n matral, producton, and usag but also systmatc rrors lk modl rrors n tst st up, strss calculatons, damag hypothss, as wll as statstcal uncrtants. 05 Th Authors. Publshd by Elsvr td. Ths s an opn accss artcl undr th CC BY-NC-ND lcns 05 Th Authors. Publshd by Elsvr td. (http://cratvcommons.org/lcnss/by-nc-nd/4.0/). Pr-rvw Pr-rvw undr undr rsponsblty rsponsblty of th of Czch th Czch octy octy for Mchancs for Mchancs. Kywords: og-normal dstrbuton, Cornll rlablty ndx, xtra safty dstanc, uncrtants, scattr, Wöhlr curv. Introducton Thr s a contnuous nd n mprovng durablty rqurmnts for dsk braks by makng thm mor corrlatd to th actual customr usag. Th potntal hr s to dffrntat th rqurmnts,.g. allowng offr of a lght wght brak for lss dmandng applcatons. Rlablty assssmnt wth rspct to fatgu falurs s a dffcult task, on th strngth sd bcaus of th larg scattr n fatgu lf, on th load sz du to hghly varyng product usr profls. An assssmnt procdur must thrfor b smpl nough to b abl to quantfy vagu nput nformaton, and b sophstcatd nough to b a * Corrspondng author. Tl.: +46 3 33335. E-mal addrss: Bngt.Johannsson@volvo.com 877-7058 05 Th Authors. Publshd by Elsvr td. Ths s an opn accss artcl undr th CC BY-NC-ND lcns (http://cratvcommons.org/lcnss/by-nc-nd/4.0/). Pr-rvw undr rsponsblty of th Czch octy for Mchancs do:0.06/j.prong.05.0.030
0 Bngt Johannsson t al. / Procda Engnrng 0 ( 05 ) 9 6 usful ngnrng tool for mprovmnts. Th mthod s a dvlopmnt of th concpt ntroducd n [] that was basd on th classcal load/strngth mthod for fatgu at varabl ampltud. Th mthod uss scalar rprsntatvs for load and strngth that ar dfnd as quvalnt ampltuds, and stmats thr prdcton uncrtants for rlablty masurs. Th mthod may b sn as a frst ordr, scond momnt rlablty ndx accordng to th thory n []. Th mthod was prvously dscrbd n dtal n [3].. Th oad-trngth Rlablty Mthod.. Th Equvalnt oad and trngth Varabls In fatgu, th srvc load can b sn as a random procss of a tm varyng load n ach pont of th structur to b analysd. Th strngth of th structur also dpnds on th tm varyng charactrstcs of th load and thus a thory of damag accumulaton s ndd to obtan th dsrd on-dmnsonal masurs and to rlat thm to fatgu lf. Th stablshd thory for th usrs s th Basqun quaton combnd wth th Palmgrn-Mnr cumulatv damag rul. Ths thory wll b usd hr wth crtan practcal modfcatons for th actual purpos of gttng a usful rlablty ndx. Th two man dffrncs n th prsnt mthod compard to th tradtonal us of damag accumulaton thory ar ) matral or componnt strngth s assumd to b dtrmnd from varabl ampltud tsts to avod larg modl rrors causd by th cumulatv damag rul, and ) a fctv nduranc lmt s dfnd whch sutably s chosn to corrspond to th rqurd fatgu lf of th structur. Th followng varant of th Basqun quaton s usd [4], N n. () Hr n s th cycl numbr corrspondng to th nduranc lmt and s th load ampltud. A propr choc for stl constructons s. Th xponnt s assumd to b a matral or componnt charactrstc and s stmatd by tsts or by xprnc. Th proprty s rgardd as a random varabl that vars btwn spcmns, rflctng dffrnt mcrostructural faturs and manfstd by th obsrvd scattr n fatgu tsts. Ths random varabl s dfnd as th quvalnt strngth and s modlld as a log-normal varabl, whr th dstrbuton paramtrs rprsnt a crtan componnt typ or a spcfc matral. Th strngth for a componnt s stmatd from laboratory tst rsults. For spctrum tsts th appld load conssts of a varabl ampltud tm hstory that for damag analyss s dscrbd by ts ampltud spctrum basd on a cycl countng procdur [5], gvng th numbr of appld load ampltuds,. Accordng to th Palmgrn-Mnr cumulatv damag assumpton usng () th followng xprsson s obtand: D M M M n n N whr s th lf accordng to () for th load ampltud and M s th total numbr of countd cycls. Falur occurs whn D quals unty and th valu of for a spcfc spcmn s: / M ~, () n whch s an obsrvaton of th random quvalnt strngth varabl. For th applcaton of th load/strngth modl, a comparabl proprty for a srvc load spctra s ndd. Ths s constructd n a smlar way from th calculatd damag for th spctrum, M Tt DT t N T n. (3) Hr w sum ovr all countd ampltuds at for th drvn dstanc, and s th targt lf by mans of, for nstanc, drvng dstanc. Now, dfn th quvalnt load strss ampltud as,
Bngt Johannsson t al. / Procda Engnrng 0 ( 05 ) 9 6 / M Tt q, (4) T n and t can b sn from (3) that, whch quals unty whn quals. Th dstanc btwn th logarthms of ths two random proprts rprsnts a safty margn that wll b usd for th rlablty ndx. Th quvalnt load s modlld as a log-normal random varabl... Th Uncrtanty Masur of trngth In (), th quvalnt strngth for a crtan spcmn was dfnd, whr th random proprty s th accumulatd damag to falur, whl th spctrum s supposd to b known and th valu s a fxd constant. Th standard dvaton of th logarthmc obsrvd quvalnt strngths can b calculatd as, n m ln ~,, s ln ~ m, (5) n, n whr n s th numbr of fatgu tsts. From ths stmatd standard dvaton, th targt s to obtan a masur of prdcton uncrtanty for th rlablty ndx. Ths wll b don by usng a propr 95% prdcton ntrval for a futur assssmnt of th strngth, gvn th tst. Undr th assumpton that th logarthm of th quvalnt strngth s normally dstrbutd, such a prdcton ntrval can b calculatd from th tudnt-t dstrbuton, m s t. (6) 0.05, n n Hr, th t-valu compnsats for th uncrtanty n th standard dvaton stmat and th trm undr th root sgn taks th uncrtanty n th man valu nto account. nc th t-valu approachs.96 whn th numbr of tsts ncrass to nfnty, th actual uncrtanty componnt for th rlablty ndx s dfnd as, s t0.05, n s t0.05, n,, (7) t0.05, n.96 n whch thrby s a masur corrspondng to a standard dvaton of th prdctd quvalnt strngth. Ths ntal uncrtanty masur ncluds th scattr of th spcmns, th uncrtanty n th stmatd man, and th uncrtanty n th varanc stmat. In cas th tsts ar prformd on a random choc of componnts, also gomtrcal scattr du to tolrancs ar ncludd. Th uncrtanty du to th xponnt, nfluncs both strngth and load and wll b addd sparatly. Othr uncrtants that can contrbut to addtonal varaton n th stmat of strngth ar dscrbd n [3]. In total, th uncrtanty masur of strngth s calculatd by a smpl addton of th corrspondng varancs of log strngth,.3. Th Uncrtanty Masur of oad p. (8), Th load varaton, by mans of rlablty, orgnats from th populaton of usrs. Ths may b dntfd as customrs, markts, mssons or ownrs dpndng on th applcaton. For ach populaton, dffrnt rlablty ndcs can b calculatd, th mportant ssu s that t s mad clar what populaton th rlablty ndx rlats to. Onc th populaton s dfnd, th man and varanc of th corrspondng load nd to b stmatd. Prfrably ths ar obtand from drct masurmnts of loads n srvc. Ths usually dmands larg masurmnt campagns, whr th samplng of customrs, nvronmnts and markts dmands grat car whn plannng th campagns, s [6]. From masurmnts on a spcfc populaton t s possbl to calculat quvalnt loads accordng to (4) for ach sampl and fnd th man and th standard dvaton of th logarthmc transformaton of th populaton,
Bngt Johannsson t al. / Procda Engnrng 0 ( 05 ) 9 6 m n ln q, n n, s ln q, m n If th numbr of sampls n s small thr wll b an uncrtanty n th stmatd standard dvaton and our uncrtanty masur must b adjustd as n th strngth cas, agan assumng a normal dstrbuton for our logarthmc proprty, t0.975, n, s (0).96 n If th load masurmnts hav bn mad on a tst track or f th masurmnt campagn s known not to b a random sampl of th actual populaton thr may b an unknown bas n th stmats and by judgmnt mor uncrtanty componnts should b ntroducd,,,,3, () whr ach dffrnt componnt s stmatd as th coffcnt of varaton of quvalnt load,.. basd on a prcntag judgmnt..4. Th Prdctv afty Indx Basd on th logarthmc varant of th Cornll rlablty ndx [] and th proprts drvd abov, th prdctv safty ndx for fatgu can b formulatd as, m m P, () D whch s dmandd to b lss than a spcfd valu. In many applcatons such an ndx s ntrprtd as a statstc followng a normalzd normal dstrbuton and th dmand s spcfd as a maxmum probablty of falur. Howvr, normally th bhavour of th random varabls n quston s hghly uncrtan n th tals and such probablts may b strongly msladng. Thrfor w hr propos anothr formulaton of th rlablty ndx (9) m m. (3) d Hr, th dmand s that th stmatd dstanc from th falur mod,, should b lss than th sum of two trms. Th frst on rprsnts a dtrmnstc xtra safty dstanc, basd on judgmnts about srousnss of falur causs and costs. Th scond trm s ntrprtd as a propr masur of our statstcal uncrtanty, basd on all possbl sourcs. Th choc of th numbr n front of th prdcton uncrtanty s du to th assumpton that a normalty assumpton s assumd to b accurat wthn th cntral 95% of th dstrbuton. It should b notd that s a masur of th xtra safty dstanc for th partcular componnt. Th lvl of xtra safty dstanc dpnds on how safty crtcal th componnt s. Braks ar consdrd safty rdundant componnts manng that, n gnral, th xtra safty dstanc should b. 3. oad and trngth Masurmnts 3.. oad Masurmnts Th data from fld tsts customrs opratng n varous road condtons and opratng cycls wr loggd. For ach customr, th brak prssur data wr xtractd for th front braks and th valus wr xtrapolatd to th full targt vhcl lf for th applcaton. Th front braks ar consdrd to b xposd to mor svr loads. Mostly havy duty trucks hav bn usd whn collctng data. In total, loggd brak prssur data from about 40 fld tst customrs hav bn consdrd. 3.. trngth Masurmnts Thr xsts a st of dmands for corrspondng to applcatons n th valuaton of dsc braks durablty.
Bngt Johannsson t al. / Procda Engnrng 0 ( 05 ) 9 6 3 3... Havy duty dmand 3-4- A standard dmand applcabl to all havy duty applcatons s calld 3-4-. Th nam of dmand s rflctng th dstrbuton of dffrnt brak applcatons snc among 8 brak applcatons 3 s at 5% of th maxmum clampng forc and brak torqu, 4 ar at 50% and s at th maxmum lvls. Th block s rpatd untl mllon brak applcatons hav bn mad. For ths dmand,.5% s don n th oppost drcton corrspondng to rvrs brakng. 3... Havy duty dmand 66- Anothr dmand applcabl to componnt tstng for all havy duty applcatons s calld 66-. Th nam of ths dmand s rflctng th numbr of forward (66) and th numbr of rvrs () brakng. Ths s a constant ampltud dmand that can b prformd on dffrnt load lvls to stmat man valu, scattr and th slop of th Wöhlr curv. 4. Cas tudy: Estmaton of Paramtrs n Rlablty Indx and Comparson to tandard Dmands A larg st of customr data wth rlvant nformaton such as brak prssur, vhcl spd, gar tc., has bn gathrd n fld masurmnts and loggng of fld tst trucks. Th brak prssur sgnals, whch ar consdrd to b vry wll corrlatd wth th clampng forc of th brak and brak momnt, ar analysd n trms of duty (psudo-damag) and compard to th currnt rqurmnts. Th duty valus ar calculatd usng ran flow count and a sngl slop n th -N curv. Usng ths valus, an quvalnt load can b calculatd to compar wth th Rlablty Indx mthod dscrbd abov. 4.. Estmaton of oads rvc brak spctra wr xtractd for ach customr and th rsults wr xtrapolatd to th vhcl lf typcal for customr applcaton. Th rsults ar shown n Fgur blow. As can b sn from th fgur thr s a grat varaton among th customrs. Morovr, t can b notd that thr s a varaton among th customrs wthn ach customr catgory. Brak prssur (Bar) 0 9 8 7 6 5 4 3 Applcaton Applcaton Applcaton 3 Applcaton 4 Applcaton 5 Applcaton 6 0 0 0 0 0 0 3 0 4 0 5 0 6 0 7 Accumulatd numbr of brak applcatons xtrapolatd to Targt Vhcl f Fg. rvc brak spctra for all customr data. Th customr data ar xtrapolatd to th vhcl lf typcal for a crtan applcaton. Each fld masurmnt load spctrum has bn usd to calculat ts quvalnt load brak prssur ampltud, usng
4 Bngt Johannsson t al. / Procda Engnrng 0 ( 05 ) 9 6 th targt lf of on mllon km and th fatgu stmatd xponnt. Th rsult was wth th standard dvaton. 4.. Brak trngth Evaluaton A numbr braks whr tstd to falur usng th 66- tst squnc.,05 og Brak prssur (bar) 0,95 0,9 0,85 4,50 4,75 5,00 5,5 5,50 og Cycls to falur Fg.. Fatgu tst rsults. Th quvalnt strngth for ach spcmn tst s stmatd usng Eq. () from th numbr of cycls to falur and th load ampltud gvn by Fg.. Th slop of -N curv s assumd to b In ordr to compar strngth stmatd from constant ampltud loadng (66-) and block loadng (3-4-) dscrbd n nxt scton w nd to scal th damag accumulatd by ach spcmn accordng to Mnr sum by factor of 0.6 [7, 8]. Th obsrvd quvalnt strngths ar: (5.39, 4.66, 5.6, 5.0, 5.5). Thr logarthmc avrag s m ~ ln. 64, and standard dvaton s 0. 07. In th cas of 3-4- tst cod, th quvalnt strngth s agan calculatd from Eq. (). Th man quvalnt strngth s stmatd as =.75. 4.3. Estmaton of uncrtants Whn th quvalnt strngth s stmatd usng th rsults of 66- tst cod, th uncrtanty n th stmat can b valuatd usng Eq. (5) and Eq. (6). 0.07.77, 66 0.08..96 5 In cas of 3-4- tst, usually only fw tsts ar don, thus th uncrtanty should b stmatd n a dffrnt way. Th strngth dsprson s unknown pror to th tst. A standard assumpton wthn fatgu thory s that th standard dvaton of th logarthm (wth bass 0) of th numbr of cycls untl falur s at most 0.. A far consrvatv assumpton s, thrfor, to st th standard dvaton to 0.. nc th numbr of tsts wll b lmtd th corrcton factor for th lmtd numbr of sampls to stmat th man has to b don. Howvr, snc w assum that th standard dvaton s known th t-factor s not ndd. Assumng that mnmum numbr of sampls qual to s usually tstd n th rg, th dsprson for th strngth can thrfor b calculatd as:, 3 0. 0.5. 4 Furthr, th load uncrtanty s
Bngt Johannsson t al. / Procda Engnrng 0 ( 05 ) 9 6 5 0.9.0, 0.3..96 4 Du to th lack of xprmntal dsgn, th masurmnts do not ncssarly rprsnt th actual customr populaton. Furthrmor, th componnts that wll b mountd nto futur vhcls wll b usd by futur customrs and thr bhavour may dffr from th currnt customrs. In ordr to tak ths uncrtants nto account, an xtra varaton of 5% n load s addd. 0.05, 0.03 3 Th addtonal uncrtanty n th slop n th -N curv s assumd to b 5% or n othr words whch roughly corrsponds to unformly dstrbutd sprad n th slop of (s xampl of calculatons n [3]). Th uncrtanty componnts ar addd by thr squars, 4.4. Th rlablty ndx 0.08 0.03 0., 0.5 0.03 0. 6, 66 34 0.03 0.9 0. 0.9, 05. In our xampl w hav no spcfd targt for th xtra dstanc, but w can fnd out how larg ths dstanc s for th actual componnt, m m d.64 0.93 d 0.3 0.7 0.63 d 66 d66.75 0.93 0.3 0.8 0.64 66 d34 d 34 d34 0.08 0.8 Th statstcal uncrtanty rprsntng a 95% prdcton ntrval lavs an xtra safty dstanc of 0.076 rspctvly 0.8 for th two mthods. 5. Dscusson Th prsntd scond momnt rlablty ndx s formulatd by mans of logarthms of th quvalnt strngth and load varabls masurd n ngnrng unts such as knm or bar. It s ntndd to nclud all possbl uncrtants n th fatgu lf assssmnt, ncludng both scattr and possbl modl rrors. It assums a normal dstrbuton of th logarthmc dstanc, but n ordr to compnsat for th unknown tals of th tru dstrbuton, an xtra safty factor s addd basd on non-statstcal consdratons. Th cas study s an xampl of how th mthod can b usd, combnng statstcal obsrvatons and ngnrng judgmnts. Th rsult shows that th total uncrtanty s domnatd by th larg varaton wthn th usags of th truck. Morovr, th uncrtants on th stmats of th strngth ar qut sgnfcant rsultng n % rspctvly 3 % for two mthods. Th varaton n quvalnt load s about 30%. Ths facts xplan th rsultng small xtra safty dstanc, and suggst a mor thorough nvstgaton and modllng of th usag profls of customrs. It s possbl to rduc th uncrtanty n th stmat of th strngth by ncrasng th numbr of tst spcmn. Thrfor, furthr consdratons must b don n ordr to fnd sutabl ruls for sttng durablty rqurmnts by mans of th dtrmnstc xtra safty dstanc.
6 Bngt Johannsson t al. / Procda Engnrng 0 ( 05 ) 9 6 Rfrncs [] Karlsson, M.; Johannsson, B.; vnsson, T.; d Maré, J.: Vrfcaton of safty crtcal componnts, Prsntaton at th VDI confrnc: Trucks and Buss solutons of rlablty, sustanabl nvronmnt and transport ffcncy, Böblngn, Grmany, Jun 005. [] Madsn, Krnk, nd: Mthods of structural safty, Prntc-Hall, Nw Jrsy, UA, 986. [3]T. vnsson, M. Karlsson, B. Johannsson, J. D Mar, P. Johannsson.: Prdctv afty ndx for Varabl Ampltud Fatgu f, cond Int Conf on Matral and Componnt Prformanc undr Varabl Ampltud oadng, Darmstadt, Grmany (009), Vol II, pp. 73-73. [4] P. Johannsson, M. pckrt, Gud to oad Analyss for Durablty n Vhcl Engnrng, frst d., John Wly & ons, td., UK, 04. [5] ATM. tandard practss for cycl countng n fatgu analyss, ATM E 049-85. Annual book of ATM standards, (999) Vol. 03.0, pp. 70-78. [6] Karlsson, M.:Classfcaton of truck nvronmnts for fatgu assssmnts AE Tchncal Papr 008-0-, 008. [7] C. Brgr, K.-G. Eultz, P. Hulr, K.-. Kott, H. Naundorf, W. chutz, C.M. onsno, A. Wmmr, H. Znnr.: Btrbsfstgkt n Grmany an ovrvw, Intrnatonal Journal of Fatgu, Vol 4 (00), pp. 603 65. [8] J. chjv.: Fatgu of tructurs and Matrals, Kluw Acadmc Publshrs, 00.