ASSESSMENT OF STOCHASTIC FATIGUE FAILURES BASED ON DETERMINISTIC FUNCTIONS

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1 h Inernonl Conference on Prolsc Sfe Assessen nd Mngeen (PSAM ) ~7 Ocoer, 6 Sheron Grnde Wlkerhll Seoul, Kore ASSESSMENT OF STOCHASTIC FATIGUE FAILURES BASED ON DETERMINISTIC FUNCTIONS Hossen Sl, Mohd Pourgol-Mohd, Sed Kd Shnd Unvers of Technolog: New Shnd Town, Trz, Irn, Posl Code: 55/996, h_sl@su.c.r Shnd Unvers of Technolog: New Shnd Town, Trz, Irn, Posl Code: 55/996, pourgolohd@su.c.r Shnd Unvers of Technolog: New Shnd Town, Trz, Irn, Posl Code: 55/996, s_kd@su.c.r To cpure he sscl nure of fgue crck growh, n sochsc odels hve een proposed. One of he os coon odels n prolsc frcure echncs nlss proposed Yng nd Mnnng. Ths odel convers he exsng deernsc fgue odel o sochsc one ddng rndo fcor. In hs sud, four dfferen deernsc odels re proposed for fgue lfe predcon odels. These odels hve ore ccurc wh less requred oun of copuon n coprng wh he exsng ones. I s shown h he proposed odels could e used n he Yng nd nnng for eer resuls. Kewords: fgue flure, sochsc crck growh nlss, prolsc frcure echncs, deernsc funcon, rndo process. I. INTRODUCTION When echncl srucure s sujeced o cclc sress or srn, fer enough ccles, fgue crck s ned on croscopcll scle, nd f ccle lod connues, hese cro-crck growhs o croscopc sze, nd fnll srucure fls s he cclc lods re ppled[]. Ths pe of flures clled fgue flures. Fgue flure s one of he os coon flures n ellc srucures. Snce echncl proper of ellc erls s non-hoogeneous, srucurl ppled lod hs scll rndo nure nd soe oher resons; fgue flure s rndo phenoenon nd, herefore, needs o e suded sochscll. Along wh he developen of frcure echncs n deernsc fgue nlss, he so-clled prolsc frcure echncs hs recenl receved good enon[]. One of he porn ssues n he prolsc frcure echncs nlss les n he prolsc odelng of fgue crck growh phenoenon[]. One of he os coon odels n prolsc frcure echncs nlss proposed Yng nd Mnnng[4].Ths ehod convers deernsc fgue odel (such s Prs-Erdogn lw, Eler s lw or n oher lw) o sochsc one ddng rndo fcor[]. Yng nd Mnnng s odel re verfed Wu nd N [] wh usng of experenl d. The jor reson for dopng Yng nd Mnnng s odel Wu nd N nl les n s generl. Deernsc pr of hs odel s ver porn,.e. deernsc funcon us e ch ver well wh he edn rend of he experenl d. Wu nd N showed h Yng nd Mnnng s odel could e odfed f sule deernsc funcon e eploed []. Yng nd Mnnng [5] nd C.C.N [6] re presened power nd polnol funcon s deernsc funcon. In hs sud for frs e four oher funcons re nroduced sed on frcure echncs odels nd curve fng echncs. These funcons hve ore ccurc or less oun of copuon hn forer ones nd re verfed Wu nd N experenl d nd Afgrow sofwre sulon. II. YANG AND MANNING STOCHASTIC MODEL In Yng nd Mnnng s sochsc fgue crck growh odel deernsc fgue crck growh hve een rndozed o e []

2 h Inernonl Conference on Prolsc Sfe Assessen nd Mngeen (PSAM ) ~7 Ocoer, 6 Sheron Grnde Wlkerhll Seoul, Kore X ( ) L( K,R,K x, S, ) d () Where L ndces generl non-negve deernsc crck growh re funcon h, s enoned efore, cn e Prs-Erdogn lw, Eler s lw or n oher lw, ΔK s he sress nens fcor rnge, R s he sress ro, K x s he xu sress nens fcor n lod ccle, S s he xu sress level, nd () s he crck lengh e. In he ove equon rndo fcor X() s dded no he deernsc crck growh equon o ke no ccoun he sscl vrl of crck growh re. Afer exensve sud, Yng nd Mnnng suggesed h X() could eer e odeled s sonr lognorl rndo process hvng en vlue of. nd sndrd devon of σ x. Under hs crcusnce, he followng norl rndo process cn e nroduced [] Z() = ln X() () whch should hve en vlue of zero nd sndrd devon of Z ln( X ) () The edn servce e for crck o grow fro sze o cn e oned fro ( ) dv Lv ( ) (4) To ke he rndo pr no consderon, he followng negron cn lso e perfored dv X () d Lv ( ) (5) If new rndo process W() s defned s W ( ) X ( ) d (6) hen W() cn lso e ssued pproprel s lognorl rndo process. Therefore he followng ssoced norl rndo vrle Y() cn e defned Y() = ln W() (7) whch s ssued o hve en vlue μ Y () nd sndrd devon σ Y (). Ther vlues re reled o he en vlue nd sndrd devon of W()[]. Fnll he prol h crck sze () wll exceed n gven crck sze n he servce nervl (,) cn e derved s e Y ( ) ln[ ( )] P ( )( ) P[ ( ) ] F ( )( ) FW ( )[ ( )] { } (8) () Y The ove prol s frequenl referred o s crck exceednce prol[]. As seen Yng nd Mnnng s odel s needed o defne ppropre deernsc funcon L. Yng nd Mnnng nd C.C.N proposed power nd polnol funcons, respecvel h wll e dscussed n ler pr. III. POWER FUNCTION A sple nd well-known deernsc relon predced fgue crck growh s power lw ndced Prs- Erdogn[7]:

3 h Inernonl Conference on Prolsc Sfe Assessen nd Mngeen (PSAM ) ~7 Ocoer, 6 Sheron Grnde Wlkerhll Seoul, Kore C ( K) (9) d where C nd re erl consns. For nfne ple wh cener hrough crck ΔK defned s [8] K () where Δσ s he ppled sress rnge. Therefore, for consn plude lodng rnge nd usng of equon () n (9), Prs-Erdogn lw ecoes: d C ( ) Q, Q C( ) nd () However, when lrge oun of d s suded, soe dscrepnc s oserved n crck propgon heor. Wlker [9] proved he Prs-Erdogn lw kng no ccoun sress ro effec s: Cw ( K) d ( R) w ( ) w () where γ s erl consn. Slrl, for consn plude lodng rnge nd usng of equon () n (), Wlker lw ecoes: Cw ( ) ( ) w d ( R) w Q C ( ) w w, Q ( ) w ( R) nd w () As seen, he equons () nd () re que slr nd oh re power funcon wh generl for of L() = Q. Afer nvesgon of crck propgon n fsener holes of rcrfs under specru lodng, Yng nd Mnnng hve suggesed he followng sple for [5] X ( ) Q[()] (4) d n whch Q nd re consns o e evlued fro he crck growh oservon. For hs funcon edn servce e nd norl rndo process ecoes dv ( ) Qv Q( ) (5) Z ( ) ln X ( ) ln ln ( ) lnq d (6) IV. POLYNOMIAL FUNCTION C.C.N [6] sud on rell of rcrf srucures dscovered he sch of power funcon wh he edn rend of he experenl resul. Therefore, polnol funcon ws proposed nsed of power funcon s: d X ( ){p q() r[()] } (7) In hs crcusnce edn servce e nd norl rndo process ecoes

4 h Inernonl Conference on Prolsc Sfe Assessen nd Mngeen (PSAM ) ~7 Ocoer, 6 Sheron Grnde Wlkerhll Seoul, Kore dv ( r A q) G ( ) n p qv rv G Aq G ra, Z G pr q ( ) ln X ( ) ln ln{p q() r[()] } 4 nd A r q (8) (9) d Unlke power funcon, polnol funcon s so coplced nd needs ore oun of copuon. In he nex secon soe new funcon wll e nroduced h hve ore ccurc or less coplc hn hese wo funcons. These funcons re cegorzed n wo prs: ) Funcons sed on clsscl frcure odels ) Funcons sed on curve fng echnque V. FUNCTIONS BASED ON CLASSICAL FRACTURE MODELS V.A. Ronl Funcon Neher of Prs-Erdogn nor Wlker odels could descre cro-crck phse. Forn [] proposed odel h could descred hrd sge of fgue crck growh plo. Forn odel followed : C ( K) f f d ( R) K K C () Where K c s frcure oughness of he srucure erl. Slrl, for consn plude lodng rnge nd usng of equon () n (), Forn lw ecoes () ( ) f d C f Q d ( R) K ( ) Q Q C, Q C ( ) f, Q ( R ) KC nd Q f () B ddng rndo fcor n ronl funcon, new odfed Yng nd Mnnng s Model ecoes Q X () d Q Q () Therefore, edn servce e nd norl rndo process ecoes Q Q v Q Q ( ) dv [ ] [ ] ().5.5 Qv Q( ) Q(.5 ) Z ( ) ln X ( ) ln ln( Q Q ) ln( Q) ln (4) d Ths odel s ppropre for nsle fgue frcure (phse III of fgue crck growh plo). As seen hs funcon s less coplced hn polnol funcon. V.B. Glol funcon Bsed on clsscl frcure echncs sress nens fcor rnge defned s [8] K ( ) f ( ) (5) Where f() s geoerc fcor. Ths fcor for crcked ple s 4

5 h Inernonl Conference on Prolsc Sfe Assessen nd Mngeen (PSAM ) ~7 Ocoer, 6 Sheron Grnde Wlkerhll Seoul, Kore C f () 4 nd (6) w n whch C nd w re consns geoerc coeffcens nd ple s wdh, respecvel. For nfne ple wh cener hrough crck hese coeffcens re C, C C C C 4 f ( ) (7) whch s led o equon (). Neher of power nor ronl funcons consders hs fcor. Therefore, he second suggesed funcon n hs sud s soclled glol funcon h defned s L( ) L ( ) L ( ) (8) Eher of power of ronl funcon could e used s L (). L () s dependence of geoerc fcor nd L (). For exple eplong of power funcon for fne cener hrough crcked ple, L () ecoes L ( ) [f()] [.8( ).88( ).5( ) ] w w w (9) For hs cse edn servce e nd norl rndo process ecoes.56 ( ) dv { [ ] [ ]} Qv Q ( v ) Z ( ) ln X ( ) ln lnq ln ( ) ln () d I s necessr o enoned for splfcon n copuon, Tlor s explnon s used o clcule negrl (). As seen hs funcon s ore coplced hn forer ones; nsed hs hgh level of ccurc nd s ore ppropre for coplced geoeres. () VI. FUNCTIONS BASED ON CURVE FITTING TECHNIQUE VI.A. The funcon n for of e e x x The frs funcon h s nroduced n hs pr hs followng for d( ) X ( )[ ] d Q e Q e () Medn servce e nd norl rndo process of hs funcon ecoes v v () ( ) ( Q e Q e ) dv Q e Q e Q e Q e Z ( ) ln X ( ) ln ln( Qe Qe ) (4) d VI.B. The funcon n for of x 5

6 h Inernonl Conference on Prolsc Sfe Assessen nd Mngeen (PSAM ) ~7 Ocoer, 6 Sheron Grnde Wlkerhll Seoul, Kore The second funcon s nroduced n followng for d( ) X ( )[ ] d Q Q (5) Medn servce e nd norl rndo process of hs funcon ecoes Q ( ) ( Qv Q) dv ( ) Q( ) (6) Z ( ) ln X ( ) ln ln( Q Q) d (7) VI.C. Vldon Experenl d of [] re used for vldon of hese wo funcons. Wu nd N used copc enson (CT) specens were cu fro 4-T5 lunu llo ple. The densons of he specens were 5. wde nd. hck. The pre-crckng es sred crck lengh of 5. nd exended o he lengh of 8.. Snusodl sgnls wh xu of 4.5 kn, nu of.9 kn were used s npu lods. Resuls re shown n Tle. TABLE Vldon of Curve Fng Bsed Funcons wh Experenl D Funcon R ( ) (ccles) Error (%) Q (Under) p q r (Over) Q e Q e Q Q (Over) (Over) As shown n Tle hese wo funcons hve ver low error nd cn e used s deernsc funcon n Yng nd Mnnng s odel. VII. RESULTS AND DISCUSSIONS AFGROW sofwre sulon s used for coprson of presened funcons. AFGROW s coercl sofwre h s developed for sulon of erospce srucure s fgue flures. Specens used n sulon were recngulr ple wh 54 wde,.54 hck nd cener hrough crcked. The lengh of crck ws.54. These specens were chosen fro lunu 4. Appled lod hd xu vlue of.4 kn nd sress ro of.5. Resuls re shown n Tle. TABLE Coprson of Proposed Funcons Funcon R ( ) (ccles) Error (%) Q (Under) p q r (Over) Q Q Q Q e Q e (Under) (Over) Under nd over refers o less nd ore predcon h ccure e, respecvel. 6

7 h Inernonl Conference on Prolsc Sfe Assessen nd Mngeen (PSAM ) ~7 Ocoer, 6 Sheron Grnde Wlkerhll Seoul, Kore Funcon R ( ) (ccles) Error (%) Q Q [ ] (Under) Q (Under) As seen n Tle, power funcon (he frs row of Tle ) hs good predcon of edn servce e. However, generl funcon (he ls row of Tle ) s proved hs predcon 4%, however ncresed oun of copuons. The es predcon of edn servce e s n he 5 h row. Ths funcon s proved edn servce e predcon s 94% wh los se oun of copuons. The 4 h funcon hs slr predcon of edn servce e o polnol funcon, nsed hs ver low oun of copuon. VIII. CONCLUSION Fro experence gned n recen sudes s found h Yng nd Mnnng s sochsc fgue crck growh odel s ver generl nd versle. To ppl hs odel, one cn sr o r spler fgue crck growh funcon. If he odel fls o expln he d ssfcorl, he r o f edn crck growh curve ore ccurel usng noher funcon []. In ddon o he forer presened funcons (power nd polnol), n hs sud four dfferen funcons s nroduced. These funcons hve ore ccurc or less oun of copuon hn forer ones. REFERENCES [] N. E. Dowlng, Mechncl ehvor of erls: engneerng ehods for deforon, frcure, nd fgue, (99). [] G. C. Sh, J. Provn, Prolsc frcure echncs nd rell: Sprnger Scence & Busness Med, (). [] W. Wu, C. N, A sud of sochsc fgue crck growh odelng hrough experenl d, Prolsc Engneerng Mechncs, Vol. 8, No., pp. 7-8, (). [4] J. Yng, S. Mnnng, Sochsc crck growh nlss ehodologes for ellc srucures, Engneerng Frcure Mechncs, Vol. 7, No. 5, pp. 5-4, (99). [5] J. Yng, S. Mnnng, A sple second order pproxon for sochsc crck growh nlss, Engneerng Frcure Mechncs, Vol. 5, No. 5, pp , (996). [6] C. N, Forulon of Polnol Sochsc Fgue Crck Growh Model, n Proceedng of, Trns Tech Pul, pp [7] P. Prs, F. Erdogn, A crcl nlss of crck propgon lws, Journl of sc engneerng, Vol. 85, No. 4, pp. 58-5, (96). [8] E. E. Gdouos, Frcure echncs: n nroducon: Sprnger Scence & Busness Med, (6). [9] K. Wlker, The effec of sress ro durng crck propgon nd fgue for 4-T nd 775-T6 lunu, n: Effecs of envronen nd coplex lod hsor on fgue lfe, Eds.: ASTM Inernonl, (97). [] R. G. Forn, V. Kerne, R. Engle, Nuercl nlss of crck propgon n cclc-loded srucures, Journl of sc Engneerng, Vol. 89, No., pp , (967). 7

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