Efficient probabilistic methods for real-time fatigue damage prognosis

Transcription

1 Aul Coerece o the Proostcs d Helth Meet Socet Ecet probblstc ethods or rel-te tue de prooss Yb d Yo Lu Deprtet o Cvl & Evroet Eeer Clrso Uverst Potsd Y lu@clrso.edu ABSTRACT A eerl probblstc tue crc rowth predcto ethodolo or ccurte d ecet de prooss s proposed ths pper. Ths ethodolo cossts two jor prts. Frst the relstc rdo lod s trsored to uvlet costt pltude lod process bsed o recetl developed echs odel. Ths trsorto vods the ccle-b-ccle clculto o tue crc rowth uder vrble pltude lod. Follow ths verse rst-order relblt ethod IFORM s used to evlute the tue crc rowth t rbtrr relblt level. Iverse FORM ethod does ot rure lre uber o ucto evlutos copred to the drect Mote Crlo sulto. Coputtol cost s sctl reduced d the proposed ethod s ver sutble or rel-te de prooss. uercl eples re used to deostrte the proposed ethod. Vrous eperetl dt uder vrble pltude lod re collected d odel predctos re copred wth eperetl dt or odel vldto. ITRODUCTIO Ftue lure s oe o ost coo lure odes or structures or copoets e.. rcrts d rotorcrts S. K. Bhu M. Sujt & M. A. Vetsw 8. The structures eperece deret lod spectrus dur the etre tue le. The ppled tue cclc lod o structures S. Ths s ope-ccess rtcle dstrbuted uder the ters o the Cretve Coos Attrbuto 3. Uted Sttes Lcese whch perts urestrcted use dstrbuto d reproducto edu provded the orl uthor d source re credted. Poer 3 s stochstc ture whch ches the stress pltude d stress rto throuhout the etre le o the structure. Oe b chllee tue crc rowth predcto s the proper cluso o rdo lod hstor eects becuse deret lod sueces duce deret lodtercto eects S. Mheevs & G. Gl 9. It ccelerte uderlod or decelerte overlod the tue crc rowth. Dur the lst ew decdes studes hve bee perored to vestte the retrdto o crc rowth cused b sle or ultple overlods D. M. Corbl & P. F. Pc 973; J. R. Moht B. B. Ver & P. K. R 9; K. T. Veteswr Ro & R. O. Rtche 988. Deret odels hve bee developed: eld zoe odels O. E. Wheeler 97; E. R. Wllebor J Wood RA 97 crc closure odels W. Elber 97; A. U. d. Ko 98; J. C. ew 98; A. R strp eld odels v. d. L. H. de Ko AU 98; J. C. ew 98; A. R d two-preter pproch A. H. orooz G. Gl & S. Lbert 7; A. H. orooz G. Gl & S. Lbert 8.These odels ocus o deret epltos o lod tercto eect d used deret tpes o vrble bloc lod or odel vldto. Relstc rdo lod cses d the sttstcl descrpto o lod tercto eects hve ot bee ull vestted the pst. Due to the coplcted d oler ture o rdo lod tercto ccleb-ccle sulto s rured or ech deret lod hstor d s coputtoll epesve or probblstc lss whch usull rures lre uber o Mote Crlo sultos. I vew o ths the objectve o ths pper s to develop ecet probblstc odel or tue crc rowth predcto whch s bsed o the sttstcl descrpto o the ppled rdo lod. The e de s to derve uvlet stress level bsed o the sttstcl descrpto o the rdo lod such s the probblstc dstrbuto o ppled stress re d stress rto. The the vrble pltude lod crc

2 Aul Coerece o the Proostcs d Helth Meet Socet rowth proble s reduced to uvlet costt pltude crc rowth proble whch retl clttes the terto or crc leth predcto. Ftue relblt s te-depedet relblt proble d deret etrcs c be used to descrbe the rdo ture o tue de. Oe coo pproch s to clculte the relblt/lure probblt t speced le Y. Lu & S. Mhdev 7; Y. Lu & S. Mhdev 9; Y. Lu & S. Mhdev 9b. Ths pproch s ver useul or eeer des st tue sce the servce le s usull ve des probles. Both sultobsed ethod d the FORM ethod c be used or ths purpose Y. Lu & S. Mhdev 9. Aother pproch s the probblstc le predcto.e. the re le estto t speced relblt level. Ths esure s ver useul or de prooss d codto-bsed tece d s the ocus o the curret stud. Drect Mote Crlo sulto c be used or probblstc tue predcto but s tecosu or lre scle pplctos. Ecet ethods or the probblstc tue le predcto re the e objectve o the proposed stud d the verse relblt ethod s proposed or ths purpose. The verse relblt ethods hve bee developed d used or the relblt-bsed des optzto RBDO proble A. Der Kureh Y. Zh & C.- C. L 994. The RBDO proble clcultes the des vrbles uder speced relblt level whch s slr to the probblstc tue le predcto proble.e. clculte the le vrble uder speced relblt level. Ths s the otvto the curret stud to vestte the verse relblt ethod or tue prooss. Ths pper s orzed s ollows. Frst the proposed uvlet stress s derved or soe specl vrble pltude hstores wthout cosder the lod tercto eect. uercl eples re deostrted to show the esblt o the proposed uvlet stress level cocept. et sll te scle tue crc rowth odel s brel dscussed d s used to obt correcto ter the proposed uvlet stress level or lod tercto eect. Follow ths ucertt qutcto o terl vrblt s perored d ecet uercl lorth or probblstc crc rowth s pleeted or tertve serch the verse FORM clculto. Model predctos re copred wth est eperetl dt or odel vldto. Fll soe coclusos d uture wor re ve bsed o the curret vestto. Equvlet stress level wthout lod tercto. Bsc cocept o the uvlet stress level There re severl deret tue crc rowth odels such s For s odel. E. Dowl 7 sro odel ASA EIFS-bsed tue crc rowth odel Y. Lu & S. Mhdev 9b d two-preter pproch A. H. orooz et l. 7; A. H. orooz et l. 8. Deret odels ocus o deret spects d wll ve deret predctos. A eerc ucto o crc rowth rte curve c be epressed s d / d R Where d re the crc leth d the tue le respectvel. s the stress re d R s the stress rto. Eq. c be reorulted s d R d For rbtrr tue ccle the creet o crc leth c be obted b tert both sdes o Eq.. d 3 R where s the crc sze t the be o ccle d s the crc sze t the be o ccle. The totl tue le uder rbtrr rdo lod hstor s the suto o d c be wrtte s totl d 4 R where s the tl crc sze d s crc leth t tue ccle. The o ths pper s to d uvlet crc rowth process uder costt pltude lod whch produces slr crc leth wth tht o the true rdo lod cse. I ths del crc rowth process the stress level s costt d s the proposed uvlet stress level ESL. The uvlet stress level c be epressed s totl d 5 R The uvlet stress level c be obted b ul Eq. 4 d Eq. 5 s d Δ d 6 σ R R Eq. 6 s the proposed uvlet stress level clculto d t c be ppled to deret tpes o crc rowth odels. For rbtrr uctos o. The ltcl soluto s ot ppret d dscussos o soe specl cses re ve below.

3 Aul Coerece o the Proostcs d Helth Meet Socet. Specl cses o uvlet stress level The stress re stress rto d correspod tue ccle re rdo vrbles the probblstc crc rowth lss uder eerl vrble pltude lod codtos. Severl specl cses re dscussed rst. Cse : ed stress rto Stress Stress b F. lod hstor: vrble pltude lod b uvlet lod A sple cse or ed R rto s llustrted rst or two deret lod ccles s show F.. The sold le represets rbtrr vrble pltude lod two ccles d the dshed le represets the uvlet lod. Both the vrble d costt lods re uder the se costt stress rto. These two lods hve the se tl d l crc sze. The tue crc rowth cused b the uvlet stress re s the se s tht cused b the two rbtrr lod ccles Eq. 7. The crc creets or the true lod hstor d the uvlet costt pltude lod c be epressed s totl d Δ d σ R R Lod totl R d Lod 7 To urther spl the dscusso oded Prs lw s used to derve the ltcl soluto o the uvlet stress. The oded Prs lw s epressed s d d ΔK R ΔK C 8 where C d re the tt preters the Prs lw. Preter C s epressed s eerc ucto o ppled stress rto to clude the R rto eect. For the cse the stress rto s ed d the C s costt. Us Eq. 7 d the oded Prs lw the tue le c be rewrtte s ΔK ΔK where Δ K ΔK ΔK ΔK C ΔK ΔK C π ΔK d dδk π π. Eq. 9 c be rewrtte s R ΔK ΔK C π π C π I ths cse ol two lod ccles re cosdered. A reltoshp betwee the uvlet stress level d the vrble lod c be developed cob Eq. 7 d Eq. s p p 9 where p d p re the occurrece probblt o these two vrble lods. Ths soluto c be esl eteded to t vrble lod. p Cse : ed stress re The bove dscusso s or vrble lod cses wth ed stress rto. Aother sple cse s llustrted here whch the stress re s ed. As etoed bove the preter C the oded Prs lw s the ucto o stress rto.e. C R. Uder costt stress re the tue le c be rewrtte s: R ΔK ΔK R π π R π 3 Frstl ol two vrble lod hstores re cosdered. A reltoshp betwee uvlet stress d vrble lod c be bult: 3

4 Aul Coerece o the Proostcs d Helth Meet Socet R totl R R totl R 4 R R p R p R where p d p re the occurrece probblt o the two stress rto. For ore eerl cses o tue ccles the uvlet stress rto c be descrbed s R p R 5 Cse 3: vr stress re d stress rto Above two specl cses ssue oe o the ppled lod preters e.. stress re or stress rto s ed. A eerl cse s dscussed here both o the re rdo vrbles. A jot dstrbuto o the s rured or the dervto. I ths cse the crc rowth or rbtrr ccle s detered b the ollow uto. Δ d Δ R π R Δ Δ totl R π ΔK 6 Follow the se procedure the eerl uvlet stress c be epressed s R totl R p R 7 where p R Δ σ s the jot dstrbuto o stress re d stress rto. Eq. 7 s the eerlzed uvlet stress level epresso wthout cosder the lod tercto eect. The proposed uvlet stress level ths Secto s ot brd-ew cocept. E. Dowl 7. I the curret stud the probblstc dstrbutos o stress re d stress rto re terted whch s es or uderstd d pplcto. th d crc rowth d coupl eect hs to be cosdered. I ths secto the prevous developed uvlet stress odel s eteded to clude the lod tercto eect such s the overlod retrdto d uderlod ccelerto. The odcto s bsed o recetl developed sll te scle orulto o tue crc rowth d lod tercto correcto ucto. 3. Sll-Te-Scle STS odel A ew sll te scle odel hs bee developed b Lu d Lu Z. Lu & Y. Lu. Ths ethod s bsed o the creetl crc rowth t te stt wth ccle d s deret ro the clsscl reversl-bsed tue lss. Ths ethodolo s bsed o the tercto o orwrd d reverse plstc zoe d hs bee vldted uder vrble pltude lods such s cobto o overlod d uderlod. The sll te scle ethod s brel descrbed below d the detled dervto c be oud the reerred pper Z. Lu & Y. Lu. F. Schetc represetto o crc tp eoetr The developed creetl crc rowth odel s schetcll show F.. The odel s developed bsed o the eoetrc reltoshp betwee the crc tp ope dsplceet CTOD d the stteous crc rowth etcs. Cosder the eoetr o crc tps t two te stts t d t dt the reltoshp betwee the CTOD creet d δ d the crc rowth d c be epressed s ctθ d dδ Cdδ 8 ctθ where C d θ s the crc tp ope le CTOA. It should be oted tht Eq. 8 ssues the tesl crc rowth. 3 Equvlet stress level cosder lod tercto Secto dscussed the uvlet stress level wthout cosder lod tercto. It s well ow tht the eor eects. Mheevs & G. Gl 9 ests or tue 4

5 Aul Coerece o the Proostcs d Helth Meet Socet F.3 Illustrto o orwrd d reverse plstc zoe uder vrble pltude lod Crc rowth predcto c be perored b drectl terto o Eq. 8. The lod tercto eect s odeled b the tercto o orwrd d reversed plstc zoe odel. A schetc eplto s show F. 3. The blue reo d ore reo represet the curret orwrd plstc zoe d the reverse plstc zoe ter overlod. The lt stte ucto s used to detere the crc rowth ree dur the te terto.e. show s Eq. 9: ol Δrp ol r 9 I overlod est the hstor lre reversed plstc zoe occurs hed o the crc tp d retrds the crc rowth. 3. Respose ucto or lod tercto The e de o ths pper s the ecet clculto o probblstc tue crc rowth. As dscussed the Secto oe b chllee or tue lss or rdo lod cse s how to hdle the lod suece eects. The trdtol ethods re us ccle-b-ccle-bsed sulto whch s coputtoll epesve or probblstc lss. To spl the clculto the developed sll te scle odel s used to d respose ucto o the ppled rdo lod. I the curret vestto overlod spectrus re used to deostrte ths respose ucto costructo. For overlod spectrus correcto ter s dded to hdle the lod tercto eect o the proposed uvlet stress level odel. The uvlet stress level cosder lod tercto eect s deed s ηδ where ol The reverse plstc zoe ter overlod σ Δ σ s the uvlet stress level cosder Δ σ s clculted the lod tercto eect d Curret orwrd plstc zoe sze Δr p ol us Eq. 7 wthout cosder the lod tercto ter. η s the coecet or the lod tercto eect d s ucto o the ppled overlod rto R OL d the occurrece probblt P OL o the overlod ccles. r Becuse η does ot hve phscl e ethod slr to DOE des o eperet cobed wth reresso lss s used to costruct the respose ucto. Frst R OL tes the vlue o..5 d.7 respectvel. At ech vlue o R OL the P OL tes the vlue o d.5. I the curret stud oler ucto s show Eq. s used to t the sulto results. B η ROL POL Alo POL ROL s eerc ucto d c be tted us the uercl sulto results. A d B re tt preters d uls to. d.38 respectvel. F. 4 shows the tt curve or three deret overlod rtos. A coprso o the tted curve wth sulto results s show F. 4 b. Ftt curve results Ftt curve results Overlod Rto. Overlod Rto.5 Overlod Rto.7 Ftt ucto Occurrece probblt Overlod Rto. Overlod Rto.5 Overlod Rto.7 Idetcl predcto Sulto results b F.4 Coprso betwee tt pots wth sulto results 4 uercl eples A uercl eple o tue crc rowth s show F.5. The terl s Al-775 wth tl crc.7. The probblt dest ucto PDF o stress re s show F. 5 whch ollows orl dstrbuto wth e vlue MP d stdrd devto MP. ccle rdo lod s show F.5 b. The stress rto s costt ul to -.5. Us the tue odel dscussed Secto the tue crc predcto hs bee perored us the uvlet stress level. A 5

6 Aul Coerece o the Proostcs d Helth Meet Socet coprso hs bee show F. 5 c. The dshed le s the tue crc rowth curve uder the true rdo lod hstor; d the sold le s the tue crc rowth uder uvlet stress re. Althouh soe sll dscrepc o tue crc rowth curve c be see the se l crc sze c be obted.. Dest or stress re Stress rto Ftue le ccle.4-3. Rdo lod process Equvlet stress level b Stress re Crc leth Lod MP Crc leth Ftue le ccle Rdo lod process Equvlet stress level b Ftue le ccles c F.5 dstrbuto o the rdo lod b rdo lod hstor ed stress rto c Ftue crc rowth Dest or stress rto c F.6 dstrbuto o the rdo lod b rdo lod hstor ed stress re c Ftue crc rowth curves uder rdo lod. A slr uercl eple s perored or the ed stress re cse d s show F. 6. The lod ollows orl dstrbuto s show F.6 b. The hstor o ccle rdo lod s show F.6. The rdo lod hs the costt stress re ul 5 MP but the stress rtos re rdo vrbles. The e vlue d stdrd devto o stress rto re - d.3 respectvel. Ftue crc rowth predcto hs bee doe us uvlet stress pltude. Good reeet o the l crc sze c be observed F. 6 c Stress rto Ftue le ccles Stress re MP Stress rto 6

7 Aul Coerece o the Proostcs d Helth Meet Socet Lod MP Crc Leth Ftue le ccle b Rdo lod process Equvlet stress level Ftue le ccles c F.7 Jot dstrbuto o the rdo lod b rdo lod hstor c Ftue crc rowth curves uder rdo lod. The lst uercl eple s ppled to the eerl cse where both stress re d stress rto vres. A jot dstrbuto Guss o stress re d stress rto s show F. 7. The e vlue d stdrd devto o stress re re MP d s MP. The e vlue d stdrd devto o stress rto re - d.3 respectvel. The correlto coecet betwee stress re d stress rto s.. The rdo 5 lod hstor s show F. 7 b. Ftue crc predcto hs bee perored or ths jot-dstrbuted stress re d stress rto. Stsctor results re obted 5 Iverse FORM ethod Above dscusso uses Mote Crlo sulto or probblstc crc rowth predcto. To urther creses the coputtol ecec ltcl relblt ethod s used to vod lre uber o sultos the drect MC ethod. A detled coprso ccle-b-ccle sulto uvletstress-level wth MC sulto uvlet-stress-level wth IFORM hs bee doe to deostrte the ecec o proposed ethod Secto 6. The Detls o Iverse FORM ethod s show below. 5. Iverse FORM ethodolo The rst-order relblt ethod s wdel used uercl techque to clculte the relblt or lure probblt o vrous eeer probles J. Che & Q. S. L 9; S. Thordhl & P. Wlles 8; D. V. Vl M. G. Stewrt & R. E. Melchers 998. M studes hve bee reported o sttc lure probles us the FORM ethod L. Czelj B. Mvo & H. Resch-Opper 994; T. H. Ss & D. A. Brr 996; S. Thordhl & P. Wlles 8. It hs bee ppled to tue probles to clculte the te depedet relblt T. Y. K K. H. Chu & S. Y. Ts 998; M. Lo. u & Q.-. Y 995; Y. Lu & S. Mhdev 9; Y. & Y. Lu ccepted. Ule the FORM ethod A. Hldr & S. Mhdev ; Y. Lu Mhdev S 9 the verse FORM ethod tres to solve the uow preters uder speced relblt or lure probblt level whch s ore sutble or probblstc le predcto.e. re le estto correspod to tret relblt level. Lt stte ucto s rured or the ltcl relblt ethod. A eerc lt stte ucto s epressed s Eq. s ucto o two sets o vrbles d. s the rdo vector d represets terl propertes lods d evroetl ctors etc. s the de vrble vector e.. te d sptl coordtes. The lt stte ucto s deed the stdrd orl spce Eq. d the o- Guss vrbles wll be dscussed lter. The lt stte ucto deto s slr to the clsscl FORM ethod A. Hldr & S. Mhdev. The soluto or the uow preters eeds to sts the relblt costrts whch re descrbed Eq. b d Eq. c. β s the relblt de whch s deed s the dstce ro or to the ost probble pot MPP the stdrd orl spce. The lure probblt P c be clculted us the cuultve dstrbuto ucto CDF Φ o the stdrd Guss dstrbuto. uercl serch s rured to d the optu soluto whch stses the lt stte ucto Eq. d. Detls o the eerl verse FORM ethod d cocept c be oud A. Der Kureh et l : b : βt ret c : p Φ β t ret P < 5% d : 5% P The overll objectve o the verse FORM ethod s to d o-etve ucto sts ll costrt codtos speced Eq.. Thus the uercl serch lorth c be used to d the solutos o the uow preters. uercl serch lorth s developed to tertvel solve the Eq.. The 7

8 Aul Coerece o the Proostcs d Helth Meet Socet 8 serch lorth s epressed s Eq. 3 ter tertos. d d d 3 where d d d re the serch drectos correspod to deret ert uctos d c be clculted us Eq. 4 [ ] [ ] d d et t et t r r β β 4 d re the weht o ucto d c be clculted s 5 Where: [ ] 3 β 6 The coverece crtero or the uercl serch lorth s ε 7 where ε s sll vlue d dctes tht the reltve derece betwee two uercl solutos s sll eouh to esure the coverece. It s oted tht the bove dervto ssues the rdo vrbles re stdrd Guss vrbles. Ths pper uses the trsorto ethod proposed b Rcwtz d Fessler R.. F. Rcwtz B Jue 976 to trsor the o-guss vrbles to ther uvlet stdrd orl spce beore the use o the verse ethod. The rdo vrble trsorto c be epressed s [ ] { } [ ] Φ Φ Φ F F F σ μ σ μ φ σ σ μ φ σ 8 where Φ d F re the cuultve dstrbuto uctos CDF o the stdrd orl d orl o-orl vrbles respectvel. φ d re the probblt dest ucto PDF o the stdrd orl d orl o-orl vrbles respectvel. Ths trsorto wors well or the tue proble the curret vestto sce the dstrbutos o rdo vrbles re ot hhl sewed. For hhl sewed dstrbuto the trsorto proposed b Rcwtz d Fessler R.. F. Rcwtz B 978 c be used sted. I prctcl pplctos the crc rowth predcto wth cert codece boud s usull rured or rs ssesset d decso. I codece s speced the crc rowth correspod to the speced codece bouds c be clculted us the bove etoed verse FORM ethod. 6 Probblstc crc predcto d coprso wth eperetl dt Prevous dscusso dd ot clude uch ore ucerttes th those o the terl propertes. Prevous dscussos do ot clude the ucerttes ro terl propertes d ol ocused o the e predcto. Mote Crlo sulto s used or probblstc crc rowth lss d the tt preters the costt pltude lod test re ssued to be rdo vrbles. These rdo vrbles represet the terl ucerttes. The costt pltude lod test s show F. 8 or Al-775 d tted sttstcl dstrbuto o terl preters s lsted Tble. The eperetl dt re reported T. R. Porter 97. A sur o the propertes o the speces used or the collected eperetl dt re lsted Tble. F. 8 Ftue crc rowth or Al-775 uder deret stress rtos Ftue Crc Growth.E-8.E-7.E-6.E-5.E-4.E-3 Delt K /ccle d/d s-^/ R. R. R.33 R.5 R.75

9 Aul Coerece o the Proostcs d Helth Meet Socet Tble Stochstc coecet o d tue lt stress Mterl preter e std. rto Al 775-T6.3.5 C 7.7E -.8E - K c 5 5 C 7.96E -.88E - K c 5 5 Tble Geoetr d terl propertes o plte speces Spece terl 775-T6 Ultte streth σ u MP 575 Yeld streth σ MP 5 Modulus o elstct E MP 696 Plte wdth 35 Plte thcess 4. Re. T. R. Porter 97 Eperetl dt o Al 775-T6 T. R. Porter 97 uder two blocs lod spectru re used or odel vldto. A schetc llustrto o the two blocs lod s show F. 9. p d F. 9 cotrols the uber o ccles t the hh pltude d the low pltude respectvel. Eht sets o eperetl deret bloc lods two costt d s vrble pltude lods re used or odel vldto d re plotted F.9. p d vlues or ech spectru lod re show the leed. Stress s σ σ F. 9 Schetc llustrto o the two blocs lod T. R. Porter 97 I F. the tue crc rowth predcto hs bee perored or Al-775-T6. Both oe thousd sples o Mote Crlo sulto uvlet stress level d Iverse FORM ethod re used to clculte the probblstc tue crc dstrbuto. I F. the tue crc rowth predcto results or Al-775 hve bee show or costt pltude lod cses b d 6 vrble pltude lod cses c~h. The trles show F. re the eperetl dt T. R. Porter 97. The sold les σ p Oe bloc o lod d the dshed les represet ed predcto o tue crc rowth d 9% codece bouds us Mote Crlo sulto respectvel. The dots show F. re or the results o ed d 9% codece bouds us the verse FORM ethod. It s show tht the verse FORM results ree well wth Mote-Crlo sulto or the ed tue le predcto d 9% codece bouds. Both ethods cpture the jor treds o tue le curves d sctters. Crc leth Crc leth Crc leth Crc leth Eperetl Dt Med predcto 9% boudres Iverse FORM Eperetl Dt Med Predcto 9% Boudres Iverse FORM b Eperetl Dt Med Predcto 9% boudres Iver FORM c Eperetl Dt Med Predcto 9% boudres Iverse FORM d p Kloccles p Kloccles p Kloccles p Kloccles 9

10 Aul Coerece o the Proostcs d Helth Meet Socet Crc leth Crc leth Crc leth Crc leth Eperetl Dt Med Predcto 9% boudres Iverse FORM Eperetl Dt Med Predcto 9% boudres Iverse FORM Eperetl Dt Med Predcto 9% boudres Iverse FORM Eperetl Dt Med Predcto 9% boudres Iverse FORM e p Kloccles p Kloccles p Kloccles p Kloccles h F. Coprso o the predcted results wth the eperetl dt o Al 775-T6 uder two bloc lod T. R. Porter 97 A ver lre sctter o tue crc predcto c be observed F.. Ths s becuse lre vrce o the two put rdo vrbles. I the curret stud the rdo vrbles re ssued to be depedet d the correlto eects wll be cluded uture stud. A sur o coputto te us three deret pproches re show the Tble 3. The coputtos re perored us the se PC: dul core tel 66 wth 4b rs d wdows 7 OS. MATLAB 9b s the pror used the curret stud. For ccle-bccle sulto the coputto te or sulto sle ru s bout 5 hours. sple ccle-b-ccle MC sulto s estted. It c be esl observed tht the MC sulto us the uvlet stress level s uch ore ecet. It s show tht the ost ecet oe s the uvlet stress level cocept wth the verse FORM. The coputtol te s severl tudes less th both ccle-b-ccle d uvlet-stress-level-bsed MC sulto. Thus the proposed ethod uvlet stress level wth verse FORM s ver useul or rel-te de prooss d s potetll tl de tolerce des. It should be oted tht lre uber o MC sulto s rured or ver low lure probblt e.. oe llo sples or.% lure probblt to crese the sulto ecec. I tht cse the rto o coputtol te o the proposed ethod d the drect MC wll be eve lrer. Due to the etree lo coputtol te drect MC sultos wth lre uber o sulto sples were ot perored. Tble 3 Sur o coputto te Approch Coputto te Ccle-b-ccle sulto sple MC sulto 5 hours estted Equvlet stress level sple MC sulto hours Equvlet stress level Iverse FORM 5 secods 7 Cocluso I ths pper ecet probblstc ethodolo s proposed or tue crc rowth prooss. The proposed ethod sples the tue crc rowth lss uder eerl rdo lods d does ot eed ccle-b-ccle clculto. Altcl verse FORM ethod vods lre uber o sultos d urther ehces the coputtol ecec. The dvte es t ver sutble rel-te de prooss d decso. Etesve eperetl dt or Al-775-T6 uder two bloc lod spectru re used to deostrte the vldto o the proposed ethodolo. Geerll the odel

11 Aul Coerece o the Proostcs d Helth Meet Socet predctos ree wth eperetl observtos well. Curret stud ocused o two bloc lod spectrus. Future stud s rured to eted the proposed ethod to structurl sste pplctos uder eerl rdo lods. Ole prooss wll be vestted bsed o the proposed ethodolo. ACKOWLEDGMET The reserch reported ths pper ws supported prt b the ASA ARMD/AvSP IVHM project uder RA 9AY54A. The support s rteull cowleded. REFERECES Bhu S. K. Sujt M. & Vetsw M. A. 8 Ftue lure o rcrt copoets. Eeer Flure Alss Che J. & L Q. S. 9 Relblt lss o lo sp steel rch brde st wd-duced stblt lure dur costructo. Jourl o Costructol Steel Reserch Czelj L. Mvo B. & Resch-Opper H. 994 Applcto o rst d secod order relblt ethods the set ssesset o crced ste eertor tub. ucler Eeer d Des Corbl D. M. & Pc P. F. 973 O the luece o sle d ultple pe overlods o tue crc propto 775-T65 luu. Eeer Frcture Mechcs de Ko AU v. d. L. H. 98. Predcto o tue crc rowth rtes uder vrble lod us sple crc closure odel. Asterd: LR MP 83U Der Kureh A. Zh Y. & L C.-C. 994 Iverse relblt proble. Jourl o Eeer Mechcs ASCE 5. Dowl. E. 7. Mechcl behvor o terls : eeer ethods or deorto rcture d tue. Upper Sddle Rver J Lodo: Perso Pretce Hll ; Perso Educto. Elber W. 97. The scce o rcture crc closure. Phldelph. Hldr A. & Mhdev S.. Probblt relblt d sttstcl ethods eeer des. ew Yor ; Chchester [Eld]: Joh Wle. K T. Y. Chu K. H. & Ts S. Y. 998 Ftue relblt evluto or coposte ltes v drect uercl terto techque. Itertol Jourl o Solds d Structures Ko A. U. d. 98. A sple crc closure odel or predcto o tue crc rowth rtes uder vrble-pltude lod. Frcture Mechcs: Thrteeth Coerece ASTM STP 743 pp : Aerc Socet or Test d Mterls. Lo M. u. & Y Q Cuultve tue de dc tererece sttstcl odel. Itertol Jourl o Ftue Lu Y. & Mhdev S. 7 Stochstc tue de odel uder vrble pltude lod. Itertol Jourl o Ftue Lu Y. & Mhdev S. 9 Ecet ethods or te-depedet tue relblt lss. AIAA Jourl Lu Y. & Mhdev S. 9b Probblstc tue le predcto us uvlet tl lw sze dstrbuto. Itertol Jourl o Ftue Lu Y. Mhdev S 9 Ecet ethods or te-depedet tue relblt lss. AIAA Jourl Lu Z. & Lu Y. Sll te scle tue crc rowth lss. Itertol Jourl o Ftue Mheevs S. & Gl G. 9 Elstc-plstc tue crc rowth lss uder vrble pltude lod spectr. Itertol Jourl o Ftue Moht J. R. Ver B. B. & R P. K. 9 Predcto o tue crc rowth d resdul le us epoetl odel: Prt II ode-i overlod duced retrdto. Itertol Jourl o Ftue ASA Ftue crc rowth coputer pror ASGRO Verso 3.-Reerece ul. JSC- 67B ASA Ldo B. Johso Spce Ceter Tes. ew J. C. 98. A crc closure odel or predct tue crc rowth uder rcrt spectru lod. Phldelph orooz A. H. Gl G. & Lbert S. 7 A stud o the stress rto eects o tue crc rowth us the ued two-preter tue crc rowth drv orce. Itertol Jourl o Ftue orooz A. H. Gl G. & Lbert S. 8 Predcto o tue crc rowth uder costt pltude lod d sle overlod bsed o elsto-plstc crc tp stresses d strs. Eeer Frcture Mechcs Poer S. 3 Cclc plstct d vrble pltude tue. Itertol Jourl o Ftue Porter T. R. 97 Method o lss d predcto or vrble pltude tue crc rowth. E. Frct. Mech Rcwtz R.. F. B 978 Structurl Relblt Uder Cobed Rdo Lod Sueces. Coputers & Structures

12 Aul Coerece o the Proostcs d Helth Meet Socet Rcwtz R.. F. B Jue 976 ote o Dscrete Set Chec Whe Us o-orl Stochstc Models or Bsc Vrbles. Lod Project Wor SessoMITCbrdeMA. R A.. A stte-spce odel o tue crc rowth or rel-testructurl helth eet. Dtl Avocs Sstes Coereces Vol. pp. 6C/ - 6C/8. Ss T. H. & Brr D. A. 996 Assess ucertt subsurce solute trsport: ecet rst-order relblt ethods. Evroetl Sotwre Thordhl S. & Wlles P. 8 Probblstc odell o overlow surchre d lood urb dre us the rst-order relblt ethod d preterzto o locl r seres. Wter Reserch Vl D. V. Stewrt M. G. & Melchers R. E. 998 Eect o reorceet corroso o relblt o hhw brdes. Eeer Structures - 9. Veteswr Ro K. T. & Rtche R. O. 988 Mechss or the retrdto o tue crcs ollow sle tesle overlods: behvor luu-lthu llos. Act Metllurc Wheeler O. E. 97 Spectru lod d crc rowth J. Bsc E. Trs. ASME Wllebor J E. R. Wood RA 97. A Crc Growth Retrdto Model Us Eectve Stress Cocept. Wrht-Ptterso Ar Force Bse Oho: Ar Force Flht Dcs Lbortor. Y. & Lu Y. ccepted Iverse rstorder relblt ethod or probblstc tue le predcto o coposte ltes uder ultl lod. ASCE Jourl o Aerospce Eeer. derees ro Toj Uverst Ch. Dr. Lu s eber o ASCE d AIAA d serves o severl techcl cottees o probblstc ethods d dvced terls. Yb : rdute reserch ssstt deprtet o cvl eeer t Clrso Uverst. He receved hs B.S. deree cvl Eeer ro Toj Uverst Ch 3 d the he wored s structurl eeer Shh d Arch Des Cop. Sce 7 he strted hs stud t Clrso Uverst d ot hs M.S. deree 9 d cotued hs PHD deree. Hs reserch terests re probblstc prooss relblt lss d sste relblt. Yo Lu: ssstt Proessor the deprtet o cvl d evroetl eeer. Hs reserch terests clude tue d rcture lss o etls d coposte terls probblstc ethods coputtol echcs d rs eet. He copleted hs PhD t Vderblt Uverst d obted hs Bchelors d Msters